Foldable sequence

The various folding mechanisms expected in foldable sequences may be classified in tenns of the (Sj,. We have already shown that sequences that fold extremely rapidly have very small values of Gj,. Based on our study of several model proteins as well as analysis of real proteins we classify the folding kinetics of proteins in the following [7]. 

Hagihara, Y. and Kim, P. S. (2002) Toward development of a screen to identify randomly encoded, foldable sequences. Proc Natl Acad Sci U S A 99, 6619-6624... 

In line with our desire to outline only the most basic physical principles, we will discuss just one question how many foldable sequences are there The full importance of this question will become apparent in Chapter 14 on the origin of life problem. For now let s just make an estimate. For that, we have to understand better the formula (10.2). We said Eg was the lowest energy (ground state) of a random sequence heteropolymer. But... 

Another similarity, as well as the difference with real proteins become apparent if we compare the number of foldable sequences with the total number of possible conformations. The former number, as we have just seen, is not greater than 98,514 the latter number (conformations of the lattice 27-mer filling the 3 3 3), as we mentioned already (Section 10.10), is 103,346. The conclusion is that there are some sequences which can fold into two (or maybe even more) different conformations This is indeed similar to the story we told in Section 10.7. Of course, the analogy with proteins is limited, for instance, in proteins the number of sequences is larger than the number of possible conformations. Still, the snake cube is a great fun for someone thinking about proteins ... 

Figure 1 One-dimensional representation of (a) a rough energy landscape that is typical of frustrated sequences. The landscape is characterized by numerous low-energy minima separated by high-energy barriers. The energetic bias 8E to the global minimum is of the same magnitude as the roughness of the surface AE (5E AE). (b) A funneled energy landscape of a foldable sequence. Here 5E AE and the native state is both thermodynamically and kinetically accessible.

In the end, protein design is the choice of sequences. The number of possible sequences is astronomical - 20 or 10 ° possible sequences exist for a small protein of length 100. On the other hand, it is believed that only between 500 and somewhat more than 1000 distinct protein families, and thus folds, exist.How many sequences then exist for a given fold A. Finkelstein has estimated that the number of foldable random sequences is on the order of one in 10 and this number appears to be supported by experiments attempting to express random sequences. If we exclude detectable homology - a sequence of length 100 has on the order of 10 similar sequences at a level of >20% identity - we still remain with 10 ° distinct, foldable sequences for each fold. Thus the mapping of sequence space into structure space is extremely redundant. 

The key question we want to answer is what are the intrinsic sequence dependent factors tliat not only detennine tire folding rates but also tire stability of tire native state It turns out tliat many of tire global aspects of tire folding kinetics of proteins can be understood in tenns of tire equilibrium transition temperatures. In particular, we will show tliat tire key factor tliat governs tire foldability of sequences is tire single parameter... 

Figure C2.5.10. The figure gives tire foldability index ct of 27-mer lattice chains witli sets containing different number of amino acids. The sets are generated according to scheme described in [27], The set of 20 amino acids is taken as a standard sample. Each sequence witli 20 amino acids is optimized to fulfil tire stability gap [5]. The residues in tire standard samples are substituted witli four different sets containing a smaller number of amino acids [27]. The foldability of tliese substitutions is indicated by tire full circles. The open diamonds correspond to tire sequences witli same composition. However, tire amino acids are chosen from tire reduced representation and tire resultant sequence is optimized using tire stability gap [5].

The reader should of course notice that this lattice model is very closely related to the coarse grained view of proteins that we discussed earlier in this Chapter. Physicists often call lattice models toys — but this is actually just a joke of restraint humor is the best way to avoid pompous seriousness which is incompatible with science. Lattice proteins is a very serious business. They are used to test the theories, to get the hints on how to improve the theories. For instance, we mentioned that only properly selected sequences are foldable this idea was tested with lattice models and beautifully proved right. 

This leads us to the question are all of the possible almost 17 million sequences foldable The answer is, of course, no. For instance, if there are even two [Pg.217]

Each of these may be evaluated with an objective function that quantifies how well a given design objective is fulfilled, either in absolute terms, or relative to known protein structures in the database. But it is not at all clear in which context which objective should have precedence and how to scale the relative importance of each. Essentially, each of these requires a relative weight and there is not much to suggest how this should be sensibly chosen. Even in case it should be possible to satisfy all of these recognized principles simultaneously, it remains to be demonstrated whether such a synthetic sequence will actually encode a foldable polypeptide. 

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