Hardy–Weinberg principle and for the heterozygotes. In the absence of selection, mutation, genetic drift, or other forces, allele frequencies "p" and "q" are constant between generations, so equilibrium is reached. The principle is named after G. H. Hardy and Wilhelm Weinberg, who first demonstrated it mathematically. Hardy's paper was focused on debunking the then-commonly held view that a dominant allele would automatically tend to increase in frequency; today, confusion between dominance and selection is less common. Today, tests for Hardy-Weinberg genotype frequencies are used primarily to test for population stratification and other forms of non-random mating. Consider a population of monoecious diploids,
Tajima's D introgression. A randomly evolving DNA sequence contains mutations with no effect on the fitness and survival of an organism. The randomly evolving mutations are called "neutral", while mutations under selection are "non-neutral". For example, you would expect to find that a mutation which causes prenatal death or severe disease to be under selection. When looking at the human population as a whole, we say that the population frequency of a neutral mutation fluctuates randomly (i.e. the percentage of people in the population with the mutation changes from one generation to the next, and this percentage is equally likely to go Small population size Small population size Small populations can behave differently from larger populations. They are often the result of a population bottlenecks from larger populations, leading to loss of heterozygosity and reduced genetic diversity and loss or fixation of alleles and shifts in allele frequencies. A small population is then more susceptible to demographic and genetic stochastic events, which can impact the long-term survival of the population. Therefore, small populations are often considered at risk of endangerment or extinction, and are often of conservation concern. The influence of stochastic variation in demographic (reproductive and mortality) rates is much higher for small populations Genetic divergence divergence, and subsequently reproductive isolation, are the outcomes of multiple small mutations over evolutionary time. Genetic divergence Genetic divergence is the process in which two or more populations of an ancestral species accumulate independent genetic changes (mutations) through time, often after the populations have become reproductively isolated for some period of time. In some cases, subpopulations living in ecologically distinct peripheral environments can exhibit genetic divergence from the remainder of a population, especially where the range of a population is very large (see parapatric speciation). The genetic differences among divergent populations can involve silent mutations (that have no effect on Panmixia Panmixia Panmixia (or panmixis) means "random mating". A "panmictic population" is one where all individuals are potential partners. This assumes that there are no mating restrictions, neither genetic nor behavioural, upon the population, and that therefore all recombination is possible. The Wahlund effect assumes that the overall population is panmictic. In genetics, random mating involves the mating of individuals regardless of any physical, genetic, or social preference. In other words, the mating between two organisms is not influenced by any environmental, hereditary, or social interaction. Hence, potential mates have an equal chance of being selected. Random mating is a factor Allee effect a population of finite size. In small populations, demographic stochasticity will decrease the population growth rate, causing an effect similar to the Allee effect, which will increase the risk of population extinction. Whether or not demographic stochasticity can be considered a part of Allee effect is somewhat contentious however. The most current definition of Allee effect considers the correlation between population density and mean individual fitness. Therefore, random variation resulting from birth and death events would not be considered part of Allee effect as the increased risk of extinction is not a consequence of the changing fates of individuals within Random match possibility Random match possibility Random Match Probability (RMP) is a kind of measure in population genetics to measure the probability of an unrelated person, randomly picked out of the general population and matching the genotype derived from the evidence. A genotype has a number of alleles, and each allele has a frequency in a certain race. This frequency is the RMP of that allele. To calculate the RMP of a certain genotype, multiply the frequencies of the alleles in the genotype together. This will give the RMP of the genotype. In forensic science, the evidence from the crime scene is gathered Genetic load recent "travelling wave" models of rapid adaptation derive a term called the "lead" that is equivalent to the substitutional load, and find that it is a critical determinant of the rate of adaptive evolution. Inbreeding increases homozygosity. In the short run, an increase in inbreeding increases the probability with which offspring get two copies of a recessive deleterious alleles, lowering fitnesses via inbreeding depression. In a species that habitually inbreeds, e.g. through self-fertilization, recessive deleterious alleles are purged. Combinations of alleles that have evolved to work well together may not work when recombined with a different suite of coevolved alleles, Fixed allele population. These random fluctuations within the allele frequencies can lead to the fixation or loss of certain alleles within a population. To the right is an image that shows thorough successive generations; the allele frequencies fluctuate randomly within a population. The smaller your population size, the faster fixation or loss of alleles will occur. However, it should be noted that all populations are driven to allele fixation and it is inevitable, it just takes varying amounts of time for this to occur due to population size. Some other causes of allele fixation are inbreeding, as this decreases the genetic variability Philopatry local gene complex that is finely adapted to the local environment. Another proposed benefit is the reduction the cost of meiosis and recombination events. Under this hypothesis, non-philopatric individuals would be maladapted and over multi-generational time, philopatry within a species could become fixed. Evidence for the optimal-inbreeding hypothesis is found in outbreeding depression. Outbreeding depression involves reduced fitness as a result of random mating, which occurs due to the breakdown of coadapted gene complexes by combining allele that do not cross well with those from a different subpopulation. However, it is important to note that outbreeding depression becomes more detrimental Dual inheritance theory frequency of cultural traits in a population may be subject to random fluctuations due to chance variations in which traits are observed and transmitted (sometimes called "sampling error"). These fluctuations might cause cultural variants to disappear from a population. This effect should be especially strong in small populations. A model by Hahn and Bentley shows that cultural drift gives a reasonably good approximation to changes in the popularity of American baby names. Drift processes have also been suggested to explain changes in archaeological pottery and technology patent applications. Changes in the songs of song birds are also thought to arise Genetic variation Genetic variation Genetic variation means that biological systems – individuals and populations – are different over space. Each gene pool includes various alleles of genes. The variation occurs both within and among populations, supported by individual carriers of the variant genes. Genetic variation is brought about, fundamentally, by mutation, which is a permanent change in the chemical structure of chromosomes. Genetic recombination also produces changes within alleles. Genetic variation among individuals within a population can be identified at a variety of levels. It is possible to identify genetic variation from observations of phenotypic variation in either quantitative traits (traits that Fixation (population genetics) Fixation (population genetics) In population genetics, fixation is the change in a gene pool from a situation where there exists at least two variants of a particular gene (allele) in a given population to a situation where only one of