# Effect of environmental noise on charge diffusion in DNA:
Towards modeling its potential epigenetic impact in live processes

M. Rossini<EMAIL_ADDRESS>Institute for Complex Quantum Systems,
Ulm University, 89069 Ulm, Germany.  Integrated Quantum Science and Technology
(IQST), Germany.  O. Ammerpohl  Institute of Human Genetics, Ulm University
$\&$ Ulm University Medical Center, 89081 Ulm, Germany.  R. Siebert  Institute
of Human Genetics, Ulm University $\&$ Ulm University Medical Center
Integrated Quantum Science and Technology (IQST), Germany.  J. Ankerhold
Institute for Complex Quantum Systems, Ulm University, 89069 Ulm, Germany.
Integrated Quantum Science and Technology (IQST), Germany.

###### Abstract

Charge diffusion through desoxyribonucleic acid (DNA) is a physico-chemical
phenomenon that on the one hand is being explored for technological purposes,
on the other hand is applied by nature for various informational processes in
life. With regard to the latter, increasing experimental and theoretical
evidence indicates that charge diffusion through DNA is involved in basic
steps of DNA replication and repair, as well as regulation of gene expression
via epigenetic mechanisms such as DNA methylation or DNA binding of proteins.
From the physics point of view, DNA supports a metallic-like behavior with
long-range charge mobility. Nevertheless, particularly considering a living
environment, charge mobility in DNA needs to take into account omnipresent
noise and disorder. Here, we analyze quantum diffusion of single charges along
DNA-inspired two-dimensional tight-binding lattices in presence of different
sources of intrinsic and environmental fluctuations. It is shown that double-
strand lattices, parametrized according to atomistic calculations of DNA
sequences, offer a complex network of pathways between sites and may give rise
to long-distance coherence phenomena. These effects strongly depend on carrier
type (electrons, holes), the energetic profile of the lattice (composition) as
well as the type of noise and disorder. Of particular interest are spatially
correlated low-frequency fluctuations which may support coherent charge
transfer over distances of a few sites. Our results may trigger further
experimental activities aiming at investigating charge mobility in DNA both in
the native in-vivo context as well as on artificial platforms.

##  I Introduction

The last decade has seen impressive activities to design structures on
nanoscales based on atomic and solid-state platforms. Also living nature
contains a series of substances and building blocks which have been subjected
by molecular chemistry tools to assemble basic molecular units to larger
structures and templates. One of the most prominent substances in this regard
is desoxyribonucleic acid (DNA), the polymer composed of two complementary
polynucleotide chains that represent the universal genetic code of living
species. Its comparably simple building principle based on only four
elementary bases (nucleotides) abbreviated with letters A, C, G, and T, gives
rise to a plethora of two- and three-dimensional meta-structures. The Watson-
Crick complementarity of DNA, where A pairs only with T and C only with G,
leads to their assembly into a ladder structure which is then further folded
into a double helix. The ladder consists of the two DNA strands each having a
5’ to 3’ orientation but placed in forward and reverse orientation through the
complementary base pairing. From the point of technological exploration of
DNA, biochemistry is able to design sequences of variable lengths and tailored
compositions out of these four bases. The binding-specific properties have
made DNA also the ideal construction set to build tailored nano-structures,
for example, in the field of DNA origami dna-origami2006 ; zhan2021 .

Particular attention DNA has received in the context of molecular electronics
where its conduction properties are explored e.g. when linked to metallic
leads dekker2000 ; wang2018 . As typical for many organic molecular
aggregates, long-range charge transfer has been observed, mediated through
$\pi$-orbitals with a typical bandgap between the highest occupied molecular
orbital (HOMO) and the lowest unoccupied level (LUMO) between 3-4 eV depending
on the sequence of nucleobases. However, the DNA-core is coated by a
negatively charged sugar-phosphate backbone and in biological contexts
embedded in aqueous mediaLiepinsh1992 ; McDermott2017 ; DubouDijon2016  which
may mask the nucleotide-based charge transfer. In the latter situation, in
designed DNA oligomers in solution bound to optically excited donor molecules
relatively long-lived charge mobility (up to ms) over relatively long
distances (up to 2 nm) was observed in a series of experiments kohler1 ;
kohler2 ; kohler3 . Other experiments seem to provide convincing evidence that
charge mobility in DNA can be long-range (up to tens of nm) depending on
perturbations in $\pi$-stacking of the bases. One suggested mechanism combines
quantum mechanical delocalization over a few bases with hopping-like diffusion
between those coherent islands.

Apparently, the challenge here lies in the intricate interplay of metallic-
like behavior (coherent charge transfer) with disorder and noise induced by
surrounding degrees of freedom. The goal of this work is to explore this
interplay for single charge transfer along DNA double strands within a
simplistic model in terms of tight-binding lattices. We consider shorter DNA
oligomers where quantum coherence is expected to survive and, most
importantly, parametrize the tight-binding structure with parameters
originating from atomistic simulations hawke_tight-binding_2010 .

