GUILLOUD Frédéric Messai Malek

Amis Karine

A LOW COMPLEXITY COHERENT CPM RECEIVER WITH MODULATION INDEX ESTIMATION

Keywords: CPM, modulation index estimation, persurvivor processing, reduced-complexity, Viterbi decoding, modulation index mismatch

In this paper we address the problem of low-complexity coherent detection of continuous phase modulation (CPM) signals. We exploit the per-survivor-process technique to build a reduced-state trellis and apply a Viterbi algorithm with modified metrics. In the case where the modulation index can vary, we propose a maximum-likelihood (ML) estimation of the modulation index and compare the performance of the resulting structure with a non-coherent receiver structure of the state of the art. Simulations on an additive white Gaussian noise (AWGN) channel both for binary and M-ary CPM show the efficiency of the proposed receiver.

INTRODUCTION

In applications with low-cost transmitters, the RF front end may suffer from non linearities. In this case, continuous phase modulations (CPM) [START_REF] Anderson | Digital Phase Modulation[END_REF], which have a constant amplitude, are a good candidate to limit the transmitted signal distorsion. A CPM is perfectly defined by the symbol alphabet, the frequency pulse and the modulation index. When the modulation index varies, either randomly in a continuous interval as for the AIS analog transmitter [START_REF] Bonacci | Advanced concepts for satellite reception of AIS messages (Toulouse Space Show[END_REF] or, intentionally as for the Bluetooth standard [START_REF]Specification of the Bluetooth System[END_REF], the priority is given to non coherent detection due to its robustness. Discriminator detectors [START_REF] Simon | Differential versus limiterdiscriminator detection of narrow-band fm[END_REF] are the most used for their low computation cost, at the expense of severely degraded error rate performance. Efficient noncoherent receiver structures such as the one proposed in [START_REF] Lampe | Noncoherent sequence detection receiver for bluetooth systems[END_REF] by Lampe et al. achieve better error rate performance with a reasonable increase of complexity. The best error rate performance is obtained through coherent maximum likelihood sequence estimation (MLSE). Based on the trellis state description of the CPM, the Viterbi algorithm performs optimally, but suffers from a high complexity. Complexity reduction has been studied in [START_REF] Svensson | A class of reduced-complexity viterbi detectors for partial response continuous phase modulation[END_REF], [START_REF] Simmons | Low complexity decoders for constant envelope digital modulations[END_REF], [START_REF] Svensson | Reduced state sequence detection of partial response continuous phase modulation[END_REF], and [START_REF] Kaleh | Simple coherent receivers for partial response continuous phase modulation[END_REF]. The per survivor processing (PSP) has been applied in [START_REF] Miller | Detection of CPFSK signals using per survivor processing[END_REF] in a specific case (binary full-response CPFSK) to reduce the trellis state number and hence the computation cost. However, like the Viterbi algorithm, even a slight modulation index variation yields a severe error rate degradation. In [START_REF] Zarei | An iterative detection algorithm for coded cpfsk signals with irrational modulation index[END_REF], the PSP has also been applied to soft-in soft-out (SISO) CPFSK detection with an irrational modulation index. Furthermore the modulation index mismatch between the transmitter and the receiver is taken into account in the metric computation, enhancing the robustness of the resulting modified SISO BCJR algorithm. However a perfect knowledge of the modulation index is assumed. In this paper, we generalize the principle of the PSP approach for state number reduction and metric modification for modulation index mismatch correction to any kind of CPM. We introduce a maximum-likelihood estimation of the modulation index in the resulting coherent receiver. We compare the proposed coherent receiver with the non-coherent structure of [START_REF] Lampe | Noncoherent sequence detection receiver for bluetooth systems[END_REF] taking into account the spectral efficiency loss involved by the ML modulation index estimation. Simulations carried out on an AWGN channel, for both binary (Bluetooth and AIS standards parameters) and non-binary CPM (Wireless M-Bus standard parameters), show the efficiency of the proposed receiver. The paper is organized as follows: we first briefly introduce the system model in Section 2. Then we define the generalized receivers for CPM signals for any modulation index as well as the modulation index estimation in Section 3. The applications and simulations are given in Section 4, followed by a conclusion and some perspectives in Section 5.

SYSTEM MODEL

CPM signal Model

The complex baseband CPM signal is defined as:

s(t, α) = E T e jφ(t,α) , ( 1
)
where E is the average symbol energy, T is the symbol duration and φ(t, α) the information-carrying phase given by:

φ(t, α) = 2πh tx ∞ i=0 α i q(t -iT ). (2)
α = {α i } denotes the information sequence. The information symbols α i are assumed to be independent and identically distributed and to take values in the M -ary alphabet M = {±1, ±3, ..., ±(M -1)}. h tx is the modulation index used at the transmitter. The phase response q(t) is defined on [0, +∞) and satisfies :

q(t) = 1 2 , t ≥ LT. (3)
Using the properties of q given in (3), the informationcarrying phase during the n-th time interval, [nT, (n + 1)T ], n ∈ N, can be written as:

φ(t, α) = 2πhtx ∞ i=0 αiq(t -iT ), = πhtx n-L i=0 αi + 2πhtx n i=n-L+1 αiq(t -iT ), = θtx,n + φtx,n(t). (4)

Coherent receiver

From (4) we observe that the modulated signal over the n-th time interval depends both on the phase state denoted by θ tx,n and on the L most recent symbols, i.e (α n , α n-1 , ..., α n-L+1 ).

For a rational modulation index h tx = ktx ptx , the phase state θ tx,n modulo 2π can take only p tx or 2p tx different values for even and odd k tx , respectively. Therefore, the phase evolution can be described by a finite-state machine, where each state is represented by an L-dimensional vector (θ tx,n , α n-1 , α n-2 , ..., α n-L+1 ) and where the number of such states is p tx M L-1 . We assume that the signal is transmitted over a Gaussian channel. The equivalent baseband received signal, denoted by r(t), is defined as:

r(t) = s(t, α) + n(t), (5)
where n(t) is a realization of a zero-mean wide sense stationary complex circularly symmetric Gaussian noise, independent of the signal, and with double-sided power spectral density 2N 0 over the bandwidth of s(t, α). The MLSEdetector aims at maximizing the scalar product between r(t) and all the realizations of s(t, α). Assuming N d transmitted symbols, the MLSE estimation of the information symbols α 0 , α 1 , ..., α N d -1 is given by:

(α 0 , α1 , , ..., αN d -1 ) = arg max α∈M N d ℜ N d T 0 r(t)s * (t, α)dt , (6)
where ℜ(x) denotes the real part of x. As

N d T 0 r(t)s * (t, α)dt = N d -1 n=0 (n+1)T nT r(t)e -jφ(t, α) dt,
the computation of all scalar products over [0, N d T ] is equivalent to the sum of all scalar products over a symbol duration given the previous phase state θ tx,n . We introduce and denote by BM (n) i,j the branch metric corresponding to the scalar product associated to the transition

from state σ i = θ i tx,n , α n-1 , α n-2 , ..., α n-L+1 to state σ j = θ j tx,n+1 , α n , α n-1 , ..., α n-L+2
, where α n is the symbol related to this transition during the n-th time interval. It is calculated as follows:

BM (n) i,j = ℜ{ (n+1)T nT r(t)e -j(θ i tx,n +φtx,n(t)) dt}. (7)
The Viterbi algorithm is based on the recursive computation of the accumulated metric at state σ i and time instant nT denoted by CM

(n) i
and defined as:

CM (n) i = max j CM (n-1) j + BM (n) i,j
. At the end, the MLSE baseddecision rule is given by the sequence α yielding the maximum over i of CM (N ) i . We assume that the modulation index h tx is not perfectly known at the receiver. The coherent detection requires an estimate of h tx . In that case, N e pilot symbols are inserted at the beginning of the frame. Denoting by H the set of possible values for h tx , the ML estimate of h tx denoted by ĥtx , is computed as:

ĥtx = arg max hi∈H ℜ NeT 0 r pilots (t)S i pilots * (t)dt , (8)
where r pilots (t) is the received signal and S i pilots is the modulated training sequence with h i .

PROPOSED RECEIVERS

Generalized Non-coherent CPM receiver

In this section we generalize to M-ary CPM the non-coherent sequence detection receiver for Bluetooth systems referred to as adaptive NSD (Non-coherent sequence detection) and developed in [START_REF] Lampe | Noncoherent sequence detection receiver for bluetooth systems[END_REF] . Non-coherent receivers are designed to solve the problem of robustness to modulation index variation when a perfect estimation cannot be guaranteed. In [START_REF] Lampe | Noncoherent sequence detection receiver for bluetooth systems[END_REF] a trellis is defined with a reduced number of states. The states corresponding to [nT, (n + 1)T ] depend on (α n , . . . , α n-L+1 ) only. The phase state θ tx,n is determined by employing the per-survivor processing approach (PSP) [START_REF] Raheli | Persurvivor processing: a general approach to MLSE in uncertain environments[END_REF]. In order to be robust to phase variations, an adaptive phase reference recursively updated q ref (n) is added. So, the new expression of the branch metric becomes:

BM (n) i,j =ℜ{ q (i) ref (n) * (n+1)T nT r(t). e -j(θ i tx,n +2πhtx n k=n-L+1 α k q(t-kT )) dt}, (9)
where the phase reference is updated as follows:

q (j) ref (n + 1) = βq (i) ref (n) + (1 -β) (n+1)T nT r(t). e -j(θ i tx,n +2πhtx n k=n-L+1 α k q(t-kT )) dt, (10)
where q

(k)

ref is the phase reference related to the kth state and i is the previous state of the state j. The parameter β, 0 ≤ β < 1 acts as a forgetting factor. However, the estimation of modulation index h tx in the NSD receiver is done with the help of the first N e received symbols to adaptively choose the modulation index yielding the maximum cumulative metric (9): α1 , α2 , ..., αNe , ĥtx = arg max α1, α2,..., αNe ,hi∈H

ℜ NeT 0 r(t) S i seq (t) * q * ref (t)dt |h i (11)
where S i (t) corresponds to the modulation of (α 1 , . . . , αNe ) with h tx = h i . After the estimation period ĥtx is used for detecting the remaining sequence. This receiver does not require the insertion of pilots.

