PhD Candidate Xavier Tellier
email: xavier.tellier@enpc.fr

Cyril Douthe

Professor Laurent Hauswirth

Professor Olivier Baverel

Researcher

Linear Weingarten membranes with funicular boundaries

Keywords: Structural design, architectural geometry, Linear-Weingarten surfaces, membranes, funicular structures, static equilibrium, minimal surfaces, fabrication-aware design, planar quad meshes

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

In the past decades, double curvature structural envelopes have become increasingly popular. Double curvature structures are attractive for several reasons. Firstly, their shape are expressive, and can give a unique character to a building. Secondly, curved shapes can be highly efficient mechanically. A remarkable example is the concrete shell of the CNIT (Paris), which spans 52m with a thickness of only 8cm (Figure 1). Three types of structures are nowadays popular for constructing a double curvature envelope: tensile membranes, gridshells and funicular vaults. Historically, membranes were arguably the first non-dome type of double-curvature structure. The exploration of their formal potential was helped by the study of minimal surfaces. These surfaces, so named because they span a boundary with minimal area, correspond to the shape naturally taken by a soap film at equilibrium without external loading. An extensive experimental program was launched by Frei Otto's team [START_REF] Bach | IL18 Forming Bubbles[END_REF] to explore possible forms. In particular, this work led to the design of iconic structures such as the Olympic Stadium Munich (Figure 1).

Many recent research projects in structural engineering focus on how to reduce the environmental impact of buildings. One strong line of research in this respect is the study of funicular structures (see for example [START_REF] Akbarzadeh | On the equilibrium of funicular polyhedral frames and convex polyhedral force diagrams[END_REF][START_REF] Block | Thrust Network Analysis. Exploring Three-dimensional Equilibrium[END_REF][START_REF] Konstantatou | Polarities in structural analysis and design: n-dimensional graphic statics and structural transformations[END_REF][START_REF] Ohlbrock | Combinatorial equilibrium modeling[END_REF]). Funicular structures can resist one given load without bending, with pure axial forces. This makes them highly efficient. As a result, they allow to build a structure with a relatively low amount of materials. They can also be used to build with materials that have low or medium performance [START_REF] Block | Tile vaulted systems for low-cost construction in Africa[END_REF], thus allowing to use more local or less polluting materials. One particular family of funicular surfaces are constant mean curvature (CMC) surfaces. These surfaces are funicular for a uniform pressure load, and correspond to the equilibrium shape of an inflated soap bubbles. Their formal potential for curved architectural envelopes was also demonstrated in [START_REF] Bach | IL18 Forming Bubbles[END_REF]. Applications to fabrication-aware design of gridshells were proposed in [START_REF] Tellier | Discrete CMC surfaces for doubly-curved building envelopes[END_REF]. Right: Cable net of the Olympic Stadium, Münich

Minimal and CMC surfaces, despite being rich groups of surfaces, can sometimes offer a design space that is too limited for a given project. However, they belong to a much wider family of surfaces: the Linear-Weingarten (LW) surfaces, which are the topic of the present paper. This family of surfaces includes minimal surfaces and CMC, but also developable surfaces, pseudo-spheres, and a large array of surfaces in-between. A LW surface is characterized by the fact that, at any point, the mean and Gaussian curvature satisfy an affine relation, with constant coefficients throughout the surface:

𝑎𝐻 + 𝑏𝐾 = 𝑐 / 𝑎, 𝑏, 𝑐 ∈ ℝ (1) These surfaces offer an interesting alternative to popular free-form design methods such as NURBS surfaces, as they offer both mechanical and geometrical properties. These properties will be detailed in section 2. Despite the wealth of knowledge gathered by the differential geometry community on LW surfaces, their application for design purposes have not been studied, mostly because no generation method was available. We introduced in (Tellier, Douthe, Baverel, & Hauswirth, n.d.) an algorithm to generate elliptical LW surfaces fitting closed boundary curves. However, two topics have not been addressed yet.

Firstly, the proposed method could just deal with fixed boundaries, and was therefore not able to model shells with openings bordered by funicular arches, or the geometry of membraned restrained by cables. This paper addresses this shortfall in section 3. Dynamic relaxation is used to find a funicular boundary compatible with the stress field of a LW surface. Secondly, the use of LW for structural design was not explored. Sections 4 investigates their potential for membrane, gridshell and funicular vault design. This paper is an extended version of the conference proceedings presented in [START_REF] Tellier | Linear Weingarten surfaces for conceptual design[END_REF].

2

Properties of Linear Weingarten surfaces 2.1 Mechanical properties [START_REF] Rogers | On the equilibrium of shell membranes under normal loading. Hidden integrability[END_REF] showed an interesting link between LW surfaces and shell mechanics. LW surfaces correspond to equilibrium shapes of membranes under uniform pressure. Membranes are highly efficient mechanical structures: they are shell structures that can resist a given load with pure in-plane stresses, without bending stresses. Furthermore, for a LW membrane subject to uniform pressure of amplitude c, membrane stresses are known explicitly, and principal stresses are aligned with principal curvature directions:

N 11 = a + b 2 k 2 ; N 12 = 0 ; N 22 = a + b 2 k 1 (2)
Where directions 1 and 2 are the principal curvature directions, 𝑘 1 and 𝑘 2 are the associated principal normal curvatures, and N ij are the components of the membrane stress tensor N. These equations can be written in intrinsic form:

𝐍 = 𝑎𝐈 + 𝑏 2 𝐒 ̂ (3)
Where 𝐈 is the identity of the tangent plane of the surface and 𝐒 ̂ is the curvature tensor (known as shape operator in surface theory), rotated by 90° in the tangent plane (in differential geometry notations: 𝐒 ̂= J𝐒).

A particular case is obtained when 𝑐 = 0: the LW surfaces then correspond to self-stressed membranes, like the ones showed in Figure 8. The fact that stresses are known analytically allows to give quick structural feedback to the designer, without having to transfer the geometry to an FE software.

Geometrical properties

Curvature lines of a surface can be used in many different ways to rationalize the fabrication of a double-curvature structure. In particular, they can be used to design gridshells covered with planar quadrangles and with nodes without geometrical torsion [START_REF] Pottmann | Architectural geometry[END_REF]. On an arbitrary surface, curvature lines tend to form an erratic network. For example, Figure 2 shows the curvature lines on the roof of the Visconti court in Le Louvres, designed by architects Ruddy Riciotti ad Mario Bellini. The configuration is badly suited for a gridshell: there is a high variation in the size and aspect ratio of faces, beams have a poor mechanical alignment, and the pattern is subjectively unaesthetic -consequently, a triangular pattern was chosen. On LW surfaces, beyond the mechanical properties discussed in the previous section, curvature lines tend to form a fair network. This property is due to the fact they are offsets of constant mean curvature surfaces [START_REF] Eisenhart | A treatise on the differential geometry of curves and surfaces[END_REF]. In particular, there cannot be singularities of index ½ like the ones highlighted with red dots in Figure 2 [START_REF] Hartman | Umbilical Points and W-Surfaces[END_REF]. A second important geometrical property is that equation ( 1) is a 2 nd order partial differential equation on surface coordinates. Hopf [START_REF] Hopf | Differential Geometry in the Large[END_REF] showed that if 𝑎² + 𝑏𝑐 > 0, this equation is elliptic. This means that we can solve the following Dirichlet problem: generating a LW surface on a given closed boundary curve. This type of problem is often encountered when covering an existing atrium.

Generation

In this section, we propose a generation method of LW surfaces with cable boundaries. The resolution is iterative, and based on triangular meshes. Interior points and cable points are treated differently.

Interior points

In [START_REF] Tellier | Linear Weingarten surfaces for conceptual design[END_REF], we proposed a first method to generate LW surfaces with fixed boundaries. We then introduced a more robust method in [START_REF] Tellier | Morphogenesis of curved structural envelopes under fabrication constraints[END_REF]). This method relies on a discrete model of LW surfaces. Discrete LW surfaces are defined as the triangular meshes 𝑇 which are the critical points of a linear combination E 1 of the area, total mean curvature, and volume:

E 1 (𝑇) = a 2 Area(𝑇) + bH tot (𝑇) + cV𝑜𝑙(𝑇) (4)
Equivalently, these meshes verify at each vertex 𝑣: a H ⃗⃗ (v) + b K ⃗ ⃗ (v) = c n ⃗ (5) where:

• H ⃗⃗ (v) = -3 2‖𝐴 (𝑣)‖
∇ 𝑝 Area(T) is the mean curvature vector of vertex v (𝐴 (𝑣) is the vector area of vector of the polyline joining the vertices adjacent to 𝑣);

• K ⃗ ⃗ (v) = -3 2‖𝐴 (𝑣)‖ ∑ 𝜃 𝑒 𝑒 ‖𝑒 ‖ 𝑒 ~𝑣
is the Gaussian curvature vector ot vertex 𝑣. The sum is taken over the edges 𝑒 adjacent to 𝑣, and depends on the dihedral angle 𝜃 𝑒 of each edge;

• n ⃗ is the normal vector at 𝑣, defined as a unit vector aligned with the vector area 𝐴 (𝑣). A simple gradient descent is not robust to generate LW surfaces, as functional 𝐸 1 is not convex. In [START_REF] Tellier | Morphogenesis of curved structural envelopes under fabrication constraints[END_REF], we proposed an algorithm based on iterative projections. It uses the partition of elliptical LW surfaces in two families, called type I and type II. We will use this method for interior points (i.e. points not located on the mesh boundary) in this article.

Cable boundary

Tensile membranes are usually held at their boundaries by cables. In this section, we investigate how this can be modelled with LW surfaces, and how to generate the shape of LW surfaces bordered by cables. This question is also of interest for the design of funicular shells with openings, for which openings are realized with funicular arches.

This problem can be understood as a generalization of the Euler-Plateau problem introduced in [START_REF] Giomi | Minimal surfaces bounded by elastic lines[END_REF]. This problem considers minimal surfaces for which the boundary is an elastica, i.e. a curve of imposed length minimizing the integral of its squared curvature. [START_REF] Pérez | Computational design and automated fabrication of Kirchhof-Plateau surfaces[END_REF] generalized this problem to Kirchoff rods, which are minimizers of a sum of a curvature and a torsional energy. Our problem is a generalization of the one of Giomi in a sense that we do not consider minimal, but LW surfaces. We are however interested only in boundaries curves without bending energy, as we aim at obtaining funicular shapes.

