The impact of long-range dispersal on gene surfing

Significance

The expansion of a population into a new area occurs frequently in nature. Examples include epidemic outbreaks, invasions by nonnative species, and colonization of regions rendered hospitable by climate change. Many expansions are driven by long-range migration of organisms or seeds borne by air, water, or traveling hosts. These events amplify the impact of random chance on genetic variation: A lucky individual which reaches unoccupied territory first can have more descendants than subsequent arrivals, even without advantageous mutations. We model the patterns of variation generated by random long-range migration events and study their effects on genetic diversity. Our results elucidate the evolutionary consequences of long-range dispersal and are important for correctly interpreting sources of variation in geographically structured genomic data.

Abstract

Range expansions lead to distinctive patterns of genetic variation in populations, even in the absence of selection. These patterns and their genetic consequences have been well studied for populations advancing through successive short-ranged migration events. However, most populations harbor some degree of long-range dispersal, experiencing rare yet consequential migration events over arbitrarily long distances. Although dispersal is known to strongly affect spatial genetic structure during range expansions, the resulting patterns and their impact on neutral diversity remain poorly understood. Here, we systematically study the consequences of long-range dispersal on patterns of neutral variation during range expansion in a class of dispersal models which spans the extremes of local (effectively short-ranged) and global (effectively well-mixed) migration. We find that sufficiently long-ranged dispersal leaves behind a mosaic of monoallelic patches, whose number and size are highly sensitive to the distribution of dispersal distances. We develop a coarse-grained model which connects statistical features of these spatial patterns to the evolution of neutral diversity during the range expansion. We show that growth mechanisms that appear qualitatively similar can engender vastly different outcomes for diversity: Depending on the tail of the dispersal distance distribution, diversity can be either preserved (i.e., many variants survive) or lost (i.e., one variant dominates) at long times. Our results highlight the impact of spatial and migratory structure on genetic variation during processes as varied as range expansions, species invasions, epidemics, and the spread of beneficial mutations in established populations.

Range expansions have occurred in the history of many species, from plants (1) to avian (2), aquatic (3, 4), and terrestrial (5, 6) animals, including humans (7). Over geological timescales, they have been driven by climactic changes such as glacial advance and melting in the northern hemisphere (8, 9). More recently, anthropogenic climate change and human-mediated introduction of invasive species have driven the expansion of species into new territory (10). These expansion events impact the genetic makeup of the population, in ways that are dramatically different from population expansions without spatial structure (11). In particular, neutral mutations occurring during range expansions leave behind signatures that are otherwise associated with selection, such as sweeps through the population (12), allelic gradients (13), and reduction in local genetic diversity (14–16). Understanding the patterns of neutral variation left behind by range expansions is crucial for disentangling the role of spatial structure from selection in determining genetic diversity (17).

Much of our current understanding of neutral evolution during range expansions is derived from situations where individuals migrate a short distance between generations (12–16, 18). In this case, the population advances through a wave of roughly constant speed separating occupied and unoccupied regions of space. Crucially, only individuals that happen to be close to the advancing front contribute to future generations, and large swaths of the population after the expansion can be traced back to a few individuals at the edge of the originating population, a phenomenon termed gene surfing. The resulting neutral variation shows a characteristic pattern: Local diversity is strongly reduced as neutral variants segregate into uniform regions, called sectors, in which a single allele dominates. Nevertheless, in a radial expansion, sectors of different variants persist at long times as a result of which global diversity is maintained, as seen in Fig. 1 A–D. These patterns persist under moderate levels of gene flow due to subsequent diffusion following the initial advance (18, 19).

Fig. 1.

Patterns of neutral diversity under different fat-tailed dispersal kernels. (A–C) Snapshots from a single simulation of a range expansion with kernel exponent $\mu =4$ at different population sizes $N$, starting from a homeland with $N_0=79$, $q=2$, $p_0=1/2$ (cyan circle). Demes occupied by neutral alleles A and B are colored light and dark, respectively. (D) Evolution of the fraction $p$ belonging to allele A, for 100 independent simulations (shaded curves), with the simulation from A–C shown as a solid red curve. (E–H and I–L) Same as A–C with $\mu =2.6$ and $\mu =1.5$, respectively.

Most organisms, however, experience some amount of long-range dispersal (20, 21). Pollen, seeds, and microorganisms are dispersed over long distances by wind and water or by wandering or migratory animals whose excursions influence their own evolution as well. The resulting distributions of dispersal distances, also called dispersal kernels, are often “fat tailed”: They do not have a characteristic cutoff length scale and fall off slower than exponentially with distance. Empirical measurements of dispersal kernels have shown that fat-tailed kernels arise in the spreading behavior of numerous species (22). Theoretical analyses have established that fat-tailed dispersal kernels accelerate expansion dynamics (20, 23), allowing the size of the expanding population to grow faster than linearly with time, and strongly influence population structure by breaking up the wave of advance associated with short-ranged spreading (24–29).

Although long-range dispersal is recognized as being consequential for range expansions (11, 30–32), its precise effects on genetic diversity are not fully understood. Whereas it is recognized that dispersal leads to monoallelic patches (24–27), the conditions for patches to dominate over sectors have not been identified. Furthermore, the structural characteristics and dynamics of the patch patterns and their subsequent impact on neutral diversity have not been systematically studied. As a result, even the basic question of whether neutral variation in the originating population is maintained during dispersal-driven range expansions is unresolved. Simulation studies involving mixtures of two Gaussian (i.e., non–fat-tailed) dispersal kernels with different mean distances have shown support for contrasting effects of increasing the weight of the broader dispersal kernel on neutral diversity (33) (the so-called “embolism effect”). At low levels, founder events ahead of the expanding front of the population wipe out diversity, but at higher levels, diversity in the expanding region is maintained by serial reintroduction of variants from the interior of the population. However, other studies have argued that a reduction in diversity due to the embolism effect occurs only for dispersal along corridors and for thin-tailed dispersal kernels (34), whereas fat-tailed kernels ought to generically enhance genetic diversity (30, 34–37).

