Our approaches lie in a number of different areas, but the aims in all cases lie in developing novel mathematical and computational methods to enable us to better our understanding of biological systems. Specific areas in which we are interested are outlined below.

### Coarse graining approaches

Whilst modelling biological systems, we are often presented with information at a fine level of description (stochastic, individual-based), while we want to study the behaviour at a macroscopic coarse-grained (continuum, population) level. Computational methods for obtaining an approximation to the macroscopic evolution of the system without explicitly obtaining macroscopic equations have been developed in collaboration with Professor Ioannis Kevrekidis (Princeton University) - the so-called equation-free methods. Equation-free methods can be viewed as a computational superstructure wrapped around a microscopic model of a biological system. They have been applied to problems where the macroscopic dynamics are either deterministic, or stochastic. The efficiency of equation-free methods can be further improved if one knows some extra information about the problem. For example, in some morphogenesis applications, we are able to easily derive approximate mean-field partial differential equation which can be used to design the so-called equation-assisted methods. Equation-free methods for problems with continuous symmetries (travelling or self-similar behaviour) can be designed more efficiently and accurately if one takes the corresponding symmetry into account.

Other work in this area involves investigation of the role of various cues in directing cell migration on growing domains. We study individual-based models for describing cell movement and domain growth, and use moment-closure methods to draw correspondence with a macroscopic-level PDE describing the evolution of cell density. The individual-based models are formulated in terms of random walkers on a lattice; domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs.

We also study approaches to coarse graining off-lattice cell-based models, which have become an increasingly used tool in the modelling of cell populations. In the off-lattice framework one specifies cell-cell force interactions at the individual cell scale and, typically, uses numerical simulations to investigate model behaviour. However, in some biological systems of interest, such as tumour spheroids or intestinal crypts, there are often many thousands of cells in a given simulation, hence relating model behaviour to the underlying model parameters can be problematic. In certain contexts it is possible to use the governing equations at the discrete scale to derive governing PDEs that describe how cell densities vary at the population scale. Analysis of the governing PDEs then allows one to describe qualitative features of the simulation results and efficiently explore parameter space.

Please contact Professor Radek Erban or Dr Ruth E. Baker for more details.

#### Key references in this area

**A. Singer, R. Erban, I. Kevrekidis and R. Coifman**(2009). Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps. Proc. Natl. Acad. Sci. USA 106:16090-16095. (preprint)**R. E. Baker, C. A. Yates and R. Erban**(2010). From microscopic to macroscopic descriptions of cell migration on growing domains. Bull. Math. Biol. 72:719-762. (eprints)**P. J. Murray, A. Walter, A. G. Fletcher, C. M. Edwards, M. J. Tindall and P. K. Maini**(2011). Comparing a discrete and continuum model of the intestinal crypt.*Phys. Biol.*8:026011. (eprints)**R. E. Baker and M. J. Simpson**(2012). Models of collective cell motion for cell populations with different aspect ratio: diffusion, proliferation and travelling waves.*Physica A*391(14):3729–3750. (eprints)

### Multiscale modelling of biological systems

Systems biology is about gaining a quantitative and predictive understanding of how function emerges at the organism level due to dynamic processes occurring at multiple lower scales involving complex interactions between sub-systems and sub-components. This implies the use of mathematical and computational models ranging from gene through to organ or whole system levels incorporating dynamic descriptions of multiple chemical, biochemical and physical processes. In collaboration with experimentalists based at the Oxford Centre for Integrative Systems Biology (OCISB) and elsewhere, we are building integrative models of well-defined biological systems, with a view to explaining (quantitatively) how specific functions (motion, cell division, death, particular response to particular stimulus. etc.) arise at the system level. As part of this work we are contributing to the development of a general framework for the simulation of multi-scale, computationally demanding problems arising in systems biology.

More information on this can be found at the Chaste project website.

#### Key references in this area

**J. Pitt-Francis, P. Pathmanathan, M. O. Bernabeu, R. Bordas, J. Cooper, A. G. Fletcher, G. R. Mirams, P. J. Murray, J. M. Osborne, A. Walter, S. J. Chapman, A. Garny, I. M. M. van Leeuwen, P. K. Maini, B. Rodríguez, S. L. Waters, J. P. Whiteley, H. M. Byrne and D. J. Gavaghan**(2009). Chaste: A test-driven approach to software development for biological modelling. Comp. Phys. Comm. 180:2452-2471. (eprints)

### Stochastic reaction-diffusion processes

Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species, for example, problems in developmental biology, genes and enzymes. Such reaction-diffusion processes can be mathematically modelled using either deterministic partial-differential equations or stochastic simulation algorithms. Stochastic models provide a more detailed understanding of the reaction-diffusion processes. Such a description is often necessary for the modelling of biological systems where small molecular abundances of some chemical species make deterministic models inaccurate or even inapplicable. Stochastic models are also necessary when biologically observed phenomena depend on stochastic fluctuations, for example, switching between two favourable states of the system. There are several stochastic (molecular-based or mesoscopic) approaches for modelling chemical reactions and molecular diffusion. Coupling models of these two fundamental processes together offers several challenging mathematical problems. The goal of this research area is to design reliable, correct and efficient methods for the stochastic simulation of reaction-diffusion processes in biology.

Please contact Professor Radek Erban for more details. Lecture notes on stochastic modelling of reaction-diffusion processes can be found here.

Key references in this area

**R. Erban and S. J. Chapman**(2009). Stochastic modelling of reaction-diffusion processes: algorithms for bimolecular reactions. Phys. Biol. 6:046001. (eprints)**R. Erban and S. J. Chapman**(2007). Reactive boundary conditions for stochastic simulations of reaction-diffusion processes. Phys. Biol. 4:16-28. (eprints)**M. Flegg, S.J. Chapman and R. Erban**(2012). The two-regime method for optimizing stochastic reaction–diffusion simulations.*J. Roy. Soc. Interface*. 9:859-868.**J. Lipkova, K. Zygalakis, J. Chapman and R. Erban**(2011). Analysis of brownian dynamics simulations of reversible bimolecular reactions.*Siam J. Appl. Math.*71(3):714-730. (eprints)

### Stochastic simulation algorithms

The stochastic simulation algorithm was introduced by Gillespie and in a different form by Kurtz. Part of the focus of this research is to come up with efficient formulations of the stochastic simulation algorithm which will be generally applicable to a variety of reaction systems. There have also been many attempts at accelerating the algorithm without deviating from the behavior of the simulated system. The crux of the explicit τ -leaping procedure is the use of Poisson random variables to approximate the number of occurrences of each type of reaction event during a carefully selected time period, τ . This method is acceptable providing the leap condition, that no propensity function changes “significantly” during any time-step, is met. Using this method there is a possibility that species numbers can, artificially, become negative. The other aspect of this research is to attempt to gain efficiency from the algorithms by careful classification of critical reactions or refactoring of τ -leaping methods including considering methods like multiscale.

#### Key references in this area

**C. A. Yates and K. Burrage**(2011). Look before you leap: A confidence-based method for selecting species criticality whilst avoiding negative populations in tau-leaping. J. Chem. Phys. 134, 084109. (eprints)**S. Cotter, K. Zygalakis, I. Kevrekidis and R. Erban**(2011). A constrained approach to multiscale stochastic simulation of chemically reacting systems.*J. Chem. Phys*. 135(9):094102.