Enhanced diffusivity in microscopically reversible active matter
EEnhanced diffusivity in microscopically reversible active matter
Artem Ryabov
1, 2, 3, ∗ and Mykola Tasinkevych
2, 3, † Charles University, Faculty of Mathematics and Physics,Department of Macromolecular Physics, V Holešovičkách 2, 180 00 Praha 8, Czech Republic Departamento de Física, Faculdade de Ciências,Universidade de Lisboa, 1749-016 Lisboa, Portugal Centro de Física Teórica e Computacional, Faculdade de Ciências,Universidade de Lisboa, 1749-016 Lisboa, Portugal
We apply a generalized active Brownian particles (ABP) model to derive diffusion coefficient ofa stochastic nanoswimmer subject to an external force. We assume the underlying self-propulsionmechanism is based upon chemical reactions. We show that diffusion coefficient grows quadraticallywith the amplitude of the force regardless of its directions. The effect is a direct consequence of thefundamental principle of microscopic reversibility imposed upon the mechanism of self-propulsionand as such is universal. This result may shed new light on the interpretation of the experiments onenhanced diffusion of enzymes in the presence of their substrates, and provide a straightforward waytowards experimental verification of the used model and its basic assumptions.
Active colloids can convert free energy of host mediumdirectly into mechanical energy of translation or rota-tion [1]. This provides an opportunity to use such particlesas microrobots [2] in several advanced applications such asa transport of microcargo in lab-on-chip devices [3], drugdelivery [4, 5], biosensing [6], environmental remediation[7, 8], or assembly of microtstructures [9]. From a basicscience perspective, active colloids exhibit novel typesof emergent collective behavior, such as “living crystals”[10] or motility-induced phase separation of a system ofactive Brownian particles with only repulsive interactions[11–13], behaviors which are not accessible in traditionalpassive colloids.The by now standard system is composed of a colloidalsphere, half-covered by a thin layer of platinum, so-calledJanus colloid, and dispersed in a solution of water andhydrogen peroxide. Platinum catalyzes the decomposi-tion of hydrogen peroxide into oxygen and water whichleads to propulsion based on self-induced gradients of,e.g., solute concentration [14–16] or electric potential [17–21]. The basic mechanisms of autonomous motion ofcatalytically active colloids is now well understood andis based on self-phoresis [16, 22–24]. Researchers alsodemonstrated feasibility and high efficiency of enzyme-powered self-propulsion of microparticles [25]. Up to date,active colloids with sizes as small as ∼ a r X i v : . [ c ond - m a t . s o f t ] F e b FIG. 1. Schematic illustration of the catalytic nanoswim-mer. The catalytic region of the swimmer, depicted in black,promotes the forward reaction step of the decomposition of asubstrate molecule (small black circle) into a product (smallwhite circle) molecule. The rate of the forward steps is k + andeach step causes particle displacement by δr along the parti-cle orientation n . The reversed reaction steps with the rate k − transform the product molecules back into the substratedriving particle translation by δr in the − n direction. active propulsion [46], derivation of the phoretic velocity[47], discussion of motility-induced phase separations [48],and active heat engines [49].In the present work, we focus on the fundamental as-sumption of the microscopic reversibility and demonstrateits impact on the particle diffusive dynamics. We discussexperimentally relevant consequences of this assumptionand their measurability based on the transient and sta-tionary dynamics of a nano-sized stochastic swimmerspropelled by reversible chemical reactions. In the firstpart, we show that the effective diffusion coefficient is (i)intrinsically larger than the one predicted by ABP modeland (ii) it can be further enhanced by an applied externalforce. On the contrary, in the second part, we argueagainst using steady state properties, such as the usuallymeasured stationary position distribution in an opticaltrap, for verifying the significance of the microreversibil-ity. Our analysis provides a way towards straightforwardexperimental tests of the used model and its assumptions.The results can be particularly relevant for understandingthe puzzling enhancement of the enzymes diffusivities. Thermodynamically consistent active propulsion.—
