Kinetic Roughening and Energetics of Tetragonal Lysozyme Crystal

CRYSTAL GROWTH & DESIGN

Kinetic Roughening and Energetics of Tetragonal Lysozyme Crystal Growth Sridhar

Gorti,*,†

Elizabeth L.

Forsythe,‡

and Marc L.

2004 VOL. 4, NO. 4 691-699

Pusey†

National Aeronautics and Space Administration, Huntsville, Alabama 35812, and BAE SYSTEMS, Physical and Biological Sciences Laboratory, NASA/MSFC, Huntsville, Alabama 35812 Received August 28, 2003;

Revised Manuscript Received March 29, 2004

ABSTRACT: Lysozyme crystal growth rates over 5 orders of magnitude in range can be described using a layerby-layer model in which growth occurs by 2D nucleation on the crystal surface. Upon the basis of the 2D nucleation model of layer growth, the effective barrier for growth was determined to be γ ) 1.3 ( 0.3 × 10-13 erg/molecule, corresponding to a barrier of 3.2 ( 0.7 kBT, at 22 °C. For solution supersaturation, ln c/ceq g 1.9 ( 0.2, the nucleation model would not predict or consistently estimate the highest observable crystal growth rates. As such, a kinetic roughening hypothesis in which crystal growth occurs by a continuous mode was implemented for all growth rate data obtained above ln cr/ceq g 2. That is, independent of the solution conditions that vary with buffer pH, temperature, or precipitant concentration, crystal growth occurs by the continuous addition of molecules anywhere on the crystal surface, above a roughening solution supersaturation. The energy barrier, Ec, for the continuous growth process is 6.1 ( 0.4 × 10-13 erg/molecule or 15 ( 1 kBT at 22 °C. Introduction Experimental measurements of tetragonal lysozyme crystal growth rates have been performed for a range of solution supersaturations under varied conditions of temperature, pH, and precipitant concentration.1-5 To understand the observed changes in growth rates under the various conditions investigated, a thermodynamic model relying on descriptions of the physical state of protein molecules in solution was proposed.6-9 In the model, an aggregate distribution was assumed for molecules in solutions as a predictor of crystal growth rates. The thermodynamic model is informative in understanding solutal processes assumed for nucleation and crystal growth. However, the model does not directly address the physical mechanisms of protein crystal growth, the principal quantity measured. This work thus provides a description for the changes in observed crystal growth rates in terms of well-established physical theories for crystal growth. Current efforts not withstanding,10-12 the field of investigation involving the description of the physical characteristics of crystal growth is mature.13,14 From these efforts, it is understood that the rate of growth of a planar crystal surface is dependent on the disequilibrium (∆µ) or excess supersaturation (σ ) ln c/ceq) of a given solution. The magnitude of the disequilibrium or excess supersaturation determines the mode or mechanism by which a crystal surface grows. The observed crystal growth rates may exhibit either a nonlinear or linear dependence on supersaturation. Supersaturation can be varied by either the concentration of attaching molecules or temperature, both beyond an equilibrium value (solubility or liquidus). Thus, at different solute concentrations, c, or temperatures, T, beyond an equilibrium concentration ceq(T), crystals * To whom correspondence should be addressed. E-mail: [email protected]. † National Aeronautics and Space Administration. ‡ BAE SYSTEMS, Physical and Biological Sciences Laboratory, NASA/MSFC.

grow by different modes or mechanisms, defined as either layer-by-layer or continuous growth. For solution conditions in the vicinity of ceq(T) or “low” supersaturation, the planar crystal surface growth occurs by step generation and incorporation of growth units at steps or kinks, layer growth. Far from ceq(T) or “high” supersaturation crystal surface growth occurs by the incorporation of growth units anywhere on the surface of a crystal, continuous growth. Transitioning from layer to continuous growth, as the solution supersaturation is varied, is defined as the kinetic roughening transition.10-19 Few systems have exhibited a transition from a nonlinear or Becker-Do¨ring (B-D) to linear or continuous mode (C-M) of growth as solution conditions exceed a critical supersaturation or solute concentration, σc or cr, respectively.16-19 Convincing evidence of either surface roughening, or a thermal roughening transition TR for protein crystals has yet to be provided. In this paper, data are presented to lend support to the hypothesis that macromolecular crystal growth rates exhibit a kinetic roughening transition at a supersaturation beyond a critical value σc. At the outset, Materials and Methods describes procedures used to obtain lysozyme 110 face growth rates, as well as rounding of lysozyme crystals beyond the roughening transition. The theoretical models used in the interpretation of experimentally measured quantities are provided. Details of the results of the crystal growth measurements and analyses making use of the kinetic roughening hypothesis are presented. Next follows a discussion section that reviews the experimental and analytical results obtained within the framework of well-established physical theories for crystal growth. Finally, the conclusion section summarizes the relevant findings regarding the 110 face growth rates. Materials and Methods Preparation of Lysozyme. Chicken egg-white lysozyme was purchased from Sigma and purified following protocols

10.1021/cg034164d CCC: $27.50 © 2004 American Chemical Society Published on Web 06/19/2004

