Download Hardy-Weinberg Equation

Hardy-Weinberg Equation
begins
p
Hi, I’m p and I represent the
frequency of the Dominant
allele
If you don’t remember the genetic terminology, here is a little reminder.
Individuals like humans, are diploid which means that they have two
copies of each chromosome.
Even though humans have a total of 46 chromosomes, there are
actually 23 chromosome pairs.
Each chromosome has many alleles that represent genetics for a
particular trait. Ex blonde hair, blue eyes
Since you get two copies of each chromosome, all the alleles are
also paired. When they are paired they determine how you
look– your phenotype. You can get combinations as follows:
TWO DOMINANT ALLELES = DOMINANT PHENOTYPE
A DOMINANT AND A RECESSIVE = DOMINANT PHENOTYPE
TWO RECESSIVE ALLELES= RECESSIVE PHENOTYPE
In a population, there are many people with many different traits, but if only one
trait is considered, it becomes much more simple.
Given one trait, in a simple model, there are only two possible alleles, either
dominant or recessive
Previously, you learned about punnett squares and how to calculate
probabilities for phenotypes given a cross between two individuals.
Ex Brown hair is dominant -B --- Blonde hair is recessive =b
Given the cross Bb x Bb determine the punnett square
B
BB
Bb
B
b
Bb
bb
b
Here the phenotypes
are ¾ brown hair and
¼ blonde hair.
RECALL THE
MENDELIAN 3:1
RATIO
Now consider that rather than this being one cross it represents the crosses for a
large number of individuals in the population.
IN THIS POPULATION:
The people with homozygous brown hair are BB
The people with heterozygous brown hair are Bb
= pp
=pq
The people with homozygous blonde hair are bb
=qq
From the previous punnett square, you might say that 75 are brown hair and 25
are blonde hair. This is the 3:1 ratio and 75+25 = 100 total population.
p
Hi I’m
q Meet my partner
I’m back to substitute in for the dominant allele
He is here to substitute in for the recessive allele
p
pp
pq
p
q
pq
qq
q
Given that 75 people are
dominant brown hair
pp+pq+pq=75
REMEMBER this includes
both heterozygotes and
homozygotes
Given that 25 people
are recessive blonde
hair qq=25
A DINNER SCIENCE DISCUSSION 1908
Reginald Punnett -- inventor or
punnett square analysis for genetics
Godfrey Hardy – eminent
mathematician
I say Godfrey old boy, do you think there could
be a way of determining the occurrence of
certain alleles in the population utilizing my
punnett squares ? In addition, could it help
determine if alleles change in a population?
Godfrey wrote the following on his napkin.
p+q=1
This means that together the total brown and blonde alleles are 100%
P2 + 2pq + q2 =1 This is the combination of the algebra from the previous slide
where pp+pq+pq+qq =1 and 1 is 100%
REMEMBER pp is mathematically pxp or p2
There are 2 pxq values so this is shortened to 2pq
And finally qq the recessive allele is mathematically q2
The Birth of the Hardy-Weinberg Principle
The allele frequencies of a
population will not change from
generation to generation, as long
as the following conditions are
met:
• the population is very large
• mating opportunities are equal
• no mutations occur
• no migration occurs
• no natural selection occurs—all
individuals have an equal chance
of reproductive
success
Do you remember the
equations yet? Here is a
reminder below
p+q=1
P2 + 2pq + q2 =1
Given the hair colour scenario, populations
essentially will not change in their genetic
composition if they obey Hardy-Weinberg
In this picture, the percentage of blonde haired
and brown haired individuals will remain
constant even though people will freely marry
(called random mating) and have children
within this population. In addition, the
population may expand, but the percentages
will remain the same.
The reason for this is the genetics. A specific
gene is found in a certain number of people. This
is called the gene frequency
p
Hi I’m
q
Hi, I’m
And I’m here to help you. I represent the gene
frequency of the Dominant allele. I, however am not as
useful as my friend q for solving Hardy Weinberg
problems . Be really careful if you use me!
And you should always try to determine my value first.
Even if they give you p, use the equation p+q=1 to
determine me. . I represent the gene frequency of the
Recessive allele
Now we are ready to start solving some introductory H-W
word problems. Follow these important tips!
q
p
You are given q if you see the words
recessive alleles or genes or if you are given
the recessive gamete allele or gene.
You are given p if you see the words dominant
alleles or genes or if you are given the
dominant gamete allele or gene
Lets try a sample simple problem
In a population blonde hair is recessive to brown. The gene frequency of
the blonde allele is 0.2 Determine the frequency of the brown allele.
