9540
J. Phys. Chem. B 2008, 112, 9540–9545
Brownian Pump in Nonlinear Diffusive Media Bao-quan Ai*,† and Liang-gang Liu‡ Institute for Condensed Matter Physics, School of Physics and Telecommunication Engineering, South China Normal UniVersity, 510006 Guangzhou, China, and Faculty of Information Technology, Macau UniVersity of Science and Technology, Macao ReceiVed: February 25, 2008; ReVised Manuscript ReceiVed: April 19, 2008
A Brownian pump in nonlinear diffusive media is investigated in the presence of an unbiased external force. The pumping system is embedded in a finite region and bounded by two particle reservoirs. In the adiabatic limit, we obtain the analytical expressions of the current and the pumping capacity as a function of temperature for normal diffusion, subdiffusion, and superdiffusion. It is found that important anomalies are detected in comparison with the normal diffusion case. The superdiffusive regime, compared with the normal one, exhibits an opposite current for low temperatures. In the subdiffusive regime, the current may become forbidden for low temperatures and negative for high temperatures. I. Introduction A living cell interacts with its extracellular environment through the cell membrane, which acts as a protective permeability barrier for preserving the internal integrity of the cell. However, cell metabolism requires controlled molecular transport across the cell membrane, a function that is fulfilled by a wide variety of transmembrane proteins, acting as passive and active transporters.1 This task is performed by some molecular devices located across the membrane which work under nonequilibrium conditions. There are two typical molecular devices. The first machine is a kind of door mechanism which opens and closes under some perturbation leading particles across the membrane from the larger concentration reservoir to the lower one. The second one is the molecular pump. This device is also located in the cell membrane and transports particles across the membrane against the concentration gradients at both sides using chemical energy. The idea of applying the ratchet mechanism to model pumps has already appeared in the literature.2–6 Prost and co-workers2 studied the transport of an asymmetric pump with a simple twolevel model and quantified how vectorial symmetry plus dissipation creates a macroscopic motion, even in the absence of any externally applied gradient. Astumian and Derenyi3 investigated a chemically driven molecular electron pump in which charge can be pumped through a tiny gated portal from a reservoir at low electrochemical potential to one at the same or higher electrochemical potential by cyclically modulating the portal and gate energies. Kosztin and Schulten4 studied the fluctuation-driven molecular transport through an asymmetric potential pump and three transport mechanisms: driven by the potential gradient, by an external periodic force, and by nonequilibrium fluctuations. A nonadiabatic electron heat pump was investigated by Rey and co-workers.5 They presented a mechanism for extracting heat metallic conductors based on the energy-selective transmission of electrons through a spatially asymmetric resonant structure subject to ac driving. Recently, Sancho and Gomez-Marin6 presented a model for a Brownian pump powered by a flashing ratchet mechanism. The pumping * Corresponding author. E-mail:
[email protected]. † South China Normal University. ‡ Macau University of Science and Technology.
Figure 1. Scheme of a Brownian pump. A spatially asymmetric potential U0(x) (defined in eq 2) is embedded in a finite region (membrane) and bounded by two particle reservoirs of concentrations F0 and F1. L and λ are the length and the asymmetric parameter of the potential, respectively. The particles are driven by an unbiased external force F(t), defined by eq 3.
device was embedded in a finite region and bounded by particle reservoirs. Their emphasis is on finding what concentration gradient the pump can maintain. All of the above examples have been formulated within a standard Brownian framework, for which diffusion properties are normal, that is, with the mean quadratic displacement growing linearly with time t. However, it is well-known that there are media signaled by a tµ growth of the squared dispersion with µ * 1. An important class is given by the “porous medium” equation,7–10 which, in the one-dimensional problem and in the absence of external forces, can be cast in the form
∂tF )
∂[ ∂2 U′(x, t) F(x, t)] + D 2 Fµ(x, t) ∂x ∂x
(1)
where F is the density of the diffusing substance, x is a dimensionless coordinate representing a bond length, angle, or any other chemical or physical state variable, t is dimensionless
10.1021/jp8017259 CCC: $40.75 2008 American Chemical Society Published on Web 07/10/2008
Brownian Pump in Nonlinear Diffusive Media
Figure 2. Current J versus temperature T for different values of F1 at Q ) 1, λ ) 0.9, F0 ) 0.5, F0 ) 1.0, and µ ) 1.
