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Geometric deep learning has demonstrated a great potential in non-Euclidean data analysis. The incorporation of geometric insights into learning architecture is vital to its success. Here we propose a curvature-enhanced graph convolutional network (CGCN) for biomolecular interaction prediction. Our CGCN employs Ollivier-Ricci curvature (ORC) to characterize network local geometric properties and enhance the learning capability of GCNs. More specifically, ORCs are evaluated based on the local topology from node neighborhoods, and further incorporated into the weight function for the feature aggregation in message-passing procedure. Our CGCN model is extensively validated on fourteen real-world bimolecular interaction networks and analyzed in details using a series of well-designed simulated data. It has been found that our CGCN can achieve the state-of-the-art results. It outperforms all existing models, as far as we know, in thirteen out of the fourteen real-world datasets and ranks as the second in the rest one. The results from the simulated data show that our CGCN model is superior to the traditional GCN models regardless of the positive-to-negative-curvature ratios, network densities, and network sizes (when larger than 500).

Abstract With extraordinarily high efficiency, low cost, and excellent stability, 2D perovskite has demonstrated a great potential to revolutionize photovoltaics technology. However, inefficient material structure representations have significantly hindered artificial intelligence (AI)-based perovskite design and discovery. Here we propose geometric data analysis (GDA)-based perovskite structure representation and featurization and combine them with learning models for 2D perovskite design. Both geometric properties and periodicity information of the material unit cell, are fully characterized by a series of 1D functions, i.e., density fingerprints (DFs), which are mathematically guaranteed to be invariant under different unit cell representations and stable to structure perturbations. Element-specific DFs, which are based on different site combinations and atom types, are combined with gradient boosting tree (GBT) model. It has been found that our GDA-based learning models can outperform all existing models, as far as we know, on the widely used new materials for solar energetics (NMSE) databank.

Abstract Molecular representations are of fundamental importance for the modeling and analysing molecular systems. The successes in drug design and materials discovery have been greatly contributed by molecular representation models. In this paper, we present a computational framework for molecular representation that is mathematically rigorous and based on the persistent Dirac operator. The properties of the discrete weighted and unweighted Dirac matrix are systematically discussed, and the biological meanings of both homological and non-homological eigenvectors are studied. We also evaluate the impact of various weighting schemes on the weighted Dirac matrix. Additionally, a set of physical persistent attributes that characterize the persistence and variation of spectrum properties of Dirac matrices during a filtration process is proposed to be molecular fingerprints. Our persistent attributes are used to classify molecular configurations of nine different types of organic-inorganic halide perovskites. The combination of persistent attributes with gradient boosting tree model has achieved great success in molecular solvation free energy prediction. The results show that our model is effective in characterizing the molecular structures, demonstrating the power of our molecular representation and featurization approach.

Summary: Molecular representation learning plays an important role in molecular property prediction. Existing molecular property prediction models rely on the de facto standard of covalent-bond-based molecular graphs for representing molecular topology at the atomic level and totally ignore the non-covalent interactions within the molecule. In this study, we propose a molecular geometric deep learning model to predict the properties of molecules that aims to comprehensively consider the information of covalent and non-covalent interactions of molecules. The essential idea is to incorporate a more general molecular representation into geometric deep learning (GDL) models. We systematically test molecular GDL (Mol-GDL) on fourteen commonly used benchmark datasets. The results show that Mol-GDL can achieve a better performance than state-of-the-art (SOTA) methods. Extensive tests have demonstrated the important role of non-covalent interactions in molecular property prediction and the effectiveness of Mol-GDL models. Motivation: Geometric deep learning (GDL) has demonstrated huge power and enormous potential in molecular data analysis. However, a great challenge still remains for highly efficient molecular representations. Currently, covalent-bond-based molecular graphs are the de facto standard for representing molecular topology at the atomic level. Here, we demonstrate that molecular graphs constructed only from non-covalent bonds can achieve similar or even better results than covalent-bond-based models in molecular property prediction. This demonstrates the great potential of novel molecular representations beyond the de facto standard of covalent-bond-based molecular graphs. Based on the finding, we propose molecular geometric deep learning (Mol-GDL). In our Mol-GDL, molecular topology is modeled as a series of molecular graphs, each focusing on a different scale of atomic interactions. In this way, both covalent interactions and non-covalent interactions are incorporated into the molecular representation on an equal footing.

