Probabilistic approximations of ODEs based bio-pathway dynamics

Bio-chemical networks are often modeled as systems of ordinary differential equations (ODEs). Such systems will not admit closed form solutions and hence numerical simulations will have to be used to perform analyses. However, the number of simulations required to carry out tasks such as parameter estimation can become very large. To get around this, we propose a discrete probabilistic approximation of the ODEs dynamics. We do so by discretizing the value and the time domain and assuming a distribution of initial states w.r.t. the discretization. Then we sample a representative set of initial states according to the assumed initial distribution and generate a corresponding set of trajectories through numerical simulations. Finally, using the structure of the signaling pathway we encode these trajectories compactly as a dynamic Bayesian network. This approximation of the signaling pathway dynamics has several advantages. First, the discretized nature of the approximation helps to bridge the gap between the accuracy of the results obtained by ODE simulation and the limited precision of experimental data used for model construction and verification. Second and more importantly, many interesting pathway properties can be analyzed efficiently through standard Bayesian inference techniques instead of resorting to a large number of ODE simulations. We have tested our method on ODE models of the EGF-NGF signaling pathway [1] and the segmentation clock pathway [2]. The results are very promising in terms of accuracy and efficiency.