Chapter 7 Dynamical Behaviours of Single Feedback Systems: Analysis, Results and
7.1 Systems with a Single Feedback Loop
7.1.7 Effects of Turnover Parameters on Bifurcation Pattern and System Dynamics
system’s bifurcation characteristics and implications on system dynamics. Since A and B are symmetrical expressions, K1thresh is also symmetrical with respect to the degradation as well as
the synthesis rates. This means that all system species equally affect the system’s bifurcation characteristics in spite of the fact that the feedback loop is acting only on the first reaction of the pathway.
About the synthesis rates, the threshold value K1thresh changes proportionally with these
parameters. Increase in the production of any of the model species therefore gives rise to a more oscillatory-prone system, indicated by a larger oscillatory region in the two-parameter K1
the synthesis rate of any system species equivalently results in exactly the same K1thresh curve
as raising the synthesis rate of any other species by the same proportion. Bifurcation patterns are therefore conserved under these quantitatively different changes. This knowledge is potentially useful in many cases. For example, it can aid the engineering of synthetic circuits with desirable targeted dynamical behaviour; since one could effectively choose appropriate points for perturbation to attain the desired dynamics. It can also help in the process of parameter estimation and optimisation of synthetic circuits.
Unlike the synthesis rates, the degradation rates affect system dynamics in a more complex and highly nonlinear manner: increasing the degradation rates do not always result in a more oscillatory system. As we identified previously, each degradation rate has its own oscillatory range. Raising it within this range initially results in an increasingly more oscillatory system but as it reaches the high end of its range, the system becomes less oscillatory. This effect is not only demonstrated in Figure 7.2 but also on Figure 7.7 when we assess how change in one degradation parameter (kd1) influences the bifurcation pattern of K1 against another
degradation parameter (kd2): the oscillatory region starts small, expands and shrinks as kd1
moves along its axis. Therefore, very low or very high degradation rates generally tend to stabilise the system.
We have discussed previously the notion of an optimal kd1 for oscillation (section 7.1.5),
which is a legitimate measure when K1 is used solely as an indicator for oscillatory capability:
the optimal kd1 gives the largest oscillatory range for K1. The 3D bifurcation diagram in Figure
7.7 of K1 vs. kd1 and kd2 further indicates an optimal pair (kd1,kd2) corresponding to the
surface’s peak with highest K1.However, when multiple parameters are used as indicators for
the likelihood of the system exhibiting oscillations, the notion of optimality is compromised. For example, Figure 7.7 (left panel) shows that although the range of K1 and the overall area of
the oscillatory region reduce as kd1 increases from 1.3 to 3, the range of kd2 available for
oscillation actually slightly expands. This suggests that optimality should be treated as a relative concept in system dynamics.
Figure 7.6.
(a) Comparison of the K1 vs. n1 bifurcation diagrams for different scenarios. K1 vs. n1 bifurcation
diagram for the base parameter set, k1=1, k2=2, k3=1, kd1=1.2, kd2=3, kd3=2, (solid); when a synthesis rate
k1 is doubled (dashed); and 10-times increased (dot). Note that exactly the same effects are also
obtained for changing other synthesis rates.
(b) Effect of changing k1 on the K1 vs. kd1bifurcation diagram. The oscillatory region, indicated by O,
expands as the synthesis rate k1 increases (parameter values n1=10, k2=2, k3=1, kd1=1.2, kd2=3, kd3=2 and
k1 = 0.5, 1, 5 were used for graphing).
Figure 7.7.
Effect of changing kd1 on the K1 vs. kd2 bifurcation diagram. We plot on the same graph the oscillatory
region of three scenarios with increasing kd1 of 0.7, 1.3 and 3. The corresponding curves of these regions
Figure 7.8.
3D plot of B for different combinations of kd1 kd2 and kd3.We generate here 500 random combinations of
the parameters (kd1, kd2,kd3) in the range (0, 5). Each combination has its corresponding B value
calculated and represented by a ball such that the value of B is indicated by the size of the ball. Clearly, most of the small balls (small B) are concentrated in the vicinity of the kd1=kd2=kd3 line (red) while large
When examining parameter effects on the threshold value of the Hill coefficient (B), our analysis reveals that comparable degradation rates across model species (kd1≈ kd2 ≈ kd3) leads to
minimum B and thus the minimum RGS for n1; whereas if one is many folds greater than
another (kdi>>kdj, i j, ∈
{
1, 2,3}
), B will be high, resulting in a large RGS (see Appendix B1.1.6 in for justification). This is demonstrated explicitly in Figure 7.8 in which we generate a large number of random triples (kd1, kd2,kd3) and display their corresponding B values.Clearly, most of the small balls - indicating small B – tend to concentrate in the vicinity of the diagonal line (kd1=kd2=kd3, in red) while large balls - indicating large B – tend to be far away
from this line and sit near the edges of the cube. This suggests a way to enhance system stability by unbalancing the degradation rates of molecular species, preferably, towards either lower or higher values.