Data from the analytical ultracentrifuge can have two kinds of systematic signal offsets: 1) a background profile that changes with radius but is common to all scans, which is referred to as time-invariant (TI noise) (possible sources are dust or scratches on the window in absorbance optics, or the systematic optical pathlength variations due to imperfections in some of the optical components of the interference optics); 2) a radially constant offset which can change in each scan (referred to as radial invariant, or RI, noise), such as the integral fringe shift and 'jitter' commonly encountered in interference optical data.

It has been shown that pseudo-absorbance data (generated without reference sector from intensity scans, thereby allowing to perform two separate sedimentation experiments in each cell) can also be described by including these systematic signal offsets (ref 4).

Traditionally, RI noise has been corrected for by 'alignment' of the scans over a range of radial values (for example, in the air-to-air region of the cell), and TI noise was eliminated by analyzing pairwise differences of scans (for example, in the dc/dt transformation or in the Dc approach). Although these approaches are suitable for many cases, sometimes they are not optimal because the alignment over a limited radial interval may not completely eliminate RI noise, the pairwise scan subtraction slightly increases the statistical noise, or because they do not allow to inspect directly the raw data versus model data.

Therefore, Sedfit uses an
algebraic noise decomposition strategy (described in Ref 1)
that is completely based on direct least-squares modeling of the
sedimentation data of interest. No increase in the statistical noise is
produced, and importantly, explicit estimates for the magnitude of the
systematic noise components are obtained. **The best-fit time-invariant
noise component of the data is shown as a black line in the data plot.
Also, the estimate of the systematic noise contributions allows to subtract the
systematic noise from the data** for a detailed inspection of the data
and the fit (see the menu Display-> subtract all systematic noise from raw data).

In the following, two derivations for the algebraic systematic noise decomposition are introduced. First, a very general mathematical approach is described which is based on simple linear least squares. Less mathematically inclined readers can jump immediately to a more intuitive illustration of the result from the mathematical treatment. Finally, examples are given of the noise decomposition for interference and absorbance data.

**The step-by-step tutorial
includes the treatment of systematic noise in interference data, as well as the example
of using Sedfit. **In
general, for using this approach, only the TI
and RI noise fields need
to be checked in the parameter box
(which is the default for analyzing interference data.)

**Mathematical Approach for Systematic Noise Analysis**

First, we consider the least-squares problem of modeling the data points *a _{s,i}*
from a set of scans with radial points

According to the principle of least-squares modeling, we want to find the
minimum of the squared difference between the data points and the model, summed over all data
points (i.e. all radial indices *i* and all scans *s*). The
parameters to be optimized are those of the Lamm equation models (*s _{k}*,

(Eq. 1)

The key to solving Eq. 1 efficiently is to divide the problem up it into separate minimization problems for the nonlinear parameters and for each of the linear parameters (Ref 3). It will be seen that the linear parameters can be directly solved, if we assume any particular value for the nonlinear parameters. Because the linear parameters are so easy and fast to calculate (as will be seen), we can afford to reevaluate them each time when we test any set of the nonlinear parameters. It is shown in the next paragraphs how we can eliminate the linear parameters, leaving an ordinary non-linear least-squares problem with a reduced number of parameters, and importantly, without any systematic noise parameters.

As in elementary calculus, the minimum of a function is characterized by a
vanishing derivative. Applying this principle to the double sum of Eq. 1, we can set the
partial derivative
of the sums in Eq. 1 with respect to any particular parameter *b _{j}* equal to zero. Since
the baseline parameters are uncorrelated, (

(Eq. 2)

(note that the
square has disappeared when taking the derivative, and that we have divided the
equation by 2). The second equation is obtained by executing the summation,
using the abbreviations (with the total number
of scans *n _{s}*) and . It can
be seen that corresponds to the average signal at
a radial point

(Eq. 3)

which does not contain any of the original noise parameters.

In fact, we can apply the same strategy to all linear parameters including
the Lamm equation loading concentrations *c _{k}*: taking the partial
derivative with respect to any of the concentrations, lets say

(Eq. 4)

This equation can be directly solved by matrix algebra. This is very similar to the size-distribution calculation, where the parameters of the entire distribution are linear (this is further developed in the tutorial on size-distribution analysis).

! Instead of the standard algebra routines for linear equations, we can make sure that we do not get negative concentration values by using the algorithm NNLS from Lawson and Hanson, Ref 2, which we have modified for use with normal equations.

