Signatures of Arithmetic Simplicity in Metabolic Network Architecture
[['The prominent role of metabolism in any biological process and the fact that a large portion of the environmental factors shaping living systems are ultimately metabolic in nature , suggest that strong selective forces have been acting on metabolic networks throughout the history of life .', 'In laboratory evolution experiments [1]–[3] one can witness mostly short term metabolic adaptations , affecting metabolic enzyme regulation and fine tuning of kinetic parameters .', "However , especially during major transitions , such as the early stages of life's appearance or the rise of oxygen in the Earth's atmosphere , selective forces must have shaped the metabolic wiring itself [4] .", 'Comparative genomics can provide top-down insight into some long-term evolution of metabolic pathways [5] , [6] .', 'In addition , studies of prebiotic chemistry scenarios have suggested possible seeds of biochemical organization from a bottom-up perspective [7]–[10] .', 'Yet , whether the long term evolution of metabolism was dominated by unpredictable frozen accidents , or by inevitable network optimization processes , remains a fundamental open question .', 'In a 1961 review , Baldwin and Krebs suggested that biochemical network topologies may reflect the adaptation toward optimally efficient metabolic strategies , and that manifold use of certain molecules may be a crucial element of this adaptation , as “it is indeed a general principle of evolution that multiple use is made of given resources . ”[11] .', 'Some computational studies have proposed that the topology of specific metabolic pathways may have evolved towards maximal efficiency [12] , minimal number of steps [13] , or that network properties may reflect optimal organization [14] .', 'Here we seek to address this problem by exploring a system that can reach a level of complexity comparable to the one observed in the union of all known metabolic pathways , yet is simple enough to allow efficient computation and analytical calculations .', 'In addition , we wish to explore at an ecosystem-level the potential role of “metabolic multi-tasking” , as suggested by Baldwin and Krebs .', 'The increasing evidence of abundant horizontal gene transfer in the history of life suggests that this question may be indeed especially relevant at the ecosystem level , where the interchange of genetic information might have created a free economy of enzymes among simple organisms , allowing for the emergence of species that share common molecular tools [15] .', 'Recent metagenomic studies of microbial consortia [16] also suggest the question of whether metabolic functions , more than individual species distributions , might be directly dependent on environmental conditions .', 'Hence it is possible that hallmarks of metabolic optimality in metabolic network wiring may be observable at the level of global ( multi-species ) metabolic networks [17] , [18] , more than at the individual species level .', 'Do specific molecules , reactions or pathway topologies appear to be universally useful in a biochemical network , i . e . , relevant for maximally efficient completion of several possible metabolic tasks , possibly across multiple organisms ?', 'In the present work , we combine the study of an extremely simple artificial chemistry [19]–[22] with recent systems biology approaches [23] , [24] to systematically compute pathways that are optimal for an array of elementary metabolic tasks , converting an input molecule into an output one .', 'Behind the apparent complexity of the ensuing pathways , we identify recurring , modularly organized categories of network topologies , and analytically predictable trends in pathway length .', 'In addition , we observe the emergence of “universal metabolic tools” across all optimal pathways .', 'Finally , despite the huge gap in the underlying chemical rules , we find that some properties of real metabolic pathways are consistent with the patterns detected in the model , suggesting that fundamental optimality principles may have played a role in shaping biochemical networks .'], ['Our artificial chemistry consists of a set of N possible molecules {a1 , a2 , a3 , … , aN} , that can participate in reversible ligation/cleavage reactions of the form ai + aj ↔ ak , with i+j\u200a=', 'k . This model could be viewed as the simplest possible string-based artificial chemistry [19] .', 'The reaction network RN that includes all metabolites up to length N and all possible reactions ( of the order of N2/4 , see Methods ) between them ( Fig . 