modelling and identification of dynamical gene interactions
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Modelling and Identification of
dynamical gene interactionsRonald Westra, Ralf Peeters
Systems Theory Group
Department of Mathematics
Maastricht University
The Netherlands
westra@math.unimaas.nl.
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
GG22
GG11
GG44
GG55
GG66
GG33
Experimental method: gene knock-out
GG22
GG11
GG44
GG55
GG66
GG33
Stochastic Gene Expression in a Single CellM. B. Elowitz, A. J. Levine, E. D. Siggia, P. S. SwainScience Vol 297 16 August 2002
How deterministic is gene regulation?
A B
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
GG22
GG11
GG44
GG55
GG66
GG33
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 = (aij) - aij denotes the coupling between gene i and gene j:
aij > 0 stimulating, aij < 0 inhibiting, aij = 0 : no coupling
Diagonal terms aii denote the auto-relaxation of isolated and expressed gene i
Relation between connectivity matrix A and the genetic pathway of the system
GG22
GG11
GG44
GG55
GG66
GG33
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 expressionsa : m agents
(a) denotes the effect of secondary interactions between agents
Agent Ai 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 aij and bij are zero.
Identification of the linear implicit model
Estimate the unknown matrices A and B from a finite number – M – of samples on times {t1, t2, .., tM} of observations of inputsu and observations y:
{(u(t1), y(t1)), (u(t2), y(t2)), .., (u(tM), y(tM))}
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
L2-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
L1- 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).
L1-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 :
Mrequired log N
The L1-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 L1-minimization for the nonlinear explicit model
What would the application of this approach yield for directapplication for the explicit nonlinear model discussed before?
Reconstruction with L1-minimization
From the explicit nonlinear model one obtains series:
{(x(t1), a(t1)), (x(t2), a(t2)), .., (x(tM), a(tM))}
For the L1-approach only the terms:
{x(t1), x(t2), .., x(tM)}
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 Westrawestra@math.unimaas.nl
Ralf Peeters ralf.peeters@math.unimaas.nl
Systems Theory GroupDepartment of MathematicsMaastricht UniversityPO box 616NL6200MD MaastrichtThe Netherlands
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