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Functional Integrals for the Parallel and Eigen Models
of Virus Evolution
Jeong-Man ParkThe Catholic University of Korea
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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
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Evolutionary Moves
Immunoglobin mutations in CDR regions
DNA polymerases regulating somatic hypermutation
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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
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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
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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
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The Crow-Kimura (parallel) model
Genome state
Hamming distance Probability to be in a given genome state
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Creation, Annihilation Operators 1 ≤ i,j ≤ N, a,b = 1,2 Commutation relations
Constraint
State nji = 1 or nj
i = 0
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State Vector Dynamics
Rewrite
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Spin Coherent State State
Completeness
Overlap
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Final State Probability Probability Trotter Factorization
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Partition Function
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Introduce the spin field
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z integrals performed
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Partition Function
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Saddle Point Approximation
Stationary point
Fitness
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Fluctuation Corrections
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Fitness to O(1/N)
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Eigen Model
Probability distribution
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Hamiltonian & Action
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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