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Table of Contents
Welcome
Whats New?
TECHNICAL QUESTIONSDynamic Light Scattering
What is a correlation curve?
How is the correlation curve measured?
Whats the relevance of the frictional coefficient?
What is the hydrodynamic radius (RH)?
Is it better to use longer acquisition times?
Whats the difference in the intensity, number, and mass distributions?
Static Light Scattering
What is the Rayleigh ratio?
How is the Rayleigh ratio measured?Whats the relevance of toluene in static light scattering?
Is the Rayleigh ratio of toluene wavelength dependent?
What are the units of the parameters in static light scattering?
When is multi-angle data necessary for MW determination?
Data Interpretation
What is the baseline?
What is the SOS?
What are number fluctuations?
Whats the difference between the old and the new Regularization?
Sample Preparation
Do samples need to be filtered?
Will filtering perturb the sample?
Why is the salt content relevant?
Calculations
What is the Regularization Method?
Can you get molecular weight information from DLS data?
How is the axial ratio calculated?
How is the mass distribution of radii calculated?
Are the %Mass calculations valid for all systems and conditions?
Sample Properties
Whats the significance of the refractive index () in DLS?
Is the refractive index increment (d/dC) wavelength dependent?
Is the solvent refractive index (o) wavelength dependent?
Are o and d/dC temperature dependent?
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Is it alright to estimate solvent viscosity values?
Software & Utilities
What is the _____ value of ______?
LABORATORY REPORTS
January 2000
Summary of Results
Lysozyme UV Analysis
Instrumental Error Effects in UV Analysis
Effects of Acquisition Time on DLS Results
Effects of Concentration on DLS Results
Defining a Quality Control Parameter
Maximum Acceptable CNF Value
Quality Control Specifications
May 2000
Summary of Results
Cell Comparison UV Measurements
Cell Comparison Beers Law Region for Lysozyme
Calculating Minimum Concentrations
Eppendorf UVettes - QC Analysis
Bovine Serum Albumin (BSA) UV Study
Effects of Laser Power on Scattering Intensity
DLS Study with Polystyrene Standards
Enhancing DynaPro Sensitivity
APPLICATION NOTES
Surfactants & Micelles
Characterization of Surfactant and Micellar Systems
Determination of Critical Micelle Concentration
Measurement of Mixed Micelle Size by DLS
Vesicle Size Distribution Analysis
Surfactants as Chaotropic Agents in Protein Systems
Representative Publications for Surfactant and Micelle Systems
Polymers & Polypeptides
Biological and Synthetic Polymer Characterization
TC Measurements of Recombinant Hydrophobic Polypeptides
Electrostatic Effects on Polyelectrolyte DLS Measurements
Polysaccharide Fractionation by Filtration
Characterization of Low Molecular Weight Peptides
Representative Publications for Polymeric Systems
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Protein Characterization
Protein Characterization
Conformation of Trimeric Form of Migration Inhibitory Factor
Insulin Quaternary Structure as a Function of pH
pH Dependent Monomer-Dimer Equilibrium Constants for GARTMonitoring Protein Thermal Stability - Glycosylated Hb Melting Points
Representative Publication in Protein Characterization
MANUALS & TUTORIALS
Utility ReadMe Files
PSI Archive Setup Instructions
PSI Books Setup Instructions
REFERENCESBackground References
Application References
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What is a correlation curve?
Photon Correlation Spectroscopy (PCS), also known as Dynamic Light Scattering (DLS), is
concerned with the investigation of correlation of photons. In a typical light scatteringexperiment single photons are detected with a single photon counting device. This detector
transforms the signal from a single photon into an electronic signal, basically a 1 or a 0,
depending on whether a photon was or was not detected in a certain time interval. The objective
of PCS is to find any peculiar properties of the scattered signal which can be used to
characterize and describe the seemingly random "noise" of the signal, and the correlation curve
is used to achieve this objective.
A typical way to describe a signal is by way of its autocorrelation. The autocorrelation
function of the signal from the scattered intensity is the convolution of the intensity signal as a
function of time with itself. In more abstract terms, if the detected intensity is described as a
function I(t), then the autocorrelation function of this signal is given by the followingexpression, where is the shift time.
The above function is also called the intensity correlation function, and it is used to describe the
correlation between the scattering intensities measured at t = 0 and some later time (tn = t0 + ).
Consider for example, the schematics shown below, where the frame represents the scattering
volume, the dark blue circle represents the particle, and the light blue region represents the
volume within which the particle must be contained. The scattering intensity at the detector is
dependent upon the position of the particle relative to the detector. If we consider very small
time increments (micro-seconds), then at t0 (t = 0) there is a finite volume of space within which
the particle can move during the first time increment. For reasons that will become evident
later, well call this finite volume of space the diffusional volume. At t = t1, the particle has
diffused a single step. Since the exact path is unknown, the diffusional volume or the volume of
space within which the particle must reside considering all paths has expanded. The change in
particle position is accompanied by a change in the measured scattering intensity at the detector.
However, this new scattering intensity is still strongly correlated with that measured at t0 due to
the finite diffusional volume and the correlation between the current and initial positions. As
the shift time increases, the diffusional volume also increases. The increased diffusional volume
is accompanied by a decrease in the correlation between the current and initial particle
positions. The correlation is still present however, as long as the diffusional volume is finite.
At longer times (frame 6), the diffusional volume becomes infinite, and the correlation between
the initial and current particle position and the respective scattering intensities is lost.
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The correlation in particle position at small shift times is contained within the measured
intensity correlation function, an example of which is shown below. In the absence of any
applied forces, the particle position is dictated by the degree of Brownian motion. As such, the
measured intensity correlation curve is an indirect measure of the particles diffusion coefficient.
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How is the correlation curve measured?
The intensity correlation function is defined as shown in the following expression.
The shift time () is often referred to as the delay time, since it represents the delay between the
original and the shifted signal. In reality, the continuous intensity correlation function can not
be measured. It can however, be approximated with discrete points obtained by a summation
over the duration of the experiment. This summation is performed by the correlator, a circuit
board composed of various logic chips and operational amplifiers which continuously multiplies
and adds measured intensity values.
A defining parameter for a correlator is its speed, i.e. its ability to add up numbers fast, or
inversely expressed, the smallest delay time that can be handled. This delay time () determines
the smallest accessible time of G(). Hence, G() is the first numerical point in the intensity
correlation curve that can be experimentally measured for a given correlator. Using a time
increment of , a grid of experimental data values can be obtained. At the start of the
experiment, t0 = 0. The first time then is at t1 = , the second is at t2 = 2, and so on. As an
array, the stream of numbers could be displayed in table format as shown below, with the
experiment ending at tn = n.