This section is divided in four parts. First, we derive the geometrical equations verified by the boundary, and how the cable tension varies. Based on the equilibrium equations, we propose a discrete model of the boundary. We then introduce a generation method using dynamic relaxation. Finally, the generation method is validated on a test case.

Equilibrium and geometry

One convenient method to generate LW surfaces bordered by cables would be to minimize an energy composed of the sum of an energy associated with the LW surface and the elastic energy of the cable: 𝐸 = 𝐸 𝐿𝑊 + 𝐸 𝑐𝑎𝑏𝑙𝑒 . However, the variational formulations of LW surfaces in the literature address only surfaces without boundaries. We therefore use an equilibrium approach to treat the boundary.

Equilibrium of a membrane cable

Figure 3 Equilibrium of a boundary cable

Let us consider a cable parametrized by arc length 𝑠 with an associated Darboux frame (𝐭, 𝐯, 𝐧), where 𝐭 is the tangent vector to the cable, 𝐧 is the surface normal, and 𝐯 is a unit vector in the tangent plane orthogonal to the boundary (Figure 3). The cable has an a tension T and is subjected only to membrane stresses (with stress tensor 𝐍), other forces are supposed to be negligible. The equilibrium of the cable can then be written:

d ds (T𝐭) + 𝐍. 𝐯 = 0
This can be developed into:

dT ds 𝐭 + T d𝐭 ds + N vv 𝐯 + N tv 𝐭 = 0
We can express the derivative of the tangent vector 𝐭 using the derivatives of the vectors of the Darboux frame:

d𝐭 ds = k g 𝐯 + k n 𝐧
Projecting the equilibrium equation along 𝐯, 𝐭 and 𝐧, we obtain successively: T k g + N vv = 0 (6)

dT ds = -N tν (7) k n = 0 (8)
We note that, in the case of a sliding cable, 𝑁 𝑡𝜈 = 0, so 𝑑𝑇 𝑑𝑠 = 0 : the tension in the cable is constant.

LW membrane stresses at boundary

Let us now express the LW membrane stresses at the boundary. We recall that the corresponding membrane stress tensor is:

𝐍 = a 𝐈 + b 2 𝐒 ̂
For a unit vector 𝐯 of the tangent plane perpendicular to the boundary: ) 𝒆 𝟐 . We obtain:

𝐍. 𝐯 = a 𝐯 + b 2 𝐒 ̂.
𝐯. 𝐍. 𝐯 = a + b 2 (cos θ sin θ) [ k 2 0 0 k 1 ] ( cos θ sin θ ) = a + b 2 (k 2 cos 2 θ + k 1 sin 2 θ) = a + b 2 (k 2 sin 2 (θ - π 2 ) + k 1 cos 2 (θ - π 2 )) 𝑁 𝑣𝑣 = 𝑎 + 𝑏 2 𝑘 𝑛 (9)
where 𝑘 𝑛 is the normal curvature of the boundary. We observe that the normal membrane stress component due to the term 𝑏𝐾 in 𝑎𝐻 + 𝑏𝐾 = 𝑐 is proportional to the normal curvature of the boundary, which vanishes at equilibrium. The shear component is:

𝐭. 𝐍. 𝐯 = a + b 2 (sin θ -cos θ) [ k 2 0 0 k 1 ] ( cos θ sin θ ) = b 2 (k 2 -k 1 ) cos θ sin θ N tv = b 2 τ g (10)
where 𝜏 𝑔 is the geodesic torsion of the boundary (see [START_REF] Do Carmo | Differential geometry of curves and surfaces[END_REF] for the derivation of 𝜏 𝑔 = (𝑘 2 -𝑘 1 ) cos 𝜃 sin 𝜃).

Geometry of a cable attached to a LW membrane

By combining the equilibrium equations of the cables with the LW stresses, we find that the geometry of the boundary must fulfil two geometrical differential equations.

The first one was already derived from the equilibrium of the cable (equation 8):

𝑘 𝑛 = 0 (8)
It implies that the cable follows an asymptotic line on the surface. For the second one, we first observe using equation ( 9) that the normal stress at the boundary is 𝑁 𝑣𝑣 = 𝑎. Equation ( 6) implies:

𝑘 𝑔 = 𝑎 𝑇 This equation can be combined with the equation 7 to remove variable 𝑇. It has different geometrical implications depending on whether 𝑏 is null or not.

• If 𝒃 = 𝟎 (i.e. the surface is CMC), we obtain from equation ( 10) that the shear is null:

𝑁 𝑡𝜈 = 0
Consequently, equation ( 6) yields:

𝑑𝑇 𝑑𝑠 = 0
As a result:

𝑑𝑘 𝑔 𝑑𝑠 = 0 (12)
This means that the geodesic curvature of the boundary is constant. Not all CMC surfaces admit an asymptotic line with constant geodesic curvature. For example, one can find such lines on a helicoid, but not on a catenoid. The presence of a boundary cable has therefore a strong impact on the surface geometry.

• If 𝒃 ≠ 𝟎, we have to consider separately the case of a sliding cable and the case of a cable attached to the membrane: -Sliding cable: 𝑁 𝑡𝜈 = 0 implies that :

𝜏 𝑔 = 0 (13)
𝜏 𝑔 = 0 means that the cable follows a curvature line. Since both the normal curvature and the torsion of the boundary are null, it is necessarily planar.

- Fixed cable: b 2 τ g = -N tv = - dT ds = -a d ds ( 1 k g ) 𝑏 2 𝜏 𝑔 = 𝑎 𝑘 𝑔 2 𝑑𝑘 𝑔 𝑑𝑠 (14)
Similarly to the case 𝑏 = 0, the quantity

k g ′ τ g k g 2
is not constant for an arbitrary asymptotic line of an arbitrary LW surface.

Discrete model

We propose to discretize boundary cables as series of particles connected by springs. This section describes the discrete model of forces acting on particles for a non-sliding cable. Each particle is subject to a tension from the cable and to membrane stresses (Figure 4). Cable forces are computed from the elastic elongation of the cable: 𝒇 𝒄𝒂𝒃𝒍𝒆 = 𝑘(𝑙 1 -𝑙 1,𝑖 ) 𝒖 𝟏 + 𝑘(𝑙 2 -𝑙 2,𝑖 ) 𝒖 𝟐 Where:

•

𝐮 𝒊 is the unit vector from P to 𝑃 𝑖 :

𝐮 𝟏 = PP 1 ⃗⃗⃗⃗⃗⃗⃗⃗ ‖PP 1 ⃗⃗⃗⃗⃗⃗⃗⃗ ‖ ; 𝐮 𝟐 = PP 2 ⃗⃗⃗⃗⃗⃗⃗⃗ ‖PP 2 ⃗⃗⃗⃗⃗⃗⃗⃗ ‖
• 𝑙 1 and 𝑙 1,𝑖 are respectively the current and the initial length of line 𝑃𝑃 1 .

Membrane forces are computed by integrated the stresses calculated in the previous section over a portion of cable of length

‖PP 1 ⃗⃗⃗⃗⃗⃗⃗⃗ ‖+‖PP 2 ⃗⃗⃗⃗⃗⃗⃗⃗ ‖ 2
centered at P:

𝒇 𝒎𝒆𝒎𝒃𝒓𝒂𝒏𝒆 = -𝑎 ∇ p A - 𝑏 2 𝑇 𝑔 𝒕
Where:

• ∇ p A is the gradient of the summed areas of the mesh triangles adjacent to 𝑃. This term corresponds to the normal component of membrane stresses 𝑁 𝑣𝑣 = 𝑎 (equation 6.9) integrated over the length of the portion of cable. We ignore the term 𝑏 2 𝑘 𝑛 (𝒕) of equation 6.9, because it vanishes at convergence anyway.

• 𝐭 is the tangent vector of the boundary curve, computed as :

𝐭 = P 2 -P 1 ‖P 2 P 1 ‖
• 𝑇 𝑔 is the halved torsion angle from the normal at P2 to the normal at P1, computed as:

𝑇 𝑔 = 1 2 arcsin(det (𝐧 𝟏 𝐩 , 𝐧 𝟐 𝐩 , 𝐭))
Where 𝐧 𝐢 𝐩 is the projection of the normal vector 𝐧 𝐢 at Pi onto the plane (P,t,n). The normal vector 𝐧 is simply computed as the normal of the plane (𝑃 𝑃 1 𝑃 2 ). To model a sliding cable, one just needs to write the membrane force as 𝒇 𝒎𝒆𝒎𝒃𝒓𝒂𝒏𝒆 = -𝑎 ∇ p A .

Note 1: This way of computing the normal requires to special treatment if boundary points are aligned (or nearly aligned).

A more robust scheme would be to approximate locally the surface by a paraboloid. Note 2: The calculation of the torsion requires a consistent orientation of the boundary curve. This orientation is determined by the orientation of the tangent vector 𝐭.

Figure 4 Equilibrium of vertex of a boundary cable

Dynamic relaxation

We look for the equilibrium position of the boundary cables with a dynamic relaxation approach [START_REF] Otter | Dynamic relaxation[END_REF]. This procedure is done simultaneously with the projection approach for the non-boundary points, as will be explained in section 3.3. The motion of a particle P of a cable is given by: 𝑚𝑃 ̈= 𝒇 𝒄𝒂𝒃𝒍𝒆 + 𝒇 𝒎𝒆𝒎𝒃𝒓𝒂𝒏𝒆 This equation can for example be discretized as [START_REF] Douthe | Etude de structures élancées précontraintes en matériaux composites , application à la conception des gridshells[END_REF]:

𝑃 𝑖,𝑡 = 𝑃 𝑖,𝑡-1 + 𝑃 ̇𝑖,𝑡-1 Δ𝑡 + Δ𝑡² 𝑚 𝑖 ( 𝒇 𝒄𝒂𝒃𝒍𝒆 + 𝒇 𝒎𝒆𝒎𝒃𝒓𝒂𝒏𝒆 )
The interior points are computed by projections at each iteration (as explained in section 3.1), so they do not conserve their momentum. In order to have a consistent model for both boundary and interior points, we ignore particle momentum at the boundaries by setting 𝑃 ̇𝑖,𝑡-1 = 0. This is equivalent to applying a so-called kinematic damping at each step. Our numerical scheme therefore reads:

𝑃 𝑖,𝑡 = 𝑃 𝑖,𝑡-1 + Δ𝑡² 𝑚 𝑖 ( 𝒇 𝒄𝒂𝒃𝒍𝒆 + 𝒇 𝒎𝒆𝒎𝒃𝒓𝒂𝒏𝒆 )

We use Δ𝑡 = 1 and a mass proportional to the length of cable, with constant lineic mass:

𝑚 𝑖 = 𝜌 ‖PP 1 ⃗⃗⃗⃗⃗⃗ ‖ + ‖PP 2 ⃗⃗⃗⃗⃗⃗ ‖ 2
Where 𝜌 is adjusted manually to obtain proper speed while avoiding oscillations (which appear near equilibrium if a value too low is chosen).