J. Phys. Chem. B, Vol. 112, No. 31, 2008 9541
Figure 5. Currents j(F0), -j(-F0), and J versus temperature T for µ ) 2.0 at Q ) 1, λ ) 0.9, F0 ) 0.5, F0 ) 1.0, and F1 ) 1.0. The C temperatures for the two inflections are TFC0 ) 0.225 and T-F ) 0.725. 0
Figure 3. Concentration ratio F1/F0 as a function of temperature T for J ) 0 at Q ) 1, λ ) 0.9, F0 ) 0.5, and µ ) 1. Figure 6. Concentration ratio F1/F0 as a function of temperature T for J ) 0 at Q ) 1, λ ) 0.9, F0 ) 0.5, F0 ) 1.0, and µ ) 1.5, 2.0, and 4.0.
The present work is extended to the study of the Brownian pump for the case of anomalous diffusion. We emphasize finding how particles can be pumped through a finite region from a particle reservoir at low concentration to one at the same or higher concentration in nonlinear diffusive media. II. General Analysis
Figure 4. Current J as a function of temperature T for different values of µ at Q ) 1, λ ) 0.9, F0 ) 0.5, F0 ) 1.0, and F1 ) 1.0.
time, and D is the diffusive constant. U(x,t) is the potential. The mean quadratic deviation follows the law 〈x(t)2〉 ∝ t2/(1+µ) without a potential. The transport is subdiffusive for µ > 1, normal for µ ) 1, and superdiffusive for 0 < µ < 1. Many physical systems are well described by this class of processes: percolation of gases in porous media,11 dispersion of biological populations,12 grain segregation,13 fluxes in plasma,14 and nonextensive statistics.15
We consider a Brownian pump which is located in a finite region and transports particles across the barrier against the concentration gradients by using an unbiased external force. Here, we model the device as a pumping system in which overdamped Brownian particles move in an asymmetric finite potential in the presence of an unbiased external force. The potential is embedded in a finite region [0, L]
{
x , 0 e x < λL λL U0(x) ) L-x Q , λL e x e L (1 - λ)L Q
(2)
where Q is the amplitude of the potential and λ is its asymmetric parameter. F(t) is an unbiased external force and satisfies
9542 J. Phys. Chem. B, Vol. 112, No. 31, 2008
F(t) )
{
F0,
-F0,
1 nτ e t < nτ + τ 2 1 nτ + τ < t e (n + 1)τ 2
Ai and Liu
j(x, t) ) -U′(x, t) F(x, t) - D (3)
where τ is the period of the unbiased force and F0 is its magnitude (see Figure 1). The correlated anomalous diffusion can be described through the following nonlinear differential equation:8–10
∂ ∂2 ∂j(x, t) ∂ F(x, t) ) [U′(x, t) F(x, t)] + D 2 Fµ(x, t) ) ∂t ∂x ∂x ∂x (4) U(x, t) ) U0(x) - F(t)x
(5)
where (x,t) is a dimensionless 1 + 1 space time and D ) kBT/ η. kB is the Boltzmann constant, T is the temperature, and η is the friction coefficient. The prime stands for the derivative with respect to the space variable x. j(x,t) is the probability current, and F(x,t) is the particle concentration. This nonlinear equation yields anomalous diffusion when µ * 1, subdiffusion for µ > 1, and superdiffusion for 0 < µ < 1. A. Normal Diffusion. For the normal diffusive case µ ) 1, eq 4 recovers the ordinary Fokker-Planck equation and j(x, t) satisfies
