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2013 107 Carlos Borau Zamora Multiscale computational modeling of single cell migration in 3D Departamento Director/es Ingeniería Mecánica García Aznar, José Manuel Kamm, Roger Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Carlos Borau Zamora MULTISCALE COMPUTATIONAL MODELING OF SINGLE CELL MIGRATION IN 3D Director/es Ingeniería Mecánica García Aznar, José Manuel Kamm, Roger Tesis Doctoral Autor 2013 Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Multiscale computational modeling of single cell migration in 3D DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Computational Mechanics by Carlos Borau Zamora Faculty advisors José Manuel García Aznar Roger D. Kamm Multiscale in mechanical and biological engineering Aragon Institute of Engineering Research Department of Mechanical Engineering, University of Zaragoza Spain & Departments of Biological and Mechanical Engineering, Massachusetts Institute of Technology United States of America Zaragoza, September 2013 To learn something new, take the path that you took yesterday. ~John Burroughs I SINOPSIS La migración celular es un proceso complejo, orquestado por factores químicos y biológicos, por la microestructura y por las propiedades mecánicas de la matriz extracelular. Este fenómeno es fundamental para el desarrollo de tejidos en los organismos pluricelulares, y como seres humanos, nos acompaña durante toda la vida, desde el mismo momento de la concepción hasta la muerte. Juega un papel fundamental durante el desarrollo embrionario determinando la formación de los diferentes órganos (morfogénesis) y es clave en todos los procesos regenerativos como la renovación de la piel, la respuesta inflamatoria o la cicatrización de heridas. Sin embargo, también contribuye al desarrollo de procesos patológicos como la metástasis, el retraso mental, la osteoporosis o enfermedades vasculares entre otros. Es por ello de vital importancia el conocer los mecanismos fundamentales que controlan la migración celular con el fin de tratar de manera efectiva las diferentes patologías, así como avanzar en el trasplante de órganos y el desarrollo de tejidos artificiales. Así pues, el objetivo de esta Tesis es el desarrollo de modelos a distintas escalas y centrados en diversos aspectos de la migración, de manera que faciliten la compresión de fenómenos específicos y sirvan como guía para el diseño de experimentos. Dada la complejidad y las grandes diferencias respecto a la migración colectiva, todos los modelos y análisis de esta Tesis se centran en células individuales. En primer lugar se ha estudiado la migración tridimensional de una célula individual embebida en una matriz extracelular donde su velocidad y orientación se consideran reguladas por estímulos mecánicos. Para ello se ha desarrollado un modelo mecanosensor basado en elementos finitos y se ha analizado el comportamiento celular en función de diferentes rigideces y condiciones de contorno a escala celular. A medida que el trabajo ha progresado, los resultados del modelo unidos a nuevos avances científicos publicados en este ámbito, han reforzado la idea de que el mecansimo mecanosensor juega un papel crítico en los procesos que dirigen la migración celular. Por ello, se ha necesitado un estudio más profundo de este fenómeno para lo que se ha utilizado un modelo mucho más detallado a escala intracelular. Así pues, se ha explorado la estructura interna del citoesqueleto y su comportamiento ante cambios mecánicos en la matriz extracelular, utilizando un modelo discreto de partículas basado en dinámica Browniana con el que se ha simulado la formación de una red de actina (polimerización) entrecruzada con proteínas y motores II Multiscale computational modeling of single cell migration in 3D moleculares. En concreto, se ha estudiado el comportamiento activo de estos motores y su papel como sensores de estímulos mecánicos externos (mecanosensores) de manera que los resultados obtenidos con este modelo “micro” han permitido validar las hipótesis del modelo previo. Consecuentemente, se ha revisado el modelo mecánico y se le ha añadido dependencia temporal, obteniendo un modelo continuo capaz de predecir respuestas celulares macroscópicas basadas en el comportamiento de los componentes microestructurales. En otras palabras, esta simplificación ha permitido la introducción de la respuesta macroscópica emergente obtenida del comportamiento dinámico de la microestructura, disminuyendo enormemente el coste computacional y por tanto permitiendo simulaciones a mayores escalas espacio-temporales. A continuación se han introducido las nuevas hipótesis en un modelo probabilístico de migración a escala celular basado en elementos finitos que permite al mismo tiempo el estudio de factores tanto a escala macroscópica (velocidades, trayectorias) como a escala celular (orientación, área de adhesión, tensiones celulares, desplazamientos de la matriz etc.). Adicionalmente, este modelo es sensible no sólo a la mecánica sino a las condiciones fluido-químicas del entorno, las cuales han sido analizadas igualmente mediante simulaciones por elementos finitos. Con todo esto, los modelos desarrollados todavía no incluyen una descripción detallada de procesos importantes envueltos en la migración celular como la protrusión de la membrana, la polimerización de actina en el frente celular o la formación de adhesiones focales. Por lo tanto, para completar la Tesis, se ha desarrollado un modelo continuo basado en diferencias finitas que permite el estudio del comportamiento dinámico del lamelipodio y el papel fundamental que juegan la polimerización de actina, los motores moleculares y las adhesiones focales (FAs) en el frente celular durante la migración. Palabras clave: mecanobiología, simulación computacional, elementos finitos, mecanosensor, migración celular, citoesqueleto, redes de actina, motores moleculares, polimerización, lamelipodio. Contents-Indice IX 3.2. Model features .................................................................................................. 85 3.2.1. Formation of an active actin network ....................................................... 85 3.2.2. Parallelization ........................................................................................... 86 3.2.3. Mechanics of actin filaments, ACPs and motors ...................................... 88 3.2.4. Dynamic behaviors of ACPs and motors .................................................. 89 3.2.5. Boundary conditions of the 3D computational domain ............................ 90 3.3. Results .............................................................................................................. 91 3.3.1. Network morphology and stress evolution depend on substrate stiffness 91 3.3.2. Network stiffness tracks the generated stress ........................................... 93 3.3.3. Effects of motor concentration .................................................................. 93 3.3.4. Effects of unbinding and walking behaviors of motors ............................ 94 3.4. Discussion ........................................................................................................ 97 4. TIME DEPENDENT MECHANOSENSING .................................................. 105 4.1. Introduction .................................................................................................... 107 4.2. Constitutive law.............................................................................................. 109 4.2.1. Definition of internal variables ............................................................... 110 4.2.2. Additive decomposition of the displacement field ................................. 111 4.2.3. Cell forces ............................................................................................... 112 4.2.4. Time-dependent response ....................................................................... 114 4.3. Problem description and results ..................................................................... 118 4.3.1. Simulated experimental assay ................................................................. 118 4.3.2. Cell contraction and force generation under different extracellular stiffnesses .............................................................................................................. 119 4.3.3. Simulating step changes in extracellular stiffness .................................. 121 4.4. Parameter sensitivity analysis ........................................................................ 123 4.5. Extension to 3D problems .............................................................................. 126 4.5.1. Geometry and model adaptation ............................................................. 127 X Multiscale computational modeling of single cell migration in 3D 4.5.2. Material definition and boundary conditions .......................................... 128 4.5.3. Results ..................................................................................................... 128 4.6. Discussion ...................................................................................................... 130 5. MODELING 3D MIGRATION AT THE CELL SCALE: A PROBABILISTIC APPROACH ............................................................................... 135 5.1. Introduction .................................................................................................... 137 5.2. Model description ........................................................................................... 138 5.3. Numerical implementation ............................................................................. 138 5.3.1. Macroscopic FE Analysis: evaluation of the environment stimuli ......... 139 5.3.1.1. Modeling chemotaxis and flow through a porous medium ............. 140 5.3.1.2. Modeling mechanotaxis ................................................................... 141 5.3.2. External stimuli and cell dynamics determine cell migration ................. 143 5.3.2.1. Cell stress ......................................................................................... 145 5.3.2.2. Flow and chemical concentration .................................................... 146 5.4. Results ............................................................................................................ 148 5.4.1. Microfluidic simulation .......................................................................... 148 5.4.2. Effects of ECM stiffness ......................................................................... 149 5.4.2.1. ECM degradation ............................................................................. 154 5.4.3. Migration................................................................................................. 155 5.4.4. Modeling a porous ECM ......................................................................... 156 5.5. Discussion and conclusions ............................................................................ 159 6. LAMELLIPODIUM DYNAMICS .................................................................... 165 6.1. Introduction .................................................................................................... 167 6.2. Model description ........................................................................................... 168 6.2.1. Forces and kinematics ............................................................................. 169 6.2.2. Actin treadmilling ................................................................................... 170 6.2.3. Cyclic contractions .................................................................................. 171 Contents-Indice XI 6.3. Model predictions ........................................................................................... 172 6.3.1. Decrease of actin polymerization ............................................................ 173 6.3.2. Myosin inhibition .................................................................................... 174 6.3.3. Decrease of FA friction ........................................................................... 176 6.4. Experimental assays ....................................................................................... 180 6.4.1. Results ..................................................................................................... 181 6.5. Materials and methods ................................................................................... 182 6.5.1. Mathematical approach ........................................................................... 183 6.5.2. Experimental methods ............................................................................ 184 6.5.3. Measurements ......................................................................................... 184 6.5.3.1. Actin flow and edge position ........................................................... 184 6.5.3.2. Actin and membrane speeds correlation .......................................... 186 6.6. Discussion ...................................................................................................... 188 7. CLOSURE ........................................................................................................... 191 7.1. Summary ........................................................................................................ 192 7.2. Conclusions .................................................................................................... 197 7.3. Original contributions .................................................................................... 201 7.3.1. Publications in peer-reviewed journals ................................................... 202 7.3.2. Congress and conference contributions .................................................. 203 7.4. Future lines of work ....................................................................................... 205 APPENDIX A: SOME COMPUTATIONAL ALGORITHMS ............................. 209 A.1 Finding a point inside an irregular hexahedron .............................................. 209 A.2 Computing cell orientation using FE shape functions ................................... 211 A.3 Cell aspect ratio using voxels ......................................................................... 213 A.4 Building a porous mesh .................................................................................. 215 A.5 Making 3D movies ......................................................................................... 216 A.6 Measuring actin flow with matrix convolution .............................................. 217 XII Multiscale computational modeling of single cell migration in 3D A.7 Function correlation and Fourier analysis ...................................................... 221 APPENDIX B: INFLUENCE OF MYOSIN-HEADS ............................................. 225 B.1 Influence of the time step ............................................................................... 226 B.2 Influence of the number of heads ................................................................... 227 B.3 Influence of rebinding .................................................................................... 228 B.4 Minifilament properties .................................................................................. 229 APPENDIX C: ADDITIONAL MODEL VALIDATIONS .................................... 233 C.1 Model randomness ......................................................................................... 233 C.2 Mechanical testing .......................................................................................... 235 C.3 Fluid-chemical factors testing ........................................................................ 238 C.4 Probability functions sensitivity analysis ....................................................... 240 C.5 Fluid-chemical simulation with cell body embedded in a porous ECM ........ 243 BIBLIOGRAPHY ....................................................................................................... 247 List of figures XIII LIST OF FIGURES MAIN CHAPTERS Figure 1.1: Factors influencing cell migration. ................................................................. 36 Figure 1.2: The migration cycle. ....................................................................................... 38 Figure 1.3: Major types of in vivo ECM. .......................................................................... 40 Figure 1.4: Experimental methods. ................................................................................... 43 Figure 1.5: Examples of migration models. ...................................................................... 46 Figure 1.6: Experimentation-Modeling feedback. ............................................................ 48 Figure 2.1: Schematic diagram of cell mechanical components. .................................... 58 Figure 2.2: Acto-myosin contraction dependence. .......................................................... 61 Figure 2.3: CSK reorientation and definition of cell’s front/back. .................................. 63 Figure 2.4: Computational algorithm. .............................................................................. 67 Figure 2.5: Parameter sensitivity analysis ....................................................................... 70 Figure 2.6: Simulated cases. ............................................................................................. 71 Figure 2.7: Migration trajectories and cell speeds. ........................................................... 73 Figure 2.8: External forces modulate migration direction. ............................................... 75 Figure 3.1: Fibroblast on a micropilar substrate and actin layers of a cell. ...................... 84 Figure 3.2: Parallelization scheme. ................................................................................... 87 Figure 3.3: Initial 3D network and cross-sections. ........................................................... 92 Figure 3.4: Different network organization. ..................................................................... 92 Figure 3.5: Effects of substrate stiffness on network stress and strain. ............................ 94 Figure 3.6: Meassurement of network stiffness. ............................................................... 95 Figure 3.7: Influences of motor concentration. ................................................................. 95 Figure 3.8: Influences of zero-force unbinding rate of motors. ........................................ 96 Figure 3.9: Effects of mechanical sensitivity of motor unbinding and walking. .............. 97 Figure 3.10: Influences of mechanical sensitivity of motor unbinging and walking. ...... 98 Figure 3.11: Mechanisms limiting network contraction. ................................................ 100 Figure 3.12: Effects of actin concentration and actin filament length. ........................... 102 Figure 3.13: Control case reduced to Hill’s equation. .................................................... 103 Figure 4.1: Mechanical models to describe the contractile cell response. ...................... 108 Figure 4.2: System schemes for measuring cell mechanosensing properties. ................ 110 XIV Multiscale computational modeling of single cell migration in 3D Figure 4.3: Model temporal response. ............................................................................ 114 Figure 4.4: Renormalization of AM system contraction depending on the definition of the acto-myosin force. ...................................................................................................... 118 Figure 4.5: Cell force evolution and plateau values depending on substrate stiffness. .. 120 Figure 4.6: Rate of force build-up, speed of shortening and cell power. ........................ 121 Figure 4.7: Cell response to step-changes in substrate stiffness. .................................... 123 Figure 4.8: Sensitivity analysis of parameters involved in motor stalling evolution. .... 125 Figure 4.9: Sensitivity analysis of the cell force evolution depending on actin and passive stiffness. ........................................................................................................................... 126 Figure 4.10: Effects of maximum slippage distance. ...................................................... 126 Figure 4.11: Experimental setup simulation in 3D. ........................................................ 127 Figure 4.12: Cell contraction between two microplates. ................................................ 129 Figure 4.13: Saturation force vs. substrate stiffness. ...................................................... 129 Figure 5.1: Scheme of the iterative loop. ........................................................................ 139 Figure 5.2: Geometry of the microfluidic device and details of domain and cell mesh. 141 Figure 5.3: Mechanosensing scheme for 3D and different cell parts. ............................ 143 Figure 5.4: Schematic example of voxel addition process. ............................................ 147 Figure 5.5: Fluid chemical analysis in a 3D microdevice. .............................................. 149 Figure 5.6: Cell response for homogeneous ECMs. ....................................................... 151 Figure 5.7: ECM stiffness gradients and theoretical cell stress. ..................................... 151 Figure 5.8: Migration trajectories and computed speeds. ............................................... 152 Figure 5.9: Cell stress and ECM displacements. ............................................................ 152 Figure 5.10: Cell shape factor and spread area. .............................................................. 153 Figure 5.11: Cell speeds and ECM degradation. ............................................................ 154 Figure 5.12: Cell migration under different environmental conditions. ......................... 156 Figure 5.13: Example of a porous ECM voxel-mesh. .................................................... 157 Figure 5.14: Cell stress and displacements in a porous ECM. ....................................... 158 Figure 5.15: Cell response in a porous ECM. ................................................................. 159 Figure 5.16: Different software used. ............................................................................. 162 Figure 6.1: Cell edge dynamics ...................................................................................... 168 Figure 6.2: Lamellipodium model scheme. .................................................................... 169 Figure 6.3: Actin concentration along lamellipodium. ................................................... 171 Figure 6.4: Cyclic scheme of lamellipodium dynamics. ................................................ 172 Figure 6.5: Leading edge position with decreased polymerization. ............................... 174 List of figures XV Figure 6.6: Leading edge position with myosin inhibition. ............................................ 175 Figure 6.7: Model cycles after myosin force inhibition. ................................................. 176 Figure 6.8: Cortactin intensity and actin density along lamellipodium .......................... 177 Figure 6.9: Model predictions vs. measured data. .......................................................... 178 Figure 6.10: Leading edge position with vinculin knocked out. .................................... 179 Figure 6.11: Lamellipodium width and actin speed relationship. ................................... 180 Figure 6.12: Actin and protein concentration at the lamella border. .............................. 180 Figure 6.13: Actin and membrane speeds during protrusion/retraction cycles. ............. 183 Figure 6.14: Actin and membrane periodicity. ............................................................... 185 Figure 6.15: Averaged periodicity. ................................................................................. 185 Figure 6.16: Actin-membrane velocities phase delay. .................................................... 186 Figure 6.17: Actin flow and membrane speed measurement. ......................................... 187 Figure 6.18: Actin and membrane speeds over time. ..................................................... 187 APPENDIX A Figure A.1: Point-finder script scheme. .......................................................................... 210 Figure A.2: Decision tree of the point-finder script. ....................................................... 211 Figure A.3: Geometric scheme of cell aspect ratio calculation. ..................................... 213 Figure A.4: Porous meshes with different porosity and pore size. ................................. 216 Figure A.5: Rendering of voxel structures with VMD. .................................................. 217 Figure A.6: Convolution of functions. ............................................................................ 218 Figure A.7: Region selection for image processing. ....................................................... 219 Figure A.8: Gaussian low-pass filters. ............................................................................ 219 Figure A.9: Image after applying low-pass filters. ......................................................... 220 Figure A.10: Original and band-passed image. .............................................................. 220 Figure A.11: Actin speed measurement. ......................................................................... 221 Figure A.12: Cross-correlation of functions. .................................................................. 221 Figure A.13: Example of membrane and actin speeds correlation fit. ........................... 223 XVI Multiscale computational modeling of single cell migration in 3D APPENDIX B Figure B.1: Scheme of myosin minifilament. ................................................................. 226 Figure B.2: Influence of time step on unbinding rate of single motors. ......................... 226 Figure B.3: Influence of time step at zero-force and several heads. ............................... 227 Figure B.4: Influence of the number of heads on filament kinetics. .............................. 228 Figure B.5: Effect of force and number of heads. .......................................................... 228 Figure B.6: Influence of rebinding probability and number of heads. ........................... 229 Figure B.7: Bell’s equation fit with multiple myosin heads and different rebinding probabilities...................................................................................................................... 230 APPENDIX C Figure C.1: Angle histograms and final cell positions in purely random migration. ..... 234 Figure C.2: Angle histograms and final positions in forced plus random migration. .... 234 Figure C.3: Cell speeds and migration path lengths. ...................................................... 236 Figure C.4: Cell shape factor and spread area for different stress levels. ...................... 237 Figure C.5: Cell mean and effective speeds for different fluid-chemical conditions. ... 239 Figure C.6: Migration trajectories for different flow-chemical conditions. ................... 239 Figure C.7: Effect of the number of factors on the probability functions. ..................... 241 Figure C.8: Effect of the alignment on the probability functions. .................................. 242 Figure C.9: Effect of the sensitivity constants on the probability functions. ................. 242 Figure C.10: Effect of the adding/removing rate on the probability functions. ............. 242 Figure C.11: Cell geometry smoothing. ......................................................................... 243 Figure C.12: Tetrahedral mesh used in the simulations. ................................................ 244 Figure C.13: Flow velocity and ECM chemical concentration. ..................................... 244 Figure C.14: Pressure gradient along the ECM and cell surface. ................................... 245 Figure C.15: Autologous chemokine concentration for different flow speeds. .............. 245 List of tables XVII LIST OF TABLES Table 2-1: Major parameters used in the model ............................................................... 68 Table 3-1: Main parameters used in the simulations ...................................................... 104 Table 4-1: Parameter list. ................................................................................................ 134 Table 5-1: Parameters for the fluid-chemical and mechanical analysis .......................... 144 Table 5-2: Constant parameters of the probability functions .......................................... 145 Table 6-1: Model parameters. ......................................................................................... 