17 Models to study gene flow

Dr. Subhendu K Acharya

Contents

• Introduction
• Gene flow
• The Island model
• Two-Way Gene Flow model
• Kin Structure Model
• Stepping stone model
• Direct measures of gene flow: Genetic marker based analysis
• Summary

Insight into the human past can be generated by analysing modern genetic diversity because genetic variation has not arisen independently in each individual, but has a shared history that has been shaped by different processes, such as natural selection and population expansion. All these processes are part of human evolutionary process. In population genetics evolutionary process means studying the process of change in allele frequencies within a population in time and space.

There are generally two types of models in studying the process of change in allele frequencies within a population. While in one type of model we look into the mathematical approximation of population, their interactions and mating structures in other model we study the process of mutation and recombination.

The first types of model that explain mathematically is Hardy-Weinberg equilibrium which describes the relationship between allele frequencies and genotype frequencies in a randomly mating population. Hardy-Weinberg equilibrium explains that a population is at equilibrium if

There is

·         No selection or random mating

·         No mutation

·         No overlap between generations

·         No migration or gene flow

·         No substructure

If any of these assumptions is broken then it confirms to the fact that the population is not at Hardy-Weinberg equilibrium. But in real conditions, one or more of these influences are typically present. So the Hardy–Weinberg principle gives us an ideal condition against which we analyse the effects of these factors.

So here we discuss one such influencing factor i.e. gene flow in the population. Or in other way how the process of migration in population and its effect affect of allele frequencies.

Gene flow has two major effects on a population. First it introduces new alleles to the population. Since mutation generally is a rare event, a specific mutant allele may arise in one population and not in another. Gene flow spreads these new alleles to other populations and. Like mutation, is a source of genetic variation for the recipient population. Second, when the allele frequencies of migrants and the recipient population differ, gene flow changes the allele frequencies within the recipient population. Through exchange of genes, different populations remain similar, and thus migration is a homogenizing force that tends to prevent populations from accumulating genetic differences among them. Lowered levels of gene flow mean that the two subpopulations have allele and genotype frequencies that tend to be independent through time.

Gene flow:

Gene flow is defined as the  introduction of receiving population. This process happens by interbreeding. Gene flow introduces new alleles to the population which further leads to increased variability and new combinations of traits. In humans, the gene flow takes place by migration. Migration is the movement from one occupied area to another. Gene flow is the outcome of a migrant contributing to the next generation in their new location. Thus to observe gene flow directly we not only need to monitor the movement of migrants but also their reproductive success.

Unlike genetic drift, mutation and selection, migration cannot change species wide allele frequencies, but it is capable of changing allele frequencies in the population. So it plays an important role in human genetic diversity. Gene flow counteracts genetic differentiation and is modelled within the framework of a larger, subdivided metapopulation.

A. The island model

The island model or the continent-island model is best explained by understanding the process of gene flow during one-way migration. It assumes one very large population where allele frequency changes only slowly over time connected by gene flow with a small population where migrants make up a finite proportion of the individuals present each generation. The continent receives the gene flow but the continent population is assumed to be large enough that immigration has a negligible effect on allele frequency. In this case a specific degree of migration occurs from the mainland to an island, but in such type of migration is strictly in one direction without any migration occurring in the reverse way from the island to the mainland. This can explain the effect on allele frequencies of the island as the effect of mainland. Symbol m is used for illustratation of the rate of migration. This rate of migration is the ratio of alleles in the island to that of from the mainland for every generation of the population.

The assumptions of island model include:

No geographic substructure apart from the division into islands

Each population persists indefinitely

No mutation

No selection

Each population has reached equilibrium between mutation and drift

The migrants are the randomly moving population from the home island

As the contribution towards the mainland gene pool is represented by m it is explained that the island is the sole contributor to the allele proportion (1-m) of the island (it is because the mainland and the island proportions do not make it to 1).

This is explained in Figure 2.

Figure2: The island model.

In figure 2 it is shown that the process of gene flow is taking place in single direction- from the source or larger population to small island population. Here the allele frequency of island does not affect the source or mainland population while it happens in the reverse direction.

