## 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 *v*_{0}/*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*)/on the abscissa and*α*−1)**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

**Fig. 1**. Primary specific velocity plot. The red and blue dots represent the intercepts of the extrapolated lines with the axes at abscissa = 0 and 1, respectively.

**Fig. 2**. Secondary plot of the intercepts (**a**) on the ordinate at abscissa = 0.

**Fig. 3**. Secondary plot of the intercepts (**b**) on the ordinate at abscissa = 1.

### Warning

The specific velocity plot is a graphical *diagnostic* method. It should not be used for statistical purposes because *σ*/(1+*σ*) = *v*_{0}/*V*. Therefore, errors in *v*_{0} appear in both the dependent and the independent variable invalidating regression analysis that requires stochastic independence of the two variables. Linear regression can be used to avoid subjective drawing of straight lines but without statistical rights.

In the specific velocity plot there is no whitewashing of bad data and experimental errors are shown as they are. Without inventing alternatives, repeating experiments as necessary is the best way to cure the problem.

## 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 σ^{0} and *σ ^{0}*/(1+

*σ*

^{0}) 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}.

**Table 1**. Values/ranges of the relative critical substrate concentration and of the critical specific velocity for the basic modifier mechanisms.

Mechanism | σ^{0 (1)} | σ^{0}/(1+σ^{0})^{ (2)} |
---|---|---|

LSpI | −∞ | 1 |

LCaI | 0 | 0 |

LMx(Sp>Ca)I | <0 | >1 |

LMx(Sp<Ca)I | <0 | <0 |

LMx(Sp=Ca)I | −1 | undefined ^{(3)} |

HSpI | undefined ^{(3)} | 1 |

HCaI | 0 | 0 |

HMx(Sp>Ca)I | <0 | >1 |

HMx(Sp<Ca)I | <0 | <0 |

HMx(Sp=Ca)I | −1 | undefined ^{(3)} |

HMxD/(I/A) | >0 | (0,1) ^{(4)} |

HCaA | 0 | 0 |

HMx(Sp>Ca)A | <0 | <0 |

HMxD(A/I) | >0 | (0,1) ^{(4)} |

HSpA | undefined ^{(3)} | 1 |

HMx(Sp<Ca)A | <0 | >1 |

HMx(Sp=Ca)A | 1 | undefined ^{(3)} |

^{(1)} Relative critical substrate concentration

^{(2)} Critical specific velocity

^{(3)} Undefined means that the operations imply division by zero

^{(4)} In the mechanisms HMxD(I/A) and HMxD(A/I) the critical specific velocity falls in an experimentally accessible range of the specific velocity plot

## 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.

**Table 2**. Initial velocities, as *v*/[E]_{t}, for the numerical example

[S] | [X] = 0 | [X] = 10 | [X] = 28 | [X] = 60 | [X] = 110 | [X] = 200 |
---|---|---|---|---|---|---|

24 | 11.18 | 9.40 | 7.57 | 5.82 | 4.70 | 3.67 |

56 | 19.60 | 17.40 | 14.43 | 11.44 | 9.19 | 7.19 |

112 | 28.34 | 25.26 | 21.46 | 17.32 | 14.40 | 11.76 |

192 | 34.41 | 30.89 | 26.90 | 22.27 | 18.82 | 15.40 |

280 | 38.05 | 35.18 | 30.21 | 25.47 | 21.56 | 18.08 |

392 | 41.60 | 31.67 | 33.02 | 28.24 | 23.79 | 20.12 |

**Fig 1**. Calculation of *k*_{cat} and *K*_{m} from the initial velocities set in **Table 2**. (**a**) Direct linear plot [3] with the 95% confidence interval calculated according to [4]. (**b**) Fit of the Michaelis-Menten equation.

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 *v*_{0}/*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.

**Table 3**. The data in **Table 1** transformed as *σ*/(1+*σ*) and *v*_{0}/*v*_{X}

σ/(1+σ) | [X] = 10 | [X] =28 | [X] = 60 | [X] = 110 | [X] = 200 |
---|---|---|---|---|---|

0.222 | 1.189 | 1.477 | 1.921 | 2.379 | 3.046 |

0.400 | 1.126 | 1.358 | 1.713 | 2.133 | 2.726 |

0.571 | 1.122 | 1.321 | 1.636 | 1.068 | 2.410 |

0.696 | 1.114 | 1.279 | 1.545 | 1.828 | 2.234 |

0.769 | 1.082 | 1.260 | 1.494 | 1.765 | 2.105 |

0.824 | 1.104 | 1.260 | 1.473 | 1.749 | 2.068 |

**Fig. 2**. Specific velocity plot for the data in Table 3. (**a**) Primary plot, (**b, c**) secondary plots. Click the thumbnail to enlarge the right part of **Fig. 2a**

### Calculate the parameters

With the information in **Fig. 2** the sought parameters *α*, *β*, *K*_{X}, and *α**K*_{X} can be calculated as shown in **Table 4**.

**Recommended:** The methods for calculating the parameters in red color are recommended. From several simulated sets of data with various added errors, a better estimate of *α**K*_{X} is obtained by multiplying *α* from row 9 with *K*_{X} from row 3.