The correlator now finds out how related these intensity numbers are to each other. It does this
by comparing each number to its neighbor a defined time interval later. This comparison is
carried through the duration of the experiment. As a summation, the comparison can be
expressed as follows, where the upper summation limit is given by the appropriate indexbelonging to the largest available summation term for k.
The above expression is very general and holds true for both linear and arbitrary delay times. To
understand the meaning of this summation, lets assume we would like to find G(t1) = G() .
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According to the previous equation, G(t1) is defined as:
In the same fashion, G(t2) is defined as:
In table format, the correlation function values can then be written as given below.
The series of G(t) values is typically normalized, such that the value for very large time (as t
) is G() = 1. For standard light (with Gaussian statistics), this normalization imposes a
theoretically limit of G(t0) = 2 at t = 0. However, only carefully optimized optical systems can
ever achieve such a high intercept or amplitude.
For a random signal and a large measurement time, all G values would add up to roughly thesame number. The signal would then be described as uncorrelated, and the resultant intensity
correlation curve would look like random fluctuations about the baseline, similar to intensity
correlation curve for H2O shown in the following figure.
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For light scattered from diffusing molecules however, the intensity correlation curve exhibits an
exponential decay, indicating that the signal is correlated.
For typical diffusion processes the correlation function has the form of 1 plus an exponential
decay function. The decay constant, in the following expression, is representative of the
diffusional properties of the particle under examination. Hence, evaluation of leads to the
particle diffusion coefficient, which in turn is used to calculate the hydrodynamic radius via the
Stokes Einstein equation.
G(t) = 1 + exp(- t )
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What's the relevance of the frictional coefficient?
In the Stokes model for motion of a hard sphere through a viscous liquid, the resistance to
motion is related to the shearing force between adjacent solvent planes moving at differentvelocities. The shearing force per unit area (Fs/A) at each slippage plane is described with the
following differential equation, where m is the mass, A is the surface area, dv/dt is the
acceleration, is the viscosity of the medium, and dv/dy is the velocity gradient across the
shearing planes.
For stick boundary conditions, where the limiting assumption that the fluid layer adjacent to the
particle has the same velocity as the particle is made, solution of the above equation gives thetotal force of viscous resistance (Fvis) as shown below, where R is the particle radius and vs is
the limiting steady state velocity of the fluid when y becomes very large.
The proportionality factor in the above expression is defined as the frictional coefficient (f = 6
R), and it is the frictional coefficient that is used to calculate the radius of a particle from the
measured diffusion coefficient (D) in dynamic light scattering studies.
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What is the hydrodynamic radius (RH)?
In dynamic light scattering experiments, the radius (R) of the particle is calculated from the
diffusion coefficient (D) via the Stokes-Einstein equation, where k is the Boltzmann constant, Tis the temperature, is the solvent viscosity, and f = 6R is the frictional coefficient for acompact sphere in a viscous medium.
By definition then, the DLS measured radius is the radius of a hypothetical hard sphere thatdiffuses with the same speed as the particle under examination. This definition is somewhatproblematic with regard to visualization however, since hypothetical hard spheres are
non-existent. In practice, macromolecules in solution are non-spherical, dynamic (tumbling),and solvated. As such, the radius calculated from the diffusional properties of the particle isindicative of the apparent size of the dynamic hydrated/solvated particle. Hence theterminology, hydrodynamic radius.
The effects of asymmetry and hydration on the radius calculated from the Stokes-Einsteinequation can be examined using lysozyme. From the crystallographic structure, lysozyme canbe described as a 26 x 45 prolate ellipsoid with an axial ratio of 1.731, a MW of 14.7 kDa,and a partial specific volume of 0.73 mL/g. Considering only the mass and partial specificvolume (Vp), the equivalent radius for a sphere with the same mass as lysozyme (RMass) is 1.62nm, considerably smaller than the 2.25 nm rotational radius (RRot) determined by rotating the
protein about the geometric center.
The Perrin or shape factor (F) is defined as the ratio of the apparent frictional coefficient for anellipsoid to that of a sphere of identical volume, and can be used to account for the non-sphericalshape of lysozyme. For a prolate ellipsoid with an axial ratio of 1.731, the Perrin factor is1.022. As shown below, the non-spherical shape of lysozyme leads to an increase in theexpected radius of the particle, designated here as RVol.
The hydrodynamic radius (RH) includes effects arising from both shape and hydration. For
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lysozyme, the measured RH is 1.90 nm. The 2.4 increase in RH compared to RVol isconsistent with the thickness (2.8 ) of a single layer of water, suggesting that thehydrodynamic radius includes a single hydration layer.
The various size parameters discussed above are represented in the following figure. As seen
here, asymmetry and hydration lead to an increase in the measured size of lysozyme compared tothat calculated for a hypothetical hard sphere of the same mass. However, this measuredhydrodynamic radius is still 15 % smaller than the rotational radius.
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Is it better to use longer acquisition times?
The number of photons striking the detector during the course of the experiment is directly
correlated with the acquisition or sample time. The more photons collected, the less noise in theautocorrelation curve. Hence, longer acquisition times are synonymous with smoother
correlation curves and enhanced confidence in the experimental results. Because of the
correlation with photons collected, longer acquisition times are usually required at low sample
concentrations, where the signal to noise (analyte to solvent) intensity ratio is inherently small.
There is a down side to long acquisition times. At longer acquisition times the likelihood of a
dust event is significantly increased. If a dust particle enters the scattering volume during the
course of measurement, the correlation curve is shifted upward proportionally, and baseline
evaluation becomes problematic. As a rule of thumb then, you should keep your acquisition
time as short as possible; but keep an eye on the noise in the correlation curve and the baseline
values.
On the upside, noise in the correlation curve can also be reduced via statistical averaging, i.e. by
collecting multiple runs at short acquisition times. This approach to noise averaging is
particularly useful for samples that arent really as clean as they should be, even though
significant data outlier editing is typically required. The technical support staff at Protein
Solutions Inc is currently developing a DYNAMICS software utility that calculates the optimum
acquisition time and number of measurements for samples of a given molecular weight and
concentration. Check with your Protein Solutions sales representative for availability.
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What's the difference between the intensity, number, and mass distributions?
According to static light scattering theory, the scattering intensity of the ith particle (Ii) in a
distribution of particles is proportional to both the molecular weight (Mi) and the concentration
as given in the simplified form of the Rayleigh expression shown below, where Ci is the weight
or mass concentration, Ni is the number concentration, and K is a constant.
The size or radius (R) of the particle can be estimated from the molecular weight via the partial
specific volume (Vp) as shown in the following expression, where NA is Avogadros number
and x is a shape parameter equal to 3 for spheres and 2 for coils.
