Mechanochemical active ratchet

Self-propelled nanoparticles moving through liquids offer the possibility of creating advanced applications where such nanoswimmers can operate as artificial molecular-sized motors. Achieving control over the motion of nanoswimmers is a crucial aspect for their reliable functioning. While the directionality of micron-sized swimmers can be controlled with great precision, steering nano-sized active particles poses a real challenge. One of the reasons is the existence of large fluctuations of active velocity at the nanoscale. Here, we describe a mechanism that, in the presence of a ratchet potential, transforms these fluctuations into a net current of active nanoparticles. We demonstrate the effect using a generic model of self-propulsion powered by chemical reactions. The net motion along the easy direction of the ratchet potential arises from the coupling of chemical and mechanical processes and is triggered by a constant, transverse to the ratchet, force. The current magnitude sensitively depends on the amplitude and the periodicity of the ratchet potential and the strength of the transverse force. Our results highlight the importance of thermodynamically consistent modeling of chemical reactions in active matter at the nanoscale and suggest new ways of controlling dynamics in such systems.


Introduction
Chemistry and physics of micron-and nano-sized selfpropelled particles are rapidly growing fields exploring optimal designs and fundamental concepts of self-propulsion mechanisms.A current intense discussion focuses on a possible self-propulsion of individual catalytic macromolecules that could serve as basic components for artificial nanomachines [1][2][3][4][5][6][7][8][9].
In contrast, strong fluctuations are unavoidable at the nanoscale, where dimensions of active particles are on par with those of intracellular molecular motors like kinesins, myosins, and ribosomes [34][35][36], and artificial molecular machines and pumps [37][38][39].Active motion at these scales can be caused by a comparably small number of reactions and is strongly influenced by thermal noise.Large fluctuations of the magnitude and the direction of selfpropulsion velocity hinder the development of methods to efficiently control the dynamics of nanoswimmers.
Here, we use a minimal thermodynamically consistent model of particle's self-propulsion [40][41][42][43] to demonstrate a novel physical mechanism for rectifying and guiding trajectories of chemical nanoswimmers.The mechanism benefits from the presence of strong fluctuations in the direction of self-propulsion velocity and it arises as a direct consequence of the main premise that the chemical reactions driving the active motion comply with the principle of microscopic reversibility (MR) [44][45][46][47].The resulting rectification effect can be used to guide and sort active nanoparticles based on their propulsion mechanisms, because a corresponding reference model without the MR, as well as passive Brownian particles, does not exhibit this type of rectified motion.

Results and Discussion
Stochastic Markovian models become thermodynamically consistent if rates of elementary transitions obey the local detailed balance [48].For an overdamped Brownian motion subjected to an external force F (r) and to thermal fluctuations represented by the zero-mean Gaussian white noise ξ(t), the local detailed balance condition is arXiv:2208.13929v2[cond-mat.stat-mech]16 Nov 2023 included in the Langevin equation for the particle velocity, ṙ = µF (r)+ √ 2D ξ(t), by requiring that the translational diffusion constant D is related to the mobility µ via the fluctuation-dissipation relation D = µk B T , T being the temperature of ambient fluid.
In addition to the force-and thermally-driven movements, the velocity of a self-propelled nanoparticle is influenced by its chemically-powered active dynamics.The particle center of mass position r(t) = (x(t), y(t)) then obeys the Langevin equation where the first term on the right-hand side represents the active velocity of magnitude v a (r, t) oriented in the direction of n(t), see Fig. 1(a).The vector n(t) = (cos ϕ(t), sin ϕ(t)), giving the orientation of the nanoparticle, performs rotational diffusion [49], i.e., the angle ϕ(t) undergoes the Brownian motion with D r being the rotational diffusion constant and ξ r (t) a zero-mean Gaussian white noise.
The speed v a (r, t) corresponds to the chemically-driven particle motion in the direction of n(t).If the kinetics of (c) Sketch of a trajectory of the nanoparticle diffusing in the potential V (x) (dark red marks the potential minima, light orange the maxima) and subjected to a constant force with the amplitude f ⊥ > 0 acting in the y-direction and with no constant force component in the x-direction (f ∥ = 0).The nonzero perpendicular bias f ⊥ can induce a net particle motion along the x-axis.This special consequence of the mechanochemical coupling term in va(r, t) is illustrated in Fig. 2. If f ∥ > 0, the direction of the motion can be reversed by changing f ⊥ as shown in Fig. 3.
