Enzyme Technology
Effects of solute diffusion on the kinetics of immobilised enzymes
In order for an immobilised enzyme to catalyse a reaction,
the substrates must be able to diffuse through the solution to the catalytically
active sites and the products diffuse away into the bulk solution. The driving
force for the net diffusive process is due to the concentration gradients,
solutes moving in the direction of higher to lower concentration. The substrates
approach the surface of the enzyme particles through the surrounding thin
stagnant unstirred layer of solution and then diffuse into any pores where they
may encounter active enzyme. The net movement of the solutes is described in
terms of these two steps;
- external diffusion where the
transport of substrates towards the surface, and products away, is in series
with the catalytic conversion of substrates to products occurring at the surface
(the processes being consecutive)
- internal diffusion
where the transport of the substrates and products, within the pores of
immobilised enzyme particles, is in parallel with the catalysed reaction (the
processes being simultaneous).
The concentration gradients caused by diffusion and partition
are shown diagrammatically in Figure 3.10. The rate
of a reaction catalysed by an immobilised enzyme (v) is normally lower than
the rate due to the same amount of free enzyme in solution (vfree).
This is due to the controlling necessity for the substrate to diffuse from the
bulk phase to the catalytic surface. The substrate concentration within the
microenvironment ([S]) is lower than that in the bulk ([S0]) due
to its depletion by the reaction. The change in reaction rate can be expressed
quantitatively by introducing the effectiveness factor (h),
where:
h= v/vfree
(3.9)
The effectiveness factor generally
lies between 0 and 1 and is dependent on the bulk substrate concentration,
amongst other factors (see later). It may sometimes be greater than unity due to
non-isothermal operation, because of partition or inhibitory effects, or if the
immobilised enzyme is stabilised relative to the free enzyme over the time
course of its assay.
Figure 3.10. Schematic diagram showing the
concentration gradients of substrate and product that may be produced around a
porous particle of an immobilised enzyme. (a) concentration gradient due solely
to reaction and internal diffusion within the particle; (b) as (a) but with
additional concentration gradients due to partition of substrate and product
into the microenvironment; (c) as (a) but with an additional concentration
gradients due to external diffusion to the surface of the particle; (d)
concentration gradients due to the combined effects of partition and diffusion.
The partition boundary layer is normally about a thousand-fold thinner than the
diffusive boundary layer. The actual concentration gradients will not show the
sharp discontinuities which are shown here for simplicity.
The effect of
external diffusion on the rate of an enzyme catalysed reaction may be simply
derived, assuming (1) Michaelis-Menten kinetics, (2) the enzyme is immobilised
to a flat impervious support, and (3) the absence of any partitioning or
electrostatic effects. If the reaction is occurring under steady state
conditions, the rate of increase of product within the bulk of the solution must
equal the rates of three consecutive processes; the rate at which substrate
diffuses to the surface, the rate of enzymic catalysis, and the rate at which
the product diffuses away from the surface. The steady state assumption is
generally valid if the volume of the bulk of the solution is sufficiently large
such that the variation in [S0] with time may be ignored. Immobilised
enzymes are open systems, where both energy and material are exchanged through
the boundary with the environment. This allows steady state operation even at
very high enzyme loading. In any circumstances, it is clear that the substrate
concentration at the catalytic surface cannot continuously increase or decrease
at a substantial rate compared with the rate of reaction.
Therefore, from
equation 1.8,
(3.10)
where A is the total surface area and
Vmax is the maximum rate of reaction catalysed by unit area of
surface. Combining equations 1.8 and 3.10 gives,
(3.11)
As e varies with
[S0], the
immobilised enzyme will no longer show hyperbolic kinetics. Eadie-Hofstee plots
(see Figure 3.18) derived from the relationship 3.11 are not linear because they
obey the transformation equation,
(3.12)
As this also
involves a term in e which varies, equalling unity when v equals
Vmax at the intercept with the vertical v axis but which may be much
lower when v is much less than Vmax at the intercept with the
horizontal v/[S0] axis.
