molecular-level thermodynamic and kinetic parameters for the self

doi:10.1006/jmbi.2000.4171 available online at http://www.idealibrary.com on J. Mol. Biol. (2000) 303, 667±678

Molecular-level Thermodynamic and KineticParameters for the Self-assembly of ApoferritinMolecules into Crystals

S.-T. Yau1, Dimiter N. Petsev1, Bill R. Thomas1,2 and Peter G. Vekilov1*

1Department of ChemistryUniversity of Alabama inHuntsville, HuntsvilleAL 35899, USA2Universities Space ResearchAssociation, Marshal SpaceFlight Center, HuntsvilleAL 35875, USA

E-mail address of the [email protected]

0022-2836/00/050667±12 $35.00/0

The self-assembly of apoferritin molecules into crystals is a suitablemodel for protein crystallization and aggregation; these processesunderlie several biological and biomedical phenomena, as well as for pro-tein and virus self-assembly. We use the atomic force microscope in situ,during the crystallization of apoferritin to visualize and quantify at themolecular level the processes responsible for crystal growth. To evaluatethe governing thermodynamic parameters, we image the con®guration ofthe incorporation sites, ``kinks'', on the surface of a growing crystal. Weshow that the kinks are due to thermal ¯uctuations of the molecules atthe crystal-solution interface. This allows evaluation of the free energy ofthe intermolecular bond f � 3.0 kBT � 7.3 kJ/mol. The crystallization freeenergy, extracted from the protein solubility, is ÿ42 kJ/mol. Publisheddeterminations of the second virial coef®cient and the protein solubilitybetween 0 and 40 �C revealed that the enthalpy of crystallization is closeto zero. Analyses based on these three values suggest that the main com-ponent in the crystallization driving force is the entropy gain of thewater molecules bound to the protein molecules in solution and releasedupon crystallization. Furthermore, monitoring the incorporation of indi-vidual molecules in to the kinks, we determine the characteristic fre-quency of attachment of individual molecules at one set of conditions.This allows a correlation between the mesoscopic kinetic coef®cient forgrowth and the molecular-level thermodynamic and kinetic parametersdetermined here. We found that step growth velocity, scaled by themolecular size, equals the product of the kink density and attachmentfrequency, i.e. the latter pair are the molecular-level parameters forself-assembly of the molecules into crystals.

# 2000 Academic Press

Keywords: ferritin; self-assembly; molecular resolution; interaction energy;solvent entropy
*Corresponding author
Introduction

Self-assembly of apoferritin molecules into crys-tals represents a suitable model system for studiesof crystallization and related phenomena of bio-logical macromolecules. Despite recent advances innuclear magnetic resonance techniques (Warren,1998; WuÈ trich, 1995), methods based on the diffrac-tion of X-rays, electrons, or neutrons by proteincrystals are still widely used for protein structuredeterminations. Claims that protein crystallizationis a major rate-limiting step in structure determi-nations abound even in recent literature (Chayen

ing author:

et al., 1996; Ducruix & Giege, 1992; McPherson,1999; Weber, 1997). Beyond protein single crystalgrowth, progress in various biochemical and bio-medical research and production tasks is impededby the lack of insight into protein nucleation andgrowth mechanisms. For instance, the slow dis-solution rate of protein crystals is used to achievesustained release of medications, such as insulinand interferon-a (Brange, 1987; Long et al., 1996;Matsuda et al., 1989; Peseta et al., 1989; Reichertet al., 1995). If the administered dose consists of afew, larger, equidimensional crystallites, steadymedication release rates can be maintained forlonger periods than for doses comprised of manysmaller crystallites. Other biomedical applicationsinclude pathological conditions related to the for-

# 2000 Academic Press

668 Molecular Parameters for Apoferritin Crystallization

mation of crystals or less ordered solid aggregatesin the human body. Often cited examples are thecrystallization of hemoglobin C and the polymeriz-ation of hemoglobin S that cause, respectively, theCC and sickle cell diseases (Eaton & Hofrichter,1990; Hirsch et al., 1985). Crystallization of the pro-teins in the eye lens is associated with the pathol-ogy of cataract formation (Berland et al., 1992).

The molecules of the chosen model protein, apo-ferritin are the hollow shells of ferritin, the mainnon-heme iron storage protein in the cytosol(Harrison & Arosio, 1996; Massover, 1993; Theil,1987). The molecules have a convenient quasi-spherical shape and in the presence of Cd2� orother divalent metal cations easily form octahedralcrystals of the cubic F432 space group with the lat-tice constant a � 18.4 nm. Contrary to typical pro-tein crystallization cases, where the electrolyteserves to screen the repulsion between the simi-larly charged protein molecules, in the apoferritincase Cd2� is involved in speci®c bonds betweenthe molecules (Hempstead et al., 1997). This speci-®city makes the case of apoferritin crystallization arelevant ®rst-approximation model of other self-assembling systems: viri, protein complexes, etc.The symmetry of the environment of a molecule ina crystal makes quantitative insight easier to obtainand comprehend.