Consider an active nanoswimmer whose propulsion mech-anism is based on a chemical reaction that releases afree energy ∆ G r in each reaction step. The individualstep causes a displacement along the particle orientation n , say, from the initial particle position r to r + n δr .Assuming further that the particle moves in an externalpotential V ( r ) (a potential in optical tweezers; the oneof gravitational, or electrostatic forces), in each step thework δW = V ( r + n δr ) − V ( r ) is done against the externalforce F = −∇ V .Microscopic reversibility of the underlying chemicalreactions [41] guarantees that to each forward processwith the displacement + δr , there exists a reversed onewith the displacement − δr , and the rates of the forward( k + ) and reversed ( k − ) processes obey the local detailed balance condition k + k − = e (∆ G r − δW ) /k B T , (1)where k B denotes the Boltzmann constant and T is theabsolute temperature of an ambient fluid.In addition to the active chemically-powered dynam-ics, we assume that the particle performs a passive over-damped Brownian motion and that its orientation n changes over time due to a rotational diffusion.The resulting stochastic motion of the particle centerof mass r can be described by the Langevin equation [50] d r dt = µ F ( r ) + √ D ξ ( t )+ n ( t ) h u + µ a F n ( r , t ) + p D a ξ n ( t ) i , (2)where the first line on the right-hand side represents thepassive Brownian motion, the second corresponds to themicroscopically reversible active propulsion.In particular, the mobility µ is related to the diffusioncoefficient D by the fluctuation-dissipation theorem D = µk B T , setting the magnitude of the thermal noise ξ ( t ).We will focus on the motion in xy -plane, hence ξ ( t ) =( ξ x ( t ) , ξ y ( t )) T . Noises ξ i ( t ) are zero-mean delta-correlatedGaussian processes: h ξ i ( t ) i = 0, h ξ i ( t ) ξ j ( t ) i = δ ij δ ( t − t ), i, j = x, y .The second line of (2) contains an instantaneous activepropulsion velocity. Its direction points along the particleorientation n ( t ), which undergoes a rotational Brownianmotion with the diffusion coefficient D r . Setting n ( t ) =(cos φ ( t ) , sin φ ( t )) T , we have dφ/dt = √ D r ξ r ( t ), where ξ r ( t ) is a delta-correlated zero-mean Gaussian white noise.The magnitude of active propulsion velocity [terms insquare brackets in (2)] consists of: the constant u propor-tional to the reaction free energy ∆ G r per unit particledisplacement δr , ∆ G r = ( u/µ a ) δr . Here, division by themobility µ a ensures that u has dimensions of velocity.The projection of F ( r ) onto the particle orientation isdenoted as F n ( r , t ) = n ( t ) · F ( r ), and the Gaussian whitenoise ξ n ( t ), h ξ n ( t ) i = 0, h ξ n ( t ) ξ n ( t ) i = δ ( t − t ), repre-sents fluctuations in a net number of chemical reactions.The noise amplitude is controlled by the diffusion coef-ficient D a obeying the fluctuation-dissipation theorem D a = µ a k B T . All the four noise processes, ξ r ( t ), ξ n ( t ), ξ x ( t ), and ξ y ( t ), are statistically independent on the time-and the length-scales where the Langevin equation (2)holds.Notice that the limit µ a → D a →
0) transformsEq. (2) to the Langevin equation for the minimal ABPmodel [51], with the active velocity of constant magnitude u . The ABP model, which does not obey the microscopicreversibility, will be considered as a reference case in thefollowing analysis. Comparing results derived from (2)with the ones for µ a = 0 will emphasize effects stemmingfrom the assumption of microscopic reversibility. External force enhances diffusivity.—
Let us consider aconstant external force, e.g. of electrostatic, gravitational,or optical nature, acting in x direction, F = ( F, n ( t ), F n ( t ) = F cos φ ( t ) , (3)determines the orientation-dependent component of theactive velocity magnitude in Eq. (2). We further assumethat x (0) = 0, y (0) = 0, and a random initial particleorientation φ (0) ∈ [0 , π ). Then, after averaging andintegrating the Langevin equations (2) we obtain h x ( t ) i = (cid:16) µ + µ a (cid:17) F t, (4) h y ( t ) i = 0 . (5)The constant force F breaks the isotropy of the particlemotion compared to the force-free ( F = 0) case. Con-sequently, the average mean particle position is driftingalong the direction of the acting force. Compared to thepassive Brownian motion and to the ABP model [for bothmodels we have µ a = 0 in (4)], the total mobility alongthe force direction is enhanced by µ a /
