Heat, Earth, and Sun

Friday I started earth science with the 5-7 graders at Peregrine
School. We started half an hour late because of the all-school
discussion of the meteor strike over Russia.  So I squeezed a lot into
35 minutes before a shortened recess break.  We reviewed the structure
of the Earth and then we talked about the three different ways heat
flows: conduction, convection, and radiation (which in this context is
just another word for light; it does not mean ionizing radiation,
which is what you need to protect your DNA from).  I brought a torch
and a saucepan to make the discussion of conduction more concrete:
cookware designers want the bottom to conduct heat very well so that
the food is heated evenly, but they want the handle to conduct heat
poorly so that you don't burn yourself.  Then I added water to segue
to convection.  Because hot fluids rise, convection occurs whenever a
fluid is heated from below, which occurs in very diverse contexts:
boiling water on the stove, fluid rock in Earth's mantle, and the
movement of air in the atmosphere.

Next, I drew a Sun far from our diagram of Earth, and I asked how heat
gets from the Sun to the Earth.  It can't be conduction or convection,
because empty space can't do either of these.  It's radiation (light).
So we observed thermal radiation (the light emitted by an object by
virtue of its temperature), noting the brightness and color of a light
bulb at different temperatures (achieved by changing the voltage).  We
analyzed the color in detail by looking through diffraction gratings
to make rainbows from the white light, and noting which color in the
rainbow was brightest.  The pattern that emerges is: raising the
temperature makes the light bluer, and makes it much brighter.  We
think of red hot as being about the hottest temperature we ever
encounter, but really white hot is even hotter (the light is a mixture
of red, green, and blue), and blue hot is even hotter than that.  (The
ocean and sky are blue because they scatter the blue light from the
Sun, not because they are emitting light.)  Even objects at room
temperature emit thermal radiation, but that light is "redder than
red" or infrared.  These kids had played with an infrared camera
before, so I didn't bring one, but we discussed their IR camera
experience in this new light.  (Read this post to get the basics of
the IR camera experience.)

The last point I made before recess break: Earth's temperature is a
balance between the energy it gets from the Sun and the energy
(infrared light) it emits into space.  To maintain a roughly stable
temperature, it must emit as much as it gets.  We would examine that
balance in more detail after the break.  During the break, I had a
trick to keep them thinking about this subject: I brought a parabolic
mirror, pointed it at the Sun, and we entertained ourselves setting
things on fire.

After the break, before moving on, I felt they needed more practice with
conduction, convection, and radiation, so I had them work in groups to design
thermoses.  We put together ideas from the different groups to arrive at a
consensus design which minimizes conduction, convection, and radiation.

Back to the main thread: I noted how the parabolic mirror gathered energy from
the Sun over a largish area and concentrated it on a small area.  If we
measured the power (energy per second) falling over one square meter
(about twice the area of the mirror), we would find that it's about
one kilowatt, or 1 kW.  I brought a 1 kW hair dryer to make that more
concrete.  We then talked about night vs day, and how the Sun is
fairly low in the sky during part of the day, and concluded that the
average power from the Sun on 1 square meter of Earth would be more
like 300 W.  So each square meter of Earth should emit about 300 W of
infrared light in order to maintain a stable temperature.

Recall that power emitted ("brightness") increases strongly as the
temperature of an object increases.  So if the temperature of that
square meter of Earth is low, it will emit less than it absorbs, and
that will raise its temperature.  But if the temperature goes up very
high, it will emit more than it absorbs, and the temp will come down.
We ought to be able to calculate the temp which is just right so that
it emits exactly 300 W.  This is where we returned to the computer
programming that the kids are loving so much.  Most of these kids are
not familiar with algebra, but they can (with lots of guidance from
me) write a loop over a range of plausible temperatures and print out
the power emitted at each temperature.

To do this, I had to give them the equation for power (in watts)
emitted as a function of temperature: 0.0000000567 T⁴, where T is in
Kelvins.  That led to a discussion of Fahrenheit vs Celsius vs Kelvin.
Fahrenheit is defined so that water freezes at 32 degrees and boils at
212, a 180-degree difference; Celsius is defined so that water freezes
at 0 degrees and boils at 100.  Therefore, each Celsius degree is
"bigger" by 180/100 or 9/5.  Therefore Fahrenheit = 9/5 Celsius + 32.
Kelvin = Celsius + 273 (I explained about absolute zero), so
Fahrenheit = 9/5 (Kelvin-273) + 32.  Admittedly, most students didn't
follow all these steps, but at least one did, and I told the others to
just use this to convert while focusing on the logical steps needed to
carry out their program.

