Towards a P Systems Pseudomonas Quorum Sensing Model

Towards a P Systems Pseudomonas Quorum Sensing Model Luca Bianco1, Dario Pescini2, Peter Siepmann3, Natalio Krasnogor3, Francisco J. Romero-Campero4, and Marian Gheorghe5 1Department of Computer Science, University of Verona Strada Le Grazie 15, 37134 Verona, Italy [email protected] 2Dipartimento di Informatica, Sistemistica e Comunicazione Universit`a degli Studi di Milano-Bicocca Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy [email protected] 3School of Computer Science and Information Technology University of Nottingham Jubilee Campus, Nottingham, NG81BB, UK {Peter.Siepmann, Natalio.Krasnogor}@nottingham.ac.uk 4Research Group on Natural Computing Department of Computer Science and Artificial Intelligence University of Seville, Avda. Reina Mercedes, 41012 Sevilla, Spain [email protected] 5Department of Computer Science, The University of Sheffield Regent Court, Portobello Street, Sheffield S1 4DP, UK [email protected] Abstract. Pseudomonas aeruginosa is an opportunistic bacterium that exploits quorum sensing communication to synchronize individuals in a colony and this leads to an increase in the effectiveness of its virulence. In this paper we derived a mechanistic P systems model to describe the behavior of a single bacterium and we discuss a possible approach, based on an evolutionary algorithm, to tune its parameters that will allow a quantitative simulation of the system. 1 Introduction The quorum sensing is a particular form of cell-to-cell communication in bacteria which exploits the concentration of a particular molecule, called signal, to “sense” the population density of the colony. The quorum sensing regulatory network is used by the individuals of the colony for collective synchronization and therefore for a coherent control over the gene expression. In Pseudomonas aeruginosa this mechanism is responsible for the effectiveness of the virulence of this bacterium [14,23,10,9]. In fact, a single bacterium starts to express his virulence factors only when it senses that the bacteria population has reached a certain threshold level such that the host response will be inadequate. The activation of a complex cellular response is what distinguishes the quorum sensing as a communication regulatory circuit from other density dependent responses such as the metabolization or detoxification of small molecules. The simplest quorum sensing network known in Gram-Negative bacteria is also the first one ever discovered [25,17]. It has been found in the Vibrio fischeri bacterium, also known as Photobacterium fischeri and is nowadays considered as the paradigm of this cell communication process. In this network two proteins and one signalling molecule are involved. The Rproteinis a transcriptional regulator, while the Iproteinis the synthase for the signalling molecule, also referred to as the autoinducer. An important role is also played by the confinement of the bacterial colony. The fact that the autoinducer molecule is not dispersed in the environment allows its diffusion inside the individuals and therefore its concentration sensing. At low cell densities the Iproteinsynthesizes the autoinducer at a basal rate and the signal freely diffuses outside the bacterium. The concentration of the signal inside each bacterium is increased by the combined effect of the confinement and the increase of the population. At this point, the binding of the Rprotein with the autoinducer becomes more likely. The binding of the signal molecules activates the Rproteintranscriptional regulator. Since the I gene is the target of the Rprotein, the bacterium starts to produce more and more signal. The regulation network signal autoinduces its transcription. In this way the high concentration of the autoinducer coordinates the transcription of all the genes that are target of the Rprotein. The quorum sensing in Pseudomonas aeruginosa is more complex, nevertheless intriguing, since this bacterium uses two different quorum sensing systems which interact with each other. The aim of this work is to provide a P system model [15,16] of the bacterium Pseudomonas aeruginosa quorum sensing focusing on the communication mechanisms. The parameters of the model will be tuned using an evolutionary algorithm. Our long term aim is to reproduce the characteristic behavior of the quorum sensing in Pseudomonas aeruginosa, namely, the switch between two distinct stable steady solutions: the first describing the behavior of the nonquorated bacterium (i.e., with low levels of autoinducer), the second modeling its quorated behavior (i.e., the behavior obtained with high concentration of the autoinducer molecule). Once the model will be entirely defined several simulations with different strategies [6,20,18] will be run. First of all, we address the modeling of the internal dynamics of one single bacterium, tuning its kinetic constants in a way ensuring its non-quorated behavior. At a later stage, we intend to exploit compartmentalization of P systems to model a colony of bacteria each of them internally specified according to the same set of kinetic constants. In this respect we will extend the current model to a Population P systems approach [3] that has been already used to express some aspects of quorum sensing in bacterium Pseudomonas aeruginosa [22] and for self-assembly problems [4]. 