Decompositions of large-scale biological systems based on dynamical properties
Nicola Soranzo
2
Fahimeh Ramezani
1
Giovanni Iacono
0
Claudio Altafini
0
Associate Editor: Martin Bishop
0
SISSA, via Bonomea 265, 34136 Trieste,
Italy
1
Max-Planck-Institut fr Informatik
, Stuhlsatzenhausweg 85, 66123 Saarbrcken,
Germany
2
CRS4 Bioinformatica, Loc. Piscina Manna, 09010 Pula (
CA
),
Italy
Motivation: Given a large-scale biological network represented as an influence graph, in this article we investigate possible decompositions of the network aimed at highlighting specific dynamical properties. Results: The first decomposition we study consists in finding a maximal directed acyclic subgraph of the network, which dynamically corresponds to searching for a maximal open-loop subsystem of the given system. Another dynamical property investigated is strong monotonicity. We propose two methods to deal with this property, both aimed at decomposing the system into strongly monotone subsystems, but with different structural characteristics: one method tends to produce a single large strongly monotone component, while the other typically generates a set of smaller disjoint strongly monotone subsystems. Availability: Original heuristics for the methods investigated are described in the article. Contact: The Author 2011. Published by Oxford University Press. All rights reserved. For Permissions, please email:
1 INTRODUCTION
One of the outstanding challenges that Systems Biology is currently
facing is to provide the right tools for the investigation of the
dynamical behavior of the large-scale networks used to represent
complex biological systems, such as gene regulatory networks,
signaling pathways and chains of metabolic reactions. Even if
our knowledge of the interactions among the molecular species
involved in these systems is growing at a fast pace, the details
of the dynamics that they describe are seldom available and often
unlikely to be obtainable in a near future. What is often more
plausible to assume is that only an influence graph is available for
these networks (Fages and Soliman, 2008; Klamt et al., 2006). An
influence graph is a signed graph where an edge represents the action
of a variable on another variable, and the signs may have the meaning
of activatory/inhibitory action, or may simply represent the signature
of the Jacobian linearization of a non-linear vector field which is
unknown but sign constant over the entire state space (common
forms of the kinetics, such as mass action and MichaelisMenten,
normally obey to this condition). In choosing this level of detail for
our networks, we are guided by an abundant literature, see e.g. Fages
and Soliman (2008); Huber et al. (2007); Klamt et al. (2006); Milo
et al. (2002); Papin et al. (2005); Shen-Orr et al. (2002); Thieffry
(2007). Important dynamical problems that can be investigated on
an influence graph include:
(1) compute the equilibria of the system (Soul, 2003);
(2) investigate the stability properties of the dynamics (Deangelis
et al., 1986; Quirk and Ruppert, 1965);
(3) identify the largest open-loop subsystem of a given system
(Ispolatov and Maslov, 2008);
(4) study the monotonicity and strong monotonicity properties of
the dynamics (Sontag, 2007); and
(5) select a minimal intervention set for medical treatment (Klamt
et al., 2006).
In this article, we are interested in the problems (3) and (4) of the
list above.
In graph theoretical terms, finding the largest open-loop
subsystem corresponds to identifying a maximum-size directed
acyclic graph (DAG) within a network by dropping all feedback
loops. In the computer science literature, this is called the minimum
feedback arc set problem, and it is well known to be NP-hard (Karp,
1972). Although several heuristic methods are already available for it
(Festa et al., 1999; Ispolatov and Maslov, 2008), the novel algorithm
we propose in this article has the advantage that available a priori
knowledge on the open-loop part of the system can be easily taken
into account when computing a maximal DAG. We will show in
the large-scale examples of Section 6 that the performances of our
algorithm are comparable to those of the best heuristics.
In a series of papers by E. Sontag and colleagues (DasGupta
et al., 2007; Maayan et al., 2008; Sontag, 2007), it was shown
that influence graphs can be used to study an important property
of dynamical systems, namely monotonicity (Kunze and Siegel,
1994, 1999; Smith, 1988, 1995; Sontag, 2007). Monotone systems
have nice properties of order in their dynamical behavior. For
example, they neither admit stable periodic orbits nor chaotic
behavior. Moreover, for strongly monotone systems [i.e. monotone
systems whose graph is irreducible, see Smith (1995); Sontag
(2007)], Hirsch theorem states that almost all bounded solutions
converge to the set of equilibria (Hirsch, 1983). The concept
is particularly attracting for biological networks, because it is
well known that these systems, though complex, have indeed
outstanding stability properties, are la (...truncated)