9/14/2012
pH
Dependence of Bovine GammaB Crystallin 2nd Virial Coefficient
Kaho Long
Advisors: Dr. George M. Thurston, Dr. John Andersen
Rochester Institute of Technology, Department of Physics, Rochester, NY, 14623
Abstract
We used static light scattering to measure the 2nd
virial coefficient of the eye lens protein, bovine
γB-crystallin, which exhibits liquid-liquid phase
separation related to cataract disease. The
measured molecular weight was (2.3 x 104 ± 1.2 x
103) grams/mole (mean ± std. dev.), within two
standard deviations of the sequence value,
20992.7 grams/mole. At pH 4.4, where no phase
separation occurs, we measured the repulsive,
dimensionless 2nd virial coefficient, B2= 8.1 ± 1.0,
exceeding the values of 4 and 5.4 that would
correspond to hard spheres and hard sphere
dimers, respectively. At pH 5.4, our
measurements suggest B2<0, consistent with net
attractive interactions and phase separation at
this pH, though higher concentration
measurements are needed to quantitate B2 at this
pH. These measurements suggest bovine γBcrystallin has a crossover from attractive to
repulsive interactions near pH 5.
Results & Analysis
Figure 3A
Nuclear and cortical material from calf lenses.
Homogenated and centrifuged cytoplasm.
Size, ion exclusion chromatography to purify γB.
Concentrations measured by UV spectroscopy.
Acknowledgements
I would like to thank Dr. George M. Thurston for being
there to answer my questions and providing support in all
manners, Dr. John Andersen for his help in teaching me the
statistical mechanics, Dr. David Ross for lending a helpful
ear, Rafael Herschberg my habitual lab partner, and the
Department of Physics of RIT for providing me this
opportunity.
References and Further Reading
[1] Peter Schurtenberger, Richard A. Chamberlin, George M. Thurston, John A.
Thomson, and George B. Benedek. Observation of critical phenomena in a proteinwater solution. Physical Review Letters., 63:2064–2067, 1989.
[2] W. Kaye and J. B. McDaniel. Low-angle laser light scattering- rayleigh factors and
depolarization ratios. Applied Optics, 13:1934–1937, 1974.
[3] Aleksey Lomakin, Neer Asherie, and George B. Benedek. Monte carlo study of
phase seperation in aqueous protein solutions. Journal of Chemical Physics.,
104:1646–1656, 1995.
[4] D.A. McQuarrie. Statistical Mechanics. Harper and Row, 1976.
[5] B. L. Neal, D. Asthagiri, and M. A. Lenhoff. Molecular origins of osmotic second
virial coefficients of proteins. Biophysical Journal, 80:613–625, 2001.
[6] C. Tanford. Physical Chemistry of Macromolecules. Wiley, 1961.
Figure 3B
Figure 1
Dimensionless
Second virial
coefficient of 0
pH 5.4,
negative
dimensionless
Second virial
coefficient
Because the protein is shaped like Figure 3B where the red and blue respectively represent –kB T and
+kB T contours of screened potential near γB-crystallin protein at pH 7, we have to consider
evaluating this expression fully :
1 2π π 2π 2π π ∞
2


pH 4.5,
Dimensionless
Second virial
coefficient of 8
B2 (T ) = −
16 π 2
∫ ∫ ∫ ∫ ∫ ∫ (exp−βW ( r,θ , φ, α, β, γ ) −1)r drsinθ dθ dφ dα sin β dβ dγ
0 0 0 0 0 0
We instead began with a more simple model that is illustrated in Figure 3A; we must walk before we
run. We postulated that the 2nd virial coefficient, which is a function of the interaction, can be
expressed using the probabilities fa and (1- fa) that patch types A and B face one another. The
expression then becomes:
2
2
 
βE
βE
βE

 3
B2 (T ) = 41− ( fa ) ( e

AA
−1) + (1− fa ) ( e
BB
−1) + 2 (1− fa ) ( fa ) ( e
AB
−1)( λ −1)

where fa is the fraction of the surface covered with charge property A, and EAA, EBB, and EAB are the
energies associated with A facing A, B facing B, and A facing B.
Materials & Methods
•
•
•
•
Simplified Patchy Square Well
Measured Dimensionless Second Virial Coefficient
∆<I>
 n 
2
The excess Rayleigh Ratio was calculated using ∆R = < Iref > Rref  nref  , where < I > is the average
count rate of the light scattered by the sample minus that of the buffer, and < Iref > is the
average count rate of the light scattered by a toluene sample minus the dark count.
Kc
1
The curve in Figure 1 is given by ∆R = M (1+ 2B2φ + ) where B2 , the dimensionless second
w
virial coefficient, is proportional to the negative of the second derivative of the excess
Rayleigh ratio with respect to the concentration. That means positive curvature gives a
negative 2nd virial coefficient, while negative curvature gives a positive 2nd virial coefficient.
Dimensionless Second Virial Coefficient for Square Well
Figure 2
The dimensionless virial coefficient for the square well
potential following the same assumptions that Lomakin
et al. made in their paper is[3]:


B2 ( T ) = 41− ( e −1) ( λ −1)
.
In the patchy interaction model we will adopt, the well
depth depends on which patches are facing one
another. For an attractive interaction, like with unlike,
there is a negative well depth and for a repulsive
interaction, like with like, there is a positive well depth.
βE
3
Calibrated Patchy Square Well
We first assumed that λ would be
the same as the value determined by
Lomakin et al. We then assumed
that the effective like-like and likeunlike interactions could be given a
nearly common value, close to kB T,
where |EAA| = |EBB | = |EAB |. The
equal coverage of positive and
negative patches at pH 7.1
simulated by Martini et al,
combined with the pH 7.1 2nd virial
coefficient, fixes |EAA| = 1.95. Then,
keeping the value of |EAA|, we note
that at nearly 100% coverage with
one charge type, the model predicts
a 2nd virial coefficient of 7, in
agreement with our measured value
of (8.1 ± 1.) at pH 4.5 where almost
all of the surface of the protein is
positively charged.
Figure 4
Measured
value of 8.1
±1.0
Hard
Sphere
value of 4
Crossover
value
of 0
Lomakin et
al. value of
-5.72
Conclusions
- Measured molecular weight (22659 ± 1220) (grams/mole)
- Measured a dimensionless second virial coefficient at pH 4.5, B2 = (8.1 ± 1.)
- At pH 5.4, negative second virial coefficient, crossover pH between pH 5.4 and 4.4
- Crossover pH may have well depth 1.93kbT and the fraction of surface with charge property A is near 83%
- Models used were simple, experiments ran into roadblocks with filters and time, hopefully further research will bypass those blocks
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