∂ F(x, t) ∂x
(6)
If F(t) changes very slowly with respect to t, namely, its period is longer than any other time scale of the system, there exists a quasistatic state. In the steady state, the concentration is just a function of space and the flux becomes a constant j. Thus, the density F(x) has the formal solution
[
F(x) ) exp -
x U′(y) j x dz]{c0 - ∫0 dz exp[∫0 dy]} ∫0x U′(z) D D D
(7) The unknown constant c0 and j can be found by imposing the left reservoir concentration F0 ≡ F(0) and the right concentration F1 ≡ F(L) as fixed boundary conditions. We can find that c0 ) F0 and
[ ]} [ ] { [ ]}
{
F0L D j(F0) ) U0(x) - F0x L exp dx 0 D F0L D F0 - F1 exp ) I(F0) D D F0 - F1 exp -
∫
where
I(F0) )
[ (
(8)
) ]
Q - F0λL λLD exp -1 + Q - F0λL D D(λ - 1)L + Q - F0(λ - 1)L
[ ( ) ( exp -
F0L Q - F0λL - exp D D
)]
(9)
The average current is
J)
Figure 7. Current J as a function of temperature T for different values of µ at Q ) 1, λ ) 0.9, F0 ) 0.5, F0 ) 1.0, and F1 ) 1.0.
1 τ
∫0τ j(F(t)) dt ) 21 [j(F0) + j(-F0)]
(10)
We studied the pumping capacity using a similar method in ref 6. Here we consider the situation in which J tends to 0, which corresponds to the case in which the pump maintains the maximum concentration difference between the two reservoirs across the barrier with no net leaking of particles. From eqs 8–10, we can obtain
I(F0) + I(-F0) F1 ) F0 eF0L/DI(F ) + e-F0L/DI(-F ) 0 0
(11)
B. Anomalous Diffusion. When µ in eq 4 is not equal to 1, we have
j(x, t) ) -U′(x, t) F(x, t) - D
∂ µ F (x, t) ∂x
(12)
In this case, the explicit expression for F(x,t) cannot be extracted from the above equation by using the same method as in the case of µ ) 1. Here, we use the method presented by Zhao and co-workers10 to derive the expressions for j and F. Let us assume
Figure 8. Currents j(F0), j(-F0), and J versus temperature T for µ ) 0.1 at Q ) 1, λ ) 0.9, F0 ) 0.5, F0 ) 1.0, and F1 ) 1.0. The temperature at the inflection is TFC0 ) 4.5.
Fm(x, t) ) G(x, t) Fµ(x, t),
jm(x, t) ) G(x, t) j(x, t) (13)
where G(x,t) is a factor function. We assume that there exists the proper Fm(x,t) and jm(x,t) satisfying diffusion law:
Brownian Pump in Nonlinear Diffusive Media
jm(x, t) ) -D
J. Phys. Chem. B, Vol. 112, No. 31, 2008 9543
∂ F (x, t) ∂x m
(14)
I(F0) )
From eqs 13 and 14, we have
j(x, t) ) -DFµ(x, t)
)
∂[ ∂ ln G(x, t)] - D Fµ(x, t) (15) ∂x ∂x
Comparing eq 12 with eq 15, we have
[
φµ(x, t) G(x, t) ) exp D
∫0L [Ψ(x)] (1-µ) dx µ
1 (µ - 1) (1 - AλL) (1-µ) - 1 + A
[
1 1 (1 - µ) (C + BL (1-µ) - (C + BλL) (1-µ) (26) B
[
]
(µ - 1)(Q - F0λL) DµλL (µ - 1)[Q + F0(1 - λ)L] B) Dµ(1 - λ)L
(16)
∫ U′(x, t) F1-µ(x, t) dx
(17)
It is obvious that φµ represents an effective F-dependent potential resulting from the nonlinear diffusive media. For a quasistatic state, we can get the static solution of eq 4:
Fs(x) ) [Ψ(x)]1/(µ-1) +
(18)
where [f]+ ) max{f,0} and
Ψ(x) ) 1 -
A)
µ-1 U(x) Dµ
(19)
and
C)1-
θ(F0) )
µD φµ(x, t) ) ln[Ψ(x)] 1-µ
(20)
] [ (
)
]
φµ(x, t) ∂ φµ(x, t) µ j(x, t) ) -D exp exp F (x, t) D ∂x D
