Vol.:(0123456789) 1 3 Journal of Membrane Computing https://doi.org/10.1007/s41965-019-00012-3 REVIEW PAPER Metabolic computing Vincenzo Manca 1 Received: 14 January 2019 / Accepted: 22 March 2019 © Springer Nature Singapore Pte Ltd. 2019 Abstract The paper reviews some aspects of MP grammars, a particular type of P systems (M stands for Metabolic) consisting of multiset rewriting rules, which were introduced in the context of Membrane Computing, for modeling biological dynamics. The main features of MP theory are recalled, such as the control mechanisms based on regulation functions, MP graphs, representation of oscillatory dynamics, regression algorithms, and MP modeling. Finally, the computational universality of MP grammars is proved by means of Minsky’s register machines. Keywords Discrete dynamics · Multiset grammars · MP regression algorithms · Metabolic computing 1 Introduction MP systems are discrete dynamical systems introduced in the context of membrane computing [1] and investigated for more than 20 years. Preliminary results were developed since the end of 1990 years [2–7]. Related approaches to MP theory and frst investigations on MP regression (see later on) were investigated in [8–18]. Applications of MP systems in modeling biological systems, theoretical founda- tions, and efcient MP regression algorithms were investi- gated in [19–30]. MP systems are essentially multiset rewriting rules with functions that defne the quantities of transformed elements. The attribute MP comes from P systems (multiset rewrit- ing rules distributed in compartments) introduced by Păun [31–34], where the focus is on metabolic processes. MP regression is a peculiar aspect of MP theory, which pro- vides methods that determine MP grammars able to generate observed time series of an observed dynamics. MP regres- sion algorithms rely on the methods of algebraic manipula- tion and Least Square Evaluation, or on statistical methods, or genetic algorithms, by providing very accurate solutions [1, 20, 24, 26, 35–43]. Public software platforms based on MP theory are available, which provide examples and docu- mentation [44, 45]. In the present paper, we mainly consider MP grammars for their computational aspect, which provides a natural notion of distributed computation where the program is encoded by a graph expressing the transformations and the regulations of a fnite number of multiset rewriting rules. 2 MP grammars An MP grammar G is a discrete dynamical system based on a set X of variables, and a state space constituted by the assignments of values to variables of X. Let ℕ be the set of natural numbers. Assuming variables in some order, if X is a fnite set of n ∈ ℕ variables, the set of possible states of G coincides with the set ℝ n of real vectors of dimension n. A dynamics function  G is associated with G that provides a next-state function, which changes the variable values, according to an increase–decrease variation specifed by all the rules (if a variable does not occur in a rule, its value remains unchanged). Namely, a reading of “MP” is the basic Minus–Plus mechanism of the rules of an MP grammar. A multiset over X is a function assigning a natural number, called multiplicity, to every x ∈ X . The formal defnition of MP grammar follows. Defnition 1 An MP grammar G is given by a structure [24]: G =(X, R, Φ), * Vincenzo Manca vincenzo.manca@univr.it 1 Department of Computer Science, and Center for BioMedical Computing, University of Verona, Verona, Italy