How is calculus helpful for biology majors?

Question

It's common for majors in biology to take calculus courses, and many calculus textbooks (and calculus professors) try to cater to these students by including applications to biology.

My question is, in what specific ways is a calculus course actually helpful for biology majors?

For example, are there any courses typically taken by biology majors that involve ideas from calculus? If so, what ideas come up? Do any biology courses actually require students to take derivatives, compute integrals, or solve differential equations?

I'm also curious in what ways a two-semester calculus course could be made more helpful for biology majors. For example, would it help to cover basic multivariable functions and partial derivatives? Applications of calculus to probability and statistics? Systems of differential equations? Fourier series?

Answer 1

I'm an old-school biologist (animal physiology) who works with mostly cell biologists. I sent out an email to a bunch of grad students and postdocs I work with. Here is the data so far:

1. Senior undergrad, pharmacology major: absolutely no calculus used in biology courses. She actually laughed when I asked her.
2. Grad student: Undergrad biophysics course used modeling with differential equations. Graduate class in systems cellular biology used modeling with differential equations.
3. Grad student: Undergrad physical chemistry used calculus, no biology
4. Grad student: none, other than watching some derivatives and integrals in an engineering-level physics. Suggests perhaps a course on bioinformatics might use calculus.
5. Grad student: none. Suggests systems biology might have some.
6. Grad student: none. Some algebra for bacterial growth curves.
7. Postdoc: no actual calculus used, but calculus helpful for understanding diffusion of molecules in space

I will add to the list (open-source data!) as emails come in, but it seems safe to say that calculus is rarely used by biology students outside of calculus class.

Answer 2

I happen to have revised our calculus syllabus for first year biology majors about one year ago (in a French university, for that matter). I benefited a lot from my wife's experience as a math-friendly biologist.

The main point of the course is to get students able to deal with quantitative models. For example, my wife studied the movement of cells under various circumstances.

A common model postulates that the average distance $d$ between two positions of a cell at times $t_0$ and $t_0+T$ is given by $$d = \alpha T^\beta$$ where $\alpha>0$ is a speed parameter and $\beta\in[\frac12,1]$ is a parameter that measures how the movement fits between a Brownian motion ($\beta=\frac12$) and a purely ballistic motion ($\beta=1$).

This simple model is a great example to show how calculus can be relevant to biology.

My first point might be specific to recent French students: first-year students are often not even proficient enough with basic algebraic manipulations to be able to do anything relevant with such a model. For example, even asking to compute how $d$ changes when $T$ is multiplied by a constant needs to now how to deal with exponents. In fact, we even had serious issues with the mere use of percentages.

One of the main point of our new calculus course is to be able to estimate uncertainties: in particular, given that $T=T_0\pm \delta T$, $\alpha=\alpha_0\pm\delta\alpha$ and $\beta=\beta_0\pm\delta\beta$, we ask them to estimate $d$ up to order one (i.e. using first-order Taylor series). This already involves derivatives of multivariable functions, and is an important computation when you want to draw conclusions from experiments.

Another important point of the course is the use of logarithms and exponentials, in particular to interpret log or log-log graphs. For example, in the above model, it takes a (very) little habit to see that taking logs is a good thing to do: $\log d = \beta\log T+\log \alpha$ so that plotting your data in log-log chart should give you a line (if the models accurately represent your experiments).

This then interacts with statistics: one can find the linear regression in log-log charts to find estimates for $\alpha$ and $\beta$. But then one really gets an estimate of $\beta$ and... $\log\alpha$, so one should have a sense of how badly this uncertainty propagates to $\alpha$ (one variable first-order Taylor series: easy peasy).

The other main goal of the course is to get them able to deal with some (ordinary) differential equations. The motivating example I chose was offered to me by the chemist of our syllabus meeting.

A common model for the kinetics of a chemical reaction $$A + B \to C$$ is the second-order model: one assumes that the speed of the reaction is proportional to the product of the concentrations of the species A and B. This leads to a not-so-easy differential equation of the form $$ y'(t) = (a-y(t))(b-y(t)).$$ This is a first-order ODE with separable variables. One can solve it explicitly (a luxury!) by dividing by the second member, integrate in $t$, do a change of variable $u=y(t)$ in the left-hand-side, resolve into partial fractions the rational fraction that comes out, and remember how log is an antiderivative of the inverse function (and how to adjust for the various constants the appeared in the process). Then, you need some algebraic manipulations to transform the resulting equation into the form $y(t) = \dots$. Unfortunately and of course, we are far from being able to properly cover all this material, but we try to get the student able to follow this road later on, with their chemistry teachers.

