Rigidity analysis of protein biological assemblies and periodic crystal structures

Background We initiate in silico rigidity-theoretical studies of biological assemblies and small crystals for protein structures. The goal is to determine if, and how, the interactions among neighboring cells and subchains affect the flexibility of a molecule in its crystallized state. We use experimental X-ray crystallography data from the Protein Data Bank (PDB). The analysis relies on an effcient graph-based algorithm. Computational experiments were performed using new protein rigidity analysis tools available in the new release of our KINARI-Web server http://kinari.cs.umass.edu. Results We provide two types of results: on biological assemblies and on crystals. We found that when only isolated subchains are considered, structural and functional information may be missed. Indeed, the rigidity of biological assemblies is sometimes dependent on the count and placement of hydrogen bonds and other interactions among the individual subchains of the biological unit. Similarly, the rigidity of small crystals may be affected by the interactions between atoms belonging to different unit cells. We have analyzed a dataset of approximately 300 proteins, from which we generated 982 crystals (some of which are biological assemblies). We identified two types of behaviors. (a) Some crystals and/or biological assemblies will aggregate into rigid bodies that span multiple unit cells/asymmetric units. Some of them create substantially larger rigid cluster in the crystal/biological assembly form, while in other cases, the aggregation has a smaller effect just at the interface between the units. (b) In other cases, the rigidity properties of the asymmetric units are retained, because the rigid bodies did not combine. We also identified two interesting cases where rigidity analysis may be correlated with the functional behavior of the protein. This type of information, identified here for the first time, depends critically on the ability to create crystals and biological assemblies, and would not have been observed only from the asymmetric unit. For the Ribonuclease A protein (PDB file 5RSA), which is functionally active in the crystallized form, we found that the individual protein and its crystal form retain the flexibility parameters between the two states. In contrast, a derivative of Ribonuclease A (PDB file 9RSA), has no functional activity, and the protein in both the asymmetric and crystalline forms, is very rigid. For the vaccinia virus D13 scaffolding protein (PDB file 3SAQ), which has two biological assemblies, we observed a striking asymmetry in the rigidity cluster decomposition of one of them, which seems implausible, given its symmetry. Upon careful investigation, we tracked the cause to a placement decision by the Reduce software concerning the hydrogen atoms, thus affecting the distribution of certain hydrogen bonds. The surprising result is that the presence or lack of a very few, but critical, hydrogen bonds, can drastically affect the rigid cluster decomposition of the biological assembly. Conclusion The rigidity analysis of a single asymmetric unit may not accurately reflect the protein\u27s behavior in the tightly packed crystal environment. Using our KINARI software, we demonstrated that additional functional and rigidity information can be gained by analyzing a protein\u27s biological assembly and/or crystal structure. However, performing a larger scale study would be computationally expensive (due to the size of the molecules involved). Overcoming this limitation will require novel mathematical and computational extensions to our software

Comparative analysis of rigidity across protein families

We present a comparative study in which 'pebble game' rigidity analysis is applied to multiple protein crystal structures, for each of six different protein families. We find that the main-chain rigidity of a protein structure at a given hydrogen bond energy cutoff is quite sensitive to small structural variations, and conclude that the hydrogen bond constraints in rigidity analysis should be chosen so as to form and test specific hypotheses about the rigidity of a particular protein. Our comparative approach highlights two different characteristic patterns ('sudden' or 'gradual') for protein rigidity loss as constraints are removed, in line with recent results on the rigidity transitions of glassy networks

Rigidity and flexibility of biological networks

The network approach became a widely used tool to understand the behaviour of complex systems in the last decade. We start from a short description of structural rigidity theory. A detailed account on the combinatorial rigidity analysis of protein structures, as well as local flexibility measures of proteins and their applications in explaining allostery and thermostability is given. We also briefly discuss the network aspects of cytoskeletal tensegrity. Finally, we show the importance of the balance between functional flexibility and rigidity in protein-protein interaction, metabolic, gene regulatory and neuronal networks. Our summary raises the possibility that the concepts of flexibility and rigidity can be generalized to all networks.Comment: 21 pages, 4 figures, 1 tabl

Phase behaviour of DNA in presence of DNA-binding proteins

To characterize the thermodynamical equilibrium of DNA chains interacting with a solution of non-specific binding proteins, a Flory-Huggins free energy model was implemented. We explored the dependence on DNA and protein concentrations of the DNA collapse. For physiologically relevant values of the DNA-protein affinity, this collapse gives rise to a biphasic regime with a dense and a dilute phase; the corresponding phase diagram was computed. Using an approach based on Hamiltonian paths, we show that the dense phase has either a molten globule or a crystalline structure, depending on the DNA bending rigidity, which is influenced by the ionic strength. These results are valid at the thermodynamical equilibrium and should therefore be consistent with many biological processes, whose characteristic timescales range typically from 1 ms to 10 s. Our model may thus be applied to biological phenomena that involve DNA-binding proteins, such as DNA condensation with crystalline order, which occurs in some bacteria to protect their chromosome from detrimental factors; or transcription initiation, which occurs in clusters called transcription factories that are reminiscent of the dense phase characterized in this study.Comment: 20 pages, 9 figures, accepted for publication at The Biophysical Journa