After substitution, the simplified Rayleigh expression can be re-written in the following form,
which relates the scattering intensity to the mass and number concentrations of the ith particle.
The mass and number distributions are then calculated from the intensity distribution as shown
below.
As seen in the following figure, the mean values for the %Mass and %Number distributions are
shifted towards lower values as a consequence of the Ri dependence in the above equations.
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What is the Rayleigh ratio?
The Rayleigh ratio is defined as the ratio of the scattered (Is) to the incident (Io) light intensity,
with the scattered intensity being summed over all space. Because of geometric limitationsimposed by the restriction of measuring the intensity at a given angle () and distance (r) fromthe scattering volume, the working expression for the Rayleigh ratio is that shown below, where
is is the scattering intensity measured at fixed r and .
The scattering intensity is dependent upon a number of solvent and particle parameters, principle
of which are the polarizability () of the particle and the permittivity (o) of the solvent. Hence,
the Rayleigh ratio can also be written as shown below, where is the particle density, NA is
Avogadros number, M is the particle molecular weight, and o is vacuum wavelength of theincident radiation.
Using the Clausius-Mosetti equation to relate the polarizability to the permittivity of a molecule,
the Rayleigh equation can be reduced to the commonly used form shown below, where C is the
weight concentration, M is the weight average molecular weight, A2 is the 2nd virial coefficient,
indicative of solute-solvent interactions, and K is the optical constant.
The vacuum wavelength, solvent refractive index (o), and analyte specific refractive index
increment (d/dC) are incorporated into the optical constant as shown below.
The above two equations are those typically used for molecular weight determinations of small
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particles in static light scattering experiments. All of the variables in the optical constant (K)
are either known or can be easily measured. As such, a plot of KC/Rvs. C in the absence ofparticle interactions should be linear, with an intercept equivalent to 1/M and a slope
proportional to the 2nd virial coefficient (A2).
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How is the Rayleigh ratio measured?
The Rayleigh ratio (R) is defined as the relative scattering intensity measured at an angle and
a distance r from the scattering center. As shown in the following equation, where is is the r anddependent scattering intensity and Io is the incident intensity, one has simply to measure is, Io,
r and to determine the Rayleigh ratio for a solution of particles. However, this simplicity is
strongly attenuated in practical application, where the measurement of Io, r, and cannot be
accomplished with the precision necessary for accurate determination of the Rayleigh ratio.
An alternate approach is to compare the scattering intensity of the sample analyte to that of a
well characterized standard, such as toluene, with a known Rayleigh ratio. The expression for R
can be rearranged to the form shown below.
If the scattering intensities of the sample and toluene are measured under the same experimental
conditions, the Rayleigh ratio of the analyte can then be determined from the ratio of intensities
as shown below, where the A and T subscripts represent the analyte and toluene respectively and
the terms are included to account for differences in the refractive indexes of the toluene and
sample solutions.
Since it is the photon count rate (CR) rather than the scattering intensity that is measured at the
detector in typical light scattering instrumentation, the expression for RA is converted to the
working expression shown below.
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What's the relevance of toluene in static light scattering?
In static light scattering experiments, the concentration dependent scattering intensity of the
sample solution can be used to determine the absolute molecular weight (M) and 2nd virialcoefficient (A2) of the analyte under examination.
In the above expression, K is an optical constant, incorporating wavelength and refractive index
effects, P() is the shape factor, accounting for angle dependent multiple scattering from large
particles, and Ris the Rayleigh ratio of the scattered intensity to the incident intensity. For
particles with diameters more than 10 times smaller than the wavelength of the incident
radiation, the shape factor goes to 1. The Rayleigh ratio is calculated using the followingexpression, where is is the residual scattering intensity (sample - solvent), Io is the incident
intensity, is the scattering angle, and r is the distance from the scattering volume to the
detector.
The above expression can be rewritten in the form shown below, where the sample dependent
parameters are grouped on the left hand side of the equality sign and the instrument dependent
parameters are grouped on the right.
The instrument dependent parameters (Io, , and r) are difficult to measure with the precision
required for acceptable molecular weight and 2nd virial coefficient calculations. Rather than
attempting to measure these parameters directly, the typical approach in static light scattering
experiments is to calibrate the instrument parameters using a standard with a well defined
Rayleigh ratio value. Toluene is one of the generally accepted standards for instrument
calibration in static light scattering experiments. If the scattering intensities of the sample and
toluene are measured under identical experimental conditions, the Rayleigh ratio of the analyte
can then be determined from the ratio of intensities as shown below, where the A and T
subscripts represent the analyte and toluene respectively.
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The terms are included in the above expression to account for differences in the refractive
indexes of the toluene and sample solutions. There is some debate as to whether or not therefractive index terms should be included. For low concentration aqueous protein solutions, the
inclusion of the terms functions to scale the toluene Rayleigh ratio by a factor of 0.8.
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Is the Rayleigh ratio of toluene wavelength dependent?
The wavelength dependence of the Rayleigh ratio (R) is embodied in the definition shown
below, where is the particle density, NA is Avogadros number, M is the particle molecularweight, o is the wavelength of the incident radiation, is the polarizability of the particle, and o is the permittivity of the solvent.
As evident in the above expression, the Rayleigh ratio should vary linearly with -4. Thistheoretical linear dependence is consistent will experimental results, exemplified in the
following figure, which shows the measured Rayleigh ratio of toluene at 25 oC and a 90o
scatting angle as a function of -4.
For the DynaPro line of molecular sizing instruments, the laser wavelength is around 830 nm.
The Rayleigh ratio for toluene at 830 nm and 25 oC is 4.23 x 10-6 cm-1. For other wavelengths,
the toluene R90 value can be predicted using the following expression, where the units for the
Rayleigh ratio and the wavelength are cm-1 and nm respectively.
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What are the units in static light scattering?
The key to simplifying the units in static light scattering experiments is to recognize that the
units must be the same on both sides of the equality sign. When the Rayleigh equation is writtenin the standard form shown below, it is easily seen that the collection of units on the left hand
side of the equation must ultimately reduce to units of inverse molecular weight, i.e. mol/g.
The reduction to mol/g can be accomplished using the units listed in the following table.
The dimensional analysis is shown below. As seen in the final expression, incorporation of the
units from the above table reduces the units in the Rayleigh equation to those of inverse
molecular weight on both sides of the equality sign.
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When is multi-angle data necessary for MW determination?
