Enzyme Technology
Simple kinetics of enzyme action
It is
established that enzymes form a bound complex to their reactants (i.e., substrates) during the course of their catalysis and prior to
the release of products. This can be simply illustrated, using the mechanism
based on that of Michaelis and Menten for a one-substrate reaction, by the
reaction sequence:
Enzyme + Substrate(Enzyme-substrate complex)
Enzyme + Product
[1.7]
where k+1, k-1 and
k+2 are the respective rate constants, typically having values
of 105 - 108 M−1 s−1, 1 - 104
s−1 and 1 - 105 s−1 respectively; the sign
of the subscripts indicating the direction in which the rate constant is acting.
For the sake of simplicity the reverse reaction concerning the conversion of
product to substrate is not included in this scheme. This is allowable (1) at
the beginning of the reaction when there is no, or little, product present,
or (2) when the reaction is effectively irreversible. Reversible reactions are
dealt with in more detail later in this chapter. The rate of reaction (v) is
the rate at which the product is formed.
(1.1)
where [ ] indicates the molar concentration
of the material enclosed (i.e., [ES] is the concentration of the enzyme-substrate
complex). The rate of change of the concentration of the enzyme-substrate
complex equals the rate of its formation minus the rate of its breakdown,
forwards to give product or backwards to regenerate substrate.
therefore:
(1.2)
During the course of the
reaction, the total enzyme at the beginning of the reaction ([E]0, at
zero time) is present either as the free enzyme ([E]) or the ES complex ([ES]).
i.e.
[E]0 = [E] +
[ES] (1.3)
therefore:
(1.4)
Gathering terms together,
this gives:
(1.5)
The
differential equation 1.5 is difficult to handle, but may be greatly simplified
if it can be assumed that the left hand side is equal to [ES] alone. This
assumption is valid under the sufficient but unnecessarily restrictive steady
state approximation that the rate of formation of ES equals its rate of
disappearance by product formation and reversion to substrate (i.e., d[ES]/dt is
zero). It is additionally valid when the condition:
(1.6)
is valid. This occurs during a substantial part of the
reaction time-course over a wide range of kinetic rate constants and substrate
concentrations and at low to moderate enzyme concentrations. The variation in
[ES], d[ES]/dt, [S] and [P] with the time-course of the reaction is shown in
Figure 1.2, where it may be seen that the simplified equation is valid
throughout most of the reaction.
Figure 1.2.
Computer simulation of the progress curves of d[ES]/dt (0 - 10−7 M
scale), [ES] (0 - 10−7 M scale), [S] (0 - 10−2 M
scale) and [P] (0 - 10−2 M scale) for a reaction
obeying simple Michaelis-Menten kinetics with k+1
= 106
M−1 s−1, k-1 = 1000 s−1,
k+2 =
10 s−1, [E]0 = 10−7 M and [S]0 = 0.01 M.
The simulation shows three distinct phases to the
reaction time-course, an initial transient phase which lasts for about a
millisecond followed by a longer steady state phase of about 30 minutes when
[ES] stays constant but only a small proportion of the substrate reacts. This is
followed by the final phase, taking about 6 hours during which the substrate is
completely converted to product.
is much less than [ES] during both of the latter two
phases.
The
Michaelis-Menten equation (below) is simply derived from equations 1.1 and 1.5,
by substituting Km for
. Km is known as the
Michaelis constant with a value typically in the range
10−1 - 10−5 M. When k+2<<k-1,
Km equals the dissociation constant (k-1/k+1) of
the enzyme substrate complex.
(1.7)
or, more simply
(1.8)
where
Vmax is the maximum rate of reaction, which occurs
when the enzyme is completely saturated with substrate (i.e., when [S] is very
much greater than Km, Vmax equals k+2[E]0, as the maximum value [ES] can have is
[E]0 when [E]0 is less than [S]0). Equation 1.8
may be rearranged to show the dependence of the rate of reaction on the ratio of
[S] to Km,
(1.9)
and the rectangular hyperbolic nature of the
relationship, having asymptotes at v = Vmax and [S] = -Km,
(Vmax-v)(Km+[S])=VmaxKm
(1.10)
The substrate concentration in these
equations is the actual concentration at the time and, in a closed system, will
only be approximately equal to the initial substrate concentration
([S]0) during the early phase of the reaction. Hence, it is usual to
use these equations to relate the initial rate of reaction to the initial, and
easily predetermined, substrate concentration (Figure 1.3). This also avoids any
problem that may occur through product inhibition or reaction reversibility (see
later).
