A fractal analysis of analyte–estrogen receptor binding and dissociation kinetics using biosensors: environmental effects

Journal of Colloid and Interface Science 263 (2003) 420–431 www.elsevier.com/locate/jcis

A fractal analysis of analyte–estrogen receptor binding and dissociation kinetics using biosensors: environmental effects Harshala D. Butala and Ajit Sadana ∗ Chemical Engineering Department, University of Mississippi, MS 38677-1848, USA Received 6 March 2002; accepted 21 March 2003

Abstract A fractal analysis is used to model the binding and dissociation kinetics between analytes in solution and estrogen receptors (ER) immobilized on a sensor chip of a surface plasmon resonance (SPR) biosensor. Both cases are analyzed: unliganded as well as liganded. The influence of different ligands is also analyzed. A better understanding of the kinetics provides physical insights into the interactions and suggests means by which appropriate interactions (to promote correct signaling) and inappropriate interactions such as with xenoestrogens (to minimize inappropriate signaling and signaling deleterious to health) may be better controlled. The fractal approach is applied to analyte–ER interaction data available in the literature. Numerical values obtained for the binding and the dissociation rate coefficients are linked to the degree of roughness or heterogeneity (fractal dimension, Df ) present on the biosensor chip surface. In general, the binding and the dissociation rate coefficients are very sensitive to the degree of heterogeneity on the surface. For example, the binding rate coefficient, k, exhibits a 4.60 order of dependence on the fractal dimension, Df , for the binding of unliganded and liganded VDR mixed with GST–RXR in solution to Spp-1 VDRE (1,25-dihydroxyvitamin D3 receptor element) DNA immobilized on a sensor chip surface (Cheskis and Freedman, Biochemistry 35 (1996) 3300–3318). A single-fractal analysis is adequate in some cases. In others (that exhibit complexities in the binding or the dissociation curves) a dual-fractal analysis is required to obtain a better fit. A predictive relationship is also presented for the ratio KA (= k/kd ) as a function of the ratio of the fractal dimensions (Df /Df d ). This has biomedical and environmental implications in that the dissociation and binding rate coefficients may be used to alleviate deleterious effects or enhance beneficial effects by selective modulation of the surface. The KA exhibits a 1½-order dependence on the ratio of the fractal dimensions for the ligand effects on VDR–RXR interaction with specific DNA.  2003 Elsevier Inc. All rights reserved. Keywords: Analyte–estrogen receptor binding; Dissociation; Biosensors; Fractals

1. Introduction The impact of many chemical and particulates used either directly or as by-products on the environment is not properly understood. Some of these chemicals exert toxic effects shortly after exposure, whereas the effects of others are felt only after long exposure and in a more subtle fashion. Some known chemicals that act as endocrine disruptors include environmental estrogens that lead to feminization of wildlife [1] and reptiles [2], declining sperm count in men [3], and increasing incidence of breast and testicular cancer [4]. The pesticide methoprene binds to retinoic acid (vitamin A) receptors and leads to developmental abnormalities; the phytochemical diethylstilbestrol and the pesticide DDT interact with estrogen receptors * Corresponding author.

E-mail address: [email protected] (A. Sadana). 0021-9797/03/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/S0021-9797(03)00338-2

and lead to developmental and reproductive disorders as well as hormone-dependent cancers; and dioxin or 2,3,7,8tetrachlorodibenzo-p-dioxin (TCDD) has been implicated in developmental defects and in tumor formation [5]. In essence, the effects of many other environmental chemicals as well as their mechanisms of action are poorly defined. Also, the endocrine disruptors may exert their deleterious effects by (a) mimicking or partially mimicking the actions of steroid hormones, estrogens, and androgens, by (b) blocking, preventing, and altering hormonal binding of hormone receptors, or influencing cell signaling pathways, and by (c) altering the production and modifying the function of hormone receptors [6]. Chawla et al. [7] have recently reviewed nuclear receptors and lipid physiology. They emphasize (a) the importance of nuclear receptor signaling in maintaining the normal physiological state and (b) how inappropriate signaling has implications in pathological disorders that include

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reproductive biology, inflammation, cancer, diabetes, and cardiovascular diseases. They classify nuclear receptors into endocrine receptors (that have high-affinity hormonal lipids as ligands), adopted orphan receptors (that have low-affinity dietary lipids as ligands), and orphan receptors (whose ligands are unknown). In essence, lipids in our diet not only are nutritionally relevant but also serve as precursors for ligands for nuclear receptors. The maintenance of normal cellular function and normal health depends on the interaction of multiple extracellular and intracellular signals. The receptors on these cells recognize normal analytes as well as environmental signals and respond accordingly. The response of these receptors to phytochemicals may either enhance, decrease, or in some way alter the normal response, which would eventually lead a disruption of the homeostatic networks. It is thus essential to better understand the response of cellular receptors to phytochemicals in order that their deleterious effects may be minimized. This may be done, for example, either by minimizing the binding response or the binding rate coefficient by, for example, changing the cellular surface, or if the binding has taken place, by somehow increasing the dissociation phase or the dissociation rate coefficient. Other means by which the binding action is minimized are also possible. It is reasonable to anticipate that there will be differences in the susceptibility of the human population to different phytochemicals and that genetic factors may play a significant role in these varying susceptibilites. It would be appropriate to delineate and link the genetic factors to the different susceptibilities to the varied phytochemicals. This would significantly assist in the control of and in the understanding of the onset of some intractable diseases that have been linked to phytochemicals. One of course recognizes that environmental chemicals will act by different mechanisms, some known, some partially known, and some unknown. In order that one may begin or promote the understanding of these mechanisms one should obtain a better understanding of the binding as well as the dissociation mechanisms. A large number of in vitro assays have been developed to study binding interactions between estrogen and its receptors. A better knowledge of the interaction of estrogens with their receptors can lead to the development of new therapies aimed at modulating these specific activities. One example of this would be the development of selective estrogen receptor modulators (SERMs) to down-regulate (reducing cellular levels of ER by inducing degradation) estrogen-specific cells in breast, uterine, and ovarian tissues [8]. Pennisi [9] has indicated that estrogen is particularly intriguing in that it protects women from heart attacks and oesteoporosis, but contributes to the development of breast and uterine cancers. The answer presumably lies in the fact that this is a receptor that recognizes different response elements (such as with estrogen and raloxifene). One should design drugs that block estrogen’s unwanted side effects and mimic its beneficial ones.