Assuming that all of the surface is equally
accessible, the rate of flow of substrate to the surface has been found to be
proportional both to the surface area and the difference in substrate
concentration between the bulk of the solution and the microenvironment next to
the surface. It is given by the relationship,
(3.13)
The proportionality constant
(kL) is known as the (local liquid phase) mass transfer
coefficient (with units of m s−1) which depends upon the
diffusivity of the substrate and the effective distance between the surface and
the bulk phase.
(3.14)
where DS is the substrate diffusivity (diffusion
coefficient) in free solution (with units of m2 s−1) and d
is the effective thickness of the unstirred layer through which the
substrate must diffuse. The diffusivity is defined by Fick's law as the rate at
which unit mass of the compound (m) travels through a unit surface area (A) due
to a concentration gradient of unit density (r)
change.
(3.15)
depends upon the molecular weight and dimensions of
the substrate, and the temperature, viscosity and composition of the liquid
phase (Table 3.6). In general terms, the higher the molecular weight and the
solution viscosity, the lower will be the diffusivity. d
may be
effectively regarded as a distance although it is not precisely defined by the
thickness of the unstirred stagnant layer as liquid motion may be detectable at
distances less than d from the surface. Under stagnant flow conditions
d is equal to the particle radius for spherical particles. It depends on
the hydrodynamic conditions, being reduced by increases in the rate of stirring
and consequent increasing particle-fluid relative velocity.; this reduction in d
causing an increase in the mass transfer coefficient (kL).
For example, using particles of 400 mm diameter, d
has been
found to be 5 mm when used in a packed bed reactor (see
Chapter 5) at a
reactant stream flow rate of 1 m s−1. Dependent upon the conditions,
d also may vary with the diffusivity, and the density and viscosity of
the liquid; increasing with increasing diffusivity and viscosity but reducing
with increasing density difference between the immobilised biocatalyst and the
medium. It may be halved by the use of ultrasound, which may be particularly
useful for larger particles. For the small biocatalytic particles normally used,
the maximum reduction in d that is achievable by increasing the
turbulence of the solution around the immobilised enzyme is about ten fold.
The
rate of diffusion is significantly affected where partitioning occurs. For
charged molecules, this will depend on the electrostatic potential gradient in
addition to the concentration gradient. The electrostatic potential gradient
causes apparent changes in both d
and DS within the immediate
vicinity of the surface (i.e., at distances very much smaller than d).
This may be particularly relevant in the case of hydrogen ion diffusion (see
later) as hydrogen ions move rapidly through solutions across electrostatic
gradients due to their ability to rapidly change their association with water
molecules.
Table 3.6 Diffusivity of
molecules in aqueous solution at 20°C
Substance
|
Molecular weight
|
Diffusivity
(m2 s−1)
|
Oxygen
|
32
|
21.0 x10−10
|
Glucose
|
180
|
6.7 x10−10
|
Sucrose
|
342
|
4.5 x10−10
|
Inulin
|
5,200
|
2.3 x10−10
|
Albumin
|
67,000
|
0.7 x10−10
|
Urease
|
480,000
|
0.3 x10−10
|
Bushy stunt virus
|
10,700,000
|
0.1 x10−10
|
Due to the consecutive
nature of the process the rate of enzymic reaction (given by v in equation
3.10)
must equal the rate of diffusion of the substrate to the catalytic surface
(given by v in equation 3.13); the terms in A cancelling out.
(3.16)
This
equation is quadratic with respect to the microenvironmental substrate
concentration ([S]), the value of which is very difficult to establish by
independent means. It may be simplified under extreme values of [S] relative to
the Km of the enzyme for the substrate.If [S] is much greater than
Km, the left-hand side of equation 3.16 simplifies to give just
Vmax
(3.17)
where vA is the rate of reaction
catalysed by unit area of the immobilised enzyme surface. It follows that the
rate of the reaction is equal to the maximum rate of reaction of the
non-immobilised enzyme when [S], and hence [S0], is much greater than
the Km. Often, however, it is found that [S] is much less than
Km. Under such conditions, equation 3.16 gives
(3.18)
collecting terms in [S],
(3.19)
(3.20)
From
equation 3.10, when [S] is much less than Km,
(3.21)
Substituting for [S] from
equation 3.20,
(3.22)
(3.23)
The relative values of the two
components of the denominator, in equation 3.23, determine whether the reaction
is controlled primarily by the diffusion of the substrate (the kL
term) or by the catalytic ability of the immobilised enzyme (the
Vmax/Km term). The comparison may be made by means of the
introduction of a substrate modulus (m, also known as the
Damkφhler number and is the dimensionless ratio of
reaction velocity to transport velocity) defined by,
(3.24)
where
Vmax is the maximum rate of reaction catalysed by unit surface area.
