J. Biochem. 83, 385-394 (1978)
Rate Equation for Amylase-Catalyzed Hydrolysis, Transglycosylation and Condensation of Linear Oligosaccharides and Amylose Ryuichi MATSUNO, Toshihiko SUGANUMA, Haruko FUJIMORI, Kazuhiro NAKANISHI, Keitaro HIROMI, and Tadashi KAMIKUBO Department of Food Science and Technology, Faculty of Agriculture, Kyoto University, Sakyo-ku, Kyoto, Kyoto 606 Received for publication, August 22, 1977
In order to analyze complex experimental results of amylase-catalyzed reactions, which include transglycosylation and condensation as well as hydrolysis, with amylose and linear oligosaccharides as substrates, a general equation governing the concentrations of substrate and product species in a reaction mixture is derived based on the subsite theory. The equation is devised so as to follow the location in products of radioactive glucose which was originally located at a certain position in a substrate. The reaction scheme assumed is as follows: 1) Enzyme can bind any substrate reversibly to form an enzyme-substrate complex, ES. 2) From the ES complex, a glucosyl bond or C^OH bond at the reducing end of the substrate is cleaved to form a reactive intermediate, releasing a product of shorter chain or HSO. 3) The reactive intermediate binds an acceptor molecule reversibly to form a ternary complex from which the product of transglycosylation or condensation is produced. The binding constants between enzyme and substrate, and between reactive intermediate and acceptor can be estimated from the subsite affinities. The rate constants of catalytic processes such as cleavage of a glucosyl bond and C'-OH bond, hydrolysis and transglycosylation are assumed to be independent of the degree of polymerization as well as of the binding mode of the substrate. The applicability of this scheme and rate equation was tested successfully with the Takaamylase A-catalyzed reaction of maltohexaose labeled with 14C at the reducing end.
The subsite theory (1-6) proposed for the hydrolysis of linear oligosaccharides and amylose catalyzed by amylases successfully accounts for the dependence of the Michaelis constant and molecular activity on the degree of polymerization. Another important property described by this theory is the action pattern of amylases for linear substrates, i.e., it predicts not only the ratio of possible binding modes between substrate and enzyme but also the probabilities of various modes of Vol. 83, No. 2, 1978
cleavage of the substrate. By the nature of the theory, it is applicable to the analysis of product distribution resulting from the hydrolysis of a given linear substrate by amylases (Matsuno, R., et ai, unpublished results.). Conversely, it is possible to estimate the relative values of subsite affinities from the analysis of products formed from reducing endlabeled substrates (3-6). Recent experimental findings (6-14) have suggested that some amylases possess activities of 385
386
R. MATSUNO et al.
transglycosylation and/or condensation as well as hydrolysis. For instance, in the case of the reaction of Taka-amylase A from Asp. oryzae with maltotriose labeled with "C at the reducing end (12), the ratio of non-labeled glucose to maltose as well as that of labeled glucose to maltose in the reaction mixture showed a marked dependency on the initial substrate concentration. Further, in the case of the reaction of maltose catalyzed by saccharifying a-amylase from Bacillus subtilis (13,14), a clear lag phase for glucose formation appeared at high maltose concentration, and a strong sigmoidal dependence of the reaction rate on substrate concentration (77,13,14) were observed. Since maltooligosaccharides longer than the substrate were also detected, it was concluded that these results were a consequence of a complex reaction mechanism involving transglycosylation and/or condensation as well as hydrolysis. In order to analyze quantitatively the results mentioned above, a general rate equation for amylase-catalyzed reactions of linear substrates, including hydrolysis, transglycosylation and condensation, is derived in this paper. The equation is generally applicable to any kind of glycosyl hydrolase which catalyzes hydrolysis, transglycosylation and condensation, including amylase and lysozyme; for the latter, similar treatments have been reported by Chipman (75) and Tada and Kakitani (16). By solving the equation, not only can concentrations of substrate and products be obtained as functions of time, but also the location in any product species of labeled glucose which is originally located at a certain position in the initial substrate can be identified. The latter can be accomplished by introducing a special method for naming molecules, together with a law of composition or combination.
quence of residues, even for the same degree of polymerization. Thus it is necessary to define a system for naming a molecule having a particular degree of polymerization and sequence. Naming of Molecules—Let us consider a linear heteropolymer composed of Nc kinds of residues and having a degree of polymerization n. Each kind of residue is distinguished by an "identification number" ranging from zero to Nt — \. The numbers are arranged linearly according to the sequence of the residues in the molecule. This gives an n-figure number, /, in the number system with a radix of Nc. Then, / is decimalized. By this procedure, the heteropolymer can be represented by the numbers n and /, as S(n, I). Example: Oligosaccharides consisting of nonlabeled glucose G(0) and labeled glucose G*(l), Nc=2. G-G*-G*-G-G-G* >S(/7,/)=S(6,011001), / i n binary system: > = S ( 6 , 25), / in decimal system.
Law of Composition (Combination) —In order to judge the species of product produced by a reaction, a law of composition is introduced. a) Condensation between S(/jlF /J and S(n2, k) in this order: Sfo, /,) S(nt, lt)=S(n1+nt,
I, X #,»• + /,)
( 1)
b) Cleavage of S(w, /) between /^th and nt +1 th residues:
where /j and lt must satisfy l=l1xNc"-"i+li. It is the integer part of ljNcn-"i and I1=1—I1XNC"-"K Example: Condensation between G*-G-G* and G-G*. — G*-G-G*-G-G*
METHODS
S(3, 5) S(2, ])=S(3+2, 5x2 I + l)=S(5, 21)
A linear polymer composed of many kinds of residues is considered. For instance, in the case of oligosaccharides consisting of labeled and nonlabeled glucose, the number of kinds of residues is two. When such a polymer is subjected to hydrolysis, transfer and condensation, various products are formed which differ in the degree of polymerization and also in composition and/or se-
Note that these methods can be applied to the case of any linear heteropolymer, such as DNA (Nc=4), oligopeptides (/V«=20), etc. Binding Constant between Substrate and Enzyme —The binding constant between substrate and enzyme is obtained by means of the subsite theory (7, 2). The assumption introduced in the theory is that the active site of an enzyme consists of several /. Biochem.
