M Schemmer

A Johnson

R Photopoulos

I Bouchoule

A Monte Carlo wavefunction description of losses in a 1D Bose gas and cooling to the ground state by quantum feedback

The effect of atom losses on a homogeneous one-dimensional Bose gas lying within the quasicondensate regime is investigated using a Monte Carlo wavefunction approach. The evolution of the system is calculated, conditioned by the loss sequence, namely the times of individual losses and the position of the removed atoms. We describe the gas within the linearized Bogoliubov approach. For each mode, we find that, for a given quantum trajectory, the state of the system converges towards a coherent state, i.e. the ground state, displaced in phase space. Provided losses are recorded with a temporal and spatially resolved detector, we show that quantum feedback can be implemented and cooling to the ground state of one or several modes can be realized.

In [1], the effect of atom losses on a one-dimensional quasi-condensate was investigated. The authors have shown that, within a linearized approach and for a large enough initial temperature, one expects the temperature of the low lying modes to decrease in time, in agreement with recent experimental results [2]. The fluctuations induced by the loss process due to the discrete nature of atoms is however responsible for a heating, limiting the temperature which can be achieved. More precisely, one expects that the temperature asymptotically converges towards gρ(t) where g is the coupling constant and ρ the linear atomic density [1,3]. In particular, excitations in the phononic regime, i.e. of frequency much smaller than gρ(t)/ , never enter the quantum regime: their mean occupation number stays very large such that they lie in the Raighley-Jeans regime. This heating only occurs if one ignores the results of the losses, or, equivalently, if one takes the the partial trace on the state of the reservoir in which losses occur, ending up with the Master equation for the system's density matrix. If on the other hand, one records the losses, more information is gained on the system and the analysis made in [1] is no longer sufficient.

feedback -lattice amplitude

Bose gas detectors lost atoms δx

M 1 M 2 M N V(x)
lattice potential

x FIG. 1. A one-dimensional Bose gas with atom losses and a spatially resolved single atom detectors system. The information on the atom loss can be used to create a feedback loop on the atoms via a lattice potential. The amplitude of the lattice potential is controlled by a processing unit which uses the information gained from the atom losses.

In this paper, we assume the losses are monitored with a spatially and temporally resolved detector and we describe the evolution of the system using a Monte Carlo wavefunction analysis. The measurement back action leads to an evolution of the system conditioned by the result of the loss process, namely a given history of losses. Averaging over the different possible histories, the results of [1] are recovered. The analysis proposed in this paper however not only presents an alternative picture conveying more physical insight, but it also opens the road to the realization of measurement based quantum feedback: controlled dynamics, conditioned on the monitored losses, allows to reach lower temperatures. In this paper, we show that feedback on a given mode of the system could in principle allow to cool this mode to the ground state. In particular we show that phononic excitations can be brought to the quantum regime. State preparation using information inferred from losses has already been used to prepare a well defined phase between two condensates [4], with a Monte Carlo wavefunction approach providing a very clear understanding of the mechanism [5]. Manipulation of cold atomic clouds by quantum feedback has been proposed in many theoretical papers using dispersive light-atom interaction [6,7], while feedback has been implemented for internal degrees of freedom [8][9][10].

Discretization of the problem. We consider a onedimensional Bose gas with contact repulsive interactions of coupling constant g, such that the Hamiltonian writes, in second quantization,

H = - 2 2m ˆdxψ + ∂ 2 ψ/∂x 2 + g 2 ˆdxψ + (x)ψ + (x)ψ(x)ψ(x).
We assume the gas is submitted to atom losses, the loss mechanism being a single atom process described by a loss rate Γ. For instance, magnetically trapped atoms could be submitted to a radio-frequency field that would transfer atoms to an untrapped state, as realized experimentally in [2]. Atoms could also be ionized by laser fields [11], or expelled by collision with fast electrons [12].

FIG. 2. A typical loss sequence during a time interval ∆t, for a single cell. The associated quantum trajectory followed by the system's wavefunction is given by Eq. 2.

We moreover assume the lost atoms are detected one by one with position-resolved detectors, as sketched in Fig. (1). We note ρ the mean linear density and we assume the gas lies within the quasi-condensate regime such that the atomic density fluctuations are small and their characteristic length scale, equal to the healing length l h = / √ mg ρ, is much larger than the mean interparticle distance [13]. We discretize space in N cells of length δx, containing a large mean atom number n = δxρ and with small relative fluctuations. We furthermore assume that δx is large enough such that the fluctuations are large compared to unity [14]. The state of the gas may be expanded (as long as one is not interested in length scales smaller than δx) on the Fock basis of each cell

|ψ = n1,n2,...,n N c n1,n2,...,n N |n 1 , n 2 , . . . , n N . (1)
Time is also discretized in intervals ∆t small compared to the time scales involved in the longitudinal dynamics of the gas. This allows to consider, during ∆t, the sole effect of losses first and then the effect of the free evolution. We will first concentrate on the effect of losses. Since losses do not introduce correlations between different cells, it is relevant to consider the case of a single cell first.

Monte Carlo description of losses in a single cell. Considering a single cell, the initial state writes |ψ(t) = n c n |n . Let us split ∆t in elementary time steps of length dt, small enough so that the probability to have an atom lost during dt is small. According to the Monte Carlo wavefunction procedure [15], if no atoms are detected during a time step dt, then the state of the system evolves according to the non Hermitian Hamiltonian H eff = -i Γa + a/2, which ensures the decrease of the probability of highly occupied states. If on the other hand a lost atom has been detected, the new state is obtained by the application of the jump operator â, which annihilates an atom in the cell. Let us now assume M atoms have been lost from the cell between time t and time t + ∆t, at times t 1 < t 2 < • • • < t M , as sketched in Fig. (2). By successively applying the procedure described above, we construct the quantum trajectory followed by the system and we find

|ψ(t + ∆t) = e -iH eff (t+∆t-t M )/ âe -iH eff (t M -t M -1 )/ . . . âe -iH eff (t1-t)/ |ψ(t) .
(2) With the normalization chosen here, the probability of the loss sequence is (dtΓ) M ψ(t + ∆t)|ψ(t + ∆t) / ψ(t)|ψ(t) . From Eq. ( 2), we find that the Fock state coefficients c n (t + ∆t) = n|ψ(t + ∆t) write

c n (t + ∆t) = f {ti} (n + M )c n+M (t), (3)
where the function f {ti} (n) depends on the loss sequence.

Assuming ∆t is small enough so that M is much smaller than the mean atom number in the cell, itself much larger than one, f {ti} (n), for a given M , becomes almost independent on the time sequence {t i } and can be approximated by

f {ti} (n) f M (n) = n M/2 e -Γnt/2 . ( 4
)
Within this approximation, the probability of the sequence is

n |c n | 2 (Γndt) M e -Γn∆t .
Summing over all possible sequences with M lost atoms, we find that the probability to have M lost atoms is

P (M ) = n |c n | 2 (Γn∆t) M e -Γn∆t /M ! .
For a given initial atom number n, we recover the expected Poissonian distribution. In the limit nΓ∆t 1, the typical number of losses is much larger than 1 and the function f M (n) can be approximated by the Gaussian

f M (n) A M e -(n-M/(Γ∆t)) 2 (Γ∆t) 2 4M , (5)
where A M is the normalization factor. Using the fact that the number of lost atoms M , is typically equal to nΓ∆t and presents small relative fluctuations, Eq. ( 5) further approximates to

f M (n) A M e -(n-M/(Γ∆t)) 2 Γ∆t 4 n . (6)
The same approximations lead to a mean number of lost atoms M tr = n Γ∆t, with a variance M 2 tr -M 2 tr = Γ∆tn, where the symbol tr indicates that averaging is done here over many different quantum trajectories.

Generalization to all cells and Bogoliubov decomposition. The results above can immediately be generalized to the case of several cells. If M i denotes the number of lost atoms in the cell i, the probability amplitude of the Fock state |n 1 , n 2 , . . . , n N is, up to a global normalization factor,

c n1,n2,...,n N (t + ∆t) = c n1+M1,n2+M2,...,n N +M N (t) i e -(n i -M i /(Γ∆t)) 2 Γ∆t 4 n (7)
Since the atom number per cell is typically very large and present fluctuations large compared to unity, one can approximate discrete sums on n i by continuous integrals and treat the n i as continuous variables.

