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I am confused as to how to compute the effective population size $N_e$ of a theoretical structured population. Let's consider here a simple case study.

Imagine a 2-deme metapopulation. Each deme is of constant size $N$ with a (forward and backward) migration rate $m$. Mating is random within each deme.

What is the effective population size of such population? In other words, how fast does the population loses heterozygosity through time?

## Effective population size/adult population size ratios in wildlife: a review

The effective population size is required to predict the rate of inbreeding and loss of genetic variation in wildlife. Since only census population size is normally available, it is critical to know the ratio of effective to actual population size ( N e /N ). Published estimates of N e /N (192 from 102 species) were analysed to identify major variables affecting the ratio, and to obtain a comprehensive estimate of the ratio with all relevant variables included. The five most important variables explaining variation among estimates, in order of importance, were fluctuation in population size, variance in family size, form of N used (adults υ. breeders υ. total size), taxonomic group and unequal sex-ratio. There were no significant effects on the ratio of high υ. low fecundity, demographic υ. genetic methods of estimation, or of overlapping υ. non-overlapping generations when the same variables were included in estimates. Comprehensive estimates of N e /N (that included the effects of fluctuation in population size, variance in family size and unequal sex-ratio) averaged only 0·10–0·11. Wildlife populations have much smaller effective population sizes than previously recognized.

## THEORY

**Large metapopulation model:** Consider a population that is subdivided into a large number of local populations or demes. The total number of demes is *D*, and these are arbitrarily labeled 1 through *D.* Deme *i* has diploid size *N _{i}*, or, equivalently, haploid size 2

*N*. In either case 2

_{i}*N*copies of each genetic locus reside within deme

_{i}*i.*The results presented below apply in a straight-forward way to haploid organisms and to diploid monoecious organisms with the additional assumption that migration and recolonization are gametic rather than zygotic. This leads to an apparent factor of two difference of terms involving

*k*below, relative to results of previous authors, but this is not a meaningful difference. N agylaki (1998) has shown that results for zygotic migration will be equivalent to those for gametic migration as long as the effective number of migrants each deme accepts each generation is not too small.

Each deme receives migrants from other demes in the population and is also subject to extinction/recolonization. If a deme goes extinct, it is recolonized immediately. Thus, there are no empty habitat patches in this model as there often are in ecological models of metapopulations *e.g.*, see H anski (1997). This assumption is unnecessary as long as the total number of extant demes remains constant from one generation to the next (P annell and C harlesworth 1999). Deme *i* receives *M _{i}* (haploid) migrants each generation. That is, a fraction

*M*/(2

_{i}*N*) of deme

_{i}*i*is replaced by migrants every generation. The other portion, 1 −

*M*/(2

_{i}*N*), is derived from the previous generation of deme-

_{i}*i*individuals. Reproduction within each deme occurs according to the Wright-Fisher model (F isher 1930 W right 1931). The parameter

*M*is the scaled backward migration rate for deme

_{i}*i.*Correspondingly,

*E*is the scaled extinction/recolonization rate, so

_{i}*E*/(2

_{i}*N*) is the pergeneration probability that deme

_{i}*i*goes extinct. If deme

*i*goes extinct, it is recolonized by

*k*individuals, which immediately restore the deme to its original size of 2

_{i}*N*gene copies. This step also occurs according to the Wright-Fisher model. That is, the 2

_{i}*N*descendants are obtained by sampling with replacement from the

_{i}*k*colonists. It is important to note that the subscripts of

_{i}*N*,

*M, E*, and

*k*refer to individual demes, not to the classes of demes introduced below.

An example, with *K* = 3 classes of demes, of the population structure assumed throughout this work. Within each of the three classes, or regions, there are many demes. Arrows depict the movement of lineages, by migration and/or extinction/recolonization, both within and among regions.

The population is assumed to comprise *K* different types of demes, which may represent different geographic regions. Demes of type *i* make up a fraction β* _{i}* of all demes thus, ∑ i = 1 K β i = 1 . In addition, demes of type

*i*receive a portion

*m*of their gametes via migration from demes of type

_{ij}*j*, where 1 ⩽

*j*⩽

*K*. In total, each deme of type

*i*is a fraction m j = ∑ j = 1 K m i j of its gametes replaced by migrants every generation. Thus,

*M*/(2

_{j}*N*) =

_{j}*m*for every type

_{i}*i*deme,

*j.*Migrants into a type

*i*deme might have come from another deme of type

*i.*They may also originate in the same deme they migrate to, although the effect of this is negligible when the number of demes is large. Demes of type

*i*go extinct with probability

*e*each generation and are recolonized by a mixture of gametes from the different classes of demes in proportions e i 1 ∕ e i , e i 2 ∕ e i , … e i K ∕ e i ( ∑ j = 1 K e i j = e i ) . When a lineage is a migrant or a colonist from a deme of type

_{i}*i*, it is equally likely to have come from each of the β

*type*

_{i}D*i*demes. The structure of this model is depicted in Figure 1.

**Separation of timescales and the structure of genealogies:** As in W akeley (1998, 1999, 2000, 2001), the results presented here will hold for metapopulations that are composed of a large number of demes. In particular, the sample size must be much smaller than the number of demes in the population (*n* ⪡ *D*). This does not appear to be an unrealistic assumption for some metapopulations in nature and is one that is commonly made in theoretical studies of metapopulations (H anksi 1997). It leads to a separation of timescales in the ancestral process of a sample, which is similar to that found in studies of partial selfing (N ordborg 1997, 1999 N ordborg and D onnelly 1997). A useful convergence theorem, derived in context of partial selfing, was found by M öhle (1998a). Consider the genealogy of a sample from such a population. The separation of timescales is a consequence of the fact that at any given time in the past, the overwhelming majority of demes in the population will not contain any lineages ancestral to the sample. Demes that do contain ancestral lineages are called occupied demes (W akeley 1999), and the fraction of these in the population is never >*n*/*D.* Two kinds of events differ vastly in rate. The first is migration and extinction/recolonization events in which the source deme is occupied. The second is coalescent events within demes and migration or extinction/recolonization events in which the source deme is unoccupied. Events of the second type dominate the history of the sample because they are approximately *D* times more likely than events of the first type.

Given this, it is necessary to distinguish sample configurations in which every lineage is in a separate deme from those in which at least one deme contains multiple lineages. When at least one deme contains multiple lineages, migration events and extinction/recolonization events will send lineages to unoccupied demes and coalescent events will join together lineages within demes until each remaining lineage is in a separate deme. This scattering phase takes a negligible amount of time compared to the waiting time to the next relevant event, which is a migration or extinction/recolonization event to an occupied deme. At least one event of this type must occur before another coalescent event can happen. In fact, if *n* ⪡ *D*, so many will occur that the movement of the lineages among unoccupied demes by migration and extinction/recolonization will reach a statistical equilibrium before two lineages will have the chance to coalesce. This is the essence of M öhle' s (1998a) result and of the strong-migration limit (N agylaki 1980). As shown below, the collecting phase is a Kingman-type coalescent process with a characteristic effective size. Thus, the structure of genealogies is two-fold. First there is a one-time stochastic sample size adjustment, the scattering phase, which results in greater relatedness within than between demes then the bulk of the history is spent in a collecting phase coalescent process. Figure 2 illustrates the structure of genealogies under this approximation for a sample from one deme.

**Scattering phase:** Consider the recent history of a sample ** n** = (

*n*

_{1}, … ,

*n*) taken from

_{d}*d*different demes. The total sample size is n = ∑ j = 1 d n i . Following the genealogy of the sample from deme

*i*, it will take on the order of 2

*N*generations for the scattering phase to be complete, fewer if

_{i}*M*and/or

_{i}*E*are large. Let n i ′ represent the number of lineages remaining of the sample

_{i}*n*from deme

_{i}*i*at the end of the scattering phase. When there are

*j*lineages in the deme, the scaled rates of migration, coalescence, and extinction/recolonization are

*jM*,

_{i}*j*(

*j*− 1)/2, and

*E*, respectively. The value of n i ′ will depend upon how many migration events occurred before the deme experiences an extinction/recolonization event and on the number of colonist-parents there are of the lineages that exist at the time of this event. The probability that the extinction event occurs when there are

_{i}*j*lineages, and not before, is given by P E ( j ∣ n i ) = < 2 E i ∕ j 2 M i + ( j − 1 ) + 2 E i ∕ j ∏ l = j + 1 n i 2 M i + ( l − 1 ) 2 M i + ( l − 1 ) + 2 E i ∕ l , 2 ≤ j ≤ n i ∏ l = 2 n i 2 M i + ( l − 1 ) 2 M i + ( l − 1 ) + 2 E i ∕ l , j = 1 . >(1) The first case specifies that

*n*−

_{i}*j*migration or coalescent events occur, which leaves

*j*lineages in deme

*i*, and then an extinction/recolonization event occurs. In the second case, the scattering phase for deme

*i*ends, with each remaining lineage in a separate deme, before an extinction/recolonization event has occurred.

