The compositional and evolutionary logic of metabolism

Wood–Ljungdahl. The WL pathway [90–92, 94] consists of a sequence of five reactions that directly reduce one CO₂ to a methyl group, a parallel reaction reducing CO₂ to CO, and a final reaction combining the methyl and CO groups with each other and with a molecule of CoA to form the thioester acetyl-CoA. The reactions are shown below in figure 5, and discussed in detail in section 4. The five steps reducing CO₂ to −CH₃ make up the core pathway of folate (vitamin B₉) chemistry and its archaeal analogue, which we consider at length in section 3.2. The reduction to CO, and the synthesis of acetyl-CoA, are performed by the bi-functional CO-dehydrogenase/acetyl-CoA synthase (CODH/ACS), a highly conserved enzyme complex with Ni–[Fe₄S₅] and Ni–Ni–[Fe₄S₄] centers [99–102]. Methyl-transfer from pterins to the ACS active site is performed by a corrinoid iron–sulfur protein (CFeSP) in which the cobalt-tetrapyrrole cofactor cobalamin (vitamin B₁₂) is part of the active site [103, 104].

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Phylogenetically, WL is a widely distributed pathway, found in a variety of both bacteria and archaea, including acetogens, methanogens, sulfate reducers, and possibly anaerobic ammonium oxidizers [76]. The full pathway is found only in strict anaerobes, because the CODH/ACS is one of the most oxygen-sensitive enzymes known [105, 106]. However, as we have argued in [22], the folate-mediated reactions form a partly-independent sub-module. This module combines with the equally-distinctive CODH/ACS enzyme to form the complete WL pathway, but can serve independently as partial carbon-fixation pathways even in the absence of the final step to acetyl-CoA (see figure 5). In this role it is found almost universally among deep bacterial clades. In addition to its being highly oxygen sensitive, recent results [309] suggest that the CODH/ACS is also sensitive to sulfides and perhaps other oxidants, a point to which we will return in section 3.4.2 when discussing evolutionary divergences between the complete and incomplete forms of WL.

All carbon fixation pathways in extant organisms employ some essential and apparently unique enzymes and most also rely in essential ways on certain cofactors. For example, the 3-hydroxypropionate pathway relies on biotin for reactions shared with (or homologous to) those in fatty acid synthesis. The reductive citric-acid cycle relies on reduced ferredoxin [107], a simple iron–sulfur enzyme, and on thiamin in its reductive carbonyl-insertion reaction [108], and also on biotin for its β-carboxylation steps [109, 110] (all of these examples will also be discussed along with the pathways in which they are used below). Among the uses of cofactors in carbon-fixation pathways, however, the function provided by pterin cofactors in WL is distinct and arguably the most complex. (Pterin is a name referring to the class of cofactors including folates and the methanopterins, which are both derived from a neopterin precursor.) Whereas most cofactors act as transfer agents cycling between two or three states, pterins undergo elaborate multi-step cycles, mediating capture of formate, reduction of carbon bound to one or two nitrogen atoms and transfer of formyl, methylene, or methyl groups. This diversity of roles has led to the folate pathway being termed the 'central superhighway' of C₁ chemistry [93]. The distinctive use of cofactors within WL continues with the dependence of the acetyl-CoA synthesis on cobalamin, a highly reduced tetrapyrrole capable of two-electron transfer [103]. In this sense the simple network topology of direct C₁ reduction seems to require a more elaborate dependence on cofactors than is seen in other pathways.

Reductive citric-acid cycle. The reductive citric-acid (reductive TCA or rTCA) cycle [111, 96] is the reverse of the oxidative Krebs cycle. It is a sequence of eleven intermediates and eleven reactions, highlighted in figure 4, which reduce two molecules of CO₂, and combine these through a substrate-level phosphorylation with CoA, to form one molecule of acetyl-CoA. In the cycle, one molecule of oxaloacetate grows by condensation with two CO₂ and is reduced and activated with CoA. The result, citryl-CoA, undergoes a retro-aldol cleavage to regenerate oxaloacetate and acetyl-CoA. Here we separate the formation of citryl-CoA from its subsequent retro-aldol cleavage, as this is argued to be the original reaction sequence, and the one displaying the closest homology in the substrate-level phosphorylation with that of succinyl-CoA [112, 113]. A second arc of reactions, sometimes termed anaplerotic [5], then condenses two further CO₂ with acetyl-CoA to produce a second molecule of oxaloacetate, completing the network-autocatalytic topology and making the cycle self-amplifying. The distinctive reaction in the rTCA pathway is a carbonyl insertion at a thioester (acetyl-CoA or succinyl-CoA), performed by a family of conserved ferredoxin-dependent oxidoreductases which are triple-Fe₄S₄-cluster proteins [108]. The cycle is found in many anaerobic and microaerophilic bacterial lineages, including Aquificales, Chlorobi and δ- and -proteobacteria.

Enzymes from reductive TCA reactions are very widely distributed among bacteria, where in addition to complete cycles they support fermentative pathways that break cycling and use intermediates such as succinate as terminal electron acceptors [68]. The distribution of full rTCA cycling correlates with clades whose origins are placed in a pre-oxygenic earth, while fermentative TCA arcs and the oxidative Krebs cycle are found in clades that by phylogenetic position (such as γ-, β- and α-proteobacteria) and biochemical properties (membrane quinones and intracellular redox couples involving them [114]) arose during or after the rise of oxygen. The co-presence of enzymes descended from both the reductive and oxidative cycles [67, 69, 115] in members of clades that at a coarser level are known to straddle the rise of oxygen, such as Actinobacteria and proteobacteria [114], may provide detailed mechanistic evidence about the reversal of core metabolism from ancestral reductive modes to later derived oxidative modes.

Functionally, the transition to an oxidizing earth is complex in two ways. First, oxygenation is unlikely to have occurred homogeneously within the oceans [116–118]. For bacteria not harbored in sheltered environments, this likely created complex needs to responsive phenotypes. Second, the co-presence of strong oxidants with geochemical reductants could in some cases provide an energy source during oxygenation, but as we see on the current Earth as well as in the history of banded iron, oxidants eventually scavenge reducing equivalents from most environments, likely leaving fermentative pathways as a fallback energy source for many organisms. It is plausible that both of these geochemical responses served as pre-adaptations enabling the complex host-associated lifecycles of bacteria and archaea descended within clades that straddled this transition.

Earlier work [71] that also recognized the central position of TCA reactions has attempted to argue for an ancestral oxidative TCA cycle. However, the optimal-path part of this argument, and similar path-length optimality arguments in [55, 70, 72], apply broadly to intermediary metabolism as a consequence of its limited reaction types or cross-linking as explained in section 2.1, without implying directionality. Furthermore, the functional criterion of acetate oxidation to produce reducing equivalents to drive ATP production through oxidative phosphorylation relies on the assumption that the organic interconversion of acetate to CO₂ was the main redox couple of early metabolism. Prior to the rise of oxygen, however, we find many inorganic electron acceptors such as elemental sulfur in redox couples with inorganic electron donors such as H₂ and not with organics of stoichiometry CH₂O [119–121]. For these reasons, combined with the strict absence of oxidative TCA as a plausible genetically reconstructed form in deep-branching clades, we find arguments for an ancestral rTCA cycle [7, 41, 122], replaced possibly via fermentative intermediates, by a later oxidative Krebs cycle, more convincing.

Dicarboxylate/4-hydroxybutyrate cycle. The dicarboxyate/4-hydroxybutyrate (DC/4HB) cycle [94, 123], illustrated in figure 6 is, like rTCA, a single-loop network-autocatalytic cycle, but has a simpler form of autocatalysis in which acetyl-CoA rather than oxaloacetate is the network catalyst. Only two CO₂ molecules are attached in the course of the cycle to form acetoacetyl-CoA, which is then thioesterified at the second acetyl moiety and cleaved to directly regenerate two molecules of acetyl-CoA. An extra copy of the network catalyst is thus directly regenerated (with suitable CoA activation) without the need for anaplerotic reactions. The cycle has so far been found only in anaerobic crenarchaeota, but within this group it is believed to be widely distributed phylogenetically [94, 123]. The first five reactions in the cycle (from acetyl-CoA to succinyl-CoA) are identical to those of rTCA. The second arc of the cycle begins with reactions found also in 4-hydroxybutyrate and γ-aminobutyrate fermenters in the Clostridia (a subgroup of Firmicutes within the bacteria), and terminates in the reverse of reactions in the isoprene biosynthesis pathway. The DC/4HB pathway thus uses the same ferredoxin-dependent carbonyl-insertion reaction used in rTCA (though only at acetyl-CoA), along with distinctive reactions associated with 4-hydroxybutyrate fermentation. In particular, the dehydration/isomerization sequence from 4-hydroxybutyryl-CoA to crotonyl-CoA is performed by a flavin-dependent protein containing an [Fe₄-S₄] cluster, and involves a ketyl-radical intermediate [124, 125].

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3-hydroxypropionate bicycle. The 3-hydroxypropionate (3HP) bicycle [106], highlighted in figures 6 and 7, has the most complex network topology of the fixation pathways, using two linked cycles to regenerate its network catalysts and to fix carbon. The network catalysts in both loops are acetyl-CoA and the outlet for fixed carbon is pyruvate. The reactions in the cycle begin with the biotin-dependent carboxylation of acetyl-CoA to form malonyl-CoA, from the fatty-acid synthesis pathway, followed by a distinctive thioesterification [126] and a second, homologous carboxylation of propionyl-CoA (to methylmalyl-CoA) followed by isomerization to form succinyl-CoA. The first cycle then proceeds as the oxidative TCA arc, followed by retro-aldol reactions also found in the glyoxylate shunt (described below). A second cycle is initiated by an aldol condensation of propionyl-CoA with glyoxylate from the first cycle to yield β-methylmalyl-CoA, which follows a sequence of reduction and isomerization through an enoyl intermediate (mesaconate) similar to the second cycle of rTCA or the 4HB pathway. This complex pathway was discovered in the Chloroflexi and is believed to represent an adaptation to alkaline environments in which the CO₂/HCO⁻₃ (bicarbonate) equilibrium strongly favors bicarbonate. All carbon fixations proceed through activated biotin, thus avoiding the carbonyl insertion of the rTCA and DC/4HB pathways. While topologically complex, the bicycle makes extensive use of relatively simple aldol chemistry, which we will argue in section 3.2.4 made its evolutionary innovation less improbable than the topology alone might suggest.

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3-hydroxypropionate/4-hydroxybutyrate cycle. The 3-hydroxypropionate/4-hydroxybutyrate (3HP/4HB) cycle [127], shown in figure 6, is a single-loop pathway in which the first arc is the 3HP pathway, and the second arc is the 4HB pathway. Like DC/4HB, 3HP/4HB uses acetyl-CoA as network catalyst and fixes two CO₂ to form acetoacetyl-CoA. The pathway is found in the Sulfolobales (Crenarchaeota), where it combines the crenarchaeal 4HB pattern of autotrophic carbon fixation with the bicarbonate adaptation of the 3HP pathway. Like the 3HP bicycle, the 3HP/4HB pathway is thought to be an adaptation to alkalinity, but because the 4HB arc does not fix additional carbon, this adaptation resulted in a simpler pathway structure than the bicycle.

Calvin–Benson–Bassham cycle. The Calvin–Benson–Bassham (CBB) cycle [128, 129] is responsible for most known carbon fixation in the biosphere. In the same way as WL adds only the distinctive CODH/ACS reaction to an otherwise very-widely-distributed folate pathway [22], the CBB cycle adds a single reaction to the otherwise-universal network of aldol reactions among sugar-phosphates that make up the gluconeogenic pathway to fructose 1,6-bisphosphate and the reductive pentose phosphate pathway to ribose and ribulose 1,5-bisphosphate³. The distinctive CBB reaction that extends reductive pentose-phosphate synthesis to a carbon fixation cycle is a carboxylation performed by the ribulose 1,5-bisphosphate carboxylase/oxygenase (RubisCO), together with cleavage of the original ribulose moiety to produce two molecules of 3-phosphoglycerate. The Calvin cycle resembles the 4HB pathways in regenerating two copies of the network catalyst directly, not requiring separate anaplerotic reactions for autocatalysis. In addition to carboxylation, RubisCO can react with oxygen in a process known as photorespiration [130–132] to produce 2-phosphoglycolate (2PG), a precursor to glyoxylate that is independent of rTCA-cycle reactions. The CBB cycle is widely distributed among cyanobacteria, in chloroplasts in plants and in some secondary endosymbionts.

The glyoxylate shunt. Although it is not an autotrophic carbon-fixation pathway, the glyoxylate shunt (or glyoxylate bypass) is of interest because it shares intermediates and reactions with many of the above fixation pathways, and because it resembles a fixation pathway in certain topological features. The pathway is shown in figure 7. All aldol reactions that can be performed starting from rTCA intermediates appear in this pathway, either as cleavages or as condensations. In addition to condensation of acetate and oxaloacetate to form citrate (CIT), these include cleavage of isocitrate (ISC) to form glyoxylate and succinate, and condensation of glyoxylate and acetate to form malate (MAL). The shunt is a weakly oxidative pathway (generating one H₂-equivalent from oxidizing succinate to fumarate), and is otherwise a network of internal redox reactions. It is therefore a very widely-used facultative pathway under conditions where carbon for biosynthesis, more than reductant, is limiting.

Two of the arcs of the shunt overlap with arcs in the oxidative Krebs cycle, but the entire pathway is a bicycle much like the 3HP-bicycle, sharing many of the same intermediates, but running in the opposite direction. Oxidative pathways such as the Krebs cycle are ordinarily catabolic, and hence not self-maintaining. The glyoxylate shunt may be regarded as a network-autocatalytic pathway for intake of acetate, using MAL as the network catalyst and regenerating a second molecule of MAL from two acetate molecules. This may be part of the reason that the shunt is up-regulated in the Deinococcus–Thermus family of bacteria in response to radiation exposure [133], providing additional robustness from network topology under conditions when metabolic control is compromised.

The central energetic costs of carbon-fixation pathways are associated with carboxylation reactions in which CO₂ molecules are added to the growing substrate, and the subsequent reactions in which the carboxyl group is reduced to a carbonyl [98]. In isolation these reactions require ATP hydrolysis, but these costs can be avoided in several ways. In some cases a thioester intermediate is used to effectively couple together a carboxyl reduction and a subsequent carboxylation, allowing the two reactions to be driven by a single ATP hydrolysis. An unfavorable (endergonic) reaction can also be coupled to a highly favorable (exergonic) reaction, allowing the reactions to proceed without ATP hydrolysis.

