J. Phys. Chem. 1996, 100, 19075-19081
19075
Adiabatic Theory for Fluctuation-Induced Transport on a Periodic Potential R. Dean Astumian Departments of Surgery and of Biochemistry and Molecular Biology, UniVersity of Chicago, MC 6035, 5841 S. Maryland AVenue, Chicago, Illinois 60637 ReceiVed: June 4, 1996; In Final Form: October 3, 1996X
Nonequilibrium fluctuations can provide energy to drive directed motion along a periodic surface. In this paper a model for fluctuation driven transport based on a rate theory approach is presented. Simple equations for the flow and efficiency of transport as functions of the fluctuation frequency and of the magnitude of an externally applied force are obtained.
Introduction
SCHEME 1
Much theoretical work has recently been devoted to fluctuation-induced motion along a periodic “washboard” potential.1-4 This has been strongly motivated by consideration of experiments on single molecular motor complexes such as actin moving on myosin (muscle)5 and kinesin moving along microtubules.6 The theoretical treatments1-4 were done on the basis of a continuum approach and involved solving the diffusion equation for the probability density of a Brownian particle along a periodic potential. Some earlier work on fluctuation-induced flow focused on the effects of thermodynamic perturbations of parameters, such as the membrane potential, on enzyme catalysis, and on transport of solutes across membranes.7-10 This was motivated by the experimental demonstration that an oscillating11 or randomly fluctuating electric field12 can drive transport by a molecular pump. It was shown that fluctuations of the rate constants such that the driving force (chemical affinity) was zero at every instant in time could lead to net flow. These studies were done in the context of thermodynamics and chemical kinetics. Here we consider a simple one-dimensional periodic potential and treat the case that fluctuations are slow compared to intrawell relaxation, which allows us to connect the kinetic treatments and those based on a Fokker-Planck-Smoluchowski (diffusion) equation. Consider a general periodic potential shown in Scheme 1. Within a period there are several wells and barriers. Potentials such as this can arise in many different contexts. For example, the potential could describe the potential energy of mean force for a charged protein (such as a molecular motor) moving along a periodic backbone of a polymer comprised of several dipolar monomers or equally could be descriptive of an enzyme cycling through multiple intermediate states in carrying out catalysis. In a more physical context the potential could describe the barriers for an electron to move through a transitor with several different junctions. The physics is that of a particle moving in a viscous medium acted on by both the thermal noise and the gradient of the local potential.13 There are two ways of treating the motion of an ensemble of classical particles moving along the potential. The first involves a continuum approach in which the equation of continuity is14
∂tP(x,t) ) -∂xJ
(1)
where ∂t and ∂x denote partial differentiation with respect to time and position, respectively, and P(x,t) is the probability X
Abstract published in AdVance ACS Abstracts, November 15, 1996.
