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# Acceleration of evolutionary spread by long-range dispersal

Edited* by Herbert Levine, Rice University, Houston, TX, and approved September 2, 2014 (received for review March 12, 2014)

## Significance

Pathogens, invasive species, rumors, or innovations spread much more quickly around the world nowadays than in previous centuries. The speedup is caused by more frequent long-range dispersal, for example via air traffic. These jumps are crucial because they can generate satellite “outbreaks” at many distant locations, thus rapidly increasing the total rate of spread. We present a simple intuitive argument that captures the resulting spreading patterns. We show that even rare long-range jumps can transform the spread of simple epidemics from wave-like to a very fast type of “metastatic” growth. More generally, our approach can be used to describe how new evolutionary variants spread and thus improves our predictive understanding of the speed of Darwinian adaptation.

## Abstract

The spreading of evolutionary novelties across populations is the central element of adaptation. Unless populations are well mixed (like bacteria in a shaken test tube), the spreading dynamics depend not only on fitness differences but also on the dispersal behavior of the species. Spreading at a constant speed is generally predicted when dispersal is sufficiently short ranged, specifically when the dispersal kernel falls off exponentially or faster. However, the case of long-range dispersal is unresolved: Although it is clear that even rare long-range jumps can lead to a drastic speedup—as air-traffic–mediated epidemics show—it has been difficult to quantify the ensuing stochastic dynamical process. However, such knowledge is indispensable for a predictive understanding of many spreading processes in natural populations. We present a simple iterative scaling approximation supported by simulations and rigorous bounds that accurately predicts evolutionary spread, which is determined by a trade-off between frequency and potential effectiveness of long-distance jumps. In contrast to the exponential laws predicted by deterministic “mean-field” approximations, we show that the asymptotic spatial growth is according to either a power law or a stretched exponential, depending on the tails of the dispersal kernel. More importantly, we provide a full time-dependent description of the convergence to the asymptotic behavior, which can be anomalously slow and is relevant even for long times. Our results also apply to spreading dynamics on networks with a spectrum of long-range links under certain conditions on the probabilities of long-distance travel: These are relevant for the spread of epidemics.

Humans have developed convenient transport mechanisms for nearly any spatial scale relevant to the globe. We walk to the grocery store, bike to school, drive between cities, or take an airplane to cross continents. Such efficient transport across many scales has changed the way we and organisms traveling with us are distributed across the globe (1⇓⇓⇓–5). This has severe consequences for the spread of epidemics: Nowadays, human infectious diseases rarely remain confined to small spatial regions, but instead spread rapidly across countries and continents by travel of infected individuals (6).

Besides hitchhiking with humans or other animals, small living things such as seeds, microbes, or algae are easily caught by wind or sea currents, resulting in passive transport over large spatial scales (7⇓⇓⇓⇓⇓⇓⇓⇓–16). Effective long-distance dispersal is also widespread in the animal kingdom, occurring when individuals primarily disperse locally but occasionally move over long distances. And such animals, too, can transport smaller organisms.

These active and passive mechanisms of long-range dispersal are generally expected to accelerate the growth of fitter mutants in spatially extended populations. However, how can one estimate the resulting speedup and the associated spatiotemporal patterns of growth? When dispersal is only short range, the competition between mutants and nonmutated (“wild-type”) individuals is local, confined to small regions in which they are both present at the same time. As a consequence, a compact mutant population emerges that spreads at a constant speed, as first predicted by Fisher (17) and Kolmogorov et al. (18): Such selective sweeps are slow and dispersal is limited. In the extreme opposite limit in which the dispersal is so rapid that it does not limit the growth of the mutant population, the competition is global and the behavior the same as for a fully mixed (panmictic) population: Mutant numbers grow exponentially fast. It is relevant for our purposes to note that in both the short-range and extreme long-range cases, the dynamics after the establishment of the initial mutant population are essentially deterministic.

When there is a broad spectrum of distances over which dispersal occurs, the behavior is far more subtle than that of either of the well-studied limits. When a mutant individual undergoes a long-distance dispersal event—a jump—from the primary mutant population into a pristine population lacking the beneficial mutation, this mutant can found a new satellite subpopulation, which can then expand and be the source of further jumps, as shown in Fig. 1 *B* and *C*. Consequently, long-range jumps can dramatically increase the rate of growth of the mutant population. Potentially, even very rare jumps over exceptionally large distances could be important (14). If this is the case, then the stochastic nature of the jumps that drive the dynamics will be essential.