Individual pathways employ such couplings to varying degrees, resulting in a range of ATP costs associated with carbon fixation. At the low end, WL eliminates nearly all use of ATP through its unique pathway chemistry, requiring only a single ATP in the synthesis of pyruvate from CO₂. This ATP is associated with the attachment and activation of formate on folates. Reducing CO₂ to free formate prior to attachment, and further reducing the activated formate on folates prior to incorporation into growing substrates saves one ATP associated with carboxylation. Additional ATP costs are saved by coupling the endergonic reduction of CO₂ to CO to the subsequent exergonic synthesis of acetyl-CoA. Finally, the activated thioester bond of acetyl-CoA allows the subsequent carboxylation to pyruvate to also proceed without additional ATP. In methanogens even the ATP cost of attaching formate to folates has been eliminated by modifying the structure of tetrahydrofolate (THF) to that of H₄MPT [22], enabling a membrane-bound iron–sulfur system to serve as energy source. Similarly, rTCA has high energetic efficiency as a result of extensive reaction coupling, requiring only two ATP to synthesize pyruvate from CO₂ [94, 98]. Two ATP are saved by coupling carboxyl reductions to subsequent carboxylations using thioester intermediates, and an additional ATP is saved by coupling the carboxylation of α-ketoglutarate to the subsequent carbonyl reduction leading to ISC.

At the high end of energetic cost of carbon-fixation are pathways that couple unfavorable reactions less effectively, or not at all, or even hydrolyze ATP for reactions other than carboxylation or carboxyl reduction. Both the DC/4HB pathway and the 3HP bicycle decouple one or more of the thioester-mediated carboxyl reduction + carboxylation sequences of the kind used in rTCA, and neither couples endergonic carboxylations to exergonic reductions. As a result, DC/4HB requires five ATP and the 3HP bicycle seven ATP to synthesize pyruvate from CO₂ [94, 98]. The 3HP/4HB pathway has the highest cost of any fixation pathway, with nine ATP required to synthesize pyruvate from CO₂. This is partly because it also decouples thioester-mediated carboxyl reduction + carboxylation sequences, and partly because pyruvate is synthesized by diverting and ultimately decarboxylating succinyl-CoA [77, 94, 98]. Finally, CBB is also at the high end in terms of cost, requiring seven ATP to synthesize pyruvate from CO₂. Although this pathway avoids the cost of carboxylation reactions by coupling them to exergonic cleavage reactions, CBB is the only fixation pathway that invests ATP hydrolysis in chemistry other than carboxylations or carboxyl reductions, thereby increasing its relative cost [98].

The apparent diversity of six known fixation pathways is unified by the role of the citric-acid cycle reactions, and secondarily by that of gluconeogenesis and the pentose-phosphate pathways. Figure 4 showed the C, H, O stoichiometry for a network of reactions that includes all six known pathways. Here, by stoichiometry we refer to the mole-ratios of reactants and products for each reaction, with molecules represented by their CHO constituents, and attached phosphate or thioester groups omitted. Where phosphorylation or thioesterification mediates a net dehydration, we have represented the dehydration directly in the figure. The network contains only 35 organic intermediates, because many intermediates and reactions appear in multiple pathways. Hydroxymethyl-glutarate and butyrate are also shown, to indicate points of departure to isoprene and fatty acid synthesis, respectively.

In figure 4 the TCA cycle and the gluconeogenic pathway are highlighted. Beyond being mere points of departure for alternative fixation pathways and for diversifications in intermediary metabolism, they are invariants under diversification because they determine carbon flow among the universal precursors of biosynthesis.

Almost all anabolic pathways in extant organisms originate in one of five intermediates in the TCA cycle—acetate (as acetyl-CoA), pyruvate, oxaloacetate, succinate (or succinyl-CoA) or α-ketoglutarate—which have been dubbed the 'pillars of anabolism' [51]. Succinyl-CoA can serve as the precursor to pyrroles (metal-coordinating groups in many cofactors)—mainly in α-proteobacteria and mitochondria—but these are more commonly made from α-ketoglutarate via glutamate in what is known as the C5 pathway [134]. A phylogenetic analysis of these pathways confirms that the C5 pathway is the most plausible ancestral route to pyrrole synthesis [135]. (An even stronger claim has been made, from a similar combination of biochemical and phylogenetic inputs, that it is a likely relic of the RNA world [136].) Thus as few as four TCA intermediates provide the organic inputs to all anabolic pathways. Figure 8 shows the major molecule classes associated with each intermediate. The only exceptions to this universality, which form a biosynthetic sequence, are glycine, serine, and a few compounds synthesized from them; this sequence can be initiated directly from CO₂ outside the pillars (see figure 5). This observation becomes key in reconstructing the evolutionary history of carbon-fixation (see section 3.3). The gluconeogenic pathway then forms a similarly unique bridge between the TCA intermediate pyruvate (in the activated form phosphoenolpyruvate) and the network of sugar-phosphate reactions known as the pentose-phosphate pathway.

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Carbon-fixation pathways must reach all four (or five) of the universal anabolic starting compounds. They may do this either by producing them as pathway intermediates, or by means of secondary reactions converting pathway intermediates into the essential precursors. The degree to which a pathway passes through all essential biosynthetic precursors may suggest its antiquity. In metabolism-first theories of the origin of life [6], the limited set of compounds selected and made available in high concentration by proto-metabolism determined the opportunities for further biosynthesis, thus establishing themselves as the precursors of anabolism.

Among the five network-autocatalytic fixation pathways, the CBB pathway is unique in not passing through any universal anabolic precursors. When used as a fixation pathway, CBB reactions must thus be connected to the rest of anabolism through several reactions in the glycolytic pathways connecting 3-phosphoglycerate (3PG) to pyruvate. Pyruvate is then connected to the remaining precursors through partial TCA sequences. The glycolytic pathway is the primary connection of CBB to the anabolic precursors, but 2-phosphoglycolate (2PG) produced during photorespiration may also be converted to glyoxylate and subsequently to glycine and serine (see figure 5). Glycine synthesis from photorespiration is not itself a carbon-fixing process, but rather a salvage pathway to compensate for poor discrimination of the enzyme RuBisCO. 2PG is produced from ribulose-1,5-bisphosphate when O₂ replaces CO₂ in the RuBisCo uptake reaction. This toxin inhibits RuBisCO, and would require excretion if it could not be recycled, leading to a net loss of carbon from the pentose-phosphate network. However, most RuBisCO uptake events do fix CO₂, and the carbon circulating in the pentose-phosphate pathway in CBB organisms is the product of these successful fixation cycles. Thus, photorespiration with glycine salvage amounts to a variant elaboration of the fixation pathway to include null cycles, and the connection of this more complex process to the precursor set.

Among the remaining loop-fixation pathways, only rTCA passes through all five anabolic pillars. Through its partial overlap with rTCA, DC/4HB passes through four, excluding α-ketoglutarate. The 3HP-bicycle further bypasses oxaloacetate, while the 3HP/4HB loop and WL include only acetyl-CoA. All of the latter pathways require anaplerotic reactions in the form of incomplete (either oxidative or reductive) TCA arcs; when these combine (in various ways) with WL carbon fixation, they are known collectively as the reductive acetyl-CoA pathways.

The most parsimonious explanation for the universality of the TCA arcs as anaplerotic reactions is lock-in by downstream anabolic pathways, to which metabolism was committed by the time carbon-fixation strategies diverged. This can also be understood as a direct extension of the metabolism-first assumption that anabolic pathways themselves formed around proto-metabolic selection of the rTCA intermediates⁴. (A similar but later form of commitment has been argued to convert basal gene regulatory networks in metazoan development into kernels, which admit no variation and act as constraints on subsequent evolutionary dynamics [137, 138].) If lock-in provides the correct interpretation of TCA universality, then much of the burden of accounting for the inventory of small metabolites is shifted away from Darwinian selection for function in a post-RNA world, and onto constraints of biosynthetic simplicity and network context. We show below that phylogenetic reconstruction is consistent with a selective role for rTCA cycling in the root metabolism of cellular life, though only as part of a larger network than the modern rTCA cycle.

Figure 6 shows the sub-network from figure 4 containing the four loop-autotrophic carbon fixation pathways that pass through some or all universal precursors, together with reactions in the glyoxylate shunt. The four loop pathways are shown in four colors, with the organic pathway-intermediates (but not environmental precursors or reductants) highlighted.

The figure shows that these pathways re-use intermediates by combining entire pathway segments. The combinatorial assembly of these segments is possible because they all pass through acetate (as acetyl-CoA), succinate (usually as succinyl-CoA), and all except the second loop of the 3HP bicycle pass through both. Thus the conserved reactions among the autocatalytic loop carbon-fixation pathways are shared within strictly preserved sequences, which have key molecules as the boundaries at which segments may be combined.

In addition to the re-use of complete reactions in pathway segments, variant carbon-fixation pathways have extensively re-used transformations at the level of local functional groups. The network of figure 6 is arranged in concentric rings, in which the arcs of the rTCA cycle align with the 3HP or 4HB pathways, or with the mesaconate arc of the 3HP bicycle. The 'ladder' structure of inputs and outputs of reductant (H₂) or water between these rings shows the similar stoichiometric progression in these alternative pathways. Figure 9 decomposes the aggregate network into two pairs of short-molecule and long-molecule arcs, and the mesaconate arc, and shows the pathway intermediates in each arc. The figure makes clear that, both within the arcs of the loop pathways, and between alternate pathways, the type, sequence and position of reactions are highly conserved. In particular, the reduction sequence from α-ketones or semialdehydes, to alcohols, to isomerization through enoyl intermediates, is applied to the same bonds on the same carbon atoms from input acetyl moieties in rTCA, 3HP and 4HB pathways, and to analogous functional groups in the bicycle. Finally, in the cleavage of both citryl-CoA and citramalyl-CoA, the bond that has been isomerized through the enoyl intermediate is the one cleaved to regenerate the network catalyst.

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Even the distinctive step to crotonyl-CoA in the 4HB pathway creates an aconate-type intermediate, and the enzyme responsible has high homology to the acrolyl-CoA synthetase [139, 140], whose output (acrolyl-CoA) follows the standard pattern. Only the position of the double bond breaks the strict pattern in crotonyl-CoA, and the abstraction of the un-activated proton required to produce this bond requires the unique ketyl-radical intermediate [141]. From crotonyl-CoA, the sequence to 3-hydroxybutyrate is then followed by a surprising oxidation and re-hydration, resulting in a five-step, redox-neutral, sequence. The net effect of this sequence is to shift the carbonyl group (of succinate semialdehyde, SSA) by one carbon (in acetoacetate, AcACE). Because the 4HB pathway takes in no new CO₂ molecules, this isomerization enables regeneration of the network catalysts in the same way the reduction/aldol-cleavage sequence enables regeneration for rTCA or the 3HP bicycle.

Duplication of reaction sequences in diverse fixation pathways appears to have resulted from retention of gene sets as organism clades diverged. Duplication of local-group chemistry in diverse reactions similarly appears to have resulted (at least in most cases) from retention of reaction mechanisms as enzyme families diverged. All enoyl intermediates are produced by a widely diversified family of aconitases [142], while biotin-dependent carboxylations are performed by homologous enzymes acting on pyruvate and α-ketoglutarate [110], and substrate-level phosphorylation and thioesterification are similarly performed by homologous enzymes on CIT and succinate in rTCA [112, 113]. Similar to the synthesis of citryl-CoA we separate here the carboxylation of α-ketoglutarate from the subsequent reduction of oxalosuccinate to ISC—performed by a single enzyme in most organisms—because it is argued to be the ancestral form [143, 144]. However, the thioesterification of propionate in the 3HP pathway is performed by distinct enzymes in bacteria and archaea, an observation that has been interpreted to suggest convergent evolution [106, 127]. The widespread use of a few reaction types by a small number of enzyme classes/homologues may reflect their early establishment by promiscuous catalysts [95, 145], followed by evolution toward increasing specificity as intermediary metabolic networks expanded and metabolites capable of participating in carbon fixation diversified.

A functional identification of modules that seeks to minimize influence from historical effects (such as lock-in) has been carried out by Noor et al [72], and identifies similar module boundaries. Using as data the first three numbers of the EC classification of enzymes—which distinguish reaction types but coarse-grain over both substrate specificity and enzyme homology—they show that many pathways in core metabolism are the shortest routes possible between inputs and products. This work builds on earlier studies showing that under very simple rules, the pentose phosphate pathway uses the minimal number of steps to connect inputs to outputs [70, 146]. In each of those studies, the authors suggested that Darwinian selective pressure may have led to a such minimal pathways as optimal network connections between given pairs of metabolites, with the implication that the selection of metabolites was based on some aspects of phenotypic function aside from their network positions.

If we do not presume that phenotypic selection preceded metabolism, however, the problem of pathway optimization ceases to be one with fixed endpoints, and causation may even run from pathways to the metabolite inventory. In this view minimal pathways may have been selected because their kinetics and regulation were easier to control. Starting points of downstream intermediary metabolism could then have been selected from the intermediates made available by fixation pathways. We argue in favor of a selection hierarchy of this form in section 3.4.4: shorter fixation pathways capable of attaining autocatalytic feedback offer fewer opportunities for dilution by parasitic side-reactions [147], and (reaction chemistry otherwise being equal) require less regulatory control. They may thus be the only sustainable forms.

Where the pathways analyzed by Noor et al overlap with those we have shown, many of their minimal sequences overlap with the modules in figure 6, as well as with others in gluconeogenesis which we do not consider here. Thus, from the perspective of an emerging metabolism, it may be that historical retention of a small number of reaction types reflects facility of the substrate-level chemistry, and that this has placed time-independent constraints on evolution.

The functional-group homology shown in figure 9 allows us to separate stereotypical sequences of widely diversified reactions from key reactions that distinguish pathways. The stereotypical sequences lie downstream of reactions such as the ferredoxin-dependent carbonyl insertion (rTCA), or biotin-dependent carboxylation (3HP), which are associated with highly conserved enzymes or cofactors. The downstream reactions are also more 'elementary', in the sense that they are common and widely diversified in biochemistry, compared to the pathway-distinguishing reactions.

The observation that alternative fixation pathways are not distinguished by their internal reaction sequences, but primarily by their initiating reactions, suggests that these reactions were the crucial bottlenecks in evolution, and perhaps reflect the limiting diversity of chemical mechanisms for carbon bond formation. Mechanisms of organosynthesis in aqueous solution are especially limited by the instability of radical intermediates, which may be stabilized by association with metal centers. The distinctive use of metals in the (often highly conserved) enzymes and cofactors for these initiating reactions may thus suggest a direct link between prebiotic mineral and metal–ligand chemistry [148], and constraints inferable from the long-term structure of later cellular evolution.

Several enzyme iron–sulfur centers have been recognized [149, 150] to use strained versions of the unit cells found in Fe–S minerals, particularly Mackinawite and Greigite. These are particular instances within a wider use of transition-metal-sulfide chemistry in core-metabolic enzymes.

Pyruvate:ferredoxin oxidoreductase (PFOR), which catalyzes the reversible carboxylation of acetyl-CoA to pyruvate, contains three [Fe₄S₄] clusters and a thiamin pyrophosphate (TPP) cofactor. The [Fe₄S₄] clusters and TPP combine to form an electron transfer pathway into the active site, and the TPP also mediates carboxyl transfer in the active site [108].