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previously established.20 The final purified protein was dialyzed against 0.1 M sodium acetate (NaAc) buffer maintained at the desired pH. Protein concentrations in these solutions were determined by UV absorbance.21 Prior to performing crystal growth measurements, quantities of the above protein solutions were further adjusted to the desired concentration by the addition of buffer and/or buffer containing the precipitant sodium chloride. Growth Rate Measurements. Face growth rate measurements were collected using a computer-controlled microscopy system with a glass cell.1 The studies reported below were performed in a temperature-controlled growth chamber with cell dimensions of 50 × 20 × 0.3 mm (L × W × D). At the outset, seed crystals (10-30 µm) were nucleated in situ on the chamber windows at the same NaCl concentration and temperature used for the subsequent growth rate determination. Growth rate measurements were initiated by replenishing the mother liquor with a desired protein solution. Using optical microscopy, observations of the changes in linear dimensions of specific facets of crystals, in time, were recorded, and the slope of growth versus time represented the protein crystal growth rate. Depending upon the anticipated growth rate, 4-30 separate face growth rates were determined during each experimental run. Accordingly, for all the data shown here each crystal was used only one time. At the conclusion of each growth rate determination the crystals were dissolved and a new batch of seeds nucleated for the next experimental run. Imaging of Crystal Growth beyond the Roughening Transition. Surface roughening measurements were performed using a simple microscope system equipped with a video imaging camera and a temperature-controlled stage. Prior to conducting measurements, quantities of the purified protein solutions maintained at a specified pH were adjusted to the desired concentration by the addition of buffer and/or buffer containing the precipitant. The desired concentration of the protein solution is dependent on the solution pH and %NaCl concentration that offered an ideal range of supersaturation (1.6 < σ < 2.6) upon moderate variations in temperature. For example, the desired protein concentration was 30 mg/mL at a solution pH 5.0 and 4% NaCl. Immediately upon addition of precipitant to the purified protein solution, the mixture was filtered through a 0.2 µm Nalgene filter. The filtered solution was placed in a 0.1-mm path length microcuvette, which was subsequently sealed and placed on the temperature-controlled stage. The initial temperature of the stage was set at 20 °C to yield a supersaturation σ ∼ 1.6-2.0. Under these conditions, seed crystals were allowed to grow until crystal facets were microscopically visible under a 40× microscope objective. Upon visible observations of faceted crystals, the stage temperature was reduced to 14 °C to obtain a supersaturation σ ∼ 2.4. The protein crystals within the sealed microcuvette were allowed to grow at σ ∼ 2.4 for a period of ∼40-90 min, while images of their growth progress were continuously taken and recorded. The above cited process was thrice repeated for two different solution conditions, pH 5.0 with 4% NaCl and pH 4.6 with 3% NaCl.

Theoretical Models Analyses of Measured Growth Rates by a 2D Nucleation Model. In the layer-by-layer mode of growth, the crystal surface grows by the generation of steps (birth of a layer) and subsequent incorporation of growth units at the step edge or kinks (spreading of layer). Step generation occurs as a result of the formation of new layers, either by self-perpetual growth from spiral dislocations or the formation of 2D nucleates on the crystal surface (Becker-Do¨ring limit).10,11 For the latter case, the free energy barrier ∆Gnuc for the formation of 2D nucleates on any “smooth” surface is defined in terms of the number of macromolecules within circular nuclei, N, disequilibrium, ∆µ, and “effective”

Gorti et al.

step free energy, γ, and is given as14

∆Gnuc ) -∆µN + γx4πN

(1)

Beyond certain values for ∆µ, the free energy has a maximum at the critical nucleus size, N* expressed as:14

N* ) πγ2/∆µ2

(2)

The 2D nucleation free energy barrier at maximum is14

∆G/nuc) πγ2/∆µ

(3)

The rate of growth rates normal to the crystal surface, Vs, generated by the nucleation and advancement of circular steps is given as14

Vs(c,T) )

( )( ) [ ( ) ] [ ]

Aceq1/3 exp

2∆µ ∆µ 3kBT kBT

1/6

exp

∆µ -1 k BT

exp -

2/3

×

πγ2 (4) 3∆µkBT

where ceq(T) is the solubility concentration at a specified temperature T, kBT is the thermal energy, and ∆µ is the solution disequilibrium. Τhe “effective” barrier for growth by 2D nucleation is defined as γ ) EsΩ1/2, where Ω is the attachment unit area at a lattice site, and Es is the step energy per unit length. Implicit in the use of eq 4 is the assumption that step and kink dynamics are determined by the diffusion of growth units on terraces.14 The prefactor A in eq 4 thus accounts for the frequency with which a macromolecule attempts to overcome a barrier for step generation and advancement, related to surface diffusive processes.14 Both γ and A are unknown quantities that remain to be determined. Analyses of Measured Growth Rates by a Continuous Addition Model. Predominantly, protein crystal growth can be well described by eq 4 for a range of supersaturation and temperatures below the thermal roughening transition (TR). A crystal surface can also grow by the continuous addition of growth units (linear growth law) anywhere on the surface when solution supersaturation exceeds a critical value, σc.12,16,18,19 Under such conditions, the growth rate is expected to be proportional to the supersaturation existing between the crystal and the fluid adjacent to the surface. Thus, beyond a critical supersaturation or critical “roughening” concentration cr, the continuous mode growth rate, Vc, in terms of solute concentration, c, and temperature, T, can be expressed as12,14,16,18,19

Vc(c,T) ) B(c - cr(T)) exp[-Ec/kBT]

(5)

where the prefactor B represents the frequency with which a growth unit presents itself to a barrier by a diffusional or translational flux from the fluid adjacent to the crystal surface. Namely, the rate of crystal growth, Vc, occurs by the translation of macromolecules from the fluid adjacent to the crystal surface toward a lattice site of an area a2, where a is the linear dimension of the adatom. The frequency, B, with which particles arrive within this area or rather the number of particles arriving within this unit area per unit time by a

Energetics of Tetragonal Lysozyme Crystal Growth

Figure 1. Growth rate dependence on excess concentration. Lysozyme protein crystal (110) face growth rates in 0.1 M NaAc, 5% NaCl as a function of excess protein concentration c/ceq are shown. The symbols 9 or b rate data obtained in solutions buffered at pH of 4.0 or pH 4.4, respectively, with temperature maintained at T ) 14 °C. The symbol 0 represents growth rate data obtained in solutions buffered at a pH of 4.0 with T ) 22 °C. The solid lines represent computergenerated fit to a selected portion of the data using eq 4, which characterizes crystal growth by 2D nucleation. The dashed lines represent computer-generated fit to all corresponding data using eq 4.

