Stochastic and game-theoretic modeling of viral evolution

Transcription

1 Stochastic and game-theoretic modeling of viral evolution February, Department of Chemical and Biological Engineering University of Wisconsin Madison 1/33

2 Outline 1 2 Modeling viral reproduction as chemical reactions Deterministic and Stochastic Modeling Reaction Equilibrium 3 Modeling viral evolution as a game : Potential antivirals 2/33

3 Outline 1 2 Modeling viral reproduction as chemical reactions Deterministic and Stochastic Modeling Reaction Equilibrium 3 Modeling viral evolution as a game : Potential antivirals 3/33

4 A virus particle or virion is a smart, parasitic molecule nm sized bundle of protein and nucleic acid Does not grow in size Virion infects a host cell and hijacks its machinery Sole aim create more copies of itself General anatomy of a virion Genome (RNA or DNA) Polymerase Protein coat or shell Influenza A virion Why do we need to study viruses? cause deadly diseases. eg: Flu, AIDS, Hepatitis Study of viruses provides vaccines and antivirals 4/33

5 Study of viruses Four broad processes Transmission (extracellular) Infection Reproduction (intracellular) Viral evolution and reproduction are interrelated study of viral reproduction will help us understand viral evolution. Need for quantitative modeling Qualitative information is available for many viruses Quantitative information is required we need to know quantities Both experiments and modeling are required to build a solid understanding. 5/33

6 Intracellular processes Replication more copies of genome are created Transcription genes are copied into mrna Translation mrna is converted to viral proteins Packaging genome and proteins are packed into a new virion Budding and Release new virions bud out of host cell Influenza A reproduction cycle (Sidorenko & Reichl, 2004) Errors in reproduction Mutations, deletions in replication defective genomes are created Errors in packaging defective genomes are packaged 6/33

7 Outline 1 2 Modeling viral reproduction as chemical reactions Deterministic and Stochastic Modeling Reaction Equilibrium 3 Modeling viral evolution as a game : Potential antivirals 7/33

8 Modeling intracellular processes as chemical reactions Example: Translation of VSV genes N and P( Hensel (2007)) Traditional modeling Deterministic rate equations Does not work for low (1-100) numbers of molecules Cannot explain why identical conditions produce different outcomes N mrna + S 2 k 2 N mrna + N Protein P mrna + S 2 k 2 P mrna + P Protein M mrna + S 2 k 2 M mrna + M Protein G mrna + S 2 k 2 G mrna + G Protein L mrna + S 2 k 2 L mrna + L Protein Describes probability of being in a state instead of rate equations Works for low numbers of molecules Explains noise, phenotypic bifurcation, cell to cell variablity Problems Both approaches require rate constants from experimental data Large number of reactions = Many rate constants = More experiments Difference in timescales = Higher computational cost 8/33

9 Outline 1 2 Modeling viral reproduction as chemical reactions Deterministic and Stochastic Modeling Reaction Equilibrium 3 Modeling viral evolution as a game : Potential antivirals 9/33

10 Problem of timescales : Reaction Equilibrium A kf 2 k r 2 B B kf 1 C k f 2, kr 2 kf 1 Reversible reaction is much faster Fast reaction uses most of the simulation effort Reaction may be assumed at equilibrium at all times Simulation by Gillespie s algorithm n A =Number of A molecules, n B =Number of B molecules, n C =Number of C molecules. Reaction equilibrium assumption (REA) Deterministic Known for atleast 20 years Returns a modified set of reactions Stochastic Still under research Returns only a set of equations 10/33

