doi:10.1016/j.jmb.2003.12.019
J. Mol. Biol. (2004) 336, 43–59
Mechanisms of Homogeneous Nucleation of Polymers
of Sickle Cell Anemia Hemoglobin in Deoxy State
Oleg Galkin and Peter G. Vekilov*
Department of Chemical
Engineering, University of
Houston, Houston, TX
77204-4004, USA
The primary pathogenic event of sickle cell anemia is the polymerization
of the mutant hemoglobin (Hb) S within the red blood cells, occurring
when HbS is in deoxy state in the venous circulation. Polymerization is
known to start with nucleation of individual polymer fibers, followed by
growth and branching via secondary nucleation, yet the mechanisms of
nucleation of the primary fibers have never been subjected to dedicated
tests. We implement a technique for direct determination of rates and
induction times of primary nucleation of HbS fibers, based on detection
of emerging HbS polymers using optical differential interference contrast
microscopy after laser photolysis of CO-HbS. We show that: (i) nucleation
throughout these determinations occurs homogeneously and not on
foreign substrates; (ii) individual nucleation events are independent of
each other; (iii) the nucleation rates are of the order of 106 – 108 cm23 s21;
(iv) nucleation induction times agree with an a priori prediction based on
Zeldovich’s theory; (v) in the probed parameter space, the nucleus
contains 11 or 12 molecules. The nucleation rate values are comparable to
those leading to erythrocyte sickling in vivo and suggest that the
mechanisms deduced from in vitro experiments might provide physiologically relevant insights. While the statistics and dynamics of nucleation
suggest mechanisms akin to those for small-molecule and protein crystals,
the nucleation rate values are nine to ten orders of magnitude higher than
those known for protein crystals. These high values cannot be rationalized
within the current understanding of the nucleation processes.
q 2003 Elsevier Ltd. All rights reserved.
*Corresponding author
Keywords: sickle cell anemia; hemoglobin S polymerization; fiber
nucleation; homogeneous nucleation rate; nucleus size
Introduction
Sickle cell anemia, or homozygous sickle cell
disease, is a genetic disorder, which affects about
250,000 children worldwide every year; many of
them have multiple strokes and die before they
are two years of age.1,2 The disease is caused by an
A-to-T mutation within the sixth codon of the
b-globin coding region.3 As a result, the glutamic
acid residues at the sixth position of the two
b-subchains are replaced by non-polar valine.4 – 6
Unlike the normal hemoglobin (Hb) A, the
mutated hemoglobin has a high propensity to
polymerize when in the tense (T) state in the
venous circulation.7 – 11 The polymerization alters
Abbreviations used: Hb, hemoglobin; DIC, differential
interference contrast.
E-mail address of the corresponding author:
[email protected]
the shape and rigidity of the red blood cells,12 – 15
and triggers a sequence of pathogenic
consequences.15 – 17
Polymerization of HbS is one example of a
number of disorders caused by phase transitions
of abnormal proteins,18 which include the eye
cataract, Alzheimer’s, Huntington’s, and the prion
diseases, etc.19 – 22 Phase transitions are ubiquitous
in healthy organisms as well; examples include
the formation and modification of the
cytoskeleton,23 the suspected microcompartmentation of the cytosol,24 etc. Data on the kinetics of
phase transitions provide insights into their mechanisms and governing physical parameters and
may help in the understanding of their normal
and pathological effects in living systems.
As many other first-order phase transitions,25 – 27
the polymerization of deoxy-HbS starts with
nucleation, in which a certain number of molecules
assemble into an embryo of the new phase.28 – 30
0022-2836/$ - see front matter q 2003 Elsevier Ltd. All rights reserved.
44
Although the progress of HbS polymerization has
been monitored by a variety of techniques (see
Eaton & Hofrichter8 for a review), to date there
exists just one set of data on nucleation rates.28,31,32
These detailed determinations use an elegant
theoretical model33 to extract nucleation rates from
the evolution of the intensity scattered from
growing polymer fibers.
Here, we implement a method for direct simultaneous determination of two independent characteristics of the nucleation kinetics of HbS polymer
fibers: the homogeneous nucleation rates and the
nucleation induction times. After polymerization
is induced by laser photolysis of carbon monoxidesaturated HbS, nucleation of individual fibers is
monitored by optical microscopy with differential
interference contrast (DIC) in slides of uniform
thickness containing supersaturated solution. The
rate of homogeneous nucleation and the induction
times are determined from the evolution of the
statistics of appearance of HbS spherulites.
The results obtained using this technique shed
light on the following questions: is nucleation of
HbS fibers compatible with the general theories of
nucleation of first-order phase transitions? What
are the typical nucleation rates of the deoxy-HbS
polymers? How do the determined values compare
to those leading to red cell sickling in vivo? How do
the determined values compare to those in other,
protein and non-protein, systems? How applicable
are the mechanisms determined from in vitro determinations to the nucleation of HbS polymers in
vivo? How many molecules are in the polymer
nucleus? Are they arranged in disks of seven pairs
as in grown HbS fibers, or do they have a different
arrangement?
Results
Polymerization, nucleation, and nucleation
rate determination
In brief, the determinations of the nucleation rate
consist of the following experimental procedures.
A CO-HbS sample is held in a glass slide of 5 or
10 mm uniform thickness. An area of , 90 mm
diameter is continuously illuminated with green
laser or lamp to photolyze CO-HbS to deoxy-HbS
and sustain the deoxy form. DIC microscopy is
employed to monitor the number of HbS fibers
and spherulites appearing in the illuminated area.
Assuming that each spherulite is generated by a
single nucleation event (see arguments supporting
this assumption in this subsection) we determine
the number of nuclei in the illuminated volume
for the time elapsed after illumination starts. Then
the illuminated spot is moved to another,
randomly selected location on the slide and the
time evolution of the number of nuclei is recorded
again. In most cases, this is done 82 times at each
temperature. In some cases 200 and more determinations are carried out. The nucleation rate is
Nucleation of HbS Polymers
determined as the ratio of the mean number of
nuclei that appear for a certain time to the volume
occupied by deoxy-HbS. Exhaustive details are
provided in Materials and Methods below.
If polymerization is driven by low-to-moderate
supersaturations, the fibers do not branch and the
solid phase consists of separate HbS fibers, such
as those in Figure 1(a).34 At the later stages of
polymerization, the solution turns into a gel
consisting of straight fibers (Figure 1(b)). Since
spherulites are easier to detect than single fibers,
we work at slightly higher supersaturations,
which ensure spherulitic morphology of the HbS
polymer phase illustrated in Figure 1(c).
