Zakari Denis

Alberto Biella

Ivan Favero

Cristiano Ciuti

Permanent Directional Heat Currents in Lattices of Optomechanical Resonators

We study the phonon dynamics in lattices of optomechanical resonators where the mutually coupled photonic modes are coherently driven and the mechanical resonators are uncoupled and connected to independent thermal baths. We present a general procedure to obtain the effective Lindblad dynamics of the phononic modes for an arbitrary lattice geometry, where the light modes play the role of an effective reservoir that mediates the phonon nonequilibrium dynamics. We show how to stabilize stationary states exhibiting directional heat currents over arbitrary distance, despite the absence of thermal gradient and of direct coupling between the mechanical resonators.

-4 -2

Introduction -The emergence of persistent currents in many-body systems is tightly bound to fundamental concepts in classical and quantum physics. In classical electrodynamics, any permanently magnetized object exhibits persistent electronic currents [START_REF] Griffiths | Introduction to Electrodynamics[END_REF]. A conducting ring in the quantum coherent regime supports a permanent electric current when pierced by an external magnetic field [START_REF] Bouchiat | [END_REF]. When pairing interactions are considered, a superconductor cooled below critical temperature displays persistent currents, and a constant magnetic field builds up through any continuous loop of the material [START_REF] Tinkham | Introduction to Superconductivity[END_REF]. Systems with nontrivial topology can also give rise to persistent edge currents [START_REF] Rivas | [END_REF].

These manifestations of persistent currents involve two noticeable ingredients: (i) an external gauge field and (ii) the presence of a significant coherence extending over the entire sample [5]. Recently, it was shown that these ingredients are not strictly required, and that permanent currents in rings can instead be generated by reservoir engineering [5,6], where specific many-body quantum states with properties of interest are stabilized [7,8]. More than a mere source of decoherence, the environment becomes then a tool to generate correlated phases, sometimes with no equilibrium counterpart [9]. In this context, the study of systems driven by nonlocal dissipators has emerged, notably in relation to nonreciprocal behaviors [6]. In several nonreciprocal realizations, a direct coupling between two bosonic modes was engineered through a common ancillary degree of freedom [10][11][12]. Very recently, the concept of engineered directionality was theoretically scaled up to extended lattices, by tailoring ancilla-assisted interactions [5,[13][14][15].

Besides nonreciprocity, the coupling of independent mechanical modes to commonly shared optical modes was proposed to transport phonons between distant resonators [16], to model out-of-equilibrium quantum thermodynamics [17], and experimentally implemented to phase-lock adjacent [18] and distant [19,20] mechanical resonators. Yet, many aspects of the nonlocal quantum dynamics of extended lattices in optomechanics remain to be explored despite their potentially uncommon features.

In this work, we study analytically the effective dynamics of originally independent mechanical resonators coupled to extended lattices of driven-dissipative optical cavities. We express the general master equation of such a reservoir-coupled system and compare our predictions with a mean-field approach. Our study demonstrates that rings of lattices of optically coupled optomechanical resonators [21,22] can exhibit permanent whirling phonon currents. The latter are mediated by spatially correlated quantum fluctuations of the optical fields, in the absence of direct mechanical coupling, and triggered by proper tuning of the phase of the optical drive. The magnitude of the current is expressed analytically within a Born-Markov approximation, while the heat flow persists when mechanical resonators interact with independent thermal baths, over a wide range of temperatures.

The existence of permanent phonon currents despite the absence of thermal gradient and of direct coupling between mechanical resonators is a novel phenomenon with no counterpart in models studied so far.

Generic model -The system under consideration consists of a network of L optomechanical resonators whose optical modes are coherently driven by external laser fields. Neighboring cavities are optically coupled to one another, while mechanical modes are not. In a specific implementation with optomechanical disk resonators, optical modes are whispering gallery modes of adjacent resonators, while mechanical modes are radial breathing modes of individual disks [23]. Such resonators can be fabricated with ultralow site-to-site disorder [24]. One optomechanical cell is schematically illustrated in Fig. 1 (a).

While in the following we focus on one-dimensional (1D) chains, here for the sake of generality, we consider an arbitrary network where the coupling between adjacent photonic modes is fully specified by a L × L adjacency matrix A where A = 1 if the sites and are coupled and A = 0 otherwise. In the frame rotating at the driving frequency ω p , the unitary part of the dynamics is described by the following Hamiltonian [25] ( = 1):

Ĥtot = L =1 -∆ â † â + F â + F â † -g â † â ( b + b † ) - J 2 L , =1 A â † â + L =1 ω ( ) m b † b , (1)
where â and b are, respectively, the photonic and phononic annihilation operators of the -th resonator, ∆ = ω pω ( ) c denotes the detuning of the driving laser frequency with respect to the local bare cavity frequency ω ( ) c , F is the (complex) amplitude of the coherent drive, g is the optomechanical vacuum coupling rate and J is the hopping rate between connected optical cavities. Incoherent processes associated to local photon losses (at a rate γ ( ) c ) and phonon thermalization with their respective thermal baths (at a rate γ ( ) m ) are taken into account by means of a master equation for the system density matrix in the Lindblad form, which fully determines the system evolution,

-1 â F +1 J J γ c b g T (a) + 1 φ ( +1)φ T γ c J (a) (b)
∂ t ρ(t) = L tot ρ(t) ≡ -i[ Ĥtot , ρ(t)] + D tot ρ(t), (2)
where

D tot ρ = L =1 γ ( ) m (n + 1)D[ b ]ρ + n D[ b † ]ρ +γ ( ) c D[â ]ρ , (3)
with D[ Ô]ρ = Ô ρ Ô † -1 2 { Ô † Ô, ρ} and n the average number of thermal bosons due to the -th thermal bath.

Provided γ c 2g â † â 1/2 , as we assume in what follows, optical fluctuations are negligibly affected by the mechanics and the coupled cavities can be regarded as an extended optical reservoir which can safely be adiabatically eliminated in a wide range of the parameter space [26]. By adjusting J/γ c , one can tune the correlation length of the reservoir going from the uncoupled resonators local case (J/γ c 1) to that of a reservoir with resolved spectrum (J/γ c 1) [16,17]. Adiabatic elimination of an extended driven-dissipative reservoir -By splitting the fields into their mean-field value plus zero-mean fluctuations as â = α +ĉ and b = β + d , we can expand the Hamiltonian and the dissipator to second order in the fluctuations around mean field,

Ĥ tot L =1 -∆ ĉ † ĉ - J 2 A ĉ † ĉ + V + ω ( ) m d † d , (4)
where ∆ ≈ ∆ + 2g 2 |α | 2 /ω ( ) m (for a high mechanical quality factor), V = (G * ĉ + G ĉ † )( d + d † ), G = g α , and the mean fields {α , β } respect a self-consistency relation [27] that exactly cancels all linear terms in the fluctuation operators in the Hamiltonian. Both the amplitude and the phase of G can be tuned through the driving. The dissipator remains that of Eq. ( 3) substituting â , b → ĉ , d . In this displaced frame, it becomes clear that finite-lived (τ c = 1/γ c ) quantum optical fluctuations are not externally driven but can enter the reservoir from the mechanics through the now linear optomechanical coupling ( V ) and can then be scattered back into some distant mechanical mode or be dissipated. We formalize this intuition hereafter by looking at the reduced dynamics of the mechanical degrees of freedom.

Within the Born-Markov approximation, the lattice of optical cavities can be adiabatically eliminated (see Supplemental Material [28]) yielding the following effective Hamiltonian and dissipator for the mechanical modes:

Ĥeff m = L =1 ω ( ) m d † d + L , =1 (Ω (+) + Ω (-) ) d † d , (5) D eff m ρm = L =1 γ ( ) m (n + 1)D[ d ]ρ m + n D[ d † ]ρ m + L =1 Γ (+) D[ β(↓) ]ρ m + Γ (-) D[ β(↑) ]ρ m , (6)
where Ω Ω Ω (±) = 1 2i S (±) -S (±) † is the effective coherent coupling, determined by the spectrum of the extended reservoir:

S (±) = G i1 ±ω ( ) m 1 -B G , (7)
with

B = -J 2 A -Diag({ ∆ + i γ ( ) c 2 }
). The nonlocal dissipation rates are given by the eigenvalues of its Hermitian part, Diag({Γ (±) }) = U (±) (S (±) + S (±) † )U (±) † , where U (±) are the associated diagonalizing unitary matrices. Finally, the nonlocal jump operators are defined as β(↓) =

L =1 U (+) d , β(↑) = L =1 U (-) d † (note that in general β(+) = β(-) † )
. These results can be extended to continuous reservoirs and two-tone driven reservoirs generating multimode squeezing (see Supplemental Material [28]).

In the case of a finite 1D chain with nearest-neighbors coupling as henceforth considered, B takes the form of a tridiagonal matrix and has thus explicit inverse expressions [29]. When |∆ ± ω miγ c /2| J/2, offdiagonal elements have exponentially decreasing magnitudes, |S (±) , +p /S (±) , | ∼ (J/2) p /|∆ ± ω miγ c /2| p , so that the reservoir mainly couples neighboring mechanical modes, as expected from the finite lifetime of the optical fluctuations within the optical lattice. The range of the effective interaction can thus be selected by tuning J/γ c . In contrast to previous works, where directional nearest-neighbors couplings at each edge , are engineered by having an independent ancillary mode coupled exclusively to the sites and , we can simply rely on the finite lifetime of the mediating photon fluctuations to make the interaction short range when required.

In this effective description, the lattice of cavities modifies the dynamics of the mechanical modes by adding coherent phonon-hopping processes between previously uncoupled mechanical modes and acting as a thermal bath for L extended phononic modes { β(↓) }.

