etc. Lecture outline Traditional approach is based on algebraic models Algebraic models restrict experiment design

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1 I N N O V T I O N L E C T U R E S (I N N O l E C) Lecture outline Binding and Kinetics for Experimental Biologists Lecture 1 Numerical Models for Biomolecular Interactions Petr Kuzmič, Ph.D. BioKin, Ltd. WTERTOWN, MSSCHUSETTS, U.S.. The problem: Traditional equations for fitting biomolecular binding data restrict the experimental design. Typically, at least one component must be present in very large excess. The solution: bandon algebraic equations entirely. Use iterative numerical models, which can be derived automatically by the computer. n implementation: Software DynaFit. n example: Kinetics of forked DN binding to the proteinprotein complex formed by DNpolymerase sliding clamp (gp45) and clamp loader (gp44/62). BKEB Lec 1: Numerical Models 2 Traditional approach is based on algebraic models lgebraic models restrict experiment design Goodrich & Kugel (2006) Binding and Kinetics for Molecular Biologists EXMPLE: Determine the association rate constant for B B typical cookbook for experimental biologists molecular mechanism algebraic model Cold Spring Harbor Laboratory Press Cold Spring Harbor, NY, 2007 ISBN10: BKEB Lec 1: Numerical Models 3 BKEB Lec 1: Numerical Models 4 Experimental handbooks are full of restrictions Numerical vs. algebraic mathematical models FROM VRIETY OF LGEBRIC EQUTIONS TO UNIFORM SYSTEM OF DIFFERENTIL EQUTIONS Goodrich & Kugel (2007) Binding and Kinetics for Molecular Biologists EXMPLE: Determine the rate constant and for B B p. 34 LGEBRIC EQUTIONS DIFFERENTIL EQUTIONS p. 79 d[]/dt = [][B] [B] d[b]/dt = [][B] [B] p. 120 d[b]/dt = [][B] [B] pplies only when [B] >> [] pplies under all conditions p. 126 etc. BKEB Lec 1: Numerical Models 5 BKEB Lec 1: Numerical Models 6 1

2 dvantages and disadvantages of numerical models THERE IS NO SUCH THING S FREE LUNCH Specialized numerical software: DynaFit MORE THN 600 PPERS PUBLISHED WITH IT ( ) DVNTGE LGEBRIC MODEL DIFFERENTIL MODEL 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 time 2009 DOWNLOD does not require an initial estimate FREE for academic users is resistant to truncation and roundoff errors has a long tradition: many papers published Kuzmic (2009) Meth. Enzymol., 467, BKEB Lec 1: Numerical Models 7 BKEB Lec 1: Numerical Models 8 DynaFit software: Citation analysis Theoretical foundations: Mass ction Law PPROXIMTELY 850 JOURNL RTICLES PUBLISHED SINCE 1998 RTE IS PROPORTIONL TO CONCENTRTION(S) rate derivative MONOMOLECULR RECTIONS products rate is proportional to [] d [] / d t = k [] monomolecular rate constant 1 / time BIMOLECULR RECTIONS B products rate is proportional to [] [B] d[] / d t = d [B] / d t = k [] [B] Kuzmic, P. (1996) "Program DYNFIT for the analysis of enzyme kinetic data: pplication to HIV proteinase" nal. Biochem. 237, Kuzmic, P. (2009) "DynaFit software package for enzymology Meth. Enzymol. 467, bimolecular rate constant 1 / (concentration time) BKEB Lec 1: Numerical Models 9 BKEB Lec 1: Numerical Models 10 Theoretical foundations: Mass Conservation Law Composition Rule: Example PRODUCTS RE FORMED WITH THE SME RTE S RECTNTS DISPPER EXMPLE MECHNISM RTE EQUTIONS EXMPLE P Q d [] / d t = d [P] / d t = d [Q] / d t E E d[p] / d t = k 5 [EB] E B EB d[eb] / d t = [E] [B] [EB] COMPOSITION RULE mechanism: B DDITIVITY OF TERMS FROM SEPRTE RECTIONS d [B] / d t = [] [B] E B EB k 3 k 3 k 4 k 4 EB EB k 4 [EB] [] k 4 [EB] k 5 [EB] B C EB k 5 E P Q Similarly for other species... BKEB Lec 1: Numerical Models 11 BKEB Lec 1: Numerical Models 12 2

