External Diffusion Effects on the Kinetic Constants of Immobilized Enzyme Systems

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External Diffusion Effects on the Kinetic Constants ofImmobilized Enzyme Systems

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<ul><li><p>f. theor. Biol. (1980) 84, 259-279 </p><p>External Diffusion Effects on the Kinetic Constants of Immobilized Enzyme Systems </p><p>SUN BOK LEE AND DEWEY D. Y. RYU </p><p>The Korea Advanced Institute of Science, P.O. Box 150, Chung- Ryang-Ri, Seoul, Korea </p><p>(Received 29 May 1979, and in revised [orm 29 November 1979) </p><p>In order to further our understanding of immobilized enzyme reaction kinetics, the effects of external diffusion on the kinetic constants are studied for various reaction systems. It is shown that the variations of apparent kinetic constants with diftusional limitations are not the same as those of Michaelis-Menten kinetics, although both the inhibition kinetics and the two substrate reaction kinetics can be expressed in the form of simple Michaelis-Menten type by using apparent rate parameters. We found that there will be changes in the apparent kinetic constants depending not only on the types of enzyme reaction kinetics but also on the relative rate of diffusion or flow-through in the microenvironment. For substrate inhibition kinetics, both the apparent maximum reaction rate and the apparent Michaelis constant decrease while the apparent maximum reaction rate for product inhibition kinetics is increased and the apparent Michaelis constant decrease as mass transfer limitation is reduced. In the case of two substrate enzyme reaction kinetics, as the diffusio'nal limitation is reduced the apparent maximum reaction rate increase but the apparent Michaelis constant can increase, decrease or remain nearly constant depending on the values of relative affinity and on the fixed substrate concentrations. The results of theoretical analyses are compared with the experimental data obtained and reported previously, and a very good agreement was found. </p><p>1. Introduction </p><p>When the enzymes are immobilized, such changes as structural con- formation, microenvironment, the partitioning or electrostatic effect and mass transfer resistance could be brought about (Goldstein, 1976). Among these the effects of mass transfer limitation on the reaction kinetics have been widely studied by many workers from the viewpoint of heterogeneous catalysis. </p><p>The effects of internal diffusion in the reaction kinetics have been studied experimentally as well as theoretically (Lee, Fratze, Wun &amp; Tsao, 1976; Moo-Young &amp; Kobayashi, 1972; Engasser &amp; Horvath, 1973). In the case of external diffusion, however, the results of several workers (Lilly etal., 1966; </p><p>259 0022-5193/80/100259+21 $02.00/0 9 1980 Academic Press Inc. (London) Ltd. </p></li><li><p>260 S.B . LEE AND D. D. Y. RYLI </p><p>Wilson, Kay &amp; Lilly, 1968a, 1968b; Sharp et al., 1969; Tosa, Mori &amp; Chibata, 1971; Kobayashi &amp; Moo-Young, 1973; Gellf, Thomas, Broun &amp; Kerneves, 1974: Cho &amp; Swaisgood, 1974; Toda, 1975; Hirano, Karube, Matsunaga &amp; Suziki, 1977; Paul, Coulet, Gautheron &amp; Engasser, 1978; Kim, Lee &amp; Ryu, 1978) appear to be contradictory and conflicting. </p><p>Most of these authors cited made some endeavors to show the important factors that affect the kinetic constants of immobilized enzymes. Many more works also attempted to analyze theoretically the effects of external diffusion on the kinetic constants. For irreversible Michaelis-Menten kinetics, Hornby, Lilly &amp; Crook (1968), Kobayashi &amp; Moo-Young (1971), Shuler, Aris &amp; Tsuchiya (1972), Hamilton, Stockmeyer &amp; Colton (1973), Kobayashi &amp; Laidler (1974), and Toda (1975) derived the equations for apparent kinetic parameters. For reversible Michaelis-Menten kinetics, Lee, Kim &amp; Ryu (1979) showed that the extent the apparent kinetic :constants changes as a function of superficial velocity depends on the value of l l Vm/Km parameter for forward reaction and for reverse reaction, respectively. For