11 Reaction Equilibrium : Deterministic version Full Model dc A dt = k f 2 c A + k r 2 c B c A (t = 0) = c A0 dc B dt = k f 2 c A k r 2 c B k f 1 c B c B (t = 0) = c B0 dc C dt = k f 1 c B c C (t = 0) = c C0 Reaction equilibrium assumption: c B = K 2 c A (K 2 = k f 2 /kr 2 ) Fast Time Scale (τ = k r 2 t): Reduced Model A kf 2 B k 2 r B kf 1 C k f 2, kr 2 kf 1 dc A dτ = K2c A + c B c A (τ = 0) = c A0 dc B dτ = K2c A c B c B (τ = 0) = c B0 dc C dτ = 0 c C (τ = 0) = c C0 Slow Time Scale (t): dc A dt = k c A c A (t = 0) = (c A0 +c B0 ) (1+K 2 ) dc B dt = k c B c B (t = 0) = K 2 (c A0 +c B0 ) (1+K 2 ) dc C dt = k c A + k c B c C (t = 0) = c C0 B k C A k C k = k1 f K 2 (1 + K 2 ) 11/33

12 Reaction Equilibrium : Stochastic version A kf 2 k r 2 B B kf 1 C k f 2, kr 2 kf 1 Initial Condition : a=number of molecules of A b=number of molecules of B c=number of molecules of C x 2=net number of times reversible reaction occurs in forward direction x 1=number of times irreversible reaction occurs (a, b, c) is called state of the system x 1, x 2 are called reaction extents a(t = 0) = a 0, b(t = 0) = b 0, c(t = 0) = c 0 x 1(t = 0) = 0, x 2(t = 0) = 0 a = a 0 x 2 States are related to extents : b = b 0 x 1 + x 2 c = c 0 + x 1 Extents are restricted to : x 2 [ b 0, a 0], x 1 [0, x 2 + b 0] Chemical Master Equation (CME) in extent form d dt P(x1, x2; t) =kf 1 (b0 x1 + x2)p(x1 1, x2) kf 1 (b0 x1 + x2)p(x1, x2) + k f 2 (a0 x2)p(x1, x2 1) + kr 2 (b0 x1 + x2 + 1)P(x1, x2 + 1) `k f 2 (a0 x2) + kr 2 (b0 x1 + x2) P(x 1, x 2) 12/33

13 Reaction Equilibrium : Stochastic version Using the same approach as Haseltine and Rawlings (2005) Chemical Master Equation equation REA Equation + Algebraic Constraint d dt P(x1; t) = k `a 0 + b 0 (x 1 1) P(x 1 1; t) k `a 0 + b 0 x 1 P(x1; t) Algebraic constraint k f 2 (a0 x2 + 1)P A(x 2 1 x 1) + k r 2 (b0 x1 + x2 + 1)P A(x x 1) `k f 2 (a0 x2) + kr 2 (b0 x1 + x2) P A (x 2 x 1) = 0 equation Algebraic constraint Contains only slow extent x 1 Represents Z k C; Z = A or B Uses the same Gillespie s algorithm Represents the reaction equilibrium assumption Analogous to c B = K 2c A Requires another auxiliary simulation 13/33

14 Compare deterministic & stochastic versions For (a 0, b 0, c 0) = (1, 1, 1) and k = 1, Deterministic Stochastic Model B k C Z k C Admissible State space A k C (1,1,1), (1,0,2), (0,1,2), (0,0,3) Algebraic constraint P(1, 1, 1) e 2t 2K 2 (1+K 2 ) 2 e 2t (1,1,1), (2,0,1), (0,2,1), (1,0,2), (0,1,2), (0,0,3) P(2, 0, 1) 0 1 (1+K 2 ) 2 e 2t P(0, 2, 1) 0 K 2 2 (1+K 2 ) 2 e 2t P(1, 0, 2) e t e 2t 2K 2 (1+K 2 ) (e t e 2t ) P(0, 1, 2) e t e 2t 2 (1+K 2 ) (e t e 2t ) P(0, 0, 3) e 2t 2e t + 1 e 2t 2e t + 1 Deterministic and stochastic reductions are different 14/33

15 Outline 1 2 Modeling viral reproduction as chemical reactions Deterministic and Stochastic Modeling Reaction Equilibrium 3 Modeling viral evolution as a game : Potential antivirals 15/33

16 Reaction equilibrium assumption Obtain a complete reaction network instead of algebraic constraint Reactions can be simulated using Gillespie s algorithm no need of auxiliary simulation Easier to understand and use Extend the same procedure to non-linear reactions Modeling intracellular processes REA may be used to obtain reduced set of reactions. 16/33

