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in Chat

Hi everyone!

I am PhD student in plant biology at the University of Georgia, Savannah River Ecology Lab. The mathematics I do is part hobby and part work. I don't have much formal training (in terms of courses), but I have self-studied quite a bit (undoubtedly there are plenty of gaps in my knowledge).

I primarily got interested in category theory through John Baez' papers (and some attempts at programming in Haskell). I saw John's paper Relative entropy in biological systems linked on the /r/math subreddit of reddit. I really liked that paper because it used some interesting math to change the way I conceive of evolution and ecology. From that paper, I was led to papers on using category theory to analyze/formalize diagrams in science (e.g., A compositional framework of passive linear networks). So I started to think I might be able to find applications of category theory to the systems I already study for work.

I think compositionality will be a useful idea in studying complex biological systems, especially ones where it is difficult to measure or control phenomena of interest. By describing how systems (e.g., plant tissues) composed to form bigger systems (e.g., whole plants), we can then deduce global system properties from local ones, or vice-versa. That way data, observations, and hypotheses at different scales (e.g., observations of forests vs. observations of tissues in a lab) can be related in a systematic way.

Anyways, that's how I ended up here.

## Comments

Hi Scott! It seems like we have a similar background, and are both interested in looking at applying these ideas to practical biological problems.

`Hi Scott! It seems like we have a similar background, and are both interested in looking at applying these ideas to practical biological problems.`

Oh cool! What area of biology do you work on? I mostly work on plant physiology and ecology, but occasionally I think about more general biological problems like ecological/evolutionary dynamics.

`Oh cool! What area of biology do you work on? I mostly work on plant physiology and ecology, but occasionally I think about more general biological problems like ecological/evolutionary dynamics.`

Mostly computational systems biology, and synthetic biology.

`Mostly computational systems biology, and synthetic biology.`

Hi, Scott! I'm glad you joined the course - your comments are bringing it some extra vitality. It would be great to connect category theory more firmly to biology. Right now I'm having the most luck with biochemistry: my recent series of posts on emergent conservation laws has a secret category-theoretic underpinning that I want to explain someday.

`Hi, Scott! I'm glad you joined the course - your comments are bringing it some extra vitality. It would be great to connect category theory more firmly to biology. Right now I'm having the most luck with biochemistry: my recent series of posts on [emergent conservation laws](https://johncarlosbaez.wordpress.com/2018/06/27/coupling-through-emergent-conservation-laws-part-1/) has a secret category-theoretic underpinning that I want to explain someday.`

I would like to hear the secret category-theoretic underpinning of the emergent conservation laws as that was an interesting idea. There's a lot of utility in just the biochemistry, and not just in molecular biology. Biologists, chemists, and geologists study how elements move through ecosystems (biogeochemistry) and biochemical reactions make up a big part of how those elements move.

Where the category theoretic ideas might have serious applications is being able to link the small and large scale. Take for example photosynthesis. Photosynthesis is a big and complicated series of reactions (and differs a little bit in different plant species), but most of those reactions are only serve to characterize a simpler "pseudo reaction" CO2 + H2O + Photon --> Sugar, which is the composition of those reactions. So understanding/describing that compositionality and properties of it could lead to insights in biogeochemistry too. I think there are similar potential applications in physiology since physiology of most organisms is mostly a mixture of biochemistry and biophysics

I think it would also be interesting to revisit some of Howard Odum's ideas (unfortunately the lesser known of the Odum brothers) with ideas from category theory / networks theory. Despite having a degree in biology (with a lot of ecology too), I had never heard of his work before seeing it referenced in one of your online talks! His approach to modeling ecosystems using electric circuit analogies and diagrams suggests that category-theoretic techniques might have some application.

`I would like to hear the secret category-theoretic underpinning of the emergent conservation laws as that was an interesting idea. There's a lot of utility in just the biochemistry, and not just in molecular biology. Biologists, chemists, and geologists study how elements move through ecosystems (biogeochemistry) and biochemical reactions make up a big part of how those elements move. Where the category theoretic ideas might have serious applications is being able to link the small and large scale. Take for example photosynthesis. Photosynthesis is a big and complicated series of reactions (and differs a little bit in different plant species), but most of those reactions are only serve to characterize a simpler "pseudo reaction" CO2 + H2O + Photon --> Sugar, which is the composition of those reactions. So understanding/describing that compositionality and properties of it could lead to insights in biogeochemistry too. I think there are similar potential applications in physiology since physiology of most organisms is mostly a mixture of biochemistry and biophysics I think it would also be interesting to revisit some of Howard Odum's ideas (unfortunately the lesser known of the Odum brothers) with ideas from category theory / networks theory. Despite having a degree in biology (with a lot of ecology too), I had never heard of his work before seeing it referenced in one of your online talks! His approach to modeling ecosystems using electric circuit analogies and diagrams suggests that category-theoretic techniques might have some application.`

My interest in Howard Odum's work is part of why I put a lot of work into formulating electrical circuit theory in terms of categories. Electrical circuits are just one of many kinds of networks for which a rigorous set of analogies exist, and category theory can handle all these networks!

