Received: April, 1993. The correlation ratchet: a novel mechanism for generating directed motion by ATP hydrolysis. C. S. Peskin, G. B. Ermentrout, G. F. Oster.
Received: April, 1993
The correlation ratchet: a novel mechanism for generating directed motion by ATP hydrolysis C. S. Peskin, G. B. Ermentrout, G. F. Oster Introduction Progressive enzymes are macromolecules which hydrolyze ATP while moving unidirectionally along a linear macromolecular “track”. Examples include motor molecules such as myosin, kinesin and dynein, RNA and DNA polymerases, and chaperonins. We propose a specific mechanical model for transduction of phosphate bond energy during ATP hydrolysis into directed motion. This model falls within the class we call “Brownian ratchets” since the model cannot function without the Brownian motion and since it derives its directionality from the rectification of such motion. (Peskin et al., 1994; Peskin et al., 1993; Simon et al., 1992). The model depends on two features of the motor system. First, there is a molecular asymmetry in the binding between the motor and its track which determines the direction the motor will move (Magnasco, 1993). Second, the nucleotide hydrolysis cycle pumps energy into the system by providing an alternating sequence of strong and weak binding states. The novelty of this mechanism is how these effects of ATP hydrolysis conspire to bias the Brownian motion of the motor. Removing either feature (alternation or asymmetry), or removing the thermal motion of the enzyme eliminates the transduction of the chemical cycle into directed motion. We call this mechanism a “correlation ratchet”, since its operation depends upon a correlation between the spatial position of the molecular motor and its binding state (strong/weak). When this correlation is lost the motor ceases to function (Feynman et al., 1963). We first present the model equations, and then supply an intuitive explanation for how the ratchet works.
Biasing diffusion with ATP hydrolysis Consider the situation illustrated in Figure 1. As the motor diffuses along the polymer it encounters periodically spaced binding sites, described by the potential V(x) = V(x + L), which exerts a force -dV(x)/dx on the motor. The crucial feature of these binding sites is their asymmetry; for kinesin, there is some evidence that this asymmetry is partly a property of the motor head, rather than the track; however, since both microfilaments and microtubles are intrinsically asymmetrical, it probably involves structural asymmetries in both proteins (Stewart et al., 1992). Since the binding between the motor and the track is electrostatic, the size of the binding site is limited by the Debye length (< 1 nm), and so is much less than the distance between sites (5-10 nm)(Cordova et al., 1990; Vale et al., 1989). Therefore, for illustrative purposes we will use the piecewise linear potential shown in Figure 1. As the motor diffuses, it continuously hydrolyzes ATP, which alternates the binding affinity between two values: weak (V0 ≈ 1 k B T) and strong (V1 ≈ 5-10 kB T. kB is Boltzmann’s constant, T the absolute temperature; kB T ≈ 4.1 pN-nm at room temperature). Denote by x(t) the position of the enzyme on its track; the stochastic equation governing the motion is:
In: Cell Mechanics and Cellular Engineering, V. C. Mow, et al. (eds.) New York: Springer-Verlag, 1995.
dx(t) D = − A(t) ⋅ V'(x) − λ + 2D ⋅ w(t) 42 43 dt k BT amplitude Force due to load 1white noise velocity binding potential
(1)
where D is the diffusion coefficient of the enzyme, λ is a constant load force resisting the motion, and w(t) is uncorrelated white noise (Doering, 1990) . The hydrolysis cycle controls the magnitudes, A(t), of the binding force. We model this as a 2-state Markov chain whose states are z = 0 (weak binding) and z=1 (strong binding); there is no difficulty in including more states, (Leibler; Huse, 1993) but this is not necessary to illustrate the principle. The rate constants for the transitions are k10 = r·f, k01 = r·(1-f), where f is the fraction of time in the weakly bound state and r ∼ 1/mean cycle time. Figure 2 shows a numerical simulation of a typical trajectory of equation (1). The motor moves stochastically to the right, towards the steep end of the potential wells—i.e. opposite to the direction one would expect if it were a macroscopic ratchet. Its mechanical effectiveness can be characterized by the load-velocity curve shown in Figure 2 (see below).
