Part of the problem with defining the shape of a protein is that we start with nothing but a point set, and the "shape" of a set of discontinuous points is poorly defined. The problem is, what do we mean by shape? As you saw above, the shape of a molecule depends on what is being used to measure it. To handle this ambiguity, we will introduce a method of shape calculation based on a parameter, α, which will determine the radius of a spherical probe that will define the surface. The method defines a class of shapes, called
α-shapes
for any given point set. It allows fast, accurate, and efficient calculations of volume and surface area.
α-shapes are a generalization of the
convex hull . Consider a
point set S. Define an α-ball as a sphere of radius α. An α-ball is empty if it contains no points in S. For any α between zero and infinity, the
α-hull of S is the complement of the union of all empty α-balls.
For α of infinity, the α-shape is the convex hull of S.
For α smaller than the 1/2 smallest distance between two points in S, the α-shape is S itself.
For any α in between, one can think of the α-hull as the largest polygon (polyhedron) or set thereof whose vertices are in the point set and whose edges are of length less than 2α. The presence of an edge indicates that a probe of radius α cannot pass between the edge endpoints.
Computing the alpha-shape: delaunay triangulation
A
triangulation of a three-dimensional point set S is any decomposition of S into non-intersecting tetrahedra (triangles for two-dimensional point sets).
The
Delaunay triangulation of S is the unique triangulation of S satisfying the additional requirement that no sphere circumscribing a tetrahedron in the triangulation contains any point in S. Although it is incidental to α-shapes, it is worth noting that the Delaunay triangulation maximizes the average of the smallest angle over all triangles. In other words, it favors relatively even-sided triangles over sharp and stretched ones.
The Delaunay triangulation of a point set is usually calculated by an incremental flip algorithm as follows:
The points of S are sorted on one coordinate (x, y, or z). This step is not strictly necessary but makes the algorithm run faster than if the points were in arbitrary order.
Each point is added in sorted order. Upon adding a point:
The point is connected to previously added points that are "visible" to it, that is, to points to which it can be connected by a line segment without passing through a face of a tetrahedron.
Any new tetrhedra formed are checked and flipped if necessary.
Any tetrahedra adjacent to flipped tetrahedra are checked and flipped. This continues until further flipping is unnecessary, which is guaranteed to occur
This algorithm runs in worst case
O(n^2) time, but expected
O(n^(3/2)) time. Without the sort in the first step, the expected case would be
O(n log n) . A full description and analysis of Delaunay triangulation algorithms is given in
[1] , chapter 9.
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
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Source:
OpenStax, Geometric methods in structural computational biology. OpenStax CNX. Jun 11, 2007 Download for free at http://cnx.org/content/col10344/1.6
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