Protein unfolding and refolding as transitions through virtual states

October 2014 EPL, 108 (2014) 28002 www.epljournal.org doi: 10.1209/0295-5075/108/28002 Protein unfolding and refolding as transitions through virtual states L. L. Bonilla1,A.Carpio 2and A. Prados3 1G. Mill´an Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid E-28911, Legan´es, Spain 2Departamento de Matem´atica Aplicada, Universidad Complutense de Madrid - E-28040, Madrid, Spain 3F´ısica Te´orica, Universidad de Sevilla - Apartado de Correos 1065, E-41080, Sevilla, Spain received 24 May 2014; accepted in final form 26 September 2014 published online 13 October 2014 PACS 87.15.Cc – Folding: thermodynamics, statistical mechanics, models, and pathways PACS 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion PACS 87.14.et – Generic models (lattice, HP, etc.) Abstract – Single-molecule atomic force spectroscopy probes elastic properties of titin, ubiquitin and other relevant proteins. We explain bioprotein folding dynamics under both length- and force-clamp by modeling polyprotein modules as particles in a bistable potential, weakly connected by harmonic spring linkers. Multistability of equilibrium extensions provides the characteristic sawtooth force-extension curve. We show that abrupt or stepwise unfolding and refolding under force-clamp conditions involve transitions through virtual states (which are quasi-stationary domain configurations) modified by thermal noise. These predictions agree with experimental observations. Copyright c EPLA, 2014 Introduction. – The study of single molecules may explain the function of many molecular assemblies found in cells [1–4]. Tissue elasticity in living organisms results from the extension and recoil of proteins fastened to rigid structures that move under force. Polyproteins or modular proteins, such as titin, that plays an important role in muscle contraction [3], ubiquitin and other relevant proteins [5–8], comprise a number of repeated single-protein domains joined by short peptide linkers. To reduce the variety of single-protein domains, artificial homopolyproteins comprising a number of identical protein modules (thereby having the same mechanical properties) have been engineered by using molecular biology techniques [5]. A simple version of tissue elasticity appears in most single-molecule experiments, like atomic force microscopy (AFM), in which a biomolecule is chained to rigid platforms whose motion is controlled [8]. As the polyprotein is pulled, one or more modules unfold at a typical force that measures its mechanical stability. It should be stressed that the unraveling of a domain is a stochastic event and may occur in a certain range of forces. These length-controlled experiments deliver a sawtooth forceextension curve (FEC) [5,6,9–12]. Similar curves are obtained by stretching nucleic acids [9,10,13,14] and other biomolecules [6,15]. When the FEC is swept at a finite rate, stochastic jumps between folded and unfolded states may be observed [9,11,14,16], and the unfolding force increases with the extension rate. In a typical force-clamp experiment, the force is first raised, kept at a large value until all domains become unfolded and then abruptly lowered to a smaller value [17,18]. Immediately after the force increment, abrupt or stepwise unfolding of the polyprotein follows [19,20]. On the other hand, after the force is lowered, refolding is similar for single-module proteins [18] and for homopolyproteins [17]; the folding events do not show traces of sequential folding for polyproteins (see fig. S2 of ref. [17] for the only reported exception showing stepwise folding of a twomodule protein). Berkovich et al. [18] interpret the results of their single-module protein experiments using a simple Langevin equation model that includes an effective bistable potential for a range of the applied force. The sawtooth FEC is well understood: force jumps are already present in equilibrium when length is controlled [12,21,22]. However, force-clamp experiments are not: Slight changes of the forces lead to completely different behaviors, suggesting that polyproteins are operating near critical conditions therein. We now put forward and motivate a simple model able to explain some 28002-p1 L. L. Bonilla et al. aspects of polyprotein folding