1 ) can be thought of as the underlying chemistry based on which specialized metabolic tasks could emerge .', 'Here we were concerned with pathways , within the RN network , that can optimally perform a given metabolic task .', 'In particular , we searched for optimal solutions to the problem of producing a specific end-product ( e . g . , aj , with output flux vout ) from a single available nutrient ( e . g . , ai , with input flux vin ) .', 'We define an optimal pathway as one that satisfies the following conditions:', '( i ) it allows a steady state solution ,', 'i . e .', ', a mass-conserving flow from input to output;', '( ii ) it has maximal yield , and no waste [25] , such that vout\u200a=\u200avin·j/i; and', '( iii ) it has the fewest reaction steps possible .', 'A pathway satisfying these conditions is termed a minimal balanced pathway ( MBP ) between ai and aj , and will be denoted ai ⇒ aj .', 'MBPs ( also referred to below as optimal pathways ) can be thought of as the pathways that are most efficient for a specific metabolic task , in the sense that they require the smallest possible number of different enzymes for producing the maximal possible yield [12] , [26] , [27] .', 'Despite the simplicity of our artificial chemistry , identifying the MBPs between all possible input-output pairs in a given artificial chemistry RN is a challenge for large N . We implement three algorithms to approach this problem: a mixed integer linear programming ( MILP ) akin to flux balance analysis ( FBA ) [28]; an algorithm that uses enumeration of elementary flux modes [23]; and finally an iterative algorithm that gradually assembles new MBPs from already identified simple ones ( see Methods ) .', 'The three algorithms differ mainly in their scalability , and in their capacity to predict multiple degenerate solutions ( see Table S1 ) .', 'A partial overview of the results of our calculations is shown in Fig . 2A and Fig . S1 ( see Tables S2 and S3 for a comprehensive list of MBPs ) .', 'Behind the apparent complexity of the topologies encountered in each of the different pathways , it is possible to observe the recurrence of three fundamental categories: each MBP functions either as a pure “addition chain” [29] , where smaller metabolites are progressively added together to build the target molecule , or as an “addition-subtraction chain” , in which metabolites are both synthesized and degraded within the pathway .', 'Addition and addition-subtraction chains are concepts borrowed from the field of cryptography , whose relevance to our question will become apparent later .', 'There is also a third , smaller category of cyclical pathways that cannot proceed unless a certain intermediate molecule is already present in the system .', 'These pathways are autocatalytic cycles ( Fig . 2B ) that very much resemble autocatalytic cycles found in real biochemistry , such as the reverse TCA cycle [7] , or the formose reaction [30] .', 'Our results show that autocatalytic cycles can be simultaneously optimal for multiple tasks ( Fig . 2B ) , suggesting that such types of structure may have a fundamental evolutionary advantage in a biological context .', 'In addition to the recurrence of these topological categories among MBPs , we find that some specific structures are used repeatedly , often in a modular fashion ( Fig . 2C ) .', 'Specifically , many simple MBPs are used hierarchically as a toolkit for the construction of progressively more complex MBPs ( data not shown ) , similar to what has been observed in real metabolic networks [15] , [31] , [32] .', 'This modular architecture of recurring graph types provides a topological signature of optimally efficient pathways in our idealized chemistry .', 'Since these pathways are chosen based on their minimal length , one may expect that a systematic analysis of all MBP lengths will display additional distinctive properties .', 'Indeed , pathway lengths increase roughly logarithmically with the size of the input ( or output ) molecule ( Fig . 3 and S3 ) , with superimposed sharp jumps .', 'For example , the task a9 ⇒ a6 can be performed in 2 steps , but the neighbor task a9 ⇒ a7 requires a minimum of 6 steps .', 'Moreover , while most MBPs have only one or a few optimal realizations , selected instances display a peak in possible redundant solutions ( Fig . S2 ) , usually due to interconversions between molecules of similar size ( e . g . , ax ⇒ ax+1 ) , or to the inherent complexity of a specific molecule ( e . g . , a7 ⇒ aj ) .', 'These regular patterns suggest that it may be possible to reproduce the MBP length curves without having to actually compute the MBPs .', 'A similar search for patterns associated with minimal steps had been previously encountered in the mathematics of addition-subtraction chains , of high importance in cryptography [29] .', 'These are integer sequences , beginning with 1 , in which the i-th entry is either the sum or difference of any two previous entries in the sequence .', 'These chains are often used in calculating large exponents of numbers [33] .', 'For example , calculating n128 can either be performed in 127 multiplications ( n × n = n2 , n2 × n = n3 , … , n127 × n = n128 ) or in a chain of 7 exponent multiplications ( n × n = n2 , n2 × n2 = n4 , n4 × n4 = n8 , … , n64 × n64 = n128 ) .', 'The latter can be further simplified by tracking the sums of the exponents in each calculation , which form an addition chain ( 1 , 2 , 4 , 8 , 16 , 32 , 64 , 128 ) .', 'Shortest addition-subtraction chains are commonly used to calculate very large numbers in the fewest number of steps , thus speeding up computation time .', 'These are often applied to methods in cryptography