A Predictive Approach Using Fractal Analysis for Analyte–Receptor Binding and Dissociation Kinetics for Surface Plasmon Resonance Biosensor Applications

Journal of Colloid and Interface Science 229, 628–640 (2000) doi:10.1006/jcis.2000.7065, available online at http://www.idealibrary.com on

A Predictive Approach Using Fractal Analysis for Analyte–Receptor Binding and Dissociation Kinetics for Surface Plasmon Resonance Biosensor Applications Anand Ramakrishnan and Ajit Sadana1 Chemical Engineering Department, University of Mississippi, Post Office Box 1848, University, Mississippi 38677-1848 Received March 28, 2000; accepted June 19, 2000

A predictive approach using fractal analysis is presented for analyte–receptor binding and dissociation kinetics for biosensor applications. Data taken from the literature may be modeled, in the case of binding using a single-fractal analysis or a dual-fractal analysis. The dual-fractal analysis represents a change in the binding mechanism as the reaction progresses on the surface. A singlefractal analysis is adequate to model the dissociation kinetics in the examples presented. Predictive relationships developed for the binding and the affinity (kdiss /kbind ) as a function of the analyte concentration are of particular value since they provide a means by which the binding and the affinity rate coefficients may be manipulated. Relationships are also presented for the binding and the dissociation rate coefficients and for the affinity as a function of their corresponding fractal dimension, D f , or the degree of heterogeneity that exists on the surface. When analyte–receptor binding or dissociation is involved, an increase in the heterogeneity on the surface (increase in D f ) leads to an increase in the binding and in the dissociation rate coefficient. It is suggested that an increase in the degree of heterogeneity on the surface leads to an increase in the turbulence on the surface owing to the irregularities on the surface. This turbulence promotes mixing, minimizes diffusional limitations, and leads subsequently to an increase in the binding and in the dissociation rate coefficient. The binding and the dissociation rate coefficients are rather sensitive to the degree of heterogeneity, D f,bind (or D f1 ) and D f,diss , respectively, that exists on the biosensor surface. For example, the order of dependence on D f,bind (or D f1 ) and D f2 is 6.69 and 6.96 for kbind,1 (or k1 ) and k2 , respectively, for the binding of 0.085 to 0.339 µM Fab fragment 48G7L 48G7H in solution to p-nitrophenyl phosphonate (PNP) transition state analogue immobilized on a surface plasmon resonance (SPR) biosensor. The order of dependence on D f,diss (or D f,d ) is 3.26 for the dissociation rate coefficient, kdiss , for the dissociation of the 48G7L 48G7H –PNP complex from the SPR surface to the solution. The predictive relationships presented for the binding and the affinity as a function of the analyte concentration in solution provide further physical insights into the reactions on the surface and should assist in enhancing SPR biosensor performance. In general, the technique is applicable to other reactions occurring on different types of biosensor surfaces and other surfaces such as cell-surface reactions. °C 2000 Academic Press 1

To whom correspondence should be addressed.

0021-9797/00 $35.00

C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.

Key Words: predictive approach; fractals; kinetics; biosensors; binding; dissociation.

INTRODUCTION

Understanding biological processes at the molecular level requires two basic approaches: structural and functional analyses. Under ideal conditions these should complement each other and provide a complete picture of the molecular processes. Electron microscopy, sequence analysis, mass spectroscopy, X-ray, and electron diffraction studies are routinely employed as structural techniques. These provide information about the atomic organization of individual as well as interacting biomolecules, but these have the disadvantage of being static and frozen in time. Functional investigation techniques such as affinity chromatography, immunological techniques, and spectrophotometric techniques give valuable information on the conditions and the specificity of the interaction, but are (a) unable to follow a process in time or (b) too slow to be rendered suitable for most biospecific interactions. Moreover, these techniques demand some kind of labeling of interactants, which is undesirable as it may interfere with the interaction and this will necessitate purification of the interactants in large quantities. A promising area in the investigation of biomolecular interactions is the development of biosensors. These biosensors are finding applications in the areas of biotechnology, physics, chemistry, medicine, aviation, oceanography, and environmental control. These sensors or biosensors may be utilized to monitor the analyte–receptor reactions in real time (1). In addition, such techniques as the surface plasmon resonance (SPR) biosensor do not require radiolabeling or biochemical tagging (2), are reusable, have a flexible experimental design, provide rapid and automated analysis, and have a completely integrated system. The SPR in combination with mass spectrometry (MS) exhibits the potential to provide a proteomic analysis (3). In addition to evaluating affinities and interactions, the SPR can be utilized to determine unknown concentrations, to determine specificity, for kinetic analysis, to check for allosteric effects, and to compare binding patterns of different species.

628

PREDICTIVE APPROACH USING FRACTAL ANALYSIS

Rao et al. (4) utilized the SPR to analyze the binding of vancomycin and its dimer to the C-terminal of D-Ala-D-Ala (DADA) group of gram-positive bacterial walls. This inhibits the cross-linking of the cell walls and is the reason for vancomycin’s activity (5, 6). Rao et al. (4) compared the binding of vancomycin and a synthetic divalent region of vancomycin to N α -Ac-L-Lys-D-Ala-(N α -Ac-KDADA) groups presented in self-assembled monolayers (SAMs). They emphasize that this illustrates the application of SPR as an analytical technique. Satoh and Matsumoto (7) utilized the SPR biosensor to analyze the binding of lectin molecules immobilized on a SPR surface using hydrazide groups. The hydrazide groups were attached to the surface of the CM5 sensor chip by reaction between the activated carboxy groups of the sensor surface and the hydrazide groups of adipic acid dihydrazide (ADHZ). Corr et al. (8) utilized the SPR biosensor to analyze the binding of T lymphocytes bearing αβ cell receptors (TCRs) to a major histocompatability complex (MHC)-encoded class I or class II peptide. The TCRs were covalently linked to the dextran-modified matrix of an SPR biosensor surface. Finally, Kolomenskii et al. (9) utilized the SPR biosensor for a competition assay, and Larrichia-Robio et al. (10) analyzed epitope mapping of apolipoprotien β-100 using a SPR-based biosensor. The SPR biosensor is an instrument gaining increasing popularity in analyzing a wide variety of reactions in different areas. However, the BIAcore biosensor that is based on the SPR principles of detection is almost prohibitively expensive. A cost reduction would significantly expand its user base as well as the number and types of analytes which may be used for possible detection. The importance of gaining a better understanding of the mode of operation of biosensors (for example, the SPR biosensor) to improve their sensitivity, stability, and specificity has been emphasized. For the binding reaction to occur, one of the components must be bound to or immobilized to the surface. This often leads to mass transfer limitations and subsequent complexities. Nevertheless, the solid phase immunoassay technique represents a convenient method for the separation and/or detection of reactants (for example, antigen) in a solution because the binding of antigen to antibody-coated surface (or vice versa) is sensed directly or rapidly. Thus, there is a need to characterize the reactions occurring at the biosensor surface in the presence of diffusional limitations that are inevitably present in these types of systems. It is essential to characterize not only the associative or binding reaction (by a binding rate coefficient, kbind or kads ) but also the desorption or dissociation reaction (by a desorption rate coefficient, kdes or kdiss ). This significantly assists in enhancing the biosensor performance parameters, such as reusability, multiple use for the same analyte, and stability, in addition to providing further insights into sensitivity, reproducibility, and specificity of the biosensor. The ratio of kdiss to kbind (equal to K ) may be used to further characterize the biosensor–analyte–receptor sys-