The rationale behind this analysis is guided by the following question: To
which extent are noise and disorder, particularly correlated disorder,
supportive for long-range charge transfer through DNA also in an ambient
living environment? There is compelling evidence that nature explores charge
fluctuations and transfer in processes such as DNA replication and repair, as
well as in the epigenetic regulation of gene expression, where epigenetic
regulation refers to stable alterations in gene expression potential that
arise during development and cell proliferation Jaenisch2003 . For example,
charge transfer within and across DNA strands is key for the function of some
DNA binding cellular proteins including polymerases or sulfur-iron containing
DNA repair enzymes OBrien2017 . Charge transfer can alter the higher-order DNA
structure and consequently its accessibility to transcription factors.
Similarly, charge localization/delocalization has the potential to alter
binding efficiencies of DNA sequence- or structure recognizing proteins.
Epigenetic modifying enzymes like DNA methyl transferases have been shown to
interrupt charge transfer by flipping out bases from the nucleotide stack
Isbel2022 ; Levo2014 . Thus, charge transfer through DNA constitutes an
informational mechanism of living beings which does not alter the DNA sequence
itself and, thus, can be regarded as epigenetic layer.

Double-stranded tight-binding models have been used previously
lambropoulos_tight-binding_2019 ; Herb2024 ; albuquerque_dna-based_2014 ;
Chargemigration ; bittner_lattice_2006  to study the quantum dynamics of
electronic excitations along nucleic acid sequences, however, not accounting
for the impact of disorder, noise, and fluctuations induced by external
(water, solvent, presence of counterions, electrodes, substrate, phonons), and
internal (nuclei, base-pair sequence, geometry) factors omnipresent in an
ambient living surrounding. In the following, we therefore analyze various
sources, namely, thermal quantum fluctuations in the extreme limits of purely
local and purely collective property as well as spatially uncorrelated and
correlated disorder. To our knowledge, the current literature on charge
dynamics on DNA-like models only sporadically considers the effects of any
source of noise or disorder, whereas these phenomena are ubiquitous when
realistically considering any biologically related model. We study simple
sequences that are already relevant for epigenetic purposes: Hence, this work
aims to align with a new line of research that combines the efforts of the
physics, chemistry, and epigenetic communities, as proposed by Siebert et
alSiebert2023 , with the potential to uncover the possible relevance of
quantum phenomena for DNA dynamics during the life of cells.

The paper is organized as follows: We introduce the tight-binding model in
Sec.II, and recall some relevant aspects of the Lindblad formalism in Sec.III.
while the following sections are devoted to showing the results of our
simulations in the different scenarios we were interested in investigating,
both analyzing unitary, dissipative (Lindblad-like) and disordered dynamics.
Finally, in Sec.VIII we provide an overview of the work described in this
paper, together with an outlook of the refinements that can be made to further
improve the analysis we have developed.

Figure 1: : Schematic representation of the TB Ladder Model (LM) used to mimic
the structure of DNA.

##  II The model

We consider a model of the geometry and structure of DNA in form of a double-
strand tight-binding ladder, called Ladder Model (LM) which has been
introduced previously Chargemigration ; bittner_lattice_2006  and is now well-
established, see for example lambropoulos_tight-binding_2019 . The LM is
indeed a minimal model that can address the influence of both base pairing and
base stacking contributions in the energetics of the system. It consists of
two tight-binding chains, denoted by the index $l\in L,\ \ L=\\{1,2\\}$, where
individual sites on each chain $l$ (nucleotides) are denoted by $1\leq j\leq
n$ with localized electronic (hole) states $|l,j\rangle$. The LM is
parametrized by on-site energies of each base, $\epsilon_{j,l}$, intra-strand
hopping integrals between successive bases, $w_{j,l}$, and inter-strand
hopping integrals, $h_{j}$, see Fig. 1 for a schematic representation.
Accordingly, the Hamiltonian reads

$H_{LM}=H_{\rm loc}+H_{\rm tun}$ (1)

consisting of two parts: A local part describing on-site energies, $H_{\rm
loc}$, and a non-local (tunneling) part describing interactions among
different sites of the double chain, $H_{\rm tun}$, i.e.

$H_{\rm loc}=\sum_{l\in L}\sum_{j=1}^{N}\epsilon_{j,l}\ket{j,l}\bra{j,l}$ (2)

and

$\displaystyle H_{\rm tun}$ $\displaystyle=$ $\displaystyle\sum_{l\in
L}\sum_{j=1}^{N-1}w_{j,l}\ket{j,l}\bra{j+1,l}+h.c.$ (3)
$\displaystyle+\sum_{j=1}^{N}h_{j}\ket{j,1}\bra{j,2}+h.c.\,.$

Figure 2: : Distribution of eigenenergies (in units of $w$) of the system as a
function of the total number of sites N. As can be shown analytically, all the
values are ranging between -2.5 and 2.5.

The LM allows for a straightforward solution in case of infinite chains and
site-independent parameters $\epsilon$, $w$, $h$ throughout
bittner_lattice_2006 . Even though this is a rather simplistic limit, it is
nevertheless intriguing to briefly recall the main results. Assuming all
parameters to be real-valued and positive and putting the lattice constant
$a=1$, one finds energy bands

$E_{\pm}(k)=\epsilon+2w\cos(k)\pm h\,,\ \ \ \ \ -\pi\leq k\leq\pi,$ (4)

with width $w$ and spacing $h$. They correspond to delocalized Bloch waves
with quasi-momentum $-\pi\leq k\leq\pi$.