For the simulation of island model regarding the gene flow in one way direction it is necessary to know the allele frequencies distribution in the island and mainland at the initial level. In the present

explanation P0 stands for the allele frequency of the island at the initial level (Gen0). P stands for the frequency of the mainland allele. In this model only gene flow is being considered. As the mainland is not receiving any gene flow from island, the allele frequency for mainland (P) remains same one generation to next generation. For calculating island allele frequency of the next generation, the ratio of alleles from island and the alleles from the mainland by the frequencies of their alleles are multiplied. This in the Generation 1 will be calculated as:

P1 = (1 — m)po + mP       (eq-1)

In the above equation m stands for the rate of migration of alleles out of mainland. . As no changes takes place in P (resulting mP remaining unchanged) the island allele frequency can be predicted after the one way of gene flow for an additional generation which can be put forth by replacing p1 for p0 in the previously explained equation. This comes out as:

p2 = (1 — m)p1 + mP          (eq-2)

figure 3 represents the results of gene flow in one way direction in island model. The island’s allele frequency at the initial level as p0 = 0.8. The allele frequency in mainland in contant propotion is P = 0.2. The migration rate for generations is m =0.5.

Figure 3: Island model and Pattern of allele frequencies change through generations. This graph demonstrates that due to the migration of mainland alleles to the island, the allele type of island gradually becomes similar to mainland.

Here, the expected change in allele frequency due to a single generation gene flow can also be extended to predict allele frequency in the island population over an arbitrary number of generations. Such outcome is completely determined by a constant rate of gene flow and has no random process such as genetic drift to introduce chance variation. The predictions of the continent island model are consistent with the fact that the continent population has a constant allele frequency over time, the island population should eventually reach an identical allele frequency when the two are mixed.

The continent island model shows that the process of gene flow alone is capable of bringing population to the same allele frequency. Identical allele frequencies between or among populations is really a lack of population structure of panmixicia. So the continent island can be thought of as a demonstration that gene flow acting in absence of the other processes will eventually results in panmixicia.

B.  Two-way gene flow model

The two-way gene flow model is different from island model in the way that it explains the gene flow in both way- mainland to island and island to mainland, while island model allows gene flow only in one direction from main land to island. Here it is important to state that the gene frequency changes in both the populations of mainland and island in case of two way gene flow model while it does not change in mainland, the source population is island model. The Two-Way Gene Flow model seeks to remove one obvious flaw of the island model i.e. the lack of geographical substructure. The model assumes equal rates of migration between subpopulations. Here it is noteworthy that both the models show that even very low rates of migration between sub populations are capable of retarding their genetic differentiation. The two way model mainly shows that the equilibrium allele frequencies in the two subpopulations are the average allele frequency of the total population when the two migration rates are equal. Such dimension is also important when there are large numbers of subpopulations, a result that will be useful to remember when considering the gene flow process in an island model in combination with another process such as genetic drift. The figure demonstrating the two way pattern of gene flow shows the migration rate as same for both the population. . The rate of migration is m for the populations-X and Z in the given explanation. Here m represents alleles’ ratio between the two populations while 1-m is the received allele’s proportion by each population.

Figure 4: Gene flow in two way direction. The figure demonstrates the two populations X and Z having alleles proportions of m and 1-m from other population and from their own stock respectively.

Considering the rate of gene flow for each generation as m, for t generation in population X and Z the allele frequencies as Px and Pz the allele frequency in the t+1 (next) generation for population X can be estimated by multiplying the proportion of alleles provided to population X with the allele frequencies of each population:

PXt+1= (1-m)PX1+mPZt     (eq-3)

Similar procedure is used to derive the allele frequency for the other population, B in the next generation:

PZt+1= mPX1+(1-m)PZ1     (eq-4)

Let us consider the allele frequencies of PX0 at initial level is 1 and 0 for PZ0; the rate of gene flow (m)

as 0.1 for each generation. Further, the allele frequency for the A population after a generation can be calculated as:

PX1 = (1 — 0.1)(1.0) + (0.1)(0.0) = 0.9

While the frequency of alleles in B population can be computed as:

PZ1= (0.1) (1.0) + (1 — 0.1)(0.0) = 0.1

It is observed that the two populations have got the allele frequencies which are close each other than previous. Here along with new allele frequencies gene flow can be extended to next generation, where the next generation will carry the allele frequencies:

PX2=(1 — 0.1)(0.9) + (0.1)(0.1)=0.82

and

PZ1=(0.1)(0.9) + (1 — 0.1)(0.1)= 0.18

as illustrated in figure 5 the above process will continue to further generations with the allele frequencies becoming similar increasingly. Here it is note worthy that the alleles become same in totality after a period of 20 generation approximately.

