price equation

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

The Price equation shows that a change in the average amount $z$ of a trait in a population from one generation to the next ($\backslash Delta\; z$) is determined by the covariance between the amounts $z\_i$ of the trait for subpopulation $i$ and the fitnesses $w\_i$ of the subpopulations, together with the expected change in the amount of the trait value due to fitness, namely $\backslash mathrm\{E\}(w\_i\; \backslash Delta\; z\_i)$:

:$\backslash Delta\{z\}\; =\; \backslash frac\{1\}\{w\}\backslash operatorname\{cov\}(w\_i,\; z\_i)\; +\; \backslash frac\{1\}\{w\}\backslash operatorname\{E\}(w\_i\backslash ,\backslash Delta\; z\_i).$

Here $w$ is the average fitness over the population, and $\backslash operatorname\{E\}$ and $\backslash operatorname\{cov\}$ represent the population mean and covariance respectively. 'Fitness' $w$ is the ratio of the average number of offspring for the whole population per the number of adult individuals in the population, and $w\_i$ is that same ratio only for subpopulation $i$.

If the covariance between fitness ($w\_i$) and trait value ($z\_i$) is positive, the trait value is expected to rise on average across population $i$. If the covariance is negative, the characteristic is harmful, and its frequency is expected to drop.

The second term, $\backslash mathrm\{E\}(w\_i\; \backslash Delta\; z\_i)$, represents the portion of $\backslash Delta\; z$ due to all factors other than direct selection which 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:

{{Blockquote|text=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 {{mvar|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. |author=G.R. Price (1972) |source=Fisher's fundamental theorem made clear}}

Suppose we are given four equal-length lists of real numbersThe lists may in fact be members of any field (i.e. a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do $n\_i$, $z\_i$, $n\_i\text{'}$, $z\_i\text{'}$ from which we may define $w\_i=n\_i\text{'}/n\_i$. $n\_i$ and $z\_i$ will be called the parent population numbers and characteristics associated with each index i. Likewise $n\_i\text{'}$ and $z\_i\text{'}$ will be called the child population numbers and characteristics, and $w\_i\text{'}$ will be called the fitness associated with index i. (Equivalently, we could have been given $n\_i$, $z\_i$, $w\_i$, $z\_i\text{'}$ with $n\_i\text{'}=w\_i\; n\_i$.) Define the parent and child population totals:

:

and the probabilities (or frequencies):{{cite journal |last1=Frank |first1=Steven A. |date=1995 |title=George Price's Contributions to Evolutionary Genetics |url=https://www.researchgate.net/publication/15724182 |journal=J. Theor. Biol. |volume=175 |issue= 3|pages=373–388 |doi= 10.1006/jtbi.1995.0148|pmid=7475081 |bibcode=1995JThBi.175..373F |access-date=Mar 19, 2023}}

:

Note that these are of the form of probability mass functions in that $\backslash sum\_i\; q\_i\; =\; \backslash sum\_i\; q\_i\text{'}\; =\; 1$ and are in fact the probabilities that a random individual drawn from the parent or child population has a characteristic $z\_i$. Define the fitnesses:

:$w\_i\backslash ;\backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash ;n\_i\text{'}/n\_i$

The average of any list $x\_i$ is given by:

:$E(x\_i)=\backslash sum\_i\; q\_i\; x\_i$

so the average characteristics are defined as:

:

and the average fitness is:

:$w\backslash ;\backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash ;\backslash sum\_i\; q\_i\; w\_i$

A simple theorem can be proved:

$q\_i\; w\_i\; =\; \backslash left(\backslash frac\{n\_i\}\{n\}\backslash right)\backslash left(\backslash frac\{n\_i\text{'}\}\{n\_i\}\backslash right)\; =\; \backslash left(\backslash frac\{n\_i\text{'}\}\{n\text{'}\}\backslash right)\; \backslash left(\backslash frac\{n\text{'}\}\{n\}\backslash right)=q\_i\text{'}\backslash left(\backslash frac\{n\text{'}\}\{n\}\backslash right)$

so that:

:$w=\backslash frac\{n\text{'}\}\{n\}\backslash sum\_i\; q\_i\text{'}\; =\; \backslash frac\{n\text{'}\}\{n\}$

and

:$q\_i\; w\_i\; =\; w\backslash ,q\_i\text{'}$

The covariance of $w\_i$ and $z\_i$ is defined by:

:$\backslash operatorname\{cov\}(w\_i,z\_i)\backslash ;\backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash ;E(w\_i\; z\_i)-E(w\_i)E(z\_i)\; =\; \backslash sum\_i\; q\_i\; w\_i\; z\_i\; -\; w\; z$

