Information Technology Reference
In-Depth Information
dv
i
dt
=
2
g(u
i
)v
i
(3.11)
d!
i
dt
= "(
4
g(u
i
)!
i
) + 2d(!
e
!
i
)
(3.12)
N
X
d!
e
dt
d
e
N
=
(!
i
!
e
)
(3.13)
i=1
where N is the total number of cells, u
i
and v
i
represent the proteins from which
the toggle switch is constructed in the i-th cell, w
i
represents the intracellular, and
w
e
the extracellular AI concentration. The mutual inuence of the genes is dened
by the functions:
w
1 + w
:
1
1 + v
;
1
1 + u
;
f(v) =
g(u) =
h(w) =
Here ; and are the parameters of the corresponding activatory or inhibitory
Hill functions.
In the Eqs. (3.10)-(3.13), the dimensionless parameters
1
and
2
regulate the
repressor operation in the toggle switch,
3
denotes the activation due to the AI,
and
4
the repression of the AI. The coupling coecients in the system are given
by d and d
e
(intracellular and extracellular) and depend mainly on the diusion
properties of the membrane, as well as on the ratio between the volume of the cells
and the extracellular volume [Kuznetsov et al. (2004)]. If the parameter " is small
("1), as in our case, the evolution of the system splits into two well-separated
time-scales, a fast dynamics of u
i
; v
i
and w
e
, and a slow dynamics of w
i
. Due to
the presence of multiple time scales, the system can produce relaxation oscillations.
The particular organization of the intercellular signaling mechanism in this case
allows coupling to be organized through the slow recovery variable in the genetic
network. As is known from oscillation theory, such coupling has the phase-repulsive
property and can be referred to as inhibitory. On the other hand, local coupling
of limit cycles via inhibitory variables has been reported to yield a coexistence of
dierent stable attractors [Volkov and Stolyarov (1991, 1994)], thus leading typically
to multirhythmicity.
The main manifestation of multistability in systems of globally coupled oscilla-
tors is clustering, dened as a dynamical state characterized by the coexistence of
several subgroups, where the oscillators exhibit identical behavior. Oscillator clus-
tering has been proved theoretically for identical phase oscillators [Okuda (1993)],
observed experimentally for salt-water oscillators [Miyakawa and Yamada (2001)]
and electrochemical oscillators [Wang et al. (2001); Kiss and Hudson (2003)]. For a
detailed recent review of synchronization in oscillatory networks see [Osipov et al.
(2007)]. As already mentioned in the repressilator case, the eects of multirhyth-
micity and multistability can be very important in understanding of evolutionary
mechanisms behind cell dierentiation and genetic clocks.
