Emergence of stable and fast folding protein structures

The number of protein structures is far less than the number of sequences. By imposing simple generic features of proteins (low energy and compaction) on all possible sequences we show that the structure space is sparse compared to the sequence space. Even though the sequence space grows exponentially with N (the number of amino acids) we conjecture that the number of low energy compact structures only scales as ln N. This implies that many sequences must map onto countable number of basins in the structure space. The number of sequences for which a given fold emerges as a native structure is further reduced by the dual requirements of stability and kinetic accessibility. The factor that determines the dual requirement is related to the sequence dependent temperatures, T_\theta (collapse transition temperature) and T_F (folding transition temperature). Sequences, for which \sigma =(T_\theta-T_F)/T_\theta is small, typically fold fast by generically collapsing to the native-like structures and then rapidly assembling to the native state. Such sequences satisfy the dual requirements over a wide temperature range. We also suggest that the functional requirement may further reduce the number of sequences that are biologically competent. The scheme developed here for thinning of the sequence space that leads to foldable structures arises naturally using simple physical characteristics of proteins. The reduction in sequence space leading to the emergence of foldable structures is demonstrated using lattice models of proteins.Comment: latex, 18 pages, 8 figures, to be published in the conference proceedings "Stochastic Dynamics and Pattern Formation in Biological Systems

Mechanical unfolding of RNA hairpins

Mechanical unfolding trajectories, generated by applying constant force in optical tweezer experiments, show that RNA hairpins and the P5abc subdomain of the group I intron unfold reversibly. We use coarse-grained Go-like models for RNA hairpins to explore forced-unfolding over a broad range of temperatures. A number of predictions that are amenable to experimental tests are made. At the critical force the hairpin jumps between folded and unfolded conformations without populating any discernible intermediates. The phase diagram in the force-temperature (f,T) plane shows that the hairpin unfolds by an all-or-none process. The cooperativity of the unfolding transition increases dramatically at low temperatures. Free energy of stability, obtained from time averages of mechanical unfolding trajectories, coincide with ensemble averages which establishes ergodicity. The hopping time between the the native basin of attraction (NBA) and the unfolded basin increases dramatically along the phase boundary. Thermal unfolding is stochastic whereas mechanical unfolding occurs in "quantized steps" with great variations in the step lengths. Refolding times, upon force quench, from stretched states to the NBA is "at least an order of magnitude" greater than folding times by temperature quench. Upon force quench from stretched states the NBA is reached in at least three stages. In the initial stages the mean end-to-end distance decreases nearly continuously and only in the last stage there is a sudden transition to the NBA. Because of the generality of the results we propose that similar behavior should be observed in force quench refolding of proteins.Comment: 23 pages, 6 Figures. in press (Proc. Natl. Acad. Sci.