aa r X i v : . [ phy s i c s . b i o - ph ] J u l Synthetic Mechanochemical Molecular Swimmer
Ramin Golestanian ∗ Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK andLaboratoire de Physico-Chimie Th´eorique, UMR CNRS Gulliver 7083, ESPCI, 75231 Paris Cedex 05, France (Dated: October 31, 2018)A minimal design for a molecular swimmer is proposed that is a based on a mechanochemicalpropulsion mechanism. Conformational changes are induced by electrostatic actuation when specificparts of the molecule temporarily acquire net charges through catalyzed chemical reactions involvingionic components. The mechanochemical cycle is designed such that the resulting conformationalchanges would be sufficient for achieving low Reynolds number propulsion. The system is analyzedwithin the recently developed framework of stochastic swimmers to take account of the noisy en-vironment at the molecular scale. The swimming velocity of the device is found to depend on theconcentration of the fuel molecule according to the Michaelis-Menten rule in enzymatic reactions.
PACS numbers: 87.19.rs, 07.10.Cm, 82.39.–k
One of the aims of nanotechnology is to be able to makesynthetic molecular devices that could propel themselvesthrough fluidic environments and perform targeted taskssuch as delivery of therapeutic agents or carrying out me-chanical work [1]. At such small scales, one cannot applystandard deterministic strategies used in engineering atthe macroscopic scales, as the dynamics of any devicewill be overwhelmed by fluctuations of thermal or otherorigins. The right strategy is to find a way to imposea bias on these fluctuations such that they will averageout to our desired behavior, as can be learned from theexample of biological molecular motors [2].In addition to enduring the noisy environment, molec-ular swimmers would also need to overcome the funda-mental difficulty posed by the governing hydrodynamicrules at low Reynolds number conditions: they shouldutilize a non-reciprocal sequence of deformations to breakthe time reversal symmetry in their periodic motions andachieve swimming [3]. This is a nontrivial task if one isto use a minimal set of degrees of freedom [4–11], whichwill most likely be what can be afforded in a human-engineered device.There has been a significant recent development alongthis line with the advent of a number of experimentallymade micron-scale prototypes of such low Reynolds num-ber swimmers. Using magnetic actuation of micron-sizedlinked magnetic beads [12, 13] or manipulation of par-ticles using optical tweezers [14], it was demonstratedthat low Reynolds number propulsion can be achieved atthe micro-scale via non-reciprocal motion. In these ex-periments, the actuation mechanisms were externally en-forced (using oscillating magnetic fields or optical traps),which means that strictly speaking the devices were notself-propelled swimmers. Moreover, it is not clear if suchactuation mechanisms could be scaled down to be ap-plicable to molecular devices. This means that we arein need of new strategies to be able to program non-reciprocal molecular deformations.While other external triggers—such as laser pulses that could induce conformational changes—could in principlebe pursued [1], it would be ideal to be able to couple theconformational changes to a local source of free energyvia a chemical reaction, like motor proteins [15]. Thiswill allow the mechanism to potentially accommodatearbitrary degrees of complexity, and avoid interferencebetween different parts of the system. To my knowledge,this scenario has not been considered so far in the litera-ture for microscopic swimmers, although there have beendemonstrations of mechanical actuation of elastic gelsby using oscillatory chemical reactions (at macroscopicscales) [16]. A strategy based on catalysis of a chemicalreaction has already been used for making self-propelledcolloidal particles, which utilize non-equilibrium inter-facial (phoretic) interactions towards their net motion[17, 18]. While this mechanism of motility has shown tobe extremely powerful, exploring other possible strate-gies would help create a diverse range of approaches thatcould be used in the future to optimize devices for spe-cific purposes and overcome