Evolution has an order

Miraia S. Chiou

Survey · Biology

Evolution has an order

Explore selection, mutation, lineage histories, and the Beta distributions of wandering populations.

2026-09-08 · 18 min read · 3 interactive demos

Imagine two cultures of cells arriving at the same crossroads. One encounters a change in reproductive conditions, then acquires a new heritable variant. The other acquires the variant first, then encounters the change. They have experienced the same two kinds of event. Should they end with the same population?

Now imagine opening their family trees. A thousand cells carry a variant in each culture. In one, all thousand descend from a single mutation. In the other, the variant arose repeatedly. The head count agrees. What else would we need to know?

Evolution invites us to ask about transformations, reachable possibilities, and ancestry. These are lively places to explore Epistemic Physics. The positive cone can describe a reproducing population. A lineage record can preserve the history of a copied trace. A Beta distribution can describe either an observer’s uncertainty or a population’s actual wandering. Let us follow those connections through three experiments.

Reproduction tilts a population through the cone

Choose a population with two heritable types. Let $x$ and $y$ denote their nonnegative amounts. These could be two variants carried by cells with one copy of the relevant gene. Different ancestral lineages may carry the same variant; we will open that distinction later.

For one reproductive episode, choose positive weights $w_{+}$ and $w_{-}$ :

x^{'} = w_{+} x, y^{'} = w_{-} y .

A weight of two doubles that type’s amount; a weight of one-half halves it. These are deterministic amounts, so fractions are allowed. Our later island experiment will count whole cells.

Introduce the sum and difference

\begin{aligned} A & = x + y, \\ P & = x - y, \\ M^{2} & = A^{2} - P^{2} = 4 x y . \end{aligned}

The pair $(P, A)$ lies inside the cone $A \geq | P |$ . The coordinate $M = 2 \sqrt{x y}$ labels hyperbolas inside it. For this construction, all three coordinates have units of population amount.

Now separate the reproductive weights into

\begin{aligned} g & = \sqrt{w_{+} w_{-}}, \\ η & = \frac{1}{2} \log \frac{w_{+}}{w_{-}} . \end{aligned}

The common factor $g$ describes a uniform scaling, while $η$ , called rapidity, describes the relative advantage. Define the matrix

B (η) = (\begin{matrix} \cosh η & \sinh η \\ \sinh η & \cosh η \end{matrix}) .

Substituting $w_{+} = g e^{η}$ and $w_{-} = g e^{- η}$ gives

(\begin{matrix} A^{'} \\ P^{'} \end{matrix}) = g B (η) (\begin{matrix} A \\ P \end{matrix}) .

This matrix is an active Lorentz boost: it moves the population vector in a fixed coordinate system. Our sign convention makes positive $η$ favor $x$ . The hyperbolic functions are $\cosh η = (e^{η} + e^{- η}) / 2$ and $\sinh η = (e^{η} - e^{- η}) / 2$ .

The boost preserves $A^{2} - P^{2}$ ; the accompanying scale changes it by $g^{2}$ . Consequently,

M^{'} = g M .

Here is a population process with an exact cone transformation. The coordinates describe the amounts we supplied. To turn those amounts into an observer’s evidence trials would require an observation model.

Odds travel by addition in rapidity

The fraction of the population carrying $x$ is

\begin{aligned} p & = \frac{x}{x + y}, \\ v & = \frac{P}{A} = 2 p - 1. \end{aligned}

For an interior population, with both types present, define

ρ = artanh v = \frac{1}{2} \log \frac{x}{y} .

The symbol $artanh$ means the inverse hyperbolic tangent. Reproduction adds a fixed amount to this coordinate:

\begin{aligned} ρ^{'} & = ρ + η, \\ v^{'} & = \tanh (ρ + η) . \end{aligned}

The apparent bend of selection on the frequency axis becomes a straight step in half-log odds. Set $r = w_{+} / w_{-}$ , the relative reproductive weight. The population and inference equations align beautifully:

Reproducing population	Observer comparing two hypotheses
Current odds $p / (1 - p)$	Current odds $q / (1 - q)$
Multiply by relative weight $r$	Multiply by likelihood ratio $L$
$p^{'} / (1 - p^{'}) = r p / (1 - p)$	$q^{'} / (1 - q^{'}) = L q / (1 - q)$
Add $\frac{1}{2} \log r$ to population rapidity	Add $\frac{1}{2} \log L$ to belief rapidity

Here $q$ is the observer’s probability for the first hypothesis, and $L$ compares how well the hypotheses predict an observation. Harper develops the relation between replicator dynamics, Bayesian updating, and information geometry. Our two-type calculation makes its odds correspondence explicit. Read Harper’s paper.

