## Modelling the influence of RKIP on the ERK signaling pathway using the stochastic process algebra PEPA

#### by Tony Garnock-Jones / @leastfixedpoint

The focus of the talk was *analysis*, rather than simulation.

##### PEPA

PEPA is a stochastic
algebra. PEPA looks like CCS, but instead of parallel composition with
handshaking on actions, you have a *cooperation operator*,
causing two processes to run in parallel synchronising on a certain
subset of their shared actions. This gives you *n*-ary
interactions — compare π with its binary interactions. PEPA was
originally used in performance modelling for telecommunications
networks.

PEPA programs are translated (via an SOS definition (?)) into labelled multi-transition graphs, and from there further transformed into Continuous Time Markov Chains (CTMCs). The experiments described in the talk were thus not simulations — CTMCs are static.

Some of what was learnt from the use of PEPA in the telco industry:

- compositionality was very important — no longer any need to rewrite the whole system for each small change
- equivalences are important when manipulating models — without a formal equivalence, you cannot prove the model is still the same after your simplification. (Simplifications are necessary to reduce the state space and thus improve tractability.)
- it turns out to be possible to syntactically identify certain
classes of efficiently-implementable models! So just by looking at
the PEPA process description it is possible to judge some aspects of
the model's complexity. (eg. "product form" of CTMCs)
</ul>
##### The ERK Signalling Pathway

The pathway is an extracellular signalling mechanism, relaying messages between cell nuclei. It is often studied since a breakdown in the pathway can lead to cancer. Only a small part of the full pathway was analysed — the part inbound from the cell membrane to the nuclear membrane.

The hypothesis that was being tested was that RKIP is a regulator of the ERK pathway. (The results appeared to supported the hypothesis, according to the graphs from the paper...)

In modelling the pathway, since the final target is a CTMC, there's no way of representing concentrations of reagents, since those values are continuous. If you're approximating using a few discrete concentrations you must be careful that you don't get combinatorial explosion in state space — it must be kept finite and small. The solution adopted was to have only two concentrations for reagents:

*high*and*low*. In the "high" state, various reactions were allowed to proceed; in the "low" state they were inhibited or modified appropriately. Essentially "high" meant "enough to proceed" and "low" meant "zeroish".Explicit representation of the two concentration-approximations was only required in

*one*of the two models studied, the reagent-centric one. The pathway-centric one implicitly encoded concentration in the states of each sub-pathway.There was some interesting commentary on the differences between the two models — the pathway-centric one was easier to assemble and

*much*easier to get right; each model also emphasised different aspects of the reactions, the reagent-centric one apparently "feeling" fine-grained, the pathway-centric one more coarse-grained.The lack of proper concentration representation means that indirect methods had to be applied to adjust initial conditions for performing experiments with the model. Essentially instead of increasing or decreasing concentrations of RKIP, the rate of the first step in the RKIP subpathway was increased or decreased, respectively, which kind of effectively does the same thing for the purposes of these experiments.

A few miscellaneous notes on the talk:

- Sum is not nondeterministic, because of the introduction of a race policy.
- Bisimulation can be used to demonstrate equivalence of models intended to represent the same system; this was put to use in the RKIP/ERK project in deciding that the two alternative representations (reagent-centric and pathway-centric) were equivalent.
- ODEs are
*non-structural*— they do not contain any information on the structure of the pathways in the system, they just allow computation of the concentrations of the reagents. Compare to stochastic π, which is structural — detailing the reaction pathways — and also quantitative enough to be able to provide useful concentration data. - Complexes are explicitly represented in the graphs of reagents and pathways. </ul>