629

tem. In essence, the analysis of only the binding step in a sense is incomplete, and the analysis of the binding and the dissociation step provides a more complete picture of the analyte–receptor reaction on the surface. The details of association/dissociation of the analyte (antibody or substrate) to a receptor (antigen or enzyme) immobilized on a surface are of tremendous significance in the development of immunodiagnostic devices as well as biosensors (11). The analysis to be presented is, in general, applicable to ligand–receptor and analyte–receptorless systems for biosensor and other applications (e.g., membrane–surface reactions). External diffusional limitations play a role in the analysis of immunodiagnostic assays (12–17). The influence of diffusion in such systems has been analyzed to some extent (14, 18–26). The influence of partial (27) and total (28–30) mass transport limitations on analyte–receptor binding kinetics for biosensor applications is available. The analysis presented for partial mass transport limitation (27) is applicable to simple one-to-one association as well as to cases in which there is heterogeneity of the analyte or the liquid. This applies to the different types of biosensors utilized for the detection of different analytes. In most of the above analyses only the association or the binding of the analyte to the receptor is analyzed. Apparently, until now, the dissociation kinetics (of the analyte–receptor complex on the surface) has not been discussed or presented in great detail, except perhaps by the software program provided by the SPR manufacturer (BIAcore AB, UK). The program provides values of the association and dissociation constants based on certain model assumptions without any regard to the degree of heterogeneity that may exist on the surface. This article attempts to include heterogeneity in the analysis by examining both the association and the dissociation phases of the analyte–receptor kinetics on the biosensor surface. This provides a more complete and realistic picture for the analyte–receptor SPR biosensor system. Kopelman (31) indicated that surface diffusion-controlled reactions that occur on clusters or islands are expected to exhibit anomalous and fractal-like kinetics. These fractal kinetics exhibit anomalous reaction orders and time-dependent (e.g., binding or dissociation) coefficients. Fractals are disordered systems with the disorder described by nonintegral dimensions (32). Kopelman (31) further indicated that as long as surface irregularities show scale invariance that is dilatational symmetry, they can be characterized by a single number, the fractal dimension. The fractal dimension is a global property and is insensitive to structural or morphological details (33). Markel et al. (34) indicated that fractals are scale, self-similar mathematical objects that possess nontrivial geometrical properties. Furthermore, these investigators indicated that rough surfaces, disordered layers on surfaces, and porous objects all possess fractal structures. A consequence of the fractal nature is a power-law dependence of a correlation function (in our case analyte–receptor complex on the surface) on a coordinate (e.g., time). This fractal nature or power-law dependence is exhibited during both the

630

RAMAKRISHNAN AND SADANA

association (or binding) and the dissociation phases. In other words, the degree of roughness or heterogeneity on the surface affects both the association or binding of the analyte to the receptor on the surface and the dissociation of the analyte–receptor complex on the surface. The influence of the degree of heterogeneity on the surface may affect these two phases differently. Also, since this is a temporal reaction, and presumably the degree of heterogeneity may be changing with (reaction) time, there may be two (or more) different values of the degree of heterogeneity for the association and the dissociation phases. Fractal aggregate scaling relationships have been determined for both diffusion-limited and diffusion-limited cluster aggregation processes in spatial dimensions 2, 3, 4, and 5 (35). Fractal dimension values for the kinetics of antigen–antibody binding (36, 37) and analyte–receptor binding (38) are available. A fractal analysis of analyte–receptor binding kinetics for SPR applications is available (39). We now extend these ideas to the dissociation phase as well. We delineate the role of surface roughness in the speed of response, specificity, stability, sensitivity, and the regenerability or reusability of SPR biosensors. This analysis of both the binding and dissociation reactions should provide further physical insights into the interactions occurring at the SPR interface. This should eventually help minimize the cost of the SPR biosensor. We obtain values of the fractal dimensions and the rate coefficient values for the association (binding) as well as the dissociation phases. A comparison of the values obtained for these two phases for the different biosensors analyzed and for the different reaction parameters in each case should significantly assist in enhancing the relevant biosensor performance parameters. The noninteger orders of dependence obtained for the binding and dissociation rate coefficient(s) on their respective fractal dimension(s) further reinforce the fractal nature of these analyte–receptor binding/dissociation systems. THEORY

An analysis of the binding kinetics of the antigen in solution to antibody immobilized on the biosensor surface is available (37). The influence of lateral interactions on the surface and variable rate coefficients is also available (40). Here we present a method of estimating fractal dimensions and rate coefficients for both the association and the dissociation phases for analyte–receptor systems utilized in SPR biosensors.

In general, k1 depends on time, whereas k 0 = k1 (t = 1) does not. Kopelman (31) indicated that in three dimensions (homogeneous space), b = 0. This is in agreement with the results obtained in classical kinetics. Also, with vigorous stirring, the system is made homogeneous and b again equals zero. However, for diffusion-limited reactions occurring in fractal spaces, b > 0; this yields a time-dependent rate coefficient. The random fluctuations in a two-state process in ligand binding kinetics have been analyzed (41). The stochastic approach can be used to explain the variable binding rate coefficient. These ideas may also be extended to the dissociation rate coefficient. The simplest way to model these fluctuations is to assume that the binding (or the dissociation) rate coefficient is the sum of its deterministic value (invariant) and the fluctuation (z(t)) (41). This z(t) is a random function with a zero mean. The decreasing and increasing binding rate coefficients can be assumed to exhibit an exponential form (42). A similar statement can also be made for the dissociation rate coefficient. Sadana and Madagula (40) analyzed the influence of a decreasing and an increasing binding rate coefficient on the antigen concentration when the antibody is immobilized on the surface. These investigators noted that for an increasing binding rate coefficient, after a brief time interval, as time increases, the concentration of the antigen near the surface decreases, as expected when lateral interactions are present or absent. The diffusionlimited binding kinetics of antigen (or antibody or substrate) in solution to antibody (or antigen or enzyme) immobilized on a biosensor surface has been analyzed within a fractal framework (36, 37). Furthermore, experimental data presented for the binding of human immunodeficiency virus (HIV) (antigen) to the antibody anti-HIV immobilized on a surface show a characteristic ordered “disorder” (43). This indicates the possibility of a fractal-like surface. It is obvious that the above biosensor system (wherein either the antigen or the antibody is attached to the surface) along with its different complexities, including heterogeneities on the surface and in solution, diffusion-coupled reactions, and time-varying adsorption (or binding), and even dissociation rate coefficients, may be characterized as a fractal system. The diffusion of reactants toward fractal surfaces has been analyzed (44). Havlin (45) has briefly reviewed and discussed these results. Here we extend the ideas to dissociation reactions as well (that is the dissociation of the analyte–receptor complex on the surface) with particular emphasis on the SPR biosensor.

