Appears in Proceedings of CMSB 2008, Rostock, Germany,
LNCS 5307, pp.103–122. Springer-Verlag, 2008
The Continuous π-Calculus:
A Process Algebra for Biochemical Modelling
Marek Kwiatkowski and Ian Stark
Laboratory for Foundations of Computer Science
School of Informatics, The University of Edinburgh, Scotland
{M.Kwiatkowski,Ian.Stark}@ed.ac.uk
Abstract. We introduce the continuous π-calculus, a process algebra for mod-
elling behaviour and variation in molecular systems. Key features of the language
are: its expressive succinctness; support for diverse interaction between agents via
a flexible network of molecular affinities; and operational semantics for a continu-
ous space of processes. This compositional semantics also gives a modular way to
generate conventional differential equations for system behaviour over time. We
illustrate these features with a model of an existing biological system, a simple
oscillatory pathway in cyanobacteria. We then discuss future research directions,
in particular routes to applying the calculus in the study of evolutionary properties
of biochemical pathways.
1 Introduction
This research aims to develop computational methods for studying the Darwinian
evolution of biochemical pathways. We work in the framework introduced by Regev
et al. [1,2,3], who identified the π-calculus process algebra as a promising formalism
for biological modelling. We modify it in a way that allows us to mention quantitative
parameters explicitly, and makes the interaction network of the agents more flexible
(see §1.1 below). To take advantage of this quantitative information, we develop a
novel operational semantics through a compositional description of continuous system
behaviour in terms of real vector spaces (§2.2). We illustrate these concepts with an
example of a concrete biological system, a simple oscillatory pathway in cyanobacteria
(§7). Finally, we discuss the possibilities of answering questions related to the evolution
of pathways using process-algebraic techniques such as model checking and behavioural
equivalences (§4).
Reliable models and simulations of evolution on the molecular level would have
wide applications, from pure evolutionary theory to drug design. Our particular in-
terest is in the ubiquitous phenomenon of mutational robustness, which has recently
received attention as a cross-level organisational principle of biological systems [4,5].
Its understanding, especially on the level of gene regulation and cellular signalling, is
an important challenge; and, moreover, one where a computational approach may give
essential assistance in tackling the complexity of the systems involved.
We motivate the use of process algebras in this context as follows: firstly, they
have already been successfully used to model biochemical networks (see §1.2 below).
http://www.inf.ed.ac.uk/~stark/continuous-pi.html
Marek Kwiatkowski and Ian Stark
Secondly, much of the genetic variation results in qualitative or quantitative changes in
the interaction network; process-algebraic descriptions of networks enable us to express
this variability easily, in syntactic terms. Finally, to model evolution we need means to
express fitness and related concepts such as neutrality [6,5]; to this end we plan to
use well-developed process-algebraic techniques like model checking and behavioural
equivalences (§4).
The main contribution of this paper is the continuous π-calculus (cπ), a process al-
gebra designed specifically to model biomolecular systems in an evolutionary context,
with an original semantics in terms of real vector spaces. It also offers a fully modular
and compositional method of generating a set of ordinary differential equations (ODEs)
governing a given system. Moreover, we give a process-algebraic model of a biomolec-
ular system of considerable interest — a primitive bacterial circadian clock recently
recreated in vitro [7].
In the remainder of this section we introduce cπ by means of small examples (§1.1)
and then very briefly recall related work in the fields of computational and theoretical
biology (§1.2). Section 2 is a formal presentation of the calculus, while §3 contains the
cπ model of the KaiC circadian clock [8], with graphs showing its oscillatory behaviour.
In the final section we present and discuss the future directions of our research.
1.1 Key Features
The classic π-calculus is well-described in existing texts [9,10,11]. Here we focus on
the key distinguishing features of our continuous π, and follow with two small examples
to illustrate them.
1. Every cπ process is a parallel (k) combination of species. Species are very similar
to classic π-processes. Every species in a given process is equipped with a real
number, to be thought of as the concentration of the substance described by the
species. Thus, every cπ process represents a complete molecular system at a certain
point in time.
