Biology Reference
In-Depth Information
pattern jointly established by the levels of X
1
and X
2
, changes over time:
S
(
t
)
[X
1
(
t
), X
2
(
t
)].
The change of
S
is visualized as the movement of the state
S
in the two-dimensional
state
space
spanned by the X
1
and the X
2
axis. The dynamics of this 2-gene network is constrained
because X
1
and X
2
alter their levels in a coordinated manner
5
as prescribed by the network
interactions (explained in
Fig. 5.1A
). As a consequence, most (X
1
,X
2
)-configurations, or states
S
, of the network are not stable but experience a
'
'
of change until all the regulatory
forces are balanced. In this case of mutual repression the dynamics is such that one typically
finds two discretely distinct stable stationary states,
S
1
and
S
2
(asterisks indicate
force
'
stability
'
).
These stationary states are stable
attractors
because in the
state space
of all possible gene
expression patterns they attract nearby states (
Fig. 5.1A
).
21
As argued above they represent
the two alternative cell types. The term
is used to describe such dynamical
behaviour in which the very same circuit can chose to be in either one of the two attractor
states
'
bistability
'
depending on which
'
basin of attraction
'
in the state space was its initial state,
[X
1
(
t
0
), X
2
(
t
0
)].
22
Bistability and the associated inverse expression patterns of X
1
and X
2
in the two attractor
states have been used to explain binary cell fate decisions in which an immature
uncommitted multipotent cell can make a decision to commit to either one of two lineages
(
S
(
t
0)
5
5
). For instance, in a type of blood cell progenitor that faces the decision to become a
cell of the erythroid or the myeloid lineage (roughly, red versus white blood cell lineages),
the erythroid lineage exhibits the GATA1
HIGH
/PU.1
LOW
expression pattern, whereas the
reciprocal myeloid lineage, has the inverse GATA1
LOW
/PU.1
HIGH
pattern. The model can
also incorporate another feature observed in many of such bistable networks: the two
antagonist fate-determining factors not only repress each other (
Fig. 5.1A
), but often also
seem to activate their own expression (
Fig. 5.1B
). This constitutes a different network
architecture, hence a different dynamical behavior. Various mathematical models of the
cross-inhibition
'
fates
'
auto-stimulation circuit motif indicate that this departure from the
classical bistable circuit structure (
Fig. 5.1A
) typically leads to the
stabilization
of the
symmetrical bipotent state
S
0
, which becomes a locally stable attractor.
23
Thus, we actually
have a tristable system (
Fig. 5.1B
). A cell differentiation corresponds to the process in which
a progenitor cell at central state (X
1
MID
/X
2
MID
) transitions to either the left attractor with
X
1
HIGH
/X
2
LOW
expression pattern or the right attractor with X
1
HIGH
/X
2
LOW
expression pattern
(
Fig. 5.1C
).
85
We have here without equations given the intuition of network dynamics and the attractor
states it produces. The above two-gene system, widely used in nature to control binary cell
fate decisions, produces three attractor states, and thus constitutes a toy model for a
multistable system. State transitions during cell fate commitment and subsequent
differentiation can be viewed as transitions between attractor states
which so far are simply
attracting points in a state space. But in which direction do spontaneous, noise-driven
attractor transitions occur? This question of relative (meta)stability of attractors is of central
biological significance, for it determines natural cell fate choice by multipotent progenitor
cells and the ease of artificially induced fate switches (reprogramming). To conceptualize and
quantitatively define attractor transitions and their rates, it is instrumental to introduce the
notion of
of attractors. This is
the reason for the concept of potential-like landscapes which has been proposed even before
the specific notion of gene networks and of attractor states in their dynamics.
'
barriers
'
(height) between attractors and the
'
relative depths
'
THE QUASI-POTENTIAL LANDSCAPE
Biological systems of course are not just 2-gene circuits but high-dimensional dynamical
systems that exhibit a large number of stable steady-states (attractors). Dealing with systems
with more than two attractors, which is rather uncommon in synthetic systems, is where the
landscape metaphor particularly comes in handy. This will prompt us to leave the domain
