Interfacial energetics of globular–blood protein adsorption to a hydrophobic interface from aqueous-buffer solution

Abstract

Adsorption isotherms of nine globular proteins with molecular weight (MW) spanning 10–1000 kDa confirm that interfacial energetics of protein adsorption to a hydrophobic solid/aqueous-buffer (solid–liquid, SL) interface are not fundamentally different than adsorption to the water–air (liquid–vapour, LV) interface. Adsorption dynamics dampen to a steady-state (equilibrium) within a 1 h observation time and protein adsorption appears to be reversible, following expectations of Gibbs' adsorption isotherm. Adsorption isotherms constructed from concentration-dependent advancing contact angles θ_a of buffered-protein solutions on methyl-terminated, self-assembled monolayer surfaces show that maximum advancing spreading pressure, ${\Pi }_{\text{a}}^{\text{max}}$, falls within a relatively narrow $10<{\Pi }_{\text{a}}^{\text{max}}<20\text{mN}{\text{m}}^{-1}$ band characteristic of all proteins studied, mirroring results obtained at the LV surface. Furthermore, Π_a isotherms exhibited a ‘Traube-rule-like’ progression in MW similar to the ordering observed at the LV surface wherein molar concentrations required to reach a specified spreading pressure Π_a decreased with increasing MW. Finally, neither Gibbs' surface excess quantities [Γ_sl−Γ_sv] nor Γ_lv varied significantly with protein MW. The ratio {[Γ_sl−Γ_sv]/Γ_lv}∼1, implying both that Γ_sv∼0 and chemical activity of protein at SL and LV surfaces was identical. These results are collectively interpreted to mean that water controls protein adsorption to hydrophobic surfaces and that the mechanism of protein adsorption can be understood from this perspective for a diverse set of proteins with very different composition.

1. Introduction

Protein adsorption is one of the most fundamental, unsolved problems in biomaterials surface science (Vogler 1993, 1998). Practical importance of the problem is related to the fact that protein adsorption is among the first steps in the acute biological response to materials that dictates biocompatibility, and hence utility in medical-device applications. As a consequence of these scientific and pragmatic factors, the protein-adsorption problem has attracted considerable research attention from diverse fields of inquiry ranging from biomaterials to physics. An undercurrent flowing through much of this research seems to be that the amount of protein adsorbed to a surface is primarily controlled by short-range, pair-wise interactions between protein molecules and adsorbent surface (consider, for example, the RSA model of protein adsorption). Adsorbed protein is frequently assumed to be irreversibly surface bound in a monolayer arrangement (e.g. Dee et al. 2002 and citations therein). However, presumption of irreversible adsorption remains controversial in the literature (Vogler 1998) and multi-layering of protein has been experimentally demonstrated in a number of cases (Graham et al. 1979; Brynda et al. 1984; Jeon et al. 1992; Claesson et al. 1995; Vogler 1998; Chen et al. 2003; Zhou et al. 2004). Furthermore, the apparent specificity/selectivity of adsorption from multi-component solutions is frequently attributed to variations in protein molecular structure that give rise to differential interactions with a particular adsorbent.

It is our contention that this view of protein adsorption to surfaces does not properly account for the role of water in the process and, in so doing, fails to discern unifying trends in protein adsorption (Krishnan et al. 2003, 2004a,b, 2005b,c). For example, literature illustrations depict protein and adsorbent surfaces without juxtaposing hydration layers, one layer for protein and one for surface, and do not contemplate how these layers are displaced or coalesced as protein and surface come into close contact. Many modern computational models probing surface–protein interactions regard water as a complicating feature that can be ignored for the sake of reasonable computational time (see Vasquez et al. 1994; Cramer & Truhlar 1999; Head-Gordon & Hura 2002 and citations therein). When water is included in such models, it is usually only those molecules directly adjacent to the protein that comprise the ‘bound-water layer’, classically measured by δ in grams-water-per-gram-protein (Durchschlag et al. 2001; Garcia de la Torre 2001; Harding 2001), where δ∼0.35 g g⁻¹ is found to be a representative average value (Durchschlag & Zipper 2001). This protein-bound water layer falls well short of the volume which must be displaced when a protein molecule approaches a hydrated adsorbent surface. That is to say, since two objects cannot occupy the same space at the same time, a volume of interfacial water at least equal to the partial specific volume v^o (0.70≤v^o≤0.75 cm³ g⁻¹ protein) of the adsorbing protein must move (Chalikian & Breslauer 1996). If protein adsorbs in multi-layers, then clearly much more water must be displaced. Some or all of this interfacial water is bound to the adsorbent surface to an extent that varies with surface energy (Vogler 1998, 2001). Consequently, protein adsorption is found to scale with water wettability (Vogler 1992a,b; Vogler et al. 1993), underscoring need to incorporate surface hydration explicitly into protein–adsorption models. Indeed, accounting for water in protein adsorption has become a significant preoccupation of quartz crystal microbalance (QCM) practitioners because QCM not only measures adsorbed protein mass but also ‘trapped’ (Hook & Kasemo 2001) or ‘intra-layer’ (Hook et al. 1998) or ‘hydrodynamically coupled’ (Hook et al. 2002) water.

We have made use of a simplified ‘core-shell’ model of globular proteins in which spheroidal molecules are represented as a packed core surrounded by a hydration shell. The core has a radius r_v that scales with molecular weight (MW^1/3) and the hydration shell has a thickness such that the ensemble radius R=χr_v equals the hydrodynamic radius (Krishnan et al. 2003), where χ is taken to be a generic factor for all proteins. Calibration to human serum albumin (FV HSA) dimensions reveals that R=1.3r_v (30% larger than r_v) and contains about 0.9 g water g⁻¹ protein. Hence, the hydration layer accommodated by this model is ca 3× greater than δ. Calibrated to neutron-reflectivity (NR) of albumin adsorption to surfaces, this model suggests that protein saturates the hydrophobic surface region by packing to nearly face-centred-cubic (FCC) concentrations wherein hydration shells touch but do not overlap (Krishnan et al. 2003). We propose that osmotic repulsion among hydrated protein molecules limits interphase capacity. Stated another way, protein adsorption is limited by the extent to which the hydrophobic interface can be dehydrated through displacement of interfacial water by adsorbing protein. Accordingly, protein adsorption is viewed as being more about solvent than protein itself; a perspective in sharp contrast to the prevailing paradigm mentioned above.

This water-oriented perspective on protein adsorption presents a considerable simplification of the protein–adsorption process and, as a result, a tractable quasi-thermodynamic theory can be sketched out for a phenomenon that would otherwise be overwhelmingly complex for more than just a few proteins in solution. We find that this theory naturally explains the experimentally observed ‘Traube-rule progression’ in which molar concentrations required to fill the liquid–vapour (LV) surface follow a homology in protein size, consistent with packing hydrated spheroidal molecules within this space (Krishnan et al. 2003). A relatively straightforward set of ‘mixing rules’ follow directly, stipulating both concentration and weight-fraction distribution of proteins adsorbed to the LV surface from multi-component aqueous solutions such as blood plasma or serum (Krishnan et al. 2004a). These mixing rules rationalize the long-known-but-otherwise-unexplained observations that: (i) LV interfacial tension γ_lv of blood plasma and serum is nearly identical, in spite of the fact that serum is substantially depleted of coagulation proteins such as fibrinogen; and (ii) γ_lv of plasma and serum derived from human, bovine, ovine, and equine blood is practically identical, even though there are substantial differences in plasma proteome among these species (Krishnan et al. 2005).

This paper discloses results of an investigation of globular–blood protein adsorption to a well-defined, hydrophobic solid/aqueous-buffer (solid–liquid, SL) interface. Methyl-terminated, self-assembled thiol monolayers (SAMs) on gold-coated semi-conductor-grade silicon wafers exhibiting water contact angles θ_a∼110° are used as test substrata. Time-and-concentration-dependent contact angles measure adsorption energetics of (globular) proteins spanning three decades in MW in a manner parallel to the above-cited studies of protein adsorption to the LV surface. We find that the basic pattern observed at the LV surface is repeated at the hydrophobic SL surface, supporting our contention that water is the significant controller of protein adsorption to surfaces.

2. Material and methods

2.1 Purified proteins and synthetic surfactants

Table 2 compiles pertinent details on proteins and surfactants used in this work. Protein purity was certified by the vendor to be no less than the respective values listed in column 4 of table 2, as ascertained by electrophoresis (SDS-PAGE or IEP). Mass, concentration, and molecular weights supplied with purified proteins were accepted without further confirmation. Issues associated with protein purity, especially contamination with surfactants, and the potential effect on measured interfacial tensions have been discussed elsewhere (Krishnan et al. 2004b). The single value given in table 2 (column 5) for physiological concentration of human proteins applied in this work was middle of the range listed by Putnam (Putnam 1975). Serial dilutions of protein stock solutions (usually 10 mg ml⁻¹) were performed in 96-well microtiter plates by (typically) 50 : 50 dilution in phosphate buffered saline solution (PBS; 0.14 M NaCl, 0.003 M KCl) prepared from powder (Sigma Aldrich) in distilled–deionized (18.2 MΩ cm⁻¹) water using procedures detailed in Krishnan et al. (2004b). Between 24 and 30 dilutions were prepared in this manner, covering a dynamic range between 10⁻¹⁰ and 1% (w/v), taking care to mix each dilution by repeated pipette aspiration and avoiding foaming of concentrated solutions.

Table 2.