As in previous work (e.g. Briehl10), a basic
assumption in the nucleation rate determinations
below is that each spherulite is generated by a
single primary nucleation event. The relative error
introduced by this assumption is equal to the probability of having two nucleation events within the
slide area occupied by one spherulite. This probability can be evaluated as the ratio of the area
occupied by all spherulites in the slide to the total
area, in which nucleation can occur. As shown
below, nucleation rate data are extracted from
images of slides with fewer than ten spherulites,
each occupying an area of , 10 mm £ 1 mm. The
probability of having a second spherulite
hidden under or within one of the ten is about
(10/5000)2 £ 10 ¼ 4 £ 1025.
Homogeneous nucleation or nucleation on
a substrate?
For tests of whether the primary nucleation of
the observed spherulites is facilitated by the upper
and lower slide covers, or dust particles in the
solution, we record the locations of appearance of
the spherulites in four repetitive identical runs at
the same area of a slide. The runs were separated
in time by 20 –30 minutes, allowing complete dissolution of the spherulites nucleated in the
preceding run. With slide thickness of 5 mm or
10 mm, buoyancy-driven convection due to
possible thermal gradients is suppressed.35,36 Thus,
possible particles in the solution could only be
moved by Brownian diffusion,
pffiffiffiffiffiffiffiffiffiffiffiin which the
expected displacement Dx < 4DDt < 10 mm for a
, 1 mm particle with diffusivity D , 1029 cm2 s21
over Dt ¼ 1200 s. The predicted displacement is
comparable to the observed sizes of the spherulitic
domains, i.e. if such particles serve as substrates
for heterogeneous nucleation, the locations of the
spherulites in subsequent runs would overlap.
Figure 2 shows that no correlation exists between
the locations and orientations of the spherulites in
the four runs. We conclude that the nucleation
events that lead to the spherulites are not
facilitated at particular spots on the slide covers or
by particles larger than 1 mm in the solution.
For further evidence for the mechanism of
primary nucleation, we carried out several determinations of the mean (over , 100 identical runs)
45
Nucleation of HbS Polymers
Figure 1. Morphologies of polymer fibers observed during deoxy-HbS polymerization visualized in 10 mm slides at
22 8C using differential interference contrast microscopy. (a) Single fibers , 20 nm thick observed at HbS concentration
170 mg ml21. (b) Gel composed of long single fibers, conditions the same as in (a), long observation times.
(c) Spherulites observed at HbS concentrations 200 mg ml21 and higher. (d) Thick gel with no individual fibers
discernible at long polymerization times with HbS concentration of 250 mg ml21.
number of nuclei that appear in solutions held in
slides of 5 mm and 10 mm thickness after identical
time periods and with special care to ensure equal
temperatures of the two runs, for details, see
below. We found that the mean number of nuclei
in the illuminated area in 10 mm slides is double
the number in 5 mm slides, i.e. the number of
nuclei generated over a certain time is proportional
to the solution volume. Furthermore, we observed
that spherulites are never firmly attached to the
surface of a glass slide. The latter two observations
rule out heterogeneous nucleation on the glass
surfaces.
Statistics of nucleation
Figure 2. Locations and orientation of spherulites in
four consecutive runs made at the same area on the
slide, with full dissolution of fibers between the runs.
T ¼ 22 8C; CHbS ¼ 232 mg ml21.
The time required for complete dissolution of the
spherulites is usually comparable or longer than
the time of spherulite nucleation and growth.
Hence, repetitive runs to determine the nucleation
statistics are carried out by illuminating randomly
selected different areas in the slide.
Figure 3(a) shows that as spherulites grow, they
get close to one another, fill the illuminated area,
and interfere with the processes of further nucleation. Hence, we only count spherulites before their
regions overlap. Furthermore, as shown below, the
nucleation rates are determined from the earliest
part of the data on spherulite numbers, where the
numbers increase linearly with time.
In Figure 3(b), we show determinations of the
number of spherulites n in the illuminated area as
46
Nucleation of HbS Polymers
the distributions of the numbers of nuclei in
Figure 4 with Poisson’s law:
PðnÞ ¼
Nn
expð2NÞ
n!
ð1Þ
where n is the number of nuclei that appear in
volume V during nucleation time t, N is the mean
number of nuclei in all runs with the same V and t
reflected in the distribution.
For each distribution, we calculate Nfit from the
best fit to equation (1) and the algebraic mean
of the
data in the distribution N. The expression
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 2 N=Nfit Þ was evaluated and its closeness to 0
was used as a criterion for the correspondence of
the actual distribution of the data to Poisson’s
law.37 In all cases, it was , 0.065. For further tests
of the goodness of the fit to the Poisson’s law,
equation (1), we calculate the parameter x2:37
x2 ¼
Figure 3. Variation of the number of spherulites n
appearing for time t at T ¼ 22 8C and HbS concentration
of 232 mg ml21. (a) Illustration of spherulite counting
procedures for determination of n. (b) Six dependences
nðtÞ at identical conditions.
a function of the time, t, elapsed after CO-HbS
photolysis. These determinations were performed
at identical conditions, within the same slide. We
see that even in the limited number of runs displayed, the delay times before the appearance of
the first spherulites vary from 8 s to 13 s, the
number of spherulites, for instance at 16 s, varies
between one and five, and the shape of the n(t)
dependence is unpredictable. These three facts
illustrate the inherently stochastic nature of the
processes of fiber nucleation.
In a stochastic process, the number of nuclei that
appear in a certain volume is a random variable:
repetitions of an experiment under identical conditions could give, for instance, one, ten, or no
nuclei. Representative statistical distributions of
the number of nuclei resulting from 200 experiments under identical conditions are presented in
Figure 4.
To verify if the individual nucleation events are
independent of each other, a prerequisite for the
applicability of the general nucleation theories, we
determine the nucleation statistics. We compare
X ½Fn 2 PðnÞNfit 2
PðnÞNfit
n
ð2Þ
where Fn are the measured frequencies from
Figure 4 and P(n) are Poissonian best-fit values. In
most such determinations, for instance the data in
Figure 4, the x2 values correspond to confidence
levels in the Poissonian character of the
distributions of 90% or better; in a few cases the
confidence level was as low as 80%. The relative
error
in determination of N is evaluated as
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N=ðnumber of runsÞ: Thus, for 100 runs yielding
on the average one nucleus, it is 0.1.
Comparisons of the distributions of the
spherulites in five series of 200 repetitive runs
under identical conditions show (data not displayed) that although the number of spherulites is
random, the statistical distributions corresponding
to experiments performed under the same conditions are reproducible. All five distributions
have the same shape, follow Poisson’s law, and
have very similar mean values.
