Steric Considerations in the Covalent Binding of Myoglobin to Thin

Allegheny College, Department of Chemistry, Meadville, Pennsylvania 16335. Langmuir , 1999, 15 (17), pp 5578–5583. DOI: 10.1021/la981123o. Publicati...
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Langmuir 1999, 15, 5578-5583

Steric Considerations in the Covalent Binding of Myoglobin to Thin Films Alice A. Deckert,* Christopher Farrell,† Jason Roos,† Rebecca Waddell, and Audria Stubna Allegheny College, Department of Chemistry, Meadville, Pennsylvania 16335 Received August 28, 1998. In Final Form: May 12, 1999 Myoglobin is shown to bind to mixed thin films of docosyl mesylate and methyl docosanoate. Myoglobin can be attached to films containing docosyl mesylate; however, films containing only methyl docosanoate exhibit only minimal myoglobin attachment. This indicates that myoglobin is specifically attached to the docosyl mesylate sites on the film surface. Surface coverage of myoglobin is dependent on the percentage of docosyl mesylate in the film. A maximum myoglobin coverage of approximately 1.2 × 1012 myoglobin/cm2 is estimated for films composed of between 22% and 30% docosyl mesylate.

Introduction The fabrication of devices for detecting trace quantities of a biologically active analyte in complex biological solutions has become a goal of many researchers.1-31 Such * To whom correspondence should be addressed. † Current address: Johns Hopkins University, Baltimore, MD. (1) Anzai, J.; Lee, S.; Osa, T. Chem. Pharm. Bull. 1989, 37, 3320. (2) Tatsuma, T.; Tsuzuki, H.; Okawa, Y.; Yoshida, S.; Wtatnabe, T. Thin Solid Films 1991, 202, 145. (3) Karymov, M. A.; Kruchinin, A. A.; Tarantov, Yu. A.; Balova, I. A.; Remisova, L. A.; Sukhodolov, N. G.; Yanklovich, A. I.; Yorkin, A. M. Sens. Actuators, B 1992, 6, 208. (4) Storri, S.; Santoni, T.; Minunni, M.; Mascini, M. Biosens. Bioelectron. 1998, 13, 347. (5) Fromherz, P. Biochim. Biophys. Acta 1971, 225, 382. (6) Sriyudthsac, M.; Yamagishi, H.; Moriizumi, T. Thin Solid Films 1988, 160, 463. (7) Anzai, J.; Hashimoto, J.; Osa, T.; Matsuo, T. Anal. Sci. 1988, 4, 247. (8) Hamachi, I.; Nakamura, K.; Fujita, A.; Kunitake, T. J. Am. Chem. Soc. 1993, 115, 4966. (9) Firestone, M. A.; Shank, M. L.; Sligar, S. G.; Bohn, P. W. J. Am. Chem. Soc. 1996, 118, 9033. (10) Samuelson, L. A.; Kaplan, D. L.; Lim, J. O.; Kamath, M.; Marx, K. A.; Tripathy, S. K. Thin Solid Films 1994, 242, 50. (11) Owaku, K.; Goto, M.; Ikariyama, Y.; Aizawa, M. Anal. Chem. 1995, 67 (9), 1613. (12) Nikolelis, D. P.; Krull, U. J. Anal. Chim. Acta 1992, 257, 239. (13) Hamachi, I.; Noda, S.; Kunitake, T. J. Am. Chem. Soc. 1991, 113, 9625. (14) Rusling, J. F.; Nassar, A.-E. F. J. Am. Chem. Soc. 1993, 115, 11891. (15) Anzai, J.-I.; Tezuka, S.; Osa, T.; Nakajima, H.; Matsuo, T. Chem. Pharm. Bull. 1987, 35 (2), 693. (16) Aizawa, M.; Owaku, K.; Matsuzawa, M.; Shinohara, H.; Ikariyama, Y. Thin Solid Films 1989, 180, 227. (17) Ohashi, E.; Tamiya, E.; Karube, I. J. Membr. Sci. 1990, 49, 95. (18) Steizie, M.; Sackmann, E. In Biosensors Applications in Medicine, Environmental Protection and Process Control; Schmid, B. D., Scheller, F., Eds.; GBF Monographs Vol. 13; Braunschweig, Germany, 1991; pp 339-346. (19) Vogel, A.; Hoffmann, B.; Sauer, Th.; Wegner, B. In Biosensors Applications in Medicine, Environmental Protection and Process Control; Schmid, B. D., Scheller, F., Eds.; GBF Monographs Vol. 13; Braunschweig, Germany, 1991; pp 347-356. (20) Lee, S.; Anzai, J.; Osa, T. Bull. Chem. Soc. Jpn. 1991, 64, 2019. (21) Ottenbacher, D.; Kindervater, R.; Gimmel, P.; Klee, B.; Jahnig, F.; Gopel, W. Sens. Actuators, B 1992, 6, 192. (22) Charych, D. H.; Nagy, J. O.; Spevak, W.; Bednarski, M. D. Science 1993, 261, 585. (23) Reichert, A.; Nagy, J. O.; Spevak, W.; Charych, D. J. Am. Chem. Soc. 1995, 117, 829. (24) Spevak, W.; Nagy, J. O.; Charych, D. H. Adv. Mater. 1995, 7 (1), 85. (25) Barraud, A. Vacuum 1990, 41, 1624. (26) Bilitewski, U.; Drewes, W.; Schmid, R. D. Sens. Actuators, B 1992, 7, 321.

