the kinetics of nucleation and growth of sickle cell hemoglobin fibers

doi:10.1016/j.jmb.2006.10.001 J. Mol. Biol. (2007) 365, 425–439

The Kinetics of Nucleation and Growth ofSickle Cell Hemoglobin Fibers

Oleg Galkin1, Ronald L. Nagel3 and Peter G. Vekilov1,2⁎
1Department of Chemical andBiomolecular Engineering,University of Houston,Houston, TX 77204-4004,USA2Department of Chemistry,University of Houston,Houston, TX 77204-4004,USA3Department of Medicine(Division of Hematology),Albert Einstein College ofMedicine, Comprehensive SickleCell Center, The Bronx, NY,10461, USA
E-mail address of the [email protected]

0022-2836/$ - see front matter © 2006 E

Polymerization of sickle cell hemoglobin (HbS) in deoxy state is one of thebasic events in the pathophysiology of sickle cell anemia. For insight into thepolymerization process, we monitor the kinetics of nucleation and growthof the HbS polymer fibers. We define a technique for the determination ofthe rates J and delay times θ of nucleation and the fiber growth rates R ofdeoxy-HbS fibers, based on photolysis of CO-HbS by laser illumination. Wesolve numerically time-dependent equations of heat conductance and COtransport, coupled with respective photo-chemical processes, duringkinetics experiments under continuous illumination. After calibrationwith experimentally determined values, we define a regime of illuminationensuring uniform temperature and deoxy-HbS concentration, and fast(within <1 s) egress to steady conditions. With these procedures, data on thenucleation and growth kinetics have relative errors of <5% and arereproducible within 10% in independent experiments. The nucleation ratesand delay times have steep, exponential dependencies on temperature. Incontrast, the average fiber growth rates only weakly depend ontemperature. The individual growth rates vary by up to 40% underidentical conditions. These variations are attributed to instability of thecoupled kinetics and diffusion towards the growing end of a fiber. Theactivation energy for incorporation of HbS molecules into a polymer isEA=50 kJ mol− 1, a low value indicating the significance of thehydrophobic contacts in the HbS polymer. More importantly, the contrastbetween the strong θ(T) and weak R(T) dependencies suggests that thehomogenous nucleation of HbS polymers occurs within clusters of aprecursor phase. This conclusion may have significant consequences forthe understanding of the pathophysiology of sickle cell anemia and shouldbe tested in further work.

© 2006 Elsevier Ltd. All rights reserved.

Keywords: sickle cell anemia; hemoglobin S polymerization; nucleationkinetics
*Corresponding author
Introduction

Sickle cell anemia1,2 is a genetic disease caused by asingle mutation from glutamate to valine at the sixthsite of the hemoglobin β chain,3 resulting in pro-duction of abnormal hemoglobin, HbS.4 Despite thelocal character of the mutation, the clinical manifes-tations of the disease are very diverse and range fromlife-threatening to asymptomatic conditions.1,2

Although stem-cell transplantation and gene ther-apy are promising the hope of a cure,5 in practice

ng author:

lsevier Ltd. All rights reserve

these treatments may be costly and not readilyavailable in developing countries, where the inci-dence of the disease is highest. Thus, a search forother means of therapeutic intervention is stillongoing.2

One of the basic events in sickle cell anemia is thepolymerization of HbS inside the red blood cells,which occurs upon deoxygenation of hemoglobin.6

HbS polymerization is a first order phase transitionin solution7 and this links sickle cell anemia to otherprotein condensation diseases: eye cataract, Alzhei-mer's, Huntington's, and the prion diseases.8

Significant efforts invested in the investigation ofthe polymerization mechanisms brought a wealth ofinformation and a feeling of understanding of thephysico-chemical fundamentals of the process,9 yet

d.

mailto:[email protected]

426 Kinetics of HbS Fibers Nucleation and Growth

no effective antisickling drug was found.10 Thecurrent treatment strategies do not include attemptsto directly inhibit hemoglobin sickling.11 A majorobstacle for the antisickling approach has been thehigh concentration of hemoglobin inside the ery-throcytes, which is incompatible with acceptableconcentrations of a drug targeting most HbSmolecules.12 Thus, further studies of the polymer-ization mechanisms are needed. These studiesshould be aimed at providing suggestions of anti-sickling mechanisms, effective at concentrations ofthe respective antisickling agents orders of magni-tude lower than those of HbS.The process of polymerization has been divided

into homogeneous nucleation of the polymer fiber,growth of the fibers, and heterogeneous nucleationof new fibers on top of exiting ones, leading tospherulites and eventually to a thick gel.13–15 Wefocus on the homogeneous nucleation: it has anexponential dependence on HbS concentration andis susceptible to control via the solution composition.Insight into nucleation of first order phase transi-tions comes either from direct imaging of the nucleistructures and their evolution16,17 or from data onthe kinetics of nuclei formation.18–20 We choose thesecond method as a step in the investigation of thekinetics of polymer nucleation, and here we presentdeterminations of homogeneous nucleation delaytimes, nucleation rates, and rates of growth of fibers,at different concentrations of HbS and temperatures.In a previous article21 we introduced an experi-

mental setup and procedures for such determina-tions. While conceptually straightforward, theexperiment protocol is challenging and requiresfine-tuning of numerous parameters. Here, weanalyze the factors that influence the determinationprocedures and define the limits of the technique.We discuss additional features and improvements tothe experiment protocol, including automated dataprocessing, which greatly reduces the experimenttime and effort. Finally, we discuss data on thekinetics of nucleation and growth of the HbSpolymer fibers at different concentrations andtemperatures, which have high accuracy and canserve as a basis for conclusions on the mechanismsof homogenous nucleation of the HbS polymerfibers and their growth.

