## Mathematics in the Plant Sciences Study Group

The very mention of differential equations or any other associated type of mathematics can be enough to have many biologists break into a cold sweat! I think it’s fair to comment that the prevailing opinion in the past has been that maths is ‘too complicated’ and that it doesn’t have ‘any relevance’ for their work**.**

However this idea is slowly changing with the appearance of more mathematic modeling in biological manuscripts and the need for cross-disciplinarity in grant applications. In basic terms, a partial differential equation aims to measure how parameters changes with respect to one another: which is exactly what any biologist is always studying. Therefore your research should, to a greater or lesser extent, be relevant for mathematical interrogation.

Therefore in order to facilitate interactions between biologists and mathematics, the ‘Mathematics in the Plant Sciences Study Group (MPSSG)’ was inaugurated in 2007 as part of the initial funding for the Centre for Integrative Biology (CPIB) at Nottingham University. Over the past 9 years these study groups have been extremely successful and have even directly led to a publication, which modeled the biomechanical parameters that are involved in anther opening (Nelson *et al*, New Phytologists, 2012, doi: 10.1111/j.1469-8137.2012.04329).

The 7^{th} MPSSG was held in Nottingham between Jan 4^{th}-7^{th} and took the same successful format as in other years. At the start of a meeting, three biologists presented a problem connected to their research, which had been selected by the organisers as being amenable for inclusion in this process.

As described below, these problems varied across the breadth of plant sciences so as to attempt to discriminate between the type of mathematics that might be applied to each case. After the initial presentations the biologists moved to seminar rooms where they were joined by a small group of mathematicians who are interested in modeling their problem. Over the course of the study group this evolved nicely as the mathematicians settled on a project that they were most interested in investigating. Most of these researchers were based in Nottingham although there were visitors from other UK and International universities.

On day one the discussions resembled a formal seminar session where the biologist led the mathematicians through the background of their problem. This is an interesting process as the mathematicians obviously had a very limited knowledge of each specific system, even though they might have some experience in modeling biological problems.

I observed this process for two of the study groups and it struck me that there was a tipping point at which time the mathematicians had taken onboard enough information about the problem to start to develop their own ideas. Suddenly at this point there were partial differential equations written on the whiteboard as the process began in earnest……

From this time onward the biologist plays a more advisory role whereas the mathematicians put in the hard work of transferring their initial equations into their analysis-program of choice. This is not a rapid process as the next two days are spent discussing and refining these models before the findings are presented on the final afternoon, just four days after most of the attendees will have first learnt about the topic!

The overall aim of the study groups is to begin to develop a model that can mathematically represent what is observed within the biological information.

In the initial stages of the process, I was somewhat surprised by how freely the mathematical ideas were discussed. In my experience, many biology-based discussions are often constrained by the need for an absolute requirement for the presence/absence/expression of a particular protein or gene in order to develop a hypothesis. However the mathematicians were unencumbered by this requirement and rather were able to throw ideas around, seeing what, if anything, would eventually stick. This ultimately might lead them blindly down closed alleyways but not always. Importantly it was clear that these discussions made the biologists look at their problem in a different way and each of the participating speakers had a list of future experiments that had resulted from these discussions.

More detail about the three problems presented in this study group can be found on the MPSSG website but in summary:

– Biosynthesis of Casparian strips: How to build a micron-scale bridge out of a lignin polymer? Presented by Dr Guilhem Lenaic (University of Aberdeen).

The molecular participants that play a role in the generation of the casparian strip (CS) are not well studied. Guilhem works in the lab of GARNet chair Professor David Salt and introduced the genes that are involved in this process and importantly, the patterns of CS that develop in different mutant plants. Simply put, the CS is a band of lignin that surrounds root endodermal cells ‘*like an belt’* and plays an important role in water uptake, which is of clear importance not only for the growth of an individual plant but has broader relevance for issues of water use efficiency. The CS develops through a series of patterning steps that have at been, in part, genetically determined. Therefore the mathematicians attempted to develop a model that generated these patterns by parametrising the activity of different proteins as well as a monolignol lignin precursor molecule.

Ultimately they made some process developing both 1D and 2D models although, as with all the problems from the study group, there was still work to do. Arguably the best line from the presentation, used to account for a stray equation and was described as “classic-MPSSG”, was “*oh yes…..that line was written on the whiteboard by John King…..but then he left….*”.

– Modeling Leaf-Sheath Interactions in Grasses. Presented by Dr Douglas Cook (NYU Abu Dhabi).

This problem introduced the biomechanical interaction between a maize sheath and the leaf that wraps itself around the sheath, providing protection and mechanical strength to the underlying tissue. The leaf surrounds the sheath prior to internode elongation, so when the sheath start to elongate there is a frictional interaction between sheath and leaf. This can result in ‘slippage’ when the forces between sheath and leaf build up. Dr Cook mentioned that can is an economical important parameter as without the protective influence of the leaf, the maize stalk is weak and can be more easily bent.

Initially the mathematicians presented this problem as being equivalent to the interaction between two cylindrical tubes. Over the course of the workshop a number of models were developed, not all of which were successful but which highlighted the important trial-and-error nature of many of these interactions. A nice analogy for this process was the comparison to the seismic activity that builds up between tectonic plates. A favourite comment from the early discussions occurred when the sheath was described as an “*infinite cylinder*”…. certainly not the type of language commonly heard when discussing plant science problems!

– Hyperspectral image analysis of plants, presented by Dr Andrew French (University of Nottingham)

Imaging with a hyperspectral camera (that records a complete breakdown of the reflectance spectra for each pixel in an image) provides a researcher with a large amount of data (5seconds=1Gb) but much of this is extraneous for use in any one particular biological problem. In this problem Dr French presented the idea that this type of data could be used to detect minute changes in the colour of leaves that are a prelude to visible signs of disease. Therefore the task for the mathematicians was to devise a way of analysing the spectral data to define a new indice that represents a relationship between wavelengths that has the most utility for this type of analysis. I did not directly observe any of the interactions of this group but they appeared to make good progress. In their final presentation the description of their methods was the most mathematically dense of the three problems (at least to a naïve observer such as myself). However the group had clearly put together some ideas that could be developed in future.

The final step in the process is to collate the information from each study group into a report, which will be published on the CPIB website. Later on this will hopefully yield further interactions and potential grant writing opportunities between the biologists and mathematicians involved in the meeting.

Please look out for a summary of the final reports in the next edition of the GARNish newsletter, to be published in the summer. From a biologists perspective I would encourage anyone who is wary of interacting with mathematicians, computer scientists or modellers to bite the bullet and make that connection. Even though your system ultimately might not be the most amenable to a modeling approach, the interaction will help you develop ideas that will undoubtedly drive your research forward in unexpectedly directions. Basically it is a no-lose interaction!

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