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In 1943, Salvador Luria and Max Delbrück grew separate Escherichia coli B cell cultures from small equal inocula, without the challenging bacteriophage. They used nutrient broth or an asparagine–glucose cell culture medium. On nutrient agar plates, they first spread excess bacteriophage and then the bacterial sample centrally, leaving an outer margin. After incubation, resistant biological colonies provided the readout; counts were usually made at 24 and 48 hours. Their control plated repeated samples from one shared culture. Independently grown cultures varied much more than this sampling comparison. That contrast supported resistant lineages arising before challenge: early mutations could produce many resistant descendants before natural selection revealed them.^[ref]

This was evidence about the timing of heritable resistance, not DNA sequencing of the responsible genes. It does not establish uniform mutation rates or exclude environmentally induced mutations; the paper itself discusses physiological dependence and delayed expression.^[ref]

Now construct a deliberately small mathematical model of the timing effect. Start with one sensitive biological cell at generation $g=0$. Every biological cell divides into two in each of five synchronous rounds, with no death or growth difference. There are $N_g=2^g$ biological cells immediately after round $g$, ending with $N_5=32$. In each scenario, exactly one newly formed daughter changes to resistance immediately after round $g$; its descendants inherit resistance without reversion or further mutations. Resistance is expressed immediately, and the final challenge allows every resistant biological cell, and no sensitive one, to form one separate biological colony.

The descendant count obeysAt the moment of the event there is one resistant biological cell; each remaining round doubles its descendants. This proves the formula by repeated multiplication, with terminal case $R_5=1$. An event after round 1 gives $R_1=16$ resistant and 16 sensitive biological cells; an event after round 4 gives $R_4=2$ resistant and 30 sensitive biological cells. Both scenarios contain one mutation event. A culture with no event gives zero resistant biological colonies under these assumptions. The difference is family size, rather than the number of events.

For comparison, suppose each of the 32 final sensitive biological cells independently acquires resistance only on challenge, with probability $p=1/16$ and no subsequent multiplication before counting. Then the resistant count has a binomial distribution, with expected value $32p=2$ and variance $32p(1-p)=15/8$. This alternative mathematical model has no pre-challenge resistant family to amplify. Its independent trials are an assumption of the comparison, not a consequence of using physically distinct biological cells. The clonal scenarios explain how a jackpot can occur; they do not assign probabilities to the event times or estimate a real mutation rate.

// Target: biology.bigb