# Numbers of Mutations within Multicellular Bodies: Why It Matters

## Abstract

**:**

## 1. Introduction

## 2. The Luria–Delbrück Problem

## 3. Average Number of Mutants

#### 3.1. Intuitive Summary

#### 3.2. Details

## 4. Variation in Number of Mutants

## 5. Distributions with Fat Tails

## 6. Distribution of Mutants

#### 6.1. Fréchet Distribution

#### 6.2. Approximate Fit

#### 6.3. Intuition

## 7. Mutation Rate

## 8. Genetics of Human Populations

## 9. Likelihood

## 10. Somatic Mosaicism and Disease

#### 10.1. Mutant Cells and the Risk of Disease

#### 10.2. Variation between Individuals

## 11. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Number of mutants in a growing population depends on the time of the first mutation. (

**a**) When the first mutation happens in the last cell division, then only one cell $\left(m={2}^{0}\right)$ carries that mutation. In this case, there are $p={2}^{3}$ different final cells that could be mutated in the final round of cell division, so such mutations confined to a single cell will be relatively common. (

**b**) When the first mutation happens in the second to last cell division, then two cells $\left(m={2}^{1}\right)$ carry that mutation. In this case, there are $p={2}^{2}$ different final groups of cells that could be mutated, so such mutations confined to two cells will be $1/2$ as common as for singleton mutations. (

**c**) When the first mutation happens in the third to last cell division, then four cells $\left(m={2}^{2}\right)$ carry that mutation. In this case, there are $p={2}^{1}$ different final groups of cells that could be mutated, so such mutations confined to four cells will be $1/4$ as common as for singleton mutations. Focusing on single mutational events, every doubling for the final number of mutant cells decreases the probability of occurrence by $1/2$. This discrete-time model matches Haldane’s formulation of the Luria–Delbrück problem [8], used here for illustration because of its simplicity. Most modern analyses use a continuous time formulation [7]. In the text, the mixture of references to the discrete and continuous formulations leads to a mixture of comments about mutation probabilities and mutation rates. When the underlying formulation is ambiguous, I use mutation rate.

**Figure 2.**The probability of observing a particular number of mutants in the final population. (

**a**) Probability declines by $1/2$ for each doubling in the number of mutants in the final population. (

**b**) A log–log plot of probability versus mutant number has a slope of minus one. This example follows from a discrete binary pattern of cellular division with a single mutational event, as in Figure 1. The actual log–log slope for a Luria–Delbrück process is approximately $-(1+e/2)\approx -2.36$ (from Equation (5)). Part of the increase arises because multiple mutational events cause a more rapid decline at the extreme of the earliest mutations and greatest final mutant numbers.

**Figure 3.**Cumulative probability distribution for the number of neutral mutants. Each population starts with one cell and grows to N cells. Mutations occur at rate u. The blue curves show the Luria–Delbrück distribution calculated by the simu.cultures computer simulation of the R package rSalvador [13]. The orange curves show the Fréchet distribution in Equation (3). Reprinted from from Figure 1 of Frank [14]. See that article for further details. The mathematical reason for the close match between the Luria–Delbrück distribution and the Fréchet distribution arises by linking two separate studies. Kessler and Levine [15] showed that the Luria–Delbrück distribution converges to a Lévy $\alpha $-stable distribution for large $Nu$. Separately, Simon [16] showed the close match between the Lévy $\alpha $-stable distribution and the Fréchet distribution. Using the Fréchet distribution provides a benefit because no explicit mathematical expression exists for the Lévy $\alpha $-stable probability distribution.

**Figure 4.**Log-likelihood for the parameter $Nu$ given the values of the data shown in the legends of each plot. A constant was added to all log-likelihood values to shift the curves up so that the peaks are at a value of three. The zero values give the log-likelihood confidence intervals for which the most likely value is ${e}^{3}\approx 20$ times more likely that the lowest values shown at the edges of the intervals. (

**a**) For a single observation of $m={10}^{5}$, the most likely value is approximately $Nu={10}^{4.05}$. (

**b**) Log-likelihood curves for different data combinations. The first (blue) curve has a single observed value, whereas the other two curves arise from pairs of observed values. These likelihood curves derive from the Fréchet approximation in Equation (3).

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Frank, S.A. Numbers of Mutations within Multicellular Bodies: Why It Matters. *Axioms* **2023**, *12*, 12.
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**AMA Style**

Frank SA. Numbers of Mutations within Multicellular Bodies: Why It Matters. *Axioms*. 2023; 12(1):12.
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**Chicago/Turabian Style**

Frank, Steven A. 2023. "Numbers of Mutations within Multicellular Bodies: Why It Matters" *Axioms* 12, no. 1: 12.
https://doi.org/10.3390/axioms12010012