Purified proteins and surfactants.

name of protein/surfactant (acronym)		molecular weight (kDa)	as-received form (mg ml−1)	purity (electrophoresis) or activity	physiologic concentration, mg/100 ml (nominal value)	vendor
ubiquitin (Ub)	prep 1	10.7	powder	98%	10–20 (15)	Sigma Aldrich
prep 2	95%	EMD Biosciences
thrombin (FIIa)	35.6	powder	1411 NIH units mg−1	n/a	Sigma Aldrich
human serum albumin fraction V (FV HSA)	prep 1	66.3	powder	98%	3500–5500 (4500)	MP Biomedicals
prep 2	powder	98%
prothrombin (FII)	72	powder	7.5 units mg−1 protein	5–10 (7.5)	Sigma Aldrich
factor XII (FXII)	prep 1	78	solution (2.1)	95%	(4)	Hematologic Technologies
prep 2	solution (5.5)
human IgG (IgG)	160	powder	97%	800–1800 (1300)	Sigma Aldrich
fibrinogen (Fb)	340	powder	70% clottable protein	200–450 (325)	Sigma Aldrich
complement component C1q (C1q)	400	solution (1.1)	single band by immunoelectrophoresis	10–25 (17.5)	Sigma Aldrich
α2-macroglobulin (α mac)	prep 1	725	powder	98%	150–350 (250)	Sigma Aldrich
prep 2	Sigma Aldrich
prep 3	MP Biomedicals
human IgM (IgM)	prep 1	1000	solution (0.8)	98%	60–250 (155)	Sigma Aldrich
prep 2	solution (5.1)	single band by immunoelectrophoresis	MP Biomedicals
sodium dodecyl sulphate (SDS)	0.28	powder	n/a	n/a	Sigma Aldrich
Tween-20 (TWN20)	1.23	neat	n/a	n/a	Sigma Aldrich

2.2 Surfaces

Methyl-terminated, self-assembled monolayer surfaces (SAMs) were prepared according to standard methods of surface engineering (Nuzzo & Allara 1983; Allara & Nuzzo 1985; Nuzzo et al. 1987, 1990; Porter et al. 1987). Briefly, silicon wafers were pre-cleaned in hot 1 : 4 H₂O₂ (30%)/H₂SO₄ followed by rinsing with distilled–deionized H₂O and absolute ethanol. Gold-coated wafers were prepared by vapour deposition of chromium and gold (99.99% purity) from resistively heated tungsten boats onto clean 3 in. diameter silicon wafers at about 1×10⁻⁸ torr base pressure in a cryogenically pumped deposition chamber. The sample was not allowed to rise above ca 40 °C during the evaporation. Film thicknesses, monitored with a quartz crystal oscillator, were typically 15 and 200 nm for chromium and gold, respectively. Chromium was deposited prior to gold to enhance adhesion to the substrate. After deposition, the chamber was backfilled with research-grade nitrogen. Gold-coated samples were removed and immersed in 1 mM solutions of 1-hexadecanethiol (CH₃(CH₂)₁₅SH) in ethanol, contained in glass jars at ambient temperature, for at least 3 days.

The alkanethiol (Aldrich Chemical Co., Milwaukee, WI) and ethanol (commercial reagent-grade) were used as-received, without further purification. Samples were stored in the thiol solution until use and were rinsed with ethanol just prior to an experiment.

2.3 Tensiometry and goniometry

Liquid–vapour interfacial tensions required by this work were measured by Pendant Drop Tensiometry (PDT) as described in Krishnan et al. (2003, 2004b). Tilting-Plate Goniometry (TPG) was performed using a commercial-automated goniometer (First Ten Angstroms, Inc., Portsmouth, VA). Advancing contact angles (θ_a) applied in this work have been verified to be in statistical agreement with those obtained by Wilhelmy Balance Tensiometry (WBT) and Captive-Drop Goniometry as detailed in Krishnan et al. (2005b). Receding angles (θ_r) were shown to be not as reliable as θ_a and, as a consequence, only θ_a was analysed in this work. The TPG employed a Tecan liquid-handling robot to aspirate 12 μl of solutions contained in a 96-well microtiter plate prepared by the serial-dilution protocol mentioned above. The robot was used to reproducibly transfer the tip with fluid contents into a humidified (99+% RH) analysis chamber and dispense 10 μl drops of protein solution onto the surface of test substrata (see below) held within the focal plane of a magnifying camera. These and all other aspects of TPG were performed under computer control. Proprietary algorithms supplied by the vendor were used to deduce contact angles from drop images captured at a programmed rate by a frame grabber. Typically, 600 images were captured at a rate of one image every 6 s following 20 s delay to permit vibrations of the expelled drop to dampen. Drop evaporation rates within the humidified chamber deduced from computed-drop volumes (based on image analysis) were observed to vary with solute concentration, generally ranging from approximately 25 nl min⁻¹ for pure water to 10 nl min⁻¹ for solute solutions greater than 0.1 w/v%. The impact of this evaporation rate over the 60 min time frame of the experiment was apparently negligible, as gauged from the behaviour of purified surfactants discussed in §4. Precision of θ_a was about 0.5° based on repeated measurement of the same drop. The analysis chamber was thermostated to a lower-limit of 25±1 °C by means of a computer-controlled resistive heater. Upper-temperature limit was not controlled but rather floated with laboratory temperature, which occasionally drifted as high as 29 °C during summer months. Thus, reported θ_a values were probably not more accurate than about 1° on an inter-sample basis considering the small, but measurable, variation of water interfacial tension with temperature. This range of accuracy was deemed adequate to the conclusions of this report which do not strongly depend on more highly accurate θ_a that is difficult to achieve on a routine basis. Instead, veracity of arguments raised herein depend more on a breadth of reliable measurements made across the general family of human proteins.

Test substrata were held on a rotating, tilting-plate platform driven by stepper motors under computer control. Substrata were allowed to come to equilibrium within the sample-chamber environment for no less than 30 min before contact angle measurements were initiated. The platform was programmed to tilt at 1° s⁻¹ from horizontal to 25° after the drop was deposited on the surface by the robot. The optimal (incipient rolling) tilt angle was found to be 25 and 15° for solutions of proteins and surfactants, respectively. The first 20 images monitored evolution of the advancing angle. At the end of the 1 h θ_a measurement period, the platform was programmed to return to horizontal and rotate 15° to the next analysis position along the periphery of the semi-conductor wafer. This process was repeated for all dilutions of the protein under study so that results reported for each protein were obtained on a single test surface, eliminating the possibility of substratum-to-substratum variation within reported results.

θ_a measurements by TPG employed in this work were verified against WBT and found to agree within a percentage difference of 2.5±1.9% for 50°<θ_a<120° (Krishnan et al. 2005b). It is worthwhile mentioning in this context that WBT itself is inappropriate for studies of protein adsorption at the SL interface (at least as applied herein) because: (i) the technique requires thin plates that are difficult to two-side coat with gold for thiol-SAM preparation, (ii) WBT, generally, requires high solution volumes (ca 10 ml) that greatly exceed availability of purified proteins, and (iii) the moving three-phase line deposits solute (protein or surfactant) at the SV interface making interpretation of the Gibbs' surface excess parameter [Γ_sl−Γ_sv] highly ambiguous (Vogler 1993). Overall, we have found the tilting-plate method applicable to measuring adsorption, at least for hydrophobic surfaces, and suitable for 1 h equilibration times if a humidified chamber is used to control evaporation (Vogler 1992a,b). However, it was observed that SAM surfaces were slightly unstable and subject to ‘hydration’ that led to a systematic decrease in water/PBS contact angles with time. These hydration dynamics were observed to be more pronounced on SAM surfaces that had been incubated for long periods (greater than 3 d) in the 100% RH atmosphere of the PDT analysis chamber (not shown). However, we do not believe this slight but apparently unavoidable attribute of SAMs on silicon wafers negatively affects the veracity of conclusions based on final, steady-state Π_a measurements made at ca 1 h analysis time.

2.4 Computation and data representation

Computational, statistical and theoretical methods used in this work have been discussed in detail elsewhere (Vogler 1992a,b, 1993; Krishnan et al. 2003, 2004b, 2005c). Briefly, time-dependent θ_a data corresponding to protein dilutions (see above) were recovered from TPG files and correlated with concentrations, leading to a matrix of results with row values representing concentration and time (in seconds) as column values. It was, generally, observed that θ_a isotherms were sigmoidal in shape when plotted on logarithmic-concentration axes (Vogler 1992a, 1993), with well-defined low-concentration asymptote ${\theta }_{\text{a}}^{\text{o}}$ and high-concentration asymptote ${\theta }_{\text{a}}^{\text{o}}$ (see figure 1). Successive non-linear least-squares fitting of a four-parameter logistic equation $[{\theta }_{\text{a}}=({\theta }_{\text{a}}^{\text{o}}-{\theta }_{\text{a}}^{\prime })/(1+{(\text{ln}{C}_{\text{B}}^{\Theta /2}/\text{ln}{C}_{\text{B}})}^M)+{\theta }_{\text{a}}^{\prime }]$ to contact angle isotherms data for each time within the observation interval quantified parameters ${\theta }_{\text{a}}^{\text{o}}$ and ${\theta }_{\text{a}}^{\prime }$ with a measure of statistical uncertainty. Fitting also recovered a parameter measuring concentration-at-half-maximal-change in θ_a, $\text{ln}{C}_{\text{B}}^{\Theta /2}$ (where Θ/2=1/2Θ^max and ${\Theta }^{\text{max}}\equiv {\theta }_{\text{a}}^{\text{o}}-{\theta }_{\text{a}}^{\prime }$), as well as a parameter M that measured steepness of the sigmoidal curve. This multi-parameter fitting to concentration-dependent θ_a data was a purely pragmatic strategy that permitted quantification of best-fit protein and surfactant characteristics but is not a theory-based analysis (Vogler 1992a,b, 1993; Krishnan et al. 2003, 2004b, 2005a). Three-dimensional representations of time-and-concentration-dependent θ_a data were created in Sigma Plot (v.8) from the data matrix discussed above and overlain onto fitted-mesh computed from least-squares fitting. Two-dimensional representations were created from the same data matrices at selected observation times. Measured θ_a were converted to advancing adhesion tension τ_a=γ_lv cos θ_a for general interpretation (Vogler 1993), where γ_lv was the interfacial tension of the contact-angle fluid. Adhesion tensions, ${\tau }_{\text{a}}^{\text{o}}={\gamma }_{\text{lv}}^{\text{o}}\text{cos}{\theta }_{\text{a}}^{\text{o}}$ (pure saline) and ${\tau }_{\text{a}}^{\prime }={\gamma }_{\text{lv}}^{\prime }\text{cos}{\theta }_{\text{a}}^{\prime }$ (at the minimum contact angle observed ${\theta }_{\text{a}}^{\prime }$) were computed with fitted parameters ${\gamma }_{\text{lv}}^{\text{o}}$ and ${\gamma }_{\text{lv}}^{\prime }$ reported in Krishnan et al. (2003, 2004b) for the proteins under investigation. Smoothed adhesion–tension isotherms (τ_a versus ln C_B) were computed from smoothed θ_a using smoothed γ_lv values computed from best-fit parameters reported in Krishnan et al. (2003, 2004b). Likewise, smoothed spreading pressure isotherms (Π_a versus ln C_B) were computed from smoothed τ_a curves, where ${\Pi }_{\text{a}}\equiv ({\tau }_{\text{a}}-{\tau }_{\text{a}}^{\text{o}})$ (table 1).