In fact, I would love to be able to do more quantitative analysis of differential equations, but it is difficult to teach since it quickly goes beyond a few recipes. For example, I would like them to become able to tell in a glimpse the variations of solutions to $$y'(t)=a\cdot y(t)-b \sqrt{y(t)}$$ (a model of population growth for colonies of small living entities organized in circles, where death occur mostly on the edge - note how basic geometry makes an appearance here to explain the model) in terms of the initial value. Or to be able to realize that solutions to $$y'(t)=\sqrt{y(t)}$$ must be sub-exponential (and what that even means...). For this kind of goals, one must first aim to basic proficiency in calculus.

To sum up, dealing with any quantitative model needs a fair bit of calculus, in order to have a sense of what the model says, to use it with actual data, to analyze experimental data, to interpret it, etc.

To finish with a controversial point, it seems to me that, at least in my environment, biologists tend to underestimate the usefulness of calculus (and statistics, and more generally mathematics) and that improving the basic understanding of mathematics among biologists-to-be can only be beneficial.

Answer 3

Most bio majors won't need calculus in their bio classes. They will take chemistry classes in which understanding rates of change is useful, so:

• partial derivatives will help them.

More importantly, many bio majors will work in quantitative areas in the life sciences, where data science is key. Think of developing drugs from chemical compounds, or clinical tests of drugs, or genomics. A calculus class with this in mind will definitely include:

• The normal curve -- since the expression $$\frac{1}{\sigma\sqrt{2\pi}}\Large e^{\Large-(x-\mu)^2/2\sigma^2}$$ and its integrals, which are ubiquitous in statistical thinking, will not become natural to them in any other way.

• Transforming data with log and exp, e.g. reading log-log plots.

• Different ways to visualize functions, e.g. contour graphs.

Answer 4

I am not a biologist, and this question asks for the contribution of a biologist, nevertheless I might be contribute about the practice in our university in Budapest.

We have a special two-semester calculus-type mathematics course for biologists developed together with biology departments. The curriculum is:

• First semester:

  • complex numbers, matrices, eigenvalues, Leslie model
  • elements of one-and higher dimensional calculus (very quickly, mostly through examples)
  • discrete dynamical systems

• Second semester:

  • differential equations (mostly geometric theory with phase diagrams on computer), Lotka-Volterra model
  • elements of probability theory

This looks extremely quick for a mathematician but we have to solve somehow the problem that some parts of biology need deep mathematical results but there is no time to develop the theory.

Later and in the master/PhD program they can choose specialized courses held by biologists about game theory in ecology and population models (based on Lotka-Volterra type models), disease transition or tumor growth models use heavy ODE theory.

Added: Here are some links to Hungarian course materials (at least the literature is in English).

population biology

evolutionary game theory

Answer 5

An all-inclusive neurobiology class, which is normally appropriate for upper-division undergraduates, will present the physiology of excitable membranes.

Modeling at this level can be as simple as the Nernst equation for the equilibrium potential of a particular ionic species:

^{http://en.wikipedia.org/wiki/Nernst_equation}

By taking into account ion permeability, the Goldman–Hodgkin–Katz equation can be used to illustrate the reversal potential for a given membrane:

^{http://en.wikipedia.org/wiki/Goldman_equation}

Neither of these models uses calculus explicitly, but more advanced students (especially those who are interested in computational modeling) can be introduced to the Hodgkin-Huxley model:

^{http://en.wikipedia.org/wiki/Hodgkin%E2%80%93Huxley_model}

As mentioned in some of the other answers, a thorough knowledge of statistics is incredibly useful to students pursuing undergraduate research or those with plans to continue their education, but the aforementioned example is an opportunity for students to directly employ differential equations-based models in the undergraduate biology curriculum.

Answer 6

One division of Biology that can be quite mathematical is Ecology and Evolutionary Biology. There are definitely courses that require calculus and differential equations quite similar to what you would teach an engineer for example. From what I understand this can come as quite a surprise to the biology students who go into Ecology because they like the outdoors and the plants/animals. But if you want to understand something like how it is possible that different animals can occupy what seems like the same evolutionary niche, then mathematical models really are the best way to do it.

