I've recently taken a mathematical biology course (emphasizing systems biology) but don't understand the applications for the tools I learned about using; suppose we have a system of ODE's based on an enzyme reaction, for example, giving something that looks like the following:
\begin{align*}\frac{d[E]}{dt} &= -k_1[E][S] + k_{-1}[ES] + k_2[ES]\\
\frac{d[S]}{dt} &= -k_{1}[E][S] + k_{-1}[ES]\\
\frac{d[ES]}{dt} &= k_1[E][S] – k_{-1}[ES] – k_2[ES]\\
\frac{d[P]}{dt} &= k_2[ES]
\end{align*}
(this is just an example, I'm not too concerned with any particular system in this question, but am more comfortable with enzyme reactions at the moment). I got a pretty strong foundation in (a) finding fixed points/steady states of something like that and (b) determining whether those fixed points are sources, sinks, etc. by doing certain analyses, but I never understood why those things matter; I understand that these show the general `behavior' of the above reaction, but not much on how to use this information. Consider the reaction curve specifically (i.e., reaction rate vs substrate that's usually a hyperbolic but sometimes sigmoid curve). Is someone able to give an example of a not-too-complicated system (i.e., preferably close to the enzyme reaction shown above) and explain how we'd relate identification and stability of fixed points to the reaction curve for the system? Or if fixed points aren't used for this specific reason, just an example or two of what conclusions about this reaction we could come to from this information?
Additionally, but not as important to me at this stage, I know that bifurcations lead to destruction/creation of fixed points and affect stability, which relates to the answers which will be given for my above question; is there any other reason we care about bifurcations in relation to data we'd expect to see from an enzyme reaction experiment, besides affecting our fixed points?
Thank you in advance—my background is in pure math but I'd really like to be able to understand why we care about these concepts in an experimental setting.
Best Answer
In essence, it seems like you're looking to understand why people use dynamical systems theory to model processes in biology. To a large extent, the answer is complexity.
When biochemical reactions take place, they usually involve dozens (if not hundreds) of reagents whose reaction kinetics are interconnected in complicated, usually nonlinear ways. Attempting to obtain an analytical understanding of the functions describing the concentrations of said reagents over time is, as a result, usually a colossal (and futile) effort.
However, dynamical systems theory offers us an avenue for obtaining some of the most important properties of such a reaction network; the equilibrium states, their stability, and how they change when altering the system.
For example, it is very easy to obtain the equilibrium states of the reagent concentrations (fixed points) by equating the right-hand side of your dynamical equations to zero. However, this doesn't answer important practical questions about these equilibrium states; namely, how do I know my initial concentrations will always go down into that equilibrium state? What if there are two equilibrium statesโhow do I know which initial biochemical conditions will wind up in which equilibria? What does the shape of the reaction curve look like as the system approaches equilibrium?
Specifically regarding your question about reaction curves, here's an example of some information you can get from dynamical systems theory: you'll find that the decay into the equilibrium state is dominated by the linear part of the right-hand side of your dynamical equations evaluated at the corresponding fixed points when it exists. In other words, in most biological systems, decays will appear exponential when sufficiently close to the equilibrium, which is why exponential models are so ubiquitous for biochemical reactions.
Dynamical systems theory provides rigorous and fairly accessible answers to these types of questions even in the most complicated systems. And in more synthetic or alterable systems, we can also ask questions like "How do I tweak this system so that this equilibrium state stops being an equilibrium state?"; this is where bifurcation theory comes in, as it precisely studies the change of stability of fixed points (among other things).
Lastly, some biochemical reaction networks exhibit exotic non-equilibrium behaviors like sustained nonlinear oscillations, which happens in everything from glycolysis to the genetodynamics associated with the circadian rhythm. Dynamical systems theory has our back with these as well, which show up as limit cycle structures in that theoretical framework (and can therefore be studied by asking the same questions as above for the equilibrium states).
Hope this helps!