Directory UMM :Data Elmu:jurnal:B:Biosystems:Vol59.Issue1.2001:

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A single-fractal analysis of cellular analyte – receptor binding

kinetics utilizing biosensors

Anand Ramakrishnan, Ajit Sadana *

Chemical Engineering Department,Uni6ersity of Mississippi,134Anderson Hall,Mississippi,MS38677-9740,USA

Received 18 January 2000; received in revised form 16 October 2000; accepted 30 October 2000

Abstract

A fractal analysis of a confirmative nature only is presented for cellular analyte – receptor binding kinetics utilizing biosensors. Data taken from the literature can be modeled by using a single-fractal analysis. Relationships are presented for the binding rate coefficient as a function of the fractal dimension and for the analyte concentration in solution. In general, the binding rate coefficient is rather sensitive to the degree of heterogeneity that exists on the biosensor surface. It is of interest to note that examples are presented where the binding coefficient,k exhibits an increase as the fractal dimension (Df) or the degree of heterogeneity increases on the surface. The predictive

relationships presented provide further physical insights into the binding reactions occurring on the surface. These should assist in understanding the cellular binding reaction occurring on surfaces, even though the analysis presented is for the cases where the cellular ‘receptor’ is actually immobilized on a biosensor or other surface. The analysis suggests possible modulations of cell surfaces in desired directions to help manipulate the binding rate coefficient (or affinity). In general, the technique presented is applicable for the most part to other reactions occurring on different types of biosensor or other surfaces. © 2001 Elsevier Science Ireland Ltd. All rights reserved.

Keywords:Cellular analyte – receptor binding; Fractals; Biosensors; Binding rate coefficient

www.elsevier.com/locate/biosystems

1. Introduction

Sensitive detection systems (or sensors) are re-quired to distinguish a wide range of substances. Sensors find application in the areas of biotech-nology, physics, chemistry, medicine, aviation, oceanography and environmental control. These sensors or biosensors can be used to monitor analyte – receptor reactions in real time (Myszka

et al., 1997). The importance of providing a better understanding of the mode of operation of biosensors to improve their sensitivity, stability and specificity has been emphasized (Scheller et al., 1991). A particular advantage of this method is that no reactant labeling is required. However, for the binding interaction to occur, one of the components has to be bound or immobilized on a solid surface. This solid surface may, for example, be a biosensor or cell surface. This often leads to mass transfer limitations and subsequent complexities.

* Corresponding author. Tel./fax: +1-601-232-7023.

E-mail address:[email protected] (A. Sadana).

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Weiping et al. (1999) have very recently indi-cated that the assembly and architecture of molecules at an interface will significantly influ-ence the reactions at the surfaces of biosensors and other biomaterials. The solid-phase im-munoassay technique represents a convenient method for the separation and/or detection of reactants (e.g., antigen) in a solution because the binding of antigen to an antibody-coated surface (or vice versa) is sensed directly and rapidly. There is a need to characterize the reactions oc-curring at the biosensor surface as well as other receptor-coated surfaces (such as cell surfaces) in the presence of diffusional limitations that are inevitably present in these types of systems. It is our intention to further develop the knowledge on analyte – receptor binding kinetics for biosensor applications, and to extend and apply it to provide insights into cellular analyte – receptor re-actions. Some examples where analyte – receptor reactions are analyzed by biosensors are presented.

van Cott et al. (1994) emphasize that there is a critical need to develop serologic tools predictive of antibody function. This applies both to in vitro as well as to in vivo studies. For example, these authors emphasize that antibodies directed to-wards the V3 loop of the envelope glycoprotein gp 20 of HIV-1 is of importance due to its preva-lence in natural infection and its ability to neutral-ize HIV-1 in vitro. Thus, these authors utilneutral-ized surface plasmon resonance and biosensor technol-ogy to analyze the binding and dissociation kinet-ics of V3-specific antibodies with biosensor matrix immobilized recombinant-gp 120. They emphasize that biosensor immobilized V3 peptides were found to mimic their conformational structure in solution.

Alphaviruses pose a significant threat to human health and cause a wide variety of diseases, such as arthralgia, myalgia, and encephalitis (Brynes and Griffin, 1998). These authors emphasize that a better understanding of the cellular receptors used by the alphaviruses would provide a clearer insight into the pathogenesis of these viruses, besides leading to the design of effective (‘live-at-tenuated’) vaccines against them. The binding of Sindbis virus to cell surface heparan sulfate

im-mobilized on a biosensor surface was analyzed. They indicate that glycosaminoglycan heparan sulfate participates in the binding of Sindbis virus to cells. In its absence, the binding of virus to the cell is diminished, though it still does occur.

The influence of a synthetic peptide adhesion epitope as an antimicrobial agent has been ana-lyzed recently using a biosensor (Kelly et al., 1999). These authors indicate that an early step in microbial infection is the adherence of binding of specific microbial adhesins to the mucosa of dif-ferent tracts, such as oro-intestinal, nasorespira-tory, or genitourinary. Utilizing a surface plasmon resonance biosensor they attempted to inhibit the binding of cell surface adhesin of

Streptococcus mutans to salivary receptors in vitro. They utilized a synthetic peptide, p1025, which corresponded to residues 1025 – 1044 of the adhesin. The two residues Q1025 and E1037 that contributed to the binding were identified by site-directed mutagenesis. They indicate that this tech-nique of utilizing peptide inhibitors of adhesion may be utilized to control other microorganisms in which adhesins are involved.

Though in the analysis to be presented, we will emphasize cellular reactions occurring on biosen-sor surfaces, the analysis is in general applicable to ligand – receptor and analyte – receptorless sys-tems for biosensor and other applications. Exter-nal diffusioExter-nal limitations play a role in the analysis of immunodiagnostic assays (Giaver, 1976; Bluestein et al., 1987; Eddowes, 1987, 1988; Place et al., 1991; Glaser, 1993; Fischer et al., 1994). The influence of diffusion in such systems has been analyzed to some extent (Stenberg and Nygren, 1982; Nygren and Stenberg, 1985; Sten-berg et al., 1986; Place et al., 1991; Sadana and Sii, 1992; Sadana and Beelaram, 1994, 1995).

Kopelman (1988) indicates that surface diffu-sion-controlled reactions that occur on clusters or islands are expected to exhibit anomalous and fractal-like kinetics. These kinetics exhibit anoma-lous reaction orders and time-dependent (e.g., binding) rate coefficients. Fractals are disordered systems and the disorder is described by non-inte-gral dimensions (Pfeifer and Obert, 1989). These authors further indicate that as long as surface irregularities show scale invariance that is

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dilata-tional symmetry they can be characterized by a single number, the fractal dimension. This means that the surface exhibits self-similarity over cer-tain length scales. In other words, the structure exhibited at the scale of the basic building blocks is reproduced at the level of larger and larger conglomerates. The fractal dimension is a global property and is insensitive to structural or mor-phological details (Pajkossy and Nyikos, 1989; Markel et al., 1991). These authors indicate that fractals are scale self-similar mathematical objects that possess non-trivial geometrical properties. Furthermore, these authors indicate that rough surfaces, disordered layers on surfaces, and porous objects all possess fractal structure.

