Channel proteins, that selectively conduct molecules across cell membranes,
often exhibit an asymmetric structure. By means of a stochastic model, we argue
that channel asymmetry in the presence of non-equilibrium fluctuations, fueled
by the cell's metabolism as observed recently, can dramatically influence the
transport through such channels by a ratchet-like mechanism. For an
aquaglyceroporin that conducts water and glycerol we show that a previously
determined asymmetric glycerol potential leads to enhanced inward transport of
glycerol, but for unfavorably high glycerol concentrations also to enhanced
outward transport that protects a cell against poisoning.Comment: REVTeX4, 4 pages, 3 figures; Accepted for publication in Phys. Rev.
  Let

B. Alberts

D. Lu

Ioan Kosztin

Klaus Schulten

P. Grayson

R. T. Voegele

S. Levin

English

arXiv

URL:http://link.aps.org/doi/10.1103/PhysRevLett.93.238102
DOI:10.1103/PhysRevLett.93.238102Channel proteins that selectively conduct molecules across cell membranes often exhibit an asymmetric structure. By means of a stochastic model, we argue that channel asymmetry in the presence of nonequilibrium fluctuations, fueled by the cell's metabolism as observed recently, can dramatically influence the transport through such channels by a ratchetlike mechanism. For an aquaglyceroporin that conducts water and glycerol, we show that a previously determined asymmetric glycerol potential leads to enhanced inward transport of glycerol, but for unfavorably high glycerol concentrations also to enhanced outward transport that protects a cell against poisoning.We acknowledge financial support from the
University of Missouri Research Board, the Institute for Theoretical Sciences, and the Department of Energy (DOE Contract No. W-7405-ENG-36) for I. K., and the National Institutes of Health (NIH 1-R01-GM067887-01) for K. S

Kosztin, Ioan

Schulterr, K. (Klaus)

