Emergence of homochirality in far-from-equilibrium systems: mechanisms and role in prebiotic chemistry

Since the model proposed by Frank (Frank FC, Biochem Biophys Acta 1953;11:459-463), several alternative models have been developed to explain how an asymmetric non-racemic steady state can be reached by a chirally symmetric chemical reactive system. This paper explains how a stable non-racemic regime can be obtained as a symmetry breaking occurring in a far-from-equilibrium reactive system initiated with an initial imbalance. Departing from the variations around the original Frank's model that are commonly described in the literature, i.e. open-flow systems of direct autocatalytic reactions, we discuss recent developments emphasizing both an active recycling of components and an autocatalytic network of simple reactions. We will present our APED model as the most natural realization of such thermodynamic openness and non-equilibrium, of recycling and of network autocatalysis, each of these in prebiotic conditions. The different experimental and theoretical models in the literature will be classified according to mechanism. The place and role of such self-structured networks responsible for the presence of homochirality in the primitive Earth will be detailed.     

Autocatalytic sets of proteins

This article investigates the possibility that the emergence of reflexively autocatalytic sets of peptides and polypeptides may be an essentially inevitable collective property of any sufficiently complex set of polypeptides. The central idea is based on the connectivity properties of random directed graphs. In the set of amino acid monomer and polymer species up to some maximum length, M, the number of possible polypeptides is large, but, for specifiable "legitimate" end condensation, cleavage and transpeptidation exchange reactions, the number of potential reactions by which the possible polypeptides can interconvert is very much larger. A directed graph in which arrows from smaller fragments to larger condensation products depict potential synthesis reactions, while arrows from the larger peptide to the smaller fragments depict the reverse cleavage reactions, comprises the reaction graph for such a system. Polypeptide protoenzymes are able to catalyze such reactions. The distribution of catalytic capacities in peptide space is a fundamental problem in its own right, and in its bearing on the existence of autocatalytic sets of proteins. Using an initial idealized hypothesis that an arbitrary polypeptide has a fixed a priori probability of catalyzing any arbitrary legitimate reaction to assign to each polypeptide those reactions, if any, which it catalyzes, the probability that the set of polypeptides up to length M contains a reflexively autocatalytic subset can be calculated and is a percolation problem on such reaction graphs. Because, as M increases, the ratio of reactions among the possible polypeptides to polypeptides rises rapidly, the existence of such autocatalytic subsets is assured for any fixed probability of catalysis. The main conclusions of this analysis appear independent of the idealizations of the initial model, introduce a novel kind of parallel selection for peptides catalyzing connected sequences of reactions, depend upon a new kind of minimal critical complexity whose properties are definable, and suggest that the emergence of self replicating systems may be a self organizing collective property of critically complex protein systems in prebiotic evolution. Similar principles may apply to the emergence of a primitive connected metabolism. Recombinant DNA procedures, cloning random DNA coding sequences into expression vectors, afford a direct avenue to test the distribution of catalytic capacities in peptide space, may provide a new means to select or screen for peptides with useful properties, and may ultimately lead toward the actual construction of autocatalytic peptide sets.     

The role of competition in creating multiple steady states and favouring evolution of prebiotic polymers

In a prebiotic context the assumption is made that a mutation leads to the formation of a new polymer displaying autocatalytic and cross-catalytic properties. By the use of a simple chemical model it is shown that, despite these favourable attributes, the new species either remains at low concentration or accumulates, unless a competition is imposed between the new polymer and its precursor. In the latter case, the well-known situation of multiple steady states and hysteresis is established and the new “improved” species, under specific conditions, may become the dominant one. The need of competition suggests an early compartmentation of the system.     

A Quasistationary Analysis of a Stochastic Chemical Reaction: Keizer’s Paradox

For a system of biochemical reactions, it is known from the work of T.G. Kurtz [J. Appl. Prob. 8, 344 (1971)] that the chemical master equation model based on a stochastic formulation approaches the deterministic model based on the Law of Mass Action in the infinite system-size limit in finite time. The two models, however, often show distinctly different steady-state behavior. To further investigate this “paradox,” a comparative study of the deterministic and stochastic models of a simple autocatalytic biochemical reaction, taken from a text by the late J. Keizer, is carried out. We compute the expected time to extinction, the true stochastic steady state, and a quasistationary probability distribution in the stochastic model. We show that the stochastic model predicts the deterministic behavior on a reasonable time scale, which can be consistently obtained from both models. The transition time to the extinction, however, grows exponentially with the system size. Mathematically, we identify that exchanging the limits of infinite system size and infinite time is problematic. The appropriate system size that can be considered sufficiently large, an important parameter in numerical computation, is also discussed.     

