Eukaryogenesis as an Algorithmic Phase Transition

The problem

Gene lengths and protein lengths across the tree of life are lognormally distributed, which is the signature of a multiplicative random process. Muro and co-workers push that observation to a striking conclusion: gene length has grown exponentially since the last universal common ancestor, its variance follows a power law in its mean (Taylor’s law), and at a mean gene length of about 1 500 base pairs protein length stops growing while genes keep growing by accumulating non-coding sequence. They read that threshold as a second-order phase transition, complete with a peak in fluctuations and an “easy-hard-easy” pattern in the evolutionary cost per nucleotide, and they place it about 2.6 billion years ago, when eukaryotes appear. It is statistical physics applied to a genomic dataset, and every step can be redone with public data.

Basis paper

E. M. Muro, F. J. Ballesteros, B. Luque and J. Bascompte, “The emergence of eukaryotes as an evolutionary algorithmic phase transition”, PNAS 122, e2422968122 (2025), doi:10.1073/pnas.2422968122. Open access under CC BY, with a supporting-information document that spells out the fits and the simulations.

The data are annotated genomes from Ensembl and reference proteomes from UniProt, both freely downloadable; the paper analyses tens of thousands of them, but the key figures already emerge from a few hundred well-annotated species.

The path

1. Study. Stochastic multiplicative processes and why they produce lognormals (the central limit theorem in the logarithm); Taylor’s law and scale invariance; the language of continuous phase transitions (order parameter, critical point, critical slowing down, fluctuation peaks); maximum-likelihood fitting and model comparison for heavy-tailed distributions; enough genomics to know what a gene, a coding sequence and an intron are.

2. Reproduce. Download gene and protein length tables for a chosen set of species, fit lognormals and check the fits, plot variance against mean gene length and recover the Taylor exponent, plot mean protein length against mean gene length and locate the plateau, compute the non-coding fraction and compare with the predicted curve. Then write the multiplicative growth simulation described in the supporting information and confirm it produces the same distributions. Each figure of the paper becomes a checkpoint.

3. Find the gaps and go beyond.

• How much does the critical length depend on annotation quality and on which species are included? Resample and see how stable it is.
• Is the lognormal really the best description, or do gamma or power-law tails do as well? Compare models properly.
• Does the fluctuation peak near the threshold survive when the analysis is restricted to one kingdom, or to genomes of similar age?
• Extend the simulation with the coding/non-coding switch at the threshold and test whether it reproduces the human intron and exon length distributions.
• Ask whether other genome-scale quantities (number of genes, genome size, GC content) show the same transition.

What you need

Statistical physics and probability from the degree; Python with NumPy, SciPy, pandas and Matplotlib; willingness to handle large text files from biological databases. No wet-lab work. Suitable for one student, or two if the simulation and the data analysis are split.