From the University of Arizona course catalog (that link will require some clicking around, sorry):

ECOL 447 - Introduction to Theoretical Ecology Population growth and density dependence; predation; competition and apparent competition; coexistence mechanisms: niches, spatial and temporal variation; food web concepts and properties; applications. Emphasis on understanding through models and examples. Prerequisite: Calculus I

Answer 7

Some years ago I taught a one-semester course on mathematics for pharmacy students. (They also got a semester of statistics in another course.) I looked at some of the second and third-year prescribed books for the pharmacy degree and they had quite a lot of calculus in them. Physical pharmacy: rates of diffusion of various things. Interpreting the elimination of a drug given orally from the body by looking at measurements in the blood at different times: the drug goes first into the stomach and then into the bloodstream, so you end up with two coupled DEs (or even three, if some organ or tissue is acting as a reservoir). Chemistry: in pharmacy you are generally dealing with weak acids and weak alkalis, so the situation is considerably more complicated than in the usual beginning chemistry.

Certainly things like semi-log plots occurred quite a lot - not exactly calculus, but often taught with it. And we did teach the trapezoidal rule!

There wasn't any other maths/stats as such except the two one-semester courses in the Pharmacy program. They did a lot of chemistry and biology, and specialised courses on Pharmacy topics. This course was in Australia.

I'm a bit surprised at the Pharmacology major mentioned above.

And I would say that anyone who is good at both maths and biology has got some fantastic opportunities.

Answer 8

Differential equations are used to model e.g. predator/prey interactions in ecology, spread of diseases in epidemiology.

Much of (molecular) biology is chemical reaction kinetics, again calculus/differential equations.

[The above just as somebody with an interest in biology in general, no formal relation to the subject.]

Answer 9

1. Mathematics courses encourage analytical thinking in a way that may be helpful for biology majors.

2. There is some argument that calculus ought to be more widely known within the biology community. For example, see the following infamous paper, which has gained over 200 citations according to Google scholar:

   M.M. Tai, A Mathematical Model for the Determination of Total Area Under Glucose Tolerance and Other Metabolic Curves. Diabetes Care, Vol 17, Issue 2, 152–154.

   The "mathematical model" discussed in the trapezoid rule, which is often covered in second-semester calculus courses.

Answer 10

I know I am a little late to the party on this question, but when I read this question, I felt I could add some information of value. First, I am not a biologist, but I have taken a course in Mathematical Biology and Ecology where a wide range of topic were covered. Additionally, there are two good resources that show and discuss the math involved in Biology one is a two volume set. The books are Mathematical Biology I: An Introduction and Spatial Models and Biomedical Applications by JD Murray and Mathematical Models in Biology by Leah Edelstein-Keshet. Another book I own which isn't entire Biology based but does have Biology in it is Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering By Steven Strogatz.

Some of the topics may be mentioned in another post but I will still list them for completeness.

Topics that require Calculus based mathematical maturity are:

1. Continuous Population Models for Single Species $$ \frac{dN}{dt}=\text{birth}-\text{deaths}+\text{migration} $$
2. Discrete Population Models for a Single Species $$ N_{t+1}=N_tF(N_t)=f(N_t) $$
3. Models for Interacting Populations \begin{align} \frac{dN}{dt}&=N(a-bP)\\ \frac{dP}{dt}&=P(cN-d) \end{align}
4. Reaction Kinetics $$ S+E\mathrel{\mathop{\rightleftharpoons}^{k_1}_{k_{-1}}} SE\to P+E $$
5. Biological Oscillators and Switches $$ \frac{d\mathbf{u}}{dt}=\mathbf{f}(\mathbf{u}) $$
6. Perturbed and Coupled Oscillators and Black Holes (Not in space) $$ \frac{d\mathbf{u}}{dt}=\mathbf{f}(\mathbf{u},\lambda) $$
7. Dynamics of Infectious Diseases: SIR models \begin{align} \frac{dS}{dt}&=-rSI\\ \frac{dI}{dt}&=rSI-aI\\ \frac{dR}{dt}&=aI \end{align}
8. Reaction Diffusion, Chemotaxis, and Nonlocal Mechanisms $$ \frac{\partial}{\partial t}\int_Vc(\mathbf{x},t)dv=-\int_S\mathbf{J\cdot ds}+\int_Vfdv $$
9. Oscillators-Generated Wave Phenomena and Central Pattern Generators

These next topics are a little more difficult and require knowledge of PDEs but an advanced undergrad could handle this

