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Colloidal dispersions of α-crystallin proteins. II.
Dynamics : a maximum entropy analysis of photon correlation spectroscopy data
P. Licinio, M. Delaye, A.K. Livesey, L. Léger
To cite this version:
P. Licinio, M. Delaye, A.K. Livesey, L. Léger. Colloidal dispersions of α-crystallin proteins. II.
Dynamics : a maximum entropy analysis of photon correlation spectroscopy data. Journal de Physique, 1987, 48 (7), pp.1217-1223. �10.1051/jphys:019870048070121700�. �jpa-00210546�
Colloidal dispersions of 03B1-crystallin proteins. II. Dynamics : a maximum entropy analysis of photon correlation spectroscopy data
P. Licinio (a,**), M. Delaye (a), A. K. Livesey (b,*) and L. Léger (c)
(a) Laboratoire de Physique des Solides, Université Paris-Sud, 91405 Orsay Cedex, France
(b) Lure, Bât. 209 C, Université Paris-Sud, 91405 Orsay Cedex, France
(c) Collège de France, Laboratoire de Physique de la Matière Condensée, 11, pl. M.-Berthelot, 75231 Paris Cedex, France
(Reçu le 2 fovrier 1987, accepté le 12 mars 1987)
Résumé. 2014 Nous avons étudié par spectroscopie par battements de photons la dynamique de dispersions
colloidales de protéines 03B1-cristallines. Cette étude a été faite dans la limite des petits vecteurs d’onde q (c’est-à-
dire pour q ~ qm, où qm désigne la position du pic primaire du facteur de structure statique) et pour des concentrations allant jusqu’à c = 0,280 g/cm3. Nous avons utilisé la méthode d’entropie maximum pour extraire de la fonction de corrélation de la lumière diffusée la distribution de temps de relaxation de la
dispersion. Nous pouvons résoudre au moins 3 composantes diffusives. La plus rapide correspond au processus de diffusion mutuelle. La suivante peut être identifiée à une composante de self-diffusion puisqu’elle correspond au coefficient de diffusion d’un marqueur, coefficient mesuré directement par recouvrement de fluorescence après photoblanchiement d’un réseau de frange. La 3e composante peut être interprétée simplement comme la diffusion d’« amas » supramoléculaires réversibles dont la taille serait, dans la gamme de concentration où cette composante a été détectée (0,100 à 0,280 g/cm3), 10 à 100 fois plus grande que les 03B1- cristallines. En résumé, les résultats décrits dans cet article devraient permettre une analyse systématique des
divers phénomènes dynamiques et en particulier des effets hydrodynamiques à N corps.
Abstract. 2014 Using photon correlation spectroscopy, the dynamics of colloidal dispersions of 03B1-crystallin proteins is investigated in the low q limit (i.e. for wave-vectors q ~ qm, where qm is the maximum of the static structure factor) up to a concentration c = 0.280 g/cm3. The maximum entropy method is then used to extract from the intensity correlation function of the scattered light the distribution of relaxation times of the
dispersion. At least three components can be unambiguously resolved, which are shown to be diffusive. The fastest one corresponds to the process of mutual diffusion. The next one is identified as self diffusion by comparison with the tracer diffusion coefficient measured directly by fluorescence recovery after fringe pattern photobleaching. The third one can be most simply interpreted as the diffusion of reversible supramolecular
clusters whose size would be 10 to 100 times larger than 03B1-crystallin particles in the concentration range where this third component has been detected (0.100 to 0.280 g/cm3). In summary, the results outlined here should allow a systematic investigation of various dynamic phenomena in particular many body hydrodynamic effects.
Classification
Physics Abstracts
82.70 - 36.20 - 05.40
1. Introduction.
In the first part of this article [1], it was shown that
the structure of colloidal a-crystallin dispersions can
be adequately described by a liquid state formalism
with simple « direct » interactions between the colloi-
dal particles. However, the dynamics of colloidal
dispersions are more complex than those of one-
component liquids because they involve indirect
hydrodynamic interactions transmitted via the veloci- ty field of the solvent. In dilute dispersions, these hydrodynamic interactions are dominated by two body interactions and are thus quite easily described [2, 3]. However, in concentrated dispersions, many-
body effects become overwhelmingly important, making the description of hydrodynamic interactions much more delicate. Recent theoretical advances have been devoted to the systematic treatment of
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048070121700
1218
such many particle effects [4-6]. These references
have allowed the calculation, for hard sphere or
screened Coulombic interactions and over an inter- mediate time range (the time-scale over which the
spatial configuration of the particles does not apprec- iably change) [7], of the variation with concentration of dynamical properties such as the mutual diffusion and the self-diffusion coefficients [6, 8-9]. These new
theories need to be tested experimentally with well
controlled colloidal systems.
