Growing a leaf mathematically

[Graham853]

Quick Summary

A leaf with a larger area collects more light, but requires a bigger network of veins to deliver water and nutrients to every part. A 'good' leaf maximizes area while minimizing transport costs. The paper below describes a model in which these conflicting aims are balanced against each other. I recommend starting by looking at the pictures on pages 12 and 13, and the attached image.

I have implemented Xia's algorithm, and can watch leaves 'grow'. I can write an illustrated article, but don't know how to do animations.

The Formation of a Tree Leaf by Qinglan Xia https://pdodds.w3.uvm.edu/research/papers/others/2007/xia2007a.pdf

About the author

I am a mathematician and programmer with a particular interest in mathematical biology, now retired. I know quite a lot about abstract trees and branching processes but I'm not any kind of expert on real trees or leaves.

For more... http://www.indriid.com https://www.researchgate.net/profile/Graham-Jones-5

Target medium

If no one cooperates with me, it will end up as a non-interactive article (PDF probably), with accompanying code for others to play with.

I can imagine it as an interactive blog post, with a web page where readers can adjust parameters and watch some leaves grow, or just watch pre-calculated examples. I'd need someone to do the web programming. I can also imagine more ambitious presentations, if anyone has sufficient enthusiasm.

More details

I think Xia's paper, which looks quite intimidating at first sight apart from the pictures, can be greatly simplified to make a good exposition. In particular, the all-important transport cost function F(G) is easier to understand if you think of it in terms of building up information at each vertex in the tree, instead of lots of formulas. I have written a version of Xia's algorithm in R.

The network of veins inside a leaf is a sort of mini-tree with its own 'root' at the base of the leaf. In graph theory terms, it is rooted tree (a tree with one distinguished vertex). There is a unique directed path from the root to every other vertex. Each edge has a weight which represents the cross-sectional area of a vein.

The algorithm for growing a leaf alternates between adding new cells at the boundary of the leaf and optimizing the structure of the veins. Cells are added where the extra cost of supplying them is small enough. The optimization of the vein structure uses a local search near each cell.

The transport cost function F(G) has two parameters, alpha and beta. Roughly speaking, alpha controls the savings from using a bigger vein to transport materials together, and beta controls the cost of bends in the paths that lead from the root to everywhere else in the leaf. By varying alpha and beta, different shapes of leaves can be generated.

I regard this model (and many others in mathematical biology) as a caricature. It captures some important features of how real leaves grow, while ignoring or distorting other aspects.

I came across Qinglan Xia's model in John Baez's blog. https://johncarlosbaez.wordpress.com/2013/02/15/prospects-for-a-green-mathematics/ You might prefer his description to mine. At any rate, I want to quote this bit: "Unlike approaches that merely create pretty images resembling leaves, Xia presents an algorithmic model, simplified yet illuminating, of how leaves actually develop. It is a network-theoretic approach to a biological subject, and it is mathematics—replete with lemmas, theorems and algorithms—from start to finish."

Contact details

Email preferred. See the bottom of the home page of my website. http://www.indriid.com

[yuhan2680]

I`m a high school student who makes video with manim sometimes,and am intresting in biology, could I do it? My e-mail is 2565376223@qq.com and phone number is (+86)13544062680. And, is Xia a chinese?I wants to connect with him and know more about the model of leaf.

[Graham853]

I have uploaded what I have done so far here http://indriid.com/workingnotes2022.html There are two R scripts and a PDF which is the main article.

@yuhan2680, Thanks for your interest! I was also contacted via email by two high school students and one Ph.D. student from Turkey who are interested in doing something. I don't mind what people do with my work, but probably best to avoid duplication. I'll email them in case they don't notice this.

Qinglan Xia's homepage is here https://www.math.ucdavis.edu/~qlxia/

[nifets]

Hi! I would be interested in helping with the making of a web page with interactive animations, if you still need someone. You can check my github repo's for examples of what I've done in the past.

[Graham853]

Thanks for the offer. I see you've been growing trees in ArborGen. I think the Turkish folk will do something, but they're happy for me to work with others too, so I'll leave to you.