Path integral approaches to subnetwork description and inference

Abstract

Path integral formalism is a powerful tool borrowed from theoretical physics to build dynamical descriptions, yet its potential is largely unexplored in the context of complex networks, such as the ones common in systems biology. In this PhD thesis, I present di erent mathematical frameworks based on path integrals to capture the time evolution of interacting continuous degrees of freedom, e.g. biochemical concentrations. The generality of path integral approaches enables us to tackle several questions related to modelling and inference for dynamics. We first develop a novel mean field approximation, the Extended Plefka Expansion, for stochastic di erential equations exhibiting generic nonlinearities. The key element is the definition of “e ective” fields which map an interacting dynamics into the “most similar” non-interacting picture, i.e. the one producing the same average observables. In the resulting picture, couplings between variables are replaced by a memory and a coloured noise. We next apply this setup to the case in which part of the network is observed and part is unknown. We study the accuracy of prediction of the unobserved dynamics as a function of the number of observed nodes and other structural parameters of the system. The Extended Plefka Expansion is expected to become exact in the limit of infinite size networks with couplings of mean field type, i.e. weak and long-ranged. We show this explicitly for a linear dynamics by comparison with other methods relying on Random Matrix Theory. We finally appeal to path integrals to design “reduced” models, where equations are referred solely to some selected variables (subnetwork) but still carry information on the whole network. This model reduction strategy leads to substantially higher quantitative accuracy in the prediction of subnetwork dynamics, as we demonstrate with an example from the protein interaction network around the Epidermal Growth Factor Receptor.

Date of Award	2016
Original language	English
Awarding Institution	King's College London
Supervisor	Peter Sollich (Supervisor) & Reimer Kuhn (Supervisor)