devices have been termed biosensors and can generally be broken down into three parts. The first part consists of a biological recognition system such as an enzyme, antibody, or antigen. The biological recognition system gives the device the specificity necessary to detect only a given analyte or class of analytes from a complex solution. The recognition system is thus unique for each biosensor design. The second component of most biosensors is the matrix or support used to immobilize the recognition system. The recognition system is immobilized in order to create a mechanically and thermally stable solid-state device with a convenient shelf life. The support system employed must allow the analyte to interact with the recognition system and must not render the recognition system inactive. In many cases the support provides an ordered template for immobilization of the recognition system thus enhancing the selectivity of the resultant device.27-32 Each biosensing application uses a unique recognition system. This often requires developers of biosensors to find novel methods for the immobilization step each time a new assay is developed. Thus the development of a simple and general method for the fabrication of thermally and mechanically robust devices employing a wide range of biological recognition systems is needed.2,4,8,11-12,27-31 Several such methods have been developed which either require multiple fabrication or synthesis steps,1-4 do not allow for control of protein coverage,5-6 require protein engineering,8-10 or can only be applied to a limited set of recognition systems.11,28 One general method for the fabrication of organized protein-lipid structures which involves hydrogen-bonding or electrostatic interactions has been developed.27-31 In this manuscript, we describe a potentially general method for covalent attachment of proteins to thin films. Covalent bonding interactions between the protein and the lipid support matrix provide films with a high degree of mechanical and thermal stability. Our general approach is to employ nucleophilic substitution (SN2) chemistry on (27) Kunitaki, T. Thin Solid Films 1996, 284-285, 9. (28) Decher, G.; Hong, J.-D. Ber Bunsen-Ges. Phys. Chem. 1991, 95, 1430. (29) Lvov, Y.; Ariga, K.; Kunitake, T. Chem Lett. 1994, 12, 2323. (30) Lvov, Y.; Ariga, K.; Ichinose, I.; Kunitake, T. J. Am. Chem. Soc. 1995, 117, 6117. (31) Lvov, Y.; Ariga, K.; Ichinose, I.; Kunitake, T. J. Chem. Soc., Chem. Commum. 1995, 2313. (32) Coulet, P. R. J. Membr. Sci. 1992, 68, 217.

