Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

48, 1, pp. 219-231, Warsaw 2010

EIGEN MODES OF THE DOUBLE DNA CHAIN HELIX
VIBRATIONS

Katica R. (Stevanović) Hedrih

Mathematical Institute SANU, Belgrade, University of Nǐs, Faculty of Mechanical Engineering,

Serbia; e-mail: katica@masfak.ni.ac.rs; khedrih@eunet.rs; khedrih@sbb.co.rs

Andjelka N. Hedrih

University of Belgrade, Faculty of Technology and Metalurgy, Belgrade, Serbia

e-mail: sandjelka@bankerinter.net

Differentmodels of two coupled homogeneousDNA chain vibrations are pro-
posed in the literature. By using as the basic approach to the DNA mathe-
matical modelling published byN. Kovaleva, L.Manevich in 2005 and 2007,
we consider a linearized model to obtain main chain subsystems of the do-
uble DNA helix. Analytical expressions of the circular eigen frequencies for
the homogeneous model of the double DNA chain helix are obtained. The
corresponding vibration eigen modes and possibilities of the appearance of
resonant regimes as well as dynamical absorption under external excitations
are considered.Two sets of normal eigen coordinates of the doubleDNAcha-
in helix for separation of the system into two uncoupled chains are identified.
This may correspond to the base pair order in complementary chains of the
DNA double helix in a living cell.

Keywords:DNA,eigenmainchains, eigencircular frequency,vibrationmodes

1. Introduction

DNA is a biological polymerwhich can exist in different forms (A,B, Z,E, . . .)
but only B form can be found in live organisms. Chemically, DNA consists of
two long polymers of simple units called nucleotides, with backbones made of
sugars and phosphate groups joined by ester bonds. To each sugar one of four
types ofmolecules called bases is attached. Two bases on opposite strands are
linked via hydrogen bonds holding the two strands of DNA together. It is the
sequence of these four bases along the backbone that encodes information the
mechanical properties of DNA are closely related to its molecular structure



220 K.R. Hedrih, A.N. Hedrih

and sequence, particularly the weakness of hydrogen bonds and electronic
interactions that hold the strands of DNA together compared to the strength
of bonds within each strand. Every process which binds or reads DNA is able
touse ormodify themechanical properties ofDNAfor purposes of recognition,
packaging and modification. It is important to note that, the DNA found in
many cells can be macroscopic in length – a few centimeters long for each
human chromosome. Consequently, cells must compact or ”package” DNA to
carry itwithin them (Bryant et al., 2003; Gore et al., 2006; Volkenstein, 1975).
Knowledge of the elastic properties of DNA is required to understand the

structural dynamics of cellular processes such as replication and transcription.
Binding of proteins and other ligands induces a strong deformation of the

DNA structure.
The aim of ourworkwas tomodel theDNAdynamics (vibrations of DNA

chains) as a biological system in specific boundary conditions that are possible
to occur in a life system during regular function of a DNAmolecule.

2. Mechanical properties of DNA determined experimentally

Experimental evidence suggests that DNA mechanical properties, in particu-
lar intrinsic curvature and flexibility, have a role in many relevant biological
processes.
For small distortions, DNA overwinds under tension (Gore et al., 2006).

Lowering of the temperature does increase the DNA curvature. Curved DNA
sequences migrate more slowly on polyacrylamide gels than their non-curved
counterparts possessing the same length. The anomaly in gel mobility is rela-
ted to the extent of DNA curvature (Tsai and Luo, 2000). The DNA double
helix is much more resistant to twisting deformations than bending deforma-
tions, almost all of the supercoiling pressure is normally relieved by writhing
(Arsuaga et al., 2002). The twist angle of the helix has been shown to depend
on sequence when the molecule is in solution, both by the effects on super-
coiling parameters when short segments of the known sequence are inserted
into closed circular DNA (Peck andWang, 1981; Tung andHarvey, 1984) and
by the nuclease digestion patterns of DNA adsorbed on surfaces (Behe et al.,
1981; Tung and Harvey, 1984).
Asabiomolecule,DNAalsohas electronic properties.WhenDNA isplaced

in vacuum it shares characteristics with semiconductors. In a solution, DNA
transfers electrons via a differentmechanism (Westerhoff andMerz Jr., 2006).
Under low tension, DNA behaves like an isotropic flexible rod. At higher ten-
sions, the behaviour of over- and underwound molecules is different. In each



Eigen modes of the double DNA... 221

Fig. 1. Model of DNA duble helix fromwww.wikipedia.org

case, DNA undergoes a structural change before the twist density necessary
for buckling is reached (Bryant et al., 2003).

