Mutation and Recombination: How they differ—Lessons from mathematical analysis Author: Roman V. Belavkin (Faculty of Science and Technology, Middlesex University, London NW4 4BT, UK) Abstract: Mutation and recombination of DNA strings are two most important mechanisms used by biological organisms during replication that allow them to inherit old and evolve new traits.  Mutation is a random substitution of some letters in the parent string by any other letters from the alphabet.  Recombination, on the other hand, is a substitution of some letters in one parent string by the letters from another string.  Inspired by Fisher’s geometric approach to study beneficial mutations, we have analysed probabilities of beneficial mutation and crossover recombination.  We consider mutations and recombinations that reduce the distance to an optimum as beneficial.  Geometric and combinatorial analysis has revealed new interesting differences and properties of these probabilities.  While mutation can potentially reach any part of the search space, the probability of beneficial mutation decreases with distance to an optimum, and the optimal mutation radius or rate should also decrease resulting in a slow-down of evolution near the optimum.  Crossover recombination, on the other hand, acts in a subspace of the search space defined by the current population of strings.  However, probabilities of beneficial and deleterious crossover are balanced, and their characteristics, such as variance, are translation invariant in a Hamming space, suggesting that recombination may complement mutation and boost the rate of evolution near the optimum. These results have been recently published in the Annals of Mathematics and Artificial Intelligence: Belavkin, R.V. (2025). Analysis and Optimization of Probabilities of Beneficial Mutation and Crossover Recombination in a Hamming Space. http://dx.doi.org/10.1007/s10472-025-09987-5 Previous work was in collaboration with Christopher Knight, Rok Krasovec,

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