While crystallization of proteins, as any othermaterial, occurs by the ordered addition of mol-ecules, the resolution of the experimental tech-niques employed until recently to study theseprocesses was signi®cantly coarser than the mol-ecular sizes (Forsythe et al., 1999; Li et al., 1999b;Vekilov et al., 1998). Only recently images of pro-tein crystal surfaces growing from solution wererecorded at molecular (Kuznetsov et al., 1999;Rashkovich et al., 1998; Yip et al., 1998) and evensub-molecular (Malkin et al., 1999) resolutions.These studies provided ®rst qualitative data on themolecular structure of the growth steps and of sur-face point defects.

The coarseness of the applied techniques deter-mined the lack of microscopic data on the pro-cesses of molecular self-assembly into aggregatesor crystals. Hence, the objective of the investi-gations reported here was to obtain quantitativemolecular-level insight into these processes. Theemployed method is in situ monitoring of the pro-cesses of molecular self-assembly with the atomicforce microscope (Yau et al., 2000; Yau & Vekilov,2000).

Using this method with the chosen proteinsystem, we ®rst consider the components of thedriving force for the processes of aggregation andcrystallization. From determinations of the magni-tude of the thermal ¯uctuations of the un®nishedlayers on the surface of growing crystals (Gibbs,1961) we aim to extract the free energy of the bondbetween the protein molecules in the crystal, asdone for metals and semiconductors (Kitamuraet al., 1993; Kuipers et al., 1993; Poensgen et al.,1992; Swartzentruber, 1998). Using recent data on

solubility and the solute interactions obtained as afunction of temperature we could distinguishbetween enthalpy and entropy contributions(Petsev et al., 2000a). We will also try to delineateentropy contributions to the crystallization drivingforce associated with the protein ordering fromthose due to the changes in the solvent state.Ordering and disordering of the solvent oftenoccurs in processes involving biological macromol-ecules and may represent a signi®cant fractions ofthe interaction potential between separate mol-ecules, or between parts of a molecule (Eaton et al.,1997; Eisenberg & Crothers, 1979).

Furthermore, we will attempt to determine themolecular-level kinetic constants for binding ofsolute protein molecules to the growth sites on thecrystal surface. By ®tting them to independentlydetermined mesoscopic and macroscopic kineticgrowth constants, we will test the applicability ofthe theoretical models for crystallization of biologi-cal macromolecules (Chernov & Komatsu, 1995a).More important, such quantitative molecular-levelinsight has the potential to suggest a rational wayof controlling the kinetics of self-assembly by vari-ation of the molecular properties.

Results

Kinks and kink density

Under all conditions used in the experimentsreported here, the crystals as seen in the opticalmicroscope attached to the AFM had the typicaloctahedral shapes with sharp edges. Accordingly,the AFM images in Figure 1 and all Figures belowindicate growth by layer generation and spreadingto cover the whole facet. This growth mode hasbeen reported for many protein crystals (Durbin &Carlson, 1992; Malkin et al., 1996; Yip & Ward,1996). In a previous AFM study of apoferritin crys-tallization (Malkin et al., 1996), rough interfacescorresponding to the normal growth mode wereseen that are typically found on metal and semi-conductor crystals growing from the melt (Van derEerden, 1994). This discrepancy is likely due toeither the high protein and precipitant levels usedin that earlier study, or to the high impurity con-tent in the commercial apoferritin preparations(Thomas et al., 1997) employed by Malkin et al.(1996).

The structures of a (111) crystal face and of agrowth step are shown in Figure 1. The edges ofun®nished layers on the crystal surface are thegrowth steps. The layers spread by the attachmentof molecules to the steps. If we consider a moleculeadsorbed on crystal surface away from a step, it isbound to at most three underlying molecules.However, a molecule attached at the step such asone in Figure 1(b) is bound to ®ve molecules (threein the underlying layer and two in the step)belonging to the crystal. Molecules in position 2 inFigure 1(b) are bound to six crystal molecules, i.e.half the number of neighbors that a molecule has

Figure 1. Molecular structures of a growth step on anapoferritin crystal at protein concentration of 70 mg/ml,corresponding to supersaturation s � 1.1. Dark green,lower layer; yellow, advancing upper layer. (a) Lowerresolution view. Adsorbed impurity clusters and surfacevacancies are indicated. (b) Higher resolution view.Three different types of molecular positions at a step aremarked with Arabic numbers, for details, see the text.Bonds with molecules belonging to the top crystal layerare marked with red. Green arches and Roman numbersmark potential growth sites, ``kinks''.