Defining $\backslash Delta\; z\_i\; \backslash ;\backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash ;\; z\_i\text{'}-z\_i$, the expectation value of $w\_i\; \backslash Delta\; z\_i$ is

:$E(w\_i\; \backslash Delta\; z\_i)\; =\; \backslash sum\; q\_i\; w\_i\; (z\_i\text{'}-z\_i)\; =\; \backslash sum\_i\; q\_i\; w\_i\; z\_i\text{'}\; -\; \backslash sum\_i\; q\_i\; w\_i\; z\_i$

The sum of the two terms is:

:$\backslash operatorname\{cov\}(w\_i,z\_i)+E(w\_i\; \backslash Delta\; z\_i)\; =\; \backslash sum\_i\; q\_i\; w\_i\; z\_i\; -\; w\; z\; +\; \backslash sum\_i\; q\_i\; w\_i\; z\_i\text{'}\; -\; \backslash sum\_i\; q\_i\; w\_i\; z\_i\; =\; \backslash sum\_i\; q\_i\; w\_i\; z\_i\text{'}\; -\; w\; z$

Using the above mentioned simple theorem, the sum becomes

:$\backslash operatorname\{cov\}(w\_i,z\_i)+E(w\_i\; \backslash Delta\; z\_i)\; =\; w\backslash sum\_i\; q\_i\text{'}\; z\_i\text{'}\; -\; w\; z\; =\; w\; z\text{'}-wz\; =\; w\backslash Delta\; z$

where

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

Consider a set of groups with $i\; =\; 1,...,n$ that are characterized by a particular trait, denoted by $x\_\{i\}$. The number $n\_\{i\}$ of individuals belonging to group $i$ experiences exponential growth:$$\{dn\_\{i\}\backslash over\{dt\}\}\; =\; f\_\{i\}n\_\{i\}$$where $f\_\{i\}$ corresponds to the fitness of the group. We want to derive an equation describing the time-evolution of the expected value of the trait:$$\backslash mathbb\{E\}(x)\; =\; \backslash sum\_\{i\}p\_\{i\}x\_\{i\}\; \backslash equiv\; \backslash mu,\; \backslash quad\; p\_\{i\}\; =\; \{n\_\{i\}\backslash over\{\backslash sum\_\{i\}n\_\{i\}\}\}$$Based on the chain rule, we may derive an ordinary differential equation:$$\backslash begin\{aligned\}$$

{d\mu\over{dt}} &= \sum_{i} {\partial \mu\over{\partial p_{i}}}{dp_{i}\over{dt}} + \sum_{i} {\partial \mu\over{\partial x_{i}}}{dx_{i}\over{dt}} \\

&= \sum_{i} x_{i}{dp_{i}\over{dt}} + \sum_{i} p_{i}{dx_{i}\over{dt}} \\

&= \sum_{i} x_{i}{dp_{i}\over{dt}} + \mathbb{E}\left( {dx\over{dt}} \right)

\end{aligned}A further application of the chain rule for $dp\_\{i\}/dt$ gives us:Summing up the components gives us that:$$\backslash begin\{aligned\}$$

{dp_{i}\over{dt}} &= p_{i}\left(f_{i} - \sum_{j}p_{j}f_{j}\right) \\

&= p_{i}\left[f_{i} - \mathbb{E}(f)\right]

\end{aligned}

which is also known as the replicator equation. Now, note that:

$$\backslash begin\{aligned\}$$

\sum_{i} x_{i}{dp_{i}\over{dt}} &= \sum_{i} p_{i}x_{i}\left[f_{i} - \mathbb{E}(f)\right] \\

&= \mathbb{E}\left\{ x_{i}\left[f_{i}-\mathbb{E}(f)\right] \right\} \\

&= \text{Cov}(x,f)

\end{aligned}Therefore, putting all of these components together, we arrive at the continuous-time Price equation:$$\{d\backslash over\{dt\}\}\backslash mathbb\{E\}(x)\; =\; \backslash underbrace\{\backslash text\{Cov\}(x,f)\}\_\{\backslash text\{Selection\; effect\}\}\; +\; \backslash underbrace\{\backslash mathbb\{E\}(\backslash dot\{x\})\}\_\{\backslash text\{Dynamic\; effect\}\}$$

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:

:$w\backslash ,\backslash Delta\; z\; =\; \backslash operatorname\{cov\}\backslash left(w\_i,\; z\_i\backslash right)$

which can be restated as:

:$\backslash Delta\; z\; =\; \backslash operatorname\{cov\}\backslash left(v\_i,\; z\_i\backslash right)$

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."