At this point, we have formally calculated all our linear parameters, but their numerical value still depends on the particular set of nonlinear parameters, i.e. on the Lamm equation solutions in our sedimentation model. We can write the remaining problem of minimizing those nonlinear parameters as

(Eq. 5)

with the abbreviation

What this means is that we optimize the non-linear parameters (*s _{k}*,

What about RI noise? Initially, the problem of modeling the boundary
with both TI noise and RI noise appears more complex, but the problem can be
reduced to the case of TI noise only. If we denote the baseline offset of
each scan as b_{s}, we can include an
additional term in Eq. 1:

(Eq. 6)

There is a slight complication because the b_{s}
and the *b _{i}* are not independent: imagine adding a value of 1 to
all b

Like before, we can set the partial derivative of the sum in Eq. 6 with
respect to b_{s} zero in order to find the
minimum. The sum over the scans disappears, since only one term will be
dependent on any particular b_{s} , and
because it is independent of radius, we can pull b_{s}
out of the summation as product of the number of radial points (n_{r})
and b_{s} . The remaining equation,
when solved for b_{s} looks like:

(Eq. 7)

In the lower equation, we have introduced an abbreviation for the 'scan-average' (i.e. summation over all radial points and division by the number of radial points) for the data and for each of the Lamm equation solutions. If we insert this result into Eq. 6, we obtain

(Eq. 8)

which is identical to Eq. 1 for TI noise only, except that we have substituted the data and the boundary model by difference between the data and its 'scan-average'. Because Eq. 8 is formally equivalent to Eq. 1, we can solve it in the same way.

It should be noted that the boundary parameters can be optimized either in the original data space, such as indicated in Eq. 5, or in a difference expression, such as the time-difference in Eq. 3 (where the reference for calculating the time-difference is an 'average scan') or a 'scan-difference' in Eq. 8. Both ways are mathematically equivalent. In some cases, e.g. in the size-distribution analysis, the difference expressions are much easier to handle in subsequent calculations (i.e. normal equations and regularization). These difference expressions capture the entire available information on the sedimentation. Explicit calculations of the systematic noise parameters can be done using Eq. 7 and 2 after all the sedimentation parameters are optimized.

If we look at just the problem of TI noise and simplify the notation, we can
say that we consider our measured data **A** as a result of some
sedimentation process, described by a boundary model **L**, superimposed by a
time-invariant noise **b**_{r}, plus some statistical noise **e**:

(Eq. 9) **A** = **L** + **b**_{r} + **e**

The time-derivative methods take the strategy of transforming the data into D**A**/Dt
as an approximation of d**A**/dt, whereas the time-difference method would
model D**A** with differences D**L**
of the boundary model. In both, the idea is that the time-invariant
component vanishes in the difference

(Eq. 10) D**A** = **A**_{1}
- **A**_{2} = **L**_{1} - **L**_{2} + (**b**_{r}
- **b**_{r}) + **e** + **e**
= D**L** + 1.4**e**

The statistical noise is increased, and in both cases, we do not gain any
information on the noise **b**_{r}.

In contrast, if we look at Eq. 2, it becomes clear that Sedfit applies the concept of a 'radial-average' scan and a 'radial-average' boundary model (i.e. at each radial value, average all scans and boundary models, respectively):

(Eq. 11)

(n denotes here the total number of scans). This average scan still
contains the time-invariant noise **b**_{r}. It can be
determined fairly precise if we have a large number of scans, because by
averaging the statistical noise is reduced with a factor equal to the inverse
square-root of the total number of scans.

Eq. 3 from above now tells us that we should model not pairwise differences of scans, but instead the difference between any particular scan and the average scan:

(Eq. 12)

With a large number of scans, this leaves the statistical noise virtually
unchanged. (To be more precise, because Sedfit
fits directly the raw data, the noise is actually not amplified at all, but
additional degrees of freedom are introduced by considering the TI noise in the
model.) Further, once we have determined the best-fit parameters of our model **L**,
we can calculate directly the time-invariant noise as difference between the
average data and the average boundary model:

(Eq. 13)

This gives us explicit values for **b**_{r }, which - if desired -
we can subtract from the original data in order to better inspect the measured
sedimentation process.

