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Evolutionary Computing in the Study of Bio/Chemical Mechanisms

Utilizing Evolutionary Computing to find initial parameter estimates for bio/chemical mechanisms using the differential evolution model selection strategy. Case study: Inhibition of Lethal Factor protease by curcumin.

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Evolutionary Computing in the Study of Bio/Chemical Mechanisms

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  1. Evolutionary Computingin the Study of Bio/Chemical Mechanisms Petr Kuzmič, Ph.D.BioKin, Ltd. Problem: Finding initial parameter estimates Solution: Differential Evolution (DE) Model selection strategy:"Supermodel Evolution" Example: Inhibition of Lethal Factor protease by curcumin

  2. The task of mechanistic enzyme kinetics SELECT AMONG MULTIPLE CANDIDATE MECHANISMS initial rate competitive ? competitive ? uncompetitive ? concentration computer mixed type ? DATA MODELS Select most plausible model Supermodel Evolution in Bio/Chemical Kinetics

  3. Most models in bio/chemical kinetics are nonlinear LINEAR VS. NONLINEAR MODELS Linear Nonlinear y = A + kx y = A [1 - exp(-kx)] y = 2 +5x y = 2 [1 - exp(-5x)] Supermodel Evolution in Bio/Chemical Kinetics

  4. We need initial estimates of model parameters NONLINEAR MODELS REQUIRE INITIAL ESTIMATES OF PARAMETERS ESTIMATED PARAMETERS initial rate k+1 k-1 k+2 k+3 k-3 A GIVEN MODEL concentration DATA computer k+1 k-1 k+2 k+3 k-3 BEST-FIT PARAMETERS Supermodel Evolution in Bio/Chemical Kinetics

  5. Anthrax bacillus CUTANEOUS AND INHALATION ANTHRAX DISEASE Supermodel Evolution in Bio/Chemical Kinetics

  6. Inhibitor? Lethal Factor (LF) protease from B. anthracis CLEAVES MITOGEN ACTIVATED PROTEIN KINASE KINASE (MAPKK) Supermodel Evolution in Bio/Chemical Kinetics

  7. Neomycin B: an aminoglycoside inhibitor A POTENT INHIBITOR OF LF PROTEASE Neomycin B Streptomyces fradiae Supermodel Evolution in Bio/Chemical Kinetics

  8. Neomycin B mechanism: mixed-type noncompetitive NEOMYCIN B INHIBITION FOLLOWS A COMPLEX MOLECULAR MECHANISM Kuzmic et al. (2006) FEBS J.273, 3054-3062. Supermodel Evolution in Bio/Chemical Kinetics

  9. Curcumin: an natural product inhibitor INHIBITION MECHANISM UNKNOWN curcumin cumin Cuminum cyminum L. Supermodel Evolution in Bio/Chemical Kinetics

  10. LF protease inhibition by curcumin: raw data SUBSTRATE INHIBITION (MAXIMUM ON SUBSTRATE SATURATION CURVE) [I] = 0 [I] = 22 mM [I] = 44 mM initial reaction rate [I] = 88 mM substrate concentration Supermodel Evolution in Bio/Chemical Kinetics

  11. Two separate problems to solve A PREREQUISITE FOR MODEL DISCRIMINATION = FITTING INDIVIDUAL CANDIDATE MODELS 1. Focus on a single reactionmechanism: Given a model (rate equation), find the best-fit kinetic constants 2. Focus on multiple reactionmechanisms: a. Repeat 1. for all candidate models (mechanisms) b. Select the most plausible model Supermodel Evolution in Bio/Chemical Kinetics

  12. The mixed-type inhibition model CONTAINS FIVE KINETIC CONSTANTS DATA ESTIMATED PARAMETERS Km = ? Ks = ? Ki = ? Kis = ? kcat = ? MODEL computer BEST-FIT PARAMETERS Supermodel Evolution in Bio/Chemical Kinetics

  13. First major difficulty: Sensitivity to initial estimates TRADITIONAL DATA-FITTING: RESULTS DEPEND ON THE INITIAL GUESS Km = 1 Ks = 1 Ki = 1 Kis = 1 kcat = 1 ESTIMATE #1 Km = Ks = Ki ~ Kis = kcat = 2000 0.6  0.0001 5100 "uncompetitive" ? Supermodel Evolution in Bio/Chemical Kinetics