10.1021/la981123o CCC: $18.00 © 1999 American Chemical Society Published on Web 06/18/1999

Binding of Myoglobin to Thin Films

Langmuir, Vol. 15, No. 17, 1999 5579

Scheme 1. Expected Nucleophilic Substitution Reaction Shown in Cartoon Forma

a The large circle represents the protein with the attached nucleophile represented by NH. A portion of the film with one docosyl mesylate flanked by two methyl docosanoate molecules is shown. Arrows indicate the donation of a pair of electrons from the nucleophile to the R carbon of the mesylate. This results in a new bond and the displacement of methylsulfonate.

a thin film. A cartoon of the expected surface reaction is depicted in Scheme 1. By engineering a thin film that contains a good leaving group such as a mesylate, we hope to make the film susceptible to attachment of substrates which contain any of a wide variety of nucleophiles. Several amino acids contain primary and secondary amines in their side chains which can act as nucleophiles. Since a number of amine groups are likely to be accessible on the surface of any given protein or biological recognition system, attachment of nearly any polypeptide is theoretically possible with this approach. Attachment of myoglobin was chosen as a proof of concept because myoglobin is well-characterized, relatively small, and water soluble and absorbs light in the visible region of the spectrum, making detection by visible spectroscopy possible. In addition, heme proteins can be used for many device applications due to their optical, electrical, and enzymatic properties.9,13-14 Experimental Section Synthesis of Docosyl Mesylate. The docosyl mesylate was prepared from 1-docosanol and excess methyl sulfonyl chloride. A 5 g mass of 1-docosanol was placed in a 250 mL three-necked flask fitted with a dropping funnel, a drying tube, and a condenser. Approximately 50 mL of pyridine and 20 mL of chloroform were added, and the mixture was stirred until nearly all the 1-docosanol was dissolved. The mixture was cooled by placing the flask in an ice bath. Approximately 5 mL of methyl sulfonyl chloride was dropped into the reaction mixture with stirring over a 10 min period. A white solid was formed upon mixing. The reaction mixture was allowed to stir overnight at 0 °C. The resulting mixture was poured over approximately 150 mL of crushed ice and acidified with 6 M HCl. The white solid formed during the reaction was filtered out by gravity filtration. The filtrate was separated into an aqueous and an organic layer. The aqueous layer was washed three times with chloroform. The organic layer was washed three times with 1 M HCl, twice with 10% sodium bicarbonate solution, and once with brine. All the resulting organic layers were combined and dried over magnesium sulfate. After the solvent was extracted by rotary evaporation, a whitishyellow solid remained. The crude product was recrystallized from petroleum ether to produce white crystals. Both the original white solid filtered from the crude reaction mixture and the white crystals harvested from the organic layer had melting points of

71-72 °C. Analysis by NMR of the two products revealed that they were both docosyl mesylate. The two products were combined and recrystallized from petroleum ether to result in approximately 5 g of pure docosyl mesylate. Purity was assumed from the absence of any evidence of contamination in the 1H NMR and from the narrow melting point range of 71.5-72 °C. The proton NMR showed resonances at 0.9 ppm (triplet, 3 protons), 1.3 ppm (multiplet, 38 protons), 1.75 ppm (multiplet, 2 protons), 3.0 ppm (singlet, 3 protons), and 4.2 ppm (triplet, 2 protons). These resonances corresponded to the terminal methyl group, the 19 methylene groups next to the terminal methyl, the methylene group beta to the mesylate, the methyl group on the mesylate, and the methylene group alpha to the mesylate, respectively. Film Fabrication and Analysis. Mixtures of docosyl mesylate and methyl docosanoate (cartoon structures shown in Scheme 1) having mole percents of docosyl mesylate between 0 and 100% were dissolved in chloroform to a total concentration of about 1 mg/mL. A mixture of the surfactant molecules was spread at the air-water interface on a NIMA Langmuir-Blodgett trough containing high-purity water from a Barnstead E-pure system. The feedwater for the E-pure system was distilled by reverse osmosis, irradiated with UV light, filtered, and deionized. The final high-purity water had a resistivity greater than 18 MΩ‚cm. The layer was compressed at the air-water interface with a speed of 20 cm2/min until the surface pressure was 30 mN/m. A quartz slide which had been rendered hydrophobic by adsorption of 1,1,1,3,3,3-hexamethylsilizane was lowered to rest horizontally at the air-film-water interface. The film was then expanded at a speed of 20 cm2/min until the surface pressure was close to 0 mN/m. This procedure ensured that only a single layer would be transferred to the quartz slide and that the film on the quartz substrate would be terminated with the mesylate and methyl ester groups. The quartz slide was then lifted off the air-water interface. Film transfer to the slide was monitored by compressing the layer at 20 cm2/min and noting the total reduction in film area. This resulted in a transfer ratio (reduction in film area/area of slide) of close to 100%. Film transfer was repeated for the other side of the quartz slide. When both sides of the quartz substrate contained a monolayer film, the quartz slide was exposed to a 0.1 mM solution of horse heart myoglobin (from Sigma) in pH 7 Tris buffer at 37 °C for 10 s. After the 10 s exposure, the film was removed from the myoglobin solution and rinsed thoroughly with pH 7 Tris buffer and then with highpurity water. The slide was allowed to dry under a gentle stream of compressed air. After drying, another monolayer film was