Periodic 1D optomechanical lattice -We now exploit the effective description derived above to study the emergence of persistent directional heat currents in an experimentally relevant model: a ring composed of L sites. To this aim, the cavities are driven individually with the same intensity but with a site-dependent phase such that |F | = F and Arg(F ) = φ with φ = 2πn/L and n ∈ Z, which creates a homogeneous phase gradient around the ring [30,31]. This situation is schematically illustrated in Fig. 1 (b). Following Eqs. ( 5) and ( 6), the unitary part of the mechanical effective dynamics is governed by:

Ĥeff m = (ω m + J (+) 0 + J (-) 0 ) d † d + ± 1≤p<L J (±) p 2 d † +p d e ∓iφ×p + H.c. , (8)
where

J (±) p = k e ikp L |G| 2 (±ω m + ∆ + J cos k) (±ω m + ∆ + J cos k) 2 + (γ c /2) 2 (9)
is the real-valued amplitude of the effective complex coupling between p-distant modes. The second line involves two sets of directional couplings, noted by ±. This can be understood from second order perturbation theory by examining the two mechanics-mechanics scattering processes having finite overlap f | V +p V |i and preserving the total energy:

f |G +p ĉ +p d † +p × G ĉ † d |i and f |G +p ĉ † +p d † +p × G ĉ d |i .
The magnitude of each of these directional hopping channels, and thus the net effective flux of phonons, can be adjusted via the drive detuning ∆. This dependence is complex in general, as shown in Fig. 2 for J (+) p . For this figure, as for all the following ones, parameters are

L = 8, φ = 2π/L, |α| 2 = 100, g/ω m = 2•10 -3 , γ c /ω m = 1•10 -1 , γ m /ω m = 1•10 -3 ,
and n = 100. The incoherent part of the effective dynamics is given by Eq. ( 6) by substituting γ

( ) m , n → γ m , n; β(↓) , β(↑) → dk , d † -k ; Γ (+) , Γ (-) → Γ k (+ω m ), Γ k (-ω m ).
The Fourier modes being defined as dk = 1

√ L e -ik d with k ∈ {n × 2π/L} L-1 n=0 and Γ k (ω) = |G| 2 γ c (ω + J cos(k + φ) + ∆) 2 + (γ c /2) 2 . ( 10
)
In contrast to the single resonator case [32], our system has L Stokes sidebands at ∆ (-) k = ω m -J cos(kφ) and L anti-Stokes sidebands at ∆ (+) k = -ω m -J cos(kφ), that can be employed to respectively amplify or cool collective mechanical modes. In Fig. 3 we show the k-space asymmetry between the incoherent gain and loss rates for φ = 0 around the lowest Stokes and anti-Stokes sidebands. Depending on the detuning, the engineered optical reservoir acts onto the system either by absorbing collective excitations with pseudomomentum k ∼ -φ (jump operator dk ) or by creating excitations with opposite momentum k ∼ +φ (jump operator d † -k ). Let us stress that this is not the result of the optical driving being at resonance with any particular k mode as it holds when the dissipation rate is of the order of the width of the optical lattice's spectrum (J ∼ γ c ). In such a regime, the concept of resonance has no longer any operative meaning.

Let us now investigate the steady state properties of this effective model by diagonalizing the Liouvillian in the Fourier mode basis as Ĥeff

m = k ω k d † k dk and D eff m ρm = k Γ (↓) k D[ dk ]ρ m + Γ (↑) k D[ d † -k ]ρ m , with ω k = ω m + ± |G| 2 (±ω m + ∆ + J cos(k ± φ)) (±ω m + ∆ + J cos(k ± φ)) 2 + (γ c /2) 2 , (11) Γ (↓) k = γ m (n + 1) + Γ k (+ω m ) ; Γ (↑) k = γ m n + Γ k (-ω m ). ( 12
)
As can be seen in Eqs. ( 11) and ( 12), both the unitary and the dissipative parts of the Liouvillian are no longer even in k space for finite φ, as a result of having explicitly broken the parity symmetry of the coupling to the reservoir. In particular, with a driving laser operated around the lowest anti-Stokes sideband ( ∆ ≈ -Jω m ) and within the resolved sideband regime (γ c ω m ), to first order in J/γ c the system has a noneven dispersion relation of the form ω k = cst. + 4(|G|/γ c ) 2 J cos(k + φ) with a ground state at finite momentum k GS = +φ.

Permanent cavity-mediated directional heat current -The optical mean-field phase gradient yields a permanent directional heat flow around the ring of disks. Indeed, as discussed in the Supplemental Material

C = - k 1≤p< L/2 ± J (±) p sin(p(k ± φ)) d † k dk . ( 13
)
The expectation value of this operator can be determined experimentally by measuring the thermal populations d † k dk , for example, via the mechanical noise spectrum around the L collective mechanical frequencies ω k measured at the output of some local resonator. For example in optomechanical disk resonators a secondary optical mode, such as a higher-order whispering gallery mode of the disk, could be used for that purpose. For our effective model, we get:

C ss = - k 1≤p< L/2 ± J (±) p sin(p(k ± φ)) Γ (↓) k /Γ (↑) -k -1 . (14)
The net permanent heat current whirling around the ring is thus simply Q C = ω m C ss . The amount of this heat transported over a phonon lifetime is shown in single phonon energy units in Fig. 4 as a function of ∆/ω m and J/γ c . Its sign (propagation direction) depends crucially on the detuning. Indeed, the effective coherent coupling can be regarded as an optical spring effect in k space and, as such, it changes sign when crossing a sideband. In Fig. 5 (a), we show the behavior of contributions

Q → +p = ω m
 → +p to the total flow as a function of J when the detuning ∆ is adjusted to follow its maximum (dash-dotted line of Fig. 4). Interestingly, Q C is nonmonotonic in J/γ c . For J γ c , optical fluctuation quanta mediating the heat transport are short lived (τ c 1/J) and are thus destroyed before reaching sites farther than their nearest neighbors. This implies that the only sizable contribution is that flowing by local steps in the clockwise direction. Conversely, for J γ c , optical fluctuation quanta can be scattered farther across the optical lattice before being destroyed by the cavity losses and the permanent heat flow is supported on supplementary directed graphs (see Fig. 5 (b)). In this case, a nonlocal anticlockwise flow contributes to the nonmonotonic dependence on J/γ c of the net current. Fig. 5 (c) shows how longer-range correlations get gradually triggered as the J/γ c ratio is increased following the lowest anti-Stokes sideband (see arrow in Figure 4).

Q/ω m × γ -1 m Q → +1 Q C Q → +2 Q → +3

Conclusion -

We have studied the emergence of spatial correlations and permanent directional heat currents across lattices of optomechanical resonators whose mechanical modes are originally uncoupled. In our picture, quantum fluctuations of the optical fields mediate effective long-range interactions between mechanical sites of both coherent and dissipative nature, whose range is tunable via the correlation length of the reservoir. A remark-able feature is the possibility to flow arbitrary phonon streams in directions and topologies that seem to contradict common thermodynamic intuition, for example, a permanent phonon heat flow can be generated in the absence of thermal gradient.

More generally, our investigation provides a first instance of a broader class of physical situations for which a weak coupling to an extended reservoir suffices to alter dramatically the fate of an initially trivial set of inde-pendent modes. The here presented effective description introduces an analytical tool for understanding quantum systems interacting via extended close-to-Markovian reservoirs, a realm yet to be fully explored. by computing the error δ = σ effσ MF 2 / (σ eff + σ MF )/2 2 . As shown in Fig. 5 (a) and (c) of the main text, and Fig. S1, the analytical results obtained from the effective theory match the numerical solution of the linearised dynamics in a wide regime of parameters.

III. ADDITIONAL REMARKS ON THE PECULIAR FEATURES OF OUR PROPOSED OPTOMECHANICAL CONFIGURATION AND THEORETICAL EFFECTIVE DESCRIPTION

In our letter, we have examined peculiar heat transport properties mediated by correlated fluctuations of a lattice of optical modes. We have shown that the weak coupling of initially-uncoupled modes to a common Markovian reservoir with finite correlation length, a situation relevant beyond the specific optomechanical system hereby described, induces dissipative as well as coherent processes that drastically alter the fate of the system. In particular, in a ring geometry, we identify permanent gauge currents whirling around the ring of optomechanical resonators. This result is original as the generation of these currents happens in the absence of any direct coupling between resonators and in the presence of thermal relaxation with local baths at non-cryogenic temperatures. In this scenario, the action of the extended reservoir is twofold: it builds coherence between distant thermal modes and allows for parity-breaking scattering events between distant modes. In this section, we discuss more details concerning the differences of this configuration with respect to previous interesting works in the literature.

The coupling of N independent mechanical modes to N -1 independent optical ones was considered in Refs. [16,17] and shown to allow one to generate reconfigurable interactions between distant resonators with great flexibility. In this setup, each "reservoir" optical mode couples to all mechanical ones with the same phase. The engineered interaction between the various mechanical modes is thus symmetric, Ĥeff = ij S ij b † i bj , with S ij = S ji ∈ R, and thus generates no gauge heat currents, which are the central point of our work.

Ref. [13] proposes two methods, namely (i) to time-modulate the coherent mechanical populations of local mechanical modes or (ii) to implement a wavelength conversion scheme in order to generate a synthetic gauge field for a lattice's photons instead of phonons. Apart from the difference in the nature of the bosonic carriers, approach (i) is completely different from our configuration that does not require modulation of the populations. Both approaches (i) and (ii) in Ref. [13] would require the presence or the engineering of two-site direct couplings between the optical modes of the lattice in order to generate a photon current. In contrast, in this work no direct mechanical coupling is involved in phonon transport.

In Ref. [5], the authors give a detailed description of persistent currents across spin chains. In that reference, in contrast to our model, the effect is achieved through a proper reservoir engineering of two-site non-reciprocal couplings [6] between adjacent lattice sites.

In Ref. [31], the authors examine singular transport properties across an open chain of optomechanical resonators with a gradient of optical mean-field phases similar to the one of our letter but with nearest-neighbor coupling between both optical and mechanical modes. In that work there is a direct phonon-phonon coupling, which is responsible for the peculiar tilting of the band structure of the chain. In their setup nonreciprocal transport properties show a crucial dependence on the asymmetric gaps of the system, around which excitations have a hybrid phonon-photon nature, as opposed to the system described in our letter where the gap is not resolved (|gα| < γ c ) and the transported excitations are of pure phononic nature.

Furthermore, the theoretical method we have applied for the proposed configuration differs significantly from Refs. [5,13,31] by introducing the concept of extended reservoir. Indeed, we have first provided a general description of the effective dynamics of a set of system modes in local contact with an extended reservoir (a situation that is not restricted to optomechanics) in terms of nonlocal coherent interaction terms as well as nonlocal dissipative processes. We have shown the generality of the approach in Section V of the present SM by deriving an effective many-mode squeezing Liouvillian for initially uncoupled mechanical modes from the elimination of a two-tone driven lattice of optical cavities. This description allows us to derive the analytical expression of the current circulating around the ring as a function of the optical phase gradient and the system parameters.

IV. CONTINUITY EQUATION IN A GENERAL SETTING

In the effective model discussed in the main text, all sites are potentially mutually coupled via the reservoir so the current operators are to be carefully defined.

To do so, let us consider a lattice system as defined by some graph G = (Σ, E) with vertices Σ, edges E and one of its subsystems G[Ω] defined as the subgraph supported on some subset of vertices Ω ⊆ Σ. Furthermore, let us denote ÔΩ = i∈Ω Ôi some extensive observable Ô. Then, ÔΩ must satisfy a continuity equation of the form

∂ t ÔΩ = - (i,j)∈∂Ω O i→j + σO Ω = L † ÔΩ , (S17)
where the boundary ∂Ω of the subsystem G[Ω] was defined as the set of edges ∂Ω = {(i, j) ∈ E : i ∈ Ω, j / ∈ Ω} directed from the subsystem to the rest of the system. σO Ω is a source term and L † is the adjoint Liouvillian [START_REF] Breuer | The Theory of Open Quantum Systems[END_REF] driving the operator dynamics. 1≤p< L/2  → +p as done in the main text which only vanishes in the absence of permanent currents, thus serving as a witness of existence of permanent currents.

V. EFFECTIVE MULTI-MODE SQUEEZING FROM A TWO-TONE-DRIVEN EXTENDED RESERVOIR

Let us consider the general Liouvillian defined in Eqs. ( 1) and (3) of the main text. By a two-tone driving of the cavities so as to have α i (t) = α (+) i e -iω (i) m t + α (-) i e +iω (i) m t (in the frame rotating at ω p ) as optical mean-field solutions,

FIG. 1 .