and refolding under either length or force control. It is inspired in mathematically similar spatially discrete models for charge transport in weakly coupled semiconductor superlattices (SLs) [23]. SLs also have a sawtooth current-voltage curve (similar to FEC in polyproteins) under voltage bias (voltage is analogous to extension, current to force). A related model for shape-memory alloys [24] has been recently reworked to analyze the FEC of biomolecules [21,22]. Moreover, behavior resembling stepwise unfolding is observed in overdamped Frenkel-Kontorova (FK) chains (which have bistable on-site potentials and nearest-neighbor harmonic coupling) [25,26] and in chains with a bistable nearest-neighbor snap-spring potential that become the FK model [27]. Model. – The time scale for stress relaxation inside a module is much smaller than the time scale of a typical unfolding/refolding event and, therefore, we can assume instantaneous mechanical equilibrium inside each module at the time scale of unfolding/refolding events. Then each module of extension uis modeled as a particle in a bistable potential whose minima represent folded (enthalpic minimum) and unfolded (entropic minimum) states [18], see fig. 1. The following effective potential is shown to provide a good description of single-module proteins at temperature Tand zero external force [18]: V(u)=U01−e−2b(u−Rc)/Rc2−1 +kBTLc 4P1 1−u Lc−1−u Lc +2u2 L2 c.(1) For ubiquitin the applied force ranges from 10 to 120 pN, P=0.28 nm (persistence length), Lc=30nm,U0= 200 pN ·nm (∼48kBT), Rc=4nm,b=2,T= 300 K. In AFM experiments, the polyprotein is tethered to two platforms and stretched so its geometry is quasi–onedimensional. If forces ±Fare applied to the ends of the modular protein and the j-th module extends from xjto xj+1,withuj=xj+1 −xj, the potential energy due to the force is Fx1−FxN+1 =−FN j=1 uj.As part of the tertiary structure of the polyprotein, modules are weakly interconnected in a structure-dependent way. This weak interaction acts on the unfolding/refolding time scale and tries to bring the extensions of the modules to a common value corresponding to global mechanical equilibrium. This crucial feature to explain sequential unfolding is absent in simpler models that do not assign different elongations to different modules [18]. As a simplification, we assume that neighboring modules (j−1,j)interactvia aspringpotentialk 2(uj−uj−1)2=k 2(xj+1 −2xj+xj−1)2, where k=0.0065[F]/Lcis the spring constant. The modules satisfy overdamped Langevin equations: γ˙uj=−∂ ∂ujG(u,F,T)+√2Dγξ j(t),(2) Fig. 1: (Colour on-line) Force field V(u)vs. uand the three solutions of V(u)=F,u(1)(F)<u (2)(F)<u (3)(F)intheforce range Fm<F <F M.Therein,FM(about 104 pN or 1.04 in dimensionless variables) and Fm(about 10.4 pN or 0.104) are the local maximum and minimum forces, with corresponding extensions uMand um, respectively. The unit of force is [F] = 100 pN, and Lc= 30 nm. Inset: potential V(u)−Fu of ubiquitin, at F= 10, 50 and 100 pN (from top to bottom). where G(u,F,T)=N j=1V(uj)−Fuj+k 2(uj−uj−1)2 is the overall potential for a N-module protein, and D= kBT/γ = 1000 nm2/s, ξj(t), kB,andTare the diffusion coefficient, zero-mean delta-correlated independent identically distributed white noises, the Boltzmann constant and the temperature, respectively. Assuming infinitely rigid springs connect the protein to AFM cantilever and platform, u0=u1,uN+1 =uN. Deterministic dynamics. – The stochastic nature of the unfolding/refolding events is well documented by experiments. However, the mere existence of stepwise unfolding indicates that a deterministic scaffolding lies below such events. To understand them, we first reveal the reason for their existence by studying the dynamics of our system without noise and later explain the important modifications noise brings to the picture. In the absence of noise and with time-independent length or force, the system (2) has stable stationary configurations of folded and unfolded modules with respective extensions u(1)(F)andu(3)(F) in the metastability region Fm<F<F M, defined in fig. 1. Configurations with only one domain wall, separating domains of folded and unfolded modules, are stationary (pinned) wave fronts. For a given number of unfolded modules, these pinned wave fronts minimize the linkers