robust nonlinear regression in enzyme kinetic parameters

Research ArticleRobust Nonlinear Regression in EnzymeKinetic Parameters Estimation

Maja MarasoviT1 Tea MarasoviT2 andMladenMiloš1

1Faculty of Chemistry and Technology University of Split Ruđera Boskovica 35 21000 Split Croatia2Faculty of Electrical Engineering Mechanical Engineering and Naval Architecture University of SplitRuđera Boskovica 32 21000 Split Croatia

Correspondence should be addressed to Tea Marasovic tmarasovfesbhr

Received 18 October 2016 Accepted 6 February 2017 Published 5 March 2017

Academic Editor Murat Senturk

Copyright copy 2017 Maja Marasovic et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Accurate estimation of essential enzyme kinetic parameters such as 119870119898 and 119881max is very important in modern biology To thisdate linearization of kinetic equations is still widely established practice for determining these parameters in chemical and enzymecatalysis Although simplicity of linear optimization is alluring these methods have certain pitfalls due to which they more oftenthen not result in misleading estimation of enzyme parameters In order to obtain more accurate predictions of parameter valuesthe use of nonlinear least-squares fitting techniques is recommended However when there are outliers present in the data thesetechniques become unreliable This paper proposes the use of a robust nonlinear regression estimator based on modified Tukeyrsquosbiweight function that can provide more resilient results in the presence of outliers andor influential observations Real andsynthetic kinetic data have been used to test our approach Monte Carlo simulations are performed to illustrate the efficacy andthe robustness of the biweight estimator in comparison with the standard linearization methods and the ordinary least-squaresnonlinear regression We then apply this method to experimental data for the tyrosinase enzyme (EC 114181) extracted fromSolanum tuberosum Agaricus bisporus and Pleurotus ostreatus The results on both artificial and experimental data clearly showthat the proposed robust estimator can be successfully employed to determine accurate values of119870119898 and 119881max

1 Introduction

Enzymes are molecules that act as biological catalysts andare responsible for maintaining virtually all life processesMost enzymes are proteins although a few are catalyticRNA molecules Like all catalysts enzymes increase the rateof chemical reactions without themselves undergoing anypermanent chemical change in a process They achieve theireffect by temporarily binding to the substrate and in doingso lowering the activation energy needed to convert it to aproduct The study of the rate at which an enzyme works iscalled enzyme kinetics and it is often regarded as one of themost fascinating research areas in biochemistry [1]

Mathematically the relationship between substrate con-centration and reaction rate under isothermal conditions formany of enzyme-catalyzed reactions can be modeled by theMichaelis-Menten equation [2]

V = 119881max119904119904 + 119870119898 (1)

where V denotes a reaction rate 119904 is a substrate concentration119881max is the maximum initial velocity which is theoreticallyattained when the enzyme has been ldquosaturatedrdquo by an infiniteconcentration of a substrate and 119870119898 is the Michaelis con-stant representing a measure of affinity of the enzyme-substrate interaction By definition 119870119898 is equal to the con-centration of the substrate at half maximum initial velocityThe Michaelis constant 119870119898 is an intrinsic parameter ofenzyme-catalyzed reactions and it is significant for its biologi-cal function [3]

Three most common methods available in the literaturefor determining the parameters of Michaelis-Menten equa-tion based on a series of measurements of velocity V as a func-tion of substrate concentration are Lineweaver-Burk plotalso known as the double reciprocal plot Eadie-Hofstee plotand Hanes-Woolf plot All three of these methods are lin-earizedmodels that transform the originalMichaelis-Mentenequation into a form which can be graphed as a straight line

HindawiJournal of ChemistryVolume 2017 Article ID 6560983 12 pageshttpsdoiorg10115520176560983

2 Journal of Chemistry

Lineweaver-Burk [4] (LB) plot still the most popularand favored plot amongst the researchers is defined by anequation

1V= 1119881max

+ 119870119898119881max

1119904 (2)

The 119910-intercept in this plot is 1119881max the 119909-intercept insecond quadrant represents minus1119870119898 and the slope of the lineis119870119898119881max

Eadie-Hofstee [5] (EH) plot is a semireciprocal plot of Vversus V119904 The linear equation has the following form

V = 119881max minus 119870119898 V119904 (3)

where the 119910-intercept is 119881max and the slope is119870119898In Hanes-Woolf [6] (HW) plot 119904V is plotted against 119904

The linear equation is given by

119904V= 119870119898119881max

+ 1119881max119904 (4)

where the 119910-intercept is 119870119898119881max and the slope is 1119881maxIn all of the above-described linear transformations

linear regression is used to estimate the slope and interceptof the straight line and afterwards119870119898 and119881max are computedfrom the straight line parameters Although these methodsare very useful for data visualization and are still widelyemployed in enzyme kinetic studies each of them possessescertain deficiencies which make them prone to errors Forinstance Lineweaver-Burk plot has the disadvantage of com-pressing the data points at high substrate concentrations intoa small region and emphasizing the points at lower substrateconcentrations which are often the least accurate [7] The 119881-intercept in Lineweaver-Burk plot is equivalent to inverse of119881max due to which any small error in measurement gets mag-nified Similarly the Eadie-Hofstee plot has the disadvantagethat V appears on both axes thus any experimental error willalso be present in both axes In addition experimental errorsor uncertainties are propagated unevenly and become largerover the abscissa thereby givingmoreweight to smaller valuesof V119904 Hanes-Woolf plot is the most accurate of the threehowever itsmajor drawback is that again neither ordinate norabscissa represents independent values both are dependenton substrate concentration

In order to reduce the errors due to the linearizationof parameters Wilkinson [8] proposed the use of least-squares nonlinear regression for more accurate estimationof enzyme kinetic parameters Nonlinear regression allowsdirect determination of parameter values from untrans-formed data points The process starts with initial estimatesand then iteratively converges on parameter estimates thatprovide the best fit of the underlying model to the actual datapoints [9 10] The algorithms used include the Levenberg-Marquardt method the Gauss-Newtonmethod the steepest-descent method and simplex minimization Numerous soft-ware packages such as Excel MATLAB and GraphPrismnowadays include readily available routines and scripts toperform nonlinear least-squares fitting [11 12]

Least-squares nonlinear regression has been criticized forits performance in dealing with experimental data This ismainly due to the fact that implicit assumptions related withnonlinear regression are in general not met in the contextof deviations that appear as a result of biological errors(eg variations in the enzyme preparations due to oxida-tion or contaminations) andor experimental errors (egvariations in measured volume of substrates and enzymesimprecisions of the instrumentation) With the presence ofoutliers or influential observations in the data the ordinaryleast-squares method can result in misleading values for theparameters of the nonlinear regression and estimates may nolonger be reliable [13]

In this paper we propose the use of robust nonlinear re-gression estimator based on modified Tukeyrsquos biweight func-tion for determining the parameters of Michaelis-Mentenequation using experimentalmeasurements in enzyme kinet-ics The main idea is to fit a model to the data that givesresilient results in the presence of influential observationsandor outliers To the best of our knowledge this is the firststudy that examines the use of this technique for applicationin Michaelis-Menten enzyme analysis We employ MonteCarlo simulations to validate the efficacy of the proposedprocedure in comparison with the ordinary least-squaresmethod and Eadie-Hofstee Hanes-Woolf and Lineweaver-Burk plots In addition we illustrate the viability of ourmethod by estimating the kinetic parameters of tyrosinasean important enzyme widely distributed in microorganismsanimals and plants responsible for melanin production inmammal and enzymatic browning in plants extracted frompotato and two edible mushrooms

The remainder of the paper is organized as followsSection 2 provides a brief overview of the robust estimationmodel Section 3 describes the experimental setup used inthis research and the diagnostics that will be used to evaluatethe effectiveness of the proposed procedure in determinationof enzyme kinetic parameters Results are discussed in Sec-tion 4 Finally Section 5 summarizes the paper with a fewconcluding remarks

