Role of molecular charge and hydrophilicity in regulating the kinetics of crystal growth

Abstract

The composition of biologic molecules isolated from biominerals suggests that control of mineral growth is linked to biochemical features. Here, we define a systematic relationship between the ability of biomolecules in solution to promote the growth of calcite (CaCO₃) and their net negative molecular charge and hydrophilicity. The degree of enhancement depends on peptide composition, but not on peptide sequence. Data analysis shows that this rate enhancement arises from an increase in the kinetic coefficient. We interpret the mechanism of growth enhancement to be a catalytic process whereby biomolecules reduce the magnitude of the diffusive barrier, E_k, by perturbations that displace water molecules. The result is a decrease in the energy barrier for attachment of solutes to the solid phase. This previously unrecognized relationship also rationalizes recently reported data showing acceleration of calcite growth rates over rates measured in the pure system by nanomolar levels of abalone nacre proteins. These findings show that the growth-modifying properties of small model peptides may be scaled up to analyze mineralization processes that are mediated by more complex proteins. We suggest that enhancement of calcite growth may now be estimated a priori from the composition of peptide sequences and the calculated values of hydrophilicity and net molecular charge. This insight may contribute to an improved understanding of diverse systems of biomineralization and design of new synthetic growth modulators.

Organisms are able to achieve rapid rates of mineralization while also selectively inhibiting or completely blocking the growth of biomineral faces. However, the mechanisms that allow for such a fine degree of control are poorly understood. Some calcifiers use the formation of amorphous precursors (1–3), whereas others appear to use classical crystal growth processes (4, 5). The widely accepted view of the latter mechanism is that macromolecular modifiers in solution are capable of directing growth morphology, but have only neutral or inhibitory effects on growth rate. Recently, this dogma was challenged by showing that low nanomolar levels of proteins isolated from abalone nacre actually increased rates of calcite growth by as much as 5-fold by accelerating the kinetics of molecular step propagation across mineral surfaces (6, 7).

Additional support for this perspective was provided by atomic force microscopy (AFM) studies of the dependence of growth on levels of linear peptides containing one to six aspartic acids (8). That study focused on inhibition occurring at high peptide concentrations and the role of water in the interactions of these peptides with steps during growth. However, further examination of that data also shows that lower concentrations of all of these molecules accelerated growth by small amounts that scaled with chain length (8). Because low levels of both peptides and proteins accelerate calcite growth rates, we now postulate that the degree of rate enhancement might be proportional to specific physicochemical variables such as molecular size or net charge.

A probable relationship between the chemical features of biomolecules and crystal growth control is supported by many observations showing that proteins isolated from sites of biomineralization in tissues are unusually enriched in highly acidic amino acids, notably aspartic acid and also glutamic acid (9–13). The Asp-rich motifs have been postulated to function by preferentially binding to cations such as Ca²⁺ (14) during the controlled formation of minerals such as calcite (12). Under physiological conditions, the carboxyl groups of aspartic acid are negatively charged and are believed to engage in electrostatic interactions with Ca²⁺ ions at the nascent crystal surface (15). Moreover, the charge and stereochemistry of exposed amino acid side chains are known to mediate specific interactions with calcite steps and surfaces (16, 17). However, although favored interactions and morphological consequences have been demonstrated, the mechanistic and kinetic roles of these molecules in modifying growth, particularly through growth acceleration, are less well understood.

In the present investigation, we sought to better define the relationship between chemical structure and the regulation of calcite growth by examining the effects of low concentrations of Asp-dipeptides and linear aspartic acid-rich peptides (18) that are intermediate in size when compared with the very small peptides (8) and proteins (6, 7) known to accelerate calcite growth. We use the results of these studies to show that there is a link between the net molecular charge and hydrophilicity of biomolecules and the acceleration of calcite growth.

