An Accurate Approximation for the Expected Site Frequency Spectrum in a Galton-Watson Process under an Infinite Sites Mutation Model

Abstract

If viruses or other pathogens infect a single host, the outcome of infection often hinges on the fate of the initial invaders. The initial basic reproduction number R₀, the expected number of cells infected by a single infected cell, helps determine whether the initial viruses can establish a successful beachhead. To determine R₀, the Kingman coalescent or continuous-time birth-and-death process can be used to infer the rate of exponential growth in an historical population. Given M sequences sampled in the present, the two models can make the inference from the site frequency spectrum (SFS), the count of mutations that appear in exactly k sequences (k = 1,2,…,M). In the case of viruses, however, if R₀ is large and an infected cell bursts while propagating virus, the two models are suspect, because they are Markovian with only binary branching. Accordingly, this article develops an approximation for the SFS of a discrete-time branching process with synchronous generations (i.e., a Galton-Watson process). When evaluated in simulations with an asynchronous, non-Markovian model (a Bellman-Harris process) with parameters intended to mimic the bursting viral reproduction of HIV, the approximation proved superior to approximations derived from the Kingman coalescent or continuous-time birth-and-death process. This article demonstrates that in analogy to methods in human genetics, the SFS of viral sequences sampled well after latent infection can remain informative about the initial R₀. Thus, it suggests the utility of analyzing the SFS of sequences derived from patient and animal trials of viral therapies, because in some cases, the initial R₀ may be able to indicate subtle therapeutic progress, even in the absence of statistically significant differences in the infection of treatment and control groups.

1. Introduction

The theory in this article is strongly motivated by the practical observation that in infection, the success of the invasive population often hinges on the fate of the first arrivals. In the viral infection of a single host, the invasive population often descends from a small set of founder viruses. In some cases, it even descends from a single founder, e.g., about 80% of human immunodeficiency virus (HIV) infections have a single founder (Keele et al. 2008, Haaland et al. 2009, Love et al. 2016). The initial basic reproduction number R₀, in viral infection the expected number of cells that a single infected cell infects in the next generation (Giorgi et al. 2010), contributes fundamentally to the chances of an invasive population establishing a successful beachhead. On one hand, if R₀ < 1, the invasive population falls below replacement and dies. On the other hand, if R₀ is slightly greater than 1, the invaders have a small but positive chance of survival, and if R₀, is large, the invaders are likely to flourish. Thus, whether preventing infection or mitigating its impact by reducing an initial viral load, the initial R₀, the basic reproduction number at the start of infection, could in principle provide a measure for setting therapeutic goals and benchmarking therapeutic progress.

Presently, if a researcher wishes to assess therapeutic progress by demonstrating a decrease in host infectability, only one type of datum has direct pertinence, i.e., the binary datum corresponding to whether or not infection occurs (Pegu et al. 2013, Strbo et al. 2013, Gordon et al. 2016). Animal trials testing viral therapies have therefore developed ingenious designs (Regoes et al. 2005, Nolen et al. 2015), primarily to squeeze binary data for the statistical power required to detect subtle changes in infectability. An estimate of R₀ would provide an additional measure of therapeutic success, beyond the binary datum of infection or viral extinction. A quantitative analysis estimating the initial R₀ would therefore help bypass a stubborn bottleneck in the systematic development of viral therapies.

Unfortunately, the initial R₀ is usually not directly accessible, because the initial viral population is typically latent, i.e., below the threshold of detectability. Although the literature for HIV gives the estimates R₀ ≈ 6 (Stafford et al. 2000) or R₀ ≈ 8 (Ribeiro et al. 2010), e.g., these estimates pertain to viremia (i.e., after HIV becomes detectable in blood) (Fiebig et al. 2003), which starts on average about 10 days after the founders infect (Kahn and Walker 1998). The lower limit of HIV detection in blood is about 20 viruses/ml (Kosaka et al. 2017), so a viremia implies that the total blood volume (5L) contains at least about 10⁵ viruses. When HIV is detected in blood, therefore, the viral invasion has long since secured its beachhead. Indeed, if current estimates (R₀ ≈ 6 or R₀ ≈ 8) of the basic reproduction number in viremia were pertinent during the initial infection, a single infected cell would probably ensure successful infection. Note, however, that the invasive routes with the highest estimates of per-act HIV transmission risk are (per 10,000 exposures) blood transfusion (9250), mother-to-child transmission (2260), and receptive anal intercourse (138). Other routes have an HIV transmission risk less than 63 per 10,000 exposures (see Table 1 in (Patel et al. 2014)). Because HIV transmission is so uncertain, a typical HIV founder virus probably faces a much more hostile initial environment than the current direct estimates of R₀ in viremia suggest.

Although detection thresholds prevent direct measurement of the initial R₀, genetic researchers have shown that the nucleic acids of present-day humans retain footprints from the population history (e.g., (Durrett and Limic 2001, Durrett 2008)). Similarly, viral sequences sampled during early viremia may be informative about the initial R₀. Figure 1 illustrates the concept, and the remainder of this article refines the qualitative insight there. In particular, Figure 1 suggests that mutations appearing in two or more sampled sequences contain information about the initial R₀ of an expanding population. Although the viral applications motivate the analysis, the theory presented here has broad applicability to inferring the early demographics of populations with very few founders.

Figure 1: Two viral ancestries, illustrating the effect of the initial R₀ on samples.

Figure 1 illustrates two hypothetical viral ancestries. The circles represent viruses. In the two ancestries, time runs downward, and for simplicity each viral population at the bottom has a single founder at the top (multiple founders introduce technical complications, but do not change the concept illustrated). The present population at the bottom contains some large number Z of individuals, from which M sequences are sampled. The ancestry on the left has a large initial R₀, so the founder has Z₁ daughters, where for illustration, we assume Z₁ is much larger than M (M − 1) / 2. Thus, any two of the M sampled sequences are likely to descend from different daughters (many standard references on the “Birthday problem” implicitly prove this statement). If so, the sampled sequences cannot share any mutations away from the founder sequence. In contrast, the ancestry on the right has a small initial R₀, so the founder has only Z₁ = 2 daughters. If either daughter’s sequence has a mutation away from the founder sequence, about half the M sampled sequences share the mutation. Thus, mutations appearing in more than one of the M sampled sequences are informative about the initial R₀. Sampled mutations are conveniently partitioned into a “site frequency spectrum”, i.e., the numbers η_m of the sites where a mutation appears in exactly m of the M sampled sequences (m = 1,2,…,M).