The bifunctional carbon monoxide dehydrogenase/acetyl-CoA synthase (CODH/ACS) enzyme that catalyzes the final acetyl-CoA synthesis reaction in the WL pathway employs even more elaborate transition-metal chemistry. Like PFOR, this enzyme uses [Fe₄S₄] clusters for electron transfer, but its active sites contain additional, more unusual metal centers. The CODH active site contains an asymmetric Ni-[Fe₄S₅] cluster on which CO₂ is reduced to CO [99], while the ACS active site contains a Ni-Ni-[Fe₄S₄] cluster on which CO (from CODH) and a methyl group from folates are joined to form acetyl-CoA [100–102]. It was originally thought that a variant form of the ACS active site contains a Cu-Ni-[Fe₄S₄] cluster [151, 152], but it was subsequently shown that the Cu-containing cluster represents an inactivated form of ACS [101]. Similarly, it has also been shown that the open form of the Ni-Ni ACS may switch to a closed, inactivated, form by exchanging one of the nickel atoms for a zinc atom [100].

Finally, methyl-group transfer to the ACS active site mediated by the corrinoid iron–sulfur protein (CFeSP) containing the cofactor cobalamin also involves elaborate metal chemistry [103, 104]. In accepting the methyl-group from folates, the cobalt in cobalamin becomes oxidized from the Co⁺ to the Co^{3 +} state. Donation of the methyl group to the ACS active site restores the Co⁺ state, while in turn oxidizing the active nickel in the ACS active site from the Ni⁰ to the Ni^{2 +} state. Release of acetyl-CoA then reduces the active nickel back to the Ni⁰ state, allowing the cycle to start over [104].

Perhaps not surprisingly, all these examples of metal-cluster enzymes concern catalysis not just of the formation of C–C bonds, but of the incorporation of the small gas-phase molecule CO₂. In general, enzymes involved in the processing of small gas-phase molecules (including H₂ and N₂) are among the most unique enzymes in biology—all but one of the known Nickel-containing enzymes belong to this group [153]—always containing highly complex metal centers in their active sites [154–159]. This indicates both the difficulty of controlling the catalysis of these reactions, and the importance of understanding their functions in the context of the emergence of metabolism [150].

The network closures that retain carbon flux and enable autocatalysis in rTCA, DC/4HB, and 3HP/4HB pathways are all topologically rather simple, and are quite similar due to the homology among most of the pathway intermediates. Their module boundaries also are all defined by acetate and succinate, and at least in the case of acetate, were probably facilitated by its multiple pre-existing roles as the redox-drain of the rTCA cycle [41] and the starting point for both isoprenoid and fatty-acid lipid biosynthesis.

In contrast, the topology of the 3HP-bicycle appears complex, and perhaps an improbable solution to the problem of recycling all carbon flux through core pathways. This form of complexity arises from the requirement to complete an autocatalytic network topology while avoiding reactions based on CO₂ in favor of those based on HCO⁻₃. It is thus different from the topological complexity within the bowtie, where dense cross-linking in the core arises at the intersection of many minimal pathways. If we are to argue that the emergence or evolution of network closures such as that in the bicycle is facilitated by a form of modularity, it must exist at the level of reaction mechanisms that render the evolutionary innovation of such topologties plausible. For the 3HP bicycle and the related glyoxylate shunt—and to a lesser degree also for rTCA—the mechanism of interest is the aldol reaction.

The aldol reaction is an internal oxidation-reduction reaction, which means that it exploits residual free energy from organosynthesis, and also that it can take place independently of external electron donors or acceptors. Many aldol reactions are also kinetically facile, occurring at appreciable rates without the aid of catalysts. We therefore expect that among compounds capable of participating in them, aldol reactions would have been common in the prebiotic world, providing opportunities for pathway generation. Since their diversity is difficult to suppress except by special mechanisms [160], we expect that potential aldol reactions among metabolites would either have become regulated (perhaps through phosphoryl occupation of hydroxyl groups) or else incorporated into actively-used biochemical pathways.

Aldol reactions are important generators of diversity in organic chemistry, notorious for the very complex network known as the formose reaction [161–164], initiated from formaldehyde and glycolaldehyde. Many aldol reactions are possible for sugars, and the reductive pentose phosphate pathway is indeed a network of selected aldol condensations and cleavages among sugar-phosphates [87].

Fewer aldol reactions are possible among intermediates of the rTCA cycle and their homologues such as methyl-malate (MML) or citramalate in other carbon-fixation pathways, but all possibilities are indeed used either in intermediary metabolism or in carbon fixation. The complex topology of the 3HP bicycle therefore also suggests that a diverse inventory of pathway segments were available at the time of its emergence. Figure 7 shows the overlap between the bicycle and the closely-related glyoxylate shunt, which is thus a possible precursor to the bicycle. In both pathways, the network topologies that regenerate all carbon flux or achieve autocatalysis are created by aldol reactions. The retention of carbon within the shunt appears to be a reason for its widespread distribution and frequent use [133, 165, 166], even when energetically more efficient pathways such as the Krebs cycle exist as alternatives within organisms. The re-use of reaction mechanisms on different substrates is a distinct form of simplicity and redundancy that we consider more generally in section 5.

A unique form of re-use is found for the sequence of reactions that directly reduce one-carbon (C₁) groups on pterin cofactors. We have argued elsewhere [22] that even when a complete, autotrophic WL pathway is not present due to the loss of the oxygen-sensitive CODH/ACS enzyme, the direct C₁-reduction sequence on pterins is often still present and being used as a partial fixation pathway. The reaction sequence supplies the diverse methyl-group chemistry mediated by S-adenosyl-methionine (SAM), and the direct synthesis of glycine and serine from methylene groups, reductant and ammonia. Serine then serves as a precursor to cysteine and tryptophan. The pathway may exist in either a complete (8-reaction) or a previously unrecognized but potentially widespread (7-reaction) form that involves uptake on N⁵ rather than N¹⁰ of THF [22] (see figure 5.)

The widely distributed and diversified form of direct C₁ reduction functions much as auxiliary catabolic pathways function in mixotrophs [5], operating in parallel to an independent 'primary' fixation pathway, with the primary and the direct-C₁ pathway supplying carbon to different subsets of core metabolites. In many cases where the CODH/ACS is lost, this loss disconnects the primary and direct-C₁ pathway segments, creating the novel feature of a disjoint carbon fixation pathway. The existence of parallel fixation pathways in autotrophs had previously been recognized only in one (relatively late-branching) γ-proteobacterium, the uncultured endosymbiont of the deep-sea tube worm Riftia pachyptila, which was found to be able to use both the rTCA and CBB cycles [167]. In that case, however, the two pathways are not disjointed, but rather connected through intermediates in the glycolytic/gluconeogenic pathways. In addition, the capacity for using either cycle is thought to reflect an ability to adapt to variation in the availability of environmental energy sources, with an apparent up-regulation of the more efficient rTCA cycle under energy-poor conditions [167]. Our phylogenetic reconstruction [22], however, indicates that parallel disjoint pathways were the majority phenotype in the deep tree of life, in which a reductive C1 sequence to glycine and serine is preserved in combination with rTCA in Aquificales and Nitrospirae, with CBB in Cyanobacteria, with the 3HP bicycle in Chloroflexi (all bacteria), and with DC/4HB in Desulfurococcales and Acetolobales, and the 3HP/4HB cycle in Sulfolobales (all archaea). In contrast, the full WL pathway is found only in a subset of lineages of bacteria (especially acetogenic Firmicutes) and archaea (methanogenic Euryarcheota).

Apparently as a result of the flexibility enabled by parallel carbon inputs to core metabolites, the direct C₁ reduction sequence is more universally distributed than any of the other loop-networks (whether paired with C₁ reduction or used as exclusive fixation pathways), or than the complete WL pathway. The status of the pterin-mediated sequence as a module appears more fundamental than its integration into the full WL pathway, and comparable to the arcs identified within rTCA, which may function as parts of fixation pathways or alternatively as anaplerotic extensions to other pathways. The two types of pathways also serve similar functional roles in our phylogenetic reconstruction of a root carbon-fixation phenotype, as the key components enabling and selecting the core anabolic precursors.

The reductive synthesis of glycine furnishes a potent reminder of the importance of taking evolutionary context into account when interpreting results from studies of metabolism. The goal of understanding human physiology and disease states has historically been a major driver in the study of biochemistry and metabolism. Although microbial biochemistry is currently better understood (because it is simpler) than human biochemistry [168], model systems and interpretations have continued to emphasize heterotrophy. An example of the interpretive bias that can result is the common reference to the reductive citric acid cycle as the 'reverse' citric acid cycle, despite its likely being the original form as we (and many others) have argued. Similarly, the 'glycine cleavage system' (GCS) was originally studied in rat and chicken livers [169], before being recognized as phylogenetically widespread. The distribution of this system is now known to be nearly universal across the tree of life (with methanogens being the main systematic exception, for reasons explained elsewhere [22]), suggesting that it was present already in the LUCA. The lipoyl-protein based system has long been known to be fully reversible [169–171], and has nearly neutral thermodynamics at physiological conditions [98]. Thus, the LUCA could have used this system either to synthesize or to cleave glycine. From this perspective the former possibility (synthesis) seems a more likely interpretation, even without additional data. From a perspective less strongly focused on heterotrophs, the choice between these alternatives might have become clear much sooner.

The essential invariance across the biosphere of the seven sub-networks listed above allows us to represent all carbon-fixation phenotypes in terms of the presence or absence, connectivity and direction of these basic modules. In this representation, metabolic innovation at the modular level retains the character of individual discrete events, even if the pathway segments involved incorporate multiple genes. In cases where multiple genes must be acquired to constitute a module, this innovation may take place at higher levels of metabolism, after which their incorporation as fixation pathways appears as a single innovation. For example, most of the reaction sequences used in the autotrophic 4HP pathway appear in diversified forms in fermentative secondary metabolism from hydroxybutyrate or aminobutyrate, which is both outside the rTCA/folate core and ecologically heterotrophic. It is plausible (and we think likely [22]) that these pathways were recruited for autotrophy from an organism similar to Clostridium kluyveri. (See also discussions in section 3.4.2).

Because the module boundaries are defined by particular (often universal) molecular species (e.g., acetyl-CoA, succinyl-CoA, and ribulose-1,5-bisphosphate) it often remains true that innovation can be traced to the change in single genes. This is true for the loss of the CODH/ACS from acetyl-CoA phenotypes, the innovation of RubisCO in CBB bacteria, or the loss of substrate-level phosphorylation to acetyl-CoA or succinyl-CoA in acetogens. A case with only slightly greater complexity is the apparently repeated, convergent evolution of an oxidative pathway to form serine from 3-phosphoglycerate (3PG), which involves three common and widely diversified reactions: a dehydrogenation, a reductive transamination and a dephosphorylation. The evolution of this bridge pathway creates a secondary connection between the previously disjoint carbon-fixation pathways described in section 3.2.5. As a full evolutionary reconstruction (described next) shows, such a bridge may permit subsequent loss of direct-C₁ reduction as a fixation route, as in the proteobacteria, or it may release a constraint, permitting change in pterin cofactor chemistry as in methanogens.

At the module level, we may represent changes in carbon fixation pathways between closely-related phenotypes in terms of single connections, disconnections or overall changes of direction within the subsets of the seven modules which are present. The change of direction within modules is usually complete, even if it is partial or intermediate at the level of whole pathways. An example is the switch from autotrophic rTCA to fermentative TCA using a reductive small-molecule arc and an oxidative large-molecule arc [68]. Such fermentative pathways may alternate with fully oxidative TCA (Krebs) cycling, and they often occur in organisms that carry homologues to genes for both oxidative and reductive pathway directions [67, 69, 115].

An important exception to this pattern is the partial reversal of the formyl-to-methylene sequence on folates, between its carbon-fixation role and its role in the catabolic cleavage of glycine. We refer in [22] to the module formed by combining the GCS with the methylene-serine transferase as the glycine cycle. The combination of the complex free energy landscape provided by the folates [93] with the reversibility and nearly neutral thermodynamics of the glycine cycle [98, 169] permits a high degree of flexibility within this module. Carbon can enter either directly through CO₂, through serine (from 3PG), or through glycine (from glyoxylate), and from any of these sources it may be redirected to all of C₁ chemistry. The topology of the main reaction sequence is preserved in all of the above cases of reversal, though new enzymes or cofactors may be recruited to reverse some reactions.

A representative example of complete module reversal (and in this case, complete cycle reversal) enabled by reductant and cofactor substitution is given by the relation between reductive and oxidative TCA cycles. The electron donor in rTCA, reduced ferredoxin, is replaced by lipoic acid as an electron acceptor in the Krebs cycle, in the TPP-dependent oxidoreductase reaction. The enzymes catalyzing the retro-aldol cleavage of rTCA, which have undergone considerable re-arrangement even within the reductive world [112, 113], are further replaced by the oxidative citryl-CoA synthetase. Finally, the change from fumarate reduction to succinate oxidation may require a substitution of membrane quinones [114]. Yet the underlying carbon skeletons over the whole pathway are completely retained, and apart from some details of reaction ordering for thioesters, and possibly the use of phospho-enol intermediates, the energetic side groups are also the same.

The small number of modules that contribute to carbon fixation, and the even smaller number of 'gateway' molecules that serve as interfaces between most of them, permit free recombination into many phenotypes satisfying the constraints of autotrophy. An important consequence of free recombination is that a constraint of overall autotrophy only enforces network completeness—the existence of some connection between gateway molecules. Because there exist multiple modules that can be used to satisfy these constraints, autotrophy alone therefore does not lock in dependences within networks over distances longer than the modules themselves. Homology across intra-modular reaction sequences—especially if it is due to catalytic promiscuity—further weakens any lock-in effect created by selection for metabolic completeness. Through these mechanisms modularity promotes innovation-sharing [172] and rapid and reliable adaptation [18] to environmental conditions, but reduces standing variation among individuals sharing a common environment.

However, despite the potential for free recombination in principle, distinct carbon-fixation pathway modules have very different couplings to the chemical environment, as we reviewed in section 3. The genome distributions reported in [22] show that they also have very uneven phylogenetic distribution. For example, TCA arcs and intermediates, as well as direct C₁-reduction, are nearly universally distributed, while the 3HP arcs are restricted to specific bacterial or archaeal clades living in alkaline environments. Finally, we note that not all module combinations consistent with autotrophy have been observed in extant organisms.

By combining these observations it is possible to arrange autotrophic phenotypes on a graph according to their degree of similarity, and to assign environmental factors as correlates of phenotypic changes over most links. The graph projects onto a tree with very high parsimony and therefore requires invoking almost no horizontal gene transfer or convergent evolution from distinct lineages. Instead, all divergences may be interpreted as independent simple innovations driven by environmental factors. Finally, the directionality of these links (divergences) and the overlap of the tree with bacterial and archaeal phylogeny motivates a natural choice of root. The lack of reticulation in a tree of innovations in autotrophy—at first surprising when compared to highly-reticulated gene phylogenies [73] covering the same period—becomes sensible as a record of invasion and adaptation to new chemical environments by organisms capable of maintaining little long-standing variation.

