Probabilistic Approximations of Bio-pathway Dynamics

Understanding the functioning of complex biological systems is a major challenge. To address this challenge, quantitative mathematical models are needed to capture the dynamics of various intra (and inter)-cellular processes. Here we focus on signaling pathways which constitute the building blocks of many of the intracellular processes that govern the behavior of cells. A standard formalism used to model bio-pathways is a system of ordinary differential equations (ODEs). The equations describe specific bio-chemical reactions, while the variables typically represent concentration levels of molecular species (genes, RNAs, proteins). Bio-pathways usually involve a large number of molecular species and bio-chemical reactions. Hence the corresponding ODE model will involve many variables and parameters and the values of many of the parameters (rate constants) will be unknown. Further, the initial concentration levels of the various species and rate constants will often be available only as intervals of values as also the experimental data reporting the measured concentration levels of a small number of proteins at a few time points. In addition, the data will often be gathered using a cell population. Consequently, when numerically simulating the ODEs model, one must resort to Monte Carlo methods to ensure that sufficiently many point values from the relevant intervals of values are being sampled. As a result, tasks such as model validation, parameter estimation and sensitivity analysis will require the generation of a huge number of trajectories. We propose a probabilistic approach to approximate the deterministic signaling pathway dynamics specified as a system of ODEs. It consists of pre-computing and storing a representative sample of trajectories induced by the system of ODEs. After discretizing the value space suitably, we use Baysian network (BN) models to compactly represent these trajectories by exploiting the dependencies/independecies in the pathway structure. As a result, a variety of analysis questions concerning the pathway dynamics traditionally addressed using Monte Carlo simulations can be converted to Bayesian inference and solved much more efficiently. The BN representation is, in essence, a succinct representation of an associated Markov chain. Hence formal verification techniques developed for Markov chains and other models [2] also become applicable.