Here, we study the evolution of neutral diversity in a simplified model of range expansions with dispersal events drawn from fat-tailed kernels. By analyzing a class of dispersal kernels which spans the two extremes of well-mixed growth and wavelike spreading, we obtain a comprehensive picture of neutral evolution in dispersal-accelerated range expansions. We find that long-range dispersal breaks up radial sectors into monoallelic patches, or blobs, but only if the kernel is sufficiently fat tailed (Fig. 1). The characteristic size of these blobs relative to the overall size of the population can vary widely, reflecting qualitative differences in the growth dynamics for different dispersal kernels. For the broadest kernels, the spatial distribution of alleles approaches a highly fragmented speckle pattern (Fig. 1 I–L). These patterns depart strongly from the prevailing paradigm of sectors as the spatial signature of range expansions (11, 14, 18).

We also investigate how global diversity is impacted by the breakup of sectors into blobs and speckles. By studying the growth of the typical number and size of blobs as the range expansion progresses, we show that fat-tailed kernels display the entire range of possible outcomes for neutral variation: Depending on the exponent characterizing the tail of the kernel, the initial diversity can be almost perfectly preserved, or completely lost, as the expansion progresses. Strikingly, we find that long-range dispersal can in some cases erode genetic diversity compared to short-range dispersal, through a mechanism that differs fundamentally from the previously documented embolism effect.

Model

We consider growth into isotropic space from a compact initial population of size $N_0$, confined to some region of space which we call the homeland. Each individual belongs to one of $q$ allelic identities, all of which are neutral relative to each other. The population is allowed to grow into an empty range, which has a finite local carrying capacity. For concreteness, we enforce the local carrying capacity by breaking up the range into a discrete regular lattice of demes, with each one able to accommodate a fixed number of individuals. We work in the limit that a newly colonized deme reaches its carrying capacity on much shorter timescales than the timescales associated with migration. In this limit, we need only to consider two occupancy levels: Demes are either completely empty or completely full, and each occupied deme has the allelic identity of the first individual that enters it. Once a deme is occupied, the local population replenishes itself constantly without changing its allelic identity and continues to send out offspring at a fixed total rate, with migration distances $r$ drawn randomly from a jump kernel $J\left(r\right)$. The migration direction is uniformly chosen from all available directions. Although our analysis can be applied to a wide range of fat-tailed kernels, we focus on the specific form $J\left(r\right)=\mu r^{-\left(\mu +1\right)}$ in this work. The kernel exponent $\mu$ quantifies the heaviness of the tail of the dispersal kernel, with higher values corresponding to jump distributions that fall off more steeply with distance. Power-law kernels of this type encompass a broad swath of population growth dynamics, ranging from effectively well mixed ($\mu \to 0$) to effectively diffusive ($\mu >d+1$) (38, 39). We henceforth refer to the population size as the number of demes, and generations are defined by the average time between migration events out of an occupied deme.

Before describing the results of multiallele simulations, we briefly summarize the known behavior of range expansions in power-law growth kernels for a monoallelic population (38) (Table 1). The value $\mu =d+1$ separates two qualitatively different behaviors: For $\mu >d+1$, the range expansion occurs via the advance of a constant-speed front separating occupied from unoccupied regions and is similar to expansion driven by short-ranged jump kernels. For $\mu <d+1$, long-range dispersal events become consequential, and the radial size of the population grows faster than linearly with time. A second threshold at $\mu =d$ separates power-law growth of the radial size with time for $d<\mu <d+1$ from stretched-exponential growth for $0<\mu <d$. As $\mu$ approaches zero, the population growth approaches the exponential behavior of a well-mixed population: Spatial structure becomes essentially irrelevant in this limit.

Table 1.

Core growth asymptotics and characteristic blob size

Kernel exponent	Asymptotic scaling of core radius $\ell \left(t\right)$	Characteristic blob size, $s*\left(r\right)$
$\mu <d$	$\exp \left(B_{\mu }{t}^{\eta }\right)$	$r^{\frac{\mu +d}{2d}}$
$\mu =d$	$\exp \left(c{\log }^{2}t\right)$	$2^{\frac{1}{4d}}{e}^{-\sqrt{\frac{\log 2\log r}{d}}}r$
$d<\mu <d+1$	$t^{1/\left(\mu -d\right)}$	$2^{\frac{-1}{\mu -d}}r$
$\mu =d+1$	$t\log t$ (1D)	—
$\mu >d+1$	$t$	—

The asymptotic behavior of $\ell \left(t\right)$ (from ref. 38) is cataloged, omitting distance and timescales for $\ell$ and t, respectively. $B_{\mu }\approx 2d\log \left(2\right)/{\left(\mu -d\right)}^2$, $\eta =\log \left[2d/\left(d+\mu \right)\right]/\log 2$, and $c=1/\left[4d\log \left(2\right)\right]$. Also shown is the characteristic blob size $s*\left(r\right)\equiv \ell \left[{\ell }^{-1}\left(r\right)/2\right]$, introduced in the text for jump-driven growth ($\mu <d+1$).

Results

Long-Range Dispersal Breaks Up Sectors into Monoallelic Blobs.

Fig. 1 shows snapshots from simulations of range expansions for three different kernel exponents. The narrowest kernel, $\mu =4$, corresponds to the regime of growth which asymptotically approaches a constant-speed advancing front at long times. In this regime, the population quickly coarsens into monoallelic sectors as has been well characterized for growth due to short-ranged kernels (14, 16, 18) as seen for a representative example in Fig. 1 A–C. At early times, boundaries between sectors of different alleles can annihilate due to the random wandering of sector boundaries from straight radial rays, but at later times, established sectors are stable against annihilation and the allelic fractions become essentially frozen in time (19, 40), up to fluctuations due to the random wandering of the boundaries between sectors. These dynamics are reflected in the evolution of allelic fractions of individual simulations, which settle to a near-constant value at long times (Fig. 1D).