Mg2+ can induce or enhance curvature in DNA fragments and helps sta-
bilize several types of DNA structures (Brukner et al., 1994). The fraction of
bentmolecules seen byEMor SFMwas higher in the presence of cationic me-
tals.DNA length varied in solutionwithdifferent ionic forces. It is significantly
longer in solution with a lower ionic force (Frontali et al., 1979).

3. DNA models by Kovaleva and Manevich

Anumber of mechanical models of the DNAdouble helix have been proposed
till today. Different models focuse on different aspects of the DNA molecule
(biological, physical and chemical processes in which DNA is involved). They
show that in a double DNA helix, a localised excitation (breather) can exist,
which corresponds to predominant rotation of one chain and small perturba-
tion of the second chain using the coarse-grained model of the DNA double
helix. Each nucleotide is represented by three beads with interaction sites
corresponding to the phosphate group, group of sugar ring, and the base (Ko-
valeva et al., 2007).

Kovaleva et al. (2007) pointed out that solitons and breathers play a func-
tional role in DNA chains. In the model, the DNA backbone is reduced to a
polymeric structure and the base is covalently linked to the center of the su-
gar ring group, thus a DNAmolecule with N nucleotides corresponds to 3N



222 K.R. Hedrih, A.N. Hedrih

Fig. 2. (a) ”Toymechanical” model of DNA. a, DNA is modeled as an elastic rod
(grey) wrapped helically by a stiff wire (red) (Gore et al., 2006); (b) model scheme
of a double helix on six coarse-grained particles (Kovaleva et al., 2007); (c) fragment
of the DNA double chain consisting of three AT base pairs. Longitudinal pitch of
the helix a=3.4Å, transverse pitch h=16.15Å (Kovaleva andManevich, 2005)

interaction centers. Apart from its well-known role as the cellular storehouse
of information, DNA is now being used to construct rigid scaffolds in one,
two and three dimensions on the nanoscale. This field is termed Structural
DNA Nanotechnology. It seeks to use the base complementarily design prin-
ciple of DNA to create ordered superstructures from a set of DNA sequences
that selfassemble into regular, well-defined topologies on the nanoscale (An-
selmi et al., 2005). Starting from a coarse-grained off-lattice model of DNA
and using cylindrical coordinates, the authors derived simplified continuum
equations corresponding to vicinities of gap frequencies in the spectrum of
linearised equations of motion. It is shown that the obtained nonlinear con-
tinuum equations describing modulations of normal modes admit spatially
localised solitons which can be identified with breathers. The authors formu-
lated conditions of the breathers existence and estimated their characteristic
parameters. The relationship between the derivedmodel andmore simple but



Eigen modes of the double DNA... 223

widely usedmodels is discussed. The analytical results are comparedwith the
data of numerical study of discrete equations of motion (see Fig.2b).
Kovaleva and Manevich (2005) presented the simplest model describing

opening of the DNA double helix. Corresponding differential equations are
solved analytically usingmultiple-scale expansions after transition to complex
variables. The obtained solution corresponds to localised torsional nonlinear
excitation – breather. Stability of the breather is also investigated.
In their work, Kovaleva and Manevich (2005) considered B form of the

DNA molecule, the fragment of which is presented in Fig.1b. The lines in
the figure correspond to the skeleton of the double helix, black and grey rec-
tangles show the bases in pairs (AT and GC). Let us focus our attention on
rotationalmotions of thebases around the sugar phosphate chains in the plane
perpendicular to the helix axis.
The authors deal with the planar DNA model in which the chains of the

macromolecule formtwoparallel straight linesplacedat adistance h fromeach
other, and the bases can make only rotary motions around their own chain,
being all the time perpendicular to it. The authors accepted as generalized
(independent) coordinates ϕk,1 the angular displacement of the k-th base of
the first chain, and as generalized (independent) coordinates ϕk,2 the angular
displacement of the k-th base of the second chain.Then, byusing the accepted
generalized coordinates ϕk,1 and ϕk,2 for k-thbases ofboth chains in theDNA
model, the authors derived a system of differential equations describing DNA
model vibrations in the following forms