Figure 1.

Advancing contact angle isotherms in three-dimensional (θ_a as a function of analysis time (drop age) and logarithmic (natural) solution concentration C_B) and two-dimensional (θ_a as a function of logarithmic solution concentration C_B at selected times) formats comparing Tween-20 (panel a,TWEEN-20, table 3), prothrombin (panel b, FII, table 3) and immunoglobulin-M (panel c, IgM, preparation 2, table 3) adsorption to a methyl-terminated SAM surface. In each case, solute concentration C_B is expressed in pmol l⁻¹ (pM) on a natural logarithmic scale. Symbols in two-dimensional panels represent time slices through three-dimensional representations (filled circle=0.25 s, open circle=900 s, filled triangles=1800 s and open triangles=3594 s; annotations in panel a indicate maximum and half-maximum contact angle reduction, ${\theta }_{\text{a}}^{\text{max}}$ and $(1/2){\theta }_{\text{a}}^{\text{max}}$, respectively. Notice that adsorption kinetics dominated IgM adsorption whereas steady-state was achieved within about 1000 s for FII, and nearly no adsorption kinetics is detected for Tween-20. Note also decrease in ${\theta }_{\text{a}}^{\text{o}}$ with time, attributed to slow hydration of the SAM surface (panel b, arrow annotation; see §4 for more discussion).

Table 1.

Glossary of symbols.

CB	bulk solution concentration (mol/volume)
$C_{\text{B}}^{\text{max}}$	bulk solution concentration at limiting interfacial tension or contact angle (mol/volume)
CI	interphase concentration (mol/volume)
$C_{\text{I}}^{\text{max}}$	maximal interphase concentration (mol/volume)
$C_{\text{B}}^{\Theta /2}$	bulk solution concentration at half-maximal-change in contact angle (mol/volume)
Csl	independent measure of protein adsorption
χ	proportionality constant, χ≡R/rv
ϵ	packing efficiency
$\mathrm{\Delta }{G}_{\text{ads}}^{\text{o}}$	free energy of protein adsorption
γlv	liquid–vapour (LV) interfacial tension (mN m−1)
γsl	solid–liquid (SL) interfacial tension (mN m−1)
γsv	solid–vapour (SV) interfacial tension (mN m−1)
${\gamma }_{\text{lv}}^{\text{o}}$	low-concentration asymptote of a concentration-dependent γlv curve (mN m−1)
${{\gamma }^{\prime }}_{\text{lv}}$	high-concentration asymptote of a concentration-dependent γlv curve (mN m−1)
Γlv	apparent Gibbs' surface excess calculated at the liquid–vapour (LV) interface (moles/area)
[Γsl−Γsv]	apparent Gibbs' surface excess calculated at the solid–liquid (SL) interface (moles/area)
M	parameter fitted to concentration-dependent γlv or θa curve
μ	activity-corrected chemical potential
P	partition coefficient, P≡CI/CB
Πa	advancing spreading pressure (mN m−1)
${\Pi }_{\text{a}}^{\text{max}}$	maximum advancing spreading pressure (mN m−1)
rv	protein radius (cm)
R	effective radius (cm), R≡χrv
RT	product of universal gas constant and Kelvin temperature (ergs mol−1)
S	parameter computed from slope of θa isotherm $S=(-1/RT)(\mathrm{\Delta }{\theta }_{\text{a}}/\mathrm{\Delta }\text{ln}{C}_{\text{B}})$ (moles/area)
τa	advancing adhesion tension (mN m−1)
${\tau }_{\text{a}}^{\text{o}}$	low-concentration asymptote of a concentration-dependent τa curve (mN m−1); ${\tau }_{\text{a}}^{\text{o}}={\gamma }_{\text{lv}}^{\text{o}}\text{cos}{\theta }_{\text{a}}^{\text{o}}$
${\tau }_{\text{a}}^{\prime }$	high-concentration asymptote of a concentration-dependent τa curve (mN m−1); ${\tau }_{\text{a}}^{\prime }={{\gamma }^{\prime }}_{\text{lv}}\text{cos}{\theta }_{\text{a}}^{\text{o}}$
θa	advancing contact angle (deg.)
${\theta }_{\text{a}}^{\text{o}}$	low-concentration asymptote of a concentration-dependent θa curve (deg.)
${\theta }_{\text{a}}^{\prime }$	high-concentration asymptote of a concentration-dependent θa curve (deg.)
${\theta }_{\text{a}}^{*}$	advancing contact angle at half-maximal change in θa isotherm ${\theta }_{\text{a}}^{*}=({\theta }_{\text{a}}^{\text{o}}+{\theta }_{\text{a}}^{\text{o}})/2$ (deg.)
Ω	total interphase thickness (cm)

3. Theory

3.1 Adsorption isotherms

Adsorption of surface-active solutes (surfactants, where the term includes both synthetic detergents and proteins) can affect LV, SV or SL interfacial tensions, thus producing a change in measured contact angles θ as given by the Young equation $\tau \equiv {\gamma }_{\text{lv}}\text{cos}\theta ={\gamma }_{\text{sv}}-{\gamma }_{\text{sl}}$, where τ is adhesion tension and γ is the interfacial tension at the interface denoted by subscripts. Thus, contact angles can be used to monitor adsorption to solid surfaces (Vogler 1992a,b, 1993) and citations therein. Contact-angle isotherms are graphical constructions that monitor effects of adsorption by plotting advancing contact angles θ_a against ln C_B (see figure 1 for examples), where surfactant bulk-phase concentrations C_B range from 10⁻¹⁰ to 1% (w/v, see §2). Contact-angle isotherms were sequentially interpreted in terms of adhesion tension (τ_a versus ln C_B) and spreading pressure (Π_a versus ln C_B) isotherms, where τ_a≡γ_lv cos θ_a, ${\Pi }_{\text{a}}=({\tau }_{\text{a}}-{\tau }_{\text{a}}^{\text{o}})$, γ_lv is the LV interfacial tension of the fluid at C_B, and ${\tau }_{\text{a}}^{\text{o}}$ is the adhesion tension of pure buffer (${\gamma }_{\text{lv}}^{\text{o}}=71.97\text{mN}{\text{m}}^{-1}$ at 20 °C). We monitored time dependence of all three isotherm forms but herein interpret only final measurements that achieve or approach steady-state (equilibrium). Issues associated with adsorption reversibility are discussed in §4. Secure interpretation of measured θ_a in terms of τ_a depends on accurate knowledge of γ_lv at the bulk-phase surfactant concentration in equilibrium with SL and LV interfaces. Thus, solute depletion of the bulk phase by adsorption may require correction of as-prepared bulk-phase concentration C_B. However, agreement between (uncorrected) tensiometry and instrumental methods of measuring adsorption for surfactants (see table 4) suggests solute-depletion was not a serious issue for surfactant standards. Likewise, for the case of protein adsorption, it can be concluded from a simple calculation that solute depletion was not a serious problem requiring correction (Krishnan et al. 2005c).

Table 4.

Gibbs' surface excess.

		apparent surface excessa (pmol cm−2)			comparison to literature values
name of protein/surfactant (acronym)		[Γsl−Γsv]	Γlv	{[Γsl−Γsv]/Γlv}	Csl (pmol cm−2)	[Γsl−Γsv]/Csl	technique (citation)
ubiquitin (Ub)b	prep 1	224	179±27	1.3	—	—	—
	prep 2	193	1.1
thrombin (FIIa)		308±34	1.7±0.3
human serum albumin FV HAS	prep 1	145±18	0.8±0.2	2.4	60	XR (Sheller et al. 1998)
	prep 2	196±21	1.1±0.2	80
prothrombin (FII)		146±17	0.8±0.2	—	—	—
factor XIIb	prep 1	136	0.8	—	—	—
	prep 2	153	0.9
human IgG (IgG)		198±37	1.1±0.3	4.5	44	QCM (Zhou et al. 2004)
		2.9	66	SAW (Zhou et al. 2004)
complement component C1q (C1q)		117±28	0.7±0.2	—	—	—
α2-macroglobulinb (α mac)	prep 1	130	0.7	—	—	—
	prep 2
	prep 3
human IgM (IgM)	prep 1	222±42	1.2±0.3	—	—	—
	prep 2	101±27	0.6±0.2
sodium dodecyl sulphate (SDS)	276±14	342±10	1.2±0.2	280	0.98	SPR (Sigal et al. 1997)
Tween-20 (TWN20)	120±16	455±17	3.8±0.1	120	1.00	SPR (Sigal et al. 1997)

^aApparent [Γ_sl−Γ_sv] or Γ_lv is computed without activity correction (see §3).

^bParameters are graphical estimates of fitted parameters (see §4).

3.2 Gibbs' surface excess

Practical use of concentration-dependent contact angles as a measure of adsorption to the SL interface has been discussed at length elsewhere (Vogler 1992a,b, 1993) and citations therein. Briefly, for the purposes of this paper, the amount of solute adsorbed to SV and SL interfaces is measured by the Gibbs' surface excess quantities Γ_sv and Γ_sl, respectively, in units of moles/area (the subscript ‘a’ specifying advancing contact angles is not carried in Γ symbology for the sake of notational compactness). The difference [Γ_sl−Γ_sv] (but not separate excess parameters) can be computed from data comprising contact-angle isotherms using equation (3.1)

$$[{\Gamma }_{\text{sl}}-{\Gamma }_{\text{sv}}]=-\left\{\frac{[{\gamma }_{\text{lv}}\text{sin}{\theta }_{\text{a}}]}{RT}\left(\frac{\text{d}{\theta }_{\text{a}}}{\text{d}\text{ln}{C}_{\text{B}}}\right)+[{\Gamma }_{\text{lv}}\text{cos}{\theta }_{\text{a}}]\right\},$$ (3.1)

where dθ_a/d ln C_B is the slope of a contact-angle isotherm. ${\Gamma }_{\text{lv}}=-(1/RT)(\text{d}{\gamma }_{\text{lv}}/\text{d}\text{ln}{C}_{\text{B}})$ is the surface excess at the LV interface determined from separate measurement of concentration-dependent γ_lv of the solute under study (Krishnan et al. 2003). This form of the Gibbs' adsorption isotherm is appropriate for a single, isomerically pure non-ionizing solute or a polyelectrolyte in swamping salt concentrations of buffer salts (Rosen 1978; Krishnan et al. 2003). It is also important to stress that [Γ_sl−Γ_sv] and Γ_lv values obtained without correcting concentration C_B for solute activity are ‘apparent’ surface excess values that can substantially deviate from actual surface excess calculated from (dθ_a/dμ) and (dγ_lv/dμ), where μ is activity-corrected chemical potential (Frommer et al. 1968; Strey et al. 1999; Krishnan et al. 2003). However, previous work suggests that the discrepancy between apparent and actual Γ_lv is roughly constant for the proteins of this study and apparent surface excess was about 56× larger than actual surface excess (Krishnan et al. 2003). We thus assume that apparent [Γ_sl−Γ_sv] is also ca 56× larger than actual, activity-corrected surface excess because the ratio {[Γ_sl−Γ_sv]/Γ_lv}∼1 (see below). Comparison to instrumental measures of adsorption confirms this factor (see table 4).

