Sicily, May 03-04 (2008)

Conduction properties of DNA molecular wires

Rafael Gutierrez

Giovanni Cuniberti

Rodrigo Caetano

Bo Song

Institute for Materials Science and Max Bergmann Centre for Biomaterials

Collins Nganou

DNA: a complex systemDNA: a complex systemWhich physical factors are important for transport?Which physical factors are important for transport?

environmentenvironment

internal vibrationsinternal vibrations base-pair sequence(electronic structure)

metal-molecule contactmetal-molecule contact

accuracy

SizeNr. atoms

model Hamiltonians

Dynamical Effects

Static Deformations30

200

∞

DFTBModels MolecularDynamics

accuracy

SizeNr. atoms

Static Deformations30

200

∞

DFTBModels MolecularDynamics

I. Bridging first-principle and model Hamiltonian approaches: Parameterization

Benchmark: twisting of Poly(GC)

• Motivation: R. Di Felice et al. work on G-stacks• tHOMO-HOMO=f() for a GC-dimer

Ef

GC1 GC2

t

I. Bridging first-principle and model Hamiltonian approaches: Parameterization

φ (degrees)

2t

(eV)

d = 3.4 Å

(a)

(b)

(a) DFTB(b) Y. Berlin et al. CPC 3, 536 (2002)

I. Bridging first-principle and model Hamiltonian approaches: Parameterization

Twisting-stretching in Poly(GC)

• Electrical current during the stretching-twisting processΓ >> |t|

Γ ~ |t|

Γ < |t|

Molecular Computing Group http://www-MCG.uni-r.de

I. Bridging first-principle and model Hamiltonian approaches

?

d φ

l l

HOMO(GC)1-HOMO(GC)2 coupling

accuracy

SizeNr. atoms

Dynamical Effects

30

200

∞

DFTBModels MolecularDynamics

Idea: map DFTB-based electronic structure onto TB-Hamiltonian along MD trajectory

)(),( 1, tVtE jjj

.....

})({}),({ 1, jjj VPEP ( ) ( )j lE t E t Probability distributions Correlation functions

, 1 1( ) ( )( . .)j j j j j j j tunnelingj j

H E d d V d d ht Ht c

II. Model Hamiltonian and dynamical effects:short poly(GC) wires in a solvent

DFTB

DFTB

DFTB

DFTB

II. Model Hamiltonian and dynamical effects: short poly(GC) wires and time series

Parameters variation time scale ~ fs

The electron will “feel” the average of the parameters over the coarse graining time (related to tunneling time)

The rate of electrons going through the DNA for a current inorder of 1 nA is 10 e/ns

II. Model Hamiltonian and dynamical effects:adiabatic approximation and time scales

II. Model Hamiltonian and dynamical effects:short poly(GC) wires in a solvent

Average current through a G-pathway

Current strongly depends on charge „tunneling time“ tun

...1(t) (t) (t)

V1(t) V7(t)

tun

Lower bound

( ) 1j tV t fs

( )

1

1( ) ( , , )( )

N t

l lt t tj

I V I V VN t

II. Model Hamiltonian and dynamical effects:short poly(GC) wires in a solvent

...1(t) (t) (t)

V1(t) V7(t)

Probability distributions P for j(t)

Gaussian distribution(for reference)

DNA frozen

II. Model Hamiltonian and dynamical effects:short poly(GC) wires in a solvent

...1(t) (t) (t)

V1(t) V7(t)

Probability distributions P for Vj(t)

Gaussian distribution(for reference)

DNA frozen

n.n. electronic coupling mainly depends on internal DNA dynamics

II. Model Hamiltonian and dynamical effects:Linear chain coupled to bosonic bath

' ' ' ' '

( )

2( ) ( ) ( ) ( ) (1 ( )) ( ) ( )

( ) iEt t

ieI V dE dE Tr f E G E E E f E G E E Eh

E dt e e

Electrical current on lead =L,R

, 1,j j jV

Time average quantities

II. Model Hamiltonian and dynamical effects:Fluctuation-Dissipation relation

...1(t) (t) (t)

V1(t) V7(t)

0

( ) ( )

2( ) tanh cos( ) ( )2 B

C t J

J dt t C tk T

Relation between correlation functions C(t) and spectral density of the bosonic bath J() is given by FD theorem

II. Model Hamiltonian and dynamical effects:Influence of correlation times for a generic C(t)

2

1( ) , 5, 11 ( )

C tt

Gap reduction

II. Model Hamiltonian and dynamical effects:Gap reduced with

=100 fs

=1 fs

0( , ) j jJd

reorganization energy

II. Model Hamiltonian and dynamical effects:Strength of dynamical disorder

2

1( ) , 10 , 11 ( )

C t fst

II. Model Hamiltonian and dynamical effects:MD-derived correlation function

Fit to algebraic functions4

21

1( ) ( ) (0)1 ( )j

j

j

C t t t

II. Model Hamiltonian and dynamical effects:Fourier transforms of ACF for the onsite energies

( ) cos( ) ( )C dt t C t

DNA base dynamics:C=N and C=C stretch vibrations?

see e.g. Z. Dhaouadi et al., Eur. Biophys. J. 22, 225 (1993)

water modes

II. Model Hamiltonian and dynamical effects:MD-derived correlation function

...1(t) (t) (t)

V1(t) V7(t)0

( ) ( )

2( ) tanh cos( ) ( )2 B

C t J

J dt t C tk T

II. Model Hamiltonian and dynamical effects:Stochastic model Hamiltonians

How to formulate and solve a model Hamiltonian which directly uses MD informations

(t) is a random variable describing dynamical disorder (time series drawn from MD simulations)

II. Model Hamiltonian and dynamical effects:Stochastic model Hamiltonians

Formal solution for the disorder-averaged Green function, assuming Gaussian fluctuations:

Only the two-times correlation function (second order cumulant) is required !A simple case:

correlation function 2

1( )

1

C tt

Toy model: single site with dynamical disorder

II. Model Hamiltonian and dynamical effects:Stochastic model Hamiltonians

Disorder-averaged transmission T(E)

II. Model Hamiltonian and dynamical effects:Stochastic model Hamiltonians

Limits: 0

White noise

Adiabatic limit

Scaling of the transmission at the Fermi level with thecorrelation time (single site model)

I.Bridging first-principle and model Hamiltonian approaches: “static“ parameterization of minimal models

II.Bridging molecular dynamics and model Hamiltonians:„dynamical“ parameterization of minimal models

III.In progress: length and base sequence dependencies solution of random Hamiltonians contact effects

Current (and prospective) research lines