Since the gas lies in the quasicondensate regime, its Hamiltonian is well approximated by the Bogoliubov Hamiltonian [13]. For a homogeneous system, the Bogoliubov modes are obtained from the Fourier decomposition. More precisely, let us introduce the Fourier quantities

   n ki,c = 2 N j cos(k i jδx)n j n ki,s = 2 N j sin(k i jδx)n j (8)
Here k i = i2π/L, where i is a integer taking values between 1 and (N -1)/2. We introduce in the same way the operator nki,c and nki,s . The Bogoliubov Hamiltonian acts independently on each Fourier mode and, for a given mode (k, r), where r stands for c or s it writes, up to a constant term,

Ĥk,r = A k n2 k,r + B k θ2 k,r , (9)
where the phase operator θk,r is the operator conjugated to nk [16], A k = (g/2 + 2 k 2 /(8mρ))/δx and B k = k 2 ρδx/(2m) and the mean particle density ρ = n/δx. The frequency of the mode is

ω k = 2 √ A k B k / .
Let us now investigate the effect of losses in the Bogoliubov basis. The state |n 1 , n 2 , . . . , n N is also an eigenstate of each operator nki,r , where r stands for c or s, with eigenvalue n ki,r . We thus use the notation |n 1 , n 2 , . . . , n N = |{n ki,r } , where {n ki,r } is a short notation for n k1,c , n k1,s , n k2,c , n k2,s . . . , n k N ,c , n k N ,s . The state of the system then writes

|ψ = ˆ i,r dn ki,r c{n k i ,r } |{n ki,r } , (10)
where c{n k i ,r } = c n1,n2,...,n N . The modification of the state of the system after a time ∆t due to atom losses is then, according to Eq. ( 7),

c{n k i ,r } (t+∆t) = c{n k i ,r +M k i ,r } (t) i e - (n k i -M k i ,r /(Γ∆t)) 2 Γ∆t 4 n , (11)
where M ki,c = 2/N j M j cos(k i x j ) and M ki,s = 2/N j M j sin(k i x j ). We used the facts that, here, on one hand side the variances of each Gaussian in Eq. ( 7) are all equal and on the other hand side the density profiles of Bogoliubov modes are orthogonal, namely the transformation between the basis n i and n k,r is orthogonal. The statistics of the different quantum trajectories gives a Gaussian distribution for M ki,r with M ki,r tr = Γ∆t n k,r and M 2 ki,r tr -M ki,r 2 tr = Γ∆tn. Eq. (11) shows that the losses affect each Fourier component, i.e. each Bogoliubov mode, independently.

If the initial state is at thermal equilibrium, different Bogoliubov modes are uncorrelated. The free evolution, under the Bogoliubov Hamiltonian as well as the effect of losses, do not introduce correlations between modes and one can consider each mode independently. In the following, we consider a given mode of momentum k and we will omit the subscript c or s, since the upcoming considerations apply for both.

Evolution of a given Bogoliubov mode: Wigner representation. Here we consider a given mode, described by the two conjugate variables n k and θ k . A convenient representation of the state of the system is its Wigner function W , a two-dimensional real function, whose expression, as a function of the density matrix D of the state, is

W (n k , θ k ) = 1 2π 2 ˆdadbe i(an k -bθ k ) T r De (-ian k +ib θk ) . ( 12
)
The effect of losses during ∆t, in the n k representation, are given by Eq. ( 11), and transform W into the new Wigner function function W according to

W (n k , θ k ) = Γ∆t 2π 3/2 n ´dθ W (n k + M k , θ) e -Γ∆t 2 n (n k -M k /(Γ∆t)) 2 e -2 n Γ∆t ( θ-θ) 2 (13)
The multiplication by a Gaussian function along the n k axis shifts the distribution towards M k /(Γ∆t), the value for which M k tr is equal to the recorder value M k . It also decreases the width in n k , which reflects the gain of knowledge acquired on n k by the detection of the number of lost atoms. The associated convolution along the axis θ k increases the width in θ k , and ensures preservation of uncertainty relations. The thermal state of the Bogoliubov Hamiltonian has a Gaussian Wigner function. Since the Gaussian character is preserved by Eq. ( 13) and by the free evolution, the state of the system stays Gaussian. W is then completely determined by its center R = ( n k , θ k ) and its covariance matrix

C = n 2 k -n k 2 n k θ k -n k θ k n k θ k -n k θ k θ 2 k -θ k 2 (14)
As shown in appendix A, to first order in ∆t, the transformation in Eq. ( 13) changes R and C to R and C with

C = C + Γ∆t n -C 2 11 -C 11 C 12 -C 11 C 12 -C 2 12 + 1 4 (15)
and

R = R - Γ∆t n k 0 -dξ 1 -C 11 /n -C 12 /n (16)
Here we introduced dξ = M k -Γ∆t n k . According to the statistic of trajectories, dξ is a Gaussian variable centered on 0 and of variance dξ 2 tr = Γ∆tn. The above equations account for the evolution of the state under the sole effect of atom losses. One should then implement the evolution under the Hamiltonian (9), which amounts to a simple rotation of the Wigner function in phase space and acts independently on C and R [START_REF]R(ω k t) = cos ω k t B/A sin ω k t -A/B sin ω k t cos ω k t with ω k = 2 √ AB[END_REF]. Finally, one can compute the long term evolution iteratively following the procedure above, knowing, at each time interval ∆t, the number of atoms lost in each cell, M i , from which dξ is computed.

Evolution of the correlation matrix. Eq. (15) shows that the evolution of the correlation matrix is the same for all possible quantum trajectories, and general statements can be made. Let us first consider a very slow mode such that one can ignore the free evolution. Then C 12 stays at 0 during the evolution and time integration of Eq. ( 15) on long times gives

C 11 n(t)/(1 -e -Γt ), C 22 (1 -e -Γt )/(4n(t)) (17)
where n(t) = n(t = 0)e -Γt is the time-dependent mean atom-number per cell. The system thus goes towards a state of minimal uncertainty, as expected, since more and more information is acquired on the system. Let us now consider the other limit of a mode of very high frequency. Then the free evolution of the system ensures, at any time, C 12 0 and the equipartition of the energy between the 2 degrees of freedom. Thus

A k C 11 B k C 22 E c /2 where E c = A k C 11 + B k C 22
is the contribution of the correlation matrix to the energy. We then find

d( E c /( ω k )) dt = Γ eff (-( E c /( ω k )) 2 + 1/4), ( 18
)
where Γ eff = Γ/ 1 + 4mg ρ/( 2 k 2 ). Γ eff depends on time via the exponential decrease of ρ due to losses. At long times, E c /( ω k ) goes to 1/2, such that the state of the system, as long as only the matrix C is concerned, evolves towards the ground state. If one assumes the excitation is initially in the phononic regime, however, we show in the appendix B that E c approaches ω k only once the decrease of ρ has already promoted the excitation to the particle regime. Thus phononic excitations can not reach the quantum regime. The situation is different if the decrease of ρ is compensated by the following time dependence of g:

g(t) = g(t = 0)e Γt . ( 19
)
Then Γ eff and ω k are constant and an excitation lying in the phononic regime stays in the phononic regime during the whole loss process and, as long as the C matrix is concerned, is cooled to the ground state. In the following, we will assume g is modified according to Eq. [START_REF] Langen | [END_REF]. Averaging over trajectories. If the loss events are not recorded, then only the quantities averaged over all possible trajectories are meaningful. If the Wigner distribution is initially centered around 0, it will stay centered at 0. Let us investigate its evolution over a time ∆t. For a given quantum trajectory, i.e. a given value dξ, the losses modifies the correlation matrix according to Eq. ( 15) as well as the center R, which acquires the non zero value R = -dξ(1 -C 11 /n; -C 12 /n). One then has

n 2 k st (t + ∆t) = n 2 k (t) -Γ∆t/nC 2 11 + dξ 2 1 - n 2 k n 2 (20)
and The mode k = 0.1 √ mg ρ/ is considered, with the initial temperature Ti = 1.1g ρ and a loss rate Γ = ω k /400. Scattered blue points give the evolution of the center R. We verified that C12 stays small while, at any time, A k C11 B k C22 Ec/2, as expected for such a large ω k . The black solid-circle, of radius √ EC , represents the final RMS width of the Wigner function. For comparison, the red dashed-circle, of radius √ E, where E is the energy obtained after averaging over 100 trajectories, gives the rms width of the averaged phase space distribution. The coordinates are given in the frame rotating according to the free evolution: namely, the plotted quantity

θ 2 k st (t+∆t) = θ 2 k (t)+Γ∆t/n(1/4-C 2 12 )+(dξC 12 /n) 2 , (21)
is R = (n k , θk ) = R -1 (ω k t)R [17].
where the subscript st specifies this holds for a single trajectory. Averaging over all possible trajectories, we then find, using dξ 2 tr = Γ∆tn, that losses modify the variances according to

n 2 k (t + ∆t) -n 2 k (t) = -2Γ∆t n 2 k (t) + Γ∆tn(t). ( 22
)
and

θ 2 k (t + ∆t) -θ 2 k (t) = Γ∆t/(4n). (23)
As expected, Eq. ( 22) and ( 23) are equal to those obtained using a master equation description of the loss process [1,[START_REF] Johnson | [END_REF]. Due to the diffusive process experienced by R, an increased rate in both equations limits the decrease of the mode energy. For phonons, and assuming the loss rate is small compared to the mode frequency, we show in appendix C that the temperature asymptotically goes towards g ρ(t)/2. This value of the asymptotic temperature T ∞ is particular to the case of an homogeneous gas with a coupling constant evolving according to Eq. [START_REF] Langen | [END_REF]. For constant g the asymptotic temperature is T ∞ = g ρ [1,[START_REF] Johnson | [END_REF]. For a gas trapped in a harmonic potential we expect T ∞ to scale as gρ p , where ρ p is the peak density. The proportionality factor has not been derived yet, but since the averaged density is smaller than ρ p , one naively expects that T ∞ is smaller than gρ p . Experimentally T ∞ has not been iden-tified, while temperatures as low as 0.25 gρ p have been reported for a harmonically confined gas [2].