FIGURE 5: it shows the gene flow in two ways. Here the allele frequencies at initial level: px = 1. 0 and pz=0.0, gene flow rate (m) as 0.1

So, the change in allele frequency from migration depends on two factors, the population of the migrants in the final population and the difference in allele frequency between the migrants and the residents. If no difference exists in the allele frequency of migrants and residents, then we can see that the change in allele frequency is zero. Populations must differ in their allele frequencies for migration to affect the makeup of the gene pool. With continued migration the source of migrating population and the destination population become increasing similar by their allele frequencies. As a result, the change  in allele  frequencies  due  to  migration decreases.  Eventually,  allele  frequency in  the  two populations will be equal without any further change. However this will happen, only when, other factors besides migration do not influence allele frequencies.

C.  Kin-Structured Migration Model

Gene flow over a period of time, as explained in the two way model, brings the interacting population closer at genetic level. This change process can be rapid if the gene flow rate is higher. The role of gene flow in population homogenization helps in reducing the population differences at genetic level. But the above assumption will be correct only in case, the reproducing individuals are randomly migrating from the source population. But here it is worth mentioning that related individuals may also migrate. For more clear explanation, it can be said that in many instances, the entire members of the families migrate to the new population. In such situation with the migrating folk as relatives, it is called kin structured migration.

Figure 6: example of Kin structure migration

The kin structured migration and its effects towards population differences have closely been observed by Alan Fix, an anthropologist. The role of gene flow is mainly to minimize the population level differences while the process under kin structured migration acts in reverse direction by slowing down it which gives a result of opposite with increase in difference between population groups. To understand the kin structured migration the observation of Fix (1978) among the Semai Senoi, a Malaysian agricultural population in the Malaya Peninsula brought the findings that though split in the population occur, the new groups were very much kin in relation. Fix in his analysis found that in spite of reduced difference at genetic level between the studied Semai Senoi groups due to kin structured migration, the effect of such force at actual level will be less subject to condition that migration took place in a random fashion. Such kin structured migration pattern has also been observed in various human population groups such as in S American Yanoinama, in American (north) Plymoth etc. But it is not a common phenomenon worldwide.

Figure 7: An example of Kin Structure migration model showing relationship of European to non-European populations.

D: Stepping stone model:

Although the island model assumes that the amount of gene flow among all populations is identical, a population organized into discrete subpopulations can also experience isolation by distance. The stepping stone model approximates the phenomenon of isolation by distance among discrete subpopulations by allowing most or all gene flow to be only between neighboring subpopulations. This gene flow pattern produces an allele frequency clumping effect among the subpopulations qualitatively very similar. Kimura and Weiss in their analysis showed that the correlation between the two alleles sampled at random from two subpopulations depends on (i) the distance between the subpopulations and (ii) the ratio of flow of gene among neighboring colonies and gene glow of long distance where alleles are exchanged among sub-populations at random distances. As expected for isolation by distance, the correlation between allelic states decreases with increasing distance between subpopulations. Interestingly the correlation between allelic states drops off more rapidly with distance when subpopulations occupy two dimensions than when they occupy one dimension.

The above discussed models for gene flow based on migration type are mathematically tractable and can be generalized in other species. Further in case of humans we in most of the cases have detailed data on migration rates, migration distances and marital distances. The uneven pattern of most human habitation falls between the models of discrete subpopulations and uniform continuity assumed by the stepping stone and isolation by distance models respectively. But at the same time migration process often include long distance movements as well as smaller scale mating choices. The choices to migrate is taken by individuals on the basis of multiple ‘push’ and ‘pull’ factors, so that migration rates are rarely , if ever, symmetric between two population; they are often age structured, sex biased and related to one another.

Direct measures of gene flow: Genetic marker based analysis

Percentage analyses are considered direct measures of gene flow since they reveal and measure the pattern of gamete movement at the scale over which the candidate parents are sampled.

