Working under confinement
aa r X i v : . [ phy s i c s . b i o - ph ] N ov EPJ manuscript No. (will be inserted by the editor)
Working under confinement
Paolo Malgaretti , , a , Ignacio Pagonabarraga , and J.M. Rubi Institut f¨ur Intelligente Systeme, Heisenbergstr. 3 D-70569 Stuttgart Germany IV. Institut f¨ur Theoretische Physik, Universit¨at Stuttgart, Pfaffenwaldring 57, D-70569Stuttgart, Germany Departament de Fisica Fonamental, Universitat de Barcelona, C. Mart´ı i Franques 1,Barcelona, Spain
Abstract.
We analyze the performance of a Brownian ratchet in thepresence of geometrical constraints. A two-state model that describesthe kinetics of molecular motors is used to characterize the energeticcost when the motor proceeds under confinement, in the presence of anexternal force. We show that the presence of geometrical constraintshas a strong effect on the performance of the motor. In particular, weshow that it is possible to enhance the ratchet performance by a propertuning of the parameters characterizing the environment. These resultsopen the possibility of engineering entropically-optimized transport de-vices.
Introduction
Transport of particles at small scales is frequently mediated by the presence of ob-stacles that particles may find along their trajectories. The effect of the obstacles isto diminish the space accessible to the particles. Since the number of microstates isproportional to the volume of the accessible space, the entropy varies along the tra-jectories and the transport proceeds through an entropic potential landscape. Thistransport referred to as entropic transport [1,2,3,4] exhibits peculiar characteristics,very different from those for the case of unbounded systems [5,6,7,8]. The case oftransport of particles in narrow and tortuous channels where the presence of obsta-cles and irregularities of the boundaries alter their trajectories is a typical example.Motion of particles in the interior of a living cell [9] or through an ion channel [10,11],diffusion in zeolites [12] and in microfluidic devices [6,13,14,15], and folding of pro-teins modeled as motion of the state of the protein through a phase space funnel-likeregion [16] are cases in which the system proceeds in a bounded region. Confinementis a source of continuous dynamic changes and consequently of modifications of theglobal properties of a system.A peculiar scenario arises when Brownian particles diffuse in a ratchet potential.The lack of a detailed balance principle gives rise to motion of the particles, a situationnot found when the particles are subjected to constant forces. Brownian ratchetsshow different dynamical behaviours for single particles [17] and for particle collectivebehavior [18,21] in an homogeneous medium. Interestingly, it has been shown that a e-mail: [email protected] Will be inserted by the editor the interplay between the ratchet mechanism and the geometrical constraint cangive rise to rectification even when the ratchet alone would not do so. Such a noveldynamical scenario, namely cooperative rectification [23], has been shown to producenet currents, even in the case of a symmetric ratchet potential and even currentinversion, i.e. a particle moving against the direction an asymmetric ratchet per sewould provide [23,24].Even though cooperative rectification has been shown to give rise to intriguingdynamic scenarios, up to now the question about its performance, i.e. the energyconsumption per unit displacement of cooperative rectification has not been yet ad-dressed. Here we shall discuss whether cooperative rectification is costly when com-pared to the bare ratchet mechanism or if the presence of entropic barriers can enhancethe overall performance of the Brownian ratchet. In particular, we will address twodifferent scenarios, namely, the case in which the Brownian ratchet displaces againsta drag force and the case in which, on the top of the drag force, Brownian ratchetsare also pulling against a conservative force. While in the latter case a good definitionof the performance is provided by the thermodynamic efficiency, in the former casesuch a definition is lacking and therefore we will rely on an ad hoc observable thatallows us to capture the energy cost per unit displacement of the molecular motor.
