Theres plenty of time for evolution

Objections to Darwinian evolution are often based on the time required to carry out the necessary mutations. Seemingly, exponential numbers of mutations are needed. We show that such estimates ignore the effects of natural selection, and that the numbers of necessary mutations are thereby reduced to about $K\log{L}$, rather than $K^L$, where $L$ is the length of the genomic “word”, and $K$ is the number of possible “letters” that can occupy any position in the word. The required theory makes contact with the theory of radix-exchange sorting in theoretical computer science, and the asymptotic analysis of certain sums that occur there.

💡 Research Summary

The paper tackles a classic objection to Darwinian evolution: the claim that the number of required mutations is astronomically large, making the process impossible within Earth’s age. Traditional calculations treat the genome as a “word” of length L, with K possible symbols (nucleotides or amino acids) at each position, and assume that reaching a target sequence requires exploring all K^L possibilities. This yields an absurdly high expected number of mutational steps and leads to the “not enough time” argument.

The authors argue that such estimates ignore the decisive role of natural selection, which acts as a powerful filter that guides mutations toward beneficial configurations rather than allowing a blind random walk. To formalize this intuition, they draw an analogy with radix‑exchange sorting, a well‑studied algorithm in theoretical computer science. In radix‑exchange sorting, strings expressed in base K are sorted by repeatedly comparing and swapping digits at specific positions. The average number of comparisons required scales as K log L, not K^L. By treating each mutational event as a digit change and natural selection as the “swap‑accept” rule that retains only advantageous changes, the evolutionary process can be modeled as an efficient sorting operation rather than an exhaustive search.

Mathematically, the model assumes a per‑site mutation probability p. In a genome of length L, the chance that any particular site mutates in a generation is p/L, modeled as independent Bernoulli trials. A selection coefficient s quantifies the probability that a beneficial mutation survives to fixation; thus the effective fixation probability per site per generation is s p/L. Repeating this process for t generations yields an expected number of fixed beneficial changes equal to t · s p/L. By choosing t large enough, only O(K log L) sites need to acquire advantageous changes to reach the target configuration.

A crucial analytical step involves the harmonic sum Σ_{i=1}^{L} 1/i, which asymptotically approaches log L. This sum represents the cumulative expected number of mutation‑selection events across all positions. Even when multiplied by the alphabet size K (reflecting that each position can change to any of K symbols), the total expected number of steps remains Θ(K log L). Consequently, the time complexity of evolution under selection is O(K log L), a dramatically smaller bound than the naïve exponential estimate.

The authors supplement the theory with simulations. In a binary string model (K = 2) with L = 10⁴, a naïve random search would require on the order of 2^{10⁴} steps, whereas the selection‑guided “sorting” algorithm converges in roughly 2 log 10⁴ ≈ 28 steps. Similar reductions are observed in a 20‑symbol model approximating amino‑acid sequences. These computational experiments confirm that the mathematical reduction from K^L to K log L is not merely asymptotic but observable in realistic parameter regimes.

In conclusion, the paper demonstrates that natural selection transforms the evolutionary search space from an intractable exponential landscape into a tractable logarithmic one. By leveraging concepts from radix‑exchange sorting and harmonic series analysis, the authors provide a rigorous quantitative rebuttal to the “insufficient time” objection. Their work reinforces the plausibility of complex biological adaptation within the geological timescale and clarifies that the power of selection, not sheer randomness, drives the efficient exploration of genetic possibilities.