# “My lab is in my mind”

### Prof. Stevens, what scientific topics are you working on right now?

As a mathematician, I am also collaborating with biologists. In this respect I am interested in developmental processes, and especially in those that are fascinating mathematically too. We construct mathematical models – in other words, equations, – which, ideally, provide explanations or predictions for biological phenomena. An example is, how initially observed symmetries during embryogenesis break down. In their early stages embryos almost look like little spheroids. With further cell divisions, this three-dimensional symmetry is broken, and invaginations start to develop. Questions we address are the following: To what extent are these changes of symmetry genetically predetermined? To what extent does this biological system develop its own dynamics while the embryo is growing? Mathematics is an ideal language for understanding and describing such non-linear dynamics in space and time.

Some topics I have been studying for a while now are the motility of microorganisms – like bacteria in our intestines, the motion of cells and the motion of molecules. When cells detect chemical signals and move towards higher concentrations of these signals, this is called chemo-taxis. Our mathematical models describe pattern formation in cell populations, which occurs due to this widespread type of behavior. At what point in time can how many cells be found at a certain spatial position? What is the expected pattern in healthy cell populations? What do anomalies look like? Could their space-time structure depend on pathological changes in cell motion? By analyzing our mathematical models, we try to find precisely those parameters, which play a decisive role for such processes. Chemo-taxis also played an important role during the evolution of single cells to multicellular organisms.

### What characterizes you personally as a scientist?

Biology has always fascinated me, mathematics even more. After high-school, when I decided to study at university, I therefore clearly opted for mathematics, but with biology as a secondary subject, even though I did not see a direct connection between the two subjects at the time. Today it is obvious to me that if a problem is interesting from a biological point of view, then there is almost always a mathematical question behind it, which is really fascinating. Of course, I naturally find mathematical problems within mathematics itself, but biology has always been very stimulating for me as a mathematician and for the mathematical questions I am asking.

### What is your greatest aim as a scientist?

What we are lacking is a sound continuum mechanics for biological tissues, similar to the continuum mechanics in physics. This describes how materials, solids, fluids and gases change and deform subject to internal and external forces. A number of approaches do already exist for a similar description of tissues, but a well-founded mathematical formulation is missing. One of the problems is that cells divide. If you model each individual cell, you may want to define a given number of cells as a reference configuration for the changes, which are about to follow. But this given number of cells increases as soon as one of the cells undergoes cell division. So suddenly there is one more element to be considered, which did not exist in the reference configuration before. Currently, it is not clear at all how a good mathematical model for such problems has to be set up. But I’m convinced that at some point there must – and will – be a solution for this.

Further, I’d like to know whether the geometry of certain biological structures can adopt important functions. Mitochondria, for example, have a strongly folded membrane. Such an enlarged surface naturally provides more space for biochemical reactions of signals or stimuli. However, this membrane structure is repeatedly restructured. Why? Wouldn’t it be much more efficient for the membrane to stay unchanged? Nature may have had a good reason for this constant restructuring of the membrane geometry.

### What’s your favorite toy for research – and what is it able to do?

My lab is in my mind. For working, I typically just need some sheets of paper and a pen. One can understand an awful lot by mathematical thought experiments. Although one may not get too many concrete numbers, one can obtain a lot of very helpful estimates. I develop equations for biological phenomena and analyze their solutions qualitatively. I am searching for critical phenomena, patterns which are forming, typical scalings and symmetries. How could solutions look like, how can they definitely not look like, and what does this mean for the mathematical model and for biology? Therefore my math is applied, even without computer simulations.

### Can you remember your happiest moment as a scientist?

I experience a lot of happy moments in research again and again, but the most defining moment I remember was when finishing my diploma thesis in number theory. With this thesis I wanted to prove to myself that I had become a mathematician, meaning that I was able to answer a mathematical question that hadn’t been solved in the existing literature before. Although it was a very small contribution, it made me very happy. And it laid an important personal basis for the much larger challenges that came afterwards.

### And what was your biggest frustration?

That was during my PhD-thesis. I had changed my research subject from pure mathematics to applied mathematics, namely partial differential equations. For the subject of my thesis I also needed further knowledge in stochastics. Therefore, I had to master a lot of mathematical techniques that were new to me, in a short amount of time. This touched limits. But it was worth it.

### What big scientific question would you like to have an answer to?

When I was a postdoc, I expected that our subject, namely mathematics in the life-sciences, would have advanced further than it already has by now. I thought that biology, mathematics, and medicine would be as tightly intertwined as mathematics and physics. Historically, mathematics has made a number of important predictions in physics for phenomena, which could not be explored experimentally. Mathematics has contributed a lot to our actual understanding of physics and the functioning of physical systems. This opportunity does, in my opinion, exist for biology and medicine too. Mathematics needs very strong and versatile young scientists in order for this to happen, and we have to ask the right biomedical questions for such an approach.

As an ad hoc example, we could, for instance, make more predictions about metabolic processes than we are managing to do at the moment. There are very good examples, but we only have a few so far. For large bio-chemical reaction networks it is important to know what their central components are, what the important scales are, and when they play which role. I’m convinced that mathematics can provide important and fundamental contributions here.