For chains of finite length $N$ the quasi-momentum takes discrete values,
i.e.,

$k_{n}=\frac{2\pi}{N}\left(n-\frac{N}{2}\right)\,,\ \ \ \ \ n=0,...,N$

and the expression in Eq.(4) remains valid upon substituting $k\to k_{n}$.
Accordingly, molecular energy bands related to HOMO and LUMO cover a range
from $E_{\rm min}=E_{-}(\pi)$ to $E_{\rm max}=E_{+}(0)$ and support coherent
long-range charge transfer through the chain. For chains of growing length the
expected density of states with pronounced maxima emerges. To illustrate the
situation, we display in Fig. 2 an example with parameters $\epsilon=0$,
$w=1$, and $h=0.5$ and varying length $N$.

A realistic modelling has to include environmental degrees of freedom
consisting e.g. local of vibrational modes as well as fluctuating modes of
lower frequency stemming from the DNA backbone. Since a complete microscopic
description is not feasible, one has to resort to effective description in
terms of various kinds of noise models. In the sequel, we will start with
broadband dynamical noise forces corresponding to local dissipation and then
proceed with various kinds of nonlocal sources for decoherence.

##  III Dissipative quantum dynamics

For broadband dynamical quantum noise, we follow the standard approach to
describe the impact of thermal reservoirs on the quantum dynamics within the
tight binding structure in case of weak interaction. The reasoning here is
that charge transfer through DNA has been shown to relatively long-ranged and
relatively fast compared to sequential processes, thus indicating relatively
long-range coherence. From now on we will consider the dynamics of a single
charge/quasi-particle along the chain.

Due to weak coupling a perturbative treatment applies to derive an effective
equation of motion for the reduced density operator $\rho(t)={\rm
tr}_{R}\\{W(t)\\}$, where all environmental degrees of freedom are traced out
from the total density operator $W$. More specifically, the total compound is
described by a Hamiltonian of the form $H_{tot}=H_{S}+H_{R}+V$ with
$H_{S}=H_{LM}$, $H_{R}$ modeling the reservoir modes, and $V$ capturing a
bilinear coupling between LM and reservoir, i.e., $V=\sum_{p}A_{p}\,X_{p}$
with hermitian system operators $A_{p}$ and collective reservoir operators
$X_{p}$. We note that the DNA-reservoir coupling can be very different in
nature: For example, a uniform coupling, where all sites interact with the
same reservoir, can be realized by setting $X\equiv X_{i}$ for all sites
$i=1,\ldots,N$ with nonlocal system couplings $A_{\nu}$, while in a fully
local coupling scheme site-dependent operators $X_{i}\neq X_{j}$ ($i\neq j$)
appear with local operators $A_{i}$, while. All kinds of intermediate cases
are possible as well, of course, but for the sake of clarity we will consider
these limiting cases only.

Under the assumption of factorizing initial states
$W(t)=\rho(t)\otimes\rho_{R}(t)$ with reservoir equilibrium at inverse
temperature $\beta=1/k_{\rm B}T$, i.e. $\rho_{R}=\exp(-\beta H_{R})/Z_{R}$
[$Z_{R}={\rm tr}\exp(-\beta H_{R})$], and a time scale separation between the
relaxation of the reduced system and the decay of equilibrium reservoir-
reservoir correlations $\langle X_{\nu}(t)X_{\mu}(0)\rangle_{\beta}$, a Born-
Markov treatment is valid. Together with a rotating wave approximation, this
leads to

$\dot{\rho}(t)=-i\left[H_{S},\rho(t)\right]+\mathcal{L}[\rho(t)]$ (5)

with Lindblad superoperators of the form

$\displaystyle\mathcal{L}[\rho]$ $\displaystyle=$
$\displaystyle\sum_{\alpha,\beta}\sum_{\nu}\gamma_{\nu}(\omega_{\alpha\beta})\big{[}A_{\nu}(\omega_{\alpha\beta})\rho(t)A_{\nu}^{\dagger}(\omega_{\alpha\beta})-$
(6)
$\displaystyle-\frac{1}{2}\\{A_{\nu}(\omega_{\alpha\beta})A^{\dagger}_{\nu}(\omega_{\alpha\beta}),\rho(t)\\}\big{]}\,.$

Here, the operators $A_{\nu}$ and $A^{\dagger}_{\nu}$, inducing transitions
between eigenstates of the system Hamiltonian
$H_{S}\ket{\alpha}=E_{\alpha}\ket{\alpha}$, are represented in the form

$A_{\nu}(\omega_{\alpha\beta})=\ket{\alpha}\bra{\alpha}A_{\nu}\ket{\beta}\bra{\beta}$
(7)

with transition frequencies $\omega_{\alpha\beta}=E_{\alpha}-E_{\beta}$. The
interaction with the reservoirs is characterized by transition rates

$\gamma_{\nu}(\omega_{\alpha\beta})=\int_{-\infty}^{\infty}dt\ {\rm
e}^{i\omega_{\alpha\beta}t}\langle X_{\nu}(t)X_{\nu}(0)\rangle_{\beta}\,,$ (8)

with the assumption that cross-correlations between different reservoirs are
absent. Note that here any bath-induced renormalization of the transition
frequencies is included in $H_{S}$. Given these rates, we can specify that the
above time evolution equations are consistent if
$\gamma(\omega_{\alpha\beta})\hbar/(k_{\rm B}T)\ll 1$ and
$\gamma(\omega_{\alpha\beta})/\omega_{\alpha\beta}\ll 1$.

