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From the Delaunay triangulation the α-shape is computed by removing all edges, triangles, and tetrahedra that have circumscribing spheres withradius greater than α. Formally, the α-complex is the part of the Delaunay triangulation that remains after removing edges longer than α. The α-shape is the boundary of the α-complex.

Pockets can be detected by comparing the α-shape to the whole Delauney triangulation. Missing tetrahedra represent indentations, concavity, and generally negative space in the overall volume occupied by the protein. Particularly large or deep pockets may indicate a substrate binding site.

Weighted alpha shapes

Regular α-shapes can be extended to deal with varying weights (i.e., spheres with different radii, such as different types of atoms) . The formal definitions become complicated, but the key idea is to use a pseudo distance measure that uses the weights.Suppose we have two atoms at positions p1 and p2 with weights w1 and w2. Then the pseudo distance is defined as the square of the Euclidean distance minus the weights. The pseudo distanceis zero if and only if two spheres centered at p1 and p2 with radii equal to sqrt(w1) and sqrt(w2) are just touching.

Pseudo distance to account for atoms of different sizes.

Calculating molecular volume using α-shapes

The volume of a molecule can be approximated using the space-filling model, in which each atom is modeled as a ball whose radius is α, where α is selected depending on the model being used: Van der Waals surface, molecular surface, solvent accessible surface, etc. Unfortunately, calculating the volume is not as simple as taking the sum of the ball volumes because they may overlap. Calculating the volume of a complex of overlapping balls is non-trivial because of the overlaps. If two spheres overlap, the volume is the sum of the volumes of the spheres minus the volume of the overlap, which was counted twice. If three overlap, the volume is the sum of the ball volumes, minus the volume of each pairwise overlap, plus the volume of the three-way overlap, which was subtracted one too many times in accounting for the pairwise overlaps. In the general case, all pairwise, three-way, four-way and so on to n-way intersections (assuming there are n atoms) must be considered. Proteins generally have thousands or tens of thousands of atoms, so the general n-way case may be computationally expensive and may introduce numerical error.

Three overlapping discs (balls if three dimensional). Calculating the total area (volume if three dimensional) of the balls requires summing the areas of each ball, then subtracting out the pairwise intersection areas, since each was counted once for each ball it is inside. Then the intersection area of the three balls must be added back because, although it was added three times initially, it was also subtracted once in each of the three pairwise intersections. In the general case, with n balls, all of which may overlap, intersections of odd numbers of balls are added, and intersections of even numbers of balls subtracted, to calculate the total area or volume.

α-shapes provide a way around this undesirable combinatorial complexity , and this issue has been one of the motivating factors for introducing α-shapes. To calculate the volume of a protein, we take the sum of all ball volumes, then subtract only those pairwise intersections for which a corresponding edge exists in the α-complex. Only those three-way intersections for which the corresponding triangle is in the α-complex must then be added back. Finally, only four-way intersections corresponding to tetrahedra in the α-complex need to be subtracted. No higher-order intersections are necessary, and the number of volume calculations necessary corresponds directly to the complexity of the α-complex, which is O(n log n) in the number of atoms.

An example of how this approach works is given on page 4 of the Liang et al. article in the Recommended Reading section below. A proof of correctness and derivation is also provided in the article. Surface area calculations, such as solvent-accessible surface area, which is often used to estimate the strength of interactions between a protein and the solvent molecules surrounding it, are made by a similar use of the α-complex.

  • H. Edelsbrunner, D. Kirkpatrick, and R. Seidel. [PDF] . "On the Shape of a Set of Points in the Plane." IEEE Transactions on Information Theory, 29(4):551-559, 1983. This is the original α-shapes paper (caution: the definition of α is different from that used in later papers--it is the negative reciprocal of α as presented above).
  • H. Edelsbrunner and E.P. Mucke. [PDF] . "Three-dimensional Alpha Shapes." Workshop on Volume Visualization, Boston, MA. pp 75-82. 1992. This article shows how to extend α-shapes to three-dimensional point sets.
  • J. Liang, H. Edelsbrunner, P. Fu, P.V. Sudhakar, and S. Subramaniam. [PDF] . Analytical shape computation of macromolecules: I. molecular area and volume through alpha shape. Proteins: Structure, Function, and Genetics, 33:1-17, 1998. This is a paper on using α-shapes to speed up volume and surface area calculations for molecular models.
  • H. Edelsbrunner, M.Facello and Jie Liang. [PDF] . On the definition and the construction of pockets in macromolecules. Discrete and Applied Mathematics, 88:83-102, 1998.


Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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