A FRACTAL ANALYSIS OF BINDING AND DISSOCIATION KINETICS OF GLUCOSE AND RELATED ANALYTES ON BIOSENSOR SURFACES ATUL M. DOKE AND AJIT SADANA CHEMICAL ENGINEERING DEPARTMENT AND COMPOSITE STRUCTURES AND NANOENGINEERING RESEARCH GROUP UNIVERSITY OF MISSISSIPPI UNIVERSITY, MS MAESC CHRISTIAN BROTHERS UNIVERSITY, MEMPHIS MAY 11, 2005

FRACTAL ANALYSIS USED TO MODEL THE BINDING AND DISSOCIATION KINETICS OF GLUCOSE AND RELATED ANALYTES ON BIOSENSOR SURFACES. VALUES OF THE FRACTAL DIMENSIONS PREDICT THE DEGREE OF HETEROGENEITY ON THE BIOSENSOR SURFACE.

INTRODUCTION MONITORING AND CONTROL OF GLUCOSE LEVELS IN THE BLOOD IS CRITICAL FOR DIABETES DIABETES IS ONE OF THE MOST PREVALENT AND COSTLY DISEASES IN THE WORLD 17 MILLION PEOPLE IN THE U.S. ARE ESTIMATED TO HAVE DIABETES (PEI ET AL., 2004) 16 MILLION PRE-DIABETICS (YONZON ET AL., 2004)

AMERICAN DIABETIC ASSOCIATION (2003) INDICATES THAT THE ECONOMIC ANNUAL COST OF DIABETES IS $132 BILLION EPIDEMIC PROPORTIONS WORLD HEALTH ORGANIZATION (WHO) ESTIMATES THAT THE NUMBER OF DIABETICS WILL DOUBLE FROM 150 MILLION TO 300 MILLION BY THE YEAR 2025.

WE RE-ANALYZE USING FRACTAL ANALYSIS THE DIFFUSION-LIMITED BINDING KINETICS DATA OF BIOMEDICAL ANALYTES SUCH AS GLUCOSE (LEEGSMA-VOGT ET AL., 2004; HSIEH ET AL., 2004) URIC ACID, ACETAMINOPHEN, AND ASCORBIC ACID USING BIOSENSORS THE LAST THREE ARE INTERFERENTS IN THE DETECTION OF GLUCOSE IN NO WAY ARE WE INDICATING THAT THE FRACTAL ANALYSIS IS BETTER THAN THE ORIGINAL ANALYSIS

FRACTALS ARE SCALE SELF-SIMILAR OBJECTS THAT POSSESS NON-TRIVIAL GEOMETRIES. THEY ARE ‘SIZE-LESS.’ THE FRACTAL DIMENSION PROVIDES A QUANTITATIVE MEASURE OF THE DEGREE OF HETEROGENEITY ON THE BIOSENSOR SURFACE THE ANALYSIS PROVIDES VALUES OF THE BINDING RATE COEFFICIENT AND THE FRACTAL DIMENSION ON THE BIOSENSOR SURFACE

THEORY SINGLE-FRACTAL ANALYSIS BINDING RATE COEFFICIENT HAVLIN INDICATES THAT THE DIFFUSION OF A PARTICLE (ANALYTE [Ag]) FROM A HOMOGENEOUS SOLUTION TO A SOLID SURFACE (E.G., RECEPTOR [Ab] COATED SURFACE) ON WHICH IT REACTS IS GIVEN BY:

DISSOCIATION RATE COEFFICIENT THE DIFFUSION OF THE DISSOCIATED PARTICLE (RECEPTOR [Ab] OR ANALYTE [Ag] ) FROM THE SOLID SURFACE (E.G., ANALYTE [Ag]-RECEPTOR [Ab] COMPLEX COATED SURFACE) INTO SOLUTION MAY BE GIVEN AS A FIRST APPROXIMATION BY: (Ab.Ag) ≈ - k t (3- Df,diss) / 2 = t p (t > t diss )

