Bulletin ojMathematical Biology, Vol 42, pp. 2 3 9 ~5 5 Pergamon Press Ltd, 1980. Printed in Great Britain

C H E M I C A L M O D I F I C A T I O N O F E N Z Y M E S : CRITICAL E V A L U A T I O N O F T H E G R A P H I C A L C O R R E L A T I O N B E T W E E N R E S I D U A L E N Z Y M E ACTIVITY A N D N U M B E R O F G R O U P S M O D I F I E D

• EVELYN STEVENS and ROBERTA F. COLMAN Depar tmen t of Chemistry,

Universi ty of Delaware, Newark, Delaware 19711

In the study of chemical modification of enzymes and other biologically active proteins, plots of fractional residual activity as a function of number of groups modified per enzyme molecule are often used to establish a correlation between the chemical modification and enzyme inactivation reactions and to determine the stoichiometry of the modification reaction. This paper presents a critical examination of the underlying theoretical framework of such graphs. Whereas these plots are usually presented as linear functions, it is shown here that the general equation describing the relationship between inactivation and modification contains an exponential term; therefore, in the general case, the plot is actually a curve. It is suggested that caution be exercised in the interpretation of such plots and that equations such as those derived in the text be used to fit theoretical curves to the data, in order to maximize the information gained from chemical modification experiments.

1. Introduction. Among the methods used to gain an understanding of the mechanism of enzyme catalysis, selective chemical modification of amino acid residues is one of the most frequently used (Glazer, 1975; Means and Feeney, 1971; Singer, 1967). The reagents and/or reaction conditions are chosen so as to minimize both the number of classes and total residues reacting while maximizing the effect on enzyme activity. The goal for a given reagent is that it react with a single "essential" amino acid at the active site, with concomitant total inactivation of the enzyme. Two of the criteria used to define an amino acid as "essential" are that it be modified at the same rate as the rate of enzyme inactivation, and that the number of residues modified be a small fraction of the total number of that type of amino acid present in the enzyme. (Other criteria, such as

239

240 EVELYN STEVENS AND ROBERTA F. COLMAN

protection by substrate, will not be considered here.) Levy et al. (1963) proposed a simple method for determining the minimum number of groups reacting. Ray and Koshland (1961, 1962) treated both the kinetic and stoichiometric aspects of group modification, and Rakitzis (1977) has more recently expanded on certain of the models of Ray and Koshland. These methods can be applied in a straightforward manner, with due con- sideration for any limitations, when accurate rates of inactivation and chemical modification are obtained. It is often the case that rates of modification are either difficult to measure, or only measurable for a small fraction of the total reaction; e.g. the assay used may not be sufficiently sensitive to reveal a small extent of modification, and precipitation of reagent and/or protein may restrict measurement of an extensive degree of modification.

A graphic method that is widely used (see, for instance, Bergh~iuser, 1975; Kantrowitz and Lipscomb, 1976; Roberts and Switzer, 1978) ap- parently to circumvent such difficulties, consists in plotting the fractional enzyme activity at a given time as a function of the number of groups modified per enzyme molecule or subunit at that time (Figure 1). The plot is identical to a Tsou plot (Tsou, 1962) for the particular case in which essential and non-essential groups react at the same rate and the number of essential groups per molecule or subunit is one. (For discussion of this method, see Paterson and Knowles, 1972, and Rakitzis, 1978.) Plots such as Figure 1 are usually interpreted by extrapolating the straight-line portion of the curve to zero activity; the intercept on the abscissa is presumed to give the maximum number of essential groups. The assump- tions on which such t reatment is based are generally not stated. In view of the widespread use of the plots, we believe that their underlying theoretical basis needs to be re-examined.

The purpose of this paper is two-fold: (1) to derive a theoretical framework, with explicitly stated assumptions, for some of the cases one might expect to encounter in the course of chemical modification of enzymes or other biologically active proteins; and (2) to show how the plots can be used to determine the stoichiometry of the reaction, rate constants and ratios of rate constants under favorable circumstances, and draw attention to some misinterpretations that can arise if certain assump- tions are not taken into consideration.