A consequence of the fractal nature is a power-law dependence of a correlation function (in our case the analyte – receptor on the cell or biosensor surface) on a coordinate (e.g., time). Rizhalla et al. (1999) have recently analyzed the influence of the fractal character of model substances on their reactivity at solid – liquid interfaces. These authors emphasize that surface characteristics and com-plexity significantly determine and control the het-erogeneous reactions at interfaces. They emphasize that fractal systems accommodate structure within structure. They occupy more space than ordered systems. In effect, the fractal dimension is a measure of the space-filling ability of a system. It is worth mentioning that fractal growth phenomena are prevalent in natural sys-tems (Viscek, 1989), and the build-up of the ana-lyte – receptor complex on a biosensor (cellular) surface is an example of this process.

We present in this paper a fractal analysis of the binding of an analyte in solution to a cellular receptor immobilized on biosensor and other sur-faces. The emphasis is to promote the understand-ing of cell-surface reactions. The fractal approach is not new and has been used previously in the studies of immunosensors and phenomena in membranes (Tam and Tremblay, 1993). This is an early attempt at a more extended application of fractal analysis to the investigation of analyte – re-ceptor binding kinetics for biosensors with the eventual goal of providing a better understanding of these reactions on cell surfaces. Similarities with immunoassay kinetics are also discussed

whenever appropriate. Our analysis is performed on data available in the literature. The fractal analysis is one way by which one may elucidate the time-dependent binding rate coefficients and the heterogeneity that exists on the biosensor or cell surface.

2. Theory

An analysis of the binding kinetics of antigen in solution to antibody immobilized on a biosensor surface is available (Sadana and Sii, 1992). There are similarities in the binding of antigen in solu-tion to antibody immobilized on a biosensor sur-face and cellular antigen – antibody or, in general, analyte – receptor binding. This applies also to the case when the (cellular) receptor is immobilized on a biosensor surface (van Cott et al., 1994; Brynes and Griffin, 1998; Kelly et al., 1999). Here we present a method of estimating actual fractal dimension values for cellular antigen – antibody, or in general analyte – receptor binding kinetics observed in biosensor applications. The selection of the binding data is constrained by whatever is available in the literature. Specific systems were chosen that their binding behavior could be de-scribed by a single-fractal analysis.

The fractal dimension represents the state of disorder of the system. In our case, it represents the degree of heterogeneity on the cellular surface. When a single-fractal analysis is utilized that means that there is a single degree of heterogene-ity on the surface, and this remains constant during the course of the reaction. The binding of the analyte in solution to the receptor on the surface is characterized by a single binding rate coefficient that remains constant during the course of a reaction. In essence, a single-fractal analysis is a two-parameter model.

2.1. Variable binding rate coefficient

Kopelman (1988) has recently indicated that classical reaction kinetics are sometimes unsatis-factory when the reactants are spatially con-strained on the microscopic level by either walls, phase boundaries, or force fields. Such

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heteroge-neous reactions, e.g., bioenzymatic reactions that occur at interfaces of different phases exhibit frac-tal orders for elementary reactions and rate coeffi-cients with temporal memories. In such reactions, the rate coefficient exhibits a form given by

k1=k´t−b_, ₀₀_b₀_{1 (}_t_E₁₎ ₍₁₎

Note that Eq. (1) fails at short times. In gen-eral, k1 depends on time, whereas k%₌_k1₍_t₌₁₎ does not. Kopelman (1988) indicates that in three dimensions (homogeneous space), b equals zero. This is in agreement with the results obtained in classical kinetics. Also, with vigorous stirring, the system is made homogeneous, and bagain equals zero. However, for diffusion-limited reactions oc-curring in fractal spaces,b\0; this yields a time-dependent rate coefficient.

The fractal dimension is a measure of the de-gree of heterogeneity on the surface. An increase in the surface roughness leads to an increase in the fractal dimension on the surface. The diffu-sion-limited binding kinetics of antigen (or anti-body or substrate) in solution to antianti-body (or antigen or enzyme) immobilized on a biosensor surface has been analyzed within a fractal frame-work (Sadana and Beelaram, 1994, 1995). One of the findings, e.g., is that an increase in the surface roughness or fractal dimension leads to an in-crease in the binding rate coefficient. Further-more, experimental data presented for the binding of HIV virus (antigen) to the antibody immobi-lized on a surface displays a characteristic ordered ‘‘disorder’’ (Anderson, 1993). 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, which include heterogeneities on the surface and in solution, diffusion-coupled reaction, time-vary-ing adsorption or bindtime-vary-ing rate coefficients, etc. can be characterized as a fractal system. Recently, the fractal analysis has been extended to include analyte – receptor and analyte – receptorless (protein) systems (Sadana et al., 1995). These ideas have also been very recently extended to the single-strand ssRNA – ssDNA or ssDNA – ssDNA binding observed in hybridization kinetics (Sadana and Vo-Dinh, 1998). For example, only

when the nucleotides on the single stand ribonu-cleic acid (RNA) are complementary to their counterparts on the single-strand deoxyribonu-cleic acid (DNA) does the binding take place.

The diffusion of reactants towards fractal sur-faces has been analyzed (De Gennes, 1982; Pfeifer et al., 1984a,b; Nyikos and Pajkossy, 1986). Havlin (1989) has briefly reviewed and discussed these results. This author (Havlin, 1989) presents an equation that may be utilized to describe the build-up of the analyte – receptor (in our case the analyte – receptor binding complex on cellular sur-faces with the receptor actually immobilized to a biosensor) on a surface during the binding reac-tion. This equation is given below.

2.2. Single-fractal analysis

Havlin (1989) indicates that the diffusion of a particle (in our case an analyte) from a homoge-neous solution to a solid surface (receptor on a cellular surface or immobilized to a biosensor surface) where it reacts to form an analyte – recep-tor complex is given by

[analyte-receptor]8

t(3−Df)/2₌_tp _t₎_tc_time, _t _{less than} _tc

t1/2

tctc time, t greater than tc,

where the analyte – receptor represents the binding complex formed on the surface. Here, Df is the fractal dimension of the surface, andtis time. Eq. (2a) indicates that the concentration of the product [analyte – receptor] on a solid fractal sur-face scales at short and intermediate times scales as analyte – receptortp _{with the coefficient} _p₌

(3−Df)/2 at short time scales and p=1/2 at intermediate time scales. This equation is associ-ated with the short-term diffusional properties of a random walk on a fractal surface. Note that in perfectly stirred kinetics on a regular (non-fractal) structure (or surface), k1, is a constant, i.e., it is independent of time. In other words, the limit of regular structures (or surfaces) and the absence of diffusion-limited kinetics lead tok1being indepen-dent of time. In all other situations, one would

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expect a scaling behavior given byk1k%_t−b_with

−b=pB0. Also, the appearance of the coeffi-cient,p, different fromp=0 is the consequence of two different phenomena, i.e., the heterogeneity (fractality) of the surface and the imperfect mixing (diffusion-limited) condition.