17 Outline 1 2 Modeling viral reproduction as chemical reactions Deterministic and Stochastic Modeling Reaction Equilibrium 3 Modeling viral evolution as a game : Potential antivirals 17/33

18 : Parasites of viruses Defective Interfering Particle (DIP) DIP is a virion which has incomplete or defective genome Replication errors create incomplete genomes Defective genomes cannot replicate in absence of complete genomes DIP parasitizes complete virus DI virus uses gene products created by complete virus Defective genomes replicate much faster Yield of complete virus decreases drastically Other parasites of viruses Satellite RNA and viruses Covirus E.g.: LMSC (satellite RNA) and STNV (satellite virus) which parasitize Tobacco Mosaic virus (TNV) E.g.: Tobacco Rattle virus (TRV), Pea Enation Mosaic virus (PENV), SV40-DIs 18/33

19 : Antiviral use of DIPs Examples Influenza A ( Dimmock and Marriott (2006)) : DI viruses save mice from Influenza A induced death (in vivo) SV40/SV40-DI : Pair of complementary DIPs eliminate wt SV40 (in vitro) LCMV/LCMV-DI : DIPs remove any trace of infection (in vitro) Counter example DIPs cannot always cure infection In VSV/VSV-DI system, DIPs and complete virus coevolve perpetually ( DePolo et al. (1987)) Designing a DI particle Design a DI particle which beats the complete virus Examples of such attempts Nelson and Perelson (1995): Describes interaction of a genetically designed DIP to cure HIV-1 infection Weinberger et al. (2003): Provides a theoretical citerion for designing crhiv-1 parasites that can persist in vivo 19/33

20 Outline 1 2 Modeling viral reproduction as chemical reactions Deterministic and Stochastic Modeling Reaction Equilibrium 3 Modeling viral evolution as a game : Potential antivirals 20/33

21 Consider an evolutionary where there are n s different kinds of genetic variations of a virus, x i (r) number of virions of i th variation at time r, and A ij represents the payoff obtained by species i when it interacts with j n p = P i=n s i=1 x i (0) = total number of virions at time r. of x(r) x i (r + 1) = [n px i (r) (Ax(r)) i x(r) Ax(r) ] i = 1,..., ns For example, in case of virus-dip co-evolution i th variation is a DIP j th variation is a complete virus When virions of i th and j th virion co-infect a cell, A ij represents the number of progeny DIPs A ji represents the number of progeny complete viruses Assuming single-hit interference 1, A ji = 0. 1 one DIP stops production of complete viruses 21/33

22 Outline 1 2 Modeling viral reproduction as chemical reactions Deterministic and Stochastic Modeling Reaction Equilibrium 3 Modeling viral evolution as a game : Potential antivirals 22/33

23 Virus-DIP interaction as a game of General features Two complete viruses co-infect a cell both create progenies Complete virus and a DIP co-infect a cell much more DIPs are created Two DIPs co-infect a cell no progenies are created This corresponds to a game of, where virus acts as a cooperator (C) and DIP acts as a defector (D). Payoff Matrix = Player 1 C D Player 2 C D «3, 3 0, 5 5, 0 1, 1 Two-player game Both players have two choices to cooperate (C) or to defect (D) Defection is a dominant strategy 23/33

24 game in viruses Turner and Chao (1999) Competition between RNA phage φ6 and its successor φh2 game of Payoff was calculated as the relative change in number of virions before and after infection Higher payoff meant more progeny creation Iterated Repeated rounds of game between the same two players Simply defecting at every round is not the dominant strategy 24/33

25 Iterated (IPD) Iterated allows us to form complicated strategies. For example, Always Cooperate (ALLC) : Cooperates at every round Always Defect (ALLD) : Defects at every round Tit-for-tat (TFT) : Cooperates if opponent cooperated in the last round, otherwise defects Here, a strategy represents a particular genetic variation or profile of a virus ALLC represents a complete virus which does not evolve it keeps on producing gene products which are usable by defectors ALLD represents a DIP which encodes no useful genes it fully exploits the complete virus TFT could represent a continuously evolving virus n s genetic variations = n s strategies n p players 25/33

