Quantitative full time course analysis of nonlinear enzyme cycling kinetics

Quantitative full time course analysis ofnonlinear enzyme cycling kineticsWenxiang Cao & Enrique M. De La Cruz

Department of Molecular Biophysics and Biochemistry, Yale University, New Haven, Connecticut 06520, USA.

Enzyme inhibition due to the reversible binding of reaction products is common and underlies the origins ofnegative feedback inhibition in many metabolic and signaling pathways. Product inhibition generatesnon-linearity in steady-state time courses of enzyme activity, which limits the utility of well-establishedenzymology approaches developed under the assumption of irreversible product release. For more than acentury, numerous attempts to find a mathematical solution for analysis of kinetic time courses withproduct inhibition have been put forth. However, no practical general method capable of extractingcommon enzymatic parameters from such non-linear time courses has been successfully developed. Here wepresent a simple and practical method of analysis capable of efficiently extracting steady-state enzymekinetic parameters and product binding constants from non-linear kinetic time courses with productinhibition and/or substrate depletion. The method is general and applicable to all enzyme systems,independent of reaction schemes and pathways.

Product release from enzyme active sites is often reversible and rebinding is common in many enzymesystems1–4. Liberated product(s) can effectively compete with substrate binding to enzyme active sitesand inhibit enzyme cycling. Such product inhibition generates non-linearity in steady-state time courses

of enzyme-catalyzed product formation, which limits the utility of established approaches developed under theassumptions of irreversible product dissociation and constant substrate concentration.

The important enzyme cycling kinetic parameters, kcat and Km, represent the maximal enzyme turnover rateand substrate concentration-dependence of enzyme activation respectively, and define an enzyme’s performance,specificity, efficiency and proficiency5,6. These parameters can in principle be obtained from analyzing non-lineartime courses displaying product inhibition (and/or substrate depletion). However, complex non-linear rateequations and their integrations render such analysis laborious and impractical for most investigators workingon enzymes and their catalyzed reactions. Instead, interference from product inhibition during steady-stateenzyme cycling is deliberately avoided by limiting analysis exclusively to initial velocities determined at earlytime regimes of product formation3,5–11, or through the use of coupled enzyme assays that catalytically removeproduct12–14. However, such experimental modifications may be subject to large uncertainty, particularly if theydo not eliminate the factors contributing to product inhibition. For example, enzymes that bind product(s) withhigher affinity than substrate are not readily amenable to analysis of initial velocities13,15–17, and enzymes thatbinds product(s) tightly13,18 can effectively compete with coupled assays at removing product from catalyticreaction, rendering such strategies inefficient13. Single turnover kinetic analysis19 can overcome the difficultiesof product inhibition, but such measurements typically require prohibitively high enzyme concentrations, par-ticularly if substrate binding is weak. Such cases inevitably require analysis of the entire non-linear time course ofsteady-state product formation, explicitly accounting for contributions arising from product inhibition and/orsubstrate depletion13,20–24.

Numerous efforts have aimed to develop methods of enzyme kinetic analysis accounting for product inhibi-tion4,5,20,25–37. However, no general, direct and practical method of extracting enzyme cycling parameters fromnon-linear time courses has resulted from these efforts8,21,22,36. For example, the integrated form of the rateequation employed in the pioneering works of Henri38 and Michaelis and Menten6,11 considers product inhibition,but it is a complex implicit function of time with superposition of a linear and a logarithmic function that poseextreme challenges in curve fitting even with modern computers. In addition, many of these analyses assumeproduct(s) bind reversibly, individually, and exclusively to free enzyme in an off-pathway reaction6,11,38. However,multiple products can also bind simultaneously (e.g. ADP and Pi of an ATPase), and product rebinding to anenzyme’s active site in most cases is on-pathway.

A practical approach for analysis of non-linear enzyme activity time courses with product inhibition and/orsubstrate depletion involves numerical integration of rate equations using kinetic simulations6,39,40. Global fitting

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Received20 May 2013

Accepted27 August 2013

Published13 September 2013

Correspondence andrequests for materials

should be addressed toE.M.D.L.C. ([email protected])

SCIENTIFIC REPORTS | 3 : 2658 | DOI: 10.1038/srep02658 1

and kinetic simulations can potentially reveal the fundamental rateand equilibrium constants of multi-step pathways such as those ofenzyme catalyzed reactions without the need for assumptions tosimplify analysis or derivation and analytical integration of rateequations40. However, such simulations do not explicitly providethe familiar enzyme cycling parameters of interest (e.g. kcat andKM), and the utility of kinetic simulations can be restricted by a lackof investigator knowledge of the catalytic pathway(s). Specifically,calculation of kcat and KM values from intrinsic rate constantsrequires knowledge of the reaction mechanism since the kcat andKM are composites of the elementary reaction constants and thusvary among different enzyme mechanisms14,17,41–43. In addition,simulations rarely if ever yield unique solutions and require inde-pendent measurements of reaction constants to constrain free para-meters during fitting40. As such, analysis of non-linear enzymeactivity time courses is a challenge. It would be very useful todetermine enzyme cycling parameters more accurately whileproviding more information (e.g. the extent of product inhibition)if practical methods of analysis were available to extract initial velo-city and product inhibition parameters directly from full non-linearenzyme activity time courses.