2 Robust Nonlinear Regression

Nonlinear regression same as linear regression relies heavilyon the assumption that the scatter of data around the idealcurve follows at least approximately a Gaussian or normaldistributionThis assumption leads to the well-known regres-sion goal to minimize the sum of the squares of the verticaldistances (aka residuals) between the points and the curveIn practice however this assumption does not always holdtrue The analytical data often contains outliers that can playhavoc with standard regression methods based on the nor-mality assumption causing them to produce more or lessstrongly biased results depending on the magnitude ofdeviation andor sensitivity of the procedure It is not unusualto find an average of 10 of outlying observations in data setof some processes [14]

Outliers are most commonly thought to be extreme val-ueswhich are a result ofmeasurement or experimental errorsBarnett and Lewis [15] provide a more cautious definition of

Journal of Chemistry 3

the term outlier describing it as the observation (or subset ofobservations) that appears to be inconsistentwith the remain-der of the dataset This definition also includes the observa-tions that do not follow themajority of the data such as valuesthat have been measured correctly but are for one reason oranother far away from other data values while the formu-lation ldquoappears to be inconsistentrdquo reflecting the subjectivejudgement of the observer whether or not an observation isdeclared to be outlying

The ordinary least-squares (OLS) estimate 120573LS of theparameter vector 120573 = [1205731 1205732 120573119899] is obtained as thesolution of the problem

120573LS = argmin120573

1119899119899sum119894=1

119903119894 (119909 120573)2

= argmin120573

1119899119899sum119894=1

(119910119894 minus 119891 (119909119894 120573))2 (5)

where 119899 denotes the number of observations 119909 = [x1 x2 xn]119879 is a 119899 times 119901matrix whose rows are 119901-dimensional vectorsof predictor variables (or regressors) 119910 = [1199101 1199102 119910119899]119879 isa 119899 times 1 vector of responses and 119891(119909119894 120573) is model functionSince all data points are attributed the same weights OLSimplicitly favors the observations with very large residualsand consequently the estimated parameters end up distortedif outliers are present

In order to achieve robustness in copingwith the problemof outliers Huber [16] introduced a class of so-called 119872-estimators for which the sum of function 120588 of the residuals isminimized The resulting vector of parameters 120573119872 estimatedby an119872-estimator is then

120573119872 = argmin120573

119899sum119894=1

120588 (119903119894120590) (6)

The residuals are standardized by a measure of dispersion120590 to guarantee scale equivariance (ie independence withrespect to the measurement units of the dependent variable)Function 120588(sdot) must be even nondecreasing for positivevalues and less increasing than the square

The minimization in (6) can always be done directlyHowever often it is simpler to differentiate 120588 function withrespect to 120573 and solve for the root of the derivative Whenthis differentiation is possible the 119872-estimator is said to beof120595-type Otherwise the119872-estimator is said to be of 120588-type

Let 120595 = 1205881015840 be the derivative of 120588 Assuming 120590 is knownand defining weights 119908119894 = 120595(119903119894120590)119903119894 the estimates 120573119872 canbe obtained by solving the system of equations

119899sum119894=1

1199082119894 1199032119894 = 0 (7)

The weights are dependent upon the residuals the residualsare dependent upon the estimated coefficients and the esti-mated coefficients are dependent upon the weights Henceto solve for 119872-estimators an iteratively reweighted least-squares (IRLS) algorithm is employed Starting from some

initial estimates 120573(0) at each iteration 119905 until it converges thisalgorithm computes the residuals 119903(119905minus1)119894 and the associatedweights 119908(119905minus1)119894 = 119908[119903(119905minus1)119894 ] from the previous iteration andyields new weighted least-squares estimates

21 Objective Function Several 120588 functions can be usedHere we opted for Tukeyrsquos biweight [17] or bisquare functiondefined as

120588 (119911119894) =

11988826 (1 minus [1 minus (119911119894119888 )2]3) if 10038161003816100381610038161199111198941003816100381610038161003816 le 119888

11988826 if 10038161003816100381610038161199111198941003816100381610038161003816 gt 119888 (8)

where 119888 is a tuning constant and 119911119894 = 119903119894120590The corresponding 120595(119911119894) function is

120595 (119911119894) = 119911119894 [1 minus (119911119894119888 )

2]2 if 10038161003816100381610038161199111198941003816100381610038161003816 le 1198880 if 10038161003816100381610038161199111198941003816100381610038161003816 gt 119888

(9)

Tukeyrsquos biweight estimator has a smoothly redescending120595 function that prevents extreme outliers to affect thecalculation of the biweight estimates by assigning them azero weighting As can be seen in Figure 1 the weights forthe biweight estimator decline as soon as 119911 departs from 0and are 0 for |119911| gt 119888 Smaller values of 119888 produce moreresistance to outliers but at the expense of lower efficiencywhen the errors are normally distributedThe tuning constantis generally picked to give reasonably high efficiency innormal case in particular 119888 = 4685produces a 95efficiencywhen the errors are normal while guaranteeing resistance tocontamination of up to 10 of outliers

In an application an estimate of the standard deviationof the errors is needed in order to use these results Usually arobustmeasure of spread is used in preference to the standarddeviation of the residuals A common approach is to take = MAD06745 where MAD is the median absolutedeviation Despite having the best possible breakout pointof 50 the MAD is not without its weaknesses It exhibitssuperior statistical efficacy for the contaminated data (iethe data that contains extreme scores) however when thedata approaches a normal distribution the MAD is only37 efficient Furthermore it is ill-suited for asymmetricaldistributions since it attaches equal importance to positiveand negative deviations from location estimate

Hence the scale parameter 120590 is computed usingRousseeuw-Croux estimator 119876119899 [18]

119876119899 = 119889 10038161003816100381610038161003816119909119894 minus 11990911989510038161003816100381610038161003816 119894 lt 119895(119896)

(10)

where 119889 is a calibration factor and 119896 = ( ℎ2 ) asymp ( 1198992 ) 4 whereℎ = 1198992 + 1 is roughly half the number of observationsThe estimator 119876119899 has the optimal 50 breakdown point itis equally suitable for both symmetrical and asymmetricaldistributions and considerably more efficient (about 82)than the MAD under a Gaussian distribution


minus4 0 4

minus1

0

1

minus4 0 40

1

2

3

4

minus4 0 40

1휌(z) 휓(z) w(z)

Figure 1 Tukeyrsquos biweight estimator objective 120595 and weight functions for 119888 = 4685

3 Experimental Setup

To illustrate the efficacy of the proposed approachweuse bothartificial data generated from the Monte Carlo simulationsand the experimental data for the tyrosinase enzyme (EC114181) extracted from three different sources

31 Monte Carlo Simulations Simulation studies are usefulfor gaining insight into the examined algorithms strengthsand weaknesses such as robustness against number ofvariable factors There are three outlier scenarios and a totalof 18 different situations considered in this researchThe datasets are generated from the model

119910119894 = (1205791119909119894)(1205792 + 119909119894) + 120598119894 119894 = 1 2 119899 (11)

where regression coefficients are fixed 120579 = [80 240] for each119894 = 1 2 119899 The explanatory variables 119909119894 are set to 100 sdot 119894and a zeromean unit variance randomnumberwithGaussiandensity 120598119894 is added as measurement error

The factors considered in this simulation are (1) level ofoutlier contamination 10 or 20 (2) sample size small(119899 = 10) medium (119899 = 20) or large (119899 = 50) and (3)distances of outliers from clean observations 10 standarddeviations 50 standard deviations or 100 standard devia-tions There are 1200 replications for each scenario and allsimulations are carried out in MATLABThe 3 scenarios andthe 18 situations considered in this research are summarizedin Table 1

These simulated data are then used to estimate thevalues of 119870119898 and 119881max using different fitting techniquesThe mean estimated values of 119870119898 and 119881max for a particularscenario and fitting technique are subsequently calculatedby averaging 119870119898 and 119881max values obtained in each of the1200 trials The estimator efficacy is assessed in terms of itsbias precision and accuracy Bias is defined as an absolutedifference between mean estimated parameter values andknown parameter values

Bias =10038161003816100381610038161003816100381610038161003816100381610038161003816sum119873119895=1 120579119894119895119873 minus 120579119894

