bioRxiv preprint doi: https://doi.org/10.1101/499251; this version posted December 17, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. Metabolic Network Reductions Mojtaba Tefagh∗y Stephen Boyd∗ December 13, 2018 Abstract 1 Genome-scale metabolic networks are exceptionally huge and even efficient algorithms can take a 2 while to run because of the sheer size of the problem instances. To address this problem, metabolic 3 network reductions can substantially reduce the overwhelming size of the problem instances at hand. 4 We begin by formulating some reasonable axioms defining what it means for a metabolic network 5 reduction to be \canonical" which conceptually enforces reversibility without loss of any information 6 on the feasible flux distributions. Then, we start to search for an efficient way to deduce some of 7 the attributes of the original network from the reduced one in order to improve the performance. 8 As the next step, we will demonstrate how to reduce a metabolic network repeatedly until no more 9 reductions are possible. In the end, we sum up by pointing out some of the biological implications 10 of this study apart from the computational aspects discussed earlier. 11 Keywords: systems biology; metabolic network analysis; canonical metabolic network reduction; 12 quantitative flux coupling analysis; QFCA. 13 Author summary 14 Metabolic networks appear at first sight to be nothing more than an enormous body of reactions. The 15 dynamics of each reaction obey the same fundamental laws and a metabolic network as a whole is the 16 melange of its reactions. The oversight in this kind of reductionist thinking is that although the behavior 17 of a metabolic network is determined by the states of its reactions in theory, nevertheless it cannot 18 be inferred directly from them in practice. Apart from the infeasibility of this viewpoint, metabolic 19 pathways are what explain the biological functions of the organism and thus also what we are frequently 20 concerned about at the system level. 21 ∗Information Systems Laboratory, Department of Electrical Engineering, Stanford University, Stanford, CA yCorrespondence: [email protected] 1 bioRxiv preprint doi: https://doi.org/10.1101/499251; this version posted December 17, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. Canonical metabolic network reductions decrease the number of reactions substantially despite leaving 22 the metabolic pathways intact. In other words, the reduced metabolic networks are smaller in size while 23 retaining the same metabolic pathways. The possibility of such operations is rooted in the fact that the 24 total degrees of freedom of a metabolic network in the steady-state conditions are significantly lower 25 than the number of its reactions because of some emergent redundancies. Strangely enough, these 26 redundancies turn out to be very well-studied in the literature. 27 1 Introduction 28 About two decades ago, the first genome-scale metabolic network reconstruction of the cellular metabolism 29 of an organism was published [EP99] shortly after the first genome was sequenced. From that time on, 30 the ever-increasing advances in the high-throughput omics technologies have allowed for the more and 31 + + more comprehensive reconstructions of exponentially growing sizes [RVSP03, FFF 03, DHP04, FSP 06, 32 + + + + + + + + JP07, FHR 07, DBJ 07, MSM 07, HSD 08, PMSF09, dODQP 10, OCN 11, MOMM 12, KMPG 12, 33 + + + AHW13, TSF 13, SSH 16, BSZ 18]. These reconstructions have numerous applications in contextual- 34 ization of high-throughput data, guidance of metabolic engineering, directing hypothesis-driven discovery, 35 interrogation of multi-species relationships, and network property discovery [OPP09]. However, the vast 36 amount of data for some organisms can be a two-edged sword which makes many essential tasks in the 37 metabolic network analysis computationally intractable. 