The expression used to describe the scattering of a dilute solution of particles is shown below,
where K is an optical constant, C is the particle concentration, Ris the Rayleigh ratio ofscattered to incident light intensity, M is the weight average molecular weight, A2 is the 2nd
virial coefficient, Rg is the radius of gyration, is the vacuum wavelength of the incident
radiation, and is the scattering angle.
The angular dependent portion of the second term arises from interference effects due to
multiple scattering from a single particle. For particles much smaller than the wavelength of the
incident radiation, this term goes to zero, and the angular dependence of the scattered light
vanishes. Under these conditions, the absolute molecular weight is determined from the
concentration dependence of the Rayleigh ratio, and angular dependent data is redundant. For
larger particles, it is still the concentration dependence that leads to the molecular weight, but
interference effects must be accounted for. It is at this point that multi-angle instruments
become necessary. In preface to the question of what is much smaller, the figure below, which
shows the dependence of the interference factor on the wavelength to particle size ratio at
different scattering angles is included here. As a rule of thumb, the size cutoff for angle
independent Rayleigh scattering is Rg 20*. Typical wavelengths for the DynaPro series of
instruments are on the order 800 nm, corresponding to an upper size limit for single angle
particle MW determination of circa 80 nm diameter.
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What is the baseline?
The baseline in the Cumulants Datalog is the measured value of the last coefficient in the
correlation curve. For ideal, noise free dynamic light scattering experiments the baseline shouldbe 1.000, indicating that there has been sufficient diffusion during the acquisition such that the
location of the particle is no longer correlated with its initial position. The DynaPro line of
molecular sizing instruments utilizes mono-modal fiber technology to minimize the introduction
of stray light or background noise. As such its not difficult to achieve baseline tolerances on
the order of 0.005.
Since the baseline value reported in the Cumulants Datalog is a measured, rather than a
calculated value, it is independent of the algorithm used to deconvolute the size and distribution
from the correlation curve. It is therefore a useful parameter for filtering out bad or outlying
correlation curves. One should remember however, that a good baseline value doesnt
necessarily mean the correlation curve is also good. Consider for example the following figure,which shows a 3 second acquisition time correlation curve for a dusty sample. The discontinuity
in the curve is indicative of number fluctuations caused by a dust particle passing through the
beam during the course of the acquisition. Even though this correlation curve is an obvious
outlier, the measured baseline is 1.000. Hence data filtering based solely on the measured
baseline value would leave this correlation curve unmarked, and lead to erroneous data
interpretation.
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What is the SOS?
The SOS is the sum of squares difference between the measured correlation curve and the best
fit curve calculated using the Cumulants method of analysis, where a dust and noise freemonomodal (single distribution) low polydispersity (narrow distribution) sample is assumed.
The SOS then is a quantitative value representative of the residual or the goodness of fit
between the measured and theoretical data (see figures below).
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In the DynaPro line of light scattering instruments, the SOS value for each correlation curve is
reported in the Datalog within the DYNAMICS software. Any deviation of the measured curve
from the theoretical will lead to increased SOS values. Sources of deviation could be noise,
dust, multi-modal samples, large polydispersity, etc. Hence, there is no absolute SOS value thatone should look for, particularly since the SOS value is also dependent upon the number of
correlation channels included in the calculations (see correlation channel cutoffs in the
DYNAMICS Data Filtering window).
The best way to use the SOS values in the Cumulants Datalog is to look for inconsistencies. As
an example: Because of background noise at low acquisition times, the SOS value for a dust
free sample might be on the order of 10 - 50. If dust passes through the beam during the
collection of a single correlation curve (during 1 acquisition time), the calculated SOS value will
jump by as much as 1 or 2 orders of magnitude. So the SOS value is a good indicator of
whether of not a dust free correlation curve has been collected. Depending upon how dirty, i.e.
dusty, the sample is, filtering the SOS values at around 500-1000 is typically sufficient to filterout bad correlation curves arising from the presence of dust during the acquisition. But these
numbers should not be taken as absolute or even as a rule of thumb. For longer acquisition
times, the cutoff could be much lower. In a like fashion, multi-modal samples could lead to
very large SOS values, as evident in the following figures.
The figures below show the Cumulants Datalog, the correlation curves for Measurements #1 and
#2, and the Regularization histograms for a PEGylated hemoglobin sample, before and after data
filtering based on inconsistencies in the SOS values. A discontinuity is evident in the
correlation curve for Measurement #2 (the same discontinuity is seen in Measurement #3),
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suggesting the presence of dust. The elevated SOS values in the Datalog for Measurements #2
and #3 are a consequence of the discontinuities in the correlation curves. Prior to data filtering,
the Regularization histogram indicates a bi-modal system. After Marking the dusty
measurements (#2 and #3), the Regularization histogram shows a mono-modal system with a
relatively low polydispersity.
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What are number fluctuations?
Number fluctuations are caused by a non-constant particle concentration in the scattering
volume. In light scattering experiments, the scattering intensity is proportional to both the MWand the concentration of the particles in the solution. As a consequence of Brownian motion,
the particles within the scattering volume are moving, and the recombined scattering light from
these moving particles leads to constructive and destructive interference effects or short time
scale total intensity fluctuations at the detector. In dynamic light scattering, these intensity
fluctuations are correlated and then transformed to give the diffusion coefficient and
subsequently the hydrodynamic radius. During the acquisition of the correlation curve however,
the particle concentration within the scattering volume must be constant. If the particle
concentration fluctuates, then the average total intensity will be skewed or weighted by the
variation in concentration, and the resultant correlation curve will be discontinuous (see figure
below).
Number fluctuations are particularly problematic with high sensitivity light scattering
instruments such as the DynaPro-MS800. In order to produce a high power density in the
scattering volume, the laser beam in the DynaPro-MS800 is focused down to a diameter of 16
m. While the increased power density increases the sensitivity of the instrument to sampleconcentration, the smaller scattering volume increases the likelihood of number fluctuations if
larger particles are present (see figures below). As a direct consequence of the smaller
scattering volume, the upper size limit for the DynaPro-MS800 is around 100 nm (diameter).
For the larger size range DynaPro instruments such as the MS/X and LSR, the laser beam
diameter is focused to 100 m. This larger beam diameter increases the scattering volume, and
while some sensitivity to concentration is lost, a much larger size range is allowed, with the
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MS/X and LSR instruments being able to accurately measure particle sizes up to 2 m diameter.
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What's the difference between the `old' and the `new' Regularization?
In DYNAMICS software versions 5.26.36 and greater, the Regularization algorithm has been
replaced with a new and improved Regularization algorithm. The fundamental difference in thetwo algorithms is that the regularization parameter in the new algorithm has been diminished
slightly to be consistent with the smoothing parameters in other correlation curve analysis
algorithms such as DynaLS and/or CONTIN.