S0022-3654(96)01614-0 CCC: $12.00
density at position x at time t. J is the flow of probability. In an overdamped medium the constituitive relation is
J ) -D[P(x,t)(∂xU(x) - F) + ∂xP(x,t)]
(2)
where here D is the diffusion coefficient, and we have taken kBT as our unit of energy. The function U(x) is the strictly periodic part of the potential, and F is a macroscopic force acting to drive flow to the left or right. Equations 1 and 2 are typically solved for the steady state ∂tP ) 0 subject to periodic boundary conditions P(x) ) P(x+L), where L is the period. In an isothermal system, if all of the quantities in eq 2 are timeindependent, the direction of flow is given by the sign of F, and if F is zero, P is given by a Boltzmann distribution P(x) ∝ exp[-U(x)] and the flow is zero. An alternate picture analogous to lattice dynamics (and to chemical kinetics) is obtained by considering “hopping” between the wells, with transition probabilities governing the change in occupancy of the wells. This model leads to a set of linear first-order ordinary differential equations for the well occupancies ni
dn1/dt ) Rmnm + β2n2 - (R1 + β1)n1 dni/dt ) Ri-1ni-1 + βi+1ni+1 - (Ri + βi)ni
i ) 2, ..., m - 1
dnm/dt ) Rm-1nm-1 + β1n1 - (Rm + βm)nm
(3)
where the R’s are the transition probabilities for a hop to the right and the β’s are the transition probabilities for a hop to the left. The flux across the barrier separating the ith and (i + 1)th well is
Ji,i+1 ) Rini - βi+1ni+1
(4)
At steady state, the time derivatives in eq 3 are all zero, and all J’s are the same so calculation of the stationary well occupancies, and hence the net flux, is simply a matter of linear algebra. Connection between the kinetic and continuum picture is established by Kramers’ theory for the transition probabilities15
Ri ) Aie∆U+i; βi ) Aie∆U-i © 1996 American Chemical Society
(5)
19076 J. Phys. Chem., Vol. 100, No. 49, 1996
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where ∆U+i and ∆U-i are the differences between the potential energy at the bottom of the ith well and at the top of the barrier to the right and to the left, respectively, and Ai is a frequency factor that reflects the intrawell relaxation time. This can be approximated as Ai ) D/Li2, where D is the diffusion coefficient and Li is the distance between the peaks surrounding the ith well.16 The macroscopic force F acting on the motion is thus
F ) L-1 ln
( ) n
Ri
∏ i)1 β
(6)
i
A major assumption necessary for the validity of a kinetic or lattice dynamic theory is that within a well local equilbrium holds; i.e., the probability obeys a Boltzmann distribution law locally. Thus, in order for a kinetic description to be valid, externally enforced changes on the system must occur slowly compared with any intrawell relaxation frequency Ai, but not necessarily slower than Ai exp(∆Ui) for any of the barriers. In this paper, we examine the effect of a two-state fluctuation in the energy profile for a very simple case with only two wells in a period to illustrate several basic principles relating to fluctuation driven flow along a periodic potential, as well as to make connection between recent work on “ratchet” potentials driven by nonequilibrium fluctuations1-4 which have been discussed as models for molecular motors and work on transport and catalysis driven by nonequilibrium fluctuations acting on membrane transporters (molecular pumps).7-10 First, let us consider a physically explicit four-state model for a membrane pump in order to develop a qualitative feel for how oscillations or fluctuations can cause transport even without a macroscopic force. Molecular Pumps Molecular pumps convert energy from ATP hydrolysis to do concentration work against an (electro)chemical gradient of ions17 or other small molecules such as sugars or amino acids. An external oscillating electric field can drive the ion pump Na+K+ ATPase even under conditions where ATP is not hydrolyzed.11 The rate of transport depends nonmonotonically on the frequency and amplitude of the field. More recently, it has been shown that a random12 external field can drive transport, with the observed behavior similar to that seen for the case of ac fields. In that experiment the lifetime of the field in the positive or negative state γ-1 was Poisson distributed (random telegraph noise), and the switching of the field was given by a rate process γ
Va-V γ
where V is the applied voltage. These results suggest that at least part of the energy for carrying out the uphill transport comes from the applied field. Since the time average of the field is zero, some sort of rectification mechanism must be at work, but the nonmonotonic dependence of the transport rate on the frequency of the field suggests a more subtle mechanism than simple rectification of a fluctuating net force. The behavior of the system has been modeled in terms of a four-state mechanism for transport of a substance across a membrane7,18 shown in Figure 1. While many pumps transport ions, here, for simplicity, we take the transported substance S to be electrically neutral, and so the overall chemical affinity of the transport reaction is independent of the electric potential difference across the membrane. The cartoon emphasizes two things. First, the macromolecule in the membrane (typically a
Figure 1. (a) Schematic representation of a four-state model for catalytic transport of an uncharged substance S across a membrane. Even conformational transitions E1 T E* 2 and E4 T E* 3 involve a change in the dipole moment of the molecule (shown as movement of the charged arm z) as well as a change in the exposure of the binding site from the outside to inside facing conformation. The “E” form has a high affinity for substrate, and the “E*” form has a low affinity. (b) Illustration of an analogy to a pump and dump scheme for optical excitation of a chemical reaction. The dashed line represents the onedimensional energy profile for the reaction (at equilibrium) in the abscence of an electric field. The green curve shows the effect of a positive outside potential (with z ) -1), and the red curve shows the effect of a negative outside potential. The designation loading and unloading should be reversed in (b).