DUAL-FRACTAL ANALYSIS BINDING RATE COEFFICIENT IN THIS CASE, THE CONCENTRATION OF THE ANALYTE-RECEPTOR COMPLEX ON THE RECEPTOR-COATED SURFACE IS GIVEN BY: t (3- Df1,bind) / 2 = t p1 (t < t1) (Ab.Ag) ≈ t (3- Df2,bind)/2 = t p2 (t1 <t <t2) = t c t 1/2 (t > t c )

RESULTS LEEGSMA-VOGT ET AL. (2004) INDICATE THE POTENTIAL OF BIOSENSOR TECHNOLOGY IN CLINICAL MONITORING AND IN EXPERIMENTAL RESEARCH THEY EMPHASIZE THAT BIOSENSORS MAY BE USED FOR THE CONTINUOUS ONLINE MONITORING OF GLUCOSE AND LACTATE. THIS WOULD FACILITATE THERAPEUTIC INTERVENTIONS WHEN NEED BE.

PROBES PLACED AT DIFFERENT LOCATIONS: (A) PLASMA INSULIN (B) PLASMA GLUCOSE (C) ADIPOSE TISSUE INTERSTITIAL GLUCOSE (D) CONNECTIVE TISSUE INTERSTITIAL GLUCOSE

A B C D FIG:1

FIG 1A: BINDING AND DISSOCIATION OF INSULIN IN PLASMA DUAL-FRACTAL ANALYSIS REQUIRED FOR BINDING SINGLE-FRACTAL ANALYSIS REQUIRED FOR DISSOCIATION FIG 1B & C: BINDING AND DISSOCIATION OF GLUCOSE IN PLASMA AND ADIPOSE TISSUE INTERSTITIAL GLUCOSE SAME AS ABOVE FIG 1D: CONNECTIVE TISSUE INTERSTITIAL GLUCOSE SINGLE-FRACTAL ANALYSIS IS ADEQUATE TO DESCRIBE THE BINDING AND THE DISSOCIATION KINETICS

TABLE Ia: Binding Rate Coefficients for Glucose in Plasma, in Connective Tissue, and in Adipose Tissue, and Insulin in Plasma (Leegsma-Vogt et al., 2004) ANALYTELOCATIONkk1k1 k2k2 kdkd InsulinPlasma1.855 ± ± ± ± GlucosePlasma0.033± ± ± ± GlucoseInterstitial Adipose tissue ± ± ± ± GlucoseInterstitial Connective tissue ±0.067 na ±0.0081

TABLE Ib Fractal Dimensions for Glucose in Plasma, in Connective Tissue, and in Adipose Tissue, and Insulin in Plasma (Leegsma-Vogt et al., 2004) CompoundLocationDfDf D f1 D f2 D fd InsulinPlasma ± ± GlucosePlasma ± ±0.136 GlucoseInterstitial Adipose tissue ± ± ± ±0.101 GlucoseInterstitial Connective tissue ±0.066 na ±0.146

k d = ( ± )D fd ±0.753 k 1 / k d = ( ± 1.912) ( D f1 /D fd ) 1.371±0.104 k 2 / k d = (4.698 ± 2.022) (D f2 /D fd ) 1.628±0.2034

HSIEH ET AL. (2004) HAVE RECENTLY DETECTED GLUCOSE USING GLUCOSE/GALACTOSE BINDING PROTEIN (GGBP) AS THE RECEPTOR IMMOBILIZED ON A SURFACE PLASMON RESONANCE (SPR) BIOSENSOR SURFACE THESE AUTHORS INDICATE THAT THE DETECTION OF LOW MOLECULAR WEIGHT ANALYTES SUCH AS GLUCOSE (180 Da) BY A SPR BIOSENSOR IS DIFFICULT SINCE THE MOLECULES HAVE INSUFFICIENT MASS TO PROVIDE A MEASUREABLE CHANGE IN THE REFRACTIVE INDEX

GGBP IS A PERIPLASMIC BINDING PROTEIN. UPON BINDING OF ITS LIGAND PROTEINS, GLUCOSE OR GALACTOSE GGBP EXHIBITS A HINGE-TWIST CONFORMATIONAL CHANGE. THIS CONFORMATIONAL CHANGE MAY BE USED TO DETECT THE BINDING OF GLUCOSE AND GALACTOSE.

