# Universal Asymptotic Clone Size Distribution for General Population Growth

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## Abstract

Deterministically growing (wild-type) populations which seed stochastically developing mutant clones have found an expanding number of applications from microbial populations to cancer. The special case of exponential wild-type population growth, usually termed the Luria–Delbrück or Lea–Coulson model, is often assumed but seldom realistic. In this article, we generalise this model to different types of wild-type population growth, with mutants evolving as a birth–death branching process. Our focus is on the size distribution of clones—that is the number of progeny of a founder mutant—which can be mapped to the total number of mutants. Exact expressions are derived for exponential, power-law and logistic population growth. Additionally, for a large class of population growth, we prove that the long-time limit of the clone size distribution has a general two-parameter form, whose tail decays as a power-law. Considering metastases in cancer as the mutant clones, upon analysing a data-set of their size distribution, we indeed find that a power-law tail is more likely than an exponential one.

## Keywords

Luria–Delbrück Branching process Clone size Cancer## 1 Introduction

Cancerous tumours spawning metastases, bacterial colonies developing antibiotic resistance or pathogens kickstarting the immune system are examples in which events in a primary population initiate a distinct, secondary population. Regardless of the scenario under consideration, the number of individuals in the secondary population, and how they are clustered into colonies, or *clones*, is of paramount importance. An approach which has offered insight has been to bundle the complexities of the initiation process into a mutation rate and assume that the primary, or *wild-type*, population seeding the secondary, or *mutant*, population is a random event.

This method was pioneered by microbiologist Salvador Luria and theoretical physicist Max Delbrück (Luria and Delbrück 1943). In their Nobel prize winning work, they considered an exponentially growing, virus susceptible, bacterial population. Upon reproduction, with small probability, a virus resistant mutant may arise and initiate a mutant clone. This model was contrasted with each wild-type individual developing resistance upon exposure to the virus with a constant probability per individual. By considering the variance in the total number of mutants in each case, they demonstrated that bacterial evolution developed spontaneously as opposed to adaptively in response to the environment.

In the original model of Luria and Delbrück, both wild-type and mutant populations grow deterministically, with mutant initiation events being the sole source of randomness. Lea and Coulson (1949) generalised this process by introducing stochastic mutant growth in the form of the pure birth process and were able to derive the distribution of the number of mutants for neutral mutations. This was again extended by Bartlett (1955) and later Kendall (1960), who considered both populations developing according to a birth process. An accessible review discussing these formulations is given by Zheng (1999).

Recent developments have focused on cancer modelling, where usually mutant cell death is included in the models. The main quantity of interest in these studies has been the total number of mutant cells. Explicit and approximate solutions appeared for deterministic, exponential wild-type growth, corresponding to a fixed size wild-type population (Angerer 2001; Dewanji et al. 2005; Iwasa et al. 2006; Komarova et al. 2007; Keller and Antal 2015), and fully stochastic wild-type growth either at fixed time or fixed size (Durrett and Moseley 2010; Antal and Krapivsky 2011; Kessler and Levine 2015). An exciting recent application has been to model emergence of resistance to cancer treatments (Kessler et al. 2014; Bozic et al. 2013; Bozic and Nowak 2014). The current study continues in this vein with our inspiration being primary tumours (wild-type) seeding metastases (mutant clones).

Interestingly, in the large time small mutation rate limit, the clone size distribution at a fixed wild-type population size coincides for stochastic and deterministic exponential wild-type growth (Kessler and Levine 2015; Keller and Antal 2015). The intuition behind this observation is that a supercritical birth–death branching process converges to exponential growth in the large time limit, and, for a small mutation rate, mutant clones are initiated at large times. So asymptotically the two methods are equivalent, but the deterministic description of the wild-type population has twofold advantages: (i) the calculations are much simpler in this case (Keller and Antal 2015), and (ii) the method can be easily generalised to arbitrary growth functions. This is the programme that we develop in the present paper.

The present work differs from previous approaches in two ways. Firstly, motivated by populations with environmental restrictions, we move away from the assumption of exponential wild-type growth, a setting which has received limited previous consideration as discussed in Foo and Michor (2014). We shall first review and extend results for the exponential case and then provide explicit solutions for power-law and logistic growth. Next, we present some general results which are valid for a large class of growth functions. This extends the classic results found in Kendall (1948), Athreya and Ney (2004), Karlin and Taylor (1981), Tavare (1987) and recent work in Tomasetti (2012), Houchmandzadeh (2015) who considered the wild-type population growth rate to be time-dependent but coupled with the mutant growth rate. Secondly, rather than the total number of mutants, our primary interest is on the distribution of mutant number in the clones initiated by mutation events. This complements Hanin et al. (2006), which allowed deterministic wild-type and mutant growth, and the treatment of clone sizes for constant wild-type populations found in Dewanji et al. (2011). While we focus on clone sizes, we demonstrate that the distribution for the total number of mutants follows as a consequence, and hence, results hold in that setting also.

