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Petr Kuzmič , Ph.D. BioKin, Ltd. WATERTOWN, MASSACHUSETTS, U.S.A.

I N N O V A T I O N L E C T U R E S ( I N N O l E C ). Binding and Kinetics for Experimental Biologists Lecture 1 Numerical Models for Biomolecular Interactions. Petr Kuzmič , Ph.D. BioKin, Ltd. WATERTOWN, MASSACHUSETTS, U.S.A. Lecture outline.

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Petr Kuzmič , Ph.D. BioKin, Ltd. WATERTOWN, MASSACHUSETTS, U.S.A.

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  1. I N N O V A T I O N L E C T U R E S (I N N O l E C) Binding and Kinetics for Experimental BiologistsLecture 1Numerical Models for Biomolecular Interactions Petr Kuzmič, Ph.D.BioKin, Ltd. WATERTOWN, MASSACHUSETTS, U.S.A.

  2. Lecture outline • The problem:Traditional equations for fitting biomolecular binding datarestrict the experimental design. Typically, at least one componentmust be present in very large excess. • The solution:Abandon algebraic equations entirely. Use iterative numerical models,which can be derived automatically by the computer. • An implementation:Software DynaFit. • An example:Kinetics of forked DNA binding to the protein-protein complex formedby DNA-polymerase sliding clamp (gp45) and clamp loader (gp44/62). BKEB Lec 1: Numerical Models

  3. molecular mechanism algebraic model Traditional approach is based on algebraic models A typical “cookbook” for experimental biologists Cold Spring Harbor Laboratory Press Cold Spring Harbor, NY, 2007 ISBN-10: 0879697369 BKEB Lec 1: Numerical Models

  4. Algebraic models restrict experiment design Goodrich & Kugel (2006) Binding and Kinetics for Molecular Biologists EXAMPLE: Determine the association rate constant for A + B  AB BKEB Lec 1: Numerical Models

  5. Experimental handbooks are full of restrictions Goodrich & Kugel (2007) Binding and Kinetics for Molecular Biologists p. 34 p. 79 p. 120 p. 126 etc. BKEB Lec 1: Numerical Models

  6. Applies under all conditions Applies only when [B] >> [A] Numerical vs. algebraic mathematical models FROM A VARIETY OF ALGEBRAIC EQUATIONS TO A UNIFORM SYSTEM OF DIFFERENTIAL EQUATIONS k1 EXAMPLE: Determine the rate constant k1 and k-1for A + B AB k-1 ALGEBRAIC EQUATIONS DIFFERENTIAL EQUATIONS d[A]/dt = -k1[A][B] + k-1[AB] d[B]/dt = -k1[A][B] + k-1[AB] d[AB]/dt = +k1[A][B] – k-1[AB] BKEB Lec 1: Numerical Models

  7. DIFFERENTIALMODEL + + + - - - - - Advantages and disadvantages of numerical models THERE IS NO SUCH THING AS A FREE LUNCH ADVANTAGE ALGEBRAICMODEL - - - + + + + + can be derived for any molecular mechanism can be derived automatically by computer can be applied under any experimental conditions can be evaluated without specialized software requires very little computation timedoes not require an initial estimateis resistant to truncation and round-off errors has a long tradition: many papers published BKEB Lec 1: Numerical Models

  8. http://www.biokin.com/dynafit DOWNLOAD Specialized numerical software: DynaFit MORE THAN 600 PAPERS PUBLISHED WITH IT (1996 – 2009) 2009 FREE for academic users Kuzmic (2009) Meth. Enzymol.,467, 247-280 BKEB Lec 1: Numerical Models

  9. DynaFit software: Citation analysis APPROXIMATELY 850 JOURNAL ARTICLES PUBLISHED SINCE 1998 Kuzmic, P. (1996) "Program DYNAFIT for the analysis of enzyme kinetic data: Application to HIV proteinase" Anal. Biochem.237, 260-273. Kuzmic, P. (2009) "DynaFit - A software package for enzymology“ Meth. Enzymol.467, 247-280. BKEB Lec 1: Numerical Models

