Download PowerPoint-presentatie - Maastricht University

Modelling and Identification
of
dynamical gene interactions
Ronald Westra, Ralf Peeters
Systems Theory Group
Department of Mathematics
Maastricht University
The Netherlands
[email protected].
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
G1
G3
G2
G4
G5
G6
Experimental method:
gene knock-out
G1
G3
G2
G4
G5
G6
How deterministic
is gene regulation?
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
A
B
Conclusions from this experiment
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.
2. Modelling
dynamical gene regulation
Implicit modeling:
Model only the relations between the genes
G1
G3
G2
G4
G5
G6
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
G1
G3
G2
G4
G6
coupling from gene
5 to gene 6 is a(5,6)
G5
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
EXAMPLE
Autocatalytic synthesis
Agent Ai catalyzes its own 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
Linear Implicit Model
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
Identification of the 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.
Challenge for identifying 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 inputs
u and observations y:
{(u(t1), y(t1)), (u(t2), y(t2)), .., (u(tM), y(tM))}
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 direct
application 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 Westra
[email protected]
Ralf Peeters
[email protected]
Systems Theory Group
Department of Mathematics
Maastricht University
PO box 616
NL6200MD Maastricht
The Netherlands