contribution to energy and are therefore the most stable configurations. By slowly increasing the protein length with time and decreasing it after all modules unfold, there appears the sawtooth FEC in the main panel. The system moves over the stationary branches (as many as polyprotein modules) in the inset of fig. 2. In the pulling (respectively, pushing) process, the system sweeps the branch where it was when the 28002-p2 Protein unfolding and refolding as transitions through virtual states 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 L F Fig. 2: (Colour on-line) Force-extension curve obtained by first solving numerically the zero-noise Langevin equations with sudden length increases ΔL=0.005 applied at times tj=jΔt, with Δt=0.25, until all modules are unfolded at time tM.After this time, we reverse this procedure by applying length decrements −ΔLat times tM+j=tM+jΔtuntil we return to the initial state having all modules folded. Time unit: [t]=γLc/[F]=1.24 ms. Inset: force vs. length curves for the stationary solutions of the zero-noise Langevin equations with 8 modules. force variation started until it reaches the limit of stability FM(respectively, Fm), and then jumps to the next branch having one more (respectively, less) unfolded module. The small upward jumps (refolding events) in the pushing process have been observed in experiments, see figs. 1C and 1D of ref. [28]. Thermal noise introduces fluctuations in this folding/refolding diagram and changes the maxima and minima of fig. 2. In general, the FEC lies between the adiabatic limit at zero temperature in the main panel of fig. 2, and the quasistatic limit discussed in ref. [12]. In force-clamp experiments, the force first suddenly increases from F|t=0 ≡Fin to a peak value Fp, remains there for a given time, then abruptly decreases to Ff. Depending on Fp, the polyprotein length increases either abruptly (large Fp, as in fig. 3(a), (b)) or in a succession of length jumps (smaller Fp, as in fig. 4(a), (b)). Depending on Ff, modules may simultaneously refold or remain unfolded at a smaller length. We will show that this behavior arises because Fpand Ffare close to the critical forces FMand Fm, respectively. Virtual states with extensions uMor um shown in fig. 1 play a crucial role. To understand the sudden unfolding in fig. 3(a), we assume that initially all modules are equally folded, take Fp just above the local maximum FMand Ffjust below the local minimum Fm, see fig. 3(b). Protein unfolding and refolding occur as passages through virtual states. For F=Fp, no stable folded state exists. Thus, all modules jump in a short time after the force increase to Fpto the 0 5 10 15 20 25 30 2 4 6 t L (a) 0 5 10 15 20 25 30 0 0.5 1 t F (b) Fig. 3: (Colour on-line) (a) Protein length response to the force variation shown in (b). Initially Fin =0anduj=u(1)(0) for all j.PeakFp=1.1 (110 pN) and final Ff=0.1 (10 pN) forces are slightly larger than FMand slightly below Fm,respectively. (c) Protein unfolding after Fin →Fp: all the modules stretch to the virtual state with extension uM, spend some time there, then simultaneously unfold to reach extensions u(3)(Fp). (d) Refolding stage: modules simultaneously contract to um, stay there a long time ∼π√2/(Fm−Ff)V (um), and refold to u(1)(Ff). The virtual state has a much more noticeable effect on the length response curve during refolding: uM−u(1)(F)is small for all Fwhile u(3)(F)−umis not, see fig. 1. virtual state with extension uM=u(1)(FM). The modules remain there for a certain time (that is larger the smaller Fp−FMis; infinite if Fp=FM), until all modules unfold simultaneously to acquire extensions u(3)(Fp). A subsequent sudden decrease to Ffjust below the local minimum of the force field where u(2)(Fm)=u(3)(Fm)=ummakes all modules collapse simultaneously to the folded state in a three-stage sequence, as observed in experiments. Similarly, all modules first fall to the virtual (unfolded) state with extension um=u(3)(Fm), stay there for a long time, then abruptly refold to u(1)(Ff), see fig. 4(a). Had we chosen Ff>F m, all modules would have remained in the stable unfolded state u(3)(Ff). They would have folded simultaneously and rapidly to u(1)(Ff)hadFfbeen smaller than but not close to Fm. This two-stage behavior is similar to experimental observations, see fig. 9 of [29]. Of course, thermal noise is present in experiments (T=0), each module may spontaneously change from folded to unfolded state and back, and the system may end up either in the unfolded or the folded state for different realizations of the experiment. See the next section for more details. 