Molecular Parameters for Apoferritin Crystallization 669

in the crystal, while those in position 3-to seven.When a molecule attaches to position 2, the mol-ecular con®guration at the step providing sixneighbors to an incoming molecule is reproducedand the surface free energy of the crystal is notincreased. Hence, it has been postulated that onlysuch locations on the step serve as growth sites

(Stranski, 1928; Stranski & Kaischew, 1934; Volmer,1939). The density of such sites, called ``kinks'', hasbeen introduced as a fundamental variable thatdetermines the ability of the crystal to incorporatesolute molecules and grow (Burton, 1951; Chernov,1984).

From Figure 1 and �15 other similar images, wedetermine the kink density along a step by count-ing the molecules between two kinks, nk. Forinstance, in Figure 1(b) nk between kinks i and ii isthree molecules, while between kinks iii and iv istwo. The distribution of nk for the growth con-ditions in Figure 1 is shown in Figure 2(b). Com-paring Figure 2(a), (b) and (c), we see that the nk

distributions are nearly the same near equilibrium,as well as at very high supersaturations, providedthat the distance between the steps is higher than�200 nm.

Figure 3 shows the distribution of nk along stepsspaced about ten molecules (0.13 mm) apart, asopposed to 0.5 to 1 mm in Figures 1 and 2. Thekink density and step meandering are lower, indi-cating interactions between neighboring steps.Such interactions may occur through the solutionor through the layer of adsorbed apoferritin mol-ecules on the crystal surface and consist of compe-tition between the steps for nutrient supply.Competition for supply effectively reduces thesupersaturation to which a step is exposed. How-ever, the results in Figure 2 indicate that the kinkdensity is independent of the supersaturationwithin very broad limits. Hence, competition forsolute supply is not the interaction underlying theseen decrease in kink density. Other interactiontypes include step-step repulsion associated withthe entropy loss of closely spaced steps (Williams& Bartelt, 1991), or overlapping of the relaxationelastic ®elds of neighboring step edges(Houssmandzadeh & Misbah, 1995; Marchenko &Parshin, 1980). Our observations are evidence thatsuch interactions may be present even for proteincrystal surfaces.

Note that kink density is affected by the pre-sence of surface point defects, such as the vacanciesand adsorbed clusters shown in Figure 1(a) (for theidenti®cation of those clusters as molecular dimers,see (Petsev et al., 2000b)). These defects act as stop-pers: straight step segments as long as eight mol-ecules, similar to those on step 2 in Figure 3(a),form and the step propagation is locally delayed(Yau et al., 2000). For the statistics in Figures 2 and3(b) we did not consider step segments aroundsuch stoppers.

Molecular frequency of attachment to kinksites and kinetics of step propagation

While the kink density is a thermodynamicgrowth variable that characterizes the af®nity ofthe crystal to the solute molecules, the kinetics ofincorporation are re¯ected by the ¯ux of moleculesinto a growth site. To monitor these fast incorpor-ation events, we disabled the slow scanning axis of

Figure 2. Distribution of number of molecules betweenkinks on steps located >0.5 mm apart, obtained fromimages similar to Figure 1 at the three supersaturations sindicated in the plots, the mean values of the distributionsfor each case are also shown. The protein concentrationscorresponding to these s's are (a) 25 mg/ml, (b) 70 mg/mland (c) 1 mg/ml.

Figure 3. (a) Structure of steps and (b) kink distri-bution for closely spaced steps at apoferritin concen-tration of 70 mg/ml, supersaturation s � 1.1. Clusterssimilar to the one in Figure 1 are seen on the terracesbehind steps 3, 4, and 5 in (a).


the AFM. The advance of a step site is shown inFigure 4. Figure 5 shows that step motion is notinhibited or accelerated at the location of scanning,i.e. the chosen scanning parameters ensured thatstep propagation was not affected by scanningover the same line for approximately threeminutes.

Despite the relatively high solution supersatura-tion s � 1.1, the time trace in Figure 4 reveals notonly 25 arrivals to but also 22 departures of mol-ecules from the monitored site. All arrivals anddepartures of molecules to and from the monitoredsite involve single molecules. Therefore, in contrastto claims of preformed multiple-molecule growthunits for the protein lysozyme (Li et al., 1999a,b;Nadarajah & Pusey, 1997), apoferritin and ferritincrystal growth by the attachment of singlemolecules.

