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 we have a population whose elements are labeled i. Element i has fitness wi and zi is some character of element i whose evolution we wish to study. The Price equation states:



w\Delta{z}=cov\left(w_i,z_i\right)+E\left(w_i\Delta z_i\right) $$

where w is the average fitness and &Delta;z is the change in the average character. cov(wi,zi) is the covariance of the characteristics with respect to the fitness in the population and E(wi,&Delta;zi) is the expectation of the fitness times the change in the characteristic.

If the specific case that characteristic zi is set to the fitness wi, 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.

Proof of the Price Equation
To prove the Price equation, we will need the following definitions. If n i is the number of occurrences of a pair of real numbers x i and y i, then:


 * The average or expectation value of x is:

E(x_i)\equiv \frac{\sum_i x_i n_i}{\sum_i n_i}(1) $$


 * The covariance between x i and y i is:

\textrm{cov}(x_i,y_i) \equiv \frac{\sum_i n_i~[x_i-E(x_i)]~[y_i-E(y_i)]}{\sum_i n_i} = E(x_iy_i)-E(x_i)E(y_i)(2) $$

The notation  = E(x i ) will also be used when convenient.

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 z i and let n i be the number of organisms in that group. The total number of organisms is then n where:



n = \sum_i n_i\, $$

The average value of the characteristic is z where:



z = \frac{\sum_i z_i n_i}{n}(3) $$

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



w_i = \frac{n_i'}{n_i}(4) $$

with average fitness of the population being



w = \frac{\sum_i w_i n_i}{n}~(5) $$

The total number of children is n'  where:



n' = \sum_i n'_i\, $$

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



z' = \frac{\sum_i z'_i n_i'}{n'}~(6) $$

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


 * $$\textrm{cov}(w_iz_i)=\textrm{E}(w_iz_i)-wz\,$$

and


 * $$\textrm{E}(w_i\Delta z_i)=\textrm{E}(w_iz'_i)-\textrm{E}(w_iz_i)\,$$

so that


 * $$\textrm{cov}(w_iz_i)+\textrm{E}(w_i\Delta z_i)=\textrm{E}(w_iz'_i)-wz\,$$

but from Equation 1 we have


 * $$\textrm{E}(w_iz'_i)=\frac{\sum_i w_iz'_in_i}{n}$$

and from Equation 4 we have:


 * $$\frac{\sum_iw_iz'_in_i}{n}=\frac{\sum_i z'_in'_i}{n'}~\frac{n'}{n}$$

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


 * $$\textrm{cov}(w_iz_i)+\textrm{E}(w_i\Delta z_i)=wz'-wz=w\Delta z\,$$

The 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 and we have a simplified version of the Price equation:



w\Delta z = \textrm{cov}\left(w_i,z_i\right) $$

which can be restated as:



\Delta z = \textrm{cov}\left(v_i,z_i\right) $$

where v i is the fractional fitness: v i = w i /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 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. Lets 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. Suppose we begin with the following description of a parent population composed of 3 types: (i = 0,1,2) with sightedness values z i = 3,2,1:
 * {| class="wikitable"

Using Equation 4, we then have for the child population (assuming the character z i doesn't change)
 * align="center"|i
 * align="center"|0
 * align="center"|1
 * align="center"|2
 * align="center"|n i 
 * align="center"|10
 * align="center"|20
 * align="center"|30
 * align="center"|z i 
 * align="center"|3
 * align="center"|2
 * align="center"|1
 * }
 * align="center"|2
 * align="center"|1
 * }
 * {| class="wikitable"

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:
 * align="center"|i
 * align="center"|0
 * align="center"|1
 * align="center"|2
 * align="center"|n i 
 * align="center"|30
 * align="center"|40
 * align="center"|30
 * align="center"|z i 
 * align="center"|3
 * align="center"|2
 * align="center"|1
 * }
 * align="center"|2
 * align="center"|1
 * }



\Delta z \equiv z'-z = 1/3 $$

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



\Delta z = \textrm{cov}\left(w_i,z_i\right)/w = 1/3 $$

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:



w(z'-z)=\langle w_i z_i\rangle - wz $$

For the second generation, we have:



w'(z' '-z')=\langle w'_i z'_i\rangle - w'z' $$

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  which we need to go on to calculate z' '  for the second generation. w' and < w' i z' i > can both be thought of a characters of the first generation, so we can use the Price equation to calculate them as well:


 * $$w(w'-w)=\langle w_i^2\rangle - w^2$$

w(\langle w'_i z'_i\rangle-\langle w_i z_i\rangle)=\langle w_i ^2 z_i\rangle - w\langle w_i z_i\rangle $$

We now need the five 0-generation variables w, z, < wi zi >, < w2i > and < w2i z i > which must be known before we can proceed to calculate the three first generation variables w' , z' ,  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 (< w ni > and < wn i 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


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=[0,1,2]. 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 i-i 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


 * $$b\left(\frac{n_0^2/2+n_0n_1/2+n_1^2/8}{n_0+n_1}\right)$$ type 0 children (unaffected)
 * $$b\left(\frac{n_0n_1/2+n_1^2/4}{n_0+n_1} \right)$$ type 1 children (carriers)


 * $$b\left(\frac{n_1^2/8}{n_0+n_1} \right)$$ 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:


 * $$w_0=ab\left(\frac{n_0^2/2+n_0n_1/2+n_1^2/8}{n_0(n_0+n_1)}\right)$$
 * $$w_1=b\left(\frac{n_0n_1/2+n_1^2/4}{n_1(n_0+n_1)} \right)$$
 * $$w_2=0\,$$

