- Email: [email protected]
A general solution for the steady-state kinetics of immobilized enzyme systems
0092-8240/8453.00+0.00 Pergamon Press Ltd. © 1984 Society for Mathematical Biology
Bulletin of Mathematieal Biology Vol. 46, No. 2, pp. 229-234, 1984. Printed in Great Britain
A GENERAL SOLUTION FOR THE STEADY-STATE KINETICS OF IMMOBILIZED ENZYME SYSTEMS [] D. C. REES
Department of Chemistry and Biochemistry, Molecular BiologyInstitute, Universityof California, Los Angeles,CA 90024, U.S.A. A power series solution is presented which describes the steady-state concentration profiles for substrate and product molecules in immobilized enzyme systems. Diffusional effects and product inhibition are incorporated into this model. The kinetic consequences of diffusion limitation and product inhibition for immobilized enzymes are discussed and are compared to kinetic behavior characteristic of other types of effects, such as substrate inhibition and substrate activation.
i. Introduction. Diffusional effects may significantly influence the reaction kinetics of immobilized enzyme systems (reviewed in Goldstein, 1976; Laidler and Bunting, 1.980). Concentration gradients may be generated within these systems owing to consumption and creation of substrate and product molecules, and to diffusional resistances in the channels connecting the external solution and the active site of the enzyme. Most mathematical treatments of these effects have been derived for the limiting case of either very low or very high substrate concentrations, or else require iterative numerical calculations, in order to obtain concentration profiles. In this paper we derive a general power series solution for substrate and product concentration profiles across a planar enzyme membrane which is valid for all substrate concentrations. 2. Model The immobilized enzyme system is modeled as an infinite plane of thickness 2d, perpendicular to the x axis and centered about x = 0. At steady state, rates of change of substrate and product concentrations due to enzyme activity and diffusion are of equal magnitude and opposite sign. For the reaction scheme: kl
E + S----~ES k_ 1
k2
k3
>EP ~---.~. E + P,
(2.1)
k_~
where E, S and P represent enzyme, substrate and product molecules respectively, the steady-state reaction velocity, v, in the absence of diffusion may be calculated (Segel, 1975): 229
230
D.C. REES
k(E t ) (S)
v=
(2.2)
Km + (S) + ( ~ i ) (P) where k = k2k3/(k2 + k3), K m = ((k3(k2 + k-l)/(kx(k2 + k3)), Ki = k3/k-3 and (Et) is the total enzyme concentration. In this mechanism product inhibition by P is competitive, and characterized by an inhibition constant Ki. Combining this expression with Fick's second law for diffusion yields the basic differential equations describing this system (Doscher and Richards, 1963):
zS
2p Ox2
k(Et )
k(Et ) Dv
(S)
(2.3)
(S) Km + ( S ) +
/--(K--7;--./(P) \/~i/
where Ds and D v are the substrate and product diffusion coefficients, respectively, inside the membrane. This treatment neglects the possible influence of boundary diffusion layer effects on the concentration profiles. Experimentally, these effects may be minimized by stirring the solution external to the membrane. The symmetry of the model requires that S and P be even functions of x, so that S(x) and P(x) may be quite generally written as: co
S(x) = Y anx2n (2.4)
P(x) = ~ bnx2n. n=O
Substitution of (2.4) into (2.3) leads to a relationship between an and b n : Ds
bn = - - - - a n ,
Dv
(2.5)
which is valid for n / > 1. The boundary conditions at x = +d are given by S(d) = So and P(d) = Po. Assuming that the volume surrounding the membrane is quite large, product molecules which are formed will be highly diluted upon diffusion from the membrane, so we will take Po = 0.
SOLUTION FOR STEADY-STATE KINETICS OF IMMOBILIZED ENZYME SYSTEMS
231
By substituting equations (2.4) and (2.5) into (2.3), recursion relationships may be derived for an:
D'----~an-l-
°_1 (1 - Lm s) i=x y ( 2 ( n - - i ) ( 2 ( n - - i ) - - 1))an_iai Ki Dp]
an =
/Cm
. tz.o)
(2n)(2n--1)IKm+ao+-~i bo ] bn may be derived from an using equation (2.5). These expressions depend on ao and bo (the substrate and product concentrations at x = 0), instead of the external concentrations So and Po. Given ao and bo, however, So and Po may be calculated from equations (2.6) and (2.4). As a result, this computational approach requires that the rate equations be solved 'inside-out' (i.e. starting at the center of the membrane, and proceeding to the surface), rather than the more conventional 'outsidein' approach. This poses no computational problem when the variation in reaction rate over a range of So is required (as in the calculations described in Section 3), since a range ofao may be selected which generates the desired range in So. If the value of the reaction rate for a particular So concentration is required, however, then an iterative scheme must be adopted to determine the appropriate value of do. In the absence of product inhibition (Ki -+ ~), the concentration gradient depends only on the value of ao. In the presence of product inhibition, however, it is necessary to find the appropriate value of bo for a given ao. With the boundary condition Po = 0, the following relationship between a0 and bo may be obtained: bo = - ~ (So -- ao).
