Modeling::Dimensionless Model

Dimensionless process

In order to make a further analysis on stability of the system, sensitivity of parameters, feedback factors-we manipulate all the arguments and parameters to make them dimensionless. Analysis of this part is crucial since parameters in vivo experiment may be different and even at odds with modeling ones but a proper dimensionless can reveal the mathematical essence of our model.

Considering the Hill equation in the simplification DDEs, A1_c1 and K_M1/ρ₁ should be the same order of magnitude, thus K_M1/ρ₁ is a well measurement of quantities of A1_c1. We have:

Parameters

Parameters in equations are listed below.

Table 4 Parameters in Dimensionless Model

Results

Sensitivity analysis

In order to find out the key parameters which will affect stability of the system at most, we need to make a sensitivity analysis on each. At first, we did brief and instinctive analyses on each parameter as follows. τ₁^* and τ₂^* represent time delay in cell 1 and cell 2 respectively, which have been discussed in part 2, have little influence on stability of system but intend to affect the oscillation period merely. Parameters a and b refer to feedback factors indirectly, which have not been discussed before, we will see how a and b affect our system later. We have clarified that parameter u is equivalent to μ/γ, thus, u is directly decided by the dilution rate of signal molecules 3OC12HSL and 3OC6HSL in environment and will inevitably influence stability of oscillation. As for m and n, they are inseparably connected to the Hill parameters whose sensitivity have been analyzed in part 2, so we can deduce that m and n are both sensitive parameters to our system.

Here we mainly did sensitivity analyses on parameters m, n and u. Parameters were set fundamentally as Table 4 shows.

Simulation results reveal that the system can oscillate stably only when u<5.3(fixed the other two sensitive parameters), 11.934.4. In other words, to ensure the stability of oscillation, the dilution rate cannot be too high, while the promoter 2 and 4 which affect m and n should be chosen appropriately.

Figure 11 Sensitivity analyses results (On u)

Figure 12 Sensitivity analyses results (On m)

Figure 13 Sensitivity analyses results (On n)

Stability analysis

Although we have done sensitivity analyses on some predominant parameters and acquired fabulous results, these analyses were all based on unary composites, holding only a single subject. We are not content with only doing sensitivity analyses, which merely care about single-in-single-out outcomes but not considering binary relation in systematic concept. So we made a bifurcation analysis on binary parameter (u,m) adopting the same method as what we have done in part 2.

Table 3 critical points (u,m) for oscillation

Depicting those points into an axis, we got the bifurcation line, which indicates the parameters’ value range when our system can oscillate stably is in the area marked by ‘bistable’ as follows.

Figure 14 Bifurcation analysis on (u,m)

Feedback analysis

By changing parameters a and b, which is equivalent to varying types of feedback introduced, we got simulation results as follows (In order to manifest more clearly, the parameter v was set larger, thus each cell’s feedback effect would put greater influence on the whole system).

Figure 15 System with feedback

The analysis shows that only with a negative feedback mechanism could the overall system be working as an oscillator. When a=b=0, the system contains no artificial negative feedback, but there may be some inherent negative feedback within the system.