We iterate until a position is found where ‖𝒇 𝒄𝒂𝒃𝒍𝒆 + 𝒇 𝒎𝒆𝒎𝒃𝒓𝒂𝒏𝒆 ‖ < 𝜺. This algorithm converged in all the tested cases. Once convergence is reached, equilibrium is guaranteed. Mode advanced numerical schemes could be developed to deal with boundary and interior points in a more uniform approach. This could be achieved by deriving a variational formulation of the cable-boundary problem.

Validation

Once the algorithm has converged, the geometry of boundaries can be validated by checking that the two differential equations ( 8) and ( 14) are verified. 𝑘 𝑛 = 0 is automatically verified at equilibrium: since 𝒕,𝒖 𝟏 and 𝒖 𝟐 are always coplanar, the membrane force ∇ p A belongs to that plane at equilibrium. The tangent plane of the membrane, oriented by 𝒕 and ∇ p A, contains then the circle fitting 𝑃𝑃 1 𝑃 2 : the curvature of the cable is purely geodesic, so the normal curvautre is null. Equation ( 14) can be validated by calculating the geodesic curvature at each point. We evaluate it by taking the inverse of the radius of the circle fitting 𝑃𝑃 1 𝑃 2 . We then use a finite difference scheme to evaluate the derivative:

dk g ds = k g (P 2 ) -k g (P 1 ) l 1 + l 2 Figure 5 shows how equation ( 14) is verified for a LW surface bordered by cables anchored at four points. The ratio

𝑏 2 𝜏 𝑔 𝑎 𝑘 𝑔 ′ 𝑘 𝑔 2 ⁄
is theoretically 1. In practice, its value is not exactly 1, but remains between 0.9 and 1.1, except at cable extremities (blue dots) and at one point near the middle of the cables (red dots) for which the numerator and denominator are near zero. The fact that the value is not exactly 1 can be mostly explained by the fact that torsion and geodesic curvature are calculated by different discrete models. Therefore, the generation method yields a cable position at equilibrium with the membrane tension, with a precision that is sufficient for our purpose.

Global resolution

The projection algorithm for interior points and the dynamic relaxation are used simultaneously in an interactive manner to deform a mesh towards a LW shape with valid cable boundaries. A user has four parameters at hand: the LW parameters 𝑎, 𝑏 and 𝑐, and the cable stiffness 𝑘. The global resolution is summarized in Figure 6. Iterations are stopped if vertex displacements are below a certain value. This iterative algorithms was implemented by scripting the software Rhino/Grasshopper. All the numerical images of this section were generated from that script. During this iteration process, the initial mesh might undergo large deformations. To avoid ending up with highly anisotropic faces, a remeshing tool was implemented. It performs an edge flipping (see for example [START_REF] Botsch | Polygon mesh processing[END_REF]) when a vertex is found inside the circumscribed of a neighbour triangle (more precisely, if the projection of the vertex is in the circle). Remeshing is manually triggered by the user.

Design applications

Tensile membranes

Tensile membranes allow to cover wide spans with a very low weight and limited cost. Formerly reserved for temporary structures such as circus tents, the recent technical advances in materials make it possible to ensure their durability for up to several decades.

Existing form finding methods

Tensile membranes have no out-of-plane stiffness, and their weight is negligible compared to external loads. Therefore, their shape is entirely determined by the geometry of their boundary and the distribution of the prestressing (plus the tension in valley cables if some are included). The geometrical design of a tensile membrane therefore amounts to determining these parameters. There are two main families of methods for this purpose.

The first is to build a minimal surface on the desired boundaries. Minimal surfaces can easily be created with wire and soap. There are also many easy-to-use algorithms for generating a minimal surface. These algorithms are based on the minimization of the area of a mesh. The resulting shapes correspond to membranes in which the prestress is isotropic and uniform. These surfaces are mechanically efficient, but offer a very limited design freedom.

The second is to prescribe pretension stresses using methods such as force densities [START_REF] Schek | The force density method for form finding and computation of general networks[END_REF], finite elements, or the updated reference strategy [START_REF] Bletzinger | Computational methods for form finding and optimization of shells and membranes[END_REF]. These methods allow to explore all the possible self-stressed forms on a given boundary, but require more work from the designer: The orientation of the principal stresses must be defined, trials and errors must be made on the values and directions of the pretension.

Linear Weingarten membranes

As discussed in section 2, LW surfaces with c=0 correspond to equilibrium shapes of self-stressed membranes. Therefore, our generation method can offer a new way of finding forms. With a=1, these surfaces verify:

𝐻 + 𝑏𝐾 = 0 (7) A designer therefore has one degree of freedom (the parameter b) to design a surface WL on a given edge; unlike with minimal surfaces, which are entirely determined by their boundary. Equation 2shows that the parameter b modifies the prestressing ratio between the two main directions of curvature. Figure 7 and Figure 8show the effect of parameter b on the shape of a membrane on a fixed boundary. When 𝑏 → ±∞, the surface tends towards a developable surface (depending on the boundary, the surface might need to split in several portions to become developable).

Figure 7 LW meshes generated on the same boundary with different values of parameter b Figure 8 Series of "big top" membranes generated on the same boundary with different values of b

Our generation method is therefore a form-finding tool which is intermediate (in terms of design freedom and required labour) between minimal surfaces and the more advanced methods discussed in section 4.1.

Choice of parameter b

The parameter b affects the following aspects, which are often critical for the design of a tense membrane.

Height of canvas

By adjusting the value of b, one can change the average height of the surface. This effect is shown in Figure 10, where a minimal surface and a LW surface are built on the same two contours.

Water ponding

Tensile membranes must have a minimum slope in order to prevent the formation of water ponds. Parameter b also allows to modify this maximum slope, as shown in Figure 10.

Expressivity

Minimal surfaces tend to be relatively flat over large portions, this effect can be seen in Figure 9. By modifying parameter b, one can control the prestressing ratios (as shown in Figure 10), and curve these flat areas.

Stability

The Gaussian curvature of a surface characterizes its non-flatness. If too low, the surface is too flat, and may oscillate or even beat with the wind. Parameter b allows to modify the distribution of the Gaussian curvature to avoid this problem, as shown in Figure 10. Coming back to the LW surface of Figure 4, we observe in the top view that boundary cables are not circular. Geodesic curvature is higher near the low points. This is expected as the tension is higher in the direction of the diagonal between the high points.

Gridshells

Gridshells are shells composed of a network of beams. In order to simplify the fabrication of such structures, it is often desirable to align the beams with principal curvature directions of the underlying surface (Pottmann et al., 2007): the surface can then be cladded by planar glass panels, only four beams meet at the nodes (unlike triangular meshes with complex 6-valent nodes), and the nodes are torsion-free, i.e. the median planes of the beams meet on a common axis. This property, illustrated in Figure 12, simplifies significantly the fabrication of the nodes, which are the costliest elements of this type of structure.

As discussed in section 2, curvature lines on LW surfaces have two properties. Firstly, they form a regular pattern, and secondly they are aligned with principal stresses under uniform pressure loading (assuming membrane-compatible supports are provided). When beams are not aligned with principal stresses, the grid is subjected to shear forces. Contrary to triangular lattices, quadrangular ones have a shear mechanism. Strength and rigidity in shear is usually obtained by one of these two means: adding diagonals (e.g. cables) or having clamped beam-beam connections. The second option is usually cheaper and preferred by architects, because cables tend to obstruct the view. However, it is impractical if shear forces are too high. In LW gridshells, the shear is null for one major load case: uniform wind load. LW gridshells are therefore good candidates to design cable-free gridshells. Future case studies will be performed to assess the variation of the orientation of the principal stresses under asymmetrical loads.

Figure 12 shows how a LW gridshell can be used to design an alternative structure for the glass roof of the British Museum in London, whose geometry is based on a triangular mesh. A surface surprisingly close to the real one can be obtained with a LW surface. The beam orientation is obtained by remeshing the LW triangular mesh following principal curvature directions.

Funicular vaults

Figure 14 shows two LW vaults which are funicular in tension under a uniform upwards pressure load. The top image shows a CMC surface constructed on a circular boundary. Only a portion of the boundary is modelled as a cable. The resulting surface is not spherical, and is somehow shaped like a water droplet. When subjected to a downwards pressure load, the cable becomes a compression funicular polygon, which can be materialized by an arch with very low bending and tension resistancesuch as a masonry arch.

The surface at the bottom is bordered by cables which are anchored at six points. Forces are transmitted only to the cornersthis is often a constraint for covering of existing atriums. Interestingly, a small dome emerges at the angle of the L.

Single curvature panels

The method can be used to generate developable surfaces. This type of surface is useful to clad an envelope with thin flexible panels such as single layer tempered glass or metal sheets. Developable surfaces are a special type of Linear-Weingarten surfaces: they verify 𝑎 = 𝑐 = 0. For a given boundary, depending on the sign of b, two surfaces can be obtained by the algorithm. Together, they form the convex envelope of the curves. Therefore, the method is restricted to convex developable surfaces. This is discussed more in details in (Tellier et al., n.d.). An example is shown in Figure 15.

A curve is drawn on a sphere, it delimits two portions of spheres, the upper one is triangulated. The right picture shows the result of the optimization (mesh edges are not displayed). The final shape is composed of portions of planes, portions of cylinders, and portions of general developable surfaces.

Figure 15 Developable surface generated from a curve on a sphere

Conclusion

In this paper, we presented how Linear Weingarten surfaces can be used as a design tool for double curvature structures. These surfaces, previously confined to the field of mathematics, have interesting mechanical, manufacturing, and aesthetic properties. We proposed a method to generate LW surfaces with both fixed and funicular boundaries. The method combines a projection method of the interior points with a dynamic relaxation for the cable points. Compared to other form-finding methods, Linear Weingarten membranes offer an interesting design freedom with a relatively low preprocessing work required from a designer. It is therefore well suited for a conceptual design tool. This study also allowed to advance the understanding of funicular membrane shapes. We showed how Linear Weingarten can be used to design self-stressed membranes, gridshells, and funicular vaults.