An example of a genealogy of sample size eight from a single deme. In this case, during the scattering phase, there are two migration events (to some unoccupied demes that are not pictured) and then an extinction/recolonization event with *k* = 2 in which all of the lineages remaining in the deme are descended from a single common ancestor. The coalescent collecting phase of the three resulting lineages is shown above. The relative duration of the scattering phase is greatly exaggerated for purposes of illustration.

To know how many lineages remain at the end of the scattering phase, we must first distinguish histories that involve different numbers of migration events. The probability that *x* migration events occur and the extinction event occurs when there are *j* lineages is given by P ME ( x , j ∣ n i ) = < 2 E i ∕ j 2 M i + ( j − 1 ) + 2 E i j S n , j ( x + 1 ) ( 2 M i ) x ∏ l = j + 1 n i 2 M i ( l − 1 ) + 2 E i ∕ l , 2 ≤ j ≤ n i S n , 1 ( x + 1 ) ( 2 M i ) x ∏ l = 2 n 2 M i + ( l − 1 ) + 2 E i ∕ l , j = 1 >(2) in which S n i , j ( x + 1 ) = coefficient of ( 2 M i ) x in ∏ l = j + 1 n i [ 2 M i + ( l − 1 ) ] . (3) These coefficients can be generated recursively, S j , i ( l ) = ( j − 1 ) S j − 1 ( l ) + S j − 1 , i ( l − 1 ) , (4) starting with S i , i ( 1 ) = 1 , and S j , 1 ( l ) are unsigned Stirling numbers of the first kind. The source deme of each migrant is determined by the stochastic migration process described above.

Given that an extinction event occurs when there are *j* lineages remaining in deme *i*, the probability that these have *y* colonist-parents in the propagule of size *k _{i}* is given by G i [ y ∣ j ] = S j ( y ) ∏ l = 0 y − 1 ( k i − l ) k i j . (5) Equation 5 is the usual backward Wright-Fisher process see, for example, W atterson (1975). That is, the number of parents of the

*j*lineages has the same distribution as the number of nonempty cells when

*j*balls are thrown randomly into

*k*boxes. The coefficients, S j ( y ) , are Stirling numbers of the second kind. The source deme of each colonist-parent is determined by the stochastic extinction/recolonization process described above.

_{i}When an extinction/recolonization event occurs, as long as *D* is large, each lineage will have a different source deme and the scattering phase will end for deme *i.* Thus, this model is a general version of S latkin' s (1977) migrant pool model. At the other extreme is S latkin' s (1977) propagule-pool model in which all *y* lineages in (5) would have the same source deme, chosen randomly according to some probability function. If this were the case, the scattering phase would continue, but with the scaled coalescent, migration, and extinction/recolonization rates of the source deme. If there was no migration, the propagule-pool model would always give n i ′ . W ade and M c C auley (1988) proposed an intermediate model, in which a fraction, ϕ, of the lineages would follow the propagule-pool model and the other 1 − ϕ would follow the migrant pool model. These and more complicated schemes could be modeled within the present framework but are not pursued here. When an extinction/recolonization event occurs the scattering phase is over for the sample from that deme. If *k _{i}* is equal to one or if the rate of extinction/recolonization is low, the present results will be identical to those of a propagule-pool model.

If there are *x* migration events and *y* colonist-parent lineages in the sample from deme *i*, then the scattering phase for deme *i* ends with n i ′ = x + y lineages each in separate demes. The probability function for n i ′ is P [ n i ′ ∣ n i ] = ∑ j = 1 n ∑ x = 0 n − j P ME ( x , j ∣ n i ) G i [ n i ′ − x ∣ j ] , (6) where we define *G _{i}*[

*y*|

*j*] to be equal to zero if

*y*is <1 or >

*j.*Because events occur independently in different demes, the joint probability function of all the n i ′ is given by P [ n ′ ∣ n ] = ∏ i = 1 d P [ n i ′ ∣ n i ] . (7) The collecting phase of the history then begins with n ′ = ∑ i = 1 d n i ′ lineages, each in separate demes.

**Collecting phase:** The distribution among deme types of the *n*′ lineages that enter the collecting phase will depend on the particular outcome of the scattering phase for each deme's sample. Let *r _{i}* be the number of lineages that are in type

*i*demes. The vector (

*r*

_{1}, … ,

*r*) then denotes the configuration of the lineages among the different types of demes. The total number of lineages is equal to r = ∑ i = 1 K r i and at the start of the collecting phase we have

_{K}*r*=

*n*′. Here it is shown that the time to a coalescent event does not depend on the starting value of (

*r*

_{1}, … ,

*r*) and is exponentially distributed as in Kingman's coalescent. First, as in W akeley (2001), the time until two lineages are in the same deme is shown to be exponentially distributed. The coalescent result follows from this and the fact that the number of times two lineages must be in the same deme before a common ancestor event occurs is geometrically distributed.

_{K}Note that, from the perspective of a single lineage in a singly occupied deme, a migration event and an extinction/recolonization event are indistinguishable. Both simply move the lineage to another deme. Therefore, it is sufficient during the collecting phase to consider the combined effect of migration and extinction/recolonization: *h _{ij}* =

*m*+

_{ij}*e*. It is assumed that

_{ij}*h*is small, on the order of the reciprocal of the deme size. Thus, squared and higher-order terms in

_{ij}*h*, which represent the movement of two or more of the lineages in a single generation, will be ignored. Note also that here there is no difference between migrant-pool and propagule-pool recolonization.

_{ij}Looking back to the immediately previous generation, there are two kinds of events: changes in the configuration, (*r*_{1}, … , *r _{K}*), and the movement of a lineage into an occupied deme. Events of the first kind occur with probability P < ( … , r i − 1 , … , r j + 1 , … ) → ( … , r i , … , r j , … ) >= ( r j + 1 ) h j i . (8) The movement of a lineage into an occupied deme of type

*i*occurs with probability b i , ( r 1 … r K ) = r i h i i ( r i − 1 β i D ) + ∑ j : j ≠ i r j h j i ( r i β i D ) . (9) Clearly the first kind of event is much more likely to occur when

*D*is large relative to

*r.*The probability that the sample configuration is unchanged is equal to P < ( r 1 , … , r K ) → ( r 1 , … , r K ) >= 1 − ∑ i = 1 K r i ∑ j : j ≠ i h i j − ∑ i = 1 K b i , ( r 1 , … , r K ) (10) ≈ 1 − ∑ i = 1 K r i ∑ j : j ≠ i h i j . (11) As in W akeley (2001), the essence of the separation of timescales is that, when

*r*⪡

*D*, an equilibrium for (

*r*

_{1}, … ,

*r*) is reached with respect to (8) and (11) before any event of the type in (9) occurs. Then, the waiting time to a movement event that places two lineages into the same type

_{K}*i*deme is the average of (9) over the stationary distribution of (

*r*

_{1}, … ,

*r*). M öhle (1998a) provided a convergence theorem for processes such as this, which is used implicitly below.

_{K}Consider first the movement of just one lineage among demes in the population. This is determined by the matrix ** Q**, which has off-diagonal entries

*q*=

_{ij}*h*. The diagonal entries are q i i = 1 − ∑ j : j ≠ i h i j , which it is important to note are not equal to

_{ij}*h*defined above. The

_{ii}*h*do not directly affect the equilibrium configuration because such moves do not take the lineage into a different class of demes. Standard matrix theory shows that as long as the matrix

_{ii}**is ergodic,**

*Q**i.e.*, irreducible and aperiodic, a stationary distribution will exist. As N agylaki (1998) notes, ergodicity in itself probably does not rule out very many plausible biological scenarios. Ergodicity requires only that lineages can eventually get from any deme type to any other and that lineages have some chance of staying in their current type of deme. If

*f*is the equilibrium probability that a lineage is in a deme of type

_{i}*i*, we have f i = ∑ j = 1 K f j q j i (12) or, equivalently, f i ∑ j : j ≠ i q i j = ∑ j : j ≠ i f j q j i , (13) where we may assume 0 <

*f*< 1 and ∑ i = 1 K f i = 1 . The quantity

_{i}*f*can be interpreted as the average relative amount of time a lineage spends in demes of type

_{i}*i.*

The stationary distribution of the full configuration (*r*_{1}, … , *r _{K}*) is multinomial: p ( r 1 , … , r K ) = r ! r 1 ! … r K ! f 1 r 1 … f K r K . (14) This is proved by induction over time. Using (8) and (11), we have p ( r 1 , … , r K ) = ∑ i = 1 K ∑ j : j ≠ i p ( … , r i − 1 , … , r j + 1 , … ) ( r j + 1 ) q j i + p ( r 1 , . . , r K ) ( 1 − ∑ i = 1 K r i ∑ j : j ≠ i q i j ) (15) or equivalently, p ( r 1 , … , r K ) ∑ i = 1 K r i ∑ j : j ≠ i q i j = ∑ i = 1 K ∑ j : j ≠ i p ( … , r i − 1 , … , r j + 1 , … ) ( r j + 1 ) q j i . (16) If the stationary distribution of (

*r*

_{1}, … ,

*r*) is given by (14), then p ( r 1 , … , r K ) ∑ i = 1 K r i ∑ j : j ≠ 1 q i j = ∑ i = 1 K ∑ j : j ≠ i p ( r 1 , … , r K ) r i f j ( r j + 1 ) f i ( r j + 1 ) q j i (17) = p ( r 1 , … , r K ) ∑ i = 1 K r i f i ∑ j : j ≠ i f j q j i , (18) which is true because the

_{K}*f*satisfy (13). The stationary distribution (14) is unique since the Markov chain is ergodic and has a finite number of states.

