Multidimensional Quantum Mechanical Modeling of Electron Transfer

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Multidimensional Quantum Mechanical Modeling of Electron Transfer and Electronic Coherence in Plant Cryptochromes: The Role of Initial Bath Conditions David Mendive-Tapia, Etienne Mangaud, Thiago Firmino, Aurélien de la Lande, Michele Desouter-Lecomte, Hans-Dieter Meyer, and Fabien Gatti J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b10412 • Publication Date (Web): 07 Dec 2017 Downloaded from http://pubs.acs.org on December 12, 2017

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The Journal of Physical Chemistry

Multidimensional Quantum Mechanical Modeling of Electron Transfer and Electronic Coherence in Plant Cryptochromes: the Role of Initial Bath Conditions David Mendive-Tapia,∗,†,‡ Etienne Mangaud,¶ Thiago Firmino,§ Aurélien de la Lande,§ Michèle Desouter-Lecomte,§ Hans-Dieter Meyer,‡ and Fabien Gatti∗,k Institut Charles Gerhardt Montpellier, UMR 5253, CNRS-UM-ENSCM, CTMM, Université Montpellier, CC 15001, Place Eugène Bataillon, 34095 Montpellier, France, Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, INF 229, D-69120 Heidelberg, Germany, Laboratoire Collisions Agrégats Réactivité, UMR 5589, IRSAMC, Université Toulouse III Paul Sabatier, F-31062, Toulouse, France, Laboratoire de Chimie Physique, CNRS, Université Paris-Sud, Université Paris Saclay, Orsay F-91405, France, and Institut des Sciences Moléculaires d’Orsay, UMR-CNRS 8214, Université Paris-Sud, Université Paris Saclay, Orsay F-91405, France E-mail: [email protected]; [email protected]

∗

To whom correspondence should be addressed Institut Charles Gerhardt Montpellier, Université Montpellier ‡ Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg ¶ Laboratoire Collisions Agrégats Réactivité, Université Toulouse III Paul Sabatier § Laboratoire de Chimie Physique, Université Paris-Sud k Institut des Sciences Moléculaires d’Orsay, Université Paris-Sud †

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Abstract A multi-dimensional quantum mechanical protocol is used to describe the photoinduced electron transfer and electronic coherence in plant cryptochromes without any semi-empirical, e.g. experimentally obtained, parameters. Starting from a two-level spin-boson Hamiltonian we look at the effect that the initial photo-induced nuclear bath distribution has on an intermediate step of this biological electron transfer cascade for two idealized cases. The first assumes a slow equilibration of the nuclear bath with respect to the previous electron transfer step that leads to an ultrafast decay with little temperature dependence; whilst the second assumes a prior fast bath equilibration on the donor potential energy surface leading to a much slower decay, which contrarily displays a high temperature dependence and a better agreement with previous theoretical and experimental results. Beyond Marcus and semi-classical pictures these results unravel the strong impact that the presence or not of equilibrium initial conditions has on the electronic population and coherence dynamics at the quantum dynamics level in this and conceivably in other biological electron transfer cascades.

Introduction The theoretical modeling of Electron Transfer (ET) in biological processes is a particularly challenging topic that arises an uncanny contraposition, which is that one has to deal with a macromolecular size problem yet significantly impacted by quantum-mechanical effects. 1 For example, long quantum coherences have been observed experimentally in complex chemical systems such as photosynthetic systems, 2,3 which has raised fundamental questions about the relation of these coherences with functionality and its enhancement 4 or how to build realistic models 5 to describe them. Here we are interested in cryptochromes, a class of flavoproteins that are involved in regulating the circadian rhythms in a variety of organisms 6,7 and suspected to play a major role in animal magneto-reception 8–14 (i.e. the ability of migratory birds to detect earth’s mag-