So each group wrote a Python script to check from 1 to 1000 Kelvins,
at each step printing out the power emitted and the Fahrenheit
temperature.  It turns out that 26 F is the right temperature for 300
W.  Is this a reasonable answer?  We discussed the approximations
involved (primarily albedo, using snow as an example).  Then we tried
representing this information graphically.  Instead of scanning a list
of numbers to find the right temperature, I taught them how to make a
graph of power emitted vs temperature.  We then added a horizontal
line at 300 W, and the temp at which the line intersects the curve is
the "right" temp.  I really want to work on graph-making and
-interpreting skills, so we discussed the labels we should put on each
axis, and how to summarize the plot in words.

As a teaser for next week, a slightly more rigorous calculation shows
that Earth's global average temperature should be even colder than 26
F.  The reason we are not in fact that cold is that our atmosphere
intercepts some of the outgoing infrared light and turns it back to
the surface: the greenhouse effect.  There is a natural greenhouse
effect which makes our planet livable.  The kids had of course heard
of the greenhouse effect and global warming, so they were able to see
right away that the problem is not the greenhouse effect per se; it is
that we are adding to the natural greenhouse effect, resulting in too
much of a good thing.  More on that next week.

The original plot we made:

and a zoom in to the important part:

Great Balls of Fire

This is a great thing about a small school: they can quickly
reconfigure to take advantage of learning opportunities.  I visited
Peregrine School for other reasons on Friday morning, but a meteor had
just injured over 1,000 people in Russia.  Students in all grades had
been assembled for chorus anyway, so right afterward I explained a bit
about the meteor and took many, many questions. 

I'm not going to write much here; the New York Times ran
excellent coverage (check out the pictures and video there, as well as the
articles listed under "Related").  If you just want to see some video,
here is a good collection of about four short videos which capture
different aspects such as the brightness of the flash, the eeriness
of the shadows, and the loudness of the boom.  After showing a few
quick videos to the kids, I really didn't say much; I just took
questions.  This was a great idea, because the kids eagerly presented
me with a fantastic variety of questions.

This was a great example of how the scientist-in-residence idea can
work well for a school.  Schools and scientists should do more to
cultivate long-term relationships with each other.

[If you want to know why there are rocks in space, here is a great visualization
of how very small grains of rock came together to form bigger rocks and
eventually planets in the early days of the solar system; some rocks still haven't
yet slammed into a planet because space is really, really big.]

Toy DNA Analysis Part III: Eve of Reconstruction

After the first and second activities of the morning, there was not
much more than 30 minutes left.  I wanted to do an activity with
mitochondrial DNA, so I went over the background first. (They had seen
much of the following earlier in the year, but the review turned out
to be necessary.)

Each cell has a nucleus which contains DNA, surrounded by the bulk of
the cell ("cytoplasm") which has various structures ("organelles") for
performing various functions.  One type of organelle is mitochondria,
when help you turn oxygen into energy.  Each cell has many
mitochondria, and here is the amazing thing: they have their own DNA!
They are not built according to instructions recorded in the DNA of
the nucleus; they simply reproduce by dividing asexually, as if they
were self-contained cells within the cell.  When the cell itself
divides, each daughter gets half the cytoplasm and therefore half the
mitochondria.  It is thought that mitochondria were once independent
bacteria, which learned to cooperate so well with other cells that
they took up residence.  That's pretty amazing!  Another amazing fact
is that all creatures on Earth share the same DNA code.  We are all
related, even humans and yeast.  (Example: if you put the DNA letters
for human insulin in yeast, the yeast understand those instructions
perfectly and makes human insulin.)

When a human egg cell is fertilized, the sperm carries in half the
nuclear DNA to complement the mother's half of the nuclear DNA.  But
the egg has an enormous amount of ctyoplasm and the sperm contributes
none.  So your mitochondrial DNA is an exact replica of your mother's,
and of her mother's, and of HER mother's....there is no shuffling with
each generation as we have with nuclear DNA.  Thus, mitochondrial DNA
makes it much easier to test whether you are a direct descendant
(through an all-female line) of, say, Cleopatra.  (A similar thing can
be done with Y chromosomes and all-male lines of inheritance.)
Furthermore, by mapping the geographical distributions of
mitochondrial DNA, we can trace out migrations of women over time.
(Ditto for Y chromosomes and men.)

