*Models
represent reality.
Models constructed for different
purposes represent different aspects of
reality but no model represents every aspect.
*

*All models simplify reality in some way. Model*

airplanes may look like real planes but can’t fly,

others fly but don’t look real

airplanes may look like real planes but can’t fly,

others fly but don’t look real

*and wind tunnel*

planes act like real planes and still can’t

fly.

planes act like real planes and still can’t

fly.

*The model we use depends*

on what we need

it to do.

on what we need

it to do.

**Mathematical models**

capture aspects of reality with equations.

The mathematical model

*5 – 1 = 4
*

*represents*

a boy with five apples

who ate one and the same model

could represent many other things.

Complex models represent complex

ideas and everything rests on

assumptions about how

the world works.

a boy with five apples

who ate one and the same model

could represent many other things.

Complex models represent complex

ideas and everything rests on

assumptions about how

the world works.

Iterative mathematical models

represent situations in which some

quantitative process is repeated over and

over again, in a loop, *and we want to know
how the consequences of that process develop
over time. *

*The iterative mathematical model*

below is general in the sense that it can represent

any number of situations.

below is general in the sense that it can represent

any number of situations.

*It represents a*

girl with some number of apples who

ate some number each day until

they were gone

girl with some number of apples who

ate some number each day until

they were gone

*and we want*

to know how long

that took.

to know how long

that took.

*Apples = 10
Days = 0 *

*Ration = 1 apple each day*

*Start eating
Apples = Apples – Ration*

**if (Apples > 0)
**

*Apples = Apples – Ration*

Days = Days + 1

go to Start Eating

Days = Days + 1

go to Start Eating

**if (Apples = 0)
**

*go to Stop Eating*

*Stop Eating*

* Print Days*

*The answer
depends on *

*the number*

of apples at the start,

of apples at the start,

*the ration,*

whether she

whether she

*ate apples every day,*

*varied*

the ration from day to day,

the ration from day to day,

*or a horse ate*

the rest of her apples.

the rest of her apples.

*All of this would*

be easy to model with the same basic

iterative structure, no matter

the details.

be easy to model with the same basic

iterative structure, no matter

the details.

*If the model
were complex enough or
needed many iterations ‘through
the ‘day loop’, it would be easier, faster,
and more reliable to run it on a computer
than to do it by hand. *

*Computers are*

fast, good at keeping track of

things, and usually don’t

make mistakes unless

you tell them to.

fast, good at keeping track of

things, and usually don’t

make mistakes unless

you tell them to.

Hardy-Weinberg

Genetic Equilibrium Theory

allows us to predict the genetic

composition of populations over time, given

certain assumptions about *migration, mutation rates,
*

*mating patterns, population size*

*and natural selection*

– –

– –

*mainly that none of them are changing. As long*

as those ideal assumptions are met, t

as those ideal assumptions are met, t

*he model*

predicts that genetic structure remains

constant from generation

to generation.

predicts that genetic structure remains

constant from generation

to generation.

*According
to the theory, allele
frequencies in gene pools
and genotype frequencies in
populations remain constant if there
are no natural selection, mutation, or
migration, etc. *

*We know the real world*

is not constant in these ways, so the value of

the theory is not to show the real world is not

ideal.

is not constant in these ways, so the value of

the theory is not to show the real world is not

ideal.

*We already know that.*

*The value of the*

theory, as of theories in general, is to help us

understand how

from the

theories represent.

theory, as of theories in general, is to help us

understand how

**the real world differs**from the

**imaginary worlds**theories represent.

*In this case,
Hardy-Weinberg Theory
helps us understand how the genetic
structure of populations changes over time
in response to specific selection pressures. *

*The*

theory’s predictions provide a yardstick and help

us see where, how, and whether the real world

violates those simple assumptions and shows

where the assumptions are wrong.

theory’s predictions provide a yardstick and help

us see where, how, and whether the real world

violates those simple assumptions and shows

where the assumptions are wrong.

