# Assumptions of Hardy-Weinberg rule

The post Solving Hardy Weinberg problems offers an easy explanation of Hardy-Weinberg rule. The current top answer explicitly does not talk about the assumptions of Hardy-Weinberg. A model makes sense only if one is able to tell its domain of definition and therefore it is critical to understand the assumptions underlying Hardy-Weinberg assumptions.

What are the assumptions of Hardy-Weinberg rule?

Here are the assumptions to the standard Hardy-Weinberg rule (HWr) formulation

$$p^2 + 2p(1-p) + (1-p)^2 = 1$$

# Assumptions

The locus of interest is bi-allelic

This is an obvious assumptions, however it is very easy to extrapole HWr to any number of loci.

Organisms are diploid

This is another obvious assumption that is very easy overcome. This also means that HWr does not hold for loci present on sexual chromosomes as those chromosomes do not display a simple diploid behaviour.

Only sexual reproduction occurs

If some individuals are able to undergo non-sexual reproduction, then the rule does not hold anymore. See also Panmixia as these two assumptions are related.

Panmixia

Panmixia is often called "random mating". Panmixia is the state where each individual is equally to mate with any other individual in the population (including itself). There is therefore no population structure and no mate choice.

Let $$N$$ be the number of individuals in the population, in absence of selection (see If selection, it must be on fecundity alone), the probability to mate with any given individual (including itself) is $$frac{1}{N}$$. You will note that it requires that individuals are able to self (but not clone as reproduction must still be sexual).

If selection, it must be on fecundity alone

If selection occurs, then it must on fecundity (right before reproduction). If selection occurs on survival during the lifetime, then the selected genotypes will be in excess in comparison to Hardy-Weinberg expectations.

If migration, it must be right before reproduction only

For the same reason as above, if there is migration, then it must be right before fecundity. If migration occur during the lifetime, then it will have a very similar effect than selection. Selection would increase the frequency of selected genotypes above HR expectations. Migration would increase the frequency of incoming genotypes above HR expectations.

Non-overlapping generations

This means that everyone reproduce in the exact same time and die right afterward. Very few species would qualify for such assumptions.

Population of infinite size

If previous assumptions already seemed hard to meet in the real world, the assumption of infinite population is absolutely impossible to meet.

Deviation from this assumption will cause deviations from expectations. Such deviations are often tested via a Chi squared goodness-of-fit test.

Note by the way that if the population is of infini size, there is no mutation, no migration and no selection, then there is no evolution.

Other assumptions

Biology is a science of complex systems. There are always other assumption one might want to consider. For example I did not talk about sex-specific selection or sex-specific mutation rate. But I am hoping that with the above I went over the most important assumptions.

# So what's the point of HWr?

All models are wrong but some are useful. There exist no real life example of a population that perfectly fit into HWr assumptions but it does not mean the model isn't useful. Actually most population approximate HW expectations quite well. You will note also that it is by understanding how a scenario lead to a specific observation that we can interpret how derivation from the expectations can be achieved.

So yes, HWr is one of the most basic and most important rule in population genetics. It is so basic, that I would hardly name it in any fancy way as it is just the simple application of basic probability theory. See Solving Hardy Weinberg problems for more info.

## 5 Conditions for Hardy-Weinberg Equilibrium

One of the most important principles of population genetics, the study of the genetic composition of and differences in populations, is the Hardy-Weinberg equilibrium principle. Also described as genetic equilibrium, this principle gives the genetic parameters for a population that is not evolving. In such a population, genetic variation and natural selection do not occur and the population does not experience changes in genotype and allele frequencies from generation to generation.

### Key Takeaways

• Godfrey Hardy and Wilhelm Weinberg postulated the Hardy-Weinberg principle in the early 20th century. It predicts both allele and genotype frequencies in populations (non-evolving ones).
• The first condition that must be met for Hardy-Weinberg equilibrium is the lack of mutations in a population.
• The second condition that must be met for Hardy-Weinberg equilibrium is no gene flow in a population.
• The third condition that must be met is the population size must be sufficient so that there is no genetic drift.
• The fourth condition that must be met is random mating within the population.
• Finally, the fifth condition necessitates that natural selection must not occur.