The dictionary lets us carry mathematical tools between reproduction and inference while remembering which process supplies the multiplier.

A missing lineage is a missing route

Suppose $x = 0$ . Multiplying it by four, forty, or four million leaves it at zero. Pure selection can favor a type without creating an instance of it.

On the cone, that population lies on the edge $P = - A$ . An active boost preserves the edge. In frequency coordinates, selection takes the form

\begin{aligned} S_{r} (p) & = \frac{r p}{1 - p + r p}, \\ S_{r} (0) & = 0, S_{r} (1) = 1. \end{aligned}

The set of types already present is called the population’s support. In this multiplicative model, support is preserved. An interior population approaches an edge under repeated disadvantage, but finite positive weights keep a positive amount positive.

Mutation changes the possibilities. Choose a symmetric mutation episode in which a fraction $u$ of each type changes into the other:

M_{u} (p) = u + (1 - 2 u) p .

For $0 < u < 1 / 2$ , a population containing only $y$ moves from $p = 0$ to $p = u$ . The previously empty type becomes available for subsequent selection. Ancestry continues through that change: the new $x$ cells descend from $y$ cells even though their type label changes.

In cone coordinates, this mutation episode has an especially simple action:

A^{'} = A, P^{'} = (1 - 2 u) P .

It moves horizontally toward the center of the cone. The total population amount stays fixed while the set of occupied types can expand.

This suggests an inviting connection to learning. Reweighting an existing list of hypotheses resembles selection. Introducing a new hypothesis resembles opening a previously unavailable direction. A future model of creative learning could give these operations separate rules and ask when a learner should search for another possibility. Feedback offers another way to shape available states, explored in A cell chooses what to remember.

Swap the events

Mutation changes the population on which selection acts. To see the consequence, start at $p = 0.1$ , choose $u = 0.1$ , and give $x$ relative reproductive weight $r = 4$ .

If mutation happens first,

0.1 \overset{M_{0.1}}{\to} 0.18 \overset{S_{4}}{\to} \frac{36}{77} \approx 0.468 .

If selection happens first,

0.1 \overset{S_{4}}{\to} \frac{4}{13} \overset{M_{0.1}}{\to} \frac{9}{26} \approx 0.346 .

The final fractions differ by about 12.14 percentage points. The newly supplied $x$ population receives the reproductive advantage only in the first sequence. Mathematicians call this noncommutativity: exchanging the order of two transformations can change their result.

Before trying the demo, make two predictions. Will the difference survive if $x$ starts absent? Will two fixed selection steps also depend on their order?

An experiment in evolutionary order

Swap the events, follow the population

Start with 100 units of population. The x lineage has fraction p; the other lineage has fraction 1 − p. Move a card earlier or later. Predict the final x fraction before looking at the gold endpoint.

1Mutation: switch 10%

Move 10% of each lineage into the other.
2Selection: x × 4

Multiply x by 4 and keep y unchanged.

Your orderReversed order

Both pictures follow the same states, numbered 0, 1, 2. The cone edges mean one lineage is absent. Lines connect event endpoints; they do not add an assumed process between events. The cone rescales to fit both routes.

Your final x fraction: 46.75%
Reversed x fraction: 34.62%
Final amounts x / y: 72.0 / 82.0
Final cone P / A: -10.0 / 154.0

Swapping the events changes the final x fraction by 12.14 percentage points. Selection acts on whichever population mutation has already supplied, or will supply later.

Chosen model: deterministic lineage amounts, positive fixed reproductive weights, and symmetric mutation. For x × 4 and y × 1, common scale g = 2 and active boost rapidity η = 0.693. Mutation preserves A and multiplies P by 0.8. Fractional amounts represent this deterministic population model.

Open the sequence experiment in the playground.

