The importance of intercalation in the covalent binding of benzo(a)pyrene diol epoxide to DNA

J. theor. Biol. (1990) 142, 113-122

The Importance of Intercalation in the Covalent Binding of Benzo(a)pyrene Diol Epoxide to DNA MICHAEL C. MACLEoD

Department of Carcinogenesis, Science Park-Research Division, University of Texas M.D. Anderson Cancer Center, Box 389, Smithville, T X 78957, U.S.A. (Received 15 May 1989, and accepted in revised form 18 August 1989) The primary mode of non-covalent interaction of the strong carcinogen, benzo(a)pyrene diol epoxide, with DNA is through intercalation. It has variously been suggested that intercalative complexes may be prerequisite for either covalent binding or DNA-catalysed hydrolysis of the epoxide or both. Geacintov [Geacintov, N. E. (1986). Carcinogenesis 7, 589.] has recently argued that intercalation is important in covalent binding and presented theoretical constructs consistent with this proposal. A more general theoretical model is presented here which includes the possibilities that either catalysis of hydrolysis or covalent binding of benzo(a)pyrene diol epoxide DNA can occur (a) in an intercalation complex, or (b) without formation of a detectable, physically bound complex. It is shown that a variety of possible mechanisms formulated under this general theory lead to equations for overall reaction rates and covalent binding fractions which are all of the same form with respect to DNA concentration dependence. A consequence of this is that experimental studies of the dependence of hydrolysis rates and covalent binding fractions on DNA concentration do not distinguish between the various possible mechanisms. These findings are discussed in relation to the interactions of benzo(a)pyrene diol epoxide with chromatin in cells. Introduction

The covalent interaction o f diol epoxides derived from carcinogenic polycyclic aromatic hydrocarbons with D N A in cells is thought to be important in the initiation of carcinogenesis. Because of this, much study has been devoted to the interactions of these diol epoxides with purified D N A as a model for the in vivo interactions. With 7r,8t-dihydroxy-9t,10t-oxy-7,8,9,10-tetrahydrobenzo(a)pyrene (BPDE-I) as the primary model c o m p o u n d , these studies have demonstrated three major classes of interaction (Abramovich et al., 1985; Geacintov, 1988; Geacintov et al., 1982; MacLeod & Selkirk, 1982; MacLeod et al., 1982; Michaud et al., 1983): physical binding to D N A via intercalation complexes, covalent binding to D N A and an enhanced rate of epoxide hydrolysis, catalysed by DNA. An obvious question is whether either of the latter two processes proceed by way of an intercalated intermediate. Several experimental findings suggest indirectly that catalysed hydrolysis may depend on intercalation. Thus, neither intercalation nor catalysed hydrolysis display enantioselectivity (Michaud et al., 1983; MacLeod & Zachary, 1985b), while covalent binding is highly enantioselective (Meehan & Straub, 1979). Inclusion of increasing concentrations of monovalent or divalent cations in reaction mixtures inhibits intercalation and catalysed hydrolysis with roughly equivalent ionic strength 113

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© 1990 Academic Press Limited

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dependencies (Geacintov et aL, 1982; MacLeod & Selkirk, 1982; Michaud et aL, 1983), while the extent of covalent binding is less affected (MacLeod et aL, 1982). However, in the latter case the inhibition of hydrolysis may directly affect the fraction of diol epoxide which is available for covalent binding, thus preventing a clear interpretation of the data. Early experimental studies demonstrated that the dependence of observed hydrolysis rates on DNA concentration resembled a typical binding isotherm and exhibited a limiting value at high DNA concentration (Geacintov et al., 1982). Similarly, the fraction of diol epoxide binding eovalently to DNA also reached a maximum value with increasing DNA concentration, amounting to 5-10% binding for (+)-BPDE-I (MacLeod et al., 1982). Geacintov (Geacintov et aL, 1982) showed that hydrolysis rate data could be fit by an equation of the form: kobs ~--

lCo+ a D I+K'D"