2. The factor 1/2complies with our intuition since the rotational diffusionof n ( t ) modifies the magnitude of F n ( t ), whose maximalvalue F occurs for φ ( t ) = 0.In an experiment, Eq. (4) can be used to measure µ a provided the mobility µ is known. For example, itcan be determined based on the Stokes relation 1 /µ =6 πηR , where η is the dynamic viscosity and R denotesthe hydrodynamic radius of the active particle.To characterize diffusivity of the particle, we have deter-mined the second moments of its coordinates. The deriva-tion involves integration of the Langevin equations (2)and averaging of exact results for x ( t ) and y ( t ). Theprocedure itself is straightforward, yet algebraic manipu-lations within its individual steps are somewhat involvedhence we have moved the full derivation into the Supple-mental Material. Here, we discuss three most strikingproperties of the final results for h x ( t ) i and h y ( t ) i : First,the diffusive spreading is isotropic. Second, its extent isincreased compared to µ a = 0 case. Third, the increasecan be controlled by the magnitude F of the externalforce.Contrary to the behavior of the mean particle position,Eqs. (4) and (5), where the mean displacement is parallelto the force F , the diffusive spreading of the positionprobability density function (PDF) around the mean po-sition is the same in all directions regardless the directionof F . In particular, variances of the coordinates x ( t ) and y ( t ) are equal at all times: h x ( t ) i − h x ( t ) i = h y ( t ) i − h y ( t ) i . (6)The mean squared displacement (MSD) of the particleposition measured relative to its mean is then given byMSD( t ) = h r ( t ) i − h r ( t ) i = 2 h y ( t ) i . (7) To justify the last equality in (7) note that the isotropyrelation (6) means the MSD is given as a sum of twoidentical terms: MSD( t ) = h x ( t ) i − h x ( t ) i + h y ( t ) i −h y ( t ) i . Since h y ( t ) i = 0, we have h y ( t ) i = h x ( t ) i −h x ( t ) i , and MSD( t ) = 2 h y ( t ) i .The exact expression for h y ( t ) i valid for arbitrary t reads h y ( t ) i = " D + D a + u D r + ( µ a F ) D r t + (cid:18) uD r (cid:19) (cid:0) e − D r t − (cid:1) + (cid:18) µ a F D r (cid:19) (cid:0) e − D r t − (cid:1) , (8)From an experimental perspective [30–32], particularlyimportant are the short- and the long-time limits of thisresult.At large observation times t , Eq. (8) predicts a lineargrowth of MSD, MSD( t ) ≈ D t , with a slope determinedby an effective diffusion coefficient D = lim t →∞ MSD( t )4 t = D + D a u D r + ( µ a F ) D r . (9)As compared to the standard ABP model (correspondingto µ a = D a = 0 case) which does not account for themicroscopic reversibility, there are two extra contributionsto D , both increasing the particle diffusivity. The term D a / F , i.e., even for F = 0. This term was clearly missingin an analysis of enzyme diffusivity based on the ABPmodel [52]. The diffusivity of the ABP model shouldbe understood as a lower bound in real-world situations.The actual diffusivity of an active particle with a mi-croreversible propulsion is always higher than this lowerbound.The last term on the right-hand side of Eq. (9) is evenmore remarkable than D a /
2. It can lead to a direct exper-imental verification of the present model by varying themagnitude of an externally applied force. At the sametime, as it depends on the squared force amplitude, itsuggests that diffusivity of an active particle is sensitiveeven to local random forces with zero mean but non-zerovariance. For example, in experiments where enzymesmove in the vicinity of a confining surface the fluctua-tions of local enzyme-surface (electrostatic and/or vander Waals) interactions, due to the surface roughness, asestimated by F can give a non-negligible effect on theobserved diffusivity.The second experimentally relevant limit of result (8)is when the measurement time satisfies t (cid:28) D − . Inthis regime observed trajectories are not long enoughcompared to the time needed for the director to thermalize,resulting in a ballistic active motion withMSD( t ) ≈ (cid:18) D + D a (cid:19) t + " u + (cid:18) µ a F (cid:19) t (10)where we omit terms of the order ( D r t ) and higher.Here, the term linear in time is again increased by D a / F and the active velocity u , whichdetermine the magnitude of t term. Stationary states in confining potentials.—