Generalized coherent CPM receiver

We first assume a perfect knowledge of the modulation index h tx . The proposed receiver relies on the decomposition of h tx in the form h tx = h rx + ∆h with h rx being a rational number. The key idea is to use the Viterbi algorithm with modified branch and state metrics on a trellis designed from h rx . It takes into account a phase difference proportional to ∆h and computed on a per survivor processing basis [START_REF] Raheli | Persurvivor processing: a general approach to MLSE in uncertain environments[END_REF]. The PSP application for state number reduction purpose was introduced in [10], [START_REF] Zarei | An iterative detection algorithm for coded cpfsk signals with irrational modulation index[END_REF] in the case of binary full response CPFSK. In [START_REF] Zarei | An iterative detection algorithm for coded cpfsk signals with irrational modulation index[END_REF], the branch metric computation takes into account ∆h, and is applied in a SISO algorithm namely the BCJR [START_REF] Bahl | Optimal decoding of linear codes for minimizing symbol error rate (corresp.)[END_REF]. In this section we generalize the PSP branch metric computation of [START_REF] Bahl | Optimal decoding of linear codes for minimizing symbol error rate (corresp.)[END_REF] to any kind of CPM modulation, whatever the modulation indices h tx and h rx , the value of L or the frequency pulse g(t) are. Going back to the information-carrying phase expression given in (4), we express it as a function of h rx and ∆h :

φ(t, α) = πhtx n-L i=0 αi + 2πhtx n i=n-L+1 αiq(t -iT ) = πhrx n-L i=0 αi + π∆h n-L i=0 αi + 2πhrx n i=n-L+1 αiq(t -iT ) + 2π∆h n i=n-L+1 αiq(t -iT ) = θrx,n + φrx,n(t) + λ i n + 2π∆h n i=n-L+1 αiq(t -iT ) (12)
with

θ rx,n = πh rx n-L i=0 α i , φ rx,n (t) = 2πh rx n i=n-L+1 α i q(t -iT ) and λ i n = π∆h n-L i=0 α i .
The first two terms in ( 12) are tracked by the Viterbi algorithm and the third term is the resulting phase difference which is built up at every symbol. This accumulation is calculated using the PSP technique by associating to each state an additional parameter λ i n . The last term of ( 12) is calculated at the output of the matched filter. We can write the following relation between the corrected and the reference phase states.

θ i tx = λ i n + i krx prx π, i = {0, 1, 2, ..., 2prx -1} = λ i n + θ i rx . ( 13
)
The only additional task that needs to be performed is the PSP update of the phase difference for each state, λ i n . The update equation is given by:

λ i n+1 = λ j * n + α (j * ,i) π∆h ( 14
)
where j * is selected from the M previous states of σ i , as the index of the maximum cumulative metric and α (j * ,i) is the corresponding data symbol involved in the transition from state σ j * to σ i in the receiver trellis structure. We define a new expression of the branch metric σ j → σ i used in the proposed algorithm:

θ j rx (λ j n ), α n-1 , α n-2 , ..., α n-L+1 αn -→ θ i rx (λ i n+1 ), α n , α n-1 , α n-2 , . . . , α n-L+2
. The branch metric is then calculated as follows:

BM (n) j,i =ℜ{e -jλ j n (n+1)T nT r(t). e -j(θ j rx +2πhrx n k=n-L+1 α k q(t-iT ))
e -j2π∆h n k=n-L+1 α k q(t-kT ) dt}

In the simulations, h tx is estimated from N e pilot symbols by applying Equation [START_REF] Svensson | Reduced state sequence detection of partial response continuous phase modulation[END_REF]. The branch metrics are computed by replacing ∆h by ∆h in (15).

APPLICATIONS AND SIMULATIONS

The proposed algorithms can be used in all cases where the original Viterbi algorithm fails to achieve the best trade-off between performance and complexity. In all simulations we take an oversampling of 8 samples per symbol interval and we denote by N d the data length. For the coherent detection, we take into account the spectral efficiency loss in the computation of E b N0 .

Bluetooth

The first example corresponds to the Bluetooth standard which employs the binary Gaussian frequency-shift keying (GFSK) modulation with a frequency pulse length L = 3 and a frame length N d = 2748. The modulation index is assumed to vary in a finite alphabet H known at the receiver [START_REF] Lampe | Noncoherent sequence detection receiver for bluetooth systems[END_REF], [START_REF] Ibrahim | Bluetooth receiver design based on laurent's decomposition[END_REF].

For the modulation index estimation, the best training sequence has all symbols equal to 1, since it allows to have the maximum decorrelation between the reference signals S i seq . In Figure 1 we compare the coherent receiver to the adaptive NSD. We fix h tx = 0.35 and h rx = 2 3 . h rx = 2 3 ensures the best state number reduction. We fix the forgetting factor β = 0.9 and N e = 50 in the case of NSD and let N e vary in the case of the coherent receiver. We observe that the coherent receiver requires at least N e = 15 to outperform the NSD. For N e = 20 its performance coincides with the optimum MLSE one and exhibits a gain of 1 dB compared to NSD for a BER = 10 -4 . The second application case is the random variation of the modulation index around its nominal value as in AIS systems due to an analog low-cost transmitter [START_REF] Bonacci | Advanced concepts for satellite reception of AIS messages (Toulouse Space Show[END_REF]. The AIS standard uses a binary GMSK modulation with BT = 0.4. At the transmitter, we generate randomly the transmission modulation index h tx in the interval [0.5 -0.035 0.5 + 0.035]. At the reception we discretize this interval with a step µ h to estimate h tx in the same manner as in [START_REF] Svensson | Reduced state sequence detection of partial response continuous phase modulation[END_REF]. For the NSD receiver we fix the forgetting factor β = 0.9, µ h = 0.001 and N e = 50 and let µ h and N e vary for the coherent receiver. The MLSE performance is plotted as a lower bound. The coherent receiver is built with h rx = 2 3 . From Figure 2, we see that for µ h = 0.001 and N e = 40 the coherent receiver outperforms the NSD receiver with a gain of 0.3 dB and a loss of only 0.2 dB compared to the lower bound for a BER = 10 -4 .

Wireless M-Bus

The third application case is the demodulation of quaternary GFSK signals. This application is known as "Wireless M- Bus" which is a European Standard for wireless Metering BUS [START_REF]Specification of Wireless M-bus System[END_REF]. To lower the cost and the power consumption, an analog transmitter can be implemented on the meters, yielding index modulation variation as for the AIS case. The best training sequence has all symbols equal to 3. In the quaternary modulation case, we found from several simulations that the value of h rx = 2 5 is the best choice for a compromise between performance and complexity. The performance of Figure 3 are obtained by averaging over several values of h tx taken randomly from the interval [0.5 -0.035 0.5 + 0.035]. For µ = 0.001 and N e = 50 we obtain near optimal performance with a negligible loss of 0.3 dB compared to the optimum MLSE and a gain of 0.2 dB compared to the NSD receiver one for a BER = 10 -4 .

CONCLUSIONS AND PERSPECTIVES

In this paper, we have extended the PSP application and branch metric modification of trellis-based coherent receivers, initially proposed for binary CPFSK, to detect M-ary CPM signals with any modulation index and phase response. We have applied this generalization to cases where the modulation index is unknown at the receiver and has to be estimated. We have simulated the error rate performance in three different applications, namely Bluetooth, Automatic Identification System (AIS) and the Wireless Metering Bus European standard. The error rate performances have been compared to the performance of an adaptive non-coherent receiver proposed for Bluetooth detection, which is, to the best of the authors' knowledge, one of the most promising receiver for the Bluetooth standard, where robustness against unknown modulation index is a considerable advantage. For this com-parison, we have extended the adaptive non-coherent receiver to non-binary CPM. The comparison shows that, despite the unknown modulation index, the coherent receiver outperforms the non-coherent receiver.

9 Fig. 1 .

 91 Fig. 1. Bluetooth detection for a transmission with h tx = 0.35 and a coherent reception with h rx = 2 3 with N d = 2748

Fig. 2 .

 2 Fig. 2. AIS detection for a transmission with random modulation index h tx and a coherent reception with h rx = 2 3 with N d = 256

Fig. 3 .

 3 Fig. 3. Wireless M-Bus detection for a transmission with random modulation index h tx and a coherent reception with h rx = 2 5 with N d =1000 quaternary symbols


Damien Biau
email: damien.biau@ensam.eu

Leonardo Rigo

Disentangling chaotic laminar and turbulent flow components

Transition to turbulence generally involves successive bifurcations from steady laminar solution to spatio-temporal chaos. For the case of slightly curved duct flows, the scale separation allows to separate the Navier-Stokes equations into two sets, namely the laminar solution and turbulent fluctuations. The laminar flow is sensitive to the weak curvature and is governed by the Dean number, while the small scale turbulent disturbances are insensitive to the curvature and are governed by the Reynolds number. The laminar solution is subject to super critical bifurcation leading to temporal chaos for increasing Dean number. On the other hand, the turbulent fluctuations show a sub-critical bifurcation depending on the Reynolds number. The separation between laminar, temporally chaotic solution and turbulent disturbances allows to separate the secondary flow in accordance to Prandtl's classification. The frequency analysis of the respective head losses illustrates the nonreciprocal influence of the temporal-chaos onto the spatio-temporal turbulence.

Duct flows have been investigated for more than two centuries with the aim of tackling a wide variety of problems, from water supply to blood circulation [START_REF] Sutera | The history of poiseuille's law[END_REF]. In 1845, Stokes obtained an analytical solution for a gravitydriven flow in an inclined tube [START_REF] Stokes | On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids[END_REF]. That solution was obtained assuming a steady state, namely a laminar flow. The Stokes solution gives a head loss per unit of length proportional to the bulk velocity, in agreement with observations for relatively low velocities. The departure from the Stokes solution for larger velocities was explained in 1883 by Osborne Reynolds who showed that two very different regimes of flow are present, laminar and turbulent [START_REF] Reynolds | Xxix. an experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels[END_REF]. Moreover, he showed that the transition is characterised by a single dimensionless number, called the Reynolds number by Sommerfeld [START_REF] Rott | Note on the history of the reynolds number[END_REF].