For relatively hydrophobic surfaces exhibiting θ_a>60° and under experimental conditions that avoid inadvertent mechanical deposition of solute at the (SV) interface, as through drop movement on the surface or evaporation for examples, it has been shown that Γ_sv∼0 and [Γ_sl−Γ_sv]→Γ_sl (Vogler 1992a,b, 1993). Under the additional restrictions that: (i) solute activities at SL and LV interfaces are approximately equal and (ii) Γ_sl∼Γ_lv, it can be expected that {[Γ_sl−Γ_sv]/Γ_lv}∼1. Experimental results confirm that these stringent physical conditions prevail and it is, therefore, concluded that apparent [Γ_sl−Γ_sv]∼Γ_sl for proteins reported herein.

3.3 Theory of protein adsorption

Previous work disclosed a theory of protein adsorption to the LV surface (Krishnan et al. 2003) that appears to be directly applicable to adsorption to the SL surface with little-or-no modification because apparent [Γ_sl−Γ_sv] can be directly interpreted in terms of Γ_sl, as discussed above. This protein-adsorption theory was based on two related experimental observations and implications thereof; namely: (i) a surprisingly slight variation in concentration dependence of γ_lv among the same diverse globular proteins studied herein (tables 2 and 3) and (ii) a substantially constant, MW-independent value of the apparent Gibbs' surface excess Γ_lv=179±27 pmol cm⁻². This work demonstrates parallel behaviour at the hydrophobic SL surface with: (i) only modest variation in Π_a isotherms (tables 2 and 3) and (ii) a substantially constant value of the apparent Gibbs' surface excess [Γ_sl−Γ_sv]=175±33 pmol cm⁻² (table 4, figure 5). Protein adsorption theory asserts that these experimental observations are outcomes of a relatively constant partition coefficient P that entrains protein within a three-dimensional interphase separating surface regions from bulk phases (bulk-solution from bulk-vapour for the LV surface or bulk-solution from bulk-solid for the SL surface). This ‘Guggenheim’ interphase treatment, which is especially relevant to the adsorption of large solutes such as proteins, is to be contrasted with the more familiar two-dimensional interface ‘Langmuir’ paradigm in which the surface is construed to be a planar area with negligible thickness (see Vogler 1998 for more discussion). The three-dimensional interphase is proposed to thicken with increasing protein size because volume occupied by adsorbed-protein molecules scales in proportion to MW according to the well-known relationships among MW, solvent-exposed area, volume, and packing density (Richards 1977). As a consequence of these relationships, molar interphase concentrations C_I of larger proteins are lower than that of smaller proteins at constant P≡C_I/C_B. In fact, C_I varies inversely with MW and this leads directly to the Traube-rule-like ordering for proteins mentioned in §1. Protein size and repulsion between molecules within the three-dimensional interphase place an upper bound on maximal interphase concentration denoted $C_{\text{I}}^{\text{max}}$. Interphase saturation occurs at $C_{\text{I}}^{\text{max}}$ and corresponds to the bulk concentration $C_{\text{B}}^{\text{max}}$ at which the limiting adhesion tension ${\tau }_{\text{a}}^{\prime }$ is achieved (i.e. the concentration at maximum spreading pressure ${\Pi }_{\text{a}}^{\text{max}}=({\tau }_{\text{a}}^{\prime }-{\tau }_{\text{a}}^{\text{o}})$). Calibration of theory to neutron-reflectometry (Lu et al. 1999) and quasi-electric light scattering (Helfrich 1998; Helfrich & Jones 1999) of albumin adsorbed to the LV surface at $C_{\text{I}}^{\text{max}}$ suggests that hydrated spheroidal protein molecules achieve nearly FCC densities or, equivalently, that core proteins pack with an efficiency factor ϵ∼0.45. $C_{\text{B}}^{\text{max}}$ is an experimental parameter that can be estimated from concentration-dependent θ_a curves (see appendix A) and is related to $C_{\text{I}}^{\text{max}}$ through the partition coefficient $P\equiv C_{\text{I}}^{\text{max}}/C_{\text{B}}^{\text{max}}$. Equation (3.2) states relationships among packing densities, molecular dimensions (MW), and $C_{\text{B}}^{\text{max}}$ in the form of a logarithmic expression that is convenient to apply to concentration-dependent θ_a data:

$$\text{ln}{C}_{\text{B}}^{\text{max}}=\text{ln}(C_{\text{I}}^{\text{max}}/P)=\text{ln}(9.68\times {10}^{11})-\text{ln}\text{MW}+\text{ln}(ϵ/P)=-\text{ln}\text{MW}+[27.6+\text{ln}(ϵ/P)].$$ (3.2)

Table 3.

Steady-state protein adsorption parameters.

name of protein/surfactant (acronym)		${\theta }_{\text{a}}^{\prime }$ (deg.)	${\theta }_{\text{a}}^{\text{o}}$ (deg.)	$\text{ln}{C}_{\text{B}}^{\Theta /2}$ PPT (pM)	M (dimensionless)	${\tau }_{\text{a}}^{\text{o}}$ (mN m−1)	${\tau }_{\text{a}}^{\prime }$ (mN m−1)	${\Pi }_{\text{a}}^{\text{max}}$ (mN m−1)	$\text{ln}{C}_{\text{B}}^{\text{max}}$ (pM)
ubiquitin (Ub)a	prep 1	100.9±0.5	75	19 (17)	—	−14	7	21	19
	prep 2	102.2±0.9	75	19 (17)	—	−15	12	27	19
thrombin (FIIa)		99.8±0.5	84.6±0.9	17.5±0.2 (13.9±0.2)	−25.0±8.5	−12.3±0.6	4.5±0.7	16.7±0.9	15.1±0.2
human serum albumin FV HSA	prep 1	103.3±0.8	88.3±0.8	15.9±0.3 (11.7±0.3)	−14.1±5.7	−16.3±0.9	1.4±0.6	17.7±1.2	13.6±0.3
	prep 2	104.5±0.8	88.5±0.6	15.7±0.3 (11.5±0.3)	−11.6±3.0	−17.7±0.9	1.2±.5	18.9±1.1	13.7±0.3
prothrombin (FII)		100.6±0.5	86.5±0.9	15.1±0.4 (10.8±0.4)	−10.1±2.7	−12.9±0.6	2.6±0.7	15.6±0.9	13.2±0.4
factor XIIa	prep 1	102.9±0.5	94.8±1.0	15.6±0.5 (11.3±0.5)	−17.9±1.2	−15.6	−3.1	12.5	12.7±0.5
	prep 2	102.0±0.4	88.2±0.8	15.7±0.4 (11.3±0.4)	−10.9±3.3	−14.6	1.2	15.8	13.6±0.4
human IgG (IgG)		103.7±0.7	94.9±1.4	15.1±0.9 (10.1±0.9)	−6.9±4.7	−16.8±0.9	−4.4±1.3	12.4±1.5	13.3±0.9
complement component C1q (C1q)		102.6±0.4	95.3±0.7	15.6±0.4 (9.6±0.4)	−12.1±5.6	−15.6±0.5	−5.0±0.7	10.6±0.8	11.4±0.4
α2-macroglobulina (α mac)	prep 1	101.9±0.5	86	19 (13)	—	−15	4	19	17
	prep 2	100.2±0.9
	prep 3	103.2±0.5
human IgM (IgM)	prep 1	102.7±0.6	91.3±1.6	15.5±0.5 (8.7±0.5)	−7.4±2.9	−15.7±0.7	−1.1±1.4	14.6±1.6	11.3±0.5
	prep 2	102.4±0.6	87.8±2.0	15.9±0.6 (9.2±0.7)	−4.9±1.6	−15.4±0.7	1.9±1.7	17.3±1.9	12.6±0.8
sodium dodecyl sulphate (SDS)		100.1±1.9	56.0±2.3	17.7±0.4 (18.9±0.4)	−17.3±4.6	−12.5±2.3	18.7±1.1	31.2±2.6	21.4±0.4
Tween-20 (TWN20)		97.1±0.6	65.1±0.7	16.4±0.3 (16.2±0.1)	−23.4±3.3	−8.9±0.8	14.6±0.5	23.5±0.6	17.8±0.1

^aParameters are graphical estimates of fitted parameters (see §4).

Figure 5.

Apparent Gibbs' surface excess scaled by protein MW at the solid–liquid (SL) ([Γ_sl−Γ_sv], panel a) and the liquid–vapour (LV) interfaces (Γ_lv, panel b) for multiple protein preparations (open circle=preparation 1, filled circle=preparation 2, filled triangle=preparation 3; see tables 2 and 4). Panel c plots the ratio of the surface excess parameters yielding {[Γ_sl−Γ_sv]/Γ_lv}∼1. Insets expand low-MW region and dashed lines represent arithmetic mean of the respective surface excess values listed in table 4 (see appendix B for sample calculations). Apparent Γ_lv (panel b) is reproduced (Krishnan et al. 2003) for comparison to [Γ_sl−Γ_sv]. Apparent surface excess [Γ_sl−Γ_sv] and Γ_lv, as well as the ratio {[Γ_sl−Γ_sv]/Γ_lv}, were found to be independent of protein MW (see §§3 and 5).

