A new mathematical model for genetic linkage and recombination of haploid virus populations

Linkage disequilibrium (LD), logarithm of the odds (LOD), and recombination frequency (Rf) have long been measured independently. However, their relationship in viral haploid recombination has not been fully understood due to the unavailability of population-wide sequencing data until recently. Here, we provide the first mathematical model of haploid LOD and Rf, LODF and RfF, using recombinant single nucleotide polymorphism pairs (RSPs) as natural genomic markers of LD in monkeypox/mpox virus (MPXV) Ib sequences during the 2023-2025 outbreak. The recombinant haplotype allele frequency (Af) includes recombinant progeny (AfF for haploid population), while RfF is the measured non-progeny AfF when recombination breaks linkage. LODF and RfF are in a decaying exponential relationship, showing that recombination events lower LODF, and newly arising RSPs are strongly favored in the population. Our data reveals that Haploview's LODH vs. AfH and our LODF vs. AfF increase in a growing power relationship, suggesting that 8,247 RSPs out of 8,587 RSPs show strong evidence of LD and are related to historical recombination. We derived a new mathematical model to determine the maximum bound of historical recombination/progeny frequency, AfH,max, at any given LODH. This serves as critical reliability marker for RSPs, because AfH,max tests the strength of the binomial connection between LOD and Af. Taken together, observing recombination drive retention of RSPs containing high LD indicates that LODF and AfF have a strong association in the MPXV Ib viral population. These different LD methods provide powerful and complementary tools for scoring genetic linkage of haploid population genetics and evolution.