I wrote about these analogies in my blog series on network theory:

Part 27 - Getting differential equations from circuit diagrams. Available on Azimuth.

Part 28 - The analogy between electronics and mechanics. Available on Azimuth.

Part 29 - Analogies between the mechanics of translation, the mechanics of rotation, electronics, hydraulics, thermal physics, chemistry, heat flow and economics. Flow versus effort. Available on Azimuth.

Then Brendan Fong, who was my student at the time, added the category theory! His thesis is a great place to start:

The Algebra of Open and Interconnected Systems, Ph.D. thesis, University of Oxford, 2016. (Blog article here.)We've subsequently developed this further:

It took us many years to finish this paper!

`My interest in Howard Odum's work is part of why I put a lot of work into formulating electrical circuit theory in terms of categories. Electrical circuits are just one of many kinds of networks for which a rigorous set of analogies exist, and category theory can handle all these networks! I wrote about these analogies in my blog series on network theory: * Part 27 - Getting differential equations from circuit diagrams. Available <a href = "https://johncarlosbaez.wordpress.com/2013/04/03/network-theory-part-27/">on Azimuth</a>. * Part 28 - The analogy between electronics and mechanics. Available <a href = "https://johncarlosbaez.wordpress.com/2013/04/10/network-theory-part-28/">on Azimuth</a>. * Part 29 - Analogies between the mechanics of translation, the mechanics of rotation, electronics, hydraulics, thermal physics, chemistry, heat flow and economics. Flow versus effort. Available <a href = "https://johncarlosbaez.wordpress.com/2013/04/23/network-theory-part-29/">on Azimuth</a>. Then Brendan Fong, who was my student at the time, added the category theory! His thesis is a great place to start: * Brendan Fong, <i><a href = "https://arxiv.org/abs/1609.05382">The Algebra of Open and Interconnected Systems</a></i>, Ph.D. thesis, University of Oxford, 2016. (Blog article <a href = "https://johncarlosbaez.wordpress.com/2016/10/23/open-and-interconnected-systems/">here</a>.) We've subsequently developed this further: * John Baez and Brendan Fong, <a href = "https://arxiv.org/abs/1504.05625">A compositional framework for passive linear networks</a>. (Blog article <a href = "https://johncarlosbaez.wordpress.com/2015/04/28/a-compositional-framework-for-passive-linear-networks/">here</a>.) It took us many years to finish this paper!`

Hi John, thanks for the links! I should read Brendan Fong's thesis. I wish I could find a good summary of Howard Odum's work (that doesn't skimp on the math). Combining traditional community and population ecology ideas (like the rate equations in the Stochastic Mechanics book) with physics-based ideas (tracking energy, matter flows, thermodynamics) has been a longterm goal in ecology (well theoretical ecology at least).

You might interested in this paper: Allen et al. (2005) Using L-systems for modeling source–sink interactions, architecture and physiology of growing trees: the L-PEACH model . This is a model of plant physiology rather than ecology like Odum's stuff. But here again there's a heavy use of circuits and dynamics on graphs.

You might heard of L-systems before, but roughly the idea is that the plant is modeled structurally as a graph (with some spatial embedding). Resource transport is simulated by treating the graph as a circuit, and then the graph changes over time to capture growth (through graph rewriting).

There are more recent versions (See L-Almond or IMapple). These models come from people with a more computer science background, so there's heavy emphasis on computer simulation and not necessarily on theorem-proofs (which biologists don't really have much appetite for either).

One of my hopes in taking the Applied Category theory course was to be able to develop ideas/theorems/heuristics to analyze these types of models. Ideally, then produce experimental predictions that can be tested and/or provide insight beyond a few particular models.

`Hi John, thanks for the links! I should read Brendan Fong's thesis. I wish I could find a good summary of Howard Odum's work (that doesn't skimp on the math). Combining traditional community and population ecology ideas (like the rate equations in the Stochastic Mechanics book) with physics-based ideas (tracking energy, matter flows, thermodynamics) has been a longterm goal in ecology (well theoretical ecology at least). You might interested in this [paper: Allen et al. (2005) Using L-systems for modeling source–sink interactions, architecture and physiology of growing trees: the L-PEACH model ](https://cloudfront.escholarship.org/dist/prd/content/qt9rn2x7t8/qt9rn2x7t8.pdf). This is a model of plant physiology rather than ecology like Odum's stuff. But here again there's a heavy use of circuits and dynamics on graphs. You might heard of L-systems before, but roughly the idea is that the plant is modeled structurally as a graph (with some spatial embedding). Resource transport is simulated by treating the graph as a circuit, and then the graph changes over time to capture growth (through graph rewriting). There are more recent versions (See [L-Almond](https://www.actahort.org/books/1160/1160_7.htm) or [IMapple](https://ieeexplore.ieee.org/document/7818293/)). These models come from people with a more computer science background, so there's heavy emphasis on computer simulation and not necessarily on theorem-proofs (which biologists don't really have much appetite for either). One of my hopes in taking the Applied Category theory course was to be able to develop ideas/theorems/heuristics to analyze these types of models. Ideally, then produce experimental predictions that can be tested and/or provide insight beyond a few particular models.`