  10. MONOMOLECULAR REACTIONS rate is proportional to [A] - d [A] / d t = k [A] monomolecular rate constant1 / time BIMOLECULAR REACTIONS rate is proportional to [A]  [B] - d [A] / d t = - d [B] / d t = k [A]  [B] bimolecular rate constant1 / (concentration  time) Theoretical foundations: Mass Action Law RATE IS PROPORTIONAL TO CONCENTRATION(S) “rate” … “derivative” BKEB Lec 1: Numerical Models

  11. - d [A] / d t = + d [P] / d t = + d [Q] / d t COMPOSITION RULE ADDITIVITY OF TERMS FROM SEPARATE REACTIONS mechanism: d [B] / d t = + k1 [A] - k2 [B] Theoretical foundations: Mass Conservation Law PRODUCTS ARE FORMED WITH THE SAME RATE AS REACTANTS DISAPPEAR EXAMPLE BKEB Lec 1: Numerical Models

  12. EXAMPLE MECHANISM RATE EQUATIONS d[P] / d t = d[EAB] / d t = Composition Rule: Example + k+5 [EAB] + k+2 [EA][B] - k-2 [EAB] + k+4 [EB][A] - k-4 [EAB] - k+5 [EAB] Similarly for other species... BKEB Lec 1: Numerical Models

  13. Rate terms: Rate equations: Input (plain text file): E + S ---> ES : k1 ES ---> E + S : k2 ES ---> E + P : k3 k1 [E]  [S] k2 [ES] +k1 [E]  [S] -k2 [ES] -k3 [ES] k3 [ES] A "Kinetic Compiler" HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS -k1 [E]  [S] d[E] / dt = +k2 [ES] +k3 [ES] d[ES] / dt = Similarly for other species... BKEB Lec 1: Numerical Models

  14. Rate terms: Rate equations: Input (plain text file): E + S ---> ES : k1 ES ---> E + S : k2 ES ---> E + P : k3 k1 [E]  [S] k2 [ES] k3 [ES] System of Simple, Simultaneous Equations HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS "The LEGO method" of deriving rate equations BKEB Lec 1: Numerical Models

  15. DynaFit can analyze many types of experiments MASS ACTION LAW AND MASS CONSERVATION LAW IS APPLIED IN THE SAME WAY EXPERIMENT DYNAFIT DERIVES A SYSTEM OF ... chemistrybiophysics Ordinary differential equations (ODE)Nonlinear algebraic equations Nonlinear algebraic equations Kinetics (time-course) Equilibrium binding Initial reaction rates enzymology BKEB Lec 1: Numerical Models

  16. Example: DNA + clamp / clamp loader complex DETERMINE ASSOCIATION AND DISSOCIATION RATE CONSTANT IN AN A + B  AB SYSTEM Typical email from a Ph.D. student: Courtesy of Senthil Perumal, Penn State University (Steven Benkovic lab) BKEB Lec 1: Numerical Models

  17. Example: Experimental setup ALL COMPONENTS PRESENT AT EQUAL CONCENTRATIONS • pre-mix sliding clamp (C) + clamp loader (L) to form C.L complex; • add DNA solution; • observe the formation of C.L.DNA ternary complex over timefinal concentrations:100 nM clamp100 nM loader100 nM DNA ... gp45 labeled with Cy5 acceptor dye ... gp44/62 ... primer labeled with Cy3 donor dye C.L complex has estimated Kd = 1 nM, so C.L  C+L dissociation upon adding DNA should be negligible Courtesy of Senthil Perumal, Penn State University (Steven Benkovic lab) BKEB Lec 1: Numerical Models

  18. Example: Raw data JUST BECAUSE THE DATA FIT TO A MODEL DOES NOT MEAN THAT THE MODEL IS CORRECT! raw fluorescence F fit to F = A0 + A1 exp (-k t) fluorescence exponential model fits the data well but it is theoretically invalid! time Courtesy of Senthil Perumal, Penn State University (Steven Benkovic lab) BKEB Lec 1: Numerical Models

