The Basic Mechanisms of Inhibition and Nonessential Activation

Abstract

The 17 enzyme–modifier interaction mechanisms identified by taxonomic criteria are by no means only theoretical concepts. They are all represented in thousands of reports, though their real identity may not be recognized at a glance. One factor is inconsistent nomenclature and another is collecting mechanisms in pools of undifferentiated special cases. Representative examples that span several branches of the biological sciences are discussed in this chapter for the 17 basic mechanisms highlighting the methods used by the authors in data interpretation. Substrates and reaction products in the role of modifiers are discussed within the mechanisms to which they belong.

The vast majority of textbooks (even the most recent ones) continue the ‘romance’ with the double-reciprocal plot, in spite of the severe way it is affected by experimental errors.

Adams KAH, Storer AC, Cornish-Bowden A (1984). J Chem Educ 61:527

Appendix

5.1.1 Rate Equations for Linear Specific Inhibition and Linear Specific Activation

In linear specific inhibition, the association between enzyme and inhibitor (X) is in equilibrium because EX is a dead-end complex . Therefore, using the Cha simplification of the King–Altman method for paths in equilibrium, this step can be represented as node (A), while the second node (B) is constituted by the single species ES as shown in Scheme 5.9a.

Scheme 5.9

Linear specific inhibition (a) and essential activation (b). A and B represent the nodes of the mechanisms

In deriving the rate equation for essential activation the association of E and X is also considered to be in rapid equilibrium. This assumption is partly justified by the fact that essential activators are often either rapidly reacting protons or small ions. The values of the nodes are as shown in Scheme 5.9b. The numbering of constants for linear specific activation is maintained coherent with the topology of the general modifier mechanism.

Values of the nodes:

$$\displaystyle\begin{array}{rcl} & & \mathrm{LSpI:}\qquad E_{A} = k_{-1} + k_{2}\;,\quad E_{B} = k_{1}^{{\prime}}\left [\mathrm{S}\right ] {}\\ & & \mathrm{LSpA:}\qquad E_{A} = k_{-5} + k_{6}\;,\quad E_{B} = k_{5}^{{\prime}}\left [\mathrm{S}\right ]\;. {}\\ \end{array}$$

Fractions of species in the nodes:

$$\displaystyle\begin{array}{rcl} & & \mathrm{LSpI:}\qquad f_{E} = \dfrac{1} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}}\;,\quad f_{EX} = \dfrac{\left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}}\;,\quad f_{ES} = 1 {}\\ & & \mathrm{LSpA:}\qquad f_{E} = \dfrac{1} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}}\;,\quad f_{EX} = \dfrac{\left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}}\;,\quad f_{ESX} = 1\;. {}\\ \end{array}$$

Values of the nodes corrected for fractions of the species within the nodes:

$$\displaystyle\begin{array}{rcl} & & \mathrm{LSpI:}\,\,\,E_{A} = \left (k_{-1} + k_{2}\right )f_{ES} = k_{-1} + k_{2}\;,\quad E_{B} = k_{1}f_{E}\left [\mathrm{S}\right ] = \dfrac{k_{1}\left [\mathrm{S}\right ]} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}} {}\\ & & \mathrm{LSpA:}\,\,\,E_{A} = \left (k_{-5} + k_{6}\right )f_{ESX} = k_{-5} + k_{6}\;,\quad E_{B} = k_{5}f_{EX}\left [\mathrm{S}\right ]=\dfrac{k_{5}\left (\left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}\right )\left [\mathrm{S}\right ]} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}} \;.{}\\ \end{array}$$

Fractions of nodes B, needed for calculating the velocity; the fractions in nodes A are not needed for this purpose:

$$\displaystyle\begin{array}{rcl} & & \mathrm{LSpI:}\qquad f_{B} = \dfrac{E_{B}} {E_{A} + E_{B}} = \dfrac{ \dfrac{k_{1}} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}}\left [\mathrm{S}\right ]} {k_{-1} + k_{2} + \dfrac{k_{1}} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}}\left [\mathrm{S}\right ]} {}\\ & & \mathrm{LSpA:}\qquad f_{B} = \dfrac{E_{B}} {E_{A} + E_{B}} = \dfrac{\dfrac{k_{5}\left (\left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}\right )\left [\mathrm{S}\right ]} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}} } {k_{-5} + k_{6} + \dfrac{k_{1}\left (\left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}\right )\left [\mathrm{S}\right ]} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}} }\;. {}\\ \end{array}$$

