Zonal Markov model of convective transport in the PCR problem in a microtube

Abstract

The polymerase chain reaction (PCR) in sealed microtubes is driven by natural convection, creating thermocycling without external mechanical components. Numerical simulation of convective flow produces velocity and temperature fields on a computational mesh of several hundred thousand cells, whose direct use for reagent transport analysis proves challenging. We propose an approach based on partitioning the computational domain into quasi-homogeneous zones using spectral clustering on a similarity graph that accounts for temperature, spatial proximity, flow direction, and velocity shear of adjacent cells. Transition probabilities between zones are determined from volume fluxes through common boundaries, forming a Markov chain. The stationary distribution gives the limiting fraction of time a DNA molecule spends in each zone. Validation via particle tracing showed a mean absolute deviation of 2.7% and Pearson correlation of 0.928, confirming model adequacy. The Markov approach provides direct computation of the stationary distribution, whereas its estimation by particle tracing would require modelling long trajectories.