Biology Reference
In-Depth Information
As long as we are within the same attractor, the Freidlin-Wentzell potential is exactly twice
as large as the potential from the normal decomposition,
V
AB
5
U
norm
B
U
norm
A
2
ð
2
Þ
.
When a
transitions from one attractor to another, the Freidlin-Wentzell potential
only accounts for the uphill
'
ball
'
which is two times that of
U
norm
. Once it goes over
the saddle point (point
X
s
in
Fig. 5.2
), the Freidlin-Wentzell potential
V
is zero for the
remaining
'
energy,
'
path. The same applies for a ball that transitions through
many
attractors in between: the Freidlin-Wentzell potential is equal to the sum of all
uphill potential between two points. All downhill paths contribute nothing to the
Freidlin-Wentzell potential.
'
free-fall
'
HOW TO OBTAIN A TRAJECTORY ON THE QUASI-POTENTIAL
LANDSCAPE FOR TRANSITION BETWEEN TWO ATTRACTORS
The landscape notion offers a new vista that captures the intuition of state transitions
by displaying this process as a jump between valleys. This picture has recently become
fashionable to illustrate cell-type reprogramming and epigenetic regulation.
15,30
However,
the link to the underlying gene regulatory network and the theoretical description of its
global dynamics has not been articulated.
To demonstrate how the formal conceptualization of the landscape as a product of the
GRN allows us in principles to build a landscape and determine the transition trajectories;
we describe below five steps which one has to take in a typical setting in biology
considering the paucity of data about the system determinants.
STEP 1: BUILD THE GENE REGULATORY NETWORK (GRN)
FROM INCOMPLETE INFORMATION
To modulate the cell phenotype by triggering an attractor switch, we obviously need to
know the structure of the underlying dynamical system, i.e. the gene regulatory network.
Unfortunately, only for a few biological systems is there sufficient information of a GRN
available that permits the straightforward translation of GRN specification into a complete
dynamical system in the form of
Eq. 5.1
. Much effort is currently spent in such
91
'
system
identification
based on data integration, inference from gene expression profile dynamics,
or direct molecular characterization of gene regulatory interactions using high-throughput
experimentation such as ChipSeq.
31,32
Nevertheless, the obtained information pertains to
the network architecture and remains sketchy. Not a single gene regulatory function that
maps the inputs on the promoter to the output (gene expression), as needed to write the
systems equations (
Eq. 5.1
), is even remotely known for any mammalian gene. Hence most
mathematical modeling approaches that treat GRNs as dynamical systems have to use an
educated guess from qualitative experiments (e.g. overexpression studies to determine
whether a regulator is inhibiting or activating) based on information in the literature from
isolated experiments. Given the sparseness of information, mathematical models have been
limited to small subnetworks (
'
'
circuits
'
) of a handful of genes.
STEP 2: TRANSLATE THE GRN INTO MATHEMATICAL FORMULA AND
IDENTIFY FUNCTIONING ATTRACTORS IN GENETIC LANDSCAPE
Assuming a GRN has been constructed for a given biological system, we can use a mathematical
formalism, such as Boolean network or ODEs, to breathe life into the static GRN structure
by translating it into mathematical equations that animate the dynamics of the GRN. The
central challenge is to define the
at each network node: to map the values of
the inputs to a gene
X
, that is the presence and activity of its upstream regulators at a given
time into its change of the expression level,
X
(
t
). As mentioned above, almost no explicit
molecular-level knowledge exists for the nature of the transfer functions in mammals, and thus
they must be inferred from individual experiments that describe causal relationships.
'
transfer function
'
