Studying and understanding the relations between discrete and
continuous modeling techniques is considered one of the most
important problems in computational system biology. This will be the
focus of our presentation, which is more of a work in progress than
a statement of results.

In particular, we concentrate on the relations between process
algebra paradigm (mainly stochastic $\pi$ calculus), and a class of
ordinary differential equations (DE) called S-Systems.

The aim of this study, in the long run, is to devise an hybrid
modeling approach, connecting together the simplicity of internal
description of systems (a feature of process algebras) and the
quantitative power of external modeling (in terms od DE).

A more concrete goal is to analyze the relationships between these
two techniques in terms of parameters and their determination. In
particular, we are working at identifying methodologies to estimate
stochastic rates for $\pi$-processes, given the differential
equations, and viceversa.

As a side step in this direction, we defined an ``extension'' of
$\pi$-calculus, by integrating into it the computational power of
Constraints. In particular, we defined a stochastic version of
Concurrent Constraint Programming (sCCP) with both synchronous and
asynchronous communication. The first motivations for the use of
this process algebra are its flexibility and the fact that it is
easy to program. But, even more important, it leads to a ``clever''
definition of parameters, i.e. the programmer has complete freedom
in setting (non-constant) rates for transitions.



In our aim to clarify how information may flow from stochastic
process algebras to DE, we started by taking a closer look into
Gillespie algorithm. In fact, Gillespie approach can be seen as a
simulation of a Continuous Time Markov Chain (CTMC), where the state
space is the set of tuples of integers counting the different
molecules present in the system, and where the rate of transitions
are determined by summing up the basic rates of all possible
reactions. This clearly shows why Gillespie Algorithm works when
applied to process algebras (it simulates exactly the CTMC induced
by the reduction semantic), enabling an easy implementation of a
simulator for CCP (metainterpreter in Prolog, 500 lines). Moreover,
this point of view gives a way to formalize the concept of
observables in the realm of stochastic simulations (in
$\pi$-calculus) of biological systems. In fact, what we observe are
changes in concentration, or better the changes in the number of
instances of some processes. Once we have a formal description of
traces in this space of observables, we can compute a set of traces
and use this information to train parameters for a set of
differential equations (S-Systems) representing the same system. In
this way, we may be able to identify the underlying mathematical law
governing the phenomenon under consideration. This method is
feasible to be extended by adding to the set of $\pi$-calculus
traces also experimental data (modulo rescaling).



The work done up to this point not only extends expressive power of
the computational machinery over $\pi$-calculus, but it also allows
a natural passage from differential equations to process algebras.
In fact, if we have the possibility of defining arbitrary rate
functions, we can use the information contained in S-Systems by
writing simple sCCP processes whose rates are controlled by the
expressions appearing in the differential equations. As a simple
case study, we considered repressilator. But a more interesting
question emerges here: in this last passage, we calculated rates for
processes with a fixed structure. It would be much more interesting
to be able to define also the logical control structure of
processes, starting from a DE description. This essentially means to
find the best set of agents whose collective behaviour best
approximates the DE's one. A path towards this goal may pass through
symbolic dynamical systems.