## Section: New Results

### Modeling biological systems: The calculus

Participants : Jesus Aranda, Sylvain Pradalier, Frank Valencia.

Nano-devices are molecular machines synthesized from molecular subcomponents whose functions are combined in order to perform the function of the machine. An important and characteristic feature of these devices is their intrinsic compositional nature. Therefore process-algebra formalisms are natural candidate for their modeling. In his PhD thesis [14] , Pradalier has introduced a dialect of the -calculus, the nano -calculus and has illustrated its relevance for the modeling and simulation of nano-devices with an example stemming from the collaboration with the chemistry department of bologna: the [2]RaH rotaxane. Pradalier has modeled it in nano and has simulated its behaviour under various conditions of concentration. He was then able to show that some classical assumption about kinetic rates were not correct any longer in this setting. The -calculus has many advantages for the modelling of biochemical systems. In particular it is compact, easily reusable and modifiable and biological-like and thus easier to learn for biochemists. On the other hand the -calculus, also often used to model biochemical systems, has a much more developed theory and more available tools. Pradalier has then investigated the possibility of encoding the nano -calculus into the stochastic -calculus, and has found a translation that satisfies strong correctness properties. Furthermore, Pradalier has considered the chemical master equation, which describes probabilistically the possible behaviours of the system over time in terms of a differential equation on the probability to be in a given state at a given instant. Pradalier has introduced a notion of equivalence based on the chemical master equation and has proved that it corresponds exactly to the notion backward stochastic bisimulation. This results establishes a bridge between a chemical semantics and a computer semantics, and it also constitutes a first step towards a metrics for biochemistry. Finally Pradalier has investigated the relative expressive power of the synchronous and asynchronous stochastic -calculus, for which he has used the encodability of the nano -calculus.