The good correspondence of the experimentally
determined distributions to Poisson’s law shows
that all individual nucleation events are independent of each other. Thus, the statistics of nucleation
of the sickle cell hemoglobin polymers indicate that
under the conditions probed here it follows the
general laws of nucleation of first-order phase
transitions.
The homogeneous nucleation rates
Figure 4 shows that with increasing nucleation
time t the distribution of the number of spherulites
shifts to the right, changing from near-exponential
at short times to near-Gaussian at long times.
Figure 5 shows that the time dependencies of
mean number of spherulites found from 82 independent runs have a characteristic shape: after an
initial delay period, when no nuclei are seen, the
number increases linearly with time. These linear
parts of the dependencies in Figure 5 correspond
to steady-state nucleation, characterized by the
Nucleation of HbS Polymers
47
Figure 4. Statistical distributions of the number of nuclei at times of the evolution of a polymerizing sample
indicated on the plots at T ¼ 25 8C, and CHbS ¼ 232 mg ml21. Bars represent experimental determinations, lines
represent fits with the Poisson distribution.
nucleation rate J. The last part of the dependencies
reflects saturation due to growth of spherulites
and overlapping of their supply fields. These data
were not included in determinations of the
nucleation rates. Note that all points in Figure 5
Figure 5. The dependence of mean number of nuclei,
determined by averaging over 82 runs on time at the
temperatures
indicated
in
the
plot
and
CHbS ¼ 232 mg ml21. The relative error in each data
point is ,0.1. The linear portion of each dependence is
fitted with a straight line, whose slope is taken as the
homogeneous nucleation rate. The intercept of the
straight line with the time axis determines the apparent
nucleation delay time.
were taken from images similar to the first five
frames in Figure 3(a), i.e. where the interaction
between spherulites is not apparent. Thus, the
time dependencies of the averaged numbers of
nucleated spherulites offer a stricter criterion for
non-interacting spherulites, from which nucleation
rates can be extracted.
The nucleation rate J, i.e. the number of nuclei
that appear in a unit solution volume within a
unit time, is determined for each dependence in
Figure 5 as the slope of the straight line
characteristic of the steady regime of nucleation.
Figure 6(a) shows the dependence of the homogeneous nucleation rate J on temperature T for a
solution of deoxy-HbS with concentration
C ¼ 232 mg ml21 ¼ 3.32 mM in 0.15 M potassium
phosphate buffer with pH ¼ 7.35. The values in
Figure 6(a) come from two independent
experiments, starting from solution preparation, as
discussed in Materials and Methods below. The
good correspondence between the two data sets is
evidence for the reproducibility of the determinations by the technique used here.
As expected, the nucleation rate J in Figure 6(a)
is
a
strong,
exponential
function
of
temperature. The values of nucleation rate from
this and other runs at similar HbS concentrations
and temperatures are in the range 105 –
108 cm23 s21. Converting to units mM s21 as
others,31,32 J ¼ 108 cm23 s21 ¼ 1.66 £ 10210 mM s21,
48
Nucleation of HbS Polymers
using the expressions for non-interacting hard
spheres from Hill:39 for justification, see Minton:42,43
B2 ¼
8V
;
2M
B5 ¼
35:3V 4
;
5=4M
B7 ¼
65:9V 6
7=6M
B3 ¼
15V 2
;
3=2M
B6 ¼
B4 ¼
24:48V 3
;
4=3M
47:4V 5
;
6=5M
ð4Þ
The specific volume V was determined from fits to
re-plotted data on sedimentation equilibria;41 best
fits are obtained with V ¼ 0:79 cm3 g21, close to
the values for other proteins44,45 and to the Hb
value reported.8 Since the virial coefficients for
non-interacting spheres are temperature-independent, the only temperature dependence in equation
(3) comes from Ceq ðTÞ:8
Ceq ðTÞ ¼ 0:319 2 0:000883ðT 2 273:15Þ
Figure 6. The dependencies of the homogeneous
nucleation rate J in (a) and the nucleation delay time in
(b) on temperature at the HbS concentration indicated
in the plots. Data in (a) and (b) have relative errors in
the range 0.05– 0.11, determined by the relative errors of
the points in Figure 5 and the their deviations from
linearity. The error in temperature is ,0.5 deg. C, see
discussion of temperature non-uniformity in Materials
and Methods.
with
log J ¼ 29:78:
The
published
determinations31,32 were carried out at higher HbS
concentrations and in the 25 – 35 8C temperature
range. If one extrapolates the decrease of J
(denoted with f0 31,32) with decreasing HbS
concentration from 4.26 mM ¼ 275 mg ml21 to
3.92 mM ¼ 253 mg ml21 at 25 8C reported by Cao
& Ferrone32 to the concentration that we use, a
value close to ours obtains.
The driving force for polymerization
To evaluate the thermodynamic supersaturation
HbS
(m is the
Dm=kB T ¼ ðmHbS
polymer 2 msolution Þ=kB T
chemical potential of the HbS in the respective
phase, kB is the Boltzmann constant and T is the
absolute temperature) at the determination temperatures, we use the virial expansion38 – 40 limited
to the sixth-order term as justified:8,41
DmðC; TÞ
C
¼ ln
kB T
Ceq ðTÞ
þ
6
X
iþ1
Biþ1 ½Ci 2 Ceq ðTÞi ð3Þ
i
i¼1
where C is the deoxy-HbS concentration, and Ceq is
its value at equilibrium with the polymers. The
values of the virial coefficients Bi are obtained
þ 0:000125ðT 2 273:15Þ2
ð5Þ
We find that the supersaturation levels are
moderate: for C ¼ 232 mg ml21 as in Figure 6(a),
Dm=kB T does not exceed 1.7 even at the highest
temperature probed.
Nucleation induction time
An advantage of the method used here is that it
allows independent determinations of the
nucleation rate and of the nucleation delay time.
As Figure 5 shows, we determine the nucleation
delay time from the intersection point with the
time axis of the dependence of the mean number
of nuclei on time. Another possibility is to average
the individual delay times in plots similar to those
in Figure 3, which would yield a different characteristic of nucleation kinetics. As shown by
Kashchiev,46 the delay time resulting from the
former method contains the nucleation induction
time u, and hence, this is the method used here.
