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((AfC submission|t||ts=20230731172757|u=Rhodydog|ns=118|demo=))<!-- Important, do not remove this line before article has been created. -->

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:

[[File:Single Covalent Protein Cycle.pdf|thumb|A single phosphorylation cycle. <math> v_i </math> are the reaction rates.]]

<math>
\begin{align}
\frac{dA}{dt} &= v_2 - v_1 \\[6pt]
\frac{dAP}{dt} &= v_1 - v_2 \\
\end{align}
</math>

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 <math>A</math> and <math>AP</math>. The moiety, <math>A</math>, is conserved during its transformation to <math>AP</math> and in its conversion from  <math>AP</math> to <math>A</math>. Therefore the total mass of moiety <math>A</math> in the system is fixed and doesn't change in time as the system evolves. In other words <math>A + AP = A_T</math> where <math>A_T</math> is the fixed total mass of moiety <math>A</math>. Mathematically it means that there is only one independent variable. If we designate the independent variable to be <math> AP </math>, then the dependent variable will be <math> A </math> and can be computed using a simple rearrangement of the conservation law:

<math> A = A_T - AP </math>

where <math> A_T </math> is the total mass in the cycle. If we first assume linear mass-action kinetics on the forward and reverse limbs we can write:

<math>
\begin{align}
\frac{dA}{dt} &= k_2 AP - k_1 A \\[6pt]
\frac{dAP}{dt} &= k_1 A - k_2 AP \\
\end{align}
</math>

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

<math> 
\begin{align}
AP = \frac{k_1}{k_1 + k_2}  \\[6pt]
A = \frac{k_2}{k_1 + k_2}  \\
\end{align}
</math>

Any input to the cycle can be modeled as changes to <math>k_1</math>. We can therefore plot the steady-state concentration of <math>AP</math> as a function of the input <math> k_1 </math>. This is shown in the plot to the right  below. 

[[File:PlotAandAPVersusk1.svg|thumb|350x350px]]

The response is a rectangular hyperbola (cf. [[Michaelis–Menten kinetics|Michaelis-Menten equation)]] hyperbolic. As the stimulus increases, <math>AP</math> increases with a corresponding drop in <math>A</math> due to mass conservation.

Another way to look at this result is to consider the sensitivity of <math>AP</math> to changes in <math>k_1</math>. There are various ways to do this but the most obvious is to evaluate the derivative, <math>dAP/dk_1</math>. Better still is to evaluate the scaled derivative since this eliminates units and converts the response into a more intuitive relative change:

<math>C^{AP}_{k_1} = \frac{dAP}{dk_1}\frac{k_1}{AP}</math>

This response can be interpreted as the percentage change in <math>AP</math> given a percentage change in <math>k_1</math>. The steady -state equation for the concentration of <math>AP</math> can be differentiated and scaled to give:

<math>C^J_{k_1} = \frac{k_2}{k_1 + k_2}</math>

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 <math>k_1</math> will always generate less than a 1% change in <math>AP</math>.


Look at sensitivity

Look at saturable kinetics, cite Goldbeter work on 

[[Ultrasensitivity#Saturation mechanisms (Zero-order ultrasensitivity)|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/ERK pathway|MAPK cascade]] contains two such double cycles.   

[[File:Double Covalent Protein Cycle.pdf|thumb|A double phosphorylation cycle. <math> v_i </math> are the reaction rates.|397x397px]]

== References ==
<!-- Inline citations added to your article will automatically display here. See en.wikipedia.org/wiki/WP:REFB for instructions on how to add citations. -->
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