A new multi-wavelength model-based method for determination of enzyme kinetic parameters 395

J. Biosci. 35(3), September 2010

1. Introduction1. Introduction

Kinetic studies on enzymes are among the most important Kinetic studies on enzymes are among the most important

tools for understanding biological interactions at the tools for understanding biological interactions at the

molecular level and to obtain information about kinetic molecular level and to obtain information about kinetic

parameters. Details of the kinetics are important because parameters. Details of the kinetics are important because

they provide essential information about how an enzyme they provide essential information about how an enzyme

will behave or respond in a given situation. Simple enzyme will behave or respond in a given situation. Simple enzyme

kinetics is generally described by the Michaelis–Menten kinetics is generally described by the Michaelis–Menten

equation (Eq. 1) (Baily and Ollis 1986; Beedham 2002).equation (Eq. 1) (Baily and Ollis 1986; Beedham 2002).

where where vv is the enzyme reaction rate, is the enzyme reaction rate, SS is substrate is substrate

concentration, concentration, VVmaxmax

is the maximum reaction rate, and is the maximum reaction rate, and KKmm

is the Michaelis–Menten constant. By rearranging Eq. (1), is the Michaelis–Menten constant. By rearranging Eq. (1),

different linear forms of the Michaelis–Menten equation different linear forms of the Michaelis–Menten equation

including the Lineweaver–Burk equation (Eq. 2) can be including the Lineweaver–Burk equation (Eq. 2) can be

derived: derived:

Plotting of Eq. (2) as Plotting of Eq. (2) as 1/v1/v versus versus 1/1/[[SS00] (known as a ] (known as a

Lineweaver–Burk plot) gives an appropriate linear plot that Lineweaver–Burk plot) gives an appropriate linear plot that

is commonly used for determination of numerical values of is commonly used for determination of numerical values of

VVmaxmax

and and KKmm..

The spectroscopic determination of enzyme activity The spectroscopic determination of enzyme activity

through a Lineweaver–Burk plot is usually carried out by through a Lineweaver–Burk plot is usually carried out by

monitoring the consumption of substrate or the production monitoring the consumption of substrate or the production

of metabolite at a selected wavelength. The method of metabolite at a selected wavelength. The method

http://www.ias.ac.in/jbiosci J. Biosci. 35(3), September 2010, 395–403, © Indian Academy of Sciences 395

A new multi-wavelength model-based method for determination of

enzyme kinetic parameters

MOHAMMAD-HOSSEIN SOROURADDIN1, KAVEH AMINI

1,#, ABDOLHOSSEIN NASERI1, JAVAD VALLIPOUR

1,

JALAL HANAEE2 and MOHAMMAD-REZA RASHIDI

2,*

1Department of Analytical Chemistry, Faculty of Chemistry, University of Tabriz, Tabriz, Iran

2Department of Medicinal Chemistry, School of Pharmacy, Tabriz University of Medical Sciences,

51664-14766 Tabriz, Iran

#Present address: Department of Chemistry, York University, Toronto, ON, M3J 1P3, Canada

*Corresponding author (Fax, +98 411 3344798; Email, rashidi@tbzmed.ac.ir)

Lineweaver–Burk plot analysis is the most widely used method to determine enzyme kinetic parameters. In the

spectrophotometric determination of enzyme activity using the Lineweaver–Burk plot, it is necessary to fi nd a

wavelength at which only the substrate or the product has absorbance without any spectroscopic interference of the

other reaction components. Moreover, in this method, different initial concentrations of the substrate should be used

to obtain the initial velocities required for Lineweaver–Burk plot analysis. In the present work, a multi-wavelength

model-based method has been developed and validated to determine Michaelis–Menten constants for some enzyme

reactions. In this method, a selective wavelength region and several experiments with different initial concentrations

of the substrate are not required. The absorbance data of the kinetic assays are fi tted by non-linear regression coupled

to the numeric integration of the related differential equation. To indicate the applicability of the proposed method, the

Michaelis–Menten constants for the oxidation of phenanthridine, 6-deoxypenciclovir and xanthine by molybdenum

hydroxylases were determined using only a single initial concentration of the substrate, regardless of any spectral

overlap.

[Sorouraddin M-D, Amini K, Naseri A,Vallipour J, Hanaee J and Rashidi M-R 2010 A new multi-wavelength model-based method for determination

of enzyme kinetic parameters; J. Biosci. 35 395–403] DOI 10.1007/s12038-010-0045-z

Keywords. Enzyme kinetics; Lineweaver–Burk plot analysis; Michaelis–Menten constants; multi-wavelength model-based method;

spectrophotometry

vV S

K Sm

=+

max [ ]

[ ]0

0

, (1)

1 1 1

0v VKV Sm= +

max max [ ], (2)

Mohammad-Hossein Sorouraddin et al.396

J. Biosci. 35(3), September 2010

is based on the measurement of the initial velocity of is based on the measurement of the initial velocity of

different concentrations of the substrate at a wavelength different concentrations of the substrate at a wavelength

at which only the subat which only the substrate or the product has absorbance strate or the product has absorbance

without any spectroscopic interference of the other reaction without any spectroscopic interference of the other reaction

components.components.

