Damping of elementary excitations in one-dimensional dipolar Bose gases

Hadrien Kurkjian

TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, België

Zoran Ristivojevic

Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, 31062 Toulouse, France (Dated: March 13, 2020) In the presence of dipolar interactions the excitation spectrum of a Bose gas can acquire a local minimum. The corresponding quasiparticles are known as rotons. They are gaped and do not decay at zero temperature.

Here we study the decay of rotons in one-dimensional Bose gases at low temperatures. It predominantly occurs due to the backscattering of thermal phonons on rotons. The resulting rate scales with the third power of temperature and is inversely proportional to the sixth power of the roton gap near the solidification phase transition. The hydrodynamic approach used here enables us to find the decay rate for quasiparticles at practically any momenta, with minimal assumptions on the exact form of the interparticle interactions. Our results are an essential prerequisite for the description of all the dissipative phenomena in dipolar gases and have direct experimental relevance.

Introduction.-At low pressures and temperatures, helium-4 is a remarkable quantum liquid that is superfluid. Landau characterized the latter state by a dissipationless flow of macroscopic objects at small velocities [START_REF] Lifshitz | Statistical physics[END_REF]. Another particular feature of the superfluid helium is seen in its spectrum of elementary excitations. While at lowest momenta it is linear, the spectrum possesses a local minimum. The corresponding quasiparticles are known as rotons and have the wavelengths that practically coincide with the mean interparticle distance. Since the interaction between helium atoms is strong, the roton can be visualized as yet undeveloped Goldstone mode due to an instability toward the crystallization [START_REF] Nozières | Is the Roton in Superfluid 4He the Ghost of a Bragg Spot?[END_REF]. However, such so-called supersolid state that unifies superfluidity with crystalline order has not been so far observed in helium, despite some controversies [START_REF] Kim | Probable observation of a supersolid helium phase[END_REF][START_REF] Day | Low-temperature shear modulus changes in solid 4 He and connection to supersolidity[END_REF][START_REF] Kim | Absence of Supersolidity in Solid Helium in Porous Vycor Glass[END_REF].

Another system that shows some similarities with superfluid helium are dipolar Bose gases. They can be realized with atoms possessing large dipolar moments, such as chromium, dysprosium, and erbium. Bose-Einstein condensates of those atoms are realized [START_REF] Griesmaier | Bose-Einstein Condensation of Chromium[END_REF][START_REF] Beaufils | Alloptical production of chromium Bose-Einstein condensates[END_REF][START_REF] Lu | Strongly Dipolar Bose-Einstein Condensate of Dysprosium[END_REF][START_REF] Aikawa | Bose-Einstein Condensation of Erbium[END_REF], which opened new avenues for studying various phenomena that originate from the dipolar interaction [START_REF] Lahaye | The physics of dipolar bosonic quantum gases[END_REF][START_REF] Baranov | Condensed Matter Theory of Dipolar Quantum Gases[END_REF]. Some of them are the striped states [START_REF] Wenzel | Striped states in a many-body system of tilted dipoles[END_REF], the quantum droplets [START_REF] Ferrier-Barbut | Observation of Quantum Droplets in a Strongly Dipolar Bose Gas[END_REF][START_REF] Chomaz | Quantum-Fluctuation-Driven Crossover from a Dilute Bose-Einstein Condensate to a Macrodroplet in a Dipolar Quantum Fluid[END_REF], and the elusive supersolid state [START_REF] Böttcher | Transient Supersolid Properties in an Array of Dipolar Quantum Droplets[END_REF][START_REF] Chomaz | Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum Gases[END_REF][START_REF] Tanzi | Observation of a Dipolar Quantum Gas with Metastable Supersolid Properties[END_REF][START_REF] Natale | Excitation Spectrum of a Trapped Dipolar Supersolid and Its Experimental Evidence[END_REF][START_REF] Guo | The low-energy Goldstone mode in a trapped dipolar supersolid[END_REF][START_REF] Tanzi | Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas[END_REF].

Trapped dipolar Bose gases can exhibit a quasiparticle spectrum with a roton minimum [START_REF] O'dell | Rotons in Gaseous Bose-Einstein Condensates Irradiated by a Laser[END_REF][START_REF] Santos | Roton-Maxon Spectrum and Stability of Trapped Dipolar Bose-Einstein Condensates[END_REF]. This occurs because the dipolar interaction cannot be described only by a shortrange pseudopotential, but it must also include an anisotropic long-range part, in order to correctly describe the low-energy scattering between bosons [START_REF] Baranov | Condensed Matter Theory of Dipolar Quantum Gases[END_REF]. The quasiparticle spectrum in weakly-interacting Bose gases is determined by the Bogoliubov theory [START_REF] Bogolubov | On the theory of superfluidity[END_REF] and depends on the Fourier transform of the pseudopotential. Since it is described by the two parameters, one for the short-range and the other the long-range part, when they are properly tuned, the local minimum can develop in the spectrum. A recent experiment [START_REF] Chomaz | Observation of roton mode population in a dipolar quantum gas[END_REF] have confirmed the presence of rotons in the dipolar Bose gas.

The current understanding of the properties and the dynam-ics of dipolar gases is limited due to the lack of dissipative mechanisms in most theoretical descriptions. Without dissipation, one can neither describe the post-quench relaxation observed in Ref. [START_REF] Chomaz | Observation of roton mode population in a dipolar quantum gas[END_REF], nor predict the thermodynamic quantities related to ergodicity, such as the gas viscosity [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF][START_REF] Enss | Viscosity and scale invariance in the unitary Fermi gas[END_REF] and loss of phase coherence [START_REF] Ilzhöfer | Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms[END_REF][START_REF] Kurkjian | Brouillage thermique d'un gaz cohérent de fermions[END_REF]. In a well-isolated gas, dissipation arises primarily from the interactions between the quasiparticles, which allow the system to reach equilibrium. A prerequisite to understand the dissipative dynamics is thus to compute the quasiparticle lifetime, which is the purpose of this work.

At zero temperature, rotons and all quasiparticle excitations at lower momenta in a dipolar Bose gas are stable. Being slower than the sound velocity, those quasiparticles cannot emit phonons due to the conservation laws of momentum and energy. Such scenario resembles to the absence of Cherenkov radiation at small velocities. However, the quasiparticles do decay at finite temperature. The dominant process for the damping of a subsonic quasiparticle involves its scattering with another thermally excited quasiparticle, where two new quasiparticles become created [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF][START_REF] Gangardt | Quantum Decay of Dark Solitons[END_REF][START_REF] Matveev | Scattering of hole excitations in a one-dimensional spinless quantum liquid[END_REF][START_REF] Kurkjian | Landau-Khalatnikov phonon damping in strongly interacting Fermi gases[END_REF][START_REF] Castin | Landau Phonon-Roton Theory Revisited for Superfluid 4 He and Fermi Gases[END_REF][START_REF] Nicolis | Mutual interactions of phonons, rotons, and gravity[END_REF]. The resulting rate typically scales as a power-law of temperature.

In this paper we study the damping of energetic quasiparticles (including rotons) in a one-dimensional dipolar Bose gas. This process is controlled by the backscattering of thermal phonons. We find the low-temperature rate that scales as the third power of temperature, T 3 . The hydrodynamic approach employed here to describe the interaction between phonons and energetic quasiparticles is not limited to the weak-coupling regime, and does not require a specific form of the interaction potential. However, in order to describe roton damping, we must require specific interactions that lead to the formation of a roton minimum. Near the solidification phase transition, when the roton gap ∆ is much smaller than the other relevant energy scales (except temperature), we find a rate diverging as 1/∆ 6 . Our results pave the way to a description of the post-quench relaxation dynamics of a dipolar Bose gas [START_REF] Van Regemortel | Prethermalization to thermalization crossover in a dilute Bose gas following an interaction ramp[END_REF]. arXiv:2003.05919v1 [cond-mat.quant-gas] 12 Mar 2020

q 0 k 0 k l T k k Hard phonons k q0 k T / c Maxons Rotons ck Thermal phonons k T / c Supersonic excitations (v k > c) ∆ T
Resonance of single-phonon emission process q q q q Figure 1. Rotonic excitation spectrum of a one-dimensional dipolar Bose gas obtained from the Bogoliubov theory [START_REF] Sinha | Cold Dipolar Gases in Quasi-One-Dimensional Geometries[END_REF]. The damping of energetic excitations of the energy k T , i.e., hard phonons, maxons, and rotons is controlled by the backscattering off thermal phonons. Our theory is valid until quasiparticles become supersonic with velocities v k > c (right vertical red line).