Thus, given M aligned viral sequences sampled simultaneously from a single infected host, the aim of this article is to reconstruct as a function of R₀ the site frequency spectrum (SFS) described in Figure 1. Before proceeding to the mathematical abstractions of Section 2, we first establish the parameter ranges relevant to an important application. In studies sampling HIV gp120 sequences from patients, M typically ranged from 16 to 30 per patient (Lee et al. 2009). The gp120 gene is about 2550 nt long, and (with crossovers neglected) each HIV replication averages ε ≈ 2.16 × 10⁻⁵ point mutations/base/replication (Giorgi et al. 2010). On average, therefore, each RNA replication entails μ ≈ 0.0551 mutations in gp120. (The three significant figures represent an unrealistic precision but retain consistency with the literature.)

Studies in human genetics suggest some methods for estimating R₀ from the SFS in gp120 sequence data, but the methods are not directly suited to viral infection in a single host. To elaborate, consider the following idealized model of an initial HIV infection, focusing first on a typical initial infected cell in a host. HIV lyses the cell, releasing about 10³ – 10⁴ viral particles (Chen et al. 2007, De Boer et al. 2010). As noted above, the initial R₀ is likely smaller than R₀ = 8, so a typical viral particle has a miniscule chance of infecting. Given an initial environment, if viral particles infect independently, the count of infected daughter cells approximately follows an Poisson distribution whose mean equals the unknown initial R₀ (see, e.g., (Arratia et al. 1989a, Arratia et al. 1989b)). Thus, the viruses from the initial infected cell infect Z₁, daughter cells, where Z₁ is a Poisson random variate with mean R₀. Estimates of the replication time for HIV range from 1.76 days to 4.2 days (Love et al. 2016), 2 days being a reasonable approximation (Markowitz et al. 2003). We therefore model the time-intervals between the lysis of a mother cell and her infected daughter cells as independent random variates with a gamma distribution, approximating the mean by 2 days and the standard deviation by 0.24 = 2 – 1.76 days (1.76 days is the minimum estimate of the HIV replication time, so the resulting gamma distribution is likely less tightly concentrated around its mean than the true random HIV replication time). The gamma distribution approximating the random HIV replication time therefore has shape and rate parameters (n, λ) = (69.4, 34.7) (i.e., its mean nλ⁻¹ = 2 and variance nλ⁻² ≈ 0.24²). The reproductive cycle begins anew with the lysis of each infected daughter. This idealized but relatively realistic model of HIV reproduction is a Bellman-Harris process (Bellman and Harris 1948); call it the “Gamma model”, to emphasize that its parameters and distribution are chosen to model HIV (see Figure 2).

Figure 2: A diagram schematically illustrating the Gamma and Delta models.

Figure 2 illustrates two hypothetical models of viral ancestries. As in Figure 1, black circles represent the lysis of infected cells, and each ancestry starts from a single founder at the top, with time running downward. The numbers in the circles count the generations away from the founder, with the single founder in generation 0. In the Gamma model on the left, the lysis of each infected cell gives birth to a random number of viral daughters whose progeny lyse cells at random times chosen independently from a gamma distribution. The graph on the extreme left illustrates the relevant gamma distribution. The Delta model on the right is like the Gamma model, except that each cell lysis gives birth to viral daughters who lyse their cells simultaneously, in synchronous generations at time-intervals equal to the mean generation time in the Gamma model.

The theory in Section 2 exploits the Delta model illustrated in Figure 2 (a Galton-Watson branching process, in discrete-time) to provide analytic approximations to the SFS of the Gamma model. To make the Delta model comparable to the Gamma model, the count of daughters in the Delta model is also a Poisson variate with mean R₀, and the deterministic replication time in the Delta model is 2 days, the mean replication time in the Gamma model. As the models’ names suggest, the synchronous generations in the Delta model therefore substitute a Dirac delta distribution (i.e., a distribution having a single atom of probability 1, a distribution as tightly concentrated around its mean as possible) for the gamma distribution in the Gamma model. For comparison with simulations of the Gamma model, a birth-and-death process with constant birth and death intensities and the Kingman coalescent model with an exponentially expanding population also provide analytic expressions for the SFS. All approximations in this article are uncontrolled, so (often without comment) we rely on the simulations of the Gamma model with parameters relevant to HIV gp120 to assess the accuracy of SFS approximations. Other parameters regimes, which require separate assessment, are beyond the immediate practical purview of this article.

The organization of this article is as follows. Section 2 (Theory) approximates the mean SFS in the Delta model as a function of the basic reproduction number R₀ and compares it to the SFS derived from the Kingman coalescent for an expanding population or from continuous-time birth-and-death processes. Section 3 (Methods) describes simulation of the Gamma and Delta models. It also discusses the dependence of the SFS variance on the mutation rate. Section 4 (Results) examines the accuracy of the various approximations when the simulations use the parameters relevant to HIV gp120. Section 5 is our Discussion, which examines the few (superable) difficulties impeding direct data analysis with the theory presented here.

2. Theory

An Approximation for the Site Frequency Spectrum of a Galton-Watson Process

With the Delta model of Figure 2 in mind, consider a discrete-time branching process starting with one individual in the generation 0. Let ${\tilde{\mathcal{G}}}_g$ denote generation g of the branching process (g = 0,1,…), let ${\tilde{Z}}_g$ count the individuals in ${\tilde{\mathcal{G}}}_g$ (e.g., ${\tilde{Z}}_0=1$), let ${\tilde{\mathcal{I}}}_{g,i}\in {\tilde{\mathcal{G}}}_g$ denote individual i in generation g (i = 1,2,…, ${\tilde{Z}}_g$; g = 0,1,…), and let ${\tilde{\mathcal{I}}}_{g,i}$ have ${\tilde{X}}_{g,i}$ daughters. Thus, ${\tilde{Z}}_{g+1}={\sum }_{i=1}^{{\tilde{Z}}_g}{\tilde{X}}_{g,i}$. The $\left\{{\tilde{X}}_{g,i}\right\}$ are mutually independent, identically distributed non-negative integer random variates, i.e., the process is a Galton-Watson process.