Results and Discussion

General outline of the experiment procedures

A slide with a thin layer of carbomonoxy sicklecell hemoglobin, CO-HbS, is continuously photo-lyzed13,14 by a Nd3+:YAG laser beam with wave-length λ=532 nm with tunable total power Ibetween 2 and 200 mW. The photolysis produces aregion of deoxy-HbS with diameter of about 50 μm.If, at the chosen temperature, the deoxy-HbSconcentration is above its solubility with respect tothe polymers, the solution is supersaturated and

fibers and then spherulites form. Their evolution ismonitored using a microscope with differentialinterference contrast optics22 and a CCD camera.Processing of 80 to 200 time sequences of imagesallows extraction of the homogeneous nucleationrates and the delay times of nucleation, as well as thegeometric characteristics of the spherulites.21

The choice of experimental parameters is criticalfor the experiment procedure. The external para-meters under our control are the total intensity oflaser beam, its diameter, the thickness of the solutionsample in a slide, and the concentration of the HbSsolution. The two solution parameters, whichdetermine the nucleation and growth rates, aretemperature and fraction of deoxy-HbS, i.e. thedegree of photolysis of CO-HbS. The energy of thelaser light used to produce deoxy-HbS is absorbedby the solution and leads to local overheating.Lower overheating and better temperature unifor-mity are obtained at lower illumination intensity. Incontrast, the degree of photolysis reaches the desired95% faster and is more uniform at higher laserpower.Additional complications arise: while the laser

intensity and its distribution are steady, the spatialprofiles of the fraction of deoxy-HbS and tempera-ture are time-dependent. The fraction of deoxy-HbSdepends on the local concentration of CO in thesolution. The carbon monoxide released by the HbSmolecules diffuses out of the illuminated area andthe rate of CO diffusion determines the fraction ofdeoxy-HbS. The HbS solution is held by glass plates,which do not absorb the illuminating wavelength,and is surrounded by HbS solution at a temperatureset by a circulating water bath. The conduction ofheat by the glass and the surrounding solutiondetermines the time-dependence of the temperaturefield in the solution.Below, we discuss procedures, which allow us to

find ranges of the control parameters, where thecontradictory requirements on temperature anddegree of photolysis are satisfied. Before that, wediscuss procedures to account for variations in pHand met-HbS content. (In met-Hb the Fe2+ in theheme is oxidized to Fe3+. Because of its differentconformation, met-HbS is excluded from the HbSpolymers).

Influence of sodium dithionite on solution pH

Addition of sodium dithionite, Na2S2O4, is acommon way to reduce met-hemoglobin to deoxy-hemoglobin and remove traces of oxygen from thesolution,23 following complex reaction pathways.24,25

We found that to completely reduce the amount ofmet-HbS present in a solution with total CHbS=230–250 mg ml−1≅0.004 M, one needs sodiumdithionite concentration of ∼0.05 M, similar to thatpublished.13,22 These concentrations are comparableto the buffer concentration of 0.15 M. The oxidationof the dithionite ion produces sulfuric and sulfurousacids24,25 and they appreciably lower the solution pHfrom its initial value of 7.35, see Figure 1. Because the

Figure 2. Simulated spectra of mixtures of met-HbSand CO-HbS normalized to 1 at 539 nm (the extinctioncoefficients of the different hemoglobin species fromSalvati & Tentory27). Fraction of met-HbS indicated onthe plot. Insert: same data in linear coordinates. Verticalbroken line at 630 nm denotes wavelength at which met-HbS content is evaluated.

Figure 1. Dependence of pH of 0.15 M potassiumbuffer solution on concentration of added sodiumdithionite. Circles, in glove box under He atmosphere;squares, in open air. In the He atmosphere, the reductionof dithionite yields less acidic species and this leads tolower deviation of pH from the initial value. At higherdithionite concentrations reduction becomes autocatalyticand reaches its maximum extent regardless of theatmosphere.24,25

427Kinetics of HbS Fibers Nucleation and Growth

solubility of HbS depends on pH,26 reproducibility ofthe final pH is obtained by adding the same amountof dithionite in all experiments. In addition, sinceunreacted sodium dithionite may lead to continuouspH changes during an experiment, to remove theunreacted reagent and ensure pH stability, we brieflyexpose the solution sample to air prior to sealing theslide.

Measurement of the met-HbS content

Nucleation data are useful if they are obtained insolutions, which contain negligible amounts of HbSin states other than carbomonoxy. The maincomponent, remaining after thorough conversionof oxy and deoxy-HbS to CO-HbS is met-HbS. Sincemet-HbS has a significantly different structure thandeoxy-HbS, it is not incorporated into the polymerand dilutes the HbS concentration in the solution.The amount of met-HbS is quantified from theabsorption spectrum of the solution, however,amounts of met-HbS as large as 10% change thespectrum insignificantly as judged by the plot inlinear coordinates in the inset of Figure 2. Wequantify the amount of met-HbS from the solutionabsorbance at 630 nm. Figure 2 shows simulatedspectra of mixtures of met-HbS with CO-HbS nor-malized to 1 at 539 nm (the extinction coefficients ofHbwere taken from27). We only work with solutionscontaining <5% met-HbS, for which the normalizedoptical density at 630 nm is <0.023, Figure 2.

Simulation of the temperature profile

Solution absorption of the laser light, used tophotolize CO-HbS, leads to an increase of tempera-ture in the illuminated area. The resulting spatialprofile of the temperature in the slide is non-uniform

due to non-uniform intensity profile of the laserbeam. Furthermore, heat diffuses out of the illumi-nated area and this significantly increases thetemperature non-uniformity. To understand howdifferent experiment parameters affect the tempera-ture regime inside the slide with solution, we solve aregular three-dimensional equation of heat transfer:

j kjTð Þ þ q ¼ UCpBTBt

where k is the thermal conductivity of the conduct-ing material, solution or glass, q is a heat source, ρand Cp are the density and the specific heat of theconducting material, and∇ is the gradient operator.This equation ignores convective transfer of heat, i.e.via solution flow that may arise in a temperaturegradient in the Earth's gravity field. Our justificationis that the region of high horizontal temperaturegradients surrounds the region, from which data arecollected, and is removed from it by tens of microns,see below. In addition, with the thin solution slidesemployed (5 or 10 μm), convective flows aresignificantly suppressed.28

