Yacine El

Jean-Baptiste Pomet

Stéphane Régnier

Laetitia Giraldi

Yacine E Faris

Comparison of optimal actuation patterns for flagellar magnetic micro-swimmers

Keywords: Optimal control, Low-Reynolds Swimmers, Magnetic micro-swimmers, Soft Robotics

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

INTRODUCTION

Robotic micro-swimmers have the potential to conduct small-scale operations such as targeted drug delivery [START_REF] Qiu | Magnetic helical microswimmers functionalized with lipoplexes for targeted gene delivery[END_REF]; [START_REF] Patra | Intelligent, self-powered, drug delivery systems[END_REF]), and minimally invasive medical diagnosis and surgery [START_REF] Mack | Minimally invasive and robotic surgery[END_REF]; [START_REF] Fusco | An integrated microrobotic platform for on-demand, targeted therapeutic interventions[END_REF]). The design and way of locomotion of these devices is based on biological swimming micro-organisms, and wireless actuation methods are often chosen over built-in energy sources because of the difficulties of miniaturizing the latter. In particular, the actuation of partially magnetized micro-swimmers via external time-varying magnetic fields has proven to be an effective way to induce propulsion at the micro-scale [START_REF] Dreyfus | Microscopic artificial swimmers[END_REF]; [START_REF] Ye | Rotating magnetic miniature swimming robots with multiple flexible flagella[END_REF]; [START_REF] Qiu | Magnetic helical microswimmers functionalized with lipoplexes for targeted gene delivery[END_REF]). In this paper, we focus on flexible magnetic microswimmers that are based on flagellar cells in their design. By magnetizing the tail [START_REF] Dreyfus | Microscopic artificial swimmers[END_REF]) or the head [START_REF] Khalil | Magnetosperm: A microrobot that navigates using weak magnetic fields[END_REF]; [START_REF] Oulmas | 3d closedloop motion control of swimmer with flexible flagella at low reynolds numbers[END_REF]) of the swimmer, the propulsion of the swimmer in a straight line is obtained in experimental settings by applying the superposition of a static magnetic field to orient the swimmer in the desired direction and an oscillating magnetic field perpendicular to the swimmer, which leads to a planar beat of its tail. Dynamic models of flexible microswimmers (whether robotic or biological) are often based on hydrodynamics and continuum mechanics (elasticity), and expressed with partial differential equations that are numerically expensive and thus often reduced to the planar case [START_REF] Lowe | Dynamics of filaments: modelling the dynamics of driven microfilaments[END_REF]; [START_REF] Tornberg | Simulating the dynamics and interactions of flexible fibers in stokes flows[END_REF]). This complexity makes such models ill suited for the numerical resolution of optimal control problems related to the actuation of flexible micro-swimming robots such as minimal-cost trajectory planning. We present a simple and numerically cheap dynamical model of a flagellar micro-swimmer in 3D and use it to investigate optimal magnetic actuation patterns for horizontal propulsion. After discretizing the shape of the tail of the swimmer, we use Resistive Force Theory [START_REF] Gray | The propulsion of seaurchin spermatozoa[END_REF]) to approximate the hydrodynamics of the system , generalizing the planar 'N-link Swimmer' models from [START_REF] Moreau | The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella[END_REF] and [START_REF] Alouges | Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers[END_REF][START_REF] Alouges | Can magnetic multilayers propel artificial microswimmers mimicking sperm cells?[END_REF]. We numericaly solve the optimal control problem of finding the actuating magnetic field that maximizes the horizontal displacement of the swimmer during a fixed time. We investigate four different types of constraints on the control : firstly, we consider the feasible controls to be the superposition of static orientating field along the prescribed swimming direction and a time-varying orthogonal actuating field, which leads to a single-input optimal control problem. Secondly, we consider the control to be a twodimensional magnetic field where both components are time-varying. Lastly, we consider two three-dimensional magnetic fields that leads to out-of-plane optimal trajectories: one with a static orientating component and two orthogonal time-varying fields, and one where all three components are time-varying. One of the main conclusions that stems from the numerical solutions of these optimal control problems is the periodicity of the optimal magnetic fields in the all cases (modulo edge effects) and the periodicity of the deformation undergone by the tail of the swimmer when actuated by these magnetic fields. Furthermore, we show that all the solutions out-perform by large the standard sinusoidal fields used in experiments, especially the nonplanar solutions. Another result shown by the simulations is that static orientating magnetic fields are not necessary for horizontal propulsion, and that propulsion speed is significantly improved where all the magnetic field components are free to be optimized. This paper is organized as follows : first, we describe our swimmer model and derive its dynamics then, we formulate the maximum-displacement optimal control problems, assuming that we want the micro-robot to swim in the x-direction and present the numerical solutions of the problems. Lastly, we discuss influence of the number of links of the tail of the swimmer on the solutions of the optimal control problems.

3D FLAGELLAR MAGNETIC MICRO-SWIMMER MODEL

The micro-swimmer model used in this study is a 3D generalization of the planar swimmer models of [START_REF] Moreau | The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella[END_REF] and [START_REF] Alouges | Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers[END_REF][START_REF] Alouges | Can magnetic multilayers propel artificial microswimmers mimicking sperm cells?[END_REF]. We consider a swimmer of the form of a magnetized spherical head attached to an articulated chain of N slender rods of length l that can rotate with respect to each other with elastic torques that tend to keep them aligned. We assume that the swimmer is immersed in an unbounded domain of a viscous fluid and that it swims at a low Reynolds number. Under these assumptions, we neglect the inertial effects on the swimmer in favor of the viscous effects and use the Resistive Force Theory (RFT) framework [START_REF] Gray | The propulsion of seaurchin spermatozoa[END_REF]) to simplify the fluid-structure interaction.

2.1 Kinematics of the swimmer Fig. 1. Reference and local frames of the discrete-shape model. The swimmer's head frame is oriented relative to the reference frame. For each link i, the corresponding local frame R i is oriented relative to R h .