the alleles remains. In the absence of mutation or heterozygote advantage, any allele must eventually be lost completely from the population or fixed (permanently established at 100% frequency in the population). Whether a gene will ultimately be lost or fixed is dependent on selection coefficients and chance fluctuations in allelic proportions. Fixation can refer to a gene in general or Genetic diversity human-caused population effects. A genetic bottleneck can occur when a population goes through a period of low number of individuals, resulting in a rapid decrease in genetic diversity. Even with an increase in population size, the genetic diversity often continues to be low if the entire species began with a small population, since beneficial mutations (see below) are rare, and the gene pool is limited by the small starting population. This is an important consideration in the area of conservation genetics, when working toward a rescued population or species that is genetically-healthy. Random mutations consistently generate genetic variation. A mutation Fixation (population genetics) fixed, whereas deleterious alleles are more likely to be lost. In populations that are shrinking in size, selection coefficients are not as effective. Thus, there is a higher probability of beneficial alleles being lost and deleterious alleles being fixed. This is because if a beneficial mutation is rare, it can be lost purely due to chance of that individual not having offspring, no matter the selection coefficient. In growing populations, the average individual has a higher expected number of offspring, whereas in shrinking populations the average individual has a lower number of expected offspring. Thus, in growing populations it is Population genetics structure in a sexually reproducing, diploid species, is to see whether genotype frequencies follow Hardy-Weinberg proportions as a function of allele frequencies. For example, in the simplest case of a single locus with two alleles denoted A and a at frequencies "p" and "q", random mating predicts freq(AA) = "p" for the AA homozygotes, freq(aa) = "q" for the aa homozygotes, and freq(Aa) = 2"pq" for the heterozygotes. In the absence of population structure, Hardy-Weinberg proportions are reached within 1-2 generations of random mating. More typically, there is an excess of homozygotes, indicative of population structure. The extent of this Thrifty gene hypothesis the islanders by a founder effect among a small initial colonising population, and was able to spread because of a selective advantage it conferred within that small group. Hence, in small populations under particular environmental conditions it may be feasible that the 'thrifty gene' idea is correct. It remains to be seen if rare variants that fill the gap in the missing heritability estimates are also 'thrifty genes' or if they are rare chance events sustained by drift, as implicated for the common variants currently linked to obesity and type 2 diabetes. Thrifty gene hypothesis The thrifty gene hypothesis, or Inbreeding the increased proportion of deleterious homozygotes exposes the allele to natural selection, in the long run its frequency decreases more rapidly in inbred populations. In the short term, incestuous reproduction is expected to increase the number of spontaneous abortions of zygotes, perinatal deaths, and postnatal offspring with birth defects. The advantages of inbreeding may be the result of a tendency to preserve the structures of alleles interacting at different loci that have been adapted together by a common selective history. Malformations or harmful traits can stay within a population due to a high homozygosity rate, and this will cause a Polygenic adaptation Polygenic adaptation Polygenic adaptation describes a process in which a population adapts through small changes in allele frequencies at hundreds or thousands of loci. Many traits in humans and other species are highly polygenic, i.e., affected by standing genetic variation at hundreds or thousands of loci. Under normal conditions, the genetic variation underlying such traits is governed by stabilizing selection, in which natural selection acts to hold the population close to an optimal phenotype. However, if the phenotypic optimum changes, then the population can adapt by small directional shifts in allele frequencies spread across all the variants that affect the Founder effect This represents a prevalence of 1 in 58, compared with a worldwide prevalence of around 1 in 4,000. The abnormally high rate of twin births in Cândido Godói could be explained by the founder effect. Founder effect In population genetics, the founder effect is the loss of genetic variation that occurs when a new population is established by a very small number of individuals from a larger population. It was first fully outlined by Ernst Mayr in 1942, using existing theoretical work by those such as Sewall Wright. As a result of the loss of genetic variation, the new population Population bottleneck smaller population size after a bottleneck event, the chances of inbreeding and genetic homogeneity increase, leading to the potential for inbreeding depression to occur. Smaller population size can also cause deleterious mutations to accumulate A slightly different form of a bottleneck can occur if a small group becomes reproductively (e.g. geographically) separated from the main population, such as through a founder event where for example a few members of a species successfully colonize a new isolated island, or from small captive breeding programs such as animals at a zoo. Alternatively, invasive species can undergo population bottlenecks through founder events when Founder effect Founder effect In population genetics, the founder effect is the loss of genetic variation that occurs when a new population is established by a very small number of individuals from a larger population. It was first fully outlined by Ernst Mayr in 1942, using existing theoretical work by those such as Sewall Wright. As a result of the loss of genetic variation, the new population may be distinctively different, both genotypically and phenotypically, from the parent population from which it is derived. In extreme cases, the founder effect is thought to lead to the speciation and subsequent evolution of new Hill–Robertson effect epistatic effects of carrying both, individuals of genotype "AB" will have a greater selective advantage than "aB" or "Ab" individuals, and "AB" will hence go to fixation. However, if there is no recombination, "AB" individuals can only occur if the latter mutation ("B") happens to occur in an "Ab" individual. The chance of this happening depends on the frequency of new mutations, and on the size of the population, but is in general unlikely unless "A" is already fixed, or nearly fixed. Hence one should expect the time between the "A" mutation arising and the population becoming fixed for "AB" Small population size than large ones. Stochastic variation in demographic rates causes small populations to fluctuate randomly in size. This variation could be a result of unequal sex ratios, high variance in family size, inbreeding or fluctuating population size. The smaller the population the greater the probability that fluctuations will lead to extinction. One demographic consequence of a small population size is the probability that all offspring in a generation are of the same sex, and where males and females are equally likely to be produced (see sex ratio), is easy to calculate: it is given by formula_1 (The chance of all animals The Neutral Theory of Molecular Evolution of a limited population size, can also cause a change in allele frequencies over time that can look like Darwinian evolution while actually being an entirely random or as Kimura puts it "neutral" process. In this scenario a relatively small population can lose neutral alleles through the random deaths or migrations of individuals that have them. It may appear to an onlooker that one trait is being selected for over another but in actuality it is a neutral process that is not necessarily undergoing selection as it would in Darwinian evolution. Within the neutral theory, selective constraint is a type Minor allele frequency Minor allele frequency Minor allele frequency (MAF) refers to the frequency at which the "second most common" allele occurs in a given population. SNPs with a minor allele frequency of 0.05 (5%) or greater were targeted by the HapMap project. MAF is widely used in population genetics studies because it provides information to differentiate between common and rare variants in the population. As an example, a recent study sequenced the whole genomes of Sardinian individuals. The authors classified the variants found in the study in three classes according to their MAF. It was observed that rare variants (MAF < 0.05) Polymorphism (biology) is the number of heterozygotes (see Hardy-Weinberg equilibrium). 