OCCAM Common Room (RI2.28)

Mathematical models of chemotactic movement of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular signaling chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s [Keller and Segel, J. Theor. Biol., 1971]. The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities that ar biologically unrealistic. Here we present a microscopic model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We show that this model permits travelling wave solutions and predicts the formation of other bacterial patterns such as radial and spiral streams. We also present connections of this microscopic model with macroscopic models of bacterial chemotaxis. This is joint work with Radek Erban, Benjamin Franz, Hyung Ju Hwang, and Kevin J.

Painter.

Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this talk we discuss efficient, scalable techniques for solving fractional-in-space reaction diffusion equations combining the finite element method with robust techniques for computing the fractional power of a matrix times a vector. We shall demonstrate the methods on a number examples which show the qualitative difference in solution profiles between standard and fractional diffusion models.

Actin polymerization in vivo is regulated spatially and temporally by a web of signalling proteins. We developed detailed physico-chemical, stochastic models of lamellipodia and filopodia, which are projected by eukaryotic cells during cell migration, and contain dynamically remodelling actin meshes and bundles. In a recent work we studied how molecular motors regulate growth dynamics of elongated organelles of living cells. We determined spatial distributions of motors in such organelles, corresponding to a basic scenario when motors only walk along the substrate, bind, unbind, and diffuse. We developed a mean field model, which quantitatively reproduces elaborate stochastic simulation results as well as provides a physical interpretation of experimentally observed distributions of Myosin IIIa in stereocilia and filopodia. The mean field model showed that the jamming of the walking motors is conspicuous, and therefore damps the active motor flux. However, when the motor distributions are coupled to the delivery of actin monomers towards the tip, even the concentration bump of G-actin that they create before they jam is enough to speed up the diffusion to allow for severalfold longer filopodia. We found that the concentration profile of G-actin along the filopodium is rather non-trivial, containing a narrow minimum near the base followed by a broad maximum. For efficient enough actin transport, this non-monotonous shape is expected to occur under a broad set of conditions. We also find that the stationary motor distribution is universal for the given set of model parameters regardless of the organelle length, which follows from the form of the kinetic equations and the boundary conditions.

Particle-based stochastic reaction-diffusion models have recently been used to study a number of problems in cell biology. These methods are of interest when both noise in the chemical reaction process and the explicit motion of molecules are important. Several different mathematical models have been used, some spatially-continuous and others lattice-based. In the former molecules usually move by Brownian Motion, and may react when approaching each other. For the latter molecules undergo continuous time random-walks, and usually react with fixed probabilities per unit time when located at the same lattice site.

As motivation, we will begin with a brief discussion of the types of biological problems we are studying and how we have used stochastic reaction-diffusion models to gain insight into these systems. We will then introduce several of the stochastic reaction-diffusion models, including the spatially continuous Smoluchowski diffusion limited reaction model and the lattice-based reaction-diffusion master equation. Our work studying the rigorous relationships between these models will be presented. Time permitting, we may also discuss some of our efforts to develop improved numerical methods for solving several of the models.

In this talk, I will introduce stochastic models to describe the state of the chemical networks using continuous-time Markov chains.
First, I will talk about the multiscale approximation method developed by Ball, Kurtz, Popovic, and Rempala (2006). Extending their method, we construct a general multiscale approximation in chemical reaction networks. We embed a stochastic model for a chemical reaction network into a family of models parameterized by a large parameter N. If reaction rate constants and species numbers vary over a wide range, we scale these numbers by powers of the parameter N. We develop a systematic approach to choose an appropriate set of scaling exponents. When the scaling suggests subnetworks have di erent time-scales, the subnetwork in each time scale is approximated by a limiting model involving a subset of reactions and species.

After that, I will briefly introduce Gaussian approximation using a central limit theorem, which gives a model with more detailed uctuations which may be not captured by the limiting models in multiscale approximations.

Next, we consider modeling of a chemical network with both reaction and diffusion.
We discretize the spatial domain into several computational cells and model diffusion as a reaction where the molecule of species in one computational cell moves to the neighboring one. In this case, the important question is how many computational cells we need to use for discretization to get balance between e ective diffusion rates and reaction rates both of which depend on the computational cell size. We derive a condition under which concentration of species converges to its uniform solution exponentially. Replacing a system domain size in this condition by computational cell size in our stochastic model, we derive an upper bound
for the computational cell size.

Finally, I will talk about stochastic reaction-diffusion models of pattern formation. Spatially distributed signals called morphogens influence gene expression that determines phenotype identity of cells. Generally, different cell types are segregated by boundary between
them determined by a threshold value of some state variables. Our question is how sensitive the location of the boundary to variation in parameters. We suggest a stochastic model for boundary determination using signaling schemes for patterning and investigate their effects on the variability of the boundary determination between cells.

Diffusive process with discontinuous coefficients provide significant computational challenges. We consider the solution of a diffusive process in a domain where the diffusion coefficient changes discontinuously across a curved interface. Rather than seeking to construct discretizations that match the interface, we consider the use of regularly-shaped meshes so that the interface "cuts'' through the cells (elements or volumes). Consequently, the discontinuity in the diffusion coefficients has a strong impact on the accuracy and convergence of the numerical method. We develop an adjoint based a posteriori error analysis technique to estimate the error in a given quantity of interest (functional of the solution). In order to employ this method, we first construct a systematic approach to discretizing a cut-cell problem that handles complex geometry in the interface in a natural fashion yet reduces to the well-known Ghost Fluid Method in simple cases. We test the accuracy of the estimates in a series of examples.

• Chong Luo - Microscopic models for planar bistable liquid crystal device
• Laura Gallimore - Modelling Cell Motility
• Yi Ming Lai - Stochastic Oscillators in Biology

• Thomas März - Calculus on surfaces with general closest point functions
• Jay Newby - Modeling rare events in biology
• Hugh McNamara - Stochastic parameterisation and variational multiscale

• Graham Morris - 'Topics in Voltammetry'
• James Lottes - 'Algebraic Multigrid for Nonsymmetric Systems'
• Amy Smith - 'Multi-scale modelling of blood flow in the coronary microcirculation'