BINDING AND DISSOCIATION OF 100 MICROMOLAR GLUCOSE IN SOLUTION TO THIOL-COUPLED E149C GGBP IMMOBILIZED ON A BIOSENSOR CHIP. THE GGBP WAS ENGINEERED TO BIND IN THE PHYSIOLOGICAL RANGE BY MUTATION AT ADDITIONAL SITES (HSIEH ET AL., 2004) E149 IS ONE SUCH MUTATION SITE

DUAL-FRACTAL ANALYSIS IS REQUIRED TO ADEQUATELY DESCRIBE THE BINDING KINETICS SINGLE-FRACTAL ANALYSIS IS REQUIRED TO DESCRIBE THE DISSOCIATION KINETICS

CAI ET AL. (2004) HAVE RECENTLY DEVELOPED A WIRELESS, REMOTE- QUERY GLUCOSE BIOSENSOR. THE SENSOR USES A RIBBON-LIKE MASS-SENSITIVE MAGNETOELASTIC SENSOR AS A TRANSDUCER. ANALYZED 1-15 mMOL/L GLUCOSE IN SOLUTION (1, 4, 7, 10, AND 15)

A B C

D E

A DUAL-FRACTAL ANALYSIS IS REQUIRED TO MODEL THE BINDING KINETICS IN EACH OF THE ABOVE CASES

Table III: Rate Coefficients and Fractal Dimensions for the Binding of Different Concentrations of Glucose in Solution (in mmol/L) to a Coating on a Magnetoelastic Sensor with a pH-Sensitive Polymer (Cai et al., 2004) Analyte (Glucose) mmol/L kk1k1 k2k2 DfDf D f1 D f ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.039

CONCLUSIONS A FRACTAL ANALYSIS IS USED TO MODEL THE BINDING AND DISSOCIATION KINETICS OF BIOMEDICAL ANALYTES LIKE CONNECTIVE TISSUE INTERSTITIAL GLUCOSE, ADIPOSE TISSUE INTERSTITIAL GLUCOSE, AND GLUCOSE AND INSULIN IN PLASMA ON A BIOSENSOR SURFACE. NUMERICAL VALUES ARE OBTAINED FOR THE BINDING AND THE DISSOCIATION RATE COEFFICIENTS.

THE BINDING AND DISSOCIATION RATE COEFFICIENTS ARE LINKED TO THE DEGREE OF HETEROGENEITY OR THE ROUGHNESS (FRACTAL DIMENSION) PRESENT ON THE BIOSENSOR SURFACE. THE DUAL-FRACTAL ANALYSIS IS USED ONLY WHEN THE SINGLE-FRACTAL ANALYSIS DID NOT PROVIDE AN ADEQUATE FIT.

THE VALUES OF THE BINDING RATE COEFFICIENT, k LINKED WITH THE DEGREE OF HETEROGENEITY EXISTING ON THE BIOSENSOR SURFACE PROVIDES A COMPLETE PICTURE OF THE REACTION KINETICS ON THE BIOSENSOR CHIP SURFACE. IT IS SUGGESTED THAT THE ROUGHNESS ON THE BIOSENSOR SURFACE LEADS TO TURBULENCE WHICH ENHANCES MIXING AND DECREASES DIFFUSIONAL LIMITATIONS (MARTIN ET AL., 1991). THIS LEADS TO AN INCREASE IN THE BINDING RATE COEFFICIENT ON THE BIOSENSOR CHIP SURFACE.