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The main analytical methods for endogeneous and synthetic estrogens have been based on gas and liquid chromatography techniques in conjunction with mass spectrometry. The disadvantage of these techniques is that the estrogens that are to be measured have to be derivativized in order to be volatile (for gas chromatographic analysis) and to improve detection sensitivity. In spite of modifications of these methods, they are still not sensitive enough and are not able to detect very low levels of estrogens. The development of biosensors and biosensor-based assays could provide an attractive alternative to currently existing analytical techniques for the detection of estrogens in blood, urine, and other body fluids. The BIACORE biosensor based on the surface plasmon resonance (SPR) principle is being increasingly used nowadays in a wide variety of areas as an important screening tool to monitor biomolecular interactions in real time. This biosensor has been widely used in many studies involving the binding of estrogens to ER, DNA, and other inhibitors and modulators. The advantage of using the SPR biosensor is that the interaction between estrogens and their receptors can be monitored in real time. Suen et al. [10] have used the BIACORE biosystem to characterize the interaction between the steroid receptor coactivator-3 (SRC-3) and estrogen receptors ERα and ERβ. Cheskis et al. [11] have used real-time interaction analysis to study the kinetics of human (h) ER binding to DNA in the absence and in the presence of 17β-estradiol and other inhibitors. Cheskis and Freedman [12] studied the interaction of nuclear receptors such as the 1,25-dihydroxyvitamin D3 receptor to DNA using the SPR biosensor. These authors have also studied the interaction between the ER and DRIP205 (domain receptor interacting protein). DRIP205 is a part of the DRIP coactivator complex, which plays an important role in the in vitro transcription process. Wong et al. [13] very recently indicated that estrogen receptors stimulate transcription by forming a preinitiation complex. They emphasize that the coactivators involved in this preinitiation complex do not alter the basal levels of the transcription process. These coactivators, however, are involved in the rate-limiting step for nuclear activation and repression. Thus, these authors [13], like Suen et al. [10], analyzed the role of estrogen receptors (ER) α and β with the SRC (steroid receptor coactivator) family of coactivators. Wong et al. [13] further emphasize that these coactivators exhibit multiple modes of action. Graumann and Jungbauer [14] indicate that steroid hormone receptors form multiprotein complexes with a variety of heatshock proteins (hsp) [15]. Hon et al. [16] also very recently indicated that hsp90- and hsp60-type chaperones are involved in the assembly and in the control of nuclear receptor complexes. This hsp90 is expressed abundantly in the cells, is involved in quite a few cellular processes, and is a core member of the nuclear receptor-chaperone complex. These authors emphasize that the loss of hsp90 reduces the

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transcriptional effects on its collaborators even in the presence of coactivators. In a biosensor-based assay the molecule to be detected (analyte) is present in the solution and the appropriate receptor is immobilized on a solid surface. The interaction between the analyte and the receptor on the solid biosensor surface is detected either by a change in the refractive index (in SPR instruments) or by changes in the fluorometric intensity, ultraviolet light intensity, etc. The SPR biosensor protocol analyzes the binding (and dissociation where applicable) kinetic curves using classical saturation models involving analyte–receptor binding using 1:1, 1:2, etc. ratios, generally under diffusion-free conditions and assuming that the receptors are homogeneously distributed over the sensor surface. Computer programs and software that come with the equipment provide values of the binding (and the dissociation) rate coefficients. Though a careful analysis and experimental protocol may eliminate or minimize the influence of diffusional limitations; realistically speaking, it is more appropriate to include a heterogeneous distribution of receptors on the sensing (or the cellular) surface. The system by its design is heterogeneous (for example, the receptors immobilized on the biosensor surface may exhibit some heterogeneity, that is, surface roughness), and often other factors such as mass transport limitations (unless they are carefully eliminated or minimized) play a significant role and further complicate the design (especially its kinetic aspects) of the assay or the correct interpretation of the assay results. One possible way of accounting for the presence of diffusional limitations and the heterogeneity that exists on the surface is by using fractals. A characteristic feature of fractals is self-similarity at different levels of scale. Fractals exhibit dilatational symmetry. Fractals are disordered systems, and the disorder is described by nonintegral dimensions [17]. Fractals have previously been used to analyze the binding and dissociation kinetics of a variety of analyte–receptor systems [18]. Fractals are particularly useful for this type of analysis because they help characterize the heterogeneity that exists on the surface by a lumped parameter, the fractal dimension. In this manuscript we provide an alternate analysis of (a) the influence of different ligands during the binding and dissociation phases involved in VDR–RXR interaction with specific DNA [11] and (b) the interactions of ER, either unliganded or liganded, with 17β-estradiol in solution to wild-type and mutated SRC3 [12]. Binding and dissociation rate coefficients, as well as fractal dimension values for the binding and the dissociation phases, will be provided for the above analyte–receptor systems wherever applicable. We offer the fractal analysis as an alternative analysis to help improve the understanding; we do not imply that this is better than the original (SPR-based software) analysis. The dissociation rate coefficient plays an important role, depending on one’s eventual goal. Germain [19], in a recent review of the adaptive immune system, indicates that the rate of ligand T-cell antigen receptor (TCR) dissocia-

tion may affect signal quality. Fast dissociation rate (off-rate) coefficients may initiate signaling, but do not allow a complex to form (if the co-receptor is far away), and signaling is aborted at an incomplete stage. Ligands with low dissociation rate coefficients permit a ternary complex to form with the co-receptor and permit the signaling process to proceed. However, this applies to “correct signaling.” In the case of xenoestrogens one needs to minimize the signals sent by them via receptors on the cell surface, and thus fast dissociation rate coefficients should be better.