Here, kobs is the apparent first-order rate constant at the given concentration of DNA, D, ko is the first-order rate constant in the absence of DNA, and K' is approximated by the association constant for intercalative binding. This equation holds when the initial DNA concentration is much greater than the initial BPDE-I concentration, a condition which is easily attained experimentally. The interpretation of the constant or, whose value is determined by curve-fitting, depends on the model for the underlying processes. Several theoretical reaction schemes have been presented in an effort to understand these experimental results, especially with respect to the DNA concentration dependence of the two processes. The most thorough treatment has been provided by Geacintov (1986) and has been reviewed recently (Geacintov, 1988). Although it was concluded that several models, including the "'two-domain model" (Meehan & Bond, 1984) and the "common intermediate model" (Geacintov et aL, 1984), were consistent with the data, the results were interpreted to favor the idea that covalent binding is dependent on intercalation. However, several logical possibilities have not been treated in previous models. It is my purpose here to present a more general model for the interactions, of which the previous models represent special cases. An interesting finding which emerges from this exercise is that models in which neither covalent binding nor catalysed hydrolysis depend directly on intercalation predict a DNA concentration-dependence which is equivalent to the predictions of the previous models. Thus, experimental determinations of these parameters cannot be used to distinguish between the various possible models. General Model: Identical Sites

I will first present a simplified but general model for the interaction of an epoxide, E, with DNA, D, in aqueous solution in which the DNA is assumed to contain multiple identical, non-interacting sites. This will later be further generalized to encompass non-identical sites. As shown in Scheme I, it is postulated that a non-covalent complex, C, can form between E and D; the association constant for

INTERCALATION

AND

COVALENT

115

BINDING

SCHEME I ko

E

k E+D.

kT

°"'C

,T k I

~T

E+D

It 2

~T

E+D

,

A

kaa ] kc ~A

this process, Ka, is the ratio of the forward and backward rate constants, ko,/k~,. It has been well documented that in the case of diol epoxides and dihydrodiols derived from benzo(a)pyrene, the major physical complex formed is an intercalation complex (Geacintov, 1988), and the rate constants for this process are fast compared to the other rate constants involved (Geacintov et al., 1982; Meehan & Bond, 1984). The complex is postulated to give rise to products with rate constants kT (tetraols) and k c (covalent adducts). Tetraols are also formed in the DNA-independent hydrolysis reaction with rate constant ko. Geacintov's studies (Geacintov et al., 1984) have shown that inclusion in this model of an activated complex, C * , which is a precursor to both tetraols and adducts, does not alter the basic form of the final equations for the DNA concentration dependence of the overall reaction rate and product distribution. For simplicity we have left this step out. At this point, the model is similar to previous formulations. To obtain a more general model, we postulate two alternative reactions which do not involve measurable physical complexes between E and D but which do lead to the same spectrum of products. These are characterized by bimolecular reaction rate constants kl leading to tetraols and k2 leading to covalent adducts. Precedent for such reactions exists in the literature. Small molecules such as ellagic acid (Sayer et aL, 1982) and 2-mercaptoethanol (Michaud et al., 1983) form adducts with BPDE-I without the involvement of experimentally demonstrable physical complexes, and catalysis of hydrolysis by buffer ions (e.g. phosphate and Tris) probably doesn't involve physical complexes. Following Scheme I, the rate equations for the various species can be written down by inspection: dE

- - = - ( k o + k i D + k 2 D + k o , D ) E + koffC dt

(1)

dC - - = ko, D E - (kof~+ kT + k c ) C dt

(2)

dT dt dA dt

- koE + k ~ D E + k T C

(3)

-- k 2 D E + kcC.

(4)

Following the methods outlined by Geacintov et al. (1982), we assume that kon, ko, >> an assumption for which there is experimental support (Geacintov et al., 1982; Meehan & Bond, 1984), and that r, the ratio of total epoxide molecules ko, k h k2, k c , kT,

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to potential binding sites, is small so that the DNA concentration, D, is constant. A solution for eqns (1) and (2) is: E ( t) = E ( + O) e -ko~J where E ( + 0) is the free epoxide concentration at a time soon after mixing such that physical complex formation has taken place but there has been no appreciable product formation. If E r is the total epoxide added to the reaction mixture, then E (+ 0) can be expressed in terms of the association constant for intercalative binding, K~, giving E ( + O) = - -ET I + KaD

and

E(t)

Er I + K~D

-

e

_kobs t

(5)

As detailed by Geacintov et al. (1982), the methods discussed by Bernasconi (1976) with the assumptions given above lead to the following expression for kobs" kobs --

ko+ (kl + kz)D + (kc + k r ) K ~ D 1 + K~D

(6)

The use of K~ in this expression is an approximation in which certain rate constants are assumed to be negligible. As argued by Geacintov, the fact that experimentally the data are well fit by eqn (6) with the use of an independently determined value for K~ provides justification for these assumptions (Geacintov et al., 1982, Geacintov, 1986). Substitution of (5) into (2), (3), and (4) allows solutions to be found for these equations also: ErKaD e_kob~t I+KoD