Substituting this in equation 3.22,
(3.25)
The relationships between the rates
of reaction and the immobilised enzyme and bulk substrate concentrations are
shown in Figures 3.11 and 3.12. When
kL is much greater than
Vmax/Km (i.e., at zero m, when mass transport is
capable of a much faster rate than that of the enzyme catalysed reaction), the
overall rate of the process is under the kinetic control of the enzyme, which is
then as effective as the free enzyme (i.e., m
= 1). This allows the
simplification of equation 3.23 to yield
(3.26)
However, when kL is much less than Vmax/Km, (i.e., at high
substrate modulus, when mass transport is much slower than the intrinsic rate of
the enzyme catalysed reaction), the overall rate of the process is under the
control of the rate of diffusion of the substrate. Equation 3.23 then simplifies
to give:
(3.27)
This
last relationship, apart from its obvious utility as a method for determining
mass transport coefficients, is very important for the proper understanding of
the behaviour of immobilised enzymes under the diffusional control of external
transport of the substrate. The rate of reaction is shown to be independent of
the activity of the enzyme. This means that it is not affected by changes in the
pH, temperature (except as it may affect viscosity) or ionic strength of the
solution, nor is it affected by the presence of inhibitors or activators. If,
however, the ratio Vmax/Km is reduced substantially by
changes in these conditions to approach the value of kL, the
relationship shown in equation 3.27 will no longer hold true. As diffusional
limitations approach a total controlling influence, the behaviour of the enzyme
with respect to these factors gradually changes from that of a free enzyme in
solution to the state of being unaffected. For example, increasing diffusional
control causes a broadening of the pH-activity profile and a lowering of the
activation energy (and the related Q10), both of which will be more
apparent at lower substrate concentrations. Under such conditions, the bulk
substrate concentration that gives half the maximum rate of reaction ([S1/2], equivalent to the apparent Km,
Kmapp) is higher than the Km of the free enzyme (see
Figure 3.12). This is in contrast to the case of the free enzyme where [S1/2] is identical to the Km. The equivalent
microenvironmental substrate concentration giving half-maximal reaction rate
remains equal to the Km.
Figure 3.11. Variation in the rate of reaction
catalysed by an immobilised enzyme with its concentration. The relationship
shows three phases; (a) kinetic control by the enzyme, extrapolated (-----) to
show the activity of equivalent amounts of free enzyme; (b) mixed intermediate
control; (c) control by the rate of external transport of substrate.
Figure 3.12. This shows the variation in the rate
of reaction catalysed by an immobilised enzyme and the dimensionless bulk
substrate concentration (b0, which equals
[S0]/Km) with the substrate modulus (m, defined by
equation 3.24). (a) free enzyme; (b) m = 1; (c) m = 3; (d)
m = 10; (e) m = 100. Also drawn are the maximum rates of
substrate diffusion to the surface, given by vA/Vmax = kLb0 (b',
c', d' and e',
showing the effect of decreasing mass transfer coefficients, kL =
1.00, 0.33, 0.10 and 0.01, corresponding to curves b, c, d and e respectively).