2. Theory Havlin [20] has reviewed and analyzed the diffusion of reactants towards fractal surfaces. The details of the theory and the equations involved for the binding and the dissociation phases for analyte–receptor binding are available [21]. The details are not repeated here; just the equations are given to permit easier reading. These equations have been applied to other biosensor systems [21,22]. Here we will attempt to apply these equations to the reactions outlined in the previous paragraph (in Section 1). The basic idea is to fold everything into the lumped parameter, the fractal dimension, Df . This lumped parameter takes care of the chemical, orientational, and geometrical details. It is also assumed that the kinetic rate depends only on the geometric (static) parameter, Df . It would also be useful to indicate the low and high limits in angstroms of the fractal character to be presented. 2.1. Single-fractal analysis 2.1.1. Binding rate coefficient Havlin [20] indicates that the diffusion of a particle (analyte) from a homogeneous solution to a solid surface (e.g., receptor-coated surface) on which it reacts to form a product (analyte–receptor complex) is given by


(3−Df ,bind)/2 = t p (analyte–receptor) ∼ t 1/2 t

(t < tc ), (t > tc ).

(1a)

Here Df,bind or Df (used later on in the manuscript) is the fractal dimension of the surface during the binding step, tc is the crossover value. In essence, one assumes that enough binding has not taken place to force the system into the homogeneous standard case. In other words, diffusion never becomes standard. 2.1.2. Dissociation rate coefficient The diffusion of the dissociated particle (receptor or analyte) from the solid surface (e.g., analyte–receptor complex coated surface) into solution may be given, as a first approximation, by

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(analyte–receptor)   t (3−Df 1,bind )/2 = t p1 ∼ t (3−Df 2,bind )/2 = t p2  1/2 t

(t < t1 ), (t1 < t < t2 = tc ), (t > tc ).

(1b)

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 complex on the surface. Henceforth, its concentration only decreases. As in the case of binding, standard diffusion does not come into play during the dissociation phase. 2.2. Dual-fractal analysis 2.2.1. Binding rate coefficient In this case, the analyte–receptor complex is given by (analyte–receptor) ∼ −k  t (3−Df,diss )/2

(t > tdiss).

(1c)

2.2.2. Dissociation rate coefficient In this case the dissociation rate coefficient is given by (analyte–receptor)
(3−Df 1,diss )/2 ∼ −t (3−Df 2,diss )/2 −t

(tdiss < t < td1 ), (td1 < t < td2 ).

(1d)

3. Results A fractal analysis will be applied to the data obtained for analyte–receptor binding taken from the literature for different biosensor systems. The fractal analysis is only one possible approach to analyzing the diffusion-limited binding kinetics assumed to be present in the systems analyzed. The parameters thus obtained would provide a useful comparison of the two different receptor–analyte biosensor systems analyzed. Understandably, alternate expressions for fitting the data are that include saturation, first-order reaction, and no diffusion limitations are available, but these expressions are apparently deficient in describing the heterogeneity that inherently exists on the surface. Another advantage of this technique is that the analyte–receptor binding (as well as the dissociation reaction) is a complex reaction, and the fractal analysis via the fractal dimension and the rate coefficient provides a useful lumped parameter(s) analysis of the diffusion-limited reaction occurring on a heterogeneous surface. In the classical situation, to demonstrate fractality, one should make a log–log plot, and one should definitely have a large of data. It may also be useful to compare the fit to some other forms, such as an exponential form or one involving saturation. At present, we do not present any independent proof or physical evidence of fractals in the examples presented. It is a convenient means (since it provides a lumped parameter) to make the degree of heterogeneity that exists

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on the surface more quantitative. Thus, there is some arbitrariness in the fractal model to be presented. 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. The Langmuir approach was originally developed for gases [23]. Consider a gas at pressure p in equilibrium with a surface. The rate of adsorption is proportional to the gas pressure and to the fraction of the surface. Adsorption will only occur when a gas molecule strikes a bare site. Researchers in the past have successfully modeled the adsorption behavior of analytes in solution to solid surfaces using the Langmuir model even though it does not conform to theory. Rudzinski et al. [24] indicate that other appropriate “liquid” counterparts of the empirical isotherm equations have been developed. These include counterparts of the Freundlich [25], Dubinin–Radushkevich [26], and Toth [27] empirical equations. These studies, with their known constraints, have provided some “restricted” physical insights into the adsorption of adsorbates on different surfaces. The Langmuir approach may be utilized to model the data presented if one assumes the presence of discrete classes of sites (for example, double exponential analysis as compared to a single exponential analysis, as mentioned earlier). Lee and Lee [28] indicate that the fractal approach has been applied to surface science, for example, adsorption and reaction processes. These authors emphasize that the fractal approach provides a convenient means to represent the different structures and morphology at the reaction surface. These authors also emphasize using the fractal approach to develop optimal structures and as a predictive approach. There is no nonselective adsorption of an analyte. Our analysis, at present, does not include this nonselective adsorption. We do recognize that, in some cases, this may not 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 heterogeneity on the surface, since, by its very nature, nonspecific adsorption is more heterogeneous than specific adsorption. 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. For a first-order reaction, as expected, an increase in the heterogeneity on the surface due to nonspecific binding would lead to lower values of the (specific) binding rate coefficient. The deletion of this nonspecific binding from the analysis would lead to (artificially) higher values of the binding rate coefficient for first-order reactions. Our reactions are, in general, higher than first-order. Sadana and Chen [29] have shown that for reaction orders higher than one a certain amount of heterogeneity is beneficial for the binding rate coefficient. There is apparently an optimum range. This is due to steric factors. Thus, depending on whether one is inside

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(a)

(b)

(c)

(d)

(e) Fig. 1. Binding and dissociation phases for the interaction of unliganded VDR–RXR in solution to Spp-1 VDRE DNA immobilized on a sensor chip surface [12]: (a) unliganded VDR mixed with GST–RXR; (b) liganded (1,25-(OH)2 D3 ) VDR mixed with GST–RXR; (c) mixed liganded receptors (0.291 µM of VDR and 0.871 µM GST–RXR); (d) both liganded receptors (Vit.D3 and 9-cis-RA); (e) one receptor liganded, (9-cis-RA) GST–RXR, and one receptor unliganded, VDR. When only a solid line (—) is used in a figure that implies that a single-fractal analysis is adequate either for the binding or for the dissociation phase(s). When both a dashed line (- - -) and a solid line (—) are used in a figure, then the dashed line represents a single-fractal analysis and the solid line represents the dual-fractal analysis for both the binding and the dissociation phases.

or outside this optimum range, the deletion of nonspecific binding in the analysis would lead to either an increase or a decrease in the binding rate coefficient. In other words, if one is in the optimum range for a particular reaction order, then the presence of nonspecific binding would lead to higher values of the (specific) binding rate coefficient. In this case, the deletion of the nonspecific binding leads to lower than real-life values of the binding rate coefficient.