diffusional flux from a distance ξ, in terms of particle concentration c, is B ) NADa2/ξMw, where D is the diffusion coefficient, NA is Avogadro’s number, and Mw is the molecular weight of the macromolecule. When the characteristic distance ξ chosen for the translation of macromolecules from the fluid adjacent to the crystal surface is taken as the mean interparticle spacing, prefactor B is then given by

B)

NADa2 1.6Mwφ-1/3

where φ is the macromolecular volume fraction. The continuous mode growth rate of crystals for the conditions considered is described by the expression:

Vc(c,T) )

NADa2 1.6Mwφ-1/3

(c - cr(T)) exp[-Ec/kBT] (6)

The unknown quantities in eq 6 to be determined from experiments are cr(T) and Ec. Results Analyses of Growth Rate Data and Implementation of a Roughening Transition. The (110) face growth rate data obtained in solutions comprised of 0.1 M NaAc pH 4.0 with 5% NaCl as a function of excess protein concentration, c/ceq, at two different temperatures 14 and 22 °C, are exhibited in Figure 1. The equilibrium concentration, ceq, at which crystal growth ceases, used in the determination of the excess protein concentration for the solution conditions represented in Figure 1, had been determined previously.22,23 As shown, the growth rate data ranged over 5 orders of magnitude as a function of excess lysozyme concentration, from which a selected portion of the data were modeled

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Figure 2. Dependence of growth rate on temperature. Lysozyme protein crystal (110) face growth rates in 0.1 M NaAc pH 4.6, 3% NaCl are displayed as a function of temperature. The symbols O, b, 0, or 9 correspond to solution protein concentrations of 20, 30, 40, or 50 mg/mL, respectively. The dashed lines are computer-generated fit to the natural logarithm of eq 4 that best characterize crystal growth as layerby-layer mode. The solid lines are obtained using the natural logarithm of the corrected form of eq 6 that best characterizes crystal growth by a continuous mode beyond a critical roughening concentration cr.

accurately (solid lines in Figure 1) using a parametric relation for the unknown quantities in eq 4. The parametric relation for eq 4 yielded a best fit to selected portions of the growth rate data with an average effective step energy γ ) 1.5 ( 0.1 × 10-13 erg/molecule. Although the magnitude of obtained values for γ are reasonable, eq 4 failed to predict the measured growth rate data beyond c/ceq ∼ 11, where significant deviations between measured and theoretically predicted values occurred. The inability of eq 4 to predict crystal growth rates investigated over 5 orders of magnitude in range (dashed lines in Figure 1) is further exemplified when all the measured growth data for a given condition are considered in the analyses, and gave γ(all) ) 1.0 ( 0.1 × 10-13 erg/molecule. The failure of eq 4 to predict accurately the measured growth rates becomes more marked when (110) face growth rates are obtained as a function of temperature. Figure 2 displays (110) face growth rates obtained as a function of temperature with solution conditions of 0.1 M NaAc maintained at pH 4.6 with 3% NaCl, for four different protein concentrations. As shown, the observed growth rate data increases expectedly with respect to decreasing temperatures due to the steady diminishment of ceq over the range of temperatures investigated.22,23 Also observed, however, is an inflection in the growth rate data that is not characteristic of the steady diminishment of ceq with respect to decreasing temperatures, but varies systematically with the solution concentration.22,23 On the basis of the kinetic roughening hypothesis, growth rate data exhibited in Figure 2 were thus analyzed using parametric relations for the unknown quantities of the natural logarithms of both eqs 4 and 6, in respective regions attributable to the separate growth processes. The dashed and solid lines are computer-generated fit to the natural logarithm of eqs 4 and 6 that best characterize crystal growth in the B-D and C-M limits, respectively. For all protein concentrations (20, 30, 40, and 50 mg/mL), the effective

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step free energy was γ ) 1.2 ( 0.5 × 10-13 erg/molecule, and the energy barrier associated with continuous growth Ec was 6.1 ( 0.3 × 10-13 erg/molecule at a crossover concentration determined to be cr ∼ 7.7 ( 0.5ceq. Dependence of the Energetics of Crystal Growth on Temperature. In the above analyses, a causal relationship between the critical crossover concentration and the equilibrium concentration was realized. Of significant concern is then an independent estimation of the dependence of not only the energy barrier associated with continuous growth, but also the crossover concentration under a variety of solution conditions. Both Ec and cr can be independently determined from measurements of crystal growth rates with respect to increasing supersaturation. Estimates for Ec and cr can be obtained either by a direct application of eq 6 or extrapolation to zero growth from linear regression analyses of growth rate measurements obtained at high supersaturation.16 Figure 3a represents the temperature-dependent (110) face growth rates of lysozyme crystals as a function of protein concentration. The growth rate data were obtained at temperatures of 4, 9, 14, and 20 °C in solutions of 0.1 M NaAc buffer at pH 4.6 in the presence of 3% NaCl. The solid lines in Figure 3a represent fit to data obtained at “high” concentrations using a parametric relation for eq 6, enabling the determinations of the magnitudes of both cr and Ec. The magnitude of cr varied markedly within the temperature range investigated, whereas the magnitude of the values obtained for Ec was invariant within the error of estimation. The average values of Ec obtained from analyses of data shown in Figure 3a are shown in Table 1. The dashed lines in Figure 3a represent fit to a selected portion of the data, where possible, using a parametric relation for eq 4, and the average values for γ obtained from analyses are also shown in Table 1. Although the magnitude of Ec was determined to be numerically invariant, within 10% error of estimation using eq 6, an actual dependence of Ec on temperature becomes more readily evident when growth rate data are exhibited as a function of reduced concentration, c - cr. Figure 3b represents the temperature dependent (110) face growth rates of lysozyme crystals as a function c - cr, where the zero point demarcates the crossover concentration determined for a given solution condition. The growth data were obtained at temperatures of 4, 6, 9, 12, 14, 18, or 20 °C in solutions of 0.1 M NaAc buffer adjusted to pH 4.6 in the presence of 3% NaCl, as the solution concentration was increased beyond cr. The solid lines in Figure 3b represent linear fit to data determining changes in the slope or kinetic coefficient, R ) B exp(-Ec/KBT), within the temperature range 4-12 °C, where the magnitude in R increased from ∼2.3 × 10-8 to ∼2.8 × 10-8 (cm4/s), as the temperature was increased. Beyond 12 °C, however, a decrease in the values obtained for R from ∼2.8 × 10-8 to ∼1.0 × 10-8 (cm4/s) is observed as the temperature was increased to 20 °C (dashed lines in Figure 3b). Dependence of the Energetics of Crystal Growth on pH. The dependence of (110) face growth rates of lysozyme crystals in 0.1 M NaAc buffer within the pH range of 4.0 to 5.4 at a temperature of 14 °C is shown