Experimentally, scattering techniques such as photon correlation spectroscopy (PCS), fringe pat-
tern photobleaching recovery or spin echo neutron scattering are currently used in colloid science to measure mutual and/or self-diffusion coefficients. c-
crystallin proteins have been shown to preserve a well-defined quaternary structure and to give stable dispersions over a very large concentration range. In addition, their « direct » interactions have been
explored experimentally [1]. These particles, there- fore, provide us with a well controlled system to characterize dynamic properties of colloids.
In this article we present a PCS study of a- crystallin dispersions performed in a wave-vector range (q) well below the maximum (qm) of the static
structure factor S(q). These dispersions were pre-
pared at 2 ionic strengths and for concentrations up to 0.280 glcm3. As c increases, the temporal corre-
lation detected in the scattered intensity extend over longer and longer time ranges. We analyse these
correlations using the maximum entropy method
(MEM), whose application to PCS has already been comprehensively described in reference [10]. This
method allows us to extract the mutual and self diffusion coefficients and their c-variation. In addi- tion to these rapid diffusive components, slower components are found at large volume fractions in the PCS autocorrelation functions.
2. Experimental procedure
2.1 PREPARATION OF COLLOIDAL a-CRYSTALLIN DISPERSIONS. - The preparation of a-crystallin dispersions at two different ionic strengths (I) has
been described in detail in the previous article [1].
PCS experiments are very sensitive to the presence of dust particles or large molecular aggregates in the dispersion. Each preparation was thus ultracen- trifuged for 30 min at 6 000 g prior to experiments.
2.2 PHOTON CORRELATION SPECTROSCOPY (PCS).
- The PCS set up is classical and has been described in reference [11]. An Argon laser beam (Ao =
5 145 A, SPECTRA PHYSICS 165) is passed through a spatial filter and focused on the glass tube containing the a-crystallin dispersion (beam width
= 200 J.L, incident power = 40 mW). The sample
tube is mounted in an index matching fluid at the
centre of a temperature regulated (20.5 °C ± 0.1 °C )
cylindrical glass vessel. A beamstop prism is placed
inside the vessel to prevent laser reflection at the output. The optical detector (imaging lens pinhole (=z 1 coherence area) + photo-multiplier) is mounted
on a goniometric arm rotating around the center of
the sample. The scattering angle 0 was varied from
30° to 150° corresponding to a q range
from 8.8 x 104 cm-1 to 3.3 x 105 cm-1 for a disper-
sion of concentration c = 0.152 glcm3 and of refrac- tive index n = 1.361.
The photo-pulses are fed through a pre-amplifier
into a digital autocorrelator (ATNE = Applications Technologiques Nouvelles en Electronique) with scaling and clipping devices. The correlator builds,
in real time, the autocorrelation function
of the scattered intensity i (t ) over 48 linearly spaced
channels at t = p. At with p = 1 to 48 and At the
chosen sample time. However, for concentrations greater than a few %, the relaxation times span too great a range to be measurable with only 48 linearly spaced channels. Consequently C (t ) are successively
accumulated (up to = 107 counts) for a series of sample times At = 2k Ato and then matched together (least squares affinity and translation) wich gives an approximation to equal log t sampling. Ato is typically
chosen as 2.5 >s. The integer k is increased up to 6
(for concentrations c of a few %) or even to 19 (at large c) so as to ensure that all decays have effectively died out. Finally the matched C (t ) curve
is plotted as a function of log t.
2.3 MAXIMUM ENTROPY DATA ANALYSIS. - In the data analysis, we assume that the scattered field autocorrelation function 91 (t) can be simply written
as an integral over a distribution of relaxation times :
This rough assumption presents the advantage of isolating various modes of relaxation but does ignore
the possible coupling between these modes. Under this assumption, with an « homodyne » detection
and a Gaussian field [12], the intensity correlation function C (t) becomes:
From the experimentally measured C (t ) we wish to
recover the distribution G (T ) of relaxation times T
describing the dynamics of our colloidal dispersion.
Since our measured data set, C (t ), is noisy and finite
in extent, there is strictly an infinite set of G ( T )
which agree with our data within the error bars. We then choose that distribution which maximizes the
Shannon-Jaynes entropy
We have shown [10] that this choise has many
compelling features : it introduces minimum correla- tions in G (T ), and G ( T ) is smooth, positive, unique
and robust to noise. In order to ensure our recovered distribution agrees with our data, we maximize (4) subject to the constraint that the Laplace transform g (t) (Formula (2)) obeys the following chi-squared
constraint
where
9" : is the calculated Laplace transform of G ( T ) at
the k-th observed time
gk : 0 is the observed value and o-k 2 its variance M : is the number of observations.