/Jp
(2.7)
Since both So and bo depend on ao, an iterative scheme was adopted for obtaining consistent values of these parameters. Initially, bo is set to 0. For a giv6n ao, an are calculated from equation (2.6). The value of So [= S(d)] may be obtained from equation (2.4). For computational purposes, the series expression for S(d) was terminated after N terms, when:
[and~Vl < e
(2.8)
with e = 10 -4. An estimate of bo is then given from (2.7). Using this value for bo, new an coefficients are evaluated, and this cycle is iterated until convergence is obtained (here, when shifts in bo are less than e" So).
232
D.C. REES
3. Results and Discussion.
Compared to more conventional numerical methods (such as finite difference algorithms) for solving the reactiondiffusion equation (2.3), power series solutions have the advantage of providing explicit representations for expressions requiring derivatives or integrals of S(x) and P(x). For example, derivatives of S(x) would be required when concentration profile gradients are determined by the variation in refractive index across a membrane. In addition, the power series formulation permits direct calculation o f S(x) and P(x) at arbitrary values o f x in the membrane. As a result, integrals involving S(x) and P(x) may be easily adapted for solution using powerful Gaussian quadrature algorithms. An important example in this category is the velocity, V, of an enzyme-catalyzed reaction in a membrane, where
1 ca k(E~) S(x)dx V = ~ J-a K m + S(x) + (Km/Ki) P(x)"
(2.9)
In an application o f these methods, the dependence of the reaction velocity on the various parameters o f the enzyme membrane model were examined through test calculations. The variation in V with D, is illustrated in Figure 1 in the form of a Lineweaver-Burk plot o f 1/V vs 1/So for a series of values for Ds. When an enzyme reaction obeys Michaelis-Menten kinetics, apparent 0.04
0.03
"
0.02
-
A 0 Q t,9 v ~>
-
a
b c
0,01.
0.00 0.0
0.2
0.4 1/So
0.6" (mM
-1
0.8
1.0
)
Figure 1. Lineweaver-Burk plots illustrating the effect of varying Ds on the velocity of an immobilized enzyme-catalyzed reaction. For curve (a) D s = 3 × 10-~ cm2/sec; curve (b) D s = 5 × 10-7 cm2/sec; curve (c)D s = 1 X 10 -6 cm2/sec; curve (d) diffusion-free (solution) rate. All other parameters and constants are defined in the text.
SOLUTION FOR STEADY-STATE KINETICS OF IMMOBILIZED ENZYME SYSTEMS
233
values for both k and Km may be determined from Lineweaver-Burk plots since the slope o f the curve is Kin~k, while the ordinate intercept is 1/k. For the calculations illustrated in Figure 1, k = 102/sec, Km = 1 raM, (Et) = 10 mM and d = 10 -4 cm. Reaction velocities were calculated for the absence of product inhibition by taking the limit o f Ki -+ ~ in equations (2.6) and (2.9). As Ds decreases, the Lineweaver-Burk plots become increasingly curved. At sufficiently high substrate concentrations, however, the diffusion limitation is overcome, and the plots coincide with the diffusion-free curve. As a consequence, values of k and Km free o f diffusional effects may be determined in this high substrate concentration regime, assuming no other complicating effects are present. Product inhibition also leads to biphasic Lineweaver-Burk plots, but these plots curve in an opposite fashion to that observed with diffusional effects. Figure 2 illustrates the influence o f variations in Ki on the reaction velocity. In this figure, k = 102/sec, K m = 1 mM, (Et) = 10 mM and Ds = Dp = 10 -6 cm2/sec. The curvature o f the plots becomes more pronounced as Ki decreases. As the substrate concentration increases, V approaches the limiting solution value, k, but the apparent Km values determined under these conditions will be too large. Although diffusional effects and product inhibition influence the shape i
0.04
i
i
0.03 a
"G o c0 v >
b 0.02
•
0.01
•
c j J
0.00
0.0
0.2
0.4 1/8o
0.6
0.8
1,0
(ram -1 )
Figure 2. Lineweaver-Burk plots illustrating the effect of varying Ki on the velocity of an immobilized enzyme-catalyzed reaction. For curve (a) Ki = 0.1 mM; curve (b) Ki = 0.3 mM; curve (c) Ki = 1.0 mM; curve (d) no product inhibition; curve (e) diffusion-free (solution) rate. All other parameters and constants are defined in the text.