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netic field). We focus on plant cryptochromes from Arabidopsis thaliana (AtCry) for which crystal structures are available. Despite having very versatile biological functions depending on the organisms in which they are expressed, cryptochromes form a family of ubiquitous proteins that share strong structural similarities. 6,7 In AtCry the chromophore Flavin Adenine Dinucleotide (FAD) triggers upon photo-excitation, transfer of an electron from the 8 nearby tryptophan residue W400 to form the [FAD·– + W·+ 400 ] radical pair. Protonation of

the FAD during this first step is currently under debate and it is dependent on the specific cryptochrome under study (i.e. insects, plants and even among migratory birds four different cryptochromes exist in their eyes). 7,8 In any case after this charge separation has taken place, two other tryptophan residues — W377 and W324 — are closely aligned to the now positively charged W·+ 400 , thus allowing an ET cascade. Table 1 summarizes the respective experimental and theoretical available time-constants (τ ), which we discuss below. Table 1: Experimental and theoretical rates associated with each individual electron transfer step in AtCry at 300 K. For an exponential decay N (t) = N0 e−λt the inverse of the decay constant τ = λ1 indicates the time at which the population is reduced to 1/e times its initial value. The interval of values for the W400 −→ FAD MT rate is due to the different possible ATP binding and protonation states in which the charge separation can take place. 19 Time-constant τ (ps) Electron transfer

Experiment 15

MT 16,17

NAMD 18

HEOM 16

ET1 W400 −→ FAD* ET2 W377 −→ W·+ 400 ET3 W324 −→ W·+ 377

0.4 4 – 15 30 – 50

2.0 – 101.1 4.97 3.03

5.26 14.29

0.82 1.20

Femtosecond broad-band transient absorption measures 15 show two decays with timeconstants of 0.4 ps and 31 ps. The first was associated with the initial ET1 W400 −→ FAD* step and the second with the two subsequent transfers within the tryptophan triad. After a multi-exponential fit this work also provided two time-constants intervals that could ·+ potentially be assigned 18 to these ET2 W377 −→ W·+ 400 (4 –15 ps) and ET3 W324 −→ W377

(30 – 50 ps) steps (Table 1). Comparison with experiment is however delicate as it is arguable how much of the signal corresponds to the ET and how much to vibrational cooling. 20



Regarding this there has been extensive theoretical work in recent years dedicated to the phenomenological modeling of magnetic interactions in radicals and their viability as a biochemical compass; 21–26 yet, the complementary modeling from an ab initio perspective is also an important and timeless achievement; 27 specially since Marcus Theory (MT) lacks a description of quantum electronic coherences and the significantly large system-bath couplings in AtCry 16 advice for computationally demanding non-perturbative techniques such as NonAdiabatic Molecular Dynamics (NAMD), 18,28–30 the Feynman path integral approach, 31 Hierarchical Equations Of Motion (HEOM) 32–35 16 or the Multi-Layer (ML) Multi-Configuration Time-Dependent Hartree (MCTDH) method. 36 In particular the computed ET rates at the MT and NAMD level are in good agreement with the experimental kinetics obtained from femtosecond spectroscopy, nonetheless counter-intuitively when the quantum dissipative dynamics were treated fully taking into account quantum mechanical effects by the current popular HEOM together with a spin-boson Hamiltonian, 16 the obtained rates were too fast (see Table 1). Therefore having in mind the ever-growing need for an accurate modeling of quantum mechanical effects in biology, in this work we focus on the ET2 W377 −→ W·+ 400 step in AtCry and show the dramatic influence that the choice of initial bath conditions can have on the subsequent quantum dissipative dynamics and that the aforementioned time-scale discrepancy can be explained through this effect.

Theory and methods We will focus this section around the simplified sketch shown in Fig. 1, in which the upper part (1a) shows a nuclear representation of the FAD cofactor and the triad of tryptophan residues, while the lower part (1b) shows a diagram of the electronic levels. For the sake of simplicity, the successive ETs are described by a single nuclear coordinate in which the ground state |FAD + W400 i is radiatively coupled to the excited state |FAD∗ + W400 i, which ·+ is electronically coupled to the charge separation state |FAD·− + W400 i that is in turn elec-


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·+ tronically coupled to the first acceptor state |FAD·− + W377 i and so on. Then if a laser

pulse is used to optically transfer population from |FAD + W400 i to |FAD∗ + W400 i we can assume a vertical instantaneous transition for the electronic degrees of freedom, 37 which will then evolve in time (represented by a series of red arrows). However, the initial nuclear disposition of the bath will find itself in a state out of equilibrium, since the minimum geom·+ etry of the |FAD + W400 i and any of the charge-separated electronic states |FAD·− + W400 i, ·+ ·+ |FAD·− + W377 i and |FAD·− + W324 i would be different (represented by a series of Gaussian

functions in Fig. 1(bottom)), thus the nuclear bath distribution must change and evolve in time as well.