Havlin (1989) indicates that the crossover value may be determined byrc2tc. Above the

charac-teristic length,rc, the self-similarity of the surface is lost. Above, tc, the surface may be considered homogeneous, and ‘regular’ diffusion is now present. 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 in disor-dered media.

It is worthwhile developing a relationship be-tween surface roughness (measured by a fractal exponent, p) and the rate of binding. This is in view of the different (statistical) fractal growth laws that are prevalent in nature. These laws include invasion percolation, kinetic gelation, dif-fusion-limited aggregation (DLA) (Viscek, 1989). These laws or models permit the computer simu-lation of the shape and growth of natural pro-cesses. For example, in the DLA model introduced by Witten and Sander (1981), a ran-domly diffusing particle (seed) collides with a surface and stops. Another particle (from far away) diffuses to the surface and arrives at a site close (adjacent) to the first particle and stops. Another particle follows and so on. In this way clusters are generated and exhibit the random branching and open structure, which are self-simi-lar in nature.

From Eq. (2a) to obtain the rate of binding one takes the time derivative of both sides to yield d[analytereceptor]/dt=kpt

p−1

(2b) This indicates that the rate of binding is directly dependent on the binding rate coefficient, k and the fractal exponent,p. It is worthwhile determin-ing the maximum rate of binddetermin-ing, by settdetermin-ing d2_{[analyte – receptor]}_/_d_t2₌_{0. This yields}

kp(p−1)tp−2₌_0. _(2c)

This is the location of the stationary point, and yieldsp=1. On substituting this in Eq. (2b) yields d[analyte – receptor]/dt=k. This occurs at time,

t=0 which is intuitively correct. It is apparently difficult to confirm the nature of the stationary point by taking higher-order derivatives. Perhaps, another way is possible.

3. Results

At the outset, it is appropriate to indicate that the mathematical approach is straightforward. It is assumed that the fractal approach applies. This may be a limitation. This is one possible explana-tion to analyze the diffusion-limited binding ki-netics assumed to be present in the cellular systems. The parameters thus obtained provide a useful comparison of the different situations. The cellular analyte – receptor binding reaction ana-lyzed is a complex reaction, and the fractal analy-sis via the fractal dimension and the binding rate coefficient provide a useful lumped parameter(s) analysis of the diffusion-limited situation.

Houshamand et al. (1999) have utilized the bacteriophage T7 to characterize the epitopes of the monoclonal antibodies F4, F5, and LT1 di-rected against mouse polyomavirus large T-anti-gen. These authors utilized an optical biosensor instrument to analyze the kinetics of phage parti-cle monoclonal antibody complex formation and dissociation. The authors were careful to use a C1 sensor chip with a flat surface to reduce steric hindrance of binding. They do indicate that the very large size of the phage analyte did lead to diffusional limitations.

All the data analyzed has been taken from the literature. For example, in Fig. 1 the solid squares are the experimental points available in the litera-ture. In this case, it is from the Houshamand et al. (1999) paper. The solid line is our contribution to the analysis wherein we use the fractal analysis to fit the data (Eq. (2a)). In essence, more data regarding different cellular analyte – receptor bind-ing kinetics may be obtained if the (cellular) re-ceptor is immobilized on a biosensor or chip surface and the corresponding (cellular) analyte is in solution. In order to approach the nature of or

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Fig. 1. Binding of 80 nM large T-antigen (15ml) in solution to mAbF4 immobilized on a sensor surface (Houshamand et al., 1999). Influence of mAbF4 concentration (mM): (a) 0, (b) 5.0, (c) 10 and (d) 20. Binding of 80 nM large T-antigen (30ml) in solution to mAbLT1 immobilized on a sensor surface (Houshamand et al., 1999). Influence of mAbLT1 concentration (mM): (e) 0, (f) 50, (g) 200 and (h) 800.

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mimic the cellular surface the biosensor or chip surface may be coated by an appropriate material. Another example of this follows later on in the manuscript wherein the binding of cell surface HLA-A2 in solution to b2m immobilized on an aminosilane (AS) or carboxymethylated dextran (CMD) surface is analyzed (Morgan et al., 1998). Fig. 1(a) – (d) shows the binding of 80 nM large T-antigen in solution to different concentrations (0 – 20 mM) of mAbF4 immobilized on a sensor chip. A single-fractal analysis is sufficient to ade-quately describe the binding kinetics. Table 1 shows the values of the binding rate coefficient,k

and the fractal dimension, Df obtained using Sigmaplot (1993) to fit the data. The equation [analytereceptor]=kt

(3−Df)/2 _(2d)

was used to obtain the value of k and Df for a

single-fractal analysis. The Sigmaplot program provided the values of the parameters presented in Table 1. The values of the parameters presented in Table 1 are within 95% confidence limits. For example, the value of k reported for the binding of 80 nM large T-antigen (15ml) in solution to 5.0

mM mAbF4 peptide immobilized on a sensor chip is 31.291.61. The 95% confidence limits indicates that 95% of thekvalues will lie between 29.6 and 32.8.

In all fairness, this is one possible way by which to analyze this analyte – receptor binding data. One might justifiably argue that appropriate mod-eling might be achieved by using a Langmuirian or other approach. The Langmuir approach was

originally developed for gases (Thomson and Webb, 1968). Consider a gas at constant pressure,

p, in equilibrium with a surface. The rate of adsorption is proportional to the gas pressure and to the fraction of the uncovered surface. Adsorp-tion 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. (1983) indicate that other appro-priate ‘‘liquid’’ counterparts of the empirical isotherm equations have been developed. These include counterparts of the Freundlich (Dabrowski and Jaroniec, 1979), Dubinin-Radushkevich (Oscik et al., 1976), and Toth (Ja-roniec and Derylo, 1981) empirical equations. These studies with their known constraints have provided some ‘restricted’ physical insights into the adsorption of adsorbates on different surfaces. The Langmuirian approach may be utilized to model the data presented if one assumes the pres-ence of discrete classes of sites. One might also attempt to model the data to be presented using a fractal (statistical) growth process such as DLA, kinetic gelation, or invasion percolation. How-ever, at present, no such attempt is made.

