# Parameter Range and Sensitivity Analysis

## Goal

To figure out the sensitive parameters in this oscillation system.

## Methods

1. Parameter range determination;
2. Use local sensitivity method to analyze theoretically;
3. Determine vital parameters for parameter sweep;

## Result

To see how different concentration of both IPTG and Arabinose can affect the period of the oscillator and the range of period, we solve the DDEs in various values of IPTG and Arabinose using R language.

(a)AraC period map

(b)AraC period Contour

Fig 1.(a)AraC period map with IPTG from 0mM to 10mM, step is 1mM, Arabinose from 0% to 10%, step is 0.1%. (b)Contour of period concerning Arabinose and IPTG. In both figures, color shows the values of period.

The period map can be divided into two areas according to IPTG concentration: 'mountain'(0mM~5mM) and 'plain' (5mM~10mM). These areas can be more clearly seen in contour. The difference between largest and smallest period is approximately 6 minutes, which is insignificant comparing with the scale of period. In 'mountain' area, when IPTG concentration is fixed, the period increases alongside with Arabinose concentration; on the other hand, when Arabinose concentration is fixed, the period increases at first when IPTG concentration rises, then it decreases when IPTG concentration keep on rising. However, in 'plain' area, the period remain steady against either IPTG or Arabinose change.
Due to the computational expenses and our limited computing power, we set a rather large step. Though coarse, we can still grab the big picture of how IPTG and Arabinose can affect the period of the oscillator. We also examined a specific area in a small step size.

Fig 2.AraC period of area whose IPTG is 0~2mM, step is 0.1mM and Arabinose is 4~5%, step is 0.1%. The edge is less sharp than the one above.

By examining specific area in a smaller step size, we found that surface of period is actually rather smooth, which suggests that a large step size does not limit its representativeness, since spline interpolation is used in plotting the discrete data and high accuracy is guaranteed by smoothness of the function.
Then we examined the derivative of AraC's period with respect to both Arabinose and IPTG.

(a)Derivative of AraC's period with respect to Arabinose

(b)derivative of AraC's period with respect to IPTG

Fig 3. derivative of AraC's period with respect to both Arabinose and IPTG. Derivatives are calculated using the data from the bigger step period map.

These figures further suggest the asymmetry behavior of Arabinose and IPTG in terms of affecting AraC's period. Lastly, we examined the role of lag $\tau$ in the period of AraC.

Fig 4.AraC's period against lag $\tau$

In these figures, we could clearly see that AraC's period increases linearly as lag increases. To sum up, the range of period is rather limited while is fixed. In other word, it's rather stable against Arabinose and IPTG changes. On the other hand, we can see that among Arabinose, IPTG and $\tau$, $\tau$ has the biggest influence on AraC's period, which should be concerned firstly while adjusting the period of the oscillator.