- Email: [email protected]
One- and two-dimensional dynamics of elastically coupled Brownian motors
Available online at www.sciencedirect.com
Physica A 325 (2003) 62 – 68
www.elsevier.com/locate/physa
One- and two-dimensional dynamics of elastically coupled Brownian motors Akito Igarashia;∗ , Hiromichi Gokoa , Shinji Tsukamotob a Department
of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan b Sumitomo Electric Industries, Osaka 541-0041, Japan Received 24 October 2002
Abstract Brownian motors, in particular, “ratchet” models have been studied actively as models for molecular motors. In this paper, a 2ashing ratchet model for elastically coupled particles in an asymmetric periodic potential is introduced and studied as a model for molecular motors in muscle. In our one-dimensional model, velocity of motors and e3ciency of energy conversion are investigated, and in our two-dimensional model we investigate the dynamics of the model by computer simulation and reproduce qualitatively the results of biological experiments in a motility assay. c 2003 Elsevier Science B.V. All rights reserved. PACS: 05.40.Jc; 02.50.Ey; 87.15.Aa Keywords: Brownian motor; Coupled ratchet; Molecular motor
1. Introduction Brownian motors, or “ratchet” [1] models, have attracted much attention as models for molecular motors [2,3]. Without any directed force these models perform transport, where only thermal noise and a proper asymmetric potential are enough to produce the macroscopic motion of the particles toward a particular direction. Coupled Brownian motors have also been studied from the interest to investigate the eAects of mutual interaction among motors. For instance, JCulicher et al. [4] studied coupled “2ashing” ratchets, where the particles feel two kinds of potentials alternatively. On the other ∗
Corresponding author. Fax: +81-75-753-4919. E-mail address: [email protected] (A. Igarashi).
c 2003 Elsevier Science B.V. All rights reserved. 0378-4371/03/$ - see front matter doi:10.1016/S0378-4371(03)00184-5
A. Igarashi et al. / Physica A 325 (2003) 62 – 68
63
hand, CsahIok et al. [5] studied a coupled “rocking” ratchet model, where a periodic external force is applied to the system. Reimann et al. [7] introduced another model of interacting Brownian particles and found some new collective phenomena. The authors have been studying the e3ciency of the coupled “2ashing” ratchet model [6], and so on. In this paper, we pay special attention to acto-myosin motors for muscle contraction, where two kinds of Jlaments, mainly consisting of actin and myosin molecules, respectively, play an important role [8,9]. These Jlaments slide past each other using energy of ATP hydrolysis and muscle contraction results from it. If we regard the particles as myosin molecules and the interaction potential as the one between actin and myosin molecules in the ratchet models, they can be investigated as models for acto-myosin motors. In muscle contraction, however, actin molecules do not work independently but work correlatively with neighbor ones; therefore, it is natural to take into account mutual interaction among particles in coupled ratchets as models for acto-myosin motors. First, we investigate the coupled model in one-dimensional space, where the particles move only in one direction. This is the actual situation of muscle contraction. The velocity of motors and the e3ciency of energy conversion are obtained from our computer simulation in this case. Next, we extend the ratchet model to a two-dimensional space, in order to explain biological experiments in a motility assay [10–12], where, if actin Jlaments are put on a cover glass coated with myosin proteins and ATP is added in this system, they start to move because they interact with the myosin proteins with the use of the energy produced by ATP hydrolysis. Among properties of this system, we shed light mainly on two points. One is a manner for the actin Jlaments to move. It is reported that they advance zig-zag, swaying its body just like a snake. We try to reproduce qualitatively the distribution of the change of their advancing direction [11]. The other is that the longer the distance between two nearest-neighbor motors (which corresponds to a higher density of myosin molecules) is, the larger the velocity of the Jlament becomes as in the experiment [12].
