Biology Reference
In-Depth Information
Because the Freidlin-Wentzell action function is usually too complicated to be found
analytically,
Eq. 5.16
can be rewritten in discrete form to find an approximate solution,
as shown below:
"
#
t
X
X
M
1
n
2
2
x
k
1
1
i
x
i
1
2
ð
2
f
k
1
1
i
f
i
Þ
V
AB
ð
X
ð
t
ÞÞ
5
Δ
(5.19)
2
2
Δ
t
i
1
k
5
1
5
Here the total time over the trajectory X(
t
) is divided into
M
t
. The time
integral of the action function is approximated with the sum of actions in the small time
segments. The rate
dx
dt
1 equal parts
Δ
2
is approximated with the first-order difference equation
x
k
1
1
x
i
2
i
t
.
To find the least-action trajectory in discrete form, initially two attractors are connected with
a straight line and the conjugate gradient method (CG) is used to minimize the action
function
V
AB
(
X
(
t
)).
48
Δ
STEP 5: DESIGN THE PERTURBATIONS NEEDED TO
DRIVE THE TRANSITION BASED ON THE LAP
When the LAP is calculated among the designated attractors, one will note that some
genes never change their expression values while others are significantly modified as the
cell moves along the LAP. Thus, the LAP serves as the first filter to select the genes whose
expression needs to be modified to most efficiently drive the transition and which genes
can be left unchanged. Besides identifying the list of genes that change during the
state transition, the LAP also provides information on the detailed temporal profile of
the expression behavior of each gene. Ideally, if we can find a
'
which exactly modifies gene expression levels such that they collectively follow the LAP,
this perturbation would be the most efficient way to induce that state transition.
However, in reality this can currently hardly be realized because of a series of problems
associated with experimentation, including intrinsic noise of the cell state (which
changes the position of
'
perturbation recipe
93
(
t
)), time delays between manipulation (transfection, drug
treatment) and desired gene expression change, the lack of precise control of expression
levels, etc.
x
Thus, if the LAP computation is to benefit reprogramming it will be in cases where the
cursory initial direction of the computed LAP trajectory deviates significantly from what is
intuitive. Such a deviation could be implemented qualitatively to improve transition
efficiency.
EXAMPLE: STATE TRANSITION IN BLOOD CELL AND PANCREAS
CELL DIFFERENTIATION AND REPROGRAMMING
The ODE Model of Blood Cell Differentiation and Reprogramming
Here, we use our example of blood cells for which a qualitative description was made
earlier to demonstrate how to find the Freidlin-Wentzell least action path. Based on the
cross-inhibition and self-activation between GATA1 and PU.1 (
Fig. 5.1B
), the governing
equations can be written as:
8
<
n
b
1
x
n
θ
dx
dt
5
F
1
ð
x
;
y
Þ
5
a
1
b
1
k
1
x
x
n
1
y
n
2
n
a
1
1
n
b
1
1
θ
θ
(5.20)
n
b
2
:
y
n
θ
dy
dt
5
F
2
ð
x
;
y
Þ
5
a
2
b
2
k
2
y
y
n
1
x
n
2
n
a
2
1
n
b
2
1
θ
θ
