The structure of DNA as deduced from fiber X-ray crystallography

X-ray diffraction patterns obtained experimentally for fibers, together with their chemical structures, can be analyzed theoretically in terms of an integral equation. The partially unknown electron density function can be solved by iteration. This mathematical technique has been applied with success to study the secondary structures of DNA fibers.     

真骨鱼卵膜孔奇异结构的数学建模

本文通过对泥鳅、大银鱼、鲂鱼等真骨鱼卵膜孔形态结构的研究,把目前已发现的奇异结构分为涡旋卷吸型、沟脊降落型、降旋混合型,并定量地建立了它们的数学模型．然后,又通过对卵膜孔周围流场的动力学分析,确证真骨鱼精子入卵除化学因素外,还存在着不容忽视的数学物理因素──机械动力作用．这两者相得益彰,形成了一个完整的精子引导系统．     

Commentary: model building, quantitative testing, and model comparison

The final goal is to create mathematical models that are based on our current knowledge of the underlying physiology and that explain all of the experimental data available. To do this, we suggest a consideration of several potential mathematical structures in the formulation of models and the formal comparison of these various structures with other models in the literature. However, when making these comparisons, one must pay careful attention to the systems being modeled and the data sets chosen to represent those systems.     

Surface growth kinematics via local curve evolution

A mathematical framework is developed to model the kinematics of surface growth for objects that can be generated by evolving a curve in space, such as seashells and horns. Growth is dictated by a growth velocity vector field defined at every point on a generating curve. A local orthonormal basis is attached to each point of the generating curve and the velocity field is given in terms of the local coordinate directions, leading to a fully local and elegant mathematical structure. Several examples of increasing complexity are provided, and we demonstrate how biologically relevant structures such as logarithmic shells and horns emerge as analytical solutions of the kinematics equations with a small number of parameters that can be linked to the underlying growth process. Direct access to cell tracks and local orientation enables for connections to be made to the underlying growth process.     

A mathematical framework for protein structure comparison

Comparison of protein structures is important for revealing the evolutionary relationship among proteins, predicting protein functions and predicting protein structures. Many methods have been developed in the past to align two or multiple protein structures. Despite the importance of this problem, rigorous mathematical or statistical frameworks have seldom been pursued for general protein structure comparison. One notable issue in this field is that with many different distances used to measure the similarity between protein structures, none of them are proper distances when protein structures of different sequences are compared. Statistical approaches based on those non-proper distances or similarity scores as random variables are thus not mathematically rigorous. In this work, we develop a mathematical framework for protein structure comparison by treating protein structures as three-dimensional curves. Using an elastic Riemannian metric on spaces of curves, geodesic distance, a proper distance on spaces of curves, can be computed for any two protein structures. In this framework, protein structures can be treated as random variables on the shape manifold, and means and covariance can be computed for populations of protein structures. Furthermore, these moments can be used to build Gaussian-type probability distributions of protein structures for use in hypothesis testing. The covariance of a population of protein structures can reveal the population-specific variations and be helpful in improving structure classification. With curves representing protein structures, the matching is performed using elastic shape analysis of curves, which can effectively model conformational changes and insertions/deletions. We show that our method performs comparably with commonly used methods in protein structure classification on a large manually annotated data set.     

Survey of research in the theory of homogeneous structures and their applications

This work presents some of the author's reflections, the purpose of which is to give an orientation on the present stage and recent developments in the investigations of the theory of homogeneous structures and their applications. This survey appears to be desirable at this time because a number of research workers in mathematical and theoretical biology are becoming interested in making contributions in the theory of homogeneous structures and their applications. It is hoped that this survey will provide a stimulus and some introduction to the subject. The choice of topics is dictated largely by my predilections, and in no way shall I attempt to illustrate the vast and rapidly increasing field of applications of mathematical techniques or even, more generally, the mathematical mode of thinking in the enormous variety of problems in the theory of homogeneous structures and its applications, including biology. Indeed, to present anything like a comprehensive picture would require bulky and manifold volumes of very heterogeneous contributions. However, I shall list a number of biological and engineering fields where techniques of homogeneous structures have been employed and give the reader a short indication of selected bibliographic aids.     

Receptive field families

It is generally agreed upon that size invariance and the absence of spurious resolution are two requirements that characterize well behaved spatial samping in visual systems. We show that these properties taken together constrain the structure of receptive fields to a very large degree. Only those field structures that arise as solutions of a certain linear partial differential equation of the second order prove to satisfy these general constraints. The equation admits of complete, orthonormal families of solutions. These families have to be regarded as the possible receptive field families subject to certain symmetry conditions. They can be transformed into each other via unitary transformations. Thus a single representation suffices to construct all the others. This theory permits us to classify the possible linear receptive field structures exhaustively, and to define their internal and external interrelations. This induces a principled taxonomy of linear receptive fields.     

