How close is close enough?
Part II
Mendel vs 1000 Ideal Worlds
Build the world in BioBIKE
biobike.csbc.vcu.edu
This demonstration is best viewed as a slide show,
enabling you to simulate a session and make
changes in cursor position more obvious.
To do this, click Slide Show on the top tool bar, then View show.
Define an ideal genetic world
How?
- Define genotype
male-genotype
female-genotype
p
p
Pp
p
Pp
Define an ideal genetic world
How?
- Define genotype
- Define meiosis
- Define gamete
Define an ideal genetic world
How?
- Define genotype
- Define meiosis
- Define gamete
- Define joining of gametes
to form progeny
Define an ideal genetic world
How?
- Define genotype
- Define meiosis
- Define gamete
- Define joining of gametes
to form progeny
- Define how color is
determined
Define an ideal genetic world
- Define genotype
- Define meiosis
- Define gamete
- Define joining of gametes
to form progeny
- Define how color is
determined
- Define Mendel's
experiment
929
crosses
Define an ideal genetic world
- Define genotype
- Define meiosis
- Define gamete
- Define joining of gametes to form progeny
- Define how color is determined
- Define Mendel's experiment
Go to the BioBIKE Portal
biobike.csbc.vcu.edu
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Enter a log in name
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Function palette
Workspace
The BioBIKE environment is divided
into three areas as shown. You'll bring
functions down from the function
palette to the workspace, execute
them, and note the results in the
results window
Results window
Two very important buttons on the
function palette:
HELP!
On-line help (general)
PROBLEM
Something went wrong?
Tell us!
Two very important buttons in the
workspace:
Undo (return to workspace
before last action)
Redo (Get back the
workspace you undid)
male-genotype
female-genotype
p
What now?
The first step in making the ideal world
is to define what are male- and femalegenotypes.
Mouse over the DEFINITION button
p
p
Click DEFINE to bring down the
DEFINE function.
A DEFINE function box is now in
the workspace.
Before continuing with the problem,
let's consider what function boxes
mean.
General Syntax of BioBIKE
Function-name
Argument
(object)
Keyword
object
The basic unit of BioBIKE is the
function box. It consists of the
name of a function, perhaps one or
more required arguments, and
optional keywords and flags.
A function may be thought of as a
black box: you feed it information,
it produces a product.
Flag
General Syntax of BioBIKE
Function-name
Argument
(object)
Keyword
object
Flag
Function boxes contain the
following elements:
• Function-name (e.g. SEQUENCE-OF or LENGTH-OF)
• Argument: Required, acted on by function
• Keyword clause: Optional, more information
• Flag: Optional, more (yes/no) information
General Syntax of BioBIKE
Function-name
Argument
(object)
Keyword
object
Flag
… and icons to help you work with
functions:
•
Option icon: Brings up a menu of keywords and flags
•
Action icon: Brings up a menu enabling you to execute
a function, copy and paste, information, get help, etc
Clear/Delete icon: Removes information you entered
or removes box entirely
•
General Syntax of BioBIKE
Function-name
Argument
(object)
Keyword
object
Flag
… and icons to help you work with
functions:
•
Option icon: Brings up a menu of keywords and flags
•
Action icon: Brings up a menu enabling you to execute
a function, copy and paste, information, get help, etc
Clear/Delete icon: Removes information you entered
or removes box entirely
•
And now back to our show…
The DEFINE function asks for two things:
the name of the variable to be defined and
the value it is to be given. Let's call the
variable male-genotype. The value of
male-genotype will be "Pp"
Click on the variable argument box to open
it up for entry…
… and type male-genotype
(remembering to connect the words
with a hyphen), closing the box
afterwards by pressing Tab.
Tab closes the entry box and
automatically opens the next one (if
it exists).
Type the genotype "Pp" (be sure to
include the quotation marks) and
press Enter.
The definition is now complete.
But it will not take effect until
the function is executed
Click the Action icon.
… and click Execute.
Notice that a new VARIABLES
button appears. We'll use it later to
access the newly defined variable.
Notice also that the definition is
confirmed by showing it in the
Results Window
Repeat what you did, this time
defining female-genotype in the
same way
Now to define meiosis.
Mouse over the DEFINITION button
and click DEFINE, just as you did
before.
Click the name box…
…type in meiosis, and press Enter.
If you like, you can give a short
summary in English what your
function does, that is, what is
meiosis.
To do this, mouse over the summary
icon…
… click summary, and enter the
summary of the function in the text
box that appears. Finally, press Enter
to accept the summary.
How to define meiosis?
It requires the diploid genotype of
the organism and then chooses one of
the chromosomes to go into a
haploid gamete.
