Analyte-receptor binding kinetics for biosensor applications

The diffusion-limited binding kinetics of antigen (analyte), in solution with antibody (receptor) immobilized on a biosensor surface, is analyzed within a fractal framework. Most of the data presented is adequately described by a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot. A single example of a dual-fractal analysis is also presented. It is of interest to note that the binding-rate coefficient (k) and the fractal dimension (D_f) both exhibit changes in the same and in the reverse direction for the antigen-antibody systems analyzed. Binding-rate coefficient expressions, as a function of the D_f developed for the antigen-antibody binding systems, indicate the high sensitivity of thek on the D_f when both a single- and a dual-fractal analysis are used. For example, for a single-fractal analysis, and for the binding of antibody Mab 0.5β in solution to gpl20 peptide immobilized on a BIAcore biosensor, the order of dependence on the D_f was 4.0926. For a dual-fractal analysis, and for the binding of 25-100 ng/mL TRITC-LPS (lipopolysaccharide) in solution with polymyxin B immobilized on a fiberoptic biosensor, the order of dependence of the binding-rate coefficients, k₁ and k₂ on the fractal dimensions, D_f1 and D_f2, were 7.6335 and-11.55, respectively. The fractional order of dependence of thek(s) on the D_f(s) further reinforces the fractal nature of the system. Thek(s) expressions developed as a function of the D_f(s) are of particular value, since they provide a means to better control biosensor performance, by linking it to the heterogeneity on the surface, and further emphasize, in a quantitative sense, the importance of the nature of the surface in biosensor performance.     

Antigen-antibody binding kinetics for biosensor applications

The diffusion-limited binding kinetics of antigen (or antibody) in solution to antibody (or antigen) immobilized on a biosensor surface is analyzed within a fractal framework. The fit obtained by a dual-fractal analysis is compared with that obtained from a single-fractal analysis. In some cases, the dual-fractal analysis provides an improved fit when compared with a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot (San Rafael, CA). These examples are presented. It is of interest to note that the state of disorder (or the fractal dimension) and the binding rate coefficient both increase (or decrease, a single example is presented for this case) as the reaction progresses on the biosensor surface. For example, for the binding of monoclonal antibody MAb 49 in solution to surface-immobilized antigen, a 90.4% increase in the fractal dimension (D_f1 toD _f2 ) from 1.327 to 2.527 leads to an increase in the binding rate coefficient (k₁ to k₂) by a factor of 9.4 from 11.74 to 110.3. The different examples analyzed and presented together provide a means by which the antigen-antibody reactions may be better controlled by noting the magnitude of the changes in the fractal dimension and in the binding rate coefficient as the reaction progresses on the biosensor surface.     

Analyte-receptor binding kinetics for different types of biosensors

A fractal analysis is presented for analyte-receptor binding kinetics for different types of biosensor applications. Data taken from the literature may be modeled using a single-fractal analysis, a single- and a dual-fractal analysis, or a dual-fractal analysis. The latter two methods represent a change in the binding mechanism as the reaction progresses on the surface. Predictive relationships developed for the binding rate coefficient as a function of the analyte concentration are of particular value since they provide a means by which the binding rate coefficients may be manipulated. Relationships are presented for the binding rate coefficients as a function of the fractal dimension D _f or the degree of heterogeneity that exists on the surface. When analyte-receptor binding is involved, an increase in the heterogeneity on the surface (increase in D _f ) leads to an increase in the binding rate coefficient. It is suggested that an increase in the degree of heterogeneity on the surface leads to an increase in the turbulence on the surface owing to the irregularities on the surface. This turbulence promotes mixing, minimizes diffusional limitations, and leads subsequently to an increase in the binding rate coefficient. The binding rate coefficient is rather sensitive to the degree of heterogeneity, D _f , that exists on the biosensor surface. For example, the order of dependence on D _f1 is 7.25 for the binding rate coefficient k ₁ for the binding of a Fab fragment of an antiparaquat monoclonal antibody in solution to an antigen in the form of a paraquat analog immobilized on a sensor surface. The predictive relationships presented for the binding rate coefficient and the fractal dimension as a function of the analyte concentration in solution provide further physical insights into the binding reactions on the surface, and should assist in enhancing biosensor performance. In general, the technique is applicable to other reactions occurring on different types of surfaces, such as cell-surface reactions.     

