Download Functional Integrals for the Parallel and Eigen Models of

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



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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