Case 3: inference for one unknown TF as only regulator

We finally validate the inferential process for Model 3.7, as applied to data simu- lated as in Figure 3.7. In this scenario, only the unobserved TF B is assumed to dynamically influence transcription of the child gene.

Here we tackle the bimodality issue by performing inference for the two modes

independently. In particular, we run two parallel chains, one adopting a HN(102)

prior with support [0,₁), and the second chain adopting a HN(102_{) prior with}

support ( ₁,0], for log(RB/R0). The two chains cover the full parameter support,

and we are also guaranteed that both modes are visited. The prior densities and the MCMC scheme are specified as in the case which assumes both LHY and the unobserved TF B as regulators (see Section 4.1.2).

The algorithm is run for 2.5⇥105 iterations, we discard a burn-in of 105 iter-

ations, and thin the posterior samples by recording one sample every 200 iterations. Again, initial conditions for all the parameters are randomly drawn from the prior densities, and convergence is monitored via visual inspection of the trace plots.

Figures 4.12 (a) and (b) show the parameter posterior densities, as estimated on 10 i.i.d. replications of simulated data from scenario A of Figure 3.7, and for the true and alternative mode, respectively. Figures 4.13 (a) and (b) show analogous plots for scenario B of Figure 3.7. Focusing first on simulation scenario A, the

Figure 4.11: Comparison of one SSA simulation from the full set of reactions of Table 1.1

(red), and 50 di↵usion approximation simulations according to Model 3.4, mean (blue) _±2

SD (shaded blue). Parameters for the SSA are set equal to: R0= 50 molecules/h,RLHY =

2.5 molecules/h, RB = 2⇥102molecules/h,RLHY,B = 50 molecules/h,k+c = 0.8,k c = 1.2

(scenario A, left); R0 = 60 molecules/h, RLHY = 35 molecules/h, RB = 2.5 molecules/h,

RLHY,B = 0.5 molecules/h, k+c = 1.2, k c = 1.2 (scenario B, right). Common parame-

ters: k+LHY =k+B = 3⇥10 4molecules, k LHY =k B = 6 molecules. LHY and TF B

are simulated assuming parameters ⌫LHY(t) = [2.88⇥102,8.7,1.68⇥102,21.6,1.26⇥102]

(in molecules per hour) with switch times SwtA = [27,40,50,61] (in hours), ⌫B(t) =

[21,1.71⇥102,2.91⇥102,9,90,1.68⇥102,21] (in molecules per hour) with switch times SwtB = [21,28,45,52,56] (in hours), respectively. µM = 0.5 h 1, ↵M = 40 h 1, µP = 0.34 h 1_,µ

M_Mg = 1.2 h 1 andXRN AP c= 10 molecules. Values are summed over 100 cells.

Parameters for the di↵usion approximation simulations are set as in Figure 3.5.

true parameters values are generally included in the 95% HPDIs, although we find

poor mixing of the chain for log(E[XMg(0)]). Considering simulation scenario B,

estimation seems to be more challenging, and in particular the rescaled dissociation

coefficient parameter log(K_B0 Kc) falls inside the HPDIs in 4 cases out of 10 at level

95%, and 6 cases out of 10 at level 99%. We also note that the true value of the log

ratio log(RLHY,B, RB) is included in the HPDIs in 8 out of 10 cases at level 95%.

We believe that this result is due to the posterior correlation structure, and indeed we notice in Figure 4.14 (b) that the first peak of the unobserved TF B smoothing density median tends to be underestimated, leading to two simulation data-points not included in any HPDI.

Comparing this situation with scenario A, we can see in Figure 4.14 (a) that the first peak is better estimated, but the second cycle seems to be preceded by the median fit by about 1-2 hours. This mismatch is however generally compensated by the variability, and we observe that in scenario A the smoothing density of the unobserved TF does not include the true value at worst in 6 out of 10 cases, at four time-points. In one replicate we have also obtained very wide HPDIs for the unobserved TF (not shown for plotting purposes).

The remaining plots of Figures 4.14 (a) and (b), show that, as expected, the two modes provides anti-phase smoothing profiles for the unobserved TF, as well as that scenario B tends to perform better than scenario A in terms of inclusion of the unobserved mRNA true simulation profile in the 95 % HPDIs of the smoothing density. We again attribute this mismatch to the approximate handling of the unobserved TF.

Despite the fact that the 95 % HPDIs do not always provide the expected

empirical coverage, the model o↵ers the possibility to infer, at least approximately,

the phase and relative amplitudes of the two cycles of the unobserved TF, which is our main objective.

(a) Mode 1 (True)

(b) Mode 2 (Alternative)

Figure 4.12: Kernel density estimates of the marginal posterior densities of the model parameters posterior densities, excluding the mean and variance of the initial

condition of the Fourier coefficients. Model 3.7, as applied to data simulated accord-

ing to scenario A of Figure 3.7. MCMC samples for 10 independent replications. The red vertical line is at the true value, and the prior density is also superimposed in red.

(a) Mode 1 (True)

(b) Mode 2 (Alternative)

Figure 4.13: Kernel density estimates of the marginal posterior densities of the model parameters, excluding the mean and variance of the initial condition of the

Fourier coefficients. Model 3.7, as applied to data simulated according to scenario

B of Figure 3.7. MCMC samples for 10 independent replications. The red vertical line is at the true value, and the prior density is also superimposed in red.

(a) Scenario A

(b) Scenario B

Figure 4.14: Unobserved TF B inference (smoothing), left panels: posterior median (magenta) and 95 % HPDIs (lower: blue; upper:cyan). True simulated child unob- served TF B superimposed (red). Unobserved child mRNA inference (smoothing), right panels: posterior median (black) and 95 % HPDIs (lower: blue; upper:cyan). True simulated child mRNA superimposed (red). True mode (top panels in each subfigure), and alternative mode (bottom panels in each subfigure). MCMC sam- ples for 10 independent simulations from Model 3.7, as applied to the simulation scenarios of Figure 3.7, with the exception of the unobserved TF smoothing profile for the true mode in scenario A: one posterior profile has extremely wide HPDIs, and is thus excluded for plotting purposes.