173 Capítulo I. Resumen 5 de multiples factores: mecánicos, químicos y biológicos entre otros (Cukierman et al., 2002, Even-Ram and Yamada, 2005, Zaman et al., 2006). Por ejemplo, la influencia de la deformación de un sustrato (Tensotaxis), su rigidez y topografía (Durotaxis) han sido ampliamente investigadas (Beloussov et al., 2000, Lo et al., 2000, Saez et al., 2007, Baker et al., 2009, Hakkinen et al., 2011), mostrando que las células tienden a migrar hacia zonas de la matriz con mayores deformaciones o de mayor rigidez, donde las adhesiones focales son más estables y permiten la generación de mayores esfuerzos (Lo et al., 2000, Cukierman et al., 2001, Schwarz and Bischofs, 2005). Las células también responden a gradientes químicos (Chemotaxis) en el tejido o fluido circundantes (Rappel et al., 2002, Zhelev et al., 2004), moviéndose hacia la fuente o alejándose de ella según la sustancia. Además, las células individuales de una monocapa tienden a orientarse en la dirección local de máxima tensión normal transmitida a través de múltiples uniones célula-célula (Plithotaxis)(Trepat and Fredberg, 2011). Gradientes de potencial eléctrico (Galvanotaxis), condiciones del fluido o gradientes de concentración de ligandos (Haptotaxis) son guías adicionales de la migración celular actualmente bajo intenso estudio (Zhao et al., 2002, Curtze et al., 2004, Li et al., 2005, Hofmann et al., 2006, Polacheck et al., 2011). Entender la migración como un proceso integrado requiere la consideración de múltiples estructuras formadas a su vez por numerosos elementos que interaccionan y se coordinan entre sí espacial y temporalmente. Sólo cuando esta integración sea completamente comprendida y tenida en cuenta, la alteración genética o la utilización de fármacos podrán ser realmente eficaces, con el consecuente impacto en los tratamientos de enfermedades conocidos hasta ahora. I.1.2. Experimentación Gracias al desarrollo de nuevas técnicas y tecnologías, la experimentación sobre migración celular se ha acelerado notablemente en los últimos años. Aún así, el aislamiento de los efectos producidos por un estímulo concreto in vivo todavía presenta numerosas dificultades. A consecuencia de ello, los ensayos in vitro han emergido como potentes herramientas para el estudio de la movilidad celular, permitiendo examinar factores específicos implicados en la migración. Debido a la vital importancia del cáncer, estos ensayos in vitro se utilizan de manera extendida para intentar entender los mecanísmos que conducen el movimiento de las células tumorales (normalmente 6 Multiscale computational modeling of single cell migration in 3D estímulos mecánicos y químicos) con el objetivo de mejorar la eficacia de las terapias que se utilizan actualmente. El micro-entorno de un tumor está normalmente formado por una red dinámica de proteínas extracelulares bañadas en fluido intersticial así como grupos de células asociadas que incluyen fibroblastos, células endoteliales o derivadas de la médula osea (revisado en (Joyce and Pollard, 2009)). Estas células conectivas remodelan la matriz generando señales mecánicas y químicas que reciben las células tumorales. Debido a la naturaleza dinámica y a la variedad de componentes que conforman este micro-entorno, la investigación de estímulos individuales es realmente compleja y requiere que las propiedades mecánicas y químicas puedan ser adaptadas con precisión y repetibilidad. Los ensayos in vitro proporcionan dicho control y están específicamente preparados para el estudio de factores particulares de interés. A continuación se muestra como ejemplo una breve clasificación de los diferentes métodos experimentales utilizados en la actualidad (Polacheck et al., 2012): (i) Ensayos de micropipeta: un micro-manipulador se usa para inyectar soluciones químicas en los alrededores de la célula o para inyectar factores de crecimiento. Los gradientes generados, sin embargo, son transitorios y difíciles de cuantificar (Soon et al., 2005). (ii) Cámaras de Boyden (o Transwell): una membrana rígida y porosa se coloca entre dos cámaras de cultivo. Las células se colocan en suspension en la cámara superior y migran a través del filtro en respuesta a un gradiente químico generado por la diferencia de concentraciones entre los medios superior e inferior (Boyden, 1962). (iii) Micropatrones: las células se cultivan sobre patrones de diferente geometría, tamaño y tipo de superficie en sustratos 1D, 2D y 3D. Estos ensayos ofrecen escalabilidad y la posibilidad de cultivar grandes poblaciones de células. Sin embargo, no suelen incluir flujo y la aplicación de estímulos mecánicos o químicos localizados no es generalmente posible (von Philipsborn et al., 2006). (iv) Ensayos de durotaxis: las células se cultivan sobre sustratos planos de rigidez variable donde la respuesta celular (fuerzas de tracción, area de adhesion y dirección de migración) se monitoriza a lo largo del tiempo (Lo et al., 2000). Capítulo I. Resumen 7 (v) Ensayos de regeneración de heridas: la migración colectiva de células se puede estudiar generando espacios vacíos (heridas) en una capa de células, donde las dinámicas de cierre (velocidad, morfología, etc) pueden ser monitorizadas. Estos ensayos están típicamente limitados a sustratos 2D con condiciones uniformes, aunque diferentes gradientes podrían ser creados mediante técnicas de microfabricación (Simpson et al., 2008). (vi) Ensayos en matrices 3D: los biomateriales de última generación permiten la creación de entornos 3D con características físicas y químicas modificables y controlables. Las células se cultivan en un medio 3D y migran dependiendo de la arquitectura, la rigidez, el tamaño de los poros, la concentración de ligandos etc. (Sabeh et al., 2009). (vii) Microfluidica: avances recientes en tecnologías de fabricación han hecho posible la creación de microdispositivos que permiten el control del microentorno celular. Con estos aparatos, es muy sencillo establecer campos eléctricos, gradientes químicos y diferentes condiciones de flujo intersticial que pueden ser controlados y ajustados libremente en los diferentes microcanales que conforman el dispositivo (Chung et al., 2010, Polacheck et al., 2011). En resumen, los ensayos con micropipeta, cámaras de Boyden y dispositivos de microfluídica permiten el control de gradiantes bioquímicos. Los ensayos de durotaxis, matrices 3D y nuevamente los dispositivos de microfluídica, facilitan el control de las características biofísicas como la rigidez de la matriz o el flujo intersticial. Por otro lado, los ensayos de regeneración de heridas y micropatrones permiten el control de las distancias intercelulares, mientras que sólo estos últimos ofrecen un control sobre la topografía del sustrato. I.1.3. Modelado de la migración celular Existe una cantidad inmensa de trabajos computacionales y modelos matemáticos en relación con aspectos específicos o generales de la migración celular. De hecho, la importancia que ha demostrado en numerosas enfermedades y procesos patológicos ha supuesto una gran inversión, provocando que un gran número de matemáticos, ingenieros 8 Multiscale computational modeling of single cell migration in 3D y físicos hayan redireccionado sus investigaciones al campo de la biofísica celular. Como resultado, los detalles y la sofisticación de los modelos de migración han aumentado notablemente. Sin embargo, a pesar de que cada modelo ofrece características únicas, se pueden clasificar en tres grandes grupos dependiendo del principal área de interes: modelos de protrusión celular, modelos de adhesión y retracción, modelos de la célula completa. A continuación se describen brevemente los principios físicos básicos sobre los que se basa cada grupo, así como sus posibilidades e interacciones con otros modelos. En cualquier caso, diferentes publicaciones revisan ésta y otras posibles clasificaciones, ofreciendo una visión más amplia y balanceada del estado del arte de modelos biofísicos (Flaherty et al., 2007, Carlsson and Sept, 2008, Rangarajan and Zaman, 2008, Mogilner, 2009). Modelos de protrusión celular La protrusión es probablemente el fenómeno más estudiado y mejor comprendido de todas las fases de la migración celular. Como se ha dicho anteriormente, el evento clave de la migración basada en actina es la polimerización de filamentos contra la membrana celular. Muchos trabajos matemáticos se han centrado en este tema, aunque pueden distinguirse según si consideran un único o múltiples filamentos de actina o si usan una aproximación contínua (Carlsson and Sept, 2008). Los modelos de un único filamento pueden identificar mecanismos plausibles para la generación de esfuerzos, prediciendo la fuerza de saturación y estableciendo una relación de fuerza-velocidad. Normalmente consideran el crecimiento del filamento por difusión de monómeros de actina. Fluctuaciones Brownianas de la membrana y del extremo del filamento crean huecos que son rellenados por nuevos monómeros. De hecho, si la concentración de monómeros es suficientemente alta, su incorporación impide que la membrana vuelva a su posición original, un proceso conocido como “rueda Browniana” (Brownian-ratchet) (Peskin et al., 1993). Estos tipos de modelos han sido extendidos para incluir conexiones entre los extremos de los filamentos y la membrana, lo cual puede establecer nuevos mecanismos en la generación de fuerzas afectando a la relación de fuerza-velocidad (Dickinson and Purich, 2002, Zhu and Carlsson, 2006). Los modelos de múltiples filamentos tratan la polimerización simultánea de muchos filamentos, incluyendo las interacciones entre ellos y con las proteínas de entrecruzamiento. La propulsión de patógenos intracelulares y la protrusión del lamelipodio y los filopodios celulares han sido los problemas que mayor atención han Capítulo I. Resumen 9 recibido en este tipo de modelos. Por ejemplo, los filopodios y los entramados de actina empujando contra la membrana (Atilgan et al., 2005, Atilgan et al., 2006) o el movimiento de obstáculos debido al crecimiento de una red de actina (Carlsson, 2001) han sido estudiados con modelos que incluyen procesos a nivel molecular como el crecimiento de los filamentos de actina en los extremos positivos y la depolimerización en los negativos, la formación de ramas, su ruptura etc. Aproximaciones incluso más detalladas tienen en cuenta las interacciones de la actina con los obstáculos que se encuentra o la variación espacial de la concentración de la misma (Alberts and Odell, 2004). Los modelos continuos describen el entramado de actina utilizando parámetros más simples como la concentración o el número de filamentos en contacto con la membrana. Estos métodos permiten predecir comportamientos de sistemas mucho más grandes y en escalas de tiempo mayores, a costa de tratar los eventos moleculares a partir de un comportamiento general promedio. Igual que en los modelos de múltiples filamentos, las aproximaciones contínuas se suelen centrar en la protrusión del lamelipodio o los filopodios celulares (Mogilner and Edelstein-Keshet, 2002, Dawes et al., 2006) así como la propulsión intracelular de patógenos (Gerbal et al., 2000, Mogilner and Oster, 2003). Sus predicciones incluyen las propiedades del dominio de actina como su grosor, su concentración, la relación fuerza-velocidad, la velocidad de protrusión durante la expansión del lamelipodio, las propiedades elásticas y difusivas de la membrana o la distribución de las longitudes de los filamentos y sus orientaciones, todas las cuales son observables experimentalmente. Modelos de adhesión-retracción A pesar del mayor dinamismo de los eventos que ocurren en el frente de avance celular, los mecanismos involucrados en la formación de adhesiones con el sustrato y la retracción de las zonas traseras son igualmente importantes. Aunque algunos tipos de células utilizan otros tipos de retracción, la retracción basada en miosina es la más común. De hecho la miosina II ha sido incluida en numerosos modelos como la causa principal de retracción y contracción celular (Ahmadi et al., 2005, Kruse et al., 2005). En cualquier caso, las fibras de estrés y los módulos contráctiles no pueden producir movimiento o esfuerzos sin una unión mecánica que los transmita al sustrato. Por tanto, la formación de adhesiones focales y la interacción célula-matriz han sido objeto de modelos, unos bajo la 10 Multiscale computational modeling of single cell migration in 3D premisa de que las fuerzas aplicadas sobre los filamentos tienen un efecto directo en la polimerización (Shemesh et al., 2005) y otros teniendo en cuenta la interacción entre las fibras de estrés y las integrinas (Novak et al., 2004). Modelos de célula completa Considerando el cuerpo celular completo existe una variedad de modelos tan amplia como los tipos posibles de célula. Estos modelos consideran normalmente fenómenos dinámicos y mecánicos incluyendo procesos explícitos a escala molecular como la polimerización de actina, o menos concretos como la desestabilización de adhesiones focales debida a fuerzas. Igualmente, la mecánica celular se puede tratar explícitamente o considerarse de forma fenomenológica. Los métodos pueden ser continuos (Gracheva and Othmer, 2004), discretos (Dokukina and Gracheva, 2010) o basados en elementos finitos (Bottino et al., 2002) entre otros. Además, diferentes niveles de detalle son incluidos en cada modelo, con escalas de tiempo y longitud muy diferentes entre ellos. Muchas de estas aproximaciones han estado comúnmente centradas en migración 2D, no sólo por simplicidad sino por la falta de datos de alta calidad en movimientos 3D. Sin embargo, el número de modelos 3D ha estado creciendo gradualmente, aunque con aproximaciones muy diversas. Algunas de ellas predicen la migración de células individuales (Zaman et al., 2005, Borau et al., 2011, Schluter et al., 2012) mientras otras simulan comportamientos colectivos (Ouaknin and Bar-Yoseph, 2009, Arciero et al., 2011). Según sus hipótesis principales se pueden clasificar en: modelos basados en fuerzas, modelos estocásticos, modelos de esferoides celulares, y estudios de Monte Carlo. En los primeros, la dinámica de migración se basa en el balance de fuerzas entre la parte trasera y delantera de la célula, así como en las fuerzas de protrusión y la fuerza de arrastre que se opone al movimiento debido a la viscosidad de la matriz (Zaman et al., 2005). El handicap de estos modelos es que sólo predicen la migración de células individuales y no tienen en cuenta cambios en la forma de la célula o en las propiedades de la matriz debido a la degradación. Por otro lado, los modelos estocásticos (o de caminos persistentes aleatorios) son capaces de predecir el movimiento de poblaciones (Parkhurst and Saltzman, 1992), pero no incluyen efectos dinámicos como las fuerzas de tracción o de arrastre, ni incorporan las propiedades del sustrato. Los modelos de multiesferoides celulares, se basan principalmente en gradientes de presión producidos por la proliferación y muerte de células (Pettet et al., 2001). Combinando movimientos aleatorios, presión y actividad química de los agregados celulares, estos modelos son Capítulo I. Resumen 11 adecuados para el estudio de tumores, aunque no incluyen otros estímulos mecánicos importantes como la densidad, rigidez o porosidad de la matriz. Finalmente, los modelos de Monte Carlo que utilizan entramados cuadrados y un conjunto de reglas simples, permiten simulaciones muy rápidas, haciéndolos apropiados para el estudio de patrones migratorios a largo plazo (Zaman, 2007, Zaman et al., 2007). Su mayor problema es la naturaleza cualitativa de los parámetros estudiados como la polarización celular o los efectos mecánicos de la matriz. I.2. Objetivos y metodología Basados en los argumentos expuestos hasta ahora, el objetivo de esta Tesis es el desarrollo de modelos númericos y computacionales para simular diversos aspectos de la migración como las interacciones célula-matriz o la respuesta mécanica de la célula en función de las condiciones del entorno. Con este propósito, a lo largo de la Tesis se han utilizado aproximaciones continuas, discretas y/o basadas en elementos finitos. El trabajo principal y los objetivos parciales se describen a continuación:  Estudiar el efecto de factores mecánicos (rigidez del sustrato, geometría y condiciones de contorno) en la migración celular en matrices 3D. Esto implica el desarrollo de un modelo de elementos finitos macroscópico basado en el fenómeno mecanosensor que incluya las fases principales del ciclo migratorio.  Explorar la estructura interna del citoesqueleto celular y su respuesta a la rigidez de la matriz extracelular. Se ha utilizado un modelo computacional de partículas basado en dinámica Browniana para polimerizar una red de actina tridimensional entrecruzada con proteínas. Además, se han incluido motores moleculares para analizar su comportamiento dinámico y su papel como elementos mecanosensores. A su vez, los resultados a escala micro se han utilizado para validar las hipótesis del modelo mecanosensor.  Incorporar los hallazgos obtenidos con el modelo intracelular en el modelo macroscópico. La estabilización temporal de la red de actina provocada por el bloqueo gradual de los motores moleculares se introduce en el modelo mecanosensor a través de una función de regulación continua que reproduce la saturación de las fuerzas generadas por el sistema de actina-miosina. 12 Multiscale computational modeling of single cell migration in 3D  Estudiar las condiciones fluído-químicas a las que está sujeta una célula en un medio poroso, simulando para ello un sistema de microfluídica utilizando elementos finitos.  Mejorar la comprensión de la respuesta celular a diversos factores simultáneos. Con este propósito, se ha desarrollado un modelo probabilístico de voxels basado en elementos finitos para simular la migración de una célula individual dependiendo de estímulos mecánicos y fluido-químicos.  Analizar el comportamiento dinámico del lamelipodio celular, incluyendo el papel que juegan la Vinculina, los motores moleculares y las adhesiones focales en las funciones exploratorias de la célula, desarrollando para ello un modelo de diferencias finitas. I.3. Organización de la Tesis La Tesis se estructura en siete capítulos y tres apéndices. Aunque los contenidos de cada capítulo son tratados individualmente, gran parte del material está interrelacionado y por lo tanto se requiere la consideración del trabajo en conjunto para apreciar el cuadro completo. Más específicamente la estructura de la Tesis se resume a: Capítulo I (actual): contiene un resumen de la Tesis escrito en castellano que incluye los objetivos de la misma, la organización del documento, las conclusiones generales, las contribuciones científicas y las posibles líneas futuras. Capítulo 1: es el capítulo equivalente al actual (en inglés) y sirve como introducción incluyendo el estado del arte, motivaciones, objetivos y organización de la tesis. Capítulo 2: inicialmente presenta el fenómeno mecanosensor así como los modelos existentes que relacionan este proceso con la movilidad celular. Después describe un modelo de migración celular individual en 3D en el que se supone que el mecanismo mecanosensor es el principal regulador del movimiento direccional. A continuación se detallan las principales hipótesis, las fases consideradas (mecanosensora, de orientación del citoesqueleto y de migración) y los métodos numéricos utilizados. Finalmente se comparan los resultados obtenidos con experimentos de la bibliografía y se somete el modelo a un análisis de sensibilidad para comprender mejor su comportamiento. Capítulo I. Resumen 13 Capítulo 3: introduce el papel dinámico del citoesqueleto celular en los procesos migratorios y describe la importancia de los motores moleculares en las redes de actina como elementos mecanosensores. Seguidamente, describe un modelo computacional de partículas basado en dinámica Browniana, detallando los métodos utilizados para la formación de las redes, la paralelización del problema y las condiciones de contorno consideradas. Para concluir se comparan los resultados computacionales con experimentos recientes. Capítulo 4: está dedicado enteramente a la revisión de las hipótesis del modelo mecanosensor desarrollado en el Capítulo 2, introduciendo en la respuesta celular una dependencia temporal basada en los resultados obtenidos en el modelo de partículas descrito en el Capítulo 3. Esta nueva aproximación se utiliza para simular experimentos relacionados con la detección de la rigidez del sustrato, incluyendo cambios bruscos en la misma y la saturación de fuerzas en el equilibrio. Este capítulo incluye también un análisis de sensibilidad y describe la posibilidad de extensión del modelo teórico a geometrías 3D más complejas. Capítulo 5: reúne las hipótesis consideradas hasta ahora y las incluye en un nuevo modelo de migración a escala celular. Este capítulo enfatiza la importancia de los múltiples estímulos que aparecen en entornos 3D y describe un modelo probabilístico basado en voxels y elementos finitos que simula la respuesta celular dependiendo de diversos factores como la rigidez de la matriz extracelular, la tensión, el flujo o las condiciones químicas. Primero presenta la simulación con elementos finitos de un dispositivo de microfluídica real. Después se evalúa el comportamiento de las funciones de probabilidad que determinan el comportamiento celular. Finalmente el capítulo resume las implicaciones del modelo y sus posibles aplicaciones. Capítulo 6: describe el papel crítico de los componentes dinámicos del lamelipodio durante los ciclos de protrusión-retracción de la membrana celular. El capítulo comienza reseñando los últimos descubrimientos de las estructuras que conforman el frente celular así como los modelos existentes en relación al tema. A continuación, se presenta un modelo basado en diferencias finitas utilizado para validar experimentos de cualquier fenómeno involucrado en el lamelipodio en general, y de los efectos de la Vinculina en el flujo retrógrado de actina en particular. 14 Multiscale computational modeling of single cell migration in 3D Capítulo 7: cierra la Tesis con un resumen de todo el trabajo realizado incluyendo las conclusiones más relevantes de cada capítulo. Adicionalmente, presenta una lista con las contribuciones originales conseguidas (publicaciones y congresos) y propone una serie de líneas futuras para la continuación del trabajo. I.4. Resumen El principal objetivo de la tesis es el desarrollo de modelos numéricos computacionales que permitan una mejor comprensión de los mecanismos que controlan la migración celular. Con este propósito, se han usado diversas aproximaciones contínuas, discretas o basadas en elementos finitos. Por ejemplo, para estudiar el efecto de los factores mecánicos (rigidez del sustrato, geometría y condiciones de contorno) sobre la migración en matrices tridimensionales, se ha desarrollado un modelo de migración de elementos finitos basado en el mecanismo mecanosensor incluyendo las fases principales del ciclo migratorio. Este modelo, descrito en el Capítulo 2, es capaz de simular el movimiento preferencial de una célula individual bajo diferentes condiciones mecánicas, prediciendo velocidades y patrones de migración similares a los obtenidos experimentalmente. En esta aproximación, el comportamiento mecánico de todo el cuerpo celular se simplifica a dos muelles en paralelo con un actuador en serie (equivalente a un tercer muelle precomprimido), lo cual implica un aumento de la fuerza ejercida por la célula conforme aumenta la rigidez del sustrato hasta alcanzar la saturación. Además, utilizando un balance de fuerzas junto con la hipótesis de que la fricción de la matriz aumenta con la rigidez (debido a mayores densidades del material), se captura el comportamiento bifásico de la velocidad migratoria. El modelo se ha utilizado también para simular un experimento concreto en el que una micro-aguja se introduce en el sustrato en los alrededores de la célula tirando o empujando hacia ella, modificando el estado tensional y, por tanto, provocando que la célula migre y se reoriente de una forma determinada. Las simulaciones sugieren que la célula en realidad no siente directamente las fuerzas, sino la tracción o compresión espacial de las zonas que la rodean causadas por cargas externas. Por tanto, si la micro-aguja estira el sustrato, las fuerzas contráctiles de la célula se oponen a las de la aguja, sintiendo una mayor rigidez en esa dirección y tendiendo a reorientarse y moverse ella. En cambio, si la micro-aguja empuja, la compresión de la célula ocurre en el mismo sentido que la de la aguja y por tanto siente menor rigidez y Capítulo I. Resumen 21 organización y adaptación de su estructura determina el comportamiento de la célula. Para comprender mejor estos mecanismos, se requiere un modelado más detallado de las redes de actina teniendo en cuenta sus principales componentes (proteínas de entrecruzamiento y motores moleculares). 4. Las células se contraen para evaluar las propiedades mecánicas de su entorno. Las simulaciones sugieren que la célula en realidad no siente directamente las fuerzas externas, si no el estado de deformación espacial que sufre el sustrato al que está anclada.  Modelado de redes dinámicas de actina entrecruzadas con proteínas 1. Al menos dos componentes son necesarios para simular una red de actina: filamentos de actina y proteínas de entrecruzamiento. Simplemente incluyendo el comportamiento dinámico de estas proteínas así como de los motores moleculares, las redes de actina simuladas exhiben comportamientos macroscópicos similares a los observados experimentalmente, indicando que las propiedades microscópicas de cada componente individual determinan la respuesta global de la red. 2. Los motores moleculares juegan un papel central como mecanosensores. De entre todos los factores estudiados, aquellos afectando la dinámica de los motores son los que más han influenciado la respuesta de la red. 3. La tensión ejercida por la célula es proporcional a la rigidez de la matriz extracelular en sustratos blandos y constante (se satura) para altas rigideces. La existencia de esta transición puede explicarse mediante los mecanismos que producen el bloqueo de los motores moleculares, constituyendo por lo tanto un posible mecanismo mecanosensor: (i) todos los puntos de anclaje (unión del motor con la actina) de los alrededores están ocupados por otros motores o proteínas, (ii) las fuerzas soportadas son demasiado altas, (iii) se ha llegado al extremo del filamento de actina. 4. Los modelos de partículas son especialmente útiles para estudios detallados de cualquier tipo de fenómeno relacionado con las redes de actina como el mecanismo mecanosensor, u otros como el fenómeno de rigidización /reblandecimiento del citoesqueleto, la fluidificación de su estructura o la formación de anillos de actina entre otros. Sin embargo, estos modelos conllevan altos costes computacionales incluso utilizando técnicas de 22 Multiscale computational modeling of single cell migration in 3D paralelización, haciéndolos inadecuados para simulaciones completas del citoesqueleto.  Respuesta temporal del proceso mecanosensor 1. La dependencia temporal es esencial para capturar comportamientos de célula reales. Introduciendo esta dependencia por medio de una variable interna que representa la evolución del bloqueo de motores es suficiente para reproducir resultados observados experimentalmente, permitiendo simulaciones que duran segundos en lugar de semanas, haciendo por tanto la aproximación adecuada para su incorporación en modelos de elementos finitos con mayores dimensiones y escalas temporales. En cualquier caso, salvo en condiciones muy específicas (y no biológicas) como cambios súbitos de la rigidez extracelular, la respuesta mecanosensora de las células a largo plazo está gobernada únicamente por el equilibrio mecánico. 2. El bloqueo de motores aumenta exponencialmente hasta que satura, con un tiempo de relajación independiente de la rigidez del sustrato. Aunque experimentalmente se ha encontrado que la relajación es dependiente de la rigidez a pequeñas escalas de tiempo, el modelo ha confirmado que esta dependencia es despreciable a largo plazo. 3. El comportamiento mecanosensor del cuerpo completo de la célula se puede representar mediante dos muelles en paralelo y un actuador en serie con uno de los muelles. A diferencia de lo expuesto anteriormente, este actuador es equivalente a un muelle pre-comprimido, cuya longitud de equilibrio es llevada a cero mediante una función de relajación. 4. Esta configuración tri-muelle permite distinguir entre las respuestas activas y pasivas de la célula. Al añadir la rigidez de la actina en serie con el actuador, es posible capturar la linealidad de la fuerza celular con respecto a la rigidez del sustrato para bajas rigideces y su saturación para altas. Además, el ratio entre la rigidez activa y pasiva permite regular la respuesta celular bajo diferentes condiciones mecánicas, jugando la rigidez de la actina un papel fundamental especialmente en sustratos rígidos y permitiendo la simulación de procesos más complejos como rupturas en la red del citoesqueleto o la rigidización por formación de fibras de estrés. Capítulo I. Resumen 23 5. Se ha demostrado experimentalmente que la reacción celular ante cambios bruscos de la rigidez del sustrato se produce casi instantáneamente, confirmando que la adaptación del citoesqueleto es un proceso puramente mecánico como sugiere el modelo.  Simulación de la migración en función de múltiples estímulos 1. El establecimiento de leyes fenomenológicas es necesario para el estudio de mayores escalas espacio-temporales. Al discretizar tanto la célula como el sustrato mediante voxels y considerar que cada voxel celular se comporta siguiendo el modelo mecanosensor descrito previamente, el modelo de migración es capaz de predecir, en escalas temporales del orden de horas, aspectos macroscópicos del movimiento celular como trayectorias, velocidades, área de adhesión, relación de aspecto, tensiones etc. 2. Al definir el comportamiento individual de cada voxel celular a través de funciones de probabilidad, el comportamiento macroscópico emerge de manera natural. 