I · Explore the probabilities

The Probability Simulator

How likely is it to reach a particular sequence of L positions, each holding one of K building blocks? The answer depends entirely on the process you assume. Here are three, each with its own assumptions. A random-sequence probability is not the probability of evolution, and the gap between the models shows why.

These models ask how a new sequence is reached. Whether existing sequences decay as mutations pile up is a separate question: see Genetic entropy.

Model 1 · Random sampling. Every trial is a brand-new random sequence, and nothing is kept between trials. This is the calculation behind most "one in 10N" figures.
Model 2 · Cumulative selection. Copies of the best sequence so far are mutated, and the best copy is kept. It is an idealised ratchet towards a target fixed in advance, as in Dawkins' weasel.
Model 3 · Population genetics. A finite population reproduces; mutations arise at a measured rate; selection favours fitter genotypes, but chance drift can lose good mutations and fix neutral ones.

Dawkins' weasel. Richard Dawkins' illustration of cumulative selection (The Blind Watchmaker, 1986): a 28-character target, with every correct letter rewarded. Its target is fixed in advance, which is the main objection raised against it, and Dawkins himself called it misleading in that respect.

The sequence

Set by the typed target.

Possible combinations KL = 2728 = 1.2 × 10⁴⁰

Only the exact target counts as success.

Population and time
Generation time
Mutation

E. coli ≈ 2 × 10⁻¹⁰ (Lee et al. 2012); humans ≈ 1.2 × 10⁻⁸ (Kong et al. 2012); RNA viruses roughly 10⁻⁶ to 10⁻⁴ (Sanjuán et al. 2010).

Selection and the landscape
Do intermediate sequences have function?

On: the parent is kept unless an offspring is at least as good (1+λ). Off: the best offspring always replaces it, even if worse (1,λ).

At the start, of all 728 possible single changes: 3.85% beneficial, 0.00% harmful, 96.15% neutral.

Fitness effects (Model 3)not used by this model

Mutations elsewhere in the genome are classified with the proportions below.

Perfeito et al. (2007) measured a mean of about 1% for new beneficial mutations in E. coli.

Estimates differ widely. Sanford (2018 lecture), citing Gerrish & Lenski 1998, puts it near 1 in a million. Perfeito et al. (2007) measured beneficial mutations at about 10⁻⁵ per genome per generation in E. coli, roughly 1% of all mutations (Lee et al. 2012: ~10⁻³). The lowest setting means none.

Neutral: 69.9%. Most fitness-affecting mutations are harmful (Eyre-Walker & Keightley 2007).

Chance of success within 10,000 generations100%approximate, from the upper bound on time
Expected generations to success≤ 92.4≈ 92.4 years at the chosen generation time
Changes to a specific other block per position, ν = μ/(K−1)1 in 650
Functional steps still needed28
Positions per step, r1
Chance an offspring takes the next step (first step)1.43%
Generations needed (fitness-level bound)92.4

The formula used

ν = μ / (K − 1)

sm = 1 − (1 − m·νr·(1 − μ)L−r)N

E[generations] ≤ Σm 1 / sm

μ
chance each position mutates in an offspring
ν
chance it becomes one particular other block
r
positions that must all be right before a step counts (now 1)
m
functional steps still incomplete
N
offspring per generation
sm
chance that at least one offspring completes a step without breaking anything

The fitness-level method (Wegener 2002) gives an upper bound when gains are preserved. Each generation the best offspring is kept; the target and the scoring are fixed in advance.

The search space

The space is too large to draw to scale, so it is shown as powers of ten. Blind sampling would need about 1.2 × 10³⁴ times more trials than are available. Selection does not sample the space at random. That is why Models 2 and 3 can give very different answers.

The same problem, three models

Same K, L, target and population, three sets of assumptions. Generations available: 10,000. The gap between the bars is the point: the probability of one random sequence is not the probability of evolution, and an idealised ratchet with a fixed target is not evolution either. Which model, if any, describes a real case is an empirical question.

Monte Carlo

Run Model 2 yourself

Each run uses the seed amber-1859, so the same settings and seed always give the same result. Change the seed to see the run-to-run variation the averages hide.