(21)
[
]
φµ(x, t) dx ) D
[ (
∫0x ∂y∂ exp
)
]
φµ(y, t) µ F (x, t) dy D (22)
When the system reaches the steady state, we can get the steady density Fµ(x):
[ ]
exp Fµ(x) )
[ ]
φµ(0) µ φµ(x) j x F (0) exp dx D D 0 D (23) φµ(x) exp D
∫
[ ]
The constant j can be found by imposing the left reservoir concentration F0 ≡ F(0) and the right concentration F1 ≡ F(L) as fixed boundary conditions:
j(F0) )
θ(F0)D µ [F - M(F0)Fµ1 ] I(F0) 0
(24)
µ⁄(1-µ) µ-1 F0L Dµ
(25)
where
[
M(F0) ) 1 + and
]
1, 0,
Ψ(x) > 0, for all values of x otherwise
1 J ) [j(F0) + j(-F0)] 2
(27)
(28)
From eqs 24–28, we can obtain the maximum concentration ratio for J ) 0:
Fµ1
Integrating over x from 0 to x, we have
∫0x exp
{
Similarly, the average current is
So we can rewrite eq 15 as
j D
(µ - 1)Q Dµ(1 - λ)
It must be pointed out that when Ψ(x) is not always positive there exists a cutoff of probability. A cutoff condition (Tsallis cutoff) yields regions with null probability. This is because a cutoff of the stationary solution eq 18 restricts the attainable space. Since the probability of particles visiting the special regions is null, the particles cannot pass across the barrier; then there is no current. In order to describe this cutoff, a function θ(F0) is defined by
From eqs 17–19, we can obtain
[
]
with
where
φµ(x, t) )
]
Fµ0
)
θ(F0) I(-F0) + θ(-F0) I(F0) (29) θ(F0) I(-F0) M(F0) + θ(-F0) I(F0) M(-F0)
III. Results and Discussion Our study focused on the current and the maximum concentration ratio at J ) 0 for normal diffusion, subdiffusion, and superdiffusion. For simplicity, we take kB ) 1, η ) 1, and L ) 1 throughout the study. A. Normal Diffusion, µ ) 1. In Figure 2, we present the current as a function of temperature T for normal diffusion (µ ) 1.0). For lower values of F1 the current is larger. When F1 is greater than F0, the current is negative for high temperatures. When T f 0, the particles cannot pass over the potential barrier and the current tends to zero. When T f ∞, the ratchet effect disappears, the transport is dominated by the concentration difference and the particles move to the left. Therefore, there exists an optimized value of T at which the current takes its positive maximum value. Note that a similar behavior is also presented in ref 6. Figure 3 shows the ratio F1/F0 (J ) 0) as a function of temperature T for normal diffusion (µ ) 1). When T f 0, no particle can pass over the barrier; thus F1/F0 f ∞. As temperature T is increased, the ratchet effect reduces and the pumping capacity decreases. Surprisingly, the temperature corresponding to the maximum current is not the same as the temperature at which the concentration ratio for zero current is maximum. This is because zero current induces the maximum concentration ratio, not the minimum one.
9544 J. Phys. Chem. B, Vol. 112, No. 31, 2008
Ai and Liu
Figure 9. Concentration ratio F1/F0 as a function of temperature T for J ) 0 at Q ) 1, λ ) 0.9, F0 ) 0.5, F0 ) 1.0, and µ ) 0.5, 0.6, 0.7, 0.8, and 0.9.