The selection-only control has a clean answer:

\begin{aligned} S_{a} (S_{b} (p)) & = S_{a b} (p) \\ = S_{b} (S_{a} (p)) . \end{aligned}

Both steps multiply the odds, and multiplication of positive numbers commutes. Their intermediate populations can differ, even when the endpoints coincide. The corresponding one-dimensional boosts add their rapidities in either order. When reproductive conditions vary across generations, the question of how to distribute a population’s options leads toward How life prepares for an uncertain world.

This is useful experimental thinking: identify the operations whose order might matter, then find a control whose algebra predicts equality. Turning mutation off also removes the mutation-selection difference. One matching endpoint at one starting frequency, however, does not establish that two transformations commute for every population.

The cone holds the current state. A record of the transformations preserves how the population arrived there. Evolutionary history becomes visible when we keep both.

A clock made from changing distinguishability

There is another geometric excursion nearby. For several types, let $p_{i}$ be type $i$ ’s fraction and $f_{i}$ its instantaneous growth rate. Define the mean rate $\bar{f} = \sum_{i} p_{i} f_{i}$ . The continuous replicator equation is

{\dot{p}}_{i} = p_{i} (f_{i} - \bar{f}),

where the dot means a derivative with respect to time. A type gains frequency when its rate exceeds the population’s mean. This is the replicator model discussed in Harper’s treatment.

To measure how much the frequency distribution changes, use the categorical Fisher metric:

d s^{2} = \sum_{i} \frac{d p_{i}^{2}}{p_{i}} .

This metric measures local statistical distinguishability between nearby distributions. On the occupied support, substituting the replicator equation yields

\begin{aligned} {(\frac{d s}{d t})}^{2} & = \sum_{i} \frac{{\dot{p}}_{i}^{2}}{p_{i}} \\ = \sum_{i} p_{i} (f_{i} - \bar{f})^{2} \\ = {Var}_{p} (f) . \end{aligned}

Thus the population’s Fisher speed equals the standard deviation of its growth rates. A larger spread of rates moves its composition faster in this particular geometry. Integrating the speed counts accumulated distinguishable change along a trajectory.

Keep the two geometric rulers beside each other:

Cone ruler	Fisher ruler
Coordinates $A, P$ describe amounts	Coordinates $p_{i}$ describe proportions
Quadratic form $d A^{2} - d P^{2}$	Positive metric $\sum_{i} d p_{i}^{2} / p_{i}$
A boost preserves the cone quadratic form	Replicator speed measures changing distinguishability

The same population admits both descriptions. Each ruler asks its own question.

One mutation can become a crowd

A population count hides a branching story. Luria and Delbrück used fluctuations between bacterial cultures to investigate that story in 1943. Their resistant-colony counts varied far more between separately grown cultures than simple sampling would predict. Their analysis connected those large fluctuations with mutations arising before exposure to the selecting virus: an early event could leave a large family of resistant descendants. Read the original fluctuation paper.

Build a miniature version. One founder divides into two cells, each divides again, and every lineage continues at the same rate. After $G$ divisions, the living population contains $2^{G}$ cells. If one cell first carries a mutation at division $k$ , and all its descendants inherit it, that origin contributes

D_{k} = 2^{G - k}

mutant descendants at division $G$ .

The exponent is the remaining time for reproduction. Move an origin one division earlier and double its final family. After twenty divisions, an origin at division two leaves $2^{18} = 262,144$ descendants; an origin at division eighteen leaves four. Timing makes a jackpot without requiring a higher mutation rate in the lucky culture.

The next experiment lets you place the events yourself. Compare the two arrangements with four mutant cells, then let all the lineages grow further.

Choose when a lineage changes

One mutation, a growing family

Rebuild a family history. Gold squares mark the final cells you want to mutate. Place mutation origins on the tree, then check whether their descendants match the pattern.

Make the left four leaves inherit one change

Invitation: use at most 1 separate origin. This budget is a puzzle choice.

Free exploration and timing presets

Hide the puzzle targets

Choose a generation

Selected: generation 1, branch 1. 8 final descendants come from this cell.

Population counts and extra generations

Unchanged lineageMutant lineageMutation origin ringSelected node ring

Living cells after 4 divisions: 16
Living mutant cells: 0
Separate mutation origins: 0
Mutant fraction: 0.00%

0 mutant cells are 0 real living cells. For the question “How many separate first mutations happened?”, follow the 0 origin rings. Change between the two “4 cells” experiments to keep abundance fixed while changing its ancestry.