Here we present a simple and practical method for determiningenzyme cycling parameters from enzyme activity time courses dis-playing non-linearity due to product inhibition and/or substratedepletion. In the Supplementary Information, we provide the theor-etical derivation of the exact solution, and first and second ordermathematical approximations of the exact solution of the non-linearproduct formation rate equation describing enzyme cycling,accounting for product inhibition as well as substrate depletion.The integrated forms of the rate equation allows for fitting of non-linear kinetic time courses, thereby permitting extraction of initialenzyme velocities and de-convolution of product inhibition and sub-strate depletion simultaneously. Since the first order approximationhas proven in our hands to converge rapidly in practice, here wefocus on implementation of the first order approximation to theanalytical solution. The method described is general under the firstorder approximation condition (see Discussion) and can be used forall enzyme systems independent of reaction schemes. For caseswhere the first order approximation is inadequate, we also providemathematical equation of the second order approximation(Supplementary Information).

ResultsThe following equation (Eq. 1; from Eq. A23 derived inSupplementary Information) describes time courses of steady-stateenzyme-catalyzed product ([P]) formation and can be used to fitboth linear and non-linear, steady-state enzyme kinetic time (t)courses (Figures 1A and 2A):

½P�~ v0

g1{e{gtð Þ: ð1Þ

The fits of the data with Eq. 1 yield two parameters: v0, the initialenzyme cycling velocity (e.g. no product inhibition or substratedepletion); and g, a new term describing the reduction in cyclingvelocity that causes non-linearity in time courses. This term is therelaxation rate constant of v0 and when time courses become linear atg , 0, Eq. 1 simplifies to the familiar [P] 5 v0t. The initial velocity(v0) values as a function of substrate concentration are used to deter-mine the steady-state cycling parameters, kcat and KM, according toestablished formalisms5.

The value of g varies with substrate concentration and indicatesthe extent of non-linearity in enzyme activity time courses, whileserving as a valuable diagnostic parameter to de-convolute the con-tributions of product inhibition and substrate depletion. Values of g. t 21, where t is the time range of data acquisition, characterizenon-linear time courses and indicate that product inhibition and/or

substrate depletion are contributing significantly to the observeddata set. Conversely, values of g = t 21, indicate linearity andfulfillment of the initial velocity approximation commonly used insteady-state enzyme kinetic analysis. The substrate concentration-dependence of g reveals the origins of non-linearity – values of g thatdecrease with substrate concentration reflect a regime where

Figure 1 | Analysis of Michaelis’ and Menten’s experimental data.(A). Time courses of product formation from Michaelis’ and Menten’s

experiment 1 at [S]0 of 0.333 (& black), 0.167 (. red), 0.0833 (m blue),

0.0416 (b magenta), 0.0208 (¤olive), 0.0104 (c navy) and 0.0052 M

(.violet). The solid lines through the data points represent the best-fits to

the product inhibition/substrate depletion equation (Eq. 1). Data at times

longer than 150 min were also included in the fit but are not shown for

clarity. (B). [S]0-dependence of the initial enzyme cycling velocity (filled

black circles) with the standard error (black bars) obtained from the best-

fits shown in Figure 1A. The initial velocities (filled cyan stars) obtained by

Michaelis and Menten from fitting data points acquired at an early time

regime to a straight line are also plotted for comparison. To convert the

observed optical rotation change to product concentration, we used the

following relation derived from the conversion used by Michaelis and

Menten: change in optical rotation 5 1.313 mu5 55.62uM21 mu5 42.36uM21 is the optical rotation of one molar sucrose substrate, obtained by

linear fit of the optical rotation vs. initial sucrose substrate concentration (t

5 0) according to Table 1 in Michaelis’ and Menten’s original paper6,11.

The solid line represents the best-fit to a rectangular hyperbola (Eq. A9),

yielding values for Vm (i.e. kcat[E]tot) 5 0.73 (60.03) mM min21, KM 5 13

(63) mM and Vm/KM 5 0.055 (60.0006) min21, similar to the Vm 5 0.76

(60.05) mM min21, KM 5 16.7 mM and Vm/KM 5 0.0454 (60.0032)

min21 reported by Michaelis and Menten6,11. Uncertainty bars in Panels B

and C represent the standard errors from the corresponding fits. (C). [S]0-

dependent enzyme cycling velocity reduction rate constant g. The solid line

through the data points is for visualization only. Substrate concentration

regimes where time courses will display linear and non-linear behavior are

indicated.

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SCIENTIFIC REPORTS | 3 : 2658 | DOI: 10.1038/srep02658 2

substrate depletion dominates, while values of g that increase withsubstrate concentration indicate a regime where product inhibitiondominates (see Supplementary Information). Values of g . t 21 thatvary little with substrate concentration arise when both productinhibition and substrate depletion contribute similarly to the degreeof observed nonlinearity in enzyme cycling kinetics. Note that g /[E]tot, which quantitatively predicts how reducing the enzyme con-centration reduces both substrate depletion and product inhibitioneffects.

As an example demonstrating the utility and simplicity of thisanalysis method, we reanalyze Michaelis’ and Menten’s original data

of invertase-catalyzed hydrolysis of sucrose to fructose and glu-cose6,11. Michaelis’ and Menten’s published data6,11 are well fitted tothe transient product formation equation developed here (Eq. 1;Figure 1A). The initial velocities (v0) obtained from fitting data toEq. 1 (black filled circles) match those obtained by Michaelis andMenten (filled cyan stars) at small and large substrate concentrationswhen the time courses have less curvature. Conversely, at intermedi-ate substrate concentrations, where the time courses display largecurvature (Figure 1B), the initial velocities obtained with ourapproach are slightly higher than those determined by Michaelisand Menten. This is understandable, given that when drawing astraight line through approximately linear data points at early timeto determine the initial velocity, it is difficult to avoid underestima-tion when the time course curvature is large. The steady-state cyclingparameters, kcat and KM, determined from the substrate concentra-tion-dependence of the initial velocity (v0) are comparable to those ofMichaelis and Menten (Figure 1B). However, because the non-lin-earity in Michaelis and Menten’s data occurs at low substrate con-centrations and diminishes at high substrate (Figures 1A and 1C),the majority of non-linearity in the data originates from substratedepletion rather than product inhibition, as the authors originallyconcluded6,11.