TRY IT
ANSWER Since you are given q=0.2 you simply utilize the
simple formula p+q=1 to determine that p =0.08
Now, extending on this same question, you could be asked to determine
the percentage of individuals who are actually blonde haired in the
population. You may think you already know, but you are wrong if you
say 0.2 because that is a gene frequency not a number of individuals!
In order to determine the number of individuals, you must utilize the
variables from the second equation. P2 + 2pq + q2 =1
P2 + 2pq is the decimal percent of number of brown hair people DOMINANT
q2 is the decimal percent of the number of blonde hair people (RECESSIVE)
THIS IS WHAT WE WERE ASKED FOR IN THE QUESTION!
TRY IT
ANSWER
Since q=0.2 q2 =0.04 Now multiply by 100% and you have 4% blondes
Now try this challenging? question. If the percentage of blondes was
determined to be 4% what is the percentage of brunettes in our population?
TRY IT
ANSWER Well the answer is very simple if you think clearly and do not get
caught up in the math. If 4% are blondes all the others are brunettes so it is
100%-4% = 96%
Now there is the hard way of determining this answer and eventually you
will have to know how to utilize this aspect of the formula so here is an
alternate solution:P2 + 2pq + q2 =1
P2 + 2pq = decimal percent of brunettes
Substituting in the unknowns p= 0.8 q=0.2
0.64 + 2(.16) = 0.96 x100% = 96% brunettes
Therefore the percentage of brunettes in the population is 96%
Ok so good so far. Now lets try to interpret some of the other information, once
again without getting caught up in the math. What percentage of brunettes in the
population are heterozygous brown hair?
TRY IT
ANSWER The math is already done for you from the alternate solution
above. You , of course, would have to do this if you did the shortcut.
2pq = 2(.16) = 0.32 x 100% = 32% heterozygous brunettes
INTERESTING FACTS
Godfrey Hardy developed the equations just a
little before Weinberg who also independently
developed the same formula.
Hardy thought the equations so simple that he
hardly thought to publish and thus claim his
discovery. Only at the insistence of Punnett
did he do so. It was lucky he did, for even
though he was an eminent mathematician,
this is the only equation he is widely known
for today.
P
Hi again, I’m
And I am the value given for the
dominant gene frequency
Now meet my big brother
2
p
Number of dominant HOMOZYGOUS
individuals only. Be careful if you use
me! I am not the same as the number of
dominant individuals!
Here, p2 is used to indicate the number of DOMINANT
homozygous individuals. This may also be rephrased in a
deceiving way, so watch out. It could say; The frequency of
homozygous black individuals—where black is the dominant
allele. This is still the p2 value. You must also be aware that it is
very unusual to get p2 initially in the word problem, but it is
possible.
Remember, that once you have p2 you could obtain p by simply
square rooting the p2 value. Ex p2 = 0.81 p = 0.9
If you are given the number of dominant
individuals in a question you are actually given:
2
P
+ 2pq
You don’t want to try to solve this so any
time you get the number of dominant
individuals you MUST determine the
number of recessive individuals (q2)
before you start the question
How you ask? Well if there are 70/100 people with dominant
dark hair, there must be 30 people with light hair.
These 30 /100 are the recessive individuals (q2)
q
Remember, that as a general rule
you should always start any H-W
problem with me! I represent the
frequency of the recessive allele
Hi I’m
Now meet my big brother
2
q
Use me if it says the number
of recessive individuals.
Now, you will know that you are given a q2 value if the question stipulates
the number of recessive individuals. Anyone who expresses the recessive
phenotype must be a double recessive without exception. So you know that
you have the q2 value if the question is phrased like; The number of
individuals with white fur is 16 out of a population of 100. The q2 value
would thus be 0.16. The next step is always to determine the q value.
NO CALCULATORS STARTING HERE PLEASE!!!!
Now lets do some more problems, but I don’t want you to use your calculator
right now. It is important to get your brain thinking, and for you not become
reliant on the machine. In order to prepare your brain for the shock of actually
having to do some math calculations, lets try practising.
The square root of 16 is? Try it!
The square root of 9 is?
Try it!
The square root of 0.16 is? Try it!
= 4
=3
=0.4
The square root of 0.09 is? Try it!
The square root of 49 is?
Try it!
=0.3
=7
The square root of 0.49 is? Try it!
= 0.7
If q2 = 90/1000, change the value to a decimal
Now solve for q in the above line
If q2 = 36/100, change the value to a decimal
Now solve for q in the above line
If q2 = 160/1000, change the value to decimal
Now solve for q in the above line
If q= 3/5, change the value to a decimal
If q2 = 4/25 change the value to a decimal
Now solve for q in the above line
Try it!
Try it!
Try it!
Try it!
Try it!
Try it!
Try it!