Even though Eq.(5) appears as an operator equation, implicit in its derivation
is the representation of the density in the eigenstate basis of $H_{S}$.
Making this explicit leads to two sets of equations, one for the diagonal
elements (populations) $P_{\alpha}=\langle\alpha|\rho|\alpha\rangle$ which
reads

$\displaystyle\dot{P}_{\alpha}(t)$ $\displaystyle=$
$\displaystyle\sum_{\beta,\nu}\gamma_{\nu}(\omega_{\alpha\beta})|\bra{\alpha}A_{\nu}(\omega_{\alpha\beta})\ket{\beta}|^{2}P_{\beta}(t)-$
(9)
$\displaystyle-\gamma_{\nu}(\omega_{\alpha\beta})|\bra{\alpha}A_{\nu}(\omega_{\beta\alpha})\ket{\beta}|^{2}P_{\alpha}(t)\,,$

and one for the off-diagonal elements (coherences)

$\dot{\rho}_{\alpha\beta}(t)=-\left(i\omega_{\alpha\beta}+\Gamma_{\alpha\beta}\right)\rho_{\alpha\beta}(t),\
\ \alpha\neq\beta\,.$ (10)

Here, rates are given by

$\Gamma_{\alpha\beta}=\frac{\Gamma_{\alpha\alpha}+\Gamma_{\beta\beta}}{2}+\frac{\gamma(0)}{2}\,,$
(11)

where
$\Gamma_{\alpha\alpha}=\sum_{\beta\neq\alpha,\nu}\gamma_{\nu}(\omega_{\alpha\beta})$.

Often it is very efficient and also very powerful to model reservoirs as
quasi-continua of harmonic modes with spectral distribution $J(\omega)$. This
is always possible if the fluctuation properties of a reservoir are Gaussian
which is the natural situation for reservoirs with a macroscopic number of
degrees of freedom. The impact of such reservoirs onto dedicated systems is
then completely captured by second cumulants which in turn are determined by
the spectral density $J(\omega)$ and temperature. Accordingly, the above
transition rates can be expressed as

$\gamma_{\nu}(\omega)=J_{\nu}(\omega)\left[\coth(\tfrac{\omega\beta}{2})-1\right]$
(12)

with detailed balance property
$\gamma_{\nu}(-\omega)=J_{\nu}(\omega)\left[\coth(\omega\beta/2)+1\right]$.

The second choice for the dissipators is to model the system-bath interaction
as a coupling of each site with individual reservoirs (local dissipation).
This can be done by making explicit the definition of the projection operators
over each site as a linear combination of the system eigenstates, i.e.,

$A_{\nu,IJ}=\ket{I}\bra{I}A_{\nu}\ket{J}\bra{J}=\sum_{n,m}\alpha_{nm}\ket{\alpha_{n}}\bra{\alpha_{n}}A_{\nu}\ket{\alpha_{m}}\bra{\alpha_{m}}\,,$
(13)

and then using these dissipators for the equations following Eq. 7. Note that
this representation in terms of the eigenstate basis avoids a simpler and
somewhat inconsistent representation in the local site basis. This latter
representation is only valid if the tunneling terms in the total Hamiltonian
(1) are absent, i.e. in the trivial case. In the following, we will define
this last choice of dissipators (see Eq.13) as Local Dissipators (LD), while
we will refer to the first introduced set of dissipators (Eq.7) as Global
Dissipators (GD).

Finally, we consider ohmic-type spectral bath distributions of the form

$J(\omega)=\eta\frac{\omega\Omega^{2}}{\Omega^{2}+\omega^{2}}$ (14)

with a coupling rate $\eta$ and a large cut-off frequency $\Omega$.

##  IV DNA-parametrized lattices: Charge diffusion in presence of quantum
noise

To investigate charge dynamics on DNA strands, we make now use of specific
parameter sets to mimic three different possible sequences of bases, namely, a
degenerate sequence 5’-G-G-G-G-G-3’ (and complementary 5’-C-C-C-C-C-3’), a
sequence 5’-G-C-G-C-G-3’ (complementary 5’-C-G-C-G-C-3’), and a sequence
5’-G-C-A-C-G-3’ (complementary 5’-C-G-T-G-C-3’). These short DNA-oligomers are
chosen to cover both trivial sequences (dG - dC in the present work), which
are already well studied in the literature, and more complex ones, whose
resulting dynamics present interesting novel features and which are of
potential relevance for epigenetic processes, particularly DNA methylation.
DNA methylation usually occurs at the cytosine (i.e. C) in so-called CpG
dinucleotides (where p denotes the binding phosphate). Thus, in addition to a
poly-G stretch, a sequence with alternating CGs in both strands was chosen.
This sequence constitutes a reduced representation of a so-called “CpG
Islands”. Such CpG islands are located in particular in promoter regions of
many genes, particularly house-keeping genes. The cytosines in the CpG context
can be methylated by sequence-specific DNA methyltransferases, a process
during which charge transfer has been experimentally shown to be interrupted
by flipping the cytosine to be modified out of the nucleotide stack. DNA
methylation of CpG islands is frequently associated with the silencing of the
expression of the linked gene Bird1986 ; Edwards2017 ; Ammerpohl2009 . For
comparison, in the sequence 5’-CGTGC-3’ after a first CpG, the sequence was
interrupted by a T and inverted to a GpC, in which the G is not targeted by
the DNA methyltransferases and, thus, not modified.

Corresponding tight-binding parameters are taken according to the work of
Hawke hawke_tight-binding_2010  et al., which provides, to our current
knowledge, the most comprehensive collection of data needed for our purposes
based on Linear Combination of Atomic Orbitals (LCAO) calculations.