In addition to improving the algorithm, Protein Solutions, Inc. has also added a resolution slider
to the Regularization histogram display in DYNAMICS. If the user has apriori information
concerning the number of distributions, the resolution slider can be used to diminish the
smoothing parameter even further and perhaps resolve multiple distributions from a single
peak. The figures below show the resultant Regularization histograms at various resolution
levels, using DLS data derived from a mixture of PEGylated and non-PEGylated chymotrypsin
in 0.1 M NaCl after incubation for 3 hours at pH 8.0. At the lowest resolution level, thecomponents are unresolved and the histogram shows a single, highly polydisperse peak
(ignoring the noise peak at 0.06 nm). The second figure shows the distribution at the resolution
level for the old Regularization algorithm, and indicates two populations, one with mean radius
of 3.58 nm and another with a mean radius of 25.5 nm. The former is consistent with the size of
the PEGylated protein measured in the absence of the non-PEGylated component. The third
figure shows the histogram distribution at the optimal resolution for the new Regularization
algorithm. For this system, the results are the same, regardless of which Regularization
algorithm is used. In the fourth figure, the resolution slider is set to the maximum level, and the
resultant histogram indicates three different populations. Subsequent chromatographic analysis
verified the presence of all three components, with the 1st peak being composed of
oligio-peptide by-products of chymotrypsin self-digestion, the 3rd peak being aggregates of thesame, and the middle peak being the non-digested PEGylated chymotrypsin.
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While the above example indicated no significant difference between the old and newRegularization algorithms, there are occasions when the two algorithms can yield what looks to
be different results. The figure below shows an example of this. The old Regularization
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algorithm indicates four populations. Because of the reduction in the smoothing parameter, the
new Regularization algorithm is able to resolve the third peak into two distinct populations,
yielding five peaks. However, if the resolution in the new algorithm is reduced by one
increment on the slider (Unresolved New in figure below), the results from the new algorithm
are consistent with those from the old.
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Do samples need to be filtered?
The scattering intensity of a sample is proportional to both the concentration and the molecular
weight of the particles in the solution. For high molecular weight particles, the contribution tothe total sample scattering intensity can be quite large. Hence, light scattering experiments are
extremely sensitive to dust. As a general rule of thumb then, samples should always be filtered.
The effects of dust on light scattering results are dependent upon a number of factors, primary of
which are the concentration and the molecular weight of the analyte. The figures below show
the DLS results and sample correlation curves for lysozyme solutions at 0.07 g/L (1st figure) and
0.7 g/L (2nd figure), both prepared from unfiltered 0.1 M NaCl. In the first figure, for the low
concentration sample, two measurements are marked as outliers (red). The correlation curves
for these two runs are shifted upward, indicative of a small number of very large particles, i.e.
dust, entering the scattering volume during the course of the measurement. In the second figure,
for the high concentration sample, no outliers are noted. While the dust particles are stillpresent, their contribution to the total scattering intensity is insignificant, compared to that of the
concentrated analyte.
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The figure below shows the DLS results for a high molecular weight 40 nm polystyrene
microsphere standard at low concentration in the same unfiltered 0.1 M NaCl. As with the
higher concentration lysozyme sample, the scattering contribution from the dust is insignificant
and no outliers are noted. In this sample however, it is the high molecular weight analyte that
diminishes the importance of removing dust via filtration.
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Will filtering perturb the sample?
Depending upon the size of the analyte and the size of the filter pore, filtering can indeed perturb
the sample. The figure below shows the DLS radius by Cumulants and DynaLS size distributionresults for an unfiltered poly(-cyclodextrin) sample. As evident in the DynaLS histogram, thesample is bi-modal (2 peaks) and polydisperse.
The concentration dependence of the measured hydrodynamic radius for the large particle isshown below. The solid and open symbols are used to indicate filtered (100 nm Anotop) and
unfiltered samples respectively. For concentrations 10.5 g/L, the measured radii are 61 and 75
nm respectively for the filtered and unfiltered samples. The elevated values at 20.95 g/L are
likely a result of interparticle interactions, which slow down the diffusion and lead to an
increase in the apparent hydrodynamic radius.
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The 20% difference in radii between the filtered and unfiltered samples is a consequence of the
removal of higher molecular weight particles during the filtering process. The figure belowshows the effects of filter pore size on the distribution of diameters for the poly(-cyclodextrin)sample. The dashed lines represent the pore cutoffs for the filters. As the filter pore size is
decreased, the larger polymers are clipped out of the solution. The result is a decrease in the
mean diameter of the larger particles and an increase in the contribution to the scattering
intensity attributed to the smaller particles at circa 6 nm. Hence the better resolution of the two
particle sizes with filtration.
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The size distributions shown previously are intensity distributions. If a flexible coil model,
where the molecular weight is proportional to R2, is assumed with regard to particle shape, the
intensity distributions can be converted to a %Mass distributions. The %Mass distribution for
the filtered 10.5 g/L poly(-cyclodextrin) sample (200 nm filter) is shown below.
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As evident in the above figure, only 7% of the total mass of the poly(-cyclodextrin) sample canbe attributed to the large particle. This system exemplifies the effects of large particle scattering
in dynamic light scattering studies. Even though there is only a small amount of the large
particle present, 97% of the total scattering intensity is attributed to that particle. In the absence
of filtering, information would have been lost, due to the inability to resolve the small, low
polydispersity particle from the larger and more polydisperse one.
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Why is the salt content relevant?
In dynamic light scattering experiments, it is the diffusion coefficient (D), rather than the
hydrodynamic radius, that is directly measured. The hydrodynamic radius is calculated fromthe diffusion coefficient via the Stokes-Einstein equation. Any factors influencing the
translational diffusion coefficient, whether or not they have any direct influence on the radius,
will have an apparent effect on the calculated radius. At high sample concentrations for
example, particle interactions can lead to a decrease in the diffusion coefficient and a subsequent
increase in the calculated radius. Electrostatic interactions between the particles can lead to
similar effects, even at low sample concentration. Electrostatic interactions can be shielded on
the other hand, by the addition of salt. Hence the relevance of salt content in dynamic light
scattering experiments.
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What is the Regularization Method?
Protein Solutions, Inc. would like to thank Dr. Maria Ivanova and Dr. Ulf Nobbmann for their
contribution of the following article.
In Photon Correlation Spectroscopy (PCS), single photons are detected and auto-correlated.
