Charge and Exciton Transport in DNA

Abstract

Rajesh Shresth Doctor of Philosophy in Chemistry

Tulane University

Alexander L. Burin, Chair

The rate of charge transfer and the degree of excited state delocalization in homogeneous DNA bases are investigated theoretically and computationally. In the ﬁrst part molecular dynamics simulation is used to calculate the reorganization energy associated with charge transfer. The reorganization energy includes high frequency quantum vibrations only fractions of which contribute to charge transfer. Small polaron theory is used to rescale the quantum vibration contribution to reorganization energy and the subsequent charge transfer rate calculated using this quantum corrected reorganization energy is in some case in better agreement with the experiment. Also the RESP charges of AT and GC base pairs used in molecular dynamics is compared with the Mullikan charges obtained from DFT calculation. The RESP charges show that the oxidized AT base pair is more delocalized than oxidized GC base pair. This trend is opposite of what is observed for Mullikan charge obtained from DFT calculation. In the second part degree of excited state delocalization in homogeneous GC strand is calculated. Exciton coupling value between two adjacent GC base pairs that dictates the degree of delocalization is evaluated by ﬁtting Van Vleck sum rule model to the experimental absorption and circular dichroism data. Furthermore the reorganization energy associated with exciton formation is calculated computationally and is used along with exciton coupling value to numerically obtain the degree of excited state delocalization in homogenous GC strand.

Dedicated to my parents Acknowledgments

I would like to express my deepest gratitude to my advisor, Dr. Alexander Burin for his excellent guidance and continuous encouragement during my time at Tulane University. I have been very fortunate to have a graduate advisor who gave me freedom to venture on my own. My development as a researcher and completion of this dissertation would not have been possible without his support. I would also like to thank the Department of Chemistry at Tulane University and especially the members of my dissertation committee. Furthermore, I would like to ac- knowledge my co-workers, John Leveritt and Arkady Kurnosov, Sarah Tesar and Gale Blaustein for insightful discussion and collaborating on research projects. I appreciate the ﬁnancial support I received during my graduate studies at Tulane University. I thank Tulane University School of Science and Engineering and Department of Chemistry, the IBM Computational Science Fellowship Program and the National Science Foundation. I am fortunate to have a tight-knit extended family. They have helped me overcome setbacks and stay focused on my graduate study. I am forever indebted to them for their unending love and support. I am also grateful to all my friends for providing support and friendship that I needed.

ii Contents

I Molecular Dynamics Study of Charge Transfer in DNA 2 1.1 Background: ...... 3 1.2 Theoretical Model: ...... 4 1.3 Computational Model: ...... 9 1.3.1 Force Field: ...... 9 1.3.2 Numerical Algorithm: ...... 10 1.3.3 Solvent: ...... 11 1.3.4 Charge Derivation: ...... 12 1.3.5 Periodic Boundary Condition: ...... 13 1.3.6 Non-Bonded Interaction: ...... 13 1.3.7 Barostat: ...... 14 1.3.8 Thermostat: ...... 15 1.4 Calculation and Result: ...... 16 1.4.1 Monitoring MD simulation: ...... 17 1.4.2 Reorganization energy calculation: ...... 28 1.4.3 Decomposition of reorganization energy: ...... 29 1.4.4 Time Correlation Function: ...... 34 1.4.5 Rate of hole transfer: ...... 37 1.5 Conclusion: ...... 39

II Exciton Delocalization in DNA 41 2.1 Background: ...... 42 2.2 Theory: ...... 43 2.3 Experimental Data and Calculations: ...... 47 2.3.1 Exciton Coupling Calculations: ...... 51

iii 2.3.2 Inhomogeneous Broadening Calculation: ...... 52 2.3.3 Exciton Delocalization Length: ...... 57 2.4 Conclusion: ...... 61

A Charges for driving force calculations: 62 A.1 Neutral charges in GC and AT base pair used in MD: ...... 63 A.2 GC RESP charge used for driving force calculation: ...... 64 A.3 GC DFT charge used for driving force calculation: ...... 65 A.4 AT RESP charge used for driving force calculation: ...... 66 A.5 AT DFT charge used for driving force calculation: ...... 67

B Code for calculating hole transfer in identical adjacent DNA bases. 68 B.1 Fortran code to calculate driving force ...... 69 B.2 Matlab/Octave code for calculatin reorganization energy: ...... 74 B.3 Matlab/Octave code to calculate correlation function ...... 76 B.4 Matlab/Octave code to rearrange the charges...... 77

C Detailed equation used for calculating driving force 79

D Program in Matlab to create a structure of GC dimer 82

E ZINDO spectrum of monomer and dimer GC Watson-Crick base pair 84 E.1 Excitation energies and oscillator strengths of GC base pair ...... 85 E.2 Excitation energies and oscillator strengths of stacked GC Dimer ...... 87

F Program in Scilab to calculate delocalization length 90

Bibliography 96

iv List of Tables

1.1 Reorganization energy ...... 29 1.2 Decomposition of reorganization energy diﬀerent two frequency components 34 1.3 Decay Constants ...... 36 1.4 Characteristic relaxation time ...... 37 1.5 Classical rate of hole transfer ...... 37 1.6 Quantum corrected hole transfer rate ...... 38 1.7 Correlation between charge localization and reorganization energy . . . . . 39 2.8 Reorganization Energy associated with GC S0 − S1excitation...... 55 2.9 Delocalization length and its dependence on site-energy...... 60

A.1 Neutral charges in GC and AT base pair used in MD: ...... 63 A.2 GC RESP charge used for driving force calculation...... 64 A.3 GC RESP charge used for driving force calculation...... 65 A.4 GC RESP charge used for driving force calculation...... 66 A.5 GC RESP charge used for driving force calculation...... 67

v List of Figures

1.1 Convergence in time of the energy of the system ...... 18 1.2 Convergence in time of the temperature of the system ...... 19 1.3 Convergence in time of the pressure of the system ...... 21 1.4 Convergence in time of the volume of the system ...... 22 1.5 Convergence in time of the density of the system ...... 23 1.6 Convergence in time of the RSMD of the system ...... 25 1.7 Fluctuation in driving force in DNA ...... 26 1.8 Fluctuation in driving force in DNA ...... 27 1.9 The normalized spectral density functions J(ω)...... 30 1.10 The normalized spectral density functions J(ω)...... 31 1.11 The normalized spectral density functions J(ω)...... 32 1.12 The normalized spectral density functions J(ω)...... 33 1.13 Time Correlation Function ...... 35 1.14 Time Correlation Function ...... 35 2.15 Skeletal structure of C12−linked DNA hairpins...... 48 2.16 Absorption spectra of C12−linked DNA hairpins ...... 49 2.17 Circular dichroism spectra of C12−linked DNA hairpins...... 50 2.18 Schematic diagram of exciton coupling ...... 53 2.19 Electronic states of GC associated with absorption and ﬂuorescence cycle. . 54 2.20 Gaussian ﬁtting of absorption of spectrum of a GC hairpin ...... 56

C.1 DNA base numbering convention used in our molecular dynamics simulation. 80

vi 1

Introduction

In 1953, Waston and Crick showed that DNA has double helix spiral staircase like structure with base pairs acting as rungs. Soon after that discovery Eley and Spivey suggested that pi-stacking interaction between the adjacent bases could allow DNA to conduct charge [1]. However, subsequent experimental progress in the understanding of the electronic property of DNA languished due to difficulty in obtaining DNA in desired quantity and sequence from natural sources; and efficient synthesis of DNA was not possible for two more decades until the breakthrough in phosphoramidite chemistry in early 1983 [2]. Interest in this field was rekindled after the pioneering work of Barton et al. in 1993, where she showed photoinduced long range DNA mediated charge transfer between non-covalently bound metallointercalators [3]. Since then there have been numerous theoretical and experimental studies in charge and energy transfer property of DNA and the interest in the field is growing rapidly due the possibility of using it in nano-electronics [4], electrochemical biosensors [5], harnessing solar energy [6], repairing oxidative damage in DNA [7], economical DNA sequencing [8]etc. This dissertation “Charge and Energy transfer in DNA,” is a theoretical and computational investigation of charge transfer rate and energy transfer parameter in homogenous DNA sequence. In part I, the quantum effect in the absolute rate of charge transfer between homogeneous adjacent DNA bases is examined. Understanding of hole transfer rate has application in desigining nanowires. In part II, the exciton coupling paramater is calculated by fitting our theroy to an exprimental data. Excion coupling paramater eluci- dates the excited state delocalization behavior of DNA and its understanding is important to use DNA for harnessing solar energy and repairing photodamage in DNA. 2

Part I

Molecular Dynamics Study of Charge Transfer in DNA 3

1.1 Background:

Theoretical studies of charge transfer in DNA are generally conducted using semiclassical Marcus’ theory [9], where coupling between donor and acceptor in the pre- exponential term is treated quantum mechanically and the reorganization energy in exponential term is treated classically. Marcus equation has a form that looks very much like an Arrhenius equation for the temperature dependence of rate of chemical reaction.

k = A.e−Ea/kbT (1.1) where Ea is the activation energy that needs to be overcome by thermal energy kbT for the reaction to occur. For the charge transfer reactions Marcus showed that the activation energy itself depends on solvent reorganization energy and Gibbs free energy diﬀerence between the reactant and the product

2 − (λ+4G) k = A.e 4λkbT (1.2)

Here ∆G is the Gibbs free energy of reaction, the prefactor A depends upon the strength of the electronic interaction between donor and acceptor and the type of charge transfer, and λ is the reorganization energy that accounts for the structural reorganization of donor- acceptor molecules and the surrounding solvent molecules. Furthermore, λ = λs + λi, where λi is the internal reorganization energy and accounts for the structural reorganization of donor-acceptor molecules and λsis the solvent reorganization energy. For polar molecule like water λs is large and dominates the charge transfer process. There has been lot of effort to calculate the solvent reorganization energy. The various method that are used to calculate solvent reorganization energy can be classified into two parts namely implicit and explicit solvent methods. In several papers [10][11][12][13], the solvent reorganization energy have been calculated treating the solvent implicitly as a continuum dielectric medium. Improvement in implicit continuum model has been made by adopting sophisticated means such as dividing the solvent environment around the solute into several heterogeneous zones with different dielectric constant [14]. Alterna- tively, the availability of supercomputers has made it possible to use molecular dynamics 4

(MD) methods to treat solvent implicitly. Computationally friendly approach such as non-polarizable force field been carried out to calculate the solvent reorganization energy explicitly [15][16][17] . To account for electronic polarization economical method such as multiplying by some scaling factor [18] has been used. For more accurate calculation computationally demanding polarization force field has been used [19][20]. Our collaborators at Northwestern University in Chicago have calculated the absolute rate of hole transfer between adjacent homogenous base pairs. Using transient absorption spectroscopy and fitting its to a kinetic model they calculated the absolute rate of hole −1 −1 hopping in poly(GC) and poly(AT) to be kGC = 4.1 ns and kAT = 1.2 ns respectively [6] [21] [22]. There have been theoretical and computational efforts to calculate the hole hopping rate in homogeneous bases. Despite these efforts the solvent reorganization energy is overestimated and also the hole hopping rate for poly(GC) is slower than poly(AT) [18][17], which is opposite of what experiments indicate. In this study we performed molecular dynamics in three different DNA strands and used temperature dependent

ﬂuctuation in driving force to estimate the reorganization energy kGC < kAT . The driving force have high frequency component that don’t contribute to classical charge transfer process. Small polaron theory [23][24][25] is used to rescale the high frequency component of the classical reorganization energy. The reorganization energy so obtained is smaller than previously reported [18][17] and the rate of hole transfer in some case is in good agreement with the experiment.

1.2 Theoretical Model:

Following Marcus model we consider charge transfer between two electronic states having energy diﬀerence ∆. The system can be described by the Hamiltonian

n − n H = ∆ 2 1 (1.3) 1 2 where operator n1and n2are two electronic states with population states either 0 for empty or 1 for charged state, and their sum is n1 + n2 = 1. Energy of the system 4 ∆ when n1is populated is − 2 , when n2is populated it is 2 and the energy diﬀerence is ∆ 5

. Transition between these two states in dark condition cannot occur without the help of the environment because of the energy conservation law. Marcus introduced reaction coordinate X responsible for the interaction of the environment with charge transfer.

n − n H = −X 2 1 (1.4) 2 2 X is introduced in such way that it is equal to zero (< X >= 0) at equilibrium and in the absense of charge transfer event i.e n1 = n2. Expanding the Hamiltonian involving reaction coordinate near the equilibrium value of X up to its ﬁrst non-vanishing term one can get,

X2 H = (1.5) env 4λ

n − n X2 H = H + H + H = (∆ − X) 2 1 + (1.6) 1 2 env 2 4λ This suggests a small deviation of reaction coordinate from equilibrium. Equilibrium energy of the system when states n1and state n2 are populated

−∆ − λ E = 2 (1.7) 1 2

∆ − λ E = 2 (1.8) 2 2 Classical transition between these two states can take place in the intermediate state

Eintwhen they have equal energy i.e. X = ∆.

∆2 E = (1.9) int 4λ The activation energy of transition from state to state is given by the diﬀerence

(∆ + λ)2 E = E − E = (1.10) A int 1 4λ Substitution of this activation energy in Arrhenius equation gives us the well known Marcus formalism as stated in equation (1.2). For hole transfer in homogeneous bases 6 that we are studying ∆ = 0 so the reorganization energy is entirely the activation energy. This activation energy is overcome by random fluctuations of the environment and hence the spontaneous hole transfer is observed. The random fluctuation of donor- acceptor energy mirrors this environment fluctuation energy and is distributed according to Boltzmann. The reorganization energy can then be derived from the variance of this < X2 >distribution as follows,

2 − X P (X) ∼ e 4λkB T (1.11)

< X2 > λ = λi + λs = (1.12) 2kBT We can extract the variance < X2 >from MD simulation to calculate solvent reorganiza- 2 tion energy λs. < X > itself can be obtained from the square of the diﬀerence in the electrostatic energy between neutral and positively charged adjacent base pairs. We call this parameter < X >, a driving force

Donor M + Acceptor M 0  1 X X qi qj X X qi qj X =  −  (1.13) ε + |r0 − r | ∞ i=1 j6=i,j=1 ri − rj i=1 j6=i,j=1 i j

N 2 1 X 2 < X >= |Xi| (1.14) N i=1 where M indicates all the atoms used in the simulation and N refers to total number of + 0 data steps collected in the simulation. qi and qi are the positive and neutral charges on base pairs adjacent to each other and ε∞ is the electronic polarization constant of the solvent. The multipole contribution to the electrostatic potential is small and neglected [14][26]. This method of calculating reorganization energy is standard, however, its’ applica- bility to DNA at room temperature is questionable. The solvent includes water, ions and DNA backbones and DNA base pairs that are not involved in charge transfer. The solvent reorganization energy is entirely classical; however at room temperature high frequency vibration of hydrogen, carbon and nitrogen bonds etc that are quantum mechanical in nature are included in the donor-acceptor potential gap [25]. Classical treatment of vibrational contribution leads to overstatement of reorganization energy that needs to be 7 corrected. Assuming these high frequency modes are oscillatory in nature we can use Fourier transform to get rid of them.