We associate to the head of the swimmer the moving frame R head = (O h , e h

x , e h y , e h z ), where O h is the center of the head. The orientation of each link i is represented by the moving frame R i = (O i , e i

x , e i y , e i z ), where O i is the extremity of the i-th link. We define R head ∈ SO(3) as the rotation matrix that allows the transformation of coordinates from the fixed reference frame to R head and the matrices R i ∈ SO(3), for i = 1 • • • N , as the relative rotation matrix that transforms coordinates from R head to R i . R head is parametrized by a (X -Y -Z) rotation sequence (θ x , θ y , θ z ) and each relative rotation matrix R i is parametrized by a Y -Z rotation sequence : (φ i y , φ i z ) relative to the head's frame. Using these notations, the swimmer is represented by two sets of variables : The Position variables: (X, Θ) where

X = (x h , y h , z h ) ∈ R 3 and Θ = (θ x , θ y , θ z ) ∈ [0, 2π] 3 , (1)
and the 2N Shape variables, denoted by

Φ = (φ 1 y , φ 1 z , • • • , φ N y , φ N z ) ∈ [0, 2π] 2N .
(2)

Dynamics of the swimmer

The RFT framework neglects the global interactions between the swimmer and the surrounding fluid in favor of the local anisotropic friction of the slender body with the surrounding fluid. This results in explicit expressions of the density of force applied by the fluid to the swimmer that are linear with respect to ( Ẋ, Θ, Φ). More precisely, We consider a hydrodynamical drag force F h head acting on the head of the swimmer that is proportional to its velocity in each of the direction of the head's frame's vectors R head and a hydrodynamical torque on the head that is proportional to its angular velocity :

F h head = -R head D H R T head Ẋ T h head = -k R Ω head .
(

) 3
where k H, , k 1 H,⊥ , and k 2 H,⊥ are positive real numbers that represents the hydrodynamic drag coefficients of the head along the e h

x , e h y and e h z direction, D H is the matrix

  k H, 0 0 0 k 1 H,⊥ 0 0 0 k 2 H,⊥   ,k R is the rotational drag coefficient
of the head and Ω head is the angular velocity vector of the head of the swimmer.

To compute the hydrodynamic effects on each link i ∈ (1, • • • , N ) of the tail, we consider the hydrodynamic force density f i (s) acting on a point x i (s) parametrized by its arclength s. Following Resistive Force Theory, the expression of f i (s) reads :

f i (s) = -k (V i (s).e i x )e i x -k ⊥ (V i (s).e i y )e i y -k ⊥ (V i (s).e i z )e i z ( 4
)
where k and k ⊥ are respectively the parallel and perpendicular drag coefficients of the swimmer and V i (s) is the velocity of x i (s).

The drag force on link i, F h i , and hydrodynamic torques on each link i (about the extremity O j of link j derive from the force densities as follows:

F h i = l 0 f i (s)ds , (5)
and

∀j ∈ (1 • • • N ) T h i,Oj = l 0 (x i (s) -O j ) × f i (s)ds , ( 6
)
where l is the length of each link.

We consider that the head of the swimmer is magnetized along the e h x axis. Denoting by M the magnetization vector of the head and considering an external homogeneous time-varying field B(t), the following torque is applied to the swimmer

T mag = M × B(t). (7)
The acceleration terms in the dynamics are neglected due to the Low Reynolds assumption [START_REF] Yates | How microorganisms move through water: the hydrodynamics of ciliary and flagellar propulsion reveal how microorganisms overcome the extreme effect of the viscosity of water[END_REF]), thus, the balance of forces and torques applied on the swimmer gives:

             F h head + N i=1 F h i =0 3 , T h head + N i=1 T h i,H = -T mag , (8)
which leads to 6 independent equations. In addition to these equations, we take the internal contributions of the tail and its elasticity into account by adding the balance of torque on each subsystem consisting of the chain formed by the links i to N for i = 1 • • • N . These 3N equations reduce to 2N non-trivial equations by taking only the components perpendicular to the link k when calculating the sum of the torques from k to N . The elasticity of the tail is discretized by considering a restoring elastic moment T el i at each joint O i that tends to align each pair (i, i + 1) of adjacent links with each other:

T el i = k el e i x × e i-1 x . (9)
Thus, the dynamics of the swimmer are described the following system of 2N + 6 equations :

                                                           F h head + N i=1 F h i = O 3 , T h head + N i=1 T h i,H = -T mag , N i=1 T h i,1 .e 1 y = -T el 1 .e 1 y , N i=1 T h i,1 .e 1 z = -T el 1 .e 1 z , . . . . . . T h N,N .e n y = -T el n .e n y , T h N,N .e n z = -T el n .e n z . (10)
Following Resistive Force Theory, and the definition of the drag densities, the hydrodynamic contributions (left-hand side of the previous system) are linear with respect to the angular and translational velocities, thus, the previous equation can be rewritten matricially in the form :

M h (Θ, Φ)   Ẋ Θ Φ  = B(X, Θ, Φ). (11)
, where M h (Θ, Φ) ∈ R (2N +6)×(2N +6 represents the hydrodynamical effects on the swimmer and the right-hand side B(X, Θ, Φ) ∈ R 2N +6 corresponds to the magnetic and elastic contributions on the swimmer.

The previous equation can be rewritten as a control system, where the dynamics of the swimmer are affine with respect to the components of the actuating magnetic field viewed as a control:

  Ẋ Θ Φ  = F 0 (Θ, Φ) + (B x (t) B y (t) B z (t)) F 1 (Θ, Φ) F 2 (Θ, Φ) F 3 (Θ, Φ) (12
) where the vector fields F 0 , • • • , F 3 are functions of the columns of (M h ) -1 and of the magnetic and elastic constants.

SWIMMING WITH SINUSOIDAL FIELDS

The most commonly used actuating strategy for straight swimming in experiments is to apply the superposition of a static magnetic field to orient the swimmer in the desired direction and a perpendicular sinusoidal field to actuate the swimmer. This induces a planar symmetric beating of the tail of the swimmer, allowing a horizontal displacement. This actuation strategy will be used as a benchmark to compare with the optimal solutions. The hydrodynamical and elastic parameters of the model were fitted on experimental data (El Alaoui-Faris et al. ( 2020)) .for the characterization of the swimmer. Using these parameters, we are able to simulate the velocity-frequency response curve of the swimmer (Fig. 3), using a magnetic field of the form :

B(t) = (B x , B y sin(2πf t), 0) . (13)
The velocity-frequency response curve of the swimmer shows a behaviour typical of flagellar low Reynolds swimmers, as it shows an increase of the swimming velocity until a cut-off frequency (1.5 Hz) where the velocity decreases slowly beyond this value.

OPTIMAL CONTROL PROBLEMS

The purpose of this paper is to find the time-varying mag-

netic field B = B x (t) B y (t) B z (t)
that maximizes the horizontal displacement of the swimmer at a fixed time t f . Denoting by Z(t) the state vector

X(t) Θ(t) Φ(t)
and rewriting equation ( 12) as Ż(t) = f (Z(t), B(t)), this optimal control problem is written as :

         max x(t f ) Ż (t) = f (Z(t), B(t)) Z(0) = 0 B(t) ∈ C , ( 14
)
where C is the set of constraints on the magnetic field and tf is the final time fixed at 3s . We investigate four optimal control problems depending on the types of constraints on the actuating magnetic field, taking in each case the same bounds on the magnetic field intensities.

(1) In the first optimal control problem (OCP1), we consider the admissible controls as the superposition of a static orientating field along the x axis and a time-varying field along the y axis :

C 1 = {B(t), B x (t) = 2.5mT, |B y | ≤ 10mT, B z (t) = 0} (2)
In the second optimal control problem (OCP2), both the x and y components of the magnetic field are time-varying.

C 2 = {B(t), B z (t) = 0, ||(B x (t), B y (t))|| ≤ 10mT } (3)
In the third problem (OCP3), we consider the admissible controls to be the superposition of a static orientating field along the x axis and a time-varying actuating field in the y -z plane.