The tree of autotrophic carbon-fixation phenotypes from [22] is shown in figure 11. All nodes in the tree satisfy the constraint that all five universal anabolic precursors plus glycine can be synthesized directly from CO₂. We have defined parsimony by requiring single changes over links at the level of pathway modules, as explained above, rather than at the level of single genes, in cases where the two criteria differ. (This definition separates the evolution of genetic backgrounds, such as 4-hydroxybutyrate fermentation, from the events at which organisms came to rely on complete pathways for autotrophy.)

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A complete-parsimony tree for the known phenotypes is not possible, so we chose a tree in which the only violations are duplicate innovation of serine synthesis from 3-phosphglycerate (3-PG), and duplication or transfer of the short-molecule 3HP pathway. The synthesis of serine from 3-PG involves reactions—the dehydrogenation of an alcohol to a carbonyl, the transamination of this carbonyl to an amino group and a dephosphorylation—that are common throughout metabolism and are performed by highly diversified enzyme families. We have therefore regarded multiple occurrences of this event as not attaching a large probability penalty to parsimony violation. We make a similar judgment for the 3HP pathway. This pathway contains two key biotin-carboxylase enzymes, one of which (acetyl-CoA carboxylase) is also part of fatty acid synthesis, which is suggested to have been present already in the LUCA [173]. Sequence analysis of propionyl-CoA carboxylase has in turn been used to suggest convergent evolution as an explanation for the multiple occurrence of this enzyme across bacterial and archaeal domains [127]. The remaining reactions in this pathway are again common metabolic reactions performed by highly diversified enzyme families. An alternative hypothesis is transfer: this complete pathway occurs in environments shared by the bacteria and archaea that harbor it, and this environment also contains a stressor (alkalinity) that may induce gene transfer [174]. Thus, both gene transfer and convergence are plausible explanations for why this phenotype should be paraphyletic.

Any tree in which either of these phenotypes was made monophyletic would require more extensive parsimony violations than the tree we chose, involving innovations for which convergence or transfer are also less plausible. Such trees would require major sub-branches to contain both bacterial and archaeal members, and within these, repeated divergences of major domain-specific differences. Common descent would fail to account for the exclusivity of rTCA-based phenotypes within bacteria, and among archaea either (non-THF) pterin-based one-carbon chemistry, or isoprene-related hydroxybutarate reactions would be required to have arisen several times. It is corroborating evidence for this parsimony argument that the tree we propose preserves the monophyly of bacteria and archaea, and is consistent with the most robust signals in purely statistical gene phylogenies, including the greatest congruence of firmicutes with the archaea, among the bacterial branches [175, 176].

The nodes in the tree of figure 11 are all phenotypes of extant organisms, with one important exception, which is the node between the Aquificale branch and the Firmicute/Archaea branch. Aquificales and all phenotypes descending from them lack the CODH/ACS enzyme, while firmicutes and archaea lack one or more ATP-dependent acyl-CoA (citryl-CoA or succinyl-COA) synthases. Therefore, if we seek a connected tree of life, two changes—the gain of one enzyme and loss of the other—are required to connect these branches. Since any organism lacking both enzymes could not fix carbon autotrophically, we have chosen the order of gain and loss so that the intermediate node has both the CODH/ACS and the acyl-CoA synthases. It therefore has both a complete WL pathway and an autocatalytic rTCA loop, connected through their shared intermediate acetyl-CoA. Losses (but not re-acquisitions) of either of these enzymes occur at multiple points on the tree, and both have likely explanations in either environmental chemistry or energetics. For this reason and several others given below, although a parsimony tree is (a priori) unrooted, we will regard the joint WL/rTCA phenotype as not only a bridging node but the root of the tree of autotrophs.

There is one further, unproblematic exception to the assignment of extant phenotypes to nodes in the tree, which is the insertion of an acetogenic phenotype with a facultative oxidative pathway to serine at the root of the Euryarchaeota. Since methanogens use this pathway, and since an acetogenic pathway lacking oxidative serine synthesis is the most plausible ancestral form for all archaea as well as for Firmicutes within the bacteria, we infer that such an intermediate state did or does exist. This fixation pathway is consistent with forms observed in extant organisms, and these proposals would be supported if such a phenotype were to be discovered or to result from reclassification of genes in an existing organism.

In the evolution of carbon fixation from a joint WL/rTCA root, the primary division is between the loss of the CODH/ACS, resulting in rTCA loop-fixation phenotypes, and the loss of the acetyl-CoA or succinyl-CoA synthetases, resulting in acetogenic phenotypes. Very low levels of oxygen permanently inactivate the CODH/ACS, so its loss is probable even under microaerobic conditions. Although the dominant mineral buffers for oxygen in the Archaean remain a topic of significant uncertainty [117, 177–179], it appears unlikely that molecular oxygen was the toxin responsible for loss of the CODH/ACS much before the 'great oxidation event' (GOE)⁵. Therefore the sensitivity of the CODH/ACS to sulfides or perhaps other oxidants [309] remains a possibly important factor in the early divergences of carbon fixation.

Alternatively, among strict WL-anaerobes, the loss of citryl-CoA or succinyl-CoA synthetase saves one ATP per carbon fixed, and all acetogenic phenotypes break rTCA cycling only through the loss of one or the other of these enzymes. We therefore interpret the loss of rTCA cycling as a result of selection for energy efficiency. The failure to regain either of these enzymes by acetogens which subsequently also lost the CODH/ACS is perhaps surprising given the inferred homology of the ancestral citryl-CoA and succinyl-CoA synthetases [112, 113], but explains the absence of rTCA cycling in either Firmicutes or any Archaea.

The remaining autotrophic phenotypes are derived from either rTCA cycling or acetogenesis in natural stages due to plausible environmental factors. Oxidative serine synthesis (from 3PG) is associated with the rise of the Proteobacteria, whose differentiation in many features tracks the rise of oxygen and the transition to oxidizing rather than reducing environments. RubisCO and subsequently photorespiration arise within the Cyanobacteria. The innovation of the 3HP bicycle from the malonate pathway arises within the Chloroflexi. In the Firmicutes (bacteria), 4-hydroxybutyrate (or closely related 4-aminobutyrate) fermentation is more or less developed. Closure of the fermentative arcs to form a ring, again driven by elimination of the CODH/ACS [22] leads to the DC/4HB pathway in Crenarchaeota, which is then specialized in the Sulfolobales to the alkaline 3HP/4HB pathway. The Euryarchaeota are distinguished by the absence of an alternative loop-fixation pathway to rTCA, so that all members are either methanogens or heterotrophs.

Similarly, the innovation of the 3HP pathways, using biotin, emerges as a specialization to invade extreme but relatively rare environments. A particularly interesting case is the modification of folates in archaea, leading from THF in ancestral nodes to tetrahydromethanopterin in the methanogens, which enables initial fixation of formate (formed by hydrogenation of CO₂) in an ATP-free system [22, 93]. The root position of rTCA explains the preservation of rTCA arcs both in reductive acetyl-CoA pathways, and in anaplerotic appendages to other fixation pathways, and the root position of direct C₁ reduction explains its near-universal distribution.

A tree is by construction a summary statistic for the relations among the phenotypes which are its leaves or internal nodes. It is not inherently a map of species descent, and takes on that interpretation only when common ancestry is shown to explain the conditional independence of branches given their (topological) parent nodes. This caution is especially important for the interpretation of figure 11, which shows high parsimony in the deepest branches where horizontal gene transfer is generally believed to have been most intense [78, 79]. We have argued that this behavior is consistent in a tree of successive optimal adaptations to varied environments, by organisms that could maintain little persistent variation. Violations of parsimony that are improbable by evolutionary convergence contain information about contact among historically separated lineages. Under this interpretation the separation is primarily environmental, with the subsequent contact identifying ecological co-habitation. As explained above, the possible transfer of genes for the 3HP pathway is especially plausible, as the organisms involved may have shared the same extreme (alkaline) environments and been under common selection pressure, which when severe is known to accelerate rates of gene transfer [174, 187].

Our methods in [22] include flux-balance analysis of core networks, where the boundaries of analysis are defined to be carbon input solely from CO₂ and the output of the universal precursors we have listed as the interface between carbon fixation and anabolism, as shown in figure 1. We do not model cellular-level mechanisms of either regulation or heredity, nor full downstream intermediary metabolism. Our system of metabolic flux constraints therefore does not distinguish between individual species and ecosystems. It does not, of course, exclude the possibility of representing individual organisms. The general agreement with robust phylogenetic signatures from many different genomic phylogenies [73, 175, 188] may thus still suggest a dominant role for vertical descent among autotrophic organisms (and not merely consortia) in the early evolution of carbon fixation.

The joint WL/rTCA network was introduced into figure 11 to produce a connected tree containing only autotrophic nodes. It also gives the most parsimonious interpretation of the nearly universal distributions of both reductive C₁ chemistry and of citric acid cycle components, and receives further circumstantial support from the identifiability of both plausible and specific environmental driving forces for most subsequent branches. The constraints which jointly required the insertion of a linked WL/rTCA network at the root have led us to propose a kind of redundancy not found in extant fixation pathways. Either WL or rTCA alone is self-maintaining (in a modern organism) so a network that incorporates both is redundantly autocatalytic. This is an important and speculative departure from known phenotypes, but it can be argued to have conferred a selective advantage under the more primitive conditions of early cells, because the pathway topology itself possesses a form of inherent robustness. The redundant network topology of the root phenotype would have allowed it to better cope with both internal and external perturbation in an era when regulation and kinetic control were probably less sophisticated and refined than they are today. In that respect it is a more plausible phenotype for a universal ancestor than any modern network.

The enhanced robustness of the joint network follows from the interaction of short-loop and long-loop autocatalysis. The threshold for autocatalysis in the rTCA loop, fragile against parasitic side reactions or unconstrained anabolism, is supported and given a recovery mode when fed by an independent supply of acetyl-CoA from WL. In turn, the production of a sufficient concentration of folates to support direct C₁ reduction, fragile if the long biosynthetic pathway is unreliable, is augmented by additional carbon fixed in rTCA. These arguments are topological, and do not make specific reference to whether the catalysts for the underlying reactions are enzymes. They may provide context for (perhaps multi-stage) models of transition from primordial mineral catalysis [74, 189] to the eventual support of carbon fixation by biomolecules.

Figure 12 shows a numerical solution for the flux through a minimal version of the joint WL/rTCA network, with lumped-parameter representations of parasitic side reactions and the net free energy of formation of acetate. (The exact rate equations used, and their interpretation, are developed in appendix B.) In the absence of a WL 'feeder' pathway, rTCA has a sharp threshold for the maintenance of flux through the network as a function of the free energy of formation of its output acetate. The existence of such a sharp threshold depending on the rate of parasitism, below which the cycle supports no transport, has been one of the major sources of criticism of network-autocatalytic pathways as models for proto-metabolism [190]. When WL is added as a feeder, however, the threshold disappears, and some nonzero flux passes through the pathway at any negative free energy of formation of the outputs. The existence of a pathway that supports some organosynthetic flux at any positive driving chemical potential—dependent on external catalysts but not contingent on the pathway's own internal state—has been one of the major reasons WL has been favored as a protometabolic pathway by molecular biologists and chemists [33, 74, 77].

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The reason (beyond evidence from reconstructions) that we regard a linear pathway modeled on the modern acetyl-CoA pathways as an incomplete answer to the needs of incipient metabolism is that it offers an avenue for production of organics, but does not by itself offer a chemical mechanism for the kind of selection and concentration of fluxes that is equally central to the sparse network of extant core metabolism [191]. Chemical self-amplification, if it can be demonstrated experimentally, is the most plausible mechanism by which the biosphere can concentrate all energy flows and material cycles through a small, stable set of organic compounds. It supplies the molecules that are within the loop—and secondarily those that are made from loop intermediates—above the concentrations they would have in a Gibbs equilibrium distribution, as a result of flow through the network. The fact that self-amplification is permitted to act in the model of figure 12, even below the chemical-potential difference where the rTCA loop alone is self-sustaining, provides a mechanism by which the loop intermediates could have been supplied in excess in the earliest stages of the emergence of metabolism. We return in section 6 to a related form of robustness and selection, which applies as anabolic pathways begin to form from loop intermediates.

A surprising observation suggested by our reconstructed history is how conservative the biosphere has been in its intermediate stages of innovation, as a consequence of geochemical niche diversity on earth. Except for the root node, we have not needed to invoke extinct ancestral forms to explain extant diversity, an argument that even Darwin [192] expected to be required frequently for cases where modern, optimized forms outcompeted their more primitive ancestors and erased direct evidence about the past. The one case where we do invoke an essential extinct ancestral form is the root node, and its character suggests reasons why it should have become extinct that are more chemically basic and biochemically consequential than the secondary physiological or ecological distinctions that modern evolutionists use to explain extinct ancestral forms. The topological robustness of the root node comes at the combined costs of sub-optimal energy efficiency and oxidant sensitivity. The fitness advantage to shedding either of these costs would have increased significantly as organisms obtained more sophisticated macromolecular components and correspondingly greater control over their internal chemistry, lessening (and ultimately removing) the selective advantage of a redundant carbon-fixation strategy even in the absence of external biogeochemical perturbations.

Without the ability to culture and analyze a population of LUCA organisms, the amount we can conclude from mathematical analyses of general network properties is of course less, but it is still within the range commonly used to assess proposals for early metabolism. For example: proposals for autocatalysis in geochemical networks with crude catalysts are routinely criticized on the basis of their shared topological feature of feedback and its associated threshold fragility [190]. These criticisms emphasize that parasitic side-reactions are a likely problem, although the corollary that autocatalysis is generally ruled out requires the strong claim that side-reactions are a problem in all plausible environments, which extends well beyond current experimental knowledge. In the opposite direction (and in this case based on particular and well-understood side-reactions), it is argued for the formose network [160] that without some severe pruning mechanism, the reactions are too productive, creating mixtures too complicated to be relevant to biochemistry. General arguments of both kinds contribute to a negative hypothesis behind the hope [193] that catalytic RNA will be a sufficient solution to both problems of productivity and selection. But the larger points in both arguments are important, and should be applied at many other places in hypotheses about the emergence and early elaboration of metabolism. They emphasize that both robustness and selectivity are needed features of any mechanism responsible for the earliest cellular organosynthesis. At the same level of generality as they are raised as criticisms, these criteria create meaningful distinctions between topologies for early carbon-fixation pathways, to which the lumped-parameter model of our root node gives a quantative form.

The biosynthesis of cofactors involves some of the most elaborate and least understood organic chemistry used by organisms. The pathways leading to several major cofactors have only very recently been elucidated or remain to be fully described, and their study continues to lead to the discovery of novel reaction mechanisms and enzymes that are unique to cofactor synthesis [203–205]. While cofactor biosynthetic pathways often branch from core metabolic pathways, their novel reactions may produce special bonds and molecular structures not found elsewhere in metabolism. These novel bonds and structures are generally central in their catalytic functions.

Structurally, many cofactors form a class in transition between the core metabolites and the oligomers. They contain some of the largest directly-assembled organic monomers (pterins, flavins, thiamin, tetrapyrroles), but many also show the beginnings of polymerization of standard amino acids, lipids or ribonucleotides. These may be joined by the same phosphate ester bonds that link RNA oligomers or aminoacyl-tRNA, or they may use distinctive bonds (e.g. 5'–5' esters) found only in the cofactor class [206].