To demonstrate the robustness and accuracy of the technique, wealso analyze simulated data of a multi-step enzyme reaction pathway(Figure A1 in Supplementary Information) with arbitrarily assignedreaction rate constants. The steady-state cycling parameters obtainedfrom analysis of the non-linear time courses with Eq. 1 (Figures 2Aand 2B), yield essentially identical values as predicted from theory(Eq. A10 in Supplementary Information) and the reaction rate con-stants. Non-linearity in this case originates predominantly fromproduct inhibition (i.e., g , t 21 at low substrate concentrationsprogresses to g . t 21 at higher substrate concentration; Figure 2C).

Analysis of non-linear, steady-state kinetic time courses by thismethod permits determination of the product binding affinity,according to established protocols of enzyme inhibition5 (here theproduct is acting as an enzyme inhibitor) without the need for addi-tional experiments. To do this, one must first plot the velocity versussubstrate concentration at different product concentrations(Figure 3). These plots can be readily generated by using the follow-ing observed velocity (vobs) equation (Eq. 2; Eq. A28 in Supple-mentary Information):

vobs~d½P�dt

~v0{g½P� ð2Þ

where the v0 and g values are obtained from analysis of non-linearproduct formation time courses with Eq. 1. We generated such plotsof observed velocity (vobs) versus initial substrate concentration usingMichaelis and Menten’s original data (Figure 1) at [P] 5 0 and 5 mM(Figure 3A). Following established analysis methods of steady-stateenzyme inhibition5, global fitting of the two data sets (Figure 3A) wasperformed to the following equation for mixed (competitive anduncompetitive) inhibition by a single inhibitor while accountingfor substrate mass conservation ([S] 5 [S]0 2 [P])5:

vobs~Vm½S�

(1z½P�=Kic)KMz(1z½P�=Kiu)½S�

~Vm(½S�0{½P�)

(1z½P�=Kic)KMz(1z½P�=Kiu)(½S�0{½P�)

ð3Þ

With fixed [P] values and unconstrained competitive inhibition bind-ing constant Kic, uncompetitive inhibition constant Kiu, maximal velo-city Vm and KM values, the global fit yields Kic 5 47 (69) mM and Kiu

, 300 mM. The latter is very weak, indicating product inhibitionoriginates exclusively from competitive binding with substrate.

In separate experiments, Michaelis and Menten determinedenzyme-product binding affinities of 58 mM (fructose, KP1) and

Figure 2 | Analysis of non-linear enzyme kinetic time courses.(A). Simulated enzyme cycling time courses according to Figure A1 with

initial [substrate] (from bottom to top) of 10, 20, 40, 80, 160, 320, 640, and

1280 mM, and the following fundamental reaction rate constants: k11 5

0.06 mM s21, k21 5 25 s21, k12 5 0.1 s21, k22 5 0.3 s21, k13 5 10 s21, k23

5 1 mM s21, k14 5 10 s21, k24 5 0.1 mM s21, k15 5 10 s21, k25 5 0.1 mM

s21, k16 5 9.7 s21, and k26 5 0.8 mM s21. The solid lines through the

simulated data represent the best-fits to the product inhibition/substrate

depletion equation (Eq. 1). (B). [S]0-dependence of the initial enzyme

cycling velocity. The solid line represents the best-fit to a rectangular

hyperbola (Eq. A9), yielding values of Vm 5 0.098 (60.0005) mM s21, KM

5 442.8 (64) mM, which agree well to the values of Vm 5 0.097 mM s21,

KM 5 412.2 mM predicted from the fundamental rate constants (Eq. A10).

(C). [S]0-dependent enzyme cycling velocity reduction rate constant g. The

solid line through the data points is for visualization only. Substrate

concentration regimes where time courses will display linear and non-

linear behavior are indicated.

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SCIENTIFIC REPORTS | 3 : 2658 | DOI: 10.1038/srep02658 3

88 mM (glucose, KP2)6,11. The overall affinity of the two products(KP1P2, under conditions of [P1] , [P2] ? [E]), can be calculatedwith the following equation:

KP1P2~½P1�z½P2�ð Þ½E�½EP1�z½EP2�

~½P1�z½P2�ð Þ½E�½E�½P1�

KP1

z½E�½P2�

KP2

*2½P1�½E�

½E�½P1�1

KP1

z1

KP2

� �~2KP1 KP2

KP1zKP2

:

ð4Þ

Using Michaelis and Menten’s measured product affinities, the cal-culated KP1P2 is 70 mM, which is about 1.5 fold greater than the Kic

value of 47 (69) mM obtained using the method described abovewith Eq. 3. The fact that the obtained Kic value agrees well with thebinding affinities of 58 mM for fructose almost within the uncer-tainty indicates that inhibition by two competing products is domi-nated by the one with tighter affinity.

Similarly, we also produced plots of observed velocity (vobs) versusinitial substrate concentration using Eq. 2 for the simulated data(Figure 2) at two product concentrations [P] 5 0 and 3.5 mM;(Figure 3B). Global fitting of the two curves (Eq. 3) yields Kic 5

7.5 (60.6) mM, which is more than two folder tighter than the overallaffinity of 21.4 mM for the two products binding to the enzymecalculated (Eq. 4) from the individual binding affinities of 12 mM(k16/k26) and 100 mM (k14/k24) (Figure 3B and SupplementaryInformation Figure A1), but agrees reasonably well with the tighteraffinity of the two. The extremely weak uncompetitive binding affin-ity (Kiu) of 3 3 1018 mM obtained from global fitting is consistentwith the lack of an enzyme-substrate-product state in the scheme on

which the simulated data is based (Supplementary InformationFigure A1).

