Properties of the method

The characteristics of the specific velocity plot [1], [2, Chapter 3] are

• Based on the general modifier mechanism by Botts and Morales
• Purely graphical method without statistical pretenses
• Normalized, dimensionless dependent and independent variables
• The kinetic mechanisms can be determined by inspection of the graphics
• With a set of accurately determined initial velocities, α = K_Ca/K_Sp , and β (factor by which the catalytic constant is changed) can be calculated graphically with high precision
  
• Reliable estimation of the enzyme-modifier dissociation constant, K_X
• Suitable for analyzing the behavior of all seventeen basic modifier mechanisms
• The method cannot be used in case of substrate inhibition or activation because the substrate behaves at the same time as modifier

The specific velocity equation (derivation in [1]) is

(1)  

All symbols are defined at the end of the Mechanisms section. The name of the plot derives from σ/(1+σ), known as specific velocity. σ/(1+σ) can also be written as [S]/([S]+K_m) but the notation in (1) is retained for consistency with the use of the ratio σ = [S]/K_m in several applications [2].

Information from the primary plot

The primary plot is constructed with v₀/v _X as dependent variable and σ/(1+σ) as independent variable (Fig. 1). The colored solid lines in Fig. 1 represent a typical, experimentally accessible range of substrate concentrations, i.e. σ/(1+σ) from 0.2 to 0.8, corresponding to substrate concentrations from 0.2×K_m to 5×K_m. For practical reasons, two ordinates are drawn at the abscissa values 0 and 1.

• Independently of the mechanism, the curves of the primary plot are always straight lines
• The intercepts of the extrapolated experimental lines (dashed) on the
  0-ordinate and the 1-ordinate are called a and b, respectively
• With the exception of the three balanced mechanisms (α = 1), for which the plot consists of lines parallel to the abscissa, the extrapolated lines of the remaining 14 mechanisms intersect at a common point with coordinates
  (α − β)/(α −1) on the abscissa and 1 on the ordinate
• The slopes are negative if α > 1 (specific or predominantly specific character)
• The slopes are positive if α < 1 (catalytic or predominantly catalytic character)
• Slopes = 0 characterize the balanced mechanisms (α = 1)

Information from the secondary plots

The values a/(a−1) and b/(b−1) are calculated and plotted against 1/[X] as shown in Figs. 2 and 3 and their expressions are

(2)  

(3)  

Properties

• The secondary plots are always straight lines
• The straight lines of both secondary plots intersect the ordinate at the value 1 for linear inhibition mechanisms
• The straight lines of both secondary plots intersect the ordinate at values >1 for hyperbolic inhibition mechanisms
• The straight lines of both secondary plots intersect the ordinate at values <1 for nonessential activation mechanisms
• Using slopes and intercepts of the plots plots, α, β and K_X can be calculated
• The value of K_X can be used as initial value, while the values α and β can be constrained in the final polish by nonlinear regression analysis

The critical substrate concentration

Putting back into (1) [S]/K_m in place of σ, rearranging and solving for [S], we obtain the expression (4) for the critical substrate concentration, [S]⁰, defined as the concentration of substrate for which the velocity is the same in the presence and in the absence of modifier

(4)  

Similarly, the relative critical substrate concentration is defined by (5)

(5)  

For 14 basic modifier mechanisms  α ≠ 1 and the straight lines in the specific velocity plot intersect at a common point, which represents the critical specific velocity,6 defined as

(6)  

The critical specific velocity (6) can be measured from the specific velocity plot as the common intersection point of all straight lines. It is very useful in the diagnosis of the mechanisms and for calculating α and β.

Table 1 lists the values or ranges of σ⁰ and σ⁰/(1+σ⁰) for the 17 basic modifier mechanisms. Only for mechanisms HMxD(I/A) and HMxD(A/I) the critical specific velocity falls in the experimentally accessible range. For the remaining mechanisms  it is either undefined or it can be obtained by extrapolation.
In mechanism HMxD(I/A), the modifier inhibits the enzyme at [S] < [S]⁰, has no effect at [S] = [S]⁰, and activates the enzyme at [S] > [S]⁰. In mechanism HMxD(A/I), the modifier activates the enzyme at [S] < [S]⁰, has no effect at [S] = [S]⁰, and inhibits the enzyme at [S] > [S]⁰.

Numerical example

We consider now the same set of initial velocities used for illustrating the dependencies of kinetic parameters on modifier concentration in Raw data. First we calculate K_m and V  for the reaction in the absence of modifier using the velocities in the column [X] = 0 of Table 2. With these data: a) we construct the direct linear plot [3], and b) we fit the Michaelis-Menten equation (Fig. 1). Since the velocities are expressed as v/[E]_t the output is k_cat, not V. Both methods yield overlapping results, but for continuing our discussion we adopt the statistically-robust results from the direct linear plot, i.e. k_cat = 49.8 and K_m = 84.

The data in Table 1 can now be transformed into those necessary for constructing the specific velocity plot, i.e. the substrate concentrations are normalized as [S]/K_m and the velocities as v₀/v_X (Table 3). These operations are  performed quickly with the help of a spreadsheet.

The specific velocity plot and the two replots are shown in Fig. 2. In the primary plot, since the data are affected by errors, the five lines do not intersect at a common point. Instead, the maximal number of possible intersections is n(n − 1)/2, where n is the number of lines, one for each modifier concentration: 10 intersections in this example. The median of the 10 intersection points is shown in Fig. 2a as blue dot: it has the coordinates 1.45, 1.01.

Diagnosis of the mechanisms

You have constructed the specific velocity plot and secondary plots with your data and now you wish to establish to which mechanism they apply. If you have already determined the values of α and β, the mechanism can be assigned by inspection of the taxonomic tree (Fig. 3). Otherwise follow the  method described below.

1. With your data construct the specific velocity plot and the secondary plots
2. The straightforward way to assign the result to a given mechanism is to use the dichotomous keys in [2, Chapter 10]
3. As an alternative, use Fig. 4 to identify  the shapes of the primary specific velocity plot (red numbers 1 to 12)  and those of the two secondary plots (blue letters A to N)
   
4. No calculations are necessary for assigning the mechanism