The polymerization exhibited within cofactors is distinguished from that of oligomers by its heterogeneity. Srinivasan and Morowitz [51] have termed cofactors 'chimeromers', because they often include monomeric components from several molecule classes. Examples are CoA, which includes several peptide units and an ATP; folates, which join a pterin moiety to para-aminobenzoic acid (PABA); quinones, which join a PABA derivative to an isoprene lipid tail; and a variety of cofactors assembled on phosphoribosyl-pyrophosphate (PRPP) to which RNA 'handles' are esterified.

We may understand the border between small and large molecules, where most cofactors are found, as more fundamentally a border between the use of heterogeneous organic chemistry to encode biological information in covalent structures, and the transition to homogeneous phosphate chemistry, with information carried in sequences or higher-order non-covalent structures. The chemistry of the metabolic substrate is mostly the chemistry of organic reactions. Phosphates and thioesters may appear in intermediates, but their role generally is to provide energy for leaving groups, enabling formation of the main structural bonds among C, N, O and H. One of the striking characteristic scales in metabolism is that its organic reactions, the near-universal mode of construction for molecules of 20 to 30 carbons or less, cease to be used in the synthesis of larger molecules. Even siderophores, among the most complex of widely-used organic compounds, are often elaborations of functional centers that are small core metabolites, such as CIT [207, 208]. Large oligomeric macromolecules are almost entirely synthesized using the dehydration potential of phosphates [209] to link monomers drawn from the inventory [50] of small core metabolites. Many cofactors have structure of both kinds, and they are the smallest molecules that as a class commonly use phosphate esters as permanent structural elements [210].

Finally, cofactors are distinguished by structure–function relations determined mostly at the single-molecule scale. The monomers that are incorporated into macromolecules are often distinguished by general properties, and only take on more specific functional roles that depend strongly on location and context [211, 212]. In contrast, the functions of cofactors are specific, often finely tuned by evolution [93], and deployable in a wide range of macromolecular contexts. Usually they are carriers or transfer agents of functional groups or reductants in intermediary metabolism [213]. Nearly half of enzymes require cofactors as coenzymes [210, 213]. If we extend this grouping to include chelated metals [214, 215] and clusters, ranging from common iron–sulfur centers to the elaborate metal centers of gas-handling enzymes [104, 150], more than half of enzymes require coenzymes or metals in the active site.

The universal reactions of intermediary metabolism depend on only about 30 cofactors [213] (though this number depends on the specific definition used). Major functional roles include (1) transition-metal-mediated redox reactions (heme, cobalamin, the nickel tetrapyrrole F₄₃₀, chlorophylls⁶), (2) transport of one-carbon groups that range in redox state from oxidized (biotin for carboxyl groups, methanofurans for formyl groups) to reduced (lipoic acid for methylene groups, SAM, coenzyme-M and cobalamin for methyl groups), with some cofactors spanning this range and mediating interconversion of oxidation states (the folate family interconverting formyl to methyl groups), (3) transport of amino groups (pyridoxal phosphate, glutamate, glutamine), (4) reductants (nicotinamide cofactors, flavins, deazaflavins, lipoic acid and coenzyme-B), (5) membrane electron transport and temporary storage (quinones), (6) transport of more complex units such as acyl and amino-acyl groups (pantetheine in CoA and in the acyl-carrier protein (ACP), lipoic acid, thiamine pyrophosphate), (7) transport of dehydration potential from phosphate esters (nucleoside di- and tri-phosphates), and (8) sources of thioester bonds for substrate-level phosphorylation and other reactions (pantetheine in CoA).

Cofactors fill roles in network or molecular catalysis below the level of enzymes, but they share with all catalysts the property that they are not consumed by participating in reactions, and therefore are key loci of control over metabolism. Cofactors as transfer agents are essential to completing many network-catalytic loops. In association with enzymes, they can create channels and active sites, and thus they facilitate molecular catalysis. An example of the creation of channels by cofactors is given by the function of cobalamin as a C₁ transfer agent to the nickel reaction center in the acetyl-CoA synthase from a corrinoid iron–sulfur protein [216–218]. An example of cofactor incorporation in active sites is the role of TPP as the reaction center in the pyruvate-ferredoxin oxidoreductase (PFOR), which lies at the end of a long electron-transport channel formed by Fe–S clusters [108]. Through the limits in their own functions or in the functional groups they transport through networks, they may impose constraints on chemical diversity or create bottlenecks to evolutionary innovation. The previous sections have shown that many module boundaries in carbon fixation and core metabolism are defined by idiosyncratic reactions, and we have noted that many of these idiosyncrasies are associated with specific cofactor functions.

Cofactors, as topological hubs, and participants in reactions at high-flux boundaries in core and intermediary metabolism, are focal points of natural selection. The adaptations available to key atoms and bonds include altering charge or pKa, changing energy level spacing through non-local electron transport, or altering orbital geometry through ring strains. Divergences in low-level cofactor chemistry may alter the distribution of functional groups and thereby change the global topology of metabolic networks, and some of these changes map onto deep lineage divergences in the tree of life. A well-understood example is the repartitioning of C₁ flux from methanopterins versus folates [22, 93]. The same adaptation that enables formylation of methanopterins within an exclusively thioester system, where the homologous folate reaction requires ATP, reduces the potential for methylene-group transfer, and necessitates the oxidative formation of serine from 3PG in methanogens, which is not required of acetogens.

Most research on the origin of life has focused either on the metabolic substrate [6, 219] or catalysis by RNA [193], but we believe the priority of cofactors deserves (and is beginning to receive) greater consideration [89, 220]. In the expansion of metabolic substrates from inorganic inputs, the pathways to produce even such complex cofactors as folates et alia are comparable in position and complexity to those for purine RNA, while some for functional groups such as nicotinamide [89] or chorismate are considerably simpler. Therefore, even though it is not known what catalytic support or memory mechanisms enabled the initial elaboration of metabolism, any solutions to this problem should also support the early emergence of at least the major redox and C- and N-transfer cofactors. Conversely, the pervasive dependence of biosynthetic reactions on cofactor intermediates makes the expansion of protometabolic networks most plausible if it was supported by contemporaneous emergence and elaboration of cofactor groups. In this interpretation cofactors occupy an intermediate position in chemistry and complexity, between the small-metabolite and oligomer levels [89]. They were the transitional phase when the reaction mechanisms of core metabolism came under selection and control of organic as opposed to mineral-based chemistry, and they provided the structured foundation from which the oligomer world grew.

We argue next that a few properties of the elements have governed both functional diversification and evolutionary optimization of many cofactors, especially those associated with core carbon fixation. We focus on heterocycles with conjugated double bonds incorporating nitrogen, and on the groups of functions that exploit special properties of bonds to sulfur atoms. The recruitment of elements or special small-molecule contexts constitutes an additional distinct form of modularity within metabolism. Like the substrate network, cofactor groups often share or re-use synthetic reaction sequences. However, unlike the small-molecule network, cofactors can also be grouped by criteria of catalytic similarity that are independent of pathway recapitulation. For example, alkyl-thiol cofactors, which comprise diverse groups of molecules, all make essential use of distinctive properties of the sulfur bond to carbon, which appear nowhere else in biochemistry. As an example involving elements in specific contexts, a large group of cofactors employing C-N heterocycles all arise from a single sub-network whose reactions are catalyzed by related enzymes, and the transport and catalytic functions performed by the heterocycles are distinctive of this cofactor group.

Members of the folate family carry C₁ groups bound to either the N⁵ nitrogen of a heterocycle derived from GTP, an exocyclic N¹⁰ nitrogen derived from a PABA, or both. The two most common folates are THF, ubiquitous in bacteria and common in many archaeal groups, and tetrahydromethanopterin (H₄MPT), essential for methanogens and found in a small number of late-branching bacterial clades. Other members of this family are exclusive to the archaeal domain and are structural intermediates between THF and H₄MPT. Two kinds of structural variation are found among folates, as shown in figure 15. First, only THF retains the carbonyl group of PABA, which shifts electron density away from N¹⁰ via the benzene ring and lowers its pKa relative to N⁵ of the heterocycle. All other members of the family lack this carbonyl. Second, all folates besides THF incorporate one or two methyl groups that impede rotation between the pteridine and aryl-amine planes, changing the relative entropies of formation among different binding states for the attached C₁ [22, 93, 221].

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Folates mediate a diverse array of C₁ chemistry, various parts of which are essential in the biosynthesis of all organisms [93]. The collection of reactions, summarized in figure 5, has been termed the 'central superhighway' of one-carbon metabolism. Functional groups supplied by folate chemistry, connected by interconversion of C₁-oxidation states along the superhighway, include (1) formyl groups for synthesis of purines, formyl-tRNA, and formylation of methionine (fMet) during translation, (2) methylene groups to form thymidilate, which are also used in many deep-branching organisms to synthesize glycine and serine, forming the ancestral pathway to these amino acids [22], and (3) methyl groups which may be transferred to SAM as a general methyl donor in anabolism, to the acetyl-CoA synthase to form acetyl-CoA in the WL pathway, or to coenzyme-M where the conversion to methane is the last step in the energy system of methanogenesis.

The variations among folates, shown in figure 15, leave the charge, pKa and resulting C–N bond energy at N⁵ roughly unaffected, while the N¹⁰ charge, pKa and C–N bond energy change significantly across the family. This charge effect, together with entropic effects due to steric hindrance from methyl groups, can sharply vary the functional roles that different folates play in anabolism.

The biggest difference lies between THF and H₄MPT. In THF, the N¹⁰ pKa is as much as 6.0 natural-log units lower than that of N⁵ [222]. The resulting higher-energy C-N bond cannot be formed without hydrolysis of one ATP, either to bind formate to N¹⁰ of THF, or to cyclize N⁵-formyl-THF to form N⁵,N¹⁰-methenyl-THF (see figure 5). This latter reaction is the mirror image of the cyclization of N¹⁰-formyl-THF, and as we will argue below, a plausibly conserved evolutionary intermediate in the attachment of formate onto folates. After further reduction, the resulting methylene is readily transferred to lipoic acid to form glycine and serine, in what we have termed the 'glycine cycle' [22] (the lipoyl-protein based cycle on the right in figure 5).

In contrast, in H₄MPT the difference in pKa between N¹⁰ and N⁵ is only 2.4 natural-log units. The lower C–N¹⁰ bond energy permits spontaneous cyclization of N⁵-formyl-H₄MPT, following (also ATP-independent) transfer of formate from a formyl-methanofuran cofactor. Through this sequence, methanogens fix formate in an ATP-independent system using only redox chemistry. The initial free energy to attach formate to methanofuran is provided by the terminal methane released in methanogenesis (the Co-M/Co-B cycle in figure 5). The resulting downstream methylene group, however, has too little energy as a leaving group to transfer to an alkyl-thiol cofactor, so methanogens sacrifice the ability to form glycine and serine by direct reduction of formate.

The reconstructed ancestral use of the 7–9 reactions in figure 5 is to reduce formate to acetyl-CoA or methane. However, the reversibility of many reactions in the sequence, possibly requiring substitution of reductant/oxidant cofactors, allows folates to accept and donate C₁ groups in a variety of oxidation states, from and into many pathways including salvage pathways. Methylotrophic proteobacteria which have obtained H₄MPT through horizontal gene transfer [195, 196] may run the full reaction sequence in reverse. They may use either H₄MPT to oxidize formaldehyde or THF to oxidize various methylated C₁ compounds, in both cases leading to formate, or other intermediary oxidation states (from THF) as inputs to anabolic pathways. In many late-branching bacteria, some archaea and eukaryotes, the THF based pathway may run in part oxidatively and in part reductively, through connections to either gluconeogenesis/glycolysis or glyoxylate metabolism. In these organisms serine (derived through oxidation, amination and dephosporylation from 3-phosphoglycerate) or glycine (derived through amination of glyoxylate) become the sources of transferable methyl groups in anabolism. This versatility has preserved the folate pathway as an essential module of biosynthesis in all domains of life, and at the same time has made it a pivot of evolutionary variation.

The structural and functional variation within the folate family illustrates the way that selection, acting on cofactors, can create large-scale re-arrangements in metabolism, enabling adaptations that are reflected in lineage divergences. The free-energy cascade described in the last section, linking ATP hydrolysis, the charge and pKa of the N¹⁰ nitrogen, and the leaving-group activity of the resulting bound carbon for transfer to alkyl-thiol cofactors or other anabolic pathways, is a fundamental long-range constraint of folate-C₁ chemistry. A comparative analysis of gene profiles in pathways for glycine and serine synthesis, explained in [22], shows that while the constraint cannot be overcome, its impact on the form of metabolism can vary widely depending on the structure of the mediating folate cofactor.

The annotated role for ATP hydrolysis in WL autotrophs is to attach formate to N¹⁰ of THF, initiating the reduction sequence. However, many deep-branching bacteria and archaea show no gene for this reaction, while multiple lines of evidence indicate that THF nonetheless functions as a carbon-fixation cofactor in these organisms [22]. In almost all cases where an ATP-dependent N¹⁰-formyl-THF synthase is absent, an ATP-dependent N⁵-formyl-THF cycloligase [223, 224] is found. This is another case where a broad evolutionary context allows an alternate interpretation. N⁵-formyl-THF cycloligase was originally discovered in mammalian systems, where its function has been highly uncertain and hypothesized to be the salvage mechanism as part of a futile cycle [223, 224], before being found to be widespread across the tree of life [22]. If we deduce by reconstruction, however, that ancestral folate chemistry operated in the fully reductive direction, and that in H₄MPT systems formate is attached at the N⁵ position, while in THF systems formate is attached at the N¹⁰ position, the widespread distribution of the cycloligase takes on a different possible meaning. It is plausible that the N⁵-formyl-THF cycloligase allows a formate incorporation pathway that is an evolutionary intermediate between the commonly recognized pathway using THF and its evolutionary derivative using H₄MPT (see figure 5). The ATP-dependent cycloligase produces N⁵,N¹⁰-methenyl-THF from N⁵-formyl-THF, which may potentially form spontaneously due to the higher N⁵-pKa [224]. ATP hydrolysis is thus specifically linked to the N¹⁰-carbon bond, which is the primary donor for carbon groups from folates. Methanogens, in contrast, escape the dependence on ATP hydrolysis by decarboxylating PABA before it is linked to pteridine to form methanopterin (see figure 15), but they sacrifice methyl-group donation from H₄MPT to most anabolic pathways, making methanogenesis viable only in clades that evolved the oxidative pathway to serine from 3-phosphoglycerate.

We noted in section 3.4 that the elimination of one ATP-dependent acyl-CoA synthase in acetogens reduces the free energy cost of carbon fixation relative to rTCA cycling. The decoupling of the formate-fixation step on methanopterins from ATP hydrolysis is a further significant innovation, lowering the ATP cost for uptake of CO₂. This divergence of H₄MPT from THF, and a related divergence of deazaflavins from flavins (see figure 16), follow phylogenetically (and we believe, were responsible for) the divergence of the methanogens from other euryarcheota [22].

