Hartmut Hetzler
email: hetzler@uni-kassel.de

Felix Boy

S Y Mpo

O Tatin

Internal dissipation and self-excited ocillations in rotating machinery: internal friction vs. internal viscous damping

Keywords:

Destabilization due to co-rotating internal viscous damping is a very well known phenomenon. However, most practical applications in engineering do rather exhibit frictional damping (stemming from joints, etc.) instead of velocity proportional dissipation. This contribution aims on investigating the basic effects of internal damping due to co-rotaing frictional dissipation and the interaction with external damping. It is found that adding internal friction damping alters the dynamical behavior significantly: the threshold rotation speed depending on D e /D i known from the linear problem is replaced by self-excited vibrations due to the frictional damping which occur between the resonance speed and the linear threshold speed and which grow hyperbolically as the linear threshold speed is approached. However, the result of the linear analysis is still meaningful since it marks the maximum speed until which finite amplitudes are to be expected.

INTRODUCTION

Among dynamical stability problems in general, selfexcited vibrations are major issues in rotordynamical applications which may lead to severe problems in acoustics or even limit operational regimes and lifetime [START_REF] Bolotin | Nonconservative problems of the theory of elastic stability[END_REF], [START_REF] Tondl | Some problems of rotor dynamics[END_REF], [START_REF] Childs | Turbomachinery rotordynamics: phenomena, modeling, and analysis[END_REF], [START_REF] Yamamoto | Linear and nonlinear rotordynamics: a modern treatment with applications[END_REF]. Apart from self-excitation due to rotor-fluid interaction in fluid-film bearings whip [START_REF] Tondl | Some problems of rotor dynamics[END_REF], [START_REF] Bolotin | Nonconservative problems of the theory of elastic stability[END_REF], [START_REF] Boyaci | Analytical bifurcation analysis of a rotor supported by floating ring bearings[END_REF] or annular sealings [START_REF] Childs | Turbomachinery rotordynamics: phenomena, modeling, and analysis[END_REF], [START_REF] Gasch | Rotordynamik[END_REF], vibrations stemming from co-rotating dissipation (so called "internal damping") are a common source for self-excitated vibrations [START_REF] Gasch | Rotordynamik[END_REF], [START_REF] Muszynska | Rotor-To-Stationary Element Rub-Related Vibration Phenomena in Rotating Machinery-Literature Suryey[END_REF], [START_REF] Childs | Turbomachinery rotordynamics: phenomena, modeling, and analysis[END_REF], [START_REF] Kellenberger | Erzwungene Biegeschwingungen einer anisotrop gelagerten Scheibenwelle mit Kreiselwirkung und Drehträgheit, äußerer und innerer Dämpfung[END_REF].

Apart from material damping, co-rotating dissipation usually stems from shrink fits, joint connections or inter-laminar friction in rotor-stacks of electric machines. Historically, it was this latter application which led to the discovery of the destabilizing effect of internal damping [START_REF] Newkirk | Measurement of internal friction in a revolving deflected shaft[END_REF], [START_REF] Kimball | Internal friction theory of shaft whirling[END_REF], [START_REF] Ehrich | Shaft whirl induced by rotor internal damping[END_REF]. In the context of this contribution it is interesting to note that the first contributions on this dynamical problem already used the term "internal friction" -however, since then almost all investigations had been restricted to the influence of various types of linear or non-linear viscous damping and accordingly the wording changed from "internal friction" to "internal damping". Reviewing the literature revealed only very few studies on the effect of co-rotating dissipation due to dry friction. Most of them focussed on forced vibrations [START_REF] Ishida | Forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping (1/2 order subharmonic oscillations and entrainment -in german)[END_REF], [START_REF] Fischer | Some considerations of modelling internal friction in rotor shaft connections[END_REF] while the influence on stability and self-excitation has only be touched superficially [START_REF] Siegl | Das Biegeschwingungsverhalten von Rotoren, die mit Blechpaketen besetzt sind (Lateral Vibrations of Rotors with Laminar Stacks -in German)[END_REF].

Recent experiments emphasize that an appropriate modelling of the dissipation in rotor stacks of electric machinery demands for modelling approaches involving dry friction [START_REF] Luchscheider | Development of a contact and a material model of laminated stacks[END_REF], [START_REF] Luchscheider | Experimentelle und numerische Identifikation eines homogenisierten Materialmodells für Blechpakete elektrischer Maschinen (Experimental and numerical Identification of a homogenized Material Model for Metal Sheet Stacks of Electric Machinery[END_REF].

Not only for rotordynamical instabilities, but even for stability theory in general the impact of dry friction has not yet been investigated into detail. While the literature on dynamical effects of joints in the context of forced vibrations is quite rich (cf. [START_REF] Jp Den | Mechanical vibrations[END_REF], [START_REF] Klotter | Theorie der reibschwingungsdämpfung[END_REF], [START_REF] Marui | Forced vibration of a base-excited single-degree-of-freedom system with coulomb friction[END_REF], [START_REF] Gaul | Nonlinear dynamics of structures assembled by bolted joints[END_REF] for instance), over a long time only very few publications had been devoted to the impact of dry friction on the bifurcation behavior and it only recently began to attract more attention [START_REF] Tondl | Quenching of self-excited vibrations: effect of dry friction[END_REF], [START_REF] Yabuno | Analysis of the van der Pol System With Coulomb Friction Using the Method of Multiple Scales[END_REF], [START_REF] Hetzler | On the effect of nonsmooth coulomb friction on hopf bifurcations in a 1-dof oscillator with self-excitation due to negative damping[END_REF], [START_REF] Hetzler | On the effect of non-smooth coulomb damping on flutter-type self-excitation in a non-gyroscopic circulatory 2-dof-system[END_REF], [START_REF] Hetzler | Bifurcations in autonomous mechanical systems under the influence of joint damping[END_REF]. However, the significant impact of Coulomb-type damping to stability problems has already been proved experimentally ( [START_REF] Yabuno | Effect of coulomb damping on buckling of a simply supported[END_REF], [START_REF] Yabuno | Analysis of the van der Pol System With Coulomb Friction Using the Method of Multiple Scales[END_REF] for instance). This contribution aims on investigating the influence of co-rotating dissipation due to dry friction, i.e. internal frictional damping, on stabiliy and bifurcation of a generic elastic rotor. To this end, the classical model problem [START_REF] Gasch | Rotordynamik[END_REF] is extended by adding rotor-fixed Coulomb elements. Subsequently, this model problem is investigated with respected to stationary steady state solutions where the Coulomb elements stick, as well as periodic steady state solutions.

