Founder Events and Frequency Shifts in Small Populations

Why Founder Size Matters

A small population established by a few individuals carries only a sample of the genetic variation in its source population. By chance, the founders may carry more or fewer copies of an allele than its source frequency would suggest. That initial sampling difference can leave the new population with a frequency quite unlike the source, even when there is no selection favoring either allele.

The key point is that the frequency after founding is random across possible founder groups. Its expected value can equal the source frequency while an individual founding event still produces a substantial shift. The amount of variation depends both on the number of founders and on the allele's starting frequency.

Definition: Consider a biallelic locus in a source population where one allele has frequency $p$. If $N_F$ diploid founders are sampled, let $K_F$ be the number of copies of that allele among their $2N_F$ allele copies. The realized founder frequency is $\hat p_F=K_F/(2N_F)$. A founder-frequency shift is $\hat p_F-p$.

The Sampling Model

For the basic calculation, assume the founders are sampled at random from a large source population, and that the $2N_F$ allele copies are independent draws with probability $p$ of carrying the allele. This is the usual binomial sampling model for diploid founders when allele copies in the source gene pool are independent. Assume also that the source frequency is fixed for this calculation and that no selection, mutation, or migration changes the sampled copies during founding.

Under these assumptions, $K_F$ counts successes in $2N_F$ independent trials, each with success probability $p$. Thus $K_F\sim\mathrm{Binomial}(2N_F,p)$, and the realized frequency is the sampled count divided by the number of sampled copies.

Theorem: Under the independent sampling model, $$ E(\hat p_F)=p,\qquad \operatorname{Var}(\hat p_F)=\frac{p(1-p)}{2N_F},\qquad SE(\hat p_F)=\sqrt{\frac{p(1-p)}{2N_F}}. $$ The standard error describes the standard deviation of realized founder frequencies across repeated founding events with the same $p$ and $N_F$. It is also the standard deviation of the founder-frequency shift, since $p$ is fixed.

The expectation says that the sampling procedure has no systematic direction: over many hypothetical repetitions, the mean founder frequency is $p$. It does not say that a particular set of founders will have frequency $p$. The variance quantifies how widely founder frequencies tend to vary around that expectation. The shift itself has expected value zero and variance $p(1-p)/(2N_F)$.

Why the Formula Holds

Write $X_j=1$ if sampled allele copy $j$ carries the allele and $X_j=0$ otherwise, for $j=1,\ldots,2N_F$. Each indicator has $E(X_j)=p$ and $\operatorname{Var}(X_j)=p(1-p)$. The founder count is $K_F=\sum_j X_j$, so $\hat p_F=(1/(2N_F))\sum_j X_j$.

By linearity of expectation, the mean of the sum is the sum of its means. By independence, the variance of the sum is the sum of its variances. Dividing the count by $2N_F$ divides its variance by $(2N_F)^2$. Therefore:

E(\hat p_F) =\frac{1}{2N_F}\sum_{j=1}^{2N_F}E(X_j) =\frac{2N_Fp}{2N_F} =p, $$ $$ \operatorname{Var}(\hat p_F) =\frac{1}{(2N_F)^2}\sum_{j=1}^{2N_F}\operatorname{Var}(X_j) =\frac{2N_Fp(1-p)}{(2N_F)^2} =\frac{p(1-p)}{2N_F}.

Taking the square root of the variance gives the standard error. Increasing $N_F$ reduces the variance in inverse proportion to founder number and the standard error in proportion to $1/\sqrt{N_F}$. The factor $p(1-p)$ is largest at $p=0.5$, so sampling variation in frequency is greatest for an allele at intermediate frequency, when founder size is fixed.

Worked Example: A Rare Allele in Ten Founders

Suppose a source population has allele frequency $p=0.20$, and a hypothetical new population is founded by $N_F=10$ diploid individuals. Under the stated model, there are $2N_F=20$ independently sampled allele copies.

The expected founder frequency is $p=0.20$. Its variance and standard error are:

\operatorname{Var}(\hat p_F) =\frac{0.20(1-0.20)}{2(10)} =\frac{0.16}{20} =0.00800, $$ $$ SE(\hat p_F)=\sqrt{0.00800}=0.08944.

Thus, across repeated hypothetical founding events, the standard deviation of the founder frequency is about 0.0894, or 8.94 percentage points. This is a measure of sampling spread, not a guarantee that a particular founder group will be within 8.94 percentage points of 0.20.

The probability that the allele is entirely absent is the probability that all 20 sampled copies are the other allele:

P(K_F=0)=(1-p)^{20}=0.80^{20}=0.01153.

So under this model, about 1.15% of repeated groups of ten founders would contain no copies of this allele, despite its source frequency being 0.20.