Antigen-antibody diffusion-limited binding kinetics for biosensors

A fractal analysis is made for antigen-antibody binding kinetics for different biosensor applications available in the literature. Both types of examples are considered wherein: (1) the antigen is in solution and the antibody is immobilized on the fiberoptic surface, and (2) the antibody is in solution and the antigen is immobilized on the fiberoptic surface. For example, when the antibody is immobilized on the surface, an increase in the antigenClostridium botulinum toxin A concentration in solution leads to (1) a decrease in the fractal dimension value or state of disorder, and (2) a higher rate constant for binding on the fiberoptic surface. An analysis of the effect of the influence of different parameters on the fractal dimension values for a particular example, such as varying treatments or incubation procedures, helps provide insights into the conformational states and reactions occurring on the fiberoptic surface. The analysis of the different examples taken together provides novel physical insights into the state of “disorder” and reactions occurring on the surface. Such types of analysis should help contribute toward manipulating the reactions occurring on the fiberoptic surfaces in desired directions.     

Size exclusion chromatography on porous fractals

Summary Some porous packings used in chromatography have been claimed to be fractals with a scale of sizes a<l<L, where a is a molecular size and L is the size of the largest pores. For a fractal porous packing, the excluded volume for molecules in solution in the vicinity of the packing surface is directly related to D_f, the fractal dimension of the pore surface (2<D_f<3). Since retention in size exclusion chromatography is itself directly related to this excluded volume, the fractal nature of the packing provides a model of retention in this technique. According to this model there is a linear relationship between log R_s and log(1-K_d), where R_s is the hydrodynamic radius of the solute macromolecules and K_d the distribution coefficient. The fractal dimension is derived from the slope of this plot. Size exclusion chromatographic retention data have been analyzed according to the model. It is found that some HPLC packings are fractals with fractal dimensions ranging from about 2.15 to 2.6, depending on the material. Such a large range of D_f values indicates large variations in the selectivities and domains of applications of the different packings. For some classical gel filtration chromatographic gels, the fractal retention model does not seem to apply.Presented at the 17th International Symposium on Chromatography, September 25–30, 1988, Vienna, Austria.     

Fractal analysis of pit morphology of Inconel alloy 600 in sulphate,nitrate and bicarbonate ion-containing sodium chloride solution at temperatures of 25–100 °C

Pit morphology of Inconel alloy 600 in sulphate (SO₄ ^2-), nitrate (NO₃ ^-) and bicarbonate (HCO₃ ^-) ion-containing 0.5 M sodium chloride (NaCl) solution was analysed in terms of fractal geometry as functions of solution temperature and anion concentration using the potentiostatic current transient technique, scanning electron microscopy, image analysis and ac-impedance spectroscopy. Potentiostatic current transients revealed that the pitting corrosion is facilitated by the increase in solution temperature, irrespective of anion additives, and that it is hindered by the increase in NO₃ ^- and HCO₃ ^- ion concentration, regardless of solution temperature. Above 60 °C, it was also found that the addition of SO₄ ^2- ions impedes pit initiation, but enhances pit growth. The value of fractal dimension D _f of the pits increased with increasing solution temperature and with decreasing NO₃ ^- and HCO₃ ^- ion concentration. Moreover, the value of D _f increased above 60 °C with increasing SO₄ ^2- ion concentration. This is caused by the increase in the ratio of pit perimeter to pit area, implying the formation of pits with micro-branched shape due to the acceleration of the local attack in the pits. From the decrease of the depression parameter with increasing solution temperature, it is inferred that the roughness of the pits increased with increasing solution temperature. In addition, the depression parameter was found to increase with increasing NO₃ ^- and HCO₃ ^- ion concentration. But, above 60 °C, in the case of SO₄ ^2- ion addition, the depression parameter decreased with increasing SO₄ ^2- ion concentration. From the experimental findings, the three-dimensional pit morphology is discussed in terms of the values of D _f of the pits and the depression parameter, with respect to anion concentration and solution temperature.     