28002-p3 L. L. Bonilla et al. 0 10 20 30 40 50 2 4 6 t L (a) 0 10 20 30 40 50 0 0.5 1 t F (b) Fig. 4: (Colour on-line) (a) Protein length response to the force variation shown in (b) with Fin =0.2, Fp=1such that Fc2<F p<F M,Ff=0.1 (20, 100 and 10 pN, respectively). Initial extensions: uj=u(1)(Fin)forj>1, u1=u(3)(Fin). (c), (d): sketches of the unfolding of the active module at the domain wall, which here moves to the right, leading to a new quasi-stationary configuration with one more unfolded module. The long common time between jumps is π√2/(Fp−Fc2)|V (u∗)|. The stepwise length increase in fig. 4(a) stems from a more complicated dynamical behavior, appearing for Fp<F Mbut close thereto as in fig. 4(b). Let us consider an embryonic wave front configuration, see figs. 4(c) and (d). Such stationary configurations become unstable and the resulting wave front moves with nonzero velocity if the force is on any of the two depinning intervals (Fm,F c1)and(Fc2,F M), whose widths vanish proportionally to the elastic constant kin eq. (1); see the appendix and also refs. [25,26]. New virtual states appear, those corresponding to the Peierls critical forces for wave front depinning [25–27], Fc1and Fc2, and more involved transitions through them play a crucial role. See [30] for general depinning transitions in random media. After Fabruptly increases to Fpin the depinning interval Fc2<F p<F M, the modules evolve to a virtual quasi-stationary state for F=Fc2, with only one unfolded module of extension u(3)(Fc2). The adjacent module has an extension u∗, slightly larger than that of the others, u(1)(Fc2), see fig. 4(c). This is the active module: it is the only one whose extension changes noticeably, slowly increasing from u∗until, at a precise time, it suddenly unfolds to u(3)(Fc2). Simultaneously, the next module becomes active attaining extension u∗, see fig. 4(d). This saltatory motion of the wave front continues until all the modules unfold to u(3)(Fc2), with all the time steps having thesamelength.Inthenextsection,weseethatthermal noise makes the steps have different lengths, as observed in the experiments, see for instance fig. 10 of [29]. An initial embryonic wave front configuration may be attained in two ways: i) For Fin between Fmand Fp< FM, we put the system in a configuration with only one unfolded module, that is, a point close to the bottom of the second branch in fig. 2; ii) all the modules are folded, but one of them has a slightly larger protein length, e.g. the one attached to the AFM cantilever. Accordingly, its potential is V(μ1u1), μ1<1, and the corresponding local maximum of the force field occurs at μ1FM<F Mwith a larger extension uM/μ1.Ifμ1FM<F p<F M,this module unfolds first, creates the wave front, and sequential stepwise unfolding follows. Stochastic dynamics. – Considering white-noise forces, the threshold forces change because the modules may unfold (refold) for peak (final) forces smaller (larger) than FM(Fm). In particular, for Fpclose enough to FM, the folded configuration becomes thermodynamically metastable (it corresponds to a local minimum while the unfolded configuration corresponds to the absolute minimum thereof). The same is true (with the roles reversed) for Ffclose to Fm: the unfolded configuration becomes metastable (local minimum) and the folded configuration becomes stable (absolute minimum). The escape time from the metastable states is finite at finite temperature and it becomes infinite only in the zero temperature limit (deterministic case). As the energy barrier between the unfolded and folded configuration vanishes for Fp→F− M(Ff→F+ m), these escape times are expected to become smaller the closer the peak (final) force is to FM(Fm). Note that Kramers rate theory requires large energy barriers (in units of kBT) for separation of time scales corresponding to intra- and inter-well dynamics [31]. This condition no longer holds for Fvery close to FM and Fm. Similarly to the previous discussion, the time intervals spent in the virtual