When $\mu <d+1$, long-ranged jumps become consequential for growth, and the resulting patterns are markedly different (Fig. 1 E–G). Alleles still segregate into monoallelic regions, but these do not form radial sectors. Instead, the population is composed of a mosaic of blobs of varying size, each of which has an irregular boundary but is roughly isotropic in shape. At each time point, a core region surrounding the homeland can be identified within which all sites are occupied. Also visible are isolated clusters of occupied sites, separated from the core by empty sites, which were colonized by the offspring of a single migrant that landed far from the bulk of the population at earlier times. These satellite outbreaks, visible at the outer edges of the population in each snapshot, are a characteristic feature of jump-driven growth in the presence of long-range dispersal (38, 41). The patchiness of jump-driven range expansions arises from the continued accumulation of satellite outbreaks. The resulting disruption of sectors when $\mu <d+1$ will impact many known genetic consequences of range expansions, as we describe in Discussion.

Upon comparing patterns for the two kernels displaying blobs in Fig. 1, the patterns at the broadest kernel, $\mu =1.5$ (Fig. 1 I–K), show finer and more numerous blobs compared to the intermediate kernel with $\mu =2.6$ (Fig. 1 E–G). Relative to the population size, blobs also appear to get finer as the range expansion progresses for $\mu =1.5$. Finally, the evolution of the allele fractions (Fig. 1 H and L) shows different characteristics for the two kernels: Each trajectory exhibits stronger variations over time for the intermediate kernel, $\mu =2.6$, and the distribution of stochastic outcomes covers a broader range of fractions. In the next section, we connect these qualitative observations to an analysis of the typical size of blobs relative to the population size in the jump-driven growth regime.

A Hierarchy of Doublings in Time Determines the Characteristic Size of Blobs.

Although the early establishment of sectors in constant-speed range expansions (such as in Fig. 1 A–C) is stochastic, their subsequent growth over time is tied to the radial population growth and is essentially deterministic, up to random wandering of sector boundaries which becomes insignificant at long times (the growth in transverse fluctuations of the boundaries is overcome by the linear expansion of the circumference with time). By contrast, the placement and size of the monoallelic regions in the jump-driven range expansions (Fig. 1 E–K; see Fig. 2A for complete time evolution of a 1D simulation) are stochastic at all stages of growth. However, the characteristic sizes of blobs incorporated into the growing core at different times (or equivalently, at different radial distances) follow a distinct pattern in the vicinity of the marginal point $\mu =d$. This pattern was first revealed in ref. 38, and the key features, applicable in all dimensions, are illustrated for a 1D range expansion in Fig. 2. The growth of the core radius $\ell$ as a function of time $t$ from the point of seeding is constrained by a self-consistency condition: The satellites themselves grow from the accumulation of satellites at smaller scales and must therefore follow the same growth rule $\ell \left(t\right)$ as the core itself (Fig. 2B). As a result, when the growth of $\ell \left(t\right)$ is faster than linear but slower than exponential (which is true in both the stretched-exponential and power-law growth regimes of Table 1), the distribution of satellite clusters which join the core at time $t$ is peaked at size $\ell \left(t/2\right)$, as shown schematically in Fig. 2C.

Fig. 2.

Time-doubling hierarchy for jump-driven growth and consequent ancestral pattern. (A) Portion of a range from simulated population growth on a 1D lattice with $\mu =1$, starting from a homeland with $N_0=100$ and $p_0=1/2$ occupying lattice sites [−50,49]. Lattice sites are colored by their occupancy as a function of time. The dashed line indicates the coarse boundary of the region that is completely occupied at a particular time, which is termed the core region. (B) Coarse-grained schematic of core growth due to successive mergers of satellite clusters, such as in A. The function $\ell \left(t\right)$ governing the growth of the core region must also govern the growth of the satellite clusters, which grow according to the same underlying stochastic process. (C) For jump-driven growth under kernels with $\mu$ close to $d$, a self-consistency argument tying the core and satellite growth rules reveals a time-doubling hierarchy between the core and satellites at its outer edge: The typical satellite merging with the core at time $t_0$ was seeded by a dispersal event of length $\ell \left(t_0\right)$ which occurred at time of order $t_0/2$ and gave rise to a satellite which grew for a period of order $t_0/2$. This hierarchy defines the characteristic blob scale $s*\left(r\right)$. (D) Spatial relationships between pairs of individuals (red and green discs) and their MRCA (diamonds) reveal the time-doubling hierarchy in jump-driven expansions. Solid lines schematically trace out the genealogy of each pair to the MRCA.

The hierarchy connecting core size scales at $\ell \left(t\right)$ and $\ell \left(t/2\right)$ in jump-driven growth was developed and used in ref. 38 to obtain the asymptotic behavior summarized in Table 1, as well as the short-time approach to asymptotics. Signatures of the hierarchy were also observed in patterns left behind after parallel adaptation due to multiple mutations in a population experiencing long-range dispersal (41). In the context of neutral diversity in range expansions, this hierarchy links the sizes $s\left(r\right)$ of monoallelic regions to their distance $r$ from the center of the range expansion: We expect $s\left(r\right)\sim \ell \left[{\ell }^{-1}\left(r\right)/2\right]$. The asymptotic growth forms (Table 1) predict qualitatively different relationships for the size of the largest satellite clusters in the population relative to the core population itself in the various growth regimes. When $\mu <d$, we expect $s\left(r\right)/r\sim r^{\left(\mu -d\right)/\left(2d\right)}$, a decreasing function of $r$: The characteristic size of the largest clusters shrinks relative to the core size itself as the range expansion advances. By contrast, when $d<\mu <d+1$, we expect $s\left(r\right)/r\sim 2^{-1/\left(\mu -d\right)}$, a constant as the core size increases. The representative patterns for $\mu =2.6$ and $\mu =1.5$ in Fig. 1 are consistent with these qualitative features. When $\mu =2.6$, the characteristic sizes of the blobs at the edge of the core do not vary significantly relative to the core size, whereas for $\mu =1.5$ the blobs appear smaller relative to the core as the simulation progresses.

