Biology Reference
In-Depth Information
ordering of (metastable) attractor states of the system in
Eq. 5.1
. Specifically, for
U
to be
of meaning for this purpose we require that
U
:
(i)
satisfies
dU
x 6¼ x
and
dU
x
5
x
(where
x
are the stable steady-states,
which can be either fixed points or limit cycle) according to the stability theory of
Lyapunov;
27
(ii)
is related to the Freidlin-Wentzell potential
V
that in turn expresses the
0 for
0 for
dt
,
dt
5
'
(LAP) between two states and hence, captures the barrier height
U
AS
.Thus,
U
permits the
computation of the transition rate.
26
Note that (
i
) expresses the stability property of attractor
state in a nonlocal sense; and (
ii
) refers to the relationship of any two points.
'
the least action path
The Quasi-Potential Function for Nonintegrable Systems
Since high-dimensional, nonequilibrium systems generally are not gradient systems, i.e.
Eq. 5.1
is usually not satisfied:
d
x
dt
5
F
ðxÞ 6¼
2
r
U
(5.4)
By contrast, one can enforce a partial notion of a quasi-potential
U
, if we write the driving
force as a sum of two terms:
d
x
dt
5
F
ðxÞ
52
r
U
F
r
(5.5)
1
where F
r
is the
beyond the component of the driving force that takes the
form of a potential gradient. Thus, we decompose the nongradient vector field F(x), which
needs to be finite and smooth (at least twice differentiable), into two components: one
that is the gradient of some
'
remainder
'
function
U
and the second that represents the
remainder of the driving forces. The question then is: what is the physical meaning of these
terms? Given that there are infinite ways of decomposing a vector field into a sum of two
fields, uniqueness of decomposition must come from imposing constraints through which
we can incorporate the physical meaning. Our objective here is to find a decomposition
such that the quasi-potential difference
'
potential-like
'
88
of the barrier between
the two attractors represents exactly the associated state transitions, whereas the
Δ
U
that manifests the
'
height
'
'
remainder
'
of the driving force F
r
will not contribute to the efforts needed for the transition.
The Normal Decomposition
One imposed constraint is that the gradient term is perpendicular to the
force
F
\
(hereafter the notation
U
norm
, F
\
instead of the general
U
, F
r
implies that these two
vector field components are perpendicular to each other):
'
remainder
'
U
norm
(5.6)
Then, one can show that
U
norm
is a Lyapunov function
27
and satisfies our condition (i)
above. This is shown below for a two-dimensional system by taking its time derivative
along any trajectory driven by the dynamic system in
Eq. 5.1
:
F
52
r
F
\
1
dU
norm
dt
U
norm
@
U
norm
@
5
@
1
@
_
_
x
y
x
y
U
norm
@
U
norm
@
U
norm
@
U
norm
@
5
@
2
@
1
@
2
@
F
y
\
F
x
\
x
1
y
1
(5.7)
x
y
2
2
U
norm
@
U
norm
@
52
@
@
U
norm
1
ðr
;
F
\
Þ
2
x
y
