Biopolymers exist within living cells as far-fromequilibrium metastable polymers. Living systems must constantly invest energy for biopolymer synthesis. In the earliest stages of life on Earth, the complex molecular machinery that contemporary life employs for the synthesis and maintenance of polymers did not exist. Thus, a major question regarding the origin of life is how the first far-from-equilibrium polymers emerged from a prebiotic "pool" of monomers. Here, we describe a proof-of-principle system, in which L-malic acid monomers form far-from-equilibrium, metastable oligoesters via repeated, cyclic changes in hydration and temperature. Such cycles would have been associated with day−night and/or seasonal cycles on the early Earth. In our model system, sample heating, which promotes water evaporation and ester bond formation, drives polymerization. Even though periodic sample rehydration and heating in the hydrated state promotes ester bond hydrolysis, successive iterations of wet−dry cycles result in polymer yields and molecular weight distributions in excess of that observed after a single drying cycle. We term this phenomenon a "polymerization ratchet". We have quantitatively characterized the "ratchet" of our particular system. Ester bond formation rates and oligoester hydrolysis rates were determined for temperatures ranging from 60 to 95°C. Based on these rates, a mathematical model was developed using polycondensation kinetics, from which conditions were predicted for oligoester growth. This model was verified experimentally by the demonstration that L-malic acid monomers subjected to multiple wet−dry cycles form oligoesters, which reach a steady-state concentration and mean length after several cycles. The concentration of oligoesters that persist between subsequent steady-state cycles depends on the temperature and durations of the dry and wet phases of the cycle. These results provide insights regarding the potential for very simple systems to exhibit features that would have been necessary for initiation of polymer evolution, before the emergence of genomes or enzymes.

The method of moments has been proposed as a potential means to reduce the dimensionality of the chemical master equation (CME) appearing in stochastic chemical kinetics. However, attempts to apply the method of moments to the CME usually result in the so-called closure problem. Several authors have proposed moment closure schemes, which allow them to obtain approximations of quantities of interest, such as the mean molecular count for each species. However, these approximations have the dissatisfying feature that they come with no error bounds. This paper presents a fundamentally different approach to the closure problem in stochastic chemical kinetics. Instead of making an approximation to compute a single number for the quantity of interest, we calculate mathematically rigorous bounds on this quantity by solving semidefinite programs. These bounds provide a check on the validity of the moment closure approximations and are in some cases so tight that they effectively provide the desired quantity. In this paper, the bounded quantities of interest are the mean molecular count for each species, the variance in this count, and the probability that the count lies in an arbitrary interval. At present, we consider only steady-state probability distributions, intending to discuss the dynamic problem in a future publication.

Applying the method of moments to the chemical master equation appearing in stochastic chemical kinetics often leads to the so-called closure problem. Recently, several authors showed that this problem can be partially overcome using moment-based semidefinite programs (SDPs). In particular, they showed that moment-based SDPs can be used to calculate rigorous bounds on various descriptions of the stochastic chemical kinetic system's stationary distribution(s)-for example, mean molecular counts, variances in these counts, and so on. In this paper, we show that these ideas can be extended to the corresponding dynamic problem, calculating time-varying bounds on the same descriptions.

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