2. Theory.

General assumptions. Since most chemical modifications are performed with a considerable excess of reagent over enzyme, it will be assumed throughout, unless otherwise stated, that

1.0

CHEMICAL MODIFICATION OF ENZYMES

I I I I

241

0.8

0.6

0 I . g

0.4

0.2

) 1.0 2 .0 5.0 4.0 NUMBER OF GROUPS MODIFIED/MOLE ENZYME

Figure 1. Graphic method for determining maximum number of "essential" groups. E0=initial enzyme activity. Et=activity at time t. The maximum

number of "essential" groups in this case is interpreted to be 3.

(1) the inactivation reaction is pseudo-first order with respect to enzyme;

(2) the modification reaction is pseudo-first order with respect to any given class, n, of amino acid residue. The classes of residues may be constituted by different amino acids, as in the case of an alkylating reagent that reacts with lysine, cysteine and histidine under given conditions (n = 3); or classes may represent forms of differing reactivity of the same amino acid, for example, "exposed" and "buried" tyrosines (n=2).

For the sake of simplicity, it will further be assumed that

(3) the reagent is a single chemical species that does not undergo secondary reactions during the course of the inactivation/modification reactions;

(4) both the inactivation and modification reactions go to completion;

(5) inactivation can be correlated kinetically with the modification of a single class of residue;

(6) there is no interaction between classes or members of a class of amino acid residues.

242 E V E L Y N S T E V E N S A N D R O B E R T A F. C O L M A N

Rate of enzyme inactivation. The rate of inactivation, Vinaet, is given by

dE Vinact = - - d~-= k inac t [E]

where [E] is the concentration of active enzyme and kinact is the observed rate constant for inactivation. Rearrangement and integration between limits 0 and t for time and E o and Et for E, gives the expression

In ~ = - kinactt (1)

which permits calculation of kinac t from a semi-log plot of Et/E o a s a function of time.

Rate of chemical modification. Several modes of chemical modification will be considered, depending on the number of classes of amino acids modified and the relation between the rate of chemical modification and the rate of inactivation.

I. One class of amino acid modified (n = 1)

1.1. Rate of modification equals rate of inactivation. The rate of modifi- cation, Vmod, is given by

dx Vrnod = ~ - = kmod(Xm--X )

where dx/dt, the rate of appearance of modified groups, is proportional to the concentration of unmodified residues of the given class; i.e. the difference between the maximum number of groups per mole of enzyme available for reaction, Xm, and the number of groups per mole already modified, x. The measured rate constant is kmo d. Rearrangement and integration between limits 0 and t for time, and the corresponding limits 0 and x t for x,

give

if " dx f f - - - - kmoddt

o X m - - X o

In x " - x t - krnod t. (2) X m

CHEMICAL MODIFICATION OF ENZYMES 243

If the rates of inactivation and modification are equal,

and, from (1),

o r

kinac t = kmo d

In xm - xt _ kinac t t = In E~ x m E o '

E t = 1 x t . (3) Eo X m

Et/Eo is a straight-line function of x t, with a negative slope equal to 1/x m.

When x ,=O (i.e. no groups modified, at t=O), E ] E o = I . When Et /Eo=O,

x ~ = x m (i.e. all groups available are modified). The interpretation is that xm represents the maximum number of groups "essential" for activity.

If a straight line is not obtained, one or more of the general assumptions must not be valid. Cases in which n > 1 are discussed in Section II; cases in which Vmod@Vinac t (assumption 5 not valid) are dealt with in subsection 1.2. In Figure 1, the generally accepted interpretation would be that the assumptions are valid for almost the entire course of the reaction, and that the maximum number of "essential" groups is 3. As will be seen, this is an approximation, and not the only possible interpretation.

