Statistical mechanics of immunity from genes to populations

Student thesis: Doctoral ThesisDoctor of Philosophy


In this thesis we provide three different models of processes related to immunity. Firstly, we describe a model of the T-cell mediated anti-tumour immune response. We describe the anti-tumour immune response as a set of cellular kinetic rate equations. T-cells are assumed to act as binary agents (active/inactive) that evolve with time according to a noisy linear threshold function. Using the Kramers-Moyal expansion of the master equation, we are able to describe the system as a closed set of ordinary differential equations. We find that there is a critical value of the ratio of helper to cytotoxic T-cells that depends on the expression of MHC-I in tumour cells. We demonstrate the effect that this interplay has on the efficacy of the helper/cytotoxic T-cell ratio and MHC-I expression as prognostic markers.

Secondly, we model infectious disease outbreak in populations where individuals are vaccinated with a vaccine that reduces the likelihood to transmit the disease to a small but finite value. We study the SIR model on networks with nodes that belong to one of several sub-populations with different individual transmissibility and apply the cavity method to derive equations for the risk that a node in a network will cause outbreak. We show that the threshold for outbreak in populations split into vaccinated and unvaccinated sub-populations, will depend upon the proportion of the population that is vaccinated, as well as the density of links between the vaccinated and unvaccinated sub-populations. Furthermore, we show that the cavity method can be used to derive the distribution of risk in populations with heterogeneity in individual transmissibility, and in some cases provide an exact expression for this distribution.

The immune system is comprised of many different cell types, with different functions. Within an organism, different cell types have the same genetic make-up, but they differ in the set of genes that are expressed. The last model presented in this thesis is concerned with determining the conditions that are necessary for gene regulatory networks to support a diverse set of stable gene expression profiles, corresponding to different cell types. We consider a bipartite model of gene regulatory networks consisting of genes and transcription factors. Gene expression is modelled as a Boolean variable (on/off), that evolves according to synchronous Glauber dynamics. Genes receive regulatory signals from transcription factors which are in turn synthesised by expressed genes. Our work focuses particularly on self-regulation, where genes that contribute to the synthesis of a transcription factor may be regulated by that same transcription factor. We extend the dynamical cavity method for systems with self-interactions and multi-node interactions to derive a computationally feasible scheme for the dynamical analysis of our model. We show that networks with bidirectional, multi-node interactions support multiple diverse gene expression profiles, suggesting that self-regulation is an important feature of multi-cellular life.
Date of Award1 May 2023
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorFranca Fraternali (Supervisor) & Alessia Annibale (Supervisor)

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