Membrane systems and stochastic simulation algorithms for the modelling of biological systems

Membrane Computing is a branch of computer science that was born after the introduction of Membrane Systems (or P systems) by a seminal paper by Gh. Paun. Membrane systems are computing devices inspired by the structure and functioning of living cells as well as from the way the cells are organized in tissues and higher order structures. The aim of membrane computing is to abstract computing ideas and models imitating these products of natural evolution. A typical membrane system is composed by a number of regions surrounded by membranes; regions contains multisets of objects (molecules) and rules (cellular processes) that specify how objects must be re-written and moved among regions. In spite of the fact that the initial primary goal of membrane systems concerned computability theory, the properties of membrane systems such as compartmentalisation, modularity, scalability/extensibility, understandability, programmability and discreteness promoted their use for an important task of the current scientific research: the modelling of biological systems (the topic “systems biology, including modelling of complex systems” has now appeared explicitly in the Seventh Framework Programme of the European Community for research, technological development and demonstration activities). To accomplish this task some features of membrane systems (such as nondeterminism and maximal parallelism) have to be mitigated while other properties have to be considered (e.g. description of the time evolution of the modelled system) to ensure the accurateness of the results gained with the models. Many approaches for the modelling and simulation of biological systems exist and can be classified according to features such as continuous/discrete, deterministic/stochastic, macroscopic/mesoscopic/microscopic, predictive/explorative, quantitative/qualitative and so on. Recently, stochastic methods have gained more attention since many biological processes, such as gene transcription and translation into proteins, are controlled by noisy mechanisms. Considering the branch of modelling focused at the molecular level and dealing with systems of biochemical processes (e.g. a signalling or metabolic pathway inside a living cell), an important class of stochastic simulation methods is the one inspired by the Gillespie's stochastic simulation algorithm (SSA). This method provides exact numerical realisations of the stochastic process defined by the chemical master equation. A series of methods (e.g. next reaction method, tau leaping, next subvolume method) and software (StochKit and MesoRD), belonging to this class, were developed for the modelling and simulation of homogeneous and/or reaction-diffusion (mesoscopic) systems. A stochastic approach that couples the expressive power of a membrane system (and more precisely of Dynamical Probabilistic P systems or DPPs) with a modified version of the tau leaping method in order to quantitatively describe the evolution of multi-compartmental systems in time is the tau-DPP approach. Both current membrane systems variants and stochastic methods inspired by the SSA lack the consideration of some properties of living cells, such as the molecular crowding or the presence of membrane potential differences. Thus, the current versions of these formalisms and computational methods do not allow to model and simulate all those biological processes where these features play an essential role. A common task in the field of stochastic simulations (mainly based on numerical rather than analytical solutions) is the repetition of a large number of simulations. This activity is required, for example, to characterise the dynamics of the modelled system and by some parameter estimation or sensitivity analysis algorithms. In this thesis we extend the tau-DPP approach taking into account additional properties of living cells in order to expand tau-DPPs modelling (and simulation) capabilities to a broader set of scenarios. Within this scope, we also exploit the main European grid computing platform as a computational platform usable to compute stochastic simulations, developing a framework specific to this purpose, able to manage a large number of simulations of stochastic models. In our formalism, we considered the explicit modelling of both the objects' (or molecules) and membranes' (or compartments) volume occupation, as mandated by the mutual impenetrability of molecules. As a consequence, the dynamics of the system are affected by the availability of free space. In living cells, for example, molecular crowding has important effects such as anomalous diffusion, variation of reaction rates and spatial segregation, which have significant consequences on the dynamics of cellular processes. At a theoretical level, we demonstrated that the explicit consideration of the volume occupation of objects and membranes (and their consequences on the system's evolution) does not reduce the computational universality of membrane systems. We achieved this aim showing that is it possible to simulate a deterministic Turing machine and that the volume required by the membrane systems that carry out this task is a linear function of the space required by the Turing machine. After this, we presented a novel version of both membrane systems (designated as Stau-DPPs) and stochastic simulation algorithm (Stau-DPP algorithm) considering the property of mutual impenetrability of molecules. In addition, we made the communication of objects independent