Conse-quences of the microscopic reversibility on the particlediffusion can be in principle verified in time-resolved ki-netic experiments. In view of recent experimental activi-ties focused on the broken detailed balance and entropyproduction in the steady states [53], we now consider howthe microreversibility is reflected in the behavior of astationary PDF for the particle position.The joint PDF p ( r , φ, t ) of the particle position andthe director orientation at time t , corresponding to theLangevin equation (2), satisfies the Fokker-Planck equa-tion ∂ t p ( r , φ, t ) = [ L + L a + L r ] p ( r , φ, t ) . (11)The three operators on the right-hand side correspond tothree evolution mechanisms in the model. The passivediffusion in the force filed F is described by the standardFokker-Planck operator L p = µ ∇ · [ k B T ∇ p − F p ] , (12)The active propulsion mechanism is represented by thesecond operator L a p = µ a ∂ n [ k B T ∂ n p − F n p ] − u ∂ n p, (13)which incorporates a drift and diffusion along the directionof n ; the directional derivative ∂ n is defined by ∂ n = n · ∇ = cos( φ ) ∂ x + sin( φ ) ∂ y . The last operator in (11)describes the rotational diffusion of the director n . Itreads L r p = D r ∂ φφ p. (14)To begin with the analysis of steady state PDFs sat-isfying [ L + L a + L r ] p st ( r , φ ) = 0, it is enlightening toconsider a system with a negligible constant part of theactive velocity u ≈ µ a . Such a limit could berealized, in principle, by a nanoswimmer with an approxi-mately isotropic distribution of the catalytic sites on itssurface. Here, the stationary PDF at leading order in u is given by the Gibbs canonical distribution p st ( r , φ ) ≈ Z exp [ − βV ( r )] , (15)where Z is a normalization constant.This limit, most strikingly highlights the qualitativedifference between the stationary and dynamical char-acteristics of the nanoswimmers. Indeed, when the sta-tionary PDF (15) has no signature whatsoever whetherthe particle is active or not, as it does not depend on µ a and the PDF itself is identical to the one for a passiveBrownian motion. Contrary, the dynamic quantities–theeffective diffusion coefficient in Eq. (9) and the terms inthe small t expansion in Eq. (10)–bare a strong evidenceof both the particle activity and the reversibility of thechemical driving via their dependence on µ a and D a .Another experimentally relevant regime of parame-ters, where the stationary PDF is likewise given bythe equilibrium-like Eq. (15) arises for nanoswimmerswith a finite u and large rotational diffusion coefficient D r (cid:29) u /D . Here, similar qualitative conclusions hold. Arigorous derivation of (15) in this case can be carried outanalogously as it has been done for the ABP model [54].In the limit of small rotational diffusivity, i.e., whenthe rotational diffusion time D − r is much larger than thetypical time max r (cid:2) µ ∇ V ( r ) (cid:3) − needed by the particle toreach some local minimum of V ( r ), we obtain at leadingorder p st ( r , φ ) ≈ Z exp [ − βV ( r ) + ˜ u ( x cos( φ ) + y sin( φ ))] , (16)with the renormalized self-propulsion velocity ˜ u = u/ ( µ + µ a ) being the only difference from what one would obtainby using standard ABP model with no microreversibil-ity [54]. Likewise, the PDF for the particle position r ,obtained by integration over φ of Eq. (16), attains theform p st ( r ) ≈ π I (˜ u | r | ) exp[ − βV ( r )] /Z , which qualita-tively resembles the one for the ABP model as derivedfor the radially symmetric traps in Ref. [54]; where I isthe modified Bessel function of the first kind. The soleeffect of the microreversibility here is to reduce the meanactive velocity (as compared to the ABP model) whenthe particle climbs against the external potential V ( r ). Itis rather surprising that the more involved Fokker-Planckequation (11) yields results which up to the renormal-ization of u are identical with the standard ABP model[ABP corresponds to µ a = 0 in (13)]. For the sake ofcompleteness, all the individual steps of the derivation ofEq. (16) are summarized in the Supplemental Material. Implications for experiments.—
We now evaluate themagnitude of the diffusion enhancement ∆ D due to theparticle activity by rewriting it as follows∆ D = u D r + D a µ a F ) D r = u D r + uδr k B T ∆ G r + u D r (cid:18) F ∆ G r /δr (cid:19) , (17)where the first term on the r.h.s., ∆ D ≡ u / D r , isthe enhancement in the standard ABP model, and thesum of the last two terms, which we denote ∆ D m , issolely due to the effects of the microreversibility. Forthe parameters in Eq. (17) we use the values reported in[31, 32] for enhanced diffusion of urease: ∆ G r ≈ k B T , u ≈ µm/s , δr ≈ D r ≈ s , which renders∆ D m ≈ (16 + 15 ( F/ µm /s to be compared withthe bare diffusion coefficient measured in the absence ofsubstrate D ≈ µm /s . If we assume the external force F ∼ ∼ e separated by the distance ∼ D m /D ≈ .