(7)

T(t)-

(ko+kiD+krK~D)Er (1 - e -ko.~') ko+ (k; + k2)D + (kc + k r ) K ~ D

(8)

A(t) -

(k2D+kcK~D)ET (l_e_kob ,)" ko+ (k~ + k2)D + (kc + k r ) K ~ D

(9)

C(t)

As the reaction approaches completion (viz as t goes to co), e -kobJ goes to zero and eqn (9) can be arranged to give J~°v=

A ( t = oo) k2D + kcKaD E---~-ko+(kl+k2)D+(kc+kr)KaD"

(10)

f~ov is the fraction of diol epoxide added to the reaction which ultimately becomes covalently bound, and hence is a directly measureable parameter. Similarly, kob~, given by eqn (6), is experimentally measurable. The general form of (6) is kob~=

ko+ a D 1 + K~D"

(11)

where a = kt + k 2 + ( k c + k-r)K,. Similarly, the general form of (10) is ~D Ao~ - - ko+ a D '

(12)

INTERCALATION

AND

COVALENT

117

BINDING

where B = k2 + kcKa. Several laboratories have documented conditions under which hydrolysis rate data is fit well by eqn (11) (Geacintov et al., 1982; MacLeod et al., 1989), and we have found that eqn (12) adequately describes experimental measurements of covalent binding (MacLeod & Tang, 1985; MacLeod & Zachary, 1985b). It is clear from this analysis that the question of whether or not covalent adduct formation is dependent on intercalation depends on the relative values of the constants k2 and K~kc. Similarly, the relative values of k~ and Kakr determine the importance of intercalation in DNA-catalysed hydrolysis. It is instructive to consider several limiting cases: C A S E A. N E I T H E R

HYDROLYSIS

BY T H E

NOR

ADDUCT

NON-INTERCALATED

FORMATION

OCCUR

PATHWAY

This is equivalent to setting kl = k2 = 0, and corresponds to the original formulation of Geacintov (Geacintov et al., 1982). Use of the activated complex formulation in the current model would result in equivalence to Geacintov's more recent, common intermediate model (Geacintov, 1986). Equations (6) and (10) above in this case become analogous to eqns (2) and (4) in Geacintov (1986). CASE

B. N E I T H E R

HYDROLYSIS

BY T H E

NOR

ADDUCT

INTERCALATED

FORMATION

OCCUR

PATHWAY

This is equivalent to setting kc = kT = 0, that is, postulating that the intercalation complex is "non-productive." In this case, the dependence of kobs on the factor 1 / ( I + K a D ) remains and naturally leads to a plateau for kobs. This is now due to an increasing "protection" o f epoxides from hydrolysis as a larger fraction of the potential reactants become intercalated at higher D N A concentrations. This model has been presented previously in preliminary form (MacLeod & Zachary, 1984). These are the limiting cases; obviously, mixed models in which one process depends on intercalation and the other process does not are also possible. The two-domain model of Meehan (k2 = kr = 0; Meehan & Bond, 1984) is one example of this kind of model; others are logically possible. The fact that the limiting value o f f , or at high D N A concentration is much less than one, rules out models in which kl = kr = 0, since this would lead to f~ov= 1 in the limit of high DNA concentration. However, as long as at least one of the rate constants for each process is non-zero, the corresponding version of eqns (6) and (10) will have a set of constants which fits the experimental data. Thus, no model makes a unique prediction of DNA concentration dependence, and consequently such data cannot prove or disprove any particular model.

General Model: Multiple Sites It is clear that the assumption of a single class of identical, non-interacting D N A sites is an over-simplification. For example, there are ten possible dinucleotide sites in each groove of the double helix, each of which may have different abilities to interact with diol epoxides. Both catalysis of hydrolysis and covalent binding fractions depend strongly on base composition (MacLeod & Zachary, 1985a);