3. La consideración o no del volumen de una sóla célula en la simulación de un microdispositivo completo no afecta a los resultados globales. Por tanto, salvo que se modelen grandes poblaciones celulares, el cuerpo de la célula se puede despreciar y la solución del estado estacionario se puede considerar constante durante las simulaciones de migración, permitiendo desacoplar el cálculo mecánico y ahorrando coste computacional. Aun así, debido a la normalmente irregular geometría de las células reales, los efectos del flujo y la concentración química a lo largo de la superficie celular deberían ser tenidos en cuenta para una consideración completa de dichos factores. 4. Comparado con una aproximación contínua, la geometría de una matriz porosa afecta a la migración, guiando a la célula a través de los poros y modificando el proceso mecanosensor. Aunque una matriz contínua es suficiente para capturar tendencias migratorias generals, la consideración de factores como la guía por contacto o la presión hidrostática sería necesaria para obtener simulaciones realistas de migraciones 3D. 24 Multiscale computational modeling of single cell migration in 3D  Simulación del lamelipodio y el efecto de las adhesiones focales 1. Una aproximación 1D es adecuada para simular el lamelipodio celular y estudiar el papel de la vinculina como “embrague” del flujo retrógrado de actina. De hecho, un modelo simple es suficiente para predecir comportamientos generales como los ciclos de protrusión/retracción de la membrana, la activación periódica de la miosina o los efectos cuando la polimerización de actina o la actividad de los motores moleculares son alterados mediante agentes químicos. 2. La convolución de matrices y la correlación de funciones son herramientas matemáticas de gran potencial que pueden ser aplicadas para la medida y el análisis de experimentos mediante tratamiento de imágenes. 3. Desactivar la vinculina en una célula altera las condiciones de fricción y polimerización en el lamelipodio. Asumiendo que la polimerización aumenta pero manteniendo una cantidad total de actina, el modelo es capaz de predecir los efectos en las células deficitarias de vinculina (Vcl-KO) observados experimentalmente. 4. Desactivar la vinculina provoca un incremento sustancial en la velocidad del flujo de actina. Las células de control (WT) así como las células con vinculina pre-activada (PA) presentan flujos más lentos mientras que las células con vinculina cuyo dominio de unión con la actina está desactivado (dAB, PAdAB) muestran velocidades intermedias, sugiriendo que la interacción actina-vinculina es el determinante fundamental del flujo retrógrado. 5. La disminución de la fricción produce lamelipodios más largos y mayores velocidades de flujo, mientras que la periodicidad de las contracciones no se ve afectada. Por tanto, este periodo puede ser considerado proporcional a la longitud del lamelipodio e inversamente proporcional a la velocidad de la actina. 6. En general, teniendo en cuenta las hipótesis utilizadas en el cálculo, el desfase negativo entre las curvas de velocidad de la actina y la membrana demuestra que los movimientos de la membrana siguen a los de la actina. Ya que este fenómeno concreto no ha sido mostrado previamente en ningún trabajo científico, más experimentos son necesarios para confirmarlo. Capítulo I. Resumen 25 I.6. Contribuciones científicas A continuación se describen brevemente las principales contribuciones originales de esta Tesis:  Formulación de un modelo de migración individual en 3D basado en el mecanismo mecanosensor, es decir, dependiente de las condiciones mecánicas de la matriz extracelular y de las cargas externas.  Simulaciones númericas de elementos finitos para determinar patrones migratorios, fuerzas y velocidades para diferences condiciones de contorno y fuerzas externas.  Adaptación de un modelo computacional de partículas basado en dinámica Browniana para estudiar el fenómeno mecanosensor en una red de actina entrecruzada con proteínas.  Extensión del modelo mecanosensor para incluir la respuesta temporal basada en los resultados del modelo de partículas.  Análisis de la influencia de los parámetros mecánicos y estudio de la respuesta del sistema ante cambios bruscos de la rigidez del sustrato.  Desarrollo de un modelo fenomenológico probabilístico basado en voxels y elementos finitos para el estudio de la migración celular en 3D bajo diferentes condiciones de contorno y la actuación de múltiples factores externos: mecánicos y fluído-químicos.  Simulaciones de elementos finitos en 3D de un dispositivo de microfluídica para determinar el flujo y el gradiente químico a través de un material poroso.  Formulación de un modelo continuo para describir el comportamiento del lamelipodio y estudiar la influencia de las adhesiones focales en el flujo retrógrado de actina.  Adaptación de una herramienta para medir el flujo de actina y el movimiento de la membrana a partir de microscopía de fluorescencia (Fluorescent Speckle Microscopy).  Desarrollo de una herramienta para analizar la periodicidad de los ciclos de protrusión/retracción de la membrana y su correlación con la velocidad de la actina. 26 Multiscale computational modeling of single cell migration in 3D  Análisis del efecto de diferentes sustancias químicas que inhiben o incrementan la polimerización de actina, la actividad de la miosina o la fuerza de las adhesiones focales. I.6.1. Publicaciones A continuación se enumeran las publicaciones realizadas durante el desarrollo de la presente Tesis en revistas internacionales. 1. BORAU, C., KAMM, R.D. & GARCÍA-AZNAR, J.M. 2011. Mechano-sensing and cell migration: a 3D model approach. Physical Biology, 8, 066008. 2. BORAU, C., KIM, T., BIDONE, T., GARCIA-AZNAR, J.M. & KAMM, R.D. 2012. Dynamic mechanisms of cell rigidity sensing: insights from a computational model of actomyosin networks. PLoS One, 7, e49174. 3. BORAU, C., GARCIA-AZNAR, J.M. & KAMM, R.D. 2013. A time-dependent phenomenological model for cell mechano-sensing. (Enviado) 4. BORAU, C., POLACHECK, W.J., GARCIA-AZNAR, J.M. & KAMM, R.D. 2013. Probabilistic voxel-FE model for single cell motility in 3D (En proceso) 5. BORAU, C., THIEVESSEN, I., GARCIA-AZNAR, J.M. & FABRY, B. 2013. Modeling lamellipodium dynamics and effects of vinculin on rearward actin flow. (En proceso). I.6.2. Contribuciones a congresos El trabajo desarrollado durante la tesis ha sido expuesto en diferentes congresos de ámbito nacional e internacional. Los más relevantes son: 1. BORAU, C., DOWEIDAR, M.H., GARCÍA-AZNAR, J.M., Mechanosensing and Cell Migration: 3D modeling. XXVIII Congreso Anual de la Sociedad Española de Ingeniería Biomédica (CASEIB), Madrid (Spain) 2010. 2. GARCÍA-AZNAR, J.M., BORAU, C., KAMM, R. D., Three-Dimensional modelling of cell migration due to active mechanosensing. TERMIS-EU, Galway (Ireland), 2010. Capítulo I. Resumen 27 3. GARCÍA-AZNAR, J.M., BORAU, C., KAMM, R. D., Modeling of cell migration and mechano-sensing in 3D. 17th Congress of the European Society of Biomechanics (ESB), Edinburgh (UK), 2010 4. BORAU, C., KAMM, R. D., KIM, T, GARCÍA-AZNAR, J.M., Exploring Rigidity Sensing and Acto-myosin based Contractility: a 3-D Actin Network Computational Model. 4th European Cell Mechanics Meeting (CellMech). Amsterdam (Netherlands) 2011. 5. GARCÍA-AZNAR, J.M., BORAU, C., KAMM, R. D., Modelling mechanosensing in cell-material interaction: implications for tissue engineering. TERMIS-EU Annual Meeting, Granada (Spain), 2011. 6. BORAU, C., KAMM, R. D., KIM, T, GARCÍA-AZNAR, J.M., Computational Model of 3-D Cross-linked Actin Networks: mechanosensing behaviour of cells. II International Conference on Particle-based Methods. Fundamentals and Applications (PARTICLES). Barcelona (Spain), 2011. 7. GARCÍA-AZNAR, J.M., BORAU, C., KIM, T, KAMM, R. D., Computational Model of 3-D Cross-linked Actin Networks: Mechanosensing Behaviour of Cells. II International conference on particle based methods Fundamentals and applications (PARTICLES), Barcelona (Spain), 2011. 8. GARCÍA-AZNAR, J.M., BORAU, C., KAMM, R. D., Models of cell migration: implications for tissue engineering. II International Conference on Tissue Engineering (ICTE), Lisbon (Portugal), 2011. 9. BORAU, C., KAMM, R. D., KIM, T, GARCÍA-AZNAR, J.M., Acto-myosin system Contraction: computational study of 3D actin networks and their response to extracellular matrix stiffness. I Reunión del Capítulo Nacional Español de la Sociedad Europea de Biomecánica (ESB-Cap Esp), Zaragoza (Spain), 2011. 10. KAMM, R. D., BORAU, C., KIM, T, GARCÍA-AZNAR, J.M., Computational insights into cytoskeletal rheology. Biophysical Society Meeting, Baltimore, MD, (USA), March, 2011. 11. KAMM, R. D., BORAU, C., KIM, T, GARCÍA-AZNAR, J.M., Computational models of the cytoskeleton and a look forward to simulation of mechanotransduction. Microscale Modeling in Biomechanics and Mechanobiology, Ericeira (Portugal), 2011. 28 Multiscale computational modeling of single cell migration in 3D 12. KAMM, R. D., BORAU, C., KIM, T, GARCÍA-AZNAR, J.M., A Computational, Brownian Dynamics Simulation of Cytoskeletal Mechanics, The 1st KIAS Conference on Subcellular Dynamics, Seoul (Korea), 2011. 13. KAMM, R. D., BORAU, C., KIM, T, GARCÍA-AZNAR, J.M., The unique biomechanical properties of the cell: Insights from computational modeling. Nanyang Technological University, 2011. 14. BORAU, C., KAMM, R. D., KIM, T, GARCÍA-AZNAR, J.M., Cell cytoskeleton dynamics: mechanosensing properties. I3A -Jornada de jóvenes investigadores, Zaragoza (Spain), 2012. 15. GARCÍA-AZNAR, J.M., BORAU, C., KIM, T, KAMM, R. D., Mechanosensing: from a discrete to a continuum approach. 8th European Solid Mechanics Conference, Graz (Austria), 2012. 16. BORAU, C., KAMM, R. D., GARCÍA-AZNAR, J.M., Computational model of 3D single cell migration: a mechano-chemical approach. 18th Congress of the European Society of Biomechanics (ESB), Lisbon (Portugal), 2012. 17. KAMM, R. D., BORAU, C., KIM, T, GARCÍA-AZNAR, J.M., Biomechanical properties of the cell: Insights from computational modeling. Society of Physical Regulation and Biomolecular Modeling, San Juan (Puerto Rico), 2012. 18. KAMM, R. D., BORAU, C., KIM, T, GARCÍA-AZNAR, J.M., The unique biomechanical properties of the cell: Insights from computational modeling. Worcester Polytechnic Institute (WPI), Worcester, MA (USA), 2012. 19. KAMM, R. D., BORAU, C., KIM, T, GARCÍA-AZNAR, J.M., The unique biomechanical properties of the cell: Insights from computational modeling. International Federation for Medical & Biological Engineering (IFMBE), Beijing (China), 2012. 20. GARCÍA-AZNAR, J.M., BORAU, C., BIDONE, T., KIM, T, KAMM, R. D., Modeling 3D actin networks: dynamics of cell mechanosensing. III International Conference on Particle-based Methods Fundamentals and Applications, Stuttgart (Germany), 2012 21. BORAU, C., THIEVESSEN, I. GARCÍA-AZNAR, J.M., FABRY, B., Lamellipodium dynamics and rearward actin flow depend on vinculin. 19th Congress of the European Society of Biomechanics (ESB), Patras (Greece), 2013. (Candidato a los premios de la ESB para estudiantes) Capítulo I. Resumen 29 I.7. Desarrollo futuro La variedad de modelos presentados en esta Tesis es sólo una pequeña contribución a la comprensión del comportamiento celular en el proceso de migración. Los resultados y conclusiones obtenidas durante este trabajo provocan a su vez nuevas preguntas y abren posibles líneas de investigación futuras. Las más importantes se describen a continuación:  Extensión de los modelos de migración individual para simular poblaciones celulares Los modelos presentados en los Capítulos 2 y 5 son útiles para aislar los efectos específicos de ciertos parámetros y de su influencia en la migración celular individual. Sin embargo, las interacciones célula-célula dan lugar a complejos cambios en la estructura y procesos de tejidos pluricelulares, como por ejemplo la regeneración de la piel o la vascularización. Por tanto, la incorporación de tales interacciones entre distintas células extendería enormemente el potencial de los modelos. Aunque algunos aspectos técnicos deberían ser revisados, los modelos actuales están, de hecho, preparados para la inclusión de múltiples células.  Estudio de diferentes estructuras de actina relevantes para el comportamiento celular Extisten numerosas estructuras formadas en su mayor parte por actina implicadas en diferentes procesos celulares que podrían ser estudiadas con el modelo de partículas de dinámica Browniana descrito en el Capítulo 3. Por ejemplo las fibras de estrés (compuestas por filamentos de actina entrecruzados de forma paralela con proteínas y motores de miosina) son cruciales para distribuir las fuerzas celulares a lo largo del citoesqueleto, y juegan un papel fundamental en la contracción celular o la morfogénesis entre otros procesos. Los anillos de actina que aparecen en la citoquinesis (la división del citoplasma durante la proliferación celular) podrían también ser simulados con el citado modelo. En estos anillos, la miosina constriñe la membrana de la célula hasta que ocurre la escisión. Fenómenos como la hidrólisis de ATP o estructuras adicionales como los microtúbulos deberían ser incorporadas para analizar debidamente este proceso. Igualmente se podrían estudiar los arcos de actina que se forman en el frente de la 30 Multiscale computational modeling of single cell migration in 3D célula y que sirven de elementos estructurales donde se forman y maduran las adhesiones focales, que a su vez forman una base firme sobre la cual polimeriza la actina y empuja la membrana en movimientos exploratorios de protrusión. Combinando el modelo de partículas con las hipótesis teóricas descritas en el Capítulo 6, permitiría un análisis muy detallado del comportamiento del lamelipodio y sus componentes.  Desarrollo de las interacciones célula-matriz Las interacciones célula-matriz se producen a través de receptores transmembrana (integrinas) y adhesiones focales, actuando como transductores de señales y presentando comportamientos muy complejos y dinámicos. Su simulación en detalle no se ha considerado a lo largo de esta Tesis. Por ejemplo, en el modelo de migración descrito en el Capítulo 5, se considera que la totalidad de la superficie celular se adhiere al sustrato y por tanto todo los voxels del perímetro ejercen fuerzas de contracción. En realidad, sólo algunas partes de las células se adhieren firmemente al sustrato y sólo algunas de ellas forman adhesiones focales capaces de ejercer fuerzas apreciables. Por tanto, un modelo detallado de la formación y maduración de adhesiones focales sería interesante para comprender realmente su papel en la migración celular. Además, mecanismos como la polimerización de actina o la presión hidrostática interna, especialmente importantes en materiales porosos y con poros pequeños, tampoco han sido incluidas en los modelos de esta Tesis, lo cual sería necesario para un modelado completo y realista de migración en 3D.  Distinguir las diferentes partes de la célula En las simulaciones 3D de los Capítulos 3 y 5, se han diferenciado a grandes rasgos tres partes de la célula: núcleo, citoplasma y córtex. De una forma muy simplificada, se han considerado diferencias como el nivel de tensión máxima o la ausencia de componente contráctil en el núcleo (modelado como un material pasivo). Aunque esto se ha hecho así inicialmente para no complicar innecesariamente el modelo, los códigos han quedado preparados para su posterior desarrollo. Por ejemplo, la rigidez de la actina podría relacionarse con la polimerización de actina (o la densidad de actina) obtenida en experimentos. La concentración de proteínas de entrecruzamiento podría también determinar la Chapter 1. Introduction 37 traction sites for migration as the cell moves forward over them, and they are disassembled at the cell rear, allowing it to detach and contract (Figure 1.2C). Note that although this picture is common among different cell types, the specific details may differ to a large extent. For example, slow-moving cells such as fibroblasts clearly show these steps in contrast with fast-moving cells such as neutrophils, where the observation of the different phases is not as obvious. In addition, cell’s migratory behavior strongly depends on its environment, being the main cause of discrepancies between in vivo and in vitro cultures. 1.1.2. Numbers matter: collective versus single cell migration As just stated, cell migration is commonly understood as the movement of individual cells that undergo polarized extension-contraction cycles coupled with adhesion to and deadhesion from the surrounding substrate. However, at least one additional major principle is relevant to cell translocation in or on tissues: the movement of cell groups, sheets, or strands consisting of multiple cells that are mobile yet simultaneously connected by cell-cell junctions (Friedl et al., 2004)(Figure 1.1). Collective cell dynamics give rise to complex changes in multicellular tissue structures, including epithelial regeneration, the sprouting of vessels and ducts in angiogenesis and branching morphogenesis, and the deregulated invasion of cell masses during cancer progression and consecutive tissue destruction (Ilina and Friedl, 2009). In this way, collective cell migration shares similarities but also important differences to individually migrating cells. For instance, similarly to single-cell migration, collective cell movement results from actin polymerization and contractility coupled to cell polarity, however in the latter case, cells remain coupled by cell-cell junctions at the leading edge as well as in lateral regions and inside the moving cell group. Therefore, collective cell migration differs from single-cell migration in the simultaneous coordinated polarization of (often many) cells at the leading edge of the cell group; the translocation of cells through physical coupling and drag force; the activity of actin-rich lamellae in multiple cells along or underneath the cell collective; the secondary remodelling of the extracellular matrix along the migration track, leading to the formation of a basement membrane or the widening of a 3D track (macropatterning) to encompass an increasing volume of the cell mass; and the coordinated retraction of multiple cells at the rear end of the group (Friedl and Gilmour, 2009). 38 Multiscale computational modeling of single cell migration in 3D 1.1.3. Microenvironment: 2D versus 3D migration Mechanisms regulating cell motility have been extensively studied in 2D. Studies of cell migration in 3D cell culture systems and in vivo have revealed several differences when compared with cell migration flat surfaces, including their morphology and mechanical and signaling control (Friedl et al., 2012). Furthermore, 3D research has been rapidly growing as it entails a better representation of the microenvironmental conditions in living tissues. In fact, the desire to look within living organisms has led to the development of advanced technologies for real-time in vivo imaging. Nevertheless, in vitro models are still indispensable to isolate and define specific contributions of single factors to the overall process. Figure 1.2: The migration cycle. With the permission of Ridley et al., 2003 Scheme showing the main steps of cell migration: A) Cell polarization B) Protrusion and adhesion formation. C) Rear retraction. Chapter 1. Introduction 39 Three are the major types of ECM through which cells migrate in vivo (Even-Ram and Yamada, 2005): tightly packed basement membrane (Figure 1.3A), dense connective tissue (Figure 1.3B) and loose connective tissue (Figure 1.3C). To mimic these environments, the most commonly ECMs used in vitro are: collagen gels, cell-derived matrix (CDM) from fibroblasts, fibrin gels and basement membrane extract (BME or Matrigel) (Hakkinen et al., 2011). All of them are useful for exploring differences in cell morphology, signaling, adhesions and motility, which vary even for the same type of ECM depending on specific properties or whether the cells are plated on top or inside. For instance, ligand concentration on 2D surfaces is often low, whereas 3D ECMs usually contain tighltly packed clusters of ligands. In addition, cells on 2D substrates present marked lamellipodia and filopodia activity at the leading edge, mainly used for exploratory functions. In 3D, however, the more elongated and very thin morphology of cells entails that some structures can appear less obvious. For instance, focal adhesions are strong and clearly visible at the leading edge in flattened cells on 2D substrates, whereas cells in 3D exhibit adhesions all over their surface. Signalling is also affected by dimensionality: higher Rac activity is associated with spontaneous changes of cell migration direction on 2D surfaces (Pankov et al., 2005), much less persistent than cells in 3D ECMs. Furthermore, changes in substrate compliance may alter the distribution of integrins and several related signaling molecules (Katsumi et al., 2005). In sum, matrix organization, composition and biological activity likely modulate both normal and pathological cell migration and invasion. 3D ECM models together with advanced imaging techniques are contributing to an increasing understanding of physiological and pathological motility processes. Some of these systems permit to isolate and study the contribution of single components to the migration process; however, it is necessary to keep in mind that oversimplification of complex microenvironments in terms of structure, content and cellular interactions may lead to unreliable observations, thus reducing the relevance of the analysis. 40 Multiscale computational modeling of single cell migration in 3D Figure 1.3: Major types of in vivo ECM. Adapted from Even-Ram and Yamada, 2005 A) Dense collagen meshwork of the basement membrane visualized by scanning electron microscopy. B) Loose areolar connective tissue of mesentery showing collagen fibers, elastic fibers and fibroblasts. C) Irregular dense connective tissue of nipple skin with irregularly oriented densely packed collagen fibers. 1.1.4. Distinct modes of 3D cell migration How different cell types employ different mechanisms to efficiently move through structurally and chemically diverse 3D environments is not well understood. Despite the evidence that migrating cells express various proteases that are able to degrade ECM components, it is unclear whether 3D migration absolutely depends on proteolytic events. In fact, recent studies of cancer cell migration in 3D environments showed that metastatic cells can switch between adhesion-dependent mesenchymal (elongated) and adhesionindependent amoeboid (rounded) cell motility, driven by actin polymerization and actomyosin contraction, respectively (Wolf et al., 2003, Lammermann and Sixt, 2009). Currently, blebbing is the only known alternative to actin driven protrusion. Membrane blebs are actin filament-free cellular extensions, generated after cortex rupture or actinmembrane breakage by intracellular hydrostatic pressure. This pressure inflates the membrane until a new actin cortex is reassembled allowing the cycle to restart (Charras and Paluch, 2008). In addition, lobopodia (large blunt cylindrical protrusions) was recently found to be the predominant protrusion type of mesenchymal cells migrating in physiological 3D environments (Petrie et al., 2012), presenting features of both blebs and lamellipods. Interestingly, they also found that in 3D environments with nonlinear elastic features, lamellipodial locomotion dominates, whereas in a regimen showing linear elasticity, fibroblasts rather use lobopodia. Nevertheless, how cells show stable functional polarity, distinguish ECM elasticity or which features feed back on the extracellular environment to switch between migration modes are still open questions. Chapter 1. Introduction 41 1.1.5. Experimental research Experimental works on the cell migration field have grown enormously during the past decades, largely thanks to the development of new technologies and techniques which have accelerated even more the process. Nevertheless, isolating the effects of individual stimulus on cell migration in vivo still presents important difficulties. As a result, in vitro models have emerged as powerful tools for investigating cell motility, permitting to examine in detail specific factors guiding cell migration. Due to the vital importance of cancer, these in vitro assays are commonly used to understand the mechanisms driving tumor cell migration (usually chemical and mechanical clues) and aim to improve the efficacy of cancer therapy. The tumor microenvironment is comprised of a dynamic network of ECM proteins bathed in interstitial fluid and a host of associated cells including endothelial cells, fibroblasts, bone marrow-derived and infiltrating immune cells (reviewed in (Joyce and Pollard, 2009)). These stromal cells remodel the ECM and provide mechanical and chemical signals to the tumor cells. Due to the dynamic nature and the variety of components comprissing the tumor microenvironment, the investigation of the effect of individual stimuli on migration requires that the mechanical and chemical properties can be tuned precisely with reproducibility. In vitro assays provide such control and are well suited for dissecting the signaling pathways that govern cell migration in response to a specific factor of interest. In particular, different experimental methods to assay tumor cell migration in vitro are shown in Figure 1.4. A further description of these methods can be found in (Polacheck et al., 2012). As a brief summary, they can be classified in (viii) Micropipette assays: a micromanipulator-controlled pipette is used in the vicinity of the cell to inject a chemoattractant solution into the culture medium to establish a growth-factor gradient. These chemokine gradients, however, are transient and difficult to quantify (Soon et al., 2005). (ix) Boyden (or Transwell) chambers: a stiff porous membrane is incorporated between two cell culture chambers. Cells are seeded in suspension in the top chamber and migrate through the filter in response to a chemokine gradient, which is established by the different culture medium concentrations in the top and bottom chambers (Boyden, 1962). 42 Multiscale computational modeling of single cell migration in 3D (x) Micropatterning: cells are seeded on patterns of different geometry, size and surface coatings in 1D, 2D and 3D substrates. These assays offer scalability and the capability to culture large cell numbers, however, as they don’t usually include fluid flow, the application of localized mechanical stimuli or chemical gradients is generally not possible (von Philipsborn et al., 2006). (xi) Durotaxis assays: cells are seeded on 2D substrates of variable stiffness where cell responses (traction forces, spread area, and migration direction) are monitored (Lo et al., 2000). (xii) Wound healing assays: collective cell migration can be studied by generating an area without cells (wound) in a cellular monolayer, where dynamics of wound closure are monitored. These assays are tipically limited to 2D substrates with uniform conditions, although stimulus gradients can be created using modified microfabrication techniques (Simpson et al., 2008). (xiii) 3D ECM assays: novel biomaterials enable the creation of 3D environments with tunable chemical and physical parameters of the ECM. Cells are seeded inside the 3D medium and migrate depending on the ECM architecture, stiffness, pore size, and ligand concentration (Sabeh et al., 2009). (xiv) Microfluidics: recent advances in fabrication technology have made it possible to create novel microdevices that allow a precise control of the cellular microenvironment. With these assays it is facile to stablish electrical fields, cytokine gradients or interstitial flow, which can be tuned by adjusting the potential, chemokine concentration or hydrostatic pressure at the different microchannels (Chung et al., 2010, Polacheck et al., 2011). In short, Micropipette, Boyden chamber and microfluidics assays enable control of biochemical gradients whereas Durotaxis, 3D ECM and microfluidics assays enable control of biophysical forces (ECM stiffness and interstitial flow). Wound healing and micropatterning assay enable control of intercellular distances, whereas only micropatterning assays enable control of substrate topography. Chapter 1. Introduction 43 Figure 1.4: Experimental methods. With the permission of Polacheck et al., 2012 Experimental methods for investigating factors that influence tumor cell migration. Triangles indicate growth-factor (purple) and ECM stiffness (black) gradients. Blue arrows indicate direction of tumor cell migration, and pressure gradients are indicated by the shades of green. 