To quantitatively test the hierarchy of blob-core sizes predicted above, we measured spatial relationships among pairs of individuals and their most recent common ancestor (MRCA). As Fig. 2D illustrates, the time-doubling hierarchy predicts that a pair of individuals centered at distance $r$ are likely to belong to the same satellite if their separation is much less than $s\left(r\right)$ and hence have an MRCA located at a distance of order $r$ from the origin. By contrast, individuals separated by distances much larger than $s\left(r\right)$ are likely to belong to independently growing satellites which were seeded by different long jumps from the core; their MRCA is located at distances much closer to the origin. In the full evolution, the radial position $r_m$ of the MRCA of a pair of occupied sites separated by a distance $s$ about a distance $r$ from the origin is a stochastic variable. However, we expect the average of the MRCA positions of many such pairs to fall from $r$ to 0, over a separation distance of order $s^{*}\left(r\right)\equiv \ell \left[{\ell }^{-1}\left(r\right)/2\right]$.

Measurements of pair–MRCA relationships in 1D simulations confirm these expectations (Fig. 3; see Materials and Methods for details). In all growth regimes, the average MRCA position ${\overline{r}}_m$ falls with separation $s$ over a characteristic decay scale which grows with increasing center–pair distance $r$ from the origin, consistent with satellite sizes growing larger as the range expansion progresses (Fig. 3 A–C). Upon rescaling the pair separation with the proposed blob scale $s^{*}\left(r\right)$, data for different values of $r$ collapse onto curves that depend only on the jump kernel (Fig. 3D), with a sharp fall in mean MRCA distance when $s/s^{*}\sim 1$. The data collapse reflects the relevance of the time-doubling hierarchy in setting the characteristic scales underlying the stochastic blob size distributions during the range expansion. Note that merely rescaling pair separation with the center–pair distance does not lead to collapse of the data curves in the stretched-exponential and marginal growth cases (Fig. 3 A and B, Insets), since the characteristic blob size falls relative to the size of the entire population as the expansion proceeds. By contrast, for power-law growth the ratio $s^{*}\left(r\right)/r$ is independent of $r$, so rescaling $s$ with $r$ also causes the data curves to coincide (Fig. 3 C, Inset).

Fig. 3.

The time-doubling hierarchy is revealed in MRCA positions of pairs of sites in 1D simulations. (A–C) Dependence of the average MRCA radial position ${\overline{r}}_m$ on the separation $s$ between the pair in outbreaks starting from a single individual at the origin, with each curve corresponding to pairs whose mean position lies at a center–pair distance $r$ from the origin. In A–C the leftmost curve corresponds to $r=\mathrm{4,096}$ and subsequent $r$ values are related to the leftmost curve by successive multiples of 2 ($r=\mathrm{8,192}$, 16,384, …, 1,048,576). Values of $r/{\overline{r}}_m$ close to one signify that the pair belongs to the same satellite, whereas pairs split across different originating satellites have $r/{\overline{r}}_m\ll 1$. A–C correspond to stretched-exponential ($\mu =0.6$), marginal ($\mu =1$), and power-law ($\mu =1.5$) growth, respectively. Insets show the same curves, with the pair separation rescaled by the radial position. (D) Curves from A–C rescaled by the blob size scale $s*\left(r\right)=\ell \left[{\ell }^{-1}\left(r\right)/2\right]$ for each value of $r$.

A Coarse-Grained Model of Blob Replication Predicts Distinct Outcomes for Neutral Diversity in Different Growth Regimes.

Next, we investigate the effect of the time-doubling hierarchy of blob sizes on the global neutral genetic diversity as the range expansion progresses. Since each satellite originated from a single founder, the isolated growth of satellites acts as a coarsening mechanism which locally reduces diversity. However, all individuals in the core can contribute long-distance migration events, so the seeding of new satellites provides a mechanism to maintain global genetic diversity in the population. The competition between coarsening and diversification determines the fate of neutral diversity during jump-driven growth.

We now develop a semideterministic model for the evolution of the average neutral heterozygosity in jump-driven range expansions, which combines the deterministic placement and growth of satellite domains with random draws of the allelic identity of each domain. The jump-driven growth dynamic has two consequences for the dynamics of seeding and coarsening: 1) The allelic identity of a typical satellite domain joining the core at time $t$ is determined by a seed drawn from the gene pool of the core at time $t/2$. 2) The seed contributes its allelic identity to the entire satellite with size of order $\ell \left(t/2\right)$, which sets the scale of the coarsening at time $t$. These facts suggest that the evolution of heterozygosity is best described over “generations” that involve doublings in time, not population size. Specifically, in the deterministic approximation, the state of the core at time $t$ provides sufficient information to generate the core at time $2t$. The sizes of the satellites added to the population between $t$ and $2t$ are set by geometry and the form of $\ell \left(t\right)$ in the deterministic approximation, but the random drawing of seeds from the core introduces stochasticity to the process. This hierarchy can be formalized in an urn-like model of core growth through the accumulation of satellites, which produces stochastic outcomes which can be numerically studied (SI Appendix, section 1).

