An Integrated Reaction-Transport Model for DNA Surface Hybridization: Implications for DNA Microarrays

Abstract

DNA microarrays have the potential to revolutionize medical diagnostics and development of individualized medical treatments. However, accurate quantification of scantily expressed genes and precise measurement of small differences between different treatments is not currently feasible. A major challenge remains the understanding of physicochemical processes and rate-limiting steps of hybridization of complex mixtures of DNA targets on immobilized DNA probes. To this end, we developed a mathematical model to describe the effects of molecular orientation and transport on the kinetics and efficiency of hybridization. First, we calculated the hybridization rate constant based on the distance between the complementary nucleotides of the target and probe DNA. The surface reaction rate was then integrated with translational and rotational transport of target DNA to the surface to calculate the kinetics of hybridization. Our model predicts that hybridization of short DNA targets is diffusion limited but long targets are kinetically limited. In addition, for DNA targets with wide size distribution, it may be difficult to distinguish between specific binding of long targets from nonspecific binding of short ones. Our model provides novel insight into the process of DNA hybridization and suggests operating conditions to improve the sensitivity and accuracy of microarray experiments.

Appendix A

At z = r _g the flux, β, entering the microscopic diffusion layer is equal to the rate of reaction:

$$ D_{\infty } \left. {\frac{{\partial C}} {{\partial z}}} \right|_{{z = r_{{\text{g}}} }} = \beta D_{\infty } = \overline{k} \left. C \right|_{{z = r_{{\text{g}}} }} $$

(A1)

Note that the flux, β entering the reaction zone layer is the same as the flux far from the surface (z→∞) (see Eq. 20). From Eq. (27) we substitute the dimensionless variables $\xi = C \mathord{\left/ {\vphantom {C {\beta r_{{\text{g}}} }}} \right. \kern-\nulldelimiterspace} {\beta r_{{\text{g}}} }$ and $\eta = z \mathord{\left/ {\vphantom {z {r_{{\text{g}}} }}} \right. \kern-\nulldelimiterspace} {r_{{\text{g}}} }$ into Eq. (A1) to obtain:

$$ \frac{{\overline{k} r_{{\text{g}}} }} {{D_{\infty } }} = \frac{1} {{\left. \xi \right|_{{\eta = 1}} }} $$

(A2)

The value of $\left. \xi \right|_{{\eta = 1}}$ is obtained as the intercept of the linear part of ξ(η) curve (Fig. 4) to η = 1.