other enzyme reaction kinetics, the effects of external diffusion on the kinetic constant have not been clearly established up to the present. </p><p>In view of these developments and some confusion, our study on the effects of external diffusion was undertaken in order to further our under- standing of reaction kinetics of immobilized enzymes. In this paper, the external diffusion effects on the kinetic constants were studied theoretically and the confusion due to conflicting results was clarified. </p><p>2. The System </p><p>It is assumed that the system is at a steady state, the enzymes are immobilized On a nonporous support material, and the diffusion coefficient is constant. </p><p>For the reaction system that a substrate, s, is converted to a product p, the diffusive fluxes can be expressed as </p><p>D ds Js = S-~y (I) </p><p>4 =Dpd~ (2) with the boundary conditions </p><p>s=S, ! p=P </p><p>s=S, p=P </p><p>aty =0 </p><p>atY =8 </p></li><li><p>DIFFUSION EFFECTS ON ENZYME KINETICS 261 </p><p>where the mass fluxes toward the surface of the immobilized enzyme are positive and the origin of the co-ordinate system is at the surface. Under steady state, J, = -Jp =3", and the Equations (1) and (2) yield the following expression </p><p>J = kLs (S - -S ) = kreCP-P) (3) </p><p>where kLs = Ds/ 8 and kt.p = Dp/ &amp; For the two substrate enzyme reaction system the diffusive flux can be </p><p>expressed as </p><p>J = ku(S , - S,) = kLi(Si -- 4 ) (4) </p><p>which is analogous to equation (3). </p><p>3. Theoretical Examination of the Mass Transfer Effect on the Kinetic Constants </p><p>(A) METHOD OF ANALYSIS </p><p>Some workers defined the apparent Michaelis constant as that cor- responds to a value at whicl~ the rate of enzyme reaction is half of the maximum reaction rate (Goldman, Kedem &amp; Katchalski, 1971; Sundaram et al., 1972; Kobayashi &amp; Laidler, 1974), although it cannot describe the overall reaction rate exactly (Horvath &amp; Engasser, 1974; Kobayashi &amp; Laidler, 1974). In general, for the homogeneous enzyme reaction system, the maximum reaction rate is defined as the reaction rate at a high substrate concentration or S-~ co and the Michaelis constant is the substrate concen- tration at which the reaction rate is half the maximal. If these concepts are applied to the heterogeneous system, then we can define that </p><p>VJtrn(app ) = lim R (5) $~o0 </p><p>I ! </p><p>gm(app) ---- S la t R =0.5 V:,.pp, (6) </p><p>where R represents the reaction rate of immobilized enzymes. This type of analysis was found to be very useful if the trends or variations of kinetic constants with diffusional limitations are to be determined, especially for the complex enzyme reaction system. </p><p>It is now convenient to introduce the following dimensionless variables, </p><p>R s v" , O r = - - /2 ,= </p><p>= V'., K'.,.' kLsK'~ </p><p>where, ~" represents the dimensionless reaction rate, o" the dimensioniess substrate concentration, and Ix the mass transfer modulus. </p></li><li><p>262 s .B . LEE AND D. D. Y. RYLI </p><p>Equations (5) and (6) becomes </p><p>a*= lim ~" (7) o- -~ oO </p><p>/3* = r s o* (8) </p><p>where a* and/3* represent the dimensionless apparent maximum reaction rate and the apparent Michaelis constant, respectively. </p><p>We will now examine the effects of diffusional limitations on the kinetic parameters for various enzyme reactions by using equations (7) and (8). The purpose of this examination is to find how a* and/3* can be affected as the mass transfer modulus, it, changes. </p><p>(B) ONE SLIBSTRATE ENZYME REACTION SYSTEMS </p><p>( i) Michaelis-Menten kinetics </p><p>When the enzymes are immobilized, the enzyme reaction takes place at the surface and the reaction rate for this heterogeneous enzyme reaction system can be expressed as </p><p>v 'g g = K" +-----ff (9) </p><p>where, R is the observed global rate of heterogeneous reaction per unit area. Equation (9) may be rewritten in terms of measurable bulk substrate concentration instead of surface concentration </p><p>/ / V,nS </p><p>R =K~ +S" (10) </p><p>From equation (3), S = S - ( J / k~) and substitution into equation (9) yields </p><p>~" = (11) 1+o. - ~rtx </p><p>in dimensionless form since the mass transfer flux, J, will be equal to the rate of reaction, R, under steady state. Equation (11) yields </p><p>= ~ {( 1 +/.t + o.) - [(1 + tt + o.)2 _ 4bto.] 1/2} (12a) </p><p>= 2o'{(1 +/z +o-)+[(1 +/z + o.)2- 4/zo.]1/2}-1 (12b) </p><p>since 0-&lt; s r-&lt; 1. From equation (12), for Michaelis-Menten kinetics </p><p>a*=l : (13) /3* = 1+0.5/~ (14) </p></li><li><p>DIFFUSION EFFECTS ON ENZYME KINETICS 263 </p><p>From these equations one finds that the apparent maximum reaction rate, a*, is constant and the apparent Michaelis constant, /~*, increases with diffusional limitations (i.e. with increasing t~). An expression similar to equation (14) was previously defined as apparent Michaelis constant by Kobayashi &amp; Laidler (1974), although the relationship holds only for the one substrate Michaelis-Menten type enzyme reaction system. </p><p>(ii) Substrate inhibition </p><p>The substrate inhibition kinetics expressed as </p><p>v 'g R= </p><p>K" +S+ ~ ' S /Ki~ </p><p>can be rewritten in a dimensionless form </p><p>(15) </p><p>where, </p><p>~" = 1 +,7-(tL +Ks(o"- ~'tz) 2 (16) </p><p>K" K,= </p><p>K~," </p><p>Equation (16) yields a cubic equation for ~" and its analytical solution is complex. !n Fig. 1 the Lineweaver-Burk plot is shown at Ks = 0.05 using the result of computer simulation. There is little change in K~', with the variation of t* while V~, and K~, vary similarly to those of Michaelis-Menten kinetics. </p><p>(iii) Product inhibition </p><p>There are, in general, three types of product inhibition, that is, competi- tive, non-competitive, and anti-competitive inhibitions (Laidler &amp; Bunting, 1973). For competitive product inhibition kinetics, </p><p>v 'g R Kin(1 +WK,p)+g (17) </p><p>can be expressed as </p><p>o ' -6 ~" = (18) </p><p>(1 + ~: -t- gd~') + o- - ~'tz </p><p>where </p><p>P K" k~ ~ = ~',.p, andKp =K~p kLp </p></li><li><p>264 </p><p>4 &amp; </p><p>3 </p><p>S. B. LEE AND D. D. Y. RYU </p><p>t ! = i 085 f t t t 0 10 </p><p>o" </p><p>Flo. I. Effect of diffusion on the Lineweaver-Burk plot of the substrate inhibition kinetics when K, = 0.05. </p><p>and ~ represents the dimensionless initial product concentration and Kp the relative value of product inhibition constant when the mass transfer coefficient of substrate and product has the same order of magnitude. </p><p>Equation (18) becomes </p><p>1 ~" = 2(1 - Kp)/z {(1 +/.t +tr + so)-[(1 +/z +tr +sr -Kp)l.ttr]]/2}. </p><p>(19) From equation (19), </p><p>a*= lim ~'= 1 (20) o-~oo </p><p>~* = o'l~=o.s = 1 +~+0.5/.t(1 + K~,). (21) </p><p>From this analysis, one finds that a* is constant and fl* decreases as/z decreases. The result of evaluation of parameters a * and fl* are summarized in Table l(a) according to the types of product inhibition. If there is no external diffusion limitation, namely/z = 0, a* and fl~' are reduced to the values shown in Table l(b), which are identical to the solution of homo- geneous enzyme reaction system. </p></li><li><p>DIFFUSION EFFECTSON ENZYME KINETICS </p><p>TABLE 1 </p><p>Values of or* and [3" for product inhibition kinetics </p><p>265 </p><p>(a) With diffusional limitations </p><p>Inhibition type a* /3' </p><p>Competitive </p><p>Non-competitive </p><p>Anti-competitive </p><p>1 1 +6+0.5/,t(1 +Kp) </p><p>1 2 Kpp,2X3+(I +,~-Kp)p.X2- (l +6+p.)X I'4(1+6) +4Kptz-(l +6)] Ktd.tX2 +(l + 6)X - 1 </p><p>2_K_~_~ 1-4(1 + 6)2 +4Kp~_ (1 + 6)] Kp ~2X3+(l+6)v'X2-(I+~)X KplzX2+ (1 +6)X- 1 </p><p>(b) Without diftusional limitations 9 . </p><p>Inhibition type a*l,,.o /3"1,=o </p><p>Competitive 1 1 + 6 Non-competitive (1 + 6) -t 1 Anti-competitive (1 + 63-1 . (l + 6)-I </p><p>P K~ Where, 6 = K-~.p ' Kp = K~p' X = 0-5 a* </p><p>For non-compet i t ive product inhibition kinetics, c~* and [3* are plotted against tz in Fig. 2. F rom this figure we find that a* will increase and fl* will decrease as Ix decreases. In addition, we find that a* increases more sharply as Kp increases (or inhibition constant is smaller) but [3* decreases more sharply as Kp decreases. Similar results are obtained for ant i -competit ive inhibition reaction kinetics. </p><p>(C) TWO SUBSTRATE ENZYME REACTION SYSTEM </p><p>For two substrate enzyme react ionsystems equation (22) can be used (Laidler &amp; Bunting, 1973; Paul et al., 1978), </p><p>V,.S;$~ R K , , , iK , ,q+K, , , f i ,+K ' .