2. As usual in the π-calculus, communication is through named channels. However,
in contrast to most π derivatives, there are no co-names. Instead, any name can
in principle communicate with any other — potentially more than one — and
for any two names it is specified whether they can communicate and at what
rate. Biologically, names are intended to model distinctive reaction sites, with the
communication rate between two names corresponding to the rate constant of the
biochemical reaction between sites. The relevant technical device to manage this
information is an affinity network. One consequence of this approach is that every
new name must come with the information about its communication potential.
Also, whenever we consider a particular process P , we assume some given affinity
network on the free names of P .
The reason for this approach is two-fold. First, it makes sense in an evolutionary
context to abandon the strict correspondence of sites and co-sites, and hence the
symmetry of names and co-names. Second, the affinity networks give a convenient
collection of parameters for varying the model: in particular, those important for
questions of evolvability and robustness.
The Continuous π-Calculus: A Process Algebra for Biochemical Modelling
3. All cπ reactions have at most two substrates and are governed by the Law of
Mass Action (following, for example, the observations of [12, p.298]). It is worth
stressing, however, that more complex kinetic behaviour can emerge as we build
up larger processes from smaller ones. Nonetheless, the system dynamics remains
purely deterministic: for every process P we derive a term
dP
dt
, denoting the speed
and direction of the temporal evolution of P .
4. To model spontaneous monomolecular reactions, such as degradation, or confor-
mation changes, we use silent actions labelled with real numbers denoting reaction
rate constants.
5. Molecular complexes are represented by parallel components within the shared
scope of one or more private names, following Regev [2]. As usual for the
π-calculus, communication between shared private names within the complex gives
rise to silent actions; and these in turn model spontaneous actions like complex
dissociation.
As a first simple example, consider two molecules, A and B,
a
b
a
k
1
— b
k
1
Fig. 1: A very simple
affinity network and
its textual rendering.
that can bind to each other (at rate k
1
) and subsequently
unbind (at rate k
2
). As noted above, we model complexation
as scope extrusion and decomplexation as interaction on
a shared private channel. This gives rise to the following
cπ definition of species A and B:
A
df
= (ν u
k
2
— v)(ahui.v.A) (1)
B
df
= b(x).x.B (2)
with the global affinity network of Fig. 1.
Here the public names a and b represent protein interaction sites, and a communica-
tion event between these two names models binding of these sites. The prefix (ν u
k
2
— v)
declares a new affinity network consisting of two private names, u and v , that can com-
municate with each other at the rate k
2
. Species A and B can react by a communication
event on the public a
k
1
— b channel, with the private name u sent via a, received on b,
and then substituted for x in x.B. This extends the scope of the network to encompass
the remaining parts of A and B and so form complex C:
A | B
τ ha,bi
−→ C
df
= (ν u
k
2
— v)(v.A | u.B) (3)
If we mix species A and B together in concentrations c
1
and c
2
, then we obtain a
cπ process (c
1
· A k c
2
· B). The formal semantics of this process reflect mass-
action dynamics: the complex C is produced in proportion to the product of substrate
concentrations, while the substrates themselves (A and B) deplete similarly. If we
compute the semantics (an appropriate
d·
dt
term) as described in §2.2, we see that this is
indeed the case:
d(c
1
· A k c
2
· B)
dt
= k
1
c
1
c
2
· C − k
1
c
1
c
2
· A − k
1
c
1
c
2
· B . (4)
Marek Kwiatkowski and Ian Stark
b
a
a
0
k
1
k
2
(a) Affinity network
A
df
= a.A
+
A
+
df
= a
0
.A
++
+ τ @k
3
.A
K
df
= b.K
P = c
1
· A k c
2
· K
(b) Species and process definitions
Fig. 2: A simple cπ system with a non-trivial affinity network modelling discriminative binding
of the kinase K (via site b) to the molecules A and A
+
(at sites a and a
0
, respectively). The
definition of the inactive A
++
species is omitted.
Similarly, when we consider a solution of complexes in concentration c
3
and derive the
semantics of the process (c
3
· C), we observe that the complex dissolves to give back
substrates A and B at the expected rate:
d(c
3
· C)
dt
= −k
2
c
3
· C + k
2
c
3
· A + k
2
c
3
· B . (5)
As another example, consider a molecule A that can exists in three states: unphospho-
rylated, phosphorylated and doubly phosphorylated; denoted A, A
+
and A
++
respec-
tively. Furthermore, suppose that kinase K (the phosphorylating agent) is more effective
at the initial phosphorylation step A → A
+
than at the subsequent one A
+
→ A
++
,
having reaction rate constants k
1
> k
2
. Finally, assume that A
+
(only) can sponta-
neously relax back to A at a rate k
3
. Figure 2 shows a cπ model for this system, and in
particular a process P representing an initial state where only A and K are present, at
concentrations c
1
and c
2
respectively.
