STUDIES ON THE DIFFERENTIAL EQUATIONS OF ENZYME KINETICS

Efficient and unbiased estimates of the parameters of the differential system, as well as simultaneous fiducial limits, are obtained through an (eventually weighted) least-squares fitting to a Taylor expansion of the concentration of the products of the reaction.

model under investigation, splits irreversibly into the enzyme and the products of the reaction P. The corresponding chemical scheme is k I k 2 E + S ES k_ 1 E+P where the k's are velocity constants.Let e, s and x be the concentration of free enzyme, free substrate and activated complex respectively.Then the differential sys- shown in [I] that of the two classical approximations of (1.1)-(1.2),that due to Haldane and Briggs and known as the steady-state approximation is preferable, but its domain of validity is limited to very small values of e or s which points to the o o need of a more direct approach.Anyway this approximation allows one to estimate only the two combinations of parameters k2e (the v of classical enzyme kinetics theo- max ry) and the Michaelis constant K M (k_l + k2)/kl" Thus, even assuming that the Briggs-Haldane scheme would be used in conjunction with the extrapolation technique suggested in [i] and therefore made more reliable, it remains that it is intrinsi- cally limited and leads to discard much of the information contained in a kinetic curve.
In practice the interest is focused on the initial portion of this curve because a feature not restricted to experiments carried out with enzymes parasite reactions may occur later and mask the phenomenon under scrutiny.In the Briggs- Haldane method (and in the Michaelis-Menten one as well) one uses exclusively the linear segment of the graph reaction velocity versus time.However the information sought, i.e. the numerical value of the parameters of the differential system is con tained in part in the non-linear portion of the curve and it is there generally speaking that is whatever may be the dependent variable selected that the analysis must be extended.
Thus the purpose of this paper is to describe a technique suitable for the fitting of a set of experimental data to the solution of (1.1)-(1.2) and for obtain- ing estimates, with their fiducial limits, of the parameters e and k's.The discus- o sion is conducted in terms of quantities easily evaluated by a digital computer.

THE FUNDAMENTAL REGRESSION.
Let there be t. (i 1,2,...,n) the times at which a certain variable is mon- itored in order to define the kinetic curve of an enzymatic reaction.As usual we take t as the (statistically) independent variable; we select as dependent variable the concentration of the products of the reaction p, which satisfies (2.1) dp/dt -(ds/dt + dx/dt) k2x Taking into account the initial conditions p(O) x(O) 0 one obtains, near the origin, p in the form P k2[X'o t2/2" + x"t3/3'o + + x(i)ti+i/(i+l)!o+ " '']   where the indicate derivatives with respect to time, and the problem is reduced to the evaluation of x (i) (i kleoS > O) and later exhibits an inflexion point (at the time T of [i]).Thus, at the beginning of the reaction at least, the function p(t) is strongly non-linear and therefore presents the features required for the analysis under consideration.
In order to facilitate the computer processing of the equations to be devel- oped, we relabel the parameters as follows, The solution p p(t;KI,K2,K3,K4) is obtained by means of the standard Taylor expansion method.
one obtains a Taylor approximation in the form P KiK2K4So[t2/2! bt3/3!+ b2 + c)t4/4!(bd 3KiKp2K4So)t5/5!] (2.3) Assume now that we know a set of approximate values K (i 1 2 3 4) of the parameters.The effect of small variations AK. around these values on p considered as a function of the K. 's is given by 4 4 in the domain D discussed in Section 5 and with i p (i 1,2,3,4).We have The preceding regression constitutes the fundamental regression of the problem.
-ill be noticed that it is mathematically constrained (as opposed to statistical- ly) to pass through the origin.Hence there remain n 4 degrees of freedom.

