From Brownian to Deterministic Motor Movement in a DNA-Based Molecular Rotor

Molecular devices that have an anisotropic periodic potential landscape can be operated as Brownian motors. When the potential landscape is cyclically switched with an external force, such devices can harness random Brownian fluctuations to generate a directed motion. Recently, directed Brownian motor-like rotatory movement was demonstrated with an electrically switched DNA origami rotor with designed ratchet-like obstacles. Here, we demonstrate that the intrinsic anisotropy of DNA origami rotors is also sufficient to result in motor movement. We show that for low amplitudes of an external switching field, such devices operate as Brownian motors, while at higher amplitudes, they behave deterministically as overdamped electrical motors. We characterize the amplitude and frequency dependence of the movements, showing that after an initial steep rise, the angular speed peaks and drops for excessive driving amplitudes and frequencies. The rotor movement can be well described by a simple stochastic model of the system.

O ne of the major differences between nanoscale devices and macroscopic machines is the dominance of Brownian motion at small scales and, thus, the presence of large thermal fluctuations that are superimposed on the desired movement of the devices.Biology has evolved intricate mechanisms for the operation of its molecular machines that allow the "rectification" of undirected Brownian movement to power locomotion and other processes at the nanoscale.−9 Ratchet systems have a periodic but anisotropic potential landscape, which can be switched by an external field or a chemical reaction.In biology, ratchet-like mechanisms are thought to be involved in the movement of DNA and RNA polymerases, 10−13 the synthesis of ATP by F0−F1 ATPase, 14,15 and linear transport motors such as kinesin. 16There have been various attempts to implement synthetic Brownian motors in physical and chemical systems, e.g., using optically trapped microparticles, 17,18 microfluidic systems, 19 or synthetic molecular motors driven by light. 20It has been noted, however, that externally driven systems behave fundamentally different than chemically driven molecular machines, the latter of which are constrained by the principle of microscopic reversibility. 21,22ecently, the DNA origami technique 23−25 was utilized to create a DNA-based rotary device with explicitly designed ratchet-like obstacles. 26In this study, the origami rotor was driven out of equilibrium using an electrical field, whose direction was periodically switched from one direction to its opposite (i.e., by 180°) and back.A similar mechanism had been previously studied theoretically. 27,28The rotor displayed directional rotation with the direction of movement determined by the relative orientation of the electric field and the origami structure.Importantly, the nonrotating field by itself would not lead to any biased rotor movement, but its superimposition with the potential of the origami structure does.
We had previously characterized the intrinsic potential landscapes of various DNA origami rotor structures, in which a DNA rotor arm was attached to a pivot point on a DNA origami base plate via single-stranded DNA connectors. 29Due mainly to a slight structural bending of the base plate, the rotors almost always displayed two preferred orientations, inadvertently resulting in a rotatory energy landscape with two minima.We surmised that an externally applied nonrotating electric field that is misaligned with these intrinsic potential minima would create an anisotropic effective energy landscape, which would allow the device to convert Brownian motion into directed rotatory motion when it is driven out of equilibrium.This is in contrast to previous studies, in which we had actively moved the rotor arms with a rotating electrical field. 30,31n the following, we demonstrate that when periodically switching the DNA origami rotors with sufficiently low external fields, the device indeed behaves like a Brownian motor.For high fields, the device transitions from the Brownian to a deterministic regime, where its steps are clocked by the external frequency and where it behaves like an "overdamped electromotor".
Design and Characterization of the DNA Origami Rotors.Our rotor structure 29,31 consists of a 55 nm × 55 nm stator base plate with a DNA six-helix bundle (6HB)�the rotor arm�attached to its center (Figure 1a).The stator is immobilized on the substrate via biotin−streptavidin linkages, while the 463 nm long rotor arm is connected to the stator via a flexible joint.The rotor arm consists of two subunits, the first of which has a length of 50 nm and includes the joint and the connection to the stator.The second subunit comprises a 6HB with a length of 413 nm and is attached to the first subunit as an extension.Previously, 31 the joint consisted of two single strands of DNA, which were shown to wind around each other during rotation of the arm until it stalled.To facilitate unlimited unidirectional rotation, we enzymatically cut one of the connecting strands, as indicated in Figure 1a. Figure 1b shows AFM and TEM images of the resulting DNA nanostructure.