Chosen model: synchronous binary division, equal reproduction and survival, inherited mutation with no reversion. An origin at division k leaves 2^(G − k) descendants at division G. A cell already carrying the mutation cannot acquire a new first origin of that same change.

Open the mutation-family experiment in the playground.

The useful count follows the question:

Biological question	Quantity to follow
How many mutant cells are alive now?	Living mutant descendants
What fraction of the population carries the variant?	Mutant descendants divided by all living cells
How many separate first mutations occurred?	Distinct mutation origins in the genealogy
How often does mutation occur per opportunity?	Origins together with the relevant division opportunities and detection model

Every descendant is a real cell. Shared ancestry tells us which cells inherited the same event. If we model mutation opportunities on separate branches as independent, the origin events become the units for that model; the labels alone do not establish statistical independence.

Epistemic Physics gives this distinction a natural home. A tracelet can retain its source identity while appearing in several question-specific records. A descendant’s existence can matter directly to an abundance question and also point back to an ancestral event. Copying a physical trace can increase its prevalence while preserving its origin.

This opens a rich possibility: track a population’s physical multiplication and its evidential ancestry together. Selection would change which families dominate observation, while the lineage record would preserve how many origins those families represent.

When the population itself wanders into a Beta

So far, populations followed deterministic rules or a prescribed tree. Small populations add another ingredient: chance in which individuals leave offspring. Genetic drift is the resulting random change in variant frequencies.

Choose the following continuous approximation, a Wright-Fisher diffusion:

\begin{aligned} d p & = [u (1 - p) - w p] d t \\ + \sqrt{\frac{p (1 - p)}{N}} d W_{t} . \end{aligned}

Here $p$ is the current $x$ frequency, $u > 0$ is the mutation rate toward $x$ , $w > 0$ is the rate away from it, and $N$ sets the population drift scale. Time $t$ is measured in our chosen generation units. The process $W_{t}$ is standard Brownian motion, whose increments have variance equal to the elapsed time.

The first term gives the average motion from mutation. The second supplies random wandering, stronger in smaller populations. This normalization matters: population-genetic conventions can use different population sizes and time units, changing the factors attached to mutation parameters. Song and Steinrücken give the neutral diffusion and its Beta stationary weight in a scaled convention. See their original derivation.

Let us derive the normalized equilibrium for the equation above. Write $a (p) = u (1 - p) - w p$ for its average drift and $b (p) = p (1 - p) / N$ for its variance rate. A stationary density $π (p)$ with zero probability current satisfies

\begin{aligned} J (p) & = a (p) π (p) \\ - \frac{1}{2} \frac{d}{d p} [b (p) π (p)] \\ = 0. \end{aligned}

Probability current describes probability flowing through a frequency value. Solving this equation gives

\begin{aligned} \frac{d \log π}{d p} & = \frac{2 N u - 1}{p} \\ - \frac{2 N w - 1}{1 - p} . \end{aligned}

Integrate and normalize:

\begin{aligned} π (p) & = \frac{p^{α - 1} (1 - p)^{β - 1}}{B (α, β)}, \\ α & = 2 N u, β = 2 N w . \end{aligned}

The Beta function $B (α, β) = \int_{0}^{1} z^{α - 1} (1 - z)^{β - 1} d z$ supplies the total area, making $π$ a probability density.

Under this recurrent-mutation diffusion, the stationary frequency follows $Beta (2 N u, 2 N w)$ . Its mean is $u / (u + w)$ . With equal mutation rates, shapes below one favor frequencies near the edges; shapes above one produce a central peak. Both sides below one can give an edge-heavy population distribution despite a mean of one-half.

What would those shapes look like as an archipelago of living populations?

Population wandering and an observer's notebook

Many islands, two meanings of a Beta

Visit an island, sample its cells, then place your estimate before opening the census. Evolution changes the islands; sampling changes your notebook.

Generation 0

Field station: island 1

The population is frozen while you sample. Collect ten cells to begin.

Your estimate: 0.50

Across-island distribution, mutation settings, and observer notebook

N = 40 cells per island. Mutation probabilities per offspring: y → x, u = 0.005; x → y, w = 0.005. Changing the mutation setting restarts the populations. Repeating the same seed and actions reproduces the same paths.