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intercalation has also been reported to depend on base composition (Chen, 1984; Geacintov, 1988). Indeed, the level of covalent binding of BPDE-I to particular deoxyguanosine residues in defined DNA sequences is strongly sequence dependent (Boles & Hogan, 1986; Kootstra et al., 1989). Thus, a more complete model would postulate n types of interaction sites in DNA, each with its own set of rate constants for each of the processes; this is diagrammed in Scheme II. The concentration of each type of site is indicated by D , while its corresponding complex is C~. The total DNA concentration, D, is assumed to be the sum of the concentrations of the individual sites, D = ~-~=1 D , the total complex concentration, C, is the sum of the n individual complexes, C =Y.i= 1 C, and again experimental conditions are chosen so that the value of r is small enough that for each class of site, D~ is constant. We maintain the restriction that the sites are non-interacting. If the association constant for the ith site is Ko~, then the overall association constant, Kapp-Y." Ko~. By . -i=l • inspection, the differential equations describing the system are: dE E kl,iD i+ _ k2.iDi+ _ kon iJ~i "-[- - k o f f . i C i dt

~-i

dCt dt-

i=l

i=l

"

i=l

ko..iD, E - (koff.l+ kr., + kc.l) C,

dC2 d t - k~o.~D~E - ( k~.~ + k~,~ + k~.~) C:

dC. dt - k o . . . D . E - ( ko~.,, + kT,,, + kc.,,) C .

dA dt

~ k2,D,E+ T. kc,Ci. " i=l "

(13)

i=,

SCHEME

II

ko kan E + D I .

I

E + D2.

kEI

"" C~

koa.t

E T

E + D,

,T

k,., T

k2,1

E+D

,A

I I kc'z • A

k~.2 kT,2I " C2 T k°~'2 I kC'2

kl~ E+D

2

, T

E+D:

k,, "", A

E+D,,

k, "", A

k c,2 ~A

E+D..

' ' C. k°a'"

]

, T

k¢,n ,A

E+D,,

k ~,,, • T

INTERCALATION

AND

COVALENT

BINDING

119

We assume a solution to this set of linear differential equations of the form: E-

Er

e_ko,s,

(14)

1 + KappD0 c, =

ErKa•iDi e 1+ KappDo

k b t

o~.

(15)

As detailed in the Appendix, this leads to the following expressions:

ko+ ~ k,.,D, + ~ kT.iKa.,D, kobs--

i=1

i=1

1 + KappD0

(16)

( k2.,+ kc.,K,.,) D, i=1

f~o,,=

(17)

ko+ ~ (k,.,+kr.iK..,)D,+ ~ (~.i+kc.,K..i)Di i=1

i=1

Intuitively, each of the terms in the denominator of eqn (17) can be thought of as being proportional to the flux of BPDE-I molecules along each of the pathways diagrammed in Scheme II. The numerator contains those fluxes which lead to covalent adducts, and the ratio of these fluxes should give the final fraction of BPDE-I bound covalently to the DNA. Again, the equations describing the model are of the form specified by eqns (11) and (12), but with the addition of the appropriate terms in the definitions of a and/3. Discussion

Previous theoretical studies of these interactions have included the tacit assumption that the irreversible reactions between diol epoxides and DNA, covalent binding and catalysed hydrolysis, proceed from reversibly bound complexes, either intercalation complexes or possible "outside-bound" complexes. Precedent for the involvement of physical complexes in the catalysis of BPDE-I-hydrolysis exists in the interaction of riboflavin-5'-phosphate with BPDE-I (Wood et al., 1982). The present results demonstrate that bimolecular reactions between free epoxide molecules and reaction centers on the surface of the DNA molecule could be involved in these processes without changing the form of the equations for DNA concentration dependence of the experimentally observed quantities, f~ov and /Cobs- ThUS, other types of measurements will be needed to determine which set of reaction pathways is the more relevant. Attempts to decide this question by altering reaction conditions (ionic strength, pH, solvent polarity, DNA base composition) will probably not be successful because rate constants in each of the postulated pathways are likely to be affected, especially since in each of the reactions the activated intermediate is likely to be a triol carbonium ion. This "kinetic ambiguity" is a general finding of transition state theory, and the inability of kinetic measurements to uniquely determine pathways has been stated forcefully by Jencks (1969: 186-187).