Fig. 1(e) – (h) shows the binding of 80 nM large T-antigen in solution to different concentrations (0 – 800 mM) of mAbLT1 immobilized on a sensor chip. A single-fractal analysis is sufficient to ade-quately describe the binding kinetics. Table 1(b) shows the values of the binding rate coefficient,k Table 1

Binding rate coefficients and fractal dimensions for the binding of phage particle to monoclonal antibody immobilized over a CM5 sensor surface (Houshamand et al., 1999) – a single-fractal analysis

Analyte in solution/receptor on surface Binding rate Fractal coefficient (k) dimension (Df)

1.7790.07 26.5593.75

(a) 80 nM large T-antigen (15ml)/0mM mAb F4 immobilized on sensor surface

80 nM large T-antigen (15ml)/5mM mAb F4 immobilized on sensor surface 31.1691.61 2.0090.02 2.0390.02 80 nM large antigen (15ml)/10mM mAb F4 immobilized on sensor surface 31.8291.53

2.2390.03 80 nM large antigen (15ml)/20mM mAb F4 immobilized on sensor surface 35.5192.28

27.8890.79

(b) 80 nM large T-antigen (30ml)/0mM mAbLT1 immobilized on sensor surface 1.8190.03 75.8491.79

80 nM large T-antigen (30ml)/50mM mAbLT1 immobilized on sensor surface 2.4990.02 2.7590.02 165.0194.37

80 nM large T-antigen (30ml)/200mM mAbLT1 immobilized on sensor surface

303.3990.5

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Fig. 2. Influence of the fractal dimension,Dfon the binding rate coefficient,k: (a) Increase in the binding rate coefficient,kwith

fractal dimension,Dffor the binding of 80 nM large T-antigen in solution to mAbF4 immobilized on a sensor surface, (b) Increase

in the binding rate coefficient,k with fractal dimension, Df for the binding of 80 nM large T-antigen in solution to mAbLT1

(peptide) immobilized on a sensor surface. Influence of the mAbF4 (peptide) concentration on the binding rate coefficient,k, (c) Increase in the binding rate coefficient,k with the concentration of mAbF4 (peptide) immobilized on a sensor surface and (d) Increase in the fractal dimension,Dfwith the concentration of mAbF4 (peptide) immobilized on a sensor surface.

and the fractal dimension, Df obtained using Sigmaplot (Scientific Graphic Software, Jandel Scientific, 1993). Once again it is of interest to note that an increase in the fractal dimension, Df

on the surface leads to an increase in the binding rate coefficient,k.

Table 1 and Fig. 2(a) indicate that an increase in the fractal dimension,Dfleads to an increase in the binding rate coefficient,k. For the binding of 80 nM large T-antigen in solution to different concentrations of mAbF4 immobilized on a sen-sor chip, an increase in the fractal dimension, Df

by about 20.6% from a value of 1.77 to 2.23 leads to an increase in the binding rate coefficient,kby about 33.7% from a value of 26.6 to 35.5. For these runs the binding rate coefficient is given by

k=(12.990.06)Df1.2690.03_. _(3a)

The above equation predicts the binding rate coefficient, k, values presented in Table 1 reason-ably well. The availability of more data points would more firmly establish this relation, espe-cially in between those presented and over a wider range. In this case, the binding rate coefficient, k

is not very sensitive to the fractal dimension, Df, as noted by the low value of the exponent. The graph is linear and this is reflected in the value of the exponent being close to one (equal to 1.26). Also, the appearance of the linearity is due to the narrow range of Df presented. There is an initial degree of heterogeneity that exists on surface, and this determines the value of k. It is this degree of heterogeneity on the surface that leads to the temporal binding rate coefficient. For a single-fractal analysis, it is assumed that this degree of heterogeneity remains constant during the reac-tion, exhibiting a single Df value.

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For the binding of 80 nM large T-antigen in solution to different concentrations of mAbLT1 immobilized on a sensor chip, an increase in the fractal dimension, Df by about 60.7% from a

value of 1.81 to 2.91 leads to an increase in the value of the binding rate coefficient,kby a factor of 10.9 from a value of 27.9 to 303. For these runs the binding rate coefficient is given by

k=(1.5090.621)Df4.7090.95_. _(3b)

The above equation predicts the binding rate coefficient, k, values presented in Table 1(b) rea-sonably well. There is some deviation in the data. This is reflected in the error estimate for the coefficient as well as in the exponent. The availability of more data points would more firmly establish this relation. However, one is constrained by the availability of data in the literature. Note the very high value of the expo-nent. This, once again, underscores that the bind-ing rate coefficient, k, is very sensitive to the degree of heterogeneity that exists on the surface. On comparing Eq. (3a) and Eq. (3b) one notes that the binding rate coefficient, k is more sensi-tive to the degree of heterogeneity on the surface for mAbLT1 compared to mAbF4. This is be-cause the exponent dependence of k on Df for mAbLT1 and for mAbF4 is 4.70 and 1.26, respectively.

A better fit could be obtained if an equation of the form

Df=a[wild type fragment]

+c[wild type fragment]d _(3c)

was used. However, in this case the equation just uses more parameters. In this case, more data points would definitely be required. The coeffi-cients a, b, c, and d are to be determined by regression.

Fig. 2(c) shows that the binding rate coefficient,

kincreases as the concentration of mAbF4 immo-bilized on the sensor chip increases for the bind-ing of 80 nM large T-antigen in solution to mAbF4. In the 0 – 20 mM of mAbF4 concentra-tion range analyzed, the binding rate coefficient,k

is given by

k=(2.6490.97)[mAbF4]0.09490.037_. _(4a)

This predictive equation fits the values of the binding rate coefficient, k presented in Table 1 reasonably well. More data points are required over a wider range of mAbF4 concentration to define more clearly the above predictive equation. The exponent dependence of the binding rate coefficient on the mAbF4 concentration is rather low, and this indicates that the binding rate coeffi-cient is very weakly dependent on the mAbF4 concentration immobilized on the sensor chip.

Fig. 2(d) shows that the fractal dimension, Df

increases as the mAbF4 concentration immobi-lized on the sensor chip increases. In the 0 – 20mM mAbF4 concentration range analyzed, the fractal dimension, Df is given by

Df=(1.7590.055)[mAbF4]

0.07590.031_. _(4b)

This predictive equation fits the values of the fractal dimension, Df presented in Fig. 2(d) rea-sonably well. More data points are required over a wider range of mAbF4 concentration to define more clearly the above predictive equation. The exponent dependence of the fractal dimension on the mAbF4 concentration is rather low, and this indicates that the fractal dimension is very weakly dependent on the mAbF4 concentration immobi-lized on the sensor chip. As noted above, there is a similar very weak dependence of the binding rate coefficient,kon the mAbF4 concentration. It appears that the mAbF4 concentration on the sensor chip surface in the range analyzed barely affects the fractal dimension and the binding rate coefficient on the surface.