List of figure captions

Figure 1

 1 Figure 1 Double curvature structural envelopes. Left: The concrete shell of the CNIT, Paris (©Alexandre Prevot). Right: Cable net of the Olympic Stadium, Münich

Figure 2

 2 Figure 2 Curvature lines of the roof of the Visconti court in Le Louvre (reproduced with permission from (Wallner & Pottmann, 2011))

Figure 5 LW

 5 Figure 5 LW Verification of the geometry of boundary cables

Figure 6

 6 Figure 6 Combined resolution of interior points, cables and fixed points.

Figure 9

 9 Figure 9 Left: Minimal surface. Right: LW surface on the same boundary.

Figure 10

 10 Figure 10 Comparison of the geometry and prestressing of a minimal and a LW membrane

Figure 11

 11 Figure 11 Examples of LW membranes bordered by cables

Figure 12

 12 Figure 12 Proposition of an alternative geometry for the glass roof of the British Museum. The proposed geometry has flat panels, torsion-free nodes, and perfect mechanical beam orientation under uniform wind load. (bottom left picture: Romain Mesnil).

Figure 13

 13 Figure 13 highlights the role of parameter b in the form of a gridshell. The geometry of the boundary of the Chadstone gridshell is replicated (height variations of the boundary are neglected). The upper figure shows a constant mean curvature surface (b=0) generated on this boundary. Heights are indicated at key points. The shape looks remarkably close to the one of the real gridshell. When b is increased, we observe that the height difference between hill tops and saddle points decreases: the crest line becomes more level. The opposite behaviour is observed when decreasing b: the higher variation of height makes the surface more expressive.

Figure 13

 13 Figure 13 Linear-Weingarten surfaces generated on a boundary similar to the one of the Chadstone gridshell. Heights with respect to the plane containing the boundary line are indicated in blue. (Picture © Seele). Compared to the case b=0 (CMC surface), increasing b levels out the height, while decreasing b increases the height difference between saddle points and hill tops.

Figure 14

 14 Figure 14 Examples of LW surfaces bordered by funicular arches

Figure 1

 1 Figure 1 Double curvature structural envelopes. Left: The concrete shell of the CNIT, Paris (©Alexandre Prevot). Right: Cable net of the Olympic Stadium, Münich Figure 2 Curvature lines of the roof of the Visconti court in Le Louvre (reproduced with permission from (Wallner & Pottmann, 2011)) Figure 3 Equilibrium of a boundary cable Figure 4 Equilibrium of vertex of a boundary cable Figure 5 LW Verification of the geometry of boundary cables Figure 6 Combined resolution of interior points, cables and fixed points.Figure 7 LW meshes generated on the same boundary with different values of parameter b Figure 8 Series of "big top" membranes generated on the same boundary with different values of b Figure 9 Left: Minimal surface. Right: LW surface on the same boundary.Figure 10 Comparison of the geometry and prestressing of a minimal and a LW membrane Figure 11 Examples of LW membranes bordered by cables Figure 12 Proposition of an alternative geometry for the glass roof of the British Museum. The proposed geometry has flat panels, torsion-free nodes, and perfect mechanical beam orientation under uniform wind load. (bottom left picture: Romain Mesnil).Figure 14 Examples of LW surfaces bordered by funicular arches

Figure 7

 7 Figure 1 Double curvature structural envelopes. Left: The concrete shell of the CNIT, Paris (©Alexandre Prevot). Right: Cable net of the Olympic Stadium, Münich Figure 2 Curvature lines of the roof of the Visconti court in Le Louvre (reproduced with permission from (Wallner & Pottmann, 2011)) Figure 3 Equilibrium of a boundary cable Figure 4 Equilibrium of vertex of a boundary cable Figure 5 LW Verification of the geometry of boundary cables Figure 6 Combined resolution of interior points, cables and fixed points.Figure 7 LW meshes generated on the same boundary with different values of parameter b Figure 8 Series of "big top" membranes generated on the same boundary with different values of b Figure 9 Left: Minimal surface. Right: LW surface on the same boundary.Figure 10 Comparison of the geometry and prestressing of a minimal and a LW membrane Figure 11 Examples of LW membranes bordered by cables Figure 12 Proposition of an alternative geometry for the glass roof of the British Museum. The proposed geometry has flat panels, torsion-free nodes, and perfect mechanical beam orientation under uniform wind load. (bottom left picture: Romain Mesnil).Figure 14 Examples of LW surfaces bordered by funicular arches

𝐯

  In order to obtain the component 𝑁 𝑣𝑣 of the membrane stress tensor, we express 𝐯 in the basis of principal curvature directions: 𝐯 = cos 𝜃 𝒆 𝟏 + sin 𝜃 𝒆 𝟐 where 𝒆 𝟏 and 𝒆 𝟐 are unit vectors aligned with principal directions. The vector 𝐭 can

	be expressed in that same basis: 𝐭 = cos (𝜃 -	𝜋 2	) 𝒆 𝟏 + sin (𝜃 -	𝜋 2

Acknowledgements

This work is supported by Labex MMCD (http://mmcd.univ-paris-est.fr/). The authors would like to thank Johannes Wallner for his help on discrete geometrical models.


Marta Salvador Castell

Maksym Golub

Nicolas Martinez

Jacques Ollivier

Judith Peters

Philippe Oger

The first study on the impact of osmolytes in whole cells of high temperature-adapted microorganisms

The hyperthermophilic piezophile Thermococcus barophilus displays a strong stress response characterized by the accumulation of the organic osmolyte mannosylglycerate during growth under sub-optimal pressure conditions (0.1 MPa). Taking advantage of this known effect, the impact of osmolytes in piezophiles in an otherwise identical cellular context was investigated, by comparing T. barophilus cells grown under low or optimal pressures (40 MPa). Using neutron scattering techniques, we studied the molecular dynamics of live cells of T. barophilus at different pressures and temperatures. We show that in presence of osmolytes, cells present a higher diffusion coefficient of hydration water and an increase of bulk water motions at high temperature. In absence of osmolytes, the T. barophilus cellular dynamics is more responsive to high temperature and high hydrostatic pressure. These results give therefore clear evidences for a protecting effect of osmolytes on proteins.