Protein folding tames chaos

Protein folding produces characteristic and functional three-dimensional structures from unfolded polypeptides or disordered coils. The emergence of extraordinary complexity in the protein folding process poses astonishing challenges to theoretical modeling and computer simulations. The present work introduces molecular nonlinear dynamics (MND), or molecular chaotic dynamics, as a theoretical framework for describing and analyzing protein folding. We unveil the existence of intrinsically low dimensional manifolds (ILDMs) in the chaotic dynamics of folded proteins. Additionally, we reveal that the transition from disordered to ordered conformations in protein folding increases the transverse stability of the ILDM. Stated differently, protein folding reduces the chaoticity of the nonlinear dynamical system, and a folded protein has the best ability to tame chaos. Additionally, we bring to light the connection between the ILDM stability and the thermodynamic stability, which enables us to quantify the disorderliness and relative energies of folded, misfolded and unfolded protein states. Finally, we exploit chaos for protein flexibility analysis and develop a robust chaotic algorithm for the prediction of Debye-Waller factors, or temperature factors, of protein structures

Mechanistic insights into allosteric regulation of the A2A adenosine G protein-coupled receptor by physiological cations.

Cations play key roles in regulating G-protein-coupled receptors (GPCRs), although their mechanisms are poorly understood. Here, 19F NMR is used to delineate the effects of cations on functional states of the adenosine A2A GPCR. While Na+ reinforces an inactive ensemble and a partial-agonist stabilized state, Ca2+ and Mg2+ shift the equilibrium toward active states. Positive allosteric effects of divalent cations are more pronounced with agonist and a G-protein-derived peptide. In cell membranes, divalent cations enhance both the affinity and fraction of the high affinity agonist-bound state. Molecular dynamics simulations suggest high concentrations of divalent cations bridge specific extracellular acidic residues, bringing TM5 and TM6 together at the extracellular surface and allosterically driving open the G-protein-binding cleft as shown by rigidity-transmission allostery theory. An understanding of cation allostery should enable the design of allosteric agents and enhance our understanding of GPCR regulation in the cellular milieu

Principles for optimal cooperativity in allosteric materials

Allosteric proteins transmit a mechanical signal induced by binding a ligand. However, understanding the nature of the information transmitted and the architectures optimizing such transmission remains a challenge. Here we show using an {\it in-silico} evolution scheme and theoretical arguments that architectures optimized to be cooperative, which propagate efficiently energy, {qualitatively} differ from previously investigated materials optimized to propagate strain. Although we observe a large diversity of functioning cooperative architectures (including shear, hinge and twist designs), they all obey the same principle {of displaying a {\it mechanism}, i.e. an extended {soft} mode}. We show that its optimal frequency decreases with the spatial extension $L$ of the system as $L^{-d/2}$, where $d$ is the spatial dimension. For these optimal designs, cooperativity decays logarithmically with $L$ for $d=2$ and does not decay for $d=3$. Overall our approach leads to a natural explanation for several observations in allosteric proteins, and { indicates an experimental path to test if allosteric proteins lie close to optimality}.Comment: 11 pages, 9 figures in the main text, 9 pages 9 figures in the supplemental materia

Large scale rigidity-based flexibility analysis of biomolecules

KINematics And RIgidity (KINARI) is an on-going project for in silico flexibility analysis of proteins. The new version of the software, Kinari-2, extends the function- ality of our free web server KinariWeb, incorporates advanced web technologies, emphasizes the reproducibility of its experiments, and makes substantially improved tools available to the user. It is designed specifically for large scale experiments, in particular, for (a) very large molecules, including bioassemblies with high degree of symmetry such as viruses and crystals, (b) large collections of related biomolecules, such as those obtained through simulated dilutions, mutations, or conformational changes from various types of dynamics simulations, and (c) is intended to work as seemlessly as possible on the large, idiosyncratic, publicly available repository of biomolecules, the Protein Data Bank. We describe the system design, along with the main data processing, computational, mathematical, and validation challenges under- lying this phase of the KINARI project

CRC 1114 - Report Membrane Deformation by N-BAR Proteins: Extraction of membrane geometry and protein diffusion characteristics from MD simulations

We describe simulations of Proteins and artificial pseudo-molecules interacting and shaping lipid bilayer membranes. We extract protein diffusion Parameters, membrane deformation profiles and the elastic properties of the used membrane models in preparation of calculations based on a large scale continuum model