76. However, a word of caution is in place regardingthese estimation, thus using the values provided abovewe find ∆ D /D ≈
6, which is way too large. In addition, δr ≈ δr ≈ F correctionterm as ≈
15 ( F/ µm /s , which increases the lowerbound for the force amplitude to have some significanteffects to 40pN. Conclusions and perspectives.—
Understanding boththe individual and the collective dynamics of active par-ticles represents a major challenge for theory. The prob-lem is particularly difficult due to the complex out-of-equilibrium nature of the particle-solute and inter-particleinteractions with a subtle interplay of self-induced chemi-cal and hydrodynamic fields [55–58], which calls for noveltheoretical approaches.For the case of micron-sized colloids, individual prop-erties are rationalized with the aid of continuum hydro-dynamic description yielding satisfactory agreement withexperiments [59–61]. Concerning the many-body phenom-ena minimal ABP model has been widely applied. Ithas been studied both via numerical simulations [11–13]and theoretically, as a starting point to construct coarse-grained continuum active field theories [62–68], whosephase behavior has been analysed at the mean-field level[64, 66, 67] and beyond [68].At the nanoscale, more and more experimental evi-dences are being accumulated that enzymes exhibit self-propulsion during their catalytic activity. The origin ofthis self-propulsion, however, remains highly debated. Re-garding the collective behavior of nano-propellers, Speckand coworkers have considered the influence of the mi-croreversibility on the motility-induced phase separationof hard discs and found no impact on their collectivebehavior [48].In contrast, here we report several significant effects ofthe microreversibility at the level of individual stochas-tic trajectories of nanoswimmers. The effective diffusioncoefficient of a nanoswimmer scales as a second power ofthe amplitude of the external force independently of theforce direction, Eq. (9). Moreover, the microreversibilityalso dictates enhanced diffusion for a hypothetical freeactive particle which is symmetrically covered by a cat-alyst, Eq. (9) with u = 0 , F = 0. The last predictionresonates with recent experimental studies of Granickand coworkers [69–71], who have observed increased diffu-sion coefficients of reactants during some common organicchemical reactions, not necessarily enzyme-driven. Theyhave suggested that the transduction of reaction free en- ergy into translational motion is a generic effect, providedthe free energy release rate exceeds some threshold.The demonstrated here sensitivity of the effective diffu-sion coefficient to the coupling between the chemical andmechanical degrees of freedom can contribute to the ba-sic understanding of the enhanced diffusivity of enzymes.Indeed, the latter operate on time scales where the fluctu-ations in chemical reactions shall be relevant [in contrastto micron-sized colloids powered by tenths of thousandsof reaction steps per “elementary displacement” δr ] andthus correctly including the chemo-mechanical coupling iscrucial in this case. Our results emphasize several uniquefeatures of the ABP model with the microreversibilityincluded, which can be directly tested in experiments, andwe hope that this study will stimulate such activities inorder to give a closer look on the behavior of nanoswim-mers in external fields or in the presence of local fieldsdue to, e.g. surface roughness. Acknowledgments.
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