10038161003816100381610038161003816100381610038161003816100381610038161003816 (12)

where119873 is the total number of replications in the simulatedscenario The term precision refers to the absence of randomerrors or variability It is measured by the coefficient of

Table 1 The 18 situations considered in the simulations

Scenario Situation Sample size Outliers 120590

1

1

10

10 102 20 103 10 504 20 505 10 1006 20 100

2

7

20

10 108 20 109 10 5010 20 5011 10 10012 20 100

3

13

50

10 1014 20 1015 10 5016 20 5017 10 10018 20 100

variation (119862V) that is the standard deviation expressed as apercentage of the mean

119862V = 120590120579119894119895 sdot 100 (13)

The prediction accuracy is defined as the overall distancebetween estimated values and true values The accuracy ismeasured by a normalized mean squared error (NMSE) thatis the mean of the squared differences between the estimatedand the known parameter values normalized by a mean ofestimated data

NMSE = 1119873119873sum119895=1

(120579119894119895 minus 120579119894120579119894119895 )

2

(14)

where again 119873 is the total number of replications in thesimulated scenario

32 Enzyme Data Sets Tyrosinase (EC 114181) is a ubiq-uitous enzyme responsible for melanization in animalsand plants [19 20] In the presence of molecular oxygen


Table 2 Input experimental kinetic data sets

Concentration [120583mol] Reaction rate [120583molmin]Ab Po Potato

25 00243 11 times 10minus4 0000450 00292 26 times 10minus4 00011100 00546 29 times 10minus4 00018250 01388 37 times 10minus4 00023500 01726 69 times 10minus4 000301000 02374 131 times 10minus4 001012500 03023 269 times 10minus4 001245000 03395 496 times 10minus4 003757500 03515 503 times 10minus4 0048510000 03652 607 times 10minus4 00569

this enzyme catalyzes hydroxylation of monophenols to 119900-diphenols (cresolase activity) and their subsequent oxidationto 119900-quinones (catecholase activity) The latter productsare unstable in aqueous solution further polymerizing toundesirable brown red and black pigments Tyrosinase hasattracted a lot of attention with respect to its biotechnologicalapplications [21] due to its attractive catalytic ability as thecatechol products are useful as drugs or drug synthons

For the purposes of present study tyrosinase wasextracted from potato (Solanum tuberosum) and two speciesof common edible mushrooms Agaricus bisporus (Ab) andPleurotus ostreatus (Po) All the source materials werepurchased from the local green market in Split CroatiaEnzyme extraction was prepared with 100mL of cold 50mMphosphate buffer (pH 60) per 50 g of a source material Thehomogenates were centrifuged at 5000 rpm for 30min andsupernatant was collected The sediments were mixed withcold phosphate buffer and were allowed to sit in cold condi-tion with occasional shaking Then the sediments containingbuffer were centrifuged once again to collect supernatantThese supernatants were subsequently used as sources ofenzyme

The tyrosinase activity was determined spectrophotomet-rically at room temperature (119905 = 25∘C) and 120582 = 475 nmmeasuring the conversion of L-DOPA to red coloured oxi-dation product dopachrome [22] The reaction mixturemdashobtained after adding a 50 120583L of enzyme extract to a cuvettecontaining 12mL of 50mM phosphate buffer (pH 60) and08mL of 10mM L-DOPAmdashwas immediately shaken and theincrease in absorbance was measured for 3 minutes Thechange in the absorbance was proportional to the enzymeconcentration The initial rate was calculated from the linearpart of the recorded progress curve One unit of enzyme wasdefined as the amount which catalyzed the transformationof 1 120583mol of L-DOPA to dopachrome per minute under theabove conditions The dopachrome extinction coefficient at475 nm was 3600Mminus1 cmminus1

To determine the values of 119870119898 and 119881max for tyrosinaseexperimental kinetic data summarized in Table 2 was gath-ered by measuring enzyme activity in a cuvette where 50 120583Lof enzyme solution was added to 2mL of 50mM phosphate

buffer (pH 60) containing various concentrations of L-DOPA (0ndash10mM) In this case the estimator performance isevaluated by computing mean absolute error (MAE) that isthe mean of the absolute differences between the observedreaction rate V119894 and the expected reaction rate calculatedusing estimates of119870119898 and 119881max at a concentration 119904119894

MAE = 1119899119899sum119894=1

(V119894 minus 119881max sdot 119904119894119904119894 + 119870119898 ) (15)

where 119899 is the number of experimental data points Meanabsolute error is regularly employed quality measure thatprovides an objective assessment of how well the variousestimated values of 119870119898 and 119881max fit the untransformedexperimental data

4 Results and Discussion

41 Parameter Estimation Using Simulated Data Figures 2ndash5provide the summary of our simulation outcomes for differ-ent sample sizes different contamination levels and differentoutlier distances By examining the simulation results itis evident that modified robust Tukeyrsquos biweight estimatoroutperforms all other four alternative fitting techniques withrespect to bias coefficient of variation and normalized meansquare error yielding both accurate and precise estimates of119870119898 and 119881max at all test conditions For example looking atthe set of values obtained for a small sample size (119899 = 10)with aminimal level of contamination present in the data andminimal outlier scatter (Situation 1 Table 1) we observe thatas per the RNR estimator the average estimated values of119870119898and119881max are 24008plusmn1439 and 80plusmn15 When EH HW andLB plots were used the data produced 22438plusmn3801 24102plusmn4408 and 23747plusmn5205 respectively as average estimates of119870119898 and 7818plusmn446 798plusmn485 and 7942plusmn615 respectivelyas119881max WhenOLS estimator was applied the correspondingaverage values of 119870119898 and 119881max were 23871 plusmn 4004 and7986 plusmn 439 respectively If the reported standard deviationsare scaled by dividing them with an appropriate mean theresulting coefficients of variation of119870119898 and119881max are 169 and57 (EH) 183and 61 (HW) 219and 77 (LB) 168and 55 (OLS) and 6 and 19 (RNR) respectively Thusit is revealed that though all three Hanes-Woolf Lineweaver-Burke and ordinary least-squares methods have a low bias(Figures 2 and 4) and produce the results that are in aclose proximity of the values obtained by robust nonlinearregression method their estimates are much more impreciseand as such are of a limited utility Figure 6 shows the plots offitted reaction curves for the randomly selected replicationsof situations with small sample size (Situations 1ndash6)

For a medium sample size (119899 = 20) with the same levelsof contamination and outlier scatter (Situation 7 Table 1)the analysis for the RNR approach yielded almost identicalresults that is average 119870119898 and 119881max estimates as 23968 plusmn93 and 80 plusmn 068 respectively The EH HW and LB plotsestimated the average 119870119898 as 22883 plusmn 2639 23801 plusmn 3812and 23852plusmn4921 respectively and119881max as 7909plusmn212 796plusmn273 and 796plusmn407 respectivelyThe average kinetic param-eters that is119870119898 and119881max obtained by the OLSmethod were


2 4 6 8 10 12 14 16 180

50

100

150

200

250

300

Situation

EH

2 4 6 8 10 12 14 16 180

100

200

300

400

500

600

700

Situation

HW

2 4 6 8 10 12 14 16 18200

210

220

230

240

250

260

270

Situation

RNR

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

LB

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

OLS

Figure 2 Mean estimated values (black dots) and standard deviations of Michaelis constant119870119898 for different simulated scenarios Red linesdenote true parameter value

23875plusmn2843 and 7988plusmn205 respectively Again if the stan-dard deviations are scaled by an appropriate mean the result-ing coefficients of variation of 119870119898 and 119881max are 115 and27 (EH) 16and 34 (HW) 206and 51 (LB) 119and 26 (OLS) and 39 and 08 (RNR) respectively