 1 FIG. 1. (a) Schematic representation of a single optomechanical cell and its nearest-neighbor optical couplings. â ( b ) is the optical (mechanical) mode of index . (b) Ring of optomechanical disk resonators. Each site is optically driven with a phase that varies as φ, being the site number. Optical modes are coupled while mechanical ones are not.

FIG. 2 .

 2 FIG. 2. Left panel: absolute value of the effective coherent coupling rate J (+) p between mechanical sites as a function of their distance p and the nonlinear detuning ∆. The hatched regions correspond to negative values. Three horizontal section cuts (labeled as A, B, C) are plotted in the right panel. Here J/γc = 2.

8 FIG. 3 .

 83 FIG.3. Gain Γ -k (-ωm) (dashed) and loss Γ k (+ωm) rates induced by the engineered reservoir for various J/γc. ∆ = +ωm -J for the gain rate and ∆ = -ωm -J for the loss rate.

  [28], the continuity equation satisfied by the phonon number operator i[ Ĥeff m , d † d ] =p ( → -p +  → +p ), with  → +p = -± J (±) p 2i ( d † +p d e ∓iφp -H.c.), induces the following definition for a net circulating current operator: C = L =1 1≤p< L/2  → +p . In k space, it reads:

FIG. 4 .

 4 FIG. 4. Absolute value of the predicted net permanent heat current |QC | in units of ωm × γm around a ring of cavitycoupled optomechanical resonators as a function of the intercavity coupling and the detuning. The hatched regions correspond to negative values. The system is unstable in the gray region.

FIG. 5 .

 5 FIG. 5. (a) Contributions Q → +p = ωm  → +p and net directional heat flow QC = p< L/2 Q → +p along the dash-dotted line of Fig. 4 as predicted by our effective theory (lines) and mean field (circles). (b) Sketch of the two leading contributions in (a). (c) Gradual triggering of off-diagonal coherence g (1) , +p = d † d +p /( d † d d † +p d +p ) 1/2 along the lowest anti-Stokes sideband ∆ = -J -ωm.

  FIG. S1. Imaginary part of the steady-state single-particle density matrix predicted by the effective theory as given by Eq. S16 for L = 8, φ = 2 × 2π/L, |α| 2 = 100, ∆ = -J -ωm, g/ωm = 1 • 10 -2 , γc/ωm = 1 • 10 -1 , γm/ωm = 1 • 10 -3 , n = 10 and J/γc = 1 (a) and J/γc = 5 (b). Relative errors δ are 1.0% (a) and 0.7% (b). (c) δ as a function of the effective optomechanical coupling and the inter-cavity coupling for L = 8, φ = 2π/L, |α| 2 = 100, ∆ = -J -ωmγc/2, γc/ωm = 1 • 10 -1 , γm/ωm = 1 • 10 -3 and n = 100. δ ≥ 5% in the dashed region. (d) Relative error δ associated to the Fig. 4 of the main text.

  Now let us split the adjoint Liouillian as L † = L O † C +L O † NC into a part conserving the total amount of Ô (L O † C ÔΣ ) and the rest L O † NC = L † -L O † C . By definition, ∂Σ = {∅} so the current contribution vanishes for Ω → Σ (∂ t ÔΣ = 0 + σO Σ ). Therefore, ∀Ω ⊆ Σ: discussed hereby, once linear terms are absorbed into a static coherent displacement, the Louvillian conserving the total internal energy ÛΣ= i∈Σ ω m d † i di is the effective Hamiltonian L U † C • = i[ Ĥeff m ,• ] while the effective dissipator acts as a source/sink. FromL U † C Û = -1≤p<L ( U → +p + U → -p ),(S19)one obtains the definition of heat current used in the main text U→ +p = ω m  → +p = -± J (±) p 2i ( d † +p d e ∓iφp -H.c). Moreover, if L U NC ρm (t → +∞) = 0,as is the case in the main text, then lim t→+∞ Tr[ρ m L U † C Û ] = 0, i.e. either U → ±p t→∞ = 0 (no permanent current) or U → +p t→∞ = -U -p→ t→∞ (permanent current). To discriminate between these two cases, one can define a directional circulating current U C = ω m L =1

We thank D. Rossini for discussions. This work was supported by ERC via Consolidator Grants NOMLI No. 770933 and CORPHO No. 616233, and by ANR via the project UNIQ.

Supplemental Material: Permanent directional heat currents in lattices of optomechanical resonators

I. ADIABATIC ELIMINATION OF EXTENDED DRIVEN-DISSIPATIVE RESERVOIRS A. Discrete reservoir

Let us consider some generic system S whose dynamics is governed by a Liouvillian L S and a reservoir R described by a Liouvillian L R . The corresponding two sets of degrees of freedom are weakly coupled via a Hamiltonian term of the form V = λ i Ri ⊗ Ŝi where λ is a scale bookmark and where { Ri } i and { Ŝi } i act respectively on the reservoir and on the system Hilbert spaces. We associate sets of ladder operators {â i } i and { bi } i to the reservoir and the system, respectively. The vacuum is then displaced to a stable mean-field solution {α i , β i } i towards which the reservoir is driven

)) and the resulting Liouvillians are expanded to second order in the reservoir's ladder operators. By construction, the reservoir becomes thermal-like, in the sense that all its correlation functions decay exponentially in time, and can be traced-out by means of the Born-Markov procedure [START_REF] Breuer | The Theory of Open Quantum Systems[END_REF]. We first perform a Born approximation, by assuming that the state of the reservoir and the system, initially ρ(t 0 ) = ρR (t 0 ) ⊗ ρS (t 0 ), remains separable upon time-evolution ρ(t) ≈ ρR (t) ⊗ ρS (t). This yields first the following non-Markovian master equation for the reduced density matrix of the system ρS (t

where it was assumed for simplicity that V (t → +∞) = 0 (e.g. V normal-ordered in {â i } i ) and where

The limit in the last line is taken as we only aim at describing the dynamics at a timescale larger than the relaxation time of the reservoir. For consistency, we also assume that the system was weakly coupled to its environment so that

We then perform a Markov approximation by neglecting the effect of the reservoir on the system on times of the order of the reservoir's correlation time τ c , yielding

By decomposing the coupling operators in the eigenoperator basis of ĤS as Ŝi = α ŝi (ω α ), where the {ŝ i (ω)} i are such that [ ĤS , ŝi (ω)] = -ωŝ i (ω) and {ω α } α denotes the set of all possible transition energies between eigenstates of the system Hamiltonian ĤS , and, by neglecting counter-propagating terms (consistent with the assumption made in the previous paragraph), one finally gets

Its Lindblad form can be made explicit by identifying the Hermitian and anti-Hermitian components of the reservoir spectrum

in terms of which Eq. (S5) reads:

Assuming that the reservoir fluctuations remain Gaussian, for any choice of { Ri } i , S (α) can be computed from the covariance matrix:

where  = [. . . , âi , â † i , . . .] T and B is the Bogoliubov operator defined by the Bogoliubov-like equation i∂ t  = B Â. In particular, for some generic linear form Ri = t i âi + H.c. as in the main text, one obtains:

where T is given by the direct matrix sum T = Diag . . . , t i ti , . . . . In the main text, no squeezing of the reservoir's fluctuations on top of mean-field was considered so that G(τ

, where the simpler Bogoliubov operator was such that i∂ t âi = B ij âj .

B. Continuous reservoir

The same procedure can be applied to continuous reservoirs. For example, let us consider the case of some translational-invariant reservoir defined by the squeezed Gaussian fluctuations of a free condensate around its meanfield solutions:

where the modes Ψ(r) = [ ψ(r), ψ † (r)] T satisfy some Bogoliubov equation i∂ t Ψ(r) = dr B(rr ) Ψ(r ) [36], where the Bogoliubov operator typically carries some dependence on the mean fields accounting for the nonlinearity of the model. This continuous set of degrees of freedom is put in contact with some discrete set of mechanical modes via an interaction Hamiltonian

with some general choice of local coupling R(r) = t T (r) • Ψ(r). Under the above-discussed approximations, the system's effective master equation has the same expression as in the previous subsection, the only difference being the expression of the reservoir spectrum S (α) ij = R+ dτ e iωατ G(r i , r j ; τ ), which here takes the form of the following convolution:

The covariance C(r; 0) = Ψ(r) ΨT (0) is to be evaluated from the steady-state mean-field solution and B(k) = dre -ik•r B(r).

II. BENCHMARKING THE EFFECTIVE DESCRIPTION

In order to benchmark our effective description, we compute the exact steady-state covariance matrix of both optical and mechanical fluctuations for the linearised model described by Eq. ( 4) and extract the exact mean-field single-particle density matrix σ mn = d † m dn ss as given by:

where φ(t) = [ĉ 1 (t), ĉ † 1 (t), . . . , d1 (t), d † 1 (t), . . .] T , and compare it to the single-particle density matrix of the effective description given explicitly by

in the spirit of reference [37], the coupling Hamiltonian in Eq. ( 4) reads in the interaction picture of the mechanical modes:

where subdominant terms (O(gĉ † ĉ d)) were neglected. As in the above reference, we define G (±) i = g i α (±) i and take |G (±) i | = G (±) , ∀i. By neglecting the rapidly varying last term and defining mechanical Bogoliubov modes βi = cosh re iθi di + sinh re iϕi d † i , where r = G (+) /G (-) , θ i = Arg(G (+) i ) and ϕ i = Arg(G (-) i ), the effective coupling becomes -η ĉi β † i + H.c. with η = √ G (-)2 -G (+)2 . By using the result of the main text, one gets the following effective Liouvillian by tracing out the optical degrees of freedom:

with

and

For instance, by having J γ c so that next-to-nearest-neighbors terms can be dropped (see Fig. S1 (a)) and choosing the phase of the drives so as to have θ +1θ = ϕ +1ϕ = α +1, [2π] and θϕ = ν, we obtain:

The system is thus subject to multi-mode squeezing as long as the system remains stable. By combining this with the engineered complex tight-binding interaction, one could in principle obtain a dissipative version of the bosonic Kitaev-Majorana described in reference [14].


Morgane Daurat

Christophe Nguyen

Sofia Dominguez Gil

Vincent Sol

Vincent Chaleix

Clarence Charnay

Laurence Raehm

Khaled El Cheikh

Alain Morère

Michele Bernasconi

Andrea Timpanaro

Marcel Garcia

Frédérique Cunin

Jochen Roessler

Jean-Olivier Durand

Magali Gary-Bobo

Mannose 6-phosphate receptor targeted with porphyrin-based periodic mesoporous organosilica nanoparticles for Rhabdomyosarcoma theranostic

Recently, we described porphyrin-based periodic mesoporous organosilica (PMO) nanoparticles synthesized from a large functional octatriethoxysilylated porphyrin precursor. The framework of the nanoparticles was formed by J-aggregates of porphyrins allowing two-photon excitation photodynamic therapy (TPE-PDT) and NIR imaging. In the present work, these PMO were grafted with polyethylene glycol (PEG) moieties and an analogue of mannose 6-phosphate functionalized on anomeric position (AMFA). AMFA are known to efficiently target mannose 6-phosphate receptors (M6PR) which are over-expressed in various cancer cell lines (breast, prostate). Here, we demonstrated that M6PR was also over-expressed in rhabdomyosarcoma cells (RMS) and could be efficiently targeted with PMO-AMFA allowing TPE imaging and TPE-PDT of RMS cells. Furthermore the same treatment did not affect healthy myoblasts which do not over-express M6PR suggesting a specificity in its biomedical action.