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Figure 1. Typical results for the integrated absorbance versus the number of layers in the films for three film compositions. The inset shows a typical set of visible spectra obtained for every five layers of a 25-layer 25% docosyl mesylate-myoglobin sandwich film. Solid lines represent the linear-least-squares fit to each set of data. The three film compositions and the corresponding symbols are as follows: (b) 25% docosyl mesylate; (4) 12.5% docosyl mesylate; (9) 0% docosyl mesylate.

Deckert et al.

Figure 2. Results from all film compositions are compiled and graphed as absorbance per layer versus mole fraction of docosyl mesylate in the film. These data show a clear local maximum near a film composition of 25% docosyl mesylate. Error bars are shown for the standard deviation between multiple trials. Data points with no error bars are results of single trials. The solid line represents the nonlinear best fit to the analytical model for the homogeneous surface described in the text.

deposited on both sides of the slide and the myoglobin exposure was repeated. In this way sandwich films of up to 30 layers were fabricated. All film fabrication procedures were carried out in a class 1000 clean room or a laminar flow hood providing class 1000 clean air at the work surface. Multiple layers were required so that the presence of myoglobin could be detected using UV-visible spectroscopy. After each application of five sandwich layers, a UV-visible spectrum of the film was obtained. The appearance of the myoglobin Soret band at 410 nm was monitored as indication that myoglobin had been incorporated into the multilayer films. The integrated absorbance between 350 and 500 nm was used as a quantitative measure of the relative amount of myoglobin in the multilayer film.

Results Figure 1 shows the results of plotting the integrated absorbance versus the number of layers in the film for three different film compositions. The inset shows the visible spectrum obtained for every five layers of the 25layer, 25% docosyl mesylate-myoglobin sandwich film. The inset clearly shows that the myoglobin Soret band at 410 nm increases with increasing number of layers which have been exposed to the myoglobin solution. The increase in integrated absorbance with number of layers is shown graphically by the closed circles. The open triangles represent the results for a film consisting of 12.5% docosyl mesylate. The closed squares represent the results for a film consisting of 0% docosyl mesylate. A linear-leastsquares fit to these data results in a slope of 0.0078 per layer for the closed circles, 0.0044 per layer for the open triangles, and 0.0016 for the closed squares. These slopes are a relative measure of the amount of myoglobin bound to the film per layer. Importantly, the intercept of each linear-least-squares fit is zero to within the standard error of the fit, as expected. This corroborates the fact that myoglobin is being incorporated into the layers of the film in a controlled manner. In addition, the closed squares show very little increase in integrated absorbance as multiple layers are built up. The minimal amount of myoglobin attachment in the 100% methyl ester films indicates that the myoglobin is interacting specifically with the mesylate groups on the surface of the film. As a control, films of 25% docosyl mesylate and 75% methyl docosanoate were exposed to pH 7 Tris buffer at

Figure 3. Limiting area for Langmuir films as a function of film composition. A slight increase in film area is seen upon progression from 0% to 100% docosyl mesylate films. No significant nonlinear trend is observed in the data. The solid line represents the linear-least-squares fit to the data.

