Stochastic Approaches in P Systems for Simulating Biological Systems

Stochastic Approaches in P Systems for Simulating Biological Systems Paolo Cazzaniga1, Dario Pescini1, Francisco J. Romero-Campero2, Daniela Besozzi3, Giancarlo Mauri1 1Universit`a degli Studi di Milano-Bicocca Dipartimento di Informatica, Sistemistica e Comunicazione Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy cazzanipa/pescini/[email protected] 2University of Sevilla Department of Computer Science and Artificial Intelligence Avda. Reina Mercedes s/n, 41012 Sevilla, Spain [email protected] 3Universit`a degli Studi di Milano Dipartimento di Informatica e Comunicazione Via Comelico 39, 20135 Milano, Italy [email protected] Summary. Different stochastic strategies for modeling biological systems with P systems are reviewed in this paper, such as the multi-compartmental approach and dynamical probabilistic P systems. The respective results obtained from the simulations of a test case study (the quorum sensing phenomena in Vibrio Fischeri colonies) are shown, compared and discussed. 1 Introduction The aims of this work are (i) to review some stochastic approaches that were recently defined in the framework of P systems as new tools for the modeling of biological systems, (ii) to compare their respective results in simulating the behavior of bacteria colonies, and finally (iii) to discuss their advantages, as well as their limits and possible improvements. The models used to describe biological systems are usually huge and complex although few copies of each reactant are present inside them. This is the reason why, in this framework, the classical approach with differential equations is not suitable. This has driven the direction of research to look for experimental evidences of the stochastic noise in these kinds of phenomena, and the importance of this behavior is now clearly stated in the literature, as reported in [4]. Here, we briefly report some notes on a well known and used stochastic algorithm, which is also the underlying structure at the core of the stochastic strategies 146 P. Cazzaniga, D. Pescini, F.J. Romero-Campero, D. Besozzi, G. Mauri reviewed in this paper. Namely, the exact stochastic simulation algorithm (SSA), introduced by Gillespie in [3], is applicable to well-stirred chemical reaction systems contained inside a single fixed volume, at a constant temperature. A biochemical system of this type is assumed to contain Nmolecular species S1, . . . , SNinteracting through Mreactions channels R1, . . . , RM, with reaction constants c1, . . . , cM which only depend on the physical condition of the system (e.g., the temperature) and on properties of the molecules. At time t,Xi(t) molecules of the species Si are assumed to be present inside the system, for any i= 1, . . . , N. The SSA algorithm is based on the fundamental hypothesis that cρδt, for any ρ= 1, . . . , M, is the average probability that a particular reactants combination for reaction Rρwill react in the next time interval δt. According to the current state of the system, this assumption allows to derive P(τ, ρ)dτ, that is, the probability determining (1) the next reaction that will occur in the volume Vduring the next differential time interval (t+τ, t +τ+dτ), and (2) that such reaction will be Rρ. Thus, the SSA algorithm computes both which reaction will be the next one in the system, and when it will be actually applied. The explicit derivation of P(τ, ρ) = P1(τ)P2(ρ) involves the definition of the propensity function aρ=hρcρ, where hρis the number of distinct reactant molecules combinations and cρis the stochastic rate constant associated to reaction Rρ. By defining a0=PM ρ=1 aρ, the explicit expressions for the probabilities are P1(τ) = a0e−a0τ,(1) P2(ρ) = aρ a0 ,(2) which allow to retrieve the information for when (τ) and which one (ρ) will be the next applied reaction from the probability density function P(τ, ρ), just by extracting two random numbers r1and r2from the uniform unit-interval distribution and then taking τ=1 a0 ln ³1 r1´(3) as the time of the next reaction, and choosing the integer corresponding to the next reaction by evaluating ρ−1 X j=1 aj< r2a0≤ ρ X j=1 aj.(4) The SSA algorithm has been widely used as the seed and the main reference model for many stochastic simulations in biochemistry and biology, and it is implemented in many of the most popular cellular simulators within the Systems Biology Markup Language project [13], such as E-Cell, Cellware, Dizzy, Stocks, SBW. Given this brief review of the Gillespie algorithm, the rest of the paper is structured as follows. In Section 2 we recall some stochastic strategies defined Stochastic Approaches in P Systems for Simulating Biological Systems 147 so far in the area of P systems, namely the multi-compartmental approach and sequential/parallel dynamical probabilistic P systems. Then, in Section 3 we report the simulations on a biological case study, the quorum sensing in Vibrio Fischeri, performed by the different stochastic approaches. We also compare the obtained results in Section 4. We conclude with some final remarks and ideas for future work. 2 Different Strategies for Stochastic Simulations in P Systems In this section we recall some stochastic strategies, based on SSA, appeared so far in P systems for the modeling of biological or biochemical systems. In Section 3 we will then compare their respective outcomes on a specific biological system, used as a test case study. 