Jk,1ϕ̈k,1−
Kk,1

2
[sin(ϕk+1,1−ϕk,1)− sin(ϕk,1−ϕk−1,1)]+

+Kαβrα(rα−rβ)sinϕk,1+

−Kαβ
1

4

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2 sin(ϕk,1−ϕk,2)= 0

(3.1)

Jk,2ϕ̈k,2−
Kk,2

2
[sin(ϕk+1,2−ϕk,2)− sin(ϕk,2−ϕk−1,2)]+

+Kαβrα(rα−rβ)sinϕk,2+

+Kαβ
1

4

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2 sin(ϕk,1−ϕk,2)= 0

where Jk,1 is the axial mass moment of inertia of the k-th base of the first
chain; Jk,2 is the axial mass moment of mass inertia of the k-th base of the
second chain, and the dot denotes differentiation with respect to time t. For
the base pair, the axial mass moments of inertia are equal to Jk,1 = mαr

2
α,

Jk,12 = mβr
2
β
. The base mass mα, length rα, and the corresponding axial



224 K.R. Hedrih, A.N. Hedrih

mass moment of inertia Jk,1 =mαr
2
α for all possible base pairs was assumed

as in Zhang et al. (1994). The fourth terms in the previous systemof equations
describe interaction of theneighbouringbases along each of themacromolecule
chains. Theparameter Kk,i (i=1,2) characterises the energy of interaction of
the k-th base with the (k+1)-th one along the i-th chain i=1,2. There are
different estimations of rigidity. For calculations, the most appropriate value
is close Kk,i =K =6 ·10

3kJ/mol.

4. Consideration of the basic DNA model – linearised
Kovaleva-Manevich’s DNA model

Let us investigate an oscillatory model of DNA, considered by Kovaleva and
Manevich (2005) and presented in the previous Section by a system of dif-
ferential equations (3.1) expressed by generalized (independent) coordinates
ϕk,1 and ϕk,2 for k-th bases of both chains in the DNAmodel.

For the beginning, it is necessary to consider the corresponding linearised
system of previous differential equations in the following form

Jk,1ϕ̈k,1−
Kk,1

2
[(ϕk+1,1−ϕk,1)− (ϕk,1−ϕk−1,1)]+Kαβrα(rα−rβ)ϕk,1+

−Kαβ
1

4

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2(ϕk,1−ϕk,2)= 0

(4.1)

Jk,2ϕ̈k,2−
Kk,2

2
[(ϕk+1,2−ϕk,2)− (ϕk,2−ϕk−1,2)]+Kαβrα(rα−rβ)ϕk,2+

+Kαβ
1

4

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2(ϕk,1−ϕk,2)= 0

or in that form

2Jk,1
Kk,1
ϕ̈k,1− [(ϕk+1,1−ϕk,1)− (ϕk,1−ϕk−1,1)]+

2Kαβrα(rα−rβ)

Kk,1
ϕk,1+

−

Kαβ

2Kk,1

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2(ϕk,1−ϕk,2)= 0

(4.2)

2Jk,2
Kk,2
ϕ̈k,2− [(ϕk+1,2−ϕk,2)− (ϕk,2−ϕk−1,2)]+

2Kαβrα(rα−rβ)

Kk,2
ϕk,2+

+
Kαβ

2Kk,2

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2(ϕk,1−ϕk,2)= 0



Eigen modes of the double DNA... 225

For the case of homogeneous systems we can take into consideration that
Jk,1 = Jk,2 = J and Kk,1 =Kk,2 =K.

By changing the generalized coordinates ϕk,1 and ϕk,2 for k-th bases of
both chains in theDNAmodel into the following new ones ξk and ηk through
the relationships

ξk =ϕk,1−ϕk,2 ηk =ϕk,1+ϕk,2 (4.3)

The previous system of differential equations (4.2) acquires the following form

2J

K
ξ̈k− ξk+1+2ξk

[
1+
Kαβrα(rα−rβ)

K
−

Kαβ

2K

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2
]
+

−ξk−1 =0 (4.4)

2J

K
η̈k−ηk+1+2ηk

(
1+
Kαβrα(rα−rβ)

K

)
−ηk−1 =0

where k=1,2, . . . ,n.