37 °C for 10 s. There was no detectable absorbance in the 410 nm region for films consisting of up to 30 layers. These data indicate that the absorbance in the 410 nm region observed for the films exposed to 0.1 mM myoglobin solution can be attributed to myoglobin that has been incorporated into the film. A similar plot to those shown in Figure 1 was constructed for eight other film compositions ranging from 0% to 100% docosyl mesylate. The slope values were determined and used as a relative measure of the amount of myoglobin per layer in each film. These data are compiled in Figure 2. Figure 2 plots the integrated absorbance per layer as a function of the percentage of docosyl mesylate in the film. Error bars are shown for the standard deviation between multiple trials. These data clearly show that film compositions between 22% and 30% docosyl mesylate are optimal for attachment of myoglobin. A Beer-Lambert plot for myoglobin in pH 7 Tris buffer was constructed. These data resulted in an integrated molar absorptivity of 2.88 × 106 L/mol cm or a molecular absorption coefficient of 4.8 × 10-15 cm2/molecule. Figure 3 shows the limiting areas taken from the pressure-area isotherms for film compositions ranging

Binding of Myoglobin to Thin Films

from 0% to 100% docosyl mesylate. These data are not averages; rather, they are the results from single trials. The limiting areas of the films are scattered about 25 Å2 and range from 20 to 36 Å2. Discussion Figure 1 indicates that myoglobin is attached to these thin films by a specific interaction with the mesylate group at the film surface. The expected substitution reaction occurs when a nucleophile on the protein donates a pair of electrons to the R carbon of the docosyl mesylate. The formation of this new bond between the protein and the long carbon chain results in the displacement of the methylsulfonate. The methylsulfonate can abstract the proton from the attached protein and become methylsulfonic acid. This reaction is shown in cartoon form in Scheme 1. Since the methyl docosanoate does not contain a good leaving group, it is not susceptible to nucleophilic attack. The data clearly show that myoglobin is interacting specifically with the docosyl mesylate and not with the methyl docosanoate. The most likely interaction is the formation of a covalent bond between the myoglobin and the long carbon chain. This conclusion is strengthened by the observation that 30 min of sonication in buffer does not result in the loss of any myoglobin bound to these films, as measured by the visible absorption spectrum before and after sonication. At this point we cannot discern which nucleophile or nucleophiles on the myoglobin have been substituted for the mesylate groups. Experiments are currently underway to investigate the attachment of BOC-protected amino acids to these films. Preliminary data show that BOChistidine binds to these films under mild conditions but that BOC-phenylalanine does not bind. BOC-histidine contains a nucleophilic imine nitrogen in its side chain and would be expected to undergo SN2 chemistry with these films. BOC-phenylalanine does not contain any nucleophilic side chain groups and would not be expected to undergo SN2 chemistry with these films. In addition, recent experiments in our lab have established that the myoglobin incorporated in these films retains its full peroxidase activity when o-methoxyphenol is used as a substrate.34 We are very encouraged by these results and feel that this film design will be useful as a general platform for attaching a wide variety of biological recognition systems under mild conditions. The measured absorption coefficient was used to estimate the coverage of myoglobin for the 25% docosyl mesylate films. The average integrated absorbance (0.0058/ layer) for the 25% and 31% docosyl mesylate films was divided by the absorption coefficient obtained from the Beer-Lambert plot.35 This calculation results in an estimated coverage of 1.2 × 1012 myoglobin/cm2. The dimensions of myoglobin in the solid state, as determined by X-ray crystallography, are 4.5 nm × 3.5 nm × 2.5 nm. Thus, in the lowest density crystal plane there are approximately 6.4 × 1012 myoglobin/cm2. Even at the maximum coverage attained in these experiments, the myoglobin surface density is only about 18% of the closepacked arrangement observed in lowest density plane of the crystalline form. These coverages have recently been confirmed by our lab with the more sensitive surface plasmon resonance (spr) technique. Using spr, coverages (33) Okusa, H.; Kurihara, K.; Kunitake, T. Langmuir 1994, 10, 3577. (34) Janesko, B.; Deckert, A. A. Effect of Immobilization on Myogolobin’s Peroxidase Activity, in preparation. (35) Miyauchi, S.; Arisawa, S.; Arise, T.; Yamamoto, R. Thin Solid Films 1989, 180, 293.