This type of data collection does not allow obser-vations of the neighboring sites at the step. Wecannot distinguish between attachment/detach-ment from molecules in the kinks or at the steps.Still, we notice that the residence times t betweenthese events fall into either t 4 1 second or t > 5seconds. In Figure 4 we have six events of thesecond type and 19 events of the ®rst. Their ratio isroughly equal to the kink density along the step,suggesting that the long-time events may beattachments/detachments to/from a kink, position

2 in Figure 1(b), while the short ones may be sight-ings of molecules at the step edge, position 1 inFigure 1(b). Furthermore, molecules may enter theline of observation due either to molecular diffu-sion along the step or to exchange with either theterrace between the steps or the adjacent solution.While the latter results in step propagation andgrowth, the former is a process that only involvesrearrangement of molecules already belonging tothe crystal and may not be associated with growth.In earlier work (Yau et al., 2000), we analyzed thetime correlation function of the step position, asdone before for steps on metal and semiconductorsurfaces (Alfonso, 1992; Bartelt, 1990; Ihle et al.,1998; Kuipers et al., 1993; Poensgen et al., 1992).These analyses indicate that the trace in Figure 4predominantly re¯ects exchange of moleculesbetween the step and its environment.

This conclusion allows us to extract fromFigure 4 a net frequency of attachment of mol-ecules to kinks. From the net attachment of threemolecules for 162 seconds and the probability ofviewing a kink of 1/nk � 1/3.5, we get

Figure 4. Incorporation of molecules into steps at apo-ferritin concentration of 70 mg/ml, s � 1.1. Pseudo-image recorded using scanning frequency of 3 Hz withthe y-scan axis disabled at time �0 shows displacementof one step site. Red contour traces step position. Redarrows indicate attachment and detachment events withresidence time >1 second, blue double-sided arrows;with residence time <1 second, for details, see the text.Appearance of 1/2 molecule attachments at times >80seconds, highlighted in green, is due to events at aneighboring site that enters image due to scanner

Figure 5. Surface scan immediately after Figure 4,arrow indicates location of monitoring in Figure 4. Clus-ters and vacancies similar to those in Figure 1 seen onlower terrace.


f � 0.065 sÿ1, or one molecule per approximately15 seconds.

The step velocities v averaged over image collec-tion times of approximately 40-50 seconds areshown in Figure 6. The data sets at the ®rstfour concentrations ®t well the proportionality(Chernov & Komatsu, 1995a,b):

v � bCe�C=Ce ÿ 1� �1�where � 1/4 a3 � 1.56 � 10ÿ18 cm3 is the crystalvolume per molecule, C is the apoferritin concen-trations and Ce � 23 mg/ml (see Materials andMethods) is the concentration at equilibriumbetween crystal and solution, or the solubility. Theaveraged step kinetic coef®cient, or the macro-scopic kinetic constant of growth that emergesfrom equation (1), is b � 6 � 10ÿ4 cm/s. The valuesof the step velocity v at the highest apoferritin con-centration C � 1 mg/ml, C/Ce ÿ 1 � 42 in Figure 6are lower by about 30 % from those expected basedon the above value of the step kinetic coef®cient.The likely reason is the competition for supplybetween closely spaced steps at the higher step

density, as evidenced for other protein systems(Land et al., 1997; Vekilov et al., 1995, 1996).

Discussion

Energy of a kink and the intermolecularbond energy

The lack of dependence of the kink density onthe thermodynamic supersaturation suggests thatthe kinks are not created by nucleation of molecu-lar rows along a step: such nucleation would resultin a steep dependence of kink density on super-saturation (Chernov et al., 1999; Teng et al., 1998).Hence, the kink density 1/nk appears to be anequilibrium property of this surface even duringgrowth in a supersaturated environment. In thiscase, the number of molecules between the kinksnk is solely determined by the balance of molecularinteractions and thermal ¯uctuations in the topcrystal layer (Gibbs, 1961; Kuipers et al., 1993;Swartzentruber, 1998), and should be a function ofthe energy w needed to create a kink. Derivationshave shown (Burton, 1951) that the average nk:

�nk � 1=2 exp�w=kBT� � 1 �2�where T is the absolute temperature and kB is theBoltzmann constant. From the value of nk inFigure 2, w � 1.6 kBT. It is quite surprising that thisvalue of w is only slightly lower than the energy ofkinks on Si crystals (Swartzentruber et al., 1990):one would expect the strong covalent bonds in theSi crystal lattice to lead to signi®cantly higher kinkenergies. Quite surprisingly, for the orthorhombicform of lysozyme w � 7.4 kBT (Chernov et al.,1999). This signi®cantly higher value leads to an

Figure 6. Step velocities v determined by comparingstep positions within 44-47 seconds, plotted as functionof concentration supersaturation (C ÿ Ce) �Cÿ1

e . �, indi-vidual determinations: data scatter re¯ects the stochasticnature of crystal growth at the molecular level; &, aver-age v for each concentration. Continuous line corre-sponds to shown step kinetic coef®cient b.


extremely low kink density with nk as high as 400-800, and step propagation limited by the rate ofkink generation (Chernov et al., 1999).