Abstract Accelerated materials development with machine learning (ML) assisted screening and high throughput experimentation for new photovoltaic materials holds the key to addressing our grand energy challenges. Data-driven ML is envisaged as a decisive enabler for new perovskite materials discovery. However, its full potential can be severely curtailed by poorly represented molecular descriptors (or fingerprints). Optimal descriptors are essential for establishing effective mathematical representations of quantitative structure-property relationships. Here we reveal that our persistent functions (PFs) based learning models offer significant accuracy advantages over traditional descriptor based models in organic-inorganic halide perovskite (OIHP) materials design and have similar performance as deep learning models. Our multiscale simplicial complex approach not only provides a more precise representation for OIHP structures and underlying interactions, but also has better transferability to ML models. Our results demonstrate that advanced geometrical and topological invariants are highly efficient feature engineering approaches that can markedly improve the performance of learning models for molecular data analysis. Further, new structure-property relationships can be established between our invariants and bandgaps. We anticipate that our molecular representations and featurization models will transcend the limitations of conventional approaches and lead to breakthroughs in perovskite materials design and discovery.

Abstract Hodge theory reveals the deep intrinsic relations of differential forms and provides a bridge between differential geometry, algebraic topology, and functional analysis. Here we use Hodge Laplacian and Hodge decomposition models to analyze biomolecular structures. Different from traditional graph-based methods, biomolecular structures are represented as simplicial complexes, which can be viewed as a generalization of graph models to their higher-dimensional counterparts. Hodge Laplacian matrices at different dimensions can be generated from the simplicial complex. The spectral information of these matrices can be used to study intrinsic topological information of biomolecular structures. Essentially, the number (or multiplicity) of k-th dimensional zero eigenvalues is equivalent to the k-th Betti number, i.e., the number of k-th dimensional homology groups. The associated eigenvectors indicate the homological generators, i.e., circles or holes within the molecular-based simplicial complex. Furthermore, Hodge decomposition-based HodgeRank model is used to characterize the folding or compactness of the molecular structures, in particular, the topological associated domain (TAD) in high-throughput chromosome conformation capture (Hi-C) data. Mathematically, molecular structures are represented in simplicial complexes with certain edge flows. The HodgeRank-based average/total inconsistency (AI/TI) is used for the quantitative measurements of the folding or compactness of TADs. This is the first quantitative measurement for TAD regions, as far as we know.

An important challenge in the study of complex systems is to identify appropriate effective variables at different times. In this paper, we explain why structures that are persistent with respect to changes in length and time scales are proper effective variables, and illustrate how persistent structures can be identified from the spectra and Fiedler vector of the graph Laplacian at different stages of the topological data analysis (TDA) filtration process for twelve toy models. We then investigated four market crashes, three of which were related to the COVID-19 pandemic. In all four crashes, a persistent gap opens up in the Laplacian spectra when we go from a normal phase to a crash phase. In the crash phase, the persistent structure associated with the gap remains distinguishable up to a characteristic length scale ϵ* where the first non-zero Laplacian eigenvalue changes most rapidly. Before ϵ*, the distribution of components in the Fiedler vector is predominantly bi-modal, and this distribution becomes uni-modal after ϵ*. Our findings hint at the possibility of understanding market crashs in terms of both continuous and discontinuous changes. Beyond the graph Laplacian, we can also employ Hodge Laplacians of higher order for future research.

