Self-similar Rayleigh-Taylor mixing with accelerations varying in time and space.

As a ubiquitous paradigm of instabilities and mixing that occur in instances as diverse as supernovae, plasma fusion, oil recovery, and nanofabrication, the Rayleigh-Taylor (RT) problem is rightly regarded as important. The acceleration of the fluid medium in these instances often depends on time and space, whereas most past studies assume it to be constant or impulsive. Here, we analyze the symmetries of RT mixing for variable accelerations and obtain the scaling of correlations and spectra for classes of self-similar dynamics. RT mixing is shown to retain the memory of deterministic conditions for all accelerations, with the dynamics ranging from superballistic to subdiffusive. These results contribute to our understanding and control of the RT phenomena and reveal specific conditions under which Kolmogorov turbulence might be realized in RT mixing.

Topology and Dynamics of Transcriptome (Dys)Regulation.

RNA transcripts play a crucial role as witnesses of gene expression health. Identifying disruptive short sequences in RNA transcription and regulation is essential for potentially treating diseases. Let us delve into the mathematical intricacies of these sequences. We have previously devised a mathematical approach for defining a "healthy" sequence. This sequence is characterized by having at most four distinct nucleotides (denoted as nt≤4). It serves as the generator of a group denoted as fp. The desired properties of this sequence are as follows: fp should be close to a free group of rank nt-1, it must be aperiodic, and fp should not have isolated singularities within its SL2(C) character variety (specifically within the corresponding Groebner basis). Now, let us explore the concept of singularities. There are cubic surfaces associated with the character variety of a four-punctured sphere denoted as S24. When we encounter these singularities, we find ourselves dealing with some algebraic solutions of a dynamical second-order differential (and transcendental) equation known as the Painlevé VI Equation. In certain cases, S24 degenerates, in the sense that two punctures collapse, resulting in a "wild" dynamics governed by the Painlevé equations of an index lower than VI. In our paper, we provide examples of these fascinating mathematical structures within the context of miRNAs. Specifically, we find a clear relationship between decorated character varieties of Painlevé equations and the character variety calculated from the seed of oncomirs. These findings should find many applications including cancer research and the investigation of neurodegenative diseases.

Group Structure as a Foundation for Entropies.

Entropy can signify different things. For instance, heat transfer in thermodynamics or a measure of information in data analysis. Many entropies have been introduced, and it can be difficult to ascertain their respective importance and merits. Here, we consider entropy in an abstract sense, as a functional on a probability space, and we review how being able to handle the trivial case of non-interacting systems, together with the subtle requirement of extensivity, allows for a systematic classification of the functional form.

Mathematical Geometry and Groups for Low-Symmetry Metal Complex Systems.

Since chemistry, materials science, and crystallography deal with three-dimensional structures, they use mathematics such as geometry and symmetry. In recent years, the application of topology and mathematics to material design has yielded remarkable results. It can also be seen that differential geometry has been applied to various fields of chemistry for a relatively long time. There is also the possibility of using new mathematics, such as the crystal structure database, which represents big data, for computational chemistry (Hirshfeld surface analysis). On the other hand, group theory (space group and point group) is useful for crystal structures, including determining their electronic properties and the symmetries of molecules with relatively high symmetry. However, these strengths are not exhibited in the low-symmetry molecules that are actually handled. A new use of mathematics for chemical research is required that is suitable for the age when computational chemistry and artificial intelligence can be used.

The functional motions and related key residues behind the uncoating of coxsackievirus A16.

The coxsackievirus A16 (CVA16) is a highly contagious virus that causes the hand, foot, and mouth disease, which seriously threatens the health of children. At present, there are still no available antiviral drugs or effective treatments against the infection of CVA16, and thus it is of great significance to develop anti-CVA16 vaccines. However, the intrinsic uncoating property of the capsid may destroy the neutralizing epitopes and influence its immunogenicity, which hinders the vaccine developments. In the present work, the functional-quantity-based elastic network model analysis method developed by our group was extended to combine with group theory to investigate the uncoating motions of the CVA16 capsid, and then the functionally key residues controlling the uncoating motions were identified by our functional-quantity-based perturbation method. Several motion modes encoded in the topological structure of the capsid were revealed to be responsible for the uncoating of CVA16 particle. These modes predominantly contribute to the fluctuation of the gyration radius of the capsid. Then, by using the perturbation method, four clusters of key sites involved in the uncoating motions were identified, whose perturbations induce significant changes in the fluctuation of the gyration radius. These key residues are mainly located at the 2-fold channels, the quasi 3-fold channels, the bottom of the canyons, and the inter-subunit interfaces around the 3-fold axes. Our studies are helpful for better understanding the uncoating mechanism of the CVA16 capsid and provide potential target sites to prevent the uncoating motions, which is valuable for the vaccine design against CVA16.