Rate equations:

$$\displaystyle\begin{array}{rcl} \mathrm{LSpI:}\qquad v& =& k_{2}\,f_{ES}\,f_{B}\left [\mathrm{E}\right ]_{\mathrm{t}} = \dfrac{k_{2}\left [\mathrm{E}\right ]_{\mathrm{t}} \dfrac{k_{1}} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}}\left [\mathrm{S}\right ]} {k_{-1} + k_{2} + \dfrac{k_{1}} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}}\left [\mathrm{S}\right ]} \\ & =& \dfrac{k_{2}\left [\mathrm{E}\right ]_{\mathrm{t}}\left [\mathrm{S}\right ]} {\dfrac{k_{-1} + k_{2}} {k_{1}} \left (1 + \dfrac{\left [\mathrm{X}\right ]} {K_{\mathrm{Sp}}}\right ) + \left [\mathrm{S}\right ]} {}\end{array}$$

(5.38)

$$\displaystyle\begin{array}{rcl} \mathrm{LSpA:}\qquad v& =& k_{6}\,f_{ESX}\,f_{B}\left [\mathrm{E}\right ]_{\mathrm{t}} = \dfrac{k_{6}\left [\mathrm{E}\right ]_{\mathrm{t}}\dfrac{k_{5}\left (\left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}\right )} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}} \left [\mathrm{S}\right ]} {k_{-5} + k_{6} + \dfrac{k_{5}\left (\left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}\right )} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}} \left [\mathrm{S}\right ]} \\ & =& \dfrac{k_{6}\left [\mathrm{E}\right ]_{\mathrm{t}}\left [\mathrm{S}\right ]} {\dfrac{k_{-5} + k_{6}} {k_{5}} \left (1 + \dfrac{K_{\mathrm{Sp}}} {\left [\mathrm{X}\right ]} \right ) + \left [\mathrm{S}\right ]}\;. {}\end{array}$$

(5.39)

The rate equations for LSpI and LSpA differ for the term in the denominator that multiplies the Michaelis constant, i.e., $1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Sp}}$ and $1 + K_{\mathrm{Sp}}\left /\right. \left [\mathrm{X}\right ]$, respectively. Thus the apparent Michaelis constant of LSpI depends linearly on $\left [\mathrm{X}\right ]$, while it is linear with $1\left /\right. \left [\mathrm{X}\right ]$ in LSpA.

5.1.2 Rate Equation for Linear Catalytic Inhibition

In linear catalytic inhibition, ESX is a dead-end complex that operates at equilibrium when the system is in steady-state (Scheme 5.10). Using the reasoning above for linear specific inhibition and activation, the steps to obtain the rate equation are shown below without comments.

Scheme 5.10

Linear catalytic inhibition. A and B represent the nodes of the mechanism

$$\displaystyle\begin{array}{rcl} E_{A}& =& k_{-1}^{{\prime}} + k_{ 2}^{{\prime}}\;,\quad E_{ B} = k_{1}\left [\mathrm{S}\right ] \\ f_{E}& =& 1,\quad f_{ES} = \dfrac{1} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Ca}}}\;,\quad f_{ESX} = \dfrac{\left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Ca}}} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Ca}}} \\ E_{A}& =& \left (k_{-1}^{{\prime}} + k_{ 2}^{{\prime}}\right )f_{ ES} = \left (k_{-1} + k_{2}\right ) \frac{1} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Ca}}}\;,\quad E_{B} = k_{1}\,f_{E}\left [\mathrm{S}\right ] = k_{1}\left [\mathrm{S}\right ] \\ f_{B}& =& \dfrac{E_{B}} {E_{A} + E_{B}} = \dfrac{k_{1}\left [\mathrm{S}\right ]} {\left (k_{-1} + k_{2}\right ) \dfrac{1} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Ca}}} + k_{1}\left [\mathrm{S}\right ]} \\ v& =& k_{2}\,f_{ES}\,f_{B}\left [\mathrm{E}\right ]_{\mathrm{t}} = \dfrac{k_{2}\left [\mathrm{E}\right ]_{\mathrm{t}} \dfrac{k_{1}} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Ca}}}\left [\mathrm{S}\right ]} {\left (k_{-1} + k_{2}\right ) \dfrac{1} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Ca}}} + k_{1}\left [\mathrm{S}\right ]} \\ & =& \dfrac{ \dfrac{k_{2}\left [\mathrm{E}\right ]_{\mathrm{t}}} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Ca}}}\left [\mathrm{S}\right ]} {\dfrac{k_{-1} + k_{2}} {k_{1}} \left ( \dfrac{1} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Ca}}}\right ) + \left [\mathrm{S}\right ]} = \dfrac{ \dfrac{k_{2}\left [\mathrm{E}\right ]_{\mathrm{t}}} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Ca}}}\left [\mathrm{S}\right ]} { \dfrac{K_{\mathrm{m}}^{0}} {1 + \left [\mathrm{X}\right ]\left /\right. K_{\mathrm{Ca}}} + \left [\mathrm{S}\right ]}\;. {}\end{array}$$

(5.40)