3 "Kinetic Compiler" System of Simple, Simultaneous Equations HOW DYNFIT PROCESSES YOUR BIOCHEMICL EQUTIONS HOW DYNFIT PROCESSES YOUR BIOCHEMICL EQUTIONS k 3 E S E.S E P k 3 E S E.S E P "The LEGO method" of deriving rate equations Input (plain text file): Rate terms: Rate equations: Input (plain text file): Rate terms: Rate equations: E S > ES : k1 [E] [S] d[e ] / dt = [E] [S] [ES] k 3 [ES] E S > ES : k1 [E] [S] ES > E S ES > E P : k2 : k3 [ES] k 3 [ES] d[es ] / dt = [E] [S] [ES] k 3 [ES] ES > E S ES > E P : k2 : k3 [ES] k 3 [ES] Similarly for other species... BKEB Lec 1: Numerical Models 13 BKEB Lec 1: Numerical Models 14 DynaFit can analyze many types of experiments MSS CTION LW ND MSS CONSERVTION LW IS PPLIED IN THE SME WY Example: DN clamp / clamp loader complex DETERMINE SSOCITION ND DISSOCITION RTE CONSTNT IN N B B SYSTEM EXPERIMENT DYNFIT DERIVES SYSTEM OF... chemistry biophysics enzymology Kinetics (timecourse) Equilibrium binding Initial reaction rates Ordinary differential equations (ODE) Nonlinear algebraic equations Nonlinear algebraic equations Typical from a Ph.D. student: Courtesy of Senthil Perumal, Penn State University (Steven Benkovic lab) BKEB Lec 1: Numerical Models 15 BKEB Lec 1: Numerical Models 16 Example: Experimental setup LL COMPONENTS PRESENT T EQUL CONCENTRTIONS Example: Raw data JUST BECUSE THE DT FIT TO MODEL DOES NOT MEN THT THE MODEL IS CORRECT! 1. premix sliding clamp (C) clamp loader (L) to form C.L complex; 2. add DN solution; 3. observe the formation of C.L.DN ternary complex over time final concentrations: 100 nm clamp 100 nm loader 100 nm DN... gp45 labeled with Cy5 acceptor dye... gp44/62... primer labeled with Cy3 donor dye fluorescence raw fluorescence F fit to F = 0 1 exp (k t) exponential model fits the data well but it is theoretically invalid! C.L complex has estimated K d = 1 nm, so C.L CL dissociation upon adding DN should be negligible time Courtesy of Senthil Perumal, Penn State University (Steven Benkovic lab) BKEB Lec 1: Numerical Models 17 Courtesy of Senthil Perumal, Penn State University (Steven Benkovic lab) BKEB Lec 1: Numerical Models 18 3

4 Example: natomy of DynaFit scripts DYNFIT SOFTWRE IS DRIVEN BY TEXT SCRIPTS MINITURE COMPUTER PROGRMS Example: DynaFit tutorial YOUR FIRST DYNFIT DTNLYSIS SESSION TUTORIL keywords sections 1. Start DynaFit 2. Select menu File... Open or press CtrlO 3. Navigate to file./courses/bkeb/lec1/ab/fit001.txt 4. Select menu File... Try or press CtrlT arbitrary names initial estimates This is the initial estimate 5. Select menu File... Run or press CtrlU Wait several seconds to finish the analysis 6. Select menu View... Results in External Browser Navigate in the output files BKEB Lec 1: Numerical Models 19 BKEB Lec 1: Numerical Models 20 Example: Detailed explanation BIT OF THEORY Molecularity and reaction order IN PRCTICE WE ENCOUNTER ONLY ZERO, FIRST, ND SECONDORDER RECTIONS 1. Reaction order ORDER PHYSICL MENING NOTTION DYNFIT NOTTION 2. Units and dimensions (scaling) 3. The DynaFit model for biomolecular kinetics zero constantrate influx or efflux X > : v 4. Initial estimates of model parameters firstisomerization or unimolecular dissociation of monomolecular one molecule k 1 B B C > B : k1 C > B : k1 second bimolecular binding (association) of two molecules B C B > C : k2 BKEB Lec 1: Numerical Models 21 BKEB Lec 1: Numerical Models 22 Reversible reactions and reaction mechanisms DYNFIT CN HNDLE N RBITRRY NUMBER OF ELEMENTRY RECTIONS IN MECHNISM Dimensions of rate constants CREFUL BOUT DIMENSIONS OF RTE CONSTNTS! DIMENSIONL NLYSIS REVERSIBLE RECTION k top 1 B C DYNFIT NOTTION B <==> C : k1 k2 left B B forward and reverse reaction rates: v = [] [B] v = [B] quantity v [X], dimension concentration / time concentration? MULTISTEP MECHNISM B B C k 4 k 3 k 5 C X DYNFIT NOTTION B <==> B : k1 k2 C <==> C : k3 k4 C > X : k5 EXMPLE dimensional analysis of (bimolecular association rate constant) v {conc.} / {time} 1 v = [] [B] = = [] [B] {conc.} {conc.} {conc.} {time} BKEB Lec 1: Numerical Models 23 BKEB Lec 1: Numerical Models 24 4