(a)

(b)

Figure 1: Photo-induced electron transfer in AtCry: (a) Nuclear representation of AtCry (b) Diagram of the photo-induced electron transfer process. The green arrow represents the optical excitation, red arrows represent the electron transfers as defined in Table 1 and the Gaussian functions represent the equilibrium positions of the nuclear degrees of freedom.

Naturally, the bath initial conditions from one ET step to another are trivially defined when performing one simulation taking into account all involved ET steps in the chain si-



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multaneously. However this is unfeasible for a fully exact quantum mechanical description in AtCry with the current computational resources as we would enter into the picosecond/nanosecond 18 time-scale. One way to bypass this limitation is to decouple each ET from the others, which is a drastic approximation equivalent to assuming that each tryptophan is electronically coupled only with the neighbouring ones and that the transfers occur in a sequential manner. In the case of AtCry the first is a reasonable approximation given the larger distance between W400 and W324 than between W400 and W377 or W377 and W324 (Fig. 1(a)). The second is however a much more extreme conjecture that is more likely for the first two ETs due to their significant disparity in time-scale (0.4 and [4 - 15] ps). Hence in the following, we discuss the modelling of a single intermediate step in AtCry while focusing on the ET2 W377 −→ W·+ 400 step and considering three aspects: (i) the construction of a two-state spin-boson Hamiltonian (ii) the definition of initial conditions for the bath and (iii) the propagation of the respective quantum dynamics using the ML-MCTDH method. First, we make use of a two-level spin-boson model in mass and frequency weighted ˆ S is coupled to a nuclear coordinates and atomic units (¯h = 1) where the electronic part H ˆ B through a linear coupling interaction H ˆ BS : bath made of a set of N harmonic oscillators H ˆ =H ˆS + H ˆB + H ˆ BS H   HD HDA  ˆS =  H   = HD |Di hD| + HA |Ai hA| + HDA (|Di hA| + |Ai hD|) HDA HA ˆB = H

N X ωi i=1 N X

ˆ BS = H

i=1

2

(ˆ p2i

+

xˆ2i ) |Di hD|

+

N X ωi i=1

2

(ˆ p2i + xˆ2i ) |Ai hA|

(1) (2)

(3)

N

X di di ωi xˆi |Di hD| − ωi xˆi |Ai hA| 2 2 i=1

(4)

This and related models for condensed phase photo-induced curve crossing processes have ˆ S consists in a two-by-two symbeen applied and developed in numerous works. 38–46 Here H metric matrix characterized by the Donor (D) and Acceptor (A) quantities HD (energy of the


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donor state), HA (energy of the acceptor state) and HDA (electronic coupling between the donor and acceptor states). The harmonic potential wells are assumed to have the same frequency (ωi ), position (ˆ xi ) and momentum (ˆ pi ) coordinates in both electronic states, but their equilibrium positions are displaced for each state by a length di leading to linear vibronic couplings ki = ωi d2i with the electronic system. 47,48 The corresponding parameters for ET2 , HD = 0.0, HA = −511.63 and HDA = 19.666 meV, were extracted from the fluctuations of the diabatic energy gaps along molecular dynamics simulations reported in Ref. [ 17]. In short, constrained DFT 19,49 was used to define (quasi-diabatic) charge transfer states and coupled to a hybrid QM/MM scheme to carry out MD simulations on the potential energy surfaces of each charge transfer step depicted in Fig. 1b in a system comprising a total of 110 000 atoms and including around 34 000 water molecules (we refer the reader to Ref. [ 16] for a detailed description). The spectral density was then computed for ET2 from the correlation function of the energy gap obtained with a femtosecond resolution 50 and expressed as:

J(ω) = ≈

N πX 2 k δ(ω − ωi ) 2 i=1 i 5 X

pl h

l=1

ω + Ωl

2

+

(5)

Γ2l

ω ih

ω − Ωl

2

+

Γ2l

i

(6)

where the upper part corresponds to the definition in dimensionless coordinates as extracted from the MD data, while the lower to the respective Lorentzian fitting of each peak 51–53 as illustrated in Fig. 2a. Once a reference spectral density has been generated, it can be rediscretized for an arbitrary number of bath modes. This can be done either directly from the MD data or using the Lorentzian fitting in order to compare with previous HEOM dissipative dynamics in Ref. [ 16]. Benchmark calculations described in the Supporting Information show how in general a finite, but large number of vibrational Degrees Of Freedom (256 DOF) is sufficient to reach convergence.