~ , .+~ (221 </p><p>and such type of two substrate enzyme reaction kinetics as p ing-pong bi-bi reaction mechanism can be analyzed by the method shown below. From </p></li><li><p>266 </p><p>a~ , I </p><p>0 6 </p><p>5 </p><p>4 </p><p>2 </p><p>I </p><p>S. B. LEE AND D. D. Y. RYU </p><p>2 4 6 8 10 </p><p>0.1 </p><p>9 0 .5 </p><p>! | ! i ! . l , I </p><p>0 # 2 4 6 8 tO </p><p>FIG. 2. Effect of diffusionai limitations on the apparent kinetic constants for the non- competitive product inhibition kinetics. The variations of a* and /3* with mass transfer modulus p, were plotted at the different values of K'p when ~ = 0. </p><p>equations (4) and (22), we find that </p><p>(o-, - ~3 (o'j - g'ui) ~" = 1 + (o-, - ~'t~i) + (o-j - (m) + (o-~ - r - t 'm) </p><p>where, </p><p>(23) </p><p>R Sl S~ ~" = V" cri K ' , ' o ' /= k" ' </p><p>v2 v" IX~=K~,kL~ and/z i= k , 9 LrK,.j </p><p>Experimentally, the kinetic constants of two substrate enzyme reaction system are determined while one substrate concentration is fixed. If we fix the concentration of Si of o'z, then </p><p>o'i - - ~ ~ i a~ = lim ~= </p></li><li><p>DIFFUSION EFFECTS ON ENZYME K INET ICS 267 </p><p>giving </p><p>a T = 2o'i{(1 +/zi+o'~)+, (1 +/xt +o'~)2-4/z,.o'i]'/2} -1. (24) </p><p>From equation (23) and (24),/3" can be obtained by eliminating o-~. </p><p>/3? = (rj </p><p>2(1 -- Ot ~)2/Xi = 1+0.5 a* tq (1 -a ] ' ) (2 -a* ) / z ,+2" (25) </p><p>The expressions for a* and/3* can also be derived simply by changing the subscript. </p><p>Now, we introduce the another dimensionless parameter, K, which is defined as </p><p>#, k_~ K ' , 6) K=- -= 0--</p></li><li><p>268 s .B . LEE AND D. D. Y. RYU </p><p>(4) f l~ decreases at K = 0 as in the M ichae l i s -Menten kinetics. When o-~ is smal l f l* can increase, decrease , or remain near ly constant wi th the values of K. When o't becomes very la rge /~* decreases as/ . t decreases . This is an oppos i te to the changes in f l* . As an example , a* and/~* are p lo t ted against p. in Fig. 3 when cr~ = tr i = 1"0. </p><p>~ I .5 (a) 2-0 </p><p>1.5 </p><p>~', o </p><p>0-5 </p><p>I 0 5 10 0 </p><p>(b) </p><p>5 10 </p><p>"01 2.0 (e) </p><p>1.5 </p><p>,5 K=o ~;,o </p><p>0.5 </p><p>(d) </p><p>0.5 o~ O. 7.~5" 1.0 </p><p>o 5 10 o 5 I0 </p><p>FIG. 3. Effect of diffusional limitations on the apparent kinetic constant for the two substrate enzyme reaction kinetics. The variations of a~, a} ~,/3* and fl~' with mass transfer modulus were plotted at the different values of K when o-~ = o- t = 1.0. </p></li><li><p>DIFFUSION EFFECTS ON ENZYME KINETICS 269 </p><p>4. Discussion </p><p>Based on the results of this study we found that, when a soluble enzyme is immobilized, there will be changes in the kinetic constants which may either go up or down depending on the types of enzyme kinetics as well as on the relative rate of diffusion or flow-through in the microenvironment. </p><p>In the case of one substrate enzyme reaction, from equations (9) and (10) we find that the observed rate parameters are </p><p>V~ = V',. (31) </p><p>K '.. = K~ (S/g) = K'~ (1 - J / kcsS) -t (32) </p><p>and similarly, </p><p>K ,", = K} , (S / g) = K}, (1 - Y/ kLsS) -1 (33) </p><p>K}~, = g~p (P/P) = K~ (1 +Y/kLpP) -t (34) </p><p>regardless of inhibition type. The changes in the kinetic constants are due to the differences between the substrate or product concentrations in the bulk phase and those at the surface since the rate of diffusion is directly related to the substrate or product corlcentration. The variations of the surface concentration of substrate (7') and product (v) with the mass transfer rate are shown in Fig. 4(a) and (b). It shows that the surface concentration of substrate increases while that of product decreases as the mass transfer...</p></li></ul>

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