The argument (input) to the function
is the genotype. Click the argument
box…
… and enter the word genotype, a
symbol that will represent whatever
genotype the function is given. Then
press Enter.
What does meiosis do?
You'll define this by putting the
appropriate action into the form box.
Click that box….
…how to choose a chromosome at
random? BioBIKE has a CHOOSEFROM function that will serve our
needs.
Mouse over the ALL button…
The ALL button provides BioBIKE
functions in alphabetical order.
Mouse over the C menu item…
…and click CHOOSE-FROM.
CHOOSE-FROM what?
From the genotype!
Click the sequence box…
… and type genotype and press
Enter.
The definition of meiosis is now
complete. To make it active, mouse
over the Action Icon…
…and click Execute.
Notice that a new FUNCTIONS
button appears. We'll use it in a
moment to access the newly defined
function.
Notice also that the definition is
confirmed by showing it in the
Results Window
Try out your new function.
Mouse over the FUNCTION button
and click Meiosis.
Click on the genotype box of
Meiosis.
Provide it with male-genotype, by
mousing over the VARIABLES
button and clicking male-genotype.
Execute your function as you do any
other function, by mousing over the
Action Icon…
…and clicking Execute.
The result of meiosis now appears in
the Results Window.
Execute the function a few more
times to get a feel for what it does.
With Meiosis in hand, you can now
define the male and female gametes.
Bring down a DEFINE box as you
have previously and set the variable
as male-gamete. (Be sure to
remember the hyphen – you can't put
spaces within variable names).
Then click the value box.
We want to define male-gamete
as the product of meiosis working on
male-genotype.
You can bring down a Meiosis box
into the value box, but it's easier to
cut/paste the box you already have.
Mouse over the Action Icon of the
Meiosis function in the Workspace.
…and click cut.
Then mouse over the Action Icon of
the value box…
…and click paste.
Now you can execute the definition.
To define female-gamete, click
on the word male-gamete, and
when the box is open for editing
(turns white), change male to
female. Press Enter to accept this
change.
Make the same change to malegenotype and execute the function.
Now to define the joining of the male
and female gametes to form a progeny.
Bring down a DEFINE box, and type
progeny as the name of the variable.
Click on the value box. We need a
function that will join the gametes.
Fortunately, BioBIKE has such a
function. Find it using the ALL button.
To enter the male-gamete and
female-gamete into the two
arguments of JOIN, click each
argument box and find the variable on
the VARIABLES menu.
Once the function is complete, execute
it several times.
Why is the behavior different with this
function than you saw when you
executed Meiosis several times?
Now we need to define a function that
analyzes a genotype and decides what
color it determines.
Bring down a DEFINE-FUNCTION
box as before and fill in the holes.
But how to make the decision?
IF the genotype is pp then the
flower is white. Otherwise the
flower is purple.
BioBIKE makes such decisions
using the IF-TRUE function,
available from the FLOWLOGIC menu.
IF-TRUE looks at a condition (in this
case whether the genotype is the same
as "pp") and THEN does something if
the condition is true ELSE it does
something else.
Let's first consider the condition. Click
that box…
…and bring down the SAME function,
available from the FLOW-LOGIC
button, Logical-Comparison submenu.
We want to know if the genotype is the
same as "pp".
Click on x, type genotype, and press
Tab. Then type "pp" into the y box.
BioBIKE will ordinarily ignore the
distinction between upper and lower
case. So the difference between "pp"
and "Pp" will be ignored…
…unless we tell BioBIKE to consider
case.
Mouse over the Option Icon…
…and click CASE-SENSITIVE
If the genotype indeed is the same as "pp", THEN the color should
be "white". Type "white" in the THEN box.
ELSE the color should be "purple". Type "purple" in the ELSE box.
(Be sure to press Tab or Enter after each entry)
Finally, execute the completed DEFINE-FUNCTION.
Bring down a COLOR-OF box from the FUNCTIONS
menu, and execute the function a few times with different
genotypes (within quotation marks). Are the results as you
expect?
Now to teach BioBIKE how to do a cross.
Bring down a new DEFINE-FUNCTION.
Call the function Make-Flower.
It will return a flower from a Pp x Pp cross.
It won't need any required arguments, but
how many functions in the body will be
required to do the job? Let's think…
We need to define male-gamete and femalegamete. That's two functions.
We need to combine them into the progeny.
We need to determine the color of the
progeny.
That's four. So we need four holes. Mouse
over the Options Icon of the Body Form…
…and click Add two more forms.
Then mouse over the Options Icon again
and click Add another to get a total of four
forms to fill in.
Click on the first form and DEFINE malegamete as before (or cut and paste the existing
box).
Then put DEFINE female-gamete in the
second form.