The autocorrelation function is the convolution of the intensity signal as a function of time with
itself. If the detected intensity is described as a function I(t), then the autocorrelation function of
this signal is given by
The above function is also called the intensity correlation function. The shift time is oftenreferred to as the delay time since it represents the delay between the original and the shifted
signal. In reality, the continuous intensity correlation function can not be measured. It can
however, be approximated with discrete points obtained by a summation over the duration of the
experiment, where the upper limit of the summation is given by the appropriate index belonging
to the largest available summation term for k.
The above general expression holds true for both linear and arbitrary delay times. Here, we arenot concerned with the specific technicality of obtaining the correlation function. That job is
left to the digital correlator, usually a plug-in card or an external USB device connected to a
personal computer. The correlator produces the normalized intensity correlation function and
makes the measured values available for further interpretation. The remainder of this article
then, focuses on the interpretation of the numerical values obtained from the correlator, namely
an array of intensity correlation function values Gk and the corresponding delay time values k.
If the intensity statistics of the measured signal are gaussian (which is true for all diffusion and
for most random processes) then the Siegert relation holds true. The Siegert relation states that
the normalized intensity autocorrelation function can be expressed as the sum of 1 and the
square of the field autocorrelation function g (scaled with a coherence factor expressing theefficiency of the photon collection system).
This equation can equivalently be written in the discrete form with the index k. For ideal
detection, the coherency factor () = 1. As a first approximation then, we can ignore the
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coherency factor, and develop a table of gk and k values.
Here the negative root value is only used for the unphysical case of G < 1 to approximate the
experimental noise of real data. This method imposes less bias than either restricting data to the
only the positive root or completely eliminating the outliers.
The ideal field correlation function of hypothetical identical diffusing spheres is given by a
single exponential decay function with decay rate determined by the diffusion coefficient and
the wave vector of the scattered light. The main objective of the data inversion consists offinding the appropriate distribution of exponential decay functions which best describe the
measured field correlation function. In mathematical circles this problem is known as a
Fredholm equation of the first kind with an exponential kernel. It is also known as an ill-posed
problem, since relatively small amounts of noise can significantly alter the solution of the
integral equation.
The fitting function for gk then consists of a summation of single exponential functions which is
constructed as a grid of exponentials with decay rate i .
The factor Ai is the area under the curve for each exponential contribution, and represents the
strength of that particular ith exponential function. The best fit is found by minimizing the
deviation of the fitting function from the measured data points, where a weighing factor k isincorporated to place more emphasis on the good, rather than the noisy, data points.
The weights are proportional to the intensity correlation function values, i.e. correlation functionvalues at small times have a higher weight than those at large times. Consider for example, the
following sets of figures. In the upper figure, the correlation curve baseline for measurement #1
is noisy. This noise is absent in the lower curve (measurement #12). In the absence of a
weighing factor, the noise could be interpreted as decays arising from the presence of very
large particles. As a direct consequence of the weighting factor however, the subtle variations in
the baseline for the upper correlation curve are correctly interpreted as noise by the
Regularization algorithm, and no significant differences are observed in the resultant size
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distributions for the two curves.
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The normal procedure to find the solution of Ais in the grid of fitted gk expressions is to
minimize the deviation 2 with respect to each Ai, and then find the solutions from the resultingsystem of equations. In other words, if we construct the solution out of N exponential functions
there will be N differentiations of the following form.
Each of the terms in the above differential contains a sum over k. The whole equation system
can therefore conveniently be re-expressed in matrix form as shown below, where the Y-vector
turns out to be a convolution of the experimental data with the kernel (= the grid matrix or the
exponential decays) and the matrix W consists of a convolution of the kernel with itself.
The standard procedure to solve this equation is to find the eigenfunctions and eigenvalues, and
construct the solution as a linear combination of the eigenfunctions. However, when the
eigenvalues are small, a small amount of noise can make the solution extremely big. Hence the
previous ill-posed problem classification. To overcome the problem, a stabilizer () is added tothe system of equations. This parameter is called a regularization parameter, and with the
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incorporation of this term we are performing a first order regularization.
The above expression is called a first order regularization because the first derivative (in Ai) isadded to the equation system. The regularization parameter determines how much emphasis
we put on this derivative or the degree of smoothness in the solution. If is small, it has little
influence and the solution can be quite choppy; whereas larger will force the solution to be
very smooth. It is this value or degree of smoothness that the user controls with theresolution slider in the new Regularization algorithm in the DYNAMICS version 6 software.
In addition to the smooth solution constraint, it is also required that the solution be physical; that
is, all Ai be non-negative and uniformly constrained. In short, these additional constraints
require that the summation of all Ai yields a finite number.
With the above constraints, Z is minimized by requiring that the first derivatives with respect to
Ai be zero. As indicated previously, this minimization corresponds to solving a system of linear
equations in Ai. The solution of Ai values is found using an iterative approach called the
gradient projection method.
A spherical particle with radius Ri will produce a correlation function with decay rate iaccording to the following expression, where D is the translational diffusion coefficient, q is the
scattering vector, kB is the Boltzmann constant, T is the absolute temperature, is the viscosity, is the solvent index of refraction, is the laser wavelength, and is the scattering angle.
The normalized display of Ai vs. Ri is the %Intensity distribution of the Regularization
histogram. The mean radius of a particular sub peak is the intensity weighted average, and is
obtained from the histogram using the following expression.
The polydispersity () (or standard deviation), indicative of the distribution in the sub peak, isalso obtained directly from the histogram.
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Can you get molecular weight information from DLS data?
The molecular weight of a particle can be estimated from dynamic light scattering data using
MW vs. RH calibration curves developed from standards of known molecular weight and size.However, because of the correlation between the specific volume (1/) of a particle and its
tertiary conformation, a universal calibration curve has yet to be developed, although numerous
calibration curves, each particular to a specific family or type of macromolecule are available.
Hence, in order to select an appropriate MW vs. RH calibration curve, some a prioriinformation
regarding the conformation of the particle must be assumed. The calibration curves included in
the latest development version of the Dynamics software package are shown below. For
reference purposes, it is noted that all of the proteins in the Protein family are globular;
Pullulans are linear polysaccharides; Ficolls are densely branched polysaccharides; and
Dendrimers are CO2 starburst type polymers, described as spherical, with a density that
increases with radial distance from the core.
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How is the axial ratio calculated?
The volume of a hypothetical sphere of homogeneous density can be calculated as shown below,
where Rsph is the spherical radius, MW is the molecular weight, V (bar) is the partial specific
volume (1/), and NA is Avogadros number.
The radius of the sphere can be written as follows.
Assuming stick boundary conditions, the frictional coefficient of this sphere (fo), can then be
expressed as
The above equation is also valid for non-spherical systems, with the exception that the measured
hydrodynamic radius (RH) is used, rather than the hypothetical spherical radius.