Confined Brownian ratchets
Brownian ratchets have been widely used as models to understand how molecu-lar [25,26] as well artificial [27,28,29] motors operate. To analyze their performanceunder confinement, we shall consider that particles move in a periodic channel whosecross-sectional area varies along the x -direction and is constant along z . The spaceavailable to the center of mass of a particle with radius R is 2 h ( x ) L z , where L z is thewidth along the z -direction and h ( x ) = h − R + h sin (cid:20) πL ( x + ∆φ ) (cid:21) (1)being h ( x ) + R the half-width of the channel along the y -direction and ∆φ the phaseshift between the periodicity of the channel and of the ratchet potential.As a ratchet model, we use the two-state model [31] that provides a simple frame-work to describe molecular motor motion. According to this model, a Brownian par-ticle jumps between two states, i = 1 ,
2, (strongly and weakly bound) that determineunder which potential, V i =1 , , it displaces [31]. A choice of the jumping rates ω , that break detailed balance, together with an asymmetric potential of the bound sate, V ( x ), determines the average molecular motor velocity v = 0. The conformationalchanges of the molecular motors introduce an additional scale that competes withrectification and geometrical confinement. Infinitely-processive molecular motors re-main always attached to the filament along which they move and are affected bythe geometrical restrictions only while displacing along the filament; accordingly, wechoose channel-independent binding rates ω ,p ( x ) = k . On the contrary, highlynon-processive molecular motors detach frequently from the biofilament and diffuseaway; this effect is accounted for a channel-driven binding rate, ω ,np ( x ) = k /h ( x ).Motors jump to the weakly bound state only in a region of width δ around the minimaof V ( x ), with rate ω = k . Accordingly, the motor densities in the strong(weak)states, P along the channel follow [31] ∂ t P , ( r ) + ∇ · J , = ∓ [ ω ( x ) P ( r ) + ω ( x ) P ( r )] (2) ill be inserted by the editor 3 where J , ( r ) = − D (cid:2) ∇ P , ( r ) + P , ( r ) ∇ βU , ( r ) (cid:3) (3)stands for the current densities in each of the two states in which motor displaces, β = 1 / ( k B T ) corresponds to the inverse thermal energy for a system at temperature T , k B stands for Boltzmann’s constant, and U , , which is periodic, U , ( x, y, z ) = U , ( x + L, y, z ), reads U , ( x, y, z ) = (cid:26) V , ( x ) | y | ≤ h ( x ) & | z | ≤ L z / ∞ | y | > h ( x ) or | z | > L z / V , ( x ) = (cid:26) ∆V sin πL x bound state ∆V otherwise (5)is the potential stemming from the interaction of the motor with the filament char-acterized by a depth ∆V .We shall assume that motor distribution equilibrates much faster in the cross sec-tion of the channel than along it. It is then possible to project the 3 D convectiondiffusion equation onto an effective 1 D equation governing the dynamics of particledensity along the longitudinal direction. Such a procedure was first introduced by Ja-cobs [1] and subsequently improved by Zwanzig [2] and by [3,32,33,34] and tested [35]in a variety of scenarios [4,36,37,38]This regime is fulfilled for a smoothly varying-section channels, ∂ x h ≪
1, in whichthe particle distribution reaches local equilibrium in the cross section almost imme-diately. We can then approximate the profile of the probability distribution function, P ( x, y, t ), assuming it equilibrates in the cross section of the channel, and write P , ( x, y, z, t ) = P , ( x, t ) e − βV , ( x,y ) e − βA , ( x ) (6)where e − βA , ( x ) = Z L z / − L z / Z h ( x ) − h ( x ) e − βV , ( x,y ) dydz (7) A , ( x ) corresponding to the potential of the mean force that can be identified withthe free energy in the coarse-grained description. Depending on the motor internalstate, two free energies, A , ( x ) = V , ( x ) − k B T S , ( x ), account for the interplaybetween the biofilament interaction and the channel constraints.By integration of Eq. 2 over dy, dz we then obtain˙ P , ( x, t ) = ∂ x D [ β P , ( x, t ) ∂ x A , ( x ) + ∂ x P , ( x, t )] , (8)which encodes both the confining as well conservative potentials given by Eq. 4 inthe free energy A , ( x ). Since all quantities of interest are independent of z , withoutloss of generality we can assume R L z / − L z / dz = 1. Defining the average, x -dependent,contribution to the energy coming from conservative potentials as W , ( x ) = e βA , ( x ) Z h ( x ) − h ( x ) V , ( x, y ) e − βV , ( x,y ) dy, (9)we can define the entropy along the channel as T S , ( x ) = W , ( x ) − A , ( x ) usingEq.( 7), leading to S , ( x ) = ln Z h ( x ) − h ( x ) e − βV , ( x,y ) dy ! + βW , ( x ) . (10) Will be inserted by the editor
In the linear regime, reached when βV , ( x, y ) ≪
1, Eq. 10 yields: S , ( x ) ≃ ln(2 h ( x )) (11)where entropy has a clear geometric interpretation, being the logarithm of the space,2 h ( x ), accessible to the center of mass of the tracer. Accordingly, we introduce theentropy barrier, ∆S , defined as ∆S , = ∆S = ln (cid:18) h max h min (cid:19) , (12)which represents the difference in the entropic potential evaluated at the maximum, h max , and minimum, h min channel aperture. Eq. 12, together with Eq. 1, showsthat the geometrical information coming from both the channel and the particle areencoded in ∆S . Effective performance
When molecular motors move in a crowded environment, their stepping performancesare strongly affected by the geometrical constraints [24]. A question that has notbeen addressed up to now is how the energy consumed by the motor is affected byconfinement.
