Peter Rashkov

A Model of Oscillatory Protein Dynamics in Bacteria

Spatial oscillations of proteins in bacteria have recently attracted much attention. The cellular mechanism underlying these oscillations can be studied at molecular as well as at more macroscopic levels. We construct a minimal mathematical model with two proteins that is able to produce self-sustained regular pole-to-pole oscillations without having to take into account molecular details of the proteins and their interactions. The dynamics of the model is based solely on diffusion across the cell body and protein reactions at the poles, and is independent of stimuli coming from the environment. We solve the associated system of reaction–diffusion equations and perform a parameter scan to demonstrate robustness of the model for two possible sets of the reaction functions.

Pattern Formation in a Reaction-Diffusion System with a Singularity

Systems of reaction-diffusion equations are widely used to model morphogenesis (organisation of forms and patterns in living organisms) in different biological contexts. A particular application describes the formation of animal coat patterns and distribution of structures on the epidermis. The patterns of interest are either an arrangement of local maxima or a stripe-like distribution of chemical concentrations over the domain of interest that persist over time. In mathematical terms, a pattern is a spatially inhomogeneous solution of the reaction-diffusion system which is asymptotically stable. The common feature of systems displaying pattern formation is the presence of a steady state that is asymptotically stable to spatially-homogeneous perturbations but asymptotically unstable to spatially-inhomogeneous perturbations.We study whether it is possible to relax this assumption with an analysis of a reaction-diffusion system with a singularity that is a slight modification of a model for hair follicle spacing . We prove existence of global solutions for the reaction-diffusion system and examine stability of spatially inhomogeneous stationary solutions.

A model for spatio-temporal dynamics in a regulatory network for cell polarity.

Cell polarity in Myxococcus xanthus is crucial for the directed motility of individual cells. The polarity system is characterised by a dynamic spatio-temporal localisation of the regulatory proteins MglA and MglB at opposite cell poles. In response to signalling by the Frz chemosensory system, MglA and MglB are released from the poles and then rebind at the opposite poles. Thus, over time MglA and MglB oscillate irregularly between the poles in synchrony but out of phase. A minimal macroscopic model of the Mgl/Frz regulatory system based on a reaction-diffusion PDE system is presented. Mathematical analysis of the steady states derives conditions on the reaction terms for formation of dynamic localisation patterns of the regulatory proteins under different biologically-relevant regimes, i.e. with and without Frz signalling. Numerical simulations of the model system produce either a stationary pattern in time (fixed polarity), periodic solutions in time (oscillating polarity), or excitable behaviour (irregular switching of polarity).

Modelling Protein Oscillations in Myxococcus xanthus

Spatio-temporal oscillations of proteins in bacterial cells play an importantrole in fundamental biological processes. Motility of the rod-shaped bacterium Myxococcus xanthus is due to two motility systems: an A-motilitysystem and a type-4 pili system [1]. Both motility systems depend on the correct localisation of regulatory proteins at the cell poles which set up apolarity (front-to-back) axis. The oscillatory motion of the individual cel lresults from dynamic inversion of the polarity axis due to a spatio-temporal oscillation of the regulatory proteins between the cell poles. A mathematical framework for a minimal macroscopic model is presented which produces self-sustained oscillations of the protein concentrations. The mathematicalmodel is based on a reaction-diusion system and is independent of external triggers. Necessary conditions on the reaction terms leading to oscillating solutions are derived theoretically. Several possible cases are studied based on dierent rates of interaction between the regulatory proteins. The interaction laws are then chosen according to mathematical analysis to produce dierent spatio-temporal oscillation patterns [2]. The dierent scenarios are numerically tested for robustness against parameter variation. Finally,possible extensions of the model will be addressed.This is joint work with B. A. Schmitt, S. Dahlke, P. Lenz, L. Sgaard-Andersen. This work has been supported by the Centre for Synthetic Microbiologyin Marburg, promoted by the LOEWE Excellence Program ofthe state of Hessen, Germany References [1] P. Lenz, L. Sgaard-Andersen, Temporal and spatial oscillations in bac-teria, Nat. Rev. Microbiol. 9 565{577, 2011. [2] P. Rashkov, B.A. Schmitt, L. Sgaard-Andersen, P. Lenz, S. Dahlke, Amodel of oscillatory protein dynamics in bacteria, Bull. Math. Biol. 742183{2203, 2012.