Maximizing output and recognizing autocatalysis in chemical reaction networks is NP-complete

Assessment of Stoichiometric Autocatalysis across Element Groups

Autocatalysis has been proposed to play critical roles during abiogenesis. These proposals are at odds with a limited number of known examples of abiotic (and, in particular, inorganic) autocatalytic systems that might reasonably function in a prebiotic environment. In this study, we broadly assess the occurrence of stoichiometries that can support autocatalytic chemical systems through comproportionation. If the product of a comproportionation reaction can be coupled with an auxiliary oxidation or reduction pathway that furnishes a reactant, then a Comproportionation-based Autocatalytic Cycle (CompAC) can exist. Using this strategy, we surveyed the literature published in the past two centuries for reactions that can be organized into CompACs that consume some chemical species as food to synthesize more autocatalysts. 226 CompACs and 44 Broad-sense CompACs were documented, and we found that each of the 18 groups, lanthanoid series, and actinoid series in the periodic table has at least two CompACs. Our findings demonstrate that stoichiometric relationships underpinning abiotic autocatalysis could broadly exist across a range of geochemical and cosmochemical conditions, some of which are substantially different from the modern Earth. Meanwhile, the observation of some autocatalytic systems requires effective spatial or temporal separation between the food chemicals while allowing comproportionation and auxiliary reactions to proceed, which may explain why naturally occurring autocatalytic systems are not frequently observed. The collated CompACs and the conditions in which they might plausibly support complex, "life-like" chemical dynamics can directly aid an expansive assessment of life's origins and provide a compendium of alternative hypotheses concerning false-positive biosignatures.

Multispecies autocatalytic RNA reaction networks in coacervates

Robust localization of self-reproducing autocatalytic chemistries is a key step in the realization of heritable and evolvable chemical systems. While autocatalytic chemical reaction networks already possess attributes such as heritable self-reproduction and evolvability, localizing functional multispecies networks within complex primitive phases, such as coacervates, has remained unexplored. Here, we show the self-reproduction of the Azoarcus ribozyme system within charge-rich coacervates where catalytic ribozymes are produced by the autocatalytic assembly of constituent smaller RNA fragments. We systematically demonstrate the catalytic assembly of active ribozymes within phase-separated coacervates-both in micron-sized droplets as well as in a coalesced macrophase, underscoring the facility of the complex, charge-rich phase to support these reactions in multiple configurations. By constructing multispecies reaction networks, we show that these newly assembled molecules are active, participating both in self- and cross-catalysis within the coacervates. Finally, due to differential molecular transport, these phase-separated compartments endow robustness to the composition of the collectively autocatalytic networks against external perturbations. Altogether, our results establish the formation of multispecies self-reproducing reaction networks in phase-separated compartments which in turn render transient robustness to the network composition.

The hierarchical organization of autocatalytic reaction networks and its relevance to the origin of life

Prior work on abiogenesis, the emergence of life from non-life, suggests that it requires chemical reaction networks that contain self-amplifying motifs, namely, autocatalytic cores. However, little is known about how the presence of multiple autocatalytic cores might allow for the gradual accretion of complexity on the path to life. To explore this problem, we develop the concept of a seed-dependent autocatalytic system (SDAS), which is a subnetwork that can autocatalytically self-maintain given a flux of food, but cannot be initiated by food alone. Rather, initiation of SDASs requires the transient introduction of chemical "seeds." We show that, depending on the topological relationship of SDASs in a chemical reaction network, a food-driven system can accrete complexity in a historically contingent manner, governed by rare seeding events. We develop new algorithms for detecting and analyzing SDASs in chemical reaction databases and describe parallels between multi-SDAS networks and biological ecosystems. Applying our algorithms to both an abiotic reaction network and a biochemical one, each driven by a set of simple food chemicals, we detect SDASs that are organized as trophic tiers, of which the higher tier can be seeded by relatively simple chemicals if the lower tier is already activated. This indicates that sequential activation of trophically organized SDASs by seed chemicals that are not much more complex than what already exist could be a mechanism of gradual complexification from relatively simple abiotic reactions to more complex life-like systems. Interestingly, in both reaction networks, higher-tier SDASs include chemicals that might alter emergent features of chemical systems and could serve as early targets of selection. Our analysis provides computational tools for analyzing very large chemical/biochemical reaction networks and suggests new approaches to studying abiogenesis in the lab.

Intrinsic and Extrinsic Thermodynamics for Stochastic Population Processes with Multi-Level Large-Deviation Structure

A set of core features is set forth as the essence of a thermodynamic description, which derive from large-deviation properties in systems with hierarchies of timescales, but which are not dependent upon conservation laws or microscopic reversibility in the substrate hosting the process. The most fundamental elements are the concept of a macrostate in relation to the large-deviation entropy, and the decomposition of contributions to irreversibility among interacting subsystems, which is the origin of the dependence on a concept of heat in both classical and stochastic thermodynamics. A natural decomposition that is known to exist, into a relative entropy and a housekeeping entropy rate, is taken here to define respectively the intensive thermodynamics of a system and an extensive thermodynamic vector embedding the system in its context. Both intensive and extensive components are functions of Hartley information of the momentary system stationary state, which is information about the joint effect of system processes on its contribution to irreversibility. Results are derived for stochastic chemical reaction networks, including a Legendre duality for the housekeeping entropy rate to thermodynamically characterize fully-irreversible processes on an equal footing with those at the opposite limit of detailed-balance. The work is meant to encourage development of inherent thermodynamic descriptions for rule-based systems and the living state, which are not conceived as reductive explanations to heat flows.