The time u is a measure of the rate of transformation of the molecular distribution in a metastable single-phase solution to the distribution in a
solution in which nucleation of the new phase
proceeds at a steady rate.47,48
The delay times are plotted as a function of temperature in Figure 6(b). The data in the two series
diverge, while the reproducibility of the nucleation
rates in Figure 6(a) is preserved. This discrepancy
illustrates the independence of these two characteristics of the nucleation kinetics; an understanding
of the physical phenomena underlying it is still
missing at his point. The delay times are the sums
of two components: the nucleation induction times
u and the instrument detection times. The
instrument detection times are determined by the
resolution of the microscope of , 0.5 mm and
the growth rate of the spherulites. We measure the
spherulite growth rates as 1.15– 2.5 mm s21 at 19 8C
49
Nucleation of HbS Polymers
and C ¼ 232 mg ml21 and hence the detection
times are , 1 s. Thus, the apparent delay times in
Figure 6(b) are good approximations to the
respective u.
TDS ¼ 7:4 kcal mol21 at 25 8C,8 resulting in
The
concentration
Dse ¼ 1:72 £ 10222 J K21.
C0 ¼ 232 mg ml21 ¼ 2.18 £ 1024 m23 and ðkB =Dse Þ
ln C0 ø 4:5: Combining this value with the slope
from Figure 7, we get for the nucleus size:
The nucleus size
np < 11:4
The nucleation theorem has been used to
determine the nucleus size from data on the
dependence of the homogeneous nucleation rate J
on supersaturation Dm.46,49,50 For non-isothermal
data sets, where supersaturation Dm is varied via
temperature, a more recent form of the nucleation
theorem applies. The resulting method employs
two data sets: the homogeneous nucleation rate J
and the nucleation induction time u:
np ¼ kB
d
kB
½T lnðJ uÞ þ
lnðC0 Þ
dDm
Dse
ð6Þ
where C0 is the concentration of single HbS
molecules in the solution, and Dse is the entropy
of HbS polymerization.46 Equation (6) is derived
assuming that Ju and C0 are in the same units,
such as m23.
Figure 7 displays the nucleation kinetics data
from Figure 6 plotted in the coordinates of
equation (6), i.e. the product T lnðJuÞ with J
rescaled in m23 s21 units, as a function of the supersaturation Dm rescaled with kB : The theories related
to the nucleation theorem do not treat the intercept
of the dependence of T lnðJuÞ on Dm. Thus, while it
is likely that the different intercepts of the two
lines in Figure 7 are linked to the deviations in
induction times between the two series in
Figure 6(a), the physical phenomena that underlie
these differences are unclear to us. However, for
the purposes of determination of the nucleus size,
it is sufficient that the slopes of the linear fits are
6.9 for both data series in Figure 7.
To evaluate the second term on the righthand side of equation (6), we use that
Figure 7. Toward determination of the nucleus size.
Plots of the data on nucleation rate J and nucleation
induction time u from Figure 6 in coordinates of the
nucleation theorem, i.e. T lnðJuÞ as a function of thermodynamic supersaturation Dm rescaled with the
Boltzmann constant kB :
ð7Þ
i.e. sizes of 11 or 12 molecules are within the error
limits of this determination.
Discussion
Nucleation of HbS polymer fibers
The polymerization of deoxy-HbS has been discussed in terms of a double nucleation model.29
The first step in polymerization is homogeneous
nucleation, in which single fibers are randomly
generated in the bulk of a supersaturated solution.
As the fibers grow, they serve as substrates for the
nucleation of new fibers.10 This leads to thickening
and branching of the fibers.8,28 – 30 As with
numerous other phase transformations proceeding
via nucleation,51 – 54 the rate of polymerization was
found to be a very strong, likely exponential
function of the concentration of HbS. Depending
on the range of concentrations probed, the strong
dependence has been described as tenth to 20th
power55 to 30th and 50th power.32
The nucleation of secondary fibers on an existing
fiber is often referred to as “heterogeneous
nucleation”.8,28 – 30 Typically, the term heterogeneous nucleation is used for nucleation on a
foreign substrate.46,56,57 Nucleation on a substrate is
energetically different from that in bulk of the old
phase; the wetting of the substrate by the new
phase is expected to decrease the nucleation barrier
and render heterogeneous nucleation faster by
many orders of magnitude. In practice heterogeneous nucleation is widespread and includes
nucleation on foreign molecules (impurities),
microscopic particles (dust or bubbles), etc. It may
occur concurrently with homogeneous nucleation
and care must be taken in data interpretation to
distinguish between homogeneous and heterogeneous nucleation. In the second stage of HbS
polymerization, the substrate is the newly appearing phase, i.e. the substrate is not a foreign object,
and it is available only after HbS polymerization
has started. In another deviation from “classical”
heterogeneous nucleation, it is not always faster
than homogeneous nucleation; at low-to-moderate
supersaturations single un-branched fibers, such
as those in Figure 1(a), are often observed.9,10,58
There have been some early claims that sickling
is akin to gelation of a colloid suspension59
supported by the fact that completely polymerized
samples behave rheologically as a gel.13 This
interpretation was contradicted by the finding
that unlike gels, the polymer has constant
density and structure, independent of external conditions for polymerization,60,61 indicating that HbS
50
Nucleation of HbS Polymers
polymerization is a first-order phase transition.62,63
In further analogy to first-order phase transitions,
the changes of enthalpy, entropy, and molar
volume
upon
polymerization
have
been
determined.8 It has recently been emphasized that
this and other phase transitions, in which an
ordered solid (crystal, fibril, polymer) emerges for
a dilute and disordered fluid (gas or solution)
should be viewed as a transition along two order
parameters,64 density and structure.65 Thus, the
fibers and spherulites in Figure 1(a) and (c) have a
higher concentration of HbS molecules than the
surrounding solution, and the molecular arrangement is ordered.
Homogeneous nucleation or nucleation on
a substrate
The results in Figure 2 do not provide evidence
against heterogeneous nucleation on particles or
pieces of the cell membrane smaller than 1 mm, or
on protein molecules larger than hemoglobin; each
of these could serve as centers of heterogeneous
nucleation.46,56,57 Indeed, if these entities have sizes
of 100 nm or smaller, they could move about the
solution volume driven by Brownian diffusion
and in subsequent runs induce nucleation at different locations. If their concentrations were constant
they would not lead to dependence of the number
of nucleated spherulites on the solution volume.
However, dynamic light-scattering characterization
of Hb solutions prepared following procedures
identical with those used here showed a single
species of a size between 5 nm and 6 nm,66 compatible with the 5.5 nm known for hemoglobin.67,68
The high limit of the detection range of these
determinations is above 1 mm. In view of the high
sensitivity of light-scattering to larger species (e.g.
Eisenberg & Crothers69), we conclude that the
solutions that we use do not contain species or
particles that could serve as heterogeneous nucleation centers. Thus, the spherulites observed here
are generated via homogeneous nucleation.
