# Mutant Number Laws and Infinite Divisibility

## Abstract

**:**

## 1. Introduction

- (i)
- Normal cell numbers increase exponentially fast: ${N}_{t}={N}_{0}{e}^{\nu t}$.
- (ii)
- Mutation occurs randomly at a rate proportional to ${N}_{t}$. Specifically, there is a mutation rate r (per unit time per bacterial cell) such that a mutation event occurs in the interval $(t,t+dt)$ with probability $r{N}_{t}dt+o\left(dt\right)$, i.e., a normal cell converts to a resistant type. There is no mutation with probability $1-r{N}_{t}dt+o\left(dt\right)$.
- (iii)
- Mutation events create mutant clones which grows independently of each other according to a linear birth process with split rate $\mu $, i.e., a binary splitting or Yule process. The relative growth rates of normal to mutant cells is denoted by $\gamma =\nu /\mu $.

## 2. Infdiv Distributions and Deterministic Mutant Growth

**Remark**

**1.**

**Definition**

**1.**

**Example**

**1.**

**Fact**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Definition**

**2.**

**Definition**

**3.**

**Fact**

**2.**

**Remark**

**4.**

**Theorem**

**1.**

- (a)
- $K=1$ (equivalently, $\nu t<\gamma log2$), in which case its mass function is non-increasing iff $m\le 1$; i.e.,$$\nu t\le log(1+\nu /r{N}_{0});$$
- (b)
- $K\ge 2$ and $\gamma \ge {\gamma}^{\ast}\approx 0.3663$, in which case its mass function is non-increasing iff $\theta (1-{2}^{-\gamma})\le 1$, i.e.,$$\nu t\le {\left[log\frac{\nu /r{N}_{0}}{1-{2}^{-\gamma}}\right]}^{+}.$$

**Proof.**

**Theorem**

**2.**

- (a)
- $t<L$, in which case the mass function is non-increasing iff $m\le 1$, i.e.,$$\gamma t\le \frac{log(1+\nu /r{N}_{0})}{log2},$$
- (b)
- $t\le L<2t$ and $\gamma t\le 2L$, in which case its mass function is non-increasing iff $\theta \le 2$, i.e.,$$\gamma t\le Llog(\nu /r{N}_{0}).$$
- (c)
- The mutant number distribution is not SD otherwise.

**Proof.**

## 3. Bondesson Classes and the Generalised Lea-Coulson Model

**Definition**

**4.**

**Fact**

**3.**

- (a)
- If X has the DF $F\in ME$, then $X\stackrel{d}{=}\epsilon Y$, where $\epsilon $ has an exponential distribution and $Y\ge 0$ is independent of $\epsilon $.
- (b)
- If $F\in ME$, then it is infdiv.
- (c)
- The DF $F\in ME$ iff$$-log\widehat{F}\left(\zeta \right)={\int}_{0}^{\infty}\frac{\zeta}{y(y+\zeta )}b\left(y\right)dy,$$$$0\le b\left(y\right)\le 1\phantom{\rule{2.em}{0ex}}\mathit{and}\phantom{\rule{2.em}{0ex}}{\int}_{0}^{1}{y}^{-1}b\left(y\right)dy<\infty .$$

**Definition**

**5.**

**Fact**

**4.**

- (a)
- If $F\in BO$, then its Laplace exponent has the form$$c\left(\zeta \right)=a\zeta +{\int}_{0}^{\infty}\frac{\zeta}{y(y+\zeta )}B\left(dy\right),$$
- (b)
- The class $BO$ is the smallest set of distributions containing $ME$ and which is closed under convolution and weak limits.