(a)

(b)

(c)

Figure 2: (a) Fitting of the average spectral density function obtained from constrained DFT/MM molecular dynamics simulations by a series of Lorentzian functions as usually used in the HEOM method. (b) Absolute value of the normal mode displacements |di | (logarithmic scale). (c) Decomposition of the spectral density into contributions from the water molecules, tryptophan residues participating in the redox process, the remaining atoms belonging to the protein and a cross term between these that has been traced to the correlation between water and protein. 16

Second, regarding the selection of initial conditions we define two idealized cases in order to study the effect that the initial photo-induced nuclear bath distribution has on an intermediate step. We point out that we do not aim at accurately mimicking the bath after


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ET1 , rather to define two limiting case scenarios to be employed in conjunction with the previously parameterized two-state spin-boson Hamiltonian and consistent with its approximations. These are illustrated in Fig. 3 through a sketch of the ground |FAD + W400 i ·+ ·+ (green curve) and excited electronic states |FAD·− + W400 i and |FAD·− + W377 i (purple and

blue curves respectively) in the space of two bath coordinates for an arbitrary mode: “Q1 ” ·+ describing the ET1 charge dissociation from |FAD + W400 i to |FAD·− + W400 i and “Q2 ” de·+ ·+ scribing the subsequent ET2 step from |FAD·− + W400 i to |FAD·− + W377 i as previously

parameterized in the spin-boson Hamiltonian. Initially, the bath shall be equilibrated on the minimum of the neutral ground state |FAD + W400 i surface (green, bottom). Then after vertical excitation at the Franck-Condon (FC) region, if we consider that the ET1 step happens too fast to become relevant in terms of the bath’s movement, the bath will find itself in non-equilibrium with respect to the excited state potential energy surfaces as shown by the arrow A. It should be noticed that a vertical excitation will land at a point of the surface crossing between charge-separated ·+ ·+ electronic states |FAD·− + W400 i and |FAD·− + W377 i if these are strongly localized on the

respective tryptophan moieties. This is the case in radical cations such as benzene, 54,55 due to symmetry (Jahn-Teller effect), or bis(methylene)-adamantyl (BMA) 56,57 and spiropyran, 58 due to the coupling between donor and acceptor is virtually zero as a result of the perpendicular orientation between the charge-bearing groups. 59 In the context of AtCry, this strong localisation of the electronic states is an approximation relying on the weak electronic coupling regime derived from the parameterization of the Hamiltonian (Eq. 2), which is commonly found in biological systems as the protein provides steric constraints that hinder the interaction among moieties. For ET2 HDA amounts to a few tens of meV (158.6 cm−1 ). We also highlight that strictly speaking, we do not have direct access to the neutral ground state minimum geometry in the constrained DFT calculations, however, the lowest energy geometry on the crossing seam can be used instead as it is the nearest configuration with an evenly distributed charge among donor and acceptor. The respective electronic excitation is



represented by the arrow B and will be referred through the article as initial non-equilibrium bath. On the contrary, if the initial non-equilibrium bath is considered to have enough time ·+ to equilibrate on the |FAD·− + W400 i donor state during the ET1 step, it shall be centred

on the minimum of this surface (purple curve), which is represented by the arrow C and will be referred to as initial equilibrium bath.

Figure 3: Sketch of the two idealized initial bath conditions employed in this work for AtCry.

Last, we can think of ML-MCTDH as a method that employs fully flexible time-dependent functions that follow the variational equations derived from the Dirac-Frenkel principle. This is a powerful tool for the computation of efficient quantum dynamics in high-dimensional systems that formally converges to the exact solution, 60 yet it can treat more degrees of freedom than usual quantum dynamics. 61 36,62–66 For example, along the lines of the present work Kühn et. al. has recently used this method in combination with the Frenkel exciton Hamiltonian to obtain numerically exact quantum dynamics simulations of the Fenna-MatthewsOlson complex at zero temperature from an experimental spectral density. 67,68 We have also recently extended the ML-MCTDH implementation with the inclusion of temperature (e.g. random phase thermal wavefunction approach) 69,70 and a semi-automatic algorithm for an ACS Paragon Plus Environment

