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:

$\begin{aligned} \frac{d A}{d t} & = v_{2} - v_{1} \\ \frac{d A P}{d t} & = v_{1} - v_{2} \end{aligned}$

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 $A P$ . The moiety, $A$ , is conserved during its transformation to $A P$ and in its conversion from $A P$ 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 $A + A P = A_{T}$ 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 $A P$ , then the dependent variable will be $A$ and can be computed using a simple rearrangement of the conservation law:

$A = A_{T} - A P$

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:

$\begin{aligned} \frac{d A}{d t} & = k_{2} A P - k_{1} A \\ \frac{d A P}{d t} & = k_{1} A - k_{2} A P \end{aligned}$

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

$\begin{aligned} A P = \frac{k_{1}}{k_{1} + k_{2}} \\ A = \frac{k_{2}}{k_{1} + k_{2}} \end{aligned}$

Any input to the cycle can be modeled as changes to $k_{1}$ . We can therefore plot the steady-state concentration of $A P$ 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, $A P$ increases with a corresponding drop in $A$ due to mass conservation.

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

$C_{k_{1}}^{A P} = \frac{d A P}{d k_{1}} \frac{k_{1}}{A P}$

This response can be interpreted as the percentage change in $A P$ given a percentage change in $k_{1}$ . The steady-state equation for the concentration of $A P$ 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 $A P$ .

Look at sensitivity

Look at saturable kinetics, cite Goldbeter work on

mechanistic dependent derivation

this is some text 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.

References

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