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We investigate the dynamics of a pair of rigid rotating helices in a viscous fluid, as a model for bacterial flagellar bundle and a prototype of microfluidic pumps. Combining experiments with hydrodynamic modelling, we examine how spacing and phase difference between the two helices affect their torque, flow field and fluid transport capacity at low Reynolds numbers. Hydrodynamic coupling reduces the torque when the helices rotate in phase at constant angular speed, but increases the torque when they rotate out of phase. We identify a critical phase difference, at which the hydrodynamic coupling vanishes despite the close spacing between the helices. A simple model, based on the flow characteristics and positioning of a single helix, is constructed, which quantitatively predicts the torque of the helical pair in both unbounded and confined systems. Finally, we show the influence of spacing and phase difference on the axial flux and the pump efficiency of the helices. Our findings shed light on the function of bacterial flagella and provide design principles for efficient low-Reynolds-number pumps. 

Abstract Protamine proteins dramatically condense DNA in sperm to almost crystalline packing levels. Here, we measure the first step in the in vitro pathway, the folding of DNA into a single loop. Current models for DNA loop formation are one-step, all-or-nothing models with a looped state and an unlooped state. However, when we use a Tethered Particle Motion (TPM) assay to measure the dynamic, real-time looping of DNA by protamine, we observe the presence of multiple folded states that are long-lived (∼100 s) and reversible. In addition, we measure folding on DNA molecules that are too short to form loops. This suggests that protamine is using a multi-step process to loop the DNA rather than a one-step process. To visualize the DNA structures, we used an Atomic Force Microscopy (AFM) assay. We see that some folded DNA molecules are loops with a ∼10-nm radius and some of the folded molecules are partial loops—c-shapes or s-shapes—that have a radius of curvature of ∼10 nm. Further analysis of these structures suggest that protamine is bending the DNA to achieve this curvature rather than increasing the flexibility of the DNA. We therefore conclude that protamine loops DNA in multiple steps, bending it into a loop.