Traditionally, human microbiology has been strongly built on the laboratory focused culture of microbes isolated from human specimens in patients with acute or chronic infection. These approaches primarily view human disease through the lens of a single species and its relevant clinical setting however such approaches fail to account for the surrounding environment and wide microbial diversity that exists in vivo. Given the emergence of next generation sequencing technologies and advancing bioinformatic pipelines, researchers now have unprecedented capabilities to characterise the human microbiome in terms of its taxonomy, function, antibiotic resistance and even bacteriophages. Despite this, an analysis of microbial communities has largely been restricted to ordination, ecological measures, and discriminant taxa analysis. This is predominantly due to a lack of suitable computational tools to facilitate microbiome analytics. In this review, we first evaluate the key concerns related to the inherent structure of microbiome datasets which include its compositionality and batch effects. We describe the available and emerging analytical techniques including integrative analysis, machine learning, microbial association networks, topological data analysis (TDA) and mathematical modelling. We also present how these methods may translate to clinical settings including tools for implementation. Mathematical based analytics for microbiome analysis represents a promising avenue for clinical translation across a range of acute and chronic disease states.

With the great advancements in experimental data, computational power and learning algorithms, artificial intelligence (AI) based drug design has begun to gain momentum recently. AI-based drug design has great promise to revolutionize pharmaceutical industries by significantly reducing the time and cost in drug discovery processes. However, a major issue remains for all AI-based learning model that is efficient molecular representations. Here we propose Dowker complex (DC) based molecular interaction representations and Riemann Zeta function based molecular featurization, for the first time. Molecular interactions between proteins and ligands (or others) are modeled as Dowker complexes. A multiscale representation is generated by using a filtration process, during which a series of DCs are generated at different scales. Combinatorial (Hodge) Laplacian matrices are constructed from these DCs, and the Riemann zeta functions from their spectral information can be used as molecular descriptors. To validate our models, we consider protein-ligand binding affinity prediction. Our DC-based machine learning (DCML) models, in particular, DC-based gradient boosting tree (DC-GBT), are tested on three most-commonly used datasets, i.e., including PDBbind-2007, PDBbind-2013 and PDBbind-2016, and extensively compared with other existing state-of-the-art models. It has been found that our DC-based descriptors can achieve the state-of-the-art results and have better performance than all machine learning models with traditional molecular descriptors. Our Dowker complex based machine learning models can be used in other tasks in AI-based drug design and molecular data analysis.

Microtubules are one of the most important components in the cytoskeleton and play a vital role in maintaining the shape and function of cells. Because single microtubules are some micrometers long, it is difficult to simulate such a large system using an all-atom model. In this work, we use the newly developed convolutional and K-means coarse-graining (CK-CG) method to establish an ultra-coarse-grained (UCG) model of a single microtubule, on the basis of the low electron microscopy density data of microtubules. We discuss the rationale of the micro-coarse-grained microtubule models of different resolutions and explore microtubule models up to 12-micron length. We use the devised microtubule model to quantify mechanical properties of microtubules of different lengths. Our model allows mesoscopic simulations of micrometer-level biomaterials and can be further used to study important biological processes related to microtubule function.

With the great significance of biomolecular flexibility in biomolecular dynamics and functional analysis, various experimental and theoretical models are developed. Experimentally, Debye-Waller factor, also known as B-factor, measures atomic mean-square displacement and is usually considered as an important measurement for flexibility. Theoretically, elastic network models, Gaussian network model, flexibility-rigidity model, and other computational models have been proposed for flexibility analysis by shedding light on the biomolecular inner topological structures. Recently, a topology-based machine learning model has been proposed. By using the features from persistent homology, this model achieves a remarkable high Pearson correlation coefficient (PCC) in protein B-factor prediction. Motivated by its success, we propose weighted-persistent-homology (WPH)-based machine learning (WPHML) models for RNA flexibility analysis. Our WPH is a newly-proposed model, which incorporate physical, chemical and biological information into topological measurements using a weight function. In particular, we use local persistent homology (LPH) to focus on the topological information of local regions. Our WPHML model is validated on a well-established RNA dataset, and numerical experiments show that our model can achieve a PCC of up to 0.5822. The comparison with the previous sequence-information-based learning models shows that a consistent improvement in performance by at least 10% is achieved in our current model.