MODEL

Equations of motion

The classical model problem to investigate the basic mechanism of self-excitation due internal damping and the influence of external damping is a Jeffcott rotor (Laval rotor) consisting of a massless shaft (lateral stiffness c) in rigid bearings, to which a rigid disk (mass m) is symmetrically mounted (cf. fig. 1) and rotates about a principle axis of inertia. Due to the aligned mounting dynamic unbalance due to gyroscopic effects will not occur. Moreover it is assumed that the mounting point M of the disk coincides with the center of mass G: hence, excitation due to static unbalance will not occur, neither. The shaft rotates with a constant angular velocity Ω. The influence of gravity is not taken into account. In addition to the conservative forces, the rotor is subjected to viscous external damping (coefficient of viscous damping d e ) due to motion with respect to the environment as well as internal dissipation due to relative motion with respect to the rotor. For the classical model this latter internal dissipation is modelled as co-rotating linear-viscous damping (coefficient d i ). For the present study, the classical model is extended by adding corotating Coulomb elements (amplitude R of the friction force) to the internal damping. For the analysis two frames of reference are introduced (cf.figure 2): one inertial frame I = {O, [ e x , e y , e z ]} and a co-rotating relative frame R = {O, [ e ξ , e η , e ζ ]} rotating with the shaft. In the following, for time derivatives leading supercripts will denote the observer system with respect to which the derivative will be calculated: thus, R d dt () will denote time derivatives with respect to an observer moving with system R.

Both internal friction and internal viscous damping depend on the relative velocity v rel = R d dt ( r M ) of the mounting point M with respect to the co-rotating frame of reference R. Using Euler's differentiation formula one finds the velocity relation

˙ r M = d r M dt I = v rel + Ω × r M ,
where Ω = Ω e z is the vector of the angular velocity. With this, the internal viscous damping force can be written as

F di = -d i v rel = -d i ˙ r M -Ω × r M .
The internal friction force is modeled using Coulomb's friction law. Accounting for sticking and sliding, the force reads

F r =    -R v rel || v rel || , || v rel || = 0 F s , || F r || < R ∧ || v rel || = 0 (1)
where the first line represents the sliding friction force and F s is the a priori unknown reaction force while sticking. The set F ∋ F s of admissible stiction forces represents a closed circle of radius R within the (F ξ , F η ) -plane (cf. figure 3).

F ξ F η R R -R -R sliding stiction F F Figure 3.
Set F of admissible stiction forces and the approximation set F in the (F ξ , F η ) -plane.

In order to find closed form approximations of stationary solutions the friction law (1) is simplified to

F r ≈ ˜ F r ∈ -R Sign( ξ) e ξ + Sign( η) e η
for the following investigations. Formally, this can be interpreted as a first order polyhedral approximation of the friction cone. Here, the set-valued Sign function defined as

Sign(x) = sign(x) , |x| = 0 [-1, 1] , |x| = 0 is used [27].
This simplified friction law implies a rectangular set F of stiction in the (F ξ , F η ) -plane, as it is also shown in figure 3. From the graph it can be seen that this approximation set is slightly larger than the original one and contains it, i.e. F ⊂ F. Thus, it will describe the friction accurately for most cases and will overestimated it only for some cases: the effect and the interpretation of this simplification will be discussed within the results section.

For the forces mentioned above, the equations of motion become

m ¨ r M + d e ˙ r M + d i ˙ r M -Ω × r M + c r M = Fr .
Introducing the nondimensional time τ = ω 0 t with ω 2 0 = c m they can be rewritten as

r ′′ M + 2(D e + D i ) r ′ M + r M -2D i Ω × r M = Fr c ,
where D e = de 2mω0 , D i = di 2mω0 and () ′ = d/dτ () is the derivative with respect to the dimensionless time τ . Introducing ρ = R c and refering to the co-rotating frame of reference R, the equations of motion read

ξ ′′ η ′′ + 2(D e + D i ) -2ν 2ν 2(D e + D i ) ξ ′ η ′ + 1 -ν 2 -2D e ν 2D e ν 1 -ν 2 ξ η ∈ -ρ Sign(ξ ′ ) Sign(η ′ ) , ( 2
)
where ν = Ω ω0 is the normalized rotational speed and ξ, η are the coordinates of r M with respect to the co-rotating frame. Due to the set-valued character of the right-hand side these equations allow for two kinds of solutions: equilibria (sticking) at the one hand side, and sliding motions on the other hand.

Sticking friction elements: Equilibria in the co-rotating frame

If the co-rotating friction damper elements stick permantly the relative motion vanishes, i.e. ξ ′′ = ξ ′ = η ′′ = η ′ = 0 must hold: this corresponds to an equilibrium solution in the co-rotating frame R. Since sticking may occur not only at one position but within an interval of positions, the equilibrium solutions belongs to a set, i.e.

