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Living organisms are replete with rhythmic and oscillatory behavior at all levels, to the extent that oscillations have been termed as a defining attribute of life. Recent studies of synthetic oscillators that mimic such functions have shown decayed cycles in batch-mode reactions or sustained oscillatory kinetics under flow conditions. Considering the hypothesized functionality of peptides in early chemical evolution and their central role in current bio-nanotechnology, we now reveal a peptide-based oscillator. Oscillatory behavior was achieved by coupling coiled-coil-based replication processes as positive feedback to controlled initiation and inhibition pathways in a continuously stirred tank reactor (CSTR). Our results stress that assembly into the supramolecular structure and specific interactions with the replication substrates are crucial for oscillations. The replication-inhibition processes were first studied in batch mode, which produced a single damped cycle. Thereafter, combined experimental and theoretical characterization of the replication process in a CSTR under different flow and environmental (pH, redox) conditions demonstrated reasonably sustained oscillations. We propose that studies in this direction might pave the way to the design of robust oscillation networks that mimic the autonomous behavior of proteins in cells (e.g., in the cyanobacterial circadian clock) and hence hint at feasible pathways that accelerated the transition from simple peptides to extant enzymes.

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In this paper, we report an open system consisting of three self-replicating peptides, in which peptide 1 inhibits the duplex template of peptide 2, peptide 2 inhibits duplex 3, and peptide 3 inhibits duplex 1 to complete the negative feedback loop. This interacting chemical network yields oscillations in the concentrations of all species over time and establishes a possible mechanism for pre-biotic chemical systems organization. The first focus of our analysis is the effect of altering rates of duplex formation and inhibition on oscillations. We then examine the autocatalytic rate constant in the symmetric and asymmetric cases.

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In our search to understand complex oscillation in discrete dynamic systems, we modify the Ricker map, where one parameter is also a dynamic variable. Using the bistable behavior of the fixed point solution, we analyze two response functions that characterize the change of the dynamic parameter. The 2D map sustains different types of burst oscillations that depend on the response functions. In either case, the parameter values yield a slow dynamic variable required to observe bursting-type oscillations.

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One of the grand challenges in contemporary systems chemistry research is to mimic life-like functions using simple synthetic molecular networks. This is particularly true for systems that are out of chemical equilibrium and show complex dynamic behaviour, such as multi-stability, oscillations and chaos. We report here on thiodepsipeptide-based non-enzymatic networks propelled by reversible replication processes out of equilibrium, displaying bistability. Accordingly, we present quantitative analyses of the bistable behaviour, featuring a phase transition from the simple equilibration processes taking place in reversible dynamic chemistry into the bistable region. This behaviour is observed only when the system is continuously fueled by a reducing agent that keeps it far from equilibrium, and only when operating within a specifically defined parameter space. We propose that the development of biomimetic bistable systems will pave the way towards the study of more elaborate functions, such as information transfer and signalling.

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We have been studying simple prebiotic catalytic replicating networks as prototypes for modeling replication, complexification and Systems Chemistry. While living systems are always open and function far from equilibrium, these prebiotic networks may be open or closed, dynamic or static, divergent or convergent to a steady state. In this paper we review the properties of these simple replicating networks, and show, via four working models, how even though closed systems exhibit a wide range of emergent phenomena, many of the more interesting phenomena leading to complexification and emergence indeed require open systems.

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Bistability and bifurcation, found in a wide range of biochemical networks, are central to the proper function of living systems. We investigate herein recent model systems that show bistable behavior based on nonenzymatic self-replication reactions. Such models were used before to investigate catalytic growth, chemical logic operations, and additional processes of self-organization leading to complexification. By solving for their steady-state solutions by using various analytical and numerical methods, we analyze how and when these systems yield bistability and bifurcation and discover specific cases and conditions producing bistability. We demonstrate that the onset of bistability requires at least second-order catalysis and results from a mismatch between the various forward and reverse processes. Our findings may have far-reaching implications in understanding early evolutionary processes of complexification, emergence, and potentially the origin of life.

