Modelling and Identification of dynamical gene interactions Ronald Westra, Ralf Peeters Systems Theory Group Department of Mathematics Maastricht University The Netherlands

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 ?

1. How deterministic is gene regulation? Main concepts: Genetic Pathway and Gene Regulatory Network

What defines the concepts of a genetic pathway and a gene regulatory network and how is it reconstructed from empirical data ?

Genetic pathway as a static and fixed model G2 G1 G4 G5 G6 G3

Experimental method: gene knock-out G2 G1 G4 G5 G6 G3

Stochastic Gene Expression in a Single Cell M. B. Elowitz, A. J. Levine, E. D. Siggia, P. S. Swain Science Vol August 2002 How deterministic is gene regulation?

AB

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

2. Modelling dynamical gene regulation

Implicit modeling: Model only the relations between the genes G2 G1 G4 G5 G6 G3

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

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

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)

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

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.

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.

the n-vector x denotes the n gene expressions, the m-vector a denotes the densities of the agents involved.

x : n gene expressions a : m agents

 (a) denotes the effect of secondary interactions between agents

Agent A i catalyzes its own synthesis: EXAMPLE Autocatalytic synthesis

Complex nonlinear dynamics observed in all dimensions x and a – including multiple stable equilibria.

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

3. Identification of gene regulatory networks

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

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

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

Notice: 1.the problem is linear in the unknown parameters A and B 2.the problem is under-determined as normally M << N 3.the matrices A and B are highly sparse

L 2 -regression? This approach minimizes the integral squared error between observed and model values. This approach would distribute the small scale of the interaction (the sparsity) over all coefficients of the matrices A and B Hence: this approach would reconstruct small coupling coefficients between all genes – total connectivity with small values and no zeros

L 1 - or robust regression This approach minimizes the integral absolute error between observed and model values. This approach results in generating the maximum amount of exact zeros in the matrices A and B Hence: this approach reconstructs sparse coupling matrices, in which genes interact with only a few other genes It is most efficiently implemented with dual linear programming method (dual simplex).

L 1 -regression Example: Partial dual L1-minimization (Peeters,Westra, MTNS 2004) Involves a number of unobserved genes x in the state space Efficient in terms of CPU-time and number of errors : M required  log N

The L 1 -reconstruction ultimately yields the connectivity matrix A of the linear implicit model hence the genetic pathway of the gene regulatory system.

Reconstruction of the genetic pathway with partial L 1 -minimization for the nonlinear explicit model What would the application of this approach yield for direct application for the explicit nonlinear model discussed before?

Reconstruction with L 1 -minimization From the explicit nonlinear model one obtains series: {(x(t 1 ), a(t 1 )), (x(t 2 ), a(t 2 )),.., (x(t M ), a(t M ))} For the L 1 -approach only the terms: {x(t 1 ), x(t 2 ),.., x(t M )} are required.

Sampling

Reconstruction of coupling matrix A

Conclusions from applying the L1-approach to the nonlinear explicit model 1. The reconstructed connectivity matrix - hence the genetic pathway - differs among different stable equilibria 2. In practical situations to each stable equilibrium there belongs one unique connectivity matrix - hence one unique genetic pathway

Discussion And Conclusions

Discussion * In practice, one unique genetic pathway will be found in one stable state, caused by the dominant eigenvalue of convergence * knock-out experiments can cause the system to converge to another stable state, hence what is reconstructed? * How realistic is the assumption of equilibrium for a gene regulatory network? Mostly the system swirls around in non-equilibrium state

Conclusions * The concept of a genetic pathway is useful (and quasi unique) in one equilibrium state but is not applicable for multiple stable states * A genetic regulatory network is a dynamic, nonlinear system and depends on the microscopic dynamics between the genes and operons involved

Ronald Westra Ralf Peeters Systems Theory Group Department of Mathematics Maastricht University PO box 616 NL6200MD Maastricht The Netherlands