Spectroscopic methods are, in general, sensitive and Spectroscopic methods are, in general, sensitive and

well suited for studying enzyme systems in which the well suited for studying enzyme systems in which the

reaction components are accessible in pure form and there reaction components are accessible in pure form and there

is no spectral overlap in responses. However, many systems, is no spectral overlap in responses. However, many systems,

particularly those with similar components, do not meet particularly those with similar components, do not meet

these conditions, and there have been some diffi culties in these conditions, and there have been some diffi culties in

studying such systems. These quantifi cation diffi culties studying such systems. These quantifi cation diffi culties

arise from spectral overlap, and chemometric methods arise from spectral overlap, and chemometric methods

have been found to be mathematically suitable for the have been found to be mathematically suitable for the

resolution of overlapping spectral bands in such quantitative resolution of overlapping spectral bands in such quantitative

determinations (Sorouraddin determinations (Sorouraddin et al.et al. 2008). In such cases, 2008). In such cases,

more information can be extracted if multivariate (multi-more information can be extracted if multivariate (multi-

wavelength) spectrophotometric data are analysed by means wavelength) spectrophotometric data are analysed by means

of an appropriate chemometric method. Model-based (hard of an appropriate chemometric method. Model-based (hard

modelling) methods include traditional least-squares curve-modelling) methods include traditional least-squares curve-

fi tting approaches. These methods are based on a previous fi tting approaches. These methods are based on a previous

postulation of a chemical model in which a set of species is postulation of a chemical model in which a set of species is

defi ned according to their kinetic constants. These values are defi ned according to their kinetic constants. These values are

then refi ned using least-squares minimization.then refi ned using least-squares minimization.

Multivariate kinetic fi tting begins by postulating a Multivariate kinetic fi tting begins by postulating a

chemical model. The model is encoded into strings of text chemical model. The model is encoded into strings of text

representing the proposed chemical model and imputed into representing the proposed chemical model and imputed into

a computer program. An intelligent model parser extracts a computer program. An intelligent model parser extracts

the number of species, their names, their corresponding the number of species, their names, their corresponding

stoichiometric coeffi cients, and the number of reactions. stoichiometric coeffi cients, and the number of reactions.

The program produces a list of parameters and their The program produces a list of parameters and their

reaction coeffi cients followed by construction of a system reaction coeffi cients followed by construction of a system

of simultaneous ordinary differential equations (ODEs) of simultaneous ordinary differential equations (ODEs)

which describe the change in concentration of each species which describe the change in concentration of each species

over time (Dyson over time (Dyson et al.et al. 2003). This approach is completely 2003). This approach is completely

general so that a system of ODEs of arbigeneral so that a system of ODEs of arbitrary complexity trary complexity

can be automatically generated for can be automatically generated for any number of coupled any number of coupled

reactions. When the initial concentrations of the species in reactions. When the initial concentrations of the species in

the chemical model are known, the differential equations the chemical model are known, the differential equations

can be integrated to yield the concentration of each species can be integrated to yield the concentration of each species

at any desired time.at any desired time.

This model allows determination of the kinetic This model allows determination of the kinetic

parameters of the reaction, concentration profi les, and parameters of the reaction, concentration profi les, and

interpolation and extrapolation outside of the conditions interpolation and extrapolation outside of the conditions

used for its determination (Maeder and Zuberbühler 1990; used for its determination (Maeder and Zuberbühler 1990;

Maeder and Neuhold 2006; Maeder and Neuhold 2007; Maeder and Neuhold 2006; Maeder and Neuhold 2007;

Puxty Puxty et al.et al. 2008). It is also able t 2008). It is also able to o determine the pure determine the pure

component spectra of all non-steady-state absorbing species component spectra of all non-steady-state absorbing species

in the reaction, provided that no linear dependence exists in the reaction, provided that no linear dependence exists

between the spectral data and the concentration profi les of between the spectral data and the concentration profi les of

those species.those species.

Modelling of enzymatically driven pModelling of enzymatically driven processes is needed to rocesses is needed to

control certain parameters of interest, such as the evolution control certain parameters of interest, such as the evolution

of substrates and productof substrates and product concentrations, the rate constants concentrations, the rate constants

and the spectra of the involved species.and the spectra of the involved species.

Hard modelling has been discussed widely in the Hard modelling has been discussed widely in the

literature and its application in spectroscopic data and literature and its application in spectroscopic data and

chemical kinetic reaction analysis has been addressed by chemical kinetic reaction analysis has been addressed by

several authors (Maeder and Zuberbühler 1990; Molloy several authors (Maeder and Zuberbühler 1990; Molloy

et al.et al. 1999; Maeder and Neuhold 2006, 2007; Puxty 1999; Maeder and Neuhold 2006, 2007; Puxty et al.et al.

2008; Vosough 2008; Vosough et al.et al. 2008). In addition, a mixed hard- and 2008). In addition, a mixed hard- and

soft-modelling approach has been previously proposed by soft-modelling approach has been previously proposed by

Amigo Amigo et al.et al. (2006a, b) to study and monitor enzymatic (2006a, b) to study and monitor enzymatic

systems in biological fl uids. systems in biological fl uids.