N−1 1 X 2πa N N N X(wa) = X(t) ∗ exp[−iωat], where wa = , a = − , + 1, ....., (1.15) N t=0 N ∗ dt 2 2 2

N P 2 N |Xa| X a λ = λa = (1.16) a 2kBT where X(wa) is the ﬂuctuation in diﬀerence between donor-acceptor energy in frequency domain. To rescale the high frequency component from reorganization energy calculation we use expression from polaron theory [23][25][24][27].

N s X ~ωa 1 λ = λa (1.17) ~ωa a=1 2kBT sinh 2kB T 2 |Xa| λa = 2kBT Here λs is the quantum corrected total reorganization energy and the sum is taken over all the data points. To calculate the charge transfer rate we need to identify whether the charge transfer process is adiabatic or non-adiabatic. Following the formalism of Rips and Jortner [28] the transition between adiabatic and non-adiabatic limit is given by adiabaticity parameter KA.

2 4π V τL ε∞ κA = , where τL = τD (1.18) ~ λ εs where V , λ, τDand τL represent electronic coupling, total reorganization energy, Debye dielectric relaxation time and longitudinal dielectric relaxation time respectively. Debye dielectric relaxation time can be experimentally determined from dielectric loss measure- ment [29]. For KA >>1 the process is considered solvent controlled adiabatic process and the rate expression is given by [28]

s s " s # −1 λ λ k = τL exp − (1.19) 16πkBT 4kBT 8

The longitudinal dielectric relaxation timeτL can be extracted from the MD data using the time auto-correlation of potential energy gap f(τ) [30][31]. For the Gaussian, Markovian relaxation process the time correlation function is exponential [32]. We can calculate the longitudinal relaxation time by exponential ﬁt of the time auto-correlation function.

T 1 f(τ) ≡< X(t) ∗ X(t + τ) >= lim X(t) *X(t + τ) dt (1.20) T →∞ T ˆ 0

− t f(t) ∼ e τL (1.21) where X(t) is the time varying potential gap energy and τ is lag time. For K 1 the charge transfer process is non-adiabatic and the rate is given by[28] [25]

2π 1 " λs # k = V 2 √ exp − (1.22) ∗ s ~ 4πλ kBT 4kBT

Here V∗ is the quantum corrected eﬀective exciton coupling and is related to the classical exciton coupling as follows;

V∗ = V exp(-Z) (1.23)

 2 N !   X λa ~ωa ~ωa Z = tanh 1 −    (1.24)  ~ωa  2 ωa 4kBT 4kBT sinh( ) a=1 ~ 4kB T

Substituting V∗ in equation 1.22 we can arrive at the following condensed expression for non-adiabatic charge transfer[25].

2π 1 " λt # k = V 2 √ exp − (1.25) s ~ 4πλ kBT 4kBT

N ! t X 4kBT ~ωa λ = λa tanh (1.26) a=1 ~ωa 4kBT 9

1.3 Computational Model:

Molecular Dynamics (MD) simulation was used to calculate the thermodynamics and conformational properties of DNA. Following Ergodic hypothesis which states that time average is equal to ensemble average one can use MD to simulate system at equilibrium. To obtain time average MD uses classical force ﬁeld and is based on Newton’s second law of motion F = ma, where F ,m and a stands for force, mass and acceleration respectively. Knowledge of position of each atom and the force exerted on it allows one to evolve the system in time. Starting position of chemical species of interest is obtained from X-ray crystallography data and the force is obtained from derivative of force-ﬁeld that is parameterized for that particular chemical species.

1.3.1 Force Field:

The force ﬁeld in Amber11 package has a general functional form composed of bonded and non-bonded terms. The bonded terms consist of bond lengths, bond angles and dihedral angles while the non-bonded terms consist of Van der Waals and electrostatic terms. All these terms are parameterized to reproduce available experimental or quantum computational result. For parametrizing macromolecule such as DNA, building blocks such as deoxyadenosine, deoxythymidine, deoxyguanosine and deoxycytidine are parameterized. So when simulating DNA one can simply combines these fragments to get the DNA of desired bases and length. The equation for a general non-polarizable force ﬁeld U(rN ) is expressed as follows:

N N N U(r ) = U(r )bonded + U(r )non−bonded (1.27)

N 1 X 2 1 X 2 X U(r )bonded = kb(r − r0) + ka(θ − θ0) + Un [1 + cos(nω − γ)] 2 bonds 2 angles torsion

! N X X qiqj X X Aij Bij U(r )non−bonded = + 12 − 6 i i>j rij i i>j rij rij Bonded Term: 10

• The ﬁrst expression in the bonded term is a harmonic potential that approximates

energy associated with bond stretching. Here kb represents force constant and r0 is the equilibrium bond length.

• The second expression in the bonded term is a harmonic potential that approximates the energy associated with the angular deformation of the bonding orbitals. Here

ka represents angular force constant and θ0 is the equilibrium bond angle.

• The last part in the bonded term is a periodic potential that represents the energy associated with torsional motion of atoms separated by three covalent bonds. Here

Un represents the barrier to rotation of dihedral angle, n describes the number of minima in the rotational potential energy surface, ω is the improper dihedral angle and γ is the dihedral angle for which torsional potential energy surface has the ﬁrst energy minima.

Non-Bonded Term:

• The ﬁrst expression in non-bonded term is the sum of two particle coulomb potential.

qi and qj are the partial charges in two atoms i and j, r is the distance between them and ε is the dielectric constant of the medium.

• The second expression in the non-bonded term is the Lennard-Jones potential. Here −12 rij describes the Pauli repulsion term due to orbital overlap at distances shorter −6 than equilibrium bond length and rij is the long range attractive term that captures

dispersion interaction. Aij and Bij are Lennard-Jones constants that describe well depth.

1.3.2 Numerical Algorithm:

In Molecular Dynamics time evolution of the system is done by repeatedly calculating the potential energy of the system and integrating the Newton’s second law of motion. Due to the complicated nature of the force ﬁeld no analytical solution exist so one uses numerical techniques to solve the classical equations of motion. In numerical method 11 one assumes that the position, velocity and acceleration can be approximated by Taylor expansion.

1 r (t + ∆t) = r(t) + v(t)δt + a(t)δt2 + .. (1.28) 2!

1 v (t + ∆t) = v(t) + a(t)δt + b(t)δt2 + .. (1.29) 2!

1 a (t + ∆t) = a(t) + b(t)δt + c(t)δt2 + .. (1.30) 2! where r is the position of an atom, v is its velocity, a is the acceleration, b and c are the third and fourth order derivative of coordinate with time. Several numerical methods such as Verlet [33], Leap-frog [34], Velocity Verlet[35] etc have been developed over the years to integrate the Newton’s equations of motion. These numerical methods diﬀer in way they discretize the coordinate and the velocity space of the system. In Amber 11 package Velocity Verlet algorithm is used for simulation. In this algorithm the velocity and position of the atoms are calculated at the same time step so unlike Leap-frog method there is no need to store value of two diﬀerent time to propagate the system. Also compared to Verlet algorithm this method allows explicit treatment of velocity and hence the calculated kinetic energy is highly precise. Following is the Velocity Verlet algorithm with r, v and a having meanings as above.

1 r (t + ∆t) = r(t) + v(t)∆t + a(t)∆t2 (1.31) 2

1 v (t + ∆t) = v(t) + [a(t) + a(t + ∆t)]∆t (1.32) 2

1.3.3 Solvent:

We used non-polarizable TIP3P solvent for our simulation. It is a 3-site rigid water model with ﬁxed charges assigned to all three atoms. In this model the bulk property of water is simulated using only non-bonded interactions that are modeled using Coulomb and Lennard-Jones potentials as shown below. 12

! X X Kcqiqj X X σij σij EAB = + 4 12 − 6 (1.33) i i>j rij i i>j rij rij

Here the ﬁrst term is the Coulomb term where Kc is the electrostatic force constant, qi and qj are the partial charges in two atoms and r is the distance between them. The second therm is Lennard-Jones potentials where σ describes the distance at which the inter- particle potential is zero, is the depth of the potential well, r−12 and r−6 are repulsive and attractive terms respectively. In second term, the H-atoms bonded to oxygen have zero Van der Walls radii so the Lennard-Jones terms only applies to interaction between atoms in diﬀerent molecules [36]. In TIP3P model the OH distance is 0.9573, HOH angle 0 −1 is 104.52, σ is 3.15061A, ε is 0.6364 kJmol , qH is 0.417 and qO is -0.834 [37].

1.3.4 Charge Derivation:

In molecular dynamics simulation fixed partial charges obtained using quantum mechanical calculation are assigned to atoms in a molecule. In AMBER, MD simulations are carried out using restrained electrostatic potential (RESP) [38] charges that are derived from least squares fit of the charges derived from ab initio Hartree-Fock (HF) 6-31G* calculation that best reproduces the electrostatic potential of the molecule. In HF calculation atomic charges is derived from Mulliken population analysis of a molecule in a minimum energy conformation. In MD a molecule samples several accessible orientation phase-space and assigning one fixed charge to all these conformations is tricky. This is addressed using RESP charge where the electrostatic potential around the molecule of interest is mapped at a large number of points defined by 4 shells of surfaces at 1.4, 1.6, 1.8, and 2.0 times the Van der Waals (VDW) radii and fitting the partial charge to minimize the error function under the constrain that the sum of the particle charge is zero.

!2 X X qi Err(q) = φj(r) − (1.34) i rij

Here Err(q) is the residual that is minimized for each charge parameter qi, φj(r) is the molecular electrostatic potential at distance j calculated using HF method and rij is the distance between an atom and the grid point in the electrostatic surface. The partial 13 charges obtained in this manner tend to produce unrealistically large charges in the heavy atoms in organic molecule such carbon that is not on the surface. In an attempt to redress this issue, the residual is restrained using a hyperbolic penalty function for any non-zero charge in heavy atoms. Additionally partial charges on three hydrogen atoms in methyl group that are NMR equivalent are set to be equal.

1.3.5 Periodic Boundary Condition:

To model an experiment realistically it requires solvent bath of thousands of molecules, which makes molecular dynamics simulation prohibitively expensive so one often uses limited number of solvent molecule that is computationally tractable. In this small solvent system the surface tension is very high, which can adversely affect the simulation. In addition, during MD simulation the outer solvent molecule tend to boils off into space that causes further loss of solvent molecules and aggravates the surface effects. To avoid this unphysical rigid boundaries AMBER uses periodic boundary condition (PBC). In PBC atoms in the central box are replicated all direction throughout the space to form an infinite lattice. For example if ri is the position of atoms in the central box then using PBC mirror images of the atoms in central box are created in all the other boxes.

rimage = ri + l.a + m.b + n.c (1.35) where a, b and c are the edge of the central box and l, m and n are any vectors in integer space. This allows the atoms that boil oﬀ from one side of the box to appear from the opposite side with the same velocity and thus keeps the total number of atom in the central box constant throughout the simulation. In using PBC one sets a limitation called the “minimum image criterion” that requires that each atom interact with only the nearest images of other N − 1 particles.

1.3.6 Non-Bonded Interaction:

Non-bonded interaction calculation is the most computationally expensive process in a simulation as it scales as O(N 2). The inverse distance dependent Coulomb interaction 14 is long-ranged and extend beyond the boundary of the box. To address this problem AMBER uses Particle Mesh Ewald (PME)[39] method in which the 1/r electrostatic term is divided into rapidly converging direct (real) space sum and reciprocal space sum that are easier to calculate.

N 1 N erfc(β) N erf(β) X = X + X (1.36) i

1.3.7 Barostat:

The pressure P of the N-body classical system can be calculated using Clausius Virial theorem as shown below;

N 2 1 X P = (K − rijFij) (1.37) 3V 2 i

dP (t) P − P (t) = 0 dt τp where P0 is the desired pressure and τp is the barostat relaxation time. In constant pressure simulation the desired pressure P0 is maintained by adjusting the volume of the 15 simulation box. The the dimension of the simulation box volume is adjusted by a scaling factor η, which is given as;

4t η = 1 − γ(P0 − P (t)) (1.38) τp Here 4t is the integrator time step and γ is the isothermal compressibility factor of the system who’s exact value is not important provided the time step and the coupling constant are chosen wisely. The scaling is done for all components of atomic coordinate concomitant to the change in the volume ∆V of the simulation box.

1.3.8 Thermostat:

The temperature T of N-body classical system is calculated using equipartition theorem as follows;

N 1 X 2 T = mivi (1.39) Nf kb i=1 where Nf is the number of degrees of freedom, kb is Boltzmann constant, miis the mass of an atom and viis its velocity. To maintain temperature AMBER uses Langevin dynamics, which deals with each atom separately, balancing a small friction term with Gaussian noise in such a way that ﬂuctuation-dissipation theorem is satisﬁed. For a system of N particles with momentum p the Langevine equation is give by;

pi = fi − γpi + R(t) (1.40)

Here fi is the force acting on atom i due to the interaction potential, γ is a coefficient of friction and R(t) is a random force that fluctuates very rapidly in comparison with the time step ∆t and the characteristic time 1/γ. It does not depend on the positions and velocities. R(t) is related to coefficient of friction via time correlation function shown below;

0 0 < R(t) R(t ) >= 6kBT γδ(t − t ) (1.41) 16 where δ is the Dirac delta function. The physical origin of R(t) is the collisional force of an atom with other molecules.

