Complements to Enzyme–Modifier Interactions

Abstract

Starting from the definitions of the basic modifier mechanisms, extensions to more complex systems are not always straightforward but can be achieved if the underlying theories are well understood and used correctly. The mathematical aspects can become more complex but the labor is compensated by rewarding results. This chapter fulfills a double purpose by showing the way for managing system of increasing complexity illustrating the theory with practical examples, and by adding methods that complement those described in the preceding chapters.

⋯ readers who understand the methods used to discriminate between the simple cases are well equipped to adapt them to special experimental circumstances and to understand the more detailed discussions found elsewhere.

A. Cornish-Bowden (2012) Fundamentals of Enzyme Kinetics, Wiley-Blackwell. p. 190

Appendix

4.1.1 Derivation of the Tight-Binding Rate Equation for the General Modifier Mechanism

Redirected from Sect. 4.4. The reactant concentrations in a rate equation represent free concentrations. However, in the presence of tight-binding, disambiguation between free and total concentrations is made by appending a ‘t’ (total), and where necessary an ‘f’ (free) to the symbols of concentration. We start the derivation of the tight-binding rate equation of the general modifier mechanism by putting in evidence the free modifier concentration while rearranging (3.35) discussed in Chap. 3

$$\displaystyle{ v_{\mathrm{X}} = \dfrac{\left (1+\sigma \right )\left (1 +\beta \dfrac{\left [\mathrm{X}\right ]_{\mathrm{f}}} {\alpha K_{\mathrm{X}}} \right )} {1 +\sigma +\dfrac{\left [\mathrm{X}\right ]_{\mathrm{f}}} {K_{\mathrm{X}}} + \dfrac{\sigma \left [\mathrm{X}\right ]_{\mathrm{f}}} {\alpha K_{\mathrm{X}}} }v_{0}\,, }$$

(4.51)

from which $\left [\mathrm{X}\right ]_{\mathrm{f}}$ can be extracted as:

$$\displaystyle{ \left [\mathrm{X}\right ]_{\mathrm{f}} = \dfrac{\left (1+\sigma \right )\left (v_{0} - v_{\mathrm{X}}\right )} {v_{\mathrm{X}}\left ( \dfrac{1} {K_{\mathrm{X}}} + \dfrac{\sigma } {\alpha K_{\mathrm{X}}}\right ) - v_{0} \dfrac{\beta } {\alpha K_{\mathrm{X}}}\left (1+\sigma \right )}\,. }$$

(4.52)

To proceed, an expression of the total modifier concentration is needed as the sum $\left [\mathrm{EX}\right ] + \left [\mathrm{ESX}\right ] + \left [\mathrm{X}\right ]_{\mathrm{f}}$, and $\left [\mathrm{EX}\right ] + \left [\mathrm{ESX}\right ]$ can be obtained considering the rate (4.51) at saturating modifier, in which case all enzyme molecules are present in these two complexes with product generation occurring only by decomposition of ESX into EX + P:

$$\displaystyle{ v_{\infty } = \dfrac{\left (1+\sigma \right ) \dfrac{\beta } {\alpha K_{\mathrm{X}}}} { \dfrac{1} {K_{\mathrm{X}}} + \dfrac{\sigma } {\alpha K_{\mathrm{X}}}}v_{0}\,. }$$

(4.53)

The denominator of (4.53) represents now EX + ESX, while the denominator of (4.51) represents $\mathrm{E} + \mathrm{ES} + \mathrm{EX} + \mathrm{ESX}$. According to the distribution equations of Cleland [6], dividing the first by the second of these denominators and multiplying by $\left [\mathrm{X}\right ]_{\mathrm{f}}\left [\mathrm{E}\right ]_{\mathrm{t}}$ yields $\left [\mathrm{EX}\right ] + \left [\mathrm{ESX}\right ]$:

$$\displaystyle{ \left [\mathrm{EX}\right ] + \left [\mathrm{ESX}\right ] = \dfrac{\left (\dfrac{\left [\mathrm{X}\right ]_{\mathrm{f}}} {K_{\mathrm{X}}} + \dfrac{\sigma \left [\mathrm{X}\right ]_{\mathrm{f}}} {\alpha K_{\mathrm{X}}} \right )\left [\mathrm{E}\right ]_{\mathrm{t}}} {1 +\sigma +\dfrac{\left [\mathrm{X}\right ]_{\mathrm{f}}} {K_{\mathrm{X}}} + \dfrac{\sigma \left [\mathrm{X}\right ]_{\mathrm{f}}} {\alpha K_{\mathrm{X}}} }\;. }$$

(4.54)