DiscussionThe general method described here provides a direct and practicalapproach to analyze nonlinear enzyme kinetic time courses withcontributions from product inhibition and/or substrate depletion,independent of the specific enzyme reaction mechanism and cata-lytic pathway. Compared to other integrated forms of the nonlinearrate equation used to analyze entire, non-linear enzyme kinetic timecourses4–6,11,20,25–37, the method described here is practical andstraightforward to use, and rapidly converges during parameteroptimization in data analysis. The method extracts meaningfulenzyme cycling kinetic (kcat and KM) parameters from entire non-linear time courses, which is more accurate than analysis of early timepoints based on the initial velocity assumption with little productformation3,5–11. Moreover, the method reveals the origin of nonli-nearity in enzyme kinetic time courses and provides the overallproduct binding affinity of the enzyme. As such, the formalism basedon determining the true initial velocity and the parameter g fromanalysis of full time courses provides physically meaningful inter-pretations of observed nonlinearity in enzyme cycling time courses.

The first order approximation of the exact solution of the non-linear rate equation implemented here simplifies the complicatedtime-dependent change in enzyme cycling rate to an exponentialfunction. When the product concentration is much smaller thanthe initial substrate concentration (i.e. [P] = [S]0), as is commonlythe case in the experimentally observed region of steady-state timecourses, the change in turnover rate follows an exponential decay andthe first order approximation is reliable. In cases where the decay inturnover rate deviates from an exponential, the second orderapproximation (presented in the Supplementary) or higher may berequired for analysis. For example, when the Km for substrate cata-lysis is much tighter than product(s) binding affinity and the timecourses are exceedingly long such that most of the substrate is con-verted to product, higher order corrections may be needed.

We emphasize here that the method introduced here is applicableonly for steady-state enzyme activity (i.e. the system must haveachieved more than one turnover per enzyme). Accordingly, pre-steady-state/transient kinetic behaviors, including burst and lagphases, are not described by the method presented in this work.Similarly, the approach does not apply to the cases where enzyme-catalyzed reaction time courses do not reflect the steady-state beha-vior of the enzyme under study (e.g., deficiencies in coupled assaysused for detection, which must be carefully evaluated with appropri-ate control experiments44).

MethodsKinetic simulations were performed with Tenua (available free at http://bililite.com/tenua/) or Kintek Global Kinetic Explorer (Kintek Co., Texas). Curve-fitting wascarried out with software Origin (www.originlab.com).

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AcknowledgementsThis work was partially supported by National Institutes of Health Grant RO1-GM097348awarded to E.M.D.L.C. We thank Dr. Michael J. Bradley for discussions and editing of themanuscript.

Author contributionsW.C. and E.M.D.L.C. designed the study and wrote the paper. W.C. performed themathematical derivation and analysis.

Additional informationSupplementary information accompanies this paper at http://www.nature.com/scientificreports

Competing financial interests: The authors declare no competing financial interests.

How to cite this article: Cao, W. & De La Cruz, E.M. Quantitative full time course analysisof nonlinear enzyme cycling kinetics. Sci. Rep. 3, 2658; DOI:10.1038/srep02658 (2013).

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SCIENTIFIC REPORTS | 3 : 2658 | DOI: 10.1038/srep02658 5

1

Quantitative full time course analysis of nonlinear enzyme cycling kinetics

Wenxiang Cao1 & Enrique M. De La Cruz

1, *

1 Department of Molecular Biophysics and Biochemistry, Yale University, New Haven,

Connecticut 06520, USA.

Supplementary Information

Theory.

Exact solution: A minimal mechanism describing the catalytic cycle of many enzymes (e.g.

hydrolases (EC3) including ATPases 1-3

and nucleotide pyrophosphatase/phosphodiesterase

(NPP) family enzymes 4) is comprised of a single substrate (S) that is converted by enzyme (E)

into two products (P1 and P2) that dissociate independently (Figure A1). This scheme differs

from that assumed by Michaelis and Menten for the glucose hydrolase activity of invertase 5,6

in

that it explicitly considers substrate to product transformation 7-9

.

Figure A1. Reaction scheme describing the catalytic cycle of enzymes in which a single

substrate (S) is converted by enzyme (E) into two products (P1 and P2) that dissociate

independently.

2 1 E P + P

1 2 1 2

1 2E + S E S E P P

k k

k k

1 2 E P + P

1 2E + P + P

2

For the chemical reaction depicted in Figure A1, there are 4 independent differential

equations:

23 1 2 4 2 3 2 1 4 2

[ ][ ] [ ][ ] [ ][ ] [ ]

d E Pk E P P k E P k E P P k E P

dt A1’

15 1 2 6 1 5 1 2 6 1

[ ][ ] [ ][ ] [ ][ ] [ ]

d E Pk E P P k E P k E P P k E P

dt A2’

1 22 3 2 1 5 1 2 2 3 5 1 2

[ ][ ] [ ][ ] [ ][ ] [ ]

d E P Pk E S k E P P k E P P k k k E P P

dt A3’

1 2 1 2 1 2

[ ][ ][ ] [ ] [ ]

d E Sk E S k E P P k k E S

dt A4’

