Vol. 47 No. 1/2000 259–268 QUARTERLY

It is not always clear that some equations affected by complicated factors can, actually, be interpreted as a ratio of two polynomials of first degree and so that they can be, in general, represented by rectangular hyperbolas. In this paper we present an easy procedure to rearrange those equations into Michaelis-Menten-type equations and so to make the aspects of these rectangular hyperbolas more clear, particularly for researchers familiar with general biochemistry. As an example, the method is applied to transform the classical rate equation of the Cleland's Ordered Uni Bi enzyme mechanism.

terpreted as functions that correspond to hyperbolic graphs.
The Michaelis-Menten equation is a good example of a frequently used hyperbolic function (v = ¦([S])): in which v is the initial velocity in an enzyme-catalyzed reaction; V max is the maximal velocity, i.e. the velocity attained at very high concentration of substrate [S]; K m is the Michaelis constant and corresponds to the concentration of substrate at which v = V max /2.This equation follows a form that is called here equation of type 1: The rectangular hyperbola (y versus x) described by an equation of type 1, crosses the origin of the axes and the equations for the horizontal and the vertical asymptotes are y = a and x = -b, respectively.
From the mathematical point of view it is known that in functions of the form where n 0 , n 1 , d 0 and d 1 are constants, the plot y versus x is also a rectangular hyperbola provided that d 1 ¹ 0 and n 0 d 1 ¹ n 1 d 0 .However, when the constants are complicated factors, it is not always clear that equations of type 2 actually represent hyperbolic functions.
In the course of an investigation on the effect of a modifier on the rate of an enzyme-catalyzed reaction (ref.[1]; accompanying paper), we handled complex equations of type 2 and arrived at a general approach making possible to transform those equations into another type (type 3; see below) that, because they resemble the classical Michaelis-Menten equation, more clearly show that they represent rectangular hyperbolas.

EQUATION OF TYPE 3
Equations of type 2 can be interpreted as a ratio of two polynomials of first degree and they can be converted into Michaelis-Menten-type equations (type 3) by adding and subtracting n 0 /d 0 : or, in general, By comparing eqns.(4) and type 3, the parameters (h, a, and b) of the type 3 equation can be deduced: Equation of type 3 describes a rectangular hyperbola where h is the value of the ordinate of the point where the hyperbola intercepts the vertical axis.The equation for the horizontal asymptote is y = a+h (i.e.y = n 1 /d 1 ) and that for the vertical asymptote is x = -b.The value of b can also be described as the value of x at which y = h + a/2.The value of a is the difference between two values: the ordinate of the point where the horizontal asymptote intercepts the vertical axis and h (a = n 1 /d 1 -h ).
Assuming b as a positive value, as it occurs in the biological phenomena referred to above, one of the advantages of an equation of type 3 is that the sign of the parameter a indicates when the hyperbola is ascendant (a > 0) or when it is descendant (a < 0); another advantage is that an equation of type 3 is more easily turned into a linear form (see below).
The procedure shown in eqns.(2-4) is valid when d 0 ¹ 0; i.e., when the vertical asymptote of the hyperbola is not the vertical axis itself.However, even in the case d 0 = 0, the equation for the vertical asymptote of the hyperbola can be obtained by evaluating the mathematical limit in an equation of type 3: when d 0 tends to zero, b tends to zero.
In the particular case in which n 1 = 0 in an equation of type 2, the horizontal asymptote of the rectangular hyperbola is the horizontal axis itself and the procedure shown in eqns.
(2-4) is valid; in this case, in the corresponding equation of type 3, the parameters h and a have the same absolute value but opposite signs.
As stated in the introduction, the equation of type 2 represents a hyperbola when d 1 ¹ 0 and n 0 d 1 ¹ n 1 d 0 .Actually, if the previous conditions do not apply, the equation corresponds to a straight line.In the case d 1 = 0, the value of the ordinate at the origin is n 0 /d 0 and the value of the slope is n 1 /d 0 .In the case n 0 d 1 = n 1 d 0 , it is not so obvious that equation of type 2 represents a straight line.However, if its transformation into an equation of type 3 is attempted, following the procedure described above, it results in a = 0, and so this straight line, that is parallel to the horizontal axis, may be viewed as a limit case between an ascendant rectangular hyperbola (a > 0) and a descendant one (a < 0).
As an example, eqn. ( 5) is here rearranged into eqn.(6) (numbered 5 and 5a, respectively, in ref. [1]) The above procedure, as such or with minor variants, may be of general application.As an example, it is here applied to classical enzyme reaction rate equations deduced by Cleland [2].