Assuming that ϵ/P is constant for all proteins within this study, equation (3.2) predicts a linear relationship between $\text{ln}{C}_{\text{B}}^{\text{max}}$ and ln MW with a slope of −1 (Krishnan et al. 2003). A value for the unknown ratio ϵ/P can be extracted from the intercept (see §4).

4. Results

4.1 SAM stability

Pure PBS buffer contact angles on SAMs were observed to monotonically decrease with observation time while interfacial tension γ_lv (measured by PDT) remained constant, as shown in figure 2 (compare open and closed circles). Specifically, it was observed that ${\theta }_{\text{a}}^{\text{o}}$ of a pure PBS droplet slowly decreased with time from the initial value of $108°<{\theta }_{\text{a}}^{\text{o}}<106°$ at t=0 to $104°<{\theta }_{\text{a}}^{\text{o}}<102°$ at t=1 h, where ${\theta }_{\text{a}}^{\text{o}}$ is the pure buffer contact angle. The range of reported results corresponds to all 17 methyl-terminated SAM surfaces analysed during the course of this work. This phenomenon attributed to SAM ‘hydration’ apparently affected time-dependent measurement of protein-solution contact angles because we observed that the whole contact angle isotherm (θ_a versus concentration) slowly shifted lower with time (see figure 1b, annotation). Steady-state spreading pressure Π_a isotherms effectively correct for the SAM hydration effect in the adsorption measurement by normalizing to final ${\tau }_{\text{a}}^{\text{o}}$, that is ${\Pi }_{\text{a}}=({\tau }_{\text{a}}-{\tau }_{\text{a}}^{\text{o}})$. A similar strategy was applied to analysis of protein adsorption kinetics, as further illustrated in figure 2 (compare closed triangles and open triangles). At any time t, reduction in pure PBS contact angle due to hydration (closed circles, figure 2) was added to the recorded θ_a for a protein-containing solution (closed triangles) to ‘correct’ observed θ_a for the hydration effect (open triangles). This correction procedure typically eliminated the long-term drift in θ_a observed for protein-containing solutions (see filled triangles, figure 2, for example), suggesting that protein adsorption kinetics had, in fact, dampened within the 1 h observation period; as had been generally observed for adsorption of these same proteins at the LV surface (Krishnan et al. 2003, 2004b).

Figure 2.

Advancing PBS contact angles ${\theta }_{\text{a}}^{\text{o}}$ (left axis, closed circles) on 1-hexadecane thiol SAMs on gold decrease monotonically with observation time while liquid–vapour interfacial tension, γ_lv (right axis, open circles) remains constant, suggesting time-dependent ‘hydration’ of the SAM surface. SAM hydration also affects θ_a adsorption isotherms shown in figure 1 (arrow annotation, panel b). SAM hydration dynamics were separated from protein adsorption kinetics by ‘correcting’ observed change in θ_a (closed triangles, corresponding to 40 mg ml⁻¹ albumin in PBS) for the decrease in ${\theta }_{\text{a}}^{\text{o}}$ observed in control experiments with pure buffer (yielding open triangles).

4.2 General aspects of the data

Table 3 compiles quantitative results of this work. Replicate protein preparations were studied for Ub, FV HSA, FXII, IgM and α₂-macroglobulin. Different vendors were used as a means of controlling for discrepancies that might arise from sourcing (table 2). Contact angle parameters ${\theta }_{\text{a}}^{\text{o}}$, ${\theta }_{\text{a}}^{\prime }$, $\text{ln}{C}_{\text{B}}^{\Theta /2}$ and M listed in columns 2–5 of table 3 are the mean fitted values corresponding to final 25 θ_a curves recorded within the 60 min time frame of the TPG experiment. Listed error is standard deviation of this mean. Corresponding adhesion tensions ${\tau }_{\text{a}}^{\text{o}}$ and ${\tau }_{\text{a}}^{\prime }$ (columns 6 and 7) were computed from ${\theta }_{\text{a}}^{\text{o}}$ and ${\theta }_a^{\prime }$ values, respectively, with uncertainty estimates computed by propagation of error in θ_a and γ_lv measurements (§2). Maximum ‘spreading pressure’ ${\Pi }_{\text{a}}^{\text{max}}\equiv ({\tau }_{\text{a}}^{\prime }-{\tau }_{\text{a}}^{\text{o}})$ (column 8) was computed directly from aforementioned τ_a values and associated uncertainty again estimated by propagation of error. Only computed estimates of ${\tau }_{\text{a}}^{\text{o}}$, ${\tau }_a^{\prime }$ and ${\Pi }_{\text{a}}^{\text{max}}$ parameters are provided for FXII since the required γ_lv values were graphical estimates (Krishnan et al. 2003, 2004b). Parameters for ubiquitin and α₂-macroglobulin are also graphical estimates from the steady-state, concentration-dependent θ_a curve since surface saturation was not reached within solubility limits for low-MW proteins at the SL interface (as discussed in appendix C). Therefore, firm values could not be ascertained by statistical-fitting procedures described in §2.

4.3 Adsorption reversibility

Fully reversible adsorption is technically challenging to unambiguously prove. Assumption of reversible adsorption, and hence achievement of thermodynamic equilibrium applied herein, is supported by the following experimental observations.

1. Concentration-dependent γ_lv and θ_a of proteins spanning 3 decades in MW (referred to as ‘protein’ or ‘proteins’ below) were like those obtained with small-molecule surfactants in that both followed expectations of Gibbs' adsorption isotherm (Vogler 1992a,b), with a linear-like decrease in γ_lv and θ_a as a function of concentration expressed on a logarithmic concentration axis. Surface excess values (Γ_lv and [Γ_sl−Γ_sv]; see §4d) computed from Gibbs' isotherm for surfactant standards agreed with instrumental methods of analysis within experimental error. Surface excess values for proteins adsorbed to LV and SL surfaces were statistically identical.

2. Concentration-dependant γ_lv and θ_a continuously decreased as a function of solution concentration, well past the concentration required to fill the surface at theoretical monolayer coverage anticipated for irreversible adsorption.

3. Proteins were observed to be weak surfactants with a commensurately low partition coefficient deduced from concentration-dependent γ_lv and θ_a measurements. Free energy of protein adsorption to hydrophobic LV and SL surfaces calculated from partition coefficients agree with values measured by hydrophobic interaction chromatography (Chen et al. 2003).

4. Quantitative aspects of protein and surfactant standards adsorbed to hydrophobic LV and SL surfaces were identical within experimental error. Protein adsorption to hydrophobic LV and SL surfaces followed a ‘Traube-like’ ordering wherein the molar concentration required to achieve an arbitrary spreading pressure decreased in regular progression with MW.

5. Competitive-protein adsorption experiments at hydrophobic LV and SL surfaces demonstrate protein displacement that follows a simple mass-balance exchange.

These lines of evidence support our contention that protein adsorption was reversible under the experimental conditions applied herein and corroborate the conclusion drawn by other investigators employing very different experimental methods that irreversible adsorption is not an inherent property of proteins (Graham & Phillips 1979; Castillo et al. 1985; Brash 1987; Duinhoven et al. 1995; Kamyshny et al. 2001); see also Vogler (1998) for a review and citations therein.

4.4 Contact-angle isotherms

Time-and-concentration-dependent θ_a for the non-ionic surfactant Tween-20 (MW=1226 Da), and purified proteins, prothrombin (FII; MW=72 kDa) and IgM (MW=1000 kDa) are compared in figure 1 in both three-dimensional (θ_a as a function of time and concentration) and two-dimensional (θ_a as a function of concentration at specified times) representations. Note that the logarithmic-solute-concentration ordinate ln C_B in figure 1 is expressed in picomolarity units (pM, 10⁻¹² mol solute L⁻¹ solution; see §2 for computational and data representation details). Examining first three and two dimensional representations of Tween-20 surfactancy (figure 1a) which serves as a reference compound, it was observed that the θ_a curve was sigmoidal in shape, with a well-defined low-concentration asymptote ${\theta }_{\text{a}}^{\text{o}}$ and a high-concentration asymptote ${\theta }_{\text{a}}^{\prime }$. In this latter regard, Tween-20 exhibited concentration-limiting behaviour that is typically interpreted as achievement of a critical micelle concentration, at least for surfactants. This paper provides no evidence of micelles, for either proteins or surfactants, and so only acknowledges a limiting behaviour at which further increase in solute concentration did not measurably change θ_a. Smooth curves through the data of figure 1 result from least-squares fitting of the four-parameter logistic equation described in §2.

Results for all proteins were similar to the surfactant standard Tween-20 (as illustrated for FII and IgM in figure 1b,c, respectively) in that sigmoidal-shaped θ_a isotherms connected low- and high-concentration asymptotes. Significantly more pronounced time dependence in θ_a was observed for proteins, however, especially for intermediate concentrations (in addition to the hydration effects mentioned above). These dynamics were undoubtedly due to rate-limiting, mass-transfer and adsorption steps that slowly brought large macromolecules to LV and SL interfaces relative to the small-molecule reference compound Tween-20 for which only limited dynamics were observed. Observation of time-dependence was important in this particular work only in so far as data demonstrate that θ_a dynamics dampen within the time frame of experimentation, achieving or approaching steady-state (equilibrium) within the 1 h observation window. In fact, data collected in table 3 refers only to steady-state measurements. The bulk-solution concentration at which the limiting ${\theta }_{\text{a}}^{\prime }$ occurs $(\text{ln}{C}_{\text{B}}^{\text{max}})$ is of theoretical interest in this work and was estimated from fitted parameters compiled in table 3, as described in appendix A.

4.5 Adhesion tension and spreading pressure isotherms

Figure 3 traces sequential interpretation of steady-state (1 h drop age), concentration-dependent θ_a data (panel a) in terms of concentration-dependent τ_a (panel b) and spreading pressure Π_a (panel c) for human serum albumin (FV HSA). Steady-state (equilibrium) spreading pressure isotherms Π_a were used as the basis of comparison of protein adsorption for the compounds listed in table 2. Figure 4 collects Π_a isotherms for selected proteins spanning the molecular weight range 10<MW<1000 kDa showing only smoothed curves for the sake of clarity, but representative θ_a, τ_a and Π_a isotherms with authentic data are amply illustrated in figures 1 and 3. The dynamic range of Π_a∼20 mN m⁻¹ was similar to that observed for these proteins at the LV surface and ${\Pi }_{\text{a}}^{\text{max}}$ fell within a relatively narrow 10 mN m⁻¹ band for the diverse set of proteins studied (Krishnan et al. 2003). Furthermore, the same ‘Traube-rule’ ordering of protein adsorption observed at the LV interface was repeated at the SL interface in that high-MW proteins reduce Π_a to any arbitrary value at lower molarity than low-MW proteins, as suggested by the horizontal arrow annotation on figure 4.