To streamline the subscripted notation R₀ for mathematical manipulation, define $r=\mathbb{E}{\tilde{Z}}_1=R_0$, e.g., $\mathbb{E}{\tilde{Z}}_g=r^g$. The case r > 1 (a supercritical Galton-Watson process) is of greatest interest in the viral application, because (as the Introduction indicates) the sampling in viremia entails a large viral population. Let $\tilde{P}(s)=\mathbb{E}{s}^{{\tilde{Z}}_1}$ denote the probability generating function (pgf) of the number of daughters in ${\tilde{\mathcal{G}}}_1$, so $r=\mathbb{E}{\tilde{Z}}_1={\tilde{P}}^{\prime }(1)$. The Galton-Watson process goes extinct if ${\tilde{Z}}_g={\tilde{Z}}_{g+1}=\dots =0$ for some g. The extinction probability 0 ≤ q < 1 satisfies $q=\tilde{P}(q)$ (i.e., an individual’s lineage goes extinct if and only if all its daughters’ lineages go extinct).

The Delta model is a Galton-Watson process where the $\left\{{\tilde{X}}_{g,i}\right\}$ share a Poisson distribution with mean R₀, so the relevant pgf is

$${\tilde{P}}_{\Delta }(s)=\tilde{P}(s)=\mathbb{E}{s}^{{\tilde{Z}}_1}=\sum _{k=0}^{\infty }\mathbb{P}\left({\tilde{Z}}_1=k\right)s^k=\sum _{k=0}^{\infty }{e}^{-r}\frac{r^k}{k!}{s}^k=e^{-r}\sum _{k=0}^{\infty }\frac{{(\mathit{rs})}^k}{k!}=e^{r(s-1)}.$$ (1)

Perhaps predictably, the approximations below depend only on $r={\tilde{P}}^{\prime }(1)$, and no other feature of $\tilde{P}(s)$.

Fix some large generation G (where G → ∞ later), and let ${\tilde{\mathcal{I}}}_{g,i}$, have ${\tilde{Z}}_{g,i;G}$ descendants in ${\tilde{\mathcal{G}}}_G$ (g = 0,1,…, G; $i=1,2,\dots ,{\tilde{Z}}_g$). For example, ${\tilde{Z}}_{G,i;G}=1$ for every $i=1,2,\dots ,{\tilde{Z}}_G$ (because by convention, every individual in ${\tilde{\mathcal{G}}}_G$ is her own descendant), and ${\sum }_{i=1}^{{\tilde{Z}}_g}{\tilde{Z}}_{g,i;G}={\tilde{Z}}_G$ for every g = 0,1,…,G (because every individual in ${\tilde{\mathcal{G}}}_G$ has exactly one ancestor in ${\tilde{\mathcal{G}}}_g$). Sample M individuals with replacement from ${\tilde{\mathcal{G}}}_G$, and let ${\tilde{S}}_{g,i;G}$ count the descendants of ${\tilde{\mathcal{I}}}_{g,i}$ in the sample from generation G. (Sampling with replacement is mathematically more convenient than sampling without replacement, but as with other uncontrolled approximations here, simulation results are required to assess its accuracy.) Each sampled individual has exactly one ancestor in ${\tilde{\mathcal{G}}}_g(g=0,1,\dots ,G)$, so the sample from ${\tilde{\mathcal{G}}}_G$ implicitly samples ancestors from ${\tilde{\mathcal{G}}}_g$, with ${\sum }_{i=1}^{{\tilde{Z}}_g}{\tilde{S}}_{g,i;G}=M$ for every g = 0,1,…,G.

If the Galton-Watson process survives long enough to be sampled at some sufficiently large G (in the Introduction, e.g., the viral population has become detectable in blood), all sampled ancestors in early generations had lineages that for any practical purpose have survived forever. For brevity, call ancestors whose lineages survive forever “immortal”; all others, “doomed”. As another uncontrolled approximation, delete all doomed individuals from the Galton-Watson process. The individuals remaining, if any, form another Galton-Watson process, the so-called skeleton process of immortal individuals.

To rephrase, if an individual is doomed, for G large enough samples from ${\tilde{\mathcal{G}}}_G$ never contain any of her descendants. To focus on relevant individuals, therefore, we replace the original Galton-Watson process with the skeleton process and drop the over-tildes from the associated quantities ${\mathcal{I}}_{g,i}$, Z_g, Z_g,i;G, S_g,i;G, etc. To avoid the triviality of an empty skeleton process, all probabilities and expectations below implicitly condition on survival of the original process. The Appendix shows that the skeleton process has the same basic reproduction number $r={\tilde{P}}^{\prime }(1)=P^{\prime }(1)$ as the original process. Because the approximations below depend only on the basic reproduction number r, dropping over-tildes and replacing the original process with the skeleton process does not impact the analysis.

To introduce genealogical conventions into the skeleton process, for linguistic convenience let ${\mathcal{I}}_{g,i}$ be both her own ancestor and her own descendant. In contrast, let the strict partial order ${\mathcal{I}}_{g,i}<{\mathcal{I}}_{g^{\prime },i^{\prime }}$ denote that ${\mathcal{I}}_{g,i}$ is an ancestor of ${\mathcal{I}}_{g^{\prime },i^{\prime }}$ and g < g′. If ${\mathcal{I}}_{g,i}<{\mathcal{I}}_{g^{\prime },i^{\prime }}$, and g′ = g + 1, therefore, ${\mathcal{I}}_{g,i}$ is the mother of ${\mathcal{I}}_{g^{\prime },i^{\prime }}$ and ${\mathcal{I}}_{g^{\prime },i^{\prime }}$ is the daughter of ${\mathcal{I}}_{g,i}$.

To introduce mutations into the genealogy of the skeleton process, note that a daughter may differ in nucleic acid sequence from her mother. To compare the sequence differences, conceptually we align the relevant sequences from every individual ${\mathcal{I}}_{g,i}$ (g = 0,1,…; i = 1,2,…,Z_g). Let the Poisson random variate ξ_g,i (e.g., (Giorgi et al. 2010)) count the novel point mutations in ${\mathcal{I}}_{g,i}$, namely, the alignment columns where ${\mathcal{I}}_{g,i}$ has a different letter from her mother (e.g., ξ_g,i = 0 if ${\mathcal{I}}_{g,i}$ and her mother have the same nucleic acid sequence). Assume an infinite sites model (Kimura 1969), which restricts the mutations allowed: if ${\mathcal{I}}_{g,i}$ and her mother differ in alignment column j (i.e., ${\mathcal{I}}_{g,i}$ bears a novel mutation in column j), no other mother-daughter pair differs in column j. Thus, each column contains at most two different letters, and all individuals bearing a mutation (relative to ${\mathcal{I}}_{0,1}$, the founder) descend from a single ancestor ${\mathcal{I}}_{g,i}$. Assume that $\mathbb{E}{\xi }_{g,i}=\mu$, i.e., the expectation of the count of novel mutations in a daughter sequence is some fixed constant μ.

