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Exciting out-of-equilibrium states in macromolecules through light pumping

In the present paper we address the problem of the energy downconversion of the light absorbed by a protein into its internal vibrational modes. We consider the case in which the light receptors are fluorophores either naturally co-expressed with the protein or artificially covalently bound to some of its amino acids. In a recent work [Phys. Rev. X 8, 031061 (2018)], it has been experimentally found that by shining a laser light on the fluorophores attached to a protein the energy fed to it can be channeled into the normal mode of lowest frequency of vibration thus making the subunits of the protein coherently oscillate. Even if the phonon condensation phenomenon has been theoretically explained, the first step - the energy transfer from electronic excitation into phonon excitation - has been left open. The present work is aimed at filling this gap.


I. INTRODUCTION
The activation of out-of-equilibrium collective intramolecular vibrations of a model protein has been recently reported in Ref. [1]. This phenomenon has been induced by light pumping, realised by shining a laser light on an aqueous solution of BSA (Bovine Serum Albumin) protein molecules each one carrying a few fluorophores covalently attached to their Lysine residues. The fluorophores were excited with a blue light at 4880Å and then they re-emitted a broadband fluorescence radiation peaked at 5190Å, thus the difference between the absorbed and re-emitted photon energies resulted in a concentration of an average energy of 0.19 eV at the fluorophores sites which thus became "hot points" on each protein. A continuous energy supply of this kind was experimentally found effective to excite the vibrational modes of the proteins and, with an energy supply rate exceeding a suitable threshold, this eventually led to a phonon condensation phenomenon into the lowest vibrational frequency. The relevance of this out-of-equilibrium collective molecular vibrations consists in the possibility of activating long-range electrodynamic interactions between bio-macromolecules [2]. The reason is that, at thermal equilibrium, a macromolecule vibrates incoherently with a broad spectrum of modes, whereas the action of an external source of energy promoting a phenomenon of phonon condensation can induce the coherent motion of the molecular subunits, so that, the resulting collective vibration can bring about a large oscillating dipole moment. Under this condition long-range and resonant (thus selective) electrodynamic forces can be activated. In turn, these electrodynamic forces could help explaining the astonishing efficiency of the impressively complex biochemical machinery at work in living cells [3], where the different actors (proteins, DNA and RNA) find their cognate partners and targets in the right place, at the right time and in the right sequence in an overcrowded environment (the cytosol). Electrodynamic resonant/selective forces are the only possible one to act at a long distance, all the others (chemical bonds, Van der Waals and electrostatic forces) are in fact either intrinsically acting at very short distances, or are screened by the freely moving small ions in the cytosol. Actually, this is a longstanding theoretical scenario [4][5][6] which, for several reasons, has been discarded. However, the upgrade of Fröhlich's theoretical proposition in [1,2] and the experimental outcomes reported in [1], represent a first crucial leap forward to ascertain whether the above mentioned hypotheses can be given experimental confirmation or refutation that can be attempted with the nowadays available technology [7,8].
Of course we are faced with the problem of understanding what might replace the laser action in living cells. There are several possible candidates to play the role of external energy suppliers, for instance, the hydrolysis of Adenosine Triphosphate (ATP) releases a highly energetic phosphate group, redox reactions and mitochondria produce weak UV photons that might excite Tryptophan and Tyrosine amino acids [9,10] in proteins, as well nucleotides of DNA and RNA. Also an anisotropic momentum transfer operated by water molecules or ions could make the job [11]. In either cases of metabolically generated photons or of ion collisions (phosphate stemming from ATP hydrolysis or other) we can assume that the external energy input for a biomolecule occurs through the generation of "hot points", as in the case of light activated fluorophores, and mediated by either radiative or collisional electronic excitation. In what follows, we aim at better understanding, qualitatively and quantitatively, how part of the photon energy received through electronic excitation of the fluorophores attached to a biomolecule (protein) is converted into vibrational energy of the chain of subunits (amino acids) composing it. As we shall see, it is found that only a fraction of the initially available electron energy is released to the phonons of a biomolecule. The quantitative estimate of this energy transfer process is very important for a better assessment of the physical conditions which are necessary to activate the intramolecular collective vibrations.