C 3 = {B(t), B x = 10mT ||(B z (t), B y (t))|| ≤ 10mT } (4)
In the last optimal control problem (OCP4), we consider the general case where all components of the magnetic field are time-varying.

C 4 = {B(t), ||(B x (t), B z (t), B y (t))|| ≤ 10mT }

NUMERICAL SOLUTIONS

We use a direct method for solving the aforementioned optimal control problems. The optimal control solver ICLOCS [START_REF] Falugi | Imperial college london optimal control software (iclocs)[END_REF]) is used for the resolution.

It discretizes the problem into a non-linear programming problem which is then solved by the interior-point solver IPOPT [START_REF] Wächter | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming[END_REF]). Fig. 3 shows the displacements of the swimmer under each optimal solution, and under the sinusoidal field at the optimal frequency (1.5 Hz). We see that non-planar solutions largely outperforms planar solutions. None of the optimal solutions display singular arcs. Accordingly, and following Pontryagin's maximum principle, the solution of the single input problem OCP1 is a sequence of bang arcs and the solutions of the other problems are continuous.

Planar Solutions

For OCP1 and OCP2, the dimension of the dynamic system reduces to N+3 because of the fact that the constraints on the controls leads to a planar trajectory in the x -y plane. The numerical solution for OCP1 takes the form of a sequence of Bang arcs, as seen in Fig. (4,(a) ) whereas the solution of OCP2 is continuous ((b), (c)).

In both cases, the optimal actuation patterns lead to a trajectory where swimmer oscillates around the x axis while moving in the x-direction (see Fig. 5).

The shape of both optimal trajectories is similar to the trajectory of the swimmer under the sinusoidal field. Interestingly enough, the optimal magnetic fields are periodic in both cases, and induce a periodic deformation of the swimmer, as seen in Fig. 6. Both solutions out-perform the reference sinusoidal actuation in terms of horizontal speed.

In practice, the solution of OCP-1 shows that having an orthogonal actuating field (in addition to the static orientating field) in the form of a square signal is more efficient than an actuating sinusoidal field. The solution of the second planar problem (OCP-2) shows that orientating fields are not necessary for straight swimming and that actuating a flagellar magnetic swimmer with two time-varying components leads to a substantial increase in swimming speed.

Non-planar solutions

The non-planar optimal magnetic fields solutions of OCP3 and OCP4 leads to trajectories ,shown in Fig. 8, where the swimmer revolves around the horizontal axis.

As seen in Fig. 3, the non-planar actuation patterns largely out-performs the planar ones, which shows the necessity of allowing flagellar swimmers to go out-of-plane in order to swim at a maximal propulsion speed. Similarly to the planar case, the optimal solution is periodic apart from transient states near the initial and final times (see Fig. 7) and induce a periodic 3D deformation of the tail of the swimmers, as shown in the phase planes of Fig. 9. Fig. 9. Shape variables of the swimmer when actuated by the non-planar optimal magnetic fields (OCP3 and OCP4). Phase portrait of the relative angles in the (φ i y -φ i z ) for each link i of the tail. 5.3 Influence of the number of links on the trajectory and shape of the swimmer An analysis of the influence of the number of links of the model on the optimal trajectories and magnetic fields have been made for OCP4. We define the relative trajectory error E tr as :

E tr = ||X N -X N -1 || ∞ ||X N || ∞ , (15)
where X N is the optimal trajectory using N links for the tail. Similarly, we define the relative solution error E sol as :

E sol = ||B N -B N -1 || ∞ ||B N || ∞ , ( 16
)
where B N is the optimal magnetic field using N links for the tail.

Fig. 10 shows the evolution of the relative trajectory and solution errors for N = (1 • • • 9). We notice that the optimal magnetic field and the optimal trajectory only marginally changes when increasing the number of links beyond 3. This shows that a 3-linked tail is enough for the optimization of the actuation of flagellar magnetic swimmers, which further emphasizes the low computational cost of our model. Fig. 10. Relative optimal trajectories and solution errors for an increasing number of links of the tail of the swimmer.

CONCLUSION AND PERSPECTIVES

In this paper, we have presented a simplified 3D dynamical model for flagellar micro-swimmers based on Resistive Force Theory and shape discretization, which circumvents the numerical drawbacks of continuum mechanics-based models. This model was used to investigate different planar and non-planar optimal actuation strategies for magnetic micro-swimmers that out-perform the commonly used sinusoidal actuation. In particular, the non-planar actuation strategies lead to a novel 3D trajectory and are significantly more efficient than the planar ones. Another result stemming from the numerical simulations is the investigation of optimal actuation strategies that do not rely on a static orientating magnetic field, as allowing all the components of the magnetic field to be time-varying leads to an increase in swimming speed. However, these actuation methods lead to larger oscillations of the swimmer around the swimming direction, which could be an inconvenient for some experimental settings such as path following.

Beyond the present study, the simplified model presented in our model allows for easy numerical optimization studies for micro-swimmers such as parameter fitting or optimal motion planning for magnetic swimmers. It has been successfully used to optimize the displacements of an experimental swimmer in (El Alaoui-Faris et al. ( 2020)). It can also be easily adapted to model biological flagellar micro-swimmers. Ongoing work includes optimization of the swimming speed for more complex cost functions (maximization of efficiency, path following ..) and adapting the model for swimming in complex environments (for example in a narrow channel or in presence of a wall).

Theoretical study of the optimal control problems is also a perspective of this work.

FrequencyFig. 2 .

 2 Fig. 2. Simulated velocity-frequency response curve of the swimmer.

Fig. 3 .

 3 Fig. 3. Comparison of x-displacements associated with the solutions of OCP1-4 and with the sinusoidal actuation

Fig. 4 .Fig. 5 .

 45 Fig. 4. Solution of the planar optimal control problems. (a) : y-component of the solution of OCP-1 compared with the sinusoidal field at optimal frequency (1.5Hz). (b): x-component of the solution of OCP-2. (c) : ycomponent of the solution of OCP-3.

Fig. 6 .

 6 Fig. 6. Shape angles of the swimmer for the solutions of both planar problems OCP-1 and OCP-2.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Solution of the non-planar optimal control problems. (a) : y-component of the solution of OCP-3. (b): z-component of the solution of OCP-2. (c) : x-component of the solution of OCP-4. (d) : y-component of the solution of OCP-4. (e) : zcomponent of the solution of OCP-4.