234
D.C. REES
of Lineweaver-Burk plots in a distinctive manner, other types o f unrelated kinetic phenomena m a y also generate apparently similar plots. For example, substrate inhibition effects [where the binding o f more than one substrate molecule to the enzyme leads to inhibition (Cleland, 1979)] generates the same type of curved Lineweaver-Burk plot as those observed for diffusional limitation in immobilized enzymes. In systems exhibiting substrate activation (where the binding o f more than one substrate molecule leads to activation of the enzyme) the Lineweaver-Burk plot resembles that seen for product inhibition in immobilized enzymes. Furthermore, combinations of these effects may lead to intermediate behavior, such as the apparently normal Michaelis-Menten kinetics exhibited by curve c in Figure 2, even though this system suffers from both diffusion and product inhibition effects. As a consequence, it is essential to account for many different types of effects before true catalytic parameters may be extracted from kinetic data for immobilized enzyme systems.
LITERATURE Cleland, W. W. 1979. "Substrate Inhibition." Methods Enzymol. 63, 500-513. Doscher, M. and F. Richards. 1963. "The Activity of an Enzyme in the Crystalline State: Ribonuclease S." J. biol. Chem. 238, 2399-2406. Goldstein, L. 1976. "Kinetic Behavior of Immobilized Enzyme Systems." Methods Enzymol. 44,397-443. Laidler, K. and P. Bunting. 1980. "The Kinetics of Immobilized Enzyme Systems." Methods Enzymol. 64,227-248. Segel, I. 1975. Enzyme Kinetics. New York: Wfley-Interscience. RECEIVED 2-17-83 REVISED 6-28-83
Bulletin of Mathematieal Biology Vol. 46, No. 2, pp. 229-234, 1984. Printed in Great Britain
A GENERAL SOLUTION FOR THE STEADY-STATE KINETICS OF IMMOBILIZED ENZYME SYSTEMS [] D. C. REES
Department of Chemistry and Biochemistry, Molecular BiologyInstitute, Universityof California, Los Angeles,CA 90024, U.S.A. A power series solution is presented which describes the steady-state concentration profiles for substrate and product molecules in immobilized enzyme systems. Diffusional effects and product inhibition are incorporated into this model. The kinetic consequences of diffusion limitation and product inhibition for immobilized enzymes are discussed and are compared to kinetic behavior characteristic of other types of effects, such as substrate inhibition and substrate activation.
i. Introduction. Diffusional effects may significantly influence the reaction kinetics of immobilized enzyme systems (reviewed in Goldstein, 1976; Laidler and Bunting, 1.980). Concentration gradients may be generated within these systems owing to consumption and creation of substrate and product molecules, and to diffusional resistances in the channels connecting the external solution and the active site of the enzyme. Most mathematical treatments of these effects have been derived for the limiting case of either very low or very high substrate concentrations, or else require iterative numerical calculations, in order to obtain concentration profiles. In this paper we derive a general power series solution for substrate and product concentration profiles across a planar enzyme membrane which is valid for all substrate concentrations. 2. Model The immobilized enzyme system is modeled as an infinite plane of thickness 2d, perpendicular to the x axis and centered about x = 0. At steady state, rates of change of substrate and product concentrations due to enzyme activity and diffusion are of equal magnitude and opposite sign. For the reaction scheme: kl
E + S----~ES k_ 1
k2
k3
>EP ~---.~. E + P,
(2.1)
k_~
where E, S and P represent enzyme, substrate and product molecules respectively, the steady-state reaction velocity, v, in the absence of diffusion may be calculated (Segel, 1975): 229
230
D.C. REES
k(E t ) (S)
v=
(2.2)
Km + (S) + ( ~ i ) (P) where k = k2k3/(k2 + k3), K m = ((k3(k2 + k-l)/(kx(k2 + k3)), Ki = k3/k-3 and (Et) is the total enzyme concentration. In this mechanism product inhibition by P is competitive, and characterized by an inhibition constant Ki. Combining this expression with Fick's second law for diffusion yields the basic differential equations describing this system (Doscher and Richards, 1963):
zS
2p Ox2
k(Et )
k(Et ) Dv
(S)
(2.3)
(S) Km + ( S ) +
/--(K--7;--./(P) \/~i/