Using information retrieved from losses detection: quantum feedback. If the losses are recorded, such that at each time interval ∆t, the values M j are recorded, the trajectory followed by the center of the Wigner distribution, R, can be computed exactly, and the heating associated to the diffusion process seen in equations ( 23) and ( 22) can be compensated for. One strategy is to perform, during the whole time-evolution, a quantum feedback on the system, based on the knowledge acquired via the atom losses, in order to prevent the center of the Wigner distribution to drift away of the phase space center. Let us here, as an illustration, assumes one is interested in a given mode k, c. The most simple back action is to submit the atomic cloud to a potential V (x) = A(t) cos(kx), where the computed amplitude A(t) depends on the recorded history of the losses. Such a potential could be realized, for instance, using the dipole potential experienced by atoms in laser field. The cosine modulation of the laser intensity can be realized using an optical lattice, or using a spatial light modulator. The contribution to the Hamiltonian of this potential is Ĥfb = A(t) N/2n k,c . In order to counteract the diffusion process of R due to the loss process, one could adjust A(t) such that the feedback Hamiltonian is

Ĥfb = -ν θ k,c nk,c (24)
where, at each time interval, θ k,c is computed by integrating the equations of motion including the effect of losses, the free evolution, and the feedback process. This Hamiltonian acts as an active damping, the damping rate ν preventing θ k to drift far from the phase space center. The free evolution Hamiltonian, by coupling the two degrees of freedom, will ensure that neither θ k nor n k drift away. For a large enough damping rate ν, the contribution of R to the energy of the mode is expected to be negligible compared to the contribution of the covariance matrix C and, according to Eq. ( 18), one expects to reach the ground state. We present numerical results illustrating such a scenario below. Before presenting numerical results, let us identify the relevant quantities governing the dynamics. Introducing the reduced variable ñk = n-1/2 n k and θk = n1/2 θ k , as shown in appendix D, we find as expected that the cell size δx drops out of the problem and, provided time is rescaled by 1/Γ, the dynamics of the mode of wave vector k is solely governed by the dimensionless parameters ω k /Γ, 2 k 2 /(mg ρ) and ν/Γ. The relevant measurement signal, for the time interval ∆t, is then Mk = 2/N at ´dxm(x) cos(kx), where N at is the total atom number and m(x) is the number of lost atoms per unit length. mode, averaged over quantum trajectories, both in the absence and in the presence of feedback. In the absence of feedback, the energy converges towards the expected value gρ/2. If the feedback scheme is implemented, we observe that the energy in the mode reaches much smaller values. For a large feedback strength ν, the drift of the center is almost completely prohibited and the mode is cooled to its ground state. Discussion. In conclusion, we proposed a description of the effect of losses in a many-body system through a Monte Carlo wavefunction approach, and we showed that quantum feedback by monitoring losses could be used to cool down selected modes of a quasi-condensate to vanishing temperatures. This work could be extended in many directions. In view of practical implementation, the sensitivity of the feedback mechanism on the exact knowledge of the system parameters should be investigated. Assuming, as is done in this paper, the system parameters are known exactly, the larger the feedback strength, the better the cooling. In presence of uncertainties, a too large feedback strength will induce heating as it will not match the exact dynamics. Additionally, in most experimental situations the quasi-condensates, trapped in a shallow longitudinal potential, are non homogeneous. Then, the effect of losses depends on the spatial coordinate. Moreover, the linearised description should use, instead of the sinusoidal modes, the spatial density profiles of the Bogoliubov modes, which are not necessarily orthogonal. These issues complicate the picture. Losses might then induce correlations between modes [7]. Another concern is the coupling between modes, which exists beyond the linearized approach considered here. Such coupling is present for instance in the Gross-Pitaevskii equation, which is a classical field approximation of the Lieb-Liniger model. However, longlived non-thermal states with different Bogoliubov modes experiencing long life-time [START_REF] Langen | [END_REF] have been reported, which indicates small coupling between modes and the possibility to cool down a particular Bogoliubov mode. Finally, note that this cooling process is not limited to 1D systems.

Aknowledgment. The authors thanks K. Mølmer for inspiring discussions. M. S. gratefully acknowledges support by the German Academic Scholarship Foundation.

FIG. 3 .

 3 FIG.3. Evolution of the phase-space distribution for a single trajectory from time t = 0 to t = 8/Γ in absence of feedback. The mode k = 0.1 √ mg ρ/ is considered, with the initial temperature Ti = 1.1g ρ and a loss rate Γ = ω k /400. Scattered blue points give the evolution of the center R. We verified that C12 stays small while, at any time, A k C11 B k C22 Ec/2, as expected for such a large ω k . The black solid-circle, of radius √ EC , represents the final RMS width of the Wigner function. For comparison, the red dashed-circle, of radius √ E, where E is the energy obtained after averaging over 100 trajectories, gives the rms width of the averaged phase space distribution. The coordinates are given in the frame rotating according to the free evolution: namely, the plotted quantity is R = (n k , θk ) = R -1 (ω k t)R[START_REF]R(ω k t) = cos ω k t B/A sin ω k t -A/B sin ω k t cos ω k t with ω k = 2 √ AB[END_REF].

  Fig. (3), shows the phase-space evolution of a single quantum trajectory, for a mode lying in the phononic regime, in absence of quantum feedback. Fig. (4) shows the time-evolution of the energy in this

  FIG. 4. Simulation of the loss process for different feedback strengths ν. Plotted is the time-evolution of the energy in the mode k, averaged over 500 quantum trajectories, the parameters being those of Fig. (3). Without feedback the energy converges to g ρ/2 (horizontal dotted line); lower energies are obtained with feedback and the ground state, of energy ω k /2 (horizontal dashed line) is reached for large enough ν.

	ν/Γ =0
	ν/Γ =1
	ν/Γ =10
	ν/Γ =400
	hωk/2
	g ρ/2
	E/(hω k )

Appendix A: Effect of losses on the Wigner representation

Here we consider a given mode and we will omit the subscripts k, r to make our notations lighter. We also introduce σ 2 = n/(Γ∆t) and q 0 = M k /(Γt). Eq. (12) writes, in representation n, W (n, θ) = 1 π ˆdu u + n|D|un e -2iuθ . (A1)

The effect of losses, given by Eq (11) transforms the Wigner function of the mode to

we then find Eq. ( 13).

Let us now consider a Gaussian state. Its Wigner function writes

where

Let us now consider the effect of the multiplication of W by e -1 2σ 2 (n-q0)) 2

, as well as the shift along n by M k . From Eq. (A3), we find

and

where B = C-1 and B = C -1 . From Eq. (A5), we obtain

Injecting σ = n/(Γ∆t) and expanding to first order in ∆t, one gets

Here we used the fact that C11 = C 11 and C12 = C 12 . This equation also takes the form of Eq. ( 15). Let us now consider the center of the distribution. Multiplying the left and right hand parts of Eq. (A6) by C and injecting (A8), we deduce

Neglecting terms beyond first order in ∆t, we obtain

Injecting M k = Γ∆ n k + dξ, we recover Eq. ( 16).

Appendix B: Evolution of EC for constant g.

We assume ω k Γ such that Eq. ( 18) is valid. Note that the condition ω k Γ also ensures adiabatic following, namely the time evolution of the Hamiltonian parameters A k and B k preserves the ratio E C /( ω k ), such that Eq. ( 18) holds both for a constant g and a timevarying g. Let us introduce the variable y = Ec g ρ and rewrite Eq. ( 18) in the form

(B1) For y = 1 we see that y (t) ≥ 0 and therefore y(t) has to be an increasing function at y = 1. It follows that for all initial conditions y(0) ≥ 1 the energy E C stays greater than g ρ . This implies in particular that, as long as an excitation stays in the phononic regime (i.e. its frequency stays much smaller than g ρ/ ), it stays in the high temperature regime, namely E C /( ω k ) 1.