Although the systematic noise decomposition allows us to conveniently perform direct boundary modeling on interference data, and to obtain estimates of the time-invariant noise which we can subtract this from the data, there are two fundamental limitations that should be kept in mind:

First, the calculated systematic noise is only an estimate, which is model dependent (Eq. 2 and 13). For different models, the estimate will be different. Only if we have a pretty good fit of the sedimentation data can we assume the calculated noise to reflect the true baseline signal, as it might be measured, for example, in an experiment with a water blank. This is no problem, however, since, except in special cases (ref 5), the true baseline offset is not a meaningful quantity that we want to derive from our experiment. The important consequence of this, however, is that after subtracting our best-fit systematic noise estimate from the data, we should continue to allow for corrections to the systematic noise be made in the course of further modeling.

Second, as outlined above, there is no fundamental difference in the effect of systematic noise
on the data analysis from a time-difference analysis. Because there is the
unknown systematic radial-dependent baseline, the only source of information is
the *change of signal with time*. In fact, as can be seen in Eq. 3
and 12, the parameters of the boundary model (L) are fitted to a time-difference
of the data. In contrast to time-difference (or time-derivative) methods
previously described, we do not take any particular scan as a reference for
calculating the boundary movement in time, but we take the average scan as a
reference, which is statistically advantageous. Nevertheless, by putting
our equations in a form that is invariant with respect to adding
the baseline parameters b, we have introduced the degrees of freedom in the
analysis that make it equivalent to a time-difference method. These
degrees of freedom can lead to correlation with (and leaving sometimes
underdetermined) the sedimentation of small molecules. It is obvious, that
this correlation will be worse for smaller data subsets. (For example, in
a g*(s) analysis, there can be correlation with the low s-value part of the
distribution, in particular with small data sets where the meniscus is not
cleared [this is not as much a problem with other boundary models].)

It should be noted that this second limitation is separate from the actual estimation of the systematic noise b, as modeling of the sedimentation parameters with Eq. 12 precedes the calculation of the baseline parameters.

Fortunately, these issues are usually not problematic. In particular, if large data sets are used, the baseline is usually well-determined, and the correlation of the degrees of freedom with small s-values is low.

**Example of the Noise
Decomposition**

The following example shows how interference optical data can be decomposed into the sedimentation part, the time-invariant noise, the radial-invariant noise, and the statistical noise:

The left figure shows in the top panel (A) the raw sedimentation data from an experiment with myoglobin. Panel B below shows the results of fit with a single-component Lamm equation solution (black) considering RI noise (shown in red as a function of time), and TI noise (blue). The bottom Panel C shows the remaining statistical noise, which is nearly random and has an rms value of 0.0065 fringes. A similar decomposition is shown in the middle Figure for the fit of data from an IgG sample with monomeric and dimeric IgG modeled as a two-component Lamm-equation fit, with statistical noise of rms = 0.0072 fringes. In the right Figure is the result from modeling the mixture of the myoglobin and IgG sample with a 3 component Lamm equation solution (using the pre-determined values for s and M). Here, we get a decomposition with rms noise of 0.0051 fringes. More details on these examples are described in Ref. 1.

The noise parameters are well-determined, and interference optical data are well-described by the combination of TI noise, RI noise, and statistical noise. An example of how the systematic noise can be subtracted from the data can be found here.

TI noise can also be useful when working with absorbance data. In Panel A below are sedimentation data of myoglobin, in Panel B the residuals of a single-species Lamm equation fit, which has an rms of 0.013. A close inspection of the noise in B reveals that it is not entirely random, but has some time-invariant features. If we model the data with TI noise (Panel C), we arrive at much more random residuals, with an rms value of 0.009.

More details can be found in Ref. 1.

**More examples can be found in the step-by-step tutorial,
which includes the treatment of systematic noise in interference data, as well
as in the example of using Sedfit. **

(1)
P. Schuck and B. Demeler (1999) Direct sedimentation analysis of
interference-optical data in analytical ultracentrifugation. * Biophysical
Journal*
76:2288-2296.

(2) C.L. Lawson and R.J. Hanson. (1974) Solving Least Squares Problems. Prentice-Hall, Englewood Cliffs, New Jersey

(3)
Ruhe, A., and P. �. Wedin. (1980) Algorithms for separable nonlinear least
squares problems. *SIAM Review.* 22:318-337.

(4) S.R. Kar, J.S. Kinsbury, M.S. Lewis, T.M. Laue, and P. Schuck (2000)
Analysis of transport experiments using pseudo-absorbance data. * Analytical
Biochemistry*
285:135-142.

(5) P. Schuck (1999) Sedimentation equilibrium analysis of interference optical data by systematic noise decomposition. Analytical Biochemistry 272:199-208.