In the present study, a new multi-wavelength model-In the present study, a new multi-wavelength model-

based method was developed and validated to study the based method was developed and validated to study the

kinetics of different enzyme reactions through analysing kinetics of different enzyme reactions through analysing

the reaction progress of only a single concentration of the the reaction progress of only a single concentration of the

substrate. The advantages of the proposed model over earlier substrate. The advantages of the proposed model over earlier

known methods are simplicity and rapid calculation of the known methods are simplicity and rapid calculation of the

kinetic parameters. For example, only a single concentration kinetic parameters. For example, only a single concentration

of the substrate is used to calculate the kinetic parameters, of the substrate is used to calculate the kinetic parameters,

instead of the several initial concentrations required in instead of the several initial concentrations required in

the Lineweaver–Burk method. Furthermore, by using the the Lineweaver–Burk method. Furthermore, by using the

proposed multi-wavelength method, the necessity of a proposed multi-wavelength method, the necessity of a

selective wavelength for the substrate or metabolite will selective wavelength for the substrate or metabolite will

not arise at all. To evaluate the applicability of the proposed not arise at all. To evaluate the applicability of the proposed

method, the conversion of xanthine to uric acid by xanthine method, the conversion of xanthine to uric acid by xanthine

oxidase (EC 1.17.3.2: xanthine oxygen oxidoreductase), and oxidase (EC 1.17.3.2: xanthine oxygen oxidoreductase), and

the oxidation of phenanthridine to phenanthridinone, and the oxidation of phenanthridine to phenanthridinone, and

6-deoxypenciclovir to penciclovir catalysed by aldehyde 6-deoxypenciclovir to penciclovir catalysed by aldehyde

oxidase (EC 1.2.3.1: aldehyde:oxidase (EC 1.2.3.1: aldehyde: oxygen oxidoreductase) oxygen oxidoreductase)

(fi gure 1) were investigated and the results were compared (fi gure 1) were investigated and the results were compared

with those obtained from the traditional method of with those obtained from the traditional method of

Lineweaver–Burk plot analysis. Lineweaver–Burk plot analysis.

2. Materials and methods2. Materials and methods

2.12.1 Chemicals Chemicals

All experiments were performed with analytical reagent-All experiments were performed with analytical reagent-

grade chemicals. Phenanthridine, phenanthridinone, grade chemicals. Phenanthridine, phenanthridinone,

xanthine and uric acid were purchased from Sigma-Aldrich xanthine and uric acid were purchased from Sigma-Aldrich

(Dorset, UK). Penciclovir (purity 99%) was purchased (Dorset, UK). Penciclovir (purity 99%) was purchased

from Livzon Pharmaceutical Group Co., Ltd. (Changzhou, from Livzon Pharmaceutical Group Co., Ltd. (Changzhou,

China). 6-Deoxypenciclovir was synthesized asChina). 6-Deoxypenciclovir was synthesized as

described previously (Harnden described previously (Harnden et al.et al. 1987). All other 1987). All other

chemicals were purchased from Merck (Darmstadt, chemicals were purchased from Merck (Darmstadt,

Germany).Germany).

2.22.2 Apparatus and software Apparatus and software

Spectrophotometric measurements were carried out at 37°C Spectrophotometric measurements were carried out at 37°C

using a Shimadzu 2550 UV/VIS spectrophotometer that was using a Shimadzu 2550 UV/VIS spectrophotometer that was

controlled by the Shimadzu UV Probe personal software controlled by the Shimadzu UV Probe personal software

A new multi-wavelength model-based method for determination of enzyme kinetic parameters 397

J. Biosci. 35(3), September 2010

package including kinetics software. The instrument was package including kinetics software. The instrument was

connected to a Shimadzu cell temperature control unit. connected to a Shimadzu cell temperature control unit.

A rectangular cuvette with a 1 cm path length was used A rectangular cuvette with a 1 cm path length was used

and the total volume used for the spectrophotometric and the total volume used for the spectrophotometric

measurements was constant (3.0 ml). All spectra were saved measurements was constant (3.0 ml). All spectra were saved

and performed in the ASCII format. The spectrophotometric and performed in the ASCII format. The spectrophotometric

data were transferred to a PC for subsequent manipulation data were transferred to a PC for subsequent manipulation

by a homemade program. The data were handled using the by a homemade program. The data were handled using the

MATLAB software (7.0).MATLAB software (7.0).

2.32.3 Preparation of partially purifi ed molybdenum Preparation of partially purifi ed molybdenum

hydroxylaseshydroxylases

Mature male Sprague–Dawley rats (150–250 g, Tabriz Mature male Sprague–Dawley rats (150–250 g, Tabriz

University of Medical Sciences, Iran) were used in this University of Medical Sciences, Iran) were used in this

study. They were fed with a standard laboratory diet and study. They were fed with a standard laboratory diet and

allowed food and water allowed food and water ad libitumad libitum. The temperature and . The temperature and

humidity were kept at 18±1ºC and 50%, respectively, and humidity were kept at 18±1ºC and 50%, respectively, and

the lighting cycle was 7:00–19:00 h light and 19:00–7:00 h the lighting cycle was 7:00–19:00 h light and 19:00–7:00 h

dark. Animals were handled with humane care in accordance dark. Animals were handled with humane care in accordance

with the guidelines of the National Institutes of Health. with the guidelines of the National Institutes of Health.

