2.2. Numerical Simulations
Numerical simulations of a single hybridization well were performed with COMSOL Multiphysics (FEMLAB), a finite element software package, which uses two geometries and three application modes to solve the formula. The first geometry is three-dimensional and represents the channel connecting the micro well. Within this geometry, the model used two application modes: incompressible Navier–Stokes, and convection and diffusion to model the transport of the target DNA. The second geometry is two-dimensional and simulates the reaction surface with the diffusion application mode. It is assumed that the hybridization reactions take place on a 2D surface (bottom surface of the micro-wells) whereas bulk flow of incoming target DNA is resolved on a 3D geometry. To correctly model the multi-scale physics, extrusion-coupling variables were defined. These variables couple the mass transport from 3D bulk flow using a surface boundary condition to the hybridization reactions on a 2D surface. In COMSOL Version 4.2 and later, instead of using extrusion-coupling variables, a Surface Reaction interface can be used with a single 3D geometry. This interface resolves surface coverage of the adsorbed species, with a mass source or sink automatically coupled to the surface boundary condition in the bulk mass transport equation. The meshing around the channel was not greater than 10 μm and around the hybridization well it was smaller than 0.2 μm. In all simulations, an “insulation” boundary condition was chosen for the non-reactive surfaces at the top and the bottom of the microchannel except where the hybridization occurred. The model is solved in two steps using different solvers. First, it solves the incompressible Navier–Stokes application mode with a non-linear solver followed by the time-dependent solver to simultaneously solve the convection and diffusion and the diffusion application modes. The model predicts the concentration of duplexes formed due to interactions between the target DNA and immobilized probes inside the hybridization well.
In order to correlate the duplex concentration obtained through our model with the hybridization signal intensities, we used Equation (5) [26], (5) I=L1+e−d/θ where I is log fluorescence intensity, d is log dye concentration, θ defines the spread and slope of the linear range of the curve and the “background” level (set to 1.0 for Cy3 dye), L is the upper limit of the dynamic range (set to 3.0). Preset values were taken from the study [26]. For our simulation, we assumed that there was only one dye molecule attached per molecule of target DNA because it was end-labeled. The complete algorithm and the data flow used for our model are illustrated in Figure 2.