Appendix C: Asymptotic temperature for non-recorded losses

We consider a mode k (we omit the index r for simplicity) and we assume averaging is done over trajectories. Then evolution of the variances of n k and θ k due to the loss process are given in Eq. ( 22) and ( 23). Let us consider the quantity Ẽ = H k /( ω k ). We assume the loss rate is small enough so that the free evolution under the Hamiltonian (9) ensures equipartition of the energy between the two quadratures, namely at each time

Note that this is equivalent to the condition of adiabatic following. Then the modification of Ẽ under the loss process is

The evolution under the Hamiltonian (9), provided the adiabatic following condition is satisfied, does not modify Ẽ. Thus Eq. (C1) gives the total time evolution of Ẽ, and it is valid both when A k and B k depend on time and when they do not depend on time. In this paper, we consider the situation given by Eq. [START_REF] Langen | [END_REF], where the exponential decrease of ρ is compensated by a time dependence of g such that K is time independent. Then Eq. (C1) evolves at long times to

For phononic modes, for which k 2 gρ, one has K 2 √ gρ/k. Then Ẽ goes to √ gρ/(2k) at long times, which gives

This energy is very large compared to ω k . Thus the excitation lies in the high temperature limit and its temperature is T E gρ/2. Note that, in the case g is constant, then K depends on time and solving Eq. (C1) with the time-dependant value of K gives that E converges to g ρ = g ρ0 e -Γt , as derived in [1].

Appendix D: Equations in reduced variables

Here we derive the evolution equations for the reduced variables ñk = n -1/2 0 n k and θk = n 1/2 0 θ k . We note R and C the associated mean vector and covariant matrix. Taking into account the exponential decrease of n 0 , Eq. ( 15) and Eq.( 16) give


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Conformational and thermal characterization of left ventricle remodeling post-myocardial infarction

Keywords:

Adverse cardiac remodeling after myocardial infarction (MI) causes impaired ventricular function and heart failure. Histopathological characterization is commonly used to detect the location, size and shape of MI sites. However, the information about chemical composition, physical structure and molecular mobility of peri-and infarct zones post-MI is rather limited. The main objective of this work was to explore the spatiotemporal biochemical and biophysical alterations of key cardiac components post-MI. The FTIR spectra of healthy and remote myocardial tissue shows amides A, I, II and III associated with proteins in freeze-died tissue as major absorptions bands. In infarcted myocardium, the spectrum of these main absorptions was deeply altered. FITR evidenced an increase of the amide A band and the distinct feature of the collagen specific absorption band at 1338 cm -1 in the infarct area at 21 days post-MI. At 21 days post-MI, it also appears an important shift of amide I from 1646 cm -1 to 1637 cm -1 that suggests the predominance of the triple helical conformation in the proteins. The new spectra bands also indicate an increase in proteoglycans, residues of carbohydrates in proteins and polysaccharides in ischemic areas. Thermal analysis indicates a deep increase of unfreezable water/freezable water in peri-and infarcted tissues. In infarcted tissue is evidenced the impairment of myofibrillar proteins thermal profile and the emergence of a new structure. In conclusion, our results indicate a profound evolution of protein secondary structures in association with collagen deposition and reorganization of water involved in the scar maturation of periand infarct zones post-MI.

Introduction

Adverse myocardial remodeling is associated with poor patient outcomes in the context of ischemic heart disease and/or myocardial infarction (MI), cardiac hypertrophy, and cardiomyopathy disease processes [START_REF] Drazner | Increased left ventricular mass is a risk factor for the development of a depressed left ventricular ejection fraction within five years: the cardiovascular health study[END_REF][START_REF] Lam | Size, shape, and stamina: the impact of left ventricular geometry on exercise capacity[END_REF][START_REF] Mann | Mechanisms and models in heart failure: the biomechanical model and beyond[END_REF]. In particular, adverse cardiac remodeling after MI causes impaired ventricular function and heart failure (HF). The incidence of HF after MI is determined by the size of the infarcted area, the infarct wound healing, and chronic left ventricular (LV) remodeling [START_REF] Pfeffer | Ventricular remodeling after myocardial infarction. Experimental observations and clinical implications[END_REF][START_REF] Opie | Controversies in ventricular remodelling[END_REF][START_REF] Dorn | Novel pharmacotherapies to abrogate postinfarction ventricular remodeling[END_REF]. LV remodeling after MI involves changes in extracellular matrix (ECM) composition and organization [START_REF] Jugdutt | Ventricular remodeling after infarction and the extracellular collagen matrix: when is enough enough?[END_REF]. Differential regional remodeling of ECM plays a crucial role in global LV structural remodeling after MI [START_REF] Zannad | Limitation of excessive extracellular matrix turnover may contribute to survival benefit of spironolactone therapy in patients with congestive heart failure: insights from the randomized aldactone evaluation study (RALES). Rales investigators[END_REF][START_REF] Mann | Inflammatory mediators and the failing heart: past, present, and the foreseeable future[END_REF] causing LV dilatation and dysfunction, disability and death [START_REF] Weber | Remodeling and reparation of the cardiovascular system[END_REF][START_REF] Jugdutt | Remodeling of the myocardium and potential targets in the collagen degradation and synthesis pathways[END_REF]. The main components of ECM are collagen molecules, specially collagen type I (85%) and type III (11%) [START_REF] Jugdutt | Ventricular remodeling after infarction and the extracellular collagen matrix: when is enough enough?[END_REF][START_REF] Jugdutt | Remodeling of the myocardium and potential targets in the collagen degradation and synthesis pathways[END_REF]. Collagen type I is mainly associated with thick fibers that confer tensile strength and resistance to stretch and deformation while collagen type III is associated with thin fibers that confer resilience. All ECM components play a pivotal role for cardiac function.

Histopathological characterization is commonly used to detect the location, size and shape of MI sites. However, the information about chemical composition, physical structure and molecular mobility of peri-and infarct zones post-MI is rather limited. Spectrophotometric techniques such as Fourier transform infrared (FTIR) spectroscopy give structural information useful to identify key biochemical and biophysical alterations in tissues. In this sense, FTIR is a powerful technique that has provided accurate results with a high reproducibility in many areas including cardiovascular research [START_REF] Kuwahara | Cardiac remodeling associated with protein increase and lipid accumulation in early-stage chronic kidney disease in rats[END_REF][START_REF] Marzec | Vascular diseases investigated ex vivo by using Raman, FT-IR and complementary methods[END_REF]. The main advantages of FITR are that data analysis can be performed directly on tissues, without the need of staining and the high speed for spectra acquisition (only few minutes). Another suitable technique to characterize biological tissues at the submicronic scale is Differential Scanning calorimetry (DSC). This is a standard method to evaluate freezable and unfreezable water in tissues [START_REF] Aktas | Determination of freezable water content of beef semimembranous muscle DSC study[END_REF], and to assess protein thermal stability and conformational changes [START_REF] Fodor | Cardiac-specific Nkx2.5 homeodomain: conformational stability and specific DNA binding of Nkx2.5(C56S)[END_REF][START_REF] Samouillan | Comparison of chemical treatments on the chain dynamics and thermal stability of bovine pericardium collagen[END_REF]. It is peculiarly well-suited to evaluate the thermal stability of purified collagens in solution or in their aggregated form [START_REF] Miles | The kinetics of the thermal denaturation of collagen in unrestrained rat tail tendon determined by differential scanning calorimetry[END_REF] or directly in cardiovascular tissues [START_REF] Samouillan | Characterization of aneurysmal aortas by biochemical, thermal, and dielectric techniques[END_REF] and biomaterials [START_REF] Lee | A multi-sample denaturation temperature tester for collagenous biomaterials[END_REF]. It has been also successfully applied to characterize protein components of muscle tissues, such as myosin and its subfragments, actine and sarcoplasmic proteins [START_REF] Wright | Differential scanning calorimetric study of muscle and its proteins: myosin and its subfragments[END_REF][START_REF] Dergez | DSC study of glycerol-extracted muscle fibers in intermediate states of ATP hydrolysis[END_REF][START_REF] Tamilmani | Thermal analysis of meat and meat products[END_REF]. The aim of current investigation was to explore the spatiotemporal biochemical and biophysical alterations of key cardiac components post-MI.

Material and methods

Mouse model of MI

Male C57/Bl6 mice (25-30 g; Charles River Laboratories, Inc.) were used for this study. MI was induced as previously described [START_REF] Bayes-Genis | Human progenitor cells derived from cardiac adipose tissue ameliorate myocardial infarction in rodents[END_REF]. Briefly, each animal was intubated and anesthetized with a mixture of O 2 / isoflurane and mechanically ventilated. The heart was exposed and the left anterior descending coronary artery was permanently occluded. Animals were sacrificed at 1 day (n = 6), 10 days (n = 14) and 21 days (n = 15) post-operation and hearts were excised and segmented in three parts (remote, peri-and infarcted area) following the scheme indicated in Fig. 1A. Detailed confocal microsopy images of each part are shown in Fig. 1B. Myocardial samples were frozen in liquid nitrogen for molecular and lipid analysis. Six animals of each group were processed for histological analysis, being the heart arrested in diastole with cardioplegic solution (68.4 mM NaCl, 59 mM KCl, 11.1 mM glucose, 1.9 mM NaHCO 3 , 29.7 mM 2,3-butanedione monoxime, 1000 U heparin), excised, fixed, cryopreserved in 30% sucrose in phosphatebuffered saline, embedded in Tissue-Tek O.C.T. (Sakura®), and snapfrozen in liquid nitrogen-cooled isopentane. Three control mice were sacrificed to obtain healthy hearts used for vibrational characterization. All animal handling procedures were approved by the Institutional Animal Care (Cardiovascular Research Center) and Use Committee and conformed to the Guide for the Care and Use of Laboratory Animals of the Institute of Laboratory Animal Research (NIH Pub. No. 86-23, Revised 1996).