Results

The dependence of molecular step speed on peptide concentration was measured by using in situ AFM to image growth on the (104) face of calcite in solutions at a fixed supersaturation, σ, of 0.92 (Fig. 1). The supersaturation is defined as σ = ln(a_Ca++a_CO3=/K_sp), where a denotes the species activity, and K_sp denotes the equilibrium solubility constant at 25°C. From previous measurements in our group, calcite step velocity dependence on σ, v(σ), remains linear up to a σ of at least 1.2 in a pure system, although some nonlinearity exists for σ < 0.3 (19). Low concentrations of all dipeptides induced increases in step velocities above those observed in peptide-free control experiments and the larger Asp-rich 27-mer peptides induced much larger increases in step velocities (Fig. 2). However, at higher concentrations, step acceleration was reversed with step speeds decreasing rapidly to zero, an inhibitory effect discussed by Elhadj et al. (8).

Fig. 1.

Representative in situ AFM images (5.6 × 2.8 μm) illustrate the steady-state morphology of calcite growth hillock. (a) In the absence of growth modifiers, the hillock structure exhibits the c-glide plane axis of symmetry and four step directions of the symmetrically equivalent obtuse and acute steps. (b and c) In the presence of increasing concentrations of Asp–His step edges become roughened. (d) Illustration shows how the orientation of the carbonate groups with respect to the (104) surface gives rise to the direction-specific differences in the step edge structure of the obtuse and acute step risers.

Fig. 2.

Experimental measurements of step propagation rate along the obtuse direction versus peptide concentration for dipeptides and 27-mer peptides (a) and aspartates of increasing molecular size (b). Normalized propagation rates are shown as step velocity in the presence of peptide impurities, v, over the step velocity without impurities, v₀. Note that, when plotted on linear-log scale diagram (which here spans six orders of magnitude), the apparent trends shown do not fully reflect the relative magnitude of the slopes between peptides.

The step acceleration portions of growth curves were further analyzed to evaluate the relative effects of low concentrations of 11 peptides on growth velocity. The effect on growth rate observed with each peptide may be expressed as a step velocity ratio, v/v₀, where v₀ is the step speed for the pure system, and v is the step speed in the presence of peptides. For purposes of analysis, we used the experimental data from Fig. 2 and the data from refs. 6 and 7 to calculate the rate enhancement at an arbitrary concentration of 0.1 μM, (v/v₀)_0.1μM, by linear interpolation. Table 1 shows that 0.1-μM levels of the peptides containing as many as six aspartic acids enhanced growth by 0.5–15% over the control velocity. The v/v₀ values at 0.1 μM for larger synthetic linear 27-mer peptides, (Asp₃–Gly)₆–Asp₃ and (Asp₃–Ser)₆–Asp₃, showed much greater increases in step velocity of 64% and 44%, respectively.

Table 1.

Physical properties and the normalized step acceleration, (v/v₀)_0.1μM of obtuse steps for the peptides and proteins shown in Figs. 3 and 4

Peptide/protein	Molecular weight, g/mol	Hydrophilicity, kJ/mol	Net charge,pH 8.5, unit charge	Step acceleration, v/v00.1μM	Step acceleration, ln[(v/v0)0.1μM]	Reference
Asp-Gly	190	12.6	−1.70	1.008	0.0080
Asp-Leu	246	5.0	−1.70	1.005	0.0050
Asp-His	270	10.9	−1.69	1.014	0.014
Asp-Glu	262	25.1	−2.70	1.033	0.032
Asp-1	133	12.6	−1.70	1.02	0.020	8
Asp-2	248	25.1	−2.70	1.05	0.049	8
Asp-4	478	50.2	−4.70	1.08	0.077	8
Asp-5	593	62.8	−5.70	1.10	0.095	8
Asp-6	708	75.3	−6.70	1.15	0.14	8
(Asp3Ser)6Asp3	2,957	271.1	−21.70	1.44	0.36
(Asp3Gly)6Asp3	2,777	259.8	−21.70	1.64	0.49
AP7-N	3,225	−12.6	−3.86	1.03	0.030	7
AP24-N	3,373	62.8	−3.41	1.12	0.11	7
AP8-β	7,800	401.7	−23.93	1.80	0.59	6
AP8-α	8,700	399.2	−28.90	2.53	0.93	6

Summary gives data obtained in present study and from previous investigations.