Damping of energetic quasiparticles.-We consider a gas of bosons with a dipolar interaction between particles in a (quasi-)one-dimensional geometry (studied theoretically in Ref. [START_REF] Sinha | Cold Dipolar Gases in Quasi-One-Dimensional Geometries[END_REF] and recently realized experimentally [START_REF] Chomaz | Observation of roton mode population in a dipolar quantum gas[END_REF]). We assume that the gas is prepared at low temperature, T ∆, mc 2 , where mc 2 denotes the characteristic energy scale for phonons. By m is denoted the mass of particles, c is the sound velocity, while the Boltzmann constant is set to unity. Subsonic quasiparticles with the dispersion k are characterized by the velocity v k = d k /hdk that is smaller than c. They can decay only due to scattering off thermally excited quasiparticles. At very high momenta, the quasiparticles become supersonic, see Fig. 1. In this case they can decay already at T = 0 by emitting phonons, which is not precluded by the conservation laws [START_REF] Tan | Relaxation of a High-Energy Quasiparticle in a One-Dimensional Bose Gas[END_REF][START_REF] Ristivojevic | Decay of Bogoliubov excitations in one-dimensional Bose gases[END_REF].

The collisionless damping rate [39] associated to a subsonic quasiparticle of the energy k T (see Fig. 1) can be computed using the Fermi golden rule:

Γ k = 2π h q,q ,k |A if | 2 L 2 δ( k + hω q -k -hω q )n q (1 + n q ). (1)
Here ω q = c|q| is the phonon frequency, L is the system size, while n q = 1/[exp(hω q /T ) -1] denotes the Bose occupation factor [START_REF]The thermal population of energetic quasiparticle modes is negligible in the studied low-temperature case[END_REF]. In Eq. ( 1), A if is the transition amplitude from the initial state,

|i = γ † k b † q |0 with an excitation present in the mode k, to the final state |f = γ † k b † q |0
. By b † and γ † are denoted the bosonic creation operators for the phonon and the energetic quasiparticle, respectively. The delta function in Eq. ( 1) accounts for the energy conservation.

The transition amplitude A if can be computed with minimal assumptions, in particular not assuming that the gas is weakly interacting, using quantum hydrodynamics [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF][START_REF] Matveev | Scattering of hole excitations in a one-dimensional spinless quantum liquid[END_REF][START_REF] Castin | Landau Phonon-Roton Theory Revisited for Superfluid 4 He and Fermi Gases[END_REF][START_REF] Nicolis | Mutual interactions of phonons, rotons, and gravity[END_REF]. Within this theory, it is sufficient to consider the cubic residual interaction among phonons given by the Hamiltonian

Ĥph = q hω q b † q bq + q,q A 3 (q, q ) √ L b † q+q bq bq + H.c. . (2)
The matrix elements A 3 follows from the hydrodynamic equations of motion [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF][START_REF] Kurkjian | Three-Phonon and Four-Phonon Interaction Processes in a Pair-Condensed Fermi Gas[END_REF] and have the form

A 3 (q, q ) = mc 2 √ 32ρ qq (q + q ) q 3 0 ρ 2 c 2 d dρ c 2 ρ + sgn(qq )
+ sgn(q(q + q )) + sgn(q (q + q )) ,

where ρ is the (mean) fluid density and q 0 = mc/h. The hydrodynamics describes the energetic quasiparticles perturbed by the phonon field within a local density approximation as

Ĥqp = 1 2 [ (p, ρ + δ ρ(r)) + pv(r) + H.c.] . (4)
Here (p, ρ) is the Hamiltonian of the unperturbed quasiparticle in first quantization, where r is its position and p its momentum operator. By δ ρ and v are denoted, respectively, the density and the superfluid velocity perturbations caused by the phonons. Expanded to a quadratic order at small δ ρ ρ and expressed in second quantization, Eq. ( 4) becomes

Ĥqp = k k γ † k γk + k,q A 1 (k, q) √ L γ † k+q γk bq + H.c. + k,q,q A 2 (k, q, q ) L γ † k+q-q b † q γk bq . ( 5
)
The phonons and the energetic quasiparticle are coupled by

A 1 (k, q) = ρ|q| 2q 0 ∂ ρ k + ∂ ρ k+q 2 + hc 2mρ|q| q k + q 2 , A 2 (k, q, q ) = ρ |qq | 2q 0 ∂ 2 ρ k + ∂ 2 ρ k+q-q 2 . ( 6
)
Note the symmetry towards the exchange

k ↔ k = k + q in A 1 and k ↔ k = k + q -q in A 2
, which is a consequence of the hermiticity of the Hamiltonian (4). Computing the transition amplitude in second-order perturbation theory (not forgetting the contribution of three-phonon residual interaction [START_REF] Petković | Dynamics of a Mobile Impurity in a One-Dimensional Bose Liquid[END_REF][START_REF] Castin | Erratum: Landau Phonon-Roton Theory Revisited for Superfluid 4 He and Fermi Gases[END_REF]), on the mass shell we obtain

A if = A 2 (k, q, q ) + A 1 (k -q , q )A 1 (k -q , q) k -hω q -k -q + A 1 (k, q)A 1 (k , q ) hω q + k -k+q + 2A 3 (q , q -q )A 1 (k, q -q ) hω q -hω q-q -hω q + 2A 3 (q, q -q)A 1 (k , q -q) k -k -hω q-q δ k+q,k +q . (7)
To compute the damping rate (1) at low temperature, we need the amplitude [START_REF] Beaufils | Alloptical production of chromium Bose-Einstein condensates[END_REF] expressed at the leading order in small q and q . The energy conservation constraint of Eq. ( 1) has non-trivial solutions only for the backscattering events, i.e., at qq < 0. It leads to the relation

q = -q × (c -v k )/(c + v k ) + O(q), kq > 0, (c + v k )/(c -v k ) + O(q), kq < 0. (8)
Since the quasiparticle scattering off a phonon experiences a small energy change, we expand it to second order around the initial energy k as

k (ρ) = k (ρ)+hv k (ρ)(k -k)+ h2 (k -k) 2 2m * (k, ρ) +O(k -k) 3 .
(9) In the limit q → 0, the on-shell amplitude takes the form

A if = hc |qq |Y k δ k+q,k +q /2ρ, where Y k = ρ 2 mc 2 (c 2 -v 2 k ) (c 2 -v 2 k )∂ 2 ρ k -(∂ ρ k )∂ ρ (c 2 -v 2 k ) + h2 c 2 k 2 m * ρ 2 - (∂ ρ k ) 2 m * - 2hck ρ (v k ∂ ρ c -c∂ ρ v k ) , (10)
in agreement with Ref. [START_REF] Matveev | Scattering of hole excitations in a one-dimensional spinless quantum liquid[END_REF]. Equation ( 10) has a divergence when v k approaches c. At high momenta this occur after the roton minimum when the quasiparticle becomes supersonic, see Fig. 1. Such singularity physically denotes the possibility for the decay by a single phonon emission. The same type of divergence in Y k also exists when the quasiparticle approaches the phonon regime. Those two thresholds are the boundaries of validity of our hydrodynamic approach (see the vertical red lines in Fig. 1).

Using the calculated on-shell amplitude we can now evaluate the damping rate (1). To the leading order in temperature we obtain

Γ k ∼ T →0 T 3 Y 2 k 8πh 3 c 2 ρ 2   c J c-v k c+v k c + v k + c J c+v k c-v k c -v k   , (11)
where Y k is given by Eq. ( 10), while the dimensionless

J(x) = x +∞ 0
dq q2 [1+ ñ(xq)]ñ(q), with ñ(q) = 1/(e q -1). The rate (11) is our main result. It shows that the lowtemperature quasiparticle decay rate universally scales with the third power of temperature. The expression [START_REF] Baranov | Condensed Matter Theory of Dipolar Quantum Gases[END_REF] is general and independent of the particle interaction strength. The k-dependent part of the rate is contained in Γ k and depends on the specific details of the spectrum. We study now Eq. ( 11) in more details.

Hard phonon regime.-Since T mc 2 , there exists a regime of "hard phonons" with the characteristic energy T k mc 2 , such that our hydrodynamic approach applies. We consider the general form of the spectrum of the energetic quasiparticle, i.e., a hard phonon,

k = hc(ρ)|k| 1 + γ(ρ) 8
k 2 q 2 0 + O (k/q 0 ) 4 . ( 12
)
Here the coefficient of the small correction term is negative, γ(ρ) < 0, which ensures that the excitation branch is subsonic (i.e., concave) and that the zero-temperature decay by phonon emission [START_REF] Ristivojevic | Decay of Bogoliubov excitations in one-dimensional Bose gases[END_REF] does not occur. Expanding Eq. ( 10) at small k we obtain Y k = -Ak/q 0 + O(k/q 0 ) 2 , where

A = 1 -2ρ c c -ρ 2 c 2 c 2 -ρ 2 c c + ρ γ γ 1 + ρ c c . (13)
By prime we denote the derivative with respect to ρ. Using J(x) ∼ 2ζ(3)x at x → ∞, the rate (11) becomes