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of between 20% and 22% of a close-packed layer were obtained. These results could indicate that the surfacebound myoglobin adopts a configuration with a larger cross section than that in the crystalline form. On the other hand, these results could indicate that the random arrangement of myoglobin on the surface of the film does not allow for myoglobin to close pack. Similar results have been observed for other systems and have been attributed to the steric bulk of the protein combined with the decreasing accessibility of the active surface sites as the density of surface sites increases.36-39 The possibility that the mixed films are not homogeneous at all compositions was investigated by measuring the limiting area for Langmuir films over the entire composition range. These films were formed at the airwater interface under the same conditions as the films that were ultimately transferred to the quartz substrates. Figure 3 reveals that the limiting area of the films does not change significantly with film composition. These data can be interpreted in two ways. Either the films form completely homogeneous, ideal mixtures, or the films consist of segregated domains.40 These two possibilities for the film morphology lead to two distinct analytical models. Models There appear to be two opposing factors influencing the attachment of myoglobin to these films which lead to the local maximum observed in the data presented in Figure 2. The total number of active mesylate sites on the surface clearly increases with increasing percentage of docosyl mesylate in the film. However, the accessibility of these sites is expected to decrease as the mesylate groups are crowded closer together. Qualitatively, these two opposing factors will result in an absorption maximum such as that found in Figure 2. The decrease in accessibility of the surface sites can be envisioned in two ways, depending on the morphology of the surface. The docosyl mesylate is just slightly longer and larger than the methyl docosanoate. Thus, a surface with a homogeneous mixture of mesylate and methyl ester sites becomes more crowded with less accessible mesylate sites as the percentage of docosyl mesylate in the film increases. By the same reasoning, on a surface consisting of segregated domains of mesylate and methyl ester sites, the mesylate sites at the edges of the domains are more accessible than those in the interior of the domains. For films consisting of low percentages of docosyl mesylate, the mesylate domains would be small with high edge-to-interior ratios. As the domains increase in size and coalesce, the edge-to-interior ratios would decrease. For films consisting of 100% docosyl mesylate, essentially all mesylate sites are the less accessible interior sites. An analytical expression that models the homogeneous surface is derived from two assumptions: (1) The film consists of a homogeneous random distribution of docosyl mesylate and methyl docosanoate so that the surface density of mesylate sites increases linearly with the percentage of docosyl mesylate in the film (x). (2) The fraction of unhindered, active sites decreases exponentially (36) Sasaki, S.; Kai, E.; Miyachi, H.; Muguruma, H.; Ikebukuro, K.; Ohkawa, H.; Karube, I. Anal. Chim. Acta 1998, 363, 229. (37) Bieniarz, C.; Husain, J.; Barnes, G.; King, C. A.; Welch, C. J. Bioconjugate Chem. 1996, 7, 88. (38) Aplin, W. Eur. J. Biochem. 1980, 110, 295. (39) Yoshinaga, K.; Kondo, A.; Higashitani, K.; Kito, T. Colloids Surf., A 1993, 77, 101. (40) Wang, S.; Li, Y.; Shao, L.; Rameriz, J.; Wang, P. G.; Leblanc, R. M. Langmuir 1997, 13, 1677.

