Functional Integrals for the Parallel and Eigen Models of Virus Evolution Jeong-Man Park The Catholic University of Korea
Outline Evolutionary moves Preliminary concepts The parallel model & the Eigen model Coherent states mapping to functional integral 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 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 n j i = 1 or n j 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