FIG. 1. (a) Energy landscape for a random heteropolymer. Notice that the presence of low energy states that are completely dissimilar is a direct consequence of the small energy bias toward the native state relative to the roughness of the landscape, (b) Funnel-like energy landscape for a minimally frustrated heteropolymer. A clearly favored native structure can be observed in the bottom of this funnel. Because of this dominant bias, all the other low energy states are similar to the native one.

and not by the bias toward the native conformation. This temperature is called the glass temperature, Tg. Minimally frustrated sequences require sufficient bias to have the folding temperature larger than the glass temperature. Therefore this competition between energetic bias toward native conformation and roughness is fundamental in determining the folding mechanism, and it leads to a diversity of folding scenarios that are discussed elsewhere (2). All these ideas are further explored later in this paper.

Sequences with a good folding funnel not only are fast folders at temperatures around the folding temperature but, most importantly, they are robust folders. Robustness is an essential property in biology. Minor variations in the folding environment such as small changes in pH, temperature, denaturant concentration, or, even more interesting, variations because of mutations may affect the native configuration in favor of other low-energy structures. If these other low-energy structures are similar to the folded one, the consequences are minor. The “new” native conformation is very similar to the “old” one. The observed linear dependence between logarithms of the folding/unfolding rates and the folding free energy is a direct indication that this is the case for proteins (2731). Frustrated sequences, on the other hand, not only are slow folders but also may have the structure of their native state drastically changed under minor variations of the conditions described above.

This diversity of scenarios suggested by the landscape theory and the funnel concept can be observed by simulations of protein folding in computer models. Such simulations can be carried out at many different levels. Ideally they should be at the atomistic level but, because of computational limitations, this approach has limited itself to insights into local aspects of folding (32, 33) and characterizing ensembles of states for unfolded proteins (3437). Thus minimalist models have been of major importance in our understanding of protein folding. Lattice models have been the center of these studies. They include the simple ones exploited in the early 1980s (5, 38, 39), and more recently in studies by several other groups (8, 12, 15, 16, 20, 4046). These models have really improved our present understanding of protein folding. Off-lattice models have also been studied (4754), but little has been done in this landscape context, making this point the focus of this paper.

In addition to simulations, new experiments have been devised to probe early folding events and to explore the landscape of small fast-folding proteins (NMR dynamic spectroscopy, protein engineering, laser-initiated folding, and ultrafast mixing; see, for example, refs. 13, 14, 28, and 5567, 85). Fast-folding proteins fold on millisecond timescales and have a single domain—i.e., they have a single, well defined, funnel (68). The combination of landscape theory, simulations, and this new family of experiments is providing the basis for a quantitative understanding of the protein folding mechanism.

In this paper we show results for an off-lattice minimalist model where we explore the behavior of two folding sequences with the same native structure, but with one containing a higher degree of frustration. A quantitative landscape framework for quantifying differences between good and bad folding sequences emerges from this comparison. Because most of the existent landscape analysis has been performed for lattice simulations, we present in the next section a summary of some selected results in the lattice to help with our discussion of the off-lattice simulations.

A Summary of Lattice Minimalist Models

Minimalist models of protein folding must contain all the features necessary to understand the folding mechanism. In its simplest version a heteropolymer must contain at least two kinds of monomers whose interactions obey some simplified interaction rule—i.e., heteropolymers may be thought as a necklace of beads of two or more kinds. The question to be answered is what sequences of beads are able to fold into a unique three-dimensional structure. In an effort to mimic the hydrophobic effect, Dill and collaborators (12) proposed the first set of interactions, called the HP model, where the interactions between H (hydrophobic) groups are attractive and all the other ones are zero. Another popular model, which is used for our simulations of 27-mers in a cubic lattice, is the one where the interactions between nearest neighbor beads of the same color are more favorable (strong attractive interaction) than the ones between beads of different colors (weak interaction). Sequences built with two kinds of beads are called two-letter code, three kinds of beads are three-letter codes, and so on.

The low-energy states of heteropolymers composed of random sequences of two or more kinds of beads are collapsed states that try to maximize the number of contacts between beads of the same color. The polymeric nature of the chain, however, prohibits all favorable interactions from being satisfied simultaneously, and some contacts occur between beads of different color. These are clearly frustrated interactions, because the polymer would rather have the maximum number of favorable interactions. Thus different low-energy states may have different structures with a different set of frustrated contacts.

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