To relate these findings for synthetic peptides to effects of native proteins, we also calculated interpolated values for (v/v₀)_0.1μM from the data obtained in the studies of Fu et al. (6) and Kim et al. (7). It should be noted that step velocity data in these studies were obtained at slightly different supersaturation and ionic strength conditions (6, 7). The most potent additive was AP8-α, an 8-kDa, highly acidic protein extracted from the mineralized tissue of abalone (10). At the 0.1-μM level of this protein the acceleration was 150%, a value 10-fold more than for Asp-6, the most potent of the very small peptides (Table 1).

Discussion

Table 1 shows that the rate-enhancing ability of these biomolecules is related to their overall acidity. This finding suggests two possible correlations, one with biomolecule charge and the other with hydrophilicity, the latter being a measure of the extent of solvation around biomolecules in an aqueous solvent. First, we consider the correlation with molecular charge. Fig. 3 shows that ln[(v/v₀)_0.1μM] scales approximately linearly with net negative charge for the entire series of compounds from a single Asp residue to synthetic polypeptides to full native proteins. The data for AP7-N, the only hydrophobic peptide overall, deviates from the latter trend, suggesting that ln[(v/v₀)_0.1μM] may also correlate with increasing hydrophobicity.

Fig. 3.

Logarithm of step velocity enhancement, (v/v₀)_0.1μM, versus calculated peptide net charge. The normalized step enhancements are reported as the step velocity interpolated at a 0.1-μM biomolecule concentration relative to the control (Table 1). Comparisons to Fig. 2 show the growth rate enhancements are directly proportional to the initial slope of the experimental data (see Materials and Methods). Values of net charge represent the ionization state of the peptides at the experimental pH of 8.5 and were calculated based on the pK_a of functional groups on the individual amino acids of the peptides (Table 1). Legend is color-coded according to synthetic 27-mer peptides, biomineral-associated whole proteins, and synthetic 30-mers of the active portion of the nacre-specific protein sequences.

To understand the potential mechanism of enhancement, the factors that control step speed must be considered. At the driving force used in this study, the dependence of step speed on calcium activity exhibits linear kinetics (20). That is,

where a and a_e are the actual and equilibrium solute activities and β is a constant known as the kinetic coefficient (20). Data on v(σ) in the presence of a fixed concentration (10⁻⁴ M) of Asp-1 impurity shows that v(σ) is near-linear to linear (21), especially in the range of 0.20 < σ < 1.07 (r² > 0.99) [see supporting information (SI) Fig. 5]. For the other impurities used in this study we assume that v(σ) remains linear, especially considering the very low concentrations (≈10⁻⁴ M) that apply to the regime of growth acceleration (Fig. 2). Thus, the assumption that v(σ) is linear will produce negligible error in the evaluation of v or β from Eq. 1.

In the present case, because the ratio of calcium to carbonate activities is fixed at unity in the bulk solution, a and a_e can be replaced by the activities of Ca²⁺. For v/v₀ to exceed unity Eq. 1 shows that either a_e must decrease or β must increase. But at a supersaturation of 0.92, the value of a_e is already 2.5 times less than a. Even if a_e were decreased to zero, the increase in a − a_e could not exceed ≈1.3 times and thus would be insufficient to account for the observed enhancement of >1.6 times measured here for (Asp₃–Gly)₆–Asp₃ and as much as five times reported for AP8-α protein (6). Moreover, because acidic peptides form complexes with Ca²⁺, the value of a_e is more likely to shift to even higher values. From this analysis we conclude that the velocity enhancement is, in fact, because of an increase in β and the enhancement v/v₀, is approximately given by β/β₀.

The magnitude of β is controlled by two primary factors, the first is the density of kink sites along the step n_k and the second is the net probability of attachment to a site, which we write as exp(−E_k/kT), where E_k is an effective barrier to attachment at a kink. In other words, β ∼ n_k exp(−E_k/kT). In the regime of linear kinetics, the kink density is constant (22), although changes in kink density from step impurities cannot be excluded. For the purposes of this analysis, kink density is assumed to be independent of σ. This assumption is equivalent to that already made above on the linearity of v(σ) in systems with impurities because v(σ) would become nonlinear if n_k depended on σ (Eq. 1). Moreover, for calcite in this regime the step is likely to be rough, i.e., the kink site density has its maximum value and impurities can only decrease this value by blocking active kink sites (23). This kink blocking effect is suggested at higher concentrations by the drop in v/v₀ (Fig. 2). Thus, in the regime of acceleration, ln(v/v₀) ∝ ΔE, where ΔE is the difference between E_k in the pure and peptide-bearing systems. This analysis, although crude, leads us to hypothesize that enhancement is a catalytic process in which the barrier to reaction is reduced in the presence of the peptides.