If we assume ®rst-neighbor interactions only, wecan evaluate the intermolecular bond energy, f.When a molecule is moved from within the stepon a (111) face of a face centered cubic (fcc) crystal(position 3 in Figure 1(b)) to a location at the step(position 1 in Figure 1(b)), four kinks are created.For this, seven bonds (four in the top layer andthree with molecules from the underlying layer)are broken, and ®ve are formed. Then, w � f/2and f � 3.2 kBT � 7.8 kJ/mol.

The intermolecular bonds in ferritin and apofer-ritin crystals involve two chains of bonds Asp-Cd2�-Glu between each pair of adjacent molecules(Hempstead et al., 1997; Lawson et al., 1991). Theabove value of f seems signi®cantly lower thanthe typical coordination bond energies. This lowvalue may stem from the need to balance Cd2�

coordination with the aminoacid residues and withthe water species (H2O and OHÿ), or from spatialconstrains imposed by the other aminoacid resi-dues involved in the intermolecular contacts.

Second and third neighbor effects

Some of the protein-protein molecular inter-actions, such as the electrostatic, may have rangelonger than the diameter of one molecule(Beresford-Smith et al., 1985). Then, interactionswith second and third neighbors will contribute tothe interactions of a molecule with a site on

the crystal surface. This will affect the dynamics ofthe kinks and bias the determination of theintermolecular bond energy from kink densitymeasurements.

To account for possible second and third neigh-bor effects in the crystal as well as in the evolvingmolecular row, we carried out model calculations.We considered an ordered layer of spherical mol-ecules deposited on a octahedral face of a fcc crys-tal. As in many previous works with proteins(Sear, 1999; ten Wolde & Frenkel, 1997), weapproximate the interaction energy between themolecules with a Lennard-Jones type potential.Then, the energy of interaction of a molecule withanother molecule labeled i at a distance ri will be:

Ui � er�

ri

� �12

ÿ2r�

ri

� �6" #

�3�

Here, e is the depth of the potential well and r*,the position of the minimum, is chosen as the crys-tallographic distance between the molecules,13 nm. The function Ui(ri/r*) is plotted inFigure 7(b). Furthermore, we assume that the total``potential'' energy of a molecule can be calculatedas superposition of the pair interactions with allother molecules in the of equation (3). In this way,all molecules in the crystal will be in the energyminima of their interactions with the other mol-ecules in the crystal. Note that unlike (Hagen& Frenkel, 1994; ten Wolde & Frenkel, 1997), wetake the position of the minimum r* from theexperimental data and do not optimize the crystalstructure.

We are only interested in the dynamics of kinksat a step. Hence, the energy of interaction with themolecules of the substrate provides a constantadditive to the energy of the molecules under con-sideration and we can limit the calculations to thetop layer. Furthermore, we found that the contri-bution of all neighbors beyond the second shell isnegligible. As a result, in the calculations weaccount for up to fourth neighbors. This allows usto limit the modeled layer to an island of 4 � 10molecules. Choosing e� 3 kBT, we get the potentialrelief of the interaction of a molecule with thisisland shown as 1 in Figure 7(a).

To simulate the formation of kinks, we remove amolecule from the edge of the island, forming twokinks, and calculate the new energy relief. Theenergy spent for this removal is equal to thedeepest minimum in 2 in Figure 7(a). Anothermechanism of kink fomation is by the attachmentof molecules to the step to form con®guration 3.The corresponding energy gain is equal to thedepth of the wells near the middle of the longisland edge in 1. The net sum of the gain and theloss, needed to calculate the average energy toform a kink is � 6.2 kBT.

As a next step, we remove a molecule next tothe vacancy in 2 to obtain the con®guration 4 anddeposit a molecule in the minimum above the

Figure 7. Model results. (a) Contour plot of the poten-tial energy U of a single molecule in the vicinity of anisland deposited on an octahedral face of a face-centeredcubic crystal. �U between two contours is kBT. Individ-ual molecules belonging to the island represented by cir-cles; the appearance of a thick line that follows thecontour of the island is a result of steep gradient of Ubetween the minimum and the molecules. (b) Plot ofequation (3) with Ui in kBT units; arches indicate sur-faces of adjacent molecules.


molecule attached at the edge in 3 producingthe con®guration 5. The net loss from these twooperations equals the difference between the topminimum in the vacancy in 4 and at the used sitein 3. We get 0.2 kBT. Then we remove a moleculenext to the double vacancy in 4 to produce 6 andattach it in line with the previous two molecules atthe island periphery in 5 to produce 7. The net con-tribution of this operation, calculated in the samefashion as the others, is negligible. It is not surpris-ing that we found same result for removing afourth molecule from the vacancy cluster in 6 andadding it to the line of attached molecules in 7.