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Nevertheless, intercalation is clearly an important factor in the overall reactions with purified DNA. The saturation of kobs with increasing [ D N A ] is due to intercalation whether the complexes are " p r o d u c t i v e " or not, and the saturation offcov with increasing [ D N A ] depends directly on the concentration-dependent hydrolysis term. In the nuclei of eukaryotic cells, D N A does not exist as free molecules but as complexes with nuclear proteins. The most abundant nuclear proteins are the histones which form a well defined primary complex with D N A , the nucleosome, which represents the basic repeating unit of chromatin ( M c G h e e & Felsenfeld, 1980). We have shown that the interaction of BPDE-I with two such complexes, either long-soluble chromatin (Dock & MacLeod, 1986) or mononucleosomes (MacLeod et al., 1989) is quantitatively very different from the interaction with purified D N A . Thus, the Ka's for intercalation are decreased by factors o f 20-30 in the p r o t e i n - D N A complexes, and there is an equivalent decrease in the concentration d e p e n d e n c e o f hydrolysis: the constant a in eqn (11) is decreased by approximately the same factor. However, there is very little decrease in the value of fcov at high D N A concentration, although the kinetics are appreciably slower. It is easily seen from eqn (12) that the limiting value of f~ov at high D N A concentration is simply r ~ a. T h e experimental results imply that the value o f f l must also be decreased by a factor of 20-30 in nucleosomal complexes. Since fl is given by k2 + kcK~, the decrease in fl could be due to the decrease in K~, which we have measured, or to a decrease in k2. The latter possibility is not unlikely, since much of the D N A is interacting directly with the histones. Thus, the experimental data again are insufficient to distinguish between the various models. Although there does exist non-nucleosomal D N A in nuclei, the current results do not suggest that possible differences in association constants would lead directly to differences in covalent binding, which of course could be biologically very significant. However, in contrast to the reaction conditions obtained in our in vitro model systems, nuclei are quite inhomogeneous spatially, and the dynamics of carcinogen interaction will almost certainly depend on spatial inhomogeneities in BPDE-I concentration in different regions of the nucleus. Regions with higher levels of intercalation may act as sinks for BPDE-I, thus increasing the local concentration. I thank P. LeBreton for helpful comments on the manuscript, N. Geacintov for communicating results prior to publication, and J. Mayhugh for expert secretarial assistance. This work has been supported by grant CA-35581 from the National Cancer Institute. REFERENCES ABRAMOVICH,M., PRAKASH,A. S., HARVEY, R. G., ZEGAR, I. S. & LEBRETON, P. R. (1985). A comparison of the intercalative binding of non-reactive benzo(a)pyrene metabolites and metabolite model compounds to DNA. Chem.-Biol. Interactions 55, 39-62. BERNASCONI,C. F. (1976). Relaxation Kinetics. New York: Academic Press. BOLES, T. C. & HOGAN, M. E. (1986). High-resolution mapping of carcinogen binding sites on DNA. Biochemistry 25, 3039-3043. CHEN, F.-M. (1984). Sequence specific binding of tetraols of benzo(a)pyrene diol epoxide to DNA in neutral and acidic solutions. Carcinogenesis 5, 753-758. DOCK, L. & MACLEOD, M. C. (1986). Interaction of (+)-7r,8t-dihydroxy-9t,10t-oxy-7,8,9,10-tetrahydrobenzo(a)pyrene with purified rat liver chromatin. Carcinogenesis 7, 589-594.