2. Using a method invented by l'Heretier and Teissier, Dobzhansky bred populations in "population cages", which enabled feeding, breeding and sampling whilst preventing escape. This had the benefit of eliminating migration as a possible explanation of the results. Stocks containing inversions at a known initial frequency can be maintained in controlled conditions. It was found that the various chromosome types do not fluctuate at random, as they would if selectively neutral, but adjust to certain frequencies at which they become stabilised. With "D. persimilis" he found that the caged population followed the Genetic hitchhiking fixed for reasons connected to selection. Genetic hitchhiking Genetic hitchhiking, also called genetic draft or the hitchhiking effect, is when an allele changes frequency not because it itself is under natural selection, but because it is near another gene that is undergoing a selective sweep and that is on the same DNA chain. When one gene goes through a selective sweep, any other nearby polymorphisms that are in linkage disequilibrium will tend to change their allele frequencies too. Selective sweeps happen when newly appeared (and hence still rare) mutations are advantageous and increase in frequency. Neutral or even slightly deleterious Weak selection are large but the significance of the mutation is minor. An example could be a change in pigmentation. Another reason could be that the phenotypic differences between wild type and mutant are actually small, such as tail length variation. In either case, the significance of the mutation, which is determined by the environment creating the selective pressure, is low in comparison to other mutations. Hence, almost near neutral mutations result in phenotypes that are weakly selected. Weak selection creates a situation in which the evolutionary dynamics governing the phenotype frequencies in a population are mainly driven by random fluctuations. Hence Quantitative genetics random fertilization, the combination occurs with a frequency of p x p (= p). Similarly, the zygote aa occurs with a frequency of q. Heterozygotes (Aa) can arise in two ways: when p male (A allele) randomly fertilize q female (a allele) gametes, and "vice versa". The resulting frequency for the heterozygous zygotes is thus 2pq. Notice that such a population is never more than half heterozygous, this maximum occurring when p=q= 0.5. In summary then, under random fertilization, the zygote (genotype) frequencies are the quadratic expansion of the gametic (allelic) frequencies: formula_1. (The "=1" states that the frequencies are Synthetic lethality interaction in wildtype populations of "Drosophila". If the combination of genetic events results in a non-lethal reduction in fitness, the interaction is called synthetic sickness. Although in classical genetics the term synthetic lethality refers to the interaction between two genetic perturbations, synthetic lethality can also apply to cases in which the combination of a mutation and the action of a chemical compound causes lethality, whereas the mutation or compound alone are non-lethal. Synthetic lethality is a consequence of the tendency of organisms to maintain buffering schemes that allow phenotypic stability despite genetic variation, environmental changes and random events such as Weak selection Weak selection Weak selection in evolutionary biology is when individuals with different phenotypes possess similar fitness, i.e. one phenotype is weakly preferred over the other. Weak selection, therefore, is an evolutionary theory to explain the maintenance of multiple phenotypes in a stable population. Weak selection can only be used to explain the maintenance of mutations in a Moran process. A Moran process is one in which birth and death are paired events, and therefore population size remains constant. If the population size was increasing, both wild type and mutant phenotypes can proliferate and the weak selection for one phenotype results Haplotype convergence Haplotype convergence Haplotype convergence is the unrelated appearance of identical haplotypes in separate populations, through either convergent evolution or random chance. Haplotype convergence is rare, due to the sheer odds involved of two unrelated individuals independently evolving the exact same genetic sequence in the site of interest. Thus, haplotypes are shared mainly between very closely related individuals, as the genetic information in two related individuals will be much more similar than between unrelated individuals. Substitution bias further increases the likelihood of haplotype convergence, as this increases the probability of mutations occurring at the same site. Sequences may also diverge from Pleiotropy chromosome 12 that codes for enzyme phenylalanine hydroxylase , that affects multiple systems, such as the nervous and integumentary system. Other examples of pleiotropy are albinism, sickle cell anemia, and certain forms of autism and schizophrenia. Pleiotropy not only affects humans, but also animals, such as chickens and laboratory house mice, where the mice have the "mini-muscle" allele. Pleiotropic gene action can limit the rate of multivariate evolution when natural selection, sexual selection or artificial selection on one trait favors one allele, while selection on other traits favors a different allele. Some gene evolution is harmful to an organism. Genetic Mutation rate or a single gene, mutation rates are changing. When the mutation rate in humans increases certain health risks can occur, for example, cancer and other hereditary diseases. Having knowledge of mutation rates is vital to understanding the future of cancers and many hereditary diseases. Different genetic variants within a species are referred to as alleles, therefore a new mutation can create a new allele. In Population genetics, each allele is characterized by a selection coefficient, which measures the expected change in an allele's frequency over time. The selection coefficient can either be negative, corresponding to an expected decrease, positive, corresponding Quantitative trait locus mutation can occur at random within breeding populations, which is a central premise of his model of selection in nature. Later in his career, Castle would refine his model for speciation to allow for small variation to contribute to speciation over time. He also was able to demonstrate this point by selectively breeding laboratory populations of rats to obtain a hooded phenotype over several generations. Castle's was perhaps the first attempt made in the scientific literature to direct evolution by artificial selection of a trait with continuous underlying variation, however the practice had previously been widely employed in the development Polygenic adaptation of these models focus on adaptation through genetic changes at a single locus and they generally assume large changes in allele frequencies. The concept of polygenic adaptation is related to classical models from quantitative genetics. However, traditional models in quantitative genetics usually abstract away the contributions of individual loci by focusing instead on means and variances of genetic scores. In contrast, population genetics models and data analysis have generally emphasized models of adaptation through sweeps at individual loci. The modern formulation of polygenic adaptation in population genetics was developed in a pair of 2010 review articles. Polygenic adaptation is presumed Genetic divergence Genetic divergence Genetic divergence is the process in which two or more populations of an ancestral species accumulate independent genetic changes (mutations) through time, often after the populations have become reproductively isolated for some period of time. In some cases, subpopulations living in ecologically distinct peripheral environments can exhibit genetic divergence from the remainder of a population, especially where the range of a population is very large (see parapatric speciation). The genetic differences among divergent populations can involve silent mutations (that have no effect on the phenotype) or