Define an indicator random variate for k-fold sampling of the ancestor ${\mathcal{I}}_{g,i}$ (g = 0,1,…,G; i = 1, 2,…, Z_g): I_g,i;k = 1 if S_g,i;G = k; and I_g,i;k = 0 otherwise. With the theoretical preliminaries in hand, consider now the random variate

$${\eta }_m=\sum _{g=1}^G\sum _{i=1}^{Z_g}{\xi }_{g,i}{I}_{g,i;m}$$ (2)

(m = 1,…,M), which counts novel mutations over all individuals ${\mathcal{I}}_{g,i}$, but only those novel mutations displayed by exactly m of the M sampled individuals. Under the infinite sites model, each novel mutation occurs in a different column, so from the perspective of the sampled sequences, Eq (2) counts all columns containing m letters mutated away from the founder, i.e., η_m counts the distinct mutations that appear in exactly m sampled sequences. Thus, η = (η₁, η₂,…, η_M) constitutes the SFS in Figure 1.

The approximation in Eq (7) below for the expected SFS $\mathbb{E}\eta =(\mathbb{E}{\eta }_1,\mathbb{E}{\eta }_2,\dots ,\mathbb{E}{\eta }_M)$ is our central result, derived as follows. As in the so-called Delta method (Ver Hoef 2012), expand

$$f(X)=f(\mathbb{E}X)+f^{\prime }(\mathbb{E}X)(X-\mathbb{E}X)+\frac{1}{2}{f}^{″}(\mathbb{E}X){(X-\mathbb{E}X)}^2+\cdots$$ (3)

into a Taylor series around $\mathbb{E}X$. Truncate the resulting series before $\frac{1}{2}{f}^{″}(\mathbb{E}X){\sigma }^2(X)$ to yield a linear function, and take expectations to yield the uncontrolled approximation $\mathbb{E}f(X)\approx f(\mathbb{E}X)$. As a preliminary,

$$\mathbb{E}\left[\frac{Z_{g,i;G}}{Z_G}\right]=\mathbb{E}\left[\frac{1}{Z_g}\sum _{i=1}^{Z_g}\frac{Z_{g,i;G}}{Z_G}\right]=\mathbb{E}\left[\frac{1}{Z_g}\right]\approx \frac{1}{\mathbb{E}{Z}_g}=r^{-g},$$ (4)

where the first equality follows from the exchangeability of $\left\{{\mathcal{I}}_{g,i}:i=1,2,\dots ,Z_g\right\}$; the second, from the fact that every individual in the G -th generation has exactly one ancestor in the g -th generation (g ≤ G); and the approximation, from a linear truncation applied to f (X) = 1/ X. A second application of a linear truncation to f (X) = X^m (1−X)^M−m with X = Z_g,i;G / Z_G shows that

$$\mathbb{P}(S_{g,i;G}=m)=\mathbb{E}\left[\left(\begin{array}{c}\hfill M\hfill \\ \hfill m\hfill \end{array}\right){\left(\frac{Z_{g,i;G}}{Z_G}\right)}^m{\left(1-\frac{Z_{g,i;G}}{Z_G}\right)}^{M-m}\right]\approx \left(\begin{array}{c}\hfill M\hfill \\ \hfill m\hfill \end{array}\right){(r^{-g})}^m{(1-r^{-g})}^{M-m}.$$ (5)

Assume that the implicit sampling {S_g,i;G} in ${\mathcal{G}}_g$ depends only weakly on the size Z_g of ${\mathcal{G}}_g$.

$$\begin{gathered} \mathbb{E}{\eta }_m=\mathbb{E}\left[\sum _{g=1}^G\sum _{i=1}^{Z_g}{\xi }_{g,i}{I}_{g,i;m}\right]=\mu \sum _{g=1}^G\mathbb{E}\left[\sum _{i=1}^{Z_g}\mathbb{E}\left[I_{g,i;m}\mid Z_g\right]\right] \\ \approx \mu \sum _{g=1}^G\mathbb{E}\left[\sum _{i=1}^{Z_g}\mathbb{P}(S_{g,i;G}=m)\right]=\mu \sum _{g=1}^G{r}^g\mathbb{P}(S_{g,1;G}=m), \end{gathered}$$ (6)

where the second equality uses the mean $\mathbb{E}{\xi }_{g,i}=\mu$, the independence properties of the {ξ_g,i} (another model assumption), and a property of the expectations conditioned on Z_g; the third (approximate) equality depends on the above assumption of weak dependence; and the final equality uses both the Malthusian population growth $\mathbb{E}{Z}_g=r^g$ of the skeleton process and the exchangeability of $\left\{{\mathcal{I}}_{g,i}:i=1,2,\dots ,Z_g\right\}$ (so $\mathbb{P}\left(S_{g,i;G}=m\right)=\mathbb{P}\left(S_{g,1;G}=m\right)$).

Together, therefore, Eqs (2)–(6) approximate $\mathbb{E}{\eta }_m$ as a function of r : for m ≥ 1,

$$\mathbb{E}{\eta }_m\approx \mu \left(\begin{array}{c}\hfill M\hfill \\ \hfill m\hfill \end{array}\right)\sum _{g=1}^G{\left(r^{-g}\right)}^{m-1}{(1-r^{-g})}^{M-m}={\mathbb{E}}_{\Delta }{\eta }_m,$$ (7)

where the equality on the right of Eq (7) defines the (approximate) Delta SFS ${\mathbb{E}}_{\Delta }{\eta }_m$.

