Two-dimensional random walk Two-dimensional random walk and critical behavior ofand critical behavior of

double-strand DNAdouble-strand DNA

G. N. Hayrapetyan1, E. Sh. Mamasakhlisov1, V. F. Morozov1, Vl. V. Papoyan1,2, S. S. Pogosyan2, V. B. Priezzhev2

1 Department of Physics, Yerevan State University, Yerevan, Armenia 2 Bogolubov Laboratory of Theoretical Physics, JINR, Dubna, Russia

helicalregion

coilregion

hydrogen bond

•Poland-Sheraga type model•Ising-like sequence of base

pairs•Peyrard-Bishop type model•molecular dynamics•renormalization group

helix-coil transition in a double-stranded homopolynucleotide

(melting phenomenon)

“minimal” model

3D

1. A molecule of DNA is considered as a two random chains which begin from the same point. Complementary pairs of nitrogen bases are able to create hydrogen bonds, and to each binding will correspond an intersection of two random chains.

2. Thereafter, we watch the vectors that connects the end of one of the random chains to the end of another. Projecting that's vectors onto the planes which are perpendicular to axis of molecule of DNA, we obtain a two dimensional random walk.

3. For convenience we consider random walk of the end of vector on quadratic lattice. In term of random walks the return to the origin will correspond to binding between complementary pair of nitrogen bases.

2D

3D

generating function for the first return

where fm is the probability of the first

return on the m-th step.

generating function for the any return

where pm is the probability of any

return on the m-th step.

generating function for the any return

to the origin on the quadratic lattice

statistical weight of contact with origin

k<1 (U>0) repulsion of particle at origin k>1 (U<0) attraction of particle at origin

k>1 (U<0)

The internal energy per step, in units Tk<1 (U>0)

E = 0

statistical weight of contact with origin U < 0

k > 1 ( T < Tc )

kc =1 Tc

“Minimal” model

k < 1 ( T > Tc )

δS(m) = -c log m c = 1

Density of the free energy

T Tc

T Tc- 0

infinite order phase transition

“loop factor” c = 1

k > 1 ( T < Tc )

Helicity degree & correlation function

T Tc- 0

ξ (T Tc- 0)

L = N / 2