Motor properties from persistence: a linear molecular walker lacking spatial and temporal asymmetry

As conceived, SKIP comprises a linear arrangement of four gated track-binding modules (A and B in figure 1(a)) where A (B) modules only bind a (b) sites on a linear track in the presence of the activating ligand, l_a (l_b). This dependence of track-binding on ligand occupancy of each foot module can be thought of akin to ATP-powered motors, whose track binding is modulated by occupancy of the ATP site [1, 2]. The separation between identical track-binding modules, d, is equal to the site separation on the track, while the separation between the central protein modules A₂ and B₂ is chosen to be shorter (e < d). This shorter central link is critical in achieving directional persistence [26]. The track consists of a periodic linear arrangement of binding motifs a₁–a₂–b₁–b₂ where the two a (b) motifs are identical (figure 1(a)).

While the following model is quite general, it is helpful to have a specific experimental implementation of the SKIP walker in mind. For example, modules A and B can be thought of as two different, ligand-gated DNA-binding repressor proteins. The physical properties of such repressor proteins that are relevant to their use as artificial motor modules have been described in detail for our previous design, the tumbleweed [24]. The linear track may then be implemented as a synthetic DNA molecule with specific binding sites a and b [24, 32]. The spacing between adjacent sites on the track, d, is set at 10 nm (table 1) to accommodate repressor binding [24]. The binding sites for repressors on DNA have palindromic sequences, hence the DNA track will be symmetric. Using a microfluidic device to temporally change the concentration of ligands [33], the binding and unbinding of modules A and B to the track can be orchestrated in arbitrary temporal order.

SKIP motility is then driven by changing the concentration of the ligand(s) bathing the track (figure 1(b)). Hence SKIP is essentially a clocked walker where each ligand condition is maintained for a period known as the ligand pulse time (τ_LP) and set by the repetitive sequence: (l_a, 0)-(l_a, l_b)-(0, l_b)-(l_a, l_b). This series of four ligand pulses is sufficient to power one complete forward cycle of SKIP, as illustrated in figure 1(b).

Consider the specific starting condition where initially the two A modules are bound to the track in the presence of ligand l_a only (figure 1(b), state (i)). When ligand l_b is added, only protein module B₁ is able to bind to the track, and only at the b₁ site nearest the bound A modules, due to the geometrical constraint that e < d (see figure 1(b), state (ii)). This geometrical constraint is one key aspect of SKIP's persistence mechanism, and was inspired by the restricted diffusional search of kinesin's forward head (which, in kinesin's case, follows a power stroke). On removing ligand l_a, SKIP remains attached to the track by a single module B₁ (figure 1(b), state (iii)).

We call state (iii) in figure 1(b) the vulnerable state, and the future behavior of the walker is dependent on what happens at this stage: it may step forward, pause or reverse. If module B₂ binds to the track at the adjacent b₂ site, then SKIP has taken a half step forwards, moving a distance of 2d along the track in the time of 2τ_LP (figure 1(b), state (iv)). Here, the binding of B₂ occurs through diffusion, while in kinesin, the 'throw-forward' motion that positions the head for forward binding is driven by the allosteric power stroke [2].

If no track binding has occurred by the time ligand l_a is reintroduced, SKIP may yet succeed in forward-binding (figure 1(b), state (iv)) or take one of three additional actions. The first is that the motor may pause by returning to the initial state (figure 1(b), state (ii)) if either A module binds to its previous track site (figure 1(b), P1 and P2). If this happens, SKIP is out of sync with the ligand pulses and only after an additional 2τ_LP can it resume forward motion. Alternatively, the motor may reverse its direction of travel, which happens if motor module A₁ binds to the track at the a₂ site nearest the bound B₁ motor module (figure 1(b) R). We observe this initial binding always to be followed by binding of A₂ at the a₁ site in the following (l_a, 0) ligand pulse when load is present, thus leading to reversal and a mirror image of state (i).