2 An Initial Model The first stage of our investigation is intended to describe the quorum sensing related network of each bacterium to capture its main features into a mechanistic model. The quorum sensing internal pathway of each bacterium is taken from models discussed in [11,9] and a graphical representation of all elements involved in it, as well as some relevant relationships between them, are depicted in Figure 1. LasR LasR lasR lasI LasI 3O Vfr rsaL RsaL Fig. 1. The Pseudomonas quorum sensing model analyzed here (from [9]). Note that double arrows denote reversible reactions, bold ones the degradation process and the empty ones the inhibitory process. According to this model, the quorum sensing pathway comprises two interconnected signalling cascades. The main elements involved in the first one are proteins LasR,RsaL,LasI (as well as the genes involved in their production), the autoinducer molecule 3-oxo-C12-HSL and the active complex LasR-3-oxo-C12HSL. The key elements of the second system are the proteins RhlR and RhlI (as well as the genes involved in their production), the autoinducer molecule C4HSL and the active complex RhlR-C4-HSL. The first one of the two signalling cascades is called las system because it was shown to regulate the expression of LasB elastase. This pathway regulates other virulence factors such as LasA protease, exotoxin A, alkaline protease A as well as the expression of at least two genes of the xcp secretory pathway. The las pathway is positively controlled by GacA and Vfr whereas it is inhibited by RsaL that, in turn, is positively regulated by the active complex LasR-3-oxo-C12-HSL and whose role is to repress the transcription of the lasI gene. The second signalling system involved in the model is named rhl system because it controls the expression of rhamnolipid via the production of rhlAB operon. The autoinducer molecule in this case is C4-HSL and the active complex is RhlR-C4-HSL. It has been shown that this cascade is necessary for the production of some virulence factors like LasB elastase and LasA protease, as well as pyocyanin, cyanide and alkaline protease. For this reason this signalling system is also known as vsm (virulence secondary metabolites). Although the corresponding autoinducing molecules are highly selective (and thus not interchangeable at all), several interconnections between the las and the rhl pathways of the quorum sensing in Pseudomonas aeruginosa are known. One link between them has been already mentioned and it is constituted by the LasB elastase, that needs both LasR-3-oxo-C12-HSL and RhlR-C4-HSL for its production. More interestingly, the las system is at a higher level in the hierarchical regulatory cascade, in fact LasR-3-oxo-C12-HSL can activate the expression of the rhlR gene. In addition, the active complex LasR-3-oxo-C12HSL canbindtoRhlR preventing it to form the complex RhlR-C4-HSL. 2.1 The Differential Equation Model Many models for the quorum sensing in the Pseudomonas aeruginosa are presented in literature and usually they approach the phenomenon from two different angles. The first one describes the colony behavior by summarizing individual dynamics as a state change avoiding a precisely detailed representation of each of the bacterium quorum sensing networks [24,1]. The second one describes in a more detailed fashion the quorum sensing pathway for each bacterium with the purpose to model the emergent behavior of the whole colony [11]. We think that the P system framework is particularly suitable for this second approach. In fact, the modularity, the compartmentalization, the hierarchical structure and the rewriting rules (all features of P systems [16]) allow a convenient description of this reality. In [11] a model of the las signalling system has been devised, but no description is given of the rhl system. The graphical description of the quorum sensing pathway depicted in Figure 1 has been translated into the set of eight differential equations presented in the next page. The correspondence between differential equation symbols and elements in the pathway are summarized in Table 1. The production of the activated complex Pby means of the autoinducer and the LasR protein (whose expression is given by the product of the constitutive elements concentrations with a rate kRA:kRARA)isanexampleofhowcooperative contributions are obtained in the differential equations approach by means of the mass action law. Basal rates productions and degradations are also taken into account, an example of the former being the k1relement giving the basal production of LasR protein (R), while an example of the latter is the degradation of the active complex (P) represented by the element kPP. The production of messenger RNAs from the corresponding genes is modeled with a Michaelis-Menten-like dynamics depending on the concentration of the promoting factor, as it happens in the case of the production of lasR and rsaL mRNAs (respectively