_{i}The total rate of events that put two lineages together in a deme of type *i* is the average of (9) over the stationary distribution of (*r*_{1}, … , *r _{K}*), that is, g r , i = ∑ < ( r 1 , … , r K ) : ∑ i = 1 K r 1 = r >p ( r 1 , … , r K ) b i , ( r 1 , … , r K ) . (19) Equation 19 is just the expectation of

*b*

_{i,(r1,…,rK)}over the multinomial distribution (14). Using (13) and the fact that

*q*=

_{ij}*h*for

_{ij}*j*≠

*i*, we have g r , i = ( r 2 ) 2 f i 2 h i β i D (20) in which h i = ∑ j = 1 K h i j . The total rate of events that put two lineages into the same deme, regardless of the type, is the sum of (20) over all types of demes: g r = ∑ i = 1 K g r , i . Given that such an event occurs, the probability that the two lineages are in a deme of type

*i*is equal to

*g*/

_{r,i}*g*. Then, once two lineages are in the same deme, they either have a common ancestor or they again wind up in separate demes, either by migration or through an extinction/recolonization event. If they wind up in separate demes, there will be another exponentially distributed waiting time with rate

_{r}*g*before two lineages are in one deme and again have a chance to coalesce.

_{r}Because each of the β* _{i}D* demes of type

*i*is equally likely to be the one that contains the two lineages, the overall chance that the two will have a common ancestor is equal to 〈 1 + E ∕ k 2 M + 1 + E 〉 i = 1 D β i ∑ < j : j ∈ Ω i >1 + E j ∕ k j 2 M j + 1 + E j , (21) where Ω

*is the set of labels of the*

_{i}*D*β

*demes of type*

_{i}*i.*As in W akeley (2001), the number of movement events to occupied demes that must occur before a common ancestor event happens is geometrically distributed with probability of success equal to the average of (21) over the distribution

*g*/

_{r,i}*g*. Because the waiting time between these events is exponential with rate

_{r}*g*, it follows (W akeley 1999) that the time to coalescent event among the

_{r}*r*lineages is exponentially distributed with rate ( r 2 ) 2 D ∑ i = 1 K f i 2 h i β i 〈 1 + E ∕ k 2 M + 1 + E 〉 i . (22) When a coalescent event occurs, the number of lineages decreases by one and the process continues.

This shows that the collecting phase is a Kingman-type coalescent process and is thus independent of the starting distribution of lineages among deme types, (*r*_{1}, … , *r _{K}*). The effective size of this coalescent process is given by 1 2 N e = 2 D ∑ i = 1 K f i 2 h i β i 〈 1 + E ∕ k 2 M + 1 + E 〉 i . (23) An equation like (23) can provide a framework for understanding the determinants of the effective population size. W akeley (2001) discusses the effects of different factors in the case of a migration-only model. It is important to note that (23) determines the rate in a coalescent process that occurs in the history of every sample. This is different than the traditional effective sizes, which are descriptions of the equilibrium behavior of genetic drift in the population. However, (23) is by definition an inbreeding effective size and is essentially the same as the various effective metapopulation sizes that others have discussed in detail for example, see W hitlock and B arton (1997). For purposes here, the significance of (23) is twofold. First, it will not be possible to differentiate among many different parameters of the model because they are buried in the composite parameter,

*N*

_{e}. Second, the collecting-phase coalescent result holds for many specific population structures.

**Analytic predictions about DNA sequence polymorphisms:** Because the collecting phase is a coalescent process, a natural way to incorporate neutral mutations is to define θ = 4*N*_{e}*u*, where *N*_{e} is given by (23) and *u* is the neutral mutation rate at some genetic locus. With θ defined in this way, the history of one particular kind of sample (*n*_{1} = 1, *n*_{2} = 1, … , *n _{d}* = 1) will conform to the standard neutral coalescent model. All the usual coalescent results, for example, those found in T avaré (1984), will apply directly to this sample when θ = 4

*N*

_{e}

*u.*Predictions about levels of genetic variation for other samples will have to be averaged over the possible outcomes of the scattering phase and weighted by their probabilities as given in (7). If

*S*(

**) is the number of segregating sites in the multideme sample,**

*n***= (**

*n**n*

_{1}, … ,

*n*), then P [ S ( n ) = i ] = ∑ n ′ P [ S ( n ′ ) = i ] P [ ( n ′ ∣ n ) ] . (24) Under W atterson' s (1975) infinite sites mutation model,

_{d}*P*[

*S*(

**′) =**

*n**i*] may be given by T avaré' s (1984) equation (9.5). Summing over all possible values of

*i*gives the corresponding equation for the expectation of

*S*(

**), E [ S ( n ) ] = ∑ n ′ E [ S ( n ′ ) ] P [ n ′ ∣ n ] , (25) in which E [ S ( n ′ ) ] = θ ∑ i = 1 n ′ − 1 1 ∕ i (W atterson 1975). In the case of migration only, integral representations of**

*n**P*[

*S*(

**′)] and**

*n**E*[

*S*(

**′)] allow the sums in (24) and (25) to be evaluated, resulting in somewhat simpler expressions (W akeley 2001), but this does not appear possible here.**

*n*The most basic prediction of this model, or of any subdivided population model, is that levels of polymorphism will be higher among than within demes. The simplest case, two sequences sampled either from the same deme or from two different demes, illustrates this point. If the two are from different demes, the expected number of pairwise differences will be equal to θ, identically for any pair of demes. A randomly chosen pair will have this same expectation because the chance of randomly sampling the same deme twice will be low when the number of demes is large. The expected number of differences between a pair of sequences from deme *i* will be equal to this, θ, times the probability that the scattering phase for this sample ends with two lineages. Call these expected values π* _{T}* and π

*, respectively. We can define an inbreeding coefficient for the deme, F i = π T − π i π T (26) = 1 + E i ∕ k i 2 M i + 1 + E i , (27) which is simply the probability that the two lineages coalesce during the scattering phase. The inbreeding coefficient for the deme will be small when the migration rate is large or when the extinction/recolonization rate and the propagule size are both large. It will be large when the rates of migration and extinction/recolonization are both small or when the extinction/recolonization rate is large and the propagule size is small. It is important to note that here θ is assumed to remain constant and that these differences in*

_{i}*F*represent the possible differences among sample demes. If θ changes as well,

_{i}*i.e.*, if demes do not differ in their characteristics, then the conclusions are different (P annell and C harlesworth 1999 W akeley 2001).

Equations 24 and 25 provide results for arbitrary samples. For example, Figure 3 plots (25) for a sample of five sequences from each of two different demes over a broad range of migration and extinction/recolonization rates. Shown are results for two different propagule sizes: *k* = 1 in Figure 3a and *k* = 10 in Figure 3b. In both cases, θ = 10. As expected, the model predicts that samples from demes with larger backward migration rates will be more polymorphic. If within each deme all five lineages share a common ancestor by the end of the scattering phase, the collecting phase begins with just two lineages and the expected number of segregating sites will be equal to 10.0. At the other extreme, if, for example, migration is very frequent, the scattering phase could end with 10 sequences all in different demes. The collecting phase would then begin with 20 lineages, and the expected number of segregating sites will be equal to 28.3. These extremes are very nearly realized in Figure 3. In contrast to migration, the effect of changes in the rates of extinction/recolonization in the two demes depends on the propagule size. When the propagule size is small (Figure 3a), increases in the rate of extinction/recolonization make the expected number of segregating sites smaller, whereas when the propagule size is large (Figure 3b), increases in extinction/recolonization rates have a similar effect to increase in the migration rates, which is to increase levels of polymorphism. This is just restatement, within the framework of the coalescent, of S latkin' s (1977) conclusions about the dual roles of extinction/recolonization.