the underlying reactions comply with the principle of MR, then, on the mesoscopic timescale, where an infinitesimal time interval dt covers many chemically-driven particle jumps, we may approximate the speed by [42] v a (r, t) ≈ u + µ c n(t) • F (r) + 2D c ξ c (t). (3) Here, u is the constant mean active speed in the zero-force case (F = 0), the dot  3) starting from the mesoscopic jump-diffusion model can be found in Ref. [42].The central assumption of this derivation is that the rates of motion-powering chemical reactions satisfy the local detailed balance condition.This is equivalent to assuming MR of the kinetics of these reactions, which means that their underlying microscopic dynamics are symmetric with respect to the time-reversal [48].A general aim of the current work is to demonstrate the impact of individual terms in (3) on the dynamics of an active nanoparticle diffusing in external force fields.According to Eq. ( 3), the active speed depends on the component n(t) • F (r) of the external force along n(t).This term describes the mechanochemical coupling [50] and allows to sculpt a free energy landscape of the macromolecule by the action of external forces.The coupling is utilized in various single-molecule force spectroscopies [11,51,52], biosensing [53] and underlies the ability of molecular motors to exert forces and torques [38,[54][55][56][57][58].
The force-dependence of v a (r, t) is a direct consequence of MR of the self-propulsion mechanism and it has prominent impacts on particle dynamics [42,43].We shall use this dependence to design a new method of nanoparticle guiding, which is inaccessible in systems lacking the mechanochemical coupling, like in the case of micron-sized particles with constant active speed, see Eq. ( 10) below.
In particular, we report on a ratchet effect that occurs if the particle is subjected to the external force determined by the constant components f ∥ and f ⊥ , and by the derivative of the asymmetric λ-periodic potential shown in Fig. 1(b).A trajectory of the particle moving in V (x) and being acted upon by f ⊥ > 0 is sketched in Fig. 1(c).
The asymmetric sawtooth-like potential V (x) is qualitatively similar to potentials used in various theoretical and experimental studies of ratchets [59][60][61][62][63][64][65][66][67][68][69][70][71].A distinc-tive feature of the ratchet mechanism discussed below is that the emergent current of particles in the x-direction is induced by the transverse force f ⊥ applied perpendicular to it in the y-direction.Although here we focus on the two-dimensional case, in an actual experiment, the force f ⊥ can point in any direction in the yz-plane.
Assuming that the force in Eq. ( 4) can be accurately adjusted in an experiment, we discuss the dependence of the ratchet effect on f ∥ , f ⊥ , and V 0 .The phenomenological model parameters D, D r , D c , and u will be kept fixed.All results reported below are obtained assuming λ = 100 nm.To get the results, we have integrated Eqs. ( 1) and ( 2) numerically, see Supplementary Note 1 and 2 for technical details.
The values of the diffusion constants D and D r were chosen having catalytically active enzymes in mind [72][73][74].Thus, D was approximated by the Stokes-Einstein equation D = k B T /6πηR h , and D r = k B T /8πηR 3 h valid for a sphere of radius R h , η is the fluid dynamic viscosity.With R h = 15 nm, T = 300 K, and η = 8.53 × 10 −4 Ns/m 2 , which corresponds to the dynamic viscosity of water at this temperature, we find Contrary to D and D r , the behavior of the phenomenological parameter D c remains poorly understood.To approximate its value, we have used D c ≈ (δr) 2 (k + + k − )/2 [42], where k + is the rate of the reaction causing the nanoparticle displacement δrn (at F = 0), and neglected the rate k − of the reversed reaction, which causes the displacement by −δrn.For k + = 10 5 s −1 and δr = 5 nm, We note that the very existence of the reported ratchet effect follows from the condition D c > 0 and does not depend on the actual value of D c .Equation ( 11) derived below will demonstrate that the ratios D c f ⊥ /k B T and µV 0 /λ are the two key parameters that control the occurrence and the strength of the ratchet effect.Their values can be changed in an experiment by adjusting the force f ⊥ , the barrier height V 0 , and the period λ.
