Authors:

Vilda Purutcuoglu (Department of Statistics, Middle East Technical University, Ankara, Turkey) and Ernst Wit (Department of Mathematics and Statistics, Lancaster University, Lancaster, UK)

Abstract:

The MAPK/ERK pathway is one of the major signal transaction system which regulates the growth control of all eukaryotes. In our previous study [2], we model this pathway by using quasi reaction list which consists of 51 proteins and 66 reactions indicating the activation of the system and the degradation of the growth factor's receptor (EGFR). In inference of the stochastic rate constants of associated reactions, we implement the Euler approximation, which is known as the discretized version of a diffusion process, within the Bayesian framework. In estimation via Euler, we update reaction rates and states at discrete time points sequentially. The updates of rates are found via random walk algorithm. But since the diffusion terms of states depend on rate constants in a non-linear way, sampling the candidate states values of the system needs more special MCMC method like Metropolis-within-Gibbs (MWG) technique [3]. In this study our aim is to investigate the application of exact, rather than MWG, sampling by using variational approximation [1]. For this purpose we define an approximate distribution by adding variational parameters in transition kernels and initial state probabilities of the tractable substructure of the true observation matrix used in the estimation. The underlying substructure is generated in such a way that the necessary links between states are decoupled, thereby, Gibbs sampling can be applicable. The lost of information from removing the links is, then, regained by linking the updates of variational parameters at time t. These free parameters are calculated by minimizing the Kullback-Leibler divergence between the true and approximate distribution of states.

References:

[1.] M. I. Jordan (ed.), Z. Ghahramani, T. S. Jaakkola, and L. K. Saul, ``An introduction to variational methods for graphical models", Learning in Graphical Models, Cambridge, MIT Press, 1999.

[2.] V. Purutcuoglu and E. Wit, ``Bayesian inference for the MAPK/ERK pathway by considering the dependency of the kinetic parameters", 2007 (under revision).

[3.] D. J. Wilkinson, ``Stochastic Modelling for Systems Biology", Chapman and Hall/CRS, 2006.