2. The model Our model consists of elastically coupled particles which move under the in2uence of a two-dimensional potential. The elastically coupled particles can be regarded as a model of an actin Jlament moving on the myosin molecules. They feel stochastically either a 2at potential (W1 ) or an asymmetric two-dimensional periodic potential (W2 ). In muscle contraction an actin molecule is attached to or detached from a myosin molecule in ATP hydrolysis repeatedly. When the particle is in the attached state, it feels the potential W2 . W1 is a 2at potential for the state where the actin and the myosin are detached, and no force is exerted between them. It is also assumed that the particles are put in heat bath represented by a white noise. If we consider the system of N particles, the equation of motion of the ith (1 6 i 6 N ) particle whose position
64
A. Igarashi et al. / Physica A 325 (2003) 62 – 68
is xi is
9W2 (xi ) √ + 2Di (t) ; (1) 9xi where we consider an overdumped case for a friction constant . D stands for temperature and the components of i (t) denote mutually independent white noises of zero mean and correlations i (t)t j (s) = ij (t − s)E, where E stands for a unit matrix. fi is the elastic force exerted on the ith particle, where the longitudinal and lateral spring constant are k and A for the natural length of the springs a. In these equations, all parameters are dimensionless, and we set U , D and to be 1:0, 0:2 and 1:0, respectively, in our simulation. a is set to be 1:0 if not mentioned. The boundary conditions of particles are set to be free. With the factor hi (t) of the second term of Eq. (1), we make the particles alternatively feel an asymmetric potential W2 and a 2at potential W1 . We set hi (t) as a colored random modulation that rules the time-dependent change expected 0 or 1 with the correlation time . Therefore, the particles are subjected to the 2at potential W1 if hi (t) is 0 and to W2 if hi (t) is 1. Since the correlation time of hi (t) is , the potential felt by a particle changes stochastically with the correlation time . In our simulation, is set to be 1.0. Thereby, we usually change only three parameters in our simulation, that is, k, A and N . Only in Section 4, we also change the value of a. Next, we consider the asymmetric two-dimensional periodic potential W2 expressing the interaction between actin and myosin Jlaments. Since actin Jlaments are polar, these Jlaments have asymmetry, in other words they have particular directions to which they easily move. In order to take into account this fact in our model, we give a particular direction to each particle corresponding to a part of an actin Jlament which interacts with a myosin. The direction of a particle is decided from the position of the neighboring two particles. That is, the direction of the ith particle is decided to be that of the straight line which links the neighboring two particles, from the (i − 1)th to the (i + 1)th particle. The two particles located at the end of the actin Jlament have only a single neighboring particle; therefore, we deJne its direction as that of the lines which links itself to the neighboring one. By this method we can assign a “natural” direction to each particle, since roughly speaking, each direction deJned above is approximately the direction of the tangential line of the Jlament, in which the actin Jlament is thought to have asymmetry in space and to move easily. In order to determine this potential, Jrst, for simplicity, we consider a twodimensional periodic potential 1 U W0 (x) = sin(x) + sin(2x) + sin(y) ; (2) 4 x˙i = fi − hi (t)
where x = (x; y) and U represents the depth of the potential well. This potential has asymmetry only in the x direction and symmetry in the y direction. As for W2 of each particle, we rotate W0 to make the asymmetric direction of the potential (the x direction) coincident with that of each particle. Although one may think that we should introduce the direction of myosin proteins as well in our model, they, in the biological experiments [10–12], with which a cover glass is coated are located in random directions, that is, they are regarded as having
A. Igarashi et al. / Physica A 325 (2003) 62 – 68
65
no particular directions on average. Therefore, we do not consider the directions of myosin molecules in our model.
3. One-dimensional model In this section, we restrict the particle motion only in the x direction and calculate the velocity and e3ciency with computer simulation. Fig. 1 shows the velocity v vs. k at D = 0; 0:1; 0:2; 0:3, and has a maximum at k 4:0 for D = 0. The velocity decreases as the temperature increases. For D = 0:3 the velocity is almost independent of the spring constant k. The in2uence of the springs is weak in comparison with that of thermal diAusion for high temperature D = 0:3. Sekimoto [13] deJned the e3ciency for thermal ratchet models with a load Fext . If we deJne v as an average velocity of the model, the expression of the e3ciency is =
Fext v ; Pin
(3)
where Pin is the power input in a unit time [13]. We show in Fig. 2 the e3ciency deJned by Eq. (3). It has a maximum as a function of Fext and at a certain threshold it has a value less than 0, which means that the model moves in the −x direction. The e3ciency for a single particle is also shown in Fig. 2, and is much lower than that for the coupled model. Muscle is known to have much more e3ciency than in our simulation. From this point, our model do not succeed to reproduce the real situation. This is mainly because of the simplicity of the model. τ = 1.0, U = 1.0, N = 20
0.08
D D D D
0.06
=0 = 0.1 = 0.2 = 0.3
v
0.04
0.02
0
-0.02
0
10
20
30
40
k Fig. 1. v vs. k. The lower the temperature is, the faster the particles move in general. For high temperature, the peak almost disappears since the particles tend to move more randomly.