The outline of this work is as follows. We define our model in Sect. 2, utilising formalism introduced in Karlin and Taylor (1981), and demonstrate a mapping between the mutant clone size distribution and the distribution for the total number of mutants. The exact time-dependent size distribution is given for exponential, power-law and logistic wild-type growth function in Sect. 4. Section 5 pertains to universal features of the clone size distribution and contains our most significant results. There, for a large class of wild-type growth functions, we demonstrate a general two-parameter distribution for clone sizes at large times. The distribution has power-law tail behaviour which corroborates previous work (Iwasa et al. 2006; Durrett and Moseley 2010; Williams et al. 2016). Large time results are also given for the mean and variance of the clone sizes under general wild-type growth. Adopting the interpretation of the wild-type population as the primary tumour and mutant clones as metastases, we test our results regarding the tail of the distribution on empirical metastatic data in Sect. 6. Section 7 considers alternative methods to ours, and we give some concluding remarks in Sect. 8.

## 2 Model

### 2.1 The Birth–Death Process

*t*, with \(Z_{0}=1\). The forward Kolmogorov equation for the distribution is given by

*k*a geometric distribution with a modified zero term

### 2.2 Mutant Clone Size Distribution

*t*, and we let the number of wild-type individuals be denoted by \(n_{\tau }\) for \(0\le \tau \le t\). Since mutants are produced by wild-type individuals, the rate of mutant clone initiations will be proportional to the product of \(n_{\tau }\) and the mutation rate \(\mu \). More precisely, the process of clone initiations is an inhomogeneous Poisson process (Karlin and Taylor 1998) with intensity \(\mu n_{\tau }\). Let the Poisson random variable \(K_{t}\) denotes the number of clones that have been initiated by

*t*, which has mean

*T*, and as we must have \(T\le t\), the density of

*T*is given by

We make the following remarks on the above. (i) The mutation rate \(\mu \) does not appear in the density for initiation times in (6); hence mutant clone sizes are independent of the mutation rate and thus all following results regarding clone sizes will be also. (ii) The integral in (8) is a convolution, and as convolutions commute, we may swap the arguments of the integrand functions (\(n_{\tau }{\mathcal {Z}}_{t-\tau }\leftrightarrow n_{t-\tau }{\mathcal {Z}}_{\tau }\)). (iii) If we start with \(n_{0}\) wild-type individuals, so the wild-type follows \(m_{\tau }=n_{0}n_{\tau }\), then both the numerator and denominator in (6) will have a factor of \(n_{0}\), which cancel. So henceforth, apart from when \(n_{0}=0\) (used occasionally for analytic convenience), we set \(n_{0}=1\) without loss of generality. (iv) By similar logic, a positive random amplitude for the wild-type growth function, i.e. \(m_{\tau }=Xn_{\tau }\) for a general positive random variable *X*, would also cancel, and so our results on clone sizes hold in that case also.

## 3 Mapping Distributions: Clone Size to Total Mutant Number

*t*. Then, \(B_{t}\) is the sum of \(K_{t}\) generic clones

*iid*random variables specifying the clone sizes. As such, \(B_{t}\) is a compound Poisson random variable, and hence its generating function is

*Large Population-Small Mutation*limit (Keller and Antal 2015), where \(\theta =\mu n_{t}\) is kept constant. Then, for exponential wild-type growth, \(n_{\tau }=e^{\delta \tau },\) (or exponential-type, see Sect. 5), the expected number of initiated clones, \({\mathbb {E}}(K_t)\), tends to \(\theta /\delta \) for large times. Hence, we see that

### Proposition 1

## 4 Finite Time Clone Size Distributions

### 4.1 Exponential Wild-Type Growth

### 4.2 Power-Law Wild-Type Growth

*i*defined in “Appendix A”. Details of the derivation are given in “Appendix C”. For immortal mutants, the mass function may be explicitly written as

Note for \(\rho \ge 1\), \(n_{0}=0\) which, while useful for analytic tractability, is unrealistic. This can be overcome by letting \(n_{\tau }=n_{0}+\tau ^{\rho }\). Then, by splitting the integral in the generating function (9) and using the above analysis, one can obtain the mass function for any \(n_{0}\). However, for practical purposes, the contribution of \(n_{0}\) is negligible.