*You specify*

the starting point and the selection pressures

the starting point and the selection pressures

*and predict what will happen, given the*

rest of the assumptions. The model

shows

rest of the assumptions. The model

shows

*you whether*

it would do

that.

it would do

that.

*What a way to learn! *

*It’s fast,
easy, fun, *

*and tests*

*your*

knowledge of genetics and population

genetics

knowledge of genetics and population

genetics

*to the core.*

*Do it by yourself, do it*

in groups, but do it. I’ll surely be doing it while

I wonder what to ask you on the final. Could I be

any clearer about how much I think using this model

will help you do well in this course?

in groups, but do it. I’ll surely be doing it while

I wonder what to ask you on the final. Could I be

any clearer about how much I think using this model

will help you do well in this course?

*Anyone who*

bet against my

bet against my

*giving you an opportunity to*

shine

shine

*in genetics, population genetics, and*

natural selection

natural selection

*on the final could*

not have been sitting in your

seat all year, because

you know

not have been sitting in your

seat all year, because

you know

*better.*

**The Hardy-Weinberg
Simulation Model **is an iterative, looping,

computer-based

*mathematical model of population*

genetics

genetics

*using the Hardy-Weinberg equations*

*to model changes in a population*

*under*

conditions of natural selection

conditions of natural selection

*that you specify.*

*The model
is easy to run. *

*When you*

run it,

run it,

*it asks you for two kinds of*

information about a simple, imaginary,

2 allele, 1 gene locus genetic system.

information about a simple, imaginary,

2 allele, 1 gene locus genetic system.

*It wants to*

know the

know the

**genotype frequencies***at time zero for*

the 3 possible genotypes,

the 3 possible genotypes,

*pp, pq, and qq, which are*

the proportions

the proportions

*of the 3 genotypes in the population.*

*The 3 values must sum to 1,*

*or 100% of the genes*

*in the population*

*for that locus,*

*and they must*

be

be

*in the proportions*

*specified by*

the binomial equation,

the binomial equation,

*p*

^{2}+ 2pq + p^{2}= 1.*(Try a different ratio and see what happens.)*

**Genotypes**

are what genes you have.

**Phenotpes **are the way you are, *and
to the extent that how you are *

*is inherited*

genetically,

genetically,

*phenotypes reflect genotypes.*

*The*

model also needs to know

model also needs to know

**selection pressures***against phenotypes, expressed as mortality rates.*

*What proportion of each phenotype dies each generation*

without reproducing?

without reproducing?

*If the p allele is*dominant

*over*

q,

q,

*pp and pq have the same phenotype,*

*the same*

exposure to natural selection,

exposure to natural selection,

*and the same*

mortality rates but if neither dominates

the other they are different

phenotypically.

mortality rates but if neither dominates

the other they are different

phenotypically.

*So when you
assign mortality rates, you
are also deciding whether the genetic
systems you investigate involve dominance,
which is one of the things those numbers mean.
*

*If neither allele dominates, there are 3 phenotypes,*

3 exposures to natural selection, 3 mortalities, and

a different outcome of the model.

3 exposures to natural selection, 3 mortalities, and

a different outcome of the model.

*You could predict*

that simple result from your understanding of

simple Mendelian genetics without running

the model, but test the prediction with

the model

that simple result from your understanding of

simple Mendelian genetics without running

the model, but test the prediction with

the model

*to be sure you*

understand it.

understand it.

*There is much
more to be learned with
the model, though. Some results
will confirm your understanding but
some will surprise you, stop you, and make
you scratch your head for a while. Then
try this, that, and the other thing
with the model until you
understand it. *

*The test of your understanding
is the accuracy of your predictions,
so don’t skip that essential step.*

*Using those
simple parameters, the model
iterates *

*through the Hardy-Weinberg*

equations to predict 50 generations of changes

in the genetic composition of the population you

define and illustrates the results as graphs and

tables of numbers.

equations to predict 50 generations of changes

in the genetic composition of the population you

define and illustrates the results as graphs and

tables of numbers.

*Before I turn you loose*

to play with the model you need to

understand how it works.

It’s simple.

to play with the model you need to

understand how it works.

