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From polymers to proteins – novel phases of short compact tubes 3 0 0 2 Jayanth R. Banavar† & Amos Maritan‡ n a J 4 1 † Department of Physics, 104 Davey Laboratory, The Pennsylvania State University, Uni- ] h versity Park, Pennsylvania 16802 c e m - t ‡ International School for Advanced Studies (S.I.S.S.A.), Via Beirut 2-4, 34014 Trieste, a t s INFM and the Abdus Salam International Center for Theoretical Physics, Trieste, Italy . t a m - d n o c [ 1 A framework is presented for understanding the common character of proteins. v 0 Proteins are linear chain molecules – however, the simple model of a polymer 2 2 1 viewed as spheres tethered together does not account for many of the observed 0 3 characteristics of protein structures. We show here that the notion of a tube 0 / t of non-zero thickness allows one to bridge the conventional compact polymer a m phase with a novel phase employed by Nature to house biomolecular structures. - d n We build on the idea that a non-singular continuum description of a tube (or a o c : sheet) of arbitrary thickness entails discarding pairwise interactions and using v i X appropriately chosen many body interactions. We suggest that the structures of r a folded proteins are selected based on geometrical considerations and are poised at the edge of compaction, thus accounting for their versatility and flexibility. We present an explanation for why helices and sheets are the building blocks of protein structures. 1 Contents I Introduction 3 II Quantum chemistry scores a major success 5 III A physics approach leads to a disconnect between the compact polymer phase and the novel phase adopted by protein structures 6 IV Protein backbone viewed as a tube 8 V Strings, sheets and many-body interactions 9 VI Marginally compact tubes 13 VII Building blocks of protein structures 17 VIII Consequences of the tube picture 20 IX Studies of short tubes 22 X Summary and Conclusions 23 2 I. INTRODUCTION Recent years have witnessed gigantic leaps in the field of molecular biology culminating in the sequencing of the human genome as reported in two historic issues of Science (Volume 291, issue 5507) and Nature (Volume 409, issue 6822) in 2001. Base pairing and the remark- able structure of the DNA molecule (Watson and Crick, 1953) provide a very efficient means of storing and replicating genetic information. The principal role of genes is to serve as a template for the synthesis of m-RNAs that, in turn, are “translated” by ribosomes into the polypeptide chains which then fold into active proteins. These proteins are the work horse molecules of life. They not only carry out a dizzying array of functions but also they interact with each other and play a role in turning the genes on or off. There is little variability in the structure of the information carrying molecule, DNA. On the other hand, there are several thousand geometries which folded proteins can adopt and these structures determine the functionality of the proteins (Branden and Tooze, 1999; Creighton, 1993; Fersht, 1998). Proteins are the basic constituents of all living cells. Some familiar examples of proteins are hemoglobin, which delivers oxygen to our tissues, actin and myosin which facilitate the contraction of our muscles, insulin which is secreted in the pancreas and signals our body to store excess sugar and antibodies that fight infection. Marvelous machines within the cell known as ribosomes make proteins by stringing together little chemical entities called amino acids into long linear chains. There are twenty types of amino acids which differ only in their side chains. The protein backbone as well as some of the side chains are hydrophobic and shy away from water, while other side chains are polar and yet others have charges associated with them. Our focus is on small, globular proteins, which, under physiological conditions, fold rapidly and reproducibly (Anfinsen, 1973) in a cooperative fashion into a somewhat compact state inorder toexpelthewater fromthecoreofthefoldedstructure, which predominantly houses 3 the hydrophobic amino acids. Thus there is an effective attraction between the hydrophobic amino acids arising from their shared tendency to avoid the water. For proteins, form determines function. The structure of the protein in its folded state (also called its native state structure) controls its functionality (Branden and Tooze, 1999; Creighton, 1993; Fersht, 1998). The rich variety of amino acids allows for many sequences to have the same native state structure. Thus even though our human body may have more than 100,000 proteins, it is believed that the number of distinct folds that they adopt in their native state, is only a few thousand in all (Chothia, 1992). Furthermore, these folds are beautiful (Levitt and Chothia, 1976; Chothia, 1984) – they are not just any compact form but, rather, are made up of building blocks of helices and sheet-like planar structures with tight loops connecting these secondary motifs (see Figure 1). In 1939, J. D. Bernal (1939) stated the challenge associated with the protein problem: Any effective picture of protein structure must provide at the same time for the common character of all proteins as exemplified by their many chemical and physical similarities, and for the highly specific nature of each protein type. Despite many advances in experiments on proteins and the advent of powerful computers, the problem has remained largely unsolved. The key components of the problem are protein folding and design: protein folding entails the prediction of the folded geometry of a protein given its sequence of amino acids while the design problem involves the prediction of the amino acid sequence which would fold into a putative target structure. It is probably not too surprising that progress has been somewhat limited because, until now, there has not been any simple unifying framework for understanding the common character of all proteins. The principal aim of this colloquium is to address this issue. Such a framework must provide an explanation for the relatively small number of protein native structures, for why the building blocks of protein structures are helices and sheets, for the highly cooperative nature of the folding transition of small globular proteins and for the versatility and flexibility