Biochemical Reaction Rules with Constraints

Abstract

We propose React(C), an expressive programming language for stochastic modeling and simulation in systems biology, that is based on biochemical reactions with constraints. We prove that React(C) can express the stochastic pi-calculus, in contrast to previous rule-based programming languages, and further illustrate the high expressiveness of React(C). We present a stochastic simulator for React(C) independently of the choice of the constraint language C. Our simulator must decide for a given reaction rule whether it can be applied to the current biochemical solution. We show that this decision problem is NP-complete for arbitrary constraint systems C, and that it can be solved in polynomial time for rules of bounded arity. In practice, we propose to solve this problem by constraint programming.

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Biochemical Reaction Rules with Constraints
Mathias John12, C´ edric Lhoussaine12,
Joachim Niehren13, and Cristian Versari4
1BioComputing, Lifl (Cnrs Umr8022) & Iri (Cnrs Usr3078)
2University of Lille 1
3Inria, Lille
4University of Bologna
Abstract. We propose React(C), an expressive programming language
for stochastic modeling and simulation in systems biology that is based
on biochemical reactions with constraints. We prove that React(C) can
express the stochastic π-calculus, in contrast to previous rule-based pro-
gramming languages, and further illustrate the high expressiveness of
React(C). We present a stochastic simulator for React(C) independently
of the choice of the constraint language C. Our simulator decides for a
given reaction rule whether it can be applied to the current biochemical
solution. We show that this decision problem is NP-complete for arbi-
trary constraint systems C and that it can be solved in polynomial time
for rules of bounded arity. In practice, we propose to solve this problem
by constraint programming.
1Introduction
The paradigm of chemical reactions is predominant in programming languages
used for modeling and simulation in systems biology [6,9,3,19,1]. Chemical re-
actions are advantageous in that they can be given both, a continuous semantics
in terms of ordinary differential equations (odes) as well as a stochastic seman-
tics in terms of continuous time Markov chains (ctmcs). While odes describe
deterministically the average dynamics of molecule populations, ctmcs describe
the probabilities and speed of molecular interactions in an individual-based man-
ner. The continuous semantics of a system of chemical reactions is an abstraction
of its more precise stochastic semantics.
Biochemical reactions in the κ-calculus are widely accepted as a useful mod-
eling language for systems biology [4,12,5,6]. The underlying idea is to model
biochemical reactions as graph rewrite rules. The following rewrite rule, for in-
stance, states that a C-molecule with a free binding site 1 can be linked to
an A-B complex by using the free binding site 1 of A (while the complex uses
binding sites 2 of A and 1 of B).
A(1 + 2y),B(1y),C(1)
4.5
− − → (νx)A(1x+ 2y),B(1y),C(1x)
The stochastic rate 4.5 determines the distribution of the speed of this interac-
tions according to the law of mass-action.
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The alternative paradigm of agent-based modeling languages attracted also
much interest for modeling and simulation in systems biology. It underlies the
stochastic π-calculus and its many extensions, [18,21,14,17,23,11], BioAmbi-
ents [20], BlenX or beta-binders [22], etc. The close relationship between model-
ing languages of both paradigms was first pointed out by Cardelli [2]. He iden-
tified the fragment of the stochastic π-calculus without ν-binders with systems
of chemical reactions with the same ctmcs in order to obtain a continuous se-
mantics of π-calculus processes in terms of odes. So far, however, there exists
no positive result showing the expressiveness of the stochastic π-calculus for any
language of chemical reactions. There exists a result for the π-calculus without
stochastic semantics, which was shown equally expressive to the join calculus, a
programming language based on chemical reaction rules, by Fournet et. al. [7].
Unfortunately, the encoding presented cannot be adapted to a stochastic setting
in any obvious manner. Conversely, Danos and Laneve [6] showed that binary
reaction rules of the κ-calculus without ν-binders on the left-hand side can be
encoded in the π-calculus. The stochastic semantics is preserved as shown in [15].
A main limitation of the κ-calculus compared to the π-calculus is the restriction
to graph rather than hypergraph rewriting.
In this paper we present React(C), a language of biochemical reaction rules
with constraints. React(C) extends on the κ-calculus by hyperedge rewriting in
particular. The graph rewrite rule from the κ-calculus above, for instance, can
be written as follows, where free is a constant standing for a free binding site:
A(free;y),B(y),C(free)
4.5
− − → (νx) A(x;y),B(y),C(x)
Note that names of binding sites are identified by positions in React(C). The
usefulness of hypergraph rewriting can be illustrated at modeling compartments,
where the natural idea is to attribute each molecule by its compartment’s name,
i.e., to introduce a hyperedge per compartment that links all its molecules. One
can then constrain reactions to happen only within a same compartment. For
instance, consider a dimerization reaction between two A-molecules in the same
compartment x:
A(x,free),A(x,free)
3.1
− − → (νz)A(x;z),A(x;z)
Variables with numeric values and arithmetic constraints are supported by
React(C) too. These are useful for modeling dynamic volumes of compartments
and to cope with alternative kinetics as those of the π-calculus. Simpler finite do-
main constraints were proposed for biological modeling in [13] and in BioCham.
