Lumpability abstractions of rule-based systems

Abstract

The induction of a signaling pathway is characterized by transient complex formation and mutual posttranslational modification of proteins. To faithfully capture this combinatorial process in a mathematical model is an important challenge in systems biology. Exploiting the limited context on which most binding and modification events are conditioned, attempts have been made to reduce the combinatorial complexity by quotienting the reachable set of molecular species, into species aggregates while preserving the deterministic semantics of the thermodynamic limit. Recently we proposed a quotienting that also preserves the stochastic semantics and that is complete in the sense that the semantics of individual species can be recovered from the aggregate semantics.
In this paper we prove that this quotienting yields a sufficient condition for \emph{weak lumpability} and that it gives rise to a backward Markov bisimulation between the original and aggregated transition system. We illustrate the framework on a case study of the EGF/insulin receptor crosstalk.

Full text

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G.Ciobanu, M.Koutny (Eds.): Membrane Computing and
Biologically Inspired Process Calculi 2010 (MeCBIC 2010)
EPTCS 40, 2010, pp. 142–161, doi:10.4204/EPTCS.40.10
Lumpability Abstractions of Rule-based Systems∗
Jerome Feret
LIENS (INRIA/´ENS/CNRS)
Paris, France
feret@ens.fr
Thomas Henzinger
Institute and Science of Technology
Vienna, Austria
thenzinger@ist.ac.at
Heinz Koeppl
School of Computer and Communication Sciences
EPFL
Lausanne, Switzerland
heinz.koeppl@epfl.ch
Tatjana Petrov
School of Computer and Communication Sciences
EPFL
Lausanne, Switzerland
tatjana.petrov@epfl.ch
The induction of a signaling pathway is characterized by transient complex formation and mutual
posttranslational modification of proteins. To faithfully capture this combinatorial process in a math-
ematical model is an important challenge in systems biology. Exploiting the limited context on which
most binding and modification events are conditioned, attempts have been made to reduce the com-
binatorial complexity by quotienting the reachable set of molecular species, into species aggregates
while preserving the deterministic semantics of the thermodynamic limit. Recently we proposed a
quotienting that also preserves the stochastic semantics and that is complete in the sense that the
semantics of individual species can be recovered from the aggregate semantics. In this paper we
prove that this quotienting yields a sufficient condition for weak lumpability and that it gives rise to a
backward Markov bisimulation between the original and aggregated transition system. We illustrate
the framework on a case study of the EGF/insulin receptor crosstalk.
1Introduction
Often a few elementary events of binding and covalent modification [31] in a biomolecular reaction
system give rise to a combinatorial number of non-isomorphic reachable species or complexes [17, 18].
Instances of such systems are signaling pathway, polymerizations involved in cytoskeleton maintainance,
the formation of transcription factor complexes in gene-regulation.
For such biomolecular systems, traditional chemical kinetics face fundamental limitations, that are
related to the question of how biomolecular events are represented and translated into a mathematical
model [24]. More specifically, chemical reactions can only operate on a collection of fully specified
molecular species and each such species gives rise to one differential equation, describing the rate of
change of that species’ concentration. Many combinatorial systems do not permit the enumeration of all
molecular species and thus render their traditional differential description prohibitive. However, even if
one could enumerate them, it remains questionable whether chemical reactions are the appropriate way
to represent and to reason about such systems.
As the dynamics of a biomolecular reaction mixture comes about through the repeated execution of a
few elementary events one may wonder about the effective degrees of freedom of the reaction mixture’s
dynamics. If the velocity of all events – or their probabilities to occur per time-unit per instance –
∗J´ erˆ ome Feret’s contribution was partially supported by the ABSTRACTCELL ANR-Chair of Excellence. Heinz Koeppl
acknowledges the support from the Swiss National Science Foundation, grant no. 200020-117975/1. Tatjana Petrov acknowl-
edges the support from SystemsX.ch, the Swiss Initiative in Systems Biology.