  19. keywords sections arbitrary names initial estimates Example: Anatomy of DynaFit scripts DYNAFIT SOFTWARE IS DRIVEN BY TEXT “SCRIPTS” - MINIATURE “COMPUTER PROGRAMS” BKEB Lec 1: Numerical Models

  20. Example: DynaFit tutorial YOUR FIRST DYNAFIT DATA-ANALYSIS SESSION TUTORIAL • Start DynaFit • Select menu “File ... Open” or press Ctrl+O • Navigate to file./courses/bkeb/lec-1/a+b/fit-001.txt • Select menu “File ... Try” or press Ctrl+TThis is the initial estimate • Select menu “File ... Run” or press Ctrl+UWait several seconds to finish the analysis • Select menu “View ... Results in External Browser”Navigate in the output files BKEB Lec 1: Numerical Models

  21. Example: Detailed explanation A BIT OF THEORY • Reaction order • Units and dimensions (scaling) • The DynaFit model for biomolecular kinetics • Initial estimates of model parameters BKEB Lec 1: Numerical Models

  22. uni-molecular monomolecular bimolecular Molecularity and reaction order IN PRACTICE WE ENCOUNTER ONLY ZERO-, FIRST-, AND SECOND-ORDER REACTIONS PHYSICALMEANING DYNAFITNOTATION ORDER NOTATION zero- constant-rateinflux or efflux X --> : v k1 A B first- isomerization ordissociation ofone molecule A --> B : k1C --> A + B : k1’ k’1 A B + C second- binding(association) of two molecules A + B --> C : k2 k2 A + B C BKEB Lec 1: Numerical Models

  23. top left DYNAFITNOTATION MULTI-STEP MECHANISM +B k1 A AB +C k2 k4 k3 k5 AC X Reversible reactions and reaction mechanisms DYNAFIT CAN HANDLE AN ARBITRARY NUMBER OF ELEMENTARY REACTIONS IN A MECHANISM DYNAFITNOTATION REVERSIBLE REACTION k1 A + B <==> C : k1 k2 A + B C k2 A + B <==> AB : k1 k2A + C <==> AC : k3 k4 AC ---> X : k5 BKEB Lec 1: Numerical Models

  24. EXAMPLE dimensional analysis of k1 (bimolecular association rate constant) {conc.} / {time} 1 v v = k1 [A] [B] k1 = = [A] [B] {conc.}  {conc.} {conc.}  {time} Dimensions of rate constants CAREFUL ABOUT DIMENSIONS OF RATE CONSTANTS! DIMENSIONAL ANALYSIS quantity v [X] k1, k2 dimension concentration / time concentration ? forward and reverse reaction rates: k1 A + B AB v = k1 [A] [B] k2 v = k2 [AB] BKEB Lec 1: Numerical Models

  25. Dimensions of rate and equilibrium constants SUMMARY k1 1/sec 1/(M sec) M 1/M k2 dissociation rate constant association rate constant dissociation equilibrium constant association equilibrium constant A + B AB k2 k1 Kd = k2 /k1 Ka = k1 /k2 k3 k3  rate constant  rate constant  equilibrium constant  equilibrium constant 1/sec 1/sec -- -- A A’ k4 k4 K = k3 /k4 K = k4 /k3 BKEB Lec 1: Numerical Models

  26. = 1 µM-1s-1 = 106 M-1.s-1 concentration units of rate constants: µM time units of rate constants: sec 100 nM time, s signal, eV 0.52 0.3526 0.56 0.3484 0.60 0.3485 0.64 0.3454 0.68 0.3499 0.72 0.3502 ... Example: Units (scaling) of rate constants ALL UNITS ARE ARBITRARY BUT MUST BE IDENTICAL THROUGHOUT THE ENTIRE SCRIPT! [mechanism] DNA + Clamp.Loader <==> Complex : kon koff [constants] kon = 1 ? koff = 1 ? [concentrations] DNA = 0.1 Clamp.Loader = 0.1 [responses] Complex = 1 ? [data] file ./.../d1-edit.txt offset auto ? BKEB Lec 1: Numerical Models