It should be noticed that curves b, c, d and e are bounded both by curve a and
by the lines b', c', d' and e' respectively. Increased diffusional control extends
the range of linearity at lower substrate concentrations, but the same
Vmax is reached in all cases if a sufficiently high substrate
concentration can be achieved (see Figure 3.27a).
vA may be substituted by Vmax/2 in
equation 3.13 when [S] equals Km, giving:
(3.28)
Therefore:
(3.29)
Therefore:
(3.30)
The
Km apparently increases with the substrate modulus. This causes a
reduction in the apparent specificity. The introduction of the dimensionless
substrate concentrations b (= [S]/Km) and b0
(= [S0]/Km) into equation 3.16 gives,
(3.31)
Substituting m from equation
3.24,
(3.32)
This may be simplified at low b0, when
b approaches zero,
(3.33)
and,
(3.34)
Therefore, if the value of m is known, the
concentration of the substrate at the surface of the immobilised enzyme may be
obtained.
The most important
factors to arise from this analysis concern the consequences of immobilisation
on the effective catalytic ability of the enzyme. It is clear that the
effectiveness factor (e) must vary with both b0 and
m, being reduced by low b0 and high
m; this
relationship being illustrated in Figure 3.13. The conditions that produce an
increased probability of external diffusional control over the rate of an
immobilised enzyme catalysed reaction may be summarised as follows:
- High
enzyme loading on the surface.
- Low bulk substrate concentration.
- Low
substrate diffusivity.
- Low Km.
- High enzymic specificity.
- Low
rate of stirring or mixing.
- Flat surfaces (e.g., large average particle
diameter; see Figure 3.14).
Figure 3.13. The combined effect of the bulk
substrate concentration (b0) and substrate modulus (m)
on the effectiveness factor (h). The plateau (a) is an area of kinetic
control, the surface dropping through an area of intermediate control (b) to an
area of diffusional control (c).
Figure 3.14. Schematic diagram showing the effect
of particle diameter on the effectiveness factor (e) of immobilised
enzymes. small surface concentration of enzyme;
--------- high
surface concentration of enzyme.
Diffusion-free enzyme kinetics can be simply determined by
decreasing the loading of enzyme on the immobilisation support or by lowering
the temperature. It is often necessary to determine the kinetic parameters of an
immobilised enzyme in the presence of external diffusional effects. This may be
in order to investigate the effect of immobilisation on the intrinsic stability
or activity of the enzyme. The intrinsic Vmax can be determined if
sufficiently high microenvironmental substrate concentrations can be achieved
(see Figure 3.27a) but determination of the intrinsic Km depends upon
knowledge of the microenvironmental substrate concentration. This may be
graphically determined if the mass transfer coefficient is known (Figure
3.15).
Figure 3.15. Diagram illustrating a method for the
determination of the intrinsic kinetic parameters (Vmax and
Km). The value of kL may be determined from the tangent of
the experimental curve at the origin (line a). For each experimentally
determined (rate : bulk substrate concentration) data point (x, on curve b) a
line is drawn of gradient -kL from the appropriate position on the
horizontal axis. The microenvironmental substrate concentration is given by the
intercept of this line with the vertical position representing the reaction
rate. These graphically determined (rate : microenvironmental substrate
concentration) data points (o, on curve c) may be used to calculate the
intrinsic kinetic parameters.
The substrate concentration
gradient, within the microenvironment, is only perfectly linear next to flat
surfaces (see equation 3.13). The more usual situation is that of enzymes
attached to curved surfaces (e.g., spherical particles, the inside of cylindrical
tubes or the outside of cylindrical fibres). These are found to produce
non-linear concentration profiles (Figure 3.16). This is due to the substrate
molecules approaching the surface through convergent or divergent pathways (e.g., substrate molecules diffusing towards the surface of a spherical particle pass
through successively decreasing volumes and areas, the concentration gradient
increasing with decreasing radius, in order to retain the same flux of molecules
throughout).
Figure 3.16. Concentration profiles of substrate
molecules approaching the curved surface of immobilised enzymes, where r is the
radius of curvature. (a) enzyme attached to spherical particles; (b) enzyme
attached to the outside of cylindrical fibres; (c) enzyme attached to the inside
of cylindrical tubes. r/d = 9;
---------- r/d = 0.11.
In each case, m = 10−3, 1 and
103 for the top,
middle and bottom pair of curves, respectively.