Variable Binding Rate Coefficient Kopelman (31) has indicated that classical reaction kinetics is sometimes unsatisfactory when the reactants are spatially constrained on the microscopic level by walls, phase boundaries, or force fields. Such heterogeneous reactions, e.g., bioenzymatic reactions, that occur at interfaces of different phases exhibit fractal orders for elementary reactions and rate coefficients with temporal memories. In such reactions, the rate coefficient is given by 0 −b

k1 = k t

.

[1]

Single-Fractal Analysis Binding rate coefficient. Havlin (45) indicated that the diffusion of a particle (analyte (Ag)) from a homogeneous solution to a solid surface (e.g., receptor (Ab)-coated surface) on which it reacts to form a product (analyte–receptor complex (Ag · Ab)) is given by ½ (Analyte · Receptor) ∼

t (3−Df,bind )/2 = t p (t < tc ) . [2a] t 1/2 (t > tc )

631

PREDICTIVE APPROACH USING FRACTAL ANALYSIS

Here Df,bind is the fractal dimension of the surface during the binding step. Equation [2a] indicates that the concentration of the product Ab · Ag(t) in a reaction Ab + Ag → Ab · Ag on a solid fractal surface scales at short and intermediate time scales as [Ab · Ag] ∼ t p with the coefficient p = (3 − Df,bind )/2 at short time scales and p = 1/2 at intermediate time scales. This equation is associated with the short-term diffusional properties of a random walk on a fractal surface. Note that in a perfectly stirred kinetics on a regular surface (nonfractal) structure (or surface), k1 is a constant; that is, it is independent of time. In other words, the limit of regular structures (or surfaces) and the absence of diffusion-limited kinetics lead to kbind being independent of time. In all other situations, one would expect a scaling behavior given by kbind ∼ k 0 t −b with −b = p < 0. Also, the appearance of the coefficient p different from p = 0 is the consequence of two different phenomena, i.e., the heterogeneity (fractality) of the surface and the imperfect mixing (diffusionlimited) condition. Havlin (45) indicated that the crossover value may be determined by rc2 ∼ tc . Above the characteristic length, rc , the selfsimilarity is lost. Above tc , the surface may be considered homogeneous, since the self-similarity property disappears, and “regular” diffusion is now present. For the present analysis, tc is arbitrarily chosen. For the purpose of this analysis, we assume that the value of tc is not reached. One may consider the analysis to be presented as an intermediate “heuristic” approach in that in the future one may also be able to develop an autonomous (and not time-dependent) model of diffusion-controlled kinetics. Dissociation rate coefficient. We propose that a mechanism similar to the binding rate coefficient, is involved (except in reverse) in the dissociation step. In this case, the dissociation takes place from a fractal surface. The diffusion of the dissociated particle (receptor (Ab) or analyte (Ag)) from the solid surface (e.g., analyte (Ag)–receptor (Ab) complex coated surface) into solution may be given, as a first approximation, by (Analyte · Receptor) ∼ −k 0 t (3−Df,diss )/2 (t > tdiss ).

[2b]

Here Df,diss is the fractal dimension of the surface for the dissociation step, and tdiss represents the start of the dissociation step. This corresponds to the highest concentration of the analyte– receptor on the surface. Henceforth, its concentration only decreases. Df,bind may or may not be equal to Df,diss . Equation [2b] indicates that during the dissociation step, the concentration of the product Ab · Ag(t) in the reaction Ag · Ab → Ab + Ag on a solid fractal surface scales at short and intermediate time scales as [Ag · Ab] ∼ −t p with the coefficient p now equal to (3 − Df,diss )/2 at short time scales and p = 1/2 at intermediate time scales. In essence, the assumptions that are applicable in the association (or binding) step are applicable in the dissociation step. Once again, this equation is associated with the short-term diffusional properties of a random walk on a fractal surface. Note that in a perfectly stirred kinetics on a regular surface (nonfractal) structure (or surface), kdiss is a constant;

that is, it is independent of time. In other words, the limit of regular structures (or surfaces) and in the absence of diffusionlimited kinetics leads to kdiss being independent of time. In all other situations, one would expect a scaling behavior given by kdiss ∼ −k 0 t −b with −b = p < 0. Once again, the appearance of the coefficient p different from p = 0 is the consequence of two different phenomena, i.e., the heterogeneity (fractality) of the surface and the imperfect mixing (diffusion-limited) condition. The ratio K = kdiss /kbind not only provides physical insights into the analyte–receptor system but also is of practical importance since it may be used to help determine (and possibly enhance) the regenerability, reusability, stability, and other SPR biosensor performance parameters. Dual-Fractal Analysis Binding rate coefficient. The single-fractal analysis we have just presented is extended to include two fractal dimensions. At present, the time (t = t1 ) at which the first fractal dimension “changes” to the second fractal dimension is arbitrary and empirical. For the most part it is dictated by the data analyzed and the experience gained by handling a single-fractal analysis. A smoother curve is obtained in the “transition” region, if care is taken to select the correct number of points for the two regions. In this case, the analyte–receptor complex (Ag · Ab) is given by  (3−D f1,bind )/2 = t p1 (t < t1 )  t (3−Df2,bind )/2 = t p2 (t1 < t < t2 = tc ) . (Analyte · Receptor) ∼ t   1/2 t (t > tc ) [2c] Dissociation rate coefficient. Once again, we propose that a mechanism similar to the binding rate coefficient(s), is involved (except in reverse) in the dissociation step. In this case, the dissociation takes place from a fractal surface. The diffusion of the dissociated particle (receptor (Ab) or analyte (Ag)) from the solid surface (e.g., analyte (Ag)–receptor (Ab) complex coated surface) into solution may be given as a first approximation by ½ (Analyte · Receptor) ∼

−t (3−Df1,diss )/2 (tdiss < t < td1 ) . [2d] −t (3−Df2,diss )/2 (td1 < t < td2 )

Note that different combinations of the binding and dissociation steps are possible as far as the fractal analysis is concerned. Each of these steps or phases can be represented by either a single- or a dual-fractal analysis. For example, the binding or the association phase may be adequately described by a singlefractal analysis. Then it is not necessary that the dissociation step also be represented by a single-fractal analysis. It is quite possible that the dissociation step may need to be adequately described by a dual-fractal analysis. Also, the association or the binding step may be adequately described by a dual-fractal analysis. Then, the dissociation phase may be adequately described by either a single- or a dual-fractal analysis. In effect, four