In order to determine the critical Reynolds number for the flow, the linear stability analysis of the laminar solution appeared promising, especially after the successful study by Taylor on the stability analysis of a viscous liquid contained between two rotating cylinders [START_REF] Taylor | Viii. stability of a viscous liquid contained between two rotating cylinders[END_REF]. However, for pipe flows, preliminary studies [START_REF] Salwen | Linear stability of poiseuille flow in a circular pipe[END_REF] provided evidence that the laminar solution is stable in a regime where turbulence is generally observed. In fact, Meseguer & Trefethen [START_REF] Meseguer | Linearized pipe flow to reynolds number 10 7[END_REF], with a very efficient numerical method, showed that circular pipe flows are asymptotically stable up to Reynolds number 10 7 , for any infinitesimal perturbations. This inconsistency between linear stability results and the observations led to question the single mode asymptotic analysis, namely the long time behavior considering the least stable mode. However, the non self-adjoint nature of the linearized equations can lead to a significant transient energy growth before the eventual stable asymptotic behavior takes place. Concretely, small velocity perturbations in the direction normal to the shear redistribute the streamwise momentum, that lift-up effect leads to an algebraic growth of kinetic energy of three-dimensional disturbances. Such new formulation of the problem moves the issue on the initial condition. One solution is to look for the optimal perturbation, namely the one that maximises the energy growth. This was introduced by Butler & Farrell [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF] for plane shear flow and Schmidt & Henningson [START_REF] Schmid | Optimal energy density growth in hagen-poiseuille flow[END_REF] for pipe flow. The optimal perturbations consist in streamwise elongated vortices which generate streamwise streaks. The streaks are then amplified, while the vortices are damped by viscosity. In absence of nonlinear feed-back to regenerate the vortices, the perturbations eventually evolves to asymptotic modal behaviour.

In order to determine the threshold of the initial perturbation energy above which the linear amplification is supplemented by nonlinear bootstrap, optimal perturbations have been extended to nonlinear equations for square duct flow [START_REF] Biau | An optimal path to transition in a duct[END_REF]. However, parametric studies with direct numerical simulations on random initial perturbations have shown that the threshold is not unique for a given Reynolds number. In fact, the border separating the basin of attraction of the laminar state and the turbulence was found to be fractal [START_REF] Schmiegel | Fractal stability border in plane couette flow[END_REF]. Hence the definition of a critical Reynolds number depends on the receptivity process and needs a statistical approach.

Recently, Avila et al. [START_REF] Avila | The onset of turbulence in pipe flow[END_REF] considered turbulence as statistically sustained when the lifetime of the spreading turbulence is greater than the lifetime of the decaying turbulence. In order to determine the onset of sustained turbulence, Avila et al. [START_REF] Avila | The onset of turbulence in pipe flow[END_REF] obtained these two characteristic time scales, averaged over many experiments and simulations for various Reynolds numbers. As expected, the spreading probability increases with the Reynolds number while the probability to relaminarize decreases with it. Hence, the intersection of these two curves, at Re = 2040 ± 10, marks the transition between transient and sustained turbulence in pipe flow. In contrast to the classical Landau-Ruelle-Takens view that turbulence arises from an increase in the temporal complexity of fluid motion, Avila et al. [START_REF] Avila | The onset of turbulence in pipe flow[END_REF] show the transition as a spatial proliferation of chaotic domains.

The velocity field can be decomposed into a laminar part U plus the turbulent fluctuations u . That sepa-ration is trivial for straight duct flows since the laminar solution remains stationary and the non streamwise components are nil for all Reynolds numbers. In the present article we consider a slightly square curved duct, which introduces a new parameter: the curvature ratio = h/R 1 (see the scheme in figure 1). The curvature ratio is thus zero for a straight duct and tends to unity for a duct elbow. Usual non-dimensional set of Cartesian coordinates (x, y, z) and velocity components u = (u, v, w) are adopted in streamwise, wall-normal and spanwise directions respectively. In this work the definitions for the Reynolds and Dean numbers are the following: Re = U b h/ν, De = Re h/R, where U b is the bulk velocity and ν is the dynamic viscosity. The square duct configuration presents the advantage of having only a limited number of symmetries as compared to circular pipe flows. Despite these differences, the dynamics are in general analogous to those in circular pipes, for example the square duct flow is also linearly stable for any Re [START_REF] Biau | An optimal path to transition in a duct[END_REF][START_REF] Tatsumi | Stability of the laminar flow in a rectangular duct[END_REF]. For overviews on the subject, see Berger et al. [START_REF] Berger | Flow in curved pipes[END_REF], who present an extensive review on laminar flow in curved pipes, while Kalpakli Vester et al. [START_REF] Vester | Turbulent flows in curved pipes: recent advances in experiments and simulations[END_REF] concentrate on the research concerning turbulent and transitional flows while providing a thorough historical review.

The aim of the present work is to understand the interaction between two separate phenomena that take place at the same time in turbulent flows in weakly bent pipes: the super-critical bifurcation to chaos of the laminar state (governed by the Dean number) and the sub-critical transition to turbulence (governed by the Reynolds number).

The weak curvature permits the decomposition of Navier-Stokes equation into a laminar set, sensitive to curvature effects and a turbulent set, not explicitly dependent on the curvature ratio. Indeed, the curvature influences the turbulent fluctuations only through the laminar part, which may be temporally complex. The model is obtained in the low curvature assumption, based on scales separation, neglecting the terms lower than h/R.

The laminar component is solution of the following equations [START_REF] Rigo | Flow in a weakly curved square duct: Assessment and extension of Dean's model[END_REF]:

V y + W z = 0 U t + V U y + W U z = - dP 0 dx + Re -1 ∆ 2 U V t + V V y + W V z - De Re U 2 = -P y + Re -1 ∆ 2 V W t + V W y + W W z = -P z + Re -1 ∆ 2 W (1)
with ∆ 2 = ∂ 2 /∂y 2 + ∂ 2 /∂z 2 . The curvature is represented uniquely by the centrifugal term of the cylindrical Navier-Stokes equations, -(De/Re U ) 2 . Validations of this simplified model against available data in the literature are reported in [START_REF] Rigo | Flow in a weakly curved square duct: Assessment and extension of Dean's model[END_REF]. The laminar solution is streamwise homogeneous, nonetheless the curvature may induce streamwise travelling-waves with very large streamwise scale and low amplitude. For the results shown in this article, the influence of these travelling-waves are negligible, thus we keep a two-dimensional laminar flow for simplicity.

The turbulent fluctuations satisfy the equations:

∇ • u = 0 u t + (U + u ) • ∇u + u • ∇U = - dp 0 dx e x -∇p + ∇ 2 u Re (2)
This decomposition should be distinguished from the classical Reynolds Averaged Navier-Stokes equations. In RANS equations, mean flow is forced by the turbulent fluctuations, which is not the case for the laminar equations [START_REF] Sutera | The history of poiseuille's law[END_REF]. As a result, the present model allows to investigate separately the curvature and turbulent effects.

Equations ( 1) and ( 2) are completed with no-slip boundary conditions on the four walls and a periodic boundary condition in the streamwise direction. Two different streamwise pressure gradients are present. The laminar pressure gradient -dP 0 /dx is adjusted in such way that the non-dimensional laminar bulk velocity is unitary (U b = yz U dy dz = 1). The pressure gradient of the fluctuations -dp 0 /dx is adjusted to ensure that u b = 0. The equations are solved with a spectral method in space and a backward Euler scheme in time [START_REF] Rigo | Flow in a weakly curved square duct: Assessment and extension of Dean's model[END_REF].

The equations (1) for the laminar component involve two control parameters: the Reynolds number Re and the Dean number De. Nonetheless, it is possible to express the same equations only in terms of the Dean number by using boundary-layer type scaling (see [START_REF] Rigo | Flow in a weakly curved square duct: Assessment and extension of Dean's model[END_REF]). Here, we keep the same scaling for the two systems (1) and (2) for convenience.

Thus, the laminar flow is fully characterised by the Dean number in the low curvature limit. Winters [START_REF] Winters | A bifurcation study of laminar flow in a curved tube of rectangular cross-section[END_REF] presented for the first time a bifurcation diagram of the problem. Wang and Yang [START_REF] Wang | Bifurcation and stability of forced convection in curved ducts of square cross-section[END_REF] provided a very detailed bifurcation diagram and identified regions of periodic and chaotic oscillations. The first bifurcation from a steady regime to an unsteady, periodic regime occurs at De = 128 through a saddle-node bifurcation. Starting from De = 193 we observe two periods which are incommensurable, characteristic of an aperiodic regime. Then the system quickly evolves into chaos from De ≈ 195 [19].

In the following we consider two Dean numbers: 160 and 220, associated respectively to periodic and chaotic laminar states. In figure 2, we show the temporal evolution of the velocity difference δu between two trajectories initially infinitesimally close. For the periodic regime (De=160), the difference remains small while for chaotic regime an exponential increase is observed, illustrating the sensitive dependence upon initial conditions for the laminar solution at De=220. The Reynolds number is fixed at 2500, large enough to observe developed turbulence. Although the bifurcation for laminar state state is supercritical, the turbulent state remains sub-critical, as observed for straight ducts, indeed, simulations with a linearized version of the equations 2 result in vanishing fluctuations.