For m ≥ 2, convergence of the geometric series ${\sum }_{g=1}^{\infty }{\left(r^{-g}\right)}^{m-1}$ shows that the sum ${\mathbb{E}}_{\Delta }{\eta }_m$ in Eq (7) converges as G → ∞. For m = 1, ${\mathbb{E}}_{\Delta }{\eta }_m$ has a very different behavior:

$$\mu \mathit{MG}-{\mathbb{E}}_{\Delta }{\eta }_1=\mu M\sum _{g=1}^G\left[1-{(1-r^{-g})}^{M-1}\right].$$ (8)

Now,

$$1-{(1-r^{-g})}^{M-1}=(M-1)\underset{1-r^{-g}}{\overset{1}{\int }}{x}^{M-2}\mathit{dx}\le (M-1)\underset{1-r^{-g}}{\overset{1}{\int }}\mathit{dx}=(M-1)r^{-g},$$ (9)

so the sum on the right of Eq (8) increases with G and is bounded above by the finite sum ${\sum }_{g=1}^{\infty }(M-1)r^{-g}$ of a geometric series. Thus, the difference on the left of Eq (8) increases to a finite limit as G → ∞. Thus, ${\lim }_{G\to \infty }{\mathbb{E}}_{\Delta }{\eta }_1∕(\mu \mathit{MG})=1$.

For m ≥ 2, the second factor in the middle expression of Eq (7) expands to yield an incidental alternative expression,

$$\begin{gathered} {\mathbb{E}}_{\Delta }{\eta }_m=\mu \left(\begin{array}{c}\hfill M\hfill \\ \hfill m\hfill \end{array}\right)\sum _{g=1}^G\sum _{j=0}^{M-m}\left(\begin{array}{c}\hfill M-m\hfill \\ \hfill j\hfill \end{array}\right){\left(r^{-g}\right)}^{m-1}{(-r^{-g})}^j \\ =\mu \left(\begin{array}{c}\hfill M\hfill \\ \hfill m\hfill \end{array}\right)\sum _{j=0}^{M-m}\left(\begin{array}{c}\hfill M-m\hfill \\ \hfill j\hfill \end{array}\right){(-1)}^j\sum _{g=1}^G{r}^{-(m-1+j)g} \\ =\mu \left(\begin{array}{c}\hfill M\hfill \\ \hfill m\hfill \end{array}\right)\sum _{j=0}^{M-m}\left(\begin{array}{c}\hfill M-m\hfill \\ \hfill j\hfill \end{array}\right){(-1)}^j\frac{1-r^{-(m-1+j)G}}{r^{m-1+j}-1}, \end{gathered}$$ (10)

whereas

$${\mathbb{E}}_{\Delta }{\eta }_1=\mu M\left[G-\sum _{j=1}^{M-1}\left(\begin{array}{c}\hfill M-1\hfill \\ \hfill j\hfill \end{array}\right){(-1)}^{j-1}\frac{1-r^{-\mathit{jG}}}{r^j-1}\right],$$ (11)

For computing, because the final sum in Eq (10) alternates in sign, it is less reliable numerically than Eq (7). For m = M, however, Eq (10) provides a convenient check on programming errors:

$${\mathbb{E}}_{\Delta }{\eta }_M=\mu \frac{1-r^{-(M-1)G}}{r^{M-1}-1}.$$ (12)

To compare the above approximations to other models of interest, consider a continuous-time birth-and-death branching process with mutation rate μ, birth rate b, and death rate d, yielding a basic reproduction number r = b / d. Let N₁ be the total population at the time t₁ of sampling M individuals. For m ≥ 2 (our main interest), if N₁ is large, the approximate expected SFS is

$$\mathbb{E}{\eta }_m^{\prime }\approx \frac{\mu M}{1-r^{-1}}\frac{1}{m(m-1)}={\mathbb{E}}_C{\eta }_m$$ (13)

independent of t₁ and the exact value of N₁, where $\mathbb{E}{\eta }_m^{\prime }=S_{\text{sample}}\left(m,\mu ,t_1\mid M\right)$ in the authors’ notation of Eqs (27) and (28) of (Ohtsuki and Innan 2017). According to those authors, Kingman’s coalescent for an exponentially growing population (e.g., (Griffiths and Tavare 1998)) yields the same approximation. (See also Theorem 2 of (Durrett 2013) about the Moran model.) Accordingly, the right side of Eq (13) defines ${\mathbb{E}}_C{\eta }_m$, the (approximate) coalescent SFS.

To relate Eq (13) to Eq (7), let G → ∞. For m ≥ 2, the Appendix shows that

$$\underset{r\downarrow 1}{\lim }\frac{{\mathbb{E}}_{\Delta }{\eta }_m}{{\mathbb{E}}_C{\eta }_m}=\underset{r\downarrow 1}{\lim }\frac{\mu \left(\begin{array}{c}\hfill M\hfill \\ \hfill m\hfill \end{array}\right)\sum _{g=1}^{\infty }{\left(r^{-g}\right)}^{m-1}{(1-r^{-g})}^{M-m}}{\frac{\mu M}{1-r^{-1}}\frac{1}{m(m-1)}}=1,$$ (14)

i.e., the ratio of the Delta and coalescent SFSs approaches 1 as r decreases to 1.

In many models of exponentially expanding populations (e.g., the birth-and-death process (Champagnat and Henry 2016)), the moments of η₂,…, η_M converge to fixed, finite values at infinite times. Here in a Galton-Watson process, the SFS approximations ${\mathbb{E}}_{\Delta }{\eta }_m(m\ge 2)$ display a similar convergence as G → ∞. Accordingly, Section 4 presents numerical results only for the limit G → ∞ in Eq (7).

3. Methods

A technique for reducing variance (Hammersley and Handscomb 1964) also reduces the programming effort in simulating the SFS for the Gamma and Delta models, as follows. For either model, let A_m count the non-founding ancestors with m descendants in the sample, and call A = (A₁, A₂, …, A_M) the ancestral sample frequency spectrum. (Note that the related term “allele frequency spectrum” is sometimes used synonymously with SFS, but it might be preferable to reserve it (e.g., (Griffiths and Pakes 1988)) for the analogous variate under the infinite alleles model (Kimura and Crow 1964).) For any model with synchronous generations (e.g., the Delta model),

$$A_m=\sum _{g=1}^G\sum _{i=1}^{Z_g}{I}_{g,i;m},$$ (15)

so

$$\sum _{m=1}^M{\mathit{mA}}_m=\sum _{m=1}^{M}m\sum _{g=1}^G\sum _{i=1}^{Z_g}{I}_{g,i;m}=\sum _{g=1}^G\sum _{i=1}^{Z_g}\sum _{m=1}^M{\mathit{mI}}_{g,i;m}=\sum _{g=1}^{G}M=\mathit{MG},$$ (16)

because ${\sum }_{m=1}^M{\mathit{mI}}_{g,i;m}$ counts the descendants of ${\mathcal{I}}_{g,i}$ in the sample. Some equivalent of the following result is doubtless stated elsewhere.

