Fitting Michaelis-Menten directly to substrate concentration data

Fitting Michaelis-Menten directly tosubstrate concentration dataMurray Hannah & Peter Moate

OutlineEnzyme Kinetics

OutlineEnzyme KineticsMichaelis−Menten equation

OutlineEnzyme KineticsMichaelis−Menten equationTraditional parameter estimation methods

OutlineEnzyme KineticsMichaelis−Menten equationTraditional parameter estimation methodsMore direct estimation strategiesDirect fit to rate dataNumerical solutionsSemi-parametric

OutlineEnzyme KineticsMichaelis−Menten equationTraditional parameter estimation methodsMore direct estimation strategiesDirect fit to rate dataNumerical solutionsSemi-parametricA GenStat procedure, MICHAELIS

Enzymes‘Little’ molecular machines that catalize biochemical reactions

An enzyme reactionk 1S + E 0 ⇋ E 1k−→2E0 + Pk −1

An enzyme reactionk 1S + E 0 ⇋ E 1k−→2E0 + Pk −1◮ Substrate S

An enzyme reactionk 1S + E 0 ⇋ E 1k−→2E0 + Pk −1◮ Substrate S◮ Empty Enzyme E 0

An enzyme reactionk 1S + E 0 ⇋ E 1k−→2E0 + Pk −1◮ Substrate S◮ Empty Enzyme E 0◮ Occupied Enzyme E 1

An enzyme reactionk 1S + E 0 ⇋ E 1k−→2E0 + Pk −1◮ Substrate S◮ Empty Enzyme E 0◮ Occupied Enzyme E 1◮ Product P

Enzyme kineticsScientists wish to◮ predict reaction rate as a function of substrateconcentration

Enzyme kineticsScientists wish to◮ predict reaction rate as a function of substrateconcentration◮ predict concentration as function of time

Enzyme kineticsScientists wish to◮ predict reaction rate as a function of substrateconcentration◮ predict concentration as function of time◮ predict reaction rate as a function of time

k 1S + E 0 ⇋ E 1k−→2E0 + Pk −1

Reaction rates . . .k 1S + E 0 ⇋ E 1k−→2E0 + Pk −1

Reaction rates . . .dSdt= −k 1 SE 0 + k −1 E 1k 1S + E 0 ⇋ E 1k−→2E0 + Pk −1

k 1S + E 0 ⇋ E 1k−→2E0 + Pk −1Reaction rates . . .dSdt= −k 1 SE 0 + k −1 E 1dE 0dt= −k 1 SE 0 + k −1 E 1 + k 2 E 1

k 1S + E 0 ⇋ E 1k−→2E0 + Pk −1Reaction rates . . .dSdt= −k 1 SE 0 + k −1 E 1dE 0dt= −k 1 SE 0 + k −1 E 1 + k 2 E 1dE 1dt= k 1 SE 0 − k −1 E 1 − k 2 E 1

k 1S + E 0 ⇋ E 1k−→2E0 + Pk −1Reaction rates . . .dSdt= −k 1 SE 0 + k −1 E 1dE 0dt= −k 1 SE 0 + k −1 E 1 + k 2 E 1dE 1dt= k 1 SE 0 − k −1 E 1 − k 2 E 1dPdt= k 2 E 1

Reaction rates . . .dSdt= −k 1 SE 0 + k −1 E 1dE 0dt= −k 1 SE 0 + k −1 E 1 + k 2 E 1dE 1dt= k 1 SE 0 − k −1 E 1 − k 2 E 1dPdt= k 2 E 1Constraints . . .E T = E 0 + E 1 ⇒ dE Tdtk 1S + E 0 ⇋ E 1k−→2E0 + Pk −1= 0,

Reaction rates . . .dSdt= −k 1 SE 0 + k −1 E 1dE 0dt= −k 1 SE 0 + k −1 E 1 + k 2 E 1dE 1dt= k 1 SE 0 − k −1 E 1 − k 2 E 1dPdt= k 2 E 1Constraints . . .E T = E 0 + E 1 ⇒ dE TdtdE 1dt≃ 0k 1S + E 0 ⇋ E 1k−→2E0 + Pk −1= 0,

... which give...dSdt= (k 2E T S( )k−1 +k 2k 1+ S= V maxSK m + S

... which give...dSdt= (k 2E T S( )k−1 +k 2k 1+ S= V maxSK m + Sk 1S + E 0 ⇋ E 1k−→2E0 + Pk −1