Morgan et al. (1998) have very recently ana-lyzed cell surface HLA class I interactions using an IAsysTM _{biosensor surface. The authors}

em-phasize that this system gives a truer reflection of the events occurring in vivo. They analyzed the kinetics of binding and dissociation of the HLA class I heterotrimetric complexes on whole cells to human b2-microglobulin (b2m). b2m were immo-bilized on the biosensor surface, and the HLA-A2 expressing (T2) cells were in solution. Fig. 3(a) shows the binding of the 4×104 _{721.221 cells in}

complete medium (containing supplements/

growth factors) in solution tob2m immobilized to

the CMD-coated cuvettes at 37°C. A single-frac-tal analysis is sufficient to adequately describe the

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binding kinetics. Table 2 shows the values of the binding rate coefficient, k and the fractal dimen-sion, Df obtained using Sigmaplot (38). Eq. (2c) was used to obtain the value of k and Df for a single-fractal analysis.

Fig. 3(b) shows the binding of 4×104 _{T2 cells}

in complete medium in solution to b2m immobi-lized to the CMD-coated cuvettes at 37°C. Once again, a single-fractal analysis is sufficient to ade-quately describe the binding kinetics. Table 2 shows the values of the binding rate coefficient,k

and the fractal dimension, Df obtained. Fig. 3(c)

shows the binding of 4×104 _{T2 cells in}

incom-plete medium in solution tob2m immobilized on a

CMD matrix. Here too, a single-fractal analysis is sufficient to adequately describe the binding kinet-ics. Table 2 shows the values of the binding rate coefficient, k and the fractal dimension, Df

obtained.

It is of interest to note that as one goes from the incomplete medium to the complete medium, both the fractal dimension, Df and the binding rate coefficient,kexhibit increases. On going from the incomplete medium to a complete medium there is a 11.6% increase in the fractal dimension from a value of Df equal to 1.81 to 2.24, and an

increase in the binding rate coefficient, k by a factor of 2.88 from a value of 2.24 to 6.44. A

Fig. 3. Binding of different types of cells in medium tob2m immobilized to different surfaces: (a) binding of 4×104721.221 cells

in complete medium in solution tob2m immobilized to the CMD matrix, (b) binding of 4×104T2 cells in complete medium in

solution to b2m immobilized to the CMD matrix, (c) binding of 4×104 T2 cells in incomplete medium in solution to b2m

immobilized to the CMD matrix, (d) binding of 4×104 _{T2 cells in solution to}_b

2m immobilized on a sensor chip, (e) binding of

4×104 _{T2 cells pre-incubated with a HLA-A2 specific peptide (AE41) in solution to}_b

2m immobilized on a sensor chip and (f)

binding of 4×104_{T2 cells pre-incubated with a HLA-A2 non-specific peptide (AE43) in solution to}_b

2m immobilized on a sensor

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Table 2

Binding rate coefficients and fractal dimensions for the binding of class I HLA complexes on whole cells to humanb2-microglobulin

(b2m) or BSA (Morgan et al., 1998)

Binding rate

Analyte in solution/receptor on surface Fractal

dimension (Df)

coefficient (k) 0.02890.003

(a) 4×l04_{721.221 cells in complete medium}_/_{CMD-coated cuvettes immobilized with} _0.85₉_0.07

b2m

4×l04_{T2 cells in complete medium}_/_{CMD-coated cuvettes immobilized with}_b

2m 6.4590.09 2.2490.01

2.0290.03 1.8190.01 4×104_{T2 cells in incomplete medium}_/_{carboxylated dextran-coated cuvettes immobilized}

withb2m

(b) 4×104_{T2 cells in incomplete medium}_/_{CMD-coated cuvettes immobilized with}_b

2m 5.3890.12 2.1690.02

1.7190.07

4×104_{T2 cells in incomplete medium}₊_{HLA-A2 specific peptide (AE41)}_/_CMD-coated _1.84₉_0.03

cuvettes immobilized withb2m

5.0790.04

4×104_{T2 cells in incomplete medium}₊_{HLA-A2 non-specific peptide} _1.93₉_0.006

(AE43)/CMD-coated cuvettes immobilized withb2m

b2m

4×104_{721.221 cells in complete medium}_/_{AS-coated cuvettes immobilized with}_b

2m 0.1890.02 1.2590.065

4×104_{T2 cells in complete medium}_/_{CMD-coated cuvettes immobilized with}_b

2m 6.7390.08 2.2690.009

2.7290.009 11.590.14

4×104_{T2 cells in complete medium}_/_{AS-coated cuvettes immobilized with}_b 2m

25.690.25

(d) 4×104_{T2 cells in complete medium}_/_{AS-coated cuvettes immobilized with BSA} _2.64₉_0.008

4.8290.33 2.4190.06 4×104_{T2 cells in complete medium}_/_{caboxymethylated dextran-coated cuvettes}

immobilized with BSA

4×104_{721.221 cells in complete medium}_/_{AS-coated cuvettes immobilized with BSA} _24.7₉_0.48 _2.59₉_0.02

similar result was obtained for the binding of large T-antigen in solution to either mAbF4 or mAbLT1 immobilized on a sensor chip (Thomson and Webb, 1968). Apparently, an increase in the degree of heterogeneity (Df) on the cellular surface

leads to an increase in the binding rate coefficient,

Morgan et al. (1998) wanted to determine if an exogenous peptide could change the binding of cell surface class I molecules in a sequential de-pendent manner. These authors analyzed the binding of T2 cells, T2 cells pre-incubated with a HLA-A2 non-specific peptide (AE43), and T2 cells pre-incubated with a HLA-A2 specific pep-tide (AE41) in solution to b2m immobilized on a

surface. Fig. 3(d) shows the binding of 4×104

T2 cells in solution at 37°C tob2m immobilized on a sensor chip. A single-fractal analysis is sufficient to adequately describe the binding kinetics. Table 2 shows the values of the binding rate coefficient,

k and the fractal dimension, Df obtained using Sigmaplot (Scientific Graphing Software, Jandel Scientific, 1993). Eq. (2d) was used to obtain the value of kand Df.

Fig. 3(e) shows the binding of 4×104

T2 cells pre-incubated with a HLA-A2 specific peptide (AE41) in solution to b2m immobilized on a

sen-sor chip. Once again, a single-fractal analysis is sufficient to adequately describe the binding kinet-ics. Table 2 shows the values of the binding rate coefficient,kand the fractal dimension,Df. It is of interest to note that the addition of the specific peptide (AE41) to T2 cells in the pre-incubated mixture leads to a decrease in the fractal dimen-sion and to a decrease in the binding rate coeffi-cient when compared to the case when T2 cells alone are used. A 15.3% decrease in the fractal dimension from a value of 2.16 to 1.84 as one goes from the T2 case to the T2+AE41 case leads to a decrease in the binding rate coefficient,

k by a factor of 68.2% from a value of 5.38 to a value of 1.71.