Theorem 1: Consider an infinite sites model where the novel mutation counts in every daughter of every mother are independent Poisson variates with fixed mean μ (e.g., the Gamma or Delta model). Given A, the coordinates of η are independent Poisson variates, with η_m having mean μA_m.

Proof: The coordinates of η are determined by the novel mutation counts in disjoint ancestral sets of sizes A. Thus, independence of mutation implies each η_m is the sum of A_m Poisson variates of mean μ, with all Poisson variates independent. The sum of A_m Poisson variates of mean μ is a Poisson variate η_m of mean μA_m, where the variates {η_m} are independent. □

Theorem 1 implies that under the Gamma or Delta model, $\mathbb{E}{\eta }_m=\mu \mathbb{E}{A}_m$. In addition, the law of total variance yields

$${\sigma }^2({\eta }_m)=\mathbb{E}[{\sigma }^2({\eta }_m\mid \mathbf{A})]+{\sigma }^2(\mathbb{E}[{\eta }_m\mid \mathbf{A}])=\mathbb{E}[\mu A_m]+{\sigma }^2(\mu A_m)=\mu \mathbb{E}{A}_m+{\mu }^2{\sigma }^2(A_m),$$ (17)

where the second equality follows from Theorem 1 and the fact that the variance of a Poisson variate equals its mean. Under the Gamma or Delta model, therefore, the simulation of ancestries alone suffices to estimate the first two moments of η_m. On the right side of Eq (17), the first term is sometimes called the mutational variance; the second, the evolutionary variance. Besides variance reduction, the simulation of A clarifies the relative contributions of the mutational and evolutionary variance in Eq (17) by displaying their dependence on μ.

For different r > 1, simulation yielded 1000 realizations of the Gamma and Delta models. As a check on using the skeleton process in Section 2, each realization started with one founder and propagated the population, simply restarting it with another founder if the population became extinct. The realization continued until the population reached a threshold of 6000 live individuals. Here, 6000 is an arbitrary large number, chosen on the (possibly irrelevant) basis that a neutral model in coalescent theory estimated the effective size of the viral population in HIV patients between 2000 and 6000 (Seo et al. 2002). As a check on using sampling with replacement in Section 2, each realization sampled the 6000 live individuals uniformly without replacement.

4. Results

Typically, the sample means and sample standard deviations of the SFS simulated from the Delta and Gamma models were visually indistinguishable, and the differences between the models’ results were always subtle. Using the Gamma model, Figure 3 exemplifies some other numerical trends. Typically for fixed m, the sample mean SFS $\widehat{\mathbb{E}}{\eta }_m$ simulated from the Gamma model increased with the sample number M, while the sample standard deviation decreased (see plots pertinent to η₂ in Figure 3a, b, c, and d for M = 8,32,128,512), whereas for fixed M, $\widehat{\mathbb{E}}{\eta }_m$ decreased with the number m of mutations in an alignment column, while the sample standard deviation increased (see plots pertinent to η₂, η₃, and η₄ in Figure 3d, e, and f for M = 512). Unlike Eq (13) for the coalescent SFS ${\mathbb{E}}_C{\eta }_m$, Eq (7) for the Delta SFS ${\mathbb{E}}_{\Delta }{\eta }_m$ generally provided accurate approximations to the sample mean $\widehat{\mathbb{E}}{\eta }_m$ of the Gamma SFS in all simulations, except possibly where it crossed over from $\widehat{\mathbb{E}}{\eta }_m$ to ${\mathbb{E}}_C{\eta }_m$ as r ↓ 1, e.g., at r = 1.1 for large M (e.g., Figure 3c, d, e, and f, where M = 512 and k = 2,3,4); or for small r for M = 128 and k = 2 (in Figure 3c).

Figure 3. Selected plots of the SFS η_m vs. the basic reproduction number r = R₀.

In Figure 3, all axes are logarithmic. All X-axes share the same scale (likewise for all Y-axes). All results displayed use the HIV value μ = 0.0551. In each subfigure, Eq (13) for the coalescent SFS ${\mathbb{E}}_C{\eta }_m$ yields the black curve joining triangular points. Because the black curves are therefore all translates of f(r) = 1/(1−r⁻¹), the translated shape provides a ready reference for comparing subfigures. In each subfigure, Eq (7) for the Delta SFS ${\mathbb{E}}_{\Delta }{\eta }_m$ yields the red curve joining square points. Generally, each red curve obscures a gold curve joining circular points. The gold curve corresponds to the sample mean $\widehat{\mathbb{E}}{\eta }_m$ estimating $\mathbb{E}{\eta }_m$ from simulations of the Gamma model, with the error bars giving the sample standard deviation. Figure 3a, b, c, and d show plots pertinent to η₂ (i.e., η_m for m = 2) for different sample numbers M = 8,32,128,512, whereas Figure 3d, e, and f show plots pertinent to η₂, η₃, and η₄ for M = 512. For M = 512 in Figure 3d, e, and f, close inspection of the leftmost point (at r = 1.1) displays ${\mathbb{E}}_{\Delta }{\eta }_m$ crossing over from $\widehat{\mathbb{E}}{\eta }_m$ to ${\mathbb{E}}_C{\eta }_m$ as r decreases to 1, as in Eq (14). Figure 3c also shows ${\mathbb{E}}_{\Delta }{\eta }_2$ crossing over from $\widehat{\mathbb{E}}{\eta }_2$ to ${\mathbb{E}}_C{\eta }_2$, albeit more subtly, for M = 128.

Figure 3 is representative of simulation results, with two exceptions. First, the inequality ${\mathbb{E}}_{\Delta }{\eta }_m\le {\mathbb{E}}_C{\eta }_m$ failed occasionally at large values of m, with some neighboring values of r yielding comparable but probabilistically independent numerical exceptions. Second, for fixed r, $\widehat{\mathbb{E}}{\eta }_m$, ${\mathbb{E}}_{\Delta }{\eta }_m$ and ${\mathbb{E}}_C{\eta }_m$ all typically decreased with increasing m (as in Figure 3d, e, and f). A notable exception occurred for $\widehat{\mathbb{E}}{\eta }_m$ and small r (e.g., r = 1.1), however, because the founder ${\mathcal{I}}_{0,1}$ of the branching process then gives rise to a long, unbranched initial lineage. The lineage eventually leads to the most recent common ancestor of the sample, so its mutations occur in all samples, making $\widehat{\mathbb{E}}{\eta }_M$ noticeably larger than $\widehat{\mathbb{E}}{\eta }_{M-1}$. As approximations, neither ${\mathbb{E}}_{\Delta }{\eta }_M$ nor ${\mathbb{E}}_C{\eta }_m$ appear to account adequately for mutations in a long, unbranched initial lineage.