The affinity network in Fig. 2(a) indicates that site b can react with either site a or
site a
0
: this will capture the double action of the kinase. Figure 2(b) gives the definitions
of the species involved in the system. The first equation states that species A can be
transformed into A
+
on interaction at the site a. The second states that A
+
can either
interact on a
0
and be transformed into A
++
or convert back to A at rate k
3
without any
external agent — here “+” models the mutually exclusive choice of alternatives. In the
third equation, kinase K can interact at site b and then return to its initial state; recall
that according to the affinity network, this interaction might be with site a (on A) or a
0
(on A
+
). The final line defines the initial state of the system, with A and K present at
the specified concentrations.
This is a dynamic model, with P evolving in a continuous fashion. At any time
instant we can, using the methods of §2.2, formally derive the vector
dP
dt
specifying the
gradient of this temporal evolution. In particular, in the initial state we have:
dP
dt
= k
1
c
1
c
2
· A
+
− k
1
c
1
c
2
· A . (6)
The Continuous π-Calculus: A Process Algebra for Biochemical Modelling
1.2 Related Work
The application of the π-calculus to biology is due to Regev and Shapiro [3], who
identified the fundamental correspondence between cellular processes and concurrent
computations [13]. They proposed modelling molecules as π-processes, and the use
of parallel composition to express the fact that such molecules act independently. In
this framework, names denote molecular interaction sites and communication models
interaction. A further important abstraction was the use of private names to model
molecular complexes and compartments.
Further refinements of this framework addressed the introduction of quantitative
information into the model and on modelling compartments more directly. This led,
for example, to the development of the biochemical stochastic π-calculus [1] and
BioAmbients [14]. Other process algebras, such as PEPA [15] have also been applied
to model biological systems [16]. Although seen mainly as simulation engines, these
formalisms have also been used to perform static analysis of the model [17].
Using process algebras as a high-level descriptive language, Calder et al. [18] have
shown how PEPA models can generate both discrete (stochastic) and continuous (ODE)
behavioural specifications for the same system. Unlike raw ODEs then, a process algebra
model is not itself the behaviour, but can be used to generate it. We believe that this
abstraction step is important in modelling variation, to identify how emergent behaviour
depends on changes in a process or its parameters.
Meanwhile, the interest in mutational robustness has been growing amongst biolo-
gists for the past 15 years. A recent monograph on the subject [5] identifies the explosion
of high-throughput techniques as an important factor for this interest; the other is the
importance of this concept in the context of systems biology [19]. The methods applied
to study this phenomenon range from pure mathematics [20] to exhaustive computations
[21].
2 The Continuous π-Calculus
In this section we set out a formal syntax and mathematical semantics for the continuous
π-calculus. Both syntax and semantics have two “layers”: of species corresponding to
individual molecules, and processes to populations of these. It is important to keep in
mind, however, that none of these terms should invoke associations with their meaning
in the context of evolutionary theory.
2.1 Syntax
Definition 1. Take N a fixed, countably infinite set of names, denoted by lower-case
letters a, b, x, y, . . . Vectors of names are denoted by ~a, ~x etc.; these may be of zero
length.
Definition 2. A prefix is a syntactic expression of the form a(
~
b; ~y) (a communication
prefix) or τ@k (spontaneous or silent prefix), where all elements of ~y are distinct, τ /∈ N
is a fixed symbol and k ∈ R
≥0
. We use symbols like π, π
0
, etc. to denote prefixes.
![Fig. 6: The phosphorylation cycle of a single KaiC molecule. The two allosteric forms have opposite (de-)phosphorylation tendencies: phosphorylation proceeds from left to right, dephosphorylation from right to left. The potential to flip between the conformations closes the cycle. Nondominant reactions are indicated with dotted arrows. Adapted from [8, Fig. 1(B)].](/figures/fig-6-the-phosphorylation-cycle-of-a-single-kaic-molecule-dlbq5180.webp)