It’s simple.

*When you
click to run the program,
you get a page of numbers and
those at the top are all you need
to know for now. *

**There are**

**exactly six of them. You**

**need to understand**

**all six**.

*Lines 1 to
3 are the proportions of
each phenotype in the population
dying before reproducing each generation,
*

*the*

of natural selection to shape gene pools. You can

change any of those numbers, as you like. N

**mortality rates**that estimate the powerof natural selection to shape gene pools. You can

change any of those numbers, as you like. N

*ote that*

unless you change it, the model assumes 10% of pp

genotypes die before reproducing each generation, and

100% of other 2 genotypes survive and reproduce.

unless you change it, the model assumes 10% of pp

genotypes die before reproducing each generation, and

100% of other 2 genotypes survive and reproduce.

*There*

is no magic in that starting point, other than that it should

be interesting to you genetically. Basically, the model

must start somewhere, and that’s it. It’s up to you to

use it as a tool. Its value is in how you set it up,

which depends on what you want to know.

That depends on what you don’t know

– – your ignorance.

is no magic in that starting point, other than that it should

be interesting to you genetically. Basically, the model

must start somewhere, and that’s it. It’s up to you to

use it as a tool. Its value is in how you set it up,

which depends on what you want to know.

That depends on what you don’t know

– – your ignorance.

*Working the*

model is a great way

model is a great way

*to*

learn

learn

*what you*

know.

know.

*In addition
to mortality rates, you
can change *

**starting conditions**, or the pp’s and qq’s that define populations and gene pools. You’ll set those at time zero (generation 0, line 7), which is how and when things start out.

depends on the selection pressures

that you impose through

mortality rates.

**How they change**depends on the selection pressures

that you impose through

mortality rates.

*If you
don’t know how to
*

*calculate p’s and q’s from*

*pp’s, pq’s, and qq’s,*

*don’t worry*

about it at first.

about it at first.

*Excel will fix it for*

you, but the

you, but the

*first generation of your*

graphs

graphs

*will take a jump. Learn to*

*do*

it yourself, save Excel the trouble,

it yourself, save Excel the trouble,

*and your graphs will be*

*more*

beautiful and more useful

learning tools.

beautiful and more useful

learning tools.

*After you
set those 6 numbers,
Excel calculates the rest of
the values automatically with Hardy
-Weinberg math, all 50 generations in a
heartbeat, and draws all the graphs for you,
just from those six numbers. *

*You could do it*

yourself by hand if you needed to, and please

don’t think you won’t get a chance to

yourself by hand if you needed to, and please

don’t think you won’t get a chance to

*demon-*

strate that ability on the final,

strate that ability on the final,

*but having*

Excel turn the crank for you saves

tons of time learning and helps you

learn much more deeply and

remember it longer

as well.

Excel turn the crank for you saves

tons of time learning and helps you

learn much more deeply and

remember it longer

as well.

*To run the
Hardy Weinberg Simulation
Model, Microsoft Excel must be on your
computer (it is part of MS Office).
If so, just click *HWSim

*.*

*Set the 6
numbers and
look at the results.
—————————
Selection against pp (ppMort)?
Selection against pq (pqMort)?
Selection against qq (qqMort)?
—————————————
Frequency of pp in the population?
*

*Frequency of pq in the population?*

*Frequency of qq in the population?*

——————————————-

——————————————-

*Excel calculates everything automatically.*

*Here’s how the model would work if
you were turning its crank by hand*

* Calculate survival rates
from mortality rates.
*

*ppSurv = 1 – ppMort*

*pqSurv = 1 – pqMort*

*qqSurv = 1 – qqMort*

*Enter Generation Loop*

*Gen = 0*

*Start **Generation Loop*

**Apply mortality**

*pp = pp * ppSurv
*

*pq = pq * pqSurv*

*qq = qq * qqSurv*

**Calculate post-mortality genetics.**

*SumSurv = pp + pq + qq
*

*pp = pp/SumSurv*

*pq = pq/SumSurv*

*qq = qq/SumSurv*

*p = pp + pq/2*

*q = qq + pq/2*

**Calculate offspring genetics**

*pp = p * p
*

*pq = p * q * 2*

*qq = q * q*

*Save genetics: pp, pq, qq, p, q*

*Administer Generation Loop*

if (Gen < 50)