Fractal dimension and phase transition of graphene oxide (GO) doped polyacrylamide

Graphene Oxide (GO)- Polyacrylamide composites prepared between 5 and 50 μl GO were performed by Fluorescence Spectroscopy. The phase transition performed on the composites was measured by calculating the critical exponents, β and γ, respectively. In addition, fractal analysis of the composites was calculated by a fluorescence intensity of 427 nm. The geometrical distribution of GO in the composites was calculated based on the power law exponent values using scaling models. While the gelation proceeded GO plates first organized themselves into a 3D percolation cluster with the fractal dimension (D_f) of the composite, D_f = 2.63, then After it goes to diffusion limited clusters with D_f = 1.4, its dimension lines up to a Von Koch curve with a random interval of D_f = 1.14.     

Characterising latex particles and fractal aggregates using image analysis

The fractal nature of latex particles and their aggregates was characterised by image analysis in terms of fractal dimensions. The one- and two-dimensional fractal dimensions, D ₁ and D ₂, were estimated for polystyrene latex aggregates formed by flocculation in citric acid/phosphate buffer solutions. The dimensional analysis method was used, which is based on power law correlations between aggregate perimeter, projected area and maximum length. These aggregate characteristics were measured by image analysis. A two-slopes method using cumulative size distributions of aggregate length and solid volume has been developed to determine the three-dimensional fractal dimension (D ₃) for the latex aggregates. The fractal dimensions D ₁, D ₂ and D ₃ measured for single latex particles in distilled water agreed well with D ₁ = 1, D ₂ = 2 and D ₃ = 3 expected for Euclidean spherical objects. For the aggregates, the fractal dimension D ₂ of about 1.67 ± 0.04 (±standard deviation) was comparable to the fractal dimension D ₃ of approximately 1.72 ± 0.13 (±standard deviation), taking the standard deviations into account. The measured three-dimensional fractal dimension for latex aggregates is within the fractal dimension range 1.6–2.2 expected for aggregates formed through a cluster-cluster mechanism, and is close to the D ₃ value of about 1.8 indicated for cluster formation via diffusion-limited colloidal aggregation. Received: 28 September 1998 Accepted: 29 October 1998     

Der geschwindigkeitsbestimmende Schritt bei der Decarboxy-lierung von 2,4-Dihydroxybenzoesäure in mässig konzentrierter wässeriger Salzsäure

The decarboxylation kinetics of 2,4-dihydroxybenzoic acid have been studied in 0.1–8 N aqueous HCl at 50°. At low HCl concentrations, the observed first order rate constant, k, increases with increasing acidity of the solution. In solutions with 3.5–6 N HCl, k remains constant. The D₂O solvent isotope effect decreases from ^kH₂O/^kD₂O = 2.0 in 1N HCl to 1.3 in 5 N HCl, and it remains unchanged at 1.3 if the HCl concentration is increased further to 8 N. It is concluded that an increase of the acidity of the solution causes a change of the rate determining step from slow proton transfer to rate limiting C? C bond cleavage.     

Viscoelastic scaling in polymer gels

New scaling laws for chain networks are derived to describe the fundamental relationships between the viscosity exponent (k), viscoelastic exponent (m), stretched exponent (β), spatial dimension (d). fractal dimension (d_f), and a universal constant (γ). The scaling of the total number of monomers and the radius of gyration is defined by d_f. We have discovered γ = m/β to be a universal constant which relates the shear modulus of a polymer gel melt to the shear modulus near the glass transition. Analyzing the size-dependent shear viscosity, we have determined γ = 3d_fcd/(7d−5d_fc) = 2.647 for d = 3 where d_fc is the fractal dimension of critical clusters at the gel point. By using γ, the present theory extends previous work pertaining to systems near the sol-gel transition, and shows how properties far from the critical point can be explained. The theoretical prediction is in good agreement with viscoelastic measurements.     

Fractal Dimensions of Coals and Cokes

Using adsorption data, we get formulas for the calculation of fractal dimensions: log[A_CO2(DP)/A_N2(BET)] = −5.3984(2 −D₁)/2 and log[A_CO2(BET)/A_N2(BET)] = −4.9569(2 −D₂)/2. The fractal dimensions (D) of 27 coals and 2 cokes have been obtained. TheDof coals decreased with the increase of f_aand reached a maximum at H/C equal to 0.66 (orC_dafabout 86%). The fractal dimension is relative to ash and volatiles of coal:D= 2.2237 + 0.6249V_daf+ 0.8863A_d. The relationship betweenDof coal coke and its conversions (X) obeys the following equation:D = aexp(−bX) +c.     