### 4.3 Constant Size Wild-Type

*s*we obtain the probabilities

Constant immigration may imply a constant size source; hence, mutants with equal birth and death rates (i.e. evolving as a critical branching process) are particularly interesting. This case yields analogous formulas to those above but \({\mathcal {S}}_t\) is replaced with the expression given in (5).

### 4.4 Logistic Wild-Type Growth

Starting from a population of one and having a carrying capacity *K*, logistic growth is given by \(n_{\tau }=\frac{K e^{\lambda \tau }}{K+e^{\lambda \tau }-1}\). We assume neutral mutations, i.e. \(\lambda \) is also the wild-type growth rate. Integrating the growth function gives \(a_{t}=\frac{K}{\lambda }\log \big (\frac{e^{\lambda t}}{n_{t}}\big ).\)

*C*is an integration constant. Therefore, the generating function is

### 4.5 Monotone Distribution and Finite Time Cut-Off

We conclude this section by demonstrating general features that exist in the clone size distribution at finite times. Again proofs are provided in “Appendix C”. Firstly, we see that, regardless of the particular wild-type growth function, the monotone decreasing nature of the mass function for the birth–death process is preserved in the clone size distribution.

### Proposition 2

As long as \(n_{\tau }\) is positive for some subinterval of [0, *t*], then for \(k\ge 1\) we have \( {\mathbb {P}}(Y_{t}=k+1)< {\mathbb {P}}(Y_{t}=k)\) for any finite \(t>0\).

Whether \( {\mathbb {P}}(Y_{t}=0)\ge {\mathbb {P}}(Y_{t}=1)\) depends on \(n_{\tau }\) and *t*, but the inequality is typically true for long times. Note that in contrast, the mass function of the total number of mutants is not monotone in general (Keller and Antal 2015).

Now restricting ourselves to the \(\lambda >0\) case, as an example, consider the mass function when the size of the wild-type population is constant, which is given by (16), and specifically for \(k\ge 1\). For any moderate *t*, \({\mathcal {S}}_{t}^{-1}\) is typically close to unity but for large *k*, \({\mathcal {S}}_{t}^{-k}\) will become the dominant term in the mass function, dictating exponential decay. We term this a cut-off in the distribution which occurs at approximately \(k= O (e^{\lambda t})\). It is an artefact of the mass function for the birth–death process (3). Hence, we will have (at least) two behaviour regimes for the mass function for finite times. Here, we show that this cut-off exists generally for finite times.

### Theorem 1

Note that \({\mathcal {S}}_{t}>1\) for finite *t*, and \({\mathcal {S}}_{t}\rightarrow 1\) exponentially fast for large *t*. Hence, the cut-off will disappear for long times and the subexponential factor, discussed in detail in Sect. 5, will completely determine the tail of the distribution. Also notice that the power-law cases, \(n_{\tau }=\tau ^{\rho }\), for \(\rho \ge 1\) are not covered as, to make the analysis tractable, they artificially start at \(n_{0}=0\). However, the generating function in this case (13) also has its closest to origin singularity at \({\mathcal {S}}_{t}\) so the cut-off exists there also.

## 5 Universal Large Time Features

Here, we give results regarding the large time behaviour of our model which is relevant in many applications and also provides general insight. In many applications, the cut-off location (\(k=O(e^{\lambda t}\))) is so large that the distribution at or above this point is of little relevance, and hence, for this purpose the limiting approximations now discussed are of particular interest. Using the notation of Theorem 1, this section investigates the large time form of \(\varTheta _{t}(k)\). The proofs for the results presented in this section can be found in “Appendix D”. We highlight the power-law decaying, “fat” tail found in each case. Henceforth, we again assume \(\lambda >0\), i.e. a supercritical birth–death process.

### 5.1 General Wild-Type Growth Functions

To give general results, we introduce the following assumption which will be assumed to hold for the remainder of this section.

### Assumption 1

- (i)
\(n_{\tau }=0\) for \(\tau <0\), continuous for \(\tau > 0\) and right continuous at \(\tau =0\).

- (ii)
\(n_{\tau }\) is positive and monotone increasing for \(\tau > 0\).