Introduction

Rhabdomyosarcomas (RMS) are the most common soft tissue sarcomas of childhood. There are two major subtypes, the embryonal and the alveolar (most aggressive) RMS. Despite intensified, multimodality treatments, the overall survival of high-risk population has remained at 5% to 20% over the last decades. This rate is insufficient, therefore new innovative selective treatments based on nanomedicine are urgently needed. Very few examples of nanoparticles for treatment of RMS have been described in the literature. Abraxane and liposomal vincristine have been investigated in RMSxenografted mice, 1, 2 magnetic drug targeting was reported in rat models of RMS, 3 chitosan nanoparticles have been used

to silence TGF1 (transforming growth factor) in cells with small hairpin RNA 4 or mesoporous silica nanoparticles functionalized with polyethyleneimine have been used in cells to silence ELMO1, a protein involved in cell migration.

None of these nanoparticles were functionalized with a biomolecule to target RMS. Targeting an overexpressed receptor in cancer research is very important to differentiate between malignant and normal tissues. Gold nanoparticles have been functionalized with TAT peptide to target RMS cells, and induced cell death 6 but TAT peptide is not specific for RMS.

Liposomes loaded with tubulin binding epothilone B agent were developed in order to target RMS. These liposomes have been shown to inhibit endothelial and tumor cell viability to the same extends as the free drug. Furthermore, potent antitumor growth activity could be demonstrated in vivo by using vascular endothelial cell targeting. Indeed the RGD peptide was attached on the surface of the liposomes. These liposomes interact with the integrin receptor and show a beneficial effect in comparison to the untargeted formulation, by increasing cumulative survival of mice bearing RMS.

However, the RGD peptide is not sufficiently specific for RMS as integrins are also over-expressed in all the vascular endothelial cells of children. Identification of cation-independent mannose 6-phosphate receptor (M6PR) overexpression on prostate cancer cells 8 led us to study the expression level of M6PR on other types of cancer such as RMS. In the present paper, overexpression of M6PR on RMS cells compared to healthy myoblasts was studied and confirmed. We therefore decided to target RMS cells using M6PR pathway and more particularly using analogues of mannose 6-phosphate functionalized on anomeric moiety (AMFA) grafted on nanoparticles.

We recently synthesized very efficient porphyrin-based periodic mesoporous organosilica nanoparticles (PMOsPOR NPs, 250 nm diameter) showing interconnected cavities (from 10 to 80 nm) for breast cancer treatment using two-photon excited photodynamic therapy (TPE-PDT). 9 For this, a large functional octatriethoxysilylated porphyrin precursor was condensed without any silica source. The framework of the nanoparticles was formed with J-aggregates of porphyrins inside the pore of the walls allowing two-photon excitation (TPE). 10 TPE is of particular interest for biological applications because it allows a deep penetration of the near-infrared beam down to 2 cm in soft tissues and a high spatiotemporal resolution for imaging and cancer detection. Importantly, TPE-PDT has demonstrated a high potential for cancer therapy, in particular, the treatment of small-sized tumor.

11

Therefore, we presented here the combination of PMO with AMFA to target M6PR overexpressed in RMS cells of both phenotypes, embryonal and alveolar subtypes for TPE-PDT of RMS. The strong therapeutic potential and selectivity of PMO grafted with PEG and AMFA were demonstrated.

Results and discussion

Overexpression of M6PR on RMS cells

The first step of this work was the study of M6PR expression on RMS cells with embryonal and alveolar subtype and healthy myoblasts. For this, the protein expression level of M6PR in 3 different RMS cells lines and in healthy myoblasts was analysed by Western blot (Figure 1). As showed in Figure 1A, the protein expression level of M6PR is clearly higher in RMS cell lines than in healthy myoblasts.

The quantification of M6PR level by Image J software corrected by the expression level of an invariant (β-actin) indicated that in mean, there was a 7 fold-higher M6PR expression in RMS cells than in healthy cells. To confirm and to expand these data, we performed proteomic analysis of the membrane fraction of several RMS cell lines. Healthy myoblasts were used as control (Figure 1C). These results confirmed a higher expression of M6PR in RMS cell lines of both alveolar and embryonal subtype. This suggested that the increase in M6PR expression could be considered as a biomarker of RMS development.

Figure 1. M6PR expression in different cell lines. (A)

The cell extracts (25 µg) were analyzed by Western blots using human M6PR or β-actin antibodies. β-actin is a control for total protein loading. (B) Quantification with Image J software of 300 kDa M6PR protein corrected by β-actin expression. RMS-YM cells are considered as 100%. (C) Proteomics analysis of M6PR expression in a panel of RMS cell lines. Membrane proteins were enriched with two-step centrifugation protocol and analysed by HPLC-MS. iTop3 values were calculated from the sum of the intensity of the 3 most intense peptides of each Leading Razor Protein.

Synthesis of PMO grafted with PEG and AMFA

Porphyrin-based PMO nanoparticles were first synthesized following our recently described procedure (Figure 2A). 9 We then investigated the functionalization of PMO with AMFA (Figure 2B). For this, aminoundecyltriethoxysilane and PEG triethoxysilane were first grafted on the surface of PMO following a method we published on mesoporous silica nanoparticles (MSN) of 20 nm diameter. 19 Then AMFA was grafted on the surface of functionalized PMO. Here, the AMFA used was the M6C-PhSq (Figure 2B), and the reaction of its squarate moiety with the amino groups led to a covalent attachment of AMFA on the surface of PMO. The different steps of the functionalization were monitored by DLS and zeta potential. PMO nanoparticles showed a negative zeta potential (-38.3 mV, after extraction of the surfactant) in agreement Si-O -groups on the surface. The DLS showed 245 nm hydrodynamic diameter. After functionalization with NH 2 groups and PEG, the hydrodynamic diameter increased to 310 nm and zeta potential was +18.3 mV (after extraction of the surfactant). The increased diameter and charge reversal showed the successful functionalization of PMO-NH 2 . AMFA grafting level was quantified by HPLC demonstrating that 66 µg of AMFA were grafted per mg of PMO. This reaction led to a 399 nm hydrodynamic diameter (Figure 2C) and a negative zeta potential of -22.9 mV in agreement with the anchoring of the carbohydrate moiety on the surface of the nanoparticles. TEM imaging showed homogenous dispersion and a large amount of small-sized nanoparticles (Figure 2D).

Biocompatibility of PMO on different cell lines

A cytotoxic study was carried out with increasing concentrations of PMO grafted with PEG and AMFA, for a 3 days incubation time. This study was performed on one RMS and one healthy cell line: RMS-YM and SK-1111, respectively. These data demonstrated that for both cell lines, the batch of PMO, which were grafted with PEG and AMFA (PMO-AMFA), did not show any cytotoxicity (Figure 3). In contrast, pristine PMO (here called PMO) and PMO-PEG/NH 2 generated slight cell death. In fact, in RMS-YM cells, the incubation with PMO at 100 µg.mL -1 induced 30% cell death (Figure 3A). In SK-1111, PMO and PMO-PEG/NH 2 at 100 µg.mL -1 induced approximately the same percentage of cell death (Figure 3B). However, up to 40 µg.mL -1 no significant cell death was observed in any cell lines. So this concentration was chosen for further biological experiments.

Study of TPE-PDT and fluorescence imaging potential

The biological activity and more particularly the ability of PMO-AMFA to target and treat RMS cells under near infrared two-photon excitation was studied (Figure 4). For this RMS-YM cancer cells and SK-1111 healthy cells were incubated with 40 µg.mL -1 of PMO, PMO-PEG/NH 2 or PMO-AMFA for 16 h. Then, the cells were irradiated or not at 800 nm for a short time (3 x 1.57 s) at maximal laser power (Figure 4A-B). PMO and PMO-PEG/NH 2 induced 62% and 57% RMS-YM cell death, respectively, and 50% and 51% on SK-1111 healthy cells. Interestingly, PMO-AMFA was highly efficient on cancer cells by inducing 94% RMS-YM cell death after TPE. In contrast, PMO-AMFA did not induce any significant phototoxicity on healthy cells. This result demonstrated the importance of the targeting and the specificity of PMO-AMFA for RMS.

The M6PR involvement in the active endocytosis of PMO-AMFA was demonstrated by the addition in culture medium of M6P, which is M6PR natural ligand. This experiment is based on a competitive inhibition of PMO-AMFA endocytosis by M6P. RMS-YM cells were incubated for 4 h with PMO-AMFA in the presence or absence of M6P excess (10 mM). Under TPE-PDT PMO-AMFA induced 58% cancer cell death. Importantly, the addition of M6P inhibited totally RMS-YM cell death suggesting that M6P prevented the internalization of PMO-AMFA through the M6PR pathway (Figure 4C). This result was confirmed by two-photon fluorescence imaging demonstrating an almost total disappearance of luminescent red dots of PMO-AMFA, in presence of M6P (Figure 4D). To study the subcellular localisation of PMO-AMFA in RMS-YM cells, a confocal imaging experiment on living cells was performed with 2 different incubation times (4 h and 16 h), in presence of cell membrane (CellMask) and nucleus (Hoechst) staining. The figure 5A showed that 4 h incubation time was sufficient for PMO-AMFA detection mainly at cell membranes. After 16 h incubation time, PMO-AMFA were, largely internalized inside the cells. This observation was confirmed by flow cytometry experiments of RMS-YM cells incubated with 40 µg.mL -1 PMO-AMFA for 1 to 24 h. Figure 5B showed the increasing amount of stained cells, depending on the increase of the incubation time.

Experimental Material

Cetyltrimethylammonium bromide (CTAB, 99%), sodium hydroxide (NaOH, 97%), toluene, and ammonium nitrate (NH 4 NO 3 ) were purchased from Sigma-Aldrich. Ethanol (EtOH) was purchased from Carlo Erba. Aminoundecyltriethoxysilane and PEGtriethoxysilane were purchased from SIKEMIA. p-[N-(2-Ethoxy-3,4-dioxocyclobut-1-enyl)amino]phenyl 6-deoxy-7-hydroxycarbonylα-D-manno-heptopyranoside [M6C-PhSq] was synthesized as described by E. Bouffard et al. 12 PMO were synthetized following the protocol previously described by C. Mauriello Jimenez et al.

8

Analytical techniques

UV-vis absorption spectra were recorded on a Hewlett-Packard 8453 spectrophotometer using correction factors supplied by the manufacturer. TEM analysis was performed on a JEOL 1200 EXII instrument. Dynamic light scattering analyses were performed using a Cordouan Technologies DL 135 Particle Size Analyzer instrument and analyzed with NanoQ software. IR spectra were recorded on a Perkin-Elmer 100 FT spectrophotometer. Zeta potential measures were performed with a Malvern Zetasizer NanoSeries Instrument.