An important issue in the studies of the sickle
cell hemoglobin polymerization is whether the
nucleation of the primary fibers is enhanced at particular sites on the membrane of the red blood
cells.15,70 Before this issue can be addressed in a
quantitative manner, benchmark data on the truly
homogeneous nucleation of HbS polymers is
needed. An added advantage of data on homogeneous nucleation stems from the fact that its
rate depends on fewer parameters than the rate of
nucleation on foreign substrates. Hence, homogeneous nucleation data serve as a firmer
foundation for conclusions about the nucleation
mechanisms.
The nucleation rates
The values of J in Figure 6(a) are comparable to
those occurring in the cytosol of non-permanently
sickled erythrocytes. Indeed, a red blood cell
whose volume is ,94 femtoliters < 10210 cm23
spends from 10 s to 20 s in the venous circulation.71,72
During that time, one or a few nucleation
events,11,70,73 followed by fast growth (as shown
above, growth rates of the polymers are a few
mm s21) and branching, may lead to sickling of the
cell. Thus, the nucleation rates of sickling of nonpermanently sickled cells are of the order of
109 cm23 s21.
Note that the conditions in the red cells are
different from those in our experiments. Most
importantly, the HbS concentrations in the sickle
red cells are in the range 320 – 460 mg ml21, with a
consistent mean of , 360 mg ml21 and this should
result in significantly faster nucleation rates.74 The
high HbS concentration is likely compensated by
the partial oxygenation of the HbS:73 due to the
residual oxygen vapor pressure of , 40 Torr, 70%
of the Hb is in oxy-form.75 Furthermore, the red
cell cytosol contains several compounds in concentrations comparable to that of HbS,76 – 79 whose
effects on polymerization and in particular on
nucleation have only partly been studied.80 In
addition, the role of the inside surface of the red
cell membrane on the nucleation is unknown.11,15
Despite these factors that may affect nucleation in
vivo and take it along pathways different from
those followed in vitro, the similarity of the
nucleation rates suggests that the mechanisms of
nucleation deduced from in vitro experiments are
a relevant first step in the understanding of
polymerization in the red cells of sickle cell anemia
patients.
The values of J of sickle cell hemoglobin
polymers are seven to ten orders of magnitude
higher than those of formation of other ordered
solid phases of proteins, summarized in Table 1.
In fact, the HbS polymer nucleation rates are
similar to the nucleation rates of water droplets at
atmospheric pressure and room temperature:54 the
phase-transition rates of small molecules are
Table 1. Homogeneous nucleation rates J and ranges of concentration C and supersaturation Dm=kB T during determinations for four proteins
Protein
Lysozyme
Insulin
Hemoglobin C
Apoferritin (estimates)
C (mg ml21)
Dm=kB T
J (cm23 s21)
30– 80
1– 3
10– 20
0.1–1
3–3.7
3.2– 3.7
2.5–3
2.2– 3.9
0.01–1
0.01–0.1
,0.01
0.1 –1
Ref.
50,83,99
L. Filobelo, unpublished
66
100,101
51
Nucleation of HbS Polymers
expected to be significantly faster. While the
protein concentrations in the determinations in
Table 1 were significantly lower than the concentrations of HbS in the determinations discussed
here, the supersaturation levels were higher. Since
the nucleation rate is expected to be a linear function of concentration and exponential function of
supersaturation,46,81,82 the huge discrepancy cannot
be solely attributed to the action of protein concentration. One could argue that the nucleation rates
for proteins crystals were determined under conditions optimized for crystal perfection and are
not the maximum possible for a given protein;
often, multiple protein crystals form in a container.
Even if one allows for 103 £ faster nucleation of
crystals under “unfavorable” conditions, this still
leaves a discrepancy of four or five orders of
magnitude between the two types of nucleation.
This discrepancy cannot be understood at this
point. Still, we would like to offer two hypotheses,
to be tested in further work. One can speculate
that it may be linked to the lower, one-dimensional
translational symmetry of the HbS polymers.
Polymerization is a phase transition occurring
along two order parameters, density and structure,
and arguments have been presented suggesting
that such transitions occur when a structure
fluctuation is superimposed on a density
fluctuation.83 – 86 Thus, it is feasible that a lower
degree of symmetry of the new phase could lead
to lower barriers for the second stage of the
process, ordering, occurring within a density
fluctuation.
Another possibility also linked to the superposition of density and structure fluctuations is
that the high nucleation rates of HbS polymers
stem from the high fluidity of the concentrated
hemoglobin solution within the quasi-droplet of
the density fluctuation. As expected for suspensions of non-interacting particles, the viscosity
of hemoglobin solutions at concentrations similar
to those in the red cell cytosol is , 10 cP
(1 P ¼ 1021 Pa s),87,88 i.e. barely above that of the
solvent, while solutions of other proteins with concentrations , 200 mg ml21 are very viscous or gellike. While the high fluidity of the concentrated
solutions should lead to linearly faster kinetics of
ordering, it also leads to faster rate of probing of
various structures, i.e. of density fluctuations
within the quasi-droplet of the density fluctuation,
and this may result in exponentially faster rate of
finding of the “right” structure. In support of this
hypothesis, we note that dense liquid phases of
HbS enhance nucleation and growth of HbS
polymers.34 This is in contrast to the viscous dense
liquid phases of other proteins,44,89 which suppress
nucleation83 and growth90 of protein crystals.
determined by others:28,29,31,32,91 the latter two
represent expectancy times until a polymer reaches
a point from which light is scattered, or at which
the solution turbidity is monitored. As such, they
are linked in a complex way to the mean time
separating nucleation events in steady state.46 As
discussed above, the induction time u characterizes
the transformation of the molecular distribution in
the meta-stable single-phase solution to the distribution in a solution in which nucleation proceeds
at a steady rate.47,48
Theoretically, the nucleation induction time u can
be evaluated as:47,48,82
u ¼ 2=3pZ2 f p
ð8Þ
The parameter Z, the Zeldovich factor, is also an
important component of the pre-exponential
factor of the nucleation rate expression.47,81 It is
proportional to the radius of curvature of the freeenergy barrier for nucleation around its maximum;
its lack of dimension stems from the scaling of the
free energy with kB T and using the number of
molecules in the near-critical cluster as a spatial
variable. The Zeldovich factor accounts for the
deviations of the cluster size distribution from that
in the state of forced equilibrium in a nucleating
system, assumed in earlier nucleation theories.