Symmetry and its transition in phyllotaxis.

Symmetry is an important component of geometric beauty and regularity in both natural and cultural scenes. Plants also display various geometric patterns with some kinds of symmetry, of which the most notable example is the arrangement of leaves around the stem, i.e., phyllotaxis. In phyllotaxis, reflection symmetry, rotation symmetry, translation symmetry, corkscrew symmetry, and/or glide reflection symmetry can be seen. These phyllotactic symmetries can be dealt with the group theory. In this review, we introduce classification of phyllotactic symmetries according to the group theory and enumerate all types of phyllotaxis, not only major ones such as spiral and decussate but also minor ones such as orixate and semi-decussate, with their symmetry groups. Next, based on the mathematical model studies of phyllotactic pattern formation, we discuss transitions between phyllotaxis types different in the symmetry class with a focus on the transition into one of the least symmetric phyllotaxis, orixate, as a representative of the symmetry-breaking process. By changes of parameters of the mathematical model, the phyllotactic pattern generated can suddenly switch its symmetry class, which is not constrained by the group-subgroup relationship of symmetry. The symmetry-breaking path to orixate phyllotaxis is also accompanied by dynamic changes of the symmetry class. The viewpoint of symmetry brings a better understanding of the variety of phyllotaxis and its transition.

Analytical symmetry detection in protein assemblies. II. Dihedral and cubic symmetries.

Protein assemblies are often symmetric, as this organization has many advantages compared to individual proteins. Complex protein structures thus very often possess high-order symmetries. Detection and analysis of these symmetries has been a challenging problem and no efficient algorithms have been developed so far. This paper presents the extension of our cyclic symmetry detection method for higher-order symmetries with multiple symmetry axes. These include dihedral and cubic, i.e., tetrahedral, octahedral, and icosahedral, groups. Our method assesses the quality of a particular symmetry group and also determines all of its symmetry axes with a machine precision. The method comprises discrete and continuous optimization steps and is applicable to assemblies with multiple chains in the asymmetric subunits or to those with pseudo-symmetry. We implemented the method in C++ and exhaustively tested it on all 51,358 symmetric assemblies from the Protein Data Bank (PDB). It allowed us to study structural organization of symmetric assemblies solved by X-ray crystallography, and also to assess the symmetry annotation in the PDB. For example, in 1.6% of the cases we detected a higher symmetry group compared to the PDB annotation, and we also detected several cases with incorrect annotation. The method is available at http://team.inria.fr/nano-d/software/ananas. The graphical user interface of the method built for the SAMSON platform is available at http://samson-connect.net.

Maximum likelihood estimates of pairwise rearrangement distances.

Accurate estimation of evolutionary distances between taxa is important for many phylogenetic reconstruction methods. Distances can be estimated using a range of different evolutionary models, from single nucleotide polymorphisms to large-scale genome rearrangements. Corresponding corrections for genome rearrangement distances fall into 3 categories: Empirical computational studies, Bayesian/MCMC approaches, and combinatorial approaches. Here, we introduce a maximum likelihood estimator for the inversion distance between a pair of genomes, using a group-theoretic approach to modelling inversions introduced recently. This MLE functions as a corrected distance: in particular, we show that because of the way sequences of inversions interact with each other, it is quite possible for minimal distance and MLE distance to differently order the distances of two genomes from a third. The second aspect tackles the problem of accounting for the symmetries of circular arrangements. While, generally, a frame of reference is locked, and all computation made accordingly, this work incorporates the action of the dihedral group so that distance estimates are free from any a priori frame of reference. The philosophy of accounting for symmetries can be applied to any existing correction method, for which examples are offered.