1.1.6. Cell migration modeling There exist a vast amount of computational or mathematical models involving specific or general aspects of cell migration. The importance that cell migration has demonstrated in many pathological processes and diseases has caused great investment and a growing influx of mathematicians, engineers, and physicists into the field of cellular biophysics. As a result, the details and sophistication of migration models has grown concomitantly, however, and despite each model is oftenly unique, they can be classified into three principal types depending on the main area of interest: cell protrusion, cell adhesion and retraction, whole-cell models. Here, a brief summary explaining their physical basis, capabilities, and relationship to other models is presented. Nevertheless, several reviews cover this and other classifications providing a wider and more balanced picture (Flaherty et al., 2007, Carlsson and Sept, 2008, Rangarajan and Zaman, 2008, Mogilner, 2009). 1.1.6.1. Protrusion models Protrusion is probably the most studied and best understood of all locomotion steps. As described before, the key event underlying actin-based cell migration is the generation of force by the polymerization of actin filaments against the cell membrane. Many mathematical works have focused on this issue and they can usually be distinguished 44 Multiscale computational modeling of single cell migration in 3D depending on whether they consider a single actin filament, multiple filaments or they use a continuum approach (Carlsson and Sept, 2008). Single-filament growth models can identify plausible mechanisms for force generation, predicting the stall force and establishing a force-velocity relationship. Usually, they consider filament growth to occur by passive monomer diffusion. Brownian fluctuations of the membrane or filament tip make room for new monomer addition. If the free-monomer concentration is high enough, the addition of these new monomers prevent the membrane from returning to its previous position, a process commonly known as “Brownian-ratchet” (Peskin et al., 1993). These types of models have been extended to include attachments between the filament tip and the membrane which can stablish what additional mechanisms may be active in force generation and evaluate the effects on the force-velocity relation (Dickinson and Purich, 2002, Zhu and Carlsson, 2006). Multiple-filament models treat the simultaneous polymerization of many actin filaments, including filament-filament interactions and actin-binding proteins. Propulsion of intracellular pathogens, lamellipodial and filopodial protrusion have been the problems receiving the most attention with this kind of models. For instance, filopodia and branched actin networks growing against a membrane (Atilgan et al., 2005, Atilgan et al., 2006) or obstacle propelling by the growth of an actin network (Carlsson, 2001) have been studied with models including molecular-level processes such as barbed-end growth, pointed end depolymerization, branch formation, debranching, barbed-end capping, and barbed-end uncapping. Even more detailed approaches taking into account filament– obstacle attachments or spatial variation of actin concentration have been used to study this phenomena (Alberts and Odell, 2004). Continuum models describe the actin gel using coarse-grained properties such as the actin concentration or the number of filaments in contact with the membrane. These methods allow predictions for larger systems over longer times at the cost of treating molecular-level events in an averaged fashion. Alike the multiple-filament models, continuum approaches have focused on lamellipodial and filopodial protrusion (Mogilner and Edelstein-Keshet, 2002, Dawes et al., 2006) as well as intracellular pathogen propulsion (Gerbal et al., 2000, Mogilner and Oster, 2003). Their predictions comprise the properties of the actin gel such as its thickness and spatial density distribution, the force–velocity relation for pathogen propulsion, the protrusion velocity for lamellipodial extension, the elastic and diffusion properties of the membrane, and the distributions of filament lengths and orientations, all of which are experimentally observable. Chapter 1. Introduction 45 1.1.6.2. Adhesion and retraction models In spite of the more dynamic events occurring at the leading edge of motile cells, mechanisms involved in formation of adhesions to the substrate and retraction of the rear are equally important. Although some types of cells use non-myosin-based retraction, Myosin II has been generally implicated in retraction of the trailing edge of cells as well as breaking or removal of focal adhesions, and it has been included in several modeling works as the main cause of cell contractility (Ahmadi et al., 2005, Kruse et al., 2005). Furthermore, contractile bundles or stress fibers cannot provide work or movement without a mechanism for adhering to the underlying substrate. Therefore, the formation of focal adhesions and the mechanical cell-environment interactions have been modeled in detail with the underlying premise that forces applied to a filament have a direct effect on its polymerization (Shemesh et al., 2005) and also taking into account the interaction between the stress fibers and the integrins (Novak et al., 2004). 1.1.6.3. Whole-cell models Considering the whole cell, there exist a variety of migration models as large as the number of cell types. These models usually consider dynamical and/or mechanical phenomena including explicit molecular-scale processes such as the polymerization of actin, or less specific physical based ones such as the change in stability of a focal adhesion due to force. Similarly, cell mechanics may be treated explicitly or may be included in a phenomenological manner. Methods differ from continuum (Gracheva and Othmer, 2004) to discrete (Dokukina and Gracheva, 2010) approaches as well as finite element calculations (Bottino et al., 2002). In addition, different levels of detail are described, with time and length-scales varying significantly. Many of these studies have usually focused on 2D migration, not only for simplicity but due to the lack of high quality data of cell movement in 3D. Nevertheless, the number of 3D migration models has been gradually increasing, although focused on different aspects of cell motility. Some of them predict individual cell migration (Zaman et al., 2005, Borau et al., 2011, Schluter et al., 2012) while others simulate collective behavior (Ouaknin and Bar-Yoseph, 2009, Arciero et al., 2011). According to their main assumptions they can be grouped in: force based dynamics models, stochastic models, multi-cell spheroid migration and Monte Carlo studies. In the former ones, migration dynamics is accounted for by the traction forces at both the front and rear end of the cell and forces due to viscous drag and 46 Multiscale computational modeling of single cell migration in 3D cell protrusion into the ECM (Zaman et al., 2005). Imbalances of these forces produce cell migration. The drawback of these models is that they only predict migration of single cells, not taking into account changes in cell shape or ECM properties due to degradation. On the other hand, stochastic models of persistent random walks are able to predict population behavior (Parkhurst and Saltzman, 1992), however, they do not include dynamic effects such as traction or drag, neither incorporating the ECM properties. Multicell spheroid migration models are mainly based on pressure gradients produced by proliferation and death of cells (Pettet et al., 2001). Combining random walks, pressure and chemotactic activity of cell aggregates make them suitable to study tumours, but fail to take into account mechanical cues such as ECM density, porosity or stiffness. Finally, Monte Carlo models using square lattices and a set of simple rules allow faster simulations thus providing long-term migration patterns (Zaman, 2007, Zaman et al., 2007). The main handicap is the qualitative nature of the studied parameters such as cellmatrix interface, cell polarization or ECM mechanical effects. Figure 1.5: Examples of migration models. A) Protrusion model of filopodia growth by Atilgan et al. 2006. B) Adhesion-retraction model of focal-adhesion mechanosensitivity by Shemesh et al. 2005 C) Whole-cell model of nematode crawling by Bottino et al. 2002. 2. COUPLING MECHANOSENSING AND CELL MIGRATION: A 3D MODEL APPROACH This chapter describes a 3D single cell migration model assuming mechanosensing as the leading mechanism driving cell motility. The model formulation and main hypothesis are presented, as well as its numerical implementation using FEM. In spite of the simplifications and limitations of this approach, the outputs are in agreement with recent literature, finding that substrate stiffness, boundary conditions and external forces, regulate specific and distinct cell movements. In addition the model lays the foundation for the development of this Thesis, and its main features and results are gathered in (Borau et al., 2011). 2 Chapter 54 Multiscale computational modeling of single cell migration in 3D Contents 2.1. Introduction ...................................................................................................... 55 2.2. Model formulation ............................................................................................ 57 2.2.1. Mechanosensing mechanism .................................................................... 58 2.2.2. CSK adaptation ......................................................................................... 62 2.2.3. Migration................................................................................................... 63 2.3. Numerical implementation ............................................................................... 66 2.4. Model outputs ................................................................................................... 68 2.4.1. Parameter sensitivity ................................................................................. 69 2.4.2. Sample of calculation ................................................................................ 70 2.4.2.1. Traction forces and cell speed ........................................................... 74 2.4.3. External forces .......................................................................................... 75 2.5. Discussion and conclusions .............................................................................. 76 Chapter 2. Coupling mechanosensing and cell migration: a 3D model approach 55 2.1. Introduction As it has been shown so far, cell motility is generally guided by environmental signals or cues from the surrounding microenvironment (Even-Ram and Yamada, 2005, Ehrbar et al., 2011). These cues reflect the physical–chemical nature of the ECM and its binding to transmembrane receptors, allowing cells to probe the mechanical properties of their environment and react in a specific way (Ingber, 2010). This ability of cells to sense ECM stiffness or pre-strain enables them to regulate their mechanical response and, therefore, its characterization is crucial for understanding their directional migration. It has become clear, in the last few years, that cells sense their surroundings by extending lamellipodia and filopodia that attach to the substrate, then exerting contractile forces in order to explore the mechanical properties of their environment (Discher et al., 2005). These active forces are generated by myosin motors and are transmitted to the ECM by means of transmembrane proteins (integrins) that often cluster to form focal adhesions (Bershadsky et al., 2003, Yang and Zaman, 2007, Yang and Zaman, 2010). The influence of the stiffness and topography of the ECM is critical to this process, and has been thoroughly investigated (Lo et al., 2000, Engler et al., 2004a, Yeung et al., 2005, Saez et al., 2007, Ghibaudo et al., 2008, Baker et al., 2009, Janmey et al., 2009, Mitrossilis et al., 2009, Sanz-Herrera et al., 2009, Ehrbar et al., 2011, Harunaga and Yamada, 2011, Trichet et al., 2012, Ghassemi et al., 2012). One important finding is that cells prefer to migrate to the stiffer part of the ECM or substrate (Lo et al., 2000, Cukierman et al., 2001, Bischofs and Schwarz, 2003, Schwarz and Bischofs, 2005). Focal adhesions are more stable and contraction forces increase on stiffer substrates (Lo et al., 2000, Ghibaudo et al., 2008, Mitrossilis et al., 2009, Cukierman et al., 2001, Trichet et al., 2012, Ghassemi et al., 2012). However, whether cells are able to sense stress or strain is still unclear (Freyman et al., 2002, Saez et al., 2005, De et al., 2008). Furthermore, the application of external stress/strain on the cell also stimulates focal adhesion formation and subsequent strengthening (Riveline et al., 2001, Kaverina et al., 2002) and therefore, this stress can trigger molecular reorganization and CSK adaptation. Cell migration studies have primarily focused on migration on 2D substrates. These studies have helped to elucidate the mechanisms by which cells migrate, interact with the substrate or change their speed. However, when completely surrounded by the ECM, cells experience a different environment and some factors that were not present in 2D appear, 56 Multiscale computational modeling of single cell migration in 3D such as the role of volume exclusion (Simpson et al., 2010) and relevant differences in cell morphology, adhesions and signaling (Fraley et al., 2010, Hakkinen et al., 2011). Therefore, to fully understand how cells migrate in vivo, it is necessary to study the movement of cells in 3D environments. In recognition of this need, experimental 3D cell movement studies (Friedl and Brocker, 2000, Cukierman et al., 2002, Vickerman et al., 2008, Chung et al., 2012) and modeling efforts (Zaman et al., 2005, Harjanto and Zaman, 2010, Groh and Wagner, 2011) have been growing in number in the recent past. In fact, although most models for cell migration have been developed for 2D conditions, more recently, 3D models have appeared. Among the possible classifications described in the previous chapter, all of these models may be generally grouped as either continuum or discrete approaches. On one hand, most continuum models are based on reaction diffusion equations for cells and diffusive chemicals (Manoussaki, 2003, Moreo et al., 2008, Häcker, 2011). In general terms, these continuum approaches have the following limitations: they are valid only when there are weak cell-cell interactions and they underestimate volume exclusion effects that are present in 3D (Simpson et al., 2010). On the other hand, discrete models have also been developed. These are of two main types: lattice models and interactionforce based models. The former ones can be subdivided into two approaches: cellular automata models (Chopard et al., 2010) and cellular Potts models (Chen et al., 2007, Merks and Koolwijk, 2009). Their main limitations are that they do not include the role of cell mechanical properties and they are mainly phenomenologically based approaches, requiring many nonphysical parameters to be determined from experiments. Finally, the so-called force-based dynamics method (Zaman et al., 2005) normally uses single cells as basic units, each of which is characterized by its location and orientation, its state of stress and the active forces it can exert in response to the local microenvironment. Knowing this for each cell, the velocity of each individual cell can be evaluated through the equilibrium of forces. Normally, the differences among models are based on the consideration of different kinds of forces that define this equilibrium. Using this approach, a mathematical model for cell movement in multicellular systems has been developed (Palsson, 2001), incorporating viscoelastic properties of cells to simulate 3D cell movement during aggregation and the slug stage of Dictyostelium discoideum, embryogenesis, limb formation and wound healing. Three main forces are considered in this work: passive, active and a viscous drag force. The active force is due to a chemotactic signal and the passive force takes into account the elastic interactions Chapter 2. Coupling mechanosensing and cell migration: a 3D model approach 57 between neighboring cells. Other force-based dynamics approaches (Zaman et al., 2005, Harjanto and Zaman, 2010) have been used to model the movement of individual cells in 3D, considering a more realistic cell–matrix interaction and taking into account the receptor–ligand adhesivity. A recent study on single-cell migration in 3D has been presented, based on this approach (Groh and Wagner, 2011), where chemotaxis as well as contact guidance are considered to regulate cellular movement. Although all these models take into account many different effects through the forces that regulate cell migration, none of them have incorporated mechanosensing. This mechanism is associated with the active contractile forces that cells exert on their surroundings in order to probe the mechanical environment (Discher et al., 2005, Sen et al., 2009). This chapter describes a 3D single cell migration model in which mechanosensing is the main regulator of the directional movement. The main mechanically relevant components of the cell are taken into account, as well as the contraction forces exerted on the ECM/substrate and the major phases in cell migration. These phases are grouped in: (i) mechanosensing through cell contractility, (ii) cell polarization/adaptation that can influence directional cell motility by means of the formation of the leading and trailing cell edges and (iii) the cell movement in 3D. With these assumptions, the model qualitatively predicts some features such as cell movement tendencies, traction forces and cell speeds in several substrates with different stiffnesses and under different mechanical constraints. The premise of this work is that a better understanding of all these features provides new possibilities to guide and regulate tissue regeneration, and is therefore useful for the design of new biomaterial scaffolds aiming at optimizing mechanical conditions to control cell migration for tissue engineering applications or medical device designs. 2.2. Model formulation The model mainly focuses on the mechanical interaction of an individual cell with the ECM/substrate. The migration of a single cell is computed to isolate the mechanical inputs from other possible intercellular interactions. Chemical or other factors are not taken into account, thereby, the cell movement is only regulated by the mechanical properties of the matrix and the cell–matrix interactions. As previously described, when a cell migrates, complex-coupled and cyclic mechanisms are activated regulating its movement, which includes polarization, protrusion and adhesion, contraction of the cell 58 Multiscale computational modeling of single cell migration in 3D body and retraction of the rear. The cell embedded within the gel extends filopodia and/or lamellipodia possibly to sense the surrounding mechanical conditions. Contractile forces are exerted to evaluate the substrate stiffness, and consequently the substrate (and the cell) are stressed and strained (Buxboim et al., 2010). According to this, the cell’s CSK adapts, and the cell becomes directionally polarized. More focal adhesions develop at the front than at the rear. Detachment of rear adhesions leads to the imbalance of traction forces that they can support, subsequently resulting in the contraction and forward movement of the cell. In this model these mechanisms are simplified into three fundamental ones: mechanosensing, CSK adaptation and cell movement. Although they occur simultaneously, for simplicity, they are considered here independent but interrelated mechanisms. In fact, these three different mechanisms are coupled assuming that all of them are regulated by the contraction stress exerted by the cell on the ECM/substrate during the mechanosensing phase. Figure 2.1: Schematic diagram of cell mechanical components. Actin bundles are modeled as spring (Kact) in series with the acto-myosin (AM) contractile system. These components act in parallel with the passive elements of the cell cytoskeleton (Kpas). 2.2.1. Mechanosensing mechanism The CSK is a dynamic structure that maintains cell shape (Wang et al., 2001), protects it and plays an important role in different processes such as cell division (Heng and Koh, 2010), intracellular transport (Damania et al., 2010) or cell locomotion (Burnette et al., 2011). The behavior of this structure is complex and various approaches have been Chapter 2. Coupling mechanosensing and cell migration: a 3D model approach 59 adopted to model its interesting rheological (Astrom et al., 2008, Astrom et al., 2009, Kim et al., 2009a, Kim et al., 2009b, Kim et al., 2011) and contractile properties (Deshpande et al., 2006, McGarry et al., 2009, Borau et al., 2012). In this work, and based on a previous approach (Moreo et al., 2008), a simplified model is proposed. The cell body is represented using a spring-like structure with a contractile actuator (Figure 2.1). The cellular elements responsible for the cell mechanics behavior considered here are the actin bundles ( act K ), the actomyosin contractile apparatus (AM) and the passive mechanical stiffness of the rest of the cell whose main contribution comes from the cytoskeleton microfilaments and the membrane ( pas K ). The CSK is linked with the ECM/substrate through focal adhesions and transmembrane integrins that are assumed perfectly rigid. The stress effectively transmitted by the cell to the ECM ( cell p ) depends on the cell strain ( cell  ). This strain can be decomposed into two parts: the strain in the actin bundles ( a  ) and the strain in the contractile AM system ( c  ). Therefore: cell c a     (2.1) and since the actin bundles are modeled as a spring with a stiffness act K , a  can be expressed as: c a act p K   (2.2) The stress that the cell effectively transmits to the ECM/substrate corresponds to the sum of the contribution of the active actin–myosin component and the stress absorbed by the passive elements ( c p and m p respectively): cell cell c m c c pas cell ( ) ( )p p p p K        (2.3) The active stress contribution of the actin–myosin motors ( c p ) is related to the extension/contraction (overlap) between the actin–myosin filaments ( c  )(Rassier et al., 1999). Here, according to the classical Huxley’s law (Huxley, 1957), the following form is assumed: 60 Multiscale computational modeling of single cell migration in 3D     c min max c min min c min cc max max c c max max max c 0 0 () 0 0 p pp                               (2.4) Hence, four different zones can be distinguished as a function of the value of ( c  ) as shown in Figure 2.2A. Without external loads, the mechanical equilibrium is governed by the AM contractile system that contracts the cell body (Zone 2: min c 0   ) and causes tension on the cell surroundings. Under too high external compression (Zone 1: c min   ), the AM contractile system is not able to exert contraction. However, when there are external loads causing cell extension (Zones 3 and 4), cell contraction is compensated decreasing the AM contractile action until reaching max  where its force contribution equals 0. In summary, combining equations (2.1), (2.2), (2.3) and (2.4), equation (2.3) can be rewritten and the magnitude of cell p can be expressed as follows:     pas cell cell min act max min cell pas cell min cell max act act min max cell cell act max max cell pas cell max act cell max act max max pas cell max cell () K Kp K p K Kp pKp K p K Kp K                                       (2.5) which is plotted in Figure 2.2B together with c cell ()p  . Although the description thus far has been in terms of a scalar stress, it is now extended to three dimensions using cell p to compute an effective stress tensor cell σ . It is assumed that the cell occupies a spherical space, with constant shape and volume. However, the cell is exerting anisotropic forces on its immediate surroundings depending on the temporal evolution of the CSK polarization. Considering this mechanosensing model, the total cell stress tensor cell σ is a function of the total cell strain tensor cell ε and Chapter 2. Coupling mechanosensing and cell migration: a 3D model approach 61 consists of two terms related to the orientation of the CSK. One term is isotropic and dependent on the volumetric strain ( cell  )(trace of the cell strain tensor cell ε ) considering that the cell body contracts in all directions. The other is anisotropic depending on the direction of the CSK polarization ( pol d ), and the longitudinal deformation in the polarization direction ( T cell pol cell pol    dεd ). Hence, the following expression defines the stress behavior of the cell: cell cell cell pol pol cell cell ( ) ( )pp    σ d d I (2.6) where I is the identity second-order tensor and pol d is the direction of polarization of the CSK which is described in the following subsection. Figure 2.2: Acto-myosin contraction dependence. A) Function of cc ()p  . Zone 1: passive behavior, external loads compress the cell. Zone 2: contraction, the cell exerts contractive forces to sense its surroundings. Zone 3: tension, the cell is being stretched by external loads but still contracts itself against those forces. Zone 4: passive behavior, external loads stretch the cell. B) Dependence of c p and cell p on cell  corresponding to equation (2.5) and for the parameters collected in Table 2-1. The global mechanical equilibrium has to be fulfilled in the cell body and in the substrate, which is modeled as a linear elastic solid to a first approximation. The ECM or substrate stress tensor ( subs σ ) must be in equilibrium with the external forces applied ( ext f ) e.g. a needle inserted in the matrix exerting pulling or pushing forces:     cell cell subs ext subs 0 in in       σ σf (2.7) 62 Multiscale computational modeling of single cell migration in 3D To solve these mechanical equilibrium equations, the finite element method (FEM) is used, as detailed in next sections. 2.2.2. CSK adaptation Here, it is hypothesized that the mechanosensing mechanism permits the cell to detect the principal directions (  d ) of the cell strain tensor cell ε , and that the CSK is reoriented according to them. Initially, the cell is assumed to be an isotropic and homogeneous body embedded in the ECM. Consistent with previous experimental observations (Schwarz and Bischofs, 2005), the active actin-myosin fibers are assume to develop preferentially parallel to the direction of lower principal strain or higher principal stress. Hence, the cell is assumed to align gradually with the closer principal strain direction (see Figure 2.3A) as follows:     pol pol pol pol ii dtt dt          d d d d d (2.8) where the index i denotes the time step and  d is the principal strain direction, which forms the lower angle (  ) with the previous polarization direction pol i d . The symbol  is the constant of reorientation (min−1) and determines how fast the vectors align. Once this direction is defined, it is updated: 1 pol pol pol ii  d d d (2.9) Note that at the end of each time step, 1 pol i d is normalized, becoming the unitary vector pol i d for the next time step. The rate of reorientation (  ) is assumed to be 1/30 min−1, which means that when  d and pol d are perpendicular, the cell is able to orientate its CSK π/4 degrees in a single time step to align with  d . It has been observed in experiments (Wang et al., 2000, Hayakawa et al., 2001, Yoshigi et al., 2005) that cells remodel their cytoskeleton on a timescale of tens of minutes to hours. This timescale has been also used in other modeling works (De et al., 2007). Chapter 2. Coupling mechanosensing and cell migration: a 3D model approach 69 2.4.1. Parameter sensitivity The model has the capability of capturing a wide variety of behaviors by appropriate selection of the model parameters. A sensitivity analysis is performed in order to illustrate some types of behavior that can be produced and also to better understand the relative importance of the major parameters. All the reference values used in all the simulations are in Table 2-1. The actin stiffness ( act K ) is a critical factor which determines the magnitude of the forces ( cell p ) exerted during mechanosensing. By virtue of the model construction, it also determines the magnitude of traction forces and therefore the velocity. It is interesting to note that the dependence of cell speed on substrate stiffness is bi-modal, having a peak at substrate stiffness of about 20–30 kPa (Figure 2.5A). Traction forces increase with substrate stiffness until saturationand decrease for lower values of act K (Figure 2.5B). Consequently, low values of act K lead to slower cell motion over the entire range of substrate stiffness in comparison with the reference value (Figure 2.5A). The reason is that increasing the value of act K leads to lower AM overlap, reducing contraction and causing higher forces with the same substrate stiffness. Therefore cell speed increases accordingly. Interestingly, the value of pas K , although also important, plays a significant role only when its value is similar to the substrate stiffness (data not shown). As described in equation (2.12), the viscosity is assumed to increase linearly with substrate stiffness, and its value strongly affects cell behavior. In fact, viscosity saturates, but it happens outside the rigidity range of this study (Saez et al., 2005). The speed is very sensitive to the value of 0  for the range of stiffness tested, since it defines the minimum viscosity in softer substrates. If this factor is decreased, the cell can reach higher velocities in compliant substrates (Figure 2.5C). In addition, the velocity peak is displaced so that maximum speeds are achieved in softer substrates compared to the control case. Consequently, increasing 0  produces opposite effects. Nevertheless, the slope vis c also significatively affects the velocity, more markedly at high stiffness. This effect is due to the saturation of forces in stiffer substrates. As can be deduced from equation (2.16), if the viscosity keeps increasing with stiffness while the force ( cell p ) remains mostly unchanged the cell speed tends to decrease, to lower values the faster the 70 Multiscale computational modeling of single cell migration in 3D viscosity increases (higher vis c ). Hence, if this factor is decreased, the velocity increases and the point of maximum speed is displaced to a higher substrate stiffness (Figure 2.5D). Figure 2.5: Parameter sensitivity analysis A) Cell speed ( trac v ) and B) traction force ( trac F ) depending on substrate stiffness for several values of actin stiffness ( act K ). C) Cell speed depending on substrate stiffness for different values of minimum viscosity ( 0  ) and D) viscosity slope ( vis c ). 