The animals were killed between 9:00 and 10:00 am The animals were killed between 9:00 and 10:00 am

by cervical dislocation. Livers were immediately excised, by cervical dislocation. Livers were immediately excised,

placed in ice-cold isotonic potassium chloride solution placed in ice-cold isotonic potassium chloride solution

(1.15% KCl w/v) containing 0.1 mM EDTA. The livers (1.15% KCl w/v) containing 0.1 mM EDTA. The livers

were then homogenized on ice in a homogenizer fi tted with were then homogenized on ice in a homogenizer fi tted with

a tefl on pestle. Partially purifi ed enzyme was prepared from a tefl on pestle. Partially purifi ed enzyme was prepared from

the liver homogenate by heat treatment and ammonium the liver homogenate by heat treatment and ammonium

sulphate precipitation as described elsewhere (Johnson sulphate precipitation as described elsewhere (Johnson et et

al.al. 1985). 1985).

2.42.4 Procedure Procedure

Enzyme processes are highly infl uenced by factors such Enzyme processes are highly infl uenced by factors such

as pH, the selected buffer and its concentration, enzyme as pH, the selected buffer and its concentration, enzyme

concentration and working temperature. Working conditions concentration and working temperature. Working conditions

that guarantee reproducibility of the measurements have to that guarantee reproducibility of the measurements have to

be established. The pH of the reaction solution was fi xed at be established. The pH of the reaction solution was fi xed at

7.4 using Sorenson buffer and the temperature of the system 7.4 using Sorenson buffer and the temperature of the system

was kept at 37ºC.was kept at 37ºC.

2.4.12.4.1 Determination of the kinetic parameters using the Determination of the kinetic parameters using the

Lineweaver–Burk method: Lineweaver–Burk method: As an initial step in the study of As an initial step in the study of

the overall reactions, the kinetic constants for the individual the overall reactions, the kinetic constants for the individual

reactions of oxidation of phenanthridine, 6-deoxypenciclovir reactions of oxidation of phenanthridine, 6-deoxypenciclovir

and xanthine were determined by means of steady-state and xanthine were determined by means of steady-state

velocity studies of reactions with different substrate velocity studies of reactions with different substrate

concentrations. The kinetic parameters for the metabolism concentrations. The kinetic parameters for the metabolism

of each compound were determined by monitoring the of each compound were determined by monitoring the

conversion of each substrate to the corresponding metabolite conversion of each substrate to the corresponding metabolite

at 37ºC and at the appropriate wavelength. at 37ºC and at the appropriate wavelength.

For phenanthridine, as a specifi c substrate of aldehyde For phenanthridine, as a specifi c substrate of aldehyde

oxidase (Rashidi oxidase (Rashidi et al.et al. 1997), the 1997), the KKmm and and VV

maxmax were assayed were assayed

by monitoring the production of phenanthridinone atby monitoring the production of phenanthridinone at

322 nm by aldehyde oxidase. The experiment was carried out 322 nm by aldehyde oxidase. The experiment was carried out

with initial concentrations of 8–30 with initial concentrations of 8–30 μμM of phenanthridine. M of phenanthridine.

In the case of 6-deoxypenciclovir, which is metabolized by In the case of 6-deoxypenciclovir, which is metabolized by

aldehyde oxidase (Rashidi aldehyde oxidase (Rashidi et al.et al. 1997), the initial velocity 1997), the initial velocity

of penciclovir formation from different concentrations of of penciclovir formation from different concentrations of

the substrate (100–300 the substrate (100–300 μμM) was determined at 310 nm. The M) was determined at 310 nm. The

kinetic constants for the conversion of xanthine, a specifi c kinetic constants for the conversion of xanthine, a specifi c

substrate of xanthine oxidase (Rashidi and Nazemiyeh substrate of xanthine oxidase (Rashidi and Nazemiyeh

2010), to uric acid was obtained through monitoring product 2010), to uric acid was obtained through monitoring product

formation at 295 nm following oxidation of xanthine at formation at 295 nm following oxidation of xanthine at

concentrations of 10–40 concentrations of 10–40 μμM. The concentration of the M. The concentration of the

enzyme solution used in the enzyme kinetic assays was 0.2 enzyme solution used in the enzyme kinetic assays was 0.2

mg/ml. All reactions were carried out in Sorenson phosphate mg/ml. All reactions were carried out in Sorenson phosphate

buffer (67 mM, pH 7.4, containing 0.1 mM EDTA) for 5 buffer (67 mM, pH 7.4, containing 0.1 mM EDTA) for 5

min by adding 100 min by adding 100 μμl enzyme fraction to 2.9 ml of substrate l enzyme fraction to 2.9 ml of substrate

solution and the initial oxidation rate of each reaction was solution and the initial oxidation rate of each reaction was

calculated from the linear part of the course of the reaction. calculated from the linear part of the course of the reaction.