Target
METHINKS·IT·IS·LIKE·A·WEASEL
Current parent
ULQ·WTZMSFYY·NGTVPWQDWPKHUBP
Generation0
Best: wrong positions28
StatusReady
Methodology and limitations

How this works, and what it leaves out

Method

  • Problem. The target and the starting sequence are drawn from the seed, unless you type a target. Success means coming within d positions of the target. A "functional step" is a block of r consecutive positions that are all correct; fitness counts complete blocks.
  • Model 1 is exact: P = 1 − (1 − p)N·G, computed in logarithms so that probabilities far below 10−300 stay correct. Its Monte Carlo draws whole random sequences and compares the distribution of wrong positions with the exact binomial distribution.
  • Model 2 is a (1+λ) evolutionary algorithm with λ = N offspring when gains are preserved, and (1,λ) when they are not. Each position of an offspring mutates with probability μ to a uniformly chosen other block; ties are broken at random, so neutral drift is allowed. The closed form is the fitness-level upper bound (Wegener 2002).
  • Model 3 is a haploid, asexual Wright–Fisher population tracked as genotype classes. Each generation N offspring are drawn in proportion to fitness (1 + s)steps + beneficial(1 − sd)harmful. Mutation then changes one target position in a fraction 1 − (1 − μ)L of individuals, and adds background mutations at rate U = μ × (rest of genome), split into beneficial, neutral and harmful. Binomial draws are exact for small expected counts and use a normal approximation for large ones. A run ends when most of the population carries a successful sequence.
  • The closed form for Model 3 is the origin–fixation approximation (Gillespie 1984; Orr 2005), with Kimura's fixation probability and a sweep time of 2 ln N / s. Steps needing several coordinated changes use a branching-process tunnelling rate (after Iwasa et al. 2004; Weissman et al. 2009), added to the rate of fixing neutral intermediates one at a time.
  • Checked against simulation. Each closed form was compared with the mean of 20–60 seeded Monte Carlo runs. Model 1 agreed to within sampling error. The Model 2 bound was within about 25% when every step counts (r = 1), but up to several hundred times too slow for r = 3, because it ignores neutral drift (hence "upper bound"). Model 3 was compared on time until half the population succeeds, against the formula minus half a sweep. Single steps agreed to within about 10%. Steps needing two or three changes came within a factor of about 1.4–1.8, with the simulation slower. With many overlapping steps the simulation was about twice as fast, because sweeps overlap (clonal interference).
  • Checked against published work. With the parameters of Sanford et al. (2015) (10,000 diploid individuals, 10−8 per nucleotide, 10% benefit, 20-year generations), Model 3 gives 1.7 million years for one specific nucleotide; the paper reports 1.5 million (and 15.2 vs 15.9 million years at a 1% benefit). For strings of 2–8 nucleotides Model 3 gives 10–20 times longer than the paper's main table. The paper simulated a 10,000-fold raised mutation rate and scaled the result back linearly. Its own extrapolation to the true rate for a 5-nucleotide string (about 40 billion years) is close to Model 3's 28 billion.
  • Reproducibility. The seed text is hashed (xmur3) into the state of an sfc32 generator, so a run is identical on every device. Everything runs in your browser.

Limitations

  • A fixed target. Every model asks how long it takes to reach a sequence chosen in advance, which is how probability arguments are usually framed. Evolution has no target: many different sequences might serve, and which counts as "success" is decided by the environment, not in advance. The tolerance d is only a crude stand-in for that.
  • An idealised landscape. Fitness here is additive across blocks. Real landscapes have strong interactions between mutations (epistasis), and most mutational paths can be closed off (Weinreich et al. 2006); some intermediate steps may be deleterious, or useful for a different function (Lenski et al. 2003).
  • Left out: sex and recombination, diploidy and dominance, population structure, changing environments, insertions, deletions, gene duplication and horizontal gene transfer. Mutations are equally likely at every position and to every block.
  • Browser limits. The Monte Carlo caps population size, offspring and generations (and says so when your settings exceed them). Real mutation rates are far too low for a browser to simulate, which is why the closed forms matter.
  • Approximations in Models 2 and 3 can be off by a large factor outside their range of validity; the notes under each result say when that applies.
  • What the models cannot answer: whether a particular biological structure actually had functional intermediates, which is an empirical question taken up in the case studies, or anything about the origin of life, since every model here starts with a replicating population.
Your conclusions

Before you move on

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