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.

Suppose there is a population of n individuals over which the amount of a particular characteristic varies. Those n individuals can grouped by the amount of the characteristic that each displays. In this case, at most there will be n groups of n distinct values of the characteristic, and at least there will be 1 group of a single shared value of the characteristic. Index each group with i so that the number of members in the group is n_i and the value of the characteristic shared among all members of the group is z_i. Now assume that having z_i of the characteristic is associated with having a fitness w_i where the product w_in_i represents the number of offspring in the next generation. Denote this number of offspring from group i by n_i' so that w_i = n_i' / n_i. Let z_i' be the amount of the characteristic displayed by the offspring from group i. Denote the amount of change in characteristic in group i by Δz_i defined by

Now take z to be the average characteristic value in this population and z' to be the average characteristic value in the next generation. Define the change in average characteristic by Δz. That is,

Note that this is not the average value of Δz_i. Also take w to be the average fitness of this population. The Price equation states:

where the functions and are respectively defined in Equations (1) and (2) below but are like the sample versions of the expected value and covariance operators from probability (see Sample mean and covariance). Note that this is really a difference equation relating the average value of a characteristic in one generation to the average value of the characteristic in the very next generation. In fact, assuming that w is not zero, it is often useful to write it as

In the specific case that characteristic z_i = w_i (i.e., the fitness is the characteristic of interest), then Price's equation reformulates Fisher's fundamental theorem of natural selection.

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.

The Price equation also has applications in economics.

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Proof of the Price equation

To prove the Price equation, the following definitions are needed. If n_i is the number of occurrences of a pair of real numbers x_i and y_i, then:

• The average or expected value of the x_i values is:

• The covariance between the x_i and y_i values is:

The notation will also be used when convenient.

Suppose there is a population of organisms all of which have a genetic characteristic described by some real number. For example, high values of the number represent an increased visual acuity over some other organism with a lower value of the characteristic. Groups can be defined in the population which are characterized by having the same value of the characteristic. Let subscript i identify the group with characteristic z_i and let n_i 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 defined as:

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 n′_i. Primes will be used to denote child parameters, unprimed variables denote parent parameters.

The total number of children is n' where:

The fitness of group i will be defined to be the ratio of children to parents:

with average fitness of the population being

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

where z′_i are the (possibly new) values of the characteristic in the child population. Equation (2) shows that:

Call the change in characteristic value from parent to child populations Δz_i so that Δz_i = z'_i − z_i. As seen in Equation (1), the expected value operator is linear, so

Combining Equations (7) and (8) leads to

but from Equation (1) gives:

and from Equation (4) gives:

Applying Equations (5) and (6) to Equation (10) and then applying the result to Equation (9) gives the Price Equation:

Simple Price equation

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

which can be restated as:

where v_i is the fractional fitness: v_i= w_i/w. This simple Price equation can be proven using the definition in Equation (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: Evolution of sight

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

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

Using Equation (4), the child population (assuming the character z_i 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 z′_i= z_i):

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 the definition in Equation (2), the simple Price equation for the character z can be written:

For the second generation:

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 〈w '_i z '_i 〉 which needs to be calculated z ' '  for the second generation. w '  and 〈w '_i z '_i 〉 can both be thought of characteristics of the first generation, so the Price equation can be used to calculate them as well:

The five 0-generation variables w, z, 〈w_i  z_i 〉, 〈w²_i 〉, and 〈w²_i z_i 〉, which must be known before proceeding to calculate the three first generation variables w ', z ', and 〈w '_i z '_i 〉, which are needed to calculate z '  for the second generation. It can be seen that in general the Price equation cannot be used to propagate forward in time unless there is a way of calculating the higher moments (〈wⁿ_i 〉 and 〈wⁿ_i z_i 〉) 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: Evolution of sickle cell anemia

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.

Let z_i=i be the number of sickle-cell genes that organisms of type i have so that z_i = 0 or 1 or 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 i–i matings is n²_i/[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 n_i=[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:

To find the concentration n₁ of carriers in the population at equilibrium, with the equilibrium condition of Δ z=0, the simple Price equation is used:

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

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.

The situation has been effectively determined for the infinite (equilibrium) generation. This means that there is 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:

Example: sex ratios

In a 2-sex species or deme with sexes 1 and 2 where z₁ = 1, z₂ = 0, z is the relative frequency of sex 1. The simplest model of the sex ratio is achieved by assuming that the fitness of an individual is proportional to the relative frequency of the sex with which it mates. Consider proportionality constants a and b such that w₁ = a(1 − z) and w₂ = bz. This gives w = (a + b)z(1 − z) and cov(w_i,z_i) + wz = E(w_i,z_i) = az(1 − z), so cov(w_i,z_i) = az(1 − z) − (a + b)z²(1 − z) = z(1 − z)(a − (a + b)z). Hence, Δz = a / (a + b) − z so that z' = a / (a + b).

Full Price equation

The simple Price equation was based on the assumption that the characters z_i do not change over one generation. If it is assumed that they do change, with z_i 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.

Examples

Evolution of altruism

To study the evolution of a genetic predisposition to altruism, altruism will be defined as the genetic predisposition to behavior which decreases individual fitness while increasing the average fitness of the group to which the individual belongs. First specifying a simple model, which will only require the simple Price equation. Specify a fitness w_i by a model equation:

where z_i is a measure of altruism, the az_i 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:

where var(z_i) is the variance of z_i which is just the covariance of z_i with itself:

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.

Now assuming 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. n_ij will specify the number of individuals of type ij. Let z_ij be the degree of altruism expressed by individual j of group i towards the members of group i. Let's specify the fitness w_ij by a model equation:

The a z_ij term is the fitness the organism loses by being altruistic and is proportional to the degree of altruism z_ij that it expresses towards members of its own group. The b z_i term is the fitness that the organism gains from the altruism of the members of its group, and is proportional to the average altruism z_i expressed by the group towards its members. Again, in studying study altruistic (rather than spiteful) behavior, it is expected that a and b are positive numbers. Note that the above behavior is altruistic only when az_ij >bz_i. Defining the group averages:

and global averages:

It can be seen that since the z_i and z_i are now averages over a particular group, and since these groups are subject to selection, the value of Δz_i = z′_i−z_i 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. 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.

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, A_m, B₀, and B_m. 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 A_m are both A-alleles, but organisms with the A₀ gene produce offspring with A₀-genes only, while organisms with the A_m gene produce (1−3m) offspring with the A_m gene, and m organisms of the remaining three gene types. Likewise, B₀ and B_m are both B-alleles, but organisms with the B₀ gene produce offspring with B₀-genes only, while organisms with the B_m gene produce (1−3m) offspring with the B_m gene, and m organisms of the remaining three gene types.

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

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

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.

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. There are two cases:

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 z_i does not change, the average mutabilities will be:

with these definitions, the simple Price equation now applies.

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. (1995). "George Price's contributions to Evolutionary Genetics". Journal of Theoretical Biology 175: 373-388.

• 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.

• Langdon, W. B. (1998). "Evolution of GP Populations: Price's Selection and Covariance Theorem". Genetic Programming and Data Structures: 167-208.

• Price, G.R. (1970). "Selection and covariance". Nature 227: 520-521.

• Price, G.R. (1972). "Fisher's "fundamental theorem" made clear". Annals of Human Genetics 36: 129-140.

• Van Veelen, M. (2005). "On the use of the Price equation". Journal of Theoretical Biology 237: 412-426.