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Figure 3. Crystal growth dependence on temperature. In panel a, the temperature-dependent (110) face growth rates of lysozyme protein crystals as a function of protein concentration are shown. The symbols b, O, 9, or 0 represent growth rate data obtained at temperatures of 4, 9, 14, or 20 °C in solutions of 0.1 M NaAc buffer at pH 4.6 in the presence of 3% NaCl, respectively. The solid lines, in panel a, represent fit to data obtained at “high” concentrations using a parametric relation for eq 6. The dashed lines in panel a represent fit to a selected portion of the data, where possible, using a parametric relation for eq 4. In panel b, the (110) face growth rates of tetragonal lysozyme crystals are exhibited as a function of the rescaled protein concentration. The symbols b, O, 9, 0, 2, [, or ] represent growth rate data obtained at temperatures of 4, 6, 9, 12, 14, 18, or 20 °C, respectively, in solutions of 0.1 M NaAc buffer adjusted to pH 4.6 in the presence of 3% NaCl. The solid and dashed lines represent linear fit to most data that best characterize the dependence of growth rates on temperature. In all cases, the error bars represent the standard deviation of measured values.

in Figure 4. In panel a, growth rate data obtained at pH of 4.0, 4.4, and 5.2 are displayed as a function of protein concentration. The solid lines in Figure 4a represent fit to data obtained at “high” concentrations using a parametric relation for eq 6 determining the magnitude of Ec and cr. The dashed lines in Figure 4a represent fit to a selected portion of the data using a parametric relation for eq 4. Both the magnitudes of cr and Ec varied marginally within the pH range investigated, within the 10% error of estimation. The average values of Ec obtained from analyses of data shown in Figure 4a, as well as data not displayed, are given in Table 1. In panel b, growth rate data are displayed as a function of the reduced protein concentration, c - cr. As shown, the magnitude of the kinetic coefficient, R, the magnitude in R decreased from ∼1.6 × 10-8 to ∼4.8 × 10-8 (cm4/s), or a factor of ∼3 decrease as the pH was increased. The magnitude of this decrease corresponds to an ∼1kBT increase in the energy barrier for continu-



Table 1. Values for the Energy Barrier of Continuous Growth, the Effective Step Free Energy and Critical Crossover Supersaturation solution condition

Ec (erg/molecule)

γ erg/molecule)

σc

3% NaCl, pH 4.6 4% NaCl, pH 5.0 5% NaCl, pH 5.0 3% NaCl, 6 °C, pH 4.0-5.4 5% NaCl, 14°C, pH 4.0-5.4 5% NaCl, 22 °C, pH 4.0-5.4 8 °C, pH 4.4, 2-7% NaCl 14 °C, pH 4.8, 2-7% NaCl 18 °C, pH 4.0, 2-5% NaCl

6.1 ( 0.3 × 10-13 6 ( 1 × 10-13 7 ( 1 × 10-13 6.0 ( 0.1 × 10-13 6.0 ( 0.2 × 10-13 5.9 ( 0.2 × 10-13 5.8 ( 0.4 × 10-13 6.2 ( 0.4 × 10-13 6.1 ( 0.5 × 10-13

1.2 ( 0.1 × 10-13

1.8 ( 0.2 2.0 ( 0.1 2.0 ( 0.2 1.7 ( 0.2 1.8 ( 0.2 1.9 ( 0.2 1.8 ( 0.2 2.0 ( 0.1 1.9 ( 0.2

ous growth, Ec, as the pH varied from 4.0 to 5.2. Upon comparison of growth rate data obtained in the pH range 4.0 to 5.2 at a temperature of 22 °C (data not shown), an ∼1kBT increase in the energy barrier for continuous growth, Ec, was also observed. Dependence of the Energetics of Crystal Growth on Precipitant Concentration. The dependence of precipitant concentration on (110) face growth rates of lysozyme protein crystals is shown in Figure 5. In panel a, growth rate data obtained at a temperature of 8 °C in solutions of 0.1 M NaAc buffered at pH 4.4 in the presence of NaCl ranging in concentration from 2% to 7% are shown. The solid lines in Figure 5a represent

Figure 4. Crystal growth dependence on pH. The (110) face growth rates of tetragonal lysozyme crystals obtained in 0.1 M NaAc with 5% NaCl at temperatures of 14 °C and varying pH are shown as a function of protein concentration, c, and reduced concentration, c - cr, in panels a and b, respectively. In panel a, the symbols b, O, 9, 0, [, or ] represent pH 4.0, 4.2, 4.4, 5.0, 5.2, or 5.4, respectively. The solid lines, in panel a, represent fit to data obtained at “high” concentrations using a parametric relation for eq 6. The dashed lines in panel a represent fit to a selected portion of the data using a parametric relation for eq 4. In panel b, the symbols b, O, 9, 0, [, ], 2, or 4 represent data obtained at pH of 4.0, 4.2, 4.4, 4.6, 4.8, 5.0, 5.2, or 5.4, respectively. In all cases, the error bars represent the standard deviation of measured values.