where Ds and D v are the substrate and product diffusion coefficients, respectively, inside the membrane. This treatment neglects the possible influence of boundary diffusion layer effects on the concentration profiles. Experimentally, these effects may be minimized by stirring the solution external to the membrane. The symmetry of the model requires that S and P be even functions of x, so that S(x) and P(x) may be quite generally written as: co
S(x) = Y anx2n (2.4)
P(x) = ~ bnx2n. n=O
Substitution of (2.4) into (2.3) leads to a relationship between an and b n : Ds
bn = - - - - a n ,
Dv
(2.5)
which is valid for n / > 1. The boundary conditions at x = +d are given by S(d) = So and P(d) = Po. Assuming that the volume surrounding the membrane is quite large, product molecules which are formed will be highly diluted upon diffusion from the membrane, so we will take Po = 0.
SOLUTION FOR STEADY-STATE KINETICS OF IMMOBILIZED ENZYME SYSTEMS
231
By substituting equations (2.4) and (2.5) into (2.3), recursion relationships may be derived for an:
D'----~an-l-
°_1 (1 - Lm s) i=x y ( 2 ( n - - i ) ( 2 ( n - - i ) - - 1))an_iai Ki Dp]
an =
/Cm
. tz.o)
(2n)(2n--1)IKm+ao+-~i bo ] bn may be derived from an using equation (2.5). These expressions depend on ao and bo (the substrate and product concentrations at x = 0), instead of the external concentrations So and Po. Given ao and bo, however, So and Po may be calculated from equations (2.6) and (2.4). As a result, this computational approach requires that the rate equations be solved 'inside-out' (i.e. starting at the center of the membrane, and proceeding to the surface), rather than the more conventional 'outsidein' approach. This poses no computational problem when the variation in reaction rate over a range of So is required (as in the calculations described in Section 3), since a range ofao may be selected which generates the desired range in So. If the value of the reaction rate for a particular So concentration is required, however, then an iterative scheme must be adopted to determine the appropriate value of do. In the absence of product inhibition (Ki -+ ~), the concentration gradient depends only on the value of ao. In the presence of product inhibition, however, it is necessary to find the appropriate value of bo for a given ao. With the boundary condition Po = 0, the following relationship between a0 and bo may be obtained: bo = - ~ (So -- ao).
/Jp
(2.7)
Since both So and bo depend on ao, an iterative scheme was adopted for obtaining consistent values of these parameters. Initially, bo is set to 0. For a giv6n ao, an are calculated from equation (2.6). The value of So [= S(d)] may be obtained from equation (2.4). For computational purposes, the series expression for S(d) was terminated after N terms, when:
[and~Vl < e
(2.8)
with e = 10 -4. An estimate of bo is then given from (2.7). Using this value for bo, new an coefficients are evaluated, and this cycle is iterated until convergence is obtained (here, when shifts in bo are less than e" So).
232
D.C. REES
3. Results and Discussion.
Compared to more conventional numerical methods (such as finite difference algorithms) for solving the reactiondiffusion equation (2.3), power series solutions have the advantage of providing explicit representations for expressions requiring derivatives or integrals of S(x) and P(x). For example, derivatives of S(x) would be required when concentration profile gradients are determined by the variation in refractive index across a membrane. In addition, the power series formulation permits direct calculation o f S(x) and P(x) at arbitrary values o f x in the membrane. As a result, integrals involving S(x) and P(x) may be easily adapted for solution using powerful Gaussian quadrature algorithms. An important example in this category is the velocity, V, of an enzyme-catalyzed reaction in a membrane, where
1 ca k(E~) S(x)dx V = ~ J-a K m + S(x) + (Km/Ki) P(x)"
(2.9)
In an application o f these methods, the dependence of the reaction velocity on the various parameters o f the enzyme membrane model were examined through test calculations. The variation in V with D, is illustrated in Figure 1 in the form of a Lineweaver-Burk plot o f 1/V vs 1/So for a series of values for Ds. When an enzyme reaction obeys Michaelis-Menten kinetics, apparent 0.04
0.03
"
0.02
-
A 0 Q t,9 v ~>
-
a
b c
0,01.