Kinetic parameters including Kinetic parameters including KKmm and and VV

maxmax were calculated were calculated

from a double reciprocal (Lineweaver–Burk) plot.from a double reciprocal (Lineweaver–Burk) plot.

Figure 1. Oxidative conversion of phenanthridine to

phenanthridinone, 6-deoxypenciclovir to penciclovir, and xanthine

to uric acid

NNH

O

Phenanthridine Phenanthridinone

AO

HN

NN

N

O

O

HN

NH

NH

HN

O

O

O

XO

Xanthine Uric acid

N

NN

N

H2N

OHOH

N

NH

N

N

O

OHOH

H2N

AO

6-Deoxypenciclovir Penciclovir

2.4.22.4.2 Determination of the kinetic parameters using Determination of the kinetic parameters using

the multi-wavelength model-based method: the multi-wavelength model-based method: For the For the

multi-wavelength model-based method, only one concen-multi-wavelength model-based method, only one concen-

tration (10 tration (10 μμM as the fi nal concentration) of each comp-M as the fi nal concentration) of each comp-

ound (phenanthridine, 6-deoxypenciclovir and xanthine) ound (phenanthridine, 6-deoxypenciclovir and xanthine)

was used and the reaction was started by adding thewas used and the reaction was started by adding the

enzyme fraction. Each experiment was repeated three enzyme fraction. Each experiment was repeated three

times.times.

The oxidation of phenanthridine was monitored by The oxidation of phenanthridine was monitored by

repeated scanning between 210 and 260 nm and the spectra repeated scanning between 210 and 260 nm and the spectra

were recorded every 15 s for 20 min. were recorded every 15 s for 20 min.

In the case of 6-deoxypenciclovir, the wavelength range In the case of 6-deoxypenciclovir, the wavelength range

was set at 250–350 nm and the spectra of the progress of the was set at 250–350 nm and the spectra of the progress of the

reaction were recorded every 30 s for 40 min. reaction were recorded every 30 s for 40 min.

The spectra of progress of the oxidation reaction of The spectra of progress of the oxidation reaction of

xanthine to uric acid were recorded at 240–325 nm everyxanthine to uric acid were recorded at 240–325 nm every

30 s for 20 min.30 s for 20 min.

The spectral data obtained were converted into ASCII The spectral data obtained were converted into ASCII

fi les for further data analysis with the hard-modelling fi les for further data analysis with the hard-modelling

algorithm.algorithm.

2.4.32.4.3 Protein determination: Protein determination: Protein concentration was Protein concentration was

determined spectrophotometrically by a bicinchoninic acid determined spectrophotometrically by a bicinchoninic acid

kit using bovine serum albumin as the standard (Smith kit using bovine serum albumin as the standard (Smith et et

al.al. 1985). 1985).

3. Theory and algorithm3. Theory and algorithm

A multivariate (multi-wavelength) model-based method A multivariate (multi-wavelength) model-based method

is a rather complex procedure; however, when there is a is a rather complex procedure; however, when there is a

need to produce a large number of data matrices such as need to produce a large number of data matrices such as

spectral data at many wavelengths, it is considered as a very spectral data at many wavelengths, it is considered as a very

important and valuable method. For a spectroscopically important and valuable method. For a spectroscopically

monitored enzyme process, the matrices in the resolution monitored enzyme process, the matrices in the resolution

model can be expressed as: model can be expressed as:

DD = = CACA,, (3)(3)

where where DD is the collected spectra as a function of time, is the collected spectra as a function of time, CC (the (the

columns) is the kinetic concentration profi les of the different columns) is the kinetic concentration profi les of the different

species in the process, and species in the process, and AA (the rows) is the related pure (the rows) is the related pure

spectra. spectra.

For a given matrix For a given matrix DD, the best rate constants defi ning the , the best rate constants defi ning the

matrix matrix CC, as well as the best molar absorptivities collected , as well as the best molar absorptivities collected

in the matrix in the matrix AA could be determined easily using the model- could be determined easily using the model-

based data-fi tting method. The quality of the fi t is represented based data-fi tting method. The quality of the fi t is represented

by the matrix of residuals, by the matrix of residuals, RR; ;

RR = = DD––CA.CA. (4)(4)

The mechanism for fi rst-order metabolism reactions can be The mechanism for fi rst-order metabolism reactions can be

proposed according to this reaction scheme:proposed according to this reaction scheme:

SS + + EE ↔ [ ↔ [ESES] → ] → PP + + EE,, (5)(5)

where where SS, , EE, , SESE and and PP are substrate, enzyme, intermediate are substrate, enzyme, intermediate

and product, respectively. and product, respectively.