It is important to note that the SKIP cycle, the track and the ligand-pulse sequence (l_a, 0)-(l_a, l_b)-(0, l_b)-(l_a, l_b) are all symmetric, and the track is not modified by SKIP stepping. A direction of motion is defined only by the initial binding configuration of SKIP. As we will show below, a SKIP in forward (reverse) facing configuration has a large likelihood of continuing to move in the same direction for many steps in sequence. It is also important to note that stepping is entirely diffusive as no power stroke is present in the model. The motor is powered by changes in the chemical potential of the ligand bath, the corresponding energy input of which is quantified below.

To investigate the behavior of SKIP, we simulated its motion using three-dimensional coarse-grained Langevin dynamics [24, 34]. SKIP is represented by a freely jointed linear tetramer composed of three stiff bonds connecting four spheres, each representing a track-binding module. We use the indices j = 1...4 to designate the four SKIP repressors: 1 ≡ A₁, 2 ≡ A₂, 3 ≡ B₁ and 4 ≡ B₂ (figure 1(a)). Let Δx_i^(j) be the change in the value of the ith coordinate (i = 1, 2, 3) of the jth track-binding module over an incremental time, Δt, at time t. The overdamped Langevin equation can then be written as follows:

$$\begin{eqnarray}&&\Delta x_{i}^{(j)}=F_{i}^{(j)}\Delta t/\gamma +{{\left( 2kT\Delta t/\gamma \right)}^{1/2}}\zeta _{i}^{(j)}\left( t \right),\end{eqnarray} \tag{ 1 }$$

where F_i^(j) is the ith component of the sum of an internal conservative force, F_Ci^(j), and an external force, F_Ei^(j), on the jth track-binding module at time t. γ is the drag coefficient for each monomer and is given by the Stokes–Einstein equation in terms of the radius, r, of the monomeric sphere and the viscosity, η, of the buffer: $\gamma =6\pi \eta r$ . The last term in equation (1) models thermal noise. ζ_i^(j)(t) is a random number taken from a Gaussian distribution with zero mean where

$$\begin{eqnarray}&&\left\langle \zeta _{i}^{(j)}\left( t \right)\cdot \zeta _{i^{\prime} }^{(j^{\prime} )}\left( t^{\prime} \right) \right\rangle ={{\delta }_{ii^{\prime} }}{{\delta }_{jj^{\prime} }}\delta \left( t-t^{\prime} \right)\end{eqnarray} \tag{ 2 }$$

F_Ci^(j) acts on each monomer and is the negative gradient of a potential given by W_H + W_SB + W_LJ. Here W_H is a harmonic potential which defines the lengths d and e of the three bonds between the track-binding modules of SKIP. W_SB is the specific binding potential present at all binding sites on the track. The track is taken to lie along the x₁-axis and the specific binding potential for a track-binding module of type j to its specific recognition sequences is given by ${{W}_{{\rm SB}}}\left( {{r}_{j}} \right)=-{{V}_{{\rm SB}}}{\rm exp} \left( -r_{j}^{2}/d_{b}^{2} \right)$ for r_j < d_b. Here V_SB is the strength of the specific binding interaction, d_b is its effective range and r_j is the distance between track-binding module j and the nearest corresponding recognition sequence on the track. To model the case where a ligand is absent from solution, this specific binding potential is turned off by making W_SB(r_j) = 0. W_LJ is the excluded volume between two track-binding modules and is simulated by a repulsive Lennard–Jones interaction in terms of the minimum distance, σ, between the centers of two spherical monomers representing the track-binding modules.

The components of the external force, F_Ei^(j), which is used to model a load force on SKIP, are taken to be parallel to the track and act only on monomers A₂ and B₂. Unless otherwise stated, the load force is applied in the negative direction (to the left in figure 1(b)), and the total external force, F_ext (the sum of the forces on A₂ and B₂) is reported. This load force is taken to represent an external force applied, for example, by optical tweezers or by an imposed fluid flow at constant velocity, and is distinct from a Stokes drag force caused by an actual load tethered to SKIP, which results in qualitatively different and more complex behavior.

The specific numerical parameters used in the simulations are listed in table 1, and are motivated by realistic values for an experimental implementation using repressor proteins and a DNA track (see [24] and above).

