Residual Dipolar Couplings

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Residual Dipolar Couplings

• By Homayoun Valafar

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Residual Dipolar Couplings
B0

• RDC arises from interaction of a magnetic dipole
  and an external magnetic field.
• Normally the average is reduced to zero.
• Induced partial alignment can reintroduce
  residual dipolar couplings (RDC).
• Analysis of RDC data can provide orientational
  constraints that can lead to structure
  determination.
• Analysis of RDC data can provide an insight into
  internal dynamics of proteins.

H
?
r
N
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Hyperplanes and Linear Equation

• An equation of the form
  is said to be a linear equation of Nth
  dimension.
• This equation represents a Nth dimensional
  hyperplane.

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Solution to a Fully Determined System of Linear
Equations

• A system of linear equations can be represented
  as Axd.

• If A is non-singular (invertable) then the
  solution to this system can be obtained by xA-1d

Rank(A) 3 Dim(Null (A))0 A-1 does exist xA-1d
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Solution Space for Over/Under Determined System
of Equations
Over-determined
Under-determined
Under-determined
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Effects of Error
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Solution to Undetermined Systems

• Over-determined and under-determined systems need
  to be treated differently.
• Various algorithms can be utilized to solve
  systems of linear equations.
• Singular Value Decomposition is one such
  algorithm that provides a solution to both under
  and over determined system of equations.

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Singular Value Decomposition

• SVD can be utilized in solving a system of linear
  equations while providing some diagnostic
  information (fast, robust and diagnostic).
• Theorem
• Any M ? N matrix A (such that M ?N) can be
  written as the product of three matrices as shown
  below

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SVD for Over-determined Systems

• SVD returns the value of x that minimizes the
  least mean square error as the solution to an
  over-determined system of linear equations.
• This may result a distortion of the solution due
  to the addition of one inaccurate measurement.

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SVD for Under-determined Systems

• Since A is singular, Ax0 has nontrivial
  solutions (null space). Therefore any
  xxoptxnull will satisfy Axb.
• AxA(xoptxnull)A xoptA xnullb0b
• For under-determined system of equations SVD
  returns the value of x that has the smallest
  magnitude. This guarantees the exclusion of any
  contribution from the null space of the system.

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What is a Tensor?

• A detailed definition of a tensor is somewhat
  abstract.
• Generally, a tensor is a collection of numbers
  which obey certain transformation laws.
• Tensors are used to describe fundamental laws of
  physics, engineering, science and mathematics
  independent of the coordinate systems.

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Order of a Tensor

• A scalar1x1, vector3x1 and matrix3x3 are examples
  of 0, 1st and 2nd order tensors.
• In a three dimensional space, an nth order tensor
  will be represented by 3n number of parameters.

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Order Tensor and Dipolar Couplings

• Partial alignment of a molecule gives rise to the
  observation of dipolar couplings.
• Analysis of dipolar coupling measurements can
  provide us with an order tensor.
• Order tensor describes the degree of alignment of
  a given molecule in relation to a reference frame.

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Order Tensor and Dipolar Couplings
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Matrix Formulation of Dipolar Coupling
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SVD in Order Tensor Analysis(For Over-Determined
Systems)?

• The general outline of application of SVD to
  order tensor analysis is as following

Randomly generate D within error
Start
No
Is s valid (traceless)?
Create the linear equations
Solve for x using pseudo-inverse sA-1D
Yes
Diagonalize and output order tensor
Setup AsD
Back calculate D using DAs
Using SVD create pseudo-inverse of A
No
Is D within error of D?
Yes
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Information Content of an Order Tensor

• Two distinct set of information are encoded
  within a given order tensor.
• These two are the degree of order and the
  direction of order.
• Any given order tensor S can be rewritten in a
  diagonalized form given by SR?R-1 where ? is a
  diagonal matrix and R is an orthonormal matrix.

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Dont talk anymore!!!
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Order and Orientational Information

• ? is a diagonal matrix that consist of at most
  three (at least 2) non zero elements.
• These elements indicate the degree of order along
  each of the main axes in the principal axis
  system.
• R defines a rotational transformation that
  relates the principle axis system to the
  molecular frame.

Alignment of the molecule relative to an
arbitrary frame.
Arbitrary molecular frame
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End
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Eigenvalue Eigenvectors

• Ae?e, ? is the eigenvalue and e is the
  corresponding eigenvector.

Vector v
y
y
Av
x
x
Eigenvectors
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Solving for Eigenvalue and Eigenvector

• Analytically, this can be done by solving the
  characteristic polynomial of the matrix A.
• Computationally, this is a very inefficient
  method.
• Other methods such as Jacobi transforamation,
  Householder reduction and QR/QL are more
  efficient.

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Jaccobi Transformation

• Need to find the diagonal form of A AR-1?R,
  R contains the right eigenvectors and elements of
  ? are the eigenvalues.

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Jacobi Transformation Matrix
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Solution Space for a System of Linear Equations

• All x such that Ax0 constitute the Null space of
  A.

Rank(A) 1 Dim(Null (A))2
Rank(A) 2 Dim(Null (A))1
Rank(A) 3 Dim(Null (A))0
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System of Linear Equations and Linear Algebra

• An equation of the form
  is said to be a linear equation of Nth
  dimension.
• This equation represents a Nth dimensional
  hyperplane.
• A system of these linear equations can be
  represented as Axd.

• If A is non-singular (invertable) then the
  solution to this system can be obtained by xA-1d

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Calculus of Hyperplanes

• The constants that define a hyperplane have
  special spatial meanings.
• For illustration purposes we will stick to 3D.