1.4 Calculation and Result:

We obtained the structure of 12 strand long DNA poly(GC)12, poly(AT )12 and mixed

(AT )5(GC)2(AT )5 sequence using NAB module available in AMBER 11 molecular dynamics package. Then the module Leap in the AMBER 11 was used put the DNA structure was in a TIP3P water box. The dimension of water box was 20 angstrom from each end of DNA molecule. To neutralize system 24 sodium ions were added close to the phosphate atoms in DNA, where the electric potential is highly negative.Then the leap module was used to generate topology and parameter. Simulation was performed using Sander’s module with parm99+parmBSC0 force field [40] and SHAKE algorithm turned on to keep hydrogen atoms fixed. In all three simulations periodic conditions, cutoff at 10 A for non-bonded interaction, 2 fs time-step for MD simulation and 1fs time-step for production were set. This 1 fs production time-step would allow us to collect all possible vibrational motion in our molecular dynamic simulation including the stretching mode of hydrogen (C-H, N-H, O-H) which vibrate in the order of few femtoseconds [41].

• In the first step of simulation we minimized the energy of our DNA+TIP3P water system. The starting structure generated using NAB and Leap module tend to have overlapping atoms and doesn’t correspond to minima in the force field. We performed a 3000 steps of energy minimization (steepest decent +conjugate gradient algorithm) of the water molecules while keeping the DNA fixed. It was followed by 3000 steps of energy minimization of the entire system.

• Next step was heating the system from 0K to 300K during 40 ps of dynamics using Particle mesh Ewalds (PME) periodic boundary condition. To avoid wild ﬂuctua- tions we put a weak positional restraint in DNA that work by restraining the atoms of DNA to conform to the initial structure structure through the use of a harmonic potential. Langevin temperature equilibration scheme was used ensure the temperature increased evenly across the system. The heating is done at ﬁxed volume 17

because the calculation of pressure is inaccurate at low temperature.

• Next we ran 6ns of equilibrium simulation at constant temperature of 300K and at constant pressure (NPT) of 1 atmospheres as this resembles the experimental condition more closely. Our simulation time of 6ns adequate for our model as it exceeds the water relaxation time by at least three order of magnitude and also exceeds the charge transfer rate between adjacent homogeneous bases [6] [21]. We ﬁnally collected 100,000 data steps to be used for reorganization energy calculation.

1.4.1 Monitoring MD simulation:

During the simulation we periodically monitored a number of system properties to ensure that our system was in equilibrium. These properties we monitored as follows;

• Energy: In this energy-time plot in figure 1.1the positive valued green line is kinetic energy and the negative valued red and blue lines are total energy and potential energy respectively. The salient feature of this plot is that all of the energies increased during the first 40 ps, corresponding to the heating from 0 K to 300 K at fixed volume. After 40 ps the kinetic energy is a flat line for the rest of the MD implying that the Langevin temperature thermostat, which couples with kinetic energy, was working correctly. The potential energy, and consequently the total energy as total energy is the sum of potential and kinetic energy, decreases as the system relaxes when the DNA restraint is switched off after 40 ps and moved to a constant pressure (NPT) portion of the simulation. Then the potential and total energy levels off and remains constant for the remainder of our simulation indicating that the relaxation was successful and that an equilibrium achieved.

• Temperature: In ﬁgure 1.2 the temperature increases from 0 K to 300 K over a period of about 40 ps as expected since we heated the system in constant volume phase of our simulation. The temperature then ﬂuctuates about 300K for the remainder of the simulation indicating the temperature regulation algorithm was successful. 18

Figure 1.1: Convergence in time of the energy of the system. The green, blue and red lines represent kinetic, potential and total energy respectively. 19

Figure 1.2: Convergence in time of the temperature of the system 20

• Pressure: For the first 40 ps we ran a constant volume simulation and during that period the pressure wasn’t evaluated so in the plot 1.3 we initially see a flat line at zero atmosphere. After 40 ps we switched to constant pressure and allowed the volume of the box to change. This causes the pressure to drop sharply and becomes negative. The negative pressure can be thought of as a force acting to decrease the box size and the positive pressure as a force acting to increase the box size. In the graph the pressure fluctuates a lot during the simulation this is because the pressure-volume isotherm of a water is extremely steep; meaning a slight instantaneous change in volume out of equilibrium results in a large fluctuation in pressure. On average the fluctuations in pressure stabilize around 1 atm during the course of the simulation and indicates that the pressure equilibration was achieved.

• Volume: In the ﬁgure 1.4 we can see that the volume of the system initially decreases as we switch to NPT part of the simulation at 40 ps. The system relaxes smoothly to equilibrium volume and then ﬂuctuates around equilibrium volume during the course of the simulation, suggesting that the volume equilibration was successful.

• Density: The density-time plot 1.5 mirrors volume-time plot 1.4. After 40 ps the density of the system increases corresponding to the decrease in volume we observed in our system. The density then equilibrates to the value of about 1.03 g/cm.3The density of pure water at 300 K is about 1.0g/cm.3and adding a 12 strand long DNA and 24 Na+ ions should increase the density as observed in the simulation. The suggest that the density equilibration was achieved.

• RMSD: We measured the root mean squared deviation (RMSD), which is the measure of the scalar distance between atoms of the same type for two structures. We chose 6 heavy atoms (P,O3’,O5’,C3’,C4’,C5’) in the backbone of the DNA to compare the spatial deviation between the starting structure (t = 0) and the conformations obtained from the time evolution of the original structure. A rigid superposition which minimizes the RMSD is performed and only this minimum RMSD 21

Figure 1.3: Convergence in time of the pressure of the system 22

Figure 1.4: Convergence in time of the volume of the system 23

Figure 1.5: Convergence in time of the density of the system 24

value is reported. The RMSD equation is as follows; v u N u 1 X RMSD(v, w) = t kvi − wik (1.42) N i where v and w are the initial and final position of the atom and the summation goes to the total number of atoms N. In the graph 1.6 we see that for the first 40 ps the RMSD value of the 6 atoms in DNA backbone remains low due to the restraint in DNA during the constant volume part of the simulation. After removing the restraints at 40 ps the DNA relaxes as reflected in the sharp increase in RMSD value of the DNA backbone atoms. After that the RMSD settles to a value of 0.27 nm and is fairly stable for the rest of the simulation. This fluctuation of DNA about equilibrium structure during the MD suggests that our simulation was successful. This was further corroborated by visually inspecting the structure of DNA using a visualization package called Visual Molecular Dynamics (VMD) 1.8.7 [42].

Once we established that our simulation was successful we collected 100,000 data points with 1fs time step. Then we proceeded to calculate donor-acceptor potential energy gap of initial and final charge transfer state of the hole transfer reaction at the same nuclear position using equation 1.13, which we call driving force. Equation 1.13 requires partial charge of atoms in DNA bases. We used two different methods to calculate partial charge namely RESP charge and DFT B3LYP/6-311+G(d,p) B3LYP/6-311+G(d,p) charge for neutral and positively charged GC and AT base pairs. The RESP charge on AT + and GC+was calculated using a RED [43] program that performs HF 6-31G* calculation and automatically does the restrained least square fitting of Mulliken charge in just one input step. We subtracted the positive partial charge in DNA bases from a neutral partial charge of respective DNA bases to obtain the map of a hole delocalized on DNA bases. This hole was then added to the neutral resp charge used for MD and the resulting perturbed charge was used for driving force calculation. The time series plot of the driving force calculated using RESP and Mulliken charges from DFT B3LYP/6-311+G(d,p) B3LYP/6- 311+G(d,p) calculation is shown in figures 1.7 and 1.8 respectively. In the plot we can see that the driving force fluctuates in time between -1 to 1 eV and has a zero mean which makes calculating variance (equation 1.14) much simpler. 25

Figure 1.6: Convergence in time of the RSMD of the system 26

Figure 1.7: Fluctuation in potential energy diﬀerence between donor and acceptor electrostatic energy (driving force) in (GC)12 (red), (AT )12 (green) and (AT )5(GC)2(AT )5 (blue) DNA strands calculated using RESP charge. 27

Figure 1.8: Fluctuation in potential energy diﬀerence between donor and acceptor electrostatic energy (driving force) in (GC)12 (red), (AT )12 (green) and (AT )5(GC)2(AT )5 (blue) DNA strands calculated using DFT B3LYP/6-311+G(d,p) B3LYP/6-311+G(d,p) 28

1.4.2 Reorganization energy calculation:

We calculated the reorganization energy for all three DNA strands from the time series data using equation 1.12. We account for electronic polarization in an ad hoc fashion by using the value of ∞ = 1.77 for electronic polarization of water. To calculate quantum scaling we converted the time series driving force data to frequency domain using fast Fourier transform (FFT). The reorganization energy calculated from frequency domain driving force data using equation 1.16 matched the reorganization value obtained from time series data, indicating that our FFT transform was accurate. In table 1.1 we compare the reorganization energy obtained using DFT B3LYP/6-311+G(d,p) B3LYP/6- 311+G(d,p) and RESP charge with the value reported in literature that were calculated using molecular dynamics method. The classical reorganization energies 1.39 eV and 1.16 eV were obtained using RESP charge for poly(GC)12 and poly(AT )12 respectively are close to the reorganization energy 1.41 and 1.21 for poly poly(GC)12, poly(AT )12 respectively reported by Kubar et. al.[18] using RESP charge, but diﬀerent reorganization energy calculation scheme. Hypertan scaling of the classical reorganization energy 1.03 of poly (AT) obtained using RESP charge is close to the value of 1.08 reported by Steinbrecher et al.[17] using QM/MM method. The classical reorganization energy obtained using DFT B3LYP/6-311+G(d,p) B3LYP/6-311+G(d,p) charge has the eﬀect of increasing the classical reorganization energy of poly(AT) and decreasing the classical reorganization energy of poly(GC). The hypertan λt and hypersine λs scaling of classical reorganization energy calculated using DFT B3LYP/6-311+G(d,p) B3LYP/6-311+G(d,p) and RESP charge are similar. It should be noted that λs is the relevant reorganization energy for Marcus classical theory while λt appears in the exponent of non-adiabatic reorganization energy and is a main value related to charge transfer rate to compare to other theories. On average hypertan scaling reduces the classical reorganization energy by about 0.28 eV and hypersine scaling reduces it by about 0.16 eV. For reorganization energy calculated using both RESP and DFT B3LYP/6-311+G(d,p) B3LYP/6-311+G(d,p) charge the trend is

λGC2AT10 > λGC)12 > λAT12 . The higher reorganization of poly(AT )5(GC)2(AT )5 compared to poly(GC)12 strand can be understood in terms of the more propeller twist motion of poly(AT) strand compared to poly (GC) strand [44]. 29

Table 1.1: Reorganization energy calculated using DFT B3LYP/6-311+G(d,p) B3LYP/6- 311+G(d,p) and RESP charges and comparison with values in literature.

Base Pair λcla λcla λcla(RESP ) λtanh(RESP ) λsinh(RESP ) λclassical(DFT ) λtanh(DFT ) λsinh(DFT ) Kubar et al. Steinbrecher et (eV) (eV) (eV) (eV) (eV) (eV) (eV) al. (eV)

poly(GC12) 1.41 1.15 1.39 1.20 1.10 1.47 1.28 1.14

poly(AT12) 1.21 1.08 1.16 1.05 0.89 0.93 0.81 0.72

poly(GC2AT10) 1.42 1.24 1.11 1.51 1.33 1.19

Reorganization energy in frequency domain: In ﬁgures 1.9, 1.10, 1.11 and 1.12we can see the spectrum associated with oscillations of poly(GC)12, poly(AT )12 and poly(AT )5(GC)2(AT )5 sequence calculated using RESP and DFT B3LYP/6-311+G(d,p) B3LYP/6-311+G(d,p) charges. We can roughly divide the spectral contribution into two frequency regimes; low frequency translation, rotational and librational motion of water superimposed with the translational motion of DNA bases involving rise, shift, slide, twist etc that lie between 0 − 1000 cm−1 and the high frequency regime above 1000 cm−1that corresponds to complicated coupled stretching, scissoring and bending modes of the atoms in the bases [45]. There is no high frequency component above 2000 cm−1 because we used TIP3P rigid water model and ran MD with SHAKE algorithm keeping hydrogen bond ﬁxed.

1.4.3 Decomposition of reorganization energy:

Decomposition of spectral density is tabulated in table 1.2 where can see that the majority of reorganization energy contribution comes from low frequency (0-1000cm−1) mode of DNA and water. Hypersine and hypertan functions scale the classical reorganization energy diﬀerently. For frequency range of 0-1000cm−1 there is on average 6% and 10% reduction in reorganization energy value with hypertan and hypersine rescaling respectively. And for frequency range of 1001-2000cm−1 there is on average 45% and 78% reduction in reorganization energy value with hypertan and hypersine rescaling respectively. This shows that our quantum rescaling function is working properly as mostly high frequency components are scaled. It is interesting to note that in poly(GC)12 −1 and poly(AT )5(GC)2(AT )5 the high frequency (1001 − 2000cm ) reorganization energy 30

Figure 1.9: The normalized spectral density functions. Hypertan (green) scaling of classical driving force spectrum (blue) calculated using RESP charge in three DNA strands. 31

Figure 1.10: The normalized spectral density functions. Hypersine(red) scaling of classical driving force spectrum (blue) calculated using RESP charge in three DNA strands. 32

Figure 1.11: The normalized spectral density functions. Hypertan (blue) scaling of classical driving force spectrum (black) calculated using DFT B3LYP/6-311+G(d,p) charge in three DNA strands. 33

Figure 1.12: The normalized spectral density functions. Hypersine (red) scaling of classical driving force spectrum (black) calculated using B3LYP/6-311+G(d,p) charge in three DNA strands. 34

Table 1.2: Decomposition of reorganization energy diﬀerent two frequency components

λcla(RESP ) Base Pair Freq λtanh(RESP ) λsinh(RESP ) λclassical(DFT ) λtanh(DFT ) λsinh(DFT ) (eV) (eV) (eV) (eV) (eV) (eV) 0-1000 1.19 1.10 1.04 1.22 1.15 1.09 GC12 1001-2000 0.19 0.13 0.05 0.25 0.13 0.05 0-1000 0.92 0.91 0.83 0.76 0.73 0.69 AT12 1001-2000 0.24 0.14 0.06 0.17 0.09 0.04 0-1000 1.24 1.15 1.08 1.27 1.20 1.14 GC2AT10 1001-2000 0.18 0.09 0.03 0.24 0.13 0.05

component is almost equal in both DFT B3LYP/6-311+G(d,p) and RESP charge cases. Their diﬀerence in reorganization energy is entirely from the low frequency(0−1000 cm−1) component. This makes a stronger case for attributing higher reorganization energy of poly(AT )5(GC)2(AT )5 to faster propeller twist motion of poly(AT ) compared to poly(GC) , as propeller twist motion lies in the low frequency region.