University of Missouri: MOspace

PRL 93, 238102 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending3 DECEMBER 2004Fluctuation-Driven Molecular Transport Through an Asymmetric Membrane ChannelIoan Kosztin1 and Klaus Schulten21Department of Physics & Astronomy, University of Missouri, Columbia, MO 65211, USA2Beckman Institute and Department of Physics, University of Illinois, Urbana, IL 61801, USA(Received 20 August 2004; published 29 November 2004)0031-9007=Channel proteins that selectively conduct molecules across cell membranes often exhibit anasymmetric structure. By means of a stochastic model, we argue that channel asymmetry in thepresence of nonequilibrium fluctuations, fueled by the cell’s metabolism as observed recently, candramatically influence the transport through such channels by a ratchetlike mechanism. For anaquaglyceroporin that conducts water and glycerol, we show that a previously determined asymmetricglycerol potential leads to enhanced inward transport of glycerol, but for unfavorably high glycerolconcentrations also to enhanced outward transport that protects a cell against poisoning.DOI: 10.1103/PhysRevLett.93.238102 PACS numbers: 87.16.Uv, 05.10.Gg, 05.40.–acytoplasmperiplasmx/L0       0.2      0.4      0.6      0.8        1U(x) [k  T]B 8 6 4 2 0-2-4-6FIG. 1 (color online). Section through the glycerol conductionpathway in GlpF and the corresponding asymmetric PMF(solid curve) reported in [8].Living cells interact with their extracellular environ-ment through the cell membrane, which acts as a protec-tive permeability barrier for preserving the internalintegrity of the cell. However, cell metabolism requirescontrolled molecular transport across the cell membrane,a function that is fulfilled by a wide variety of trans-membrane proteins, acting as passive and active trans-porters [1]. Channel proteins as passive transportersfacilitate the diffusion of specific molecules across themembrane down a free energy gradient. Active transport-ers conduct molecules along or against the free energygradient consuming for that purpose external energy.However, the plasma membrane of living cells is subjectto a variety of nonequilibrium, i.e., nonthermal processes,e.g., interaction with the cytoskeleton and with activemembrane proteins [2,3]. In this Letter we want to arguethat in an active plasma membrane, even passive channelproteins can act as active transporters by consumingenergy from nonequilibrium fluctuations fueled by cellmetabolism [4].The Escherichia coli glycerol uptake facilitator (GlpF)is an aquaglycerol channel protein, which transports bothwater and glycerol molecules, but excludes charged sol-utes, e.g., protons [5]. The recently determined 3D struc-ture of GlpF at atomic resolution [6] (Fig. 1) has providedmuch insight into the underlying microscopic mechanismof molecular transport and selectivity through GlpF [7].In particular, molecular dynamics (MD) studies [8–11]established that water and glycerol diffusion throughGlpF is single file, and for biologically relevant periplas-mic glycerol concentrations, correlation effects betweenconsecutive glycerol molecules are negligible due to theirlarge spatial separation. The corresponding potential ofmean force (PMF) [8] that guides the transport of glyc-erol through the channel is highly asymmetric reflectingthe atomic structure of GlpF (see Fig. 1), with a prominentpotential well at the external (periplasmic) side and aconstriction region with several pronounced potentialbarriers towards the internal (cytoplasmic) side of thechannel [6,12]. Besides GlpF, there are several other04=93(23)=238102(4)$22.50 238102channels that exhibit similar spatial asymmetry [13–15], and in spite of recent efforts, no biological functioncould be attributed to the asymmetry [16]. Here we pro-pose and demonstrate that under realistic physiologicalconditions, the asymmetry of GlpF furnishes active glyc-erol transport through a ratchetlike mechanism. In gen-eral, the ratchet effect refers to the generation of directedmotion of Brownian particles in a spatially periodic andasymmetric (ratchet) potential in the presence of nonequi-librium fluctuations and/or externally applied time-periodic force with zero mean [17–19]. Since cell mem-branes are subject to nonequilibrium fluctuations, whichspan a wide range of time and length scales [2,4], oneexpects a ratchet effect contribution to the transportthrough asymmetric channel proteins, such as GlpF,even if the PMF is nonperiodic. Assuming simplified,heuristic models, the ratchet effect has been invokedbefore to explain the functioning of active biomolecules,e.g., motor proteins and