Figure 1.
A basic set of two-component homeostatic controller motifs with two implementations of integral control.
(a) Compound is the homeostatic controlled variable and
is the controller or manipulated variable [22]. The motifs fall into two classes termed as inflow and outflow controllers, dependent whether their compensatory fluxes
add or remove
from the system. In motifs outlined in gray the controller compound
inhibits the compensatory flux, while in the other motifs
activates the compensatory flux. (b) middle figure shows a standard control engineering flow chart of a negative feedback loop, where the negative feedback results in the subtraction of the concentration of
(blue line) from
's set-point (red line) leading to the error
. The error feeds into the integral controller (brown box). The controller output (the integrated error) is the concentration of
(green line) which regulates the process that creates
. The perturbations which affect the level of
are indicated in orange color. (b) left panel shows the structure of negative feedback (outflow) controller 5. The colors correspond to those of the control engineering flow chart. For example, the set-point (red) is given by the ratio between removing and synthesis rates of
, while the integral controller (brown) is related to the processing kinetics of
, in this case
is removed by zero-order [22], [43]. (b) right panel shows the same outflow controller (motif 5). The only difference is that the integral controller is now represented by a first-order autocatalytic formation (indicated by brown dashed arrow) and a first-order removal with respect to
[45].
Figure 2.
Representation and kinetics of conservative oscillators based on motif 2 and motif 5.
(a)–(c) “Goodwin's oscillator” (motif 2). Conservative oscillations occur when and
; the latter condition introduces integral feedback and thereby robust homeostasis [22], [43]. (b) Conservative oscillations in
and
, with
,
,
,
,
,
,
,
. Initial concentrations:
,
. At time t = 50.0
is changed from 1.0 to 3.0. (c)
,
, and frequency as a function of the perturbation
. While the frequency increases and
decreases with increasing
,
is kept at its set-point
. (d)–(f) Harmonic oscillator representation of motif 5. Conservative (harmonic) oscillations occur when
(or
) and
. (e) Harmonic oscillations in
and
, with
(the perturbation),
,
,
,
,
, and
. At time t = 50.0
is changed from 1.0 to 3.0. Initial concentrations:
,
. (f)
,
, and frequency as a function of the perturbation
. Typical for the harmonic oscillator is the constancy of the frequency upon changing
values.
increases with increasing
, while
is kept at its set-point
.
Figure 3.
A limit-cycle model of controller motif 2.
(a) Reaction scheme. Rate equations:
;
;
. (b) Homeostatic response of the model for three different perturbations (
values). For time
between 0 and 50 units,
, for
between 50 and 100 units,
, and for
between 100 and 150 units,
. In the oscillatory case
at time
is given as
(ordinate to the right) showing that
is under homeostatic control despite the fact that
peak values may be over one order of magnitude larger than the set-point. (c)
,
, and frequency values as a function of
. Simulation time for each data point is 100.0 time units. Note that
is kept at
independent of
. Rate constant values (in au):
,
,
,
,
,
,
, and
. It may further be noted that the degradation kinetics with respect to
are no longer zero-order as required in the conservative case (Figs. 2a–c). Initial concentrations in (b):
,
, and
. Initial concentrations in (c) for each data point:
,
, and
.
Figure 4.
A limit-cycle model of controller motif 5.
(a). Rate equations: ;
;
. (b) Homeostatic behavior in
illustrated by three different perturbations (
values). At time
is changed from 4.0 to 10.0, and at
is changed from 10.0 to 20.0 (indicated by solid arrows). The set-point of
is given as
. Rate constant values:
is variable,
,
,
,
,
,
, and
. Initial concentrations:
,
, and
. (c)
,
, and frequency values as a function of
showing that
is kept at the set-point independent of
. Rate constants as in (b). Initial concentrations for each data point:
,
, and
. Simulation time for each data point is 10000.0 time units.
Figure 5.
Quasi-harmonic behavior of motif 5 oscillator (Fig. 4a).
For time , a perfect overlay between the numerical calculation of
(blue color) and the single harmonic
(black color) is found, where
,
,
,
, and
.
and
represent the numerically calculated amplitude and period length, respectively.
was adjusted to give a closely matching overlay. Other rate constant values (numerical calculations):
,
,
,
,
,
, and
. Initial concentrations:
,
, and
. At times
and
(solid arrows)
is changed to respectively 5.0 and 10.0. For these
values the amplitude of
has reached its maximum, which is twice the value of the set-point. (b)
,
,
, and frequency as a function of
. Simulation time for each data point is 1000.0 time units. (c) Demonstration of limit-cycle behavior of the quasi-harmonic oscillations. Same initial conditions as in (a) with
, and
. (d) Same system as in (a), but at times
and
(solid arrows)
is changed and kept to 0.1. The oscillations are efficiently quenched, but
remains under homeostatic control.
