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In recent years, elastic network models (ENM) have been widely used to describe low-frequency collective motions in proteins. These models are often validated and calibrated by fitting mean-square atomic displacements estimated from x-ray crystallography (B-factors). We show that a proper calibration procedure must account for the rigid-body motion and constraints imposed by the crystalline environment on the protein. These fundamental aspects of protein dynamics in crystals are often ignored in currently used ENMs, leading to potentially erroneous network parameters. Here we develop an ENM that properly takes the rigid-body motion and crystalline constraints into account. Its application to the crystallographic B-factors reveals that they are dominated by rigid-body motion and thus are poorly suited for the calibration of models for internal protein dynamics. Furthermore, the translation libration screw (TLS) model that treats proteins as rigid bodies is considerably more successful in interpreting the experimental B-factors than ENMs. This conclusion is reached on the basis of a comparative study of various models of protein dynamics. To evaluate their performance, we used a data set of 330 protein structures that combined the sets previously used in the literature to test and validate different models. We further propose an extended TLS model that treats the bulk of the protein as a rigid body while allowing for flexibility of chain ends. This model outperforms other simple models of protein dynamics in interpreting the crystallographic B-factors.

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Proteins that perform mechanical functions in living organisms often exhibit exceptionally high strength and elasticity. Recent studies of the unfolding of single protein molecules under mechanical loading showed that their strength is mostly determined by their native topology rather than by thermodynamic stability. To identify the topologies of polymer molecules that maximize their resistance to unfolding, we have simulated the response of cross-linked polymer chains under tensile loading and have found that chain configurations that maximize the unfolding work and force involve parallel strands. Chains with such optimal topologies tend to unfold in an all-or-none fashion, in contrast to randomly cross-linked chains, most of which exhibit low mechanical resistance and tend to unfold sequentially. These findings are consistent with AFM studies and molecular mechanics simulations of the unfolding of β-sheet proteins. In particular, parallel strands give rise to the high strength of the immunoglobulin-like domains in the muscle protein titin.

A boundary element method for solving three-dimensional linear elasticity problems that involve a large number of particles embedded in a binder is introduced. The proposed method relies on an iterative solution strategy in which matrix-vector multiplication is performed with the fast multipole method. As a result the method is capable of solving problems with N unknowns using only O(N) memory and O(N) operations. Results are given for problems with hundreds of particles in which N"O(10).

We derive asymptotic expansions for the Green functions associated with coercive difference equations on general lattices. These expansions lead to rigorous methods for approximating the lattice Green functions by polyharmonic rational functions.

We have used kinetic Monte Carlo simulations to study the kinetics of unfolding of cross-linked polymer chains under mechanical loading. As the ends of a chain are pulled apart, the force transmitted by each crosslink increases until it ruptures. The stochastic crosslink rupture process is assumed to be governed by first order kinetics with a rate that depends exponentially on the transmitted force. We have performed random searches to identify optimal crosslink configurations whose unfolding requires a large applied force (measure of strength) and/or large dissipated energy (measure of toughness). We found that such optimal chains always involve cross-links arranged to form parallel strands. T he location of those optimal strands generally depends on the loading rate. Optimal chains with a small number of cross-links were found to be almost as strong and tough as optimal chains with a large number of cross-links. Furthermore, optimality of chains with a small number of cross -links can be easily destroyed by adding cross-links at random. The present findings are relevant for the interpretation of single molecule force probe spectroscopy studies of the mechanical unfolding of "loadbearing" proteins, whose native topology often involves parallel strand arrangements similar to the optimal configurations identified in the study.

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