Rui Yu

Yan Zhang

Mihail Barboiu
email: mihail-dumitru.barboiu@umontpellier.fr

Marie Maumus

Danièle Noël
email: daniele.noel@inserm.fr

Christian Jorgensen

Suming M Li
email: suming.li@umontpellier.fr

Biobased pH

Keywords: O-carboxymethyl chitosan, Imine chemistry, Dynamic hydrogel, Self-healing, pH responsive, Cytocompatibility

come

Introduction

In the past decades, hydrogels have been widely studied as biomaterials for various applications in drug delivery, wound dressing and tissue engineering due to their outstanding properties such as biocompatibility [START_REF] Qu | Antibacterial adhesive injectable hydrogels with rapid self-healing, extensibility and compressibility as wound dressing for joints skin wound healing[END_REF], biodegradability [START_REF] Ghobril | The chemistry and engineering of polymeric hydrogel adhesives for wound closure: a tutorial[END_REF], mechanical properties [START_REF] Van Vlierberghe | Biopolymer-based hydrogels as scaffolds for tissue engineering applications: a review[END_REF], and stimuliresponsive properties [START_REF] Zhang | Synthesis of multiresponsive and dynamic chitosan-based hydrogels for controlled release of bioactive molecules[END_REF]. Hydrogels consist of a cross-linked network [START_REF] Lv | Hygroscopicity modulation of hydrogels based on carboxymethyl chitosan/Alginate polyelectrolyte complexes and its application as pH-sensitive delivery system[END_REF][START_REF] Su | Synthesis and characterization of novel carboxymethyl chitosan grafted polylactide hydrogels for controlled drug delivery[END_REF] which can absorb large amounts of water [START_REF] Yang | Highly elastic and ultratough hybrid ionic-covalent hydrogels with tunable structures and mechanics[END_REF] or physiological fluids [START_REF] Dimatteo | In situ forming injectable hydrogels for drug delivery and wound repair[END_REF] while maintaining their three-dimensional structural integrity. Hydrogels can be classified into two categories according to the preparation approach: chemical gels or irreversible gels, and physical gels or reversible gels [START_REF] Iftime | Salicyl-imine-chitosan hydrogels: Supramolecular architecturing as a crosslinking method toward multifunctional hydrogels[END_REF][START_REF] Stewart | Stable and ordered amide frameworks synthesised under reversible conditions which facilitate error checking[END_REF]. Chemical gels are formed by irreversible covalent bonding, whereas in physical gels, the polymeric chains are held together by chain entanglements and/or supramolecular hydrophobic, ion-pair or hydrogen bonding interactions.

Dynamic hydrogels or dynagels are a dynamic system on both the molecular and supramolecular levels [START_REF] Burdick | Moving from static to dynamic complexity in hydrogel design[END_REF][START_REF] Marin | Out-of-water constitutional self-organization of chitosan-cinnamaldehyde dynagels[END_REF]. They are able to reversibly exchange their components [START_REF] Sreenivasachary | Gelation-driven component selection in the generation of constitutional dynamic hydrogels based on guanine-quartet formation[END_REF], responding to external stimuli such as pH [START_REF] Zeng | A drug-self-gated mesoporous antitumor nanoplatform based on pH-sensitive dynamic covalent bond[END_REF], ions [START_REF] Arnal-Hérault | Amplification and transcription of the dynamic supramolecular chirality of the guanine quadruplex[END_REF][START_REF] Rotaru | G-Quartet hydrogels for effective cell growth applications[END_REF], and temperature [START_REF] Zhang | Injectable and body temperature sensitive hydrogels based on chitosan and hyaluronic acid for pH sensitive drug release[END_REF]. Among the various dynamic reactions (Zhang & Barboiu, 2015a), imine bond formation is considered as the most promising strategy to generate dynamic materials with modulable properties. In fact, imine chemistry allows to implement reversible rearrangements of the components in a multivalent material which can bind bioactive molecules, cells or present self-healing properties [START_REF] Chao | Dynamic covalent polymer networks based on degenerative imine bond exchange: tuning the malleability and self-healing properties by solvent[END_REF]. This dynamic constitutional framework is composed of linear and/or multi-armed components reversibly interconnected via imine bonds and containing stimuli-responsive functional groups [START_REF] Zhang | Constitutional dynamic materials toward natural selection of function[END_REF]. Chitosan (CS), a polysaccharide obtained from alkaline hydrolysis of chitin found in the exoskeleton of crustaceans, presents remarkable properties such as biocompatibility, biodegradability, low toxicity, low cost or immune-stimulatory activity [START_REF] Ali | A review on chitosan and its nanocomposites in drug delivery[END_REF].

Chitosan is a good candidate for in-situ dynamic reversible crosslinking via its amino groups present along the polymer chain with aldehydes [START_REF] Marin | Imino-chitosan biodynamers[END_REF], alginate [START_REF] Qin | Convenient one-step approach based on stimuli-responsive sol-gel transition properties to directly build chitosanalginate core-shell beads[END_REF], or gelatin [START_REF] Qiao | Molecular interactions in gelatin/chitosan composite films[END_REF], resulting in the formation of pHresponsive and biodegradable hydrogels. The water solubility of chitosan depends on many factors, in particular pH of the medium, chain length and degree of acetylation [START_REF] Varum | Water-solubility of partially N-acetylated chitosans as a function of pH: effect of chemical composition and depolymerisation[END_REF]. The carboxymethylation of the D-glucosamine moieties of chitosan generates O-carboxymethyl chitosan (CMCS) which is readily soluble in water at neutral pH, thus allowing uses in tissue engineering, drug delivery, wound dressing and food industry.

Crosslinking of CS or CMCS with aldehydes via imine bond formation along polymeric chains proceeds with very low yield in aqueous medium, but is significantly improved in hydrogels or in solid state films with dynamic properties [START_REF] Marin | Imino-chitosan biodynamers[END_REF]. It is well known that no continuous cross-linked networks are formed when monoaldehydes [START_REF] Iftime | Salicyl-imine-chitosan hydrogels: Supramolecular architecturing as a crosslinking method toward multifunctional hydrogels[END_REF][START_REF] Marin | Imino-chitosan biodynamers[END_REF] or dialdehydes [START_REF] Yu | A novel pH-induced thermosensitive hydrogel composed of carboxymethyl chitosan and poloxamer cross-linked by glutaraldehyde for ophthalmic drug delivery[END_REF] are used for cross-linking. The resulting hydrogels exhibit good swelling behaviors, but disordered micro-structure and weak mechanical strength.

In contrast, dynamic hydrogels exhibiting pH and temperature-responsive swelling behaviors, strong mechanical performance, and self-healing behavior have been obtained by using 3-armed [START_REF] Deng | Macroscopic organohydrogel hybrid from rapid adhesion between dynamic covalent hydrogel and organogel[END_REF] or 4-armed aldehydes [START_REF] Huang | Strong and rapidly self-Healing hydrogels: potential hemostatic materials[END_REF]. Nevertheless, the toxicity of aldehydes, and especially of glutaraldehyde [START_REF] Bhatia | Traumatic injuries[END_REF][START_REF] Ghobril | The chemistry and engineering of polymeric hydrogel adhesives for wound closure: a tutorial[END_REF] restricts their use for biomedical applications, and imposes the necessity of finding new biocompatible crosslinking agents.