632

RAMAKRISHNAN AND SADANA

combinations are possible: single-fractal (association)–singlefractal (dissociation); single-fractal (association)–dual-fractal (dissociation); dual-fractal (association)–single-fractal (dissociation); dual-fractal (association)–dual-fractal (dissociation). Presumably, it is only by the analysis of a large number of association–dissociation analyte–receptor data from a wide variety of systems that this point may be further clarified. RESULTS

At the outset it is appropriate to indicate that a fractal analysis will be applied to the data obtained for analyte–receptor binding and dissociation data for the SPR biosensor system. This is one possible explanation for analyzing the diffusion-limited binding and dissociation kinetics assumed to be present in all of the SPR systems analyzed. The parameters thus obtained would provide a useful comparison of different situations. Alternate expressions involving saturation, first-order reaction, and no diffusion limitations are possible, but they are apparently deficient in describing the heterogeneity that inherently exists on the surface. The analyte–receptor binding as well as the dissociation reaction is a complex reaction, and the fractal analysis via the fractal dimension (either Df,bind or Df,diss ) and the rate coefficient for binding (kbind ) or dissociation (kdiss ) provide a useful lumped parameter(s) analysis of the diffusion-limited situation. Also, we do not present any independent proof or physical evidence of the existence of fractals in the analysis of these analyte–receptor binding/dissociation systems except by indicating that it has been applied in other areas and that it is a convenient means for making more quantitative the degree of heterogeneity that exists on the SPR surface. Thus, in all fairness, this is one possible way to analyze this analyte–receptor binding/dissociation data. One might justifiably argue that appropriate modeling may be achieved by using a Langmuirian or other approach. The Langmuirian approach has a major drawback because it does not allow for or accommodate the heterogeneity that exists on the surface. Furthermore, it is appropriate to indicate that all the systems to be analyzed are rough and may display fractal dimensions

different from two. However, roughness by itself does not guarantee that the surface is fractal. Thus, other models should also be kept in mind. For example, Avnir and co-workers (46, 47) in analyzing a set of highly dispersed adsorbents indicated that only a few of them display fractal characteristics. Also, one needs to be cautioned that even though the kinetics exhibit a noninteger time exponent, that does not guarantee that the adsorbent is fractal. Dewey (48) indicated that the protein bulk has a fractal dimension, Df , slightly less than three. The protein can therefore be viewed as a collapsed polymer. This view is consistent with the surface fractal dimension of globular proteins (Df in the range of 2.1 to 2.2). This is not much greater than the fractal dimension value of two for smooth surfaces (49, 50). These types of noninteger time exponent kinetics may also be a result of surface heterogeneity (51–53). Other possible explanations for noninteger kinetics include the Elovich equation (54), progressive roughening of the surface as adsorption proceeds (55, 56), and energy heterogeneity (induced heterogeneity as discussed by Porter and Tompkins (57)) as well as fixed heterogeneity (53). The above references do indicate what effect(s) control the kinetics in each case. In our analysis to be presented, however, we assume that fractal kinetics applies for each of the examples presented. Patten et al. (58) analyzed the binding of different concentrations of the Fab fragment 48G7L 48G7H in solution to the p-nitrophenyl phosphonate (PNP) transition state analogue immobilized on a BIAcore (SPR) biosensor surface. These authors wanted to better understand the evolution of immune function. Figure 1a shows the curves obtained using Eq. [2c] for the binding of 0.339 µM 48G7L 48G7H in solution to the PNP immobilized on the BIAcore biosensor surface, and the dissociation of the 48G7L 48G7H –PNP complex from the same surface, and its eventual diffusion in to solution. A dual-fractal analysis is required to adequately describe the binding kinetics (Eq. [2c]), and a single-fractal analysis (Eq. [2b]) is sufficient to describe the dissociation kinetics. Table 1 shows the values of the binding rate coefficient, kbind , the dissociation rate coefficient, kdiss , the fractal dimension for binding, Df,bind , and the fractal dimension for

TABLE 1 Fractal Dimensions and Binding and Dissociation Rate Coefficients Using a Single- and a Dual-Fractal Analysis for the Binding of Fab 48G7L 48G7H (Analyte) in Solution to p-Nitrophenyl Phosphonate (PNP) Transition State Analogue (Receptor) Immobilized on a BIAcore Biosensor Surface (58) 48G7L 48G7H (in µM)/PNP on surface

kbind

k1,bind

k2,bind

kdiss

Df,bind

Df1,bind

Df2,bind

Df,diss

0.339 0.169 0.085 0.042 0.021

— — — 6.31 ± 0.24 0.529 ± 0.00

65.8 ± 5.91 29.8 ± 1.8 12.5 ± 0.4 — —

357 ± 4.3 186 ± 5.2 39.7 ± 1.7 — —

3.47 ± 0.13 1.93 ± 0.14 2.73 ± 0.19 2.10 ± 0.12 1.73 ± 0.09

— — — 1.57 ± 0.02 1.00 ± 0.00

2.04 ± 0.10 1.80 ± 0.06 1.58 ± 0.04 — —

2.91 ± 0.02 2.67 ± 0.04 2.13 ± 0.07 — —

1.70 ± 0.05 1.44 ± 0.09 1.65 ± 0.09 1.62 ± 0.08 1.89 ± 0.06

PREDICTIVE APPROACH USING FRACTAL ANALYSIS

633

FIG. 1. Binding and dissociation of different concentrations (in µM) of fab fragment 48G7L 48G7H (analyte) in solution to p-nitrophenyl phosphonate (PNP) transition state analogue (receptor) immobilized on a BIAcore biosensor surface (58): (a) 0.339, (b) 0.169, (c) 0.085 (binding modeled as a dual-fractal analysis), (d) 0.042, and (e) 0.021 (binding modeled as a single-fractal analysis). In all cases dissociation modeled as a single-fractal analysis.

dissociation, Df,diss . The values of the binding and dissociation rate coefficient(s) and the fractal dimension(s) for association or adsorption (or binding) and dissociation presented in Table 1 were obtained from a regression analysis using Sigmaplot (59) to model the experimental data using Eq. [2c], wherein [analyte · receptor] = k1,bind t p and k2,bind t p for the binding step(s), and Eq. [2b] wherein [analyte · receptor] = −kdiss t p for the dissociation step. The binding and dissociation rate coefficient values presented in Table 1 are within 95% confidence limits. For example, for the binding of 0.339 µM 48G7L 48G7H in solution to the PNP-immobilized surface, the reported value of k1,bind value is 65.8 ± 5.91. The 95% confidence limits

indicate that 95% of the k1,bind values will lie between 59.9 and 71.7. This indicates that the values are precise and significant. The curves presented in the figures are theoretical curves. Figures 1b, 1c, 1d, and 1e show the binding of 0.169, 0.085, 0.042, and 0.021 µM 48G7L 48G7H in solution to a PNPimmobilized surface and the dissociation of the 48G7L 48G7H – PNP complex from the surface. For the binding of 0.169 and 0.085 µM 48G7L 48G7H in solution to the PNP-immobilized surface, a dual-fractal analysis is required to adequately describe the binding kinetics. For the binding of lower concentrations of 0.042 and 0.021 µM 48G7L 48G7H in solution to the PNPimmobilized surface, a single-fractal analysis is sufficient to