Summing the total energy expenditure, we get6.4 kBT as the energy needed to generate fourkinks. The average energy for formation of onekink w is 1.6 kBT, adjusted to the experimentallyfound value by selecting the appropriate e. If weassume that the depth of the Lenard-Jones well e isa good equivalent to the intermolecular bondenergy f, we see that accounting for higher neigh-

bor effects, at least in the ®rst approximation usedhere, decreases this value from 3.2 to 3 kBT, i.e. by�7 %.

Enthalpy, entropy and free energyfor crystallization

The intermolecular bond energy f contains bothenthalpy and entropy components. The enthalpyones are due to the ion-mediated, hydrogen, andother bonds between the molecules, while theentropy components stem from the net release orbinding of water and other small molecules uponcrystallization (Eisenberg & Crothers, 1979, p. 164).With this in mind, we can write an expression forthe free energy for crystallization as:

�G� � �H�bond ÿ T�S�bond ÿ T�S�protein �4�

Here �H�bond ÿ T�S�bond are contributions associ-ated with f, and �S�protein is the loss of entropy ofthe protein molecules. A crude estimate of �S�protein

and of the relative weights of the two entropy con-tributions can be obtained by comparing the stan-dard free energy change for apoferritincrystallization �G� determined from the solubility,to the value corresponding to the intermolecularbond free energy f.

The calculate �G�, we convert the solubility ofthe apoferritin at the chosen crystallization con-ditions 23 mg/ml, see Methods below, to molarconcentration Ce � 5.2 � 10ÿ8 M. At equilibriumbetween crystal and solution, for the apoferritin:

�G �G��crystal�ÿ�G��solution� �NAkBT ln�gCe�� 0 �5�

where NA is the Avogadro number, g is the proteinactivity coef®cient, and the product NAkB � R isthe universal gas constant (Atkins, 1998). Hence:

�G� � G��crystal� ÿ G��solution� � NAkBT ln�gCe��6�

(Eisenberg & Crothers, 1979). Using g � 1 for sol-utions in which the attraction and repulsionbetween the molecules are practically balanced, seeMETHODS below, �G� � ÿ 42 kJ/mol.

To get �H�bond ÿ T�S�bond from f � 3kBT � 7.3 kJ/mol, we have to multiply f byZ1/2 � 6, the half number of neighbors in thecrystal lattice (two molecules partake in a bond, ina fcc lattice Z1 � 12) and, accounting for thesign, we get ÿ44 kJ/mol. The closeness of�H�bond ÿ T�S�bond to �G� indicates the insigni®-cance of �S�protein for the free energy of crystalliza-tion. This insigni®cance may stem from the hugenumber (103 ÿ 104) of water molecules that may beassociated to a single apoferritin molecule in thesolution. In this respect, the investigated proteinsolution system appears similar to colloid systems,in which the ratio between the numbers of thecolloid particles and the solvent molecules is often


similar and �S� due to the aggregation of colloidparticles is low (Poon et al., 1996).

To evaluate �H�bond we used data on the tem-perature dependencies of the protein solubility andof the second virial coef®cient in the range from 0to 40 �C (Petsev et al., 2000a). According to theresults of that study, neither of solubility, not thesecond virial coef®cient has noticeable dependenceon temperature. Therefore, the enthalpy of crystal-lization and the related energy of pair interactionsin the solution are close to zero. In combinationwith the insigni®cance of �S�protein, this allows usto conclude that crystallization is mostly driven bythe maximization of the entropy of the solvent.Such disordering may stem from a signi®cant dis-ordering of the water and other solvent com-ponents bound to the protein molecules in thesolution. A similarity can be traced to the processesthat underlie hydrophobic attraction that governsmany processes in nature (Israelachvili, 1995),including some stages of protein folding (Eatonet al., 1997).

Molecular-level parameters underlying themacroscopic growth kinetics

The results on the incorporation of moleculesinto a growth site in Figure 4 indicate that, even atthe relatively high s � 1.1, incorporation of mol-ecules into the crystal is extremely slow and occursat macroscopic time scales. Furthermore, the ratioof 25 arrivals and 22 departures, mostly due toexchange with the medium, indicates that crystalgrowth is a very selective process even at suchhigh supersaturations. If foreign or misorientedmolecules attach to a step, they have signi®cantchances of detaching into the solution and freeingthe space for a proper molecular attachment. Thismay underlie the puri®cation upon crystallization,utilized in various technological processes.

The step kinetic coef®cient b, extracted from thestep velocity data averaged over 40-50 seconds inFigure 6 is b � 6 � 10ÿ4 cm/s. This value is com-parable to values of other, faster growing proteins,indicating that the low net ¯ux and step velocitiesof this protein are caused by the low solution con-centration, re¯ected in the molecular density ratiobetween the solution and crystal, Ce � 5 � 10ÿ5.For comparison, the value of Ce for a anotherwell studied protein, lysozyme growing from a50 mg/ml solution is as high as 0.02 (Vekilov &Rosenberger, 1996).