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improved efficiency (e.g. ML-spawning). 71 The first allows us to perform more realistic simulations, while the second provides us with a simplified way of numerically converging large dimensional Hamiltonians. When temperature was included, the initial wavepacket was obtained by propagating in imaginary time a wavefunction (e.g. defining either the initial non-equilibrium or equilibrium bath conditions) with equal amplitude and random phases until the desired thermal wavefunction at a finite temperature was obtained. In the random phase thermal wavefunction approach 69,70 a finite temperature wavefuntion is obtained by propagating in imaginary time up to t = β/2 where β =

1 , kB T

kB is the Boltzmann’s

constant and T is the temperature. Since there are two coupled electronic states in the spin-boson Hamiltonian, their transfer coupling needs to be neglected during the imaginary time relaxation.

Results and discussion Throughout the article we will analyse the electronic population and coherence dynamics. That is the diagonal (real and positive) and the imaginary part of the off-diagonal (complex) h i elements of the density matrix. These were computed as ρmn (t) = T rQ |mi hn| ρˆ(Q, t) where the trace is over all nuclear degrees of freedom Q = (x1 , ..., xN ) and ρmm corresponds to the population of the m state and ρmn with m 6= n corresponds to the coherence between the m and n states. The density operator was defined as ρˆ(Q, t) = |Ψ(Q, t)i hΨ(Q, t)|) where Ψ(Q, t) is the multidimensional wavefunction obtained through ML-MCTDH computations. In the initial non-equilibrium bath, the wavepacket was initially centered on xi = 0 and t = 0, which corresponds to the middle point between the two constrained quasi-diabatic states minima and situation B in Fig. 3. Meanwhile in the initial equilibrium bath, the wavepacket was initially centered on xi = −di /2 and t = 0, which corresponds to the minimum of the ·+ |FAD·− + W400 i state and situation C in Fig. 3 (See the Supporting Information for more

details about the spin-boson model).



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Influence of the different regions of the spectral density We begin with an analysis of the role of the individual parts of the spectral density for the ET2 step at 0 Kelvin. In order to achieve this, we compare the quantum dynamics obtained without any approximations regarding the spectral density defining the bath (e.g. as extracted from the MD simulations, see Eq. 5) with a series of truncated models where each of the peaks in the spectral density function were systematically neglected one at a time.

(a)

(b)

(c)

(d)

·+ step at 0 K systematically neFigure 4: Computed properties for the ET2 W377 −→ W400 glecting individual peaks of the spectral density function: (a) Diabatic population dynamics (initial equilibrium bath) (b) Diabatic population dynamics (initial non-equilibrium bath) (c) Electronic coherence dynamics (initial equilibrium bath, imaginary part) (d) Electronic coherence dynamics (initial non-equilibrium bath, imaginary part). The models are labeled accordingly to the numbering of peaks in Fig. 2a.

On the one hand, Fig. 4a-4c illustrates the diabatic population and electronic coherence


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dynamics (e.g. red, labeled “12345”) for the initial equilibrium bath case. This will be our reference to be compared with the respective truncated models labelled accordingly to the numbering of peaks in Fig. 2a. As we can see, the population decay is completely null at 0 K. Also peak number 1 dominates by damping the transfer rate since only when we remove the corresponding modes we observe a significant population transfer and coherence dephasing (“12345” (red) vs “2345” (blue)). On the other hand, for the initial non-equilibrium bath limit the respective reference diabatic population and electronic coherence dynamics are shown in Fig. 4b-4d (e.g. red, labeled “12345” as well). Here in contrast we observe that the reference population profile already shows a continuous transfer that saturates around 0.5 ps without full decay (the saturated population value is around 0.5), while the imaginary part of the electronic coherence shows damped oscillations. In this case the differences between reference and truncated profiles highlight the importance of peaks 1, 2 and 4 as their removal accelerates the population transfer at the expense of damping the electronic coherence oscillations. Interestingly, the influence of the different regions of the spectral density are not fixed, rather dependent on the initial state of the bath. In the case of a full equilibration low-frequency modes strongly damp the population transfer, while for the non-equilibrium scenario low and relatively high frequency modes have a noticeable role to play. Following this, we illustrate another of the advantages of the present protocol thanks to the fact that ML-MCTDH simulations can employ any shape of spectral density function; thus bypassing the need of approximating it as a sum of Lorentzian peaks in order to keep the bath in a numerically convenient form for the HEOM algorithm. In general any realistic spectral density can be represented as a sum of Lorentzians provided one includes enough terms, however for the particular case of ET2 in AtCry a maximum of three out of the five peaks in the spectral density function could be taken into account via HEOM in Ref. [ 16]. When including more Lorentzians the computation turned out to be numerically unreachable (9 366 819 auxiliary matrices had to be used at the 80th order), thus limiting the accuracy of