ξ 0 η 0 ∈ E = -ρ 1 -ν 2 -2D e ν 2D e ν 1 -ν 2 -1 Sign(0) Sign(0) .
Since this is an affine mapping of the convex and polygonal set Sign(0), the sticking set E will also be a convex polygone. Moreover, since the mapping is given by an orthogonal matrix, it will only comprise scaling and rotation but without any distorting. Thus, depending on the mapping parameters ν and D e and the friction parameter ρ, the stiction set E will be scaled and rotated square: figure 4 shows an example for (ν, D e ) = (0.8, 0.1) . In order to visualise the influence of the parameters (ν, D e , ̺) on the size of E figure 5 shows the relative length OP/̺ versus the rotational speed ν for different values of D e . Please note that the size of E is the same shape as the magnification factor of a forced 1DoF-oscillator and that the equilibrium solutions refer to the rotor-fixed frame of reference. Thus, an observer from the inertial frame I will observe oscillations with a constant amplitude

A 0 ≤ √ 2 OP

Sliding frictional elements: periodic solutions in the co-rotating frame

Numerical simulations indicate, that there also exists a stable limit cycle for ν > 1 if internal friction is present.

Here, Galerkin's method will be used to find an approximation to this stationary solution. In the sense of a truncated first order Fourier-series approximations of stationary periodic solutions may be searched using

ξ ≈ ξ = A ξ cos(ωτ ), (3) η ≈ η = A η cos(ωτ + ψ), (4)
where A ξ , A η > 0; ω, ψ ∈ R. Here, the formula for the first coordinate does not contain a phase shift since for autonomous systems this phase angle may always be eliminated by shifting the time datum. Introducing (3), ( 4) into the equation of motion (2) yields the residuum r = r(A ξ , A η , ω, ψ). Then, Galerkin projection over one oscillation period using the ansatz functions as weighting functions yields

A ξ 1 -(ω 2 + ν 2 ) + 2A η (sin ψω -D e cos ψ) ν = 0 (5) A η 1 -(ω 2 + ν 2 ) + 2A ξ (sin ψω + D e cos ψ) ν = 0 (6) -A ξ (D e + D i ) ω +A η (+ cos ψω + D e sin ψ) ν -sign(ω) 2ρ π = 0 (7) -A η (D e + D i ) +A ξ (-cos ψω + D e sin ψ) ν -sign(ω) 2ρ π = 0 (8)
Equations ( 5)-( 8) show some type of pairwise antisymmetry: from this one may observe that

A ξ = A η =: A (9) cos ψ = 0 → sin ψ = ±1 (10) → ψ 1 = +π/2 , ψ 2 = -π/2 (11)
are obvious solutions. The phase shift implies circular orbits, which is a reasonable since with regard to the symmtry of the system. Using these first results, two more equations for the unknowns A and ω are obtained

A[(D e + D i )ω -D e ν sin ψ] = -sign(ω) 2ρ π ( 12
)
ω 2 -2(ν sin ψ) ω -(1 -ν 2 ) = 0. (13)
Solving for A and ω yields four solutions. However, since A > 0 had been assumed only two admissible solutions remain, having the same amplitude A but opposite angular velocity and sign of phase angle, i.e.

1) (A, ω, ψ = π/2) 2) (A, -ω, ψ = -π/2).
The corresponding approximate solutions read

1) ξ = A cos ωτ 2) ξ = A cos ωτ η = A cos(ωτ + π/2) η = A cos(ωτ -π/2) = -A sin ωt = A sin ωt.
Further analysing these two analytical results reveals that the solutions 1) and 2) both describe the same backward whirling motion within the co-rotating frame, as shown in figure 6.

ξ η ω > 0 ψ = -π/2 = ω < 0 ψ = +π/2 ξ = A cos ωτ η = A cos(ωτ + ψ) Figure 6
. Backward whirl of stationary periodic solutions with respect to the co-rotating frame.

In the following, solution 2) for ψ = -π 2 will be referenced since the sign of ω directly expresses the backward whirling motion in an obvious way. For this, amplitude, frequency and range of validity of the limit cycle is given by

A LC = 2ρ π 1 D e + D i (1 -ν) ( 14
)

ω LC = 1 -ν, 1 < ν < 1 + D e D i . ( 15
)
Please note that this is the angular frequency of motions with respect to the co-rotating frame. For an inertial observer, the co-rotating back-ward whirl will compensate the rotational speed of the rotor and oscillations with the constant angular frequency I ω LC = 1 will be seen.

RESULTS

Within this section, the analytical approximations shall be compared to numerical simulations and analysed from a physical point of view. The discussion starts with a short review of results for internal viscous damping only (ρ = 0, D i = 0), known from classical textbooks [START_REF] Gasch | Rotordynamik[END_REF], [START_REF] Childs | Turbomachinery rotordynamics: phenomena, modeling, and analysis[END_REF], [START_REF] Yamamoto | Linear and nonlinear rotordynamics: a modern treatment with applications[END_REF]. Subsequently the system with internal friction only (ρ > 0, D i = 0) is investigated. Finally the interaction between internal friction and internal viscous damping (ρ > 0, D i > 0) is analysed. In addition, the effect of approximating the circular set of equilibria E by a rectangular shape Ẽ will be addressed.