Although evolutionary spread with long-range jumps has been simulated stochastically in a number of biological contexts (6, 8, 19⇓⇓⇓–23), few analytic results have been obtained on the ensuing stochastic dynamics (19, 24⇓–26). Most analyses have resorted to deterministic approximations (27⇓⇓⇓⇓⇓⇓⇓–35), which are successful for describing both the local and global dispersal limits. However, in between these extreme limits, stochasticity drastically changes the spreading dynamics of the mutant population. This is particularly striking when the probability of jumps decays as a power law of the distance. Just such a distance spectrum of dispersal is characteristic of various biological systems (36⇓⇓⇓⇓–41), including humans (3). We will show that the behavior is controlled by a trade-off between frequency and potential effectiveness of long-distance jumps and the whole spectrum of jump distances can matter. The goal of this paper is to develop the theory of stochastic spreading dynamics when the dispersal is neither short range nor global.

Long-distance dispersal can occur either on a fixed network or more homogeneously in space. For simplicity, we focus on the completely homogeneous case and then show that many of the results also apply for an inhomogeneous transportation network with hubs between which the long-distance jumps occur. For definiteness, we consider for most of the paper the evolutionary scenario of the spread of a single beneficial mutation, but, by analogy, the results can be applied to other contexts, such as the spread of infectious disease or of invasive species.

## Basic Model

The underlying model of spatial spread of a beneficial mutant is a population in a *d*-dimensional space with local competition that keeps the population density constant at *r* away of *s*, over the original population. A lattice version of this model is more convenient for simulations (and for aspects of the analysis): Each lattice site represents a “deme” with fixed population size, *s*, it survives stochastic drift to establish, it will take over the local population. When the total rate of migration between demes is much slower than this local sweep time, the spatial spread is essentially from demes that are all mutants to demes that are all of the original type.

Short jumps result in a mutant population that spreads spatially at a roughly constant rate. However, with long-range jumps, new mutant populations are occasionally seeded far away from the place from which they came, and these also grow. The consequences of such long jumps are the key issues that we need to understand. As we shall see, the interesting behaviors can be conveniently classified when the jump rate has a power-law tail at long distances, specifically, with *μ* needed for the total jump rate to be finite). Crudely, the behavior can be divided into two types: linear growth of the radius of the region that the mutants have taken over and faster than linear growth. In Fig. 1, these two behaviors are illustrated via simulations on 2D lattices. In addition to the mutant-occupied region at several times, shown are some of the longest jumps that occur and the clusters of occupied regions that grow from these. In Fig. 1*A*, there are no jumps that are of comparable length to the size of the mutant region at the time at which they occur, and the rate of growth of the characteristic linear size *B* and *C*,

## Results

### Simulated Spreading Dynamics.

We have carried out extensive simulations of a simple lattice model, in which the sites form either a one-dimensional, of length *L*, or two-dimensional *s*) that the mutant establishes a new population: We define **x** nucleates a mutant population near **y**. In each computational time step, we pick a source and target site randomly such that their distance *r* is sampled from the (discretized) jump distribution *SI Text*, section *SI1* for more details on the simulation algorithm.

The growth of mutant populations generated by our simulations is best visualized in a space–time portrait. Fig. 2 *A* and *B* shows the overlaid space–time plots of multiple runs in the regimes *μ*. For

To explain these dynamics in detail, we develop an analytical theory that is able to predict not only the asymptotic growth dynamics but also the crucial transients.

### Breakdown of Deterministic Approximation.

Traditionally, analyses of spreading dynamics start with a deterministic approximation of the selective and dispersal dynamics—ignoring both stochasticity and the discreteness of individuals. To set up consideration of the actual stochastic dynamics, we first give results in this deterministic approximation and show that these exhibit hints of why they break down.

