Inference for single experiments - Cry1-luc data analysis

Chapter 6 Inference for spatio-temporal Cry1-luc data from the SCN

6.2 Cry1-luc data analysis

6.2.2 Inference for single experiments

We present the results of the estimation process by plotting the posterior median estimates of the parameters of interest, along with a measure of their variability, in a spatial fashion across the SCN. We ideally want to both visualise patterns of spatial variation, and have an estimate of their reliability. At coordinates corresponding to

the analysed 2⇥2 pixel boxes, we therefore plot a dot with colour scale proportional

to posterior density median of each parameter, and size inversely proportional to

the corresponding coefficient of quartile variation (see e.g. Bonett, 2006). Note that

proportionalities of both colour and size can be di↵erent for di↵erent parameters,

in order to obtain visually interpretable plots. In analogy with Figures 6.4 and 6.5,

the background is given byCry1-luclight intensity at comparable times of the first

circadian cycle across the three experiments.

We note in Figure 6.6 high values ofn, with respect to the known number of

binding sites, as well as low values ofµMg with respect to the prior of Yamaguchi

et al. (2003). These parameters are indeed believed to compensate for model approximation, as our model represents only a simplified representation of the more

complex process outlined in Chapter 5. We also observe a low degree of spatial vari-

ation forR0,Kpc, if we ignore a proportion of extreme estimates characterised by

a higher dispersion, which we again attribute to model approximation. The remain- ing parameters seem to exhibit a more evident spatial structure, with similar values clustering together, as well as exhibiting decreasing or increasing trends as we move from central to more peripheral locations. In Figure 6.7 we first observe an increase

in the median estimates of ✏ from central to more external locations, suggesting

that peripheral locations are characterised by a lower signal to (measurement) noise ratio. Central locations tend also to show higher responsiveness of the promoter, as

indicated by a high estimate ofn, as well as a higher mean and standard deviation

of the delay, and scale. These estimates point in the direction of a system compris-

ing a higher intrinsic noise in the central area of the SCN, with longer and noisier delays in the feedback cycles. This result is particularly interesting if considering that the core area of the SCN, which covers approximately the lower half of the SCN, is the recipient of light impulses coming from the retina; we may put forward the hypothesis that intrinsic noise contributes to the higher responsiveness of the upper core SCN to external inputs, although we note that such a result would also

imply di↵erent mechanisms to be taking place between lower and upper core SCN

regions. The link between intrinsic noise and responsiveness to external signals,

is supported by mathematical studies on the e↵ect of noise on oscillatory systems,

and in particular Steuer et al. (2003) show that, for a given amplitude of the in- put, noisy systems may exhibit higher amplitudes in the outputs than deterministic ones. Moreover, studies have found that intrinsic noise has a key role in generating circadian oscillations as a consequence of extra-cellular signalling, when individual cell clocks are disrupted by mutation of BMAL1 (Ko et al., 2010).

Finally, comparison among the three experiments show similarities in parameter estimates. In order to obtain a clearer and more robust picture of the inferred spatial dynamics, a natural step is to perform a meta-analytic study, which we formulate via a Bayesian hierarchical model. This is presented in Section 6.3.

(a) ✏= 0.01.

(b) ✏= 0.05.

Figure 6.2: Kernel densities estimates of the model parameters posterior densities, excluding the parameters of the initial condition. Model 5.26, with unobserved state initial condition as in Equation 5.27, applied to the simulated data of Figure 5.5,

for the two simulation scenarios assuming ✏ = 0.01 (top) and ✏ = 0.05 (bottom),

m = 1. MCMC samples for 10 independent replications. The red vertical line is

at the true value, and the prior density is also superimposed in red. Prior for the degradation rate from Yamaguchi et al. (2003).

(a) ✏= 0.01.

(b) ✏= 0.05.

Figure 6.3: Kernel densities estimates of the model parameters posterior densities, excluding the parameters of the initial condition. Model 5.26, with unobserved state initial condition as in Equation 5.27, applied to the simulated data of Figure 5.5,

for the two simulation scenarios assuming ✏ = 0.01 (top) and ✏ = 0.05 (bottom),

m = 1. MCMC samples for 10 independent replications. The red vertical line is

at the true value, and the prior density is also superimposed in red. Prior for the degradation rate centred close to the true simulation value.

Figure 6.4: Amplitude of the harmonic corresponding to the leading frequency of

the observed Cry1-luc light intensities, for selected locations in the right half of

the SCN, and averaged over 2⇥2 pixels blocks. Points at the selected locations.

Experiment 1 (left), 2 (middle) and 3 (right). Data from Hastings lab. at MRC, Cambridge.

Figure 6.5: Phase of the harmonic corresponding to the leading frequency of the

observedCry1-luclight intensities, for selected locations in the right half of the SCN,

and averaged over 2_⇥2 pixels blocks. Points at the selected locations. Experiment

Figure 6.6: Posterior estimates forR0,Kpc,nand µMg; colour scale proportional

to the median, size inversely proportional to the coefficient of quartile variation.

Samples are transformed in the original scale to compute the interquartile coeffi-

cient of variation. The size scale may vary between di↵erent parameters for plotting

purposes, but are equal for the same parameter across the three experiments. The MCMC estimation algorithm for Model 5.26, with unobserved state initial condition

as in Equation 5.27, is run onCry1-lucintensities averaged over 2_⇥2 pixel boxes,

at the plotted points locations. The chains of two locations in experiment 3 show very poor mixing, and their estimates are therefore not included for plotting pur-

poses (very low coefficient of quartile variation). Data from Hastings lab. at MRC,

Figure 6.7: Posterior estimates forE[⌧], SD[⌧], ✏ and , colour scale proportional