where the calculated exponents can have on the order of thousands to tens of thousands of bits [34] .', 'The pathways explored in our model resemble optimal addition-subtraction chains .', 'For example , the problem of obtaining a128 from a1 is formally equivalent to the addition chain example described above .', 'However while typical addition-subtraction chains start with the number 1 , in our MBPs we explore minimal paths starting from any molecule ai ( i≥1 ) .', 'As described in detail in the Methods , we extended previous work on addition-subtraction chains [29] , [35] to derive the following analytical estimate of the length of MBPs: ( 1 ) where L ( i ,', 'j ) is the number of reactions in the MBP with input ai and output aj , and gcd ( i ,', 'j ) is the greatest common divisor of i and', 'j . As seen in Fig . 3 , Eq .', '( 1 ) reproduces the corresponding pathway lengths obtained by computing individual MBPs .', 'This agreement implies that the number of reaction steps needed to construct an efficient metabolic pathway between two metabolites in our artificial chemistry can be roughly estimated from Eq .', '( 1 ) .', 'The only feature that determines the pathway lengths is the complexity of the input and output molecules .', 'We can now ask whether similar minimal pathway length signatures are discernible in real metabolic networks .', 'To cope with the gap in complexity between our model and real chemistry , we mapped real metabolic networks onto a single atom backbone [13] , [14] .', 'For example , the aldolase reaction , which cleaves fructose-1 , 6-bisphosphate ( C6H14O12P2 ) into dihydroxyacetone phosphate ( C3H7O6P ) and glyceraldehyde-3-phosphate ( C3H7O6P ) , can be mapped onto a carbon atom backbone , becoming simply C6 ↔ C3 + C3 ( see Methods ) .', 'This reaction is now formally analogous to the a6 ↔ a3 + a3 reaction in the idealized chemistry .', 'Upon performing this mapping onto a carbon atom backbone , we ask whether the structure of real metabolic networks allows interconversions that use the minimal , logarithmic number of steps found for the artificial chemistry ( Fig .', '3 and Eq .', '( 1 ) ) .', "Specifically , we identified all shortest pathways between any two carbon compounds in Escherichia coli's metabolic network .", 'This was performed using two methods .', 'The first was an explicit use of elementary flux modes as done in the artificial chemistry .', 'As with the artificial chemistry method , this has the advantage of finding all of the shortest pathways that connect any two carbon compounds , but is limited in computational scope to a smaller network .', "Because of this limitation , we used the network of E . coli's central carbon metabolism [36] , [37] , modified to remove cofactors and reactions that do not affect carbon transfer ( see Methods ) .", 'After finding all minimal elementary flux modes that connect every pair of carbon compounds in the network , we reduced those compounds to their carbon content alone , as described above .', 'We determined , for each input compound , the length of the shortest elementary flux mode that reaches its closest molecule with j carbons; then , for each value of i , we averaged these path lengths over all input molecules with i carbons .', 'The results show that the lengths of the E . coli elementary flux modes correlate with the lengths of the corresponding artificial chemistry MBPs and with the analytical predictions , though the actual E . coli values are overall larger than the artificial chemistry ones ( Fig . 4 ) .', 'This last fact , as discussed later , may be due , for example , to energetic constraints , or to the higher complexity of real organic chemistry .', 'The second method is aimed at identifying all shortest pathways between any two carbon compounds in the whole genome-scale metabolic network of Escherichia coli [38] , for which it is still infeasible to apply the elementary flux mode analysis .', 'For this , we implemented a heuristic approach to analyze the set of shortest pathways between every pair of metabolites .', 'We first determined , for each input compound , the minimal path length to reach its closest molecule with j carbons; then , for each value of i , we averaged these path lengths over all input molecules with i carbons .', 'The results ( Fig . 3 and S3 ) show that these E . coli minimal path lengths approximately follow the predicted logarithmic trend .', 'For some curves ( e . g . the one with C5 as an input ) , the specific peaks and valleys of the predicted function are closely followed by the E . coli network .', 'While this does not prove that MBPs are indeed used in real metabolic networks , it suggests that the logarithmic strategy of MBPs is embedded in their architecture .', 'However , because this second method focuses on shortest paths across a metabolite-to-metabolite network rather than on flow-conserving MBPs , the predicted values are likely an underestimate of the number of reactions necessary to construct one metabolite from another in a mass-conserved manner .', 'So far , we have analyzed the properties of individual MBPs in our idealized chemistry , as well as