This equilibrium assumption requires the action of
a mechanism of decay of supercritical clusters,
while in fact supercritical clusters undergo
uninhibited growth.81 Typical estimates are Z <
0:1 at low supersaturations typical of phase transitions with small molecules.82 For large biological
macromolecules, for which the characteristic
supersaturations during nucleation are severalfold the thermal energy, Z < 0:01 are likely.92 Note
that evaluations of Z using equation (6) are
necessarily only approximate: the small size of the
HbS polymer nuclei suggests that the addition or
detachment of single molecules to near critical
clusters leads to significant changes in its thermodynamic characteristics. Thus, continuous considerations, such as those behind equation (6), can
only provide semi-quantitative understanding of
the nucleation processes.
The parameter f p is the frequency of attachment
of molecules to the critical cluster (nucleus). It can
be evaluated from the rate of growth of
, 1 mm s21, see determination above, as f p ¼ 180
molecules s21. Using these values of f p and Z; we
get u , 12 s, with shorter u if higher attachment
rates apply. Although the induction times in the
two series of determinations in Figure 6(b) differ,
especially at low T values, for both series they are
of the same order of magnitude as this estimate.
In further agreement with the predictions of the
Zeldovich theory, they decrease as T is increased.
The nucleation induction time
The nucleus size
The delay time and the nucleation induction
time u determined here are inherently different
from the nucleation time lag or the 1/10 time
The most important characteristic of the nucleation process is the nucleus size np : The nucleus is
52
a cluster that has equal probabilities to grow and
dissolve and is thus in unstable equilibrium at the
peak of the free energy landscape along the nucleation reaction pathway.25,26 The size of the nucleus
largely determines the value of the free energy at
this peak, the free energy barrier for nucleation.
The nucleation theorem examines how the nucleation barrier varies in response to supersaturation
changes and shows that the respective derivative
allows determination of the nucleus size.46,49,93,94
Recently, it has been pointed out that the
nucleation theorem relies on very few confining
assumptions about the nucleation process, and is
thus a general, system-independent nucleation
law.93,94
The earliest estimates of the nucleus size for HbS
polymerization were based on determinations of
the delay time td of polymerization monitored via
the solution turbidity.95 In a quasi-chemical
approach, the apparent reaction order m was determined as a slope of the plot of logðtd Þ versus logðCÞ;
obtaining a value around 32. In subsequent
investigations, the necessity to account for the
non-ideality of HbS solutions was acknowledged,
bringing the reaction order to 9 or 6.96 Still, determinations of the nucleus size n p from the reaction
order remained a difficult problem, with estimates
ranging from np ¼ m þ 1 to np ¼ 2m:
In another series of studies, the apparent
reaction order was determined from the time
evolution of the HbS polymerization monitored
with light-scattering.30,31,91 Homogeneous nucleation rates were extracted from the distributions of
the time to achieve one-tenth of the final reaction
yield,33 assuming that the primary nucleation of
HbS polymers is the only source of stochasticity of
the data. The apparent reaction order was 47(^ 5).
The authors found that no simple relation of the
reaction order of the tenth-time to molecular parameters exists and employed the double nucleation
model to determine the size of nuclei as 7.31 Note
that this result, obtained at higher supersaturation,
does not contradict our finding: the nucleus size is
expected to decrease at increasing supersaturation.
Thus, the determination of nucleus size using the
nucleation theorem offers advantages: the nucleation theorem is a general nucleation law, which
does not rely on assumptions about the kinetic
pathway or the structure of critical nucleus.
The data used in the evaluation of np are the
dependencies of J and u on temperature and the
thermodynamic properties of the HbS solution.
The determination of the nucleus size of the HbS
polymers calls attention to an interesting point: np
is smaller then 14, the number of HbS molecules
in the cross-section of the fiber. Obviously, such
small nuclei cannot contain one, or two, or several
14-member “disks”. The question of whether the
nucleus contains 10– 12 molecules from the 14 in
the cross-section, or five or six pairs of molecules
in one of the Wishner – Love double strands,97 or
another ordered, or an altogether disordered configuration is an open one at this point.
Nucleation of HbS Polymers
Perspectives for future work
A line of intriguing and relevant questions
arising from analogies to recent findings on the
nucleation mechanisms of crystals50,83 – 85 and on
the phase behavior of hemoglobin mutants34 have
not been addressed here. Some of these are related
to the fundamentals of the nucleation mechanisms:
is the HbS polymer nucleus ordered or disordered?
Does nucleation occur in one or two steps, i.e. is
there a disordered precursor to an ordered HbS
critical cluster? What is the exact shape and structure of the critical cluster? Others are related to
the participation of the cell membrane and other
red cell cytosol components in the nucleation
process: is HbS polymer nucleation affected by
any of the red cell cytosol components present in
concentration comparable to those of HbS?76,77 Is
the nucleation of HbS polymers enhanced at
certain areas of the inside of the red cell
membrane?
Materials and Methods
Solution preparation
Hemoglobin S was isolated from the blood of sickle
cell anemia patients kindly provided by Dr R. E. Hirsch
from Albert Einstein College of Medicine. The blood
was washed in isotonic 0.9% (w/v) NaCl solution and
centrifuged to remove white blood cells. The red blood
cells were lyzed with deionized water and the solution
was centrifuged again to remove cell membranes. The
remaining Hb solution was filtered through a 0.45 mm
Sterivex-HV filter. To purify HbS, the solution was
diluted into a large volume of 20 mM Tris buffer (pH
8.5), and loaded onto a Q-Sepharose FPLC column (XK
50, Amersham Biosciences). Separation of different Hb
variants was achieved by elution using a gradient of
NaCl. The purity of the fractions was checked by gel
electrophoresis (PhastSystem, Amersham Biosciences).
After the purification procedure, the gels showed a
single band of HbS. The sample was concentrated by
centrifugation in Centricon Plus-20 filters (Millipore).
For polymerization experiments, the solution was
dialyzed into 0.15 M potassium phosphate buffer (pH
7.35), using Slide-A-Lyzer Cassettes (Pierce). The
solution was finally concentrated to about 270 mg ml21.
After purification, Hb is in the oxy form. The quantum
yield of O2-Hb is 50 times lower than that of CO-Hb98 so
CO-Hb is better suited for use in photolysis experiments.
To prepare CO-Hb we used the following procedure:
1 ml of stock solution was placed for one hour in a ten
liter glove box with an atmosphere of 100% CO saturated
with water vapor at room temperature. While exposed to
CO, the sample was carefully and mildly stirred four
times. During this step, HbS molecules lose oxygen and
bind CO. After this the atmosphere in the glove box
was changed to 100% He, saturated with water. The
sample was again mildly stirred four times. The second
step reduces the concentration of free CO in the solution,
allowing for full photolysis of Hb at lower laser power in
the subsequent experiments. After this procedure, the
concentration of the stock solution was determined, the
53
Nucleation of HbS Polymers
solution was stored as stock under liquid nitrogen and
used as needed.