1.4.4 Time Correlation Function:

To investigate the charge transfer in adiabatic regime we need to find the characteristic solvent relaxation time. We define it using the driving force autocorrelation function. The formal definition of time auto-correlation function in equation 1.20 is for infinitely long period of time T. In our simulation we collected 100 ps of data so we used the discrete variant of time auto-correlation functionf(τ) as follows;

1 N−τ f(τ) = X X(t) ∗ X(t + τ) (1.43) N − τ n=0 Here we can see that increasing the lag time τ reduces the useful length of data that can be averaged to obtain accurate correlation function. So in our calculation we chose the maximum τ value of 1 ps. The time-correlation function for all three strands calculated using RESP and DFT B3LYP/6-311+G(d,p) charge are shown in ﬁgures 1.13 and 1.14 respectively. 35

Figure 1.13: Time Correlation function (Black curve) ﬁtted with double exponential decay function (red curve) of three DNA strands derived using RESP charge.

Figure 1.14: Time Correlation function (Black curve) ﬁtted with double exponential decay function (red curve) of three DNA strands derived using DFT B3LYP/6-311+G(d,p) derived charges. 36

Next we fitted time correlation functions using several exponential decay functions. The best fit was obtained using two exponential decay function with different exponents as shown below.

c(t) = Ae−t/τ1 + (1 − A)e−t/τ2 (1.44)

Here A is the pre-exponential factor describing the probability of relaxation. τ1and τ2are decay times describing two different relaxation pathways. The values of A,τ1and τ2 for DNA strands are shown in table 1.3.There is an ultrafast decay component at about 20 fs and a longer decay component at about 2 ps that is typical of the solvation dynamics of charged solute in rigid water model [46][47]. Hynes et al. [48] in their DNA solvation study report similar ultrafast decay constant of ∼ 60 fs, longer decay rate of ∼ 1 − 2 ps. Majority of correlation decay in our simulation is from fast dynamics in both RESP and DFT B3LYP/6-311+G(d,p) charge cases. The ultrafast relaxation is mostly attributed to the libration motion of water [46][47][48][49] and the slower decay originates mostly from the interaction of the DNA base with the water molecules and the counterions [48]. The correlation decay rate doesn’t seem to be dependent on the charge distribution in solute, as there is only small difference in decay rate in DFT B3LYP/6-311+G(d,p) and RESP case; similar to the results reported in literature [46][47], where the difference correlation function between charged and uncharged solute was small.

Table 1.3: The ﬁrst three columns with decay constant values is for the DNA strands that used resp charge and the next three is for the DNA strand that used DFT B3LYP/6- 311+G(d,p) charge.

GC12(RESP ) AT12(RESP ) GC2AT10(RESP ) GC12(DFT ) AT12(DFT ) GC2AT10(DFT ) A 0.75 0.88 0.69 0.76 0.85 0.69

τ1 (ps) 0.02 0.03 0.014 0.018 0.029 0.015

τ2(ps) 1.64 3.68 1.23 1.59 3.18 1.34

−1 The rate becomes adiabatic for poly(AT )12(DFT ) and has the value of 1.44 ns , which is close to the experiment. 37

A single characteristic longitudinal relaxation time τL was obtained for each time correlation function shown in ﬁgures 1.13 and 1.14 by averaging their respective decay rate constants as follows;

1 A (1 − A) = + (1.45) τL τ1 τ2 where A is the probability that decay occurs with the average life time of τ1. The result is shown in table 1.4, where we can see the average relaxation life time is about 0.02 ps.

Table 1.4: Characteristic relaxation time

Decay Constant GC12(RESP ) AT12(RESP ) GC2AT10(RESP ) GC12(DFT ) AT12(DFT ) GC2AT10(DFT )

τk(ps) 0.023 0.03 0.019 0.024 0.034 0.022

1.4.5 Rate of hole transfer:

We calculated classical and quantum rate of hole transfer for both adiabatic and non- adiabatic process using equation 1.22 and 1.19 respectively. Following tables shows the rate of hole transfer in three diﬀerent DNA strands.

Table 1.5: Classical rate of hole transfer Classical rate of hole transfer

GC12(RESP ) AT12(RESP ) GC2AT10(RESP ) GC12(DFT ) AT12(DFT ) GC2AT10(DFT )

KA 1.34 0.79 1.31 1.27 1.19 0.49 Adiabatic (ns−1) 0.75 0.42 0.06 0.04 3.5 0.02 Non − adiabatic (ns−1) 0.10 0.33 0.08 0.05 4.1 0.01

The adiabaticity parameter is ~ 1 for all the DNA strands in both classical and quantum corrected hole transfer rates so it not conclusive whether the rate is adiabatic or non-adiabatic. In general the classical hole transfer rate is an order of magnitude bigger than the quantum corrected hole transfer rate. Compared to experimental hole transfer rate which is in the order of picoseconds the classical hole transfer rate between adjacent 38

bases is factor of 10 smaller sans AT12(DFT ) strand. This deviation of the rate from the experimental value is because of the inclusion of the high frequency component in the classical reorganization energy.

Table 1.6: Quantum corrected of hole transfer rate

GC12(RESP ) AT12(RESP ) GC2AT10(RESP ) GC12(DFT ) AT12(DFT ) GC2AT10(DFT )

KA 1.71 1.03 1.68 1.27 1.54 1.24 Adiabatic (ns−1) 1.1 5.04 1.01 0.76 23.48 0.48 Non − adiabatic (ns−1) 0.71 1.33 0.48 0.32 15.14 0.20

−1 The rate becomes adiabatic for poly(AT )12(DFT ) and has the value of 1.44 ns , which is close to the experiment.

In quantum corrected hole rate transfer table we can see that for both poly(GC)12(RESP ) and poly(GC)12(DFT ) the non-adiabatic hole transfer rate is about an order of magnitude lower than the experimental value of 4.1 ns−1. Similar, order of diﬀerences in rate are observed for non-adiabatic mixed poly(AT )5(GC)2(AT )5(RESP ) and poly(AT )5(GC)2(AT )5(DFT ) strands. Among the three strands the trend in hole transfer rate is GC2AT10< GC12

DFT B3LYP/6-311+G(d,p) charges (i.e. delta positive charge from quantum calculation added to MD charge) in GC and AT bases show higher sum of squared charges in GC pair. The sum of squared charges gives a measure of charge delocalization, and higher the delocalization lower the reorganization energy [12]. The sum of squared DFT B3LYP/6- GC AT X 2 X 2 311+G(d,p) charges is higher in AT than GC (i.e q+ = 2.86< q+ = 3.68). However, adding the delta positive DFT B3LYP/6-311+G(d,p) charges to MD charges reverses this GC AT X 2 X 2 trend (i.e q+ = 7.49> q+ = 5.04). This suggest that if we were to run MD calculation using DFT B3LYP/6-311+G(d,p) charge the reorganization energy of GC pair would be lower than that of AT pair due to relatively higher delocalization of charge in oxidized GC. We believe this is a reason why our calculation of rate of charge transfer from MD simulation is opposite in trend (i.e AT>GC) compared to the experiment (GC >AT). The AMBER MD package we use to run our simulation comes with a force ﬁeld that is optimized for neutral RESP charge and switching to DFT B3LYP/6-311+G(d,p) charge would require reevaluating the force ﬁeld parameters which is a serious undertaking and beyond the scope of this research.

Table 1.7: Correlation between charge localization and reorganization energy

P 2 P 2 Base Pair λclassical qi λclassical qi (eV) (RESP) (eV) (DFT ) poly(GC) 1.39 7.42 1.47 7.49 poly(AT ) 1.16 5.81 0.93 5.04 poly(GCAT ) 1.42 7.42 1.51 7.49

1.5 Conclusion:

We used molecular dynamics to study the reorganization energy and rate of hole transfer in three different 12 base pairs long DNA strands. There was a decrease in reor- 40 ganization energy due to quantum scaling and the subsequent hole transfer rate calculated for poly(AT) is in good agreement with the experiment. We also compared the effect of charge (i.e. RESP and DFT B3LYP/6-311+G(d,p)) and base pairs (AT, GC and mixed) on reorganization energy. We found that reorganization energy is directly correlated to charge delocalization as measure by sum of squared charges. GC+base pair has higher sum of charge square and the reorganization energy of poly(GC) strand is greater than poly(AT) by ~0.4 eV. The reorganization energy in mixed (AT5GC2AT5) is slightly greater than poly(GC) which can be attributed to the larger propeller twist motion of poly(AT) strand. The reorganization energy of poly(AT) and poly(GC) pair that we calculated using RESP charge is similar to the value reported by Kubar et. al. who also used same hole calculation technique as we did but calculated reorganization energy in different way by running MD simulation of initial and final state and then calculating the reorganization energy as the difference of these two states. Our method much easier to use as we only have to run the simulation once. Also since we use molecular coordinate from equilibrium MD calculation and calculate reorganization energy from fluctuation in electrostatic energy between donor and acceptor alone, comparing effect of charges on reorganization energy become a much simpler task. Our rates of hole transfer in poly(AT) and poly(GC) show opposite trend than the experimental result of Lewis et al. We think this discrep- ancy arises in our MD calculation and in the MD calculation of Kubar et. al. because the RESP charges incorrectly reports higher charge localization in GC compared to AT. This is opposite of what is obtained in DFT B3LYP/6-311+G(d,p)/6-311+G(d,p) calculation, which is a more accurate higher level quantum calculation. 41

Part II

Exciton Delocalization in DNA 42

2.1 Background:

Interest in the excited state of DNA regarding whether the singlet excited states of DNA are localized on single bases or delocalized over a certain number of them can be traced far back as half a century ago [50][51][52][53]. Understanding the nature of this ultrafast excited state motion in DNA is of utmost importance to tease out the mechanism of UV-induced photodamage that can lead to cancer [54][55]. Studies in mechanism of radiation induced damage in DNA have mainly focused on understanding the migration of charge through stacked base pairs once the charge is separated [56][57][58]. Available studies in excited state dynamics of DNA have given mixed result ranging from strongly coupled excited state [59][60][61][62], to excited state localized on single base [53] and to excimer state [52]. Our collaborator Dr. Lewis at Northwester University have performed absorption and CD spectroscopy on GC hairpin ranging from 2 to 4 bases. The absorption spectrum of the GC hairpin are blue-shifted as number of bases increases. This indicates that the proximity of the base pairs in DNA that facilitates pi-pi interaction allows delocalized exciton state with shared oscillator strength between adjacent bases. The delocalized excited state is caused by the resonance energy transfer between monomers. The strength of this energy transfer is determined by the exciton coupling between the monomers serving as chromophores [60]. This exciton coupling causes shift in the absorption peak of DNA. For biological molecule in water this shift in absorption peak is cloaked by inhomogeneous broadening in water which is very large i.e. W >0.2 eV [63] compared to a typical exciton coupling value for DNA bases i.e. V0 =0.04 eV for AT [60] . Despite strong inhomogeneous broadening of the spectrum of bio-polymers in water, an optical spectrum of DNA in some instances is still sensitive to the interaction of its monomers [63] [64]. Exciton coupling is very sensitive to the structural and electronic conﬁguration of bases in DNA and it strength changes as DNA undergoes conformational adjustment in response to the alteration to its environment. In the case of DNA interacting with other molecule, tracing the change in exciton coupling strength of DNA would allow us to peek into the interaction dynamics of DNA, which is not well understood. The exciton 43 coupling strength can be calculated using an absorption spectrum alone[60]. However, CD spectrum is much more sensitive than absorption spectrum to the change in coupling strength [61]. This is especially true for DNA where the chromophore is planar and forms a long chain of parallel stacks so that the coupling strength uniquely deﬁnes the CD spectrum [63]. Here we extract exciton coupling parameter from experimental absorption and CD data using Van Vleck’s sum rule [65]. After the exciton coupling strength calculation it is natural to wonder whether exciton is localized within a single GC base or delocalized over number of base pairs. A general rule is if the inhomogeneous broadening much greater than coupling W V0an exciton is localized in a single chromophore and if V0/W ∼ 1 then it is delocalized over number of chromophores [66]. We calculate the inhomogeneous broadening using electronics structure method and then used the coupling and inhomogeneous broadening value to numerically calculate delocalization length.

2.2 Theory:

We can describe circularly polarized electromagnetic radiation using vector potential.

~ i(~q·~z−wt) x y Alt/rt(r, t) = Re[A0e (ˆe ∓ ieˆ )] (2.46)

x y x y where Alt/rt(r, t) is the vector potential for left (ˆe − ieˆ ) and right (ˆe + ieˆ ) circularly polarized light, A0 is the amplitude of the wave vector propagating in ~z direction, , w is the frequency, ~q = nrw/c is the wave vector, nr is the refractive index of the medium, c is the velocity of light and eˆx and eˆy are unit vectors in x and y directions, respectively. Experimentally the absorption of light is approximated from the decrease in intensity of transmitted light with the sample of thick. ness L.

I A = − log( ) = − log(e−αLN ) = αLN (2.47) I0 where α is the absorption cross-section area and N is the number density of the sample. CD spectrum is measured as the diﬀerence in absorbtance A of left and right polarized light and is historically speciﬁed in terms of ellipticity angle. 44

∆A = AL − AR = ∆αL−RLN (2.48)

E − E ln(10) tan Θ ≈ Θ = L R = ∆A (2.49) EL + ER 4

Here EL and ER are the magnitudes of the electric ﬁeld vectors of left-circularly and the right-circularly and polarized light, respectively and the angle Θ is in degrees. Interaction of electromagnetic radiation with electrons in molecules results in absorption of energy. This interaction can be represented as follows[67].

1 Vˆ = − drA~ (r) ˆj(r) (2.50) lt,rt c ˆ lt,rt ˆ where Vlt,rtis matrix element, ˆj is an operator of an electron current density. In case of a polymer made of n monomer with no electronic overlap this interaction can be written as follows as the sum of interaction with each monomer i.