2.4.2. Sample of calculation A 3D ECM is simulated, which consists of a rectangular cuboid of the following dimensions: 1400 × 700 × 700 μm, with two different rigidities ( 20.04E MPa, 10.001E MPa) under different boundary conditions listed below for each of the three examples analyzed here (Figure 2.6). Case 1: The stiffer side is constrained (fixed nodes on the left surface) and the softer side is free of external loads (on the right surface). The remaining four surfaces are also free of external loads. Chapter 2. Coupling mechanosensing and cell migration: a 3D model approach 71 Case 2: Both sides are constrained (fixed nodes on left and right surfaces). The remaining four surfaces are free of external loads. Case 3: The stiffer side is free of loads (on the left surface), whereas the softer side is constrained (fixed nodes on the right surface). The remaining four surfaces are also free of external loads. Figure 2.6: Simulated cases. The size of the computational domain is 1400 × 700×700 μm. Black arrows represent the direction of migration. Dotted lines represent zones where a change in the migration direction is observed. Circles in each case represent schematically the initial cell position for each subcase. A regular mesh of 85750 hexahedron elements (C3D8) is used. Simulated time is 9 h, whereas the computational time is 30 min. Although only some results corresponding to different initial positions in each case are shown (subcases), many simulations have been performed, finding similar and consistent patterns. At least ten repetitions per subcase were tested. The computed trajectories of migration were all different due to the stochastic behavior; however, the general trend was consistent for each subcase. Hence, for clarity, only one of the examples for each subcase is plotted in Figure 2.7. As the boundary conditions change along the x-axis, the principal results are discussed focusing on migration in the x-direction. Note that all the subcases are referred as c-casenumberletter. In the first case, no matter where the cell is initially placed, either in the softer side (c1a) or the stiffer side (c1b), since it always moves toward the constrained side, which also has the highest Young’s modulus ( 2 E ). The cell never migrates from the stiffer side to the softer one (Figure 2.7). Note that the randomness seen in the trajectories causes the cell to deviate from a straight line, and move out of the x–y plane. These computational results are consistent with experiments (Lo et al., 2000), where they found that cells tend to move from a soft substrate to a stiffer one, but not in the opposite direction. 72 Multiscale computational modeling of single cell migration in 3D In the second case, as the soft side is also constrained, there exists a zone (dotted line in the figures) where the cell changes its migration direction. A cell placed close enough to the soft side constraint (c2a) moves toward that boundary. In other cases (cells placed further from the boundary), the tendency is to move toward the left constraint, whether the cell is initially on the softer (c2b) or stiffer side (c2c) (Figure 2.7). In the third example, three zones can be distinguished (separated by dotted lines). From right to left, a cell placed in the right zone, near the constraint (c3a), migrates directly toward it as in the previous case. A cell placed in the intermediate zone, either on the softer side (c3b) or the stiffer side (c3c), moves to the left. If the cell reaches the left zone (c3c), it stops its advance and moves backward. Once again in the intermediate zone, the cell migrates to the left, crosses to the left zone and moves backward, repeating this process randomly, but indefinitely. Thus, in a zone with no differences in mechanical properties and in the absence of other stimuli (such as chemistry, flow, cell–cell interactions) random cell migration would predominate (Petrie et al., 2009). The cell would migrate randomly within that zone but would not deviate far. In the same way, a cell initially placed in the left zone (c3d) moves away from the free side to the interior of the ECM, but once it crosses to the intermediate zone, it reverses and randomly migrates as in the previous subcase, becoming trapped around this interface. This change in the cell’s migration pattern always occurs at similar x-coordinates, but at different yand zcoordinates due to the randomness of the cellmovement (Figure 2.7). Chapter 2. Coupling mechanosensing and cell migration: a 3D model approach 73 Figure 2.7: Migration trajectories and cell speeds. Computed trajectories projected on y–z (top right), x–y (bottom left) and x–z (top left) planes. The initial position of each case is highlighted with a square and the corresponding label. I indicates a zone where cells become trapped due to specific mechanical conditions. II denotes the interface separating the regions of different stiffnesses. Note that in all simulations, the cell starts at the same y–z point, but are plotted displaced to identify clearly the different cases. Black arrows refer to cells crossing the interface denoted by II. 74 Multiscale computational modeling of single cell migration in 3D 2.4.2.1. Traction forces and cell speed The model predicts that traction forces increase with higher substrate stiffness until saturation (Figure 2.5B), which has previously been reported in experiments (Mitrossilis et al., 2009, Webster et al., 2011, Trichet et al., 2012, Ghassemi et al., 2012). It also predicts a biphasic dependence of cell migration speed on substrate compliance, as reported in (Peyton and Putnam, 2005) and modeled in (Dokukina and Gracheva, 2010). In all the analyzed examples, the cell exerts higher forces while moving in the stiffer substrate. When a cell moving in the softer substrate approaches and crosses an interface where Young’s modulus changes (as happens in c1a, c2b and c3b), the traction force increases abruptly. When the new value is reached, it remains nearly constant as the cell migrates further into the stiffer zone. Traction forces are about 0.015 μN on the more compliant side (0.001 MPa) and 0.055 μN on the stiffer one (0.04 MPa) which correspond with speeds of 0.21 μm min−1 and 0.67 μm min−1, respectively (Figure 2.7 bottom). The work developed in (Lo et al., 2000) in a 2D substrate with different rigidities shows good agreement with the values of traction forces and cell speeds in the range of their study. They measured a maximum traction stress of 1.09 ± 0.34 kPa and a maximum cell speed of 0.54 ± 0.13 μm min−1 for a 0.03 MPa substrate stiffness. The corresponding computational values in the present model are 0.662 kPa and 0.67 μm min−1, respectively. Similar speed ranges were found in (Peyton and Putnam, 2005) with a maximal of 0.72 ± 0.06 μm min−1 for 0.021 MPa substrate stiffness. In addition, they suggested that optimal stiffness for maximum migration is shifted depending on the concentration of the ECM protein covalently attached to the substrate. However, it is necessary to keep in mind that 2D and 3D speeds are being compared. The recent literature (Fraley et al., 2010, Hakkinen et al., 2011) demonstrates the low correlation between 2D and 3D motility, suggesting that 2D studies are poor predictors of 3D speeds. Nevertheless, they found 3D cell speeds similar to those obtained in this model (specifically 0.3–0.8 μm min−1 in (Fraley et al., 2010) and 0.2–0.7 μm min−1 in (Hakkinen et al., 2011). A direct quantitative comparison of cell speeds is difficult, due to the significant variability observed in the experiments as a function of the cell type, substrate composition or morphology. In fact, in (Hakkinen et al., 2011) they studied the cell behavior in four different ECMs (cell-derived matrix, matrigel, collagen, fibrin) and they concluded that considering the molecular composition of the matrix is crucial for a 3D Chapter 2. Coupling mechanosensing and cell migration: a 3D model approach 75 cell migration study. Similar conclusions can also be found (Harley et al., 2008, Peyton et al., 2011, Ehrbar et al., 2011), which studied the influence of geometrical and mechanical properties of the microenvironment on 3D migration. Interestingly, in spite of different conditions and cell types, similar ranges (comparable with the model results) of cell speeds were found (~0.1–0.26 μm min−1 in (Harley et al., 2008), ~0.1–0.8 μm min−1 in (Peyton et al., 2011) and ~0.2–1.0 μm min−1 in (Ehrbar et al., 2011)). 2.4.3. External forces All the previous results correspond to isolated cells under different mechanical conditions, focusing on the boundary conditions and the elasticity of the ECM. Some additional simulations were performed aiming to understand the effect of applying external loads on specific locations inside the matrix surrounding a single cell. In particular, relevant in this respect is the work of (Lo et al., 2000), where they demonstrated that inserting a micro-needle near the cell and stretching/pushing it, can modify its behavior and even change completely its migration direction. Figure 2.8: External forces modulate migration direction. A) Left: scheme corresponding to case 1, showing the relative position of a micro-needle and the direction of the applied force. Right: migration direction depending on traction/compression magnitude. B) Experiment from Lo et al. showing the change in cell polarization and migration in response to a pulling/ pushing micro-needle. Image adapted from (Lo et al., 2000). In order to test the model under external forces, the insertion of a micro-needle was simulated by means of the application of one local external force applied at a distance of 76 Multiscale computational modeling of single cell migration in 3D 40 μm from the cell, which was maintained constant throughout the simulation. The conditions of the substrate were exactly the same as those used in the case 1 (specifically c1a, as shown in Figure 2.8A). It was found that with a sufficient level of applied force, the micro-needle was able, as shown by (Lo et al., 2000), to change the polarization and the direction of the cell movement (Figure 2.8B). In the simulated case, in normal conditions, the cell tends to migrate toward the stiffer substrate (left). When the stretching (or pushing) force exceeds a certain threshold value (specifically 0.005 μN), the cell changes its migration trend. Interestingly, stretching and pushing values were equivalent. In addition, as could be expected, similar results were obtained varying the distance between the cell and the microneedle, where further distances are equivalent to lower forces. Nevertheless, further research is needed on this issue, being a key point for instance in cell-cell interaction phenomena. 2.5. Discussion and conclusions Although cell migration phenomena involve many different and complex mechanisms, here, a simplified model capable of simulating the preferential movement of an individual cell in 3D under different mechanical conditions is presented. This simplification is based on the hypothesis that mechanosensing is one of the main regulatory mechanisms to direct cell movement. In fact, three relevant phenomena are considered: mechanosensing, CSK remodeling and migration, and their corresponding equations are solved separately and sequentially (since the mechanosensing defines the CSK remodeling and both define the migration). First, during mechanosensing, the stress equilibrium between the cell, substrate and external forces is satisfied. Depending on the mechanical properties and boundary conditions, different strain/displacement fields and values of forces exerted by the cell on the ECM/substrate ( cell p ) are obtained in each step. With these data, the CSK remodels and reorients, updating the internal variable that describes the preferential orientation of the cell ( pol d ). Once the values of cell p and pol d are obtained, the traction forces are evaluated, and by satisfying the equilibrium of forces acting on the cell, its speed is computed. Consistent with this model is the observation that traction forces increase with substrate stiffness (Mitrossilis et al., 2009, Webster et al., 2011, Trichet et al., 2012, Chapter 2. Coupling mechanosensing and cell migration: a 3D model approach 77 Ghassemi et al., 2012). Without external loads, the cell strain is always negative (contractile) since the AM system is always active to reduce the dimensions of the cell. Therefore, stiffer substrates lead to lower values of strain (closer to zero), higher values of cell p and consequently to higher values of the traction exerted ( trac F ). In the presented cases, cell speed is higher in the stiffer substrate. However, the elastic modulus of the stiffer side ( 2 E ) was selected to reach maximum velocitieswith the reference values used in the model. Using a higher value of stiffness would lead to lower speeds, which could be even lower than in the softer substrate due to the increase in viscosity and the saturation of forces (Figure 2.5A,B). Of course, a stiffer matrix would also tend to be more difficult to enzymatically degrade and may have different transport properties and density of adhesive ligand, all of which could influence the migration speed. It is important to note here that the presented calculations only examine mechanical effects, thereby down-playing other additional factors. Recent experimental works (Fraley et al., 2010, Hakkinen et al., 2011) have quantitatively demonstrated the main differences between 2D and 3D cell migration. The lack of correlation between 2D and 3D motility suggests that focal adhesion proteins may regulate motility in a matrix in a manner fundamentally different from that in planar cell motility. Nevertheless, 2D studies are still useful and, in some respects, comparable to 3D (directionality, number of adhesions, axial ratio and even adhesion area (Hakkinen et al., 2011)). Here, some of the model results and predictions are compared with the experimental data on 2D developed in (Lo et al., 2000). These experiments consist of 2D substrates with two different rigidities, where some isolated cells (only interacting with the ECM) are embedded. Their findings indicate that cells placed in the softer part of the substrate tend to migrate toward the stiffer part and cross the interface which separates the substrates, whereas cells placed in the stiffer zone do not cross this interface. This suggests, as proposed here, that cells are capable of sensing the mechanical properties of their surroundings and tend to move toward stiffer substrates. As the only forces acting on the substrate are those exerted by the cell itself, its movement is governed by the local mechanical environment, the boundary conditions and the mechanical properties of the ECM. In fact, it has been investigated in recent experiments (Harley et al., 2008, Fraley et al., 2010, Peyton et al., 2011, Ehrbar et al., 2011, Hakkinen et al., 2011) how microarchitecture, local mechanical properties and molecular composition influence cell migration behavior. The main assumption of the model is that the cell aligns with the 78 Multiscale computational modeling of single cell migration in 3D direction of principal strain and moves according to the relative displacements between the cell body and its centroid, which depend on imposed boundary conditions and local changes in substrate stiffness. The recent literature suggests that elasticity, boundary conditions and perhaps embedded fibers can modulate the apparent elasticity of matrices that cells are likely to sense (Buxboim et al., 2010). In all the shown cases, cell migration follows this criterion, in agreement with those experimental observations (Figure 2.7). For example in the second case, there are two zones with an interface located in the middle of the softer side, where the local displacement field is modified due to the right constraint. When a cell is located near that constraint, the cell senses it and moves to the right, whereas if located further away, it moves to the left, where Young’s modulus is higher ( 2 E ) and there is also a constraint. Note how in the third case, where the stiffer side (left) is free of constraints, a third zone appears. The substrate displacement field obtained by the mechanosensing analysis, reaches a minimum in the middle of the stiffer substrate. As a result, the opposing gradients of displacements cause a cell to move randomly around this location. This happens when the cell reaches a zone in the substrate with similar mechanical conditions in all directions. With no differences in local mechanical properties (and absence of other stimuli such as chemistry, flow, cell–cell interactions, etc), the cell would not be able to decide where to move and random migration would be predominant (Petrie et al., 2009). The magnitude of cell speed for the reference values of parameters used in the model ranges from 0.2 to 0.7μm min−1, showing good agreement with experimental data (Lo et al., 2000, Peyton and Putnam, 2005, Harley et al., 2008, Fraley et al., 2010, Peyton et al., 2011, Ehrbar et al., 2011, Hakkinen et al., 2011). The wide range of parameters used in the model, allows adapting it to different conditions and/or experiments, taking into account the limitations discussed below. The model is also used to study the case in (Lo et al., 2000), where local forces are applied in the cell surroundings to understand their role on the preferential movements of single cells. A blunted micro-needle is introduced in the substrate near the cell and moves toward or away from the cell to modify the local state of stress in the ECM. Experiments show that the cell moves toward the pulling forces, and away from the pushing forces. As it has been shown in the results section, the model is able to predict this effect and helps to explain it. If the microneedle pulls, the compressive forces of the cell oppose the needle forces. Hence, the local displacement field changes in a way that the cell senses the forces induced by the needle Chapter 3. Modeling cytoskeleton dynamics: rigidity-sensing of actin networks 85 Merely by modeling actin and myosin activity in the absence of proteins related to adhesion complexes, both equilibrium and dynamic behaviors are predicted, indicating that actomyosin machinery can function as a stand-alone mechanism for the mechanosensing of cells. 3.2. Model features A previous agent-based Brownian Dynamics model (Kim et al., 2009b) is used to simulate active cross-linked actin networks as systems that generate force as well as sense surrounding mechanical conditions. In this approach, actin filaments, actin cross-linking proteins (ACPs), and molecular motors and their local interactions are explicitly taken into account. To facilitate understanding of the results predicted in this study, its main features are briefly presented. 3.2.1. Formation of an active actin network Before going into the modeling details, it is useful to introduce how real actin networks are formed and how nature regulates polymerization dynamics. Actin exists as a globular monomer (G-actin) and as a filamentous polymer (Factin), which is in fact a string of G-actin subunits. Although the filament is often described as a single helix of monomers, it can also be thought of as consisting of two protofilaments, held together by lateral contacts, which wind around each other as two parallel strands of a helix, with a twist repeating every 37 nm (Alberts et al., 2008). The polymerization of actin filaments proceeds in three sequential phases: lag nucleation, elongation and steady state. In the first phase, G-actin aggregates slowly into short, unstable oligomers. These oligomers can act as a seed or nucleus, which in the second phase rapidly elongate into a filament by the addition of monomers onto both ends. All the subunits within the filament have the same orientation, giving it a structural polarity and making the two ends of the polymer different. The kinetic rate constants for association and dissociation ( on k and on k respectively) are much greater at one end than at the other. The more dynamic end, where both growth and shrinkage are fast, is called the plus (barbed) end and the other end is called minus (pointed) end. Furthermore, as a consequence of the nucleotide hydrolysis that accompanies polymer formation, the critical concentration of free monomers at each end changes ( cc CC   ). Thus, polymerization and growing proceed until the third phase, where the concentration of free 86 Multiscale computational modeling of single cell migration in 3D monomer reaches a value above c C but below c C . At this steady state, called treadmilling, the subunits undergo a net assembly at the plus end and a net disassembly at the minus end at an identical rate. There is a net flux of subunits through the polymer, but it maintains constant length. In this work, active actin networks with motors are generated in a similar fashion to previous studies (Kim et al., 2009b). G-actins, passive ACPs, and motors are assembled into a network via reversible reactions in a 3D cubical domain with periodic boundary conditions in all directions. ACPs and motors can exist in three states: monomeric (free), inactive (partially bound), and active (bound to two filaments) states. Note that following the initial formation of the network, monomeric ACPs and motors are implicitly considered via their local concentration and second-order reaction equations. After concentrations of G-actin, ACPs, and motors reach a dynamic steady state, residual Gactins are deleted with actin assembly/disassembly deactivated for simplicity. A geometrically identical network is used in all simulations to isolate the effects of the stiffness of the surrounding medium and other parameters. To vary the concentration of motors, they are removed from networks or added as monomers at the beginning. The average filament length ( f L ) is ~2 μm, actin concentration, A C , is 12 μM, density of ACPs, ACP R (= ACP C / A C ), is 0.01, and the initial width of the cubical domain is 5.0 μm. Density of motors, M R (= M C / A C ), is 0.02 unless specified. 3.2.2. Parallelization The number of particles involved in the simulations is relatively high (tens of thousands), however, what makes the codes computationaly heavy is the huge amount of time increments (tens of millions) needed. This is due to the small time scale (of the order of nanoseconds, see Table 3-1) at which the dynamic processes such as binding or unbinding occur and the desire of simulating long periods of time (hundreds of seconds). Thus, a parallelized approach is compulsory. Parallel computing operates in the principle that large problems can be divided into smaller ones, which are then solved concurrently. It is, indeed, a powerful tool which permits to face requirements otherwise unthinkable with current technology. Multipleinstruction multiple-data systems (MIMD) are those in which a collection of autonomous processors operate on their own data streams. The most commonly used method of programming in MIMD systems is message passing. The processes coordinate their Chapter 3. Modeling cytoskeleton dynamics: rigidity-sensing of actin networks 87 activities by explicitly sending and receiving messages. For the computation of the model, the Message Passing Interface (MPI) is used. Note that it is not a new programming language, but a collection of functions, macros, or a library that can be used in C programs. It assumes that all the processes are statically allocated, i.e., the number of processes (p) is set at the beginning of the execution and do not change during the calculation. Each process is assigned a unique integer rank (0, 1, … , p-1) where rank 0 acts usually as “master process” gathering and managing the data from other processes. This approach is called single-program, multiple data (SPDM) (Pacheco, 1997). In the upcoming calculations, long-range interaction forces are not present. Thus, the most convenient parallelization strategy is spatial subdivision with load balancing, that is to say, maintaining a similar load in each CPU (Rapaport, 2002). Therefore the initial domain is divided in p equal-sized parts (Figure 3.2), each of them corresponding to a different process. Each CPU performs computations for the particles belonging to its subdomain. However, it is important to notice, that all the information of particles located within a region in adjacent subdomains must be transferred for synchronization and calculation of forces. This is called the overlapping region, and its size strongly affects the computational cost, since the time required for data communication directly depends on this factor. Regarding to the load balancing, the size of each subdomain (and therefore the number of particles belonging to it) is uploaded periodically to maintain, as long as possible, a similar number of calculations in each CPU. Figure 3.2: Parallelization scheme. The network is divided in p equal-sized parts (black lines) each of them corresponding to a different CPU. The overlapping region (yellow shade) transfers the information of contained particles between subdomains for syncronization. Size of subdomains is uploaded periodically to balance the loads of the CPUs. 88 Multiscale computational modeling of single cell migration in 3D 3.2.3. Mechanics of actin filaments, ACPs and motors Actin filaments comprise cylindrical segments of length 140 nm ( 0,A r ), and both the ACPs and motors are represented by two arms parallel to each other spanning between cross-linked actin filaments a distance of 70 nm ( 0,ACP 2r ) and 140 nm ( 0,M 2r ), respectively. Motions of the network components are governed by the Langevin equation: 2B 2 dd mdt dt     rr FF (3.1) where m is the mass of each element (actin, ACP, or motor), r is the element’s location,  is the friction coefficient, t is time, B F is a thermal force satisfying the fluctuationdissipation theorem, and F is a net deterministic force including extension, bending, and repulsive forces. Since inertia of all elements is negligible on the length and time scales of interest, positions of the elements are updated using the Euler integration scheme:       B 1 t t t t       r r F F (3.2) where ∆t is a time step. Extension and bending of the cylindrical segments constituting actin filaments, ACPs, and motors are computed using simple quadratic potentials, denoted by subscripts “s” and “b” respectively:     2 s s 0 1 2 U r r r   (3.3)     2 b b 0 1 2 U      (3.4) where r is bond length, s  is extensional stiffness,  is bending angle, b  is bending stiffness, and the subscript 0 denotes an equilibrium (zero-force) value. As in previous studies (Kim et al., 2009b), bending stiffnesses are introduced to restrict actin filament bending ( b,A  ), keep the two arms of ACP ( b,ACP1  ) or motor ( b,M1  ) parallel, and maintain the right angle between the axis of a filament and the arm of ACP ( b,ACP2  ) or motor ( b,M2  ). Specific values of the geometrical and mechanical parameters are listed in Table 3-1. In addition, the repulsive force is responsible for volume-exclusion effects by Chapter 3. Modeling cytoskeleton dynamics: rigidity-sensing of actin networks 89 which actin filaments cannot pass through each other, which is calculated by the following harmonic potential, r U , depending on the minimum distance, 12 r , between two cylindrical segments (Kim et al., 2009b):   2 r 12 c 12 c r 12 12 c 1if () 2 0 if r r r r Ur rr       (3.5) where r  is the strength of repulsive effects, and c r is the diameter of cylindrical segments. Then, the repulsive force is distributed to the two ends of the actin segment based on the relative location on the segment where 12 r is measured. 