Jeffamine is a polyetheramine composed of poly(propylene oxide) (PPO) and/or poly(ethylene oxide) (PEO) blocks with primary amino groups attached to the chain ends. It is widely used as a macromonomer to prepare PEO-based hydrogels as its water solubility facilitates reaction in aqueous medium [START_REF] Zimmermann | Novel hydrogels as supports for in vitro cell growth: poly (ethylene glycol) and gelatin-based (meth) acrylamidopeptide macromonomers[END_REF]. In this work, bifunctional Jeffamine ED-2003 was linked to benzene-1,3,5-tricarbaldehyde via imine formation, yielding a constitutional dynamer Dy that can be used as a water soluble cross-linking component. CMCS based hydrogels were then prepared by mixing CMCS and Dy aqueous solutions through a "green" synthetic route. The chemical structures of the hydrogels, their morphological, rheological and swelling properties, as well as their self-healing behaviors were evaluated and discussed. The cytocompatibility of hydrogels was assessed by co-culture in the presence of human mesenchymal stromal cells (MSCs) to evaluate their potential as scaffold in cartilage engineering.

Experimental section

2.1 Materials: Benzene-1,3,5-tricarbaldehyde (BTA) from Manchester Organics and O,O'-Bis(2aminopropyl) PPO-b-PEO-b-PPO (Jeffamine ® ED-2003, Mn 1.9 × 10 3 ) from Sigma Aldrich were used without purification. CMCS (Mn 2 × 10 5 Da, degree of deacetylation 90 %, degree of carboxymethylation 80 %) was purchased from Golden-shell Biochemical Co., Ltd. Methanol (96%), citric acid (≥99.5%), disodium hydrogen phosphate dodecahydrate (≥99%), boric acid (≥99.5%), borax(≥99%) were of analytical grade, and obtained from Sigma Aldrich.

2.2 Synthesis of dynamer Dy: Typically, BTA (162 mg, 1 mmol), Jeffamine (1.90 g, 1 mmol) are added in 30 mL methanol, and the reaction mixture was stirred at 70 °C for 4 h. After evaporation of the solvent, 20 mL Milli Q water was added, yielding a homogeneous dynamer solution of 5 × 10 -2 M as calculated from the remaining aldehyde groups.

2.3 Preparation of CMCS-Dy hydrogels: CMCS (1.06 g, 5 mmol calculated from D-glucosamine units) was dissolved in 50 mL Milli-Q water at room temperature, yielding a transparent solution of 1× 10 -1 M. CMCS and dynamer solutions were mixed at different ratios to a total volume of 12 mL, followed by ultra-sonication for 1 min to remove trapped bubbles. Gelation then proceeded at 37 °C for 24 h, yielding a CMCS-based hydrogel.

Freeze-drying was performed as follows so as to conserve the original structure. As-prepared hydrogels were placed in small vials and immersed in liquid nitrogen (-196 °C) for instantaneous freezing. The vials were then placed in a 500 mL round-bottomed flask which was fixed on LABCONCO® freeze dryer. The hydrogels were freeze-dried for 24 h before analyses.

2.4 Characterization: 1 H NMR spectroscopy was carried out using Bruker NMR spectrometer (AMX500) of 300 MHz. CDCl 3 or D 2 O was used as the solvent. 5 mg of sample were dissolved in 0.5 mL of solvent for each analysis. Chemical shifts were recorded in ppm using tetramethylsilane (TMS) as internal reference. The morphology of freeze dried hydrogels was examined using scanning electron microscopy (SEM, Hitachi S4800). The samples were subjected to gold coating prior to analysis. Fourier-transform infrared spectroscopy (FT-IR) was performed with Nicolet Nexus FT-IR spectrometer, equipped with ATR diamant Golden Gate.

Structural stability of Dy:

The stability of the dynamer Dy was evaluated in D 2 O under neutral and acidic conditions since imine bond formation is reversible at low pH. D 2 O solutions at pH of 1, 3, and 5 were prepared by addition of trifluoroacetic acid. NMR spectra were registered just after dynamer dissolution and after 7 days.

2.6 Rheology: The rheological properties of hydrogels were examined with Physical MCR 301 Rheometer (Anton Paar). Hydrogels prepared in Milli-Q water were placed on a cone plate (diameter of 4 cm, apex angle of 2 °, and clearance 56 µm). Measurements were made in the linear visco-elastic range as a function of time, strain, or frequency.

2.7 Swelling: The swelling behavior of hydrogels was evaluated in buffer solutions at various pH values. Solutions from pH 1 to pH 7 were prepared using 0.1 × 10 -3 M citric acid solution and 0.2 × 10 -3 M disodium hydrogen phosphate solution, whereas solutions of pH 8 and 9 were prepared using 0.2 × 10 -3 M boric acid solution and 0.5 × 10 -4 M borax solution. Freeze-dried gels were immersed in a buffer, and taken out at different time intervals. The swollen hydrogels were weighed after wiping surface water with filter paper, freeze-dried for 24 h, and weighed again. The swelling ratio and mass loss ratio of hydrogel were calculated according to equation

(1) and equation ( 2), respectively:

Swelling ratio % = (𝑀 𝑠 -𝑀 𝑑 ) 𝑀 𝑑 × 100
(1)

Loss ratio % = (𝑀 0 -𝑀 𝑑 ) 𝑀 0 × 100 (2)
Where M 0 is the initial mass of xerogel, M s is the wet mass of the swollen hydrogel, and M d is the dried mass of the swollen hydrogel after lyophilization.

2.8 Self-healing experiments: Various hydrogel samples were prepared in Milli-Q water, and in pH = 7 and pH = 8 buffers. Some of them were dyed yellow with 5 µL of lucigenin, or dyed red with Rhodamine. 3 different approaches were applied to examine the self-healing behavior of hydrogels: 1) a hole with diameter around 3 mm was punched at the circle center of the sample;

2) samples were split into two pieces, and then a yellow piece was put together with a transparent piece immediately at 37 °C ; 3) injection of transparent and red samples on the surface of a Petri dish to observe color changes.

2.9 Cell cultures and Cytotoxicity: 2 mL of CMCS at 100 mmol and 1 mL of dynamer at 50 mmol were mixed. Human MSCs isolated from adipose tissue (AT-MSCs) or from bone marrow (BM-MSCs) were added at a concentration of 1 x 10 6 cells / mL. As control, AT-MSCs or BM-MSCs were seeded on 96-wells TCPS (Tissue Culture Polystyrene System) plates at 5 x 10 3 cells per well, or were embedded in 3 mg / mL rat collagen type I hydrogel (Corning) at 1 × 10 6 cells / mL. 50 µL of cell laden solution were loaded in each well of 96 wells ultra-low adhesion plates (Corning). After 2 h gelation in an atmosphere at 37 °C, 5 % CO 2 and 95 % of humidity, 100 µL of proliferative medium (αMEM containing 10% fetal calf serum, 100 µg/mL penicillin/streptomycin, 2 × 10 -3 M glutamine, 1 ng / mL of basic fibroblast growth factor) were added on the top of the hydrogel. MSCs were cultured for 7 days with medium change at day 3.

After 1 or 7 days culture, the cell viability was analyzed by confocal microscopy (Leica) after staining the live cells in green and the dead cells in red using the live/dead assay kit (Invitrogen).