The Perrin or shape factor (F) then is defined as the ratio of the measured frictional coefficient
to the frictional coefficient for a hypothetical sphere of the same radius.
The above equation is that used in the Dynamics axial ratio calculator, where RH, Mw, and V
(bar) are input values and F is an output value.
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Once the frictional ratio is determined, the axial ratios for oblate and prolate ellipsoids of
revolution are interpolated from the data given in the following figure (see Cantor and
Schimmel reference).
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How is the mass distribution of radii calculated?
The mass distribution is calculated directly from the intensity distribution of radii via a
simplified form of the Rayleigh equation developed from static light scattering theory.According to static theory, the scattering intensity of the ith particle is proportional to both the
molecular weight (Mi) and the weight concentration (Ci) of the particle.
The molecular weight of the ith particle can be estimated from the expression shown below,
where vp is the partial specific volume and x is a shape parameter equal to 3 for spherical
particles and 2 for coils.
After substitution, the Rayleigh expression can be rewritten in the following form.
The %Mass of the ith particle is then defined as:
As evident in the above equations, the calculated %Mass is strongly dependent on the radius of
the particle. This dependence is exemplified it the following figures, which show the
Regularization distribution of radii displayed by %Intensity and %Mass for an 8000 molecular
weight poly(ethylene glycol) sample in 0.1 M NaCl. Comparison of the figures reveals a 10%
difference in the mean radius values for the two profiles, indicative of the 1/R2 dependence in
the %Mass calculations for coils.
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Are the %Mass calculations valid for all systems and conditions?
The complete expression describing the scattering intensity of a dilute solution of particles is
shown below, where Io is the intensity of the incident radiation with a vacuum wavelength of o, o is the solvent refractive index, d/dC is the particle refractive index increment, A2 is the
2nd virial coefficient, P() is a shape factor that ranges from 0 to 1, is the scattering angle, and
r is the distance from the scattering center to the detector.
In the %Mass calculations within the DYNAMICS software package, this equation is simplified
to the following form, where it is assumed that d/dCi is independent of particle size, A2Ci
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What's the significance of the refractive index () in DLS?
In typical dynamic light scattering studies, it is the autocorrelation function, an example of
which is shown below, that is measured during the course of an experiment. The correlationfunction can be fit to an exponential form, or more correctly, a summation of exponentials, with
the decay constants (i) containing the information needed to calculate the diffusion coefficients
(Di) of the system of particles under examination.
The diffusion coefficient is calculated using the expression shown below, where o is thesolvent refractive index, is the vacuum wavelength of the incident radiation, (= 1/) is the
inverse of the exponential decay time, and is the scattering angle. As seen here, the solvent
refractive index is introduced into the dynamic light scattering calculations through the
scattering vector (K).
A common misconception is that the subscript in o indicates that the refractive index to be
used is that for the pure/base solvent. Such is not the case. The scattering vector is dependentupon the dielectric properties of the medium, with o entering the expression via the
Clausius-Mosetti equation, which relates the polarizability to the permittivity of a molecule.
Hence, the refractive index that should be used is that for the medium within which the particle
is diffusing, i.e. the base solvent + any additives.
The CRC Handbook of Chemistry and Physicscontains a significant collection of temperature
and wavelength dependent refractive index values for aqueous mixtures of various
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Is the refractive index increment (d/dC) wavelength dependent?
In Chapter 6 ofLight Scattering From Polymer Solutions(Ed. M.B. Huglin; Academic Press:
New York, 1972), Huglin includes an extensive list of d/dC values (100 pages). According toHugglins empirical observations, the specific refractive index increment varies directly with -2. In order to verify this prediction, ten proteins were randomly selected from Huglins table
and checked. The figure below exemplifies the d/dC dependence on -2 for all of the proteins,with the BSA and Lactoglobulin data being the outer limits of the entire set of ten. As evident
below, the slope [(d/dC)/-2] is independent of protein type.
The effects of ionic strength on d/dC for BSA and -lactoglobulin at pH 7 are indicated in the
figure shown below. As seen here, (d/dC)/-2 is also independent of the ionic strength,although the addition of salt does lead to a decrease in the absolute d/dC value. Similar trends
were observed for all of the proteins in the original set of ten. (It is noted, that all of the proteins
are reported to be stable with regard to conformational changes under the cited conditions.)
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The independence of (d/dC)/-2 on protein type and solution salt content is likely a result of
the globular nature of the proteins examined. The refractive index is correlated with the densityof the media. For globular proteins, the specific volume (1/) is relatively independent of
protein type (vsp = 0.73 cm3/g). Hence the similarity between the d/dC and (d/dC)/-2values for the globular proteins examined here. As shown in the figure below however,
significant changes in both d/dC and (d/dC)/-2 are observed when the macromoleculesare coiled, rather than globular.
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Ideally, d/dC should be measured under the appropriate solution conditions and at the specified
wavelength, prior to beginning any static light scattering experiment. In the absence of theideal, one could estimate the protein d/dC value using the equation shown in the following
figure, where the symbols are the same as those used in previous figures. The equation is that of
the best fit line (black) for all ten of the protein data sets.
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The dashed lines are included in the above figure for estimating error. If solution conditions and
protein type are ignored, i.e. if the equation is used, the error in d/dC is 3%. If thewavelength dependence is ignored, i.e. via the use of a single d/dC value for all proteins underall solution conditions and wavelengths, the error in d/dC can jump to upwards of 10-15%.
With those observations in mind, it is instructional to consider the relative error in molecular
weight arising from d/dC error in static light scattering measurements. The Rayleigh equation
for static light scattering can be written as shown below, where is the solvent refractive index,
Ris the Rayleigh ratio of scattering intensities, is the wavelength of the incident radiation,NA is Avogadros number, A2 is the 2nd virial coefficient, and C is the analyte weight
concentration.
The derivative with respect to d/dC of the above expression can be written as follows.
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After rearrangement, the relative error in the molecular weight can then be expressed in the form
shown below. As evident in this equation, the relative d/dC error is amplified by a factor
proportional to the 2nd virial coefficient, the concentration, and the molecular weight, when
calculating the molecular weight from total intensity light scattering data.
Static light scattering measurements using ovalbumin, with M 43 kDa and A2 0.0005, havebeen conducted at Protein Solutions, Inc. The highest concentration was 0.02 g/L. Using these
parameters, the propagation factor (1 + 2A2CM) in the relative error equation is 1.0009 1.Under these conditions then, the relative error in calculated molecular weight is equivalent to the
relative error in the refractive index increment, implying that the 3% difference in d/dC for
globular proteins, due to different solvent conditions, is negligible. The 10 - 15% error in
d/dC arising from wavelength effects however, could be problematic, depending upon the
exact solution conditions employed and the properties (M and A2) of the particle under
examination.