Absence of external forces
When the motor does not displace against an applied external force, the standarddefinition of efficiency , namely the mechanical work performed over the energy con-sumed is not adequate since the mechanical work performed vanishes. In this case, itis more convenient to compute the amount of energy consumed to perform one stepforward [19,20,21,22].The two state model is particularly insightful because it clearly identifies themechanism where energy is consumed. Then, an unambiguous definition of the energyconsumption per unit step can be defined. Each jump of the molecular motor betweenthe bound and the weakly bound state costs an energy, ∆V , see Eq. 5, equal to theratchet potential depth. Therefore knowing the average number of jumps needed bya motor to perform one step froward, we can calculate the energy consumption perunit step. From Eq. 2, we can introduce Ω , the average number of jumps a motorperforms per unit time between the strongly and weakly bound states, Ω = Z L w off ( x ) p ( x ) dx = Z L w on ( x ) p ( x ) dx Having obtained this quantity, we can define the average number of jumps betweenthe two states needed to perform a single step as Γ = ΩLv (13)where v is the average motor velocity. Γ , a dimensionless parameter, is proportionalto the energy consumed by a molecular motor to perform a single step. Here we refer to the efficiency as obtained from the first law of thermodynamicsill be inserted by the editor 5 -0.3-0.2-0.1 0 0.1 0.2 0.3 0 0.2 0.4 0.6 0.8 1 L v / D ∆φ (a) -0.1 0 0.1 0.2 0.3 0.4 1 10 L v / D ∆ S(b)
Fig. 1.
Rectification of a processive (circles), non-processive (triangles) Brownian particlemoving due to the two state model in a symmetric channel. (a): particle velocity, in units of D /L , with D = D ( R = 1), as a function of the phase shift ∆φ for different values of theparameter ∆S = 1 . , . , .
94 (the larger the symbol size, the larger ∆S ), for ∆V = 0 . ω , /ω , = 0 .
01. (b): Processive (circles), non-processive (triangles) Brownian motorvelocity, in units of D /L , as a function of ∆S upon variation of particle radius R (solidlines, for h = 1 . , h = 0 . h (solid points, for R = 1 , h = 0 .
2) or h (open points, for R = 1 , h = 1 .
25) for ∆φ = 0 . , . ∆φ ). Reprinted withpermission from Malgaretti, Pagonabarraga and Rubi J. Chem. Phys. 138, 194906 (2013).Copyright (2013) by the American Physical Society. Kinetics under external forces
When molecular motors are stepping against an externally applied force, it is possibleto define a proper efficiency as the ratio between the work performed against theconservative force, | f | L over the energy consumed, ∆V Γ , when the motor displacesa single period of the potential, L . Therefore we can define the efficiency as η = | f | L∆V Γ = | f | v∆V Ω (14)that can be easily recognized as the power transformed in mechanical work over thepower injected that allows the molecular motor to jump between its two internalstates. Cooperative rectification
In order to analyze the interplay between the ratchet potential and the geometricalconstraints, in the following we assume that L = 1. The most striking feature ofthat interplay takes place when both the ratchet potential and the channel shape aresymmetric [23]. Despite the fact that in this case none of the mechanisms can rectifyindependently, rectification can be observed [23,24].Fig. 1.a shows that, even though the ratchet potential and the channel are bothsymmetric along the longitudinal direction, a net particle current develops. In partic-ular, such a current sets when the ratchet potential and the channel corrugation areout of registry. Cooperative rectification is quite sensitive to the entropic barrier ∆S .As shown in Fig. 1.b the net current can be modulated and its sign can be invertedby tuning the amplitude of ∆S at least for processive motors . Similar behavior has been obtained for non processive motors when the ratchet potentialis asymmetric, as shown in reference [23,24] Will be inserted by the editor -1 Γ ∆ S 10 -1 Γ ∆ S Fig. 2. Γ as a function of the entropic barrier ∆S for different values of the phaseshift where lighter lines stands for larger values of ∆φ for a processive motor (left, ∆φ = 0 . , . , . , . , . ∆φ = 0 . , . , . , .
4) inthe case of a symmetric ratchet potential, characterized by β∆V = 10, and symmetric chan-nel shape. In the case of processive motors velocity changes sign upon increase of ∆S asmarked in the figure: filled (open) points stands for negative (positive) velocities. -1 Γ β∆ V -1 ∆ S -1 Γ β∆ V -1 ∆ S Fig. 3. Γ as a function of the ratchet potential depth, ∆V , for processive (left) and nonprocessive (right) symmetric molecular motors moving along a symmetric channel, for aphase shift ∆φ = 0 .
3. The different curves correspond to different degrees of confinement,quantified by the magnitude of the entropic barrier, ∆S = 0 . , . , . , . , . , .
6; thelighter the color of the curve the larger the value of ∆S . Inset: Γ as a function of ∆S for β∆V = 0 . , . , . , , , ,
10; the lighter the color of the curve the larger the value of δV . Working under confinement
No external force
We will first analyze the motion of a motor in the absence of an applied external forcein a symmetric channel, and will analyze the efficiency of motor transport for botha symmetric and an asymmetric ratchet potential. In the former case, cooperativerectification emerges when the phase shift between the ratchet and the channel arenot in register. In the latter scenario we will assess how the modulation of the netmotor motion due to the geometrical constraints affects its efficiency.
Symmetric channel and ratchet potential
In this case, we expect cooperative rectification to generate net fluxes when the ratchetpotential and the channel are not in registry. Fig. 2 shows the dependence of Γ on ill be inserted by the editor 7 -4 -3 -2 -1 -3 -2 -1 | < v > | -3 -2 -1 -3 -2 -1 | < v > | Fig. 4.