Similarly for a large sample size (119899 = 50) with thesame levels of contamination and outlier scatter (Situation 13Table 1) the values of 119870119898 and 119881max are estimated as 23267 plusmn2238 and 7971 plusmn 098 (EH) 2432 plusmn 2988 and 7999 plusmn 111(HW) 24697plusmn7833 and 8015plusmn343 (LB) 24068plusmn1919 and8008plusmn086 (OLS) and 23962plusmn719 and 8001plusmn031 (RNR)respectively In this case the resulting coefficients of variationof 119870119898 and 119881max are 96 and 12 (EH) 123 and 14(HW) 317and 43 (LB) 8and 11 (OLS) and 3and04 (RNR) respectively It should be noted that all thewhilein all of the aforementioned cases the Eadie-Hofstee method

has coefficients of variation that are highly comparableto those based on ordinary least-squares method the EHestimated119870119898 and119881max values aremuch further away from thetrue values than the estimates obtained byHanes-Woolf ordi-nary least-squares and robust nonlinear regressionmethods

With the increase of the contamination level and theoutlier scatter the average estimates of 119870119898 and 119881max valuesas per linear plots and the OLS method begin to deviatesignificantly However the modified robust Tukeyrsquos biweightestimator is able to keep the errors in check and produce theresults that are highly comparable andmuch closer to the trueparameter values

Numerically by looking at the estimated values it is hardto tell which of the selected estimators has the overall best per-formance nevertheless with the help of the normalizedmeansquare error method we can see the values of parameters for


EH HW LB OLS RNR EH HW LB OLS RNR



0

2

4

6

8

10

12

02468

10121416

05

1015202530354045

0

20

40

60

80

100

120

140

0

10

20

30

40

50

60

0

100

200

300

400

500

600

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50

Figure 3 Normalized mean square error of119870119898 for different scenarios

which the error is minimum Thus from Figures 3 and 5 wemay say that the best (most accurate) values of 119870119898 and 119881maxare obtained in situation 17 for which the minimum errorvalues are 00541 and 85 times 10minus4 respectively (as obtained byRNR) This proves that the estimated values are more or lesssimilar to the true valuesTheworst error values of 16982 and01517 for RNR method are obtained in situation 4 (Figures3 and 5) In all other situations the normalized mean squareerrors for RNR method are less than 1 and 01 respectively(Figures 3 and 5) This shows the credibility and the robust-ness of the proposedmodified Tukeyrsquos biweight estimator rel-ative to other methods when outliers or influential observa-tions are present in the data If we compare the robust nonlin-ear regression method with ordinary least-squares methodwe find that the RNRmethod normalizedmean square errors

are on average more than 10 times lower than the normalizedmean square errors produced by the OLS method

42 Parameter Estimation Using Experimental Data Theviability of the proposed robust estimator was also tested byusing the experimental kinetic data for tyrosinase enzymeThe corresponding 119870119898 and 119881max values produced bydifferent estimationmodels are given in Table 3 Upon closerinspection and analysis of these values it can be observedthat in case of Ab mushroom and potato tyrosinase thekinetic values that is 119870119898 and 119881max yielded by the RNRmethod (599 120583mol and 038 120583molmin for Ab mushroomand 10740 120583mol and 0118 120583molmin for potato resp)are much closer to the values yielded by HW plot(555 120583mol and 0381 120583molmin for Ab mushroom and


2 4 6 8 10 12 14 16 1855

60

65

70

75

80

85EH

Situation

2 4 6 8 10 12 14 16 1876

77

78

79

80

81

82

83

Situation

RNR

2 4 6 8 10 12 14 16 1860

65

70

75

80

85

90

95

Situation

HW

2 4 6 8 10 12 14 16 18556065707580859095

100

Situation

LB

2 4 6 8 10 12 14 16 186065707580859095

100

Situation

OLS

Figure 4 Mean estimated values (black dots) and standard deviations of maximum initial velocity 119881max for different simulated scenariosRed lines denote true parameter value

Table 3 Kinetic parameters119870119898 and 119881max values and mean absolute errors for tyrosinase extracted from different sources estimated usingdifferent methods

Source Parameter MethodLB HW EH OLS RNR

Ab119870119898 26309 55518 41409 54479 59941119881max 02484 03812 03470 03767 03798MAE 00511 00071 00137 00064 00048

Po119870119898 35991 35921 99227 57003 66305119881max 00017 00079 00042 00095 00099MAE 00013 26 times 10minus4 75 times 10minus4 17 times 10minus4 15 times 10minus4

Potato119870119898 12160 10659 20934 25720 10740119881max 00208 01104 00394 02086 01180MAE 00139 00022 00087 00018 00014


EH HW LB OLS RNR EH HW LB OLS RNR0

01

02

03

04

05

06

07

0



02040608

112141618

0

05

1

15

2

25

3

1

2

3

4

5

6

0

05

1

15

2

25

3

35

1

2

3

4

5

6

7

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100

Figure 5 Normalized mean square error of 119881max for different scenarios

10659 120583mol and 011120583molmin for potato resp) thanby OLS method (545120583mol and 0376 120583molmin for Abmushroom and 25720120583mol and 0209120583molmin forpotato resp) Furthermore it is interesting to note thatthe parameter values yielded by LB plot (119870119898 263 120583mol and119881max 0248120583molmin for Ab mushroom 119870119898 360 120583mol and119881max 0002 120583molmin for Po mushroom and 119870119898 1216 120583moland 119881max 0021 120583molmin for potato) for all three tyrosinasesource are very far from the values yielded by other fourestimation methods Figures 7(a) 7(b) and 8(a) show thecurves fitted to the experimental data using the modifiedTukeyrsquos biweight estimator in comparison with the standardlinearization methods and the ordinary least-squares non-linear regression The mean absolute errors between thepredicted reaction rates and the actual data are plotted inthe right graph in Figure 8 Particularly the mean errors for

RNR method are 00048 15 times 10minus4 and 00014 respectivelywhich shows a good fit of the achieved model

5 Conclusion

When an enzymatic reaction follows Michaelis-Mentenkinetics the equation for the initial velocity of reaction asa function of the substrate concentration is characterized bytwo parameters the Michaelis constant 119870119898 and the maxi-mum velocity of reaction 119881max Up to this day these param-eters are routinely estimated using one of these differentlinearizationmodels Lineweaver-Burke plot (1V versus 1119904)Eadie-Hofstee plot (V versus V119904) and Hanes-Wolfe plot(119904V versus V) Although the linear plots obtained by thesemethods are very illustrative and useful in analyzing thebehavior of enzymes the common problem they all share


0 100 200 300 400 500 600 700 800 900 10000

1020304050607080

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

Concentration (nM)

Concentration (nM)Concentration (nM)

Concentration (nM)

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

RNROLSEH

HWLB

RNROLSEH

HWLB

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

102030405060708090

100

Reac

tion

rate

(mm

olm

inm

g)

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100

Figure 6 Curves fitted using different fitting techniques for random replications of situations 1ndash6

is the fact that transformed data usually do not satisfy theassumptions of linear regression namely that the scatter ofdata around the straight line follows Gaussian distributionand that the standard deviation is equal for every value ofindependent variable

More accurate approximation of Michaelis-Mentenparameters can be achieved through use of nonlinear least-squares fitting techniques However these techniques requiregood initial guess and offer no guarantee of convergence tothe global minimum On top of that they are very sensitiveto the presence of outliers and influential observations inthe data in which case they are likely to produce biasedinaccurate and imprecise parameter estimates

In this paper a robust estimator of nonlinear regressionparameters based on a modification of Tukeyrsquos biweightfunction is introduced Robust regression techniques havereceived considerable attention in mathematical statisticsliterature but they are yet to receive similar amount ofattention by practitioners performing data analysis Robustnonlinear regression aims to fit amodel to the data so that theresults are more resilient to the extreme values and are rela-tively consistent when the errors come from the high-taileddistribution The experimental comparisons using both realand synthetic kinetic data show that the proposed robustnonlinear estimator based on modified Tukeyrsquos biweightfunction outperforms the standard linearization models and


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

00501

01502

02503

03504

Concentration (휇mol)

Reac

tion

rate

(휇m

olm

in)

RNROLSEH

HWLB

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7


Reac

tion

rate

(휇m

olm

in)

times10minus3

RNROLSEH

HWLB

(b)

Figure 7 Curves fitted using different fitting techniques for tyrosinase extracted from Agaricus bisporus (a) and Pleurotus ostreatus (b)mushrooms

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

006

RNROLSEH

HWLB


Reac

tion

rate

(휇m

olm

in)