The following reaction rule for instance states that a polymerase bound at the
DNA nucleotide with position x can advance to the next nucleotide at position
y = x + 1 if x belongs to the finite domain {1,...47} \ {37}. The speed is given
by the law of Mass action with a stochastic rate of 4.5.
∀x ∈ {1,...47} \ {37}. Pol(x),Dna(y)
Finally, React(C) enables general kinetics beside Mass action and Michaelis-
Menten. Note that BioCham [3] and sbml [9] support both these kinetics but
neither variables nor ν-binders.
if y=x+1 then 4.5 else 0
− − − − − − − − − − − − − − − − − → Dna(x),Pol(y)
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Our main technical contribution is a proof that React(C) can indeed express
the stochastic π-calculus, even if restricted to binary reaction rules with equality
constraints and arithmetics on real numbers. This result is relevant since the π-
calculus is the usual yardstick for the expressiveness of concurrent languages. Our
result also illustrates that React(C) can express extensions of the stochastic π-
calculus with constraints such as π@ [23] (without priorities) and the attributed
π-calculus [11]. This means that all previous models in these languages carry over
to React(C) in a systematic manner. Hyperedges and constraints thus provide
the missing link between rule-based and agent-based modeling languages.
We present a stochastic simulation algorithm for React(C) that is indepen-
dent of the choice of the constraint language C. Our simulator must decide for
a given reaction rule whether it can be applied to the current biochemical solu-
tion. We show that this decision problem is NP-complete for arbitrary constraint
systems C and that it can be solved in polynomial time for rules of bounded
arity. In practice, we propose to solve this problem by constraint programming.
Our hardness proof relies on hypergraphs, so it does not apply to the κ-
calculus. Indeed, the so called rigidity property of the κ-calculus (Lemma 3 of
[5]) fails for React(C). Rigidity states that a matching of a connected pattern
is entirely determined by matching only a single one of its molecules. It implies
that the matching problem for the κ-calculus restricted to rules with connected
patterns can be solved in P-time. The general case with multiple connected
components remains open. We leave it also open whether the scalable simulation
algorithm for the κ-calculus from [5] can be lifted to React(C) in any sense.
Outline. We start with a small language React=of biochemical reaction rules
with equality constraints in Section 2 and show that it can express the stochastic
π-calculus in Section 3 and 4. The main remaining problem not discussed so far, is
to link stochastic mass-actions semantics with redex based stochastic semantics
as in the stochastic π-calculus. The full language React(C) is presented in Section
5 and our simulation algorithm based on constraint programming in Section 6.
2 Reaction Rules with Equality Constraints
We present a small language of biochemical reaction rules with equality con-
straints React=that can express the stochastic π-calculus. We equip React=
with a stochastic semantics that follows the usual law of Mass action.
We assume a signature A of molecule names A ∈ A, each of which has a fixed
arity in ar(A) ∈ N0. We also assume an infinite set N of (link) names ranged
over by x,y and write ˜ x for a possibly empty sequence of names x1;...;xn. A
molecule a is a term A(˜ x) with n = ar(A).
We define the biochemical solutions of React=in Fig. 2 as terms s that
are constructed from molecules A(˜ x), the composition operator s,s?, and the
empty solution 0. We often think of a biochemical solution as a (hyper-) graph
of molecules that are linked by (hyper-) edges. For instance A(x),B(x) describes
the graph with two nodes A(x) and B(x) linked by a single edge named x. Such
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Solutionss ::= A(˜ x) | s,s?| 0
e ::= if x1=x2 then e1 else e2
where A ∈ A, ˜ x ∈ N
where x1,x2 ∈ N
and d ∈ R+
where fn(r) = fn(s)
Rate expressions
| e1+ e2 | e1∗ e2 | d
r ::= s
0
Reaction rules
e− → (ν˜ x)s?
d− → s?
Reactionsρ ::= s
Fig.1. Syntax of reaction rules of language React=
Precongruence(s1,s2),s3 ≈ s1,(s2,s3)
s ≈ s?
s1,s2 ≈ s2,s1
s,0 ≈ s
Congruence
σ : N → N injective
s ≡ s?σ
Fig.2. Precongruence and congruence on solutions
graphs do neither depend on the order of molecules nor on the concrete choice
of link names. For instance, the same graph is obtained by solutions B(y),A(y)
and A(x),B(x).
