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Polar and Cylindrical Coordinates01:22

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The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. So, when describing rotation, the polar coordinate system is generally used.
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Equations of Motion: Rectangular Coordinates and Cylindrical Coordinates01:21

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Understanding the motion of particles is a fundamental aspect of classical mechanics, and the choice of the coordinate system plays a pivotal role in unraveling the complexities of their dynamics.
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Deformation in a Circular Shaft01:10

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One of the distinctive characteristics of circular shafts is their ability to maintain their cross-sectional integrity under torsion. In other words, each cross-section continues to exist as a flat, unaltered entity, simply rotating like a solid, rigid slab. To understand the distribution of shearing stress within such a shaft, consider a cylindrical section inside this circular shaft. This section has a length of L and a radius of R, with one end fixed. The radius of the cylindrical section is...
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Centroid for the Paraboloid of Revolution01:16

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The paraboloid of revolution is an axially symmetric surface generated by rotating a parabola around its axis. This shape has several applications in mechanical engineering due to its advantageous structural properties, such as strength against stress concentration points and rotational symmetry.
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Curvilinear Motion: Polar Coordinates01:27

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In polar coordinates, the motion of a particle follows a curvilinear path. The radial coordinate symbolized as 'r,' extends outward from a fixed origin to the particle, while the angular coordinate, 'θ,' measured in radians, represents the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the particle.
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Gauss's Law: Cylindrical Symmetry01:20

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A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
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Updated: Sep 14, 2025

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Accurate helical parameter estimation based on cylindrical unrolling.

Mingtao Huang1, Jinying Ma2, Xiaoyu Fu2

  • 1Key Laboratory for Protein Sciences of Ministry of Education, School of Life Sciences, Tsinghua University, Beijing 100084, China; Tsinghua-Peking Joint Center for Life Sciences, Beijing 100084, China; Beijing Frontier Research Center for Biological Structure, Beijing 100084, China; State Key Laboratory of Membrane Biology, School of Life Sciences, Tsinghua University, Beijing 100084, China; School of Life Sciences, Tsinghua University, Beijing 100084, China; Department of Electronic Engineering, Tsinghua University, Beijing 100084, China; Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China.

Structure (London, England : 1993)
|July 18, 2025
PubMed
Summary
This summary is machine-generated.

A new method, HELIS, accurately estimates helical parameters for macromolecular assemblies. This tool simplifies structure determination for flexible and heterogeneous helical structures visualized by cryo-electron tomography.

Keywords:
cryo-electron tomographyhelical parameterhelical structureparameter estimation

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Area of Science:

  • Structural biology
  • Biophysics
  • Biochemistry

Background:

  • Helical structures are essential for cellular functions like scaffolding and signaling.
  • Accurate estimation of helical parameters is critical for determining the structure of these assemblies.
  • Existing layer-line-based methods face challenges with flexible and heterogeneous helical structures.

Purpose of the Study:

  • To introduce a novel method, HELIS (helix is simple), for precise helical parameter estimation.
  • To provide a software package implementing the HELIS method for practical application.
  • To offer a comprehensive tool for analyzing helical assemblies visualized by cryo-electron tomography.

Main Methods:

  • HELIS employs a cylindrical unrolling approach, treating helical structures as rolled 2D crystals.
  • It derives helical parameter estimation from simplified 2D reciprocal-lattice measurements.
  • Auxiliary algorithms are included for tracing curved filaments, determining polarity, and in situ helical reconstruction.

Main Results:

  • HELIS demonstrates high accuracy in estimating helical parameters.
  • The method is applicable to diverse helical assemblies, including those with flexibility and heterogeneity.
  • The HELIS software package facilitates comprehensive helical structure determination.

Conclusions:

  • HELIS offers a robust and accurate solution for helical parameter estimation.
  • This method overcomes limitations of traditional approaches, particularly for challenging biological samples.
  • HELIS is a valuable tool for advancing the structural analysis of macromolecular assemblies.