Functional Integrals for the Parallel and Eigen Models

of Virus Evolution

Jeong-Man ParkThe Catholic University of Korea

Outline

Evolutionary moves Preliminary concepts The parallel model & the Eigen model Coherent states mapping to functional inte

gral Saddle point limit Gaussian fluctuations: The determinant Conclusions and extensions

Evolutionary Moves

Immunoglobin mutations in CDR regions

DNA polymerases regulating somatic hypermutation

Evolutionary Moves

Evolution of drug resistance in bacteria (success of bacteria as a group stems from the capacity to acquire genes from a diverse range of species)

Mutations in HIV-1 protease and recombination rates

Preliminary Concepts

Fitness For immune system: binding constant For protein evolution: performance In general

Temporal persistence Number of offspring

Sequence Space N letters from alphabet of size l l = 2, 4, 20 reasonable N can be from 10 to 100,000

General Properties Distribution of population around peak Mutation: increases diversity Selection: decreases diversity Error threshold: > c delocalization

Mutation Mutation error occur in two ways

Mutations during replication (Eigen model) Rate of 10-5 per base per replication for viruses

Mutations without cell division (parallel model) Occurs in bacteria under stress Rate not well characterized

The Crow-Kimura (parallel) model

Genome state

Hamming distance Probability to be in a given genome state

Creation, Annihilation Operators 1 ≤ i,j ≤ N, a,b = 1,2 Commutation relations

Constraint

State nji = 1 or nj

i = 0

State Vector Dynamics

Rewrite

Spin Coherent State State

Completeness

Overlap

Final State Probability Probability Trotter Factorization

Partition Function

Introduce the spin field

z integrals performed

Partition Function

Saddle Point Approximation

Stationary point

Fitness

Fluctuation Corrections

Fitness to O(1/N)

Eigen Model

Probability distribution

Hamiltonian & Action

Conclusions

We have formulated Crow-Kimura and Eigen models as functional integrals

In the large N limit, these models can be solved exactly, including O(1/N) fluctuation corrections

Variance of population distribution in genome space derived

Generalizations Q > 2 K > 1 Random replication landscape Other evolutionary moves