## THE HARDY-WEINBERG THEOREM

English mathematician Godfrey II. Hardy and German physician Wilhelm Weinberg independently derived a mathematical model in 1908. This model explains what happens to the frequency of alleles in a population over time. Their combined ideas became known as the Hardy-Weinberg theorem. It states that If certain assumptions are met, evolution will not occur because the allelic frequencies will not change from generation to generation, even though the specific mixing•of alleles in individuals may vary. The assumptions oethe hardy-Weinberg theorem are as follows:

I. The population size must be large. Gene frequency does not change by chance in large population size.

1. Mating within the population must be random. Every individual must have an equal chance to mate with any other Individual in the population. In non-random mating, some Individuals can more reproduce than others. Then natural selection may occur. Therefore. Mating should be random. Individuals cannot migrate into, or out of. the population Migration may introduce new genes into the gene pool. It can add or delete copies of existing genes.

4. Mutations must not occur. Or mutational equilibrium must exist. Some mutations take place from wild type allele to a mutant form. Some mutations occur from the mutant form back to the wild type. If both these mutations are balanced then mutational equilibrium is established. Thus no new genes are introduced in to the population from this source. These assumptions stop the change in allelic frequencies. Therefore, evolution does not occur. These assumptions are restrictive. A real population cannot meet these conditions. Therefore, evolution occurs in most populations. However, Hardy-Weinberg theorem provides a useful theoretical framework. The changes in gene frequencies in populations can be examined by this theorem.

## Assumptions of Hardy-Weinberg rule - Biology

The post Solving Hardy Weinberg problems offers an easy explanation of Hardy-Weinberg rule. The current top answer explicitly does not talk about the assumptions of Hardy-Weinberg. A model makes sense only if one is able to tell its domain of definition and therefore it is critical to understand the assumptions underlying Hardy-Weinberg assumptions.

What are the assumptions of Hardy-Weinberg rule?

Here are the assumptions to the standard Hardy-Weinberg rule (HWr) formulation

The locus of interest is bi-allelic

This is an obvious assumptions, however it is very easy to extrapole HWr to any number of loci.

Organisms are diploid

This is another obvious assumption that is very easy overcome. This also means that HWr does not hold for loci present on sexual chromosomes as those chromosomes do not display a simple diploid behaviour.

Only sexual reproduction occurs

If some individuals are able to undergo non-sexual reproduction, then the rule does not hold anymore. See also Panmixia as these two assumptions are related.

Panmixia is often called "random mating". Panmixia is the state where each individual is equally to mate with any other individual in the population (including itself). There is therefore no population structure and no mate choice.

Let \$N\$ be the number of individuals in the population, in absence of selection (see If selection, it must be on fecundity alone), the probability to mate with any given individual (including itself) is \$frac<1>\$. You will note that it requires that individuals are able to self (but not clone as reproduction must still be sexual).

If selection, it must be on fecundity alone

If selection occurs, then it must on fecundity (right before reproduction). If selection occurs on survival during the lifetime, then the selected genotypes will be in excess in comparison to Hardy-Weinberg expectations.

If migration, it must be right before reproduction only

For the same reason as above, if there is migration, then it must be right before fecundity. If migration occur during the lifetime, then it will have a very similar effect than selection. Selection would increase the frequency of selected genotypes above HR expectations. Migration would increase the frequency of incoming genotypes above HR expectations.

Non-overlapping generations

This means that everyone reproduce in the exact same time and die right afterward. Very few species would qualify for such assumptions.

Population of infinite size

If previous assumptions already seemed hard to meet in the real world, the assumption of infinite population is absolutely impossible to meet.

Deviation from this assumption will cause deviations from expectations. Such deviations are often tested via a Chi squared goodness-of-fit test.

Note by the way that if the population is of infini size, there is no mutation, no migration and no selection, then there is no evolution.

Other assumptions

Biology is a science of complex systems. There are always other assumption one might want to consider. For example I did not talk about sex-specific selection or sex-specific mutation rate. But I am hoping that with the above I went over the most important assumptions.

All models are wrong but some are useful. There exist no real life example of a population that perfectly fit into HWr assumptions but it does not mean the model isn't useful. Actually most population approximate HW expectations quite well. You will note also that it is by understanding how a scenario lead to a specific observation that we can interpret how derivation from the expectations can be achieved.

So yes, HWr is one of the most basic and most important rule in population genetics. It is so basic, that I would hardly name it in any fancy way as it is just the simple application of basic probability theory. See Solving Hardy Weinberg problems for more info.

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## Assumption 2: A Closed Population

Emigration and immigration -- that is, transfer of individuals into and out of a population -- can change the frequency of alleles. Emigrating individuals might take more of one allele out of a population, and immigrating individuals might come from a population with a different proportion of alleles. This is known as genetic flow, and it is common for most populations, with notable exceptions such as remote islands, deep caves or mountaintops. A population at Hardy-Weinberg equilibrium is assumed to have no genetic flow, or to be a completely closed population.