... which give...dSdt= (k 2E T S( )k−1 +k 2k 1+ S= V maxSK m + SK m = k −1+k 2k 1k 1S + E 0 ⇋ E 1k−→2E0 + Pk −1= substrate-enzyme "affinity"

... which give...dSdt= (k 2E T S( )k−1 +k 2k 1+ S= V maxSK m + Sk 1S + E 0 ⇋ E 1k−→2E0 + Pk −1K m = k −1+k 2k 1= substrate-enzyme "affinity"V max = k 2 E T = maximal reaction rate

... which give...dSdt= (k 2E T S( )k−1 +k 2k 1+ S= V maxSK m + Sk 1S + E 0 ⇋ E 1k−→2E0 + Pk −1K m = k −1+k 2k 1= substrate-enzyme "affinity"V max = k 2 E T = maximal reaction ratedSdt→ V max , S → ∞,S = K m ⇔ dSdt= V max2

The Michaelis-Menten equation.v = V maxSK m + SMaud Menten and Leonor Michaelis, 1913

How good is the steady-state approximation?

How good is the steady-state approximation?http://online.redwoods.cc.ca.us/...

How good is the steady-state approximation?MM approx compared to actual DE

Traditional estimation of V maxand K mLinearize v = V maxS, Linear regression, or graphicalK m + S

Traditional estimation of V maxand K mLinearize v = V maxS, Linear regression, or graphicalK m + S◮ 1v = 1V max+ KmV max1SLineweaver Burk, 1939

Traditional estimation of V maxand K mLinearize v = V maxS, Linear regression, or graphicalK m + S◮ 1v = 1V max+ KmV max1SLineweaver Burk, 1939◮vS = VmaxK m− 1K mv Scatchard, 1949

Traditional estimation of V maxand K mLinearize v = V maxS, Linear regression, or graphicalK m + S◮ 1v = 1V max+ KmV max1SLineweaver Burk, 1939◮◮vS = VmaxK m− 1K mv Scatchard, 1949Sv = KmV max+ 1V maxS Woolf (Haldane, 1957)

Traditional estimation of V maxand K mLinearize v = V maxS, Linear regression, or graphicalK m + S◮ 1v = 1V max+ KmV max1SLineweaver Burk, 1939◮◮vS = VmaxK m− 1K mv Scatchard, 1949Sv = KmV max+ 1V maxS Woolf (Haldane, 1957)◮ or by non-linear regression. (e.g. FITCURVE [CURVE=ldl])

Traditional estimation of V maxand K mLinearize v = V maxS, Linear regression, or graphicalK m + S◮ 1v = 1V max+ KmV max1SLineweaver Burk, 1939◮◮vS = VmaxK m− 1K mv Scatchard, 1949Sv = KmV max+ 1V maxS Woolf (Haldane, 1957)◮ or by non-linear regression. (e.g. FITCURVE [CURVE=ldl])v i and S i derived from experiment i = 1 . . . k

Fit directly to concentration data, S vs t?

Fit directly to concentration data, S vs t?Require S(t), the solution to dSdt= V maxSK m + S

Fit directly to concentration data, S vs t?Require S(t), the solution to dSdt= V maxSK m + SBut no closed form solution for S(t) exists!

Solution to S(t){ ( )}S0 S0 − V max tS(t) = K m W expK m K m

Solution to S(t){ ( )}S0 S0 − V max tS(t) = K m W expK m K mwhere Lambert’s W : y = W (x) such that ye y = x

Solution to S(t){ ( )}S0 S0 − V max tS(t) = K m W expK m K mwhere Lambert’s W : y = W (x) such that ye y = x(Fig. from Barry et al., Math Comp Sim, 2000.)