1. Biological Waves: Single Species Models $$ \frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2} $$
2. The Use of Fractals
3. Multiple-Species Waves $$ \frac{\partial\mathbf{u}}{\partial t}=\mathbf{f(u)}+D\nabla^2\mathbf{u} $$
4. Spatial Pattern Formation with Reaction Diffusion Systems
5. Bacterial Patterns and Chemotaxis $$ \nabla\cdot\mathbf{J}_c=\nabla\cdot[\chi(n,c)\nabla c] $$
6. Mechanical Theory of Vascular Network Formations $$ \frac{\partial n}{\partial t}=-\nabla\cdot\frac{\partial\mathbf{u}}{\partial t} + \nabla\cdot\nabla\cdot(\mathbf{D(\epsilon)}n) $$
7. Epidermal Wound Healing \begin{align} f(n)&=\lambda c_0\frac{n}{n_0}\frac{n_0^2+\alpha^2}{n^2+\alpha^2}\\ f(n)&=\frac{\lambda c_0}{n_0}n \end{align}
8. Neural Models of Pattern Formations $$ \frac{\partial n}{\partial t} = f(n)+\int_Dw(x-x')[n(x',t)-1]dx' $$
9. Geographic Spread and Control of Epidemics \begin{align} \frac{\partial S}{\partial t}&=-rIS+D\nabla^2S\\ \frac{\partial I}{\partial t}&=rIS-aI+D\nabla^2I \end{align}

Answer 11

When you want to discuss the rate something happens, you will find that differential equations of calculus are helpful.

Some examples in biology:

1. population growths: dx/dt = Rx, describes unlimited/exponential growth of a population that could be rabbits, cells, etc.

2. kinetics of a chemical reaction: reversible [A][B] <-> [AB]. d[AB]/dt = k1*[A][B]-k2[AB] formation rate of d[AB]/dt slows as you use up [A] and [B]

Answer 12

One important application of calculus in biology is called the predator-prey model, which determines the equilibrium numbers of predator and prey animals in an ecosystem.

It's actually an application of "differential equations" but you will need calculus to "get there."

Answer 13

re: Senior undergrad, pharmacology major: absolutely no calculus used in biology courses. She actually laughed when I asked her. This truly unbelievable. I can find no Pharmacokinetics text that does not use AUC = Area Under the Curve, a Calculus concept if ever there was one. How can you be a pharmacologist without knowing about Bioavailability, a concept defined in terms of AUC? My guess that she just didn't realise what AUC actually meant. Sad. But this isn't just restricted to Professionals. I have seen inserts in prescription medicines, meant to be read by the uninitiated, referencing "the AUC to Infinity" (!) (I would include a scan but I don't know how to insert an image)

Answer 14

Calculus is seldom helpful for biology majors, if "helpful" means useful in a utilitarian, professional sense. The vast majority of biology majors are going into allied health fields: they intend to be doctors, pharmacists, physical therapists, vets, optometrists, and dentists. These professions are not like engineering, in which calculus is used from day to day. Here in California, the UC system decided ca. 1997 to start requiring biology majors to take calculus-based physics. The motivation was pretty transparent: they had too many biology majors (the major was "impacted"), and they wanted to get rid of some. This is similar to the fact that in 19th century Britain, if you wanted to be a military officer, you had to pass a test on Greek and Latin.

Does this mean that future military officers have nothing to gain from learning ancient Greek, or that future dentists have nothing to gain from taking calculus? Absolutely not. It simply means that for the future dentist, learning calculus is one possible ingredient in that quaint notion of a general education. It's a way of gaining broad knowledge about the world and getting experience in varied intellectual pursuits and ways of thinking.

For comparison, it may be helpful to ask the similar question of whether biology coursework is helpful to biology majors. A lot of it clearly is not, if helpful is used in the sense of day-to-day professional utility. For example, biology majors learn about the reproduction of ferns and club mosses, which is likely to be of very little practical utility to an optometrist.

Answer 15

There is at least one very good reason to know calculus as a Biologist. There was a certain paper published, I don't know the details, but could probably look it up, by a biologist to a bio journal detailing how to calculate the area under a curve by using this amazing approximation using rectangles and trapezoids. This of course was peer reviewed and hailed as a major advance for some part of bio that constantly needed to do this. The story goes on to say that the biologist knew that this came for math somewhere, but so many other biologists wanted to use the technique and needed something to cite, so he published the paper. However, the issue remains: Biologists, didn't know basic integration. I'm sure you could find this story online. I am not sure if it is valid, but I find it likely to be in part true at the very least. So being a respectable scientist is a good enough reason to learn something like calculus.

Answer 16

At the end of the day, all science is 'applied mathematics'...without the math supporting your observations, you greatly limit yourself in your chosen field. Can you get through life in a science career without math? Sure...if all you care about is qualitative observations. With post-trig math knowledge though (e.g.- Calculus, Differential Equations, Linear Algebra, etc)...you're provided a deeper, quantitative understanding of your chosen field.