As a first result, we consider the curvature effect of the duct on internal flow patterns at De=160. The organised fluid motions can be regarded as comprising two components, a primary streamwise flow and a secondary flow. These secondary flows are distinguished into first and second kind after Prandtl. The first kind secondary flow is a consequence of curvature. The centripetal acceleration -(De/Re U ) 2 is in equilibrium with the vertical pressure gradient -∂P/∂y. Because of the side walls, the pressure is non-homogeneous in the lateral direction -∂P/∂z = 0 which induces a cross-flow. That mechanism is responsible of the so-called Dean two cells vortices [START_REF] Dean | note on the motion of fluid in a curved pipe[END_REF][START_REF] Dean | Lxxii. the stream-line motion of fluid in a curved pipe (second paper), London Edinburgh[END_REF][START_REF] Dean | Note on the motion of fluid in a curved pipe[END_REF]. Moreover, when the radial pressure gradient is strong enough, it can give rise to additional Taylor-Görtler vortices on the concave outer wall illustrated in figure 3, that shows the time averaged laminar component of the flow. The presence of additional vortices results in complex laminar states which experience successive bifurcations between steady, unsteady and temporally chaotic regimes governed by the Dean number. The vortices of the second kind are a consequence of turbulence, as discovered by Prandtl in 1927. In turbulent regimes the average velocity field contains nonzero transverse components induced by the Reynolds shear stress [START_REF] Bradshaw | Turbulent secondary flows[END_REF]. Secondary motions in square ducts are known to come in the form of eight counter-rotating eddies bringing high-momentum fluid from the duct core towards the corners [START_REF] Gavrilakis | Numerical simulation of low-reynoldsnumber turbulent flow through a straight square duct[END_REF][START_REF] Huser | Direct numerical simulation of turbulent flow in a square duct[END_REF]. Vortices of the second kind also occur in the presence of curvature; figure 4 observation is made possible by the introduction of the decomposition between the unsteady laminar and turbulent components. If such decomposition is not applied the laminar and turbulent effects can be hardly discrimi-nated, which prevents the observation of the second kind motions. In a bent pipe their shape is topologically similar to the one of a straight duct, with the presence of eight streamwise vortices, that differ in position and size. The two pairs closer to the outer wall present vortices similar in size, whose separator is not aligned with the duct bisector as in the straight case, but has a different angle. This feature is even mode evident for the two pairs closer to the inner wall, in which the upper vortex is much stronger than the lower one.

The laminar-turbulent decomposition also permits to analyse the pressure drop driving the flow. Both curvature and turbulent fluctuations increase the drag, and therefore also the pressure drop required to ensure a unitary bulk velocity. In figures 5 and 6, the temporal spectra of the laminar component -dP 0 /dx and the turbulent component -dp 0 /dx are depicted for De=160 and 220 respectively. At De=160, the laminar pressure drop spectrum presents the characteristics of a periodic signal, with sharp peaks at the fundamental frequency and the harmonics, while the turbulent signal shows a rather broadband spectrum. Figure 6 illustrates a different behaviour, the laminar part is characterised by two non commensurable frequencies (and their harmonics) while the turbulent part shows frequency synchronisation with the dominant laminar frequencies.

In conclusion, the laminar-turbulent decomposition allows to distinguish between the effects of curvature and turbulent fluctuations. Moreover, the model also reveals an alternative scenario for the transition to turbulence. In contrast to the classical Landau-Ruelle-Takens view that turbulence arises from an increase in the temporal complexity of fluid motion, starting with steady laminar flow, as described in [START_REF] Kühnen | Subcritical versus supercritical transition to turbulence in curved pipes[END_REF][START_REF] Canton | Critical point for bifurcation cascades and featureless turbulence[END_REF]; here we show that in a configuration with two control parameters the transition to chaos can be independently observed for both laminar and turbulent components.

FIG. 1 .

 1 FIG. 1. Scheme and notations.

FIG. 2 .

 2 FIG.2. The separation between two neighbouring trajectories plotted against time for two Dean numbers. At De=220, the sensitivity to initial condition illustrates the chaotic nature of the laminar solution.

1 FIG. 3 .

 13 FIG. 3. Time averaged cross flow vortices. Laminar solution, Re=2500, De=160.

1 FIG. 4 .

 14 FIG. 4. Time averaged cross flow vortices. Turbulent part, Re=2500, De=160.

FIG. 5 .

 5 FIG.5. Spectrum, laminar periodic (red discontinuous line) vs turbulent (blue continuous line), at Re=2500, De=160.

FIG. 6 .

 6 FIG.[START_REF] Salwen | Linear stability of poiseuille flow in a circular pipe[END_REF]. Spectrum, laminar chaotic (red discontinuous line) vs turbulent (blue continuous line), at Re=2500, De=220.


Elham Faraji
email: elham.faraji2012@gmail.com

Philip Kurian
email: pkurian@howard.edu

Roberto Franzosi
email: roberto.franzosi@unisi.it

Stefano Mancini
email: stefano.mancini@unicam.it

Irena Cosic
email: icosic@amalnaconsulting.com

Drasko Cosic
email: dcosic@amalnaconsulting.com

Giulio Pettini
email: pettini@fi.unifi.it

Marco Pettini
email: pettini@cpt.univ-mrs.fr

Electrodynamic forces driving DNA-enzyme interaction at a large distance

In the present paper we address a model of DNA-protein selective interaction. The biochemical model is chosen on the basis of existing literature, both theoretical and experimental. This consists of a restriction enzyme, EcoRI, binding selectively to a 6 base-pair subsequence of an oligonucleotide to cleave it. A signature of the specific partnership of the two molecules is first obtained by applying the so-called Resonant Recognition Model. Then the same system is investigated by means of a model inspired to the standard Davydov and Holstein-Fröhlich models describing the electron motion along a biomolecule. Starting with a model Hamiltonian written in second quantization, the Time Dependent Variational Principle is used to work out the dynamical equations of the system. The time-Fourier spectra of the electron currents numerically computed for the DNA fragment and for the EcoRI enzyme, respectively, are multiplied to get a crossspectrum which displays a sharp peak of co-resonance. The remarkable result is the replacement of this sharp peak with a broad and noisy frequency pattern when the recognition sequence GAATTC on the DNA is randomized. This sequence-dependent charge transfer phenomenology is suggestive of a potentially rich variety of selective electrodynamic interactions of DNA molecules and transcription factors under the action of electron excitation.

I. INTRODUCTION

Progress in molecular and cellular biology is consistently linked to a better knowledge of the structure of, and functional interplay between biomolecules such as DNA, RNA and proteins.

Even though these building blocks of the living matter display no apparent systematic spatial order, it is well known that all the relevant biochemical processes follow precisely timed sequences, in other words display a dynamical order. DNA/RNA-interacting proteins (e.g., helicases, polymerases, nucleases, recombinases) modulate essential transaction processes involving nucleic acids to achieve DNA duplication and repair, gene expression and recombination, with an astonishing efficiency. Such an astonishing efficiency raises a fundamental question from a physical point of view. With biochemical reactions mostly being stereo specific, two (or more) reacting partners have to come in close contact and exhibit a definite spatial orientation to initiate particular reactions. Then, how do the various actors in a given biochemical process efficiently find each other (i.e., how does a protein effectively recruits the appropriate co-effector partner(s) or selectively connects with its DNA/RNA target(s) in a crowded cyto/nucleoplasm environment)? In other words, what are the physical forces that bring all these players at the right place, in the right order and in a reasonably short time to sustain cellular function and ultimately cellular life? The classical way to tackle these issues invokes Brownian motion (actually other proposals have been put forward, like facilitated diffusion, but these alternative explanations do not apply to bulk recruitment of molecules). At physiological temperature, ubiquitously distributed water molecules undergo chaotic motion, colliding with larger/heavier fluid components. On the latter, the neat outcome from simultaneous hits is a force of both random intensity and direction. Hence, large molecules move in a diffusive way throughout the cellular spaces and sooner or later shall encounter their cognate partners. Is this truly a good answer to the problem formulated here? Many doubts arise when one tries to estimate diffusion driven activation for some of the biochemical processes mentioned above, in fact free diffusion is considerably slowed down in the crowded cellular space [START_REF] Gori | Random walk of passive tracers among randomly moving obstacles[END_REF]. Moreover, the discrepancy between the observed reaction rates in cells and the predictions of strict random diffusion modelling are being recently questioned [START_REF] Banani | Biomolecular condensates: Organizers of cellular biochemistry[END_REF][START_REF] Berry | Physical principles of intracellular organization via active and passive phase transitions[END_REF][START_REF] Sweetlove | The role of dynamic enzyme assemblies and substrate channelling in metabolic regulation[END_REF][START_REF] Kulaeva | Distant activation of transcription: Mechanisms of enhancer action[END_REF][START_REF] Wang | Exploring the mechanisms of genomewide long-range interactions: Interpreting chromosome organization[END_REF][START_REF] Barton | Substrate channelling as an approach to cascade reactions[END_REF]. Therefore, it is timely to reconsider a longstanding hypothesis, put forward in the 1960s by H. Fröhlich [START_REF] Fröhlich | Long Range Coherence and Energy Storage in Biological Systems[END_REF] who surmised that certain biochemical reactions could be accelerated by selective electrodynamic forces acting over long distances. This hypothesis would explain a number of phenomena in living matter, such as the extraordinary efficiency of enzymatic reactions [START_REF] Fröhlich | The Extraordinary Dielectric Properties of Biological Materials and the Action of Enzymes[END_REF], of the molecular technological and theoretical reasons, no formal confirmation (or refutation) of this hypothesis of electrodynamic interactions between biomolecules has been validated until recently. After a thorough theoretical revisitation of Fröhlich's theory [START_REF] Preto | On the role of electrodynamic interactions in long-distance biomolecular recognition[END_REF], an experimental feasibility study [START_REF] Nardecchia | Detection of long-distance electrostatic interactions between biomolecules by means of Fluorescence Correlation Spectroscopy[END_REF][START_REF] Nardecchia | Experimental detection of long-distance interactions between biomolecules through their diffusion behavior: Numerical study[END_REF] and the experimental observation of an out-of-equilibrium phonon condensation in a model protein [START_REF] Nardecchia | Out-of-equilibrium collective oscillation as phonon condensation in a model protein[END_REF] which is a necessary condition [START_REF] Preto | On the role of electrodynamic interactions in long-distance biomolecular recognition[END_REF] to activate intermolecular electrodynamic interactions, a first experimental evidence of the activation of these forces has been provided very recently [START_REF] Lechelon | Experimental evidence for long-distance electrodynamic intermolecular forces[END_REF].