Results and discussion

3.1 Synthesis of dynamer: A water soluble dynamer Dy was first synthesized by reaction of BTA as the core structure and bifunctional diamine, Jeffamine ® ED-2003 (Mn=1900) as the watersoluble linker at a molar ratio of 1:1 via reversible imine bond formation, as shown in Scheme 1.

Thus, it remains in average one aldehyde group per molecule of BTA for further cross-linking reaction with the amine groups of CMCS.

1 H NMR spectroscopy was used to monitor the formation and stability of imine bonds during the synthesis of the dynamer. The effect of reaction time on the formation of dynamer was investigated. No difference was observed on the 1 H NMR spectra of samples up to 72 h reaction (Fig. S1, Supporting Information), thus implying that equilibrium was reached after 4 h reaction. Therefore, the dynamer obtained after 4 h reaction was selected for further studies. Moreover, the dynamer apparently remained unchanged for a week in pure D 2 O and in acidic D 2 O at pH 3 and 5, while became highly hydrolyzed in strongly acidic medium at pH 1 (Fig. 2). The spectra obtained in D 2 O and at pH 3 and 5 remain unchanged even after 7 days (Fig. S2, Supporting Information). In contrast, major changes are observed on the spectrum of dynamer at pH 1. The signal a (10.21 ppm) belonging to free aldehydes becomes much more intense, indicating that imine bonds are hydrolyzed back to aldehydes. Therefore, the dynamer Dy seems stable at neutral and slightly acidic pH, but unstable at strongly acidic pH. As previously observed for PEGylated networks, Jeffamine chains could have a protecting effect against the hydrolysis of imine bonds, favoring the imine formation in slightly acidic or neutral media [START_REF] Catana | Dynamic constitutional frameworks for DNA biomimetic recognition[END_REF]. CMCS and 5 × 10 -2 M dynamer aqueous solutions to a total volume of 12 mL. Gelation proceeded at 37 °C for 24 h. The D-glucosamine to dynamer molar ratio varied from 1:1 to 8:1, as shown in Table 1. [mmol]

[mg]

[w/v %] [mmol] c) [mg] d) [w/v %]
Gel1 b) The concentration of CMCS solution is 100 mM. Calculations are made on the basis of the average molar mass of 212 g/mol obtained for D-glucosamine, taking into account the degree of deacetylation of 90 % and the degree of carboxymethylation of 80 %;

c) The concentration of Dy solution is 50 mM calculated from the remaining aldehyde groups. In a typical reaction, 1 mmol BTA (162 mg) reacts with 1 mmol Jeffamine (1900 mg) to form a dynamer. As BTA has 3 aldehydes and Jeffamine 2 amines, there remains theoretically 1 mmol of aldehydes in the dried dynamer. Addition of 20 mL water yields a dynamer solution of 50 mM.

d) The amount of Dy in solution is obtained from the initial quantities of BTA and Jeffamine.

FTIR was used to confirm the formation of imine bonds during the synthesis of the dynamer Dy and CMCS based hydrogel. As shown in Fig. 3, the characteristic bands of aldehyde and imine bonds are observed at 1698 and 1645 cm -1 on the spectra of Dy, respectively. The dried gels present all the characteristic bands of CMCS and dynamer: a large band in the 3200-3500 cm -1 assigned to free OH and NH 2 groups, an intense band at 1595 cm -1 assigned to carboxyl groups of CMCS, and two strong signals at 2850 and 1100 cm -1 attributed to C-H and C-O stretching in the dynamer, respectively. With increasing D-glucosamine to dynamer molar ratio, the aldehyde band progressively disappears at ratios above 4:1, while the imine band merges with that of carboxyl groups at 1595 cm -1 which turns more intense. These findings confirm that hydrogels are formed because of imine formation between aldehyde and amine groups. 3.3 Rheological studies. The rheological properties of hydrogels were investigated under various conditions. CMCS and Dy aqueous solutions were mixed in situ on the plate of rheometer, and changes of the storage modulus (G) and loss modulus (G) were followed as a function of time at 37°C (Fig. 4a). For all samples, at the beginning of experiment the storage modulus is lower than the loss modulus (G < G), which illustrates a liquid-like behavior of the starting mixture.

After an induction time, both G and G begin to increase, G increasing faster than G. A crossover point between G and G is detected, indicating sol-gel transition. As shown in Figure 4a, the gelation time decreases from 600 s for Gel1-1 to a minimum of 360 s for Gel4-1, and then increases to 660 s for Gel8-1. In fact, gelation occurs by crosslinking via imine bonds formation and is thus dependent on the ratio between amine groups of CMCS and aldehyde groups of the dynamer. In Gel1-1, there are less amine groups than aldehyde ones, as calculated by taking into account the degree of deacetylation of 90 %. Thus, gelation is relatively slow. Gelation is progressively improved for Gel2-1 and Gel4-1, as the concentration of amine groups increases.

Gelation is not optimal for Gel2-1, since unreacted aldehyde groups are detected by FTIR after 24 h at 37 °C (Fig. 3). In contrast, optimal imine bond formation is achieved for Gel4-1 as aldehyde groups are no longer detectable. Nevertheless, with further increase of Dglucosamine/dynamer ratio to 6:1 and 8:1, the gelation becomes longer as there are less aldehyde groups available for imine formation. and c) G and G changes of Gel4-1 as a function of frequency at 25 °C, and strain of 1%. All hydrogels are prepared in Milli-Q water.

Rheological measurements performed at 25 °C illustrate the viscoelastic behaviors of as prepared hydrogels. The storage modulus of all gels slightly decreases (less than 20 % of the initial value)

when increasing the strain up to 20 % (Fig. 4b), indicating that the hydrogels are stable in this strain range with viscoelastic behavior. On the other hand, the modulus increases with increasing D-glucosamine to dynamer molar ratio from 1:1 to 4:1, reaching a maximum value of c.a. 1200

Pa at 4:1. In contrast, higher D-glucosamine to dynamer ratios of 6:1 and 8:1 result in decrease in modulus because there are less aldehydes available for crosslinking in Gel6-1 and Gel8-1 compared to Gel4-1. These findings well agree with storage modulus (G) and loss modulus (G) changes versus time in Fig. 4a, confirming that optimal crosslinking is achieved with Gel4-1. In order to investigate the stability of the hydrogels, a frequency sweep over a range from 0.01 to 50 Hz was carried out at a fixed strain of 1 %. Taking Gel4-1 as an example (Fig. 4c), the storage modulus G is always much higher than the loss modulus G. G remains nearly unchanged, whereas G exhibits some fluctuations with increasing frequency. The other hydrogels exhibit similar behaviors (Fig. S3, Supporting Information). These rheological results well corroborate with the formation of highly stable covalent networks, in contrast to physical hydrogels whose storage and loss moduli are dependent on the frequency [START_REF] Li | Rheology and drug release properties of bioresorbable hydrogels prepared from polylactide/poly (ethylene glycol) block copolymers[END_REF][START_REF] Zhang | Novel thymopentin release systems prepared from bioresorbable PLA-PEG-PLA hydrogels[END_REF]. It is generally admitted that hydrogels with G' below 2000 Pa are 'soft' materials suitable for specific tissue engineering applications (brain, cartilage, muscle, etc).