The ligand pulse time, τ_LP, was varied between 0.03 and 1.152 ms. The lower limit is set by the time SKIP needs to diffuse from the state with two modules bound (states (i) or (iv) in figure 1(b)) to that with three modules bound (states (ii) or (v) in figure 1(b)). When τ_LP > 0.03 ms, SKIP is processive under zero load (see below). It was not feasible to model τ_LP > 1.152 ms, timescales that would more closely mimic kinesin's step time (∼10 ms at saturating ATP) [35, 36]. The τ_LP accessible in experiments can be expected to be ≫1 ms.

For all simulations, the motor is started in state (ii) (figure 1(b)). Changes in ligand concentrations are modeled by turning the corresponding binding potentials on or off. Binding of a monomer occurs if it happens to diffuse within a distance d_b of an active binding site. The value of V_SB has been chosen large enough to ensure that, once bound, a monomer does not unbind until the corresponding specific binding potential is turned off.

Langevin simulations were also used to model the behavior of SKIP under feedback control, where feedback was used to reverse the direction of the load force following reversal of SKIP on its track. The calculation of the work done by SKIP under feedback control is performed in the following manner in the Langevin simulations. We begin by assuming that SKIP is in the forward-facing configuration (either states (ii) or (v) in figure 1(b)) at ligand pulse n under a total rearward (load) force F_ext < 0. Then there are three possibilities.

• (i)  
  If SKIP advances two lattice spacings to the next forward-facing configuration (equivalent to states (v) or (ii) in figure 1(b)) at ligand pulse n + 2, the work done by SKIP against the rearward force is given by F_ext*2d, which is added to the total work performed so far.
• (ii)  
  If SKIP reverses its direction via the configuration shown in figure 1(b) R, its configuration after a full ligand cycle (at ligand pulse n + 4) becomes reverse facing (i.e., a mirror image of either states (ii) or (v) in figure 1(b)), and the position of SKIP's center of mass is changed, on average, by 1.062d in the reverse direction, based on the geometry of the motor. In this case the work done by the rearward force is 1.062d* F_ext. This is subtracted from the total work performed so far and the external force is reversed at this point in the simulation run, i.e., F_ext is replaced by −F_ext.
• (iii)  
  If SKIP's position is unchanged via one of the configurations shown in figure 1(b) P1 and P2 after two τ_LP (pausing), no work was done.

If SKIP is in a reverse-facing configuration at ligand pulse n with a forward external force, the same three possibilities exist for work output in the mirrored transitions.

Monte Carlo simulations were undertaken to investigate the role of the vulnerable state on the processivity of SKIP and to characterize the effect of feedback on motor efficiency. The probabilities of the four possible transitions from the vulnerable state (state (iii) in figure 1(b)) were used as input: P_F, P_P1, P_P2 and P_R, for forward (iii)–(iv), pausing and reverse transitions. These probabilities were determined from the Langevin simulations for each force and value of τ_LP under investigation (see figure 2(a)). If the forward transition occurred, we assumed that this transition always leads to productive forward motion, so the position of SKIP was changed by 2d upon forward translocation, the time was advanced by 2τ_LP, and the choice among P_F, P_P1, P_P2 and P_R recurred for the next step. In this case, the work performed by SKIP was F_ext*2d, taken to be positive since this motion occurred against the applied force. If a pause transition was selected, the position of SKIP remained unchanged, no work was performed, and the time was updated by 4τ_LP, as a full cycle was needed before forward motion would become in sync with the externally regulated ligand supply. Finally, if a reverse transition was selected, we assumed this would always lead to a reverse-facing SKIP configuration (mirror image of state (i)), and consequently the walker's position was changed by −1.062d. The time was then updated by 4τ_LP, the time required to achieve this full reorientation of the walker. The work performed by SKIP was −|F_ext|*1.062d, and at this point, the direction of the external force was reversed, so that it again applied a load to SKIP, whose direction was taken to be the 'new' forward.