" n # ˆ 1 X i(~q·~z−wt) ˆx ˆy Vlt,rt = −Re dr A0e (ji ∓ iji ) (2.51) c ˆ i=1

" n i(~q·~z−wt) # ˆ X A0e i~q·~ρ ˆx ˆy Vlt,rt = −Re dρe (ji ∓ iji ) (2.52) i=1 c ˆ

Here ~ri and ~zi represent the average coordinate of the monomer i and ~ρ = ~r − ~ri. We can substitute this expression for the light matter interaction in the Fermi Golden rule to obtain the transition rate from ground state|0 > to excited states |α >.

2π D E 2 ˆ Γlt,rt = a|Vlt,rt|0 δ(Eα − E) (2.53) ~

2π A2 n D E 0 X ik(~zi−~zj ) ˆx ˆy ˆx ˆy Γlt,rt = 2 δ(Eα − E)e ((ji ∓ iji )(q))0α((ji ± iji )(−q))α0 (2.54) ~ c i,j=1;α

x ∗ i~q·~ρˆx where where ((ji )(q))0α = dρψ0e ji (p)ψα, α numerates the electronic excited states of the monomer and ψ is the´ molecular wave function of the monomer. 45

Absorption cross-section area is deﬁned as the ratio of the rate of energy (E) absorption over the energy ﬂux (I) through a unit area and it related to the transition rate Γ as follows;

EΓ α(E) = (2.55) lt,rt I The energy ﬂux (I) is given by the absolute value of the Poynting vector S.

n c I = |S| = r E2 (2.56) 4π 0

Here nr is the refractive index of the medium, c is the velocity of ligh and E0 is the amplitude of the electric ﬁled. Subsituting the expression for Γ and I we can write absorption cross-section as follow;

2π2 1 E n D E X ik(~zi−~zj ) ˆx ˆy ˆx ˆy α(E)lt,rt = 2 δ(Eα − E)e ((ji ∓ iji )(q))0α((ji ± iji )(−q))α0 ~ cnr ω i,j=1;α (2.57) Subsituting this value of α in equation 2.47 we can obtain the general expression for the absorption of light of photon energy E = ~w in the linear regime by a sample of thickness L and concentration N as follows;

2π2 LN E n D E X ik(~zi−~zj ) ˆx ˆy ˆx ˆy Alt,rt(E) = 2 δ(Eα−E)e ((ji ∓ iji )(q))0α((ji ± iji )(−q))α0 ~ ln(10)cnr ω i,j=1;α (2.58) For absorbance we can ignoring the small products ~q · ~p and ~q · zˆ because wave length of light is much bigger than size of the molecule. In this approximation there is no difference between left and right hand term. Conformational averaging over the random orientation of the molecule with respect to the light propagation direction, eˆz can be done using the identity ((êz · a)(êz · b) = a · b/3) which gives usva factor of 1/3 in the final expression.

2 n 4π LN X D 0α α0E A(E) = E δ(E − Ea)ˆµi µˆj (2.59) 3~ ln(10)nrc i,j=1;α i ˆ where µˆ = ω dρj is the dipole moment operator. ´ 46

0 The eﬀect of polarization appears when one expands the term eiq.(~zi− ~zj )+iq(~ρ−~ρ ) ≈ 0 1 + (~zi − ~zj) + iq(~ρ − ~ρ ) in equation 2.58. Both second and third term can be diﬀerent for left and right polarized light and ellipticity angle of CD spectrum can be expressed summing the average of the these two terms i.e Θ = Θ1 + Θ2.

2 n π LN 2 X D 0α α0 E Θ1 = − 2 2 E δ(E − Ea)~rij(ˆµi × µˆj ) (2.60) 3c ~ i,j=1;α

2 n 2π LN 2 X D ˆ0α α0 0αˆα0E Θ2 = − 2 2 E δ(E − Ea)(ii mˆ j +m ˆ i ij (2.61) 3c ~ i,j=1;α ˆ 1 ˆ where ~rij = (~zi − ~zj), ii = iωµˆi and mˆ = 2c ~ρ × jdρ is the magnetic moment of a monomer. ´ Next we describe the exciton coupling between the monomers using a tight binding Hamiltonian[68].

X + X + H = Vijci cj + φici ci (2.62) i6=j i + Here ci and ci are annihilation and creation operators in monomer site i and φi is the random monomer excitation energy. This random excitation energy can be characterzied by a Gaussian distribution fuction P (φ) with conﬁgurational average φ =< φi >. Vij is the coupling between monomer i and j. This coupling value is largest for the adjacent monomers and its value decreases as distance between monomers increases. For DNA the distance betweent the adjacent monomers is comparable to the size of the monomer so coupling term cannot be reduced to just dipole-dipole interaction and must include various multipole moments.

All monomer posses transition dipole moment µi and magnetic moments mi having identical absolute value µ0 and m0. The operator of transition dipole and magnetic moment can be expressed as

+ µˆi = µi(ci + cj) (2.63)

+ mˆ i = mi(ci − cj) (2.64) 47

Combining equations 2.59 and 2.63 and then integrating we get

∞ A(E) 4π2 LN nµ2 dE = abs 0 (2.65) ˆ E 3~ ln(10)nrc 0 Similarly, combining equation 2.64, 2.60 and 2.61 integrating we get

∞ 2 n 2 n Θ(E) π LNcd X 4π LNcd X ˆ dE = − 2 2 rijVijµi × µj − 2 Vijjimˆ i (2.66) ˆ E 3c ~ i6=j 3c~ i,j=1,α 0 where We consider the fact that in DNA all monomers have planar structure and they are parallel to each other. This causes the dot products of the matrix elements for transition currents and magnetic moments to be zero and only the ﬁrst term in equation 2.66 survives. We now take the ratio of remaining ﬁrst term of equation 2.66 over 2.65 [61].

∞ Θ(E) dE E 0 n − 1 MCD V0 ´∞ = − ln(10) d sin(φ)nr (2.67) A(E) 2n Mabs c~ dE E ´0 where Mabsand MCD are molar concentrations. ·180·1000 Rearranging the above equation and using Ψ = Θ π for conversion to experimental unit of mill-degrees, d = 3.4Å for the distance between two adjacent base pair, 0 and φ = 36 and nr = 1.33 for the refractive index of water and the experimental ratio of MCD ∼ 10, we get following expression for nearest neighbor exciton coupling. MAbs

∞ Ψ(E) dE E n 0 V0 ≈ −0.11 ´∞ (2.68) n − 1 A(E) dE E ´0 2.3 Experimental Data and Calculations:

The synthesis and characterization of C12−linked poly(GC) hairpin M80-M82 as shown in ﬁgure 2.15 was done in Lewis group at Northwestern University using the method reported previously[69]. Lewis group previously reported stable stilbene-linked hairpins 48

[70], however for our purpose of calculating exciton coupling for singlet excitation the dodecane (C12) linked hairpins are better because unlike stilbene-linked hairpins the dodecane linker does not have any absorption band that overlaps with the singlet S0 to S1 low energy absorption of DNA base.

Figure 2.15: Skeletal structure of C12−linked DNA hairpins.

The absorption spectra of GC hairpins M80-M82 are shown in figure 2.16. The absorption spectrum is normalized for the concentration of number of bases and we see slight hypochromism with increasing number of bases as expected for DNA [71][60]. There are two peaks in absorption spectrum in figure 2.16; the low-energy absorption peak at the wave-length of about 260 nm (4.5 eV) corresponding to singlet S0 to S1 excitation is well separated from the high-energy excitation around 200 nm (6.1 eV) by energy of the order of 1 eV, which is much greater than the inhomogeneous broadening that are typically of the order of 0.4 eV [61]. This allows us to use sum rule to calculate the exciton coupling associated with singlet S0 to S1 excitation. CD spectra of GC hairpins M80-M82 are shown in figure 2.17. Positive cotton effect at lower wave length is seen with zero crossing at 270 nm for hairpin M80 and zero crossing at 245 for M81 and M82. This positive cotton effect can be understood by realizing that DNA is right-handed and CD spectra is calculated as difference in left and right 49

-M80

1.29

-M81

-M82 -1 cm -1

0.86 M -4 x 10 x /n

0.43

0.00

200 250 300 350

Wavelength, nm

Figure 2.16: .Base pair normalized absorption spectra of C12−linked DNA hairpins in 10mM phosphate buﬀer (pH 7.2) containing 100mM NaCl 50 circularly polarized light. At shorter wavelength in the vicinity of zero crossing point the dipole moments of interacting bases are aligned almost in parallel, creating a combined exciton state [60][61]. Here the right-hand polarized light is most strongly absorbed. On the other hand the longer wavelength represents interacting bases dipole moment that are directed almost opposite of each other, which allows for the strong absorption of left-hand polarized light and hence the positive cotton eﬀect.

Figure 2.17: Base pair normalized Circular dichroism spectra of C12−linked DNA hairpins in 10mM phosphate buﬀer (pH 7.2) containing 100mM NaCl 51

2.3.1 Exciton Coupling Calculations:

To calculate exciton coupling we need to obtain the numerator and denominator in equation 2.68 by numerically integrating absorption and CD spectra respectively for each M80 to M82 GC hairpins. The absorption spectra is integrated over the low energy absorption peak that is associated with first electronic excitation. In the absorption spectrum we can see that the low energy absorption mixes with the high energy spectrum at about 230 nm (5.4 eV) which causes a problem in accurately integrating the low energy absorption spectrum.This can be addressed by realizing that the low energy spectrum is symmetrical about the maximum and only the high energy side of this spectrum is convoluted with second absorption peak while the lower energy side of this absorption maximum remains essentially unaffected by the high energy absorption at 200 nm. Therefore instead of doing the full integration over the absorption spectrum of first excited state we can do double integration over half the spectrum on the longer wavelength side. We used Origin software [72] to do the integration and got following results for M80-M82 hairpins. ∞ abs A(E) Ix = dE E 0 abs ´ abs abs IM80= 0.16, IM81= 0.22 and IM82= 0.21 Integrating over the ellipticity angle of CD spectra we run into the same problem of contamination of low-energy spectrum as we saw for absorption spectra. To utilize the same solution that we used in integrating absorption spectrum we redefine CD absorption energy variable as E = E0 + ξ, where E0 is the energy corresponding to zero circular dichroism. Then the double integration over half the lower energy CD spectra is given as ; ∞ 0 CD ψ(E) ψ(E0+ξ) I = dE ≈ 2 dξ 2 ξ x E (E0+ξ) ´0 −∞´ Again using Origin software we performed the integration on the lower energy positive half of the CD spectra for each of the three hairpins. CD CD CD IM80= 0.043, IM81=0.09 and IM82= 0.076 Computing these results in equation 2.68 and using number of base pairs n = 2, 3 and 4 for M80, M81 and M82 respectively we get following exciton coupling. M80 M81 M82 V0 = 0.057 eV, V0 = 0.06 eV and V0 =0.052 eV. We corroborated this sum rule calculation using semi-empirical Zerner’s Intermediate 52

Neglect of Differential Overlap (ZINDO) [73] method available in Gaussian software [74]. We optimized the structure of GC using density function theory (DFT) with B3LYP/6- 311-G(d,p) basis sets. DFT optimization of GC dimer without the backbone distorts the structure so much that they are no longer Watson-Crick base pairs. To get around this problem we took the coordinate of the DFT optimized single GC pair and translated and rotated it by 3.4 Å and 360 respectively using the script in appendix D. This was followed by excited state ZINDO calculation on the optimized GC monomer and GC dimer. First ten excited states of GC monomer and dimer is shown in appendix E. ZINDO calculation of single pair shows that the first bright state is around 3.82 eV and the second bright state is around 4.26 eV and the third bright state is around 4.42 eV. The oscillator strength of second bright state is more than twice as much bigger than first bright state and a factor of 10 bigger than the third bright state. So the high oscillator strength of second bright state appears to overshadow the lower frequency first bright state in the experiment. Compared to ZINDO spectrum of monomer GC the ZINDO spectrum of GC dimer shows a clear splitting of the excited state into a higher energy and lower energy state. In the appendix E we can see the excited state 4 corresponding to excitation energy of 4.3 eV of single GC spectrum splitting into state 7 and state 9 of GC dimer spectrum corresponding to excitation energy of 4.17 eV and 4.28 eV respectively. The oscillator strength of state 9 is factor of ∼ 10 bigger than that of state 7 which shows blue shift of the absorption spectrum in GC dimer. Exciton coupling is calculated as half of the difference of the energy of the higher (state 9) and the energy of the lower split state (state 7) as shown in figure 2.18, which we found to be V0 = 0.058 eV. The average exciton coupling value from fitting the experimental data is V0 = 0.056 eV and considering the the error limits of ZINDO [75] this value is in excellent agreement with coupling value from ZINDO.

2.3.2 Inhomogeneous Broadening Calculation:

Stokes shift is the diﬀerence between max absorption (hf1) and max emission (hf2) energy and is related to the reorganization energy by the relation[76]; 53

Figure 2.18: Schematic diagram of exciton coupling. Monomers A and B can interact in excited state S1and split to higher and lower energy state. Direction of the dipoles of the monomers determine the shift in energy. The red shift is due to the head to tail dipole alignment and blue shift is due to the head to head dipole alignment.

(hf − hf ) λ = 1 2 (2.69) 2

hf1 = hf2 + λ1 + λ2

(λ + λ ) λ = 1 2 (2.70) 2

Here λ1 and λ2 are the energies associated with relaxation of Franck–Condon states as shown in the figure 2.19. We used time dependent density function theory (TDDFT) available in G09 package to calculated the energy of the four states involved in absorption and emission cycle shown in figure 2.19. Running density function theory (DFT) and its time dependent variation (TD-DFT) calculations with B3LYP/6-31G+(d,p) [77] force field in both gas phase and in recently developed State Specific Polarizable Continuum Model in TDDFT (PCM-TDDFT)[78][79] we were able to calculated internal and solvent reorganization energy associated with 54

Figure 2.19: Electronic states of GC associated with absorption and fluorescence cycle. absorption-emission cycle. The excitation and emission energy and the associated Stokes shift of GC base in both gas and solvent phase are shown in table 2.8. The excitation energy of 4.67 eV calculated computationally is very close to the experimental excitation energy of 4.8 eV. The excited state and ground state Stokes shift of GC base pair in vacuum are almost the same, however in water the ground state Stokes shift is almost twice as much than the excited state Stokes shift. We find the internal reorganization energy associated with the excitation of first excited state to be 0.21 eV and the solvent reorganization energy associated with the same excitation to be slightly higher with the value of 0.24 eV. 55

Table 2.8: Reorganization Energy associated with GC S0 − S1excitation.