For the constant part of the speed (3), we have used u = 10 nm/s.The reported results, however, are insensitive to the actual value of u, as we have checked several alternative values of u between 0 and 10 3 nm/s.The negligible effect of u on the translational dynamics results from the large D r , Eq. ( 7), characteristic for small particles (D r ∼ 1/R 3 h ) as compared to the slower translational motion whose speed is determined by D in (6) (D ∼ 1/R h ), u and by D c (8).This is in sharp contrast to the motion of micron-sized active particles that move ballistically for significant periods of time due to relatively large u and small D r [1,[75][76][77][78][79][80].6)-( 8), λ = 100 nm, and u = 10 nm/s.Symbols represent simulated ⟨vx⟩ of the active nanoparticle with the microscopically reversible self-propulsion mechanism (MR ANP) illustrated in Fig. 1.Dashed lines show corresponding results for the active Brownian particle (ABP) model with constant active speed (10).The three dashed lines overlap.
We quantify the ratchet effect in terms of the mean velocity ⟨v x ⟩ of the particle motion in the x-direction.It is defined as the average over N tr simulated trajectories, where x i (t max ) is the final x-coordinate of the i-th trajectory.We have used D r t max = 10 3 , N tr = 10 4 , and checked that increasing t max and N tr does not affect the displayed results.Figure 2 shows ⟨v x ⟩ for f ∥ = 0, i.e., when there is no net force in the x-direction.Symbols mark ⟨v x ⟩ of the active nanoparticle with the microscopically reversible self-propulsion mechanism (MR ANP) described by the active speed v a (r, t) in (3).Dashed lines represent ⟨v x ⟩ of the so-called active Brownian particle (ABP) model [76,78,79,[81][82][83][84][85][86][87][88][89][90][91][92][93] which is a paradigmatic model for micron-sized active colloidal particles [1,[75][76][77][78][79] and is characterized by the constant active speed The ABP model is formally obtained from the MR ANP model after neglecting the mechanochemical coupling and the noise in Eq. (3), i.e., by setting µ c = D c /k B T = 0.The simulated mean velocities ⟨v x ⟩ for these two models are plotted in Fig. 2 as the functions of the perpendicular force f ⊥ for three representative values of the barrier height V 0 : (i) V 0 = 1 k B T , light-blue symbols, where the barrier height is equal to the energy of thermal fluctuations.Thus, the asymmetric potential V (x) has only a minor influence on the dynamics and ⟨v x ⟩ of MR ANP is vanishingly small.(ii) V 0 = 6 k B T , yellow symbols.Here, thermal fluctuations can induce hopping-like transitions of particles over the barriers of V (x).As a result of the ratchet effect, MR ANP exhibits a nonzero mean velocity in the negative direction of the x-axis (⟨v x ⟩ ≤ 0).There exists an optimal value of f ⊥ leading to a maximal negative velocity of MR ANP.In the limits of small (f ⊥ → 0) and large (f ⊥ → ∞) perpendicular forces, ⟨v x ⟩ vanishes.
(iii) V 0 = 10 k B T , red symbols.In this case, the barriers of V (x) are large compared to the energy of thermal fluctuations.The cross-well transitions of MR ANP are driven dominantly by the active velocity.The ratchet effect becomes the most pronounced out of the three cases.
In Fig. 2, the three dashed lines marking ⟨v x ⟩ for the ABP model overlap and follow closely the line ⟨v x ⟩ = 0 for all f ⊥ .
These observations suggest that the ratchet effect should be caused by the force-dependence of the active speed (3).Can we understand this effect qualitatively, e.g., by relating it to some of the well-understood ratchet mechanisms?It turns out that this is not only possible but is indeed an insightful step as it will lead to yet another way of control over the motion of the nanoparticle.
For the considered values of parameters, the Langevin equation for the x-coordinate of MR ANP (and for f ∥ = 0) approximately reduces to where the effective enhanced mobility μ = µ + µ c /2, and the enhanced diffusivity D = D + D c /2 (compared to µ and D), satisfy the fluctuation-dissipation relation D = μk B T .Indeed, due to the fast rotational diffusion, we can neglect u in Eq. (3) as discussed above, yielding v a (r, t) , for the x-component of the active velocity.Since ϕ is a fast process and µ ≫ µ c , cos 2 ϕ can be replaced by its mean value, cos 2 ϕ ≈ 1/2.The noise term is then simplified accordingly: cos ϕ , and is added to the thermal noise √ 2D ξ x , which eventually leads to (11); see Supplementary Note 3 for more technical details and tests of validity of this approximation.
Equation (11) describes the dynamics of a basic model of a Brownian motor, the so-called overdamped tilting ratchet [59,64,66,94,95].Discussion of the dynamics governed by Eq. ( 11) can provide a further physical understanding of the mechanism leading to the reported ratchet effect.Below, we focus on qualitative arguments and intuitive interpretation of the effect.For a thorough mathematical analysis of various classes of tilting ratchet

FIG. 3.
