Mathematics and Computation for the Systems Biology of Cells UvA/SCS

The aim of the project is to develop, implement, and validate mathematical and computational techniques for the systems biology of the cell. Biologists and mathematicians together will formulate realistic mathematical models of metabolic and regulatory networks including intrinsic spatial non-homogeneity. Depending on the cellular phenomenon considered, models and methods of appropriate temporal and spatial scales will be developed and can then be applied: models in the form of ordinary differential equations and methods for system reduction; multi-adaptive computational methods for partial differential equations (PDEs) for moderate spatial and temporal variability within a cell or an organelle; particle models describing the interaction of individual molecules and computational methods for the evaluation of the dynamic behavior; and methods for integration of these different approaches into a single simulation.

The planned outcome of the project are computational and mathematical algorithms, implemented in auto-adaptive computational models, and simulation results for the functioning of living cells. Research focus

1. system reduction techniques for ordinary differential equations (ODEs) of the type that arises in chemical networks (simplification and modularization in the chemical `dimension')
2. particle-based methods for modelling of features with high spatial variability or low number of molecules
3. multi-adaptive numerical methods for the efficient solving of reaction-diffusion PDEs with varying spatial and temporal scales, and space dependent chemical schemes
4. methods that allow 1-3 to be integrated into a single simulation, in order to take advantage of simplification and modularization wherever and whenever possible

The focus within the UvA/SCS group will be on the topics 2 and 4. In this part of the project we will develop a three-dimensional particle model for bulk and surface reaction-diffusion. Examples of relevant systems are reactions at the surface of cellular membranes. For the reaction-diffusion behaviour we will use a Lattice-Boltzmann approach (LBE) or, if fluctuations and correlations cannot be neglected, a Lattice Gas Automaton (LGA). An important challenge is the matching of microscopic coefficients to macroscopic measurements; in addition, the inclusion of a large number of reactive species in an LGA model presents practical difficulties.

The particle models developed in this work package are destined to be modules in a larger simulation: Most of the cell will be described by a PDE (or even an ODE), but parts of the cell will be represented by a particle model where this is necessary to obtain the required resolution. At the interface between the PDE-based and particle-based regions concentrations and fluxes should be continuous at the macroscopic level. We will investigate two representations of this model coupling, one with a surface interface and one with a zone of overlap.

For the ultimate goal of a self-organizing simulation, which treats different parts of the cell with different methods according to local requirements, a measure is needed to quantify the degree of local spatial heterogeneity and therefore the degree of necessity of a particle-based representation. To determine such a measure we will make comparisons of the performance of PDE and particle-based methods using the case study of protein patches in membranes. In the PDE-based approach a fine grid near the membrane-bound protein will be required, while in the particle-based modelling techniques (LBE and LGA) the reactions in the cytosol will be computationally expensive. We expect to deduce a quantitative relation between the degree of protein aggregation and the relative advantage of a particle-based method.

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