Γ k ∼ T →0 k/q0→0 32ζ(3) 9π A 2 q 2 0 γ 2 k 2 T 3 h3 c 2 ρ 2 . ( 14
)
Equation ( 14) becomes very large at small k, which reflects the denominator in Eq. [START_REF] Lahaye | The physics of dipolar bosonic quantum gases[END_REF]. Our approach requires the quasiparticle energy correction δ k = k -hc|k| of Eq. ( 12) not to be thermally smeared, i.e., δ k T . Therefore the rate ( 14) applies for phonons of wavevectors satisfying q 1/3 th q 2/3 0 |k| q 0 with the thermal wavevector q th = T /hc. To study the regime of k comparable to q 1/3 th q 2/3 0 , our phonon Hamiltonian (2) should be further expanded to describe four-phonon interactions [START_REF] Kurkjian | Landau-Khalatnikov phonon damping in strongly interacting Fermi gases[END_REF], while the phonon dispersion should be now taken with the correction, as in Eq. [START_REF] Wenzel | Striped states in a many-body system of tilted dipoles[END_REF]. In fact, this regime has been studied microscopically [START_REF] Lin | Thermalization of acoustic excitations in a strongly interacting one-dimensional quantum liquid[END_REF][START_REF] Ristivojevic | Decay of Bogoliubov quasiparticles in a nonideal one-dimensional Bose gas[END_REF]. For k = αq 1/3 th q 2/3 0 and α independent of temperature, the T 3 power law for Γ k breaks down. However, quite remarkably, one recovers Eq. ( 14) (specified to the Bogoliubov dispersion, see below) by taking the limit α → ∞ (see Eq. ( 8) in Ref. [START_REF] Lin | Thermalization of acoustic excitations in a strongly interacting one-dimensional quantum liquid[END_REF]). This excludes the existence of an intermediate scaling law between the regimes k ≈ q 0 (of which Eq. ( 14) gives the low-k limit) and k ≈ q 1/3 th q 2/3 0 (of which Eq. ( 8) in Ref. [START_REF] Lin | Thermalization of acoustic excitations in a strongly interacting one-dimensional quantum liquid[END_REF] gives the high-k limit).

Case of the rotons.-Rotons are a special case of the energetic quasiparticles considered above, which have a purely quadratic dispersion around the roton minimum k 0 :

k (ρ) = ∆(ρ) + h2 [k -k 0 (ρ)] 2 2m * (ρ) + O(k -k 0 ) 3 . ( 15
)
The effective mass m * is positive for rotons, but the present discussion also applies to maxons, which have a similar quadratic dispersion but a negative mass, m * < 0. For such quadratic dispersion, using J(1) = π 2 /3, the damping rate (11) acquires a simpler form

Γ k0 ∼ T →0 π 12 Y 2 k0 T 3 h3 c 2 ρ 2 , (16)
where one can use Eq. ( 10) at k = k 0 or its alternative version

Y k0 = ρ 2 mc 2 dµ dρ 2 d 2 ∆ dµ 2 - d∆ ρdρ - 1 m * d∆ cdρ 2 + h2 (k 0 /ρ -dk 0 /dρ) 2 m * . (17)
The correspondence with the general form ( 10) is seen using

∆ = k0 , d∆/dρ = ∂ ρ k0 , d 2 ∆/dρ 2 + (hdk 0 /dρ) 2 /m * = ∂ 2 ρ k0 , v k0 = 0, and hdk 0 /m * dρ = -∂ ρ v k | k=k0
, which follows from partially deriving Eq. ( 15) with respect to ρ and k . The sound velocity c is related to the chemical potential of the gas µ via the relation mc 2 = ρdµ/dρ.

Weak coupling limit.-At weak interaction, one can calculate analytically the quasiparticle spectrum [START_REF] Sinha | Cold Dipolar Gases in Quasi-One-Dimensional Geometries[END_REF][START_REF] Ristivojevic | Decay of Bogoliubov excitations in one-dimensional Bose gases[END_REF] which takes the Bogoliubov form,

k = E k (E k + 2ρg k ), (18)
where E k = h2 k 2 /2m is the kinetic energy of a free boson. By g k is denoted the Fourier transform of the effective two particle interaction. Postponing the discussion about its specific form for the following paragraph, we now assume that the spectrum has the characteristic form with the roton minimum, see Fig. 1. Equation ( 18) enables us to compute the relevant derivatives with respect to the density:

ρ∂ ρ k = 2 k -E 2 k 2 k , ρ 2 ∂ 2 ρ k = - ( 2 k -E 2 k ) 2 4 3 k , ρ∂ ρ v k = v k 2 k + E 2 k 2 2 k - hk m E k k . (19)
The roton damping rate ( 16) now depends only on ∆ and k 0 , besides the thermodynamic parameters T, c, and ρ which already appears in Eq. ( 16). Since those quantities have been measured experimentally, our expression of Γ k0 can be tested without any assumption on g k . At ∆ → 0, such that T ∆ E k0 , mc 2 , we find

Γ k0 ∼ T →0 ∆→0 π 192 (1 + R) 2 E 8 k0 (mc 2 ) 2 ∆ 6 T 3 h3 c 2 ρ 2 , (20)
which diverges as ∆ -6 . In Eq. ( 20) we have introduced the dimensionless parameter

R = ∆/m * c 2 . ( 21
)
It has a finite nonzero limit at ∆ → 0. We finally notice that the Bogoliubov spectrum (18) leads to A = 3(γ -1)/2γ, with γ = 1 + 2ρmg 0 /h 2 , which should be substituted in Eq. ( 14). The quasiparticle spectrum k in units of the transverse trapping energy hω (red solid line) and their rescaled damping rate h3 ρ 2 c 2 Γ k /T 3 (blue solid line) in a quasi-one-dimensional dipolar Bose gas as a function of the wavevector k in units of 2/l = 2 √ mω/h. The dotted vertical black lines show the maxon and roton extrema. The dashed black curve is the hard-phonon asymptote of Eq. ( 14).

Effective interaction potential.-For the application of our theory on the realistic model, we consider cylindrically symmetric quasi-one-dimensional Bose gas characterized by the transverse trapping frequency ω in both transverse directions, and the average dipole moment along the direction of motion x [START_REF] Sinha | Cold Dipolar Gases in Quasi-One-Dimensional Geometries[END_REF]. In Fourier space, the effective one-dimensional interaction potential acquires the form

g k = g 1D - 4αd 2 l 2 1 -ǩ2 e ǩ2 Γ 0, ǩ2 , (22)
where g 1D is the effective one-dimensional contact coupling constant describing short-range interactions (including the short-range behavior of the dipolar interactions [START_REF] Deuretzbacher | Groundstate properties of few dipolar bosons in a quasi-onedimensional harmonic trap[END_REF][START_REF] Deuretzbacher | Erratum: Ground-state properties of few dipolar bosons in a quasi-onedimensional harmonic trap[END_REF]), ǩ = kl/2 where l = h/ √ mω is the harmonic-oscillator length, d is the dipole moment, while Γ denotes the incomplete gamma function. The dipole precesses at high frequency around x-axis, which leads to an effective dipolar strength αd 2 with -1/2 < α < 1 [START_REF] Giovanazzi | Tuning the Dipolar Interaction in Quantum Gases[END_REF]. For the interaction potential [START_REF] Santos | Roton-Maxon Spectrum and Stability of Trapped Dipolar Bose-Einstein Condensates[END_REF], the sound velocity satisfies mc 2 = ρ(g 1D -V d ), where V d = 4αd 2 /l 2 . The low-energy stability condition requires g 1D > V d . A roton minimum appears at k 0 (which solves d k /dk| k=k0 = 0) for g 1D < 0 and V d,min < V d < V d,max < g 1D . The lower bound V d,min is reached at the phase transition ∆ = k0 = 0 and the upper bound V d,max when the dispersion has an inflexion point (when simultaneously d k /dk| k=k0 = 0 and d 2 k /dk 2 | k=k0 = 0). In Fig. 2 we plot the quasiparticle dispersion and the rescaled damping rate as functions of the wavevector, at the interaction strengths ρg 1D = -hω and ρV d = -1.144hω. The rate Γ k diverges at low k as predicted by Eq. ( 14) as well as at k l l/2 0.44 when the excitations become supersonic (v k l = c). It varies several order of magnitude between the maxon and the roton regimes, the latter being far more susceptible to the decay. Figure 3 shows the roton damping rate as a function of the dipolar interaction strength ρV d , or equivalently of the roton gap ∆, fixing the short-range coupling constant ρg 1D = -hω. The roton minimum exists for -1.182hω < ∼ ρV d < ∼ -1.143hω and the gap varies from 0 to about 0.13hω. The rate Γ k is a monotonic decreasing function of ∆ and diverges as ∆ -6 when ∆ → 0 as Eq. ( 20) predicts.