Assuming that the laser intensity is cylindricallysymmetric with respect to the optical axis and thatthe illuminated area is far from the sides of the slide,we neglect the angular dependence of the tempera-ture field and reduce the above equation to:

1rB

BrkrBTBr

� �þ B

BzkBTBz

� �þ q ¼ UCp

BTBt

; ð1Þ

where z is the vertical axis of symmetry. Figure 3(a)shows the model geometry of a slide, which iscomposed of two layers of glass with different thick-nesses and a layer of HbS solution sandwiched be-tween them. We approximate the thermal properties

Figure 3. Simulations of the temperature field inside aslide. (a) Schematic of the composite geometry of a slideused in simulation; dimensions not to scale. Axis z isdirected along the optical axis of microscope. (b) Exampleof a temperature profile obtained from equation (1) withthe laser beam radius of 25 μm. Temperature andhorizontal scales indicated on plot.


of the HbS solution with those of water and use thefollowingvalues of thematerials' parameters:ρwater=1000 kg m−3, Cp,water=4181.8 J kg− 1 K− 1, kwater=0.58 J s− 1 m− 1 K− 1, ρg;ass=2500 kg m−3, Cp,glass=600 J kg− 1 K− 1, kglass=1.05 J s

− 1 m− 1 K− 1. Since laserlight is only absorbed by the HbS solution, the heatsource is defined only for the solution layer as:

q z; rð Þ ¼ F rð Þ � A zð Þ ¼ I2pj2 e

� r2

2r2

�OD ln10l

10�ODð1�z=lÞ; ð2Þ

where F(r) is the density of laser intensity for aGaussian beam with total intensity I and radius σand A(z) accounts for the fraction of radiationabsorbed at a depth z. The optical density of thesolution at the wavelength of the illuminationλ=532 nm is:

OD ¼ E532nmHb CHbSl; ð3Þ

where εHb532nm is the extinction coefficient of HbS,CHbS

is its concentration, and l is the thickness of the slide.The boundary conditions were set so that the edgesof the slide were at room temperature T=300 K. Wesolved equation (1) with these parameters andboundary conditions using the finite elementmethod implemented in the package Femlab (Com-sol, Inc.) in the geometry depicted in Figure 3(a). Anexample of the resulting temperature distributions inthe slide is shown in Figure 3(b).Since the heat source term q in equation (1) is

proportional to the total laser intensity I, equation(2), this I linearly scales the temperature distribu-tion T(r,z), but does not affect its shape. Wepresent the simulation results as overheating ΔT,which is the increase of the solution temperaturedue to the absorption of laser light. Unlessexplicitly specified, in the simulations below weassume I=10 mW, the value used in the kineticsdeterminations.First, we estimate the time needed to reach a

steady temperature value after the laser beam isturned on. Figure 4(a) shows the evolution of ΔT,normalized to the steady-state value. This ΔT issampled at the center of the illuminated area, at r=0.Figure 4(a) shows that 95% of the overheating isreached within ∼10 ms. This is significantly shorterthan the shortest time for appearance of HbS fibersof ∼1 s and longer in our experiments.To evaluate the spatial inhomogeneity of the

temperature profile, we compare in Figure 4(b) theradial distribution of temperature resulting fromtwo heat sources: one with a uniform distribution ofintensity and one with a Gaussian distribution, asthe laser beam employed in the experiments, seeFigure 4(b). This results, also in Figure 4(b), showthat in both cases ΔT at the center of the slide is atleast ∼2× higher than that at the edge of theilluminated region. In addition, the overheatingextends to and slowly decays outside the illumi-nated area. In vertical direction, i.e. along the z-coordinate, the inhomogeneity of ΔT is less pro-nounced, reaching ∼7% at the top and bottom glasssurfaces (Figure 4(c)).In Figure 5 we simulate the response of ΔT to

variations in the experiment parameters, assumingGaussian or uniform intensity distributions. Figure5(a) shows that the ΔT at the center of theilluminated area is nearly proportional to thelaser beam diameter. Figure 5(b) shows that ΔTis also nearly proportional to the total opticaldensity of the solution, defined in equation (3).Figure 5(c) shows that if the total optical density iskept constant, e.g. by reducing the HbS concentra-tion proportionally to the thickness increase,variations of the slide thickness induce ΔT varia-tions of <3%.These simulation results show that the overheat-

ing is mostly determined by the total optical densityof a slide and the laser beam diameter, and that it isalmost independent of slide thickness. To optimizethe experiments parameters, we assume HbS con-centration as in the nucleation kinetics experiments

Figure 4. Simulation results for overheatingΔT due toabsorption of laser light. (a) Dependence of overheatingΔT at the center of the illuminated area, r=0, z=2.5 μm, ontime after switching on the laser. (b) Radial profiles ofoverheating ΔT in the middle of the slide, z=2.5 μm,under uniform (black broken line) and Gaussian (graybroken line) intensity profiles, depicted with respec-tive continuous lines. (c) Vertical spatial profile of over-heating ΔT at r=0, calculated assuming Gaussian laserbeam with intensity I=10 mW and radius σ=25 μm,CHb= 250 mg ml− 1, slide thickness l=5 μm. ΔTmax is thesteady-state value of overheating.