66
A. Igarashi et al. / Physica A 325 (2003) 62 – 68 0.015 coupled particles single particle
0.013 0.011 0.009
0.007 0.005 0.003 0.001 -0.001 -0.003 -0.005
0
0.02
0.04
0.06
0.08
0.1
0.12
Fext Fig. 2. Load Fext vs. e3ciency at D = 0:1 and k = 4:0. The e3ciency has a maximum at Fext 0:06 for the coupled system, and it is less than 0 for Fext ¿ 0:11, which means that the model moves in −x direction. E3ciency for a single particle is lower than that for the coupled model.
4. Two-dimensional model In this section, we investigate full two-dimensional dynamics of our model. First, we investigate the distribution of an angle between the velocities of coupled particles at two diAerent times, s and s + T . This angle (0◦ 6 6 180◦ ) is deJned as follows: −1 v(s + T ) · v(s) : (4) = cos v(s + T )v(s) In our simulation, T is set to be 500. We calculate the velocity vector of the coupled particles at time s, v(s), by measuring the diAerence between the coordinates of the center of mass of the particles (X; Y ) at s + T and s. That is, v(s) and its amplitude v(s) are deJned as X (s + T ) − X (s) Y (s + T ) − Y (s) v(s) = ; v(s) = |v(s)| : (5) ; T T Fig. 3 shows the distribution of for N = 20 and 80, respectively. The longer the actin Jlament of our model becomes, the more clearly the distribution concentrates near angle 0. That is, for a large number of the particles, they move stably in the almost same direction. This result is coincident with that of the biological experiment quite well [11]. Finally, we investigate the velocity v as a function of the natural length of the spring, a (Fig. 4). In biological experiment [12], it is found that if the density of myosin molecules spread on a cover glass is increased, the velocity of the actin becomes faster in some degree and saturates for higher densities. We reproduce this result by
A. Igarashi et al. / Physica A 325 (2003) 62 – 68
100 number of events
number of events
50 40 30 20 10 0
67
0
30
60
80 60 40 20 0
90 120 150 180
0
30
60
90 120 150 180
Fig. 3. Distribution of for k = 4:0. N = 20 for the left Jgure and N = 80 for the right Jgure. The more the number of the particles becomes, the more stably they move in the almost same direction.
0.02
v
0.015
0.01
k = A = 10 k = A = 12
0.005
k = A = 14 0
0
4
8
a
12
16
Fig. 4. Velocity as a function of the natural length of the springs, a, for N × a (which is the total length of the coupled particles) = 80. Three lines in this Jgure correspond to the results of the diAerent values of k = A (=10:0, 12.0 and 14.0).
changing a and Jxing the total length of the particles in our simulation. As the natural length of the spring, a, becomes shorter (which means that more particles exist per unit length), a smaller number of wells of the interaction potential is expected to exist per a spring. Therefore, for a small value of a, our model is regarded as in a state of a low density of myosin molecules on the cover glass in the biological experiment [12]. Regardless of the elasticity, k and A, of the model, our results show the similar feature to those in the biological experiment.
