We present a simple physical model that demonstrates that the native-state folds of proteins can emerge on the basis of considerations of geometry and symmetry. We show that the inherent anisotropy of a chain molecule, the geometrical and energetic constraints placed by the hydrogen bonds and sterics, and hydrophobicity are sufficient to yield a free-energy landscape with broad minima even for a homopolymer. These minima correspond to marginally compact structures comprising the menu of folds that proteins choose from to house their native states in. Our results provide a general framework for understanding the common characteristics of globular proteins. P rotein folding (1-5) is complex because of the sheer size of protein molecules, the twenty types of constituent amino acids with distinct side chains, and the essential role played by the environment. Nevertheless, proteins fold into a limited number (6, 7) of evolutionarily conserved structures (8, 9). It is a familiar, yet remarkable, consequence of symmetry and geometry that ordinary matter crystallizes in a limited number of distinct forms. Indeed, crystalline structures transcend the specifics of the various entities housed in them. Here, we ask the analogous question (10): is the menu of protein folds also determined by geometry and symmetry?We show that a simple model that encapsulates a few general attributes common to all polypeptide chains, such as steric constraints (11-13), hydrogen bonding (14-16), and hydrophobicity (17), gives rise to the emergent free-energy landscape of globular proteins. The relatively few minima in the resulting landscape correspond to putative marginally compact nativestate structures of proteins, which are assemblies of helices, hairpins, and planar sheets. A superior fit (18, 19) of a given protein or sequence of amino acids to one of these predetermined folds dictates the choice of the topology of its native-state structure. Instead of each sequence shaping its own free energy landscape, we find that the overarching principles of geometry and symmetry determine the menu of possible folds that the sequence can choose from.Following Bernal (20), the protein problem can be divided into two distinct steps: first, analogous to the elucidation of crystal structures, one must identify the essential features that account for the common characteristics of all proteins; second, one must understand what makes one protein different from another. Guided by recent work (21,22) that has shown that a faithful description of a chain molecule is a tube and using information from known protein native-state structures, our focus, in this paper, is on the first step: we demonstrate that the native-state folds of proteins emerge from considerations of symmetry and geometry within the context of a simple model.We model a protein as a chain of identical amino acids, represented by their C ␣ atoms, lying along the axis of a selfavoiding flexible tube.

Molecular dynamics simulations in simplified models allow one to study the scaling properties of folding times for many proteins together under a controlled setting. We consider three variants of the Go models with different contact potentials and demonstrate scaling described by power laws and no correlation with the relative contact order parameter. We demonstrate existence of at least three kinetic universality classes that are correlated with the types of structure: the alpha-, alpha-beta-, and beta- proteins have the scaling exponents of approximately 1.7, 2.5, and 3.2, respectively. The three classes merge into one when the contact range is truncated at a reasonable value. We elucidate the role of the potential associated with the chirality of a protein.

We consider six different secondary structures of proteins and construct two types of Go-type offlattice models: with the steric constraints and without. The basic aminoacid-aminoacid potential is Lennard Jones for the native contacts and a soft repulsion for the non-native contacts. The interactions are chosen to make the target secondary structure be the native state of the system. We provide a thorough equilibrium and kinetic characterization of the sequences through the molecular dynamics simulations with the Langevin noise. Models with the steric constraints are found to be better folders and to be more stable, especially in the case of the β-structures. Phononic spectra for vibrations around the native states have low frequency gaps that correlate with the thermodynamic stability. Folding of the secondary structures proceeds through a well defined sequence of events. For instance, α-helices fold from the ends first. The closer to the native state, the faster establishment of the contacts. Increasing the system size deteriorates the folding characteristics. We study the folding times as a function of viscous friction and find a regime of moderate friction with the linear dependence. We also consider folding when one end of a structure is pinned which imitates instantaneous conditions when a protein is being synthesized. We find that, under such circumstances, folding of helices is faster and of the β-sequences slower.

Mechanical stretching of the I27 domain of titin and of its double and triple repeats are studied through molecular dynamics simulations of a Go-like model with Lennard-Jones contact interactions. We provide a thorough characterization of the system and correlate the sequencing of the folding and unraveling events with each other and with the contact order. The roles of cantilever stiffness and pulling rate are studied. Unraveling of tandem titin structures has a serial nature. The force-displacement curves in this coarse-grained model are similar to those obtained through all atom calculations.

The effect of temperature on mechanical unfolding of proteins is studied using a Go-like model with a realistic contact map and Lennard-Jones contact interactions. The behavior of the I27 domain of titin and its serial repeats is contrasted to that of simple secondary structures. In all cases thermal fluctuations accelerate the unraveling process, decreasing the unfolding force nearly linearly at low temperatures. However differences in bonding geometry lead to different sensitivity to temperature and different changes in the unfolding pattern. Due to its special native state geometry titin is much more thermally and elastically stable than the secondary structures. At low temperatures serial repeats of titin show a parallel unfolding of all domains to an intermediate state, followed by serial unfolding of the domains. At high temperatures all domains unfold simultaneously and the unfolding distance decreases monotonically with the contact order, that is the sequence distance between the amino acids that form the native contact.