## GENETIC ANALYSIS

### 7 POPULATION GENETICS UPHOLDS DARWINISM

Mendel's hypothesis of inheritance of discrete factors that are not diluted should have resolved a major difficulty that Darwin encountered. Shortly after the publication of his Origin of Species, in 1867, Fleeming Jenkins showed that, adopting Darwin's theory of inheritance by mixing pangenes, would wash out any achievement of natural selection (see Hull [1973 , 302-350]). Hugo de Vries and especially William Bateson, considered Mendel's Faktoren as indicated by his hypothesis of inheritance to provide a rational basis for the theory of evolution. Although as early as in 1902 Yule showed that, given small enough steps of variation, the Mendelian model reduces to the biometric claim [ Yule, 1902 ], this was largely ignored in the bitter disputes between the Mendelians and the Biometricians [ Provine, 1971 ], (see Tabery [2004 ]). Hardy's [1908] proof that in a large population, the proportion of heterozygotes to homozygotes will reach equilibrium after one generation of random mating (provided no mutation or selection interfered), developed in the same year by Weinberg [ Stern, 1943 ], became the basic theorem of population genetics — the Hardy-Weinberg principle . It took, however, another decade for R. A. Fisher to convince that the continuous phenotypic biometric variation reduces to the Mendelian model of polygenes [ Fisher, 1918 ]. Thus, finally the way was cleared to examine the Darwinian theory of natural evolution on the basis of Mendelian genetic analysis, not only in vivo but also in papyro. As formulated by Fisher in his fundamental theorem of natural selection: “The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time” [ Fisher, 1930 , 37].

Whereas Fisher examined primarily the effects of selection of alleles of single genes in indefinitely large population under the assumption of differences in genotypic fitness, J. B. S. Haldane concentrated on the impact of mutations on the rate and direction of evolution of one or few genes (and the influence of population size) [ Haldane, 1990 ]. Sewall Wright in his models of the dynamics of populations wished to be more “realistic”, and stressed the influence of finite population size, the limited gene flow between subpopulations, and the heterogeneity of the habitats in which the population and its subpopulations lived [ Wright, 1986 ].

Experimentally, the main British group, led by E. B. Ford adopted a strict Mendelian reductionist approach, emphasizing largely the effects of selection on single alleles of specific genes (the evolution of industrial melanism in moths, the evolution of mimicry in African moth species, the evolution of seasonal polymorphisms in snails, etc.). The American geneticists, especially Dobzhansky and his school, concentrated more on problems of whole genotypes, such as speciation (Sturtevant) and chromosomal polymorphisms (Dobzhansky) in Drosophila.

The triumph of reductionist Mendelism was at the 1940s with the emergence of the “New Synthesis” that defined natural populations and the forces that affect their evolution in terms of gene alleles’ frequencies [ Huxley, 1943 ]. This notion dominated population genetics for the next decades. Attempts to emphasize the role of non-genetic constraints, such as the anatomical-physiological factors (e.g. by Goldschmidt [1940 ]), or the environmental (and evolutionary-historical) constraints (for example by Waddington [1957 ]) were largely overlooked.

The introduction of the analysis of electrophoretic polymorphisms [Hubby and Lewontin, 1966 Lewontin and Hubby, 1966 ] allowed a molecular analysis of allele variation that was also largely independent of the classical morphological and functional genetic markers (see also Lewontin [1991 ]). Although genes were still treated as algebraic point entities, inter-genic interacting system, such as “linkage disequilibrium” were considered [ Lewontin and Kojima, 1960 ]. The New Synthesis was, however, seriously challenged when it was realized that a great deal of the variation at the molecular level was determined by stochastic processes, rather than because of differences in fitness [ Kimura, 1968 King and Jukes, 1969 ].

This assault on the notion of the New Synthesis was intensified when, in 1972 Gould and Eldridge, two paleontologists, suggested a model of evolution by “punctuated equilibrium”, or long periods of little evolutionary change interspersed with (geologically) relatively short period of fast evolutionary change. Moreover, in the periods of (relatively) fast evolution large one-step “macromutational” changes were established [ Eldredge and Gould, 1972 ]. Although it could be shown that analytically the claims of punctuated equilibrium could be reduced to those of classical population genetics [Charlesworth et al., 1982] , these ideas demanded re-examination of the developmental conceptions that, as a rule, could not accept one-step major developmental changes since these called for disturbance in many systems and hence would have caused severe disturbances in developmental and reproductive coordination.