II. DEFINITION OF THE MODEL
In Ref. [1] the external source of energy driving the phonon condensation was modeled (by one of us among the others) as high temperature heat bath. This was done to reformulate in a classical framework the Wu-Austin [12] quantum model leading to the original Fröhlich rate equations of Ref. [4]. We now aim at refining this part of the model in view of a better understanding of the basic excitation mechanism that can bring a macromolecule out of thermal equilibrium.
In both cases of photo-excitation and, presumably, of ionic collisions, the excitation mechanism is supposed to be mediated by the molecular electron cloud. Therefore, the model describing the phenomenon that we want to investigate is borrowed from the standard Davydov and Holstein-Fröhlich models [13][14][15] to account for electron-phonon interaction. Hence, the following energy operator is assumedĤ where the first termĤ el is the electron energy operator withB n andB † n the annihilation and creation operators for the electron at any site n (n = 1, 2, , ..., N ) which labels the amino acid along the protein. The term E 0B † nB n accounts for the initial "bare" electron energy distributed on several lattice sites according to initial shape of the electron wavefunction. The constant J is the nearest neighbour coupling energy of the hopping electron across two neighbouring amino acids. In this model we have considered only a longitudinal chain of amino acids. The electron moving from the excited fluorophore interacts on its way with almost free electrons in each amino acid, and it may just make a disturbance which will allow a next electron to continue on the trip. It would be then more a disturbance traveling than a single electron, but the net effect will be the same of a traveling electron. Thus the term B † nB n B † nB n has been introduced to take into account non-linear effects due to the interaction between the electron in motion along the chain and the electrons of the substrate of amino acids. In particular, the term takes into account effects related to the Coulombic repulsion between the traveling electron and the charges localized on the amino acids. The averaging is intended as the expectation value ofB † nB n on the dynamically evolving state of the system. The second termĤ ph in (1) is the phonon energy operator wherep n andû n are momentum and position operators for longitudinal displacements of amino acids at site n, respectively. Furthermore, M and Ω are average values of the mass of the amino acids of a protein and of the spring constants of two neighbouring amino acids, respectively. The quartic term is a correction stemming from the power series which gives the harmonic term at the lowest order expansion around the minimum of interparticle interaction potential (typically nonlinear, as is the case, for example, of the Van der Waals potential). This term is responsible for phonon-phonon interaction, absent in the harmonic approximation.
Finally, the third termĤ int in (1) is the electron-phonon interaction operator where χ is the energy coupling parameter.

III. DERIVATION OF THE DYNAMICAL EQUATIONS WITH TDVP
In order to derive from the model Hamiltonian (1) the corresponding dynamical equations, we make a simplifying ansatz about the state vectors by assuming the following factorization |ψ = |Ψ |Φ (5) in which |Ψ describes an electron given a single quantum excitation and supposed to be free to propagate along the chain of N amino acids composing a protein where |0 el is the vacuum state of the Amide-I oscillators, and We then set where β n (t) and π n (t) are the average values of the longitudinal displacement and momentum of an amino acid, respectively.
To derive dynamical equation we now resort to the time-dependent variational principle (TDVP) in quantum mechanics. TDVP is a formulation of the time-dependent Schrödinger equation through variation of an action functional. The Schrödinger equation is obtained by requiring that the action functional be stationary under free variation of the time-dependent state. According to this principle, we define a new wave function |φ in terms of |ψ in Eq. (5) as where S(t) is a time-dependent phase factor (S(t) ∈ R), which will be determined in a selfconsistent manner and the normalization condition is φ|φ = 1. The wave function |φ satisfies the Schrödinger equation which according to Eq. (9) becomes Integrating, we obtain We can now derive the equations of motion by requiring that the action with the Lagrangian to be stationary From Eqs. (5), (6), and (7) we write and then arrive at Thus the Lagrangian (13) becomes where H(C n , C * n , β n , π n ) = ψ(t)|Ĥ|ψ(t) .
Imposing the condition (14), we get from which it results The expectation value of the Hamiltonian is So, from Eq. (21) we have