Analytical progress can be made through additional simplifying assumptions. First, we ignore the spatial structure of the core between doublings. Instead, we treat the core at each doubling as a spatially homogeneous mix of alleles from which seeds are randomly drawn. Second, rather than attempting to capture the time evolution of all alleles, we track the global heterozygosity $H$, a commonly used metric of population-level genetic variation. The heterozygosity is defined as the probability that a random pair sampled from the population will have alleles of different identity. To be concrete, we consider a doubling which evolves a homogeneous core of radial size $\ell \left(t_0\right)$ to size $\ell \left(2t_0\right)$. In the deterministic approximation, the doubling requires the addition of $g_1$ satellites, where $g_1\sim {\left[\ell \left(2t_0\right)/\ell \left(t_0\right)\right]}^d$ is the number of patches of the $t_0$ population which fit into the core at time $2t_0$ (Fig. 2D). The allelic identities of the new satellites are stochastically determined by random draws from the population within $\ell \left(t_0\right)$. The heterozygosity $H_1$ of the new population is therefore a stochastic random variable, but its expected value can be related to the initial heterozygosity $H_0$ by evaluating the probability that a random draw of two individuals from the new population is biallelic. This probability is zero if both individuals are drawn from the same new satellite and $H_0$ in all other situations. Ignoring the variation in size of the new satellites (see SI Appendix, section 1 for more details), the expected heterozygosity after one doubling is

$$\mathbb{E}\left[H_1\right]\approx \left(1-\frac{1}{g_1}\right)H_0\equiv f_1{H}_0,$$ [1]

where $i\in \left\{\mathrm{1,2}\dots ,g_1\right\}$ indexes the new satellites.

Eq. 1 captures the balance between coarsening and seeding in the deterministic approximation to jump-driven growth, with the influence of the growth dynamic manifested in the factor $g_1$. The value $f_1$ is always less than one because of the coarsening due to monoallelic satellites; however, it can be close to one if many small satellites are seeded in each doubling, so that $g_1\gg 1$ for a large number of satellites. Upon comparing Eq. 1 with the evolution of heterozygosity $H_1=\left(1-1/N\right)H_0$ for one generation of a haploid Wright–Fisher model with population size $N$, we may interpret $g_1$ as the effective population size determining the strength of genetic drift during the first “generation” which takes the population from time $t_0$ to $2t_0$.

To connect the evolution described above to the growth of a population out of a spatially homogeneous, compact homeland of population size $N_0$, we interpret the homeland as a core which has grown to a radius ${\ell }_0\equiv {\left(N_0/{\omega }_d\right)}^{1/d}$, where ${\omega }_d$ is the volume of the unit $d$-dimensional sphere (${\omega }_1=2$, ${\omega }_2=\pi$). The growth rule $\ell \left(t\right)$ can be inverted to define a characteristic time $t_0\equiv {\ell }^{-1}\left({\ell }_0\right)$ which sets the timescale over which heterozygosity is tracked in the coarse-grained model. The evolution of the average heterozygosity for growth over times $t\gg t_0$ is given by the cumulative effect of $n\approx {\log }_2\left(1+t/t_0\right)$ successive doublings, each indexed by $m$,

$$\mathbb{E}\left[H_n\right]=H_0\prod _{m=1}^n\left(1-\frac{1}{g_m}\right)=H_0\prod _{m=1}^n{f}_m,$$ [2]

where $g_m$ is the number of satellites generated in the $m$th doubling. Over long times, therefore, the neutral diversity in the effective population of satellites is determined by the behavior of the product ${\prod }_{m=1}^n{f}_m$, which in turn is determined purely by geometry and the form of the growth rule $\ell \left(t\right)$.

We now investigate the evolution of the average heterozygosity predicted by Eq. 2 for the asymptotic growth forms of Table 1. For kernels with $\mu <d$, the asymptotic growth rule is $\ell \left(t\right)\propto \exp \left(B_{\mu }{t}^{\eta }\right)$, where $\eta ={\log }_2\left[2d/\left(d+\mu \right)\right]$ approaches one as $\mu \to 0$ (the well-mixed limit) and zero as $\mu \to 1$. For successive doublings starting from a homeland of size ${\ell }_0=\ell \left(t_0\right)$, we find $g_m\sim {\left[\ell \left(2^m{t}_0\right)/\ell \left(2^{m-1}{t}_0\right)\right]}^d={\ell }_0^{d\left(1-2^{-\eta }\right)2^{\eta m}}$. Away from $\eta \approx 0$ (i.e., from $\mu \to d$), the number of satellites per doubling grows extremely fast ($\sim e^{e^m}$) with the number of doublings. As a result, $f_m\approx 1-1/g_m$ quickly approaches one with increasing $m$, leading to fast convergence of the product in Eq. 2 to a constant value (see SI Appendix, section 1 for details). The coarse-grained model therefore predicts that neutral diversity can be preserved at long times in this model: Complete fixation of one allele is avoided at long times, and a finite average heterozygosity is reached within a few doublings.

In contrast to the stretched-exponential growth rule, a power-law growth rule $\ell \left(t\right)\propto t^{1/\left(\mu -d\right)}$ has no intrinsic size scale: The relative geometry of the new satellites added during a deterministic doubling is independent of the size of the core. As a consequence, $f_1$ is independent of the size of the core, and $f_m=f_1\equiv f$ remains constant during subsequent doublings. Eq. 2 then predicts that the heterozygosity $H_n=f^n{H}_0$ decays exponentially with doublings. Unlike the stretched-exponential growth regime, the effective population of monoallelic satellites does not grow fast enough to evade fixation of a single allele over many doubling generations. The effective population of satellites loses genetic diversity in a process akin to genetic drift, due to the existence of a constant “effective population size” $N_{\text{e}}\sim g\sim {\left[\ell \left(2t\right)/\ell \left(t\right)\right]}^d=2^{d/\left(\mu -d\right)}$. We emphasize that the lapse of one generation for the effective population of satellites corresponds to a doubling of time, in contrast to a fixed increment of time for ordinary genetic drift.

Kernels with $\mu =d$ represent a marginal situation between power-law and stretched-exponential growth (Table 1), with a core growth $\ell \left(t\right)\sim \exp \left({\log }^{2}t\right)$ that is faster than any power law yet slower than stretched-exponential growth at long times. Repeating the above analysis for growth out of a homeland of radius $\ell \left(t_0\right)$, we find that the number of satellites and hence the effective population size grow exponentially in the number of doublings $m$ as $g_m\sim 2^{m/2}\sqrt{t_0}$. This growth is fast enough for the average neutral heterozygosity, Eq. 2, to converge to a finite value which lies between $H_0$ and zero (SI Appendix, section 2). In contrast with the power-law growth regime, the growth of the effective population size allows neutral diversity to be partially preserved over long times; however, the growth of $g_m$ with doublings and hence the convergence of the average heterozygosity are significantly slower than for the stretched-exponential regime.