protein) that catalyzes the reaction is conformationally flexible and can exist in several different states with different chemical properties. In the figure, the cylinder (the “E” state) is able to bind substrate from the outside, and the rectangle (i.e., the “E*” state) can bind substrate from the inside. The affinity for S may be very different in these two states. Second, by drawing the binding site as a mobile charge z, the fact that the transition between these different states may involve movement of charge or change of dipole moment is emphasized. Thus, even though the overall affinity is independent of the membrane potential difference, the equilibria for the individual steps E1 T E* 2 and E4 T E*3 do depend on the membrane potential difference. At equilibrium, the concentration of substrate is the same on the two sides of the membrane, and for simplicity we work in units where this concentration is 1. Let the affinity for binding S on the left to the “E” state be much larger than 1 and the affinity for binding S on the right to the “E*” state be much less than 1. This situation results in states E1 and E* 3 being much more populated than states E* 2 and E4 at equilibrium, as represented by the size of the colored circles in Figure 2b for t < t0. Because S is uncharged, a dc electric field applied across the membrane will not cause net transport, but the relative concentrations of the transporter states readjust since the different states have different electrical properties. What if we apply an oscillating electric field? For simplicity, we take a square wave field of large amplitude (but with a time average value of zero) shown in Figure 2a, with z ) -1.
Fluctuation-Induced Transport on a Periodic Potential
J. Phys. Chem., Vol. 100, No. 49, 1996 19077
Figure 2. (a) Time sequence of a stimulating periodic membrane potential. The time ∆t () t3 - t2, etc.) and the time 1/f ) t5 - t1. (b) Schematic illustration of the relative state probabilities (E1 is blue, E* 2 is yellow, E* 3 is green, and E4 is red) at different times during a cycle of the applied field. For t > t1 the enzyme cycle is entrained by the applied field and follows the sequence E* 3 f E4 f E1 f E* 2 f E* 3 etc., transporting substrate from outside to inside, even against a concentration gradient. (c) Time plots of the enzyme state probabilities calculated by solving the differential equations describing the kinetics of the system. (d) Plot of the cycling rate per enzyme (in s-1) vs log(f) where f is in s-1 for several values of b, the ratio of the affinity on the outside to the affinity on the inside. The rate constants used in the calculations were R1 ) 10/φ, φ, R2 ) b, R3 ) 10 φ, R4 ) b, β1 ) 1, β2 ) 10 b, β3 ) 1, and β4 ) 10b/φ, where φ ) exp(∆ψ/kBT), and all values are in s-1. Note that the chemical force, ln(∏Ri/βi) ) 0 at every instant in time.
