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S. G. J. Mochrie, Yale University
When biology and premedical students arrive in introductory physics courses, they are often skeptical about the relevance of physics and mathematics to their academic and professional goals. Often, students’ skepticism is reinforced by the topics presented in introductory physics, which generally owe more to tradition than to what is most scientifically important and/or relevant to biology. A destructive feedback loop can then result: low student interest leads to low student engagement, which leads to low student performance, which leads instructors to “dumb down” the course, which further reduces student interest, leading to even less engagement, and so on.
By contrast, biology is experiencing an ongoing transformation into a quantitative science as eloquently described by Bialek and Botstein in a seminal 2004 paper :
“Dramatic advances in biological understanding, coupled with equally dramatic advances in experimental techniques and computational analyses, are transforming the science of biology. The emergence of new frontiers of research in functional genomics, molecular evolution, intra- cellular and dynamic imaging, systems neuroscience, complex diseases, and the system-level integration of signal-transduction and regulatory mechanisms require an ever-larger fraction of biologists to confront deeply quantitative issues that connect to ideas from the more mathematical sciences.”
Biology departments are now increasingly hiring faculty with physics backgrounds, and biological physics is now a major subfield of physics, represented in physics departments across the country and in the APS. In recognition of this new biology, a number of reports [2, 3, 4, 5] have highlighted the increasing importance of quantitative skills for students who are planning biomedical careers, and the need to modify and augment undergraduate biology and premedical education accordingly. Recently, the biology community, in Vision and Change in Undergraduate Biology Education (VCUBE), has specified a number of core competencies that all undergraduate biology students should possess . These competencies and examples of how each competency might be demonstrated in practice are summarized in VCUBE’s Table 2.1, reproduced here as Table 1 (on the following page).
Inspection of Table 1 reveals that undergraduate biology education in the twenty-first century must embrace quantitative and mathematical approaches. Remarkably, many of the specified competencies are those that physicists seek for students to acquire in physics classes. In this context, the two-semester Introductory Physics for Life Sciences (IPLS) sequence, currently required for premedical students and biological science majors, presents a natural platform where these students could encounter quantitative and mathematical descriptions of biological and physiological phenomena for the first time. Where better than IPLS to first develop problem-solving strategies? Where better than IPLS to first develop the ability to use quantitative reasoning? Where better than IPLS to start developing the ability to use modeling and simulation? Where better than IPLS to start developing the ability to apply physical laws to biological dynamics? Where better than IPLS to start developing the ability to incorporate stochasticity into biological models? Indeed, VCUBE can be read by physicists as a call to transform IPLS into an engaging and exciting subject that is appreciated as essential to every biologist’s undergraduate education, in striking contrast to biology students’ current preconceptions. If physics departments ignore this coming change in the nature of under- graduate biology, biology students will nevertheless have to acquire the required competencies. It is then possible to envision biology departments replacing their physics requirements with “quantitative biology” requirements, taught outside of physics, with ensuing negative consequences for physics departments.
Now in its fifth year, we have developed and taught the first semester of a calculus-based IPLS class at Yale, that re-imagines the introductory physics syllabus to effectively channel a number of VCUBE’s competencies via a selection of biologically and medically relevant topics. This class is one of four introductory physics classes offered at Yale. The others comprise a course at a similar overall mathematical level, that follows a traditional syllabus, and two higher-level courses aimed at physics majors-to-be. Since 2010, approximately 580 students have taken IPLS. 66% are female; 34% are male. They are ethnically diverse: 40% self-identify as white, 60% self-identify as non-white, including significant numbers of students from groups underrepresented in STEM disciplines (e.g. 10% African American or African and 10% Hispanic). Most arrive possessing considerable biological and chemical sophistication, as a result of prior biology and chemistry classes. Their mathematical skills, however, are rusty, but improve tremendously over the course of the year. More than 80% identify themselves as premedical students. In many cases, they are involved in biological or biomedical research, and in medically-related volunteer work. After graduation, many go on to highly-ranked medical and graduate schools. All of them take the class in order to fulfill the physics requirement for medical school and/or the physics requirement of their major. It is the final physics course that they will ever take, and so there is no rationale to postpone compelling material for later.