Actual islands nowDiffusion equilibrium reference

Gold and blue heights are probabilities integrated over each bin, joined at bin centers. This keeps edge-heavy Beta shapes visible without clipping their infinite endpoint densities. Rose marks the selected island's actual p. The last bin includes p = 1. Gold is a diffusion equilibrium prediction; the finite islands are shown at their current generation.

Bin 0 to 0.1: 0 of 96 islands. Diffusion equilibrium probability: 24.0%.

Generation: 0
Islands at p = 0 or 1: 0 / 96
Diffusion Beta shapes α / β: 0.40 / 0.40
Mean x fraction across islands: 0.500

Here α = β = 0.4. The diffusion's long-run law favors the edges: islands often have one common lineage and one scarce lineage. A population at an edge can return when mutation creates the missing type.

Follow one island, then sample it

Selected island

Pause evolution at the current generation. An observer starts with a uniform Beta(1, 1) belief about this island's p, then samples cells with replacement. Each independent draw is one trial. The observer's blue distribution appears on the frequency chart above.

Actual island fraction p: 0.500
Observed x / y draws: 0 / 0
Observer Beta shapes: 1 / 1
Observer posterior mean: 0.500

Sampling leaves the population unchanged. Selecting another island or advancing evolution begins a new notebook. Finite reproduction uses K′ ∼ Binomial(40, q), q = u(1 − p) + (1 − w)p. The reference diffusion uses dp = [u(1 − p) − wp]dt + √[p(1 − p)/40]dW, giving α = 80u and β = 80w. Its continuous stationary law approximates this discrete chain in the diffusion regime; it is not the chain's exact finite-population law.

Open the island experiment in the playground.

The demo simulates a finite chain. If an island currently has fraction $p$ , mutation gives the offspring probability

q = u (1 - p) + (1 - w) p .

Its next $x$ count is $K^{'} \sim Binomial (N, q)$ : draw $N$ offspring independently, each carrying $x$ with probability $q$ . The next frequency is $K^{'} / N$ , restricted to a grid of possible counts.

The gold curve uses integrated probabilities from the continuous diffusion equilibrium. Weak mutation and an appropriate population scale connect the two models through a diffusion approximation. The finite chain retains discrete endpoints and uses offspring variance $q (1 - q) / N$ per generation; its exact stationary law need not be Beta. Starting every island at one-half also creates a transient before any equilibrium comparison becomes informative.

The observer keeps a different notebook

Freeze one island and sample cells with replacement. For an observer using a continuous Bernoulli model, start with a uniform $Beta (1, 1)$ prior for the probability of drawing $x$ . Let $D$ denote the record of $m$ independent draws, of which $k$ are $x$ . The posterior is

p ∣ D \sim Beta (1 + k, 1 + m - k) .

Those shapes count the chosen prior plus observed draws. The diffusion shapes combine mutation rates and population scale. The two Betas can share a graph because both use a frequency axis, while their probabilities answer different questions:

The gold law describes the population frequencies expected across an ensemble at diffusion equilibrium.
The blue law describes the observer’s uncertainty about one frozen island’s sampling probability.

Collecting more draws changes the blue law while leaving the island unchanged. Advancing generations changes the physical population, so the demo starts a new sampling notebook. An observer who explicitly used the known finite count grid could instead place a discrete prior over those counts. Following populations through observations made in different places adds another layer, explored in How to read a living landscape.

This is a particularly useful habit when reading a distribution: ask what is random, and under which experiment. The same mathematical family can organize physical variation and incomplete knowledge about physical variation.

Ask evolution a question about timing

We can now imagine an experiment designed around sequence. Choose two modeled interventions, prepare matching initial populations, and compare both orders. Keep a selection-only control whose endpoint equality is predicted. Record genealogies as well as final type counts. Finally, compare several replicate populations to distinguish their evolutionary variation from sampling uncertainty within each one.

An intriguing research direction would treat these transformations as the basic vocabulary of an evolving evidence system. Reproduction amplifies a lineage. Mutation opens a type. Drift explores available frequencies. Observation creates question-specific tracelets from the resulting population. Each operation has a place in the history, and exchanging places may change what can happen next.

The invitation is to keep that history visible. When two populations look alike, open their family trees. When two interventions look interchangeable, swap them. When a Beta curve appears, ask whether it describes the wandering world or the observer learning about it.

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