Cheskis and Freedman [12] analyzed the influence of ligands on the interaction of the human 1,25-dihydroxyvitamin D3 receptor (VDR) (a nuclear hormone receptor) with the retinoid X receptor (RXR). This, they indicate, occurs at the level of protein–protein and protein–DNA interactions. Figure 1a shows the binding of unliganded VDR mixed with GST (glutathione-S-transferase)-RXR in solution to Spp-1 VDRE (1,25-dihydroxyvitamin D3 receptor element)

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Table 1 Rate coefficients for the binding and dissociation phases for ligand effects on VDR–RXR interaction with specific DNA [12] kd

KA = k/kd

k

k1

k2

(a) Unliganded VDR mixed with GST–RXR/Spp-1 VDRE DNA immobilized on sensor chip surface

336 ± 12.2

na

na

517 ± 4.5

0.65

Liganded (1,25-(OH)2 D3 ) VDR mixed with GST–RXR/Spp-1 VDRE DNA immobilized on sensor chip surface

960 ± 24.0

825 ± 8.4

1260 ± 2.4

199 ± 5.1

na

(b) Mixed unliganded receptors (0.291 µM of VDR and 0.871 µM of GST–RXR)/Spp-1 VDRE DNA immobilized on sensor chip surface

30.0 ± 0.87

na

na

57.4 ± 2.5

0.52

Both liganded receptors (Vit.D3 and 9-cis-RA)/Spp-1 VDRE DNA immobilized on sensor chip surface

115 ± 6.0

na

na

158 ± 0.97

0.73

One receptor liganded (9-cis-RA) GST–RXR and one receptor unliganded, VDR/Spp-1 VDRE DNA immobilized on sensor chip surface

36.7 ± 1.6

na

na

50.4 ± 0.99

0.73

Analyte in solution/receptor on surface

Table 2 Fractal dimensions for the binding and dissociation phases for ligand effects on VDR–RXR interaction with specific DNA [12] Df

Df 1

Df 2

Df d

Df /Df d

(a) Unliganded VDR mixed with GST–RXR/Spp-1 VDRE DNA immobilized on sensor chip surface

2.34 ± 0.02

na

na

2.90 ± 0.01

0.81

Liganded (1,25-(OH)2 D3 ) VDR mixed with GST–RXR/Spp-1 VDRE DNA immobilized on sensor chip surface

2.82 ± 0.01

2.72 ± 0.01

2.93 ± 0.003

2.76 ± 0.03

na

(b) Mixed unliganded receptors (0.291 µM of VDR and 0.871 µM of GST–RXR)/Spp-1 VDRE DNA immobilized on sensor chip surface

1.83 ± 0.02

na

na

2.60 ± 0.04

0.70

Both liganded receptors (Vit.D3 and 9-cis-RA)/Spp-1 VDRE DNA immobilized on sensor chip surface

2.48 ± 0.03

na

na

2.8 ± 0.005

0.86

One receptor liganded (9-cis-RA) GST–RXR and one receptor unliganded, VDR/Spp-1 VDRE DNA immobilized on sensor chip surface

2.07 ± 0.03

na

na

2.69 ± 0.02

0.77

Analyte in solution/receptor on surface

DNA immobilized on a sensor chip surface. A single-fractal analysis is adequate to describe the binding as well as the dissociation kinetics. The values of (a) the binding rate coefficient, k, and the fractal dimension, Df , for a single-fractal analysis and (b) the dissociation rate coefficient, kd , and the fractal dimension for dissociation, Df d , are given in Table 1. The values of the binding and the dissociation rate coefficient(s) and the fractal dimension for the binding and the dissociation phase presented in Tables 1 and 2 were obtained from a regression analysis using Corel Quatro Pro 8.0 [30] to model the data. Equations (1a) and (1b) were used where (analyte–receptor) = kt (3−Df )/2 for a single-fractal analysis for the binding phase, and (analyte–receptor) = −k  t (3−Ddiss,f )/2 for the dissociation phase. The binding and the dissociation rate coefficient values presented in Tables 1 and 2 are within 95% confidence limits. For example, for the binding of unliganded VDR mixed with GST–RXR in solution to Spp-1 VDRE DNA immobilized on a sensor chip surface, the estimated binding rate coefficient (k) value is 336 ± 12.2. The 95% confidence limit indicates that the k value lies between 324 and 348. This indicates that the values are precise and significant. The equilibrium coefficient

(KA , defined by k/kd ) value is also given in Table 1. It is equal to 0.65. Figure 1b shows the binding of liganded (1,25-(OH)2D3 ) VDR mixed with GST–RXR in solution to Spp-1 VDRE DNA immobilized on a sensor chip surface. In this case a dual-fractal analysis is required to adequately describe the binding kinetics, and a single-fractal analysis is required to describe the dissociation kinetics. The values of the binding and the dissociation rate coefficients, as well as the fractal dimensions for the binding and the dissociation phase, are given in Tables 1 and 2. It is of interest to note that as one goes from the unliganded case to the liganded case, a single- and a dual-fractal analysis are required, respectively, to describe the binding kinetics. This indicates that there is a change in the binding mechanism when a ligand is used and when it is absent, at least in this case. Figure 1c shows the binding of mixed unliganded receptors (0.291 µM of VDR and 0.871 µM GST–RXR) in solution to Spp-1 VDRE DNA immobilized on a sensor chip surface. Once again, a single-fractal analysis is required to adequately describe the binding as well as the dissociation kinetics. The values of the binding and the dissociation rate