(a)

EH HW LB OLS RNR0

001

002

003

004

005

006

(b)

Figure 8 Curves fitted using different fitting techniques for tyrosinase extracted frompotato (a)Mean absolute errors for different regressionmodels used (b)

ordinary least-squares method and yields superior resultswith respect to bias accuracy and consistency when thereare outliers or influential observations present in the data

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work reported in this paper was supported by CroatianScience Foundation Research Project ldquoInvestigation of Bioac-tive Compounds from Dalmatian Plants Their AntioxidantEnzyme Inhibition and Health Propertiesrdquo (IP 2014096897)

References

[1] H J Fromm and M Hargrove Essentials of BiochemistrySpringer Berlin Germany 2012

[2] K A Johnson and R S Goody ldquoThe original Michaelisconstant translation of the 1913 Michaelis-Menten paperrdquoBiochemistry vol 50 no 39 pp 8264ndash8269 2011

[3] A Cornish-Bowden ldquoThe origins of enzyme kineticsrdquo FEBSLetters vol 587 no 17 pp 2725ndash2730 2013

[4] H Lineweaver and D Burk ldquoThe determination of enzyme dis-sociation constantsrdquo Journal of the American Chemical Societyvol 56 no 3 pp 658ndash666 1934

[5] G S Eadie ldquoThe inhibition of cholinesterase by physostigmineand prostigminerdquo The Journal of Biological Chemistry vol 146pp 85ndash93 1942

[6] C S Hanes ldquoStudies on plant amylasesrdquo Biochemical Journalvol 26 no 5 pp 1406ndash1421 1932


[7] A Ferst Structure and Mechanism in Protein Science A Guideto Enzyme Catalysis and Protein Folding Freeman 2000

[8] G N Wilkinson ldquoStatistical estimations in enzyme kineticsrdquoThe Biochemical journal vol 80 pp 324ndash332 1961

[9] H Motulsky and A Christopoulus Fitting Models to BiologicalData Using Linear and Nonlinear Regression GraphPad Soft-ware Inc 2003

[10] C Cobelli and E Carson Introduction toModeling in Physiologyand Medicine Academic Press 2008

[11] S R Nelatury C F Nelatury and M C Vagula ldquoParameterestimation in different enzyme reactionsrdquo Advances in EnzymeResearch vol 2 no 1 pp 14ndash26 2014

[12] G Kemmer and S Keller ldquoNonlinear least-squares data fittingin Excel spreadsheetsrdquo Nature Protocols vol 5 no 2 pp 267ndash281 2010

[13] C Lim P K Sen and S D Peddada ldquoRobust nonlinearregression in applicationsrdquo Journal of the Indian Society ofAgricultural Statistics vol 67 no 2 pp 215ndash234 291 2013

[14] R A Maronna R D Martin and V J Yohai Robust StatisticsTheory and Methods Wiley Series in Probability and StatisticsJohn Wiley amp Sons New York NY USA 2006

[15] V Barnett and T Lewis Outliers in statistical data Wiley Seriesin Probability and Mathematical Statistics Applied Probabilityand Statistics John Wiley amp Sons Ltd Chichester EnglandThird edition 1994

[16] P J Huber ldquoRobust estimation of a location parameterrdquo Annalsof Mathematical Statistics vol 35 no 1 pp 73ndash101 1964

[17] F Mosteller and J W Tukey Data Analysis and RegressionAddison-Weasley 1977

[18] P J Rousseeuw and C Croux ldquoAlternatives to the median abso-lute deviationrdquo Journal of the American Statistical Associationvol 88 no 424 pp 1273ndash1283 1993

[19] M R Loizzo R Tundis and F Menichini ldquoNatural and syn-thetic tyrosinase inhibitors as antibrowning agents an updaterdquoComprehensive Reviews in Food Science and Food Safety vol 11no 4 pp 378ndash398 2012

[20] S Y Lee N Baek and T-G Nam ldquoNatural semisynthetic andsynthetic tyrosinase inhibitorsrdquo Journal of Enzyme Inhibitionand Medicinal Chemistry vol 31 no 1 pp 1ndash13 2016

[21] K U Zaidi A S Ali S A Ali and I Naaz ldquoMicrobialtyrosinases promising enzymes for pharmaceutical food bio-processing and environmental industryrdquoBiochemistry ResearchInternational vol 2014 Article ID 854687 16 pages 2014

[22] Z Yang and F Wu ldquoCatalytic properties of tyrosinase frompotato and edible fungirdquo Biotechnology vol 5 no 3 pp 344ndash348 2006

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


Lineweaver-Burk [4] (LB) plot still the most popularand favored plot amongst the researchers is defined by anequation

1V= 1119881max

+ 119870119898119881max

1119904 (2)

The 119910-intercept in this plot is 1119881max the 119909-intercept insecond quadrant represents minus1119870119898 and the slope of the lineis119870119898119881max

Eadie-Hofstee [5] (EH) plot is a semireciprocal plot of Vversus V119904 The linear equation has the following form

V = 119881max minus 119870119898 V119904 (3)

where the 119910-intercept is 119881max and the slope is119870119898In Hanes-Woolf [6] (HW) plot 119904V is plotted against 119904

The linear equation is given by

119904V= 119870119898119881max

+ 1119881max119904 (4)

where the 119910-intercept is 119870119898119881max and the slope is 1119881maxIn all of the above-described linear transformations

linear regression is used to estimate the slope and interceptof the straight line and afterwards119870119898 and119881max are computedfrom the straight line parameters Although these methodsare very useful for data visualization and are still widelyemployed in enzyme kinetic studies each of them possessescertain deficiencies which make them prone to errors Forinstance Lineweaver-Burk plot has the disadvantage of com-pressing the data points at high substrate concentrations intoa small region and emphasizing the points at lower substrateconcentrations which are often the least accurate [7] The 119881-intercept in Lineweaver-Burk plot is equivalent to inverse of119881max due to which any small error in measurement gets mag-nified Similarly the Eadie-Hofstee plot has the disadvantagethat V appears on both axes thus any experimental error willalso be present in both axes In addition experimental errorsor uncertainties are propagated unevenly and become largerover the abscissa thereby givingmoreweight to smaller valuesof V119904 Hanes-Woolf plot is the most accurate of the threehowever itsmajor drawback is that again neither ordinate norabscissa represents independent values both are dependenton substrate concentration

In order to reduce the errors due to the linearizationof parameters Wilkinson [8] proposed the use of least-squares nonlinear regression for more accurate estimationof enzyme kinetic parameters Nonlinear regression allowsdirect determination of parameter values from untrans-formed data points The process starts with initial estimatesand then iteratively converges on parameter estimates thatprovide the best fit of the underlying model to the actual datapoints [9 10] The algorithms used include the Levenberg-Marquardt method the Gauss-Newtonmethod the steepest-descent method and simplex minimization Numerous soft-ware packages such as Excel MATLAB and GraphPrismnowadays include readily available routines and scripts toperform nonlinear least-squares fitting [11 12]

Least-squares nonlinear regression has been criticized forits performance in dealing with experimental data This ismainly due to the fact that implicit assumptions related withnonlinear regression are in general not met in the contextof deviations that appear as a result of biological errors(eg variations in the enzyme preparations due to oxida-tion or contaminations) andor experimental errors (egvariations in measured volume of substrates and enzymesimprecisions of the instrumentation) With the presence ofoutliers or influential observations in the data the ordinaryleast-squares method can result in misleading values for theparameters of the nonlinear regression and estimates may nolonger be reliable [13]

In this paper we propose the use of robust nonlinear re-gression estimator based on modified Tukeyrsquos biweight func-tion for determining the parameters of Michaelis-Mentenequation using experimentalmeasurements in enzyme kinet-ics The main idea is to fit a model to the data that givesresilient results in the presence of influential observationsandor outliers To the best of our knowledge this is the firststudy that examines the use of this technique for applicationin Michaelis-Menten enzyme analysis We employ MonteCarlo simulations to validate the efficacy of the proposedprocedure in comparison with the ordinary least-squaresmethod and Eadie-Hofstee Hanes-Woolf and Lineweaver-Burk plots In addition we illustrate the viability of ourmethod by estimating the kinetic parameters of tyrosinasean important enzyme widely distributed in microorganismsanimals and plants responsible for melanin production inmammal and enzymatic browning in plants extracted frompotato and two edible mushrooms