1.3 ( 0.1 × 10-13 1.0 ( 0.2 × 10-13 1.3 ( 0.1 × 10-13 1.4 ( 0.3 × 10-13 1.4 ( 0.3 × 10-13 1.4 ( 0.3 × 10-13 1.5 ( 0.5 × 10-13

fit to data obtained at “high” concentrations using a parametric relation for eq 6 determining the magnitude of Ec. The dashed lines in Figure 6a represent fit to a selected portion of the data using a parametric relation for eq 4. The magnitude of cr as well as values obtained for Ec varied systematically within the precipitant

Figure 5. Crystal growth dependence on precipitant concentration. The (110) face growth rates of tetragonal lysozyme crystals obtained in 0.1 M NaAc with a pH of 4.4 at temperatures of 8 °C and varying NaCl concentrations are shown as a function of protein concentration, c, and reduced concentration, c - cr, in panels a and b, respectively. In panel a, the symbols b, O, 9, 0, or [ represent growth rate data obtained at NaCl concentrations of 2, 2.5, 3, 5, or 7%, respectively. The solid lines, in panel a, represent fit to data obtained at “high” concentrations using a parametric relation for eq 6 determining the magnitude of Ec. The dashed lines in panel a represent fit to a selected portion of the data using a parametric relation for eq 4. In panel b the symbols b, O, 9, 0, [, ], or 2 represent a growth rate data obtained in the presence of 2, 2.5, 3, 3.5, 4, 5, or 7% NaCl, respectively. The solid lines are linear fit to respective data corresponding to a marked, ∼2kBT, increase in the barrier as the ionic concentration is reduced. In all cases, the error bars represent the standard deviation of measured values.

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Figure 7. Roughening of a lysozyme crystal surface. Smooth (a) and rough (b) surfaces of lysozyme protein crystals are exhibited. The protein crystal observed in (a) was grown initially in a sealed sample cell at a solution supersaturation σ ∼ 1.6. After some period of growth, the solution supersaturation was adjusted to ∼2.4 by lowering the temperature. After prolonged growth (∼90 min) microscopically rough surfaces were observed. The initial size of the lysozyme crystal presented in (a) is ∼15 µm, which subsequently had grown by ∼3-4µm in (b).

Figure 6. The behavior of the crossover concentration with respect to temperature, pH, and %NaCl. Crossover concentrations for lysozyme crystal growth by continuous addition, in panel a, as a function of temperature (b), pH (O), and precipitant concentrations (9). In panel b, the supersaturation values obtained for data presented in panel a, are exhibited in the same format.

concentration range investigated. The average values of Ec obtained from analyses of data shown in Figure 5a, as well as data not displayed are given in Table 1. In Figure 5b, growth rate data are displayed as a function of the reduced protein concentration. The solid lines in Figure 5b are linear fit to respective data indicating a progressive diminishment in the kinetic coefficient R from ∼0.8 × 10-8 (cm4/s) to ∼6.1 × 10-8 (cm4/s) as the precipitant concentration is reduced from 7 to 2%, respectively. Independent of the solution temperature or pH presented (data not shown), the magnitude of R diminished by a factor of ∼8-9 as the NaCl concentration is varied from 7 to 2%, corresponding to an ∼2kBT increase in the energy barrier for continuous growth as the ion concentration is reduced. Moreover, a nonlinear, asymptotic increase in the kinetic coefficient R as the precipitant concentration was diminished below 4% (w/v) was also observed. Dependence of the Roughening Concentration, cr, on Solution Conditions. With the implementation of a kinetic roughening hypothesis, the crossover concentration, cr, defining the upper limits of the B-D growth process had been determined for a variety of conditions. The magnitude of variations in the estimates for cr was dependent on the experimental parameter temperature (T), precipitant concentration (NaCl), or solution pH under consideration. The dependence of cr(T, NaCl, pH) data under corresponding conditions of T, NaCl, and pH is exhibited in Figure 6a. Of the three conditions, the behavior of cr with respect to variations in precipitant concentration (solid squares in Figure 6a)

at three different temperatures exhibited the most notable trends. That is, for a given temperature, the crossover concentrations determining the threshold for crystal growth from B-D to C-M limit decreased asymptotically as a function of precipitant concentration. As the kinetic coefficient R also varied markedly with the precipitant concentration, it could then be postulated that the magnitude of cr would reflect the change in the attachment probability. In fact, Figure 6b, which displays the critical supersaturation σc ) ln(cr(T, pH, NaCl)/ceq(T, pH, NaCl)) for a variety of conditions of solution temperature T, pH, and NaCl concentration, indicates that a single crossover supersaturation (1.9 ( 0.2) determines the threshold for crystal growth from B-D to C-M limit. Experimental Verification of the Formation of Rough or Rounded Surfaces. The implementation of a kinetic roughening hypothesis gave rise to an unexpected result of a functional dependence of the crossover concentration, cr, on solubility, ceq, which is independent of the solution parameter being varied. If indeed the assumption of kinetic roughening hypothesis was valid, lysozyme crystals grown beyond the critical supersaturation must exhibit rough or rounded surfaces, which must also be independent of the solution parameter being varied. In the example shown in Figure 7, the initial solution conditions were pH 5.0, 4% NaCl with a protein concentration of 30 mg/mL. At the outset, the temperature of the stage was set to 20 °C, to yield a supersaturation σ ∼ 1.6, to obtain seed crystals that were allowed to grow until crystal facets were microscopically visible under a 40× microscope objective. Upon visible observations of faceted crystals, the temperature of stage was reduced to 14 °C to obtain a supersaturation σ ∼ 2.4. The protein crystals within the sealed microcuvette were then allowed to grow for a period of ∼90 min, while images of their growth progress were continuously taken and recorded. Figure 7 exhibits a lysozyme crystal initially grown at 20 °C for the condition below σc (a) and subsequently grown at a solution condition above σc at 14 °C (b). A smooth faceted crystal is observed in Figure 7a, whereas microscopically