Fig. 3(f) shows the binding of 4×104 _{T2 cells}

pre-incubated with a HLA-A2 non-specific pep-tide (AE43) in solution to b2m immobilized on a sensor chip. Once again, a single-fractal analysis is sufficient to adequately describe the binding kinet-ics. Table 2 shows the values of the binding rate

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Fig. 4. Influence of the fractal dimension,Dfon the binding

rate coefficient,k: (a) increase in the binding rate coefficient,k

with an increase in the fractal dimension,Dffor the binding of

HLA-A2 expressing (T2) cells in solution tob2m immobilized

on a sensor chip and (b) increase in the binding rate coeffi-cient,k with an increase in the fractal dimension,Df for the

binding of T2 cells in solution tob2m immobilized on a sensor

chip.

chip in the presence of a complete and an incom-plete medium, and in the absence of a medium, the binding rate coefficient, k is given by

k=(0.07390.007)Df

5.6190.13_. _(5a)

Fig. 4(a) shows that the above equation fits the binding rate,kvalues presented in Table 2 reason-ably well. The availability of more data points would more firmly establish this relation. Note the very high value of the exponent. This once again underscores that the binding rate coeffi-cient, k is very sensitive to the degree of hetero-geneity that exists on the surface.

For the binding of T2 cells in solution (either alone, or pre-incubated with either a non-specific (AE43) or specific peptide (AE41) tob2m

immobi-lized on a sensor chip, the binding rate coefficient,

k is given by

k=(0.06490.051)Df5.9394.98_. _(5b)

Fig. 4(b) shows that the above equation fits the binding rate coefficient, k values presented in Table 2 reasonably well. There is deviation in the data. This is indicated in the estimated values of the exponent as well as the coefficient. More data points are required to more firmly establish a quantitative relationship in this case. Perhaps, one may require a functional form for kthat involves more parameters as a function of the fractal di-mension. One possible form could be

k=a[Df]

b₊_c_[_D

d_. _(5c)

Here a, b, c, and d are the coefficients to be determined by regression. This functional form may describe the data better, but one would need, as indicated above, more data points to justify the use of a four-parameter equation.

Morgan et al. (1998) have also analyzed the binding between cell surface HLA-A2 and b2m immobilized either on an AS or a CMD surface. coefficient,kand the fractal dimension,Df. It is of

interest to note that the addition of the non-spe-cific peptide (AE43) to T2 cells in the pre-incu-bated mixture once again, leads to a decrease in the fractal dimension,Dfand to a decrease in the

binding rate coefficient, k when compared to the case when T2 cells alone are used. A 5.8% de-crease in the fractal dimension from a value of 5.38 to 5.07 leads to a decrease in the binding rate coefficient, k by 10.6% from a value of 2.16 to 1.93.

For the binding of HLA-A2 expressing (T2) cells in solution to b2m immobilized on a sensor

Fig. 5. Binding of different types of cells tob2m immobilized to different surfaces (Morgan et al., 1998): (a) binding of 4×104

721.221 (non-HLA class I expressing) cells in complete medium in solution tob2m immobilized on a CMD surface, (b) binding of

4×104 _{721.221 cells in complete medium in solution to} _b

2m immobilized on an AS surface, (c) binding of 4×104 T2 cells in

complete medium in solution tob2m immobilized on a dextran-coated surface, (d) binding of 4×104T2 cells in complete medium

in solution tob2m immobilized on an AS-coated surface, (e) binding of 4×104T2 cells in complete medium in solution to BSA

immobilized on an AS surface, (f) binding of 4×104_{T2 cells in solution to BSA immobilized on a dextran-coated cuvette and (g)}

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These authors indicate that the two surfaces im-part different biochemical properties to the bind-ing. The AS surface is planar, and it permits the ligand to closely approach the receptor on the surface. The CMD surface is a flexible hydrogel, and this according to the authors may inhibit the analyte to closely approach the receptor on the surface.

Fig. 5(a) shows the binding of 4×104 _721.221

(non-HLA class I expressing) cells in complete medium in solution to b2m immobilized on a CMD surface. Once again, a single-fractal analy-sis is sufficient to adequately describe the binding kinetics. The values of the binding rate coefficient,

kand the fractal dimension,Dfare given in Table 2. Fig. 5(b) shows the binding of 4×104 _721.221

cells in complete medium in solution to b2m im-mobilized on an AS surface. A single-fractal anal-ysis is sufficient to adequately describe the binding kinetics. The values of the binding rate coefficient,

kand the fractal dimension,Dfare given in Table 2. For the results presented in Fig. 5(a) and (b), it is of interest to note that as one goes from the CMD surface to the AS surface, the fractal di-mension, Df increases by a factor of 2.16 from a

value of 0.085 to 0.184, and the binding rate coefficient, k exhibits only a very small change (decrease) in value from 1.27 to 1.25.

Fig. 5(c) shows the binding of 4×104 _{T2 cells}

in complete medium in solution to b2m immobi-lized on a dextran-coated surface. A single-fractal analysis is sufficient to adequately describe the binding kinetics. The values of the binding rate coefficient, k and the fractal dimension, Df are given in Table 2. It is of interest to compare the binding of T2 (HLA-A2 expressing) cells and 721.221 cells (non-HLA class I expressing) tob2m immobilized on a dextran surface. As one goes from the 721.221 cells to the T2 cells there is an increase in the fractal dimension by a factor of 1.78 from a value of 1.27 to 2.26, and an increase in the value of the binding rate coefficient,kby a factor of 79.2 from a value of 0.085 to 6.73. Note that the fractal dimension and the binding rate coefficient exhibit changes in the same direction. Apparently, in the case of the T2 cells (when compared to the 721.221 cells), an increase in the

degree of heterogeneity on the surface (increase in

Df) leads to a very significant increase in the binding rate coefficient.

Fig. 5(d) shows the binding of 4×104 _{T2 cells}

in complete medium in solution to b2m immobi-lized on an AS-coated surface. Once again, a single-fractal analysis is sufficient to adequately describe the binding kinetics. The values of the binding rate coefficient and the fractal dimension are given in Table 2. Once again, as done above it is of interest to compare the binding of T2 and 721.221 cells to b2m immobilized on an AS sur-face. As one goes from the 721.221 cells to the T2 cells, there is, an increase in the fractal dimension by a factor of 2.2 from a value of 1.25 to 2.71, and to a very significant increase in the binding rate coefficient, kby a factor of 62.5 from a value of 0.184 to 11.5. The above analysis indicates that, at least for the data presented, the binding rate coefficient for the T2 cells is much higher than that of the 721.221 cells for the CMD and the AS surfaces analyzed. This may be due to the HLA-A2 expressing ability of the T2 cells and the non-HLA class I expressing 721.221 cells or due to some other factors, besides the degree of het-erogeneity that exists on the surface.

