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Modelling and Identification of dynamical gene interactions Ronald Westra, Ralf Peeters Systems Theory Group Department of Mathematics Maastricht University The Netherlands westra@math.unimaas.nl. Slide 2 Themes in this Presentation How deterministic is gene regulation? How can we model gene regulation? How can we reconstruct a gene regulatory network from empirical data ? Slide 3 1. How deterministic is gene regulation? Main concepts: Genetic Pathway and Gene Regulatory Network Slide 4 What defines the concepts of a genetic pathway and a gene regulatory network and how is it reconstructed from empirical data ? Slide 5 Genetic pathway as a static and fixed model G2 G1 G4 G5 G6 G3 Slide 6 Experimental method: gene knock-out G2 G1 G4 G5 G6 G3 Slide 7 Stochastic Gene Expression in a Single Cell M. B. Elowitz, A. J. Levine, E. D. Siggia, P. S. Swain Science Vol 297 16 August 2002 How deterministic is gene regulation? Slide 8 Slide 9 Slide 10 AB Slide 11 Slide 12 Elowitz et al. conclude that gene regulation is remarkably deterministic under varying empirical conditions, and does not depend on particular microscopic details of the genes or agents involved. This effect is particularly strong for high transcription rates. These insights reveal the deterministic nature of the microscopic behavior, and justify to model the macroscopic system as the average over the entire ensemble of stochastic fluctuations of the gene expressions and agent densities. Conclusions from this experiment Slide 13 2. Modelling dynamical gene regulation Slide 14 Implicit modeling: Model only the relations between the genes G2 G1 G4 G5 G6 G3 Slide 15 Implicit linear model Linear relation between gene expressions N gene expression profiles : m-dimensional input vector u(t) : m external stimuli p-dimensional output vector y(t) Matrices C and D define the selections of expressions and inputs that are experimentally observed Slide 16 Implicit linear model The matrix A = (a ij ) - a ij denotes the coupling between gene i and gene j: a ij > 0 stimulating, a ij < 0 inhibiting, a ij = 0 : no coupling Diagonal terms a ii denote the auto-relaxation of isolated and expressed gene i Slide 17 Relation between connectivity matrix A and the genetic pathway of the system G2 G1 G4 G5 G6 G3 coupling from gene 5 to gene 6 is a(5,6) Slide 18 Explicit modeling of gene-gene Interactions In reality genes interact only with agents (RNA, proteins, abiotic molecules) and not directly with other genes Agents engage in complex interactions causing secondary processes and possibly new agents This gives rise to complex, non-linear dynamics Slide 19 An example of a mathematical model based on some stoichiometric equations using the law of mass actions Here we propose a deterministic approach based on averaging over the ensemble of possible configurations of genes and agents, partly based on the observed reproducibillity by Elowitz et al. Slide 20 In this model we distinguish between three primary processes for gene-agent interactions: 1.stimulation 2.inhibition 3. transcription and further allow for secondary processes between agents. Slide 21 the n-vector x denotes the n gene expressions, the m-vector a denotes the densities of the agents involved. Slide 22 x : n gene expressions a : m agents Slide 23 (a) denotes the effect of secondary interactions between agents Slide 24 Agent A i catalyzes its own synthesis: EXAMPLE Autocatalytic synthesis Slide 25 Slide 26 Slide 27 Slide 28 Slide 29 Slide 30 Slide 31 Slide 32 Slide 33 Complex nonlinear dynamics observed in all dimensions x and a including multiple stable equilibria. Slide 34 Conclusions on modelling More realistic modelling involving nonlinearity and explicit interactions between genes and operons (RNA, proteins, abiotic) exhibits multiple stable equilibria in terms of gene expressions x and agent denisties a Slide 35 3. Identification of gene regulatory networks Slide 36 the matrices A and B are unknown u(t) is known and y(t) is observed x(t) is unknown and acts as state space variable Linear Implicit Model Slide 37 the matrices A and B are highly sparse : Most genes interact only with a few other genes or external agents i.e. most a ij and b ij are zero. Identification of the linear implicit model Slide 38 Estimate the unknown matrices A and B from a finite number M of samples on times {t 1, t 2,.., t M } of observations of inputs u and observations y: {(u(t 1 ), y(t 1 )), (u(t 2 ), y(t 2 )),.., (u(t M ), y(t M ))} Challenge for identifying the linear implicit model Slide 39 Notice: 1.the problem is linear in the unknown parameters A and B 2.the problem is under-determined as normally M