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GEACINTOV, N. E. (1986). Is intercalation a critical factor in the covalent binding of mutagenic and tumorigenic polycyclic aromatic diol epoxides to DNA. Carcinogenesis 7, 759-766. GEACINTOV, N. E. (1988). Mechanisms of reaction of polycyclic aromatic epoxide derivatives with nucleic acids. In: Polycyclic Aromatic Hydrocarbon Carcinogenesis: Structure-Activity Relationships. (Yang, S. K. & Sitverman, B. D., eds) Vol If, pp. 181-206. Boca Raton: CRC Press. GEACINTOV, N. E., HIBSHOOSH, H., IBANEZ, V., BENJAMIN, M. J. & HARVEY, R. G. (1984). Mechanisms of reaction of benzo(a)pyrene-7,8-diol-9,10-epoxide with DNA in aqueous solutions. Biophys. Chem. 20, 121-133. GEACINTOV, N. E., YOSH1DA, H., IBANEZ, V. & HARVEY, R. G. (1982). Noncovalent binding of 7fl,8ot-dihydroxy-9a, lOa-epoxytetrahydro-benzo(a)pyrene to deoxyribonucleic acid and its catalytic effect on the hydrolysis of diol epoxide to tetrol. Biochemistry 21, 1864-1869. JENCKS, W. P. (1969). Catalysis in Chemistry and Enzymology, New York: McGraw-Hill. KOOTSTRA, A., LEW, L. K., NAIRN, R. S. & MACLEOD, M. C. (1989). Preferential modification of GC boxes by benzo(a)pyi'ene-7,8-diol-9,10-epoxide. Molec. Carcinogen. 1, 239-244. MACLEOD, M. C., MANSFIELD, B. K. & SELKIRK, J. K. (1982). Covalent binding of isomeric benzo(a)pyrene diol-epoxides to DNA. Carcinogenesis 3, 1031-1037. MACLEOD, M. C. & SELKIRK, J. K. (1982). Physical interactions of isomeric benzo(a)pyrene diolepoxides with DNA. Carcinogenesis 3, 287-292. MACLEOD, M. C., SMITH, B. & LEW, L. K. (1989). Interaction of an ultimate carcinogen, benzo(a)pyrene diol epoxide, with nucleosomal core particles: apparent lack of protection of DNA by histone proteins. Molec. Carcinogen. l, 245-252. MACLEOD, M. C. & TANG, M.-S. (1985). Interactions of benzo(a)pyrene diol-epoxides with linear and supercoiled DNA. Cancer Res. 45, 51-56. MACLEOD, M. C. & ZACHARY, K. (1984). DNA-catalysed detoxification of carcinogenic and noncarcinogenic diol epoxides of benzo(a)pyrene. Proc. Am. assoc. Cancer Res. 25, 84. MACLEOD, M. C. & ZACHARY, K. ( 1985a ). Involvement of the exocyclic amino group of deoxyguanosine in DNA-catalysed carcinogen detoxification. Carcinogenesis 6, 147-149. MACLEOD, M. C. & ZACHARY, K. (1985b). Catalysis of carcinogen-detoxification by DNA: Comparison of enantiomeric diol epoxides. Chem.-Biol. Interactions 54, 45-55. MCGHEE, J. D. & FELSENFELD, G. (1980). Nucleosome structure. Annu. Reo. Biochem. 49, 1115-1156. MEEHAN, T. & BOND, D. M. (1984). Hydrolysis of benzo(a)pyrene diol epoxide and its covalent binding to DNA proceed through similar rate-determining steps. Proc. hath. Acad. Sci. U.S.A. 81, 2635-2639. MEEHAN, T. & STRAUB, K. (1979). Double-stranded DNA stereoselectively binds benzo(a)pyrene diol epoxides. Nature, Lond. 277, 410-412. MICHAUD, D. P., GUPTA, S. C., WHALEN, D. L., SAYER, J. M. & JERINA, D. M. (1983). Effects of pH and salt concentration of the hydrolysis of a benzo(a)pyrene 7,8-diol-9,10-epoxide catalyzed by DNA and polyadenylic acid. Chem.-Biol. Interactions 44, 41-52. SAYER, J. M., YAGI, H., WOOD, A. W., CONNEY, A. H. & JERINA, D. M. (1982). Extremely facile reaction between the ultimate carcinogen benzo(a)pyrene-7,8-diol-9,10-epoxide and ellagic acid. J. Am. chem. Soc. 104, 5562-5564. WOOD, A. W., SAYER, J. M., NEWMARK, H. L., YAGI, H., MICHAUD, D. P., JERINA, D. M. & CONNEY, A. H. (1982). Mechanism of the inhibition of mutagenicity of a benzo(a)pyrene 7,8-diol-9,10-epoxide by riboflavin Y-phosphate. Proc. nam. Acad. Sci. U.S.A. 79, 5122-5126. APPENDIX Differentiation of the postulated solutions [eqns (14) and (15)] yields the following expressions for the derivatives of the concentration variables: dE d-t-=-

k°bsE

dC, dt

-

kobsC,

If the postulated solutions are correct solutions, each of these derivatives can then be equated to the corresponding expression for the derivative obtained from the

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original set of differential equations:

- kobsC, = kon.,D,E - (konj + kr., + kc.,) C, The entire set of equations can then be summed and both sides of the resulting equation divided by (-)E: k r.,C,

~ kc.,C,

Substitution o f K~.~Di for C J E and KappDo for ~7= 1 K~jD~ yields the text expression (16) for kobs. To obtain the expression for f~o,, the postulated solutions can be substituted into (13), the differential equation for dA/dt:

O A _ ~ k2.,D,E + ~. kc.,C, dt i=l i~l

(13)

dA dt This expression can be integrated, giving:

--

A-

l+KappDo

~. k2 iDi + _ kc.iK.,iDi

i=l

'

kobs

i=l

(1 - e - k°b,')

Upon substitution of (16) for kobs, rearrangement, and taking the limit as t goes to oo this becomes (17).