The standard model: internal viscous damping only

The classical approach to model internal damping is based on co-rotating linear viscous damping without accounting for dry friction, i.e. ρ = 0, D i = 0. In this case, the steady state solution to (2) is the trivial solution (ξ 0 , η 0 ) = (0, 0) -thus, the set E degenerates to a single point. Moreover, it is found that this trivial solution loses its stability at a threshold speed of ν crit = 1 + De Di . This stability threshold is outlined in figure 7. Further analysis shows, that the steady state loses its stability due to a Hopf-bifurcation: thus, for subcritial speeds perturbations will lead to oscillations with decaying amplitude, while for supercritical speeds the system will oscillate with growing amplitude. Eventually, the growth of amplitude will be limited by some nonlinearity (e.g. nonlinear stiffness or damping for instance): this aspect is not incorporated in the model presented here but is extensively discussed in standard literature on bifurcations.

Internal friction only

For the case of viscous external damping and purely frictional internal dissipation -i.e. vanishing of the viscous part of the internal damping -equation ( 14) for stationary amplitudes simplifies to

A LC Di=0 = 2 π ρ D e , ∀ν ≥ 1. ( 16
)
Obviously, beyond the critical speed ν =1self-excited vibrations of finite amplitude will occur. for limit cycle amplitudes is indicated by a solid line and is compared to results obtained from a numerical time simulations which is marked by stars. The corresponding limit cycle frequency ω LC observed from the co-rotating frame R is shown in figure 10. The dark grey region outlines the set of co-rotating equilibria E where the frictional elements stick.

It is found, that the analytical prediction A LC Di=0 for the stationary limit-cycle amplitudes gives very accurate results, apart from the region immediately after the bifurcation at ν =1 . This is due to the fact that close to the bifurcation point the limit cycle orbits are not circular and thus the chosen Galerkin ansatz may only yield rough approximations (cf. fig. 11). Further numerical simulations indicate, that this is related to the chosen polygonal approximation of the friction characteristic (cf. 3) which overestimates the friction for some kinematic constellations. However, the analytical approximation is very accurate in most cases and always yields an upper limit for the model discussed here.

Concerning dynamics it is found that for subcritical speeds ν<1 the equilibrium set E is attractively stable: any inital condition will converge to this set and settle down to a point in E. Since the notion of "equi- librium" refers to the co-rotating frame R, an observer from the inertial system I will measure oscillations at rotor frequency ν and with a constant amplitude, which corresponds to the distance of the rest position within E from the origin O.

As the critical speed ν crit = 1 is exceeded, the equilibrium set loses stability and the system state converges to an attractively stable limit cycle. For the present case of vanishing viscous internal damping, the limit cycle amplitude is constant for almost all speeds and only depends on ν near to the bifurcation point. This friction induced limit cycle excists for all supercritical speeds. For an inertial observer, this limit cycle will have the constant amplitude A LC and the oscillations frequency I ω LC = 1.

Internal dissipation due to co-rotating friction and viscous damping

Considering internal viscous damping in addition to the co-rotating frictional dissipation limit cycle amplitudes are given by the full equation ( 14) and do only exist within the speed range 1 < ν < 1 + De Di , i.e. limit cycles will only exist between the resonance speed and the linear stability margin. The corresponding amplitudes are outlined examplarily for the parameter set D e = 0.1, D i = 0.02, r = 0.05 in figures 12 (details) and in figure 13 (overview). For these parameter values, the linear stability margin -and thus the upper limit of the existence intervall of A LC is at ν = 3. The oscillation frequency does not depend on D i and thus corresponds to that found before, cf. figure 10.

As already found in sect. 2.2 for the case of only co-rotating frictional damping, a self-excited limit cycle appears at the critical speed ν = 1. However, the limit cycle amplitude now is no longer constant over the rotor speed, but increases hyperbolically as it approaches the pole at ν = 1 + De Di . Once this singularity has been trespassed, limit cycles due to internal friction do no longer exist and the amplitudes will be limited by some other nonlinearities, which are not contained within the model discussed here. Obviously, the frictional part of the internal damping not only induces self-excited vibrations, but also strongly interacts with the viscous part of the internal damping. From a physical point of view, this seems reasonable according to the following line of reasoning: for the classical linear problem, the internal damping D i may feed energy into the system and eventually lead to self-excited vibrations as the linear stability threshold ν crit = 1 + De Di is passed. Internal damping due to friction produces a self-excited limit cycle once the resonance speed ν = 1 has been passed. As the rotational speed is increased, the internal viscous damping D i will feed more and more energy per cycle into the system, thus leading to larger and larger limit cycle amplitudes. Within a certain range, the frictional dissipation and the external viscous damp-ing are able to compensate the energy feed-in by the viscous internal damping. However, once the threshold of the linear stability problem has been trespassed, the viscous internal damping can no longer be compensated by neither the viscous external damping nor the frictional damping. By this, it is plausible that also for the problem comprising frictional and viscous internal damping, the stability margin of the linear problem still has a meaning.

CONCLUSIONS

Within this contribution the classical problem of selfexcitation of elastic rotors due to co-rotating internal dissipation has been reviewed and extended in order to also account for non-viscous frictional forces contributing to the internal damping. To this purpose the classical Jeffcott rotor (Laval-rotor) model with viscous internal damping and viscous external damping has been extended by adding co-rotating dry friction forces. Numerical simulations indicated limit cycles for operational conditions above the critical speed. This observation has been proved by analytically approximations for sticking and sliding friction elements using Galerkin's method.

• Due to the set-valued character of stiction forces the unique steady state solution of the smooth problem turns into an entire set E of possible solutions for sticking friction elements. The size of this set behaves similar to the magnification function of a forced 1DoF-system and may become very large near the critial speet at ν = 1.

• As long as the system rests at a sticking solution within E the external observer will measure oscillations of constant amplitude having the rotational frequency of the rotor. Based on frequency, such solutions could mistakenly be interpreted as unbalance excitation.

