How to Start a Career in Mathematical Biology

Mathematical biology is a multidisciplinary field of research where questions that arise in biology are addressed with mathematics. In this sense, it is very much like mathematical physics, which has a long history of advancing physics, while at the same time has given rise to new areas of research in mathematics. However, biology is far more complex than physics. Take, for example, a single cell with its millions of proteins and DNA molecules: It has to navigate its life by defending itself while supporting the organ to which it belongs, continuously sensing its ever-changing environment and keeping in contact with other cells. And there are billions of billions of such cells even in a mouse, or smaller animals.

There has been gigantic progress in the biological and biomedical sciences in recent decades, generating an immense amount of data. The need to extract knowledge from these data has been a challenge and an opportunity for mathematics to step in. This, in fact, was the reason why I founded, in 2002, the Mathematical Biosciences Institute (MBI) at The Ohio State University. The MBI, which was one of the eight institutes funded by the National Science Foundation at the time, brought biologists and mathematicians together in many workshops, and hosted short- and long-term visitors, and long-term postdocs, to discuss and collaborate on research projects in mathematical biology. Based on my experience in working in this field over the last 20 years, I would like to advise graduate students on how to start a career in mathematical biology, focusing primarily on how to proceed with the first project, toward a PhD thesis. I assume that you have already been exposed to mathematical biology at the undergraduate level, in terms of simple models and numerical simulations. Such material can be found in the textbooks 1, 2 that we are using at OSU.

The field of mathematical biology is immensely broad. It is important that you choose a mentor who is actively engaged in research in mathematical biology and, preferably, not too narrow. Expect your mentor to suggest reading material, books and research articles (they are mostly online). Your mentor is expected to meet with you frequently, to respond to your questions, and to hold informal discussions that will arise from your readings. Such discussions will naturally lead to biological questions that have the potential to be addressed with mathematics. Addressing one of these questions, I expect, could become a project for your PhD thesis. So here is some advice how to develop this project, illustrated by an example of a model represented by a dynamical system.

1.

Build a chemical reaction network based on the biological mechanism

You need to build a chemical reaction network as a basis for addressing your biological question. This necessitates a solid background on the biological details related to your question, and only then decide which variables (e.g., cells, proteins) have to be included in order to address the question, and which may be dropped; it is important not to make the model unnecessarily complicated, but also not to oversimplify the biology. Draw a reaction network with the selected variables as nodes, and with edges indicating their functions (activation, inhibition, catalyzation, etc.). For example, let X and Y be two types of proteins, a sharp arrow from node X to node Y indicates that X activates Y, while an arrow with blocked edge indicates that X deactivates (or blocks) Y.

2.

Develop a mathematical model

To develop a mathematical model on the basis of a biological network, you need to represent each edge by a mathematical expression. For example, a sharp arrow from cells Y to proteins X means that Y produces X, and we write

$$\begin{equation*} \frac{dX}{dt}=aY-bX, \end{equation*}$$

where a is the rate of production and b is the rate of degradation of X (every protein eventually degrades). If a drug Z blocks the degradation of X, we replace $b$ by $\tfrac{b}{1+cZ}$, where $c$ is the “strength”of the drug.

Decisions have to be made, based on the particular situation, as to what level of detail to incorporate into the model. An example of a more complex case is the enhancement of reaction by ligands. Here, let L, R, and P denote a ligand, receptor, and product respectively. When the ligands L outside cells X attach to proteins R on the surface (membrane) of cells X and, as a result, R changes its structure and starts a chain reaction (signaling cascade) of proteins that eventually lead to the production, by X, of some new proteins P. How do we model this process? The answer depends on what is relevant to the question we are addressing. If all we need to know is that L caused the production of P by X, then we represent this process by writing,

$$\begin{equation*} \frac{dP}{dt}=a\frac{XL}{K+L}, \end{equation*}$$

where a is the rate by which X can “eat” the proteins L (with R), and the ‘eating’ expression is the famous Michaelis-Menten functional response. If the biological question involves space (e.g., tissue growth), then we need to use a PDE model which includes diffusion of some of the variables.

3.

Parameter estimates

The mathematical model has many parameters, and they have to be correctly estimated before the model can be reliably used. Some parameters can be easily estimated. For example, consider the case $\frac{dX}{dt}=a-bX$, where X is a protein with constant source a. The half-life s of X is defined by the relations

$$\begin{equation*} \frac{dX}{dt}=-bX,\quad X(s)=\frac{X(0)}{2}, \end{equation*}$$

so that $b=\tfrac{ln2}{s}$. Since the quantity s is recorded for many known substances, the parameter b can be reliably determined. As for the parameter a, we try to find the average density $X_0$ of $X$, and then infer $a=bX_0$, assuming the biological system is at average state.