RATE EQUATION FOR AMYLASE REACTION
independent subsites, the subsite affinities of which are simply additive, and the binding constant can be estimated from the sum of the subsite affinities of the subsites which are occupied in the relevant binding mode. In the case of a heteropolymer composed of Nc kinds of residues, each subsite has Nc different subsite affinities corresponding to each kind of Nc residue. Thus, the binding constant of a heteropolymer S(/z, I) is, in general, determined when n, I and the binding mode are given. However, in the case of a linear homopolymer such as amylose or maltooligosaccharide, even if it contains labeled glucose, each subsite has only one subsite affinity. In this case, only n and the binding mode need be specified for the determination of the binding constant. The discussion will now be concentrated only on amylase with linear maltooligosaccharides as substrates. Figure 1 shows the active site of an amylase consisting of m subsites numbered from 1 to m. The catalytic site is located between the rth and r + l t h subsites. When a linear oligosaccharide with degree of polymerization n is bound to the amylase as shown in Fig. I, i.e., the reducing end binds to the /th subsite, the binding mode is specified by the number j and the binding constant is represented by K(n,j), regardless of productive and non-productive binding modes. In the case of binding between subsites and a substrate without participation of the reducing end residue, we add imaginary subsites numbered from m+1 next to the mth subsite. The binding mode is then again specified by the subsite
387
number j of the imaginary subsite at which the reducing end is situated. In this case, j exceeds m. Let At be the subsite affinity of /th subsite, then K(n,j) is given by the following equation, using the subsite theory (1, 2). A:(n,;)=0.018exp (SA.,/RT)
(3)
where SA.j is the sum of subsite affinities of the subsites occupied, and is given by, j= Y, ZVi-ZW(-Ai where (4)
l , ujt=j—i In Appendix I, the proof of Eq. 4 is shown. In the case of an exo-enzyme such as glucoamylase and /)-amylase, the subsite affinity of the first subsite is taken as - c o . Reaction Scheme—Figure 2 shows the scheme assumed for amylase-catalyzed reactions which include hydrolysis, transglycosylation and condensation of maltooligosaccharides and amylose {12-14). The enzyme binds any x-mer substrate ST reversibly to form an enzyme-substrate complex ESi in any possible binding mode. A ternary enzyme-substrate complex in which two substrate molecules are bound to the enzyme (ESXSW) is also taken into account. From a productive binding constant K(n,j) complex, a glucosyl bond is cleaved with a rate constant k, to form a reactive intermediate E'Si, G—-G-G—-G-— which may presumably be an "enzyme-carbonium ion intermediate" complex (17), releasing a product A. Sx-X. As a first approximation, we adopt the A j m simple assumption that ki is constant irrespective Fig. 1. Schematic representation of the active site of the degree of polymerization as well as of the structure of an amylase and binding mode of a linear binding mode of the linear substrate (1, 2).1 substrate. The subsite is numbered by i (from 1 to m) integers counting from the non-reducing end side (left) 1 This assumption is supported by chemical evidence and Ai is the corresponding subsite affinity. The wedge between the rth and r+lth subsites signifies the that the rate of acid-catalyzed hydrolysis of a-1,4 linkages is independent of the degree of catalytic site. The binding constant of n-mer oligo- glucosidic polymerization of the substrate. For enzyme-catalyzed saccharide, the reducing end of which is bound to the reactions, the assumption is equivalent to considering /th subsite, is represented by K(n,f). j can exceed m. that the subsites are independent of each other. This point has been discussed by several authors (3-6). For details, see the text.
—-8
Vol. 83, No. 2, 1978
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R. MATSUNO et al.
be classified into two types, A and B, according to whether the r + l t h subsite is occupied or not occupied by non-reducing end residue of ST. Then, E'SX and E'S,Sy of B type react with water to form S t with a rate constant kH through hydrolysis. From E'SiSy of A type, S J+y is Sx-z or HjO produced with a rate constant kT through transglycosylation. \ X G-Gk From a kind of non-productive complex, ESX, in which the rth subsite is occupied by the reducing end residue of Sx, a reactive intermediate E'SX is formed with a rate constant ke releasing an H,O molecule. Then, E'SX and E'S x S y of B type release Sx through hydrolysis, and E'S x S y of A type releases S x+y through condensation. In this case, z is equal to x and S x _ t is H 2 O. G-G- G-G-G G-G-G-.G-G-G The rate equation is derived from this scheme I I \r\ I I I under the following assumptions; Fig. 2. Scheme for amylase-catalyzed reactions which 1) Rapid equilibria between E and ESX and include hydrolysis, transglycosylation and condensation of maltooligosaccharides and amylose. E and S repre- between E'S, and E'S,Sy. 2) Steady state for reaction intermediates, sent enzyme and substrate, respectively. ESX, enzymesubstrate complex; E'Si, reactive intermediate; E'SiSy, i.e., the rate of formation of reaction intermediate ternary complex formed between E'Si and acceptor equals that of its breakdown. S y ; x, y, and z, degree of polymerization of substrate. 3) The catalytic rate constants k,, kc, kH, For condensation, z equals x. ki, k,, kH, and kr are and k are independent of the degree of polyT rate constants of cleavage of glucosyl and C'-OH bonds, merization as well as of the binding mode of of hydrolysis and of transglycosylation, respectively. oligosaccharides. K(.x,Ji) a n d K(y,ji) are binding constants between enzyme and x- and .y-mer substrates in binding modes specified by j \ and j t , respectively, where j \ and j \ are the number of the subsite at which the reducing end of each substrate is situated. G- signifies a glucose residue in the form of reactive intermediate.