It's good to ask the kids a few questions to see how well they
understand.  In this case, a girl said she was sorry for boys because
they had no mitochondria.  So we discussed that issue again: everyone
has mitochondria (that's how they turn oxygen into energy) but boys
won't pass theirs on to their kids.  Moms really do contribute more
than half, as immortalized by this song.

But there can be mutations. It turns out they're fairly rare in
mitochondria, probably because most mutations would be fatal very
early on.  But they do happen.  So if we gather mitochondrial DNA from
a large sample of people, we will find sequences that differ by a
little bit.  We should be able to trace the mutations backward and
reconstruct ancestor DNA.  For example, if we saw sequences GATTACA,
GATTACT, and AATTACA, we might guess that the ur-grandmother, many
generations back, of all three people had the sequence GATTACA.  One
mutation somewhere along the line would explain the people with
GATTACT, a different mutation somewhere else along the line would
explain the people with AATTACA, and the people who had never
experienced a mutation along their line would still have GATTACA.
They hypothesis of, say GATTACT being the ancestor is much less likely
because it requires that there was one mutation to make it GATTACA and
then, in the line with this mutation already present, there was
a second mutation making AATTACA.

So here's the problem I posed to the kids: reconstruct the ancestor of
these sequences:

CATTACGACT     
GAATACGACA      
GATTACAACT      
GATTACGACA      
GATTACGACT      
GATTATAACT
GATTCCAACT      
GTTTCCAACT      

Go ahead: print these out and cut them into strips, try to arrange
them as leaves of a tree, and guess what the branches and trunk have
to look like.

Some groups were lost, and so I tried to work it out with them on the
board, starting by making a guess about the immediate ancestor of one
very similar pair.  It turned out this was probably a bad guess,
because once we had worked out two hypothetical ancestors of two
different pairs, those two hypothetical ancestors seemed to have very
little in common, whereas we would have expected them to look similar
enough that we could guess a hypothetical original ancestor which
spawned them both.  Just as I was realizing that we were almost out of
time, another group handed me a sheet of paper in which they had
worked it all out.  The lesson I drew for everyone: don't be afraid to
take a guess, work out the consquences of that guess, and if it
doesn't work, scrap that guess and start over.  That's what science is
all about! (See the first minute or so of this video.)  Just because
lunchtime was coming up fast does not mean that we had done anything
wrong.  The wrong thing would be to continue pushing a guess which
doesn't explain all the evidence.

If I do this activity again, I would print out very large copies of
the sequence so I could rearrange them easily on the board (writing
with chalk does not lend itself to rearrangement).  Or I would print
it at regular size and use a document camera.  I would probably also
walk them through a simplified example first as I did in writing this
post.  Another idea I just had is to try representing the information
differently.  Perhaps a color code instead of letters would make
things just jump out. 

I saved a few minutes for the coolest part of this: because we know
that mutations happen about once every 10,000 years, we can use this
as a clock.  In my simplified example, you have to reverse-engineer
three mutations to get back to a common ancestor which explains all
the data.  That makes 30,000 years.  In real life with real data, you
have to go back 200,000 years, but you can do it.  That means that
there was one female about 200,000 years ago from whom every human
alive today has inherited their mitochondrial DNA; she is called
mitochondrial Eve.  This doesn't mean that other females living at
that time didn't contribute to people alive today; they surely did,
through their nuclear DNA.  But mitochondrial Eve is the only one who
has an unbroken female line to anyone alive today.  And a similar
argument identifies "Y-chromosomal Adam" who lived around 100,000
years ago.  We are all intimately related!

Toy DNA Analysis Part II

After the first activity and recess break, I posed the challenge of
finding out which animal is most closely related to the hippopotamus.
I gave them short (400-base) DNA sequences of various mammals and
asked them to figure out which was most similar to the hippo's.
Again, I just made up these short sequences to keep it simple.  In
real life, there are a lot more complications just as varying number
of chromosomes, billions of base pairs, etc.