2.1 Multi-Compartmental Approach In this section we briefly summarize the multi-compartmental algorithm, introduced in [1], [2] and lately applied in [7]. The aim of this method is to exploit the P systems topology, namely, the hierarchical structure of its compartments. The core of this approach, based on the Gillespie one, is the creation of an appropriate common list for all the applicable rules occurring in all the membranes, at each time step. The sorting of this list (over time) then allows to choose always the first reaction that has to be applied in the whole system. This approach is intrinsically sequential and inherits the precision of the SSA, but with the additional feature of working with different compartments and moving objects among compartments (by means of the classical communication rules in P systems). Let us consider a P system Π= (O, Lab, µ, M1, M2, . . . , Mn, R1, . . . , Rn), where each membrane initially contains the multiset Mmand is labeled with a symbol in Lab (different membranes can have the same label), and with the set of rules Rm, 1 ≤m≤n. Each membrane mis considered a compartment enclosing a volume, therefore the index ρof the next rule to be used inside membrane m, and its waiting time τ, will be computed using the classical Gillespie algorithm, which returns the triple (τm, ρm, m). The procedure used in the multi-compartmental approach is the following: •Initialization ◦set time of the simulation t= 0; ◦for each membrane min µcompute a triple (τm, ρ, m) using the SSA; construct a list containing all these triples; ◦sort the list of triples (τm, ρ, m) according to τm(in decreasing order); •Iteration ◦extract the first triple (τm, ρ, m) from the list; 148 P. Cazzaniga, D. Pescini, F.J. Romero-Campero, D. Besozzi, G. Mauri ◦set time of the simulation t=t+τm; ◦update the waiting time for the rest of the triples in the list by subtracting τm; ◦apply the rule ρonly once, accordingly changing the number of objects in the membranes affected by the application of the rule; ◦for each membrane m0affected by the application of the rule ρ, remove the corresponding triple (τ0 m0, ρ0, m0) from the list; ◦for each membrane m0affected by the application of the rule ρ, run again the Gillespie algorithm for the new context in m0to obtain the triple (τ00 m0, ρ00, m0), the next rule ρ00 to be used inside membrane m0, and its waiting time τ00 ρ0; ◦add the new triples (τ00 m0, ρ00, m0) in the list, sort this list according to each waiting time and iterate the process. •Termination ◦Terminate the procedure when time treaches or exceeds a preset maximal time of simulation. As a general remark about the multi-compartmental approach, we can say that it is useful when one wants to simulate small systems that involve several compartments, each one with a homogeneous distribution of molecules. Exploiting the P systems structure, and possibly defining “virtual” compartments (e.g., regions corresponding to the cellular membrane bilayers) besides the real biological ones, it might be possible to reproduce the previous outlined conditions for many biological processes, thus opening P systems to a vast area of applications. This approach encapsulates the Gillespie algorithm in each compartment, the resulting description of the dynamics of each membrane is exact and the overall description of the system is accurate. It is well known that Gillespie algorithm does not scale linearly with the number of objects and the number of rules defined in the system, and this property is obviously inherited also by the multi-compartmental approach. Even a different programming version for the code (i.e., distributed over a set of processors) would not significantly reduce this difficulty, since the structure of the algorithm itself forces to apply one rule at each step. In fact, once that the rule has been fired, the computation takes place only in those membranes affected by the application of this rule. This limits the benefits of a distributed coding only in the initialization stage of the algorithm, where all the membrane compute the propensity function in order to obtain the triples to be inserted in the list. 2.2 Dynamical Probabilistic P