Estimation strategies using (S, t) data:◮ Transform (S, t) to rate data, (v, S)

Estimation strategies using (S, t) data:◮ Transform (S, t) to rate data, (v, S)◮ Purely numeric solution to S(t)

Estimation strategies using (S, t) data:◮ Transform (S, t) to rate data, (v, S)◮ Purely numeric solution to S(t)◮ Analytic expansion for Lamberts W

Estimation strategies using (S, t) data:◮ Transform (S, t) to rate data, (v, S)◮ Purely numeric solution to S(t)◮ Analytic expansion for Lamberts W◮ A semi-parametric method

Transform (S, t) to (v, S ∗ )?v i = − S i+1 − S iand Si ∗ = S i + S i+1t i+1 − t i 2

Transform (S, t) to (v, S ∗ )?v i = − S i+1 − S iand Si ∗ = S i + S i+1t i+1 − t i 2

Note that{ }Si+1 − S iVar(V i ) = Vart i+1 − t i= 2σ2δi2 , where δ i = t i+1 − t i

Note that{ }Si+1 − S iVar(V i ) = Vart i+1 − t i= 2σ2δi2 , where δ i = t i+1 − t iUnequal time intervals δ i ⇒ unequal variance.

Fitting v = V maxS ∗K m +S ∗ , with weights = δ 2 i

Transforming (S, t) to (v, S ∗ ) data:Allows us to fit MM directly, but ...◮ Heteroscedasticity, due to unequal δ i .

Transforming (S, t) to (v, S ∗ ) data:Allows us to fit MM directly, but ...◮ Heteroscedasticity, due to unequal δ i .◮ Correlated errors, MA(1), induced by differencing:V i = − S i+1 − S iδ i, ⇒ Cov(V i , V i+1 ) = − σ2δ i δ i+1

Transforming (S, t) to (v, S ∗ ) data:Allows us to fit MM directly, but ...◮ Heteroscedasticity, due to unequal δ i .◮ Correlated errors, MA(1), induced by differencing:V i = − S i+1 − S iδ i, ⇒ Cov(V i , V i+1 ) = − σ2δ i δ i+1◮ Errors in variables:Var(S ∗ ) = σ22

Direct fit to (S, t) data: S(t) numeric solution?◮ E.g., S(t) by Euler’s method:

Direct fit to (S, t) data: S(t) numeric solution?◮ E.g., S(t) by Euler’s method:◮ Note, the slope at t isv(t) = − V maxS(t)K m + S(t)

Direct fit to (S, t) data: S(t) numeric solution?◮ E.g., S(t) by Euler’s method:◮ Note, the slope at t is◮v(t) = − V maxS(t)K m + S(t)Estimate S(t +h) ≃ S(t) + v(t) × h

Direct fit to (S, t) data: S(t) numeric solution?◮ E.g., S(t) by Euler’s method:◮ Note, the slope at t is◮◮v(t) = − V maxS(t)K m + S(t)Estimate S(t +h) ≃ S(t) + v(t) × hStart with initial value, S(0)

Direct fit to (S, t) data: S(t) numeric solution?◮ E.g., S(t) by Euler’s method:◮ Note, the slope at t isv(t) = − V maxS(t)K m + S(t)◮ Estimate S(t +h) ≃ S(t) + v(t) × h◮ Start with initial value, S(0)◮ Iterate, gives ˜S(t) for t = 0, h, 2h, ...

Direct fit to (S, t) data: S(t) numeric solution?◮ E.g., S(t) by Euler’s method:◮ Note, the slope at t isv(t) = − V maxS(t)K m + S(t)◮ Estimate S(t +h) ≃ S(t) + v(t) × h◮ Start with initial value, S(0)◮ Iterate, gives ˜S(t) for t = 0, h, 2h, ...◮ interpolate ˜S(ti )

Direct fit to (S, t) data: S(t) numeric solution?◮ E.g., S(t) by Euler’s method:◮ Note, the slope at t isv(t) = − V maxS(t)K m + S(t)◮ Estimate S(t +h) ≃ S(t) + v(t) × h◮ Start with initial value, S(0)◮ Iterate, gives ˜S(t) for t = 0, h, 2h, ...◮ interpolate ˜S(ti )Minimizen∑i=1[S i − ˜S(t i ; V max , K m )] 2w.r.t V max , K m

Euler’s solution to S(t)Simple to program, fits MM directly to (S, t), but ...◮ Computationally intensive

Euler’s solution to S(t)Simple to program, fits MM directly to (S, t), but ...◮ Computationally intensive◮ Uncertain accuracy

Euler’s solution to S(t)Simple to program, fits MM directly to (S, t), but ...◮ Computationally intensive◮ Uncertain accuracy◮ Runge-Kutta methods

Analytic numeric solution for S(t)?◮{ ( )}S0 S0 − V max tS(t) = K m W expK m K mwhere Lambert’s W : y = W (x) such that ye y = x