3.2.4. Dynamic behaviors of ACPs and motors Each motor in the simulation is assumed to correspond to a single myosin minifilament consisting of multiple myosin II molecules. Motors in the active state walk along actin filaments toward a barbed end at a rate, w k [s-1], depending on the extensional force acting on the arm, ss FU :       80,A 0,M s s w,1 s w,2 w,1 w,2 w,3 BB ws 80,A w,1 w,2 w,3 3.6 10 / if and 0 exp exp 3.6 10 / else rr r F t F t F t d d d k T k T kF r d d d                              (3.6) where w d ’s and w  ’s are time constants and mechanical sensitivities for walking of motors (Table 3-1), respectively, and t is a unit vector locally tangent to an actin segment in the direction of a pointed end. Although motors in this study mimic a myosin minifilament consisting of numerous myosin II molecules, equation (3.6) and the values of w d ’s and w  ’s are adopted from a single-molecule experiment examining myosin V under 1 mM ATP. The intention was to model generalized motor activity, and myosin V was chosen because it has been extensively characterized. In fact, the load-dependent walking rate of the minifilament (myosin II) is still qualitatively similar to that of myosin V, justifying the use of equation (3.6) for roughly mimicking myosin minifilament behavior. Nevertheless, a specific study of minifilament kinetics depending on the number of myosin heads was performed (Appendix B) concluding that, after adjustment, 90 Multiscale computational modeling of single cell migration in 3D myosin V and II behaviors are computationally similar. As seen in that equation, only tension ( 0,M rr ) directed to a pointed end ( s0Ft ) affects w k , resulting in a stall force, ~4 pN, beyond which motors cease walking. In addition, as in (Kim et al., 2011), ACPs and motors are able to unbind in a force-dependent manner following Bell’s equation:   us 0 u0 us B 0 u0 exp if if F k r r kF kT k r r         (3.7) where 0 u k is the zero-force unbinding rate coefficient for ACPs ( 0 u,ACP k ) or motors ( 0 u,M k ), and u  is the mechanical sensitivity for unbinding of ACPs ( u,ACP  ) or motors ( u,M  ) (Table 3-1). Note that although unbinding of motors is also one of the phases of walking, these two events are considered separable for systematic analysis. If the arm of motors reaches the barbed end of a filament by walking, it remains there until it unbinds. 3.2.5. Boundary conditions of the 3D computational domain After obtaining the network, actin filaments crossing the domain boundaries are severed and permanently clamped with periodic boundary conditions deactivated in all directions. During the measurement of strain and stress, the boundaries also act as sticky surfaces to take the binding between actin filaments and membrane into account; if either end of an actin filament is located within 30 nm of a boundary, the end is irreversibly clamped. Normal stress (  ) on each boundary is the sum of normal forces exerted by actin filaments clamped on the boundary, divided by actual area.  is used to compute movement of the boundaries in simulation; assuming that the domain is surrounded by an elastic medium with identical Young’s modulus, E , on all boundaries. Each boundary (assumed planar) experiencing  is displaced a distance corresponding to a strain  / E . Simulations begin with zero stress on all boundaries and proceed over time for 200 s. At this point, the network reaches a steady state stress in most cases, which is here defined to be the plateau stress. However, in a few cases, stress continues to slowly rise even after 200 s. Chapter 3. Modeling cytoskeleton dynamics: rigidity-sensing of actin networks 91 3.3. Results Here, the role of molecular motors as rigidity sensors is investigated and the contractile (normal) stress and strain of actomyosin networks tethered to 3D cubical domains is predicted. These are examined as a function of the various kinetic parameters and concentrations of motors as well as different elasticity of the surrounding medium. 3.3.1. Network morphology and stress evolution depend on substrate stiffness The initial network (Figure 3.3A) starts from a zero stress condition. Due to motor activity, and depending on substrate stiffness ( E ), the network shrinks to different extents at different rates. Lower E leads to shrunk and concentrated networks with highly bent actin filaments (Figure 3.3B and Figure 3.4-left). On the other hand, higher E prevents the domain contraction, forming heterogeneous networks with highly stretched filaments (Figure 3.3B and and Figure 3.4-right). These differences in network morphology have been reported in experiments where they found, for different cell types, that F-actin networks tend to be denser and less organized on more compliant substrates (Bordeleau et al., 2012, Blakney et al., 2012). Stress (  ) in all cases rapidly increases at the beginning although the rate of increase gradually falls, rising at a much slower rate by ~200 s in most cases (Figure 2A). Recognizing that stress continues to rise after this time, but constrained by computational resources from extending the calculations further, the value of stress at 200 s is used as a reference, and it is denoted as the “plateau stress”, p  . For E < 3 kPa, p  is proportional to E but becomes relatively constant for E ≥ 3 kPa, which corresponds well to literature (Lo et al., 2000, Mitrossilis et al., 2009, Mitrossilis et al., 2010, Saez et al., 2005) (Figure 3.5B). The maximum of p  is ~420 Pa. The initial slope of stress, 0  , measured at t < 10 s increases swiftly for E < 3 kPa and slower for E > 3 kPa (Figure 3.5D). The initial strain rate, 0  (= 0  / E ), decreases with greater E (Figure 3.5C); since contraction is associated with energy expenditure to overcome the internal friction and the rupture of cross-links, cells contracting against softer substrates will experience larger energy dissipation, leading to the slower rise in stress. The “plateau strain”, p  (strain at plateau stress), decreases with greater E falling below 0.05 for E > 10 kPa (Figure 3.5E). 92 Multiscale computational modeling of single cell migration in 3D p  and p  with various E show a first zone where p  rapidly changes, followed by a period of slower increase, which agrees well with (Rassier et al., 1999). Figure 3.3: Initial 3D network and cross-sections. A) The initial network is generated using a polymerization model and consists of actin filaments (cyan) crosslinked by ACPs (green) and molecular motors (red). Details of motors and ACPs are magnified. B) Cross-sections of the network at t = 200 s for three different values of E showing morphology and the magnitudes of extensional forces ( s F ). Softer substrates lead to a condensed network, whereas stiffer substrates contract very little, resulting in a heterogeneous network with tensed filaments. Figure 3.4: Different network organization. Inner view of a network surrounded by a compliant substrate (left) and a stiff one (right). Softer substrates lead to packed and more homogeneous network organization compared with stiffer ones which bring heterogeneous formations with long straight filaments crossing the network. The mechanical power was also measured, 00 PV   where 0  is initial stress corresponding to 0  , and V is the instantaneous volume of the domain. P exhibits a bimodal dependence on 0  , having a peak at E ~ 0.6 kPa (Figure 3.5F). At this peak, the network exerts 40% of the maximum 0  with intermediate 0  (~0.015 s-1), compared to cases with other E . Chapter 3. Modeling cytoskeleton dynamics: rigidity-sensing of actin networks 93 3.3.2. Network stiffness tracks the generated stress It has been recently found that cell stiffness tracks substrate stiffness over a range of stiffnesses before reaching a constant value (Tee et al., 2011). The steady-state stiffness of networks was measured at each E . Network stiffness ( n E ) was found to be proportional to (and nearly equal to) p  over the entire range of E , but proportional to E only up to a value of E ~ 3 kPa (Figure 3.5B). This tendency is consistent with the direct proportionality between prestress and G’ (or K’) of passive actin networks observed in experiments (Gardel et al., 2006). It is worth it to note that in order to measure the stiffness of networks, differential sinusoidal normal displacement of amplitude 280 nm was applied to the networks and the responding stress was calculated. For the purpose of this calculation, all the motor and actin cross-linking dynamics was deactivated to probe the instantaneous network stiffness, avoiding any progressive time-dependent changes in the network. Under these conditions, the networks exhibited a predominantly elastic response as indicated by the small phase delay between the applied strain and the responding stress (Figure 3.6). The network stiffness was then calculated by dividing the amplitude of stress by that of strain. 3.3.3. Effects of motor concentration Motor density is varied by adjusting the initial concentration of motors in the network, . In all cases,  increases with E and then exhibits a much slower rate of increase at high E (Figure 3A), but compared to the control case ( M R = 0.02), the tendency is less clear in the other cases, especially for low M R where the dependence between p  and E weakens. For low M R (< 0.02), p  tends to be higher with greater M R , in good agreement with literature (Dou et al., 2007, Kovacs et al., 2004, Mitrossilis et al., 2009). With a maximum at M R = 0.02 for most E , p  drops for higher M R . High contractile activity of the network enhances the rate of stress generation. Thus, 0  and 󰇗 increase for all values of E until M R reaches the optimal level explained above (Figure 3.7B and C). M R 94 Multiscale computational modeling of single cell migration in 3D Figure 3.5: Effects of substrate stiffness on network stress and strain. A) Time evolution of  at different E . Numbers in the legend indicate the values for E .  increases rapidly at first but reaches a nearly constant plateau value ( p  ) at ~200 s regardless of E . B) p  (circles) and network stiffness ( n E , triangles) as functions of E . p  monotonically increases for 3E kPa but saturates for 3E kPa. The network stiffness shows the same tendency as p  for all E . C) Contraction speed ( 0  ) as function of E . The network contracts rapidly with low E but more slowly as E increases. D) Initial rate of stress increase ( 0  ) with different E . 0  increases following ~ 0.55 E for E < 1 kPa and ~ 0.16 E for E > 1 kPa. E) p  and corresponding strain ( p  ) at various E . Higher p  corresponds to lower p  . For high E , p  asymptotically approaches 0. F) A relation between normalized power (P) and initial stress ( 0  ). P becomes maximal at E ~ 0.6 kPa, generating 40% of the maximum 0  and intermediate 0  of ~0.015 s-1. Each color within the symbols in E and F indicates the value of E in A with the same line color. 3.3.4. Effects of unbinding and walking behaviors of motors Motor unbinding is explored by varying the zero-force unbinding rate ( 0 u,M k ) and the processivity ( u,M  ). On the other hand, motor walking is studied by varying the sensitivity ( w  ) which is equivalent to variation of the stall force. Although higher 0 u,M k results in more frequent unbinding, it does not necessarily lead to lower p  since unbinding can also help stalled motors due to blocking effects to bind to other binding sites so that they can keep walking. However, too frequent Chapter 3. Modeling cytoskeleton dynamics: rigidity-sensing of actin networks 101 Mitrossilis et al., 2010, Webster et al., 2011). Although the time required to reach p  is somewhat different between ~200 s in these simulations and ~600 s in experimental studies, it is common that the time needed to reach p  is relatively independent of E in both, as found in (Mitrossilis et al., 2009) (Figure 3.5A). The inverse proportionality between p  and E (Figure 3.5E), consistent with experiments (Rassier et al., 1999), supports that cells on stiff substrates tend to rearrange intracellular structures rather than deforming the substrate, as in (Engler et al., 2004b, Munevar et al., 2004, Engler et al., 2004a, Richert et al., 2004). Numerical results of 0  depending on E (Figure 2C) and of 0 ()P  (Figure 3.5F), show that the emergent behavior of the network follows Hill’s equation for muscle contraction (Hill, 1938). The 0  ( 0  ) curve was fit (Figure 3.13) with the relation    00 a b c     where a = 10.0 Pa, b = 0.0103 s-1, and c = 1.0023 Pa/s. In experiments using various types of muscle cells (McMahon, 1985) and myoblasts cells (Mitrossilis et al., 2009), introducing a shape factor max max r a F b V   0.25 normalized the data. For the control case, it was obtained 1 0,max ra   0.12 and 2 0,max rb   0.4. The values of these factors were found to be regulated by parameters, such as 0 u,M k , u,M  and w  although normalizing the data leads to similar curves (Figure 3.10A-D). Further insights were provided regarding the effects of motor concentration ( M R ). It was observed p  attains a maximum at M R = 0.02 (Figure 3A). This could be attributed to the limited binding sites for motors on actin filaments in the simulations. However, note that M R in this model corresponds to the concentration of multimerized myosin II structures in cells rather than that of individual molecules. Therefore, M R = 0.02 is actually a very high density of large motor structures, and thus cells are likely to have such an optimal R due to blocking effects between the large aggregates of myosins. A refined model including multiple myosin heads per motor and multiple binding sites per actin segment would help to clarify this issue. 102 Multiscale computational modeling of single cell migration in 3D Figure 3.12: Effects of actin concentration and actin filament length. A) p  monotonically increases with A C . In these simulations, ACP R is constant at 0.01, but M R decreases with higher A C since M C is fixed at 0.24 μM, corresponding to the constant number of motors. B) Fraction of motors stalled due to: (i) high applied forces (black), (ii) blocking (gray), or (iii) arrival at barbed ends of filaments (white) at steady state as a function of A C . At low A C , ~50% of motors are not stalled since many of them lie in the inactive state due to lack of network percolation. As A C increases, motors are more likely to be stalled due to high forces as opposed to blocking. C) p  increases dramatically as f L is increased. D) Fraction of motors stalled as a function of f L . Legend is shared with B. As f L increases, more motors are stalled due to attaining their maximum force while fewer motors are stalled due to arrival at barbed ends. The number of motors stalled due to blocking remains nearly constant regardless of f L . Considering the variable extent of multimerization of myosin II molecules into a minifilament or thick filament, the effects of the zero-force rate of motor unbinding ( 0 u,M k ) were evaluated (Figure 3.8) and the mechanical sensitivity for unbinding ( u,M  ) and walking ( w  ) on network contraction. Motors with high u,M  more readily unbind, and those with high w  are more likely to be stalled at small forces. p  , 0  and 0  are higher for lower u,M  (Figure 3.9A-C) or for lower w  (Figure 3.9D-F) although the Chapter 3. Modeling cytoskeleton dynamics: rigidity-sensing of actin networks 103 overall trend of the curves is conserved well in both cases. By contrast, with high 0 u,M k , p  is substantially reduced while 0  and 0  are not affected (Figure 3.8A-C). Figure 3.13: Control case reduced to Hill’s equation. The network shrinks faster for softer substrates, developing less stress while slower shrinkage leads to higher stress. Values for the constants a, b, and c in Hill’s equation    00 a b c     are 10.0 Pa, 0.0103 s-1, and 1.0023 Pa/s respectively. Note that 0  and 0  were measured at t = 10 s. In summary, this work elucidated one mechanism by which cells can modulate their properties and respond to the surrounding environment via cytoskeleton contractility, using an agent-based computational model. Although the model is based on molecularlevel processes, macroscopic behaviors of the active cross-linked actin networks agree well with the response of cells probed in experimental quantitative studies (Zaman et al., 2006, Discher et al., 2005, Lo et al., 2000, Ruegg et al., 2002). It was found that the biphasic relation between substrate stiffness and the level of generated forces (Engler et al., 2004b, Munevar et al., 2004, Engler et al., 2004a, Richert et al., 2004) is attributable to a transition from stalling due to steric hindrance or “blocking” in soft substrates to that due to stall forces in stiff substrates. In addition, the results showed that in response to increases in substrate stiffness, the contraction rate of cells increases while the corresponding contraction velocity decreases, also consistent with experiments (Webster et al., 2011). All of these suggest that actomyosin contractility is one plausible standalone mechanism capable of contributing directly to cell mechano-sensing (Mitrossilis et al., 2009), consistent with various experimental findings that myosins are crucial for cells to sense surrounding matrix elasticity (Engler et al., 2006, Trichet et al., 2012), and that cell responses to rigidity of the external matrix reflect adaptation of the actomyosin machinery to load following Hill’s relation (Mitrossilis et al., 2009). 104 Multiscale computational modeling of single cell migration in 3D Variable Symbol Value Diameter of cylindrical actin segments c r 7.0×10-9 [m] (0.05) Length of cylindrical actin segments 0,A r 1.4×10-7 [m] (1.0) Time step t 2.3×10-8 [s] (1.0×10-5 ) Strength of repulsive force r  4.2×10-4 [N/m] (2,000) Extensional stiffness of actin s,A  4.2×10-3 [N/m] (20,000) Bending stiffness of actin b,A  2.64×10-19 [N m] (63.75) Number of actins per segment c N 20 Length of a single arm of ACP 0,ACP r 3.5×10-8 [m] (0.25) Extensional stiffness of ACP s,ACP  4.3×10-4 [N/m] (2,000) Bending stiffness 1 of ACP b,ACP1  1.04×10-18 [N/m] (250) Bending stiffness 2 of ACP b,ACP2  4.142×10-18 [N m] (1,000) Length of a single arm of motor 0,M r 7.0×10-8 [m] (0.5) Extensional stiffness of motor s,M  4.23×10-4 [N/m] ] (2,000) Bending stiffness 1 of motor b,M1  1.04×10-18 [N/m] (250) Bending stiffness 2 of motor b,M2  4.142×10-20 [N m] (10) Substrate stiffness E 0.16 - 40.96 [kPa] Concentration of actin A C 1.2×10-5 [M] Ratio of CACP to CA ACP R 0.01 Ratio of CM to CA M R 0.02 (unless specified) Zero-force unbinding rate coefficient of ACP 0 u,ACP k 0.115 [s-1]a Sensitivity of ACP unbinding u,ACP  1.04×10-10 [m]a Zero-force unbinding rate coefficient of motor 0* u,M k 2×10-5 [s-1]b§ Sensitivity of motor unbinding * u,M  2.6×10-10 [m]b§ Sensitivity 1 of motor walking * w,1  1.3×10-8 [m]c Sensitivity 2 of motor walking * w,2  2.2×10-9 [m]c Time constant 1 of motor walking w,1 d 1.1×10-3 [s]c Time constant 2 of motor walking w,2 d 8.0×10-2 [s]c Time constant 3 of motor walking w,3 d 5.9×10-3 [s]c Table 3-1: Main parameters used in the simulations Numbers in parentheses are corresponding dimensionless values as defined in the text. “*” on symbols indicates reference values of parameters studied in the sensitivity analysis. a(Ferrer et al., 2008), b(Guo and Guilford, 2006), c(Uemura et al., 2004). Values marked by “§” are adopted from given literature with adjustment based on assumption that motors in this study consist of many myosin II molecules. 4. TIME DEPENDENT MECHANOSENSING This chapter revises the assumptions of the mechanosensing model described in Chapter 2, introducing a temporal dependence of the cell response based on experimental findings from literature and the results predicted by the Brownian-dynamics model detailed in Chapter 3. This new approach, which is initially simplified to 1D, is used to simulate recent experiments of cell mechanosensing involving sudden changes in substrate compliance and force saturation with time and stiffness. Finally, the model is extended and applied to different 3D geometries simulating the same experiments and the results are compared to the previous simplified approach. 4 Chapter 106 Multiscale computational modeling of single cell migration in 3D Contents 4.1. Introduction .................................................................................................... 107 4.2. Constitutive law.............................................................................................. 109 4.2.1. Definition of internal variables ............................................................... 110 4.2.2. Additive decomposition of the displacement field ................................. 111 4.2.3. Cell forces ............................................................................................... 112 4.2.4. Time-dependent response ....................................................................... 114 4.3. Problem description and results ..................................................................... 118 4.3.1. Simulated experimental assay ................................................................. 118 4.3.2. Cell contraction and force generation under different extracellular stiffnesses .............................................................................................................. 119 4.3.3. Simulating step changes in extracellular stiffness .................................. 121 4.4. Parameter sensitivity analysis ........................................................................ 123 4.5. Extension to 3D problems .............................................................................. 126 4.5.1. Geometry and model adaptation ............................................................. 127 4.5.2. Material definition and boundary conditions .......................................... 128 4.5.3. Results ..................................................................................................... 128 4.6. Discussion ...................................................................................................... 130 Chapter 4. Time dependent mechanosensing 107 4.1. Introduction Currently, there exists much interest in characterizing the mechanical properties of living cells as a material (Kasza et al., 2007). In particular, cells behave completely different when supporting loads (passive) than when exerting forces (active) (Kollmannsberger and Fabry, 2011, Ronan et al., 2012, Ujihara et al., 2012). As stated in previous chapters, cells are constantly pulling on the extracellular matrix (ECM) in order to evaluate the mechanical environment and accordingly respond, adjusting their properties. This mechanism that cells use to sense rigidity has been attributed to two main contributions: cytoskeleton contractility and adhesion complexes (Discher et al., 2005). However, recent experimental works have shown the predominant role of cell contractility. In fact, some of these experiments (Mitrossilis et al., 2009, Mitrossilis et al., 2010, Fouchard et al., 2011) suggest that contractility at cell scale due to acto-myosin response to load is the main hypothesis to understand rigidity sensing mechanism. Others, however, have presented strong evidences that the rigidity-sensing mechanism in cell migration is not only locally driven by focal adhesion growth, but also mediated by a larger-scale mechanism originating in the cytoskeleton (Trichet et al., 2012). Many different experimental (Kobayashi and Sokabe, 2010, Trichet et al., 2012, Ghassemi et al., 2012) and computational works (McGarry et al., 2009, Zemel et al., 2010b, Vernerey and Farsad, 2011) have been developed in order to characterize the mechanical active response of cells under different rigidity conditions of the extracellular environment. In particular, novel experiments using uni-axial loading conditions with a precise control of the mechanical properties of the extracellular environment have been rising. In this sense, Mitrossilis et al (Mitrossilis et al., 2009) have recently developed a single cell traction force experiment with a custom-made parallel microplate setup. In this study, the authors adhere single cells to two parallel glass microplates coated with fibronectin. One plate was rigid, whereas the other was flexible and used as a nanoNewton force sensor (i.e., a spring of calibrated stiffness). A computer-controlled detection of the flexible plate deflection allowed quantifying real-time single cell traction forces. Therefore, using this setup they were able to measure the cell response as a function of the plate rigidity. In a first phase, the forces increased at different rates, depending on the plate stiffness. After approximately 10 minutes, the forces saturated reaching a plateau. This plateau force was observed to depend upon the stiffness of the flexible plate as long as the stiffness is less than 60nN/μm. However, at higher stiffness 108 Multiscale computational modeling of single cell migration in 3D values, the plateau force achieved a maximum value of ≈300 nN that was independent of the stiffness. Moreover, using this measurement system is facile to induce a step change in the extracellular stiffness in order to evaluate the viscoelastic response of cell contraction to this change. In fact, recently, several authors carried out this experiment, concluding that contracting cells are able to adapt to the stiffness step change on a short timescale of 10’s of seconds, showing practically an instantaneous response (Mitrossilis et al., 2010, Crow et al., 2012). Figure 4.1: Mechanical models to describe the contractile cell response. A) the two-spring approach (Schwarz et al., 2006), B) active matter theory (Marcq et al., 2011), C) the three-spring approach (Moreo et al., 2008) and D) the three-spring and a dashpot approach (Crow et al., 2012). Collectively, these experiments constitute a set of benchmark data that can be used to validate the predictive potential of models to simulate mechanosensing. The first computational model developed to simulate the mechanosensory role of cells was proposed by Schwarz et al (Schwarz et al., 2006) and was based on a two-spring approach (Figure 4.1A). This model is able to predict saturation force phenomena, although the plateau force is independent of extracellular stiffness. Different “three-spring” approaches have been also proposed to model mechanosensing phenomenon (Figure 4.1B). In particular, Moreo et al. (Moreo et al., 2008) presented a three-spring model with a linear stiffness-dependent actuator, which was capable of predicting cell contractility in Chapter 4. Time dependent mechanosensing 109 response to changes in extracellular matrix stiffness (Figure 4.1C). In fact, this is the model used and described in Chapter 2 of this Thesis. However, the mechanical approach is static, whereas real cell contractility is time-dependent as has been extensively exhibited experimentally (Mitrossilis et al., 2009). To overcome this limitation several authors have proposed a different theoretical model based on active matter theory (Zemel et al., 2010b, Marcq et al., 2011). Indeed, Zemel et al (Zemel et al., 2010b) established a new model in order to understand the stiffness-dependent orientation of stress fibers in adherent cells. Marcq et al (Marcq et al., 2011) focused their modeling work on the analysis of the temporal cell response to substrate rigidity, but, they did not study the response of their model to sudden changes in substrate rigidity. Recently, Foucard and Vernerey (Foucard and Vernerey, 2012) investigated the viscoelastic behavior of stress fibers, but their primarily addressed the dependence of stress fiber elasticity on stretching frequency. Deshpande et al. combined mechanics with time-dependent chemical signaling (Deshpande et al., 2006) in order to predict the role of focal adhesions and stress fiber concentration in the development of force by cells. Finally, Crow and co-workers (Crow et al., 2012) proposed a three-spring model including a dashpot and an independent actuator contracting at a constant velocity (Figure 4.1D), although their approach was focused on capturing the instantaneous cell response rather than the long term behavior. Notably, with this scheme the authors were able to adjust a mechanical law to successfully simulate step changes induced in the extracellular rigidity; however, their model is not able to predict temporal saturation of forces under different stiffness. Therefore, to fully predict this benchmark of experiments, a novel one-dimensional constitutive law for cell contractility and force generation, capable of reproducing some important features of cell response to extracellular stiffness is proposed in this chapter. 