2

Introduction

Only few decades ago, it was thought that most extreme environments were too hostile to harbour life. However, since the major discoveries of the 1970s [START_REF] Brock | Sulfolobus: A New Genus of Sulfur-Oxidizing Bacteria Living at Low PH and High Temperature[END_REF][START_REF] Yayanos | Isolation of a Deep-Sea Barophilic Bacterium and Some of Its Growth Characteristics[END_REF][START_REF] Pikuta | Microbial Extremophiles at the Limits of Life[END_REF] , it is admitted that possibly no place on Earth is too extreme to support cellular life. Indeed, nowadays, it is difficult to imagine limits for life on Earth and only one environmental condition seems limiting: the presence of liquid water. Amongst extreme biotopes, the most intriguing may be the deep-hot biosphere, where organisms experience high hydrostatic pressures (HHP) and high temperatures (HT) simultaneously. In this context, HHP refers arbitrarily to pressures above 10 MPa. It is among the prokaryotes (cells lacking a nucleus), and especially within the archaea domain that we find organisms adapted to the most extreme environmental conditions. The cellular adaptation to extreme conditions has been largely studied by different approaches from genomics, to protein and lipid studies [START_REF] Campanaro | Laterally Transferred Elements and High Pressure Adaptation in Photobacterium Profundum Strains[END_REF][START_REF] Oger | The Many Ways of Coping with Pressure[END_REF][START_REF] Oger | Adaptation of the Membrane in Archaea[END_REF][START_REF] Brininger | The More Adaptive to Change, the More Likely You Are to Survive: Protein Adaptation in Extremophiles[END_REF] . It has allowed the identification of several adaptive routes that may, or may not be, shared between the different environmental stressors. For example, the sequence, and consequently the structure, of proteins is modified to respond to the specificities of some environment stressors (salinity, temperature) in order to maintain protein function and the molecular dynamics in the so-called "corresponding state principle" window [START_REF] Vihinen | Relationship of Protein Flexibility to Thermostability[END_REF][START_REF] Jaenicke | Stability and Stabilization of Globular Proteins in Solution[END_REF] . Today, we are starting to understand the molecular basis of the structural and genetic adaptation to high temperature in protein of thermophiles. This implies the substitution of a set of preferred amino-acids in the sequence of the proteins. However, extensive whole genome studies were unsuccessfull to identify the structural and genetic adaptation to extreme pressure to date. Recently, we have proposed the first molecular model for the adaptation to HHP in the archaeal order of the Thermococcales. Thermococcales are hyperthermophiles isolated from hydrothermal vent systems that can be found from the surface to the deepest part of the oceans (4500 m below sea level) and grow optimally at temperatures ranging from 75 to 105 °C. In our study, we compared Thermococcus kodakarensis, a piezosensitive (sensitive to HHP) archaeon to Thermococcus barophilus, a piezophile (requiring HHP for life) model species [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF] . Although pangenomic studies have failed to identify the genetic basis of the structural adaptation to HHP in these organisms, using incoherent neutron scattering on whole cells, we could demonstrate that the structural proteome adaptation to HHP occurs through variations of molecular dynamics in the proteome and in the hydration shell [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF] . In contrast to the Le Chatellier principle [START_REF] Chatelier | Sur Un Enoncé Général Des Lois Des Équilibres Chimiques[END_REF] , it is expected that high pressure adapted a. Address here. b. Address here. c. Address here. † Footnotes relating to the title and/or authors should appear here. Electronic Supplementary Information (ESI) available: [details of any supplementary information available should be included here]. See DOI: 10.1039/x0xx00000x proteins will resist increasing pressure better than piezosensitive ones to preserve the required protein flexibility of the active site. However, we found that proteins from the piezophile T. barophilus have a higher pressure sensitivity, undergo a phase-like transition with increasing pressure to reach their functional state and have a lower hydration shell contribution and dynamics. Furthermore, in contrast as well with expectations from high pressure studies, we showed that piezophilic cells perceive sub-optimal pressures as a stress, and counteract the effect of low-pressure stress by accumulating the organic osmolyte, mannosylglycerate (MG) [START_REF] Cario | Molecular Chaperone Accumulation as a Function of Stress Evidences Adaptation to High Hydrostatic Pressure in the Piezophilic Archaeon Thermococcus Barophilus[END_REF] . To maintain the appropriate cell turgor and to restore the cell volume, all organisms accumulate low-molecular-weight osmolytes that are mainly organic solutes in response to cold, heat, pH and HHP stresses [START_REF] Gao | Crowders and Cosolvents-Major Contributors to the Cellular Milieu and Efficient Means to Counteract Environmental Stresses[END_REF] . These solutes are amino acids and derivatives, polyols, sugars and derivatives, methylamines, and methylsulfonium compounds, which fall into only a few chemical categories: amino acids and derivatives such as trimethylamine oxide (TMAO) and small carbohydrates such as sugars, sugar derivatives, polyols and cyclitols [START_REF] Wood | Osmosensing and Osmoregulatory Compatible Solute Accumulation by Bacteria[END_REF][START_REF] Empadinhas | Diversity and Biosynthesis of Compatible Solutes in Hyper/Thermophiles[END_REF][START_REF] Al-Ayoubi | Influence of Cosolvents, Self-Crowding, Temperature and Pressure on the Sub-Nanosecond Dynamics and Folding Stability of Lysozyme[END_REF] . Some solutes, such as betaine, are widespread. To date, MG has only been detected in organisms thriving in hot environments [START_REF] Empadinhas | Diversity and Biosynthesis of Compatible Solutes in Hyper/Thermophiles[END_REF] . Due to their stronger interaction with water than with macromolecules, many of these osmolytes help to stabilize macromolecular structures by enhancing water structure [START_REF] Gao | Crowders and Cosolvents-Major Contributors to the Cellular Milieu and Efficient Means to Counteract Environmental Stresses[END_REF] . Therefore, they are accumulated to counteract the destabilizing effects of environmental stresses on cell components [START_REF] Empadinhas | Diversity and Biosynthesis of Compatible Solutes in Hyper/Thermophiles[END_REF][START_REF] Borges | Comparative Study of the Thermostabilizing Properties of Mannosylglycerate and Other Compatible Solutes on Model Enzymes[END_REF] . Although the mechanisms by which osmolytes help stabilize the proteome under low pressure stress in T. barophilus have not been explored yet, it is reasonable to assume that they will act as previously shown for other stresses, and in this particular case, as has been shown for MG, e.g. by rigidifying the macromolecular structures [START_REF] Balny | High Pressure Effects on Biological Macromolecules: From Structural Changes to Alteration of Cellular Processes[END_REF][START_REF] Martin | Solute Accumulation in the Deep-Sea Bacterium Photobacterium Profundum[END_REF] . Since most investigations only consider parts of cells separately [START_REF] Fitter | Internal Molecular Motions of Bacteriorhodopsin: Hydration-Induced Flexibility Studied by Quasielastic Incoherent Neutron Scattering Using Oriented Purple Membranes[END_REF][START_REF] García Sakai | Current Opinion in Colloid & Interface Science Quasielastic Neutron Scattering in Soft Matter[END_REF][START_REF] Ishai | Influence of Ions on Water Diffusion -A Neutron Scattering Study[END_REF][START_REF] Stadler | Determination of Conformational Entropy of Fully and Partially Folded Conformations of Holo-and Apomyoglobin[END_REF][START_REF] Grimaldo | Hierarchical Molecular Dynamics of Bovine Serum and above Thermal Denaturation †[END_REF] , important effects resulting from the interplay of the different molecules or the crowding of the cellular milieu influencing the adaptation mechanisms might be overseen. Crowding is defined as the exclusion of volume by other polymers or the same polymer (self-crowding), which induces attraction between molecules caused by the lack of free space surrounding them. It is therefore a characteristic of concentrated solutions, such as a biological cell. Crowding and osmolytes are now recognized as important parameters of protein stability [START_REF] Al-Ayoubi | Influence of Cosolvents, Self-Crowding, Temperature and Pressure on the Sub-Nanosecond Dynamics and Folding Stability of Lysozyme[END_REF][START_REF] Erlkamp | Influence of Pressure and Crowding on the Sub-Nanosecond Dynamics of Globular Proteins[END_REF] . In the case of piezophiles, it is therefore crucial to determine their role in stress response and HHP adaptation. To reproduce these intracellular conditions, one can either employ self-crowding, which allows to reproduce the high cytoplasmic protein concentrations (around 300 mg/ mL [START_REF] Ellis | Macromolecular Crowding: An Important but Neglected Aspect of the Intracellular Environment[END_REF][START_REF] Balny | High Pressure Effects on Biological Macromolecules: From Structural Changes to Alteration of Cellular Processes[END_REF] ), but fails at capturing the diversity and complexity of proteins present; or use crowding agents such as osmolytes, which simulate well the interaction with water molecules but also fail at simulating the complexity of intracellular media. Despite these limitations, several studies have demonstrated the protecting effect of osmolytes and crowding on the HHP tolerance of proteins in vitro [START_REF] Erlkamp | Influence of Pressure and Crowding on the Sub-Nanosecond Dynamics of Globular Proteins[END_REF][START_REF] Luong | Pressure -A Gateway to Fundamental Insights into Protein Solvation, Dynamics, and Function[END_REF] . To extend and confirm this data in a meaningful context, a whole cell approach has been designed to provide a third alternative for studying the effect of high concentrations in a realistic biological context [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF][START_REF] Tehei | Adaptation to Extreme Environments: Macromolecular Dynamics in Bacteria Compared in Vivo by Neutron Scattering[END_REF][START_REF] Jasnin | In Vivo Measurement of Internal and Global Macromolecular Motions in Escherichia Coli[END_REF][START_REF] Tehei | Neutron Scattering Reveals Extremely Slow Cell Water in a Dead Sea Organism[END_REF][START_REF] Marty | Neutron Scattering: A Tool to Detect in Vivo Thermal Stress Effects at the Molecular Dynamics Level in Micro-Organisms[END_REF][START_REF] Vauclare | Molecular Adaptation and Salt Stress Response of Halobacterium Salinarum Cells Revealed by Neutron Spectroscopy[END_REF][START_REF] Foglia | Water Dynamics in Shewanella Oneidensis at Ambient and High Pressure Using Quasi-Elastic Neutron Scattering[END_REF] . These studies use neutron scattering to determine the dynamics inside the cells, which is the method of choice because as it does not induce radiation damage to cells while it allows to separate the molecular dynamics of cell constituents or water directly inside in vivo cells [START_REF] Bée | QuasiElastic Neutron Scattering: Principles and Applications in Solid State Chemistry[END_REF] . In contrast to studies on the impact of crowding on pure proteins, cell studies showed that the release of crowding constraints on proteins leads to an increase in rigidity and a decrease in HHP sensitivity. In a way similar to HHP adaptation in piezophiles, the hydration water layer is decreased when crowding is decreased [START_REF] Golub | The Effect of Crowding on Protein Stability, Rigidity, and High Pressure Sensitivity in Whole Cells[END_REF] , demonstrating a first link between protein adaptation and the impact of crowding or osmolytes on proteins. In the experimental setup of the Golub study [START_REF] Golub | The Effect of Crowding on Protein Stability, Rigidity, and High Pressure Sensitivity in Whole Cells[END_REF] , the variations in crowding were generated by cell lysis. In the present study, we take advantage of the accumulation of osmolytes by the piezophilic archaeon T. barophilus under low pressure stress [START_REF] Cario | Molecular Chaperone Accumulation as a Function of Stress Evidences Adaptation to High Hydrostatic Pressure in the Piezophilic Archaeon Thermococcus Barophilus[END_REF] in order to test the impact of the presence of osmolytes in a relevant cellular context (Figure 1).

Experimental section

Sample preparation

Cell samples were prepared for neutron scattering immediately before the start of beam-time. Cells of T. barophilus were cultivated in 4 litres of Thermococcales' rich medium [START_REF] Zeng | Pyrococcus CH1, an Obligate Piezophilic Hyperthermophile: Extending the Upper Pressure-Temperature Limits for Life[END_REF] , under anoxic conditions at 358 K until late exponential phase. The first sample (Tba_Ø) was grown under optimal conditions, e.g. at 40 MPa in Poly Ethylene Terephthalate (PET) bags. Under these culture conditions, T. barophilus cells do not accumulate osmolytes. The second sample (Tba_MG) was cultivated under low-pressure stress, e.g. at atmospheric pressure. Under these culture conditions, cells are known to accumulate MG [START_REF] Cario | Membrane Homeoviscous Adaptation in the Piezo-Hyperthermophilic Archaeon Thermococcus Barophilus[END_REF] . Cells were concentrated by centrifugation at 14000 g for 30 mn, and washed twice with saline solution. To avoid possible damage to the cell during sample preparation for the high pressure cell, all manipulations were performed at 4°C, a temperature which blocks all metabolic activity in Thermococcus cells, and increases their tolerance to oxygen. Thus, both samples were constituted of live T. barophilus cells. Samples were kept under anoxic atmosphere. For neutron experiments, about 600 mg of cell pellets were transferred under anoxic atmosphere as Figure 1. Schematic representation of differences between the two samples of T. barophilus, used in this study. Tba_Ø (right) is grown under optimal pressure and temperature conditions. Tba_MG (left) is grown at optimal temperature but atmospheric pressure. Since atmospheric pressure is perceived as a stress by T. barophilus, it will accumulate significant levels of mannosylglycerate, an organic osmolyte, to maintain the functionality of cellular macromolecules.

Please do not adjust margins Please do not adjust margins described previously [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF] to the HP cell sample insert [START_REF] Peters | New Pressure Cells for Membrane Layers and Systems in Solutions up to 100°C[END_REF] . In brief, the cell pellet was transferred to the insert using a syringe and a blunted 0.9mm needle and deposited at the bottom of the sample holder to avoid the formation of gas bubbles. The sample geometry in this insert is 10 mm x 59 mm and 1 mm thickness. The residual space was filled with 2.5% NaCl saline solution in D2O to guarantee a homogeneous pressure transmission. A cryostat was used to precisely control and change the temperature. The HHP transmitting medium in the heated HHP stick was Fluorinert 39 which is completely inert and which was separated from the sample by a HHP separator [START_REF] Lelièvre-Berna | MPa Sample Stick for Studying Liquid Samples or Solid-Gas Reactions down to 1.8 K and up to 550 K[END_REF] . Neutron data is acquired on live cells. Under the conditions used for data acquisition (see details below), cells remain alive, but are metabolically inactive below 343 K. At the highest temperature tested, 358 K, cells are potentially active but in absence of a suitable carbon source, no activity was detected in the lab, and thus no metabolic activity should occur within the HP cell. Cell survival was estimated by microscopic observation of the fraction of lyzed cells, as a proxy for cell survival. No significant viability decrease was observed at the end of experiments. This is similar to previous experimentation [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF] showing that there is no loss of viability during irradiation, and that samples are indistinguishable in terms of molecular dynamics before and after data acquisition.