IV. DEFINITION OF THE PHYSICAL PARAMETERS FOR NUMERICAL SIMULATIONS
Let us see how to make a physically reasonable choice of the coupling parameters entering the Hamiltonian. We borrow from Ref. [16,17] the estimates of the interaction energy between an electron and each of all the 20 amino acids. The average value of these interaction energies is ∆E = 0.74 eV with a dispersion σ E = 0.47 eV. As a first rough picture of an electron hopping across the sequence of amino acids constituting a protein we can consider the electron of energy E 0 moving in a periodic sequence of square potential barriers of height V 0 = 0.74 eV and of width a = 4.5Å, the average distance between two nearest neighboring amino acids [13]. We can then weigh the electron hopping operators between neighbouring sites with the probability P (n → n ± 1) of tunnelling from one potential well to the nearest ones. This is achieved by computing the transmission coefficient Moreover, the coefficient of the electron hopping term in the Hamiltonian has to be a characteristic energy scale of the process, thus a natural choice is to set J ∝ ∆E T , then, assuming that an electron is initially excited at any given point of the chain of amino acids and that it has the same probability of moving to the left or to the right, we add a factor 1/2 so that finally we have J = 1 2 ∆E T . Now, assuming E 0 = 0.19 eV as initial value of the electron energy, we find J = 0.0585 eV, whereas assuming that only a fraction η ∈ [0, 1] of the maximum available energy is kept by the electron, for example for η = 0.5, we find J = 0.031 eV. For what concerns the electron-phonon coupling constant χ, we make a rough estimate of its value as χ = ∆E/∆x = σ E /∆x = σ E /a = 0.47eV /4.5Å 100 pN.
By rescaling time and lengths as t = ω −1 τ and β n = Lb n , respectively, where L = √ ω −1 M −1 , the following dimensionless dynamical equations are obtained and the dimensionless expression of the Hamiltonian is where In order to perform numerical integration of the dynamical equations it is useful to introduce the variables so that Eqs.(24) becomė By denoting with B n [b(t), q(t), p(t)] the r.h.s. of Eq. (30) we have which can be rewritten in the formḃ Equations (28) and (29) and the above system have been numerically integrated by combining a finite differences scheme and a leap-frog scheme as follows where Q n [b(t), q(t), p(t)] and P n [b(t), q(t), p(t) are the r.h.s. of Eqs.(28) and (29), respectively.
By using sufficiently small time steps ∆t the desired precision of energy conservation can be attained.
About the initial conditions, we aim at simulating a physical situation where each photon absorbed by a fluorophore attached to a protein releases -in the average -0.19 eV of energy to the surrounding electron cloud. This energy is the difference between the energies of the absorbed photon of 4880Å and that of the re-emitted one as fluorescent radiation of 5150Å. We assume, as already stated above, that the effect of a single photon excitation is to make one electron moving across the protein by tunnelling through a sequence of potential barriers. In the experiments to which we are referring [1] each protein is labelled with 5-6 fluorochromes, and a laser light is continuously shined on the labelled proteins, therefore what we are after is modelling an elementary process and assuming, in a first approximation, a property of additivity of the same elementary process. In other words, if more than one electron is activated we assume that the resulting physical effect is the sum of a single electron effect. As a consequence, the electron initial condition is assumed to be described by a wavefunction C n (t = 0) centered at the site n = n 0 at time t = 0 [13]:  where σ 0 = 3ΩJ/χ 2 .
Then, coming to the initial conditions of the phonon component of the system, we assume a thermalized macromolecule at room temperature, that is at T = 310K. At equilibrium, the energy equipartition theorem for the Hamiltonian (3) reads where k B is the Boltzmann constant. At thermal equilibrium, energy is equally shared among all the degrees of freedom and, in particular, between kinetic and potential energies, therefore at t = 0 the velocities and the displacements have been initialized with random values of zero mean and fulfilling the conditions expressed in dimensionless form.
In Table I the values chosen for the physical parameters are reported. These are: the initial excitation energy E 0 , an average value of the mass M of the amino acids, the dipole-dipole coupling constant J, the elasticity constant Ω used in the numerical studies of Ref. [13], and the electronphonon coupling χ. In Table I also the corresponding dimensionless values of the same physical quantities are reported, these are obtained by using (26) and the frequency ω = 10 13 s −1 .