Corresponding data are collected in Tab.1 and 2.111We are aware that in the
literature one can find different results for the parameters we are using in
this model, which do not always agree with each other. Moreover, we
acknowledge the fact that in the epigenetic field, there are DNA sequences
that are of the utmost importance for the purpose they serve, which are
therefore good candidates for the analysis we want to perform. We mention in
passing that even with advanced DFT techniques only relatively short chains
can be treated, typically no longer than 3-4 bases.

After an evaluation of the dynamics resulting from such a setting in a closed
environment, we introduce the two types of Lindblad operators (LD and GD)
introduced in Sec.III to simulate decoherence for charge diffusion. All the
resulting dynamics (unitary and open quantum) are plotted in Figs.3, 4 and 5.
The time-evolution of gross coherences for the two Lindblad-like evolutions
are shown in the insets of the same figures mentioned above.

Figure 3 shows the quantum diffusion of a charge along a ladder consisting of
five sites with G (top) - C (bottom) nucleotides. Initially, the charge is
localized at the leftmost site on the upper strand. As expected, in the
unitary case (grey line) the particle delocalizes relatively quickly along the
upper G strand with almost no transfer to the complementary C strand though.
This is in agreement with results found in the literature and is due to the
large differences in on-site energies between the G and C sites, which are
more than an order of magnitude larger than the energies associated with the
overlap of the molecular orbitals of the two bases (see Tabs.1 and 2).

The dissipative dynamics is similar to the dissipative dynamics along a
generic uniform chain, as expected. However, it is interesting to note the
differences between different dissipators: The GD partially preserves the
coherence of the state even after for longer times, unlike the LD. On the
other hand, the charge dynamics induced by the LD is able to populate sites
that are not accessible in the unitary case and also in the GD scenarios.

Figure 4 displays a more intriguing behavior of the quantum diffusion along
the chain. Here, an alternating top chain 5’-G-C-G-C-G-3’ with complementary
’letters’ on the bottom side (5’-C-G-C-G-C-3’) is studied. One clearly sees
how, in the unitary dynamics, the topography of on-site and hopping energies
force the charge to tunnel between guanine bases on one strand to the guanine
bases on the opposite strand through cytosine bases. This process, known also
as super-exchange, is a first example of non-trivial dynamics arising from
tight-binding models parametrized according to the DNA energy topology. A
similar phenomenon can be seen in Fig.5 for the 5’-G-C-A-C-G-3’ (5’-C-G-T-
G-C-3’) chain, where it is additionally favorable for the charge to tunnel
even through the A-T base pair acting as a barrier in the middle of the chain.
These data suggest that even in short and relatively simple DNA sequences, the
quantum dynamics is prone to an interplay between quantum delocalization and
back-scattering due to a non-trivial energy landscape along the chain.

A huge difference between the quantum dynamics in the LD and GD schemes
appears along chains of the 5’-G-C-G-G-G-3’ and 5’-G-C-A-C-G-3’ types, as seen
in Figs. 4 and 5. If reservoirs couple through local operators, we see
complete destruction of coherence with a coherence measure $\sigma(t)$ quickly
dropping below the value of $0.5$. In addition, we can observe population
dynamics along the chain that differs substantially from the corresponding
unitary evolution: While the unitary evolution is dominated by tunnelling
through neighboring sites, thus populating almost only alternating G sites, LD
dynamics quickly starts to populate almost equally the entire chain, thus
disrupting this structure of the charge distribution. In essence, thermal
reservoirs kill coherent dynamics on a shorter time scale than the one
required for the chain to show long-range quantum mobility.

The dynamics resulting from a GD system-bath coupling scheme presents a very
different behavior though. While we are still witnessing a gradual suppression
of coherence, both achieve higher values of coherence in their dynamics, and
their trends suggest that coherence will not collapse to zero for long times.
Alongside these results, we can also see major differences in the population
dynamics. Indeed, while interferences and revivals along the chain are
gradually washed out, only those specific sites that are also populated in the
unitary dynamics discussed above get populated. Thus, the features arising
from the topography of energy barriers (quantum tunneling) remain basically
unchanged.

The main finding of this analysis can be summarized as follows: The way
broadband reservoirs couple to DNA-type double-strand lattices with non-
trivial energy landscape, namely, either nonlocal GD or local LD, has
tremendous impact on the predicted charge diffusion process. It is not proven
in the literature which coupling scheme is more suitable to mimic realistic
DNA, in particular, there is no evidence to assume that DNA is predominantly
locally coupled to environmental degrees of freedom at the single base level.
Now, our analysis suggests that a more subtle interplay between charge
mobility along DNA lattices and quantum effects is required for a realistic
description, in agreement with first experimental findings that report the
existence of coherent ’islands’ of up to 4-5 bases Genereux2009 ; Boon2002 ;
Barton2019 .