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coefficients and the fractal dimensions for the binding and the dissociation phase are given in Tables 1 and 2. Note also that the dissociation rate coefficient, kd , decreases by a factor of 9.05 from a value of 517 (unliganded case) to 57.4 (mixed unliganded case). This has implications as far as signal quality is concerned if one uses Germain’s [19] arguments. High dissociation rate (off-rate) coefficients may initiate signaling but do not allow a complex to form (if the co-receptor is far away), and signaling is aborted at an incomplete stage. Ligands with low dissociation rate coefficients permit the ternary complex to form with the co-receptor and permit the signaling to proceed. Furthermore, it is of interest to compare the KA values obtained in these two cases. As one goes from the unliganded to the mixed unliganded case, there is a 20.7% decrease in the KA value from a value of 0.65 to 0.515, respectively. Teixera [31] has recently analyzed the influence of nanostructure and biomimetic surfaces on cell behavior. This author indicates the importance of ridges and grooves on cell behavior on examining the membranes in the cell lining of the cornea. Lauffenberger [32] has recently emphasized the importance of cellular diagnostics. He indicates the need for the development of design parameters for diagnostic procedures. Cells have the ability to modulate their surface heterogeneity. Subsequently, they can modulate their dissociation rate coefficients and permit signaling either to proceed or not. Appropriate signaling from analyte (estrogen) receptors should be allowed to proceed. Inappropriate and deleterious signaling, such as between xenoestrogens and cellular receptors, should be prevented. This may be done by increasing the surface heterogeneity for the dissociation reaction, leading to a high dissociation rate coefficient. This minimizes inappropriate signaling and consequently minimizes the damage. If the cell were to lose this ability to change its surface heterogeneity due to disease or other physiological changes, this then would prevent the cell from minimizing damage due to inappropriate signaling. Any insights provided into the mechanistic actions of such analyte–receptor reactions is useful. Figure 1d shows the binding of both liganded receptors (Vit.D3 and 9-cis-RA) in solution to Spp-1 VDRE DNA immobilized on a sensor chip surface. The binding and the dissociation phases may both be adequately modeled using a single-fractal analysis. The values of (a) the rate coefficients and (b) the fractal dimensions for the binding and the dissociation phases are given in Table 2. It is of interest to compare the values of the binding and the dissociation rate coefficients for the unliganded and the liganded cases. There is an increase in the binding rate coefficient (k) value by a factor of 3.88 from a value of 29.6 to 115 as one goes from the mixed unliganded case to the case when both receptors are liganded. Similarly, there is an increase in the dissociation rate coefficient, kd , by a factor of 2.75 from a kd value equal to 57.4 to a value of 158 as one goes from the mixed unliganded case to the case when both receptors are liganded. The equilibrium rate coefficient (KA ) also exhibits

an increase in its value (by 41.5%) from a value of 0.515 to a value of 0.729 on going from the mixed unliganded case to the case when both receptors are liganded. Figure 1e shows the binding and the dissociation curves when one receptor (GST–RXR) is liganded and when receptor (VDR) is unliganded. In this case, too, the binding and the dissociation phases may be adequately described by a single-fractal analysis. The values of the binding and the dissociation rate coefficients and the fractal dimensions for the binding and the dissociation phases are given in Tables 1 and 2. Note that when only one receptor is liganded, then the values of the binding and the dissociation rate coefficients fall in between the cases when both receptors are unliganded and when both receptors are liganded. In other words, the presence of one or two ligands, at least in this case, leads to increases in the binding and in the dissociation rate coefficients. The presence of one ligand leads to moderate increase in the binding and in the dissociation rate coefficients when compared to the unliganded case. However, there are substantial increases in the binding and in the dissociation rate coefficients when one goes from a single liganded receptor to the case when both receptors are liganded. At least, this is observed for the data presented here. This may or may not be true for the other cases available in the literature. Tables 1 and 2 indicate that when a single-fractal analysis applies for the binding phase for the case when two receptors are used, an increase in the fractal dimension, Df , leads to an increase in the binding rate coefficient, k. For the data presented in Tables 1 and 2, the binding rate coefficient, k, is given by k = (1.65 ± 0.523)Df4.60±1.28.

(2a)

For the data presented, the binding rate coefficient, k, is very sensitive to the degree of heterogeneity that exists on the surface, as noted by the high value of the exponent. Tables 1 and 2 indicate that when a single-fractal analysis applies for the dissociation phase and once again when two receptors are used, an increase in the fractal dimension, Df d , leads to an increase in the dissociation rate coefficient, kd . For the data presented in Tables 1 and 2, the dissociation rate coefficient, kd , is given by kd = (0.0015 ± 0.00045)Df11.2±4.88 . d

(2b)

For the data presented in Tables 1 and 2, the dissociation rate coefficient, kd , is very sensitive to the degree of heterogeneity on the surface as noted by the very high value of the exponent. It is also of interest to note that the KA (= k/kd ) value changes as the degree of heterogeneity changes on the surface. Since we are looking at two different phases, the binding and the dissociation phase, it is appropriate to associate the fractal dimension for the binding phase, Df , with the binding rate coefficient, k, and the fractal dimension for the dissociation phase, Df d , with the dissociation rate coefficient, kd . Tables 1 and 2 indicate that the KA

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value increases with an increase in the fractal dimension ratio, Df /Df d . For the four data points presented in Tables 1 and 2, the KA may be given by KA = (0.953 ± 0.116)(Df /Df d )1.57±0.775.