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J. Feret, T. Henzinger, H. Koeppl, T. Petrov 143
are different for all complexes (w.r.t. modification) and pairs of complexes (w.r.t. binding) to which
the events can apply to, then the degrees of freedom would equal to the number of molecular species.
However, due to the local nature of physical forces underlying molecular dynamics, the kinetics of most
events appear to be ignorant with respect to the global configuration of the complexes they are operating
on. More provocatively, one may say that even if there would be variations of kinetics of an event from
one context to another, experimental biology does not – and most likely never will – have the means to
discern between all different contexts. For instance, fluorescence resonance energy transfer (FRET), may
report on a specific protein-binding event and even its velocity, however we have no means to determine
whether the binding partners are already part of a protein complex – not to speak of the composition and
modification state of these complexes. To this end, molecular species remain elusive and appear to be
inappropriate entities of descriptions.
To align with the mentioned experimental insufficiencies and with the underlying biophysical lo-
cality, rule-based or agent-based descriptions were introduced as a framework to encode such reaction
mixtures succinctly and to enable their mathematical analysis [8, 3]. Rules exploit the limited context
on which most elementary events are conditioned. They just enumerate that part of a molecular species
that is relevant for a rule to be applicable. Thus, in contrast to chemical reactions, rules can operate on a
collection of partially specified molecular species. Recently, attempts have been made to identify the set
of those partially specified species that allow for a self-consistent description of the rule-set’s dynamics
[11, 6]. Naturally, as partially specified species – or fragments – in general encompass many fully speci-
fied species, the cardinality of that set is less than of the set of molecular species. These approaches aim
to obtain a self-consistent fragment dynamics based on ordinary differential equations. It represents the
dynamics in the thermodynamic limit of stochastic kinetics when scaling species multiplicities to infinity
while maintaining a constant concentration (multiplicity per unit volume) [22]. In many applications in
cellular biology this limiting dynamics is an inappropriate model due to the low multiplicities of some
molecular species – think of transcription factor - DNA binding events. We recently showed that the
obtained differential fragments cannot be used to describe the finite volume case of stochastic kinetics
[12]. Exploiting statistical independence of events we instead derived stochastic fragments that repre-
sent the effective degrees of freedom in the stochastic case. Conceptually, the procedure replaces the
Cartesian product by a Cartesian sum for statistically independent states. In contrast to the differential
case, stochastic fragments have the important property that the sample paths of molecular species can be
recovered from that of partially specified species.
We believe that interdisciplinary fields, such as systems biology, can move forward quickly enough
only if the well-established knowledge in each of the disciplines involved is exploited to its maximum,
i.e. if the standardized, well-established theories are recognized and re-used. For that reason, in this
paper we translate our abstraction method ([12]) into the language of well-established contexts of ab-
straction for probabilistic systems – lumpability and bisimulation. Lumpability is mostly considered
from a theoretical point of view in the theory of stochastic processes [19, 13, 28, 26, 27, 4]. A Markov
chain is lumpable with respect to a given aggregation (quotienting) of its states, if the lumped chain pre-
serves the Markov property [21]. A sound aggregation for any initial probability distribution is referred
to as strong lumpability, while otherwise it is termed weak lumpability [4, 29]. Approximate aggregation
techniques for Markov chains of biochemical networks are discussed in [14]. Probabilistic bisimulation
was introduced as an extension to classic bisimulation in [23]. It is extended to continuous-state and
continuous-time in [9] and, for the discrete-state case, to weak bisimulation [2]. For instance, in [9]
the authors use bisimulation of labelled Markov processes, the state space of which is not necessarily
discrete, and they provide a logical characterization of probabilistic bisimulation. Another notion of
weak bisimulation was recently introduced in [10]. Therein two labeled Markov chains are defined to

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144 Lumpability Abstractions of Rule-based Systems
be equivalent if every finite sequence of observations has the same probability of occurring in the two
chains. Herein we recognize the sound aggregations of [12] as a form of backward Markov bisimulations
on weighted labeled transition systems (WLTS), and we show it to be equivalent to the notion of weak
lumpability on Markov chains.
The remaining part of the paper is organized as follows. In the Section 2, we introduce weighted
labeled transition systems (WLTS) and we assign it the trace density semantics of a continuous-time
Markov chain (CTMC). Moreover, we define the Kappa language, and we assign a WLTS to a Kappa
specification. Based on the notion of the annotated contact map we briefly summarize in Section 3 the
general procedure to compute stochastic fragments, as it is offered in [12]. In Section 4, we introduce
the characterizations of sound and complete abstractions on WLTS as a backward Markov bisimulation.
Moreover, we define it being equivalent to the weak lumpability on Markov chains. Finally, we provide
in Section 5 results for the achieved dimensionality reduction for a rule-based model of the crosstalk
between the EGF/insulin signaling pathway [6]. This mechanistic model comprises 76 rules giving rise
to 42956 reactions and 2768 molecular species.
2 Preliminaries
The stochastic semantics of a biochemical network is modelled by a continuous-time Markov chain
(CTMC). The main object that we will use in the analysis is the weighted labelled transition system
(WLTS) on a countable state space. We will assign a WLTS to a given Kappa specification, and we
manipulate that object when reasoning about abstractions.
2.1 CTMC
We will observe the CTMC that is generated by the weighted labelled transition system (WLTS) on a
countable state space. We define the CTMC of a WLTS, by defining the Borel σ-algebra containing all
cylinder sets of traces [20] that can occur in the system, and the corresponding probability distribution
among them. We also introduce the standard notation of a rate matrix, which we will use when analysing
the lumpability and bisimulation properties in Sec. 4.
Definition 1. (WLTS) A weighted-labelled transition system W is a tuple (X ,L,w,π0), where
• X is a countable state space;
• L is a set of labels;
• w : X ×L ×X → R+
• π0: X → [0,1] is an initial probability distribution.
We assume that the label fully identifies the transition, i.e. for any x ∈ X and l ∈ L, there is at most one
x?∈ X , such that w(x,l,x?) > 0. Moreover, we assume that the system is finitely branching, in the sense
that (i) the set {x ∈ X | π0(x) > 0} is finite, and (ii) for arbitrary ˆ x ∈ X , the set {(l,x?) ∈ L ×X |
w(ˆ x,l,x?) > 0} is finite.
The activity of the state xi, denoted a : X → R+
a(xi) :=∑{w(xi,l,xj) | xj∈ X, l ∈ L}.
ThedefinitionofaWLTSimplicitelydefinesatransitionrelation→⊆X ×X , suchthat(xi,xj)∈→,
if and only if there exists a non-zero transition from state xito state xj, i.e. the total weight over all labels
is strictly bigger then zero, written∑{w(xi,l,xj) | l ∈L}>0. Moreover, we can differentiate the initial
set of states I ⊆ X , such that their initial probabilities are positive, i.e. I = {x ∈ X | π0(x) > 0}.
0is the weighting function that maps two states and a label to a real value;
0is the sum of all weights originating at xi, i.e.