1.2. Rate s o f modi f ica t ion and inac t iva t ion not equal (general a s sumpt ion 5

not valid). A ratio of rate constants, r, can be defined, such that

kinact r - - -

kmod "

The time course of inactivation is given by equation (1) and the time course of chemical modification by equation (2), which can be rewritten

r In xm - xt - - kinac t t. Xm

Comparing with equation (1):

r In X m - - X t

X m

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CHEMICAL MODIFICATION OF ENZYMES 245

II. Simultaneous modification of different classes of amino acids (n > 1).

II.1. General case. Given a protein with n classes of reactive amino acids, the overall rate of modification, Vmoa, will be the sum of the rates of reaction of the individual classes:

Vmod = ~ d ( x n ) - - ~ dt kin°d" [(Xn)m -- (Xn)] '

where (x,),, is the maximum number of groups available for reaction in each class, (x,) is the number of groups reacted in each class, and kmod. is the rate constant for chemical modification of class n. For each class of residue, the rate equation can be rearranged and integrated between limits 0 and t for time, and the corresponding limits 0 and (x,,)t for (x,). The resulting expressions are analogous to equation (2):

In (x.) , . - (x.)t (x.)~ - k~od.t (5)

and the overall modification reaction can now be described by the sum of n such equations:

(X.)m (X.)~ )_ln - - ~ kmo d t. (6) (~.).~ °

If one of the rates of modification, e.g. Vmoal , is equal to the rate of inactivation, then

kmod~ ~ kinac t.

If, in addition, (xl)t can be measured independently of all other x,'s (e.g. different amino acid residues), equation (5) written for (Xl) can be compared with equation (1), the expression for inactivation, and one obtains

o r

In (x l )m- (xl) ' = - kinact t = I n E~t (xl)m Eo

E~ _ 1 (x~)~ Eo (x~)~ (V)

246 EVELYN STEVENS AND ROBERTA F. C O L M A N

Equation (7) is analogous to equat ion (3), and the properties of the plot EJEo vs (xl)~ have been described.

In practice, however, the quanti ty measured may not be (xl)t, but - the sum of all x,'s:

?l

Z(x.),=xT,

where x r is the sum of all modified residues at time t (e.g. a given amino acid, with different reactivity depending on its environment in the enzyme). To obtain EJEo as a function of x T, the following transformations can be made: Since x r is the sum of all x,'s,

(x l ) ,=Xr- (x,),. (8) n = 2

Each individual (x,)t can be calculated from equat ion (5)

(x . ) ,= (x.)=(1 - e x p ( - kmod.t)). (9)

Substituting equat ion (9) into equat ion (8) and the result into equat ion (7):

L [(x,)m(1 - -exp(- - kmod t))] E-J-t = 1 -~ ,=2 XT Eo

(10)

Inspection of equat ion (10) reveals that a plot of EJEo as a function of x T will, in general, be an exponential curve, since t (time) is an indepen- dent variable. At zero time, x r = 0 , the term containing the exponential reduces to zero, and EJE o= 1. When EJE o=0 ( t= oe), the exponential exp ( - kmodt ) =0 , and equat ion (10) becomes:

0=1-+ ,=2 Xr (xx) m (xl) m. (11)

To simplify equat ion (11), the partial sum of x,'s can be written as a function of the max imum number of groups that will eventually react (xM):

so that

CHEMICAL MODIFICATION OF ENZYMES , 247

(x.)m = x u - (xl)m. (12) , = 2

Substituting in equation (11)

0 = l q (X )m

• . X r = X ~ ( a t t = o o ) .

In other words, extrapolation of the curve to zero activity yields the maximum number of groups available for reaction.

Representative plots for different values of n, (x,),~, and kmod. are shown in Figures 3-6 and will be discussed below.

1.0

O.B

0.6

o

0 . 4

0.2

0 i 0 0.5 1.0 I.,5 2.0 2.5

IT

~ \.. , \ \k L I L

Figure 3. Plot of equation (10), for n = 2 ; (x l )m=(x2)~=l; kmodj=kinact; and krnodjkmodl as indicated for each solid curve. Dashed lines (---) are the lower and upper limiting curves; dotted lines ('--) are extrapolation of curves to Et/E o

= 1 and EJEo = O.

II.2. S p e c i a l c a s e s .

11.2.1. A l l r a t e s e q u a l (kmoal = kmod~ . . . . . kmod° = kinact, ). Since all kmod 'S are equal, different classes of groups cannot be distinguished, a n d even if in

248 EVELYN STEVENS AND ROBERTA F. COLMAN

I.O ~ ~ 0 '~ ~ ~ [

~,

0.6 ~

0.4 '""~'

0 0 1.0 2.0 3.0 4.0 5.0 6.0 XT

Figure 4. P lot of equa t ion (10), for n = 2 ; ( x l ) , , = l , (xz ) , ,=4 ; kmoa =kinact; and kmoa2/kmod,, as indicated for each solid curve. Dashed lines (---) are the lower and upper l imiting curves; dot ted lines ( . - ) are ext rapola t ion of curves to

Et/Eo = 1 and E,/Eo =0 .