rand s), the first modeled by Vr P Kr+Pand the second by Vs P Ks+P. The production of lasI mRNA (l) is also down-regulated by the presence of RsaL protein (S)andthis is modeled by Vl P Kl+P 1 KS+S, in which the Michaelis-Menten-like dynamics is attenuated by an inversely proportional function of the RsaL concentration. dP dt =kRARA −kPP dR dt =−kRARA +kPP−kRR+k1r dA dt =−kRARA +kPP+k2L−kAA dL dt =k3l−kLL dS dt =k4s−kSS ds dt =Vs P Ks+P−kss dr dt =Vr P Kr+P−krr+r0 dl dt =Vl P Kl+P 1 KS+S−kll+l0 (1) Unfortunately, no value is known for the 21 kinetic constants present in the set of differential equations (1). To overcome this problem, in [11] several simplifying assumptions are considered, that lead to fewer equations and fewer parameters as well. In the following we will describe a possible parameter estimation strategy to tackle this problem (see Section 4). The idea is to relay to this differential equations system as a “synthetic bio-experiment” used to confront our model to. 2.2 A First P Systems Model Several attempts to simulate the quorum sensing in bacteria are present in P systems literature [5,19], but, as far as we know, none of them deals with the Pseudomonas aeruginosa bacterium. Here we describe a direct P systems translation of the differential equation model previously discussed [11]. Formally, the Pseudomonas P system is Π=(A, μ, w, R) Table 1. Variable-concentration correspondence between the differential formulation and the graphical description of the quorum sensing model of Pseudomonas aeruginosa (from [11]) Variable Concentration RLasR A3-oxo-C12-HSL PLasR-3-oxo-C12-HSL LLasI SRsaL rlasR mRNA llasI mRNA srsaL mRNA where: –A={geneR,geneL,R,A,P,L,S,r,l,s}is the alphabet; –μ=[] 0is the membrane structure: since we address the single bacterium case, it contains the cellular membrane only; –w=geneR geneL is the initial configuration that comprises only LasR and LasI genes, thus is represented as the string; –R={r1,···,r 18}is the set of the rules: r1:geneR −→ geneR +r r2:r−→ λ r3:r−→ r+R r4:P−→ P+r r5:R+A−→ P r6:P−→ R+A r7:P−→ P+s r8:s−→ λ r9:S−→ λ r10 :s−→ s+S r11 :P−→ P+l r12 :l−→ l+L r13 :l−→ λ r14 :geneL −→ geneL +l r15 :L−→ λ r16 :L−→ L+A r17 :A−→ λ r18 :R−→ λ Note that, symbols in Acorrespond to the variables of the differential equation and their correspondence to the biological reality is given in Table 1. Two new elements (i.e., geneR and geneL) are introduced, which account for the genes involved in the basal production of the LasR and LasI mRNAs. Each one of the rules in Ris directly obtained from the differential description of the considered quorum sensing model. For examples, we can see that rule r1 models the basal production of the LasR mRNA, while rule r2expresses its degradation, moreover rules r5and r6describe the reversible reaction of the complex Pformation by starting from its fundamental constituents Rand A. Due to the different level of abstraction in the representation of different parts of the model (as in the case of the Michaelis-Menten-like kinetics that are modelled with a higher level of abstraction than other components of the system), we cannot directly apply mechanistic algorithms [2] to this model. For this reason, we will apply to this set of rules only the strategy known as Metabolic Algorithm (for details refer to [6]), whose simulation results, together with some numerical solutions of the set of differential equations (1), are shown in Section 2.3 for different choices of parameters. The metabolic algorithm simulation needs to specify a set of reaction maps, each one associated in a one-to-one manner to the rules of R. Reaction maps [6] are functions defined over the state of the system (i.e., multiplicity or concentration of all elements of the system depending on the case), that are used by the Metabolic algorithm to allocate objects to rules. For example, as we will see in a while, Fr1, that is the reaction map of rule r1, is simply the constant rate of production of LasR mRNA. We can have more complicated reaction maps, as in the case of rule r4that takes into account the Michaelis-Menten-like production of the LasR mRNA elicited by the LasR-3oxo-C12-HSL complex. As in the case of the rules, that specify the physical interactions and connections between the elements of the modeled reality, we can obtain this information from the differential equation formulation. The set of reaction maps employed in our simulations are the following: Fr1=r0Fr2=kr Fr3=k1Fr4=Vr Kr+P Fr5=kRA Fr6=kP Fr7=Vs Ks+PFr8=ks Fr9=k4Fr10 =kS Fr11 =Vl (Kl+P)·(KS+S)Fr12 =k3 Fr13 =klFr14 =l0 Fr15 =kLFr16 =k2 Fr17 =kAFr18 =kR (2) Note that all reaction maps are constant apart from three of them. We have already discussed the meaning of the reaction map associated to rule r4; analogous considerations hold for Fr7as well. More interesting is the reaction map associated to rule r11 that takes into account the inhibitory effect of RsaL protein on the production of the lasI mRNA. Remarkably, the method allows the current description of different parts of the system at different abstraction levels; moreover it is still applicable if all reaction maps are constant, a condition required by mechanistic algorithms. In the following some simulation results are shown, as well as the numerical solution of the differential equation system, for some chosen parameters. 