Immunofluorescence and histology

Mouse heart cryosections were incubated with primary antibodies against vimentin (2 μg/mL; Abcam, Cambridge UK), elastin (1 μg/mL; Abcam, Cambridge UK) and cardiac troponin I (cTnT; 2 μg/mL; Abcam, Cambridge UK). Nuclei were counterstained with 4′,6-diamidino-2phenylindole (DAPI) and results were analyzed with an Axio-Observer Z1 (Zeiss, Oberkochen, GE) laser confocal microscope. A polarized light microscope was used to measure collagen types I (predominant in infarcted myocardium) and III (predominant in remote myocardium) on Sirius red-stained sections from both infarct and remote areas. Twelve randomly selected fields from remote (6 images) and infarcted (6 images) zones were taken with a light microscope for polarization (DMi8, Leica, Wetzlar, GE) [START_REF] Nicoletti | Left ventricular fibrosis in renovascular hypertensive rats. Effect of losartan and spironolactone[END_REF]. Masson's trichromic stain was used to distinguish the muscle fibers (in red, predominant in remote myocardium) and collagen (in blue, predominant in infarcted myocardium).

RNA extraction, cDNA synthesis and gene expression analyses by Real-Time Reverse Transcriptase-Polymerase Chain Reaction

Frozen cardiac tissues from animals sacrificed at 1 day (N = 6), 10 days (N = 6) and 21 days (N = 6) were pulverized using a mortar and a pestle in liquid nitrogen to maintain the integrity of the RNA. An aliquot of tissue (20 mg) was weighed and total RNA was isolated by TriPure™ isolation reagent (Roche Molecular Biochemicals) according to the manufacturer's instructions. RNA yield and quality were assessed by 1% agarose gel electrophoresis. Reverse transcription was performed using 1.5 μg of total RNA and the High Capacity cDNA Reverse Transcription kit (Applied Biosystems). The cDNA was stored at -20 °C.

Gene expression analyses of collagen type I (Rn00801565_g1; Applied Biosystems, Foster City, CA) collagen type III (Rn01437682_m1; Applied Biosystems, Foster City, CA) and elastin (Mm00514670_m1; Applied Biosystems, Foster City, CA) mRNA were performed at mRNA level by semi-quantitative Real-Time Reverse Transcriptase-Polymerase Chain Reaction (RT-PCR). Arbp (Mm.PT.58.43894205, IDT Conda, Integrated DNA Technologies, Inc) was used as internal control gene amplification. RT-PCR was performed using 1 μL of reverse transcription products and 10 μL of TaqMan Universal PCR Master Mix (Applied Biosystems, Foster City, CA), 1 μL of 20 × assays and 8 μL of nuclease-free water. After gentle mixing, the mixture was transferred into a RT-PCR microplate.

RT-PCR was performed in a PCR-7600HT sequence detection system (Abiprism, Applied Biosystems, Foster City, CA) at 50 °C for 2 min, 95 °C for 10 min, 40 cycles at 95 °C for 15 s and 60 °C for 1 min. The relative levels of gene expression were quantified and analyzed using SDS 2.4 software. The real-time value for each sample was averaged and compared using the C T method, where the amount of target RNA (2-ΔΔCT) was normalized to the endogenous control (ΔCT).

Lipid extraction and semi-quantitative analysis of cholesteryl ester, free cholesterol and triglyceride content of cardiomyocytes and myocardium

One portion of myocardial-pulverized tissue (7 mg) was weighed and homogenized in NaOH 0.1 M. The protein content of the extracts was quantified by Pierce BCA Protein Assay (Thermo Fisher Scientific, Waltham, MA) to normalize lipid content. An aliquot of tissue extract (3 mg) was mixed with water (1 mL) and dichloromethane-methanol (1:2) (3 mL). After intense vortexing, dichloromethane (1 mL) was added and vortex again. The mixture was centrifuged at 3500 rpm for 15 min and the organic phase was collected. The sample was concentrated by evaporating the organic solvent under a N 2 stream to prevent lipid oxidation.

Cholesteryl ester (CE), free cholesterol (FC) and triglyceride (TG) content in the lipid extract was analyzed by thin layer chromatography (TLC). Lipid extracts were loaded on silica G-25 plates (DC-Fertigplatten SIL G-25 UV 254 ). A mixture of cholesterol, cholesterol palmitate, triglyceride, diglyceride and monoglyceride was applied to each plate as standards. The first phase consisting of a mixture of heptane:diethylether:acetic acid (74:21:4 v/v/v), separated the different lipid of the sample. The second phase, consisting in heptane, dragged remaining lipids.

The spots corresponding to CE, TG and FC were quantified by densitometry using ChemiDoc system and Quantity-One software (Bio-Rad, Hercules, CA).

Vibrational characterization

One portion (5 mg) of each segmented frozen part and of healthy myocardium was freeze-dried and used for vibrational characterization. Fourier transform infrared spectroscopy/attenuated total reflectance (FTIR/ATR) spectra of freeze-dried tissues were acquired using a Nicolet 5700 FTIR (Thermo Fisher Scientific, Waltham, MA) equipped in ATR device with a KBr beam splitter and a MCT/B detector. The ATR accessory used was a Smart Orbit with a type IIA diamond crystal (refractive index 2.4). Samples 1 mm 2 in surface were directly deposed on the whole active surface of the crystal. For each sample, 80 interferograms were recorded in the 4000-450 cm -1 region, co-added and Fourier-transformed to generate an average spectrum of the segmented heart part with a nominal resolution of 2 cm -1 using Omnic 8.0 (Thermo Fisher Scientific, Waltham, MA). A single-beam background spectrum was collected from the clean diamond crystal before each experiment and this background was subtracted from the spectra. Spectra were then subjected to ATR and baseline corrections and normalized in the amide II region. These spectra were next used in calculation of integrated bands intensities and their ratios. We employed integration method with a linear baseline drawn trough the peak edges, and the spectrum below this line was integrated over the wavenumber range of the band. Second derivatives and Fourier self-deconvolution (using k = 1.7 and half-width = 13.5 cm -1 ) were used to enhance the chemical information present in overlapping infrared absorption bands of spectra. All spectra processing was performed using Omnic 8.0. Spectra presented for each group were calculated by averaging the spectra of all the samples within each group.

Differential scanning calorimetry

Analyses were performed using a DSC Pyris calorimeter (Perkin Elmer, Waltham, MA). The calorimeter was calibrated using Hg and In as standards, resulting in a temperature accuracy of ±0.1 °C and an enthalpy accuracy of ±0.2 J/g. Fresh samples, 5-10 mg in weight, were set into hermetic aluminium pans and equilibrated at the initial temperature during 5 min before cooling at -100 °C at 10 °C/min. Then the samples were heated at 10 °C/min until 90 °C. After a cooling at -100 °C at 10 °C/min, the samples were heated again at 10 °C/min until 90 °C (2nd heating). After completing the DSC measurements, pans were reweighted to check that they had been correctly sealed. The sample pans were pierced and dried to constant weight at 195 °C for 10 min to determine the sample dry weight and the total water weight.

Statistical analysis

Quantitative values are shown as means ± SEM. All experiments were performed with at least four replicates for each group and for at least three independent repeats. A one-way analysis of variance (oneway ANOVA) and Tukey post-hoc analysis for multiple comparisons were performed to compare the means of the samples. The p-value was also used to determine the significant difference between groups. It was considered statistically significant threshold of P value b0 .05.