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We investigated the force-dependent probability of different steps in the walker's cycle by Langevin simulations, recording first-passage times (FPT) for transitions from state (ii) (see figure 1(b)) following the removal of ligand l_a, which results in the unbinding of A-type track-binding modules and SKIP entering the vulnerable state (iii). Figure 2(a) shows a representative set of force-dependent probabilities for τ_LP = 0.576 ms. At low external force, the diffusive search process is fast enough that the probability of executing the forward step during τ_LP remains near unity, even as this rearward force increases. When the force exceeds about 5kT/2d for this τ_LP, the forward probability decreases rapidly with increasing rearward force as the transition from states (iii) → (iv) is increasingly inhibited within τ_LP. This provides access to other processes (pause 1, pause 2 and reversal, labeled P1, P2 and R in figure 1(b)) in the subsequent ligand environment (l_a, l_b).

To characterize the force at which the forward probability starts to decrease we define F₉₉ as the force at which the forward transition decreases to a 99% probability. As a proxy for a stall force, we define F_S as the force at which forward and reverse transitions are equally probable (figure 2(a)). The values of F₉₉ and F_S depend on τ_LP: for longer ligand pulse durations, the motor has more chance to forward-bind, and can execute this transition under increasing load (figure 2(b)). The shapes of the transition probability curves in figure 2(a) only change substantially as τ_LP becomes shorter than the FPT: at sufficiently short times, SKIP may not successfully move forward even at zero load (P_F < 1), and there is a small but finite probability of detachment (supporting figure 1 available at stacks.iop.org/NJP/17/055017/mmedia).

We note that for very high rearward forces, pausing is predicted from the probabilities (figure 2(a)) to dominate, keeping the walker stationary on the track. This is also visible in the initial part of the trajectory for F_ext = −2 pN in figure 2(c). It is worth noting that SKIP's force tolerance is remarkably high for a purely diffusive motor. For the value of τ_LP = 0.576 ms, we find F₉₉ ≅ −1 pN or ∼ −5 kT/2d, where kT/2d is the natural force scale of a diffusive motor with SKIP's step size, 2d. We discuss a comparison between SKIP and the natural protein motor kinesin at the conclusion of this paper.

As predicted by the force-dependent transition probabilities, SKIP behaves as a persistent walker under low rearward force, stepping processively in the forward direction established by its initial bound configuration on the track (figure 2(c) and supporting movie available at stacks.iop.org/NJP/17/055017/mmedia). It occasionally pauses, however, before resuming forward motion or reversing. The latter is shown in figure 2(c) (F_ext = −6.4 kT/2d or −1.3 pN). As F_ext increases in magnitude, SKIP is more likely to undergo pause and reversal, doing so earlier in its run. Finally, under sufficiently large forces (for τ_LP = 0.576 ms, |F_ext| > ∼ 8 kT/2d or ∼1.6 pN), pausing dominates as predicted by figure 2(a), leading to long-lived stationary paused states before the walker eventually reverses direction (e.g. |F_ext| = 2 pN in figure 2(c)). Under any force, once the direction of motion is aligned with the external force, the motor undergoes unidirectional processive motion assisted by this force (figure 2(c)).

The persistent run length of SKIP is remarkably high and depends both on the load force, F_ext, and τ_LP. Figure 2(d) shows the mean persistence time (number of 2τ_LP) and distance (number of 2d steps) as a function of load force for τ_LP = 0.576 ms. Both persistence times and lengths are exponentially distributed at all forces, consistent with expected Poisson-type behavior, though their timescales differ at high forces. The mean run length and time increase exponentially as load force decreases below 6.5 kT/2d (1.3 pN), with SKIP taking hundreds of unidirectional steps when |F_ext| = |F₉₉| before reversing direction. We emphasize that this is achieved only by persistence of the initial direction, with no asymmetry in the track design or ligand pulse sequence. At higher load force the mean persistence time appears to plateau while the mean persistence length goes to zero. Here the run length is limited by stalling. Below 4 kT/2d (0.8 pN), P_F ≈ 1, whereby SKIP continued to move forwards for the duration of the simulations. For comparison, kinesin, considered a processive molecular motor, takes about 150 steps before detaching from its microtubule track [35, 37].