ENZYME TRACK p(x) ~ exp[-V(x)/k BT] FLUX
0 r·(1-f)
r·f 1
Weak 1 kT binding
Strong 10 kB T binding
8 nm Figure 1. A protein diffusing along a linear polymer encounters electrostatic binding sites spaced 8 nm apart. In order to prevent diffusing away from the track the enzyme is either sleeve-shaped (RNA polymerase, GroEL), or works in tandem with another enzyme (myosin, kinesin). The ATP cycle alternates the binding affinity between strong and weak binding states with rates r·f and r·(1-f), where f is the fraction of time in the weak binding state and r is the overall cycle rate. The depth of the strong binding state is about 10 kBT, while the weak binding state is about 1 kBT. The binding potential wells are spaced L = 8nm apart and have asymmetry α = (L1-L2)/(L1+L2), where L1 and L2 are the lengths of the left and right sides of the well, respectively. While in the strong binding state the protein equilibrates in the deep well so that its position is given by a Boltzmann distribution, p(x) ∼ exp[-V(x)/kBT], where p(x) is the probability distribution of the enzyme’s location at the moment of switching into the weak binding state. The asymmetry of this distribution is the source of the diffusion bias towards the steep side of the potential ratchet.
The explanation for the propulsive force lies in the statistics of the process. Consider the state of the enzyme when it is strongly bound. If it remains in the well long enough to thermally equilibrate (< 10-9 s), its probability distribution relaxes to the Boltzmann form. That is, the probability, p(x), of being at position x is p(x) ∼ exp[-A1V1 (x)/k B T], which is asymmetric because V(x) is. When the well is suddenly lifted to its weakly bound level, this probability cloud begins to diffuse away. On a population basis, there is more motion towards
-2-
the steep side of the potential, since the probability gradient is steepest there. For the well shown in Figure 1, motion is to the right. If the switching rate of the well is approximately the mean time to diffuse between wells (1/r ∼ L2/2D), then the next well to the right will have a good opportunity to capture those enzymes that have moved in its direction. Because of the exponential Boltzmann factor the bias is quite sensitive to the asymmetry in the well, and considerable motion can be generated with only modest asymmetries.
50 40 30 x [nm] 20 10 0 0
0.1
0.2
t [sec]
0.3
0.4
0.5
Figure 2. A typical sample path for equation (1). The time scale for a diffusion step is much shorter than the ATP cycle time; therefore, in order to compute a sample path the viscosity of the medium must be artificially increased so that D = 10 3 nm2 /s. The other parameters employed in the simulation are: V0 = 0.5 kB T, V1 = 10 k B T, L = 8 nm, L1 = 0.8 nm, L2 = 0.2 nm, r = 100/s, f = 0.1, load = 0.
We analyze the statistical behavior of equation (1) by converting it to a pair of diffusion (Fokker-Planck) equations (Doering, 1990): ∂ p0 ∂ ∂p0 D(λ+ A0 V'(x)) =− − p0 − r(1 − f)p 0 + rf p1 −D ∂t ∂x ∂x k BT
(2)
∂ p1 ∂ ∂p D(λ+ A1 V'(x) ) = − −D 1 − p1 + r(1− f)p 0 − r fp 1 ∂t ∂x ∂x k BT
(3)
where p0(x,t) and p1(x,t) are the probability density functions for the states 0 and 1, respectively. The steady state solution to equations (2) and (3) yields the load-velocity curve shown in Figure 3. Although Figure 3 is computed for a fixed value of r, we remark that the force and velocity developed by the motor both vanish when the hydrolysis rate, r, is zero or in the limit r → ∞. Dimensional analysis of equations (2) and (3) shows that the mechanical behavior of the motor is governed by four dimensionless parameters: β = rL2 /D measures the diffusion time between binding sites relative to the ATP cycle time, and three parameters describing the depths and asymmetry of the binding site: V0 ≡ V0 /kBT, V1 ≡ V1 /kBT, and α ≡ (L1 -
-3-
L2 )/(L1 +L2 ), where L 1 and L2 are the lengths of the shallow and steep sections of the potential, respectively.