Adenine | Thymine | Guanine | Cytosine
---|---|---|---
-4.4 eV | -4.9 eV | -4.5 eV | -4.3 eV
Table 1:  On site energies of the DNA bases for the charge transfer model hawke_tight-binding_2010  DNA bases | Tr. integrals | DNA bases | Tr. integrals
---|---|---|---
AA | 16 | AT | 7
AC | -3 | AG | 1
TT | -30 | TA | -7
TC | 22 | TG | -17
GG | 20 | GA | 30
GT | -32 | GC | 43
CC | -47 | CA | -12
CT | 63 | CG | 15
A-T | -9 | G-C | 16
Table 2:  Intra-chain and inter-chain (last row) transfer integrals for the neighbours DNA bases and base pairs in the charge transfer model (in meV) hawke_tight-binding_2010 . Note that in general the values per base coupling do not commute, reflecting the nature of DNA of having one preferred direction along the strand. Figure 3: : Population dynamics in the 5’-GGGGG-3’ double-chain of an electron with unitary (grey line) and dissipative time evolution in the presence of a common bath (red line for global dissipation, blue line for local dissipation). In the bottom-central panel, an inset shows the time-evolution of the system coherence during the process. Figure 4: : Population dynamics in the 5’-GCGCG-3’ double-chain of an electron with unitary (grey line) and dissipative time evolution in the presence of a common bath (red line for global dissipation, blue line for local dissipation). In the bottom-central panel, an inset shows the time-evolution of the system coherence during the process. Figure 5: : Population dynamics in the 5’-GCACG-3’ double-chain of an electron with unitary (grey line) and dissipative time evolution in the presence of a common bath (red line for global dissipation, blue line for local dissipation). In the bottom-central panel, an inset shows the time-evolution of the system coherence during the process. Adenine | Thymine | Guanine | Cytosine
---|---|---|---
-8.3 eV | -9.0 eV | -8.0 eV | -8.8 eV
Table 3:  On site energies of the DNA bases for the hole transfer model hawke_tight-binding_2010  DNA bases | Tr. integrals | DNA bases | Tr. integrals
---|---|---|---
AA | -8 | AT | 68
AC | 68 | AG | -5
TT | -117 | TA | 26
TC | -86 | TG | 28
GG | -62 | GA | -79
GT | 73 | GC | 80
CC | -66 | CA | 5
CT | -107 | CG | -1
A-T | -12 | G-C | -12
Table 4:  Intra-chain and inter-chain (last row) transfer integrals for the
neighbours DNA bases and base pairs in the hole transfer model (in meV)
hawke_tight-binding_2010 . Note that in general the values per base coupling
do not commute, reflecting the nature of DNA of having one preferred direction
along the strand.

##  V Comparing electron and hole dynamics

The methods we have applied so far can also be used to study the quantum
diffusion of holes (lack of an electron in the HOMO band) along DNA-mimicking
structures. To do so, we need to choose a different set of parameters to mimik
the behavior of this type of quasi-particles on double-chain sites. This set
is given in Table 3 and 4 and is also taken from Hawke et al hawke_tight-
binding_2010 . In particular, we can compare these results with those
simulating the dynamics of an electronic charge along the same chain.
Interestingly, the combined electron-hole dynamics may give rise to the
formation of excitons which is discussed by Herb et al in Herb2024 .

The results of the simulations are shown in Fig.6, where for clarity only the
results for the 5’-GCACG-3’ sequence are shown, as these are the most
relevant. We have already commented that in this scenario the electron is able
to tunnel through the DNA sites to populate the guanine site on the right side
of the chain. The hole dynamics, however, shows different tunneling
characteristics. Namely, being able to populate the two guanine sites on the
left side of the chain in a similar way to the electron dynamics, its behavior
changes dramatically on the right side, where the hole can only occupy the
guanine site to the right of the thymine site in the center of the chain.

How electrons and holes populate the right-hand side of this chain can
potentially be a strong source of resistance to exciton annihilation. In fact,
the probability that an electron will relax back into the orbital from which
it was excited depends strongly on the overlap of wavefunctions characterizing
the electron and hole states at any point in time during the dynamics. Thus,
if they are "trapped" far away from each other, the probability for excitonic
annihilation is greatly reduced. As a possible consequence, long-lived
excitation phenomena as observed in different experimental setups kohler1 ;
kohler2 ; kohler3  could be explained in this way. These results suggest that
investigating more thoroughly electron-hole relaxation could be critical to
finally being able to link yet not completely understood experimental results
with theoretical modelling.

Figure 6: : Population dynamics in the 5’-GCACG-3’ double-chain for unitary
time evolution and for both electron (blue) and hole (orange) transfer. In the
bottom-central panel, an inset shows the time-evolution of the system
coherence during the process for both electron and hole. Figure 7: :
Population dynamics in the 5’-GCACG-3’ double-chain for static disorder and
global Lindblad dissipation time evolution in electronic transfer. In the
bottom-central panel, an inset shows the time-evolution of the system
coherence during the process for all the dynamics plotted. Figure 8: :
Population dynamics in the 5’-GCACG-3’ double-chain for static disorder and
global Lindblad dissipation time evolution in hole transfer. In the bottom-
central panel, an inset shows the time-evolution of the system coherence
during the process for all the dynamics plotted.

##  VI Impact of static disorder

Here we consider another source of decoherence with the tendency for
localization of charges which can arise from static disorder in the local
energy distribution of the DNA bases along the double strand. As suggested by
Gutiérrez et alGutirrez2009 , dynamic fluctuations in the energetics of the
system can strongly affect how a particle propagates along the DNA chain, and
this effect can also differ between electron and hole dynamics. We describe
this by running several different dynamical simulations over the same DNA
sequence, 5’-GCACG-3’, where the site energies $\epsilon_{j,l}$ in (2) are
replaced by $\epsilon_{i,l}+\xi_{i,l}$ with $\xi_{i,l}$ being a random
variable sampled from a Gaussian distribution with a mean equal to
$\epsilon_{i,l}$ and a variance to be set as an external parameter. We use two
different values for the variance, the weaker one resembling energetic
fluctuations typical of body temperature processes (about 27 meV), and the
second, inspired by Gutiérrez et alGutirrez2009  (360 meV), aiming to include
even more sources of disorder, such as interactions with the backbone and
solvent degrees of freedom.

