Two-state flashing molecular pump
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un epl draft Two-state flashing molecular pump
A. Gomez-Marin (a) and J. M. Sancho
Facultat de F´ısica, Universitat de Barcelona - Diagonal 647, Barcelona, Spain, EU
PACS – Fluctuation phenomena, random processes, noise, and Brownian motion
PACS – Stochastic analysis methods
PACS – Theory, modeling, and simulations
Abstract. - Here we study a pumping device capable of maintaining a density gradient anda flux of particles across a membrane. Its driving mechanism is based on the flashing ratcheteffect powered by the random telegraph process in the presence of thermal fluctuations. UnlikeBrownian motors, the concentrations at both reservoir boundaries need to be implemented asboundary conditions. The residence transition rates of the dichotomic flashing are related to thebinding and hydrolysis of ATP molecules. The model is exactly solved and explored. The pumpenergetics is discussed and the relevant parameter values are tuned within a biological scale.
Introduction.
Active transport processes are ubiq-uitous in living matter. In fact, the cell cannot survivewhen specific out-of-equilibrium tasks inside cease, mainlycarried out by proteins such as kinesin, RNA-isomerase,dynein or rotatory ATP-ase. For instance, in order tomaintain a suitable osmotically regulated state, the pump-ing of ions across the cell membrane is a crucial job that isperformed by molecular pumps, which are, in turn, pow-ered by the hydrolysis of ATP molecules. Stemming fromthe studies on active transport in noisy environments [1],the stochastic features of pumps were originally addressedin [2, 3] and applied to chemically driven electron pump-ing [4]. An experiment on artificial ion pores showing recti-fication was presented in [5], where the observations whereinterpreted by a model containing flashing ratchets. Alsoin [6], experimental realizations of synthetic nanoscopicdevices transporting potassium ions against their concen-tration gradient when stimulated with external field fluc-tuations were built. Recently it has been directly demon-strated that the ratchet mechanism is indeed of use ina biological system where molecular transport driven byfluctuations occurs in a membrane channel [7]. Other the-oretical works have studied the gating kinetics of ionicchannels, based on the effect of non-equilibrium fluctua-tions combined with the ratchet effect [8]. A Browniandevice with cyclic steps accounting for its configurationalchanges has been shown to reproduce the main features ofexperiments on the Na,K-ATP-ase ion pump [9]. Moving (a)
Current address: EMBL-CRG Systems Biology Unit, Center forGenomic Regulation, UPF - Barcelona, Spain, EU. to more simplistic descriptions, the study of active trans-port in an idealized Brownian pump focusing on the con-centration gradient created and maintained at both sidesof a membrane has been analytically and numerically dis-cussed in [10].Despite its relevance for modeling molecular pumps, theratchet effect and its transport features against a concen-tration gradient have drawn very little attention. In thisLetter we introduce a simple, yet revealing, two-state theo-retical model for a machine which pumps particles againsta concentration gradient. The model is inspired by re-cent experimental works where the operating cycle of achannel has been shown to involve two steps (opening andclosing) related to the ATP binding and hydrolysis respec-tively [11,12]. The pumping device we present consists of aratchet potential, embedded in a membrane and boundedby particle reservoirs, which exhibits dichotomic fluctua-tions controlled by the ATP concentration. We do notfocus on the precise structural details of the pump, but doit at the level of energetics. We consider the asymmetricratchet profile as an energy barrier. In the steady state,we exactly calculate the concentration ratio between bothreservoirs and the particle flux. We relate the performanceof the device with ATP concentration, study its energeticsand finally place it in a biological context.
The model.