634

RAMAKRISHNAN AND SADANA

FIG. 2. Influence of 48G7L 48G7H (analyte) concentration in solution on (a) binding rate coefficient, k1 (or k1,bind ), (b) binding rate coefficient, k2 (or k2,bind ), (c) affinity, kd /k1 , and (d) affinity, kd /k2 .

adequately describe the binding kinetics. This indicates that there is a change in the binding mechanism as one goes from the lower to the higher 48G7L 48G7H concentration in solution since for the lower concentrations a single-fractal analysis is adequate, whereas for the higher concentrations a dual-fractal analysis is required. The dissociation kinetics for each of the 48G7L 48G7H concentrations utilized may be adequately described by a singlefractal analysis. Patten et al. (58) indicate that nonspecific binding has been eliminated by subtracting out sensorgrams from negative control surfaces. Our analysis, at present, does not include this nonselective adsorption or binding. We do recognize that, in some cases, this may be a significant component of the adsorbed material and that this rate of association, which is of a temporal nature, would depend on surface availability. If we were to accommodate the nonselective adsorption into the model, there would be an increase in the degree of heterogeneity on the surface, since by its very nature nonspecific adsorption is more heterogeneous than specific adsorption or binding. This would lead to higher fractal dimension values since the fractal dimension is a

direct measure of the degree of heterogeneity that exists on the surface. Table 1 indicates that k1,bind increases as the 48G7L 48G7H concentration in solution increases. Figure 2a shows the increase in k1,bind (or k1 , to avoid the double subscript nomenclature) with an increase in 48G7L 48G7H concentration in solution. In the 48G7L 48G7H concentration range (0.021 to 0.339 µM) analyzed k1,bind is given by k1,bind = (244 ± 8.8)[48G7L 48G7H ]1.20±0.04 .

[3a]

The fit is reasonable. More data points are required to establish this equation more firmly. Nevertheless, Eq. [3a] is of value since it provides an indication of the change in k1,bind as the 48G7L 48G7H concentration in solution changes in the range analyzed. The fractional exponent dependence indicates the fractal nature of the system. Table 1 indicates that k2,bind increases as the 48G7L 48G7H concentration in solution increases. Figure 2b shows the increase in k2,bind (or k2 ) with an increase in 48G7L 48G7H concentration

PREDICTIVE APPROACH USING FRACTAL ANALYSIS

in solution. In the 48G7L 48G7H concentration range (0.085 to 0.339 µM) analyzed, k2,bind is given by k2,bind = (2306 ± 1046)[48G7L 48G7H ]1.59±0.38 .

[3b]

The fit is reasonable. There is some scatter in the data. More data points are required to establish this equation more firmly. Nevertheless, Eq. [3b] is of value since it provides an indication of the change in k2,bind as the 48G7L 48G7H concentration in solution increases. The fractional exponent dependence once again indicates the fractal nature of the system. k2,bind is slightly more sensitive than k1,bind on the 48G7L 48G7H concentration in solution as noted by their exponent values (1.59 and 1.20, respectively). The binding rate coefficients exhibit an order of dependence on the analyte concentration between 1 and 2. The ratio kdiss /k1,bind (or kd /k1 ) or kdiss /k2,bind (or kd /k2 ) is of importance since it provides a measure of affinity of the receptor for the analyte, in this case the PNP immobilized on the BIAcore biosensor surface for the 48G7L 48G7H in solution. For the dualfractal analysis examples presented, Fig. 2c shows the decrease in kd /k1 as the 48G7L 48G7H concentration in solution increases. In the 0.085 to 0.339 µM 48G7L 48G7H concentration range in solution analyzed, the ratio kd /k1 is given by kd /k1 = (0.015 ± 0.008)[48G7L 48G7H ]−1.02±0.42 .

[3c]

Figure 2c shows that the ratio kd /k1 decreases as the 48G7L 48G7H concentration in solution increases. There is scatter in the data, and this is reflected in the error estimates for both the exponent and the coefficient. More data points are required to more firmly establish this relation. Nevertheless, the equation is of value since it provides an indication of the affinity and its quantitative change with a change in the 48G7L 48G7H concentration in solution. The dependence of kd /k1 is clearly nonlinear even though the exponent estimate is close to one. However, the error estimate for the exponent does have a value of ±0.42, which underscores the nonlinearity dependence. At the lower 48G7L 48G7H concentrations the ratio kd /k1 is higher, at least in the concentration range analyzed. Thus, if affinity is of concern, and one has the flexibility of selecting the analyte concentration to be analyzed, then one should utilize lower concentrations of 48G7L 48G7H . For the dual-fractal analysis examples presented, Fig. 2d shows the decrease in kd /k2 as the 48G7L 48G7H concentration in solution increases. In the 0.085 to 0.339 µM 48G7L 48G7H concentration range in solution analyzed, the ratio kd /k2 is given by kd /k2 = (0.0015 ± 0.0032)[48G7L 48G7H ]−1.41±0.77 . [3d] Figure 2d shows that the ratio kd /k2 decreases as the 48G7L 48G7H concentration in solution increases. There is quite a bit of scatter in the data, and this is reflected in the error estimates for both the exponent and, especially, the coefficient.

635

More data points are definitely required to more firmly establish this relation. Nevertheless, the equation is of value since it provides an indication of the affinity and its quantitative change with a change in the 48G7L 48G7H concentration in solution. The dependence of kd /k2 on 48G7L 48G7H concentration is clearly nonlinear. At the lower 48G7L 48G7H concentrations the ratio kd /k2 value is higher, at least in the concentration range analyzed. Thus, if affinity is of concern, and one has the flexibility of selecting the analyte concentration to be analyzed, then one should utilize lower concentrations of 48G7L 48G7H . Table 1 and Fig. 3a show that the binding rate coefficient k1 (or k1,bind ) increases as the fractal dimension Df1 (or Df1,bind) increases. In the 48G7L 48G7H concentration range (0.085 to 0.339 µM) analyzed, k1 is given by 6.69±0.06 . k1 = (0.578 ± 0.006)Df1

[4a]