(∆/ ω) 6 ρ 2 c 2 Γ k /T 3 ∆/ ω ρV d / ω
Discussion.-The T 3 behavior of the damping rate is perhaps our most easily testable prediction. This power law is specific to one-dimensional systems, and it would change to T 5 in the two-dimensional case, [START_REF]This follows from a simple power-counting[END_REF] and to T 7 in three dimensions [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF][START_REF] Castin | Landau Phonon-Roton Theory Revisited for Superfluid 4 He and Fermi Gases[END_REF][START_REF] Fåk | Roton-Phonon Interactions in Superfluid 4 He[END_REF]. Attempts to observe the T 7 behavior of the roton-phonon damping rate in superfluid helium failed to resolve it from the roton-roton damping rate [START_REF] Fåk | Roton-Phonon Interactions in Superfluid 4 He[END_REF], which follows an activation law in e -∆/T . In our case, the T 3 law is more favorable, and the capacity to tune the gap offered by ultracold gases can also be used to engender a large prefactor (see, e.g., Fig. 3). We note that the regime ∆ T is largely within the reach of the state of the art experiments since a temperature of 500 nK was reached in Ref. [START_REF] Ilzhöfer | Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms[END_REF]. For a gap of about 10 2 Hz, this gives ∆/T ≈ 10 -3 . Bragg-spectroscopic measurements capable of detecting the linewidths of the elementary modes have also been demonstrated in Ref. [START_REF] Petter | Probing the roton excitation spectrum of a stable dipolar bose gas[END_REF]. Let us finally notice that an attempt to describe the quasiparticle damping rate for a two-dimensional dipolar Bose gas is made in Ref. [START_REF] Natu | Landau damping in a collisionless dipolar Bose gas[END_REF] using the Beliaev-like process where a quasiparticle decays into two others, which resulted in an exponentially suppressed rate at low temperatures. However, our theory adopted to the two-dimensional case would give a universal T 5 power law.

In this paper we studied the damping of quasiparticles of the dipolar Bose gas with rotons, which cannot decay at T = 0 except at large wavevectors. This should be contrasted with the weakly-interacting Bose gas with short-range interactions which has a convex spectrum and thus the decay occurs al-ready at T = 0 [START_REF] Ristivojevic | Decay of Bogoliubov excitations in one-dimensional Bose gases[END_REF]. At weak interaction, the two cases have an overlap at very large wavevectors, where the damping rate at T = 0 approaches a constant value [START_REF] Tan | Relaxation of a High-Energy Quasiparticle in a One-Dimensional Bose Gas[END_REF][START_REF] Ristivojevic | Decay of Bogoliubov excitations in one-dimensional Bose gases[END_REF].

This research was supported by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement number 665501.

  Figure 2.The quasiparticle spectrum k in units of the transverse trapping energy hω (red solid line) and their rescaled damping rate h3 ρ 2 c 2 Γ k /T 3 (blue solid line) in a quasi-one-dimensional dipolar Bose gas as a function of the wavevector k in units of 2/l = 2 √ mω/h. The dotted vertical black lines show the maxon and roton extrema. The dashed black curve is the hard-phonon asymptote of Eq. (14).

Figure 3 .

 3 Figure 3. Rescaled roton (blue curve) and maxon (orange curve) damping rates (∆/hω) 6 ρ 2 c 2 Γ k /T 3 as functions of the roton gap (lower x-axis) or the dipolar strength ρV d (upper x-axis) at fixed ρg1D = -hω. The dashed black line is the corresponding ∆ → 0 asymptote of the roton damping rate obtained from Eq. (20).


Coralie F Daussy

Mathilde Galais

Baptiste Pradel

Véronique Robert-Hebmann

Sophie Sagnier

Sophie Pattingre

Martine Biard-Piechaczyk

Lucile Espert
email: lucile.espert@irim.cnrs.fr

HIV-1 Env induces pexophagy and an oxidative stress leading to uninfected CD4 + T cell death

Keywords: apoptosis, autophagy, HIV-1 Env, peroxisomes, ROS antibodies, AF: auranofin, AP: anti-proteases, ART: antiretroviral therapy, BafA1: bafilomycin A1, BECN1: beclin 1, CAT: catalase, CD4: CD4 molecule, CXCR4: C-X-C motif chemokine receptor 4, DHR123: dihydrorhodamine 123, Env: HIV-1 envelope glycoproteins microtubule associated protein 1 light chain 3, NAC: N-acetyl-cysteine, PARP reactive oxygen species, siRNA: small interfering ribonucleic

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

HIV-1 infection is characterized by a progressive decline in the number of CD4 + T lymphocytes, ultimately leading to acquired immuno-deficiency syndrome/AIDS, in untreated patients [START_REF] Alimonti | Mechanisms of CD4+ T lymphocyte cell death in human immunodeficiency virus infection and AIDS[END_REF][START_REF] Fauci | The human immunodeficiency virus: infectivity and mechanisms of pathogenesis[END_REF]. Importantly, the majority of dying CD4 + T cells are not productively infected but are bystander uninfected cells [START_REF] Finkel | Apoptosis occurs predominantly in bystander cells and not in productively infected cells of HIV-and SIV-infected lymph nodes[END_REF][START_REF] Herbeuval | CD4+ T-cell death induced by infectious and noninfectious HIV-1: role of type 1 interferon-dependent, TRAIL/DR5-mediated apoptosis[END_REF][START_REF] Varbanov | Mechanisms of T-cell depletion in HIV-1 infection triggered by viral proteins[END_REF][START_REF] Westendorp | Sensitization of T cells to CD95-mediated apoptosis by HIV-1 Tat and gp120[END_REF]. Among the several mechanisms identified to explain this depletion [START_REF] Ameisen | Cell dysfunction and depletion in AIDS: the programmed cell death hypothesis[END_REF][START_REF] Jekle | In vivo evolution of human immunodeficiency virus type 1 toward increased pathogenicity through CXCR4-mediated killing of uninfected CD4 T cells[END_REF][START_REF] Meyaard | Programmed death of T cells in HIV-1 infection[END_REF][START_REF] Terai | Apoptosis as a mechanism of cell death in cultured T lymphoblasts acutely infected with HIV-1[END_REF], it has been demonstrated that HIV-1 envelope glycoproteins, composed of gp120 and gp41 associated in trimers (here called Env), play a major role. Indeed, Env expressed at the surface of infected cells is able to induce apoptosis in bystander CD4 + T cells after its interaction with the CD4 receptor and the co-receptor, either CCR5 or CXCR4 [START_REF] Garg | Role of HIV Gp41 mediated fusion/hemifusion in bystander apoptosis[END_REF][START_REF] Perfettini | Mechanisms of apoptosis induction by the HIV-1 envelope[END_REF].

Our team has demonstrated that Env, and specifically the gp41 fusogenic function, triggers a massive induction of macroautophagy (hereafter referred to as autophagy) in bystander CD4 + T cells [START_REF] Denizot | HIV-1 gp41 fusogenic function triggers autophagy in uninfected cells[END_REF][START_REF] Espert | Autophagy is involved in T cell death after binding of HIV-1 envelope proteins to CXCR4[END_REF]. Importantly, this mechanism is necessary for the subsequent apoptosis, suggesting that Env-mediated autophagy could specifically degrade a component crucial for cellular survival [START_REF] Espert | Autophagy is involved in T cell death after binding of HIV-1 envelope proteins to CXCR4[END_REF]. Interestingly, we and others, have also shown that Env triggers an oxidative stress in bystander CD4 + T cells, characterized by a strong production of ROS (Reactive Oxygen Species) [START_REF] Garg | HIV gp41-induced apoptosis is mediated by caspase-3-dependent mitochondrial depolarization, which is inhibited by HIV protease inhibitor nelfinavir[END_REF][START_REF] Molina | Proteomic analysis of the cellular responses induced in uninfected immune cells by cell-expressed X4 HIV-1 envelope[END_REF].

Autophagy is a catabolic pathway involved in several cellular functions such as homeostasis, development, proliferation and immunity [START_REF] Choi | Autophagy in human health and disease[END_REF]. It can be divided in three major steps: (i) the initiation step, with the formation of a double-membrane structure named a phagophore, (ii) the expansion step, where this structure expands engulfing portions of cytoplasm and then closes to form a double-membrane vesicle called an autophagosome, and (iii) the maturation step, where the autophagosomes fuse with lysosomes in order to degrade and recycle the sequestered material. More than 30 genes and several signaling platforms are involved in autophagosome formation, such as the BECN1 (beclin 1)-containing class III phosphatidylinositol 3-kinase complex, acting at the initiation step of the process, and two ubiquitination-like conjugation systems leading to the covalent conjugation of Atg8-family proteins to a lipid, phosphatidylethanolamine, involved in the expansion step of phagophore membranes [START_REF] Ktistakis | Digesting the Expanding Mechanisms of Autophagy[END_REF]. Autophagic degradation can be highly selective using proteins called autophagy receptors, such as SQSTM1/p62 (sequestosome 1) [START_REF] Green | To be or not to be? How selective autophagy and cell death govern cell fate[END_REF]. These proteins bind to both the phosphatidylethanolamine-conjugated Atg8-family proteins and to substrates, allowing their targeting to phagophores [START_REF] Johansen | Selective Autophagy: ATG8 Family Proteins, LIR Motifs and Cargo Receptors[END_REF].

Autophagy and oxidative stress are intimately connected and the level of ROS production is tightly regulated. Low amounts of ROS are involved in the activation of survival signalling pathways [START_REF] Martindale | Cellular response to oxidative stress: signaling for suicide and survival[END_REF], whereas an excessive ROS production activates cell death [START_REF] Green | Cell biology. Metabolic control of cell death[END_REF]. By degrading organelles involved in ROS production, such as damaged mitochondria, selective autophagy balances excessive ROS production [START_REF] Filomeni | Oxidative stress and autophagy: the clash between damage and metabolic needs[END_REF]. However, when the levels of ROS become excessive, cell death is initiated either by apoptosis or necrosis [START_REF] Green | Cell biology. Metabolic control of cell death[END_REF]. Alternatively, autophagy controls the cellular oxidative status by its involvement in peroxisome homeostasis.

