From: Alan McGowen <alanm@igc.apc.org>
Newsgroups: sci.environment
Date: 15 Nov 92 23:04 PST
Subject: ECO CENTRAL 21
Nf-ID: #N:cdp:1466601912:000:6505
Nf-From: cdp.UUCP!alanm    Nov 15 23:04:00 1992


Note: ECOCENTRAL REDIVIVUS -- ECO CENTRALS hereby resume their 
irregular publication with number 21.  

ECO CENTRAL 21

                    Hardy-Weinberg Equilibrium

In physics, kinematics precedes dynamics in that it is necessary 
to be able to measure positions, momenta, moments of inertia and 
other instantaneous properties before it is possible to talk 
about how they evolve in time. Evolutionary biology is the study 
of the effects of genetic changes over time, and this genetic 
"dynamics" must also be founded on a stationary "kinematics". In 
biology stationary states are equilibria of yet deeper processes: 
all living systems are driven by the inexorable laws of 
thermodynamics towards nonliving states, and any stable property 
which they exhibit requires an explanation in terms of biological 
equilibrium processes. 

Sometimes in science a "kinematic" reference frame is important 
for explanation even though no ideal example may exist in nature. 
In physics, the inertial reference frame is an enormously useful 
approximation, despite the fact that with finer measurement, 
every reference frame available is found to be accelerated: the 
earth spins and orbits the sun, the sun orbits the galactic 
center, and the galaxy apparently participates in some grand 
filamentary flow. No actual frame is unaccelerated, yet the ideal 
inertial frame is the basis of the successful development of 
classical mechanics. Precedent exists, therefore, for a reference 
frame which is mathematically useful, and which obtains 
approximately in large numbers of cases, but which may not hold 
with absolute precision in any real case. 

Hardy-Weinberg equilibrium is such a reference frame. It is the 
state of evolutionary stasis --  what happens, with certain 
idealizations, when natural selection can be ignored. It is 
therefore the starting point for population genetics.

We consider a population of sexually reproducing organisms. These 
organisms are *diploid*: each one has, at some stage of its life, 
chromosome pairs and hence *two* potentially different variants 
of the gene at each chromosomal locus. Different variants of the 
same gene are called *alleles*. The sex cells or *gametes* 
produced by these organisms have only one member of each  
chromosome pair of the diploid stage, and thus only half of its 
genetic complement. This is called a *haploid* condition. The 
haploid gametes unite to form diploid *zygotes* in a process 
called "mixis", which is bio-speak for "do it". "Mixis" is 
the act of sex.

In many organisms the act of sex -- mixis -- is not very 
intimate. When corals do it they all let go -- males and females 
-- at the same time. Male and female gametes float up in the 
water column and form a slick on the surface, uniting in mad 
abandon to form zygotes. This is a pretty close approximation to 
a completely random mating of all the gametes in a population -- 
a situation called *panmixis*. A population which regularly 
mates in such a random manner is called *panmictic*. Hardy-
Weinberg equilibrium applies to panmictic populations. It is the 
"inertial reference frame" of population genetics. 

We consider a single gene locus within a panmictic population, 
and suppose that there are two different alleles, called A and a, 
which can occur at that locus. Each diploid organism has either 
AA, Aa, or aa at its two loci (we identify Aa and aA). Each of 
these *genotypes* has a certain frequency in the population: let 
the frequency of AA be FAA, that of Aa be FAa and that of aa be 
Faa. Clearly,

1) FAA + FAa + Faa = 1

since those are the only possibilities. 

Further let the total frequency of all the A alleles in the 
population be p (= FAA + 1/2 FAa) , and hence the frequency of 
all the a alleles is q = 1-p (= Faa + 1/2 FAa).

The frequencies FAA, FAa, and Faa are those prior to a panmictic 
mating event. We let them have any value subject to the 
constraint 1). Now let mating occur, forming a gamete pool of As 
and as. During mating the following happens:

Each AA contributes an A with probability 1

Each Aa contributes an A with probability 1/2 and an a with 
probability 1/2.

Each aa contributes an a with probability 1.

The result is that 

P(A) = P(A | AA)FAA + P(A | Aa)FAa = FAA + 1/2 FAa = p
P(a) = P(a | Aa)FAa + P(a | aa)Faa = 1/2 FAa + Faa = q 

During mating, the probability that a given zygote forms with two 
As is

P(AA) = FAA (next generation) = p^2

Likewise

P(Aa) = FAa (next generation) = 2 pq 

P(aa) = Faa (next generation) = q^2


This means that the original distribution FAA, FAa, Faa is 
completely wiped out in a single generation, no matter what it 
was, leaving the next generation at the *Hardy-Weinberg 
equilibrium* of p^2: 2pq : q^2. If we see a population  
with this distribution in its diploid stage it is at Hardy-
Weinberg equilibrium. This is what we expect to see if perfect 
panmixis -- random mating -- occurs, and also (and this is 
crucial) IF THERE IS NO DIFFERENTIAL SURVIVAL of the different 
genotypes between the formation of zygotes and their next mating 
stage, nor of the different gametes before union, nor any 
DIFFERENTIAL FERTILITY in the production of gametes. These are 
the factors of natural selection.

In the absence of natural selection then, a panmictic population 
should be at Hardy-Weinberg equilibrium. Conversely, if a 
population is *not* at Hardy-Weinberg equilibrium, we can 
conclude that it is either not panmictic -- some forms of 
selection are at work in the union of gametes -- or else 
differential survival of genotypes or differential survival or 
production of gametes is occurring. In either case, natural 
selection of some kind is at work: failure of Hardy-Weinberg 
equilibrium is evidence of natural selection.

In practice, widely-distributed populations are typically not at 
perfect Hardy-Weinberg equilibrium. Semi-disjoint populations 
which are each at Hardy-Weinberg equilibrium approach joint 
equilibrium at a rate determined by the *gene flow* between them. 
Failure of Hardy-Weinberg equilibrium between disjoint 
populations is common and indicates *geographical variation*. 
Failure of equilibrium within contiguous populations can also 
occur, for example it can arise from the evolution of different 
mating types. The latter event -- which often occurs in 
populations which are geographically contiguous but nevertheless 
are becoming ecologically isolated -- can presage speciation.

-----------
Alan McGowen

Newsgroups: sci.environment