  14. First major difficulty: Sensitivity to initial estimates TRADITIONAL DATA-FITTING: RESULTS DEPEND ON THE INITIAL GUESS Km = 100 Ks = 100 Ki = 100 Kis = 100 kcat = 100 ESTIMATE #2 Km < Ks = Ki = Kis = kcat = 0.000001 0.004 0.002 0.02 54000 Supermodel Evolution in Bio/Chemical Kinetics

  15. First major difficulty: Sensitivity to initial estimates TRADITIONAL DATA-FITTING: RESULTS DEPEND ON THE INITIAL GUESS Km = 10 Ks = 100 Ki = 10 Kis = 100 kcat = 100 ESTIMATE #3 Km = Ks = Ki = Kis = kcat = 10 110 40 14 68 mixed-type ? Supermodel Evolution in Bio/Chemical Kinetics

  16. First major difficulty: Sensitivity to initial estimates TRADITIONAL DATA-FITTING: RESULTS DEPEND ON THE INITIAL GUESS Km = 10 Ks = 100 Ki = 10 Kis = 100 kcat = 100 Km = Ks = Ki = Kis = kcat = 10 110 40 14 68 ? ESTIMATE #3 Is this the best we can do with the mixed-type model? Supermodel Evolution in Bio/Chemical Kinetics

  17. The crux of the problem: Finding global minima SUM OF SQUARED DEVIATIONS S(data - model)2 data - model = "residual" global minimum MODEL PARAMETER • Least-squares fitting only goes "downhill" • How do we know where to start? Supermodel Evolution in Bio/Chemical Kinetics

  18. Charles Darwin to the rescue BIOLOGICAL EVOLUTION IMITATED IN "DE" Charles Darwin (1809-1882) ISBN-10: 3540209506 Supermodel Evolution in Bio/Chemical Kinetics

  19. sequence of bits representing a number ...01110011000001101101110011... "KM" "kcat" • two letter alphabet • fixed length (16 or 32 bits) Biological metaphor: "Gene, allele" BIOLOGY COMPUTER gene ...AAGTCG...GTAACCGG... "keratin" four-letter alphabet variable length Supermodel Evolution in Bio/Chemical Kinetics

  20. particular combination of all model parameters 011010110110011110011010001111101101 Kis=78.9 Vmax=1.23 KM=4.56 full set of parameters "Chromosome, genotype, phenotype" BIOLOGY COMPUTER genotype ...AAGTCGGTTCGGAAGTCGGTTTA... keratin oncoprotein phenotype Supermodel Evolution in Bio/Chemical Kinetics

  21. FITNESS: agreement between the data and the model "Organism, fitness" BIOLOGY COMPUTER genotype ...AAGTCGGTTCGGAAGTCGGTTTA... keratin oncoprotein FITNESS:"agreement" with the environment Supermodel Evolution in Bio/Chemical Kinetics

  22. Vmax KM Kis high fitness medium fitness Vmax KM Kis low fitness Vmax KM Kis "Population" BIOLOGY COMPUTER Supermodel Evolution in Bio/Chemical Kinetics

  23. DE Population size in DynaFit number of population membersper optimized model parameter number of population membersper order of magnitude Supermodel Evolution in Bio/Chemical Kinetics

  24. random crossover point mother 01101011011001111001101 00011111011 father 01101011011001111001101 11100011011 "sexual mating" probability pcross child 01101011011001111001101 11100011011 Vmax KM Kis "Sexual reproduction, crossover" BIOLOGY COMPUTER Supermodel Evolution in Bio/Chemical Kinetics

  25. DE Crossover probability in DynaFit probability that child inherits father's genes, not mother's genes Supermodel Evolution in Bio/Chemical Kinetics

  26. father 01101011011 001111001101 11100011011 mutation mutant father 11100111011 001011010101 11001011001 Vmax Vmax(*) KM KM(*) Kis Kis(*) "Mutation, genetic diversity" BIOLOGY COMPUTER Supermodel Evolution in Bio/Chemical Kinetics

