Dynamics of the tug-of-war model for cellular transport

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 82, Issue 1, 2010, Pages

  Zhang, Yunxin ^a,b   Fisher, Michael E ^a  

^a University of Maryland   (United States)

^b FUDAN UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

CELLULAR TRANSPORT; DOMAINS OF ATTRACTION; FILAMENTOUS TRACKS; FUNCTION OF TIME; IMPOSED LOADS; LIVING CELL; MODE TRANSITIONS; MODEL PARAMETERS; MODES OF MOTION; MOLECULAR MOTORS; MOTOR PARAMETERS; ORDER 2; PROBABILITY FLUX; STALL FORCE; STATIONARY MODES; SYSTEM STATE;

DAMPING;

EID: 77955137909     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.82.011923     Document Type: Article

References (56)

1. D. Bray, Cell Movements: From Molecules to Motility, 2nd ed. (Garland, New York, 2001).
2. J. Howard, Mechanics of Motor Proteins and the Cytoskeleton (Sinauer Associates and Sunderland, MA, 2001).
3. J. Howard, Annu. Rev. Biophys 38, 217 (2009). 10.1146/annurev.biophys. 050708.133732
4. F. Jülicher and J. Prost, Phys. Rev. Lett. 75, 2618 (1995). 10.1103/PhysRevLett.75.2618
5. F. Jülicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69, 1269 (1997). 10.1103/RevModPhys.69.1269
6. S. Camalet, F. Jülicher, and J. Prost, Phys. Rev. Lett. 82, 1590 (1999). 10.1103/PhysRevLett.82.1590
7. T. J. Mitchison and H. M. Mitchison, Nature (London) 463, 308 (2010). 10.1038/463308a
8. A. B. Kolomeisky and M. E. Fisher, Annu. Rev. Phys. Chem. 58, 675 (2007). 10.1146/annurev.physchem.58.032806.104532
9. T. Elston and G. Oster, Biophys. J. 73, 703 (1997). 10.1016/S0006- 3495(97)78104-9
10. H.-Y. Wang, T. Elston, A. Mogilner, and G. Oster, Biophys. J. 74, 1186 (1998). 10.1016/S0006-3495(98)77834-8
11. A. B. Kolomeisky and B. Widom, J. Stat. Phys. 93, 633 (1998). 10.1023/B:JOSS.0000033246.14231.e1
12. M. E. Fisher and A. B. Kolomeisky, Proc. Natl. Acad. Sci. U.S.A. 96, 6597 (1999). 10.1073/pnas.96.12.6597
13. R. D. Vale, Cell 112, 467 (2003). 10.1016/S0092-8674(03)00111-9
14. D. B. Hill, M. J. Plaza, K. Bonin, and G. Holzwarth, Eur. Biophys. J. 33, 623 (2004). 10.1007/s00249-004-0403-6
15. C. Kural, H. Kim, S. Syed, G. Goshima, V. I. Gelfand, and P. R. Selvin, Science 308, 1469 (2005). 10.1126/science.1108408
16. G. T. Shubeita, S. L. Tran, J. Xu, M. Vershinin, S. Cermelli, S. L. Cotton, M. A. Welte, and S. P. Gross, Cell 135, 1098 (2008). 10.1016/j.cell.2008.10.021
17. N. J. Carter and R. A. Cross, Nature (London) 435, 308 (2005). 10.1038/nature03528
18. S. M. Block, Biophys. J. 92, 2986 (2007). 10.1529/biophysj.106.100677
19. A. Gennerich and R. D. Vale, Curr. Opin. Cell Biol. 21, 59 (2009). 10.1016/j.ceb.2008.12.002
20. M. Badoual, F. Jülicher, and J. Prost, Proc. Natl. Acad. Sci. U.S.A. 99, 6696 (2002). 10.1073/pnas.102692399
21. E. B. Stukalin, H. Phillips, and A. B. Kolomeisky, Phys. Rev. Lett. 94, 238101 (2005). 10.1103/PhysRevLett.94.238101
22. S. Klumpp and R. Lipowsky, Proc. Natl. Acad. Sci. U.S.A. 102, 17284 (2005). 10.1073/pnas.0507363102
23. M. J. I. Müller, S. Klumpp, and R. Lipowsky, Proc. Natl. Acad. Sci. U.S.A. 105, 4609 (2008). 10.1073/pnas.0706825105
24. M. J. I. Müller, S. Klumpp, and R. Lipowsky, J. Stat. Phys. 133, 1059 (2008). 10.1007/s10955-008-9651-7
25. Y. Zhang, Phys. Rev. E 79, 061918 (2009). 10.1103/PhysRevE.79.061918
26. In independent work, cited in, M. J. I. Müller, has also explored the large (N +, N -) or mean-field limit: see M. J. I. Müller, Ph.D. thesis, University of Potsdam, 2008; http://opus.kobv.de/ubp/volltexte/2008/ 1871/. To the degree that her results overlap those reported in, they are fully consistent; but they do not extend to the further results reported here.
27. X. Nan, P. A. Sims, P. Cheng, and X. S. Xie, J. Phys. Chem. B 109, 24220 (2005). 10.1021/jp056360w
28. V. Levi, A. S. Serpinskaya, E. Gratton, and V. Gelfand, Biophys. J. 90, 318 (2006). 10.1529/biophysj.105.067843
29. V. Soppina, A. K. Rai, A. J. Ramaiya, P. Barak, and R. Mallik, Proc. Natl. Acad. Sci. U.S.A. 106, 19381 (2009). 10.1073/pnas.0906524106
30. A. Gennerich and D. Schild, Phys. Biol. 3, 45 (2006). 10.1088/1478-3975/3/1/005
31. A. R. Rogers, J. W. Driver, and M. R. Diehl, Phys. Chem. Chem. Phys. 11, 4882 (2009). 10.1039/b900964g
32. R. Mallik, B. C. Carter, S. A. Lex, S. J. King, and S. P. Gross, Nature (London) 427, 649 (2004). 10.1038/nature02293
33. S. Toba, T. M. Watanabe, L. Yamaguchi-Okimoto, Y. Y. Toyoshima, and H. Higuchi, Proc. Natl. Acad. Sci. U.S.A. 103, 5741 (2006). 10.1073/pnas.0508511103
34. A. Gennerich, A. P. Carter, S. L. Reck-Peterson, and R. D. Vale, Cell 131, 952 (2007). 10.1016/j.cell.2007.10.016
35. I. H. Riedel-Kruse, A. Hilfinger, J. Howard, and F. Jülicher, HFSP J. 1, 192 (2007). 10.2976/1.2773861
36. A. Hilfinger, A. K. Chattopadhyay, and F. Jülicher, Phys. Rev. E 79, 051918 (2009). 10.1103/PhysRevE.79.051918