Peroxisomes are essential single membrane-limited organelles containing multiple enzymes involved in metabolic and antioxidant reactions. CAT (catalase), one of the major peroxisomal enzyme, catalyzes the transformation of hydrogen peroxide into water, therefore detoxifying the cell from this harmful component [START_REF] Islinger | The peroxisome: an update on mysteries 2.0[END_REF][START_REF] Schrader | Peroxisomes and oxidative stress[END_REF]. PEX/peroxin (peroxisomal biogenesis factor) lumenal and membrane proteins control peroxisome assembly and division. Peroxisomes acquire their various components along their biogenesis and grow towards mature peroxisomes, reaching a size between 0.5 µm and 1 µm. Selective autophagy of peroxisomes, called pexophagy, is responsible for the number and quality of these organelles [START_REF] Eberhart | Pexophagy in yeast and mammals: an update on mysteries[END_REF] by maintaining an equilibrium between their biogenesis and breakdown [START_REF] Cho | Pexophagy: Molecular Mechanisms and Implications for Health and Diseases[END_REF][START_REF] Honsho | Peroxisome homeostasis: Mechanisms of division and selective degradation of peroxisomes in mammals[END_REF].

The aim of the present study was to understand how Env-mediated autophagy could trigger apoptosis in bystander CD4 + T cells. Given that Env also induced an accumulation of ROS in target cells and considering the essential role of peroxisomes in the regulation of the oxidative status of the cell, we focused our research on the interplay between autophagy and peroxisomes upon Env exposure. Our results show that Env-mediated autophagy is responsible for ROS accumulation in bystander CD4 + T cells and demonstrate that Envmediated autophagy selectively targets mature peroxisomes.

Results

Env-mediated autophagy and ROS production are responsible for bystander CD4 + T cells apoptosis.

We previously demonstrated that Env, expressed at the cell surface, induces autophagy in target bystander CD4 + T cells, ultimately leading to apoptosis [START_REF] Espert | Autophagy and CD4 T lymphocyte destruction by HIV-1][END_REF]. These results were obtained with different types of effector cells and target cells also used in the present study (Figure 1A). These models, based on the co-culture between effector cells expressing Env and target cells that express CD4 and CXCR4, allow the analysis of the Env effect in the absence of viral replication. In the model 1, the target cells were either primary CD4 + T lymphocytes or the CEM-derived A201/CD4.403 T cell line. These target cells were co-cultured with HEK293 (human embryonic kidney cells) stably expressing Env (HEK.Env) during 48 h. The parental HEK293 cell line was used as a negative control. In the model 2, the target cells were HEK293 cells stably expressing CD4.403 and CXCR4 and are referred to as (HEK/CD4.403/CXCR4). These target cells were co-cultured for 48 h with 8.E5 cells expressing Env and harboring a single defective copy of replication incompetent HIV-1 proviral DNA. The parental CEM cell line was used as a negative control. Importantly, Env induces equivalent levels of autophagy (between 20 and 30%) and cell death (between 15 and 20%) in the two coculture models and, in both cases, Env-induced apoptosis depends on autophagy [START_REF] Denizot | HIV-1 gp41 fusogenic function triggers autophagy in uninfected cells[END_REF][START_REF] Espert | Autophagy is involved in T cell death after binding of HIV-1 envelope proteins to CXCR4[END_REF][START_REF] Espert | Differential role of autophagy in CD4 T cells and macrophages during X4 and R5 HIV-1 infection[END_REF][START_REF] Roggero | Binding of human immunodeficiency virus type 1 gp120 to CXCR4 induces mitochondrial transmembrane depolarization and cytochrome c-mediated apoptosis independently of Fas signaling[END_REF]. Moreover, it should be noted that the interaction between Env-expressing cells and target cells that express HIV-1 receptors, leads to a transient cell-to-cell fusion, also called hemifusion, or to a complete cell-to-cell fusion with the formation of syncytia, in both models. Importantly, Envmediated autophagy and cell death are triggered in both single cells and syncytia [START_REF] Perfettini | Mechanisms of apoptosis induction by the HIV-1 envelope[END_REF][START_REF] Denizot | HIV-1 gp41 fusogenic function triggers autophagy in uninfected cells[END_REF][START_REF] Garg | HIV gp41-induced apoptosis is mediated by caspase-3-dependent mitochondrial depolarization, which is inhibited by HIV protease inhibitor nelfinavir[END_REF][START_REF] Blanco | Cell-surface-expressed HIV-1 envelope induces the death of CD4 T cells during GP41-mediated hemifusion-like events[END_REF].

We first used model 1 to confirm the results obtained previously using a homogeneous time resolved fluorescence (HTRF) method. This method allows the quantitative detection of the cleaved PARP (poly(ADP-ribose) polymerase), which is a known substrate of CASP3 (caspase 3) when apoptosis is induced. As shown in Figure 1B, Env-induced cleavage of PARP is completely blocked in the presence of AMD3100, which is an antagonist of CXCR4 [START_REF] Donzella | AMD3100, a small molecule inhibitor of HIV-1 entry via the CXCR4 co-receptor[END_REF],

demonstrating that this cleavage is due to Env binding to this co-receptor. Moreover, Env-mediated cleavage of PARP is dependent on autophagy since it is abolished by addition of spautin 1 (Figure 1C) and bafilomycin A1 (BafA1) (Figure 1D), which are inhibitors of the early and late steps of autophagy, respectively.

As mentioned above, we, and others, demonstrated that effector cells expressing Env induce an oxidative stress in target bystander CD4 + T cells characterized by a production of ROS [START_REF] Garg | HIV gp41-induced apoptosis is mediated by caspase-3-dependent mitochondrial depolarization, which is inhibited by HIV protease inhibitor nelfinavir[END_REF][START_REF] Molina | Proteomic analysis of the cellular responses induced in uninfected immune cells by cell-expressed X4 HIV-1 envelope[END_REF]. Here, we evaluated the production of ROS in target cells after 48 h of coculture with effector cells (model 1) using both the dihydrorhodamine 123 (DHR123) probe and an oxyblot assay. The first assay allowed us to observe a significant increase in the production of free radicals (Fig. S1A) and the second allowed us to observe a significant increase in the oxidation of total cellular proteins (Fig. S1B). As oxidative stress induces apoptosis, we evaluated the involvement of the Env-induced production of ROS in this apoptosis. We thus co-cultured effector cells expressing Env with target CD4 + T cells, either a CD4 + T cell line (Figure 1E) or blood-purified primary CD4 + T cells (Figure 1F), in the presence of N-acetylcysteine (NAC), which is a general antioxidant molecule. Addition of NAC completely abolished Env-induced apoptosis, demonstrating that Env-mediated accumulation of ROS is responsible for this cell death. NAC also abolished the total Env-mediated bystander CD4 + T cell death as shown by Propidium Iodide internalization (Fig. S1C).

As the antioxidant properties of autophagy are well characterized, we then analyzed the effect of this process on the Env-induced ROS production. To this aim, using the model 1 of co-culture, we analyzed ROS production in target cells in conditions where autophagy is inhibited by addition of spautin 1. After 48 h, we added DHR123 to quantify ROS production. Surprisingly, we found that inhibition of autophagy leads to a decrease in ROS production (Figure 1G). These results strongly suggested that autophagy is involved in the Env-mediated accumulation of ROS, which is responsible for bystander CD4 + T cell apoptosis.

Autophagy selectively targets peroxisomal proteins.

Peroxisomes are major cellular antioxidant organelles. Their number and integrity are regulated through selective degradation by autophagy in a process called pexophagy. We hypothesized that Env-induced autophagy could degrade peroxisomes in target cells, leading to the accumulation of ROS. To test this hypothesis, we analyzed the level of two peroxisomal proteins by western blot, in target cells after their co-culture during 48 h with Env-expressing cells in different conditions. We followed the level of expression of CAT, which is one of the major peroxisomal enzymes. As CAT is also present in mitochondria, we also analyzed PEX14, which is strictly located at the peroxisomes [START_REF] Grant | The biogenesis protein PEX14 is an optimal marker for the identification and localization of peroxisomes in different cell types, tissues, and species in morphological studies[END_REF]. As shown in Figure 2A and Fig. S2A, the levels of PEX14 and CAT respectively decreased in target cells in contact with Env. AMD3100, which is known to block Env-mediated autophagy and cell death, rescued their level, confirming the involvement of Env in this effect. Next, to analyze the role of autophagy in Env-mediated PEX14 and CAT decreases, we added lysosomal protease inhibitors (AP for anti-proteases: E64d and pepstatin A) to the co-culture. We observed that the Env-mediated PEX14 and CAT decreases are mediated by lysosomal degradation (Figure 2B and Fig. S2B). No significant impact of AMD3100 or AP addition has been observed, at the basal level, on the PEX14:GAPDH ratio. Notably, we verified by RT-PCR that the observed diminution of PEX14 expression levels was not due to a difference in PEX14 mRNA production (Fig. S2C).