Analytic numeric solution for S(t)?◮{ ( )}S0 S0 − V max tS(t) = K m W expK m K mwhere Lambert’s W : y = W (x) such that ye y = x◮ Expansion to give (approximate) Lambert’s W :

Analytic numeric solution for S(t)?◮{ ( )}S0 S0 − V max tS(t) = K m W expK m K mwhere Lambert’s W : y = W (x) such that ye y = x◮ Expansion to give (approximate) Lambert’s W :◮ See Golicnik, Analytic Biochemistry (2010), andreferences therein.maximum error < 0.02% (Barry et al., 2000)

Analytic numeric solution for S(t)?◮{ ( )}S0 S0 − V max tS(t) = K m W expK m K mwhere Lambert’s W : y = W (x) such that ye y = x◮ Expansion to give (approximate) Lambert’s W :◮ See Golicnik, Analytic Biochemistry (2010), andreferences therein.maximum error < 0.02% (Barry et al., 2000)Minimizen∑i=1[S i − Š(t i; V max , K m )] 2w.r.t V max , K m

"Fit Michealis-Menten (following Golicnik)"SCALAR Vm,Km,K1,W0SCALAR epsilon; VAL=0.4586887CALC d1=12/5EXPRESSION mm[1]; VALUE=!e(x=(W0-K1-Vm*TIME)/Km)& mm[2]; VALUE=!e(x=(W0-K1)*exp(x)/Km)& mm[3]; VALUE=!e(d2=x/log(1+d1*x))& mm[4]; VALUE=!e(d2=5*log(d1*d2))& mm[5]; VALUE=!e(d3=(1+epsilon)*log(6*x/d2))& mm[6]; VALUE=!e(d4=log(1+2*x))& mm[7]; VALUE=!e(d4=epsilon*log(2*x/d4))& mm[8]; VALUE=!e(W=Km*(d3-d4)+K1)

Semi-parametric method?Represent S(t) by f (t) = Φ(t)β,e.g. beta spline

Semi-parametric method?Represent S(t) by f (t) = Φ(t)β,Requiref ′ (t) ≃ V maxf (t)K m + f (t) .e.g. beta spline

Semi-parametric method?Represent S(t) by f (t) = Φ(t)β,RequireMinimizen∑{s i − f (t i )} 2 + λi=1f ′ (t) ≃ V maxf (t)K m + f (t) .n∑i=1e.g. beta spline{f ′ (t i ) − V }maxf (t i ) 2K m + f (t i )

Semi-parametric method?Represent S(t) by f (t) = Φ(t)β,RequireMinimizeorn∑{s i − f (t i )} 2 + λi=1i=1f ′ (t) ≃ V maxf (t)K m + f (t) .n∑i=1e.g. beta spline{f ′ (t i ) − V }maxf (t i ) 2K m + f (t i )n∑∫ {{s i − f (t i )} 2 + λ f ′ (t) − V }maxf (t) 2dtK m + f (t)

Nested minimization[ n∑ ∫ {min min {s i −f (t i )} 2 +λ f ′ (t)− V }maxf (t) 2dt]{V max ,K m} βK m + f (t)i=1where f (t) = Φ(t)β, B-spline(Cao et al., Biometrics, 2011)

Semi-parametric methodFit spline to (S, t) data, conforming to MM, but ...◮ Large number of non-linear parameters.

Semi-parametric methodFit spline to (S, t) data, conforming to MM, but ...◮ Large number of non-linear parameters.◮ Need expressions for gradients (Ramsay et al., JRSS,2007.)

Semi-parametric methodFit spline to (S, t) data, conforming to MM, but ...◮ Large number of non-linear parameters.◮ Need expressions for gradients (Ramsay et al., JRSS,2007.)◮ Arbitrary tuning parameter, λ,

Semi-parametric methodFit spline to (S, t) data, conforming to MM, but ...◮ Large number of non-linear parameters.◮ Need expressions for gradients (Ramsay et al., JRSS,2007.)◮ Arbitrary tuning parameter, λ,◮ Estimate λ by cross-validation

Semi-parametric methodFit spline to (S, t) data, conforming to MM, but ...◮ Large number of non-linear parameters.◮ Need expressions for gradients (Ramsay et al., JRSS,2007.)◮ Arbitrary tuning parameter, λ,◮ Estimate λ by cross-validation◮ Programming