• For subcritical speeds ν < 1 the set E is attractive and any initial condition will converge to this set. For supercritical speeds ν > 0 the set E becomes unstable: although the system state may still remain in E (for sticking friction elements), small perturbations will lead to growing oscillations.

• As the rotor passes through ν = 1 the internal frictional dissipation gives rise to a stable limit cycle. This limit cycle exists within the interval 1 < ν < 1 + De Di . An external observer will observe oscillatons at a frequencey I ω LC = 1. The corresponding orbit will be whirling backward with respect to the co-rotating frame of reference.

• Similar to viscous internal damping, frictional internal damping may destabilize the steady state solutions. However, in contrast to viscous damping this happens quite earlier at ν = 1 and leads to finite limit-cycle amplitudes.

• As the rotation frequency ν approaches the linear stability margin ν crit = 1 + De Di the amplitude of the limit cycle grows hyperbolically. As this linear stability threshold is exceeded, the system investigated within this study is not able to predict a stationary solution anymore: the system state growths towards another nonlinear attractor.

Thus, it may be concluded that co-rotating frictional damping changes the stability and bifurcation behavior significantly. In particular, for systems with pronounced non-viscous frictional internal dissipation -as will be the case for most assembled rotors consisting of several parts with joints -the stability border ν crit of the linear system may give totally wrong threshold speeds for the occurence of self-excited vibrations.

Thus, for systems incorporating internal frictional dissipation it will rather be adequate to investigate the occurence as well as the amplitude of limit cycles. Consequently instead of the traditional discrimination between "stable" and "unstable" (as given by the linear analysis), one should rather ask whether vibrations occur and how relevant in terms of amplitude they will be for the operation of the system. beam. Proc. of DETC99, Las Vegas, NV, pages 1-12, 1999.

[27] R.I. Leine and N. van de Wouw. Stability and convergence of mechanical systems with unilateral constraints. Springer Verlag, 2008.

Figure 1 .

 1 Figure 1. Jeffcott rotor model with internal friction and internal viscous damping.

Figure 2 .

 2 Figure 2. Kinematics with inertial frame I and co-rotating frame R.

Figure 4 .

 4 Figure 4. Set of equilibrium positions (ξ 0 , η 0 ) in the (ξ, η) -plane.

Figure 5 .

 5 Figure 5. Width OP of the stiction set E depending on ν and D e .

Figure 7 .Figure 8 .

 78 Figure 7. Classic stability chart for internal viscous damping only.

Figure 9 Figure 9 .Figure 10 .

 9910 Figure 9. Equilibrium set, analytical approximation an numerical simulations results of the limit cycles vs. ν.

Figure 11 .

 11 Figure 11. Limit cycle orbits for purely frictional internal damping and viscous external damping (D i = 0 , ρ = 0.05) at ν = 1.5, ν = 1.75.

Figure 12 .Figure 13 .

 1213 Figure 12. Equilibrium set E, analytical approximationA LC an numerical results of limit cycle amplitudes vs. ν for D e = 0.1, D i = 0.02, r = 0.05 (detail).

Figure 14 .Figure 15 .

 1415 Figure 14. Oscillation amplitudes varying r and D e /D i = 2.


Leia Axelle Colin

Yvon Jaillais
email: yvon.jaillais@ens-lyon.fr

Leia Axelle

C ; Pss1

Pip5k

Phosphatidylinositol-4-Phosphate

Phosphorylethanolamine Cytidylyltransferase

FLOWERING LOCUS T; LHY, LONG ELONGATED HYPOCOTYL ; Ft

Cca

Phospholipids across scales: lipid patterns and plant development

Keywords: PI4P, phosphatidylinositol-4-phosphate, PI(4,5)P2, phosphatidylinositol-4,5-bisphosphate, PS, phosphatidylserine, PC, phosphatidylcholine, PE, phosphatidylethanolamine, PG, phosphatidylglycerol, DAG, diacylglycerol, PIPs, phosphoinositides, ER, endoplasmic reticulum, PM, plasma membrane, PH, pleckstrin homology, Lact, Lacthaderin, PLC, Phospholipase

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

Introduction

Phospholipids, also referred to as glycerophospholipids, are a class of lipids whose structure is based on a glycerol backbone that connects a hydrophobic tail -consisting of two long-chain fatty acids linked at the sn-1 and sn-2 position (diacylglycerol) -and a hydrophilic head group (Figure 1a). Because of their amphiphilic nature, phospholipids are key structural components of membrane bilayers, with the fatty acid chains aggregating together in the core of the bilayer and the hydrophilic heads on both sides. The simplest phospholipid class, with only a phosphate group at position sn-3, is phosphatidic acid (PA). The phosphate head group can be further modified to form the different classes of phospholipids, by addition of a choline, serine, ethanolamine, inositol or glycerol molecule, forming phosphatidylcholine (PC), phosphatidylserine (PS), phosphatidylethanolamine (PE), phosphatidylinositol (PI) or phosphatidylglycerol (PG), respectively (Figure 1a). PG main functions are in plastids and mitochondria and will not be discussed hereafter [START_REF] Nakamura | Plant Phospholipid Diversity: Emerging Functions in Metabolism and Protein-Lipid Interactions[END_REF]. Phosphatidylinositol can be further phosphorylated on the 3rd, 4th and/or 5th position of the inositol head group to form phosphoinositides (PIPs, also referred to as polyphosphoinositides) such as phosphatidylinositol-4-phosphate (PI4P) and phosphatidylinositol-4,5-bisphosphate (PI(4,5)P2) (Figure 1a) [START_REF] Noack | Precision targeting by phosphoinoistides: how PIs direct endomembrane trafficking in plants[END_REF].