There are two relevant time scales: ∆t, the time scale for switching the potential difference between + and -; and 1/f, the inverse of the frequency of oscillation defined by ∆ψ(t) ) ∆ψ(t+1/f). Consider a limiting case that the time scale for the conformational change between the “E” states and the “E*” states is short compared to ∆t, and let the time scale for binding and debinding of S on either side be short compared to 1/f, but long compared to ∆t. When the potential is positive on the right, the molecules in state E1 convert to state E* 2, and the molecules in state E4 convert to state E* 3 (t0 < t < t1). Because the conformational change is fast, this conversion is almost reversible. If the field remains postive on the right for a long time, chemical equilibration between states E* 2 and E* 3 occurs (t1 < t < t2). Since the affinity of the E* state for S is small, this means that S is released on the right-hand side, and almost all of the transporter molecules end up in state E* 3. Now, when the field is switched to positive on the left-hand side, the
transporters in state E* 3 are converted to state E4, and the transporters in state E* 2 are converted to state E1 (t2 < t < t3). Shortly after switching the field, state E4 predominates over state E1, but the affinity of S for the E form is very large so chemical equilibration between states E4 and E1 takes place, resulting in the binding of S on the left-hand side (t3 < t < t4). In Figure 2c we have plotted the enzyme state probabilities as a function of time. As the field oscillates back and forth, the process continues, cyclying in the order 1 f 2 f 3 f 4 f 1 with binding of S on the left and release of S on the right, resulting in the eventual formation of a concentration gradient of S. The maximum concentration gradient that can be supported is given by the ratio of the affinities on the two sides, and the thermodynamic efficiency of this process can approach 100%.19 In Figure 2d, the rate as a function of frequency f is shown for several values of the ratio of the binding affinities on the two sides of the membrane.18
19078 J. Phys. Chem., Vol. 100, No. 49, 1996 In Figure 1b, an analogy between the ac-field-induced transport mechanism and what has been termed a “pump and dump” mechanism in the literature of optical control of chemical reactions20 is illustrated. In that context, a light pulse (pump) is used to populate an electronically excited state starting from a ground state in equilibrium. The excited molecules relax on the potential surface that characterizes the new electronic state and are eventually deexcited by a “dump” pulse. The point on the potential surface where this occurs need not be the same as the point from which they were originally excited, thus resulting in a disequilibration of the ground state. In this way the formation of a ground state product can be significantly enhanced beyond the fraction that would be formed by a purely thermal adiabatic process, and indeed, a reaction can even be driven in a direction opposite to that predicted by the overall ∆G of the process. The energy for this comes from the light. For the membrane transport reaction we use an electric field rather than light. The effect can be described entirely from the point of view of thermodynamics. Nevertheless, this ac chemical approach can in general allow control of chemical reactions along pathways that would not be predominant in any constant thermodynamic environment. We have focused on the case that the field oscillates periodically in time. Nothing essential is changed, however, if the field switches between plus and minus randomly,8-10 consistent with the experiments using external random field fluctuations.12 Further, although we have considered several approximations on the time scales involved in order to arrive at a simple description of how net flow and the formation of a concentration gradient of a neutral substance is caused by a zero average oscillating field, the ability to transduce energy from a fluctuating field is much more general, and the theory has been used to fit experimental data obtained for the effect of an oscillating field on the NaK ATPase in red blood cells and other systems.21 The picture given above illustrates qualitatively how an external zero average perturbation can lead to unidirectional flow. Now, let us consider quantitatively a similar two-state model specialized for description of transport of a Brownian particle along a one-dimensional track. Two-State Mechanism for a Fluctuation Driven Molecular Motor Imagine a particle moving along a filament where the two are attracted by van der Waals and hydrophobic interactions. The particle is a protein, and the filament is made up of many identical protein molecules linked together. Although motion away from the filament is constrained, the particle can still move laterally along the filament by minute conformational changes arising from Brownian motion and accompanied by the breaking and making of many weak contacts. Typically, there will be a few sites of contact (binding sites) within a periodic unit of the filament where the particle will be localized with high probability. Occasionally, a transition over an energy barrier to a binding site on the monomer to the left or to the right will occur. In the absence of a metabolic energy source, the number of transitions to the left is equal to the number to the right. The stochastic exploration of a filament by a motor molecule can be described theoretically. We imagine a curve (potential energy profile) that describes the potential energy as a function of the position of the center of mass of the motor along the filament.
Astumian
Figure 3. Illustration of two potential surfaces for motion of a particle. Transitions between the surfaces are mediated, for example, by binding of ATP and release of ADP. Thus, the upper surface could represent the potential experienced by a molecular motor moving along a biopolymer track when ATP is bound to the molecule, and the lower surface the potential profile for the same process but without ATP bound. It is the nonequilibrium fluctuation between the two that leads to uphill flow.