E'Sj binds any y-rner substrate Sy reversibly to form a ternary complex E'S t S y in any possible binding mode. The ternary complex E'S,Sy can dC(n.l) Eodt
„ rr.
„ ,„
RESULTS AND DISCUSSION Under the reaction scheme and assumptions described in the previous section, the rate equations governing the concentrations of substrate and product species consisting of labeled and nonlabeled glucose in a reaction mixture can be derived as Eqs. 5 to 23.
r
(5)
where, C.=
(6)
n,[) = C(n. /)/(! +EoD(n)IF) 2
j-i
y-i
(7) -%- £ C,'*£1K(x.j)L(.x-j+r)R(x,j) kit
r-i
-rp-Z Ka
y-r+i
C,K(x, r)R(x, r)L(x)
(8)
i-l
J. Biochem.
RATE EQUATION FOR AMYLASE REACTION
389
F*_ = C(n./) * + F #('/,;) R(n,j)
(9 )
Fk+ = E [K(x,n+r)G(x,n,l)R(x,n+r)+K(x,x+r-n)H(x,x-n,l)R(.x,x+r-n)P(n)]
(10)
J-.+I
L
|
'
'
E
M « E C,*;(j,rrtr)^j-A-+f) i-l
(11)
J-«+l
|
E H(y,y-x, v)K(y,y-x+r) R(y,y-x+r)
where v and u must satisfy / = N c " ~ ' x v + u .
v: the integer part of IINC'~*. n:
(12)
=I—Nc"~*xv.
Fc- = - ^ C(n, I) [K{n, r+n) "f C, K(x, r) R(x, r) M(x) +K(n, r) M(n) R(n, r) "t" C, K(y, r+y)] (13) , - \
I-I
KB
Fc+ = ^-"zC(n-x,u)K(n-x,n-x+r)C(x, KH
v) K{x,r) R(x,r) M{x)
(14)
X-1
where v is the integer part of I/Nc"~' and u equals l—Nc"~* x v. D{n)=+Y,l±-K(n,j)[\ +R(n,j) + C(n, l)J{n, n,j)]+-^-[ E C, K(x, r) R(x, r)M(x) I{x, ri) N
fctK(x,r)R(x,r)M(x)]
+ -^-[ £ C.'*'£ K(x,j) R(x,j) KH
x-l
M{x) E C, K(y,y-x+r) R(y,y-x+r)]
G(x,y,v)=
£,~lC(x,Nc'Xw+v)
M(x-j+r)I(x,n)
t-r+l
(15)
(16)
H(x, v, v) = "f" C(x, ^ X i- + it-)
(17) (18)
J(x.y.h)=
m+ l
£ Kb>J)+Z'K(yJ)
(19)
E C,/(xj)+E'c,^,r+^)]/WW
(20)
E C,/(jr,y)+-^L*E*C,A:(^r+^l ^
i
(21) J (22)
,j) = \+£c,J(x,y,j) Vol. 83, No. 2, 1978
(23)
390
R. MATSUNO et at.
Eo and C(n,f) are the total concentrations (free plus bound) of enzyme and that of a species S(/J,/)> respectively. C(n,f) is the concentration of free S(n,l), t is the time, and N is the maximum length of oligosaccharide to be considered. By definition, Nc is 1 when no species contains labeled glucose, while N, is 2 when it is present. The details of derivation and physical significance of equations are given in Appendix II. In Eq. 5, the terms Fk-—Fi,+, F,-—F,+, and Fc-—Fc+ represent the contributions of hydrolysis, transglycosylation, and condensation, respectively. Since C(n,l) is not exactly equal to C(n,l), the calculation is rather complicated. Practically, however, the enzyme concentration Eo is usually low enough for EoD(n)/ JFglucan glucanohydrolase, Aspergillus oryzae) with maltohexaose labeled with 14C at the reducing end, S(6,l), is examined. It was found from a study of the reaction of maltotriose (12) that Taka-amylase A had pronounced transglycosylation activity and low condensation activity as well as hydrolysis activity. Condensation activity is not taken into consideration in the present example (kc=0) because the effect of condensation in the reaction of maltohexaose is not very important in view of the subsite structure of Taka-amylase A. The experimental results in Fig. 3(a) show the dependence of distribution of end-labeled products on the initial
Fig. 3. Product distribution in the reaction of Takaamylase A with maltohexaose labeled by " C at the reducing end, S(6, 1). Reaction products were separated by paper chromatography and subjected to measurement of the radioactivities. Details of the experimental techniques and conditions are given in reference (72). Parameters used for theoretical calculation are as follows. Subsite affinities Al to At: 0.2, 0.4, 4.9, -3.7, 0.0, 3.3, 1.2, -0.7, and 0.8 kcal/mol. Ar/ = 12,000 1/min, £ c =0.0, kijkH E + S ,+ y
kH
Non-productive complex ESw and ESwS». Reactive intermediate from productive com-
(II. 6)
E+S. ;==± ESw
. C'C
. ; g
(ii. 4) _I_
U
P
(n. 5)
(, V
rcc
i
/ rE'S
Swl+itrlE'S,S D/^fE'Sx]1) =1 +K(w,j3)C. +{kTlkH)K(y,ti C,= MM{i) (EL 19) / . Biochem.