of ∼250 mg ml−1. If the slide thickness is 5 μm, thetotal optical density OD532nm=0.062. According toFigure 5(b), with laser intensity of I=10 mW andbeam radius σ=25 μm, this yields ΔT≈2 K at thecenter of the illuminated area.Temperature uniformity is an important factor for

the quality of the nucleation kinetics data. The

nucleation rates and delay times vary by ∼5–10% iftemperature varies by 0.5 °C. While these variationsin the kinetic variables are comparable to othererrors in their determination, temperature variations>0.5 °C would lead to unacceptable uncertainty.Thus, only data collected from areas within whichtemperature variations are constrained within<0.5 °C should be considered. The ΔT distributionsin Figure 4(b) show that such areas are limited to thecenter of the illuminated area and that their radiusdepends on the total beam intensity I. For anaccurate estimate of the radius, in Figure 5(d) weshow temperature profiles calculated assumingthree different values of I: the radius decreasessignificantly with increase of I.An additional requirement on the radius of the

laser beam is set by the microscopy technique. Todiscern spherulites in a digital image, the imagedarea should contain at least 100×100 pixels; withoptical resolution of 0.5 μm, the radius of the imagedarea should be ≥25 μm. Figure 5(d) shows that thisrequirement is met by a beam with I=10 mW andσ=25 μm. Note that the choice of experimentparameters would be less constrictive with lowerHbS concentrations. For instance, with HbS concen-tration of 25 mgml−1 and corresponding OD=0.006,the laser power needed for complete photolysis issignificantly lower and the induced overheating ΔTis ∼0.1 K.

Experimental determination of temperatureprofile

Experimental determination of the temperatureprofile is necessary to calibrate the overheatingestimated by the model and to test its main features.Such determinations face three main difficulties.(1) Temperature is measured in a layer, which is5–10 μm thick. (2) The diameter of the illuminatedarea is ∼50 μm, so the temperature probe should beseveral microns in size. (3) The temperature probeshould not disturb the temperature profile by itsmere presence. To satisfy these requirements, wedesigned a technique based on the use of a heat-sensitive material with a well-defined melting pointnear the used temperature range. We use the crayonOMEGASTICK (Omega Engineering Inc.) withTmelt=40.5 °C. We measure the laser intensity I, atwhich the crayon melts, as a function of location onthe slide and solution conditions. In a previousarticle,21 we assumed that if the crayon is illumi-nated, it would disturb the heat profile and the grainof crayon was hidden behind a mask. This yielded alower value of the real ΔT because masking distortsthe temperature distribution.To test if the crayon disturbs the heat distribution,

we placed a ∼5 μm grain in a transparent bufferwithout protein and exposed it to the maximumlaser intensity of 160 mW, while keeping thetemperature below its melting temperature by0.1 K: the crayon did not melt, indicating that itdoes not absorb the illuminating wavelength anddoes not affect the solution overheating.


To quantitatively characterize the overheatingduring a nucleation experiment, we carried outtwo independent series of determinations: of theshape of the T-distribution and of the overheating atthe center of the beam ΔT(r=0). To determine theshape of the temperature profile in and around the

illuminating beam in a HbS solution, we set thetemperature of the water circulator maintaining thetemperature of the slide to ΔT*=0.5 K below themelting point of the crayon. The slide is translatedalong a straight line chosen so that the grain passesthrough the center of the illuminating beam. At eachpoint r (r is measured from the center of the beam)we determine the laser intensity I*(r), at which thecrayon melts. According to (1)–(3), the overheatingΔT* is proportional to this intensity I*(r): ΔT* ∝S(r) I*(r). Thus, the shape of the temperature profileS(r) ∝ 1/I*(r).The experimental determination of the tempera-

ture profile is compared to the calculated one inFigure 6(a). The laser beam diameter σ was 10 μm.This width is narrower than the one used in thenucleation experiments to highlight the tails outsideof the illuminated area: with wider beams, these tailsare only detectable with high laser intensity, atwhich the center fractions of the illuminated solu-tion coagulate. The overall shape is very close to thatobtained from the simulation and possesses all of itscharacteristic features: the area of increased tem-perature is wider then the photolyzing beam, andthe profile has a long exponential tail.To calibrate the temperature profile estimated by

the model, we determine the temperature over-heating at the center of a slide ΔT(r=0). For this, weplace there a grain of the temperature-sensitivecrayon and determine the temperature of the non-illuminated solution Tbackground at which the crayonmelts as different intensities of illumination I andoptical densities OD of the solution (Figure 6(b)).The overheating of the solution ΔT due to absorp-tion of the illuminating light isΔT=Tmelt–Tbackground(I, OD). As expected from (1)–(3), these dependencesare linear. The dependence of the overheating on thesolution OD (Figure 6(c)) allows us to calibrate theoverheating for a chosen setting of the laser intensityI and beam width 2σ.

Evaluation and determination of the profile ofdegree of photolysis

In addition to control of temperature, we need toachieve steady concentration of deoxygenated HbS

Figure 5. Influence of experiment parameters ontemperature overheating ΔT. (a) Dependence of thesteady-state overheating ΔTmax, defined in Figure 4(a),in the center of a slide, at r=0 and z=2.5 μm, on the radiusof laser beam with uniform intensity distribution, seeFigure 4(b). (b) Dependence of the steady-state over-heating ΔTmax in the center of the slide on the opticaldensity of the solution, illuminated by a beam withGaussian intensity distribution. (c) Dependence of over-heating ΔTmax in the center of the slide on slide thicknessat constant optical density of the solution, illuminated by abeam with Gaussian intensity distribution. (d) Profiles ofoverheating for three intensities of illumination I indicatedin the plot. Radius of the area, within which ΔT< 0.5 K, isindicated and it decreases with increasing intensity. Modelparameters are the same as in Figure 4(c).