give rise to significant morphological and/or physiological changes. Genetic divergence will always F-statistics chromosomes sampled within diploid individuals with the average number obtained when sampling chromosomes randomly from the population (excluding the grouping per individual). One can modify this definition and consider a grouping per sub-population instead of per individual. Population geneticists have used that idea to measure the degree of structure in a population. Unfortunately, there is a large number of definitions for "F", causing some confusion in the scientific literature. A common definition is the following: where the variance of p is computed across sub-populations and "p"(1−"p") is the expected frequency of heterozygotes. It is well established that the genetic diversity Heterozygote advantage advantage to heterozygotes to make it beneficial, so that it remained at dynamic equilibrium in the gene pool. Kalmus introduced flies with the ebony mutation to a wild-type population. The ebony allele persisted through many generations of flies in the study, at genotype frequencies that varied from 8% to 30%. In experimental populations, the ebony allele was more prevalent and therefore advantageous when flies were raised at low, dry temperatures, but less so in warm, moist environments. Sickle-cell anemia (SCA) is a genetic disorder caused by the presence of two incompletely recessive alleles. When a sufferer's red blood cells are Genetic purging that each individual carries on the average one of these alleles. Although about 63% of the individuals carry at least one of these lethal alleles, almost none carry two copies of the same lethal. Therefore, since lethals are considered completely recessive (i.e., they are harmless in heterozygosis), they cause almost no deaths. Now assume that population size reduces to a small value (say "N"=10), and remains that small for many generations. As inbreeding increases, the probability of being homozygous for one (or more) of these lethal alleles also increases, causing fitness to decline. However, as those lethals begin to occur Quantitative genetics This mating system, dependent upon these assumptions, became known as "panmixia". Panmixia rarely actually occurs in nature, as gamete distribution may be limited, for example by dispersal restrictions or by behaviour, or by chance sampling (those local perturbations mentioned above). It is well-known that there is a huge wastage of gametes in Nature, which is why the diagram depicts a "potential" gamete-pool separately to the "actual" gamete-pool. Only the latter sets the definitive frequencies for the zygotes: this is the true "gamodeme" ("gamo" refers to the gametes, and "deme" derives from Greek for "population"). But, under Fisher's assumptions, the "gamodeme" McDonald–Kreitman test α. The presence of slightly deleterious mutations is strongly linked to genes that have experienced the greatest reduction in effective population size. This means that soon after a recent reduction in effective population size in a species has occurred, such as a bottleneck, we observe a larger presence of slightly deleterious mutations in the protein-coding regions. We can make a direct connection with the increase in number of slightly deleterious mutations and the recent decrease in effective population size. For more information on why population size affects the tendency of slightly deleterious mutations to increase in frequency, refer to the Background selection variation". Under the theory of background selection, the paradox is resolved, because neutral variation is predicted to be purged in conjunction with deleterious mutations. Background selection also contributes to a selective explanation of the positive correlation between recombination and polymorphism across populations. In areas of high recombination, neutral loci are more likely to ‘escape' the effects of nearby selection and be retained in the population. A version of this scenario has been observed in studies of "Drosophila", where regions of low recombination in the genome exhibit low levels of genetic variation. Hudson and Kaplan showed that the difference between the Linkage disequilibrium at that locus), while at a different locus allele "B" occurs with frequency formula_3. Similarly, let formula_4 be the frequency with which both "A" and "B" occur together in the same gamete (i.e. formula_4 is the frequency of the "AB" haplotype). The association between the alleles "A" and "B" can be regarded as completely random—which is known in statistics as "independence"—when the occurrence of one does not affect the occurrence of the other, in which case the probability that both "A" and "B" occur together is given by the product formula_6 of the probabilities. There is said to be a Polygenic adaptation trait. Polygenic adaptation can occur relatively quickly (as described in the breeder's equation), however it is difficult to detect from genomic data because the changes in allele frequencies at individual loci are very small. Polygenic adaptation represents an alternative to adaptation by selective sweeps. In classic selective sweep models, a single new mutation sweeps through a population to fixation, purging variation from a region of linkage around the selected site. More recent models have focused on partial sweeps, and on soft sweeps - i.e., sweeps that start from standing variation or comprise multiple sweeping variants at the same locus. All Quantitative genetics 1, these three frequencies become respectively p, 0 and q Conversely, when f = 0, they reduce to the random-fertilization quadratic expansion shown previously. The population mean shifts the central reference point from the homozygote midpoint (mp) to the mean of a sexually reproduced population. This is important not only to relocate the focus into the natural world, but also to use a measure of "central tendency" used by Statistics/Biometrics. In particular, the square of this mean is the Correction Factor, which is used to obtain the genotypic variances later. For each genotype in turn, its allele effect is multiplied Assortative mating random mating. Disassortative mating reduces the genetic similarities within the family. Positive assortative mating occurs more frequently than negative assortative mating. In both cases, the nonrandom mating pattern result in a typical deviation from the Hardy–Weinberg principle (which states that genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences, such as "mate choice" in this case). Several hypotheses have been proposed to explain the phenomenon of assortative mating. Assortative mating has evolved from a combination of different factors, which vary across different species. Assortative mating with respect to body size Random match possibility and the genotype of DNA evidence is collected. This is then matched with the suspect. The RMP of the evidence is useful in convincing the jury, because the chances of the genotype of the evidence and that of the suspect matching is very low, and therefore it can hardly be claimed that the suspect is innocent, as the chances of it being so is very low. Random match possibility Random Match Probability (RMP) is a kind of measure in population genetics to measure the probability of an unrelated person, randomly picked out of the general population and matching the genotype Genetic variation (recessive alleles are only expressed in the less common homozygous individuals). Natural selection can also maintain genetic variation in balanced polymorphisms. Balanced polymorphisms may occur when heterozygotes are favored or when selection is frequency dependent. Genetic variation Genetic variation means that biological systems – individuals and populations – are different over space. Each gene pool includes various alleles of genes. The variation occurs both within and among populations, supported by individual carriers of the variant genes. Genetic variation is brought about, fundamentally, by mutation, which is a permanent change in the chemical structure of chromosomes. Genetic recombination also produces changes Evolution rest of their species, separation may eventually produce organisms that cannot interbreed. The second mode of speciation is peripatric speciation, which occurs when small populations of organisms become isolated in a new environment. This differs from allopatric speciation in that the isolated populations are numerically much smaller than the parental population. Here, the founder effect