from the system's structure in order to obtain a strong expressive power. After showing that the Stau-DPP algorithm can accurately reproduce particle diffusion (in a comparison with the heat equation), we presented two test cases to illustrate that Stau-DPPs can effectively capture some effects of crowding, namely the reduction of particle diffusion rate and the increase of reaction rate, considering a bidimensional discrete space domain. We presented also a test case to illustrate that the strong expressive power of Stau-DPPs allows the modelling and simulation (by means of the Stau-DPP algorithm) of processes taking place in structured environments; more precisely, we modelled and simulate the diffusion of molecules enhanced by the presence of a structure resembling the role of a microtubule (a sort of “railway” for intracellular trafficking) in living cells. Subsequently, we further extended Stau-DPPs and the respective evolution algorithm to explicitly consider the membrane potential difference and its effect over charged particles and voltage gated channel (VGC, a particular type of membrane protein) state transitions. In fact, the membrane potential difference exhibited by biological membranes plays a crucial role in many cellular processes (e.g. action potential and synaptic signalling cascades). Similarly to what we did for the Stau-DPPs, we presented the novel version of both the membrane systems (designated as EStau-DPPs) and the stochastic simulation algorithm (EStau-DPP algorithm) to capture the additional properties we had considered. In order to describe the probability of charged particle diffusion in a discrete space domain, we defined a propensity function starting from the deterministic and continuous description of charged particle diffusion due to an electric potential gradient. We showed by means of a focused test case that a model for ion diffusion between two regions, in which the number of ions is maintained at two different constant values and where an electric potential difference is available, correctly reaches the expected state as predicted by the Nernst equation. To describe the probability of transition between two VGC states, we derived a propensity function taking into consideration the Boltzmann-Maxwell distribution. We considered a model describing the state transitions of a VGC and we showed that the model predictions are in close agreement with the experimental data collected from literature. Lastly, we presented the framework to manage a large number of stochastic simulations on a grid computing platform. While creating this framework, we considered the parameter sweep application (PSA) approach, in which an application is run a large number of times with different parameter values. We ran a set of PSAs concerning the simulations of a stochastic bacterial chemotaxis model and the computation of the difference between the dynamics of one of its components (as a consequence of model parameter variation) compared to a reference dynamics of the same component. We then used this set of PSAs to evaluate the performance of the EGEE project's grid infrastructure (Enabling Grid for the E-sciencE). On the one hand, the EGEE grid proved to be a useful solution for the distribution of PSAs concerning the stochastic simulations of biochemical systems. The platform demonstrated its efficiency in the context of our middle-size test, and considering that the more intensive the computation, the more scalable the infrastructure, grid computing can be a suitable technology for large scale biological models analysis. On the other hand, the use of a distributed file system, the granularity of the jobs and the heterogeneity of the resources can present issues. In conclusion, in this thesis we extended previous membrane systems variants and stochastic simulation methods for the analysis of biological systems, and exploited grid computing for large scale stochastic simulations. Stau-DPPs and EStau-DPPs (and their respective algorithms to calculate the temporal evolution) increase the set of biological systems that can be investigated \textit{in silico in the context of the stochastic methods inspired by the SSA. In fact, compared to its precursor approach (tau-DPPs), Stau-DPPs allow the stochastic and discrete analysis of crowded systems, structured geometries, while EStau-DPPs also take into account some electric properties (membrane electric potential and its consequences), enabling, for example, the modelling of cellular signalling systems influenced by the membrane potential. In future, we plan to improve both the formalisations and the algorithms that we presented in this thesis. For example, Stau-DPPs can not model and simulate objects bigger than a single compartment, which conversely can be convenient for the analysis of big crowding agents in a tightly discretised space domain; instead, EStau-DPPs are, for instance, currently limited to the modelling of systems composed by two compartments separated by a boundary that can be assumed to act as a capacitor (e.g biological membranes). Moreover, we plan to optimize the parallel (MPI) implementation of both the Stau-DPP and EStau-DPP algorithms, which are presently based on a one-to-one relationship between processes and compartments, a limiting factor for the simulation of discrete spaces composed by a high number of compartments. Lastly, as grid computing demonstrated to be a useful approach to handle a large number of simulations, we plan to develop a solution to handle the simulations required in the context of sensitivity analysis.