Figure 6. Experimental determination of the over-heating ΔT in a slide. (a) Comparison with simulationresults: Black diamonds, experimentally determined tem-perature profile. Continuous red line, calculated tempera-ture profile. Green circles, measured profile of laserintensity; continuous green line, Gaussian fit to laserintensity profile. Radius of laser beam σ=10 μm. (b)Temperature of non-illuminated solution, at which a grainof crayon, placed in the center of illuminated area, melts,as a function of intensity of illumination I at the five valuesof the solution optical densities, OD, indicated in the plot.The difference between Tmelt=40.5 °C of crayon andtemperature plotted here equals the overheating ofsolution ΔT due to absorption of illuminating light. (c)Dependence of overheating ΔT at center of illuminatedarea on optical density of HbS solution for illuminatingintensity I=10 mW.


in the illuminated solution. This is necessary foraccurate evaluation of the supersaturation drivingHbS polymerization, which, in turn, is a prerequisitefor the use of kinetics data as a basis for conclusionson the nucleation mechanisms.We first model the profile of the degree of

photolysis f ¼ Cdeoxy hemes

CCO hemesat a time immediately

after the initiation of photolysis. These calculationsfollow the line put forth by Ferrone et al.13 Todescribe the dependence of f on laser intensity I, weuse the two-state allosteric model of Monod,Wyman, and Changeux.29,30 This model explainsthe characteristic S-shaped dependence of thefraction of deoxy hemes f on the concentration ofcarbon monoxide CCO:

f ¼ 1� KRCCOð1þ KRCCOÞ3 þ LKTCCOð1þKTCCOÞ3ð1þ KRCCOÞ4 þ Lð1þ KTCCOÞ4

ð4ÞThe model contains three basic parameters: KR andKT are the intrinsic association constants of CO withhemoglobin molecules in R and T- states, respec-tively, and L is the constant for R to T equilibrium ofunliganded Hb molecules. R and T are the twoconformational states of hemoglobin, the R-confor-mation is selected in oxy-state, and the T-conforma-tion in deoxy state.31 The association constants aredefined as:

KR ¼ konRkoffR ðIÞ KT ¼ konT

koffT ðIÞ ð5Þ

where kon are the overall association constants andkoff are the overall dissociation constants. The dis-sociation constants koff depend on laser intensity I as:

koffR;T ¼ BR;TnCh

¼ BR;TF

hvNA

1� 10�eR;TChl

lChcBR;T

FhvNA

ln10 eR;T

ð6Þwhere φ is the quantum yield of photolysis, n is thenumber of absorbed photons, Ch=4 CHbS is theconcentration of hemes; F is the density of laserintensity for a Gaussian beam defined in equation(2), hν is the energy of a photon of the laserwavelength of 532 nm, NA is Avogadro's number,εR, εT are the extinction coefficients of CO-HbS anddeoxy-HbS at the laser wavelength, l is the thicknessof the sample.The concentration of CO in the solution is

determined by its release from the hemes due tophotolysis:

CCO ¼ C0 þ g f Chemes ð7Þ

where C0 is solubility of CO in water, γ=1.4 is acorrection coefficient accounting for decrease ofvolume available for CO molecules due to thepresence of Hb molecules.13


We combine (4)–(6) and insert the result for f intoequation (7) and find the roots using Mathcad(MathSoft, Inc). We use the following parametervalues: L=6×104, konR =6×106 M− 1s− 1, konT = 1×105

M− 1s− 1,32 φR=0.47, φT=1.0,33 ε532 nm

R =13.1×103

M− 1cm− 1, ε532 nmT =8.22×103 M− 1cm− 1.27 The solu-

bility of CO in water is C0=1 mM34 and, in ourconditions, it is significantly less then the con-centration of CO released from hemoglobin mole-cules (for instance, CCO=15.5 mM if 250 mg ml−1=3.9 mM HbS solution is completely photolyzed). InFigure 7(a) and (b) we plot the spatial distributionof the fraction of deoxy hemes f(r) at different laserintensities. Note that the coefficients in equation (4)significantly depend on solution conditions35 andall available sets of values are for conditionsdifferent form those used in the nucleation experi-ments. Thus, the results in Figure 7(a) and (b) onlyhave qualitative character.Figure 7(a) shows that high levels of photolysis are

achieved only for sufficiently high laser intensities.From this point of view, it would be advantageousto work at lower concentration of HbS, e.g. CHbS=25 mg ml− 1, as in Figure7(b), where the samelaser intensities produce much higher level ofphotolysis. As can be expected, the radius of thearea of full photolysis increases with increasing laserintensity.As discussed by Ferrone et al.,13 the spatial profile f

(r) is time-dependent due to the diffusion of COaway from the illuminated area. Equations (4)–(6)show that the level of photolysis depends on localconcentration of CO at a given laser intensity I.Diffusion will decrease the concentration of COcreated at t=0, which will cause the release of moreCO molecules from hemoglobins and will increasethe fraction of deoxy hemes fwith time. Tomodel thetime dependence of f(r), we solve a diffusionequation for CCO:

BCCO

Bt¼ DCODCCO þ s; ð9Þ

where Δ=∇2 is the Laplace operator and DCO is theCO diffusion coefficient, in a cylindrical geometrywith no angular dependence and with a time-dependent source s determined by the release ofCO molecules due to photolysis:

BCCO

Bt¼ 1

rB

BrrDCO

BCCO

Br

� �� þ CCO

BfBCCO

BCCO

Bt

ð10Þ

As an initial condition for these calculations wetake the profile f(r) calculated from equations (4)–(7).Equation (10) was solved using the finite elementsmethod implemented in Femlab (Comsol Inc.).DCO in hemoglobin solutions was taken to be3×10−6 cm2 s− 136 and the time-dependent sourceterm was evaluated from equations (4)–(7).Figure 7(c) shows the time dependence of the

photolysis profile f(r) for I=5 mW. Diffusion of COcauses increase of f(r=0) and of the radius of the

photolyzed area. Again, lowering the concentrationof HbS would produce larger area of full photolysisat the same laser intensity (Figure 7(d)).Evaluation of the time-dependence of f at the

center of the illuminated area in Figure 7(e) and (f)shows that it achieves saturation within severalseconds. This time agrees with a simple estimateτ=σ2/DCO=2.08 s for the characteristic radius of thephotolyzed area σ=25 μm, and this agreementindicates that the evolution of f at times >0 isdetermined by the diffusion of CO and its time scaleis practically unaffected by the release of additionalCO from the HbS molecules.Evaluation of the dependence of f on the laser

