Additivity principles in chemistry, biochemistry, and biophysics have been used extensively for decades. Nevertheless, it is well known that additivity frequently breaks down in complex biomacromolecules. Nonadditivity within protein double mutant free energy cycles of spatially close residue pairs is a generally well-understood phenomenon, whereas a robust description of nonadditivity extending over large distances remains to be developed. Here, we test the hypothesis that the mutational effects tend to be nonadditive if two structurally well-separated mutated residues belong to the same rigid cluster within the wild type protein, and additive if they are located within different clusters. We find the hypothesis to be statistically significant with Pvalues that range from 10 −5 to 10 −6 . To the best of our knowledge, this result represents the first demonstration of a statistically significant preponderance for nonadditivity over long distances. These findings provide new insight into the origins of long-range nonadditivity in double mutant cycles, which complements the conventional wisdom that nonadditivity arises in double mutations involving contacting residues. Consequently, these results should have far-reaching implications for a proper understanding of protein stability, structure/function analyses, and protein design.

Background: The human innate immune system uses a system of extracellular Toll-like receptors (TLRs) and intracellular Nod-like receptors (NLRs) to match the appropriate level of immune response to the level of threat from the current environment. Almost all NLRs and TLRs have a domain consisting of multiple leucine-rich repeats (LRRs), which is believed to be involved in ligand binding. LRRs, found also in thousands of other proteins, form a well-defined "horseshoe"-shaped structural scaffold that can be used for a variety of functions, from binding specific ligands to performing a general structural role. The specific functional roles of LRR domains in NLRs and TLRs are thus defined by their detailed surface features. While experimental crystal structures of four human TLRs have been solved, no structure data are available for NLRs.

The time it takes for proteins to fold into their native states varies over several orders of magnitude depending on their native-state topology, size, and amino acid composition. In a number of previous studies, it was found that there is strong correlation between logarithmic folding rates and contact order for proteins that fold with two-state kinetics, while such correlation is absent for three-state proteins. Conversely, strong correlations between folding rates and chain length occur within three-state proteins, but not in two-state proteins. Here, we demonstrate that chain lengths and folding rates of two-state proteins are not correlated with each other only when all-a, all-b, and mixed-class proteins are considered together, which is typically the case. However, when considering all-a and all-b two-state proteins separately, there is significant linear correlation between folding rate and size. Moreover, the sets of data points for the all-a and all-b classes define asymptotes of lower and upper limits on folding rates of mixed-class proteins. By analyzing correlation of other topological parameters with folding rates of two-state proteins, we find that only the long-range order exhibits correlation with folding rates that is uniform over all three classes. It is also the only descriptor to provide statistically significant correlations for each of the three structural classes. We give an interpretation of this observation in terms of Makarov and Plaxco's diffusion-based topomer-search model.

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Lowest-order nondipole effects are studied systematically in double photoionization ͑DPI͒ of the He atom. Ab initio parametrizations of the quadrupole transition amplitude for DPI from the 1 S 0 state are presented in terms of the exact two-electron radial matrix elements. Analytic expressions for these matrix elements within lowest-order perturbation theory ͑LOPT͒ in the interelectron interaction are also given. The corresponding parametrizations for the dipole-quadrupole triply differential cross section ͑TDCS͒ are presented for the case of an elliptically polarized photon. A general analysis of retardation-induced asymmetries of the TDCS including the circular dichroism effect at equal energy sharing is presented. Numerical LOPT estimates of nondipole asymmetries in photoelectron angular distributions for the cases of linear and circular polarization and of the circular dichroism effect at equal energy sharing are presented. We find that experimental observation of nondipole effects at excess energies of the order of tens to hundreds of eV should be feasible in TDCS measurements. Our numerical results exhibit a nondipole forward-backward asymmetry in the TDCS for DPI of He at an excess energy of 450 eV that is in qualitative agreement with existing experimental data.

Single-photon, two-electron ionization of He is analysed, taking into account electron correlation using lowest-order perturbation theory and including all individual electron angular momenta in the final two-electron continuum. Perturbative account of electron correlation in the final state, which describes the so-called TS-1 mechanism of double photoionization, combined with a variational account of electron screening, is found to provide results for the triply differential cross section at an excess energy of 20 eV that are in excellent agreement with both absolute experimental data and results of non-perturbative calculations, for all kinematics of the process in which the TS-1 mechanism is expected to dominate.

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The distance constraint model (DCM) is a unique computational modeling paradigm that integrates mechanical and thermodynamic descriptions of macromolecular structure. That is, network rigidity calculations are used to account for nonadditivity within entropy components, thus restoring the utility of free energy decomposition. The DCM outputs a large number of structural characterizations that collectively allow for quantified stability/flexibility relationships (QSFR) to be identified. In this review, we describe the theoretical underpinnings of the DCM and introduce several common QSFR metrics. Application of the DCM across protein families highlights the sensitivity within the set of protein structure residue-to-residue couplings. Further, we have developed a perturbation method to identify putative allosteric sites, where large changes in QSFR upon rigidification (mimicking ligand-binding) detect sites likely to invoke allosteric changes.