A synergetic theory of quadrupedal gaits and gait transitions

We present a theoretical analysis of the patterns of interlimb co-ordination in the gaits of quadrupedal locomotion. Introducing as collective variables a set of relative phases that describe the co-ordination patterns, we classify gaits by their symmetry properties, which can be expressed as invariances under groups of transformations. We define dynamics of the collective variables, on which we impose symmetry restrictions. The stable observable gait patterns correspond to atractors of these dynamics. A non-trivial consequence of this theoretical viewpoint is that gait transitions can take the form of non-equilibrium phase transitions that are accompanied by loss of stability. We show how various types of such phase transitions involving hysteresis, slowing down and fluctuation enhancement can occur. Also the difference between smooth and abrupt transitions is given theoretical foundation. While existing experimental evidence is consistent with the theory developed here, we propose new experimental measures that can serve to test the present theoretical framework. Finally, the influence of underlying symmetries of the dynamics on the nature of the gait patterns and their stability is analyzed. For example, breaking of a front-hind symmetry can lead to a change from absolute to relative co-ordination in the sense of von Holst (1939, Ergebnisse der Physiologie 42, 228). Also, differential stability of straight and reverse gaits results from thus lowering the symmetry.     

From abstract topology to real thermodynamic brain activity

Recent approaches to brain phase spaces reinforce the foremost role of symmetries and energy requirements in the assessment of nervous activity. Changes in thermodynamic parameters and dimensions occur in the brain during symmetry breakings and transitions from one functional state to another. Based on topological results and string-like trajectories into nervous energy landscapes, we provide a novel method for the evaluation of energetic features and constraints in different brain functional activities. We show how abstract approaches, namely the Borsuk–Ulam theorem and its variants, may display real, energetic physical counterparts. When topology meets the physics of the brain, we arrive at a general model of neuronal activity, in terms of multidimensional manifolds and computational geometry, that has the potential to be operationalized.     

Inverse Symmetry in Complete Genomes and Whole-Genome Inverse Duplication

The cause of symmetry is usually subtle, and its study often leads to a deeper understanding of the bearer of the symmetry. To gain insight into the dynamics driving the growth and evolution of genomes, we conducted a comprehensive study of textual symmetries in 786 complete chromosomes. We focused on symmetry based on our belief that, in spite of their extreme diversity, genomes must share common dynamical principles and mechanisms that drive their growth and evolution, and that the most robust footprints of such dynamics are symmetry related. We found that while complement and reverse symmetries are essentially absent in genomic sequences, inverse–complement plus reverse–symmetry is prevalent in complex patterns in most chromosomes, a vast majority of which have near maximum global inverse symmetry. We also discovered relations that can quantitatively account for the long observed but unexplained phenomenon of -mer skews in genomes. Our results suggest segmental and whole-genome inverse duplications are important mechanisms in genome growth and evolution, probably because they are efficient means by which the genome can exploit its double-stranded structure to enrich its code-inventory.     

Morphology and the gradient of a symmetric potential predict gait transitions of dogs

Gaits and gait transitions play a central role in the movement of animals. Symmetry is thought to govern the structure of the nervous system, and constrain the limb motions of quadrupeds. We quantify the symmetry of dog gaits with respect to combinations of bilateral, fore–aft, and spatio-temporal symmetry groups. We tested the ability of symmetries to model motion capture data of dogs walking, trotting and transitioning between those gaits. Fully symmetric models performed comparably to asymmetric with only a \(22\%\) increase in the residual sum of squares and only one-quarter of the parameters. This required adding a spatio-temporal shift representing a lag between fore and hind limbs. Without this shift, the symmetric model residual sum of squares was \(1700\%\) larger. This shift is related to (linear regression, \(n=5\), \(p=0.0328\)) dog morphology. That this symmetry is respected throughout the gaits and transitions indicates that it generalizes outside a single gait. We propose that relative phasing of limb motions can be described by an interaction potential with a symmetric structure. This approach can be extended to the study of interaction of neurodynamic and kinematic variables, providing a system-level model that couples neuronal central pattern generator networks and mechanical models.     