The remainder of the paper is organized as followsSection 2 provides a brief overview of the robust estimationmodel Section 3 describes the experimental setup used inthis research and the diagnostics that will be used to evaluatethe effectiveness of the proposed procedure in determinationof enzyme kinetic parameters Results are discussed in Sec-tion 4 Finally Section 5 summarizes the paper with a fewconcluding remarks

2 Robust Nonlinear Regression

Nonlinear regression same as linear regression relies heavilyon the assumption that the scatter of data around the idealcurve follows at least approximately a Gaussian or normaldistributionThis assumption leads to the well-known regres-sion goal to minimize the sum of the squares of the verticaldistances (aka residuals) between the points and the curveIn practice however this assumption does not always holdtrue The analytical data often contains outliers that can playhavoc with standard regression methods based on the nor-mality assumption causing them to produce more or lessstrongly biased results depending on the magnitude ofdeviation andor sensitivity of the procedure It is not unusualto find an average of 10 of outlying observations in data setof some processes [14]

Outliers are most commonly thought to be extreme val-ueswhich are a result ofmeasurement or experimental errorsBarnett and Lewis [15] provide a more cautious definition of




120573LS = argmin120573

1119899119899sum119894=1

119903119894 (119909 120573)2

= argmin120573

1119899119899sum119894=1

(119910119894 minus 119891 (119909119894 120573))2 (5)



120573119872 = argmin120573

119899sum119894=1

120588 (119903119894120590) (6)




119899sum119894=1

1199082119894 1199032119894 = 0 (7)




120588 (119911119894) =

11988826 (1 minus [1 minus (119911119894119888 )2]3) if 10038161003816100381610038161199111198941003816100381610038161003816 le 119888

11988826 if 10038161003816100381610038161199111198941003816100381610038161003816 gt 119888 (8)


120595 (119911119894) = 119911119894 [1 minus (119911119894119888 )

2]2 if 10038161003816100381610038161199111198941003816100381610038161003816 le 1198880 if 10038161003816100381610038161199111198941003816100381610038161003816 gt 119888

(9)




119876119899 = 119889 10038161003816100381610038161003816119909119894 minus 11990911989510038161003816100381610038161003816 119894 lt 119895(119896)

(10)



minus4 0 4

minus1

0

1

minus4 0 40

1

2

3

4

minus4 0 40

1휌(z) 휓(z) w(z)





119910119894 = (1205791119909119894)(1205792 + 119909119894) + 120598119894 119894 = 1 2 119899 (11)




Bias =10038161003816100381610038161003816100381610038161003816100381610038161003816sum119873119895=1 120579119894119895119873 minus 120579119894

10038161003816100381610038161003816100381610038161003816100381610038161003816 (12)




1

1

10

10 102 20 103 10 504 20 505 10 1006 20 100

2

7

20

10 108 20 109 10 5010 20 5011 10 10012 20 100

3

13

50

10 1014 20 1015 10 5016 20 5017 10 10018 20 100


119862V = 120590120579119894119895 sdot 100 (13)


NMSE = 1119873119873sum119895=1

(120579119894119895 minus 120579119894120579119894119895 )

2

(14)












MAE = 1119899119899sum119894=1

(V119894 minus 119881max sdot 119904119894119904119894 + 119870119898 ) (15)






2 4 6 8 10 12 14 16 180

50

100

150

200

250

300

Situation

EH

2 4 6 8 10 12 14 16 180

100

200

300

400

500

600

700

Situation

HW

2 4 6 8 10 12 14 16 18200

210

220

230

240

250

260

270

Situation

RNR

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

LB

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

OLS











0

2

4

6

8

10

12

02468

10121416

05

1015202530354045

0

20

40

60

80

100

120

140

0

10

20

30

40

50

60

0

100

200

300

400

500

600

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50






2 4 6 8 10 12 14 16 1855

60

65

70

75

80

85EH

Situation

2 4 6 8 10 12 14 16 1876

77

78

79

80

81

82

83

Situation

RNR

2 4 6 8 10 12 14 16 1860

65

70

75

80

85

90

95

Situation

HW

2 4 6 8 10 12 14 16 18556065707580859095

100

Situation

LB

2 4 6 8 10 12 14 16 186065707580859095

100

Situation

OLS




Ab119870119898 26309 55518 41409 54479 59941119881max 02484 03812 03470 03767 03798MAE 00511 00071 00137 00064 00048


Potato119870119898 12160 10659 20934 25720 10740119881max 00208 01104 00394 02086 01180MAE 00139 00022 00087 00018 00014



01

02

03

04

05

06

07

0



02040608

112141618

0

05

1

15

2

25

3

1

2

3

4

5

6

0

05

1

15

2

25

3

35

1

2

3

4

5

6

7

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100




5 Conclusion



0 100 200 300 400 500 600 700 800 900 10000

1020304050607080

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

Concentration (nM)


Concentration (nM)

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

RNROLSEH

HWLB

RNROLSEH

HWLB

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

102030405060708090

100

Reac

tion

rate

(mm

olm

inm

g)

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100






0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

00501

01502

02503

03504


Reac

tion

rate

(휇m

olm

in)

RNROLSEH

HWLB

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7


Reac

tion

rate

(휇m

olm

in)

times10minus3

RNROLSEH

HWLB

(b)


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

006

RNROLSEH

HWLB


Reac

tion

rate

(휇m

olm

in)

(a)

EH HW LB OLS RNR0

001

002

003

004

005

006

(b)



Competing Interests


Acknowledgments


References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




120573LS = argmin120573

1119899119899sum119894=1

119903119894 (119909 120573)2

= argmin120573

1119899119899sum119894=1

(119910119894 minus 119891 (119909119894 120573))2 (5)



120573119872 = argmin120573

119899sum119894=1

120588 (119903119894120590) (6)




119899sum119894=1

1199082119894 1199032119894 = 0 (7)




120588 (119911119894) =

11988826 (1 minus [1 minus (119911119894119888 )2]3) if 10038161003816100381610038161199111198941003816100381610038161003816 le 119888

11988826 if 10038161003816100381610038161199111198941003816100381610038161003816 gt 119888 (8)


120595 (119911119894) = 119911119894 [1 minus (119911119894119888 )

2]2 if 10038161003816100381610038161199111198941003816100381610038161003816 le 1198880 if 10038161003816100381610038161199111198941003816100381610038161003816 gt 119888

(9)




119876119899 = 119889 10038161003816100381610038161003816119909119894 minus 11990911989510038161003816100381610038161003816 119894 lt 119895(119896)

(10)



minus4 0 4

minus1

0

1

minus4 0 40

1

2

3

4

minus4 0 40

1휌(z) 휓(z) w(z)





119910119894 = (1205791119909119894)(1205792 + 119909119894) + 120598119894 119894 = 1 2 119899 (11)




Bias =10038161003816100381610038161003816100381610038161003816100381610038161003816sum119873119895=1 120579119894119895119873 minus 120579119894

10038161003816100381610038161003816100381610038161003816100381610038161003816 (12)




1

1

10

10 102 20 103 10 504 20 505 10 1006 20 100

2

7

20

10 108 20 109 10 5010 20 5011 10 10012 20 100

3

13

50

10 1014 20 1015 10 5016 20 5017 10 10018 20 100


119862V = 120590120579119894119895 sdot 100 (13)


NMSE = 1119873119873sum119895=1

(120579119894119895 minus 120579119894120579119894119895 )

2

(14)












MAE = 1119899119899sum119894=1

(V119894 minus 119881max sdot 119904119894119904119894 + 119870119898 ) (15)