the description. In order to illustrate this, Fig. 5 shows a comparison of the different baths obtained by substituting each of the five peaks constituting the spectral density as extracted from MD data in Eq. 5 by the corresponding Lorentzian fitted peak in Eq. 6. For both the initial equilibrium (Fig. 5a and 5c) and non-equilibrium baths (Fig. 5b and 5d) all these lead to the correct formally “exact” result except when substituting peak number 1 (orange, labeled “lrtz1_2345”), as the use of only one Lorentzian term to represent this peak leads to difficulties describing the asymptotic behaviour towards ω → 0. The error derived from not describing properly the continuous part of the spectral density near zero frequency is small in the non-equilibrium case, but much more dramatic for an initially equilibrated bath. We thus point out that despite HEOM being formally exact and widely used to model biological phenomena, care must be taken when choosing its fitting. 72

Influence of the temperature In the initial equilibrium bath limit case, the population transfer at 0 K is completely null and thus it is necessary the inclusion of temperature to achieve a significant decay rate. One must then construct a stochastic average of several initial random phase wavefunctions in order to represent a Boltzmann distribution with finite temperature at the quantum mechanical level. Fig 6a and 6c show the individual computations and resulting averages at 298.15 K for the diabatic population decay and electronic coherence, respectively. The individual computations have no physical meaning, but they are displayed here as a witness of the fast convergence of the total reduced density (in particular the populations) with the number of initial seeds. The number of total realizations can therefore be quite low (10 as used here). As we can see there is a remarkable increase of the overall decay time-scale to the ps regime and an important impact of the temperature averaging; specially for the electronic coherence where the averaging substantially “damps” the oscillations seen for the individual components. This is not observed in the initial non-equilibrium bath case at the same temperature, which for comparison is displayed in Fig. 6b and 6d. We also notice the


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(a)

(b)

(c)

(d)

·+ Figure 5: Computed properties for the ET2 W377 −→ W400 step at 0 K systematically substituting individual peaks by the corresponding Lorentzian fitting: (a) Diabatic population dynamics (initial equilibrium bath) (b) Diabatic population dynamics (initial non-equilibrium bath) (c) Electronic coherence dynamics (initial equilibrium bath, imaginary part) (d) Electronic coherence dynamics (initial non-equilibrium bath, imaginary part). The models are labeled accordingly to the numbering of peaks in Fig. 2a.

largely different scales when comparing both electronic population (y-axis in Fig. 6a vs Fig. 6b) and electronic coherence (y-axis in Fig. 6c vs Fig. 6d) graphs. In order to investigate the effect of temperature more closely we considered the electronic population dynamics for a range of values (0-300 K). Table 2 collects the extrapolated timeconstants, while Fig. 7a illustrates the corresponding population dynamics. These show a gradual acceleration when increasing temperature that contrasts with the little temperature dependence observed for the initial non-equilibrium bath, which is displayed in Fig. 7b for comparison. Such opposite behaviours can be explained due to the fact that in the non-equilibrium bath the initial Gaussian functions representing each oscillator land at the ACS Paragon Plus Environment


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(a)

(b)

(c)

(d)

·+ Figure 6: Thermally averaged computed properties for the ET2 W377 −→ W400 step at 298.15 K: (a) Diabatic population dynamics (initial equilibrium bath) (b) Diabatic population dynamics (initial non-equilibrium bath) (c) Electronic coherence dynamics (initial equilibrium bath, imaginary part) (d) Electronic coherence dynamics (initial non-equilibrium bath, imaginary part).

surface crossings rather than the minima in Fig. 3. The derived dynamics is then highly driven by the strong non-equilibrium forces of the potential energy surfaces, leading to similar coherence oscillations at 0 and 298.15 K and thus little affected by random factors on the initial positions. Second, analysis of the spectral density components by neglecting each peak at 298.15 K revealed that peak number 1 still dominates over the initial rate decay at 298.15 K (See Supporting Information). We can understand this by looking at the expectation value of the average quantum number for a harmonic mode of vibration hni i = (eβωi − 1)−1 , which increases strongly as the frequency tends to zero. Therefore it is not surprising that low-