When the jump rate decreases exponentially or faster with distance, the spread is qualitatively similar to simple diffusive dispersal and the extent of the mutant population expands linearly in time. However, when the scale of the exponential falloff is long, the speed, *v*, is faster than the classic result for local dispersal (17, 18), *D* is independent of the characteristic length, *b*, of the jumps. A linear deterministic approximation for the mutant population density in a spatial continuum and a saddle point analysis to find the distance at which the population density becomes substantial yield, as for the conventional diffusive case, the correct asymptotic speed. The resulting expression for the speed is modified from the diffusive result by a function of the only dimensionless parameter,

We now provide a simple argument as to why the deterministic approximation drastically overshoots the stochastic spreading dynamics for broader than exponentially decaying jump kernels **R**. After a time of order **R** takes off, proportional to *b* (i.e., **R** indeed dominate for large distances *R*. (For small *b*, in contrast, the speed is much faster than *bs* because the spread is dominated by multiple small jumps: The diffusion approximation is then good.)

If

We can now understand why the deterministic approximation fails miserably for very long-range jumps. In the time, *R*, away, the expected total number of jumps to the whole region *R* or farther from the origin is only of order *R* and the deterministic approximation must fail (19, 44).

With local dispersal, the deterministic approximation is a good starting point with only modest corrections to the expansion speed at high population density, the most significant effect of stochasticity being fluctuations in the speed of the front (45). At the opposite extreme of jump rate independent of distance, the deterministic approximation is also good with the mutant population growing as *a*: These arise from early times when the population is small. It is thus surprising that in the regimes intermediate between these two, the deterministic approximation is not even qualitatively reasonable.

### Iterative Scaling Argument.

We assume that, at long times, most of the sites are filled out to some distance scale *SI Text*, section *SI3*. We call the crossover scale

In the dynamical regimes of interest, the core population grows primarily because it “absorbs” satellite clusters, which themselves were seeded by jumps from the core population. We now show that the rate of seeding of new mutant satellite clusters and the growth of the core populations by mergers with previously seeded clusters have to satisfy an iterative condition that enables us to determine the typical spreading dynamics of the mutant population.

It is convenient to illustrate our argument using a space–time diagram, Fig. 4, in which the growth of the core has the shape of a funnel. Now consider the edge of this funnel at time *T* (gray circle in Fig. 4). The only way that this edge can become populated is by becoming part of a population subcluster seeded by an appropriate long-range jump at an earlier time. To this end, the seed of this subcluster must have been established somewhere in the inverted blue funnel in Fig. 4. This “target” funnel has the same shape as the space–time portrait of the growing total population, but its stem is placed at

Now, we argue the consistency of growth and seeding requires that there is, on average, about one jump from the source to the target funnel: If it were unlikely that even one jump leads from the source to the target region, then the assumed shape for the source funnel would be too large and its edge (gray circle in Fig. 4) would, typically, not be occupied. Conversely, if the expected number of jumps was much larger than 1, then seeding would occur so frequently that a much larger funnel than the assumed one would typically be filled by the time *T*.

The condition of having, on average, about one jump from source to target region can be stated mathematically as*d* dimensions, where *d*-dimensional ball of radius **e** a unit vector in an arbitrary direction. The kernel *d*-dimensional volume of (established) jumps of size *r*. Eq. **1** mathematizes the space–time picture of Fig. 4: To calculate the expected number of jumps from the red source to the blue target funnel, we need a time integral, **x** to the target point

Note that the above intuitive picture is further sharpened in *SI Text*, section *SI2* and justified by developing rigorous bounds in *SI Text*, section *SI3*.

### Asymptotic Results for Power-Law Jumps.

We now show that the asymptotic growth dynamics are essentially constrained by the above iterative scaling argument. Specifically, although the argument is more general, we consider a power-law jump distribution*r*. The prefactor *ε* in

For the intermediate-range case **1** and **2** exhibit the asymptotic scaling solution**3**] into Eq. **1** determines the prefactor *SI Text*, section *SI2.B*. Interestingly, the value *μ* and runs from 0 to ∞ as *μ* passes through the interval from *d* to *μ* for

We now turn to the (very) long-range case, **1**] cannot be found (and the dimensional analysis argument gives nonsense). However, much can be learned by approximating [**1**] using *x* and *y* compared with **4**] is a rapidly growing stretched exponential,**4**. Note that as *SI Text*, section *SI3*.