In Figs. 7 and 8 we show the dynamics for an electron and a hole,
respectively, both on the same DNA sequence, plotted together with their
dynamics when in contact with a thermal bath modeled by global Lindblad
operators. As for the plots above, the gross coherence dynamics for these
simulations can be found in the bottom right panel’s inset. It is interesting
to note that the influence of static disorder on the dynamics of the particles
is very similar to that of the global Lindblad operators: This is due to the
fact that both sources of decoherence have the tendency to preserve the
structure of eigenstates of the system, thus both mimicking low frequency
fluctuations.

As a result, it seems evident that the main considerations we made about the
relationship between electron and hole dynamics along the 5’-GCACG-3’ strand
hold true even in the presence of static disorder, both for small and large
variations in the on-site energies, since this e-h separation effect appears
stable under this noise source.

##  VII Correlated noise sources

Finally, we turn our attention to the effect of correlated fluctuations, as
opposed to the uncorrelated ones presented in the previous section. Correlated
fluctuations on local energies of the system are implemented inspired by an
approach proposed by Liu et alLiu2016 . However, here we focus on the full
quantum dynamics along the chain. First, we introduce a measure of the spatial
correlation of the particle dynamics along the chain, defined by

$\chi(i)_{\sigma}^{d_{0}}=\frac{<P_{0}(t)P_{i}(t)>_{t}^{d_{0},\sigma}}{<P_{0}(t)P_{0}(t)>_{t}^{d_{0},\sigma}}\,.$
(15)

Here, the symbol $<\cdot>_{t}^{d_{0},\sigma}$ refers to the time average for a
given choice of parameters $\sigma$ and $d_{0}$ (be described below), and
$P_{i}(t)$ is the population over time at site $i$ of the system investigated,
which in this case is the uniform 5’-GGGGG-3’ double-strand. The reasoning to
chose 5’-GGGGG-3’ DNA double strands to demonstrate the impact of correlated
low-frequency noise on the charge dynamics in a system with an already
structurally high degree of charge mobility. The above measure allows us to
track how correlated fluctuations of on-site energies affect the long-range
charge propagation, and how such correlated noise may be
beneficial/detrimental to support long-range coherence.

The noise profile for a given chain is assumed to be constant over the time
span of a single run of the model’s dynamics. For each run, a noise profile of
degrading correlation strength with distance between sites is generated as
follows: First, a value is extracted from a normal distribution with zero mean
and variance $\sigma$ to be specified (it represents the strength of the
energy fluctuations in the systems). This value is added to the local energy
of the top left base of the system, representing its energetic disorder. The
energy fluctuation of its neighboring base is then extracted from another
Gaussian curve, now averaged around the value found for the first fluctuation
and with a variance $\sigma^{*}$ derived as follows

$(\sigma^{*})^{2}=2\sigma^{2}\left(1-C_{ij}\right)\,,$ (16)

where $C_{ij}=e^{-\frac{|i-j|}{d_{0}}}$ is the correlation strength between
the two bases. Here $d_{0}$ represents the coherence length of the correlated
noise (the longer $d_{0}$, the higher the correlation of the noise
distribution in space, with the limit $\sigma^{*}\xrightarrow{}0$ for
$d_{0}\xrightarrow{}\infty$). Thus, the further away two bases are, the larger
the value of $\sigma^{*}$ will be, resulting in noise profiles that are more
and more uncorrelated with distance. A more detailed derivation of such a
result is given in the supplementary material of Liu et alLiu2016 , including
the possibility of accounting for temporal correlations in the energy
fluctuations, which is beyond the scope of this paper.

In a previous work, Kubař et alKubar2008  showed how the average correlation
strength between neighboring G sites on a DNA strand is $C^{G}_{i+1,i}\sim
0.7$, corresponding to a value of $d_{0}\sim 2.8$. For this reason, in our
investigation, we explore different choices of the parameter $d_{0}$ ranging
from 1 to 5 and $d_{0}=10$ for neighboring bases. The intensity of the energy
fluctuations, modeled by the parameter $\sigma$, is studied under different
conditions, ranging from $\sigma=0.1\rm meV$ to $\sigma=100\rm meV$, in order
to cover both weakly and strongly perturbed regimes with respect to the
typical interaction energies of the system under study (see Tables 2 and 4).

Figure 9: : Correlation measure for the time-averaged particle dynamics along
each site (x-axis) of a double-stranded 5’-GGGGG-3’ chain, for different
values of the noise level ($\sigma$ parameter). The numerical labeling of the
basis goes left to right for both strands (G1 top left, C1 bottom left) Figure
10: : Total correlation along a chain (sum over each site) for increasing
$\sigma$ values (x-axis), for different correlation lengths. Smaller values of
$\sigma$ are evaluated more finely (logarithmic selection of values).