rough and rounded surfaces are evidenced on all faces of the crystal in Figure 7b. The result that microscopically rough surfaces were obtained when crystals were grown with solution supersaturation beyond σc is in direct support of the kinetic roughening hypothesis. Discussion The data analyses presented above utilized ∼1000 averaged values for crystal growth rates obtained from measurements of ∼10 000 individual lysozyme crystals. In addition to growth rates, several thousand independent measurements of tetragonal lysozyme solubility for the various conditions of temperature, pH, and precipitant concentration were also required to permit detailed analyses.22,23 The results of the analyses are (1) determination of the effective step free energy and attachment barriers, γ and Ec, respectively, (2) determination of a critical crossover supersaturation σc, and (3) the dependence of γ and Ec on temperature, pH, and NaCl concentration. This section thus summarizes and addresses the significance of the results. From the outset, the (110) face lysozyme crystal growth rate data obtained in the supersaturation range ∼3 < c/ceq e 11 or ∼1 < σ e 2.4 for all solution conditions and temperatures were analyzed using a 2D nucleation model. Solely on the basis of experimental determinations of the magnitude of either the supersaturation or crystal growth rate, available physical models describing crystal growth cannot, a priori, predict the mode by which a crystal grows. In the absence of independent determinations of the various modes of growth, a determining factor in the applicability of any given model relies on the correspondence between the assumed model and experimental results. For this particular investigation, the application of a 2D nucleation model describing crystal growth within a given range of supersaturation can be experimentally justified, by an independent means capable of detecting the various modes of growth at the molecular level. Namely, recent investigations using atomic force microscopy (AFM) have shown the formation of 2D islands in every case within the range 0.69 < σ < 1.79.24 However, investigations using AFM to determine precisely the mode of lysozyme crystal growth beyond σ > 2.0 have yet to be performed. Alternatively, Figure 7 provides an example for the determination of crystal growth beyond a kinetic roughening concentration, a rough or rounded crystal grown at high supersaturations is considered as direct support for the kinetic roughening hypothesis. Beyond either a certain magnitude for growth velocities or solution supersaturation, the observed deviation between theoretical and measured crystal growth rates (see Figure 1) can be viewed as an “effective” slowing of measured values. Several hypothetical reasons can been attributed for the effective slowing of measured crystal growth rates at high supersaturation: impurity incorporation,25,26 solution thermodynamic nonideality,27 macromolecular aggregate distribution,6-9 or kinetic roughening.28 Any one of these hypothetical processes may contribute to the observed effective slowing of crystal growth rates, either individually or in concert. Impurity incorporation could possibly account for the slowing of crystal growth beyond certain growth veloci-


ties, whereas solution nonidealities, aggregate distribution, or kinetic roughening are expected to become effective beyond a certain solution supersaturation. An understanding of the effective slowing down of crystal growth velocities upon invoking either the impurity incorporation or aggregate distribution hypotheses would require speculative assumptions on crystal growth or solution behavior that would render quantitative, physical description of crystal growth rates difficult. The effects of thermodynamic nonideality are, however, rather straightforward and imply a virial correction for the magnitude of the ideal solution disequilibrium:27 ∆µr ) ∆µ + 2B2kBT(c - ceq), where ∆µr is the “real” solution chemical potential and B2 is the second virial coefficient. For the case of lysozyme that exhibits attractive interactions (negative B2), the magnitude of the virial correction term B2 is on the order of ∼ -1 to -3 × 10-2 (mL/mg) for the data represented in Figure 1.29 Over the range of concentration investigated, the magnitude of correction to ∆µr varies from ∼5% at the lowest concentration of 6 mg/mL to ∼20% at 60 mg/mL. Upon implementing the virial correction to ∆µr, neither the estimation of γ ) 1.5 × 10-13 erg/molecule nor the observed deviations between measurement and theory were significantly affected (data not shown). Nevertheless, the effects of solution nonidealities are considered to become negligible when growth rates are determined upon varying the solution disequilibrium by adjusting the solution temperature.27 Growth rate data obtained by adjusting the solution temperature provide additional convincing support for the kinetic roughening hypothesis (see Figure 2). Inflections in any given rate process that are observable in an Arrhenius plot can be considered indicative of either a phase transition or a change in the energetics of rate processes. The liquid-liquid (L-L) phase transition of lysozyme solutions30-33 are expected to contribute to a diminishment of crystal growth rate, as solution temperatures or concentrations approach the solution cloudpoint. The results exhibited in Figure 2 are, however, not considered to arise as a result of L-L phase transitions. For example, the observed inflection in growth rates exhibited in Figure 2 occur ∼8 to 10 °C above the observed cloud point temperatures for respective solution concentrations. A possible remaining mechanism that gives rise to a change in the energetics of the crystal growth process is then a transition from B-D to C-M growth as solution supersaturation or solute concentration exceeds σc or cr. The combination of experimental observables: diminished growth rates beyond a critical supersaturation, inflections in an Arrhenius plot, and rounding of crystal surfaces provide considerable support for the kinetic roughening hypothesis. Indeed, models for lysozyme crystal growth by surface nucleation at low supersaturations and kinetic roughening at high supersaturations adequately describe the features of a vast array of experimentally recorded kinetics. There are, however, certain important discrepancies that arise upon interpretation of growth rate data, which require further examination. These discrepancies are (1) the dependence of Ec with respect to temperature, (2) the increase in Ec as the net titration charge is reduced, (3) the increase in Ec as the ionic concentration is reduced, (4)

