# Guinier analysis¶

This tutorial covers basic principles and best practices for doing a Guinier analysis. This is not a tutorial on how to use RAW for this type of analysis. For that, please see the RAW tutorial.

## Overview¶

Guinier’s approximation states that at low-q the scattering profile can be approximated as follows:

$I(q)\approx I(0) e^{-q^2 R_g^2 /3}$

where Rg is the radius of gyration and I(0) is the intensity at zero scattering angle (q=0). Because of the exponential in the Guinier approximation, Rg and I(0) can be determined by performing a linear fit to a plot of $$\ln(I)$$ vs. $$q^2$$, called the Guinier plot. The region chosen for the linear fit is called the “Guinier region.”

The Guinier fit provides information on the overall size of the molecule, and about the quality of your data.

## Why do we do a Guinier fit?¶

The Guinier fit is done for two reasons. First, you get the Rg and I(0) parameters. The Rg tells you about the overall size of the molecule, while I(0) depends on the molecular weight times the concentration. These parameters are useful characterizations of your molecule, and are also needed to calculate other information from the SAXS data, including molecular weight and volume.

Second, and perhaps most importantly, many of the problems that can affect your SAXS data will show up in the low q region, including:

• Aggregation
• Interparticle interactions
• Buffer mismatch (in some cases)

These data quality issues cause deviations from linearity in the Guinier region. For this reason, having a good Guinier fit is one of the primary ways we assess the quality of SAXS data. A good Guinier is a strong indicator that your data is from a monodisperse sample and is otherwise free of artifacts. If you cannot obtain a good Guinier fit, or you can only obtain a good Guinier fit by excluding a significant amount of data at the lowest q values, then your data probably has one or more of the problems listed above and usually should not be used for further analysis.

## How do we do a Guinier fit?¶

The Guinier approximation only holds when the exponential $$\exp(-q^2 R_g^2 /3)$$ is small. This means that in order to do a good Guinier fit, we need qRg to be sufficiently small. The qRg value at which the Guinier approximation starts to fail for a given scattering profile depends on the overall shape of the scatterer. Below is a figure showing the Guinier approximation (black), and the scattering intensity for a sphere, thin rod, and thin disc (all with the same Rg).

As you can see, the scattering intensity for the rod only agrees with the Guinier approximation until $$qR_g\sim 1.0$$, the sphere until $$qR_g\sim 1.3$$, and the disc until $$qR_g\sim 1.7$$. Thus, depending on the overall particle shape, you should fit different amounts of the low q data to have a good Guinier fit. (Note: the plot’s x axis is (qRg) 2, so $$qR_g\sim 1.3$$ appears as the dashed line at 1.69)

In practice, we fit both globular (sphere-like and disc-like) objects until $$qR_g\sim 1.3$$. while we fit highly extended (rod-like) objects until $$qR_g\sim 1.0$$. These values were chosen to have <10% error resulting from the deviation of actual shape from the Guinier approximation. The reason we accept that much deviation is that you also get uncertainty from fitting fewer points in your data, so there is a trade-off between how well the approximation works (fitting to smaller maximum qRg) and how well you can fit your data (fitting to larger maximum qRg).

The range of the Guinier fit is thus ideally from the earliest available q value until a maximum qRg of 1.0 or 1.3. However, given that Rg is derived from the Guinier fit, how do you determine the correct maximum q value for the end of the fit? The answer is that the Guinier fit is done iteratively:

1. Guess a starting maximum q value for the fit.
2. Calculate the Guinier fit and get Rg.
3. If qRg > 1.3 (or 1.0), reduce the maximum q. If qRg < 1.3 (or 1.0), increase the maximum q.
4. Repeat steps 2 and 3 until you converge on a final maximum q.

Most software these days will do this iterative search for you, and for good quality data will provide you with a reasonable maximum q value that may need just a bit of manual refinement.

The minimum q value of a Guinier fit is usually determined by the minimum available q value in your data, which is set by the instrument on which you make the measurement. However, it is important to have a small enough minimum q to have a reasonable range for the Guinier fit. Typically, the minimum qRg value should be $$qR_g\leq 0.65$$, though for globular systems it can be okay to have $$qR_g\leq 1.0$$. This means that the minimum q value required depends on the size of the system measured. In some cases, with particularly large systems, you may have to deliberately seek out an instrument that can measure to sufficiently low q.

If your data has quality issues at low q, which can be caused by the problems listed above, you may find that excluding those data from the fit can improve the quality of the fit. While this can be acceptable, you should proceed with caution when doing that, and always show the full data range on plots. The most acceptable case for this to happen is when the first few points are either too high or too low, but the rest of the range fits perfectly (see below for criteria for a good fit). In that case, those couple of points closest to the beamstop may have poor statistics or higher instrumental background scattering, and can usually be safely ignored.

