Energetics in a model of prebiotic evolution

Previously we reported [A. Wynveen et al., Phys. Rev. E 89, 022725 (2014)] 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.

Figure 1

Values of (a) $\beta \mu$ and $\beta \mathrm{\Delta }$ as well as (b) the ratio $\mathrm{\Delta }/\mu$ as computed on the fly during a simulation of the model in which the dynamics were continuously driven toward equilibrium determined by the instantaneous energy $E$ and polymer number $N$ through Eq. (7). Note that the ratio $N_1/N_f$ as well as $N_f$ was fixed during this and similar runs.

Figure 2

Dependence of the likelihood of producing any state as a function of $p$ and $N_1/N_f$. Except very near $N_1/N_f=0$ and $N_1/N_f=1$, this essentially maps the probability of having a viable network, with little dependence on $N_1/N_f$. The data are from at least 809 301 simulations. More details are in Appendix pp6.

Figure 3

Dependence on $N_1/N_f$ and $p$ of the likelihood of producing a nonequilibrium state in a thermally isolated system. Here we required that the steady-state entropy at the end of the run be less than 70% of the maximum possible entropy at the final $E$ and $N$, given that the food population ($N_1$ and $N_f$) was fixed. The data are from the same set of simulations which were used to generate Fig. 2.

Figure 4

Scatter plot of values of the mean and standard deviation of the nonfood polymer length produced from 14 753 thermally isolated simulations of the model. Here $p=0.0064$ and $N_1/N_f=0.33$.

Figure 5

Sample scatter plot of the distances defined in Eqs. (9) and (8) from the final state of runs at fixed positive ambient temperature. The colors indicate populations. Once sees several examples of the high population cases of LDTD (bottom left corner), LDTA (bottom right corner), and LATA (top right corner) with high populations. Here $\beta \mathrm{\Delta }=50$ corresponds to a relatively low temperature and it is likely that many states are not very dynamically active. (No dynamics cut, as discussed below, has been made). Here $p=0.00905$ and $l_{\text{max}}=8$. The data are from 17 751 simulations.

Figure 6

Scatter plot of values of the mean and standard deviation of the nonfood polymer length produced from simulations driven toward equilibrium at fixed positive temperature for $p=0.00226$, $\mathrm{\Delta }\beta =0.316$, and $l_{\text{max}}=8$. The colors indicate the values of the distances (a) $R_L$ and (b) $R_T$. Points which are red in both plots are LDTD systems. Blue in both plots indicates LATA systems. Blue in (b) and red in (a) indicate LDTA systems. The data are from 9116 simulations.

Figure 7

Probabilities of producing an entropically steady state at the indicated values of the Kauffman parameter $p$ determining the reaction network sparseness as defined in the text and the ambient inverse temperature $\beta \mathrm{\Delta }$. Here $l_{\text{max}}=8$ and states were deemed LATA if $R_L$ and $R_T$ were both greater than 0.35. (a) No cut was made to exclude states with no dynamic activity or which were not growing exponentially in population. (b) Result of subjecting the states counted in the data of (a) to the dynamics cut described in Appendix pp3. Here we required that ${\omega }_m>10\left(\frac{2\pi }{\delta t_{\text{Gillespie}}}\right)$. (c) Result of requiring that the states counted in (b) also have exponentially growing populations. Here we required that the final states be exponentially growing at a rate of one inverse Gillespie time steps or greater (see Appendix pp1). All probabilities shown take account of the probability that an artificial chemical network generated by our algorithm as described in Ref. [5] is viable as defined there and in Sec. 2.

Figure 8

Same as Fig. 7 except that $R_L<0.35$ and $R_T>0.35$ which we here term LDTA states. The data are from at least 1 101 877 simulations. More details are in Appendix pp6.