Figure 4. (a) Plot of the flux J vs frequency γ with F ) 0, U0 ) 10, and δU ) 9. (b) Plot of the stoichiometry 2J/γ (number of periodic displacements per cycle + f - f + of the fluctuation) vs frequency γ with F ) 0, U0 ) 10, and δU ) 9.
Such a curve represents a projection of the complex conformational and physical trajectory of the motor molecule onto a single “reaction” coordinate. The potential profile is spatially periodic, but in general anisotropic; i.e., there is no plane of mirror symmetry. As the particle moves, it can bind to and catalyze the conversion of chemicals such as ATP. The potential profile will in general depend on what chemical is bound to the particle, but the periodicity of the profile must of course be independent of the chemical state. The stochastic binding of ATP, conversion to ADP and Pi, and release of these products thus causes the particle to fluctuate between several different chemical states, each with a different periodic potential profile. Consider the case shown in Figure 3, where the energy profile undergoes two-state fluctuation between U+(x) and U-(x). The quantity γi+ is the rate constant for making a transition from the “+” potential to the “-” potential and γi- is the rate constant for transition from the “-” potential to the “+” potential at the ith well. The transitions between the two potentials could be caused by an external electric generator, by a far from equilibrium chemical reaction such as ATP hydrolysis, or by the absorption and emission of a photon of light. The matrix equation for the occupancy of the wells (states) is
[] [
Fluctuation-Induced Transport on a Periodic Potential
n+ 1 n+ d 2- /dt ) n1 n2
+ + -(R+ 1 + β1 + γ1 ) + (R+ 2 + β2 ) + γ1 0
][ ]
J. Phys. Chem., Vol. 100, No. 49, 1996 19079
+ (R+ 1 + β1 ) + + -(R+ 2 + β2 + γ2 ) 0 γ+ 2
γ1 0 -(R1 + β1 + γ1 ) (R2 + β2 )
0 γ2 (R1 + β1 ) -(R2 + β2 + γ2 )
n+ 1 n+ 2 n1 n2
(7)
For the specific case in Figure 3 we have R+ 1 ) R2 ) 4 exp(-U0 + δU + F/4) + R1 ) R2 ) 4 exp(-U0 - δU + F/4) + β+ 1 ) β2 ) β1 ) β2 ) 4 exp(-U0 - F/4)
(8)
where we work in energy units such that kBT ) 1, length units such that L ) 1, and time units such that the diffusion coefficient D ) 1. In going from the + to the - potential, the energy of particles in well 2 is increased by 2δU while the energy of those in well 1 is unchanged. Hence, γ+ 2 /γ2 ) exp(-2δU) and + γ1 /γ1 ) 1 fulfills the requirements for equilibrium fluctuation.8,9,22 When this condition is inserted into eq 7 the probabilities n( i obey a Boltzmann distribution at steady state, and the flux is identically zero when F ) 0 and otherwise has the same sign as F, irrespective of the time scale of the fluctuation. If, however, we enforce nonequilibrium fluctuations (in general, fluctuations for which γ+(x)/γ-(x) * exp[U+(x) U-(x)]) by, for example, the action of a nonequilibrium chemical reaction or by the absorption of light, the distribution will not obey a Boltzmann equation, and net uphill flux (left to right in Figure 3) may result. Here we treat a simple case γ+ i ) γi ) γ, for i ) 1, 2. As shown elsewhere, such a situation can occur if the fluctuations are driven by a far from equilibrium chemical reaction such as ATP hydrolysis.22 At steady state, the time derivatives in eq 7 are zero, and the occupancies of the wells in the plus and minus states can be calculated as a function of γ n+ 1 ) n2 )
γ + (e-FL/4 + eFL/4 - δU)e-U0 4Σ
+ n1 ) n2 )
γ + (e-FL/4 + eFL/4 + δU)e-U0 4Σ
(9)
where
Σ ) γ + [e-F/4 + cosh(δU)eF/4]e-U0
(10)
+ - After a bit of algebra, we find the flux J ) R+ 1 n1 + R1 n1 + + - (β2 n2 + β1 2n2 ) to be
e-U0 4Σ (11)
J ) {2γ[eF/4 cosh(δU) - e-F/4] + 4e-U0 sinh(F/2)}
The effective “stoichiometry” is the number of periods of length L traveled per cycle + f - f + of the fluctuation and is 2J/ γ. The power into the system from the fluctuations is
Pin ) 2γδUn+ 2
(12)
This assumes that there is no mechanism for conservation of energy if a particle is promoted from the + to the - surface and then back to the + surface without a transition either to the left or right. Thus, the amount of energy loss for such a cycle is 2δU. The output power includes work done against
Figure 5. Plot of J vs an externally applied force F at constant γ ) 0.1, U0 ) 10, and δU ) 9.