26 Finding the best strategy Designing a suitable DI particle is like finding a suitable strategy Lets first find the best strategy for this example The strategy space is large We restrict ourselves to Markovian strategies a player remembers only the last round of the game Markovian strategies can be stochastic or deterministic uncountable number of Stochastic strategies, only 32 deterministic strategies The best strategy depends on other strategies in the pool We restrict ourselves to only deterministic strategies only. Consider n s strategies, x i (r) players per strategy Players play an infinitely repeated game in a round-robin fashion 2 Payoffs A ij and A ji are given as A ij = A ji = lim n r lim n r P nr r=0 CMr y(0) n r P nr r=0 C M r y(0) n r 2 Every player against every other player 26/33

27 Outline 1 2 Modeling viral reproduction as chemical reactions Deterministic and Stochastic Modeling Reaction Equilibrium 3 Modeling viral evolution as a game : Potential antivirals 27/33

28 Results 1 Start with 100 players of each strategy. n s = 32, n p = /33

29 Results 2 Start with 100 players of each strategy. n s = 32, n p = Top strategies that survive Strategy Name Equilibrium number GRIM 2123 TFT 690 WSLS 216 S ALLC 2 ALLD 0 GRIM Starts with a cooperative move Keeps on cooperating until opponent defects Keeps defecting from then on (never fogives) 29/33

30 Outline 1 2 Modeling viral reproduction as chemical reactions Deterministic and Stochastic Modeling Reaction Equilibrium 3 Modeling viral evolution as a game : Potential antivirals 30/33

31 1 Repeated Prisoner s Dilemma Game Analytical derivation of Optimal Invading strategy Formulation as an optimization problem: p, p 0 = arg max x p,p v inf 0 such that x inf i = ˆn px inf i Analysis of non-markovian strategies (Ax inf ) i (x inf ) A(x inf ) i = 1,..., n s Combining intracellular modeling with Game theory A game theoretic model needs information from intracellular models. For example: intracellular equations to predict the increase in number of progenies of a DI particle which skips a particular gene Possible design of optimal therapeutic strategies. Example: Determine the optimal size or number of RNAs that need to be packaged to create a desired DI particle 31/33

32 2 Obtain payoff values from experimental data Number of DIPs and complete viruses have been experimentally obtained( Thompson et al. (2009), DePolo et al. (1987)) An approach similar to Turner and Chao (1999) may be used Inclusion of host evolution Viral infection triggers host evolution as well 32/33

33 I N. J. DePolo, C. Giachetti, and J. J. Holland. Continuing Coevolution of Virus and Defective Interfering Particles and of Viral Genome Sequences during Undiluted Passages: Virus Mutants Exhibiting Nearly Complete Resistance to Formerly Dominant Defective Interfering Particles. J. Virol., 61(2): , N. J. Dimmock and A. C. Marriott. In vivo antiviral activity: defective interfering virus protects better against virulent Influenza A virus than avirulent virus. J. Gen. Virol., 87: , E. L. Haseltine and J. B. Rawlings. On the origins of approximations for stochastic chemical kinetics. J. Chem. Phys., 123:164115, October S. Hensel. Stochastic kinetic modeling of the vesicular stomatitis virus (vsv). Master s thesis, University of Wisconsin Madison, October URL G. W. Nelson and A. S. Perelson. Modeling Defective Interfering Virus Therapy for AIDS: Coditions for DIV Survival. Mat. Biosci., 125: , K. A. S. Thompson, G. A. Rempala, and J. Yin. Multiple-hit inhibition of infection by defective interfering particles. J. Gen. Virol., 90: , P. Turner and L. Chao. in an RNA virus. Nature, 398: , L. S. Weinberger, D. V. Schaffer, and A. P. Arkin. Theoretical Design of a Gene Therapy to Prevent AIDS but not Human Immunodeficiency Virus Type 1 Infection. 77(18): , /33