Inhibitors may affect
the rate of an immobilised enzyme catalysed reaction in a different manner than
that in which they affect the free enzyme. The relative affects of reversible
competitive, uncompetitive and noncompetitive inhibitors can be understood by
consideration of the resultant changes in the apparent kinetic constants
Kmapp and Vmaxapp. If mi
represents the value of the substrate modulus (m) in the presence of the
inhibitor and using equation 3.24,for competitive inhibition, from
equation 1.84,
(3.35)
for uncompetitive inhibition, from equations
1.93 and 1.94,
(3.36)
for
noncompetitive inhibition, from equation 1.98,
(3.37)
Competitive and noncompetitive
inhibition both show a reduction in mi relative to m.
Substrate diffusional resistance will, therefore, have less affect on these
inhibited processes than on the uninhibited immobilised enzyme. This is because
such inhibited reactions are inherently slower and less likely to be controlled
by the rate of substrate diffusion. No affect is noticed in the case of
uncompetitive inhibition as this has negligible affect on the rate of reaction
at low substrate concentrations. Even in the former two cases the affect of the
degree of inhibition becomes negligible at high substrate modulus (m
> 50), when equation 3.27 holds.
Product inhibition may be more severe in the
case of immobilised enzymes due to product having to diffuse away from the site
of reaction. Its concentration is likely to be much higher within the
microenvironment than in the bulk of the solution (Figure
3.10). At the
beginning of the reaction, this product concentration will build up until the
concentration gradient to the bulk macroenvironment is sufficient to allow it to
diffuse away at an equal rate to its production by the reaction. If
kLS and kLP represent the substrate and product
mass transfer coefficients, and [P] and [P0] represent the product
concentrations in the microenvironment and bulk respectively, then from equation
3.13,
(3.38)
Under diffusional
control, [P] and [S0] are much greater than [P0] and [S] respectively.
(3.39)
Substituting
for [P] in equation 1.85 and assuming that Km is much greater than
[S],
(3.40)
The
effect of product inhibition (the second term in the denominator) depends on the
bulk substrate concentration and the ratio
KmkLS/(KPkLP) which
expresses the competition between the substrate and the product for the enzymic
surface. The build-up of the product at the surface increases with this ratio
causing a reduction in both the rate of reaction and the effectiveness factor.
The effect is greater when Km/KP or
kLS/kLP are large and m is small (<
50). kLS is usually approximately equal to kLP
but may be higher where the reaction involved produces a higher molecular weight
product from more than one substrate molecule, or substantially lower in a
depolymerisation (e.g., hydrolysis) reaction.
Substrate inhibition presents a
somewhat more complex scenario. Equations 1.96 and
3.13 may be combined to give,
(3.41)
This equation is third order with respect
to the microenvironmental substrate concentration ([S]). For this reason, it is
not surprising that multiple steady states are possible, provided that diffusion
of the substrate to the enzyme is sufficiently slow (Figure
3.17). The same bulk
substrate concentration may give two different stable concentrations within the
microenvironment; (1) a low concentration which gives a relatively fast rate of
reaction, without much substrate inhibition, and equally fast rate of inward
diffusion of substrate due to the steep concentration gradient, and (2) a much
higher concentration which gives a relatively slow rate of reaction, due to the
substrate inhibition, and equally slow rate of inward diffusion of substrate
down the relatively slight concentration gradient. A third possibility exists of
an intermediate concentration representing an unstable state which is not
naturally established and is of no practical consequence (Figure
3.17). The
choice of which stable state exists depends on the start-up conditions. Addition
of the substrate under conditions of low substrate modulus (e.g., at a low
temperature or with vigorous stirring) allowing an initially high
microenvironmental substrate concentration to be achieved and favouring the
stable state involving higher [S] as the temperature and substrate modulus are
raised. The alternative stable state may be reached by addition of the substrate
under conditions of high substrate modulus where the microenvironmental
substrate concentration is initially zero and kept low.