We labeled the tips of the rotor arms with 42 dye molecules each and used total internal reflection fluorescence microscopy (TIRFM) to record their motion.The movement was slowed by using a high viscosity buffer containing 48% (w/w) sucrose.Figure 1c displays a sample from one camera frame (exposure time of 5 ms), showing multiple rotor structures.Figure 1d shows a localization heat map of the acquired video data for a single experiment.Freely rotating structures, indicating a correctly cut, single-stranded joint, are selected for further analysis (dashed circles in Figure 1d).The center of each structure is determined (Figure 1e), and the positional coordinates of all localizations are subsequently transformed to relative polar coordinates.The rotor orientation with respect to the external camera reference frame is denoted as φ, while the integrated angular distance covered by each rotor over time is defined as φ c .Experimental details are given in the Supporting Information.
As demonstrated previously, 29,30 the rotors assume two preferred orientations in free diffusion measurements (Figure 1f).Histograms of the rotor positions provide an estimate for the equilibrium angle distribution p(φ), which in turn allows us to reconstruct the intrinsic potential energy landscape E in of the rotor on top of the stator by inverting the Boltzmann relation, i.e., E in (φ) = −k B T ln[p(φ)].The resulting potential can be well approximated by the function with the fit parameters A = 0.98, B = −2.24,and C = −0.108.This function is 2π-periodic and has minima at ϕ = 0 and ϕ = π.We arbitrarily defined the angle of the lower minimum as ϕ = 0, which we take as the reference angle of the rotor structure (the angle ϕ is measured with respect to this angle, while φ is measured with respect to the camera frame).
Brownian Motor Movement at Low External Driving.We experimentally assessed whether the DNA rotors would display directional movement (Figure 2a) by monitoring  29,31 All images and data are generated with the devices used for the present study.
individual rotors randomly oriented on a microscopic cover slide.To switch the potential, we applied a voltage protocol V(t) that periodically generated 100 ms long electrical pulses with a given field direction, followed by 100 ms long pulses in the opposite direction (i.e., V(t) = V 0 sgn(sin ω 0 t) with f 0 = ω 0 /2π = 1/(200 ms) = 5 Hz).We tracked the movement of the tips of individual rotor arms for different values of V 0 (170−400 structures per measurement) and determined the integrated angular displacement φ c (t) of the arms from the data.These were then used to calculate the effective angular velocity ω(t) of the rotor arms.We also determined the orientation of the individual rotor arm platforms with respect to the external field, characterized by the angular mismatch α, and then plotted the angular velocity of each arm as a function of this angle.Application of voltages below V 0 = 20 V did not result in noticeable directional movement of the rotors (Figure 2b), which indicated that the intrinsic energy barriers were not sufficiently modulated by the external field.Directional movement was clearly observed, however, when applying higher voltage amplitudes (Figure 2c,d).The speed of the rotors had a sinusoidal dependence on the angular mismatch (∝ sin 2α), with a maximum absolute value at α = ±π/4, and vanishing net movement for α = 0, ±π/2.As indicated, we find movement in the counterclockwise (CCW) direction (ω = ϕ ̇c > 0) for 0 ≤ α < π/2 and clockwise (CW) rotation (ω < 0) for −π/2 ≤ α < 0.
Angle Dependence of the Energy Landscape.Our experimental observations can be rationalized by considering the superposition of the externally applied bias potential with the intrinsic potential landscape of the rotor E in .In thermodynamic equilibrium, the rotor arm diffusively moves in its intrinsic potential without directional bias; i.e., clockwise and counterclockwise movements cancel each other on average.Applying an alternating (AC) electric field with orientation −α with respect to the rotor platform will seesaw the intrinsic potential by adding an oscillatory term ∝ V 0 cos(ϕ + α) × sgn(sin ω 0 t) (the argument of the cosine is ϕ + α due to the definition of the angle α, cf. Figure 2).This results in a total effective potential that is given by Here ξ represents the coupling strength that converts externally applied voltage into mechanical torsion energy.Importantly, the external field alone does not impose any directional movement on the rotor.However, the combined time-dependent potential E tot can give rise to a ratchet-like effect whose strength depends on the angular mismatch α.This can be easily understood by considering the shape of the combined potential in its two states for different angles α (Figure 3a).When the external field is in its positive or negative half-cycle, the total potential will have the shape The intrinsic potential has a minimum at ϕ ≈ 0 and, for small values of the parameter C, two neighboring maxima at ϕ ≈ ±π/2, which separate it from the other minimum at ϕ ≈ ±π.