In summary, the coarse-grained, semideterministic approximation of the jump-driven growth provides a minimal model that allows us to evaluate the competing effects of coarsening and diversification during jump-driven range expansions. The model predicts that when $\mu \le d$, the diversifying effect dominates and average heterozygosity converges to a finite value at long times. By contrast, the coarsening mechanism dominates when $d<\mu <d+1$, and diversity is steadily lost over time in a process similar to genetic drift in constant-size populations.

Heterozygosity Evolution in Simulations Is Consistent with the Coarse-Grained Model.

To test whether the predictions of the coarse-grained model hold under the full dynamics, we investigate the evolution of neutral diversity in simulations with $q=2$, for which the global diversity is given by $H=2p\left(1-p\right)$ where $p$ is the fraction of one allele, starting from its initial value $p_0=1/2$. For each kernel, hundreds of independent simulations were performed and their heterozygosities were averaged to produce the ensemble-averaged heterozygosity $⟨H⟩$. Measurements of $⟨H⟩$ from both 2D and 1D simulations are reported in Fig. 4.

Fig. 4.

Average final heterozygosity is nonmonotonic in kernel exponent. (A and B) Average heterozygosity measured from simulations at different values of kernel exponent $\mu$ for 2D (A) and 1D (B). The initial homeland size is $N_0=111$ in A and $N_0=100$ in B. The average is shown for three values of final population size. Error bars show SEM across 200 to 800 independent simulations. Insets show the ensemble-averaged heterozygosity traces against population size for a subset of the simulated kernels (labeled). Asterisks denote data points for which the largest system size was restricted to the following values due to simulation constraints: 2D, $N=6.4\times 10^7$ for $\mu =4$ and $N=5.6\times 10^7$ for $\mu =6$; 1D, $N=2.4\times 10^7$ ($\mu =1.6$), $N=7\times 10^6$ ($\mu =1.8$), $N=8\times 10^5$ ($\mu =3$), $N=6\times 10^5$ ($\mu =4$).

For both planar and linear range expansions, the measured average heterozygosities are consistent with the predicted trends. In 2D simulations (Fig. 4A), the average heterozygosity does not change significantly between final population sizes of $10^7$ and $10^8$ for $\mu <2$ and $\mu >3$, signifying convergence to a finite value (see Fig. 4 A, Inset and SI Appendix, Fig. A7 for evolution of heterozygosities with population size). However, in the range of kernel exponents $2\le \mu \le 3$, the average heterozygosity continues to drop with population growth over the entire range of simulated sizes, which is consistent with a steady loss of diversity as the population grows for this range of kernels. The results of 1D simulations (Fig. 4B and SI Appendix, Fig. A8) are qualitatively similar, but with persistent loss of diversity occurring for kernel exponent values $1\le \mu \le 2$. These results support the conclusion that coarsening leads to loss of neutral variation when $d<\mu <d+1$, whereas diversification preserves neutral variation for $\mu <d$. The preservation of partial diversity when $\mu >d+1$ is due to the known mechanism of sector formation, since the range expansion occurs via a constant-speed front in this range.

Although the qualitative difference between kernels with $\mu <d$ and $\mu >d$ is readily observed, the outcome of the marginal situation $\mu =d$ is less clear. In both 2D and 1D, the average heterozygosity is still falling up to the largest simulated population sizes (Fig. 4). However, this observation is consistent with the conclusions from the coarse-grained model. For marginal growth, the convergence of the product in Eq. 2 occurs over many time-doubling generations. In real time, this implies that the population must grow over many orders of magnitude before the average heterozygosity converges to its limiting value. In SI Appendix, Fig. A3, we show that the heterozygosity decay observed in simulations is consistent with Eq. 2, for several homeland sizes in both 1D and 2D. According to our theory, the largest simulations we have run correspond to roughly 7 time-doubling generations, whereas convergence is expected after 15 to 20 doublings. Therefore, our simulated population sizes are still many orders of magnitude too small for convergence to to be observed.

SI Appendix, section 2 also reports additional predictions for the average heterozygosity from the coarse-grained model in the stretched-exponential ($\mu <d$) and power-law ($d<\mu <d+1$) regimes. Comparisons of these predictions with simulation results (SI Appendix, Figs. A2 and A4) demonstrate that the coarse-grained model, despite its simplifications and approximations, reproduces many features of the heterozygosity trends for kernels across different regimes.

Discussion

We have demonstrated that long-range dispersal can dramatically impact the local structures and global trends of neutral genetic diversity left behind by a range expansion. Specifically, dispersal kernels with a power-law tail characterized by exponent $\mu <d+1$ show patterns of diversity that qualitatively differ from the radial sectors left behind by populations which expand through local dispersal (cf. Fig. 1). Locally, regions dominated by a single allele form nearly isotropic patches, termed “blobs,” whose typical size increases as the range expansion progresses. Upon considering the growth in blob size relative to the size of the population itself, two distinct regimes have been identified. When $d<\mu <d+1$, the typical size of the largest blobs is a constant kernel-dependent fraction of the size of the population (i.e., $s^{*}\left(r\right)/r=\text{const.}$; Table 1). By contrast, when $\mu \le d$, blob sizes shrink relative to the population as the expansion progresses, leading to a pattern of fine monoallelic speckles (Fig. 1 I–K).

As a consequence of a time-doubling hierarchy inherent to jump-driven growth, we have identified an effective population size of blobs generated during a doubling in time. The evolution of this effective population over “time-doubling generations” (which are constant increments in log time) follows distinct trends in the two regimes of jump-driven growth: The effective population size is stagnant in the first growth regime ($d<\mu <d+1$), but grows in the second regime ($\mu \le d$). As shown schematically in Fig. 5, the resulting stochastic fluctuations in allele fractions accumulate to produce very different outcomes from superficially similar growth mechanisms in a process akin to genetic drift, but where each generation is an increment in ${\log }_2\left(t/t_0\right)$ rather than a fixed time interval. Although the simulations reported above include only two distinct alleles, our coarse-grained model applies for generic initial allele numbers and frequencies, and our predictions for the evolution of heterozygosity are independent of the number of distinct alleles present (SI Appendix, section 3 and Fig. A6).

