Teaching Dynamics to Biology Undergraduates: the UCLA Experience

The goal of Intermediate Physics for Medicine and Biology, and the goal of this blog, is to explore the interface between physics, medicine, and biology. But understanding physics, and in particular the physics used in IPMB, requires calculus. In fact, Russ Hobbie and I state in the preface of IPMB that “calculus is used without apology.” Unfortunately, many biology and premed students don’t know much calculus. In fact, their general math skills are often weak; even algebra can challenge them. How can students learn enough calculus to make sense of IPMB?

A team from UCLA has developed a new way to teach calculus to students of the life sciences. The group is led by Alan Garfinkel, who appears in IPMB when Russ and I discuss the response of cardiac tissue to repetitive electrical stimulation (see Chapter 10, Section 12). An article describing the new class they’ve developed was published recently in the Bulletin of Mathematical Biology (Volume 84, Article Number 43, 2022).

There is a growing realization that traditional “Calculus for Life Sciences” courses do not show their applicability to the Life Sciences and discourage student interest. There have been calls from the AAAS, the Howard Hughes Medical Institute, the NSF, and the American Association of Medical Colleges for a new kind of math course for biology students, that would focus on dynamics and modeling, to understand positive and negative feedback relations, in the context of important biological applications, not incidental “examples.” We designed a new course, LS 30, based on the idea of modeling biological relations as dynamical systems, and then visualizing the dynamical system as a vector field, assigning “change vectors” to every point in a state space. The resulting course, now being given to approximately 1400 students/year at UCLA, has greatly improved student perceptions toward math in biology, reduced minority performance gaps, and increased students’ subsequent grades in physics and chemistry courses. This new course can be customized easily for a broad range of institutions. All course materials, including lecture plans, labs, homeworks and exams, are available from the authors; supporting videos are posted online.

Sharks and tuna, the predator-prey problem,
from Garfinkel et al.,
Bulletin of Mathematical Biology,
84:43, 2022.

This course approaches calculus from the point of view of modeling. Its first example develops a pair of coupled differential equations (only it doesn’t use such fancy words and concepts) to look at interacting populations of sharks and tuna; the classical predator-prey problem analyzed as a homework problem in Chapter 2 of IPMB. Instead of focusing on equations, this class makes liberal use of state space plots, vector field illustrations, and simple numerical analysis. The approach reminds me of that adopted by Abraham and Shaw in their delightful set of books Dynamics: The Geometry of Behavior, which I have discussed before in this blog. The UCLA course uses the textbook Modeling Life: The Mathematics of Biological Systems, which I haven’t read yet but is definitely on my list of books to read.

My favorite sentence from the article appears when it discusses how the derivative and integral are related through the fundamental theorem of calculus.

We are happy when our students can explain the relation between the COVID-19 “New Cases per day” graph and the “total cases” graph.

If you want to learn more, read the article. It’s published open access, so anyone can find it online. You can even steal its illustrations (like I did with its shark-tuna picture above).

I’ll end by quoting again from Garfinkel et al.’s article, when they discuss the difference between their course and a traditional calculus class. If you replace the words “calculus” and “math” by “physics” in this paragraph, you get a pretty good description of the approach Russ and I take in Intermediate Physics for Medicine and Biology.

The course that we developed has a number of key structural and pedagogical differences from the traditional “freshman calculus” or “calculus for life sciences” classes that have been offered at UCLA and at many other universities. For one, as described above, our class focuses heavily on biological themes that resonate deeply with life science students in the class. Topics like modeling ecological systems, the dynamics of pandemics like COVID-19, human physiology and cellular responses are of great interest to life science students. We should emphasize that these examples are not simply a form of window dressing meant to make a particular set of mathematical approaches palatable to students. Rather, the class is structured around the idea that, as biologists, we are naturally interested in understanding these kinds of systems. In order to do that, we need to develop a mathematical framework for making, simulating and analyzing dynamical models. Using these biological systems not purely as examples, but rather as the core motivation for studying mathematical concepts, provides an intellectual framework that deeply interests and engages life science students.

 

Introduction to state variables and state space. Video 1.1 featuring Alan Garfinkel.

https://www.youtube.com/watch?v=yZWG0ALL3mI

Defining vectors in higher dimensions. Video 1.2 featuring Alan Garfinkel.

https://www.youtube.com/watch?v=2Rjk0O3yWc8

The Emperor of All Maladies: A Biography of Cancer

One topic that appears over and over again throughout Intermediate Physics for Medicine and Biology is cancer. In Section 8.8, Russ Hobbie and I discuss using magnetic nanoparticles to heat a tumor. Section 9.10 describes the unproven hypothesis that nonionizing electromagnetic radiation can cause cancer. In Section 13.8, we analyze magnetic resonance guided high intensity focused ultrasound (MRgHIFUS), which has been proposed as a treatment for prostate cancer. Section 14.10 includes a discussion of how ultraviolet light can lead to skin cancer. One of the most common treatments for cancer, radiation therapy, is the subject of Section 16.10. Finally, in Section 17.10 we explain how positron emission tomography (PET) can assist in imaging metastatic cancer. Despite all this emphasis on cancer, Russ and I don’t really delve deeply into cancer biology. We should.

Last week, I attended a talk (remotely, via zoom) by my colleague and friend Steffan Puwal, who teaches physics at Oakland University. Steffan has a strong interest in cancer, and has compiled a reading list: https://sites.google.com/oakland.edu/cancer-reading. These books are generally not too technical but informative. I urge you to read some of them to fill that gap between physics and cancer biology.

The Emperor of All Maladies,
by Siddhartha Mukherjee.

