Understand Kinetics Modeling Equations/Derivation

[Incognition]

In a paper, I encountered a system modeled by six coupled first-order differential equations like so:

$$\frac{dm_i}{dt}=-m_i+\frac{\alpha}{1+p^n_j}+\alpha_0$$

$$\frac{dp_i}{dt}=-\beta(p_i-m_i)$$ , where i=1,2,3 and j=3,1,2.

According to the paper, the system has a unique steady state which becomes unstable when $$\frac{(\beta+1)^2}{\beta}<\frac{3X^2}{4+2X}$$, where X is defined $$X=-\frac{\alpha n p^(n-1)}{(1+p^n)^2}$$and p is the solution to $$p=\frac{\alpha}{1+p^n}+\alpha_0$$.

Lacking a textbook, I have had very little success in seeing how the steady state was derived. I intend to model a similar system. Can someone point me in the right way to understand these equations or show the derivation outright?

Thank you in advance.

 

[HallsofIvy]

Finding the steady state solution (also called an "equilibrium solution") is very simple. The "steady state" solution is one that doesn't change which means that the derivatives are 0. You must have
$$\frac{dm_i}{dt}=-m_i+\frac{\alpha}{1+p^n_j}+\alpha_0= 0$$
and
$$\frac{dp_i}{dt}=-\beta(p_i-m_i)= 0$$
From the second equation, we obviously have $p_i= m_i$
Putting that into the second equation, we have
$$-p_i}+\frac{\alpha}{1+p^n_j}+\alpha_0= 0$$
If we let p be the function that satisfies
$$-p}+\frac{\alpha}{1+p^n}+\alpha_0= 0$$
(just the equation above with p_n replaced by p)
then the steady state solution is p_i(t)= m_i(t)= p for all i and t.

The "hard" part is determining when the steady state solution is stable or unstable. It is stable if, when the solution is slightly off the steady state solution, it tends to go toward it, and unstable if it tends to move away. In particular, taking p to be the solution to the equation above,
if p_i>p, $\frac{dp_i}{dt}$ must be negative so the solution will tend downward toward p. Conversely, if p_i< p, then the derivative must be postive. Of course, as long as the solution is not the steady state solution, we cannot assume that all the p_i(t) are the same not that p_i= m_i. You can perhaps see from the form of the condition- a complicated equation involving X and then a complicated function defining X itself- that determining when the steady state solution is stable or unstable is not a simple problem!

 

[tacman]

The set of equations you have encountered is known as a system of coupled first-order differential equations, which is commonly used to model dynamic systems. In this particular case, the system represents a biochemical reaction network, where the variables m and p represent the concentrations of different components in the system. The parameters α, α0, and β represent the rate constants of the reactions involved.

To understand the derivation of these equations, it is important to first understand the concept of kinetics. Kinetics is the study of the rates at which chemical reactions occur and the factors that influence these rates. In this case, the equations represent the rates of change of the concentrations of the components in the system over time.

To derive these equations, the authors of the paper most likely used a combination of principles from chemical kinetics and mathematical modeling techniques. They may have also used experimental data to validate and refine their model.

In order to fully understand the derivation, it would be helpful to have a textbook or reference material on chemical kinetics and mathematical modeling. However, if you are unable to access such resources, you can try breaking down the equations and understanding each term individually. For example, you can start by looking at the first equation \frac{dm_i}{dt}=-m_i+\frac{\alpha}{1+p^n_j}+\alpha_0 and understanding what each term represents and how they are related to the overall system. You can also try to simulate the system using different values for the parameters and see how it affects the behavior of the system.

Overall, understanding the derivation of these equations may require some background knowledge in chemical kinetics and mathematical modeling. But with some effort and experimentation, you may be able to gain a better understanding of the system and use it to model a similar system for your own research.