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Encyclopedia > Price equation

The Price equation (also known as Price's equation) is a covariance equation which is a mathematical description of evolution and natural selection. The Price equation was derived by George R. Price, working in London to rederive W.D. Hamilton's work on kin selection. In 1832, while traveling on the Beagle, naturalist Charles Darwin collected giant fossils in South America. ... The Galápagos Islands hold 13 species of finches that are closely related and differ most markedly in the shape of their beaks. ... George R. Price (1922 - January 6, 1975) was a American population geneticist. ... This article is about the British biologist Bill Hamilton. ... Kin selection refers to changes in gene frequency across generations that are driven at least in part by interactions between related individuals, and this forms much of the conceptual basis of the theory of social evolution. ...


Suppose we have a population whose members or elements are indicated by i. Element i has associated with it a fitness wi and some additional character zi whose evolution we wish to study. The Price equation states: Fitness (often denoted in population genetics models) is a central concept in evolutionary theory. ...

where w is the average fitness and Δz is the change in the average character. The term cov(wi,zi) is the covariance of the characteristics with respect to the fitness in the population and E(wizi) is the expectation of the fitness times the change in the characteristic. In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ... expectation in the context of probability theory and statistics, see expected value. ...


In the specific case that characteristic zi is set to the fitness wi, then Price's equation reformulates Fisher's fundamental theorem of natural selection. In population genetics, Ronald Fishers fundamental theorem of natural selection was originally stated as: The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time. ...


Price's equation is, importantly, a tautology. It is a statement of mathematical fact between certain variables, and its value lies in the insight gained by assigning certain values encountered in evolutionary genetics to the variables. For example, the statement "if every pair of birds has two offspring, then among ten pairs of birds there will be twenty offspring" is a tautology. It doesn't really impart any new information about birds so much as it organizes our concepts about birds and their offspring. The Price equation is much more sophisticated than the above statement, but at its core, it too is a mathematically provable tautology. Within the study of logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of its parts. ...


The Price equation also has applications in economics. Face-to-face trading interactions among on the New York Stock Exchange trading floor Economics, as a social science, studies the production, distribution, and consumption of resources. ...

Contents

Proof of the Price Equation

To prove the Price equation, we will need the following definitions. If ni is the number of occurrences of a pair of real numbers xi and yi, then:

The notation 〈xi〉 = E(xi) will also be used when convenient. expectation in the context of probability theory and statistics, see expected value. ... In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ...


Suppose we have a population of organisms all of which have a genetic characteristic described by some real number z. For example we can say z measures visual acuity, with high values of z representing an increased visual acuity over some other organism with a lower value of z. We can define groups in the population which are characterized by having the same value of z. Let subscript i identify the group with characteristic zi and let ni be the number of organisms in that group. The total number of organisms is then n where:

The average value of the characteristic is z where:

Now suppose that the population reproduces, all parents are eliminated, and then there is a selection process on the children, by which less fit children are removed from the reproducing population. After reproduction and selection, the population numbers for the child groups will change to ni. Primes will be used to denote child parameters, unprimed variables denote parent parameters. The fitness of group i will be defined to be the ratio of children to parents:

with average fitness of the population being

The total number of children is n' where:

and the average value of the child characteristic will be z' where:

where zi are the (possibly new) values of the characteristic in the child population. We can see from Equations 1 and 2 that:

and

so that

but from Equation 1 we have

and from Equation 4 we have:

Applying Equations 5 and 6 to the above equation we finally have:

The Simple Price Equation

When the characters zi do not change from the parent to the child generation, the second term in the Price equation becomes zero and we have a simplified version of the Price equation:

which can be restated as:

where vi is the fractional fitness: vi= wi/w. This simple Price equation can be proven using definition 2 above. It makes this fundamental and tautological statement about evolution: "If a certain inheritable characteristic is correlated with an increase in fractional fitness, the average value of that characteristic in the child population will be increased over that in the parent population."


