Figure 1.
The reaction scheme of the chained modification model.
Schematic representation (A) and reaction diagram (B) of our model. A substrate has modification sites. Modification reactions for the substrates progress without catalyst at rates
and demodification reactions are facilitated by the catalyst at rates
.
Figure 2.
Slow logarithmic relaxation of the chained modification model.
The initial condition is set as and
for other
, and the relaxation process of the modification level is computed without input (
). The parameters are given as
,
, and
. The time courses for the modification level for different values of the catalyst concentration,
,
,
,
,
,
,
,
, and
, are plotted with different colors, where the concentration of
is fixed at
. Although exponential relaxation is observed as in first-order reactions (dotted line) when the concentration of the catalyst is sufficiently large, the relaxation is drastically slowed as the concentration of the catalyst becomes lower than that of the substrate.
Figure 3.
Change in the dependence of the catalyst on the relaxation time against
, the dissociation constant (A), against
, the heterogeneity of the dissociation constant (B), and against
, the number of modification sites (C).
is plotted against
;
is defined as the time when the summation of all of
falls below the threshold value(
) without input, starting from the initial condition (
). (A)
,
,
. (B)
,
,
. (C)
,
,
.
Figure 4.
The time course of for each
is plotted by setting the initial conditions as already described. (A)
's relax in descending order, in the same manner as in the first-order reactions. (B)
's relax in ascending order, that is, converse to the order expected from the first-order reactions. The highly modified state relaxes only after the relaxation of the less-modified
. The relaxation process consists of several plateaus, which are typically observed in the relaxation process of kinetic glass [25].
Figure 5.
Dependence of the relaxation process on the magnitude and duration of a stimulus.
(A) The relaxation time after exposure to the stimulus with various magnitudes and durations is plotted as a color map. The initial condition is given as and
for
, and the input is given as
for
and
for
. When the magnitude (
) and duration of the stimulus (
) increase,
increases continuously over an order of magnitude. The catalyst concentration is set at
of the substrate concentration. (B) Dependence of the relaxation process on the duration of stimulus exposure. The duration of stimulus exposure is changed while the magnitude is fixed at
. Here, the relaxation time increases nearly exponentially with the increase in duration for the some extent small
. When
is sufficiently long, the modification is maintained for a long time.
Figure 6.
Kinase-phosphatase model (A) and its relaxation time (B).
The relaxation times of the variables
are plotted against the total concentration of phosphatase.
is defined as the time when the summation of
of all
falls below the threshold value, after relaxation at a kinase-rich condition (
). The model shows the transition from fast exponential relaxation to slow logarithmic relaxation at the critical point (
). (When the amount of the phosphatase is lower than that of kinase (
), the relaxation time itself is shorter, whereas the logarithmic relaxation remains. Here, the stable fixed-point value of the concentration
changes to a higher value, and the relaxation time is decreased.)
Figure 7.
The extended Asakura-Honda model (A) and its slow logarithmic relaxation after exposure to an environmental stimulus (B).
After the system is relaxed in the presence of the attractant as , the system transitions to a repellant condition as
, and the relaxation process of
is computed. The parameters are given as
,
, and
. The time courses of
for different values of the catalyst concentration,
,
,
,
,
,
,
,
, and
, are plotted with different colors, where the concentration of
is fixed at
. Although exponential relaxation is observed as in the original A-H model (dotted line), when the concentration of the catalyst is sufficiently large, the relaxation is drastically slowed as the concentration of the catalyst becomes lower than that of the substrate.