Measuring complexity to infer changes in the dynamics of ecological systems under stress

Despite advances in our mechanistic understanding of ecological processes, the inherent complexity of real-world ecosystems still limits our ability in predicting ecological dynamics especially in the face of on-going environmental stress. Developing a model is frequently challenged by structure uncertainty, unknown parameters, and limited data for exploring out-of-sample predictions. One way to address this challenge is to look for patterns in the data themselves in order to infer the underlying processes of an ecological system rather than to build system-specific models. For example, it has been recently suggested that statistical changes in ecological dynamics can be used to infer changes in the stability of ecosystems as they approach tipping points. For computer scientists such inference is similar to the notion of a Turing machine: a computational device that could execute a program (the process) to produce the observed data (the pattern). Here, we make use of such basic computational ideas introduced by Alan Turing to recognize changing patterns in ecological dynamics in ecosystems under stress. To do this, we use the concept of Kolmogorov algorithmic complexity that is a measure of randomness. In particular, we estimate an approximation to Kolmogorov complexity based on the Block Decomposition Method (BDM). We apply BDM to identify changes in complexity in simulated time-series and spatial datasets from ecosystems that experience different types of ecological transitions. We find that in all cases, K_BDM complexity decreased before all ecological transitions both in time-series and spatial datasets. These trends indicate that loss of stability in the ecological models we explored is characterized by loss of complexity and the emergence of a regular and computable underlying structure. Our results suggest that Kolmogorov complexity may serve as tool for revealing changes in the dynamics of ecosystems close to ecological transitions.     

Titin and the sarcomere symmetry paradox

Titin is thought to play a major role in myofibril assembly, elasticity and stability. A single molecule spans half the sarcomere and makes interactions with both a thick filament and the Z-line. In the unit cell structure of each half sarcomere there is one thick filament with 3-fold symmetry and two thin filaments with approximately 2-fold symmetry. The minimum number of titin molecules that could satisfy both these symmetries is 12. We determined the actual number of titin molecules in a unit cell from scanning transmission electron microscopy mass measurements of end-filaments. One of these emerges from each tip of the thick filament and is thought to be the in-register aggregate of the titin molecules associated with the filament. The mass per unit length of the end-filament (17.1 kDa/nm) is consistent with six titin molecules not 12. Thus the number of titin molecules present is insufficient to satisfy both symmetries. We suggest a novel solution to this paradox in which four of the six titin molecules interact with the two thin filaments in the unit cell, while the remaining two interact with the two thin filaments that enter the unit cell from the adjacent sarcomere. This arrangement would augment mechanical stability in the sarcomere.     

TOPK and PTEN participate in CHFR mediated mitotic checkpoint

Mitotic progression is regulated by co-ordinated action of several proteins and is crucial for the maintenance of genomic stability. CHFR (Check point protein with FHA and RING domains) is an E3 ubiquitin ligase and a checkpoint protein that regulates entry into mitosis. But the molecular players involved in CHFR mediated mitotic checkpoint are not completely understood. In this study, we identified TOPK/PBK, a serine/threonine kinase and PTEN, a lipid phosphatase to play an important role in CHFR mediated mitotic transitions. We demonstrated that CHFR ubiquitinates and regulates TOPK levels, which is essential for its checkpoint function. Moreover, TOPK phosphorylates and inactivates PTEN, which in turn activates Akt that leads to proper G2/M progression. Collectively, our results reveal TOPK and PTEN as new players in CHFR mediated mitotic checkpoint.     