states become considerably longer than the deterministic times, as they are proportional to the exponential of the barrier energy in units of kBT.The stepwise deterministic unfolding of fig. 4(a) is also affected strongly by the noise, which may shorten or enlarge greatly the step duration, an effect observed in experiments [29]. Throughout this section, we consider a nondimensional temperature θ=kBT/([F]Lc)=0.0014 (corresponding to T= 300 K), independent of the damping constant. The latter only selects the time unit [t]=γLc/[F]. Firstly, let us see how thermal noise affects the refolding stage. Figures 5(a) and (b) illustrate this by depicting the length response to a force change where Fp= 1 (100 pN, just below FM=1.04) and the final forces Ffare 0.117 and 0.11 (11.7 and 11 pN), respectively, both above but close to Fm=0.104. For a given temperature, the more Ffdiffers from Fm, the higher the barrier between the unfolded and the folded configurations and the longer the time scale for the refolding process. This is clearly shown 28002-p4 Protein unfolding and refolding as transitions through virtual states 01000 2000 1 2 3 4 5 6 7 t L (a) 0100 200 1 2 3 4 5 6 7 t L (b) Fig. 5: (Colour on-line) Effect of noise on the length responses to force changes with Fp= 1 (100 pN) and (a) Ff=0.117 (11.7 pN) and (b) Ff=0.11 (11 pN). The nondimensional noise strength is θ=kBT/([F]Lc)=0.0024. Note the different global time scales in both graphs. When the force is suddenly decreased from Fpto Ff(t= 50), the system almost instantaneously falls on the unfolded state corresponding to the final force Ffin both graphs. Afterwards, the jump to the corresponding completely folded state at Ffoccurs over a much longer time scale for Ff=0.117, which is further from Fm. in fig. 5(a), in which the system refolds on a time scale that is quite longer than the one in fig. 5(b). Moreover, the length response after the force drops from Fpto its final value is quite similar in the cases of a polyprotein with 8 modules and of a single-module protein (not shown): The time scale for refolding is the same in both cases, although not all the polyprotein modules refold strictly at the same time in the case of fig. 5. Let us consider now how noise affects the unfolding stage. In fig. 6, we show two realizations of the length response to a force protocol like the one in fig. 4(b), with initial, peak and final forces Fin =0.2>F m,Fp=1,and Ff=0.108 >F m, respectively. The observed stepwise unfolding is strongly affected by noise: (a) the system evolves on a time scale roughly 100 times larger than that of fig. 4, and (b) the steps are no longer uniform. The finite value of the temperature may greatly shorten or enlarge the step duration, an effect observed in experiments [29]. The refolding stage is similar to that in fig. 5. When the force is decreased to Ffat t= 4000, all the modules remain unfolded but their length almost instantaneously decrease to the value u(3)(Ff). Then, after spending a stochastic time (longer in (b) than in (a)) in this metastable state, all the modules finally refold to the thermodynamic stable state u(1)(Ff). Similar to fig. 5, the modules do not refold strictly at the same time but the details are not appreciable due to the longer time span in fig. 6. Discussion. – In force-clamp experiments exhibiting stepwise unfolding, polyproteins are operating near critical conditions corresponding to the nonequilibrium depinning 0 1000 2000 3000 4000 5000 0 2 4 6 t L (b) 0 1000 2000 3000 4000 5000 2 4 6 t L (a) Fig. 6: (Colour on-line) Two realizations of the length responses to force changes with nondimensional values Fin =0.2 (initial), Fp=1(peak),Ff=0.108 (final). Note (a) the much longer time scale for the unfolding process, as compared to the deterministic case shown in fig. 4, and (b) the nonuniform duration of the stepwise unfolding process. The final length is the same in both cases. transition for wave fronts [26]. This may seem reminiscent of the idea that muscle materials are finely tuned to operate close to a mean-field equilibrium phase transition [32]. However, including mean-field couplings as in [32] dramatically changes the dynamics by precluding stepwise unfolding: The short-range couplings between modules are at the root of the depinning transition that causes the