Energy State Vacuum (eV) PCM Water (eV) Excitation 4.78 4.67 Emission 4.37 3.77 Excited State Stokes Shift 0.20 0.33 Ground State Stokes Shift 0.21 0.57 Reorganization energy 0.21 0.45

The position and orientation of water molecules with respect to each GC hairpins at the time of excitation differs and this results in an inhomogeneously broadened absorption spectrum that we see in figure 2.16. The extent of inhomogeneous broadening is reflected in the width of the spectrum and can be found by fitting this spectrum with Gaussian function. The width W of the experimental spectra are generally given in term of FWHM, and is related to variance σ2of Gaussian distribution as;

√ W = 8σ2 ln 2 (2.71)

From Marcus theory [9] it follows that the variance σ2 of the Gaussian energy distribution is related to reorganization energy by the equivalence;

2 σ = 2λkbT (2.72)

Substituting the value of λ = 0.45eV from the table 2.8 in equation 2.72 we get σ = 0.15eV and further substituting that value in equation 2.71 we get the extent of inhomogeneous broadening W = 0.35eV for S0 − S1 excitation, which is similar to the experimentally obtained value of W = 0.4eV reported for the excitation of AT hairpin [61]. Gaussian ﬁt of the experimental GC hairpin absorption data after conversion of unit from nm to eV as shown in ﬁgure 2.20 yielded W = 1.03eV . This factor of two bigger 56

Adj. R-Square 0.98722 Value Standard Error 2.0 xc 4.90168 0.00516 w 0.44464 0.00624 B A 1.10304 0.00984

1.5

1.0 Absorbance

0.5

0.0

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Wavelength (eV)

Figure 2.20: Gaussian fitting of absorption of spectrum of a GC hairpin. experimental reorganization energy value compared to the computational result suggest intense mixing between the low and higher energy absorption peak. Indeed the comparison of absorption spectrum of AT hairpin [61] and GC hairpin in figure 2.16 shows that the extent of mixing is more pronounced in absorption spectrum of GC hairpin. In GC hairpin the absorption profile on the high energy side of the S0 − S1 excitation absorption peak dips less than half the height of the peak before rising up as the profile of the high energy th absorption peak. In AT the same profile dips to almost 3/4 of the S0 − S1 excitation absorption peak making it amenable to extraction of inhomogeneous broadening using Gaussian fit. 57

2.3.3 Exciton Delocalization Length:

Depending on the strength of the coupling excited state wave function can be spread over a number of molecules called an exciton delocalization length which aﬀects optical response and energy transport properties of a chromophore. At low temperature this length is determined by the competition between the intermolecular exciton transfer interactions and the static disorder in the environment. At higher temperatures inelastic scattering of excited state wave function by nuclear motions (dynamic disorder) also come into play and lead to an exciton coherence length that is shorter than the delocalization length at lower temperature. Here we try to obtain the delocalization length in GC hairpin using following two numerical methods.

1. A well-known measure for the energy dependent delocalization size of an exciton in the presence of disorder is participation ratio [80][81][82], which is given as the inverse of the sum of the fourth power of the eignvectors. The exciton wave function |k > of a singly excited polymer chain can be written as superposition of the 0 monomer excited state wave functions |φi >. The coeﬃcients Cis are obtained by 0 diagonalization of the exciton Hamiltonian in equation 2.62. These Cis can give us important information about the exciton delocalization length.

N X |k >= Ci|φi > (2.73) k,i=1

N X 4 Pk = |Ck,i| (2.74) i=1 1 L = (2.75) < Pk >

Here Pk is the participation ratio and L is the inverse avearge participation ratio that gives the exciton delocalization length. The above equations can be better understood in terms of two limiting conditions. First in the completely localized state which can be found in highly disordered system where exciton is collapsed

in a single monomer, we have Pk = N, leading to localization length of L = 1/N as expected. Second in the completely delocalized state which can be found in 58

perfectly ordered chains where the exciton wave function is spread equally over all

the monomers in the chain, we have Pk = 1/N, giving expected delocalization length of L = N. Projecting the participation ratio on the diﬀerent states about the max excitation energy state we can see the eﬀect of site energy disorder on delocalization length. P δ(E − Ek)Pk Pk(E) = P (2.76) δ(E − Ek)

2 −w(E−Ek) δ(E − Ek) = e

2 PN PN 4 −w(E−Ek) k=1 i=1 |Ck,i| e 1 P (E) = 2 , where w ∼ (2.77) PN PN −w(E−Ek) 2 k=1 i=1 e W 1 L(E) = (2.78) < Pk(E) > where P(E) measures the number of monomers that on the average participate in the states that occur at energy E, w is the weight, W is inhomogeneous broadening and L(E) is energy speciﬁc exciton delocalization length.

2. Another method to calculate delocalization length is to use the Gibbs entropy formula from statistical mechanics where entropy is deﬁned as a measure of the probability that the system (exciton) is spread out over diﬀerent microstate (monomers).

N X 2 2 Sk = − |Ck,i| ln |Ck,i| (2.79) i=1

L = e (2.80)

Taking the exponential of the average entropy < Sk > gives number of states with signiﬁcant probability of being occupied or in our case the number of monomers that the exciton is delocalized. We can quickly test this deﬁnition by substituting

the value of Ci for fully localized and fully delocalized case in above equation, which yields delocalization length of 1 and N respectively as expected. Similar to ﬁrst method we can also obtain the eﬀect of site-energy disorder in delocalization length of the exciton as;

2 PN PN 2 2 −w(E−Ek) − k=1 i=1 |Ck,i| ln (|Ck,i| ) e 1 Sk(E) = 2 , where w ∼ (2.81) PN PN −w(E−Ek) 2 k=1 i=1 e W 59

L(E) = e− (2.82)

where S(E) measures the number of monomers that on the average participate in the states that occur at energy E, w is a scaling factor, W is inhomogeneous broadening and L(E) is the energy speciﬁc exciton delocalization length.

In the scilab program in appendix F we have three modules. The first module generates n × n tri-diagonal normal Hamiltonian matrix with mean E = 4.5 eV that corresponds to experimental absorption max, standard deviation of σ = 0.15 obtained from inhomogeneous broadening calculation and off diagonal matrix value V0 = 0.056 eV that corresponds to exciton coupling. The second and third module use the eignvectors from the first module and calculate delocalization length using equation 2.80 and 2.82 respectively. √ This process was repeated for N iteration until the standard error SD = σ/ Nwas less than 1%. The max delocalization length of 2.13 bases for the eigenstate corresponding to 4.9 eV excitation and length of 2.9 bases for the eigenstate corresponding to 5.05 eV excitation was obtained from inverse participation ratio and taking exponential of entropy respectively as shown in table 2.9. For AT base stack Emanuele et al.[81] numerically calculated that the excitation is delocalized over two bases. Our study shows similar value for GC delocalization length. These energies corresponding to maximum delocalization length is close to the middle of the H-band and for the energies away from the center of the band the delocalization length drops off rapidly. This is in good agreement with the general understanding that states close to the center of the one-exciton band stay more delocalized in the presence of disorder than states lying on the edge of the band. [82][80]. 60

Table 2.9: Delocalization length in the unit of number of bases and its dependence on site-energy.

Energy Delocalization Delocalization PR_Err S_Err Length (1/PR) Length exp (S) 3.4 1.15034 1.32357 0.00141385 0.00248311 3.55 1.19539 1.40306 0.00163629 0.00290351 3.7 1.25336 1.50664 0.00184312 0.00333831 3.85 1.32624 1.63532 0.00202108 0.00375811 4 1.41638 1.79133 0.00215902 0.00414138 4.15 1.52637 1.97578 0.00224693 0.0044644 4.3 1.6578 2.18562 0.00227569 0.00470138 4.45 1.80748 2.40887 0.00224105 0.00483287 4.6 1.96029 2.62061 0.00215468 0.00486205 4.75 2.08245 2.7867 0.00205793 0.00482623 4.9 2.1302 2.8811 0.00201406 0.00478087 5.05 2.08188 2.90557 0.00205691 0.00476112 5.2 1.95957 2.88918 0.00215238 0.00476633 5.35 1.80722 2.86516 0.00223731 0.00478042 5.5 1.65847 2.85027 0.00227102 0.00479167 5.65 1.52814 2.84462 0.00224295 0.00479706 5.8 1.41897 2.84322 0.00215837 0.0047987 5.95 1.32894 2.84299 0.00202654 0.00479903 6.1 1.25511 2.84296 0.0018558 0.00479908 6.25 1.19511 2.84296 0.00165366 0.00479908 6.4 1.14745 2.84296 0.00142958 0.00479908 61

2.4 Conclusion:

In this study we successfully used the sum rules to extract exciton coupling value in stacked GC bases from absorption and CD spectra of GC hairpin structure. The condensed expression of the sum rule in equation 2.68 can be used to probe conformation change during drug and GC double helix interaction. Furthermore we used newly developed State Speciﬁc PMC-TDDFT method to calculate reorganization energy and invoking Marcus theory we calculated inhomogeneous broadening. Equipped with exciton coupling and inhomogeneous broadening values we simulated the monomer transition energies by Gaussian functions resembling the absorption bands of stacked GC bases in aqueous so- lutions. From calculation of inverse participation ratio and exponential of entropy we arrived at the average exciton delocalization domain of 2.7 bases. These bases stay in coherently coupled domain for a certain period of time until the system reorganizes to solvate the polarized excited state. 62

Appendix A

Charges for driving force calculations: 63

A.1 Neutral charges in GC and AT base pair used in MD:

Table A.1:

Atoms in GC base pair MD Atom Number Charge Atoms in AT base pair MD Atom Number Charge N9 11 0.0577 N9 11 -0.0268 C8 12 0.0736 C8 12 0.1607 H8 13 0.1997 H8 13 0.1877 N7 14 -0.5725 N7 14 -0.6175 C5 15 0.1991 C5 15 0.0725 C6 16 0.4918 C6 16 0.6897 O6 17 -0.5699 N6 17 -0.9123 N1 18 -0.5053 H61 18 0.4167 H1 19 0.352 H62 19 0.4167 C2 20 0.7432 N1 20 -0.7624 N2 21 -0.923 C2 21 0.5716 H21 22 0.4235 H2 22 0.0598 H22 23 0.4235 N3 23 -0.7417 N3 24 -0.6636 C4 24 0.38 C4 25 0.1814 N1 586 -0.0239 N1 556 -0.0339 C6 587 -0.2209 C6 557 -0.0183 H6 588 0.2607 H6 558 0.2293 C5 589 0.0025 C5 559 -0.5222 C7 590 -0.2269 H5 560 0.1863 H71 591 0.077 C4 561 0.8439 H72 592 0.077 N4 562 -0.9773 H73 593 0.077 H41 563 0.4314 C4 594 0.5194 H42 564 0.4314 O4 595 -0.5563 N3 565 -0.7748 N3 596 -0.434 C2 566 0.7959 H3 597 0.342 O2 567 -0.6548 C2 598 0.5677 O2 599 -0.5881 Sum of Charge Square 7.66165183 5.44898725 64

A.2 GC RESP charge used for driving force calculation:

Table A.2:

Atoms in GC base pair Positive RESP Charge Neutral RESP Charge Delta RESP Charge Delta RESP + MD Charge N9 0.4252 0.4541 -0.0289 0.0288 C8 -0.1363 -0.2911 0.1548 0.2284 H8 0.2697 0.2368 0.0329 0.2326 N7 -0.3994 -0.4015 0.0021 -0.5704 C5 0.3633 0.1281 0.2352 0.4343 C6 0.4192 0.418 0.0012 0.493 O6 -0.437 -0.5692 0.1322 -0.4377 N1 -0.0118 -0.0188 0.007 -0.4983 H1 -0.1027 -0.0536 -0.0491 0.3029 C2 0.6171 0.5668 0.0503 0.7935 N2 -0.6629 -0.8001 0.1372 -0.7858 H21 0.3962 0.3831 0.0131 0.4366 H22 0.3962 0.3831 0.0131 0.4366 N3 -0.4658 -0.5696 0.1038 -0.5598 C4 -0.0431 -0.0912 0.0481 0.0481 H’ N1 0.9035 0.7486 0.1549 0.121 C6 -0.8167 -0.7486 -0.0681 -0.0864 H6 0.3868 0.3502 0.0366 0.2659 C5 0.0305 0.0106 0.0199 -0.5023 H5 0.154 0.1337 0.0203 0.2066 C4 0.2355 0.2112 0.0243 0.8682 N4 -0.798 -0.7443 -0.0537 -1.031 H41 0.4188 0.3904 0.0284 0.4598 H42 0.4188 0.3904 0.0284 0.4598 N3 0.0489 0.0232 0.0257 -0.7491 C2 -0.1627 -0.0708 -0.0919 0.704 O2 -0.4473 -0.4693 0.022 -0.6328 H’ Sum of Charge Square 5.22358974 5.00492386 0.18138408 7.42437025 65

A.3 GC DFT charge used for driving force calculation:

Table A.3:

Atoms in GC base pair Positive DFT Charge Neutral DFT Charge Delta DFT Charge Delta DFT + MD Charge N9 -0.245135 -0.277664 0.032529 0.032529 C8 0.22941 0.178885 0.050525 0.050525 H8 0.185449 0.122857 0.062592 0.062592 N7 -0.121869 -0.201773 0.079904 0.079904 C5 0.065579 -0.031015 0.096594 0.096594 C6 0.219587 0.245761 -0.026174 -0.026174 O6 -0.277648 -0.41089 0.133242 0.133242 N1 -0.416963 -0.453372 0.036409 0.036409 H1 0.507769 0.46072 0.047049 0.047049 C2 0.30623 0.327143 -0.020913 -0.020913 N2 -0.515846 -0.559451 0.043605 0.043605 H21 0.454684 0.373831 0.080853 0.080853 H22 0.317734 0.27487 0.042864 0.042864 N3 -0.152513 -0.288923 0.13641 0.13641 C4 -0.015923 -0.06132 0.045397 0.045397 H’ 0.340569 0.292959 0.04761 0.04761