During steady-state cycling, the steady-state condition requires all the intermediates remain

constants, i.e. in Eqs. A1’-A4’, 1 2 1 1 2[ ] [ ] [ ][ ]0

d E P P d E P d E Pd E S

dt dt dt dt . After rearranging

terms, Eqs. A1’-A4’ become

3 1 2 4 2 3 2 1 4 2[ ] [ ][ ] [ ][ ] [ ]k E P P k E P k E P P k E P A1

5 1 2 6 1 5 1 2 6 1[ ] [ ][ ] [ ][ ] [ ]k E P P k E P k E P P k E P A2

2 3 2 1 5 1 2 2 3 5 1 2[ ] [ ][ ] [ ][ ] [ ]k E S k E P P k E P P k k k E P P A3

1 2 1 2 1 2[ ][ ] [ ] [ ]k E S k E P P k k E S A4

3

Solving 2[ ]E P in Eq. A1, 1[ ]E P in Eq. A2 and [ ]E S in Eq. A4, and substituting in Eq. A3

yields:

3 1 2 4 2 5 1 2 6 11 2 1 22 3 1 5 2

1 2 3 1 4 5 2 6

2 3 5 1 2

[ ] [ ][ ] [ ] [ ][ ][ ][ ] [ ][ ] [ ]

[ ] [ ]

[ ]

k E P P k E P k E P P k E Pk E S k E P Pk k P k P

k k k P k k P k

k k k E P P

which can be rearranged and simplified to

3 3 1 5 5 22 22 3 5

1 2 3 1 4 5 2 61 2

3 4 2 1 5 6 2 11 2

1 2 3 1 4 5 2 6

[ ] [ ]

[ ] [ ][ ] [ ]

[ ][ ] [ ][ ][ ]

[ ] [ ]

k k P k k Pk kk k k

k k k P k k P kE E P P

k k P P k k P Pk k S

k k k P k k P k

. A5

Solving 2[ ]E P in Eq. A1, 1[ ]E P in Eq. A2, and [ ]E S in Eq. A4 and substituting them with

[ ]E (Eq. A5) into the mass balance equation for the total enzyme [E]tot yields:

6 1 3 51 4 2 2

1 2 3 1 4 5 2 6 1 2 3 1 4 5 2 6

61 4 2

1 2 3 1 4

1 2

1 2 2 1

[ ][ ] [ ]1 [ ] 1

[ ] [ ] [ ] [ ]

[ ] [ ]1

[ ]

[ ]

[ ] [ ] [ ] [ ] [ ] [ ]tot

k P k kk S k P kE

k k k P k k P k k k k P k k P k

kk S k P

k k k P k

E P P

E E E S E P P E P E P

3 3 1 5 5 22 2

2 3 5

1 1 2 3 1 4 5 2 6

3 4 1 2 5 6 1 21 25 2 6

1 2 3 1 4 5 2 6

3 52

1 2 3 1 4 5

[ ] [ ]

[ ] [ ] [ ]

[ ][ ] [ ][ ][ ][ ]

[ ] [ ]

1[ ] [

k k P k k Pk kk k k

P k k k P k k P k

k k P P k k P Pk k Sk P k

k k k P k k P k

k kk

k k k P k k

1 22 6

[ ]]

E P PP k

A6

and allows for 1 2[ ]E P P in Eq. A6 to be solved according to:

4

3 3 1 5 5 22 2

2 3 5

6 1 1 2 3 1 4 5 2 61 4 2

3 4 1 2 5 6 1 21 21 2 3 1 4 5 2 6

1 2 3 1 4

1 2[ ] [ ]

[ ] [ ] [ ][ ] [ ]1

[ ][ ] [ ][[ ][ ] [ ]

[ ]

[ ][ ] tot

k k P k k Pk kk k k

k P k k k P k k P kk S k P

k k P P k k P Pk k Sk k k P k k P k

k k k P k

EE P P

5 2 6

3 52

1 2 3 1 4 5 2 6

]

[ ]

1[ ] [ ]

k P k

k kk

k k k P k k P k

.A7

During steady-state cycling the two product release rates are equal and given by:

2 16 1 6 1 3 1 2 3 2 1

3 5 1 2 5 1 2 3 2 1

2 2 1 2

1 2 1 21 2

1 2 1 2

[ ] [ ][ ] [ ][ ] [ ] [ ][ ]

[ ] [ ][ ] [ ][ ]

[ ] [ ]

[ ][ ] [ ]

d P d Pk E P k E P k E P P k E P P

dt dt

k k E P P k E P P k E P P

k E S k E P P

k k k kE S E P P

k k k k

3 3 5 51 2 2 2

2 3 5

1 2 1 2 3 4 5 6

3 4 5 61 2 1 2

1 2 1 2 3 4 5 6

1

1 2

1 2

1 2 1 2

1 2

[ ] [ ][ ]

[ ] [ ]

[ ][ ] [ ][ ][ ]]

[ ] [ ]

[ ]1

[tot

k k P k k Pk k k kS k k k

k k k k k P k k P k

k k P P k k P Pk k k k SE

k k k k k P k k P k

k S

6 3 3 5 54 2 2

2 3 5

1 2 3 4 5 6 1 2 3 4 5 6

3 5 3 42 1 2

1 2 3 4 5 6 1 2

1 1 22

1 2 1 2

1 2

[ ] [ ] [ ][ ]

[ ] [ ] [ ] [ ]

[[ ]1

[ ] [ ]

k P k k P k k Pk P k kk k k

k k k P k k P k k k k P k k P k

k k k k Pk k k S

k k k P k k P k k k

5 6

3 4 5 6

1 2 1 2

1 2

][ ] [ ][ ]

[ ] [ ]

P k k P P

k P k k P k

. A8

Eqs. A2, A4, A5 and A7 were substituted in derivation of Eq. A8. Accordingly, when 0t ,

1 2[ ], [ ] 0P P and 0[ ] [ ]S S , the initial substrate concentration, and the initial steady-state

velocity simplifies to:

5

1 2 01 2 2 2 1 20 2 3 5

1 2 1 2 1 2 1 2

1 0 3 5 12 2 22 3 5

1 2 1 2 1 2 4 6

10

[ ][ ] [ ]

[ ]1 1

[ ]|

tot

t

k k Sk k k k k kS k k k E

k k k k k k k k

k S k k k kk k kk k k

k k k k k k k k

d P

dt

2 0

1 2

[ ]S

k k

1 2 0 3 5

2 3 2 51 2 3 5 1 2 1 0 2 3 5 2

4 6

[ ] [ ]

[ ]

totk k S k k E

k k k kk k k k k k k S k k k k

k k

0

0

[ ] [ ]

[ ]

cat tot

M

k E S

K S

A9

which is the familiar Briggs-Haldane equation for the initial substrate concentration-dependent

enzyme cycling in the absence of product inhibition and substrate depletion, with the enzymatic

reaction parameters

2 3 5 4 6

4 6 2 2 3 5 2 3 6 4 5

4 6 1 2 3 5 1 2

1 4 6 2 2 3 5 2 3 6 4 5

cat

M

k k k k kk

k k k k k k k k k k k

k k k k k k k kK

k k k k k k k k k k k k

.

A10

The steady-state kcat and KM expressions in Eq. A10 are identical to those obtained in the absence

of product inhibition 4.

Under conditions where there is no initial product (i.e. 1 2[ ] [ ] 0P P at t = 0), and free

substrate and product concentrations are much greater than total enzyme ( i.e.

1 2[ ], [ ], [ ] [ ]totS P P E ), the two free products are approximately equal ( 2 1[ ] ~[ ] [ ]P P P ) and

the free substrate 0 1 0[ ]~[ ] [ ] [ ] [ ]S S P S P . Eq. A8 can therefore be expressed as:

6

1 2 5 6 1 2 5 6 1 2 3 1 4

1 2 3 4 1 2 3 4 1 2 5 2 6

1 2 2 1 2 3 1 4 5 2 6

3 4 2 5 2 6 5 6 1 3 1 4 1

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ] [ ][ ]

[ ] [ ] [ ]

[ ] [ ] [ ] [ ]

tot

k k k k S k k k k P P k P k

k k k k S k k k k P P k P k Ed P

dt k k k S k k k P k k P k

k k P k P k k k P k P k k k

1 2 4 2 1 3 2 4 1 2 3 4 1 2 2 3 4 1 2 5 2 6

1 2 6 1 1 5 2 6 1 2 5 6 1 2 2 5 6 1 2 3 1 4

2

3 4 6 1 4 5 6 2 3 4

[ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

k k k P k k k k S k k k k k k k k k P P k P k

k k k P k k k k S k k k k k k k k k P P k P k

k k k P k k k P k k

5 1 2 3 5 6 1 2 1 2[ ][ ] [ ][ ]k P P k k k P P k k

1

21 2 5 6 0 1 2 5 6 3 4

21 2 3 4 0 1 2 3 4 5 6

1 2 2 0 1 2 3 4 5 6

3 4 5 6 5 6 3 4 1 2

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

tot

k k

k k k k S P k k k k P k P k

k k k k S P k k k k P k P k E

k k k S P k k k P k k P k

k k P k P k k k P k P k k k

2

2 4 1 3 2 4 0 1 2 3 4 1 2 2 3 4 5 6

2

1 2 6 1 5 2 6 0 1 2 5 6 1 2 2 5 6 3 4

23 4 6 4 5 6 3 4 5 3

[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ]

k P k k k k S P k k k k k k k k k P k P k

k k k P k k k k S P k k k k k k k k k P k P k

k k k P k k k P k k k P k

25 6 1 2[ ]k k P k k

2 30 1 2 3

2 30 1 2 3

[ ] [ ] [ ][ ]

[ ] [ ] [ ]tot

P P PE

P P P

.

A11

where some of the i and i (i = 0, 1, 2, 3) constants depend on the initial substrate concentration.

The third order polynomial function in the numerator of Eq. A11,

2 30 1 2 3[ ] [ ] [ ]P P P , has three roots, which are a positive root, r, and either two

negative roots or a complex conjugate pair of roots 10

. Therefore, Eq. A11 can be rearranged as

follows

3 1 0

2 23 3 1 0

[ ]1[ ] [ ]

[ ] [ ] [ ]tot

B P Bqd P E dt

P r P A P A

A12

7

where the new coefficients q, A0, A1, B0, and B1 are combinations of i and i (i = 0, 1, 2, 3) that

may or may not depend on the initial substrate concentration. Integrating both sides of Eq. A12

and incorporating the initial condition [P] = 0 yields:

3 012 2 2

3 3 3 1 0

1 1 1 102 2 2 2

3 0 1 0 1 0 1

21 0

[ ] ln ln[ ] 2 [ ] [ ]

2[ ]1 2arctan arctan

2 4 4 4

[ ] , for 4tot

ABq rP

r P P A P A

A B P A AB

A A A A A A

E t A A

A13

or

3 012 2 2

3 3 3 1 0

21 1 01 1

02 23 1 0

21 1 0

21 0

[ ] ln ln[ ] 2 [ ] [ ]

2[ ]1

41 1ln

2[ ]2 4 14

[ ] , for 4tot

ABq rP

r P P A P A

P

A A AA BB

PA A

A A A

E t A A

A13

Although Eqs. A13 and A13 represent exact analytical solutions of differential equation Eq. A11,

the concentration of free enzymatic product [P] is a complicated implicit function of time (t)

composed of a linear, two logarithmic and an inverse tangent or additional logarithmic function,

and thus it is difficult to extract physically meaningful parameters (e.g. kcat and KM). An

approximation approach to obtain a solution of the concentration of free product [P] that allows

for physically meaningful parameters to be extracted is as follows.

