Draft:Modeling Biochemical Cascades

A biochemical cascade, also known as a signaling cascade or signaling pathway, is a series of chemical reactions that occur within a biological cell when activated by a stimulus. This article will describe the mathematical modeling and analysis of the properties of such cascades.

Single Phosphorylation Cycle

The fundamental unit of a biochemical cascade is the phosphorylation cycle. This can either be a single phosphorylation or double phosphorylation resulting in a double cycle. The single phosphorylation cycle is shown in the adjacent figure. THis system will be modeled using a set of differential equations:

A single phosphorylation cycle. $v_{i}$ are the reaction rates.

${\begin{aligned}{\frac {dA}{dt}}&=v_{2}-v_{1}\\[6pt]{\frac {dAP}{dt}}&=v_{1}-v_{2}\\\end{aligned}}$

Note that these equations are linearly dependent since either one can be obtained from the other by multiplying by minus one. This is due to mass conservation between $A$ and $AP$. The moiety, $A$, is conserved during its transformation to $AP$ and in its conversion from $AP$ to $A$. Therefore the total mass of moiety $A$ in the system is fixed and doesn't change in time as the system evolves. In other words $A+AP=A_{T}$ where $A_{T}$ is the fixed total mass of moiety $A$. Mathematically it means that there is only one independent variable. If we designate the independent variable to be $AP$, then the dependent variable will be $A$ and can be computed using a simple rearrangement of the conservation law:

$A=A_{T}-AP$

where $A_{T}$ is the total mass in the cycle. If we first assume linear mass-action kinetics on the forward and reverse limbs we can write:

${\begin{aligned}{\frac {dA}{dt}}&=k_{2}AP-k_{1}A\\[6pt]{\frac {dAP}{dt}}&=k_{1}A-k_{2}AP\\\end{aligned}}$

using the conservation equation we can solve for the steady-state levels of $A$ and $AP$ by setting the independent differential equation to zero:

${\begin{aligned}AP={\frac {k_{1}}{k_{1}+k_{2}}}\\[6pt]A={\frac {k_{2}}{k_{1}+k_{2}}}\\\end{aligned}}$

Any input to the cycle can be modeled as changes to $k_{1}$. We can therefore plot the steady-state concentration of $AP$ as a function of the input $k_{1}$. This is shown in the plot to the right below.

The response is a rectangular hyperbola (cf. Michaelis-Menten equation) hyperbolic. As the stimulus increases, $AP$ increases with a corresponding drop in $A$ due to mass conservation.

Another way to look at this result is to consider the sensitivity of $AP$ to changes in $k_{1}$. There are various ways to do this but the most obvious is to evaluate the derivative, $dAP/dk_{1}$. Better still is to evaluate the scaled derivative since this eliminates units and converts the response into a more intuitive relative change:

$C_{k_{1}}^{AP}={\frac {dAP}{dk_{1}}}{\frac {k_{1}}{AP}}$

This response can be interpreted as the percentage change in $AP$ given a percentage change in $k_{1}$. The steady -state equation for the concentration of $AP$ can be differentiated and scaled to give:

$C_{k_{1}}^{J}={\frac {k_{2}}{k_{1}+k_{2}}}$

The most important aspect of this result is that the sensitivity is always less than or equal to one, That is a 1% change in $k_{1}$ will always generate less than a 1% change in $AP$.

Look at sensitivity

Look at saturable kinetics, cite Goldbeter work on

mechanistic dependent derivation

this is some t3 t to hold it in draft mode so that it doesn’t get deleted. 1 Jan 2024.

Double Phosphorylation Cycle

It is very common to find double phosphorylation cycles. For example, the MAPK cascade contains two such double cycles.

A double phosphorylation cycle. $v_{i}$ are the reaction rates.