Because rates of step propagation depend on the uptake of calcium and carbonate ionic units into kink sites at step edges, the relationship in Fig. 3 suggests that the physical basis for the measured rate enhancements must be related to electrostatic interactions between impurities and growth sites. Asp-rich impurities are well known to have preferential interactions with step edges rather than terraces (8, 17), which hints at possible scenarios for increasing step velocity. First, the presence of the charged peptides at the steps could increase the local potential gradient and, therefore, the electrostatic force for the cations to either incorporate into existing kink sites or attach to the step edges and create new kink sites, thereby increasing step speeds. Whatever the specific processes involved in growth enhancement are, they must involve electrostatic interactions (Fig. 3) that constitute a primary driver in these impurity-induced growth rate enhancements.

The above picture ignores the fact that CaCO₃ grows by addition of two distinct species. This consideration raises an alternative explanation based on the conventional assumption that the desolvation of Ca²⁺ is the rate-limiting step during CaCO₃ formation (24). Because negatively charged peptides can complex with cations at the steps (14), the local Ca²⁺ and CO₃²⁻ stoichiometry may be shifted toward Ca²⁺-rich, even under constant saturation, to produce an increase in step velocity (25).

Although the relationship in Fig. 3 indicates that charged impurities may interfere with the electrostatic interactions between growth units and the kink site environment, a closer inspection suggests that differences in biomolecule charge alone cannot fully explain the results. For example, whereas Asp–Leu and Asp–His have the same net charge, the latter induces a greater step enhancement (Table 1). Likewise, an explanation based solely on differences of molecular weight (Table 1) is likewise unsatisfactory; Asp-1, single amino acid, causes greater enhancement than larger dipeptides such as Asp–Leu, and Asp–Gly (also compare, for ex., a dimer, Asp-2, that produces a greater enhancement effect than AP7-N, a 30-mer with a much larger molecular weight). Moreover, these three molecules have the same net charge. The spread in the data in Fig. 3 make it clear that, in addition to molecular charge and size, other factors must play a role in growth enhancement.

Our previous studies of Asp–calcite interactions show that at the low concentrations causing growth enhancement, Asp-1, as well as polyaspartates, have weak electrostatic interactions with the surface that affect the solvation of the step edge environment. The result is a biomolecule-specific effect on the kinetics of step propagation (8), which suggests the roles of desolvation barriers to incorporating growth units within kink sites may be important. Thus, by modulating kinetic barriers to dehydration of solvated growth units, changes in growth rates may arise. To test this idea, for each peptide and protein we calculated the molecular hydrophilicity by using the residues index from Hopp and Woods (26) as described in Materials and Methods. Hydrophilicity is a property that describes the effects of biomolecules on water-mediated interactions. During crystallization of the solutes, part of the entropic and enthalpic components of the free energy change reflects the water structuring at molecular interfaces and beyond, as water accommodates the solutes, the forming crystal, and the biomolecules (27). These hydrophilic interactions thus depend on how water molecules interact with the amino acid moieties of the peptides via ionic, hydrogen, polar, and van der Waals bonds, but also depend on “hydrophobic surface patches” on parts of the peptides that present unfavorable water interactions, such as from neutral or nonpolar residues, and thus restrict water–water binding and structuring (28). Therefore, in essence, we are using hydrophilicity as a proxy for estimating the extent of water restructuring around each type of molecule in the system and, also, at the step edges.

