  Time course data are often used to study the changes to a biological process
after perturbation. Statistical methods have been developed to determine
whether such a perturbation induces changes over time, e.g. comparing a
perturbed and unperturbed time course dataset to uncover differences. However,
existing methods do not provide a principled statistical approach to identify
the specific time when the two time course datasets first begin to diverge
after a perturbation; we call this the perturbation time. Estimation of the
perturbation time for different variables in a biological process allows us to
identify the sequence of events following a perturbation and therefore provides
valuable insights into likely causal relationships.
  In this paper, we propose a Bayesian method to infer the perturbation time
given time course data from a wild-type and perturbed system. We use a
non-parametric approach based on Gaussian Process regression. We derive a
probabilistic model of noise-corrupted and replicated time course data coming
from the same profile before the perturbation time and diverging after the
perturbation time. The likelihood function can be worked out exactly for this
model and the posterior distribution of the perturbation time is obtained by a
simple histogram approach, without recourse to complex approximate inference
algorithms. We validate the method on simulated data and apply it to study the
transcriptional change occurring in Arabidopsis following inoculation with P.
syringae pv. tomato DC3000 versus the disarmed strain DC3000hrpA.
  An R package, DEtime, implementing the method is available at
https://github.com/ManchesterBioinference/DEtime along with the data and code
required to reproduce all the results.