  27. one concentration unit (in this case 1 µM) of Complex will produce an increase in the signal equal to 1.00 instrument units time, s signal, eV 0.52 0.3526 0.56 0.3484 0.60 0.3485 0.64 0.3454 0.68 0.3499 0.72 0.3502 ... Example: The “response” coefficient MOLAR “RESPONSE” = PROPORTIONALITY FACTOR LINKING CONCENTRATIONS TO SIGNAL [mechanism] DNA + Clamp.Loader <==> Complex : kon koff [constants] kon = 1 ? koff = 1 ? [concentrations] DNA = 0.1 Clamp.Loader = 0.1 [responses] Complex = 1.00 ? [data] file ./.../d1-edit.txt offset auto ? BKEB Lec 1: Numerical Models

  28. Example: Initial estimates NONLINEAR REGRESSION ANALYSIS ALWAYS REQUIRES INITIAL ESTIMATES OF THE SOLUTION the initial estimate of rate constants [mechanism] DNA + Clamp.Loader <==> Complex : kon koff [constants] kon = 1 ? koff = 1 ? [concentrations] DNA = 0.1 Clamp.Loader = 0.1 [responses] Complex = 1 ? [data] file ./.../d1-edit.txt offset auto ? optimized model parameters A VERY DIFFICULT PROBLEM: HOW TO GUESS “GOOD ENOUGH” INITIAL ESTIMATES OF RATE CONSTANTS? BKEB Lec 1: Numerical Models

  29. Example: “Good” initial estimate data model BKEB Lec 1: Numerical Models

  30. Example: “Good” initial estimate – results BKEB Lec 1: Numerical Models

  31. Example: “Bad” initial estimate model a hundred times smaller / larger data BKEB Lec 1: Numerical Models

  32. Example: “Bad” initial estimate – results BKEB Lec 1: Numerical Models

  33. DynaFit warnings from running the “bad” estimate: Example: “Good” vs. “Bad” results - comparison initial estimate sum of squares relativesum of sq. “best-fit” constants Kd, nM= k2/k1 k1 = 2.2 ± 0.5 k2 = 0.030 ± 0.015 k1 = 1 k2 = 1 0.002308 1.00 13 nM “good” k1 = 0.2 ± 3.4 k2 = 0.2 ± 0.6 1000 nM k1 = 100 k2 = 0.01 0.002354 1.02 “bad” BKEB Lec 1: Numerical Models

  34. Example: “Good” vs. “Bad” results - comparison From “good” initial estimate From “bad” initial estimate not very encouraging! BKEB Lec 1: Numerical Models

  35. LOOKING AHEAD • DynaFit offers more reliable and convenient safeguards • Global minimization • Systematic combinatorial scan • Confidence interval search Example: “Good” vs. “Bad” results - summary • Initial estimates “off” by a factor of 100 can produce misleading results. • The data/model overlay may “look good”, but the results may be invalid. • The same applies to the residual sum of squares (only 2% difference). • The only indication that something went wrong might be:a. huge standard errors of model parameters; andb. various warnings from the least-squares fitter • The simplest possible safeguard: Use several different initial estimates?Disadvantage: how do we know which multiple estimates? BKEB Lec 1: Numerical Models

  36. Summary and conclusions NUMERICAL MODELS IN BIOCHEMISTRY AND BIOPHYSICS: BETTER THAN ALGEBRAIC EQUATIONS • Numerical models are applicable to all experimental conditions.No more “large excess of this over that”. • Numerical models apply uniformly to all types of experiments:a. reaction progress (kinetics);b. equilibrium composition (binding);c. enzyme catalysis. • Numerical models can be automatically derived by computer.No more looking up algebraic equations – if they exist at all. • Main disadvantage: requirement for specialized software.But DynaFit is free to academic users. • Not a “silver bullet”! Example: the initial estimate problem.But this is not specific to numerical models (applies to algebraic models, too). BKEB Lec 1: Numerical Models

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