5 Dimensions of rate and equilibrium constants SUMMRY Example: Units (scaling) of rate constants LL UNITS RE RBITRRY BUT MUST BE IDENTICL THROUGHOUT THE ENTIRE SCRIPT! B dissociation rate constant B association rate constant dissociation equilibrium constant association equilibrium constant K d = / K a = / 1/sec 1/(M sec) M 1/M [mechanism] DN Clamp.Loader <==> Complex : kon koff [constants] = 1 µm 1 s 1 = 10 6 M 1.s 1 kon = 1? koff = 1? [concentrations] DN = 0.1 Clamp.Loader = 0.1 concentration units of rate constants: µm 100 nm time units of rate constants: sec k 3 k 4 rate constant rate constant equilibrium constant equilibrium constant k 3 k 4 K = k 3 /k 4 K = k 4 /k 3 1/sec 1/sec [responses] Complex = 1? [data] file./.../d1edit.txt offset auto? time, s signal, ev BKEB Lec 1: Numerical Models 25 BKEB Lec 1: Numerical Models 26 Example: The response coefficient MOLR RESPONSE = PROPORTIONLITY FCTOR LINKING CONCENTRTIONS TO SIGNL Example: Initial estimates NONLINER REGRESSION NLYSIS LWYS REQUIRES INITIL ESTIMTES OF THE SOLUTION the initial estimate of rate constants [mechanism] DN Clamp.Loader <==> Complex : kon koff [mechanism] DN Clamp.Loader <==> Complex : kon koff [constants] kon = 1? koff = 1? [concentrations] DN = 0.1 Clamp.Loader = 0.1 one concentration unit (in this case 1 µm) of Complex will produce an increase in the signal equal to 1.00 instrument units [constants] kon = 1? koff = 1? [concentrations] DN = 0.1 Clamp.Loader = 0.1 optimized model parameters VERY DIFFICULT PROBLEM: [responses] Complex = 1.00? [data] file./.../d1edit.txt offset auto? time, s signal, ev [responses] Complex = 1? [data] file./.../d1edit.txt offset auto? HOW TO GUESS GOOD ENOUGH INITIL ESTIMTES OF RTE CONSTNTS? BKEB Lec 1: Numerical Models 27 BKEB Lec 1: Numerical Models 28 Example: Good initial estimate Example: Good initial estimate results data model BKEB Lec 1: Numerical Models 29 BKEB Lec 1: Numerical Models 30 5

6 Example: Bad initial estimate Example: Bad initial estimate results model a hundred times smaller / larger data BKEB Lec 1: Numerical Models 31 BKEB Lec 1: Numerical Models 32 Example: Good vs. Bad results comparison Example: Good vs. Bad results comparison initial estimate sum of squares relative bestfit sum of sq. constants K d, nm = / good = 1 = = 2.2 ± 0.5 = ± nm bad = 100 = = 0.2 ±3.4 = 0.2 ± nm DynaFit warnings from running the bad estimate: From good initial estimate From bad initial estimate not very encouraging! BKEB Lec 1: Numerical Models 33 BKEB Lec 1: Numerical Models 34 Example: Good vs. Bad results summary Summary and conclusions NUMERICL MODELS IN BIOCHEMISTRY ND BIOPHYSICS: BETTER THN LGEBRIC EQUTIONS 1. Initial estimates off by a factor of 100 can produce misleading results. 2. The data/model overlay may look good, but the results may be invalid. 3. The same applies to the residual sum of squares (only 2% difference). 4. The only indication that something went wrong might be: a. huge standard errors of model parameters; and b. various warnings from the leastsquares fitter 5. The simplest possible safeguard: Use several different initial estimates? Disadvantage: how do we know which multiple estimates? LOOKING HED DynaFit offers more reliable and convenient safeguards Global minimization Systematic combinatorial scan Confidence interval search 1. Numerical models are applicable to all experimental conditions. No more large excess of this over that. 2. Numerical models apply uniformly to all types of experiments: a. reaction progress (kinetics); b. equilibrium composition (binding); c. enzyme catalysis. 3. Numerical models can be automatically derived by computer. No more looking up algebraic equations if they exist at all. 4. Main disadvantage: requirement for specialized software. But DynaFit is free to academic users. 5. 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 35 BKEB Lec 1: Numerical Models 36 6