(151b) Stochastic Simulation of Cytosolic Calcium Dynamics

Ordinary differential equations-based deterministic simulation is accurate only if large reaction volumes (and hence a large number of molecules) are considered. At the subcellular level, the stochastic behavior of chemical reactions becomes effective. This is even more important when dealing with local (spatial) variations. Many of the cellular functions are strongly dependent on the spatial variation of chemical species inside the cell. For example, the density of receptors on the cell surface of a cell is not uniform. Instead, patches of high-receptor density are found. So, to account for spatial variations, in the deterministic approach, the system is modeled as a set of partial differential equations. The resulting equations are solved using the grid-based finite elements method. The number of species molecules in each of the grids is even smaller. Thus, stochastic simulation is necessary.

Given that any of the realizations of stochastic simulation do not individually correspond to the experimental results in single cells, at best, the average and other statistical measures, such as distinct clusters if any, of many (thousands of) realizations can be compared to the corresponding properties of a small cell-population. This results in a more accurate description of the system as compared to that provided by deterministic simulation. However, the computational complexity is enormous, thus requiring the methods for accelerated stochastic simulation. Starting with the pioneering work of Gillespie (1) and several other researchers, recently several methods have been developed to tackle the complexity of stochastic simulation. One approach is the tau-leap method in which a larger time-step is taken and many reactions are fired in the chosen time interval (2). A variant of this method is the binomial tau-leap method to avoid negative populations (3, 4). In another approach, deterministic simulation, stochastic simulation and Markov model based approaches are used in an integrated manner (5). Yet a third approach partitions the reactions and species as ?slow? and ?fast? with the partial equilibrium assumption that the population of slow species is not altered by the fast reactions (6).

Here we present the application of stochastic simulation technique to a model of the regulation of the dynamics of cytosolic calcium ions (Ca2+) in mouse macrophage RAW 264.7 cells that we have developed recently (7). The (current) deterministic model consists of 19 ordinary differential equations. The flux expressions are based on simple rate laws such as law of mass action (LOMA) kinetics, Michaelis-Menten (M-M) kinetics and Hill-dynamics as well as more complex forms. We present the results of stochastic simulation and compare the response characteristics obtained in stochastic and deterministic simulation.

Literature Cited

1. Gillespie, D. T. 1976. General method for numerically simulating stochastic time evolution of coupled chemical-reactions. Journal of Computational Physics. 22:403-434.

2. Gillespie, D. T. 2001. Approximate accelerated stochastic simulation of chemically reacting systems. Journal of Chemical Physics. 115:1716-1733.

3. Chatterjee, A., D. G. Vlachos, and M. A. Katsoulakis. 2005. Binomial distribution based tau-leap accelerated stochastic simulation. J Chem Phys. 122:024112.1-7.

4. Chatterjee, A., and D. G. Vlachos. 2006. Multiscale spatial Monte Carlo simulations: Multigriding, computational singular perturbation, and hierarchical stochastic closures. J Chem Phys. 124:064110.1-16.

5. Kaznessis, Y. N. 2006. Multi-scale models for gene network engineering. Chemical Engineering Science. 61:940-953.

6. Cao, Y., D. Gillespie, and L. Petzold. 2005. Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems. Journal of Computational Physics. 206:395-411.

7. Maurya, M. R., and S. Subramaniam. 2007. A kinetic model for calcium dynamics in RAW 264.7 cells: 1. Mechanisms, parameters, and subpopulational variability. Biophys J. 93:709-28.