The first series of the previous system of equations is decoupled and in-
dependent with relations of the second series of the equations. Then we can
conclude that the new coordinates of ξk and ηk are main coordinates of DNA

chains and that we obtain two fictive decoupled single eigen chains of the DNA

liner model. This is the first fundamental conclusion as an important property

of the linear model of vibrations into double DNA helix.

Systems of differential equations (4.2) contain two separate subsystems of
differential equations expressed by the coordinates of ξk and ηk which are
main coordinates of the double DNA chain helix and separate the linear DNA
model into two independent chains. Then, it is possible to apply the trigono-
metricmethod (Rašković, 1965, 1985; Hedrih, 2006, 2008a,b) to both series of
equations (both subsystems) in the form (k=1,2, . . . ,n)

ξk =Ak cos(ωt+α)=C sinkϕcos(ωt+α)
(4.5)

ηk = Ãk cos(ωt+α)=Dsinkϑcos(ωt+α)

where

Ak =C sinkϕ Ãk =Dsinkϑ (4.6)

are amplitudes of separate eigen chains of themodel of the double DNA chain
helix, and ω circular eigen frequency of one vibration mode.

After introducing the proposed solutions into differential equations of pre-
vious separate subsystems (4.4), we obtain the following separate subsystems



226 K.R. Hedrih, A.N. Hedrih

of algebraic equations with respect to the amplitudes Ak and Ãk

−Ak+1+2Ak
{[
1+
Kαβrα(rα−rβ)

K
−

Kαβ

2K

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2
]
+

−

J

K
ω2
}
−Ak−1 =0 (4.7)

−Ãk+1+2Ãk
(
1+
Kαβrα(rα−rβ)

K
−

2J

K
ω2
)
− Ãk−1 =0

After applying the following denotations

µ−κ=
Kαβrα(rα−rβ)

K
−

Kαβ

2K

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2

κ=
Kαβ

2K

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2 (4.8)

µ=
Kαβrα(rα−rβ)

K
u=
J

K
ω2

we obtain the following simple forms of subsystems (4.7)

−Ak+1+2Ak(1+µ−κ−u)−Ak−1 =0
(4.9)

−Ãk+1+2Ãk(1+µ−u)− Ãk−1 =0

After introducing proposed solutions (4.6), the trigonometric method is ap-
plied and we obtain two equations

C sinkϕ[−2cosϕ+2(1+µ−κ−u)] = 0
(4.10)

Dsinkϑ[−2cosϑ+2(1+µ−u)] = 0

From the previous system, we obtain the following eigen numbers for both
separate eigen chains of themodel of the double DNA chain helix in the follo-
wing forms

u=2sin2
ϕ

2
+(µ−κ) u=2sin2

ϑ

2
+µ (4.11)

and the corresponding analytical expressions for circular eigen frequencies of
vibration modes of separate chains ω2

ω2s =
K

J

[
2sin2

ϕs

2
+(µ−κ)

]
ω2s =

K

J

[
2sin2

ϑs

2
+µ
]

(4.12)



Eigen modes of the double DNA... 227

5. Boundary conditions of the double DNA chain helix

Now, it is necessary to consider some boundary conditions of the s-th double
DNA chain helix in accordance with possible real situations. For that reason,
we take into account two cases of the double DNA chain helix, in which the
ends of chains are either free or fixed. Then, we can write the following boun-
dary conditions for the double DNA chain helix:
—first case: bothendsof thedoubleDNAchainhelixare free. In that situation
the first and n-th equations from the subsystems are in the form

A1(1+µ−κ−2u)−A2 =0 −An−1+An(1+µ−κ−2u)= 0 (5.1)

and

Ã1(1+µ−κ−2u)− Ã2 =0 − Ãn−1+ Ãn(1+µ−2u)= 0 (5.2)

and after applying proposed solutions (4.6) we obtain that

ϕs =
sπ

n
ϑs =

sπ

n
s=1,2, . . . ,n (5.3)