We obtain gk from C (tk) by
This requires a good estimate of B which was
obtained by measuring C (t ) to sufficiently long
times that all decays have effectively died out. The
very non-linear nature of the entropy function enables us to efficiently refine B if so required.
2.4 FRINGE PATTERN PHOTOBLEACHING RE- covERY. - To measure directly the self-diffusion coefficient D, (low q limit), we performed photo- bleaching recovery experiments at ionic strength
I = 150 mequiv for various values of c. The a-
crystallin samples were supplemented with about 5 % of a-crystallins labelled with fluorescent molecu- les. These tracers were obtained as follows :
a) incubation of 50 mg of a-crystallins dispersed
in 90 ml of phosphate buffer (pH 6.8, 150 mequiv
with 19 mg FITC (fluorescein isothiocyanate, SIG-
MA Chemical Company) in presence of 180 mg of celite (incubation time 10 min),
b) centrifugation for 30 min at 4 000 g to eliminate celite,
c) 8 times concentration with a DIAFLO YM 30 ultrafiltration membrane (AMICON Corporation),
d) gel filtration chromatography on a Biogel P2 (200-400 Mesh) BIORAD Laboratories) to separate the FITC - a-crystallin tracers from the unbound-
ed FITC. Under these conditions, fluorescence in-
tensity measurements indicated the presence of about 18 FITC molecules per o-crystallin tracer.
Photobleaching is produced and detected using an
interference pattern with a 74 tim fringe spacing
over the 1.5 mm diameter of an argon laser beam
(4 880 A, Spectra Physics 165). An intense (300 mW
incident power) interference pattern is first used during 0.5 s to bleach locally a fraction of the FITC molecules. A similar pattern but attenuated by a
factor 104 is then applied, and let oscillate spatially (in order to vary its phase relative to the bleaching pattern) at a frequency of 1.0 kHz. The optical signal produced by the fluorescence of the FITC molecules thus oscillates at twice the frequency of
the grid until bleached and non-bleached molecules
are uniformly mixed by diffusion (convection was
avoided by the use of a thin cell and a « horizontal »
geometry). Under these conditions, the amplitude of
the oscillating part of the optical signal is expected to decay exponentially with a rate proportional to the
self-diffusion coefficient. Details of the experimental
set up can be found in reference [13].
3. Results
3.1 INTENSITY CORRELATION FUNCTIONS C (t). - Figure l(a, b, c, d) shows on a semi-logarithmic plot,
a series of C (t ) curves corresponding to increasing
concentrations c, for a-crystallin dispersions of equal
ionic strength 1=17 mequiv, all measured at
(J = 90°. Figure 2(a, b, c, d) shows the effect of
decreasing the wave-vector q at constant ionic strength (I = 17 mequiv ) and concentration (c =
0.152 glcm3). This latter series enables us to deter- mine whether or not each peak results from a
diffusive process.
Figure 1a (c = 0.01 g/CM3) is very similar to a
single exponential decay and can be correctly fitted (giving random residuals) with a single exponential decay
where a is the amplitude corresponding to the single
relaxation time T 1 and b is a static stray light component [14].
The best least-squares fit to C (t ) gives
T = 82.7 iLs with a low stray light intensity (b/a = 0.04) suggesting that the centrifuge had sufficiently removed dust particles and other con-
taminants. Providing T 1 is measured at a concen-
tration low enough to make interactions negligible
one can deduce from T the hydrodynamic radius of
the a-crystallins, Rh, using the Einstein formula :
where
7J : is the solvent viscosity kB : is the Boltzmann constant T : is the absolute temperature
1220
Fig. 1. - Intensity correlation functions C (t ) as measured
from PCS (dots), and relaxation time distribution G(r) as
recovered by MEM (continuous curve), from a-crystallin dispersions at I = 17 mequiv, 0 = 90° scattering angle
and various concentrations c. a) c = 0.01 glcm3 ; b)
c = 0.152 g/CM3 ; c) c = 0.226 g/CM3 ; d) c = 0.280 g/CM3.
The time scale is logarithmic.
D : is the particle diffusion constant and
q : is the scattering wave-vector
After proper extrapolation to zero concentration
Rh is found to be = 100 A. In fact this hydrodynamic
radius is slightly larger than that of the average a-
crystallin particle since scattering measurements do favour large particles. For a size polydispersity of
- 0.10 the correction was estimated to be - 5 A, leading to an hydrodynamic radius Rh = 95 A (i.e.
D - 2.24 x 10 -7 cm2/s) for the average a-crystallin particle.