2 4 6 8 10 12 14 16 180

50

100

150

200

250

300

Situation

EH

2 4 6 8 10 12 14 16 180

100

200

300

400

500

600

700

Situation

HW

2 4 6 8 10 12 14 16 18200

210

220

230

240

250

260

270

Situation

RNR

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

LB

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

OLS











0

2

4

6

8

10

12

02468

10121416

05

1015202530354045

0

20

40

60

80

100

120

140

0

10

20

30

40

50

60

0

100

200

300

400

500

600

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50






2 4 6 8 10 12 14 16 1855

60

65

70

75

80

85EH

Situation

2 4 6 8 10 12 14 16 1876

77

78

79

80

81

82

83

Situation

RNR

2 4 6 8 10 12 14 16 1860

65

70

75

80

85

90

95

Situation

HW

2 4 6 8 10 12 14 16 18556065707580859095

100

Situation

LB

2 4 6 8 10 12 14 16 186065707580859095

100

Situation

OLS




Ab119870119898 26309 55518 41409 54479 59941119881max 02484 03812 03470 03767 03798MAE 00511 00071 00137 00064 00048


Potato119870119898 12160 10659 20934 25720 10740119881max 00208 01104 00394 02086 01180MAE 00139 00022 00087 00018 00014



01

02

03

04

05

06

07

0



02040608

112141618

0

05

1

15

2

25

3

1

2

3

4

5

6

0

05

1

15

2

25

3

35

1

2

3

4

5

6

7

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100




5 Conclusion



0 100 200 300 400 500 600 700 800 900 10000

1020304050607080

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

Concentration (nM)


Concentration (nM)

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

RNROLSEH

HWLB

RNROLSEH

HWLB

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

102030405060708090

100

Reac

tion

rate

(mm

olm

inm

g)

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100






0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

00501

01502

02503

03504


Reac

tion

rate

(휇m

olm

in)

RNROLSEH

HWLB

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7


Reac

tion

rate

(휇m

olm

in)

times10minus3

RNROLSEH

HWLB

(b)


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

006

RNROLSEH

HWLB


Reac

tion

rate

(휇m

olm

in)

(a)

EH HW LB OLS RNR0

001

002

003

004

005

006

(b)



Competing Interests


Acknowledgments


References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


minus4 0 4

minus1

0

1

minus4 0 40

1

2

3

4

minus4 0 40

1휌(z) 휓(z) w(z)





119910119894 = (1205791119909119894)(1205792 + 119909119894) + 120598119894 119894 = 1 2 119899 (11)




Bias =10038161003816100381610038161003816100381610038161003816100381610038161003816sum119873119895=1 120579119894119895119873 minus 120579119894

10038161003816100381610038161003816100381610038161003816100381610038161003816 (12)




1

1

10

10 102 20 103 10 504 20 505 10 1006 20 100

2

7

20

10 108 20 109 10 5010 20 5011 10 10012 20 100

3

13

50

10 1014 20 1015 10 5016 20 5017 10 10018 20 100


119862V = 120590120579119894119895 sdot 100 (13)


NMSE = 1119873119873sum119895=1

(120579119894119895 minus 120579119894120579119894119895 )

2

(14)












MAE = 1119899119899sum119894=1

(V119894 minus 119881max sdot 119904119894119904119894 + 119870119898 ) (15)






2 4 6 8 10 12 14 16 180

50

100

150

200

250

300

Situation

EH

2 4 6 8 10 12 14 16 180

100

200

300

400

500

600

700

Situation

HW

2 4 6 8 10 12 14 16 18200

210

220

230

240

250

260

270

Situation

RNR

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

LB

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

OLS











0

2

4

6

8

10

12

02468

10121416

05

1015202530354045

0

20

40

60

80

100

120

140

0

10

20

30

40

50

60

0

100

200

300

400

500

600

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50






2 4 6 8 10 12 14 16 1855

60

65

70

75

80

85EH

Situation

2 4 6 8 10 12 14 16 1876

77

78

79

80

81

82

83

Situation

RNR

2 4 6 8 10 12 14 16 1860

65

70

75

80

85

90

95

Situation

HW

2 4 6 8 10 12 14 16 18556065707580859095

100

Situation

LB

2 4 6 8 10 12 14 16 186065707580859095

100

Situation

OLS




Ab119870119898 26309 55518 41409 54479 59941119881max 02484 03812 03470 03767 03798MAE 00511 00071 00137 00064 00048


Potato119870119898 12160 10659 20934 25720 10740119881max 00208 01104 00394 02086 01180MAE 00139 00022 00087 00018 00014



01

02

03

04

05

06

07

0



02040608

112141618

0

05

1

15

2

25

3

1

2

3

4

5

6

0

05

1

15

2

25

3

35

1

2

3

4

5

6

7

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100




5 Conclusion



0 100 200 300 400 500 600 700 800 900 10000

1020304050607080

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

Concentration (nM)


Concentration (nM)

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

RNROLSEH

HWLB

RNROLSEH

HWLB

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

102030405060708090

100

Reac

tion

rate

(mm

olm

inm

g)

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100






0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

00501

01502

02503

03504


Reac

tion

rate

(휇m

olm

in)

RNROLSEH

HWLB

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7


Reac

tion

rate

(휇m

olm

in)

times10minus3

RNROLSEH

HWLB

(b)


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

006

RNROLSEH

HWLB


Reac

tion

rate

(휇m

olm

in)

(a)

EH HW LB OLS RNR0

001

002

003

004

005

006

(b)



Competing Interests


Acknowledgments


References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of










MAE = 1119899119899sum119894=1

(V119894 minus 119881max sdot 119904119894119904119894 + 119870119898 ) (15)






2 4 6 8 10 12 14 16 180

50

100

150

200

250

300

Situation

EH

2 4 6 8 10 12 14 16 180

100

200

300

400

500

600

700

Situation

HW

2 4 6 8 10 12 14 16 18200

210

220

230

240

250

260

270

Situation

RNR

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

LB

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

OLS











0

2

4

6

8

10

12

02468

10121416

05

1015202530354045

0

20

40

60

80

100

120

140

0

10

20

30

40

50

60

0

100

200

300

400

500

600

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50






2 4 6 8 10 12 14 16 1855

60

65

70

75

80

85EH

Situation

2 4 6 8 10 12 14 16 1876

77

78

79

80

81

82

83

Situation

RNR

2 4 6 8 10 12 14 16 1860

65

70

75

80

85

90

95

Situation

HW

2 4 6 8 10 12 14 16 18556065707580859095

100

Situation

LB

2 4 6 8 10 12 14 16 186065707580859095

100

Situation

OLS




Ab119870119898 26309 55518 41409 54479 59941119881max 02484 03812 03470 03767 03798MAE 00511 00071 00137 00064 00048


Potato119870119898 12160 10659 20934 25720 10740119881max 00208 01104 00394 02086 01180MAE 00139 00022 00087 00018 00014



01

02

03

04

05

06

07

0



02040608

112141618

0

05

1

15

2

25

3

1

2

3

4

5

6

0

05

1

15

2

25

3

35

1

2

3

4

5

6

7

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100




5 Conclusion



0 100 200 300 400 500 600 700 800 900 10000

1020304050607080

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

Concentration (nM)


Concentration (nM)

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

RNROLSEH

HWLB

RNROLSEH

HWLB

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

102030405060708090

100

Reac

tion

rate

(mm

olm

inm

g)

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100






0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

00501

01502

02503

03504


Reac

tion

rate

(휇m

olm

in)

RNROLSEH

HWLB

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7


Reac

tion

rate

(휇m

olm

in)

times10minus3

RNROLSEH

HWLB

(b)


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

006

RNROLSEH

HWLB


Reac

tion

rate

(휇m

olm

in)

(a)

EH HW LB OLS RNR0

001

002

003

004

005

006

(b)