Effect of Lipid Head Groups on Double-Layered Two-Dimensional Crystals Formed by Aquaporin-0

Aquaporin-0 (AQP0) is a lens-specific water channel that also forms membrane junctions. Reconstitution of AQP0 with dimyristoyl phosphatidylcholine (DMPC) and E. coli polar lipids (EPL) yielded well-ordered, double-layered two-dimensional (2D) crystals that allowed electron crystallographic structure determination of the AQP0-mediated membrane junction. The interacting tetramers in the two crystalline layers are exactly in register, resulting in crystals with p422 symmetry. The high-resolution density maps also allowed modeling of the annular lipids surrounding the tetramers. Comparison of the DMPC and EPL bilayers suggested that the lipid head groups do not play an important role in the interaction of annular lipids with AQP0. We now reconstituted AQP0 with the anionic lipid dimyristoyl phosphatidylglycerol (DMPG), which yielded a mixture of 2D crystals with different symmetries. The different crystal symmetries result from shifts between the two crystalline layers, suggesting that the negatively charged PG head group destabilizes the interaction between the extracellular AQP0 surfaces. Reconstitution of AQP0 with dimyristoyl phosphatidylserine (DMPS), another anionic lipid, yielded crystals that had the usual p422 symmetry, but the crystals showed a pH-dependent tendency to stack through their cytoplasmic surfaces. Finally, AQP0 failed to reconstitute into membranes that were composed of more than 40% dimyristoyl phosphatidic acid (DMPA). Hence, although DMPG, DMPS, and DMPA are all negatively charged lipids, they have very different effects on AQP0 2D crystals, illustrating the importance of the specific lipid head group chemistry beyond its mere charge.     

Hydrophobin Film Structure for HFBI and HFBII and Mechanism for Accelerated Film Formation

Hydrophobins represent an important group of proteins from both a biological and nanotechnological standpoint. They are the means through which filamentous fungi affect their environment to promote growth, and their properties at interfaces have resulted in numerous applications. In our study we have combined protein docking, molecular dynamics simulation, and electron cryo-microscopy to gain atomistic level insight into the surface structure of films composed of two class II hydrophobins: HFBI and HFBII produced by Trichoderma reesei. Together our results suggest a unit cell composed of six proteins; however, our computational results suggest P6 symmetry, while our experimental results show P3 symmetry with a unit cell size of 56 Å. Our computational results indicate the possibility of an alternate ordering with a three protein unit cell with P3 symmetry and a smaller unit cell size, and we have used a Monte Carlo simulation of a spin model representing the hydrophobin film to show how this alternate metastable structure may play a role in increasing the rate of surface coverage by hydrophobin films, possibly indicating a mechanism of more general significance to both biology and nanotechnology.     

Compressibility of protein transitions

We review the results of compressibility studies on proteins and low molecular weight compounds that model the hydration properties of these biopolymers. In particular, we present an analysis of compressibility changes accompanying conformational transitions of globular proteins. This analysis, in conjunction with experimental compressibility data on protein transitions, were used to define the changes in the hydration properties and intrinsic packing associated with native-to-molten globule, native-to-partially unfolded, and native-to-fully unfolded transitions of globular proteins. In addition, we discuss the molecular origins of predominantly positive changes in compressibility observed for pressure-induced denaturation transitions of globular proteins. Throughout this review, we emphasize the importance of compressibility data for characterizing protein transitions, while also describing how such data can be interpreted to gain insight into role that hydration and intrinsic packing play in modulating the stability of and recognition between proteins and other biologically important compounds.     

Determination of restriction sites and the nucleotide sequence surrounding the relaxation site of ColE1

The relaxation site of ColE1 has been located within the restriction fragment HpaII L, which is 148 base-pairs in length. Restriction mapping data indicate that the relaxation nick (the presumptive origin of transfer) of ColE1 is located at a distance of 250 to 300 nucleotides away from the replication origin, downstream in the direction of replication. This result is consistent with the observation made by Inselburg (1977), that the relaxation phenomenon probably does not play a direct role in vegetative replication of ColE1. The sequence of 185 nucleotides surrounding the relaxation site has been determined and this contains a translational symmetry and several 2-fold rotational symmetries. These symmetric elements may be recognition sites for proteins involved in the conjugal transfer of ColE1. The sequence further demonstrates that the relaxation site, unlike the cis A nicking site of φX174, is located in an intercistronic region. The site of the relaxation break has a 2-fold rotational symmetry.     