intensity I in Figure 7(g) shows that a decrease in thesolution concentration will significantly reduce thelaser intensity needed to achieve full photolysis.We experimentally determined the time evolution

of the fraction of deoxy HbS. We used illuminationwith λ=436 nm (produced by a band pass filter(Edmund Industrial Optics) with halfwidth 10 nm)and monitored the region within the laser beam.This wavelength is differently absorbed by deoxy-HbS and CO-HbS. The optical density of the solu-tion at this wavelength is a function of the fraction ofdeoxy hemes f.21 Visual observation revealed a darkspot, corresponding to area of deoxy-HbS, whichgrew in diameter on the timescale of seconds. Wecaptured the images with a linear response CCDcamera (Kodak Megaplus ES1.0/1260), in which theintensity recorded by a pixel is proportional to thenumber of photons reaching this pixel. These imagesallow determination of the spatial distribution of f inthe monitored area. Figure 7(h) shows profiles of f atdifferent times. The laser intensity was set to a lowI=5 mW to slow down the evolution of f and allowsufficient time resolution of the observations. Theevolution of the profiles in Figure 7(h) is similar tothe simulation results assuming similar conditionsin Figure 7(c): a region of f≈1 is reached after ∼5 s,the distribution flattens out and its diameter slowlyincreases with time.The numerical and experimental results on the

fraction of deoxy-HbS indicate that with laserillumination intensity I=10 mW, an optimal valuefrom the viewpoint of solution overheating, theinitial fraction of deoxy HbS is about 0.87 and itegresses to near 1 after about 2 s. For polymerizationkinetics experiments with CHbS in the range 200–230 mg ml− 1 the nucleation delay times are 10 s andlonger, i.e. the increase in supersaturation at t≈2 smay be insignificant. However, if one aims atcharacterization of faster kinetics, means to reachsolution steady state sooner should be considered.

Automatic processing of data on nucleation andgrowth kinetics

To quantify the kinetics of nucleation and growthof the HbS polymers at one HbS concentration, werun experiments at four to seven temperatures. Ateach temperature, 82 to 200 runs are carried out,each consisting of ∼30 images.21 Thus, a series of

Figure 7. Fraction of deoxy-HbS. In (a), (c) and (e) CHbS=250 mg ml− 1; in (b), (d) and (e) CHbS=25 mg ml− 1. (a)–(d)Simulated profiles of the fraction of deoxy hemes f. (a) and (b) Immediately after initiation of photolysis, at different laserintensity I. (c) and (d) At different times, laser intensity I=5 mW. (e) and (f) Time-dependences of the fraction of deoxyhemes f at the center of photolized region at different laser intensities. (g) Simulated dependence of the initial fraction ofdeoxy hemes f at the center of illuminated region on laser intensity I with two different HbS concentrations. Modelparameters in (a)–(g): Gaussian laser beam with radius σ=25 μm, slide thickness l=5 μm. (h) Experimentally measuredprofiles of the fraction of deoxy hemes f at different times for I=5 mW.


determinations totals to at least 10,000, often up to50,000, images.For automated processing of these large data sets,

we developed a custom software package, based ongeneral image processing routines.37 The programdetermines the total number of spherulites in each

image and their geometrical characteristics. We usedthe IDL programming language (Research SystemsInc.) to implement the following image processingsequence: (1) The first image in every run (it does notcontain any fibers or spherulites) is subtracted fromall subsequent images as a background. (2) To reduce


high-frequency noise, each difference image issmoothed with a boxcar average of specified width.(3) To remove small-scale noise and the large-scaleintensity non-uniformity, a bandpass filter is appliedwith lower and higher settings adjusted for thespherulites. (4) The image is binarized with a levelcut adjusted so that areas containing spherulites arenon-zero. (5) An area cut is performed so that featureswith areas lower than a prescribed value are set tozero. This removes spikes resulting from noise, whichusually have small area. (6) A dilation-erosion ope-ration is applied, which unifies pixels belonging to thesame spherulite. (7) The resulting features in theimage are consecutively labeled and every spheruliteis assigned a different color. (8) A mask is applied toremoveall features lyingoutside the region of interest,inwhich, for instance, the overheating and the level ofphotolysis satisfy the requirements discussed above.(9) The number of spherulites in the image is counted.Their geometrical characteristics, such as length oflongest dimension, area, and orientation are deter-mined. Figure 8 shows an example of the applicationof steps (1) through (9) to one image in a run.

Figure 8. Illustration of automatic procedure fordetecting the number of spherulites and determining theirdimensions. (a) Initial image, (b) processed image withoutlined features of detected spherulites. Determination oforientation angleϕ and spherulite length (L) are illustrated.The region of interest (ROI), in which spherulites arecounted, is shown.

Counting of spherulites stops when they grow toolarge, as judged by two criteria. (1) If the total areacovered by all spherulites is greater then a presetthreshold. (2) If the area of any single spherulite isgreater than another preset threshold. To determinethe nucleation rates and delay times, for each run theprogram composes the dependence of number ofspherulites on time. In this dependence, the numberof spherulites increases by random numbers, asillustrated in Figure 3(b) of Galkin & Vekilov.21 Thenumber of spherulites present in images from allruns at a certain time is averaged. The time depen-dence of the mean number of spherulites is plottedas shown in Figure 5 of Galkin & Vekilov.21

The growth rates of the spherulites are calculatedby comparing the lengths in the longest dimension ofspherulites of the same color in images within samerun. Negative growth rates were never obtained,and this certifies to the accuracy of the spheruliteidentification and labeling. Spherulite orientationswere determined as the angle between the longestaxis of a spherulite and the horizontal direction.These procedures allow processing all data from

one experiment without operator interferencewithin times from 10 min to 1 h. Manual processingrequires the participation of a qualified operator andtakes three to four days. For tests and verification ofthe automated procedures, several data sets wereprocessed with both methods. The rates of nuclea-tion and growth of the fibers and the nucleationdelay times stemming from both methods wereindistinguishable. The typical uncertainties of thedeterminations were 4–5% of the mean values.These error ranges are about the size of the symbolsand error bars are not plotted.