Example: The Evolution of Sight

As an example of the simple Price equation, consider a model for the evolution of sight. Suppose zi is a real number measuring the visual acuity of an organism. An organism with a higher zi will have better sight than one with a lower value of zi. Lets say that the fitness of such an organism is wi=zi which means the more sighted it is, the fitter it is, that is, the more children it will produce. Suppose we begin with the following description of a parent population composed of 3 types: (i = 0,1,2) with sightedness values zi = 3,2,1:

i 0 1 2
ni 10 20 30
zi 3 2 1

Using Equation 4, we then have for the child population (assuming the character zi doesn't change)

i 0 1 2
ni 30 40 30
zi 3 2 1

We would like to know how much average visual acuity has increased or decreased in the population. From Equation 3, the average sightedness of the parent population is z = 5/3. The average sightedness of the child population is z' = 2, so that the change in average sightedness is:

which indicates that the trait of sightedness is increasing in the population. Applying the Price equation we have (since zi= zi):

Dynamical Sufficiency and the Simple Price Equation

Sometimes the genetic model being used encodes enough information into the parameters used by the Price equation to allow the calculation of the parameters for all subsequent generations. This property is referred to as dynamical sufficiency. For simplicity, the following looks at dynamical sufficiency for the simple Price equation, but is also valid for the full Price equation.


Referring to definition 2, the simple Price equation for the character z can be written:

For the second generation, we have:

The simple Price equation for z only gives us the value of z′ for the first generation, but does not give us the value of w′ and 〈wi zi〉 which we need to go on to calculate z″ for the second generation. Characters w' and 〈wi zi〉 can both be thought of as characters of the first generation, so we can use the Price equation to calculate them as well:

We now need the five 0-generation variables w, z, 〈wi zi〉, 〈w2i〉 and 〈w2i zi〉 which must be known before we can proceed to calculate the three first generation variables w′, z′, 〈wi zi〉 which we need to calculate z″ for the second generation. With a little thought it can be seen that in general we cannot use the Price equation to propagate forward in time unless we have a way of calculating the higher moments (〈wni〉 and 〈wni zi〉) from the lower moments in a way that is independent of the generation. Dynamical sufficiency means that such equations can be found in the genetic model, allowing the Price equation to be used alone as a propagator of the dynamics of the model forward in time.


Example: The Evolution of Sickle Cell Anemia

An example of autosomal recessive inheritance. In the sickle cell case, the two parents are "carriers" who are resistant to malaria. Their children have one chance in four of inheriting both sickle cell genes and suffering sickle cell anemia, two chances in four of being a carrier themselves, and being resistant to malaria like their parents, and one chance in four of not inheriting the gene from either parent, and being susceptible to malaria.
An example of autosomal recessive inheritance. In the sickle cell case, the two parents are "carriers" who are resistant to malaria. Their children have one chance in four of inheriting both sickle cell genes and suffering sickle cell anemia, two chances in four of being a carrier themselves, and being resistant to malaria like their parents, and one chance in four of not inheriting the gene from either parent, and being susceptible to malaria.

As an example of dynamical sufficiency, consider the case of sickle cell anemia. Each person has two sets of genes, one set inherited from the father, one from the mother. Sickle cell anemia is a blood disorder which occurs when a particular pair of genes both carry the 'sickle-cell trait'. The reason that the sickle-cell gene has not been eliminated from the human population by selection is because when there is only one of the pair of genes carrying the sickle-cell trait, that individual (a "carrier") is highly resistant to malaria, while a person who has neither gene carrying the sickle-cell trait will be susceptible to malaria. Let's see what the Price equation has to say about this. Diagram showing how recessive genes are transmitted and expressed. ... Diagram showing how recessive genes are transmitted and expressed. ... Sickle-shaped red blood cells Sickle cell anemia (American English), sickle cell anaemia (British English) or sickle cell disease is a genetic disease in which red blood cells may change shape under certain circumstances. ...


Let zi=i be the number of sickle-cell genes that organisms of type i have so that zi = [0,1,2]. Assume the population sexually reproduces and matings are random between type 0 and 1, so that the number of 0–1 matings is n0n1/(n0+n1) and the number of ii matings is n2i/[2(n0+n1)] where i = 0 or 1. Suppose also that each gene has a 1/2 chance of being passed to any given child and that the initial population is ni=[n0,n1,n2]. If b is the birth rate, then after reproduction there will be

type 0 children (unaffected)
type 1 children (carriers)
type 2 children (affected)

Suppose a fraction a of type 0 reproduce, the loss being due to malaria. Suppose all of type 1 reproduce, since they are resistant to malaria, while none of the type 2 reproduce, since they all have sickle-cell anemia. The fitness coefficients are then:

We wish to find the concentration n1 of carriers in the population at equilibrium. The equilibrium condition is Δ z=0 or, using the simple Price equation:

where f=n1/n0. At equilibrium then, assuming f is not zero, we have:

In other words the ratio of carriers to non-carriers will be equal to the above constant non-zero value. In the absence of malaria, a=1 and f=0 so that the sickle-cell gene is eliminated from the gene pool. For any presence of malaria, a will be smaller than unity and the sickle-cell gene will persist.


