Waiting for Something to Happen
Poisson processes across evolution and ecology
A Poisson process is what you get when something happens at a steady rate and each occurrence pays no attention to the ones before it. You can look at it from either end. Count the events in a fixed window and you get a Poisson distribution, whose variance happens to equal its mean. Measure the gaps between events instead and you get an exponential distribution with mean \(1/\lambda\). Set the count to zero and you get \(e^{-\lambda t}\), the chance that nothing happens at all — and a lot of what follows is really a question about that.
What makes the whole thing work is memorylessness. A flower that has been open four days without a visitor is no more likely to get one tomorrow than it was on its first morning. A genus that has lasted 20 million years isn’t owed an extinction. The past leaves no mark on what comes next.
Mutation, recombination, and gene flow
| Event | Rate | Where it matters |
|---|---|---|
| Point mutation at a site | Per-site per-generation mutation rate | Mutation accumulation lines; germline rate estimates |
| Neutral substitution along a lineage | Equals the mutation rate (\(2N\mu \times 1/2N\)) | The molecular clock; population size cancels out |
| Synonymous substitution between paralogs | \(K_s\) divergence | Dating gene and genome duplications |
| Recombination breakpoints along a chromosome | Crossovers per Mb | Linkage map construction; LD decay |
Because there are only four bases, a site hit twice looks the same as one hit once, and a site that mutates and then reverts looks untouched. Observed differences level off as a result — at 0.75 under Jukes–Cantor, once sites are thoroughly saturated. Distance corrections are just a way of working backwards from what you can see to how many changes actually happened.
Genome content
| Event | Rate | Where it matters |
|---|---|---|
| Gene duplication | Roughly 0.01 per gene per My (varies widely) | Gene family expansion |
| Gene loss / pseudogenization | Duplicate half-life on the order of a few My | Fractionation after polyploidy |
| Transposable element insertion | Highly variable | Genome size evolution — but strongly bursty, see below |
Gene families work the same way with one wrinkle: every copy can duplicate or be lost, so a family of \(n\) genes gains copies at rate \(n\lambda\) and sheds them at rate \(n\mu\). Letting \(\lambda = \mu\) is not unreasonable at large scales where gains and losses are approximately in equilibrium, but plants add a complication. Whole-genome duplications double everything at once, and they are followed by large-scale losses that are systematically biased.
Macroevolution
| Event | Rate | Where it matters |
|---|---|---|
| Speciation | Per-lineage rate \(\lambda\) | With \(n\) lineages, the next split waits Exp(\(n\lambda\)). Waiting times shrink as clades grow |
| Extinction | Per-lineage rate \(\mu\) | Log-linear survivorship in the fossil record implies extinction is always possible |
| Fossilization / preservation | Per-lineage sampling rate | Fossilized birth–death models; gap statistics |
The Red Queen hypothesis builds upon the memorylessness of evolution. A long history of survival does nothing to prevent extinction in the future. Evolution is always occurring.
One thing to watch for: reconstructed phylogenies tend to bend upward near the present even when rates have been steady all along, simply because lineages that arose recently haven’t had time to die yet. A fair number of apparent recent radiations are this artifact rather than a real burst.
Ecology and life history
| Event | Rate | The cost of waiting |
|---|---|---|
| Pollinator visit to a flower | Visitation rate | Nectar, water, respiration — sets optimal floral longevity |
| Fire | Fire return interval | Delayed reproduction in fire-adapted species |
| Outcrosser pollen arrival | Pollinator reliability | Inbreeding depression — sets timing of delayed selfing |
The general theme, something worth having will turn up eventually. There is no guarantee when in a given time interval, but evolution suggests the events happen regularly enough to exert selective pressures on phenotypes and ultimately genotypes.