strategy-specific hurdles fora particular end use. For example, a swimmer that uti-lizes interfacial interactions as its main motility mech-anism is potentially vulnerable to intervention by otherinterfacial forces when it is close to boundaries or otherdevices. Moreover, one wonders why nature has not cho-sen to take advantage of the “phoretic” strategy in anyliving system; one may speculate that it could be becauseit cannot be conveniently miniaturized and robustly func-tion in maximally crowded intracellular environments.Here I propose an alternative minimal design for amolecular swimmer, which would be able to catalyzechemical reactions and use the free energy gain and prod-ucts to induce the conformational changes that would besufficient for achieving low Reynolds number propulsionin a noisy environment [19]. The present study could alsoshed some light on how the motor proteins could com-bine enzymatic action with conformational changes, as itpresents a full physical analysis of a simple model thatexplains how such a coupling could work in practice. P + (2): tR – +h (5): tR – + hQ – (6): t+ hQ – (4): tR – +hF (3): t+hF (1): t+h G + G + P + P + Q – Q – R – R – R – S S F F S FIG. 1. The 6-state chemical cycle of the three-sphere molec-ular device that comprises enzyme h at the head and enzyme t at the tail (see Fig. 2). h catalyzes F → Q − + G + and t catalyzes R − + P + → S . The proposed design for the molecular swimmer isbased on the simple three-sphere low Reynolds numberswimmer model [5, 20]. To incorporate catalytic activity,we take two enzymes, h and t , and place them at the head and the tail of the three-sphere device, respectively. Theenzymes catalyze the following chemical reactions F + h → hF → hQ − + G + → h + Q − + G + , (1) R − + P + + t → tR − + P + → t + S, (2)which would otherwise occur very slowly in the bulk.Since the head can be found in three distinct states of h , hF , and hQ − , and the tail has two distinct statesof t and tR − , the combined system can have 6 distinctchemical states. The reactions (1) and (2) then multiplywithin these states and produce the chemical cycle shownin Fig. 1.The second ingredient in the design is to take advan-tage of the presence of ionic components in the reac-tions and use electrostatic interactions to induce con-formational changes in the swimmer. If we make the middle sphere permanently negatively charged (see Fig.2), the temporarily charged states of hQ − and tR − willintroduce electrostatic repulsion that can elongate theright and left arms, respectively. This allows us to mapthe 6 chemical states shown in Fig. 1 onto the 6 dis-tinct mechanochemical states shown in Fig. 2. Thesemechanochemical states can be represented in a 3D con-figuration space whose axes correspond to deformation ofthe right arm U R , deformation of the left arm U L , andcomplexation of h with F . The reaction routes in thechemical cycle in Fig. 1 can now be used to identify thecorresponding transitions between the mechanochemicalstates in the configuration space, as shown in Fig. 2.The relevant transition rates are defined on the configu-ration space diagram in Fig. 2. I have chosen a notationwhere the rates of similar transitions are differentiatedby a prime, with the general rule that k ′ α > k α . These U L hF (1) (2) (5) (3) (4) (6) (1): (2): (3): (4): (5): (6): Q QR R R F F - U R k ’ k k p k r ’ k r k p ’ k p k r ’ k f k f k k ’ J t m h Swimming Direction ( J >0) J J J - - - - - - - - - - FIG. 2. The 6 distinct mechanochemical states of the three-sphere swimmer in the configuration space. The conforma-tional states of the swimmer are determined by the projec-tion onto the space of the deformation of the left ( U L ) andright ( U R ) arms, while binding of h with F produces distinctchemical states that have identical conformations. All possi-ble transitions described in the chemical cycle in Fig. 1 areshown by arrows with attributed rate constants. choices are made depending on whether the presence ofadditional electrostatic charges hampers or facilitates acertain reaction [21].We do not make assumptions about any specific or-dering of the various events involved in the cycle (seeFigs. 