68
A. Igarashi et al. / Physica A 325 (2003) 62 – 68
5. Conclusion In this paper, we investigate elastically coupled 2ashing ratchet model with the use of computer simulation. In one-dimensional model, the velocity and the e3ciency of energy conversion are obtained. It is found that the velocity of the particles has a maximum at a certain value of the spring constant. The eAects of the coupling between particles on the e3ciency of energy conversion are also found to be very important. The elastically coupled particles can pull heavier load per particle than a single particle, and the e3ciency per particle for the coupled model is larger than that for the single-particle model. In two-dimensional model, we Jnd that when N is relatively large, the coupled particles in the rotated potential move smoothly in a particular direction, and that the experimental results can be reproduced [11] such that the distribution of moving direction is concentrated to a certain direction as shown in Fig. 3. We also reproduce qualitatively the results for the dependence of velocity on the density of myosin molecules in a motility assay, that is, for higher density, the velocity becomes faster in some degree and saturated for higher densities. References [1] R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. I, Addison-Wesley, Reading, MA, 1966 (Chapter 46). [2] R.D. Astumian, Science 276 (1997) 917, and references cited therein. [3] P. Reimann, Phys. Rep. 361 (2002) 57, and references cited therein. [4] F. JCulicher, A. Ajdari, J. Prost, Rev. Mod. Phys. 69 (1997) 1269, and references cited therein. [5] Z. CsahIok, F. Family, T. Vicsek, Phys. Rev. E 55 (1997) 5179. [6] A. Igarashi, S. Tsukamoto, H. Goko, Phys. Rev. E 64 (2001) 051908. [7] P. Reimann, R. Kawai, C. Van Den Broeck, P. HCanggi, Europhys. Lett. 45 (1999) 545. [8] T. Yanagida, S. Esaki, A. Hikikoshi Iwane, Y. Inoue, A. Ishijima, K. Kitamura, H. Tanaka, M. Tokunaga, Philos. Trans. Ser. B 355 (2000) 441. [9] L. Stryer, Biochemistry, 4th Edition, W.H. Freeman, New York, 1995 (Chapter 15). [10] S.J. Kron, J.A. Spudich, Proc. Natl. Acad. Sci. USA 83 (1986) 6272. [11] Y. Shikata, A. Shikata, R. Shimo, H. Takada, C. Kato, M. Ito, T. Oda, K. Mihashi, Proc. Jpn. Acad. Ser. B 70 (1994) 117. [12] T.Q.P. Uyeda, S.J. Kron, J.A. Spudich, J. Mol. Biol. 214 (1990) 699. [13] K. Sekimoto, J. Phys. Soc. Jpn. 66 (1997) 1234.
Physica A 325 (2003) 62 – 68
www.elsevier.com/locate/physa
One- and two-dimensional dynamics of elastically coupled Brownian motors Akito Igarashia;∗ , Hiromichi Gokoa , Shinji Tsukamotob a Department
of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan b Sumitomo Electric Industries, Osaka 541-0041, Japan Received 24 October 2002
Abstract Brownian motors, in particular, “ratchet” models have been studied actively as models for molecular motors. In this paper, a 2ashing ratchet model for elastically coupled particles in an asymmetric periodic potential is introduced and studied as a model for molecular motors in muscle. In our one-dimensional model, velocity of motors and e3ciency of energy conversion are investigated, and in our two-dimensional model we investigate the dynamics of the model by computer simulation and reproduce qualitatively the results of biological experiments in a motility assay. c 2003 Elsevier Science B.V. All rights reserved. PACS: 05.40.Jc; 02.50.Ey; 87.15.Aa Keywords: Brownian motor; Coupled ratchet; Molecular motor
1. Introduction Brownian motors, or “ratchet” [1] models, have attracted much attention as models for molecular motors [2,3]. Without any directed force these models perform transport, where only thermal noise and a proper asymmetric potential are enough to produce the macroscopic motion of the particles toward a particular direction. Coupled Brownian motors have also been studied from the interest to investigate the eAects of mutual interaction among motors. For instance, JCulicher et al. [4] studied coupled “2ashing” ratchets, where the particles feel two kinds of potentials alternatively. On the other ∗
Corresponding author. Fax: +81-75-753-4919. E-mail address: [email protected] (A. Igarashi).
