Price equation

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. Examples of the Price equation have been constructed for various evolutionary cases. The Price equation also has applications in economics.

Statement

The Price equation shows that a change in the average amount z of a trait in a population from one generation to the next (Δz) is determined by the covariance between the amounts zi of the trait for subpopulations i and the fitnesses wi of the subpopulations, together with the expected change in the amount of the trait value due to fitness, namely E(wi Δzi):

$\Delta {z}={\frac {1}{w}}\operatorname {cov} (w_{i},z_{i})+{\frac {1}{w}}\operatorname {E} (w_{i}\,\Delta z_{i}).$ Here w is the average fitness over the population, 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 (wi) and the trait value (zi) 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(wi Δzi)) 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 (1972) referred to this as the "environment change" term, and denoted both terms using partial derivative notation (∂NS and ∂EC). This concept of environment includes interspecies and ecological effects. Price describes this as follows:

Fisher adopted the somewhat unusual point of view of regarding dominance and epistasis as being environment effects. For example, he writes (1941): ‘A change in the proportion of any pair of genes itself constitutes a change in the environment in which individuals of the species find themselves.’ Hence he regarded the natural selection effect on M as being limited to the additive or linear effects of changes in gene frequencies, while everything else - dominance, epistasis, population pressure, climate, and interactions with other species - he regarded as a matter of the environment.

— Price 1972, Fisher's 'fundamental theorem' made clear.

Proof

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 $z$ is 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 and the number of individuals in group $i$ in the next generation is represented by $n'_{i}$ . The so-called fitness $w_{i}$ of group $i$ is defined to be the ratio of the number of its individuals in the next generation to the number of its individuals in the previous generation. That is,

 $w_{i}={\frac {n_{i}'}{n_{i}}}$ $(4)\,$ Thus, saying that a group has "higher fitness" is equivalent to saying that its members produce more offspring per individual in the next generation. Similarly, $n'$ represents the total number of individuals across all groups, which can be expressed as:

$n'=\sum _{i}n'_{i}$ Furthermore, the average fitness of the population can be shown to be the growth rate of the population as a whole, as in:

 $w\;{\stackrel {\mathrm {def} }{=}}\;\operatorname {E} (w_{i})={\frac {1}{n}}\sum _{i}w_{i}n_{i}={\frac {1}{n}}\sum _{i}{\frac {n_{i}'}{n_{i}}}n_{i}={\frac {1}{n}}\sum _{i}n_{i}'={\frac {n'}{n}}$ $(5)$ Although the total population may grow, the proportion of individuals from a certain group may change. In particular, if one group has a higher fitness than another group, then the higher-fitness group will have a larger increase in representation in the next generation than the lower-fitness group. The average fitness represents how the population grows, and those groups with below-average fitness will tend to decline in proportion, while those groups with above-average fitness will tend to increase in proportion.

Along with the proportion of individuals in each group changing over time, the trait values within a single group may vary slightly from one generation to another (e.g., due to mutation). These two pressures together will cause the average value of the characteristic over the whole population to change over time. Assuming the value of $z'_{i}$ has changed by exactly the same amount for all members of the original group, in the new generation of the group, the average value $z'$ of the characteristic is:

 $z'={\frac {1}{n'}}\sum _{i}z'_{i}n_{i}'$ $(6)$ where zi are the (possibly new) values of the characteristic in group i. 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

 $\operatorname {cov} (w_{i},z_{i})+\operatorname {E} (w_{i}\,\Delta z_{i})=\left[\operatorname {E} (w_{i}z_{i})-wz\right]+\left[\operatorname {E} (w_{i}z'_{i})-\operatorname {E} (w_{i}z_{i})\right]=\operatorname {E} (w_{i}z'_{i})-wz$ $(9)$ Now, let's compute the first term in the equality above. From Equation (1), we know that:

$\operatorname {E} (w_{i}z'_{i})={\frac {1}{n}}\sum _{i}w_{i}z'_{i}n_{i}$ Substituting the definition of fitness, ${\textstyle w_{i}={\frac {n'_{i}}{n_{i}}}}$ (Equation (4)), we get:

 $\operatorname {E} (w_{i}z'_{i})={\frac {1}{n}}\sum _{i}{\frac {n'_{i}}{n_{i}}}z'_{i}n_{i}={\frac {1}{n}}\sum _{i}n'_{i}z'_{i}={\frac {n'}{n}}~{\frac {1}{n'}}\sum _{i}z'_{i}n'_{i}$ $(10)$ Next, substituting the definitions of average fitness (${\textstyle w={\frac {n'}{n}}}$ ) from Equation (5), and average child characteristics (${\textstyle z'}$ ) from Equation (6) 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 zi 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 vi is the fractional fitness: vi= wi/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

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{aligned}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{aligned}} The five 0-generation variables w, z, 〈wi  zi 〉, 〈w2i 〉, and 〈w2i zi 〉 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 (〈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.

Full Price equation

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

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 zi does not change, the average mutabilities will be:

{\begin{aligned}z&={\frac {1}{n}}\sum _{i}z_{i}n_{i}\\z'&={\frac {1}{n'}}\sum _{i}z_{i}n'_{i}\end{aligned}} 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{aligned}z&={\frac {1}{n}}\sum _{i}z_{i}n_{i}\\z'&={\frac {1}{n'}}\sum _{i}z_{i}n'_{i}\end{aligned}} with these definitions, the full Price equation now applies.

Criticism

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.