37. S. W. Grill, K. Kruse, and F. Jülicher, Phys. Rev. Lett. 94, 108104 (2005). 10.1103/PhysRevLett.94.108104
38. J. Pecreaux, J.-C. Röper, K. Kruse, F. Jülicher, A. A. Hyman, S. W. Grill, and J. Howard, Curr. Biol. 16, 2111 (2006). 10.1016/j.cub.2006.09. 030
39. The linear form Eq. was introduced independently by Grill in their treatment of mitotic spindle oscillations; however, the attachment and detachment rates of the force generating motors there, were related to an elastic linker length rather than directly to the load acting.
40. In their initial study, S. Klumpp and R. Lipowsky used the linear force-velocity relation V (F ext) = V F [1 - (F ext / F S)] for 0 ≤ F ext ≤ F S as in Eq. but, in place of Eq. took V (F ext) = 0 for F ext > F S. For assisting loads, i.e., when F ext < 0, they employed Eq..
41. D. Tsygankov and M. E. Fisher, Proc. Natl. Acad. Sci. U.S.A. 104, 19321 (2007) 10.1073/pnas.0709911104
42. D. Tsygankov and M. E. Fisher, J. Chem. Phys. 128, 015102 (2008). 10.1063/1.2803213
43. A. Kunwar and A. Mogilner, Phys. Biol. 7, 016012 (2010). 10.1088/1478-3975/7/1/016012
44. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical System, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).
45. In recent work by Müller (which the authors kindly communicated to us prior to publication) this principal conclusion regarding the exponential dependence of the probability fluxes or transition times is also found. However, as we explain further below, their calculational approach differs significantly from ours and is, we believe, of more limited applicability.
46. M. J. I. Müller, S. Klumpp, and R. Lipowsky, Biophys. J. 98, 2610 (2010). 10.1016/j.bpj.2010.037
47. We comment now that rather than identifying the underlying basins of attraction separating the distinct modes of motion, as we have done, Müller. consider only the overall switch times between uniform plus-end directed motion and the reverse. To that end, they calculate the probability flux across what we have, in Sec., termed the AB boundary in the (y, z) or (n +, n -) flow plane: see again Figs., here the strict linearity of this boundary, defined by Eq., is evident. In some cases the difference between this AB boundary and what we believe is a more appropriate boundary specification, may not be very significant numerically. But in other cases, such as seen in Figs., where there are, indeed, only two stable states, the differences are much greater. While in Fig., where there are three states stable in the mean field limit, it seems evident a priori that the AB boundary represents a less than optimal choice for of a locus across which to compute relevant fluxes.
48. The formulation used here is also employed in but the fits reported in Table provide a concrete numerical picture.
49. Another possible definition, favored in some experimental studies (see, e.g.,), follows the motion of the cargo in a fixed harmonic trap and records what might be termed a "stalling distribution," in which a mean velocity of zero is considered to be observed but under fluctuations in position and, hence, in the force actually imposed by the trap. While this is an experimentally convenient procedure in that it avoids the use of a "force clamp," which, with suitable feedback, imposes a steady load force, it seems somewhat less definitive for characterizing the intrinsic system behavior. In particular, an element of judgment is involved in deciding that, in a specific run, a further forward step and, hence "stalling" under a greater mean load will not eventually be realized as the mean velocity smoothly decreases with load. Conversely, by unequivocally identifying mean backwards motion, as in the experiments of Carter and Cross on kinesin, the value of the stall force (including a clear upper bound) is more convincingly established.
50. A. Kunwar, M. Vershinin, J. Xu, and S. P. Gross, Curr. Biol. 18, 1173 (2008). 10.1016/j.cub.2008.07.027
51. A. Yildiz, M. Tomishige, A. Gennerich, and R. D. Vale, Cell 134, 1030 (2008). 10.1016/j.cell.2008.07.018
52. C. Cho, S. L. Reck-Peterson, and R. D. Vale, J. Biol. Chem. 283, 25839 (2008). 10.1074/jbc.M802951200
53. More recently Lipowsky have extended their calculations utilizing Eq. for superstall loads. They note that the cargo stall force, F C (N), for a single motor species is substantially smaller than N F C (1) = N F S, the sum of the individual stall force, but do not report the significant sublinear variation we find: see Fig. 14.
54. R. Lipowsky, J. Beeg, R. Dimova, S. Klumpp, and M. J. I. Müller, Physica E (Amsterdam) 42, 649 (2010), 10.1016/j.physe.2009.08.010
55. this article also contains a review from the authors' perspective of the tug-of-war model.
56. See supplementary material at http://link.aps.org/supplemental/10.1103/ PhysRevE.82.011923 for describing analysis of stall forces in the tug-of-war models.