1.0

0_8

0 . 6

o w

w-

OA

0 .2

0 0

'. \ 1 i I

'-.,,, ,,,,,, ".._

1.0 2.0 3.0 x T

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extrapolat ion of straight-l ine por t ion of curves to Et/Eo = 1 and E,/E o = O_

CHEMICAL MODIFICATION OF ENZYMES

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O [aJ

0_4

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0 i I 0 2.0 4.0 6.0 8.0

X T

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extrapolation of straight-line portion of curves to E]E o = 1 and E]E o =0.

249

fact n =~ 1, experimentally nob~= 1. All groups can be treated as though they belonged to one class, so, as in 1.1, equat ion (3) applies:

Et - i XT

Eo

Et/E o is thus a straight-line function of the total number of groups reacted at any given time (XT); and the line goes th rough Et /Eo=l when XT=0 (no groups modified) and through XT---X ~ when Et/Eo=O (zero activity, max imum number of groups available for reaction). (See, for example, the straight solid lines in Figure 3 and Figure 4, where n = 2, x M = 2; and n = 2, xM = 5, respectively.)

II.2.2. Limiting cases. (a) k i n a e t = krnod ' ~Zn=2 kmodn. In equat ion (6), kmOdl becomes negligible in the sum ~" kmod t. Thus, modification of all groups except (xl) proceeds according to equat ion (6), and since x I is not modified, Et/E o remains unchanged and equal to 1. After a very long time

250 EVELYN STEVENS AND ROBERTA F_ COLMAN

(relative, dependent on the actual values of kmod.), the product kmodt is no longer negligible. Now all (x,)'s have reacted almost completely, except x~, so the exponential term in equat ion (10) approaches zero. Substi tuting equation (12) into equat ion (10):

Et 1 -~ X M - (X1)m X T

Eo

i.e. the plot of E]Eo as a function of x T is now a straight-line function of x r. When E]Eo = 1

XT=XM-- (Xl)m,

the difference between the m a x i m u m number of all reactive groups and the maximum number of groups essential for activity. When E ] E o = O,

X T ~- XM.

The number of groups essential for activity, (xl),, , can therefore be determined as the difference on the abscissa of the extreme points of a straight line. (See upper dashed lines in Figures 3 and 4.)

(b) kin,ct=kmodl>>~,"=2 kmod. In this case, at relatively short times, the sum of products kmodt not correlated with activity becomes negligible with respect to kmodt , and E ] E o can be calculated from equat ion (7). As seen before, this is a straight-line plot going through E ] E o = 1 at (x l ) t=0 , and through (xl),, at E]Eo=O. At longer times, ~,"=2kmoat is no longer negligible, and modification of the "non-essential" groups yields a straight line that coincides with the abscissa, up to a max imum xr =XM. (See lower dashed lines in Figures 3 and 4.)

II.2.3. Non-limiting cases. For all non-limiting values of kmodn , t h e plot of Et/Eo as a function of XT will yield a curve lying between the upper and lower limiting lines described in the previous section. The curved lines in Figures 3 and 4 are examples of such cases for kmoa, = kinact, kmod. ranging from 0.lkmod, to 10krnod,; for Figure 3, n = 2 and (Xl)m=(Xa)m=l, while for Figure 4, (xl)m= 1, (x2)m=4.

Figures 5 and 6 illustrate the change in shape of the curves when n = 3. Inspection of the curves shows that an addit ional limiting case (not shown) can be calculated: the plot resulting when kmod2 >~ kmodl and

C H E M I C A L M O D I F I C A T I O N OF ENZYMES 251

kmod~ ~ kmod~(kmodl= kinact)- It can readily be seen that this will be a three- segment line consisting of a straight line parallel to the abscissa, at Et/E o = 1, from XT=O to x r = (x2),, a straight line with a negative slope equal to 1/(xl),, from x r = (x2)m to x r = (Xl)m+ (X2)m, and a straight line parallel to the abscissa at E,/Eo=O, from XT= (Xl),, + (X2) m to x r = (xl),, + (X2)m -}- (X3)rn ~ X M.