When the angular mismatch is α = 0 (more generally, α = nπ (n ∈ )), the external field pulls in the direction of one minimum in one-half-cycle and in the direction of the other minimum in the other half-cycle.Depending on the magnitude of the modulation V 0 , the arm will either remain in the local minimum or transition to the other minimum.As the landscape is (almost) symmetric for α = 0, there is no directional bias for this transition.By contrast, for other angles α, one of the barriers will be reduced and the other elevated in each half-cycle of the electric field, favoring transitions between the minima to occur always in the same direction (see Figure 3a; cf.ref 5).When the external modulation is strong enough, the rotor arm can escape the minimum with the help of thermal fluctuations, leading to CCW Brownian motor-like movement for 0 < α < π/2 and CW movement for −π/2 < α < 0. A stochastic simulation of the Langevin equation for the rotor system shows the expected behavior (Figure S5).We find that the rectification effect is strongest for α = ±π/4, which is in agreement with our experimental observations (Figure S6).Transition to Quasi-Deterministic Motor Movement.Our model suggests that for high enough voltages the energy landscape will transition from a potential with two minima within [−π, π] to a potential with only one minimum (Figure 3b).When for α = π/4 the external field is increased, the potential minima and maxima that initially were at ϕ ≈ π (mod 2π) and ϕ ≈ 3π/2 (mod 2π), respectively, move toward each other and merge at ϕ = 5π/4 (mod 2π) when V 0 ≈ 4.55 k B T/ξ (Figure 3b).Thus, for strong enough fields, the energy barrier in one direction actually vanishes, and therefore the rotor arm can move without assistance of thermal fluctuations.In this regime, the overdamped system is expected to move quasideterministically within its potential landscape.
Our simple rotor design allows application of much higher voltages than the more complex multicomponent origami rotors studied previously. 26As shown in Figure 4, the rotor moves much faster when rectangular voltage signals with amplitudes above ≈100 V are applied.Furthermore, the φ c (t) traces indicate that in contrast to the Brownian regime, the rotor indeed rarely reverts its direction (Figure 4a).As anticipated, also in the deterministic case, the rotor velocity has a sinusoidal dependence ω ∝ sin 2α, and the movement of the rotor arms is observed fastest for α = π/4 (Figure 4b).
For f 0 = 5 Hz, we observed the highest speed with a rectangular signal around V 0 = 150 V, which resulted in a rotation frequency of f = ω/2π ≈ 3.1 Hz.Notably, application of higher voltages leads to a reduction in the frequency of the rotor rotation, e.g., to f ≈ 1.6 Hz at V 0 = 300 V (Figure 4c).Our model for E tot (ϕ,t) eq 3 indicates that for higher voltages, the potential landscape is completely dominated by the externally imposed potential and thus becomes almost symmetric.Simulations show that under these conditions, the rectification effect is indeed diminished (Figure S8).
As the maximum possible angular velocity is given by f 0 = 5 Hz, we only achieve ω max,exp ≈ 0.61 × ω max,theo at this frequency.This suggests that the rotor arm is not able to follow the external field in every cycle (i.e., it has ≈39% idle cycles).In order to study the frequency dependence of the rotor arm movement in more detail, we systematically changed the switching frequency f 0 for a constant voltage amplitude of V 0 = 150 V (Figure 4d,e).While the speed steadily rises when going from f 0 = 1 to 2 and finally 5 Hz, it is strongly reduced for f 0 = 10 Hz.This behavior is a consequence of the overdamped movement of the rotor arm.For higher frequencies, the rotor cannot follow the switching of the potential landscape anymore and therefore increasingly undergoes idle cycles with Δϕ = 0.The critical frequency f c is thus set by the friction coefficient γ r of the rotor arm, which is determined by its geometry and the viscosity of the medium.The frequency dependence of the rotor arm movement can be recapitulated in Brownian dynamics simulations (Figures S9−S11), which show that for large enough voltages f c ∼ V 0 /γ r .
