In the present paper, we study nonlinear dynamics of microtubules (MTs). As an analytical method, we use semi-discrete approximation and show that localized modulated solitonic waves move along MT. This is supported by numerical analysis. Both cases with and without viscosity effects are studied.

1.
S.
Zdravković
,
M. V.
Satarić
,
A.
Maluckov
, and
A.
Balaž
, “
A nonlinear model of the dynamics of radial dislocations in microtubules
,”
Appl. Math. Comput.
237
,
227
237
(
2014
).
2.
P.
Drabik
,
S.
Gusarov
, and
A.
Kovalenko
,
Biophys. J.
92
,
394
403
(
2007
).
3.
E.
Nogales
,
M.
Whittaker
,
R. A.
Milligan
, and
K. H.
Downing
,
Cell
96
,
79
88
(
1999
).
4.
M.
Remoissenet
, “
Low-amplitude breather and envelope solitons in quasi-one-dimensional physical models
,”
Phys. Rev. B
33
,
2386
2392
(
1986
).
5.
R. K.
Dodd
,
J. C.
Eilbeck
,
J. D.
Gibbon
, and
H. C.
Morris
,
Solitons and Nonlinear Wave Equations
(
Academic Press, Inc.
,
London
,
1982
).
6.
T.
Kawahara
, “
The derivative-expansion method and nonlinear dispersive waves
,”
J. Phys. Soc. Jpn.
35
,
1537
1544
(
1973
).
7.
M.
Remoissenet
and
M.
Peyrard
, “
Soliton dynamics in new models with parameterized periodic double-well and asymmetric substrate potentials
,”
Phys. Rev. B
29
,
3153
3166
(
1984
).
8.
S.
Zdravković
, “
Helicoidal Peyrard-Bishop model of DNA dynamics
,”
J. Nonlinear Math. Phys.
18
(
2
),
463
484
(
2011
).
9.
V. E.
Zakharov
and
A. B.
Shabat
, “
Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media
,”
Sov. Phys. JETP
34
,
62
69
(
1972
).
10.
A. C.
Scott
,
F. Y. F.
Chu
, and
D. W.
McLaughlin
, “
The soliton: A new concept in applied science
,”
Proc. IEEE
61
,
1443
1483
(
1973
).
11.
T.
Dauxois
, “
Dynamics of breather modes in a nonlinear “helicoidal” model of DNA
,”
Phys. Lett. A
159
,
390
395
(
1991
).
12.
T.
Dauxois
and
M.
Peyrard
,
Physics of Solitons
(
Cambridge University Press
,
Cambridge, UK
,
2006
).
13.
R.
Kohl
,
A.
Biswas
,
D.
Milovic
, and
E.
Zerrad
, “
Optical soliton perturbation in a non-Kerr law media
,”
Opt. Laser Tech.
40
,
647
662
(
2008
).
14.
A.
Biswas
and
S.
Konar
,
Introduction to Non-Kerr Law Optical Solitons
(
CRC Press, Boca Raton
,
FL, USA
,
2006
).
15.
S.
Zdravković
and
M. V.
Satarić
, “
Nonlinear Schrödinger equation and DNA dynamics
,”
Phys. Lett. A
373
,
126
132
(
2008
).
16.
S.
Zdravković
and
M. V.
Satarić
, “
Single molecule unzippering experiments on DNA and Peyrard-Bishop-Dauxois model
,”
Phys. Rev. E
73
,
021905
(
2006
).
17.
D.
Havelka
,
M.
Cifra
,
O.
Kučera
,
J.
Pokorný
, and
J.
Vrba
,
J. Theor. Biol.
286
,
31
40
(
2011
).
18.
D.
Havelka
,
M.
Cifra
, and
J.
Vrba
, “
What is more important for radiated power from cells—Size or geometry?
,”
J. Phys.: Conf. Series
329
,
012014
(
2011
).
19.
M. V.
Satarić
,
J. A.
Tuszyński
, and
R. B.
Žakula
, “
Kinklike excitations as an energy-transfer mechanism in microtubules
,”
Phys. Rev. E
48
,
589
597
(
1993
).
20.
J.
Pokorný
,
F.
Jelinek
,
V.
Trkal
,
I.
Lamprecht
, and
R.
Hölzel
, “
Vibrations in microtubules
,”
Astrophys. Space Sci.
23
,
171
179
(
1997
).
21.
J. E.
Schoutens
,
J. Biol. Phys.
31
,
35
55
(
2005
).
22.
S.
Zdravković
,
M. V.
Satarić
, and
S.
Zeković
, “
Nonlinear dynamics of microtubules—A longitudinal model
,”
Europhys. Lett.
102
,
38002
(
2013
).
23.
T.
Das
and
S.
Chakraborty
, “
A generalized Langevin formalism of complete DNA melting transition
,”
Europhys. Lett.
83
,
48003
(
2008
).
24.
C. B.
Tabi
,
A.
Mohamadou
, and
T. C.
Kofané
, “
Modulated wave packets in DNA and impact of viscosity
,”
Chin. Phys. Lett.
26
,
068703
(
2009
).
25.
K. R.
Foster
and
J. W.
Baish
, “
Viscous damping of vibrations in microtubules
,”
J. Biol. Phys.
26
,
255
260
(
2000
).
26.
D. J.
Bakewell
,
I.
Ermolina
,
H.
Morgan
,
J.
Milner
, and
Y.
Feldman
, “
Dielectric relaxation measurements of 12 kbp plasmid DNA
,”
Biochim. Biophys. Acta
1493
,
151
158
(
2000
).
27.
S.
Zdravković
,
A.
Maluckov
,
M.
Đekić
,
S.
Kuzmanović
, and
M. V.
Satarić
, “
Are microtubules discrete or continuum systems?
,”
Appl. Math. Comput.
242
,
353
360
(
2014
).
28.
A. L.
Hodgkin
and
A. F.
Huxley
, “
A quantitative description of membrane current and its application to conduction and excitation in nerve
,”
J. Physiol.
117
,
500
544
(
1952
).
29.
St.
Pnevmatikos
,
N.
Flytzanis
, and
M.
Remoissenet
, “
Soliton dynamics of nonlinear diatomic lattices
,”
Phys. Rev. B
33
,
2308
2321
(
1986
).
You do not currently have access to this content.