For each experiment a solution sample of CO-HbS was
prepared by mixing 20 ml of stock solution in 0.15 M
potassium phosphate buffer at pH 7.35 with 1 ml of
sodium dithionite in the same buffer to give final concentration of 0.055 M sodium dithionite. About 3 ml of this
solution was placed on a standard 75 mm £ 50 mm
microscope glass slide (Corning), covered with a
22 mm £ 22 mm cover glass, and sealed with MountQuick (Daido Sangyo, Japan). All of the above
procedures were performed in a glove box under He
atmosphere saturated with water. Tests showed that
when handling volumes of several microliters, control
of humidity during sample preparation is crucial. Thus,
the rate of evaporation of a 20 ml droplet of the phosphate buffer used here was found to be 2.4 ml hour21 at
relative humidity of 55% and was still 0.4 ml hour21 at
95% relative humidity.
After sealing, the spectrum of the solution in the slide
in the range of 450– 700 nm was determined to verify
that it contains 100% CO-HbS and no met-HbS. All
nucleation experiments were performed immediately
after the sample preparation; samples older than
24 hours aged markedly, having higher nucleation rates
and smaller spherulites sizes, consistent with the observations reported by Ferrone et al.28
To determine the solution concentration, we used
Drabkin’s reagent (Sigma). For concentrations of
,260 mg ml21 the solution was diluted up to 500-fold.
We measured the absorption at 540 nm using extinction
coefficient 1 ¼ 0.6614 ml mg21 cm21. The procedure of
concentration determination for HbS samples required
particular attention because of the strong dependence of
the nucleation rate on concentration. The smallest pipette
volume used here was 1 ml with an accuracy of ,1.5%
(Eppendorf Research Series 2100 pipetter). Using 500 ml
solution volumes with a 1 ml pipette with an accuracy
of 0.5% (the pipette accuracies were taken from the
manufacturers’ specifications and were independently
experimentally verified), and using the accuracy of the
photometric readout of 0.1%, we get 1.6% for the
accuracy of the concentration determination. For a
concentration of 260 mg ml21 it yields an error of
^4.2 mg ml21.
Experiment setup
The experiment setup is built around a Leica DM R
fluorescence microscope and is shown in Figure 8.
Observations of HbS fiber nucleation are made in transmitted light with DIC optics, a 63x HCX APO L U-V-I
water immersion objective and a 0.90 S1 achromatic condenser. Pictures are taken with VCC-151 color video
camera (Hitachi), FlashPoint-3D frame grabber (Integral
Technologies Inc.) and custom-made software. With the
current configuration the shortest time between frames
is 1 s.
Photolysis is achieved by a beam of a continuouswave Nd:YAG laser (LCS-DTL-316 from Power Technology Inc., l ¼ 532 nm, Itotal ¼ 2 – 200 mW). Alternatively, we use a 100 W gas discharge Hg lamp (Leica).
When laser is used, its beam is expanded by a spatial
filter assembly consisting of two objective lenses (focus
lengths F1 ¼ 16:5 mm and F2 ¼ 60 mm) and a 12 mm pinhole. The pinhole is positioned over the focus of the first
lens. The position of the second lens is varied to control
the beam expansion ratio and its divergence so that an
illuminated spot of a desired size in the working plane
of the microscope could be achieved. Besides expansion,
this arrangement provides for filtering of laser beam
non-uniformities.
The expanded and filtered beam is input through the
microscope’s light path for fluorescence excitation
illumination. The diameter of the photolyzed region
typically is , 90 mm, adjusted by a variable aperture
diaphragm. A beam splitter with a dichroic mirror and
filter is used, respectively, to direct the photolyzing
beam towards the solution sample through the specialized fluorescence objective and to reject the laser light
Figure 8. Schematic of the
experiment setup, see the text for
details.
54
Nucleation of HbS Polymers
scattered in the direction of observation. Since the filter
completely cuts off the photolysis wavelength from the
viewing path, to measure and control the diameter of
the photolyzing beam, we used a slide with Rhodamin
6G dye in ethanol. To achieve precise timing and to automate the data acquisition, a computer-controlled shutter
(Melles Griot, opening time ,few milliseconds) is used.
With slide thickness of 5 mm and diameter of the
photolyzed region of , 90 mm, the total photolyzed
volume is , 3 £ 1028 cm3. The shortest times between
captured frames are 1 s, and since we can confidently
resolve at most ten spherulites in the illuminated
volume, these parameters determine the fastest nucleation rate that we can measure as 108 cm23 s21. Faster
nucleation rates can be measured by using a faster
camera.
Slide thickness uniformity
The nucleation statistics is determined through independent runs in different locations on the slide containing HbS solution. This requires accurate determinations
of the slide thickness and verification of the thickness
uniformity throughout the slide. Furthermore, to prevent
sample overheating, low optical density of the slide
loaded with a relatively high concentration of HbS,
, 200 mg ml21 and higher, is required. This can be
achieved by using slides as thin as 5 mm or 10 mm. To
determine the uniformity of the slide thickness, we
measured the spatial distribution of slide optical density
(OD) on spatial coordinates with a Beckmann DU-68
spectrophotometer (sample thickness h is proportional
to the optical density because of Lambert – Beer law
OD ¼ 1Ch; where 1 is extinction coefficient, C is sample
concentration). For these measurements, the slide was
placed in the focus of the spectrophotometer light beam
with a spot size of 2 mm £ 0.3 mm, and scanned in two
perpendicular directions.
The determinations with slides without thickness control revealed that their thickness was extremely nonuniform (Figure 9(a)), with variations of as much as
50%. For thickness control, we used glass spheres (Duke
Scientific) placed between the slide and the cover slip.
These glass spheres are available in diameters
2.0(^ 0.5) mm, 4.9(^0.5) mm, 10.0(^ 1.0) mm. Using these
spheres as spacers, we achieve slide thickness uniformity
of ^ 2% for 10 mm spheres and within ^ 5% for 5 mm
spheres (Figure 9(b)). In most determinations, 5 mm
thick slides were used that provided for lower overheating due to absorption of the photolysis light.
Temperature of polymerization
To control the temperature of the sample we use a
50 mm £ 50 mm £ 20 mm box assembled by gluing
brass side walls to a glass slide. The top of this assembly
is left open for microscope objective access. A RTE-100
waterbath (Neslab) circulates deionized water through
the box over the slide and sets its temperature between
0 8C and 80 8C. The use of water immersion objectives
provides for high quality DIC images in this arrangement.