Systems In this section we review the stochastic approach underlying dynamical probabilistic P system (DPP, in short), introduced in [8] and lately developed and applied in [9, 10] (the code is available in the software section of the P systems web page [12]). The aim of this strategy is to capture the major features of a membrane system (compartments, communication, parallelism) and to understand how they could Stochastic Approaches in P Systems for Simulating Biological Systems 149 be exploited in realistic simulations of biological systems. In particular, while the importance and the applicability of a membrane structure has been largely investigated from a practical and a theoretical point of view, it is a matter of fact that the role of maximal parallelism in nature is still not completely understood (and one of the main motivation of this work). We recently implemented the DPP approach using the MPI (Message Passing Interface) C libraries [11] in order to: (i) spread the computation over a cluster of processors to achieve scalability, (ii) have a direct mapping of the communications among the membranes, and (iii) speed up the computation. As a direct outcome, the performances obtained by DPP with MPI implementation allow to easily and actually execute parallel processes, managing the communication between them, and thus to achieve higher scalability than the multi-compartmental approach. In the following, we summarize the algorithm used for the evolution of a DPP, referring the reader to [9], where the original version running on a single machine is described. Here we remark that, with respect to the single processor program of DPP, the use of the MPI library affects only the system communications while preserving the rest of the code. In this way each membrane evolves independently from the others for everything but the communication, such that it could reside on an independent process. Each membrane min a DPP will be considered as a single volume, and will be simulated using an MPI process. All the membranes of a DPP simultaneously evolve, and require to be synchronized at the end of each step to allow the communication process. Since in each membrane the rules are applied in a maximally parallel way, in each membrane we can divide a step in three stages: (1) the computation of the probability distribution for the rules, (2) the assignment to the rules of all objects which can be modified by rules, (3) the communication and multisets updating. The corresponding algorithm is: •Initialization ◦create all parallel processes; ◦set the current step t= 0; ◦load the set of rules and the initial multisets for each process. •Iteration (for each process) ◦make a snapshot of the current status of the system; ◦calculate the propensity functions of rules with SSA; ◦toss the rules until all the objects are exhausted, and build the relative trace using the snapshot; ◦update the system state according to the generated trace and, for each rule, do what follows: •if the rule target is here, update the multisets values according to the left and right side of the rule; 150 P. Cazzaniga, D. Pescini, F.J. Romero-Campero, D. Besozzi, G. Mauri •if the rule target is in or out, decrease the value of the local multisets according to the left side of the rule, and send the message to the process (i.e., the membrane) indicated by the target of the rule; •wait until all other processes have executed one rule; ◦check any incoming messages (i.e., objects from other membranes) and increase the value of the multisets according to the received objects; ◦increment by one the step value. •Termination ◦terminate the simulation when the current step reaches the maximal value previously fixed. Note that the waiting step in the algorithm is needed to synchronize the execution between the membranes, otherwise a process icould, for instance, check for any received messages before another process jwill send a message to i. An advantage of this approach consists in the fact that, with slight changes in the program code, it is possible to probe various degrees of parallelism. For instance, by forcing the simulator to apply one rule at each step it is possible to obtain a parallel version of the Gillespie algorithm. In this case, if in each membrane we also allow to compute the τof its rules, then each membrane evolves with its own time stream. One common step corresponds to a potentially different τin each membrane, thus creating some paradox when objects are to be communicated between two membranes. An example could be a message received in a membrane in a time that corresponds to the past time of the sending membrane. This approach will be referred hereafter as 1 rule dynamical probabilistic P system (1rDPP). Another way to probe the role of parallelism in this framework is the insertion of rules of