Molecular Level Kinetics of Growth

While the kink density is a thermodynamic growth variable that characterizes the affinity of the crystal to the solute molecules, the kinetics of incorporation are reflected by the flux of molecules into a growth site. To monitor these fast incorporation events, we disabled the slow scanning axis of the atomic force microscope. The advance of a step site is shown in Figs. 8 and 9. Area scans immediately following the trace in Figs. 8 and 9 revealed that step motion is not inhibited or accelerated at the location of scanning; that is, the chosen scanning parameters ensured that step propagation was not affected by scanning over the same line for approx 3 min.

Despite the relatively high solution supersaturation a = 1.1, the time trace in Fig. 8 reveals not only 25 arrivals to, but also 22 departures of molecules from the monitored site. All arrivals and departures of molecules to and from the monitored site involve single molecules. Thus, in contrast to claims of preformed multiple-molecule growth units for the protein lysozyme (63-65), apo-ferritin and ferritin crystal grow by the attachment of single molecules.

This type of data collection does not allow observations of the neighboring sites at the step. Hence, we cannot distinguish between attachment and detachment from molecules in the kinks or at the steps. Still, we notice that the resi-

Net -growth

Net -growth

Surface Coord Irate [n«i|

Fig. 8. Incorporation of molecules into steps at apoferritin concentration of 70 mg/mL, o = 1.1, and (C/Ce - 1) = 2.04. The pseudoimage recorded using a scanning frequency of 3 Hz with the y scan axis disabled at time = 0 shows displacement of one step site. The contour traces the step position. Red arrows indicate attachment and detachment events with a residence time >1 s, and blue double-sided arrows with a residence time <1 s; for details, see the text. The appearance of 1/2 molecule attachments at times >80 s, highlighted in green, is owing to events at a neighboring site that entered the image owing to scanner drift. (From ref. 10.)

Surface Coord Irate [n«i|

Fig. 8. Incorporation of molecules into steps at apoferritin concentration of 70 mg/mL, o = 1.1, and (C/Ce - 1) = 2.04. The pseudoimage recorded using a scanning frequency of 3 Hz with the y scan axis disabled at time = 0 shows displacement of one step site. The contour traces the step position. Red arrows indicate attachment and detachment events with a residence time >1 s, and blue double-sided arrows with a residence time <1 s; for details, see the text. The appearance of 1/2 molecule attachments at times >80 s, highlighted in green, is owing to events at a neighboring site that entered the image owing to scanner drift. (From ref. 10.)

dence times t between these events fall into either t < 1 s or t > 5 s. Figure 8 shows six events of the second type and 19 events of the first. Their ratio is roughly equal to the kink density along the step, suggesting that the longtime events may be attachments and detachments to and from a kink, and the short ones may be sightings of molecules at the step edge.

Furthermore, molecules may enter the line of observation owing to molecular diffusion along the step, or to exchange with the terrace between the steps or the adjacent solution. Whereas the latter results in step propagation and growth, the former is a process that only involves rearrangement of molecules already belonging to the crystals and that may not be associated with growth. To distinguish between the two, as done previously for steps on metal and semiconductor surfaces (28,54,66-68), we calculated the time correlation func-

Surface Coordinate [nm]

Fig. 9. Incorporation of molecules into steps on ferritin crystal at protein concentration of 70 mg/mL, corresponding to supersaturation o = A|j./kBT = 0.7, (C/Ce - 1) = 1. A pseudoimage recorded with the scan axis parallel to the step disabled at time = 0 shows the displacement of one molecular site at the step. In this imaging mode, the molecules in the upper and lower layers appear as vertical columns. The red contour traces the step position. Left shifts of this contour indicate detachment of a molecule from the monitored site, and right shifts indicate molecular attachment into the monitored site. (From ref. 31.)

tion of the step position x (in molecular size units) as ([x(t + At) - x(t)]2)At, with averaging over the respective At. In Fig. 10, it is plotted as a function of At. Theoretical analyses of the exchange of the steps with the medium at equilibrium (28,54,68-70) predict that if diffusion along the step edge dominates the advance of the step site, the cross-correlation should follow At1/4 dependence (28,54,68-70). We found no theory dealing with supersaturated conditions. However, motion of a site on the step edge is similar to Brownian motion (68,69). For Brownian diffusion, the coefficient relating ([x(t + At) - x(t)]2)At and At1/2 may vary, but the exponent 1/2 of At does not depend on the presence

Fig. 10. Time correlation curve, characterizing mean square displacement of a location at a step for a time interval At as a function of this At, corresponding to trace of step location in Fig. 8; inset: logarithmic plot. Solid squares indicate data; lines fits with exponential dependencies on time as indicated in the plots. (From ref. 25.)