Editors Note (10/13/00)
Please note that earlier versions of this document contained a unit error in the calculations of the relative error inovalbumin molecular weight and in the empirical equation for the wavelength dependence of d/dC. Wed like to
thank our friends at Wyatt Technology for bringing these mistakes to our attention (Thanks Ron - as always your
reviews and comments are greatly appreciated). The mistakes has been corrected, and the conclusions modified to
be consistent with the corrected value of the propagation factor in the relative error calculations. We apologize for
any inconvenience this oversight may have caused.
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Is the solvent refractive index (o) wavelength dependent?
The refractive index () of a medium is defined as the ratio of the speed of light in a vacuum to
that in the medium. As the light passes through an interface into a medium of higher density,say for example from a vacuum into a solvent, the decreased velocity of the radiation in the
solvent causes the light to be refracted or bent. According to Snells law, the magnitude of the
refraction, and hence the velocity of the radiation in the solvent, is dependent upon the
wavelength of the incident beam. Ergo, since the velocity of light in the solvent is wavelength
dependent, the measured solvent refractive index will be dependent upon the same.
On a molecular level, the decrease in the velocity of the radiation in the solvent arises from the
electromagnetic interactions of the light wave with the solvent molecules. If the solvent
molecules are more densely packed, the magnitude of the decrease in light velocity is
proportionate. Hence, the refractive index of the solvent is strongly correlated with the solvent
density. The dependence of the refractive index on density is described by the Lorentz-Lorenzformula shown below, where R is the molar refraction, M is the molecular weight, and is thedensity.
The figure below shows a compilation of refractive index values, extracted from the 77th edition
of the CRC Handbook for Chemistry and Physics, for pure water at different wavelengths and
temperatures. The linearity of the data at constant temperature is consistent with the Cauchy
expression, which predicts that the refractive index will be proportional to -2.
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As evident in the above figure, the solvent refractive index is also dependent on temperature.
This dependence arises from the temperature dependence of the solvent density. As the
temperature of the solvent is increased, the density decreases, leading to a subsequent increase in
the velocity of the radiation and a decrease in the measured refractive index. The equation
shown below, where T is the temperature in Celsius and is the wavelength in nm, is an
empirical expression developed from the above pure water refractive index data. While notgrounded in theory, it does simplify calculations of H2O for DLS experiments under normal
temperature and wavelength conditions.
Because of the wide use of toluene as a Rayleigh ratio standard in static light scattering
experiments, assorted temperature and wavelength dependent toluene refractive index values are
also included here. As evident in the figure below, Tol is proportional to -2, consistent with
the predictions of the Cauchy equation.
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Are o and d/dC temperature dependent?
Both the refractive index () and specific refractive index increment (d/dC) are closely
correlated with the density, and are hence sensitive to temperature. The correlation of therefractive index properties with density is evident in the figure below, which shows the
temperature dependence (0-25 oC) of d/dC for selected globular proteins. The similarity in the
(d/dC)/T tends is a direct consequence of the similarity in the partial specific volumes (1/)of the proteins.
The solvent refractive index is also temperature dependent, as evident in the following figures,
which show the reported (CRC Handbook of Chemistry and Physics) refractive index values for
pure water and toluene respectively, plotted against the square of the temperature (T2) in Celsius
at various wavelengths (). In the 1st figure, the slopes of the vs. T2 lines are independent ofwavelength, facilitating development of the empirical expression for H2O shown at the bottom
of the figure. The dependence of the refractive index on -2 is discussed elsewhere. As shownin the legend of the 2nd figure, the slopes of the vs. T2 lines are wavelength dependent for
toluene. Hence, development of an empirical expression relating the toluene refractive index to
both the temperature and the wavelength is more problematic.
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Is it alright to estimate solvent viscosity values?
The fundamental parameter extracted from the measured autocorrelation curve in dynamic light
scattering studies is the diffusion coefficient (D). The solvent viscosity () enters the picture via
the Stokes-Einstein equation for calculating the hydrodynamic radius (RH) of freely diffusing
spheres, assuming non-slip boundary conditions.
The standard error propagation expression for RH is shown below. Temperatures can be
measured quite precisely. Hence, T/T is negligible. If the questionable assumption that the
relative error in D (D/D) is negligible is also made, the best that can be expected is that the
%error in the radius is equivalent to the %error in the estimated viscosity value.
If the propagation of error ended here, small errors in the estimated viscosity would probably be
acceptable. Typically however, our clients also desire the particle molecular weight, which
cannot be directly calculated from the Stokes-Einstein equation and the measured diffusion
coefficient. As such, the molecular weight can only be estimated, using MW vs. RH calibration
curves developed from standards of know mass, size, and molecular conformation. Since these
calibration curves are empirical, error in the solvent viscosity is propagated even further, if
particle molecular weight is the desired parameter
With regard to the question then of whether or not it is OK to estimate solvent viscosity, the
answer depends upon how much error one is willing to accept in the target parameter and what
other a prioriinformation is available. If one it interested only in the radius of a system of
particles of singular size, then small errors in the viscosity might be acceptable. If however, the
solution is more complex or the target parameter is the estimated molecular weight, error in the
solvent viscosity becomes problematic. For example, the measured hydrodynamic radii for
dimeric and tetrameric insulin are 1.4 and 1.7 nm respectively. With only a 21% difference in
the size of the two inactive forms of insulin, a 10% error in the solvent viscosity can be fatal, if
one is attempting to determine the predominant form under a given set of solution or preparatory
conditions.
The CRC Handbook of Chemistry and Physicscontains an enormous collection of temperature
dependent viscosity values for solvents of various composition. The figure below shows a
compilation of 20 vs. W% (in water) data for some typical additives, extracted from the 77th
edition of the CRC. The empirical form of the 20 dependence on W% for all of the additives
presented here is a 2nd order polynomial, the coefficients of which are given in the figure legend.
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What is the _____ value of ______?
A number of solvent parameters, such as the viscosity and the refractive index, are incorporated
into the calculations in dynamic and static light scattering studies. Depending upon theconditions, error in the values for the solvent parameters can lead to significant error in the
results. Hence, questions of the form presented above are common. It would be impossible to
address all the permutations; but we can certainly try. The figure below shows a screen shot of
the new Archive utility developed at Protein Solutions, Inc. While it isnt as complete as the
CRC Handbook of Chemistry and Physics, it is always available and you can edit and/or add
your own information. This utility is currently available for download at the Protein Solutions,
Inc. web page, and will be included in the Dynamics 6.0 package.
ArchiveSetup Readme.txt