8

First order approximation analysis: Given that the normalized free product 0

[ ]1

[ ]

Px

S and

letting 644 6

0 0

' , '[ ] [ ]

kkk k

S S

(ratio between product release rate constant(s) and initial

substrate concentration) and normalized product release rate 0

1 [ ]

[ ]

dx d P

dt S dt , the differential

equation Eq. A11 of [P] becomes the following differential equation of a small variable x:

1 2 4 1 3 2

21 2 5 6 1 2 5 6 3 4

21 2 3 4 1 2 3 4 5 6

1 2 2 0 1 2 3 4 5 6

3 4 5 6 5 6 3 4 1 2

[ ]

' 1 '

' 1 ' [ ]

[ ] 1 ' '

' '

tot

k k k x k k k S

k k k k x k k k k x k x k

k k k k x k k k k x k x k Edx

dt k k k S x k k k x k k x k

k k x k x k k k x k x k k k

2

0 4 1 2 3 4 1 2 2 3 4 0 5 6

2

1 2 6 1 5 2 0 6 1 2 5 6 1 2 2 5 6 0 3 4

2 23 4 6 4 5 6 3 4 5 3 5 6 1 2

' 1 ' [ ] '

[ ] ' 1 ' [ ] '

' '

k x k k k k k k k k k x S k x k

k k k x k k k S k x k k k k k k k k k x S k x k

k k k x k k k x k k k x k k k x k k

[ ] ( )[ ]tot tot

Q xE f x E

P x . A14

where P(x), Q(x) and f(x) are functions of x defined as

( )

( )( )

Q xf x

P x A15

9

2

1 2 4 1 3 2 0 4 1 2 3 4 1 2 2 3 4 0 5 6

1 2 6 1 5 2 0 6

1 2 2 0 1 2 3 4 5 6

3 4 5 6 5 6 3 4 1 2

[ ] 1 [ ]

[ ] 1

[ ] 1 ' '

' '

' ' '

'

k k k x k k k S k x k k k k k k k k k x S k x k

k k k x k k k S k x

P x k k k S x k k k x k k x k

k k x k x k k k x k x k k k

2

1 2 5 6 1 2 2 5 6 0 3 4

2 23 4 6 4 5 6 3 4 5 3 5 6 1 2

[ ]' '

' '

k k k k k k k k k x S k x k

k k k x k k k x k k k x k k k x k k

A16

and

1 2 3 5 6 3 4 5 1 2 3 4 6 4

2

1 2 5 6 1 2 5 6 3 4

1 2 3 5 6 3 4 5 4 6 3 5

21 2 3 4 1 2 3 4 5 6

1 2 4 6 3 5

' ' '

' 1 '

' ' ' '

' 1 '

' '

k k k k k k k k k k k k k k

Q x k k k k x k k k k x k x k

k k k k k k k k k k k k x

k k k k x k k k k x k x k

k k k k k k

2

5 6

3

1 2 3 5 4 6' k k x k k k k k k x

A17

with partial derivatives to x as '( )f x ,

1 2 4 1 3 2 0 4

1 2 2 0 3 4 5 6

1 2 2 0 1 2 3 5 6

1 2 2 0 1 2 3 4 5

3 4 5 6 3 4 5 5 6 3 4 3 5 6 1 2

[ ]

' [ ] ' '

[ ] 1 '

[ ] 1 '

' '

k k k k k k S k

P x k k k S k x k k x k

k k k S x k k k k x k

k k k S x k k k x k k

k k k x k k k xk k k k x k k k k x k k

1 2 2 3 4 0 5 6

2

1 2 4 1 3 2 0 4 1 2 3 4 1 2 2 3 4 0

1 2 6 1 5 2 0 6 1 2 2 5 6 0 3 4

1 2 6 1 5 2 0 6

5

2 [ ]

[ ] 1 [ ]

[ ] 2 [ ]

[ ]

' '

' '

' '

k k k k k x S k x k

k k k x k k k S k x k k k k k k k k k x S

k k k k k k S k k k k k k x S k x k

k k k x k k k S k

k

2

1 2 5 6 1 2 2 5 6 0 3

3 4 6 4 5 6 3 4 5 3 5 6 1 2

1 [ ]' '

' ' 2 2

x k k k k k k k k k x S k

k k k k k k k k k x k k k x k k

A18

and

10

1 2 3 5 6 3 4 5 1 2 3 4 6 4 5 6

1 2 3 5 6 3 4 5 4 6 3 5

21 2 3 5 4 62 ' ' ' ' 3

' ' ' ' '

k k k k k k k k k k k k k k k k x

Q x k k k k k k k k k k k k

k k k k k k x

.A19

Since 0

[ ]1

[ ]

Px

S , and is therefore a small variable, the normalized product release rate

dx

dt in

Eq. A14 can be expanded at x = 0 using the Taylor series expansion.

2( )[ ] (0) '(0) [ ]tot tot

dxf x E f f x O x E

dt A20

where O(x2) is a very small quantity proportional to the second order of the variable x that can be

ignored in the first order approximation. In Eq. A20, the fact that x is a normalized free product

concentration that always increases (i.e. the rate, 0dx

dt ), but the increase gradually slows down

due to product inhibition or substrate depletion (i.e. the second derivative of x to t is negative)

leads to

0

2

20 0

1(0) 0

[ ]

( ) 1'(0) 0

[ ]

tot x

totx x

dxf

E dt

f x d xf

x E dt

A21

At the first order approximation, the solution to the linear differential equation Eq. A20 with the

initial condition x = 0 is

0

0

1[ ]

tvx e

S

A22

or

11

0[ ] 1 tvP e

. A23

where v0 is the initial steady-state velocity defined as

1 2 2 3 5 4 6 2 3 6 4 5 0 1 2 1 2 3 5 4 6

0 0 0

1 2 3 5 4 6 0

0

0

[ ]

0(0)[ ] [ ] [ ] [ ]

0

[ ] [ ]

[ ] [ ]

[ ]

tot tot

tot

cat tot

M

k k k k k k k k k k k k S k k k k k k k k

Qv f S E S E

P

k k k k k k S E

k S E

K S

A24

with kcat and KM expressed as in Eq. 10.

