I did several attempts at writing this, it was very hard to put it into words without pictures and the screen saver. After all that is very difficult chemical stuff if you want to go through it in detail. Maybe we can exchange e-mail addresses, that would make exchanging links and texts much easier. Ok, but now for the answer:

You mistake the �side chains� as chains like those you see in your Rosetta graphics, but that is not correct. The Rosetta graphics only shows the backbone. The side chains are better called residues as they are part of the amino acids. Every amino acid has exactly one residue. Think of the amino acids as magnetic sticks with a flag on them. You will always connect north pole and south pole of the magnets � that is the backbone. The magnets themselves are indistinguishable: the always consist of the atoms

��.[-N-C-C-O-][N-C-C-O-][N-C-C-O]�..

On each magnet you have one flag (the residue). It is always hanging on the C atom that is directly to the right of the N atom. This flag determines the nature, the name and the chemical properties of the amino acid (as the part in the backbone is indistinguishable from its neighbour). It is important to be aware that it is the stick plus the flag that makes one amino acid. The order in which the amino acids follow one another is called the sequence. This sequence is called the �primary structure�.

There are 20 such flags with different chemical properties and thus there are 20 different amino acids. Some are similar and can form bonds: lets say the flags have numbers and it is the flags 1 to 7, 8 to 14 and 15 to 20 that can form bonds to flags from the same group.

Ok, now look at your Rosetta graphics. You see only the backbone, no residues at all, that is important. If you would stretch the backbone to a long ribbon, it would have a zig-zag shape which remains; you cannot stretch that to a straight line as the angles of the bonds in the backbone are more or less fixed. So you must rotate around those bonds to change the shape. Bends are only achieved by rotating a number of adjacent bonds, not by changing angles.

Now imagine that the different side chains are trying to get near to ones that they can form bonds with. Often they can do so with close neighbours. One of the results from that are the helices you see in the graphics: as the zig-zagging remains intact, one good way to get neighbouring parts together is to form these helices. Such neighbouring structures are called the �secondary structure�. There are different sorts, but they will not bother us here.

After that secondary structure is formed, the helices, sheets, hairpins an whatever will still have unbound residues and will now continue to bend in order to form bonds to adjacent parts. When finally all bonds are optimally formed and only a few side chains are still �dangling outside� of the protein without a partner you have the final form of the protein: its 3D structure, also called the �tertiary structure�.

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The process of reaching that structure is called folding as you kind of have a ribbon that folds together. Rosetta folds proteins by sifting through numerous possibilities of how they can bend and bond and calculating the overall energies of each structure. It does not say anything on the way that structure is reached step by step in nature, that is what Folding@home does.
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Describing protein design is very hard. I'll just try to give an impression. Think of it like you would have to take a motor out of a car and you needed a tool that takes apart all screwes at once, not one by the other. That is how proteins work: snap on and do.

Now you have several parts to assemble that tool from, your amino acids. Of course you will first put all wrenches in place and then build a rig aorund that that hold all those wrenches. The wrenches are the residues that do the job, the rig is the backbone. The overall shape of the rig is determined by the residues of the other amino acids that interact with each other, not with the motor.

There are two ways to construct that rig: you may just assemble it roughly by putting the wrenches in place and assemble the rest of the amino acids around that. The angles will then not all exactly be in accordance with the zig-zag of the backbone, so it will twitch here and there, there will be tensions, but it will fit and work. That is fixed backbone design.

If you now go through a second step in which you calculate the shape your rig really takes and after that redesign it a little by using some better fitting lengths of wrenches, so that there are less tensions and twitches, you do flexible backbone design. It is most efficient if you repeat it a few times, thus reducing the tensions to a minimum.

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On �wild types�: they are the types occurring in nature. What the sentence means is that they created two new proteins. These proteins were designed in a way that they should only react with one another and not with the natural occurring type anymore. So you now have two pairs of proteins that can be in the same cell without interfering with one another.

One application is this: you want to cut out a specific part of a DNA that contains a defective gene. So you design a protein that does exactly that. Then the DNA is repaired with a functional copy of that gene. After that you want to stop the protein from cutting out the gene again, as the task is completed. So you need to design a protein that exactly fits into the first one and remains in it, thus blocking its function.

But, as there are numerous specific proteins that cut out DNA and all of them have their own special, vital task, you need to prove that you can produce a protein that will definitely only stop the other artificial protein and not the natural ones too, which might be extremely harmful. That is what they showed: the wild types bond to one another and the designed types bond to one another, but no cross-bonding. It is to be hoped that someday that can become a pretty safe method if done carefully.