RT Dissertation/Thesis
T1 Mathematical modelling, analysis and numerical simulation for a general class of gene regulatory networks
T2 Modelado, análisis matemático y simulación numérica para una clase general de redes de regulación genética
A1 Pájaro Diéguez, Manuel
K1 2404 Biomatemáticas
K1 1202 Análisis y Análisis Funcional
K1 1206 Análisis Numérico
AB The research work developed in this thesis is mainly oriented to the mathematicalmodelling of biological systems, the behaviour of which is inherently stochastic, as itis the case of gene regulatory networks. Their relevance emerges from the fact thatall necessary information for life cycle is encoded in the DNA. Consequently, thestudy of DNA expression, transcription into messenger RNA and translation intoproteins, together with their regulation becomes essential to predict cells responseto environmental signals.The inherent stochastic nature of gene expression makes these systems to be faraway from the classical kinetic limit where the (macroscopic) deterministic methodsare valid. In modelling these systems, we need to employ microscopic methods whichincorporate the underlying stochastic behaviour. The Chemical Master Equation(CME) remains at the basis for the modelling of these phenomena. However, aclosed form solution of the CME is unavailable in general, due to the large number(eventually infinity) of coupled equations. A widespread technique to approximatethe CME solution is the Stochastic Simulation Algorithm (SSA), a computationallyinvolved Monte Carlo type method.Although many numerical approximations emerge to reduce the complexity ofthe CME, we will focus on the Partial Integro-Differential Equation (PIDE) or Friedmanmodel, which represents a continuous approximation of the CME. For the onedimensional version (self-regulation), the PIDE model has an analytic solution forits steady state. This fact will allow us to characterize the regions in the space ofparameters in which the system changes its behaviour (unimodal, bimodal). Alsowe have carried out an stability analysis by means of entropy methods.Moreover, we obtain a multidimensional version of the Friedman model to handle more complex gene regulatory networks with more than one gene. The mathematicalproperties of the corresponding equation will be exhaustively analyzed, also withspecial emphasis on stability of the solution using entropy methods.In addition, we propose two semi-Lagrangian methods for the numerical solutionof the multidimensional model. The first method results very efficient and scalableto higher dimensions, as the numerical results illustrate, although in practice exhibitsfirst order convergence in time and space. Solutions provided by the proposedmethod are compared with those obtained by SSA to assess efficiency, accuracyand computational costs. For the second semi-Lagrangian method we develop thetheoretical numerical analysis, thus proving second order convergence in time andspace. This is clearly illustrated by a numerical example. However, the computationalcost of this second approach results much higher, so that the scalability tohigher dimensions seems a difficult task.All the numerical techniques have been implemented on a user friendly toolbox(SELANSI) which is detailed in the Appendix.
YR 2017
FD 2017-12-19
LK http://hdl.handle.net/11093/917
UL http://hdl.handle.net/11093/917
LA eng
NO Ministerio de Economía y Competitividad de España (BES-2013-063112)
DS Investigo
RD 05-dic-2024