Within this newly opened field, the aim of the present work is to investigate whether intermolecular electrodynamic interactions can be also put at work under different conditions of activation, but without giving up the ingredient of resonance, a crucial one for selective recruitment of the cognate partners of a biochemical reaction. To this aim we will combine and adapt the Resonant Recognition Model [START_REF] Cosic | Macromolecular Bioactivity: Is It Resonant Interaction Between Macromolecules? -Theory and Applications[END_REF], detailed below, and a recent picture of intermolecular interactions as being mediated by water dipolar waves [START_REF] Kurian | Water-Mediated Correlations in DNA-Enzyme Interactions[END_REF]. This paper is organized as follows. In Section II the RRM is quickly sketched and applied to the interaction of the EcoRI restriction enzyme with an oligonucleotide (a 66 base pairs double stranded DNA fragment) containing a cleavage sequence recognized by the enzyme. In Section III we define the model used to describe the electron motion along the DNA fragment and the enzyme separately. Section IV contains the definition of the physical parameters used in the numerical simulations of the model equations. The results of these numerical simulations are then reported in Section V. The possibility of activating water mediated DNA-EcoRI interaction is discussed in Section VI. Finally, in Section VII some concluding remarks are reported.

II. THE RESONANT RECOGNITION MODEL

The Resonant Recognition Model (RRM), which has been extensively published [START_REF] Cosic | Macromolecular Bioactivity: Is It Resonant Interaction Between Macromolecules? -Theory and Applications[END_REF][START_REF] Cosic | The Resonant Recognition Model of Macromolecular Bioactivity: Theory and Applications[END_REF][START_REF] Cosic | Macromolecular Resonances[END_REF][START_REF] Cosic | Prediction of Tubulin resonant frequencies using the Resonant Recognition Model (RRM)[END_REF], is based on discovery that crucial driving force for macromolecules (protein, DNA and RNA) activation and interaction is resonant electromagnetic energy transfer at specific frequency unique for specific activation and interaction. The RRM model is capable to calculate these frequencies from periodicities within the distribution of energy of delocalised electrons along protein, DNA and/or RNA molecules using charge velocity through these macromolecules. This concept has been applied on number of proteins, DNA and/or RNA examples [START_REF] Cosic | Macromolecular Bioactivity: Is It Resonant Interaction Between Macromolecules? -Theory and Applications[END_REF][START_REF] Cosic | The Resonant Recognition Model of Macromolecular Bioactivity: Theory and Applications[END_REF][START_REF] Cosic | Macromolecular Resonances[END_REF][START_REF] Cosic | Prediction of Tubulin resonant frequencies using the Resonant Recognition Model (RRM)[END_REF][START_REF] Cosic | Is it possible to predict electromagnetic resonances in proteins, DNA and RNA?[END_REF], as well as on some medical conditions like: Crigler-Najjar syndrome [START_REF] Cosic | The Treatment of Crigler-Najjar Syndrome by Blue Light as Explained by Resonant Recognition Model[END_REF], pain [START_REF] Cosic | Influence of Tuning Element Relief Patches on Pain as Analyzed by the Resonant Recognition Model[END_REF] and influence of environmental light to health [START_REF] Cosic | Environmental Light and Its Relationship with Electromagnetic Resonances of Biomolecular Interactions, as Predicted by the Resonant Recognition Model[END_REF]. This concept has been also experimentally tested by predicting the electromagnetic frequencies for activation of L-Lactate Dehydrogenase [START_REF] Vojisavljevic | The Effect of Electromagnetic Radiation (550nm-850nm) on I-Lactate Dehydrogenase Kinetics[END_REF] and has been tested on experimental measurements of photon emission from dying melanoma cells [START_REF] Dotta | Shifting wavelength of ultraweak photon emissions from dying melanoma cells: their chemical enhancement and blocking are predicted by Cosic's theory of resonant recognition model for macromolecules[END_REF], on photon emission from lethal and non-lethal Ebola strains [START_REF] Murugan | Cosic's Resonance Recognition Model for Protein Sequences and Photon Emission Differentiates Lethal and Non-Lethal Ebola Strains: Implications for Treatment[END_REF], as well as on differentiation of osteoblasts stem cells [START_REF] Cosic | Explanation of Osteoblastic Differentiation of Stem Cells by Photo Biomodulation Using the Resonant Recognition Model[END_REF]. These findings could be used, not only to understand biological processes and resonances in biomolecules, but also to influence these processes using either radiation or design of related molecules and conductive particles.

A. The RRM applied to the DNA-EcoRI model

In order to meet with the model considered in Refs. [START_REF] Kurian | Water-Mediated Correlations in DNA-Enzyme Interactions[END_REF][START_REF] Kettlin | Real-time enzyme kinetics monitored by dualcolor fluorescence cross-correlation spectroscopy[END_REF], we have applied here the RRM model to analyse forces driving DNA-enzyme interaction between a 66bp oligonucleotide containing a cleavage subsequence of the EcoRI enzyme and EcoRI enzyme itself, as presented in Figure 1. Through the RRM model it is possible to analyse interactions directly computationally between amino acid sequences (proteins) and nucleotide sequences (DNA and RNA), based only on matching frequencies within free electron energy distribution along these macromolecules. When we have applied the RRM model to compare 66bp oligonucleotide containing a cleavage subsequence of the EcoRI enzyme and EcoRI enzyme, we have found very strong common RRM frequency at f = 0.1699 ± 0.0152, indicating that this frequency is crucial for mutual interaction between 66bp oligonucleotide and EcoRI enzyme, as presented in Figure 2. Interestingly, we have also found that nucleotide pair C-G at the position 33, which is within the cleavage site of 66bp oligonucleotide is mostly contributing to RRM characteristic frequency at f = 0.1699 ± 0.0152 and thus represent 'hot spot' [START_REF] Cosic | Macromolecular Bioactivity: Is It Resonant Interaction Between Macromolecules? -Theory and Applications[END_REF][START_REF] Cosic | The Resonant Recognition Model of Macromolecular Bioactivity: Theory and Applications[END_REF][START_REF] Cosic | Macromolecular Resonances[END_REF] for this interaction. Furthermore, electromagnetic frequencies that correspond to RRM frequencies depend on velocity of charge through macromolecular backbone.

For the velocity of 7.87 × 10 5 m/s as proposed by RRM [START_REF] Cosic | Macromolecular Bioactivity: Is It Resonant Interaction Between Macromolecules? -Theory and Applications[END_REF][START_REF] Cosic | The Resonant Recognition Model of Macromolecular Bioactivity: Theory and Applications[END_REF][START_REF] Cosic | Macromolecular Resonances[END_REF], the corresponding electromagnetic frequency for RRM frequency f = 0.1699 ± 0.0152 would be from 160THz to 192THz. However, for the velocity of 1.2 × 10 5 m/s as proposed by Yomosa [START_REF] Yomosa | The Exciton in Protein[END_REF], the corresponding electromagnetic frequency for RRM frequency f = 0.1699 ± 0.0152 would be from 24THz to 29THz, which is in accordance with the range of frequencies calculated in the next sections of this paper.

III. DEFINITION OF A DYNAMICAL MODEL AND ITS SOLUTION

In our recent work we found a rich phenomenology of intermolecular interactions of DNA molecules under the action of an external source of energy. Depending on the initial electron excitation site and excitation energy, we saw the concentrated periodic motions of the electrons arising a well-peaked frequency spectrum of the electron current to the motions with a spread noisy frequency spectrum. This motivated us strongly to interpret Resonant Recognition Model (RRM) with the aid of an explicit modelling of the electronic motions along the backbones of interacting DNA-protein biomolecules. In order to describe these electronic motions and their electrodynamic interactions we resort to a model partly borrowed from the standard Davydov and Holstein-Fröhlich models that have been originally introduced to account for electron-phonon interaction [START_REF] Scott | Davydov's soliton[END_REF][START_REF] Fröhlich | Electrons in lattice fields[END_REF][START_REF] Holstein | Studies of polaron motion: Part I. The molecular-crystal model[END_REF]. Thus, to model the electrons moving along a given DNA sequence and along the backbone of a DNA-interacting enzyme (we will consider the EcoRI restriction enzyme), separately, the following common Hamiltonian operator for both EcoRI enzyme and DNA is assumed

Ĥ = Ĥel + Ĥph + Ĥint , (1)
with

Ĥel = N n=1 E 0 B † n Bn + B † n Bn B † n Bn + J n ( B † n Bn+1 + B † n Bn-1 ) , (2)
Ĥph = 1 2 n p2 n M n + Ω n (û n+1 -ûn ) 2 + 1 2 µ(û n+1 -ûn ) 4 , (3)
Ĥint = n χ n (û n+1 -ûn ) B † n Bn . (4)
in which Ĥel and Ĥph are respectively the electronic and phononic Hamiltonians and Ĥint indicates the electron-phonon interaction term. The coupling parameters are assumed dependent on the electron excitation site with respect to the site-independent case [START_REF] Faraji | Energy transfer to the phonons of a macromolecule through light pumping[END_REF]. Considering only a longitudinal chain of amino acids (or nucleotides), Bn and B † n denote the lowering and raising operators between the lattice site n ∈ {1, 2, . . . , N } labelling the amino acids along the EcoRI enzyme (or nucleotides along a DNA). The parameter E 0 implies the initial excitation energy of the electron according to the initial form of the electronic state vector. The nonlinear constant is the coupling energy of the interaction between the moving electron along the chain with the electrons of the substrate of amino acids (or nucleotides). The coupling parameter J n is site-dependent tunnelling term of electron across two nearest neighbouring amino acids (or nucleotides).

The momentum and position operators pn and ûn of the vibronic Hamiltonian determine the longitudinal displacements of the n-th phonon in the sequence of amino acids (or nucleotides) from their equilibrium position and the coupling term Ω n denotes the site-dependent spring parameter of two neighbouring sites. M n is the mass of the n-th amino acid of EcoRI enzyme sequence ( or nucleotide of a DNA segment) and the nonlinear coupling constant µ implies phonon-phonon interaction, absent in the harmonic approximation. Finally, the parameter χ n of the interaction Hamiltonian is the electron-phonon coupling dependent of the n-th site.