Quantification of the nucleation and growthkinetics

The time dependencies of the mean number ofspherulites typically consist of three regions (Figure9). At long times, the area occupied by spherulitesand their supply fields, in which the HbS concentra-tion is lower, is significant. This reduces the areaavailable for nucleation of new fibers and slows therate of increase of the number of spherulites. Atshort times, no spherulites are present at any runand the mean number of spherulites is zero. Atintermediate times, the mean number of spherulitesincreases linearly with time and this linearityindicates steady-state regime of homogeneousnucleation of the HbS polymer fibers.The time dependencies of the mean number of

spherulites can be used to determine two character-istics of nucleation: the nucleation delay time θ, andthe nucleation rate J. These determinations are basedon the assumption that each spherulite is generatedby a single primary nucleation event, justified byBriehl.38 The relative error introduced by thisassumption is equal to the probability of havingtwo nuclei within the slide area occupied by onespherulite. Nucleation rate data are extracted fromimages of an area of ∼2500 μm2, containing fewer

Figure 9. Illustration of the determination of thenucleation statistics, nucleation delay time and nucleationrate at the conditions shown in the plot. (a) Increase of thenumber of detected spherulites with time. Step-wise lines,individual series of images, in each subsequent image,taken once per second, the number of spherulites isunchanged or increases by one, or another whole number.Four such series, distinguished with broken, dotted, fine-doted, and dash-dotted lines, are shown. At each timestep, the numbers of the spherulites detected in 81 to 200independent series are averaged. An example of thedistribution of these numbers is shown in (b). Similarly todistributions of the numbers of spherulites at differenttimes, concentrations and temperatures, for examples, seeGalkin & Vekilov,21 the distribution is near-Poissonian,shown with continuous line in (b), indicating theindependence of the individual nucleation events fromone another. The mean number of spherulites for eachtime is plotted in (a) with filled circles. The straight linethrough these points indicates a regime of steady-statehomogeneous nucleation. The intercept with the time axisand the slope of this line determine, respectively, thenucleation delay time ϕ, and nucleation rate J, asillustrated in the plot.


than ten spherulites, each occupying an areaof <10×1 μm2. The probability of having a secondspherulite hidden under or within one of the tenis about (10/2500)2×10=1.6×10−4. This estimate issomewhat higher than the one determined byGalkin & Vekilov21 because of the narrower area ofinterest used here.

The time at which the extrapolated steady-stateline crosses the time axis is the nucleation delay timeθ19 (Figure 9). Galkin & Vekilov21 assumed that theordered HbS polymer nuclei assemble in the diluteHbS solution, following the so-called direct or one-step nucleation mechanism. With this assumption,the nucleation delay time was interpreted asZeldovich time required for transition between ametastable steady state of a homogeneous super-saturated HbS solution, and another state, in whichnucleation occurs at a steady rate. Recently, adifferent, two-step, nucleation mechanism has beenshown for many transitions from dilute solutions toordered solid phases: protein, colloid, and small-molecule crystals, and amyloid fibrils.39–44 Accord-ing to this mechanism, the formation of orderednuclei is preceded by the formation of droplets of adense liquid and occurs within these droplets.Mesoscopic metastable liquid droplets were identi-fied as likely precursors for ordered-phase nuc-leation,45,46 and the applicability of this mechanismto a variety of systems41,43,47 has been suggested.Our purpose here is not to test the applicability ofthis more complex mechanism to the nucleation ofthe HbS polymers. If the two-step mechanismoperates for HbS polymerization and the nucleationof the dense liquid droplets is the rate-limitingkinetic step, the nucleation delay time has a differentmeaning: it is reciprocal to the product of the rate ofnucleation of the HbS fibers within the dense liquiddroplets and the total volume occupied by thedroplets.48

The slope of the linear part of the time dependenceof the mean number of spherulites in Figure 9 yieldsanother nucleation characteristic. If a one-stepnucleation mechanism operates, this slope is equalto the product of the homogeneous nucleation rateJc, i.e. the number of HbS fibers appearing withinunit volume per unit time, and the total volumefrom which data are collected. If the two-stepmechanism operates, this slope is equal the productof the rate of the nucleation of the dense liquiddroplets Jl to the same total volume. Since this totalvolume is easily evaluated, we plot the ratio of theslope to the volume and denote it with J.The growth rates R determined from the increase

of the length L of the individual polymers with timein Figure 10(a) vary significantly: if a mean value isdetermined based on the traces from ∼50 polymers,the individual determinations of the growth ratediffer from the mean by up to 40% (Figure 10(b)).These variations happen within the same slide, andeven the growth rate of a single polymer increasesand drops as it grows. The latter observationsuggests that the variations do not correspond todifferent thicknesses of the fibers: the thickness canonly increase as a fiber grows due to heterogeneousnucleation of new fibers on top of it. The character-istic time scale of the variations of the growth rate isseveral seconds (Figure 10(b)). Because of thearbitrarily chosen moment of start of monitoring ofeach fiber, the growth rate of some fibers initiallyincreases, while for others it decreases. The char-

Figure 10. Variable growth rate of the polymer fibers.(a) Increase in polymer length L with time at conditionsshown in the plot. Traces for 12 different polymers areshown. At time=0, individual polymers were between 2and 5 μm in length. (b) Variations of growth rate R at samecondition as in (a). Traces for six different fibers are shownwith different symbols.