causes rapid speciation after an increase in inbreeding increases selection on homozygotes, leading to rapid genetic change. The third mode is parapatric speciation. This is similar to peripatric speciation in that a small population enters a new habitat, but differs in that there is no Single-nucleotide polymorphism in another. Within a population, SNPs can be assigned a minor allele frequency — the lowest allele frequency at a locus that is observed in a particular population. This is simply the lesser of the two allele frequencies for single-nucleotide polymorphisms. Variations in the DNA sequences of humans can affect how humans develop diseases and respond to pathogens, chemicals, drugs, vaccines, and other agents. SNPs are also critical for personalized medicine. Examples include biomedical research, forensics, pharmacogenetics, and disease causation, as outlined below. SNPs' greatest importance in biomedical research is for comparing regions of the genome between cohorts (such as Founder effect species. In the figure shown, the original population has nearly equal numbers of blue and red individuals. The three smaller founder populations show that one or the other color may predominate (founder effect), due to random sampling of the original population. A population bottleneck may also cause a founder effect, though it is not strictly a new population. The founder effect occurs when a small group of migrants that is not genetically representative of the population from which they came establish in a new area. In addition to founder effects, the new population is often a very small population, so Mendelian randomization the meiotic process. A more perfect control of conditions is scarcely possible, than that of different genotypes appearing in the same litter." --R.A. Fisher Mendelian randomization is a method that allows one to test for, or in certain cases to estimate, a causal effect from observational data in the presence of confounding factors. It uses common genetic polymorphisms with well-understood effects on exposure patterns (e.g., propensity to drink alcohol) or effects that mimic those produced by modifiable exposures (e.g., raised blood cholesterol). Importantly, the genotype must only affect the disease status indirectly via its effect on the exposure of interest. Consanguinity and uncle-niece marriages in South India. The phenomenon of inbreeding increases the level of homozygotes for autosomal genetic disorders and generally leads to a decreased biological fitness of a population known as inbreeding depression, a major objective in clinical studies. The offspring of consanguineous relationships are at greater risk of certain genetic disorders. Autosomal recessive disorders occur in individuals who are homozygous for a particular recessive gene mutation. This means that they carry two copies (alleles) of the same gene. Except in certain rare circumstances (new mutations or uniparental disomy) both parents of an individual with such a disorder will Genetic hitchhiking Genetic hitchhiking Genetic hitchhiking, also called genetic draft or the hitchhiking effect, is when an allele changes frequency not because it itself is under natural selection, but because it is near another gene that is undergoing a selective sweep and that is on the same DNA chain. When one gene goes through a selective sweep, any other nearby polymorphisms that are in linkage disequilibrium will tend to change their allele frequencies too. Selective sweeps happen when newly appeared (and hence still rare) mutations are advantageous and increase in frequency. Neutral or even slightly deleterious alleles that happen to be close Population genetics it by some simpler average rule. Empirically, beneficial mutations tend to have a smaller fitness benefit when added to a genetic background that already has high fitness: this is known as diminishing returns epistasis. When deleterious mutations also have a smaller fitness effect on high fitness backgrounds, this is known as "synergistic epistasis". However, the effect of deleterious mutations tends on average to be very close to multiplicative, or can even show the opposite pattern, known as "antagonistic epistasis". Synergistic epistasis is central to some theories of the purging of mutation load and to the evolution of sexual reproduction. Mutation Population genetics to move or spread (vagility), and tendency to remain or come back to natal place (philopatry), natural populations rarely all interbreed as may be assumed in theoretical random models (panmixy). There is usually a geographic range within which individuals are more closely related to one another than those randomly selected from the general population. This is described as the extent to which a population is genetically structured. Genetic structuring can be caused by migration due to historical climate change, species range expansion or current availability of habitat. Gene flow is hindered by mountain ranges, oceans and deserts or even man-made Underdominance construct. In this way, individuals with only one of the two constructed genes (heterozygotes) are selected against, but homozygotes with both or neither construct are genetically healthy. Analysis of this model using simple population genetics shows that successful spread of refractory genes using this engineered underdominance is possible with relatively small release of the constructed genotype into the population. A similar system of manipulation of pest populations was achieved in a population of "Drosophila melanogaster" by using a knock-down/rescue system. The genetic construct in this system employs a dsRNAi knockdown of a C-reactive protein, RpL14, as well as a rescue Polygenic adaptation female hip and waist size. Polygenic adaptation Polygenic adaptation describes a process in which a population adapts through small changes in allele frequencies at hundreds or thousands of loci. Many traits in humans and other species are highly polygenic, i.e., affected by standing genetic variation at hundreds or thousands of loci. Under normal conditions, the genetic variation underlying such traits is governed by stabilizing selection, in which natural selection acts to hold the population close to an optimal phenotype. However, if the phenotypic optimum changes, then the population can adapt by small directional shifts in allele frequencies spread across all Haldane's dilemma genes are rare because their appearance by mutation is balanced by natural selection. A sudden change occurs in the environment, for example, pollution by smoke, a change of climate, the introduction of a new food source, predator, or pathogen, and above all migration to a new habitat. It will be shown later that the general conclusions are not affected if the change is slow. The species is less adapted to the new environment, and its reproductive capacity is lowered. It is gradually improved as a result of natural selection. But meanwhile, a number of deaths, or their equivalents in lowered Variome Variome The variome is the whole set of genetic variations found in populations of species that have gone through a relatively short evolution change. For example, among humans, about 1 in every 1,200 nucleotide bases differ. However, as the human species diverged only 10,000 years ago, this variation rate is comparatively small. In practice, the variome can be the sum of the single nucleotide polymorphisms (SNPs) of the population. The Human Variome Project seeks to compile this genetic variation data worldwide. Variomics is the study of variome and a branch of bioinformatics. The blend word 'variome' is from genetic variant Habitat fragmentation they tend to support smaller populations of fewer species. Small populations are at an increased risk of a variety of genetic consequences that influence their long-term survival. Remnant populations often contain only a subset of the genetic diversity found in the previously continuous habitat. Processes that act upon underlying genetic diversity such as adaptation have a smaller pool of fitness-maintaining alleles to survive in the face of environmental change. Gene flow occurs when individuals of the same species exchange genetic information through reproduction. Populations can maintain genetic diversity through migration. When a habitat becomes fragmented and reduced in area, gene Quantitative genetics pattern. [See section on Allele and Genotype frequencies.] Here the sampling of gametes from the "potential" gamodeme is discussed in more detail. The