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obtained by integration, i.e. where there exists a function U int (
x
) with the following
properties:
@
U int
@
@
U int
@
@
U int
@
F 1 ðxÞ; ? ;
F i ðxÞ; ? ;
F n ðxÞ
(5.2)
x 1 52
x i 52
x n 52
One can obtain a potential function U int (
x
) by successive integration,
dU int
ðxÞ 52 ð
F 1
ðxÞ
dx 1
F i
ðxÞ
dx i
F N
ðxÞ
dx N
Þ;
1?1
1?1
ð F 1
(5.3)
U int
ðxÞ 52
ðxÞ
dx 1
F i
ðxÞ
dx i
F N
ðxÞ
dx n
1?1
1?1
If U exists then the driving force is the gradient of U . However, unlike the equilibrium
system, the transition rate for
is not path-independent here. Its spontaneity is
determined by the potentials of attractor states and saddle points between them. The
transition probability P x A - x B
x A - x B
U int
AS
U int
U int
and the transition
follows the least action path , 26 as discussed below. Here, x S is the saddle point between two
attractors
is related to
Δ
ðxÞ 5
ðx S Þ 2
ðx A Þ
x B , as shown in Figure 5.2 .
Biological dynamical systems with N
x A and
1 are typically nonintegrable, nongradient systems,
i.e. a function U cannot be obtained by integration, and a pure potential function in
general does not exist. Yet, there is validity in a formal notion of an equivalent quantity.
It is loosely referred to as
.
Such a function would allow us to compute
transition trajectories if properly defined such that it can serve as a quantity for the
'
quasi-potential.
'
87
Stem cells
Reprogrammed
stem cell
Progenitor cells
X B
X S
X A
Epigenetic
barrier
Terminally differentiated cells
in the 'low' attractors
FIGURE 5.2
The quasi-potential landscape as a formal concept for cell differentiation and reprogramming. The figure shows a
schematic quasi-landscape that is computed from the normal decomposition of the dynamic system, which is constructed
from a gene regulatory network. Here, for representation, the high-dimensional state space (N c 2 genes) is compressed
and projected into a two-dimensional plane, whereas the ' elevation ' represents the quasi-potential U associated with every
state. The landscape captures the global dynamics, enabling the comparison of ' relative depth ' of different attractors,
which is equivalent to Waddington ' s epigenetic landscape. X A and X B are two cell attractors in the landscape while the
saddle point X S is in between them. The downhill movement (in yellow) of the ball represents the differentiation process
from stem cells to progenitor cells, and to terminally differentiated cells. The transition paths (in red) of the ball represent
the cell reprogramming which require external stimulus and do not happen often.
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