1 and 2). For example, state (1) can proceed tostate (2) upon binding of R − to t , and can also proceedto (3) if F binds to h . Any of these two can happenat a time in a stochastic manner, and only the averagerates of these transitions are controlled. Moreover, notall of the transitions contribute constructively towardsnet swimming. For example, state (2) can proceed tostate (4) upon binding of F to h , which will take the sys-tem further down towards completing a non-reciprocalmechanical cycle, but could also go back to state (1) if P + binds to tR − , which undoes the progress made bythe previous transition towards net swimming.The dynamics of the device can be studied using astochastic description of the conformational transitionsand the relative population of the different states at sta-tionary state [20]. In stationary situation, each of thesix mechanochemical states shown in Fig. 2 is populatedwith a probability p i ( i = 1 , · · · , p + p + p + p + p + p = 1.The loops in the configuration space contribute prob-ability currents denoted as J , J , J , and J in Fig.2. Conservation of probability current leads to the fol-lowing nine equations between the probabilities and thecurrents: J + J = k p , J + J = k f [ F ] p , J = k ′ p , J − J = k ′ p , J − J = k f [ F ] p , J = k p , J = k ′ r [ R − ] p − k p [ P + ] p , J = k ′ p [ P + ] p − k r [ R − ] p , J − J = k ′ r [ R − ] p − k p [ P + ] p . The above ten linear - - (a) (b) (c) j j jf p r p = 1, r = 1 p = 0 . r = 1 p = 1, r = 0 . r = 0 . r = 0 . r = 1 p = 5 p = 1 p = 0 . FIG. 3. Dimensionless current j = J/k as a function of f = k f [ F ] /k , r = k r [ R − ] /k , and p = k p [ P + ] /k , corresponding to α = 1 . β = 1 . δ = 1 .
1, and b = 1. (a) Michaelis-Menten behavior, for γ = 1 .
1. (b) Sign change, for f = 1 and γ = 2 .
7. (c)Nonmonotonic concentration dependence, for γ = 1 . f → ∞ . equations can be solved for the ten unknowns (six prob-abilities and four currents) as functions of the concentra-tions [ F ], [ R − ], and [ P + ].The six states in the mechanochemical configurationspace of the swimmer correspond to only four indepen-dent conformations (Fig. 2). To study the coupling be-tween the kinetics of the mechanochemical cycle and thehydrodynamics of the ambient fluid, we need to consideronly the transitions between the distinct conformationalstates. Assuming that the extensions of the arms aresmall compared to their equilibrium lengths, it has beenshown that the net swimming velocity of the device iscontrolled by the surface area A enclosed by the confor-mational states that make the (1) ⇀↽ (2) ← (5) ⇀↽ (6) → (1) loop (in Fig. 2) and the net rate of sweeping this area J [20]. If we choose to make all the rates dimensionlessby k , and in particular define the dimensionless current j = J/k , we can write the average swimming velocity ofthe device as V = gDAk L j (cid:18) k f [ F ] k , k r [ R − ] k , k p [ P + ] k (cid:19) , (3)where g is a geometrical numerical prefactor of orderunity [22], D is the diameter of the spheres, and L isthe average total length of the swimmer.For simplicity, let us define the dimensionless variables α = k ′ r /k r , β = k ′ p /k p , γ = k ′ /k , δ = k ′ /k , and b = k /k , as well as the dimensionless concentrations f = k f [ F ] /k , r = k r [ R − ] /k , and p = k p [ P + ] /k . Asargued above, we expect α , β , γ , and δ to be numeri-cal factors (not too much) larger than unity. Solving theabove system of equations, we find that detailed-balanceholds (i.e. j = 0), when γ ( αβ − δ ) + ( αβ − γδ )( f + p + αr ) = 0 . (4)By tuning the concentrations f , r , and p away from val-ues that satisfy the above equality, we can drive the sys-tem away from equilibrium and obtain a nonvanishingswimming velocity ( j = 0). The concentration dependence of j reveals a numberof interesting features in the device, as shown in Fig.3. Figure 3a shows that the dependence of j and conse-quently the swimming velocity on the concentration of F adopts a Michaelis-Menten form. This behavior, which isalso observed in molecular motors [15], is inherited fromthe (multi-step enzymatic) reaction kinetics. Figure 3bshows that for sufficiently low concentrations of F andrelatively large values of γ , it is possible to have a situa-tion where j changes sign, which