Let us consider a pumping mechanismbased on a flashing potential embedded in a membrane oflength L , whose boundaries are infinite reservoirs of parti-cles with fixed mean densities ρ and ρ . The dynamics ofa particle in the membrane is determined by the followingp-1. Gomez-Marin and J. M. SanchoLangevin equation in the over-damped regime, γ ˙ x = − U ′ ( x, t ) + ξ ( t ) , (1)where U ( x, t ) is a time dependent potential (the prime de-notes position derivative) and ξ ( t ) the random force aris-ing from the thermal fluctuations. It has zero mean andits autocorrelation is given by the fluctuation-dissipationtheorem, h ξ ( t ) ξ ( t ′ ) i = 2 γk B T δ ( t − t ′ ) , (2)where γ is the friction coefficient and T the temperature ofthe environment. Whereas in the reservoirs the particlescan diffuse freely, in the membrane they experience thepotential U ( x, t ) = V ( x ) ζ ( t ) , (3)where V ( x ), as depicted in figure 1, is a piecewise linearasymmetric ratchet potential: V ( x ) = V xδL , x ∈ [0 , δL ] , (4) V ( x ) = V L − x (1 − δ ) L , x ∈ [ δL, L ] . (5) V is the energy barrier height and δ controls the spatialasymmetry by taking values between zero and unity.The flashing modulation ζ ( t ) is a stochastic forcing intime which randomly switches the ratchet potential be-tween two states [1, 2]. It is distributed according to arandom telegraph (dichotomous Markov) process, flippingbetween values ζ a = 1 (closed state) and ζ b = 0 (openstate), with residence probability transitions w a and w b respectively [13, 14]. Its mean value and correlation are h ζ ( t ) i = w a w a + w b , (6) h ∆ ζ ( t )∆ ζ ( t ′ ) i = w a w b ( w a + w b ) e − ( w a + w b ) | t − t ′ | (7)where ∆ ζ ( t ) ≡ ζ ( t ) − h ζ ( t ) i . While the asymmetric poten-tial breaks the spatial symmetry, the flashing transitionsbetween states are responsible for the breaking of detailedbalance in the system. Transition rates and ATP concentration.
Thehydrolysis of ATP molecules provides the energy input.It creates the necessary out-of-equilibrium conditions todrive molecular pumps. As a first level of approxima-tion to include the effect of ATP consumption into themodel, we relate the ATP concentration, its binding andits hydrolysis processes to both residence transition rates.This two-state picture has been described in experimentsin channels [11, 12] and it is a valid paradigm for themathematical characterization of mechano-chemical en-ergy transduction. We consider that, when the pump isin the closed state ( ζ a = 1), it is the binding of an ATPmolecule to a certain pocket center of the pump which in-duces conformational change to the open state ( ζ b = 0).The transition rate w a is thus dependent on the ATP Figure 1: Pump model: an asymmetric potential V ( x ) flashesits shape at random between a blocking (closed) and a flat(open) configuration with transitions rates w a and w b . concentration available in the surroundings, which can bemodeled as a Michaelis-Menten kinetics law: w a = w [ATP] k M + [ATP] = w /σ , (8)where k M is the so-called affinity constant and [ATP] isthe concentration of ATP molecules. Only their ratio isrelevant: σ ≡ [ATP] / k M . The rate w b is given by theinverse of the machine’s intrinsic time, which we considerconstant: w b = w . (9)In the limit of a saturating [ATP], both transition prob-abilities are equal ( w a = w b = w ) indicating that thepump cannot operate faster. Medium [ATP] values implya longer average time in the closed state. At low concen-trations, w a is small and the pump seldom starts the cycle.Diffusive leak losses dominate and its pumping capacity isthen very low. See scheme in figure 2. Numerical simulations.
As a first test to show thatthe model just introduced can indeed create and maintaina concentration gradient, we use the simulation frameworkdiscussed in [10]. This method allows for the numerical im-plementation of the reservoir boundary conditions, whichis not trivial. We then simulate non-interacting particlesfollowing the Langevin equation (1), under thermal whitenoise and the random telegraph process. In figure 3, weplot the histogram of the position of the particles, be-ing illustrative of the density profiles typically obtainedin the steady state at zero net flux. It is clear that thepump is able to generate and maintain a concentration ra-tio ( ρ = ρ ), while the edges emulate particle reservoirswith uniform concentrations. Theoretical analysis.
