Energetics in a model of prebiotic evolution

Abstract

Previously we reported [A. Wynveen et al., Phys. Rev. E 89, 022725 (2014)PLEEE81539-375510.1103/PhysRevE.89.022725] that requiring that the systems regarded as lifelike be out of chemical equilibrium in a model of abstracted polymers undergoing ligation and scission first introduced by Kauffman [S. A. Kauffman, The Origins of Order (Oxford University Press, New York, 1993), Chap. 7] implied that lifelike systems were most probable when the reaction network was sparse. The model was entirely statistical and took no account of the bond energies or other energetic constraints. Here we report results of an extension of the model to include effects of a finite bonding energy in the model. We studied two conditions: (1) A food set is continuously replenished and the total polymer population is constrained but the system is otherwise isolated and (2) in addition to the constraints in (1) the system is in contact with a finite-temperature heat bath. In each case, detailed balance in the dynamics is guaranteed during the computations by continuous recomputation of a temperature [in case (1)] and of the chemical potential (in both cases) toward which the system is driven by the dynamics. In the isolated case, the probability of reaching a metastable nonequilibrium state in this model depends significantly on the composition of the food set, and the nonequilibrium states satisfying lifelike condition turn out to be at energies and particle numbers consistent with an equilibrium state at high negative temperature. As a function of the sparseness of the reaction network, the lifelike probability is nonmonotonic, as in our previous model, but the maximum probability occurs when the network is less sparse. In the case of contact with a thermal bath at a positive ambient temperature, we identify two types of metastable nonequilibrium states, termed locally and thermally alive, and locally dead and thermally alive, and evaluate their likelihood of appearance, finding maxima at an optimal temperature and an optimal degree of sparseness in the network. We use a Euclidean metric in the space of polymer populations to distinguish these states from one another and from fully equilibrated states. The metric can be used to characterize the degree and type of chemical equilibrium in observed systems, as we illustrate for the proteome of the ribosome.

Some generic measures of the extent of chemical disequilibrium applied to living and abiotic systems

“…We denote the total number of polymers N in a sample by is N = lmax L=1 N L . However, in contrast to the situation in the dynamic simulations described in [1], the input data for calculation of equilibrium distributions are not N and E but the volumetric polymer concentration ρ = N/V where V is the solution volume and the volumetric energy density e = E/V . To take entropic account of the dilution of the experimental sample we introduce a microscopic length R 0 L ν where R 0 is a length related to the polymer persistence length and ν is an index which would be 1/2 for a random walk.…”

“…We report the numerical values for ν and R 0 used for the various systems considered in the next section. We modify the expression for the entropy used in [1] to take account of the number of ways to distribute N polymers in a volume V as follows:…”

“…b is the number of monomers available for inclusion in the polymers in the system and ν is a dimensionless number which is 1/2 for a random walk. The expression is identical to the one used in [1] except for the factor V /v p L 3ν in the degeneracy G L . Note that a similar configurational entropy factor was used by us in an earlier paper [6].…”

“…The term v p /V on the right hand side of (5) is in all cases much less than b L /L 3ν and is dropped in the numerical analysis. As before [1] we refer to this equilbrium as 'isolated'. There is no reference to an external temperature bath.…”

“…A goal is to determine how well the measures distinguish living from nonliving systems and what they reveal about the differences. The measures used here were first applied by us to the results of simulations of a Kauffman-like model as described in [1]. However we did not need to assume a detailed correspondence between that model and the experimentally observed systems described here in order to carry out the program reported in this paper.…”