Results

Analysis of the main FIRT absorption bands in cardiac tissue

Table 1 summarizes IR bands present in healthy and remote tissues and their assignments according to literature [START_REF] Staniszewska | Rapid approach to analyze biochemical variation in rat organs by ATR FTIR spectroscopy[END_REF][START_REF] Wang | Infrared imaging of compositional changes in inflammatory cardiomyopathy Vib[END_REF][START_REF] Gough | Fourier transform infrared evaluation of microscopic scarring in the cardiomyopathic heart: effect of chronic AT1 suppression[END_REF][START_REF] Bromberg | Collagen remodeling in the extracellular matrix of the cardiomyopathic Syrian hamster heart as assessed by FTIR attenuated total reflectance spectroscopy[END_REF][START_REF] Zohdi | Evidence of altered biochemical composition in the hearts of adult intrauterine growth-restricted rats[END_REF]. The major absorptions in the FTIR spectra of healthy and remote tissues (Fig. 2) are highly reproducible and are composed of amide A, amide I and amide II, mainly associated with proteins in freeze-dried tissues. The main proteins in mass of the myocardium are the cardiomyocyte myofibrillar proteins (myosin, α-actin), sarcoplasmic proteins and ECM structural proteins, peculiarly collagens and elastin. Among the five types of collagens (I, III, IV, V, VI) identified in the heart, the fibrillar collagens I and III represent more than 90% of the total collagens, acting as strength transducers and maintaining the tridimensional structure during the contraction-relaxation cycle.

Most of these proteins have overlapped absorptions bands. Nevertheless, previous studies showed that collagens (I, III, IV, V, VI) exhibit a specific triplet at 1206, 1238, 1280 cm -1 (arising from the amide III vibration) and a specific absorption at 1338 cm -1 (wagging of proline side chain) [START_REF] Staniszewska | Rapid approach to analyze biochemical variation in rat organs by ATR FTIR spectroscopy[END_REF][START_REF] Bromberg | Collagen remodeling in the extracellular matrix of the cardiomyopathic Syrian hamster heart as assessed by FTIR attenuated total reflectance spectroscopy[END_REF][START_REF] Belbachir | Collagen types analysis and differentiation by FTIR spectroscopy[END_REF]. Moreover, comparison between the FTIR spectra of muscle and its separated components -myofibrils and extracellular matrix-corroborated the absence of the 1206, 1240 and 1339 cm -1 absorption band in myofibers [START_REF] Kirschner | Monitoring of denaturation processes in aged beef loin by Fourier transform infrared microspectroscopy[END_REF]. Among these different vibrations, the 1338 cm -1 band is the only one that does not overlap with the absorption or other components like DNA, lipids or proteoglycans. It must be pointed out that such a vibration, associated with proline, is also present in the FTIR spectrum of another rich-proline protein of ECM, namely elastin. Nevertheless, our previous studies on pure collagen and elastin FTIR spectra evidenced that the absorbance of this band was significantly higher in collagen, certainly due to the specific arrangement of proline in this protein in contrast with elastin. The band at 1338 cm -1 can thus be addressed to the specific signature of the structural proteins of ECM with a strong contribution of collagens.

Moreover, the band at 1304 cm -1 , found in the FTIR spectrum of myofibers but absent in the spectra of ECM, is assigned to the specific signature of myofibers.

Other components contribute to these complex spectra, especially lipids with their classical markers in the 2800-3000 cm -1 (especially the CH 2 stretching of long hydrocarbon chains), and in the 1475-1450 cm -1 . These lipids mainly include phospholipids of the plasmatic membranes (confirmed by the presence of the C_O stretching of ester groups at 1740 cm -1 [START_REF] Dritsa | An infrared spectroscopic study of aortic valve. A possible mechanism of calcification and the role of magnesium salts[END_REF] and the PO 2 -stretching bands) that represent 20% in weight of heart , triglycerides (C_O stretching of ester groups at 1740 cm -1 ), that are known to represent 25% of the total mass of lipids and free fatty acids (C_O stretching at 1712 cm -1 and COO-stretching at 1392 cm -1 ) absorbed by the cardiac cells and used as the primary supply of energy [START_REF] Goldberg | Lipid metabolism and toxicity in the heart[END_REF]. Unsaturated lipids possess a specific marker at 3013 cm -1 .

As already mentioned, DNA also plays a part in this spectrum with the contribution to the absorption bands at 1712, 1246-1235, 1079 and 973 cm -1 , which are composite bands that cannot be used for quantification.

Proteoglycans are also present in the myocardium tissues, with the contribution to the 1079 cm -1 band and the specific band at 1226 cm -1 , overlapping with protein amide III. They are frequently ignored when examining the cardiac ECM, but could be an important determinant of residual stress [START_REF] Fomovsky | Contribution of extracellular matrix to the mechanical properties of the heart[END_REF]. Finally, glycogen and other polysaccharides contribute to the FTIR spectrum of healthy and remote myocardium, in the 1200-1000 cm -1 region, which is a 'mixed' region.

In order to quantify the variations observed in the averaged spectra, the area of the different absorption bands were computed from the individual spectrum of each tissue and the appropriate ratio of areas was performed according to the literature data [START_REF] Staniszewska | Rapid approach to analyze biochemical variation in rat organs by ATR FTIR spectroscopy[END_REF][START_REF] Wang | Infrared imaging of compositional changes in inflammatory cardiomyopathy Vib[END_REF].

Spatio-temporal alterations in FITR absorption bands related with cardiac remodeling post-MI

The major absorption bands reported in Table 1 for remote myocardium tissues are evidenced in the averaged spectra of peri-and infarcted tissues. On the averaged global spectra, the major difference concerns the increase of the amide A band in peri-and infarcted tissues (especially at 21 days post-MI) (Fig. 2A), with the enhancement of the 3390 cm -1 component associated with H-bonded OH groups [START_REF] Popescu | Structural analysis of some soluble elastins by means of FT-IR and 2D IR correlation spectroscopy[END_REF] and the distinct feature of the collagen specific absorption band at 1338 cm -1 peculiarly in the infarcted zone at 21 days post-MI (Fig. 2C), attesting of a profound remodeling of the ECM. Polarized light and confocal microscopy images show an important deposition of collagen and elastin, respectively (Fig. 3A-L) at 10 and 21 days post-MI. Accordingly, RT-PCR results show a significant increase of collagen type I (Fig. 3M), collagen type III (Fig. 3N) and elastin (Fig. 3O) at 10 days. Collagen type III and elastin mRNA expression were maintained over 21 days post-MI. In contrast, collagen type I decayed at 21 days post-MI. In agreement with these results, the indicator of the quantity of the structural proteins of the ECM (collagens and elastin) with respect to the total amount of proteins in myocardial tissues computed from the corresponding FTIR peak ratio significantly increases in infarcted zones at 10 days post-MI (Fig. 4A).

The side chain absorption was also modified in infarcted tissues, in particular the tyrosine band at 1517 cm -1 was reduced in infarct tissues (Fig. 5A). It is known that collagen has a very low tyrosin content, thus, the tyrosin reduction could be caused by an increase of the collagenic fraction in infarcted tissues as already suggested in remodeled cardiac tissues [START_REF] Bromberg | Collagen remodeling in the extracellular matrix of the cardiomyopathic Syrian hamster heart as assessed by FTIR attenuated total reflectance spectroscopy[END_REF]. Moreover, the contribution of glutamic acids and aspartic acids (present in a relatively large amount in collagen) certainly contribute to the enhanced absorption at 1566 cm -1 (Fig. 5A) previously described in remodeled tissues [START_REF] Barth | The infrared absorption of amino acid side chains[END_REF][START_REF] Rabotyagova | Collagen structural hierarchy and susceptibility to degradation by ultraviolet radiation[END_REF].

Spatio-temporal evolution of the secondary structure of proteins

Both the amide I and the amide II have been shown to be conformational sensitive, and thus are widely used to determine protein secondary structure. Second derivatives and Fourier self-deconvolution are classically used to enhance the chemical information present in overlapping infrared absorption bands of spectra, and these techniques were applied to the spectra of remote and infarcted myocardial tissues in order to follow the evolution of the proteins structures. The important shift of the amide I towards low wave number (from 1646 to 1637 cm -1 ) in peri-and infarcted zone, which is visible at 10 and 21 days post-MI (Fig. 2B) attests the important modification of the secondary structures of proteins post-MI.

As an averaging technique, IR spectroscopy simultaneously provides information concerning all proteins present in the tissue. Thus, it is not possible to attribute specific amide absorptions to peculiar secondary structures within individual proteins. However, the appearance of an amide I maximum at 1650 cm -1 in FSD spectra (Fig. 5A) was corroborated by a main minima at the same wave number in second derivative spectra in remote myocardium (Fig. 5B). These results indicate that the proteins in myocardium adopt a predominantly α-helical/random conformation. This is also confirmed by the main maximum of amide II at 1538 cm -1 in the FSD spectrum of remote tissues (and confirmed by a main minima in its second derivative spectrum), associated with α helical structures/unordered in heart [START_REF] Staniszewska | Rapid approach to analyze biochemical variation in rat organs by ATR FTIR spectroscopy[END_REF]. Another secondary structure evidenced in remote tissue is the β sheet structure with its associated vibration at 1632 cm -1 in the amide I zone and at 1555 cm -1 in the amide II zone (Fig. 5A andB).