Fig. 5(e) shows the binding of 4×104 _{T2 cells}

in complete medium in solution to bovine serum albumin (BSA) immobilized on an AS-coated sur-face. A single-fractal analysis is sufficient to ade-quately describe the binding kinetics. The values of the binding rate coefficient, k and the fractal dimension, Df are given in Table 2. Fig. 5(f)

shows the binding of 4×104

T2 cells in solution to BSA immobilized on a dextran-coated cuvette. Here too, a single-fractal analysis is sufficient to adequately describe the binding kinetics. The val-ues of the binding rate coefficient and the fractal dimension are given in Table 2. Fig. 5(g) shows the binding of 4×104 _{T2 cells in incomplete}

medium in solution to BSA immobilized on an AS-coated surface. A single-fractal analysis is sufficient to adequately describe the binding kinet-ics. The value of the binding rate coefficient, k

and the fractal dimension,Dfare given in Table 2. For the binding of T2 and 721.221 cells in complete medium and in solution tob2m immobi-lized on a CMD or an AS surface, the binding rate coefficient, k is given by

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k=(0.03290.021)Df6.1690.75_. _(6a)

Fig. 6(a) shows that the above equation fits the binding rate coefficient, k values presented in Table 2 reasonably well. There is some deviation in the data, and this is reflected in the error in the estimated value of the coefficient. The availability of more data points would more firmly establish this relation. Note the very high value of the exponent. This once again underscores that the binding rate coefficient is very sensitive to the degree of heterogeneity that exists on the surface. No theoretical explanation is offered to explain this very high exponent dependence.

For the binding of T2 cells in solution in com-plete and in incomcom-plete medium to BSA immobi-lized either to a CMD or an AS surface, the binding rate coefficient, kis given by

k=(1.57E−0790.5E−07)Df19.694.72

. (6b)

Fig. 6(b) shows that the above equation fits the binding rate coefficient, k values presented in Table 2 reasonably well. There is some deviation in the data, and this is reflected in the error of in the estimated value of the coefficient as well as in the exponent. The availability of more data points would more firmly establish this relation. Note the extremely high value of the exponent. This once again underscores that the binding rate co-efficient, k is very sensitive to the degree of het-erogeneity that exists on the surface. No theoretical explanation is offered to explain this very high exponent dependence.

4. Conclusion

A fractal analysis of a confirmative nature only of the binding of cellular analyte in solution to a cellular receptor immobilized on a biosensor sur-face provides a quantitative indication of the state of disorder (fractal dimension, Df) or the degree of heterogeneity and the binding rate coefficient,k

on the surface. TheDf value provides a quantita-tive measure of the degree of heterogeneity on the biosensor surface for the cellular analyte – receptor binding systems analyzed. Although the cellular receptor is immobilized to a biosensor surface, it does provide insights into the cellular analyte – re-ceptor binding reaction. A single-fractal analysis is utilized to provide an adequate fit. This was done by the regression analysis provided by Sigmaplot (Scientific Graphic Software, Jandel Scientific, 1993).

For the examples analyzed, the binding rate coefficient, k is related to the degree of hetero-geneity on the surface or the fractal dimension,Df

in accord with the prefactor analysis for aggre-gates (Sorenson and Roberts, 1977). The fractal dimension,Df, is not a classical independent vari-able such as analyte concentration. Nevertheless, the expressions for the binding rate coefficient obtained as a function of the fractal dimension indicate the sensitivity of k on Df. This is clearly brought out, in general, by the high order of the binding rate coefficient on the fractal dimension. For example, the order of dependence of the binding rate coefficient, k, on the fractal

dimen-Fig. 6. Influence of the fractal dimension,Dfon the binding

rate coefficient,k: (a) increase in the binding rate coefficient,k

with an increase in the fractal dimension,Dffor the binding of

T2 and 721.221 cells in complete medium and in solution to b2m immobilized on a CMD or an AS surface and (b) increase

in the binding rate coefficient,kwith an increase in the fractal dimension,Dffor the binding of T2 cells in complete and in

incomplete medium to BSA immobilized either to a CMD or an AS surface.

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sion, Df, ranges from 1.26 for the binding of 80 nM large T-antigen in solution to different con-centrations (0 – 800mM) of mAbLT1 immobilized on a sensor chip (Houshamand et al., 1999) to 119.6 for the binding of T2 cells in solution in complete and in incomplete medium to BSA im-mobilized either to a CMD or an AS surface (Jaroniec and Derylo, 1981). The expressions de-veloped for the binding rate coefficient,k, and the fractal dimension, Df, provide further physical insights into the analyte – receptor reactions occur-ring on the surface. In general, both of these types of expressions exhibit a low dependence on the analyte concentration in solution, indicating that they are not very sensitive to the analyte concen-tration in solution.

To the best of these authors’ knowledge this is the first study where the binding rate coefficient has been directly related to the fractal dimension for cellular analyte – receptor reactions occurring on biosensor surfaces. Although the reaction is occurring on a biosensor or other surfaces, the analysis provides physical insights into cellular analyte – receptor reactions. The quantitative ex-pressions developed for the analyte – receptor sys-tems should assist in understanding cellular analyte – receptor reactions, in general. More such studies are required to determine if the binding rate coefficient is sensitive to the degree of hetero-geneity that exists on the cellular surface. For cellular analyte – receptor binding reactions, this provides an extra dimension of flexibility by which these reactions may be controlled. Cells may be induced or otherwise to modulate their surfaces in desired directions. The analysis should encourage cellular experimentalists to pay increas-ing attention to the nature of the surface, and how it may be modulated to control cellular analyte – receptor reactions in desired directions.

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A.Ramakrishnan,A.Sadana/BioSystems59 (2001) 35 – 51 46

Fig. 4. Influence of the fractal dimension,Dfon the binding

rate coefficient,k: (a) increase in the binding rate coefficient,k

with an increase in the fractal dimension,Dffor the binding of

HLA-A2 expressing (T2) cells in solution tob2m immobilized

on a sensor chip and (b) increase in the binding rate coeffi-cient,k with an increase in the fractal dimension,Df for the

binding of T2 cells in solution tob2m immobilized on a sensor

chip.

chip in the presence of a complete and an incom-plete medium, and in the absence of a medium, the binding rate coefficient, k is given by

k=(0.07390.007)Df

5.6190.13_. _(5a)

For the binding of T2 cells in solution (either alone, or pre-incubated with either a non-specific (AE43) or specific peptide (AE41) tob2m

immobi-lized on a sensor chip, the binding rate coefficient,

k is given by

k=(0.06490.051)Df5.9394.98. (5b)

k=a[Df]

b₊_c_[_D

d_. _(5c)

Morgan et al. (1998) have also analyzed the binding between cell surface HLA-A2 and b2m

immobilized either on an AS or a CMD surface. coefficient,kand the fractal dimension,Df. It is of

interest to note that the addition of the non-spe-cific peptide (AE43) to T2 cells in the pre-incu-bated mixture once again, leads to a decrease in the fractal dimension,Dfand to a decrease in the

For the binding of HLA-A2 expressing (T2) cells in solution to b2m immobilized on a sensor

Fig. 5. Binding of different types of cells tob2m immobilized to different surfaces (Morgan et al., 1998): (a) binding of 4×104

