A dual process for the coupled Wright–Fisher diffusion

Abstract

The coupled Wright–Fisher diffusion is a multi-dimensional Wright–Fisher diffusion for multi-locus and multi-allelic genetic frequencies, expressed as the strong solution to a system of stochastic differential equations that are coupled in the drift, where the pairwise interaction among loci is modelled by an inter-locus selection. In this paper, an ancestral process, which is dual to the coupled Wright–Fisher diffusion, is derived. The dual process corresponds to the block counting process of coupled ancestral selection graphs, one for each locus. Jumps of the dual process arise from coalescence, mutation, single-branching, which occur at one locus at the time, and double-branching, which occur simultaneously at two loci. The coalescence and mutation rates have the typical structure of the transition rates of the Kingman coalescent process. The single-branching rate not only contains the one-locus selection parameters in a form that generalises the rates of an ancestral selection graph, but it also contains the two-locus selection parameters to include the effect of the pairwise interaction on the single loci. The double-branching rate reflects the particular structure of pairwise selection interactions of the coupled Wright–Fisher diffusion. Moreover, in the special case of two loci, two alleles, with selection and parent independent mutation, the stationary density for the coupled Wright–Fisher diffusion and the transition rates of the dual process are obtained in an explicit form.

Asymptotic behaviour of sampling and transition probabilities in coalescent models under selection and parent dependent mutations

“…This formula is a consequence of the duality relationship between the ASG and the diffusion, in fact, when such relationship is proved, it is also shown that the right hand side of (2.4) solves the recursion formula that defines the sampling probability, see for example [18,10]. Furthermore, since the sample is exchangeable, the formula can also be explained by de Finetti's representation theorem.…”

“…[25,6,8], and when selection is included, a weighted Dirichlet density, see e.g. [8,10]. Unfortunately, the PIM case is the only case where the stationary distribution is explicitly known.…”

“…The adjective 'typed' is used to put emphasis on the fact that, in this paper, the lineages of the ASG are always associated to a type, as in e.g. [8,10], whereas often, e.g. in [18], the ASG is first constructed backwards in time with no regard for types, which are superimposed afterwards on the graph.…”

Cancer's second genome: Microbial cancer diagnostics and redefining clonal evolution as a multispecies process

“…• Create microbe as one agent type in the microenvironment and cancer cell as another agent type • Allow clonal evolution of cancer cells and separate evolution of microbes in equations • Create biophysical rules accounting for spatial movement of microbes (e.g., quorum sensing) and effect of microbes on evolutionary rates, such as proliferation or death of cancer cells during drug delivery [204][205][206][207] Wright-Fisher type model • Microbial species could undergo distinct Wright-Fisher evolutionary dynamics that are independent of, or, in turn, affect cancer cell evolution • Effects of microbes present could also be interwoven into cancer cell fitness evolving under Wright-Fisher dynamics • Fitness parameter of certain cancer cell genotypes may depend on metabolites, proteins, and antigens from intracellular bacteria, which in certain cases may drive differential immunoediting between cancer cell-bearing bacteria [176,[208][209][210] Moran-type model…”

Exact simulation of coupled Wright–Fisher diffusions

“…For a more thorough interpretation of the transition density g(x, •, t) and its one-dimensional counterpart, it is worth noting that the expansion in (3.2) is derived via a duality principle for Markov processes [16], that is, from the moment dual process of the Wright-Fisher diffusion, which is also a pure death process representing lineages backwards in time; see for example [15] or [29] for a complete derivation and details. The corresponding dual (coalescent) process for coupled Wright-Fisher diffusions is derived in [19].…”