Mean encounter times for cell adhesion in hydrodynamic flow: analytical progress by dimensional reduction

PACS 82.39.-k – chemical kinetics in biological systems PACS 47.15.G-– fluid dynamics, low-Reynolds-number flows PACS 05.10.Gg – stochastic analysis methods (FokkerPlanck, Langevin, etc.) Abstract.-For a cell moving in hydrodynamic flow above a wall, translational and rotational degrees of freedom are coupled by the Stokes equation. In addition, there is a close coupling of convection and diffusion due to the position-dependent mobility. These couplings render calculation of the mean encounter time between cell surface receptors and ligands on the substrate very difficult. Here we show for a two-dimensional model system how analytical progress can be achieved by treating motion in the vertical direction by an effective reaction term in the mean first passage time equation for the rotational degree of freedom. The strength of this reaction term can either be estimated from equilibrium considerations or used as a fit parameter. Our analytical results are confirmed by computer simulations and allow to assess the relative roles of convection and diffusion for different scaling regimes of interest. Introduction. – Biological function is often based on the formation of a specific binding complex between receptor and ligand [1]. However, in order for binding to occur , a physical transport process must exist which brings the binding partners to sufficiently close proximity [2]. In many cases of interest, this transport process is rather complex. Usually it contains several coupled degrees of freedom, like a cell surface receptor moving laterally on a membrane which fluctuates in the vertical direction [3]. The efficiency of biological transport processes often can be framed as mean first passage time (MFPT) problems, for example for the gating of ion channels [4] or the arrival of a virus at the nucleus [5]. Another example of a complex transport process of large biological relevance is the receptor-mediated adhesion of cells which are carried over a ligand-coated substrate by hydrodynamic flow [6]. Here the mean encounter time between receptors and lig-ands is a measure for the efficiency of cell adhesion under the conditions of hydrodynamic flow [7]. For this system, additional complications arise from the presence of multiple length scales. For the micron-sized cell, the hydrody-namic equations result in coupling of the translational and rotational degrees of freedom. Even for high shear rates, Brownian motion is relevant because receptors and ligands are nanometer-sized objects, thus even small movements for the cell result in a large effect on the molecular level.