The first part of the course is a traditional treatment of kinematics, Newtonian mechanics (excluding rotational dynamics), and energy conservation, although we follow recent advice [6, 7] to include a careful discussion of binding energies. We rely on this material in a number of subsequent modules. These later modules, however, deviate significantly from a traditional syllabus.
We include a module on random walks to introduce students to the ubiquitous role of stochastic processes in biology. Our first random walk example is Brownian motion. We point out that Albert Einstein’s theory of Brownian motion and its subsequent confirmation by Jean Perrin were pivotal in finally convincing skeptics at the outset of the twentieth century that atoms and molecules really exist. It is remarkable that biology works in spite of Brownian motion. What is truly amazing is that sometimes biology works because of Brownian motion, as the class sees when we discuss Brownian ratchets . We also discuss that, in evolutionary biology, a random walk has a central role in evolution via “genetic drift”. Genetic drift is the random process – it can be conceived as a random walk – by which “neutral” mutations, that offer no selective (dis)advantage, become fixed in a population, leading the population’s genome to evolve, or drift, away from its initial state, over many generations, as such neutral mutations accumulate. Importantly, the theoretical underpinnings of phylogenetic analyses, that seek to establish evolutionary relationships among organisms and populations of a single organism critically rely on the properties of evolution by genetic drift.
Starting from Newton’s laws, our module on fluid mechanics focuses on laminar fluid flow in microfluidic devices and especially on the flow of blood through the circulatory system. We discuss atherosclorosis and how you blush. We also discuss fluid flows in fluid circuits , which is precisely analogous to current flows in electrical circuits. The module concludes with a discussion of the principles that underlie the physiology of your circulatory system (Murray’s Law), which demonstrates how the physics of viscous liquid friction plausibly has determined human physiology.
A final example that stands outside usual introductory physics is our module on rates of change, which focuses on how a number of interesting processes can be modeled mathematically. Several disparate processes are described by similar equations. We are thus able to point out that the solutions and understanding gained in one case can be transferred to another. When colleagues first hear our topics, some complain: “that’s not physics.” Our riposte is that the equations that describe the early time course of HIV viral load in an individual patient following infection can be re-interpreted to describe an atomic explosion; and the equations that describe the occurrence of retinoblastoma – the most common childhood eye cancer – that results from somatic mutations are formally identical to equations that describe the number of a particular species of radioactive nuclei in a certain (medically-relevant) radioactive decay chain. We explain these applications and point out their connections in the course of the module. The medical bona fides of these examples are compelling: The mathematical modeling of HIV progression in an individual was important in the development of HIV treatments [10, 11, 12]; and the age-dependence of retinoblastoma onset lead to the hypothesis of tumor suppressor genes, which proved a tremendously important step in understanding cancer . The biomedical authenticity of these examples provide a powerful lesson for our students. In this year of Ebola, our upcoming discussion of the progression of disease through a population via the SIR model  is tragically topical. This module also well positions us to discuss genetic circuits in the second semester .
Of course, a key issue for any IPLS class is the level of mathematics to use and the mathematical approach to take. Because all students in the class at Yale would previously have taken a first course in calculus, our starting point was a calculus-based class. We also decided to emphasize the use of Wolfram Alpha to facilitate mathematical manipulations of all sorts. Using Wolfram Alpha empowers students to carry out more sophisticated mathematics than otherwise. Beyond calculus, the mathematics that we employ is dictated by the topics that we seek to treat, i.e. we carried out “backwards design” . We also include a number of simulations and calculations, implemented as Wolfram Demonstrations, which students can conveniently run in their web browsers and interact with via sliders. To discuss Brownian motion and statistical mechanics, we incorporate probability and random walks; to discuss steady-state diffusion and laminar fluid flow, we use simple differential equations, which we solve by direct substitution. Our module on mathematical modeling leads us to simple linear algebra. The incorporation of eigenvalues and eigenvectors in IPLS might initially seem ambitious, but, in fact, the technical demands on the students are that they be able to take a derivative of an exponential function of time, and that they then be able to solve two simultaneous equations in two variables to find the eigenvalues and eigenvectors, both of which they are able to do. On the other hand, we avoid multi-dimensional integrals, removing a source of significant discomfort for these students. At the end of the semester, we carry out an anonymous survey to ascertain students’ opinions. Of 366 respondents in 2011-2013, when asked about the level of mathematics, it was “way too advanced for my current ability” for just 2% of the students, “advanced but manageable” for 19%, “about equal to my current ability” for 47%, “below my current ability” for 24%, and “well-below my current ability” for 8%. Concerning the level of calculus specifically, it is “too high” for 19%, “just right” for 65%, and “too low” for 16% of 474 respondents in 2010-2013. In view of these responses, we judge that the mathematics we use is appropriate.