We introduce a new term, , to describe the decay of the steady-state cycling rate due to

product inhibition or substrate depletion as:

2

' 0 0 0 ' 0'(0)[ ] [ ]

0tot tot

Q P Q Pf E E

P

which can be expressed in terms of reaction rate constants as follows:

4 6 2 2 3 3 6 4 5 5

22 3 5 4 6 3 6 4 5 0

4 6 1 2 3 3 6 3 4 6 3 4 6 4 5 5 4 5 6 4 5 6

2

1 2 3 5 4 6 4 6 0

3 5 4

1 2 1

[ ]

[ ]

[ ]

tot k k k k k k k k k k

k k k k k k k k k

k k k k k k k k k k k k k k k k k k k k k k

k k k k k k k k

k k k

S

k k E k

S

2

6 1 2 1 2 3 5

2 2 3 5 4 6 2 3 6 4 5 0

1 2 1 2 3 5

1

2

4 6

[ ]

k k k k k k k

k k k k k k k k k k k

k k k k k k

k S

k k

.A25

12

When 0 , from Eq. A23, 0[ ]P v t , i.e. the time course of product formation becomes linear

with time. Eq. A23 predicts that 0[ ]v

P

as t and the enzymatic reaction reaches

equilibrium. In other words, accumulation of product gradually decreases the steady-state

cycling rate to 0 (i.e., the substrate, enzyme and product(s) are in equilibrium, with no net

conversion of substrate(s) to product(s)).

The value of serves as a valuable experimental diagnostic for assessing time course

non-linearity, specifically identifying the origin of non-linearity (product inhibition vs. substrate

depletion). For non-linearity to be negligible and product formation to be linear across the

observed time range , << 1; that is, 1

. When [S]0 0, Eq. A25 simplifies to:

1 2 1 2 3 5

51 2 3[ ]

[ ]cat

totM

tot kE

k k k k k k K

k k k k E

. A26

Under this condition is independent of product binding and non-linearity arises from substrate

depletion, which is proportional to cat

M

k

K. When [S]0 , Eq. A25 simplifies to:

4 6 2 2 3 3 6 4 5 5

2 3 5 4 6 3 6 4 5

2 2 3 5 4 6 2 3 6 4 5

2

2

[ ]tot

k k k k k k k k k k

k k k k k k k k k

k k k k k k k k k k k

k

E

A27

Under this condition the value of and any observed non-linearity in enzyme activity time

courses are dominated by product inhibition. Note that [E]tot, which quantitatively predicts

13

how reducing the enzyme concentration reduces both substrate depletion and product inhibition

effects.

The rate equation for product formation (Eq. A20) can be rewritten using the definition of

v0 and as follows:

0

[ ][ ]

d Pv v P

dt A28

If the initial velocity v0 and rate constant of the decrease in product formation are obtained

from the analysis of a non-linear time course of product formation by Eq. A23, Eq. A28 provides

a method to calculate the velocity v at any given product concentration [P] for all the initial

substrate concentration [S]0. Subsequently, the plots of v versus [S]0 for several different product

(the inhibitor in this case) concentrations can be used to analyze the inhibitor (product) binding

constant(s) using the standard inhibition analysis technique for enzyme kinetics 7 even though

product formation time courses are non-linear, which we demonstrate in the main text.

Second order approximation analysis: The time-dependent product formation Eq. A23 was

derived from applying the first order approximation to the rate equation (Eqs. A14 and A20). For

the second order approximation, the rate equation Eq. A20 becomes

2 31( )[ ] (0) '(0) ''(0) [ ]

2tot tot

dxf x E f f x f x O x E

dt

A29

where f(x) is the second partial derivative to x. O(x3) is a small quantity proportional to the third

order of x and is ignored at the second order approximation. Integration of Eq. A29 yields

14

2 2 2

2 2 2

''(0) '(0)[ ]

, if '(0) 2 ''(0) (0) [ ] 0

''(0) '(0)[ ]

2 ''(0) '(0) tan if '(0) 2 ''(0) (0) [ ] 0[ ] 2

ttottot

tot

tottot

f x fE

Ce f f f E

f x fE

t Cf x f f f f E

E

. A30

C in Eq. A30 is an arbitrary constant to be determined by the initial conditions. is defined in

the equation. Since a tangent function is not a physically meaningful solution for the time course

of product formation, the condition 2'(0) 2 ''(0) (0) 0f f f would never occur and the solution

with a tangent function is disregarded. Eq. A30 with the initial condition x = 0 can be written as

0

0

0

2 (1 )2 (0)[ ] (1 ) [ ]

[ ]

'(0) '(0)[ ] [ ] [ ] [ ] [ ] [ ]

2 (1 )

tt

tot

t t

tot tot tot tot tot tot

t

t

ve

f S e EP

f f e eE E E E E E

v e

e

A31

where v0, and are defined in Eqs. A24, A25 and A30, respectively.

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15

3 De La Cruz, E. M., Wells, A. L., Rosenfeld, S. S., Ostap, E. M. & Sweeney, H. L. The

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