Price equation

"Price's theorem" redirects here. For the theorem in general relativity, see Richard H. Price.

In the theory of evolution and natural selection, the Price equation (also known as Price's equation or Price's theorem) describes how a trait or gene changes in frequency over time. The equation uses a covariance between a trait and fitness to give a mathematical description of evolution and natural selection. It provides a way to understand the effects that gene transmission and natural selection have on the proportion of genes within each new generation of a population. The Price equation was derived by George R. Price, working in London to re-derive W.D. Hamilton's work on kin selection. The Price equation also has applications in economics. Examples of the Price equation can be found here: Price equation examples.

Statement

File:Example of Price equation for a trait under positive selection.png

Example of Price equation for a trait under positive selection

The Price equation shows that a change in trait z (Δz) is determined by the covariance between each value of z (z_i) and fitness (w_i) and the expected change in the trait value due to fitness (w_i), (E(w_i Δz_i)):

$\Delta{z} = \frac{1}{w}\operatorname{cov}(w_i, z_i) + \frac{1}{w}\operatorname{E}(w_i\,\Delta z_i),$

where w is the mean fitness, and E and cov represent the population mean and covariance respectively.

The covariance term captures the effects of natural selection; if the covariance between fitness (w_i) and the trait value (z_i) is positive then the trait value is predicted to increase on average over population i. If the covariance is negative then the trait is deleterious and is predicted to decrease in frequency.

The second term (E(w_i Δz_i)) is more nuanced and represents factors other than direct selection that can affect trait evolution. This term can encompass genetic drift, mutation bias, or meiotic drive. Additionally, this term can encompass the effects of multi-level selection or group selection.

Price equation and group selection

The Price equation given above examines how the trait value of individuals (z_i), denoted by the subscript i, relates to the fitness of that individual (w_i).The first step in adapting the Price equation to group selection is to change the subscript on each term to reflect the average trait value (z_g), and corresponding mean fitness (w_g) of each groups (g), as opposed to each individual:

$\Delta{z} = \frac{1}{w}\operatorname{cov}(w_g, z_g) + \frac{1}{w}\operatorname{E}(w_g\,\Delta z_g)$

Giving the estimated change in a trait between groups. To incorporate changes within groups, set Δz_g to individual form of the Price equation giving:

$\Delta{z} = \frac{1}{w}\operatorname{cov}(w_g, z_g) + \frac{1}{w}\operatorname{E}(w_g(\,\frac{1}{w}\operatorname{cov}(w_{g,i}, z_{g,i}) + \frac{1}{w}\operatorname{E}(w_{g,i},\Delta z_{g,i})))$

Using this formulation, the Price equation can yield insights into the evolution of altruistic traits. By definition, altruistic traits increase group fitness while decreasing individual fitness. Therefore the covariance between the group average of an altruistic trait (z_g) and group average fitness (w_g) is positive (var(w_g,z_g)>0), and the covariance between the individual investment in an altruistic trait and that individuals fitness is negative (cov(w_i,z_i)<0). Therefore, the Price equation shows that high between group variance (var(z_g)) coupled with low within group variance (var(w_gi,z_gi)) is required for altruistic traits to spread due to natural selection.

Proof of the Price equation

Suppose there is a population of $n$ individuals over which the amount of a particular characteristic varies. Those $n$ individuals can be grouped by the amount of the characteristic that each displays. There can be as few as just one group of all $n$ individuals (consisting of a single shared value of the characteristic) and as many as $n$ groups of one individual each (consisting of $n$ distinct values 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 $w_i n_i$ 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 average amount of the characteristic displayed by the offspring from group $i$. Denote the amount of change in characteristic in group $i$ by $\Delta z_i$ defined by

$\Delta{z_i} \;\stackrel{\mathrm{def}}{=}\; z_i' - z_i$

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 $\Delta{z}$. That is,

$\Delta{z} \;\stackrel{\mathrm{def}}{=}\; z' - z$

Note that this is not the average value of $\Delta{z_i}$ (as it is possible that $n_i \neq n'_i$). Also take $w$ to be the average fitness of this population. The Price equation states:

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

where the functions $\operatorname{E}$ and $\operatorname{cov}$ are respectively defined in Equations (1) and (2) below and are equivalent to the traditional definitions of sample mean and covariance; however, they are not meant to be statistical estimates of characteristics of a population. In particular, the Price equation is a deterministic difference equation that models the trajectory of the actual mean value of a characteristic along the flow of an actual population of individuals. Assuming that the mean fitness $w$ is not zero, it is often useful to write it as

$\Delta{z} = \frac{1}{w}\operatorname{cov}(w_i, z_i) + \frac{1}{w}\operatorname{E}(w_i\,\Delta z_i)$

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

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 mean of the $x_i$ values is:

$\operatorname{E}(x_i) \;\stackrel{\mathrm{def}}{=}\; \frac{1}{\sum_i n_i}\sum_i x_i n_i$

$(1)$

• The covariance between the $x_i$ and $y_i$ values is:

$\operatorname{cov}(x_i, y_i) \;\stackrel{\mathrm{def}}{=}\; \frac{1}{\sum_i n_i}\sum_i n_i[x_i-\operatorname{E}(x_i)][y_i-\operatorname{E}(y_i)] = \operatorname{E}(x_i y_i) - \operatorname{E}(x_i)\operatorname{E}(y_i)$

$(2)$

The notation $\langle x_i \rangle = \operatorname{E}(x_i)$ 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:

$n = \sum_i n_i$

The average value of the characteristic is $z$ defined as:

$z \;\stackrel{\mathrm{def}}{=}\; \operatorname{E}(z_i) = \frac{1}{n}\sum_i z_i n_i$

$(3)$

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 total number of children is n' where:

$n' = \sum_i n'_i$

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

$w \;\stackrel{\mathrm{def}}{=}\; \operatorname{E}(w_i) = \frac{1}{n}\sum_i w_i n_i = \frac{1}{n}\sum_i \frac{n_i'}{n_i} n_i = \frac{1}{n}\sum_i n_i' = \frac{n'}{n}$

$(5)$

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

$z' = \frac{1}{n'}\sum_i z'_i n_i'$

$(6)$

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

$\operatorname{cov}(w_i, z_i) = \operatorname{E}(w_i z_i) - wz$

$(7)$

Call the change in characteristic value from parent to child populations $\Delta z_i$ so that $\Delta z_i = z'_i - z_i$. As seen in Equation (1), the expected value operator $\operatorname{E}$ is linear, so

$\operatorname{E}(w_i\,\Delta z_i) = \operatorname{E}(w_i z'_i) - \operatorname{E}(w_i z_i)$

$(8)$

Combining Equations (7) and (8) leads to

$\operatorname{cov}(w_i, z_i) + \operatorname{E}(w_i\,\Delta z_i) = \left[\operatorname{E}(w_i z_i) - wz\right] + \left[\operatorname{E}(w_iz'_i) - \operatorname{E}(w_i z_i)\right] = \operatorname{E}(w_i z'_i) - wz$

$(9)$

Now, let's compute the first term in the equality above. From Equation (1), we know that:

$\operatorname{E}(w_iz'_i) = \frac{1}{n}\sum_i w_i z'_i n_i$

Substituting the definition of fitness, ${\textstyle w_i=\frac{n'_i}{n_i}}$ (Equation (4)), we get:

$\operatorname{E}(w_i z'_i) = \frac{1}{n}\sum_i\frac{n'_i}{n_i}z'_in_i = \frac{1}{n}\sum_i n'_i z'_i = \frac{n'}{n}~\frac{1}{n'}\sum_i z'_i n'_i$

$(10)$

Next, substituting the definitions of average fitness (${\textstyle w=\frac{n'}{n}}$) from Equation (9), and average child characteristics (${\textstyle z'}$) from Equation (10) gives the Price Equation:

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

Simple Price equation

When the characteristic values 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:

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

which can be restated as:

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

where v_i is the fractional fitness: v_i= w_i/w.

This simple Price equation can be proven using the definition in Equation (2) above. It makes this fundamental 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."

Applications of the Price equation

The Price equation can describe any system that changes over time but is most often applied in evolutionary biology. The Evolution of sight provides an example of simple directional selection. The Evolution of sickle cell anemia shows how a [heterozygote] advantage can affect trait evolution. The Price equation can also be applied to population context dependent traits such as the evolution of sex ratios. Additionally, the Price equation is flexible enough to model second order traits such as the evolution of mutability. The price equation also provides an extension to Founder effect which shows change in population traits in different settlements

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:

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

For the second generation:

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

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

$\begin{align} w(w' - w) &= \langle w_i^2\rangle - w^2 \\ w\left(\langle w'_i z'_i\rangle - \langle w_i z_i\rangle\right) &= \langle w_i ^2 z_i\rangle - w\langle w_i z_i\rangle \end{align}$

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.

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.

Genotype fitness

We 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:

$\begin{align} z &= \frac{1}{n}\sum_i z_i n_i \\ z' &= \frac{1}{n'}\sum_i z_i n'_i \end{align}$

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:

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

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:

$\begin{align} z &= \frac{1}{n}\sum_i z_i n_i \\ z' &= \frac{1}{n'}\sum_i z_i n'_i \end{align}$

with these definitions, the full Price equation now applies.

Criticism of the use of the Price equation

The use of the change in average characteristic (z'-z) per generation as a measure of evolutionary progress is not always appropriate. There may be cases where the average remains unchanged (and the covariance between fitness and characteristic is zero) while evolution is nevertheless in progress.

A critical discussion of the use of the Price equation can be found in van Veelen (2005) "On the use of the Price equation" and van Veelen et al. (2012) "Group selection and inclusive fitness are not equivalent; the Price equation vs. models and statistics". A discussion of this criticism can be found in Frank (2012)

Cultural references

Price's equation features in the plot and title of the 2008 thriller film WΔZ.

The Price equation also features in posters in the computer game BioShock 2, in which a consumer of a "Brain Boost" tonic is seen deriving the Price equation while simultaneously reading a book. The game is set in the 1950s, substantially before Price's work.

See also

• Price equation examples

References

In-line references

General references