Our approach stands in contrast to the point of view that IPLS should conform to a perceived difference in culture between the physical sciences and the life sciences, namely that the life sciences have generally not embraced mathematics and quantitative analyses, in contrast to the physical sciences. Instead, our view is that future biologists will be well-served by a version of IPLS that seeks to meaningfully contribute to closing the mathematical and quantitative gap between physical and life science students. Of course, because of varying student interests, motivations, and abilities, there cannot and should not be a one-size-fits-all IPLS course, suitable for every situation. However, we believe there is a place for a course like ours at many institutions across the country. As biology faculty introduce new upper-level biology electives and major tracks that demand mathematical sophistication, in quantitative and systems biology, for example, or as they incorporate more mathematical elements into upper-level classes in ecology, epidemiology, or evolutionary biology, a class like ours will serve as an essential prerequisite.
We close with a few anonymous quotes from biology and pre-medical students about IPLS (PHYS 170):
“I just want to thank you for making physics a lot of fun for me. It means a lot. I am from a traditional Indian education system, that emphasized on learning a science in order to get into a college, and not for the pure pleasure of it all. As a result, I developed a huge aversion to physics and really delayed the taking of this class. However, my views have changed a lot. I am fascinated by this subject and the amount of scope it has. Thank you for making that possible.”
“Thank you for an awesome semester! I can’t believe I’d ever say this but I actually like physics now. Thank you for showing me how cool it can be.”
“Thank you for class this semester. It has been really great to make links between all of my science courses at Yale, and in many ways (and I am shy to admit this, but against my expectations) PHYS 170 provided the platform for just that.”
“I was skeptical about 170 at first and I mainly chose it over 180 because I didn’t have the multivariable calculus background needed to take 181. You have really turned me around. The first third of this class was pretty standard physics (mechanics, momentum), but the latter part of the semester really tied in to biological phenomena. Obviously, the models were over-simplifications of biological systems but they added another layer of understanding to concepts that I previously took as a given. The idea of diffusion as explained using probability is not something I would have ever thought of on my own. I doubt I would find that in a biology or biochemistry text- book either. I took biochemistry (MBB300) at the same time as 170 this semester and I was pleasantly surprised to see the Michaelis-Menten equation derived via different methods in both classes. It is wonderful to see the same ideas presented on a biological, chemical, and physical level.”
“This class is amazing if you genuinely like biology. If you’re a biology major because you’re premed or whatever you might not like it as much, but if you really care about biology this class is great. Physics is the future of biology, and this class gives you a taste of all the cool ways we can use quantitative techniques to describe living systems.”
Acknowledgements: I would like to express my thanks to the wonderful students who have taken the class over the last five years and to the fantastic Graduate Teaching Fellows that I have had the opportunity to work with. I am also deeply indebted to Sean Barrett, Sidney Cahn, Sarah Demers, Eric Dufresne, Jennifer Frederick, Stephen Irons, Corey O’Hern, Lynne Regan, Rona Ramos, Nick Read, William Segraves, and Paul Tipton for their invaluable advice and support, and Michael Choma, Scott Holley, and Tom Pollard for wonderful guest lectures.
Simon Mochrie is a Professor of Physics and Applied Physics at Yale University. His research leverages physical techniques to measure and understand biological mechanisms.