Parameter estimation of an enzyme kinetic system using computer algebra techniques

EISEVIER

Applied Mathematics and Computation 99 (1999) 93-98

Parameter estimation of an enzyme kinetic system using computer algebra techniques Mustafa Bayram *, Bunyamin Yildiz At&irk

8niversitesi,

Fen-Edebiyut

Fakiiltesi,

Matemutik

BXimii,

25240 Erzurum.

Turkey

Abstract A procedure for fitting enzyme kinetic data directly to the flux equation was described. It involves choosing parameters that minimize the sum of the squares of deviations due to errors in s, the substrate concentration at time t. Estimates of the standard errors of the parameters are provided using computer algebra and numerical analysis techniques. 0 1999 Elsevier Science Inc. All rights reserved.

1. Introduction Determination of enzyme kinetic parameters has traditionally been done by isolating individual enzymes in vitro and performing initial rate experiments [12,24,27,29]. Linearizations of the formulae governing enzyme behaviour allow these data to be plotted, and fitting a straight line yields the kinetic parameters of the enzyme mechanism [2-6,15,23,30]. The nature of experimental error in the determination of initial velocities of enzyme catalysed reactions was investigated [28]. The quantitative treatment of experimental data requires an adequate consideration of the reliability of the dependent and independent variables. In enzyme kinetics this problem has been investigated by numerous researchers [l, 13,14,27,3 11. The analysis of progress curves for enzyme catalysed reactions by non-linear regression has been studied [10,17,18,20,21,26]. A procedure, based on the Gauss-Newton method for non-linear regression, has been developed to obtain enzyme kinetic constants from the analysis of progress curve data. The method

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Corresponding author.

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of progress curve analysis for enzyme catalysed reactions has been extended to a two substrate, reversible reaction through the use of enzyme catalysed recycling of one of the products [ 19,3 11. A combination of the jackknife [22] and non-linear regression techniques to measurements of the rate constants for enzyme catalysed reactions has been applied [16]. A particular technique of importance to this work is the calculation of Grobner bases [l 11. These are canonical representations of systems of multivariate polynomials. A consequence is that they often permit us to solve simultaneous non-linear equations. A new approach to analysis of enzyme kinetics has previously been proposed in [7-91. Measurement of enzyme kinetic parameters is usually performed by fitting experimental data to steady state rate equations. In this paper we have used computer algebra and numerical analysis method to fit kinetic parameters to rate laws using experimental data.

2. Use of computer algebra system

A computer algebra system computes with symbols rather than numbers. Such systems are useful for manipulating formulae. A particularly useful technique is the calculation of a Griibner Basis which permits us to solve simultaneous polynomial equations. We have used REDUCE in four ways: l We have used its matrix operations to implement Reder’s algorithm for determining conservation rules. l We have used its linear algebra facilities to express all metabolite concentrations in terms of one independent concentration. l We have used a Grobner Basis calculation to obtain a steady state rate law for the system [l 11. l We have used the GENTRAN package to generate the resulting formulae as FORTRAN expressions for numerical evaluation [25].

3. Steady state rate law

We have considered glutamate dehydrogenase (GDH) catalysed reactions with the primary objective of measuring rate constants for the electron transfer reaction. The data analysed cover the transient and part of the steady state, with enzyme and L-glumate concentrations fixed at 0.018 and 25 mM, respectively. The enzyme mechanism is considered to be an ordered addition of NAD+, A, followed by addition of L-glumate, B, to enzyme, E. At sufficiently high L-glumate concentrations, its rate of association with EA will be much greater than the association rate of A with E, and we may then use the simplified mechanism,

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where P, stands for NADPH and other products. The observed absorbance changes are proportional to the total concentration of NADPH. We have not used it for further analysis of the system. Rate constant ks represents the rate limiting step of the reaction at high substrate concentrations. The following five independent equations, d[EAB] ~ = kl [E] [A] + kd[EP] - (k2 t k3) [EAB], d[E$ - dt = k3[EAB] - (k4 + kS)[EP], -

dt

= u = kS[EP],

PI = M tot- MB1 - WI 7 PI = L%3t- WY - [EPI- PI

(1)

describe the dynamics of the system, where [El,,, is total enzyme concentration. We have the following overall rate law using computer algebra techniques, ?(500k;(ks

+ 2kzk: + 2k;ka + k3 + 2k3k4 + k4)

+ o(ksk3(500ksk,P

- 500ksk3 - 12509kskl - 500k5k2 + 500k3klP

+ 500kdk,P - 1209k3k, - 12509k4kl - 500k4k2) - 9k;ek,(P

- 25) = 0. (4

Remembering

that

4Pl v=dt

(3)

we have obtained a differential equation in [P] and t. Our data are in terms of just [P] and t, so estimating the kinetic parameters is essentially an overconstrained multi-point boundary value problem. 4. Fitting parameters The logged spectrophotometric data give us Fig. 1 for P (NADPH) concentration against time. The rate laws we are fitting relate d[P]/dt to [PI. We have to options in fitting: 1. Estimate d[P]/dt from the experimental data and fit directly to the rate law. We reject this option, since derivative estimation is highly unreliable. 2. Integrate the rate law and fit directly to the data. This is much more reliable, but analytical integration of the rate law is not in general possible. However we can use numerical integration instead.

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Time(m second) Fig. 1. Graph of NADPH time course data in GDH system. The 210 points were collected spectrophotometrically. The experiment was performed at 30°C.

Table 1 Convergence path for five parameter model of Eq. (2) Parameters

Starting values

Final values

k, (M-Is-‘) k2 (s-l) k3 (S-I) k4 (s-‘1 ks (s-l)

0.150 0.733 0.604 0.630 0.435

0.214 0.120 0.320 0.701 0.614

x x x x x

10-6 102 lo* IO* 10’

f + + f +

0.190 0.220 0.251 0.340 0.025

x x x x x

1O-6 10’ lo? lo? 10’

Table 2 Goodness of fit and correlation matrix Parameters

k, k2 k3 k4 ks

k, 1BOO 0.928 -0.693 -0.560 0.162

k2

k3

t

1.000 -0.690 -0.613 0.059

1.000 0.953 0.275

1.ooo 0.530

1.000

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When fitting to integrated equations, it is important not to perform a complete numerical integration over the range and minimize residuals. Such approach is highly sensitive to errors in just the initial conditions. Instead we start with each data point, integrate to the next point and minimize the residuals obtained [9]. We obtained estimates of the five rate constants and illustrated them in Tables 1 and 2.

5. Results Computational difficulties have prevented us from successfully fitting to the composite rate law to date. However new Grobner Basis algorithms seem to allow us to handle the complete system and at the time of writing this paper it looks possible that we will now be able to fit to the composite rate law. We have obtained estimates for five rate constants by fitting to the rate law. Error estimates are obtained using the bootstrap method [22]. Very low errors are however an indication that the system is too underdetermined to affect the estimate of these parameters.

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