  In this article, we introduce a novel backward method to model stochastic
gene expression and protein level dynamics. The protein amount is regarded as a
diffusion process and is described by a backward stochastic differential
equation (BSDE). Unlike many other SDE techniques proposed in the literature,
the BSDE method is backward in time; that is, instead of initial conditions it
requires the specification of endpoint ("final") conditions, in addition to the
model parametrization. To validate our approach we employ Gillespie's
stochastic simulation algorithm (SSA) to generate (forward) benchmark data,
according to predefined gene network models. Numerical simulations show that
the BSDE method is able to correctly infer the protein level distributions that
preceded a known final condition, obtained originally from the forward SSA.
This makes the BSDE method a powerful systems biology tool for time reversed
simulations, allowing, for example, the assessment of the biological conditions
(e.g. protein concentrations) that preceded an experimentally measured event of
interest (e.g. mitosis, apoptosis, etc.).