The bottom line is that they had to take an animal, look at each
position in the sequence, ask whether that that animal's DNA at that
position was different from the hippo's, and total up the differences.
After repeating this for several animals, they would see which one had
the least differences.  After the near-chaos of the morning's first
activity, I thought it would be a good idea to go over the big picture
and assemble some pseudocode on the board based on their ideas:

get hippo dna
for each animal other than hippo:
    get this animal's dna
    compare this animal's to hippodna
    print number of differences

I emphasized basic ideas like what do we want to put inside the loop and what do we want to put outside the loop. A program like this does it:

hippodna = open('hippo.txt').read()
nletters = len(hippodna)

for filename in ('cat.txt','dog.txt','rhino.txt','bluewhale.txt','rockhyrax.txt','rhesusmonkey.txt'):
    dna = open(filename).read()
    differences = 0
    position = 0
    for letters in hippodna:
        if hippodna[position] != dna[position]:
            differences += 1
 position += 1
    print filename,'has',differences,'differences with hippo dna' 

Yes, Python experts, I know there are more efficient ways to do it but this seems most straightforward for a kid. I won't go into the detail I went into for the morning's first activity, but it was a similarly intense back-and-forth with students running into an obstacle every 30 seconds, which I tried to turn into a learning experience. They again needed emphasis on proper indenting, punctuation, remembering that if they named a variable 'difference' early on then they couldn't refer to it as 'differences' later on, etc. They again wanted to do one animal at a time rather than write a loop. But it was a really good learning experience.

This took close to an hour. At the end, we discussed how they might modify the program to see what's most closely related to some other mammal, and then a third mammal, etc, and build up a picture of how everything is related to everything else: a family tree. That was the emphasis of the morning's third activity.

Toy DNA Analysis Part I

Last time I visited the 5-7 graders at Peregrine School, I introduced
the idea of computer programming languages (specifically, Python) as a
way of automatic repetitive tasks.  That sounds boring, but I made it
interesting by building the activity around an inherently interesting
challenge---cracking a code---which happens to have a repetitive
aspect---translating each letter of a long message after you've
figured out the substitution pattern.  At the end, we had a little
time to discuss how DNA is a code, which is the connection I really
wanted to make to what they had already learned in biology.  This
week, we were poised to take it much further with three data analysis
challenges, two of them requiring Python programming.  As a result,
I'm going to split the morning's activities into three separate blog
entries.

The first challenge was to rescue a baby.  Some babies are born with a
genetic defect which does not allow their body to process the amino
acid phenylalanine.  If they don't follow a strict low-phenylalanine
diet, it builds up in their body and causes brain damage (mental
retardation) within the first year of life.  So there is a very strong
motivation to find out if the baby has this condition very soon after
it is born!  In practice, there is a blood test which does not involve
analyzing the DNA itself, but I framed the problem as one of DNA
analysis.  I made up long DNA sequences for about 15 babies, and asked
them to write a program which would look at a certain location within
each sequence, and print an alert if the sequence was not normal.  To
keep things simple, I just made up a simple condition for "normal":
the three letters starting at position 399 should be "GAT".

They had forgotten a lot in the two weeks since my first visit.  I
would normally recommend much more frequent visits for teaching
programming, but I was sick last week.  Still, they were very into it.
They wracked their brains trying to remember things, but I had to give
a refresher tutorial on for loops and if statements.  It took almost
an hour for everyone to finish completely, and there was again a
friendly competition to see who could finish (identify the sick baby)
first. 

The evolution of their programs was fascinating.  One group thought
they were done when they had something like:

dna = open('jane.txt').read()
print dna[398:401]

Even this simple program has a lot of ideas in it.  For example, doing
the subscripting correctly requires thinking about: (1) position 398
is actually the 399th position because Python starts counting at
position 0 rather than position 1; (2) 398:401 means up to but not
including 401; (3) you can't write dna[398-401] because the arithmetic
operation 398-401 yields -3, and Python will interpret this as 3
positions before the end of the sequence.  There was a lot of
interaction between me and the kids on each of these points.

In any case, when the first group had gotten this far, they thought
they were done.  They would just change jane.txt to john.txt, rerun it
to see if John was sick, repeat for Joan, etc, and just read off the
screen whether it said 'GAT' for that baby.  This would work ok for
the 15 examples I brought, but PKU (the aforementioned disease) is
present in only about 1 in 15,000 babies, so they really needed to
automate a loop over all the babies.  They initially thought I was
moving the goalposts, but the other groups agreed that this first
group was not really done.  So they started working on a loop:

names = ('jane.txt','john.txt','joe.txt')
for name in names:
    dna = open(name).read()
    print dna[398:401]