Then DEFINE progeny in the third.
Then put COLOR-OF progeny in the fourth.
Execute DEFINE-FUNCTION .
Then bring it down the new MAKE-FLOWER
function from the FUNCTIONS button and try it
out a few times.
When you're satisfied, clear the screen (x in
upper right corner).
MAKE-FLOWER is good for one cross, but
Mendel did 929. We need to repeat the
MAKE-FLOWER function 929 times.
You can repeat the function that many times
by going to the DEFINITION button…
929
crosses
… and clicking REPEAT-FUNCTION.
Click the function box and fill it with
MAKE-FLOWER… not the words but the
function. To get the function…
… go to the FUNCTIONS button.
Then click the number box and fill it with
929.
Execute the function to do Mendel’s
complete experiment in this ideal genetic
world.
And there's the results, all 929 of them.
How many of the flowers are white?
Counting 929 virtual flowers is not
something humans are good at, so let the
computer do the work…
…Bring down the COUNT-OF function
from the ALL button.
Click the query argument box, and fill it
with the word "white" (including the double
quotation marks).
Count of "white" in what? We need to
specify what should be counted.
To do that mouse over the Options Icon…
…and click the IN option.
Again,… Count of "white" in what? In this
case, it's the results, i.e. that list of "purple"
and "white" flowers.
In general, however, we're looking for the
count of "white" in the results of Mendel's
experiment.
To bring Mendel’s experiment (i.e. the
REPEAT-FUNCTION box) into the value
box, you could cut/paste as before, but this
time, try drag/drop. Click under REPEATFUNCTION for a half second and hold the
button down…
Now drag the box over to the value box…
…so that the upper left corner of the
dragged box is in the target box (the target
box will gain a red outline), then release the
mouse button.
Now the function is set up to tell you how
many flowers in Mendel’s experiment are
white.
Execute it. In fact, execute it twice.
You may get a different answer each time,
because if Mendel had repeated his
experiment with 929 flowers, he would
probably have gotten a different number of
white flowers. Each time is potentially
different.
That’s Mendel’s experiment.
Let’s package it as a function to make it
easier to manipulate.
Call the function MENDELS-EXPERIMENT
(no apostrophe’s please!), and give it a summary
that will remind you what it does.
Mendel’s experiment is precisely the COUNTOF box we just made.
To bring that box into MENDELSEXPERIMENT, drag it into the form box of the
Body section…
…and drop it there.
Now execute the function definition to add it to
your collection of functions.
After doing that, you might want to clear the
screen (red X in upper right corner).
Then Bring down MENDELS-EXPERIMENT
from your FUNCTION button and execute it a
couple of times.
Each time you execute the function, you will
probably get a different number of white flowers
counted. That’s the way experiments work.
Define an ideal genetic world
- Define genotype
- Define meiosis
- Define gamete
We’ve now accomplished everything we’ve set
out to do,… except one thing: We still don’t
know whether Mendel was right to call his
results 3:1.
- Define joining of gametes
to form progeny
- Define how color is
determined
- Define Mendel's
experiment
929
crosses
Define an ideal genetic world
Recall… he observed 224 white flowers.
- Define genotype
- Define meiosis
- Define gamete
A 3:1 ratio predicts 232¼ white flowers.
Is that close enough?
With our virtual experiment in hand, we’re
ready to define what we mean by
gametesfinally
“close enough”.
- Define joining of
to form progeny
- Define how color is
determined
- Define Mendel's
experiment
Purple
+
696¾ +
Observed: 705
Expected:
White
=
232¼ =
224
Total
929
929
Define an ideal genetic world
- Define genotype
- Define meiosis
- Define gamete
Suppose that Mendel is right. The ratio of
purple to white flowers is really 3:1, and his
deviant result was just because experiments
give different results each time, and 224 is the
result he happened to get. If he did it again, he
might get 232 white flowers.
Number of experiments
- Define joining of gametesWell, we now can do it again! If we repeat the
to form progeny
experiment many times, and his observed value
of 224 is common (as in the case below), then
- Define how color is
Mendel was justified in making his
determined
approximation.
- Define Mendel's
Replications in 3:1 world
experiment
Expected
Observed
Purple flowers
Define an ideal genetic world
We have a strategy! Just repeat his experiment
gametesand see how frequently we get 224 white
flowers (or fewer).
Replications in 3:1 world
Expected
Observed
Purple flowers
Replications in 3:1 world
Expected
Number of experiments
- Define joining of
to form progeny
- Define how color is
determined
- Define Mendel's
experiment
Number of experiments
- Define genotype
- Define meiosis
- Define gamete
On the other hand, suppose Mendel is wrong.
