A two-phase model of a de novo beneficial mutation timeline. With

• initial population size $N_e$
• mutation selection coefficient $s$
• mutation rate $\mu$ (humans $\approx 10^{-8}$)
• target allele frequency $p$

it shows the expected number of generations to establish a mutation (against genetic drift) and grow that population to the desired allele frequency p.

Importance of population size $N_e$

In this Dwarkesh interview David Reich suggests that for human populations larger than $\approx 10^6$ "every mutation that can occur does occur within a couple of generations". This is off, and does not account for the genetic drift (i.e. initial stochasticity of the process) that kills most of beneficial mutants. Indeed, for such $N_e$ regardless of $s$ the time required for a mutation to successfuly estabilish dominates:

$$ \frac{ \underbrace{\frac{1}{2N_e\mu P_{\text{est}}}}_{\text{waiting for successful mutant}} + \underbrace{\frac{\log(1/s)}{s}}_{\text{growth to establishment}} }{ \underbrace{ \frac{1}{s} \log\left( \frac{p(1-p_{\text{est}})} {p_{\text{est}}(1-p)} \right) }_{\text{deterministic growth to target}} } > 1 $$

$$ P_{\text{est}}=\frac{1-e^{-2s}}{1-e^{-4N_es}} $$

$$ p_{\text{est}}\approx\frac{1}{2N_es} $$

as derived below. Example values:

s	$N_e$ = $10^5$	$N_e$ = $10^6$	$N_e$ = $10^7$	$N_e$ = $10^8$	Breakpoint $N_e$
0.1	28.08	2.45	0.35	0.15	$2.55 \cdot 10^6$
0.025	30.53	2.71	0.48	0.26	$3.10 \cdot 10^6$
0.005	37.14	3.30	0.68	0.40	$4.62 \cdot 10^6$

This model separates the timeline into two phases:

1. Phase 1: the beneficial mutation must appear and survive early genetic drift.
2. Phase 2: after establishment, the allele grows mainly by deterministic natural selection.

Installation

conda env create -f environment.yml
conda activate allele

To run

streamlit run allele.py

Phase 1: mutation supply and escape from drift

A new beneficial mutation begins as one allele copy. In a diploid population, there are approximately $2N_e$ allele copies, so new beneficial mutant copies appear at rate $2N_e\mu$ per generation ($t$).

However, most new beneficial mutations are lost by genetic drift. The establishment probability is:

$$P_{\text{est}} = \frac{1-e^{-2s}}{1-e^{-4N_es}}$$

and therefore the rate at which successful beneficial mutations appear is $2N_e\mu P_{\text{est}}$. We can calculate the expected waiting time for a successful mutation:

$$T_{\text{wait}} = \frac{1}{2N_e\mu P_{\text{est}}}$$

Once a future-successful mutation appears, its expected early copy-number growth is $\mathbb{E}[X_t] = e^{st}$ where $X_t$ is the mutant allele copy count. We approximate the establishment threshold as

$$ X_{\text{est}} \approx \frac{1}{s} $$

So the expected time from appearance to establishment is:

$$T_{\text{appear}\to\text{est}} = \frac{\log(1/s)}{s}$$

Thus the full early phase is approximately:

$$T_{\text{phase 1}} \approx \frac{1}{2N_e\mu P_{\text{est}}} + \frac{\log(1/s)}{s}$$

In asymptotic notation:

$$T_{\text{phase 1}} = O\left( \frac{1}{N_e\mu P_{\text{est}}} + \frac{\log(1/s)}{s} \right)$$

Using the common large-population approximation $P_{\text{est}} \approx 2s$, this becomes:

$$T_{\text{phase 1}} = O\left( \frac{1}{N_e\mu s} + \frac{\log(1/s)}{s} \right)$$

Phase 1 is therefore governed by mutation supply, establishment probability, and early stochastic survival.

Phase 2: deterministic growth after establishment

After establishment, the allele is common enough that genetic drift is less dominant. Its frequency is modeled by deterministic selection:

$$\frac{dp}{dt} = sp(1-p)$$

The time to grow from establishment frequency $p_{\text{est}}$ to target frequency $p$ is then:

$$T_{\text{phase 2}} = \frac{1}{s} \log\left( \frac{p(1-p_{\text{est}})} {p_{\text{est}}(1-p)} \right) \approx O\left(\frac{1}{s}\right)$$

only scaling with $s$! This means the deterministic growth phase is mostly controlled by selection strength. If $s$ doubles, the allele spreads roughly twice as fast. If $s$ is cut in half, the spread takes roughly twice as long.