B. Subdiffusion, µ > 1. Figure 4 shows the current J versus temperature T for subdiffusion (µ > 1) at F1 ) F0 ) 1.0. The curve for normal diffusion is observed to be bell-shaped. However, as µ is increased, the curve becomes not smooth. There exist two values of temperature at which the curve has inflections: the lowest temperature to obtain a positive current; the optimized temperature to obtain a maximum current. For low temperatures, there lies a finite temperature only above which the particle can pass over the barrier; otherwise the particles will be confined in both reservoirs. For high temperatures, the current is negative. However, when T f ∞, the current will approach zero. In order to illustrate the transport behavior for subdiffusion, the currents j(F0), -j(-F0), and J as a function of T for µ ) 2.0 are shown in Figure 5. From eqs 19 and 27, we can find that there exist two values of temperature C T-F ) 0
µ-1 (Q ( F0λ) µ
(30)
only above which Ψ(x) is always positive; otherwise Ψ(x) may C be negative. Therefore, when T e T-F , the state space 0 becomes disconnected and crossings become forbidden. The local current j tends to zero. This is because a cutoff of the stationary solution of eq 18 restricts the attainable space. From Figure 5, we can see that both j(F0) and j(-F0) are zero and J C ) 0 for T < T-F ; j(F0) is positive, j(-F0) is zero, and J ) 0 C (1/2)j(F0) for TFC0 < T < T-F ; j(F0) is positive, j(-F0) is 0 C negative and J ) (1/2)[j(F0) + j(-F0)] for T > T-F . The first 0 C C inflection is at TF0 and the second one is at T-F0. Figure 6 shows the concentration ratio F1/F0 as a function of T for subdiffusion (µ ) 1.5, 2.0, and 4.0). The ratio F1/F0 decreases with increasing T. The curve is not smooth, and there exists a inflection at T ) TFC0. For low temperatures, the ratio F1/F0 decreases with increasing µ, while it increases with increasing µ for high temperatures. C. Superdiffusion, 0 < µ < 1. Figure 7 shows the current J as a function of temperature T for superdiffusion (0 < µ < 1) at F1 ) F0. It is found that the curve is not smooth and there is a inflection at which the negative current takes its maximum value. For low temperatures, the particles move to the left and the negative current increases with decreasing µ. As temperature T is increased, the current becomes positive and the positive current increases with decreasing µ.
Figure 10. Stationary probability density distribution for subdiffusion (µ ) 2.0, see Figure 5) at Q ) 1, λ ) 0.9, F0 ) 0.5. (a) T ) 0.2; (b) T ) 0.5; (c) T ) 1.0.
In order to explain the transport behavior for superdiffusion, the currents j(F0), j(-F0), and J versus temperature T for µ ) 0.1 are shown in Figure 8. We can find from eqs 19 and 27 that there is a critical value of temperature
TFC0 ) -F0
µ-1 µ
(31)
below which the local current j(F0) is zero. When T is less than TFC0, j(F0) is zero and j(-F0) is negative, so the current J ) (1/2)j(-F0) is negative. When T is greater than TFC0, j(F0) is
Brownian Pump in Nonlinear Diffusive Media greater than -j(-F0), so the current J is positive. As temperature T is increased, j(F0) tends to -j(-F0); then the current J goes to zero. Figure 9 shows the concentration ratio F1/F0 (J ) 0) as a function of temperature T for superdiffusion (µ ) 0.5, 0.6, 0.7, 0.8, and 0.9). When T is less than TFC0, j(F0) goes to zero, so the particles move to the left and F1/F0 < 1. As µ is decreased, the lowest temperature for pumping the particles to the right increases. Therefore, the pumping system in the superdiffusive regime needs a higher temperature than that in the normal diffusive regime. For the above phenomena, we focus on the stationary probability density distribution. As a example, we illustrate the unique