2.3 Simulation Results Here we show how the same model-reality can be described with two different approaches. As mentioned before, we do not have precise values for the model parameters. For this reason, as a first comparison attempt, we make a completely fictitious choice for them. As a further work, we plan to adopt some automatic way for the parameter estimation (see Section 4 for more details). The initial choice of parameters is here shown, and all the subsequent changes to this initial parameter set will be explicitly mentioned: kRA =10 kP=2 kR=5 k1=1 k2=1 kA=1 k3=1 kL=1 k4=1 kS=1 Vs=1 Ks=1 ks=0.5Vr=1 Kr=1 kr=1 r0=1 Vl=1 Kl=1 kl=1 l0=1 KS=1 (3) The lack of biological information makes this choice completely arbitrary and prevents us to compute the dynamics of the system by means of stochastic algorithms such as the Gillespie one [12,13], Dynamical Probabilistic P Systems [20] or the Multi-compartmental Gillespie [18]. In this section we compare the dynamics generated by the metabolic algorithm with the solutions obtained for the corresponding differential equation system. Figure 2 depicts the case in which parameters are chosen according to (3). The dynamics of each species reaches a steady state in both approaches, but the relative position of the species is different and this leads to two distinct system dynamics. Moreover, the time of the two systems differs; in the solution of the differential equation system this is measured in arbitrary units (due to the arbitrary choice of parameters), while in the model based on P systems the time is measured in steps of system evolution. In Figure 3 the choice of Vl=0 switches off rule r11 of the P system model and in this case the results of the two different approaches qualitatively match each other. Finally, the last choice of parameters is aimed at obtaining a quorum sensing consistent behavior, that is, in the case of a single bacterium in the environment it should not quorate and thus the concentration of the complex Pshould reach the basal rate. Accordingly, we set KRA to the value 0.1. In this case, depicted in Figure 4, the dynamics produced by the two approaches is qualitatively similar again. 0 0.5 1 1.5 2 2.5 0 5 10 15 20 25 30 P R A L S s r l 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 5000 10000 15000 20000 25000 30000 P R A L S s r l Fig. 2. Results for the quorum sensing model with parameters showed in (3) using ODE approach (left) and metabolic algorithm (right) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 5 10 15 20 25 30 P R A L S s r l 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 5000 10000 15000 20000 25000 30000 P R A L S s r l Fig. 3. Results for the quorum sensing model with Vl= 0 using ODE approach (left) and metabolic algorithm (right) 0 0.2 0.4 0.6 0.8 1 1.2 0 5 10 15 20 25 30 P R A L S s r l 0 0.2 0.4 0.6 0.8 1 1.2 0 5000 10000 15000 20000 25000 30000 P R A L S s r l Fig. 4. Results for the quorum sensing model with parameters kRA =.1usingODE approach (left) and metabolic algorithm (right) 3 Towards a Detailed P Systems Model Although the preliminary P system model described in Subsection 2.2 showed that we can obtain comparable results with the current models presented so Fig. 8. The target Michaelis-Menten concentrations and the evolved P systems ones 5 Conclusions and Further Work We have briefly described a part of the quorum sensing network in the Pseudomonas aeruginosa. Starting from a differential equations based model we have provided a P systems version of it and we compared the dynamics of the two approaches. In order to apply different simulation strategies on this intriguing phenomenon we provided a more detailed, mechanistic model which, we believe, is closer to the biological reality. The lack of biological information regarding the dynamics of the system led us to use an automatic way for estimating them by using an evolutionary algorithm approach that offers a reliable and effective method in this respect. An immediate step further, after obtaining all the parameters regulating a single bacterium dynamics, is to extend the proposed model at a colony level, exploiting the compartmentalization offered by P systems and already established population P systems models. Other important developments are related to the use of experimental data to tune the dynamics of our specifications such as to simulate real biological processes. In this respect the use of model checking methodologies, already under consideration in a paper under preparation, will contribute towards validating certain properties of the systems modeled. On long term we believe that these steps can represent the first stage toward a quantitative analysis that will hopefully lead to a successful drug design process. Acknowledgements. N. Krasnogor and P. Siepmann acknowledge the EPSRC for funding project EP/D021847/1. References 1. K. Anguige, J.R. King, J.P. Ward, and P. Williams. 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