To definitively attest the role of Env-induced autophagy on PEX14 and CAT levels, we inhibited autophagy by transfecting siRNA targeting the expression of BECN1 in target cells before their co-culture with Env-expressing cells. We previously showed that Env-mediated cell death was completely inhibited in these conditions [START_REF] Espert | Autophagy is involved in T cell death after binding of HIV-1 envelope proteins to CXCR4[END_REF]. We still observed a PEX14 decrease in target cells transfected with a control siRNA upon Env exposure, but PEX14 expression level remains unchanged when target cells, in which BECN1 was downregulated, are co-cultured with Env-expressing cells (Figure 2C). As the peroxisome number and quality are regulated by autophagy at the basal level in the cells, we checked if BECN1 siRNA could affect the expression level of PEX14 in the absence of Env. We found that siBECN1 had no significative effect on the PEX14:GAPDH ratio without Env (not shown). Of note, similar results were obtained for CAT (Fig. S2D). Taken together, these results confirmed that Env induces the autophagic degradation of peroxisomal proteins.

To complete the analysis, we checked the involvement of the SQSTM1/p62 autophagy receptor, known for its involvement in pexophagy [START_REF] Yamashita | The membrane peroxin PEX3 induces peroxisomeubiquitination-linked pexophagy[END_REF]. We transfected siRNA targeting SQSTM1/p62 in target cells and co-cultured them with Env-expressing cells. The PEX14 protein is no longer degraded in target cells where the expression of SQSTM1/p62 has been downregulated (Figure 2D). Notably, the SQSTM1/p62 siRNA had no significant effect on the PEX14:GAPDH ratio without Env (not shown).

Together, these results showed that Env-induced autophagy selectively target peroxisomal proteins in bystander CD4 + T cells.

Env-induced autophagy targets peroxisomes in target cells.

Based on the above results, we next wanted to analyze whether the entire peroxisomes were degraded by autophagy. We labelled peroxisomes in target cells with green fluorescent protein (GFP) by transfecting the GFP-SKL plasmid that contains a C-terminal tripeptide, serine-lysine-leucine (SKL), called a "peroxisomal targeting signal/PTS", targeting the GFP into peroxisomes. We confirmed that this construct stains peroxisomes by labelling PEX14 in the GFP-SKL-transfected cells. The PEX14 puncta massively colocalized with the GFP signal, confirming that the GFP-SKL construct efficiently labels peroxisomes (Figure 3A). We thus cocultured the GFP-SKL-transfected target cells with Env-expressing cells during 24 h and 48 h and quantified the GFP signal using flow cytometry. We observed a significant decrease in the GFP signal in target cells in contact with Env-expressing cells in a time-dependent manner (Figure 3B). Interestingly, the fluorescence signal was rescued by the addition of spautin 1, arguing that Env induces pexophagy. To further confirm this result, we analyzed the colocalization between GFP-SKL and MAP1LC3B/LC3B (microtubule associated protein 1 light chain 3 beta, here called LC3B), which is an ortholog of Atg8, by immunofluorescence. Target cells, transfected with the GFP-SKL construct, were co-cultured during 48 h with Envexpressing cells in presence of BafA1 to block the autophagic flux. By confocal microscopy, we observed several colocalizations between LC3B and GFP-SKL (Figure 3C). As the number of peroxisomes and autophagosomes was very high in the conditions used, we decided to quantify the colocalization events between GFP-SKL and LC3B in the total target cell volume.

To this aim, several optical slices of target cells (Z=1 µm) were acquired after their co-culture with effector cells expressing or not Env. We used the ImageJ Plugin "JaCOP" allowing the determination of the Mander's colocalization score between LC3B puncta and peroxisomes.

We observed a significant increase in the colocalization between GFP-SKL and LC3B in target cells after their contact with Env-expressing cells compared to the control (Figure 3C), confirming that peroxisomes were targeted to autophagy in cells exposed to Env. Furthermore, Env induced an increase in the colocalization events between GFP-SKL and the SQSTM1/p62 autophagic receptor (Figure 3D).

As peroxisomes respond to changes in the cellular environment by adapting their number and morphology [START_REF] Ribeiro | Peroxisome morphology in pathology[END_REF], we analyzed these last parameters in target cells after their contact with Env-expressing cells by immunofluorescence with a particular focus on their size.

To this aim, we stained peroxisomes in target cells after 48 h of co-culture with Env-expressing cells using an anti-PEX14 antibody in presence or absence of spautin 1. Peroxisomes vary in size from 0.1 to 1 μm in diameter. Although they are capable of fission, the "mature" peroxisomes, which exert their antioxidant capacity, are larger than those undergoing biogenesis. The target cells were analyzed by confocal microscopy (Figure 3E). In order to distinguish large peroxisomes from small ones (in the process of biogenesis), we used the "Cell Profiler" software and designed a protocol allowing us to measure their area in target cells.

Stacks of 1 μm between each optical slice were acquired to avoid capturing the same peroxisome twice. Each acquisition, with a confocal microscope, was performed on the total volume of the target cells. We set up a threshold of 0.7 μm in diameter to identify "mature" peroxisomes. We first quantified the mean number of total peroxisomes, and the number of large ones. We observed an increase in the total peroxisomes upon Env exposure (Figure 3F). This result is in accordance with the observed Env-induced production of ROS (Figure 1G).

Moreover, the addition of spautin 1 slightly decreased the total number of peroxisomes, which is in accordance with the reduced production of ROS observed upon spautin 1 treatment in Figure 1G. Then, based on these measurements, we established, for each image analyzed, a ratio between the number of mature peroxisomes (diameter> 0.7μm) and the number of total peroxisomes. The data were collected for each image constituting the volume of a target cell.

Interestingly, this ratio was decreased in target cells after their contact with Env-expressing cells and the addition of spautin 1 restored the ratio close to the one of control cells (Figure 3G). As demonstrated in Figure 2B, C and D, Env-induced autophagy degraded PEX14. These results, together with the LC3B and SQSTM1/p62 colocalization experiments (Figure 3C and D), strongly suggested that Env-induced autophagy degraded active peroxisomes.

Discussion

The interplay between autophagy and HIV-1 infection is complex and depends on the target cell type and on the infectious status of the cell (infected vs bystander cells) [START_REF] Espert | Differential role of autophagy in CD4 T cells and macrophages during X4 and R5 HIV-1 infection[END_REF][START_REF] Kyei | Autophagy pathway intersects with HIV-1 biosynthesis and regulates viral yields in macrophages[END_REF][START_REF] Zhou | Human immunodeficiency virus type-1 infection inhibits autophagy[END_REF]. We previously showed that Env, expressed on infected cells, induces autophagy in bystander CD4 + T cells, after its interaction with HIV-1 receptors present at their surface. Notably, this Envinduced autophagy leads to apoptosis of these target cells, mechanism likely contributing to CD4 + T cells depletion observed in HIV-1 associated physiopathology [START_REF] Espert | Autophagy is involved in T cell death after binding of HIV-1 envelope proteins to CXCR4[END_REF]. As HIV-1 Env induces the production of ROS in bystander CD4 + T cells [START_REF] Molina | Proteomic analysis of the cellular responses induced in uninfected immune cells by cell-expressed X4 HIV-1 envelope[END_REF], we first demonstrated that this oxidative stress is responsible for subsequent apoptosis. The cellular ROS levels are tightly regulated by the action of antioxidant enzymes in order to avoid their harmful effect. As we previously showed that Env-mediated autophagy is a prerequisite to apoptosis in bystander cells, we hypothesized that antioxidant pathways is targeted by autophagy and, consequently, deprive the cell of their protective action. Inhibition of autophagy by adding spautin 1 blocks apoptosis (Figure 1C), but also dramatically decreases the production of ROS in bystander cells cocultured with effector cells expressing Env (Figure 1G), strongly suggesting that Env-induced autophagy leads to an uncontrolled oxidative stress in bystander cells. ROS can be produced at different cellular sites and three major organelles are known to control it: mitochondria, endoplasmic reticulum and peroxisomes, forming the "redox triangle" [START_REF] Yoboue | Redox crosstalk at endoplasmic reticulum (ER) membrane contact sites (MCS) uses toxic waste to deliver messages[END_REF]. These three organelles are interconnected and cooperate to fight oxidative stress. Importantly, the inhibition of peroxisomal CAT, concomitantly with an increase of ROS, results in mitochondrial redox imbalance [START_REF] Ivashchenko | Intraperoxisomal redox balance in mammalian cells: oxidative stress and interorganellar cross-talk[END_REF]. Moreover, Env induces the mitochondrial pathway of apoptosis in bystander CD4 + T cells [START_REF] Roggero | Binding of human immunodeficiency virus type 1 gp120 to CXCR4 induces mitochondrial transmembrane depolarization and cytochrome c-mediated apoptosis independently of Fas signaling[END_REF]. In addition, a recent study showed that a subset of miRNAs, targeting peroxisomal proteins, are upregulated in the brain of HIV/AIDS patients with and without HIV-associated neurocognitive disorders/HAND. The authors found that these miRNAs significantly decreased peroxisome number and affect their morphology [START_REF] Xu | MicroRNAs upregulated during HIV infection target peroxisome biogenesis factors: Implications for virus biology, disease mechanisms and neuropathology[END_REF]. In this context, we focused our work on peroxisomes and demonstrate that these organelles are targeted by Env-induced autophagy. Our results suggest that peroxisomes are insufficiently recognized targets of HIV-1 infection and open new routes in the characterization of the relationships between autophagy and this virus. Obviously, we cannot exclude a role of the mitochondria and the endoplasmic reticulum in our model and it would be important to test if these organelles, as well as other antioxidant pathways, are also targeted by autophagy in the same context.