Intergenerational Care Perceptions of Older Women and Middle Adolescents in a Resource-Constrained Community in South Africa.

This study describes intergenerational care perceptions in a resource-challenged community. Ten women (aged 60+) and eight middle adolescents (3 boys and 5 girls) participated in the Mmogo-method®, a visual data-collection method. Textual data were analysed thematically, and visual data by applying Roos and Redelinghuys (2016) proposed steps. Both groups provided physical and instrumental care to the other. Older women cared for adolescents by teaching and disciplining them, while the adolescents cared for them by obtaining an education and by showing respect. Older women felt being cared for when adolescents helped them, obeyed and complied with instructions and discipline, while the youngsters expressed it when their basic needs were addressed and school attendance was enabled. Older women's expressions of caring about were vague, while the younger people detected, act and elicited reactions from the elders. The adopted care approach informed care perceptions. Joint intergenerational activities are proposed to discover care currencies and contributions of generational members.

Matrix group structure and Markov invariants in the strand symmetric phylogenetic substitution model.

We consider the continuous-time presentation of the strand symmetric phylogenetic substitution model (in which rate parameters are unchanged under nucleotide permutations given by Watson-Crick base conjugation). Algebraic analysis of the model's underlying structure as a matrix group leads to a change of basis where the rate generator matrix is given by a two-part block decomposition. We apply representation theoretic techniques and, for any (fixed) number of phylogenetic taxa L and polynomial degree D of interest, provide the means to classify and enumerate the associated Markov invariants. In particular, in the quadratic and cubic cases we prove there are precisely [Formula: see text] and [Formula: see text] linearly independent Markov invariants, respectively. Additionally, we give the explicit polynomial forms of the Markov invariants for (i) the quadratic case with any number of taxa L, and (ii) the cubic case in the special case of a three-taxon phylogenetic tree. We close by showing our results are of practical interest since the quadratic Markov invariants provide independent estimates of phylogenetic distances based on (i) substitution rates within Watson-Crick conjugate pairs, and (ii) substitution rates across conjugate base pairs.

Dinucleotide circular codes and bijective transformations.

The presence of circular codes in mRNA coding sequences is postulated to be involved in informational mechanisms aimed at detecting and maintaining the normal reading frame during protein synthesis. Most of the recent research is focused on trinucleotide circular codes. However, also dinucleotide circular codes are important since dinucleotides are ubiquitous in genomes and associated to important biological functions. In this work we adopt the group theoretic approach used for trinucleotide codes in Fimmel et al. (2015) to study dinucleotide circular codes and highlight their symmetry properties. Moreover, we characterize such codes in terms of n-circularity and provide a graph representation that allows to visualize them geometrically. The results establish a theoretical framework for the study of the biological implications of dinucleotide circular codes in genomic sequences.

Algebraic double cut and join : A group-theoretic approach to the operator on multichromosomal genomes.

Establishing a distance between genomes is a significant problem in computational genomics, because its solution can be used to establish evolutionary relationships including phylogeny. The "double cut and join" (DCJ) model of chromosomal rearrangement proposed by Yancopoulos et al. (Bioinformatics 21:3340-3346, 2005) has received attention as it can model inversions, translocations, fusion and fission on a multichromosomal genome that may contain both linear and circular chromosomes. In this paper, we realize the DCJ operator as a group action on the space of multichromosomal genomes. We study this group action, deriving some properties of the group and finding group-theoretic analogues for the key results in the DCJ theory.

A geometric approach to robust medical image segmentation.