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Living organisms often display rhythmic and oscillatory behavior. We investigate here a challenge in contemporary Systems Chemistry, that is, to construct "bottom-up" molecular networks that display such complex behavior. We first describe oscillations during self-replication by applying kinetic parameters relevant to peptide replication in an open environment. Small networks of coupled oscillators are then constructed in silico, producing various functions such as logic gates, integrators, counters, triggers, and detectors. These networks are finally utilized to simulate the connectivity and network topology of the Kai proteins circadian clocks from the S. elongatus cyanobacteria, thus producing rhythms whose constant frequency is independent of the input intake rate and robust toward concentration fluctuations. We suggest that this study helps further reveal the underlying principles of biological clocks and may provide clues into their emergence in early molecular evolution.

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In biological systems, regulation plays an important role in keeping metabolite concentrations within physiological ranges. To study the dynamical implications of self-regulation, we consider a functional form used in genetic networks and couple it to a mechanism associated with chemical self-replication. For the two-variable minimal model, we find that activation can yield chemical toggles similar to those reported for gene repression in E. coli as well as more complex dynamics.

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Over the years, the Rosenzweig-MacArthur (RM) model has been used to study simple prey-predator systems. It has been observed, however, that the RM model cannot sustain Turing patterns when using a diagonal diffusion tensor. As a result, researchers have introduced changes to the RM model that induce stable Turing patterns. In most cases, the changes have been made to the so-called response function, changing the interspecies interaction, or by adding an intraspecies interaction to the model. In this communication, we study the original RM model but we include cross-diffusion, which considers off diagonal elements in the diffusion tensor. Although cross-diffusion is well characterized in multicomponent solutions, including electrolytes, it has an apparent counterintuitive meaning in predator-prey systems. We observe, however, that in plant and fish systems, the lack of predator mobility is compensated by their ability to camouflage and attract their prey, which yields a negative cross-diffusion coefficient. We show that negative cross-diffusion is enough to trigger stable Turing patterns in the RM model.

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Chemical self-replication of oligonucleotides and helical peptides exhibits the so-called square root rate law. Based on this rate we extend our previous work on ideal replicators to include the square root rate and other possible nonlinearities, which we couple with an enzymatic sink. For this generalized model, we consider the role of cross diffusion in pattern formation, and we obtain exact general relations for the Poincare-Adronov-Hopf and Turing bifurcations, and our generalized results include the Higgins, Autocatalator, and Templator models as specific cases.

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The authors consider a minimal cross-catalytic self-replicating system of only two cross-catalytic templates that mimics the R3C ligase ribozyme system of Dong-Eu and Joyce [Chem. Biol. 11, 1505 (2004)]. This system displays considerably more complex dynamics than its self-replicating counterpart. In particular, the authors discuss the Poincare-Andronov-Hopf bifurcation, canard transitions, excitability, and hysteresis that yield birhythmicity between simple and complex oscillations.

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Chemical self-replication of oligonucleotides and helical peptides show the so-called square root rate law. Based on this rate we extend our previous work on ideal replicators to include the square root rate and other possible nonlinearities, which we couple with an enzimatic sink. Although the nonlinearity is necessary for complex dynamics, the nature of the sink is the essential feature in the mechanism that allows temporal and spatial patterns. We obtain exact general relations for the Poincare-Adronov-Hopf and Turing bifurcations, and our generalized results include the Higgins, autocatalator, and templator models as specific cases.

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In this paper, we present a three-level trophic food chain, including intraspecies interaction. In contrast with other analyses, we consider the effect on the third trophic level by the first-level parameters. The model shows complex, as well as, chaotic oscillations. Bifurcation diagrams show period doubling route to chaos and crises. Also from the forward and backwards sections of the bifurcation diagrams, we find hysteresis. This result implies the coexistence of attractors for the same parameter values. In particular, we consider the coexistence of a chaotic and a P1 attractors. Our results show that the regulation in the food chain is not exclusive to either a food-prey or prey-predator interaction, but to a more subtle food-prey-predator interaction, where, for some parameter values, a food-prey or a prey-predator regulation may dominate the system's dynamics. Finally, we consider the impact of the intraspecies interaction in the overall dynamics of the food chain.

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