By analyzing both peroxisomal proteins and entire peroxisomes, we were able to demonstrate, for the first time, that pexophagy plays a major role in a viral infection, probably by impairing the control of induced oxidative stress. Pexophagy, which occurs at the cellular basal level, is thus induced upon Env exposure. It would be important now to decipher the specific Env-induced signalling pathway triggered upstream of this phenomenon and particularly the role of the kinase ATM (ATM serine/threonine kinase). Indeed, this protein is activated by peroxisomal ROS, leading to pexophagy and involves the selective targeting of PEX5 by SQSTM1/p62 [START_REF] Tripathi | A new role for ATM in selective autophagy of peroxisomes (pexophagy)[END_REF].

Concerning HIV-1 infection, the degradation of cellular antioxidant system requires a special attention as a known side-effect of ART (Anti-Retroviral Treatment) is to induce an oxidative stress in patients [START_REF] Manda | Highly active antiretroviral therapy drug combination induces oxidative stress and mitochondrial dysfunction in immortalized human blood-brain barrier endothelial cells[END_REF][START_REF] Nagiah | Inverse association between microRNA-124a and ABCC4 in HepG2 cells treated with antiretroviral drugs[END_REF][START_REF] Wang | Roles and mechanisms of human immunodeficiency virus protease inhibitor ritonavir and other anti-human immunodeficiency virus drugs in endothelial dysfunction of porcine pulmonary arteries and human pulmonary artery endothelial cells[END_REF][START_REF] Weiss | Efavirenz Causes Oxidative Stress, Endoplasmic Reticulum Stress, and Autophagy in Endothelial Cells[END_REF]. Moreover, oxidative stress is also clearly involved in the development of HIV-1-associated disorders such as neurotoxicity, dementia and immune imbalance [START_REF] Baier-Bitterlich | Oxidative stress and apoptosis in HIV infection[END_REF][START_REF] Baruchel | The role of oxidative stress in disease progression in individuals infected by the human immunodeficiency virus[END_REF][START_REF] Couret | Reactive Oxygen Species in HIV Infection[END_REF][START_REF] Israel | Oxidative stress in human immunodeficiency virus infection[END_REF][START_REF] Ivanov | Oxidative Stress during HIV Infection: Mechanisms and Consequences[END_REF][START_REF] Legrand-Poels | Activation of human immunodeficiency virus type 1 by oxidative stress[END_REF]. Several HIV-1-associated co-morbidities related to an increased risk of cardiovascular diseases development are favored by the oxidative stress induced by the presence of the virus, such as high blood pressure, atherosclerosis and myocarditis [START_REF] Ballocca | Cardiovascular disease in patients with HIV[END_REF][START_REF] Fan | Activating the Nrf2-mediated antioxidant response element restores barrier function in the alveolar epithelium of HIV-1 transgenic rats[END_REF][START_REF] Spieker | Rapid progression of atherosclerotic coronary artery disease in patients with human immunodeficiency virus infection[END_REF].

Moreover, in this infectious context, the peroxisome depletion could be beneficial for pathogens and their escape from innate immunity. Indeed, peroxisomes have been identified as antiviral innate immunity signalling platforms since they harbor mitochondrial antiviral signaling protein at their surface and participate to type I interferon production in response to viral invasion [START_REF] Dixit | Peroxisomes are signaling platforms for antiviral innate immunity[END_REF]. Beyond this, one of the major barriers against the eradication of HIV-1 in patients is the presence of cellular reservoirs that contain integrated, replication-competent provirus, resistant to ART. Even if these treatments maintain a very low level of viral replication and have considerably ameliorated the HIV-1 infected patient life, a rebound of viremia is observed upon treatment interruption. Interestingly, a new therapy, based on the oxidative stress response, has been proposed to target the HIV-1 reservoir. Auranofin (AF) is a gold compound with a pro-oxidant function [START_REF] Becker | Thioredoxin reductase as a pathophysiological factor and drug target[END_REF] capable of inducing apoptosis of memory T cells based on their oxidative status [START_REF] Badley | Altering cell death pathways as an approach to cure HIV infection[END_REF]. AF combined with ART completely suppresses simian immunodeficiency virus/SIV viremia in macaques. More interestingly, the survival of treated macaque is maintained for 2 years after the treatment interruption without any sign of immunodeficiency [START_REF] Shytaj | Two-Year Follow-Up of Macaques Developing Intermittent Control of the Human Immunodeficiency Virus Homolog Simian Immunodeficiency Virus SIVmac251 in the Chronic Phase of Infection[END_REF]. These results indicate that apoptosis-inducing drugs, acting on oxidative stress, could be considered to reduce the HIV-1 reservoir. It would be thus interesting to analyze if AF, combined to ART, could modulate memory T cells autophagy.

In conclusion, our results open new route for future investigations at multiple levels of HIV-1-host interactions and, to our knowledge, they demonstrate for the first time a role of pexophagy during a viral infection.

Materials and Methods

Cell culture

The HEK293, HEK.Env [START_REF] Roggero | Binding of human immunodeficiency virus type 1 gp120 to CXCR4 induces mitochondrial transmembrane depolarization and cytochrome c-mediated apoptosis independently of Fas signaling[END_REF] and HEK/CD4.403/CXCR4 [START_REF] Denizot | HIV-1 gp41 fusogenic function triggers autophagy in uninfected cells[END_REF] cell lines were cultured in DMEM (Life Technologies, 61965-026) supplemented with 1% penicillin-streptomycin, 1% Glutamax, and 10% fetal calf serum (Sigma-Aldrich, F7524). The CEM T cell line was provided by the ATCC (ATCC® CCL119™). The CEM-derived A2.01/CD4.403 cell clone, which expresses a mutant form of CD4 truncated at position 403, was previously described [START_REF] Denizot | HIV-1 gp41 fusogenic function triggers autophagy in uninfected cells[END_REF] and was provided by D.R.

Littman (New York Medical College, New York, NY, USA). The 8.E5 cell line, a CEM-derived T cell line containing a single integrated copy of HIV-1 and unable to produce infectious virions, was provided by F. Barré-Sinoussi (Institut Pasteur, Paris, France). These T cell lines were cultured in RPMI 1640 medium (Life Technologies, 61870-010) supplemented with 1% penicillin-streptomycin, 1% Glutamax, and 10% fetal calf serum. Primary CD4 + T cells were purified from the blood of healthy donors by negative selection using the Human CD4 + T Cell Enrichment cocktail (StemCell Technologies, 19157). These primary cells were cultured in complete RPMI 1640 medium.

Reagents and antibodies

AMD3100, NAC, spautin 1, E64d, pepstatin A, bafilomycin A1 were purchased from Sigma-Aldrich (239820, A9165, E8640, P5318, B1793, respectively). The anti-PEX14 and anti-BECN1 antibodies (Abs) were purchased from Proteintech (105941-AP and 11306-1-AP, respectively).

The anti-SQSTM1/p62 Ab was purchased from Euromedex (GTX100685). The anti-CAT (catalase) Ab was purchased from Abcam (ab1877-10). The anti-LC3B and the anti-GAPDH Abs were purchased from Sigma-Aldrich (L7543 and G9295, respectively).

Transfections

RNA interference: ON-TARGET plus Human SMART Pool from Dharmacon were used for RNA interference targeting BECN1 (reference L-010552) and siGENOME Human siRNA -SMART Pool for SQSTM1/p62 (reference M-010230). The ON-TARGET plus Non-targeting Pool (reference D-001810-10) was used as a control. Briefly, 0.5x10 6 target cells were transfected using LipoRNAiMax (Invitrogen, P/N56531) with 20 nM siRNA according to the manufacturer's instructions. Sixteen h after transfection, target cells were co-cultured with effector cells expressing, or not, Env (1.10 6 /ml) for 48 h before analysis.

Plasmids: Target cells (0.5x10 6 ) were cultured in 6-well plates and transfected with pEGFP-SKL (Addgene, 53450; from Jay Brenman Lab) plasmids using TurboFect Transfection Reagent (Thermo-Scientific, R0531) according to the manufacturer's instructions. Transfected target cells were co-cultured with effector cells expressing or not Env for 48 h before analysis.

RT-PCR

Target cells were co-cultured with effector cells expressing, or not, Env for 48 h. RNA was purified from 10 6 target cells using the Nucleospin RNA plus (Macherey-Nagel, 740984.50), and RT-PCR was performed using the QIAGEN OneStep RT-PCR kit according to the manufacturer's instructions. The oligonucleotides used for PEX14 and GAPDH mRNA are:

PEX14

FOR: 5'-GGGCTGACAGATGAAGAGATTG-3' and PEX14 BACK: 5'-CCTGGATCTTCTGCTGCTGCTG -3' GAPDH FOR: 5'-CCCATCACCATCTTCCAG-3'; GAPDH BACK: 5'-CCTGCTTCACCACCTTCT-3'.