  27. Vmax(2) Vmax(2)-Vmax(1) Vmax(1) KM(2) KM(1) KM(2)-KM(1) Kis(2) Kis(2)-Kis(1) Kis(1) "Mutation, genetic diversity" THE "DIFFERENTIAL" IN DIFFERENTIAL EVOLUTION ALGORITHM - STEP 1 Compute difference between two randomly chosen "uncle" phenotypes uncle#1 01101011011 001111001101 11100011011 subtract uncle#2 11100111011 001011010101 11001011001 uncle#2 minus uncle #1 11100111011 001011010101 11001011001 Supermodel Evolution in Bio/Chemical Kinetics

  28. Vmax* Vmax Vmax(2)-Vmax(2) KM KM(2)-KM(1) KM* Kis* Kis(2)-Kis(1) Kis "Mutation, genetic diversity" THE "DIFFERENTIAL" IN DIFFERENTIAL EVOLUTION ALGORITHM - STEP 2 Add weighted difference between two "uncle" phenotypes to "father" father 01101011011 001111001101 11100011011 add a fraction of uncle#2 minus uncle #1 11100111011 001011010101 11001011001 mutant father 11100111011 001011010101 11001011001 Supermodel Evolution in Bio/Chemical Kinetics

  29. "Mutation, genetic diversity" THE "DIFFERENTIAL" IN DIFFERENTIAL EVOLUTION ALGORITHM EXAMPLE: Michaelis-Menten equation "father" "uncle 1" "uncle 2" KM* = KM + F (KM(1)KM(2)) "mutant father" weight (fraction) mutation rate Supermodel Evolution in Bio/Chemical Kinetics

  30. DE Mutation rate in DynaFit fractional difference used in mutations KM* = KM + F (KM(1)KM(2)) Supermodel Evolution in Bio/Chemical Kinetics

  31. DE Mutation strategies in DynaFit six different mutation strategies (1, 2, ... 6) details in the book: Supermodel Evolution in Bio/Chemical Kinetics

  32. low sum of squares 0110101101100111100110100011111011 Vmax KM Kis high fitness more likely to breed more likely to be carried to the next generation high sum of squares 0000000000111111111111100000000000 less likely to breed Vmax KM Kis low fitness less likely to be carried to the next generation "Selection" BIOLOGY COMPUTER Supermodel Evolution in Bio/Chemical Kinetics

  33. 5 Natural selection: keep child in gene pool if more fit than mother 4 Sexual reproduction:random crossover with probability Pcross 3 Mutation:random gene modification (mutate father, weight F) Basic Differential Evolution Algorithm - Summary 1 Randomly create the initial population (size N) Repeat until almost all population members have very high fitness:2 Evaluate fitness: sum of squares for all population members Supermodel Evolution in Bio/Chemical Kinetics

  34. Application to curcumin: Mixed-type mechanism THREE EXAMPLES OF POPULATION MEMBERS (POPULATION SIZE n = 845) POPUL. MEMBER #1 POPUL. MEMBER #7 POPUL. MEMBER #642 Km 65.7630 Ks 1.0488e-3 Ki 8.7928e-10 Kis 8.1374e+20 kcat 3760400 Km 3.3355e-11 Ks 7868500 Ki 106.99 Kis 4.0737e-9 kcat 29.783 Km 3713600 Ks 60218 Ki 427880 Kis 153.55 kcat 918170 Supermodel Evolution in Bio/Chemical Kinetics

  35. Application to curcumin: Mixed-type mechanism INITIAL DISTRIBUTION OF THE MICHAELIS-CONSTANT KM 24 orders of magnitude ESTIMATE # Supermodel Evolution in Bio/Chemical Kinetics

  36. Application to curcumin: Mixed-type mechanism INITIAL DISTRIBUTION OF ALL MODEL PARAMETERS Generation #1 Supermodel Evolution in Bio/Chemical Kinetics

  37. Application to curcumin: Mixed-type mechanism INTERMEDIATE DISTRIBUTION OF MODEL PARAMETERS (EVOLUTION IS ONGOING) Generation #20 Supermodel Evolution in Bio/Chemical Kinetics

  38. Application to curcumin: Mixed-type mechanism FINAL DISTRIBUTION OF MODEL PARAMETERS (EVOLUTION IS COMPLETED) Generation #125 Supermodel Evolution in Bio/Chemical Kinetics

