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Rosen's (M,R) System in Process Algebra Rosen’s (M,R) system in process algebra Gatherer and Galpin Gatherer and Galpin BMC Systems Biology 2013, 7:128 http://www.biomedcentral.com/1752-0509/7/128 Gatherer and Galpin BMC Systems Biology 2013, 7:128 http://www.biomedcentral.com/1752-0509/7/128 RESEARCH ARTICLE Open Access Rosen’s (M,R) system in process algebra Derek Gatherer1,3* and Vashti Galpin2 Abstract Background: Robert Rosen’s Metabolism-Replacement, or (M,R), system can be represented as a compact network structure with a single source and three products derived from that source in three consecutive reactions. (M,R) has been claimed to be non-reducible to its components and algorithmically non-computable, in the sense of not being evaluable as a function by a Turing machine. If (M,R)-like structures are present in real biological networks, this suggests that many biological networks will be non-computable, with implications for those branches of systems biology that rely on in silico modelling for predictive purposes. Results: We instantiate (M,R) using the process algebra Bio-PEPA, and discuss the extent to which our model represents a true realization of (M,R). We observe that under some starting conditions and parameter values, stable states can be achieved. Although formal demonstration of algorithmic computability remains elusive for (M,R), we discuss the extent to which our Bio-PEPA representation of (M,R) allows us to sidestep Rosen’s fundamental objections to computational systems biology. Conclusions: We argue that the behaviour of (M,R) in Bio-PEPA shows life-like properties. Keywords: Robert Rosen, (M,R), Metabolism-replacement, Metabolism-repair, Relational biology, Process algebra, Bio-PEPA, Computability, Turing machine Background kind. By contrast, relational biologists define complex Relational biology is a discipline founded by Robert systems as only those systems having impredicative (self- Rosen (1934–1998) [1-3], based on the previous work of referencing) components, and claim that these systems Nicolas Rashevsky (1899–1972), which considers bio- are non-computable as functions by Turing machines. For logical network structures using the mathematical tools relational biologists, complex systems are therefore quali- of category theory. Relational biology might be consid- tatively different to simple ones. ered a branch of systems biology, in that it studies the A Turing machine is a formal mathematical model of same objects as systems biology, and for similar reasons, computation, which has a notion of its internal state, an namely to improve our understanding of how complex unbounded tape and a read-write head for that tape [4]. biological processes work. However, its methods and It also has a table to determine the next internal state, conclusions are so radically different to those of conven- the symbol to be written to the tape and the direction in tional systems biology that one might almost say it con- which to move the head, all of which depend on the stitutes an alternative discourse on the subject. Systems current internal state and the current symbol. The con- biologists attempt to represent biological network sys- cept of the unbounded tape may be taken to imply limit- tems as software objects for simulation on computers. less time and space for computation. A function is These network structures may vary in their degree of (Turing-) computable if it can be evaluated by a Turing complexity, but the difference between simple and com- machine which, when given an input to the function, plex networks is treated as one of degree rather than of eventually writes the value of the function for that input on its output tape, assuming that the function is defined for the input. No time bound can be imposed on how * Correspondence: [email protected] long the Turing machine will take to output the value. 1MRC-University of Glasgow Centre for Virus Research, 8 Church Street, Glasgow G11 5JR, UK An adequate exposition of the logic of relational biol- 3Present address: Biomedical & Life Sciences, Lancaster University, Lancaster ogy is impossible within the limits of a research paper. LA1 4YQ, UK The strategy adopted here is therefore to concentrate on Full list of author information is available at the end of the article © 2013 Gatherer and Galpin; licensee BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Gatherer and Galpin BMC Systems Biology 2013, 7:128 Page 2 of 10 http://www.biomedcentral.com/1752-0509/7/128 Rosen’s Metabolism-Repair/Replacement system, conven- tionally abbreviated in brackets as (M,R), a very small, but nevertheless allegedly non-computable, network which is the central object of study in modern relational biol- ogy. Metabolism-Repair was Rosen’s own phrase, but Metabolism-Replacement is used here for reasons given by previous authors [5]. Readers interested in the con- struction of (M,R), and in particular the reasons why it is deemed to be non-computable, are referred to Louie’s re- cent book [6], and for contextual material to Rosen’s pre- vious book [7] and the recent series of reviews [5,8-11]. Only the briefest description of (M,R) is given in the fol- lowing paragraphs, referring to Figures 1 and 2, which illustrate (M,R) in two ways. Figure 2 The Goudsmit representation of (M,R) [15]. Black lines Relational biologists argue that the non-computability are metabolic productive reactions. Red dotted lines represent of (M,R) by Turing machines is achieved through a circu- catalysis – for instance f catalyses the production of B from A. Compare with Figure 1, to which this figure is equivalent. lar pathway of causal relations, termed closure to efficient causation [7]. Furthermore relational biologists seek to demonstrate that such non-computable systems, when represented as graphs, may be surprisingly small in terms have used them [12]. The abstract graph structures de- of their total number of nodes and edges. Relational biolo- veloped by relational biologists are referred to as models, gists do not deny that complex systems can be simulated, and a modelling relation is achieved between a system in but maintain that they can never truly be computed as the natural world and a model in the mathematical world functions [6,7,12]. Since even the most advanced com- when their entailment structures, meaning the totality of puters still use Turing architecture (i.e. they are random their internal causal relations, are identical. Entailment is access stored program machines that are almost Turing an important concept in relational biology. When a certain machine equivalent), the full computational analysis of pattern of system states necessarily results in a certain out- complex biological systems is therefore postulated to be come, it is said that this outcome is entailed by that cause. beyond our current computing abilities. Relational biologists describe logical entailments in their Relational biology has therefore concentrated its ef- mathematical work that represent entailments from the forts into pure mathematics to the exclusion of compu- physical world and insist, for a correct modelling relation tational analysis. Rosen once quipped that his mentor, to be obtained, that these entailments must occur in the Rashevsky, had lived in a time when computers were not model. A simulation, by contrast, represents the world in easily accessible but, even if they had been, he would not a much more approximate way than a model, without the requirement for congruence of internal causal relations. A simulation may be an excellent predictive tool for the nat- ural world, but when it goes wrong one may have no idea why, since its entailment structure is largely a work of cre- ative approximation. Relational biologists maintain that much of conventional systems biology is merely about the development of such approximate simulations [6,7,12]. Rosen also argued that a corollary of the non- computability of complex systems is the fact that they are also non-reducible to their component reactions, in that a model of a complex system cannot be constructed sim- ply by an additive assembly of smaller models of that sys- tem’s individual reaction components [7]. By contrast, M,R Figure 1 The Louie-Kercel representation of ( ) [10]. Lines whatever claims that systems biologists may make about with open arrowheads are metabolic productive reactions, for instance A produces B. Lines with filled arrowheads represent their own anti-reductionism [13], the construction of catalysis. Conventionally these point to the substrate of a metabolic software representations of biological networks requires reaction. For instance f catalyses the production of B from A, and the individual representation of component reactions in this is indicated by f pointing to A. Likewise, Φ catalyses the production code, in effect a reduction at the software level. Relational of f from B, so an open-headed arrow goes from B to f, and a filled biologists acknowledge that there are many systems that arrow from Φ to B. are reducible and algorithmically computable in this way, Gatherer and Galpin BMC Systems Biology 2013, 7:128 Page 3 of 10 http://www.biomedcentral.com/1752-0509/7/128 but none of these are truly complex, and therefore few Rosen intended (M,R) to be interpreted very generally, are really interesting from a biological point of view. A as a representation of a whole metabolic system B pro- further point of disagreement concerns the issue of duced from an entire set of nutrient sources A. Meta- mechanism. Systems biologists, and molecular biologists bolic activity in B is catalysed by a set of enzymatic in general, view biological processes as machine-like.
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