As in W akeley (1999) it may be possible to find analytic expressions for the expected number of sites segregating at particular joint frequencies among the sampled demes. In addition, because the collecting phase is a coalescent, it is relatively straightforward to incorporate changes in effective population size over time. Again, existing expressions found for changing population sizes in the context of a Kingman-type coalescent, such as those in S latkin and H udson (1991) and H ey and H arris (1999), will apply directly to the sample where *n* = *d* and can be averaged over (7) for other samples. It should also be possible to model diverging species, each of which conforms to the present metapopulation model, as in W akeley (2000). Instead of pursuing these ideas any further here, the next section describes how genealogies can be simulated easily so that these and other questions can be addressed computationally.

## 1 INTRODUCTION

Maintaining high levels of genetic diversity and keeping inbreeding low are important aspects in the management of threatened populations. For this reason the concept of genetically effective population size (*N*_{e}) plays a central role in conservation biology *N*_{e} relates to the rate at which genetic drift occurs, and in particular the rates of inbreeding and loss of genetic diversity are of concern in conservation. The past decade “has seen an explosion of interest in the use of genetic markers to estimate effective population size” (Waples, 2016 ). Little attention has been paid, however, to whether those estimates really quantify the relevant rates of genetic change when substructured populations are the focus of empirical studies.

Symbol | Definition/comments |
---|---|

s | Number of subpopulations |

t | Time measured in generations |

N _{c} | Census population size |

N _{e} | Effective population size (in general) |

x | An arbitrary subpopulation that is part of a metapopulation |

f | Coefficient of inbreeding |

f _{x} | Average inbreeding in subpopulation x |

f _{Meta} | Average inbreeding coefficient of the total metapopulation (here weighted according to subpopulation effective size). Corresponds to f_{I} in Hössjer et al., ( 2015 ) |

m | Migration rate, in the context of an island model, expressed as the proportion of individuals in each generation that are immigrants from the metapopulation as a whole (including the target population). Migration is stochastic and m reflects the binomial average |

m' | Migration rate, expressed as the proportion of individuals in each generation that are immigrants from outside the target population. Migration is stochastic and m' reflects the binomial average. In an island model, where immigrants can be conceptualized as drawn from an infinitely large pool of individuals to which all the s subpopulations have contributed equally, m and m' are related as m' = m(s–1)/s. Many texts on the island model are not explicit when defining migration, i.e., it is not always clear whether or not the immigrants include a proportion of individuals from the target population |

N _{eI} | Inbreeding effective size (in general). N_{eI} reflects the rate at which inbreeding increases inbreeding is the occurrence of homozygosity of alleles that are identical by descent, i.e., alleles that can be traced back to the exact same allele copy in an ancestor (also known as the coalescent). N_{eI} is not defined for situations where inbreeding decreases, and in a case where inbreeding stays constant we have N_{eI} = ∞ |

N _{eV} | Variance effective size (in general). N_{eV} reflects the rate of allele frequency change. The quantity of interest is the change of the standardized drift variance |

N _{eLD} | Linkage disequilibrium effective size (in general) it reflects the degree of linkage (gametic phase) disequilibrium. Mathematical treatment of N_{eLD} is complicated and not yet fully resolved. Approximate equations for N_{eLD} in a local population exist for the special case of an ideal (N_{ex} = N_{cx}) island model (Waples & England, 2011 this paper) but not for the global population |

N _{eGD} | Gene diversity effective size (in general). This quantity reflects the rate at which gene diversity, i.e., expected heterozygosity, declines. We have previously (Hössjer et al., 2016 ) referred to this N_{e} as “haploid inbreeding effective size,” but here we call it N_{eGD} to avoid confusion in the present context |

N _{eAV} | Additive variance effective size (in general) it reflects the rate at which additive genetic variation is lost due to genetic drift. N_{eAV} is very close to N_{eGD}, and in this paper we have used N_{eGD} (which is easier to compute) as a proxy for N_{eAV} |

N _{eCo} | Coalescence effective size (in general) it reflects the time for ancestral lineages to coalesce to a common ancestor. We have not focused on this effective size see the Discussion for more details on this N_{e} including its relationship to other N_{e} dealt with here |

N _{cx} | Census size of subpopulation x |

N _{ex} | Effective size of subpopulation x in isolation, i.e. when all types of N_{e} are the same, i.e. N_{ex} = N_{eI} = N_{eV} = N_{eLD} = N_{eAV} etc |

N _{eIRx} | Inbreeding effective size of subpopulation x (under prevailing migration scheme) |

N _{eVRx} | Variance effective size of subpopulation x (under prevailing migration scheme) |

N _{eLDRx} | Linkage disequilibrium effective size of subpopulation x (under prevailing migration scheme) |

N _{eAVRx} | Additive variance effective size of subpopulation x (under prevailing migration scheme) |

N _{eE} | Eigenvalue effective size (of the metapopulation as a whole). The global population will eventually reach a state where inbreeding increases at a constant rate, which results in the inbreeding effective size of the metapopulation to stay constant at a value indicated by N_{eE}. In a metapopulation where each subpopulation exchanges migrants with the rest of the system (through one or more subpopulations) the rate of inbreeding will eventually be the same (1/(2N_{eE})) in all subpopulations as well as for the system as a whole |

N _{eMeta} | Total effective size (in general) of the metapopulation as a whole (the global population) |

N _{eIMeta} | Total (global) inbreeding effective size of the metapopulation as a whole. This quantity reflects the change of f_{Meta} from generation t to t + 1. N_{eIMeta} can be viewed as a weighted average of N_{eIRx} over all subpopulations, and it will eventually approach N_{eE.} |

N _{eVMeta} | Total (global) variance effective size of the metapopulation as a whole. N_{eVMeta} eventually approaches a value very close to N_{eE}, but N_{eVRx} of a local population does not (cf. Hössjer et al., 2016 ) |

N _{eLDMeta} | Total (global) linkage disequilibrium effective size of the metapopulation as a whole. Currently, analytical as well as simulation approaches to assess this parameter are missing. In the present paper we only deal with the local form N_{eLDRx} |

N _{eAVMeta} | Total (global) additive variance effective size of the metapopulation as a whole |

Most natural populations are not completely isolated, however, but connected to others by more or less frequent migration. In contrast to the situation with isolated populations various types of *N*_{e} can be very different for a population under migration (Chesser, Rhodes, Sugg, & Schnabel, 1993 Wang, 1997a , 1997b ). Considerable work has been devoted to modelling effective sizes of subdivided populations (e.g., Maruyama & Kimura, 1980 Nunney, 1999 Tufto & Hindar, 2003 Wang & Caballero, 1999 Waples, 2010 Whitlock & Barton, 1997 Wright, 1938 ). Most of these efforts, however, have focused on a single effective size (*N*_{eI} or *N*_{eV}) using simplifying assumptions such as drift-migration equilibrium, haploid populations, or ideal demographic conditions where census and effective sizes under isolation are identical (*N*_{c} *= N*_{e}). Means for modelling several types of *N*_{e} under both equilibrium and nonequilibrium conditions and for complex metapopulations deviating from nontraditional patterns of migration have previously not been possible.

We have recently developed a general analytical framework for exploring the dynamics of many effective population sizes in more complex metapopulations (Hössjer et al., 2016 Hössjer, Olsson, Laikre, & Ryman, 2014 , 2015 ). Our approach allows modelling systems at equilibrium as well as before equilibrium has been reached, with any number of subpopulations of arbitrary census and effective size under isolation. Migration patterns are also optional, as are initial degrees of inbreeding and relatedness within and among populations. As an example, we applied this analytical tool to model the case of the wolf metapopulation on the Fennoscandian peninsula and showed that the observed unidirectional gene flow from Finland to Sweden greatly reduces the overall metapopulation inbreeding effective size. Further, gene flow from a large Russian wolf population into the Fennoscandian metapopulation has limited effect on inbreeding rates unless gene flow within Fennoscandia increases substantially (Laikre, Olsson, Jansson, Hössjer, & Ryman, 2016 ). These observations were previously unknown phenomena of direct relevance to management.

The “50/500 rule” of Franklin ( 1980 ) presents an example of a situation where it may be critical to know the particular type of *N*_{e} that is obtained when applying an estimator to genotypic data. This rule has become widely established in conservation biology, suggesting that for a single isolated population *N*_{e} *≥* 50 is needed for short-term conservation and *N*_{e} *≥* 500 for long-term conservation (Allendorf et al., 2013 Franklin, 1980 ). As detailed by Franklin ( 1980 ) the short-term rule of *N*_{e} *≥* 50 refers to an effective size quantifying the rate of inbreeding (inbreeding effective size, *N*_{eI}). The logic of the 50-rule is that too rapid inbreeding can result in excessive homozygosity for deleterious recessive alleles resulting in inbreeding depression and reduced fitness (Chapter 10 of Lynch & Walsh, 1998 ). An *N*_{eI} *≥* 50 implies that inbreeding increases by no more than 1% per generation, which is considered acceptable with respect to fitness over short time periods (Franklin, 1980 ). The long-term “*N*_{e} *≥* 500 rule” refers to an effective size relating to loss of additive genetic variation, here referred to as *N*_{eAV} (Hössjer et al., 2016 Table 1 below), and the concern here is the maintenance of sufficient levels of genetic variation for quantitative traits associated with fitness that will allow adaptation to new selective regimes (i.e., retention of evolutionary potential). Indeed, it follows from Fisher's Fundamental Theorem of Natural Selection (Price, 1972 ) that it is the amount of additive genetic variance that will determine the rate of fitness change. With *N*_{eAV} *≥* 500 the loss of such variation through drift is considered to be compensated for by new mutations (Allendorf & Ryman, 2002 Franklin, 1980 ).