In Fig. 2, we define two congruence relations on solutions. The precongruence
≈ captures order independence. It is the least equivalence relation on solutions
that renders the composition operator “,” associative and commutative with the
neutral element 0. We write [s]≈for the equivalence class of a solution s. Clearly,
we can identify precongruence classes with multisets of molecules. The weaker
congruence relation ≡ accounts for the irrelevance of concrete names in addition.
It is defined such that s ≡ s?if and only if there exists a solution s??≈ s and
an injective function σ : N → N such that s?= s??σ, i.e., the term obtained by
renaming all names x in s??to σ(x).
We write fn(s) for the set of names occurring in s. As usual, we define it-
erated compositions?0
Modulo precongruence, this term stands for the multiset with mioccurrences of
molecule ai.
Reactions ρ are terms of the form s1
solutions congruent to s,s1to some solution congruent to s,s2:
s?
d− → s2? s?
Judgements ρ ? s?
Note that the rate constant d is irrelevant here; it matters in the stochastic
semantics only, see below.
Reaction rules r are terms of the form s
as schemas that define sets of reactions, one reaction per substitution σ : N →
i=1si= 0,?n
i=1si= (?n−1
i=1si),sn, and sm=?m
i=1s. If
i=1ami
a1,...,anare pairwise distinct molecules then we define
·
?n
i=1ami
i
=?n
i
.
d− → s2. They can be applied to rewrite
(react)
1≡ s,s1
s1
s,s2≡ s?
1→ s?
2
2
1→ s?
2capture the non-deterministic semantics of reactions.
e− → (ν˜ x)s?. They are to be understood
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(cond1)
e1 ⇓ d1
if x=x then e1 else e2 ⇓ d1
(reals)d ∈ R+
d ⇓ d
(cond2)
x1 ?= x2
e2 ⇓ d2
if x1=x2 then e1 else e2 ⇓ d2
e1 ⇓ d1
e1∗ e2 ⇓ d1∗Rd2
0
(+)
e1 ⇓ d1
e1+ e2 ⇓ d1+Rd2
e2 ⇓ d2
(∗)
e2 ⇓ d2
Fig.3. Big-step evaluator of rate expressions.
(inst)
σ : fn(s) → Nσ?: {˜ x} → N\(N ∪ fn(s)) injective
s
eσ ⇓ d
e− → (ν˜ x)s?⇓σ,N sσ
d− → s?σ?σ
Fig.4. Instantiation and evaluation of reactions rules to reactions.
N. Reaction rules contain a rate expressions e. Substitutions σ instantiating the
rule are applied to e before evaluation, yielding another rate expression that
we denote by eσ. Before formalizing the semantics of reaction rules, we need to
define the values of rate expressions.
A rate expression e is a term built from constants d ∈ R+
tiplication e∗e, and rate-valued equality constraints if x1=x2then e1else e2.
We write fn(e) for the set of names occurring in e and Exprs for the set of all rate
expressions. Usual Boolean-valued constraints are subsumed, as for instance the
conjunctive equality and inequality constraints x=y ∧ y?=z by rate expression
if x=y then (if y=z then 0 else 1) else 0. In Fig. 3, we define an evaluator
for rate expressions ⇓: Exprs → R+
terminates: neither there exist program errors nor non-termination.
A reaction rule r = s
names in ˜ x with scope s?similarly to the new operator of the π-calculus. It
requires the creation of new names ˜ x with scope s?. Being new means to not
occur in the current solution to which the rule is applied. The free names of
(ν˜ x)s?are thus defined by fn((ν˜ x)s?) = fn(s?) \ {˜ x}, and the free names of the
reaction rule by fn(r) = fn(s) ∪ fn(e) ∪ fn((ν˜ x)s?).
The instantiation (Inst) of a reaction rule r = s
is defined in Fig. 4. There, we assume that N ⊆ N is a finite set of names
– this will be the set of names of the current chemical reaction – and that
σ : fn(s) → N is a substitution. Even though the domain of σ is restricted to
fn(s) we can still apply σ to r, since fn(r) = fn(s) is assumed in the syntax of
React=. As a consequence, there exist only finitely many such substitutions and
thus only finitely many rule instances to be considered (for some fixed new-name
generator). The application of σ to r is defined as follows. First, some new-name
generator σ?: {˜ x} → N \ (N ∪ fn(s)) introduces new names fresh for N and
fn(s) on r.h.s. of r, second, substitution σ is applied to the resulting rule, third,
the expression eσ is evaluated to some real number d. In this case, we say that
r can be instantiated ρ by σ and N, where ρ = sσ
0, addition e+e, mul-
0as usual. Note that this evaluator always
e− → (ν˜ x)s?uses the new operator (ν˜ x) that binds the
e− → (ν˜ x)s?to some reaction
d− → s?σ?σ, and write r ⇓σ,Nρ.
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