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Dynamics of reaction-diffusion systems in non-homogeneous media Paola Lecca, Lorenzo Demattè and Corrado Priami The Microsoft Research – University of Trento Centre for Computational and Systems Biology Objective To model the dynamics of a stochastic reaction-diffusion system in a nonhomogeneous medium. A possible way to achieve this goal is to incorporate a model of diffusion into a biochemical stochastic simulation algorithm, as Gillespie algorithm. At the mesoscopic intra-cellular scale the parameters governing the kinetics of the diffusion have to be dependent on local variables, as solute concentration and frictional forces. Modelling reaction-diffusion systems A model of a reaction-diffusion systems consists of two parts o a set of biochemical reactions which produce, transform or remove chemical species o a mathematical description of the diffusion process The great majority of mesoscopic reaction-diffusion models of intracellular kinetics usually assume that the diffusion coefficient is constant over the time and the diffusion is so fast that all concentrations are maintained homogeneous in space. However, recent experimental data on intracellular diffusion constants show that this supposition is not necessarily valid even for small prokaryotic cells. Spatial effects Spatial effects are present in many biological systems. Some examples: o mRNA movement within the cytoplasm o Ash1 mRNA localization in budding yeast o morphogen gradients across egg-polarity genes Our description of diffusion processes in a highly structured and nonhomogenous medium starts from a generalization of the Fick’s law. Generalized Fick’s law (1) The driving force leading to diffusion is the Gibbs energy difference between regions of different concentrations. The chemical potential of a species is defined as the partial derivative of the Gibbs energy with respect to the concentration of the species where is the standard chemical potential, is the ideal gas constant, the absolute temperature and the chemical activity of the species . is the activity coefficient and is a reference concentration. Generalized Fick’s law (2) The flux defined as the number of number of moles of solute which pass through a small surface per unit time per unit area is where the diffusion coefficient given by Supposing that the chemical activity of the solute only weakly depends on the concentration of the other solutes we can assume that . Generalized Fick’s law (3) In 1D- space, the rate of change of concentration of species is given by where and due to diffusion and denote the concentration of the substance . at coordinate , and Generalized Fick’s law (4) The rate of diffusion of a substance at the mesh point where, by introducing the friction coefficient of species , , we have that (Chemical activity coefficient) is at mesh point (Second virial coefficient) , A case study: chaperone-assisted protein folding Protein folding, chaperone binding, and misfolded protein accumulation take place inhomogeneously in the space. The spatial distribution of chaperones in the cytoplasm may not be uniform, and consequently the distribution of correct and faulty proteins may be not uniform. In turn, the time evolution of spatial distribution of chaperones may affect the time evolution of the spatial distribution of faulty proteins. A view of the system In a 2D space, for simplicity. Reaction event Diffusion event is the diffusion coefficient of species i at time t = 0. is the number of species that diffuse is the waiting reaction time is the number of diffusion events is the number of reaction events At time t, the event (diffusion or reaction) that succeds to occur is the fastest one (the most probable), as in a Gillespie-like approach. Most probable diffusion direction Parameters of simulation Simulation space: a grid of thus consisting of 81 cells (each cell has size nm) is considered. A 2D diffusion model is simulated and a spatially homogeneous distribution of nascent_protein and an initial null concentration of right_protein in every cell are assumed. The density (expressed in number of molecules per ). Total simulation time: Simulation results (1) Time evolution of chaperone distribution Simulation results (2) Time evolution of ”misfolded_1” protein distribution Simulation results (3) Time evolution of ”misfolded_2” protein distribution Simulation results (4) Time evolution of distribution of correctly folded proteins. Spatial average correlation The spatial correlation between the proteins and chaperones has been monitored in terms of the quantity and are the concentration of proteins and chaperones, respectively. Correlation matrices (1) Matrices of correlation between chaperone and “misfolded 1” protein concentration. Correlation matrices (2) Matrices of correlation between chaperone and “misfolded 2” protein concentration. Correlation matrices (3) Matrices of correlation between chaperone and correctly folded protein concentration. Conclusions In non-homogenoues media, constant diffusion coefficients w. r. t. the time and space are rather more than exception than a rule in living cells. This work provides a theoretical derivation of the molecular origin of these parameters. The results obtained with the simulation of the process of chaperoneassisted protein folding are in agreement with the qualitative and quantitative experimental data (A. R. Kinjo and S. Takada, Biophys. Journal, 85:3521-3531.).