**Definition**

**6.**

**Definition**

**7.**

**Fact**

**6.**

- (a)
- The discrete infdiv distribution $\left({p}_{j}\right)\in BOP$ iff $({\lambda}_{j+1}:j=0,1,\dots )$ is a Hausdorff moment sequence; specifically,$$\mathrm{\Lambda}=a+{\int}_{0}^{\infty}{\left[y(1+y)\right]}^{-1}B\left(dy\right)\phantom{\rule{1.em}{0ex}}and\phantom{\rule{2.em}{0ex}}{\lambda}_{j}=a{\delta}_{j,1}+{\int}_{0}^{\infty}{(1+y)}^{-j-1}B\left(dy\right).$$
- (b)
- A distribution in $BOP$ is a mixture of geometric distributions if $B\left(dy\right)=b\left(y\right)dy$ and $0\le b\left(y\right)\le 1$.

**Remark**

**5.**

**Remark**

**6.**

**Theorem**

**3.**

**Proof.**

**Remark**

**7.**

**Example**

**2.**

## 4. Thorin Classes and the Lea-Coulson Model

**Definition**

**8.**

**Fact**

**7.**

- (a)
- A GGC is a SD distribution for which the function k is completely monotone, $x\ell \left(x\right)=k\left(x\right)={\int}_{0}^{\infty}{e}^{-xy}U\left(dy\right)$.
- (b)
- Any GGC has a unimodal pdf f.
- (c)
- A GGC belongs to BO and its Bondesson measure is absolutely continuous with density $b\left(y\right)=T\left(y\right)$.

**Definition**

**9.**

**Fact**

**8.**

- (a)
- the PGF of a GNBC has the canonical form$$M\left(s\right)=exp\left(-a(1-s)-{\int}_{0+}^{1-}log\frac{1-u}{1-us}dV\left(u\right)\right),$$$${\int}_{(0,{\scriptstyle \frac{1}{2}})}udV\left(u\right)<\infty \phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}{\int}_{({\scriptstyle \frac{1}{2}},1)}log\left({(1-u)}^{-1}\right)dV\left(u\right)<\infty .$$
- (b)
- The r-sequence (c.f. (9)) is a Hausdorff moment sequence,$${r}_{j}=a{\delta}_{j0}+{\int}_{0}^{1}{u}^{j+1}dV\left(u\right).$$Conversely, if the r-sequence of a DID distribution has this moment representation, then it is a GNBC.
- (c)
- The GNBC class is the smallest class of discrete distributions which contains negative-binomial distributions and is closed under convolution and weak limits.
- (d)
- A GNBC is discrete unimodal and its mass function is non-decreasing iff ${\lambda}_{1}=a+{\int}_{0}^{1}udV\left(u\right)\le 1$.

**Remark**

**8.**

**Fact**

**9.**

- (a)
- A function M defined on $[0,1]$ is the PGF of a geometric mixture distribution iff it has the form$$M\left(s\right)=exp\left[-{\int}_{0}^{1}\left(\frac{1}{1-u}-\frac{1}{1-us}\right)w\left(u\right)du/u\right]$$$$0\le w\left(u\right)\le 1\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}{\int}_{{\scriptstyle \frac{1}{2}}}^{1}{(1-u)}^{-1}w\left(u\right)du<\infty .$$
- (b)
- A GNBC PGF (35) is the PGF of a geometric-mixture distribution iff $a=0$ and its representing function V satisfies $-V(0+)\le 1$, in which case $w\left(u\right)=-V\left(u\right)$.

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

- (a)
- The distribution of ${M}_{t}$ is a GNBC and hence unimodal. Its mass function is non-increasing iff$${\lambda}_{1}={e}^{-\mu t}{\int}_{0}^{t}{e}^{\mu v}N\left(v\right)dv\le 1.$$
- (b)
- The Lévy density $\ell (x,t)$ of the mixing GGC is given by$$x\ell (x,t)={\int}_{0}^{\infty}{e}^{-xy}{d}_{y}T(y,t)$$$$T(y,t)=(r/\mu )N\left(t+{\mu}^{-1}log\frac{y}{1+y}\right)H(y-({\varphi}^{-1}-1)).$$
- (c)
- The canonical form of the PGF of ${M}_{t}$ is$$M\left(s\right)=exp\left[-{\int}_{0}^{1}log\frac{1-u}{1-us}dV(u,t)\right],$$$$-V(u,t)=\left\{\begin{array}{cc}(r/\mu )N(t+{\mu}^{-1}log(1-u))\hfill & if0\le u<\varphi ,\hfill \\ 0\hfill & if\varphi \le u\le 1.\hfill \end{array}\right.$$
- (d)
- The mutant number distribution is a geometric mixture iff $rN\left(t\right)\le \mu $.