Robustness of deep learning segmentation models is crucial for their safe incorporation into clinical practice. However, these models can falter when faced with distributional changes. This challenge is evident in magnetic resonance imaging (MRI) scans due to the diverse acquisition protocols across various domains, leading to differences in image characteristics such as textural appearances. We posit that the restricted anatomical differences between subjects could be harnessed to refine the latent space into a set of shape components. The learned set then aims to encompass the relevant anatomical shape variation found within the patient population. We explore this by utilising multiple MRI sequences to learn texture invariant and shape equivariant features which are used to construct a shape dictionary using vector quantisation. We investigate shape equivariance to a number of different types of groups. We hypothesise and prove that the greater the group order, i.e., the denser the constraint, the better becomes the model robustness. We achieve shape equivariance either with a contrastive based approach or by imposing equivariant constraints on the convolutional kernels. The resulting shape equivariant dictionary is then sampled to compose the segmentation output. Our method achieves state-of-the-art performance for the task of single domain generalisation for prostate and cardiac MRI segmentation. Code is available at https://github.com/AinkaranSanthi/A_Geometric_Perspective_For_Robust_Segmentation.

SUBGROUPS: a computer tool at the Bilbao Crystallographic Server for the study of pseudo-symmetric or distorted structures.

SUBGROUPS is a free online program at the Bilbao Crystallographic Server (https://www.cryst.ehu.es/). It permits the exploration of all possible symmetries resulting from the distortion of a higher-symmetry parent structure, provided that the relation between the lattices of the distorted and parent structures is known. The program calculates all the subgroups of the parent space group which comply with this relation. The required minimal input is the space-group information of the parent structure and the relation of the unit cell of the distorted or pseudo-symmetric structure with that of the parent structure. Alternatively, the wavevector(s) observed in the diffraction data characterizing the distortion can be introduced. Additional conditions can be added, including filters related to space-group representations. The program provides very detailed information on all the subgroups, including group-subgroup hierarchy graphs. If a Crystallographic Information Framework (CIF) file of the parent high-symmetry structure is uploaded, the program generates CIF files of the parent structure described under each of the chosen lower symmetries. These CIF files may then be used as starting points for the refinement of the distorted structure under these possible symmetries. They can also be used for density functional theory calculations or for any other type of analysis. The power and efficiency of the program are illustrated with a few examples.

The zipper groups of the amyloid state of proteins.

Fibrous proteins in the amyloid state are found both associated with numerous diseases and in the normal functions of cells. Amyloid fibers contain a repetitive spine, commonly built from a pair of ß-sheets whose ß-strands run perpendicular to the fiber direction and whose side chains interdigitate, much like the teeth of a zipper. In fiber spines known as homosteric zippers, identical protein segments sharing identical packing environments make the two ß-sheets. In previous work based on atomic resolution crystal structures of homosteric zippers derived from a dozen proteins, the symmetries of homosteric zippers were categorized into eight classes. Here, it is shown through a formal derivation that each homosteric zipper class corresponds to a unique set of symmetry groups termed `zipper groups'. Furthermore, the eight previously identified classes do not account for all of the 15 possible zipper groups, which may be categorized into the complete set of ten classes. Because of their foundations in group theory, the 15 zipper groups provide a mathematically rigorous classification for homosteric zippers.

Symmetry in models of natural selection.

Symmetry arguments are frequently used-often implicitly-in mathematical modelling of natural selection. Symmetry simplifies the analysis of models and reduces the number of distinct population states to be considered. Here, I introduce a formal definition of symmetry in mathematical models of natural selection. This definition applies to a broad class of models that satisfy a minimal set of assumptions, using a framework developed in previous works. In this framework, population structure is represented by a set of sites at which alleles can live, and transitions occur via replacement of some alleles by copies of others. A symmetry is defined as a permutation of sites that preserves probabilities of replacement and mutation. The symmetries of a given selection process form a group, which acts on population states in a way that preserves the Markov chain representing selection. Applying classical results on group actions, I formally characterize the use of symmetry to reduce the states of this Markov chain, and obtain bounds on the number of states in the reduced chain.

Group theoretic particle swarm optimization for gray-level medical image enhancement.