The ratchet effect and the velocity reversal.The mean nanoparticle velocity ⟨vx⟩ in the x-direction (symbols), for the case when the nanoparticle is subjected to the nonzero parallel force f ∥ = 0.025 pN.Other parameters are the same as in Fig. 2. The black solid line represents the velocity (12) of a Brownian particle moving in a flat potential with the enhanced mobility (µ + µc/2).The two dashed lines marking corresponding results for the ABP model overlap.models, we refer to Sec. 5 of the review article [59].
In particular, Eq. ( 11) describes the dynamics of an overdamped Brownian particle diffusing in the potential V (x) subject to the time-dependent stochastic force (f ⊥ /2) sin[2ϕ(t)].If the magnitude of the force becomes large enough, the particle may start surmounting potential barriers of V (x) located in the direction of the force.Because V (x) is asymmetric, shapes of the left (moderate slope) and right (steeper slope) barrier surrounding a given well differ and the barrier-crossing is easier in one of the two directions: in (−x)-direction in the present case.After surpassing multiple barriers in both directions, the asymmetry of V (x) translates into the nonzero mean motion in the easy (−x)-direction giving a nonzero mean velocity ⟨v x ⟩ observed in Fig. 2.
The velocity ⟨v x ⟩ attains its maximum for moderate values of f ⊥ .If f ⊥ → 0, ⟨v x ⟩ vanishes because a weak force does not affect the rate of the barrier-crossings.At large f ⊥ , ⟨v x ⟩ vanishes too: When the force is extremely strong, the potential barriers of V (x) have only a minor influence on the particle motion and the effect of their asymmetry becomes negligible.Such a release of nanoparticles from deep potential wells that is induced by a strong perpendicular force f ⊥ can be utilized for controlling the direction of ⟨v x ⟩.This is demonstrated in Fig. 3 showing a reversal of the direction of ⟨v x ⟩ for particles subjected to a weak force f ∥ > 0 acting in the x-direction.
For small and moderate f ⊥ , the results shown in Fig. 3 resemble those in Fig. 2: MR ABP experiences the ratchet effect and moves on average with ⟨v x ⟩ < 0. However, at larger f ⊥ , as the barriers of V (x) become less significant for the MR ANP dynamics, the direction of the net particle motion is reversed, and ⟨v x ⟩ > 0. Eventually, as f ⊥ → ∞, ⟨v x ⟩ converges towards the solid line given by which equals to the mean velocity of an overdamped Brownian particle diffusing in a flat potential [V (x) = 0] with the enhanced mobility μ = µ + µ c /2, cf. the mobility in Eq. (11).Contrary to this complex behavior, ⟨v x ⟩ in the ABP model does not depend on f ⊥ and is vanishingly small because the ABP spends most of the time being trapped near the minima of V (x) regardless of the value of f ⊥ .
Finally, let us note that the magnitude of ⟨v x ⟩ and the range of f ⊥ values, where the transverse ratchet effect occurs, depend on the value of the phenomenological parameter µ c = D c /k B T characterizing the strength of the mechanochemical coupling.This dependence is demonstrated in Fig. 4, where blue circles correspond to µ c used in all other figures and obtained from the estimate (8) and red triangles (yellow squares) represent simulation results for µ c being 5 times (10 times) larger.Hence, for a stronger mechanochemical coupling, the ratchet effect becomes more pronounced and occurs at lower values of the transverse force f ⊥ .

Summary and Perspectives
The principle of microscopic reversibility is fundamental for nonequilibrium statistical mechanics and thermodynamics.Here, we have discussed the physical origins and fundamental properties of a novel ratchet mechanism resulting from the MR of the chemically-driven self-propulsion mechanism of an active nanoparticle.The mechanism enables efficient control of the velocity of a nanoswimmer propelled by chemical reactions.It is based on an interplay of the particle translational diffusion in an asymmetric periodic potential V (x) and the modification of its self-propulsion speed because of the mechanochemical coupling.The resulting ratchet effect manifests itself most strikingly by the emergent net particle motion in the easy direction of the ratchet potential, when a constant force f ⊥ is applied to the particle in a perpendicular direction.The mean velocity ⟨v x ⟩ of this motion can be controlled by adjusting the barrier heights V 0 of the periodic potential and the magnitude of f ⊥ .For a given V 0 , there exists an optimal value of f ⊥ , for which the mean speed |⟨v x ⟩| attains a maximum.Furthermore, if a constant force f ∥ is applied in the x-direction, the mean velocity can be reversed by changing f ⊥ .