Chemical Transformation Motifs-Modelling Pathways as Integer Hyperflows

We present an elaborate framework for formally modelling pathways in chemical reaction networks on a mechanistic level. Networks are modelled mathematically as directed multi-hypergraphs, with vertices corresponding to molecules and hyperedges to reactions. Pathways are modelled as integer hyperflows and we expand the network model by detailed routing constraints. In contrast to the more traditional approaches like Flux Balance Analysis or Elementary Mode analysis we insist on integer-valued flows. While this choice makes it necessary to solve possibly hard integer linear programs, it has the advantage that more detailed mechanistic questions can be formulated. It is thus possible to query networks for general transformation motifs, and to automatically enumerate optimal and near-optimal pathways. Similarities and differences between our work and traditional approaches in metabolic network analysis are discussed in detail. To demonstrate the applicability of the mathematical framework to real-life problems we first explore the design space of possible non-oxidative glycolysis pathways and show that recent manually designed pathways can be further optimized. We then use a model of sugar chemistry to investigate pathways in the autocatalytic formose process. A graph transformation-based approach is used to automatically generate the reaction networks of interest.

An intermediate level of abstraction for computational systems chemistry

Computational techniques are required for narrowing down the vast space of possibilities to plausible prebiotic scenarios, because precise information on the molecular composition, the dominant reaction chemistry and the conditions for that era are scarce. The exploration of large chemical reaction networks is a central aspect in this endeavour. While quantum chemical methods can accurately predict the structures and reactivities of small molecules, they are not efficient enough to cope with large-scale reaction systems. The formalization of chemical reactions as graph grammars provides a generative system, well grounded in category theory, at the right level of abstraction for the analysis of large and complex reaction networks. An extension of the basic formalism into the realm of integer hyperflows allows for the identification of complex reaction patterns, such as autocatalysis, in large reaction networks using optimization techniques.This article is part of the themed issue 'Reconceptualizing the origins of life'.

Flows, scaling, and the control of moment hierarchies for stochastic chemical reaction networks

Stochastic chemical reaction networks (CRNs) are complex systems that combine the features of concurrent transformation of multiple variables in each elementary reaction event and nonlinear relations between states and their rates of change. Most general results concerning CRNs are limited to restricted cases where a topological characteristic known as deficiency takes a value 0 or 1, implying uniqueness and positivity of steady states and surprising, low-information forms for their associated probability distributions. Here we derive equations of motion for fluctuation moments at all orders for stochastic CRNs at general deficiency. We show, for the standard base case of proportional sampling without replacement (which underlies the mass-action rate law), that the generator of the stochastic process acts on the hierarchy of factorial moments with a finite representation. Whereas simulation of high-order moments for many-particle systems is costly, this representation reduces the solution of moment hierarchies to a complexity comparable to solving a heat equation. At steady states, moment hierarchies for finite CRNs interpolate between low-order and high-order scaling regimes, which may be approximated separately by distributions similar to those for deficiency-zero networks and connected through matched asymptotic expansions. In CRNs with multiple stable or metastable steady states, boundedness of high-order moments provides the starting condition for recursive solution downward to low-order moments, reversing the order usually used to solve moment hierarchies. A basis for a subset of network flows defined by having the same mean-regressing property as the flows in deficiency-zero networks gives the leading contribution to low-order moments in CRNs at general deficiency, in a 1/n expansion in large particle numbers. Our results give a physical picture of the different informational roles of mean-regressing and non-mean-regressing flows and clarify the dynamical meaning of deficiency not only for first-moment conditions but for all orders in fluctuations.

Solving moment hierarchies for chemical reaction networks

The study of chemical reaction networks (CRN's) is a very active field. Earlier well-known results (Feinberg 1987 Chem. Enc. Sci. 42 2229, Anderson et al 2010 Bull. Math. Biol. 72 1947) identify a topological quantity called deficiency, for any CRN, which, when exactly equal to zero, leads to a unique factorized steady-state for these networks. No results exist however for the steady states of non-zero-deficiency networks. In this paper, we show how to write the full moment-hierarchy for any non-zero-deficiency CRN obeying mass-action kinetics, in terms of equations for the factorial moments. Using these, we can recursively predict values for lower moments from higher moments, reversing the procedure usually used to solve moment hierarchies. We show, for nontrivial examples, that in this manner we can predict any moment of interest, for CRN's with non-zero deficiency and non-factorizable steady states.

Complex Autocatalysis in Simple Chemistries

Life on Earth must originally have arisen from abiotic chemistry. Since the details of this chemistry are unknown, we wish to understand, in general, which types of chemistry can lead to complex, lifelike behavior. Here we show that even very simple chemistries in the thermodynamically reversible regime can self-organize to form complex autocatalytic cycles, with the catalytic effects emerging from the network structure. We demonstrate this with a very simple but thermodynamically reasonable artificial chemistry model. By suppressing the direct reaction from reactants to products, we obtain the simplest kind of autocatalytic cycle, resulting in exponential growth. When these simple first-order cycles are prevented from forming, the system achieves superexponential growth through more complex, higher-order autocatalytic cycles. This leads to nonlinear phenomena such as oscillations and bistability, the latter of which is of particular interest regarding the origins of life.