Try it!
Try it
q2=0.09
q=0.3
q2=.36
q = 0.6
q2 = 0.16
q =0.4
q= 0.6 (3/5= 6/10)
q2= .16 (4/25=16/100)
q=0.4
YOU HAVE COMPLETED ALL THE MATH
NEEDED TO COMPLETE THE INTRODUCTORY
QUESTIONS IN THE PREVIOUS SLIDE, SO....
NO CALCULATORS
PLEASE! USE YOUR
BRAIN FOR THESE
QUESTIONS.
Now it’s time to analyze the word problems and determine if you can figure
out the unknowns you are given.
All of the following represent populations in Hardy-Weinberg equilibrium
In humans, red hair is recessive to brown. The gene frequency of red hair
is 0.6. What value are you given?
Try it!
You are given q = 0.6
In rats white fur is recessive to brown. The number of white rats is 36 out
of population of 100. What value are you given?
Try it!
You are given q2 = .36
In moths, red wing spot colour is dominant to white. There are 18 white
spotted moths in a population of 200 moths. What are you given?
Try it!
You are given q2 = 18/200 or in lowest terms q2 = 0.09
REMEMBER, every time you get q2 you must convert it to q to determine the
other values! DON’T FORGET!!!!!!
What are you given? Continued with some tricks
In a bat population in H-W equilibrium, short legs are dominant to
long. The frequency of long legged individuals is 0.09 . What value
are you given?
Try it!
ANSWER You are given q2=0.09 Frequency is the trick as the
GENE FREQUENCY would be q. The indicator of this trick is that
these are individuals NOT gene frequencies
A population of flower has red, white and pink flowers. Crosses
reveal that this polymorphism is controlled by one locus and two
alleles. WW individuals have red flowers, Ww pink and ww white
flowers. If the frequency of white flowered individuals in the
population is 0.09, what is the frequency of the pink flowered
individuals if the population is in H-W equlibrium?
Try it!
ANSWER You are givenq2=0.09 thus q=0.3
Since p+q=1 p=0.7 From the equation p2+2pq+q2=1 the pink
flowers would be the heterozygotes 2pq
Substituting 2(0.7x 0.3)= 0.42 or 42%
Therefore the frequency of pink flower plants is 42%
Now, lets try some questions that walk you through the steps.
The frequency of the dominant allele in a population of 100 is 40%. What are
you given?
Try it!
You are given p = 0.4 , and you would determine q to continue with
this question. The formula you would use to determine q would
be?
Think
p+q =1 The value of q is therefore 0.6 You could then substitute
these values to calculate all the unknown population frequency
number utilizing this equation
Think
Use p2 + 2pq + q2 = 1—Determine how many in the population
are the dominant phenotype
Think
Since q=0.6 therefore q2 = 0.36 or 36%. This means that there
are 36 out of 100 with the recessive phenotype. That means
the rest must be dominant ---64 individuals are the dominant
phenotype.
You can also do it the long way and substitute into p2 + 2pq
This question is
unusual
because you
initially are
getting the
value for p.
Lets try another question with a little less of a walk through.
In cats black fur is dominant to white. If there are 84 black cats and
16 white cats, determine how many of the cats are heterozygotes?
You are given two values. What do you do?
Think think think
Remember, the
heterozygotes are
This is a little tricky, first calculate the total number of cats
black as they
84+16= 100 cats
possess the
Next. Always try to determine q
dominant allele
The number of recessive phenotype cats is 16, thus q2 is 16/100,
If q2 = .16, then q= .4 Now can you answer the original question -How many cats are heterozygotes?
Think think think
p+q=1 thus since q=0.4,therefore p =0.6
Use p2 + 2pq + q2 =1 -- The component you need from this
equation is 2pq
Just substitute 2(0.4x0.6) = .48
Thus 48% of the cats are heterozygous black
If you needed to calculate how many of the cats are homozygous
black, it would be easy at this point, just substitute
Since p=0.6 p2 = 0.36 -- 36% of the cats are homozygous black
Now you should feel a little more confident. Try this question
Assume that a population of diploid individuals is in H-W
equilibrium for a trait controlled by one locus and two alleles. If
the frequency of the recessive allele is 0.8, what is the frequency
of heterozygous individuals?
Think think think
ANSWER You are given q You just use p+q=1
to determine p=0.2 Once you have this substitute into the
relavant part of p2+2pq+q2 This is 2pq –means the decimal
percent of the heterozygotes 2(0.2 x 0.8)= 0.32
Therefore the frequency of heterozygous individuals is 0.32 or
32% of the population are heterozygotes.
