We describe one application of our sequence design procedure to defining the subset of sequence space compatible with a protein structure. For a complete list of applications, see [1-5].

Application: Defining the subset of sequence space compatible with a protein structure [5]

The sequences of naturally occuring proteins are defined by evolutionary selective pressure, which is controlled by a fine balance of function, stability and kinetics. While most random mutations of sequences are unlikely to enhance stability or function, they can be accepted by natural selection as long as they are neutral (or near neutral). As a consequence, the size of the sequence space compatible with a given protein fold is very large (although small compared to the full space a protein sequence can explore, whose size is 20^N, where N is the number of residues in the protein).

The 32,000 protein domains contained in the PDB as of March 2001 can be clustered into 564 different structural families of folds, and the size of these families are found to vary greatly [6]. A large number of these folds have a single representative, whereas other folds, such as the TIM fold or the Ig fold, have hundreds of representatives in the PDB [7]. The question arises whether these differences are a consequence of variations in function, in stability, in evolution, or in all three of the above. Our approach to provide an answer to this question is based on computational protein design. We propose to measure the size of the sequence space compatible with a protein structure.

• Two proteins of similar lengths can accomodate subsets of sequence space of different lengths



The size of the sequence space trivially depends on the size (i.e. number of residue) of the protein considered. The size of the protein however is not the main determinant in defining the sequence space. To assess the roles of other factors, we designed in computer experiments families of sequences for two proteins os similar size , 1ctf (68 residues) and 2hsp (71 residues). The protein 1ctf is a small, highly stable protein whose fold is nearly unique. On the other hand, 2hsp adopts an SH3, which is observed in many unrelated proteins in the PDB. The diversity of the families of sequences compatible with 1ctf and 2hsp is defined by an entropy measure [12]. For each protein, the calculated design entropy is compared with the observed structural entropy, computed from the family of proteins in the PDB whose structure is similar to the structure of the protein of interest.






The entropy in sequence space of each residue of 1ctf and 2hsp is plotted against the position
of the residue in the sequence of the protein. The continuous line shows the calculated design entropy, whereas
the dashed line shows the average entropy per residue derived from natural sequences known to adopt the fold of interest.





From this experiment, we note that the sequence spaces compatible with two proteins of similar size can have significantly different sizes. It is also noteworthy that the calculated design entropy correlates well on average with the observed structural entropy.




• Geometry and stability define the subset of the sequence space compatible with a protein structure



The procedure described above was repeated on 11 proteins. These proteins vay in size from 56 residues to 310 residues, and cover all four classes of protein folds. For each protein, three measures of the size of its sequence space were derived.

1. A design entropy S_des is computed from a family of 100 sequences designed by computer experiment for the protein fold.


2. A structural entropy S_str is derived from the sequences of the proteins in the PDB known to adopt the fold of the protein of interest.


3. A sequence-only entropy S_seq is derived from sequences detected to be similar to the sequence of the protein of interest, using Psi-Blast [8].
   

The three measures of entropy are compared in the figure below.





The sequence entropy [S_seq, *] and the design entropy [S_des , o] are plotted agains the structure entropy S_str for a data set of 11 proteins.
The line represents the first diagonal, i.e. the line where the different measures of entropy would be equal.
The correlation coefficient between S_seq and S_str is 0.92, whereas the correlation coefficient between S_seq and S_str is 0.99.





For most proteins, the entropy derived from sequence information compares well with the entropy derived from structural information. Proteins 1tim and 1ede are two major exceptions. In the case of tim for example, PSI-blast identifies only triose phosphate isomerases as similar to the native sequence of 1tim, whereas a large collection of sequences of proteins with the tim-barrel fold but unrelated to a triose phosphate isomerase are included in the computation of the structure entropy.

In comparison, we do find a striking correlation, both qualitative and quantitative, between the entropy derived from our designed sequences (S_des), and the entropy derived from naturally occuring sequences known to share the same fold (S_str). The designed sequences are derived from the knowledge of the 3D conformation of the backbone of the protein (its topology or geometry) and the minimization of the free energy of the sequence for that protein. We consequently state that it is the topology of a protein, its length and its stability that predominantly define the size of the sequence space that is compatible with its structure.