In econophysics, the achievements of information filtering methods over the past 20 years, such as the minimal spanning tree (MST) by Mantegna and the planar maximally filtered graph (PMFG) by Tumminello et al., should be celebrated. Here, we show how one can systematically improve upon this paradigm along two separate directions. First, we used topological data analysis (TDA) to extend the notions of nodes and links in networks to faces, tetrahedrons, or k-simplices in simplicial complexes. Second, we used the Ollivier-Ricci curvature (ORC) to acquire geometric information that cannot be provided by simple information filtering. In this sense, MSTs and PMFGs are but first steps to revealing the topological backbones of financial networks. This is something that TDA can elucidate more fully, following which the ORC can help us flesh out the geometry of financial networks. We applied these two approaches to a recent stock market crash in Taiwan and found that, beyond fusions and fissions, other non-fusion/fission processes such as cavitation, annihilation, rupture, healing, and puncture might also be important. We also successfully identified neck regions that emerged during the crash, based on their negative ORCs, and performed a case study on one such neck region.

Progesterone receptor (PR) is a member of the nuclear receptor (NR) superfamily and plays a vital role in the female reproductive system. The malfunction of it would lead to several types of cancers. The understanding of conformational changes in its ligand binding domain (LBD) is valuable for both biological function studies and therapeutically intervenes. A key unsolved question is how the binding of a ligand (agonist, antagonist, or a selective modulator) induces conformational changes of PR LBD, especially its helix 12. We applied molecular dynamics (MD) simulations to explore the conformational adaptations of PR LBD with or without a ligand or the co-repressor peptides binding. From the simulations, both the agonist progesterone (P4) and the selective PR modulator (SPRM) asoprisnil induces agonistic-like helix 12 conformations (the “closed” states) in PR LBD and the complex of LBD-SPRM is less stable, comparing to the agonist-liganded PR LBD. The results, therefore, explain the partial agonism of the SPRM, which could induce weak agonistic effects in PR. We also found that co-repressor peptides could be stably associated with the LBD and stabilize the LBD in a “semi-open” state for helix 12. These findings would enhance our understanding of PR structural and functional relationships and would also be useful for future structure and knowledge-based drug discovery.

In this paper, we introduce sequence-based multiscale modeling for biomolecular data analysis. We employ spectral clustering method in our modeling and reveal the difference between sequence-based global scale clustering and local scale clustering. Essentially, two types of distances, i.e., Euclidean (or spatial) distance and genomic (or sequential) distance, can be used in data clustering. Clusters from sequence-based global scale models optimize spatial distances, meaning spatially adjacent loci are more likely to be assigned into the same cluster. Sequence-based local scale models, on the other hand, result in clusters that optimize genomic distances. That is to say, in these models, sequentially adjoining loci tend to be cluster together. We propose two sequence-based multiscale models (SeqMMs) for the study of chromosome hierarchical structures, including genomic compartments and topological associated domains (TADs). We find that genomic compartments are determined only by global scale information in the Hi-C data. The removal of all the local interactions within a band region as large as 10 Mb in genomic distance has almost no significant influence on the final compartment results. Further, in TAD analysis, we find that when the sequential scale is small, a tiny variation of diagonal band region in a contact map will result in a great change in the predicted TAD boundaries. When the scale value is larger than a threshold value, the TAD boundaries become very consistent. This threshold value is highly related to TAD sizes. By the comparison of our results with those previously obtained using a spectral clustering model, we find that our method is more robust and reliable. Finally, we demonstrate that almost all TAD boundaries from both clustering methods are local minimum of a TAD summation function.

We establish a new theory which unifies various aspects of topological approaches for data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and hypergraphs to super-hypergraphs and establish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology. [ABSTRACT FROM AUTHOR]

Flexibility-rigidity index (FRI) has been developed as a robust, accurate, and efficient method for macromolecular thermal fluctuation analysis and B-factor prediction. The performance of FRI depends on its formulations of rigidity index and flexibility index. In this work, we introduce alternative rigidity and flexibility formulations. The structure of the classic Gaussian surface is utilized to construct a new type of rigidity index, which leads to a new class of rigidity densities with the classic Gaussian surface as a special case. Additionally, we introduce a new type of flexibility index based on the domain indicator property of normalized rigidity density. These generalized FRI (gFRI) methods have been extensively validated by the B-factor predictions of 364 proteins. Significantly outperforming the classic Gaussian network model, gFRI is a new generation of methodologies for accurate, robust, and efficient analysis of protein flexibility and fluctuation. Finally, gFRI based molecular surface generation and flexibility visualization are demonstrated. [ABSTRACT FROM AUTHOR]

