20071213 17:48:53
1
On Process Rate Semantics
Luca Cardelli
Microsoft Research
Abstract
We provide translations between process algebra and systems of chemical reactions. We show that
the translations preserve discretestate (stochastic) and continuousstate (concentration) semantics,
and in particular that the continuousstate semantics of processes corresponds to the differential
equations of chemistry based on the law of mass action. The novel semantics of processes so ob
tained equates processes that have the same state occupation dynamics, but that may have different
interaction interfaces.
1
Introduction
We study
stochastic interacting processes
: a simple compositional model of stochastic systems, with a
natural semantics in terms of continuous time Markov chains. These interacting processes can be
translated by an intuitive procedure into a set of chemical reactions from which a continuous seman
tics can be extracted in the form of Ordinary Differential Equations (ODEs). Such a translation estab
lishes a precise connection between process algebra models of biochemical systems, and more tradi
tional models based on chemistry and ODEs.
Process algebra interactions are at first sight
richer than chemical interaction, so it is not imme
diately clear that the ODEs extracted from the chem
ical translation faithfully represent the behavior of
the processes according to the processes
’
own se
mantics. The correspondence is fairly obvious when
the process interactions are
detangled
, meaning when
each interaction channel has exactly one source of
inputs and one source of outputs. Then, each interac
tion channel corresponds exactly to a chemical reac
tion between two chemical species, and in fact the
translation from chemistry back to processes pro
duces detangled systems. In general, though, process interactions can be
entangled
, meaning that there
can be many sources of inputs and outputs on each channel. This is a convenient feature that supports
compact ways of organizing models: its effectiveness is indicated by the fact that detangled system
can be N
2
bigger than corresponding entangled systems. In this paper we show that these more gener
al process models are still faithful: both the Markov and ODE dynamics of the chemical reactions ex
tracted from process models match the intrinsic dynamics of the processes themselves.
A simple example can illustrate the potential problem with such a correspondence. In this intro
duction we limit our discussion to
automata
, which are those processes th
at do not ‚
split
‛ dynamically
Figure 1
Automata and chemistry
: A
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A
B
A
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B
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!a
?a
A
A
’
A
‛
!a
a: A+B
r
A
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+B
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a: A+A
2r
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+A
‛
([email protected])
@r
([email protected])
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into more processes, and that can be conveniently drawn as transition diagrams. (Automata are not
sufficient to model all of chemistry, however, because a molecule can split into two.) In Figure 1 we
have three basic situations and their chemical interpretation as changes in molecule numbers [23][24].
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 Fall '09
 Chemistry, Chemical reaction, Rate equation

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