On the symmetries of multi-palindromic DNA sequences

There are several instances of multiple, overlapping palindromes in DNA sequences which have recently been reported. To ascertain the likelihood of specific biological function for these interdigitating symmetries it is necessary to calculate their probabilities of occurrence. While the probability of occurrence of a single palindrome in a random sequence is calculated in a straightforward fashion, the occurrence of overlapping palindromes is not so simply analyzed. In this paper a general method for handling multiple symmetries is presented. Several theorems concerning the constraints which overlapping symmetries place on each other are presented. A general result is that the probability of occurrence of a symmetry can be significantly enhanced by already existing symmetries. As examples of the theorems and methods developed, the lac operator, the lac CAP-binding site and a region of the λ left operator are examined. The occurrence of several overlapping symmetries appears to be fairly common in such sequences.     

Trends in flower symmetry evolution revealed through phylogenetic and developmental genetic advances

A striking aspect of flowering plant (angiosperm) diversity is variation in flower symmetry. From an ancestral form of radial symmetry (polysymmetry, actinomorphy), multiple evolutionary transitions have contributed to instances of non-radial forms, including bilateral symmetry (monosymmetry, zygomorphy) and asymmetry. Advances in flowering plant molecular phylogenetic research and studies of character evolution as well as detailed flower developmental genetic studies in a few model species (e.g. Antirrhinum majus, snapdragon) have provided a foundation for deep insights into flower symmetry evolution. From phylogenetic studies, we have a better understanding of where during flowering plant diversification transitions from radial to bilateral flower symmetry (and back to radial symmetry) have occurred. From developmental studies, we know that a genetic programme largely dependent on the functional action of the CYCLOIDEA gene is necessary for differentiation along the snapdragon dorsoventral flower axis. Bringing these two lines of inquiry together has provided surprising insights into both the parallel recruitment of a CYC-dependent developmental programme during independent transitions to bilateral flower symmetry, and the modifications to this programme in transitions back to radial flower symmetry, during flowering plant evolution.     

Study of the Expression Transition of Cardiac Myosin Using Polarization-Dependent SHG Microscopy

Detection of the transition between the two myosin isoforms α- and β-myosin in living cardiomyocytes is essential for understanding cardiac physiology and pathology. In this study, the differences in symmetry of polarization spectra obtained from α- and β-myosin in various mammalian ventricles and propylthiouracil-treated rats are explored through polarization-dependent second harmonic generation microscopy. Here, we report for the, to our knowledge, first time that α- and β-myosin, as protein crystals, possess different symmetries: the former has C6 symmetry, and the latter has C3v. A single-sarcomere line scan further demonstrated that the differences in polarization-spectrum symmetry between α- and β-myosin came from their head regions: the head and neck domains of α- and β-myosin account for the differences in symmetry. In addition, the dynamic transition of the polarization spectrum from C6 to C3v line profile was observed in a cell culture in which norepinephrine induced an α- to β-myosin transition.     

对称蛋白质序列与结构关系研究

进化的观点认为,蛋白质结构的对称性是基因复制和融合的结果,但是由于在长期进化过程中的氨基酸突变,绝大多数现有的蛋白质序列都失去了这种直观的重复性特征。该文简要地回顾了国际上发展的寻找蛋白质序列中重复片段的方法,重点介绍了作者自己提出的分析蛋白质序列和结构对称性的方法以及在蛋白质对称结构形成机理方面的初步工作,并系统分析了各类对称折叠子的序列与结构关系,发现它们的序列都具有隐含的与结构相同的对称性,或者说序列的对称性决定结构的对称性。     

Sequences of symmetry-breaking in phyllotactic transitions

This paper studies the transition of phyllotactic patterns by a group-theoretic approach. Typical phyllotactic patterns are represented here as dotted patterns on a cylinder, where the cylinder is regarded as the stem of a plant and the dots are points where leaves branch from the stem. We can then classify the symmetries of the alternate and opposite phyllotaxis into four types of groups, and clarify sequences of symmetry-breaking among these groups. The sequences turn out to correspond to transition paths of phyllotactic patterns found in the wild. This result shows the usefulness of classification of phyllotactic patterns based on their group symmetries. Moreover, the breaking of reflection symmetry is found to be an important rule for real phyllotactic transitions.     