Mean motor velocity, v , of processive (left) and non processive (right) sym-metric molecular motors moving along a symmetric channel as a function of Γ . Dif-ferent symbols correspond to different phase shift, quantified by ∆φ = 0 . ∆φ = 0 . ∆φ = 0 .
25 (upwards triangles), ∆φ = 0 . ∆φ = 0 . β∆V = 10. The magnitude of the entropic barrier, ∆S =0 . , . , . , . , . , , , . , . , . , . , . , . , . , .
6, is encoded in the color code, thelighter the color of the symbol the larger the value of ∆S . The dotted line is a guide for theeye highlighting the linear relation between v and Γ . the entropic barrier ∆S for different values of the phase shift. Both processive andnon processive motors show a non monotonic behavior of Γ as a function of ∆S therefore identifying an optimal value of ∆S for which Γ is minimum. Hence the spa-tial constriction induced by the channel plays a double role. On one hand geometricconstraints, jointly with the motor mechanism, are responsible for the onset of coop-erative rectification and therefore for the net current of motors. On the other handchannel corrugation affects the energetic cost of displacing along a in a corrugatedenvironment. Overall, increasing channel corrugation does not necessary lead to anincrease in the energy consumed per unit step, as shown in Fig.2. For a finite set ofvalues of ∆φ , processive motors change their displacement direction upon increasing ∆S [23], see Fig. 1. Accordingly, Γ shows a second divergence at the corresponding ∆S value . For the parameters analyzed in the Figure, non processive motors donot change their directionality and Γ only diverges for a uniform channel, ∆S = 0,when the motor does not rectify. Fig. 3 shows the dependence of Γ on the depth of theratchet potential ∆V . Increasing ∆V decreases the value of Γ till a plateau is reachedfor β∆V ≃
10. Interestingly, we have found that the value of ∆S that minimizes Γ is quite robust under variation of the ratchet potential depth, as shown in the insetsof Fig. 3.According to Eq.13, Γ depends on both the number of jumps between the twostates and the average velocity achieved by the motor. Therefore, a further insightinto the dynamics is shown in Fig. 4 that displays the inverse proportionality betweenthe average velocity v and Γ . The dependence displayed in Fig. 4 , in agreement withthe simple expectation quantified in Eq.13, indicates that the motor hopping dynam-ics between the two states is not significantly affected by the corrugated channel.Therefore the reduction in the energy consumption provided by cooperative rectifica-tion relies on a more efficient transduction of the energy injected in the system intodisplacement rather than on a reduction of the energy consumed by the motor perunit time. Γ diverges when the motor spends energy but is not displacing along the channel. Will be inserted by the editor -1 -1 Γ / Γ ∆ S 10 -1 -1 Γ / Γ ∆ S Fig. 5. Γ , normalized by Γ ( ∆S = 0) = Γ , as a function of the entropic barrier, ∆S ,for processive (left) and non processive (right) symmetric molecular motors moving along asymmetric channel, for β∆V = 10 . The different curves correspond to different values ofthe phase shift, ∆φ = 0 , . , . , . , . -1 -1 Γ / Γ β∆ V -1 -1 ∆ S -1 -1 Γ / Γ β∆ V -1 -1 ∆ S Fig. 6.
Γ/Γ as a function of the ratchet potential ∆V for processive (left) and non processive(right) asymmetric molecular motors moving along a symmetric channel, for a phase shift ∆φ = 0 .
1. The different curves correspond to different degrees of confinement, quantifiedby the magnitude of the entropic barrier, ∆S = 0 . , . , . , . , . , .
6; the lighter thecolor of the curve the larger the value of ∆S . Inset: Γ/Γ as a function of ∆S for β∆V =0 . , . , . , , , ,
10; the lighter the color of the curve the larger the value of ∆V . Symmetric channel and asymmetric ratchet potential
Molecular motors will now displace even in the absence of a spatially-varying channel.Therefore, we can define Γ as the value of Γ obtained for a flat channel, i.e. for ∆S = 0. In this way we can characterize the deviations in Γ due to the geometricalconfinement. As it has already been discussed [23,24], the presence of geometricalconstraints strongly affect the net motion of molecular motors even when motors canrectify by themselves. Fig. 5 shows the dependence of Γ on the entropic barrier ∆S .For both processive and non processive motors, Γ shows a non monotonous behavioridentifying an optimal entropic barrier for which Γ attains a minimum. Therefore thepresence of the entropic barrier not only tunes motor currents [23,24] but also allowsto reduce the energy consumed by motors in order to perform a single step.As shown in Fig. 6 the behavior of Γ/Γ as a function of ∆V is similar to thatobserved for a symmetric ratchet potential, see Fig. 3. However, in the present casethe value of ∆S for which the minimum of Γ is more sensitive to ∆V . For smallervalues of ∆V the minimum of Γ is obtained by minimizing ∆S , while for larger valuesof ∆V the minimum is obtained for non vanishing ∆S . As in the case of a symmetricratchet potential, we find an inverse proportionality relation between Γ and v , shown ill be inserted by the editor 9 -1 Γ /Γ -2 -1 | < v > | -1 Γ /Γ -2 -1 | < v > | Fig. 7.