Gen = Gen + 1

go to Enter Generation Loop

*if (Gen = 50)
*

*go to Stop Looping*

*Stop Looping*

*Draw graphs*

*Stop*

*Then study the results,
think about what they tell you,
realize what you still need to know,
and do everything again until you know
what you need to know to understand
how natural selection works in
natural and modeled systems. *

——————–

*In the
example, Excel
assumes that unless you
change things, the homozygous pp
phenotype suffers 10% pre-reproduction
mortality each generation, compared
to the others, which get off Scot free. *

—————————————

*The Allele Frequency Chart
(the tab at the table bottom)
shows that the 10%
disadvantage
specified
in the
mortalities
was enough to drive
the p gene from 50% of
the gene pool to nearly extinct.*

—————————————

**A Few Instructive Simulations**

*The following
simulations will help
you learn population genetics,
particularly how genetic systems
change over time under various
assumptions about survival
and reproduction. *

*By adjusting
those parameters,
you can run scenarios about
anything you want, so please use
these suggestions just to get you
started. After that, be my guest.
Let your imagination
be your guide. *

*The model
is especially helpful in
understanding the population
genetics of *Sickle Cell Anemia

*and other*

inherited ailments, especially as they vary

among ecologically different environments.

inherited ailments, especially as they vary

among ecologically different environments.

*The model is simple and grossly simplifies*

enormously complex realities, but you’ll

be amazed to discover that you

can model those things.

enormously complex realities, but you’ll

be amazed to discover that you

can model those things.

*Before
you run the model
under any set of parameters,
any time you run it, *

**STOP!**I

*magine*

the p, q, pp, pq, and qq curves

changing, generation by

generation, over 50

generations, the

the curves.

the p, q, pp, pq, and qq curves

changing, generation by

generation, over 50

generations, the

**shapes**ofthe curves.

*Based on
your understanding of
the genetic and ecological system
you define and the selection regime you
specify, and mindful of what you want to learn
by it, what do you imagine will happen? You
don’t need calculations for this. It’s not
about math but understanding. Just
imagine the curves and see
what happens. *

*If you
don’t know for
sure, take a guess and
s*

*ooner than you think, two*

things will happen.

things will happen.

*Your guesses*

will become more accurate, more

of the time, as you home in on under-

standing of population genetics and

natural selection.

will become more accurate, more

of the time, as you home in on under-

standing of population genetics and

natural selection.

*As your under-*

standing grows, so will your

appreciation of what you

don’t yet know.

standing grows, so will your

appreciation of what you

don’t yet know.

*You will discover
things you still need to learn
about genetics, population genetics,
and ecology. Together with your
imagination, the model will help you
find and learn them. *

*Playing with*

the model is a good way to find

out what you need

the model is a good way to find

out what you need

*to*

*study.*

*Simulations to run
*

*Selection against*

a lethal recessive allele

at various intensities.

a lethal recessive allele

at various intensities.

*Selection against
a lethal dominant allele
at various intensities.*

*Heterozygote superiority,
or selection against both homozygotes,
at various intensities.*

*Heterozygote inferiority,
or selection against heterozygotes,
at various intensities.*

*STOP!*

**After
**you run the model,

each time you run it,

**STOP!**

*and think about what just happened.*

Look at the curves.

Look at the curves.

*See if you can explain*

to yourself how they came out that way, whether

you imagined it correctly or not.

to yourself how they came out that way, whether

you imagined it correctly or not.

*You’ll be*

explaining these kinds of things

sooner than you think anyway,

so you may as well start by

explaining them

explaining these kinds of things

**to me**sooner than you think anyway,

so you may as well start by

explaining them

**to**

yourself.yourself

*Learn
everything
you can learn
from each run, then
test what you learned by
running it a different way
and predicting the result.*

—————-

**Setting up a
lethal recessive **

**simulation.**

A population begins

at genotype frequencies

25% pp, 50% pq, and 25% qq.