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We regard this example as representative of the way that innovations in cofactor chemistry more generally mediated large-scale rearrangements in metabolism, and corresponding evolutionary (and ecological) divergences of clades. Another similar example comes from the quinones, a diverse family of cofactors mediating membrane electron transport [225]. [114] found that the synthetic divergence of mena- and ubiquinone follows the pattern of phylogenetic diversification within proteobacteria. δ- and -proteobacteria use menaquinone, γ-proteobacteria use both mena- and ubiquinone, and α- and β-proteobacteria use only ubiquinone. Because mena- and ubiquinone have different midpoint potentials, it was suggested that their distribution reflects changes in environmental redox state as the proteobacteria diversified during the rise of oxygen [114, 226].

An interpretive frame for many of these observations is the proposal that metabolism is an outgrowth of geochemistry [41, 74, 149], which came under the control of living organisms [58] (see section 7 for dedicated discussion). If we wish to judge this proposal, then it is informative to look for parallels and differences between biochemical and plausible geochemical reaction sequences. The distinctive features of biochemical C₁ reduction are the attachment of formate to tuned heterocyclic or aryl-amine nitrogen atoms for reduction, and the transfer of reduced C₁ groups to sulfhydryl groups (of SAM, lipoic acid or CoM). In the mineral-origin hypothesis for direct reduction, the C₁ were adsorbed at metals and either reduced through crystal oxidation [227] or by reductant in solution. The transfer of reduced C₁ groups to alkyl-thiol cofactors may show continuity with reduction on metal-sulfide minerals. However, the mediation of reduction by nitrogens appears to be a distinctively biochemical innovation.

The common reaction mechanism unifying the purine-derived cofactors is an initial hydrolysis of both purine and ribose rings performed by cyclohydrolases assigned EC numbers 3.5.4 (see figure 14). All cyclohydrolases within this EC family are used for biosynthesis or conversions within this class of molecules. They are responsible for the synthesis of inosine-monophosphate (IMP, precursor to AMP and GMP) from 5-formamidoimidazole-4-carboxamide ribonucleotide (FAICAR), for the first committed steps in the syntheses of both folates and flavins from GTP, and for the initial ring-opening step in the synthesis of histidine from ATP and PRPP. Figure 14 shows the key steps in the network synthesizing both purines and the pterins, folates, flavins, thiamin and histidine.

The common function of the 3.5.4 cyclohydrolases is hydrolysis of rings on adjacent nucleobase and ribose groups, or the formation of cycles by ligation of ring fragments. In all cases, the ribosyl moieties come from PRPP. In the synthesis of pterins from GTP and of histidinol from ATP, both a nucleobase cycle and a ribose are cleaved. In pterin synthesis, the imidazole of guanine and the purine ribose are cleaved. In histidine synthesis, the six-membered ring of adenine is cleaved (at a different bond than the one synthesized from FAICAR), and the ribose comes from a secondary PRPP.

By far the most complex synthesis in this family is that of thiamin from aminoimidazole ribonucleotide (AIR). This sequence begins with an elaborate molecular rearrangement, performed in a single step by the enzyme ThiC [205]. (Eukaryotes use an entirely different pathway, in which the pyrimidine is synthesized from histidine and pyridoxal-5-phosphate [228].) While the ThiC enzyme is unclassified, and its reaction mechanism incompletely understood, it shares apparent characteristics with members of the 3.5.4 cyclohydrolases. As in the first committed steps in the synthesis of folates and flavins from GTP, both a ribose ring and a 5-member heterocycle are cleaved and subsequently (as in folate synthesis) recombined into a 6-member heterocycle. The complexity of this enzymatic mechanism makes a pre-enzymatic homologue to ThiC difficult to imagine, and suggests that thiamin is both of later origin, and more highly derived, than other cofactors in this family. This derived status is supported by the fact that the resulting functional role of thiamin is not performed on the pyrimidine ring itself, but rather on the thiazole ring to which it is attached, and which is likewise created in an elaborate synthetic sequence [205].

Figure 16 shows the detailed substrate re-arrangement in the sub-network leading from GTP to methanopterins, folates, riboflavin and the archaeal deazaflavin F₄₂₀. In the pterin branch, both rings of neopterin are synthesized directly from GTP, and an aryl-amine originating in PABA provides the second essential nitrogen atom. PABA is either used directly (in folates) or decarboxylated with attachment of a PRPP (in methanopterins) to vary the pKa of the amine. In contrast, the flavin branch is characterized by the integration of either ribulose (in riboflavin) or chorismate (in F₄₂₀) to form the internal rings. Two 6,7-dimethyl-8-(D-ribityl)lumazine are condensed to form riboflavin, whereas a single GTP with chorismate forms F₄₂₀.

The cyclohydrolase reactions can be considered the key innovation enabling the biosynthesis of this whole family of cofactors and, importantly, of purine RNA itself. The heterocycles that are formed or cleaved by these reactions provide the central structural components of the active parts of the final cofactor molecules. In this sense, except for TPP, the distinctions among purine-derived cofactors can be considered secondary modifications on a background structured by PRPP and C–N heterocycles. If we consider sub-networks of metabolism as producing key structural or functional components, in this case for the synthesis of cofactors, then this family draws on only two such developed sub-networks. The first of these is purine synthesis and the other is synthesis of chorismate, the precursor to PABA and the unique source of single benzene rings in biochemistry [229]. Flexibility in the ways that chorismate is modified to control electron density, and the way the benzene ring is combined with other heterocycles, contributes to the combinatorial elaboration within the family.

The following observations suggest to us the possibility that most of the purine-derived cofactors (perhaps excepting thiamin) were available contemporaneously with monomer purine RNA.

For some reactions, the abstraction of enzyme mechanisms is advanced enough to identify small-molecule organocatalysts that could have provided similar functions [230, 231]. The current understanding of cyclohydrolase proteins, however, does not suggest other simpler mechanisms by which similar reactions might first have been catalyzed, leaving us almost wholly uncertain about how RNA was first formed. Unless non-enzymatic mechanisms are discovered which are both plausible and selective, our previous arguments about the permissiveness of crude catalysts lead us to expect that, at whatever stage catalysts capable of interconverting AIR, AICAR, FAICAR and IMP first became available, pteridines would have been formed contemporaneously and possibly played a role in the elaboration of the metabolic network. (See section 5 for further discussion on promiscuous versus selective enzymes.) If the chorismate pathway (which begins in the sugar-phosphate network) had also arisen by that stage, the same arguments suggest that folates and flavins may also have been available. In this supposition we are treating the first three EC numbers as an appropriate guide to reaction mechanism without restriction of the molecular substrate. Whether the first RNA were produced in this way, or through structurally very dissimilar stages, is a currently active question [232].

As in our discussion of the root node in section 3.4.4, we consider it important to apply ubiquitously the premise that enabling network throughput and pruning network diversity were concurrent ongoing requirements in the co-evolution of substrate reactions and their catalysts. Most often [160, 190, 233], the inability to prune networks is recognized as a problem for the early formation of order. In the case of the purine-derived cofactors, it may offer both clues to help explain the structure of the biosynthetic network, and a way to break down the problem of early metabolic evolution into simpler steps with intermediate criteria for selection.

The patterns that characterize current metabolism as a recursive network expansion [200, 201] about inorganic inputs are most easily understood as a reflection of the organic-chemical possibilities opened by cofactors. Pterins, as donors of activated formyl groups, support (among other reactions) the synthesis of purines, forming a short autocatalytic loop. Similarly, flavins would have augmented redox reactions. Finally, it has long been recognized that acid/base catalysis is uniquely served by histidine, which has a pKa ≈ 6.5 on the ε-nitrogen, a property not found among any biological ribonucleotides (though possible for some substituted adenine derivatives) [234].

Within the class of GTP-derived cofactors, a sub-structure may perhaps be suggested: the dimer condensation that forms riboflavin is a hierarchical use of building blocks formed from GTP. Although simple and consisting of a single key reaction, this could reflect a later stage of refinement. It is recognized [235] that flavins are somewhat specialized reductants, both biosynthetically and functionally more specific than the much simpler nicotinamide cofactors, which plausibly preceded them [89].

4.2.6. Purine-derived cofactors selected before RNA itself, as opposed to having descended from an RNA world defined through base pairing? The overlap between RNA and cofactor biosynthesis, and the incorporation of AMP in several cofactors (where is serves primarily as a 'handle' for docking), has been noticed and given the interpretation that cofactors are a degenerated relic of an oligomer RNA world [210]. While monomer RNA is of comparable complexity to small-molecule cofactors, oligomer RNA is significantly more complex. The only significant logical motivation to place oligomer RNA prior to small-molecule cofactors, is therefore the premise that RNA base pairing and replication is the least-complex plausible mechanism supporting (specifically, Darwinian) selection and persistence of catalysts that are hypothesized to have been required for the elaboration of biosynthesis.

This is still a complex premise, however, as it requires not only organosynthesis of oligomer RNA, but also chiral selection and mechanisms to enable base pairing and (presumably template-directed) ligation [236]. A particular problem for RNA replication is the steric restriction to 3'-5' phosphate esters, over the kinetically favored 2'-5' linkage. In comparison, small-molecule catalysis by either RNA [237] or related cofactors may be considered in any context that supports their synthesis⁷. If chemical mechanisms are found which support structured organosynthesis and selection—a requirement for any metabolism-first theory of the origin of life—the default premise may favor simplicity: that heterocycles were first selected as cofactors, and that purine RNA, only one among many species maintained by the same generalized reactions, was subsequently selected for chirality, base-pairing and ligation.

Transfer of methyl or methylene groups. The S atoms of CoM, lipoic acid, and S-adenosyl-homocysteine accept methyl or methylene groups from the nitrogen atoms of pterins. Considering that transition-metal sulfide minerals are the favored substrates for prebiotic direct-C₁ reduction [147, 149, 246], a question of particular interest is how, in mineral scenarios for the emergence of carbon fixation, the distinctive relation between tuned nitrogen atoms in pterins as carbon carriers, and alkyl-thiol compounds as carbon acceptors, would have formed.

Reductants and co-reductants. CoB and CoM act together as methyl carrier and reductant to form methane in methanogenesis. In this complex transfer [150], the fully-reduced (Ni⁺) state of the nickel tetrapyrrole F₄₃₀ forms a dative bond to –CH₃ displacing the CoM carrier, effectively re-oxidizing F₄₃₀ to Ni^{3 +}. Reduced F₄₃₀ is regenerated through two sequential single-electron transfers. The first, from CoM–SH, generates a Ni^{2 +} state that releases methane, while forming a radical CoB^·–S–S–CoM intermediate with CoB. The radical then donates the second electron, restoring Ni⁺. The strongly oxidizing heterodisulfide CoB–S–S–CoM is subsequently reduced with two NADH, regenerating CoM–SH and CoB–SH.

A similar role as methylene carrier and reductant is performed by the two SH groups in lipoic acid. CoM is specific to methanogenic archaea [247], while lipoic acid and S-adenosyl-homocysteine are found in all three domains [22, 248]. Lipoic acid is formed from octanoyl-CoA, emerging from the biotin-dependent malonate pathway to fatty acid synthesis, and along with fatty acid synthesis [109], may have been present in the universal common ancestor. The previously noted universal distribution of the glycine cycle supports this hypothesis.

Role in the reversal of citric-acid cycling. Lipoic acid becomes the electron acceptor in the oxidative decarboxylation of α-ketoglutarate and pyruvate in the oxidative Krebs cycle, replacing the role taken by reduced ferredoxin in the rTCA cycle. Thus the prior availability of lipoic acid was an enabling precondition for reversal of the cycle in response to the rise of oxygen.

Carriers of acyl groups. Transport of acyl groups in the ACP proceeds through thioesterification with pantetheine phosphate, similar to the thioesterification in fixation pathways. In fatty acid biosynthesis acyl groups are further processed while attached to the pantetheine phosphate prosthetic group.

Electron bifurcation. The heterodisulfide bond of CoB–S–S–CoM has a high midpoint potential (E'₀ = −140 mV), relative to the H⁺/H₂ couple (E'₀ = −414 mV), and its reduction is the source of free energy for the endergonic production of reduced ferredoxin (Fd^{2 −}, E'₀ in situ unknown but between −520 mV and −414 mV) [249], which in turn powers the initial uptake of CO₂ on H₄MPT in methanogens. The remarkable direct coupling of exergonic and endergonic redox reactions through splitting of binding pairs into pairs of radicals, which are then directed to paired high-potential/low-potential acceptors, is known as electron bifurcation [140]. Variant forms of bifurcation are coming to be recognized as a widely-used strategy of metal-center enzymes, either consuming oxidants as energy sources to generate uniquely biotic low-potential reductants such as Fd^{2 −} [249, 250–252], or to 'titrate' redox potential to minimize dissipation and achieve reversibility of redox reactions involving reductants at diverse potentials, e.g. by combining low-potential (Fd^{2 −}, E'₀ = −420 mV) and high-potential (NADH, E'₀ = −300 mV) reductants to produce intermediate-potential reductants (NADPH, E'₀ = −360 mV) [253]. Together with substrate-level phosphorylation (SLP), electron bifurcation may be the principal chemical mechanism (contrasted with membrane-mediated oxidative phosphorylation) for interconverting biological energy currencies, and along with SLP [241], a mechanism of central importance in the origin of metabolism [254]. Small metabolites including such heterodisulfides of cofactors, which can form radical intermediates exchanging single electrons with Fe–S clusters (typically via flavins) are essential sources and repositories of free energy in pathways using bifurcation. Both electron bifurcation and the stepwise reduction of F₄₃₀ (above) illustrate the central role of metals as mediators of single-electron transfer processes in metabolism.

The similarity between the glycine cycle and methanogenesis in figure 5 emphasizes the convergent roles of alkyl-thiol cofactors. In the glycine cycle, methylene groups are accepted by the terminal sulfur on lipoic acid, and the subadjacent SH serves as reductant when glycine is produced, leaving a disulfide bond in lipoic acid. The disulfide bond is subsequently reduced with NADH. In methanogenesis, a methyl group from H₄MPT is transferred to CoM, with the subsequent transfer to F₄₃₀, and the release from F₄₃₀ as methane in the methyl-CoM reductase, coupled to formation of CoB–S–S–CoM. The heterodisulfide is again reduced with NADH, but employs a pair of electron bifurcations to retain the excess free energy in the production of Fd^{2 −} rather than dissipating it as heat [249]. Methanogenesis is thus associated with seven distinctive cofactors beyond even the set known to have diversified functions within the archaea [5], again suggesting the derived and highly optimized nature of this Euryarchaeal phenotype. The striking similarity of these two methyl-transfer systems, mediated by independently evolved and structurally quite different cofactors, suggests evolutionary convergence driven specifically by properties of alkyl thiols.

