Goldbeter–Koshland kinetics

Since equilibrium properties are searched one can write

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

From Michaelis–Menten kinetics the rate at which Z_P is dephosphorylated is known to be ${\displaystyle r_{1}={\frac {k_{1}[X][Z_{P}]}{K_{M1}+[Z_{P}]}}}$ and the rate at which Z is phosphorylated is ${\displaystyle r_{2}={\frac {k_{2}[Y][Z]}{K_{M2}+[Z]}}}$. 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 one can additionally write that [Z]₀ = [Z_P] + [Z] and one thus gets:

${\displaystyle {\begin{aligned}{\frac {d[Z]}{dt}}=r_{1}-r_{2}={\frac {k_{1}[X]([Z]_{0}-[Z])}{K_{M1}+([Z]_{0}-[Z])}}&-{\frac {k_{2}[Y][Z]}{K_{M2}+[Z]}}=0\\{\frac {k_{1}[X]([Z]_{0}-[Z])}{K_{M1}+([Z]_{0}-[Z])}}&={\frac {k_{2}[Y][Z]}{K_{M2}+[Z]}}\\{\frac {k_{1}[X](1-{\frac {[Z]}{[Z]_{0}}})}{{\frac {K_{M1}}{[Z]_{0}}}+(1-{\frac {[Z]}{[Z]_{0}}})}}&={\frac {k_{2}[Y]{\frac {[Z]}{[Z]_{0}}}}{{\frac {K_{M2}}{[Z]_{0}}}+{\frac {[Z]}{[Z]_{0}}}}}\\{\frac {v_{1}(1-z)}{J_{1}+(1-z)}}&={\frac {v_{2}z}{J_{2}+z}}\qquad \qquad (1)\end{aligned}}}$

with the constants

${\displaystyle {\begin{aligned}z={\frac {[Z]}{[Z]_{0}}};\ v_{1}=k_{1}[X];\ v_{2}&=k_{2}[Y];\ J_{1}={\frac {K_{M1}}{[Z]_{0}}};\ J_{2}={\frac {K_{M2}}{[Z]_{0}}};\ \qquad \qquad (2)\end{aligned}}}$

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

${\displaystyle {\begin{aligned}{\frac {v_{1}(1-z)}{J_{1}+(1-z)}}&={\frac {v_{2}z}{J_{2}+z}}\\J_{2}v_{1}+zv_{1}-J_{2}v_{1}z-z^{2}v_{1}&=zv_{2}J_{1}+v_{2}z-z^{2}v_{2}\\z^{2}(v_{2}-v_{1})-z\underbrace {(v_{2}-v_{1}+J_{1}v_{2}+J_{2}v_{1})} _{B}+v_{1}J_{2}&=0\\z={\frac {B-{\sqrt {B^{2}-4(v_{2}-v_{1})v_{1}J_{2}}}}{2(v_{2}-v_{1})}}&={\frac {B-{\sqrt {B^{2}-4(v_{2}-v_{1})v_{1}J_{2}}}}{2(v_{2}-v_{1})}}\cdot {\frac {B+{\sqrt {B^{2}-4(v_{2}-v_{1})v_{1}J_{2}}}}{B+{\sqrt {B^{2}-4(v_{2}-v_{1})v_{1}J_{2}}}}}\\z&={\frac {4(v_{2}-v_{1})v_{1}J_{2}}{2(v_{2}-v_{1})}}\cdot {\frac {1}{B+{\sqrt {B^{2}-4(v_{2}-v_{1})v_{1}J_{2}}}}}\\z&={\frac {2v_{1}J_{2}}{B+{\sqrt {B^{2}-4(v_{2}-v_{1})v_{1}J_{2}}}}}.\qquad \qquad (3)\end{aligned}}}$

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 phosphorylation and dephosphorylation reaction and the concentrations of the kinase and phosphatase. The solution is the Goldbeter–Koshland function with the constants from (2):

${\displaystyle {\begin{aligned}z={\frac {[Z]}{[Z]_{0}}}=G(v_{1},v_{2},J_{1},J_{2})&={\frac {2v_{1}J_{2}}{B+{\sqrt {B^{2}-4(v_{2}-v_{1})v_{1}J_{2}}}}}.\\\end{aligned}}}$

The ultrasensitivity (sigmoidality) of a Goldbeter–Koshland module can be measured by its Hill Coefficient:

${\displaystyle n_{H}={\frac {log(81)}{log(EC90/EC10)}}}$.

where EC90 and EC10 are the input values needed to produce the 10% and 90% of the maximal response, respectively.

In a living cell, Goldbeter–Koshland modules are embedded in a bigger network with upstream and downstream components. This components may constrain the range of inputs that the module will receive as well as the range of the module’s outputs that network will be able to detect. Altszyler et al. (2014) [3][4] studied how the effective ultrasensitivity of a modular system is affected by these restrictions. They found that Goldbeter–Koshland modules are highly sensitive to dynamic range limitations imposed by downstream components. However, in the case of asymmetric Goldbeter–Koshland modules, a moderate downstream constrain can produce effective sensitivities much larger than that of the original module when considered in isolation.