sampling involves random fertilization between pairs of random gametes, each of which may contain either an A or an a allele. The sampling is therefore binomial sampling. Each sampling "packet" involves 2N alleles, and produces N zygotes (a "progeny" or a "line") as a result. During the course of the reproductive period, this sampling is repeated over and over, so that the final result is a mixture of sample progenies. The result is "dispersed random fertilization" formula_19 Population fragmentation Population fragmentation Population fragmentation is a form of population segregation. It is often caused by habitat fragmentation. Fragmentation can be natural or caused by human actions. In modern times, human activity is the most common cause. Population fragmentation causes inbreeding depression, which leads to a decrease in genetic variability in the species involved. This decreases the fitness of the population for several reasons. First, inbreeding forces competition with relatives, which decreases the evolutionary fitness of the species. Secondly, the decrease in genetic variability causes an increased possibility a lethal homozygous recessive trait may be expressed; this decreases the average litter Fixed allele population is put under strong selective pressure, and only certain individuals survive. These surviving individuals have a decreased number of alleles present within their population than were present in the initial population, however these remaining alleles are the only ones left in future populations assuming no mutation or migration. This bottleneck effect can also be seen in natural disaster, as shown in the rabbit example above. Similar to the bottleneck effect, the founder's effect can also cause allele fixation. The founder effect occurs when a small founding population is moved a new area and propagates the future population. This can Quantitative genetics Quantitative genetics Quantitative genetics is a branch of population genetics that deals with phenotypes that vary continuously (in characters such as height or mass)—as opposed to discretely identifiable phenotypes and gene-products (such as eye-colour, or the presence of a particular biochemical). Both branches use the frequencies of different alleles of a gene in breeding populations (gamodemes), and combine them with concepts from simple Mendelian inheritance to analyze inheritance patterns across generations and descendant lines. While population genetics can focus on particular genes and their subsequent metabolic products, quantitative genetics focuses more on the outward phenotypes, and makes summaries only of Evolution not always be shuffled away from each other and genes that are close together tend to be inherited together, a phenomenon known as linkage. This tendency is measured by finding how often two alleles occur together on a single chromosome compared to expectations, which is called their linkage disequilibrium. A set of alleles that is usually inherited in a group is called a haplotype. This can be important when one allele in a particular haplotype is strongly beneficial: natural selection can drive a selective sweep that will also cause the other alleles in the haplotype to become more common in Fluctuating selection for reuse. It continually tries them in combination, waiting for the time when the focus of disadvantage has moved elsewhere. When this has happened, the genotypes carrying such genes spread by successful reproduction, becoming simultaneously stores for other bad genes and thus onward in continuous succession. In this conception of sex, the population is a storehouse of variation and sex is a mechanism for distributing old, minority variants once they become useful. This theory depends on fluctuating selection, as fluctuating selection dynamics make adaptive previously maladaptive variants due to ecological shifts. Fluctuating selection Fluctuating selection is a mode of natural Quantitative genetics two "same" alleles ( A and A, or a and a ) have a common origin"—or, more formally, "The probability that two homologous alleles are autozygous." Previously, the emphasis was on an individual's likelihood of having two such alleles, and the coefficient was framed accordingly. It is obvious, however, that this probability of autozygosity for an individual must also be the probability that each of its "two parents" had this autozygous allele. In this re-focused form, the probability is called the "co-ancestry coefficient" for the two individuals "i" and "j" ( "f" ). In this form, it can be used Phenotype the seeds of "Hieracium umbellatum" land in, determine the phenotype that grows. An example of random variation in "Drosophila" flies is the number of ommatidia, which may vary (randomly) between left and right eyes in a single individual as much as they do between different genotypes overall, or between clones raised in different environments. The concept of phenotype can be extended to variations below the level of the gene that affect an organism's fitness. For example, silent mutations that do not change the corresponding amino acid sequence of a gene may change the frequency of guanine-cytosine base pairs (GC content). Small population size risk of extinction focus on increasing population size as well as genetic diversity, which determines the fitness of a population and its long-term persistence. Some methods include captive breeding and genetic rescue. Stabilizing the variance in family size is an effective way can double the effective population size and is often used in conservation strategies. Small population size Small populations can behave differently from larger populations. They are often the result of a population bottlenecks from larger populations, leading to loss of heterozygosity and reduced genetic diversity and loss or fixation of alleles and shifts in allele frequencies. A small Coalescent theory SNPs along chromosomes appears to cluster in accordance with a variance to mean power law and to obey the Tweedie compound Poisson distribution. In this model the regional variations in the SNP map would be explained by the accumulation of multiple small genomic segments through recombination, where the mean number of SNPs per segment would be gamma distributed in proportion to a gamma distributed time to the most recent common ancestor for each segment. Coalescent theory is a natural extension of the more classical population genetics concept of neutral evolution and is an approximation to the Fisher–Wright (or Wright–Fisher) model Fixation (population genetics) is not too negative, there will be an appreciable chance that a mildly deleterious allele will fix. For example, in a diploid population of size formula_19, a deleterious allele with selection coefficient formula_20 has a probability fixation equal to formula_21. This estimate can be obtained directly from Kimura's 1962 work. Deleterious alleles with selection coefficients formula_20 satisfying formula_23 are effectively neutral, and consequently have a probability of fixation approximately equal to formula_24. Probability of fixation is also influenced by population size changes. For growing populations, selection coefficients are more effective. This means that beneficial alleles are more likely to become Single-nucleotide polymorphism Single-nucleotide polymorphism A single-nucleotide polymorphism, often abbreviated to SNP (; plural ), is a variation in a single nucleotide that occurs at a specific position in the genome, where each variation is present to some appreciable degree within a population (e.g. > 1%). For example, at a specific base position in the human genome, the C nucleotide may appear in most individuals, but in a minority of individuals, the position is occupied by an A. This means that there is a SNP at this specific position, and the two possible nucleotide variations – C or A – are said to Wahlund effect each population be different, i.e. formula_15. Suppose each population is in an internal Hardy–Weinberg equilibrium, so that the genotype frequencies AA, Aa and aa are "p", 2"pq", and "q" respectively for each population. Then the heterozygosity (formula_16) in the overall population is given by the mean of the two: which is always smaller than formula_17 ( = formula_18) unless formula_19 The Wahlund effect may be generalized to different subpopulations of different sizes. The heterozygosity of the total population is then given by the mean of the heterozygosities of the subpopulations, weighted by the subpopulation