Figure 9 shows, for a fixed parameter $d_{0}=3$ as a reference, the time-
averaged correlation between the charge dynamics at each site of the DNA
strand with respect to the dynamics at site $G_{0}$. The two plots show the
same scenario for different choices of sigma. The left plot displays the
degree of dynamical correlation of each site of the double chain for
increasing values of the parameter $\sigma$, from $0$ to $0.05$. The right
plot instead starts from the results for $\sigma=0.05$ and increases this
value up to the extreme choice of $\sigma=1$. First, we can see that the
correlation for the sites on the opposite strand of Cs is always zero. This
reflects the fact that due to the energy topology the particle only
delocalizes along the upper G-strand and never travels to the opposite chain.
More interestingly, however, the data demonstrate that correlations along the
G strand increase significantly with growing noise level, up to the value of
$\sigma=0.05$ (left plot in Fig. 9), and only for even larger values begin to
decrease, as would normally be expected from a measure of correlation versus
noise level (right plot in Fig. 9). The same can be observed for different
values of $d_{0}$.

To better investigate this aspect and to compare scenarios for different
values of $d_{0}$, we considered the following quantity

$\Omega(\sigma)^{d_{0}}=\sum_{i}\chi(i)_{\sigma}^{d_{0}}$ (17)

which is plotted in Fig.10. It is trivial to note that for $\sigma=0$ all
curves converge to a fixed value, since for zero fluctuations the contribution
of the $d_{0}$ parameter is nullified. Moreover, along the rest of the curves,
for larger values of $d_{0}$, the overall correlation $\Omega$ is generally
higher for each value of $\sigma$ considered, reflecting the fact that for
longer correlation lengths one expects a higher degree of correlation between
the dynamics at different sites.

More importantly, however, these results illustrate that, for any value of
$d_{0}$, the maximum degree of correlation cannot be found for $\sigma=0$, as
one would expect naively from the zero noise condition, but for a finite value
of $\sigma$. This value depends on the correlation length of the systems under
analysis. Hence, the analysis suggests that the presence of correlated static
perturbations along the DNA does not necessarily disrupt coherent propagation
but may even preserve it to some extent, in line with previous experimental
findings Genereux2009 ; Boon2002 ; Barton2019 . However, this can only be the
first step and further investigations toward understanding the true nature of
the disorder effects on charge dynamics along DNA-like chains are required to
fully understand their potential impact on DNA behavior.

##  VIII Conclusions

In this work, we investigated charge diffusion on two-dimensional DNA-inspired
lattices in the presence of various noise mechanisms, from thermal reservoirs
to spatially correlated disorder. The motivation for this study was triggered
on the one hand by recent developments for long-range charge transfer through
DNA and on the other hand by potential implications of charge
delocalization/localization for DNA hemostasis and gene regulation, with a
special focus on DNA modification by methylation.

The lattice model used here includes intra- and inter-strand site couplings
and initial states prepared on one of the sites. For DNA lattices,
parametrized according to atomistic simulations, in the absence of noise the
energetic profile along complementary chains does lead to finite populations
for spatially separated sites due to long-range tunneling related to long-
range coherences due to proper $\pi$-stacking of bases.

Weak coupling to quantum thermal reservoirs substantially modifies this
picture, however, depending on the details of the coupling mechanism (local
versus global). As a result, one observes that the assembling of specific
sequences of nucleobases allows to control charge diffusion. While oscillatory
patterns in the population dynamics die out rather quickly also for weak
dissipation, interestingly, intra-strand coherences survive on relatively long
times scales. For global quantum noise, coherences are present on much longer
time scales which may be of use in set-ups, where molecular structures are
placed in cavities.

Electron and hole diffusion are quite different in DNA-like lattices, which
can lead to complex exciton transfer in bare and weakly dissipative chains, as
further investigated in Herb2024  for unitary dynamics. This phenomenon
appears to survive in the presence of both Lindblad-like dissipation as well
as for disorder noise, and we believe that further investigation may uncover
interesting phenomena of potential biological relevance e.g. via epigenetic
mechanisms.

The impact of disorder of on-site energies of the system also revealed an
interesting behaviour: Static disorder, both at low and high intensities,
reveals dynamics similar to those induced by global dissipation which is
somehow expected. In particular, when considering static disorder at room
temperature energies, simulations suggest that coherences along the system are
not suppressed on ultrafast time scales, but rather tend to have a relatively
long-lived behaviour.

Even more interesting results have been found for the effects of correlated
disorder on the coherent propagation of an excess charge along the double
strand. They suggest that a certain degree of disorder in the energy landscape
of DNA could not only not hinder the coherent propagation of the particles,
but even act as a support for it.

So far, we have investigated a rather simple model with the aim of providing
general insight into the properties of quantum diffusion on DNA-like lattices
when exposed to bath interaction effects. This may already trigger further
studies on more specific situations where DNA is not in its native form as a
carrier of genetic information, but rather as a tool to monitor quantum states
in designed aggregates that are molecular or semiconductor-like in nature.
Looking further into the future, however, one can imagine that these platforms
could allow us to delve deeper into the charge transfer properties of specific
DNA sequences including disease-associated mutations and the relationships
between charge distributions and environmental degrees of freedom relevant to
biological mechanisms affecting gene expression but not DNA sequence like
epigenetic information.

##  IX Acknowledgements

We would like to thank Ciprian Paduariu, Björn Kubala, Jürgen Stockburger,
Sima Karimi Farsijani and Emil Chteinberg for fruitful discussions. Moreover,
we heartfully thank Dennis Herb for fruitful discussions and technical support
during the project. Financial support of the Center for Integrated Quantum
Science and Technology (IQST) and the BMBF through the project QCOMP in the
Cluster4Future QSens is gratefully acknowledged.

##  X Data availability

The data that supports the findings of this study are available within the
article.

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