There are different ways to classify phospholipids based on for example: i) their relative abundance (abundant vs minor lipids, i.e. less than 5% of total phospholipids), ii) their net charges (neutral vs anionic), iii) their metabolic fluxes (relatively stable concentration vs acute production/interconversion and fast turn-over), iv) their site of production (produced at the site of action or exported), v) their shape (conical, straight or inverted conical) or vi) their fatty acid chains (number of carbons:number of unsaturation) (Figure 1a andb). Overall, abundant phospholipids (e.g. PC, PE) are often considered as "structural" lipids in the sense that they are essential to establish a hydrophobic barrier. These phospholipids are produced by biosynthetic enzymes mainly in the endoplasmic reticulum (Figure 1b andc). Low abundant lipids (e.g. PA and PIPs) are labelled as "signaling" lipids in the sense that they mostly have regulatory effects on membrane functions. These signaling lipids are negatively charged, which enable specific interactions with proteins. In addition, they have fast turn-over and can be produced in an acute manner by lipid kinases, phosphatases or phospholipases at their site of action outside of the endoplasmic reticulum (Figure 1b andc) [START_REF] Noack | Precision targeting by phosphoinoistides: how PIs direct endomembrane trafficking in plants[END_REF].

However, the frontiers between the "structural" and "signaling" lipids are not always clear-cut.

Indeed, these categories mostly arise from considerations at the cellular scale, initially drawn from studies using single cell systems such as yeast or in vitro cell culture. As such, phospholipid biosynthesis and localization are often depicted in a single cell. However, recent evidence suggests that specific pattern of phospholipid accumulation arise at the tissue or organ level in plants, particularly for classical "signaling" lipids but also for "structural" lipids. These patterns have downstream consequences on cell behaviors ultimately impacting plant development. Here, we review three examples in which spatiotemporal phospholipid pattern impacts plant development in different ways, highlighting the importance of integrative approaches to study phospholipid biology.

Phosphatidylinositol-4,5-bisphosphate patterns and their roles in tissue differentiation

Phosphoinositides are minor phospholipids, making less than 1% of total phospholipids (Figure 1a). They have specific accumulation patterns in the different compartments of the cell and as such they determine organelle identity [START_REF] Noack | Precision targeting by phosphoinoistides: how PIs direct endomembrane trafficking in plants[END_REF][START_REF] Boutte | Modulation of endomembranes morphodynamics in the secretory/retrograde pathways depends on lipid diversity[END_REF]. Furthermore, because of their respective charges and shape, they are key determinants of membrane physicochemical parameters (Figure 1b) [START_REF] Noack | Precision targeting by phosphoinoistides: how PIs direct endomembrane trafficking in plants[END_REF]. Some of these parameters appears to be invariant between different tissues. For example, in all cell types analyzed so far, the plasma membrane is the most electronegative compartment, a feature that specifies its identity [START_REF] Simon | A PtdIns(4)P-driven electrostatic field controls cell membrane identity and signalling in plants[END_REF]. The strong electrostatic field of the plasma membrane is powered by the accumulation of PI4P in this membrane, the most abundant phosphoinositide species in plants (Figure 1a) [START_REF] Simon | A PtdIns(4)P-driven electrostatic field controls cell membrane identity and signalling in plants[END_REF]. PI4P accumulation at the plasma membrane appears to be invariant when analyzing different tissue and organ [START_REF] Simon | A PtdIns(4)P-driven electrostatic field controls cell membrane identity and signalling in plants[END_REF]. However, during cell division, PI4P strongly accumulates at the cell plate, which is therefore highly electrostatics [4] [START_REF] Caillaud | Anionic Lipids: A Pipeline Connecting Key Players of Plant Cell Division[END_REF]. The accumulation of PI4P at the cell plate correlates with the localization of PI4K in this organelle and the double mutant pi4k12 is defective in cytokinesis and phragmoplast establishment [START_REF] Lin | PI4K1 localizes at the cell plate in dividing cells and that the growth phenotype of the pi4k1 pi4k2 double mutant is rescued when PI4K1 is expressed under the control of the KNOLLE promoter, which is specific of late cytokinesis. PI4Ks are involved in the trafficking of PIN2, KNOLLE and clatrhin at the cell plate. In addition, the pi4k1 pi4k2 double mutant shows overstabilization of phragmoplast microtubules and mislocalization of MICROTUBULE-ASSOCIATED65-3 (MAP65-3), a phenotype that PI4Ks regulate in close association with its binding partner, the MAP kinase MPK4[END_REF][START_REF] Kang | Electron tomography of RabA4b-and PI-4Kbeta1-labeled trans Golgi network compartments in Arabidopsis[END_REF]. By contrast, PI(4,5)P2 is not only patterned at the level of the cell but also at the tissue/organ scales.