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J. Feret, T. Henzinger, H. Koeppl, T. Petrov145
Definition 2. (Rate matrix of a WLTS) Given a WLTS W = (X ,L,w,π0), we assign it the CTMC rate
matrix R : X ×X → R, given by R(xi,xj) =∑{w(xi,l,xj) | l ∈ L}.
The consequence is that we do not enforce R(xi,xi) = −∑{R(xi,xj) | i ?= j}, as it is usual for the
generator matrix of CTMCs. This however does not affect the transient, not the steady-state behavior
of the CTMC [1]. We do so for the following reason. When abstracting the WLTS by partitioning the
state space, we get another WLTS. If the two states x and x?which have a transition between each other
were aggregated in the same partition class ˜ x, it will result as a prolongation of the residence time in the
abstract state ˜ x, i.e. we will have a self-loop in the abstract WLTS.
If we refer to the generated stochastic Markov process, written as a continuous-time random variable
{Xt}t∈R+
the value xiat time point t. It thus holds that Pr(X0= xi) = π0(xi), and Pr(Xt+dt= xj | Xt= xi) =
R(xi,xj)dt when i ?= j, whereas Pr(Xt+dt= xi | Xt= xi) = R(xi,xi)dt+(1−∑{R(xi,xj?)dt | xj? ∈ X }),
which gives after simplification: Pr(Xt+dt= xi| Xt= xi) = 1−∑{R(xi,xj?)dt | j??= i}.
We define the cylinder sets of traces that can be observed in the system W . By observing the trace at
a certain time point, we mean observing the sequence of visited states, labels that were assigned to the
executed transitions, and time points of when the transition happened.
Definition 3. (A trace of a WLTS) Let us observe the WLTS W = (X ,L,w,π0) and its CTMC. Given a
number k in N, we define a trace of length k as τ ∈ (X ×L ×R+
τ = x0
→ x1...xk−1
If the trace τ is such that (i) π0(x0) > 0, and (ii) for all i, 0 ≤ i ≤ k, we have that w(xi,li,xi+1) > 0, then
we say that τ belongs to the set of traces of W , and we write it τ ∈ T (W ).
The ‘time stamps’ on each of the transitions denote intuitively the absolute time of the transition,
from the moment when the system was started (t = 0). We cannot assign the probability distribution to
the traces in T (W ), since the probability of any such trace is zero. We thus introduce the cylinder set of
traces over intervals of times.
Definition 4. (Cylinder set of traces) If IR is the set of all nonempty intervals in R+
cylinder set of traces τIR∈ (X ×L ×IR)k×X , such that:
τIR= x0
→ x1...xk−1
denotes the set of all traces τ = x0
→ x1...xk−1
of traces τIRis such that π0(x0) > 0, and for all i = 0,...,k−1, we have that w(xi,li,xi+1) > 0, then we
say that τIRbelongs to the cylinder set of traces of W , and we write τIR∈ TIR(W ).
Let Ω(TIR(W )) be the smallest Borel σ-algebra that contains all the cylinder sets of traces in
TIR(W ). We define a probability measure over Ω(TIR(W )) in the following way.
Definition 5. (Trace density semantics on a WLTS) Given a WLTS (X ,L,w,π0), and a number k in N,
the probability of the cylinder set of traces τIR∈ TIR(W ), specified as in expression (1), is given by:
k
∏
i=1
0, over the countable state space X . We write Pr(Xt= xi), the probability that the process takes
0)k×X , written
xk.
l1,t1
lk,t1+...+tk
→
0, then we define the
l1,I1
lk,Ik
→ xk
xk, such that ti∈ Ii, 1 ≤ i ≤ k. If the cylinder
(1)
l1,t1
lk,t1+...+tk
→
π(τIR) = π(x0
l1,I1
→ x1...xk−1
lk,Ik
→ xk) = π0(x0)
w(xi−1,li,xi)
a(xi−1)
?
e−a(xi−1)·inf(Ii)−e−a(xi−1)·sup(Ii)?
.
Note that?
a(xi−1)e−a(xi−1).
Iia(xi−1)e−a(xi−1)·tdt = e−a(xi−1)·inf(Ii)−e−a(xi−1)·sup(Ii)is the probability of exiting the state
xi−1in a time interval Ii−1, since the probability density function of the residence time of xi−1is equal to