Synthesis of octasilylated porphyrin derivative

A mixture of the octapropargyl porphyrin derivative (100 mg, 9.0 × 10 -2 mmol), bromotris(triphenylphosphine)-copper(I) ([CuBr(PPh3)3], 13 mg, 1.5 × 10 -2 mmol), and anhydrous THF (3 mL) was placed in a 10 mL sealable microwave reactor, and 3azidopropyltriethoxysilane (178 mg, 7.1 × 10 -1 mmol) was added. Then, the tube was flushed with argon and the microwave irradiation was conducted for 30 min at 100°C (maximum power 200 W). After evaporation of the solvent, octasilylated porphyrin was quantitatively obtained as a purple solid (225 mg, 7.25 × 10 -2 mmol).

Synthesis of Porous Porphyrin-Based Organosilica Nanoparticles (PMO)

A mixture of CTAB (120 mg, 0.39 mmol), distilled water (60 mL), and NaOH (0.2 M aqueous solution, 437 µL) was stirred at 80°C for 2 hours at 750 rpm in a 250 mL three-neck round bottom flask. Then, the octasilylated porphyrin (55 mg, 0.018 mmol, in 1 mL of absolute ethanol) was added, and the mixture was stirred for 30 hours at 80°C. Afterwards, the solution was cooled to room temperature while stirring. The crude mixture was centrifuged (20 000 rpm, 20 min). The supernatant was removed, and the PMO were washed with ethanol and were stored at 4°C. After extraction of the surfactant from a sample, the zeta potential was -38.3 mV (DLS, 245 nm, H 2 O).

Synthesis of PMO-PEG/NH 2 nanoparticles

After centrifugation, PMO (48.8 mg) were resuspended in 6 ml of toluene at 100°C. Then a mix of 11-aminoundecyltriethoxysilane (29.3 mg -87.8 µmoles) and PEG-triethoxysilane (36 mg -87.8 µmoles) was dissolved in 1 mL of toluene and added to the NPs. 20 µL of water were added to the reaction. The reaction was kept under stirring overnight at 100°C. Then, it was cooled down to room temperature and centrifuged for 30 minutes at 20 000 rpm. The supernatant was removed and the solid was washed with EtOH leading to 40 mg of grafted PMO-PEG/NH 2 in EtOH. DLS measurement in water showed a hydrodynamic diameter of 443 nm. Zeta potential in water gave a value of + 36.5 mV. After extraction of the surfactant from a sample, the zeta potential was +18.3 mV showing the functionalization. (DLS, 310 nm, H 2 O).

Synthesis of PMO-AMFA nanoparticles

10 mg of PMO-PEG/NH 2 were resuspended in ethanol at 50°C. A water solution (2 mL) of 3.6 mg of AMFA, more precisely a mannose 6-carboxylate with a phenyl squarate arm (M6C-PhSq), was added dropwise to the PMO. The reaction was kept under stirring at 50°C overnight. Then, the resulting mixture was cooled down to room temperature and centrifuged. The supernatant was then removed and the solid washed with a solution of ammonium nitrate in ethanol (6 g/L), water, and ethanol leading to 9.3 mg of grafted PMO-AMFA in EtOH. DLS measurement in water showed a hydrodynamic diameter of 399 nm. Zeta potential in water gave a value of -22.9 mV in agreement with the functionalization with AMFA.

AMFA grafting quantification

The amount of AMFA grafted on PMO was determined by HPLC measurement in the supernatant using projection with reference solution of AMFA at different concentrations. We calculated indirectly the grafting efficiency by separating the nanoparticles out of the suspension with ultracentrifugation at the end of the reaction, and measured AMFA concentration in the supernatant with HPLC. We performed a calibration curve with the HPLC and took a reading for our supernatant. One μL of solution of subsequent dilutions from 0.84 to 0.1 µg/mL of AMFA, prepared in water / ethanol, was injected to HPLC Agilent 1260 infinity on a Kinetex EVO column C18 1.7 µM 100 Å 50 x 2.1 mm. One µL of the supernatant was injected and gave us the quantification of non-grafted AMFA. The amount of grafted AMFA is 66 μg of AMFA per mg of PMO. The mobile phase consisted of a mixture of acetonitrile and water with trifluoroacetic acid and was eluted at a flow rate of 0.5 mL/min. The analysis was performed at 298 nm, and run time was about 11 min. The chromatographic data analysis was performed using Agilent OpenLAB Software. Linearity was determined through the construction of three calibration curves using five AMFA concentrations at 3 wavelengths specific of AMFA absorbance (298, 223 and 198 nm). Linear least squares methodology was applied to calculate the calibration equation and correlation coefficient.

Cell Culture

Rhabdomyosarcoma cell lines Rh30 were obtained from ATCC; Rh4, Rh18

Western Blotting

To study the expression level of M6PR, RMS-YM, RD, RMS, SK-1111 and MH38 were subjected to Western blot analysis. The cells were harvested, washed three times in PBS and lysed by three freeze-thaw cycles in buffer containing 50 mM HEPES, 150 mM NaCl, 1 mM EDTA, 2.5 mM EGTA, 0,1% Tween 20, 10% glycerol, 1 mM NaF, 1 mM NaNO 3 , 10 mM glycerophosphate and protease inhibitors (dilution 1:25 Complete, Roche Diagnostics). The lysates were precleared by centrifugation at 10,000 × g for 15 min at 4°C. Samples were tested for protein concentration by the Bradford method and equal amounts (25 µg) of cell extract and 1 µg bovine purified M6PR were resolved by 12% SDS-PAGE electrophoresis. After blotting, the gel was transferred onto PVDF membrane and M6PR protein was detected by probing with anti-human M6PR (cation independent) antibody (dilution 1:50000, Abcam). β-actin detected by a monoclonal mouse antibody was used as the loading control. Immunoblotting was performed using secondary antibody coupled with horseradish peroxidase and revealed by ECL detection system (Amersham).

Proteomic analysis

Membrane proteins were enriched from 80-90% confluent cells plated in six 15 cm dishes, following a two-step centrifugation protocol 13 . After gel separation, and in-gel digestion, peptides were loaded onto HPLC-MS 14 . Data acquisition was made in data dependent mode with precursor ion scans recorded in the Fourier transform detector (FT) with resolution ≥ 60000 (at m/z-400) and fragment spectra of the most intense precursor ions in an Orbitrap (QE). For protein quantification, we relied on the MaxQuant built-in label-free quantification (LFQ) algorithm 15 and also applied a top3 peptide approach 14,16 . For top3, all peptide form identifications within a sample set in the evidence output file from MaxQuant were median normalized before imputation of missing values from the normal distribution of LOG2-transformed peptides using a down shift of 1.8 and a width of 0.3 standard deviations, a left-censored imputation strategy set as default in Perseus software (version1.5.5.3) 17 . Missing value imputation was carried out when there were at least two peptide form identifications in all technical replicates from the same sample, otherwise the intensity was set to zero according to recommendations 18 . The three most intense peptide intensities were then summed to the protein group iTop3 intensity.

Cytotoxicity Measurement

For cytotoxicity analysis, RMS-YM and SK-1111 cells were seeded into a 96 well plate, 2000 cells per well in 200 µL of culture medium and allowed to grow for 24 h. Then cells were treated with increasing concentrations of nanoparticles (from 1 to 100 µg.mL -1

). Three days after treatment, a MTT assay was performed to determine the cell viability. Briefly, cells were incubated for 4 h with 0.5 mg.mL -1 of MTT (3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide; Promega) in media. The MTT/media solution was then removed and the precipitated crystals were dissolved in EtOH/DMSO (v/v). The solution absorbance was read at 540 nm in a microplate reader. All values are reported to the control values (without any treatment) which are considered at 100% of living cells.

Two-Photon Excited Photodynamic Therapy

RMS-YM and SK-1111 cells were seeded into a 384 multi-well glass-bottomed plate (thickness 0.17 mm) with a black polystyrene frame at a concentration of 1000 cells per well in 50 µL of culture medium, and allowed to grow for 24 h. Then, cells were treated with nanoparticles (40 µg.mL -1

) for 16 h. For M6P competitive inhibition assay, the cells were incubated or not with 10 mM of M6P for 10 min prior to the incubation with PMO-AMFA (40 µg.mL -1

) for 4 h. The cells were submitted or not to laser irradiation with the LSM 780 live confocal microscope (Carl Zeiss Microscope) at 800 nm by 3 scans of 1.57 s duration in 4 different areas of the well with a focused laser at a maximum laser power (lase power input 3W). The laser beam was focused by a microscope objective lens (Carl Zeiss 10×/0.3 EC Plan-Neofluar). After 2 days, the MTT assay was performed as previously described and was corrected according to the following formula: Abs control -2 × (Abs control -Abs nanoparticles).

Two-photon fluorescence imaging

RMS-YM cells were grown on tissue culture dish with cover glass bottom (FluoroDish from WPI) in complete culture medium. The next day, the cells were treated or not with 10 mM of M6P for 10 min, then the cells were incubated with PMO-AMFA (40 µg.mL -1 ) for 4 h. Fifteen minutes before the end of incubation, cells were loaded with Hoechst 33342 (5 µg.mL -1 , Invitrogen) for nuclear staining and with green CellMask (5 µg.mL -1 , Invitrogen, Cergy Pontoise, France) for membrane staining. Before visualization, cells were washed three times with cell media. Cells were examined under LSM 780 live confocal microscope (Carl Zeiss Microscope) and excited at 488 nm for membranes at 760 nm for nuclei and 800 nm for PMO-AMFA. All images were performed with a high magnification (63×/1.4 OIL DIC Plan-Apo).

Flow cytometry

RMS-YM cancer cells were seeded on 6-well plate during 24 h. Then, cells were treated or not with 40 µg.mL -1 of PMO-AMFA for 0, 1, 2, 4, 16 and 24 h. After treatment, cells were washed thrice with culture medium, harvested and centrifuged (1300 rpm, 5 min). Cell pellets were re-suspended in PBS enriched with CaCl 2 + , MgCl 2 + . Flow cytometry determination of PMO-AMFA positive cells was done by FACS Novocyte Flow Cytometer (ACEA Biosciences Inc.) with a minimum of 5 000 cells collected (excitation laser 561 nm; Filter: 660 ± 20 nm).

Statistical analysis

Statistical analysis was performed using the Student's t test to compare paired groups of data. A p-value of <0.05 was considered to be statistically significant.

Conclusions

In this study, we have demonstrated that M6PR was a new target for rhabdomyosarcoma. In fact, we have observed an over-expression of this receptor in a panel of RMS cells of different subtypes (embryonal and alveolar) allowing to differentiate between cancer cells and healthy cells. The targeting of M6PR was performed using AMFA and more precisely by the grafting of a carboxylate analogue of mannose 6-phosphate on the PMO. Altogether, the results presented here showed that PMO-AMFA were highly efficient for TPE-theranostic of RMS-YM cells by a mechanism involving M6PR dependent endocytosis. Moreover, PMO-AMFA did not show any effect on healthy cells demonstrating the specificity of the targeting toward cancer cells. Targeted TPE-PDT could be considered as a new promising therapeutic strategy for rhabdomyosarcoma.

Figure 2 .

