Jan. 1, 2003
Publications in Computational Chemistry & Drug Design by :
"The relationship between protein sequence and three-dimensional structure is one of the primary unsolved problems in biology today." - Charles Brooks, Carnegie Mellon University
P.W. Finn and L. Kavraki
Algorithmica, 25 (1999), 347-371.
Web-link : http://www.cs.rice.edu/CS/Robotics/publications.html
The rational approach to pharmaceutical drug design begins with an investigation of the relationship between chemical structure and biological activity. Information gained from this analysis is used to aid the design of new, or improved, drugs. Primary considerations during this investigation are the geometric and chemical characteristics of the molecules. Computational chemists who are involved in rational drug design routinely use an array of programs to compute, among other things, molecular surfaces and molecular volume, models of receptor sites, dockings of ligands inside protein cavities, and geometric invariants among different molecules that exhibit similar activity. There is a pressing need for efficient and accurate solutions to the above problems. Often, limiting assumptions need to be made, in order to make the calculations tractable. Also, the amount of data processed when searching for a potential drug is currently very large and is only expected to grow larger in the future. This paper describes some areas of computer-aided drug design that are important to computational chemists but are also rich in algorithmic problems. It surveys recent work in these areas both from the computational chemistry and the computer science literature.
In J.-P. Laumond and M. Overmars, editors,
Algorithms for Robotic Motion and Manipulation,
pages 435--448. A. K. Peters, 1997.
(Proc. of WAFR '96)
This paper surveys several problems and approaches in the area of computer-aided pharmaceutical drug design and draws analogies with problems from robotics and computational geometry.
Robert H. Lewis and Stephen Bridgett
Mathematics and Computers in Simulation
Volume 61, Issue 2, 1 January 2003, Pages 101-114
The Apollonius Circle Problem dates to Greek antiquity, circa 250 . Given three circles in the plane, find or construct a circle tangent to all three. This was generalized by replacing some circles with straight lines. Viéte [Canon mathematicus seu Ad triangula cum adpendicibus, Lutetiae: Apud Ioannem Mettayer, Mathematicis typographum regium, sub signo D. Ioannis, regione Collegij Laodicensis, p. 1579] solved the problem using circle inversions before 1580. Two generations later, Descartes considered a special case in which all four circles are mutually tangent to each other (i.e. pairwise). In this paper, we consider the general case in two and three dimensions, and further generalizations with ellipsoids and lines. We believe, we are the first to explicitly find the polynomial equations for the parameters of the solution sphere in these generalized cases. Doing so is quite a challenge for the best computer algebra systems. We report later some comparative times using various computer algebra systems on some of these problems. We also consider conic tangency equations for general conics in two and three dimensions.
Apollonius problems are of interest in their own right. However, the motivation for this work came originally from medical research, specifically the problem of computing the medial axis of the space around a molecule: obtaining the position and radius of a sphere which touches four known spheres or ellipsoids.
Author Keywords: Conic tangency equations; Apollonius problems; Medial axis; Polynomial system.
2. Biochemical motivation
2.1. Automated docking algorithms
2.2. The medial axis
3. Mathematical approach
4. The Cayley-Dixon-Bezout-KSY resultant method
5. Two-dimensional Apollonius results
6. Three-dimensional Apollonius results
7. General conic tangency problems
7.1. Tangency of general conics in two dimensions
7.2. Tangency of general conics in three dimensions
Fig.3: Left: medial axis (lines) for a very simple
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