ATP hydrolysis driven ion pumps[17,20]. To our knowledge, this Letter uses for the firsttime a realistic, microscopically determined PMF to in--1  2004 The American Physical SocietyPRL 93, 238102 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending3 DECEMBER 2004vestigate the precise role of nonequilibrium fluctuationsin facilitated transport. We find that as a result of channelasymmetry, glycerol uptake driven by a concentrationgradient is enhanced significantly in the presence ofnonequilibrium fluctuations. Furthermore, the ratcheteffect-caused enhancement is larger for the outward,i.e., from the cytoplasm to the periplasm, flux than forthe inward one, suggesting that nonequilibrium fluctua-tions also play an important role in protecting the interiorof the cell against excess uptake of glycerol.Glycerol transport through GlpF can be modeled interms of overdamped Brownian motion along the axis ofthe channel as a result of the concentration gradientestablished at the ends of the channel. The interaction ofa diffusing glycerol molecule with the protein, solvent,lipid, and other glycerol molecules is taken into accountthrough the PMF, Ux, as determined from steered mo-lecular dynamics simulations by employing the Jarzynskiequality [8]. The motion of a glycerol molecule insideGlpF and in the presence of an external force Ft in thestrong friction limit is described by the Langevin equa-tion _x  fx  t  Ft; (1)where  is the friction coefficient, fx  U0x is thedeterministic force derived from the PMF, and t is theLangevin force due to the equilibrium thermal fluctua-tions. As usual, t is a Gaussian white noise withhti  0 and ht0i  2D2t, where t is theDirac-delta function, and D is the effective diffusioncoefficient of a glycerol molecule inside GlpF.According to the fluctuation-dissipation theorem, D and are related through the Einstein relation D  kBT=.We assume that Ft is time-dependent, but homogeneous.It describes either an externally applied force, or someintrinsic nonequilibrium fluctuations of the system (seebelow). Because of the single file nature of the glyceroltransport through GlpF, a force applied at either end ofthe channel will be transmitted along the file withoutsignificant loss in intensity (incompressibility of thesingle file), which justifies our assumption that Ft ishomogeneous along the channel. For a periodic fx,Eq. (1) describes what is often referred to as a fluctuatingforce (tilting) ratchet [17].At this point, we introduce dimensionless units thatwill be employed throughout this Letter, unless otherwisestated. All other units can be expressed in terms of thefollowing three: length of GlpF L  L 	 4:8 nm, diffu-sion time T  D  L2=D 	 107 s, and thermal en-ergy E  kBT 	 4:28 1021 J; here kB is theBoltzmann constant, T  310 K is the physiological tem-perature, and D 	 2:2 1010 m2=s is the effective dif-fusion coefficient of glycerol inside GlpF [16]. Thus, theforce unit is F  D=L  kBT=L 	 0:9 pN. In the newunits, the Fokker-Planck equation (FPE) correspondingto Eq. (1) reads238102@tpx; t  @xJx; t; (2a)Jx; t  @xpx; t  fx  Ftpx; t; (2b)where px; tdx is the (unnormalized) probability of find-ing a glycerol molecule in x; x dx (see Fig. 1), andJx; t is the local, instantaneous flux of glycerol throughthe channel. The probability density px is related to thelocal concentration Cx by px  SxCx, with Sxthe area of the channel cross section. From the crystalstructure, one finds that the opening area at both ends ofGlpF is S0  S0 	 S1 	 100 A2 [6].GlpF can be regarded as a nanopore, which connectstwo reservoirs of glycerol molecules located at x  0(periplasm) and x  1 (cytoplasm), respectively,(Fig. 1). The glycerol uptake is driven by a concentrationgradient: glycerol concentration (and therefore p0 p0) is finite in the periplasm and vanishingly small inthe cytoplasm (p1  p1 	 0). Indeed, once glycerolenters the cytoplasm it gets phosphorylated by glycerolkinase and, as a charged particle, the product glycerolphosphate cannot leave the cell [21]. However, excessiveaccumulation of glycerol phosphate may result in cellpoisoning and death. To avoid this, the enzyme glycerolkinase is genetically turned off, preventing further glyc-erol phosphorylation [21]. Since the glycerol concentra-tion gradient across the cell membrane persists, oneexpects that glycerol uptake should continue in spite ofits potential damaging effect on the cell until this gradientvanishes (i.e., p1  p0). Below we demonstrate that chan-nel asymmetry, combined with nonequilibrium fluctua-tions, can stop glycerol uptake against the persistentconcentration gradient, keeping the cytoplasmic glycerolat a level p1 <p0. To this end, we calculate the steadyglycerol flux through GlpF in four distinct cases corre-sponding to suitable choices of Ft.Transport driven by concentration gradient[Ft  0].