Evidence for PI(4,5)P2 patterning at the tissue scale comes from quantitative imaging studies of PI(4,5)P2 sensors lines [START_REF] Stanislas | A phosphoinositide map at the shoot apical meristem in Arabidopsis thaliana[END_REF]. These transgenic lines stably express PI(4,5)P2 binding domains fused to a fluorescent protein as an indirect way to localize this phospholipid [START_REF] Simon | A multi-colour/multi-affinity marker set to visualize phosphoinositide dynamics in Arabidopsis[END_REF][START_REF] Van Leeuwen | Visualization of phosphatidylinositol 4,5-bisphosphate in the plasma membrane of suspensioncultured tobacco BY-2 cells and whole Arabidopsis seedlings[END_REF][START_REF] Vincent | A Sec14p-nodulin domain phosphatidylinositol transfer protein polarizes membrane growth of Arabidopsis thaliana root hairs[END_REF][START_REF] Platre | Guidelines for the Use of Protein Domains in Acidic Phospholipid Imaging[END_REF]. Indeed, such fusion proteins are recruited to membranes that accumulate this lipid [START_REF] Platre | Guidelines for the Use of Protein Domains in Acidic Phospholipid Imaging[END_REF]. By contrast, in the absence of its target lipid, the sensor accumulates in the cytoplasm as a default localization [START_REF] Platre | Guidelines for the Use of Protein Domains in Acidic Phospholipid Imaging[END_REF]. In some cases, the chronic absence of the biosensor cognate lipid induces its subsequent destabilization and degradation. For example, the C2 domain of the bovine Lacthaderin protein (C2Lact), a PS-binding domain, is unstable and degraded when expressed in a mutant background that lacks PS biosynthesis [13]. Quantitative 3-D imaging revealed that the PI(4,5)P2-binding Plekstrin Homology (PH) domain of Phospholipase C (PHPLC) accumulates to a low extent at the center of the shoot apical meristem in Arabidopsis, while it is stabilized in the boundary region (Figure 2a) [START_REF] Stanislas | A phosphoinositide map at the shoot apical meristem in Arabidopsis thaliana[END_REF]. By analogy to the C2Lact sensor, one may speculate that destabilization of PHPLC occurs when PI(4,5)P2 levels are low. This hypothesis was recently substantiated in cotyledons, as PHPLC accumulates about four fold when seedling are treated with a high concentration of NaCl (a condition known to induce PI(4,5)P2 synthesis [START_REF] Van Leeuwen | Visualization of phosphatidylinositol 4,5-bisphosphate in the plasma membrane of suspensioncultured tobacco BY-2 cells and whole Arabidopsis seedlings[END_REF]) or U-73122 (which inhibits PI(4,5)P2 hydrolysis by phosphoinositide-specific-PLC (PI-PLC) and thereby promotes its accumulation) [14]. Although the exact role of the PI(4,5)P2 pattern in the shoot apical meristem needs to be defined, it could be involved in the two main functions of the meristem: stem cell maintenance and organogenesis. For instance, in the clv3-17 mutant, which displays a larger central zone with more stem cells, the PI(4,5)P2 pattern tends to disappear, suggesting that destabilization of the PHPLC sensor correlates with stem cell maintenance [START_REF] Stanislas | A phosphoinositide map at the shoot apical meristem in Arabidopsis thaliana[END_REF].

The shoot apical meristem is also characterized by a typical mechanical stress pattern, with strong and highly anisotropic tensions in boundaries compared to the central zone (Figure 2a) [START_REF] Hamant | Developmental patterning by mechanical signals in Arabidopsis[END_REF]. A positive correlation between the accumulation of PI(4,5)P2 sensors and mechanical stresses in the boundary zones suggests that the PI(4,5)P2 pattern could be linked with tissue mechanics and mechanical signal transduction [START_REF] Stanislas | A phosphoinositide map at the shoot apical meristem in Arabidopsis thaliana[END_REF]. Future work will reveal how this PI(4,5)P2

pattern is established and what its function is in shoot meristem biology.

Lipid sensors are not the only clues pointing at the importance of specific PI(4,5)P2 patterns at the tissue level. Indeed, PI4P 5-kinases (PIP5K), PI(4,5)P2 5-phosphatases and PI-PLCs have tissue specific expression patterns [START_REF] Mei | Arabidopsis phosphatidylinositol monophosphate 5kinase 2 is involved in root gravitropism through regulation of polar auxin transport by affecting the cycling of PIN proteins[END_REF][START_REF] Tejos | Bipolar Plasma Membrane Distribution of Phosphoinositides and Their Requirement for Auxin-Mediated Cell Polarity and Patterning in Arabidopsis[END_REF][START_REF] Ischebeck | Phosphatidylinositol 4,5-bisphosphate influences PIN polarization by controlling clathrin-mediated membrane trafficking in Arabidopsis[END_REF][START_REF] Zhang | Knock-Down of Arabidopsis PLC5 Reduces Primary Root Growth and Secondary Root Formation While Overexpression Improves Drought Tolerance and Causes Stunted Root Hair Growth[END_REF][START_REF] Zhang | Arabidopsis Phospholipase C3 is Involved in Lateral Root Initiation and ABA Responses in Seed Germination and Stomatal Closure[END_REF][START_REF] Kanehara | Arabidopsis AtPLC2 Is a Primary Phosphoinositide-Specific Phospholipase C in Phosphoinositide Metabolism and the Endoplasmic Reticulum Stress Response[END_REF]. For example, PIP5K1, PIP5K2, PLC3 and PLC5