The other correlation functions (Figs. Ib to Id)
are much broader and suggest there is much more
structure in the correlation time distribution. For
example figure lc shows temporal correlations ex-
Fig. 2. - Intensity correlation function C (t ) as measured
from PCS (dots), and relaxation times distribution
G (T ) as recovered by MEM (continuous curve) from a- crystallin dispersions at I = 17 mequiv ; c = 0.152 g/cm 3
and various scattering angles 8. a) 0 = 150° (q = 3.21 x 105 CM- 1) ; b) 0 = 90° (q = 2.35 x 105 CM- 1); c) 6 = 60°
(q = 1.66 x 105 CM- 1) ; d) 0 = 40° (q = 1.14 x 105 cm-1). The time scale is logarithmic.
tending over 6 decades with at least one shoulder
clearly visible in the C (t) profile. In such cases,
neither formula (7) nor a two exponential curve
could fit the data within the experimental precision.
Instead, we used the maximum entropy method
(MEM) which enables us to recover a distribution of relaxation times G (T ) which correctly describes the
decay of the autocorrelation function C (t).
3.2 DISTRIBUTION OF RELAXATION TIMES. - The distribution of relaxation times G ( T ) recovered by
MEM is plotted on the same graph as the correlation function C (t ) from which it is derived (Figs. la to Id
and 2a to 2d). Each resolved peak was characterized
by its average (in log space) decay time To i taken
between two consecutive minima and its relative
amplitude ai, taken as the area under the peak i,
divided by the area of the first peak (al = 1).
At low concentrations (c = 0.01 g/CM3 ), G ( T ) can
be described by a single peak with 7-1 = 85.4 iLs in agreement (within a 3 % precision) with the fit to formula (7) above. The width of the peak reflects
both the intrinsic spread of relaxation times resulting
from the polydispersity of the solution together with broadening or smoothing effect of MEM arising
from the finite extent of and the experimental
uncertainties in the measured autocorrelation func- tion.
With increasing concentration more peaks are
resolved in the distribution of decay times. At
c = 0.226 glcm3 (Fig. lc), f.e., 3 clearly-resolved peaks can be seen at T = 36.0 fJ.S, T2 = 1.53 ms and T3 = 40.6 ms. As the concentration increases
(Fig. 1d, c = 0.280 g/cm3), the relative amplitudes
a2 and a3 of the longer relaxation times T2 and T3 continue to increase. These peaks move to longer
relaxation times (in contrast to T 1 which moves down
to a shorter relaxation).
When the wave-vector q is decreased at constant concentration (c = 0.152 g/cm 3 , I = 17 mequiv, Fig. 2) the 3 peaks, Tl, T2 and T3 are found to shift
simultaneously towards longer times while further
peaks appear (Fig. 2a to 2d). Nevertheless the
amplitude a2 remains essentially constant in the q- range investigated while a3 progressively increases.
Fig. 3. - Successive relaxation times of the G ( T ) distribu-
tions as a function of wave-vector q for an a-crystallin dispersion with c = 0.152 g/CM3 and I = 17 mequiv. Both
scales are logarithmic. The 3 components Ti, z2 and T3 are seen to be diffusive (slope = - 2 for the log-log plots of T versus q). The slowest relaxation times,
T4, are only detected at small angles 0.
The evolution of these relaxation times as a
function of q is shown in a logarithmic scale in figure 3. It is clear that T1, T2 and T3 vary as
q-2 suggesting that they arise from diffusive proces-
ses. Diffusion constants Di = 1/Ti q2 were then
found to be
The q-dependence of the slowest decays, T 4’ could not be fully characterized, as a result of the limited
orange at which these fast growing components are unambiguously detected. The few data presented here, however, would suggest that T4 is independent
of q.
PCS experiments on a series of a-crystallin dispersions at a higher ionic strength (I =
150 mequiv) were similarly analysed by MEM. The
relaxation time distributions G ( T ) display qualitat- ively the same overall structure as the 17 mequiv.
series. However it is noticeable that the decrease of T1 1 with the concentration c is much smaller for I = 150 mequiv than for I = 17 mequiv whereas the T2 and T3 peaks appear at similar positions for both
ionic strengths.
3.3 FRINGE-PATTERN PHOTOBLEACHING. - The self-diffusion coefficients, Ds, directly obtained from
fringe-pattern photobleaching experiments are given
in table I for 3 a-crystallin concentrations at I = 150 mequiv and T = 20 °C. The diffusion constants obtained from the series of PCS experiments at the
same ionic strength and temperature are also tabu- lated for comparison. D2 and Dg s agree within
Table I. - Comparison o f the various diffusion coeffi-
cients measured at different concentrations c , for a-cris-
tallin dispersions at 1 = 150 mequiv. Ds is the tracer diffusion coefficient directly measured by fringe pattern
photobleaching. V1, D2 and D3 are the diffusion coeffi-
cients deduced.from PCS data and corresponding res- pectively to the first (T,), second (i2) and third (z3) diffusive component.
Note : The error bars estimates in D1, D2 and D3 simply reflect
the widths of the peaks as recovered by MEM and are not statistical confidence limits.