Mean molecular motor velocity, v , as a function of Γ/Γ of processive (left) and nonprocessive (right) asymmetric molecular motors displacing in a spatially-varying, symmetricchannel. The different symbols correspond to different phase shifts, quantified by ∆φ =0 . ∆φ = 0 . ∆φ = 0 .
25 (upwards triangles), ∆φ = 0 . ∆φ = 0 . β∆V = 10. The magnitude of the entropic barrier ∆S = 0 . , . , . , . , . , , , . , . , . , . , . , . , . , .
6, is encoded in the color ofthe symbols, the lighter the color of the curve the larger the value of ∆S . The dotted line isa guide for the eye highlighting the linear relation between v and Γ . in Fig. 7, indicating that, analogously to the case of symmetric ratchet potential,the hopping dynamics of the molecular motor between its two internal states is notstrongly affected by the confinement imposed by the corrugated channel. Motor performance in the presence of an external force
When motors are pulling against an external force it is possible to define a thermo-dynamic efficiency, as discussed to introduce Eq. 14. We can also analyze the perfor-mance of a symmetric motor in a symmetric channel or when the ratchet potential isasymmetric. In this latter case the motor already displace when moving along a flatchannel. Hence, in this second scenario it is possible to define a “bulk” efficiency, η ,which we use as a reference to characterize the role of the geometrical constraint onmotor’s efficiency.The external force decreases the intrinsic motor velocity and it can eventuallyinvert its motion. Obviously, at the stall force the molecular motor efficiency vanishesand the motor efficiency at larger values of the force is not well defined. Hence, wewill not analyze motors efficiency beyond the stall force. Symmetric channel and ratchet potential
Since molecular motor rectification emerges from the interplay between molecularmotor and channel corrugation, symmetric molecular motors cannot displace againstthe applied force for a flat channel. As the corrugation increases there exists a finitestall force that initially increases with channel corrugation. Fig. 8 and its insets showthat both processive and non-processive motors have a maximum stall force for afinite, optimal channel corrugation. For the same reason, increasing the magnitudeof the applied force decreases the range of channel corrugations where the motordisplaces against the applied force, as shown in Fig. 8. Moreover, the insets Fig. 8 alsoshow that there exists a maximal stall force beyond which the motor cannot displaceagainst the applied force, independently of the channel corrugation. Comparing the -8 -7 -6 -5 -4 -1 η ∆ S -3 -2 -1 -1 β | f | L ∆ S -6 -5 -4 -3 -2 -1 η ∆ S -2 -1 -1 β | f | L ∆ S Fig. 8.
Molecular motor efficiency, η , as a function of the entropic barrier ∆S for pro-cessive (left) and non processive (right) symmetric molecular motors moving along aspatially-varying, symmetric channel, for a phase shift ∆φ = 0 . β∆V = 10. Thedifferent curves correspond to different values of the applied external force, β | f | L =0 . , . , . , . , . , .
02; the lighter the color of the curve the larger the magni-tude of f . Insets: stepping state of the motor. Bigger circles represent the sets of parameters ∆S and f for which the motor can step against the force, smaller squares the set of valuesfor which the motor cannot step against the force -7 -6 -5 -4 -1 η β∆ V -7 -6 -5 -4 -1 ∆ S10 -1 -1 β ∆ V ∆ S -6 -5 -4 -3 -1 η β∆ V -7 -6 -5 -4 -3 -1 ∆ S10 -1 -1 β ∆ V ∆ S Fig. 9.
Molecular motor efficiency, η , as a function of the ratchet potential depth ∆V forprocessive (left) and non processive (right) symmetric molecular motors moving along aspatially-varying, symmetric channel, for a phase shift ∆φ/ (2 π ) = 0 .
2. The different curvescorrespond to different values of the entropic barrier ∆S = 0 . , . , . , .
5; the lighter thecolor of the curve the larger the magnitude of ∆S . Upper insets: efficiency as a function of ∆S for β∆V = 1 , , ,
10 for ∆φ = 0 .
2. Lower insets: stepping state of the motor. Biggercircles represent the sets of parameters ∆S and f for which the motor can step against theforce, smaller squares the set of values for which the motor cannot step against the force. insets in Fig. 8 with their corresponding main panels, we find that the stall force ismaximized at maximum efficiency for both processive and non processive motors.Fig. 8 shows the dependence of the efficiency, η , as a function of the entropic barrier ∆S . For both processive and non processive motors we observe an optimum of theefficiency for non vanishing values of ∆S . Such a behavior derives from the nontrivialdependence of cooperative rectification shown in Fig. 2 where we can identify anoptimal value of the entropic barrier, ∆S opt that minimizes Γ . Such a behavior isretained leading to a maximum efficiency for ∆S ≃ ∆S opt . Therefore the reductionof the energy cost observed in the absence of external forces, see Fig. 2, leads to anincrease of the efficiency when motors are pulling against applied forces.Fig. 9 and its top insets show the dependence of the efficiency upon variations ofthe ratchet potential depth, ∆V for different values of ∆S . As compared to the case of ill be inserted by the editor 11 -4 -3 η / β |f|L10 -3 -2 -1 | < v > | -4 -3 -2 η / β |f|L10 -3 -2 -1 | < v > | Fig. 10.