Consistent with the Langevinequation (1), the corresponding partial differential equa-tions for the time evolution of the concentration of parti-cles in each flashing state, P a ≡ P a ( x, t ) and P b ≡ P b ( x, t ),originate from a combination of the Fokker-Planck de-scription for the continuous variable x and the Masterequation for the discrete states a and b . They readp-2wo-state flashing molecular pump Figure 2: Illustrative representation of the effect of ATP con-centration in the switching rates of the dichotomic flashingratchet mechanism. [2, 13, 14] ∂ t P a = γ − ∂ x [ V ′ ( x ) + k B T ∂ x ] P a − w a P a + w b P b ,∂ t P b = γ − ∂ x [ k B T ∂ x ] P b − w b P b + w a P a . (10)It is convenient to work with dimensionless quantities.We first re-scale space and time: z ≡ x/L and s ≡ t ( k B T ) / ( γL ). Consequently, the new transition rates are f ≡ w ( γL ) / ( k B T ) (for w a , w b and w ) and L is absorbedin the concentration distributions. We introduce the fol-lowing densities: the total density of particles ρ ≡ P a + P b and the auxiliary function π ≡ P a − P b . In the steadystate, eqs. (10) are governed by a new set of equations foreach region (indicated by the subindex i = 1 , ρ ′ i − F i [ ρ i + π i ] = − J, (11) π ′′ i − F i ( ρ ′ i + π ′ i ) = π ( f b + f a ) − ρ ( f b − f a ) , (12)where F = − v / δ and F = v / − δ ). v ≡ V /k B T measures the relative strength of the potential with re-spect to the thermal energy. J is the physical total flux ofparticles across the membrane.Substituting (11) in (12) one gets a linear ODE for ρ i ( z ), ρ ′′′ i ( z ) − F i ρ ′′ i ( z ) − ( f a + f b ) ρ ′ i ( z )+2 f b F i ρ i ( z ) = J ( f a + f b ) , (13)whose solution has the standard form ρ i ( z ) = C i e λ i z + C i e λ i z + C i e λ i z + ( J/F i ) α, (14)where α ≡ ( f a + f b ) / f b . The constants C ij are, so far,unknown parameters. Instead, the constant coefficients λ ij are found by inserting ρ i ( z ) back in (13) and solvingthe algebraic equation λ ij − F i λ ij − ( f a + f b ) λ ij + 2 f b F i = 0 . (15)Plugging (14) in equation (11) and defining s ij ≡ ( λ ij /F i −
1) and β ≡ ( f b − f a ) / f b , one finds π i ( z ) = s i C i e λ i z + s i C i e λ i z + s i C i e λ i z + ( J/F i ) β. (16) ρ ρ Figure 3: Density profile obtained directly from numericalsimulations of the Langevin equation in the steady state atzero flux.
Boundary conditions.
The formal solution of theconcentration of particles in the steady state depends onseven unknown constants (the coefficients C ij and the flux J ), whose value is determined by boundary conditions. Incontrast with models for Brownian motors, in the presentBrownian pump one should not impose normalization northe periodicity condition for the probability. Instead,the total concentrations at left and right boundaries, ρ and ρ respectively, are externally fixed: ρ (0) ≡ ρ and ρ (1) ≡ ρ . At the boundaries (in contact with the par-ticle reservoirs) the state probabilities can be factorizedas the total concentration times the stationary weightthe state, complying with P a (0) = ρ f b / ( f b + f a ) and P b (0) = ρ f a / ( f b + f a ) (and analogously at z = 1). Rewrit-ten in terms of π ( z ), this supplies two new conditions: π (0) = rρ and π (1) = rρ , where r ≡ ( f b − f a ) / ( f b + f a ).Third, ρ ( z ) and π ( z ) must be continuous functions con-necting both regions: ρ ( δ ) = ρ ( δ ) and π ( δ ) = π ( δ ).Last, the total flux J associated to ρ ( z ) has implicitly beenassumed continuous across both zones. The same holds forthe one associated to π ( z ) in (12), which in practice yieldsto impose continuity on π ′ i ( z ) − F i [ ρ i ( z ) + π i ( z )]. This lastboundary condition can be recast as a property of the π ( z )derivatives: ( F − F )[ ρ ( δ ) + π ( δ )] = π ′ ( δ ) − π ′ ( δ ) . Solution.