Fig. 4 shows that increasing positive values of molecular hydrophilicity correlate well with step propagation rates. A possible explanation for this effect comes from recent experimental studies of step dynamics during growth from solutions (29). There it was concluded that the rate-limiting step during crystallization was desolvation of the solute, where the barrier to desolvation was that associated with bulk diffusion limited by the restructuring of water. Specifically, the diffusion barrier to attachment was attributed to “ … repulsive potentials due to water structuring at hydrophobic and hydrophilic surface patches …” (29). The same diffusion-limited kinetics and potential energy barriers were also shown to control incorporation of small inorganic molecules into solids such as CaCO₃ crystals (29). Therefore, from measurements of activation energies of step incorporation of small inorganic solutes, the crossing of an energy barrier is required for the solute-to-solid phase transformation during crystallization, including that of calcite (29), for which the barrier was experimentally determined to be ≈33 kJ/mol (0.34 eV per molecule) (19).

Fig. 4.

Logarithm of normalized step enhancement velocity, (v/v₀)_0.1μM, versus calculated peptide hydrophilicity. The normalized step enhancements are reported as the step velocity interpolated at a 0.1-μM biomolecule concentration relative to the control. Peptide hydrophilicity is calculated based on the Hopp and Woods hydrophilicity scale (26) of the constituents amino acids (see Materials and Methods). Legend is color-coded according to the same scheme used in Fig. 3. Step accelerations and hydrophilicity values are reported in Table 1.

Moreover, these water-dependent interactions can have significant range (30, 31), consistent with our finding that very limited amounts of peptides produce significant velocity enhancements (Fig. 2). Based on an assumption that the concentration near the surface is the same as the average in the bulk solution, the added turnover rate of Ca²⁺ and CO₃²⁻ at the accelerated steps must be on the order of ≈10³ to 10⁴ per peptide per second. However, to effectively describe the magnitude of the impact of biomolecules on the solutes, it is more accurate to consider the distribution of biomolecules between the surface and the bulk solution because growth is controlled locally at the crystal surface (32). A Langmuir-type adsorption isotherm best predicts this distribution between the bulk solution and the surface fractional coverage of impurities (8, 33, 34). Nevertheless, the relatively low amounts of biomolecules needed to affect growth rates likely reflect the much larger sizes of the biomolecules compared with the solute ions and thus the biomolecules greater ability to perturb the solvation environment of the solute ions compared with smaller impurities (35). These enhancements are thus expected to scale with the concentration C_i of biomolecules [i.e., β(C_i)], in the growth solution up until the point of onset of inhibition as is indeed experimentally observed (Fig. 2).

The above relations suggest that the measured rate enhancements, as described by an increase in β, stem in part from a reduction in the heights of diffusive barriers, E_k, through local perturbation in the structuring of water. The amount, ΔE, by which biomolecules need to reduce E_k to produce the measured enhancements in v are then estimated from ΔE = kTln(v/v₀)_0.1μM. For 1.005 < (v/v₀)_0.1μM < 2.53 (Table 1), ΔE ranges from 0.00013 to 0.024 eV compared with E_k = 0.34 eV for the formation of calcite (19). Thus reductions in the total energy barrier <8% are required to account for the measured enhancements. The exact relationship between the local water restructuring and how biomolecules alter the energy pathway during crystallization remains unknown. However, the observed correlation in Figs. 3 and 4 suggests that this variation in diffusive barrier height, and, therefore, of the reaction rates for the formation of calcite, may be determined a priori, and in large part, by knowledge of the peptides' constituent amino acids from which the hydrophilicity and net charge values were calculated. In the case of growth enhancement, we propose that the peptides ability to perturb local water structuring is such that the barrier for attachment of the growth unit to the solid during phase transition is reduced. In other words, we propose that the release of water molecules around the solutes during incorporation in the solid is facilitated by the influence of biomolecules on water structure. Recent molecular dynamics simulations suggest that an increase in the structuring of water at the molecular interface can produce a reduction in the rate of desolvation (36, 37). Therefore, because cations are much more firmly associated with the surrounding water molecules and their desolvation is rate limiting (24), the perturbation of the structured water shell should involve that of Ca²⁺ ions in particular. This physical model is consistent with recent findings that ions on highly charged surfaces are partially dehydrated (38). In addition, accumulation of Na⁺ counterions on Apoferritin proteins was shown to influence the magnitude of the intermolecular hydration forces (39). Thus, simple inorganic cations also have the ability to disrupt the surrounding water structure, and the degree of this entropic effect has been shown to promote development of negative charges on silica surfaces (40). Moreover, it was recently found that the step kinetics on calcium oxalate monohydrate could be enhanced by the addition of small amounts (10–25 nM) of Tamm-Horsfall protein (M. Weaver, S. R. Qiu, J.J.D.Y., J.R.H., and W. H. Casey, unpublished data). Increasing the protein concentration to 75 nM, i.e., toward normal levels in urine, causes the step speed to drop, a pattern of modulation similar to that in the present studies of calcite. These observations suggest the possibility that the hydrophilicity of organic or inorganic solutes can regulate the degree of growth enhancement, in a variety of mineral systems.