3. Discussion. Plots of E]E o as a function of (x,)~ or xr are commonly used to determine or confirm the stoichiometry of chemical modification reactions performed on enzymes or other biologically active proteins, and to verify the correlation between chemical modification and inactivation. If the theoretical basis for such plots is not taken into account, there is a large probability of misinterpretation, as will be discussed below.

Cases in which curves are obtained. If all assumptions previously listed are valid, extrapolation of the curve to Et/Eo=O always yields x~, maximum number of groups of all classes available for reaction. Extrapolation of the (approximately) straight line portion of the curve to E]Eo = 1 and E]E o = 0 can give misleading results, as shown in Table I, where in many cases the "extrapolated end-point ' , calculated as indicated in footnote a to the Table, is neither xM nor any particular (x~),,. In all cases, except in Figure 2, the true value of (x 1),. = 1. It can readily be seen that the extrapolated end-point is a good approximation of the true end- point only if the rate constants for the various reactions differ by an order of magnitude or more. When kmod./kmodl is close to unity, more than one straight line can sometimes be drawn, yielding very different values of x,,, as shown in Figures 3 and 4 for a ratio of 2. In addition, the smaller the ratio of "non-essential" to "essential" groups, the closer the extrapolated value will be to the true end-point. For example, for n = 2 and kmoa2/krnodl =0.1, the extrapolated end-point is 1.53 for (X2)m/(Xl)m=4 (Figure 4) and 1.10 for (x2)m/(xl),,=l (Figure 3). For n = 3 and kmod./kmoal=lO; 0.1, the values are 1.85 for [(Xz)m+(x3),,]/(xl),,=8 (Figure 6) and 1.24 for [(x2) m + (x3)m]/(xl)~=2 (Figure 5).

The limiting cases described in Section II.2.2 (Figures 3 and 4) are difficult to distinguish from consecutive reactions. Where kinaot =kmod~2~=2 kmod., the interpretation would be that all non-essential groups react before the "essential" groups, and therefore, before in- activation occurs; where kinact = kmod >~Znn=2 kmod., the opposite would be true. In intermediate cases, it might not be possible to differentiate

252 EVELYN STEVENS A N D ROBERTA F. C O L M A N

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CHEMICAL MODIFICATION OF ENZYMES 253

kinetically between combinations of consecutive reactions, as opposed to simultaneous reactions taking place with different rate constants.

T h e parabolas in Figure 2 may be very difficult to distinguish from exponential curves if r (kinact/kmod) is not a small integer. A plot of (Et/Eo) 1/~ as a function of x r should yield a straight line, provided one chooses the correct value of r. However, whether one deals with the curve or the straight line, two points must be considered: (1) The fact that the ratio of rate constants is not equal to 1 (r :p 1) may be interpreted to mean that the modification reaction in itself is not directly responsible for inactivation. If r > 1, inactivation of the enzyme occurs prior to actual chemical modifi- cation (e.g. in a binding step). If r < 1, inactivation occurs after modifi- cation (e.g. subsequent conformational change or denaturation). (2) The end-point xM only indicates the total number of groups available for the modification reaction.

The curves in Figure 2 are true parabolas, and it is therefore not licit to draw straight lines through them and extrapolate to E]Eo=O. The examples in Table I have only been included in order to give an indication of how badly one can be misled if one does at tempt to do so.

Cases in which straight lines are obtained. A comparison of cases in which n = 1 (Figure 2) or in which n > 1 but all kmoa'S are equal (Figures 3 and 4) shows that if E]E o = f ( x r ) results in a straight line with intercepts 1 and XM, it is not possible to distinguish classes of groups. The end-point, xM, is the maximum number of groups available for reaction and may be equal to or greater than the number of groups "essential" for activity (Table I). Once a linear correlation between inactivation and chemical modification has been established, even in the seemingly unambiguous case where n = 1, xM--1, the cause of inactivation would of course still remain to be explored (e.g. direct interference with catalysis or binding, confor- mational change, steric hindrance, etc.).