For the marginal case, *μ* close to *d*: This is the source of the singular behavior of the coefficients

Our source-funnel argument obviously neglects jumps that originate from the not fully filled regions outside the core radius *SI Text*, section *SI2.E*, which also allows us to estimate the probability of occupancy outside the core region. Further, we present in *SI Text*, section *SI3* outlines of rigorous proofs of lower and upper bounds for the asymptotic growth laws in one dimension, including the slow crossovers near the marginal case. The linear growth for *SI Text*, section *SI3*. More importantly, our bounds are explicitly for the iterative scaling analyses, thereby justifying them (and future uses of them). Our bounds hence include the full crossovers for *μ* near *d* (in one dimension), rather than just the asymptotic results: As we next show, understanding these crossovers is essential.

### Crossovers and Beyond Asymptopia.

Asymptotic laws are of limited value without some understanding of their regime of validity, especially if the approach to the asymptotic behavior is slow. And such knowledge is crucially needed to interpret and make use of results from simulations.

Assuming that long jumps are typically much rarer than short jumps, they will become important only after enough time has elapsed that there have been at least some jumps of order *SI Text*, section *SI2*.*A*). For the purpose of this section, it is convenient to measure time and length in units of these elementary crossover scales.

At times much longer than 1 (in rescaled time), we expect another slow crossover close to the boundary between the stretched-exponential and power-law regimes. Thus, we must take a closer look at the dynamics in the vicinity of the marginal case, **4** develops a sharp peak at *t* and to a larger logarithmic factor associated with the narrowness of the range of integration and its dependence on *t* and *SI Text*, section *SI2.C*, are negligible if we focus on the behavior on logarithmic scales in space and time—natural given their relationships. Defining **7** yields a linear recurrence relation that can be solved exactly (*SI Text*, section *SI2.B*). Rescaling *ℓ* can be used to make

The asymptotic scaling for **3**] and yields the prefactor *δ* (compare Fig. S3). For **5**] and fixes the prefactor

The singular prefactors for *μ* near *d*. The asymptotic scaling can be observed only on times and length such that**6**]. The rapid divergence of the logarithm of the time after which the asymptotic results obtain makes it nearly impossible to clearly observe the asymptotic limits: In one-dimensional simulations this problem occurs when **8**].

Although the dynamics at moderate times will be dominated by the initial growth characteristic of the marginal case, we expect [**8**] to be a good description of the universal dynamics at large *δ* is small. The limit **8**] then reduces to a scaling form*δ* in one scaling plot (Fig. 5). To make the approximation uniformly valid both in the scaling regime and at asymptotically long times outside of it, we can simply replace, for *η* defined in [**5**]). The scaling form [**10**] will then be valid up to corrections that are small compared with the ones given in all regimes: We thus use this form for the scaling fits in Fig. 5, *Inset*, plotting data obtained for different *δ* in one scaling plot, thereby testing our solution across all intermediate asymptotic regimes.

### Heterogeneities and Dynamics on Networks.

Thus far we have considered spatially uniform systems, in which the jump probability between two points depends only on their separation. However, long-distance transport processes may be very heterogeneous. An extreme example is airplane travel, which occurs on a network of links between airports with mixtures of short- and long-distance flights, plus local transportation to and from airports. A simple model is to consider each site to have a number of connections from it, with the probability of a connection between each pair of sites a distance *r* away being *r* is *r* is

If there are insufficient numbers of connections for the heterogeneity of the network to be effectively averaged over, the behavior changes. The extreme situation is when there is a distance-independent rate for jumps along the longest connection out of a site: i.e., *S*, to get from the origin to a point *R* will reach that point in a time proportional to *S*; i.e., *η* as in the homogeneous case we have analyzed. The difference between this result and ours is only in the power-law-of-*T* prefactor of *α* dependent on the coefficient of the power-law decay of the connection probability.

If the probability of a jump along a long-distance connection decays with distance but more slowly than

For natural transport processes, the probabilities of long dispersal events will depend on both the source and the destination. If the heterogeneities are weak on large length scales, our results still obtain. However, if there are sufficiently strong large-scale heterogeneities, either in a spatial continuum or in the network structure (i.e., location of nodes and links and the jump rates along these or hub-spoke structure with multiple links from a small subset of sites), then the spatial spread will be heterogeneous even on large scales: How this reflects the underlying heterogeneities of the dispersal has to be analyzed on a case-by-case basis.

## Discussion

### Modeling Evolutionary Spread.

We have studied the impact of long-range jumps on evolutionary spreading, using the example of mutants that carry a favorable genetic variant. To this end, we analyzed a simple model in which long-range jumps lead to the continual seeding of new clusters of mutants, which themselves grow and send out more migrant mutants. The ultimate merging of these satellite clusters limits the overall growth of the mutant population, and it is a balance of seeding and merging of subclusters that controls the spreading behavior.