A curious pattern, which we note but do not attempt to interpret, is the association of non-sulfur, nitrogen-heterocycle cofactors with WL carbon fixation, contrasted with the use of sulfur-containing heterocycles in carboxylation reactions of the rTCA cycle. The non-sulfur cofactors THF and H₄MPT are used in the reactions of the WL pathway, while the biosynthetically-related but sulfur-containing cofactor thiamin mediates the carbonyl insertion (at a thioester) in rTCA [108, 255]. Biotin—which has been generally associated with malonate synthesis in the fatty-acid pathway (and derivatives such as propionate carboxylation to methyl-malonate in 3HP [109])—mediates the subsequent β-carboxylation of pyruvate and of α-ketoglutarate [110, 256, 257]. Thus the two cofactors we have identified as using sulfur indirectly to tune properties of carbon or nitrogen C₁-bonding atoms mediate the two chemically quite different sequential carboxylations in rTCA.

Biochemical subsystems driven, respectively, by redox potential or phosphoanhydride-bond dehydration potential, cannot usually be directly coupled to one another due to lack of 'transducer' reactions that draw on both energy systems. In addition to the ultimate physical constraint of limits to free energy, biochemistry also operates under additional proximate constraints from the chemical and quantum-mechanical substrates in which that free energy it is carried. The notable exception to the general lack of direct coupling between energy systems is the exchange of phosphate and sulfur groups in substrate-level phosphorylation [235] from thioesters (which may proceed in either direction depending on conditions). Although it provides a less flexible mode of coupling than membrane-mediated oxidative phosphorylation, this crucial reaction type, which occurs in some of the deepest reactions in biochemistry (those employing CoA, including all those in the six carbon fixation pathways), has been proposed as the earliest coupling of redox and phosphate [241], and the original source of phosphoanhydride potential [75] enabling pathways that require both reduction and dehydration reactions.

Phosphate concentration limits growth of many biological systems today, and phosphate concentrations appear to be even lower in vent fluids [278] than on average in the ocean, making it difficult to account for the emergence of many metabolic steps in hydrothermal vent scenarios for the origin of life. Serpentinization and other rock–water interactions that produce copious reductants—and are believed to have been broadly similar at least from the early archean to the present [74, 279]—also scavenge phosphates into mineral form. Unless new mechanisms are discovered that could have produced an increased amount of phosphate for early vents, it thus appears doubtful that phosphates were abundant in the environments otherwise most favorable to geochemical organosynthesis. What little phosphate is found in water is primarily orthophosphate, because the phosphoanhydride bond is unstable to hydrolysis. Therefore the retention of orthophosphate, and the continuous regeneration of pyrophosphate and polyphosphates [280–284], may have been essential to the spread of early life beyond relatively rare geochemical environments.

The membrane-bound ATP-synthetase, which couples phosphorylation to a variety of redox reactions [5] through proton or sodium pumping, is therefore essential in nearly all biosynthetic pathways, and must have been among the first functions of the integrated cell. Without a steady source of phosphate esters, none of the three oligomer families could exist. The ATP synthetase itself is homologous in all organisms, providing one strong argument (among many [109, 173]) for a membrane-bound last common ancestor. Proton-mediated phosphorylation (best known through oxidative phosphorylation in the respiratory chain [235]) requires a topologically enclosed space to function as a proton capacitor [274]. However, as shown by gram-negative bacteria [5] and their descendants mitochondria and plastids, which acidify the periplasmic space or thylakoid lumen, the proton capacitor need not be (and generally is not) the same compartment as the cytoplasm containing enzymatic reactions. Because the coupling of energy systems is a different function from regulating reaction rates catalytically, the phosphorylation system should not generally have been subject to the same set of evolutionary pressures and constraints as other cellular compartments, and need not have arisen at the same time. We note that, because it may have lower osmotic pressure than the cytoplasm, the acidified space required for proton-driven phosphorylation may not have required a cell wall, greatly simplifying the number of concurrent innovations required for compartmentalization, compared to those for the cytoplasm. Therefore we conjecture that proton-mediated phosphorylation could have been the first function leading to selection for lipid-bilayer compartmentalization, allowing other cellular functions to accrete at later times.

The second function of cellular compartments, and the one most emphasized in vesicle theories of the origin of life [6, 285, 286], is the enhancement of second-order reactions by collocation of catalysts and their substrates. Here we note another role that we have not seen mentioned, which is more closely related to the functions of the cell that inhibit reactions. Organisms employing autocatalytic-loop carbon fixation pathways must reliably limit their anabolic rates to avoid drawing off excess network catalysts into anabolism, resulting in passage below the autocatalytic threshold for self-maintenance, and collapse of carbon fixation and metabolism. Regulating anabolism to maintain viability and growth may have been an early function of cells.

We noted in section 3.4.4 the fragility of autocatalytic-loop pathways to parasitic side-reactions, and the way the addition of a linear pathway such as WL stabilizes loop autocatalysis in the root node of figure 11. For proto-metabolism, spontaneous abiotic side-reactions may be hazardous, if catalysts in the main fixation pathway do not sufficiently accelerate their reaction rates, creating a separation of timescales relative to the uncatalyzed background. Within the first cells, the same hazard is posed by secondary anabolism, as its reaction rates become enhanced by catalysts similar to those in the core. This fact was clearly noted already in [147]. It may thus be that the optimizations in either branch of the carbon-fixation tree were not possible until rates of anabolism were sufficiently well-regulated to protect supplies of loop intermediates or essential cofactors. Therefore, while the root node is plausible as a pre-cellular [147] or an early cellular (but non-optimized) form, either branch from it may have required the greater control afforded by quite refined cellular regulation of reaction rates. It is here that we envision a crucial role for feedback regulation at the genomic level [24, 25] as a support for the architectural stability of the underlying substrate network, prior even to its service in homeostasis in complex environments or in phenotypic plasticity.

The elements in a bipartite-graph representation of a chemical reaction or reaction network are defined as follows:

• Filled dots represent concentrations of chemical species. Each such dot is given a label indicating the species, such as
  
  used to refer to acetate in the text.
• Dashed lines represent transition states of reactions. Each is given a label indicating the reaction, as in .
• Hollow circles indicate inputs or outputs between molecular species and transition states, as in
  
  Each circle is associated with the complex of reactants or products of the associated reaction, indicated as labeled line stubs.
• Hollow circles are tied to molecular concentrations with solid lines ; one line per mole of reactant or product participating in the reaction. (That is, if m moles of a species A enter a reaction b, then m lines connect the dot corresponding to [A] to the hollow circle leading into reaction b. This choice uses graph elements to carry information about stoichiometry, as an alternative to labeling input- or output-lines to indicate numbers of moles.)
• Full reactions are defined when two hollow circles are connected by the appropriate transition state, as in
  
  describing the reductive carboxylation of acetate to form pyruvate.
• The bipartite graph for a fully specified reaction takes the form
  
  where labeled stubs are connected to filled circles by mole-lines. The bipartite-graph corresponds to the standard chemical notation for the same reaction as shown.

The mass-action kinetics for a graph such as the reductive carboxylation of acetate is given in terms of two half-reaction currents, which we may denote with the reaction label and an arbitrary sign as

$$\begin{eqnarray} j_b^{+} & = & k_b \left[ \mbox{ACE} \right] \left[ {\mbox{CO}}_2 \right] \left[ {\mbox{H}}_2 \right] \nonumber \\ j_b^{-} & = & {\bar{k}}_b \left[ \mbox{PYR} \right] \left[ {\mbox{H}}_2 \mbox{O} \right] . \end{eqnarray} \tag{ B.2 }$$

k_b and ${\bar{k}}_b$ denote the forward and reverse half-reaction rate constants. The total reaction current J_b ≡ j⁺_b − j⁻_b is related to the contribution of this reaction to the changes in concentration as

$$\begin{eqnarray} &&\dot{[ \mbox{ACE} ]} = \dot{[ {\mbox{CO}}_2 ]} = \dot{[ {\mbox{H}}_2 ]} = -J_b \nonumber \\ &&\dot{[ \mbox{PYR} ]} = \dot{[ {\mbox{H}}_2 \mbox{O} ]} = J_b , \end{eqnarray} \tag{ B.3 }$$

where the overdot denotes the time derivative. Reaction currents on graphs do not have inherent directions, reflecting the microscopic reversibility of reactions. All sources of irreversibility are to be made explicit in the chemical potentials that constitute the boundary conditions for reactions.

Each term in the mass-action rate equation may be identified with a specific graphical element in the bipartite representation. The half-reaction rate constants k_b, ${\bar{k}}_b$ are associated with the hollow circles, and the current J_b (which is the time-derivative of the coordinate giving the 'extent of the reaction') is associated with the transition-state dashed line. Concentrations, as noted, are associated with filled dots, and stoichiometric coefficients are associated with the multiplicities of solid lines.

The simplest reduction is removal of an intermediate chemical species that is the sole output to one reaction, and the sole input to another, in a linear chain. Examples in the TCA cycle include MAL and ISC, produced by reductions and consumed by dehydrations. They also include CIT itself, produced by the hydration of aconitate and consumed by retro-aldol cleavage.

For a single linear reaction as shown in figure B1, the mass-action law is

$$\begin{equation} [ \mbox{A} ] k_a - [ \mbox{B} ]{\bar{k}}_a = J_a , \end{equation} \tag{ B.4 }$$

and concentrations change as

$$\begin{eqnarray} \dot{[ \mbox{A} ]} & = & - J_a \nonumber \\ \dot{[ \mbox{B} ]} & = & J_a . \end{eqnarray} \tag{ B.5 }$$

The equilibrium constant for the reaction A → B is

$$\begin{equation} K_{ \mbox{A} \rightarrow \mbox{B} } = \frac{k_a}{{\bar{k}}_a} . \end{equation} \tag{ B.6 }$$

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For two such reactions in a chain, as shown in figure B2, the mass-action laws are

$$\begin{eqnarray} &&{[} \mbox{A} ] k_a - [ \mbox{X} ]{\bar{k}}_a = J_a \nonumber \\ &&{[} \mbox{X} ] k_b - [ \mbox{B} ]{\bar{k}}_b = J_b , \end{eqnarray} \tag{ B.7 }$$

and the conservation laws become

$$\begin{eqnarray} \dot{[ \mbox{A} ]} & = & - J_a \nonumber \\ \dot{[ \mbox{X} ]} & = & J_a - J_b \nonumber \\ \dot{[ \mbox{B} ]} & = & J_b . \end{eqnarray} \tag{ B.8 }$$

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Under the steady-state condition $\dot{[ \mbox{X} ]} = 0$, we wish to replace the equations (B.7,B.8) with a rate equation

$$\begin{equation} [ \mbox{A} ] k_{ab} - [ \mbox{B} ]{\bar{k}}_{ab} = J_{ab} \end{equation} \tag{ B.9 }$$

and a conservation law expressed in terms of J_a = J_ab = J_b. The rate constants in equation (B.9) are to be specified through a composition rule

$$\begin{equation} ( k_a , {\bar{k}}_a ) \circ ( k_b , {\bar{k}}_b ) = ( k_{ab} , {\bar{k}}_{ab} ) \end{equation} \tag{ B.10 }$$

derived from the graph rewrite. Removing [X] from the mass-action equations using $\dot{[ \mbox{X} ]} = 0$, we derive that the rate constants satisfying equation (B.9) are given by

$$\begin{eqnarray} k_{ab} & = & \frac{ k_a k_b }{ {\bar{k}}_a + k_b } \nonumber \\ {\bar{k}}_{ab} & = & \frac{ {\bar{k}}_a {\bar{k}}_b }{ {\bar{k}}_a + k_b } . \end{eqnarray} \tag{ B.11 }$$

The associated equilibrium constant correctly satisfies the relation

$$\begin{equation} \frac{k_{ab}}{{\bar{k}}_{ab}} = \frac{k_a}{{\bar{k}}_a} \frac{k_b}{{\bar{k}}_b} . \end{equation} \tag{ B.12 }$$

The composition rule (B.12) is associative, meaning that internal nodes may be removed from chains of reactions in any order, as shown in figure B3. All composition rules derived in the remainder of this appendix will be variants on the elementary rule (with additional buffered concentration variables added), so we demonstrate associativity for the base case as the foundation for other cases.

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From equation (B.12) for $( k_a , {\bar{k}}_a ) \circ ( k_b , {\bar{k}}_b )$, followed by the equivalent expressions for $( k_{ab} , {\bar{k}}_{ab} ) \circ ( k_c , {\bar{k}}_c )$, $( k_a , {\bar{k}}_a ) \circ ( k_{bc} , {\bar{k}}_{bc} )$, and$( k_b , {\bar{k}}_b ) \circ ( k_c , {\bar{k}}_c )$, we derive the sequence of reductions

$$\begin{eqnarray} k_{abc} & = & \frac{ k_{ab} k_c }{ {\bar{k}}_{ab} + k_c } \nonumber \\ & = & \frac{ k_a k_b k_c }{ {\bar{k}}_a {\bar{k}}_b + ( {\bar{k}}_a + k_b ) k_c } \nonumber \\ & = & \frac{ k_a k_b k_c }{ {\bar{k}}_a ( {\bar{k}}_b + k_c ) + k_b k_c } \nonumber \\ & = & \frac{ k_a k_{bc} }{ {\bar{k}}_a + k_{bc} } , \end{eqnarray} \tag{ B.13 }$$

and a similar equation follows for ${\bar{k}}_{abc}$. Thus we have

$$\begin{eqnarray} &&\fl [ ( k_a , {\bar{k}}_a ) \circ ( k_b , {\bar{k}}_b ) ] \circ ( k_c , {\bar{k}}_c ) = ( k_a , {\bar{k}}_a ) \circ [ ( k_b , {\bar{k}}_b ) \circ ( k_c , {\bar{k}}_c ) ] .\nonumber\\ \end{eqnarray} \tag{ B.14 }$$

Next we consider the elimination of an internal node [X] that is produced or consumed together with other products or reactants. Conservation $\dot{[ \mbox{X} ]} = 0$ implies relations among the currents of these other species as well. All remaining graph reductions that we will perform for the TCA cycle are of this kind. In some cases both the secondary product and reactant are the solvent (water), as in the aconitase reactions (repeated in TCA, 3HB, 4HB and bicycle pathways). In other cases they are reductants or inputs such as CO₂ that we consider buffered in the environment.