In the experimental situation described by Svoboda, et al. (1993), β ~ 100[1/s]·82 [nm]/7.2×105 [nm2 /s] < 10-2 . In the Appendix we show that, in the limit of small b, one can derive the following approximate expression for the unloaded velocity as a function of the reaction rate:
(4)
v = r·f(1 - f)·L·F0
where F0 is a dimensionless constant that depends only on the form of the two potential wells. 3
2 v [nm/s] 1
0 0
0.02
0.04
0.06
0.08
Load [pN] Figure 3. A load-velocity curve computed from equations (2) and (3) for D = 104 nm 2 /s, f = 0.1, r = 103 , L1 = 0.8 nm, L 2 = 0.2 nm. Velocity decreases linearly, with a stall load much less than 1 pN.
Discussion Several authors have suggested that Brownian motion plays a crucial role in the operation of molecular motors (Cordova et al., 1991; Huxley, 1957; Meister et al., 1989). Magnasco was the first to point out the importance of asymmetric potentials in biasing thermal motion (Magnasco, 1993) , and Doering et al. solved a model similar to equation (1) but with additive noise (Doering et al., 1994). We have shown here that one of the characteristic properties of ATP hydrolysis, its ability to alter the binding affinity of a macromolecule at a different site—so called, noncompetitive inhibition—can bias its diffusive motion providing the binding affinity at the track-binding site is not symmetrical. This diffusive bias generates a very weak force, incapable of driving much of a load. However, it can be turned into a powerful directional motor by coupling with a second ratchet mechanism (Peskin et al., 1994; Peskin et al., 1993; Simon et al., 1992). For example, a pair of kinesin ATPases can drive a vesicle against a considerable load (Svoboda et al., 1993) , but there is no firm evidence that a single head can do so. However, if the two heads alternate binding to the microtubule lattice then the bound head can hold fast the load while the other head diffuses to its next binding site. A model incorporating two coupled heads, symmetrical in all respects save the potential well asymmetry, can drive a load of several piconewtons. We will report on such a model in a separate publication. There are other cases in which the motion of a progressive enzyme may work by a Brownian ratchet mechanism of the type described in [2]: An RNA polymerase, by adding ribonucleotide subunits to the posterior 3’-hydroxyl end of the RNA chain, could ratchet the progressive -4-
diffusion of the enzyme (Schafer et al., 1991). The progressive action of the chaperonin ATPase GroEL in facilitating polypeptide folding may work by a similar mechanism (Ellis, 1993).
Acknowledgments GO and CSP were supported by NSF Grant FD92-20719; BE was supported by NSF Grant DMS9002028. The authors profited greatly from conversations with S. Block, D. Brillinger, C. Doering, S. Evans, W. Horsthemke, M. Magnasco, and K. Svoboda. We thank T. Ryan, K. Svoboda and P. Janmey for critical reading of the manuscript. Note: Aftter completion of this manuscript our attention was called to two papers which propose models whose mathematical aspects are similar to ours, but with quite different biological assumptions: (Ajdari; Prost, 1992; Astumian; Bier, 1993).