A Single and a Dual-Fractal Analysis of Analyte-Receptor Binding Kinetics for Surface Plasmon Resonance Biosensor Applications

The diffusion-limited binding kinetics of analyte in solution to either a receptor immobilized on a surface or to a receptorless surface is analyzed within a fractal framework for a surface plasmon resonance biosensor. The data is adequately described by a single- or a dual-fractal analysis. Initially, the data was modeled by a single-fractal analysis. If an inadequate fit was obtained then a dual-fractal analysis was utilized. The regression analysis provided by Sigmaplot (32) was used to determine if a single fractal analysis is sufficient or if a dual-fractal analysis is required. In general, it is of interest to note that the binding rate coefficient and the fractal dimension exhibit changes in the same direction (except for a single example) for the analyte-receptor systems analyzed. Binding rate coefficient expressions as a function of the fractal dimension developed for the analyte-receptor binding systems indicate, in general, the high sensitivity of the binding rate coefficient on the fractal dimension when both a single- and a dual-fractal analysis is used. For example, for a single-fractal analysis and for the binding of human endothelin-1 (ET-1) antibody in solution to ET-115-21.BSA immobilized on a surface plasmon resonance (SPR) surface (33), the order of dependence of the binding rate coefficient, k, on the fractal dimension, Df, is 6.4405. Similarly, for a dual-fractal analysis and for the binding of 10(-6) to 10(-4) M bSA in solution to a receptorless surface (direct binding to SPR surface) (41) the order of dependence of k1 and k2 on Df1 and Df2 were -2.356 and 6.241, respectively. Binding rate coefficient expressions are also developed as a function of the analyte concentration in solution. The binding rate coefficient expressions developed as a function of the fractal dimension(s) are of particular value since they provide a means to better control SPR biosensor performance by linking it to the degree of heterogeneity that exists on the SPR biosensor surface. Copyright 1999 Academic Press.     

Surface Roughness Characterization of Poly(methylmethacrylate) Films with Immobilized Eu(III) β-Diketonates by Fractal Analysis

The structural complexity of the 3-D surface of poly(methylmethacrylate) films with immobilized europium β-diketonates was studied by atomic force microscopy and fractal analysis. Fractal analysis of surface roughness revealed that the 3-D surface has fractal geometry at the nanometer scale. Poly(methylmethacrylate) (PMMA) as immobilization matrix is dense and uniform, and a tendency for formation of chain structures was observed. Fractal analysis can quantify key elements of 3-D surface roughness such as the fractal dimensions D _f determined by the morphological envelopes method of the Eu(DBM)₃ and Eu(DBM)₃ · dpp nanostructures, which are not taken into account by traditional surface statistical parameters.     

污泥活性炭的结构特征及表面分形分析

以城市污水厂污泥为主要原料添加适量添加剂, 采用ZnCl₂化学活化法制备的污泥活性炭, 借助XRD, BET法, FT-IR, SEM等现代分析测试方法结合液相吸附法, 表征结构特征和分析表面分形维数. 结果表明: 在适宜的活化温度、活化时间、ZnCl₂浓度、原料粒度等工艺条件下, 加入少量添加剂制备的污泥活性炭, 最可几孔径分布在4.16 nm左右,平均孔容0.4484～0.5122 mL•g^－1, 比表面积为634.8～748 m²•g^－1, IR峰中出现C—OH, C—H, N＝O, C＝C功能组, 孔结构是具有平行壁的狭缝状介孔结构. 由液相吸附法得到的污泥活性炭分维近似为2, 属于低分维二维表面.     