I had to repeat many times the importance of proper indentation so that it's clear
(to the computer and to a human reading the code) which actions get
repeated for each name, and which actions only get done once.  A
common mistake is

names = ('jane.txt','john.txt','joe.txt')
for name in names:
    dna = open(name).read()
print dna[398:401]

which prints only the last one, after cycling through all the names. Notice that I typed in only three names to make sure my loop would work, before bothering to type in all 15 names. In fact, I taught them how to avoid typing in any names by grabbing all names that fit a certain pattern:

import glob
names = glob.glob('j*.txt')
for name in names:
    dna = open(name).read()
    print dna[398:401]

I briefly discussed how it can be incredibly time-saving to import some functionality that someone else has already written. But back to the main point, the fastest group now thought that they were really done: this program would print "GAT" for each healthy person, and they just had to scan the output for something other than GAT. But, I asked them, how would they know which person had that abnormal sequence? They tried adding to the end of their program:

import glob
names = glob.glob('j*.txt')
for name in names:
    dna = open(name).read()
    print dna[398:401]
    print name

But this gives output like:

GAT
jane.txt
GAT
john.txt
GCT
joe.txt
GAT
jim.txt
GAT
...

and they incorrectly interpreted this as John having GCT, because they hadn't paid attention to the fact that they asked the computer to print the DNA first, and then the name. An important rule of programming is to make your output clear. Things which belong together should be printed together. A much better print statement is:

print dna[398:401], name

which keeps it to one line per baby:

GAT jane.txt
GAT john.txt
GCT joe.txt
GAT jim.txt
...

Now they really thought they were done, but I pointed out that they were looking for 1 in 15,000 babies, and they couldn't count on a human to scan a list of 15,000 lines which say 'GAT' and not miss an abnormal one. They really need to print out just the sick ones with a very clear warning:

import glob
names = glob.glob('j*.txt')
for name in names:
    dna = open(name).read()
    if dna[398:401] != 'GAT':
       print name, "is sick"

Now they were done: the output is simply

joe.txt is sick

I took a few minutes to talk about the subtleties of combining if's, for example the difference between

if dna[398] != 'G' and dna[399] != 'A' and dna[400] != 'T'

vs

if dna[398] != 'G' or dna[399] != 'A' or dna[400] != 'T'

Surprisingly (to me at least), they found this last point very easy. As I said, this whole activity took very nearly an hour. (I had saved a bit of time by preloading their computers with the data files.) It was highly interactive, and if there had been much more than three groups having another knowledgeable adult in the room probably would have been a good idea. I'm not going to pretend that as a result of this activity the kids have actually mastered even the very basics of programming, but it was good practice in logical thinking, and it's clear that with continued practice I will eventually get them doing some real data analysis. One good sign: one of the girls continued programming throughout the recess break following this activity!

DNA is Code

Today I started Friday morning sessions with the upper (5-7) graders
at Peregrine School.  I'll have about ten of these weeks before I
switch to astronomy with the 3-4 graders.  My main job with the upper
graders is to cover the geology standards which didn't get covered in
the lead-up to the Yosemite trip, but I'm going to take a few weeks
first to build on some of the ideas they learned before Christmas
break regarding genetics and DNA.  In the process, they will learn how
to analyze data and program computers.  It's ambitious, but with these
kids I have to be ambitious.

DNA is a code.  The letters themselves have no intrinsic meaning, but

they represent different specific amino acids which, when assembled in

the sequence specified by the code, fold up into a 3-d shape which is

useful for performing some function in the body.  So to introduce the

idea of "breaking the code," I started with a non-DNA message in code:

Uif Ivohfs Hbnft jt b 2008 zpvoh bevmu opwfm cz Bnfsjdbo xsjufs Tvaboof Dpmmjot. Ju jt xsjuufo jo uif wpjdf pg 16-zfbs-pme Lbuojtt Fwfseffo, xip mjwft jo uif qptu-bqpdbmzqujd obujpo pg Qbofn, xifsf uif dpvousjft pg Opsui Bnfsjdb podf fyjtufe. Uif Dbqjupm, b ijhimz bewbodfe nfuspqpmjt, fyfsdjtft qpmjujdbm dpouspm pwfs uif sftu pg uif obujpo. Uif Ivohfs Hbnft bsf bo boovbm fwfou jo xijdi pof cpz boe pof hjsm bhfe 12–18 gspn fbdi pg uif uxfmwf ejtusjdut tvsspvoejoh uif Dbqjupm bsf tfmfdufe cz mpuufsz up dpnqfuf jo b ufmfwjtfe cbuumf up uif efbui.