Mean velocity, v , as a function of the relative efficiency, η/η , of a processive (left)and non processive (right) motor for δφ = 0 . β∆V = 10. Different values of the exter-nal force β | f | L = 0 . , . , . , . , . , . , .
04 are encoded by different simbols(squares, circles, up triangles, down triangles, diamonds, pentagons, crosses respectively)while the different values of ∆S are color coded, where lighter colors stand for larger valuesof β | f | LS . absence of external force, see Fig. 3, here we have found a more sensitive dependenceof the efficiency on ∆V .When the motor step against the force, we have found that the value of ∆V maximizing efficiency is quite sensitive to the value of the entropic barrier ∆S ascan be see in the main figure and in the upper insets. Moreover processive and nonprocessive motors show a diverse relation between the efficiency and ∆V . In fact,for processive motors we observe that η saturates when increasing ∆V while for nonprocessive motors we observe a monotonous growth of η with ∆V for the range ofvalues of ∆V explored, as shown in Fig. 9.The dependence of the efficiency on the force is more involved. As shown in Fig. 8,when ∆S maximizes the efficiency, i.e. for ∆S = ∆S opt , we find that η increases withthe external force. At larger values of the external force, when it approaches the stallforce, the efficiency will decrease and will eventually vanish for the correspondingcorrugation, ∆S . Fig. 8 shows that for ∆S < ∆S opt the maximum efficiency is notalways attained for the largest of the range of applied external forces sampled, showingalready the existence of a characteristic external force for which the optimal efficiencyis achieved; this optimal force changes with the channel corrugation, ∆S .However, the relation between η and the average velocity, v , is linear, as shownin Fig. 10. Moreover, similarly to what we have seen in the previous cases , all thedata collapse on a straight line ensuring a direct proportionality between η and v .Coherently with the previous cases, we conclude that the hopping dynamics is onlyslightly affected by the geometrical confinement and the main dependence of theefficiency on ∆S comes from the velocity. Symmetric channel and asymmetric ratchet potential
When the ratchet potential is asymmetric, motors can step in a flat channel, ∆S =0. Therefore, if the external force is smaller than the stall force, we can define areference efficiency, η defined as the efficiency obtained for ∆S = 0. Fig. 11 shows thedependence of the normalized efficiency upon variation of the entropic barrier ∆S .Both processive and non processive motors show a confinement-induced efficiency We remind that η ∝ Γ − -1 η / η ∆ S 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210 -1 η / η ∆ S Fig. 11. η , normalized by the efficiency η of a motor in a flat channel, as a function of theentropic barrier ∆S for different values of the force, f , where lighter lines stands for largervalues of f , β | f | L = 0 . , . , . , . , . , .
02, for a processive motor (left) andnon processive motor (right) in the case of an asymmetric ratchet potential and symmetricchannel shape. The phase shift ∆φ is ∆φ = 0 ( ∆φ = 0 .
2) for filled (empty) points in thecase of processive motor and ∆φ = 0 . ∆φ = 0 .
5) for filled (empty) points in the case ofnon processive motors. -2 -1 -1 η / η β∆ V -1 ∆ S -3 -2 -1 -1 η / η β∆ V -1 ∆ S Fig. 12. η/η as a function of the entropic barrier ∆S for different values of the ratchetpotential ∆V = 0 . , . , . , . , , , , ,
10, where lighter lines stands for larger values of ∆V for a processive motor (left) and non processive motor (right) in the case of a symmetricratchet potential and symmetric channel shape, being ∆φ = 0 . for processive motors and ∆φ = 0 . enhancement and an optimal value of ∆S . Such a behavior is strongly affected by thephase shift as well as by the processive nature of the motors. In fact, while for ∆φ = 0( ∆φ = 0 .
2) for processive (non processive) motors we observe an enhancement in theefficiency upon increasing ∆S , for ∆φ = 0 . ∆φ = 0 .
5) we observe a reduction inthe efficiency for increasing values of ∆S .The dependence of η/η on ∆V is shown in Fig. 12. Differently from the case ofsymmetric ratchet potential, in the present case both processive and non processivemotors exhibit a monotonous increase of the efficiency with ∆V till β∆V ≃
10 where η/η shows a plateau. As in the previous case, see Fig. 9, the efficiency increases forlarger ∆S . The insets of Fig. 12 show the dependence of the efficiency as a function of ∆S for different values of ∆V . As shown in the figure, for vanishing small corrugation, ∆S → η/η →
1. However, by increasing ∆S the efficiency shows a nonmonotonous behavior and it reaches a maximum for a finite value of ∆S .Finally the relation between the efficiency and the net velocity is shown in Fig. 13.Surprisingly in the present case we find, for both processive and non processive motors, ill be inserted by the editor 13 | < v > | η/η | < v > | η/η Fig. 13.