3.4 Morphology and swelling studies: Scanning electron microscopy (SEM) was used to qualitatively assess the microstructure of the freeze-dried hydrogels. As shown in Fig. 5, all samples exhibit a sponge-like structure with open and interconnected pores. Gel4-1 apparently exhibits the most uniform porous structure with mean pore size around 150 μm and mean wall thickness of c.a 3 µm, whereas the other samples, in particular Gel1-1 and Gel8-1, present larger and irregular pore size and larger wall thickness. These findings well agree with the optimal imine formation or crosslinking of Gel4-1 since higher crosslinking leads to smaller pore size and wall thickness. Gel6-1; (i, j) Gel8-1.

The swelling behaviors of hydrogels are of major importance for the applications as drug carrier or as tissue engineering scaffold. The five samples exhibit similar swelling behaviors at a given pH value in the pH range from 3 to 9. The highly pH-sensitive swelling ratios are below 1000 % for acidic media (gray bar, pH 3/4), between 1000 % and 2000 % for neutral media (violet bar, pH 5/6/7), above 2000 % and up to 6000 % for alkaline media (orange bar, pink bar, reddish orange bar, magenta bar, and red bar, pH 8/9) as shown in Fig. 6a. Interestingly, when immersed in slightly alkaline medium at pH 8, the swelling ratio of Gel4-1 dramatically depends on the immersion time (Fig. 6c). It increases from 3130 % (green bar) after 1 h to 7050 % (red bar) after 48 h immersion. In more alkaline medium at pH 9, the variation of the swelling ratio is attenuated, from 2500 % (blue bar) after 1 h to 3500 % (green bar) after 48 h immersion (Fig. 6c).

In contrast, this exceptional time dependent swelling behavior of Gel4-1 rapidly reaches an equilibrium at 1 h for pH 3-7.

Mass loss could occur after swelling of hydrogels at various pH values, resulting from the diffusion and washing away of non-crosslinked species, including those initially present or formed by hydrolysis of imine bonds under acidic conditions. Thus the mass loss ratio reflects the crosslinking degree and the stability of hydrogels. Obviously, when immersed in acidic media at pH ≤ 5 for all hydrogels or in the whole pH range for Gel1-1 and Gel2-1 with low CMCS and high Dy contents (Fig. 6b), the mass loss ratio is above 60 % (green bar). Loss ratios below 60 % (violet and gray bars) are obtained only in neutral or alkaline media (pH ≥6) for Gel4-1 with optimal crosslinking, and Gel6-1 and Gel8-1 with decreasing Dy content (Table 1).

These findings indicate that higher Dy content and acidic medium are conducive to the mass loss of hydrogels during swelling. It is thus supposed that unconnected or incompletely connected Dy is predominant in the soluble fraction. Dy rings or homopolymers could be formed during reaction of BTA and Jeffamine (Scheme 1). These species may escape coupling with CMCS in the hydrogel preparation procedure. This assumption is consistent with IR analysis showing the presence of the aldehyde band for Gel1-1 and Gel2-1, and its absence for Gel4-1, Gel6-1 and Gel8-1 samples (Fig. 3). The reaction conditions could be improved by reducing the time and/or lowering the temperature to minimize the formation of these species. The mass loss ratio of Gel4-1 also varies with immersion time at different pH values: in the range of 20-60 % (milk white bar) up to 48 h in neutral / alkaline media at pH ≥ 6, and above 60%

(dusty blue bar) in acidic media at pH=3-5 probably because of the partial hydrolysis of imine bonds (Fig. 6d). There results indicate that freeze dried hydrogels could be interesting for uses in physiological environment owing to higher swelling and better stability.

The pH dependent swelling behaviors of hydrogels could be explained by the electrostatic interactions due to the presence of amino and carboxyl groups along CMCS chains. In fact, the pK a of amino and carboxyl groups is 6.5 and 2.7 [START_REF] Lv | Hygroscopicity modulation of hydrogels based on carboxymethyl chitosan/Alginate polyelectrolyte complexes and its application as pH-sensitive delivery system[END_REF], respectively. Thus, at acidic pH 3 and 4, there is strong electrostatic attraction between negatively charged -COO -and positively charged -NH 3 + groups, which results in shrinkage or low swelling ratio of hydrogels (Fig. 6e). With increasing pH up to 7, there are less protonated NH 3 + and ionized -COO -groups, leading to lower electrostatic attraction and higher swelling. In contrast, at pH 8, the NH 2 groups are not charged, while the electrostatic repulsion between the charged -COO -groups along CMCS chains leads to strong swelling. However, at pH 9, the electrostatic repulsion between the -COO -groups is counterbalanced by the OH -ions in solution. Consequently, the swelling is attenuated as compared to that observed at pH 8.

Changes of the micro-structure of hydrogels were observed by using SEM after 24 h swelling at two pH values. At pH 4, all freeze-dried hydrogels strongly shrink with reduced pore size and pore number (Fig. S4, Supporting Information). Noticeably, the pore size of Gel4-1 decreases from c.a 150 to 100 μm, and the wall thickness increases from c.a 3.5 μm to 15 μm, reminiscent with the contraction of hydrogels due to electrostatic attraction at acidic pH (Figure 6e). In contrast, expansion of the porous structure is observed at pH 8 (Fig. S5, Supporting Information).

The pores wall shows a cracked structure, with the thickness strongly decreasing from c.a 3.5 μm to 200-600 nm due to strong swelling of hydrogels provoked by electrostatic repulsion at basic pH (Fig. 6e).

3.5 Self-healing: CMCS-based hydrogels present interesting self-healing behaviors as evidenced by rheological recovery tests at fixed frequency of 1 Hz and at 37°C. Gel4-1 hydrogels were prepared in Milli-Q water, and in pH 7 and pH 8 buffers in order to examine the self-healing behavior under different swollen conditions. Gelation was realized at 37°C for 24 h. As shown in Fig. 7a, both the storage modulus (G) and loss modulus (G) of Gel4-1 in Milli-Q water slightly decreases until a strain of 20%. Beyond, G dramatically decreases, whereas G rapidly increases.

A crossover point of G and G values is observed at a strain of 60%. Similar profiles are observed for Gel4-1 at pH 7 with a crossover point at 35% (Fig. S6a, Supporting Information). In contrast, Gel4-1 at pH 8 exhibits lower storage modulus because of its highly swollen state as shown in Figure 6a. A crossover point of G and G is detected at 55% at pH 8 (Fig. S6c, Supporting Information). Based on the strain amplitude sweep results, continuous step strain measurements were performed to examine the rheological recovery behavior of Gel4-1. At 1 %, Gel4-1 in Milli-Q water behaves as a hydrogel since G is largely superior to G. As the oscillatory shear strain increases from 1% to 60% and is maintained at 60% for 105 s (Fig. 7b), G becomes lower than G, indicating the destruction of hydrogel structure. Both G and G immediately recover their initial values when the strain is back to 1%. Modulus recovery is observed when larger strains (100, 200, and 300%) and small strain (1%) are alternatively applied. Similar phenomena are also observed for Gel4-1 prepared in pH 7 and 8 buffers (Fig. S6b, S6d, Supporting Information).

