Applying torque to the Escherichia coli flagellar motor using magnetic tweezers

The bacterial flagellar motor of Escherichia coli is a nanoscale rotary engine essential for bacterial propulsion. Studies on the power output of single motors rely on the measurement of motor torque and rotation under external load. Here, we investigate the use of magnetic tweezers, which in principle allow the application and active control of a calibrated load torque, to study single flagellar motors in Escherichia coli. We manipulate the external load on the motor by adjusting the magnetic field experienced by a magnetic bead linked to the motor, and we probe the motor’s response. A simple model describes the average motor speed over the entire range of applied fields. We extract the motor torque at stall and find it to be similar to the motor torque at drag-limited speed. In addition, use of the magnetic tweezers allows us to force motor rotation in both forward and backward directions. We monitor the motor’s performance before and after periods of forced rotation and observe no destructive effects on the motor. Our experiments show how magnetic tweezers can provide active and fast control of the external load while also exposing remaining challenges in calibration. Through their non-invasive character and straightforward parallelization, magnetic tweezers provide an attractive platform to study nanoscale rotary motors at the single-motor level.


S1 Motor-driven rotation of a magnetic bead in a magnetic potential
The equation of motion for a magnetic bead spun around by a rotary motor in an external magnetic field in the overdamped limit, relevant in our case, is given by: 0 = !"#$%& + !"#"$ + !"#$ + !!!"#$% , Equation S1 where !"#$%& is the torque due to the interaction between the magnetic field of the magnets and the dipole of the bead; !"#"$ is the motor torque and quantity of interest; !"#$ = − !"#$ !"#$ is the drag torque on the bead, where !"#$ is the rotational drag coefficient in bulk, and !"#$ is the angular speed; and !!!"#$% = 2 ! !"#$ is the torque due to thermal fluctuations on the bead, where is a white-noise process with properties [1]: where is Dirac's delta function. We perform numerical simulations and analytical calculations of the simple model in Equation S1 for a quantitative description of our assay. In the modelling, we assume for simplicity that the motor torque is independent of its rotational speed. This assumption is motivated by the fact that the motor torque is reported to be roughly constant in the plateau of its torque-speed curve (in E.coli, the motor torque decreases by ≈ 10% between 0 and 175 Hz [2]). A change in motor torque (e.g. due to stator exchange or other external factors) would appear as a jump between curves simulated at different (constant) motor torques, and can be accounted for by these jumps.

S2 Analytical approximation for the magnetic torque
As the motor torque is the quantity of interest, the only term in Equation S1 not discussed yet is the magnetic torque. A previous report [3] has shown that the magnetic potential for a MyOne bead in magnetic tweezers is π periodic. An obvious and simple choice is then to assume ∝ − cos 2 , where = − ! , because is π periodic and has a minimum at = 0 . The analytical calculations (Supplement S3 and S4) approximate the magnetic torque accordingly as: where ! is the maximum magnetic torque, is the angular orientation of the anisotropy axis inside the magnetic bead, and ! is the orientation of the external magnetic field. In such a potential, the trap stiffness near the bead's equilibrium position ! is !"#$ = − !"#$%& = 2 ! , from which it follows that the maximum magnet torque is ! = !"#$ 2. From previous reports [3], we know that near the bead's equilibrium position, !"#$ = + , where is the effective volume of superparamagnetic nanoparticles inside the bead, is the magnetic field, and is the magnetization, which depends on the field . We make three remarks about this sine approximation for the magnetic torque: first, the potential for aligned (an approximation itself) superparamagnetic nanoparticles looks more like a skewed sine and approaches an actual sine function only in the low and high field limits; second, although the magnetic torque averaged over one revolution is zero ( sin 2 !! ! = 0 with = − ! and = − ! ), the relevant parameter here is the time-averaged magnetic torque over one revolution , which is nonzero for nonzero fields; third, the magnetic torque reduces the time-averaged motor speed, but the instantaneous motor speed at certain bead orientations actually increases compared to the drag-limited speed (Supplement S9), because at those orientations the motor torque and magnetic torque work in the same direction. In addition to this sinusoidal approximation, the analytical calculations assume deterministic behaviour, i.e., !!!"#$% = 0. This deterministic approximation (Supplement S3) converges to the stochastic solution (Supplement S4) in the low and high field limits. The effect of noise is most apparent near the critical point, stall, where motor torque and magnetic torque are similar in magnitude.