Both amide I and amide II undergo profound modifications in infarcted tissues, at 10 and 21 days post-MI, in particular with the emergence of a new band at 1637 cm -1 (marked on Fig. 5A andB). It is noteworthy that this component becomes the most intense in the FSD spectrum of infarcted tissues at 21 days post-MI, explaining the shift toward low wavenumber of the amide I of the global spectrum of infarcted tissues (Fig. 2B). Another important modification concerns the amide II band with the vanishing of the main component at 1538 cm -1 (assigned to α helical or unordered structures) and the emergence of a new main component at 1542-1545 cm -1 both in FSD and second derivative spectra (Fig. 5A andB).

Finally, there is a shift of the band at 1632 cm -1 (associated with β sheets structures) toward 1627 cm -1 in infarcted tissues (Fig. 5A), that could be attributed to an aggregation of proteins (inter molecular β sheets).

Spatio-temporal alterations in lipids post-MI

A slight decrease of the absorption of the 1740 cm -1 band associated to phospholipids and nucleic acids was detected in peri-and infarcted zones both at 10 and 21 days post-MI (Fig. 2B). This decrease is in agreement with the decrease of the absorption band of unsaturated lipids at 3013 cm -1 in the averaged spectra of infarcted tissues at 21 days post-MI (Fig. 2A).The indicator of the amount of phospholipids related to proteins drastically decreases in peri-and infarcted myocardial tissues at all stages of post-MI (Fig. 4B). The loss of phospholipids and the increase in structural proteins in the ischemic areas is illustrated by the third indicator (collagen/phospholipid ratio) (Fig. 4C) computed from distinct absorption bands and proportional to the ratio of ECM structural proteins against the lipid phase.

TLC after lipid extraction showed a significant increase of CE content in the infarcted area compared to peri-and remote areas at 10 days post-MI (Fig. 6A andB). At 21 days post-MI, the CE content is also higher in infarct areas but lacks significance. In contrast to CE, TG remained constant between the three tested zones at 10 days and has a tendency to decrease in infarct areas at 21 days post-MI (Fig. 6A andC). There was not alteration in the myocardial FC content between areas at 10 or 21 days post-MI (Fig. 6A andD).

Spatio-temporal alterations in proteoglycans and polysaccharide post-MI

The zone 1250-1200 cm -1 that corresponds to the mixed zone of proteins/phospholipids/proteoglycans/polysaccharide is altered in infarcted tissues at 21 days post-MI, with a shift of the main maximum from 1232 towards 1222 cm -1 (Fig. 2C). This shift could be due to the increase of the proteoglycans. The 1042 cm -1 band associated with carbohydrate residues of proteins (in peculiar collagen) and polysaccharides is increased in peri-and infarct zones at 21 days post-MI.

Spatio-temporal evolution in the hydric organization of myocardial tissues

In Fig. 7A are reported the DSC thermograms of fresh myocardial tissues corresponding to the first heating at 10 °C min -1 between -30 and 85 °C; for the sake of clarity only one thermogram for each type of sample was presented.

All the thermograms are characterized by an endothermic peak in the [-10;10 °C] zone corresponding to the melting of previously frozen water. This extrinsic transition is widely used to quantify the amount of total freezable water in hydrated proteins and tissues (by dividing the area of the measured endothermic peak by 334 J g -1 , corresponding to the melting enthalpy of pure ice at 0 °C) [START_REF] Aktas | Determination of freezable water content of beef semimembranous muscle DSC study[END_REF][START_REF] Panagopoulou | Glass transition and dynamics in BSA-water mixtures over wide ranges of composition studied by thermal and dielectric techniques[END_REF][START_REF] Heys | Free and bound water in normal and cataractous human lenses[END_REF] and completes thermogravimetric analyses giving the total amount of water. The term freezable water is preferred to the term 'free' waterrather reserved to vibrational or relaxational techniques -since it covers bulk water in excess but also confined water in mesopores. The amount of unfreezable water can be calculated by a simple difference. This quantification was performed for all the samples and reported in Fig. 7B.

The amounts of total water and unfreezable water were higher in peri-and infarcted tissues than in healthy ones where was higher the amount of freezable water. Here, we report three main findings, 1) phospholipid/protein ratio decreases while collagen/protein increases in peri-and infarcted zones; 2) peri-and infarcted areas are more hydrophilic than noninfarcted areas and 3) alterations in secondary structures and aggregation of proteins are important phenomena in infarcted areas.

Discussion

Here, we observed that among the different vibrations found in the FTIR spectra, the band of 1338 cm -1 was the only one that did not overlap with the absorption of other components like DNA, lipids or proteoglycans. In previous studies, the peak at 1338 cm -1 was used to estimate the collagen integrity [START_REF] Camacho | FTIR microscopic imaging of collagen and proteoglycan in bovine cartilage[END_REF] and more recently it has reported a strong correlation between FTIR absorption at 1338 cm -1 and deposition of collagen type I analyzed by immuno-histochemical staining [START_REF] Cheheltani | Fourier transform infrared spectroscopic imaging of cardiac tissue to detect collagen deposition after myocardial infarction[END_REF]. In post-MI, collagen deposition allows scar maturation [START_REF] Dewald | Of mice and dogs: species-specific differences in the inflammatory response following myocardial infarction[END_REF][START_REF] Frangogiannis | The immune system and cardiac repair[END_REF]. These findings agree with our FTIR results and confocal microscopy images showing and important collagen deposition in the peri-and infarcted areas at 10 and 21 days post-MI, when the scar is forming.

We also found in peri-and infarct areas a clear increase in the quantity of elastin, other key structural protein of the ECM. The increase in the quantity of collagen and elastin deposition in these areas was concomitant with a parallel decrease in unsaturated lipids, previously associated with necrosis and phagocytosis of membrane debris [START_REF] Lui | Modification of the extracellular matrix following myocardial infarction monitored by FTIR spectroscopy[END_REF]. In addition, the decrease of the ratio phospholipid/protein ratio that we have observed in the infarcted tissues in mice is in agreement with previous results published in Syrian hamster model [START_REF] Wang | Infrared imaging of compositional changes in inflammatory cardiomyopathy Vib[END_REF]. The higher amount of total and unfreezable water in infarcted myocardium that we have found in DSC thermograms is coherent with the enhancement of the H-bonded OH absorption in the Amide A and the decrease of phospholipid/proteins ratio quantified by FTIR on freeze-dried tissues. Taken together, these results show a strong extracellular remodeling with increased collagen and elastin and decreased phospholipid content occurring in infarcted myocardium. These results are in agreement with previous results [START_REF] Nasa | Changes in fatty acid compositions of myocardial lipids in rats with heart failure following myocardial infarction[END_REF]. In our model, the important decrease of phospholipid/protein ratio could be due to both the decrease of phospholipids from plasmatic membranes of cardiomyocytes (due to the necrosis of cardiomyocytes and phagocytosis of membrane debris in the first stage of MI) and to scar formation with collagen deposition as main component in the infarcted zone. The FSD spectra evolution confirms this assumption, with the reduction of the 1713 cm -1 band, one of the major DNA sensitive bands [START_REF] Liu | Quantitative determination of apoptosis on leukemia cells by infrared spectroscopy[END_REF], in infarcted areas.

The hydrophilic evolution of the infarcted zones is confirmed by the significant modification of the unfreezable water/freezable water ratio (from 0.11 in remote tissues to 0.57 in infarcted tissues, p b 0.05).

Unfreezable water -corresponding to the filling of the first hydration shell of proteins and other hydrophilic components, and so associated with 'bound' water-is a very minor part in remote tissues, containing more than 20% in weight of lipids that exclude water and organize it in freezable water. In our case, collagen deposition in infarcted areas leads to a significant increase of the unfreezable water. It can be explain through the collagen co-deposition with hydrophilic compounds as heparan-sulfate proteoglycans [START_REF] Lyon | Co-deposition of basement membrane components during the induction of murine splenic AA amyloid[END_REF][START_REF] Lunde | Syndecans in heart fibrosis[END_REF]. Proteoglycans could increase the hydrophilic character of the infarcted area, increasing the unfreezable water content detected by DSC thermograms. Myocardial water content is clinically used as an in-vivo marker of myocardial edema [START_REF] Scholz | NMR relaxation times in acute myocardial infarction: relative influence of changes in tissue water and fat content[END_REF][START_REF] Garcia-Dorado | Myocardial oedema: a preventable cause of reperfusion injury?[END_REF][START_REF] García-Dorado | Analysis of myocardial oedema by magnetic resonance imaging early after coronary artery occlusion with or without reperfusion[END_REF]. Traditionally it has been considered that edema occurs soon after reperfusion and dissipates at 24 h. However, Fernandez-Jimenez and collaborators have recently reported that edema in the ischemic region follows a bimodal pattern in a pig model of myocardial infarction [START_REF] Fernández-Jiménez | Pathophysiology underlying the bimodal edema phenomenon after myocardial ischemia/reperfusion[END_REF][START_REF] Fernández-Jiménez | Myocardial edema after ischemia/reperfusion is not stable and follows a bimodal pattern: imaging and histological tissue characterization[END_REF]. The first wave of edema is clearly related to reperfusion, however, according to authors, the second wave of edema could be caused by removal of cardiomyocyte debris and its replacement by water, collagen and healing. Our results support this hypothesis since we have found a significant increase of unfreezable water and proteoglycans in the ischemic regions at 21 days post-MI. Therefore, we propose that proteoglycans may participate in unfreezable water increase and ultimately post-MI edema. Indeed, previous studies already reported proteoglycans to be key for the development of pulmonar edema [START_REF] Pelosi | The extracellular matrix of the lung and its role in edema formation[END_REF][START_REF] Negrini | The role of proteoglycans in pulmonary edema development[END_REF]. Further studies are required to evaluate the role of proteoglycans on edema formation and evolution. Indeed, this is a crucial clinical question due to the pivotal role of edema on diagnosis, prognosis and therapeutics. In particular, previous studies have reported that volume of edema relates to functional recovery after reperfused acute myocardial infarction [START_REF] Kidambi | Relationship between myocardial edema and regional myocardial function after reperfused acute myocardial infarction: an MR imaging study[END_REF].