0.00 0.0
0.2
0.4 1/So
0.6" (mM
-1
0.8
1.0
)
Figure 1. Lineweaver-Burk plots illustrating the effect of varying Ds on the velocity of an immobilized enzyme-catalyzed reaction. For curve (a) D s = 3 × 10-~ cm2/sec; curve (b) D s = 5 × 10-7 cm2/sec; curve (c)D s = 1 X 10 -6 cm2/sec; curve (d) diffusion-free (solution) rate. All other parameters and constants are defined in the text.
SOLUTION FOR STEADY-STATE KINETICS OF IMMOBILIZED ENZYME SYSTEMS
233
values for both k and Km may be determined from Lineweaver-Burk plots since the slope o f the curve is Kin~k, while the ordinate intercept is 1/k. For the calculations illustrated in Figure 1, k = 102/sec, Km = 1 raM, (Et) = 10 mM and d = 10 -4 cm. Reaction velocities were calculated for the absence of product inhibition by taking the limit o f Ki -+ ~ in equations (2.6) and (2.9). As Ds decreases, the Lineweaver-Burk plots become increasingly curved. At sufficiently high substrate concentrations, however, the diffusion limitation is overcome, and the plots coincide with the diffusion-free curve. As a consequence, values of k and Km free o f diffusional effects may be determined in this high substrate concentration regime, assuming no other complicating effects are present. Product inhibition also leads to biphasic Lineweaver-Burk plots, but these plots curve in an opposite fashion to that observed with diffusional effects. Figure 2 illustrates the influence o f variations in Ki on the reaction velocity. In this figure, k = 102/sec, K m = 1 mM, (Et) = 10 mM and Ds = Dp = 10 -6 cm2/sec. The curvature o f the plots becomes more pronounced as Ki decreases. As the substrate concentration increases, V approaches the limiting solution value, k, but the apparent Km values determined under these conditions will be too large. Although diffusional effects and product inhibition influence the shape i
0.04
i
i
0.03 a
"G o c0 v >
b 0.02
•
0.01
•
c j J
0.00
0.0
0.2
0.4 1/8o
0.6
0.8
1,0
(ram -1 )
Figure 2. Lineweaver-Burk plots illustrating the effect of varying Ki on the velocity of an immobilized enzyme-catalyzed reaction. For curve (a) Ki = 0.1 mM; curve (b) Ki = 0.3 mM; curve (c) Ki = 1.0 mM; curve (d) no product inhibition; curve (e) diffusion-free (solution) rate. All other parameters and constants are defined in the text.
234
D.C. REES
of Lineweaver-Burk plots in a distinctive manner, other types o f unrelated kinetic phenomena m a y also generate apparently similar plots. For example, substrate inhibition effects [where the binding o f more than one substrate molecule to the enzyme leads to inhibition (Cleland, 1979)] generates the same type of curved Lineweaver-Burk plot as those observed for diffusional limitation in immobilized enzymes. In systems exhibiting substrate activation (where the binding o f more than one substrate molecule leads to activation of the enzyme) the Lineweaver-Burk plot resembles that seen for product inhibition in immobilized enzymes. Furthermore, combinations of these effects may lead to intermediate behavior, such as the apparently normal Michaelis-Menten kinetics exhibited by curve c in Figure 2, even though this system suffers from both diffusion and product inhibition effects. As a consequence, it is essential to account for many different types of effects before true catalytic parameters may be extracted from kinetic data for immobilized enzyme systems.
LITERATURE Cleland, W. W. 1979. "Substrate Inhibition." Methods Enzymol. 63, 500-513. Doscher, M. and F. Richards. 1963. "The Activity of an Enzyme in the Crystalline State: Ribonuclease S." J. biol. Chem. 238, 2399-2406. Goldstein, L. 1976. "Kinetic Behavior of Immobilized Enzyme Systems." Methods Enzymol. 44,397-443. Laidler, K. and P. Bunting. 1980. "The Kinetics of Immobilized Enzyme Systems." Methods Enzymol. 64,227-248. Segel, I. 1975. Enzyme Kinetics. New York: Wfley-Interscience. RECEIVED 2-17-83 REVISED 6-28-83