In addition to numerical results, we have shown that for the model parameters roughly corresponding to catalytic enzymes, the dynamics of the nanoparticle in the x-direction is equivalent to that of a Brownian tilting ratchet.This mapping provides an intuitive physical interpretation of the reported ratchet effect and demonstrates that it cannot occur in active dynamics lacking the mechanochemical coupling.In the present model, this coupling arises as a consequence of the microscopic reversibility of the fluctuating chemical kinetics powering nanoparticle self-propulsion.Further exploration of such coupled thermodynamic processes at the nanoscale can inspire novel designs for micromanipulation techniques, where macromolecules are acted upon by externally applied electromagnetic or/and mechanical forces.Additionally, testing such effects in experiments may contribute toward a resolution of the long-standing question of whether active enzymes self-propel or not.
For the values of parameters u, µ, µ c , f ⊥ , and D r used in the main text, Eq. (S27) can be considerably simplified.This is done in two steps.
First, for u ≲ 1000 nm/s we can neglect the term proportional to u in Eq. (S27).This approximation is valid for active particles with smaller hydrodynamic radii R h .It results from the large value of D r , D r ∼ 1/R 3 h , Eq. ( 7) in the main text.Thus, for u ≈ 10 nm/s, used in all figures in the main text, Eq. ( S27 Setting further f ∥ = 0 yields Eq. ( 11) in the main text.8) in the main text, triangles (squares) correspond to values of µc five (ten) times higher than the reference value.Right panel: µc hundred times higher than the reference value.

FIG. 1 .
FIG. 1. Illustration of the model.(a) An active nanoparticle (blue) whose self-propelled motion with speed va(r, t) occurs in the direction of orientation n(t) (diagonal arrow) and is powered by chemical reactions (red arrow).Assuming the microscopic reversibility of the self-propulsion mechanism gives rise to the mechanochemical coupling term [µcn(t)•F (r)] in Eq. (3) for va(r, t).The x-and y-component of the external force F (r), Eq. (4), are represented by the horizontal and vertical arrows.(b) The periodic asymmetric potential V (x) given in Eq. (5) normalized by its barrier height V0 for λ = 1.(c)Sketch of a trajectory of the nanoparticle diffusing in the potential V (x) (dark red marks the potential minima, light orange the maxima) and subjected to a constant force with the amplitude f ⊥ > 0 acting in the y-direction and with no constant force component in the x-direction (f ∥ = 0).The nonzero perpendicular bias f ⊥ can induce a net particle motion along the x-axis.This special consequence of the mechanochemical coupling term in va(r, t) is illustrated in Fig.2.If f ∥ > 0, the direction of the motion can be reversed by changing f ⊥ as shown in Fig.3.

FIG. 2 .
FIG.2.Rectified particle motion.Net particle translation induced by the mechanochemical coupling term in the active speed (3).The mean particle velocity ⟨vx⟩ in the x-direction is plotted as the function of the perpendicular force f ⊥ acting in the y-direction for three values of the barrier height V0 (colors) and f ∥ = 0. Parameters D, Dr, Dc are given in Eqs.(6)-(8), λ = 100 nm, and u = 10 nm/s.Symbols represent simulated ⟨vx⟩ of the active nanoparticle with the microscopically reversible self-propulsion mechanism (MR ANP) illustrated in Fig.1.Dashed lines show corresponding results for the active Brownian particle (ABP) model with constant active speed(10).The three dashed lines overlap.

FIG. 4 .
FIG.4.Impact of µc on the ratchet effect.The mean velocity ⟨vx⟩ of MR ANP in the x-direction for f ∥ = 0, V0 = 20 kBT , and three values of µc.Circles represent results for µc obtained from(8), triangles (squares) for five (ten) times times larger µc.Other parameters are the same as in Figs.2 and 3.
FIG. S1.Mean velocity ⟨vx⟩ in the x-direction, Eq. (S25), for f ∥ = 0, V0 = 20 kBT , and four values of the parameter µc.Other model parameters are the same as in Figs. 2 and 3 in the main text.Symbols (lines) represent results based on the numerical integration of the exact Eq. (S26) [approximate Eq. (S31)].Left panel: circles mark results for the reference value of µc obtained from Eq. (8) in the main text, triangles (squares) correspond to values of µc five (ten) times higher than the reference value.Right panel: µc hundred times higher than the reference value.