This predictive equation fits the values of the binding rate coefficient k1 presented in Table 1 reasonably well. The very high exponent dependence indicates that the binding rate coefficient is very sensitive to the degree of heterogeneity that exists on the surface. More data points are required to more firmly establish this relation. Table 1 and Fig. 3b show that the binding rate coefficient k2 (or k2,bind ) increases as the fractal dimension Df2 (or Df2,bind ) increases. In the 48G7L 48G7H concentration range (0.085 to 0.339 µM) analyzed, k2 is given by 6.96±0.15 . k2 = (0.207 ± 0.007)Df2

[4b]

This predictive equation fits the values of the binding rate coefficient k2 presented in Table 1 reasonably well. The very high exponent dependence indicates that the binding rate coefficient is very sensitive to the degree of heterogeneity that exists on the surface. More data points are required to more firmly establish this relation. It is of interest to note that the order of dependence on the surface heterogeneity is approximately the same for both k1 (exponent dependence 6.69) and k2 (exponent dependence 6.96). Table 1 and Fig. 3c show that the dissociation rate coefficient kdiss (or kd ) increases as the fractal dimension for the dissociation step Df,diss (or Df,d ) increases. For the data presented in Table 1, kdiss (or kd ) is given by 3.26±0.87 . kd = (0.574 ± 0.06)Df,d

[4c]

This predictive equation fits the values of the dissociation rate coefficient kd presented in Table 1 reasonably well. The same fitting procedure has been applied in Figs. 3a–3c. The curves fit the data points in Figs. 3a and 3b rather well. The fit in Fig. 3c is not as good. This is due to the scatter in the data, and this is reflected in the error estimates for the exponent dependence. The dissociation rate coefficient is quite sensitive to the degree of heterogeneity that exists on the surface as indicated by the

636

RAMAKRISHNAN AND SADANA

FIG. 3. Influence of (a) the fractal dimension, Df1 on the binding rate coefficient, k1 , (b) the fractal dimension, Df2 on the binding rate coefficient, k2 , (c) the fractal dimension, Dfd on the binding rate coefficient, kd , (d) the fractal dimension, Df1 on the affinity, kd /k1 , and (e) the fractal dimension, Df2 on the affinity, kd /k2 .

approximately third-order dependence on the fractal dimension, Df,d . More data points are required to more firmly establish this relation. As indicated earlier, the ratio kd /k1 or kd /k2 is of importance since it provides a measure of the affinity of the receptor for the

analyte. It would be of interest to analyze the influence of surface heterogeneity on the affinity of the receptor for the analyte. Table 1 and Fig. 3d show that the ratio kd /k1 decreases as the fractal dimension for binding Df1 increases. It would perhaps be more appropriate to obtain some sort of average of the fractal

PREDICTIVE APPROACH USING FRACTAL ANALYSIS

dimension for the binding and the dissociation steps, and then fit kd /k1 versus the appropriate average of the fractal dimensions. Since no appropriate average of the fractal dimension for the two steps is available, only the Df1 (for the binding step) is used. For the data presented in Table 1, kd /k1 is given by −5.74±2.24 . kd /k1 = (2.62 ± 1.26)Df1

[4d]

This predictive equation fits the values of the affinity presented in Table 1 reasonably well. There is some scatter in the data, which is reflected in the error estimates for the exponent dependence as well as in the coefficient. Some of the error may be attributed to not using an appropriate weighted average of the fractal dimension for the binding and dissociation phases. Nevertheless, the affinity is very sensitive to the degree of heterogeneity that exists on the surface, as noted by the high value of the exponent. More data points are required to more firmly establish this relation. Equation [4d] is of value since it provides an indication of the influence of the degree of heterogeneity on the surface on

637

the affinity. If the affinity is of primary concern, then one should apparently try to minimize the degree of heterogeneity on the surface, since this is deleterious for affinity. In other words, keep the surface as homogeneous as possible if the affinity is of primary concern. Table 1 and Fig. 3e show that the ratio kd /k2 decreases as the fractal dimension for binding Df2 increases. It would perhaps be more appropriate to obtain some sort of average of the fractal dimension for the binding and dissociation steps and then fit kd /k2 versus the appropriate average of the fractal dimensions. Once again, and as indicated above, since no appropriate average of the fractal dimension for the two steps is available, only the Df2 (for the binding step) is used. For the data presented in Table 1, kd /k2 is given by −6.67±1.64 . kd /k2 = (9.69 ± 4.45)Df2

[4e]

This predictive equation fits the values of the dissociation rate coefficient kd /k2 presented in Table 1 reasonably well. There is

.

FIG. 4. Binding and dissociation of different concentrations (in nM) of anti-β-galactosidase IgM in solution to β-galactosidase immobilized on a lipid monolayer using a SPR biosensor (60): (a) 0.63; (b) 1.25, binding modeled as a single-fractal analysis; (c) 2.5; (d) 5.0, binding modeled as a dual-fractal analysis. In all of the above cases the dissociation modeled as a single-fractal analysis.

638

RAMAKRISHNAN AND SADANA

TABLE 2 Fractal Dimensions and Binding and Dissociation Rate Coefficients Using a Single- and a Dual-Fractal Analysis for the Binding of Anti-β-galactosidase IgM in Solution to β-Galactosidase Immobilized on a Lipid Monolayer Using a Surface Plasmon Resonance (SPR) Biosensor (60) Anti-β-galactosidase concentration (in nM) in solution/ β-galactosidase on a lipid monolayer

