Ligand-Receptor Interactions in Chains of Colloids: When Reactions Are Limited by Rotational Diffusion ² Nam-Kyung Lee, ‡ Albert Johner, § Fabrice Thalmann, § Laetitia Cohen-Tannoudji, | Emanuel Bertrand, | Jean Baudry, | Je ´ro ˆme Bibette, | and Carlos M. Marques* ,§ Department of Physics, Institute of Fundamental Physics, Sejong UniVersity, 143-747, Seoul, Institut Charles Sadron CNRS UPR 22 and ULP, 6 rue Boussingault, F-67083 Strasbourg Cedex, France, and Laboratoire Colloı ¨des et Mate ´ riaux DiVise ´ s, ESPCI, ParisTech, UniVersite ´ Pierre et Marie Curie, CNRS UMR 7612, 10 rue Vauquelin, Paris F-75231 Cedex 05, France ReceiVed June 4, 2007. In Final Form: August 2, 2007 We discuss the theory of ligand receptor reactions between two freely rotating colloids in close proximity to one other. Such reactions, limited by rotational diffusion, arise in magnetic bead suspensions where the beads are driven into close contact by an applied magnetic field as they align in chainlike structures. By a combination of reaction- diffusion theory, numerical simulations, and heuristic arguments, we compute the time required for a reaction to occur in a number of experimentally relevant situations. We find in all cases that the time required for a reaction to occur is larger than the characteristic rotation time of the diffusion motion τ rot . When the colloids carry one ligand only and a number n of receptors, we find that the reaction time is, in units of τ rot , a function simply of n and of the relative surface R occupied by one reaction patch R) πr C 2 /(4πr 2 ), where r C is the ligand receptor capture radius and r is the radius of the colloid. 1. Introduction Ligand receptor pairs build lock-and-key complexes through the formation of specific noncovalent bonds. 1 They play a crucial role in cell adhesion events that allow the communication, proliferation, differentiation, and migration of cells. 2 The quantitative understanding and control of the molecular recogni- tion mechanisms is an important scientific challenge, not only in the fields of molecular and cell biology but also for immunodiagnosis, 3 a diagnosis of disease based on the detection of antigen-antibody reactions in the blood serum. Immunochemistry is often based on the precipitation of large complexes made of antibodies and antigens. 4 For instance, if an antigen has two different epitopes binding to two antibodies A and B, to reveal the presence of the antigen, one mixes the sample to be tested with particles grafted with A and B. One usually distinguishes between homogeneous and heterogeneous immu- noassays. The homogeneous assay, an old, well-established technique, is made by simultaneously mixing the three com- ponents and by monitoring the formation of small clusters with changes in the scattered light. It is currently the simplest and most straightforward test, with several hundred different tests being available for practitioners. In contrast to homogeneous tests, heterogeneous assays comprise several steps of mixing and rinsing; they achieve a much better sensitivity. The sensitivity of homogeneous assays is generally limited by the poor control of the composition of the physiological sample to test, which may contain many adherent proteins. This brings about unwanted nonspecific adhesion between colloids, the false positives, which is usually prevented by coating the beads with a protective layer that in turn reduces the specific adhesion to be detected, the false negatives. A possible strategy for improving test sensitivity consists of enhancing the specific adhesion rate by enforcing some degree of local organization among colloids. For instance, ultrasonic standing wave patterns have been used 5 for this purpose, concentrating the colloids near the nodes, with a resulting 2 orders of magnitude increase in sensitivity. Recently, Bibette et al. discovered that the combined use of magnetic fields and specially designed magnetic colloids provided unique control of the spatial arrangement of the particles. 6 Under a suitable applied magnetic field, the magnetic beads arrange into linear chains with a finely controlled, adjustable relative spacing. When the field is switched off, the beads reversibly disperse if no binding has occurred. The speed of assembly and disassembly is faster in many cases than the characteristic binding rates of functionalized particles, thus offering an unmatched tool to probe the adhesion kinetics with fast time resolution. Kinetic studies from organized particles functionalized with streptavidin and biotin are very promising. They allow one to test the kinetics of biorecognition complex formation as a function of the relevant physical parameters. Experimentally, the reaction kinetics is better studied when most colloids of the magnetic chain carry the receptors but only a few colloids bear a single ligand. Under these conditions, the formation of colloid aggregates larger than dimers is prevented, and the time evolution of dimer formation can been monitored by light diffusion techniques. The parameters associated with the ligands and the receptors involved in the reaction can now be well controlled, allowing for tuning the number or receptors or ligands per colloid or the spacer length and rigidity. The theoretical challenge that we thus face is to directly relate the measured reaction rates and the molecular ligand receptor ² Part of the Molecular and Surface Forces special issue. ‡ Sejong University. § Institut Charles Sadron. | Universite ´ Pierre et Marie Curie. (1) Alberts, B.; Johnson, A.; Lewis, J.; Raff, M.; Roberts, K.; Walter, P. Molecular Biology of the Cell, 4th ed.; Garland: New York, 2002. (2) Bongrand, P. Rep. Prog. Phys. 1999, 62, 921-968. (3) Price, C. P.; Newman, D. J. Principles and Practice of Immunoassay; Stockton Press: New York, 1991. (4) Baudry, J.; Bertrand, E.; Lequeux, N.; Bibette, J. J. Phys.: Condens. Matter 2004, 16, R469-R480. (5) Thomas, N. E.; Coakley, W. T. Ultrasound Med. Biol. 1996, 22, 1277- 1284. (6) Baudry, J.; Rouzeau, C.; Goubault, C.; Robic, C.; Cohen-Tannoudji, L.; Koenig, A.; Bertrand, E.; Bibette, J. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 16076-16078. 1296 Langmuir 2008, 24, 1296-1307 10.1021/la701639n CCC: $40.75 © 2008 American Chemical Society Published on Web 10/23/2007