— second case: both ends of the double DNA chain helix are fixed

Ak =C sinkϕ A0 =An+1 =0 Am+1 =C sin(n+1)ϕ=0

Ãk =Dsinkϑ Ã0 = Ãn+1 =0 Ãm+1 =Dsin(n+1)ϑ=0

ϕs =
sπ

n+1
ϑs =

sπ

n+1
s=1,2, . . . ,n

(5.4)
Then the analytical expressions for ω2s – circular eigen frequencies of the

vibration modes of separate chains in the double DNA chain helix are

ω2s =
K

J

[
2sin2

ϕs

2
+
Kαβrα(rα−rβ)

K
−

Kαβ

2K

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2
]

(5.5)

ω2s =
K

J

[
2sin2

ϑs

2
+
Kαβrα(rα−rβ)

K

]

—first case: both ends of the double DNA chain helix are free (see Fig.3)

ω2s =
K

J

[
2sin2

sπ

2n
+
Kαβrα(rα−rβ)

K
−

Kαβ

2K

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2
]

(5.6)

ω2s =
K

J

[
2sin2

sπ

2n
+
Kαβrα(rα−rβ)

K

]



228 K.R. Hedrih, A.N. Hedrih

Fig. 3. Double DNA chain helix in form of a multipendulum model with free ends

—second case: both ends of the doubleDNA chain helix are fixed (see Fig.4)

ω2s =
K

J

[
2sin2

sπ

2(n+1)
+
Kαβrα(rα−rβ)

K
−

Kαβ

2K

(
1−
ωαβ2

ωαβ1

)
(rα−rβ)

2
]

(5.7)

ω2s =
K

J

[
2sin2

sπ

2(n+1)
+
Kαβrα(rα−rβ)

K

]

Fig. 4. Double DNA chain helix in form of a multipendulum systemwith fixed ends

6. Concluding remarks

At the end, we can conclude that the new coordinates of ξk and ηk composed
by generalized coordinates in as ξk =ϕk,1−ϕk,2 and ηk =ϕk,1+ϕk,2 aremain
coordinates of the doubleDNAchain helix and that it is possible to obtain two
fictive decoupled and separated single eigen chains of the double DNA chain
helix linear model. This is the first fundamental conclusion as an important
propertyof the linearmodel of vibrations in thedoubleDNAhelix.Considered



Eigen modes of the double DNA... 229

asa linearmechanical system,aDNAmolecule as adoublehelixhas its circular
eigen frequencies and that is its characteristic. Mathematically, it is possible
to decuple it into two chains with their own circular eigen frequencies which
are different. Thismay correspond to different chemical structure (the order of
base pairs) of the complementary chains of DNA.We are free to propose that
every specific set of the base pair order has its circular eigen frequencies, and
it changes when DNA chains are coupled in the system of double helix. DNA
as a double helix in a living cell can be considered as a nonlinear system, but
under certain conditions its behaviour can be described by linear dynamics.

Additionally, analytical expressions for the quadrate of ωs – circular eigen
frequencies of the vibration modes of separate chains of the homogeneous
double DNA chain helix are obtained. By using these results, it is easy to
consider these values as a series of resonant frequencies under external multi-
frequency excitations, and also as the reason for the appearance of dynamical
absorbtion phenomena as well as some explanation of real processes in the
homogeneous doubleDNA chain helix. Next considerations will be focused on
small nonlinearity in thedoubleDNAchainhelix vibrations and rarenonlinear
phenomena such as resonant jumps and energy interactions between nonlinear
modes.

Acknowledgements

A part of this research was supported by the Ministry of Sciences and Techno-
logy of Republic of Serbia through Mathematical Institute SANU Belgrade Grant
ON144002 ”Theoretical and Applied Mechanics of Rigid and Solid Body. Mechanics
ofMaterials” and Faculty of Mechanical Engineering University of Nǐs.

Another part of this researchwas supported by theMinistry of Sciences andTech-

nology ofRepublic of Serbia throughFaculty of Technology andMetalurgy,Belgrade,

Grant ON142075.