The rate at which the reaction (5) occurs can be explained The rate at which the reaction (5) occurs can be explained

using the Michaelis–Menten equation (1). Since the enzyme using the Michaelis–Menten equation (1). Since the enzyme

reaction rate (ν) is defi ned as the rate of consumption of the reaction rate (ν) is defi ned as the rate of consumption of the

substrate or production of metabolite, by assuming steady-substrate or production of metabolite, by assuming steady-

state conditions, the differential states can be written as state conditions, the differential states can be written as

equations (6) and (7) to explain the variations of substrate equations (6) and (7) to explain the variations of substrate

and metabolite concentrations with time in each enzyme and metabolite concentrations with time in each enzyme

reaction: reaction:

where [where [SS] and [] and [PP] are the substrate and product ] are the substrate and product

concentrations, concentrations, KKmm

is the Michaelis–Menten constant, and is the Michaelis–Menten constant, and

VVmax max

is the maximum velocity. These equations also indicate is the maximum velocity. These equations also indicate

the changes in substrate and product concentrations as a the changes in substrate and product concentrations as a

function of time. For a given value of function of time. For a given value of KKmm, , VV

maxmax and [ and [SS], it ], it

is possible to calculate is possible to calculate CC, which contains the concentration , which contains the concentration

profi les of the absorbing species (profi les of the absorbing species (SS and and PP) by solving of Eqs. ) by solving of Eqs.

6 and 7. For any particular matrix 6 and 7. For any particular matrix CC, the best set of molar , the best set of molar

absorptivities, absorptivities, AA, can be calculated using Eq. 8:, can be calculated using Eq. 8:

AA = = CC++DD,, (8)(8)

where where CC++ is the so-called pseudoinverse of is the so-called pseudoinverse of CC. Eq. 4 can be . Eq. 4 can be

rearranged as:rearranged as:

RR = = DD – – CCCC++D.D. (9)(9)

The molar absorptThe molar absorptivities are linear parameters, which can ivities are linear parameters, which can

be eliminated during the fi tting of the non-linear parameters be eliminated during the fi tting of the non-linear parameters

((KKmm and and VV

maxmax). Therefore, ). Therefore, RR is defi ned as a function of non- is defi ned as a function of non-

linear parameters only. linear parameters only.

Since 1973, several different methods have been Since 1973, several different methods have been

developed and are available for the fi tting of any number of developed and are available for the fi tting of any number of

parameters to a given measurement (Jezewska 1973; Lopez parameters to a given measurement (Jezewska 1973; Lopez

et al.et al. 2002). According to Maeder and Zuberbühler (1990), 2002). According to Maeder and Zuberbühler (1990),

these methods can be classifi ed into two groups: (a) direct these methods can be classifi ed into two groups: (a) direct

methods, where the sum of squares, ssq, of methods, where the sum of squares, ssq, of RR is minimized, is minimized,

and (b) the Newton-Gauss–Levenberg/Marquardt (NGL/and (b) the Newton-Gauss–Levenberg/Marquardt (NGL/

M) method, where the residuals, M) method, where the residuals, RR, are used to guide , are used to guide

the iterative process toward the minimum. In contrast to the iterative process toward the minimum. In contrast to

direct methods, the NGL/M type of algorithm requires the direct methods, the NGL/M type of algorithm requires the

complete vector or matrix of residuals to drive the iterative complete vector or matrix of residuals to drive the iterative

refi nement toward the minimum. The process starts from an refi nement toward the minimum. The process starts from an

initial assumption of the constants, initial assumption of the constants, KKmm and and VV

maxmax. Then, the . Then, the

parameter vector is continuously improved by the addition parameter vector is continuously improved by the addition

of the appropriate (‘best’) parameter shift vector Δof the appropriate (‘best’) parameter shift vector ΔKKmm and and

ΔΔVVmaxmax

. The shift vector is calculated using a sophisticated . The shift vector is calculated using a sophisticated

technique that is based on the derivatives of the residuals technique that is based on the derivatives of the residuals

with respect to the parameters.with respect to the parameters.

Therefore, the generTherefore, the general steps of the algorithm can be al steps of the algorithm can be

summarized as follows:summarized as follows:

Mohammad-Hossein Sorouraddin et al.398

J. Biosci. 35(3), September 2010

−[ ]=

+d Sdt

V SK Sm

max[ ][ ]0

0

, (6)

d Pdt

V S

K Sm

[ ]=

+max [ ]

[ ]0

0

, (7)

J. Biosci. 35(3), September 2010

A new multi-wavelength model-based method for determination of enzyme kinetic parameters 399

Figure 2. UV-vis monitoring of oxidation of 10 μM of phenanthridine to phenanthridinone by rat liver aldehyde oxidase; (a) repeated

scanning of the reaction progress of phenanthridine oxidation to phenanthridinone for 20 min; (b) the pure UV-vis spectra of phenanthridine

(—) and phenanthridinone (….) in comparison with the resolved spectra of phenanthridine (····) and phenanthridinone (○○○○); (c) the

resolved concentration profi les of phenanthridine (––) and phenanthridinone (….).

Mohammad-Hossein Sorouraddin et al.400

J. Biosci. 35(3), September 2010

1. Initial estimates of 1. Initial estimates of VVmaxmax

and and KKmm

as non-linear as non-linear

parameters, parameters,

2. Solving of differential equations (Eqs. 4 and 5) 2. Solving of differential equations (Eqs. 4 and 5)

using the ODE23 command of MATLAB to obtain using the ODE23 command of MATLAB to obtain

concentration matrix concentration matrix CC,,

3. Calculation of the residual matrix using Eq. 9, and3. Calculation of the residual matrix using Eq. 9, and

4. Optimization of 4. Optimization of KKmm and and VV

maxmax using the NGL/M using the NGL/M

algorithm.algorithm.