analogous minimal length pathways in E . coli metabolism .', 'However , some fundamental aspects of the architecture of metabolism may be visible only at the ecosystem-level , namely by collectively analyzing the metabolic network obtained as the union of the metabolic maps of known individual species ( sometimes called the “meta-metabolome” ) .', 'For example , previous work using an algorithm of network expansion applied to this meta-metabolome has identified potential signatures of major evolutionary events [39] , including the metabolic transition that took place upon the great oxidation event , about two billion years ago [4] .', 'Here , we build a meta-metabolome for our idealized chemistry by considering the collection of MBPs .', 'One could imagine that each task ai ⇒ aj corresponds to a different organism , which has filled a specific metabolic niche ( availability of ai ) , and found an optimal solution ( the MBP ) for its main metabolic task ( produce aj ) .', 'The question we ask next is whether , in this ecosystem of MBPs , all metabolites and reactions are used in roughly the same number of pathways , or if specific metabolites or reactions seem to be essential for many optimal tasks , hence representing “universal tools” .', 'For this analysis we used the set of MBPs calculated on the R19 network using the MILP method .', 'One first result of this analysis is that every metabolite of an even length is used in many more MBP reactions than their odd length neighbors , compared to the underlying chemistry ( Fig . 5B ) .', 'Thus even-length metabolites are more important in that they can be used for more tasks .', 'A possible explanation for this enhanced importance comes from the logarithmic nature of the MBP path lengths .', 'For example , producing a8 from a1 requires only three doubling reactions ( a1 + a1 → a2 , a2 + a2 → a4 , and a4 + a4 → a8 ) .', 'In addition , this same pathway , with one additional reaction , can also be used to optimally produce a9 and a10 ( see Tables S2 and S3 ) , overall increasing the number of pathways in which each of those even-length intermediates is used .', 'Indeed , because similar logarithmic pathways can be used as a backbone connecting distant inputs and outputs , we expect metabolites of an even length to appear more often .', 'Similarly , one can address the relevance of each possible reaction across different MBPs .', 'The existence of ubiquitous reactions is visible in Fig . 2A and Fig . S1 , and can be more systematically assessed by plotting a usage distribution ( Fig . S2 ) .', 'The most abundant reactions – the “universal tools” in this model chemistry – are the ones that ligate two identical molecules ( e . g . a2 + a2 ↔ a4 , see Tables 1 , S4 , and S5 ) .', 'The distribution of reaction utilization follows a long-tailed distribution ( Fig . 5A ) , whose fit to a power law gives an exponent of approximately −1 . 1 ( R2\u200a=\u200a0 . 99 ) .', 'This value is close to our theoretically predicted value of −1 ( See Methods ) .', 'As in the case of the artificial chemistry network , we can now search for patterns of metabolites and reactions usage in the collective set of all metabolic reactions known in living systems , obtained from the KEGG database [17] .', 'The presence of such signatures would suggest a long-term selective advantage of molecules and reactions that are useful for multiple tasks across different organisms and environments .', 'By counting how many times each possible carbon backbone reaction is used across this biosphere-level metabolism we obtained a broad distribution , and a fit to a power-law gives the exponent of −0 . 89 , comparable with the analytically predicted value , and with that in the artificial chemistry model ( Fig . 5A and Fig . S4 ) .', 'We found that several reactions that are top ranking in their count across MBPs in the artificial network , are also at the top of the list in the KEGG-derived reactions ( Spearman correlation p-value<10−6; see also Tables 1 and S6 , S7 , S8 , S9 , S10 , and S11 ) .', 'This suggests that the RN network model , despite its simplified chemical rules , captures some fundamental features of the role of the carbon reaction backbone of real metabolic networks .', 'In addition to a preference for specific reactions , we can ask whether the spectrum of metabolite usage across the whole KEGG metabolism reflects the possible optimality criteria encountered in the model ( Fig . 5C ) .', 'The metabolite usage in the hydrogen backbone network ( see Methods ) is similar to that in the artificial chemistry: each even-length hydrogen metabolite is used more often than its odd-length neighbors ( Fig . S4A ) .', 'For the carbon backbone distribution , we see a similar descending periodic behavior , but with a periodicity of approximately 5 ( Fig . 