Perturbation Theory for Steady-State Laplacian Models of Biological Systems
OPEN ACCESS
Author / Producer
Date
2017
Publication Type
Doctoral Thesis
ETH Bibliography
yes
Citations
Altmetric
OPEN ACCESS
Data
Rights / License
Abstract
A fundamental problem in fields such as systems biology and pharmacology is to determine how biological systems change their dose-response behaviour upon external or natural perturbations. A comprehensive understanding of this relation through the effects of topologies, reactions, and parameters on the response is invaluable when designing, analysing, and identifying biological processes. However, the characterisation of said relation is frequently challenged by the complexity, topological and parametric uncertainty, multiple levels of organisation, and heterogeneity inherent to biology. We address these challenges through the lens of the classical steady-state Laplacian models, which are mathematical models with only zero and (pseudo) first order mass-action reactions. Laplacian models have found countless applications in biology since, frequently, they can be obtained from non-linear models using time-scale separation, and due to their analytical tractability, i.e. that a closed form of their steady-states always exists. Despite their apparent simplicity, Laplacian models suffer from the (super) exponential growth of the size of their steady-state expressions that makes the symbolic analysis of models of even moderate sizes practically impossible.
This thesis develops theory and methods to study how perturbations in steady-state Laplacian models of biological systems translate to dose-response relations, while accounting for biological complexity, uncertainty, and heterogeneity.
First, in Chapter 3, we lay the theoretical groundwork of the thesis by investigating the factorisation properties of an important class of polynomials, the so called Kirchhoff polynomials, which link the topology of Laplacian models, expressed through directed graphs, to their symbolic steady-state expressions. We reveal the intimate connection between combinatorial properties of the digraph representation of Laplacian models and its corresponding Kirchhoff polynomial. Specifically, we devise digraph decomposition rules corresponding to factorisation steps of the Kirchhoff polynomial and provide necessary and sufficient primality conditions for the resulting factors expressed through connectivity properties of the decomposed components. As a result we propose a linear time algorithm for the prime factorisation of Kirchhoff polynomials based on directed graph connectivity properties.
In Chapter 4 we employ the prime factorisation algorithm to develop a framework for the efficient manipulation and generation of expressions of Kirchhoff polynomials, which result from steady-state derivations for Laplacian models. To manipulate such expressions we transform them to a coarse-grained representation which can easily be symbolically handled. To generate such expressions we propose two heuristic algorithms producing compressed Kirchhoff polynomials. Thereby we demonstrate that, contrary to prior belief, Kirchhoff polynomial generation is not restricted by the (super) exponentially growing size of the polynomials but, rather, by the connectivity properties of their corresponding directed graphs.
In Chapter 5 we proceed to study the relative differences between dose-response curves produced by a reference and a perturbed steady-state Laplacian model. We exploit the connectivity properties of the directed graph representation of Laplacian models to identify equivalence classes of models, reliably reject improbable hypothetical models, and determine how perturbations in topology and parameters affect the dose-response, all in a parameter-free manner.
Finally, in Chapter 6 we formulate a framework to search for a minimal model of interferon type I differential signalling. The framework accounts for topological uncertainty by investigating an ensemble of hypothetical models and ranking them with respect to experimental dose-response data using Bayesian model comparison, and for parametric uncertainty by employing Bayesian parameter inference. Each considered model is a simple multi-scale threshold model that incorporates at its core a steady-state Laplacian submodel determining the number of active interferon receptors resulting from interferon stimulation, includes receptor number cell-to-cell variability, and produces the proportion of alive cells in a population resulting from interferon-induced activities. As a result we demonstrate that the minimal sufficient mechanisms explaining differential signalling are receptor assembly, receptor endocytosis and recycling, and inhibition by the factor USP18.
Permanent link
Publication status
published
External links
Editor
Contributors
Book title
Journal / series
Volume
Pages / Article No.
Publisher
ETH Zurich
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Organisational unit
03699 - Stelling, Jörg / Stelling, Jörg