Evolutionary Computing in 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

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

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

Supermodel Evolution in Bio/Chemical Kinetics 4 We need initial estimates of model parameters NONLINEAR MODELS REQUIRE INITIAL ESTIMATES OF PARAMETERS concentration initial rate DATA computer k +1 k -1 k +2 k +3 k -3 A GIVEN MODEL ESTIMATED PARAMETERS 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 Lethal Factor (LF) protease from B. anthracis CLEAVES MITOGEN ACTIVATED PROTEIN KINASE KINASE (MAPKK) Inhibitor?

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

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,

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

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 M [I] = 44 M [I] = 88 M substrate concentration initial reaction rate

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 reaction mechanism: Given a model (rate equation), find the best-fit kinetic constants 2. Focus on multiple reaction mechanisms: 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 computer K m = ? K s = ? K i = ? K is = ? k cat = ? MODEL ESTIMATED PARAMETERS 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 K m = 1 K s = 1 K i = 1 K is = 1 k cat = 1 K m = K s = K i ~ K is = k cat =  "uncompetitive" ? ESTIMATE #1

Supermodel Evolution in Bio/Chemical Kinetics 14 First major difficulty: Sensitivity to initial estimates TRADITIONAL DATA-FITTING: RESULTS DEPEND ON THE INITIAL GUESS K m = 100 K s = 100 K i = 100 K is = 100 k cat = 100 K m < K s = K i = K is = k cat = ESTIMATE #2

Supermodel Evolution in Bio/Chemical Kinetics 15 First major difficulty: Sensitivity to initial estimates TRADITIONAL DATA-FITTING: RESULTS DEPEND ON THE INITIAL GUESS K m = 10 K s = 100 K i = 10 K is = 100 k cat = 100 K m = K s = K i = K is = k cat = mixed-type ? ESTIMATE #3

Supermodel Evolution in Bio/Chemical Kinetics 16 First major difficulty: Sensitivity to initial estimates TRADITIONAL DATA-FITTING: RESULTS DEPEND ON THE INITIAL GUESS K m = 10 K s = 100 K i = 10 K is = 100 k cat = 100 K m = K s = K i = K is = k cat = 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 Least-squares fitting only goes "downhill" How do we know where to start? MODEL PARAMETER SUM OF SQUARED DEVIATIONS  (data - model) 2 global minimum data - model = "residual"

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

Supermodel Evolution in Bio/Chemical Kinetics 19 Biological metaphor: "Gene, allele" gene BIOLOGYCOMPUTER...AAGTCG...GTAACCGG... four-letter alphabet variable length "keratin" sequence of bits representing a number two letter alphabet fixed length (16 or 32 bits) "K M ""k cat "

Supermodel Evolution in Bio/Chemical Kinetics 20 "Chromosome, genotype, phenotype" genotype BIOLOGYCOMPUTER...AAGTCGGTTCGGAAGTCGGTTTA... keratin oncoprotein phenotype particular combination of all model parameters K M =4.56 V max =1.23 K is =78.9 full set of parameters

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

Supermodel Evolution in Bio/Chemical Kinetics 22 "Population" BIOLOGYCOMPUTER low fitness V max KMKM K is medium fitness V max KMKM K is high fitness V max KMKM KisKis

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

Supermodel Evolution in Bio/Chemical Kinetics 24 "Sexual reproduction, crossover" BIOLOGYCOMPUTER mother father "sexual mating" probability p cross child random crossover point V max KMKM K is

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 "Mutation, genetic diversity" BIOLOGYCOMPUTER father V max KMKM K is mutant father V max (*) K M (*) K is (*) mutation

Supermodel Evolution in Bio/Chemical Kinetics 27 "Mutation, genetic diversity" uncle#1 V max (1) K M (1) K is (1) uncle#2 V max (2) K M (2) K is (2) THE "DIFFERENTIAL" IN DIFFERENTIAL EVOLUTION ALGORITHM - STEP 1 Compute difference between two randomly chosen "uncle" phenotypes subtract uncle#2 minus uncle #1 V max (2) -V max (1) K M (2) -K M (1) K is (2) -K is (1)

Supermodel Evolution in Bio/Chemical Kinetics 28 "Mutation, genetic diversity" father V max KMKM K is mutant father THE "DIFFERENTIAL" IN DIFFERENTIAL EVOLUTION ALGORITHM - STEP 2 Add weighted difference between two "uncle" phenotypes to "father" add a fraction of uncle#2 minus uncle #1 V max (2) -V max (2) K M (2) -K M (1) K is (2) -K is (1) V max * KM*KM* K is *

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

Supermodel Evolution in Bio/Chemical Kinetics 30 DE Mutation rate in DynaFit fractional difference used in mutations K M * = K M + F  (K M (1)  K M (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 "Selection" BIOLOGYCOMPUTER high fitness more likely to breed V max KMKM KisKis more likely to be carried to the next generation low sum of squares low fitness less likely to breed V max KMKM K is less likely to be carried to the next generation high sum of squares

Supermodel Evolution in Bio/Chemical Kinetics 33 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 5 Natural selection: keep child in gene pool if more fit than mother 4 Sexual reproduction: random crossover with probability P cross 3 Mutation: random gene modification (mutate father, weight F)

Supermodel Evolution in Bio/Chemical Kinetics 34 Application to curcumin: Mixed-type mechanism THREE EXAMPLES OF POPULATION MEMBERS (POPULATION SIZE n = 845) Km Ks1.0488e-3 Ki8.7928e-10 Kis8.1374e+20 kcat POPUL. MEMBER #642POPUL. MEMBER #7 Km Ks60218 Ki Kis kcat POPUL. MEMBER #1 Km3.3355e-11 Ks Ki Kis4.0737e-9 kcat29.783

Supermodel Evolution in Bio/Chemical Kinetics 35 Application to curcumin: Mixed-type mechanism INITIAL DISTRIBUTION OF THE MICHAELIS-CONSTANT K M 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 K m = K s = K i = K is = k cat = The fittest "phenotype" in generation #125 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 reaction mechanism: Given a model (rate equation), find the best-fit kinetic constants 2. Focus on multiple reaction mechanisms: 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 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 R2 IN PRESS EXAMPLE

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 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 ASSUMPTIONS

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" "Competitive" mechanism is a subset of the "Supermodel" K ii  K ss   k sp = 0 K si   k ip = 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" K ii   K ss   k sp = 0 Ki  Ki   k ip = 0

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

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

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

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 51 Summary and Conclusions DIFFERENTIAL EVOLUTION HAS PROVED USEFUL IN BIO/CHEMICAL KINETICS Given a particular model, we can always find globally optimal parameters: DE avoids false minima and other pathologies in nonlinear data fitting Given a set of possible bio/chemical processes, we can "evolve" the most plausible model: - 1. Evolve a "Supermodel" - 2. Exclude redundant reaction steps - 3. Check confidence intervals 

Supermodel Evolution in Bio/Chemical Kinetics 52 Acknowledgements: Lethal Factor protease work Hawaii Biotech currently Panthera BioPharma Mark Goldman Sheri Millis Lynne Cregar Aiea, Island of Oahu, Hawaii National Institutes of Health Grant No. R43 AI U.S. Army Medical Research and Materials Command Contract No. V549P-6073