References

1. Anselmi C., De Santis P., Scipioni A., 2005, Nanoscale mechanical and
dynamical properties of DNA single molecules,Review Biophysical Chemistry,
113, 209-221

2. Arsuaga J., Tan R.K.-Z., Vazquez M., De Sumners W., Harvey S.C.,
2002, Investigation of viral DNA packaging usingmolecularmechanicsmodels,
Biophysical Chemistry, 101/102, 475-484



230 K.R. Hedrih, A.N. Hedrih

3. Behe M., Zimmerman S., Felsenfeld G., 1981, Changes in the helical
repeatof poly (dG-m5dC)andpoly (dG-dC)associatedwith theB-Ztransition,
Nature, 293, 233-235

4. Brukner, Susic S., Dlakic M., Savic A., Pongor S., 1994, Physiological
concentrations of magnesium ions induce a strong macroscopic curvature in
GGGCCC-containing DNA, J. Mol. Biol., 236, 26-32

5. Bryant Z., Stone M.D., Gore J., Smith S.B., Cozzarelli N.R., Bu-
stamante C., 2003, Structural transitions and elasticity from torque measu-
rements on DNA, Nature, 424, 338-341

6. Frontali C., DoreE., Ferrauto A., GrattonE., Bettini A., Pozzan
M.R., Valdevit E., 1979, An absolute method for the determination of the
persistence length of native DNA from electronmicrographs,Biopolymers, 18,
1353-1357

7. Gore J., Bryant Z., Ilmann M., Le M.-U., Cozzarelli N.R., Busta-
mante C., 2006, DNA overwinds when stretched,Nature, 442, 836-839

8. Hedrih (Stevanović) K., 2006, Modes of the Homogeneous Chain
Dynamics, Signal Processing, Elsevier, 86, 2678-2702, ISSN: 0165-1684,
www.sciencedirect.com/science/journal/01651684

9. Hedrih (Stevanović) K., 2008a, Dynamics of coupled systems, Nonlinear
Analysis: Hybrid Systems, 2, 2, 310-334

10. Hedrih (Stevanović) K., 2008b, Dynamics of multipendulum systems with
fractional order creep elements, Journal of Theoretical and Applied Mechanics,
46, 3, 483-509

11. Kovaleva N., Manevich L., 2005, Localized nonlinear oscillation of DNA
molecule, 8th Conference on Dynamical Systems – Theory and Applications,
Lodz, Poland, 103-110

12. Kovaleva N., Manevich L., Smirnov V., 2007, Analitical study of coarse-
grained model of DNA, 9th Conference on Dynamical Systems – Theory and
Applications, Lodz, Poland

13. Peck L.J., Wang J.C., 1981, Sequence dependence of the helical repeat of
DNA in solution,Nature, 292, 375-378

14. Rašković D., 1965,Teorija oscilacija, Naučna Knjiga, 503

15. Rašković D., 1985,Teorija elastičnosti, Naučna knjiga, 414

16. Tsai L., Luo L., 2000, A statistical mechanical model for predicting B-DNA
curvature and flexibility, J. Theor. Biol., 207, 177-194

17. Tung C.-S., Harvey S.C., 1984, A moecular mechanical model to predict
the heix twist angles of B-DNA,Nucleic Acids Research, 12, 7, 3343-3356



Eigen modes of the double DNA... 231

18. Westerhoff L.M.,Merz K.M. Jr., 2006,Quantummechanical description
of the interactions betweenDNAandwater,Journal ofMolecular Graphics and
Modelling, 24, 440-455

19. Volkenstein M.V., 1975,Biophysics, AIP, NewYork

20. Zhang P., Tobias I., Olson W.K., 1994, Computer simulation of protein-
induced structural changes in closed circularDNA, J.Mol. Biol., 242, 271-290

Postacie własne drgań podwójnego łańcucha helisy DNA

Streszczenie

W literaturze można spotkać opis różnych modeli sprzężonych drgań jednorod-
nego łańcuchaDNA.Wprezentowanej pracy rozważania oparto na zlinearyzowanym
modelu Kovalevej i Manevicha (2005, 2007) do wydzielenia głównych podukładów
łańcuchowych podwójnej helisy DNA. Uzyskano analityczne wyrażenia na częstości
własne jednorodnego modelu helisy i odpowiadające im postacie własne oraz po-
twierdzono możliwość wystąpienia rezonansów i dynamicznej absorpcji drgań przy
obecności wymuszeń zewnętrznym polem sił. Zidentyfikowano dwa zbiory współrzęd-
nych normalnych helisy DNA potrzebnych do separacji układu na dwa rozprzężone
łańcuchy. Niewykluczone, że mogą one odpowiadać rzędowi podstawowych komple-
mentarnych podwójnych łańcuchówDNAw żywej komórce.

Manuscript received April 15, 2009; accepted for print August 19, 2009