N1 -0.332439 -0.338932 0.006493 0.006493 C6 -0.161703 -0.182677 0.020974 0.020974 H6 0.175938 0.149001 0.026937 0.026937 C5 0.033072 0.027022 0.00605 0.00605 H5 0.149956 0.127489 0.022467 0.022467 C4 0.187123 0.174683 0.01244 0.01244 N4 -0.469586 -0.503765 0.034179 0.034179 H41 0.36037 0.411678 -0.051308 -0.051308 H42 0.283418 0.255581 0.027837 0.027837 N3 -0.400686 -0.360738 -0.039948 -0.039948 C2 0.391584 0.347772 0.043812 0.043812 H’ -0.445585 -0.42466 -0.020925 0.022498 Sum of Charge Square 2.820379852 2.864791529 0.089808775 7.488157216 66

A.4 AT RESP charge used for driving force calculation:

Table A.4:

Atoms in GC base pair Positive RESP Charge Neutral RESP Charge Delta RESP Charge Delta RESP + MD Charge N9 0.378 0.4441 -0.0661 -0.0929 C8 -0.1625 -0.2987 0.1362 0.2969 H8 0.2918 0.2582 0.0336 0.2213 N7 -0.4191 -0.4279 0.0088 -0.6087 C5 0.1885 0.1608 0.0277 0.1002 C6 0.5162 0.4392 0.077 0.7667 N6 -0.4653 -0.8059 0.3406 -0.5717 H61 0.3867 0.3816 0.0051 0.4218 H62 0.3867 0.3816 0.0051 0.4218 N1 -0.4776 -0.3618 -0.1158 -0.8782 C2 0.5411 0.4025 0.1386 0.7102 H2 0.0911 0.0722 0.0189 0.0787 N3 -0.4989 -0.587 0.0881 -0.6536 C4 0.1514 0.0188 0.1326 0.5126

N1 0.967 0.7856 0.1814 0.1575 C6 -1.1016 -0.9738 -0.1278 -0.3487 H6 0.4415 0.3972 0.0443 0.305 C5 0.3637 0.2564 0.1073 0.1098 C7 -0.0756 -0.0628 -0.0128 -0.2397 H71 0.0402 0.033 0.0072 0.0842 H72 0.0402 0.033 0.0072 0.0842 H73 0.0402 0.033 0.0072 0.0842 C4 0.2663 0.3177 -0.0514 0.468 O4 -0.5312 -0.5066 -0.0246 -0.5809 N3 -0.0139 0.0872 -0.1011 -0.5351 H3 0.1758 0.0407 0.1351 0.4771 C2 -0.1213 -0.1106 -0.0107 0.557 O2 -0.3995 -0.4079 0.0084 -0.5797 Sum of Charge Square 5.09354511 4.68034352 0.30014323 5.8155047 67

A.5 AT DFT charge used for driving force calculation:

Table A.5:

Atoms in GC base pair Positive RESP Charge Neutral RESP Charge Delta RESP Charge Delta RESP + MD Charge N9 -0.240091 -0.271735 0.031644 0.409644 C8 0.219354 0.172242 0.047112 -0.115388 H8 0.186156 0.129114 0.057042 0.348842 N7 -0.159693 -0.225688 0.065995 -0.353105 C5 -0.005422 -0.097847 0.092425 0.280925 C6 0.236335 0.326681 -0.090346 0.425854 N6 -0.485632 -0.568197 0.082565 -0.382735 H61 0.489575 0.388925 0.10065 0.48735 H62 0.316398 0.274138 0.04226 0.42896 N1 -0.151666 -0.250923 0.099257 -0.378343 C2 0.106013 0.066198 0.039815 0.580915 H2 0.20703 0.170466 0.036564 0.127664 N3 -0.060027 -0.182802 0.122775 -0.376125 C4 -0.185766 -0.223231 0.037465 0.188865 H’ 0.35227 0.305396 0.046874 N1 -0.424768 -0.45237 0.027602 0.994602 C6 -0.324767 -0.320879 -0.003888 -1.105488 H6 0.166476 0.136029 0.030447 0.471947 C5 0.789828 0.811826 -0.021998 0.341702 C7 -0.463668 -0.541787 0.078119 0.002519 H71 0.179847 0.166901 0.012946 0.053146 H72 0.154856 0.131652 0.023204 0.063404 H73 0.179847 0.1669 0.012947 0.053147 C4 -0.165046 -0.200112 0.035066 0.301366 O4 -0.371422 -0.356713 -0.014709 -0.545909 N3 -0.542796 -0.564389 0.021593 0.007693 H3 0.467663 0.534554 -0.066891 0.108909 C2 0.455586 0.461242 -0.005656 -0.126956 O2 -0.302829 -0.334044 0.031215 -0.368285 H’ 0.376359 0.348455 0.027904 Sum of Charge Square 3.397281409 3.678654519 0.095052502 5.042830232 68

Appendix B

Code for calculating hole transfer in identical adjacent DNA bases. 69

B.1 Fortran code to calculate driving force

PROGRAM Reorganization_Energy c This program takes charge and coordinate as inputs and calculates reorganization c energy. integer nmax, atm,n1,n2,n3,n4,n5,n6,n7,n8,pn,pnn,i,j integer count_0,count_1,count_rate,count_max parameter (nmax=100000, atm=45848) real x(atm),y(atm),z(atm),box(3),t1(atm),t2(atm),q(atm),qp(28) real p1(12),p2(12),p3(12),p4(12) common /GLOBALS/ x,y,z,t1,t2,n1,n2,n3,n4,n5,n6,n7,n8 double precision v1,v2,v3,v4,enz, ...... c Open the data ﬁle for reading and writing open (10, FILE=’../reimaged-8ns.mdcrd’, STATUS=’OLD’) open (20, File=’md-charge.dat’, STATUS=’OLD’) open (30, File=’positive-charge.dat’, STATUS=’OLD’) open (40, FILE=’atom_num.dat’,STATUS= ’OLD’) open (50, FILE=’energy.txt’,STATUS= ’UNKNOWN’) call system_clock(count_0, count_rate, count_max) ...... c Loop over the data points read (20,*) (q(i),i=1,atm) read (30,*) (qp(i),i=1,28) c Read atom number read(40,*) (p1(i), p2(i), p3(i), p4(i),i=1,12) pn=6 pnn=pn+1 n1=p1(pn) n2=p2(pn) n3=p1(pnn) 70 n4=p2(pnn) n5=p3(pnn) n6=p4(pnn) n7=p3(pn) n8=p4(pn) t1=q do 100 i=n1,n2 t1(i)= qp(1+i-n1) 100 enddo do 101 i=n7,n8 t1(i)= qp(17+i-n7) 101 enddo t2=q do 102 i=n3,n4 t2(i)= qp(1+i-n3) 102 enddo do 103 i=n5,n6 t2(i)= qp(17+i-n5) 103 enddo read(10,*) do 1 j=1,nmax read(10,19) (x(i), y(i), z(i),i=1,atm) read(10,29) box c call subroutine Call sub1(n1,n2,v1) Call sub2(n3,n4,v2) Call sub2(n5,n6,v3) Call sub1(n7,n8,v4) v=v1+v4-(v2+v3) enz=v*14.403D0/1.77D0 write (50,*),enz 71

1 enddo call system_clock(count_1, count_rate, count_max) print *,count_1*1.0/count_rate print *,count_1*1.0/count_rate - count_0*1.0/count_rate 19 format (10F8.3) 29 format (3F8.3) 39 format (5E16.8) 49 format (3f8.3) c Close the ﬁle close (10) close (20) close (30) end ...... c This subroutine calculates potential for GC+. SUBROUTINE sub1(a1,a2,vi) integer atm,n1,n2,n3,n4,n5,n6,n7,n8,a1,a2,i,j double precision vi,vj,r parameter (atm=45848) real x(atm),y(atm),z(atm),qq(atm),t1(atm),t2(atm) common /GLOBALS/ x,y,z,t1,t2,n1,n2,n3,n4,n5,n6,n7,n8 qq=t1 vi=0.0D0 do 3 i=a1,a2 vj=0.0D0 r=0.0D0 do 4 j=1,n1-1 r=sqrt((x(i)-x(j))**2.+(y(i)-y(j))**2.+(z(i)-z(j))**2.) vj=vj+qq(j)/r 4 enddo do 5 j=n1,n2 72 if (j .ne. i) then r=sqrt((x(i)-x(j))**2.+(y(i)-y(j))**2.+(z(i)-z(j))**2.) r=r*2D0 vj=vj+qq(j)/r endif 5 enddo do 6 j=n2+1,n7-1 r=sqrt((x(i)-x(j))**2.+(y(i)-y(j))**2.+(z(i)-z(j))**2.) vj=vj+qq(j)/r 6 enddo do 7 j=n7,n8 if (j .ne. i) then r=sqrt((x(i)-x(j))**2.+(y(i)-y(j))**2.+(z(i)-z(j))**2.) r=r*2D0 vj=vj+qq(j)/r endif 7 enddo do 8 j=n8+1,atm r=sqrt((x(i)-x(j))**2.+(y(i)-y(j))**2.+(z(i)-z(j))**2.) vj=vj+qq(j)/r 8 enddo vi=vi+qq(i)*vj c print *,i, ’vj=’, vj, vi 3 enddo return end ...... c This subroutine calculates potential for GC. SUBROUTINE sub2(a1,a2,vi) integer atm,n1,n2,n3,n4,n5,n6,n7,n8,a1,a2,i,j double precision vi,vj,r 73 parameter (atm=45848) real x(atm),y(atm),z(atm),qq(atm),t1(atm),t2(atm) common /GLOBALS/ x,y,z,t1,t2,n1,n2,n3,n4,n5,n6,n7,n8 qq=t2 vi=0.0D0 do 3 i=a1,a2 vj=0.0D0 r=0.0D0 do 4 j=1,n3-1 r=sqrt((x(i)-x(j))**2.+(y(i)-y(j))**2.+(z(i)-z(j))**2.) vj=vj+qq(j)/r 4 enddo do 5 j=n3,n4 if (j .ne. i) then r=sqrt((x(i)-x(j))**2.+(y(i)-y(j))**2.+(z(i)-z(j))**2.) r=r*2.0D0 vj=vj+qq(j)/r endif 5 enddo do 6 j=n4+1,n5-1 r=sqrt((x(i)-x(j))**2.+(y(i)-y(j))**2.+(z(i)-z(j))**2.) vj=vj+qq(j)/r 6 enddo do 7 j=n5,n6 if (j .ne. i) then r=sqrt((x(i)-x(j))**2.+(y(i)-y(j))**2.+(z(i)-z(j))**2.) r=r*2.0D0 vj=vj+qq(j)/r endif 7 enddo do 8 j=n6+1,atm 74

r=sqrt((x(i)-x(j))**2.+(y(i)-y(j))**2.+(z(i)-z(j))**2.) vj=vj+qq(j)/r 8 enddo vi=vi+qq(i)*vj 3 enddo return end

B.2 Matlab/Octave code for calculatin reorganization energy:

% This program calculates reorganization energy. x=load ’driving_force.txt’; % Initalizating paramaters % All unit in eV TeV=0.0000861705; KbT=300*TeV; hbar=6.58211899*10^(-16); Period=N*10^(-15); f0=2*pi/Period; cm=3.33565*10^(-11); % Hertz to cm^-1 conversion. fcm=f0*cm/(2*pi); r=(f0*hbar)/(2*KbT); r1=r/2; Average = mean(x); Maximun = max(x); Minimun = min(x); x = x - mean(x); % setting the mean to zero...... % Using builtin matlab fast fourier transform program to change data to freq. domain. 75

N=length(x); A=zeros(N,1); % matrix initilization A=ﬀt(x)/sqrt(N); A=ﬀtshift(A); ...... % Claculating hypersine function. B=zeros(N,1); k=0; for j=[-N/2:-1,1:N/2]; k = k+abs(j/j); B(k,1) = (j*r)/sinh(j*r); end ...... % Claculating hypertan function. C =zeros(N,1); m=0; for i=[-N/2:-1,1:N/2]; m = m+abs(i/i); C(m,1) = tanh(r1*i)/(r1*i); end ...... % Putting the classical and quantum part together C_VAR = (A’*A)/N; %This calculates calssical variance C_VAR_MARTIX = A.*conj(A); C_LOW_FREQ = sum(C_VAR_MATRIX(N/2+1:N/2+3031))/N; C_HIGH_FREQ = sum(C_VAR_MATRIX(N/2+3032:N))/N; Q_VAR_sinh = sum(A.*conj(A).*B)/N; % Variance after hypersine scaling. Q_VAR_tanh = sum(A.*conj(A).*C)/N; % Variance after hypertan scaling. LAMBDA_C = C_VAR / (2*KbT) LAMBDA_C_LOW_FREQ =2.0*clac1/(2*KbT) LAMBDA_C_HIGH_FREQ=2.0*clac2/(2*KbT) 76

LAMBDA_sinh=Q_VAR_sinh /(2*KbT) LAMBDA_tanh=Q_VAR_tanh/(2*KbT)

B.3 Matlab/Octave code to calculate correlation function

% This PROGRAM calculates CORRELATION FUNCTION X = load (’driving_force.txt’); N=length(X) Tmin = 0; Tmax = 1000; EX = X - mean(X); Var = (1/N)*(EX’)*EX; tau = zeros (Tmax - Tmin+1, 1); Corr = zeros (Tmax - Tmin+1, 1); time() k = 0; for k = Tmin:Tmax F = 0; F = sum(EX(1:N-k).*EX(k+1:N)); F = F/(Var*(N - k)); Corr(k+1,1) = F; tau(k+1,1) = k/1000; end time() y(:, 1) = tau; y(:, 2) = Corr; save correlation.txt y 77

B.4 Matlab/Octave code to rearrange the charges.