Because SKIP includes no asymmetry in its track design or ligand pulse sequence, persistent 'forward' and 'backward' motion are equivalent in the absence of an external force. This feature allows the interesting possibility of using SKIP as a shuttle, for example to transport cargo from one end of a track to the other end, where signals could be present to unload and pick up new cargo for reverse transportation. Such a characteristic is neither possible with biological walkers such as kinesin, whose tracks possess polarity that directs motor motion, nor with previous synthetic walkers, which either exploit track polarity or alter the track as they pass. A simple modification of the track enforces the shuttling of SKIP: each end of the track has only one a (or b) binding site, so that the walker remains in the vulnerable state (figure 1(b) (iii)), diffusively searching for the reverse-binding transition (figure 1(b), R). This introduces an element of local asymmetry into the track design. A simulation of shuttling with such a track design is shown in figure 3 (see also supporting movie available at stacks.iop.org/NJP/17/055017/mmedia), illustrating that while it can take a number of pulse sequences before SKIP successfully changes direction at the end of the track, processive shuttling motion is easily achievable.

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A modification of this design can establish a means by which to enforce the initial direction of SKIP. A 'localization beacon' could be included at one end of the track, with a recognition domain included in SKIP. In the absence of ligands l_a and l_b, SKIP would be driven to bind to the track at one end; upon introduction of ligand pulses SKIP would orient to move away from this end of the track. Inspired by a similar strategy used in a different synthetic motor system [30], incorporation of this beacon would enable control over the otherwise random initial orientation and location of SKIP on the track.

As the load force increases, SKIP is increasingly less likely to continue in a forward direction. For cargo transport against a force, this is an undesirable property. From a physics point of view, this behavior does, however, provide an intriguing possibility to explore use of feedback as a means of enhancing motor performance. From information about the walker's directionality, the external force can be reversed so that it always exerts a load, thereby providing a potential means of increasing the work done against an external force by the walker. The operation of a motor in this way corresponds to an implementation of a Maxwell demon: using some mechanism to read the state of the motor (its momentary stepping direction), an external agent would take action by adjusting the direction of a load force to maximize the work extracted from the motor. It is worth noting that—strictly speaking—feedback is intrinsically needed to be able to extract work from a motor: if the initial binding configuration is random (as it would be in the absence of a localization beacon), stepping needs to be observed in order to determine the direction of motion and to choose the direction of a load force.

As a question of fundamental physics of motors, we wish to understand the energetic implications of using feedback. How well does feedback work, and what is its thermodynamic cost in terms of efficiency?

The protocols used to implement feedback are described in section 2.2 for the Langevin simulations and in section 2.3 for the Monte Carlo simulations. Once a reversal of motor direction is detected in the simulations, the direction, but not the magnitude, of the applied force is reversed so as to apply a load force opposing the new direction of motion. Figure 4(a) shows sample trajectories obtained using this feedback protocol in Langevin simulations. Up to considerable forces of several kT/2d, the motor executes extended runs against this force in a given direction, and rarely switches provided that the ligand pulse time is large enough. At higher forces (|F_ext| larger than about 8 kT/2d, or 1.6 pN), the motor exhibits shorter runs prior to reversal, thus leading to more frequent switching of force direction. Although the motor may not move far, it performs positive work at every forward step in either direction because of this feedback. Figure 4(b) shows the accumulated work along with the time-trace of the applied force for $|{{F}_{{\rm ext}}}|=2\;{\rm pN}$ (which corresponds to the green trace in figure 4(a)). Since negative work only occurs upon reversal transitions (rare, particularly at lower forces), this leads to a considerable amount of average work per half cycle (figure 4(c)).

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The output work of SKIP operated under feedback control initially increases linearly with applied force, since reversals and pauses are rare, and so most timesteps involve F_ext*2d of work arising from forward stepping against the force (figure 4(c)). Around F₉₉, the work per half-cycle (2τ_LP) reaches a maximum before declining nonlinearly with applied force. This decline results from the increasingly likely transitions to pause and reversal configurations from the vulnerable state, which respectively contribute zero and negative work to the process. Nonetheless, due to the feedback algorithm, the average work per half-cycle remains positive up to reasonably high forces of magnitude greater than 2 pN.

