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### Title-affiliation

1 University of Athens, Physics Department, Athens, Greece

2 University of Patras, Materials Science Department, Patras, Greece

0 (no longer at Leibniz Institute for Neurobiology, Magdeburg, Germany)

small polaron hopping

A model

for the multi-phonon-assisted transport

along DNA molecules,

in the presence of disorder

Georgios Triberis 1,

Constantinos Simserides1,2,0,

Vasileios Karavolas 1

Low-T:

ionic conduction

(counterions)

but

cannot account for high-T

λ phage DNA

Experiment 1 (no contacts – resonant cavityσ = σ(T) at high / low T

High-T:

carrier excitations

across single-particle gaps

or

T-driven hopping

Alternatively

phonon-assisted

polaron hopping

a small polaron hoppingmodel

poly(dA)-poly(dT),

poly(dG)-poly(dC)

Experiment 2I-V(T) i.e. G = G(T)

at high / low T

poly(dA)–poly(dT):

hopping distance

~ 16.8 Å(5 base pairs)

poly(dG)–poly(dC):

hopping distance

~ 25 Å(7 base pairs!)

Interpret Tran et al?

not quite convincing

other possible mechanisms?

I ~ b V

fitting parameter b

(with unclear T-dependence)

activated Arrhenius law + constant

or

variable range hopping + constant

native wet-spun

calf thymus Li-DNA

Experiment 3σ = σ(T) at high T

… we have to take into accountthe system characteristics …

~ 3.4 Å

helix step ~ 34 Å

DNA double helix

(1) deoxyribose nucleic acidtwo helices =

two polynucleotide strands

nucleotide =

phosphate +

sugar+

base (A, T, C, G)

4 nucleotides

4 bases

Adenine (A)

Thymine (T)

Cytosine (C)

Guanine (G)

Minor groove

~ 1.2 nm

Major groove

~ 2.2 nm

base pairs:

Cytosine <3 Η bonds> Guanine

Αdenine <2 Η bonds> Τhymine

(phonons)

Vibrating

(phonons)

site j

site i

(2) deoxyribose nucleic acidH bond length ~ 2.9Å

“random”(non-periodic)

base sequence

(disorder one)

base pair separation

(i,i+1) ~ 3.4 Å

(helix step ~ 34 Å)

(3) deoxyribose nucleic acid

diameter ~ 20 Å

base end to end separation ~ 10.7 Å

Phosphate has

a negative charge

Phosphate

attracts counterions

e.g. K+, Na+, etc

H bond length ~ 2.9 Å

(disorder two)

Polaron

2. Electron (linear interaction)

1, 2 comparable magnitude

εi(0)

1. Phonon (parabolic … + nearest neighbors)

3. Polaron formation (Eb(i))

- Ai xi

MCM (Molecular Crystal Model) 1D ordered all i equivalent

T. Holstein, Ann. Phys. NY 8(1959) 343

GMCM (Generalized Molecular Crystal Model) 3D disordered

G. P. Triberis and L. R. Friedman, J. Phys. C: Solid State Phys. 14 (1981) 4631 high-T

G. P. Triberis, J. Non-Cryst. Solids 74 (1985) 1 low-T

ii) Percolation

UnSuccessful low-T (all experiments above)

ionic conductivity?

sth else?

What is the character of our system? What must the model take into account?(2) It is plausible that the carriers are

small (wavefunction extent)

polarons (phonon + electron).

Phonon (parabola + nearest neighbors).

- Molecular wire (quasi one dimensional character).
Each base pair can be considered as a molecular site.

(3) Disorder (random base sequence, counterions)

- multi-phonon assisted polaron hopping
few-phonon assisted hopping

analytical expressions for

σ=σ(Τ)

rm = rm(T)

Successful high-T (all experiments above) fitting σ = σ(T)

reasonable rm = rm(T)

Wij0h depends on Ei and Ej

Wij0ℓ depends on Ei or Ej

transport problem transformed to equivalent network of impedances

V. Ambegaokar, B. I. Halperin and J. S. Langer, Phys. Rev. B 4 (1971)2612

percolation condition high-T

≠

percolation condition low-T

conductivity high-T

≠

conductivity low-T

equilibrium transition probabilityequilibrium

occupation probability

intrinsic

transition rate

The analytical expressions

high-T (multi-phonon-assisted)

≠

low-T (few-phonon-assisted)

Conductivity high-T low-T

G. P. Triberis, C. Simseridesand V. C. Karavolas,

J. Phys.: Condens. Matter 17 (2005) 2681–2690

High-TTran et alG. P. Triberis, C. Simseridesand V. C. Karavolas,

J. Phys.: Condens. Matter 17 (2005) 2681–2690

High-T Yoo et alPercolation conditionhigh-Tlow-T

max

hopping

distance

max

site

energy

max

hopping

distance

max

site

energy

r (spatial dimension) = 1

ε (number of energies) = 1

r (spatial dimension) = 1

ε (number of energies) = 2

ε = number of energies in the percolation condition

r = spatial dimensions

Conductivity “Law”3Dhigh-TT-2/5Triberis and Friedman

3D low-TT−1/4Mott

2D low-TT−1/3 Pollak

1Dhigh-TT−2/3 here

1D low-TT−1/2 here and

inaccordance with

variable range hopping for localized states

Pollak, Lee, Serota et al

AcknowledgmentsFrank Angenstein(Leibniz Institute for Neurobiology, Magdeburg, Germany)Georgia Thodi(DUTh, Molecular Biology and Genetics Department, Alexandroupolis, Greece)

End.Thank you.

