Stability of Zeno Equilibria in Lagrangian Hybrid Systems

This paper presents both necessary and sufficient conditions for the stability of Zeno equilibria in Lagrangian hybrid systems, i.e., hybrid systems modeling mechanical systems undergoing impacts. These conditions for stability are motivated by the sufficient conditions for Zeno behavior in Lagrangian hybrid systems obtained in [11]—we show that the same conditions that imply the existence of Zeno behavior near Zeno equilibria imply the stability of the Zeno equilibria. This paper, therefore, not only presents conditions for the stability of Zeno equilibria, but directly relates the stability of Zeno equilibria to the existence of Zeno behavior

Dynamics and Stability of Low-Reynolds-Number Swimming Near a Wall

The locomotion of microorganisms and tiny artificial swimmers is governed by low-Reynolds-number hydrodynamics, where viscous effects dominate and inertial effects are negligible. While the theory of low-Reynolds-number locomotion is well studied for unbounded fluid domains, the presence of a boundary has a significant influence on the swimmer’s trajectories and poses problems of dynamic stability of its motion. In this paper we consider a simple theoretical model of a microswimmer near a wall, study its dynamics, and analyze the stability of its motion. We highlight the underlying geometric structure of the dynamics, and establish a relation between the reversing symmetry of the system and existence and stability of periodic and steady solutions of motion near the wall. The results are demonstrated by numerical simulations and validated by motion experiments with macroscale robotic swimmer prototypes

The action of selection on codon bias in the human genome is related to frequency, complexity, and chronology of amino acids

BACKGROUND: The question of whether synonymous codon choice is affected by cellular tRNA abundance has been positively answered in many organisms. In some recent works, concerning the human genome, this relation has been studied, but no conclusive answers have been found. In the human genome, the variation in base composition and the absence of cellular tRNA count data makes the study of the question more complicated. In this work we study the relation between codon choice and tRNA abundance in the human genome by correcting relative codon usage for background base composition and using a measure based on tRNA-gene copy numbers as a rough estimate of tRNA abundance. RESULTS: We term major codons to be those codons with a relatively large tRNA-gene copy number for their corresponding amino acid. We use two measures of expression: breadth of expression (the number of tissues in which a gene was expressed) and maximum expression level among tissues (the highest value of expression of a gene among tissues). We show that for half the amino acids in the study (8 of 16) the relative major codon usage rises with breadth of expression. We show that these amino acids are significantly more frequent, are smaller and simpler, and are more ancient than the rest of the amino acids. Similar, although weaker, results were obtained for maximum expression level. CONCLUSION: There is evidence that codon bias in the human genome is related to selection, although the selection forces acting on codon bias may not be straightforward and may be different for different amino acids. We suggest that, in the first group of amino acids, selection acts to enhance translation efficiency in highly expressed genes by preferring major codons, and acts to reduce translation rate in lowly expressed genes by preferring non-major ones. In the second group of amino acids other selection forces, such as reducing misincorporation rate of expensive amino acids, in terms of their size/complexity, may be in action. The fact that codon usage is more strongly related to breadth of expression than to maximum expression level supports the notion, presented in a recent study, that codon choice may be related to the tRNA abundance in the tissue in which a gene is expressed

Modulating Vesicle Priming Reveals that Vesicle Immobilization Is Necessary but not Sufficient for Fusion-Competence

In neurons and neuroendocrine cells, docked vesicles need to undergo priming to become fusion competent. Priming is a multi-step process that was shown to be associated with vesicle immobilization. However, it is not known whether vesicle immobilization is sufficient to acquire complete fusion competence. To extend our understanding of the physical manifestation of vesicle priming, we took advantage of tomosyn, a SNARE-related protein that specifically inhibits vesicle priming, and measured its effect on vesicle dynamics in live chromaffin cells using total internal reflection fluorescence microscopy. We show here that while in control cells vesicles undergo immobilization before fusion, vesicle immobilization is attenuated in tomosyn overexpressing cells. This in turn increases the turnover rate of vesicles near the membrane and attenuates the fusion of newcomer vesicles. Moreover, the release probability of immobile vesicles in tomosyn cells is significantly reduced, suggesting that immobilization is an early and necessary step in priming but is insufficient, as further molecular processes are needed to acquire complete fusion competence. Using tomosyn as a molecular tool we provide a mechanistic link between functional docking and priming and suggest that functional docking is the first step in vesicle priming, followed by molecular modifications that do not translate into changes in vesicle mobility

Formal and practical completion of Lagrangian hybrid systems

This paper presents a method for completing Lagrangian hybrid systems models in a formal manner. That is, given a Lagrangian hybrid system, i.e., a hybrid system that models a mechanical system undergoing impacts, we present a systematic method in which to extend executions of this system past Zeno points by adding an additional domain to the hybrid model. Moreover, by utilizing results that provide sufficient conditions for Zeno behavior and for stability of Zeno equilibria in Lagrangian hybrid systems, we are able to give explicit bounds on the error incurred through the practical simulation of these completed hybrid system models. These ideas are illustrated on a series of examples, and are shown to be consistent with observed reality

Stability and Completion of Zeno Equilibria in Lagrangian Hybrid Systems

This paper studies Lagrangian hybrid systems, which are a special class of hybrid systems modeling mechanical systems with unilateral constraints that are undergoing impacts. This class of systems naturally display Zeno behavior-an infinite number of discrete transitions that occur in finite time, leading to the convergence of solutions to limit sets called Zeno equilibria. This paper derives simple conditions for stability of Zeno equilibria. Utilizing these results and the constructive techniques used to prove them, the paper introduces the notion of a completed hybrid system which is an extended hybrid system model allowing for the extension of solutions beyond Zeno points. A procedure for practical simulation of completed hybrid systems is outlined, and conditions guaranteeing upper bounds on the incurred numerical error are derived. Finally, we discuss an application of these results to the stability of unilaterally constrained motion of mechanical systems under perturbations that violate the constraint