The need to reexamine the reductionist assumptions of genetic population analysis and to pay more consideration to constraints on the genetic determinations of intra- and extra-organismal factors coincided with the resurrection of developmental genetics. However, the major change in the analysis of evolution and development came from the molecular perspective. These allowed first of all detailed upward analysis, from the specific DNA sequences to the early products, rather than the analyses based on end-of-developmental pathway markers. Yet, arguably, the most significant development was the possibility of in-vitro DNA hybridization. This molecular extension of genetic analysis sensu stricto finally overcame the empirical impossibility to study (most) in vivo interspecific hybrids. The new methods of DNA hybridization had no taxonomic inhibitions whatsoever, and soon hybrid DNA molecules of, say mosquito, human and plant, were common subjects for research. Genetic engineering, which allowed direct genetic comparison between any species and the transfer of genes from one species to individuals of another, unrelated species, prompted the genetic analysis of the evolution of developmental process, or evo-devo.

Molecular genetic analysis of homeotic mutants, in which one organ is transformed into the likeness of another, usually a homologous one, revealed stretches of DNA that were nearly identical in other genes with homeotic effects (like the homeobox of some 180 nucleotides, that appear to be involved in when-and-where particular groups of genes are expressed along the embryo axis during development [McGinnis et al., 1984a McGinnis et al., 1984b ]). The method of determining homologies by comparing DNA sequences is nowadays done mainly in silico. As suggested many years ago [ Ohno, 1970 ], the abundance of homologous sequences in the same species genome (paralogous sequence that do not necessarily share similar functions any more) or in different species (orthologous sequences that ‘usually’ have similar functions in different species), indicate that the system's structural and functional organization have been also causal factors rather than merely consequences in the history of the process of evolution.

## The assumptions of the Hardy-Weinberg principle

There are 5 assumptions that are made when using the Hardy-Weinberg equations. These are:

1. No natural selection: There are no evolutionary pressures which may favour a particular allele.
2. Random mating: Each individual in a population mates randomly so that mating with an individual carrying a particular allele is not favoured.
3. No mutations: There are no DNA mutations occurring for the alleles which may affect their function.
4. A closed population: Individuals within the population do not leave and new individuals are not introduced to the population.
5. Large population size: The population is considered large enough, at best infinite, so that major changes in allele frequencies do not cause a genetic drift.

If any of these assumptions are not satisfied, then the principle cannot be applied.

## Population Genetics: the Hardy-Weinberg Principle

How would a researcher know if selection or drift or even mutation were altering the allele frequencies for population? In other words, can we use the mechanisms of to detect evolution happening in real populations? To do that we’d need a null expectation or a baseline against which to measure change. We call that baseline the Hardy-Weinberg equilibrium (HWE). Remember that the modern definition of evolution is a change in the allele frequencies in a population. To calculate what the alleles frequencies (p and q in the example below) should be in the absence of any evolution, we need to assume that the population is undergoing no selection, no mutation, no drift, no gene flow, and that individuals are selecting mates at random.
Also recall that each individual is a diploid, carrying two copies (alleles) of each gene. Assume that the entire population only has two variants, or alleles, for a gene for pea color. Individuals that carry at least one Y allele have yellow coloration, while those who carry two copies of the y allele are green. In the figure below, the frequency of the y allele is q, and the frequency of the Y allele is p, and p + q = 1. The Hardy-Weinberg analysis in the lower half of the figure models the result of random mating in the absence of selection, drift, mutation or migration (eg, in the absence of evolution). The progeny generation will have genotype frequencies in the following proportions:

If the population is in Hardy-Weinberg equilibrium, two things will be true:

1. allele frequencies will not change from one generation to the next (recall our definition of biological evolution), and
2. the actual genotype frequencies observed in the population will match the above predicted genotypes based on the Hardy-Weinberg Principle.

We can see that this population of pea plants appears to be in H-W equilibrium, because the proportion of YY, Yy, and yy genotypes match the H-W predictions of p^2, 2pq, and q^2, respectively.

When populations are in the Hardy-Weinberg equilibrium, the allelic frequency is stable from generation to generation and the genotype frequencies match the Hardy-Weinberg proportions. If the allelic frequency measured in the field differs from the predicted value, scientists can make inferences about what evolutionary forces are at play. (Source: OpenStax Biology)

What about another population of pea plants, composed of 300 YY plants, 100 Yy plants, and 100 yy plants? Is this second population in H-W equilibrium?
Below is a Crash Course Biology video on Population Genetics that explains Hardy-Weinberg equilibrium dynamically…using ear wax phenotype in humans.