Feb23-2019

从前，我从广泛的阅读里获得了很多好处，但读的东西基本上不能帮助我进行工作，而稍微有用一些的东西，在放松的状态下，又读不进去。

我决定每周抽出几天来广泛地看一些文章。最好是一些有趣的研究，在生物或其他的背景下应用了物理的一些工具，得到了重要的结果。这样，训练了读写的能力，也可以提升对于整个领域的了解。大概一次写一篇或两篇。

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Luria-Delbrueck Experiment

原文地址：https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1209226/pdf/491.pdf

The Luria-Delbrueck experiment aims to test two contradictory hypothesis in the occurrence of a phage-proof gene:

h1: the mutation that lead to the gene happened after infections, i.e. the immunity was acquired.

h2: the mutation was independent of the infections.

The authors noticed that the variance of the number of mutants may help testing, which was a great step leaned in history of quantitative biology. Without much experimental exquisiteness, the second hypothesis was proved.

First, under h1 the N(mutants) at one specific time (after a contact with phages) must obey a Poisson distribution, meaning that $\frac{\mathrm{average}}{\mathrm{variance}} = 1$.

Things under h2 however, are a bit more complicated. Assume that $r$ is the number of mutants and $N$ is the overall population. To make things simple, we just say that $N=N_0e^t$ and $\frac{\mathrm{d}r}{\mathrm{d}t}=aN+r$. Integrating the second equation from $t_0$ to $t$, we have $r = (t-t_0)aN$. The idea of integrating from $t_0$ is to make sure that integration begins at a time when there is at least one mutant, eliminating potential errors. Denoting number of cultures as $C$, we can write $a(N_{t_0}-N_0)C=1$, in which $N_0$ could be ignored and $t-t_0=\ln(aNC)$ be acquired. With all these above, $r = aN\ln (aNC)$.

Then we consider the average number of new mutations occurred in $[t-\tau, t-\tau+\mathrm{d}\tau]$, which should be $N_te^{-\tau}a\mathrm{d}\tau$. Since the mutation is a Poisson process, that should also be its variance, hence during the time $[t-\tau, t]$, the variance caused by these mutations grows to $N_te^{\tau}a\mathrm{d}\tau$. Integrating $\tau$ in $[0, t-t_0]$, we have that $\mathrm{var}_r=aN_t(e^{t-t_0}-1)=Ca^2N_t^2$.

Therefore, $\frac{\ln(aNC)}{aNC}$ and 1 can be compared with the data to see which was closer. It turned out to be the former, suggesting a much larger variance. That is natural, since compared to acquired immunity in which all the mutants originate at the same time, independent mutation is more like

a slot machine, where the average return from a limited number of plays is probably considerably less than the input, and improbably, when the jackpot is hit, the return is much bigger than the input.