As a principal category in the promising field of medical image processing, medical image enhancement has a powerful influence on the intermedia features and final results of the computer aided diagnosis (CAD) system by increasing the capacity to transfer the image information in the optimal form. The enhanced region of interest (ROI) would contribute to the early diagnosis and the survival rate of patients. Meanwhile, the enhancement schema can be treated as the optimization approach of image grayscale values, and metaheuristics are adopted popularly as the mainstream technologies for medical image enhancement. In this study, we propose an innovative metaheuristic algorithm named group theoretic particle swarm optimization (GT-PSO) to tackle the optimization problem of image enhancement. Based on the mathematical foundation of symmetric group theory, GT-PSO comprises particle encoding, solution landscape, neighborhood movement and swarm topology. The corresponding search paradigm takes place simultaneously under the guidance of hierarchical operations and random components, and it could optimize the hybrid fitness function of multiple measurements of medical images and improve the contrast of intensity distribution. The numerical results generated from the comparative experiments show that the proposed GT-PSO has outperformed most other methods on the real-world dataset. The implication also indicates that it would balance both global and local intensity transformations during the enhancement process.

Group theoretic particle swarm optimization for multi-level threshold lung cancer image segmentation.

Background: Image segmentation is an important step during the processing of medical images. For example, for the computer aid diagnostic systems for lung cancer image analysis, the segmented regions of tumors would help doctors in early diagnosis to determine timely and appropriate treatment possibilities and thereby improve the survival rate of the patients. However, general clinical routines of manual segmentation for large number of medical images are very difficult and time consuming, which is the challenge we aim to tackle using our proposed method. Methods: A novel image segmentation method with evolutionary learning technique named Group Theoretic Particle Swarm Optimization is proposed. It can tackle multi-level thresholding optimization problem during the segmentation process and rebuild the search paradigm according to the solid mathematical foundation of symmetric group from four designable aspects, which are particle encoding, solution landscape, neighborhood movement and swarm topology, respectively. The Kapur's entropy of multi-level thresholds is assessed as the objective function. Results: In contrast to those conventional metaheuristics methods for lung cancer image segmentation, this newly presented method generates the best performance result among them. Experimental results show that its Kapur's entropy has the value of 9.07, which is 16% higher than the worst case. Computational time is acceptable at the cost of 173.730 seconds, average level of evaluation metrics [Kappa, Precision, Recall, F1-measure, intersection over union (IoU) and receiver operating characteristic (ROC)] is over 90%, and search process of multi-level threshold combination would finally converge in the later phase of iterations after 700. The ablation study indicates that all components are significant to the contributions of our proposed method. Conclusions: Group Theoretic Particle Swarm Optimization for multi-level threshold segmentation is an efficient way to split a medical image into distinct regions and extract tumor tissues regions from the background. It maintains the balanced relationship between diversification and intensification during the search process and helps clinicians to make the diagnosis more accurately. Our proposed method processes potential medical value and clinical meanings.

Groups of Symmetries of the Two Classes of Synthetases in the Four-Dimensional Hypercubes of the Extended Code Type II.

Aminoacyl-tRNA synthetases (aaRSs) originated from an ancestral bidirectional gene (mirror symmetry), and through the evolution of the genetic code, the twenty aaRSs exhibit a symmetrical distribution in a 6-dimensional hypercube of the Standard Genetic Code. In this work, we assume a primeval RNY code and the Extended Genetic RNA code type II, which includes codons of the types YNY, YNR, and RNR. Each of the four subsets of codons can be represented in a 4-dimensional hypercube. Altogether, these 4 subcodes constitute the 6-dimensional representation of the SGC. We identify the aaRSs symmetry groups in each of these hypercubes. We show that each of the four hypercubes contains the following sets of symmetries for the two known Classes of synthetases: RNY: dihedral group of order 4; YNY: binary group; YNR: amplified octahedral group; and RNR: binary group. We demonstrate that for each hypercube, the group of symmetries in Class 1 is the same as the group of symmetries in Class 2. The biological implications of these findings are discussed.

Spin splitting in monoperiodic systems described by magnetic line groups.

In this paper we report the classification of all the 81 magnetic line group families into seven spin splitting prototypes, in analogy to the similar classification previously reported for the 1651 magnetic space groups, 528 magnetic layer groups, and 394 magnetic rod groups. According to this classification, electrically induced (Pekar-Rashba) spin splitting is possible in the antiferromagnetic structures described by magnetic line groups of type I (no anti-unitary operations) and III, both in the presence and in the absence of the space inversion operation. As a specific example, a group theoretical analysis of spin splitting in CoO (8, 8) nanotube is carried out and its predictions are confirmed byab initiodensity functional theory calculations.