Electrochemical studies of 9,10-anthraquinone interacting with hemoglobin and determination of hemoglobin

Electrochemical investigation of the interaction of 9,10-anthraquinone (AQ) with hemoglobin (HB) on a mercury electrode is reported for the first time. On addition of hemoglobin to an anthraquinone solution, both the reduction and oxidation currents decrease, with increasing peak separation. In the presence of hemoglobin, no new peaks appear, but the electrochemical parameters (standard rate constantk _s and diffusion coefficientD) change significantly. Reaction of anthraquinone with hemoglobin forms an electrochemically active complex HB-AQ. The equilibrium constant for this complex is calculated to be 3.27 × 10⁵ l/mol. A satisfactory result has been obtained for the determination of hemoglobin in clinical blood samples.     

Characterization of Titania/Silica Gel by Means of Low-Pressure Nitrogen Adsorption

Adsorbents synthesized by grafting of titania onto mesoporous silica gel surfaces at different temperatures were studied by means of nitrogen adsorption–desorption and water desorption. The pore size distribution f(R_p) of titania/silica gel depends on the titania concentration (C_TiO2) and the temperature of titania synthesis. Nonuniformity of TiO₂ phase is maximal at a low C_TiO2 value (3.2 wt.% anatase deposited at 473 K), and two peaks of the fractal dimension distribution f(D) are observed at such a concentration of titania, but at larger C_TiO2 values, only one f(D) peak is seen. More ordered filling of pores and adsorption sites by nitrogen, reflecting in the shape of adsorption energy distributions f(E) at different pressures of adsorbate, is observed for adsorbent with titania (rutile+anatase) grafted on silica gel at a higher temperature (673 K).     

Fractal approach to the CO oxidation on silica porous materials

We present an approach establishing a relation between the activation energy of heterogeneous catalytic processes and the fractal dimension of a catalyst. The approach is verified by experimental study of the CO oxidation on various porous silica and zeolite NaX. The fractal dimension of a catalyst (D_F) was calculated from the nitrogen adsorption isotherms. Our results indicate that the activation energy increases with increasing the fractal dimension of a catalyst. We show a good correspondence between theoretical and experimental results.     

An Evaluation of Cellular Analyte-Receptor Binding Kinetics Utilizing Biosensors: A Fractal Analysis

A fractal analysis is presented for cellular analyte-receptor binding kinetics utilizing biosensors. Data taken from the literature can be modeled by using (a) a single-fractal analysis and (b) a single- and a dual-fractal analysis. Case (b) represents a change in the binding mechanism as the reaction progresses on the biosensor surface. Relationships are presented for the binding rate coefficient(s) as a function of the fractal dimension for the single-fractal analysis examples. In general, the binding rate coefficient is rather sensitive to the degree of heterogeneity that exists on the biosensor surface. For example, for the binding of mutagenized and back-mutagenized forms of peptide E1037 in solution to salivary agglutinin immobilized on a sensor chip, the order of dependence of the binding rate coefficient, k, on the fractal dimension, D(f), is 13.2. It is of interest to note that examples are presented where the binding coefficient (k) exhibits an increase as the fractal dimension (D(f)) 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 us 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 coefficients (or affinities). In general, the technique presented is applicable for the most part to other reactions occurring on different types of biosensors or other surfaces. Copyright 2000 Academic Press.     

Monitoring coalescence behavior of soft colloidal particles in water by small-angle light scattering

The fractal dimension (D _f) of the clusters formed during the aggregation of colloidal systems reflects correctly the coalescence extent among the particles (Gauer et al., Macromolecules 42:9103, 2009). In this work, we propose to use the fast small-angle light scattering (SALS) technique to determine the D _f value during the aggregation. It is found that in the diffusion-limited aggregation regime, the D _f value can be correctly determined from both the power law regime of the average structure factor of the clusters and the scaling of the zero angle intensity versus the average radius of gyration. The obtained D _f value is equal to that estimated from the technique proposed in the above work, based on dynamic light scattering (DLS). In the reaction-limited aggregation (RLCA) regime, due to contamination of small clusters and primary particles, the power law regime of the average structure factor cannot be properly defined for the D _f estimation. However, the scaling of the zero angle intensity versus the average radius of gyration is still well defined, thus allowing one to estimate the D _f value, i.e., the coalescence extent. Therefore, when the DLS-based technique cannot be applied in the RLCA regime, one can apply the SALS technique to monitor the coalescence extent. Applicability and reliability of the technique have been assessed by applying it to an acrylate copolymer colloid.