- (iii)
For \(x\ge 0\) the limit \(\lim _{t\rightarrow \infty }n_{t-x}/n_{t}\) exists, is positive and finite.

We note that the cases discussed in Sect. 4 are all covered by Assumption 1. The reason for the seemingly arbitrary limit assumed in (iii) becomes clear with the following result which is an application of the theory of regular variation found in Bingham et al. (1987).

### Lemma 1

Often the long-time behaviour of the clone size distribution may be separated into \(\delta ^*>0\) and \(\delta ^*=0\), and so we give the following definition (Flajolet and Sedgewick 2009).

### Definition 1

*f*(

*x*) such that

*f*(

*x*) is of

*exponential-type*for \(\delta ^*\ne 0\) and is

*subexponential*for \(\delta ^*=0\).

Simple examples of subexponential functions are \(e^{\sqrt{t}},\, \log (t)\), \(t^{\rho }\), while \(e^{\delta t}\), \(e^{\delta t}t^{\rho }\) are of exponential-type, with \(\delta ,\rho \in {\mathbb {R}}\).

### 5.2 Mean and Variance

We now address the asymptotic properties of the clone size distribution by first discussing its mean and variance.

### Theorem 2

### 5.3 Large Time Clone Size Distribution

Turning to the distribution function, we have the following result regarding the generating function at large times.

### Theorem 3

### Corollary 1

The case of immortal mutants does not simplify the above expressions for subexponential growth, but for exponential-type growth, by applying (23) then (22) to the limiting generating function, we have the following link to the Yule-Simon distribution which appears often in random networks (Simon 1955; Krapivsky and Redner 2001).

### Corollary 2

### Corollary 3

### 5.4 Large Time Distribution for Total Number of Mutants

Finally, to conclude this section, we give the corresponding results for the total number of mutants \(B_{t}\) in the often used *Large Population-Small Mutation* limit.

### Theorem 4

## 6 Tail Behaviour in Empirical Metastatic Data

Given the above discussion we expect, for a large class of wild-type growth functions, to see power tail behaviour on approach to the exponential cut-off in the clone size distribution. We take the first steps to verify this theoretical hypothesis by analysing an empirical metastatic data. In this setting, the wild-type population is the primary tumour and mutant clones are the metastases.

Our data are sourced from the supplementary materials in Bozic et al. (2013). These data are taken from 22 patients; 7 with pancreatic ductal adenocarcinomas, 11 with colorectal carcinomas, and 6 with melanomas. One patient had only a single metastasis so we discard this data. Of the 21 remaining patients, the number of cells in a single metastasis ranged from \(6\times 10^6\) to \(2.23\times 10^9\). Our theoretical model predicts a cut-off in the distribution around \(k=e^{\lambda t}\). Taking some sample parameters from the literature, namely \(\lambda =0.069\)/day (Diaz et al. 2012), and \(t=14.1\) years (Yachida et al. 2010), this leads to a cut-off around \(k\approx 10^{154}\) cells. Due to the enormity of this value, we ignore the cut-off here. Additionally, as the minimum observed metastasis size is \(6\times 10^6\) cells, we assume that all data points are sampled from the tail of the distribution.

*N*. We test the hypothesis that \(\mathbf y \) is drawn from a power-law distribution, \({\mathbb {P}}_{1}(Y_{t}=k)=C_{1}k^{-\omega }\), versus that it is sampled from a geometric distribution, \({\mathbb {P}}_{2}(Y_{t}=k)=C_{2}p(1-p)^{k}\), where \(C_{1},\,C_{2}\) are normalising constants and

*p*is the parameter for the geometric distribution. The log-likelihood ratio is

## 7 Alternative Approaches

### 7.1 Deterministic Approximation

However, despite this agreement, the densities given by (17) for specific wild-type growth function differ significantly compared with stochastic mutant proliferation. Letting \(Y_{t}^{\mathrm {Stoch}}\) be the clone size distribution with stochastic mutant growth and \(Y_{t}^{\mathrm {Det}}\) be its deterministic approximation specified by (17), we may quantify the approximation error, at least for the moments, by the following theorem, whose proof can be found in “Appendix F”.

### Theorem 5

### 7.2 Time-Dependent Rate Parameters

Some authors Houchmandzadeh (2015), Tomasetti (2012) have previously considered the case where all rates in the system are multiplied by a time-dependent function, say \(z(\tau )\). This is relevant in the scenario where both the wild-type and mutant populations have their growth restricted simultaneously by environmental factors, for example exposure to a chemotherapeutic agent. We observe that under a change of timescale this system is equivalent to our setting with exponential wild-type growth. This is due to the following argument.