Incoherent neutron scattering

Atomic nuclei scatter neutrons coherently and incoherently, but the incoherent part, which refers to molecular dynamics, is the significant contribution when using disordered samples [START_REF] Bée | QuasiElastic Neutron Scattering: Principles and Applications in Solid State Chemistry[END_REF] as it is the case for entire cells in solution. For the hydrogen ( 1 H), the incoherent scattering cross section is much larger than for any other atomic nucleus present in biological samples [START_REF] Sears | Neutron Scattering Lengths and Cross Sections[END_REF] , therefore, the use of neutrons to be diffused on entire cells allows to determine the average motions of the hydrogens and the molecular subgroups bound to them. We used elastic incoherent neutron scattering (EINS) on the backscattering spectrometer IN13 42 and quasi-elastic neutron scattering (QENS) on the time-of-flight spectrometer IN5 43 at the Institut Laue Langevin (ILL, France). In addition to the cell pellets, we have measured the buffer, i.e. D2O added as a pressure transmitter to the HHP cell, for a later subtraction from the sample signal, and a vanadium rod, which scatters purely incoherently and is used to mimic the instrumental energy resolution. The data reduction was carried out using the Large Array Manipulation Program (LAMP) available at ILL [START_REF] Richard | Analysis and Visualisation of Neutron-Scattering Data[END_REF] and Origin (http://www.originlab.com/). Possible multiple scattering effects for a sample in a similar HHP sample container were evaluated previously and we have found that this effect was significant only at very low Q-values [START_REF] Trapp | High Hydrostatic Pressure Effects Investigated by Neutron Scattering on Lipid Multilamellar Vesicles[END_REF] . This result is also in agreement with findings by S. Busch and T. Unruh [START_REF] Busch | The Slow Short-Time Motions of Phospholipid Molecules with a Focus on the Influence of Multiple Scattering and Fitting Artefacts[END_REF] .

Elastic incoherent neutron scattering

The elastic contribution refers to an energy transfer between the neutron and the scatterer equal to zero when hydrogens perform only atomic vibrations around their equilibrium positions. The incident wavelength on IN13 42 of 2.23 Å gives an almost momentum transfer independent energy resolution of 8 µeV. IN13 is characterized by a very large momentum transfer range (0.2 Å -1 < Q < 4.9 Å -1 , where Q is the difference between the incoming and outgoing neutron wave vector) that corresponds to a space and time window of 1-30 Å and 100 ps, respectively. To determine their dynamics at extreme conditions, we have measured both samples at different temperatures (298 K, 312 K, 326 K, 345 K and 358 K) and pressures (0.1 MPa, 20 MPa, 30 MPa and 40 MPa). Due to the high thermal inertia of the HHP cell, temperature was increased from the lowest to the highest, and data acquisition at each pressure was performed during the temperature plateau. Each data point lasted from 1.5 h at 298K to 4.5 h at 358K, for a total acquisition time including HHP and T ramps of 48 h per sample. P ramps were of 3 MPa per minute. T ramps were of 0.5°C per minute. The elastic structure factor is given within the Gaussian approximation (Equation 1) [START_REF] Rahman | Theory of Slow Neutron Scattering by Liquids[END_REF][START_REF] Smith | Protein Dynamics: Comparison of Simulations with Inelastic Neutron Scattering Experiments[END_REF] :

𝑆𝑆 𝑒𝑒𝑒𝑒 (𝑄𝑄, 0 ± ∆𝐸𝐸) ≈ 𝑆𝑆 0 𝑒𝑒𝑒𝑒𝑒𝑒 � -1 3 < 𝑢𝑢 2 > 𝑄𝑄 2 � ( 1
)
where ΔE is the half-width half maximum (HWHM) of the instrumental energy resolution; <u 2 > is the atomic mean square displacement (MSD). The MSD indicates the flexibility of the sample under specific conditions of pressure and temperature. The MSD can be obtained for each temperature or pressure value by the slope of the semi-logarithmic plot of the incoherent scattering function through equation 2:

𝑢𝑢 2 >≈ -3 𝑑𝑑𝑒𝑒𝑑𝑑𝑆𝑆 𝑒𝑒𝑒𝑒 (𝑄𝑄,0±∆𝐸𝐸) 𝑑𝑑𝑄𝑄 2
(2).

Equation 2 is valid for small Q-values, when <u 2 > Q 2 ~ 1. Accordingly, we restricted the fit range to the Q regime between 0.19 and 1.47 Å -1 . The MSD is inversely proportional to the sum over the scattering intensities (summed I), however, the summed I does not require the Gaussian approximation and a broader Q-range can be used, which increases statistics.

Quasi-elastic neutron scattering

The QENS part of the incoherent neutron scattering appears as a broadening of the elastic peak resulting from the relaxation of atomic motions in the spatial and temporal windows of the instrument. QENS reveals the scattering that presents an exchange of small amounts of energy between the neutrons and the sample [START_REF] Bée | QuasiElastic Neutron Scattering: Principles and Applications in Solid State Chemistry[END_REF] .

The incoming wavelength on IN5 43 was 10 Å, which amounts to an instrumental energy resolution of 10 µeV and a momentum This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins transfer range of about 0.2 < Q < 1.1 Å -1 . Such setup permits to probe internal local movements with a space-time window of up to 32 Å and 100 ps, respectively. With samples such as whole live cells, a more thorough data analysis of water behaviour has proven impossible, due to the resolution of the instrument and the complexity of the samples. The Q-range used here has been shown previously to be the best compromise to obtain the parameters of the molecular dynamics of whole cells [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF][START_REF] Golub | The Effect of Crowding on Protein Stability, Rigidity, and High Pressure Sensitivity in Whole Cells[END_REF] . Samples were measured for 30 mn per data point at two temperatures (298 K and 358 K) and two pressures (0.1 MPa and 40 MPa).

In QENS data, a diffusional motion can be described by a Lorentzian curve. As more than one molecular population is present here and each contributes differently to the total scattering intensity, the experimental curve scattered by cells can be fitted by a sum of Lorentzian functions [START_REF] Bée | QuasiElastic Neutron Scattering: Principles and Applications in Solid State Chemistry[END_REF] . In theory, the sum could go over an infinity of Lorentzian curves, as there exists infinity multitude of dynamical populations (e.g. from water molecules, biomolecules and co-solutes) and many possible models were suggested in the past how to treat best QENS data of complex systems [START_REF] Fitter | Internal Molecular Motions of Bacteriorhodopsin: Hydration-Induced Flexibility Studied by Quasielastic Incoherent Neutron Scattering Using Oriented Purple Membranes[END_REF][START_REF] García Sakai | Current Opinion in Colloid & Interface Science Quasielastic Neutron Scattering in Soft Matter[END_REF][START_REF] Ishai | Influence of Ions on Water Diffusion -A Neutron Scattering Study[END_REF][START_REF] Stadler | Determination of Conformational Entropy of Fully and Partially Folded Conformations of Holo-and Apomyoglobin[END_REF][START_REF] Grimaldo | Hierarchical Molecular Dynamics of Bovine Serum and above Thermal Denaturation †[END_REF][START_REF] Paciaroni | Temperature-Dependent Dynamics of Water Confined in Nafion Membranes[END_REF] . However, in practice, the precision of our data does not permit to distinguish more than a few contributions in the instrumental resolution window represented by individual Lorentzian curves, which allows to restrict the sum and to extract characteristic parameters for some specific detected populations. Here, we use the model presented by Martinez et al. [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF] for the same type of samples. Therefore, we can split the signal into four different contributions: an elastic peak, one Lorentzian function corresponding to the proteome, one to the hydration water and one to the bulk water contribution and a constant background B(Q). All functions are convolved by the instrumental resolution function Sresolution (Q, ω) (Equation 3):

𝑆𝑆(𝑄𝑄,
where S1(Q, ω) and S2(Q, ω) are the structure factors [START_REF] Bée | QuasiElastic Neutron Scattering: Principles and Applications in Solid State Chemistry[END_REF] corresponding to the contributions of bulk and hydration water, respectively. These two components are characterized by a translational diffusion constant DT, a rotational diffusion constant DR and a residence time τ 50 , which describes the typical time a water molecule performs oscillatory motions around its equilibrium position before diffusing continuously.

The contribution S3 is associated with all possible motions within the biological system itself (the proteome) and presents a faster relaxation which is described by a large Lorentzian with a constant HWHM (Γproteome) [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF] . The p-values represent the fractions of each contribution: elastic, bulk water, hydration water and proteome. The sum of all populations is equal to one, with around 70-80% of total water content [START_REF] Paciaroni | Temperature-Dependent Dynamics of Water Confined in Nafion Membranes[END_REF][START_REF] Bratbak | Bacterial Biovolume and Biomass Estimations[END_REF] .

The two water populations were fitted by a convolution of two Lorentzians representing the translational diffusion and rotational diffusion that occur simultaneously. However, the rotational diffusion was small and constant for all samples [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF][START_REF] Golub | The Effect of Crowding on Protein Stability, Rigidity, and High Pressure Sensitivity in Whole Cells[END_REF] , so we neglected it. Finally, τ can be determined when HWHM tends to a constant value at high Q-values. The Q-range of IN5 did not allow to reach such high Q-values for the hydration water population and, therefore, it was not determined. We normalized each spectrum to unity in its maximum to neglect the Debye-Waller-factor, which does not introduce additional information for QENS analysis.