kbind

k1,bind

k2,bind

kdiss

Df,bind

Df1,bind

Df2,bind

Df,diss

0.63 1.25 2.5 5

6.87 ± 0.37 33.4 ± 1.8 — —

— — 24.4 ± 0.08 36.4 ± 0.15

— — 263 ± 3.3 533 ± 5.61

0.585 ± 0.087 25.1 ± 2.7 11.7 ± 1.2 39.2 ± 2.49

1.32 ± 0.06 1.79 ± 0.06 — —

— — 1.01 ± 0.00 1.02 ± 0.00

— — 2.49 ± 0.02 1.69 ± 0.02

1.03 ± 0.19 2.31 ± 0.09 1.82 ± 0.11 2.15 ± 0.06

some scatter in the data and this is reflected in the error estimates for the exponent dependence. Once again, some of the error may be attributed to not using an appropriate weighted average of the fractal dimension for the binding and the dissociation phases. Nevertheless, Eq. [4e] is of value since it provides an indication of the influence of the degree of heterogeneity on the surface on the affinity. Once again, if affinity is of primary concern, then one should apparently minimize the degree of heterogeneity on the surface, since an increasing heterogeneity is deleterious for affinity. In other words, keep the surface as homogeneous as possible if affinity is of primary concern. Cooper and Williams (60) have utilized the SPR biosensor to analyze the binding kinetics of anti-β-galactosidase in solution to β-galactosidase immobilized on a lipid monolayer. These authors emphasize that it is important to make quantitative the bivalent binding of analytes in solution to receptors on the surface for the prediction of their behavior for in vivo reactions. They further emphasize that their lipid monolayers closely resemble the surface of a cellular membrane and that their lipid membrane mimics the membrane surface behavior in vivo. Figure 4a shows the curves obtained using Eq. [2a] for the binding of 0.63 nM anti-β-galactosidase immobilized on a lipid monolayer and the dissociation (using Eq. [2c]) of the anti-βgalactosidase–β-galactosidase complex from the same surface and its eventual diffusion into solution. A single-fractal analysis is required to adequately describe the binding kinetics (Eq. [2a]), as well as the dissociation kinetics (Eq. [2c]). Table 2 shows the values of the binding rate coefficients, kbind , and the dissociation rate coefficient, kdiss , the fractal dimension for binding, Df,bind , and the fractal dimension for dissociation, Df,diss . Figure 4b shows the curves obtained for the binding and the dissociation of 1.25 nM anti-β-galactosidase in solution to 500 RU of β-galactosidase immobilized on a lipid monolayer. Once again, a single-fractal analysis is sufficient to adequately describe the binding as well as the dissociation kinetics. Figures 4c and 4d show the curves obtained for the binding and the dissociation of 2.5 and 5.0 nM anti-β-galactosidase in solution, respectively, to 500 RU β-galactosidase immobilized on a lipid monolayer. In both of these two cases a dual-fractal analysis is required to adequately describe the binding kinetics.

The dissociation kinetics in both of these cases is, once again, adequately described by a single-fractal analysis. Apparently, there is a change in the binding mechanism as one goes from the lower (0.63 and 1.25 nM anti-β-galactosidase in solution) to the higher (2.5 and 5.0 nM anti-β-galactosidase in solution). This is because the binding of the lower concentrations may be adequately represented by a single-fractal analysis, whereas the higher concentrations require a dual-fractal analysis to adequately describe the binding kinetics. Also, apparently there is no significant change in the dissociation kinetics for the 0.63 to 5.0 nM anti-β-galactosidase in solution since all of the four concentrations analyzed (0.63, 1.25, 2.5, and 5.0 nM anti-βgalactosidase) may be adequately described by a single-fractal analysis. CONCLUSION

A predictive approach using fractal analysis of the binding of analyte in solution to a receptor immobilized on the SPR biosensor surface provides a quantitative indication of the state of disorder (fractal dimension, Df,bind ) and the binding rate coefficient, kbind , on the surface. In addition, fractal dimensions for the dissociation step, Df,diss , and dissociation rate coefficients, kdiss , are also presented. This provides a more complete picture of the analyte–receptor reactions occurring on the SPR surface than an analysis of the binding step alone, as done previously (61). One may also use the numerical values for the rate coefficients for binding and the dissociation steps to classify the analyte–receptor biosensor system as, for example, (a) moderate binding, extremely fast dissociation, (b) moderate binding, fast dissociation, (c) moderate binding, moderate dissociation, (d) moderate binding, slow dissociation, (e) fast binding, extremely fast dissociation, (f) fast binding, fast dissociation, (g) fast binding, moderate dissociation, and (g) fast binding, slow dissociation. The fractal dimension value provides a quantitative measure of the degree of heterogeneity that exists on the SPR surface for the analyte–receptor systems analyzed. The degree of heterogeneity for the binding and the dissociation phases is, in general, different for the same reaction. This indicates that the

639

PREDICTIVE APPROACH USING FRACTAL ANALYSIS

same surface exhibits two degrees of heterogeneity for the binding and the dissociation reaction. Both types of examples are given wherein either a single- or a dual-fractal analysis is required to describe the binding kinetics. In some cases this applies to the same analyte–receptor system but where an experimental condition is changed. The dual-fractal analysis was used only when the single-fractal analysis did not provide an adequate fit. This was done by the regression analysis provided by Sigmaplot (59). The dissociation step was adequately described by a single-fractal analysis for all of the examples presented. In accord with the prefactor analysis for fractal aggregates (35), quantitative (predictive) expressions are developed for the binding rate coefficient, kbind , the dissociation rate coefficient, kdiss , and the affinity as a function of the fractal dimension. Predictive equations are also developed for the binding and the dissociation rate coefficient and the affinity as a function of the analyte concentration in solution. The parameter K (=kdiss /kbind ) values presented are of interest since they provide an indication of the stability, reusability, and regenerability of the SPR biosensor. Also, depending on one’s final goal, a higher or a lower K value may be beneficial for a particular analyte–receptor SPR system. The fractal dimension for the binding or the dissociation phase, Df,bind or Df,diss , respectively, is not a typical independent variable, such as analyte concentration, that may be directly manipulated. It is estimated from Eqs. [2a] and [2b] (or Eqs. [2c] and [2d]), and one may consider it as a derived variable. Note that a change in the degree of heterogeneity on the surface would, in general, lead to changes in both the binding and the dissociation rate coefficient. Thus, this may require a little thought and manipulation. The binding and the dissociation rate coefficients are rather sensitive to their respective fractal dimensions or the degree of heterogeneity that exists on the biosensor surface. This may be noted by the high orders of dependence. It is suggested that the fractal surface (roughness) leads to turbulence, which enhances mixing, decreases diffusional limitations, and leads to an increase in the binding or the dissociation rate coefficient (62). More such studies are required to determine whether the binding and the dissociation rate coefficient are sensitive to their respective fractal dimensions or the degree of heterogeneity that exists on the SPR biosensor surface. If this is correct, then experimentalists may find it worth their effort to pay a little more attention to the nature of the SPR surface and how it may be manipulated to control the relevant parameters and SPR (and other) biosensor performance in desired directions. Also, in a more general sense the treatment should also be applicable to nonbiosensor applications wherein further physical insights could be obtained. The analysis, if extendable and applicable to cellular surfaces where analyte–receptor reactions occur, should be of considerable use. This is especially true if the cellular surface heterogeneities could be modulated in desired directions, thus effectively manipulating the reaction velocity and direction of these cellular membrane surface reactions.