The determinations of the step velocities and cor-responding kinetic coef®cient allows a test of themolecular mechanism of incorporation into stepsunderlying step motion and the de®nition of b(Chernov, 1989; Chernov & Komatsu, 1995b).According to these works, the step velocity v:

v � a�1= �nk�f �7�where a is the molecular diameter, 1/nk is the kinkdensity and f is the frequency of molecular incor-

poration into kinks. Using the respective valuesa � 13 nm, nk � 3.5 and f � 0.065 sÿ1 fromFigures 2, 4, and 8 we get a(1/nk)f � 0.24 nm/s. Atthe conditions corresponding to those for Figure 4,(C/Ce ÿ 1) � 2, s � 1.1, the average step velocityfrom Figure 6 is v � 0.26 nm/s. The nearness ofthe predicted and actual values indicates that kinkdensity and net frequency of attachment to a kinkare the fundamental molecular level variables thatfully determine the rate of step propagation duringcrystal growth.

Materials and Methods

Solution preparation

Apoferritin stock solutions were separated into Mono-mer, Dimer and ``Trimer'' fractions as described in detailelsewhere (Petsev et al., 2000b; Thomas et al., 1997). Allexperiments discussed here were carried out using theMonomer fraction. The sum residual amount of Dimerand ``Trimers'' in this fraction after puri®cation was�5 % (w/w). To induce crystallization, CdSO4 wasadded to concentration in the crystallizing solution of2.5 % (w/v).

Crystallization and atomic force microscopy imaging

Droplets of crystallizing solution of �50 ml wereplaced on 12 mm glass coverslips mounted on irondisks. To avoid evaporation, the droplets were coveredwith glass covers, hermetically sealed and left overnightin a controlled temperature room at �22 �C. Typically,this lead to the formation of three to 20 crystals of sizesranging from 20 to 200 mm ®rmly attached to the glassbottom. Droplets with three to ®ve crystals were selectedand magnetically mounted on the AFM scanner. The¯uid AFM cell was ®lled with crystallizing solution andimaging commenced.

Temperature in the laboratory was stabilized to�22 �C. No additional temperature control of the sol-ution in the AFM ¯uid cell was employed. Tests byinserting a thermocouple in the crystallizing solutionrevealed that its temperature was higher by �0.5-1.0deg.C than the room temperature. The documentedinsensitivity of apoferritin crystallization to temperaturevariations (Petsev et al., 2000a) justi®es this approach totemperature control.

All images were collected in situ during the growth ofthe crystals using the less intrusive tapping imagingmode (Hansm et al., 1994; Noy et al., 1998). This allowsvisualization of adsorbed protein and impurity species(tip impact in contact imaging mode often prevents suchimaging). We also employed scans with disabled y-axis,similar to previous STM work on metals and semicon-ductors in ultra-high vacuum by (Giesen-Seibert et al.,1993; Kitamura et al., 1993; Poensgen et al., 1992). Thetechnique allows monitoring of processes with character-istic times of fractions of a second.

We used the standard SiN tips and tapping drivefrequency was adjusted in the range 25-31 kHz to theresonance value for speci®c tip used. Other scanningparameters were adjusted such that continuous imagingaffected neither the surface structure, nor the processdynamics. For veri®cation, we varied the scan sizes andthe time elapsed between image collections, and saw

Figure 8. Accuracy and resolution of atomic forcemicroscopy imaging. (a) View of [111] apoferritin crystalface. (b) Height pro®le along line in (a) allowing deter-mination of layer thickness and molecular spacing.(c) Determination of the resolution limits of the atomicforce microscopy imaging used in these studies. Realspace high-resolution image (d) Fourier transform of (c),red circles highlight high-resolution peaks; 8th orderpeaks at top and bottom of image correspond to resol-ution of �1.6 nm.


that neither the spatial nor the temporal characteristics ofthe processes changed.

To test the accuracy of the determination of lateraland vertical dimensions by the AFM, we determined the

periodicity within the molecular rows along the {110}directions and the thickness of the [111] layers.Figure 8(a) and (b) and similar analysis of many otherimages yield values for these two quantities of 13 and10.5 nm, respectively. These dimensions agree well withcalculations, based on the F432 crystal structure. Thecrystal lattice parameter is a � 18.4 nm (Hempstead et al.,1997; Lawson et al., 1991). Therefore, the distancebetween centers of molecules along the {110} directionsis (1/2) � (2a2)1/2 � 13.0 nm and the distance between[111] layers is (1/3) � (3a2)1/2 � 10.6 nm.