The above cumbersome boundary conditionscan be compactly written as a system of linear equations:see eq . (17) , from which the flux J (and also the density profile ρ ( z ) bymeans of C ij ) can be obtained as a function of the systemparameters. New constants are defined for compact nota-tion: ∆ F ≡ (1 /F − /F ) and h ij ≡ λ ij ( s ij + F /F − J = 0) such linear system has to berearranged. The ratio of concentrations ρ /ρ , which isthen the relevant observable, is easily found by imposinga minor of the coefficient matrix equal to zero (ensuringthat the solution of the problem exists):see eq . (18) . p-3. Gomez-Marin and J. M. Sancho α/F e − λ e − λ e − λ α/F s s s β/F s e λ s e λ s e λ β/F e λ δ e λ δ e λ δ − e λ δ − e λ δ − e λ δ α ∆ F s e λ δ s e λ δ s e λ δ − s e λ δ − s e λ δ − s e λ δ β ∆ F h e λ δ h e λ δ h e λ δ − λ s e λ δ − λ s e λ δ − λ s e λ δ F ∆ F C C C C C C J = ρ ρ rρ rρ (17)det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ /ρ ρ /ρ ρ /ρ e − λ e − λ e − λ s − r s − r s − r s − r ) e λ ( s − r ) e λ ( s − r ) e λ e λ δ e λ δ e λ δ − e λ δ − e λ δ − e λ δ s e λ δ s e λ δ s e λ δ − s e λ δ − s e λ δ − s e λ δ h e λ δ h e λ δ h e λ δ − λ s e λ δ − λ s e λ δ − λ s e λ δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 (18) Parameter exploration: zero flux case.
When thepump is forced to operate at vanishing particle flux, ρ and ρ become boundary conditions, whose ratio is therelevant quantity to explore as a function of the parame-ters of the system. When examining the behavior of ρ /ρ as a function of the asymmetry parameter δ , we check thatfor δ = 0 . ρ = ρ . Moreover, the solu-tion of the problem is anti-symmetric when δ and 1 − δ are interchanged. The pumping ability clearly dependson the relative energetic barrier v . For low values of v ,the ratchet can barely pump the particles because diffu-sive losses dominate, whereas as the barrier height is in-creased, the concentration ratio does so. For very highvalues of v , the fluctuating potential does not improve itseffect anymore and its pumping capacity saturates. Allthese features were verified (data not shown) and used asconsistency tests for the analytical solution.The ratio ρ /ρ as a function of the transition rates ofthe dichotomic noise is explored in figure 4. By meansof varying f we observe that there is an optimum valuewhich maximizes the particle gradient. For a very fastswitching ( f high), the particles do not have time to dif-fuse in the open configuration and cannot be biased inthe closed configuration. The opposite limit is also phys-ically intuitive. In the adiabatic case ( f small), no par-ticle gradient can be created: the flashing time-scale ismuch greater than diffusion, which prevents any possibleseparation of particles induced by the ratchet effect. Fur-ther adjustments on δ , v and σ may allow to obtain evenhigher concentration ratios, optimizing the performance ofthe pump. Figure 4 depicts how tuning both parameters f a and f b (equivalently we vary f and σ ) can increase theratio ρ /ρ . For very small [ATP], f a tends to zero. Theratchet is in the closed state and it takes a very long toopen. This leads to ρ → ρ . For high values of σ , therates are equal, f a = f b = f , and f alone controls thepumping capacity. In the intermediate region, there is amaximum distinctive of an optimal behavior, since σ can f ρ / ρ σ = 0.5 σ = 1.0σ = 0.01 Figure 4: Concentration ratio ρ /ρ versus the characteristicfrequency f for different σ values ( δ = 0 .