Mathematical aspects of protein structure determination with NMR orientational restraints

The field of structural biology is becoming increasingly important as new technological developments facilitate the collection of data on the atomic structures of proteins and nucleic acids. The solid-state NMR method is a relatively new biophysical technique that holds particular promise for determining the structures of peptides and proteins that are located within the cell membrane. This method provides information on the orientation of the peptide planes relative to an external magnetic field. In this article, we discuss some of the mathematical methods and tools that are useful in deriving the atomic structure from these orientational data. We first discuss how the data are viewed as tensors, and how these tensors can be used to construct an initial atomic model, assuming ideal stereochemistry. We then discuss methods for refining the models using global optimization, with stereochemistry constraints treated as penalty functions. These two processes, initial model building followed by refinement, are the two crucial steps between data collection and the final atomic model.     

The combinatorial distance geometry method for the calculation of molecular conformation. I. A new approach to an old problem

A new approach to the long-standing local minimum problem of molecular energy minimization is proposed. The approach relies upon a field of computer mathematics known as combinatorial optimization, together with methods of conformational analysis derived from distance geometry. The advantages over the usual numerical techniques of optimization are, first, that the algorithms derived are globally convergent, and second, that the mathematical problems involved are well-posed and suitable for study within the modern theory of computational complexity. In this paper we introduce the approach, and describe a computer program based on it.     

Mathematical descriptions of biochemical networks: stability, stochasticity, evolution

In this paper, we review some fundamental aspects, as well as some new developments, in the emerging field of network biology. The focus of attention is placed on mathematical approaches to conceptual modeling of biomolecular networks with special emphasis on dynamic stability, stochasticity and evolution.     

Design, construction, and analysis of a novel class of self-folding RNA

RNA can play multiple biological roles through use of its three-dimensional (3-D) structures. Recent advances in RNA structural biology have revealed that complex RNA 3D structures are assemblages of double-stranded helices with a variety of tertiary structural motifs. By employing RNA tertiary structural motifs together with the helices, we designed a novel class of self-folding RNA. In RNA composed of three helices (P1, P2, and P3), P1 interacts with P3 via a tetraloop-receptor interaction and P2 forms consecutive base-triples. Two designed RNAs of this class were prepared and their folding properties indicate that they form defined tertiary structures as designed. These RNAs may be used as modular units for constructing artificial ribozymes or nanometer-scale materials.     

Neural Mechanisms Underlying the Computation of Hierarchical Tree Structures in Mathematics

Whether mathematical and linguistic processes share the same neural mechanisms has been a matter of controversy. By examining various sentence structures, we recently demonstrated that activations in the left inferior frontal gyrus (L. IFG) and left supramarginal gyrus (L. SMG) were modulated by the Degree of Merger (DoM), a measure for the complexity of tree structures. In the present study, we hypothesize that the DoM is also critical in mathematical calculations, and clarify whether the DoM in the hierarchical tree structures modulates activations in these regions. We tested an arithmetic task that involved linear and quadratic sequences with recursive computation. Using functional magnetic resonance imaging, we found significant activation in the L. IFG, L. SMG, bilateral intraparietal sulcus (IPS), and precuneus selectively among the tested conditions. We also confirmed that activations in the L. IFG and L. SMG were free from memory-related factors, and that activations in the bilateral IPS and precuneus were independent from other possible factors. Moreover, by fitting parametric models of eight factors, we found that the model of DoM in the hierarchical tree structures was the best to explain the modulation of activations in these five regions. Using dynamic causal modeling, we showed that the model with a modulatory effect for the connection from the L. IPS to the L. IFG, and with driving inputs into the L. IFG, was highly probable. The intrinsic, i.e., task-independent, connection from the L. IFG to the L. IPS, as well as that from the L. IPS to the R. IPS, would provide a feedforward signal, together with negative feedback connections. We indicate that mathematics and language share the network of the L. IFG and L. IPS/SMG for the computation of hierarchical tree structures, and that mathematics recruits the additional network of the L. IPS and R. IPS.     