4.2. Constitutive law In this work, the mechanosensing model described in Chapter 2 (initially proposed by (Moreo et al., 2008)) is extended to include rate-dependent effects. The system is similar as in the previous work, consisting on two parallel springs representing the stiffness of the passive mechanical components of the cell and the actin filaments which are, furthermore, in series with the myosin motor contractile system (Figure 4.2). This approach was purely mechanical and static, so that the contractile system exerted a specific force depending on the cell strain, and thus depending on the substrate stiffness. However, the time-force 110 Multiscale computational modeling of single cell migration in 3D evolution, which plays a critical role in cell mechanosensing, was not taken into account. These rate-dependent or viscous effects have already been included into other models by means of the addition of dashpots, e.g. in the three-spring approach, however, they are not able to simulate all cell responses associated with changes in the extracellular rigidity. Hence, a different approach is here presented incorporating an internal variable that takes into account the kinetics associated to cell molecular motors, in particular, the force saturation due to motor stalling. Figure 4.2: System schemes for measuring cell mechanosensing properties. A) Experimental setup used by (Mitrossilis et al., 2009) to measure the effect of substrate rigidity on cell forces. They decouple probe elongation (i.e., force) from cell contraction using a double feedback loop which independently regulates spring and cell lengths to maintain cell-spring contact in a fixed position. In this way, the setup acts as if the cell was compressing a spring of stiffness   0 spring cell K dL dL , permitting the study of a wide range of rigidities with a single probe of stiffness 0 K . B) Model scheme used in this work to investigate stiffness-dependent cell response. In similar fashion to the experimental setup, the substrate stiffness is represented by a single spring ( subs K ). The cell body is modeled as two parallel springs, one of them in series with a contractile actuator. pas K represents the passive stiffness of different mechanical components of the cell (e.g. microtubules, membrane, cytoplasm), whereas act K stands for the rigidity of the actin filaments. The acto-myosin system (AM) is then placed in series, contracting the cell body by stretching the actin and compressing the passive components 4.2.1. Definition of internal variables To simulate the cell, a one-dimensional mechanical device consisting of two springs and a linear stiffness-dependent actuator   cc f  is considered (Figure 4.2B). The active force of the contractile actuator simulates the force provided by the actin and myosin crossbridges at the sarcomere level when shortening. In this scheme, the series element act K Chapter 4. Time dependent mechanosensing 117 words, free motors tend to form dynamic links that may reorganize the network but contribute little to the overall contraction, whereas stalled motors stabilize the cross-links leading to a more organized structure able to contract and generate high forces. Therefore, this hypothesis implies that at 0   , the cytoskeleton network is still not fully formed, and the motors have not yet had much opportunity to contract ( c0   ) and little or no force is generated ( c0f ). However, when all the motors are stalled ( 1   ), the cell cytoskeleton has achieved its maximum ability to generate force at all values of c  . Hence, when c0   and all of the motors have become stalled, the maximum force can be generated ( c max ff ); when c1   (that is, when no constrains impede contraction, subs 0K ) all motors have reached their maximum contractile displacement, and the force generating capacity goes to zero ( c0f ). In any case, 0  in equation (4.13) allows regulating the initial network state, although 00   was used in the simulations. On the other hand, the alternative definition of c f implies that the acto-myosin system is able to contract up to 1  regardless of the number of motors that have stalled, which is not fully consistent with the assumption that represents the motor stalling evolution. For instance,   t  is set to 0 (so that no motor stalls in the simulation) and a hypothetical actin network behaving as we propose is considered, the actin would permanently slide on the myosin motors, continuously remodeling the cytoskeleton network but not necessarily contracting the whole structure. In other words, if motors were not stalled (forming a more stable cross-linked network), they would tend to “walk”, sliding on actin filaments rather than transmitting contraction through the network. Additionally, the alternative approach would lead to more complicated equations as shown below. Recall the mathematical procedure described previously in this Chapter but starting from the alternative definition of   alt c max c 1 ( ) 1f t f     . The superscript “alt” is used to avoid confusion with previous equations: With this new approach and again combining combining equations (4.3) (4.4) (4.5) and (4.6), the slippage ( alt c  ) dependence on  becomes:       act pas subs max 1 alt c act pas subs max pas subs act 1 ( ) ( ) () K K K f tt K K K t f K K K         (4.15) 118 Multiscale computational modeling of single cell migration in 3D And therefore the cell force can be written as: subs act cell max pas subs max max act act act 1 act 1 () ( ) ( ) 1 K K f t f K K t f t f K K K K               (4.16) Note the presence of   t  in the denominators Figure 4.4: Renormalization of AM system contraction depending on the definition of the acto-myosin force. The current approach (left) renormalizes both the maximum slippage and maximum force with time, whereas an alternative definition (right) only renormalizes the force with important mathematical implications. In sum, this alternative definition of c f makes difficult to choose a regulatory rule (  ) with physical menaning (e.g. system approaching to its maximum force) which furthermore leads to a temporal response   t  that fulfills the experimental data, also maintaining some physical meaning (e.g. motor stalling evolution). 4.3. Problem description and results 4.3.1. Simulated experimental assay A single cell fixed between a rigid and a flexible plate is simulated in order to reproduce the experimental setup designed by Mitrossilis et al.(Mitrossilis et al., 2009) and used by different authors (Webster et al., 2011, Crow et al., 2012) to evaluate cell mechano- Chapter 4. Time dependent mechanosensing 119 sensing properties (Figure 4.2A). In this setup, the cell pulls on the two parallel plates while the plate deflection is measured. By using a double feedback loop which independently regulates spring and cell lengths to maintain cell-spring contact in a fixed position, they obtain the temporal cell response for different external rigidities. A simplified one-dimensional problem is defined to understand contractility and force generation due to cell response under different extracellular rigidities (Figure 4.2B). Two main conditions are simulated. Firstly, the temporal evolution of cell contractility and force generation under a wide range of extracellular rigidity values are evaluated. Secondly, the cellular time-response under step-changes of the extracellular stiffness is computed. 4.3.2. Cell contraction and force generation under different extracellular stiffnesses The temporal evolution of force generation, which depends on the extracellular rigidity, is shown in Figure 4.3D and Figure 4.5A. In the former, and according to equation (4.9), force develops following the evolution of motor activation. The maximum value achieved (the plateau force p f ) depends on the substrate stiffness ( subs K )(Figure 4.5B). Note that the model parameters were adjusted to obtain similar values to experimental data (Mitrossilis et al., 2009). At a given time, the force increases with stiffness, abruptly for low stiffness and smoothly for higher ones (Figure 4.5A). Qualitatively similar results were found experimentally, measuring the cell stress exerted on micropillars of different stiffness. (Trichet et al., 2012). Taking the force at t=1200 seconds as the plateau force and plotting it against substrate stiffness, it is found that for subs K < 100 nN/μm, p f rapidly increases with stiffness, saturating for higher rigidities, in good agreement with literature (Mitrossilis et al., 2009). In fact, both in the model and experiments, p f is proportional to subs K in compliant substrates ( 0.87 subs K and 0.94 subs K respectively). In addition, other authors found an initially linear relationship ( 0.93 subs K ) (Trichet et al., 2012), although their study only took into account substrate stiffness up to 80 nN/ μm (Figure 4.5B). The rate of force build-up ( dF dt , slope of the force curve) increases quickly and proportionally to the stiffness at first, and slows down as the force approaches the maximum plateau (~300 nN (Mitrossilis et al., 2009)). In relation with the 120 Multiscale computational modeling of single cell migration in 3D substrate stiffness and measuring dF dt in the first phase of contraction (t<100 s), the rate of force build-up strongly increases for compliant substrates and presents slight changes for stiffer ones (Figure 4.6A), similarly to recent experimental findings (Mitrossilis et al., 2009, Mitrossilis et al., 2010, Trichet et al., 2012). Note that in this last work (Trichet et al., 2012), as in the measurement of the plateau force, only the initial linear regime was observed since their study was focused on compliant substrates. Figure 4.5: Cell force evolution and plateau values depending on substrate stiffness. A) Cell force as a function of substrate stiffness for different times. The force increases exponentially at low stiffness, and saturates (plateau force) for higher ones. The force increases with time for all the stiffness due to the growing motor activation. B) Saturation or plateau force ( p f ) as a function of substrate stiffness in log-log scale. p f , in other words cell f measured at t=1200 s, increases proportionally to subs K ( 0.87 p subs fK ) for softer substrates, in similar fashion to the experimental finding from (Mitrossilis et al. 2009) ( 0.94 p subs fK , exp 1 in the legend), and (Trichet et al. 2012) ( 0.93 p subs fK , exp 2), and saturates for stiffer ones as in (Mitrossilis et al. 2009). The speed of shortening ( cell v ) is easily derived from the rate of force build-up as   cell subs /v dF dt K . The cell shortening is faster for compliant substrates, whereas stiffer substrates, which resist the contraction, lead to practically null velocities for subs K >1000 nN/μm (Figure 4.6B), qualitatively similar to the speed behavior found in (Mitrossilis et al. 2009). Mechanical power (P) is also computed, presenting a bi-phasic behaviour with the load and showing a peak at ~40% of the maximum generated force, following the classical behavior observed in muscles and recently found in myoblasts (Mitrossilis et al., 2009) due to acto-myosin contraction (Figure 4.6C). Chapter 4. Time dependent mechanosensing 121 Figure 4.6: Rate of force build-up, speed of shortening and cell power. A) Initial rate of force build-up ( dF dt ) as a function of substrate stiffness. In the first phase of contraction (t <100 s), the rate of force build-up strongly increases for compliant substrates and presents slight changes for stiffer ones as found by (Mitrossilis et al. 2009)(exp 1 in the legend). B) Inverse of cell speed of shortening ( cell v ) as a function of substrate stiffness. The cell contracts faster for softer substrates, cell v approaching zero for subs K >1000 nN/μm. The curve behaves similarly to experiments (Mitrossilis et al. 2009) C) Normalized mechanical power vs. normalized force. P presents a bi-phasic behaviour with the load, showing a peak at ~40% of the maximum generated force. These behaviors are qualitatively similar to those found by (Mitrossilis et al. 2009). 4.3.3. Simulating step changes in extracellular stiffness In these simulations the cell response to step-changes in substrate stiffness is evaluated with a period of 20 seconds (Figure 4.7A top plot), following the experiments carried out by Crow et al. (Crow et al., 2012) and of 100 seconds (Figure 4.7B top plot), to study the long-term response and compare the results with Mitrossilis et al. data (Mitrossilis et al., 2010). Specifically, subs K is first varied from 10 to 100 nN/μm in a first case, and then from an extremely low value (~0) to an extremely high one (∞) with a period of 20 seconds (Figure 4.7A,B). In both cases, the force generated increases faster (higher slope) when the stiffness is higher (since dF dt increases with subs K ) (Figure 4.7A,B middle plots) in good agreement with the simulated experiments (Mitrossilis et al., 2010, Crow et al., 2012). The cell height, however, decreases faster for compliant substrates (Figure 4.7A,B bottom plots). The general trend is clearly seen in the cases with extreme values of substrate stiffness. When subs K ~ ∞ the force increases very fast (Figure 4.7A middle plot), especially during the first seconds when the activation of motors is in its early phase, whereas the cell height remains constant (Figure 4.7B bottom plot). In contrast, when subs K ~ 0, the rate of force build-up is near zero but cell height rapidly decreases (Figure 4.7B bottom plot). For higher periods of stiffness step change (Figure 4.7C), the 122 Multiscale computational modeling of single cell migration in 3D system behaves similarly but the effect of substrate stiffness on the cell force variation is less relevant as time increases due to the saturation of  . The motor stalling determines the time at which the system reaches equilibrium, however, the force attained at that point depends on the load-history. For this reason, in the case with stiffness step variations from ~0 to ~∞ (Figure 4.7C), the plateau force after 500 seconds is below 200 nN, much lower than the value that would correspond to an infinitely rigid substrate (~ 300 nN). Logically the cell contraction is higher than in previous cases due to the long periods of low stiffness. Certainly, the motor stalling could also depend on substrate stiffness due, for instance, to morphology changes of the intracellular network. This could affect the force evolution after each step change, however, it has been reported that although the reasons for motor stalling depend on external stiffness (higher percentage of motors get stalled by forces for stiffer substrates whereas higher percentage of motors get stalled by blocking phenomena if softer substrates), the temporal evolution of stalling remains practically unchanged (Borau et al., 2012). Chapter 4. Time dependent mechanosensing 123 Figure 4.7: Cell response to step-changes in substrate stiffness. A) The stiffness switches from 10 to 100 nN/μm with a period of 20 seconds. The force generated increases faster (higher slope) when the stiffness is higher. The cell height, however, decreases faster for low substrate stiffness. B) The stiffness switches from 0 to ∞ with a period of 20 seconds and C) 100 seconds. The force increases at maximum rate for a completely rigid substrate, whereas the cell height remains invariable. Nevertheless, for a completely compliant substrate, there is no force development and the cell body contracts at maximum speed. 4.4. Parameter sensitivity analysis In order to find which parameters most strongly influence cell rigidity-sensing, a sensitivity analysis is performed. For this analysis, the substrate stiffness is held constant ( subs K =100 nN/μm). 124 Multiscale computational modeling of single cell migration in 3D The definition of the internal variable that describes the motor stalling evolution (  ) is the key for all the time-dependent processes simulated in this model. Thus, a proper understanding of the effects of the involved parameters is needed. The global parameter controlling the time evolution of  is the relaxation time (  ), which determines how fast the stalling events reach saturation and specifically the rising of force with time ( dF dt ). Figure 4.8A shows how  plateaus faster for lower values of  . However,  is not a free parameter, since it depends on  and max f ( equation (4.12)). While  only affects  , max f affects the mechanical equilibrium, therefore altering the magnitude of the exerted forces ( cell f ). Figure 4.8B shows how the plateau force adjusts due to variations of max f , while the time-evolution remains unchanged. This happens because  is varied in the same proportion as max f thus leading to a constant relaxation time. If however, max f is varied alone (Figure 4.8C), both the plateu force and the relaxation time are affected. High values of max f lead to higher forces, which are, furthermore, attained faster. On the other hand,  is a parameter only affecting time evolution (Figure 4.8D). Hence, changes in  are equivalent to changes in  (Figure 4.8A). The purely mechanical components of the cell ( pas K , act K ) do play an important role in the mechanical equilibrium. The actin stiffness ( act K ) appears to be the more relevant component in the mechanical system. Therefore, changing its value, leads to substantial changes in the plateau force (Figure 4.9A). As shown in Figure 4.2, this component is in series with the AM system. Thus, increasing the actin stiffness leads to lower values of c  , which in turn increases the cell force (equations (4.4) and (4.6)). However, bigger changes of pas K are needed to be reflected in cell f (Figure 4.9B). Actually, these changes in the plateau force are only noticeable when pas K has similar or greater values than act K . This is confirmed in Figure 4.9C, where both parameters are varied together obtaining similar results to Figure 4.9C, where only act K changes. Moreover, the effects of the slippage distance parameter 1  are explored. It is important to note that this parameter is independent of the cell length ( c L ). Hence, c L does not affect the results as long as 1  remains constant. However, higher values of slippage, lead to higher forces and higher deformability of the cell as shown in Figure 4.10. Other authors have taken this value as infinity (Marcq et al., 2011), which is equivalent to consider a constant force of the AM system, independent of substrate Chapter 4. Time dependent mechanosensing 125 stiffness and other parameters. This allows capturing the force-stiffness linearity only for very compliant substrates, whereas the approach presented in this chapter is able to extend the linear regime to experimental ranges, as will be discussed subsequently. Figure 4.8: Sensitivity analysis of parameters involved in motor stalling evolution. A) Evolution of stalling depending on the relaxation time (  ). For higher values of  ,  needs more time to reach plateau. B) If max f and  are varied together, the relaxation time (  ) remains constant, whereas the magnitude of exerted forces changes. C) max f affects both the relaxation time and the mechanical equilibrium. Higher values of max f correspond with higher plateu forces attained faster. D) The viscosity coefficient only affects the time-evolution. Thus, changes in  are equivalent to changes in the relaxation time. 126 Multiscale computational modeling of single cell migration in 3D Figure 4.9: Sensitivity analysis of the cell force evolution depending on actin and passive stiffness. A) Changes in act K lead to important changes in the plateau force. High values of actin stiffness decrease the contraction of the AM system, thus producing higher forces. B) Variations of the cell passive stiffness affect the force generation only when the values of pas K are similar or greater than the values of act K . C) By varying both act K and pas K at the same time, the changes in force generation are similar to those obtained when varying only act K , confirming that the actin stiffness is the predominant mechanical parameter. Figure 4.10: Effects of maximum slippage distance. Curves corresponding to subs K =100 nN/μm. Higher values of 1  enhance system contractility, leading to higher exerted forces. 4.5. Extension to 3D problems So far, the 1D approach of the mechanosensing model has demonstrated its ability to reproduce experimental data and its potential in cell motility models. However, in vivo cell response involves many complex mechanisms such as internal pressure, evolving adhesion area, steric effects, curvature and other factors that become relevant in 3D. In this section, following the simulations previously described and using a experimental set similar to the one proposed by (Mitrossilis et al., 2009), the cell contractile response is computed using a 3D approach. Appendix B 229 The maximum number of heads studied is 20 since for higher values, the computational time increases too much. In fact, with ( reb p / u p ~1) and more than 20 heads, 0 u,Fil k goes beyond 1e-10 s-1 and the complete unbinding of the myosin minifilament might never happen. The dependence of 0 u,Fil k on both reb p / u p and heads n is shown in Figure B.6. The ratio reb p / u p =0 corresponds with the results discussed in previous sections. As could be expected, at higher ratios and higher number of heads, the minifilament remains bound to the actin more time, so the rate of unbinding decreases. This behavior is consistent for different values of forces. Figure B.6: Influence of rebinding probability and number of heads. B.4 Minifilament properties The analysis conducted so far allows understanding the role of the number of myosin heads, and the influence of several parameters on the general behavior of a myosin minifilament. Using the parameters of one single myosin head in the model described in Chapter 3 would lead to low cell contraction compared to experiments. With 0 u,M 0.02k s-1 and 9 u,M 2.6e    , all the motors would unbind with insignificant forces and the whole network would collapse. Hence, it is assumed that each molecular motor actually represents a minifilament with multiple heads and therefore with lower rates and sensitivity of unbinding. Specifically, it was found that 05 u,M 2ek  s-1 and 10 u,M 2.6e    were appropriate to achieve more realistic results. To justify these adjustment, the target curve 230 Multiscale computational modeling of single cell migration in 3D (Bell’s equation using those values, Figure B.7B) was fit by varying the number of heads and the probability of rebinding of each head. Note that the probability of rebinding should be independent of the force acting on the filament, however, it is really difficult to properly fit the Bell’s curve with a constant value of reb p and for any heads n . Also note that u,Fil k (and consequently u p ) changes very quickly with the applied force and the number of heads. Hence, if reb p remains constant, it affects u,Fil k much more when reb p is comparable to u p . In other words, if reb p is low/high, u,Fil k is influenced at low/high forces when u p is also low/high. In sum, the curve becomes destabilized and it is not possible to exactly fit it with a constant value of rebinding probability. For this reason, reb p is systematically adjusted for each level of force and for different number of heads. Interestingly, with low number of heads, reb p must increase with force to fit the curve. On the other hand, to do so with high number of heads, reb p must decrease, suggesting that there exists an optimum value of heads n which fits the target curve with an almost constant value of reb p . Figure B.7: Bell’s equation fit with multiple myosin heads and different rebinding probabilities. A) The probability of rebinding ( reb p ) at each level of force and heads n that fits the Bell’s equation. The trend for low heads n is reb p increasing with force, whereas for high heads n , reb p decreases. B) Bell’s curve (target) fit for several heads n and reb p . Legend is shared with A. Appendix B 231 With all this, it can be concluded that a minifilmaent of myosin II with a number of heads between 50 and 100 and a constant probability of rebinding between 0.002 and 0.004 would be equivalent to the motor kinetics used in the Brownian-dynamics model (Chapter 3), hence justifying the use of 05 u,M 2ek  s-1 and 10 u,M 2.6e    instead of the experimental values found for single myosin V motors. 232 Multiscale computational modeling of single cell migration in 3D APPENDIX C: ADDITIONAL MODEL VALIDATIONS The migration model described in Chapter 5 studies the cell behavior depending on different environmental factors through probability functions. In this Appendix, the randomness of these probability functions is firstly tested to check the consistency of the model. Next, the mechanical conditions are isolated to properly check the model parameters, and the fluid and chemical factors are independently studied. Following, the probability functions are subjected to a sensitivity analysis. Finally, some assumptions of the microfluidic simulation are tested with detailed simulations at the cell scale. C.1 Model randomness To check whether the random functions used in the model are truly stochastic, a simple analysis is performed for validation. This study consists of 100 simulations of cell migration, each of them including 50 steps. The cell starts at the same position in all the simulations, and its final position is recorded. The angle histograms of the effective trajectories projected on the coordinate planes as well as the 3D point cloud of the final positions are shown in Figure C.1. Their homogeneous distribution demonstrates that the computed trajectories are indeed random. The same procedure is used to check the migration behavior when the cell is forced to migrate in a certain direction (manually specifying the direction of maximum stress). Specifically, the cell is forced to migrate in the x-direction, although the random direction remains activated. As previously, the angle histograms of the effective trajectories projected on the coordinate planes and the 3D point cloud are shown in Figure C.2. The directional pattern in the forced direction confirms the proper behavior of the probability functions and permits the adjustment of parameters, preparing the model for further testings. C 234 Multiscale computational modeling of single cell migration in 3D Figure C.1: Angle histograms and final cell positions in purely random migration. Angle histograms of random migration simulations projected on the coordinate planes. The random distribution confirms the stochastic nature of the cell movement. The 3D distribution (bottom right panel) shows the spherical arrangement of cell final positions (red to yellow shaded points) with respect to the initial location (black circle). Figure C.2: Angle histograms and final positions in forced plus random migration. Angle histograms of forced plus random migration simulations projected on the coordinate planes. The directional distribution confirms the proper behavior of the model, with most of the trajectories comprissed in a small range of angles with respect the forced direction (x-direction). The 3D distribution (bottom right panel) shows the accumulation of cell final positions (red to yellow shaded points) at the +X location with respect to the initial position (black circle). Appendix C 235 C.2 Mechanical testing Ideally, a cell embedded in a homogeneous ECM would feel very similar conditions along its surface. In the model, however, geometry and morphology due to the voxelbased approach affect importantly the mechanical conditions and therefore the mechanosensing process. To validate the mechanical factors used in the probability functions, it becomes necessary to perform a simpler analysis where the mechanical effects are completely isolated. Here, geometrical and morphological effects are neglected by considering a constant stress along the cell surface regardless its position and shape. Furthermore, this assumption allows skipping the FE-based mechanical calculation permitting a faster analysis and therefore the computation of many repetitions. Three different cases are considered: (i) forced movement in a specific direction, (ii) forced movement plus a random component, (iii) purely random migration. For each of these cases, three values of cell stress are taken into account: (j) high stress,1.5 kPa (corresponding with stiff ECMs, 200E kPa), (jj) intermediate stress, 0.6 kPa (corresponding with ECMs of 3E kPa), (jjj) low stress, 0.1 kPa (corresponding with compliant ECMs 0.4E kPa). Each simulation comprises 50 steps of 5 minutes and it is repeated 10 times to avoid errors due to random effects. As in Chapter 5, the model parameters are adjusted to obtain cell speeds in the experimental range observed in fibroblasts (Lo et al., 2000, Friedl and Brocker, 2000, Peyton and Putnam, 2005, Peyton et al., 2011, Hakkinen et al., 2011). Results show that in forced migration (stablishing the direction of maximum stress in the desired direction) both the effective and mean speeds increase with cell stress (equivalent to increasing ECM stiffness)(Figure C.3A). As might be expected with random effects deactivated, the effective speed is similar to the mean speed since the cell migrates in a straight fashion parallel to the forced direction. 236 Multiscale computational modeling of single cell migration in 3D Figure C.3: Cell speeds and migration path lengths. A) Cell speed increases with stress in the case of forced migration. Effective and mean speeds present similar values. B) A random direction is added to the forced one. As a result the effective speed is lower than the mean speed, although they still rise with increasing stress. C) Purely random migration. The effective speed decreases due to unidirectional movements independently of cell stress. D) Migration paths for low, E) intermediate and F) high stress. The cell migrate through longer distances when forced and random directions act together, however, the effective advanced distance from the initial point is similar to the case of forced migration for all levels of stress. This trend is also observed in Figure C.3 with travelled path lengths increasing with stress. When a random direction is added, the mean speed is increased, whereas the effective one is maintained (Figure C.3B). In the case of purely random migration, the mean speed is much higher than the effective one and its value is logically independent of cell stress (Figure C.3C). Overall, the migration path lengths present similar patterns for all levels of stress (although travelled distances increase with stress) and each subcase (Figure C.3D-F). Interestingly, the total distance in random migration is higher than in the forced case and for any cell stress. This is probably due to the the fact that forced migration makes cells elongate in the chosen migration direction, compared with the more spherical cells produced by random migration. Hence, there are few surface cell elements at the front and back parts where new voxels will likely appear/disappear. This is equivalent to a decreased probability of advance, and consequently the computed paths Appendix C 237 are shorter. Supporting this idea, the cell shape factor (or aspect ratio) is notably higher in forced migration for all levels of stress as shown in Figure C.4 (top panels). Interestingly, for high levels of stress, the shape factor becomes noisy and presents a lower mean value than in cases of lower stress. Figure C.4: Cell shape factor and spread area for different stress levels. Cell shape factor is higher in forced migration for all levels of cell stress, producing elongated cells in the chosen direction (top panels) and leading to smaller spreading areas (bottom panels) compared to other cases. This happens because the probability functions saturate at high stresses and hence the voxel appearing/disappearing probability is still high even in different directions to the forced one. Consequently, the cell becomes less elongated. In random migration cases, cells present very low aspect ratio (close to 1), which implies rounded shapes. Furthermore, this is accompanied by slightly higher adhesion area (peripheral voxel faces in contact with the ECM) in comparison with forced migration (Figure C.4 bottom panels). 