 2 Figure 2. Synthesis and characterization of PMO-AMFA. (A) Synthesis of PMO. (B) Schematic representation of coupling of the mannose 6-carboxylate with a phenyl squarate arm (M6C-PhSq) also called AMFA in order to obtained PMO-AMFA. (C) DLS in intensity of PMO-AMFA. (D) Microscopy images of PMO-AMFA by TEM at different magnifications.

Figure 3 .

 3 Figure 3. Cytotoxicity of PMO on RMS and healthy cells. (A) RMS-YM cells and (B) SK-1111 healthy cells were treated with increasing concentrations of nanoparticles. After three days of incubation, a MTT assay was performed. Data are presented as (mean ± SEM), n=2.

Figure 4 . 1 )

 41 Figure 4. TPE-PDT and fluorescence imaging with PMO. (A) Cancer cells (RMS-YM) were incubated 16 h with PMO (40 µg.mL -1 ) and irradiated at 800 nm (3 x 1.57 s). Top of (A): pictures of living cells revealed by MTT reagent. Scale bar: 100 µm. Bottom of (A): quantification of cell death. (B) Healthy cells (SK-1111) were incubated 16 h with PMO (40 µg.mL -1 ) and irradiated at 800 nm (3 x 1.57 s). (C) RMS-YN were incubated for 4 h with PMO-AMFA (40 µg.mL -1 ) in the presence or absence of M6P (10 mM). (D) confocal fluorescence imaging of RMS-YM cells incubated with PMO-AMFA (40 µg.mL -1 ) in the presence or absence of M6P (10 mM). Scale bar: 10 µm. Data are mean values ± standard deviations from 3 independent experiments. *Statistically significant versus non irradiated PMO (p < 0.05 from Student's t test).

Figure 5 .

 5 Figure 5. PMO-AMFA internalization in RMS-YM cells. (A) Confocal imaging on living cells in presence of Hoechst (nuclei stained in blue) and cell mask (membranes stained in green) after 4 h and 16 h of incubation time. Scale bar: 10 µm. (B) Quantification of PMO-AMFA in RMS-YM cells by flow cytometry experiments after increasing incubation times (from 1 h to 24 h). Bar graphs are mean ± standard deviations of 2 experiments. *Statistically significant versus control (p < 0.05 from Student's t test).

Acknowledgments

Grafting quantification was performed with technical support of the Synbio3 platform, which is supported by GIS-IBISA ITMO Cancer (Plan Cancer 2014-2019). We acknowledge the imaging facility MRI, member of the national infrastructure France-BioImaging supported by the French National Research Agency (ANR-10-INBS-04, «Investments for the future»).

Conflicts of interest

There are no conflicts to declare.


domain: spi chim phys

text:
Valentina Nigro
email: valentina.nigro@uniroma1.it

Barbara Ruzicka
email: barbara.ruzicka@roma1.infn.it

Beatrice Ruta

Federico Zontone

Monica Bertoldo

Elena Buratti

Roberta Angelini
email: roberta.angelini@roma1.infn.it

Relaxation Dynamics, Softness, and Fragility of Microgels with Interpenetrated Polymer Networks

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

In the last decades many works have been focused on the study of the glass transition in complex systems aiming at understanding differences and similarities between structural glasses (SG) and colloidal glasses (CG) [START_REF] Poon | Phase separation, aggregation and gelation in colloid polymer mixtures and related systems[END_REF][START_REF] Zaccarelli | Colloidal Gels: Equilibrium and Non-Equilibrium Routes[END_REF]. In the case of structural glasses, as the liquid is rapidly quenched below its melting temperature, it undergoes a glass transition avoiding nucleation and crystallization. In colloids the transition is instead triggered by volume fraction φ or waiting time t w (signaled by an aging phenomenon) whose increase drives the systems in an out-of-equilibrium configuration. Changing control parameters like temperature, packing fraction or aging time, the dynamics of both structural and colloidal glasses slows down enormously and is accompanied by a dramatic increase of the characteristic relaxation time (τ(T ), τ(φ ), τ(t w )) by several orders of magnitude up to the glassy state. The most direct way to access microscopic information on the relaxation dynamics is to look at the evolution of the intensity autocorrelation functions which are well described by the Kohlrausch-Williams-Watts (KWW) expression F(Q,t) ∝ exp(-(t/τ(Q)) β ) where τ is an "effective" relaxation time and β measures the distribution of relaxation times and assumes generally positive values β <1 (stretched behaviour). Recently an anomalous dynamics, characterized by a shape parameter β >1 (compressed behaviour), has emerged in out of equilibrium states of different materials like colloidal glasses [START_REF] Bellour | Aging processes and scale dependance in soft glassy colloidal suspensions[END_REF][START_REF] Bandyopadhyay | Evolution of particle-scale dynamics in an aging clay suspension[END_REF][START_REF] Schosseler | Diagram of the aging dynamicsin laponite suspensions at low ionic strength[END_REF][START_REF] Angelini | Dichotomic Aging Behavior in a Colloidal Glass[END_REF][START_REF] Angelini | Glass-glass transition during aging of a colloidal clay[END_REF][START_REF] Kwaśniewski | Anomalous dynamics at the hard-sphere glass transition[END_REF][START_REF] Angelini | Non-diffusive dynamics in a colloidal glass: Aging versus rejuvenation[END_REF][START_REF] Pastore | Differential Variance Analysis: a direct method to quantify and visualize dynamic heterogeneities[END_REF], gels [START_REF] Cipelletti | Universal Aging Features in the Restructuring of Fractal Colloidal Gels[END_REF][START_REF] Guo | Gel formation and aging in weakly attractive nanocolloid suspensions at intermediate concentrations[END_REF][START_REF] Cristofolini | Synchrotron X-ray techniques for the investigation of structures and dynamics in interfacial systems[END_REF], supercooled liq-uids (14,[START_REF] Conrad | Correlated heterogeneous dynamics in glass-forming polymers[END_REF], metallic glasses [START_REF] Ruta | Atomic-Scale Relaxation Dynamics and Aging in a Metallic Glass Probed by X-Ray Photon Correlation Spectroscopy[END_REF][START_REF] Evenson | X-Ray Photon Correlation Spectroscopy Reveals Intermittent Aging Dynamics in a Metallic Glass[END_REF], polymeric systems [START_REF] Falus | Crossover from stretched to compressed exponential relaxations in a polymer-based sponge phase[END_REF][START_REF] Narayanan | Dynamics and internal stress at the nanoscale related to unique thermomechanical behavior in polymer nanocomposites[END_REF][START_REF] Srivastava | Complex dynamics in polymer nanocomposites[END_REF], ceramics [START_REF] Balitska | Stretched-to-compressedexponential crossover observed in the electrical degradation kinetics of some spinelmetallic screen-printed structures[END_REF] and investigated also in theoretical [START_REF] Bouchaud | Anomalous Relaxation in Complex Systems: From Stretched to Compressed Exponentials[END_REF][START_REF] Nicolas | Disordered systems and neural networks deformation and flow of amorphous solids: An updated review of mesoscale elastoplastic models[END_REF] and simulation [START_REF] Morishita | Compressed exponential relaxation in liquid silicon: Universal feature of the crossover from ballistic to diffusive behavior in single-particle dynamics[END_REF][START_REF] Bouzid | Elastically driven intermittent microscopic dynamics in soft solids[END_REF][START_REF] Pastore | Many facets of intermittent dynamics in colloidal and molecular glasses[END_REF][START_REF] Wu | Stretched and compressed exponentials in the relaxation dynamics of a metallic glass-forming melt[END_REF] works. The existence of stretched and/or compressed correlation functions is currently a very debated issue also faced in this work. Despite of many studies on the glass transition in SG and CG several questions are still open: Is the structural relaxation behaviour peculiar of the specific investigated system? How does it depend on the control parameters? What is the β parameter behaviour? What happens to the diffusive dynamics of the equilibrium liquid? Combining X-ray Photon Correlation Spectroscopy (XPCS) and Dynamic Light Scattering (DLS), we answer these questions investigating the behaviour of the structural relaxation time and β exponent for a soft colloid. Soft colloids are an interesting class of glass-formers that, at variance with hard sphere-like colloids, provide a good tunability of particle softness giving rise to unconventional phase-behaviours [START_REF] Ramírez-González | Glass transition in soft-sphere dispersions[END_REF][START_REF] Vlassopoulos | Tunable rheology of dense soft deformable colloids[END_REF].

They are indeed aqueous suspensions of nanometre-or micrometre-sized hydrogel particles sensitive to external stimuli whose dimension and effective volume fraction can be varied by changing external parameters such as temperature and/or pH.

The most studied responsive microgels are those based on the thermo-sensitive poly(N-isopropylacrylamide) (PNIPAM), with a reversible Volume Phase Transition (VPT) around 305 K. These systems have been largely investigated both theoretically and experimentally [START_REF] Pelton | Temperature-sensitive aqueous microgels[END_REF][START_REF] Hellweg | Colloidal crystals made of poly(N-isopropylacrylamide) microgel particles[END_REF][START_REF] Lyon | The Polymer/Colloid Duality of Microgel Suspensions[END_REF][START_REF] Seekell | Relationship between particle elasticity, glass fragility, and structural relaxation in dense microgel suspensions[END_REF][START_REF] Behera | Effects of polydispersity on the glass transition dynamics of aqueous suspensions of soft spherical colloidal particles[END_REF][START_REF] Gnan | In Silico Synthesis of Microgel Particles[END_REF][START_REF] Philippe | Glass transition of soft colloids[END_REF]. In this work we study a PNIPAM-based microgel with a second interpenetrated polymer network (IPN) of poly(acrylic acid) (PAAc) [START_REF] Xia | Synthesis and Light Scattering Study of Microgels with Interpenetrating Polymer Networks[END_REF][START_REF] Mattsson | Soft colloids make strong glasses[END_REF][START_REF] Ma | Synthesis and characterization of Poly(Nisopropylacrylamide)/Poly(acrylic acid) semi-IPN nanocomposite microgels[END_REF][START_REF] Liu | Study of pH/temperature dual stimuli-responsive nanogels with interpenetrating polymer network structure[END_REF][START_REF] Nigro | Dynamic light scattering study of temperature and pH sensitive colloidal microgels[END_REF][START_REF] Nigro | Dynamical behavior of microgels of interpenetrated polymer networks[END_REF] that provides additional pH sensitivity, topological constraints to the particles and extra charges to the system. In this way the IPN microgel softness can be controlled by synthesis varying the amount of poly(acrylic acid).

Experimental Methods

The dynamic structure factor has been investigated at different scattering vectors Q, weight concentrations C w and PAAc content C PAAc through combined XPCS and DLS techniques.

XPCS measurements were performed at ID10 beamline of ESRF in Grenoble using a partially coherent X-ray beam with a photon energy of 21 keV. A series of scattering images were recorded by CdTe Maxipix detector (photon counting). The ensemble averaged intensity autocorrelation function g 2 (Q,t) =

I(Q,t 0 )I(Q,t 0 +t) p I(Q,t 0 ) p
, where ... p is the ensemble average over the detector pixels mapping onto a single Q value and ... is the temporal average over t 0 , was calculated by using a standard multiple τ algorithm [START_REF] Madsen | Beyond simple exponential correlation functions and equilibrium dynamics in X-ray photon correlation spectroscopy[END_REF]. XPCS data were complemented by DLS measurements performed at the CNR-ISC laboratory. The monochromatic and polarized beam emitted from a solid state laser (100 mW at λ = 642 nm) was focused on the sample placed in a cylindrical VAT for index matching and temperature control. The scattered intensity was simultaneously collected by single mode optical fibers at five different scattering angles, namely θ =30°, 50°, 70°, 90°, 110°, corresponding to different scattering vectors Q, according to the relation Q=(4πn/λ ) sin(θ /2).

