Using a recently developed inhomogeneous, macroscopic model for long DNA bound to proteins, we examine topological and geometric aspects of DNA/protein structures and dynamics on various stages of the Hin inversion pathway. This biological reaction involves exchange of DNA in a synaptic complex that brings together several DNA sites bound to Hin dimers as well as Fis enhancers. Brownian dynamics simulations in the millisecond timescale allow us to follow and analyze the DNA/protein dynamics trajectories and to examine the effects of DNA superhelicity and protein binding on various reaction steps. Analysis of the generated kinetic pathways helps explain mechanistic aspects regarding the process by which two or three protein-bound DNA sites come to close spatial proximity and show that how topological selectivity (two trapped supercoils), enhancer binding, and properties of supercoiled DNA play critical roles in regulating the inversion reaction. Specifically, a critical amount of DNA superhelicity (e.g., |σ| > 0.02) leads to an optimal interplay for the first reaction step - two-site juxtaposition - between large-scale random rearrangements of Hin-bound DNA and local slithering within branches of plectonemes. The three-site juxtaposition, the second step, is significantly accelerated by the presence of an enhancer protein that, due to severe local bending, also alters juxtaposition mechanisms, especially for superhelical density magnitude greater than around 0.04.
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