Figure 1.3. A normalised plot of the initial rate (v0)
against initial substrate concentration ([S]0) for a reaction obeying
the Michaelis-Menten kinetics (equation 1.8). The plot has been normalised in
order to make it more generally applicable by plotting the relative initial rate
of reaction (v0/Vmax) against the initial substrate
concentration relative to the Michaelis constant ([S]0/Km,
more commonly referred to as b, the dimensionless substrate
concentration). The curve is a rectangular hyperbola with asymptotes at
v0 = Vmax and [S]0 = -Km. The tangent to
the curve at the origin goes through the point (v0 =
Vmax),([S]0 = Km). The ratio
Vmax/Km is an important kinetic parameter which describes
the relative specificity of a fixed amount of the enzyme for its substrate (more
precisely defined in terms of kcat/Km). The substrate
concentration, which gives a rate of half the maximum reaction velocity, is
equal to the Km.
It has been established that
few enzymes follow the Michaelis-Menten equation over a wide range of
experimental conditions. However, it remains by far the most generally
applicable equation for describing enzymic reactions. Indeed it can be
realistically applied to a number of reactions which have a far more complex
mechanism than the one described here. In these cases Km remains an
important quantity, characteristic of the enzyme and substrate, corresponding to
the substrate concentration needed for half the enzyme molecules to bind to the
substrate (and, therefore, causing the reaction to proceed at half its maximum
rate) but the precise kinetic meaning derived earlier may not hold and may be
misleading. In these cases the Km is likely to equal a much more
complex relationship between the many rate constants involved in the reaction
scheme. It remains independent of the enzyme and substrate concentrations and
indicates the extent of binding between the enzyme and its substrate for a given
substrate concentration, a lower Km indicating a greater extent of
binding. Vmax clearly depends on the enzyme concentration and for
some, but not all, enzymes may be largely independent of the specific substrate
used. Km and Vmax may both be influenced by the charge and
conformation of the protein and substrate(s) which are determined by pH,
temperature, ionic strength and other factors. It is often preferable to
substitute kcat for k+2, where
Vmax = kcat[E]0, as the precise meaning of
k+2, above, may also be misleading. kcat is also known as
the turnover number as it represents the maximum number of
substrate molecules that the enzyme can 'turn over' to product in a set time (e.g., the turnover numbers of a-amylase, glucoamylase and glucose isomerase
are 500 s−1, 160 s−1 and 3 s−1 respectively; an
enzyme with a relative molecular mass of 60000 and specific activity 1 U
mg−1 has a turnover number of 1 s−1). The ratio
kcat/Km determines the relative rate of reaction at low
substrate concentrations, and is known as the specificity
constant. It is also the apparent 2nd order rate constant at
low substrate concentrations (see Figure 1.3), where
(1.11)
Many applications
of enzymes involve open systems, where the substrate concentration remains
constant, due to replenishment, throughout the reaction time-course. This is, of
course, the situation that often prevails in vivo. Under these
circumstances, the Michaelis-Menten equation is obeyed over an even wider range
of enzyme concentrations than allowed in closed systems, and is commonly used to
model immobilised enzyme kinetic systems (see Chapter 3).
Enzymes have evolved by
maximising kcat/Km (i.e., the specificity constant for the
substrate) while keeping Km approximately identical to the naturally
encountered substrate concentration. This allows the enzyme to operate
efficiently and yet exercise some control over the rate of reaction.
The
specificity constant is limited by the rate at which the reactants encounter one
another under the influence of diffusion. For a single-substrate reaction the
rate of encounter between the substrate and enzyme is about 108 -
109 M−1 s−1. The specificity constant of some
enzymes approach this value although the range of determined values is very
broad (e.g., kcat/Km for catalase is 4 x 107
M−1 s−1, whereas it is 25 M−1 s−1 for
glucose isomerase, and for other enzymes varies from less than 1 M−1
s−1 to greater than 108 M−1
s−1).
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This page was established in 2004 and last updated by Martin
Chaplin on
6 August, 2014
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