GGI program on       Glassy aspects of classical and quantum computation              Florence, 5 July 2019

Can adiabatic methods help 'unsolve' a hard problem

               

                Hanrong Chen         John Barton       Arup K. Chakraborty      

Mehran Kardar (MIT)

Fitness & Mutation

Elements of Population Genetics:

Fitness f  governs reproduction rates according to                 

Population mean  is defined by                                          

Mean fitness evolves as                                                     

green bullet Roland Fisher's Fundamental Theorem of Natural Selection

Affinity ω  is a trait correlated with fitness, and (including possibility of mutations) evolves as

green bullet George Price's Equation:                                                

Manfred Eigen's Quasispecies Model and the error catastrophe.

Mapping to Quantum Ising chains (Wagner, Baake, Gerisch, J Stat. Phys, 92, 1017 (1998)) (offline)

To use the  equations in Affinity Maturation, we need to describe the dependence of fitness on affinity.

Joachim Krug's GGI seminar on fitness landscapes.

An evolutionary version of the Random Energy Model (offline)  ['House of cards' model]

HIV

Human immunodeficiency virus infects CD4 cells of the immune system: (wikipedia)

HIV life cycle and associated drug targers: [P.A. Volberding & S.G. Deeks, Lancet 376, 49 (2010)]

Progress of HIV infection: (wikipedia)

HIV Fitness Landscape

HIV mutates rapidly in host, due to:

Large number of replications (1010 virions per day) [Perelson et al, Science 271, 582 (1996)] 

Transcription errors (10-4 - 10-5 per base per cycle) [Perelson et al, Science 271, 582 (1996)]

Recombination (10-5 per site per generation) [R.A. Neher & T. Leitner, PLOS Comp. Bio. 6, e1000660 (2010)]

There is large sequence diversity: [B. Korber, IAVIReport (2010)]

Patient-derived sequences can be used to construct a prevalence landscape, using a

Maximum entropy constuction.

Patient-derived sequence frequencies are outcome of many factors, incluing

Empirical evidence suggests that prevalence is a reasonable proxy for replicative fitness

[A.L. Ferguson et al, Cell Immunity 38, 606 (2013)] (gag)

Potential uses of the prevalence landscape as a proxy for the fitness landscape:

Immunogenic versus non-immunogenic proteins

Design of vaccines

Retrodicting drug resistance sites

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 additional 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 between 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. 

GGI program on       Glassy aspects of classical and quantum computation              Florence, 5 July 2019

Can adiabatic methods help 'unsolve' a hard problem

               

                Hanrong Chen         John Barton       Arup K. Chakraborty      

Mehran Kardar (MIT)