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

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:

:$w(z\text{'}\; -\; z)\; =\; \backslash langle\; w\_i\; z\_i\; \backslash rangle\; -\; wz$

For the second generation:

:$w\text{'}(z\text{'}\text{'}\; -\; z\text{'})\; =\; \backslash langle\; w\text{'}\_i\; z\text{'}\_i\; \backslash rangle\; -\; w\text{'}z\text{'}$

The simple Price equation for $z$ only gives us the value of $z\text{'}$ for the first generation, but does not give us the value of $w\text{'}$ and $\backslash langle\; w\_iz\_i\backslash rangle$, which are needed to calculate $z\text{'}\text{'}$ for the second generation. The variables $w\_i$ and $\backslash langle\; w\_iz\_i\backslash rangle$ can both be thought of as characteristics of the first generation, so the Price equation can be used to calculate them as well:

:$\backslash begin\{align\}$

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 $w$, $z$, $\backslash langle\; w\_iz\_i\backslash rangle$, $\backslash langle\; w\_i^2\backslash rangle$, and $\backslash langle\; w\_i^2z\_i$ must be known before proceeding to calculate the three first generation variables $w\text{'}$, $z\text{'}$, and $\backslash langle\; w\text{'}\_iz\text{'}\_i\backslash rangle$, which are needed to calculate $z\text{'}\text{'}$ 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 $\backslash langle\; w\_i^n\backslash rangle$ and $\backslash langle\; w\_i^nz\_i\backslash rangle$ 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.

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.

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:

:$n\text{'}\_i\; =\; \backslash sum\_j\; w\_\{ji\}n\_j\backslash ,$

which gives fitness:

:$w\_i\; =\; \backslash frac\{n\text{'}\_i\}\{n\_i\}$

Since the individual mutability $z\_i$ does not change, the average mutabilities will be:

:$\backslash begin\{align\}$

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.

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:

:$n\text{'}\_i\; =\; n\_i\backslash sum\_j\; w\_\{ij\}\backslash ,$

which gives fitness:

:$w\_i\; =\; \backslash frac\{n\text{'}\_i\}\{n\_i\}\; =\; \backslash sum\_j\; w\_\{ij\}$

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

:$z\text{'}\_j\; =\; \backslash frac\{\backslash sum\_i\; n\_i\; z\_i\; w\_\{ij\}\; \}\{\backslash sum\_i\; n\_i\; w\_\{ij\}\}$

with global characters:

:$\backslash begin\{align\}$

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.

The use of the change in average characteristic ($z\text{'}-z$) 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. For example, if we have $z\_i=(1,2,3)$, $n\_i=(1,1,1)$, and $w\_i=(1,4,1)$, then for the child population, $n\_i\text{'}=(1,4,1)$ showing that the peak fitness at $w\_2=4$ is in fact fractionally increasing the population of individuals with $z\_i=2$. However, the average characteristics are z=2 and z'=2 so that $\backslash Delta\; z=0$. The covariance $\backslash mathrm\{cov\}(z\_i,w\_i)$ is also zero. The simple Price equation is required here, and it yields 0=0. In other words, it yields no information regarding the progress of evolution in this system.

A critical discussion of the use of the Price equation can be found in van Veelen (2005),{{cite journal |last1=van Veelen |date=December 2005 |first1=M. |title=On the use of the Price equation |journal=Journal of Theoretical Biology |volume=237 |issue=4 |pages=412–426 |doi=10.1016/j.jtbi.2005.04.026 |pmid=15953618|bibcode=2005JThBi.237..412V }} van Veelen et al. (2012),{{cite journal |last1=van Veelen |first1=M. |last2=García |first2=J. |last3=Sabelis |first3=M.W. |last4=Egas |first4=M. |date=April 2012 |title=Group selection and inclusive fitness are not equivalent; the Price equation vs. models and statistics |journal=Journal of Theoretical Biology |volume=299 |pages=64–80 |doi=10.1016/j.jtbi.2011.07.025 |pmid=21839750|bibcode=2012JThBi.299...64V }} and van Veelen (2020).{{cite journal |last1=van Veelen |date=March 2020 |first1=M. |title=The problem with the Price equation |journal=Philosophical Transactions of the Royal Society B |volume=375 |issue=1797 |pages=1–13 |doi=10.1098/rstb.2019.0355 |pmid=32146887|pmc=7133513 |doi-access=free }} Frank (2012) discusses the criticism in van Veelen et al. (2012).{{cite journal |last=Frank |first=S.A. |year=2012 |title=Natural Selection IV: The Price equation |journal=Journal of Evolutionary Biology |volume=25 |issue=6 |pages=1002–1019 |arxiv=1204.1515 |doi=10.1111/j.1420-9101.2012.02498.x |pmid=22487312 |pmc=3354028}}

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.

• The breeder's equation, which is a special case of the Price equation.