We wish to find the concentration n 1 of carriers in the population at equilibrium. The equilibrium condition is &Delta; z=0 or, using the simple Price equation:



0=\textrm{cov}(w_i/w,z_i) = \frac{f(2-2a-af)}{(1+f)(2a+2f+af)} $$

where f=n 1/n 0. At equilibrium then, assuming f is not zero, we have:



f=\frac{n_1}{n_0}=\frac{2(1-a)}{a} $$

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:


 * $$ \langle w_i^2z_i \rangle = \frac{\langle w_i z_i \rangle}{z}$$
 * $$ \langle w_i^2 \rangle = \frac{-b z^2 w^2 + 2 b z^2 w \langle w_i z_i

\rangle + b z \langle w_i z_i \rangle^2 - b z^2 \langle w_i z_i \rangle^2 - 4 \langle w_i z_i \rangle^3} { b z^2 - 4 z \langle w_i z_i \rangle} $$

The Full Price Equation
The simple Price equation was based on the assumption that the characters z i do not change over one generation. If we assume 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.

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 w i by a model equation:



w_i = \frac{n'_i}{n_i} = k - a z_i + b z $$

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, we can see that:



w\Delta z = -a~\textrm{var}\left(z_i\right) $$

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



\textrm{var}(z_i) \equiv \textrm{E}(z_i^2)-E(z_i)^2 $$

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. 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:



w_{ij} = \frac{n'_{ij}}{n_{ij}} = k - a z_{ij} + b z_i $$

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, 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 az ij >bz i. We define the group averages:


 * $$n_i = \sum_j n_{ij}\,$$


 * $$z_i = \frac{\sum_j z_{ij}n_{ij}}{n_i}$$


 * $$w_i = \frac{\sum_j w_{ij}n_{ij}}{n_i}=k+(b-a)z_i$$


 * $$n_i'= \sum_j n_{ij}'=n_i(k+(b-a)z_i)\,$$


 * $$z_i'= \frac{\sum_j z_{ij}n_{ij}'}{n_i'}$$

and global averages:


 * $$n = \sum_{ij} n_{ij} = \sum_i n_i\,$$


 * $$z = \frac{\sum_{ij} z_{ij}n_{ij}}{n} = \frac{\sum_i z_in_i}{n}$$


 * $$w = \frac{\sum_{ij} w_{ij}n_{ij}}{n} = \frac{\sum_i w_in_i}{n}$$


 * $$n'= \sum_{ij} n_{ij}' = \sum_i n_i'\,$$


 * $$z'= \frac{\sum_{ij} z_{ij}n_{ij}'}{n'} = \frac{\sum_i z_i'n_i'}{n'}$$

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 &Delta; z i = z'  i -z i will not necessarily be zero, and the full Price equation will be needed.



\Delta z = \textrm{cov}(w_i/w,z_i)+\textrm{E}(w_i\Delta z_i/w)\, $$

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. Th first term is:



\textrm{cov}(w_i/w,z_i)=\left(b-a\right)\textrm{var}(z_i) $$

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 &alpha; be the amount of the first kind of food and &beta; 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: A 0, A m , B 0 , 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 &alpha;. 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 &beta;. A 0 and A m are both A-alleles, but organisms with the A 0 gene produce offspring with A 0 -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, B 0 and B m are both B-alleles, but organisms with the B 0 gene produce offspring with B 0 -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 A 0, Am, B 0 , 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)


 * {| class="wikitable"


 * align="center"|&alpha;
 * align="center"|0
 * align="center"|0
 * align="center"|0
 * align="center"|m&alpha;
 * align="center"|(1-3m)&alpha;
 * align="center"|m&beta;
 * align="center"|m&beta;
 * align="center"|0
 * align="center"|0
 * align="center"|&beta;
 * align="center"|0
 * align="center"|m&alpha;
 * align="center"|m&alpha;
 * align="center"|m&beta;
 * align="center"|(1-3m)&beta;
 * }
 * align="center"|m&alpha;
 * align="center"|m&beta;
 * align="center"|(1-3m)&beta;
 * }


 * $$\mathbf{w}=

\begin{bmatrix} \alpha & 0            &0       & 0      \\ m\alpha & (1-3m)\alpha & m\beta & m\beta \\ 0      & 0            & \beta  & 0      \\ m\alpha & m\alpha     & m\beta & (1-3m)\beta \end{bmatrix} $$

Mutators are at a disadvantage when the food supplies &alpha; and &beta;  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 &alpha; 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:



z_i = \left[0,1,0,1\right] $$

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:

n'_i = \sum_j w_{ji}n_j\, $$ which gives fitness:

w_i=\frac{n'_i}{n_i} $$ Since the individual mutability z i does not change, the average mutabilities will be:


 * $$z = \frac{\sum_i z_i n_i}{n}$$
 * $$z' = \frac{\sum_i z_i n'_i}{n'}$$

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:

n'_i = n_i\sum_j w_{ij}\, $$ which gives fitness:



w_i=\frac{n'_i}{n_i} = \sum_j w_{ij} $$

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

z'_j = \frac{\sum_i n_i z_i w_{ij} }{\sum_i n_i w_{ij}} $$ with global characters:


 * $$z = \frac{\sum_i z_i n_i}{n}$$
 * $$z' = \frac{\sum_i z_i n'_i}{n'}$$

with these definitions, the full Price equation now applies