Chapter 7. Simulation results and testing
7.4 Local sensitivity analysis
In order to maintain a robust circadian clock, experimental observations in various biological systems showed that interlocked feedback loops are the key (Cheng et al. 2001). As discussed in the previous sections, to maintain the rhythmic oscillation of circadian clock components, the effective translation of clk, per and tim genes resulting in the synthesis of their translated proteins is very important. Thus it is well known that CLK transcriptional activation loops with PER/TIM self-repression loops are vital. With the inclusion of a CWO negative feedback loop in our new models, we have the opportunity to test the role of the CWO loop in maintaining the robustness of the circadian clock. Wet-lab analysis has suggested that CWO acts as a behavioural rhythm amplifier and that it has a modulatory effect in repressing the CLK/CYC activation of E-boxes (Lim et al. 2007). Hence, it is imperative to check whether the robustness of the Drosophila circadian clock is increased by the CWO feedback loop. In the previous CWO model, to understand the effects of CWO on period oscillations, simulations where performed by increasing or decreasing (up to 10%) of the individual repressive weights of CWO in DD conditions (Fathallah-Shaykh et al. 2009). In our models, to understand the behaviour of the CWO loop better, we performed a sensitivity analysis. In order to quantify the robustness occurring due to
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Figure 7-10 Period changes in PER protein oscillation of perL and perS mutants: (a) In model A, (b) In model B, and (c) In model C.
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CWO loops, we compared the estimated local sensitivities of all parameters with and without the CWO feedback loop. To perform the sensitivity analysis without the CWO feedback loop, the oscillation of CWO protein was stopped and its value was fixed. We found in resulting simulations that, all core clock proteins and mRNA oscillations were preserved except CWO in all three models. The plots of protein oscillations in all three models without CWO loop are provided in Figure 7-11. Next, the periodic sensitivity ratios were calculated from sensitivity outputs. In order to understand the sensitivity outputs, it is necessary to understand how local sensitivity is calculated in dynamical systems based on ODEs.
In oscillatory systems local sensitivity analysis is carried out by measuring the period difference in oscillation with respect to perturbation in individual parameter values. To generate local sensitivities, 5% parametric perturbations were assigned and the sensitivities were calculated based on the following function,
, , i i j j x t S t p (8.8)which is explained as follows,
,
change in state at time ,
pert. on parameter at time
i j i th t S t j th (8.9)
where,
p
j is the parameter with parameter index j , t p
j is the period of the system. It is clear from the sensitivity function that the system robustness to parameter perturbations will increase with proportional decrease in the S value. The initial time0
t
is commonly considered to be the perturbation time
in local sensitivity analysis (Ihekwaba et al. 2005).Thus, the differential equation for the sensitivity coefficients can be written without
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Figure 7-11 Robust oscillation of proteins with CWO loop removed: (a) In model A, (b) In model B, and (c) In model C.
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; d S t f S t f dt x p (9.0)in which f x is the two dimensional matrix of all the first order partial derivatives, f
denotes the list of functions to be differentiated and p denotes the list of parameters.
In order to screen for more sensitive parameters, the general sensitivity rankings of parameters were generated in COPASI, directly from the sensitivity coefficients. To generate parameter rankings, as the parameter values might have huge ranges, standardised sensitivity values given by the following expression are used in COPASI:
, log . log j i i i j j i j p x t x t S t p x t p (9.1)Thus, sensitive parameters were shortlisted from COPASI, sensitivities were simulated as shown in Eqn 9.3 and the calculated ratios are plotted in Figure 7-12. The simulation results showed that without the CWO negative feedback loop in all three models most of the parameters showed negligible sensitivity changes, but the sensitivities of the same six parameters (a43, a47, a57, a61, a63 and a67) related to clk
and per gene components previously identified in robustness analysis yielded a maximum change of up to 80%.
In particular, the sensitivities of the transient rate of translation of per and clk mRNAs (a57 and a61) decreased more than 80% (for both per and clk) when removing the CWO feedback loop in model A and B. But the same parameter sensitivities increased by 22% and 36% (for PER and CLK respectively) in model C (Figure 7-13). Even though models A and B with respect to model C, gave opposite effects in their sensitivities to PER and CLK synthesis, one agreement they all had in common was that CWO loop removal affected the rhythmic oscillation of the two crucial clock components PER and CLK. From the sensitivity analysis results of all three models our personal judgement favours the results of the CWPT model, since CWO is known
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Figure 7-12 A 3D histogram of Local sensitivity results with CWO loop removed and normal:(a) In model A, (b) In model B, and (c) In model C.
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to play a modulatory role in oscillation of clock proteins affecting their amplitude (Lim et al. 2007; Richier et al. 2008). This result of model C is in excellent agreement with the wet-lab arguments. From these results a probable function of the CWO feedback loop can be inferred. It is possible that CWO plays an important role in modulating CLK and PER protein feedback loops, thereby increasing the circadian clock robustness.