The Structure of Autocatalytic Sets: Evolvability, Enablement, and Emergence

This paper presents new results from a detailed study of the structure of autocatalytic sets. We show how autocatalytic sets can be decomposed into smaller autocatalytic subsets, and how these subsets can be identified and classified. We then argue how this has important consequences for the evolvability, enablement, and emergence of autocatalytic sets. We end with some speculation on how all this might lead to a generalized theory of autocatalytic sets, which could possibly be applied to entire ecologies or even economies.     

Detecting autocatalytic, self-sustaining sets in chemical reaction systems

The ability of systems of molecular reactions to be simultaneously autocatalylic and sustained by some ambient 'food source' of simple molecules may have been an essential step in the origin of life. In this paper we first describe a polynomial-time algorithm that determines whether any given set of molecules, reactions and catalysations contains a subsystem that is both autocatalytic and able to be sustained from a given subset of the molecules. We also describe some combinatorial properties of this algorithm, and show how it can be used to find irreducible auto-catalysing and sustaining subsystems. In the second part of the paper we use the algorithm to investigate random catalytic networks-in particular, a model described by Kauffman. Using simulations and some analytic techniques we investigate the rate of catalysis that is required for the emergence of autocatalytic and sustaining subsystems.     

Scaling behaviour of different polymer models in dissipative particle dynamics of unentangled melts

We present a dissipative particle dynamics (DPD) study of scaling behaviour for three polymer models. The scaling behaviour is explored for the conformational and dynamic properties of unentangled polymer melts. DPD employs a bead–spring model together with an aggressive coarse-graining to represent polymers at the mesoscale. The first model studied utilises a simple soft repulsion potential for the bead–bead interactions together with a harmonic spring potential to connect beads into a polymer chain. The second model differs from the first model by replacing the harmonic spring with a finitely extensible nonlinear elastic spring. The third model uses realistic coarse-grain potentials for the bead–bead, spring and bending interactions based on the iterative Boltzmann inversion procedure and it corresponds to a mesoscopic model of polyethylene. We systematically vary the chain length and spring constant (in the case of the first and second models), and simulate the conformational properties such as the end-to-end distance or radius of gyration, and dynamic properties such as the centre-of-mass self-diffusion coefficient or viscosity. The scaling of the conformational and dynamic properties with chain length (scaling laws) is compared with the Rouse theory, which is considered as a standard theory for unentangled polymer melts. The comparison shows that simulated scaling laws typically agree with the Rouse scaling laws for the DPD polymer models with more than 10 DPD beads. For the shorter DPD polymers, deviations from the Rouse theory exist and become significant for the dynamic properties, especially for the viscosity of the polymer melts.     

Design of a binary biomarker study from the results of a pilot study

Biomarkers are increasingly used in clinical and epidemiologic studies. Prior to these studies, small pilot studies are often conducted to assess the reproducibility of the biomarker. This article discusses how the results of a pilot study can be used to design subsequent studies when the biomarker is a binary assessment. We consider situations in which the pilot study has two factors (e.g., laboratory and individual) that are either crossed or nested. We discuss how binary random-effects models can be used for estimating the sources of variation and how parameter estimates from these models can be used to appropriately design future studies. We also show that fitting a linear variance components model that ignores the binary nature of the data is a simple alternative method that results in nearly unbiased and moderately efficient estimators of important design parameters. We illustrate the methodology with data from a study assessing the reproducibility of p53 immunohistochemistry in bladder tumors.     

A simple analogical model for the selection of optical activity and of the most efficient catalysts in the course of molecular evolution

Analogical models, based on the use of simple hydrodynamic devices are proposed to discuss qualitatively and easily the properties of chemical networks of reactions implying a limited supply of a reagent usable in several parallel autocatalytic reaction pathways.This discussion shows that the racemic state can be unstable at the steady state in such systems and that the pathways involving the most efficient catalysts are selected in the course of their chemical evolution.     

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.     