of protein structures, which account 4 for the ability of the proteins to perform a wide range of functions. II. QUANTUM CHEMISTRY SCORES A MAJOR SUCCESS Linus Pauling and his collaborators (Pauling, Corey and Branson, 1951; Pauling and Corey, 1951) invoked the chemistry of covalent and hydrogen bonds to show that helices and sheets were periodically repeatable structures for which appropriately placed hydrogen bonds could provide the scaffolding. This stunning prediction was experimentally confirmed in short order. Unfortunately, theseobservationsdonotprovideacomplete explanationoftheselection of the protein folds. The difficulty arises because hydrogen bonds can equally easily form between the protein molecule and the water surrounding it. While helices and sheets are nicely stabilized by hydrogen bonds, one may construct other viable structures which do not have helices or sheets as the building blocks but yet have a large number of hydrogen bonds and hence a favorable energy. Aproteiniscomplexbecauseofthemanyfeaturesthatoneisconfrontedwith. Asmentioned before, we need to deal with twenty types of amino acids with their individual properties and, in addition, there is the crucial role played by the solvent. A first principles approach might consist of considering all the numerous atoms making up a protein and the surrounding solvent and carrying out some really heavy computer calculations to simulate the folding process. Very quickly one realizes that, with the somewhat imperfect knowledge of the interactions and the sheer magnitude of the job at hand, this approach is not too likely to yield qualitatively new insights into the protein folding problem. Furthermore, one might worry that, at best, one would be able to mimic Nature but would one obtain an understanding of Nature? 5 III. A PHYSICS APPROACH LEADS TO A DISCONNECT BETWEEN THE COMPACT POLYMER PHASE AND THE NOVEL PHASE ADOPTED BY PROTEIN STRUCTURES Let us now consider the protein problem afresh from a physics point of view and attempt to identify the key issues. It is of course possible and, one might fear, even likely that many of the details are crucial in understanding the intricate behavior of proteins. In order to make progress, we will take the approach, though, of looking at what we might imagine to be the the most essential features and adding in details as we require them. This will allow us to retain some control over our understanding and we will be able to assess, a posteriori, the relative importance of the features that we may have to incorporate. The approach is analogous to one commonly used in physics (Chaikin and Lubensky, 1995) of distilling out just the most essential features for understanding emergent phenomena. For example, one can use general geometrical and symmetry arguments to predict the different classes of crystal structures. The existence of these structures does not rely on quantum mechanics or onchemistry. They area consequence of a deeper and moregeneral mathemat- ical framework. Of course, given a chemical compound like common salt, a careful quantum mechanical study would show that sodium chloride adopts the face centered cubic lattice structure. Also, a clever grocer would use the same crystal structure for the efficient packing of fruits. Thus the structures transcend the specifics of the chemical entities that are housed in them. One might therefore seek to determine what the analogous structures are for protein native states that are determined merely by geometrical considerations. What are the bare essentials that determine the novel phase adopted by biopolymers such as proteins? Proteins are linear chains and ignoring the details of the amino acid side-chains, all proteins have a backbone. A protein folds because of hydrophobicity or the tendency of certain amino acids to shy away from water. In the folded state, therefore, one would like to have 6 a conformation which squeezes the water out from certain regions of the protein populated by the hydrophobic amino acids. As stated before, the simplest way of encapsulating such a tendency for compaction is by means of an effective attractive interaction between the backbone atoms promoting a somewhat compact native state. Anearly success ofthe physics approachwas thework of Ramachandran, Ramakrishnan and Sasisekharan (Ramachandran and Sasisekharan, 1968), as embodied in the Ramachandran plot. They showed that steric constraints, relating to or involving the arrangement of atoms in space, alone dictated that the backbone conformations of a protein lie predominantly in two regions of the space of the so called torsional angles corresponding to α-helical and β-strand conformations (see Figures 1-3). In other words, the high cost associated with the overlap of two atoms viewed as hard spheres leads to conformations which are consistent with the local structure associated with a helix or a sheet. We hit a snag in our thought experiment – careful computational studies (Hunt et al., 1994; Yee et al., 1994) have shown that the standard polymer model of chain molecules, viewed as spheres tethered together, when subjected to interactions which promote compactness, have innumerable conformations almost none of which have any secondary motifs. In contrast, proteins have a limited number of folds to choose from for their native state structure and the energy landscape is vastly simpler. In addition, the structures in the polymer phase are not specially sensitive to perturbations and are thus not as flexible and versatile as protein native state structures are in order to accommodate the dizzying array of functions that proteins perform. Indeed, there has been somewhat of a disconnect between the familiar compact polymer phase and thenovel phaseused by Natureto house biomolecules. To quote from P. J. Flory (1969): Synthetic analogs of globular proteins are unknown. The capability of adopting a dense globular configuration stabilized by self-interactions and of transforming reversibly to the random coil are peculiar to the chain molecules of globular proteins alone. 