The agreement between the shape of the work-versus-force curve from Langevin simulations and from Monte Carlo simulations provides insight into the mechanism of SKIP's motility. While the Langevin simulations inherently considered all possible transitions and changes of center-of-mass, the Monte Carlo simulations focused only on the vulnerable state, assuming that transitions from this state (figure 1(b) (iii)) were of predominant importance when explaining the force-dependent characteristics of SKIP. This assumption is validated by the reasonable agreement between the work outputs calculated using the two means of simulation (figure 4(c)). This result highlights that improvements to the performance of SKIP should focus on enhancing forward transitions from the vulnerable state under load. In kinesin, this is accomplished by a power stroke [2], whereas SKIP relies on diffusion.

To estimate the energy efficiency of SKIP we need to calculate the input energy. As described briefly in section 2.1, we conceived an implementation of SKIP where stepping is achieved by periodically changing ligand concentrations using a fluidic device. The motor is then powered by moving ligand molecules from temporal regions of high chemical potential to ones with low chemical potential.

On binding to the track, two SKIP modules, AA (or BB) remove two ligands from the high ligand concentration, l_a (or l_b). During release from the track, these two modules release the ligands to a low ligand concentration. This transfer of ligands reduces the free energy difference between ligand pulse solutions. To ensure microscopic reversibility, we now assume that the ligands l_a and l_b are present in all ligand pulses, with a concentration ratio of 10^m between the 'high' and 'low' concentration conditions (i.e., if m = 3, then there is a 1000 fold greater concentration of l_a in (l_a, l_b) than in (0, l_b)). Assuming the ligand pulses are ideal solutions, the change in Gibbs free energy per half-cycle (2τ_LP) is given by:

$$\begin{eqnarray}&&\Delta {{G}_{{\rm half}\,{\rm cycle}}}=-2mkT\cdot {\rm ln} 10.\end{eqnarray} \tag{ 3 }$$

Thus the free energy utilized by the motor is determined by the ratio of the high ligand concentration where the modules bind to ligands and attach to the track, to the low ligand concentration where the modules release the ligands and detach from the track.

The minimum input free energy will depend on details of motor implementation, including the τ_LP. Basing each track-binding module on a ligand-gated DNA-binding repressor, as per our previous design [24], would require the transfer of two ligands per module, as the repressor proteins are homodimeric. Thus the change in Gibbs free energy per half-cycle will be double the value calculated using equation (3). Our previous design of a protein-based motor that walks on a DNA track indicates that a minimum value of m = 3 ($\Delta G=113\;{\rm pN}$ nm or about 28 kT) would likely be required experimentally to enable on/off control of ligand-gated track-binding for τ_LP on the order of milliseconds to seconds [24].

SKIP's efficiency can be estimated from the simulation results and input energy calculations. Here we take efficiency to be the thermodynamic efficiency, ${{\eta }_{{\rm th}}}={{W}_{{\rm out}}}/\Delta {{G}_{{\rm in}}},$ in the absence of external feedback.

Using F₉₉ as the external force, the maximum work performed by the motor in each half cycle (2τ_LP = 1.1 ms) is on the order of 20 pN nm, or about 5 kT (table 2). The efficiency depends on the ratio between the high and low ligand concentrations as per above. If this ratio is ∼10³ (m = 3), then the resulting maximal efficiency will be ∼18% (table 2). This is not much lower than the efficiency of the motor protein kinesin (table 2) [36], whose maximum number of steps is comparable to SKIP's directional persistence at this value of the rearward force (figure 2(d)).

Table 2.  Comparison with kinesin.