Competition between Primary Nucleation and Autocatalysis in Amyloid Fibril Self-Assembly

Kinetic measurements of the self-assembly of proteins into amyloid fibrils are often used to make inferences about molecular mechanisms. In particular, the lag time—the quiescent period before aggregates are detected—is often found to scale with the protein concentration as a power law, whose exponent has been used to infer the presence or absence of autocatalytic growth processes such as fibril fragmentation. Here we show that experimental data for lag time versus protein concentration can show signs of kinks: clear changes in scaling exponent, indicating changes in the dominant molecular mechanism determining the lag time. Classical models for the kinetics of fibril assembly suggest that at least two mechanisms are at play during the lag time: primary nucleation and autocatalytic growth. Using computer simulations and theoretical calculations, we investigate whether the competition between these two processes can account for the kinks which we observe in our and others’ experimental data. We derive theoretical conditions for the crossover between nucleation-dominated and growth-dominated regimes, and analyze their dependence on system volume and autocatalysis mechanism. Comparing these predictions to the data, we find that the experimentally observed kinks cannot be explained by a simple crossover between nucleation-dominated and autocatalytic growth regimes. Our results show that existing kinetic models fail to explain detailed features of lag time versus concentration curves, suggesting that new mechanistic understanding is needed. More broadly, our work demonstrates that care is needed in interpreting lag-time scaling exponents from protein assembly data.     

Evolution of Autocatalytic Sets in Computational Models of Chemical Reaction Networks

Several computational models of chemical reaction networks have been presented in the literature in the past, showing the appearance and (potential) evolution of autocatalytic sets. However, the notion of autocatalytic sets has been defined differently in different modeling contexts, each one having some shortcoming or limitation. Here, we review four such models and definitions, and then formally describe and analyze them in the context of a mathematical framework for studying autocatalytic sets known as RAF theory. The main results are that: (1) RAF theory can capture the various previous definitions of autocatalytic sets and is therefore more complete and general, (2) the formal framework can be used to efficiently detect and analyze autocatalytic sets in all of these different computational models, (3) autocatalytic (RAF) sets are indeed likely to appear and evolve in such models, and (4) this could have important implications for a possible metabolism-first scenario for the origin of life.     

Bayesian multivariate logistic regression

Bayesian analyses of multivariate binary or categorical outcomes typically rely on probit or mixed effects logistic regression models that do not have a marginal logistic structure for the individual outcomes. In addition, difficulties arise when simple noninformative priors are chosen for the covariance parameters. Motivated by these problems, we propose a new type of multivariate logistic distribution that can be used to construct a likelihood for multivariate logistic regression analysis of binary and categorical data. The model for individual outcomes has a marginal logistic structure, simplifying interpretation. We follow a Bayesian approach to estimation and inference, developing an efficient data augmentation algorithm for posterior computation. The method is illustrated with application to a neurotoxicology study.     

Selection between multiple periodic regimes in a biochemical system: complex dynamic behaviour resolved by use of one-dimensional maps

We analyse a model biochemical system in which two autocatalytic enzyme reactions are coupled in series, in conditions where multiple stable periodic regimes coexist for the same set of parameter values. We determine how the periodic regimes are reached from different initial conditions. The structure of the attraction basins is generally simple in the case of two coexisting limit cycles (birhythmicity). This structure and the associated behaviour may, however, become highly complex. In particular, the system exhibits enhanced sensitivity to initial conditions when the boundaries of the attraction basins are fractal. In the latter case, it becomes difficult to predict the evolution towards either one of two limit cycles, a phenomenon known as final state sensitivity. We show how these complex phenomena can be explained in a unified and simple manner by means of one-dimensional return maps derived from the time evolution of the model and from fifth degree polynomial equations. We suggest experimental tests of the sensitivity to initial conditions in chemical systems presenting birhythmicity. The physiological significance of the results is discussed with respect to the sensitivity of regulatory systems admitting multiple stable biological rhythms.     

Path PPI: an integrated dataset of human pathways and protein-protein interactions

Integration of pathway and protein-protein interaction(PPI) data can provide more information that could lead to new biological insights. PPIs are usually represented by a simple binary model, whereas pathways are represented by more complicated models. We developed a series of rules for transforming protein interactions from pathway to binary model, and the protein interactions from seven pathway databases, including PID, Bio Carta, Reactome, Net Path, INOH, SPIKE and KEGG, were transformed based on these rules. These pathway-derived binary protein interactions were integrated with PPIs from other five PPI databases including HPRD, Int Act, Bio GRID, MINT and DIP, to develop integrated dataset(named Path PPI). More detailed interaction type and modification information on protein interactions can be preserved in Path PPI than other existing datasets. Comparison analysis results indicate that most of the interaction overlaps values(OAB) among these pathway databases were less than 5%, and these databases must be used conjunctively. The Path PPI data was provided at http://proteomeview. hupo.org.cn/Path PPI/Path PPI.html.     

