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Summary: Intracellular biochemical processes exhibit stochasticity that can potentially influence the phenotypic characteristics. However, stochastic simulations are prohibitive owing for most complex biological systems. The recently developed binomial τ-leap method (Chatterjee, Vlachos et al. 2005) is employed to carry out stochastic simulations of the epidermal growth factor (EGF) receptor induced mitogen activated protein (MAP) kinase cascade. Results indicate that the binomial τ-leap method is computationally 1001000 times more efficient than the exact stochastic simulation algorithm of Gillespie. Furthermore, the new method avoids negative populations and accurately captures the disparity in populations of the reaction network along with their fluctuations. Availability: Details about the binomial τ-leap algorithm, a software and manual are available at (Website). Contact: vlachos@che.udel.edu * To whom correspondence should be addressed. INTRODUCTION The role of stochasticity in biological systems is well established by several theoretical studies (Arkin, Ross et al. 1998; Morton-Firth and Bray 1998; Resat, Ewald et al. 2003; Meng, Somani et al. 2004) and experimental work (Elowitz, Levine et al. 2002). Stochasticity has been extensively studied using the exact stochastic simulation algorithm (SSA) of Gillespie (Gillespie 1976). However, SSA is limited to short times and to problems that do not exhibit a disparity in time scales. Biological networks exhibit multiple times scales, ranging from microseconds to days, and demand efficient stochastic simulation algorithms (Lok 2004). Such recently developed algorithms are reviewed in (Vlachos 2004;Turner, Schnell et al. 2004). The family of τ-leap methods, which have a mathematical foundation, is the only coarse-grained stochastic simulation method that can provide accuracy with nearly correct fluctuations. Here the binomial τ-leap method (Chatterjee, Vlachos et al. 2005), a variant of the original Poisson τ-leap method of Gillespie (Gillespie 2001), is employed to study the signaling pathway of EGF receptor (EGFR) activated MAP kinase cascade. EGFRs belong to the receptor tyrosine kinase (RTK) family of receptors and play an important role in many physiological events in cell proliferation. In this work, we have used a recently proposed signaling pathway to study the surface and internalized EGFR activated MAP kinase cascade (Schoeberl, Eichler-Jonsson et al. 2002). The published model is based on ordinary differential equations (ODEs) of 94 signaling species among 296 reactions (referred to as events in this paper) in a well mixed environment. The complexity of the model renders the MAP kinase cascade an excellent benchmark problem for testing the applicability of the binomial τ-leap method for biological systems. THE BINOMIAL τ-LEAP METHOD The binomial τ-leap method follows the stochastic evolution of N species among M reactions as an approximate Markov process. The algorithm requires the initial population size of all species (i.e., the number of molecules) and the reaction kinetics as inputs. This information is used to compute transition probabilities per unit time of all events at time t. A ‘bundle or firings’ of events, sampled from a binomial distribution, are allowed to trigger in a time interval of size τ, and the time is incremented to t+τ. This process is repeated as the algorithm leaps along the time axis. The maximum τ is chosen so it does not substantially change the populations (Gillespie 2001). This is automatically taken care of using a simple, inexpensive criterion (Chatterjee, Vlachos et al. 2005) according to which the user can control the speed-up by inputting the coarse-graining factor r of time stepping in comparison to the SSA. Previous experience indicated that r 0.2 £ provides accuracy and speed-up. The algorithm conserves mass at both the reaction and entire network levels by constraining the number of reaction firings, thereby eliminating for the first time negative populations in τ-leaping (Chatterjee, Vlachos et al. 2005). Numerical simulations using simple reaction networks (Chatterjee, Vlachos et al. 2005) demonstrated that the method is more accurate than the original