To classify the phenomena emerging from this model, we focused on jump distributions that exhibit a power-law tail. We found that, with power-law jumps, four generic behaviors are possible on sufficiently long times: The effective radius of the mutant population grows at constant speed, as a superlinear power law of time, as a stretched exponential, or simply exponentially depending on the exponent,

The breakdowns of both deterministic and diffusive expectations are indicative of the importance of fluctuations: The dynamics are dominated by very rare—but not too rare—jumps: roughly, the most unlikely that occur at all up to that time. One of the consequences of this control by the rare jumps is the relatively minor role played by the linear growth speed *v* of individual clusters due to short-range migration: In the regime of power-law growth, the asymptotic growth of the mutant population becomes (to leading order) independent of *v* although when individual clusters grow more slowly, the asymptotic regime is reached at a later time. In the stretched exponential regime, the growth of subclusters sets the crossover time from linear to stretched exponential and thus determines the prefactor in the power law that characterizes the logarithm of the mutant population size (*Crossovers and Beyond Asymptopia*).

An important feature of the spreading dynamics is that the approach to the asymptotic behavior is very slow in the vicinity of the marginal cases, as illustrated in Fig. 6. Consider, for instance, the 2D case, where we have asymptotically stretched exponential growth for

However, based on a simple geometrical argument illustrated in Fig. 4, we could show that the full crossover dynamics can be understood from a trade-off between frequency and potential effectiveness of long-distance jumps: Jumps of a given size are more abundant at late times (source funnel is large) but they are most effective at early times (blue funnel is large). As a result, the key jumps that dominate the filled regions at time *T* predictably occur near half that time. This led to a simple recurrence relation for the spread at time *T* in terms of its behavior at time **7**). Its exact solution predicts a universal crossover function for the transient dynamics near *SI Text*, section *SI3* provide further support. Understanding this crossover is essential for making sense of, and extrapolating from, simulations as asymptotic behavior is not visible until enormous system sizes even when the exponent *μ* is more than 0.2 from its marginal value.

Another benefit of the ability of the simple iterative scaling argument to capture well nonasymptotic behavior is that it can be used in cases in which the dispersal spectrum of jumps is not a simple power law, e.g., with a crossover from one form to another as a function of distance. And the heuristic picture that it gives rise to—an exponential hierarchy of timescales separated by roughly factors of 2—is suggestive even in more complicated situations. That such a structure should emerge without a hierarchal structure of the underlying space or dynamics is perhaps surprising.

### Potential Applications and Dynamics of Epidemics.

Our primary biological aim is the qualitative and semiquantitative understanding that emerges from consideration of the simple models and analyses of these, especially demonstrating how rapid spatial spread of beneficial mutations or other biological novelties can be even with very limited long-range dispersal. As the models do not depend on any detailed information about the biology or dispersal mechanisms, they can be considered as a basic null model for spreading dynamics in physical, rather than more abstract network, space.

The empirical literature suggests that fat-tailed spectra of spatial dispersal are common in the biological world (8, 13, 36⇓⇓⇓⇓–41). Because most of these are surely neither a constant power law over a wide range of scales nor spatially homogeneous, our detailed results are not directly applicable. However, as discussed above, our iterative scaling argument is more general and can be applied with more complicated distance dependence, anisotropy, or even directional migration (47). Furthermore, some of the heterogeneities of the dispersal will be averaged out for the overall spread, while affecting when mutants are likely to arrive at particular locations.

For dispersal via hitchhiking on human transport, either of pathogens or of commensals such as fruit flies with food, the apparent heterogeneities are large because of the nature of transportation networks. Nevertheless, data from tracking dollar bills and mobile phones indicate that dispersal of humans can be reasonably approximated by a power law with **8** until the key jumps fall into the exponential tail of the truncated power-law kernel, upon which linear growth (at a speed set by the exponential tail) would ensue.