Mean velocity, v , as a function of the relative efficiency, η/η , of a processive(left) and non processive (right) motor for β | f | L = 0 . , . , . , . , . , . , . ∆S =0 . , . , . , . , . , , , . , . , . , . , . , . , . , . ∆S . a non linear relation between η/η and v as opposed to all other cases analyzedpreviously. Fig. 13 shows that the maximum speed is obtained at maximum efficiency.However, when either efficiency or velocity are not maximized, we find that for a givenefficiency motors can achieve two different speeds and viceversa. The value of entropicbarriers that maximizes the efficiency, ∆S opt , identifies the threshold between twoscenarios and for ∆S < ∆S opt motors displace less efficiently as compared to motorsmoving at the same speed in a channel characterized by a larger value of ∆S . Conclusions
We have shown that the presence of geometrical constraints strongly affect the per-formance of molecular motors. In particular the efficiency, or the energy spent perunit step, shows a strong dependence on the environmental properties, encoded inthe entropic barrier.When motors are not pulling against an externally applied force there is no cleardefinition of efficiency since motors are not producing any mechanical work. However,it is possible to capture their performance by measuring the energy cost, Γ , neededto perform a single step forward. We have shown that the cooperative rectificationthat arises from the interplay between internal, out of equilibrium, transformation ofthe molecular motors and the geometrical confinement can give rise to very differentperformances according to the amplitude of the entropic barrier as well as on thephase shift between the two potentials. In the case of symmetric ratchet and channel,we have show that Γ has a non monotonous behavior for both processive and nonprocessive motors, see Fig. 2 leading to an optimal value of ∆S that minimizes theenergy required to perform a single step. Such an optimal value of ∆S depends on thephase shift between the ratchet potential and the channel as well as on the processivenature of the motor. Surprisingly, the magnitude of the spatial confinement, quantifiedby ∆S that minimizes Γ is robust against variation in the ratchet potential depth ∆V , see Fig. 3. Finally, while for processive motors we have found an optimal valueof ∆V , for a given ∆S , that minimizes Γ (see Fig. 3), for non processive motors Γ decreases monotonously for increasing values of ∆V . The inverse proportionalitybetween the molecular motor average velocity v and Γ , shown in Fig. 4, underlinesthe mild dependence of the hopping dynamics on the confinement as opposed to thestrong dependence experienced by v . For asymmetric motors, which rectify even in the absence of spatial confinement,we have found that entropic barriers affect their overall performance. The perfor-mance of both processive and non processive motors increases by properly tuning theamplitude and phase shift of the entropic barrier, as shown in Fig. 5. Therefore, thegeometrical constraints reduce the molecular motor energy consumption to displace.Although in this case Γ always decreases when increasing the binding motor poten-tial, ∆V , the dependence of Γ on ∆V , strongly depends on the processive natureof the motors. Specifically, we have seen that for non processive motors the value of ∆S minimizing Γ is quite insensitive to variations in ∆V , while the reverse holds forprocessive motors. In particular we have found that while for smaller values of ∆VΓ is minimum for ∆S = 0, when ∆V exceed a threshold the value of ∆S minimizing Γ jumps to ∆S = 0. The dependence of v on Γ , as in the previous case, shows themild dependence of he hopping dynamics on the confinement, see Fig. 7.When motors are pulling against an applied force we can properly define an ef-ficiency, η , as the ratio between the work performed over the energy consumed. Wehave seen that cooperative rectification allows symmetric molecular motors to performmechanical work against external forces. Moreover, we have found that the efficiencyincreases with the external force at the expense of a reduction of the range of valuesof ∆S for which the motor can still step against the force. For larger forces motors areunable to step against and a confinement-dependent stall force can be defined. Thepresence of an external force dramatically changes the dependence of η on ∆V forprocessive motors while non processive retain a behavior similar to that obtained for f = 0, see Fig. 3. In particular processive motors exhibit a ∆V -dependent value of ∆S for which η is maximized and the dependence of η on ∆V identifies a ∆S -dependentoptimal value for which η is maximum. On the contrary