Founder Number and Starting Frequency

The formula separates two influences. First, for a fixed starting frequency, more founders mean more sampled allele copies and less variation. Doubling the number of diploid founders halves the variance, but reduces the standard error only by a factor of $\sqrt{2}$. Second, for a fixed number of founders, variation is largest at intermediate frequencies and smaller near zero or one. However, a rare allele can still be vulnerable to loss: a small variance in frequency does not rule out a meaningful probability of sampling zero copies.

Worked Example: Comparing Small and Large Founder Groups

Consider two hypothetical founding events from a source with $p=0.50$. One group has $N_F=5$ diploid founders and the other has $N_F=50$. For each group, the expectation remains 0.50. The variances and standard errors differ:

\operatorname{Var}(\hat p_F\mid N_F=5) =\frac{0.50(0.50)}{10} =0.0250,\qquad SE=\sqrt{0.0250}=0.1581, $$ $$ \operatorname{Var}(\hat p_F\mid N_F=50) =\frac{0.50(0.50)}{100} =0.002500,\qquad SE=\sqrt{0.002500}=0.05000.

The larger group has one-tenth the variance, because it has ten times as many founders, and its standard error is smaller by a factor of $\sqrt{10}$. These are not predictions that either realized frequency will equal 0.50; they quantify how much less variable the larger group's frequency is across repeated samples.

For a direct probability comparison, in the group of five founders there are ten sampled copies. The probability of obtaining at least seven copies of the allele, corresponding to $\hat p_F\geq0.70$, is:

P(\hat p_F\geq0.70) =P(K_F\geq7) =\sum_{k=7}^{10}{10\choose k}(0.50)^k(0.50)^{10-k} =\frac{176}{1024} =0.1719.

For the group of 50 founders, the standard error is much smaller, but this particular tail probability would require summing binomial probabilities for counts $k\geq70$ out of 100. The standard-error comparison alone already shows why large frequency shifts are less typical with more founders; it does not supply that exact tail probability.

Worked Example: The Same Founder Size, Two Starting Frequencies

Compare two hypothetical alleles sampled into populations of $N_F=10$ diploid founders. The first has source frequency $p=0.05$, and the second has source frequency $p=0.50$. Each founding event samples 20 allele copies.

For the rare allele:

\operatorname{Var}(\hat p_F) =\frac{0.05(0.95)}{20} =0.002375,\qquad SE=\sqrt{0.002375}=0.04873.

For the allele at frequency 0.50:

\operatorname{Var}(\hat p_F) =\frac{0.50(0.50)}{20} =0.01250,\qquad SE=\sqrt{0.01250}=0.1118.

The intermediate-frequency allele has the larger frequency variance. But the rare allele has a substantial chance of being missing altogether:

P(K_F=0\mid p=0.05)=0.95^{20}=0.3585, $$ $$ P(K_F=0\mid p=0.50)=0.50^{20}=0.0000009537.

Thus, in this hypothetical comparison, the rare allele is absent in about 35.9% of founder samples, whereas the allele starting at frequency 0.50 is absent in fewer than one in a million. There is no contradiction: variance measures the spread of frequency values, while loss probability asks specifically whether the sampled count is zero.

Assumptions and Interpretation

The binomial formula relies on independent allele-copy sampling. If founders are related, or if their genotypes have dependence beyond the model, the copies are not independent and the variance need not equal $p(1-p)/(2N_F)$. The same warning applies if founders are selected nonrandomly from the source. In those settings, the founder composition may be systematically unrepresentative, or its sampling variation may differ from the simple calculation.

The model also treats $p$ as known and fixed. If $p$ is estimated from a finite source sample, uncertainty in that estimate is a separate source of uncertainty and is not included in the variance above. Likewise, the calculations describe sampling at founding, not subsequent changes in frequency.

Key takeaway: With independent random sampling of $2N_F$ allele copies, the founder frequency has expectation $p$ and variance $p(1-p)/(2N_F)$. A founder event can shift frequency in either direction; small founder groups make larger sampling shifts more variable, and rare alleles may be lost even when the average frequency across repeated events is unchanged.

Common Mistake

Wrong reasoning: “The expected founder frequency is $p$, so a small founder population should have approximately the source frequency.”

Correction: The expectation is an average across repeated hypothetical founder samples. It does not describe how close any one sample must be to $p$. For a single event, the variance and the exact binomial probabilities quantify the sampling uncertainty. In particular, a rare allele can have a high probability of being absent even though its expected frequency remains $p$.

Check Your Understanding

Use the independent allele-copy sampling model for each question unless stated otherwise.

A source allele has frequency 0.30. Write the expected founder frequency and variance for a founding group of 12 diploid individuals.
For a fixed source frequency, what happens to the variance and standard error if the number of diploid founders is multiplied by four?
Why can an allele with a low source frequency have a substantial probability of being lost even when the frequency variance is largest near 0.50?
For an allele with source frequency 0.10 and eight diploid founders, express the probability that the allele is absent as a power of the other-allele frequency. Then evaluate it.
Name one condition under which the binomial variance formula may not apply, and explain which part of the sampling model is violated.