Experimentally, rates are measured which contain the concentrations of the complexes. Thus, such concentrations must be deduced from the differences between the rate in the absence ($v_{0}$) of modifier, at a given modifier concentration ($v_{\mathrm{X}}$) and at saturating modifier ($v_{\infty }$), i.e., $\left (v_{0} - v_{\mathrm{X}}\right )$ and $\left (v_{0} - v_{\infty }\right )$ using the explicit expression $v_{0} = V \sigma \left /\right. \left (1+\sigma \right )$, (4.51) and (4.53):

$$\displaystyle\begin{array}{rcl} v_{0} - v_{\mathrm{X}}& =& \frac{V \sigma \left [\mathrm{X}\right ]_{\mathrm{f}}\left (\alpha +\sigma -\beta -\beta \sigma \right )} {\left (1+\sigma \right )\left (\alpha K_{\mathrm{X}} +\alpha \sigma K_{\mathrm{X}} + \left [\mathrm{X}\right ]_{\mathrm{f}}\sigma + \left [\mathrm{X}\right ]_{\mathrm{f}}\alpha \right )} \\ v_{0} - v_{\infty }& =& \frac{V \sigma \left (\alpha +\sigma -\beta -\beta \sigma \right )} {\left (1+\sigma \right )\left (\alpha +\sigma \right )} \\ \frac{v_{0} - v_{\mathrm{X}}} {v_{0} - v_{\infty }}& =& \frac{\left (\alpha +\sigma \right )\left [\mathrm{X}\right ]_{\mathrm{f}}} {\alpha K_{\mathrm{X}} +\sigma \alpha K_{\mathrm{X}} + \left [\mathrm{X}\right ]_{\mathrm{f}}\sigma + \left [\mathrm{X}\right ]_{\mathrm{f}}\alpha } = \dfrac{\dfrac{\left [\mathrm{X}\right ]_{\mathrm{f}}} {K_{\mathrm{X}}} + \dfrac{\sigma \left [\mathrm{X}\right ]_{\mathrm{f}}} {\alpha K_{\mathrm{X}}} } {1 +\sigma +\dfrac{\left [\mathrm{X}\right ]_{\mathrm{f}}} {K_{\mathrm{X}}} + \dfrac{\sigma \left [\mathrm{X}\right ]_{\mathrm{f}}} {\alpha K_{\mathrm{X}}} }\,.{}\end{array}$$

(4.55)

After multiplication by $\left [\mathrm{E}\right ]_{\mathrm{t}}$, the ratio in (4.55) is identical with (4.54) and the sought conservation equation for the modifier concentration becomes:

$$\displaystyle{ \left [\mathrm{X}\right ]_{\mathrm{t}} = \left [\mathrm{EX}\right ] + \left [\mathrm{ESX}\right ] + \left [\mathrm{X}\right ]_{\mathrm{f}} = \frac{v_{0} - v_{\mathrm{X}}} {v_{0} - v_{\infty }}\left [\mathrm{E}\right ]_{\mathrm{t}} + \left [\mathrm{X}\right ]_{\mathrm{f}}\;. }$$

(4.56)

We have now all elements for writing the rate equation of the general modifier mechanism under tight-binding conditions but it would contain $v_{\infty }$, which can be usefully replaced by other parameters to produce an expression that contains only familiar elements present in (4.51) and in the specific velocity equation (3.36). Inspection of (4.53) suggest the following, equivalent rearrangement obtained by multiplying the numerator and the denominator by α K _X:

$$\displaystyle{ v_{\infty } = v_{0}\beta \frac{1+\sigma } {\alpha +\sigma } \;. }$$

(4.57)

Furthermore, purposeful rearrangement of (4.52) gives

$$\displaystyle\begin{array}{rcl} \left [\mathrm{X}\right ]_{\mathrm{f}}& =& \dfrac{v_{0} - v_{\mathrm{X}}} {v_{\mathrm{X}} - v_{0} \dfrac{\beta } {\alpha K_{\mathrm{X}}}\left ( \dfrac{1+\sigma } { \dfrac{1} {K_{\mathrm{X}}} + \dfrac{\sigma } {\alpha K_{\mathrm{X}}}}\right )} \times \dfrac{1+\sigma } { \dfrac{1} {K_{\mathrm{X}}} + \dfrac{\sigma } {\alpha K_{\mathrm{X}}}} \\ & =& \dfrac{v_{0} - v_{\mathrm{X}}} {v_{\mathrm{X}} -\underbrace{\mathop{ v_{0}\beta \left (\dfrac{1+\sigma } {\alpha +\sigma } \right )}}\limits _{v_{\infty }}} \times \dfrac{1+\sigma } { \dfrac{1} {K_{\mathrm{X}}} + \dfrac{\sigma } {\alpha K_{\mathrm{X}}}}\;, {}\end{array}$$

(4.58)

where the term underscored by the curled brace represents $v_{\infty }$ in (4.57).

The conservation equation for the modifier’s concentration is finally obtained introducing (4.51), (4.57), and (4.58) in (4.56). The resulting expression is then used to extract $v_{\mathrm{X}}$, a quadratic equation, whose physically significant root is displayed as (4.35) in the main text.