the external force F as well as work done on the viscous medium
Pout ) JF/2 + J 2
(13)
In Figure 4 the flow J and stoichiometry 2J/γ are plotted vs log(γ) with F ) 0, and in Figure 5, the flow is plotted vs force F at constant γ ) 0.1, where in both figures U0 ) 10 and δU ) 9. With other choices, the flow-force curves can have either upward or downward curvature. At low frequency (γ < 1), the calculations based on the kinetic approach discussed here agree well with numerical calculations using the Fokker-Planck-Smoluchowski diffusion approach (to be published elsewhere). At higher frequencies (γ g 1) the local equilibrium approximation required for the kinetic approach breaks down, and the Fokker-Planck approach must be used. Three ingredients are required for fluctuation induced flow: (1) the anisotropy of the potential energy function, which ultimately reflects the anisotropy of the physical structure of the molecules and their interactions with one another; (2) thermal noise, which is necessary to allow transitions from one local potential energy minimum to another; (3) energy input (e.g., from a nonequilibrium chemical reaction) to power the nonequilibrium fluctuations and break detailed balance. In the model in Figure 3, the input energy is used to drive the unfavorable transition from state “2+” to “2- ”, followed by spontaneous relaxation to state “1-”, predominately to the right over the small kinetic barrier rather than to the left over the higher barrier. Subsequent transition to the + surface and relaxation to position “2+”, also predominately to the right, can occur spontaneously. Rescaling and Comparison with Experiment In order to convert back to laboratory units, the flux J must be multiplied by D/L, the transition frequency γ by D/L2, force F by kBT/L, and energies U0 and δU by kBT (≈4 × 10-21 J at 300 K). For the motion of a molecular motor (such as kinesin or myosin) along a biopolymer backbone (such as microtubule or actin), the spatial period L is of order 10-8 m.7,8 With a diffusion coefficient of 10-13 m2 s-1, we find the factors D/L2 ≈ 103 s-1, D/L ≈ 10-5 m/s, and kBT/L ≈ 4 × 10-13 N. We took an activation barrier of U0 ) 10kBT, and since in many cases the energy source for biological motors and energy transducing enzymes is the hydrolysis of ATP, which releases about 20kBT of energy per molecule hydrolyzed at physiological conditions, we take δU ) 9kBT as a reasonable energy change
19080 J. Phys. Chem., Vol. 100, No. 49, 1996
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to result from binding of ATP or release of ADP (i.e., 18kBT per cycle). From Figure 5 we see that at a frequency of 100 s-1 (log(γ) ) -1) the velocity at zero force is 4 × 10-7 m s-1 with a stoichiometry of about 0.6. The velocity decreases nearly linearly with increasing force to zero at a force of ≈4 × 10-12 N, in good agreement with experimental values for kinesin and myosin.5,6 The stoichiometry also decreases with increasing load. The efficiency is relatively small (