Numbering of atoms in Gaussina program is diﬀerent than in MD. The following program rearranges the atoms number and corresponding charge to match the numbering convention used in MD. X = load("charge.dat"); le=length(X) Y=zeros(le-1,1); if le==30 disp (’AT Charge’); Y(1 ) = X( 16 )+X(30); Y(2 ) = X( 17 ); Y(3 ) = X( 29 ); Y(4 ) = X( 18 ); Y(5 ) = X( 19 ); Y(6 ) = X( 20 ); Y(7 ) = X( 21 ); Y(8 ) = X( 22 ); Y(9 ) = X( 23 ); Y(10 ) = X( 24 ); Y(11 ) = X( 25 ); Y(12 ) = X( 28 ); Y(13 ) = X( 26 ); Y(14 ) = X( 27 ); Y(15 ) = 0.00000 ; Y(16 ) = X( 6 )+X(9); Y(17 ) = X( 7 ); Y(18 ) = X( 8 ); Y(19 ) = X( 2 ); Y(20 ) = X( 1 ); Y(21 ) = X( 13 ); 78

Y(22 ) = X( 14 ); Y(23 ) = X( 15 ); Y(24 ) = X( 3 ); Y(25 ) = X( 12 ); Y(26 ) = X( 4 ); Y(27 ) = X( 11 ); Y(28 ) = X( 5 ); Y(29 ) = X( 10 ); Y A=sum(Y(1:14)) T=sum(Y(16:29)) sum(Y) AT=A+T elseif le<30 disp (’GC Charge’); Y(1 ) = X( 1 )+X(16); Y(2 ) = X( 2 ); Y(3 ) = X( 15 ); Y(4 ) = X( 3 ); Y(5 ) = X( 4 ); Y(6 ) = X( 5 ); Y(7 ) = X( 6 ); Y(8 ) = X( 7 ); Y(9 ) = X( 14 ); Y(10 ) = X( 8 ); Y(11 ) = X( 9 ); Y(12 ) = X( 11 ); Y(13 ) = X( 10 ); Y(14 ) = X( 12 ); Y(15 ) = X( 13 ); 79

Appendix C

Detailed equation used for calculating driving force 80

Here we give the breakdown of the equation used for calculating driving foce. The following ﬁgure shows the barebone structure of GC base pair and numbering convention used in our molecular dynamics simulation.

Figure C.1: DNA base numbering convention used in our molecular dynamics simulation.

The atom numbers of positive base are; G6 (176-190) - C18 (19)(586-597) G7 (209-223) - C19(18)(556-567) The driving force is given by; + Edriving−force = E (6) − E(7) where, E(6) = G(6) + C(19) and E(7) = G(7) + C(18) Detailed formula used in our code to calculate the driving force:

Donor N + Acceptor N + X X 1 qi · qj X X 1 qi · qj Edriving−force = E (6) − E(7) = − i=1 j6=i=1 ε∞ rij i=1 j6=i=1 ε∞ rij

N 1 q+ · q N 1 q+ · q E(6) = E(G6) + E(C19) = X X i j + X X i j i=∀G(6) j6=i=1 ε∞ rij i=∀C(19) j6=i=1 ε∞ rij

 190 175 1 q+ · q 1 190 1 q+ · q 585 1 q+ · q E(G6) = X X i j + X i j + X i j + i=176j=1 ε∞ rij 2 j=176,j6=i ε∞ rij j=191 ε∞ rij  1 597 1 q+ · q N 1 q+ · q  X i j + X i j 2 j=586 ε∞ rij j=598 ε∞ rij  81

 597 175 1 q+ · q 1 190 1 q+ · q 585 1 q+ · q E(C19) = X X i j + X i j + X i j + i=586j=1 ε∞ rij 2 j=176 ε∞ rij j=191 ε∞ rij  1 597 1 q+ · q N 1 q+ · q  X i j + X i j 2 j=586,j6=i ε∞ rij j=598 ε∞ rij 

N 1 q · q N 1 q · q E(7) = E(G7) + E(C18) = X X i j + X X i j i=∀G(7) j6=i=1 ε∞ rij i=∀C(7) j6=i=1 ε∞ rij

 223 208 1 q · q 1 223 1 q · q 555 1 q · q E(G7) = X X i j + X i j + X i j i=209j=1 ε∞ rij 2 j=209,j6=i ε∞ rij j=224 ε∞ rij  1 567 1 q · q N 1 q · q  + X i j + X i j 2 j=556 ε∞ rij j=568 ε∞ rij 

 567 208 1 q · q 1 223 1 q · q 555 1 q · q E(C18) = X X i j + X i j + X i j + i=556j=1 ε∞ rij 2 j=209 ε∞ rij j=224 ε∞ rij  1 567 1 q · q N 1 q · q  X i j + X i j 2 j=556,j6=i ε∞ rij j=568 ε∞ rij  82

Appendix D

Program in Matlab to create a structure of GC dimer 83

%This program takes the optimzed coordinate of a single Watson-Crick GC base pair as input and translats and rotates it 3.4 Angstrom and 36 degrees respectively. Matrix tup gives the cooridinate of GC in positive z direction from the starting structure and matrix tdown gives the structure in the negative direction from the input coordinate. x=load(’coordinate.txt’); n=length(x); tx=zeros(n,3); %translate for i=1:n tx(i,:)= x(i,:)-x(24,:); end ...... % rotate + z-shift rot=[cosd(36),-sind(36);sind(36) cosd(36)]; nrot=[cosd(-36),-sind(-36);sind(-36) cosd(-36)]; a=rot*tx(:,1:2)’; b=nrot*tx(:,1:2)’; up=[a’,tx(:,3)+3.4]; down=[b’,tx(:,3)-3.4]; ...... % tranalate back for i=1:n tup(i,:)= up(i,:)-tx(1,:); tdown(i,:)= down(i,:)-tx(1,:); end 84

Appendix E

ZINDO spectrum of monomer and dimer GC Watson-Crick base pair 85

E.1 Excitation energies and oscillator strengths of GC base pair

Excited State 1: Singlet-A 3.8206 eV 324.51 nm f=0.2087 =0.000 49 -> 51 0.67158 49 -> 52 0.13175 This state for optimization and/or second-order correction. Total Energy, E(CIS) = -151.500971755 Copying the excited state density for this state as the 1-particle RhoCI density. Excited State 2: Singlet-A 3.9209 eV 316.22 nm f=0.0008 =0.000 46 -> 52 -0.18749 47 -> 51 0.28564 47 -> 52 0.52549 47 -> 54 -0.12342 47 -> 57 -0.21473 Excited State 3: Singlet-A 4.0892 eV 303.20 nm f=0.0002 =0.000 41 -> 50 0.18031 41 -> 55 -0.16723 46 -> 50 0.40866 46 -> 53 0.36390 46 -> 55 0.12943 47 -> 50 0.29053 47 -> 53 0.14647 Excited State 4: Singlet-A 4.2598 eV 291.06 nm f=0.5161 =0.000 48 -> 50 0.11203 49 -> 51 -0.13351 49 -> 52 0.66018 Excited State 5: Singlet-A 4.3174 eV 287.17 nm f=0.0034 =0.000 41 -> 50 -0.35679 41 -> 53 -0.18585 86

46 -> 50 -0.14226 46 -> 53 0.36104 46 -> 55 0.28913 47 -> 50 -0.24079 47 -> 55 0.14201 Excited State 6: Singlet-A 4.4193 eV 280.55 nm f=0.0862 =0.000 45 -> 53 0.12336 48 -> 50 0.67138 49 -> 52 -0.10189 Excited State 7: Singlet-A 4.7452 eV 261.28 nm f=0.0001 =0.000 49 -> 50 0.69724 Excited State 8: Singlet-A 4.8523 eV 255.52 nm f=0.0185 =0.000 49 -> 54 0.68435 Excited State 9: Singlet-A 5.1049 eV 242.88 nm f=0.0087 =0.000 39 -> 52 -0.19295 39 -> 54 -0.28002 41 -> 52 0.14005 41 -> 54 0.13559 44 -> 51 -0.18230 44 -> 52 0.44927 44 -> 54 0.23518 44 -> 57 0.11004 47 -> 51 0.10722 47 -> 52 -0.11563 Excited State 10: Singlet-A 5.2710 eV 235.22 nm f=0.0158 =0.000 39 -> 51 0.22169 39 -> 52 -0.18635 44 -> 51 0.48860 44 -> 54 0.28062 47 -> 51 -0.24168 47 -> 54 -0.11475 87

E.2 Excitation energies and oscillator strengths of stacked GC Dimer

Excited State 1: Singlet-A 3.7712 eV 328.77 nm f=0.0250 =0.000 97 ->101 0.42126 97 ->103 -0.14939 98 ->101 -0.20538 98 ->103 -0.39669 98 ->105 -0.25530 This state for optimization and/or second-order correction. Total Energy, E(CIS) = -303.148062926 Copying the excited state density for this state as the 1-particle RhoCI density. Excited State 2: Singlet-A 3.8190 eV 324.65 nm f=0.3062 =0.000 97 ->101 0.36010 97 ->103 0.18673 97 ->105 0.12851 98 ->101 -0.32718 98 ->103 0.37386 98 ->105 0.15380 98 ->106 0.11381 Excited State 3: Singlet-A 3.8965 eV 318.20 nm f=0.0010 =0.000 90 ->105 -0.12144 94 ->101 0.29960 94 ->103 -0.18605 94 ->105 0.49270 94 ->106 -0.10728 94 ->107 -0.12903 94 ->113 -0.22471 Excited State 4: Singlet-A 3.9685 eV 312.42 nm f=0.0009 =0.000 92 ->106 -0.14426 88

96 ->103 0.22019 96 ->105 0.21764 96 ->106 0.53089 96 ->108 -0.12698 96 ->115 -0.21808 Excited State 5: Singlet-A 4.0454 eV 306.48 nm f=0.0004 =0.000 81 ->100 0.18526 81 ->111 -0.14508 82 ->100 0.10612 92 ->100 0.43516 92 ->104 0.33312 92 ->111 0.10221 96 ->100 0.27337 96 ->104 0.11511 Excited State 6: Singlet-A 4.0780 eV 304.03 nm f=0.0002 =0.000 80 -> 99 -0.17339 80 ->109 0.16911 89 -> 99 -0.12021 89 ->102 -0.10332 90 -> 99 0.41384 90 ->102 0.34742 94 -> 99 0.24823 94 ->102 0.11020 Excited State 7: Singlet-A 4.1685 eV 297.43 nm f=0.0947 =0.000 97 ->103 0.16046 97 ->105 -0.33526 97 ->106 0.23111 98 ->103 -0.26461 98 ->105 0.29347 98 ->106 0.33746 Excited State 8: Singlet-A 4.2825 eV 289.52 nm f=0.0132 =0.000 89

81 ->100 -0.28237 81 ->104 -0.17374 82 ->100 -0.16191 92 ->100 -0.11020 92 ->104 0.39034 92 ->111 0.31592 96 ->100 -0.19259 96 ->111 0.11540 Excited State 9: Singlet-A 4.2838 eV 289.43 nm f=0.8686 =0.000 95 ->100 0.14783 97 ->103 -0.21896 97 ->105 0.31168 97 ->106 0.14732 98 ->105 -0.15983 98 ->106 0.47330 Excited State 10: Singlet-A 4.3026 eV 288.16 nm f=0.0034 =0.000 80 -> 99 0.33245 80 ->102 0.20502 89 ->102 -0.10656 90 -> 99 -0.16156 90 ->102 0.35426 90 ->109 0.28543 94 -> 99 -0.21323 94 ->109 0.10864 90

Appendix F

Program in Scilab to calculate delocalization length 91

This is the program in scilab to calculate the energy speciﬁc delocalization length. function [out] = HLoop(N, n, W, E0, v,c) ErrMax = 0.001; Ek = [(E0-8*W):.W:(E0+8*W)]’; //Ek is independent variable in the domain E0-W, E0+W Ekk = length(Ek); //disp (Ek,’EK=’) PP1 = zeros(1,Ekk); PP2 = zeros(1,Ekk); PP3 = zeros(1,Ekk); Mat1 = zeros(1,Ekk); Mat2 = zeros(1,Ekk); Mat3 = zeros(1,Ekk); p_ave = zeros(1,Ekk); s_ave = zeros(1,Ekk); p_std = zeros(1,Ekk); s_std = zeros(1,Ekk); s_err = zeros(1,Ekk); p_err = zeros(1,Ekk); count = 0; for i = 1:N; ...... [H] = Hamiltonian(n, W, E0, v); [P1,P2,D] = PartRat(H,c,Ek); PP1 = PP1 + P1; PP2 = PP2 + P2; PP3 = PP3 + D; ...... // disp (PP1) [summSS,summSS2,summNN] = Entrp(H,c,Ek) Mat1 = Mat1 + summSS ; 92

Mat2 = Mat2 + summSS2 ; Mat3 = Mat3 + summNN ; ...... count = count + 1; if count > 5 p_ave = PP1 ./ PP3 ; p_std = sqrt ( PP2 ./ PP3 - p_ave^2 ); //weighted std dev p_err = p_std ./ sqrt(N); s_ave = Mat1 ./ Mat3 ; s_std = sqrt ( Mat2 ./ Mat3 - s_ave^2 ); //weighted std dev s_err = s_std ./ sqrt(N); //disp (s_err,’s_err=’,p_err,’p_err=’) // if s_err

P22 = P22 + partrat^2 * exp(-c*(a(k)-Ek(j))^2); D1 = D1 + exp(-c*(a(k)-Ek(j))^2); end //disp(length(Ek)) //disp(P11,P22,D1) P1(1,j)=P11; P2(1,j)=P22; D (1,j)=D1; //disp(P1) end clear j k b a endfunction //*********SUBROUTINE************** //This function generates ***entropy*** function [summSS,summSS2,summNN] = Entrp(H,c,Ek) n = max(size(H)); [a,b] = bdiag(H); a = diag(a); // a = diag(a); sumSS = 0; sumSS2 = 0; sumNN = 0; summSS = zeros(1,length(Ek)); summSS2 = zeros(1,length(Ek)); summNN = zeros(1,length(Ek)); for j = 1:length(Ek) for k = 1:n; // loops the eigen value evec = b(:,k); SS = 0; for l = 1:n; // loops the eigen vector r = evec(l); 95 r2 = r^2 ; if r2 ~= 0 then // to avoid r^2=0 SS = SS - (r2) * log(r2); end end sumSS = sumSS + SS * exp(-c*(a(k)-Ek(j))^2); sumSS2 = sumSS2 + SS^2 * exp(-c*(a(k)-Ek(j))^2); sumNN = sumNN + exp(-c*(a(k)-Ek(j))^2); end summSS (1,j) = sumSS; summSS2 (1,j) = sumSS2; summNN (1,j) = sumNN; end // disp (summSS,’s=’,summSS2,’s^2=’,summNN,’Texp=’) clear a b endfunction 96

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