MATH 360 PROJECT: EMPIRICAL MODELLING OF CHEMICAL REPLICATION

	How do enzymes work? How do molecules of DNA and RNA replicate
themselves? Chemists and molecular biologists have found that the
basic process is that the enzyme or "template" molecule has a surface
with holes which are well matched to the shape of just one compound
(such as an amino acid) found in the "soup" in which the template is
located. When all the pieces of the molecule to be formed have been 
attracted to the template, the molecule returns to the surrounding
environment and the template is unchanged, ready to attract more pieces.
	One part of this procedure which is difficult to  describe is
just how the template arranges to attract just the piece it needs to
complete the molecule, since the various pieces flow past randomly. In
principle, the process is the result of the strength of the electrochemical
bonding forces, but these are extremely difficult to calculate in
complex molecules. In this project, we will try to determine empirically
how the template "decides" where to place an incoming molecule.
	We will use a computer simulation of the attraction of pieces
to a template. The environment will consist of a two-dimensional empty
rectangle 10 units wide and 20 units high; the template will be the 
bottom edge. We will represent the pieces floating by as "compounds"
consisting of four "atoms" (unit squares). The molecules to be formed
from these will be the horizontal rows (10 units wide).
	So our representation of this process is as follows. One of the
4-unit pieces will appear at the top of the environment. The template
must "decide" how to move the piece left or right or rotate it so that
it can fit in the space adjoining the template. Whenever a horizontal
row is filled up, it is removed from the environment (and the remaining
"atoms" shift down closer to the template. 


		EEEEEEEEEEEE
		E          E
		E          E       E=Edge of environment
		E          E	   C=New compound to be placed
		E  CCC     E	   A=atoms not yet part of replicated molecule
		E    C     E   	   X=desirable loaction for new compound
		E	   E	   T=template
		E          E
		E          E
		E AAA      E
		EAAAAAA XXXE
		EAAAAAAAAAXE
		EAA AAAAAAAE
		EAAAAAA AAAE
		EAAAAAAAAAAE
		EAAAA AAAAAE
		EAA AAAAAAAE
		 TTTTTTTTTT

	Now the problem is to take this simulation of the process of 
molecular replication and to figure out the pattern of how the template
can decide where to place the incoming pieces to make full molecules 
without cluttering up the environment with pieces waiting for completion.
We will say we have understood the process well when we can make the
decisions for the template for a long time without having the environment
fill up with incomplete molecules.

1. How many different "compounds" are there in this simulation? (Count
two the same if they are identical after shifting  or rotation, but
different if they would need to be flipped out of the plane to become
identical).
	How many different ways can the template choose to arrange each
piece before drawing it in closer? (The arranging is down with 90-degree
rotations and whole-unit shifts left or right).

	In the computer lab we have left a program to carry out the
simulation. You can run this progam in one of several modes. Try first
playing the role of Active Template: when each piece appears, you must
shift it left or right, or rotate it, and then drop it into place.
This will give you a sense of what constitute good practices on the part
of the computer.

2. Is it conceivable that the environment might look like the sample
illustration above at some point in the process? What kinds of configurations
are impossible in this simulation?

	Each of the different possible arrangements of a piece will
give a different resulting configuration, after the piece is dropped as
low as it can go to the template and full molecules are removed. Some
of these resulting configurations are "bad" - they leave the environment 
cluttered up with incomplete molecules that will be difficult to complete
when more pieces show up. It is the role of the template, then, to choose
the arrangement of the new piece in order to get the "best" of the possible
resulting configurations. Our goal will be to advise the template how
to recognize one configuration as better than another.

3. Describe characteristics of a configuration that make it worse than 
another. (For example, a "hole" - a vacant grid location with atoms above 
it - is one such bad characteristic).

	In the Adviser mode of the simulation, the template will indeed
automatically choose the arrangement that yields the best resulting 
configuration - but you need to tell it what "best" means. The program
will list a variety of characteristics of a configuration, such as those
you listed in Problem 3. Your job is to assign a penalty to each of those
characteristics (e.g., you could choose to give a penalty of 10 points
per hole). The program will then assess this penalty each time the
characteristic appears in the configuration. The configuration with the
fewest penalties  will be considered the best, and the template will
arrange the given piece so as to attain that configuration in the end.

4. Suppose one set of penalties has already been assigned. What would happen
if you ran the simulation again with all the penalties doubled?

5. Run the simulation in Adviser mode ten times with ten different choices
of how to assign penalties (you and your teammates can each do a few
of the runs). Note your success rate -- how long does each simulation
run? Which appear to be the most important characteristics to penalize? 

	The simulation program also allows you to create a file (using a
random process) which can then be used to make the pieces appear in the
sequence in more than one run of the simulation. The is called
"Adviser-nonrandom" mode. Use it when you wan to compare the efficacy of
more than one assignment of penalties for a given sequence of pieces.
You will need to keep a diskette in the A: drive to hold the file. 

6. Use your results from Prob. 5 to select a good assignment of penalties,
and run the simulation in Adviser-nonrandom mode with this assignment. Then,
if the results are encouraging, use the same mode, with the same file
of pieces appearing, but with somewhat different penalties assigned. Do
this five or ten times and note how long you were able to keep the 
environment clear in each case.

7. Use Minitab on your data to try to predict the "score" as a 
function of the various penalties you assigned. (Hint: by Prob.4,
you can and should keep one of the penalties the same in each of your runs.)

8. Using the linear function you create, find a better assignment of penalties
than you used in Prob. 6. (Since the function you created in Prob. 7 is
only an approximation, you should not choose an assignment which differs
from the previous ones by more than they differed from each other.)

	You may wish to repeat problems 6, 7, and 8 to improve your
performance even more.
	Keep in mind that you are optimizing the penalty assignments
for one particular file determining the appearence of the pieces. Slightly
different results may be obtained from a different file.
	You may also wish to compare your results when you advise the
template in this way to the results you obtain from choosing the arrangements
of the pieces yourself.


	Bring your best results to clas on the day of the project
presentation. We will run the simulation in Adviser-Nonrandom mode with
the same file selecting the pieces for all five teams, and compare results.