  39. Application to curcumin: Mixed-type mechanism THE SUPER-ORGANISM The fittest "phenotype" in generation #125 Km = Ks = Ki = Kis = kcat = 10 110 40 14 68 ! Yes, this is the best we can do with the mixed-type model! Supermodel Evolution in Bio/Chemical Kinetics

  40. The model selection problem remains A PREREQUISITE FOR MODEL DISCRIMINATION = FITTING INDIVIDUAL CANDIDATE MODELS  1. Focus on a single reactionmechanism: Given a model (rate equation), find the best-fit kinetic constants 2. Focus on multiple reactionmechanisms: a. Repeat 1. for all candidate models (mechanisms) b. Select the most plausible model ? Supermodel Evolution in Bio/Chemical Kinetics

  41. Model proliferation: CYP450 / reductase example COMBINATORIAL PROLIFERATION OF POSSIBLE MECHANISMS EXAMPLE Global Analysis of Protein-Protein Interactions Reveals Multiple Cytochrome P450 2E1 / Reductase Complexes Arvind P. Jamakhandi‡, Petr Kuzmič§, Daniel E. Sanders‡, and Grover P. Miller‡ Journal: Biochemistry Ms# BI-2006-003476.R2 IN PRESS Supermodel Evolution in Bio/Chemical Kinetics

  42. Model proliferation: CYP450 / reductase example COMBINATORIAL PROLIFERATION OF POSSIBLE MECHANISMS P = cytochrome P450 (2E1) R = cytochrome reductase A = catalytically active N = inactive 42 separate mechanisms were examined Supermodel Evolution in Bio/Chemical Kinetics

  43. The "Supermodel" approach CREATE AN AGGREGATE MODEL ENCOMPASSING ALL POSSIBLE INTERACTIONS 1. Consider only the most complex model 2. Evolve all parameters using "DE" 3. Identify redundant parameters by examining the final distribution of fittest phenotypes 4. Eliminate redundant parameters thereby reducing the model ("small is beautiful") the most complex (realistic) model Supermodel Evolution in Bio/Chemical Kinetics

  44. The "Supermodel" for LF inhibition by curcumin COMPILE AN AGGREGATE OF ALL POSSIBLE MOLECULAR INTERACTIONS ASSUMPTIONS • Substrate can bind with 1:1 or 2:1 stoichiometry • Inhibitor can bind with 1:1 or 2:1 stoichiometry • Substrate and inhibitor can bind at the same time • Any enzyme-substrate complex can have catalytic activity Supermodel Evolution in Bio/Chemical Kinetics

  45. The "Supermodel" includes all standard mechanisms STANDARD MECHANISMS DIFFER ONLY BY VALUES OF KINETIC CONSTANTS IN THE "SUPERMODEL" kip = 0 "Competitive" mechanism is a subset of the "Supermodel" Ksi  Kii  Kss  ksp = 0 Supermodel Evolution in Bio/Chemical Kinetics

  46. The "Supermodel" includes all standard mechanisms STANDARD MECHANISMS DIFFER ONLY BY VALUES OF KINETIC CONSTANTS IN THE "SUPERMODEL" "Uncompetitive" mechanism is a subset of the "Supermodel" kip = 0 Ki  Kii  Kss  ksp = 0 Supermodel Evolution in Bio/Chemical Kinetics

  47. "Supermodel" evolution: Curcumin inhibition of LF INITIAL DISTRIBUTION OF MODEL PARAMETERS (POPULATION SIZE = 1355) totalstability constants Supermodel Evolution in Bio/Chemical Kinetics

  48. 10-12 ZERO "Supermodel" evolution: Curcumin inhibition of LF FINAL DISTRIBUTION OF MODEL PARAMETERS Supermodel Evolution in Bio/Chemical Kinetics

  49. kp sum of squares 95% likelihood logarithm of parameter Kii Curcumin inhibition of LF: Confidence intervals ARE ALL PARAMETERS NECESSARY? Supermodel Evolution in Bio/Chemical Kinetics

  50. Curcumin inhibition of LF: Final model FULLY EVOLVED "SUPERMODEL" "Partial Mixed-Type Noncompetitive" "With Substrate Inhibition" Supermodel Evolution in Bio/Chemical Kinetics

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