Percentage analyses are also commonly used to test hypotheses about what factors influence patterns of mating among individuals. Percentage analysis measures gene flow by inferring numerous mating events within the population of candidate parents that leads to each sampled progeny or juvenile in a population. This provide estimates of quintiles such as the average distance between parents or the number of mating where both parents were within a sample area compared to the number of matings where a parent was outside that area.

Further there is usually some spatial scale at which the chances of mating are limited. This varies with the species and could be distances as small as few meters or as large as thousand kilometers depending on the range of movement of individuals and their gametes. The phenomenon of decreasing chances of mating with increasing distance separating individuals is termed isolation by distance. One effect of isolation by distance is local changes in allele frequency in a population with local regions approaching fixation or loss, akin to the impact of reducing the effective population size. Alternatively, isolation by distance can be thought of as a form of inbreeding, since restricted mating distances cause homozygocity within subpopulations to increase. The patterns of genotypes in the simulated populations bear this out, with an obvious decline in the overall frequency of heterozygote over time with isolation by distance but no such decline when there is panmexia.

The effects of gene flow have important ramifications not only for the evolution of species but also for the conservation of the species. Many species that have wide geographic ranges show variation in genetic structure over the species range. Part of the natural genetic structure of a species could include population subdivision in which populations are loosely connected to each other by gene flow. Since gene flow is important in maintaining genetic diversity, this feature of population genetic structure must be taken into account by those interested in conserving genetic identity of species.

Finally, migration of individuals into a population may alter the makeup of the population gene pool if the allele frequency of the migrants differs from that of the migrant resident population. Migration or otherwise calling it as gene flow tends to reduce genetic divergence among populations and increases the effective size of the population. The amount of migration among populations of the same species determines how much genetic sub structuring exists and whether different populations of the same species become very different from each other genetically

Summary

• Gene flow is defined as the introduction of genetic material from one population of a species to another population by changing the gene pool composition of the receiving population. Gene flow introduces new alleles to the population which further leads to increased variability and new combinations of traits. In humans, the gene flow takes place by migration. Gene flow counteracts genetic differentiation and is modelled within the framework of a larger, subdivided metapopulation.
• The higher the rate of gene flow, the more rapidly this convergence occurs. Gene flow is most often considered a homogenizing force that reduces the genetic difference between populations.
• Gene flow has two major effects on a population. First it introduces new alleles to the population. Since mutation generally is a rare event, a specific mutant allele may arise in one population and not in another. Gene flow spreads these new alleles to other populations and. Like mutation, is a source of genetic variation for the recipient population. Second, when the allele frequencies of migrants and the recipient population differ, gene flow changes the allele frequencies within the recipient population.
• The simplest place to start with understanding how gene flow works is to examine the case of one-way migration or the island model or more specifically, the continent-island model. It assumes one very large population where allele frequency changes only slowly over time connected by gene flow with a small population where migrants make up a finite proportion of the individuals present each generation.
• The Two-Way Gene Flow model addresses the issue of lack of geographical substructure. The model assumes equal rates of migration between subpopulations. The two way model mainly shows that the equilibrium allele frequencies in the two subpopulations are the average allele frequency of the total population when the two migration rates are equal. In this model the change in allele frequency from migration depends on two factors, the population of the migrants in the fine population and the difference in allele frequency between the migrants and the residents.
• In some situations, the migrants may be related. One way that migrants can be related, occurs in some small-scale human societies when part of a population splits off and then fuses with another population. There are other examples of entire families moving into a new population. When the migrants are related, we call this kin-structured migration.
• The correlation between the two alleles sampled at random from two subpopulations depends on (i) the distance between the subpopulations and (ii) the ratio of gene flow between neighboring colonies and long distance gene flow where alleles are exchanged among sub-populations at random distances.
• Percentage analyses are also commonly used to test hypotheses about what factors influence patterns of mating among individuals. Percentage analysis measures gene flow by inferring numerous mating events within the population of candidate parents that leads to each sampled progeny or juvenile in a population. This provide estimates of quintiles such as the average distance between parents or the number of mating where both parents were within a sample area compared to the number of matings where a parent was outside that area.