Probabilities of the emergence, by chance, of different versions of the breakthrough system in an O-region: a toy calculation of the upper bounds
General assumptions: an O-region contains 1022 stars and every 10th star has a habitable planet, hence 1021 habitable planets (undoubtedly, a gross over-estimation because, in reality, most stars have no planets at all, let alone habitable ones). Each planet is the size of earth and has a 10 kilometer (106 cm) thick habitable layer; hence the volume of the habitable layer is 4/3π[R3-(R-l)3] ≈ 5 × 1024 cm3, where R is the radius of the planet and l is the thickness of the habitable layer. RNA synthesis occurs in 1% of the volume of the habitable layer, i.e., a volume V ≈ 5 × 1022 cm3 is available for RNA synthesis (undoubtedly, a gross over-estimation because, in reality, there would be very few "RNA-making reactors"). Let the concentration of nucleotides in volume V and the rate of the synthesis of RNA molecules of size n (a free parameter depending on the specific model of the breakthrough stage; hereinafter n-mer) be 1 molecule/cm3/second (a gross overestimate for any sizable molecule; furthermore, the inverse dependence on n, which is expected to be strong, is disregarded). The time available after the Big Bang of the given O-region (as an upper bound) of all planets in it is 1010 years ≈ 3 × 1017 seconds. Then, the number of uniquen-mers "tried out" during the time after the Big Bang is:
S ≈ 5 × 1022 × 1021 × 3 × 1017 ≈ 1.5 × 1061.
Let us assume that, for the onset of biological evolution, a unique n-mer is required. The number of such sequences is N = 4n ≈100.6n.
Then, the expectation of the number of times a unique n-mer emerges in an O-region is: E = S/N = 1.5 × 1061/100.6n and n = log(E × 1.5 × 1061)/0.6.
Substituting E = 1, we get n ≈102 (nucleotides). Note that, because n is proportional to logS, the estimate is highly robust to the assumptions on the values of the contributing variables; e.g., a order of magnitude change in S will result in an increase or decrease of n by less than 2 nucleotides.
A ribozyme replicase consisting of ~100 nucleotides is conceivable, so, in principle, spontaneous origin of such an entity in a finite universe consisting of a single O-region cannot be ruled out in this toy model (again, the rate of RNA synthesis considered here is a deliberate, gross over-estimate).
The requirements for the emergence of a primitive, coupled replication-translation system, which is considered a candidate for the breakthrough stage in this paper, are much greater. At a minimum, spontaneous formation of:
- two rRNAs with a total size of at least 1000 nucleotides
- ~10 primitive adaptors of ~30 nucleotides each, in total, ~300 nucleotides
- at least one RNA encoding a replicase, ~500 nucleotides (low bound)is required. In the above notation, n = 1800, resulting in E -1018.
In other words, even in this toy model that assumes a deliberately inflated rate of RNA production, the probability that a coupled translation-replication emerges by chance in a single O-region is P -1018. Obviously, this version of the breakthrough stage can be considered only in the context of a universe with an infinite (or, in the very least, extremely vast) number of O-regions.
The model considered here is not supposed to be realistic by any account. It only serves to illustrate the difference in the demands on chance for the origin of different versions of the breakthrough system (see Fig. 1) and hence the connections between these versions and different cosmological models of the universe.
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