It is clear that we have been able to effectively determine the situation for the infinite (equilibrium) generation. This means that we have dynamical sufficiency with respect to the Price equation, and that there is an equation relating higher moments to lower moments. For example, for the second moments:

The Full Price Equation

The simple Price equation was based on the assumption that the characters zi do not change over one generation. If we assume that they do change, with zi being the value of the character in the child population, then the full Price equation must be used. A change in character can come about in a number of ways. The following two examples illustrate two such possibilities, each of which introduces new insight into the Price equation.


Example: The Evolution of Altruism

We want to study the evolution of a genetic predisposition to altruism. We will define altruism as the genetic predisposition to behavior which decreases individual fitness while increasing the average fitness of the group to which the individual belongs. Lets first specify a simple model, which will only require the simple Price equation. Specify a fitness wi by a model equation:

where zi is a measure of altruism, the azi term is the decrease in fitness of an individual due to altruism towards the group and bz is the increase in fitness of an individual due to the altruism of the group towards an individual. Assume that a and b are both greater than zero. From the Price equation, we can see that:

where var(zi) is the variance of zi which is just the covariance of zi with itself: In probability theory and statistics, the variance of a random variable (or equivalently, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

It can be seen that, by this model, in order for altruism to persist it must be uniform throughout the group. If there are two altruist types the average altruism of the group will decrease, the more altruistic will lose out to the less altruistic.


We will now assume a hierarchy of groups which will require the full Price equation. The population will be divided into groups, labelled with index i and then each group will have a set of subgroups labelled by index j. Individuals will thus be identified by two indices, i and j, specifying which group and subgroup they belong to. nij will specify the number of individuals of type ij. Let zij be the degree of altruism expressed by individual j of group i towards the members of group i. Let's specify the fitness wij by a model equation:

The a zij term is the fitness the organism loses by being altruistic and is proportional to the degree of altruism zij that it expresses towards members of its own group. The b zi term is the fitness that the organism gains from the altruism of the members of its group, and is proportional to the average altruism zi expressed by the group towards its members. Again, if we are going to study altruistic behavior, we expect that a and b are positive numbers. Note that the above behavior is altruistic only when azij >bzi. We define the group averages:

and global averages:

It can be seen that since the zi and zi are now averages over a particular group, and since these groups are subject to selection, the value of Δzi = zizi will not necessarily be zero, and the full Price equation will be needed.

In this case, the first term isolates the advantage to each group conferred by having altruistic members. The second term isolates the loss of altruistic members from their group due to their altruistic behavior. We know that the second term will be negative. In other words there will be an average loss of altruism due to the in-group loss of altruists, assuming that the altruism is not uniform across the group. The first term is:

In other words, for b>a there may be a positive contribution to the average altruism as a result of a group growing due to its high number of altruists and this growth can offset in-group losses, especially if the variance of the in-group altruism is low. In order for this effect to be significant, there must be a spread in the average altruism of the groups.


Example: The Evolution of Mutability

Suppose there is an environment containing two kinds of food. Let α be the amount of the first kind of food and β be the amount of the second kind. Suppose an organism has a single allele which allows it to utilize a particular food. The allele has four gene forms: A0, Am, B0, and Bm. If an organism's single food gene is of the A type, then the organism can utilize A-food only, and its survival is proportional to α. Likewise, if an organism's single food gene is of the B type, then the organism can utilize B-food only, and its survival is proportional to β. A0 and Am are both A-alleles, but organisms with the A0 gene produce offspring with A0-genes only, while organisms with the Am gene produce (1−3m) offspring with the Am gene, and m organisms of the remaining three gene types. Likewise, B0 and Bm are both B-alleles, but organisms with the B0 gene produce offspring with B0-genes only, while organisms with the Bm gene produce (1−3m) offspring with the Bm gene, and m organisms of the remaining three gene types.