A GenStat procedureProcedure MICHAELISM.C. Hannah (May, 2011)1. PurposeFits Michaelis-Menten curve to substrate concentration vs time data2. DescriptionThe Michaelis-Menten non-linear curve, for biochemical reaction rate versus substrateconcentration,dS(t)VmaxS(t)v( t)= =dt K + ,mS(t)can be fitted in GenStat using FITCURVE [CURVE=ldl]. However, in practice, data areavailable only for substrate concentration, S, at time t, and not for the reaction rate, v. Thefunction S(t), that is the solution to this differential equation, has no closed form expression.The procedure MICHAELIS fits the curve S (t)versus t, obtaining parameter estimates forV max , K m and, if required, S 0 (the initial substrate) and, also if required, an additive constantK 1 representing the quantity of non-reactive substrate. This generalized Michaelis-Mentencurve is thus,dS(t)Vmax[ S(t)− K1]= .dt K + S(t)− Km1

4. OptionsPRINT = stringPLOT = stringWINDOW = scalarTITLE = textXTITLE = textYTITLE = textWEIGHTS = variateI: strings {model, deviance, summary, estimates,correlations, fittedvalues, monitoring;default mode, summ, esti printed output (same as forFITNONLINEAR)I: requests graphs, strings {fitted, rate }; defaultfitted.I: scalarI: Title of S vs t graph; default, ‘Fitted Michaelis-Menten solution’I: Title for x-axis; default the TIME identifierI: Title for y-axis; default the CONCENTRATION identifierI: weights for use in FITNONLINEAR5. ParametersCONCENTRATION = variate I: Substrate concentration dataTIME = variate I: Times at which substrate concentration data were measuredINITIAL = variate I: optional initial values for parameters !(Vm, Km, K1, W0).Generally not required, but must be in correct order, if supplied.STEP = variate I: optional step lengths for parameters (see GenStat directivesRCYCLE and FITNONLINEAR). If supplied, these must be incorrect order, !(Vm, Km, K1, W0). A zero will hold thecorresponding parameter at its initial value.FINAL = variate O: Final parameter estimates; !(Vm, Km, K1, W0)VCOV = symmetric O: matrix, vcov for estimatesFITTED = variate O: Fitted concentrationFRATE = variate O: Fitted reaction rate

MICHAELIS CONC=S; TIME=ElapsDays

MICHAELIS CONC=S; TIME=ElapsDays357 MICHAELIS CONC=S; TIME=ElapsDaysInitial parameter valuesVm_i Km_i K1_i S0_i4.511 447.5 34.88 2627Nonlinear regression analysisResponse variate: SWeight variate: WtsNonlinear parameters: Vm, Km, K1, S0Model calculations: mm[1], mm[2], mm[3], mm[4], mm[5], mm[6], mm[7], mm[8]Summary of analysisSource d.f. s.s. m.s. v.r.Regression 4 137399158. 34349789.5 111690.71Residual 48 14762. 307.5Total 52 137413920. 2642575.4Percentage variance accounted for 100.0Standard error of observations is estimated to be 17.5.Message: the following units have large standardized residuals.Unit Response Residual1 2589.4 -2.41Estimates of parametersParameter estimate s.e.Vm 5.1627 0.0980Km 293.2 28.3K1 200.7 10.6S0 2635.97 5.48

MICHAELIS CONC=S; TIME=ElapsDays

MICHAELIS CONC=S; TIME=ElapsDaysRGRAPH INDEX=ElapsDays

MICHAELIS CONC=S; TIME=ElapsDaysRGRAPH INDEX=ElapsDaysRCHECK

MICHAELIS CONC=S; TIME=ElapsDaysRGRAPH INDEX=ElapsDaysRCHECKMICHAELIS [PLOT=#,Rate] CONC=SubConc; \TIME=ElapsDays

Fix K 1 = 0 (original Michaelis-Menten.)MICHAELIS CONC=SubConc; TIME=ElapsDays; \INITIAL=!(*,*,0,*); STEP=!(*,*,0,*)

Fix K 1 = 0 (original Michaelis-Menten.)MICHAELIS CONC=SubConc; TIME=ElapsDays; \INITIAL=!(*,*,0,*); STEP=!(*,*,0,*)Estimates of parametersParameter estimate s.e.Vm 6.550 0.322Km 850.3 99.7K1 0 *S0 2648.2 12.7

Leonor Michaelis Maud Menten, 2011