Temperature is measured with a HH 506R thermocouple
thermometer (Omega Engineering Inc.) with 0.1 K accuracy and is input to a computer using a serial port. We
checked the temperature inside the slide by placing a
small thermocouple under the cover-slip; its temperature
reading was equal to that in the circulating water.
Because our sample absorbs light at the wavelength of
Figure 9. Characterization of the slide thickness
uniformity using the proportionality of the spectrophotometrically determined optical density to local thickness.
Optical density at 540 nm, normalized to 1, is plotted as
a function of location along the slide. (a) Two slides without thickness control. (b) A slide with 10 mm spheres
used as spacers between slide and coverslip. (c) Same as
(b) but for 5 mm spacers.
photolysis, the steady-state temperature rise in the
photolyzed region is different from the temperature outside of the illuminated region. To measure this temperature we deposited in a slide a , 10 mm grain of
OMEGASTICS crayon (Omega Engineering) that melts
at 40.5 8C. To determine the temperature near the central
axis of the beam we masked half of it and positioned
the crayon grain near its center (Figure 11(a)). Then we
varied the temperature of the water circulating over the
slide and recorded the temperatures at which crayon
melted with laser power on, as in Figure 11(b), and off.
With the laser off, the crayon melted at the manufacturer-specified 40.5 8C. The overheating due to laser
light increases roughly linearly with laser power, slide
thickness and HbS concentration, and is about 4.5 deg. C
with HbS concentration of 232 mg ml21, slide thickness
of 15 mm and laser power of 10 mW. Our observations
are consistent with those reported by Ferrone et al.,28
thus, we rely on their conclusion that the typical times
to reach steady temperature are of the order of several
milliseconds, significantly shorter than our observation
times of ,1 s.
Nucleation of HbS Polymers
55
The total intensity of the photolysis illumination and
its spatial distribution determine the extent of HbS conversion to deoxy-state, its spatial distribution, and the
temperature regime in a sample. We measured the intensity profile by scanning either a 12 mm diaphragm placed
at the sample plane or a 0.5 mm diaphragm placed near
the expanding lens of the telescope. Figure 10(a) shows
the intensity profile with laser illumination; it is adequately described by a Gaussian function. Tests showed
that the beam width does not depend on the laser intensity and can be adjusted in a wide range from several
mm to 200 mm. When the gas discharge lamp was used
(Figure 10(b)), an essentially flat profile of intensity
obtained.
We determined the attenuation of the light by the
diaphragms and lenses in the optical pathway by placing
the detector of the light power meter in front of the laser,
and in the slide focal plane. We found that the
attenuation ratio is 0.52.
To determine the laser power needed to achieve full
photolysis of the CO-HbS, we carried out the following
tests. We use the fact that CO-Hb has higher extinction
coefficients at l ¼ 532 nm than deoxy-Hb. At low laser
power all Hb is in CO form and the optical density is
highest. When laser power is increased, more and more
Hb molecules lose CO and the optical density drops
until, at a certain intensity setting, it reaches the level of
deoxy-Hb. To determine the degree of photolysis, we
determined the optical density of a slide loaded with
CO-HbA (HbA was used to avoid interference from
fiber formation with the optical density measurements).
We used the illuminating pathway of the microscope, in
which a narrow band-pass interference filter (Edmund
Industrial Optics, l0 ¼ 436 nm, halfwidth 10 nm) was
mounted between the illumination lamp and the sample.
Figure 10. Intensity profiles of the photolyzing beams
in the plane of slide with HbS solution: (a) illumination
with a laser beam, a 90 mm diaphragm around the center
of the beam leaves intensity non-uniformity of ,35%;
(b) lamp illumination, 90 mm diaphragm around the
center of the beam leaves near-uniformly illuminated
area.
Figure 11. Measurement of overheating of the slide in
steady state. (a) A , 10 mm grain of OMEGASTICK
crayon with Tmelt ¼ 40:5 8C is placed near the center of
half-blocked laser beam, not visible because of the filter
that absorbs photolyzing light reflected in the direction
of observation. (b) The grain is intact at 35 8C, at 36.1 8C
a crescent-shaped molten area is detectable in the direction of the beam; at higher temperature the molten area
increases. Upon reversal of temperature changes, molten
crescent disappears at 36 8C.
The Gaussian profile of the laser beam (see below)
gives rise to a non-uniform temperature field. To limit
this non-uniformity, we expanded the beam and used a
variable aperture diaphragm to cut out the beam
periphery, where the intensity drops off. In the remaining central part of the beam, the intensity non-uniformity
was , 35%. With the above determination of the overheating due to light absorption, the temperature nonuniformity at laser power 10 mW in a 5 mm slide, the
most common arrangement used here, is ,0.5 8C.
Photolysis illumination intensity and uniformity
56
Nucleation of HbS Polymers
References
Figure 12. Dependence of transmittance T at 436 nm of
a 230 mg ml21 solution of CO-HbA on the power density
F of the photolyzing beam at 22 8C at different times Dt
after initiation of photolysis. Horizontal broken lines
mark transmittance of CO-HbA (upper line) and deoxyHbA (lower line) calculated from published spectra.
Right ordinate axis is scaled as a gauge for the fraction
of deoxy-HbA in the solution.
The intensity of the light transmitted through the sample
was measured with 1830-C power meter (Newport)
whose sensor was placed in the image plane of the
microscope, instead of the video camera in Figure 8. To
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three short-pass filters (Melles Griot, l ¼ 500 nm) and a
polarizer. A diaphragm set to an area of 16 mm diameter
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Figure 12 shows that power density of 0.3 kW cm22,
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to power density of 0.3 kW cm22, at which the dissociation of the CO-HbS into deoxy-HbS is completed
within a time period sufficiently short to only insignificantly bias the determinations of the induction times
and nucleation rates.
Acknowledgements
We thank Rhoda Elison Hirsch for numerous
helpful suggestions and for providing some of the
sickle cell and normal blood samples, Ronald
N. Nagel for patient guidance through the field of
hemoglobin diseases, Gregoire Nicolis for insight
into viscosity effects, Bill R. Thomas for the
development of a procedure for HbS isolation and
purification, Angela Feeling-Taylor for normal and
sickle cell blood samples. This work was supported
by the Office of Physical and Biological Research,
NASA, through grants NAG8-1854 and NAG81824.
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Edited by K. Nagai
(Received 14 August 2003; received in revised form 25 November 2003; accepted 2 December 2003)