c 2003 Elsevier Science B.V. All rights reserved. 0378-4371/03/$ - see front matter doi:10.1016/S0378-4371(03)00184-5
A. Igarashi et al. / Physica A 325 (2003) 62 – 68
63
hand, CsahIok et al. [5] studied a coupled “rocking” ratchet model, where a periodic external force is applied to the system. Reimann et al. [7] introduced another model of interacting Brownian particles and found some new collective phenomena. The authors have been studying the e3ciency of the coupled “2ashing” ratchet model [6], and so on. In this paper, we pay special attention to acto-myosin motors for muscle contraction, where two kinds of Jlaments, mainly consisting of actin and myosin molecules, respectively, play an important role [8,9]. These Jlaments slide past each other using energy of ATP hydrolysis and muscle contraction results from it. If we regard the particles as myosin molecules and the interaction potential as the one between actin and myosin molecules in the ratchet models, they can be investigated as models for acto-myosin motors. In muscle contraction, however, actin molecules do not work independently but work correlatively with neighbor ones; therefore, it is natural to take into account mutual interaction among particles in coupled ratchets as models for acto-myosin motors. First, we investigate the coupled model in one-dimensional space, where the particles move only in one direction. This is the actual situation of muscle contraction. The velocity of motors and the e3ciency of energy conversion are obtained from our computer simulation in this case. Next, we extend the ratchet model to a two-dimensional space, in order to explain biological experiments in a motility assay [10–12], where, if actin Jlaments are put on a cover glass coated with myosin proteins and ATP is added in this system, they start to move because they interact with the myosin proteins with the use of the energy produced by ATP hydrolysis. Among properties of this system, we shed light mainly on two points. One is a manner for the actin Jlaments to move. It is reported that they advance zig-zag, swaying its body just like a snake. We try to reproduce qualitatively the distribution of the change of their advancing direction [11]. The other is that the longer the distance between two nearest-neighbor motors (which corresponds to a higher density of myosin molecules) is, the larger the velocity of the Jlament becomes as in the experiment [12].
2. The model Our model consists of elastically coupled particles which move under the in2uence of a two-dimensional potential. The elastically coupled particles can be regarded as a model of an actin Jlament moving on the myosin molecules. They feel stochastically either a 2at potential (W1 ) or an asymmetric two-dimensional periodic potential (W2 ). In muscle contraction an actin molecule is attached to or detached from a myosin molecule in ATP hydrolysis repeatedly. When the particle is in the attached state, it feels the potential W2 . W1 is a 2at potential for the state where the actin and the myosin are detached, and no force is exerted between them. It is also assumed that the particles are put in heat bath represented by a white noise. If we consider the system of N particles, the equation of motion of the ith (1 6 i 6 N ) particle whose position
64
A. Igarashi et al. / Physica A 325 (2003) 62 – 68
is xi is
9W2 (xi ) √ + 2Di (t) ; (1) 9xi where we consider an overdumped case for a friction constant . D stands for temperature and the components of i (t) denote mutually independent white noises of zero mean and correlations i (t)t j (s) = ij (t − s)E, where E stands for a unit matrix. fi is the elastic force exerted on the ith particle, where the longitudinal and lateral spring constant are k and A for the natural length of the springs a. In these equations, all parameters are dimensionless, and we set U , D and to be 1:0, 0:2 and 1:0, respectively, in our simulation. a is set to be 1:0 if not mentioned. The boundary conditions of particles are set to be free. With the factor hi (t) of the second term of Eq. (1), we make the particles alternatively feel an asymmetric potential W2 and a 2at potential W1 . We set hi (t) as a colored random modulation that rules the time-dependent change expected 0 or 1 with the correlation time . Therefore, the particles are subjected to the 2at potential W1 if hi (t) is 0 and to W2 if hi (t) is 1. Since the correlation time of hi (t) is , the potential felt by a particle changes stochastically with the correlation time . In our simulation, is set to be 1.0. Thereby, we usually change only three parameters in our simulation, that is, k, A and N . Only in Section 4, we also change the value of a. Next, we consider the asymmetric two-dimensional periodic potential W2 expressing the interaction between actin and myosin Jlaments. Since actin Jlaments are polar, these Jlaments have asymmetry, in other words they have particular directions to which they easily move. In order to take into account this fact in our model, we give a particular direction to each particle corresponding to a part of an actin Jlament which interacts with a myosin. The direction of a particle is decided from the position of the neighboring two particles. That is, the direction of the ith particle is decided to be that of the straight line which links the neighboring two particles, from the (i − 1)th to the (i + 1)th particle. The two particles located at the end of the actin Jlament have only a single neighboring particle; therefore, we deJne its direction as that of the lines which links itself to the neighboring one. By this method we can assign a “natural” direction to each particle, since roughly speaking, each direction deJned above is approximately the direction of the tangential line of the Jlament, in which the actin Jlament is thought to have asymmetry in space and to move easily. In order to determine this potential, Jrst, for simplicity, we consider a twodimensional periodic potential 1 U W0 (x) = sin(x) + sin(2x) + sin(y) ; (2) 4 x˙i = fi − hi (t)
where x = (x; y) and U represents the depth of the potential well. This potential has asymmetry only in the x direction and symmetry in the y direction. As for W2 of each particle, we rotate W0 to make the asymmetric direction of the potential (the x direction) coincident with that of each particle. Although one may think that we should introduce the direction of myosin proteins as well in our model, they, in the biological experiments [10–12], with which a cover glass is coated are located in random directions, that is, they are regarded as having
A. Igarashi et al. / Physica A 325 (2003) 62 – 68
65
no particular directions on average. Therefore, we do not consider the directions of myosin molecules in our model.