Figure 3.17. The effect of substrate inhibition on
the rate of immobilised enzyme reactions.
immobilised enzyme vs
microenvironmental substrate concentration (or free enzyme vs bulk substrate
concentration). The line ---------- represents the rate of diffusion of the
substrate from the bulk (b0 =
8) to the microenvironment
(i.e., rate = kL(b0
- b); lower microenvironmental
substrate concentration, relative to the bulk concentration, giving the higher
rate.
Reversible reactions catalysed by immobilised enzymes may
be severely affected by the slow diffusion of the product away from the
catalytic surface. Even a slight build-up in the microenvironmental product
concentration will increase the reverse rate of reaction, severely reducing the
productivity of the enzyme. An analysis of the effect of this diffusional
resistance may be made by combining equations 1.68 and 3.38, which then describe
the resultant lowering of the rates of reaction and effectiveness factor. Figure
3.18 shows Eadie-Hofstee plots comparing the effect of reaction reversibility at
increasing external substrate modulus.
Figure 3.18. Eadie-Hofstee plots illustrating the
effect of external diffusional resistance on the kinetics of an immobilised
enzyme. Essentially non-reversible (a) and reversible (b) reactions are shown.
The reversible situation has been modelled on the glucose isomerase reaction
using values of 1.14 for the Keq and 42% and 51% (w/w, of carbohydrate
present) for the initial bulk concentration of fructose and glucose,
respectively. The lines represent the kinetically controlled reaction and
increasing substrate modulus (m), as shown. (a) corresponds to
Figure 3.12.
Diffusional resistance can lead to an apparent increase in
the stability of immobilised enzymes. The reason for this is the ceiling that
the diffusional resistance imposes upon the activity. At a high loading of
active enzyme, the activity obeys equation 3.27 and is independent of the
inherent activity of the immobilised enzyme. Under these circumstances, there is
always sufficient reaction occurring to remove the substrate as it arrives from
the bulk solution, most of the enzyme molecules being effectively redundant
(effectiveness factor close to zero). The productivity of the immobilised
biocatalyst remains constant until the specific activity of the enzyme has
decayed sufficiently that the reaction is no longer diffusionally controlled (Figure
3.19). Some of the early researchers in this field who used data
collected only during the initial diffusionally controlled period were misled
into believing that the immobilisation process was more likely to stabilise an
enzyme than is now realised.
Figure 3.19. The apparent stabilisation of an
immobilised enzyme, when diffusionally controlled.
utilisable
activity; ---------- intrinsic activity, not realised above the activity ceiling
imposed by the slower (rate-controlling) diffusion of the substrate to the
enzyme.
A special case of diffusional control
of immobilised enzyme reactions concerns hydrogen ions. Many enzyme catalysed
reactions involve the release or consumption of H+ (e.g., dehydrogenases,
peptidases and esterases). This may include any reaction involving molecules
containing ionisable groups as the pKas of such groups on the substrates
and products may differ. Sometimes the reaction may produce or consume hydrogen
ions dependent on the pH of the reaction (e.g., urease, see
Table 6.2). Hydrogen ions diffuse relatively slowly, similar to other monovalent
ions, in the absence of an electrical field. This is due to four main causes,
(1) H+ is normally hydrated (H30+, H904+)
increasing its apparent molecular weight and hence reducing its diffusion coefficient,
(2) the need for localised electroneutrality necessitates the co-diffusion of
positively and negatively charged species, again causing an increase in the
effective molecular weight (i.e., each H+ diffuses with an anionic
counterion), (3) hydrogen ions are buffered by histidine and other groups on
the immobilised enzyme particles which reduce their ability to diffuse into
the bulk liquid phase, (4) normally the H+ concentration in the bulk
of the solution is low (e.g., pH 7 is equivalent to 10−7 M H+)
which only allows the production of very small concentration gradients even
at pH differences of 1 or 2. The shallowness of H+ concentration
gradients are especially noticeable when compared with those normally encountered
for other substrates or products (greater than mM). For these reasons many reactions
may be controlled by the diffusion of hydrogen ions to, or away from, immobilised
enzyme surfaces. The effect of this diffusional control may be examined by combining
equations 1.17 and 3.13 for a reaction involving the production of H+.