The peristomatic structures of Lithobiomorpha (Myriapoda, Chilopoda): comparative morphology and phylogenetic significance

A comparative survey of the epipharynx and hypopharynx of lithobiomorph centipedes by light and scanning electron microscopy examines 18 species that sample the major groups of both families, the Lithobiidae and Henicopidae. Cladistic analysis of 11 characters of the peristomatic structures together with 29 additional morphological characters serves as a basis for interpreting the evolution of the lithobiomorph peristomatic structures. Scutigeromorpha is used for outgroup comparison in the framework of a homology scheme for the basic components of the epi- and hypopharynx. Compared to other chilopods, the monophyly of Lithobiomorpha is supported by a row of distinctive bottle-shaped gland openings at the border between the labral and clypeal parts of the epipharynx, as well as by a distinctive shape of the hypopharynx. Paired rows of elongate spines on the clypeal part of the epipharynx are an apomorphic character of Lithobiidae. The transformation of these spine rows into a few groups of branching spines is characteristic for the Monotarsobius group sensu Verhoeff. Similar groups of branching clypeal spines characterize the Anopsobiinae within Henicopidae, whereas Henicopinae possess a dense cluster of short, simple spines instead. The recently described genus Dzhungaria is resolved closer to Henicopinae than to Anopsobiinae, a hypothesis supported by a field of grooves on the medial labral part of the epipharynx. Monophyly of Henicopidae does not receive unique support from the peristomatic structures although two homoplastic characters contribute to this node; among these, the reduction of a median spine field between clypeal and labral parts of the epipharynx to a narrow transverse band also supports a close relationship between the Ezembius group and Hessebius within Lithobiidae. An Ezembius+Hessebius clade is additionally supported by the absence of a transverse bulge between the clypeal and labral parts of the epipharynx, a character otherwise present in all lithobiomorph species studied so far. Lithobius is resolved as polyphyletic, with different species being most closely related to such genera as Australobius, Hessebius and Pleurolithobius.     

Bioinformatic strategies for better understanding of immune function

Novartis Foundation sponsored a Symposium which brought together a group of experimental immunologists, theoretical immunologists, and bioinformaticians to discuss the new field of immunoinformatics. The discussion focused on immunological databases, antigen processing and presentation, immunogenomics, host-pathogen interactions, and mathematical modelling of the immune system. A main conclusion of the meeting is the critical role played by immunoinformatics in current immunology research. In particular, immunoinformatics provides a foundation for the emerging fields of systems immunology and immunogenomics.     

Surface structure of ionic liquids under an external electric field

Abstract

Surface structure and properties play an important role in many applications of ionic liquids (ILs). ILs can form unique surface structures that are very different from the bulk. In imidazolium-based ILs, for example, polar groups form ordered layer structure, while cationic alkyl chains are bundled together and point out from the surface. In many applications, ILs work under an external electric field. However, the effect of external electric field on the surface structure of ILs is still not clear. Here by using coarse-grained molecular dynamics simulation, we found that an electric field as strong as 1 V/nm is required to alter the surface structure of 1-dodecyl-3-methylimidazolium nitrate. Under a strong external electric field, layered structure disappears, and instead a sparser, more homogeneous and less orientationally ordered surface develops.     

A mathematical model for spreading cortical depression.

A mathematical model is derived from physiological considerations for slow potential waves (called spreading depression) in cortical neuronal structures. The variables taken into account are the intra- and extracellular concentrations of Na+, Cl-, K+, and Ca++, together with excitatory and inhibitor transmitter substances. The general model includes conductance changes for these various ions, which may occur at nonsynaptic and synaptic membrane together with active transport mechanisms (pumps). A detailed consideration of only the conductance changes due to transmitter release leads to a system of nonlinear diffusion equations coupled with a system or ordinary differential equations. We obtain numerical solutions of a set of simplified model equations involving only K+ and Ca++ concentrations. The solutions agree qualitatively with experimentally obtained time-courses of these two ionic concentrations during spreading depression. The numerical solutions exhibit the observed phenomena of solitary waves and annihilation of colliding waves.     

Synthesis, spectroscopy, and structures of new rhodium(I) and rhodium(III) corroles and catalysis thereby

The syntheses and molecular structures of six- and five-coordinated rhodium(III) corroles (by pyridines and a chiral amine, respectively) and the rhodium(I) complex of a chiral corrole are described, together with some interesting features in the NMR spectra of the complexes and their utilization as carbene-transfer catalysts.     

Force field dependence of NMR-Based,restrained molecular dynamics DNA structure calculations including an analysis of the influence of residual dipolar coupling restraints

Restrained molecular dynamics is widely used to calculate DNA structures from NMR data. Here, results of an in silico experiment show that the force field can be significant compared to the NMR restraints in driving the final structures to converge. Specifically, we observed that i) the influence of the force field leads to artificially tight convergence within final families of structures and ii) the precision and character of resulting structures depend on the choice of force field used in the calculations. A canonical B-DNA model was used as a target structure. Distances, dihedral angles, and simulated residual dipolar couplings were measured in the target structure and used as restraints. X-PLOR and Discover, which use force fields developed for CHARMM and AMBER programs, respectively, were tested and found to produce different final structures despite the use of identical distance and dihedral restraints. Incorporation of residual dipolar coupling restraints in X-PLOR improves convergence with the target structure and between families of structures indicating that the force field dependence can potentially be overcome if residual dipolar coupling restraints are employed.