(2c)

The KA value does increase with an increase in the (Df / Df d ) ratio. The KA value does exhibit close to a 1½ order of dependence on the (Df /Df d ) ratio. The above relation suggests that if one is interested in high KA values, then one should attempt to have high values of heterogeneity on the surface during the binding phase (Df ) and low values of the degree of heterogeneity during the dissociation phase (Df d ). This may or may not always be easy to control since the binding phase precedes the dissociation phase, and some degree of heterogeneity present during the binding phase may carry over to the dissociation phase. This, of course, depends on what the basic mechanisms involved in causing the degree of heterogeneity on the sensor chip surface. Is it (a) the roughness of the sensor chip surface itself or the immobilization of the receptors on the surface, or (b) is the roughness on the surface caused by the reaction itself, or (c) does some other, more complex mechanism occur on the surface? Also, in essence, if one wants to promote correct signaling (which is normally the case), then higher values of KA are required where (a) we have higher binding rate coefficients and (b)

427

we have lower dissociation rate coefficients that permit appropriate signaling to take place in accord with Germain’s [19] arguments. On the other hand, if one wants to minimize inappropriate signals by, for example, xenobiotics, then one should use lower KA values that exhibit either lower binding rate coefficients or higher dissociation rate coefficients. Wong et al. [13] have analyzed the binding of unliganded and liganded (with 17β-estradiol) estrogen receptors (ER) in solution to 870 RU of wild-type (wt) and mutated SRC3 (M2-SRC3) immobilized on the surface of a sensor chip using an anti-GST (glutathione-S-transferase) antibody. SRC3 is a steroid receptor activator. These authors emphasize that these coactivators are rate-limiting for nuclear receptor activation and repression, but do not change the basal levels of transcription to any significant extent. Figure 2a shows the binding of 23 nM ERα liganded with 17β-estradiol in solution to 870 RU of wild type SRC3 immobilized on the surface of a sensor chip with an anti-GST antibody. A singlefractal analysis is adequate to describe the binding kinetics, and a dual-fractal analysis is required to describe the dissociation kinetics. The values of (a) the binding rate coefficient, k, and the fractal dimension, Df , for a single-fractal analysis, and (b) the dissociation rate coefficient, kd , and the fractal dimension for dissociation, Df d , for a single-fractal analysis, and (c) the dissociation rate coefficients, kd1 and

(a)

(b)

(c)

(d)

Fig. 2. Binding and dissociation phases for the interaction of ER unliganded and liganded with 17β-estradiol (E2) to wild-type (wt) and mutated (M2) SRC3 immobilized on a sensor chip surface [13]: (a) estrogen receptor (ER) + E2/SRC3; (b) ER-E2/SRC3; (c) ER + E2/M2-SRC3; (d) ER-E2/M2-SRC3.

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Table 3 Rate coefficients and fractal dimensions for the binding and the dissociation phase for interactions of ER either unliganded or liganded with 17β-estradiol (E2) in solution to wild-type and mutated SRC3 immobilized on a sensor chip using anti-GST antibody [13] Analyte in solution/receptor on surface ER + E2/wtSRC3 ER-E2/wtSRC3 ER + E2/M2-SRC3 ER-E2/M2-SRC3

k

Df

kd

kd1

10.6 ± 0.47 12.2 ± 0.69 6.65 ± 0.39 7.23 ± 0.59

1.01 ± 0.04 1.30 ± 0.05 1.54 ± 0.06 1.75 ± 0.08

62.9 ± 7.2 30.6 ± 2.5 41.6 ± 1.6 31.4 ± 1.0

39.9 ± 2.8 20.8 ± 1.1 na na

kd2 , and the fractal dimensions for dissociation, Df d1 and Df d2 , for a dual-fractal analysis are given in Table 3. Figure 2b shows the binding of 23 nM unliganded ERα in solution to 870 RU of wild type SRC3 immobilized on the surface of a sensor chip with an anti-GST antibody. A single-fractal analysis is adequate to describe the binding kinetics, and a dual-fractal analysis is required to describe the dissociation kinetics. The values of (a) the binding rate coefficient, k, and the fractal dimension, Df , for a singlefractal analysis, and (b) the dissociation rate coefficient, kd , and the fractal dimension for dissociation, Df d , for a singlefractal analysis, and (c) the dissociation rate coefficients, kd1 and kd2 , and the fractal dimensions for dissociation, Df d1 and Df d2 for a dual-fractal analysis, are given in Table 3. Note that as one goes from the ER+E2 (17β-estradiol) case to the ER-E2 case, there is an increase in the fractal dimension for binding, Df , by 28.7% from a value of 1.01 to 1.30, and there is a corresponding increase in the binding rate coefficient, k, value by 15.14% from a value of 10.6 to 12.2. Note that changes in the binding rate coefficient and in the fractal dimension are in the same direction. Apparently, the presence of 17β-estradiol leads to a decrease in the fractal dimension (decrease in the degree of heterogeneity on the surface) and to a corresponding decrease in the binding rate coefficient. Figure 2c shows the binding of 23 nM ERα liganded with 17β-estradiol in solution to 870 RU of M2 (mutated)– SRC3 immobilized on the surface of a sensor chip with an anti-GST antibody. A single-fractal analysis is adequate to describe the binding kinetics, and a dual-fractal analysis is required to describe the dissociation kinetics. The values of (a) the binding rate coefficient, k, and the fractal dimension, Df , for a single-fractal analysis, and (b) the dissociation rate coefficient, kd , and the fractal dimension for dissociation, Df d , for a single-fractal analysis, and (c) the dissociation rate coefficients, kd1 and kd2 , and the fractal dimensions for dissociation, Df d1 and Df d2 , for a dual-fractal analysis are given in Table 3. Figure 2d shows the binding of 23 nM unliganded ERα in solution to 870 RU of M2-SRC3 immobilized on the surface of a sensor chip with an anti-GST antibody. A single-fractal analysis is adequate to describe the binding kinetics, and a dual-fractal analysis is required to describe the dissociation kinetics. The values of (a) the binding rate coefficient, k, and the fractal dimension, Df , for a single-fractal analysis, and (b) the dissociation rate coefficient, kd , and the fractal dimension for dissociation, Df d , for a single-fractal analy-