RATE EQUATION FOR AMYLASE REACTION R{x,jd and M(z) in Eqs. II. 18 and II. 19 correspond to R(x,j) and M(x) in Eqs. 23 and 21, respectively, and their physical significance is indicated by Eqs. II. 18 and II. 19. J(x,y,j) in Eq. 23 is given by Eq. 19 as J(x. y, h). J(x, y, h) represents the sum of the binding constants of possible binding modes of other substrates with degree of polymerization y when a substrate with degree of polymerization x is bound to the enzyme in binding mode h. I(x, y) in Eq. 21 is related to J(x, y, h) by the relation I(x, y)=J(x, y, r)— K(y, y+r) and is shown in Eq. 18. Equation n . 16 can be converted to Eq. II. 20, [E'S,]l\E]={kI/kB)K(x,j1)C,R(xJ1)M(z)
(II. 20)
In the same way, Eq. II. 21 can be derived from Eq. II. 15. =(k.lkH)K(x', h')C,.R(x', h (H.21) The relation between the concentration of total enzyme and that of free enzyme is obtained from Eq. II. 1. Writing =
([E'SJ+[E'S,SW]+[E'S,SJD/[E'SJ l+K(w,j3)C.+K(y,jt)CT
as L(z)/M(z), the ratio of total to free enzyme concentrations is given by,
[E]XL(z)/M(z))+([E'Sx-]/ [E])(L(XyM(x'))=F
(11.22)
where the precise form of L(z) is given by Eq. 20. Equation II. 22 corresponds to F in Eq. 8. Then, the reaction rates are given by the following equations. Rate of cleavage of productive complex: Ar;([ESx]+[ESxSw])=
, h)C.R(x. J (11.23)
Rate of hydrolysis: MIE'SJ -HE'S.S,,]) =M[E'S x(0E'S1]+[E'SIS,])/TJE'SJ) =k,K(x, h) C,R(.x, h) P(z)EjF Vol. 83, No. 2, 1978
393
where ([E'S,]+[E'SISw])/[E'SI] = H-A:()v,ys)C. is written as P(z)/M(z). P{z) is given by Eq. 22 as P(x). Rate of transglycosylation: Ar[E'S,S y ]=£,*(*,. x (kT/kH)K(y,h)C,M{z)EJF Rate of condensation: kT[E'Sx>S7]=kcK(x', x(kTlkn)K(y,jt)C,
(II. 26)
Hence, the rate equation for S(n,l) is formulated by combining Eqs. II. 23, II. 24, II. 25, and II. 26, and is given by Eq. 5. The term F4_ represents the contribution to the decrease of S(n,l) by cleavage of S(n,/) via productive complex. The first term of FM+ represents the contribution to the increase of S(n,l) by the cleavage of substrate of longer chain length than that of S(n,l) via productive complex. This corresponds to the formation of Sx_, in Fig. 2. The second term of Fi+ represents the contribution to the increase of S(n,l) via hydrolysis. G{x,n,l) and H(x,x—n,l) in Eq. 10 are given by Eqs. 16 and 17 as G(x,y,v) and H(x,y,v), respectively. G(x,y,v) is the sum of concentrations of the substrates with degree of polymerization x and having y residues of the reducing end side the same as those of a molecule %(y,v). H(x,y,v) is the sum of concentrations of the substrates with degree of polymerization x and having x—y residues of the non-reducing end side the same as those of a molecule S(x— y,v). The term F,_ represents the contribution to the decrease of S(n,l) as an acceptor of transglycosylation. The term F,+ represents that to the increase of S(n,/) via transglycosylation. The two terms in Fc- represent the contributions to the decrease of S(n,l) as an acceptor and donor of condensation, respectively. The term Fc+ represents that to the increase of S(n,l) via condensation. Finally, the total concentration of each substrate or product is the sum of the concentrations of free and bound material. C, = C+[bound substrate or product]
(II. 24)
(II. 25)
(II. 27)
This relation is shown in Eq. 7 and D(n) is given by Eq. 15.
394
R. MATSUNO et at.
The authors wish to express their gratitude to Dr. Yasunori Nitta for providing crystalline enzyme. REFERENCES 1. Hiromi, K. (1970) Biochem. Biophys. Res. Commun. 40,1-6 2. Hiromi, K., Nitta, Y., Numata, C , & Ono, S. (1973) Biochim. Biophys. Ada 302, 362-375 3. Thoma, J.A., Brothers, C , & Spradlin, J. (1970) Biochemistry 9, 1768-1775 4. Thoma, J.A., Rao, G.V.K., Brothers, C , Spradlin, J., & Li, L.H. (1971) / . Biol. Chem. 246, 5621-5635 5. Allen, J.D. & Thoma, J.A. (1976) Biochem. J. 159, 105-120 6. Allen, J.D. & Thoma, J.A. (1976) Biochem. J. 159, 121-131 7. Matsubara, S. (1961) /. Biochem. 49, 226-231 8. Hehre, E.J., Okada, G., & Genghof, D.S. (1969) Arch. Biochem. Biophys. 135, 75-89
9. Robyt, J.F. & French, D. (1970) /. Biol. Chem. 245, 3917-3927 10. Hehre, E.J., Genghof, D.S., & Okada, G. (1971) Arch. Biochem. Biophys. 142, 382-393 11. Shibaoka, T., Inatani, T., Hiromi, K., & Watanabe, T. (1975) J. Biochem. 77, 965-968 12. Suganuma, T., Ohnishi, M., Matsuno, R., & Hiromi, K. (1976) /. Biochem. 80, 645-648 13. Fujimori, H., Ohnishi, M., Sakoda, M., Matsuno, R., & Hiromi, K. (1976) FEBS Lett. 72, 283-286 14. Fujimori, H., Ohnishi, M., Sakoda, M., Matsuno, R., & Hiromi, K. (1977) / . Biochem. 82, 417-427 15. Chipman, D.M. (1971) Biochemistry 10, 1714-1722 16. Tada, H. & Kakitani, T. (1973) Bull. Chem. Soc. Japan 46, 1226-1232 17. Phillips, D.C. (1966) Sci. Amer. 215, No. 5, 78-90 18. Nitta, Y., Mizushima, M., Hiromi, K., & Ono, S. (1971) J. Biochem. 69, 567-576 19. Bender, M.L. & Kezdy, F.J. (1964) /. Amer. Chem. Soc. 86, 3704-3714
/ . Biochem.