4. Results and discussion4. Results and discussion

Determination of enzyme kinetic parameters by linear Determination of enzyme kinetic parameters by linear

forms of the Michaelis–Menten equation such as the forms of the Michaelis–Menten equation such as the

Lineweaver–Burk plot is a time-consuming process and Lineweaver–Burk plot is a time-consuming process and

requires measurement of the initial metabolism of different requires measurement of the initial metabolism of different

concentrations of a substrate at an appropriate wavelength. concentrations of a substrate at an appropriate wavelength.

Apart from the complexity of the procedure and problems Apart from the complexity of the procedure and problems

encountered in fi nding an appropriate wavelength for encountered in fi nding an appropriate wavelength for

spectroscopic assay of reaction progress, producing spectroscopic assay of reaction progress, producing

reproducible results is sometimes problematic. In the reproducible results is sometimes problematic. In the

present study, a new simple multi-wavelength model-based present study, a new simple multi-wavelength model-based

method was developed to obtain enzyme kinetic parameters. method was developed to obtain enzyme kinetic parameters.

The productivity and applicability of the proposed method The productivity and applicability of the proposed method

was evaluated by spectrophotometric study of three was evaluated by spectrophotometric study of three

different enzyme reactions. This method requires only different enzyme reactions. This method requires only

spectrophotometrical monitoring of the reaction progress spectrophotometrical monitoring of the reaction progress

of substrate transformation at one concentration by repeated of substrate transformation at one concentration by repeated

scanning in spite of any overlap between the spectra of the scanning in spite of any overlap between the spectra of the

reaction components. To evaluate the validity of the method, reaction components. To evaluate the validity of the method,

the proposed method was applied to determine the kinetic the proposed method was applied to determine the kinetic

parameters of three enzyme reactions and the results were parameters of three enzyme reactions and the results were

compared with those obtained from the Lineweaver–Burk compared with those obtained from the Lineweaver–Burk

plot. In fi gure 2, the spectrophotometric data for the plot. In fi gure 2, the spectrophotometric data for the

oxidation of phenanthridine by rat liver aldehyde oxidase oxidation of phenanthridine by rat liver aldehyde oxidase

has been illustrated. Both the Lineweaver–Burk plot and the has been illustrated. Both the Lineweaver–Burk plot and the

proposed multi-wavelength model-based method produced proposed multi-wavelength model-based method produced

very similar results (table 1) indicating the validity of the very similar results (table 1) indicating the validity of the

proposed method. The results obtained are comparable proposed method. The results obtained are comparable

to those reported by Sorouraddin to those reported by Sorouraddin et al.et al. (2009). They (2009). They

have reported 2.20 have reported 2.20 μμM and 0.037 M and 0.037 μμmol/min/mg values mol/min/mg values

(obtained from spectrofl ourometric-multivariate calibration (obtained from spectrofl ourometric-multivariate calibration

method) and 5.79 method) and 5.79 μμM and 0.047 M and 0.047 μμmol/min/mg values mol/min/mg values

(calculated by the spectrophotometric-univariate calibration (calculated by the spectrophotometric-univariate calibration

method) for method) for KKmm and and VV

maxmax, re, respectively. Meanwhile, the spectively. Meanwhile, the

simplicity of the new method and its ability to overcome simplicity of the new method and its ability to overcome

spectroscopic interference of the reaction components are spectroscopic interference of the reaction components are

two major advantages of this method over the well-known two major advantages of this method over the well-known

Lineweaver–Burk plot analysis. In addition, it has been Lineweaver–Burk plot analysis. In addition, it has been

indicated that the determination of kinetic parameters for indicated that the determination of kinetic parameters for

the oxidation of phenanthridine to phenanthridinone by the oxidation of phenanthridine to phenanthridinone by

guinea pig or rabbit liver aldehyde oxidase using univariate guinea pig or rabbit liver aldehyde oxidase using univariate

spectroscopic methods is not successful due to a lower limit spectroscopic methods is not successful due to a lower limit

of detection; however, the problem can be overcome using of detection; however, the problem can be overcome using

multivariate calibration methods (Sorouraddin multivariate calibration methods (Sorouraddin et al.et al. 2008). 2008).

Therefore, the applicability of the method developed in this Therefore, the applicability of the method developed in this

study to determine the kinetic parameters for the oxidation study to determine the kinetic parameters for the oxidation

of phenanthridine by guinea pig or rabbit liver aldehyde of phenanthridine by guinea pig or rabbit liver aldehyde

oxidase would be another advantage of the proposed oxidase would be another advantage of the proposed

method. method.