— In the steady state, the flux is constantthroughout the channel [c.f. Eq. (2a)] and, in the absenceof the external force Ft, is given byJp0; p1  A0p0  A1p1; (3a)Ai  eUiZ 10eUxdx1; i  0; 1: (3b)Since the PMF Ux vanishes on both sides of the chan-nel, i.e., U0  U0  U1  U1  0 (see Fig. 1), onehas A0  A1, and the flux is proportional to the glycerolconcentration difference, i.e., Jp0; p1  A0p0  p1 A0S0C0  C1. Hence, the flux vanishes in the absence ofa concentration gradient, and the flux is insensitive to theasymmetry of the channel. In order to produce directedtransport, one needs to drive the system out of equilib-rium, e.g., by applying an external force or by subjectingthe system to nonequilibrium fluctuations.Transport driven by potential gradient[Ft  F0  const].— The dependence of the flux onthe asymmetry of the channel is manifest when U0  U1,-2FIG. 2. (a) Ratio of outward and inward glycerol fluxes inGlpF as function of a constant force F0. (b)–(c) Transportinduced by a square-wave force in GlpF. Shown are the inward[p1  0], outward [p0  0], and equal concentration [p1  p0]fluxes vs force amplitude F0. (d) Ratio of the inner to outerglycerol concentrations vs F0 for vanishing flux.PRL 93, 238102 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending3 DECEMBER 2004i.e., in the presence of a potential gradient. The constantexternal force leads to an effective, tilted PMF, Ueffx Ux  F0x, and according to Eqs. (3) the stationary fluxreadsJF0jp0; p1  A0F0p0  A1F0p1 (4a)A0F0 Z 10eUeff xdx1; A1F0  A0F0eF0 : (4b)For a symmetric PMF, i.e., U1 x  Ux, followsA1F0  A0F0, implying that the (magnitude ofthe) flux is also symmetric jJF0jp0; p1j jJF0jp1; p0j. In general, for a generic asymmetricPMF, the inward and outward fluxes will be different.Under normal conditions when p1  0, the inward flux isJ>  A0F0p0. The same flux through an inverted chan-nel, i.e., p0; p1 ! 0; p0 and F0 ! F0 , would be0. 2. 4. 6. 8.F0-2.-1.0.1.2.3.4.5.LogT 09.57.5.3.2.1.51.11.0.99 0.950. 2. 4.F04.1.21.01(a) (b)FIG. 3 (color online). Contour density plots of the numericallyJF0; T0j0; p0=J0, and (c) equal concentration JF0; T0jp0; p0=mean switching time T0 of the RTF; J0  J0; T0jp0; 0 is the inw238102J<  A1F0p0. It can be readily checked that J_=J0 >1, i.e., the constant driving force enhances the flux whenapplied along the concentration gradient. Furthermore,according to Fig. 2(a), under identical conditions, the fluxthrough the inverted channel is always larger than theflux through the normally oriented GlpF, the ratio of thetwo increasing monotonically with F0.Transport driven by an external periodic drivingforce.— Next, we consider an external force Ftthat switches periodically between F0, F0  const(square-wave force). Although the time average of Ftis zero, this force induces a finite flux through GlpF evenin the absence of a concentration gradient. Indeed, assum-ing that one can neglect the transient in the instantaneousflux after switching Ft, the mean flux J throughthe channel can be expressed as JF0jp0; p1 JF0jp0; p1  JF0jp0; p1=2, where JF0jp0; p1 isgiven by Eq. (4a). Then, for p1  p0, J=J0 JF0jp0; p0= J0jp0; 0 is negative, and decreases mono-tonically with F0, as shown in Fig. 2(c). In this case, too,J < 0 implies that for GlpF the outward flux J< JF0j0; p0 is bigger than the inward flux J> JF0jp0; 0 as shown in Fig. 2(b), and both fluxes J_are bigger than J0, the flux in the absence of the externalforce. Furthermore, the concentration ratio p1=p0 atwhich the flux through the channel vanishes (currentreversal), i.e., JF0jp0; p1  0, is plotted as a functionof F0 in Fig. 2(d). The values p1=p0 < 1 found are ex-pected and consistent with the fact that for the same forcelevel and concentration gradient, the outward flux islarger than the inward flux.Transport driven by nonequilibrium fluctuations.—Finally, we consider the effect of nonequilibrium fluctu-ations of the cell membrane on the glycerol transportthrough GlpF. We model such fluctuations by a randomtelegraph force (RTF), i.e., a homogeneous dichotomousforce Ft, which switches between two states F0 with6. 8.20.16.12.8.6.2.0. 2. 4. 6. 8.F06.4.2.1.0.50.20.10.010.2.4.6.8.10.12.14.16.18.20.(c)evaluated relative (a) inward JF0; T0jp0; 0=J0, (b) outwardJ0 glycerol fluxes in GlpF as a function of the amplitude F0 andard flux in the absence of the RTF.-30. 2. 4. 6. 8.F0-2.