ESTIMATES AND FIDUCIAL LIMITS: BASIC THEORY.
We first assume that pexp is normally distributed.Then a least-squares solu- tion of (2.5) or more generally of a N-linear regression of the same type will fur- nish estimates of AK i (i 1,2,...,N) which are both efficient and unbiased.Hence by solving iteratively the regression until the increments are made sufficiently small, one will obtain efficient and unbiased estimates of the Ki's.Denoting by ik and z k respectively the value of i and z for the k-th experimental point, the normal equations of the problem can be written as TA q, where T is the real symmetric ma- trix of general element the vector of general element AK. and the vector of general element The total sum of squares is Zz 2 and the sum of squares accounted for by the k regression is '+q.Hence the variance S 2 is estimated from be the general element of T-I.Then the variance of K. is S2t ii and 1 as K i is normally distributed, fiducial limits for this quantity (and consequently for K.) can be obtained as where t is the value of the Student t-distribution at the selected level of confi- dence.
One could use the fiducial limits (3.2) to test statistically the concordance between the set of experimental data and the computed values of the parameters, but such a test is inefficient because these limits are not simultaneous: the probability that they be reached simultaneously is smaller than the probability corresponding to the selected t point[4].
The determination of simultaneous fiducial limits has already been treated in [2] and [3] but we give here a more streamlined derivation.The sum of squares ac- counted for by the regression &.q, used in (3.1) to estimate the variance, can be written as Q A'TA where a denotes a transposition; thus Q is a quadratic form which is distributed as 2 X with N degrees of freedom.On the other hand S 2 is x2-distributed with n N de- grees of freedom.Hence the ratio (Q/N)-'.[S2/(nN)] is distributed as F with N and n N degrees of freedom and the relation Q defines simultaneous limits for the AK0's at the level of confidence selected for F.
In many situations however, the interest is focused on the relative variations AKi/K'z rather than on the magnitude of the K.'s.zTherefore we define i AKi/Ki In the space of the i's the solutions of (3.4)-(3.5),that is feasible simul- taneous limits, are hyperellipsoids and this makes in most cases for unnecessarily complicated calculations.Now amongst all these ellipsoids there is, at a given level of confidence, one hypersphere whose radius x satisfies where K is the vector of general element K i.Let us define QK K'TK it will be noticed that QK is computed from quantities present in memory at the end of the iteration.Then (3.6) takes the form X2QK [N/(n N)]FS 2 Let x I be the value of x for the critical value F i. Thus xQ K [N/(n N)]S 2 and simultaneous relative departures of the parameters from the computed values K i by an amount no greater than x I are meaningless, whatever may be the level of confi- dence.This is to say that x I is a statistic which measures the degree of concord- ance between say the solution (2.3) of the differential system and the (normally dis- tributed) set of experimenal data.Now (3.6) can be rewritton as x 2 xF.Hence once x I is known, one can com- pute simultaneous fiducial limits at a specified level of confidence.
When the variance of pexp is not homogeneous, it is sufficient to divide both sides of (2.5) by the weight factor w Var(pexp).The introduction of w is quite es- sential: for example when the course of the reaction is followed spectrophotometri- cally, as in that case the variance of pexp exhibits dramatic variations on the do- main of absorbancies most used [5].For the calculation of w in situations of practl- cal interest, see [3].
Finally it must be emphasized that even when one exclusively analyzes sets of normally distributed pexp, it is advisable to introduce formally the weight factors w (all equal to i when the variance is homogeneous) because the scanning technique described in [3] and used to detect inhomogeneitles in a set of data analyzed with respect to a steady-state approximation formula is directly applicable to the case at hand.Assume that the k-th point is aberrant: if one depresses in turn the influence of each point by increasing its associated w factor, the x I values calculated will be more or less of the same magnitude, except for the k-th one which will be markedly smaller.One can then test the hypothesis that the k-th point is the only aberrant one of the set in the following way: the influence of that point is depressed or quasi-eliminated by increasing its w factor and the remaining points are scanned as previously.If the sole k-th point stands apart from the others, the x I values re- corded now will have more or less the same magnitude; etc.
By construction the matrix T is positive definite, but barely so in fact, and it is nearly singular (as can be verified e.g., on the system s i00, K I K 2 =.01, o K 3 .005, K 4 .008).On the other hand it must be emphasized that to some extent the condition of a linear system is relative, in the sense that, as in any linear al- gebra computation, the machine word size and the presence or the absence of a double- length accumulator are of paramount importance.Furthermore, even if the matrix of the complete system, that is to say with N 4 variables, cannot be inverted, it may happen that through the exclusion of one variable, the matrix of order 3 so obtained will prove readily invertible on the particular computer used.For instance the con- dition of the matrix T associated with the system mentioned improves when one ex- cludes either K I or K 2.
meters but in general it is advisable to handle directly the 4-parameters problem and because from a practical standpoint the most useful discussion refers to a worst case situation, we shall examine the analysis of the solution (2.3) of the same example within the domain D (see Section 5), with a machine using mantissae corresponding to 13-14D (a value typical of many a microprocessor-based system); all the operations were performed in single length mode.
The most striking feature of the matrix T is that it loses easily its positive definiteness, for example in the course of a Cholevsky triangularization.In fact the inversion by all the classical techniques involving a triangularization fails.Thus one must abandon any idea of inverting en bloc the matrix T; as a matter of fact this is of little consequence because the only computation in which the t ij strictly speaking the t ii (i 1,2,3,4) are involved in an essential way, is that of the variance of the parameters.But these variances are not much used as x I is a more convenient and realistic statistic.Thus one can dispense with the inversion of T and we consider now the lesser problem of solvimg once the normal equations by an itera- tive technique which does not require a preliminary deep transformation of the matrix of the coefficients.The standard techniques for solving a linear system with a pos- itive and non-singular matrix, such as the conjugate gradient method or the steepest descent method, all failed.
An obvious way to deal with the evanescence of the positive definiteness of T is to multiply on the left the normal equations by T but the multiplication (here the squaring) of ill-conditioned matrices is usually deemed undesirable.This formal cal- culation of T 2 is avoided in a variant described by Hestenes and Stiefel [6].This procedure proved unsuitable for our needs because huge oscillations of the terms b i of formulae (10:2) of [6] were recorded, although the stability of the coefficients a. was acceptable.This suggests to modify the method and to recalculate at each step 1 the residual vector r from its definition.This is to say that we use the following algorithm: This procedure bears little resemblance to a conjugate gradient technique, which is essentially an orthogonalization method.On the other hand the first step of {s(t),x(t)} is analytic on (5.3).Hence an upper bound of the maximum sampling time (5.4) The practical implementation of the techniques discussed in this paper and their implications regarding the experimental design of enzyme kinetics measurements will be described elsewhere.Suffice it to point out here that an immediate consequence and a major advantage of this numerical procedure is that it furnishes an es- timate of e (= K I) in the same units as those used for So.In other words, through the analysis of a kinetic curve one can obtain directly the value of the initial con- centration of an enzyme in well-defined physical units.
tem which describes the kinetics of the reaction is ds/dt -kle s + k x + klSX o -1 (1.1) dx/dt kleoS (k_l + k2)x klS x where e denotes the initial concentration of enzyme and with the initial conditions O non-linear and no close-form solution is available.It has been 576 C. MARMASSE and J. WIENER the initial value A 0. No stability problem was encountered.
The calculations are much simplified if, taking into account (2.1), one rewrites (2.2) in the non-canonical form Putting Given a certain set of experimental values Pi the classical Gauss linearization technique and compute, within the domain D, correc- tions AK. (i 1 2 3 4) to the estimates K i by means of the quadrilinear regression