238 Multiscale computational modeling of single cell migration in 3D While the values of the cell shape factor are in agreement with those found in literature (what they call cell axial ratio), the computed spread area is ~10 fold higher than experimental measurements (Hakkinen et al., 2011). As explained in Chapter 5, this area is calculated as the sum of all the cell voxel faces (each of 9 μm2) in contact with the ECM, and its value is limited by the maximum number of cell voxels (specifically it is allowed an increase of 10% of the initial volume). Hence, the difference with respect to experimental values relies basically on the overestimation produced by a voxelized structure compared to a smoother surface. For instance, a sphere of radius 15 μm has an area of ~2828 μm2, whereas when that same sphere is voxelized, its area increases to 4374 μm2. Furthermore, the discrepancy becomes higher when the cell elongates and presents small protrusions that increase considerably the computed area. Nevertheless, the general trend is similar to the quick increase and saturation of adhesion area found in cells on micropilar substrates (Trichet et al., 2012). C.3 Fluid-chemical factors testing Similarly to the previous analysis, the migration behavior is tested under isolated conditions of fluid and chemical inputs. As described in Chapter 5, the flow direction at each point as well as the chemical species concentration field are taken into account, extracting their values from a FE simulation of a whole microdevice. The validation is performed distinguishing 5 different cases: 1) Chemical gradient in x-direction. 2) Chemical gradient and fluid flow in x-direction. 3) Low chemical gradient and fluid flow in x-direction. 4) Chemical gradient and fluid flow in x-direction with cell receptors blocked 5) Low chemical gradient and fluid flow in x-direction with cell receptors blocked. As before, 10 simulations (50 steps) of each case are computed. Cases 4 and 5 are equivalent to 2 and 3 except that the probabilities regarding fluid flow are reversed (receptors blocked). In other words, cells prefer to migrate upstream. This behavior was observed experimentally (Shields et al., 2007, Polacheck et al., 2011) and the explanation is that tumor cells utilize interstitial flow to create and amplify autologous transcellular chemokine gradients. When specific receptors are blocked (e.g. CCR7), cells are unable to detect such autologous gradients and tend to migrate upstream probably due to their preference for higher pressures. Appendix C 245 Figure C.14: Pressure gradient along the ECM and cell surface. The pressure changes linearly along the porous ECM, slightly affected by the cell body inclusion. Detail of pressure along cell surface is magnified. Additional simulations aer performed to check how the flow velocity affects the concentration of autologous chemokine gradients around the cell. The same geometry and boundary conditions of the previous analysis are used but adding a normalized concentration at the cell surface. The steady state is computed for the experimental value of flow speed (Figure C.15A) and multiplying such speed 10 (Figure C.15B) and 100 fold (Figure C.15C). Although these last values may be not physiologically relevant, the computation is still useful to check the consistency of the model. As might be expected, higher flow velocities drags the chemokine factor downstream. Cell receptors would detect higher concentrations in that direction and therefore its body would align and migrate following the fluid flow. Figure C.15: Autologous chemokine concentration for different flow speeds. The experimental value of flow speed A) is multiplied 10 fold B) and 100 fold C).. 246 Multiscale computational modeling of single cell migration in 3D BIBLIOGRAPHY AHMADI, A., LIVERPOOL, T. B. & MARCHETTI, M. C. 2005. Nematic and polar order in active filament solutions. Phys Rev E Stat Nonlin Soft Matter Phys, 72, 060901. ALBERTS, B., JOHNSON, A., LEWIS, J., RAFF, M., ROBERTS, K. & WALTER, P. 2008. Molecular Biology of the Cell, Garland science. ALBERTS, J. B. & ODELL, G. M. 2004. In silico reconstitution of Listeria propulsion exhibits nano-saltation. PLoS Biol, 2, e412. ALEXANDROVA, A. Y., ARNOLD, K., SCHAUB, S., VASILIEV, J. M., MEISTER, J. J., BERSHADSKY, A. D. & VERKHOVSKY, A. B. 2008. Comparative dynamics of retrograde actin flow and focal adhesions: formation of nascent adhesions triggers transition from fast to slow flow. PLoS One, 3, e3234. ANDERSON, T. W., VAUGHAN, A. N. & CRAMER, L. P. 2008. Retrograde flow and myosin II activity within the leading cell edge deliver F-actin to the lamella to seed the formation of graded polarity actomyosin II filament bundles in migrating fibroblasts. Mol Biol Cell, 19, 5006-18. ARCIERO, J. C., MI, Q., BRANCA, M. F., HACKAM, D. J. & SWIGON, D. 2011. Continuum model of collective cell migration in wound healing and colony expansion. Biophys J, 100, 535-43. ASTROM, J. A., KUMAR, P. B. S. & KARTTUNEN, M. 2009. Aster formation and rupture transition in semi-flexible fiber networks with mobile cross-linkers. Soft Matter, 5, 2869-2874. ASTROM, J. A., KUMAR, P. B. S., VATTULAINEN, I. & KARTTUNEN, M. 2008. Strain hardening, avalanches, and strain softening in dense cross-linked actin networks. Phys Rev, 77, 051913. ATILGAN, E., WIRTZ, D. & SUN, S. X. 2005. Morphology of the lamellipodium and organization of actin filaments at the leading edge of crawling cells. Biophys J, 89, 3589-602. ATILGAN, E., WIRTZ, D. & SUN, S. X. 2006. Mechanics and dynamics of actin-driven thin membrane protrusions. Biophys J, 90, 65-76. ATIYEH, B. S., GUNN, S. W. & HAYEK, S. N. 2005. State of the art in burn treatment. World J Surg, 29, 131-48. BAKER, E. L., BONNECAZE, R. T. & ZAMAN, M. H. 2009. Extracellular Matrix Stiffness and Architecture Govern Intracellular Rheology in Cancer. Biophys J, 97, 1013-1021. BELOUSSOV, L. V., LOUCHINSKAIA, N. N. & STEIN, A. A. 2000. Tensiondependent collective cell movements in the early gastrula ectoderm of Xenopus laevis embryos. Dev Genes Evol, 210, 92-104. BERSHADSKY, A. D., BALABAN, N. Q. & GEIGER, B. 2003. Adhesion-dependent cell mechanosensitivity. Annu Rev Cell Dev Biol, 19, 677-695. BESSER, A. & SCHWARZ, U. S. 2010. Hysteresis in the cell response to timedependent substrate stiffness. Biophys J, 99, L10-2. BISCHOFS, I. B. & SCHWARZ, U. S. 2003. Cell organization in soft media due to active mechanosensing. Proc Natl Acad Sci U S A, 100, 9274-9279. BLAKNEY, A. K., SWARTZLANDER, M. D. & BRYANT, S. J. 2012. The effects of substrate stiffness on the in vitro activation of macrophages and in vivo host 248 Multiscale computational modeling of single cell migration in 3D response to poly(ethylene glycol)-based hydrogels. J Biomed Mater Res A, 100, 1375-86. BORAU, C., KAMM, R. D. & GARCIA-AZNAR, J. M. 2011. Mechano-sensing and cell migration: a 3D model approach. Phys Biol, 8, 066008. BORAU, C., KIM, T., BIDONE, T., GARCIA-AZNAR, J. M. & KAMM, R. D. 2012. Dynamic mechanisms of cell rigidity sensing: insights from a computational model of actomyosin networks. PLoS One, 7, e49174. BORDELEAU, F., MYRAND LAPIERRE, M. E., SHENG, Y. & MARCEAU, N. 2012. Keratin 8/18 regulation of cell stiffness-extracellular matrix interplay through modulation of Rho-mediated actin cytoskeleton dynamics. PLoS One, 7, e38780. BOTTINO, D., MOGILNER, A., ROBERTS, T., STEWART, M. & OSTER, G. 2002. How nematode sperm crawl. J Cell Sci, 115, 367-384. BOYDEN, S. 1962. The chemotactic effect of mixtures of antibody and antigen on polymorphonuclear leucocytes. J Exp Med, 115, 453-66. BURNETTE, D. T. B. D. T., MANLEY, S., SENGUPTA, P., SOUGRAT, R., DAVIDSON, M. W., KACHAR, B. & LIPPINCOTT-SCHWARTZ, J. 2011. A role for actin arcs in the leading-edge advance of migrating cells. Nat Cell Biol, 13, 371-U88. BUXBOIM, A., IVANOVSKA, I. L. & DISCHER, D. E. 2010. Matrix elasticity, cytoskeletal forces and physics of the nucleus: how deeply do cells 'feel' outside and in? J Cell Sci, 123, 297-308. CARLSSON, A. E. 2001. Growth of branched actin networks against obstacles. Biophys J, 81, 1907-23. CARLSSON, A. E. 2010. Dendritic actin filament nucleation causes traveling waves and patches. Phys Rev Lett, 104, 228102. CARLSSON, A. E. & SEPT, D. 2008. Mathematical modeling of cell migration. Methods Cell Biol, 84, 911-37. CIRIT, M., KRAJCOVIC, M., CHOI, C. K., WELF, E. S., HORWITZ, A. F. & HAUGH, J. M. 2010. Stochastic Model of Integrin-Mediated Signaling and Adhesion Dynamics at the Leading Edges of Migrating Cells. PLoS Comput Biol, 6. CRAIG, E. M., VAN GOOR, D., FORSCHER, P. & MOGILNER, A. 2012. Membrane tension, myosin force, and actin turnover maintain actin treadmill in the nerve growth cone. Biophys J, 102, 1503-13. CROW, A., WEBSTER, K. D., HOHLFELD, E., NG, W. P., GEISSLER, P. & FLETCHER, D. A. 2012. Contractile equilibration of single cells to step changes in extracellular stiffness. Biophys J, 102, 443-51. CUKIERMAN, E., PANKOV, R., STEVENS, D. R. & YAMADA, K. M. 2001. Taking cell-matrix adhesions to the third dimension. Science, 294, 1708-1712. CUKIERMAN, E., PANKOV, R. & YAMADA, K. M. 2002. Cell interactions with threedimensional matrices. Curr Opin Cell Biol, 14, 633-639. CURTZE, S., DEMBO, M., MIRON, M. & JONES, D. B. 2004. Dynamic changes in traction forces with DC electric field in osteoblast-like cells. J Cell Sci, 117, 27219. CHAN, C. E. & ODDE, D. J. 2008. Traction dynamics of filopodia on compliant substrates. Science, 322, 1687-91. CHARRAS, G. & PALUCH, E. 2008. Blebs lead the way: how to migrate without lamellipodia. Nat Rev Mol Cell Biol, 9, 730-6. CHEN, N., GLAZIER, J. A., IZAGUIRRE, J. A. & ALBER, M. S. 2007. A parallel implementation of the Cellular Potts Model for simulation of cell-based morphogenesis. Comput Phys Commun, 176, 670-681. Bibliography 249 CHOPARD, B., OUARED, R., DEUTSCH, A., HATZIKIROU, H. & WOLFGLADROW, D. 2010. Lattice-Gas Cellular Automaton Models for Biology: From Fluids to Cells. Acta Biotheor, 58, 329-340. CHUNG, B. G., LEE, K. H., KHADEMHOSSEINI, A. & LEE, S. H. 2012. Microfluidic fabrication of microengineered hydrogels and their application in tissue engineering. Lab Chip, 12, 45-59. CHUNG, S., SUDO, R., VICKERMAN, V., ZERVANTONAKIS, I. K. & KAMM, R. D. 2010. Microfluidic Platforms for Studies of Angiogenesis, Cell Migration, and Cell-Cell Interactions. Ann Biomed Eng, 38, 1164-1177. DAMANIA, D., SUBRAMANIAN, H., TIWARI, A. K., STYPULA, Y., KUNTE, D., PRADHAN, P., ROY, H. K. & BACKMAN, V. 2010. Role of Cytoskeleton in Controlling the Disorder Strength of Cellular Nanoscale Architecture RID B6689-2009. Biophys J, 99, 989-996. DAWES, A. T., BARD ERMENTROUT, G., CYTRYNBAUM, E. N. & EDELSTEINKESHET, L. 2006. Actin filament branching and protrusion velocity in a simple 1D model of a motile cell. J Theor Biol, 242, 265-79. DE, R., ZEMEL, A. & SAFRAN, S. A. 2007. Dynamics of cell orientation. Nat Phys, 3, 655-659. DE, R., ZEMEL, A. & SAFRAN, S. A. 2008. Do cells sense stress or strain? Measurement of cellular orientation can provide a clue. Biophys J, 94, L29-L31. DEMBO, M. & WANG, Y. L. 1999. Stresses at the cell-to-substrate interface during locomotion of fibroblasts. Biophys J, 76, 2307-2316. DESHPANDE, V. S., MCMEEKING, R. M. & EVANS, A. G. 2006. A bio-chemomechanical model for cell contractility. Proc Natl Acad Sci U S A, 103, 14015-20. DICKINSON, R. B. & PURICH, D. L. 2002. Clamped-filament elongation model for actin-based motors. Biophys J, 82, 605-17. DISCHER, D. E., JANMEY, P. & WANG, Y. L. 2005. Tissue cells feel and respond to the stiffness of their substrate. Science, 310, 1139-1143. DOKUKINA, I. V. & GRACHEVA, M. E. 2010. A Model of Fibroblast Motility on Substrates with Different Rigidities. Biophys J, 98, 2794-2803. DOU, Y., ARLOCK, P. & ARNER, A. 2007. Blebbistatin specifically inhibits actinmyosin interaction in mouse cardiac muscle. AM J Physiol-Cell Ph, 293, C1148C1153. DOUBROVINSKI, K. & KRUSE, K. 2008. Cytoskeletal waves in the absence of molecular motors. Epl, 83. EHRBAR, M., SALA, A., LIENEMANN, P., RANGA, A., MOSIEWICZ, K., BITTERMANN, A., RIZZI, S. C., WEBER, F. E. & LUTOLF, M. P. 2011. Elucidating the Role of Matrix Stiffness in 3D Cell Migration and Remodeling. Biophys J, 100, 284-293. ELLIOTT, C. M., STINNER, B. & VENKATARAMAN, C. 2012. Modelling cell motility and chemotaxis with evolving surface finite elements. J R Soc Interface, 9, 3027-44. ENGLER, A., BACAKOVA, L., NEWMAN, C., HATEGAN, A., GRIFFIN, M. & DISCHER, D. 2004a. Substrate compliance versus ligand density in cell on gel responses. Biophys J, 86, 617-628. ENGLER, A. J., GRIFFIN, M. A., SEN, S., BONNETNANN, C. G., SWEENEY, H. L. & DISCHER, D. E. 2004b. Myotubes differentiate optimally on substrates with tissue-like stiffness: pathological implications for soft or stiff microenvironments. J Cell Biol, 166, 877-887. 250 Multiscale computational modeling of single cell migration in 3D ENGLER, A. J., SEN, S., SWEENEY, H. L. & DISCHER, D. E. 2006. Matrix elasticity directs stem cell lineage specification. Cell, 126, 677-689. EVEN-RAM, S. & YAMADA, K. M. 2005. Cell migration in 3D matrix. Curr Opin Cell Biol, 17, 524-532. FABER, M., ENCULESCU, M. & FALCKE, M. 2010. Filament capping and nucleation in actin-based motility. Eur Phys J-Spec Top, 191, 147-158. FERRER, J. M., LEE, H., CHEN, J., PELZ, B., NAKAMURA, F., KAMM, R. D. & LANG, M. J. 2008. Measuring molecular rupture forces between single actin filaments and actin-binding proteins. Proc Natl Acad Sci U S A, 105, 9221-9226. FLAHERTY, B., MCGARRY, J. P. & MCHUGH, P. E. 2007. Mathematical models of cell motility. Cell Biochem Biophys, 49, 14-28. FOUCARD, L. & VERNEREY, F. J. 2012. A thermodynamical model for stress-fiber organization in contractile cells. Appl Phys Lett, 100, 13702-137024. FOUCHARD, J., MITROSSILIS, D. & ASNACIOS, A. 2011. Acto-myosin based response to stiffness and rigidity sensing. Cell Adh Migr, 5, 16-9. FRALEY, S. I., FENG, Y., KRISHNAMURTHY, R., KIM, D.-H., CELEDON, A., LONGMORE, G. D. & WIRTZ, D. 2010. A distinctive role for focal adhesion proteins in three-dimensional cell motility. Nat Cell Biol, 12, 598-604. FRANCK, C., MASKARINEC, S. A., TIRRELL, D. A. & RAVICHANDRAN, G. 2011. Three-dimensional traction force microscopy: a new tool for quantifying cellmatrix interactions. PLoS One, 6, e17833. FREYMAN, T. M., YANNAS, I. V., YOKOO, R. & GIBSON, L. J. 2002. Fibroblast contractile force is independent of the stiffness which resists the contraction. Exp Cell Res, 272, 153-162. FRIEDL, P. & BROCKER, E. B. 2000. The biology of cell locomotion within threedimensional extracellular matrix. Cell Mol Life Sci, 57, 41-64. FRIEDL, P. & GILMOUR, D. 2009. Collective cell migration in morphogenesis, regeneration and cancer. Nat Rev Mol Cell Biol, 10, 445-57. FRIEDL, P., HEGERFELDT, Y. & TUSCH, M. 2004. Collective cell migration in morphogenesis and cancer. Int J Dev Biol, 48, 441-9. FRIEDL, P., SAHAI, E., WEISS, S. & YAMADA, K. M. 2012. New dimensions in cell migration. Nat Rev Mol Cell Biol, 13, 743-7. GANZ, A., LAMBERT, M., SAEZ, A., SILBERZAN, P., BUGUIN, A., MEGE, R. M. & LADOUX, B. 2006. Traction forces exerted through N-cadherin contacts. Biol Cell, 98, 721-30. GARDEL, M. L., NAKAMURA, F., HARTWIG, J. H., CROCKER, J. C., STOSSEL, T. P. & WEITZ, D. A. 2006. Prestressed F-actin networks cross-linked by hinged filamins replicate mechanical properties of cells. Proc Natl Acad Sci U S A, 103, 1762-7. GERBAL, F., CHAIKIN, P., RABIN, Y. & PROST, J. 2000. An elastic analysis of Listeria monocytogenes propulsion. Biophys J, 79, 2259-75. GHASSEMI, S., MEACCI, G., LIU, S., GONDARENKO, A. A., MATHUR, A., ROCACUSACHS, P., SHEETZ, M. P. & HONE, J. 2012. Cells test substrate rigidity by local contractions on submicrometer pillars. Proc Natl Acad Sci U S A, 109, 532833. GHIBAUDO, M., SAEZ, A., TRICHET, L., XAYAPHOUMMINE, A., BROWAEYS, J., SILBERZAN, P., BUGUIN, A. & LADOUX, B. 2008. Traction forces and rigidity sensing regulate cell functions. Soft Matter, 4, 1836-1843. Bibliography 251 GIANNONE, G., DUBIN-THALER, B. J., DOBEREINER, H. G., KIEFFER, N., BRESNICK, A. R. & SHEETZ, M. P. 2004. Periodic lamellipodial contractions correlate with rearward actin waves. Cell, 116, 431-43. GIANNONE, G., DUBIN-THALER, B. J., ROSSIER, O., CAI, Y., CHAGA, O., JIANG, G., BEAVER, W., DOBEREINER, H. G., FREUND, Y., BORISY, G. & SHEETZ, M. P. 2007. Lamellipodial actin mechanically links myosin activity with adhesion-site formation. Cell, 128, 561-75. GOV, N. S. & GOPINATHAN, A. 2006. Dynamics of membranes driven by actin polymerization. Biophys J, 90, 454-69. GRACHEVA, M. E. & OTHMER, H. G. 2004. A continuum model of motility in ameboid cells. Bull Math Biol, 66, 167-93. GROH, A. & WAGNER, M. 2011. Biased three-dimensional cell migration and collagen matrix modification. Math Biosci, 231, 105-119. GUO, B. & GUILFORD, W. H. 2006. Mechanics of actomyosin bonds in different nucleotide states are tuned to muscle contraction. Proc Natl Acad Sci U S A, 103, 9844-9849. HÄCKER, A. 2011. A mathematical model for mesenchymal and chemosensitive cell dynamics. J Math Biol, 1-41. HAKKINEN, K. M., HARUNAGA, J. S., DOYLE, A. D. & YAMADA, K. M. 2011. Direct Comparisons of the Morphology, Migration, Cell Adhesions, and Actin Cytoskeleton of Fibroblasts in Four Different Three-Dimensional Extracellular Matrices. Tiss Eng Pt A, 17, 713-724. HARJANTO, D. & ZAMAN, M. H. 2010. Computational study of proteolysis-driven single cell migration in a three-dimensional matrix. Ann Biomed Eng, 38, 181525. HARLEY, B. A. C., KIM, H. D., ZAMAN, M. H., YANNAS, I. V., LAUFFENBURGER, D. A. & GIBSON, L. J. 2008. Microarchitecture of threedimensional scaffolds influences cell migration behavior via junction interactions. Biophys J, 95, 4013-4024. HARUNAGA, J. S. & YAMADA, K. M. 2011. Cell-matrix adhesions in 3D. Matrix Biol, In Press, Corrected Proof. HAYAKAWA, K., SATO, N. & OBINATA, T. 2001. Dynamic reorientation of cultured cells and stress fibers under mechanical stress from periodic stretching. Exp Cell Res, 268, 104-14. HENG, Y. W. & KOH, C. G. 2010. Actin cytoskeleton dynamics and the cell division cycle RID A-2215-2011. Int J Biochem, 42, 1622-1633. HILL, A. V. 1938. The heat of shortening and the dynamic constants of muscle. P Roy Soc Lond B Bio, 126, 136-195. HOFMANN, M., GUSCHEL, M., BERND, A., BEREITER-HAHN, J., KAUFMANN, R., TANDI, C., WIIG, H. & KIPPENBERGER, S. 2006. Lowering of tumor interstitial fluid pressure reduces tumor cell proliferation in a xenograft tumor model. Neoplasia, 8, 89-95. HU, K., JI, L., APPLEGATE, K. T., DANUSER, G. & WATERMAN-STORER, C. M. 2007. Differential transmission of actin motion within focal adhesions. Science, 315, 111-5. HUXLEY, A. F. 1957. Muscle structure and theories of contraction. Prog Biophys Biophys Chem, 7, 255-318. ILINA, O. & FRIEDL, P. 2009. Mechanisms of collective cell migration at a glance. J Cell Sci, 122, 3203-3208. 252 Multiscale computational modeling of single cell migration in 3D INGBER, D. E. 2010. From Cellular Mechanotransduction to Biologically Inspired Engineering. Ann Biomed Eng, 38, 1148-1161. JANMEY, P. A., WINER, J. P., MURRAY, M. E. & WEN, Q. 2009. The Hard Life of Soft Cells. Cell Motil Cytoskeleton, 66, 597-605. JOYCE, J. A. & POLLARD, J. W. 2009. Microenvironmental regulation of metastasis. Nat Rev Cancer, 9, 239-52. KASZA, K. E., ROWAT, A. C., LIU, J. Y., ANGELINI, T. E., BRANGWYNNE, C. P., KOENDERINK, G. H. & WEITZ, D. A. 2007. The cell as a material. Curr Opin Cell Biol, 19, 101-107. KATSUMI, A., NAOE, T., MATSUSHITA, T., KAIBUCHI, K. & SCHWARTZ, M. A. 2005. Integrin activation and matrix binding mediate cellular responses to mechanical stretch. J Biol Chem, 280, 16546-9. KAVERINA, I., KRYLYSHKINA, O., BENINGO, K., ANDERSON, K., WANG, Y. L. & SMALL, J. V. 2002. Tensile stress stimulates microtubule outgrowth in living cells. J Cell Sci, 115, 2283-2291. KIM, T., HWANG, W. & KAMM, R. D. 2009a. Computational Analysis of a Crosslinked Actin-like Network. Exp Mech, 49, 91-104. KIM, T., HWANG, W. & KAMM, ROGER D. 2011. Dynamic Role of Cross-Linking Proteins in Actin Rheology. Biophys J, 101, 1597-1603. KIM, T., HWANG, W., LEE, H. & KAMM, R. D. 2009b. Computational Analysis of Viscoelastic Properties of Crosslinked Actin Networks. PLoS Comput Biol, 5, e1000439. KOBAYASHI, T. & SOKABE, M. 2010. Sensing substrate rigidity by mechanosensitive ion channels with stress fibers and focal adhesions. Curr Opin Cell Biol, 22, 66976. KOESTLER, S. A., AUINGER, S., VINZENZ, M., ROTTNER, K. & SMALL, J. V. 2008. Differentially oriented populations of actin filaments generated in lamellipodia collaborate in pushing and pausing at the cell front. Nat Cell Biol, 10, 306-13. KOLLMANNSBERGER, P. & FABRY, B. 2011. Linear and Nonlinear Rheology of Living Cells. Ann Rev Mater Res, 41, 75-97. KOVACS, M., TOTH, J., HETENYI, C., MALNASI-CSIZMADIA, A. & SELLERS, J. R. 2004. Mechanism of blebbistatin inhibition of myosin II. J Biol Chem, 279, 35557-35563. KRANING-RUSH, C. M., CALIFANO, J. P. & REINHART-KING, C. A. 2012. Cellular traction stresses increase with increasing metastatic potential. PLoS One, 7, e32572. KRUSE, K., JOANNY, J. F., JULICHER, F., PROST, J. & SEKIMOTO, K. 2005. Generic theory of active polar gels: a paradigm for cytoskeletal dynamics. Eur Phys J E Soft Matter, 16, 5-16. KUNG, C. 2005. A possible unifying principle for mechanosensation. Nature, 436, 64754. KUUSELA, E. & ALT, W. 2009. Continuum model of cell adhesion and migration. J Math Biol, 58, 135-61. LADOUX, B., ANON, E., LAMBERT, M., RABODZEY, A., HERSEN, P., BUGUIN, A., SILBERZAN, P. & MEGE, R.-M. 2010. Strength dependence of cadherinmediated adhesions. Biophys J, 98, 534-42. LAI, F. P., SZCZODRAK, M., BLOCK, J., FAIX, J., BREITSPRECHER, D., MANNHERZ, H. G., STRADAL, T. E., DUNN, G. A., SMALL, J. V. & Bibliography 253 ROTTNER, K. 2008. Arp2/3 complex interactions and actin network turnover in lamellipodia. EMBO J, 27, 982-92. LAMMERMANN, T. & SIXT, M. 2009. Mechanical modes of 'amoeboid' cell migration. Curr Opin Cell Biol, 21, 636-44. LAUFFENBURGER, D. A. & HORWITZ, A. F. 1996. Cell migration: A physically integrated molecular process. Cell, 84, 359-369. LEGANT, W. R., CHOI, C. K., MILLER, J. S., SHAO, L., GAO, L., BETZIG, E. & CHEN, C. S. 2013. Multidimensional traction force microscopy reveals out-ofplane rotational moments about focal adhesions. Proc Natl Acad Sci U S A, 110, 881-6. LI, S., HUANG, N. F. & HSU, S. 2005. Mechanotransduction in endothelial cell migration. J Cell Biochem, 96, 1110-26. LIM, C. T., ZHOU, E. H. & QUEK, S. T. 2006. Mechanical models for living cells - A review. J Biomech, 39, 195-216. LINDER, S. 2007. The matrix corroded: podosomes and invadopodia in extracellular matrix degradation. Trends Cell Biol, 17, 107-17. LO, C. M., WANG, H. B., DEMBO, M. & WANG, Y. L. 2000. Cell movement is guided by the rigidity of the substrate. Biophys J, 79, 144-152. MANOUSSAKI, D. 2003. A mechanochemical model of angiogenesis and vasculogenesis. Esaim-Math Model Num, 37, 581-599. MARCQ, P., YOSHINAGA, N. & PROST, J. 2011. Rigidity sensing explained by active matter theory. Biophys J, 101, L33-5. MARTINAC, B. 2004. Mechanosensitive ion channels: molecules of mechanotransduction. J Cell Sci, 117, 2449-60. MASKARINEC, S. A., FRANCK, C., TIRRELL, D. A. & RAVICHANDRAN, G. 2009. Quantifying cellular traction forces in three dimensions. Proc Natl Acad Sci U S A, 106, 22108-22113. MCGARRY, J. P., FU, J., YANG, M. T., CHEN, C. S., MCMEEKING, R. M., EVANS, A. G. & DESHPANDE, V. S. 2009. Simulation of the contractile response of cells on an array of micro-posts. Philos T R Soc A, 367, 3477-3497. MCMAHON, T. 1985. Muscles, Reflexes, and Locomotion. Percept Motor Skill, 60, 679679. MEDEIROS, N. A., BURNETTE, D. T. & FORSCHER, P. 2006. Myosin II functions in actin-bundle turnover in neuronal growth cones. Nat Cell Biol, 8, 215-26. MERKS, R. M. H. & KOOLWIJK, P. 2009. Modeling Morphogenesis in silico and in vitro: Towards Quantitative, Predictive, Cell-based Modeling. Math Model Nat Pheno, 4, 149-171. MITROSSILIS, D., FOUCHARD, J., GUIROY, A., DESPRAT, N., RODRIGUEZ, N., FABRY, B. & ASNACIOS, A. 2009. Single-cell response to stiffness exhibits muscle-like behavior. Proc Natl Acad Sci U S A, 106, 18243-18248. MITROSSILIS, D., FOUCHARD, J., PEREIRA, D., POSTIC, F., RICHERT, A., SAINT-JEAN, M. & ASNACIOS, A. 2010. Real-time single-cell response to stiffness. Proc Natl Acad Sci U S A, 107, 16518-16523. MOGILNER, A. 2009. Mathematics of cell motility: have we got its number? J Math Biol, 58, 105-34. MOGILNER, A. & EDELSTEIN-KESHET, L. 2002. Regulation of actin dynamics in rapidly moving cells: a quantitative analysis. Biophys J, 83, 1237-58. MOGILNER, A. & OSTER, G. 2003. Polymer motors: pushing out the front and pulling up the back. Curr Biol, 13, R721-33. 254 Multiscale computational modeling of single cell migration in 3D MOREO, P., GARCIA-AZNAR, J. M. & DOBLARE, M. 2008. Modeling mechanosensing and its effect on the migration and proliferation of adherent cells. Acta Biomater, 4, 613-621. MUNEVAR, S., WANG, Y. L. & DEMBO, M. 2004. Regulation of mechanical interactions between fibroblasts and the substratum by stretch-activated Ca2+ entry. J Cell Sci, 117, 85-92. NOVAK, I. L., SLEPCHENKO, B. M., MOGILNER, A. & LOEW, L. M. 2004. Cooperativity between cell contractility and adhesion. Phys Rev Lett, 93. OUAKNIN, G. Y. & BAR-YOSEPH, P. Z. 2009. Stochastic collective movement of cells and fingering morphology: no maverick cells. Biophys J, 97, 1811-21. PACHECO, P. S. 1997. Parallel programming with MPI, Morgan Kaufmann Publishers, Inc. San Francisco California. PALECEK, S. P., LOFTUS, J. C., GINSBERG, M. H., LAUFFENBURGER, D. A. & HORWITZ, A. F. 1997. Integrin-ligand binding properties govern cell migration speed through cell-substratum adhesiveness. Nature, 385, 537-40. PALSSON, E. 2001. A three-dimensional model of cell movement in multicellular systems. Future Gener Comp Sy, 17, 835-852. PANKOV, R., ENDO, Y., EVEN-RAM, S., ARAKI, M., CLARK, K., CUKIERMAN, E., MATSUMOTO, K. & YAMADA, K. M. 2005. A Rac switch regulates random versus directionally persistent cell migration. J Cell Biol, 170, 793-802. PARKHURST, M. R. & SALTZMAN, W. M. 1992. Quantification of human neutrophil motility in three-dimensional collagen gels. Effect of collagen concentration. Biophys J, 61, 306-15. PELHAM, R. J. & WANG, Y. L. 1997. Cell locomotion and focal adhesions are regulated by substrate flexibility. Proc Natl Acad Sci U S A, 94, 13661-13665. PESKIN, C. S., ODELL, G. M. & OSTER, G. F. 1993. Cellular motions and thermal fluctuations: the Brownian ratchet. Biophys J, 65, 316-24. PETRIE, R. J., DOYLE, A. D. & YAMADA, K. M. 2009. Random versus directionally persistent cell migration. Nat Rev Mol Cell Biol, 10, 538-549. PETRIE, R. J., GAVARA, N., CHADWICK, R. S. & YAMADA, K. M. 2012. Nonpolarized signaling reveals two distinct modes of 3D cell migration. J Cell Biol, 197, 439-55. PETTET, G. J., PLEASE, C. P., TINDALL, M. J. & MCELWAIN, D. L. 2001. The migration of cells in multicell tumor spheroids. Bull Math Biol, 63, 231-57. PEYTON, S. R., KALCIOGLU, Z. I., COHEN, J. C., RUNKLE, A. P., VAN VLIET, K. J., LAUFFENBURGER, D. A. & GRIFFITH, L. G. 2011. Marrow-Derived stem cell motility in 3D synthetic scaffold is governed by geometry along with adhesivity and stiffness. Biotechnol Bioeng, 108, 1181-1193. PEYTON, S. R. & PUTNAM, A. J. 2005. Extracellular matrix rigidity governs smooth muscle cell motility in a biphasic fashion. J Cell Physiol, 204, 198-209. POLACHECK, W. J., CHAREST, J. L. & KAMM, R. D. 2011. Interstitial flow influences direction of tumor cell migration through competing mechanisms. Proc Natl Acad Sci U S A, 108, 11115-11120. POLACHECK, W. J., ZERVANTONAKIS, I. K. & KAMM, R. D. 2012. Tumor cell migration in complex microenvironments. Cell Mol Life Sci. POLLARD, T. D. & BORISY, G. G. 2003. Cellular motility driven by assembly and disassembly of actin filaments. Cell, 112, 453-65. PONTI, A., MACHACEK, M., GUPTON, S. L., WATERMAN-STORER, C. M. & DANUSER, G. 2004. Two distinct actin networks drive the protrusion of migrating cells. Science, 305, 1782-6.