Obtaining empirical estimates of effective size is crucial in the management of natural animal and plant populations to find out, e.g., if a particular population reaches any of the targets of the 50/500 rule. Rapidly growing efforts have been devoted to developing and applying methods that are based on genetic markers for estimating contemporary *N*_{e} in natural populations such estimates are used to provide practical conservation management advice (e.g., Harris et al., 2017 Kajtoch, Mazur, Kubisz, Mazur, & Babik, 2014 Rieman & Allendorf, 2001 Sarno, Jennings, & Franklin, 2015 Wennerström, Jansson, & Laikre, 2017 ), and several papers discuss and compare the performance of various approaches (e.g., Gilbert & Whitlock, 2015 Luikart, Ryman, Tallmon, Schwartz, & Allendorf, 2010 Palstra & Ruzzante, 2008 Wang, 2005 Wang, 2016 Waples, 2016 ).

Further, the most widely used estimators of *N*_{e} from genotypic data target *N*_{eV} or *N*_{eLD} (Gilbert & Whitlock, 2015 ). For isolated populations of constant size such estimates can be directly translated into *N*_{eI} or *N*_{eAV} and thus provide the rates at which inbreeding increases or additive variance is lost – the rate of particular relevance to conservation. In contrast, this may not be the case for populations under migration. Overall, the issue of which effective sizes that are estimated empirically in substructured populations when applying different estimators has, as far as we are aware, not been addressed.

In this paper we focus on exploring how *N*_{eV} and *N*_{eLD} relate to *N*_{eI} and *N*_{eAV} in metapopulations. We address the following questions: (a) When do different types of effective size follow the same dynamics to the extent that they can be used as approximate substitutes for one another in substructured populations, and how much do their dynamics differ otherwise and (b) What is the expected magnitude of bias when using estimates of *N*_{e} obtained under the assumption of isolation in situations where this conjecture is erroneous?

We find that frequently applied estimators of *N*_{e} do typically not reflect the rates of inbreeding or loss of additive genetic variation of separate subpopulations in the face of migration. We conclude that estimates of contemporary *N*_{e} from empirical data do not tell us what we need to know for efficient conservation management.

## Results

### Basic descriptives

Genetic diversity among samples was moderate to high, with expected heterozygosities per locus ranging from 0.675 (Ssa-12) to 0.917 (Ssa-171), and heterozygosity per sample ranging from 0.749 (Middle Barachois-1973) to 0.835 (Middle Barachois-1998). The total number of alleles per sample ranged from 85 (Middle Barachois-1973) to 167 (Middle Barachois-1998), with a grand total of 268 alleles available for further analyses. 17 out of 247 tests (6.9%) for Hardy–Weinberg equilibrium were significant, a result expected by chance alone (at *α*=0.05, *χ* 2 -test, *df*=1, *P*=0.24). No deviation was consistent across loci or samples. MICRO-CHECKER indicated four potential occurrences of null alleles, again not consistent across samples, and we thus conclude that the allele frequencies used for subsequent analyses were unlikely to exhibit bias due to scoring or technical errors. See Supplementary Appendix A for a complete overview of gene marker characteristics.

### Population structure

Genetic differentiation among samples, as measured by pairwise *F*_{ST} (Weir and Cockerham, 1984) was very weak to moderate. Among rivers, genetic differentiation ranged from *F*_{ST}=−0.0009 (Crabbes-2000 vs Robinsons-2001) to *F*_{ST}=0.069 (Middle Barachois-1982 vs Western Arm Brook-1982). Temporal stability among samples within rivers ranged from strong (*F*_{ST}=−0.0012, Robinsons-1982 vs Robinsons-2000) to weak (*F*_{ST}=0.0192, Middle Barachois-1973 vs Middle Barachois-1980). Significant isolation-by-distance patterns (Figure 2) were observed over the entire range sampled from the Highlands river in the south to the Western Arm Brook river in the north, as well as at the local scale (within 50 km) of the samples from the five rivers in the study system (see Table 1 and Figure 1).

Isolation-by-distance and by-time relationships over rivers within the study system (dark filled circles, *r* 2 =0.33) and over all rivers sampled (open circles, *r* 2 =0.51).

An analysis of molecular variance on the samples from the study system suggested spatial and temporal components of genetic variance of similar magnitudes. When all five rivers were considered, 1.04% of genetic variance could be attributed to variance among rivers, with 0.33% attributed to variance among temporal samples within rivers (both variance components significant *P*<0.001). However, when these analyses were repeated on the subset of samples from Crabbes, Robinsons and Middle Barachois, the temporal variance within rivers (0.29%, *P*<0.001) was slightly larger than spatial variance among rivers (0.25%, *P*=0.059). In both cases, the percentage of variance explained was thus extremely low. A similar result was obtained when repeating the latter analyses on samples grouped into age cohorts (0.30% among cohorts (*P*<0.001) and 0.26% among rivers (*P*=0.026)).

These results are closely mirrored in the Bayesian clustering analyses of STRUCTURE . Analyses of samples from Crabbes, Robinsons and Middle Barachois suggest they form a single genetic cluster (highest posterior probability for *K*=1). Overall five rivers, the results from STRUCTURE analysis suggest *K*=5 is most likely, but these clusters did not clearly correspond to the individual rivers.

Together, the results of the population structure analyses suggest a dynamic system with weakly differentiated river populations, with connectivity best characterized by a one-dimensional stepping-stone model of gene flow, indicated by the isolation-by-distance pattern observed (Figure 2). These results, therefore, suggest the use of a stepping-stone model of connectivity in subsequent analyses is appropriate.

### Effective population size (N̂e(s)) per river

Based on single samples, estimates of *N*_{e(LD)} ranged from 188 (Middle Barachois-1975) to ∞ (in several samples) (Table 2). *N*_{e(LD)} estimates also displayed considerable variation over time within the same river. Harmonic mean *N̂*_{e(LD)} per river ranged from 204 to ∞. Temporal estimates of *N*_{e(v)} (Table 3) were possible for four rivers and ranged from 38 (Flat Bay Brook) to 1110 (Middle Barachois). The moment-based estimator of Jorde and Ryman (2007) and the pseudo-likelihood approach (Wang, 2001) yielded results that were qualitatively consistent (Table 3). For each river, estimates of *N*_{e(v)} using the temporal approach of Wang and Whitlock (2003) including gene flow were smaller than those assuming closed populations (see Supplementary Appendix B), with gene flow estimates comparable to dispersal rates typically observed in Atlantic salmon (Stabell, 1984). Combining estimates of the temporal methods assuming closed populations resulted in harmonic mean *N*_{e(v)} estimates per river ranging from 54 (Flat Bay) to 849 (Middle Barachois) (Table 3).

Although the different estimators apply to slightly different time scales and may thus not be strictly comparable, they all provide independent indications of the general magnitude of contemporary *N*_{e} of the salmon populations in these rivers. Thus, composite *N̂*_{e(s)} estimates per river were 204 (Highlands), 433 (Crabbes), 496 (Middle Barachois), 1347 (Flat Bay) and 1646 (Robinsons).

### Effective metapopulation size (meta-Ne)

#### ΣN̂e(s)

First, summing the harmonic mean estimates for individual rivers gave Σ*N̂*_{e(s)}=2575 for CMR and Σ*N̂*_{e(s)}=4126 for all five study rivers. Logically, this estimate is larger when all five rivers are included than when only the three weakly differentiated central rivers (CMR) are considered.

#### Meta-N̂e(pooled)

Second, for the set of three weakly differentiated rivers (CMR), the linkage disequilibrium method gave meta-*N̂*_{e(pooled)} ranging from 706 (samples 1980–1982 pooled) to ∞ (samples 1998–2001 pooled) (Table 2). We used these two pooled samples for the temporal methods, which gave meta-*N̂*_{e(pooled)} ranging from 1071 to 2234 (Table 3). Second, we estimated ‘total’ metapopulation effective size by pooling, where possible, genetic data from all rivers. Meta-*N̂*_{e(pooled)} estimation for all five rivers, using the linkage disequilibrium method, was possible only for the combined samples from 1998 to 2001, giving meta-*N̂*_{e(pooled)}=729. Using this combined sample (1998–2001) in the temporal approach with a combined sample from CMR-1980 as *t*_{0}, gave meta-*N̂*_{e(pooled)}=443. Combining and weighting these estimates (*cf*. Waples and Do, 2010) gave meta-*N̂*_{e(pooled)}=2097 for CMR and meta-*N̂*_{e(pooled)}=665 for all five rivers. Hence, with this ‘pooling rivers’ approach, meta-*N̂*_{e(pooled)} is larger for the three central rivers (CMR) than it is for all five rivers.