In the 45-85 °C zone of remote tissues thermograms, multiple and weak endothermic events are observed corresponding to the denaturation zone of cardiac muscle proteins. Myofibrillar, sarcoplasmic proteins and collagens constitute the major class of proteins in muscle. neosynthesized collagen fibers for the scar maturation. As a matter of fact, the mature cardiac scar contents cross-linked stiff collagen forms that provide rigidity and stability to the scar [START_REF] Lopez | Role of lysyl oxidase in myocardial fibrosis: from basic science to clinical aspects[END_REF]. This specific thermal signature of the triple helix domain at 65 °C has to be connected to the specific vibrational component at 1637 cm -1 , evidenced both in the second derivative and FSD spectra in infarcted areas. It has been identified as a specific marker of the triple helix of collagen by some authors [START_REF] Petibois | Analysis of type I and IV collagens by FT-IR spectroscopy and imaging for a molecular investigation of skeletal muscle connective tissue[END_REF] and evidenced in the triple helical synthetic polypeptide (Pro-Hyp-Gly) 10 [START_REF] Kishimoto | Synthesis of poly(Pro-Hyp-Gly)(n) by direct poly-condensation of (Pro-Hyp-Gly)(n), where n = 1, 5, and 10, and stability of the triple-helical structure[END_REF], which corroborates the quantitative FTIR analysis and microscopic images of remodeled infarcted tissues. Even if the assignation to peculiar secondary structures is less straightforward for amide II, and most of all in the case of complex tissues [START_REF] Bonnier | FTIR protein secondary structure analysis of human ascending aortic tissues[END_REF], the band at 1543 cm -1 that intensifies at 21 days post-MI can be assigned to another specific marker of collagen in infarcted tissues [START_REF] Petibois | Analysis of type I and IV collagens by FT-IR spectroscopy and imaging for a molecular investigation of skeletal muscle connective tissue[END_REF]. It has been previously reported that triple helix characteristically represent fibrosis, that may only appear in old myocardial infarction and that these structures are extensively involved in the infarction process [START_REF] Yang | Histopathology mapping of biochemical changes in myocardial infarction by Fourier transform infrared spectral imaging[END_REF].

In summary, we have determined the spatiotemporal biochemical and biophysical alterations of key cardiac components in LV remodeling at 10 and 21 days post-MI. Our analysis evidence a profound evolution of protein secondary structures and an important collagen deposition in association with a significant reorganization of water in the infarcted tissue, reflecting physical processes involved in scar maturation post-MI.

Funding

The research leading to these results received funding from the Instituto de Salud Carlos III (FIS PI14/01729 and FIS PI14/01682) that was cofinanced by the European Fund for Regional Development, and CIBER Cardiovascular (CB16/11/00403), as well as from the Ministerio de Ciencia e Innovación (SAF2011-30067-C02-01), the Fundació La Marató de TV3 (201521_10, 201502_30 and 080330), the Red de Terapia Celular-TerCel (RD12/0019/0029), the Sociedad Española de Cardiología and the Fundació Privada Daniel Bravo Andreu. DdG is a recipient of CD14/0109 from Sara Borell of the Instituto de Salud Carlos III.

Transparency Document

The Transparency document associated with this article can be found, in online version.

Fig. 1 .

 1 Fig. 1. (A) Masson's trichromic staining was used to distinguish remote (in red; cardiac muscle fibers), peri-and infarcted tissues (in green; collagen). (B) Immunofluorescence staining identified the scar in the infarcted zone. Cardiac fibroblast marker vimentin is shown in green and cardiac TnI in red. Cell nuclei were counterstained with DAPI (blue). Scale bar 20 μm.

Figs. 2 .Fig. 3 .Fig. 4 .

 234 Figs.[START_REF] Lam | Size, shape, and stamina: the impact of left ventricular geometry on exercise capacity[END_REF]. Averaged FTIR spectra of healthy, remote, peri-and infarcted myocardial tissues in the 4000-2500 cm -1 window (A) in the 1800-1350 cm -1 window (B) and in the 1370-900 cm -1 window (C). N = 4. Major assignments are noted on the graphs. Amides I and III marked by dotted lines to show the shift. Frame of the specific absorbance of collagen at 1338 cm -1 . N = 4.

Fig. 5 . 6 .

 56 Fig. 5. Averaged second derivative spectra (A) and FSD spectra (B) computed from FTIR spectra of remote, peri-and infarcted myocardial tissues. Arrows indicate new bands in infarcted tissues. N = 4.

Fig. 7 .

 7 Fig. 7. DSC thermograms (A) and bar graph of the hydric organization (B) of remote, peri-and infarcted myocardial tissues (fresh tissues). Results are expressed as mean ± SEM. N = 3. ⁎⁎ P b 0.01 versus remote area.

Table 1

 1 Main absorption bands of healthy and remote myocardial tissues.

	Band position
	(cm -1 )	Assignment
	3283	Amide A
		mainly ν(N-H) mode of protein with the contribution of the
		ν(O\ \H) stretching mode in H 2 O and polysaccharides
	3072	ν(C\ \H) aromatic
	3013	ν(C_H) of unsatured lipids, triglycerides, fatty acids
	2957, 2923,	νas(CH 3 ), νas(CH 2 ), νs(CH 3 ),νs(CH 2 ),
	2871, 2852	(Gly, Pro, Hyp, Ala) of proteins + phospholipids, triglycerides
		Most representative of proteins: νs(CH 3 )
		Most representative of lipids: νas(CH 2 ) and νs(CH 2 )
	1740	ν(C_O) of ester carbonyl groups of phospholipids and
		triglycerides
	1712	ν(C_O) of nucleoside side of nucleic acids and ν(C_O) of free
		fatty acids
	1646	Amide I ν(C_O)
		Multi-component band sensitive to protein secondary
		structure
		Better resolved in second derivative or FSD spectra
	1540	Amide II ν(C-N), δ (N-H)
		Multi-component band sensitive to protein secondary
		structure
		Better resolved in second derivative or FSD spectra
	1515	Tyrosine band
	1471, 1452	δ(CH 2 ) scissoring, δ(CH 3 ) bending of lipids (mainly), proteins
	1392	νs(COO-) of free amino acids, fatty acids, δ(CH 3 )
	1338	δ(CH 2 ) wagging of proline chain
		Specific band of collagen
	1310-1202	Amide III δ plan (N\ \H) and ν(C\ \N) of proteins
		Sensitive to secondary structure
		1240 cm -1 : specific of ECM (collagen)
		1204 cm -1 : specific of ECM (collagen)
		1304 cm -1 : specific of myofibers
	1246-1235 1226	-) stretching of phospholipids, nucleic acids νas(PO 2 ν(SO 4 2-) of proteoglycans
	1200-1000	proteins,
		oligosaccharides, glycolipids
		Better resolved in second derivative or FSD spectra
		In peculiar
		1171: νas(CO\ \O\ \C) of cholesterol esters, phospholipids
		(specific of myofibers)
		1159: ν(C\ \OH) of Hyp and nucleic acids
		1127: ν(C\ \O) lactate, polysaccharides, glycogen -specific of
		myofibers
		1080: ν(C\ \O\ \C) collagen, glycogen, oligosaccharides,
		glycolipids, proteoglycans -specific of ECM
		νs(PO 2 -) nucleic acids, phospholipids
		1042: νs(CO\ \O\ \C) of carbohydrates residues and
		polysaccharides
		1120: glycogen (specific of myofibers)
	973	Phosphorylated proteins, phospholipids, nucleic acids
	930	υ(Cα-C)characteristic of α helices

ν(C\ \O), ν(C\ \C), ν(C\ \OH), ν(C\ \O\ \C) of