To further confi rm the applicability of the proposed To further confi rm the applicability of the proposed

method for other enzyme reactions, the UV spectral data for method for other enzyme reactions, the UV spectral data for

the oxidation of 6-deoxypenciclovir and xanthine were also the oxidation of 6-deoxypenciclovir and xanthine were also

analysed (fi gures 3 and 4). 6-Deoxypenciclovir is a synthetic analysed (fi gures 3 and 4). 6-Deoxypenciclovir is a synthetic

guanine derivative which is metabolized to the antiviral guanine derivative which is metabolized to the antiviral

compound penciclovir by aldehyde oxidase (Rashidi compound penciclovir by aldehyde oxidase (Rashidi et et

al.al. 1997). The results obtained from the analysis of the 1997). The results obtained from the analysis of the

collected data using the proposed method together with the collected data using the proposed method together with the

results obtained from the Linewever–Burk plot are given in results obtained from the Linewever–Burk plot are given in

table 1. Rashidi table 1. Rashidi et al.et al. (1997) have reported (1997) have reported KKmm=370 =370 μμM and M and

VVmaxmax

=0.026 =0.026 μμmol/min/mg values for 6-deoxypencyclovir mol/min/mg values for 6-deoxypencyclovir

oxidation by rat liver aldehyde oxidase. Similar results oxidation by rat liver aldehyde oxidase. Similar results

were obtained with the oxidation of xanthiwere obtained with the oxidation of xanthine to uric acid by ne to uric acid by

xanthine oxidase (table 1). The kinetic constants we obtained xanthine oxidase (table 1). The kinetic constants we obtained

are consistent with those reported by others (Escribano are consistent with those reported by others (Escribano et al.et al.

1988).1988).

The proposed multi-wavelength model-based method The proposed multi-wavelength model-based method

was successfully applied to calculate kinetic parameters was successfully applied to calculate kinetic parameters

for fi rst-order enzyme reactions using only a singlefor fi rst-order enzyme reactions using only a single

initial concentration of the substrate, regardless ofinitial concentration of the substrate, regardless of

any spectral overlap. This technique was used to measureany spectral overlap. This technique was used to measure

the the KKmm and and VV

max max values for three enzyme reactions. Invalues for three enzyme reactions. In

addition, kinetic parameters for each reaction were calculated addition, kinetic parameters for each reaction were calculated

using the traditional Lineweaver–Burk method andusing the traditional Lineweaver–Burk method and

the results obtained by the two methods were foundthe results obtained by the two methods were found

Table 1. Comparison of the kinetic parameters obtained from traditional Lineweaver–Burk plot analysis and the proposed hard-

modelling methods for the oxidation of xanthine catalysed by rat liver xanthine oxidase, and oxidation of phenanthridine and 6-

deoxypenciclovir catalysed by rat liver aldehyde oxidase*

Phenanthridine 6-Deoxypenciclovir Xanthine

Km (μM) V

max (μmol/

min/mg)

Km (μM) V

max (μmol/

min/mg)

Km (μM) V

max (μmol/

min/mg)

Lineweaver–Burk plot analysis 2.27 (3.31) 0. 45 (3.16) 316.29 (2.91) 0.90 (3.19) 3.57 (2.43) 0.47 (2.81)

Multi-wavelength model-based method 1.82 (2.34) 0.44 (1.99) 333.33 (3.12) 0.96 (3.71) 3.67 (1.76) 0.53 (2.11)

* The numbers given in parentheses represent relative standard deviations (%) (N=3).

A new multi-wavelength model-based method for determination of enzyme kinetic parameters 401

J. Biosci. 35(3), September 2010

Figure 3. UV-vis monitoring of oxidation of 10 μM of 6-deoxypenciclovir to penciclovir by rat liver aldehyde oxidase; (a) repeated

scanning of the reaction progress of 6-deoxypenciclovir oxidation to penciclovir for 40 min; (b) the pure UV-vis spectra of 6-

deoxypenciclovir (––) and penciclovir (….) in comparison with the resolved spectra of 6-deoxypenciclovir (····) and penciclovir (○○○○);

(c) the resolved concentration profi les of 6-deoxypnciclovir (––) and penciclovir (….).

Mohammad-Hossein Sorouraddin et al.402

J. Biosci. 35(3), September 2010

Figure 4. UV-vis monitoring of oxidation of 10 μM of xanthine to uric acid by rat liver xanthine oxidase; (a) repeated scanning of

the reaction progress of xanthine oxidation to uric acid for 20 min; (b) the pure UV-vis spectra of xanthine (––) and uric acid (….) in

comparison with the resolved spectra of xanthine (····) and uric acid (○○○○ ); (c) the resolved concentration profi les of xanthine (––) and

uric acid (….).

A new multi-wavelength model-based method for determination of enzyme kinetic parameters 403

J. Biosci. 35(3), September 2010

to be in very good agreement. Simplicity, low cost, andto be in very good agreement. Simplicity, low cost, and

the applicability othe applicability of the proposed method in those conditions f the proposed method in those conditions

with extensive overlapping spectra make this method with extensive overlapping spectra make this method

an effi cient tool an effi cient tool for kinetic studies of fi rst-order enzyme for kinetic studies of fi rst-order enzyme

reactions.reactions.

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MS received 23 February 2010; accepted 19 July 2010

ePublication: 5 August 2010

Corresponding editor: MARÍA ELIANO LANIO