#### Meta-N̂e(T&H) and comparisons

Third, the harmonic mean per-river *N̂*_{e(s)} values were used for the calculation of meta-*N̂*_{e(T&H)}. Assuming a one-dimensional stepping-stone model with symmetrical gene flow among neighbouring rivers, meta-*N̂*_{e(T&H)}=1835 for CMR and meta-*N̂*_{e(T&H)}=2418 for all five rivers. Thus, meta-*N̂*_{e(pooled)} and meta-*N̂*_{e(T&H)} generally gave estimates of meta-*N*_{e} that were smaller than Σ*N̂*_{e(s)}, but the extent of this difference appears related to the extent of genetic differentiation among populations. In fact, meta-*N̂*_{e(pooled)} was similar to meta-*N̂*_{e(T&H)} for CMR, whereas overall five rivers, where population structure was stronger meta-*N̂*_{e(pooled)} was only 28% of meta-*N̂*_{e(T&H)}.

### Sensitivity analysis

Simulation was used to evaluate which individual deme *N*_{e} estimates has the strongest influence on meta-*N*_{e}. First, for the CMR cluster, the estimate for Crabbes (having the smallest harmonic mean *N̂*_{e(s)}) exerts the largest influence on meta-*N̂*_{e(T&H)} (Figure 3, top). Conversely, over all five rivers, uncertainty about *N̂*_{e(s)} of the two outermost rivers (Highlands and Flat Bay Brook) has the strongest influence on meta-*N̂*_{e(T&H)} (Figure 3, bottom). Second, for the CMR group, the difference between meta-*N̂*_{e(T&H)} and meta-*N̂*_{e(pooled)} can be explained by relatively small bias in any of the individual *N̂*_{e(s)} values. Conversely, over all five rivers (upward) bias in any of the *N*_{e(s)} estimates appears unlikely to be the explanation for the reduction in meta-*N̂*_{e(pooled)} relative to meta-*N̂*_{e(T&H)}.

Relative sensitivity of meta-*N̂*_{e(T&H)} to uncertainty in deme *N*_{e} estimates, quantified by varying model input values of individual deme *N*_{e} estimates (from 50 to 150% of *N̂*_{e(s)}) for a weakly differentiated subset of rivers (top, *d*=3 consisting of Crabbes, Robinsons and Middle Barachois Brook) and for all five rivers (bottom, *d*=5). Also given are the relative magnitudes of meta-*N̂*_{e(pooled)} to meta-*N̂*_{e(T&H)} for both sets of samples (solid grey lines).

Simulations of the effects of gene flow asymmetry on meta-*N̂*_{e(T&H)} (Figure 4, left) suggest that positive density-dependent (higher dispersal from larger populations) dispersal tends to increase meta-*N̂*_{e(T&H)}, whereas negative density-dependent dispersal reduces meta-*N̂*_{e(T&H)}. Importantly, meta-*N̂*_{e(T&H)} is affected primarily not by the magnitude of gene flow, but by the degree of gene flow asymmetry among populations. These trends are similar when smaller input values of *N̂*_{e(s)} are used (Figure 4, right). Therefore, over all five rivers, concordance between meta-*N̂*_{e(pooled)} and meta-*N̂*_{e(T&H)} requires strong negative density-dependent dispersal (for example, *m*_{S}=10*m*_{L}), smaller *N̂*_{e(s)} values as input (that is, the original estimates are biased upwardly) or a combination of these two.

Sensitivity of meta-*N̂*_{e(T&H)} to gene flow asymmetry. Shown are results for various gene flow scenarios, where dispersal from small populations (*m*_{S}) is varied relative to dispersal from large population (*m*_{L}), over the range of gene flow rates commonly observed in Atlantic salmon. *m*_{S}<1.0 indicates positive density-dependent dispersal, *m*_{S}>1.0 indicates negative density-dependent dispersal and *m*_{S}=*m*_{L} indicates gene flow symmetry among neighbouring demes. Results of simulations are given for scenarios based on original *N̂*_{e(s)} values (left panel) and scenarios using 50% of *N̂*_{e(s)} as input values (right panel).

Finally, an extension of these analyses to all rivers in the study system gave meta-*N̂*_{e(T&H)}=1913. Although harmonic mean *N̂*_{e(s)} was smaller than for the situation where only five rivers were analysed, the number of demes also increased, and the resulting meta-*N̂*_{e(T&H)} remains much larger than meta-*N̂*_{e(pooled)} for the five rivers. Upward bias in meta-*N̂*_{e(T&H)} due to the omission of some (small) rivers in its original estimation may thus be unlikely to be an explanation for its large magnitude relative to meta-*N̂*_{e(pooled)}.

### Simulations of meta-Ne

Simulations using EASYPOP suggest individual population *N*_{e(LD)} estimates can be biased upwardly, but only when incoming gene flow is moderate to high (in simulations with *N*_{e(s)}=500 and *m*>0.05 or *F*_{ST}<0.01, Figure 5, top panels). Thus, the discrepancy observed between the two empirical meta-*N*_{e} estimation methods for all five rivers might not be due to inflation in meta-*N̂*_{e(T&H)} caused by a general upward bias in the underlying individual *N*_{e(s)} estimates.

Evaluation of potential bias in meta-*N*_{e} estimates. Given are comparisons between deme *N*_{e(s)} estimates and the expected value of *N*_{e(s)}=500 used in simulations (bias-*N̂*_{e(s)} expressed as the ratio *N̂*_{e(s)}/500), for a range of genetic differentiation among demes in the island model (top left) and stepping-stone model (top right). Also given are comparisons of meta-*N*_{e} estimates based on pooled genetic data (**x** meta-*N̂*_{e(pooled)}), on the model of Tufto and Hindar (2003) (• meta-*N̂*_{e(T&H)}) and on the sum of individual deme estimates ( ○ Σ*N̂*_{e(s)}) for a range of values of genetic differentiation among demes, for the island model (bottom left) and stepping-stone model (bottom right). Simulations performed in EASYPOP and empirical *N*_{e} estimates calculated using software LDNe.

Second, population subdivision may introduce downward bias into meta-*N̂*_{e(pooled)} estimates, but not into estimates of meta-*N̂*_{e(T&H)}, since the latter method explicitly accounts for structure. Results are similar for the stepping-stone and island models and will thus be presented as one (Figure 5, bottom panels). As genetic differentiation among demes increases due to reduced gene flow, meta-*N̂*_{e(pooled)} estimates decrease exponentially, compared with meta-*N̂*_{e(T&H)} estimates. For both models of population structure (stepping-stone or island), Σ*N̂*_{e(s)} were consistently larger than meta-*N̂*_{e(T&H)} (paired *t*-test, two-tailed, *P*=0.0014) but this difference appears unaffected by the extent of genetic differentiation.

Finally, the rates of loss of neutral genetic diversity observed in the simulated metapopulations generally suggest these are predicted much more closely by meta-*N̂*_{e(T&H)} than by meta-*N̂*_{e(pooled)}. These simulations therefore indicate that population subdivision (spatial structure) introduces downward bias into empirical estimation of meta-*N*_{e} when this structure is not explicitly accounted for.

## Inbreeding effective size

Alternatively, the effective population size may be defined by noting how the average inbreeding coefficient changes from one generation to the next, and then defining *N*_{e} as the size of the idealized population that has the same change in average inbreeding coefficient as the population under consideration. The presentation follows Kempthorne (1957). [11]

For the idealized population, the inbreeding coefficients follow the recurrence equation

Using Panmictic Index (1 − *F*) instead of inbreeding coefficient, we get the approximate recurrence equation

The difference per generation is

The inbreeding effective size can be found by solving

although researchers rarely use this equation directly.

### Theoretical example: overlapping generations and age-structured populations

When organisms live longer than one breeding season, effective population sizes have to take into account the life tables for the species.

#### Haploid

Assume a haploid population with discrete age structure. An example might be an organism that can survive several discrete breeding seasons. Further, define the following age structure characteristics:

Then, the inbreeding effective population size is [12]

#### Diploid

Similarly, the inbreeding effective number can be calculated for a diploid population with discrete age structure. This was first given by Johnson, [13] but the notation more closely resembles Emigh and Pollak. [14]

Assume the same basic parameters for the life table as given for the haploid case, but distinguishing between male and female, such as *N*_{0} *ƒ* and *N*_{0} *m* for the number of newborn females and males, respectively (notice lower case *ƒ* for females, compared to upper case *F* for inbreeding).