Figure 9

History of a run which produced a lifelike state according to the criteria applied. (a) Computed value of the chemical potential $\beta \mu$ and the total polymer population $N$ as a function of simulation time. (b) Values of $R_L$ and $R_T$ as a function of simulation time. (c) Macrospace polymer distribution for the final state (red), population distribution of a locally equilibrated state at the same energy and total polymer number (blue), and population distribution of a globally equilibrated state at the ambient temperature used in the dynamics algorithm (green). This was a single run with $l_{\text{max}}=6$, $p=0.00226$, and $\beta \mathrm{\Delta }=10$.

Figure 10

Polymer length distribution for the proteins in the ribosome of E. coli (red) compared with the local equilibrium distribution associated with the energy of those proteins within the model (blue) and with the thermal distribution associated with the same number of polymers at an ambient temperature of the order of that to be expected at the surface of the earth (green). In the thermal distribution, all the 52 polymers are in the largest bin, so the number of monomers is much larger and the height of the green peak is not to scale.

Figure 11

Logarithmic reaction time derivative of the polymer number and the energy for isolated systems during the entire run with $p=0.00226$ and $N_1/N_f=0.33$. The data are for 7174 systems.

Figure 12

Logarithmic derivative of the polymer number and the energy for systems coupled to a thermal bath with $p=0.00226$ and $\beta \mathrm{\Delta }=3.16$. The data are for 9343 systems.

Figure 13

Example of time variation of $N$, $E$, $S$, $\beta \mathrm{\Delta }$, $\beta \mu$, and $k_d$ during an isolated run ending in a nonequilibrium entropically steady state.

Figure 14

(a) Average logarithmic Gillespie time derivative of the polymer number $N$ and the ratio of the standard deviation of that quantity divided by its average for a series of runs at fixed ambient temperature given by $\beta \mathrm{\Delta }=0.1$, with $p=0.00226$ and $l_{\text{max}}=8$. The data are for 9600 systems. (b) Same quantities for the energy $E$.

Figure 15

Fourier transforms and the power spectrum of $C\left(\tau \right)$ of a run at fixed infinite ambient temperature ($\beta \mathrm{\Delta }=0$). (a) and (b) Values of $N$, $E$, $R_L$, and $R_T$ over the entire history of the run, indicating that it is rather close to equilibrium. (c) Plot of $C\left(\tau \right)$ calculated for the last ${10}^5$ steps of the run. (d)–(f) Real and imaginary parts and the power spectra of the Fourier transform of $C\left(\tau \right)$, with point weighting as described in the text, near the origin. Here $p=0.00226$ and $l_{\text{max}}=6$.

Figure 16

Fourier transforms and the power spectrum of a run at fixed ambient temperature $\beta \mathrm{\Delta }=1$. Panels are defined as in Fig. 15. This system is classified as LATA based on the $R$ values. Its population grew rapidly and its dynamic spectrum is qualitatively similar to the previous, equilibrium example. Here $p=0.00226$ and $l_{\text{max}}=6$. The same network that was used to generate the data in Fig. 15 was used here.

Figure 17

Fourier transforms and the power spectrum of a run at fixed ambient $\mathrm{\Delta }\beta =1$ but exhibiting a sharp peak in its the power spectrum of $C\left(\tau \right)$ around the maximum frequency $\omega =\pi$. Panels as defined as in Figs. 15 and 16 except that (d)–(f) have been centered around $\omega =\pi$. This system would qualify as LDTA on the basis of the $R$ values but its population is not growing and its dynamic spectra are very different from the previous two examples. For this run $p=0.00226$, $l_{\text{max}}=6$, and the network is different from that used to generate Figs. 15 and 16.

Figure 18

Scatter plot of values of the nonfood mean polymer length and its standard deviation produced by 41 667 simulations driven toward equilibrium at fixed positive temperature with $\beta \mathrm{\Delta }=10$, $p=0.01280$, and $l_{\text{max}}=6$. The dashed curves show the results of Eq. (E3) for the cases $l_1=6$ and $l_2=\{3,4,5\}$ with no adjustable parameters.