5C and Fig . S5B ) .', 'Hence , molecules containing carbons in a number that is a multiple of 5 are used more abundantly than other molecules across different metabolic reactions .', 'One possible explanation for this C5 periodicity is the profuse usage of adenine and nicotinamide adenine dinuculeotide compounds as energy carriers and redox balance molecules , although the removal of such compounds has little effect on the observed periodicity ( Fig . S6 ) .', 'Hence , the prominent usage of compounds with specific numbers of carbons might reflect global network optimization principles for the efficiency of multiple pathways , as observed in the artificial chemistry model .', 'The periodicity of 5 that we observe , together with the evidence displayed in Fig . 3C , may suggest that the evolutionary optimization of metabolism has been partially taking place around building blocks of five carbons , compatible with previous observations of prebiotic abundance of terpenoids [40] and pentoses [41] .', 'It is also interesting to note that an unexplained periodicity of two had been previously observed in the distribution of the number of carbons among known organic compounds [42]–[44] .', 'While our analysis is based on the distribution of usage of carbon compounds in different reactions , rather than the total count of molecules , future analyses may investigate possible connections between these trends .'], ['We have explored the potential existence of general principles underlying the evolution of metabolic network architecture .', 'Specifically , we studied the properties of pathways ( the MBPs ) that perform elementary metabolic tasks with maximal yield and minimal length in an idealized chemistry .', 'Using the results from the model chemistry , we asked whether similar signatures of optimally efficient organization could be found in real metabolic networks .', 'In computing possible MBPs , we have focused mostly on identifying modular features , on predicting their lengths , and on the statistics of usage of metabolites and reactions .', 'In the future , it may be interesting to characterize the full spectrum of degenerate MBPs for large artificial chemistries .', 'This would allow us to assess , for example , the density of specific topologies ( such as autocatalytic cycles ) , or the dependence of degeneracy on the numerical properties of input/output pairs .', 'One of our algorithms ( the elementary modes one ) can find a large number of degenerate solutions , including autocatalytic cycles .', 'This algorithm is currently not scalable , because of the difficulty of computing elementary flux modes , especially in the highly connected artificial chemistry network we have used , though very recent improvements in elementary flux mode calculations [45] might be useful for this enumeration .', 'In addition , while intuitively this approach seems to capture the full degeneracy of MBPs , this still remains to be formally proven .', 'Another intriguing possibility might be to modify our flux balance MILP approach to identifying degenerate solutions by employing integer cuts , as described in [46] .', 'Alternative avenues for optimization using Linear Programming rather than MILP could be in principle devised to reduce the complexity of calculations .', 'For example minimizing the sum of absolute values of fluxes allows for rapid calculation of pathways up to the R100 network , though in this case the ensuing pathways are not of minimal length ( see Fig . S7 ) .', 'In any case , for the purpose of the current work , we verified that degeneracy does not affect the statistics of usage of different reactions ( Fig . S8 ) .', 'Among the recurrent MBP topologies identified , we encountered numerous autocatalytic cycles .', 'The properties of autocatalytic cycles have been studied previously [8] , [11] , and their self-replication potential has been theorized to be important in the early evolution of carbon fixation [7] , [9] .', 'Autocatalytic cycles have also been shown to be kinetically stable , even in the absence of regulatory control [47] .', 'We found that some autocatalytic pathways ( e . g . the pathway from a7 to a8 ) are simultaneously optimal for multiple metabolic tasks .', 'In this specific case , the MBP a7 ⇒ a8 is also an MBP for the production of each intermediate in the cycle ( Fig . 2B ) .', 'This special property of autocatalytic cycles in our artificial chemistry may have a parallel in real metabolism .', 'For example , many metabolites in the TCA cycle ( which is autocatalytic when run in reverse [7] ) are precursors for fundamental anabolic processes [48] , [49] .', 'Similar properties can be observed in the fundamental autocatalytic cycle known as the formose reaction [30] .', 'Along with the structural details of MBPs , we also used analytical methods to estimate the length of MBPs as a function of the length of input and output molecules .', 'This estimate closely matched the lengths of the artificial chemistry pathways computed with numerical algorithms .', 'These calculations establish a new link between two apparently unrelated disciplines , namely the mathematics of addition-subtraction chains and biochemistry .', 'It will be interesting to explore in the future whether extensions to more realistic artificial chemistries can be formalized in a similar fashion .', 'Conversely , the MBP length estimate obtained for biochemical pathways may suggest new avenues in applied mathematics .', 'To determine