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I am part of a recently founded laboratory in computational biology and we are trying to get all the parts (biologists, computer scientists, mathematicians, and so on) to speak the same language, or at least one we can all understand. To do so, we want to figure out some strategies beside seminars and weekly lectures.

I am not the only one who went through this challenge, see: [1]. For example, How to explain core biological concepts like evolution and selection to a mathematician? How to explain a mathematical formalization to a biologist (beyond the very basic models the majority of biologists know, such as predator-prey and logarithmic growth)?

There must be strategies; for instance, books that are in an effective middle (mathematically rigorous, but stepped enough that a biologist can understand it). This is a long term project and we need to devise strategies to progressively "retrain" ourselves. (Bibliography suggestions are welcome.)

[1] https://liorpachter.wordpress.com/2014/12/30/the-two-cultures-of-mathematics-and-biology/

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    It is easy to explain evolution and selection to a mathematician. The hard part is to show them that the World is messy, complicated and each assumption is a very, very coarse approximation of the observed data (NOT "reality"!). In other words, that the land of biologists is very far from the Platonic Cave. – Piotr Migdal Apr 3 '15 at 22:06
  • In my experience there is sometimes an insurmountable cultural gap between biologists and mathematicians. In this case the only solution is to have someone in between who you can both talk to. Basically, if the biologist has never tried to formulate their problem mathematically because they don't see the point, then talking to a mathematician is not likely to work. Also, both sides need to be happy to accept each other's motivation. It's no good if the biologist feels they "own" the real problem. – dorothy Apr 4 '15 at 8:39
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Disclosure: I'm a mathematical biologist that came into it from the biology side.

I don't think it is necessary to retrain the biologists so they understand maths and retrain the mathematicians so they understand biologists, although these things should occur naturally to some degree with interdisciplinary work. Rather, I think it is important to understand the motivations of each 'type' and to tailor the language to the audience.

To grossly generalise, biologists are more interested in quantitative methods as a tool to answer interesting biological questions and mathematicians are more interested in the method/analysis used to answer that question.

So when mathematicians talk to biologists, they need to place less emphasis on the technical details of a model/analysis and focus on the general features. For example, if you are building a model to answer an evolutionary or ecological question, a biologist is more interested in the biological assumptions the model is making and whether or not the model is a reasonable abstraction of the biological system. In turn, the mathematician may need to explain why certain details of the system can't or shouldn't be included in the model (e.g. because they would complicate the analysis for little gain in intuitive understanding).

When biologists talk to mathematicians they need to frame their questions in a way that is conducive to a quantitative framework. If a mathematician is trying to build a model, they don't need or want to know every minute detail of a system. It's overkill and will just lead to confusion. What are the most relevant points? For example, if a biologist is interested in how the density of cows affects the density of grass in a paddock, then it isn't helpful for the biologist to give the mathematician a lesson on all the intricacies of grass growing and grass eating. It would be better if the biologist comes with a defined question, such as 'how does increasing the number of cows in a paddock affect grass regeneration?' and points out that the main elements in the system are 1. how grass grows (as some function of grass density) and 2. how grass is eaten (as some function of cow density).

If you want a book about mathematical biology that is written for biologists then I'd recommend "A Biologist's Guide to Mathematical Modeling in Ecology and Evolution" by Sarah P. Otto & Troy Day

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These kinds of interdisciplinary collaborations are becoming increasingly common. I am sure there are books covering at least 2 if not more disciplines but I think communication that is specific to the collaborative needs would be more effective. I would try to figure out what each of the party really needs to know about the other discipline. Try to find this out by simply asking over meetings, mails etc.

For the long run, depending on who is the "host" for a particular meeting or discussion, they take charge and deliver a talk about basics accompanied by slides, pointers etc.

For example, say the Mathematician is the host for a meeting about porting a program to a Beowulf cluster that does formal analysis of a particular gene expression over time (hypothetical scenario). In such a scenario, the mathematician would introduce the nitty-gritty about the formal analysis methods, things that needs to be considered and things that could be safely ignored and such.

If the computer scientist is the host in the same scenario, they will speak about the parameters needed for the program execution, why scaling up is important and issues such is numerical precision and software bugs that needs to be taken care of. Accompany the discussion with slides and provide pointers to basic concepts.

Over sufficient time, and enough communication, each party will become familiar with other party's jargons. Things may start slow but a cumulative effect will help accelerate the process as time passes.

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Have people well-versed in more than one area. They are very useful for bridging between the two camps. After a while, the people who are not as well versed in more than one area will at least gain a better understanding of what the others need to be useful, and of what is practical, what is possible but impractical, and what is impossible.

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You all know how to teach and learn the formal parts of any discipline. I'd suggest teaching each other the assumptions, and the things that don't work quite as well as they do in theory, and the jokes.

Although most feelings of "huh, that's funny" about the next discipline's work are going to be embarassing undergraduate misconceptions, always share them. Some of them are the sign of a mismatch in disciplinary assumptions and you want those to be discussed as soon as possible.

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Since you ask for potentially appropriate books, you may find Mathematical Biology: An Introduction with Maple and Matlab by Shonkwiler and Herod helpful for bridging the gap. What struck me about this book is that it takes the time to explain the biology to mathematicians, which I appreciated coming from the mathematics side. (Whether it is as successful at explaining the mathematics to biologists is more difficult for me to judge.) The final three chapters introduce genetics, genomics and phylogenetics, including a brief introduction to algebraic statistics.

I should emphasize that the book is just an introduction, so do not expect to find an in-depth examination of any of the topics. However, it could be a useful stepping stone.

As for helping biologists to understand mathematics, I think a key step is to provide motivation: Why is the mathematics useful and what can it do to aid analysis, solve biology problems, or even deepen understanding of biology itself? In particular, what worthwhile things can be achieved with mathematics that would be difficult or impossible without it? What's the payoff? A following or concurrent step is to make connections with prior knowledge. For instance, what mathematics/statistics do biologists already know and use? How do new proposed methods/algorithms/formalisms build on and improve on previous ones?

As an example of the latter, in Gilbert Strang's classic Introduction to Applied Mathematics, he masterfully introduces the Kalman filter step-by-step by starting with least squares and linear regression, then going on to weighted least squares (what can you do when you trust some observations or measurements more than others), then introducing recursive least squares (when your measurements arrive one at a time and you want adjust your model without a full recalculation each time) and finally bringing in the Kalman filter to deal with the situation when your model is non-stationary (see sections 1.4 and 2.5 of his book for details). While this example is not specific to biology (although the Kalman filter is used in systems biology), it shows the step-by-step process, starting with a familiar topic.

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