698 Crystal Growth & Design, Vol. 4, No. 4, 2004

a measurably invariant σc, and finally (5) the magnitude of ∆G/nuc at the experimentally determined σc. The findings that the energy barrier for continuous addition, Ec, becomes modified with respect to temperature (Figure 3), pH (Figure 4), and precipitant concentration (Figure 5) are not readily understood in terms of quantifiable physical models. However, it is generally considered that such complex behavior is an inherent characteristic of protein molecules. For example, growth rate data obtained beyond σc behave predictably within the temperature range of 4-12 °C (see Figure 3), where the magnitude of the kinetic coefficient, R, increases as expected. Beyond 12 °C, however, the magnitude of the kinetic coefficient, R, unexpectedly decreases as the temperature is increased. Similar nonphysical results of decreasing magnitudes of kinetic coefficients are also obtained when either the net titration charge is reduced as the solution pH is increased, or precipitant concentration is reduced. Such nonphysical findings are often described in terms of changes occurring in the “hydrophobic” nature of protein molecules. Short of developing new methods and models to precisely quantify hydrophobic interactions, understanding the hydrophobic nature of proteins remains qualitative, at best. The finding of a measurably invariant critical crossover supersaturation, σc, can, however, be readily understood in terms of the estimated values for the effective barrier, γ, for 2D nucleation. The magnitude of γ ) 1.3 ( 0.3 × 10-13 erg/molecule offers direct comparisons with previously determined values for the effective step free energy, given as γ ) 9.4 × 10-14 erg/ molecule34 and γ ) 1.4 × 10-13 erg/molecule.28 Both the reported values agree remarkably well when constraints utilized in their determination are taken into consideration. It should be noted that solution conditions of pH, buffer, and precipitant concentrations, and temperatures in the determination of γ in refs 28 and 34 are different and do not overlap the conditions presented. Upon the basis of values obtained for a vast array of conditions (see Table 1), it could be concluded that the step free energy is measurably invariant. If ∆G/nuc is minimum and invariant at the B-D to C-M crossover, a constant effective step free energy predicts the observed invariance of the critical supersaturation (see eq 3).16,18,19 For systems that exhibit kinetic roughening, it is assumed that the magnitude of ∆G/nuc is on the order of kBT for the B-D to C-M crossover.16,18,19 This assumption was necessary in the estimation of the effective step free energies γ for known values of σc, where σc ) π(γ/ kBT)2.16,18,19 Applying this assumption predicts either γ ) 3.0 × 10-14 erg/molecule or σc ) 32, both of which are nearly an order of magnitude in error from the experimentally determined values. Nevertheless, eq 3 predicts invariance of the critical supersaturation as γ is also invariant under the conditions investigated for the assumption that B-D to C-M crossover occurs at constant ∆G/nuc. In the present case, both γ and σc are defined quantities and thus enable the determination of the critical free energy maximum at the kinetic roughening transition directly as, according to eq 3: ∆Gnuc ) 6.2 ( 1.5 × 10-13 erg/molecule, an average of all data reported in Table 1. In addition to the determination of ∆G/nuc,

Gorti et al.

values obtained for γ and σc also enable, using eq 2, the determination of the critical nucleus size, N* ∼ 8 ( 5 molecules, at the B-D to C-M crossover. In theory, the value for N* obtained at the critical crossover supersaturation represents the number of molecules within an adatom that permits crystal growth.10,11,14 This value agrees remarkably well with the estimated growth unit size determined from modeling and AFM measurements.8,24 It is of interest to note that the value ∆G/nuc ) 6.2 ( 1.5 × 10-13 erg/molecule is remarkably comparable to Ec ) 6.1 ( 0.4 × 10-13 erg/molecule, the free energy barrier for continuous mode growth. On the basis of these results, it is postulated that the critical supersaturation for a crossover from B-D to C-M occurs when the limit ∆G/nuc approaches the magnitude of the free energy barrier for C-M growth and not kBT, as previously hypothesized.16,18,19 Nevertheless, detailed investigations would be required to determine the validity of the hypothesis that B-D to C-M crossover occurs as the limit ∆G/nuc approaches the magnitude of Ec, particularly at temperatures far below the thermodynamic roughening temperature, Tr. Conclusion Analyses of changes in the observed crystal growth rates in terms of well-established physical theories for crystal growth enabled the determination of the following: (i) an invariant effective step free energy γ; (ii) an invariant critical supersaturation σc; (iii) an adatom comprised of 8 ( 5 macromolecules that contributes to the continuous growth of lysozyme crystals.; (iv) an energy barrier Ec that was functionally dependent on the temperature, pH, and NaCl concentration; (v) a hypothesis that B-D to C-M occurs as ∆G/nuc approaches Ec at temperatures far below the thermodynamic roughening temperature, Tr. Although other possibilities might exist, the analyses provided the means for the possible discovery of a kinetic roughening transition. The assumption of kinetic roughening simplified analyses of growth rate data, where specific characteristics of Ec that were dependent on the temperature, pH, and precipitant concentration were observed. The specific trends in Ec require a detailed understanding of the chemical-physical characteristics of lysozyme molecules to permit reasonable interpretations. Summary of Definitions c (mg/mL) concentration of protein ceq (mg/mL) equilibrium concentration (crystal does not grow or dissolve) cr (mg/mL) crossover concentration indicating a change in growth mechanism c - cr (mg/ reduced protein concentration beyond the mL) critical crossover concentration ∆µ, σ disequilibrium ∆µ ) kBTσ or excess supersaturation, σ ) ln(c/ceq) energy barrier determining the probability Ec (erg/ molecule) of macromolecular attachment in crystal growth in the continuous mode