In that case his observed value (or worse) will
very rarely crop up (as in the other case
below).
Observed
Purple flowers
We already have the experiment.
All we have to do is to repeat it – say 100 times.
You know how to repeat a function. Go to the
DEFINITION button and get REPEAT-FUNCTION.
Now drag MENDELS-EXPERIMENT into the
function box…
…and release it.
We want to repeat the experiment a lot of times, but
too many times and execution will be painfully slow.
Insert 100 as the number of times.
Then execute the function
Well there’s our answer.
All we need to do is to count how many times the
number of white flowers is 224 or less.
But humans weren’t made to count so many
numbers. We need to filter away the numbers we
don’t want and keep the ones we do.
To do this, bring down the FILTER
function…
…and drag (or copy/paste) the
result at the bottom of the screen
into the data box….
…and drop it there.
We want the filter to discriminate
between those numbers less than or
equal to 224 and the rest, so the test
we want is <= (less than or equal
to).
Mouse over the choose test icon…
…and click <= (less than or equal to).
Now type 224 into the value box.
…and execute the function.
That’s a whole lot better.
But still I’d rather not count
(estimate, yes; count, no).
Fortunately, I have a COUNT-OF
function available. All I need to do
is go to the Action Icon of
FILTER…
…and click Surround-with.
Now I bring down COUNT-OF, as
I have done in the past…
…and FILTER is surrounded by the
COUNT-OF function.
Execute that to get the answer of…
22. 22 times out of a 100 repetitions
of his experiment, Mendel would be
expected to get a count of white
flowers at least as deviant from
expectation as he actually observed.
Almost. 224 is 8¼ flowers deviant
from the expected 232¼ ,… but so
is 240½!
We should also consider as equally
deviant those experiments that give
more than 240 white flowers.
We could construct from scratch a
different filter and count what
passes through it, but it’s much
easier to copy and modify the
current function.
So copy it…
…and go to the FILE button to
paste the copy into the workspace.
Now, alter the test so that it is…
Now, > (for greater than 240)
…and change the threshold value
from 224 to 240.
Execute…
So 22 times out of a hundred
Mendel would have gotten a
number of white flowers that is
too low, at least as deviant as the
number he actually got.
And 24 times out of a hundred he
would have gotten a number that
is too high, at least as deviant as
the number he actually got.
22 + 24 = 46% of the time…
almost half the time. Not so
unlikely!
Number of experiments
Replications in 3:1 world
Expected
24%
Observed
22%
Purple flowers
So Mendel WAS right in calling
his result 3:1. Or at least his
observed value was not out of
line with the expected value,
given normal variation.
Number of experiments
Replications in 3:1 world
Expected
24%
In considering such problems as
this one (VERY common!), you
generally don’t have access to a
computer that can do this
simulation.
Observed
22%
Purple flowers
But you do have access to
something that’s equivalent…
Statistics!
Chi-Squared Test
And Ei is the expected value
Number of experiments
Where Oi is the observed value
Replications in 3:1 world
Expected
24%
Observed
22%
Purple flowers
The chi-squared (X2) test does
through an equation exactly what
you just did on the computer.
Try it!
Application of Chi-squared test
Chi-Squared Test
And Ei is the expected value
Number of experiments
Where Oi is the observed value
Replications in 3:1 world
Expected
24%
2
2
(705
–
696¾)
(224
–
232¼)
+
X2=
696 ¾
232¼
= 0.391
Observed
22%
Purple
Purple flowers
+
696¾ +
Observed: 705
Expected:
White
=
232¼ =
224
Total
929
929
Application of Chi-squared test
Chi-Squared Test
And Ei is the expected value
Number of experiments
Where Oi is the observed value
Replications in 3:1 world
Expected
24%
Observed
22%
X2= 0.391
Now go to a web site that will
translate this number into an area
under a curve as in the figure to the
left.
Purple flowers
http://www.fourmilab.ch/rpkp/experiments/analysis/chiCalc.html
Scroll down to
the calculator…
Insert .391 for the X2 value
…and 1 for the degrees of
freedom (the flower is either
white or it is not… one
question to answer)
…then click Calculate.
Chi-Squared Test
And Ei is the expected value
Number of experiments
Where Oi is the observed value
Replications in 3:1 world
Expected
24%
Observed
22%
Purple flowers
The probability (area under the
curve) you get from the X2 test
should be within several percentage
points from what you got from
repeating the virtual experiment…
…because that’s what a X2 test is!
(so long as you do the experiment an
infinite number of times!)
Number of experiments
Replications in 3:1 world
Expected
24%
Chi-Squared Test
Observed
22%
Where Oi is the observed value
Purple flowers
And Ei is the expected value