Properties and origins of protein secondary structure.

Proteins contain a large fraction of regular, repeating conformations, called secondary structure. A simple, generic definition of secondary structure is presented which consists of measuring local correlations along the protein chain. Using this definition and a simple model for proteins, the forces driving the formation of secondary structure are explored. The relative role of energy and entropy are examined. Recent work has indicated that compaction is sufficient to create secondary structure. We test this hypothesis, using simple non-lattice protein models. 87.15.By Typeset using REVTEX 1 Recently, there has been a great deal of interest in the study of proteins from a physical perspective [1–6]. Most of these works have focused on the folding problem; i.e., how does the sequence of amino acids encode the three-dimensional structure of the protein. Although progress has been made in this area, there is still a long way to go before there is a complete understanding of how proteins fold. However, proteins have many other interesting properties. While each protein has a specific structure determined by its sequence, all proteins share several common structural features. They are highly compact, with very little free internal space. More striking is the high degree of order found, which consists of regular periodic arrangements of the main chain into one of a few universal patterns (called secondary structure). Roughly 50% of the structure of all proteins is in some form of secondary structure [7]. In this paper we define in a simple, generic way precisely what secondary structure is. This definition will be valid not only for proteins but for simpler polymers and simple protein like models. We then use it to investigate what forces are responsible for the formation of secondary structure. Although this is not directly related to the folding problem, a thorough understanding of what factors are responsible for secondary structure may aid in the study of the folding problem. There has been a great deal of past work attempting to understand the origins of secondary structure. At first it was believed that local interactions (local hydrogen-bonds or dihedral angle potentials for example) were responsible. Here, the term local means close with respect to the separation along the polymer chain. For example, a hydrogen bond between monomer i and i + 4 would be a local interaction, as would an angle potential. Several recent studies indicate that local forces may not be the dominant effect, rather compaction of the chain may be the important factor. By examining exhaustive enumerations of short chains on a lattice, Chan and Dill [8–10] found that as the compactness of the chains increased so did the percentage of secondary structure present. They also found that the maximally compact chains had roughly the same amount of secondary structure as real proteins and the proportions of helices to sheets was also approximately the same. Subsequently, Gregoret and Cohen [11] studied non-lattice models. Their results also suggest that 2 compactness does influence the amount of secondary structure, but they indicate that the effect is most pronounced at densities 30% greater than that of real proteins. In both of these studies however, local interactions were present. For example, a lattice has a specific set of allowed bond angles, which provides an effective bond angle potential. In the nonlattice work, compact chains were generated using a biased random walk in which the bond angles were chosen not from a uniform distribution but from the distribution observed in real-proteins. This also provides an effective angle potential. Therefore, it is not clear from these works whether compaction is sufficient to generate secondary structure. We wish to determine whether compaction, without local interactions, is sufficient. There are two distinct questions to keep in mind: (1) why do proteins (or other polymers) form regular structures and (2) why do proteins form particular types of secondary structure. Question one is equivalent to asking, why do proteins form helices and sheets. The second question asks, why are these helices α-helices and the sheets β-sheets. The answer to the second question certainly involves local interactions. It is the specific hydrogen bonding patterns in proteins which favor the formation of α-helices. In other polymers, different local interactions would favor other forms. For example, the structures of 179 polymers have been solved and 79 are found to be in one of 22 different types of helices [9,12]. In each polymer the specific types of local interactions determine the preferred type of secondary structure. In this work we are interested in studying the first question: what forces are responsible for formation of regular structures. Specifically we will test the previous suggestions that compaction of the chain is the key driving force. To do so we will be using models without any local interactions. However, without local interactions there is no way of knowing before hand what types of secondary structure will be formed. Most definition of secondary structure are specific to a given type of structure (i.e. α-helices), consequently one needs to know a priori what types of secondary structures will occur in order to detect their presence. To overcome this problem we developed a generic method of determining whether secondary structure is present without the need to know a priori what its specific form is. A simple way of defining secondary structure is to realize that it consists of repeating 3 patterns. Consequently the polymer chain should be correlated with itself along the chain. The correlation length should be related to the average size of secondary structures. To detect secondary structure we measure the correlations between different points along the protein chain. Specifically, let θj represent the value of the dihedral angle associated with the j α-carbon (see figure 1). We then calculate: Cθ(∆) = 〈