Goldbeter-Koshland kinetics

The Goldbeter-Koshland kinetics describe a steady-state solution for a 2-state biological system. In this system, the interconversion between these two states is performed by two enzymes with opposing effect. One example would be a protein Z that exists in a phosphorylated form Z_P and in an unphosphorylated form Z; the corresponding kinase Y and phosphatase X interconvert the two forms. In this case we would be interested in the equilibrium concentration of the protein Z (Goldbeter-Koshland kinetics only describe equilibrium properties, thus no dynamics can be modeled). It has many applications in the description of biological systems.

The Goldbeter-Koshland kinetics is described by the Goldbeter-Koshland function:

$\begin{align} z = \frac{[Z]}{[Z]_0 } = G(v_1, v_2, J_1, J_2) &= \frac{ 2 v_1 J_2}{B %2B \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}\\ \end{align}$

with the constants

$\begin{align} v_1 = k_1 [X]�; \ v_2 &= k_2 [Y]�; \ J_1 = \frac{K_{M1}}{[Z]_0 }�; \ J_2 = \frac{K_{M2}}{[Z]_0 }; \ B = v_2 - v_1 %2B J_1 v_2 %2B J_2 v_1 \end{align}$

Graphically the function takes values between 0 and 1 and has a sigmoid behavior. The smaller the parameters J₁ and J₂ the steeper the function gets and the more of a switch-like behavior is observed.

Derivation

Since we are looking at equilibrium properties we can write

$\begin{align} \frac{d[Z]}{dt} \ \stackrel{!}{=}\ 0 \end{align}$

From Michaelis–Menten kinetics we know that the rate at which Z_P is dephosphorylated is $r_1 = \frac{k_1 [X] [Z_P]}{K_{M1}%2B [Z_P]}$ and the rate at which Z is phosphorylated is $r_2 = \frac{k_2 [Y] [Z]}{K_{M2}%2B [Z]}$ . Here the K_M stand for the Michaelis Menten constant which describes how well the enzymes X and Y bind and catalyze the conversion whereas the kinetic parameters k₁ and k₂ denote the rate constants for the catalyzed reactions. Assuming that the total concentration of Z is constant we can additionally write that [Z]₀ = [Z_P] + [Z] and we thus get:

$\begin{align} \frac{d[Z]}{dt} = r_1 - r_2 = \frac{k_1 [X] ([Z]_0 - [Z])}{K_{M1}%2B ([Z]_0 - [Z])} &-\frac{k_2 [Y] [Z]}{K_{M2}%2B [Z]} = 0 \\ \frac{k_1 [X] ([Z]_0 - [Z])}{K_{M1}%2B ([Z]_0 - [Z])} &= \frac{k_2 [Y] [Z]}{K_{M2}%2B [Z]} \\ \frac{k_1 [X] (1- \frac{[Z]}{[Z]_0 })}{\frac{K_{M1}}{[Z]_0 }%2B (1 - \frac{[Z]}{[Z]_0 })} &= \frac{k_2 [Y] \frac{[Z]}{[Z]_0 }}{\frac{K_{M2}}{[Z]_0 }%2B \frac{[Z]}{[Z]_0 }} \\ \frac{v_1 (1- z)}{J_1%2B (1 - z)} &= \frac{v_2 z}{J_2%2B z} \qquad \qquad (1) \end{align}$

with the constants

$\begin{align} z = \frac{[Z]}{[Z]_0 }�; \ v_1 = k_1 [X]�; \ v_2 &= k_2 [Y]�; \ J_1 = \frac{K_{M1}}{[Z]_0 }�; \ J_2 = \frac{K_{M2}}{[Z]_0 }; \ \qquad \qquad (2) \end{align}$

If we thus solve the quadratic equation (1) for z we get:

$\begin{align} \frac{v_1 (1- z)}{J_1%2B (1 - z)} &= \frac{v_2 z}{J_2%2B z} \\ J_2 v_1%2B z v_1 - J_2 v_1 z - z^2 v_1 &= z v_2 J_1%2B v_2 z - z^2 v_2\\ z^2 (v_2 - v_1) - z \underbrace{(v_2 - v_1 %2B J_1 v_2 %2B J_2 v_1)}_{B} %2B v_1 J_2 &= 0\\ z = \frac{B - \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}{2 (v_2 - v_1)} &= \frac{B - \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}{2 (v_2 - v_1)} \cdot \frac{B %2B \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}{B %2B \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}\\ z &= \frac{ 4 (v_2 - v_1) v_1 J_2}{2 (v_2 - v_1)} \cdot \frac{1}{B %2B \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}\\ z &= \frac{ 2 v_1 J_2}{B %2B \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}. \qquad \qquad (3) \end{align}$

Thus (3) is a solution to the initial equilibrium problem and describes the equilibrium concentration of [Z] and [Z_P] as a function of the kinetic parameters of the phoshorylation and dephoshorylation reaction and the concentrations of the kinase and phosphotase. The solution is the Goldbeter-Koshland function with the constants from (2):

$\begin{align} z = \frac{[Z]}{[Z]_0 } = G(v_1, v_2, J_1, J_2) &= \frac{ 2 v_1 J_2}{B %2B \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}.\\ \end{align}$

Literature

Goldbeter A, Koshland DE (November 1981). "An amplified sensitivity arising from covalent modification in biological systems". Proc. Natl. Acad. Sci. U.S.A. 78 (11): 6840–4. doi:10.1073/pnas.78.11.6840. PMC 349147. PMID 6947258. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=349147.
Zoltan Szallasi, Jörg Stelling, Vipul Periwal: System Modeling in Cellular Biology. The MIT Press. p 108. ISBN 978-0262195485

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