Evolutionary Mean Field Dynamics

Assume a fitness landscape for viral sequence (single protein or whole genome),

 [binary (0 for consensus, 1 for mutant) of length L] including one site and two site terms.

Add an additonal cost from epitopes targeted by the host immune system (mostly T cells for HIV):

 

Use the   equations to compute the time evolution of single site mutation probabilities  

  ,

where the `mean-field approximation' from statistical physics gives

 

The essence of the `mean-field approximation' is the neglect of the correlations betweem sites, setting

Such `linkage equilibrium' is achieved in population genetics if the sequence elements are completely reshuffled post replication, i.e. for infinite recombination rate [Neher & Shraiman, Rev. Mod. Phys. 83, 1283 (2011)] (offline)


How well does this approach work in an actual patient?

Patient CH58 [Liu et al, J. Clin Invest 123, 380 (2013)] infected with HIV

Blood samples extracted at several time points to determine HIV sequences; T cell responses were also mapped.

Putative fitness landscape of p24 was constructed using maximum entropy approach on sequence diversity.

Given the many approximations involved, the results of mean-field dynamics are encouraging:

  vs.