Figure 11. Kinetics of nucleation and growth of theHbS polymers. Circles, CHbS=201 mg ml− 1; invertedtriangles, CHbS=210 mg ml− 1; squares, CHbS=230 mgml− 1. Different symbols correspond to different experi-ment series. (a) The nucleation rate J. (b) Nucleation delaytime ϕ. (c) Rate of growth HbS polymer fibers R.


acteristic time of HbS diffusion in a slide of thicknessd=10 μm is d2/D≈4 s, where D=2.7×10−7 cm2 s− 1

is the HbS diffusion coefficient at CHbS≈220 mgml− 1.49 Since convection in thin slides is almostentirely suppressed, see discussion above, thesupply of HbS molecules to the growing fiber isalmost purely diffusive. The correspondence of thecharacteristic diffusion time to the characteristictime of growth rate variations suggests that thevariations are due to the coupling of diffusivesupply to the kinetics of incorporation into thefiber: such coupling is known to readily lead tounsteady growth kinetics.50,51

The average growth rate in such coupled systemsis reduced, in comparison to the case of infinitelyfast supply of material, by a factor [1+Rdρpolymer/ρsolutionD]≈3,52 where ρsolution and ρpolymer are thenumber densities of HbS molecules in the solutionand in the HbS polymer, ρpolymer/ρsolution≈3.9 Thus,the average of the measured growth rate values isrepresentative of the intrinsic fiber growth rate, i.e.the growth rate unaffected by the transport limita-tions, and is lower than it by ∼3×.

The dependences of the HbS polymer nucleationand growth rates on HbS concentration andtemperature

Figure 11(a)–(c) shows the dependencies of thenucleation rate J, the nucleation delay time θ and theaverage growth rate R of the polymers on tempera-ture at three different HbS concentrations. Severalindependent experiments, starting from HbS isola-tion and purification, were carried out at each HbSconcentration. The variables J and θ are reproducibleto within their error ranges. The nucleation rate Jand the delay time θ have steep exponential-typedependencies on temperature. While such depen-dence is expected for J for both the direct and two-step mechanism, a steep θ(T) dependence can beinterpreted in favor of the two-step mechanism.48


Four of the five series of R data show increasing Rwith increasing T. These four series were recorded attemperatures below 35 °C, while the fifth series wasrecorded around 35 °C. Below 35 °C, HbS solubilitydecreases as temperature is increased, i.e. the super-saturation is higher at the higher temperatures.9 It ispossible that the unusual response of R to tempera-ture in the fifth series is related to the reversal of thesolubility dependence on temperature. Accountingfor the different supersaturations at different tem-peratures, we determine the Arrhenius activationenergy from each series of R data and get values of50(±5) kJ mol− 1. These values are lower than thetypical activation energies for chemical and enzy-matic reactions,53 however, they are similar to thosefound for the crystallization of several proteins.54

This similarity and the hydrophobic nature of theintermolecular contacts in the HbS polymer9 suggestthat the activation energy reflects the breaking of thewater shell at the non-polar HbS residues in thecourse of establishing hydrophobic contacts.

Materials and Methods

Solution preparation and the experiment setup formeasurements of homogeneous nucleation rates of HbSfibers have been described in detail21 and here we onlybriefly summarize them and introduce some new features.

Solution preparation

Hemoglobin S was isolated from blood, purified onchromatographic column by salt gradient elution, dia-lyzed into 0.15 M potassium phosphate buffer (pH 7.35),concentrated to about 270 mg/ml, and stored under liquidnitrogen. Before each experiment, a 20 μl sample wasexposed to carbon monoxide for 1 h and 1 μl of sodiumdithionate was added for final concentration of 0.05 M.The solution was then loaded on microscope glass slideswith 5 μm glass sphere spacers.

Slide cleaning protocol

A procedure for thorough and reproducible cleaning ofthe surfaces of the glass slide and cover slip is essential forthe following reasons. The thickness of a HbS solutionsample during the nucleation kinetics determinations isonly several microns. Thus, dust particles on the slide arewithin the depth of imaging of the microscope and arerecorded in the captured images. They interfere with theautomatic digital image processing: they may increase thebase level of noise in an image, increase the size detectionlimits of spherulites, or be misconstrued for HbSspherulites. Microscopic inspection of glass slides cleanedmanually with a sponge and detergent showed that theywere often inadequately clean, with streaks and dustparticles on the surface.After considering different ways of glass cleaning,† we

adopted the following protocol for Corning 2947 microslides. The glass slides and cover slips were washedmanually with sponge and detergent to remove any

†http:\\www.solgel.com\articles\

visible dust and were placed in a glass beaker filled with10% solution of Alconox powdered precision cleaner (pH9.5). The beaker was then sonicated in an ultrasonic bathfor 25 min at a temperature of 50 °C. After that, the slideswere washed with ample amounts of deionized water,and then sonicated for another 25 min in deionized waterat 50 °C. The slides were once again washed with copiousamount of deionized water and then left to soak in roomtemperature water for 30 min. The slides were thenpromptly removed from water, dried under a flow ofnitrogen, and immediately placed in a closed plastic box toavoid contamination with dust. The slides cleaned in thisway were used within the same day.

Determination of slide thickness and definition of areaof interest

As discussed by Galkin & Vekilov,21 to achieve uniformthickness of the slide with HbS solution, we used glassspheres (Duke Scientific) with diameter 4.9(±0.5) μmplaced between the slide and the cover slip.For definition of slide thickness uniformity, before each

experiment, we determine the variations of the thicknessof the slide by measuring the spatial distribution of opticaldensity of the slide at 532 nm: OD is proportional to theHbS concentration and the slide thickness, and we assumethat concentration is uniform. For this, we scan the slidewith solution under the microscope objective with a lateralstep of 1 mm. Then we choose an area of interest, usually afew mm2 in size, and again determine its OD(x,y). Weselect area where OD varies by <2%. The coordinates ofthe selected area are stored in the computer controlling themicroscope stage. In this way, data were collected only atlocations within the selected area.

Acknowledgements

We thank V. Uzunova for careful reading of themanuscript. This work was supported by theNational Lung, Blood and Heart Institute, NIHthrough grant number G091474.

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Edited by M. Moody

(Received 30 May 2006; received in revised form 31 August 2006; accepted 1 October 2006)Available online 5 October 2006

the kinetics of nucleation and growth of sickle cell hemoglobin fibers

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