Gaussian networkmodel (GNM) and anisotropicnetworkmodel (ANM) are some of the most popular methods for the study of protein flexibility and related functions. In this work, we propose generalized GNM (gGNM) and ANM methods and show that the GNM Kirchhoff matrix can be built from the ideal low-pass filter, which is a special case of a wide class of correlation functions underpinning the linear scaling flexibility-rigidity index (FRI) method. Based on the mathematical structure of correlation functions, we propose a unified framework to construct generalized Kirchhoff matrices whose matrix inverse leads to gGNMs, whereas, the direct inverse of its diagonal elements gives rise to FRI method. With this connection, we further introduce two multiscale elasticnetworkmodels, namely, multiscale GNM (mGNM) and multiscale ANM (mANM), which are able to incorporate different scales into the generalized Kirchhoff matrices or generalized Hessian matrices. We validate our new multiscale methods with extensive numerical experiments. We illustrate that gGNMs outperform the original GNM method in the B-factor prediction of a set of 364 proteins. We demonstrate that for a given correlation function, FRI and gGNM methods provide essentially identical B-factor predictions when the scale value in the correlation function is sufficiently large. More importantly, we reveal intrinsic multiscale behavior in proteinstructures. The proposed mGNM and mANM are able to capture this multiscale behavior and thus give rise to a significant improvement of more than 11% in B-factor predictions over the original GNM and ANM methods. We further demonstrate the benefits of our mGNM through the B-factor predictions of many proteins that fail the original GNM method. We show that the proposed mGNM can also be used to analyzeprotein domain separations. Finally, we showcase the ability of our mANM for the analysis of protein collective motions. [ABSTRACT FROM AUTHOR]

Although persistent homology has emerged as a promising tool for the topological simplification of complex data, it is computationally intractable for large datasets. We introduce multiresolution persistent homology to handle excessively large datasets. We match the resolution with the scale of interest so as to represent large scale datasets with appropriate resolution. We utilize flexibilityrigidity index to access the topological connectivity of the data set and define a rigidity density for the filtration analysis. By appropriately tuning the resolution of the rigidity density, we are able to focus the topological lens on the scale of interest. The proposed multiresolution topological analysis is validated by a hexagonal fractal image which has three distinct scales. We further demonstrate the proposed method for extracting topological fingerprints from DNA molecules. In particular, the topological persistence of a virus capsid with 273 780 atoms is successfully analyzed which would otherwise be inaccessible to the normal point cloud method and unreliable by using coarse-grained multiscale persistent homology. The proposed method has also been successfully applied to the protein domain classification, which is the first time that persistent homology is used for practical protein domain analysis, to our knowledge. The proposed multiresolution topological method has potential applications in arbitrary data sets, such as social networks, biological networks, and graphs. [ABSTRACT FROM AUTHOR]