REFERENCES 1. Myszka, D. G., Morton, T. A., Doyle, M. L., and Chaiken, I. M., Biophys. Chem. 64, 127 (1997). 2. Jonsson, U., Fagerstam, L., Ivarsson, B., Johnsson, B., Karlsson, R., Lundh, K., Lofas, S., Persson, B., Roos, H., and Ronnberg, I., Biotechniques 11, 620 (1991). 3. Williams, C., and Addona, T. A., Tibtech 18, 45 (2000). 4. Rao, J., Yan, L, Xu, B., and Whitesides, G. M., J. Am. Chem. Soc. 121, 2629 (1999). 5. Perkins, H. R., Biochem. J. 111, 195 (1969). 6. Nieto, M., and Perkins, H. R., Biochem. J. 124, 845 (1971). 7. Satoh, A., and Matsumoto, I., Anal. Biochem. 275, 268 (1999). 8. Corr, M., Salnetz, A. E., Boyd, L. F., Jelonek, M. T., Khilko, S., Al-Ramadi, B. K., Kim, Y. S., Maher, S. E., Bothwell, A. L. M., and Margulies, D. H., Science 265, 946 (1994). 9. Kolomenskii, A. A., Gershon, P. D., and Schnessler, H. A., Biosensors 2000, The Sixth World Congress on Biosensors, San Diego, California, 24–26 May, 2000. 10. Larichhia-Robio, L., Ubodi, P., Marcovina, S., Revotella, R. P., and Catapano, A., Biosensors 2000, The Sixth World Congress on Biosensors, San Diego, 24–26 May, 2000. 11. Pisarchick, M. L., Gesty, D., and Thompson, N. L., Biophys. J. 63, 215 (1992). 12. Bluestein, R. C., Diaco, R., Hutson, D. D., Miller, W. K., Neelkantan, N. V., Pankratz, T. J., Tseng, S. Y., and Vickery, E. K., Clin. Chem. 33(9), 1543 (1987). 13. Eddowes, E., Biosensors 3, 1 (1987/1988). 14. Place, J. F., Sutherland, R. M., and Dahne, C., Anal. Chem. 64, 1356 (1985). 15. Giaver, I., J. Immunology 116, 766 (1976). 16. Glaser, R. W., Anal. Biochem. 213, 152 (1993). 17. Fischer, R. J., Fivash, M., Casa-Finet, J., Bladen, S., and McNitt, K. L., Methods 6, 121 (1994). 18. Stenberg, M., Stiblert, L., and Nygren, H. A., J. Theor. Biol. 120, 129 (1986). 19. Nygren, H., and Stenberg, M., J. Colloid Interface Sci. 107, 560 (1985). 20. Stenberg, M., and Nygren, H. A., Anal. Biochem. 127, 183 (1982). 21. Morton, T. A., Myszka, D. G., and Chaiken, I. M., Anal. Biochem. 227, 176 (1995). 22. Sjolander, S., and Urbaniczky, C., Anal. Chem. 63, 2338 (1991). 23. Sadana, A., and Sii, D., J. Colloid Interface Sci. 151, 166 (1992). 24. Sadana, A., and Sii, D., Biosens. Bioelectron. 7, 559 (1992). 25. Sadana, A., and Madagula, A., Biosens. Bioelectron. 9, 45 (1994). 26. Sadana, A., and Beelaram, A., Biosens. Bioelectron. 10, 301 (1995). 27. Christensen, L. L. H., Anal. Biochem. 249, 153 (1997). 28. Matsuda, H., J. Electroanal. Chem. 179, 107 (1967). 29. Elbicki, J. M., Morgan, D. M., and Weber, S. G., Anal. Chem. 56, 978 (1984). 30. Edwards, P. R., Gill, A., Pollard-Knight, D. V., Hoare, M., Bucke, P. E., Lowe, P. A., and Leatherbarrow, R. J., Anal. Biochem. 231, 210 (1995). 31. Kopelman, R., Science 241, 1620 (1988). 32. Pfeifer, P., and Obert, M., in “The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers” (D. Avnir, Ed.), p. 11, Wiley, New York, 1989. 33. Pajkossy, T., and Nyikos, L., Electrochim. Acta 34, 171 (1989). 34. Markel, V. A., Muratov, L. S., Stockman, M. I., and George, T. F., Phys. Rev. B 43, 8183 (1991). 35. Sorenson, C. M., and Roberts, G. C., J. Colloid Interface Sci. 186, 447 (1997). 36. Sadana, A., J. Colloid Interface Sci. 190, 232 (1997). 37. Milum, J., and Sadana, A., J. Colloid Interface Sci. 187, 128 (1997). 38. Sadana, A., and Sutaria, M., Biophys. Chem. 65, 29 (1997). 39. Ramakrishnan, A., and Sadana, A., J. Colloid Interface Sci. 213, 465 (1999).

640

RAMAKRISHNAN AND SADANA

40. Sadana, A., and Madagula, A., Biotechnol. Prog. 9, 259 (1993). 41. DiCera, E., J. Chem. Phys. 95, 5082 (1991). 42. Cuypers, P. A., Willems, G. M., Kop, J. M., Jansen, M. P., and Hermens, W. T., in “Proteins at Interfaces. Physicochemical and Biochemical Studies” (J. L. Brash and T. A. Horbett, Eds.), p. 208, Amer. Chem. Soc., Washington, DC, 1987. 43. Anderson, J., Unpublished results. NIH panel meeting, Case Western Reserve University, Cleveland, Ohio, 1993. 44. Sadana, A., Biotechnol. Prog. 11, 50 (1995). 45. Havlin, S., in “The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers” (D. Avnir, Ed.), p. 251, Wiley, New York, 1989. 46. Pfeifer, P., and Avnir, D., J. Chem. Phys. 79, 3558 (1986). 47. Avnir, D., Farin, D., and Pfeifer, P., J. Chem. Phys. 79, 3566 (1988). 48. Dewey, T. G., J. Chem. Phys. 98, 2250 (1993). 49. Fedorov, B. A., Fedorov, B. B., and Schmidt, P. W., J. Chem. Phys. 99, 4076 (1993). 50. Dewey, T. G., Het. Chem. Rev. 2, 91 (1995).

51. Austin, R. H., Beeson, K. W., Eisenstein, L., Frauenfelder, H., and Gunsalus, I. C., Biochemistry 14, 5355 (1975). 52. Ansari, A., Di Lorio, E. E., Lott, D., Frauenfelder, H., Iben, I. E. T., Langer, P., Roder, H., Sauke, T. B., and Shyamsunder, E., Biochemistry 25, 3139 (1986). 53. Cerofolini, G. F., and Re, N., Langmuir 13, 990 (1997). 54. Aharoni, C., and Tompkins, F. C., Adv. Catal. 21, 1 (1970). 55. Landsberg, P. T., J. Chem. Phys. 23, 1079 (1955). 56. Cerofolini, G. F., Langmuir 13, 995 (1997). 57. Porter, A. S., and Tompkins, F. C., Proc. Roy. Soc. (London) A 217, 529 (1953). 58. Patten, P. A., Gray, N. S., Yang, P. L., Marks, C. B., Wedemayer, G. J., Boniface, J. J., Stevens, R. C., Science 271, 1086 (1996). 59. Sigmaplot, Scientific Graphing Software. User’s Manual. Jandel Scientific, San Rafael, CA, 1993. 60. Cooper, M. A., and Williams, D. H., Anal. Biochem. 276, 36 (1999). 61. Sadana, A., Biosens. Bioelectron. 14, 515 (1999). 62. Martin, S. J., Granstaff, V. E., and Frye, G. C., Anal. Chem. 65, 2910 (1991).