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(a)

(b)

(c)

(d)

·+ Figure 7: Computed properties for the ET2 W377 −→ W400 step varying the temperature and averaging over 10 seeds: (a) Initial equilibrium bath diabatic population (range [0-300] K) (b) Initial non-equilibrium bath diabatic population (0 and 298.15 K) (c) Initial equilibrium bath electronic coherence (0, 100 and 298.15 K, imaginary part) (d) Initial non-equilibrium bath electronic coherence (0 and 298.15 K, imaginary part).

frequency modes at physiological temperature have a high quantum number with a complex wavefunction structure full of nodes that affects the dynamics. To which extend these excited vibrational modes affect the quantum dynamics is determined by the value of the respective ki coupling constant in the spectral density, thus both parameters, ωi and ki , are crucial. Interestingly their ratio ki /ωi , which corresponds to the displacements di of the harmonic potential, shows that modes below 300 cm−1 are about two orders of magnitude larger than the rest (Fig. 2b). So in order to get additional insights into these low-frequency modes we performed a decomposition of the spectral density into contributions from the water molecules, tryptophan residues participating in the redox process and the remaining atoms



belonging to the protein (Fig. 2c). Below 300 cm−1 the water and protein are the main contributors, while the role of the tryptophans is negligible. The cross correlation term is rather large showing a strong correlation between terms that has been traced to the correlated dynamics of the water and protein atoms. 16 This correlation is certainly attributable to water molecules localized in the active site (one or two at the most between W377 and W224 ) and the protein surface, which is in agreement with the idea that low-frequency modes are related to protein hydration and hence, functional properties. 73 In general low-frequency modes occur due to weak force-constant groups involving global collective motions that have been related to shallow local minima on the potential energy surface 74 and are thought to be responsible for the flow of conformational energy and directed reactivity in proteins, which is in agreement with their observed high influence on the decay rate for this case. The solvation level of the tryptophans in the particular case of AtCry has been already examined in Ref. [ 16 ]. Here the solvent reorganization was found to be much larger for the ET2 step due to the larger exposure of the tryptophan pair to the solvent. This however contrasts with the Xenopus laevis (6-4) photolyase, which did not exhibit any water pocket around the second tryptophan. 75 Table 2: Extrapolated time-constants (τ ) at different temperatures assuming an initial equi·+ librium bath on the donor electronic state for the ET2 W377 −→ W400 step in AtCry. Parameters fitted after applying the random phase thermal wavefunction approach and averaging over 10 seeds. Initial equilibrium bath Temperature (K) 100 150 200 250 298.15

τ (ps) 16.7 8.7 5.6 4.1 3.3


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Outlook and final remarks Last after discussing the trends observed in the theoretical model, we outline some of the speculations that can be extracted for the ET2 process in the real protein: 1. In first place, the fact that the decay time-scale for the equilibrium scenario is in much better agreement with experiment, suggests that the ET2 step in the real protein is ·+ possibly slowed down by equilibration of the bath on the |FAD·− + W400 i donor surface.

This is in agreement with the fact that MT rates in Table 1 are relatively large and close to experiment since Marcus theory assumes a localized fully equilibrated initial state. A similar effect in the opposite direction (acceleration due to non-equilibrium conditions) has been observed in Escherichia coli DNA photolyase employing nonadiabatic QM/MM Eherenfest dynamics. 76 Here the time-scale of the ET was observed to be faster than MT, which was attributed to the ET being accelerated by a nonequilibrated conformational ensemble. 2. Within the equilibrium initial conditions, low-frequency deformations are seen to play a crucial role. As seen in the decomposition of the spectral density function (Fig. 2c) they are linked to water molecules probably localized at the protein surface. Therefore, due to hydration water dynamics are in general faster than protein side-chain relaxations, 77–79 the related modes can also be expected to achieve full equilibration sooner than the rest of the protein, which points out to a partial equilibration as a more realistic approximation of the bath initial conditions. 3. This intermediate scenario for the initial conditions being the most realistic case has important consequences for the electronic coherence oscillations. Considering the equilibrium initial conditions as the most realistic bath could mean that the electronic coherence oscillations we see for the non-equilibrium case are “damped” in reality, however, this would not be necessarily true for the mixed case leading to an intermediate rate and only a partial suppressing. The reason to expect this is that modes below ACS Paragon Plus Environment