## Criteria for a good Guinier fit¶

You are looking for four essential components in your Guinier fit:

1. $$\mathbf{q_{min}R_g<0.65}$$ .
• The minimum q of your fit, qmin, times the Rg of your fit should be less than 0.65. This criteria ensures you have enough q range to properly calculate the Rg and I(0) values. For globular particles (sphere- or disk-like), you can get away with $$q_{min}R_g<1.0$$.
2. $$\mathbf{q_{max}R_g \sim 1.3}$$ (globular) or $$\mathbf{q_{max}R_g \sim 1.0}$$ (extended).
• The maximum q of your fit, qmax, times the Rg of your fit should be less than 1.3 for globular (sphere- and disc-like) particles and less than 1.0 for extended (rod-like) particles. This ensures you remain in the linear range of the Guinier approximation for the fit.
3. The Guinier fit residuals should be flat and randomly distributed about zero.
• If your residuals have a ‘smile’ (above zero near start and end of fit, below in the middle), or a ‘frown’ (below zero near start and end of fit, above in the middle), it indicates you have non-ideal data. The ‘smile’ is characteristic of aggregation, the ‘frown’ characteristic of interparticle repulsion.
4. The fit extends to the lowest available q point.
• You shouldn’t have to excluded very many points at the start of the fit. A few is generally fine, as the points nearest the beamstop can be noisy (depending on the exact details of the measurement). Having to exclude more than 3-5 points at the low q may indicate a problem with your data.

Having a good Guinier fit is a major quality check, and a good sign that your data is from a monodisperse sample with no interparticle interactions.

## What is a bad Guinier fit, and what does it mean?¶

Non-linearities in your Guinier fit are indicative or problems in your sample. The type of non-linearity can indicate what the problem may be. The figure below gives a quick summary off the most common pathologies, more detail is available in the sections below.

### Aggregation¶

Aggregation causes a characteristic upturn at low q. This can either be caused by aggregates initially present in your sample, or by radiation induced aggregation (radiation damage). The figure below shows what that might look like in your data.

The effect of aggregation can also be clearly seen in the fit residual, where it shows up as a ‘smile’, with the residual above zero near the start and end of the fit and below zero in the middle. The figure below illustrates this.

### Repulsion¶

Repulsive interparticle interactions result in a structure factor that causes a downturn in the scattering profile at low q. This is typically caused by electrostatic interactions, and can often be remedied by either reducing the sample concentration or adding more salt to the buffer. These effects are also clearly seen in the fit residual, where it shows up as a ‘frown’, with the residual below zero near the start and end of the fit and above zero in the middle. This is shown in the figure below.

Note that the above figure is an extreme example of repulsion, the downward curve may not be that obvious.

Good SAXS data depends on subtracting away all scattering from the buffer and instrument background. If this subtraction is not good, you can end up with a downturn at low q (over subtraction) or an upturn at low q (under subtraction). This will look similar to aggregation or repulsion in the Guinier fit.

## FAQ¶

### What if I don’t know my particle shape, should I fit to a maximum qRg 1.3 or 1.0?¶

Very often, you don’t know what your particle shape is before making a SAXS measurement (in fact, this is often one of the purposes of a SAXS measurement). In that case, start out by fitting to a maximum qRg of 1.3. If that has a non-flat residual, reduce the fitting range to a maximum qRg of 1.0. If the residual becomes flat upon reducing the maximum qRg, then your particle is likely more extended than globular, and you should keep the maximum qRg at 1.0. If reducing the maximum qRg still leaves a non-flat residual, your data is showing signs of aggregation, repulsion, or some other issue.

### My Guinier fit isn’t great, can I still use my data?¶

If your Guinier fit isn’t great, typically you shouldn’t use the data. Even small amounts of aggregation (<1%) can affect things like the measured maximum dimension, and three dimensional reconstructions. While data with imperfect Guinier fits can be used in some specific cases, my general recommendation is to collect the data again.

### My Guinier fit is a bit off, how can I fix it?¶

If your Guinier fit isn’t ideal, typically the only way to fix this is to collect the data again, improving your sample/solution conditions. Below are a few things you can try to fix these issues.

For inherent aggregation:

1. Spin down your sample in a centrifuge at high speeds (~16000 g) for 5-10 minutes before data collection. In some cases, ultracentrifugation may help.
2. Use size exclusion chromatography coupled to SAXS instead of batch mode SAXS for in-line sample purification.
3. Reduce the concentration of your sample.
4. Re-purify your sample using size exclusion or ion exchange chromatography immediately before the SAXS measurement.

2. Increase the flow/oscillation speed of your sample.
3. Reduce the exposure time and/or number of exposures.
5. Attenuate the incident x-ray beam.

For repulsion:

1. Add salt to the buffer to reduce repulsion.
2. Reduce the concentration of your sample.
3. Change the pH of your buffer.