Fig. 5.

The effective population of satellites determines the fate of neutral diversity. The coarse-grained model of jump-driven growth treats a range expansion out of a well-mixed homeland as a succession of time doublings indexed by $i$, each of which adds $g_i$ monoallelic satellites drawn at random from the population after the previous doubling. The $g_i$ values are determined by the core growth rule, $\ell \left(t\right)$ (Table 1). (A) In the power-law growth regime, $g_i$ remains constant over successive time-doubling generations (Left), where the lapse of a generation in the effective population requires time to double in the actual population (i.e., generations are increments in ${\log }_2\left(t\right)$). As a result, genetic diversity is eroded over time due to chance events, in a process analogous to genetic drift in a well-mixed population with the number of individuals kept constant (Right). (B) By contrast, in the marginal and stretched-exponential growth regimes, the effective population of satellites grows with each time-doubling generation, and the growth is fast enough to mitigate drift.

The breakup of sectors into blobs and speckles has important genetic consequences. As can be seen in Fig. 1 A–C, the sector geometry generated by constant-speed range expansions leaves a signature of the direction of expansion on local genetic patterns (14, 18), which can be detected in principal-component maps of genetic variation (42). By contrast, the relation between local blob geometry and global population history in jump-driven expansions (Fig. 1 E–K) is more subtle: Individual blob shapes are isotropic and do not directly reveal the expansion direction. Blobs do get larger with increasing distance from the homeland on average, but the significant stochastic variation in blob sizes makes this signature difficult to detect. Since neighboring blobs could have been seeded by migrants from well-separated regions, jump-driven range expansions are also expected to have significantly higher levels of mixing compared to sectored expansions. This mixing reduces the positional advantage conferred to mutations which arise near the edge of the expanding population, thus mitigating the gene surfing effect (12, 13). The finer structure and increased mixing due to blobs are likely to have an impact on the evolutionary effects of geographic structure such as reduced adaptive potential (43), response to inbreeding depression (44), and expansion load (45).

Whereas we have focused on describing the expansion of a population into previously unoccupied territory, our results are applicable to other biological expansions as well. The growth dynamics of our model also apply to the spread of beneficial mutations from a localized region into an established wild-type population with a spatially uniform population density (19). In this context, our model would describe the patterns of variation caused by the spread of distinct beneficial mutations with similar fitness effects—a soft sweep (41, 46, 47)—out of a small region (the homeland) which experienced a selection pressure earlier than the rest of the population. Analogues of the patterns and mechanisms described here could also play a role in within-host viral dynamics during infections (48, 49) and in cancer metastasis (50, 51).

An outstanding question regarding the population-genetic consequences of dispersal on range expansions has been whether enhanced dispersal preserves or erodes diversity relative to short-range migration. In a previous study of mixtures of two non–fat-tailed kernels with different characteristic jump lengths, intermediate levels of longer-range jumps were shown to reduce neutral diversity relative to low or high levels in narrow corridors (33). The reduction occurred when a fortuitously placed satellite from a pioneer seed quickly filled the width of the corridor and blocked other alleles from advancing, a mechanism termed the embolism effect [this is a version of the more generic “founder takes all” mechanism (52)]. However, restriction of growth along a narrow corridor is essential for the embolism effect to wipe out genetic variation (34); in a radial expansion, embolisms would suppress diversity only within certain angular ranges, and neutral variation would persist in the form of sectors at long times. Furthermore, the embolism effect was demonstrated for kernels with a strict upper limit to the allowed jump distances, which restricted the ability of individuals from the interior of the population to contribute to diversification. For these reasons, other studies have speculated that fat-tailed kernels without an upper cutoff in dispersal distance might not experience the embolism effect and might enhance diversity relative to short-ranged dispersal in all cases (34–37).

In the context of these previous studies, a key result of our work is that even fat-tailed kernels without a cutoff can induce a loss of neutral variation. Similarly to ref. 33, we have shown that boosting long-range dispersal has a nonmonotonic effect on diversity: Intermediate kernels (i.e., with power-law exponent $d<\mu <d+1$) preserve less variation than broader ($\mu <d$) or narrower ($\mu >d+1$) kernels. However, the loss of neutral variation we observe for intermediate kernels ($d<\mu <d+1$) is fundamentally different from the embolism effect: It relies on the fat-tailed nature of the kernel, which allows a finite probability of jumps that span the whole population regardless of its size. Through a mechanism reminiscent of genetic drift, the expanding population is engulfed by satellites belonging to a single allele purely through chance. As a result, the decay of heterozygosity is observed even in 2D radial range expansions which are not confined to a corridor. Even in a linear geometry, our mechanism can be distinguished from the embolism effect. For a 1D expansion from a central source growing out in two directions, the embolism effect could support two different alleles growing out in the left and the right directions. By contrast, in our model diversity is completely eroded when $1<\mu <2$ in 1D, with only one allele surviving at very long times.

Theoretical and computational studies (12, 13, 15, 33–37) and laboratory experiments on model organisms (14), although highly simplified relative to real-world populations, have nevertheless provided heuristics and mechanisms that have guided the interpretation of field genomic data. In the context of range expansions, the concepts of gene surfing, sectors, and embolisms have been invoked to explain patterns of genetic variation in a wide range of plant (53–55), animal (56, 57), and microbial (58) populations. Our work shows that these heuristics are incomplete when long-range dispersal is present: Sectors give way to blobs and speckles with increasing levels of dispersal; the mechanism of engulfment can engender diversity loss even for fat-tailed kernels in the absence of corridors, where the embolism effect does not apply.