	Kinesina	SKIPb
Stall force	5.7 ± 4 pN	∼1 pNc
	5.1 ± 0.5 pN
Step size	8 nm	20 nm
Step time	∼10 ms	∼1 ms
Run length (steps)	125 (from [35] at 0 pN)	∼600 (at F99 and ${{\tau }_{{\rm LP}}}=0.576\;{\rm ms}$)
Maximum work/step	∼40 pN nm	20 pN nm
Input free energy/step	ATP hydrolysis:	Chemical potential:
	∼80 pN nm	113 pN nm (m = 3)d
		227 pN nm (m = 6)
		340 pN nm (m = 9)
Maximum efficiency	50%	18% (m = 3)
		8.8% (m = 6)
		5.9% (m = 9)

^aKinesin parameters based on [36] unless otherwise noted. ^bSKIP parameters based on a half-cycle (2τ_LP) where τ_LP = 0.576 ms. ^cMaximum load force is set at F_99. ^d m is the order of magnitude of the ratio between high ligand and low ligand concentrations (see text).

Of course, if left to run for a long time under a rearward force, SKIP will eventually change direction and step assisted by the external force. Under these conditions it will do negative work, eventually erasing its gains in work (and efficiency) from stepping with an opposing force. However, positive work and hence efficiency can be regained by imposing feedback just after SKIP has reversed (see section 3.4).

When using feedback, the thermodynamic cost of recording information, W_info, must be taken into account, and the thermodynamic efficiency becomes ${{\eta }_{{\rm th}}}={{W}_{{\rm out}}}/\left( \Delta {{G}_{{\rm in}}}+{{W}_{{\rm info}}} \right)$ [38]. To calculate W_info we transform the data for F_ext(t) (figure 4(b)) obtained from Langevin simulations into a binary string, representing changes in the direction of the force as '1' and unchanged force as '0', and calculate the probability p of motor reversals as the number of switches divided by the number of half-cycles. The information cost can then be determined from the Shannon entropy [39] as

$$\begin{eqnarray}&&{{W}_{{\rm info}}}=kT\left( -p{\rm ln} p-\left( 1-p \right){\rm ln} \left( 1-p \right) \right).\end{eqnarray} \tag{ 4 }$$

This represents an upper limit to the information cost, which is exact for completely uncorrelated force switching times. If the switching probability were 50%, W_info would reach a maximum kTln2 per step, which is known as the Landauer limit for information [39, 40].

Comparing W_info to W_out, we see that the information cost W_info is generally substantially smaller than W_out, except at the highest forces where pauses together with some reversals become so frequent that W_out goes to zero or becomes slightly negative, even in the presence of feedback (figure 4(c)). For small $|{{F}_{{\rm ext}}}|,$ W_info goes to zero because motor reversals are extremely rare, and the information content of measurements of the motor direction is essentially zero. Thus, in almost all cases, the information cost is minimal compared to the work output.

The shape of the force-dependent W_info curve can be interpreted in terms of the reversal probabilities. While the upper limit for W_info is kTln2, our analysis reveals that, for this τ_LP, the maximum reached for SKIP is less than half of this value (figure 4(d)). This is consistent with the reversal probability always remaining below 0.5 (figure 2(a)). The dependence of W_info on force found from analysis of Langevin force switches is similar to what would be predicted from equation (4) using the force-dependent reversal probabilities P_R from figure 2(a). As seen in figure 4(d), however, the energy cost assuming these single-step probabilities overestimates what is found through explicit simulation. This is likely caused by the contributions of pausing to the dynamics of the system, from which entry into the reversal state is not immediately possible. This reduces the average effective reversal probability per half-cycle below P_R from the vulnerable state alone.

Finally, we compare the energy scales of feedback and the free energy we estimate to run the motor. From this, we find that the motor efficiency is negligibly affected by the thermodynamic cost of feedback. This is because W_info is bounded from above by kTln2 ≈ 2.8 pN nm per step, which is much smaller than the ΔG_in of at least 113 pN nm required for the repressor-based implementation of SKIP. It can therefore be inferred from figure 4(d) that the thermodynamic efficiency is very little altered by the incorporation of W_info, and therefore has the same force dependence as the output work (figure 4(c), right-hand axis). Implementing feedback in this experimental conception of SKIP is seen to have no significant cost and substantial benefits.