Results and discussion

Experiments Intensities were summed over all scattering angles, while the MSD were extracted in a restricted Q range between 0.19 and 1.47 Å -1 . Summed I and MSD, according to equation ( 2), are presented in Figure 2a and 2b, respectively. As observed in previous experiments using the high-pressure cell, the signal is somewhat noisy due to the high neutron absorption, but significant differences can be observed between the two sample types. Some differences in summed intensities between the probed systems are identified in temperature-dependent comparisons (Figure 2), but no difference is observed within the pressure range studied here (Figure S1 & S2). Up to 325 K, the summed I decrease for both samples with increasing temperature (Figure 2a), and the dynamics of both systems increase as expected from a temperature increase (Figure 2b). The curves of both samples display two different regimes with a change of slope at ca. 325 K. For Tba_Ø, the sample grown at optimal conditions, the summed I increase slightly above 325 K, indicating that cell molecular dynamics are stabilized when cells are approaching their optimal growth temperature (358 K). In contrast, for the sample grown under pressure stress (Tba_MG) which accumulates mannosylglycerate, the summed I further decreases (the MSD increases) but at a lower rate. This change in cell dynamics is indicative of structural rearrangements of the cell's macromolecules, which in the current setup correspond mostly to proteins. It is noteworthy that such structural rearrangements have been previously reported for the T. barophilus proteome between low and optimal hydrostatic pressures [START_REF] Peters | Deep Sea Microbes Probed by Incoherent Neutron Scattering under High Hydrostatic Pressure[END_REF] , and proposed to be part of the HHP adaptation in this species. Interestingly, the critical temperature found here (325 K) is very close to the minimal growth temperature of T. barophilus (321 K) [START_REF] Marteinsson | a New Barophilic and Hyperthermophilic Archaeon Isolated under High Hydrostatic Pressure from a Deep-Sea Hydrothermal Vent[END_REF] which further substantiates a possible physiological requirement for the T. barophilus proteome as proposed earlier [START_REF] Cario | Molecular Chaperone Accumulation as a Function of Stress Evidences Adaptation to High Hydrostatic Pressure in the Piezophilic Archaeon Thermococcus Barophilus[END_REF][START_REF] Peters | Deep Sea Microbes Probed by Incoherent Neutron Scattering under High Hydrostatic Pressure[END_REF] . All these findings suggest that the proteome of this species is in a destabilized, partially nonfunctional form at temperatures below 321 K, which might explain its inability to grow below that temperature, and that this organism gets into functional intracellular dynamics only at ca. 321 K. Similarly to what has been observed for HHP [START_REF] Peters | Deep Sea Microbes Probed by Incoherent Neutron Scattering under High Hydrostatic Pressure[END_REF] , once the functional dynamics is achieved, the cell dynamics becomes insensitive to further increase in temperature probably up to its maximum growth temperature (373 K) [START_REF] Marteinsson | a New Barophilic and Hyperthermophilic Archaeon Isolated under High Hydrostatic Pressure from a Deep-Sea Hydrothermal Vent[END_REF] . The presence of MG in sample Tba_MG seems to inhibit the extent of the structural rearrangements of the proteome, which is expected if MG rigidifies and stabilizes the proteome. The MSD confirms the Please do not adjust margins Please do not adjust margins low overall dynamics sensitivity of Tba_Ø and the increase of the dynamics at higher temperatures for the Tba_MG sample (Figure 2b). As seen in the summed I, the overall dynamics of Tba_Ø is more insensitive to temperature than that of Tba_MG.

QENS

QENS data were acquired at two temperatures, 298 K and 358 K, and two pressures, 0.1 and 40 MPa on IN5 at ILL. As expected, the intensities summed over all available scattering angles revealed important changes in dynamics due to temperature, and small changes due to pressure application for both samples (Figure 3). Smaller differences are visible in the low energy transfer region (-0.2 to 0 meV) in response to HHP (Tba_Ø at 0.1 MPa vs. Tba_Ø at 40 MPa) or the presence of mannosylglycerate (Tba_Ø vs. Tba_MG, at 358 K) on molecular dynamics. The dynamical parameters that can be extracted from the QENS data all relate to translational or rotational diffusion and relaxation processes. The data was fitted according to the model of Martinez et al. [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF] with four populations: elastic contribution (pelastic), bulk water (pbulk), hydration water (phydration) and proteome (pproteome). It appeared to be the best compromise between the extraction of a minimum of valuable results to characterise the dynamics and the risk of overfitting the data by a too complex model. Each Lorentzian curve has its characteristic HWHM value as function of Q, which is directly related to each population dynamics. The best fit values along with errors obtained through error propagation are given in Table 1, and an example of the experimental data compared with a fit is given in Figure 4 (for more examples, see Figure S3 -S10).

Pressure and osmolyte accumulation effects on intracellular dynamics at 298 K. The relative contributions of the four populations for both samples (pelastic, pbulk, phydration, pproteome) and the extracted motion parameters are of the same order as those determined before by Martinez et al. [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF] which are characteristic for a piezophile microorganism (Table 1). Γproteome (0.43 ± 0.01 Please do not adjust margins Please do not adjust margins meV) is of the same order that found before (0.431 ± 0.004 meV) [START_REF] Martinez | High Protein Flexibility and Reduced Hydration Water Dynamics Are Key Pressure Adaptive Strategies in Prokaryotes[END_REF] and is specific for the T. barophilus proteome. The HWHM of the translational component for hydration and bulk water at the two temperatures studied (298 K and 358 K) versus Q 2 values are shown in Figure 5 and6. The curves present the typical signature for jump-diffusional motions [START_REF] Bée | QuasiElastic Neutron Scattering: Principles and Applications in Solid State Chemistry[END_REF] , e.g. they are almost linear in Q 2 , tending to zero at the origin and declining slightly at higher Q values. As reported previously, the hydration water presents much slower motions than bulk water [START_REF] Settles | Anomalous Diffusion of Adsorbed Water: A Neutron Scattering Study of Hydrated Myoglobin[END_REF][START_REF] Dellerue | Relaxational Dynamics of Water Molecules at Protein Surface[END_REF] .

Focusing on the hydration water (Figure 5), one can observe that for both samples, the hydration water is less mobile at HHP (DThydration : (4.5 ± 0.4)•10 -7 cm 2 s -1 vs. (5.9 ± 0.3)•10 -7 cm 2 s -1 and (2.9 ± 0.3)•10 -7 cm 2 s -1 vs. (5.7 ± 0.3)•10 -7 cm 2 s -1 for Tba_MG and Tba_Ø, respectively). As expected, if MG is protecting the proteome and is therefore interacting with its hydration shell, this decrease in dynamics is more pronounced for the less crowded sample Tba_Ø (1.97 vs 1.31 times for Tba_Ø and Tba_MG, respectively). The translational diffusion coefficient for bulk water is the same for both samples and is essentially insensitive to HHP in the pressure range tested, but due to cell crowding, it is slightly inferior (DTbulk ~ (1.85 ± 0.05)•10 -5 cm 2 /s) than that of free water at 298 K (DTfree= 2.3 •10 -5 cm 2 /s 56 ). In contrast, the residence time, τTbulk, which is related to interactions between molecules, and enhanced due to exchange mechanisms of biomolecules with hydration water, is pressure sensitive and increases with HHP. Such effect may be explained by a higher confinement due to HHP.

In summary, HHP does not have any impact on the population proportions in the sample or on the diffusion of bulk water

DTbulk. It slightly increases the residence time of bulk water τTbulk and decreases the diffusion coefficient of the hydration shell water DThydration for both samples. The most important impact of pressure and crowding is on the diffusion coefficient of hydration water, DThydration. Both, HHP and low crowding decrease DThydration. The impact of HHP is more pronounced on DThydration in the less crowded sample (Tba_Ø) which confirms at the same time the pressure effect and the importance of crowding on cells. The presence of the osmolyte MG is associated with a lowering of the impact of HHP on sample dynamics which translates into a lower decrease of the diffusion coefficient of the hydration shell water DThydration. This data is congruent with the proposed mechanism of action for MG on protein stabilization [START_REF] Borges | Comparative Study of the Thermostabilizing Properties of Mannosylglycerate and Other Compatible Solutes on Model Enzymes[END_REF][START_REF] Faria | Protein Stabilisation by Compatible Solutes: Effect of Mannosylglycerate on Unfolding Thermodynamics and Activity of Ribonuclease A[END_REF][START_REF] Faria | Protein Stabilization by Osmolytes from Hyperthermophiles[END_REF] .

Pressure and osmolyte accumulation effects on intracellular dynamics at 358 K. HT induces strong modifications of the relative contributions from the populations determined at 298 K (Table 1), which is characterized by an increase of pproteome concomitant with an increase in phydration and a decrease of pbulk.

HT mostly impacts the dynamics of the bulk water and proteome. The higher mobility of the proteome at higher temperatures is established by its higher Γproteome = 0.61 ± 0.01 meV at 358 K compared to Γproteome = 0.43 ± 0.01 meV at 298 K. Bulk water dynamics (Figure 6) increases as a function of temperature (DTbulk varies from ~1.85•10 -5 cm 2 s -1 to (5.2 ± 0.1) and (6.7 ± 0.1)•10 -5 cm 2 s -1 for Tba_MG and Tba_Ø, respectively).

The translational bulk water component is not pressure sensitive for the pressure range studied. As observed for DTbulk at 298 K, it is slower than free water at the corresponding temperature (DTfree= 7.3•10 -5 cm 2 /s at 358 K [START_REF] Simpson | Diffusion and Nuclear Spin Relaxation in Water[END_REF] ). The presence of osmolytes in sample Tba_MG is associated with a lower increase of bulk water dynamics (Figure 6a) and slower motions (DTbulk= (5.2 ± 0.1)•10 -5 cm 2 /s) in contrast to the sample without osmolytes, Tba_Ø (DTbulk= (6.7 ± 0.1)•10 -5 cm 2 /s), which further confirms that crowding impacts bulk water motions, as reported previously for lysed cells [START_REF] Golub | The Effect of Crowding on Protein Stability, Rigidity, and High Pressure Sensitivity in Whole Cells[END_REF] . This impact of osmolyte crowding on bulk waters dynamics should be visible at 298 K, but is too small to be quantified accurately (Figure 6b). As observed at low temperature, the translational diffusion coefficient of hydration water (DThydration) decreases with HHP by a factor of ca. 1.5. Moreover, the less crowded sample, Tba_Ø, presents lower dynamics than Tba_MG at room pressure (DThydration= (2.7 ± 0.4)•10 -7 cm 2 /s vs. (5.6 ± 0.4)•10 -7 cm 2 /s), which is congruent with previous reports that less crowded samples present lower DThydration [START_REF] Golub | The Effect of Crowding on Protein Stability, Rigidity, and High Pressure Sensitivity in Whole Cells[END_REF] . In summary, HT changes the contributions of the different populations detected, it increases the dynamics and the ratio of pproteome, which is accompanied by an increase of phydration and a consequent decrease of pbulk. Temperature increases dynamics as observed for DTbulk, and, furthermore, Tba_Ø presents higher translational diffusion of bulk water, which is congruent with a lower crowding of the cells. However, temperature does not increase DThydration: Indeed, Tba_Ø shows lower DThydration at 358 K than at 298 K.