To evaluate the lateral resolution limit of our AFMimaging technique, we scanned a 200 nm � 200 nmsquare on the surface of a ferritin crystal without steps.The individual molecules in Figure 8(c) are readily dis-tinguishable and some sub-molecular details are detect-able. To quantify the resolution of the imaging, wecomputed the two-dimensional Fourier transform of thisimage, using the respective module from the NanoscopeIIIa software package. The Fourier image has theexpected hexagonal symmetry, with the distancebetween the peaks in the ®rst hexagon and the center ofthe plot corresponding to resolution equal to the molecu-lar size of 13 mn. The maximum resolution, determinedfrom the location of the most distant peak in Figure 8(d),is 1.6 nm. Similar analyses with other images of compar-able size revealed that typically the resolution wasbetween 5 and 2.5 nm.

The molecular details in Figure 8(c) do not seem topossess translational symmetry. Careful observationallows identi®cation of several sets of molecules withsimilar features that are randomly distributed through-out the image. The general impression is that the mol-ecules on the surface have several rotational orientations,different from the crystallographically required orien-tation on a (111) face. We tentatively relate this apparentabsence of translational crystallographic symmetry withthe lack of consistent decoration pattern of the moleculeson surfaces frozen-hydrated ferritin crystals examined byelectron microscopy (S. Weinkauf, personal communi-cation). Such patterns have been obtained with crystalsof a number of other proteins and protein complexes(Bacher et al., 1992; Meining et al., 1995; Weinkauf et al.,1991).

Note that crystals grown under conditions identical tothose used in these studies diffract X-rays and the resol-ution limit is 1.8-1.85 AÊ (Thomas et al., 1997). Therefore,it is unlikely that any molecular rotational disorder ispresent in the crystal bulk. Assuming that Figure 8(c)suggests such disorder in the surface layer of the crystal,the mechanism by which these molecules reorient as thecrystal grows poses an intriguing problem.

The lack of translational symmetry of the submolecu-lar details in Figure 8(c) indicates that some of the ®nerfeatures of the molecules seen in Figure 8(c) will not bere¯ected by the Fourier transform in Figure 8(d). Hence,the resolution of the AFM imaging may be higher thanthe values suggested by Figure 8(d).

Solubility and driving force for crystallization

To determine the solubility of the apoferritin at thechosen temperature, pH and buffer and precipitant con-centrations, we monitored trains of growth steps andgradually decreased the apoferritin concentration in thesolution. We found that at C � 23 mg/ml, the propa-gation of the steps stops, and when the concentration islower below this value, the step movement was back-


wards and the crystal was dissolving. Repetitions of thisprocedure allowed determination of the range of thismeasurement, Ce � 23(�5) mg/cm3.

Supersaturation s in protein crystallization studieshas often been de®ned as s � ln(C/Ce) and this s hasbeen taken as a measure of the chemical potential differ-ence between solution and crystal �m in kBT units. Suchassumption implies solution ideality. To evaluate thenon-ideality of the crystallizing solutions, we employedstatic and dynamic light scattering as described byPetsev et al. (2000b). The found value of the second virialcoef®cient B2 � ÿ 2.35 � 10ÿ5 cm3 mol/g2.

Following the logic of (Guo et al., 1999), it is easy toshow that non-ideality leads to a correction in the aboveexpression for the supersaturation:

�mcr ÿ mp�=kBT � �m=kBT

� ln�C=Ce� � 2B2M�Cÿ Ce� � � � � �8�where M � 440,000 Da is the protein molecular mass.

Using the above value of B2 the correction 2 B2M(C ÿ Ce) � ÿ 0.03, i.e. � ÿ 0.8 % of ln(C/Ce) � 3.77 atC � 1 mg/ml, the highest protein concentrationemployed in the experiments. Hence the use of the sim-pli®ed de®nition of s � �m/kBT � ln(C/Ce) is justi®ed inthis case.

For other proteins under crystallizing conditions, thevalues of B2 are limited from below to � ÿ 8 � 10ÿ4 cm3

mol/g2 (George & Wilson, 1994; Guo et al., 1999;Rosenbaum & Zukoski, 1996). At higher concentrationsof proteins of higher molecular masses, the non-idealitycorrection may lead to a supersaturation signi®cantlysmaller than what the simpli®ed de®nition may suggest.

Acknowledgments

We thank C. Kundrot, S. Weinkauf, A. McPhersonand M. Machius for discussions and helpful suggestionson some aspects of this work, O. Galkin for critical read-ing of the manuscript, and L Carver for expert graphicswork. This research was supported by NHLBI, NIH(Grant No HL58038), NASA (Grants No NAG8-1354 and97 HEDS-02-50), and the State of Alabama through theCenter for Microgravity and Materials Research at theUniversity of Alabama in Huntsville.

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Edited by W. Baumeister

(Received 25 July 2000; received in revised form 5 September 2000; accepted 5 September 2000)

molecular-level thermodynamic and kinetic parameters for the self

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