Rate Equation for Amylase-Catalyzed Hydrolysis, Transglycosylation and Condensation of Linear Oligosaccharides and Amylose Ryuichi MATSUNO, Toshihiko SUGANUMA, Haruko FUJIMORI, Kazuhiro NAKANISHI, Keitaro HIROMI, and Tadashi KAMIKUBO Department of Food Science and Technology, Faculty of Agriculture, Kyoto University, Sakyo-ku, Kyoto, Kyoto 606 Received for publication, August 22, 1977
In order to analyze complex experimental results of amylase-catalyzed reactions, which include transglycosylation and condensation as well as hydrolysis, with amylose and linear oligosaccharides as substrates, a general equation governing the concentrations of substrate and product species in a reaction mixture is derived based on the subsite theory. The equation is devised so as to follow the location in products of radioactive glucose which was originally located at a certain position in a substrate. The reaction scheme assumed is as follows: 1) Enzyme can bind any substrate reversibly to form an enzyme-substrate complex, ES. 2) From the ES complex, a glucosyl bond or C^OH bond at the reducing end of the substrate is cleaved to form a reactive intermediate, releasing a product of shorter chain or HSO. 3) The reactive intermediate binds an acceptor molecule reversibly to form a ternary complex from which the product of transglycosylation or condensation is produced. The binding constants between enzyme and substrate, and between reactive intermediate and acceptor can be estimated from the subsite affinities. The rate constants of catalytic processes such as cleavage of a glucosyl bond and C'-OH bond, hydrolysis and transglycosylation are assumed to be independent of the degree of polymerization as well as of the binding mode of the substrate. The applicability of this scheme and rate equation was tested successfully with the Takaamylase A-catalyzed reaction of maltohexaose labeled with 14C at the reducing end.
The subsite theory (1-6) proposed for the hydrolysis of linear oligosaccharides and amylose catalyzed by amylases successfully accounts for the dependence of the Michaelis constant and molecular activity on the degree of polymerization. Another important property described by this theory is the action pattern of amylases for linear substrates, i.e., it predicts not only the ratio of possible binding modes between substrate and enzyme but also the probabilities of various modes of Vol. 83, No. 2, 1978
cleavage of the substrate. By the nature of the theory, it is applicable to the analysis of product distribution resulting from the hydrolysis of a given linear substrate by amylases (Matsuno, R., et ai, unpublished results.). Conversely, it is possible to estimate the relative values of subsite affinities from the analysis of products formed from reducing endlabeled substrates (3-6). Recent experimental findings (6-14) have suggested that some amylases possess activities of 385
386
R. MATSUNO et al.
transglycosylation and/or condensation as well as hydrolysis. For instance, in the case of the reaction of Taka-amylase A from Asp. oryzae with maltotriose labeled with "C at the reducing end (12), the ratio of non-labeled glucose to maltose as well as that of labeled glucose to maltose in the reaction mixture showed a marked dependency on the initial substrate concentration. Further, in the case of the reaction of maltose catalyzed by saccharifying a-amylase from Bacillus subtilis (13,14), a clear lag phase for glucose formation appeared at high maltose concentration, and a strong sigmoidal dependence of the reaction rate on substrate concentration (77,13,14) were observed. Since maltooligosaccharides longer than the substrate were also detected, it was concluded that these results were a consequence of a complex reaction mechanism involving transglycosylation and/or condensation as well as hydrolysis. In order to analyze quantitatively the results mentioned above, a general rate equation for amylase-catalyzed reactions of linear substrates, including hydrolysis, transglycosylation and condensation, is derived in this paper. The equation is generally applicable to any kind of glycosyl hydrolase which catalyzes hydrolysis, transglycosylation and condensation, including amylase and lysozyme; for the latter, similar treatments have been reported by Chipman (75) and Tada and Kakitani (16). By solving the equation, not only can concentrations of substrate and products be obtained as functions of time, but also the location in any product species of labeled glucose which is originally located at a certain position in the initial substrate can be identified. The latter can be accomplished by introducing a special method for naming molecules, together with a law of composition or combination.
quence of residues, even for the same degree of polymerization. Thus it is necessary to define a system for naming a molecule having a particular degree of polymerization and sequence. Naming of Molecules—Let us consider a linear heteropolymer composed of Nc kinds of residues and having a degree of polymerization n. Each kind of residue is distinguished by an "identification number" ranging from zero to Nt — \. The numbers are arranged linearly according to the sequence of the residues in the molecule. This gives an n-figure number, /, in the number system with a radix of Nc. Then, / is decimalized. By this procedure, the heteropolymer can be represented by the numbers n and /, as S(n, I). Example: Oligosaccharides consisting of nonlabeled glucose G(0) and labeled glucose G*(l), Nc=2. G-G*-G*-G-G-G* >S(/7,/)=S(6,011001), / i n binary system: > = S ( 6 , 25), / in decimal system.
Law of Composition (Combination) —In order to judge the species of product produced by a reaction, a law of composition is introduced. a) Condensation between S(/jlF /J and S(n2, k) in this order: Sfo, /,) S(nt, lt)=S(n1+nt,
I, X #,»• + /,)
( 1)
b) Cleavage of S(w, /) between /^th and nt +1 th residues:
where /j and lt must satisfy l=l1xNc"-"i+li. It is the integer part of ljNcn-"i and I1=1—I1XNC"-"K Example: Condensation between G*-G-G* and G-G*. — G*-G-G*-G-G*
METHODS
S(3, 5) S(2, ])=S(3+2, 5x2 I + l)=S(5, 21)
A linear polymer composed of many kinds of residues is considered. For instance, in the case of oligosaccharides consisting of labeled and nonlabeled glucose, the number of kinds of residues is two. When such a polymer is subjected to hydrolysis, transfer and condensation, various products are formed which differ in the degree of polymerization and also in composition and/or se-
Note that these methods can be applied to the case of any linear heteropolymer, such as DNA (Nc=4), oligopeptides (/V«=20), etc. Binding Constant between Substrate and Enzyme —The binding constant between substrate and enzyme is obtained by means of the subsite theory (7, 2). The assumption introduced in the theory is that the active site of an enzyme consists of several /. Biochem.