Uif cppl sfdfjwfe nptumz qptjujwf gffecbdl gspn nbkps sfwjfxfst boe bvuipst, jodmvejoh bvuipst Tufqifo Ljoh boe Tufqifojf Nfzfs. Ju xbt qsbjtfe gps jut tupszmjof boe dibsbdufs efwfmpqnfou, uipvhi tpnf sfwjfxfst ibwf opufe tjnjmbsjujft cfuxffo Dpmmjot' cppl boe Lpvtivo Ublbnj't Cbuumf Spzbmf (1999), cpui pg xijdi efmjwfs ubmft frvbmmz hsjqqjoh, zfu mftt ebsl uibo uif hfosf't psjhjobm nbtufs xpsl: Ljoh't Uif Mp oh Xbml (1979). Jo xsjujoh Uif Ivohfs Hbnft, Dpmmjot esfx vqpo Hsffl nzuipmphz, Spnbo hmbejbupsjbm hbnft boe dpoufnqpsbsz sfbmjuz ufmfwjtjpo gps uifnbujd dpoufou. Uif opwfm xpo nboz bxbset, jodmvejoh uif Dbmjgpsojb Zpvoh Sfbefs Nfebm, boe xbt obnfe pof pg Qvcmjtifst Xfflmz't "Cftu Cpplt pg uif Zfbs" jo 2008.

Uif Ivohfs Hbnft xbt gjstu qvcmjtife jo ibsedpwfs po Tfqufncfs 14, 2008 cz Tdipmbtujd, gfbuvsjoh b dpwfs eftjhofe cz Ujn P'Csjfo. Ju ibt tjodf cffo sfmfbtfe jo qbqfscbdl boe bmtp bt bo bvejpcppl boe fcppl. Bgufs bo jojujbm qsjou pg 200,000, uif cppl ibe tpme 800,000 dpqjft cz Gfcsvbsz 2010. Tjodf jut sfmfbtf, Uif Ivohfs Hbnft ibt cffo usbotmbufe joup 26 mbohvbhft, boe qvcmjtijoh sjhiut ibwf cffo tpme jo 38 uf ssjupsjft. Uif opwfm jt uif gjstu jo Uif Ivohfs Hbnft usjmphz, gpmmpxfe cz Dbudijoh Gjsf (2009) boe Npdljohkbz (2010). B gjmn bebqubujpo, ejsfdufe cz Hbsz Sptt boe dp-xsjuufo boe dp-qspevdfe cz Dpmmjot ifstfmg, xbt sfmfbtfe jo 2012.

This was a really good idea for motivating the students; they really
wanted to break the code once they realized it was doable! I elicited
from them some observations which might be useful:

• Each paragraph begins with "Uif".  They guessed pretty quickly that "Uif" might stand for "The".  This was a springboard for making  explicit what they already implicitly knew about word and letter frequencies in English.
• Numbers seem to stand for themselves: "Tfqufncfs 14, 2008" looks like a date.
• Ditto for punctuation: the periods and commas look natural, not like they stand in for something else.
• "b" is a word by itself.  Presumably it stands for either "a" or "I".

One student guessed the answer even as I was writing down the fourth

observation: simply replace each letter here with the letter which

precedes it in the alphabet.  This "b" becomes "a" and "Uif" becomes

"The".  I asked them to make some predictions to double-check this

hypothesis.  For example, this hypothesis predicts that "a" and "r"

will be uncommon because they stand for the uncommon letters "z" and

"q".  That checks out, so let's proceed.

Here's where I introduce computers.  It would be extremely tedious to

manually translate all these letters to decode the message.  But if we

write a computer program to do it, in a short time we will have a tool

that automatically translates this and all future such messages.

Computers are great for automating repetitive tasks!

Before we actually open our laptops, let's write some pseudocode.

"Code" has a new meaning here: "a set of instructions to the

computer", and pseudocode is when we write those instructions a bit

more like everyday language to help us organize and communicate our

thoughts to other humans.  To decode our message, we want to tell the

computer to do this:

for each character in the message:
    if character is a letter:
       print the previous letter
    else:
       print the same character # (because it's punctuation or a number)

(The # symbol and everything after it represent a comment about the instruction, rather than an actual instruction.)  This pseudocode illustrates two of the most important ideas in programming: looping, in which we repeat some set of instructions over and over, and if-then-else.