whether predictions of minimal MBP length in our idealized chemistry could have implications in real biochemistry , we searched for pathways of minimal length between compounds with different counts of carbon atoms in the E . coli metabolic network .', 'The complexity of real chemistry relative to our idealized system made this comparison difficult to interpret .', 'MBPs in E . coli were found to be composed of many more reactions than in the artificial chemistry .', 'This might be due to the additional requirement of energy gradients ( e . g . the phosphorylation steps ) , to the complex interdependence of multiple elements , and to the properties of stability of intermediates .', 'Previous work had addressed the question of optimality in specific metabolic pathways .', 'For example , Meléndez-Hevia and Torres [50] , used optimality criteria to infer that some metabolic pathways , most notably the pentose-phosphate pathway , can be traversed using the fewest number of reactions .', 'Heinrich and Schuster [51] , conversely , describe the identification of a series of phosphorylation/dephosphorylation and ATP consumption/production steps that maximize the flux of ATP production in central carbon metabolism pathways .', 'In contrast , our search for MBPs in the artificial chemistry model corresponds to a systematic search for maximal yield , minimal length pathways for all possible input-output relationships ( something not yet feasible for real metabolic pathways ) .', 'This allows us to infer analytical relationships and potential principles that may hold ( perhaps in an approximate way ) for virtually any evolved metabolism .', 'In real systems , various compounds are often produced as waste byproducts and simply excreted into the environment , leading to suboptimal yields of target metabolites .', 'For example , one could go in one step from a C6 compound to C5 , with a yield of 5/6 ∼83% and an additional C1 byproduct; yet , obtaining maximal yield in this transformation , would cost at least 3 more reactions .', 'One could argue that the combination of all of these criteria , including the maximization of ATP production and optimization of enzymatic catalysis may have played a key role in the evolution of modern metabolism , leading to compromise solutions .', 'Exploring pathways that produce multiple compounds from multiple inputs with the addition of thermodynamic constraints might constitute an interesting model extension for further investigation .', 'We found that the statistical properties of the usage of reactions across MBPs recapitulate the statistics of reaction usage in the union of all known metabolic pathways ( represented by the KEGG metabolic database ) .', 'Both across the set of all MBPs for the idealized chemistry , and in the KEGG metabolic map , we observed that a few reactions are used far more often than many of the others in the set .', 'Another way of determining the importance of individual reactions in the context of the global functionalities of a meta-metabolome would be to perform perturbation experiments .', 'We implemented such an experiment in our idealized chemistry , by progressively removing reactions and checking how many metabolites can still be produced .', 'Depending on whether reactions are removed in random order or in the order determined by their usage across MBPs , the outcome is quite different ( Fig . S9 ) .', 'This analysis sheds light on the importance of reactions in terms of the capacity to produce a certain output .', 'Determining to what extent real metabolic networks obey optimality principles like the ones described here will take additional effort .', 'Even if an underlying arithmetic simplicity governs idealized optimal pathways , deviations from ideal behavior should be expected .', 'For example , parallel selection pressures for energy production and biochemical stability would likely sacrifice pathway minimality .', 'However , guiding principles as the ones we are proposing could serve as reference points for future research , including circumstances in which metabolism can be different from what we are used to .', 'Using synthetic biology techniques , for example , it might be possible to redesign metabolic pathways so as to approach predicted ideal efficiencies and minimal enzyme cost [52] , [53] .', 'From a totally different perspective , in the field of astrobiology , having a prediction of possible signatures of an evolved metabolism might help select , among the molecular spectra of extrasolar planets , those possibly indicative of biogenic processes [54]–[56] .'], ['We define an artificial chemistry inspired by previous string-based artificial chemistries ( see also main text and Fig . 