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to a constant nonzero value. Another analytical expression that models the inhomogeneous surface is derived from two assumptions: (1) The film consists of randomly distributed domains of docosylmeslyate and methyl docosanoate. The percentage of mesylate sites on the edges of domains initially increases as many small domains are formed and then begins to decrease as the small domains grow and coalesce. (2) Mesylate sites at the edges of domains are more accessible than mesylate sites in the interior of domains. The integrated absorbance per layer can be calculated from the measured integrated absorbance coefficient () and the surface density of myoglobin (M). The surface density of myoglobin is equal to the surface density of mesylate sites (N) times the fraction of unhindered, active sites available for attachment (f), or

Values of b ) 0.02 and k ) 5 suggest that, for a film consisting of 100% docosyl mesylate before any exposure to myoglobin, only about 2% of the surface sites are available for myoglobin attachment due to “steric hindrance”. Likewise, for a film consisting of 25% docosyl mesylate before any exposure to myoglobin, 31% of the sites are available for myoglobin attachment. Inhomogeneous-Surface Model. The inhomogeneous surface can be modeled by breaking the surface-attached myoglobin into two categories. Those myoglobin molecules attached to mesylate sites at the edge of a domain (Me) constitute one group, and those attached to sites in the interior of a domain (Mc) constitute the second group. The absorbance can then be computed as in eq 1 by summing the contributions from edge-attached myoglobin and interior-attached myoglobin.

ABS ) M ) Nf

ABS ) [feMe + (x - fe)Mc]

(1)

Homogeneous-Surface Model. For the homogeneoussurface model, the surface density of active sites is calculated from assumption 1 as

N ) mx

(2)

The fraction of sites available for attachment (f) is calculated from assumption (2). The fraction of sites physically open at myoglobin coverage is unity minus the fraction physically blocked by myoglobin attached to the surface. The fraction of sites physically blocked is equal to the surface density of myoglobin times the area per myoglobin (A). Assumption 2 is applied by weighting the fraction of free sites with a distribution that exponentially decreases with the percentage of docosyl mesylate in the film. This leads to the following expression for the fraction of available sites:

f ) (1 - MA)(e-kx + b) ) (1 - NfA)(e-kx + b) (3) The surface density of myoglobin is found by solving eq 3 for f and multiplying the result by eq 2. The absorbance (ABS) is obtained when the resulting expression for Nf is multiplied by the integrated absorption coefficient, as shown in eq 1. Thus

ABS )

mx(e-kx + b) 1 + Amx(e-kx + b)

(4)

where x is the fraction of docosyl mesylate in the film. The measured integrated absorption coefficient () and the cross-sectional area of a myoglobin molecule (A) are obtained from experimental measurements and have values of 5 × 10-15 cm2 and 2 × 10-13 cm2, respectively. There are three adjustable parameters: m, b, and k. These parameters provide measures of the increase in total active site density for each fractional increase of docosyl mesylate in the film, the fraction of active sites unhindered for a film consisting of 100% docosyl mesylate, and a relative “steric factor”, respectively. The solid curve shown in Figure 2 shows that this quantitative model provides a reasonable fit (R2 ) 0.60) to the data with values of m ) (1.6 ( 0.5) × 1013 cm-2, b ) 0.02 ( 0.01, and k ) 5 ( 1. A value of about 2 × 1013 cm-2 for the parameter m suggests that when it is close packed in the film, the docosyl mesylate has a cross-sectional area of 500 Å2. In a typical Langmuir-Blodgett film, straight chain carboxylic acids have cross-sectional areas of about 25-28 Å2. This suggests that the model drastically underestimates the total number of active sites per square centimeter in these films.

(5)

where fe represents the fraction of mesylates sites that are on an edge and x is the fraction of docosyl mesylate in the film. Thus (x - fe) is the fraction of mesylate sites in the interior of domains. An expression for the fraction of mesylate sites on the edge of a domain is developed from the first assumption for the inhomogeneous-surface model. The function for fe must start initially at x ) 0, reach a maximum for some x between 0 and 1, and finally decrease back to 0 at x ) 1. Any function of this sort will have a second derivative that is