Steffan says that the best of these books is The Emperor of All Maladies: A Biography of Cancer, by Siddhartha Mukherjee. Ken Burns has produced a television documentary based on this book. You can listen to the trailer at the bottom of this post. If you are looking for a more technical review paper, Steffan suggests “Hallmarks of Cancer: The Next Generation,” by Douglas Hanahan and Robert Weinberg (Cell, Volume 4, Pages 646–674, 2011). It’s open access, so you don’t need a subscription to read it. He also recommends the websites for the MD Anderson Cancer Center (https://www.mdanderson.org) and the Dana Farber Cancer Institute (https://www.dana-farber.org). 

Thanks, Steffan, for teaching me so much about cancer.

Cancer: The Emperor of All Maladies, Trailer with special introduction by Dr. Siddhartha Mukherjee.
 https://www.youtube.com/watch?v=L9lIsNkfQsM

 

Siddhartha Mukherjee, The Cancer Puzzle

 https://www.youtube.com/watch?v=vLH7e724TjA&t=178s

The Rest of the Story 3

Harry was born and raised in England and attended the best schools. After excelling at Summer Fields School, he won a King’s Scholarship to Eton College—the famous boarding school that produced twenty British Prime Ministers—where he won prizes in chemistry and physics. In 1906 he entered Trinity College at the University of Oxford, the oldest university in the English-speaking world, and four years later he graduated with his bachelor’s degree.

Next Harry went to the University of Manchester, where he worked with the famous physicist Ernest Rutherford. In just a few short years his research flourished and he made amazing discoveries. Rutherford recommended Harry for a faculty position back at Oxford. He might have taken the job, but after Archduke Franz Ferdinand of Austria was assassinated in Sarajevo in June 1914, the world blundered into World War I.

Like many English boys of his generation, Harry volunteered for the army. He joined the Royal Engineers, where he could use his technical skills as a telecommunications officer to support the war effort. Millions of English soldiers were sent to fight in France, where the war soon bogged down into trench warfare.

Page 2

First Lord of the Admiralty Winston Churchill devised a plan to break the deadlock. England would attack the Gallipoli peninsula in Turkey. If the navy could fight their way through the Dardanelles, they could take Constantinople, reach the Black Sea, unite with their ally Russia, and attack the “soft underbelly” of Europe. Harry was assigned to the expeditionary force for the Gallipoli campaign.

The plan was sound, but the execution failed; the navy could not force the narrows. The army landed on the tip of the peninsula and immediately settled into trench warfare like in France. There in Gallipoli, on August 10, 1915, a Turkish sniper shot and killed 27-year-old Second Lieutenant Henry Moseley—known as Harry to his boyhood friends.

Isaac Asimov wrote that Moseley’s demise “might well have been the most costly single death of the War to mankind.” Moseley’s research using x-rays to identify and order the elements in the periodic table by atomic number was revolutionary. He almost certainly would have received a Nobel Prize if that honor were awarded posthumously.

And now you know the REST OF THE STORY. Good Day!

-----------------------------------------------------------------------------------------------------

This blog entry was written in the style of Paul Harvey’s radio show “The Rest of the Story.” My February 5, 2016 and March 12, 2021 entries were also in this style. Homework Problem 3 in Chapter 16 of Intermediate Physics for Medicine and Biology explores Moseley’s work. Learn more about Henry Moseley in my March 16, 2012 blog entry.

Does a Nerve Axon Have an Inductance?

When I was measuring the magnetic field of a nerve axon in graduate school, I wondered if I should worry about a nerve’s inductance. Put another way, I asked if the electric field induced by the axon’s changing magnetic field is large enough to affect the propagation of the action potential.

Here is a new homework problem that will take you through the analysis that John Wikswo and I published in our paper “The Magnetic Field of a Single Axon” (Biophysical Journal, Volume 48, Pages 93–109, 1985). Not only does it answer the question about induction, but also it provides practice in back-of-the-envelope estimation. To learn more about biomagnetism and magnetic induction, see Chapter 8 of Intermediate Physics for Medicine and Biology.

Section 8.6

Problem 29½. Consider an action potential propagating down a nerve axon. An electric field E, having a rise time T and extended over a length L, is associated with the upstroke of the action potential.

(a) Use Ohm’s law to relate E to the current density J and the electrical conductivity σ. 

(b) Use Ampere’s law (Eq. 8.24, but ignore the displacement current) to estimate the magnetic field B from J and the permeability of free space, μ₀. To estimate the derivative, replace the curl operator with 1/L. 

(c) Use Faraday’s law (Eq. 8.22, ignoring the minus sign) to estimate the induced electric field E* from B. Replace the time derivative by 1/T. 

(d) Write your result as the dimensionless ratio E*/E. 

(e) Use σ = 0.1 S/m, μ₀ = 4 π × 10^-7 T m/A, L = 10 mm, and T = 1 ms, to calculate E*/E. 

(f) Check that the units in your calculation in part (e) are consistent with E*/E being dimensionless. 

(g) Draw a picture of the axon showing E, J, B, E*, and L. 

(h) What does your result in part (e) imply about the need to consider inductance when analyzing action potential propagation along a nerve axon.

For those of you who don’t have IPMB handy, Equation 8.24 (Ampere’s law, ignoring the displacement current) is

∇×B = μ₀ J

and Eq. 8.22 (Faraday’s law) is

∇×E = −∂B/∂t .

I’ll leave it to you to solve this problem. However, I’ll show you my picture for part (g).

Also, for part (e) I get a small value, on the order of ten parts per billion (10^-8). The induction of a nerve axon is negligible. We don't need an inductor when modeling a nerve axon.