Bibliography – Sources Theory + Experiments

1. G. P. Triberis, C. Simseridesand V. C. Karavolas,

J. Phys.: Condens. Matter 17 (2005) 2681–2690

“Small polaron hopping transport along DNA molecules”

2. P. Tran, B. Alavi and G. Gruner,Phys. Rev. Lett. 85 (2000) 564

3. K. H. Yoo et al, Phys. Rev. Lett. 87 (2001) 198102 *

4. Z. Kutnjak et al, Phys. Rev. Lett. 90 (2003) 098101

5. H. Bottger and V. V. Bryksin (1985) Hopping Conduction in Solids (Berlin: Akademie-Verlag) p.73[a small polaron hopping model] *

6. Ch. Adessi and M. P. Anantram, Appl. Phys. Lett. 82 (2003) 2353 [counterion-induced disorder]

7. W. Zhang, O. Govorov and S. E. Ulloa, Phys. Rev. B 66 (2002) 060303 [Polarons with a twist]

8. Z. G. Yu and X. Song, Phys. Rev. Lett. (2001)86 6018 [variable range hopping ** + T-dependent localization length]

9. N. F. Mott and E. A. Davis (1971) Electronic Processes in Non-Crystalline Solids (Oxford: Oxford University Press) p. 216 **

10. S. S. Alexandre et al, Phys. Rev. Lett. 91 (2003) 108105 [ab initio poly(dC)–poly(dG)] [evidence for small polaron formation]

11. T. Holstein, Ann. Phys. NY 8(1959) 343 [Molecular Crystal Model]

12. G. P. Triberis and L. R. Friedman, J. Phys. C: Solid State Phys. 14 (1981) 4631 [Generalized MCM] disordered high-T

13. G. P. Triberis, J. Non-Cryst. Solids 74 (1985) 1 [Generalized MCM] disorderedlow-T

14. V. Ambegaokar, B. I. Halperin and J. S. Langer, Phys. Rev. B 4 (1971)2612 [equivalent network of impedances]

15. M. Pollak, 1978 The Metal–Non-Metal Transitions in Disordered Systems,ed L R Friedman andD P Tunstall (Edinburgh) p 138, 139

[percolation condition]

Bibliography –Sources Theory + Experiments

16. G. P. Triberis, Semiconductors 7 (1993) 471 [review on amorphous materials]

17. D. Emin, Adv. Phys.24 (1975)305 [microscopic velocity operator]

18. We use α−1 = 2 Å, within the range of values usually used for DNA and similar structures

J. A. Reedijk et al, Phys. Rev. Lett. 83 (1999) 3904

Z. G. Yu and X. Song, Phys. Rev. Lett. (2001)86 6018

19. P. A. Lee, Phys. Rev. Lett. 53 (1984) 2042 ; R. A. Serota, R. K. Kalia and P. A. Lee, Phys. Rev. B 33 (1986) 8441, and Pollak (15.) [T−1/2, variable range hopping** between localized states]

See also Z. G. Yu and X. Song, Phys. Rev. Lett. (2001)86 6018

20. G. P. Triberis, Semiconductors7 (1993)471 [brief review application to amorphous 3D]

Some illustrations used in this talk were processed after copying from:

1. National Human Genome Research Institute (http://www.genome.gov/).

All of the illustrations in the Talking Glossary of Genetics are freely available and may be used without special permission.

2. Wikipedia the free-content encyclopedia (http://en.wikipedia.org/).

Mechanisms proposed for strong σ=σ(T) at high-T

unistep superexchange and multistep hopping [16, 17]

[16] Jortner J 1998 Proc. Natl Acad. Sci. USA 95 12759

[17] Ratner M 1999 Nature 397 480 and references cited therein

carrier excitations across single-particle gaps [18]

[18] Tran P, Alavi B and Gruner G 2000 Phys. Rev. Lett. 85 564

bandlike electronic transport [19]

[19] Cuniberti G et al 2002 Phys. Rev. B 65 241314

variable range hopping [20]

[20] Yu Z G and Song X 2001 Phys. Rev. Lett. 86 6018

small polaron transport [21–26]

[21] Sartor V, Henderson P T and Schuster G B 1999 J. Am. Chem. Soc. 121 11027

[22] Bruinsma R et al 2000 Phys. Rev. Lett. 85 4393

[23] Conwell E M and Rakhmanova S V 2000 Proc. Natl Acad. Sci. USA 97 4556

[24] Rakhmanova S V and Conwell E M 2001 J. Phys. Chem. 105 2056

[25] Schuster G B 2000 Acc. Chem. Res. 33 253

[26] Henderson P T et al 1999 Proc. Natl Acad. Sci. USA 96 8353

It is plausible that when an electron or a hole is injected into a deformable macromolecule such as DNA,

it will induce local distortions of the structure as the latter adjusts to the excess charge and lowers

the system energy [27]; in other words a ‘polaronic’ distortion will be formed.

[27] Komineas S, Kalosakas G and Bishop A R 2002 Phys. Rev. E 65 061905

8. Z. G. Yu and X. Song, Phys. Rev. Lett. (2001)86 6018 **9. N. F. Mott and E. A. Davis (1971) Electronic Processes in Non-Crystalline Solids (Oxford: Oxford University Press) p. 216 **

Yu and Song to interpret exp1 by Tran et al proposed variable range hopping (Mott and Davis)

+ a 1D disordered system

- possible small polaron character of the carriers

- multiphonon-assisted at high-T

Conclusion:

T↓ nearest neighbor hopping to variable range hoppingof electrons between localized states

T-dependent localization length (due to thermal structural fluctuations)

without invoking ionic conduction

probability for a hop from an occupied site to a vacant site

P ~ exp[- 2αR – W/kBT]

R distance between sites α-1 localization length W the energy difference between sites

Mott (variable range hopping): competition between the two terms in the exponential

Indeed in DNA: two competing mechanisms (unistep superexchange and multistep hopping)

For details see my notes.

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