Energetics of Tetragonal Lysozyme Crystal Growth energy barrier per unit length determining Es (erg/ the probability of macromolecular attachmoleculement at a kink or step edge for crystal cm) growth in the layer mode γ (erg/ effective energy barrier, γ ) EsΩ1/2 molecule) h (cm) step height R (cm4/s) effective rate of macromolecular attachment, per milligram quantity of substance, onto a rough crystal surface σc the critical crossover supersaturation ln(cr /ceq) Ω (cm2) macromolecular attachment area at a kink or step Vs (cm/s) growth velocity or rate normal to the surface of a smooth crystal in growth by layers generated by 2D nucleation Vc (cm/s) growth velocity or rate normal to the surface of a rough crystal in growth by continuous mode

Acknowledgment. One of the authors (S.G.) is grateful to Dr. Phil Segre for numerous discussions on the subject of protein crystal growth, which contributed greatly to improvements in this manuscript, in particular, the derivations of eq 6. S.G. would also like to thank Dr. Alexander Chernov for his insightful comments and suggestions. References (1) Pusey, M. L. Rev. Sci. Instrum. 1993, 64, 3121-3125. (2) Forsythe, E. L.; Pusey, M. L. J. Cryst. Growth 1994, 139, 89-94. (3) Forsythe, E. L.; Ewing, F. L.; Pusey, M. L. Acta Crystallogr., Sect. D 1994, 50, 614-619. (4) Nadarajah, A.; Forsythe, E. L.; Pusey, M. L. J. Cryst. Growth 1995, 151, 163-172. (5) Forsythe, E. L.; Nadarajah, A.; Pusey, M. L. Acta Crystallogr., Sect. D 1999, 55, 1005-1011. (6) Li, M.; Nadarajah, A.; Pusey, M. L. J. Cryst. Growth 1995, 156, 121-132. (7) Nadarajah, A.; Pusey, M. L. Acta Crystallogr., Sect. D 1996, 52, 983-996. (8) Nadarajah, A.; Li, M.; Pusey, M. L. Acta Crystallogr., Sect. D 1997, 53, 524-534. (9) Pusey, M. L.; Nadarajah, A. Cryst. Growth Des. 2002, 2, 475-483.

Crystal Growth & Design, Vol. 4, No. 4, 2004 699 (10) Levi, A. C.; Kotrla, M. J. Phys.: Condens. Matter 1997, 9, 299-344. (11) Tartaglino, U.; Levi, A. C. Phys. A 2000, 277, 83-105. (12) Cuppen, H. M.; Meekes, H.; van Veenendaal, E.; van Enckevort, W. J. P.; Bennema, P.; Reedijk, M. F.; Arsic, J.; Vleig, E. Surf. Sci. 2002, 506, 183-195. (13) Burton, W. K.; Cabrerra, N.; Frank, C. Philos. Trans. R. Soc. A. 1951, 243, 299-358. (14) Saito, Y. Statistical Physics of Crystal Growth; World Scientific: Singapore, 1996; pp 9-10. (15) Hwa, T.; Kardar, M.; Paczuski, M. Phys. Rev. Lett. 1991, 66, 441-444. (16) Elwenspoek, M.; van der Eerden, J. P. J. Phys. A 1987, 20, 669-678. (17) Elbaum, M. Phys. Rev. Lett. 1991, 67, 2982-2985. (18) Liu, X.-Y.; van Hoof, P.; Bennema, P. Phys. Rev. Lett. 1993, 71, 109-112. (19) van Hoof, P. J. C. M.; Schoutsen, M.; Bennema, P. J. Cryst. Growth 1998, 192, 307-317. (20) Ewing, F. L.; Forsythe, E. L.; van der Woerd, M.; Pusey, M. L. J. Cryst. Growth 1996, 160, 389-397. (21) Aune, K. C.; Tanford, C. Biochemistry 1969, 8, 4579-4590. (22) Cacioppo, E.; Pusey, M. L. J. Cryst. Growth 1991, 114, 286292. (23) Forsythe, E. L.; Judge, R. A.; Pusey, M. L. J. Chem. Eng. Data 1999, 44, 637-640. (24) Rong, L.; Yamane, T.; Niimura, N. J. Cryst. Growth 2000, 217, 161-169. (25) Vekilov, P. G. Prog. Cryst. Growth 1993, 26, 25-49. (26) Rosenberger, F.; Muschol, M.; Thomas, B. R.; Vekilov, P. G. J. Cryst. Growth 1996, 168, 1-23. (27) Grant, M. L. J. Cryst. Growth 2000, 209, 130-137. (28) Kurihara, K.; Miyashita, S.; Sazaki, G.; Nakada, T.; Suzuki Y.; Komatsu, H. J. Cryst. Growth 1996, 166, 904-908. (29) Guo, B.; McDonald, H.; Asanov, A.; Combs, L. L.; Wilson, W. W. J. Cryst. Growth 1999, 196, 424-433. (30) Ishimoto, C.; Tanaka, T. Phys. Rev. Lett. 1977, 39, 474477. (31) Taratuta, V. G.; Holschbach, A.; Thurston, G. M.; Blankstein, D.; Benedek, G. B. J. Phys. Chem. 1990, 94, 21402144. (32) Broide, M. L.; Tominc, T. M.; Saxowsky, M. D. Phys. Rev. E. 1996, 53, 6325-6335. (33) Manno, M.; Xiao, C.; Bulone, D.; Martorana, V.; Biagio, P. L. S. Phys. Rev. E. 2003, 68, 011904. (34) Durbin, S.; Feher, G. J. Cryst. Growth 1986, 76, 583-592.

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Kinetic Roughening and Energetics of Tetragonal Lysozyme Crystal

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