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DLS Theory - Regularization |
As we have seen in the Principle of DLS, the technique of obtaining a single Hydrodynamic
Radius value from a DLS experiment can be achieved by the methods of Autocorrelation and Least Squares via the Stokes-Einstein equation.
However, one of the most common applications of DLS is Protein Crystallography.
Whether a protein crystal can be grown or not, depends on its purity. To determine
the purity of a sample we need more than a single Hydrodynamic Radius RH value -
we need to know the distribution of sizes present in the sample.
There are several mathematical techniques for extracting size distribution data
from the observed autocorrelation function, but the general principle is the same:
minimize a system of equations for the amplitudes of a discrete range of sizes.
In other words, define a set of sizes (a-p), and figure out how much of each best
models our observed data:
The trouble is, this turns out ot be what is called an 'ill-posed mathematical problem'
in that the exact solution won't exist, and worse still, there will be
an infinite number of approximate solutions that all fit the experimental data equally
well.
Fortunately, there are a few techniques for solving this problem - the most widely
used being the method of regularization, whereby we introduce another variable which
is a property of the solution which we are prepared to sacrifice closeness of fit
for. Naturally occurring size distributions tend to be 'smooth', whereas other theoretical
solutions tend not to be. In the method of Tikhonov regularization, we introduce
the 'smoothness' of the solution as an additional factor to take into consideration
when solving our system of equations =- the amount of smoothness is called the 'regularizer'.
This technique has proven successful in a number of physical applications including
Dynamic Light Scattering. However, there remains a key question: 'how much smoothness
are we prepared to sacrifice in favor of closeness of fit'. In mathematical terms,
this problem is called optimizing the regularizer.
Many techniques have been implemented in the past, but now that high-performance
computing is available at a very low cost, a previously unused technique is possible
to use in DLS applications. This technique is called 'L-Curve analysis'.
The basic premise is that we solve the system of equations many times with different
regularizers and pick the best one; with modern PCs we can perform this optimization
at lightening speeds and get reliable results.
 For multiple values of the regularizer,
we examine the solution yielded, and plot two properties against each other. We
plot the 'irregularity' on the Y-Axis, and the 'Residual' on the X-Axis.
Irregularity is a property that is related to the smoothness of the solution distribution
- most often the sum of the slope of the distribution curve. A smooth distribution
will have a low sum of slopes, whereas an irregular distribution will have high
slopes.
The Residual is simply the variance between the observed autocorrelation function
and the modeled autocorrelation function described by the solution under consideration.
Clearly, we want to find the solution which offers the best trade-off between these
values, and fortunately, this plot produces a convenient 'L' shaped curve, with
the 'elbow' of the curve being the optimal choice of regularizer - or mathematically,
the point at which the curvature of the graph is greatest.
This technique has proven to yield reliable distributions and accurate polydispersities,
and is used in many applications such as protein crystallography to great effect. |
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