CLELAND EQUATIONS; THE ORDERED UNI BI CASE
The following are examples of the formulas deduced by Cleland for the steady state assumption kinetics [2]: Ordered Uni Bi mechanism: Vol. 47 Michaelis-Menten-type equations 261 Ordered Bi Bi mechanism: Ping Pong Bi Bi mechanism: where k n are rate constants (whose significance is irrelevant in the context of this paper), A and B are concentrations of substrates, P and Q are concentrations of products and Et is the concentration of enzyme; v is the reaction rate for the conversion of A (+B) into P + Q.When the reaction proceeds in the opposite direction v has a negative value; in this paper P and Q will be called products independently of the macroscopic direction of the reaction.
The equations deduced by Cleland [2] can be viewed from a new perspective if the mathematical procedure described above is applied, as exemplified here for the case of the Ordered Uni Bi mechanism.
Equations (7-9) can be arranged to show more clearly that they fit the type 2 model, if the concentration of one of the substrates or one of the products replaces x.As examples, the eqns.(10-12) were obtained by rearranging eqn.( 7 (iii) variable Q (Q = x); A and P are assumed constants Equations (10-12), arranged as type 2 equations, can now be transformed into equations of type 3 by applying the procedure deduced above.The method could also be used to analyze Ordered Bi Bi (eqn.8) or Ping Pong Bi Bi (eqn.9) mechanisms.However, only the Ordered Uni Bi mechanism will be analyzed here (eqn.10).
Equation ( 13) is deduced from eqn. (10): where By analyzing eqns.(13-16) (see below), it is easy to visualize the aspect of the plot v versus A, and the way it depends on P and Q.
(1) P and Q are assumed constants; general aspects.Equation (13) shows that the plot v versus A is, in general, a rectangular hyperbola (Fig. 1; plot 1): (a) The hyperbola intercepts the vertical axis below the origin of the axes (h < 0) when neither P nor Q are zero.When P or Q are zero, h = 0 and the hyperbola crosses the origin of the axes (Fig. 1; plot 2).
(c) The equation for the horizontal asymptote (y = n 1 /d 1 ; see eqn.17) shows that, in general, the hyperbola tends to a positive value when A tends to saturation (i.e.A®¥) and so the hyperbola crosses the horizontal axis.
Vol. 47 Michaelis-Menten-type equations 263 (14) The point where the hyperbola crosses the horizontal axis (v = 0) corresponds to chemical equilibrium; the value of A at which chemical equilibrium is obtained (A eq ) depends, obviously, on the values of P and Q.The points of the hyperbola when v is positive correspond to concentrations of A at which the reaction proceeds from A into P + Q, and the points when v is negative, to concentrations of A at which the reaction proceeds from P + Q into A.
As all the three parameters h, a, and b (eqn.13) depend on both P and Q, the actual aspect of the plot v versus A depends on the values of P and Q.The effect produced by P or Q on this plot can be deduced by analyzing how the parameters h, a, and b depend on their values (eqns.14-16).Interestingly, these equations and the equation for the horizontal asymptote (17) recall equations of type 2; when they are, actually, equations of type 2, a reasoning similar to that followed above can be made and the aspect of the corresponding plots visualized intuitively.
Equations (14-17) can be rearranged using the procedure presented above: If Q is assumed constant, and a nonzero and finite quantity, from these equations it can be intuitively recognized that the plots h versus P, b versus P and n 1 /d 1 versus P are rectangular hyperbolas (see Fig. 2).As P increases from zero to saturation (i.e.P®¥) the value of h decreases from zero to a finite negative quantity, the value of b varies between two finite positive quantities, and the value of n 1 /d 1 decreases from a finite positive quantity to zero.
In absence of P, h = 0 and eqn.(13) turns into an equation of type 1.In this case, the plot v  P=3 and Q = 6 in plot 1; P = 0 and Q = 6 in plot 2; P=1000 and Q = 3 in plot 3; and P = 3 and Q = 5000 in plot 4. A, P, Q and Et have the dimension of concentration and the enzyme reaction rate (v) dimensions of (concentration time -1 ).Rate constants, A, P, Q and Et are defined as in ref. [2].In all cases units are irrelevant and arbitrary.
A versus A is an ascendant rectangular hyperbola that crosses the origin of the axes (Fig. 1; plot 2).Actually, this is just the Michaelis-Menten equation when the value of b (the apparent K m ) depends on Q (eqn.20).
At saturating concentrations of P, the plot v versus A is an ascendant rectangular hyperbola (a > 0) with a negative value of the ordinate for A = 0 (h < 0), but the horizontal asymptote is the horizontal axis itself (n 1 /d 1 = 0; see Fig. 1; plot 3).In these conditions, v is neg-ative even at high A: P is a total inhibitor for the conversion of A into P + Q and, as it does not compete with A, saturating P can not be overtaken by saturating A [2].
(3) Influence of Q on the hyperbola parameters.If Q is considered the variable in eqns.( 14) and (15), it is easy to recognize them as equations of type 2. Note that for eqn.( 14) n 0 = 0 and that eqn.( 15) is, actually obtained by an addition of a constant value and an equation of type 2. These equations can be rearranged into equations of type 3: If P is assumed constant, and a nonzero and finite quantity, from eqns.( 22) and (23) it can be intuitively recognized that the plots h versus Q and a versus Q are hyperbolas (see Fig. 2).As Q increases from zero to saturation (Q®¥) the value of h decreases from zero to a finite negative quantity, the value of a increases by a value that is equal to the decrease of h so that (a + h) remains constant; i.e. the position of the horizontal asymptote for the plot v versus A does not depend on Q (see eqn. 17 and Fig. 1).Equation (16) shows that the plot b versus Q is a straight line (d 1 = 0) the values of which for the ordinate at the origin and for the slope are always positive and depend on P (see Fig. 2).
In absence of Q, h = 0 and eqn.(13) turns into an equation of type 1 and so the hyperbola crosses the origin of the axes; the value of a is, in general, positive and so the hyperbola is as-cendant (Fig. 1; plot 2); again, it is just the Michaelis-Menten equation when the value of b (the apparent K m ; see eqn.16) and the value of a (the apparent V max ; see eqn.23) depend on P. In the particular case, in which Q is absent and P is saturating, the value of the parameter a equals zero (see eqn. 23), and the plot v versus A is no more a hyperbola but a straight line coincident with the horizontal axis.When Q tends to infinity, b tends to infinity (eqn.16) and a to a finite positive value that depends on P (see eqn. 23); this means that at saturating Q, v does not depend on A unless A is also saturating.Actually, Q is a competitive total inhibitor for the reaction A ® P + Q [2]: if Q is supposed saturating, A can not bind to the enzyme; if A is supposed saturating, Q can not bind to the enzyme [1].So, if Q is supposed saturating for all concentrations of A, - (23) the plot v versus A is a straight line parallel to and located below the horizontal axis; the equation for this straight line is v = h (Fig. 1; plot 4).If Q is supposed saturating only at low concentrations of A (higher concentrations of A dislocate Q from the enzyme) the plot v versus A is an ascendant rectangular hyperbola intercepting the vertical axis below the origin of the axes; unless P is saturating for all concentrations of A, the hyperbola crosses the horizontal axis and tends to a positive value (n 1 /d 1 ) when A tends to saturation (see Fig. 1; plot 1).
(4) Tridimensional representation.The classical Lineweaver-Burk (LB) plots can be made tridimensional by adding a third axis, representing the concentration of an inhibitor, perpendicular to both the LB axes and crossing them at point (0,0) [3].Similarly, the plot v versus A can be made tridimensional by adding a third axis representing P or Q.The axes A and P (or Q) define a horizontal plane and the axes v and P (or Q) a vertical one.In this type of graph, a family of hyperbolas representing v versus A is obtained for different concentrations of the variable product.The points where the hyperbolas cross the vertical plane draw the plot h versus P (or Q); the points where the hyperbolas cross the horizontal plane draw the plot A eq , versus P (or Q).