Conclusions

Taken together the QENS and EINS results demonstrate that Tba_Ø displays lower dynamics than Tba_MG at HT and HHP Please do not adjust margins Please do not adjust margins (Figure 7). Macromolecular components, such as lipids or nucleic acids, could affect cell dynamics. However, due to their low concentration in the cells, e.g. ca. 10% for lipids and 20% for RNAs for example, their contribution to the QENS and EINS signal, which averages the whole cell, is only marginal. Since, the significative QENS and EINS signal originates solely from proteins, which constitute more than 50% of the cell macromolecules, the lower dynamics most probably originates from the proteome reorganization as seen on the EINS data (Figure 2). The Tba_Ø DThydration value at its optimal pressure (40 MPa) and temperature (358 K), i.e. (1.8 ± 0.4)•10 -7 cm 2 /s, is very similar to that found for hydration water on other cell types at their own optimal temperatures. For example, at 310 K, red blood cells present a diffusion coefficient of ~ 1.75•10 -7 cm 2 /s 60 , E. coli at 303 K has a DThydration~ 1.1•10 -7 cm 2 /s 29 and H. marismortui presents a DThydration~ 1•10 -7 cm 2 /s at 300 K [START_REF] Tehei | Neutron Scattering Reveals Extremely Slow Cell Water in a Dead Sea Organism[END_REF] . These values indicate that regardless the cell type or their optimal growth temperature, the cell dynamics required to maintain the macromolecules functional implies that the hydration water dynamics is of the same order. This resembles the "corresponding state principle" [START_REF] Vihinen | Relationship of Protein Flexibility to Thermostability[END_REF][START_REF] Jaenicke | Stability and Stabilization of Globular Proteins in Solution[END_REF] which states that the flexibility of proteins adapted to extreme conditions should be of the same magnitude than a non-extremophile protein at ambient conditions. It is striking that the possible rearrangements of the T. barophilus proteome occurs at the lowest temperature permissive for growth, much like it was shown to occur at the lowest hydrostatic pressure which did not include a stress on cells [START_REF] Peters | Deep Sea Microbes Probed by Incoherent Neutron Scattering under High Hydrostatic Pressure[END_REF] . Thus, the lower DThydration found in Tba_Ø may reflect the structural adaptation of the T. barophilus proteome to HT as observed by EINS. Mannosylglycerate is the osmolyte accumulated under hydrostatic pressure stress in T. barophilus, i.e. at ambient pressure. Sample Tba_Ø, which is less crowded in absence of the osmolyte MG, presents some characteristics similar to that found in lysed cells, where crowding effect is highly diminished: 1) at 298 K, Tba_Ø has a more pressure-sensitive dynamics since the decrease on hydration diffusion is stronger; 2) at 358 K, Tba_Ø displays the higher translational diffusion for bulk water;

3) at highest temperature, Tba_Ø presents lower translational diffusion at both pressures studied.

In the present investigation, we have studied intact, live cells, which indicates that the accumulation of osmolytes as a response to pressure stress significantly impacts the intracellular crowding. It has been proposed to act by rigidifying the protein structure, although the exact mechanism was not fully detailed [START_REF] Faria | Protein Stabilisation by Compatible Solutes: Effect of Mannosylglycerate on Unfolding Thermodynamics and Activity of Ribonuclease A[END_REF] . All our results are congruent with this hypothesis, and further support that MG does not influence protein dynamics directly, but through a crowding effect, possibly at the hydration shell level. Our data on the T. barophilus proteome clearly demonstrates the existence of a pressure induced phase transition, from the low pressure, highly flexible and probably non-functional state, to a drastically less flexible, and biologically functional one at 400 bars. Until recently, there was no clear explanation for such an observation, since it is expected that increasing hydrostatic pressure will destabilize the structural organization of proteins. A similar behaviour has been observed in the g-D-crystallin [START_REF] Cinar | Pressure-Sensitive and Osmolyte-Modulated Liquid-Liquid Phase Separation of Eye-Lens γ-Crystallins[END_REF] .

Indeed, under its physiological conditions, this protein is properly folded and appears transparent which is consistent with its biological function in the eye lens. At lower temperature, the protein in solution appears turbid, and the proper folding is induced by increasing hydrostatic pressure. Thus, as we observe in the T. barophilus proteome, pressure seems to induce a transition from a poorly folded, non functional state to a properly folded, functional one. In the case of T. barophilus, the functional, e.g. normal state, is 400 bars and 85 °C, thus functionally, the lowering of pressure is inducing the transition to the poorly folded state. In γ-D-crystallin, this pressure-induced folding has been explained in part by more configurational freedom, less void volume and far fewer hydrophobic contacts, but interactions of all types may also contribute to this transition [START_REF] Cinar | Hydrostatic Pressure, and Osmolyte Effects on Liquid-Liquid Phase Separation in Protein Condensates: Physical Chemistry and Biological Implications[END_REF] . Although our data does not allow us to identify the cause of the phase transition observed in T. barophilus, we believe that they are the similar to those explaining the phase transition in γ-D-crystallin.

Our observation of the pressure sensitivity of the T. barophilus proteome in a very narrow pressure range is a new example in the now longer list of proteins sensitive to very small variations of hydrostatic pressure, e.g. γ-D-crystallin, lysozyme, α-elastin and Deadbox helicase [START_REF] Cinar | Hydrostatic Pressure, and Osmolyte Effects on Liquid-Liquid Phase Separation in Protein Condensates: Physical Chemistry and Biological Implications[END_REF] , which undergo pressure-sensitive liquid-liquid phase separation (LLPS). These findings are of paramount importance for our studies, although the relation between LLPS and molecular dynamics of proteins within cells

is not yet established and has to be further investigated. However, the pressure sensitivity of T. barophilus is observed within a few tens to hundreds of bar and is altered by the presence of the osmolyte MG. Regardless of the underlying mechanism, what the different systems tell us is that what drives the biological function is the corresponding principle [START_REF] Vihinen | Relationship of Protein Flexibility to Thermostability[END_REF] , which states that the physiologically relevant value a specific parameter can take belongs to a very narrow range of values, independently of the environmental conditions. For example, the flexibility of a functional protein/proteome will be similar in T. barophilus at 400 bars and 85°C, and in γ-D-crystallin at ambient pressure and 37°C. Any variation from these values, will lead to the destabilization of the proteic structure. Under moderate pressure stress, osmolytes can protect proteins against unfolding. In T. barophilus, MG is expected to interact strongly with the hydration shell water, which can be seen by enhanced translational diffusion coefficients of hydration water, which is different from what has been characterized for the well studied osmolyte TMAO, which is excluded from the hydration shell. It would therefore be interesting to investigate how MG induces such behaviour at the protein surface or in hydration water. This work further extends on previous observations that the main HHP adaptive mechanisms of piezophiles include specific populations' ratios and a decrease in hydration water diffusion. Such pressure-response is higher at Tba_Ø, in absence of MG. Moreover, at both HHP response studies at 298 K, we could detect a higher residence time of bulk water and therefore, a higher confinement due to the increase of pressure. Further

Please do not adjust margins Please do not adjust margins work on the basis for the pressure sensitivity of proteins will help in the identification of the structural adaptations required in piezophilic enzymes for life at HHP.

  were performed in December 2015 on IN13 (DOI: 10.5291/ILL -Data.8-04-763) and in November 2016 on IN5 (DOI: 10.5291/ILL -Data.8-04-783), therefore separate sets of samples were produced. The results established the reproducibility of the outcome from various samples.

Figure 2 .

 2 Figure 2. Summed Intensities (A) and MSD <u2> (B) plotted against temperature for Tba_MG (filled symbols) and Tba_Ø (empty symbols) at different pressures (0.1 MPa (black squares), 20 MPa (red circles), 30 MPa (green point up triangles) and 40 MPa (blue point down triangles).

Figure 3 .

 3 Figure 3. Normalized QENS spectra of whole Tba_MG and Tba_Ø cells at 298 K (solid lines), 358 K (dash lines) and 0.1 MPa or 40 MPa at Q = 0.87 Å-1 on a semilogarithmic scale. The figure highlights significant divergence in dynamics between the samples visible in the low energy region.

Table 1 .

 1 Fit parameter values for the two samples at different pressures (0.1 MPa and 40 MPa) and temperatures (298 K and 358 K). The fields in green show characteristic responses to HHP for piezophiles. Dark green indicates similarities between samples, light green fields point out parameters differing between samples. The fields in blue correspond to the osmolyte effect on bulk water. The fields in red are indicative of a different response to HT between samples.

Figure 4 .

 4 Figure 4. Normalized QENS spectra and fit for Tba_Ø at 298 K, 40 MPa and Q= 0.96 Å-1 on a semi-logarithmic scale. Scheme in the red circle represents the different populations studied, the ribbons represent a protein and the dots water molecules (green for hydration water and blue for bulk water).

Figure 5 .

 5 Figure 5. HWHM of the translational hydration water (represented by green dots in the left scheme) for both samples at different pressure (0.1 MPa and 40 MPa) and temperature values (a) 298 K and b) 358 K) as a function of Q2.

Figure 6 .

 6 Figure 6. HWHM of the translational bulk water component (represented by blue dots in the left scheme) for both samples at different pressure (0.1 MPa and 40 MPa) and temperature values (298 K and 358 K) as a function of Q2. a) HWHM obtained at 298 K and 358 K; b) zoom in on the 298 K region from 0.6 Å-2 < Q2 < 1.3 Å-2.

Figure 7 .

 7 Figure 7. Schematic representation of main results for both samples at high-temperature (HT, 358 K) and high hydrostatic pressure (HHP, 40 MPa). For clarity of presentation, this scheme intentionally simplifies the dynamics by neglecting the subdiffusive nature of hydration water. MG refers to mannosylglycerate. a) and b) represent the overall intracellular dynamics obtained by EINS at a time window of 100 ps: Tba_MG shows higher dynamics (dark red) than Tba_Ø at the same temperature (358 K). From c) to f): representation of translational diffusion of hydration water for both samples at HT and HHP obtained by QENS for a time window of 100 ps, more "green dots" represents higher dynamics. Tba_Ø manifests lower dynamics of hydration water than Tba_MG at HT (c,d), but also at HHP (e,f).

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 3

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 5

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 9

Acknowledgements

We are gratefully acknowledging the ILL for allocation of beam time and the CNRS -Défi Instrumentation aux Limites 2014for financing the development of the high-pressure cell. We would like to thank the two anonymous reviewers whose comments helped us improve the quality of this manuscript.

Conflicts of interest

There are no conflicts to declare.