A somewhat old-fashoned way of constructing a loop which does something 100 times is:

tasksCompleted = 0
while tasksCompleted < 100:
      doSomething
      tasksCompleted = tasksCompleted + 1

Although this is a bit old-fashioned, I thought it was worth making the loop logic very explicit for beginners. Next, we practiced some simple loops in an actual language, Python.  I gave them a file containing the coded message, named 'coded1.txt'.  I showed them how to read it in:

letters = open('coded1.txt').read()

Now letters is the name of a location in memory that contains the message.  A very simple loop is:

for character in letters:
     print character,

With this more modern type of loop, Python automatically keeps track of how many times we need to repeat.  This is easier to write, easier for others to read, and less error-prone.

So far we have just repeated the very simple action of printing that character.  By the time each group had finished this, the 45 minutes was up and we took a break. After the break, we set about putting more complicated instructions inside the loop.  We started writing something like:

for character in letters:
    if character == 'b':
       print 'a',   
    elif character == 'c':
       print 'b',

(Note that == is a test of equality, whereas = would have been an assignment, forcing them to be equal; and elif is shorthand for else if.)  We can see that this will become tedious if we have to type in an if-else block for each substitution rule, so let's find a better way.  Let's set up a dictionary named sub (short for substitution):

sub = {'b':'a',
       'c':'b',
       'd':'c',
       ...}

so now we can just say

print sub[character]

and if the character is 'b' the computer will look up the "definition" (the value, really) stored at the dictionary location of 'b', which is 'a', and print that.  (Note that the ... above is not real code, but you can guess what it stands for.)  We spent a while talking about the software concept of dictionaries and playing with them for a while before returning to the decoding task, which now looks like:

letters = open('coded1').read()
sub = {'a':'z',
       'b':'a',
       'c':'b',
       'd':'c',
       ...}

for character in letters:
    if character in sub:
        print sub[character]
    else:
        print character

As a test, we did this after typing in just some of the substitution rules.  We could see that it was going to work, but it didn't handle capitals: "Uif" became "Uhe". Instead of typing in another 26 upper-case substitution rules, we did this before entering our main loop:

for key in sub.keys():
    upperCaseKey = str.upper(key)
    upperCaseSub = str.upper(sub[key])
    sub[upperCaseKey] = upperCaseSub

str.upper is a function which makes an upper-case version of the letter which is given to it inside the parentheses.  So this block of code creates a new upper case dictionary entry and value for each pre-existing one, so we don't have to type them in manually.  (Yes, I know there are more efficient ways to do this, but I wanted to make the logic very explicit for the learners.) With that done, the race was on as each group rushed to be the first to type in all the substitution rules and read the message.  That was really fun.  If you want to see the message, you'll just have to assemble your own Python script out of the parts I've illustrated here!

Once all the groups had decoded the message, we had used about 45 minutes since the break.  A word of advice to teachers: it takes a while.  Although I explained several times how one typo or indentation mistake will torpedo the script, kids are not used to this, and they need a lot of help finding their mistakes even though I had my own example code projected up on the screen. But it was really worth spending the time.  They started to get the idea of debugging; although I often had to find the bugs for them, they started to understand the process.  I had the luxury of doing it with six kids in three groups; if you have a larger group I would try to arrange for assistants and give them some advance training.

That gave us about 12 minutes to discuss how DNA is a code.  We reviewed a bit about how letters code for amino acids, which build proteins.  Then we looked at the DNA code table and discussed by some amino acids have many codons (groups of three DNA letters) which code for them, while others have just one.  The basic idea is that some "typos" will not actually result in a mistake in the amino acid, which is a good thing; it adds robustness to the code.  And it's even more robust if the most common amino acids are the ones with the most robustness against typos, in other words have the most different codons.  That led to an interesting discussion of where DNA "typos" might come from, including cell reproduction and radiation.

Finally, I asked why we need three letters to represent an amino acid if we have all these leftover three-letter combinations.  If we have four options (G,C,T, or A) at each DNA location and we have three consecutive locations, how many total options are there? The best way to think about this, if you are not used to mathematical manipulations, is to draw a tree with four big branches representing the four options at the first decision point.  Each of those big branches then has four medium-size branches representing the four options at the second decision point.  And each of those 16 medium-size branches then then has four little branches representing the four options at the third decision point, which makes 64 little branches, representing 64 possible combinations.  There are only 20 amino acids, so with 64 code words we can have a lot of redundancy, leading to a lot of robustness against damage.  The last question we discussed was: since we have so many extra code words by using three-letter combinations, could we (robustness aside) get away with using just two DNA letters to code for an amino acid?  I'll let you figure that one out for yourself.