Poisson τ-leap method of (Gillespie 2001). RESULTS Simulations spanning 15 minutes of real time were performed for the MAP kinase cascade using the binomial τ-leap method, with different levels of coarse-graining r=0.10.5, and the SSA. Reaction dynamics and initial populations are specified in (Website). Intracellular populations are plotted against time in Figure 1 for selected species whose populations span from small to large ones. The binomial τ-leap method accurately captures the network kinetics for all times despite the large disparity in species population size in this complex network. Figure 2 compares fluctuations of the two methods using the probability density functions obtained at time t=1 min (with ensemble averages over 500 trajectories for the SSA and 1000 for the binomial τ-leap method). The mean and standard deviation for representative species of small and large populations are tabulated in the inset of Figure 2. The excellent agreement obtained indicates that not only the correct mean and standard deviation but also nearly correct higher order moments can be obtained using the binomial τ-leap method. The SSA required 3 days on a DEC alpha 833 MHz processor for the simulation depicted in Figure 1 as compared to 6–30 min taken using the τ-leap method. The binomial τ-leap is 100-1000 times more efficient than the SSA over the range of r values explored. The low computational requirements of the binomial τ-leap method enable accurate long time simulations that are beyond the reach of SSA (see example in (Website)). Due to its accuracy, robustness, simplicity and substantial speed-up, the binomial τ-leap method appears to be a particularly promising, approximate stochastic simulation method for computational biology. ACKNOWLEDGEMENTS This work was partially supported by the NSF through CTS-0312117. REFERNCES Arkin, A. P., J. Ross, et al. (1998). "Stochastic kinetic analysis of developmental pathway bifurcation in phage l-infected escherichia coli cells. " Genetics 149: 1633-1648. Chatterjee, A. , D. G. Vlachos, et al. (2005). "Binomial distribution based τ-leap accelerated stochastic simulation. " Journal of Chemical Physics In press. Elowitz, M. B. , A. J. Levine, et al. (2002). "Stochastic gene expression in a single cell. " Science 297(16): 1183-1186. Gillespie, D. T. (1976). "A general method for numerically simulating the stochastic evolution of coupled chemical reactions. " Journal of Computational Physics 22: 403-434. Gillespie, D. T. (2001). "Approximate accelerated stochastic simulation of chemically reacting systems. " Journal of Chemical Physics 115(4): 1716-1733. Lok, L. (2004). "The need for speed in stochastic simulation. " 22(8): 964-965. Meng, T. C. , S. Somani, et al. (2004). "Modeling and simulation of biological systems with stochasticity. " In Silico Biology 4: 0024. Morton-Firth, C. J. and D. Bray (1998). "Predicting Temporal Fluctuations in an Intracellular Signalling Pathway. " Journal of Theoretical Biology 192(1): 117128. Resat, H. , J. A. Ewald, et al. (2003). "An integrated model of epidermal growth factor receptor trafficking and signal transduction. " Biophysical Journal 85: 730-743. Schoeberl, B. , C. Eichler-Jonsson, et al. (2002). "Computational modeling of the dynamics of the MAP kinase cascade activated by surface and internalized receptors. " Nature Biotechnology 20: 370-375. Turner, T. E. , S. Schnell, et al. (2004). "Stochastic approaches for modelling in vivo reactions. " Computational Biology and Chemistry 28: 165-178. Vlachos, D. G. (2004). "A review of multiscale analysis: Examples from systems biology, materials engineering, and other fluid-surface interacting systems. " Adv. Chem. Eng.: submitted. Website "http://www.che.udel.edu/. " FIGURE CAPTIONS Fig. 1 Time dependent populations using the stochastic simulation algorithm (SSA; thick solid line) and the binomial τ-leap method (thin solid line, r=0.2 and squares, r=0.5). Fig. 2 Probability density functions (pdf) using the stochastic simulation algorithm (SSA, circles) and the binomial τ-leap method at time t = 1 min (squares, r=0.2; diamonds, r=0.5). The solid lines are curve fits for visual aid. Average populations and standard deviation are tabulated in the inset. Chatterjee et al. Figure 1 0 50