poorly understood stepwise unfolding observed in experiments for certain values of the peak force. Depinning of wave fronts is a general phenomenon in spatially discrete systems [25–27] and it is behind observed behavior in systems quite removed from proteins. For instance, nonlinear charge transport in weakly coupled dc voltage biased semiconductor SLs produces current-voltage curves alike FECs in length-clamp experiments (current is assimilated to force and voltage to extension) [23]. Stepwise unfolding of the force-clamp protein could have a counterpart in dc current biased SLs but the necessary experiments do not yet exist, as there are no precise current controlled experiments on these nanostructures. We have provided a unified framework involving quasistationary virtual states to understand quite different observed behaviors, such as simultaneous vs. stepwise unfolding or three-stage vs. two-stage refolding. These virtual states are the closest stable configurations that the modules can attain when the forces are outside (but close to) the metastability region, that is, the region of forces in which the unfolded and folded configurations coexist. We also predict novel behaviors, such as the sequential character of stepwise unfolding. Thus, our work opens new attainable perspectives in the experimental investigation of tethered biomolecules. In real experiments, unfolding may not be sequential if the heterogeneity in the 28002-p5 L. L. Bonilla et al. potential is larger than the intermodule spring potential energy. Then additional simulations of our model would predict the unfolding order. ∗∗∗ This work has been supported by the Spanish Ministerio de Econom´ıa y Competitividad grants FIS2011-28838- C02-01 (LLB), FIS2011-28838-C02-02 (AC), and FIS2011- 24460 (AP). Appendix: nondimensional model. – We measure force, extensions and time in the units: [F] = 100 pN, Lc=30nmand[t]=γLc/[F]=kBTLc/(D[F]), respectively. The equations of the model are ˙uj=F−V(uj)+κ(uj+1 +uj−1−2uj) +√2θξ j(t),(A.1) V(u)=μ1−e−β(u−ρ)2−1 +A1 1−u−1−u+2u2,(A.2) where μ=U0/(Lc[F]), β=2bLc/Rc,ρ=Rc/Lc,κ= kLc/[F], A=kBTLc/(4PU0), θ=kBT/([F]Lc)=0.0014 and the ξj(t) are i.i.d. zero-mean delta-correlated white noises. Note that θis independent of the diffusion constant D, which sets the unit of time [t]. Stepwise unfolding. To explain stepwise unfolding when θ=0andFhas increased abruptly to Fp∈ (Fc2,F M)fromFin, assume that one module has stretched to u(3)(Fc2) and the others to u(1)(Fc2)foracritical force Fc2(slightly below FM) such that κ[u(1)(Fc2)+ u(3)(Fc2)] = V(u∗)+2κu∗and 2κ+V(u∗)=0,for uJ=u∗. We have a wave front joining a domain with N−1 modules of extension u(1)(Fc2) and one unfolded module of extension u(3)(Fc2). Let uJbe the extension of the module adjacent to the unfolded one. In (A.1) we have uj=u(1)(Fc2)forj<Jand uj=u(3)(Fc2)forj>J. Then expanding the right-hand side of (A.1) in powers of (uJ−u∗), we obtain ˙uJ∼Fp−Fc2−1 2V(u∗)(uJ−u∗)2,(A.3) provided that u(1)(Fc2)+u(3)(Fc2)=2u∗+V(u∗) κand V(u∗)+2κ=0. Sinceκis small, u∗is close to uMand Fc2<F p<F Mare close. Then V(u∗)≈ V(uM)<0 and (A.3) has the solution uJ=u∗+ 2Γ |V (u∗)|tan[Γ(t−tJ)], where tJis a constant and Γ = (Fp−Fc2)|V(u∗)|/2. Notice that uJ(tJ)=u∗and that the tangent function becomes ±∞ when Γ(t−tJ)= ±π/2. After the argument of the tangent function reaches π/2, at t=tJ+π/(2Γ), uJjumps to u(3)(Fc2), and the point uJ−1(t) becomes active. This means that the J-th module has unfolded, the wave front has advanced one step to the left and uJ−1(t) satisfies (A.3) for |t−tJ−1|<π/(2Γ), where tJ−1=tJ+π/(2Γ). The duration of the steps between jumps of the wave front is (tj−1−tJ)=π/(2Γ). Details of the jumps and the matching between jumps are given in [25,26] for the saltatory motion of wave fronts near the depinning transition. Simultaneous unfolding and refolding. Simultaneous module unfolding and refolding imply evolution to virtual states at uj=uMand uj=um(for all j), respectively. Following a line of reasoning similar to that in the last paragraph, we find that the ujare near u∗during a long time π/2|F−F∗||V(u∗)|(F∗is FMor Fm). REFERENCES [1] Alberts B.,Cell,92 (1998) 291. [2] Oberhauser A. F. and Carri´ on-V´ azquez M.,J. Biol. Chem.,283 (2008) 6617. [3] Linke W. A.,Cardiovasc. 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