All individuals of genotype qq die

before reaching reproductive age

(lethal recessive) and all individuals of

genotypes pp and pq survive and reproduce.

*What are the genotype
and gene frequencies after 50 generations
of this kind and degree of selection?*

**Starting Frequencies
**

*pp = 0.25*

*pq = 0.50*

*qq = 0.25*

*p = 0.50*

*q = 0.50*

or any other frequencies you want.

Your question is what happens

and the same sort of thing

should happen regardless

of where you start, but

or any other frequencies you want.

Your question is what happens

and the same sort of thing

should happen regardless

of where you start, but

*test this claim,*

though; it could

be a trick.)

though; it could

be a trick.)

*Mortalities
*

*ppMort = 0.0*

*pqMort = 0.0*

*qqMort = 1.0*

*With those
starting frequencies and mortalities,
the model generates these frequencies
after 50 generations.
*pp = 0.9619

pq = 0.0377

qq = 0.0004

p = 0.9808

q = 0.0192

*This handout
and the model itself
evolved greatly over the years,
if only to keep up with technology.
*

*It is at least 13 years out of date in 2019.*

*The suggestions
I give in this handout about how
to use the model as a learning tool, especially
about intentionally using it as a way to detect weaknesses
in understanding and home in on and repair them,
directly expresses how I suggest studying
in general in
*First and Last Words from the

Trip Director

*.*

**History of the model.**

Before 1974 my students did

Hardy-Weinberg calculations

with hand-held pocket calculators.

Needless to say, it was a long, tedious,

error-prone process to run it that

way, so the model was much

less useful to them than

what came later.

*******
My students
and I used 2 Sharp EL-515
Solar-Powered calculators for 30
years at UBC and I’m still using both of them
in 2019! Unless those calculators can’t *

*subtract,*

I’ve been using

I’ve been using

*them for 45 years and they’re*

still going

still going

*strong. I wonder how long an*

abacus

abacus

*would last under the*

abuse

abuse

*they’ve seen.*

********

*I wrote my
first H-W computer model in
Fortran in 1974, to run on the first Unix
machine in Canada, a DEC 11/45 intended only
for research. *

*I brought small groups of first year*

students into our tiny Zoology Data Centre to run

simulations, sometimes late at night.

students into our tiny Zoology Data Centre to run

simulations, sometimes late at night.

*They told me what*

parameters to enter, I entered them, and the results

came out on a big, noisy, typewriter-style printer

with big paper.

parameters to enter, I entered them, and the results

came out on a big, noisy, typewriter-style printer

with big paper.

*When the PC revolution hit in*

1978 I rewrote the model in Apple Basic and

Apple Pascal and brought students into

my lab (again intended only for

research) to do their

modeling.

1978 I rewrote the model in Apple Basic and

Apple Pascal and brought students into

my lab (again intended only for

research) to do their

modeling.

*Few students
had their own computers that
early and the model kept evolving
to keep up. Someone wrote it in Fortran
again for PCs and one in C ran on
bigger computers at UBC. *

*When enough*

students had access to computers one

of my graduate students wrote it in

Excel and students ran it as

much as they wanted

at home.

students had access to computers one

of my graduate students wrote it in

Excel and students ran it as

much as they wanted

at home.

*As soon
as they could do that
the model mushroomed in value
and became *

**very**useful!

*Now, after*

45 years, anyone in the world who has

Excel and wants to understand how

natural selection operates on

populations of organisms

can run it.

45 years, anyone in the world who has

Excel and wants to understand how

natural selection operates on

populations of organisms

can run it.

*A quick Google search shows
lots of similar programs are available.
I didn’t try any of them, but some are sure to be
easier to use, not require Excel, include more features,
and have better graphics than this model, but the point
of this story is not about the model anyway. It’s about
how I think about teaching and how I’ve lived it.
*

Edited May 2022