**Proof.**

- (a)
- Observe that $\beta (y,t)$ is non-decreasing in y and that, since $\beta (\infty -,t)<\infty $, it follows from (22) that$${r}_{j}=(j+1){\lambda}_{j+1}=-(r/\mu ){\int}_{0}^{\infty}\beta (y,t){d}_{y}{y}^{-(j+1)}=(r/\mu ){\int}_{0}^{\infty}{y}^{-(j+1)}d\beta (y,t),$$
- (b)
- (c)

**Remark**

**9.**

**Theorem**

**6.**

- (a)
- It is a GNBC, hence unimodal. Its mass function is non-increasing iff$$\theta \left(1-{(1-\varphi )}^{\gamma +1}\right)\le \gamma +1.$$
- (b)
- The function V in the representation (35) is$$V\left(u\right)=-\theta {(1-u)}^{\gamma}\left(1-H(u-\varphi )\right).$$In particular, the $LDM(\theta ,\varphi ,\gamma )$ distribution is a geometric mixture iff $\theta \le 1$.
- (c)
- The GGC mixing distribution has the Thorin distribution function$$T\left(x\right)=\left\{\begin{array}{cc}\theta {\left(\frac{x}{1+x}\right)}^{\gamma}\hfill & ifx\ge {\varphi}^{-1}-1,\hfill \\ 0\hfill & otherwise.\hfill \end{array}\right.$$
- (d)
- The Lévy measure of the GGC mixing distribution has a density which has the following explicit forms:If $\varphi =1$, then$$\ell \left(x\right)=(\theta /\gamma )\mathrm{\Gamma}(\gamma +1)U(\gamma ,0,x).$$If $\gamma =1$, then$$\ell \left(x\right)=\theta {e}^{x}\left[{x}^{-1}{e}^{-x/\varphi}-{E}_{1}(x/\varphi )\right].$$

**Remark**

**10.**

**Proof.**

**Remark**

**11.**

**Discussion**

**1.**

## 5. Branching Process Models

**Theorem**

**7.**

**Proof.**

**Remark**

**12.**

## 6. Some Other Mutant Number Distributions

**Theorem**

**8.**

**Proof.**

- The probability of a mutation during $(t,t+dt)$ is $\varphi \left(t\right)dt$, where $\varphi \left(t\right)=r{N}_{t}$, but otherwise is arbitrary;
- A mutation occurring at time t induces a growing clone of size ${C}_{t}$ at the time of plating/observation. Define $p(j,t)=P({C}_{t}=j)$; and
- Mutations are classifies as type j if ${C}_{t}=j$. The number of type j mutations in a single culture is denoted by ${M}_{j}$, a random variable having a Poisson$\left({\lambda}_{j}\right)$ distribution where$${\lambda}_{j}={\int}_{0}^{T}p(j,t)\varphi \left(t\right)dt,$$

**Theorem**

**9.**

**Proof.**

**Lemma**

**1.**

**Proof**

**1**

## Funding

## Data Availability Statement

## Conflicts of Interest

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Pakes, A.G.
Mutant Number Laws and Infinite Divisibility. *Axioms* **2022**, *11*, 584.
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Pakes AG.
Mutant Number Laws and Infinite Divisibility. *Axioms*. 2022; 11(11):584.
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2022. "Mutant Number Laws and Infinite Divisibility" *Axioms* 11, no. 11: 584.
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