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Alternatively, if many rotations need to be performed at the same time (and the intermediate Cartesian coordinates are not needed), these rotations could be sorted by bond number and applied simultaneously, by noting that rotations can be performed in a cumulative way as the backbone is traversed from anchor to end atom. The ability to chain rotations around arbitrary vectors in space (i.e. not through the origin) is one of the main benefits of homogeneous transformations. For example, if two rotations need to be applied at the same time, one around bond 3 by 30 degrees and another around bond 7 by 15 degrees, the atoms between bonds 3 and 7 get updated by:

But the atoms after bond 7 are updated by:

In the above, bond n is the unit vector defined along bond n, easily computed by subtracting the coordinates of atoms n+1 and n, and then dividing by its norm. The chaining of transformations as explained above is very useful to achieve arbitrary rotations of bonds within a protein. Sections of the protein (i.e. atoms belonging to certain residues) can be updated when a dihedral rotation is performed simply by constructing the overall matrix that should affect them.

Denavit-hartenberg local frames

The previous approach, while simple and intuitive, has some shortcomings:

  • The accumulation of math operations in the rotation matrices is prone to numerical instability. After only a couple of hundred rotations of apoint, each accumulating on the other, the final position of the point may start differing significantly from its actual, intended position.As a consequence, the relative position and orientation of atoms in the protein chain will no longer be in agreement with the protein structure.In particular, bond lengths and angles will begin stretching and deviating from their physically acceptable values.
  • The actual values of the Cartesian coordinates are always stored in a particular, arbitrarily chosen frame of reference. For example, if wewanted to translate the protein, we would need to modify the Cartesian coordinates stored.
  • Once a rotation is applied, the method "forgets" the current values of the dihedral angles, which would need to be re-computed if needed. What isstored is a snapshot of the current Cartesian coordinates of each atom.

The original definition of Forward Kinematics, however, is a method to obtain the Cartesian coordinates of each atom from the current values of the internal degrees of freedom (dihedral angles in our case) at any time. In such an approach, the Cartesian coordinates need not be recomputed after every change in the dihedral angles; rather, the idea is to store the current values of the dihedral angles, and to have a procedure to reconstruct the atomic positions when needed. The advantages of this approach are:

  • A more compact representation of the variables of the problem, since the dihedral angles require less space than the (x,y,z)coordinates of each atom (the protein topology requires the values of the bond lengths and angles anyway, so the total amount of numbersto store is comparable).
  • It is not prone to numerical instability since the number of rotations performed to position an atom is always its sequence number in thechain. (Actually if the chain is thousands of residues long, some uncertainty could arise in the position of atoms far along the chain,but the relative position of consecutive atoms can still be kept under control, avoiding bond stretching).
  • Performing a dihedral rotation consists simply of adding/subtracting the rotation angle from the stored value for each angle. In particular, simultaneous rotations(i.e. rotating more than one dihedral angle at a time) which consists of multiplying many 4x4 matrices in the global method, reduces to modifying the angle values.
  • There is no explicit global coordinate frame for the protein. It can be positioned arbitrarily by prepending a position/orientation matrixto the forward kinematics computation.

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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