3. One-dimensional model In this section, we restrict the particle motion only in the x direction and calculate the velocity and e3ciency with computer simulation. Fig. 1 shows the velocity v vs. k at D = 0; 0:1; 0:2; 0:3, and has a maximum at k 4:0 for D = 0. The velocity decreases as the temperature increases. For D = 0:3 the velocity is almost independent of the spring constant k. The in2uence of the springs is weak in comparison with that of thermal diAusion for high temperature D = 0:3. Sekimoto [13] deJned the e3ciency for thermal ratchet models with a load Fext . If we deJne v as an average velocity of the model, the expression of the e3ciency is =
Fext v ; Pin
(3)
where Pin is the power input in a unit time [13]. We show in Fig. 2 the e3ciency deJned by Eq. (3). It has a maximum as a function of Fext and at a certain threshold it has a value less than 0, which means that the model moves in the −x direction. The e3ciency for a single particle is also shown in Fig. 2, and is much lower than that for the coupled model. Muscle is known to have much more e3ciency than in our simulation. From this point, our model do not succeed to reproduce the real situation. This is mainly because of the simplicity of the model. τ = 1.0, U = 1.0, N = 20
0.08
D D D D
0.06
=0 = 0.1 = 0.2 = 0.3
v
0.04
0.02
0
-0.02
0
10
20
30
40
k Fig. 1. v vs. k. The lower the temperature is, the faster the particles move in general. For high temperature, the peak almost disappears since the particles tend to move more randomly.
66
A. Igarashi et al. / Physica A 325 (2003) 62 – 68 0.015 coupled particles single particle
0.013 0.011 0.009
0.007 0.005 0.003 0.001 -0.001 -0.003 -0.005
0
0.02
0.04
0.06
0.08
0.1
0.12
Fext Fig. 2. Load Fext vs. e3ciency at D = 0:1 and k = 4:0. The e3ciency has a maximum at Fext 0:06 for the coupled system, and it is less than 0 for Fext ¿ 0:11, which means that the model moves in −x direction. E3ciency for a single particle is lower than that for the coupled model.
4. Two-dimensional model In this section, we investigate full two-dimensional dynamics of our model. First, we investigate the distribution of an angle between the velocities of coupled particles at two diAerent times, s and s + T . This angle (0◦ 6 6 180◦ ) is deJned as follows: −1 v(s + T ) · v(s) : (4) = cos v(s + T )v(s) In our simulation, T is set to be 500. We calculate the velocity vector of the coupled particles at time s, v(s), by measuring the diAerence between the coordinates of the center of mass of the particles (X; Y ) at s + T and s. That is, v(s) and its amplitude v(s) are deJned as X (s + T ) − X (s) Y (s + T ) − Y (s) v(s) = ; v(s) = |v(s)| : (5) ; T T Fig. 3 shows the distribution of for N = 20 and 80, respectively. The longer the actin Jlament of our model becomes, the more clearly the distribution concentrates near angle 0. That is, for a large number of the particles, they move stably in the almost same direction. This result is coincident with that of the biological experiment quite well [11]. Finally, we investigate the velocity v as a function of the natural length of the spring, a (Fig. 4). In biological experiment [12], it is found that if the density of myosin molecules spread on a cover glass is increased, the velocity of the actin becomes faster in some degree and saturates for higher densities. We reproduce this result by
A. Igarashi et al. / Physica A 325 (2003) 62 – 68
100 number of events
number of events
50 40 30 20 10 0
67
0
30
60
80 60 40 20 0
90 120 150 180
0
30
60
90 120 150 180
Fig. 3. Distribution of for k = 4:0. N = 20 for the left Jgure and N = 80 for the right Jgure. The more the number of the particles becomes, the more stably they move in the almost same direction.