We have shown that a rotary DNA origami nanodevice "serendipitously" acts as a Brownian motor due to its intrinsic energy landscape that contains two energy minima with a depth on the order of 1 k B T. When the rotor arm diffusively explores this energy landscape, it will randomly transition between these minima without any directional bias, leading to an overall zero net movement.Directional movement can be induced by externally applying an electric field that is switched back and forth between two opposite directions.In such a setting, the electric field alone does not provide any directional bias, but the superimposed potential generated by the intrinsic mechanical landscape and the external field does.Depending on the relative orientation of the external field and the intrinsic minima (measured by the angular mismatch α), the effective potential landscape will be more or less asymmetric.When the system is driven out of equilibrium by switching between two alternative asymmetric potentials (corresponding to the two field directions), the diffusive movement of the rotor arm can be rectified.The effect turns out to be maximal for α = ±π/4.
The overall behavior of the system in the low-voltage regime is reminiscent of a Brownian ratchet. 5The canonical flashing ratchet model considers an intrinsically asymmetric energy landscape that can be modulated or switched completely on and off.In our system, the asymmetry is created by the superposition of external and intrinsic potential.Even though our measured intrinsic potential is itself also slightly asymmetric (due to the nonvanishing term C in eq 1), simulations indicate that this is not a necessary requirement.As a caveat, we must note that our estimated intrinsic potential is only an approximation.First, the Boltzmann inversion is likely not accurate for angular positions close to the potential barriers, where we cannot collect many data points.Furthermore, we do not know whether the intrinsic landscape will change in the presence of electric fields (e.g., via structural deformation).
For low amplitudes of the external field, thermal fluctuations are required to drive the movement, and our system behaves as a Brownian motor.Accordingly, in our Langevin simulations the movement of the rotor arm ceases when the fluctuation term is set to zero (Figure S5).Notably the behavior is different for the high-voltage case, where the intrinsic landscape is distorted by the external potential to such a degree that the rotor arm deterministically moves downhill after switching the potential (Figures S4 and S7).
With an estimated rotational drag coefficient of γ r = 1.1 pN• nm•s 31 and a moment of inertia of I = ML 2 /3 = 6 × 10 −34 kg• m 2 that can be derived from the mass M of the ≈8000 DNA base-pairs (with 650 Da per base-pair) comprising the arm and its length of L ≈ 463 nm, the relaxation time t r = I/γ r is way below 1 ps.The rotor arm thus is completely overdamped and, in the quasi-deterministic regime, will move with an angular drift velocity that is given by ω drift = τ/γ r = −1/γ•dE tot /dϕ.Because of the absence of inertia, the rotor arm will immediately stop when the field is switched off.
In conclusion, we have shown that a rotary DNA origami nanodevice composed of a DNA rotor arm attached on a rigid stator plate can be repurposed as a nanoscale electromotor by externally applying a nonrotating, periodically switched electrical field whose direction is mismatched with the minima of its intrinsic energy landscape.Depending on the strength of the electric field and the corresponding modulation of the potential energy, the rotor acts as a Brownian motor, utilizing and rectifying the thermal fluctuations of the system, or as an overdamped electrical motor.Our results show that a Brownian motor can be generated from a DNA-based nanodevice without explicit design by simply utilizing irregularities in its potential landscape.One of the major challenges for future work will be realization of out-ofequilibrium systems by other means, in particular Brownian motors that are fueled by chemical reactions. 32,33The working principle of chemically driven molecular machines is fundamentally different from the externally driven system shown here, however, as they have to be able to "gate" a chemical reaction depending on their current mechanical state. 21,22ASSOCIATED CONTENT

Figure 1 .
Figure 1.Overview of the structure design and data analysis workflow.(a) DNA origami nanostructure consisting of a 55 × 55 nm 2 stator base plate, which bears a 463 nm long rotor arm connected to it via a flexible joint.The rotor consists of two subunits: a 50 nm long part (blue) connected to the stator through the joint and a 413 nm long rotor extension (yellow).The rotor is labeled at the tip with 42 fluorescent dyes (indicated by the red spot).The flexible joint consists of two single-stranded DNA domains of the scaffold, one of which is cut to enable free rotation (cf.magnified inset).(b) AFM (top) and TEM (bottom) images of the fully assembled nanostructure.In schematic images we use the pictogram shown in the inset for the structure.(c) Single camera frame of the acquired video data.The fluorescence intensity is displayed in pseudocolor.(d) Pseudocolor localization heatmap of the total acquired data of one experiment.Dashed circles show freely rotating structures that are selected for further analysis.(e) Fitting a circle to the tracking data of a single structure allows to determine the position of the stator plate in the center and transform positional coordinates of individual localizations into relative polar coordinates.(f) The angular positions explored by the arm via diffusion (blue histogram) allow the calculation of the intrinsic energy landscape of our structure (shown in red), which exhibits two energy minima separated by ≈180°.The experimental workflow was established in previous work.29,31All images and data are generated with the devices used for the present study.