References Ajdari, A.; J. Prost: Mouvement induit par un potentiel périodique de basse symétrie: diélectrophorése pulsée. C. R. Acad. Sci. Paris. 3 1 5 : 1635-1639; 1992 Astumian, D.; M. Bier: Fluctuation driven ratchets—Molecular motors. Phys. Rev. Lett. (in press): 1993 Cordova, N.; B. Ermentrout; G. Oster: The mechanics of motor molecules I. The thermal ratchet model. Proc. Natl. Acad. Sci. (USA). 8 9 : 339-343; 1991 Cordova, N.; G. Oster; R. Vale. Dynein-microtubule interactions. In: L. Peliti, eds. Biologically Inspired Physics. New York: Plenum.; 1990: p. 207-215. Doering, C. Modeling complex systems: Stochastic processes, stochastic differential equations, and FokkerPlanck equations. In: L. Nadel; D. Stein, eds. 1990 Lectures in Complex Systems. Redwood City, CA: AddisonWesley; 1990: p. 3-51. Doering, C.; W. Horsthemke; J. Riordan: Nonequilibrium fluctuation-induced transport. Phys. Rev. Lett. In press: 1994 Ellis, R.J.: Chaperonin duet. Nature. 3 6 6 : 213-214; 1993 Feynman, R.; R. Leighton; M. Sands.The Feynman Lectures on Physics. Reading, MA: Addison-Wesley; 1963 Huxley, A.F.: Muscle structure and theories of contraction. Prog. Biophys. biophys. Chem. 7 : 255-318; 1957 Leibler, S.; D. Huse: Porters versus rowers: a unified stochastic model of motor proteins. J. Cell Biol. 1 2 1 : 13571368; 1993 Magnasco, M.O.: Forced thermal ratchets. Phys. Rev. Lett. 7 1 : 1477-1481; 1993 Meister, M.; S.R. Caplan; H.C. Berg: Dynamics of a Tightly Coupled Mechanism for Flagellar Rotation: Bacterial Motility, Chemiosmotic Coupling, Protonmotive Force. Biophys. J. 5 5 : 905-914; 1989 Peskin, C.; V. Lombillo; G. Oster. A depolymerization ratchet for intracellular transport. In: M. Millonas, eds. Fluctuations and Order: The New Synthesis. New York: Springer-Verlag; 1994. Peskin, C.; G. Odell; G. Oster: Cellular motions and thermal fluctuations: The Brownian ratchet. Biophys. J. 6 5 : 316-324; 1993 Schafer, D.A.; J. Gelles; M.P. Sheetz; R. Landick: Transcription by single molecules of RNA polymerase observed by light microscopy. Nature. 3 5 2 : 444-448; 1991 Simon, S.; C. Peskin; G. Oster: What drives the translocation of proteins? Proc. Natl. Acad. Sci. USA. 8 9 : 37703774; 1992 Stewart, R.J.; J.P. Thaler; L.S.B. Goldstein: Direction of microtubule movement is an intrinsic property of the motor domains of kinesin heavy chain and Drosophila ncd protein. Proc. Natl. Acad. Sci. 9 0 : 5209-5213; 1992
-5-
Svoboda, K.; C.F. Schmidt; B.J. Schnapp; S.M. Block: Direct observation of kinesin stepping by optical trapping interferometry. Nature. 3 6 5 : 721-727; 1993 Uyeda, T.; S. Kron; J. Spudich: Myosin step size: estimation from slow sliding movement of actin over low densities of heavy meromyosin. J. Mol. Biol. 2 1 4 : 699-710; 1990 Uyeda, T.; H. Warrick; S. Kron; J. Spudich: Quantal velocities at low myosin densities in an in vitro motility assay. Nature. 3 5 2 : 307-311; 1991 Vale, R.; D. Soll; I. Gibbons: One-dimensional diffusion of microtubules bound to flagellar dynein. Cell. 5 9 : 915-925; 1989
Appendix In this Appendix we derive the following asymptotic formula for the velocity of a 1-legged ratchet in the limiting case where β ≡ rL2 /D → 0, i.e. the reaction rate is much slower than diffusion: v = R ⋅L ⋅F0 [ V ]
(A.1)
where R is the overall rate of hydrolysis, R = r·f·(1-f), and F0 [V] depends on the form of the two potential wells, V0 (x) and V1 (x). The situation is this: since the reaction rate is much slower than the time for the probability distribution of particle locations, P(x,t) to reach equilibrium, then we know that the distributions in the strong and weakly bound states are Boltzmann. Thus we need only compute the net average transfer of probability between potential wells per hydrolysis cycle. We begin with the conservation equation for an ensemble of points, c(x,t):
∂c ∂J =− ∂t ∂x
(A.2)
where ∂c 1 dV J = −D + c , ∂x k BT dX
(A.3)
and the boundary condition on c(x, t) is periodic: c(x + L, t) = c(x, t).