Given experimental points that deviate from a straight line, it is usually possible to draw a "best" straight line through them that intersects the ordinate at Et/E o = 1, and consequently the abscissa at some arbitrary x M. In performing this operation, instead of using the curve to determine how many groups or classes of groups are responsible for inactivation, one makes either of the following assumptions, in addition to those listed earlier :

(7) n > 1 and all kmod'S are equal to kinact, i.e. there are several kinetically indistinguishable classes of reactive groups, reacting at the same rate as the rate of enzyme inactivation; or

254 EVELYN STEVENS AND ROBERTA F_ COLMAN

(7') n = 1, kmod,= kin,c, i.e. there is one class of reactive group, reacting at the same rate as the rate of inactivation.

Equations such as those derived in this paper would provide theoretical lines which might better fit the data, and thus provide a more complete description of the inactivation/modification reactions, without the need for introducing these further assumptions.

Other cases. More complex but possibly common cases, such as con- secutive reactions, binding prior to reaction, etc., have not been considered in this treatment. If these types of mechanisms were believed to be operating, and plots of E / E o vs XT were called for, the equations would have to be suitably modified.

Practical considerations. Plots of Et/E o as a function of the number of groups modified per mole of enzyme have apparently been used success- fully in many cases to correlate chemical modification and inactivation processes and determine the number of groups essential for enzyme activity (see, for instance, Foster and Harrison, 1974; Levy et al., 1977; and Soon et al., 1978). However, given the difficulty in obtaining good end-points for "essential" reactive groups, and the possible ambiguity in interpreting both straight-line and curved plots, it would seem advisable to exercise caution in their interpretation. Care should be taken to obtain a sufficient number of experimental points at early and late times during the modification reaction (low and high XT) , SO as to be able to determine whether the reaction is best described by a curve or by a straight line. In either case, an attempt should be made to obtain corroborative evidence for the number of "essential" groups. In particular, when a straight line results, the maximum number of groups available for reaction (xM) may be greater than the actual number of "essential" groups (xl)m, and other types of experiments might be required to determine (xl) m. Finally, if one does obtain a curve, a fit should be made to the equation describing the most likely set of reactions, based on reasonable assumptions. The equation will contain rate constants and end-points for each type of reactive residue, and none of the information in the curve will be lost.

This work was supported by United States Public Health Service Grant AM17552.

L ITERATURE

BerghS, user, J. 1975. "A reactive arginine in adenylate kinase." Biochim_ biophys. Acta 397, 370-376.

CHEMICAL MODIFICATION OF ENZYMES 255

Foster, M. and J. H. Harrison. 1974. "'Selective chemical modification of arginine residues in mitochondrial malate dehydrogenase." Biochem. bioph)'s. Res. Commun. 58, 263-267.

Glazer, A. N., R. J. De Lange and D. S_ Sigman. 1975. Chemical Modification of Proteins: Selected Methods and Analytical Procedures. Amsterdam: North-Holland.

Kantrowitz, E. R. and W. N. Lipscomb. 1976. "An essential arginine residue at the active site of aspartate transcarbamylase." J. biol. Chem. 2ill, 2688-2695.

Levy, H. M., P. D. Leber and E. M. Ryan. 1963. "Inactivation of myosin by 2,4- dinitrophenol and protection by adenosine triphosphate and other phosphate com- pounds." J. biol. Chem_ 238, 3654-3659.

Levy, R. H., J. Ingulli and A. Afolayan. 1977. "Identification of essential arginme residues in glucose-6-phosphate dehydrogenase from Leuconostoc mesenteroides." J. biol. Chem. 252, 3745-3751.

Means, G. E. and R. E. Feeney. 1971. Chemical Modification of Proteins. San Francisco: Holden-Day.

Paterson, A. K. and J. R. Knowles. 1972. "The number of catalytically essential carboxyl groups in pepsin." Eur. J. Biochem. 31,510-517.

Rakitzis, E, T. 1977. "Kinetics of irreversible enzyme inhibition: co-operative effects." J. theor. Biol. 67, 49-59.

Rakitzis, E. T. 1978. "Kinetics of irreversible enzyme inhibition: the Tsou plot in irreversible binding co-operativity." J. theor. Biol. 70, 461-465.

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