33 and v = 10). independently improve the pumping capacity given a fixedvalue of f . Parameter exploration: non-zero flux case.
Infigure 5-top we plot the flux J as the concentration ρ isincreased (given a fixed ρ = 1) for several values of v .A restoring entropic force appears opposing the pumpingand leading to a decrease of the total flux. The decreaseof the flux is linear. When the barrier v is higher, thedecay is softer due to a more powerful functioning of thepump (at the expense of more energy). Even if the pumpis able to carry a non-zero net flux of particles against theconcentration gradient, if ρ is sufficiently increased, theparticle flux can be reversed.The analytical prediction of the flux J as a function ofthe rate f for several values of ρ (where ρ = 1) is shownin figure 5-bottom. When ρ = ρ and f = 0, the deviceis stopped and the net flux vanishes. As the pumping rateis increased, transport starts. The higher ρ , the lowerthe flux J , even becoming negative for great concentrationvalues. In that case, if the flashing rate is increased, themachine can reverse the sign of the flux back to J > ρ -0.0500.05 J v =20 v =30v =10 f -0.2-0.100.10.2 J ρ =5ρ =2ρ =1 Figure 5: Top: Total flux J as a function of concentration ρ when the effective height of the barrier v is varied ( ρ = 1, δ = 0 . f = 100 and σ = 0 . J as a functionof the flashing rate f at different concentrations ρ ( ρ = 1, δ = 0 . v = 20 and σ = 0 . flux steadily tends to zero because, when the flashing isalmost instantaneous, there is no effective pumping. Energetic characterization.
The energetic aspectsare relevant and abundant in the literature of motor mod-eling [15–17] and should definitely be addressed in theirpump counterparts. A suitable definition of efficiency forpumps is not obvious. It is unclear how to interpret andquantify the payoff achieved from the pumping process asa whole. However, from a purely energetic point of view,meaningful quantities can still be defined consistently.First, the input source of energy can be convenientlystudied as follows. Working in the overdamped approxi-mation, every time the potential is lifted (due to the ζ ( t )modulation) energy is injected into the system. This en-ergy is used to actively create the concentration profileand maintain it. When the pump flashes down to theopen configuration, no energy is injected nor recovered.In such situation, the pumping energy input per unit oftime is mathematically quantified as˙ E in = Z L dxV ( x ) w b P b ( x ) . (19)The mean power input that the dichotomic flashing in-serts into the system is a controllable feature of the modelby a judicous choice of its pameter values. An explicitexpression for the above quantity (in dimensionless vari- ables) can be obtained by splitting it into two parts, eachcorresponding to the piecewise linear potential:˙ E in v f = Z δ dz [ ρ ( z ) − π ( z )] / δ/z + Z δ dz [ ρ ( z ) − π ( z )] / − δ ) / (1 − z ) . (20)By using the state probabilities (14) and (16) we are leftwith sums of exponentials, whose integrals are trivial. Infigure 6 we show the behavior of ˙ E in at zero flux as thetypical frequency f is varied (solid line) for several valuesof v . The faster the flashing, the more energy per unit oftime is inserted in the system. Similarly, when the barrier v is increased, the power input rises very quickly.Secondly, the power output produced by the machinecan be quantified as the product ˙ E chem = J ∆ φ , where J isthe particle current and ∆ φ = k B T ln( ρ /ρ ) is the chemi-cal potential difference between both reservoirs. Note thatat zero flux, the power output vanishes, namely, the pumpis using energy but it is not performing work. This isequivalent to the stalling regime in motors, where the en-ergetic efficiency is zero since there is consumption but nowork performance.On the whole, the benefit pursued by a pump is oftennot an energetic one. For instance, maintaining a con-centration gradient at zero flux can indeed be very im-portant for biological functions. The energetic input canstill be used to constrain the biophysical tuning of themodel, which we address as follows. In connection to realbiomachines, one ought to require the input energy percycle not to be greater than the energy released from thehydrolysis of one ATP molecule, which is approximately20 k B T . Given that the typical period of the pump cycleis τ cycle = w − a + w − b , the power released by the ATP is˙ E ATP = E ATP /τ cycle which, in dimensionless units, yields˙ E ATP = f
202 + 1 /σ ≥ ˙ E in . (21)In figure 6 we plot the above expression (dashed line) asa function of the frequency f . Biophysical scale.