size. The reduction in heterozygosity can Recurrent evolution anticipated to be prime targets for evolutionary changes and therefore repeated evolution." Recurrent evolution Recurrent evolution is the repeated evolution of a particular character. Most evolution, or changes in allele frequencies from one generation to the next, is the result of drift, or random chance of some alleles getting passed down and others not. Recurrent evolution is when patterns emerge from this stochastic process when looking across populations. These patterns are of particular interest to evolutionary biologists as they can teach people about the underlying forces of evolution. Recurrent evolution is a broad term, but is usually used to describe Fixation (population genetics) formula_11, and furthermore, that the formula_12 converge to a specific value formula_13, which is the probability that the individual will have no descendants. The probability of fixation is then formula_14 since the indefinite survival of the beneficial allele will permit its increase in frequency to a point where selective forces will ensure fixation. Weakly deleterious mutations can fix in smaller populations through chance, and the probability of fixation will depend on rates of drift (~formula_15) and selection (~formula_16), where formula_17 is the effective population size. The ratio formula_18 determines whether selection or drift dominates, and as long as this ratio Fixed allele population in the Mosel Valley of Germany has been genetically characterized, and has shown to have six long-term monomorphic microsatellites. Six microsatellites were examined looking at the current population in 2008 as well as museum samples from 1895 to 1989. One of the microsatellite alleles examined has fixed within the population prior to 1895. For the current population all six microsatellites as well as all sixteen alloenzymes analyzed were fixed. Fixed alleles can often be deleterious to populations, especially when there is a small population size and low genetic variability. For example, the California Channel Island Fox ("Urocyon littoralis") has Hill–Robertson effect whose genome has only two genes, "a" and "b". If an advantageous mutant ("A") of gene "a" arises in a given individual, that individual's genes will through natural selection become more frequent in the population over time. However, if a separate advantageous mutant ("B") of gene "b" arises before "A" has gone to fixation, and happens to arise in an individual who does not carry "A", then individuals carrying "B" and individuals carrying "A" will be in competition. If recombination is present, then individuals carrying both "A" and "B" (of genotype "AB") will eventually arise. Provided there are no negative Genetic history of Europe or microsatellites. Fst is a special case of F-statistics, the concept developed in the 1920s by Sewall Wright. Fst is simply the correlation of randomly chosen alleles within the same sub-population relative to that found in the entire population. It is often expressed as the proportion of genetic diversity due to allele frequency differences among populations. The values range from 0 to 1. A zero value implies that the two populations are panmixis, that they are interbreeding freely. A value of one would imply that the two populations are completely separate. The greater the Fst value, the greater the genetic Allee effect diversity was correlated with defects that include lower sperm quality, abnormal testosterone levels, cowlicks, and kinked tails. In response, a genetic rescue plan was put in motion and several female pumas from Texas were introduced into the Florida population. This action quickly led to the reduction in the prevalence of the defects previously associated with inbreeding depression. Although the timescale for this inbreeding depression is larger than of those more immediate Allee effects, it has significant implications on the long-term persistence of a species. Demographic stochasticity refers to variability in population growth arising from sampling random births and deaths in Fixation (population genetics) fixation. This causes the surrounding areas to also be driven to fixation, even though they are not being selected for. By looking at the "Drosophila melanogaster" population data, Lee et al. found a reduced amount of heterogeneity within 25 base pairs of focal substitutions. They accredit this to small-scale hitchhiking effects. They also found that neighboring fixations that changed amino acid polarities while maintaining the overall polarity of a protein were under stronger selection pressures. Additionally, they found that substitutions in slowly evolving genes were associated with stronger genetic hitchhiking effects. Fixation (population genetics) In population genetics, fixation is the Genetic purging non-inbred population with mean fitness . Then, the size of the population reduces to a new smaller value (in fact, the effective population size should be used here), leading to a progressive increase of inbreeding. Then inbreeding depression occurs at a rate , due to (partially) recessive deleterious alleles that were present at low frequencies at different loci. This means that, in the absence of selection, the expected value for mean fitness after generations of inbreeding, would be: formula_1 where formula_2 is the population mean for Wright's inbreeding coefficient after generations of inbreeding. However, since selection operates upon fitness, mean Natural selection recombination reshuffles the alleles. The chance that such a reshuffle occurs between two alleles is inversely related to the distance between them. Selective sweeps occur when an allele becomes more common in a population as a result of positive selection. As the prevalence of one allele increases, closely linked alleles can also become more common by "genetic hitchhiking", whether they are neutral or even slightly deleterious. A strong selective sweep results in a region of the genome where the positively selected haplotype (the allele and its neighbours) are in essence the only ones that exist in the population. Selective sweeps Heteroplasmy the bottleneck is debated, with a recent mathematical and experimental metastudy providing evidence for a combination of random partitioning of mtDNAs at cell divisions and random turnover of mtDNA molecules within the cell. The mitochondrial bottleneck concept refers to the classic evolutionary term, which is used to explain an event that reduces and specifies a population. It was developed to describe why mitochondrial DNA in an embryo might be drastically different from that of its mother. When a large population of DNA is subsampled, each sample population will receive a slightly different proportion of mitochondrial genotypes. Consequently, when paired with Rare functional variant sequencing efforts. While these variants are individually infrequent in populations, there are many in human populations, and they can be unique to specific populations. They are more likely to be deleterious than common variants, as a result of rapid population growth and weak purifying selection. They have been suspected of acting independently or along with common variants to cause disease states. Rare functional variant A rare functional variant is a genetic variant which alters gene function, and which occurs at low frequency in a population. Rare variants may play a significant role in complex disease, as well as some Mendelian Loss of heterozygosity in the cancer cells. This is often done using polymorphic markers, such as microsatellites or single-nucleotide polymorphisms, for which the two parents contributed different alleles. Genome-wide LOH status of fresh or paraffin embedded tissue samples can be assessed by virtual karyotyping using SNP arrays. It has been proposed that LOH may limit the longevity of asexual organisms. The minor allele in heterozygous areas of the genome is likely to have mild fitness consequences compared to de-novo mutations because selection has had time to remove deleterious alleles. When allelic gene conversion removes the major allele at these sites organisms are likely Isolation by distance total population and reproduction occurs solely within the local population. This ecological isolation by distance, according to Wright, can create genetic differentiation among subpopulations, leading to evolutionary change. Individuals within the subpopulation are neighbors in the sense that their gametes may come together and