# What maths knowledge is required for a lab-based (molecular and cell biology) PhD?

I am currently enrolled on an MSci in Biosciences and am planning to apply to PhDs starting next year. I want to use my final summer to practise my maths skills which are severely lacking.

Because I have just three months I don't want to attempt something unrealistic, I just want to study the most important things that will enable me to work in the lab to conduct basic scientific research (measurements, concentrations, dilutions?). So far my plan is to work methodically through the book 'Fundamental Laboratory Mathematics' by Lela Buckingham. I don't know if this is the best way because I don't really know what is expected at PhD level.

The list is something like this:

• Reasonable competence at arithmetic. It helps to be able to do math in your head so you don't have to keep looking for the calculator in the middle of pipetting.
• Basic (middle school) algebra. Often when preparing solutions, you will end up with simple linear equations and have to solve for x.
• Probability and statistics are useful when designing and interpreting experiments and dealing with experimental error. This is technically not something you use while doing the experiments, but at "the drawing board", and you can collaborate with someone who is knowledgeable about these topics if you are not.
• Understanding logarithms and exponents is useful when working with enzyme kinetics, quantitative chemical reactions (pH) and exponential growth (microbe cultures and quantitative PCRs). There are usually software tools available to do this stuff for you, and doing these types of calculations "by hand" gives noticeably worse results, so it's not so much a matter of "having math skills" as understanding the general principles.
• If you work with things like biophysics, biochemistry, protein folding simulations, microscopy then a strong knowledge of modern physics (electromagnetism, quantum physics, molecular forces, fluid dynamics) will be very helpful.
• If you work in high-throughput areas like genomics, a strong knowledge of probability, statistics and combinatorics will be very helpful.
• If you work in a highly computational field, good knowledge of programming, machine learning, data analysis, AI, high performance computing will be helpful.
• If your work is about developing new devices, good knowledge of electromagnetism and electronics will be helpful.

In other words, for just "general" lab work, just middle school math is basically "enough". If you know some basic stats and probability, you're golden.

If you're going into a specialized field, you may need a lot more, but then you would know this already when going into that field.

Note: I have completely ignored things like ecology here, because the question specifies molecular and cell biology.

• For the log and exponentials, the goal should be to have an intuitive understanding of what kind of result to expect. You want to notice if you misstyped in the software or the software is not doing what you think it does. Commented Jun 5, 2023 at 9:49
• As a physicist, I would quibble with one detail of your terminology: "modern physics" is not usually taken to include electromagnetism or fluid dynamics. Commented Jun 5, 2023 at 20:02

(In many graduate programs) After you are admitted you may have a chance to take an undergraduate course in math if needed to fill in gaps in your background. So don't get stressed out if you cannot finish everything in three months.

Because I have just three months I don't want to attempt something unrealistic, I just want to study the most important things that will enable me to work in the lab to conduct basic scientific research (measurements, concentrations, dilutions?).

I am not familiar with your book, but did skim it it's TOC online. The book seems reasonable.

I would also ask your professors in your current program as well as your future program what they think you need. Ask professionally and politely and most likely, they will be impressed at yourself awareness and desire to improve a missing skill on your own (or a least view your actions favorably). They know your program and its needs better than anyone on this site. You might look at books for general audiences like Essential Math for Data Science.

Last, I hopefully you are not too stressed out about this. Doing anything is better than doing nothing, so you're off to a good start. In general, graduate faculty view self initiative as a positive attribute.

There is no one size fits all answer here. It will depend on your research project (which can still change direction even after you have started, especially if you are talking about fundamental research in the area of molecular and cell biology) which and how much math you will require.

I would argue, though, that things like concentrations and dilutions are an absolute must to be able to work with when you are in the lab. Even if nowadays you can look everything up online or have a website/program do the math for you, it is very useful to know and understand what you are doing so you can also develop a feel for when you are off by an order of magnitude. So yes, this would be basic biology applied mathematics that you need to know or your lab work will be slowed down and or suffer on a daily basis. That being said, studying this stuff over the summer will only get you so far: you become much more skilled by doing this in practice. Research internships typically prepare you a bit for what life at the bench will be like in real life.

All other things you should probably aim to learn on the job - applying basic statistics (and consulting with a statistician when you aren't sure), more advanced problems when they arrive (do you need to do quantitative modelling for your project? do you need to understand statistical significance of large genomics data analyses or population genetics? Totally different questions and different expertise required). That's the beauty about scientific research and a PhD in particular: You continue to learn every single day. Also don't forget that science is becoming increasingly collaborative, so find colleagues/collaborators with the skills you lack so you can complement each other.

I am an ex-physicist who has reviewed* several PhD memoirs of friends who were biologists. There were "experimentalists" and "theoreticians" in that cohort.

By far what they were lacking most was knowledge of statistical methods. Some were so bad that I mentally could not cope to go through the whole thesis. The defenses were very successful.

So learn statistical methods from a purely practical angle if you want to be ethical in your research. There are so many butchered statistical results that yours will probably slip through so no need to be overly stressed. Note that the ones who will read you are likely to have the same understanding of statistics as you.

This said practical statistical methods are really worth learning so that you know if you are going in the right direction before someone else does and then a lot of unrecoverable work (or time) is lost.

The silver medal goes, ex aequo, to calculus and differential equations.

Again, we are talking practical stuff here, not theory.

With calculus, you will be able to understand how various functions behave and what to look for to understand their behaviour. Expected level is high-school.

Differential equations are, I think, how the world is described (but they do not want us to know). Sooner or later you will find a "change of something is a function of the something", or twice this. I do not think you should learn how to solve them (this is tricky to say the least, in physics we used to have a year-long course just on differential equations). You should really try, however, to find someone knowledgeable so that you can get their opinion about feasibility/countability/etc.

Story time: my wife ended up with a stiff equation when working on enzymes and attempted to resolve it with (I forgot the name of the program they were using in the 90's at her biology department). It was a disaster, mostly because no one in the dream of them could detect a stiff equation (they did not know about this beast) and it would have taken them eons to compute them the way the intended. She turned to me (tadam!) and I did the computation for her and guess what - she found out that the "standard behaviour" that was in use at the time had an issue because the equation was not solved correctly. Then came the Nobel prize and everything (well, it would have come if we did not switch to different lives after that)

The bronze (and special prize) goes to learning to code with Python. Look up Jupyter, spend 2 afternoons learning Python, then a week to learn Pandas and you will have an incredibly powerful tool in your toolbox. You can become the hero of the team with this.

* and have ben traumatized by

• I don't think functional analysis is commonly studied in high-school. Calculus or just "study of functions" seem to better capture what you're after here. Commented Jun 3, 2023 at 14:21
• Can only accept one but just wanted to say thanks, your answer was extremely helpful. Commented Jun 3, 2023 at 18:14
• @Anyon: thank you, you are absolutely right (that was a failed translation from French). I corrected the answer.
– WoJ
Commented Jun 4, 2023 at 17:22
• Bayesian statistics is most important for understanding machine learning and how it is used. The traditional AP Statistics class is rather limited. Commented Jun 5, 2023 at 17:14

I would approach this in an empirical manner. Contact your Ph.D. program and ask them what they expect. You can brush up in specific areas where you do not feel confident.

Learn from people who properly use mathematical, statistical, and computer science tools. Always ask them why a particular tool is relevant and how to use it properly. Always be open to learning how new tools can help you in your research.