leads to the reversal ofthe swimming velocity. This can be understood from thecondition of detailed-balance in Eq. (4) as j is propor-tional to its left hand side. Figure 3c shows that whenthe concentration of F is at its saturation limit (see Fig.3a) the dependence of j on the concentrations of R − and P + is nonmonotonic, and suggests a strategy to optimizethe swimming velocity.There are many efficient enzymatic reactions thatcould be used in practice in such a setup. For example,Eq. (1) can represent the conversion of carbon dioxide( F = CO ) (plus water) into bicarbonate ( Q − = HCO − )and proton ( G + = H + ) catalyzed by carbonic anhydrase ( h = CA) [23]. An example for Eq. (2) could be su-peroxide anion ( R − = O · − ) being scavenged by su-peroxide dismutase ( t = SOD) by consuming a proton( P + = H + ) and producing hydrogen peroxide and oxy-gen ( S = H O + O ) [23]. Looking at Figs. 1 and2, we find that in this example k = k CA = 6 × s − [23], while k r = k SOD ∼ × M − s − [24]. Whilewe need to know all of the rates to be able to perform afull calculation, it is possible to make a rough estimateby looking at typical optimal values of j ∼ − fromFig. 3, and the above estimate for k . Using Eq. (3)with D ∼ A ∼ , and L ∼
10 nm, we find anestimate of V ∼
100 nm s − , which is comparable to theefficient molecular motors that use protofilament tracksfor their locomotion. In this molecular realization, theseenzymes could presumably play the role of the spheresthemselves, and could be attached to one another usingunstructured tails or covalently bonded polymeric teth-ers. It might also be feasible to attempt a larger scalesetup with colloidal spheres covered with adsorbed en-zymes and connected with polymeric tethers.In practice it might be difficult to fully suppress othertransitions, and we might have to deal with more compli-cated configurational spaces (e.g. involving other tran-sition routes between these states) as well as the backreactions. Moreover, realistic chemical reactions mighthave more complicated kinetics (such as the four-stagekinetics of the reaction of superoxide anion by SOD [23]that we have simplified into two stages in the above ex-ample). Such complications can be taken into account inthe calculation of the average swimming velocity using asimilar strategy [25], although they are not expected tochange the qualitative behavior of the system providedthe detailed-balance is broken.In addition to affecting the conformational transitions,thermal fluctuations will also tend to randomize the ori-entation of the swimmer, on time scales larger than therotational diffusion time τ r ∼ ηL / ( k B T ), where η is theviscosity of water, k B is the Boltzmann constant and T isthe temperature. In addition to the rotational diffusionof the swimmer, fluctuations in the density of the fuelmolecule could also induce anomalous fluctuations in themagnitude and direction of the velocity. The combina-tion of these effects could lead to a variety of differentbehaviors, similar to those found in the context of self-phoretic swimmers [26]. We note that a related enhanceddiffusion of active enzymes has recently been observed ex-perimentally [27]. The dependence of the velocity of theswimmer on concentration of the fuel in Eq. (3) also sug-gests that such artificial swimmers could be guided usingchemotaxis [18, 28, 29].In conclusion, I have proposed a design for a molecu-lar scale low Reynolds number swimmer that can use thefree energy gain and products of a nonequilibrium chem-ical reaction and undergo conformational changes thatare non-reciprocal and therefore can lead to net swim-ming under low Reynolds number conditions. While thepresent analysis has used a minimal kinetic model thatcan achieve swimming at such small scales, the schemecan be readily generalized to help analyze any realisticmechanochemical cycle corresponding to more complexmolecular devices that could be synthesized.I would like to thank A. Ajdari, D. Lacoste, A. Najafi,E. Rapha¨el, and H. A. Stone for fruitful discussions. Thiswork was supported by CNRS and EPSRC. ∗ r.golestanian@sheffield.ac.uk[1] E. R. Kay, D. A. Leigh, and F. Zerbetto, Angew. Chem.Int. 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