Math & Computational Core

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Theme D: Mathematical and Computational Biology Core Theme Leader: Nie Other Project Faculty: Lowengrub, Xin, Mjolsness Overview The equations and models describing the biological processes and interactions dealt with in Themes A-C are mainly nonlinear Partial Differential Equations (PDEs), with inclusion of stochastic effects or/and moving boundaries for some cases. Understanding the role of each biological process in a complex biological system requires us to study and analyze these equations in great detail. Because the questions we seek to answer typically involve quantitative system-level features, we face substantial challenges in analyzing those equations using current computational and analytical tools. For example, in order to investigate sensitivities of solutions to variations in input parameters computationally, one may need to perform millions of calculations to collect and analyze correlations between solution patterns and biological processes. We have performed such calculations to identify new strategies to achieve robustness of morphogen gradients at steady-state for a system involving one spatial dimension and one diffusing ligand (refs to our robustness paper). But for more complex systems involving temporal dynamics or/and higher (two and three) spatial dimensions, new and efficient numerical methods are needed. The first two projects in this theme focus on computational tools dealing with these types of systems. Most of the bio-chemical reactions proposed in previous themes have stiff reaction rate constants that consequently restrict the size of time-step for temporal updating. This is a bottle-neck for numerical simulations of most stiff reaction-diffusion equations. In the first project (D.1), we propose to develop efficient temporal algorithms for stiff reaction-diffusion equations in two- and three- spatial dimensions by reducing or even removing the timestep constraint associated with stiffness. The research plan is based on our recent success on developing semi-implicit algorithms for onedimensional systems and our preliminary studies on incorporation of this type of new method with twoand three- dimensional systems. Project A.1, a study on suppression of spatial noise, demands a large number of both temporal and steady-state calculations for PDEs with many diffusing ligands, and Project A.2, a study on developmental patterning of Zebrafish, requires speedy temporal simulations of reaction-diffusion equations in a three-dimensional geometry. Development of new temporal methods is critical to both projects in addition to Projects B.1, B.3, C.1 and C.2 that all involve similar types of equations.