Therefore, it could be concluded that dynamic hydrogels exhibit rapid recovery (self-healing)

behavior probably due to the reconstruction of reversible imine bond linkage when they are subjected to alternatively applied high and low oscillatory shear strains.

The self-healing behavior of Gel4-1 was further evidenced with four different macroscopic approaches using one transparent hydrogel sample and another one incorporating yellow lucigenin or red Rhodamine B dyes. First, a hole with diameter of 3 mm was punched at the center of a hydrogel sample prepared in Milli-Q water, and the hole disappeared after 20 min at 37°C (Fig. 7c). In a second approach, transparent and yellow hydrogel samples prepared in Milli-Q water were cut into two semicircular pieces. They became integrated after only 20 min contact at 37°C. The merged piece could be then taken off and support its own weight (Fig. 7d).

In a third approach, one transparent hydrogel and another one containing red Rhodamine B dye prepared in pH 7 buffer were crushed via injection onto a Petri dish using a syringe, and became integrated 20 min later (Fig. 7e). Almost the whole hydrogel was dyed red after 24 h. Similar phenomena were observed in a fourth approach for transparent and dyed red hydrogels prepared in pH 8 buffer (Fig. 7f), demonstrating that the color exchange may be observed via diffusion at the restored self-healed interfaces between different dynagels at pH 7 or pH 8.

These tests strongly demonstrate the outstanding self-healing properties of the dynamic hydrogels -dynagels via reconstruction of reversible imine bond crosslinking, and migration of components or constituent exchanges between different hydrogels. Importantly, the use of these hydrogels with distinct and interchangeable states at different pH conditions would be advantageous for biomedical applications such as drug delivery and tissue engineering.

3.6 Cytocompatibility of hydrogels: Human mesenchymal stromal cells (MSCs) isolated from subcutaneous adipose tissue (AT-MSCs) or from bone-marrow (BM-MSCs) were encapsulated inside Gel4-1 (1 x 10 6 cells/mL), and cultured up to 7 days in proliferative medium at 37 °C to evaluate the cytocompatibility. Compared to control conditions in 2D on TCPS plate or encapsulated in a type I collagen hydrogel, AT-MSCs and BM-MSCs exhibited a round shape in Gel4-1 and not a fibroblastic phenotype (Fig. 8). One day after inclusion in the hydrogel, the large majority of AT-MSCs (95%) and BM-MSCs (99%) were alive as indicated by the green color in confocal microscopy using live/dead assay, whereas only 66% of viability was observed in type I collagen hydrogel (Fig. 8). The two cell types survived for at least 7 days, as only 1 and 4% of dead cells were quantified for AT-MSCs and BM-MSCs, respectively (Fig. 8). These findings demonstrate the excellent cytocompatibility of the hydrogel. It is noteworthy that the size of AT-MSCs and BM-MSCs after 7 days was smaller than that at day 1, suggesting that the This concept, exploiting different physical swelling states depending on pH values, results in the definition of stimuli-responsive dynagels which self-adapt their structure in response to environmental conditions. These 'two-in-one' dynagels may find potential uses in biomedical applications in particular as scaffold in tissue engineering.

ASSOCIATED CONTENT

  Scheme 1. Synthesis route of CMCS-based dynamic hydrogel: a) chemical structures of BTA,

Fig. 1 .

 1 Fig. 1. 1 H NMR spectrum of the dynamer Dy obtained by reaction of BTA and Jeffamine in

Fig. 2 .

 2 Fig. 2. 1 H NMR spectra of the dynamer Dy in pure D 2 O and acidic D 2 O at pH = 1, 3, and 5

Fig. 3 .

 3 Fig. 3. a) FT-IR spectra and b) enlarged view of the 1550-1750 cm -1 wavelength range of

Fig. 4 .

 4 Fig. 4. a) Storage modulus (G) and loss modulus (G) changes as a function of time after mixing

Fig. 5 .

 5 Fig. 5. SEM images of freeze dried hydrogels: (a, b) Gel1-1; (c, d) Gel2-1; (e, f) Gel4-1; (g, h)

Fig. 6 .

 6 Fig. 6. a) Equilibrium swelling ratios, and b) Mass loss ratios of Gel1-1, Gel2-1, Gel4-1, Gel6-1,

Fig. 7 .

 7 Fig. 7. a) Modulus changes as a function of strain of Gel4-1 prepared in Milli-Q water; b)

  pore size was reduced without affecting cell viability. Three-D reconstruction clearly shows the homogeneous distribution of MSCs in the whole hydrogel volume, indicating that the gelation time was compatible with homogenous distribution of the cells without sedimentation (Vid. S7, Supporting Information).

Fig. 8 .

 8 Fig. 8. Cell viability of human AT-MSCs or BM-MSCs in Gel4-1, in comparison with BM-

Table 1 .

 1 Molar and mass composition of CMCS-Dy hydrogels a)

		D-	CMCS b)	Dy	Total polymer
	Sample	glucosamine/Dy			concentration
		molar ratio			[w/v %]

  ) Hydrogels are prepared by mixing CMCS and dynamer solutions at different ratios to a total volume of 12 mL;

	-1	1:1	0.4	85	0.7	0.4	810	6.8	7.5
	Gel2-1	2:1	0.6	127	1.1	0.3	608	5.0	6.1
	Gel4-1	4:1	0.8	170	1.4	0.2	405	3.4	4.8
	Gel6-1	6:1	0.9	191	1.6	0.15	304	2.5	4.1
	Gel8-1	8:1	0.96	204	1.7	0.12	243	2.0	3.7

a

ACKNOWLEDGMENT This work is supported by the scholarship from China Scholarship Council (CSC) under the Grant CSC N° 201706240281, and the Institut Européen des Membranes (Exploratory project "Biostent -Health" of the Internal IEM Call 2017). Authors acknowledge funding support from the Inserm Institute and the University of Montpellier.

Supporting Information.

The following files are available free of charge. NMR spectra of daynamers; rheology data of as-prepared hydrogels; SEM images of freeze dried as-prepared hydrogels and hydrogels after swelling in buffers (pH4, pH8); rheology data of selfhealing hydrogels made in buffers (pH7, pH8), 3-D reconstruction of MSCs after 7 days culture in Gel4-1 (AVI). ABBREVIATIONS CMCS, O-carboxymethyl chitosan; Dy, Dynamer; BTA, Benzene-1,3,5-tricarbaldehyde; PBS, phosphate buffered saline; TMS, tetramethylsilane; MSCs, human mesenchymal stromal cells.