Figure 3.

Sequential interpretation of a steady-state (3600 s drop age) contact angle adsorption isotherm for human serum albumin (FV HSA, preparation 2, table 3); panel a, advancing contact angles, θ_a; panel b, advancing adhesion tension, τ_a; panel c, advancing spreading pressure Π_a. Smoothed curves through the data serve as guides to the eye. Annotations identify low- and high-concentration asymptotes for contact angles $({\theta }_{\text{a}}^{\text{o}},{\theta }_{\text{a}}^{\prime })$, adhesion tensions $({\tau }_{\text{a}}^{\text{o}},{\tau }_{\text{a}}^{\prime })$ and maximum spreading pressure ${\Pi }_{\text{a}}^{\text{max}}$ that are used to characterize isotherms (table 3).

Figure 4.

Comparison of steady-state spreading pressure Π_a isotherms for selected proteins spanning three decades in molecular weight (table 2). Smooth curves are guides to the eye (see figures 1 and 3 for similar plots including authentic data and table 3 for statistics of fit). Molar scaling reveals an ordering among diverse proteins, similar to the ‘Traube-rule’ observed for proteins at the liquid–vapour interface wherein molar concentration required to reach a specified Π_a value decreased with increasing MW (arrow).

4.6 Apparent Gibbs' surface excess

Adsorption to the SL interface was measured through the apparent Gibbs' excess parameter [Γ_sl−Γ_sv] computed using equation (3.1) applied to contact-angle isotherms (see appendix B for example calculations). As noted in §3, the term ‘apparent’ alerts the reader to the fact that casual application of Gibbs' adsorption isotherm using C_B instead of activity treats solutes (proteins and surfactants) as isomerically pure, non-ionized polyelectrolytes (Frommer & Miller 1968) at infinite dilution with unit activity coefficients (Strey et al. 1999). Table 4 collects results for proteins and the small-molecule surfactant standards SDS and Tween-20. Γ_lv used in calculation of [Γ_sl−Γ_sv] and {[Γ_sl−Γ_sv]/Γ_lv} for surfactant standards was 342±10 and 455±17 pmol cm⁻² for SDS and Tween-20, respectively, and were measured by PDT specifically for this work. [Γ_sl−Γ_sv] for proteins were computed using the average Γ_lv=179±27 pmol cm⁻² previously reported to be characteristic of the proteins listed in table 2 (Krishnan et al. 2003). Table 4 also lists results of independent measures of adsorption, C_sl, for a few of the compounds listed in table 4 to be compared to apparent [Γ_sl−Γ_sv] measured by TPG. Note that results for small molecule surfactants SDS and Tween 20 were in good agreement with TPG (i.e. {[Γ_sl−Γ_sv]/C_sl}∼0.99±0.01; rows 10, 11 column 6). However, results for proteins (rows 1–9, column 6) were in substantial disagreement (i.e. {[Γ_sl−Γ_sv]/C_sl}=62.5±14.9). Figure 5 shows that MW dependence of apparent [Γ_sl−Γ_sv] (panel a) and Γ_lv (panel b) as well as the ratio {[Γ_sl−Γ_sv]/Γ_lv} (panel c) was flat for proteins listed in table 2 yielding {[Γ_sl−Γ_sv]/Γ_lv}∼1 (see §3).

4.7 A Traube-rule-analogue for protein adsorption and partition coefficient

Figure 6 plots $\text{ln}{C}_{\text{B}}^{\text{max}}$ data compiled in table 3 for proteins at the SL interface (panel a), and compares with results from the LV interface collected in panel b (Krishnan et al. 2003, 2004b) on natural logarithmic coordinates compatible with equation (3.2) of §3 (data corresponding to ubiquitin was estimated as described in appendix C). Protein data fell within a monotonically decreasing band, generally, consistent with the anticipation of a unit slope and positive intercept [$\text{ln}{C}_{\text{B}}^{\text{max}}=(-1.3\pm 0.2)\text{ln}\text{MW}+(19.8\pm 1.0)$; R²=78%]. Comparison to equation (3.2) revealed that ϵ/P∼4.1×10⁻⁴ from the nominal intercept value and, by assuming ϵ∼0.45 (as discussed in §3), estimated P∼1100.

Figure 6.

Relationship between the surface-saturating bulk solution concentration $C_{\text{B}}^{\text{max}}$ and protein MW (natural logarithmic scale) at the solid–liquid (SL, panel a) and liquid–vapour interfaces (LV, panel b) for multiple protein preparations (open circle=preparation 1, filled circle=preparation 2, filled triangle=preparation 3). Error bars represent uncertainty computed by propagation of experimental errors into compiled $\text{ln}{C}_{\text{B}}^{\text{max}}$ values (see table 3 and appendix A for representative calculations). Panel b is reproduced from (Krishnan et al. 2003) for the purpose of comparing the LV and SL interfaces. Linear regression through the SL data yielded [$\text{ln}{C}_{\text{B}}^{\text{max}}=(-1.3\pm 0.2)\text{ln}\text{MW}+(19.8\pm 1.0)$; R²=78%] compared to [$\text{ln}{C}_{\text{B}}^{\text{max}}=(-1.4\pm 0.2)\text{ln}\text{MW}+(21.8\pm 1.3)$; R²=72%] for the LV interface, consistent with the expectation of unit slope and a positive intercept (see §§3 and 5). Note that low-MW proteins require greater bulk-phase concentrations to saturate the interphase than higher-MW proteins.

4.8 Competitive protein adsorption

We have observed that ${\tau }_a^{\prime }$ for all of the diverse proteins studied herein fell within a relatively narrow 10 mN m⁻¹ band. However, no two proteins were found to be identical in this regard, mirroring results obtained for these same proteins adsorbed to the hydrophobic LV surface (Krishnan et al. 2003, 2004a,b). In fact, we found that this ‘interfacial signature’ could be used as a kind of tracer in competitive-adsorption experiments revealing the composition of the interphase formed by adsorption from binary protein mixtures. These mixing experiments also demonstrate that one protein can displace another, strongly indicating that proteins were not irreversibly adsorbed to the surface. Figure 7 examines time-dependent adhesion tension and spreading pressure of hIgM and FV HSA solutions mixed in various proportions at a fixed total protein concentration of 5 mg ml⁻¹; see Krishnan et al. (2004b) for more details of HSA and IgM interfacial properties. Protein-adsorption kinetics led to time-dependent τ_a (corrected for SAM hydration, see above) wherein adhesion tension was observed to quickly rise from ${\tau }_{\text{a}}^{\text{o}}\sim -20\text{mN}{\text{m}}^{-1}$ characteristic of pure PBS on the SAM surface to a steady-state (equilibrium) ${\tau }_{\text{a}}^{\prime }$ characteristic of that protein solution, as illustrated in figure 7a for 100% albumin (circles), 50 : 50 albumin : IgM (diamonds), and 100% IgM (squares). Figure 7b plots observed steady-state (1 h) spreading pressure Π_obs at varying weight-fraction albumin compositions f_alb in hIgM (expressed as percent of 5 mg ml⁻¹ total protein). These results strongly suggest that competitive adsorption between proteins leads to displacement of hIgM by albumin through a process that strictly follows the wt/v concentration of competing proteins and clearly indicate that IgM was not irreversibly adsorbed.

Figure 7.

Time-dependent adhesion tension τ_a (panel a) of pure albumin (circles), pure hIgM (squares), and a 50 : 50 mixture of albumin in hIgM (diamonds) at constant 5 mg ml⁻¹ total protein. Note that τ_a of the 50 : 50 mixture fell between the pure protein solutions. Observed spreading pressure Π_obs (panel b) followed a simple linear combining rule expressed in weight-fraction protein in the bulk phase ${\Pi }_{\text{obs}}={\Pi }_{\text{alb}}-f_{\text{IgM}}(\mathrm{\Delta }\Pi )$, where ΔΠ=(Π_alb−Π_IgM), Π_alb or Π_IgM refer to Π_obs at 100% albumin (f_IgM=0) or 100% IgM (f_IgM=1), respectively. Error bars represent standard deviation of the mean of the final 25 Π_obs values observed at 1 h equilibration time.

5. Discussion

5.1 Adsorption isotherms

Adsorption isotherms constructed from concentration-dependent contact angles (θ_a, τ_a and Π_a, see figures 1, 3 and 4) for the proteins studied herein exhibited many similarities to concentration-dependent γ_lv reported previously (Krishnan et al. 2003). Maximum spreading pressure, ${\Pi }_{\text{a}}^{\text{max}}$, fell within a relatively narrow $10<{\Pi }_{\text{a}}^{\text{max}}<20\text{mN}{\text{m}}^{-1}$ band characteristic of all proteins studied, just as observed at the LV surface. Furthermore, Π_a isotherms exhibited the ‘Traube-rule-like’ progression in MW observed at the LV surface wherein the molar concentration required to reach a specified Π_a value decreased with increasing MW. Bearing in mind the great range in MW spanned by proteins in figure 4, it is reasonable to conclude that commensurate variability in protein composition did not confer widely varying SL interfacial activity; at least not in comparison to the full range available to ordinary surfactants. The inference taken from the Traube-rule-like progression is that protein concentration required to reduce Π_a to a specified value decreases with MW in a manner loosely consistent with the addition of a generic amino-acid-building-block having an ‘average amphilicity’ that increases MW but does not radically change protein amphilicity. Otherwise, if MW increased by addition of amino-acid-building-blocks with highly variable amphilicity, then Π_a would be expected to be a much stronger function of protein MW than is observed. Thus, it appears that molar variability in Π_a is achieved by aggregating greater mass of similar amphiphilic character, as opposed to accumulating greater amphilicity with increasing MW.