The investigated samples were IPN microgels synthesized by a sequential free radical polymerization method, then purified, liophylized and diluted in water to the desired concentration under magnetic stirring for 1 day. A more detailed description of the preparation protocol is reported in Ref. [START_REF] Nigro | Local structure of temperature and ph-sensitive colloidal microgels[END_REF][START_REF] Nigro | Swelling of responsive-microgels: experiments versus models[END_REF]. Samples at different concentra- in the Q range Q=(0.006 ÷ 0.063) nm -1 below the peak of the static structure factor.

The particles radii of the investigated samples at T =311 K are: C PAAc =2.6 % R=(26

± 1) nm, C PAAc =10.6 % R=(52 ± 2) nm, C PAAc =15.7 % R=(68 ± 3) nm, C PAAc =19.2
% R=(89 ± 2) nm, C PAAc =24.6 % R=(130 ± 2) nm; they increase with PAAc content as reported in Ref. [START_REF] Nigro | Study of network composition in interpenetrating polymer networks of poly(n isopropylacrylamide) microgels: The role of poly(acrylic acid)[END_REF]. An example of the temperature behaviour of hydrodynamic radius for C PAAc =24.6 % and C w =0.01 % is reported in Fig. 1(b). At low C w the intensity scattering functions are well described by a monomodal decay capturing the particle diffusion in the high dilution limit. As the particle concentration increases a dynamical decoupling is observed and a two-step relaxation, characteristic of glass-forming systems approaching the glass transition, comes out. The monomodal decay for C w < 0.6 % and the final decay for C w ≥ 0.6 % are well described by the Kohlrausch-Williams-Watts expression (47):

Results and Discussion

g 2 (Q,t) = b[(Aexp(-(t/τ(Q)) β )) 2 + 1] (1)
where b • A 2 is the coherence factor, τ the structural relaxation time and β the shape exponent, representing a phenomenological hallmark of glass forming liquid dynamics.

The fits are shown as full lines in Fig. 1.

The normalised structural relaxation time is reported in Fig. 2(a) as a function of weight concentration C w and for different PAAc content C PAAc . Two main features stand out: firstly the normalised structural relaxation time has a smoother and smoother concentration dependence with decreasing PAAc content and secondly a divergence occurs at lower concentrations for particles with higher PAAc content. This can be related to the increased interactions due to COOH groups belonging to PAAc chains as deeply discussed in Ref. [START_REF] Nigro | Study of network composition in interpenetrating polymer networks of poly(n isopropylacrylamide) microgels: The role of poly(acrylic acid)[END_REF]. Our data are well described by the super-Arrhenius expression:

τ = τ 0 exp( D C w C w C w0 -C w ) (2)
where C w0 sets the apparent divergence, D C w controls the growth of the structural relaxation time and τ 0 is the characteristic structural relaxation time in the high dilution limit. This result points out that, by changing PAAc content, different dynamical 1.0x10 behaviours can be achieved in microgels as in the case of molecular glass formers [START_REF] Angell | The old problems of glass and the glass transition, and the many new twists[END_REF]. This is highlighted in Fig. 2 Recently, the unifying concept of fragility has been extended to colloidal suspensions [START_REF] Seekell | Relationship between particle elasticity, glass fragility, and structural relaxation in dense microgel suspensions[END_REF][START_REF] Behera | Effects of polydispersity on the glass transition dynamics of aqueous suspensions of soft spherical colloidal particles[END_REF][START_REF] Mattsson | Soft colloids make strong glasses[END_REF][START_REF] Scheer | Fragility and Strength in Nanoparticle Glasses[END_REF][START_REF] Gnan | The microscopic role of deformation in the dynamics of soft colloids[END_REF] and many efforts have been devoted to understand the effect of softness on fragility [START_REF] Mattsson | Soft colloids make strong glasses[END_REF][START_REF] Scheer | Fragility and Strength in Nanoparticle Glasses[END_REF][START_REF] Gnan | The microscopic role of deformation in the dynamics of soft colloids[END_REF][START_REF] Sengupta | Dependence of the fragility of a glass former on the softness of interparticle interactions[END_REF][START_REF] Shi | Structure, dynamics, and thermodynamics of a family of potentials with tunable softness[END_REF]. In this context the slope of the data at C w = C * w can be used to define a kinetic fragility as:

m = ∂ logτ ∂ (C w /C * w ) C w =C * w (3)
Since the pioneering study of Ref. [START_REF] Mattsson | Soft colloids make strong glasses[END_REF] the connection between softness and fragility in soft colloids is still largely debated. A recent work on PNIPAM microgels (Ref. [START_REF] Philippe | Glass transition of soft colloids[END_REF])

states that regardless of their softness, soft colloids exhibit the same fragile-like behaviour of hard colloids. Moreover a simple theoretical model (Ref. [START_REF] Scheer | Fragility and Strength in Nanoparticle Glasses[END_REF]) has found that depending on osmotic pressure, presence of ions, dimension of particles, osmotic deswelling may be significant, giving rise to strong rather than fragile glasses. Here we show that the fragility can be also finely controlled by tuning the amount of PAAc interpenetrated within PNIPAM network.

To this aim we define the swelling ratio as:

α = R swollen H R shrunken H ( 4
)
where R swollen H = R(T =297 K) and R shrunken H = R(T =311 K) (blue and red arrows in Fig. 1(b) respectively). Particles with higher softness can shrink more, providing a larger value of α that decreases with increasing PAAc content as reported in Ref. [START_REF] Nigro | Study of network composition in interpenetrating polymer networks of poly(n isopropylacrylamide) microgels: The role of poly(acrylic acid)[END_REF].

Plotting the fragility m as a function of α (Fig. 2(c)) we find a linear decrease with increasing particle softness in surprisingly agreement with very recent theoretical results on the role of deformation in soft colloids [START_REF] Gnan | The microscopic role of deformation in the dynamics of soft colloids[END_REF]. A clear picture of the relation between softness and fragility is thus provided: the higher is the PAAc content, the stiffer are the particles, yielding thus to more fragile systems. These results compared to previous works [START_REF] Philippe | Glass transition of soft colloids[END_REF][START_REF] Mattsson | Soft colloids make strong glasses[END_REF][START_REF] Scheer | Fragility and Strength in Nanoparticle Glasses[END_REF] corroborate the idea that soft colloids can give rise to both fragile and strong behaviours. In our system the interpenetration of PAAc determines an increase of topological constraints, an increase of particle dimensions and extra charges. The corresponding structural relaxation time τ is reported in Fig. 3 A similar crossover has been recently reported for PNIPAM microgels [START_REF] Philippe | Glass transition of soft colloids[END_REF] and share many similarities with the dynamics of hard colloids and structural glasses close to an out of equilibrium state. The relaxation time vs aging time t w of an hard colloid, the largely studied Laponite ® suspensions [START_REF] Bandyopadhyay | Evolution of particle-scale dynamics in an aging clay suspension[END_REF][START_REF] Angelini | Glass-glass transition during aging of a colloidal clay[END_REF][START_REF] Tanaka | Kinetics of ergodic-tononergodic transitions in charged colloidal suspensions: Aging and gelation[END_REF][START_REF] Ruzicka | Observation of empty liquids and equilibrium gels in a colloidal clay[END_REF], is reported in Fig. 3(b) from

Ref. [START_REF] Angelini | Dichotomic Aging Behavior in a Colloidal Glass[END_REF]. It evolves from an exponential behaviour τ = τ 0 exp( D t t w t ∞ w -t w ) [START_REF] Bellour | Aging processes and scale dependance in soft glassy colloidal suspensions[END_REF][START_REF] Schosseler | Diagram of the aging dynamicsin laponite suspensions at low ionic strength[END_REF][START_REF] Angelini | Dichotomic Aging Behavior in a Colloidal Glass[END_REF][START_REF] Angelini | Non-diffusive dynamics in a colloidal glass: Aging versus rejuvenation[END_REF] where t w drives the system out of equilibrium and t ∞ w sets the apparent divergence [START_REF] Ruzicka | Routes to gelation in a clay suspension[END_REF][START_REF] Saha | Investigation of the dynamical slowing down process in soft glassy colloidal suspensions: comparisons with supercooled liquids[END_REF], to a slower than Arrhenius dependence with increasing t w . Fig. 3(c) shows instead the temperature dependence of the structural relaxation time for a metallic glass [START_REF] Evenson | X-Ray Photon Correlation Spectroscopy Reveals Intermittent Aging Dynamics in a Metallic Glass[END_REF]. Also in this case, widely investigated in the last years [START_REF] Ruta | Atomic-Scale Relaxation Dynamics and Aging in a Metallic Glass Probed by X-Ray Photon Correlation Spectroscopy[END_REF][START_REF] Evenson | X-Ray Photon Correlation Spectroscopy Reveals Intermittent Aging Dynamics in a Metallic Glass[END_REF][START_REF] Ruta | Relaxation processes and physical aging in metallic glasses[END_REF], the existence of two dynamical regimes below and above the glass transition temperature T g has been observed.

After T g the weaker temperature dependence of the structural relaxation time is well described by an Arrhenius behaviour. It is worth to note that the dynamical crossover is reached by increasing concentration (Fig. 3(a)) or waiting time (Fig. 3(b)) in colloidal glasses and by decreasing temperature (Fig. 3(c)) in structural glasses. Our findings for IPN microgels suggest that a unifying scenario can be provided: the slowing down of the dynamics achieved by varying weight concentration [START_REF] Cipelletti | Universal Aging Features in the Restructuring of Fractal Colloidal Gels[END_REF][START_REF] Philippe | Glass transition of soft colloids[END_REF][START_REF] Li | Long-term aging behaviors in a model soft colloidal system[END_REF], waiting time [START_REF] Bellour | Aging processes and scale dependance in soft glassy colloidal suspensions[END_REF][START_REF] Schosseler | Diagram of the aging dynamicsin laponite suspensions at low ionic strength[END_REF][START_REF] Angelini | Dichotomic Aging Behavior in a Colloidal Glass[END_REF][START_REF] Angelini | Non-diffusive dynamics in a colloidal glass: Aging versus rejuvenation[END_REF] or temperature [START_REF] Ruta | Atomic-Scale Relaxation Dynamics and Aging in a Metallic Glass Probed by X-Ray Photon Correlation Spectroscopy[END_REF][START_REF] Evenson | X-Ray Photon Correlation Spectroscopy Reveals Intermittent Aging Dynamics in a Metallic Glass[END_REF] is accompanied by a super Arrhenius increase of the structural relaxation time followed by a slower than Arrhenius or an Arrhenius behaviour. Thus the existence of two different dynamical regimes is a general feature of soft colloids, hard colloids and structural glasses.

