Autocatalysis in kinetic chemical networks and the problem of the origin of life.

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In biology, abiogenesis or the origin of life is the natural process by which life has arisen from non-living matter, such as simple organic compounds. Life as we know it is characterized by several characteristics, such as heredity, reproduction and random mutations. It is an open question whether physico-chemical systems can share similar characteristics.

Among foremost issues is how and when template-based reproduction has emerged. In the metabolism-first approach, emphasis is put upon the ability of chemical networks to "reproduce" by autocatalysis. Catalysts are chemical species that enhance the rate of a reaction; the specificity of autocatalytic species is that they are products of the reactions they catalyze. The concept became popular in the theory of evolution through the notion of collectively autocatalytic sets introduced by Stuart Kauffman in the 80es.

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In an ongoing project with Philippe Nghe (ESPCI, Paris), we aim at a general study of the dynamical and self-organizational properties of  autocatalytic networks present in deterministic or random kinetic chemical networks. A first joint work (2022,

see also 2024) is dedicated to the characterization of autocatalysis for diluted (i.e. low-concentration) chemical networks; it is shown that both the stoechiometric autocatalysis (i.e. existence of combinations of reactions increasing the stoechiometry of all species) and  the dynamical (in general,  rate-dependent) autocatalysis, i.e. positivity of the Lyapunov exponent of the linearized system, are equivalent or strongly related to a certain topological property of the network.

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The next step in the same direction is a full characterization of Lyapunov data (eigenvalue and eigenvector components) in a multi-scale setting. Assuming scale separation -- i.e. multiplicity of kinetic scales, allowing approximate recursive resummation of transitions above dropping time scales --, a rigorous algorithm, implemented in Python, gives the order of magnitude of Lyapunov data and idealized time trajectories of all autocatalytic networks until the dilution hypothesis fails.  Our working hypothesis is that this effort may be pursued to deal with 'generic' reaction networks away from the diluted regime. 

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Our PhD student, Praneet Nandan, currently develops computational chemistry methods to derive generative chemistry from a systematic (computer-aided) search for 'arrow-pushing' mechanisms, and assess kinetic rates of prebiotic reaction mechanisms. The general aim is to couple this approach with the multi-scale one, and apply these tools to predict the time-behavior of complex reaction networks, such as those arising in the evolution through aqueous alteration of pre-accretional analogs produced in lab (work by G. Danger and his team at PIIM, Marseille), fostering collaboration with experimentalists from PEPR Astro. 

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D. Lacoste, our PhD student Arthur Genthon and I have also been working on autocalysis in formose reaction and coupled cycles using the same kind of mathematical formalism.

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Growth models. 

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The general interest is in understanding how biological populations adapt to changing environments; diversities of strategies are reflected e.g. in food-dependent gene regulation or phenotypic switching in bacteria, or innate/adaptive/CRISPR-like (acquired and heritable)  immune systems. The models we consider are Markovian; both the environment and population switch stochastically between different states. On top of that comes a state-dependent  growth rate. For this class of models, one may look for the population "phenotypic" switching rates optimizing the averaged (quenched) growth rate. The question is important to understand biological evolution, where the growth rate can be interpreted as fitness of a population. 

Thinking of growth as coming from self-catalytic reactions, and including reverse reactions, we may also recast the problem in the formalism of chemical reaction networks, and use insights from thermodynamics.

                              

In a first joint work with Luis Dinis and David Lacoste, we give a detailed mathematical and numerical analysis of mean/variance trade-off for the growth rate for a simpler model not inspired by biology but by finance, which is Kelly's model for horse races.  The idea is that an investor may not choose the growth-rate maximizing betting strategy (so-called Kelly's strategy), because it involves large fluctuations. Less risky strategies are obtained by maximizing combinations of the mean and the standard deviation depending on a free risk-aversion parameter; the outcome is a curve called Pareto front, whose general properties are investigated in connection with stochastic thermodynamics. We obtain in particular a very simple uncertainty relation for the growth rate.

The original biological problem is in some sense a non-commutative analog of Kelly's model, in which the long-time growth is computed by multiplying matrices exhibiting a Markov dependence on time. The mean growth rate of the  model has been studied in various approximations, in particular in the Kussell-Leibler limit when environmental transitions are slow compared to phenotype transitions. The case of two-by-two matrices has been solved analytically by Hufton-Lin-Galla using a PDMP (piecewise-deterministic Markov process) formalism. Using the same formalism and ideas coming from the general theory of continuous-time Markov processes, we express the variance as the maximum of a functional, and obtain a closed formula in the case of two-by-two matrices. A detailed numerical analysis of the model, together with implications for  biological populations in terms of best strategies for growth,  are discussed in a joint work with L. Dinis and D. Lacoste. It is also argued that the variance is indeed a measure of risk,  insofar as variance and extinction probability have same direction of variation.

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