Model fitting to the genetic toggle switch post aTc induction

The priors shown in Table 5.11, the hazard functions defining the model and the flow cytometry time course data were supplied to ABC-Flow. ABC-Flow was used to infer the parameter values that could produce the best fit to the experimental data. The resulting simulated time course data and posterior distributions are given in Figures 5.24 and 5.28 respectively. The model was also simulated stochastically without converting the number of molecules to fluorescence intensity in order to confirm that the model behaves like a switch. This is shown in Figure 5.24B. Fol-lowing aTc induction, the number of GFP molecules increases and the number of mCherry molecules decreases.

The inferred parameters, and the 95% credible intervals, are given in Table 5.12.

In order to validate these results I compare the 95% confidence intervals of the in-ferred parameters to values reported in the literature. First, the half-lives of the two proteins in the system, GFP and mCherry, are estimated. The half-life of a protein is given by

𝑡 =𝑙𝑛(2) 𝑘

where 𝑘 is the decay rate of the protein. The half-life of GFP was estimated to be (7.788, 43.322) hours which is consistent with the >24 hours half life that has been previously reported for the variant of GFP used here, GFPmut3b (Andersen et al. 1998). The half-life for mCherry was estimated to be (9.242, 31.507) hours which is consistent with the long half-life of mCherry, reported to be more than 24 hours (Shaner et al. 2004). We find that the inferred parameter for GFP gene expression, R_u is (1.055, 1.979) molecules h^-1, whereas the parameter for gene ex-pression of mCherry (R_v) was inferred to (8.970, 42.611) molecules h^-1. These val-ues correspond to the promoter strengths of P_LtetO-1 and P_trc2 respectively. This is in agreement with characterisation data on the strength of these two promoters, which report that there is a difference in strengths, P_trc2being a stronger promoter than P_LtetO-1 (Litcofsky et al. 2012). By simulating the system using the median in-ferred values of the system, we find that when the switch is in the GFP high state, there are approximately 600 GFP molecules in the cell. Further, we find that the values of the intensity parameters μ and σ, which represent the settings on the flow cytometer, to be inferred better for GFP than for mCherry.

The inferred parameters can be used to study the switch system present in the pKDL071 plasmid. Here I examined the effect that the values of the most well

in-ABC-Flow parameter inference for experimental data 177 Table 5.12 The inferred parameter values of the toggle switch post-aTc induction

Parameter Units 0.025 Median 0.975

R_u molecules h^-1 1.055 1.298 1.979 KL_u molecules h^-1 19.606 40.466 49.138

KD_u h^-1 0.016 0.0834 0.089

KR_u molecules^-1h^-1 0.006 0.013 0.093 R_v molecules h^-1 8.970 31.057 42.611 KL_v molecules h^-1 5.807 35.396 38.333 KR_v molecules^-1h^-1 0.095 0.386 0.947

KD_v h^-1 0.022 0.043 0.075

KI_v h^-1μM^-1 21.690 54.214 89.937

μ_GFP AU 53.424 76.229 84.055

μ_mCherry AU 91.329 108.691 176.058

σ_GFP AU 154.373 193.198 196.525

σ_mCherry AU 68.178 115.581 139.816

ferred parameters, R_uand KR_uas well as the initial conditions have on the behaviour of the system. The results are shown in Figure 5.26. First, I studied the effect of the initial conditions of the dominant protein, in this case mCherry has on the system.

In order to do that I simulated the model using the median values from the posterior distribution, and increased the initial condition value of mCherry. We do not find this to have had an effect on the final state of the system, as GFP reaches a similar value at 6 hours as the one produced using inferred initial condition values (shown in Figure 5.26B). On the other hand, decreasing the initial condition values of GFP to 10 molecules, destabilises the switch, as shown in Figure 5.26B. The switch happens at a later time point, and it does not take place for all stochastic trajectories.

The value of R_u represents the strength of the promoter driving GFP expression.

It is therefore important to understand the effect that the promoter strength has on the behaviour of the system. In order to do that, I simulated the model with values that exceeded the 95% credible region of the values inferred from ABC-Flow.

The results are shown in Figure 5.26C and D. We find that if the promoter strength is lower than the range of the 95% credible region of the inferred value then the system stops behaving like a switch. If the value is much higher, GFP reaches a much higher value after 6 hours. Further, I examined the effect that the strength of KR_u, the parameter representing the mCherry-induced GFP repression rate, has to the behaviour of the system. We find that KR_u affects the ability of the system to behave like a switch. If KR_uis set to 0.1, the switch occurs at a later time point, and

178 Bayesian model fitting for flow cytometry data

does not take place for all stochastic trajectories. If KR_uis set to much higher than the 95% credible region, the system does not switch. These findings are important in the understanding the system under study as they allow the above predictions to be made about the behaviour of the system.

ABC-Flow parameter inference for experimental data 179

Figure 5.24 (A) The post-aTc induction flow cytometry time course data (blue) and the resulting model fit from ABC-Flow (red). (B) The model simulation us-ing parameters sampled from the posterior distribution shows that this model has a high and low state for GFP. A solid line depicts the median value of the trajectories and the dotted line depict 0.025 and 0.975 quartiles. (C) The particle acceptance rate from ABC-Flow (D) The progression of epsilon threshold values at each population.

180 Bayesian model fitting for flow cytometry data

Figure 5.25 The posterior distributions of the 13 parameters fitted to post-aTc induction time course data using ABC-Flow. We find that the parameters for GFP expression (R_u) and repression (KR_u) are the most identifiable.

ABC-Flow parameter inference for experimental data 181

(A) (B)

(C) (D)

(E) (F)

GFPmCherry

Ru = 0.1

KRu = 0.1 KRu = 0.8

Ru = 5

GFP = 10 molecules mCherry = 530 molecules

Time (h) Time (h)

# Molecules# Molecules # Molecules

# Molecules # Molecules# Molecules

Time (h) Time (h)

Time (h) Time (h)

Figure 5.26 (A) Increasing the initial condition value of mCherry does not affect the state of the system. (B) When the initial condition value of GFP is decreased, the system no longer exhibits a high and low state for GFP and mCherry. (C) De-creasing parameter R_uto a value outside the 95% credible region stops the model from behaving like a switch. (D) Increasing the value of R_u, the rate representing the transcription rate of GFP, causes GFP to reach higher levels. (E) Increasing the value of KR_u, the parameter representing the mCherry-induced GFP repres-sion rate causes the switch to happen at a later time point and not all stochastic trajectories switch state. (F) If the value of KR_uis set to larger than the 95% cred-ible region, the system does not switch states. For all plots, a solid line depicts the median value of the trajectories and the dotted line depict 0.025 and 0.975 quartiles.

182 Bayesian model fitting for flow cytometry data