**Important:** The values of *α* and *β* can be used as known constants to be constrained in the fine-tuning procedure.

Go to The final polish, or Back to data analysis.

Value | Theor. | ||
---|---|---|---|

From the secondary plots | |||

1 | Slope of the a/(a−1) replot = αK_{X}/(α−β) | 47 | |

2 | Ordinate intercept of the a/(a−1) replot = α/(α−β) | 1.19 | |

3 | K_{X} (calculated as slope/intercept) | 39.5 | 39.8 |

4 | Slope of the b/(b−1) replot = αK_{X}/(1−β) | 1.51 | |

5 | Ordinate intercept of the b/(b−1) replot = 1/(1−β) | 1.49 | |

6 | β (calculated from the intercept) | 0.33 | 0.35 |

7 | αK_{X} (calculated as slope/intercept, not recommended) | 101.3 | 95.5 |

From the primary plot and β from the intercept of the b/(b-1) replot | |||

8 | Critical specific velocity (α−β)/(α−1) | 1.45 | |

9 | Reciprocal allosteric coupling constant α | 2.49 | 2.40 |

10 | αK_{X} from K_{X} in row 3 and α in row 9 (recommended) | 98.4 | 95.5 |

### 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.

- With your data construct the specific velocity plot and the secondary plots
- The straightforward way to assign the result to a given mechanism is to use the dichotomous keys in [2, Chapter 10]
- 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)

- No calculations are necessary for assigning the mechanism

**Fi****g. 3**. Taxonomy of enzyme-modifier interactions: inhibition and nonessential activation

**Fig 4.** Shapes taken by the primary specific velocity plot (**red numbers**). Locate *the shape* that matches your findings and note its **number**. The green arrows indicate the increasing direction of the modifier concentration.

**Fig 4, continued.** Shapes taken by its secondary plots (**blue letters**). Locate *the shape* that that matches your findings and note its **letter**.

**Table 5**. The unique codes that identify the 17 basic modifier mechanism according to their characteristic primary and secondary specific velocity plots. Compare your finding, such as **4**–**C** or **12**–**N**, with the codes in this table and read out the name of the mechanism.

**Mechanism of the numerical example**

The primary and secondary specific velocity plots (**Fig. 2**) have the code **2-F**. Thus the featured modification mechanism is *Hyperbolic mixed, predominantly specific inhibition* in accordance with the result from the dependence of the kinetic parameters on modifier concentration.

Acronym | Specific Velocity Plot Identification Code | Name |
---|---|---|

LSpI | 1-A | Linear specific inhibition |

LCaI | 3-A | Linear catalytic inhibition |

LMx(Sp>Ca)I | 2-B | Linear mixed, predominantly specific inhibition |

LMx(Sp<Ca)I | 4-C | Linear mixed, predominantly catalytic inhibition |

LMx(Sp=Ca)I | 5-D | Linear mixed, balanced inhibition |

HSpI | 1-E | Hyperbolic specific inhibition |

HCaI | 3-E | Hyperbolic catalytic inhibition |

HMx(Sp>Ca)I | 2-F | Hyperbolic mixed, predominantly specific inhibition |

HMx(Sp<Ca)I | 4-H | Hyperbolic mixed, predominantly catalytic inhibition |

HMx(Sp=Ca)I | 5-I | Hyperbolic mixed, balanced inhibition |

HMxD(I/A) | 6-J | Hyperbolic mixed, dual modification (inhibition → activation) |

HCaA | 7-K | Hyperbolic catalytic activation |

HMx(Sp>Ca)A | 8-L | Hyperbolic mixed, predominantly specific activation |

HMxD(A/I) | 9-M | Hyperbolic mixed, dual modification (activation → inhibition) |

HSpA | 10-K | Hyperbolic specific activation |

HMx(Sp<Ca)A | 11-G | Hyperbolic mixed, predominantly catalytic activation |

HMx(Sp=Ca)A | 12-N | Hyperbolic mixed, balanced activation |

#### Mechanisms from the specific velocity plot

#### Fingerprints of the specific velocity plots for all mechanisms

#### References

- Baici A (1981) The specific velocity plot. A graphical method for determining inhibition parameters for both linear and hyperbolic enzyme inhibitors. Eur J Biochem 119: 9-14. doi:10.1111/j.1432-1033.1981.tb05570.x, PMID:7341250
- Baici A (2015) Kinetics of Enzyme-Modifier Interactions – Selected Topics in the Theory and Diagnosis of Inhibition and Activation Mechanisms. Springer, Vienna
- Eisenthal R, Cornish-Bowden A (1974) The direct linear plot. A new graphical procedure for estimating enzyme kinetic parameters. Biochem J 139: 715-720. doi:10.1042/bj1390715, PMID:4854723
- Cornish-Bowden A, Porter WR, Trager WF (1978) Evaluation of distribution-free confidence limits for enzyme kinetic parameters. J Theor Biol 74: 163-175. doi:10.1016/0022-5193(78)90069-3, PMID:713571