Although hydrophilicity is strongly related to the ionic charge via hydrogen bond interactions with polar water molecules during solvation, it does not entirely depend on charge. Uncharged moieties within residues also contribute to reductions or increases in the overall hydrophilicity of a biomolecule. The degree to which they do so depends on a number of factors, such as their polarity and/or the hydrophobic character of their alkyl or phenyl groups. Thus, in general, the extent of velocity enhancement is the result of some combination of molecular factors, which include molecular size, charge, hydrophilicity, and secondary structure, which all can impact the trends seen in Figs. 3 and 4. However, the correlation with molecular size is clearly not as strong, and determining the degree to which there is dependence on secondary structure will require further experiments that specifically control for this molecular parameter. Nevertheless, hydrophilicity and its implied impact on the hydration layer at the growth interface appears to be an important determinant because it is well correlated with the observed rate increases.

These findings suggest a key role for local biomolecule chemistry as a microenvironmental control on rates of mineral formation. For example, because impurity contents in calcites are sensitive to growth rate (19, 41), native biomolecules may indirectly impact the compositional signatures of some biogenic calcites that are currently used to interpret formation of paleoenvironments. These findings also provide an avenue for designing the structural and chemical features of synthetic molecules that will allow modulation of growth rates with a degree of control that is currently expressed only in biogenic minerals.

Materials and Methods

Crystal Substrate and Solution Preparation.

Natural calcite crystals (Chihuahua, Mexico) were cleaved to produce ≈0.2 × 0.2 × 0.05 cm³ fresh (104) faces as substrates for calcite growth. Calcite samples were used immediately upon cleaving after a brief cleaning with a nitrogen jet to remove any debris. Growth solutions were prepared immediately before use from reagent grade calcium chloride (CaCl₂·H₂O), sodium bicarbonate (NaHCO₃), and sodium chloride (NaCl) dissolved in deionized (≥18 MΩ) and filtered water (0.2 μm). The chemistry of the baseline growth solution was carefully controlled with ionic strength fixed at 0.11 M, pH 8.50, and a supersaturation of σ = 0.92. Supersaturation is defined as σ = ln(a_Ca++a_CO3=/K_sp), where a denotes the species activity calculated from Geochemist's Workbench (Urbana, IL) for specified parameters (pH, ionic strength, and temperature), and K_sp denotes the equilibrium solubility constant at 25°C. For all experiments, the a_Ca++/a_CO3= ≈ 1.0. The dipeptides Asp–His, Asp–Leu, Asp–Glu, and Asp–Gly were obtained from Sigma-Aldrich (St Louis, MO). The (Asp₃–Gly)₆–Asp₃ and(Asp₃–Ser)₆–Asp₃ 27-mers were synthesized as described (42, 43).

In Situ AFM.