The pair of mass action equations we wish to reduce are¹²

$$\begin{eqnarray} {[} \mbox{A} ] k_a - [ \mbox{X} ] [ \mbox{C} ]{\bar{k}}_a & = & J_a \nonumber \\ {[} \mbox{X} ] [ \mbox{D} ] k_b - [ \mbox{B} ]{\bar{k}}_b & = & J_b , \end{eqnarray} \tag{ B.15 }$$

and the desired reduced form is

$$\begin{equation} [ \mbox{A} ] [ \mbox{D} ] k_{ab} - [ \mbox{C} ] [ \mbox{B} ]{\bar{k}}_{ab} = J_{ab} . \end{equation} \tag{ B.16 }$$

We first reduce equation (B.15) to the base case of the previous section, by absorbing the concentrations not to be removed into a pair of effective rate constants

$$\begin{eqnarray} {[} \mbox{A} ] k_a - [ \mbox{X} ] ( [ \mbox{C} ]{\bar{k}}_a ) & = & J_a \nonumber \\ {[} \mbox{X} ] ( [ \mbox{D} ] k_b ) - [ \mbox{B} ]{\bar{k}}_b & = & J_b . \end{eqnarray} \tag{ B.17 }$$

From these we derive a composition equation

$$\begin{equation} {[} \mbox{A} ]{\hat{k}}_{ab} - [ \mbox{B} ]{\bar{\hat{k}}}_{ab} = J_{ab} , \end{equation} \tag{ B.18 }$$

corresponding to the graph representation in figure B4. We may then define ${\hat{k}}_{ab}$ and ${\bar{\hat{k}}}_{ab}$ by the elementary composition rule (B.10)

$$\begin{equation} ( k_a , [ \mbox{C} ]{\bar{k}}_a ) \circ ( [ \mbox{D} ] k_b , {\bar{k}}_b ) = ( {\hat{k}}_{ab} , {\bar{\hat{k}}}_{ab} ) , \end{equation} \tag{ B.19 }$$

giving the transformation¹³

$$\begin{eqnarray} {\hat{k}}_{ab} & = & \frac{ k_a [ \mbox{D} ] k_b }{ [ \mbox{C} ]{\bar{k}}_a + [ \mbox{D} ] k_b } \nonumber \\ {\bar{\hat{k}}}_{ab} & = & \frac{ [ \mbox{C} ]{\bar{k}}_a {\bar{k}}_b }{ [ \mbox{C} ]{\bar{k}}_a + [ \mbox{D} ] k_b } . \end{eqnarray} \tag{ B.20 }$$

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Now removing the factors of [C] and [D] used to define the hatted rate constants,

$$\begin{eqnarray} {\hat{k}}_{ab} & = & [ \mbox{D} ] k_{ab} \nonumber \\ {\bar{\hat{k}}}_{ab} & = & [ \mbox{C} ]{\bar{k}}_{ab} , \end{eqnarray} \tag{ B.21 }$$

we obtain a direct expression for the composition rule in equation (B.18), of

$$\begin{eqnarray} k_{ab} & = & \frac{ k_a k_b }{ [ \mbox{C} ]{\bar{k}}_a + [ \mbox{D} ] k_b } \nonumber \\ {\bar{k}}_{ab} & = & \frac{ {\bar{k}}_a {\bar{k}}_b }{ [ \mbox{C} ]{\bar{k}}_a + [ \mbox{D} ] k_b } , \end{eqnarray} \tag{ B.22 }$$

which is the interpretation of the graph reduction shown in figure B5.

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Associativity for composite graphs follows from the associativity of the elementary composition rule (B.14), via the grouping (B.19). To show how this works, we demonstrate associativity for the minimal case shown in figure B6. The important features are that the graph 're-wiring' follows from composition of the rule demonstrated in figure B5, and the composition rule for rate constants permits consistent removal of the necessary factors of reagent concentrations.

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The application of the elementary reduction to remove X, corresponding to the second line in figure B6, yields equations (B.19) and (B.20). An equivalent removal of Y first (the third line of figure B6) gives

$$\begin{equation} ( k_b , [ \mbox{E} ]{\bar{k}}_b ) \circ ( [ \mbox{F} ] k_c , {\bar{k}}_c ) = ( {\hat{k}}_{bc} , {\bar{\hat{k}}}_{bc} ) , \end{equation} \tag{ B.23 }$$

with rule

$$\begin{eqnarray} {\hat{k}}_{bc} & = & \frac{ k_b [ \mbox{F} ] k_c }{ [ \mbox{E} ]{\bar{k}}_b + [ \mbox{F} ] k_c } \nonumber \\ {\bar{\hat{k}}}_{bc} & = & \frac{ [ \mbox{E} ]{\bar{k}}_b {\bar{k}}_c }{ [ \mbox{E} ]{\bar{k}}_b + [ \mbox{F} ] k_c } . \end{eqnarray} \tag{ B.24 }$$

The two equivalent rules for removing whichever internal node was not removed in the first reduction are

$$\begin{eqnarray} ( {\hat{k}}_{ab} , [ \mbox{E} ]{\bar{\hat{k}}}_{ab} ) \circ ( [ \mbox{F} ] k_c , {\bar{k}}_c ) & = & ( {\hat{k}}_{abc} , {\bar{\hat{k}}}_{abc} ) , \nonumber \\ ( k_a , [ \mbox{C} ]{\bar{k}}_a ) \circ ( [ \mbox{D} ]{\hat{k}}_{bc} , {\bar{\hat{k}}}_{bc} ) & = & ( {\hat{k}}_{abc} , {\bar{\hat{k}}}_{abc} ) . \end{eqnarray} \tag{ B.25 }$$

Composing these rules for intermediate rate constants, we may check that

$$\begin{eqnarray} {\hat{k}}_{abc} & = & \frac{ {\hat{k}}_{ab} [ \mbox{F} ] k_c }{ [ \mbox{E} ]{\bar{\hat{k}}}_{ab} + [ \mbox{F} ] k_c } \nonumber \\ & = & \frac{ ( k_a [ \mbox{D} ] k_b ) [ \mbox{F} ] k_c }{ [ \mbox{C} ]{\bar{k}}_a [ \mbox{E} ]{\bar{k}}_b + ( [ \mbox{C} ]{\bar{k}}_a + [ \mbox{D} ] k_b ) [ \mbox{F} ] k_c } \nonumber \\ & = & \frac{ k_a [ \mbox{D} ] ( k_b [ \mbox{F} ] k_c ) }{ [ \mbox{C} ]{\bar{k}}_a ( [ \mbox{E} ]{\bar{k}}_b + [ \mbox{F} ] k_c ) + [ \mbox{D} ] k_b [ \mbox{F} ] k_c } \nonumber \\ & = & \frac{ k_a [ \mbox{D} ]{\hat{k}}_{bc} }{ [ \mbox{C} ]{\bar{k}}_a + [ \mbox{D} ]{\hat{k}}_{bc} } , \end{eqnarray} \tag{ B.26 }$$

and a similar equation follows for ${\bar{\hat{k}}}_{abc}$. Converting the hatted forms to the normal reaction form produces the rate equation

$$\begin{equation} [ \mbox{A} ] [ \mbox{D} ] [ \mbox{F} ] k_{abc} - [ \mbox{C} ] [ \mbox{E} ] [ \mbox{B} ]{\bar{k}}_{abc} = J_{abc} . \end{equation} \tag{ B.27 }$$

We may directly obtain the rate constants k_abc, ${\bar{k}}_{abc}$ with the composition rule

$$\begin{eqnarray} ( k_{abc} , {\bar{k}}_{abc} ) & = & ( k_{ab} , {\bar{k}}_{ab} ) \circ ( k_c , {\bar{k}}_c ) \nonumber \\ & = & ( k_a , {\bar{k}}_a ) \circ ( k_{bc} , {\bar{k}}_{bc} ) , \end{eqnarray} \tag{ B.28 }$$

using the appropriate version of the graph-dependent evaluation rule (B.22) in each step. The resulting composition (B.28) is automatically associative, because it satisfies the conversion

$$\begin{eqnarray} {\hat{k}}_{abc} & = & [ \mbox{D} ] [ \mbox{F} ] k_{abc} \nonumber \\ {\bar{\hat{k}}}_{abc} & = & [ \mbox{C} ] [ \mbox{E} ]{\bar{k}}_{abc} \end{eqnarray} \tag{ B.29 }$$

with equation (B.26), which is associative. As a final check, the equilibrium constants in the normal reaction form satisfy the necessary chain rule

$$\begin{equation} \frac{k_{abc}}{{\bar{k}}_{abc}} = \frac{k_a}{{\bar{k}}_a} \frac{k_b}{{\bar{k}}_b} \frac{k_c}{{\bar{k}}_c} . \end{equation} \tag{ B.30 }$$

Intermediate (hatted) rate constants have been used here to show how associativity is inherited from the base case. The examples below work directly with the actual (un-hatted) rate constants, which keep the network in its literal form at each reduction.

The bipartite graph for the minimal rTCA network in CHO compounds is shown in figure B7. All networks in the text are generated by equivalent methods. Highlighted nodes are those that can be removed by the base reduction in section appendix B.2.1. Reactions are labeled with lowercase Roman letters, and relative to the elementary half-reaction rate constants, the lumped-parameter rate constants are given by

$$\begin{eqnarray} k_{de} & = & \frac{ k_d k_e }{ {\bar{k}}_d + k_e } {} {\bar{k}}_{de} = \frac{ {\bar{k}}_d {\bar{k}}_e }{ {\bar{k}}_d + k_e } \nonumber \\ k_{ij} & = & \frac{ k_i k_j }{ {\bar{k}}_i + k_j } {} {\bar{k}}_{ij} = \frac{ {\bar{k}}_i {\bar{k}}_j }{ {\bar{k}}_i + k_j } \nonumber \\ k_{ka} & = & \frac{ k_k k_a }{ {\bar{k}}_k + k_a } {} {\bar{k}}_{ka} = \frac{ {\bar{k}}_k {\bar{k}}_a }{ {\bar{k}}_k + k_a } , \end{eqnarray} \tag{ B.31 }$$

with equivalent expressions for the $\bar{k}$s. These define the elementary reactions in the reduced graph of figure B8. Here and below, we give formulae only for the forward half-reaction rate constants k. Formulae for the backward half-reaction rate constants $\bar{k}$ have corresponding forms as shown in the preceding sections.

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One further reduction that follows the elementary rule in figure B8 is removal of cis-aconitate (cAC), which involves a common factor of the solvent [H₂O]. The resulting lumped-parameter rate constants are given by

$$\begin{equation} k_{ijka} = \frac{ k_{ij} k_{ka} }{ {\bar{k}}_{ij} + k_{ka} } {} {\bar{k}}_{ijka} = \frac{ {\bar{k}}_{ij} {\bar{k}}_{ka} }{ {\bar{k}}_{ij} + k_{ka} } . \end{equation} \tag{ B.32 }$$

These lead to the graph of figure B9.

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All further graph reductions require the composition rules of appendix B.2.3, and result in changes of the input or output stoichiometries of the unreduced nodes. All highlighted compounds in figure B9 may be removed, and the resulting lumped-parameter rate constants are given by

$$\begin{eqnarray} k_{bc} & = & \frac{ k_b k_c }{ [ {\mbox{H}}_2 \mbox{O} ]{\bar{k}}_b + [ {\mbox{CO}}_2 ] k_c } \nonumber \\ k_{def} & = & \frac{ k_{de} k_f }{ [ {\mbox{H}}_2 \mbox{O} ]{\bar{k}}_{de} + [ {\mbox{H}}_2 ] k_f } \nonumber \\ k_{defg} & = & \frac{ k_{def} k_g }{ [ {\mbox{H}}_2 \mbox{O} ]{\bar{k}}_{def} + [ {\mbox{H}}_2 ] [ {\mbox{CO}}_2 ] k_g } \nonumber \\ k_{defgh} & = & \frac{ k_{defg} k_h }{ {[ {\mbox{H}}_2 \mbox{O} ]}^2 {\bar{k}}_{defg} + [ {\mbox{CO}}_2 ] k_h } \nonumber \\ k_{defghijka} & = & \frac{ k_{defgh} k_{ijka} }{ {[ {\mbox{H}}_2 \mbox{O} ]}^2 {\bar{k}}_{defgh} + [ {\mbox{H}}_2 ] k_{ijka} } . \end{eqnarray} \tag{ B.33 }$$

These define the maximal reduction of the original rTCA graph, to the graph shown in figure B10.

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The lumped-parameter rate equations for figure B10, parametrized by lumped-parameter rate constants, are

$$\begin{eqnarray} J_{bc} & = & [ \mbox{ACE} ] [ {\mbox{H}}_2 ]{[ {\mbox{CO}}_2 ]}^2 k_{bc} \nonumber \\ & & \mbox{} - [ \mbox{OXA} ] [ {\mbox{H}}_2 \mbox{O} ]{\bar{k}}_{bc} \nonumber \\ J_{defghijka} & = & [ \mbox{OXA} ]{[ {\mbox{H}}_2 ]}^4 {[ {\mbox{CO}}_2 ]}^2 k_{defghijka} \nonumber \\ & & \mbox{} - [ \mbox{OXA} ] [ \mbox{ACE} ]{[ {\mbox{H}}_2 \mbox{O} ]}^2 {\bar{k}}_{defghijka} . \end{eqnarray} \tag{ B.34 }$$

In steady state J_bc = 0 and [OXA] may be replaced with the equilibrium function

$$\begin{equation} [ \mbox{OXA} ] = \frac{k_{bc}}{{\bar{k}}_{bc}} \frac{ [ {\mbox{H}}_2 ]{[ {\mbox{CO}}_2 ]}^2 }{ [ {\mbox{H}}_2 \mbox{O} ] } [ \mbox{ACE} ] . \end{equation} \tag{ B.35 }$$

For the remainder of the appendix we replace the subscript _defghijka with designation _rTCA in currents J, half-reaction rate constants k, $\bar{k}$, and equilibrium constants K. Dimensionally, the rate constants require the concentration of OXA in the mass-action law, and so presume that the anaplerotic segment _bc has been handled.

Plugging equation (B.35) into the second rate equation of equation (B.34), and supposing [OXA] is in equilibrium with [ACE] at a (non-equilibrium) steady state for the network as a whole, we obtain the only independent mass-action rate equation for the reduced network. This is the current producing acetate:

$$\begin{eqnarray} &&\fl J_{{\rm rTCA}} = {\bar{k}}_{{\rm rTCA}} \frac{k_{bc}}{{\bar{k}}_{bc}} [ {\mbox{H}}_2 ]{[ {\mbox{CO}}_2 ]}^2 [ {\mbox{H}}_2 \mbox{O} ] [ \mbox{ACE} ] \nonumber\\ &&\times\,\left( \frac{ k_{{\rm rTCA}} }{ {\bar{k}}_{{\rm rTCA}} } \frac{ {[ {\mbox{H}}_2 ]}^4 {[ {\mbox{CO}}_2 ]}^2 }{ {[ {\mbox{H}}_2 \mbox{O} ]}^2 } - [ \mbox{ACE} ] \right) . \end{eqnarray} \tag{ B.36 }$$

The first term in parenthesis in equation (B.36) is the concentration at which acetate would be in equilibrium with the inorganic inputs, which we denote

$$\begin{equation} {[ \mbox{ACE} ]}_G \equiv \frac{ k_{{\rm rTCA}} }{ {\bar{k}}_{{\rm rTCA}} } \frac{ {[ {\mbox{H}}_2 ]}^4 {[ {\mbox{CO}}_2 ]}^2 }{ {[ {\mbox{H}}_2 \mbox{O} ]}^2 } . \end{equation} \tag{ B.37 }$$

Therefore the network response is proportional to the offset of [ACE] from its equilibrium value, with a rate constant that depends on the particular contributions of chemical potential from [CO₂] and reductant. Although the lumped-parameter rate constant in this relation appears complex, the consistency conditions with single-reaction equilibrium constants ensure that $k_{{\rm rTCA}} / {\bar{k}}_{{\rm rTCA}}$ is independent of synthetic pathway and equal to the exponential of the Gibbs free energy of formation.