The inbreeding effective number is

## Decision letter

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

The manuscript assesses the intra-host effective population size of influenza based on longitudinal deep sequencing data from a chronic influenza B infection. Using principles modeling and statistical approaches, the authors show that the short length of a typical influenza infection is the key limiting factor upon selection at the within-host level. The topic is important, as it sheds light on the interplay between the two scales of selection within- and between-host in shaping the evolution of influenza virus.

**Decision letter after peer review:**

Thank you for submitting your article "A large effective population size for within-host influenza virus infection" for consideration by *eLife*. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Aleksandra Walczak as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Georgii A. Bazykin (Reviewer #2).

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

We would like to draw your attention to changes in our revision policy that we have made in response to COVID-19 (https://elifesciences.org/articles/57162). Specifically, we are asking editors to accept without delay manuscripts, like yours, that they judge can stand as *eLife* papers without additional data, even if they feel that they would make the manuscript stronger. Thus the revisions requested below only address clarity and presentation.

The manuscript presents a study on within-host population genetics of influenza virus and in particular, inference of effective population size during chronic infection in immunocompromised patients. The topic is important as it explores the interplay between the two scales of selection: within-host and between-host selection that shape the evolution of influenza. Based on the analysis of sequence polymorphism, authors infer a relatively large effective population size

10 7 during chronic infection, in contrast to previously inferred values of

10 2 or less during transmission and in acute infections. All of the reviewers agree that the findings in this manuscript are interesting and a large effective population would have significant implications for efficacy of selection during within-host evolution of influenza. However, there are still some concerns regarding methodology, interpretation and presentation of the results which we would like to see addressed.

1) Comparison between chronic and acute infections:

The authors analyzed data from chronic influenza infections and concluded that the effective population size of the virus is high, including during acute infections. For instance, the authors argue that "the observed lack of within-host variation in typical cases of influenza can be explained by the short period of infection the stochastic effects of genetic drift do not limit the impact of positive selection". It is however not evident that the authors' estimates of effective population size from chronic infections apply to acute infections given the exponential increase and decrease of viral load that dominate the course of acute infections. In fact, it's not clear that effective population size is even a very useful concept in this case.

Also, McCrone et al., 2018, and Xue and Bloom, 2020, have both shown that within-host variation in acute infections is dominated by non-synonymous mutations, and Xue and Bloom, 2020, also document stop-codon mutations within acute infections that are rarely found at appreciable frequencies in chronic infections. These observations suggest that selection is inefficient within hosts in acute infections, contrary to the authors' claims.

Moreover, McCrone et al. see radical changes in variant frequencies over the course of a few days (Figure 2E in that work) – but lineages in chronic infections (this work) persist for many months. If the authors think that N_{e} is comparable between acute and chronic infections, how do they explain the lack of diversity observed in acute infections? One way to explain this is to maintain a high N_{e} but with strong transmission bottleneck to impose stochasticity. But as point out above, "N_{e}" is really not a well-defined quantity in this case. Alternatively, could the difference imply a lower census size in acute infections, and if so, is this consistent with differences in viral load? This issue is important in view of the proposed relevance of high N_{e} for long-term influenza evolution (e.g., last phrase of the Abstract and the last phrase of the Introduction).

Overall, the authors should acknowledge the differences between acute and chronic infections, and discuss their estimates in light of the previous observations. Moreover, it may also be helpful to revise the title to indicate that the manuscript focuses on chronic infections.

2) High N_{e} is inferred from small drift and a small rate of "substitutions" (which under the authors' terminology also account for minor changes in allele frequencies). In other words, the authors are inferring a large N_{e} based on the longer-term coexistence of multiple lineages within a host. Therefore, it would be important that the manuscript also discusses alternative explanations that could lead to such patterns of polymorphism. Importantly, as N_{e} in the manuscript is inferred from a Wright-Fisher (WF) model, violations in the underlying assumptions of the model can bias the results. For example, one can imagine that demographic effects like population structure could be responsible for long-term coexistence and survival of lineages, e.g., if each of the samples represents a mixture of persistent subpopulations? The authors seem to suggest this by analyzing clades A and B separately, Results and Discussion, second paragraph. Alternatively, could balancing selection in the host be responsible for maintaining this polymorphism (seems unlikely, but still a formal possibility)? A discussion and/or analysis of such alternative scenarios would be useful in assessing the robustness of the manuscript's findings.

3) Robustness of the analysis and proposed statistics:

a) It would be useful to have a clearer sense of the sensitivity of N_{e} to the cutoffs used. While a lot of care has gone into the choice, some diagrams showing the sensitivity of N_{e} to cutoff choice would better demonstrate the degree to which it is a function of low frequency variants in a straightforward way.

b) To estimate how N_{e} affects changes in allele frequencies, the authors simulate a single generation of Wright-Fisher evolution using initial allele frequencies from a randomly selected sample from the infection. As the equation in the subsection “Summary” indicates, populations with high-frequency alleles will experience larger changes in allele frequency at a given effective population size, so the initial distribution of allele frequencies from this randomly chosen sample can have a major effect on the expected change in allele frequencies. The authors show in Figure 2—figure supplement 1 that mutations can reach frequencies of 20-30% in neuraminidase, and in the influenza A patients analyzed in Figure 2—figure supplement 3, many mutations reach these and even higher frequencies, particularly at later points in the infection. The authors should run their Wright-Fisher simulations with different initial allele frequencies to evaluate how this choice of allele frequencies may affect estimates of effective population size.

c) The authors design statistic D to assess their estimation of N_{e}. This statistic is a sum of changes in variant frequencies across sites (subsection “Calculation of evolutionary rates”), which is then compared between data and Wright-Fisher simulations for different N_{e} values. The authors seem to suggest that D should be more robust to noise (subsection “Summary”), without providing any evidence. In particular, the authors should clearly state how the assumptions they made about recombination structure in WF simulation could impact the statistics D and the interpretation of the inferred N_{e}. From the manuscript it is not clear whether WF simulations are done at the site-wise, segment-wise, or genome-wise level, which would impact the correlation between changes in variant frequencies. For example, simulations done with high (free) recombination would expect a lower variance D compared to the case with strong linkage (data), for the same N_{e}. These points should be better clarified.

4) In Figure 1A, it is clear (and the authors also mention) that the patient's viral load drops to undetectable levels for over a month of the infection, and viral load also varies substantially while the patient is continually infected. Effective population size and census population size are not always directly related, but the authors should discuss how changing population sizes affect their estimate of effective population size and whether a single effective population size is adequate to represent the infection.

5) The authors calculate sequence distance between every pair of sequenced timepoints to reduce the influence of noise from sequencing error, but as a result, the points in Figure 2A are non-independent and may contribute to a tighter confidence interval around the evolutionary rate than is realistic. In particular, changes in variant frequencies that take place during the middle of the infection will be overcounted in these pairs and will disproportionately influence the overall estimate of evolutionary rates. When the authors estimate the evolutionary distance between consecutive timepoints and divide by the number of days between them, how well does the estimate correspond to the estimates in Figure 2? What is the variance in these estimates?

6) The regression performed in Figure 2A, C, and analogous figures may be especially influenced by the few points at the right end of the distribution, which represent evolutionary distances between points spaced further apart in time. How robust is the estimate of evolutionary rate to removal of these points, or by calculation of evolutionary rate as suggested in comment 4?

7) The authors chose to infer effective population size using variants and haplotypes on the neuraminidase and hemagglutinin segments. This is an odd choice since these regions tend to experience the strongest selection, which can strongly influence the estimates of effective population size. Selection can act on linked haplotypes across the genome in some cases, but have the authors tested to see if these results hold for other gene segments as well?

8) Why are the effective population size estimates for the clade B samples calculated separately from the clade A samples? It's not evident from the SAMFIRE inference of haplotypes that clades A and B constitute separate subpopulations it seems that they could be distinct genotypes in a well-mixed population as well, as might result from a coinfection.

9) The authors assume the generation time of 10 hours per generation for influenza B. However, if generations are longer in immunocompromised individuals, the analysis would lead to an overestimation of N_{e}. Given that the main result in this manuscript is that N_{e} is high, this possibility should at least be discussed.