38 To overcome the demands of systems biology, even while they are outpacing Moore's law [BPS13], 39 faster computational techniques are needed to enable the current methods to scale up to match the 40 progress of high-throughput data generation in a prospective manner. As a natural solution, reducing 41 the size of genome-scale metabolic networks has always been exploited to the advantage of performance 42 for predetermined computational tasks [BVM01, JS14, VPS14, ESK15, RB17]. However, all of these 43 mentioned studies assume that a set of protected reactions which must be retained in the reduced 44 metabolic network is given in advance and the other reactions are dispensable. In this way, these 45 metabolic network reductions are not agnostic to the downstream analysis and lose information on some 46 of the reactions which are irrelevant to the task at hand. 47 In the field of flux coupling analysis (FCA), from the very beginning Flux Coupling Finder (FCF) 48 [BNSM04] considers aggregating all the isozymes and removing the blocked reactions. More recently, fast 49 + flux coupling calculator (F2C2) [LDSB12] considers merging the enzyme subsets [PSnVN 99] too. Flux- 50 Analyzer [GK04] also detects conservation relations as a preprocessing step. MONGOOSE [CTRB14] 51 employs a similar loss-free network reduction to convert the input model into a canonical form. 52 Although these general-purpose reductions seem decent in a common sense approach, to the best of 53 2 bioRxiv preprint doi: https://doi.org/10.1101/499251; this version posted December 17, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. our knowledge the formal definition of a metabolic network reduction, in general, is not clear so far. 54 The aforementioned studies provide strong support for the idea that such general-purpose reductions 55 preserve some interesting attributes of metabolic networks e.g., the flux coupling features. In this article, 56 we specify a broad class of metabolic network reductions with respect to which some generally desired 57 properties of metabolic networks are invariant. 58 Outline. In x2, we briefly overview the FCA framework. In x3.1 and x3.2, we define and study single 59 and multiple sequential metabolic network reductions, respectively. Afterward in x4, we argue that the 60 reduced metabolic networks are not only interesting from the computational perspective but also they 61 are biologically interpretable. Finally in x5, we conclude by highlighting the contributions of this work 62 and pointing out areas for further research. 63 2 Background 64 m We specify a metabolic network by an ordered quadruple N = (M; R; S; I) where M = fMigi=1 denotes 65 n the set of metabolites of size m, R = fRigi=1 denotes the set of reactions of size n, S denotes the m × n 66 stoichiometric matrix, and I ⊆ R denotes the set of irreversible reactions. Since it is often the case that 67 n m, we consider n as the size of N too. 68 In the constraint-based analysis of metabolic networks, the constraints Sv = 0 and vI ≥ 0 are imposed 69 on the metabolic network by the steady-state conditions and the definition of irreversible reactions, 70 respectively. By a slight abuse of notation, vI ≥ 0 means vi ≥ 0 for all the indices i for which Ri 2 I. 71 We denote the steady-state flux cone [SSPH99] by 72 n C = fv 2 R j Sv = 0; vI ≥ 0g; The feasible flux distributions of N are then defined to be the members of C. 73 We call Rk 2 R a blocked reaction if vk = 0 for all the feasible flux distributions v 2 C. To the end 74 of this paper, whenever we assume that all the blocked reactions are removed, by this, we also assume 75 that if a reversible reaction Rk is blocked in only one direction, then this blocked direction is removed 76 too meaning that Rk is included in I subsequently. In x3.1, we will review these trivial ways of reducing 77 a metabolic network in more details. 78 In order to derive even more metabolic network reductions, we should exploit other redundancies of 79 the steady-state flux cone analogous to the case of blocked reactions where the rate of a blocked reaction 80 is always zero irrespective of what the other flux coefficients are equal to. A somewhat less obvious 81 3 bioRxiv preprint doi: https://doi.org/10.1101/499251; this version posted December 17, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. situation is when the rate of one reaction is unambiguously determined by another one as is the case in 82 the following definitions of FCA. 83 Definition 1 ([BNSM04]). For an arbitrary pair of unblocked reactions Ri;Rj 2 R: 84 Directional coupling: Ri is directionally coupled to Rj, denoted by Ri −! Rj, if for all feasible flux 85 distributions vi 6= 0 implies vj 6= 0. 86 Partial coupling: Ri is partially coupled to Rj, denoted by Ri ! Rj, if both Ri −! Rj and vice 87 versa Rj −! Ri hold. 88 Full coupling: Ri is fully coupled to Rj, denoted by Ri () Rj, if there exists a full coupling equation 89 (FCE) 90 vi = cvj c 6= 0; (1) which holds for all v 2 C.