Gene regulatory networks
: insights from statistical mechanics and network theory

Student thesis: Doctoral ThesisDoctor of Philosophy


In this thesis I present a simplified model for gene regulation, where gene expression is regulated by transcription factors (TFs), which are single proteins or protein complexes. Proteins are in turn synthesised from expressed genes, creating feedback loops of regulation. This leads to a directed bipartite network in which a link from a gene to a TF exists if the gene codes for a protein contributing to the TF, and a link from a TF to a gene exists if the TF regulates the expression of the gene. Both genes and TFs are modelled as binary variables, which indicate, respectively, whether a gene is expressed or not, and a TF is synthesised or not. I consider the scenario where for a TF to be synthesised, all of its contributing genes must be expressed. This results in an “AND” gate logic for the dynamics of TFs. By adapting percolation theory to directed bipartite graphs, in which the nodes representing TFs implement an AND gate logic, I determine the necessary conditions, in the network parameter space, under which bipartite networks can support a non-trivial multiplicity of stable gene expression patterns, under noisy conditions, as required to support multi-cellular life.

In particular, the analysis reveals the possibility of a bi-stability region, where the extensive percolating cluster is or is not resilient to perturbations. This is remarkably different from the transition observed in standard percolation theory. In addition, I consider perturbations involving single node removal that mimic gene knockout experiments. Results indicate the strong dependence of the gene knockout cascade on the logic implemented, highlighting in particular that avalanche sizes cannot be easily related to properties of gene-gene interaction networks, since the role of TFs is important to predict the cascade size.

Further, I also investigate the dynamic properties of such models, which are out of equilibrium in general. In the case of a directed tree, the dynamic cavity method provides an efficient tool to characterise the dynamic trajectory of the nodes for the linear threshold model. However, because of the computational complexity of the method, the analysis is usually limited to systems where the largest number of neighbours is small. I present an efficient implementation of the dynamic cavity method which substantially reduces the computational complexity of the method for systems with discrete couplings. This approach opens up the possibility to investigate the dynamic properties of networks with fat-tailed degree distribution. I exploit this new implementation to study properties of the non-equilibrium steady-state. I present an extension of the dynamical cavity approach to calculate the pairwise correlations induced by different motifs in the network. The results suggest that just two basic motifs of the network are able to accurately describe the entire statistics of observed correlations. Finally, I investigate models defined on networks containing bi-directional interactions. I observe that the stationary state associated with networks with symmetric or anti-symmetric interactions is biased towards the active or inactive state respectively, even if independent interaction entries are drawn from a symmetric distribution. This phenomenon, which can be regarded as a form of spontaneous symmetry-breaking, is peculiar to systems formulated in terms of Boolean variables, as opposed to Ising spins.
Date of Award1 Aug 2022
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorAlessia Annibale (Supervisor) & Reimer Kuhn (Supervisor)

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