Spatially Resolved Anomalous Kinetics of a Catalytic Reaction: Enzymatic Glucose Oxidation in Capillary Spaces Anna L. Lin,* Mark S. Feldman, and Raoul Kopelman* Department of Chemistry, UniVersity of Michigan, Ann Arbor, Michigan 48109-1055 ReceiVed: June 23, 1997 X A reaction involving the catalytic oxidation of glucose by the enzyme glucose oxidase is employed as a first spatially resolved experimental realization of a nonclassical A + C f C + Products chemical reaction. This heterogeneous kinetics experiment models the operation of optical glucose sensors. One-dimensional nonclassical reaction kinetics, manifested in anomalous rate laws, are observed. The time evolution of the density gradient of O 2 in the zone near the catalytic trap was monitored. The temporal dependence of the growth of this depletion zone M, predicted to scale as t 1/2 , was measured to be M ∼ t 0.50(0.05 . This time dependence of the growth in M is directly proportional to the anomalous, time-dependent, reaction rate coefficient. 1. Introduction Many chemical reaction processes are heterogeneous; that is, they take place at interfaces of different phases or in solid, viscous, or porous media, where particles, molecules, excitons, etc. are not freely stirred or able to diffuse in three dimensions. Examples include industrial surface catalysis, enzymatic reac- tions, and atmospheric reactions. We study here an effectively one-dimensional heterogeneous reaction of importance to biomedical science and technology, particularly to glucose sensors. Under such conditions it is usually difficult to impose mechanical stirring or to obtain complete convective mixing. Thus, a reaction occurring under heterogeneous conditions is often limited by the diffusion of the reactants. Also, in low dimensions, the diffusive self-stirring is not an effective stirring mechanism and this causes the reaction process to exhibit anomalous, dimension-dependent rates or rate coefficients. Theory 1-17 and simulations 2,3,7,18-24 have established that the kinetics of all diffusion-limited elementary reactions are highly affected by the spatial dimension in which they occur. Below some critical dimension, these reaction processes exhibit nonclassical reaction kinetics manifested in a time-dependent rate coefficient k(t) ) kt -h . This is a result of the formation of a nonrandom spatial distribution of the reactants. That is, in contrast to the Smoluchowski model, 25-30 there is a time-de- pendent growth of depletion zones. Therefore, time-resolved studies of such elementary reactions are called for. The A + C f C trapping reaction has been examined in quasi-1-D systems by physical exciton annihilation experiments in porous membranes and Vycor glass 27,28 and by the physical photobleaching experiments in capillaries and also in quasi- 2-D sample geometries. 22 The trapping reactions studied in these experiments were shown to obey the theoretical predictions of low-dimensional nonclassical reaction kinetics. However, no spatial resolution of the reaction progress was obtained. Here we employ a two-phase enzymatic, catalytic chemical reaction, which is more complex but, nevertheless, analogous to the simple trapping reaction. Specifically, we used oxygen in water (phase 1) as the reactant A and glucose oxidase embedded in an acrylamide polymer matrix as the catalytic trap C (phase 2) in an effort to observe this behavior experimentally via the chemical reaction: The experiments were run under conditions of excess glucose and water. Thus, effectively, the reaction can be written as the pseudo-monomolecular reaction Measuring the instantaneous, O 2 sensitive fluorescence, of tris- (1,10-phenanthroline)ruthenium(II) chloride, a molecular com- plex known to respond linearly to O 2 concentrations at 0.24 mM (air saturation) and below, 31 the depletion of oxygen was monitored as a function of time. Classically, the A + C f C reaction obeys the pseudo-first- order rate law where F is the concentration and k′ ) kF C is a constant, since C is not consumed by the reaction process. This leads to the integrated rate law which is equivalent to In the nonclassical kinetics formalism 4 for the trapping reaction, the rate constant k′ is replaced in the differential form of the rate law by k 1 t -h . Here h is the heterogeneity coefficient and is equal to 1 - (d s /2), where d s is the spectral dimension 30 and is given by the relation where P is the probability that a random walker will return to its origin after a certain time t. In one dimension d s ) 1 and h ) 1 / 2 ; thus a nonclassical rate coefficient of is predicted for the differential rate law. Substitution of (7) * Corresponding authors. E-mail: alin@chaos.ph.utexas.edu, kopelman@umich.edu. X Abstract published in AdVance ACS Abstracts, September 15, 1997. glucose oxidase + glucose + O 2 + H 2 O f glucose oxidase + gluconic acid + H 2 O (1) O 2 + glucose oxidase f glucose oxidase + products (2) -dF A /dt ) kF A F C ) k′F A (3) ln ( F A F A 0 29 )-k′t (4) F A )F A 0 e -k′t (5) P ∼ t -d s /2 (6) k(t) ) k′ ) k 1 t -1/2 (7) 7881 J. Phys. Chem. B 1997, 101, 7881-7884 S1089-5647(97)02221-9 CCC: $14.00 © 1997 American Chemical Society