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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 c Error threshold: > 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 nj i =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