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146Lumpability Abstractions of Rule-based Systems
2.2Kappa
We present Kappa in a process-like notation. We start with an operational semantics, then define the
stochastic semantics of a Kappa model.
We assume a finite set of agent names A , representing different kinds of proteins; a finite set of
sites S, corresponding to protein domains; a finite set of internal states I, and Σι,Σλtwo signature maps
from A to℘(S), listing the domains of a protein which can bears respectively an internal state and a
binding state. We denote by Σ the signature map that associates to each agent name A ∈ A the combined
interface Σι(A)∪Σλ(A).
Definition 6. (Kappa agent) A Kappa agent A(σ) is defined by its type A ∈ A and its interface σ. In
A(σ), the interface σ is a sequence of sites s in Σ(A), with internal states (as subscript) and binding
states (as superscript). The internal state of the site s may be written as sε, which means that either it
does not have internal states (when s ∈ Σ(A)\Σι(A)), or it is not specified. A site that bears an internal
state m ∈ I is written sm(in such a case s ∈ Σι(A)). The binding states of a site s can be specified as sε,
if it is free, otherwise it is bound (which is possible only when s ∈ Σλ(A)). There are several levels of
information about the binding partner: we use a binding label i ∈ N when we know the binding partner,
or a wildcard bond − when we only know that the site is bound. The detailed description of the syntax
of a Kappa agent is given by the following grammar:
a
N
σ
::=
::=
::=
N(σ)
A ∈ A
ε | s,σ
(agent)
(agent name)
(interface)
s
n
ι
λ
::=
::=
::=
::=
nλ
x ∈ S
ε | m ∈ I
ε | − | i ∈ N
ι
(site)
(site name)
(internal state)
(binding state)
We generally omit the symbol ε.
Definition 7. (Kappa expression) Kappa expression E is a set of agents A(σ) and fictitious agents / 0.
Thus the syntax of a Kappa expression is defined as follows:
E ::= ε | a , E | / 0 , E.
Thestructuralequivalence≡, definedasthesmallestbinaryequivalencerelationbetweenexpressions
that satisfies the rules given as follows:
E , A(σ,s,s?,σ?) , E?
E , a , a?, E?
≡
≡
≡
E , A(σ,s?,s,σ?) , E?
E , a?, a , E?
E , / 0 E
i, j ∈ N and i does not occur in E
E[i/j] ≡ E
i ∈ N and i occurs only once in E
E[ε/i] ≡ E
stipulates that neither the order of sites in interfaces nor the order of agents in expressions matters, that
a fictitious agent might as well not be there, that binding labels can be injectively renamed and that
dangling bonds can be removed.
Definition 8. (Kappa pattern, Kappa mixture) A Kappa pattern is a Kappa expression which satisfies
the following five conditions: (i) no site name occurs more than once in a given interface; (ii) each site
name s in the interface of the agent A occurs in Σ(A); (iii) each site s which occurs in the interface of
the agent A with a non empty internal state occurs in Σι(A); (iv) each site s which occurs in the interface
of the agent A with a non empty binding state occurs in Σλ(A); and (v) each binding label i ∈ N occurs
exactly twice if it does at all –there are no dangling bonds. A mixture is a pattern that is fully specified,
i.e. each agent A documents its full interface Σ(A), a site can only be free or tagged with a binding label
i ∈ N, a site in Σι(A) bears an internal state in I, and no fictitious agent occurs.