Figure 6.
Oscillator based on motif 5 with robust frequency control.
(a) Reaction scheme. Rate equations: ;
;
;
;
. (b) Demonstration of robust frequency control.
,
, and frequency are shown as functions of
. Rate constants:
,
,
,
,
,
,
,
,
,
,
,
,
, and
. Set-points for
by controllers
and
are given as
and
, respectively. Initial concentrations for each data point (black dots):
,
,
,
, and
2.7657 × 102. Gray dots show the frequency as a function of
without control by
and
. (c) System as in (b), but controller
not present. (d) System as in (b), but controller
not present. (e) Reaction scheme of oscillator, but with a constant
concentration. Rate constants otherwise as in (b). (f) Frequency as a function of
for the system described in (e) using different constant
concentrations (indicated within the graph). The homeostatic region of the frequency increases with increasing
concentrations.
Figure 7.
Oscillator based on motif 2 with robust frequency control.
(a) Reaction scheme. Rate equations: ;
;
;
;
. Shaded area indicates part of the model for which the control coefficents of the frequency/period with respect to the parameters within this area become zero when frequency homeostasis is enforced by controllers
and
. (b) Demonstration of frequency homeostasis by varying
. Black dots show the frequency when controllers
and
are active. Rate constants:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, and
. Set-points for
by controllers
and
are given as
and
, respectively. The set-point
of the outflow controller
has been set slightly higher than
for the inflow controller
to avoid integral windup and that the controllers work “against” each other [22]. Initial concentrations for each data point (black dots):
,
,
,
, and
. Gray dots show the frequency as a function of
for the uncontrolled case, i.e., in the absence of controllers
and
. (c) System as in (b), but controller
is “knocked out” by setting
and
to zero. Homeostasis occurs only at high
values when controller
is active. (d) System as in (b), but inflow controller
is inactivated by setting
and
to zero. Frequency homeostasis is observed for low
when controller
is active. At high
values the frequency homeostasis breaks down, because controller
is not present to compensate the increased outflow of
, which leads to low
values. (e) Oscillations of system in (b) illustrating frequency homeostasis. At time
(solid arrow)
is changed from 3.0 to 8.0. Initial concentrations:
,
,
,
, and
.
Figure 8.
Oscillator based on motif 2 with robust frequency control but alternative feedback regulation by and
.
(a) Reaction scheme. Rate equations: ,
;
;
;
. (b) Using different set-points
and
, the frequency (solid dots) can switch between two homeostatic frequency regimes, dependent whether
is low or high. The two regimes are separated by a transition zone. Rate constants:
,
,
,
,
,
,
,
,
,
,
,
,
,
. Initial concentrations:
,
,
,
, and
. (c) Oscillations of system in (b) illustrating frequency switch. At time
(solid arrow)
is changed from 2.0 to 8.0. Initial concentrations:
,
,
,
, and
.
Figure 9.
A limit-cycle model of controller motif 2 using autocatalysis as an integral controller.
(a) Reaction scheme. Rate equations: ;
;
. (b) Homeostatic response of the model for three different perturbations (
values). For time
between 0 and 250 units,
, for
between 250 and 500 units,
, and for
between 500 and 750 units,
.
at time
is defined as in Fig. 3. (c)
,
, and frequency values as a function of
. Simulation time for each data point is 2000.0 time units. Note that
is kept at
(solid black line) independent of
. Rate constant values (in au):
,
,
,
,
,
, and
. Initial concentrations in (b):
,
, and
. Initial concentrations in (c) for each data point are the same as in (b).
Figure 10.
A homeostatic model of cytosolic Ca2+ oscillations.
The model considers a stimulated non-excitable cell under stationary conditions using an extended version of outflow controller motif 6, where is the controller molecule. Intermediate
has been included to get limit-cycle oscillations. Rate constant
describes the total inflow of Ca2+ from the ER and from the extracellular space into the cytosol and reflects the strength of the stimulation. For the sake of simplicity the external Ca2+ concentration (
) is considered to be constant (
).
denotes cytosolic Ca2+ and its concentration. (a) Reaction scheme. Rate equations:
;
;
. Rate constants:
, variable;
;
;
;
;
;
. The homeostat's set-point for Ca
is given by
. (b)
oscillations and average cytosolic Ca
concentration,
, at different stimulations and as a function of time
. Initial concentrations:
,
;
. The quenching of oscillations at low
is due to an increased
value. (c) Period length and average cytosolic Ca2+ concentration (
) calculated after 2000 time units for different stimulation strengths (
values). Same rate constants as in (b) with
. Initial concentrations for each calculated data point:
,
;
.