So far we have solved the equationsof the model and examined the behavior of the pump asa function of dimensionless parameters. Can all that beplaced in a biophysical scale [18–21]? We use a tuningapproach that combines constraints and optimization cri-teria. Let us consider the length of the molecular pumpof the order of L ≃ nm and an effective opening sectionof S ≃ nm . At biological conditions ( T = 310 K ) thethermal energy is k B T ≃ . · − J . Given an effectiveviscosity inside the membrane two orders of magnitudebigger than that of the water, the resulting diffusion con-stant of the medium yields D = k B T /γ ≃ − m /s .This leads to a typical time-scale associated with diffu-sion of the order τ = L /D ≃ − s . Then, when theoptimal fluctuating rate f found in our explorations ismapped to dimensional units we find w ≃ ∼ Hz ,which is compatible with typical reported rates of func-tioning of the Na,K-ATP-ase pump under the effect ofp-5. Gomez-Marin and J. M. Sancho f E in E ATP v = 30v = 20v = 10 Figure 6: Input power ˙ E in as a function of the flashing rate f for three different values of v ( δ = 0 . σ = 0 . ρ = 1 and J = 0). The dashed line represents the power related to thehydrolysis of one ATP molecule per cycle. The parameter spaceof the model is tuned at the black dot according to biophysicalconstraints. an oscillating electric field [22]. Guided by the energeticcalculations developed previously, we impose that the en-ergy spent per cycle is close but not higher than the oneobtained from the ATP hydrolysis. See the black dot infigure (6). This sets the possible space of parameters. Forinstance, a plausible potential barrier V is expected to beof the order of a few tens of k B T . Higher values can lead togreater concentration ratios but their energetics is not bio-logically meaningful. In this way, by recursively adjustingthe parameter space where the pump functions optimally,bearing in mind biological constrains, the model is shownto be consistently translated into dimensional units of theorder of those of a real molecular pumps. Final remarks and conclusions.
The interplay be-tween experimental observations and theoretical modelssubstantially helped to understand the basic features ofmolecular motors. We believe that applying now the basicmechanisms of the ratchet effect to molecular pumps willcontribute to understand the fundamental working prin-ciples of protein machines embedded in cell membranes.Although the model presented here is a dramatic sim-plification of the great complexity and richness of a bi-ological pump, it contains many interesting and new el-ements that capture its basic biophysical aspects. First,the explicit implementation of concentrations as bound-ary conditions at both ends of the membrane, leading tothe characteristic quantity ρ /ρ . Second, the use of thesimulation framework developed in [10], which is proneto be applied in models whose an analytical solution isnot possible. Third, a simple coarse-grained connectionbetween transition rates and ATP binding and hydrolysisprocesses. Fourth, the energetic characterization of themodel and its connection to biophysical values. Finally,it is worth commenting on the straightforward extensionof this approach for particles of charge q in a membranepotential Φ by including an electrostatic force F q = q Φ /L in the equations of motion. The pump then needs to sup-ply extra energy, reducing the flux and the concentrationratio accordingly.On the whole, we expect that the present work will beof use as a starting point to make progress, from a phys-ical point of view, towards future investigations of activeprocesses in membranes. Acknowledgments.
We acknowledge financial sup-port from the Ministerio de Educaci´on y Ciencia of Spainunder Project FIS2006-11452-C03-01 and Grant FPU-AP-2004-0770 (A. G–M.).
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