the type A→A, which will compete for the same objects with other rules having Aas the left side, without affecting the system evolution, so that the result is a reduced consumption of these objects. Examples of these strategies will be given in Sections 3.3 and 3.4. Obviously, the main problem that affects a DPP approach is the lack of a defined common time stream for all the membranes, which only allows to derive qualitative descriptions of the modeled phenomena. A deeper investigation of this problem, and of the role of parallelism in nature, is already under development and will appear elsewhere. The added value of this approach is the independent evolution of the membranes, which is fundamental to lead to a reliable and scalable simulator for large and complex systems, such as biological ones. A pure sequential approach binds the computational efficiency of the simulation to the efficiency of the computer on which the simulation itself is running, while the possibility of sharing the work among multiple processing units allows to scale the simulation with respect to the number of the membranes involved. Stochastic Approaches in P Systems for Simulating Biological Systems 151 3 The Different Strategies in Practice In Section 3.1 we briefly describe the quorum sensing phenomena in a Vibrio Fischeri bacteria colony and the corresponding P system based model presented in [2] and [1]. In Sections 3.2, 3.3, 3.4 we apply the strategies previously described on this biological model, used as a test case study. 3.1 Quorum Sensing in Vibrio Fischeri. A Case Study The marine bacterium Vibrio Fischeri regulates the expression of certain genes in response to population density using a family of transcriptional regulators, in a process called auto-induction. In the free living state these bacteria are non luminescent, but they start to produce light when are at high cell density. The quorum sensing mechanism in Vibrio Fischeri relies on the synthesis, accumulation and subsequent sensing of a signal molecule, which we will call OHHL. When only a small number of bacteria are present, the signal is produced by the bacteria at a low level. OHHL diffuses out of the bacterial cells and into the surrounding environment. At high cell density, the signal accumulates in the area surrounding the bacteria and can also diffuse to the inside of the bacterial cells. The signal is able to interact with the LuxR protein to form the complex LuxR-OHHL, which binds to a region of DNA called the Lux Box causing the transcription of the luminescence genes, a small cluster of 5 genes, luxCDABE. Adjacent to this cluster are two regulatory genes for the transcription of LuxR and OHHL. In this sense OHHL and LuxR are said to be auto-inducers because they activate their own synthesis. In this section we present the model used in [6] to simulate a colony of bacteria of this type, in order to reproduce the mechanisms at the base of the luminescent behavior observed in nature. Using the framework of P systems, each bacterium is represented by a membrane, and the skin membrane of the system is intended to enclose the environment (the depth of the system is 2, because each bacteria is an elementary membrane). In this way we can look at the evolution of every single bacterium, as well as at the evolution of the whole system. The system used for the simulations is ΠV f (N) = (O, {e, b}, µ, (w1, e),(w2, b),...,(wN+1, b),Rb,Re),(5) where: 1. Nrepresents the number of bacteria in the colony; 2. the alphabet is: O={OHHL,LuxR,LuxR-OHHL,LuxBox,LuxBox-LuxR-OHHL}; 3. the membrane structure is: µ= [e[b]b. . . [b]b]e; 4. the initial multisets are: w1=∅, wi={LuxBox},2≤i≤N+ 1; 152 P. Cazzaniga, D. Pescini, F.J. Romero-Campero, D. Besozzi, G. Mauri 5. the rules used inside the system are: r1: [ LuxBox ]b c1 −→ [ LuxBox ,OHHL ]b r2: [ LuxBox ]b c2 −→ [ LuxBox ,LuxR ]b r3: [ LuxR ,OHHL ]b c3 −→ [ LuxR-OHHL ]b r4: [ LuxR-OHHL ]b c4 −→ [ LuxR ,OHHL ]b r5: [ LuxBox ,LuxR-OHHL ]b c5 −→ [ LuxBox-LuxR-OHHL ]b r6: [ LuxBox-LuxR-OHHL ]b c6 −→ [ LuxBox ,LuxR-OHHL ]b r7: [ LuxBox-LuxR-OHHL ]b c7 −→ [ LuxBox-LuxR-OHHL ,OHHL ]b r8: [ LuxBox-LuxR-OHHL ]b c8 −→ [ LuxBox-LuxR-OHHL ,LuxR ]b r9: [ OHHL ]b c9 −→ OHHL [ ]b r10 : [ OHHL ]b c10 −→ [ ]b r11 : [ LuxR ]b c11 −→ [ ]b r12 : [ LuxR-OHHL ]b c12 −→ [ ]b r13 : OHHL [ ]b c13 −→ [ OHHL ]b r14 : [ OHHL ]e c14 −→ [ ]e The rules from r1to r12 are placed inside each bacterium, whereas rules r13 and r14 are placed inside the environment. The following set of parameters have been chosen for running simulations: c1= 2, c2= 2, c3= 9, c4= 1, c5= 10, c6= 2, c7= 250, c8= 200, c9= 50, c10 = 30, c11 = 20, c12 = 20, c13 = 0.1·N, c14 = 1. These stochastic rate constants (expressed in hours−1) have been taken or suggested by the literature (see [6] and references therein). 