5.2 Apparent Gibbs' surface excess

Adsorption measurements by concentration-dependent contact angles were in good agreement with literature values for the surfactant standards SDS and Tween-20, as listed in table 4. Close agreement between apparent [Γ_sl−Γ_sv] and C_sl from alternative methods suggests that (i) assumptions of purity and unitary activity coefficients were reasonable for these small molecules and (ii) solute deposition at the SV interface was negligible (see §3). However, [Γ_sl−Γ_sv] for proteins were quite different than values drawn from comparable literature sources, as was observed to be the case for apparent Γ_lv (Krishnan et al. 2003). No doubt proteins violate assumptions of ideality and unitary activity coefficients (Wills et al. 1993; Knezic 2002), causing apparent [Γ_sl−Γ_sv] to deviate substantially from real, activity-corrected surface excess. Previous work showed that apparent and real Γ_lv for proteins were different by a factor of about 56 and that apparent Γ_lv was approximately constant across the span of protein MW studied (Krishnan et al. 2003). Apparent [Γ_sl−Γ_sv] was found to differ from independent measures by a factor of 62.5±14.9, as inferred from the mean {[Γ_sl−Γ_sv]/C_sl} ratio for proteins (see column 6, rows 1–9, table 4), consistent with estimates from the LV interface above. Figure 5 plots apparent [Γ_sl−Γ_sv] and Γ_lv, and the ratio {[Γ_sl−Γ_sv]/Γ_lv} as a function of MW showing that Γ_lv∼[Γ_sl−Γ_sv] and that, as a consequence, {[Γ_sl−Γ_sv]/Γ_lv}∼1. We thus conclude that [Γ_sl−Γ_sv]∼Γ_sl∼Γ_lv for the globular proteins studied herein. By contrast, {[Γ_sl−Γ_sv]/Γ_lv}=3.8±0.1 for Tween 20 (row 11, column 4, table 4) suggesting nearly 4× concentration at the SL interface over LV, consistent with results reported for Tween-80 at silanated glass surfaces (Vogler 1993).

5.3 A Traube-rule-analogue for protein adsorption and partition coefficient

A flat trend in Γ_lv and Γ_sl with MW is consistent with an interphase concentration C_I (in units of moles cm⁻³) scaling inversely with MW and an interphase thickness Ω (in units of cm) that scales directly with MW. This is because Γ=C_IΩ (when the partition coefficient P≫1) and MW dependence cancels (Krishnan et al. 2003). In other words, the interphase thickens as adsorbed proteins become larger and Gibbs' dividing plane descends deeper into the surface region (Vogler 1992a,b, 1993). Interpreted in terms of the theory of protein adsorption briefly outlined in §3, hydrated spheroidal protein molecules with net radius R scaling as a function of MW^1/3 pack into the interphase to a concentration $C_{\text{I}}^{\text{max}}$ limited by osmotic repulsion between molecules. Or stated another way, $C_{\text{I}}^{\text{max}}$ is limited by the extent to which the interphase can be dehydrated by protein displacement of interfacial water. Interphase dehydration is more related to the properties of water than the proteins themselves and so the partition coefficient P≡C_I/C_B is observed to be approximately constant among the proteins investigated.

Figure 6a plots $C_{\text{B}}^{\text{max}}$ data compiled in table 3 on logarithmic coordinates compatible with equation (3.2) of §3. Proteins fell within a monotonically decreasing band roughly consistent with the anticipation of a unit slope and positive intercept [$\text{ln}{C}_{\text{B}}^{\text{max}}=(-1.3\pm 0.2)\text{ln}\text{MW}+(19.8\pm 1.0)$; R²=78%]. A similar trend was observed for protein adsorption at the LV surface, shown in figure 6b. Interpretation of these results must take into account that the highly simplified model of adsorption treats proteins as uniform hard spheres and does not attempt to account for structural complexities of real molecules, or unfolding (denaturation) that may occur upon packing within the surface region. Hence, failure of data to quantitatively adhere to equation (3.2) is hardly surprising. Even so, results for Ub were significantly off the trend obtained at the LV surface (compare to figure 6a,b), possibly signalling that this small protein does not retain a spherical geometry at the SL surface. Clearly, more work is required to further test such speculation and expand the range of proteins explored. However, even in light of scatter in the data of figure 6a, it is of interest to estimate ϵ/P∼4.1×10⁻⁴ from the nominal intercept value and, by assuming ϵ∼0.45 (see §3), estimate P∼1100; which is within an order-of-magnitude of the P∼150 estimate from analysis of protein adsorption to the LV surface and P∼5000 from neutron reflectometry of albumin adsorption to the LV surface (Krishnan et al. 2003). Clearly, goniometry is not a good method for deducing partition coefficients, but it is of continued interest to compute protein adsorption energetics based on these rough estimates. With 10²<P<10³, the free energy of protein adsorption to the hydrophobic surface $\mathrm{\Delta }{G}_{\text{ads}}^{\text{o}}=-RT\text{ln}P$ is very modest, lying within the range $-7RT<\mathrm{\Delta }{G}_{\text{ads}}^{\text{o}}<-4RT$. This is consistent with estimates for lysozyme, myoglobin, and α-amylase adsorption to hydrophobic surfaces $(\mathrm{\Delta }{G}_{\text{ads}}^{\text{o}}\sim -5RT)$ measured by hydrophobic interaction chromatography (Chen et al. 2003). Thus, a conclusion that can be drawn, in spite of rather poor estimates of P, is that adsorption of proteins to a hydrophobic surface is energetically favourable by only small multiples of thermal energy RT and apparently does not vary significantly among proteins.

According to equation (3.2) and figure 6, low-MW proteins require greater bulk-phase concentrations to saturate the SL (or LV) interphase than higher-MW proteins. Given that $C_{\text{B}}^{\text{max}}$ values plotted in figure 6 approach 1 w/v%, it is reasonable to anticipate that extrapolated $C_{\text{B}}^{\text{max}}$ values for yet-lower-MW proteins must equal or exceed protein-solubility limits. As a consequence, surface saturation and the related limiting ${\Pi }_{\text{a}}^{\text{max}}$ is not expected for low-MW proteins at fixed P. In this regard, it is noteworthy that Π_a isotherms for low-MW proteins such as ubiquitin (10.7 kDa) fail to achieve a limiting ${\Pi }_{\text{a}}^{\text{max}}$ at any concentration below the solubility limit, as was observed for concentration-dependent γ_lv.

5.4 Competitive protein adsorption

Figure 7 is strong evidence that there is ready exchange of albumin and IgM at the SAM surface, with relative amounts of adsorbed protein following a simple linear combining rule expressed in weight-fraction protein in the bulk phase. Taken together with related observations summarized in §3, we are led to conclude that protein adsorption to hydrophobic SAM surfaces was substantially reversible under the experimental conditions employed in this work. The word ‘substantially’ is purposely used here because evidence at hand does not guarantee that every adsorbed protein molecule was reversibly bound to the surface (or within the surface region). Indeed, some unknown fraction of adsorbed protein could be irreversibly bound to surface defects which are undetected by tensiometric methods applied herein. However, given the exquisite quality of SAM surfaces and similarity of results obtained at molecularly smooth LV surfaces, this putative fraction of irreversibly bound protein must be vanishingly small.

6. Conclusions

Interfacial energetics of protein adsorption from aqueous-buffer solutions to hydrophobic methyl-terminated SAM surfaces are strikingly similar to the interfacial energetics of protein adsorption to the hydrophobic air–water surface. The observed Traube-rule-like progression in interfacial-tension reduction (γ_lv and τ_a), conserved partition coefficient P, and constant Gibbs' surface excess (Γ_lv and Γ_sl) for globular proteins spanning three decades in MW all occur because water controls the energetics of the adsorption process. Hence, protein adsorption to hydrophobic surfaces has more to do with water than the proteins themselves. A relatively straightforward theory of protein adsorption predicated on the interfacial packing of hydrated spherical molecules with dimensions scaling as a function of MW accounts for the essential physical chemistry of protein adsorption and rationalizes significant experimental observations. From this theory it is evident that displacement of interfacial water by hydrated proteins adsorbing from solution places an energetic cap on protein adsorption to hydrophobic surfaces $(-7RT<\mathrm{\Delta }{G}_{\text{ads}}^{\text{o}}<-4RT)$. This phenomenon is generic to all proteins. Thus, globular–blood protein adsorption to hydrophobic surfaces is not found to significantly vary among diverse protein types.

Variations from general trends discussed above may reflect deviations in protein geometry from simple spheres and/or tendency of some proteins to adopt a more spread/compact configuration (denature) in the adsorbed state. Indeed, there is the expectation from a burgeoning literature base that proteins ‘denature’ over time (Birdi 1989). Denaturation can include changes in molar free volume/interfacial area, loss of higher-order structure with concomitant change in specific bioactivity, and irreversible adsorption. Of course, tensiometric methods are effectively blind to these molecular processes, except insofar as denaturation may lead to time-varying interfacial tensions and contact angles. Our measurements achieved, or asymptotically approached, a well-defined steady-state within the hour observation window applied, suggesting that putative ‘denaturation processes’ either had an insignificant impact on results or occurred significantly faster/slower than the time frame of experimentation. Given the similarity in adsorption energetics to hydrophobic LV and SL surfaces among the broad array of proteins studied and the general expectation that denaturation is a slow process, we are inclined to conclude that either denaturation did not significantly affect results (perhaps accounting for small-but-measurable differences among proteins) or the denaturation effect was astonishingly similar among very different proteins. With regard to irreversible adsorption, we note that experiments examining competitive adsorption between albumin and IgM at the LV surface demonstrated protein displacement (Vroman effect) that followed a simple mass-balance exchange (Krishnan et al. 2004a), strongly suggesting that neither albumin nor IgM was irreversibly adsorbed to this surface.