phenomena of Figure 5 in Figure 10. From Figure 10, we can see that, when T < TFC0 [(a) T ) 0.2 ], both probability density distributions for (F0 are discontinuous and the particles cannot pass through this space, so no currents occur. When the C temperature is between TFC0 and T-F [(b) T ) 0.5 ], the 0 probability density distribution for +F0 is continuous, while the distribution for -F0 is discontinuous. Therefore, j(-F0) is equal to zero and j(F0) is positive. Both probability density C distributions are continuous for T > T-F [(c) T ) 1.0 ], and 0 both currents are not equal to zero. Because the probability of particles visiting the special regions is null, the particles cannot pass across the barrier; then there is no current. The phenomena in other figures also have the corresponding explanations in stationary probability density distributions. IV. Concluding Remarks In this study, we investigate a Brownian pump in nonlinear diffusive media with an unbiased external force. The pump is embedded in a finite region and bounded by two particle reservoirs. The analytical results are obtained in the adiabatic limit. In the normal diffusive regime, current is a peaked function of temperature and the concentration ratio for zero current decreases with increasing temperature. In the subdiffusive regime, current is forbidden for low temperatures and negative for high temperatures. The concentration ratio for zero current decreases with increasing temperature, also. In the
J. Phys. Chem. B, Vol. 112, No. 31, 2008 9545 superdiffusive regime, the transport compared with normal diffusion exhibits an opposite direction for low temperatures. There exists a finite temperature at which the concentration ratio takes its maximum value. For anomalous diffusion, there exist inflections in the curves for J vs T and F1/F0. This is because a cutoff of the stationary solution eq 18 restricts the attainable space. Though the model presented does not pretend to be a realistic model for a biological pumps, the results we have presented have a wide application in membrane proteins and other realistic pumps. Acknowledgment. The work was supported by the National Natural Science Foundation of China under Grant 30600122 and the GuangDong Provincial Natural Science Foundation under Grant 06025073. References and Notes (1) Alberts, B.; et al. Molecular Biology of the Cell, 4th ed.; Garland Science: New York, 2002. (2) Prost, J. J.; Chauwin, F.; Peliti, L.; Ajdari, A. Phys. ReV. Lett. 1994, 72, 2652. (3) Astumian, R. D.; Derenyi, I. Phys. ReV. Lett. 2001, 86, 3859. Astumian, R. D. Phys. ReV. Lett. 2003, 91, 118102. (4) Kosztin, I.; Schulten, K. Phys. ReV. Lett. 2004, 93, 238102. (5) Rey, M.; Strass, M.; Kohler, S.; Hanggi, P.; Sols, F. Phys. ReV. B 2007, 76, 085337. (6) Sancho, J. M.; Gomez-Marin, A. Proc. SPIE 2007, 6602, 66020B; Sancho, J. M.; Gomez-Marin, A. http://arxiv.org/abs/0709.0890v1. (7) Peletier L. A. In Application of Nonlinear Analysis in the Physical Sciences; Ammam, H., Bazley, N., Eds.; Pitman: Boston, 1981; p 229. (8) Anteneodo, C. Phys. ReV. E 2007, 76, 021102. (9) Lenzi, E. K.; Anteneodo, C.; Borland, L. Phys. ReV. E 2001, 63, 051109. (10) Zhao, J. L.; Bao, J. D.; Wei, W. P. J. Chem. Phys. 2006, 124, 024112. (11) Muskat, M. The Flow of Homogeneous Fluids through Porous Media; McGraw-Hill: New York, 1937. (12) Gurtin, M. E.; MacCamy, R. C. Math. Biosci. 1977, 33, 35. (13) Khan, Z. S.; Morris, S. W. Phys. ReV. Lett. 2005, 94, 048002. (14) Rosenau, P. Phys. ReV. Lett. 1995, 74, 1056. Compte, A.; Jou, D.; Katayama, Y. J. Phys. A 1997, 30, 1023. (15) Plastino, A. R.; Plastino, A. Physica 1995, 222, 347. Tsallis, C.; Bukman, D. J. Phys. ReV. E 1996, 54, R2197.
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