Secondary plot (kinetics)

In enzyme kinetics, a secondary plot uses the intercept or slope from several Lineweaver–Burk plots to find additional kinetic constants.[1][2]

For example, when a set of v by [S] curves from an enzyme with a ping–pong mechanism (varying substrate A, fixed substrate B) are plotted in a Lineweaver–Burk plot, a set of parallel lines will be produced.

The following Michaelis–Menten equation relates the initial reaction rate v₀ to the substrate concentrations [A] and [B]:

{\begin{aligned}{\frac {1}{v_{0}}}&={\frac {K_{M}^{A}}{v_{\max }{[}A{]}}}+{\frac {K_{M}^{B}}{v_{\max }{[}B{]}}}+{\frac {1}{v_{\max }}}\end{aligned}}

The y-intercept of this equation is equal to the following:

{\begin{aligned}{\mbox{y-intercept}}={\frac {K_{M}^{B}}{v_{\max }{[}B{]}}}+{\frac {1}{v_{\max }}}\end{aligned}}

The y-intercept is determined at several different fixed concentrations of substrate B (and varying substrate A). The y-intercept values are then plotted versus 1/[B] to determine the Michaelis constant for substrate B, $K_{M}^{B}$ , as shown in the Figure to the right.[3] The slope is equal to $K_{M}^{B}$ divided by $v_{\max }$ and the intercept is equal to 1 over $v_{\max }$ .

Secondary Plot of enzyme system Horseradish Peroxidase and o-Phenylenediamine (with hydrogen peroxide as the second substrate)

Secondary plot in inhibition studies

A secondary plot may also be used to find a specific inhbition constant, K_I.

For a competitive enzyme inhibitor, the apparent Michaelis constant is equal to the following:

{\begin{aligned}{\mbox{apparent }}K_{m}=K_{m}\times \left(1+{\frac {[I]}{K_{I}}}\right)\end{aligned}}

The slope of the Lineweaver-Burk plot is therefore equal to:

{\begin{aligned}{\mbox{slope}}={\frac {K_{m}}{v_{\max }}}\times \left(1+{\frac {[I]}{K_{I}}}\right)\end{aligned}}

If one creates a secondary plot consisting of the slope values from several Lineweaver-Burk plots of varying inhibitor concentration [I], the competitive inhbition constant may be found. The slope of the secondary plot divided by the intercept is equal to 1/K_I. This method allows one to find the K_I constant, even when the Michaelis constant and v_max values are not known.

gollark: I might be somewhat annoyed about someone not paying me a cut of that, except I didn't even invent the algorithm.

gollark: Besides, that isn't particularly evil.

gollark: It's actually ported from someone's Haskell implementation but several times faster, so you could just have NFTized output from that anyway.

gollark: I'm sure people will definitely use my fractal art program, random esolangs, deliberately inefficient matrix multiplier program, slow full text search thing, and length terminated strings for evil.

gollark: Perhaps if I had something actually useful (and userfacing) I'd not do that, but meh.

References

A. Cornish-Bowden. Fundamentals of Enzyme kinetics Rev. ed., Portland: London, England, (1995) pp. 30-37, 56-57.
J. N. Rodriguez-Lopez, M. A. Gilabert, J. Tudela, R. N. F. Thorneley, and F. Garcia-Canovas. Biochemistry, 2000, 39, 13201-13209.
The Horseradish Peroxidase/ o-Phenylenediamine (HRP/OPD) System Exhibits a Two-Step Mechanism. M. K. Tiama and T. M. Hamilton, Journal of Undergraduate Chemistry Research, 4, 1 (2005).

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[1] A. Cornish-Bowden. Fundamentals of Enzyme kinetics Rev. ed., Portland: London, England, (1995) pp. 30-37, 56-57.

[2] J. N. Rodriguez-Lopez, M. A. Gilabert, J. Tudela, R. N. F. Thorneley, and F. Garcia-Canovas. Biochemistry, 2000, 39, 13201-13209.

[3] The Horseradish Peroxidase/ o-Phenylenediamine (HRP/OPD) System Exhibits a Two-Step Mechanism. M. K. Tiama and T. M. Hamilton, Journal of Undergraduate Chemistry Research, 4, 1 (2005).