1. Functional Integrals for the Parallel and Eigen Models of Virus Evolution Jeong-Man Park The Catholic University of Korea

2. 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

3. Evolutionary Moves Immunoglobin mutations in CDR regions DNA polymerases regulating somatic hypermutation

4. 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

5. 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

6. 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

7. The Crow-Kimura (parallel) model Genome state Hamming distance Probability to be in a given genome state

8. Creation, Annihilation Operators 1 ≤ i,j ≤ N, a,b = 1,2 Commutation relations Constraint State n j i = 1 or n j i = 0

9. State Vector Dynamics Rewrite

10. Spin Coherent State State Completeness Overlap

11. Final State Probability Probability Trotter Factorization

12. Partition Function

13. Introduce the spin field

14. z integrals performed

15. Partition Function

16. Saddle Point Approximation Stationary point Fitness

17. Fluctuation Corrections

18. Fitness to O(1/N)

19. Eigen Model Probability distribution

20. Hamiltonian & Action

21. 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