Apoptosis and total cell death analysis

Target cells were co-cultured with effector cells expressing, or not, Env for 48 h. Apoptosis was analyzed in the target cells using "Cleaved PARP (Asp214) Cellular Kit" from CisBio (PerkinElmer, 64PARPEG) according to the manufacturer's instructions. FRET analysis was measured using a SPARK 10M TECAN. Propidium iodide staining was used to quantified the level of total cell death. Briefly, suspension cells were washed with phosphate-buffered saline (ThermoFisher Scientific, 14190-094), 4 μl PI was added to the cells and they were then analyzed on a COULTER EPICS XL Flow Cytometer (Beckman Coulter).

ROS analysis

DHR123 assay: Target cells were co-cultured with effector cells during 48 h. Then, cells were incubated for 15 min at 37°C in HBSS containing 0.86 mM DHR 123 (ThermoFisher, D23806) and fluorescence intensity was rapidly measured at 543 nm by flow cytometry.

Oxyblot assay: After 48 h of coculture with effector cells, target cells were lysed and the carbonyl content present in cellular proteins was determined using the "OxyBlot Protein Oxidation Detection Kit" (Sigma-Aldrich, S7150) according to the manufacturer's instructions.

Equivalent amounts of each protein fraction were analyzed. Immunoblots were visualized by ECL as described in next section.

Western blot analysis

Target cells were co-cultured with effector cells during 48 h. Target cells were then harvested and lysed directly in Laemmli buffer 2x and heated for 10 min at 95°C. Cell lysates were loaded on 12% Prosieve 50 gels (Lonza, LON50618) and transferred to polyvinylidene (PVDF) membranes. After a blocking step in PBS containing 5% casein (Sigma-Aldrich, C7078) and 0.05% Tween 20 (Sigma-Aldrich, P9416) for 1 h at room temperature, blots were incubated overnight at 4°C with the primary Ab in the blocking buffer. After 3 washes with PBS+0.05%

Tween 20, the blots were incubated for 1 h at room temperature with peroxidase-coupled antiserum (Sigma-Aldrich, goat anti-rabbit and goat anti-mouse, A0545 and A2304, respectively) diluted in blocking buffer. After further washes, the immune complexes were revealed by ECL (Clarity western ECL Substrate; Bio-Rad, 170-5061) and analyzed with a ChemiDoc camera (Bio-Rad). Quantification of protein expression was performed using the Image lab Software (Bio-Rad). Data were normalized with reference to the densitometry analysis of GAPDH.

Immunofluorescence studies

Target cells were cultured on glass slides and co-cultured with effector cells during 48 h. For the staining, target cells were first incubated with 1% paraformaldehyde for 10 min and then with methanol during 5 min for LC3B staining and incubated with 1% paraformaldehyde + 0.1% Triton X-100 (Sigma-Aldrich, T8787) for 15 min for SQSTM1/p62 staining. After a saturation step in a PBS, 0.1% saponin (Calbiochem, 558255), 1% FBS buffer, the primary Ab was incubated during 2 h at room temperature. After several washes, the secondary Ab coupled with a fluorochrome was added during 1 h. Imaging was done using a Spinning Disk Dragonfly Andor microscope (Montpellier RIO Imaging platform). Acquired images were analyzed using the Fiji plugin JaCOP. The Mander's coefficient, measuring the overlap of GFP-SKL in LC3B or SQSTM1/p62 staining, was determined on each z-stack of each acquisition before being averaged on all stacks of each cell [START_REF] Bolte | A guided tour into subcellular colocalization analysis in light microscopy[END_REF]. For the « Cell profiler » analysis, a procedure was set up to discriminate peroxisomes over their area size.

Statistical analysis

Differences were considered significant at *p < 0.05, **p < 0.01, and ***p < 0.001. Variances were measured using the Wilcoxon test on GraphPad Prism.

FiguresFigure 1 .

 1 Figures Figure 1.Env-induced autophagy is a prerequisite for apoptosis in CD4 + T cells. (A) Description of the cellular models. The experiments are based on the co-culture between effector cells expressing Env at their surface and target cells expressing CD4 and CXCR4. In the model 1, target cells are either primary CD4 + T lymphocytes purified from healthy donor blood samples or a CD4 + T cell line expressing CXCR4. Effector cells are HEK293 stably expressing Env (HEK.Env). HEK293 cells are used as negative control. In the model 2, target cells are HEK293 cells stably expressing CD4 and CXCR4. Effector cells are 8.E5 cells, which are chronically infected cells and thus express Env at their surface. The parental CEM cell line is used as a negative control. (B-D) Env-induced apoptosis depends on autophagy. Target CD4 + T cells were co-cultured for 48 h with HEK293 or HEK.Env in the presence or absence of (B) AMD3100 (1 µg/ml), (C) spautin 1 (50 μM), (D) BafA1 (50 nM). Target cells were harvested and fold increase in Env-mediated apoptosis was studied by quantifying the cleavage of PARP using HTRF. Results are from at least 3 independent experiments. *p<0.05; **p < 0.01; ***p < 0.001. (E and F) Env-induced autophagy participated in ROS production that is responsible for uninfected CD4 + T cells apoptosis. Target CD4 + T cells (either T cell line, E or primary CD4 + T cells, F) were co-cultured for 48 h with HEK or HEK.Env in the presence or absence of NAC (10 mM). Target cells were harvested and fold increase in Env-mediated apoptosis was studied by quantifying the cleavage of PARP using HTRF. Data are representative of at least three independent experiments. **p < 0.01; ***p < 0.001. (G) Target CD4 + T cells were co-cultured for 48 h with HEK293 or HEK.Env in the presence or absence of spautin 1 (50 µM). Analysis of intracellular ROS production was measured by flow cytometry using DHR 123. Data are representative of at least three independent experiments. *p<0.05; **p < 0.01.

Figure 2 .

 2 Figure 2. PEX14 is degraded by Env-induced autophagy. (A) Env induces a decrease in PEX14 expression level. PEX14 expression level was analyzed in target CD4 + T cells after 48 h of coculture with effector HEK293 cells expressing or not Env, in the presence or absence of

Figure 3 .

 3 Figure 3. Env-induced autophagy targets mature peroxisomes. (A) The SKL motif drives GFP to peroxisomes. Target cells were transfected with the GFP-SKL construct. Twenty-four h later, peroxisomes were labelled using an anti-PEX14 antibody. Images were acquired with a confocal microscope. (B) The GFP-labelled peroxisomes decrease overtime in target cells after their co-culture with effector cells expressing Env. Target cells were transfected with the GFP-SKL construct. Twenty-four h later, the transfected cells were co-cultured with effector cells in presence or not spautin 1 (50 µM). The mean fluorescence intensity (MFI) of the GFP signal was analyzed in target cells using flow cytometry after 24 and 48 h of co-culture. Data are representative of at least 3 independent experiments. *p<0.05; **p < 0.01; ***p < 0.001. (C)The colocalization between the GFP labelled peroxisomes and LC3B increases in target cells

Figure S2 .

 S2 Figure S2.CAT is a target of Env-mediated autophagy. (A) CAT expression level was analyzed in target CD4 + T cells after 48 h of co-culture with effector HEK293 cells expressing or not Env, in the presence or absence of AMD3100 (1 µg/ml). The expression level ratio of CAT was calculated between the conditions with or without Env and normalized to that obtained with anti-GAPDH Ab. (B) CAT expression level was analyzed in target CD4 + T cells after 48 h of coculture with effector HEK293 cells expressing or not Env, in the presence or absence of AP (E64d + pepstatin A, 10 µM each). The expression level ratios of CAT were calculated between the conditions with or without Env and normalized to that obtained with anti-GAPDH Ab. (C) PEX14 mRNA level was analyzed by RT-PCR in target CD4 + T cells after 48 h of co-culture with effector HEK293 cells expressing or not Env, in the presence or absence of AP (E64d + pepstatin A, 10 µM each). GAPDH mRNA was used as a control. (D) HEK/CD4.403/CXCR4 cells were cocultured during 48 h with effector cells expressing or not Env (8.E5 or CEM, respectively) after their transfection with BECN1 (siBECN1) or control (siCT) siRNAs. The expression level ratios of CAT were calculated between the conditions with or without Env and normalized to that obtained with anti-GAPDH Ab.

Figure 2 CFigure 3

 23 Figure 2

Acknowledgments

This work was supported by the "Agence Nationale de Recherche sur le SIDA et les Hépatites Virales" (ANRS), by SIDACTION, by the Centre National de la Recherche Scientifique (CNRS) and by the University of Montpellier. MG was the recipient of a fellowship from the Labex EPIGENMED. We thank Dr. Matteo Bonazzi (IRIM CNRS UMR9004 Montpellier, France) for its help on imaging analysis.

Declaration of interest statement

No conflict of interest.

Supplementary Figures