**
Determination of V _{max }and K_{m}
**

It is important to have as thorough knowledge as is
possible of the performance characteristics of enzymes, if they are to be used
most efficiently. The kinetic parameters V_{max}, K_{m} and k_{cat}/K_{m}
should, therefore, be determined. There are two approaches to this problem using
either the reaction progress curve (integral method) or the initial rates of
reaction (differential method). Use of either method dat least approximately,
the optimum conditions for the reepends on prior knowledge of the mechanism for
the reaction and, action. If the mechanism is known and complex ts), uhen the
data must be reconciled to the appropriate model (hypothesisually by use of a
computer-aided analysis involving a weighted least-squares fit. Many such
computer programs are currently available and, if not, the programming skill
involved is usually fairly low. If the mechanism is not known, initial attempts
are usually made to fit the data to the Michaelis-Menten kinetic model.
Combining equations (1.1) and (1.8),

(1.99)

which, on integration, using the boundary condition that
the product is absent at time zero and by substituting [S] by ([S]_{0} -
[P]), becomes

(1.100)

If the fractional conversion (** X**)
is introduced, where

(1.101)

then equation (1.100) may be simplified to give:

(1.102)

Use of equation (1.99)
involves the determination of the initial rate of reaction over a wide range of
substrate concentrations. The initial rates are used, so that [S] = [S]_{0},
the predetermined and accurately known substrate concentration at the start of
the reaction. Its use also ensures that there is no effect of reaction
reversibility or product inhibition which may affect the integral method based
on equation (1.102).
Equation (1.99)
can be utilised directly using a computer program, involving a weighted
least-squares fit, where the parameters for determining the hyperbolic
relationship between the initial rate of reaction and initial substrate
concentration (i.e.. K_{m} and V_{max}) are chosen in order to
minimise the errors between the data and the model, and the assumption is made
that the errors inherent in the practically determined data are normally
distributed about their mean (error-free) value.

Alternatively the direct linear plot may be used (Figure 1.10). This is a powerful non-parametric statistical method which depends upon the assumption that any errors in the experimentally derived data are as likely to be positive (i.e. too high) as negative (i.e. too low). It is common practice to show the data obtained by the above statistical methods on one of three linearised plots, derived from equation (1.99) (Figure 1.11). Of these, the double reciprocal plot is preferred to test for the qualitative correctness of a proposed mechanism, and the Eadie-Hofstee plot is preferred for discovering deviations from linearity.

**Figure 1.10.** The direct linear plot. A plot of the
initial rate of reaction against the initial substrate concentration also
showing the way estimates can be directly made of the K_{m} and V_{max}.
Every pair of data points may be utilised to give a separate estimate of these
parameters (i.e. n(n-1)/2 estimates from n data points with differing [S]_{0}).
These estimates are determined from the intersections of lines passing through
the (x,y) points (-[S]_{0},0) and (0,v); each intersection forming a
separate estimate of K_{m} and V_{max}. The intersections are
separately ranked in order of increasing value of both K_{m} and V_{max}
and the median values taken as the best estimates for these parameters. The
error in these estimates can be simply determined from sub-ranges of these
estimates, the width of the sub-range dependent on the accuracy required for the
error and the number of data points in the analysis. In this example there are 7
data points and, therefore, 21 estimates for both K_{m} and V_{max}.
The ranked list of the estimates for K_{m} (mM) is 0.98,1.65, 1.68,
1.70, 1.85, 1.87, 1.89, 1.91, 1.94, 1.96, **1.98**, 1.99, 2.03, 2.06, 1.12,
2.16, 2.21, 2.25, 2.38, 2.40, 2.81, with a median value of 1.98 mM. The K_{m}
must lie between the 4th (1.70 mM) and 18th (2.25 mM) estimate at a confidence
level of 97% (Cornish-Bowden *et al.*, 1978). The list of the estimates for
V_{max} (mM.min^{-1}) is ranked separately as 3.45, 3.59, 3.80,
3.85, 3.87, 3.89, 3.91, 3.94, 3.96, 3.96, **3.98**, 4.01, 4.03, 4.05, 4.13,
4.14, 4.18, 4.26, 4.29, 4.35, with a median value of 3.98 mM.min^{-1}.
The V_{max} must lie between the 4th (3.85 mM.min^{-1}) and 18th
(4.18 mM.min^{-1}) estimate at a confidence level of 97%. It can be seen
that outlying estimates have little or no influence on the results. This is a
major advantage over the least-squared statistical procedures where rogue data
points cause heavily biased effects.

**Figure 1.11.** Three ways in which the hyperbolic
relationship between the initial rate of reaction and the initial substrate
concentration

can be rearranged to give linear plots. The examples are
drawn using K_{m} = 2 mM and V_{max} = 4 mM min^{-1}.

(a) Lineweaver-Burk (double-reciprocal) plot of 1/v
against 1/[S]_{0} giving intercepts at 1/V_{max} and -1/K_{m}

(1.103)

(b) Eadie-Hofstee plot of v against v/[S]_{0}
giving intercepts at V_{max} and V_{max}/K_{m}

(1.104)

c) Hanes-Woolf (half-reciprocal) plot of
[S]_{0}/v against [S]_{0} giving intercepts at K_{m}/V_{max}
and K_{m}.

(1.105)

The progress curve of the reaction (Figure
1.12) can be used to determine the specificity constant (k_{cat}/K_{m})
by making use of the relationship between time of reaction and fractional
conversion (see equation (1.102).
This has the advantage over the use of the initial rates (above) in that fewer
determinations need to be made, possibly only one progress curve is necessary,
and sometimes the initial rate of reaction is rather difficult to determine due
to its rapid decline. If only the early part of the progress curve, or its
derivative, is utilised in the analysis, this procedure may even be used in
cases where there is competitive inhibition by the product, or where the
reaction is reversible.

**Figure 1.12.** A schematic plot showing the amount of
product formed (productivity) against the time of reaction, in a closed system.
The specificity constant may be determined by a weighted least-squared fit of
the data to the relationship given by equation (1.102).

The type of inhibition and the inhibition constants may be
determined from the effect of differing concentrations of inhibitor on the
apparent K_{m}, V_{max} and k_{cat}/K_{m},
although some more specialised plots do exist.