Biology Reference
In-Depth Information
of intuitive explanations as presented above and enter the realm of abstract formalism.
To stay focused, however, the practical utility for cell-type modulation will guide us in the
following discourse.
The stable steady-states (attractors) in the complex GRN represent so-called
'
dissipative
structures
that are sustained steady-states far from thermodynamic equilibrium, because
their maintenance requires continuing intake of free energy and energy dissipation.
24
Transitions between attractor states correspond to cell phenotype switches and are affected
by the
'
that separates these attractors.
These metaphoric concepts are not arbitrary but become necessary in multiattractor systems
when one has to consider attractor transitions which are frequent and integral to the
behaviors of complex biological systems. Such behavior where attractor exit is the norm and
not the feared extreme are rarely encountered in engineered systems where one is interested
in the stability around a single attractor state. In dynamical systems theory, such
'
relative depth
'
or, equivalently,
'
relative barrier height
'
'
local
'
stability is evaluated by linear stability analysis with multiple attractors that are
'
agnostic of
each other.
Thus, the landscape notion arises when one confronts the problem of nonlocal
stability and wishes to relate a set of attractors to each other with respect to their
'
'
relative
stability
and the associated transition rates. If we want to manipulate biological
functionality by prescribing specific transitions from one attractor to another, one needs to
understand the landscape topography associated with a given GRN. This goes beyond
traditional dynamical systems analysis that focuses on existence and local stability of
attractors. In the following sections, we will introduce a mathematical framework to analyze
'
'
relative stability
'
of multistable dynamic systems.
The Absence of Integrability and the Essence of a Potential Function
In a multistable system with
M
attractor states x
1
;
x
2
; ...
x
M
(where x
i
is the
N
-dimensional
state vector that defines the position of attractor
i
), one thus wishes to obtain some sense of
the
86
relative depth,
25
or more precisely, of the ordering of the attractor states with respect to
their (meta)stability through some energy-like quantity,
U
1
,
U
2
,
U
3
...
'
U
M
, associated with
each steady-state. This becomes relevant when we design biological processes associated
with trajectories that go through a certain set of attractors. Such a potential energy-like
function
U
(x) for any point x in the state space of the system could be used to determine
'
'
that could inform about the probability and direction of transitions
between attractor states in a noisy or perturbed system.
potential differences
Let us consider a deterministic system (network) of
N
variables
x
i
(e.g. the activity
of interacting genes) whose values describe the cell state
x
(
t
)
,
x
N
(
t
))
T
and whose dynamics result from how each gene influences the activity of other genes
(as invariantly predetermined by the gene regulatory network that is hard-coded in
the genome). Such dynamics is described by the first-order ordinary differential equations
(ODEs) which in general are nonlinear:
(
x
1
(
t
),
x
2
(
t
),
...
5
dx
1
dt
5
F
1
ð
x
1
;
x
2
;
?
;
x
N
Þ
dx
2
dt
5
(5.1)
F
2
ð
x
1
;
x
2
;
?
x
N
Þ
^
or in vector form:
d
x
=
dt
F
ð
x
Þ
5
F(x) represents the equivalent of
,
x
N
)
in this inertia-free system. If we have a gradient system then a potential function
U
can be
'
forces
'
acting to change the system state x(
x
1
,
x
2
,
...
