tions {xn,m}m=1,...,M of the target gene n themselves. In the complete data scenario
I consider the protein rather than the mRNA concentrations as potential regulators. To be consistent with the fundamental equation of transcription, Equation (4.1),xn,m
will always be included in either scenario; I won't mention that explicitly in the text. For consistency with the fundamental equation of transcription, Equation (4.1), I will enforce that each regulator set πn for yn contains the concentration xn of n,
symbolically xn ∈ πn. Thereby as the transcription rate yn of gene n will cer-
tainly depend on its mRNA concentration xn I add the mRNA concentrations
of gene n to the protein proles. The potential regulators for yn are then given by
{xn,m, xp,1,1. . . , xp,N?,m}m=1,...,M. However, I ignore this distinction in the method-
ological denitions, and use the term regulators generically for both scenarios.
4.3 Method Extensions
The methods that take part in this study have been previously described in Chapter 2. An exception are the following modications to the hierarchical Bayesian regression (HBR) method that take the genetic data set into special consideration. The time- varying dynamic of the plant data is under the major inuence of light and darkness that are expressed typically over 24 hours. I exploit to change-process of the HBR to model the light and dark phase as explained in Section 4.3.1 with HBR-light. Further- more I modify the HBR in such a way that change-points are applied to the amplitude of mRNA response gradients (Section 4.3.2, HBR-cps). I anticipate a substantial im- provement on the approximation of Michaelis-Menten dynamics with this approach. A simple, yet eective, method is the expansion of the explanatory data with product and non-linear terms as described in 4.3.3. Although, this does not modify the HBR method itself, I will refer to this expansion as HBR-nl.
4.3.1 Fixed change-point induced by the external light condition (HBR-light)
Since light may have a substantial eect on the regulatory relationships of the circadian clock, I divide the observations of the target variables into two segments according to a binary light phase indicator: h = 1 (light) versus h = 2 (darkness). This reects the nature of the laboratory experiments, where A.thaliana seedlings are grown in an articial light chamber whose light is switched on or o. It is straightforward to
66 Chapter 4 generalize this approach to more than two segments to allow for extended dawn and dusk periods in natural light. Given that the light phase is known, I consider the segmentation as xed, and I refer to the model as the hierarchical Bayesian regression (HBR) model with two light-induced components (HBR-light). Since I also assume that light has a substantial inuence, I do not penalize any dierences between the interaction parameters associated with the two light phases and apply the uncoupled non-homogeneous Bayesian regression model, shown in the right panel of Figure 2.1.
4.3.2 Change-points in the amplitude of the target variable (HBR-cps)
To approximate the non-linear dynamics of the Michaelis-Menten kinetics, I sort the realizations yn,1, . . . , yn,M of each target variable, yn, in increasing order to obtain
the order statistics yn,(1) ≤ . . . ≤ yn,(M).3 Applying the non-homogeneous Bayesian
regression models to the ordered realizations,yn,(1), . . . , yn,(M), then eectively yields
a segmentation of the realizations, yn,1, . . . , yn,M, with respect to the amplitude of
the target variable yn. To infer the number of change-points and the change-point
locations, I again follow Grzegorczyk and Husmeier [60] and use a point process prior, where the distance between two successive change-points, Mn,h = τn,h+1 −τn,h, is
assumed to have a negative binomial distribution with hyper-parameters p ∈ [0,1]
and k = 1, symbolically Mn,h ∼ NBIN(p,1). I apply both variants of the non-
homogeneous Bayesian regression model. The uncoupled variant is shown in the right panel of Figure 2.1, and I setmn,h=0 for allh≥0in Equation (2.11). In the coupled
variant the regression parameter vectors,wn,h(h= 1, . . . , Hn), are sequentially coupled
via Equations (2.11-2.12). I refer to these hierarchical Bayesian regression models as the change-point-divided hierarchical Bayesian regression models (HBR-cps).
4.3.3 HBR with additional non-linear terms
A straightforward extension of the HBR method is to include non-linear terms in the design matrix Xπn. In my study I tested, as an alternative to the HBR model just
described, the inclusion of quadratic and inverse terms. So for a set of regulators
πn = {A, B}, the columns of design matrix Xπn, [1, xA(m), xB(m)]
0 are replaced by
[1, xA(m), xB(m), xA(m), xB(m),1/xA(m),1/xB(m)]0, where the inverse terms are in-
cluded for a better approximation of the Michaelis-Menten kinetics, and the mixed
3For eachy
nI apply exactly the same permutation to order the realizations of the explanatory vari- ables (covariates) and thereby ensure that the segment-specic design matrices are built properly.
4.4. DATA 67