Many interesting avenues for further study can be identified. By filling up demes instantaneously and irreversibly, we have focused in this work on the genetic patterns seen immediately after colonization. We have ignored subsequent reshuffling of alleles among demes, which would blur the boundaries between blobs at later times and smear out the predicted spatial patterns over time. This smearing would impact the ability to detect the patchiness of jump-driven range expansions at later times, with the problem being more severe closer to the homeland. While previous studies have shown that gene segregation due to the range expansion persists for appreciable periods of time beyond the colonization for both short-range (18) and long-range (34) dispersal, a more refined spatial model incorporating the exchange of individuals among demes of finite population size would address the question of how long the patterns remain measurable at different distances from the homeland. (Note that the blurring of patterns near the interior during the range expansion does not impact the evolution of global diversity, since the spatial organization of alleles in the core is irrelevant to the identity of satellites generated in the time-doubling hierarchy.) Such a model could also be used to study the interplay of genetic drift within demes and the large-scale diversity evolution captured in our coarse-grained model. In addition, the applicability of our results to continuous populations without a deme structure could be studied by introducing long-range dispersal into continuum population genetics models (59, 60) and simulations (61).

Our results show that spatial constraints fundamentally alter the mathematical structures underlying neutral evolution in expanding populations. Whereas well-mixed populations map onto Markovian Pólya–Eggenberger urn models (62), the hierarchy of doublings in time eliminates Markovianness in the full jump-driven growth: The genetic makeup of individuals entering the population at time $t$ depends on the state of the population at times of order $t/2$. Furthermore, the reinforcing influence (63) of a random draw (the size of the satellite generated by a rare long-distance jump) increases over time as ${\ell }^d\left(t/2\right)$. Generalized urn-like models that incorporate these features could be useful to develop and analyze as minimal mathematical models of genetic diversity in spatially structured expanding populations. In particular, they could lead to an understanding of heterozygosity distributions, including limit distributions in situations where the long-time heterozygosity converges.

The coarse-grained model of blob evolution provides a route to understanding the genealogical structures left behind by jump-driven expansion, which is crucial for demographic inference. The effective population of satellites, over generations corresponding to doublings in time, is much simpler to describe compared to the full stochastic dynamics; analyzing the genealogical structure of this effective population in different growth regimes would be a useful first step to understanding genealogies in the full stochastic model. For instance, in the power-law growth regime ($d<\mu <d+1$), the effective population size is constant, which suggests that the genealogies of satellites are described by the Kingman coalescent under an appropriate rescaling of the time variable (essentially, ${\log }_2\left(t\right)$ must be used in place of time for the Kingman coalescent to apply to the effective population of satellites). More generally, the temporal connection between the population at time $t$ and its state at the earlier time $t/2$ is reminiscent of the mechanism underlying the seed-bank coalescent (64, 65) which might be applicable to our model in all growth regimes. Computations of genealogy statistics in the effective satellite population could be used to generate quantitative predictions for individual-based sampling statistics such as allele frequency spectra. Such predictions, which will also require evaluating the statistical relationships between samples of individuals and samples of satellites in the time-doubling hierarchy, are an exciting target for future work.

Materials and Methods

Simulations were implemented in the C++ programming language. Pseudorandom numbers were generated using the Mersenne Twister engine provided in the C++ standard library. Deme positions are quantized to an integer lattice in $d$ dimensions. The simulation keeps track of all occupied demes and the allelic identity (0 or 1) of each deme. To avoid finite-size effects without initializing enormous arrays of mostly empty demes, occupied demes were stored in unordered containers implemented using hash tables (66, 67) (specifically, unordered_map from the C++ Standard Template Library was used). Using this approach, the effective lattice size is $2^{64}$ demes in 1D and $2^{32}\times 2^{32}$ demes in 2D.

Simulations are initialized by randomly assigning allelic identities to a compact zone around the origin. In 1D, demes are filled in the range $\left(-N_0/2,N_0/2\right]$, whereas in 2D all demes are filled out to a specified distance from the origin. At each simulation step, an occupied deme is chosen at random from the population as the source for a jump attempt. The jump distance, $r$ is drawn at random from the probability distribution $J\left(r\right)=\mu r^{-\left(\mu +1\right)}$ (operationally, a random number $X$ is drawn from the uniform real number distribution between 0 and 1, and $r=X^{-1/\mu }$ then provides a variable which follows the $J\left(r\right)$ distribution). A random $d$-dimensional unit vector is also generated ($\pm 1$ in 1D and evenly distributed on the unit circle in 2D) and multiplied with $r$, following which each component is rounded to the nearest integer to obtain the candidate jump vector. The target deme for the jump attempt is obtained by adding the source deme position to the candidate jump vector. If the target deme is empty, it is filled with the allelic identity of the source; if it is occupied, the jump attempt is not successful.

The output of the simulations varied based on the measured quantity. For tracking the heterozygosity, it was sufficient to record the allele fractions at successive population sizes. Images of simulated populations required occupied deme positions and allelic identities to be recorded. Measurements of the MRCA positions were conducted in separate simulations in which the outbreak was begun from a single occupied deme, and a genealogical tree was maintained and updated at each successful jump. At the end of each simulation, the MRCA was recorded for pairs positioned at different center–pair distances from the origin.

Run times are determined by the number of failed attempts made as the simulation progresses toward a target population size. The run time for individual simulations ranged from a few minutes to 72 h and was significantly higher for kernels with $\mu >d$ compared to broader kernels. The upper limit was set by access to computational resources, and as a result, desired population sizes could not be achieved for the narrowest kernels. Ensemble-averaged heterozygosity measurements were obtained by averaging over 100 to 700 independent simulations for each set of parameters, depending on system size. MRCA distance measurements were obtained by averaging over 1,000 independent simulations for $\mu =0.6$ and $\mu =1.0$ and over 400 independent simulations for $\mu =1.4$.

Code Availability.

Code used to generate simulation data will be made available by the authors upon request.