SIAM J. MATH. ANAL. c  2006 Society for Industrial and Applied Mathematics Vol. 38, No. 5, pp. 1371–1388 CHEMICAL KINETICS ON SURFACES: A SINGULAR LIMIT OF A REACTION-DIFFUSION SYSTEM ∗ G. FIBICH † , I. GANNOT ‡ , A. HAMMER § , AND S. SCHOCHET † Abstract. We show that chemical kinetics relations can be used to describe processes that involve binding and dissociation reactions that take place on surfaces. From a mathematical per- spective, the problem we study is a singular limit of a reaction-diffusion system in which one of the variables concentrates on a lower-dimensional set in the limit, while the other continues to diffuse in a fixed domain. Key words. chemical kinetics, surface, binding, dissociation, singular limit, invariant region AMS subject classifications. 92C45, 92C50, 35K57, 35B25 DOI. 10.1137/050633767 1. Introduction. Numerous biological processes involve binding and dissocia- tion reactions that take place on surfaces. For example, in antibody-antigen interac- tions, antibodies immobilize and agglutinate infectious agents by binding to specific receptors located on the surface of antigens [1, 19, 22]. Additional examples include the binding of proteins to cell membranes either to initiate transduction of external signals into the cell (signal transduction) or to open the ion channels of the membrane (see, e.g., [18]); the binding of microbiological cultures to attachment sites on the in- ner walls of flow reactors [12]; and the phenomenon of surface plasmon resonance, which involves interactions of biopolymers with various ligands [13]. A natural way to model surface reactions is to adapt the standard chemical- kinetics approach used for reactions occurring in volumes. This means that the bind- ing rate for surface reactions is assumed to be proportional to the product of the volumetric concentration of the reactant at the surface and the surface concentration of the binding sites [18, 20]. There is a methodological problem with this approach, however, since chemical-kinetics relations are usually derived under the assumption that reactions take place in a volume, in which the two reactants are well mixed. Our goal here is to justify the use of chemical-kinetics relations for reactions that take place on surfaces. To do so, we will first construct a volumetric model in which the binding sites, and hence also the binding and dissociation reactions, take place in a narrow volumetric layer around the surface. We will then show that as the width of the binding sites layer shrinks to zero, the volumetric model reduces to a surface model, in which binding sites are located on the surface, and for which the reactions are still described by chemical-kinetics relations. From a mathematical perspective, the problem we study is a singular limit of a ∗ Received by the editors June 16, 2005; accepted for publication (in revised form) July 5, 2006; published electronically December 26, 2006. http://www.siam.org/journals/sima/38-5/63376.html † School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (fibich@math.tau. ac.il, schochet@post.tau.ac.il). ‡ Department of Biomedical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel (gannot@eng.tau.ac.il) and Program of Biomedical Engineering, Department of Elec- trical and Computer Engineering, School of Engineering and Applied Sciences, George Washington University, Washington, DC 20052 (gannoti@mail.nih.gov). § Department of Biomedical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel (amit hammer@hotmail.com). 1371