Transcript Functional Integrals for the Parallel and Eigen Models of Virus

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
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Immunoglobin mutations
in CDR regions
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DNA polymerases
regulating somatic
hypermutation
Evolutionary Moves
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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)
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Mutations in HIV-1
protease and recombination
rates
Preliminary Concepts
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Fitness
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For immune system: binding constant
For protein evolution: performance
In general
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Temporal persistence
Number of offspring
Sequence Space
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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
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Distribution of population around peak
Mutation: increases diversity
Selection: decreases diversity
c
Error threshold:  >  delocalization
Mutation
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Mutation error occur in two ways
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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
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Genome state
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Hamming distance
Probability to be in a given genome state
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Creation, Annihilation Operators
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1 ≤ i,j ≤ N, a,b = 1,2
Commutation relations
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Constraint
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State
nj
i
=1
or
nj
i
=0
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State Vector
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Dynamics
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Rewrite
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Spin Coherent State
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State
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Completeness
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Overlap
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Final State Probability
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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
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Stationary point
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Fitness
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Fluctuation Corrections
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Fitness to O(1/N)
Eigen Model
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Probability distribution
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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
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Q>2
K>1
Random replication landscape
Other evolutionary moves