(3.42)
where V* is the maximum rate
of reaction if pKa1 and pKa2 are well separated, kLH is the mass transfer coefficient of H+, [H+] is the microenvironmental concentration of
H+, and [H+,80] is the bulk concentration of H+. If the
reaction involves the consumption of hydrogen ions, the reaction obeys the
following related equation
(3.43)
The magnitude of the diffusional
resistance is given by the proton modulus
(mH), where
(3.44)
Note
the similarity in the definitions of the proton and substrate moduli (equation
3.24).
Figure 3.20. The effect of diffusional control on the
local pH and pH-activity profile of an immobilised enzyme catalysed reaction.
(a) the variation of surface pH with bulk pH at various proton moduli
(mH);
H <=
10−5; -------- mH = 10−4,
mH =
10−3, mH =
10−1 and mH = 10. The reduction in microenvironmental pH is most
noticeable at high pH due to the much lower hydrogen ion concentration
gradients, and hydrogen ion diffusion rates, present (e.g., a pH difference of 1
at pH 12 produces a concentration gradient only a millionth of that produced at
pH 6). (b) the effect of the surface pH on the pH-activity profile. Note that
mH and hence the pH-activity profile will vary with the flow
rate and degree of turbulence. If the reaction is started at pH = 10 and
mH = 0.1, the rate of hydrogen ion production is initially low
(initial rate shown) but faster than the rate that the hydrogen ions can diffuse
away. This causes a drop in the microenvironmental pH towards the optimal pH,
increasing the rate of reaction. This process continues until the concentration
gradient between the microenvironmental and bulk phases is sufficient that the
rate of diffusion equals the rate of reaction (final rate shown).
It is found that hydrogen
ions accumulate at the surface when the proton modulus is greater than about
10,000 (Figure 3.20). Such reactions may be operated at a significant rate, well
away from the pH optimum of the free enzyme. This may be very useful in cases
where such a pH allows higher substrate solubilities or a more favourable
process environment for the reaction. The diffusion of hydrogen ions may be
facilitated by conjugate acid-base pairs, even at concentrations appreciably
lower than those needed for their conventional buffering action. The reason for
this is that the 'buffers' are capable of far greater concentration gradients
than hydrogen ions, without necessarily affecting the bulk pH (Figure
3.21). In
this case, equation 3.42 must be extended to include the diffusional transport
of hydrogen ions by the conjugate acid. For a reaction producing H+
this becomes,
(3.45)
where kLHB is the mass
transfer coefficient of HB, [HB] is the micro-environmental concentration of the
conjugate acid (HB) and [HB0] is the bulk concentration of HB. At a
bulk pH of 7 about a thousand times more hydrogen ions are transported by a
millimolar buffer with a pKa of 7 than as free hydrogen ion. The
presence of these buffering ions can significantly affect the pH-activity
profile of reactions limited by hydrogen ion diffusion (Figure
3.22).
Figure
3.21. Schematic diagram showing the facilitated transport of hydrogen
ions away from an immobilised enzyme catalysing a reaction producing hydrogen
ions. The hydrogen ion concentration gradient is small due to the low
concentration of hydrogen ions in the bulk and the inability to produce a
substantial hydrogen ion concentration (e.g., pH < 3) in the microenvironment.
The buffer, represented by the conjugate pair HB/B−, removes protons
from the surface down much steeper concentration gradients, dependent on its
concentration and pKa relative to the pH of the microenvironmental and
bulk phases. The proton removal and delivery reactions occurring in these
microenvironmental and bulk phases, respectively, are shown at the bottom.
Figure 3.22. The effect of facilitated transport of
hydrogen ions away from an immobilised enzyme catalysing a reaction producing
hydrogen ions on the pH-activity profile (under conditions when m is
10). no buffer; ---------- high buffer concentration, pKa =
7; ········· low buffer concentration, pKa = 7. The buffer is only
effective, at facilitating transport, close to its pKa as at low pH
there is insufficient base present which binds the hydrogen ions and at high pH
very little binding can occur due to the unfavourable equilibria.
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