RESULTS AND DISCUSSION

Several properties of the inherited efficiencies model have been explored via Monte Carlo simulation. For a range of parameters, the increase in the catalytic capabilities of the protocell that defines non-genomic evolution has been observed. Here we describe the results of simulations aimed at assessing the role played by the details of the hydrolysis bias and the balance between the ease of creation of efficient ligases and proteases.

The bias in the action of the hydrolytic enzymes towards the destruction of less efficient peptides is expected to play an important role in non-genomic evolution. Several simulations were performed to explore this. In all cases, the number of monomers within the protocell was fixed at 1000, the maximum efficiency means ( and ) were 1000.0, the minimum efficiency means ( and ) were 1.0 and the maximum of the bias () was 1.0. The simulations were performed for 210 Monte Carlo cycles. Variations in the location of the midpoint of the bias () and of the rate of decrease of the bias () were not observed to have a qualitative effect on the behavior of the model (data not shown). Changes to the relative depth of the bias had a marked effect, however. The results of three representative simulations, for 0.05, 0.025, and 0.01, are shown in Figure 1. In these simulations, the functions governing the means of the efficiency distributions were adjusted to make the formation of ligases slightly easier than the formation of proteases (, , , ). As the minimum value of the hydrolysis bias was decreased from 0.1 to 0.001, the behavior of the protocell changed: systems with exhibited a large and sustained increase in both the average length and average catalytic efficiencies of the peptides within the protocell. In contrast, systems with showed little increase in either the average length or the average catalytic efficiencies of their peptides. A single system was simulated with and it exhibited very slight growth in the length and catalytic efficiencies of its peptides.

  
Figure 1: Results for (solid lines), 0.025 (dashed lines), and 0.01 (dotted lines). (a) The average ligation efficiency of the polymers in the protocell. (b) The average length of the polymers within the protocell. (c) The number of polymers within the protocell.

Since is the value of the hydrolysis bias for highly structured, and therefore highly efficient, peptides, its value determines the ``lifespan'' of highly efficient peptides. Large values of mean that the probability that a highly efficient peptide will be hydrolyzed is not much reduced over the probability that a peptide of average efficiency will be hydrolyzed. Thus, when highly efficient peptides are generated in a system with a large , they are hydrolyzed before their actions greatly affect the population of peptides within the protocell and the rate with which the protocell explores the space of all peptides is not changed. In contrast, for small values of , the probability that a highly efficient peptide will be hydrolyzed is much smaller than the probability that a peptide of average efficiency will be hydrolyzed. Thus, highly efficient peptides are long-lived and their presence can increase the rate at which the protocell generates new peptides. The protocell can then evolve non-genomically.

  
Figure 2: Results for (solid lines), 25 (dashed lines), and 30 (dotted lines) for . (a) The average ligation efficiency of the polymers in the protocell. (b) Average length of the polymers within the protocell. (c) The number of polymers within the protocell.

The rate at which novel peptides are generated within a protocell not only depends on the depth of the hydrolysis bias but is also sensitive to the balance between the rates of creation of small, efficient ligases and small, efficient proteases. Clearly, if highly efficient ligases are much more easily formed than efficient proteases, the protocell will fill with a diverse array of long peptides. Eventually, the protocell will burst. At the other extreme, if small, efficient proteases are much more easily formed than small, efficient ligases, the formation of long peptides will proceed slowly and any peptides formed will be hydrolyzed rapidly; the overall catalytic efficiency of the protocell will therefore remain small. A series of simulations were performed to examine the sensitivity of non-genomic evolution to slight imbalances in the ease of creation of ligases and proteases. Particular attention was paid to cases where proteases were slightly easier to produce than ligases. As before, the number of monomers within the protocell was fixed at 1000, the maximum efficiency means ( and ) were 1000.0, the minimum efficiency means ( and ) were 1.0 and the maximum of the bias () was 1.0. The minimum of the bias () was set to 0.01, the rate of decrease of the bias () to 0.065 and the midpoint of the bias decrease () to 58.0. The parameters governing the means of the hydrolysis efficiency distributions were fixed to and . The parameter governing the rate of change of the means of the ligation efficiency distributions, , was fixed at 0.230 and three values for were considered: 21, 25, and 30. The simulations were performed for 210 Monte Carlo cycles.

The results of these simulations are displayed in Figure 2. Clearly shown is the sensitive dependence of the rate of evolution on the ease of creation of ligases: as increases, the rate of improvement in the average efficiency and length of the polymers in the protocell decreases. No real improvement is seen when . These data demonstrate that the rates with which ligases and proteases are formed must be in a close balance for non-genomic evolution to occur.