7 IV. PROTEIN BACKBONE VIEWED AS A TUBE So what new feature should we incorporate next? Are the details of the amino acids important? We expect not because it is known that many sequences fold into the same native state structure (Branden and Tooze, 1999; Creighton, 1993; Fersht, 1998). At a somewhat simpler level, we recall the work of Ramachandran (Ramachandran andSasisekharan, 1968), who showed that steric interactions or the undesirability of two atoms to sit on top of each other, even when the atoms are treated as effective hard spheres, lead to certain regions of conformational space being excluded for a protein chain (Rose, 1996). The side chains of the amino acids occupy space as well, and thus it seems important to allow for room around the backbone to accommodate these atoms. So we proceed by incorporating a new ingredient – let us treat the protein backbone not as a chain of spheres but as a tube of non-zero thickness analogous to a garden hose. So how does such a tube behave if it has an effective attractive self-interaction that tends to make its conformation somewhat compact? There is some hope on the horizon, because the problem is enriched and we now have two length scales to play with, the thickness of the tube and the range of the attractive interactions. It is useful to consider what the standard model of a chain represented as tethered spheres is missing. For unconstrained particles, spheres are the simplest objects that one might consider. Of course, symmetry matters a great deal and when these spheres are replaced by asymmetric objects, one gets a host of qualitatively new liquid crystalline phases (Chaikin and Lubensky, 1995). There are two simple ingredients associated with a chain – the particles are tethered to each other (which is well-captured by the standard model of tethered spheres) and associated with each particle of the chain is a special direction representing the local direction associated with the chain as defined by the adjacent particles at that location. This selection of a local direction immediately leads to the requirement that the symmetrical spherical objects comprising the chain must be replaced by anisotropic objects (such as coins) for which one of the three directions is different from the other two. Thus, 8 if one were to think of a chain as being made up of stacked coins instead of spheres, one naturally arrives at a picture of a tube. Indeed, previous analysis (Banavar et al., 2002 (a); Banavar et al., 2002 (b)) of the native state structures of proteins has shown that a protein backbone may be thought of approximately as a uniform tube of radius 2.7˚A. Before we explore the phases associated with a tube subject to compaction, we will have an interlude where we will revisit some issues in polymer physics. V. STRINGS, SHEETS AND MANY-BODY INTERACTIONS Strings and chains have been studied over the years in the field of polymer physics. Tubes of non-zero thickness are ubiquitous – garden hoses and spaghetti populate our houses. How does one mathematically describe a tube of non-zero thickness in the continuum limit? Alas, a visit to the library confirms our worst fears, this elementary problem has not been tackled before. A continuum description of a string was put forward by Sam Edwards (Doi and Edwards, 1993) – it captures self-avoidance by means of a singular delta function repulsion between different parts of a string. The delta function describes a situation in which the repulsive interaction is infinitely strong precisely when there is an exact overlap and zero otherwise. This description is therefore valid only for an infinitesimally thin string. An associated complication is that the analysis of a continuum string requires the use of renormalization group theory to regularize the theory by introducing a lower-length scale cut-off combined with a proof that the behavior, at long length scales, is independent of this cut-off length scale. Unfortunately, the renormalization group theory analysis, in this context, is peripheral to the physics being studied. Recently, with the help of two mathematicians, Oscar Gonzalez and John Maddocks, we were able to write down a singularity-free description of manifolds such as chains or sheets (Banavar et al., 2002(c)). The solution is very simple but not intuitively obvious. In science, 9 the starting point for describing interacting matter is by means of pair-wise interactions. In order to describe your interactions with your friends, it is a good starting point to consider your pair-wise interactions with each of them – your true interaction will be different from this only because of genuine many body interactions that may be thought of as higher order corrections. With a pair-wise interaction, there is only one length scale that one can construct from a knowledge of where you are and where your friend is. This length scale is your mutual distance. One can define potential energies of interaction between you and your friend which depends on this length scale. Generically, such an interaction may be one in which if you and your friend are separated by a sufficiently long distance, you do not talk to each other and there is no interaction. There is an optimal distance between you and your friend where the interaction is at its happiest. Any closer approach leads to a higher energy with the potential energy becoming infinitely large when you sit on top of each other. Unfortunately, such an analysis is not very helpful when you and your friends (and your enemies) are formed into a conga line by someone who does not know what your personal relationships are. Let us assume that one is working again with pair-wise interactions and you are told that two people are spatially (not necessarily emotionally) close to each other. With that information alone, you will not be able to tell whether the two people are from different parts of the chain and are close to each other because they like each other or whether the two people are sworn enemies who happen to be close to each other simply because they were constrained to be next to each other in the conga line. In other words, pair-wise interactions merely provide the mutual distance but not the context in which the interacting particles exist. The basic idea behind the development of a continuum theoryof a tubeofnon-zero thickness is to discard pair-wise interactions and consider appropriately chosen three-body interactions as the basic interacting unit. The requirements for a well-founded theory are that one ought to be able to take a continuum limit on increasing the density of particles, that self- 10