During calcite growth, we imaged the steady-state morphology of atomic steps at constant supersaturation (σ = 0.92) for peptide concentrations from 0 to 0.10 M. Using established methods, calcite was overgrown onto the surface of a calcite seed crystal in an AFM flow-through cell (50 μl) that continuously supplied the input solution at a rate of 30 ml/h via a syringe pump. These flow conditions insured that calcite growth was reaction and not transport limited as demonstrated in previous studies (44). Measurements of step speeds were conducted at room temperature with a Digital Instruments Nanoscope III (Veeco, Santa Barbara, CA) operating in Contact Mode. The AFM images were collected by using scan rates of 5–20 Hz and a resolution of 512 × 512, while minimizing tip-surface force interactions during the flow-through of the growth solutions to minimize artifactual effects on step edge morphology and measured velocities (20). Negative (acute) step velocities mirrored those obtained for positive (obtuse) steps, although, as in previous reports, the speeds remained lower and showed greater variability for the acute steps (19, 41). The effects of peptides on calcite growth were measured in situ by using AFM measurements of the surface for a series of peptides bearing solutions at a fixed supersaturation σ of 0.92. All peptides were aspartate-based. The 27-mers were included to gain data concerning effects of molecular size, whereas the dipeptides were included to gain data concerning molecular charge for “hydrophobic” (Asp–Gly and Asp–Leu), “basic” (Asp–His), and “acidic” (Asp–Glu) dipeptides. Step velocity measurements were conducted for both the positive and negative step edge directions on growth hillocks that had equilibrated with each type of growth solution (20).

Calculation of Step Velocity Enhancement, Peptide Net Charge, and Hydrophilicity.

The increased growth rates were determined by measuring the step propagation velocity in control solutions (σ = 0.92) and solutions containing low levels of the peptides at the same driving force (σ = 0.92). Step velocity enhancements or accelerations, (v/v₀)_0.1μM, were calculated from step velocity measurements and determined for a common arbitrary concentration of 0.1 μM for all peptides. This normalization of the data assumes the step velocity versus peptide concentration curves are near-linear at lower peptide concentrations, thus the first two data points on each curve were used to approximate the corresponding initial slope as Δv_i/ΔC_i (nm/s per μM, where v is the step velocity, i represents a given peptide, and C_i is the peptide concentration) that was used to interpolate the value of v at 0.1 μM. The step velocity enhancements thus calculated provide a measure of the acceleration of step speed caused by the presence of the peptides in solution and allows comparison across peptides and proteins on a common concentration basis. Because at low peptide concentrations there is a weak dependence of step velocity enhancement on peptide concentration, any nonlinearity in this region would not cause significant errors in the determination of step acceleration, (v/v₀)_0.1μM. Each of the step accelerations reported represent the average over three to six measurements of step velocities from a given growth spiral. The standard deviations of the measurements are equal to or less than the size of the data points.

Net peptide charge was calculated based on knowledge of the peptide sequence, pK_a values (45) of (i) the individual residues and (ii) the peptide end groups N and C terminus. Using the well known Henderson–Hasselbach equation (46) at the fixed experimental pH 8.5, the extent of protonation, i.e., the charge associated with individual amino acids, could be calculated and their contribution to the net peptide charge summed over the entire peptide sequence. By this approach the net charge is an approximation that does not take into account, for example, the peptide 3D structure, and the effects of the local environment on pK_a. However, this approximation enables simple determination of the relative magnitude of the peptide's net charge.

The peptide hydrophilicity was calculated based on an empirical hydrophilicity scale for individual amino acids modified by Hopp and Woods (26), and derived by Levitt (47) and Nozaki and Tanford (48). This hydrophilicity scale was derived from amino acid solubility data in aqueous, mixed, and nonaqueous solvents to determine the free energy change (using the Van't Hoff equation), ΔG (kJ/mol), during the amino acids transfer from an aqueous to a nonaqueous phase (47, 48). The hydrophilicity for a given peptide was calculated as the sum of each individual amino acid hydrophilicity value making up the amino acid sequence of the peptide. Hydrophilicity values can be both positive for hydrophilic amino acids and negative for hydrophobic amino acids. Calculating hydrophilicity in this way is arbitrary and does not represent an absolute value of the peptides' hydrophilicity but, rather, is designed to predict the relative hydrophilicity of the peptides. It does not take into account, among other things, the secondary structure of the peptide that can reduce solvent exposure of hydrophobic amino acids. However, this type of approach, which simply uses a linear combination of the amino acids' hydrophilicity, was successfully used to predict antigenic sites within sequences of a broad range of antibodies and is thus mostly functional and empirical (26). Other hydrophobicity scales were used for comparison (49–53) and yielded qualitatively similar results, although not as linear as with the Hopp and Woods scaling (26).

Abbreviation

AFM

atomic force microscopy.