300 cm−1 are the main responsible for the ET slowing down when equilibrated and as shown in Fig. 4d peaks 1, 2 and 4 contribute the most to the electronic oscillations when not equilibrated. Nonetheless in order to construct such initial conditions, it is difficult to define where is the borderline for mode relaxation and to which extend it happens as these are aspects probably highly influenced by the dynamics of the ET1 W400 −→ FAD step. This is a necessary step for a reliable comparison with experiment when aiming at the simulation of individual steps in an ET multi-chain that we hope will be addressed in future work.

Conclusions ·+ In summary, we have presented a quantum mechanical modeling of the ET2 W377 −→ W400

step in AtCry. The protein and solvent environments were represented as a two-level spinboson model coupled to a large dimensional set of quantum harmonic oscillators extracted from QM/MM simulations. The respective quantum dynamics were then propagated via the ML-MCTDH method through our recent implementation of the ML-spawning algorithm 71 and including the often neglected impact of temperature. 69,70 We thus highlight that these are numerically exact quantum dynamics simulations at physiological temperature that did not require any experimentally obtained parameters. We looked at the effect that the initial photo-induced nuclear bath distribution has for two idealized cases, namely initial non-equilibrium and equilibrium baths, which show strong antagonistic dynamics governed either by non-adiabatic or thermal aspects, respectively. The latter case is in better agreement with experimental measures, NAMD simulations and MT offering an explanation for the too fast rates observed in previous HEOM simulations using the initial non-equilibrium bath condition. 16 Regarding limitations and possible shortcomings, however, we indeed remark that these are two idealized and first approach scenarios that rely on the decoupling of the ET2 step with respect to the rest and the neglect of any


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of the dynamical effects derived from the previous transfer. In other words, it is assumed either an absence of equilibration on the donor state because ET1 happens too fast to become relevant in terms of the bath’s movement or a full equilibration that “washes out” any trace of the ET1 step’s memory due to a hypothetical fast bath’s evolution. Two extreme but necessary approximations for any level of description that aims to represent a highly complex ET multi-chain taking place in a protein through a restricted two-state spin-boson model. Therefore, it will be instructive to compare the present reference results with those of more complex Hamiltonians taking into account three or more uncoupled ETs and/or removing other approximations such as a constant electronic coupling, linear system-bath coupling or the harmonicity of the modes. This last being valid only for small displacements of the protein and thus questionable in particular for an accurate description of low-frequency modes. Nevertheless, at this stage the basic picture presented here provides us with useful insights about the role that particular parts of the spectral density function have and how these roles change depending on the presence or not of equilibrium initial conditions. In particular for AtCry, low-frequency modes related with water solvent molecules are seen to play a crucial role slowing down the ET rate to the ps time-scale, however their effect is only appreciable when the bath is equilibrated on the donor surface. On the contrary when the initial bath is considered to be in non-equilibrium after vertical excitation at the FC region, the non-adiabatic forces at the electronic crossing region lead to a much faster decay on the few-hundred fs time-scale with strong electronic coherence oscillations. These results thus unravel the strong impact that the degree of initial bath equilibration has on the ET2 step in AtCry and point out to the relevance that differences in the time-scale evolution between bath (nuclear) and electronic degrees of freedom can have in a reactive mechanism; here in the context of ET in biology, but also a matter of intensive work and current debate in the context of few-fs charge migration 80–82 and electronic decoherence 83–85 in chemistry and physics. 86–88



Supporting Information Available Spin-boson model; harmonic potential displacements; spectral density Lorentzian fitting; benchmarking computations; initial non-equilibrium bath truncated models; comparison with Hierarchical Equations Of Motion (HEOM).

Acknowledgement We thank Prof. C. Meier and Dr. F. Cailliez for fruitful discussions. D. M.T thanks Dr. F. Boyrie for his help and dedicated assistance managing the computing facilities at the Institut Charles Gerhardt Montpellier UMR 5253. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA grant agreement n◦ 622876 and resources of the GENCI-CINES/IDRIS project i2016087713. We acknowledge support from the CoConicS Project (ANR-13-BS08-0013-03).

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TOC Graphic

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Multidimensional Quantum Mechanical Modeling of Electron Transfer

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