### 7.3 Poisson Process Characterisation of Tail

Complementing Corollary 3 in Sect. 5, following Tavare (1987), we can also describe the size distribution for large clones at long times via a Poisson process in the following way. Let \((Z^{(i)}(t))_{i\ge 1}\) be independent copies of the birth–death process as in Sect. 2 and \((T_{i})_{i\ge 1}\subset (0,\infty )\) be the points of a of Poisson process with intensity \(\mu n_{\tau }\), for \(\tau \ge 0\). The \(T_{i}\) represent the clone arrival times, and so \(K_t\) is the number of \(T_{i}\) less than or equal to *t*.

*iid*. The random sequence \((e^{-\lambda T_{i}}W_{i})_{i\ge 1}\) takes non-negative real values; however, if we restrict our attention to only the positive elements (that is clones that do not die), then these can be taken to be points from a non-homogeneous Poisson process. More precisely, the set \(\{\sigma _{j}\}_{j\ge 1}:=\{e^{-\lambda T_{i}}W_{i}\}_{i\ge 1}\setminus \{0\}\) are the points (in some order) from a Poisson process on \((0,\infty )\) with mean measure

### Theorem 6

## 8 Discussion

In this study, we focus on the size distribution for mutant clones initiated at a rate proportional to the size of the wild-type population. The size of the wild-type population is dictated by a generic deterministic growth function, and the mutant growth is stochastic. This shifts the focus from previous studies which have mostly been concerned with exponential, or mean exponential, wild-type growth, and considered the total number of mutants. Results for the total number of mutants are, however, easily obtained from the clone size distribution.

The special cases of exponential, power-law and logistic wild-type growth were treated in detail, due to their extensive use in models for various applications. Utilising a generating function centred approach, exact time-dependent formulas were ascertained for the probability distributions in each case. Regardless of the growth function, the mass function is monotone decreasing and the distribution has a cut-off for any finite time. This cut-off goes to infinity for large times and is often enormous in practical applications; hence, we focused on the approach to the cut-off.

We found that the clone size distribution behaves quite distinctly for exponential-type versus subexponential wild-type growth. Although the probability of finding a clone of any given size stays finite as \(t\rightarrow \infty \) for exponential-type growth, it tends to zero for subexponential type. Despite these differences, with a proper scaling, for a large class of growth functions, we proved that the clone size distribution has a universal long-time form. This long-time form possesses a power-law “fat” tail which decays as 1 / *k* for subexponential wild-type growth, but faster for exponential-type growth. This can be intuitively understood as the tail distribution represents clones that arrive early, and the chance that a clone is initiated early in the process is larger for a slower growing wild-type function. Hence, we expect a “fatter” tail in the subexponential case.

Note that although we consider the case of subexponential wild-type growth, surviving mutant clones will grow exponentially for large time, which can be unrealistic in some situations. Stochastic growth which accounts for environmental restrictions, for instance the logistic branching process, introduces further technical difficulties and is left for future work. We do note that, despite the drawbacks of deterministic mutant growth as discussed in Sect. 7, when both the wild-type and mutant populations grow deterministically as \(\tau ^{\rho }\), it is easy to see that for large times the clone size distribution still displays a power-law tail, \( \lim _{t\rightarrow \infty } t f_{Y_t}(y) = \frac{\rho +1}{\rho }y^{1/\rho -1}. \)

An underlying motivation for this work is the scenario of primary tumours spawning metastases in cancer. We test our hypothesis regarding a power-law tail in metastasis size distributions by analysing empirical data. For 19 of 21 data-sets, the power-law distribution is deemed more likely than an exponentially decaying distribution. The exponent of the power-law decay was estimated in each case and found to lie between \(-1\) and \(-2\). Interpreting this in light of our theory, either the primary tumour had entered a subexponential growth phase or, if one assumes exponential primary growth, the metastatic cells had a fitness advantage compared to those in the primary. Either way we can conclude that, for the majority of patients, the metastases grew faster than the primary tumour.

## Notes

### Acknowledgments

We thank Peter Keller, Paul Krapivsky, Martin Nowak, Karen Ogilvie, Bartlomiej Waclaw and Bruce Worton for helpful discussions. MDN acknowledges support from EPSRC via a studentship.

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