Figure 2 .
Figure 2. Brownian motor movement at low external driving.(a) Schematic explaining how the superposition of the time-dependent electric field and the intrinsic energy landscape can lead to directed movement.The field stabilizes the rotor arm in one of the two energy minima of the stator.When the field direction is switched, the arm preferentially moves in the clockwise direction for the particular arrangement shown here.(b) Left: particle tracking traces were recorded in the absence of an external field.The ensemble of all trajectories is shown in gray; an exemplary trace is highlighted in black.Right: the effective angular velocity ω for each trace is plotted against the corresponding angular mismatch α.(c) Left: exemplary rotor arm traces in the presence of an applied external field (V 0 = 80 V) are whose direction is switched back and forth with f 0 = 5 Hz.The blue trace shows a rotor moving predominantly in clockwise direction, while the orange trace displays anticlockwise rotation.The full ensemble of trajectories is shown in transparent gray.The angle α is defined as the orientation of the closest intrinsic potential minimum with respect to the direction of the external field, i.e., α > 0 in the upper scheme (CCW rotation) and α < 0 (CW rotation) in the lower scheme.Right: effective angular velocity ω for each trace on the left is plotted against the corresponding angular mismatch α. ω shows a sinusoidal dependence on α on average.The red curve is a fit to the data with ω(α) = ω fit •sin 2α.The maximum angular speed (±ω fit ) is observed for α = ±45°.(d) Plot of ω fit as a function of externally applied voltage.

Figure 3 .
Figure 3. (a) Total potential as a function of Φ for different angular mismatches α (cf.FiguresS3 and S4).The dark blue curves show E tot (Φ) for an external potential of ξV 0 = +3.5kB T, and the light blue curves correspond to the opposite polarity (ξV 0 = −3.5kB T).The intrinsic potential is shown with a gray dashed line.When for α = 0 the arm initially is in the minimum at Φ = 0 (indicated by the black spot on the dark blue curve), switching the potential will lift it to the corresponding minimum at Φ = 0 of the light blue curve.As indicated by the red arrows, it may stay at Φ = 0 or escape to either the left (CW) or right (CCW) with a probability that depends on the corresponding energy barriers (one is indicated by a black double arrow).For α = +π/4 the barrier to the left is elevated, while the barrier to the right is reduced and the arm will preferentially transition to the minimum on the right after switching (leading to CCW movement).When it does not escape from the higher minimum on the light blue curve before the potential is switched again, it may undergo an idle cycle, as indicated.For α = +π/2, transitions between the two states of the total potential also do not generate any net movement (for Brownian dynamics simulations in the potentials see the FiguresS5 and S7).(b) Total potential for α = +π/4 for external voltages below, at, and above the critical value V c = 4.55 k B T/ξ (gray dashed line: intrinsic potential).For V 0 < V c , the potential has a low and high minima close to Φ = 0, 2π and Φ = ±π.The rotor arm can thermally escape from the minima at Φ = ±π, leading to Brownian motor movement in the CCW direction.At V 0 = V c , the minima close to Φ = ±π become saddle points, and at V 0 > V c the movement to the right is always downhill until the arm reaches one of the deep minima.

Figure 4 .
Figure 4. Rotor movement at higher driving voltages.(a) Exemplary traces at 150 V applied to the external AC field at f 0 = 5 Hz.The blue trace shows deterministic movement in a clockwise direction, while the orange trace depicts a structure rotating counterclockwise.The full ensemble rotor trajectories is shown in transparent gray.(b) Effective angular velocity ω vs angular mismatch α for V 0 = 150 V.The red curve is a fit to the data with the function ω(α) = ω fit sin 2α.(c) Plot showing the dependence of ω fit on the applied voltage, combining low voltage (black crosses) and high voltage data (red crosses).(d) Frequency dependence of deterministic rotor movement at V 0 = 150 V, displaying ω as a function of α for measurements with f 0 = 1 Hz (black), 5 Hz (blue), and 10 Hz (orange) and the corresponding sine fits.(e) Plot of the angular velocity ω fit for all measured frequencies.