To complement these observations, we report in Fig. 3(d) the behaviour of the β parameter derived from the fits through Eq.( 1). Surprisingly we find that the dynamical crossover at C * w is signaled by a sharp change of the shape parameter β from stretched (β < 1) to compressed (β > 1) in the high concentrated samples that we attribute to rising stresses deriving from the high packaging of the particles. As in the case of the (b) Power exponent from fits trough Eq.5. Blue full line is the predicted behaviour from Eq.12 while green full line is the predicted behaviour using Eq.10.

C w =2.8% C w =0.8% C w =0.4% C w =0.1% C w =0.01% Q(nm -1 ) C w Q -1 Q -0.6 (s) Q -2 (a) n C w (%) (b)
structural relaxation time, the behaviour of β displays many similarities with that observed in the hard colloid of Ref. [START_REF] Angelini | Dichotomic Aging Behavior in a Colloidal Glass[END_REF] as a function of t w (Fig. 3(e)) and in the metallic glass of Ref. [START_REF] Evenson | X-Ray Photon Correlation Spectroscopy Reveals Intermittent Aging Dynamics in a Metallic Glass[END_REF] as a function of 1/T (Fig. 3(f)). For all these systems the dynamical crossover driven by C w , t w or T is always accompanied by a discontinuity of β , as also found for PNIPAM microgels [START_REF] Philippe | Glass transition of soft colloids[END_REF], corroborating the existence of a universal behaviour regardless of the control parameter and the specific interactions of the system. It is worth to note that in the first dynamical regime β decreases below 1 while in the second one it increases to values up to 1 ÷ 1.5 depending on the system. This deviation from the common stretched behaviour can be attributed to the development of internal stresses at the transition [START_REF] Angelini | Dichotomic Aging Behavior in a Colloidal Glass[END_REF][START_REF] Cipelletti | Universal Aging Features in the Restructuring of Fractal Colloidal Gels[END_REF][START_REF] Ruta | Atomic-Scale Relaxation Dynamics and Aging in a Metallic Glass Probed by X-Ray Photon Correlation Spectroscopy[END_REF][START_REF] Bouzid | Elastically driven intermittent microscopic dynamics in soft solids[END_REF][START_REF] Wu | Stretched and compressed exponentials in the relaxation dynamics of a metallic glass-forming melt[END_REF]. The peculiar values of each system depend on the specific material, on its history and/or on the preparation protocols whose optimization and control is subject of theoretical and experimental investigations.

The Q-dependence of the structural relaxation time in the range Q=(0.006 ÷ 0.063) nm -1 is reported in Fig. 4(a) at different concentrations. The data are well fitted through a power law

τ(Q) ∝ Q -n (5)
shown as solid lines in Fig. 4 . Aiming to explain this behaviour we consider that at low weight concentration C w < 0.1 % particles freely move, the diffusion process is well described by Fickian law, as in the case of Brownian motion, and the mean square displacement < r 2 (t) > has a linear time dependence:

< r 2 (t) >= 6Dt ( 6
)
where D is the particle diffusion coefficient. At increasing weight concentraton C w > 0.1 % the system starts to become spatially heterogeneous, particles are no longer free to move as in the case of pure diffusive dynamics since they are trapped in cages characterized by different persisting times. In our experiments we probe scattering vectors Q in the range Q=(0.006 ÷ 0.063) nm -1 that corresponds to length scales up to ten particles radii (R=(130 ± 2) nm) and we are thus sensitive to these heterogeneities.

As a consequence Eq. 6 fails to reproduce the data which instead can be described by a non-Fickian anomalous diffusion (60):

< r 2 (t) >= Γt β (7)
with an effective diffusion coefficient (61, 62) D(t) = Γt β -1 . The diffusive dynamics (Eq. 6) is recovered when β = 1 and D(t) = Γ = 6D. For 0 < β < 1 the dynamics is subdiffusive and for β > 1 is superdiffusive, with the special case of β = 2 corresponding to ballistic motion.

The mean square displacement, in the Gaussian approximation, is related to the intermediate scattering function as [START_REF] Hansen | Theory of Simple Liquids with Applications to Soft Matter[END_REF]:

F(Q,t) = exp(-Q 2 < r 2 (t) > /6) (8)
that, using the generalized mean square displacement of Eq. 7, becomes F(Q,t) = exp(-Q 2 Γ 6 t β ). The comparison with the KWW expression (Eq.( 1)) of the intermediate scattering function

F(Q,t) ∝ exp(-(t/τ(Q)) β ), gives (t/τ(Q)) β = Q 2 Γ
6 t β and:

(1/τ(Q)) β = Q 2 Γ/6 (9)
that provides:

τ(Q) ∝ Q -2/β (10)
In the case of diffusive dynamics with β =1 one gets the typical τ(Q) ∝ Q -2 behaviour while in the case of ballistic motion of particles with β =2 τ(Q) ∝ Q -1 . A comparison between Eq.5 and Eq.10 gives n = 2/β . This behaviour is reported in Fig. 4

τ(Q) ∝ Q -1/β (11)
Surprisingly for C w ≥ C * w a perfect agreement is now observed as shown in Fig. 4(b) (blue full line). Therefore the experimental data are well described through:

τ(Q) ∝ Q -2/β C w < C * w τ(Q) ∝ Q -1/β C w ≥ C * w ( 12
)

Conclusions

In conclusion through XPCS and DLS measurements we have shown that for IPN microgels interpenetrating different amount of PAAc into PNIPAM network originates particles with varying softness allowing to finely control the fragility of the system. Moreover, we find the existence of two dynamical regimes, with a crossover of the structural relaxation time associated to a clear change of the β exponent from stretched (β <1) to compressed (β >1). The comparison with similar behaviours found in hard colloids and in metallic glasses indicates that these are general common features of different sys-tems regardless of the control parameters (C w , t w , T) and of the specific interactions at stakes. The scattering vector dependence of the structural relaxation time in IPN microgels displays a surprising behaviour of the derived power exponent with a maximum in correspondence of the dynamical crossover. We provide an interpretation that well depict the experimental data: at very low concentration, when particles are free to move, the dynamics is well described by a Fickian diffusion, at intermediate concentrations, when particles start to be trapped in cages, an effective non Fickian anomalous diffusion has to be considered, while at the highest investigated concentrations, when the systems is going towards a glassy state, the emergence of a ballistic motion is well described within the Mode Coupling Theory. All these results contribute to the further understanding of the role of softness in the fragility, to the analogies and differences between colloidal and structural glasses and add new insights to the so far debated stretched to compressed transition. In order to further test our findings and hypothesis, more experiments on a wide range of systems will be useful.

  rhenius, a minimum for the shape of the intensity autocorrelation function and the emerging of distinct anomalous mechanisms for particle motion. This complex phenomenology can be modelled by a Fickian diffusion at very low concentrations, an effective non Fickian anomalous diffusion at intermediate values and a ballistic motion well described within the Mode Coupling Theory on approaching the glassy state.

tions were obtained by dilution at pH close to 5 . 5 .Figure 1 :

 551 Figure 1: (a) Normalized intensity autocorrelation functions of IPN microgel particles at C PAAc =24.6 %, T =311 K and Q=0.022 nm -1 as a function of concentration and (b) Temperature behaviour of the hydrodynamic radius at C w =0.01 %, blue and red arrows indicate T =297 K and T =311 K respectively.

Figure 1 shows

 1 Figure 1 shows DLS and XPCS intensity autocorrelation functions for IPN microgels with fixed PAAc content (C PAAc =24.6 %) at different concentrations C w and fixed tempera-

Figure 2 :

 2 Figure 2: (a) Normalised structural relaxation time as a function of weight concentration at the indicated PAAc contents and T =311 K. Solid lines are fits through Eq.(2). (b) Angell plot for the normalized structural relaxation time versus normalized weight concentration at T =311 K (closed symbols) compared with data at T =297 K for a sample at 24.6 % of PAAc as an example (open symbols). Solid lines are fits according to Eq.(2). (c) Fragility index m as a function of swelling ratio related to particles softness.

  Fig.1(b)), as shown for the sample at 24.6 % of PAAc (open symbols).

Figure 3 :

 3 Figure 3: (a) Structural relaxation time from Eq.(1) as a function of weight concentration for IPN microgel suspensions at C PAAc =24.6 %, T =311 K and Q=0.022 nm -1 . Full lines represent the best fits with Eq.(2) and power law behaviours at small and large C w respectively. (b) Structural relaxation time behaviour with aging time t w from Ref. (6) for colloidal suspensions of Laponite and (c) structural relaxation time behaviour with the inverse of temperature 1/T from Ref. (17) for a metallic glass. (d) β parameter from Eq.(1) as a function of weight concentration for IPN microgel suspensions. Full line is a guide to eyes. (e) β exponent as a function of aging time t w from Ref. (6, 53) for colloidal suspensions of Laponite and (f) β exponent dependence with 1/T from Ref. (16) for a metallic glass.

  (a). An interesting behaviour shows up: at low C w , τ grows exponentially up to a critical weight concentration value C * w with a behaviour τ = τ 0 exp( D C w C w C w0 -C w ). Above C * w it suddenly increases as a power law τ ∼ C α w with α = 2.6 ± 0.1 signing the existence of two different dynamical regimes.

10

 10

Figure 4 :

 4 Figure 4: (a) Structural relaxation time as a function of the scattering vectors Q for IPN microgel suspensions at C PAAc =24.6 % and T =311 K as measured through DLS and XPCS at increasing weight concentrations. Full lines represent the fits through Eq.5.(b) Power exponent from fits trough Eq.5. Blue full line is the predicted behaviour from Eq.12 while green full line is the predicted behaviour using Eq.10.

  (a). The behaviour of n as a function of weight concentration is reported in Fig.4(b). Two dynamical regimes can be distinguished below and above C * w . In the first one (C w < C * w ) n increases from values typical of diffusive dynamics (n ∼ 2), to unusual values (n ∼ 4) for which diffusive dynamics is ruled out and an increase of viscosity is observed by eyes. In the second dynamical regime (C w ≥ C * w ) n suddenly decreases down to values n ≤ 1 where the case of n ∼1 is peculiar of the ballistic motion of particles. The crossover between the two dynamical regimes is characterized by a maximum in correspondence of the critical weight concentration C * w and is also associated to the observed crossover of the structural relaxation time (Fig. 3(a)) and β parameter (Fig. 3(d))

  (b) as green line. One can observe a good agreement with the experimental data only for C w < C * w . The critical weight concentration C * w , characterized by a relaxation time of the order of seconds (Fig.3(a)), sets the transition from a fluid to an intermediate state that is visually arrested, as shown in the top photograph of Fig.3. For C w ≥ C * w the dynamics slows down progressively towards a glassy state, as witnessed by the increasing relaxation times up to values of the order of thousand seconds. Here the Gaussian approximation of Eq.8 becomes inaccurate (63) and a different scenario has to be invoked. The dynamics approaching the glassy state can be described whitin the Mode Coupling Theory (MCT) (64,65) that provides a different Q-dependence of the structural relaxation time:

Acknowledgments:

The authors acknowledge acknowledge ESRF for beamtime and support from MIUR Fare SOFTART (R16XLE2X3L).

Author Contributions

Competing Interests

The authors declare that they have no competing financial interests.