But there are usually many other parameters that cannot be found, nor inferred from available literature, and the only way to estimate them is by making some assumptions. It may be good to discuss with your mentor or biologists which assumptions are biologically acceptable and which are not. Such a case arises, for instance, in an equation of the form

$$\begin{equation*} \frac{dX}{dt}=aY-cZ-bX, \end{equation*}$$

where both a and c are unknown rates. If Y is assumed to be more efficient than Z in activating X, then you make an assumption that a is one order of magnitude greater than c, for instance, and you can reduce the situation to the case of one unknown parameter, as in the case considered above.

4.

Model validation

You first simulate the model extensively to show that the predictions agree with known experimental studies. This model validation process is essential, since there is always some uncertainty in the parameters, and perhaps even in some of the model equations. For experimental data you look either into animal studies (e.g., mouse models) or in vitro studies. Agreement with experimental data has to be at least qualitatively, but also ‘not far off’ quantitatively in terms of the range of known measurements.

Unless you are extremely fortunate, you are more than likely to find some disagreements with known literature in your first try. This often means that more than one of your uncertain parameters are wrong and need to be changed. To this end, one must make an educated guess as to which parameter is the most critical and which should be the first to change. This is a judgement call that improves with intuition and experience with the mathematical models; your mentor could be very helpful.

After you have made the changes in some parameters, you repeat the simulations. This process can take many steps, but eventually converges to a model that has shown to replicate known experimental results. Now you are ready to use the model to address the biological question.

5.

Addressing your biological question: A case study

Cancer is treated with a combination of two drugs, chemotherapy D and immune therapy G. Both drugs are injected intravenously once a month, D at day 1 of the month and G by day 14.

Question: Is this the best schedule, or are there other schedules that yield more benefits to the patient?

You begin by choosing the major players of the model. Clearly, cancer cells and the drugs D and G need to be included, but, since G activates the immune response to cancer, also the most relevant immune cells; we probably need also to include proteins (cytokines) that these cells secrete in their interactions among each other and with cancer cells.

In order to address the biomedical question, you need to quantify the benefits of treatment, i.e., the efficacy of the drugs, in mathematical terms. If your model consists of a system of PDEs, then you may take the efficacy E(D,G) to be the percentage of tumor volume reduction, by a specified time horizon T. We can now simulate the model and compute E(D,G) for different schedules, and thus generating quantitative answer to address the original question.

But you can do much more with model. You can try changing/optimizing the amount of drugs to reduce side effects, or changing the ratio D/G. You can also modify the model to fit the profile of a specific type of cancer. You may end your project with a paper that makes suggestions for future clinical trials. In [3] you will find more details and examples on how to build mathematical models.

As you have completed a few projects successfully, you have now finished your thesis and are going to receive a PhD degree. The time has come to apply for a job in a university or in some research institute. Take a look at the book “A PhD is not enough,” 4; it has good advice for early careers in the physical sciences, and some of it applies also to careers in mathematical biology. In your application for a job, you will include a CV and a short essay describing your research accomplishments and plans for future research. Your application will show that you have done innovative work in an area of interest and promise. Ask your mentor to review your application!

If your project has attracted the attention of fellow researchers, you may be invited for one or multiple job interviews, and you are going to give a one-hour job talk. It is critically important that you prepare your presentation very carefully. Giving a talk is not the same as writing a paper. Prepare a talk that will make people want to hire you. Start by describing a broad area of research, of broad interest and critical importance, with challenges and opportunities for progress. And then present your own research, as naturally embedded in this general context. Your future research plans should be ambitious, but not unrealistic. Tell a good story. Rehearse your talk in front of an audience who are not shy to give critical feedback, prepare a response to questions.

In your job interview, you are likely to be asked to describe your research in a few sentences (e.g., by the dean or chair of the department); prepare how to respond. It is sometimes more difficult to explain your work in five minutes than to explain it in one hour.

Although, in this article, I gave examples of modeling with ODE or PDE, there are many other areas of mathematics which offer PhD projects and a career in mathematical biology, including:

(i)

Algebraic methods of symbolic computations, used in genomics, proteomic, and analysis of molecular structure of genes 5;

(ii)

Knot theory, applied to folding of proteins and nucleic acids 6;

(iii)

Topological data analysis, used to generate shapes of biomolecules and cells 7;

(iv)

Computational biology, with data analysis, mathematical modeling and computational simulations, which is used to study protein-protein interaction and metabolic processes 8, could be a project more suitable in a multidisciplinary PhD program, as could be a project in artificial intelligence in biology, applied to agriculture, medicine, and bioindustry 9.