V. NUMERICAL RESULTS
All the numerical computations have been performed using an integration time step ∆t = 5 × 10 −5 entailing a very good energy conservation, with typical relative error ∆E/E 10 −5 .
The length of the chain is N = 500 rounding the number of amino acids of the protein in [1].
Figures (1) and (2) show the spatial distribution of the probability |ψ(n, t)| 2 of finding the moving electron at any site n versus time for the electron-phonon coupling χ = 100 pN and χ = 366 pN, respectively. The electron is initially centered around the site n = 250. Figure (1) shows that the electron wavefunction quickly spreads over the whole substate of amino acids, a phenomenon somewhat less pronounced in Figure (2) and to some extent counterintuitive since the latter corresponds to a stronger electron-phonon coupling.
and then, by following Ref. [18], the coordinate transformations Q m = S mn b n and P m = S mnḃn , transform the Hamiltonian (37) intõ where Of course, these oscillators are the normal modes (phonons) of the system. Then a spectral entropy S(t) is defined as where E T (t) = N m=1 E m (t) and E m (t) = (P 2 m + Ω ω 2 m Q 2 m )/2, so that the weights p m (t) are normalized. The maximum value of S(t) is attained when all the p m (t) are equal to 1/N . Thus, at equipartition, when the energy content of each normal mode is the same, entropy attains its maximum, this allows to define a normalized entropy as       Time t is measured in 10 −13 s; electron energy is given in eV. Time t is measured in 10 −13 s; electron energy is given in eV.

VI. CONCLUDING REMARKS
The parameter space of the system investigated here is of course very large, thus we have limited our investigation to a basic choice of physically meaningful parameters with respect to the topic that we aimed at better understanding. Then we have checked the robustness of the phenomenology so observed by changing some parameters, as is the case of the nonlinear coupling constants and µ, or the electron-phonon coupling constant χ. The results actually show that after having given 0.19eV of initial excitation energy to an electron, the electron wavefunction spreads through the chain by releasing to the phonons only a small fraction of the electron energy, approximately 0.02eV. This is a somewhat unexpected result but interesting because it helps in understanding why exciting a collective intramolecular oscillation of the BSA protein required a very long time. Of course, the contributions of several fluorophores add up, and the continuous illumination of the labelled proteins with an intense laser light allows to accumulate energy in the protein until the activation threshold of the coherent oscillation of all its atoms is reached and passed over.
The phonon part has been simplified with respect to the model derived by the de-quantisation of the original Fröhlich's model [4] because the model investigated here has focussed only on the mechanism of down-conversion of the energy of the photons, harvested by the protein through its fluorophores receptors, to the internal vibrations of the chain of amino acids. Although no more than 10% of energy is dissipated by electron to phonons, it seems that in the studied regime no coherent transport of information can occur on the amino acids (as sometimes one could expect in a spin chain model [19] ) due to the fact that the electron wave function spreads over all sites. The can be dynamically generated which keep a system out of thermodynamic equilibrium. Among many other systems where QSS are produced [20], let us mention a beam of fast particles interacting with the set of waves describing a physical system [21,22], a situation which is reminiscent, for example, of the above mentioned fast phosphate groups -produced by ATP hydrolysis.