Protein structural fluctuation, typically measured by Debye-Waller factors, or B-factors, is a manifestation of protein flexibility, which strongly correlates to protein function. The flexibility-rigidity index (FRI) is a newly proposed method for the construction of atomic rigidity functions required in the theory of continuum elasticity with atomic rigidity, which is a new multiscale formalism for describing excessively large biomolecular systems. The FRI method analyzes protein rigidity and flexibility and is capable of predicting protein B-factors without resorting to matrix diagonalization. A fundamental assumption used in the FRI is that protein structures are uniquely determined by various internal and external interactions, while the protein functions, such as stability and flexibility, are solely determined by the structure. As such, one can predict protein flexibility without resorting to the protein interaction Hamiltonian. Consequently, bypassing the matrix diagonalization, the original FRI has a computational complexity of O(N2). This work introduces a fast FRI (fFRI) algorithm for the flexibility analysis of large macromolecules. The proposed fFRI further reduces the computational complexity to O(N). Additionally, we propose anisotropic FRI (aFRI) algorithms for the analysis of protein collective dynamics. The aFRI algorithms permit adaptive Hessian matrices, from a completely global 3N × 3N matrix to completely local 3 × 3 matrices. These 3 × 3 matrices, despite being calculated locally, also contain non-local correlation information. Eigenvectors obtained from the proposed aFRI algorithms are able to demonstrate collective motions. Moreover, we investigate the performance of FRI by employing four families of radial basis correlation functions. Both parameter optimized and parameter-free FRI methods are explored. Furthermore, we compare the accuracy and efficiency of FRI with some established approaches to flexibility analysis, namely, normal mode analysis and Gaussian network model (GNM). The accuracy of the FRI method is tested using four sets of proteins, three sets of relatively small-, medium-, and large-sized structures and an extended set of 365 proteins. A fifth set of proteins is used to compare the efficiency of the FRI, fFRI, aFRI, and GNM methods. Intensive validation and comparison indicate that the FRI, particularly the fFRI, is orders of magnitude more efficient and about 10% more accurate overall than some of the most popular methods in the field. The proposed fFRI is able to predict B-factors for α-carbons of the HIV virus capsid (313 236 residues) in less than 30 seconds on a single processor using only one core. Finally, we demonstrate the application of FRI and aFRI to protein domain analysis. [ABSTRACT FROM AUTHOR]

Abstract Background Early life microflora is an important determinant of immune and metabolic development and may have lasting consequences. However, the mode of delivery and the effect of povidone iodine disinfection on neonatal oral microflora colonization are still unclear. The objective of the study was to understand the effects of the use of polyvidone iodine on infant’s oral microflora after transvaginal examination during delivery, provided data support for the establishment of neonatal oral microflora health. Methods A total of 20 cases of full-term neonatal delivered in October 2017 in Shenzhen Bao’an Maternity and Child Health Hospital through vaginal delivery. These neonates were randomly divided into two groups, the conventional disinfection group and the non-disinfection group. Simultaneously, 10 infants with elective cesarean section were taken as comparison. With Illumina MiSeq platform, 16S rRNA V3-V4 sequencing method was used to analyze bacterial DNA of oral secretions. Results At the phylum level, compared to the non-disinfection group, higher relative abundance of Bacteroidetes and Proteobacteria, and lower proportion of Firmicutes were observed in the cesarean section group and the disinfection group. As main composition of phylum Firmicutes, genus Lactobacillus presented extremely low in the cesarean section group and the disinfection group, whereas it was the absolute dominant bacteria in the non-disinfection group. Compared with the caesarean section group, only Lactobacillus increased in majority of the non-disinfection group. There was no increase in Lactobacillus in the disinfection group, but Prevotella, Escherichia-Shigella, Staphyloccus, and Klebsiella increased significantly. Through KEGG pathway analysis, we found that there were more harmful pathways such as staphylococcus aureus infection, viral myocarditis and sporulation in the disinfection group. Conclusions The mode of delivery affects the infant’s Lactobacillus obtained from the mother. Moreover, vulvar disinfection played an important part in the colonization of neonatal oral microbiota. And the impact of the first oral colonizers on infant health needs further follow-up investigations.

Material interfaces are omnipresent in the real-world structures and devices. Mathematical modeling of material interfaces often leads to elliptic partial differential equations (PDEs) with discontinuous coefficients and singular sources, which are commonly called elliptic interface problems. The development of high-order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades. Despite of significant advances, challenges remain in the construction of high-order schemes for nonsmooth interfaces, i.e., interfaces with geometric singularities, such as tips, cusps and sharp edges. The challenge of geometric singularities is amplified when they are associated with low solution regularities, e.g., tip-geometry effects in many fields. The present work introduces a matched interface and boundary (MIB) Galerkin method for solving two-dimensional (2D) elliptic PDEs with complex interfaces, geometric singularities and low solution regularities. The Cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure. Consequently, the interface cuts through elements. To ensure the continuity of classic basis functions across the interface, two sets of overlapping elements, called MIB elements, are defined near the interface. As a result, differentiation can be computed near the interface as if there is no interface. Interpolation functions are constructed on MIB element spaces to smoothly extend function values across the interface. A set of lowest order interface jump conditions is enforced on the interface, which in turn, determines the interpolation functions. The performance of the proposed MIB Galerkin finite element method is validated by numerical experiments with a wide range of interface geometries, geometric singularities, low regularity solutions and grid resolutions. Extensive numerical studies confirm the designed second order convergence of the MIB Galerkin method in the L8 and L2 errors. Some of the best results are obtained in the present work when the interface is C1 or Lipschitz continuous and the solution is C2 continuous. [ABSTRACT FROM AUTHOR]