S3 Average speed in the deterministic approximation
In the deterministic approximation, the effect of thermal fluctuations is ignored, i.e., !!!"#$% = 0. The noise-free equation of motion for a magnetic bead spun around by a rotary motor in an external magnetic field in the overdamped limit is given by: where each of the terms is as indicated in the main text. We substitute the appropriate parameters: where !"#$ = . When the orientation of the external magnetic field is fixed, as in our current experiments, ! !" !!! ! = !"#$ , and we rewrite the noise-free equation of motion as:

Equation S7
The speed of rotation in Hertz is then given by: and the corresponding angular speed in rad/s is:

S4 Average speed in the stochastic description
We have an analytical expression for the deterministic approximation (Equation S9), but we would also like to understand the effect of thermal fluctuations. We consider the equation of motion (Equation S6), which describes a tilted periodic potential. The periodic potential is: An external force tilts this periodic potential; in our system the tilting force is the motor torque. The tilted periodic potential is:

Equation S11
Without the assumption of a noise-free system as in Supplement S3, Stratonovich showed that the time-averaged speed is described by [6]: , Equation S12 where ! = ! !"#$ , = , and we arbitrarily choose to use ! , which is given by [6]:

Equation S13
To understand the effect of thermal fluctuations, we compare the deterministic solution (Equation S9) to the stochastic solution (Equation S12) at different temperatures, where we perform the integration in Equation S13 numerically ( Figure S1). All curves overlap in the low and high field limits, as they should. For decreasing temperatures, the curves move towards the analytical solution of Equation S9, as the thermal fluctuations reduce to the curve where they are not considered (Equation S9). The stochastic approximation (Equation S12) includes thermal effects, however, it is less convenient to use than the deterministic approximation (Equation S9) because of the integral ± (Equation S13). Both the deterministic and stochastic approximations still assume a sinusoidal description of the magnetic potential (Equation S10), which might be incorrect. Therefore we also perform numerical simulations (Supplement S5).

S5 Numerical simulation of angular traces
We numerically simulate angular traces of a magnetic bead spun around by a rotary motor in an external magnetic field. We repeat the simulations at different magnetic field strengths, and extract the average speed, the mean angle, and the standard deviation in the angle. Starting with the equation of motion, filling in all terms, and rewriting, we obtain: where !"#$%& is the magnetic torque, explained below, !"#"$ is the motor torque, and !!!"#$% = ! 2 ! !"#$ Δ , where ! is a random number drawn from a normal distribution with zero mean and unit standard deviation.
The magnetic torque !"#$%& is calculated as described previously [3]. Summarizing, we calculate the torque on a single superparamagnetic nanoparticle, which depends on the external magnetic field and the angle !" between anisotropy axis and field. To obtain the torque on a single bead, we assume the bead consists of a polymer matrix with embedded, identical, superparamagnetic nanoparticles that are all aligned in the same direction, so !"#$%& = !" !" . We extract the torque on the bead as a function of field and orientation !" from a lookup table during the simulations to reduce computation time.
Two examples of simulated angular traces are depicted in Figure S2 and Figure S3.
We simulate the angle , and we deduce the corresponding and positions using the parametric equations for an ellipse, arbitrarily selecting a length for the major and minor axes. Two examples are shown: one at 1 mT (Figure S2), where the speed is essentially only limited by drag; and one at 24 mT (Figure S3), when the motor nearly stalls. We display the and positions first to facilitate comparison with the experimental datasets ( Figure S7 and Figure S8).  Traces simulated over a wide range of fields are used to extract the average motor speed, the bead orientation during stall, and the angular fluctuations of the bead during stall versus magnet distance. These extracted quantities are plotted in Figure  S4, Figure S5, and Figure S6, respectively. These figures serve as comparison to    The blue data points are the numerically simulated data.
In Figure S5 and Figure S6, we observe that the mean and standard deviation in the angle do not go to zero for high fields, but level off at a nonzero value. This is because the trap stiffness saturates at high fields. The expected mean angle at high fields is ! ! asin !"#"$ !,!"# ≈ 5°, and the expected standard deviation at high field is ≈ 1°.
From Figure S4 and Figure S5, we observe that the analytical expressions fit the numerical simulations reasonably well. This good agreement makes it possible for us to employ the analytical expressions to compare to the experimental data instead of the numerical simulations. As a concomitant advantage, the analytical expressions are more convenient to use than the numerical simulations, because the calculations are faster.

