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Abstract

Time is one of the main factors in any kind of real-life systems. When a certain system is analysed one is often interested in its evolution with respect to time. Various phenomena can be described using a form of time-dependency. The difference between load in call-centres is the example of time-dependency in queueing systems. The process of migration of biological species in autumn and spring is another illustration of changing the behaviour in time. The ageing process in critical infrastructures (which can result in the system component failure) can also be considered as another type of time-dependent evolution. Considering the variability in time for chemical and biological systems one comes to the general tasks of systems biology [9]. It is an inter-disciplinary study field which investigates complex interactions between components of biological systems and aims to explore the fundamental laws and new features of them. Systems biology is also used for referring to a certain type of research cycle. It starts with the creation a model. One tries to describe the behaviour in a most intuitive and informative way which assumes convenience and visibility of future analysis. The traditional approach is based on deterministic models where the evolution can be predicted with certainty. This type of model usually operates at a macroscopic scale and if one considers chemical reactions the state of the system is represented by the concentrations of species and a continuous deterministic change is assumed. A set of ordinary differential equations (ODE) is one of the ways to describe such kind of models. To obtain a solution numerical methods are applied. The choice of a certain ODE-solver depends on the type of the ODE system. Another option is a full description of the chemical reaction system where we model each single molecule explicitly operating with their properties and positions in space. Naturally it is difficult to treat big systems in a such way and it also creates restrictions for computational analysis. However it reveals that the deterministic formalism is not always sufficient to describe all possible ways for the system to evolve. For instance, the Lambda phage decision circuit [1] can be a motivational example of such system. When the lambda phage virus infects the E.coli bacterium it can evolve in two different ways. The first one is lysogeny where the genome of the virus is integrated into the genome of the bacterium. Virus DNA is then replicated in descendant cells using the replication mechanism of the host cell. Another way is entering the lytic cycle, which means that new phages are synthesized directly in the host cell and finally its membrane is destroyed and new phages are released. A deterministic model is not appropriate to describe this process of choosing between two pathways as this decision is probabilistic and one needs a stochastic model to give an appropriate description. Another important issue which has to be addressed is the fact that the state of the system changes discretely. It means that one considers not the continuous change of chemical species concentrations but discrete events occuring with different probabilities (they can be time-dependent as well). We will use the continuous-time Markov Population Models (MPMs) formalism in this thesis to describe discrete-state stochastic systems. They are indeed continuous- 1 time Markov processes, where the state of the system represents populations and it is expressed by the vector of natural numbers. Such systems can have innitely many states. For the case of chemical reactions network it results in the fact that one can not provide strict upper bounds for the population of certain species. When analysing these systems one can estimate measures of interest (like expectation and variance for the certain species populations at a given time instant). Besides this, probabilities for certain events to occur can be important (for instance, the probability for population to reach the threshold or the probability for given species to extinct). The usual way to investigate properties of these systems is simulation [8] which means that a large amount of possible sample trajectories are generated and then analysed. However it can be difficult to collect a sufficient number of trajectories to provide statistical estimations of good quality. Besides the simulation, approaches based on the uniformization technique have been proven to be computationally efficient for analysis of time-independent MPMs. In the case of time-dependent processes only few results concerning the performance of numerical techniques are known [2]. Here we present a method for conducting an analysis of MPMs that can have possibly infinitely many states and their dynamics is time-dependent. To cope with the problem we combine the ideas of on-the-y uniformization [5] with the method for treating timeinhomogeneous behaviour presented by Bucholz.