Appendix A. Estimation of $C_{\text{B}}^{\text{max}}$

$C_{\text{B}}^{\text{max}}$ was calculated from the slope of an advancing contact angle θ_a isotherm Δθ_a/Δ ln C_B and fitted data (table 3) by evaluating equation (A 1) at half-maximal change in θ_a, which occurs at a bulk-phase composition $\text{ln}{C}_{\text{B}}^{\Theta /2}$ (where Θ/2=1/2Θ^max and ${\Theta }^{\text{max}}\equiv {\theta }_{\text{a}}^{\text{o}}-{\theta }_{\text{a}}^{\prime }$)

$$\mathrm{\Delta }{\theta }_{\text{a}}/\mathrm{\Delta }\text{ln}{C}_{\text{B}}=-RTS=\frac{({\theta }_{\text{a}}^{\prime }-{\theta }_{\text{a}}^{*})}{[\text{ln}{C}_{\text{B}}^{\text{max}}-\text{ln}{C}_{\text{B}}^{\Theta /2}]}=\frac{\left({\theta }_{\text{a}}^{\prime }-\left({\theta }_{\text{a}}^{\text{o}}+{\theta }_{\text{a}}^{\text{o}}/2\right)\right)}{[\text{ln}{C}_{\text{B}}^{\text{max}}-\text{ln}{C}_{\text{B}}^{\Theta /2}]}\Rightarrow \text{ln}{C}_{\text{B}}^{\text{max}}=\frac{{\Theta }^{\text{max}}}{2RTS}+\text{ln}{C}_{\text{B}}^{\Theta /2},$$ (A 1)

where the terms $S\equiv -(1/RT)(\mathrm{\Delta }{\theta }_{\text{a}}/\mathrm{\Delta }\text{ln}{C}_{\text{B}})$, ${\theta }_{\text{a}}^{*}\equiv {\theta }_{\text{a}}$ measured at $\text{ln}{C}_{\text{B}}^{\Theta /2}$ and ${\Theta }^{\text{max}}\equiv {\theta }_{\text{a}}^{\text{o}}-{\theta }_{\text{a}}^{\prime }$. All of the parameters in the RHS of equation (A 1) are derived from non-linear, least-squares fitting of θ_a isotherms to the four-parameter logistic equation described in §2. Confidence in $C_{\text{B}}^{\text{max}}$ values listed in table 3 and plotted in figure 6 was computed by propagation of the standard errors in best-fit parameters through equation (A 1), as given by equation (A 2). In consideration of all sources of experimental error, we conclude that $\text{ln}{C}_{\text{B}}^{\text{max}}$ estimates are no better than about 20%.

$${\sigma }_{\text{ln}{C}_{\text{B}}^{\text{max}}}^2={\sigma }_{\text{ln}{C}_{\text{B}}^{\Theta /2}}^2+\frac{1}{4{(RTS)}^2}\left[{\sigma }_{{\theta }_{\text{a}}^{\text{o}}}^2-{\sigma }_{{\theta }_{\text{a}}^{\prime }}^2-{\left(\frac{{\theta }_{\text{a}}^{\text{o}}-{\theta }_{\text{a}}^{\prime }}{S^2}\right)}^2{\sigma }_S^2\right],$$ (A 2)

where σs represent standard errors in $\text{ln}{C}_{\text{B}}^{\text{max}}$ and the best-fit parameters $\text{ln}{C}_{\text{B}}^{\Theta /2}$, ${\theta }_{\text{a}}^{\text{o}}$, ${\theta }_{\text{a}}^{\prime }$ and S as denoted by subscripts.

Appendix B. Estimation of [Γ_sl−Γ_sv]

The apparent Gibbs' surface excess [Γ_sl−Γ_sv] was computed from equation (3.1) of §3 for each of the proteins and surfactants listed in table 4. The following steps illustrate surface excess calculations for FV HSA (preparation 1, table 3) at the SL interface. Fit of θ_a isotherm data plotted in figure 3 yielded ${\theta }_{\text{a}}^{\text{o}}=103.3\pm 0.8$, ${\theta }_{\text{a}}^{\prime }=88.3\pm 0.8$, $\text{ln}{C}_{\text{B}}^{\Theta /2}=15.9\pm 0.3$ and M=−14.1±5.7. Inflections in the θ_a curve were located at X₁=13.7 and X₂=10.9 (dimensionless), yielding a slope estimate S′ from the finite difference with calculated uncertainty as $S^{\prime }\equiv \mathrm{\Delta }{\theta }_{\text{a}}/\mathrm{\Delta }X=-2.95\pm 0.04\text{deg}=-0.050\pm 0.007\text{rad}$, where $\mathrm{\Delta }{\theta }_{\text{a}}={\theta }_{\text{a}}{|}_{X_2}-{\theta }_{\text{a}}{|}_{X_1}$ and ΔX=X₂−X₁. Values for θ_a were calculated from the characteristic parameters above, conveniently evaluated at $\text{ln}{C}_{\text{B}}=\text{ln}{C}_{\text{B}}^{\Theta /2}$, where the logistic equation simplifies to ${\theta }_{\text{a}}={\theta }_{\text{a}}^{*}=({\theta }_{\text{a}}^{\text{o}}+{\theta }_{\text{a}}^{\prime })/2$. Thus, ${\theta }_{\text{a}}^{*}=(103.3+88.3)/2=95.8°$; $\text{sin}{\theta }_{\text{a}}^{*}=0.99$; and $\text{cos}{\theta }_{\text{a}}^{*}=-0.10$. The required term γ_lv was calculated from a comparable logistic equation for γ_lv isotherms, using LV fitted parameters (Krishnan et al. 2003, 2004b), but evaluated at $\text{ln}{C}_{\text{B}}=\text{ln}{C}_{\text{B}}^{\Theta /2}=11.7$ as

$$\left[{\gamma }_{\text{lv}}=\frac{70.8-46.2}{1+{(12.4/11.7)}^{-7.3}}+46.2\right]=61.1\text{mN}{\text{m}}^{-1}.$$

Using Γ_lv=179 pmol cm⁻² determined from Krishnan et al. (2003), equation (3.1) was computed as

$$[{\Gamma }_{\text{sl}}-{\Gamma }_{\text{sv}}]=-\{\frac{[61.1\text{sin}95.8]\text{ergs}{\text{cm}}^{-2}}{(8.31\times {10}^7)(298.15)({10}^{-12})\text{ergs}{\text{pmol}}^{-1}}(-0.05)+[179\text{cos}95.8]\}=145\text{pmol}{\text{cm}}^{-2}.$$

Uncertainty in [Γ_sl−Γ_sv] was computed by propagation of error into ΔΓ=[Γ_sl−Γ_sv] as

$${\sigma }_{\mathrm{\Delta }\Gamma }^2={\left[\frac{\text{sin}{\theta }_{\text{a}}}{RT}\frac{\text{d}{\theta }_{\text{a}}}{\text{d}\text{ln}{C}_{\text{B}}}\right]}^2{\sigma }_{\gamma }^2+{\left[\frac{{\gamma }_{\text{lv}}}{RT}\frac{\text{d}{\theta }_{\text{a}}}{\text{d}\text{ln}{C}_{\text{B}}}\text{cos}{\theta }_{\text{a}}+{\Gamma }_{\text{lv}}\text{sin}{\theta }_{\text{a}}\right]}^2{\sigma }_{\theta }^2+{\left[\frac{{\gamma }_{\text{lv}}\text{sin}{\theta }_{\text{a}}}{RT}\right]}^2{\sigma }_{S^{\prime }}^2+{\sigma }_{{\Gamma }_{\text{lv}}}^2{\text{cos}}^2{\theta }_{\text{a}},$$ (B1)

where S′≡dθ_a/d ln C_B. The σ terms for γ_lv, θ_a and S′ were computed from ${\sigma }_{{\gamma }_{\text{lv}}}^2=({\sigma }_{{\gamma }_{\text{lv}}^{\text{o}}}^2+{\sigma }_{{{\gamma }^{\prime }}_{\text{lv}}}^2)/4$ and ${\sigma }_{{\theta }_{\text{a}}}^2=({\sigma }_{{\theta }_{\text{a}}^{\text{o}}}^2+{\sigma }_{{\theta }_{\text{a}}^{\prime }}^2)/4$, where ${\gamma }_{\text{lv}}^{\text{o}}$, ${{\gamma }^{\prime }}_{\text{lv}}$ and ${\theta }_{\text{a}}^{\text{o}}$, ${\theta }_{\text{a}}^{\prime }$ are fitted parameters from γ_lv and θ_a isotherms; as described above. Uncertainty in slope ${\sigma }_{S^{\prime }}^2=({\sigma }_{{\theta }_{\text{a}}^{\text{o}}}^2+{\sigma }_{{\theta }_{\text{a}}^{\prime }}^2)/\mathrm{\Delta }{X}^2$. Thus, uncertainty in ΔΓ is given by

$${\sigma }_{\mathrm{\Delta }\Gamma }^2={\left[\frac{\text{sin}95.8}{(2.48\times {10}^{-2})}(-0.05)\right]}^2(1.88)+{\left[\frac{61.1\text{sin}95.8}{(2.48\times {10}^{-2})}\right]}^2(5.1\times {10}^{-5})+{(27)}^2{\text{cos}}^{2}95.82+{\left[\frac{61.1}{(2.48\times {10}^{-2})}(-0.05)\text{cos}95.8+179\text{sin}95.8\right]}^2(9.8\times {10}^{-5})=324.95,$$

$${\sigma }_{\mathrm{\Delta }\Gamma }=18.02,$$

where $RT=(8.31\times {10}^7)(298.15)({10}^{-12})=2.48\times {10}^{-2}\text{ergs}{\text{pmol}}^{-1}$. Thus, $[{\Gamma }_{\text{sl}}-{\Gamma }_{\text{sv}}]=145\pm 18\text{pmol}{\text{cm}}^{-2}$ as reported in table 4 (row 4 column 2).

Appendix C. Estimation of parameters for ubiquitin and α₂-macroglobulin

Parameters for ubiquitin and α₂-macroglobulin listed in tables 3 and 4 and shown in figures 4–6 were graphical estimates from the steady-state, concentration-dependent θ_a curve. Firm values could not be ascertained by statistical-fitting procedures described in §2 because surface saturation was not reached within solubility limits for this protein. Thus, well-defined high concentration asymptotes, ${\theta }_{\text{a}}^{\prime }$ were not achieved at physically realizable concentrations. Hence, θ_a measured at the highest-concentration studied was used as an estimate for ${\theta }_{\text{a}}^{\prime }$. Adhesion tensions were computed accordingly, with graphical estimates from γ_lv isotherm as ${\tau }_{\text{a}}^{\text{o}}={\gamma }_{\text{lv}}^{\text{o}}\text{cos}{\theta }_{\text{a}}^{\text{o}}$ and ${\tau }_{\text{a}}^{\prime }={{\gamma }^{\prime }}_{\text{lv}}\text{cos}{\theta }_{\text{a}}^{\prime }$. $\text{ln}{C}_{\text{B}}^{\Theta /2}$ and dθ_a/d ln C_B parameters were estimated by graphical location of inflection points on the θ_a curve. These estimates were used in the calculation of $C_{\text{B}}^{\text{max}}$ and [Γ_sl−Γ_sv] parameters, as described in appendices A and B.