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Using Component Trees to Explore Biological Structures

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Computational Methods for Three-Dimensional Microscopy Reconstruction

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

An understanding of the three-dimensional structure of a macromolecular complex is essential to fully understand its function. This chapter introduces the reader to the concept of a component tree, which is a compact representation of the structural properties of a multidimensional image (such as a molecular density map of a biological specimen), and then presents ongoing research on the use of such component trees in interactive tools for exploring biological structures. Component trees capture essential structural information about a biological specimen, irrespective of the process that was used to obtain an image of the specimen and the resolution of that image. We present various scenarios in which component trees can help in the exploration of the structure of a macromolecular complex. In addition, we discuss ideas for a docking methodology that uses component trees.

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Notes

  1. 1.

    In this chapter we distinguish between digital images and digital pictures. The difference between the two is that a digital picture has an adjacency relation but a digital image does not: A digital picture can be regarded as the result of equipping a digital image (which we may refer to as its underlying image) with an adjacency relation on the picture elements. Digital pictures which have the same underlying image but have different adjacency relations are considered to be different digital pictures. For example, in the case when the picture elements are voxels, we could have chosen the adjacency relation to be the face or edge adjacency that exists between two voxels if they share either exactly one face or exactly one edge [7]. The component tree of a digital picture will depend on its adjacency relation as well as its underlying image.

  2. 2.

    A rooted tree T is a pair (N, E), where N is a finite set of nodes and E is a set of edges. Each edge is an ordered pair of distinct nodes that are, respectively, called the parent node and the child node of the edge, and the nodes and edges satisfy the following conditions: (1) every member of N, except one element called the root, is a child node of just one edge; (2) the reflexive transitive closure of E is a partial order on N. If x and y are nodes such that x = y or x precedes y in the partial order, then x is called an ancestor of y and y is called adescendant of x. In particular, every node in N is a descendant of the root. We say x is a proper ancestor (respectively, proper descendant) of y if x is an ancestor (respectively, a descendant) of y and x ≠ y.

  3. 3.

    Component trees are very similar to the foreground history trees of [22] and the join trees of [3]. They are also related to contour trees [3]; the relationship between contour and component trees is discussed in Sect. 2.7.1 of [8].

  4. 4.

    A node of a rooted tree is called a leaf if it has no children.

  5. 5.

    In Fig. 9.2c, d, each node that has two or more children (such as the root of the subtree that is highlighted in panel d) is represented by a horizontal segment, and an edge from a node to one of its children is represented by a vertical segment whose length is proportional to the difference between the levels (see later for definition) of those two nodes.

  6. 6.

    In this chapter “just if” is used with its precise mathematical meaning—for any two statements P and Q, the statement “P just if Q” means “P is true if, and only if, Q is true.”

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Acknowledgements

The work presented here is currently supported by the National Science Foundation (award number DMS-1114901). We are grateful to José-María Carazo and Joachim Frank for their advice on this chapter based on careful reading of the originally submitted material.

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Authors and Affiliations

  1. Computer Science Ph.D. Program, Graduate Center, City University of New York, 365 Fifth Avenue, New York, NY, 10016, USA

    Lucas M. Oliveira & Gabor T. Herman

  2. Computer Science Department, Queens College, City University of New York, 65-30 Kissena Boulevard, Flushing, NY, 11367, USA

    T. Yung Kong

Authors
  1. Lucas M. Oliveira
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  2. T. Yung Kong
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  3. Gabor T. Herman
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Correspondence to Lucas M. Oliveira .

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Editors and Affiliations

  1. Department of Computer Science, The Graduate Center City University of New York, New York, New York, USA

    Gabor T. Herman

  2. Department of Biology, Biochemistry and Molecular Biophysics, Columbia University, New York, New York, USA

    Joachim Frank

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Oliveira, L.M., Kong, T.Y., Herman, G.T. (2014). Using Component Trees to Explore Biological Structures. In: Herman, G., Frank, J. (eds) Computational Methods for Three-Dimensional Microscopy Reconstruction. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-9521-5_9

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