S6 Experimental traces of the bead position
In our magnetic tweezers assay, a tracking algorithm determines the position of the bead in each of the recorded video images. We convert the extracted and positions to an angular position using a fit to the mathematical function that describes an ellipse. Two examples of such angular traces are depicted in Figure S7 and Figure  S8: one at 1 mT when the speed is essentially only limited by drag ( Figure S7) and one at 9 mT when the motor nearly stalls (Figure S8).

S7 Motor losing function
In a particular measurement, we found the motor to lose some and later on most of its torque-generating power (Figure S9). We find such a case to be exceptional. of the main text with !!!" = 400 pN·nm/rad [7,8], !"#$ is based on previous results [3], and !"#"$ as in (C). The dotted line indicates the fluctuations under a minimal stiffness, only due to the hook for !!!" = 400 pN·nm/rad.
In this case, the motor torque reduces during stall. This reduction can be seen from Figure S9(A) and (C). At the start of the experiment (blue data in the average speedversus-magnet height plot), the motor rotates at approximately 35 Hz. As the magnetic field in the sample plane increases, the average speed decreases until the motor stalls. During stall, the magnetic field strength does not affect the average speed; the speed remains 0 Hz. When reducing the magnetic field strength again, the motor only escapes from stall at lower field strength than it entered stall, indicating a reduced motor torque. Further, after escaping from stall, the motor does not recover to its initial speed, again indicating a reduced motor torque. Figure S9(C) shows that the motor enters stall (blue data) at a higher field than it escape from stall (red data). In addition, it requires higher field strengths to force the bead to align with the field than when the magnetic torque is reduced again, indicating a higher motor torque during the first part of the period of stall than during the second part.
When the magnetic field becomes less than 2 mT, the motor speed suddenly drops to less than 0.5 Hz. As if the reduced motor torque after stall was an indicator for some sort damage to the motor, the motor speed reduces just when the magnetic torque starts to become negligible (≈ 1-2 mT). The change in speed happens instantaneously on the time scale of our measurement; the average speed at magnet height 16.5 mm is 8.4 Hz, because the sudden change in speed happened during that measurement point, and we averaged over the full measurement point.

S8 Double spring system: motor-bead-magnetic trap
In our experimental system (Figure 1 of the main text), the thermal fluctuations of the magnetic bead are constrained by the magnetic trap and by the flagellar system of motor and hook. This system is described as a double spring system with two torsional springs, the magnetic trap and the hook, working in parallel ( Figure S10).
Because the two springs work in parallel, the stiffness of the system is simply the sum of the two spring constants, i.e., !"!#$% = !"#$ + !!!" . The amplitude of the angular fluctuations depends on the thermal energy ! and on the stiffness of the system !"!#$% , and is described by equipartition theorem, i.e., ! = ! !"!#$% .
In our magnetic tweezers setup, !"#$ depends on the external magnetic field, and can be varied from ≈ 0 − 10 • . The hook stiffness !!!" is independent of the magnetic field, but depends on the external torque stored in the hook; a normalsize hook has a soft initial phase with !!!" ≈ 0.4 • / up to ≈ π rotation and a more rigid phase thereafter [7,8]. Assuming the torsional stress in the hook results from a motor torque of approximately 1 • , the hook dominates the stiffness of the system at low fields, whereas at high fields, the magnetic trap dominates. Therefore the hook stiffness cannot be neglected, as is usually done for nucleic acid tethers, and should be considered when assessing the thermal fluctuations of the bead.

S9 Instantaneous speed in the deterministic approximation
In Supplement S2, the third remark about the sine approximation mentions the instantaneous speed of the motor. Here, we address the instantaneous speed of the motor in the deterministic regime in contrast to the average speed of the motor covered in Supplement S3. Rewriting Equation S.9 of Ref. [4] gives the angle of the bead: = arccot tan − ! − 4 Equation S15 where = , and , ! , and ! are as defined in Supplement S3, i.e., = − ! , ! = ! !"#$ , and ! = !"#"$ !"#$ . An example trace is plotted in Figure S11(A). The derivative of Equation S15 gives the instantaneous speed:

Equation S16
An example of an instantaneous speed trace is plotted in Figure S11(B).
At a range of magnet heights, we compute the instantaneous speed over multiple turns, and we determine the average speed, its standard deviation, its maximum, and its minimum. The results are plotted below.

Figure S12
Speeds versus magnet height. The blue data show the average speed over multiple turns and its standard deviation. The green data show the highest and lowest speeds. The red line is the analytical expression given in Equation S8 .
As already mentioned in Supplement S2, the instantaneous motor speed at certain bead orientations actually increases compared to the drag-limited speed. This happens because at those orientations the motor torque and magnetic torque work in the same direction.