{{reflist|25em|refs=

{{cite journal |last=Price |first=G.R. |author-link=George R. Price |title=Fisher's "fundamental theorem" made clear |journal=Annals of Human Genetics |year=1972 |volume=36 |issue=2 |pages=129–140 |doi=10.1111/j.1469-1809.1972.tb00764.x |pmid=4656569|s2cid=20757537 }}

}}

{{refbegin}}

• {{cite journal |last=Frank |first=S.A. |year=1995 |title=George Price's contributions to Evolutionary Genetics |journal=Journal of Theoretical Biology |volume=175 |issue=3 |pages=373–388 |url=http://stevefrank.org/reprints-pdf/95JTB-Price.pdf |doi=10.1006/jtbi.1995.0148 |pmid=7475081 |bibcode=1995JThBi.175..373F |citeseerx=10.1.1.136.7803}}
• {{cite journal |last=Frank |first=S.A. |year=1997 |title=The Price equation, Fisher's fundamental theorem, kin selection, and causal analysis |journal=Evolution |volume=51 |issue=6 |pages=1712–1729 |url=http://stevefrank.org/reprints-pdf/97Evol-Causal.pdf |doi=10.2307/2410995 |pmid=28565119 |jstor=2410995}}
• {{cite journal |last=Gardner |first=A. |year=2008 |title=The Price equation |journal=Curr. Biol. |volume=18 |issue=5 |pages=R198–R202 |doi=10.1016/j.cub.2008.01.005 |pmid=18334191 |s2cid=1169263 |doi-access=free |bibcode=2008CBio...18.R198G }}
• {{cite journal |last=Grafen |first=A. |year=2000 |title=Developments of the Price equation and natural selection under uncertainty |journal=Proceedings of the Royal Society B |volume=267 |issue=1449 |pages=1223–1227 |url=http://users.ox.ac.uk/~grafen/cv/PriceUnc.pdf |pmid=10902688 |pmc=1690660 |doi=10.1098/rspb.2000.1131}}
• {{cite book |last=Harman |first=Oren |year=2010 |title=The Price of Altruism: George Price and the search for the origins of kindness |publisher=Bodley Head |isbn=978-1-84792-062-1}}
• {{cite book |last=Langdon |first=W.B. |chapter=8.1 Evolution of GP populations: Price's selection and covariance theorem |title=Genetic Programming and Data Structures |url=https://archive.org/details/geneticprogrammi00lang |url-access=limited |year=1998 |pages=[https://archive.org/details/geneticprogrammi00lang/page/n183 167]–208|publisher=Springer |isbn=9780792381358 }}
• {{cite journal |last=Price |first=G.R. |author-link=George R. Price |year=1970 |title=Selection and covariance |journal=Nature |volume=227 |issue=5257 |pages=520–521 |pmid=5428476 |url=http://www0.cs.ucl.ac.uk/staff/ucacbbl/ftp/papers/price_nature.pdf |doi=10.1038/227520a0 |bibcode=1970Natur.227..520P|s2cid=4264723 }}
• {{cite journal |last=Price |first=G.R. |author-link=George R. Price |year=1972 |title=Extension of covariance selection mathematics |journal=Annals of Human Genetics |volume=35 |issue=4 |pages=485–490 |pmid=5073694 |doi=10.1111/j.1469-1809.1957.tb01874.x|s2cid=37828617 }}
• {{cite journal |last1=van Veelen |first1=Matthijs |first2=Julián |last2=García |first3=Maurice W. |last3=Sabelis |first4=Martijn |last4=Egas |name-list-style=amp |year=2010 |title=Call for a return to rigour in models |department=Correspondence |journal=Nature |volume=467 |issue=7316 |pages=661 |bibcode=2010Natur.467..661V |doi=10.1038/467661d |pmid=20930826|doi-access=free }}
• {{cite book |last=Day |first=T. |title=Disease Evolution |chapter=Insights from Price's equation into evolutionary epidemiology |series=DIMACS Series in Discrete Mathematics and Theoretical Computer Science |year=2006 |volume=71 |pages=23–43 |doi=10.1090/dimacs/071/02 |isbn=9780821837535}}
• {{cite web |url=http://www.evolutionandgames.com/price |title=How to quit the Price equation: An online self-help tutorial}}
• {{cite episode |series=Radiolab |airdate=14 December 2011 |title=The Good Show |season=9 |number=1 |url=http://www.radiolab.org/2010/dec/14/ |station=WNYC |location=New York}}
• {{cite arXiv |last1=Markovitch |last2=Witkowski |last3=Virgo |year=2018 |title=Chemical heredity as group selection at the molecular level |eprint=1802.08024 |class=q-bio.PE}}

{{refend}}

{{Population genetics}}

Category:Equations

Category:Evolutionary dynamics

Category:Evolutionary biology

Category:Population genetics