In a population of diploid individuals in H-W equilibrium, the
frequency of the dominant allele for a certain hereditary trait is
0.3. What percent of individuals in the next generation would be
expected to be homozygous for the dominant gene?
Think think think
ANSWER You are given p. The representation of the
HOMOZYGOUS dominant individuals is p2 from p2+2pq+q2=1
Since p=0.3 p2 = 0.09 or 9% homozygous individuals
This question is
rare because
you are initially
given the p
value and you
don’t need to
calculate q .
Usually q is
what you need
to find first!
Now for some little twists to the question. See if you can find the tricks
For a trait that is controlled by two alleles at a single locus, the frequency
of the dominant allele is 0.6. What is the percent of individuals that show
the dominant trait? Find the trick first
Think think think
ANSWER You are given p, but this time you are asked to determine
p2+2pq Can you see why? –it’s because this represents all individuals
that express the dominant allele. You must start this with the STEP 1
DETERMINE q Now continue with the solution
Think think think
ANSWER Since p=0.6 and p+q = 1 therefore q=0.4
Substitute the values into p2+2pq
0.36 + 2(0.6 x 0.4)= 0.84 or 84% express the dominant trait
In a spider, grey colour is dominant to white. There are 910 grey spiders in a
population of 1000. How many of the grey spiders are heterozygous?
Think think think
ANSWER You are given p2+2pq = 910/1000 DON’T TRY TO SOLVE
THIS –GO THE EASY WAY If 910 spiders are grey that means that 90
are white – this is q2 = 90/1000 or q2= 0.09 and thus q= 0.3 and p=0.7
Heterozygotes are represented by 2pq so substituting 2(0.7x0.3)=0.42
or 42% of the spiders are heterozygous grey
Okay, you can
use your
calculator now
if you need it.
Now try something even more difficult. Can you see the trick
If one gamete in five carries a recessive allele, what must be the frequency of
the homozygous recessive genotype in a population at H-W equilibrium?
Think think think
ANSWER The gamete alleles mean the same as the frequency of the
recessive allele. Therefore q=1/5 Now this is the same as q= 2/10 or 0.2
Shame on you if you needed your calculator Since q=0.2 thus the
frequency of the homozygous recessive is q2=0.04 or 4%
-------------------------NOW YOU CAN USE YOUR CALCULATOR----------------------Suppose that 1 in 400 people in a large population have
a recessive disorder. Apply the Hardy–Weinberg principle
to estimate the proportion of individuals who are
carriers of (i.e., heterozygous for) this disorder.
Think think think
ANSWER You are given q2=1/400 thus q2= 0.0025 and q=.05
Substitute into p+q=1 and p=0.95
Utilizing p2+2pq+q2=1 the carriers (heterozygous) are 2pq
Substitute 2(0.95x 0.05)= 0.095
Therefore 9.5% of the people are carriers.
Now let’s try to look at something closer to reality. If this crowd were in H-W
equilibrium, we could predict the percentages of blondes in all future
generations given a simple analysis. Simplified as Dark hair etc. dominant to blonde
Count how many of these people are blondes (I kid)
“Count the total number of people n the population
I count 3573 people of which 142 are blondes
Determine the future percentages of Dark haired people, blonde haired people
homozygous dark haired people, heterozygous dark haired people and
homozygous blondes.
The number of blondes is actually the number of
individuals with the recessive phenotype .
Thus q2=142/3573
If q2 = 0.03975 then q= 0.2
Utilizing p+q=1 p=0.8
Utilize p2+2pq+q2=1 to determine the percentages
p2= 0.64 or 64% homozygous dark hair
2(0.8 x0.2)= .32 or 32% heterozygous dark hair
q2 = 0.04 or 4% (homozygous) blonde hair
Also there would be 96% Dark haired people and 4%
blonde haired people, not just in this picture, but in
all future generations of this population in H-W
equilibrium.
Extension: Real Project for extra credit
Why don’t you try to measure one genetic trait in our
Cardinal Carter Secondary School population. See if you can
choose a trait that is only due to two alleles to keep it as
simple as possible. (Hair and eye colour are not good
choices due to multiple alleles if you remember this from
your grade 11 work) Conduct an actual count -- determine
p and q and predict what the percentages of the trait would
be in future generations if the population were in H-W
equilibrium. (It’s not really H-W equilibrium –discuss why)
In the future, we can resample compare your numbers and
see if the numbers are shifting or relatively stable. Hand it
in to me before the final exam if you want credit
CONGRATULATIONS YOU HAVE
COMPLETED YOUR HARDYWEINBERG TRAINING --GOOD
LUCK WITH THESE ON THE EXAM
–REMEMBER TO TRY THESE ON
THE QUIZZES PROVIDED ON THE
WEBSITE
THE END