LINEAR TRANSFORMATIONS
Several methods are commonly used to fit the experimentally obtained data to eqn.(1) by first transforming it into the equation of a straight line to which the data can be easily fitted by linear regression.Once equations of type 3 are obtained they can be easily turned into a linear form, the first step being to subtract h from both terms of the equation: ): (i) variable A (A = x); P and Q are assumed constants (ii) variable P (P = x); A and Q are assumed constants 262 R. Fontes and others 2000

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The addition of two equations that represent rectangular hyperbolas represents a rectangular hyperbola in the particular case when the vertical asymptotes of the two hyperbolas coincide.

Figure 2 .
Figure 2. Influence of the concentration of P or Q on the reaction rate parameters in the Uni Bi enzyme mechanism.At the left, plots h versus P, a versus P, b versus P, and n 1 /d 1 (= a + h) versus P assuming Q = 6 and constant; at the right, similar plots considering Q the variable and assuming P = 3 and constant.As in Fig. 1 these graphs have been drawn using the program Microsoft Excel 4.0 for Macintosh.Values of the rate constants and Et as in Fig. 1.As discussed in the text, h, a and b are parameters in an equation of type 3 that corresponds to a rectangular hyperbolic plot; n 1 /d 1 is the value of the ordinate of the point where the horizontal asymptote of the rectangular hyperbola intercepts the vertical axis.

Figure 1 .
Figure 1.Influence of concentration of the substrate A on the reaction rate (initial velocity) in the Uni Bi enzyme mechanism.Plot 1 represents the general case in which P and Q are nonzero and finite quantities; plot 2 displays the situation when P or Q are zero; in plot 3, P is saturating and Q ¹ 0; in plot 4, Q is saturating and P ¹ 0. The graph has been drawn using the program Microsoft Excel 4.0 for Macintosh, assuming the following set of values for the rate constants: k 1 = 2*10 3 , k 2 = 2*10 3 , k 3 = 3*10 3 , k 4 = 4*10 3 , k 5 = 8*10 3 , and k 6 = 6*10 3 ; the dimensions of k 1 , k 4 and k 6 are (concentration -1 time -1 ) and those for k 2 , k 3 and k 5 are (time -1 ); Et is always equal to 10 -3 .