0.02
v
0.015
0.01
k = A = 10 k = A = 12
0.005
k = A = 14 0
0
4
8
a
12
16
Fig. 4. Velocity as a function of the natural length of the springs, a, for N × a (which is the total length of the coupled particles) = 80. Three lines in this Jgure correspond to the results of the diAerent values of k = A (=10:0, 12.0 and 14.0).
changing a and Jxing the total length of the particles in our simulation. As the natural length of the spring, a, becomes shorter (which means that more particles exist per unit length), a smaller number of wells of the interaction potential is expected to exist per a spring. Therefore, for a small value of a, our model is regarded as in a state of a low density of myosin molecules on the cover glass in the biological experiment [12]. Regardless of the elasticity, k and A, of the model, our results show the similar feature to those in the biological experiment.
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A. Igarashi et al. / Physica A 325 (2003) 62 – 68
5. Conclusion In this paper, we investigate elastically coupled 2ashing ratchet model with the use of computer simulation. In one-dimensional model, the velocity and the e3ciency of energy conversion are obtained. It is found that the velocity of the particles has a maximum at a certain value of the spring constant. The eAects of the coupling between particles on the e3ciency of energy conversion are also found to be very important. The elastically coupled particles can pull heavier load per particle than a single particle, and the e3ciency per particle for the coupled model is larger than that for the single-particle model. In two-dimensional model, we Jnd that when N is relatively large, the coupled particles in the rotated potential move smoothly in a particular direction, and that the experimental results can be reproduced [11] such that the distribution of moving direction is concentrated to a certain direction as shown in Fig. 3. We also reproduce qualitatively the results for the dependence of velocity on the density of myosin molecules in a motility assay, that is, for higher density, the velocity becomes faster in some degree and saturated for higher densities. References [1] R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. I, Addison-Wesley, Reading, MA, 1966 (Chapter 46). [2] R.D. Astumian, Science 276 (1997) 917, and references cited therein. [3] P. Reimann, Phys. Rep. 361 (2002) 57, and references cited therein. [4] F. JCulicher, A. Ajdari, J. Prost, Rev. Mod. Phys. 69 (1997) 1269, and references cited therein. [5] Z. CsahIok, F. Family, T. Vicsek, Phys. Rev. E 55 (1997) 5179. [6] A. Igarashi, S. Tsukamoto, H. Goko, Phys. Rev. E 64 (2001) 051908. [7] P. Reimann, R. Kawai, C. Van Den Broeck, P. HCanggi, Europhys. Lett. 45 (1999) 545. [8] T. Yanagida, S. Esaki, A. Hikikoshi Iwane, Y. Inoue, A. Ishijima, K. Kitamura, H. Tanaka, M. Tokunaga, Philos. Trans. Ser. B 355 (2000) 441. [9] L. Stryer, Biochemistry, 4th Edition, W.H. Freeman, New York, 1995 (Chapter 15). [10] S.J. Kron, J.A. Spudich, Proc. Natl. Acad. Sci. USA 83 (1986) 6272. [11] Y. Shikata, A. Shikata, R. Shimo, H. Takada, C. Kato, M. Ito, T. Oda, K. Mihashi, Proc. Jpn. Acad. Ser. B 70 (1994) 117. [12] T.Q.P. Uyeda, S.J. Kron, J.A. Spudich, J. Mol. Biol. 214 (1990) 699. [13] K. Sekimoto, J. Phys. Soc. Jpn. 66 (1997) 1234.