721.221 (non-HLA class I expressing) cells in complete medium in solution tob2m immobilized on a CMD surface, (b) binding of

4×104 _{721.221 cells in complete medium in solution to} _b

2m immobilized on an AS surface, (c) binding of 4×104 T2 cells in

complete medium in solution tob2m immobilized on a dextran-coated surface, (d) binding of 4×104T2 cells in complete medium

in solution tob2m immobilized on an AS-coated surface, (e) binding of 4×104T2 cells in complete medium in solution to BSA

immobilized on an AS surface, (f) binding of 4×104_{T2 cells in solution to BSA immobilized on a dextran-coated cuvette and (g)}

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(3)

A.Ramakrishnan,A.Sadana/BioSystems59 (2001) 35 – 51 48

Fig. 5(a) shows the binding of 4×104 _721.221

(non-HLA class I expressing) cells in complete medium in solution to b2m immobilized on a

CMD surface. Once again, a single-fractal analy-sis is sufficient to adequately describe the binding kinetics. The values of the binding rate coefficient,

kand the fractal dimension,Dfare given in Table

2. Fig. 5(b) shows the binding of 4×104 _721.221

cells in complete medium in solution to b2m

im-mobilized on an AS surface. A single-fractal anal-ysis is sufficient to adequately describe the binding kinetics. The values of the binding rate coefficient,

kand the fractal dimension,Dfare given in Table

2. For the results presented in Fig. 5(a) and (b), it is of interest to note that as one goes from the CMD surface to the AS surface, the fractal di-mension, Df increases by a factor of 2.16 from a

value of 0.085 to 0.184, and the binding rate coefficient, k exhibits only a very small change (decrease) in value from 1.27 to 1.25.

Fig. 5(c) shows the binding of 4×104 _{T2 cells}

in complete medium in solution to b2m

immobi-lized on a dextran-coated surface. A single-fractal analysis is sufficient to adequately describe the binding kinetics. The values of the binding rate coefficient, k and the fractal dimension, Df are

given in Table 2. It is of interest to compare the binding of T2 (HLA-A2 expressing) cells and 721.221 cells (non-HLA class I expressing) tob2m

immobilized on a dextran surface. As one goes from the 721.221 cells to the T2 cells there is an increase in the fractal dimension by a factor of 1.78 from a value of 1.27 to 2.26, and an increase in the value of the binding rate coefficient,kby a factor of 79.2 from a value of 0.085 to 6.73. Note that the fractal dimension and the binding rate coefficient exhibit changes in the same direction. Apparently, in the case of the T2 cells (when compared to the 721.221 cells), an increase in the

degree of heterogeneity on the surface (increase in

Df) leads to a very significant increase in the

binding rate coefficient.

Fig. 5(d) shows the binding of 4×104 _{T2 cells}

in complete medium in solution to b2m

immobi-lized on an AS-coated surface. Once again, a single-fractal analysis is sufficient to adequately describe the binding kinetics. The values of the binding rate coefficient and the fractal dimension are given in Table 2. Once again, as done above it is of interest to compare the binding of T2 and 721.221 cells to b2m immobilized on an AS

sur-face. As one goes from the 721.221 cells to the T2 cells, there is, an increase in the fractal dimension by a factor of 2.2 from a value of 1.25 to 2.71, and to a very significant increase in the binding rate coefficient, kby a factor of 62.5 from a value of 0.184 to 11.5. The above analysis indicates that, at least for the data presented, the binding rate coefficient for the T2 cells is much higher than that of the 721.221 cells for the CMD and the AS surfaces analyzed. This may be due to the HLA-A2 expressing ability of the T2 cells and the non-HLA class I expressing 721.221 cells or due to some other factors, besides the degree of het-erogeneity that exists on the surface.

Fig. 5(e) shows the binding of 4×104 _{T2 cells}

shows the binding of 4×104

medium in solution to BSA immobilized on an AS-coated surface. A single-fractal analysis is sufficient to adequately describe the binding kinet-ics. The value of the binding rate coefficient, k

and the fractal dimension,Dfare given in Table 2.

For the binding of T2 and 721.221 cells in complete medium and in solution tob2m

immobi-lized on a CMD or an AS surface, the binding rate coefficient, k is given by

(4)

k=(0.03290.021)Df

6.1690.75_. _(6a)

For the binding of T2 cells in solution in com-plete and in incomcom-plete medium to BSA immobi-lized either to a CMD or an AS surface, the binding rate coefficient, kis given by

k=(1.57E−0790.5E−07)Df 19.694.72

. (6b)

4. Conclusion

of heterogeneity and the binding rate coefficient,k

on the surface. TheDf value provides a

quantita-tive measure of the degree of heterogeneity on the biosensor surface for the cellular analyte – receptor binding systems analyzed. Although the cellular receptor is immobilized to a biosensor surface, it does provide insights into the cellular analyte – re-ceptor binding reaction. A single-fractal analysis is utilized to provide an adequate fit. This was done by the regression analysis provided by Sigmaplot (Scientific Graphic Software, Jandel Scientific, 1993).

For the examples analyzed, the binding rate coefficient, k is related to the degree of hetero-geneity on the surface or the fractal dimension,Df

in accord with the prefactor analysis for aggre-gates (Sorenson and Roberts, 1977). The fractal dimension,Df, is not a classical independent

vari-able such as analyte concentration. Nevertheless, the expressions for the binding rate coefficient obtained as a function of the fractal dimension indicate the sensitivity of k on Df. This is clearly

brought out, in general, by the high order of the binding rate coefficient on the fractal dimension. For example, the order of dependence of the binding rate coefficient, k, on the fractal

dimen-Fig. 6. Influence of the fractal dimension,Dfon the binding

rate coefficient,k: (a) increase in the binding rate coefficient,k

with an increase in the fractal dimension,Dffor the binding of

T2 and 721.221 cells in complete medium and in solution to

b2m immobilized on a CMD or an AS surface and (b) increase

in the binding rate coefficient,kwith an increase in the fractal dimension,Dffor the binding of T2 cells in complete and in

incomplete medium to BSA immobilized either to a CMD or an AS surface.

(5)

A.Ramakrishnan,A.Sadana/BioSystems59 (2001) 35 – 51 50

sion, Df, ranges from 1.26 for the binding of 80

nM large T-antigen in solution to different con-centrations (0 – 800mM) of mAbLT1 immobilized on a sensor chip (Houshamand et al., 1999) to 119.6 for the binding of T2 cells in solution in complete and in incomplete medium to BSA im-mobilized either to a CMD or an AS surface (Jaroniec and Derylo, 1981). The expressions de-veloped for the binding rate coefficient,k, and the fractal dimension, Df, provide further physical

insights into the analyte – receptor reactions occur-ring on the surface. In general, both of these types of expressions exhibit a low dependence on the analyte concentration in solution, indicating that they are not very sensitive to the analyte concen-tration in solution.