RATE EQUATION FOR AMYLASE REACTION
independent subsites, the subsite affinities of which are simply additive, and the binding constant can be estimated from the sum of the subsite affinities of the subsites which are occupied in the relevant binding mode. In the case of a heteropolymer composed of Nc kinds of residues, each subsite has Nc different subsite affinities corresponding to each kind of Nc residue. Thus, the binding constant of a heteropolymer S(/z, I) is, in general, determined when n, I and the binding mode are given. However, in the case of a linear homopolymer such as amylose or maltooligosaccharide, even if it contains labeled glucose, each subsite has only one subsite affinity. In this case, only n and the binding mode need be specified for the determination of the binding constant. The discussion will now be concentrated only on amylase with linear maltooligosaccharides as substrates. Figure 1 shows the active site of an amylase consisting of m subsites numbered from 1 to m. The catalytic site is located between the rth and r + l t h subsites. When a linear oligosaccharide with degree of polymerization n is bound to the amylase as shown in Fig. I, i.e., the reducing end binds to the /th subsite, the binding mode is specified by the number j and the binding constant is represented by K(n,j), regardless of productive and non-productive binding modes. In the case of binding between subsites and a substrate without participation of the reducing end residue, we add imaginary subsites numbered from m+1 next to the mth subsite. The binding mode is then again specified by the subsite
387
number j of the imaginary subsite at which the reducing end is situated. In this case, j exceeds m. Let At be the subsite affinity of /th subsite, then K(n,j) is given by the following equation, using the subsite theory (1, 2). A:(n,;)=0.018exp (SA.,/RT)
(3)
where SA.j is the sum of subsite affinities of the subsites occupied, and is given by, j= Y, ZVi-ZW(-Ai where (4)
l , ujt=j—i In Appendix I, the proof of Eq. 4 is shown. In the case of an exo-enzyme such as glucoamylase and /)-amylase, the subsite affinity of the first subsite is taken as - c o . Reaction Scheme—Figure 2 shows the scheme assumed for amylase-catalyzed reactions which include hydrolysis, transglycosylation and condensation of maltooligosaccharides and amylose {12-14). The enzyme binds any x-mer substrate ST reversibly to form an enzyme-substrate complex ESi in any possible binding mode. A ternary enzyme-substrate complex in which two substrate molecules are bound to the enzyme (ESXSW) is also taken into account. From a productive binding constant K(n,j) complex, a glucosyl bond is cleaved with a rate constant k, to form a reactive intermediate E'Si, G—-G-G—-G-— which may presumably be an "enzyme-carbonium ion intermediate" complex (17), releasing a product A. Sx-X. As a first approximation, we adopt the A j m simple assumption that ki is constant irrespective Fig. 1. Schematic representation of the active site of the degree of polymerization as well as of the structure of an amylase and binding mode of a linear binding mode of the linear substrate (1, 2).1 substrate. The subsite is numbered by i (from 1 to m) integers counting from the non-reducing end side (left) 1 This assumption is supported by chemical evidence and Ai is the corresponding subsite affinity. The wedge between the rth and r+lth subsites signifies the that the rate of acid-catalyzed hydrolysis of a-1,4 linkages is independent of the degree of catalytic site. The binding constant of n-mer oligo- glucosidic polymerization of the substrate. For enzyme-catalyzed saccharide, the reducing end of which is bound to the reactions, the assumption is equivalent to considering /th subsite, is represented by K(n,f). j can exceed m. that the subsites are independent of each other. This point has been discussed by several authors (3-6). For details, see the text.
—-8
Vol. 83, No. 2, 1978
388
R. MATSUNO et al.
be classified into two types, A and B, according to whether the r + l t h subsite is occupied or not occupied by non-reducing end residue of ST. Then, E'SX and E'S,Sy of B type react with water to form S t with a rate constant kH through hydrolysis. From E'SiSy of A type, S J+y is Sx-z or HjO produced with a rate constant kT through transglycosylation. \ X G-Gk From a kind of non-productive complex, ESX, in which the rth subsite is occupied by the reducing end residue of Sx, a reactive intermediate E'SX is formed with a rate constant ke releasing an H,O molecule. Then, E'SX and E'S x S y of B type release Sx through hydrolysis, and E'S x S y of A type releases S x+y through condensation. In this case, z is equal to x and S x _ t is H 2 O. G-G- G-G-G G-G-G-.G-G-G The rate equation is derived from this scheme I I \r\ I I I under the following assumptions; Fig. 2. Scheme for amylase-catalyzed reactions which 1) Rapid equilibria between E and ESX and include hydrolysis, transglycosylation and condensation of maltooligosaccharides and amylose. E and S repre- between E'S, and E'S,Sy. 2) Steady state for reaction intermediates, sent enzyme and substrate, respectively. ESX, enzymesubstrate complex; E'Si, reactive intermediate; E'SiSy, i.e., the rate of formation of reaction intermediate ternary complex formed between E'Si and acceptor equals that of its breakdown. S y ; x, y, and z, degree of polymerization of substrate. 3) The catalytic rate constants k,, kc, kH, For condensation, z equals x. ki, k,, kH, and kr are and k are independent of the degree of polyT rate constants of cleavage of glucosyl and C'-OH bonds, merization as well as of the binding mode of of hydrolysis and of transglycosylation, respectively. oligosaccharides. K(.x,Ji) a n d K(y,ji) are binding constants between enzyme and x- and .y-mer substrates in binding modes specified by j \ and j t , respectively, where j \ and j \ are the number of the subsite at which the reducing end of each substrate is situated. G- signifies a glucose residue in the form of reactive intermediate.