The wave function |ψ(t) at any time t may be written in the Davydov ansatz by the following factorization

|ψ(t) = |Ψ(t) |Φ(t) , (5)
with the normalization condition ψ(t)|ψ(t) = 1. The state vector |Ψ(t) describes a single quantum excitation of electron propagating along a protein chain of N amino acids ( or a DNA sequence of N nucleotides)

|Ψ(t) = n C n (t) B † n |0 el , (6)
in which |0 e is the electronic vacuum state, and |Φ(t) is the vibronic wave function

|Φ(t) = e -i/ [βn(t)pn-πn(t)ûn] |0 ph , (7)
for which the expectation values for longitudinal displacement ûn and momentum pn are respec-

L(t) = i ψ(t)|∂ t |ψ(t) -ψ(t)| Ĥ|ψ(t) . (8)
Now, TDVP which is equivalent to the least action principle reads as

δS(t) = δ t 0 L(t )dt = 0. (9)
Then from the wave function [START_REF] Kulaeva | Distant activation of transcription: Mechanisms of enhancer action[END_REF] and Lagrangian (8) we have

L = n i Ċn (t)C * n (t) + 1 2 π n (t) βn (t) -πn (t)β n (t) -H(C n , C * n , β n , π n ) , (10)
in which

H(C n , C * n , β n , π n ) = ψ(t)| Ĥ|ψ(t) .
Here we fulfill the stationary action ( 9) and obtain

δS(t) = n i -Ċ * n (t)δC n (t) + Ċn (t)δC * n (t) + βn (t)δπ n (t) -πn (t)δβ n (t) -(∂ Cn H)δC n -(∂ C * n H)δC * n -(∂ βn H)δβ n -(∂ πn H)δπ n = 0, (11)
which gives the equations

i Ċn = ∂ C * n H βn = ∂ πn H πn = -∂ βn H. ( 12
)
According to the expectation value of the Hamiltonian

ψ| Ĥ|ψ = n E 0 |C n | 2 + |C n | 4 + J n (C * n C n+1 + C * n+1 C n ) + 1 2 1 Mn π 2 n + Ω n (β n+1 -β n ) 2 + 1 2 µ(β n+1 -β n ) 4 +χ n (β n+1 -β n )|C n | 2 . ( 13
)
and Eqs. ( 12) the equations of the motion are found as

i Ċn = E 0 + 2 |C n | 2 + χ n (β n+1 -β n ) C n + J n C n+1 + J n-1 C n-1 , M n βn = Ω n β n+1 + Ω n-1 β n-1 -Ω n-1 β n -Ω n β n + χ n |C n | 2 -χ n-1 |C n-1 | 2 + µ (β n+1 -β n ) 3 -(β n -β n-1 ) 3 . ( 14
)
It is worth noting that the dynamical equations worked out by means of the TDVP are formally classical but give the time evolution of actual quantum expectation values.

IV. PHYSICAL PARAMETERS FOR THE NUMERICAL COMPUTATIONS

We need to determine the physical and authentic values of the coupling parameters of the Hamiltonian to do our numerical simulations. In so doing, we borrow the quantities from

Ref. [START_REF] Cosic | Macromolecular Bioactivity: Is It Resonant Interaction Between Macromolecules? -Theory and Applications[END_REF][START_REF] Veljkovic | General model of pseudopotentials[END_REF] of the potential interaction energies between an electron and each of all the amino acids reported in table I as well as the potential energies of the interaction between an electron Table II: Electron-Ion interaction potential (EIIP) value for 20 amino acids. From Ref. [START_REF] Cosic | Macromolecular Bioactivity: Is It Resonant Interaction Between Macromolecules? -Theory and Applications[END_REF].

with each of four nucleotides presented in table II. The electron in motion with the initial energy E 0 , during its route, experiences a periodic sequence of square potential barriers of different height and of the same width a -the average distance between two nearest neighboring sites -by tunneling across the chain of amino acids constituting a protein or the sequence of nucleotides composing DNA. Such a distance is a = 4.5 Å and a = 3.4 Å respectively in EcoRI enzyme and DNA fragment. Then we can estimate roughly the electron tunneling term as J n = E 0 T n,n+1 by introducing the transmission coefficient T n,n+1 from the probability P (n → n ± 1) of tunneling from one potential barrier to the nearest one as follows

• Case 1: E 0 < E n+1 T n,n+1 = 1 + E 2 n+1 sinh 2 (β n+1 a) 4E 0 (E n+1 -E 0 ) -1 , (15)
where

β n+1 = [2m e (E n+1 -E 0 )/ 2 ] 1/2 . • Case 2: E 0 > E n+1 T n,n+1 = 1 + E 2 n+1 sin 2 (β n+1 a) 4E 0 (E 0 -E n+1 ) -1 , (16)
in which

β n+1 = [2m e (E 0 -E n+1 )/ 2 ] 1/2 .
Here m e is the mass of electron and E n+1 are the potential interaction energies between the electrons in motion and the local amino acids ( or nucleotides). Moreover, in a rough estimation we set χ n = dE/dx = (E n+1 -E n )/a as the site-dependent electron-phonon coupling.

In order to perform the numerical simulations, the dimensionless expectation value of the Hamitonian ( 13) and of the dimensionless equations of motion [START_REF] Nardecchia | Experimental detection of long-distance interactions between biomolecules through their diffusion behavior: Numerical study[END_REF] are found by rescaling time t = ω -1 τ and length

β n = Lb n where L = ( ω -1 M -1 n ) 1/2 .
Then we obtain

ψ| Ĥ|ψ = n E |C n | 2 + |C n | 4 + J n (C * n C n+1 + C * n+1 C n ) + 1 2 ḃ2 n + Ω n (b n+1 -b n ) 2 + 1 2 µ (b n+1 -b n ) 4 + χ n (b n+1 -b n )|C n | 2 , (17)
and

i dC n dτ = E + 2 |C n | 2 + χ n (b n+1 -b n ) C n + J n C n+1 + J n-1 C n-1 , d 2 b n dτ 2 = Ω n b n+1 + Ω n-1 b n-1 -Ω n-1 b n -Ω n b n + χ n |C n | 2 -χ n-1 |C n-1 | 2 + µ (b n+1 -b n ) 3 -(b n -b n-1 ) 3 , (18)
where the dimensionless parameters are

E = E 0 ω ; = ω ; J n = J n ω ; χ n = χ n √ M n ω 3 ; Ω n = Ω n M n ω 2 ; µ = µ M 2 n ω 3 . (19)
The sound speed of amino acids is V ∼ 4 Km/s from [START_REF] Scott | Davydov's soliton[END_REF][START_REF] Cruzeiro | Temperature effects on the Davydov soliton[END_REF] and of nucleotides is V = 1.69

Km/s from [START_REF] Hakim | The speed of sound in DNA[END_REF] (neglecting small local variations due to the different masses of the amino acids or the nucleotides). We apply two different analyzes for computing the spring parameter in our simulations. First, we consider the known speed of sound V = a(Ω n /M n ) 1/2 leading to the constant dimensionless parameter Ω = V 2 /a 2 ω 2 from [START_REF] Buchanan | The long reach of dipoles[END_REF] where Ω = 0.79 for amino acids

and Ω = 0.25 for nucleotides. Second, from [START_REF] Scott | Davydov's soliton[END_REF] we borrow the spring constant of amino acids Ω = 18.3N/m giving us from [START_REF] Buchanan | The long reach of dipoles[END_REF] the site-dependent dimensionless Ω n = 1.83/m n . Third, we equations in [START_REF] Cosic | Influence of Tuning Element Relief Patches on Pain as Analyzed by the Resonant Recognition Model[END_REF] with an Euler predictor-corrector to arrive

q (0) n (t + ∆t) = q n (t) + ∆t Q n [b(t), q(t), p(t)], p (0) n (t + ∆t) = p n (t) + ∆t P n [b(t), q(t), p(t)], q (1) n (t + ∆t) = q n (t) + ∆t 2 Q n [b(t), q(t), p(t)] + Q n [b(t), q (0) (t + ∆t), p (0) (t + ∆t)] , p (1)
n (t + ∆t) = p n (t) + ∆t 2 P n [b(t), q(t), p(t)] + P n [b(t), q (0) (t + ∆t)), p (0) (t + ∆t)] , b n (t + ∆t) = b n (t) + ∆t π n (t), π n (t + ∆t) = π n (t) + ∆t B n [b(t + ∆t), q (1) (t + ∆t), p (1) (t + ∆t)]. (26)
The integration of half of the set of the dynamical equations ( 25) by means of a symplectic algorithm, and half of the equations by means of the Euler predictor-corrector [START_REF] Cosic | Environmental Light and Its Relationship with Electromagnetic Resonances of Biomolecular Interactions, as Predicted by the Resonant Recognition Model[END_REF] results in the very well conservation of total energy without any shift-just with zero-mean fluctuations around a given value fixed by the initial conditions-by considering sufficiently small integration time steps ∆t. We need also to define the initial states of electron and phonon independently of the specific physical excitation mechanism. The electron wavefunction ( 6) is described by the amplitudes C n (t = 0) centered at the excitation site n = n 0 and distributed at time t = 0 [START_REF] Scott | Davydov's soliton[END_REF] as

C n (t = 0) = 1 √ 8σ 0 sech n -n 0 4σ 0 (27)
where σ 0 specifies the amplitude width. Concerning the phonon part of the system, we consider a thermalized macromolecule EcoRI enzyme and DNA fragment at room temperature T = 310 • K.

At thermal equilibrium, average kinetic and potential energies per degree of freedom are equal, and the total energy is equally shared among all the phonon modes. Accordingly, the displacements and the associated velocities have been initialized with random values of zero-mean at t = 0, then in a dimensionless form we have

|b n (0)| n = k B T ωΩ ; |π n (0)| n = k B T ω . (28)
Periodic boundary conditions have been used for the both electron and phonon part of the DNA-EcoRI interacting system and the frequency has been assumed ω = 10 13 s -1 .