kd2 344 ± 1.0 83.3 ± 1.6 na na

Df d

Df d1

Df d2

1.94 ± 0.05 1.94 ± 0.04 2.39 ± 0.02 2.41 ± 0.01

1.75 ± 0.05 1.76 ± 0.04 na na

2.55 ± 0.007 2.30 ± 0.04 na na

sis, and (c) the dissociation rate coefficients, kd1 and kd2 , and the fractal dimensions for dissociation, Df d1 and Df d2 , for a dual-fractal analysis are given in Table 3. Note that as one goes from the ER+E2 (17β-estradiol) case to the ER-E2 case, there is an increase in the fractal dimension for binding by 13.6% from a value of 1.54 to 1.75, and there is a corresponding increase in the binding rate coefficient, k, value by 8.72% from a value of 6.65 to 7.23. Note, once again, that changes in the binding rate coefficient and in the fractal dimension are in the same direction. Apparently, once again and as observed above, the presence of 17β-estradiol leads to a decrease in the fractal dimension (decrease in the degree of heterogeneity on the surface) and to a corresponding decrease in the binding rate coefficient. This is observed for the wild type as well as for the mutated form of the SRC3. Wong et al. [13] have also analyzed the binding and the dissociation of ERβ in the absence and in the presence of ligand in solution to SRC3601–762 immobilized on a sensor chip surface. Figure 3a shows the binding of unliganded ERβ in solution to SRC3601–762 immobilized on the surface of a sensor chip with an anti-GST antibody. A single-fractal analysis is adequate to describe the binding as well as the dissociation kinetics. The values of (a) the binding rate coefficient, k, and the fractal dimension, Df , for a single-fractal analysis, and (b) the dissociation rate coefficient, kd , and the fractal dimension for dissociation, Df d , for a single-fractal analysis are given in Table 4. Figure 3b shows the binding of ERβ liganded with WAY164397 in solution to SRC3601–762 immobilized on the surface of a sensor chip with an anti-GST antibody. A singlefractal analysis is adequate to describe the binding as well as the dissociation kinetics. The values of (a) the binding rate coefficient, k, and the fractal dimension, Df , for a singlefractal analysis, and (b) the dissociation rate coefficient, kd , and the fractal dimension for dissociation, Df d , for a singlefractal analysis, and (c) the dissociation rate coefficients, kd1 and kd2 , and the fractal dimensions for dissociation, Df d1 and Df d2 , for a dual-fractal analysis are given in Table 4. Figure 3c shows the binding of ERβ liganded with 17βestradiol in solution to SRC3601–762 immobilized on the surface of a sensor chip with an anti-GST antibody. A singlefractal analysis is adequate to describe the binding as well as the dissociation kinetics. The values of (a) the binding rate coefficient, k, and the fractal dimension, Df , for a singlefractal analysis, and (b) the dissociation rate coefficient, kd , and the fractal dimension for dissociation, Df d , for a singlefractal analysis are given in Table 4.

H.D. Butala, A. Sadana / Journal of Colloid and Interface Science 263 (2003) 420–431

(a)

(b)

(c)

(d)

429

Fig. 3. Binding and dissociation phases for the interaction of unliganded and liganded ERβ in solution with SRC3601–762 immobilized on a sensor chip surface [13]: (a) ERβ + no ligand; (b) ERβ + WAY164397; (c) ERβ + 17β-estradiol; (d) ERβ + ICI 182,780. Table 4 Rate coefficients and fractal dimensions for the binding and the dissociation phase for ERβ in the absence of ligand and in the presence of 1.0 µM 17β-estradiol, WAY 164397, or ICI-182,780 in solution to SRC3601–762 immobilized on a sensor chip surface [13] Analyte in solution ERβ + no ligand ERβ + WAY164397 ERβ + 17β-estradiol ERβ + ICI 182,780

k

Df

kd

kd1

kd2

Df d

Df d1

Df d2

3.20 ± 0.18 16.4 ± 0.48 29.6 ± 0.69 2.95 ± 0.02

1.27 ± 0.09 1.79 ± 0.05 1.83 ± 0.04 1.22 ± 0.01

16.5 ± 0.66 12.7 ± 1.05 21.3 ± 0.91 7.10 ± 0.21

na 8.57 ± 0.52 na na

na 35.2 ± 0.10 na na

2.37 ± 0.02 2.15 ± 0.07 2.22 ± 0.03 2.16 ± 0.03

na 1.96 ± 0.08 na na

na 2.52 ± 0.01 na na

Finally, Fig. 3d shows the binding of ERβ liganded with ICI 182,780 in solution to SRC3601–762 immobilized on the surface of a sensor chip with an anti-GST antibody. A singlefractal analysis is adequate to describe the binding as well as the dissociation kinetics. The values of (a) the binding rate coefficient, k, and the fractal dimension, Df , for a singlefractal analysis, and (b) the dissociation rate coefficient, kd , and the fractal dimension for dissociation, Df d , for a singlefractal analysis are given in Table 4. Table 4 indicates that for a single-fractal analysis for the binding of ERβ in the absence and in the presence of ligands, an increase in the fractal dimension, Df , leads to an increase in the binding rate coefficient, k. For the data presented in Tables 1 and 2, the binding rate coefficient, k, is given by k = (0.95 ± 0.25)Df5.30±0.63.

(3a)

For the data presented, the binding rate coefficient, k, is very sensitive to the degree of heterogeneity that exists on the surface, as noted by the high value of the exponent. The KA value increases with an increase in the fractal dimension ratio, Df /Df d . For the three data points presented in Table 4, the KA may be given by KA = (3.12 ± 1.47)(Df /Df d )4.06±1.16.

(3b)

The KA value does increase with an increase in the (Df / Df d ) ratio. In this case, the KA value is very sensitive to the ratio of the fractal dimensions. Once again, the above relation suggests that if one is interested in high KA values, then one should attempt to have high values of heterogeneity on the surface during the binding phase (Df ) and low values of the degree of heterogeneity during the dissociation phase (Df d ). This may or may not always be easy to control, as

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indicated above, since the binding phase precedes the dissociation phase, and some degree of heterogeneity present during the binding phase may carry over to the dissociation phase. This, of course, depends on the basic mechanisms involved in causing the degree of heterogeneity on the sensor chip surface. The arguments presented previously apply here, too, so they are not repeated here. The major difference between the present case and the case presented previously is that here the KA exhibits a much higher order of dependence on the ratio of the fractal dimensions (equal to 4.06) as compared to the previous case. No explanation is offered at present to explain this except that we are looking at two different types of system. Both of them have, however, been analyzed by the SPR biosensor system, with one of the components immobilized on the sensor chip surface. One may as well ask the same questions as posed previously to provide insights into the order of dependence exhibited by the two different systems on the degree of heterogeneity on the surface. Is it (a) the roughness of the sensor chip surface itself or the immobilization of the receptors on the surface, or (b) is the roughness on the surface caused by the reaction itself, or (c) does some other more complex mechanism occur on the surface? Answers to these questions, and others, may well provide insights into the signaling process and other reactions as well.