3.2 Multi-Compartmental Approach In this section we present the results obtained by simulating the Vibrio Fischeri colony model with the multi-compartmental approach. Two different kinds of membranes are to be initialized: the environment, and the internal membranes corresponding to bacteria. Moreover, all bacteria share the same set of rules. As previously said, every membrane evolves with the SSA dynamics, apart from the execution of communicating rules, because, in such cases, the rule can affect two membranes (i.e., the environment and one bacterium). It is clear that there are two kinds of communication rules: (1) the environment executes a communicating rule nondeterministically choosing a bacterium of the colony and sending one OHHL signal into it; (2) the bacterium applies a deterministic communicating rule always sending an OHHL signal to the environment. The multi-compartmental approach is suitable for the simulation of this model because it can trace the behavior of the single bacterium, as well as the behavior of the whole system. In the following, we present the results of the simulations for colonies consisting of 10, 300, 400, 500, and 600 bacteria. In Figure 1 the behavior of a single bacterium is shown, where the production of light (after about 20 hours of simulation time) is due to the fact that this one Stochastic Approaches in P Systems for Simulating Biological Systems 153 “guessed wrong” the concentration of the population and got up-regulated. But then, after sensing that the signal does not accumulate in the environment, it switched off its systems. Auto-quoration is a rare event but it is still possible due to the stochastic nature of the simulation algorithm, as the figure shows. In Figure 2 we show the behavior of a colony of 10 bacteria. During the simulation of 30 hours of real life, the signal inside the environment does not exceed 130 copies and there are at most 2 quorate bacteria at the same time. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 5 10 15 20 25 30 t [h] OHHL environment Quorated bacterium Fig. 1. Single Bacterium Behavior In Figures 3, 4, 5 and 6 we represent the behaviors of the other colonies (namely, with 300, 400, 500 and 600 bacteria) studied with this approach; it is possible to see that the OHHL signal starts to accumulate at the beginning of the simulation and also the number of quorate bacteria increases until the signal in the environment reaches a steady state. In Figure 7 the number of OHHL signal in the environment and the number of quorate bacteria of all the simulations are shown. It is possible to see that, as the number of the bacteria of the colony increases, the number of the quorate bacteria increases as well, but the number of OHHL signal inside the environment is about the same for all the simulations. This is due to the fact that in each bacterium the probability to send out a signal does not depend on the whole number Nof bacteria in the colony, but at the steady state they will have an internal common average concentration of OHHL, since they share the same constants. On the contrary, in the environment the probability to send a signal inside a bacterium is proportional to N(see the definition of c13). The net effect is that, at the steady state, the ratio between these probabilities is independent from the dimension of the colony. 160 P. Cazzaniga, D. Pescini, F.J. Romero-Campero, D. Besozzi, G. Mauri 0 1 2 3 4 5 6 7 8 0 200 400 600 800 1000 OHHL Quorated Fig. 12. 10 Bacteria Colony 0 50 100 150 200 250 300 0 500 1000 1500 2000 OHHL Quorated Fig. 13. 300 Bacteria Colony time of every membrane could be). However, using this approach we can share out the load of work between different nodes of a cluster. In this way, it is possible to simulate larger colonies in a smaller time, compared with the multi-compartmental approach. Stochastic Approaches in P Systems for Simulating Biological Systems 161 0 50 100 150 200 250 300 350 400 0 500 1000 1500 2000 OHHL Quorated Fig. 14. 400 Bacteria Colony 0 50 100 150 200 250 300 350 400 450 500 0 500 1000 1500 2000 OHHL Quorated Fig. 15. 500 Bacteria Colony 4 Conclusion and Future Work In this work we have shown that the stochastic strategies here presented are able to reproduce the bacterial mechanism of quorum sensing, which describes how each bacterium senses the presence of the other members of the colony. 162 P. Cazzaniga, D. Pescini, F.J. Romero-Campero, D. Besozzi, G. Mauri 0 100 200 300 400 500 600 0 500 1000 1500 2000 OHHL Quorated Fig. 16. 