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)
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.
