Nonlinear backbone torsional pair correlations in proteins

Dynamic correlations: exact and approximate methods for mutual information

MOTIVATION

Proteins are dynamic entities that undergo conformational changes critical for their functions. Understanding the communication pathways and information transfer within proteins is crucial for elucidating allosteric interactions in their mechanisms. This study utilizes mutual information (MI) analysis to probe dynamic allostery. Using two cases, Ubiquitin and PLpro, we have evaluated the accuracy and limitations of different approximations including the exact anisotropic and isotropic models, multivariate Gaussian model, isotropic Gaussian model, and the Gaussian Network Model (GNM) in revealing allosteric interactions.

RESULTS

Our findings emphasize the required trajectory length for capturing accurate mutual information profiles. Long molecular dynamics trajectories, 1 ms for Ubiquitin and 100 µs for PLpro are used as benchmarks, assuming they represent the ground truth. Trajectory lengths of approximately 5 µs for Ubiquitin and 1 µs for PLpro marked the onset of convergence, while the multivariate Gaussian model accurately captured mutual information with trajectories of 5 ns for Ubiquitin and 350 ns for PLpro. However, the isotropic Gaussian model is less successful in representing the anisotropic nature of protein dynamics, particularly in the case of PLpro, highlighting its limitations. The GNM, however, provides reasonable approximations of long-range information exchange as a minimalist network model based on a single crystal structure. Overall, the optimum trajectory lengths for effective Gaussian approximations of long-time dynamic behavior depend on the inherent dynamics within the protein's topology. The GNM, by showcasing dynamics across relatively diverse time scales, can be used either as a standalone method or to gauge the adequacy of MD simulation lengths.

AVAILABILITY AND IMPLEMENTATION

Mutual information codes are available at https://github.com/kemaldemirtas/prc-MI.git.

Cleaning the molecular machinery of cells via proteostasis, proteolysis and endocytosis selectively, effectively, and precisely: intracellular self-defense and cellular perturbations

Network coordinates of cellular processes (proteostasis, proteolysis, and endocytosis), and molecular chaperones are the key complements in the cell machinery and processes. Specifically, cellular pathways are responsible for the conformational maintenance, cellular concentration, interactions, protein synthesis, disposal of misfolded proteins, localization, folding, and degradation. The failure of cellular processes and pathways disturbs structural proteins and the nucleation of amyloids. These mishaps further initiate amyloid polymorphism, transmissibility, co-aggregation of pathogenic proteins in tissues and cells, prion strains, and mechanisms and pathways for toxicity. Consequently, these conditions favor and lead to the formation of elongated amyloid fibrils consisting of many-stranded β-sheets (N,N-terminus and C,C-terminus), and abnormal fibrous, extracellular, proteinaceous deposits. Finally, these β-sheets deposit, and cells fail to degrade them effectively. The essential torsion angles (φ, ψ, and ω) define the conformation of proteins and their architecture. Cells initiate several transformations and pathways during the regulation of protein homeostasis based on the requirements for the functioning of the cell, which are governed by ATP-dependent proteases. In this process, the kinetics of the molding/folding phenomenon is disturbed, and subsequently, it is dominated by cross-domain misfolding intermediates; however, simultaneously, it is opposed by small stretching forces, which naturally exist in the cell. The ubiquitin/proteasome system deals with damaged proteins, which are not refolded by the chaperone-type machinery. Ubiquitin-protein ligases (E3-Ub) participate in all the cellular activity initiated and governed by molecular chaperones to stabilize the cellular proteome and participate in the degradation phenomenon implemented for damaged proteins. Optical tweezers, a single-resolution based technique, disclose the folding pathway of linear chain proteins, which is how they convert themselves into a three-dimensional architecture. Further, DNA-protein conjugation analysis is performed to obtain folding energies as single-molecule kinetic and thermodynamic data.

Significance of triple torsional correlations in proteins

The free energy landscape (FEL) of a given complex molecular system is fundamentally the joint probability density of its many comprising degrees of freedom (DOFs). Computation of a complete FEL at atomistic scale is unfortunately intractable for a typical biomolecular system. The challenge of entropy calculation comes from various correlations among different DOFs. The common strategy to treat such complexity is expansion of the full correlation into various orders of local correlations. In reality, expansion is usually cut off at the second order (i.e. pairwise interactions) for protein torsional correlations without reliable estimation of the resulting error. Here, we estimated the mutual information of different torsion sets and found that triple correlations were significant for both local/distant residue pairs and consecutive backbone torsional segments. As expected, the third order approximations were found to be consistently better than the second order approximations. These findings were true for all analyzed proteins with different folds, were independent of the two different force fields utilized to generate trajectory sets, and were therefore likely to be of general importance for proteins. Additionally, binning strategies are of universal importance for numerical computation of correlations, we here provided a detailed comparison between equal-width and equal-sample binning for different bin numbers and demonstrated the impact of binning strategies on variances and biases of calculated mutual information. Our observation suggested that caution should be taken when quantitative comparison of correlations were intended between different studies with different binning strategies.

Torsional mutual information for 10 typical residue pairs calculated with full joint distributions (MI), second order expansion (MI₂), third order expansions (MI₃), and their linear recombinations (MI_LR).

How News May Affect Markets' Complex Structure: The Case of Cambridge Analytica

The claim of Cambridge Analytica, a political consulting firm, that it was possible to influence voting behavior by using data mined from the social platform Facebook created a sudden fear in its users of being manipulated; consequently, even the market price of the social platform was shocked.We propose a case study analyzing the effect of this data scandal not only on Facebook stock price, but also on the whole stock market. To such a scope, we consider 15-minutes prices and returns of the set of the NASDAQ-100 components before and after the Cambridge Analytica case. We analyze correlations and Mutual Information among components finding that assets become more correlated and their Mutual Information grows higher. We also observe that correlation and Mutual Information are mutually increasing and seem to follow a master curve. Hence, the market appears more fragile after the Cambridge Analytica event. In fact, as it is well-known in finance, an increase in the average value of correlations augments the systemic risk (i.e., all the market can collapse as a whole) and decreases the possibility of allocating a safe investment portfolio.