  In many biological systems, chemical reactions or changes in a physical state
are assumed to occur instantaneously. For describing the dynamics of those
systems, Markov models that require exponentially distributed inter-event times
have been used widely. However, some biophysical processes such as gene
transcription and translation are known to have a significant gap between the
initiation and the completion of the processes, which renders the usual
assumption of exponential distribution untenable. In this paper, we consider
relaxing this assumption by incorporating age-dependent random time delays into
the system dynamics. We do so by constructing a measure-valued Markov process
on a more abstract state space, which allows us to keep track of the "ages" of
molecules participating in a chemical reaction.
  We study the large-volume limit of such age-structured systems. We show that,
when appropriately scaled, the stochastic system can be approximated by a
system of Partial Differential Equations (PDEs) in the large-volume limit, as
opposed to Ordinary Differential Equations (ODEs) in the classical theory. We
show how the limiting PDE system can be used for the purpose of further model
reductions and for devising efficient simulation algorithms. In order to
describe the ideas, we use a simple transcription process as a running example.
We, however, note that the methods developed in this paper apply to a wide
class of biophysical systems.
