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Stochastic and Multiscale Modelling in Biology

Final Report Summary - STOANDMULMODINBIO (Stochastic and Multiscale Modelling in Biology)

The main goals of this ERC-funded project were the development, analysis and application of mathematical and computational methods in biology, with a focus on problems in which stochastic effects and multiscale phenomena play a key role.

One class of methods which were investigated were algorithms for stochastic spatio-temporal modelling in cellular and molecular biology. There are two main approaches to stochastic reaction-diffusion modelling:

(1) compartment-based methods which discretize the space into (well-mixed) compartments;
(2) molecular-based methods which compute the behaviour of each individual biomolecule.

If we go from (1) to (2), we will increase the level of detail which is included in the model, but we will also increase the computational intensity of simulations. In some applications, microscopic detail is only required in a relatively small region. We developed a methodology which is able to use a detailed modelling approach (2) in localized regions of particular interest (in which accuracy and microscopic detail is important) and a less detailed model (1) in other regions in which accuracy may be traded for simulation efficiency. By investigating how models with various levels of detail can be used in different parts of the computational domain, we developed accurate and efficient methods for stochastic simulations of reaction-diffusion processes in biology. One of these methods was implemented in software package Smoldyn which is one of the most successful software projects for particle-based stochastic spatio-temporal (Brownian dynamics) simulations of intracellular processes, as demonstrated by its growing user community. By implementing multiscale simulation methods in Smoldyn, we increased both the impact of our methodological work and the applicability of Smoldyn to more complex problems. We also applied this methodology to modelling intracellular calcium dynamics and intracellular actin dynamics.

Another class of multiscale problems which we studied concerned models which have processes (reactions) occurring on different timescales. That is, there are some reactions which occur many times on a timescale for which others are unlikely to occur at all. Considering that the biochemical system is well-mixed, the standard method for simulating these systems is called the Gillespie stochastic simulation algorithm. It simulates every single reaction that occurs in the system, and thus its use for these types of chemical systems can become intractable. We developed an approach which does not simulate all reactions, but is still able to estimate interesting characteristics of a stochastic chemical model (for example, the time to switch between favourable states of the system, or an estimation of the period of noise induced oscillations). Improvements of the computational efficiency of stochastic simulations are important whenever a modeller wants to study the dependence of a system behaviour on its parameters. We introduced and investigated approaches based on numerical solutions of high-dimensional partial differential equations for parametric analysis of stochastic chemical systems.

We have also investigated collective behaviour of several systems of locally interacting particles (individuals). The application areas included interacting bacteria, swarming locusts, ions in ion channels, liquid crystal molecules and swarm robotics. We investigated connections between microscopic (individual-based) and macroscopic (population-level) models. We were interested in coupling these models together to design efficient multiscale algorithms. For example, in the case of bacteria, we studied hybrid models of chemotaxis where bacteria are modelled as particles and extracellular molecules are described by mean-field partial differential equations for concentrations. We analyzed the formation of travelling waves and found a qualitative difference in behaviour between the hybrid model and the corresponding continuum model due to stochastic effects.