Fig. 10. Time correlation curve, characterizing mean square displacement of a location at a step for a time interval At as a function of this At, corresponding to trace of step location in Fig. 8; inset: logarithmic plot. Solid squares indicate data; lines fits with exponential dependencies on time as indicated in the plots. (From ref. 25.)

or absence of concentration/chemical potential gradients (57). Hence, we use only the exponents of At stemming from the data in Fig. 10 for further discussion.

The data in Fig. 10 do not fit a single exponential. The deviation from 1/4 at times longer than 20 s allows us to conclude that the trace in Fig. 8 likely reflects exchange of molecules between the step and interstep terraces or the adjacent solution. This conclusion allows us to extract from Figs. 8 and 9 net frequencies of attachment of molecules to kinks. For apoferritin at (C/Ce - 1) = 2, from the net attachment of three molecules for 162 s and the probability of viewing a kink of 1/nk = 1/3.5, we get (j+- j-) = 0.065 s-1, or one molecule per approx 15 s. For ferritin at (C/Ce - 1) = 1, Fig. 9 shows net growth of two molecules for 128 s, leading to an average net flux (j+ - j-) = 0.054 s-1 into the growth sites distributed with mean density = 0.28. Thus, even at the relatively high supersaturation in Figs. 8 and 9, incorporation of molecules into the crystal is extremely slow and occurs over macroscopic time scales

The step velocities v for the two proteins, determined using the three methods discussed in Subheading 2.3. are shown in Fig. 11. The data fit well the proportionality with Q = 1/4 a3 = 1.56 x 10-18 cm3—the crystal volume per ferritin or apoferritin molecule, and step kinetic coefficient P (50,71) is (6.0 ± 0.4) x 10-4 cm/s for ferritin and (6.0 ± 0.3) x 10-4 cm/s for apoferritin.

Fig. 11. Determination of kinetic coefficients for step growth for ferritin and apo-ferritin. Dependencies of the step growth rates v on the crystallization driving force (C/Ce - 1). (□, O) v-Values for, respectively, ferritin and apoferritin, from sequences of molecular resolution in situ AFM images of the advancing steps. The reason for the lower value of the point for apoferritin at (C/Ce - 1) = 42 is not well understood. This point was not used in the determination of the kinetic coefficient p. (A) Data for apoferritin extracted from disabled y-axis scans. (■) Data for ferritin from time traces of the step growth rate using laser interferometry. The straight line corresponds to the step kinetic coefficient p = 6 x 10-4 cm/s. (From ref. 40.)

Fig. 11. Determination of kinetic coefficients for step growth for ferritin and apo-ferritin. Dependencies of the step growth rates v on the crystallization driving force (C/Ce - 1). (□, O) v-Values for, respectively, ferritin and apoferritin, from sequences of molecular resolution in situ AFM images of the advancing steps. The reason for the lower value of the point for apoferritin at (C/Ce - 1) = 42 is not well understood. This point was not used in the determination of the kinetic coefficient p. (A) Data for apoferritin extracted from disabled y-axis scans. (■) Data for ferritin from time traces of the step growth rate using laser interferometry. The straight line corresponds to the step kinetic coefficient p = 6 x 10-4 cm/s. (From ref. 40.)

Since there are no sources or sinks of molecules at the step other than the attachment sites, the step growth rate v should equal ank~1(j+ - j_) (50,72). At (C/Ce - 1) = 1, at which all data on ferritin in Fig. 9 were collected, the value of the step growth rate for ferritin from Fig. 11 is v = 0.20 nm/s, equal to the product anfl(j+ - j-). For apoferritin, the average step velocity at (C/Ce - 1) = 2, a = 1.1 is v = 0.26 nm/s. The product ank_1(j+ - j-) determined at the same conditions is 0.24 nm/s.

The closeness of the values of the product anfl(j+ - j-) and measured vs indicates that the step propagation in ferritin crystallization occurs only owing to incorporation of molecules into the kinks along the steps (10,25).

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