Whether transport via a network with hubs at many scales fundamentally changes the dynamics of an expanding population of hitchhikers from that with more homogeneous jump processes depends on the nature of how the population expands. For spread of a human epidemic, there are several possible scenarios. If the human population is reasonably uniform spatially, and the chance that a person travels from, say, his or her home to another person’s home is primarily a function of the distance between these rather than the specific locations, then whether the properties of the transportation network matter depends on features of the disease. If individuals are infectious for the whole time the outbreak lasts and if transmission is primarily at end points of journeys rather than en route—for example, HIV—then the transportation network plays no role except to provide the spatial jumps. At the other extreme is whether individuals living near hubs are more likely to travel (or even whether destinations near hubs are more likely) and, more so, whether infections are likely to occur en route, in which case the structure of the transportation network—as well as of spectra of city sizes, etc.—matters a great deal. In between these limits the network (or lack of it in places) may matter for initial local spread but at longer times the network structure may effectively average out and the dynamics be more like those of the homogeneous models. The two opposite limits and behavior in between these, together with the specific network model we analyzed with jumping probabilities depending on distance even in the presence of a connection—as is true from airports—all illustrate an important point: Geometrical properties of networks alone rarely determine their properties; quantitative aspects, such as probabilities of moving along links and what exists at the nodes, are crucial.

More complicated epidemic models can be discussed within the same framework: The model discussed thus far corresponds to an SI model, the most basic epidemic model, which consists of susceptible and infected individuals only. Many important epidemics are characterized by rather short infectious periods, so that one has to take into account the transition from infected to recovered: SIR models of the interaction between susceptible (S), infected (I) and recovered (R) individuals. This changes fundamentally the geometry of the space–time analysis in Fig. 4. Whereas the target funnel remains a full funnel, the source funnel becomes hollow: The center of the population consists mostly of fully recovered individuals, whose long-range jumps are irrelevant. The relevant source population of infected individuals is primarily near the boundaries of the funnel. This leads to a break in the time symmetry of the argument. As a result, the spreading crosses over from the behavior described above to genuine SIR behavior. The SIR dynamics are closely related to the scaling of graph distance in networks with power-law distributions of link lengths (24, 46), as recently shown by 1D and 2D simulations (48, 49). In particular, the limited time of infectiousness causes wave-like spreading at a constant speed for

We expect our approach to be useful also for investigating genetic consequences of range expansions and genetic hitchhiking on spatial selective sweeps with long-distance dispersal (50⇓⇓–53). A generalization of the analysis of the phenomenon of “allele surfing,” which has been mainly analyzed assuming short-range migration so far (54⇓–56), could be used to clarify the conditions under which long-distance dispersal increases or decreases genetic diversity—both effects have been seen in simulations (57⇓–59). New effects arise due to the “patchiness” (60) generated by the proliferation of satellite clusters that are seeded by long-range jumps, as, e.g., observed in many plant species (61). A needed further step is the analysis of the interaction between multiple spreading processes, for instance generated by the occurrence of multiple beneficial variants in a population, which leads to soft sweeps or clonal interference in space (62, 63). Whereas colliding clones strongly hinder each other’s spread with only short-range migration, rare long-distance jumps may overcome these constraints (62), leading to irregular global spreading as we have found for a single clone.

Finally, our analyses naturally provide information on the typical structure of coalescent trees backward in time. For a given site, the path of jumps by which the site was colonized can be plotted as in Fig. 7. Doing this for many sites yields coalescent trees, which can reveal the key long-range jumps shared by many lineages. In general, such genealogical information is helpful for reconstructing the demographic history of a species. In the particular context of epidemics, combining spatiotemporal sampling of rapidly evolving pathogens with whole-genome sequencing is now making it possible to construct corresponding “infection” trees, and their analysis is used to identify major infection routes (64). For such inference purposes, it would be interesting to investigate the statistical properties of coalescence trees generated by simple models such as ours and how they depend on the dispersal properties, network structure, and other features of epidemic models.

## Acknowledgments

We wish to thank Rava da Silveira for useful discussions. This work was partially supported by a Simons Investigator award from the Simons Foundation (O.H.), the Deutsche Forschungsgemeinschaft via Grant HA 5163/2-1 (to O.H.), and the National Science Foundation via Grants DMS-1120699 (to D.S.F.) and PHY-1305433 (to D.S.F.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: ohallats{at}berkeley.edu.

Author contributions: O.H. and D.S.F. designed research, performed research, contributed analytic tools, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

↵*This Direct Submission article had a prearranged editor.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1404663111/-/DCSupplemental.

Freely available online through the PNAS open access option.

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