E'Sj binds any y-rner substrate Sy reversibly to form a ternary complex E'S t S y in any possible binding mode. The ternary complex E'S,Sy can dC(n.l) Eodt
„ rr.
„ ,„
RESULTS AND DISCUSSION Under the reaction scheme and assumptions described in the previous section, the rate equations governing the concentrations of substrate and product species consisting of labeled and nonlabeled glucose in a reaction mixture can be derived as Eqs. 5 to 23.
r
(5)
where, C.=
(6)
n,[) = C(n. /)/(! +EoD(n)IF) 2
j-i
y-i
(7) -%- £ C,'*£1K(x.j)L(.x-j+r)R(x,j) kit
r-i
-rp-Z Ka
y-r+i
C,K(x, r)R(x, r)L(x)
(8)
i-l
J. Biochem.
RATE EQUATION FOR AMYLASE REACTION
389
F*_ = C(n./) * + F #('/,;) R(n,j)
(9 )
Fk+ = E [K(x,n+r)G(x,n,l)R(x,n+r)+K(x,x+r-n)H(x,x-n,l)R(.x,x+r-n)P(n)]
(10)
J-.+I
L
|
'
'
E
M « E C,*;(j,rrtr)^j-A-+f) i-l
(11)
J-«+l
|
E H(y,y-x, v)K(y,y-x+r) R(y,y-x+r)
where v and u must satisfy / = N c " ~ ' x v + u .
v: the integer part of IINC'~*. n:
(12)
=I—Nc"~*xv.
Fc- = - ^ C(n, I) [K{n, r+n) "f C, K(x, r) R(x, r) M(x) +K(n, r) M(n) R(n, r) "t" C, K(y, r+y)] (13) , - \
I-I
KB
Fc+ = ^-"zC(n-x,u)K(n-x,n-x+r)C(x, KH
v) K{x,r) R(x,r) M{x)
(14)
X-1
where v is the integer part of I/Nc"~' and u equals l—Nc"~* x v. D{n)=+Y,l±-K(n,j)[\ +R(n,j) + C(n, l)J{n, n,j)]+-^-[ E C, K(x, r) R(x, r)M(x) I{x, ri) N
fctK(x,r)R(x,r)M(x)]
+ -^-[ £ C.'*'£ K(x,j) R(x,j) KH
x-l
M{x) E C, K(y,y-x+r) R(y,y-x+r)]
G(x,y,v)=
£,~lC(x,Nc'Xw+v)
M(x-j+r)I(x,n)
t-r+l
(15)
(16)
H(x, v, v) = "f" C(x, ^ X i- + it-)
(17) (18)
J(x.y.h)=
m+ l
£ Kb>J)+Z'K(yJ)
(19)
E C,/(xj)+E'c,^,r+^)]/WW
(20)
E C,/(jr,y)+-^L*E*C,A:(^r+^l ^
i
(21) J (22)
,j) = \+£c,J(x,y,j) Vol. 83, No. 2, 1978
(23)
390
R. MATSUNO et at.
Eo and C(n,f) are the total concentrations (free plus bound) of enzyme and that of a species S(/J,/)> respectively. C(n,f) is the concentration of free S(n,l), t is the time, and N is the maximum length of oligosaccharide to be considered. By definition, Nc is 1 when no species contains labeled glucose, while N, is 2 when it is present. The details of derivation and physical significance of equations are given in Appendix II. In Eq. 5, the terms Fk-—Fi,+, F,-—F,+, and Fc-—Fc+ represent the contributions of hydrolysis, transglycosylation, and condensation, respectively. Since C(n,l) is not exactly equal to C(n,l), the calculation is rather complicated. Practically, however, the enzyme concentration Eo is usually low enough for EoD(n)/ JFglucan glucanohydrolase, Aspergillus oryzae) with maltohexaose labeled with 14C at the reducing end, S(6,l), is examined. It was found from a study of the reaction of maltotriose (12) that Taka-amylase A had pronounced transglycosylation activity and low condensation activity as well as hydrolysis activity. Condensation activity is not taken into consideration in the present example (kc=0) because the effect of condensation in the reaction of maltohexaose is not very important in view of the subsite structure of Taka-amylase A. The experimental results in Fig. 3(a) show the dependence of distribution of end-labeled products on the initial
Fig. 3. Product distribution in the reaction of Takaamylase A with maltohexaose labeled by " C at the reducing end, S(6, 1). Reaction products were separated by paper chromatography and subjected to measurement of the radioactivities. Details of the experimental techniques and conditions are given in reference (72). Parameters used for theoretical calculation are as follows. Subsite affinities Al to At: 0.2, 0.4, 4.9, -3.7, 0.0, 3.3, 1.2, -0.7, and 0.8 kcal/mol. Ar/ = 12,000 1/min, £ c =0.0, kijkH E + S ,+ y
kH
Non-productive complex ESw and ESwS». Reactive intermediate from productive com-
(II. 6)
E+S. ;==± ESw
. C'C
. ; g
(ii. 4) _I_
U
P
(n. 5)
(, V
rcc
i
/ rE'S
Swl+itrlE'S,S D/^fE'Sx]1) =1 +K(w,j3)C. +{kTlkH)K(y,ti C,= MM{i) (EL 19) / . Biochem.