4. Conclusions A fractal analysis of the binding and dissociation of analyte and estrogen receptor interactions occurring on surface plasmon resonance biosensor surfaces provides a quantitative indication of the state of disorder or the degree of heterogeneity on the biosensor surface. The analysis of both the binding as well as the dissociation steps provide a more complete picture of the reaction occurring on the surface besides providing a value of the constant, KA , which is the ratio of the rate coefficient for the binding, k, and the dissociation, kd , steps. The numerical values of KA obtained may be used along with the values of the rate coefficient for binding and dissociation to classify the analyte–estrogen 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 KA value may also have biomedical implications, in accord with Germain’s [19] analysis. For appropriate signaling to occur (in cells, for example) higher KA (high binding rate coefficient and low dissociation rate coefficient values) are beneficial. However, when phytochemicals are involved, and inappropriate signaling needs to be minimized, then lower KA values are better (low binding rate coefficients and high dissociation rate coefficient values). This may be

done by selective modulation of the (cell) surface nudging the KA values in desired directions. The cell may inherently possess this ability to change the “roughness” of its surface, especially where the interactions are occurring, and thereby modulate the reactions. The fractal dimension value provides a quantitative measure of the degree of heterogeneity that exists on the surface for the analyte–estrogen receptor systems. The degree of heterogeneity for the binding and the dissociation phases is, in general different. Both types of examples are presented wherein either a single- or a dual-fractal analysis is required to describe the binding and/or the dissociation kinetics. The dual-fractal analysis is used only when the single fractal analysis did not provide an adequate fit (sum of least squares less than 0.97). This was done by regression provided by Corel Quattro Pro 8.0 [30]. In accord with the prefactor analysis for fractal aggregates [33], quantitative (predictive) expressions are developed for (a) KA as function of the ratio Df /Df d , (b) k as a function of Df , and (c) kd as a function of Df d . Depending on the final goal, for example, a higher or a lower value of KA may be beneficial for a particular analyte–estrogen receptor system. The fractal dimension for the binding or the dissociation phase is not a typical independent variable, such as analyte concentration in solution or the receptor (estrogen or other) on the biosensor surface that may be directly manipulated. It is estimated from Eqs. (1a), (1b), (1c), and (1d), as the case may be, and one may consider it as a derived variable. The predictive relationships presented for the rate coefficients as a function either of the analyte concentration in solution or of the degree of heterogeneity that exists on the surface (fractal dimension value) provide a means by which these bindings or the dissociation rate coefficients may be manipulated by changing either the analyte concentration in solution or the degree of heterogeneity that exists on the surface. Note that a change in the degree of heterogeneity on the surface would, in general, lead to limitations and leads to an increase in the binding rate coefficient [34]. In our case, this also applies to the dissociation rate coefficient. To the best of the authors’ knowledge this is the first study where the binding and the dissociation rate coefficients are directly related to the fractal dimension that exists on the biosensor surface for analyte–nuclear receptor reactions. Even though the analysis is presented for interactions occurring on biosensor surfaces, they do provide insights into reactions occurring on cellular surfaces. More such studies are required to determine if the binding and dissociation rate coefficients, along with KA , are sensitive to the degree of heterogeneity that exists on the biosensor or cellular surfaces, which may be noted by the high orders of dependence. As indicated above, the fractal surface (roughness) leads to turbulence, enhances mixing, decreases diffusional limitations, and leads to an increase in the binding rate coefficient. For this to occur the characteristic length of this turbulent boundary layer may have to extend a few monolayers above

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the sensor surface to affect bulk diffusion to and from the surface. However, given the extremely laminar flow regimes in most biosensors, this may not actually take place. A fractal surface is characterized by grooves and ridges, and this surface morphology may lead to eddy diffusion. This eddy diffusion can then help to enhance the mixing and extend the characteristic length of the boundary layer to affect the bulk diffusion to and from the surface. For cellular surfaces involving analyte–nuclear receptor binding and dissociation reactions, this provides extra flexibility by which these reactions may be controlled. Cells may be induced or otherwise (if they have the ability to do so as indicated above, or have lost the ability to do so) to modulate the degree of heterogeneity that exists on their surfaces in desired directions. The analysis should encourage cellular experimentalists, particularly those dealing with analyte–nuclear receptor reactions, to pay increasing attention to the nature of the surface and how it may be modulated to control cellular analyte–nuclear receptor reactions in desired directions. It should be borne in mind that different laboratories use different technologies or slightly different technologies or experimental designs to analyze the affinity of ligands or cofactors to target proteins (or analytes) of interest. Surely, the comparison of data between different technologies and experimental designs and conclusions therefrom should be done with great caution. Ideally, one should compare affinities of ligands or cofactors to a particular target protein (or analyte) analyzed. One recognizes that in vitro methods do not mimic the microenvironment of the target protein (or analyte) analyzed. This would have an effect on the affinity of ligands, cofactors, and other components with which the target protein interacts. In vitro methods cannot be viewed as anything other than a diagnostic tool. The present analysis is of value in that it provides pros and cons of different technologies. This makes the user of the technology aware of the quality of the data generated and what can be done to improve the analysis. References [1] J.P. Sumpter, S. Joblin, Environ. Health Perspect. 103 (1995) 173–178. [2] J.J. Bull, W.H.N. Gutzke, D. Crews, Gen. Comp. Endocrinol. 70 (1988) 425–428. [3] E. Carlsen, A. Giwercman, N. Keiding, N.E. Skakkebaek, Environ. Health Perspect. 103 (1995) 137–139.

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