V. NUMERICAL RESULTS

We have used an integration time step ∆t = 5 × 10 -6 to work out our numerical simulations with a very good energy conservation and the typical relative error ∆E/E = 10 -6 . The following analyses have reported the spectral properties of electron currents in the interaction of a DNA fragment of N = 66 nucleotides and an EcoRI restriction enzyme of N = 276 amino acids for the different initial activation energies of electron E 0 , the various initial excitation sites of electron n 0 of the probability amplitude [START_REF] Vojisavljevic | The Effect of Electromagnetic Radiation (550nm-850nm) on I-Lactate Dehydrogenase Kinetics[END_REF] and the distinct forms of the phononic spring term Ω n . We study the Fourier spectrum of the electron current activated on a segment of DNA and also DNAinteracting enzyme, separately and, from now on, use the index 1 and 2 for all the terms of DNA and EcoRI, respectively. Resorting to the standard probability current j(x, t) of the electron wave function ( 6) the electron density current is given by j(x, t) = e 2m e i (ψ ∇ψ -ψ∇ψ )

hence the average electron current, in a spatially discretized form for numerical computation, is

i 1,2 (t) = 1 l 1,2 l 1,2 0 j 1,2 (x, t)dx = e 2N 1,2 a 1,2 m e i × N 1,2 j=1 Ψ * 1,2 (x j , t) Ψ 1,2 (x j+1 , t) -Ψ 1,2 (x j-1 , t) 2 -Ψ 1,2 (x j , t) Ψ * 1,2 (x j+1 , t) -Ψ * 1,2 (x j-1 , t) 2 , (29)
where l 1,2 are the lengths, and i 1,2 are the currents flowing along the DNA fragment and the EcoRI enzyme macromolecules, respectively. In Figures. ( 3) and ( 5), we have plotted the cross Fourier spectrum of the currents ĩ * 1 (ν) ĩ2 (ν) of DNA containing the sites CTTAAG, recognized by the enzyme EcoRI, and studied whether they are specific sites, which has a fundamental role in the DNA-protein interaction. Fig. [START_REF] Berry | Physical principles of intracellular organization via active and passive phase transitions[END_REF] shows the behavior of the system when the excited electron on the DNA has the initial energy E 1,0 = 0.72 eV and its wavefunction is initially centered at the site n 1,0 = N/2, while for the restriction enzyme the initial excitation energy of the electron is E 2,0 = 0.2 eV localized at n 2,0 = N/3. Besides, as we discussed already in Section IV, we consider the dimensionless expressions of the site-dependent phononic spring Ω 1,n = 0.48/m n for the nucleotides and the constant term Ω 2 = 0.79 for the amino acids. We see the very interesting phenomenon of a clear co-resonance around 20 THz when the specific CTTAAG restriction sequence is taken into account. This result is in qualitative agreement, and possibly also in very good quantitative agreement as discussed in Section II, with the peak found by applying the RRM.

Another significant finding is that the cross spectrum becomes completely spread when the recognition sites are randomly chosen AGCTTA. Moreover, when we exchange just one nucleotide of the restriction sequence with its own complementary as CATAAG the co-resonance undergoes a little alteration and broadens by changing two nucleotides of the recognition sites in the form of GTTAAC. In Figure [START_REF] Kulaeva | Distant activation of transcription: Mechanisms of enhancer action[END_REF] the results are reported as numerical simulations obtained for the different initial conditions which confirm well the robustness of the phenomenology previously seen. Here we assume the initial electronic activation energy E 1,0 = 0.85 eV posited in the site n 1,0 = N/2 in DNA macromolecule and the ones in the DNA-interacting enzyme E 2,0 = 0.85 eV located in n 2,0 = N/3. Also, the dimensionless parameter of phononic spring in DNA fragment is assumed constant Ω 1 = 0.25 and in EcoRI enzyme is considered site-dependent Ω 2,n = 1.83/m n . The sharp peak of co-resonant spectrum of the DNA-EcoRI interaction with the characteristic site restriction sites CTTAAG happens around 29 THz that broads entirely by choosing the randomized recognition sites TCATGA. It is clear that the sharp frequency spectrum ramifies very little by exchanging only one nucleotide of the sites, cleaved by EcoRI, with its complementary as CTTATG and destroys somehow more when two nucleotides are exchanged with their complementary sites as CTATAG

1 = 2 = 0.0329 eV, µ 1 = µ 2 = 0.5, Ω 2,n = V 2 M /a 2 , Ω 1,n = V 2 M n /a 2 , J 1,n , J 2,n , χ 1,n and

VI. POSSIBLE MECHANISMS ACTIVATING LONG RANGE DNA-ENZYME INTERACTION

In principle, the results reported in the preceding section can be at the origin of a selective electrodynamic interaction between DNA and enzyme. In order to assess the actual relevance in biological contexts of the co-resonance of electron currents, a quantitative estimate of the strength of the implied interaction requires to work out a similar approach to the one reported in [START_REF] Preto | On the role of electrodynamic interactions in long-distance biomolecular recognition[END_REF] from the analytical viewpoint, and to get experimental information about the intensity of the currents and the possible mechanisms of their activation in a biological environment. These points will be tackled in future investigations, in what follows we qualitatively sketch possible scenarios to activate electrodynamic DNA-enzyme interaction.

First, given two electron currents j (1) (x, t) and j (2) (x, t), representing those of DNA and EcoRI, respectively

j (1,2) (x, t) =
e 2m e i ψ ∇ψ -ψ ∇ψ according to the D'Alembert equations (in Gaussian units and Lorenz gauge)

2 A (1) (x, t) = (4π/c) j (1) (x, t) and 2 A (2) (x, t) = (4π/c) j (2) (x, t)
the mutual interaction is described by the coupling terms

j (2) (x, t) • A (1) (x, t) and j (1) (x, t) • A (2) (x, t) .
Since the D'Alembert equation is linear, the vector potential inherits the spectral properties of the current that generates it. As a consequence, the co-resonance between the two currents j (1) (x, t) and j (2) (x, t) entails the largest values of the time averages of the interaction energies.

Second, intriguing connections exist between the models presented above, which describe electronic motions along a given DNA sequence and a given protein sequence, and the coordinated electronic fluctuations that arise from van der Waals many-body dispersion forces [START_REF] Kurian | Water-Mediated Correlations in DNA-Enzyme Interactions[END_REF][START_REF] Kurian | How Quantum Entanglement in DNA Synchronizes Double-Strand Breakage by Type II Restriction Endonucleases[END_REF][START_REF] Hermann | Nanoscale π -π Stacked Molecules Are Bound by Collective Charge Fluctuations[END_REF][START_REF] Gori | Second quantization approach to many-body dispersion interactions[END_REF] in a variety of molecular contexts. Specifically, productive insights have emerged from attempts to unify atomistic, continuum, and mean-field treatments in the quantum electronic behaviors of DNA and proteins in water [START_REF] Kurian | Water-Mediated Correlations in DNA-Enzyme Interactions[END_REF][START_REF] Gori | Second quantization approach to many-body dispersion interactions[END_REF][START_REF] Kurian | Chirality-Energy Conversion Induced by Static Magnetic Effects on Free Electrons in Quantum Field Theory[END_REF][START_REF] Stöhr | Quantum Mechanics of Proteins in Explicit Water: The Role of Plasmon-like Solute-Solvent Interactions[END_REF][START_REF] Kurian | From Micro to Macro: A Relativistic Treatment of the Chiral Energy Shifts Caused by Static Electromagnetic Effects on Free Electrons[END_REF]. Even in the presence of thermally turbulent aqueous environments, it has been shown that these collective electronic dispersion correlations can persist at several nanometers from the protein-water interface, and these correlations are energetically relevant for protein folding processes at the microsecond scale [START_REF] Stöhr | Quantum Mechanics of Proteins in Explicit Water: The Role of Plasmon-like Solute-Solvent Interactions[END_REF], and likely for even longer times in vivo.

Kurian and coworkers [START_REF] Kurian | Water-Mediated Correlations in DNA-Enzyme Interactions[END_REF][START_REF] Kurian | How Quantum Entanglement in DNA Synchronizes Double-Strand Breakage by Type II Restriction Endonucleases[END_REF] PyMOL and adapted from [START_REF] Kurian | Water-Mediated Correlations in DNA-Enzyme Interactions[END_REF].

Following standard treatments in quantum optics [START_REF] Gerry | Introductory Quantum Optics[END_REF], this interaction between the DNA radiative field and the surrounding (quasi-continuous) water dipole field can be written in the form of a Jaynes-Cummings-like Hamiltonian that scales with the number of water molecules N as

H int = √ N γ(a † S -+ aS + ), ( 30
)
where γ is the coupling constant proportional to the matrix element of the molecular dipole moment and inversely proportional to the volume square root, a † and a are the creation and annihilation operators, respectively, for the DNA radiative electric field E, and S + and S -are the raising and lowering operators, respectively, for the collective water dipole state. The quasicontinuum water dipole "field" thus takes the place of the N two-level systems described in the Tavis-Cummings model [START_REF] Tavis | Exact Solution for an N -Molecule-Radiation-Field Hamiltonian[END_REF].

It should be noted that the coupling √ N γ in Equation ( 30) between the DNA radiative field and the collective water orientational state levels [START_REF] Kurian | Water-Mediated Correlations in DNA-Enzyme Interactions[END_REF] scales with the square root of the water density ρ, which varies with temperature and pressure. However, if we consider that the number of water molecules in a (cubic) domain encompassed by infrared wavelengths 1 µm exceeds 10 billion, such sufficiently large N for the collective state can provide a protective gap against thermalization (k B T ≈ 0.02 eV at physiological temperatures) for the long-range correlations we consider.

Furthermore, the spontaneous breakdown of phase symmetry generates a field polarization (in the so-called "limit cycle" regime) that preserves gauge invariance by dynamical coherence between the matter quasi-continuum field (DNA, water, enzyme) and the phase-locked electromagnetic field (radiative field from DNA, water, enzyme).

As a toy model, we use Faraday's law of induction for the DNA double helix, considered here as a long solenoid with radius R, n turns per unit length, and current along the backbone varying as I = I 0 e -αt , where α is in general complex. For distances from the longitudinal axis r > R outside the helix-solenoid, we can estimate the induced electric field E(r, t) tangent to a circular path surrounding the cylindrically symmetric system:

|E(r, t)| = Ω 2 |e -αt | r , (31)
where Ω = |α|µnI 0 R 2 and µ is the magnetic permeability in water. From Equation (31) we can thus derive the creation and annihilation operators a † , a for the radiative field in the interaction Hamiltonian of Equation [START_REF] Cosic | Explanation of Osteoblastic Differentiation of Stem Cells by Photo Biomodulation Using the Resonant Recognition Model[END_REF].

The resulting interaction energies range between ∼ 0.1 -1 eV, populating bands in the infrared spectrum between 0 < ν < 1000 cm -1 , which overlaps with the energy scale of the collective electronic fluctuation modes in the DNA target sequence and in the enzyme when taken separately, but remains distinct from the more energetic intramolecular vibrations and purely electronic transitions of individual water molecules. These collective electronic fluctuation modes in the 0.1 -1 eV range do not couple to the rotational quantum transitions of individual water dipoles (meV scale), but rather to the emergent polarization modes present in the collective dipole network. The spectroscopic peaks for liquid water also lie completely within this range.