Let i=0,1,2,3 be the indices associated with the A0, Am, B0, and Bm genes respectively. Let wij be the number of viable type-j organisms produced per type-i organism. The wij matrix is: (with i denoting rows and j denoting columns)

α 0 0 0
(1−3m mβ mβ
0 0 β 0
mα mα mβ (1−3m

Mutators are at a disadvantage when the food supplies α and β are constant. They lose every generation compared to the non-mutating genes. But when the food supply varies, even though the mutators lose relative to an A or B non-mutator, they may lose less than them over the long run because, for example, an A type loses a lot when α is low. In this way, "purposeful" mutation may be selected for. This may explain the redundancy in the genetic code, in which some amino acids are encoded by more than one codon in the DNA. Although the codons produce the same amino acids, they have an effect on the mutability of the DNA, which may be selected for or against under certain conditions. The general structure of an α-amino acid molecule, with the amine group on the left and the carboxyl group on the right. ... RNA codons. ... The general structure of a section of DNA Deoxyribonucleic acid (DNA) is a nucleic acid that contains the genetic instructions for the biological development of a cellular form of life or a virus. ...


With the introduction of mutability, the question of identity versus lineage arises. Is fitness measured by the number of children an individual has, regardless of the children's genetic makeup, or is fitness the child/parent ratio of a particular genotype?. Fitness is itself a characteristic, and as a result, the Price equation will handle both.


Suppose we want to examine the evolution of mutator genes. Define the z-score as:

in other words, 0 for non-mutator genes, 1 for mutator genes. We have the following two cases:


Example: Genotype Fitness

Lets focus on the idea of the fitness of the genotype. The index i indicates the genotype and the number of type i genotypes in the child population is:

which gives fitness:

Since the individual mutability zi does not change, the average mutabilities will be:

with these definitions, the simple Price equation now applies.


Example: Lineage Fitness

In this case we want to look at the idea that fitness is measured by the number of children an organism has, regardless of their genotype. Note that we now have two methods of grouping, by lineage, and by genotype. It is this complication that will introduce the need for the full Price equation. The number of children an i-type organism has is:

which gives fitness:

We now have characters in the child population which are the average character of the i-th parent.

with global characters:

with these definitions, the full Price equation now applies


References

  • Frank, S.A. (1997). "The Price Equation, Fisher's Fundamental Theorem, Kin Selection, and Causal Analysis". Evolution 51 (6): 1712-1729.
  • Grafen, M. (2000). "Developments of the Price equation and natural selection under uncertainty". Proc. R. Soc. London B 267: 1223-1227.
  • Price, G.R. (1972). "Fisher's "fundamental theorem" made clear". Annals of Human Genetics 36: 129-140.
  • Langdon, W. B. (1998). "Evolution of GP Populations: Price's Selection and Covariance Theorem". Genetic Programming and Data Structures: 167-208.

Use of Price's theorem in computer science of evolutionary computation George R. Price (1922 - January 6, 1975) was a American population geneticist. ... First title page, November 4, 1869 Nature is one of the oldest and most reputable scientific journals, first published on 4 November 1869. ... George R. Price (1922 - January 6, 1975) was a American population geneticist. ...

  • Van Veelen, M. (2005). "On the use of the Price equation". Journal of Theoretical Biology 237: 412-426.
Topics in population genetics
v  d  e
Key concepts: Hardy-Weinberg law | genetic linkage | linkage disequilibrium | Fisher's fundamental theorem | neutral theory
Selection: natural | sexual | artificial | ecological
Effects of selection on genomic variation: genetic hitchhiking | background selection
Genetic drift: small population size | population bottleneck | founder effect | coalescence
Founders: R.A. Fisher | J.B.S. Haldane | Sewall Wright
Related topics: evolution | microevolution | evolutionary game theory | fitness landscape | genetic genealogy
List of evolutionary biology topics

  Results from FactBites:
 
Price equation - Wikipedia, the free encyclopedia (2524 words)
The Price equation (also known as Price's equation) is a covariance equation which is a mathematical description of evolution and natural selection.
From Equation 3, the average sightedness of the parent population is z = 5/3.
Dynamical sufficiency means that such equations can be found in the genetic model, allowing the Price equation to be used alone as a propagator of the dynamics of the model forward in time.
George R. Price - Wikipedia, the free encyclopedia (933 words)
Price was born in 1922, the younger brother of Edison.
W.D. Hamilton (1996) failed to recall when Price first contacted him, but says he Price had read Hamilton's 1964 papers on kin selection, and with no training in population genetics or statistics devised the Price equation, a covariance equation that generated the change in allele frequency of a population.
Price's contributions were then largely overlooked for twenty years; he had worked only in theoretical biology for a short time and was not very thorough in publishing papers.
  More results at FactBites »

 
 

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