RATE EQUATION FOR AMYLASE REACTION R{x,jd and M(z) in Eqs. II. 18 and II. 19 correspond to R(x,j) and M(x) in Eqs. 23 and 21, respectively, and their physical significance is indicated by Eqs. II. 18 and II. 19. J(x,y,j) in Eq. 23 is given by Eq. 19 as J(x. y, h). J(x, y, h) represents the sum of the binding constants of possible binding modes of other substrates with degree of polymerization y when a substrate with degree of polymerization x is bound to the enzyme in binding mode h. I(x, y) in Eq. 21 is related to J(x, y, h) by the relation I(x, y)=J(x, y, r)— K(y, y+r) and is shown in Eq. 18. Equation n . 16 can be converted to Eq. II. 20, [E'S,]l\E]={kI/kB)K(x,j1)C,R(xJ1)M(z)
(II. 20)
In the same way, Eq. II. 21 can be derived from Eq. II. 15. =(k.lkH)K(x', h')C,.R(x', h (H.21) The relation between the concentration of total enzyme and that of free enzyme is obtained from Eq. II. 1. Writing =
([E'SJ+[E'S,SW]+[E'S,SJD/[E'SJ l+K(w,j3)C.+K(y,jt)CT
as L(z)/M(z), the ratio of total to free enzyme concentrations is given by,
[E]XL(z)/M(z))+([E'Sx-]/ [E])(L(XyM(x'))=F
(11.22)
where the precise form of L(z) is given by Eq. 20. Equation II. 22 corresponds to F in Eq. 8. Then, the reaction rates are given by the following equations. Rate of cleavage of productive complex: Ar;([ESx]+[ESxSw])=
, h)C.R(x. J (11.23)
Rate of hydrolysis: MIE'SJ -HE'S.S,,]) =M[E'S x(0E'S1]+[E'SIS,])/TJE'SJ) =k,K(x, h) C,R(.x, h) P(z)EjF Vol. 83, No. 2, 1978
393
where ([E'S,]+[E'SISw])/[E'SI] = H-A:()v,ys)C. is written as P(z)/M(z). P{z) is given by Eq. 22 as P(x). Rate of transglycosylation: Ar[E'S,S y ]=£,*(*,. x (kT/kH)K(y,h)C,M{z)EJF Rate of condensation: kT[E'Sx>S7]=kcK(x', x(kTlkn)K(y,jt)C,
(II. 26)
Hence, the rate equation for S(n,l) is formulated by combining Eqs. II. 23, II. 24, II. 25, and II. 26, and is given by Eq. 5. The term F4_ represents the contribution to the decrease of S(n,l) by cleavage of S(n,/) via productive complex. The first term of FM+ represents the contribution to the increase of S(n,l) by the cleavage of substrate of longer chain length than that of S(n,l) via productive complex. This corresponds to the formation of Sx_, in Fig. 2. The second term of Fi+ represents the contribution to the increase of S(n,l) via hydrolysis. G{x,n,l) and H(x,x—n,l) in Eq. 10 are given by Eqs. 16 and 17 as G(x,y,v) and H(x,y,v), respectively. G(x,y,v) is the sum of concentrations of the substrates with degree of polymerization x and having y residues of the reducing end side the same as those of a molecule %(y,v). H(x,y,v) is the sum of concentrations of the substrates with degree of polymerization x and having x—y residues of the non-reducing end side the same as those of a molecule S(x— y,v). The term F,_ represents the contribution to the decrease of S(n,l) as an acceptor of transglycosylation. The term F,+ represents that to the increase of S(n,/) via transglycosylation. The two terms in Fc- represent the contributions to the decrease of S(n,l) as an acceptor and donor of condensation, respectively. The term Fc+ represents that to the increase of S(n,l) via condensation. Finally, the total concentration of each substrate or product is the sum of the concentrations of free and bound material. C, = C+[bound substrate or product]
(II. 24)
(II. 25)
(II. 27)
This relation is shown in Eq. 7 and D(n) is given by Eq. 15.
394
R. MATSUNO et at.
The authors wish to express their gratitude to Dr. Yasunori Nitta for providing crystalline enzyme. REFERENCES 1. Hiromi, K. (1970) Biochem. Biophys. Res. Commun. 40,1-6 2. Hiromi, K., Nitta, Y., Numata, C , & Ono, S. (1973) Biochim. Biophys. Ada 302, 362-375 3. Thoma, J.A., Brothers, C , & Spradlin, J. (1970) Biochemistry 9, 1768-1775 4. Thoma, J.A., Rao, G.V.K., Brothers, C , Spradlin, J., & Li, L.H. (1971) / . Biol. Chem. 246, 5621-5635 5. Allen, J.D. & Thoma, J.A. (1976) Biochem. J. 159, 105-120 6. Allen, J.D. & Thoma, J.A. (1976) Biochem. J. 159, 121-131 7. Matsubara, S. (1961) /. Biochem. 49, 226-231 8. Hehre, E.J., Okada, G., & Genghof, D.S. (1969) Arch. Biochem. Biophys. 135, 75-89
9. Robyt, J.F. & French, D. (1970) /. Biol. Chem. 245, 3917-3927 10. Hehre, E.J., Genghof, D.S., & Okada, G. (1971) Arch. Biochem. Biophys. 142, 382-393 11. Shibaoka, T., Inatani, T., Hiromi, K., & Watanabe, T. (1975) J. Biochem. 77, 965-968 12. Suganuma, T., Ohnishi, M., Matsuno, R., & Hiromi, K. (1976) /. Biochem. 80, 645-648 13. Fujimori, H., Ohnishi, M., Sakoda, M., Matsuno, R., & Hiromi, K. (1976) FEBS Lett. 72, 283-286 14. Fujimori, H., Ohnishi, M., Sakoda, M., Matsuno, R., & Hiromi, K. (1977) / . Biochem. 82, 417-427 15. Chipman, D.M. (1971) Biochemistry 10, 1714-1722 16. Tada, H. & Kakitani, T. (1973) Bull. Chem. Soc. Japan 46, 1226-1232 17. Phillips, D.C. (1966) Sci. Amer. 215, No. 5, 78-90 18. Nitta, Y., Mizushima, M., Hiromi, K., & Ono, S. (1971) J. Biochem. 69, 567-576 19. Bender, M.L. & Kezdy, F.J. (1964) /. Amer. Chem. Soc. 86, 3704-3714
/ . Biochem.