inbreeding within the subpopulation increases homozygosity. Wright’s statistical theory for isolation by distance looks at population genetic consequences measured by F-statistics where the correlation of randomly uniting gametes within a subpopulation relative to those of the total population is the FST value. formula_1 The equation takes into account the variance in the distribution (var), the allele Crossover value This has the overall effect of increasing the variety of phenotypes present in a population. The process of non-sister chromatide exchanges, including the crossover value, can be observed directly in stained cells, and indirectly by the presence or absence of genetic markers on the chromosomes. The visible "crossovers" are called "chiasmata". The large-scale effect of crossover is to spread genetic variations within a population, as well as genetic basis for the selection of the most adaptable phenotypes. The crossover value depends on the mutual distance of the genetic loci observed. The crossover value is equal to the recombination value or Disease gene identification regions that are identical in both affected individuals and unaffected siblings. This is then repeated for multiple families, thus generating a small, overlapping fragment, which theoretically contains the disease gene. Homozygosity/Autozygosity mapping is a powerful technique, but is only valid when searching for a mutation segregating within a small, closed population. Such a small population, possibly created by the founder effect, will have a limited gene pool, and thus any inherited disease will probably be a result of two copies of the same mutation segregating on the same haplotype. Since affected individuals will probably be homozygous in the regions, looking Genetic association performed using an unrelated population sample or family trios in which the quantitative trait is measured in the offspring. Genetic association Genetic association is when one or more genotypes within a population co-occur with a phenotypic trait more often than would be expected by chance occurrence. Studies of genetic association aim to test whether single-locus alleles or genotype frequencies (or more generally, multilocus haplotype frequencies) differ between two groups of individuals (usually diseased subjects and healthy controls). Genetic association studies today are based on the principle that genotypes can be compared "directly", i.e. with the sequences of the actual genomes Thrifty gene hypothesis for being "thrifty", the hypothesis argues that an individual's disease risk is primarily determined by epigenetic events. Subtle, epigenetic modifications at many genomic loci (gene regulatory networks) alter the shape of the canal in response to environmental influences and thereby establish a predisposition for complex diseases such as metabolic syndrome. There may be epigenetic inheritance of disease risk. Watve and Yajnik suggested that changing insulin resistance mediates two phenotypic transitions: a transition in reproductive strategy from "r" (large number of offspring with smaller investment in each) to "K" (smaller number of offspring with greater investment in each) (see r/K selection Tajima's D polymorphisms relative to expectation, indicating population size expansion (e.g., after a bottleneck or a selective sweep) and/or purifying selection. A positive Tajima's D signifies low levels of both low and high frequency polymorphisms, indicating a decrease in population size and/or balancing selection. However, calculating a conventional "p-value" associated with any Tajima's D value that is obtained from a sample is impossible. Briefly, this is because there is no way to describe the distribution of the statistic that is independent of the true, and unknown, theta parameter (no pivot quantity exists). To circumvent this issue, several options have been proposed. However, Polymorphism (biology) Polymorphism (biology) Polymorphism in biology and zoology is the occurrence of two or more clearly different morphs or forms, also referred to as alternative "phenotypes", in the population of a species. To be classified as such, morphs must occupy the same habitat at the same time and belong to a panmictic population (one with random mating). The term polyphenism can be used to clarify that the different forms arise from the same genotype. Genetic polymorphism is a term used somewhat differently by geneticists and molecular biologists to describe certain mutations in the genotype, such as single nucleotide polymorphisms that may Genetic purging the decline of mean fitness caused by inbreeding (the inbreeding depression for fitness). The term "purge" is sometimes used for selection against deleterious alleles in a general way. It would avoid ambiguity to use "purifying selection" in that general context, and to reserve purging to its more strict meaning defined above. Deleterious alleles segregating in populations of diploid organisms have a remarkable trend to be, at least, partially recessive. This means that, when they occur in homozygosis (double copies), they reduce fitness by more than twice than when they occur in heterozygosis (single copy). In other words, part of their Fluctuating selection Fluctuating selection Fluctuating selection is a mode of natural selection characterized by the fluctuation of the direction of selection on a given phenotype over a relatively brief period of evolutionary time. For example, a species of plant may come in two varieties: one which prefers wetter soil and one which prefers dryer soil. During a period of wet years, the wet variety will be more fit and produce more offspring, and thereby increase the frequency of wet-preferring plants. If this wet period is followed by drought, the dry variety will be selected for and its numbers will increase. As periods Founder effect generations, continue rapidly towards fixation. This consequence of inbreeding makes the colony more vulnerable to extinction. When a newly formed colony is small, its founders can strongly affect the population's genetic makeup far into the future. In humans, who have a slow reproduction rate, the population will remain small for many generations, effectively amplifying the drift effect generation after generation until the population reaches a certain size. Alleles which were present but relatively rare in the original population can move to one of two extremes. The most common one is that the allele is soon lost altogether, but the other Population genetics the continuous variation measured by the biometricians could be produced by the combined action of many discrete genes, and that natural selection could change allele frequencies in a population, resulting in evolution. In a series of papers beginning in 1924, another British geneticist, J.B.S. Haldane, worked out the mathematics of allele frequency change at a single gene locus under a broad range of conditions. Haldane also applied statistical analysis to real-world examples of natural selection, such as peppered moth evolution and industrial melanism, and showed that selection coefficients could be larger than Fisher assumed, leading to more rapid adaptive evolution Inbreeding inbreeding, since their genomes have many similarities already. This overall homozygosity becomes an issue when there are deleterious recessive alleles in the gene pool of the family. By pairing chromosomes of similar genomes, the chance for these recessive alleles to pair and become homozygous greatly increases, leading to offspring with autosomal recessive disorders. Inbreeding is especially problematic in small populations where the genetic variation is already limited. By inbreeding, individuals are further decreasing genetic variation by increasing homozygosity in the genomes of their offspring. Thus, the likelihood of deleterious recessive alleles to pair is significantly higher in a small inbreeding Galton–Watson process the numbers of different men's sons to be independent random variables, all having the same distribution. Then the simplest substantial mathematical conclusion is that if the average number of a man's sons is 1 or less, then their surname will almost surely die out, and if it is more than 1, then there is more than zero probability that it will survive for any given number of generations. Modern applications include the survival probabilities for a new mutant gene, or the initiation of a nuclear chain reaction, or the dynamics of disease outbreaks in their first generations of spread, or