1 ) .', 'One may think of molecules in this artificial chemistry as polymers ( up to a given length N ) of a monomeric unit a .', 'Since no specific assumption is made in the model about the nature of these molecules , they could equally represent aggregates or branched polymers of different sizes , as well as molecules with different counts of a specific atom .', 'A network RN =\u200a{MN , CN} is defined by the set of N molecules MN = {ai | ∀ i =\u200a1 , … , N} and the set of all possible uni-bi ligation/lysis reactions between them , CN = {ai + aj ↔ ak | ∀ i , j , and k , such that i≤j , i+j\u200a=\u200ak , and k\u200a=\u200a2 , … , N} .', 'The number of reactions in CN can be shown to grow quadratically with N . Since each reaction describes how to reversibly combine two molecules to make a larger one , there is a fixed number of ways to produce any given metabolite ai , equal to ⌊i/2⌋ .', 'The number of reactions in CN is then given by , which can be approximated by N2/4 .', 'Flux Balance Analysis ( FBA ) is a steady state constraint-based approach to study the flow of mass through metabolic networks [28] , [57] , [58] .', 'Briefly , FBA represents the metabolic network of interest as an n×m stoichiometric matrix S , whose element Sij indicates the number of molecules of metabolite i ( i\u200a=\u200a1 , … , m ) that participate in reaction j ( j\u200a=\u200a1 , … , n ) ( with a positive sign if the metabolite is produced , negative if it is consumed ) .', 'Each reaction can be associated with a flux , vj .', 'Under the assumption of a steady state the following set of mass conservation constraints on the fluxes is generated: ( 2 ) Additional constraints ( such as availability of nutrients , experimentally observed irreversibility , maximal or minimal rates , etc . ) can be imposed on the fluxes as inequalities of the form ( 3 ) where αj is the minimal allowed rate of a reaction and βj is its maximal rate .', 'Taken together , the above constraints define a convex polyhedron ( the “feasible space” ) in the n-dimensional space of fluxes .', 'Linear programming ( LP ) can be used to identify , within the feasible space , flux ve
['Introduction', 'Results', 'Discussion', 'Methods']
['Metabolic networks perform some of the most fundamental functions in living cells , including energy transduction and building block biosynthesis .', "While these are the best characterized networks in living systems , understanding their evolutionary history and complex wiring constitutes one of the most fascinating open questions in biology , intimately related to the enigma of life's origin itself .", 'Is the evolution of metabolism subject to general principles , beyond the unpredictable accumulation of multiple historical accidents ?', 'Here we search for such principles by applying to an artificial chemical universe some of the methodologies developed for the study of genome scale models of cellular metabolism .', 'In particular , we use metabolic flux constraint-based models to exhaustively search for artificial chemistry pathways that can optimally perform an array of elementary metabolic functions .', 'Despite the simplicity of the model employed , we find that the ensuing pathways display a surprisingly rich set of properties , including the existence of autocatalytic cycles and hierarchical modules , the appearance of universally preferable metabolites and reactions , and a logarithmic trend of pathway length as a function of input/output molecule size .', 'Some of these properties can be derived analytically , borrowing methods previously used in cryptography .', 'In addition , by mapping biochemical networks onto a simplified carbon atom reaction backbone , we find that properties similar to those predicted for the artificial chemistry hold also for real metabolic networks .', 'These findings suggest that optimality principles and arithmetic simplicity might lie beneath some aspects of biochemical complexity .']
['Metabolism is the network of biochemical reactions that transforms available resources ( “inputs” ) into energy currency and building blocks ( “outputs” ) .', 'Different organisms have different assortments of metabolic pathways and input/output requirements , reflecting their adaptation to specific environments , and to specific strategies for reproduction and survival .', 'Here we ask whether , beneath the intricate wiring of these networks , it is possible to discern signatures of optimal ( i . e . , shortest and maximally efficient ) pathway architectures .', 'A systematic search for such optimal pathways between all possible pairs of input and output molecules in real organic chemistry is computationally intractable .', 'However , we can implement such a search in a simple artificial chemistry , which roughly resembles a single atom ( e . g . , carbon ) version of real biochemistry .', 'We find that optimal pathways in our idealized chemistry display a logarithmic dependence of pathway length on input/output molecule size .', 'They also display recurring topologies , including autocatalytic cycles reminiscent of ancient and highly conserved cores of real biochemistry .', 'Finally , across all optimal pathways , we identify universally important metabolites and reactions , as well as a characteristic distribution of reaction utilization .', 'Similar features can be observed in real metabolic networks , suggesting that arithmetic simplicity may lie beneath some aspects of biochemical complexity .']
['computational biology/metabolic networks', 'computational biology/systems biology', 'physics/interdisciplinary physics']