1-3. Generate some ideas of what your protein may look like. These will become
your ``sanity checks" for later, once you start generating possible models 
on the computer. 

1. Make a physical
bead-on-strings model of protein T 
using colored beads of any sort threaded through.
Explore various conformations in 2D following the basic rules below. 
 Arrive at conclusions about which configurations
in space might be energetically most preferable == lowest energy. 

2. Assume that these amino-acids (AA): 
Asp, Arg, Lys, Glu, Asn, Ser, Gln are hydrophilic (polar), 
and the rest are hydropobic. See old hydrophobicity scale (Wiki). 
Assume the following rules. When 2 hydrophobic AA make a contact, the 
protein energy drops down by 2; when a hydrophobic and a hydrophilic make a contact, the drop is 1, and when two hydrophilic ones are in contact, the drop is 0.1.
That is the total energy E = Sum_ij [pairs of amino acids], where each pair 
contributes either -2, -1, or 0.1, as defined above. Assume that for the short 
protein we will play with in this class, amino-acids can 
only form pairwise constacts, that is no triplets. 
 


3. Note that you have an Arg and an ASP in your sequence, and they are 
not adjacent. Will these want to be next to each other in space? 
How would you account for it in your energy function?  Note that these 
are not just hydrophilic, there is something else special about them. 


Skim through this paper: 

Emergence of preferred structures in a simple model of protein folding.
https://www.ncbi.nlm.nih.gov/pubmed/8662562

What you are doing is a highly simplified version of what they have done. 
If you prefer, you can write a code that searches for the minimum energy 
of the protein on 2D square lattice using the same rules as in the paper, 
and present the results in lieu of the simple bead-on-strings model. 
 

4. For the part where you explore folding funnels, 
construct analytical, differentiable 
functions z = z(x, y) [or z(r, phi) if you use cylindrical coordinates ]  
that produce "realistic looking" folding funnels 
(e.g.
http://wavefunction.fieldofscience.com/2011/06/protein-folding-funnel-and-its.html
)
with great many local minima.  In fact, you might want to start with 1D 
funnels y = y(x). 

Then think which of the "standard" 
minimization method(s) might work. I highly recommend that 
you buy "Numerical recipes in C" book -- a bible in this field. Most 
of these methods are implemented in Mathematica, you can test them out easily.
Do not code them yourselves, it will be a waste of time.  
Visualize results!  Start with the simplest, such as Newton or Steepest
descent. You want to make a list of 5-10, discuss their pros and cons, 
and test them out on your funnels. 

I suggest that you do the funnel exploration concurrently with the 
"beads on string" toy modeling.  

By the end of the day, you need to have a pretty good idea which method(s) might work within 
a month at most, to allow just enough time to learn how to
fold a real protein using real scientific software AMBER. 

5. I will prepare the unfolded protein in the correct input format for AMBER. 
Your goal is to figure out which method to use, what are the control
parameters, and how to run the code. Folding will be through trial and error: 
you will try several parameters to see which ones might work. So by 
this point you will need
to have a pretty good idea what is computationally reasonable on your
supercomputer, and which method(s) might work. There is no point wasting your
precious time [and contribute to global warming] by running 
algorithms that are apriori not good enough to handle the 
"grand challenge" of computational science. 

Specifically, several differentiable functional forms E(x_i,y_i,z_i) that 
define a realistic funnel as a function of atomic 
coordinates x_i,y_i,z_i i=1,N are
available in AMBER. You will not have 
to worry about coding up anything. You can test out a couple of options for 
E(x,y,z) [set by igb=5 or igb=8 parameter], 
and choose the specific minimization option and its parameters. But 
remember to compare energies of structures only within one or the 
other energy function, that is do not compare energy value from between 
igb5 to igb8.  
You will be visualizing the folding trajectories via VMD, checking your 
simulation outputs often.