A challenging problem in systems biology is the quantitative modelling
of transcriptional regulation. Transcription factors (TFs), which are the
key proteins at the centre of the regulatory processes, may be subject
to post-translational modification, rendering them unobservable at the
mRNA level, or they may be controlled outside of the subsystem being
modelled. In both cases, a mechanistic model description of the regula-
tory system needs to be able to deal with latent activity profiles of the key
regulators. A promising approach to deal with these difficulties is based
on using Gaussian processes to define a prior distribution over the latent
TF activity profiles. Inference is based on the principles of non-parametric
Bayesian statistics, consistently inferring the posterior distribution of the
unknown TF activities from the observed expression levels of potential
target genes. The present work provides explicit solutions to the differ-
ential equations needed to model the data in this manner, as well as the
derivatives needed for effective optimisation. The work further explores
identifiability issues not fully shown in previous work and looks at how
this can cause difficulties with inference. We subsequently look at how the
method works on two different TFs, including looking at how the model
works with a more biologically realistic mechanistic model. Finally we
analyse the effect of more biologically realistic non-Gaussian noise on the
biologically realistic model showing how this can cause a reduction in the
accuracy of the inference