non processive motors showa ∆V -independent optimal value of ∆S maximizing η as we found in the case f = 0.The dependence of v on η , shown in Fig. 10 present the same behavior obtained in theprevious cases, and the hopping dynamics is almost independent on the confinement.When the ratchet potential is asymmetric, i.e. motors rectify even for ∆S = 0, wefound, as in the last case, that η can be enhanced by properly tuning ∆S and ∆φ ,as shown in Fig. 11. Moreover, both processive and non processive motors exhibita maximum in η for a ∆V -independent value of ∆S , see Fig. 12. Interestingly, forprocessive motors such a behavior is different from both the case of asymmetric ratchetand f = 0, see Fig. 6 and the case of symmetric ratchet and f = 0, see Fig. 9 andreminds that of a symmetric ratchet potential in the absence of an external force, seeFig. 3. Finally, the dependence of v on η is remarkably different from that obtainedin all the previous cases. In fact, in the present case we found that v and η are notdirectly proportional, implying a more involved dependence of the hopping dynamicson the confinement. Such a different behavior underlines that the presence of anexternal force can strongly affect the inner dynamics of the motor therefore affectingthe overall motor performance.J.M.R. and I.P. acknowledge the Direcci´on General de Investigaci´on (Spain) andDURSI for financial support under projects FIS 2011-22603 and 2014SGR922, re-spectively and financial support from Generalitat de Catalunya under program
IcreaAcad`emia . References
1. Jacobs, H., M.,
Diffusion Processes (Springer-Verlag, New York 1967)2. Zwanzig, R., J. Phys. Chem., , (1992) 39263. Reguera, D., and Rubi, J. M. , Phys. Rev. E, , (2001) 0611064. Reguera, D., Schmid, G., Burada, P. S., Rubi, J. M., Reimann, P., and H¨anggi, P., Phys.Rev. Lett., , (2006) 130603ill be inserted by the editor 155. Vazquez, M., Berezhkovskii, A., and Dagdug, L., J. Chem. Phys., , (2008) 0461016. Dagdug, L., Berezhkovskii, A. M., Makhnovskii, Y. A., Zitsereman, V. Y., and Bezrukov,S., J. Chem. Phys., ,(2011) 1011027. Burada, P.,S., H¨anggi, P., Marchesoni, F., Schmid, G. and Talkner,P., Chem. Phys. Chem. , (2009) 458. Malgaretti, P., Pagonabarraga I and Rubi, J.,M., Front. Physics , (2013) 219. Brangwynne, C.P., Koenderink, G., MacKintosh, F., and Weitz, D., Trends Cell. Biol., , (2009) 42310. Hille, B., Ion Channels of Excitable Membranes (Sinauer, Sunderland 2001)11. Calero, C., Faraudo, J., and Aguilella-Arzo, M., Phys. Rev. E, ,(2011) 02190812. Barrer, R. M., Zeolites and Clay Minerals as Sorbents and Molecular Sieves (Academic,New York 1978)13. Han, J., and Craighead, H., Science, , (2000) 102614. Altintas, E., Sarajlic, E., Bohringerb, F. K., and Fujita, H., Sensors and Actuators A, , (2009) 12315. Malgaretti, P., Pagonabarraga, I., and Rubi, J. M., Phys. Rev. Lett. ,(2014) 12831016. Dill, K., and Chan, H., Nat. Struct. Biol., , (1997) 1017. H¨anggi, P., and Marchesoni, F., Rev. Mod. Phys. ,(2009) 3818. Guerin, T., Prost, J., Martin, P. and sinoJ.-F.Joanny, Curr. Op. Cell Biol. , 14 (2010)19. Suzuki, D., Munakata, T., Phys. Rev. E , (2003) 02190620. Machura, L., Kostur, M., Talkner, P., Luczka, J., Marchesoni, F., and H¨anggi, P., Phys.Rev. E , (2004) 06110521. Malgaretti, P., Pagonabarraga, I., and Frenkel, D., Phys. Rev. Lett. , (2012) 16810122. Lervikemail, A., Bresme, F., Kjelstrup, S. and Rubi, J. M., Biophys. J. ,(2012) 121823. Malgaretti, P., Pagonabarraga, I., and Rubi, J. M., Phys. Rev. E , (2012) 01010524. Malgaretti, P., Pagonabarraga, I., and Rubi, J. M., J. Chem. Phys, , (2013) 19490625. Thomas, G., Prost, J., Martin, P., and Joanny, J.-F., Curr. Op. in Cell Biol., , (2010)1426. Astumian, R. D., Biophys. J., , (2010) 240127. Allison, A., and Abbott, D., Microelectronics Journal, , (2002) 23528. Zhu, B. Y., Marchesoni, F., and Nori, F., Phys. Rev. Lett., , (2004) 18060229. Linke, H., Xu, H., Lo, A., Sheng, W., Svensson, A., Omling, P., Lindelof, P. E., Newbury,R,. and Taylor, R. P., Physica B, ,(1999) 6130. H¨anggi, P., and Marchesoni, F., Rev. Mod. Phys., , (2009) 38731. J¨ulicher, F., Ajdari, A., and Prost, J. , RMP Colloquia , (1997) 126932. Kalinay, P., and Percus, J. K., J. Chem. Phys. , (2005) 20470133. Kalinay, P., and Percus, J. K., Phys. Rev. E, ,(2006) 04120334. Martens, S., Schmidt, G., Schimansky-Geier, L., and H¨anggi, P., Phys. Rev. E, ,(2011) 05113535. Burada, P. S., Schmid, G., Reguera, D., Rubi, J. M., and H¨anggi, P., Phys. Rev. E ,(2007) 05111136. Reguera, D., Luque, A., Burada, P. S., Schmidt, G., Rubi, J. M., and H¨anggi, P., Phys.Rev. Lett. ,(2012) 02060437. Pineda, I., Alvarez-Ramirez, J., and Dagdug L., J. Chem. Phys. ,(2012) 17410338. Kalinay, P., Phys. Rev. E89