Chiral sum frequency generation spectroscopy experiments [START_REF] Mcdermott | DNA's Chiral Spine of Hydration[END_REF] have demonstrated the exis-tence of a chiral water superstructure surrounding DNA under ambient conditions, thereby confirming that the chiral structure of DNA can be imprinted electrodynamically on the surrounding solvent. These experiments have also shown that some sequence-specific fine structure persists in this chiral spine of hydration, providing a mediating context for DNA target sequence recognition by the enzyme.

VII. CONCLUDING REMARKS

The aim of the present paper is twofold. First, we put forward a novel physical interpretation for the Resonant Recognition Model (RRM) of biomolecular interactions, emerging from a widely used electron-phonon Hamiltonian applied to alternating currents along the backbone of specific DNA target sequences. Second, the work here reported contributes to the still wide open discussion of long-distance electrodynamic intermolecular interactions, which have recently been demonstrated [START_REF] Lechelon | Experimental evidence for long-distance electrodynamic intermolecular forces[END_REF][START_REF] Buchanan | The long reach of dipoles[END_REF].

Regarding the first aim, the RRM has produced phenomenologically interesting outcomes since its conceptualization in the early '80s. The RRM in its original formulation was applied to the pair of partners of the biochemical reaction involving a DNA fragment and a restriction enzyme, EcoRI, that binds to a specific subsequence of the DNA fragment to cleave it. The interaction energies of an electron with the sequence of nucleotides composing a specific DNA fragment on the one side, and the interaction energies of an electron with the sequence of amino acids composing the EcoRI enzyme on the other side, yield two numerical sequences. The product of their Fourier spectra, or cross-spectrum, displays a sharp peak. The peak so found qualitatively witnesses to the specific relationship between the two biomolecules, though the physics behind this co-resonance still needs to be clarified. Such a clarification is provided by the co-resonance of the time-domain Fourier spectra of the alternating electron currents moving along the DNA and enzyme, respectively. These currents are worked out through quantum dynamical models describing the electron-phonon coupling, derived from standard Davydov and Holstein-Fröhlich treatments [START_REF] Scott | Davydov's soliton[END_REF][START_REF] Fröhlich | Electrons in lattice fields[END_REF][START_REF] Holstein | Studies of polaron motion: Part I. The molecular-crystal model[END_REF]. The remarkable finding is the disappearance of the co-resonance peak when the six-base-pair (bp) target recognition subsequence GAATTC on the DNA is randomized in different ways.

Regarding the second aim of the paper, the prospective relevance for biology of long-range selective and attractive intermolecular interactions was discussed in the Introduction and has recently been given experimental confirmation [START_REF] Lechelon | Experimental evidence for long-distance electrodynamic intermolecular forces[END_REF] in the presence of collective intramolecular oscillations. The question naturally arises whether the electronic degrees of freedom of electrodynamically interacting molecules can offer alternative or complementary mechanisms to activate such long-range intermolecular forces. We have presented a first step in this second direction, and the remarkable finding mentioned above motivates further investigations. In fact, at present we have considered the motion of a single electron, but we can think that under suitable excitation processes (for example, under repeated ATP hydrolysis events or near an ionic channel) definitely stronger currents can be activated, producing either direct electrodynamic current-to-current interactions, or, as intriguingly proposed in [START_REF] Kurian | Water-Mediated Correlations in DNA-Enzyme Interactions[END_REF] and discussed in the preceding section, water-mediated electrodynamical interactions between the radiative field emerging from electronic fluctuational motions in DNA and in protein, and the water dipole (matter) field in the quasi-continuum limit. Finally, the observed sequence-dependent co-resonance phenomenology for the chosen biochemical model is suggestive of a potentially rich variety of selective electrodynamic interactions of more general kind, like, for example, of DNA molecules and transcription factors under the action of electron excitation.

Figure 1 :

 1 Figure 1: Top: Sequence of an oligonucleotide (66 bp) containing a cleavage subsequence of EcoRI enzyme. Bottom: amino acid sequence of EcoRI enzyme.

Figure 2 :

 2 Figure 2: Characteristic RRM frequency for mutual interaction between 66bp oligonucleotide and EcoRI enzyme at f = 0.1699 ± 0.0152

Figure 3 :

 3 Figure 3: The cross frequency spectrum of the interaction between DNA strand with N 1 = 66 nucleotides and the EcoRI enzyme with N 2 = 276 amino acids for the initial conditions: T = 310 • K, N 0,1 = N/2, N 0,2 = N/3, E 1,0 = 110, E 2,0 = 30, 1 = 2 = 5, µ 1 = µ 2 = 0.5, Ω 2 = 0.79 and site-dependent parameters Ω 1,n = 0.48/m n , J 1,n , J 2,n , χ 1,n and χ 2,n corresponding to E 0,1 = 0.72 eV, E 0,2 = 0.2 eV,

χ 2 ,

 2 n regarding to the Equations (15) and (16); and σ 1,0 = σ 2,0 = 0.1. a) DNA containing the specific CTTAAG recognition sites, b) randomized restriction sites AGCTTA, c) exchanging only one nucleotide with its complementary CATAAG, d) exchanging two nucleotides with their complementaries GTTAAC.

Figure 4 :

 4 Figure 4: Same physical conditions of Figure 3. The cross frequency spectrum of the interaction between DNA and the EcoRI enzyme. e) The randomized restriction sites TCATGA, f) exchanging only one nucleotide with its complementary CTTAAC, g) exchanging two nucleotides with their complementaries CATATG. The frequency ν is measured in 10 13 s -1 .

Figure 5 :

 5 Figure 5: The cross frequency spectrum of the interaction between DNA strand with N 1 = 66 nucleotides and the EcoRI enzyme with N 2 = 276 amino acids for the initial conditions: N 0,1 = N/2, N 0,2 = N/3, E 1,0 = E 2,0 = 129.17, Ω 1,n = 0.25, Ω 2,n = 1.83/m n corresponding to E 1,0 = E 2,0 = 0.85 eV, Ω 1,n = V 2 M n /a 2 and Ω 2 = 18.3 N/m. The other parameters are the same as Figure (3); a) DNA containing the specific recognition sites CTTAAG, b) randomized restriction sites TCATGA, c) exchanging only one nucleotide with its complementary site CTTATG, d) exchanging two nucleotides with their complementaries CTATAG.

Figure 6 :

 6 Figure 6: Same physical conditions of Figure 5. The cross frequency spectrum of the interaction between DNA and the EcoRI enzyme. e) randomized restriction sites AGATCT, f) exchanging only one nucleotide with its complementary site CATAAG, g) exchanging two nucleotides with their complementaries GT-TATG. The frequency ν is measured in 10 13 s -1 .

  have additionally shown that such collective electronic (Drude quantum harmonic oscillator) modes are suitably fine-tuned for the synchronized catalysis of two phosphodiester bonds (∼ 0.46 eV), and that the palindromic mirror symmetry of the doublestranded DNA target sequence recognized by the enzyme (see Figure 7) allows for conservation of parity in the symmetric, site-specific cleavage of both DNA strands. By considering the radiative field E created by the collective electronic fluctuation modes in the DNA target sequence, a nonvanishing polarization density emerges spontaneously in the orientational correlations of the water dipole network through the interaction Hamiltonian H = -d e •E, where d e is the permanent electric dipole moment for a single water molecule.

Figure 7 :

 7 Figure 7: Aromatic network in EcoRI-DNA complex. Tryptophan (blue), tyrosine (purple), and phenylalanine (green) form correlated electronic dispersion networks in EcoRI, shown here in the top panel bound to its double-stranded DNA substrate, with adenine-thymine (yellow) and cytosine-guanine (orange) base pairs highlighted. Other amino acids (gray) are displayed in the context of their secondary structures within the enzyme, and in the bottom panel only one of the two EcoRI dimers is shown for clarity, to showcase the π -π stacking of the DNA bases. Image of EcoRI (PDB ID: 1CKQ) at 1.85 Å resolution created with

Acknowledgments E.F. warmly thanks the Fondazione Cassa di Risparmio di Firenze for having co-funded her PhD fellowship. P.K. acknowledges support from the U.S.-Italy Fulbright Commission, the National Science Foundation, and the Whole Genome Science Foundation. M.P. participated in this work within the framework of the project MOLINT which has received funding from the Excellence Initiative of Aix-Marseille University -A*Midex, a French "Investissements d'Avenir" programme. R. F. acknowledges the support received by the National Group of Mathematical Physics (GNFM-INdAM), and the support from the QuantERA ERA-NET Co-fund 731473 (Project Q-CLOCKS).

assume the average spring constant Ω = V 2 M /a 2 of DNA -in which M is the average masses of the nucleotides-and acquire the dimensionless site-dependent parameter Ω n = 0.48/m n for nucleotides. The expression m n represents the dimensionless mass of amino acids and nucleotides. In order to perform numerical integration of the dynamical equations it is useful to introduce the variables

then we rewrite Eqs. [START_REF] Kurian | Water-Mediated Correlations in DNA-Enzyme Interactions[END_REF] as

Substituting the r.h.s of above equation [START_REF] Cosic | Is it possible to predict electromagnetic resonances in proteins, DNA and RNA?[END_REF] with

Furthermore, we denote Q n [b(t), q(t), p(t)] and P n [b(t), q(t), p(t) respectively with the r.h.s. of Eqs.( 21) and [START_REF] Cosic | Prediction of Tubulin resonant frequencies using the Resonant Recognition Model (RRM)[END_REF] and obtain the numerical integrations by combining a finite differences scheme and a leap-frog scheme as follows

The integration scheme for b n (t) and p n (t) is a symplectic one, meaning that all the Poincaré invariants of the associated Hamiltonian flow are conserved, among these invariants there is energy.

We can not apply the simple leap-frog scheme to the equations for qn (t) and ṗn (t) due to the r.h.s.

of the equations for qn (t) explicitly depend on q n (t) and b n (t); therefore, we integrate the first two