Competing Interests


Acknowledgments


References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


2 4 6 8 10 12 14 16 180

50

100

150

200

250

300

Situation

EH

2 4 6 8 10 12 14 16 180

100

200

300

400

500

600

700

Situation

HW

2 4 6 8 10 12 14 16 18200

210

220

230

240

250

260

270

Situation

RNR

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

LB

2 4 6 8 10 12 14 16 1850

100

150

200

250

300

350

400

Situation

OLS











0

2

4

6

8

10

12

02468

10121416

05

1015202530354045

0

20

40

60

80

100

120

140

0

10

20

30

40

50

60

0

100

200

300

400

500

600

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50






2 4 6 8 10 12 14 16 1855

60

65

70

75

80

85EH

Situation

2 4 6 8 10 12 14 16 1876

77

78

79

80

81

82

83

Situation

RNR

2 4 6 8 10 12 14 16 1860

65

70

75

80

85

90

95

Situation

HW

2 4 6 8 10 12 14 16 18556065707580859095

100

Situation

LB

2 4 6 8 10 12 14 16 186065707580859095

100

Situation

OLS




Ab119870119898 26309 55518 41409 54479 59941119881max 02484 03812 03470 03767 03798MAE 00511 00071 00137 00064 00048


Potato119870119898 12160 10659 20934 25720 10740119881max 00208 01104 00394 02086 01180MAE 00139 00022 00087 00018 00014



01

02

03

04

05

06

07

0



02040608

112141618

0

05

1

15

2

25

3

1

2

3

4

5

6

0

05

1

15

2

25

3

35

1

2

3

4

5

6

7

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100




5 Conclusion



0 100 200 300 400 500 600 700 800 900 10000

1020304050607080

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

Concentration (nM)


Concentration (nM)

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

RNROLSEH

HWLB

RNROLSEH

HWLB

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

102030405060708090

100

Reac

tion

rate

(mm

olm

inm

g)

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100






0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

00501

01502

02503

03504


Reac

tion

rate

(휇m

olm

in)

RNROLSEH

HWLB

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7


Reac

tion

rate

(휇m

olm

in)

times10minus3

RNROLSEH

HWLB

(b)


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

006

RNROLSEH

HWLB


Reac

tion

rate

(휇m

olm

in)

(a)

EH HW LB OLS RNR0

001

002

003

004

005

006

(b)



Competing Interests


Acknowledgments


References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





0

2

4

6

8

10

12

02468

10121416

05

1015202530354045

0

20

40

60

80

100

120

140

0

10

20

30

40

50

60

0

100

200

300

400

500

600

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50






2 4 6 8 10 12 14 16 1855

60

65

70

75

80

85EH

Situation

2 4 6 8 10 12 14 16 1876

77

78

79

80

81

82

83

Situation

RNR

2 4 6 8 10 12 14 16 1860

65

70

75

80

85

90

95

Situation

HW

2 4 6 8 10 12 14 16 18556065707580859095

100

Situation

LB

2 4 6 8 10 12 14 16 186065707580859095

100

Situation

OLS




Ab119870119898 26309 55518 41409 54479 59941119881max 02484 03812 03470 03767 03798MAE 00511 00071 00137 00064 00048


Potato119870119898 12160 10659 20934 25720 10740119881max 00208 01104 00394 02086 01180MAE 00139 00022 00087 00018 00014



01

02

03

04

05

06

07

0



02040608

112141618

0

05

1

15

2

25

3

1

2

3

4

5

6

0

05

1

15

2

25

3

35

1

2

3

4

5

6

7

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100




5 Conclusion



0 100 200 300 400 500 600 700 800 900 10000

1020304050607080

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

Concentration (nM)


Concentration (nM)

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

RNROLSEH

HWLB

RNROLSEH

HWLB

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

102030405060708090

100

Reac

tion

rate

(mm

olm

inm

g)

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100






0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

00501

01502

02503

03504


Reac

tion

rate

(휇m

olm

in)

RNROLSEH

HWLB

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7


Reac

tion

rate

(휇m

olm

in)

times10minus3

RNROLSEH

HWLB

(b)


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

006

RNROLSEH

HWLB


Reac

tion

rate

(휇m

olm

in)

(a)

EH HW LB OLS RNR0

001

002

003

004

005

006

(b)



Competing Interests


Acknowledgments


References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


2 4 6 8 10 12 14 16 1855

60

65

70

75

80

85EH

Situation

2 4 6 8 10 12 14 16 1876

77

78

79

80

81

82

83

Situation

RNR

2 4 6 8 10 12 14 16 1860

65

70

75

80

85

90

95

Situation

HW

2 4 6 8 10 12 14 16 18556065707580859095

100

Situation

LB

2 4 6 8 10 12 14 16 186065707580859095

100

Situation

OLS




Ab119870119898 26309 55518 41409 54479 59941119881max 02484 03812 03470 03767 03798MAE 00511 00071 00137 00064 00048


Potato119870119898 12160 10659 20934 25720 10740119881max 00208 01104 00394 02086 01180MAE 00139 00022 00087 00018 00014



01

02

03

04

05

06

07

0



02040608

112141618

0

05

1

15

2

25

3

1

2

3

4

5

6

0

05

1

15

2

25

3

35

1

2

3

4

5

6

7

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100




5 Conclusion



0 100 200 300 400 500 600 700 800 900 10000

1020304050607080

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

Concentration (nM)


Concentration (nM)

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

RNROLSEH

HWLB

RNROLSEH

HWLB

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

102030405060708090

100

Reac

tion

rate

(mm

olm

inm

g)

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100






0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

00501

01502

02503

03504


Reac

tion

rate

(휇m

olm

in)

RNROLSEH

HWLB

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7


Reac

tion

rate

(휇m

olm

in)

times10minus3

RNROLSEH

HWLB

(b)


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

006

RNROLSEH

HWLB


Reac

tion

rate

(휇m

olm

in)

(a)

EH HW LB OLS RNR0

001

002

003

004

005

006

(b)



Competing Interests


Acknowledgments


References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



01

02

03

04

05

06

07

0



02040608

112141618

0

05

1

15

2

25

3

1

2

3

4

5

6

0

05

1

15

2

25

3

35

1

2

3

4

5

6

7

N = 10

N = 20

N = 50

N = 10

N = 20

N = 50

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100




5 Conclusion



0 100 200 300 400 500 600 700 800 900 10000

1020304050607080

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

Concentration (nM)


Concentration (nM)

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

RNROLSEH

HWLB

RNROLSEH

HWLB

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

102030405060708090

100

Reac

tion

rate

(mm

olm

inm

g)

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100






0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

00501

01502

02503

03504


Reac

tion

rate

(휇m

olm

in)

RNROLSEH

HWLB

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7


Reac

tion

rate

(휇m

olm

in)

times10minus3

RNROLSEH

HWLB

(b)


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

006

RNROLSEH

HWLB


Reac

tion

rate

(휇m

olm

in)

(a)

EH HW LB OLS RNR0

001

002

003

004

005

006

(b)



Competing Interests


Acknowledgments


References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 100 200 300 400 500 600 700 800 900 10000

1020304050607080

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

Concentration (nM)


Concentration (nM)

Concentration (nM)

Reac

tion

rate

(mm

olm

inm

g)

RNROLSEH

HWLB

RNROLSEH

HWLB

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Reac

tion

rate

(mm

olm

inm

g)

0 100 200 300 400 500 600 700 800 900 10000

102030405060708090

100

Reac

tion

rate

(mm

olm

inm

g)

o = 10 휎 = 10 o = 20 휎 = 10

o = 10 휎 = 50 o = 20 휎 = 50

o = 10 휎 = 100 o = 20 휎 = 100






0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

00501

01502

02503

03504


Reac

tion

rate

(휇m

olm

in)

RNROLSEH

HWLB

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7


Reac

tion

rate

(휇m

olm

in)

times10minus3

RNROLSEH

HWLB

(b)


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

006

RNROLSEH

HWLB


Reac

tion

rate

(휇m

olm

in)

(a)

EH HW LB OLS RNR0

001

002

003

004

005

006

(b)



Competing Interests


Acknowledgments


References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

00501

01502

02503

03504


Reac

tion

rate

(휇m

olm

in)

RNROLSEH

HWLB

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7


Reac

tion

rate

(휇m

olm

in)

times10minus3

RNROLSEH

HWLB

(b)


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

001

002

003

004

005

006

RNROLSEH

HWLB


Reac

tion

rate

(휇m

olm

in)

(a)

EH HW LB OLS RNR0

001

002

003

004

005

006

(b)



Competing Interests


Acknowledgments


References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

robust nonlinear regression in enzyme kinetic parameters

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