Modeling Colliding Galaxy Clusters and Dark Matter

Science is about making models.  We make a conceptual model of
something (in other words, we guess how it works), then we figure out
what that model would predict, we compare those predictions to
observations, and then we either discard, modify, or (for the time
being) accept that the model could be correct. (Recently I stumbled
across an entertaining video of Richard Feynman explaining this
method.)  Ideally, the model is conceptually simple, with as few
moving parts as possible, yet is able to match a rich variety of data.
Sometimes this modeling process is about something really fundamental,
such as how gravity works; other times the modeling process is just
about estimating something we can't measure directly.  For example, a
scale which doesn't read the correct weight can still be used to infer
your weight, as long as you have a workable model such as "reads about
ten pounds too heavy" or "reads about half the true weight."

My student Will Dawson recently finished some work which is a really nice example of the latter kind of modeling.  We have been studying colliding clusters of galaxies; our lives are too short to see the collision play out, so we have to make a model which allows us to extrapolate from the current (just post-collision) state back to the time of maximum impact.

A good analogy: given a photograph and a speed gun reading taken a
split-second after an automobile collision, reason backward from that
to infer the velocity of collision and the time since collision.  The
main difference is that the galaxy clusters mostly go right through
each other because the space between the galaxies is so big.  The hot
gas clouds filling those spaces do collide, leaving a pancake of hot
gas in the middle while the galaxies continue on (eventually to slow
down and turn around due to gravity).  We are interested in what
happens to the dark matter: the Bullet Cluster and the Musketball
Cluster show that it is mostly collisionless, but "mostly" is
something we really want to quantify, for reasons I'll explain in a
future post.

But first, we have to make a model of the collision.  Speed guns and
spectrographs can only tell how fast the galaxies are moving toward us
or away from us; they say nothing about how fast they are moving in
the transverse direction (in the plane of the sky).  To study dark
matter we need to know the full three-dimensional velocity, and we
want to know what it was at the time of collision, rather than what it
is right now, after the collision.  This is closely related to how
much time has passed since the collision (by collision I really mean
the maximum overlap of the two clusters), because the observed
separation since the collision could have been achieved by moving at a
high velocity for a short time, or moving at a low velocity for a long
time.  Making things more complicated, the separation we observe is
only the transverse part of the separation.  So a collision which
occurs along the line of sight will give us a large velocity on our
speed gun but a small apparent separation, while the same collision
viewed transversely will exhibit a small part of the velocity and a
large part of the separation.  We don't know what angle we are viewing
the system at, so the true velocity could be just about anything.

Like a rock climber inching up a chimney, the way out of this is to
push against two extremes.  If we observe any line-of-sight velocity,
the motion can't be completely transverse; in fact we can rule out a
range of near-transverse geometries because they would require an
absurdly large three-dimensional velocity.  (Absurdly large is defined
here as larger than the free-fall velocity from infinity.)  Similarly,
if we observe any transverse separation we can rule out a range of
nearly line-of-sight geometries because they would require absurdly
large three-dimensional separation.  (Absurdly large is defined here
as requiring longer than the age of the universe to reach that
separation.)

Still, we are left with a range of geometries in the middle; at the
extremes of that range the geometries aren't completely ruled out, but
look pretty unlikely.  Here Will applied another important concept:
marginalizing over all remaining possibilities.  His code ranges over
all possible geometries, tabulating how well they match the data, and
thus produces a probability distribution.  So we don't know exactly
how fast the collision was, but we can be 99% confident it is within
have a certain broad range, 90% confident it is within a certain
smaller range, etc.

This was a pretty big advance in sophistication compared to the way
previous studies had estimated the collision velocity and time since
collision.  Using this technique, Will demonstrated that what
astronomers don't know can hurt them---not knowing the angle from
which we are viewing a collision results in substantial uncertainty in
the quantities we need to know to study the behavior of dark matter.
To narrow things down to a useful range, we need additional
information about a collision.  (In the case of the Bullet Cluster,
the shock gives this information, but most collisions have not shown
an observable shock.)

Will also used his new technique to estimate that it has been about
700 million years since the Musketball Cluster collided, compare to
"only" 200 million years for the Bullet Cluster.  This is good for the
study of dark matter because people would be justifiably skeptical of
conclusions about the nature of dark matter based on just one kind of
collision (a recent, high-speed collision of massive clusters such as
the Bullet).  Studying a range of collision conditions--including less
recent, lower-speed collisions of less-massive clusters such as the
Musketball--gives us a much better chance of identifying universal
properties of dark matter with high confidence.