Biology Reference
In-Depth Information
Boolean networks in which gene expression
X
takes the value 1 or 0 (on or off) are one
of the simplest mathematical formalisms to model the network dynamics, and have been
historically widely used to gain insight into the most fundamental features of complex
system dynamics.
22,33
35
They evade the problem of having to determine the values
of parameters that are part of the description of the interactions, but face the same problem
as ODEs in that the Boolean function that integrates and maps all the inputs of a gene
at time
t
to its future expression value (output at
t
1) are virtually unknown. However, the
qualitative nature of a Boolean network allows them to more readily capture the
information obtained from qualitative overexpression/deletion experiments.
36
39
However,
since in deterministic Boolean networks unstable steady-states do not exist the landscape
has no saddles or hilltops, but consists of disjointed attractors.
21
1
By contrast, ODE models are the most widely used formalisms and benefit from a large
body of mathematical theory of dynamical systems that offer analytical tools to study the
existence and stability of steady-states. Certainly, ODEs suffer from the need to define
the functional form of regulatory relationships, i.e., how the expression state of upstream
regulatory genes map into the expression state change of the target gene and determine the
values of coefficients in the ODEs equations. One common work-around is the assumption
of a universal sigmoidal transfer function (e.g. Hill function) that maps input into output.
40
This is justified on the grounds that the mechanism of transcriptional gene activation by the
regulatory factors requires the assembly of multiprotein complexes on the promoter, which
typically generates cooperativity. But even without the latter, the stochastic nature of gene
activation and other physicochemical factors in the cell leads to sigmoidal input
output.
41
The next challenge is how to integrate all the sigmoidal transfer functions of the individual
inputs, given that for no mammalian promoter, the promoter logics, let alone of the transfer
functions are known. In this respect a variety of ad hoc argued approaches have been used.
Duff et al. recently compared several ODE approaches for the same network for the case for
a hematopoietic system. The case of pancreas cell reprogramming below offers another
example.
10,42
46
92
STEP 3: DEFINE THE BIOLOGICAL TRANSFORMATION OF INTEREST AS A
TRAJECTORY BETWEEN ATTRACTORS IN THE QUASI-POTENTIAL LANDSCAPE
Many useful manipulations of biological behaviors consist of imposing a change along
trajectories that cause cells or tissues to exit the current (possibly maladaptive) stable state
and enter into a benign attractor that may even actively contribute to a vital function of
the organism. For instance, a novel therapy envisaged for diabetes patients is to convert alpha
cells in the pancreatic islet into the distinct but developmentally neighboring insulin-secreting
pancreas beta-cells.
47
This transition between two adjacent attractors can be externally
triggered by the ectopic expression of a combination of regulatory genes or possibly by small
molecules that modulate the activities of the appropriate set of genes. Such manipulations
amount to the perturbation of a set of network nodes, which moves the state of a cell in state
space. The ease of exit from an attractor is dictated by the gene network via the ODEs and
depends on the direction of the perturbation (
'
push
'
) as computed in the next step.
STEP 4: CALCULATE THE LEAST ACTION PATH (LAP) FOR
THE TRANSITION BETWEEN CELL ATTRACTORS
Once the systems equations are defined and the relevant attractors (departure and destination
states) are identified, we can determine the nature of the perturbation such that it follows
the course of the LAP for the desired phenotype switch. This can be computed using the
Freidlin-Wentzell formalism based on the same knowledge of network specification that has
allowed us to compute the attractor states.