-1.0.1.2.3.4.5.LogT 00.980.90.80.70.60.50.40.30.20.30.40.50.60.70.80.91.0FIG. 4 (color online). Contour density plot of the concentra-tion ratio p1=p0  A0=A1 at which current reversal (J  0)takes place in GlpF subjected to RTF.PRL 93, 238102 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending3 DECEMBER 2004switching times that obey a Poisson distribution. For theRTF holds hFti  0, and hFtF0i  F20e2t=T0 , whereT0 is the mean switching time. The stationary Fokker-Planck equation in this case consists of two coupledequations, @2xpx  @xfxpx  F0px px=T0  px=T0  0, where px is the condi-tional probability density that the RTF is in theF0 state.The corresponding flux is J  p0x  fxpx F0px  const, where p  p  p is the totalprobability density, and p p p. For the feature-rich GlpF PMF (Fig. 1), the flux JF0; T0jp0; p1 A0F0; T0p0 A1F0; T0p1 needs to be computed nu-merically, e.g., by employing a matrix continued-fractionmethod. The computed flux is shown in Fig. 3 as two-dimensional density plots for F0 2 0; 8 and T0 2102; 105 (logarithmic scale). In SI units, these valuescorrespond to F0 2 0; 7:2 pN and T0 2 109; 102 s,respectively. The conclusions drawn from Fig. 3 are con-sistent with our previous findings for externally applieddeterministic forces. First, the RTF-induced asymmetrybetween the inward and outward fluxes is manifest[Figs. 3(a) and 3(b)], with bigger outward flux for thesame F0, T0, and concentration gradient. Second, theRTF-induced flux enhancement is more pronounced forslower fluctuations and for largerF0. Third, the flux in theabsence of a concentration gradient across the membraneis always outward. Thus, the ratio of the inner to outerglycerol concentrations at which glycerol uptake ceases isalways less than 1, as shown in Fig. 4.Our calculations have demonstrated that nonequilib-rium force fluctuations acting on glycerol in GlpF canhave an important effect on the glycerol uptake by a cell.On the one hand, slow, large amplitude fluctuations en-hance the concentration gradient-driven glycerol uptake,which may be vital for the cell under poor nutrientconditions. On the other hand, when glycerol is abundant,238102fluctuations provide an effective protection mechanism tothe cell by stopping glycerol uptake well before the cyto-plasmic concentration reaches the periplasmic one. Themechanism underlying this behavior is related to theratchet effect and depends sensitively on the asymmetricshape of the PMF characterizing glycerol conduction inGlpF. The effects described should be testable experimen-tally. In fact, recent experiments have demonstrated thatcell membranes are subject to slow, kHz frequency( logT0  105) nonequilibrium fluctuations that may berelated to pumping mechanisms by which cells supple-ment the passive diffusion of nutrients [4].The authors thank Emad Tajkhorshid for valuable in-sights. We acknowledge financial support from theUniversity of Missouri Research Board, the Institute forTheoretical Sciences, and the Department of Energy(DOE Contract No. W-7405-ENG-36) for I. K., and theNational Institutes of Health (NIH 1-R01-GM067887-01)for K. S.-4[1] B. Alberts et al., Molecular Biology of the Cell (GarlandScience, New York, 2002), 4th ed.[2] S. Levin and R. Korenstein, Biophys. J. 60, 733 (1991); L.Mittelman, S. Levin, and R. Korenstein, FEBS Lett. 293,207 (1991).[3] S. Ramaswamy, J. Toner, and J. Prost, Phys. Rev. Lett. 84,3494 (2000).[4] A. E. Pelling et al., Science 305, 1147 (2004).[5] M. Borgnia, S. Nielsen, A. Engel, and P. Agre, Annu.Rev. Biochem. 68, 425 (1999).[6] D. Fu et al., Science 290, 481 (2000); P. Nollert et al.,FEBS Lett. 504, 112 (2001).[7] E. Tajkhorshid et al., Science 296, 525 (2002); B. Rouxand K. Schulten, Structure 12, 1 (2004), and referencestherein.[8] M. Ø. Jensen, S. Park, E. Tajkhorshid, and K. Schulten,Proc. Natl. Acad. Sci. U.S.A. 99, 6731 (2002).[9] B. L. de Groot and H. Grubmu¨ller, Science 294, 2353(2001).[10] M. Ø. Jensen, E. Tajkhorshid, and K. Schulten, Structure9, 1083 (2001).[11] P. Grayson, E. Tajkhorshid, and K. Schulten, Biophys. J.85, 36 (2003).[12] M. S. P. Sansom and R. J. Law, Curr. Biol. 11, R71 (2001).[13] S.W. Cowan et al., Nature (London) 358, 727 (1992).[14] Y. Wang et al., J. Mol. Biol. 272, 56 (1997).[15] D. Forst, W. Welte, T. Wacker, and K. Diederichs, Nat.Struct. Biol. 5, 37 (1998).[16] D. Lu, P. Grayson, and K. Schulten, Biophys. J. 85, 2977(2003).[17] P. Reimann, Phys. Rep. 361, 57 (2002).[18] R. D. Astumian, Science 276, 917 (1997).[19] M. O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993).[20] T.Y. Tsong and T. D. Xie, Appl. Phys. A 75, 345 (2002).[21] R. T. Voegele, G. D. Sweet, and W. Boos, J. Bacteriol. 175,1087 (1993).

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