Chiral symmetry breaking: Experimental results and computer analysis of a liquid-phase autoxidation

The autoxidation of tetralin is treated as a model reaction system to define the applicability of stereospecific autocatalysis. This concept, predicting a spontaneous amplification of enantiomeric excess generated by an autocatalytic chemical reaction, is used in several theoretical models as an explanation for the origin of natural optical activity. The reaction system investigated obeys the basic criteria of these models: a chiral intermediate (tetralin hydroperoxide) is produced from an achiral substrate (tetralin) via an autocatalytic pathway where the feedback mechanism is expected to generate a state of broken chiral symmetry. In order to test the amplification capacity of this reaction a computer analysis of the kinetic scheme is performed. This simulation is derived from the known kinetic scheme of autoxidation and is validated by fitting the experimentally observed data of hydroperoxide evolution. Calculations show that this model allows powerful amplification of enantiomeric excess and a transient amplification of the optical rotation. It is also demonstrated that the model system exhibits pronounced sensitivity toward any loss of absolute configuration of the involved chiral species. Since an amplification effect results exclusively at a high degree of stereoselectivity, it is concluded that stereospecific autocatalysis is possible in systems which show template reactions, crystallization, or colloidal effects. © 1993 Wiley-Liss, Inc.     

On latent-variable model misspecification in structural measurement error models for binary response

Summary .  We consider structural measurement error models for a binary response. We show that likelihood-based estimators obtained from fitting structural measurement error models with pooled binary responses can be far more robust to covariate measurement error in the presence of latent-variable model misspecification than the corresponding estimators from individual responses. Furthermore, despite the loss in information, pooling can provide improved parameter estimators in terms of mean-squared error. Based on these and other findings, we create a new diagnostic method to detect latent-variable model misspecification in structural measurement error models with individual binary response. We use simulation and data from the Framingham Heart Study to illustrate our methods.     

Modelling autocatalytic networks with artificial microbiology

Cells can usefully be equated to autocatalytic networks that increase in mass and then divide. To begin to model relationships between autocatalytic networks and cell division, we have written a program of artificial chemistry that simulates a cell fed by monomers. These monomers are symbols that can be assembled into linear (non-branched) polymers to give different lengths. A reaction is catalysed by a particular polymer or 'enzyme' that may itself be a reactant of that reaction (autocatalysis). These reactions are only studied within the confines of the 'cell' or 'reaction chamber'. There is a flux of material through the cell and eventually the mass of polymers reaches a threshold at which we analyse the cell. Our results indicate a similarity between the connectivity of the reaction network and that of real metabolic networks. Developing the model will entail attributing increased probabilities of reactions to polymers that are colocalised to evaluate the consequences of the dynamics of large assemblies of diverse molecules (hyperstructures) and of cell division.     

Predicting the Mechanical and Electrical Properties of Nanocomposites Formed from Polymer Blends and Nanorods

By integrating different simulation techniques, we investigate the self-assembly and macroscopic properties of nanocomposites composed of nanoscale rods and a binary polymer blend. In particular, we combine a Cahn–Hilliard (CH) theory for binary mixtures and a Brownian dynamics (BD) for nanorods to create a hybrid model that allows us to determine the structural evolution of the nanocomposite. The incorporation of the nanorods into the minority phase of the phase-separating blend yields a bicontinuous morphology, where the nanorods form a percolating network within the continuous minority phase. This morphology serves as the input to a lattice spring model (LSM), which is used to determine the mechanical properties, and a finite difference model (FDM), which is used to calculate the electrical conductance of the material. We find that in this doubly percolating system, the reinforcement efficiency of the nanorods and the electrical conductivity of the material are significantly increased relative to the behavior in composites where the nanorods are randomly dispersed in a homogeneous matrix. The integration of these various techniques allow us to predict the complex nanorod/polymer morphologies as a function of the constituents' characteristics, determine the mechanical and electrical, behavior of the resultant material and consequently relate the nanoscopic structure of the mixture to the macroscopic properties of the composite.