Tyr74 is essential for the formation, stability and function of Plasmodium falciparum triosephosphate isomerase dimer

Plasmodium falciparum triosephosphate isomerase (PfTIM) is known to be functional only as a homodimer. Although many studies have shown that the interface Cys13 plays a major role in the stability of the dimer, a few reports have demonstrated that structurally conserved Tyr74 may be essential for the stability of PfTIM dimer. To understand the role of Tyr74, we have performed molecular dynamics (MD) simulations of monomeric and dimeric PfTIM mutated to glycine and cysteine at position 74. Simulations of the monomer revealed that mutant Tyr74Gly does not produce changes in folding and stability of the monomer. Interestingly, comparison of the flexibility of Tyr74 in the monomer and dimer revealed that this residue possesses an intrinsic restricted mobility, indicating that Tyr74 is an anchor residue required for homodimerization. Tyr74 also appears to play an important role in binding by facilitating the disorder-to-order transitions of loops 1 and 3, which allows Cys13 to form favorable interactions with loop 3 and Lys12 to be locked in a favorable position for catalysis. High-temperature MD simulations of the wild-type and Tyr74Gly PfTIM dimers showed that the aromatic moiety of Tyr74 is necessary to preserve the geometry and native contacts between loops 1 and 3 at the interface of the dimer. Disulfide cross-linking between mutant Tyr74Cys and Cys13 further revealed that Tyr74 stabilizes the geometry of loop 1 (which contains the catalytic residue Lys12) and the interactions between loops 1 and 3 via aromatic-aromatic interactions with residues Phe69, Tyr101, and Phe102. Principal component analysis showed that Tyr74 is also necessary to preserve the collective motions in the dimer that contribute to the catalytic efficiency of PfTIM dimer. We conclude that Tyr74 not only plays a role in the stability of the dimer, but also participates in the dimerization process and collective motions via coupled disorder-to-order transitions of intrinsically disordered regions, necessary for efficiency in the catalytic function of PfTIM.     

Molecular dynamics shows complex interplay and long-range effects of post-translational modifications in yeast protein interactions

Post-translational modifications (PTMs) play a vital, yet often overlooked role in the living cells through modulation of protein properties, such as localization and affinity towards their interactors, thereby enabling quick adaptation to changing environmental conditions. We have previously benchmarked a computational framework for the prediction of PTMs’ effects on the stability of protein-protein interactions, which has molecular dynamics simulations followed by free energy calculations at its core. In the present work, we apply this framework to publicly available data on Saccharomyces cerevisiae protein structures and PTM sites, identified in both normal and stress conditions. We predict proteome-wide effects of acetylations and phosphorylations on protein-protein interactions and find that acetylations more frequently have locally stabilizing roles in protein interactions, while the opposite is true for phosphorylations. However, the overall impact of PTMs on protein-protein interactions is more complex than a simple sum of local changes caused by the introduction of PTMs and adds to our understanding of PTM cross-talk. We further use the obtained data to calculate the conformational changes brought about by PTMs. Finally, conservation of the analyzed PTM residues in orthologues shows that some predictions for yeast proteins will be mirrored to other organisms, including human. This work, therefore, contributes to our overall understanding of the modulation of the cellular protein interaction networks in yeast and beyond.     

Generative models versus underlying symmetries to explain biological pattern

Mathematical models play an increasingly important role in the interpretation of biological experiments. Studies often present a model that generates the observations, connecting hypothesized process to an observed pattern. Such generative models confirm the plausibility of an explanation and make testable hypotheses for further experiments. However, studies rarely consider the broad family of alternative models that match the same observed pattern. The symmetries that define the broad class of matching models are in fact the only aspects of information truly revealed by observed pattern. Commonly observed patterns derive from simple underlying symmetries. This article illustrates the problem by showing the symmetry associated with the observed rate of increase in fitness in a constant environment. That underlying symmetry reveals how each particular generative model defines a single example within the broad class of matching models. Further progress on the relation between pattern and process requires deeper consideration of the underlying symmetries.