(A.5)
If the initial distribution, c(x,0), is given we wish to compute the total fllux past a given point x during the relaxation of c(x, 0) → c(x, ∞):
F(x) =
∞
∫0 J(x,t)dt
(A.6)
Integrating the conservation equation over all times yields:
∂F = c(x,0) − c(x,∞) ∂x
(A.7)
This determines F(x) up to an additive constant, which can be computed as follows. Multiplying (A.3) by exp(V(x)/ kB T ) and integrating first over x and then over t we obtain
-6-
∫
L
0
V(x) F(x) ⋅ exp dx = 0 k BT
(A.8)
which determines the additive constant. The system switches back and forth between V0 and V1 , and the corresponding equilibrium distributions in each state are obtained by setting J = 0 in (A.3):
ci (x) =
exp( −Vi (x)/ k BT ) L
∫0
exp (−Vi (x' ) / k BT )dx'
, i = 1,2
(A.9)
Let F 2 (x) be the value of F(x) for the switch from V1 → V2 , and F 1 (x) be the value of F(x) for the switch form V2 → V1 . Then ∂F c2 (x) - c1 (x) + ∂x2 = 0
(A.10a)
∂F c1 (x) - c2 (x) + ∂x1 = 0
(A.10b)
Thus F0 = F1 (x) + F2 (x) = constant. From (A10a,b) we have F1 (x) = F 1 (0) + Φ(x)
(A.11a)
F2 (x) = F 2 (0) - Φ(x)
(A.11b)
x
where Φ(x) =
∫ 0 [c (x’) - c (x’)]dx’. But from (A.8) we also have the identities 2
1
L
∫ F (x)b (x)dx = 0 ∫ F (x)b (x)dx =0 1
0
1
L
2
0
where bi = exp( Vi (x)/k B T) /
∫
L 0
(A.12)
2
exp( Vi (x')/k B T)dx' . Note that b i has the same definition as ci
except that the sign of V has been reversed. From this we conclude that L
F1(0) = −∫ Φ(x)b1(x)dx 0 L
F2 (0) = ∫ Φ(x)b2 (x)dx 0
Thus
-7-
(A.13)
F0 = F2 (0) + F1(0) = ∫ Φ(x)( b2 (x) − b1(x)) dx L
0
=
∫0 (b2 (x) − b1 (x)) ∫0 (c2(x' ) − c1(x') )dx' dx L
x
(A.14)
Now the average hydrolysis cycle time for the 2-state model is 1/τ1 + 1/τ2 =1/(r·f) + 1/(r·(1 - f)); therefore, the mean cycle rate is the reciprocal of this: R = r·f·(1-f). Since F0 ·L is the mean distance moved per cycle, so multiplying by the cycle rate, R, we obtain equation (4):
(A.15)
v = r·f(1 - f)·L·F0
Note that in this limit, the velocity is maximum at f = 1/2, and increases linearly with reaction rate, in accordance with the numerical solution to the full boundary value problem shown in Figure A1. Figure A.1 gives the impression that the velocity increases indefinitely with r, but this is not the case. As r → ∞, the velocity has to approach zero because the motor then sees only a single average potential, but this is outside the domain of validity of the asymptotic method used here, and it is also far beyond the physiological range of hydrolysis rates. Note also that in the slow reaction rate limit the velocity is independent of the diffusion coefficient, D. This is because all the motion occurs on a time scale much shorter than the hydrolysis rate, and so the system is essentially switching instantly between equilibrium states. An equation similar to (1) has been used to estimate the mean step size of single, unloaded, motors from in vitro assays wherein the velocity of a microtubule or vesicle is measured, and the stepsize inferred by dividing by the hydrolysis rate: L ≈ v/R (Uyeda et al., 1990; Uyeda et al., 1991). Aside from the experimental error of using hydrolysis rates obtained in separate assays, we see that there is an additional factor, F0, that should also be included in the relationship between v, L, and R (c.f. equation A.15). 12 10 8
v [nm/s] 6 4 2 0
0
2
4
3
6
8
10
r [x 10 1/sec] Figure A1. Velocity as a function of the hydrolysis rate, r, when β = rL 2 /D
