Reaction-Diffusion kinetics of Protein DNA Interactions

Abstract: Transcription factors need to rapidly find one specific binding site among millions of nonspecific sites on the chromosomal DNA. In this thesis I use various aspects of reaction-diffusion theory to investigate the interaction between proteins and DNA and to explain the searching, finding and binding to specific operator sites. Using molecular dynamics methods we calculate the free energy profile for the model protein LacI as it leaves a nonspecific stretch of DNA and as it slides along DNA. Based on the free energy profiles we estimate the microscopic dissociation rate constant, kdmicro ~1.45×104s-1, and the 1D diffusion coefficient, D1 ~ 0.05-0.29 μm2s-1 (2-40μs to slide 1 basepair (bp)). At a non-atomistic level of detail we estimate the number of microscopic rebindings before a macroscopic dissociation occurs which leads to the  macroscopic residence time, τDmacro ~ 48±12ms resulting in a in vitro sliding length estimate of 135-345bp.When we fit the DNA interaction parameters for in vivo conditions to recent single molecule in vivo experiments we conclude that neither hopping nor intersegment transfer contribute to the target search for the LacI dimer, that it appears to bind the specific Osym operator site as soon as it slides into it, and that the sliding length is around 40bp in the cell. The estimated in vivo D1 ~ 0.025 μm2s-1 is higher than expected from estimates of D1 based on viscosity and the atomistic simulations. Surprisingly, we were also forced to conclude that the nonspecific association for the LacI dimer appeared reaction limited which is in conflict with the free energy profile. This inconsistency is resolved by allowing for steric effects. Using reaction-diffusion theory and simulations we show that an apparent reaction limited association can be diffusion limited if geometry and steric effects are taken into account. Furthermore, the simulations show that a protein binds ~2 times faster to a DNA molecule with a helical reactive patch than to a stripe patch running along the length of the DNA. This facilitated binding has a direct impact on the search time especially in the presence of other DNA binding proteins.