BioSystems 109 (2012) 78–86 Contents lists available at SciVerse ScienceDirect BioSystems j o ur nal homep age : www.elsevier.com/locate/ biosystems Solving dynamical inverse problems by means of Metabolic P systems V. Manca ∗ , L. Marchetti University of Verona, Department of Computer Science, Strada Le Grazie 15, 37134 Verona, Italy a r t i c l e i n f o Article history: Received 26 October 2011 Received in revised form 29 December 2011 Accepted 29 December 2011 Keywords: Metabolic P systems Dynamical systems Dynamical inverse problems Stepwise regression a b s t r a c t MP (Metabolic P) systems are a class of P systems introduced for modelling metabolic processes. We refer to the dynamical inverse problem as the problem of identifying (discrete) mathematical models exhibiting an observed dynamics. In this paper, we complete the definition of the algorithm LGSS (Log- gain Stoichiometric Stepwise regression) introduced in Manca and Marchetti (2011) for solving a general class of dynamical inverse problems. To this aim, we develop a reformulation of the classical stepwise regression in the context of MP systems. We conclude with a short review of two applications of LGSS for discovering the internal regulation logic of two phenomena relevant in systems biology. © 2012 Elsevier Ireland Ltd. All rights reserved. 1. Introduction The main framework analysis for the most part of biological dynamics remains the theory of ordinary differential equations (ODEs). Metabolic P systems (MP systems), based on P˘ aun’s P systems (P˘ aun, 2002), were introduced in Manca et al. (2005) for mod- elling metabolic systems by means of suitable multiset rewriting grammars. They are essentially a particular type of finite difference recurrent equations where “fluxes” (see later) play a role analogous to that of derivatives in ODEs. This change of perspective, from a continuous to a discrete approach, provides in many cases compu- tational and modelling advantages. The following discussion and the results of the present paper intend to argument an important case showing such a kind of advantages. A Metabolic P system is essentially a multiset grammar where multiset transformations are regulated by functions (Manca, 2010; P˘ aun and Rozenberg, 2010). Namely, a rule like a + b → c means that a number u of molecules of kind a and u of kind b are replaced by u molecules of type c. The value of u is the flux of the rule application. Let us assume to consider a system at some time steps i = 0, 1, 2, . . ., t (i ∈ N, the set of natural numbers). Let us also assume that a substance x is produced by rules r 1 , r 3 and consumed by rule r 2 . If u 1 [i], u 2 [i], u 3 [i] are the fluxes of the rules r 1 , r 2 , r 3 , respectively, ∗ Corresponding author. E-mail addresses: vincenzo.manca@univr.it (V. Manca), luca.marchetti@univr.it (L. Marchetti). in the passage from step i to step i + 1, then the variation  x [i] of substance x at step i is given by:  x [i] = x[i + 1] - x[i] = u 1 [i] - u 2 [i] + u 3 [i]. (1) In an MP system, in any state the flux u l of rule r l is provided by a state function ϕ l , called regulator of the rule. A state is essentially determined by the values of the system variables, that is, substances and parameters (quantities which are not transformed by the rules). However, usually only some variables enter as arguments of regu- lators, therefore if u l = ϕ l (x, y, . . .), the arguments x, y, . . . of ϕ l will be called tuners of the regulator. Substances (also metabolites), rules, initial values and regulators define an MP grammar which is easily representable by an MP graph (Manca and Bianco, 2008). The set of the rules of an MP grammar can be also represented by a stoichiometric matrix A, which gives a sort of “matrix-like representation” of the system stoichiometry (see Fig. 1). An MP system is essentially an MP grammar equipped with a temporal interval , a conventional mole size , and substances masses, which specify the time and population (discrete) granular- ities, respectively (Manca, 2010; P˘ aun and Rozenberg, 2010). MP systems inherited from P systems the multiset rewrit- ing mechanism as their fundament, by developing a different perspective. In fact, while P systems were essentially unconven- tional computational models, MP systems are intended to generate dynamics instead of computations. Namely, their aim in modelling biological phenomena is that of finding the multiset rewriting mechanism underlying an observed biological behaviour. They were successfully applied in many modelling contexts (Manca and Marchetti, 2010b,a; Manca et al., 2011; Marchetti and Manca, 2012) 0303-2647/$ – see front matter © 2012 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2011.12.006