Finally, the magnitude of the ratio ${\mu }^2{\sigma }^2\left(A_m\right)∕\left(\mu \mathbb{E}{A}_m\right)$ of the evolutionary variance to mutational variance in σ²(η_m) from Eq (17) was surprisingly robust over different sample sizes M and m = 1,2,…,M. It decreased as a function of r and went from about 0.5 at r = 1.1 to about 0.1 at r = 1.5. Thus, in the present context of μ = 0.0551 (HIV gp120), even at r = 1.1, if the evolutionary variance were neglected and the standard deviation σ(η_m) replaced by a Poisson approximation $\sqrt{\mathbb{E}{\eta }_m}\approx \sqrt{\mu \mathbb{E}{A}_m}$ determined numerically from Eq (7), the resulting underestimate is at worst $1-1∕\sqrt{1+0.5}\approx 18\%$.

Figure 4 provides a different view of many phenomena in Figure 3. First, for M ≥ 64 and small values of m, the top dotted curves for ${\mathbb{E}}_{\Delta }{\eta }_m$ (particularly the black curve for r = 1.1) display the crossover from $\widehat{\mathbb{E}}{\eta }_m$ to ${\mathbb{E}}_C{\eta }_m$, by overestimating $\widehat{\mathbb{E}}{\eta }_m$. Second, in contrast to the corresponding dotted curves, the top two solid curves (for r = 1.1 and r ≈ 1.46) in Figure 4a for M = 8 show an increase from $\widehat{\mathbb{E}}{\eta }_{M-1}$ to $\widehat{\mathbb{E}}{\eta }_M$. As in Figure 3, the increase reflects a long, unbranched initial lineage from founder ${\mathcal{I}}_{0,1}$. Finally, for M ≥ 64, the two bottom dotted curves (for r ≈ 6.12 and r ≈ 8.14) in particular display some numerical instability at very small values of η_m. Premature numerical truncation of Eq (7) as G → ∞ does not appear to be the cause, so the instability may be inherent in the delta approximation, possibly due to the exponentiation of large values of r.

Figure 4. The expected SFS $\widehat{\mathbb{E}}{\eta }_m$ and ${\mathbb{E}}_{\Delta }{\eta }_m$, vs. m for different sample sizes M.

In Figure 4, the Y-axes are logarithmic with the same range, and the X-axes all have categories m = 2,3,…,8. All results displayed use the HIV value μ = 0.0551 . Each subfigure displays simulation results for a different sample size M. The solid curves display simulation sample means $\widehat{\mathbb{E}}{\eta }_m$ (m = 2,3,…,8) corresponding to different reproduction numbers r = R₀ in a geometric series with ratio 1.331 = 1.1³. From top to bottom, the curves correspond to r = 1.1 (black), r ≈ 1.46 (purple), r ≈ 1.95 (dark blue), r ≈ 2.59 (light blue), r ≈ 3.45 (green), r ≈ 4.59 (orange), r ≈ 6.12 (light red), and r ≈ 8.14 (dark red). The dotted curves correspond to Eq (7) for the Delta SFS ${\mathbb{E}}_{\Delta }{\eta }_m$ (implicitly connecting points for m = 2,3,…,8).

5. Discussion

The success of an invasive population often hinges on the fate of the first arrivals. Sometimes (e.g., in some viral infections), initial invaders may be very few and reproduce in near-synchronous bursts, blunting the accuracy of some population models in approximating the SFS (the Introduction and Figure 1 define the SFS verbally; Eq (2), mathematically). The Results section shows that in at least one important case, the Gamma model of HIV gp120 (a Bellman-Harris process), the simpler Delta model (a Galton-Watson process) can often yield analytic approximations indistinguishable by eye from the expected SFS. In particular, Figure 3 (c.f. particularly, m = 2 for M = 128, or m = 2,3,4 for M = 512) demonstrates that the SFS can carry information about an initial viral basic reproduction number r = R₀. The estimated R₀ can serve a quantitative measure of therapeutic progress in human and animal trials of viral therapies, particularly if the trials sample more viral sequences than they do at present.

Several technical difficulties present themselves, however. HIV researchers already recognize that multiple founders impede sequence analysis (Love et al. 2016). In addition, the invading viral population may pass through many environments, causing R₀ to vary. If the technical difficulties prove superable, however, the present theory suggests a novel practical use for extant sequence data already sampled from patient and animal trials: the SFS can benchmark subtle therapeutic progress by estimating the initial R₀, even when infection occurred and the trial had insufficient statistical power to infer therapeutic efficacy from infection data alone.

Many assumptions here are undemanding. For example, the difference between sampling with and without replacement can be neglected for light sampling (Freedman 1977). In applications to HIV, each infected cell in the sampled generation produces to 10³ – 10⁴ viral particles, and gp120 RNA averages μ ≈ 0.0551 mutations per replication, so many gp120 samples have identical sequences anyway.

Similarly, the assumption that viral sequences are sampled simultaneously is excessively stringent. For practical purposes, the most recent common ancestor of most sampled pairs often occurs early in the population’s expansion, an assumption holding for many expanding populations, regardless of whether sampling is simultaneous.

The Delta model therefore provides robust, accurate approximations to the expected SFS for the Gamma model of HIV reproduction (see Figure 3 and Figure 4). In that context, moreover, the approximations are superior to approximations based on continuous-time birth-and-death process or the Kingman coalescent for an exponentially expanding population. (Some other results on the SFS (Champagnat et al. 2012, Champagnat and Henry 2016), though not directly relevant to the Gamma model, are worth noting here.) The superiority should be unsurprising. The Gamma model mimics tight bursts of HIV replication every 2 days, much like a Delta model with synchronous generations every 2 days. In contrast, the continuous-time birth-and-death and the coalescent processes are Markovian, so by their nature, they do not mimic coordinated cell lysis as viruses burst forth from an infected cell. In addition, a small population such as a small initial viral population is likely to degrade approximations using the coalescent (Stadler et al. 2015): “…in most cases, the coalescent approximation works very well down to small population sizes (a few hundred individuals)” (Eriksson et al. 2010).

