## On a thermodynamic approach to biomolecular interaction networks

##### Abstract

We explore the direct and inverse problem of thermodynamics in the context of
rule-based modelling. The direct problem can be concisely stated as obtaining
a set of rewriting rules and their rates from the description of the energy
landscape such that their asymptotic behaviour when t → ∞ coincide. To
tackle this problem, we describe an energy function as a finite set of connected
patterns P and an energy cost function e which associates real values to each of
these energy patterns.We use a finite set of reversible graph rewriting rules G to
define the qualitative dynamics by showing which transformations are possible.
Given G and P, we construct a finite set of rules Gp which i) has the same
qualitative transition system as G and ii) when equipped with rates according
to e, defines a continuous-time Markov chain that has detailed balance with
respect to the invariant probability distribution determined by the energy
function. The construction relies on a technique for rule refinement described
in earlier work and allows us to represent thermodynamically consistent
models of biochemical interaction networks in a concise manner.
The inverse problem, on the other hand, is to i) check whether a rule-based
model has an energy function that describes its asymptotic behaviour and
if so ii) obtain the energy function from the graph rewriting rules and their
rates. Although this problem is known to be undecidable in the general case,
we find two suitable subsets of Kappa, our rule-based modelling framework
of choice, were this question can be answer positively and the form of their
energy functions described analytically.