## 5 EFFECTIVE POPULATION SIZE: MEAN CROWDING OF GAMETES AROUND PARENTS

The effective population size, *N*_{e}, is the size of the ideal population with the same rate of loss of heterozygosity per generation as a natural population of size, *N*, whose biology causes it to deviate from the theoretical ideal. The strength of random genetic drift is determined by *N*_{e} which is important in several areas of evolutionary genetics, including inbreeding (Frankham, 1995 Whitlock, 1992 , 2003 ), speciation (Bryant & Meffert, 1988 , 1990 , 1996 Carson, 1968 , 1975 Carson & Templeton, 1984 Meffert, 1999 Meffert & Bryant, 1992 Powell & Wistrand, 1978 Templeton, 1980 ), mating system evolution (McCauley & Taylor, 1997 ), conservation biology (Hedrick & Kalinowski, 2000 ) and Wright's shifting balance theory (SBT) (Goodnight & Wade, 2000 Wade & Goodnight, 1998 Wright, 1931 , 1978 ). Among the factors that reduce *N*_{e} below *N* is random variation in the number of gametes contributed by parents to the next generation.

We illustrate the relationship between mean crowding and *N*_{e} by following Crow and Kimura ( 1970 , p. 361–362, eq. 7.6.4.1). We consider a diploid hermaphrodite population with a constant size, *N*, of self-compatible, random mating adults with discrete generations and no age structure. Chance or selection result in some individuals leaving more offspring than others, raising the probability that two offspring, randomly chosen from the population, will have had the same parent. If we let *k*_{i} be the number of gametes in all the offspring that the ith parent contributes to the next generation, *k*_{.} be the mean number of gametes, and *V*_{k}, the variance among parents in gamete numbers for a nuclear diploid locus, we have

where *k** is the mean crowding of gametes around parents. If this variance is owing to selection, Equation (5-5) shows how directional natural selection, acting on one gene or one trait, increases *k** relative to *k*_{.}. This *reduces* the effective population size, *N*_{e} below *N*. In particular, as a result of sexual selection (see above), males often have a larger *k** than females, such that *N*_{e,males} < *N*_{e,females}. This has consequences for the effective population size of mitochondrial or chloroplast genes, with matrilineal inheritance, relative to the effective population size of Y-linked genes with patrilineal inheritance (Wade & Shuster, 2004 ). If the population size remains constant from one generation to the next, then *k*_{.} = 2, and the expression above reduces to the familiar, classic equation (Crow & Kimura, 1970 , p. 361, eq. 7.6.4.3): *N*_{e} = (4*N* − 2)/(*V*_{k} + 2).

## Realized migrant distributions in Wade-Goodnight experiment

The experimental protocol of Wade and Goodnight (1991) converted the variance in among demes in offspring numbers, V_{P}, into variance in relative demic fitness, V_{w}, causing V_{w}, to equal S 2 (cf. Wade and Goodnight, 1991, p. 1016). Stochastic migration (independent of relative demic fitness) was imposed on the control metapopulations (C1, C2, and C3) at the same average rate per deme as in the corresponding experimental metapopulation (E1, E2, and E3). The periodicity of migration varied among treatments. It was every generation for the E-1 and C-1 metapopulations, every second generation for E-2 and C-2, and every third generation for E-3 and C-3.

The total number of migrants per deme in the three experimental and three control treatments (50 demes per treatment cf. Wade and Goodnight, 1991) for generations one through 13 was computed from the census data and experimental protocols. Table 1 presents the total number of migrants, the mean number of migrants per deme (X), the variance in number of migrants among demes (V), and the ratio (V_{E}/V_{C}) of the variance of experimental to control metapopulations. It is not surprising that the total number of migrants is a function of the periodicity with which migration was imposed. Imposing migration every generation (E1, C1) resulted in more total migrants than imposing migration every two (E2, C2) or every three (E3, C3) generations ( Table 1 column 2). (The small differences in the E and C totals represent minor errors in executing our protocols over the 2.5 years the experiment was run.) The totals and the means per deme (X) show that each pair of control and experimental demes experienced the *same average* migration rate ( Table 1 , columns 2 and 3). The last two columns of Table 1 reveal that the variance in the total number of migrants received per deme was 3 to 5 fold higher in the experimental metapopulations with Phase III migration than it was in the controls with island model migration (variance ratio test, p < 0.001 in each case). The very large values for the variance in Phase III migration relative to island model migration are caused in part by the covariance across generations of mean demic fitness. That is, the significant populational heritability of demic fitness (Wade and Goodnight 1991) means that there was a measurable tendency of some demes (those withthe highest values of w_{i}) to be net exporters of migrants for two or more consecutive generations as well as some demes to be net importers of migrants for two or more consecutive generations (those with the lowest values of w_{i}). The heritability of demic fitness inflates the variance in migration rate among demes above the value expected from the interdemic selection differential (S) alone.

### Table 1

The mean (X) per deme, the variance (V) and the total number of migrants moved among 50 demes over the first 13 generations of each control metapopulation (C1, C2, and C3) and each experimental metapopulation (E1, E2 and E3). The ratio of the variance in migrant numbers of the experimentals to the controls, (V_{E}/V_{C}), illustrates the degree to which Phase III migration is more variable than random migration.

Metapopulation | Total Migrants | X | V | V_{E}/V_{C} |
---|---|---|---|---|

C1 | 1,244 | 24.88 | 24.35 | 4.3 |

E1 | 1,239 | 24.78 | 104.71 | |

C2 | 650 | 13.00 | 14.20 | 3.2 |

E2 | 636 | 12.72 | 44.94 | |

C3 | 544 | 10.88 | 14.15 | 4.8 |

E3 | 543 | 10.86 | 67.55 |

At generation 13, we censused every population as usual but also determined the genotype (++, +/b, and bb) of each beetle in every deme at a single locus, semi-dominant black body color mutation ( b ) which was segregating within the c-SM stock from which our metapopulations were derived. This allowed us to calculate the gene frequency, p, in every deme and its variance among demes. (These were not estimates of gene frequency but rather calculations based on total counts of all adult genotypes.) The demic frequency of p was not significantly correlated with deme size in any of the six metapopulations. From the values of p for each deme, we calculated the observed F_{ST}, the amount of genetic differentiation by random genetic drift in all six metapopulations ( Table 2 ). Observed values of F_{ST} increase from C-1 to C-2 to C-3 as expected from the imposed migration rates ( Table 2 column 3). The F_{ST} observed in the E-2 and E-3 experimentals exceeds that of their respective controls, C-2 and C-3. The same is not true, however, for C-1 and E-1. In this case, the mean number of migrants per deme up to generation 13 is nearly double that of the other two treatments, swamping the differential effect of the variance in migrant numbers. In addition, E1 had a much lower group heritability of demic fitness than E2 (Wade and Goodnight 1991).

### Table 2

The average frequency, p, of the semi-dominant, black body color allele observed within the 50 demes of each control (C1, C2 and C3) and each experimental (E1, E2 and E3) metapopulation and F_{ST}, the among-deme component of the variance in p.

Metapopulation | Frequency of b-allele | FST |
---|---|---|

C1 | 0.0488 | 0.077 |

E1 | 0.0474 | 0.066 |

C2 | 0.0620 | 0.089 |

E2 | 0.0567 | 0.118 |

C3 | 0.0248 | 0.094 |

E3 | 0.0560 | 0.161 |

## Acknowledgements

We thank M Bradley, A Dibb, D Hervieux, N McCutchen, L Neufield, C Sacchi, F Schmiegelow, M Schwartz, M Sherrington, J Skilnick, S Slater, K Smith, D Stepnisky, J Wilmshurst and J Wittington. Research was conducted under Alberta, BC, and Parks Canada, Universities of Montana, Calgary and Alberta research and collection permits. The authors thank D Ruzzante and two anonymous referees for helpful reviews of this manuscript.

### Funding statement

Support was provided by the Alberta Department of Sustainable Resource Development , British Columbia Ministry of the Environment , BC Ministry of Forests , Canadian Association of Petroleum Producers , Conoco-Philips , NSERC , Parks Canada , Petroleum Technology Alliance of Canada , Royal Dutch Shell , UCD SEED funding, Weyerhaueser Company , Alberta Innovates , Alberta Conservation Association , the Y2Y Conservation Initiative , and the National Aeronautic and Space Agency ( NASA ) under award no. NNX11AO47G .

## Effective population size

The **effective population size** is the number of individuals that an idealised population would need to have in order for some specified quantity of interest to be the same in the idealised population as in the real population. Idealised populations are based on unrealistic but convenient simplifications such as random mating, simultaneous birth of each new generation, constant population size, and equal numbers of children per parent. In some simple scenarios, the effective population size is the number of breeding individuals in the population. However, for most quantities of interest and most real populations, the census population size *N* of a real population is usually larger than the effective population size *N*_{e}. [1] The same population may have multiple effective population sizes, for different properties of interest, including for different genetic loci.

The effective population size is most commonly measured with respect to the coalescence time. In an idealised diploid population with no selection at any locus, the expectation of the coalescence time in generations is equal to twice the census population size. The effective population size is measured as within-species genetic diversity divided by four times the mutation rate μ

The concept of effective population size was introduced in the field of population genetics in 1931 by the American geneticist Sewall Wright. [2] [3]