Biology Reference
In-Depth Information
(B)
(A)
5
5
0.1
S
2
0.1
S
2
0
0
S
0
0.05
S
0
0.05
2
2
1
1
3
0
2
0
0
X2
3
S
1
1
0
0
2
0
X2
0
1
Time
2
1
X1
S
1
0
1
2
X1
Time
S
2
0.4
S
2
0.4
2
0.3
2
S
0
S
0
0.3
0.2
1
0.2
1
0.1
0
0
0.1
0
S
1
0
0
1
2
1
2
0
0
X1
Time
0
1
2
1
2
S
1
X1
Time
FIGURE 5.3
Least action paths on the quasi-potential landscape are computed based on the Freidlin-Wentzell theory. Least action path for attractor transition
(
'
transdifferentiation
'
) between the two cell attractors of the example in
Figure 5.1B
. (A) The least action path for transition from attractor S
2
-
S
1
.
Green
point is the starting point, red point is the end point. V(t) is the action function at every time step. V is the accumulative action function at each time t.
(B) The least action path for transition from attractor S
1
-
S
1
because this is a nonlinear dynamic
system. The Wentzell action function is the same as for the transition from S
2
-
S
1
because attractors S
1
and S
2
have the same
'
height
'
S
2
,whichisdifferentfromthatofpathS
2
-
in
the quasi-potential U
norm
.
94
Since the dynamics of this network is structurally quite robust, we can use simple,
symmetric values of parameters to demonstrate the network dynamics. In this example,
n
1. The quasi-potential
function
U
norm
from the normal decomposition is shown in
Figure 5.3A
. It has
three attractors designated
S
0
(progenitor state),
S
1
(erythroid lineage exhibits the
GATA1
HIGH
/PU.1
LOW
pattern), and
S
2
(myeloid exhibits the GATA1
LOW
/PU.1
HIGH
pattern).
If we want to perturb the blood cell system to cause a transition from the myeloid to the
erythroid lineage, the associated Wentzell action functions and LAP are calculated from
the numerical minimization based on
Eq. 5.19
, as shown in
Figure 5.3B
. Note that the least
action path myeloid
4,
k
1
5
k
2
5
1,
θ
a
1
5
θ
a
2
5
θ
b
1
5
θ
b
2
5
0
:
5,
a
1
5
a
2
5
b
1
5
b
2
5
5
myeloid.
Such rather counterintuitive path irreversibility is a common characteristic of nonlinear
biological systems.
erythroid is different from the reverse path erythroid
-
-
The Specific Pancreas Development GRN and the ODE Model
In the second example, we return to pancreas cell reprogramming to illustrate the above
principles of how to calculate a state transition given the partial knowledge of the
underlying GRN. We first model the normal differentiation of the main pancreas cell
lineages
as cell behavior that takes place on the quasi-potential landscape: the exocrine
and the endocrine cells, including the
islet cells. Using ODEs to describe the
mutual regulatory influence of 10 TFs involved in the differentiation of these cells during
pancreas development, we present a minimal model that qualitatively captures known
interactions and is able: (i) to recapitulate normal pancreas cell differentiation; (ii) to
predict the temporal changes of key TFs during the development of particular cell lineages;
and (iii) to predict the outcome of perturbations and hence, to help design new recipes
of reprogramming experiments.
46
β
,
δ
, and
α