Figure 3 shows also shows that as R₀ approaches 1 (e.g., R₀ = 1.1), Eq (7) for the SFS ${\mathbb{E}}_{\Delta }{\eta }_m$ from the Delta model no longer closely approximates the SFS $\mathbb{E}{\eta }_m$ for the Gamma model, but instead crosses over from $\mathbb{E}{\eta }_m$ to Eq (13) for the SFS ${\mathbb{E}}_C{\eta }_m$ derived from a coalescent or continuous-time birth-and-death process. The Appendix gives a mathematical proof of the crossover, which has foundations in viewing the continuous-time birth-and-death process as the appropriate limit of Galton-Watson processes.

For HIV gp120, μ ≈ 0.0551, so μ² is much less than μ. In Eq (17), one might suspect that consequently, the mutational variance dominates the evolutionary variance. The last paragraph of Section 4 (Results) bears out the suspicion. In fact, one incisive model of gp120 phylogeny in HIV is deterministic and explicitly neglects the evolutionary variance (Lee et al. 2009). The linear truncation in Section 2 (Theory) justifies the deterministic model (as well as making sense of it for non-integer r). The present paper adds a minor caveat to the deterministic model, however, particularly in its application to a study involving whole viral genomic sequence instead of a single protein. To explain, the ratio of the lengths of the HIV genome and of gp120 is 9200 / 2550 ≈ 3.6. The calculation at the end of Section 4 (Results) indicates that at r = 1.1, the neglect of evolutionary variance underestimates σ(η_m) by about $1-1∕\sqrt{1+3.6(0.5)}\approx 40\%$, enough to start impacting error estimates, and therefore scientific conclusions. The largest viruses have genome length around 1Mb, where indiscriminate neglect of evolutionary variance may lead to error.

To summarize, this article has presented an approximation for the expected site frequency spectrum in a Galton-Watson process with mutation. In many parameter regimes, the approximation is superior to approximations from a continuous-time birth-and-death process or a coalescent process. Although the (superable) practical problems described above prevent immediate application of the theory presented here, the present article indicates the possibility of using sequence data collected after a virus has become detectable in blood to infer the initial reproduction number R₀, with the aim of examining the efficacy of therapies for preventing or mitigating initial viral infection.

Appendix

The Skeleton Process

The Harris-Sevastyanov transformation (Harris 1948, Harris 1963) gives the pgf $P\left(s\right)=\mathbb{E}{s}^{Z_1}$ of the skeleton process:

$$P(s)=\frac{\tilde{P}\left(\left(1-q\right)s+q\right)-q}{1-q}.$$ (18)

To keep the article self-contained, we note that Eq (18) has the following heuristic justification (see, e.g., (Schuh 1982) or Part D.12 Decomposition of the Supercritical Branching Process, p.47 et seq. in (Athreya and Ney 2004)). The pgf $\tilde{P}\left(q+\left(1-q\right)s\right)=\mathbb{E}\left[{\left(q+\left(1-q\right)s\right)}^{{\tilde{Z}}_1}\right]$ deletes the founder’s doomed daughters independently with the correct probability q. The founder is either doomed (with probability q) or immortal (with probability 1 − q). If she is immortal, she is part of the skeleton process, which has pgf P(s). Thus,

$$\tilde{P}\left(\left(1-q\right)s+q\right)=q+(1-q)P(s),$$ (19)

justifying Eq (18). Eq (19) shows that $P^{\prime }(1)={\tilde{P}}^{\prime }(1)=r$, i.e., the original and skeleton processes have the same basic reproduction number.

Note that the Harris-Sevastyanov transformation reduces the variance of the offspring distribution, probably contributing to the accuracy of deterministic approximations in this article:

$${\sigma }^2(Z_1)=P^{″}(1)+r-r^2={\tilde{P}}^{″}(1)(1-q)+r-r^2={\sigma }^2({\tilde{Z}}_1)-q{\tilde{P}}^{″}(1)\le {\sigma }^2({\tilde{Z}}_1).$$ (20)

Proof of the Limit in Eq (14) for 2 ≤ k ≤ M

Consider the function f(x) = x^k−2(1−x)^M−k and partition [0,1] into subintervals with the points $x_g^{(r)}=r^{-g}(g=1,2,3,\dots )$. Let $x_g^{(0)}=1$. Almost immediately, the theory of approximating integrals by Riemann sums shows that

$$\underset{0}{\overset{1}{\int }}f(x)\mathit{dx}=\underset{r\downarrow 1}{\lim }\sum _{g=1}^{\infty }f\left(x_g^{(r)}\right)\left(x_{g-1}^{(r)}-x_g^{(r)}\right)=\underset{r\downarrow 1}{\lim }\sum _{g=1}^{\infty }f\left(r^{-g}\right)r^{-g}(r-1).$$ (21)

Substitution of f(x) into Eq (21) yields

$$\underset{r\downarrow 1}{\lim }\sum _{g=1}^{\infty }{\left(r^{-g}\right)}^{k-1}{\left(1-r^{-g}\right)}^{M-k}(r-1)=\underset{0}{\overset{1}{\int }}{x}^{k-2}{(1-x)}^{M-k}\mathit{dx}=\frac{(k-2)!(M-k)!}{(M-1)!},$$ (22)

after using standard results for the Beta integral in the middle expression. Thus,

$$\begin{gathered} \underset{r\downarrow 1}{\lim }\frac{\mu \left(\begin{array}{c}\hfill M\hfill \\ \hfill k\hfill \end{array}\right)\sum _{g=1}^{\infty }{\left(r^{-g}\right)}^{k-1}{\left(1-r^{-g}\right)}^{M-k}}{\frac{\mu M}{1-r^{-1}}\frac{1}{k(k-1)}}=\underset{r\downarrow 1}{\lim }\frac{\left(\begin{array}{c}\hfill M\hfill \\ \hfill k\hfill \end{array}\right)\sum _{g=1}^{\infty }{\left(r^{-g}\right)}^{k-1}{\left(1-r^{-g}\right)}^{M-k}(r-1)}{\frac{M}{r^{-1}}\frac{1}{k(k-1)}} \\ =\underset{r\downarrow 1}{\lim }\frac{\left(\begin{array}{c}\hfill M\hfill \\ \hfill k\hfill \end{array}\right)\frac{(k-2)!(M-k)!}{(M-1)!}}{\frac{M}{r^{-1}}\frac{1}{k(k-1)}} \\ =1. \end{gathered}$$ (23)