Examples of Differential equations in Biology

I am a mathematician currently teaching some math classes at a university. Next semester, I'll have bachelor's degree biochemistry students. I want to know where certain math tools might be needed in such a field.

Where do differential equations arise naturally in professional Biology? Maybe from some data analysis of a complex phenomenon? What scenario?

I was thinking of a scenario where, for a complex set of variables, you would take one of them, say $$x$$ and plot $$\frac{\Delta x}{\Delta t}$$ vs $$x$$ and find that the relationship was e.g. polynomial ($$\frac{\Delta x}{\Delta t} \propto x^{3}$$).

• Unless you’re teaching in Harvard or MIT, the most you can expect from biochemistry students is simple algebra. That doesn’t help you, but “Blessed are they that expect nothing…”. Jan 31 at 20:02
• @David I was interested in numerically approximating systems of differential equations to model metabolic pathways when I was an undergrad. I was not allowed to pursue this. Not because it was explicitly disallowed, but rather because I was kept too busy memorizing multisyllabic terms (a little bit of physics and graph theory would have been better IMO) that I promptly would forget after finals. Feb 1 at 2:12
• @Galen: Did those multisyllabic terms include "multisyllabic"? =P Feb 1 at 6:21
• @user21820 Who knows! I forgot! 😂🤪 Feb 1 at 6:29
• All the posts are very helpful! Thank you to everyone! I am quite familiar with the use of Differential equations in population biology, but I sensed that wouldn’t be that satisfying to biochemists. You provided me with a lot of useful references :) Feb 1 at 20:32

For biochemistry students the obvious link to differential equations is the Michaelis–Menten equation which is a simple model of the kinetics of reactions involving enzymes. Your students will be hearing a lot about Michaelis–Menten in their biochemistry classes if they haven't already. It's also very accessible to students having their first exposure to differential equations.

Considering the wider field of biology there are several major topics that revolve around differential equations. Population Biology and the related field of Population Genetics are heavily mathematical with lots of differential equations involved.

Expanding beyond ordinary differential equations there is Alan Turing's Theory of Morphogenesis which tried to explain striping and spotting in animal hides using a system of coupled partial differential equations to model reaction rates and diffusion.

• Michaelis-Menten is super obvious, can't believe I didn't think of it. Feb 2 at 22:50

Update

Hodgkin-Huxley as suggested by commenter ctwardy is a great idea too.

I would strongly recommend the book mathematical ideas in biology by J Maynard Smith.

A couple of examples that spring immediately to mind with relevance to biochemistry might be:

Less biochemical but still very fun is Lotka-Volterra modeling of predator-prey interactions.

Another book that seems to directly address your question is this one.

Generally speaking, many biochemical process would often be modelled on the macro-molecular scale by differential equations. These include rate equations for the chemical processes, as well as diffusion kinetics within the cytoplasm or across membranes.

On the larger scale of organs and entire systems, we also have alveolar gas exchange, digestion and absorption processes, renal filtration, homeostasis, and of course many others.

Logistic differential equations in modeling population dynamics is the first to come to mind.

Populations grow under the constraints of limited resources, capturing the transition from initial exponential growth to a plateau as the carrying capacity of the environment is approached.

For a detailed exploration of logistic differential equations you can refer to this comprehensive guide on Brilliant.org.

The other answers offer many examples of differential equations for initial-value problems where you solve an (most often here) an ODE or a PDE. There are also boundary-value problems in biology, for example when modelling the strengths and elasticity of various structures supporting a plant (e.g. https://doi.org/10.34133/ehs.0007) or of animal bones (e.g. https://doi.org/10.1007/s12018-015-9201-1 for human medicine, but the principle is the same for other living and fossil vertebrates) or modelling the elasticity of plant leaves based on the plant's turgor (https://doi.org/10.1111/tpj.12042). Similar simulations had also been done for individual cells. In these cases the PDEs are elliptic and are most often solved by the finite element method or similar methods.