| Abstract |
The emergence of patterns in space and time is a fundamental phenomenon displayed by all forms of life. Patterns can emerge from the interactions of chemical, diffusive and mechanical processes that occur within the same biological environment, such as the cell cytoplasm or a tissue. Biological systems however are often compartmentalised, i.e. subdivided into smaller, encapsulated volumes that can also dynamically evolve by dividing, fusing or exchanging molecules. Cells, organelles, vesicles and biomolecular condensates are all examples of how compartmentalisation allows the organisation of micro-environments where specialized biochemical processes can take place. Compartmentalisation couples the discrete and noisy nature of these biochemical processes with that of the compartment interactions. This gives rise to a joint, multiscale dynamics that can both mitigate and amplify fluctuations, shaping the heterogeneity of the compartment population towards what is necessary for its biological function. The modelling paradigm of stochastic reaction networks allows us to capture both the discreteness and stochasticity of biological systems, providing the foundations for their theoretical and computational analysis. Previous work from our group presented a theoretical framework that generalises stochastic reaction networks to include compartmentalisation and account for its effects. This makes it possible to represent processes occurring across the internal and compartmental scales, as well as their interactions, in a unified way. The framework also allows deriving equations that directly describe the stochastic evolution of the moments of a compartment population. This is of particular importance, as the Monte Carlo estimation of population summary statistics from ensembles of explicit stochastic simulations can have a high, or even prohibitive, computational cost even on simple models. Practically, however, these population moment equations can be derived and solved only for a small class of simple systems: models with non-polynomial propensity functions cannot give rise to self-enclosed systems of moment equations, and those with propensities that are polynomial but non-linear give rise to systems of equations of infinite size, requiring ad-hoc moment closure approximations to be employed, which can be difficult to derive in practice. In this thesis I first present how the formalism of stochastic reaction networks, together with its computational techniques, can be applied to model the structure of the mitotic spindle as a pattern in the organization of populations of microtubules. This organization is based on the interplay between chemical reactions, diffusion and mechanical interactions, and occurs in a single volume. A spatial discretization is used to consistently model spatial processes, such as diffusion, advection and the long-range interactions that are induced by single microtubules along their considerable length. While this system is not compartmentalised, it constitutes an example of how the stochastic reaction network formalism can be used to model phenomena that go beyond simple chemical reactions. I then tackle the problem of approximating the moment equations of compartmentalised stochastic reaction networks. I present LPAC, a principled and systematic method for deriving moment closure approximations for a broad class of compartmentalised systems, including propensities that are non-polynomial with respect to the compartment content. I show that for the considered class of systems, the moment equations involve expectations over functions that factorize into two parts, one depending on the molecular content of the compartments (content-dependent) and one depending on the compartment number distribution (state-dependent). The method exploits this structure and approximates both functions with suitable polynomial expansions, leading to a closed system of moment equations. Furthermore, I show how this method can be easily applied in software, yielding a tool that can derive and approximate population moment equations in a fully automated manner. I perform numerical experiments to validate LPAC against several compartmentalised models that are inspired by cell populations and organelle networks. I find that its accuracy depends on the characteristics of the population distribution, with high accuracy on narrow unimodal populations, while wider and multimodal populations lead to errors and numerical failures. To address this limitation, I then introduce a quadrature-based scheme to obtain better accuracy in the approximation of the content-dependent function and its expectation. I present a method that adaptively updates quadrature nodes and basis functions to track the evolution of the compartment population solely through the knowledge of a specific set of summary statistics. I finally present how this method can be integrated into a quadrature-LPAC (qLPAC). My numerical experiments validate its ability to adaptively track the expected evolution of the compartment population, and to approximate its population moments, even through a transition from unimodal to bimodal. In closing my thesis, I present how I leverage the mathematical structure of population moment equations in the design of ALPACa, a software toolbox that takes concise symbolic descriptions of compartmentalised stochastic reaction networks and automatically produces efficient approximations of their population moments, with a simple and intuitive interface. The design of ALPACa allows not only to easily switch between the LPAC and qLPAC methods, but also to easily develop and integrate new approximation techniques within the same theoretical framework. In conclusion this thesis explores how the the theoretical framework of stochastic reaction networks, and the related computational techniques, can be used to study biological phenomena that give rise to patterns, both in space and in the distribution of a population. It also shows that biological systems and their patterns can challenge existing computational techniques, inspiring the development of new methods and ultimately help us push the boundary, even if just by a tiny amount, of our scientific computing knowledge. |