This work introduces molecular nonlinear dynamics (MND) as a new approach for describing protein folding and aggregation. By using a mode system, we show that the MND of disordered proteins is chaotic while that of folded proteins exhibits intrinsically low dimensional manifolds (ILDMs). The stability of ILDMs is found to strongly correlate with protein energies. We propose a novel method for protein thermal uncertainty quantification based on persistently invariant ILDMs. Extensive comparison with experimental data and the state-of-the-art methods in the field validate the proposed new method for protein B-factor prediction. [ABSTRACT FROM AUTHOR]

Protein flexibility is an intrinsic property and plays a fundamental role in protein functions. Computational analysis of protein flexibility is crucial to protein function prediction, macromolecular flexible docking, and rational drug design. Most current approaches for protein flexibility analysis are based on Hamiltonian mechanics. We introduce a stochastic model to study protein flexibility. The essential idea is to analyze the free induction decay of a perturbed protein structural probability, which satisfies the master equation. The transition probability matrix is constructed by using probability density estimators including monotonically decreasing radial basis functions. We show that the proposed stochastic model gives rise to some of the best predictions of Debye-Waller factors or B factors for three sets of protein data introduced in the literature. [ABSTRACT FROM AUTHOR]

This work presents a few variational multiscale models for charge transport in complex physical, chemical, and biological systems and engineering devices, such as fuel cells, solar cells, battery cells, nanofluidics, transistors, and ion channels. An essential ingredient of the present models, introduced in an earlier paper [Bull. Math. Biol., 72 (2010), pp. 1562-1622], is the use of the differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain from the microscopic domain, while dynamically coupling discrete and continuum descriptions. Our main strategy is to construct the total energy functional of a charge transport system to encompass the polar and nonpolar free energies of solvation and chemical potential related energy. By using the Euler-Lagrange variation, coupled Laplace-Beltrami and Poisson-Nernst-Planck (LB-PNP) equations are derived. The solution of the LB-PNP equations leads to the minimization of the total free energy and explicit profiles of electrostatic potential and densities of charge species. To further reduce the computational complexity, the Boltzmann distribution obtained from the Poisson-Boltzmann (PB) equation is utilized to represent the densities of certain charge species so as to avoid the computationally expensive solution of some Nernst-Planck (NP) equations. Consequently, the coupled Laplace-Beltrami and Poisson-Boltzmann-Nernst-Planck (LB-PBNP) equations are proposed for charge transport in heterogeneous systems. A major emphasis of the present formulation is the consistency between equilibrium Laplace-Beltrami and PB (LB-PB) theory and nonequilibrium LB-PNP theory at equilibrium. Another major emphasis is the capability of the reduced LB-PBNP model to fully recover the prediction of the LB-PNP model at nonequilibrium settings. To account for the fluid impact on the charge transport, we derive coupled Laplace-Beltrami, Poisson-Nernst- Planck, and Navier-Stokes equations from the variational principle for chemo-electro-fluid systems. A number of computational algorithms are developed to implement the proposed new variational multiscale models in an efficient manner. A set often protein molecules and a realistic ion channel, Gramicidin A, are employed to confirm the consistency and verify the capability of the algorithms. Extensive numerical experiments are designed to validate the proposed variational multiscale models. A good quantitative agreement between our model prediction and the experimental measurement of current-voltage curves is observed for the Gramicidin A channel transport. This paper also provides a brief review of the field. [ABSTRACT FROM AUTHOR]