600 Bacteria Colony 0 100 200 300 400 500 600 0 500 1000 1500 2000 OHHL Quorated OHHL Quorated OHHL Quorated OHHL Quorated Fig. 17. Comparison between the simulations The main difference between the results obtained using the multi-mompartmental and the DPP approaches regards the simulation of the 10 bacteria colony. In the first case (Figure 2), the simulated colony behaves in the right way, that is, at most 2 bacteria got quorate (because they guessed wrong the size of the colony) and the OHHL signal does not accumulate in the environment. While Stochastic Approaches in P Systems for Simulating Biological Systems 163 using the DPPs (Figure 12), we obtained the luminescent behavior although the colony concentration is not enough to start to produce light, and the OHHL signal does not accumulate inside the environment, because the signal is immediately redistributed from the environment to the bacteria in a single parallel step. Looking at the simulations for the larger colonies performed with both approaches, it is possible to see that the number of quorate bacteria of the colonies increases as the cardinality of the colonies increases, as expected. Using the multicompartmental approach we can observe that the OHHL signal starts to accumulate inside the environment, and then it reaches a steady state value common to all the simulated colonies. The number of quorate bacteria follows the same dynamic of the signal. With the DPPs, the OHHL signal and the concentration of quorate bacteria do not share the same behavior. A smaller “constant” (average) amount of OHHL signal is present inside the environment in all the colonies, because of the aforementioned parallel redistribution of the signal, while the number of quorate bacteria increases at the beginning of the evolution and then reaches a steady state. At the moment the great advantage in using the Multi-Compartmental strategy is the defined time stream for the whole simulated system, which allows a quantitative description of the model evolution. The DPP strategy is still at a qualitative description of the system evolution, though the parallel design of this strategy leave room for large improvement (it is currently under deep development) and it is very promising for its computational performances and its scalability. For instance, the introduction of mute rules in the DPPs (i.e., A→A) allows to simulate different “levels” of parallelism. The 1 rule DPP and the DPP represent the sequential and the maximal parallel approaches, respectively, though between them there exist different levels of parallelism that could be closer to reality and thus it is worthwhile to investigate. In the case study of Vibrio Fischeri, one possible improvement for the DPP model would be to add a mute rule like OHHL →(OHHL, here) inside the environment, by means of which we can simulate the different levels of parallelism just by changing the rate constant of that rule. Moreover, adding mute rules for every symbol of the alphabet in every membrane, it is easy to simulate the whole system exploiting the parallel processes of the DPPs for every compartment. References 1. F. Bernardini: Membrane Systems for Molecular Computing and Biological Modelling. Doctoral Thesis, The University of Sheffield, Department of Computer Science, November 2005. 2. F. Bernardini, M. Gheorghe, N. Krasnogor, R.C. Muniyandi, M.J. P´erez-Jim´enez, F.J. Romero-Campero: On P systems as a modelling tool for biological systems. In Pre-Proceedings of the 6th International Workshop on Membrane Computing (WMC6) (R. Freund, G. Lojka, M. Oswald, Gh. P˘aun, eds.), TU Vienna, July 2005, 193–213. 164 P. Cazzaniga, D. Pescini, F.J. Romero-Campero, D. Besozzi, G. Mauri 3. D.T. Gillespie: Exact stochastic simulation of coupled chemical reactions. Journ. Phys. Chem., 81 (1977), 2340–2361. 4. T.C. Meng, S. Somani, P. Dhar: Modelling and simulation of biological systems with stochasticity. In Silico Biology, 4 (2004). 5. Gh. P˘aun: Membrane Computing. An Introduction. Springer-Verlag, Berlin, 2002. 6. M.J. P´erez-Jim´enez, F.J. Romero-Campero: Modelling Vibrio Fischeri’s behaviour using P systems. Systems Biology Workshop, ECAL 2005, Sept. 2005. 7. M.J. P´erez-Jim´enez, F.J. Romero-Campero: P systems, a new computationl modelling tool for Systems Biology. Transactions on Computational Systems Biology, to appear. 8. D. Pescini, D. Besozzi, C. Zandron, G. Mauri: Analysis and simulation of dynamics in probabilistic P systems. In Proceedings of DNA11 - 11th International Meeting on DNA Computing (N. Pierce, A. Carbone, eds.), LNCS, to appear. 9. D. Pescini, D. Besozzi, G. Mauri, C. Zandron: Dynamical probabilistic P systems. International Journal of Foundations of Computer Science, 17, 1 (2006), 183–204. 10. D. Pescini, D. Besozzi, G. Mauri: Investigating local evolutions in dynamical probabilistic P systems. Proceedings of SYNASC 2005–TAPS (Workshop on Theory and Applications of P Systems), IEEE Computer Press, to appear. 11. The MPI standard website http://www.mcs.anl.gov/mpi/index.html. 12. The P systems web page http://psystems.disco.unimib.it/. 13. The Systems Biology Markup Language web page http://www.sbml.org.