The Jackpot

Bloomington, Indiana, February 1943. Salvador Luria, an Italian refugee three years off the boat, is at a faculty dance. During a pause in the music, he watches a colleague feed a slot machine and lose, lose, lose, and then hit a jackpot. Luria starts thinking about the odds. He has been stuck for months on a question about bacteria, and the slot machine has just handed him the answer.

The question was old and looked unanswerable. Pour a billion E. coli onto a plate seeded with a virus — a bacteriophage that kills them on contact — and almost all die. But a few colonies grow: resistant bacteria. Where did the resistance come from? Two stories, and no obvious way to choose between them:

Plate one culture and count the resistant colonies, and the two stories predict the same thing: some number of survivors. A single number cannot separate them. Luria's slot-machine insight was that you must run the experiment many times in parallel — many separate little cultures — and look not at the average but at how wildly the cultures disagree with each other.

Here is the engine of the whole proof. Under hypothesis A, every cell independently rolls the same long-odds die at the moment of plating, so the number of survivors per culture is a sum of independent rare events — a Poisson count. A Poisson distribution has a signature so exact it is almost a fingerprint: its variance equals its mean. All the cultures cluster tightly around the same value.

Under hypothesis B, a mutation can strike at any moment during the days of growth. If it strikes late, the mutant has time to make only a handful of resistant descendants before plating. If it strikes early — while the culture is still small — that one mutant and all its progeny double, and double, and double, until by plating day there are thousands of resistant cells from a single ancestral accident. A jackpot. Most cultures get no early mutation and show few or zero survivors; a rare few hit the jackpot and show enormous counts. The cultures disagree violently. The variance dwarfs the mean.

Variance is the tell. The mean cannot separate the hypotheses; the spread between parallel cultures can, and only one story makes the spread explode.

I · The fluctuation test, run live

Below are two trays of parallel cultures. Each culture starts from a tiny inoculum and grows by binary fission to millions of cells; then it is plated whole onto phage and its resistant colonies counted. The left tray obeys hypothesis A (induced): every cell converts independently at plating with the rate set so the average comes out to m per culture. The right tray obeys hypothesis B (spontaneous): mutations occur at random generations during growth, and an early one founds a jackpot clone. Same expected event count, different survivor averages. Watch the spread.

The number that matters is the variance-to-mean ratio. For the induced tray it sits stubbornly near 1, the Poisson fingerprint, no matter how many cultures you run. For the spontaneous tray it can reach the hundreds, fluctuating as cultures accumulate and sometimes jumping when a large jackpot lands. That gap, two or three orders of magnitude wide, is what Luria and Delbrück saw in 1943, and it is not subtle. The spread alone decides the case, and it decides for spontaneous mutation.

II · The exact distribution — and why its mean is infinite

Spontaneous mutation doesn't just predict “large variance.” In the ideal limiting model it predicts a specific distribution. D. E. Lea and C. A. Coulson published its closed-form probability generating function in 1949, from a manuscript and calculations completed in 1947. The clean way to see it: the number of mutational events per culture is Poisson with mean m, and each event founds a clone whose final size s has a 1/s² tail. The model assigns a constant mutation probability per cell division, with exponentially more opportunities late in growth. An early mutant has more time to grow a large clone. In the equal-fitness stochastic growth model, a clone has size k with probability 1/(k(k+1)). The mutant count is the sum of a Poisson number of such clones, and its probabilities follow the Ma–Sandri–Sarkar recursion:

// Ma–Sandri–Sarkar (1992) recursion, m = expected mutations per culture
p₀ = e^(−m)
pₙ = (m / n) · Σ[i=0..n−1] pᵢ / (n − i + 1)

// computed a second way as a compound-Poisson convolution:
//   p = Σ[j] e^(−m) mʲ/j! · c^(*j),   c_k = 1/(k(k+1))
// the two methods agree to machine precision:
max |p_n(recursion) − p_n(convolution)| = …

Dial m below and read the distribution off exactly. Note the shape: a tall spike at zero (those are the … of cultures that never mutated), then a long, fat tail that falls off as m/n², slowly enough that the rare jackpots out at n = 100 or 1000 still carry real probability. The grey bars sample this exact clone law, separately from the binary-growth simulation in Instrument I; they approximate the exact probabilities.

Now the strange part, and the reason this distribution is famous beyond microbiology. The expected clone size is Σ k·(1/(k(k+1))) = Σ 1/(k+1) — the harmonic series, which diverges. So the mean of the Luria–Delbrück distribution is infinite. Not large: infinite. In any real culture the clone is capped at the final population N, so the mean is finite but grows like m·ln N and never settles to a fixed value — it depends on how big you grew the cultures, which is no property of the mutation rate at all.

The raw average is not an estimate of m. The ideal limiting law has no finite mean; the finite growth simulation above does. The jackpots are heavy enough to drag it wherever the largest one happens to land.

III · So they counted the empty cultures

One of the 1943 methods used a statistic the jackpots cannot corrupt. Luria and Delbrück also used an average-based estimator, but their zero method used the simplest count imaginable: the fraction of cultures with zero resistant colonies. A culture is empty exactly when no mutation occurred at all, and mutational events are Poisson(m), so

P(culture is empty) = p₀ = e^(−m)      ⇒ invert it:
m̂ = −ln( fraction of empty cultures )

// mutation events within one culture are Poisson(m).
// Among C independent cultures, the empty count is Binomial(C, e^(−m)).

Below, the two estimators race. Each click runs 40 fresh experiments of 100 parallel spontaneous cultures each and applies both: the p₀ estimator −ln(empty fraction), and the naïve sample mean. The p₀ estimate parks itself on the true m; the sample mean staggers around like a drunk, because every experiment that happens to catch a big jackpot throws it off by a factor.

The p₀ method has a known weakness: if m is large, almost no culture is empty and there is nothing to count. The modern “gold standard” is the Ma–Sandri–Sarkar maximum-likelihood fit that uses the whole distribution from Instrument II. The simpler p₀ estimator ignores jackpot sizes and is useful when m is small.

IV · What it settled, and what it didn't

The fluctuation test was the first quantitative proof that mutations in bacteria arise spontaneously and at random with respect to the selection that later reveals them — that bacteria are genetic organisms obeying the same Darwinian logic as everything else, not Lamarckian shapeshifters. It opened bacterial genetics; it is part of why Luria and Delbrück (with Alfred Hershey) shared the 1969 Nobel Prize. In 1952 Joshua and Esther Lederberg closed the case visually with replica plating: resistant colonies can be lifted off a plate that never met the phage, proving the mutants pre-existed.

What it did not prove is that all mutation is blind in every organism and circumstance forever — a caution worth keeping. Later work (the contested “adaptive mutation” results of the late 1980s, and now the targeted machinery of CRISPR-Cas immunity) shows the genome–environment relationship is richer than a clean 1943 dichotomy. There is a small irony the modern literature enjoys: the E. coli strain B that Luria and Delbrück happened to use lacks a working CRISPR-Cas system — had they picked a strain with one, they would have been studying a genuine case of heritable, environment-directed resistance, and the experiment might have read very differently. The test was right about its phage and its strain; the universe kept a subtler card in reserve.

The variance settled it. The average alone could not decide between the hypotheses. Luria and Delbrück read the proof in how much the cultures disagreed, and used both empty-culture and average-based methods to estimate the mutation rate.

The check, shown

…

Cousins on the ground

Sources