Absence of a barrier to backwards rotation of the bacterial flagellar motor demonstrated with optical tweezers

A cell of the bacterium Escherichia coli was tethered covalently to a glass coverslip by a single flagellum, and its rotation was stopped by using optical tweezers. The tweezers acted directly on the cell body or indirectly, via a trapped polystyrene bead. The torque generated by the flagellar motor was determined by measuring the displacement of the laser beam on a quadrant photodiode. The coverslip was mounted on a computer-controlled piezo-electric stage that moved the tether point in a circle around the center of the trap so that the speed of rotation of the motor could be varied. The motor generated ≈4500 pN nm of torque at all angles, regardless of whether it was stalled, allowed to rotate very slowly forwards, or driven very slowly backwards. This argues against models of motor function in which rotation is tightly coupled to proton transit and back-transport of protons is severely limited.

On the controllability of distributed systems

To “control” a system is to make it behave (hopefully) according to our “wishes,” in a way compatible with safety and ethics, at the least possible cost. The systems considered here are distributed—i.e., governed (modeled) by partial differential equations (PDEs) of evolution. Our “wish” is to drive the system in a given time, by an adequate choice of the controls, from a given initial state to a final given state, which is the target. If this can be achieved (respectively, if we can reach any “neighborhood” of the target) the system, with the controls at our disposal, is exactly (respectively, approximately) controllable. A very general (and fuzzy) idea is that the more a system is “unstable” (chaotic, turbulent) the “simplest,” or the “cheapest,” it is to achieve exact or approximate controllability. When the PDEs are the Navier–Stokes equations, it leads to conjectures, which are presented and explained. Recent results, reported in this expository paper, essentially prove the conjectures in two space dimensions. In three space dimensions, a large number of new questions arise, some new results support (without proving) the conjectures, such as generic controllability and cases of decrease of cost of control when the instability increases. Short comments are made on models arising in climatology, thermoelasticity, non-Newtonian fluids, and molecular chemistry. The Introduction of the paper and the first part of all sections are not technical. Many open questions are mentioned in the text.