are expressed in the root meristem and in vascular tissues and PIP5K2 expression is regulated by the phytohormone auxin [16-18][19,20]. Furthermore, the plasma membrane-localized 5phosphatase COTYLEDON VASCULAR PATTERN2 (CVP2) is expressed in developing vascular cells in various organs and displays exquisite expression specificity for the protophloem in the root [START_REF] Carland | Cotyledon vascular pattern2-mediated inositol (1,4,5) triphosphate signal transduction is essential for closed venation patterns of Arabidopsis foliar organs[END_REF][START_REF] Rodriguez-Villalon | Molecular genetic framework for protophloem formation[END_REF][24][25]. CVP2 and its homolog CVP2-LIKE1 (CVL1), as well as PIP5K1 and PIP5K2 are involved in venation pattern establishment in cotyledons, while plc2, plc3 and plc5 mutants have root architecture phenotypes [START_REF] Tejos | Bipolar Plasma Membrane Distribution of Phosphoinositides and Their Requirement for Auxin-Mediated Cell Polarity and Patterning in Arabidopsis[END_REF][START_REF] Zhang | Knock-Down of Arabidopsis PLC5 Reduces Primary Root Growth and Secondary Root Formation While Overexpression Improves Drought Tolerance and Causes Stunted Root Hair Growth[END_REF][START_REF] Zhang | Arabidopsis Phospholipase C3 is Involved in Lateral Root Initiation and ABA Responses in Seed Germination and Stomatal Closure[END_REF][START_REF] Carland | Cotyledon vascular pattern2-mediated inositol (1,4,5) triphosphate signal transduction is essential for closed venation patterns of Arabidopsis foliar organs[END_REF]. The cvp2 cvl1 double mutant root protophloem fails to fully differentiate, as revealed by the presence of undifferentiated "gap" cells within protophloem strands (Figure 2b) [25]. Surprisingly, CVP2 overexpression, which promotes PI(4,5)P2 processing into PI4P, also induces protophloem gap cells and phenocopies the cvp2 cvl1 double mutant, which accumulates PI(4,5)P2 (Figure 2b)

[25]. Furthermore, inducible overexpression at the plasma membrane of a highly processive PIP5K, which boosts PI(4,5)P2 production at the expense of the PI4P pool, also induces gap cells in developing protophloem (Figure 2b) [24]. Therefore, it appears that the PI(4,5)P2/PI4P balance, specifically in protophloem cells, rather than the absolute amount of PI(4,5)P2 is critical for the correct differentiation of this tissue (Figure 2b).

Cellular analyses revealed a correlation between increased vacuolar trafficking upon the alteration of the PI(4,5)P2/PI4P ratio and protophloem differentiation, suggesting that vacuole biogenesis and/or related endocytic trafficking regulated by these phosphoinositides could be linked with phloem development [24]. Furthermore, PIP5K1 and PIP5K2 are not only localized at the plasma membrane but also in the nucleus [START_REF] Tejos | Bipolar Plasma Membrane Distribution of Phosphoinositides and Their Requirement for Auxin-Mediated Cell Polarity and Patterning in Arabidopsis[END_REF][START_REF] Ischebeck | Phosphatidylinositol 4,5-bisphosphate influences PIN polarization by controlling clathrin-mediated membrane trafficking in Arabidopsis[END_REF] However, it is acidic, interacts with specific proteins and is present at low levels (i.e. from 1 to 5% of total phospholipids depending on the tissue [29]), which is a hallmark of "signaling" lipids (Figure 1a andb) [START_REF] Noack | Precision targeting by phosphoinoistides: how PIs direct endomembrane trafficking in plants[END_REF]. PS in produced in the luminal leaflet of the endoplasmic reticulum and is then actively orientated by flippases in the cytoplasmic membrane leaflet [30]. This regulation is organelle specific, which allows the accumulation of PS in the cytoplasmic leaflet of post-Golgi compartments [30]. Localization studies using PS specific biosensors revealed that it accumulates at the plasma membrane and all along the endocytic pathway in Arabidopsis [START_REF] Simon | A PtdIns(4)P-driven electrostatic field controls cell membrane identity and signalling in plants[END_REF]13]. This accumulation gradient is critical to define an "electrostatic" membrane territory within the cell, which specifies the identity of plasma membrane-derived organelles [13].

Interestingly, the relative accumulation of PS biosensors at the plasma membrane or endosomes varies during root epidermis differentiation (Figure 3a) [31]. Indeed, PS biosensors mostly accumulate at the plasma membrane in meristematic cells, while they partition more evenly between the cell surface and endosomes in differentiating epidermal cells (Figure 3a).

Experimental manipulation of the PS content by knock-out, knock-down or overexpression of the Arabidopsis PSS1 gene leads to the graded activation of plasma membrane auxin signaling pathway, which is regulated by the small GTPase RHO-OF-PLANTS6 (ROP6) [31,32].

Interestingly, PS is dispensable for the proper targeting of ROP6 to the plasma membrane [31].

However, super resolution microscopy and single molecule imaging revealed that PS is required for the auxin-induced clustering and stabilization of ROP6 in plasma membrane nanodomains (Figure 3b andc). ROP6 stabilization in nanoclusters is required for downstream auxin signaling. This effect is dose-dependent, with no stabilization of ROP6 in nanodomains in the absence of PS and therefore no downstream signaling, even when ROP6 is constitutively active (Figure 3c). By contrast, ROP6 is hyper stabilized in nanodomains in response to auxin in PSS1 overexpressing plants (Figure 3c). Accordingly, these plants have the same phenotypes as plants expressing a constitutive active version of ROP6, including the chronic inhibition of endocytosis and a hypergravitropic root response [31].

One of the future challenges will be to decipher how is the PS pattern developmentallycontrolled during cell differentiation, a process which could involve lipid flippases or regulation of PS transfer at membrane contact sites, which are close (below 30nm) and stable apposition of two membranes involves, among other functions, in lipid exchanges [30, [START_REF] Chung | PI4P/phosphatidylserine countertransport at ORP5and ORP8-mediated ER-plasma membrane contacts[END_REF][START_REF] Moser Von Filseck | Phosphatidylserine transport by ORP/Osh proteins is driven by phosphatidylinositol 4-phosphate[END_REF][START_REF] Sohn | PI(4,5)P2 controls plasma membrane PI4P and PS levels via ORP5/8 recruitment to ER-PM contact sites[END_REF][START_REF] Wang | An equal opportunity collaboration between lipid metabolism and proteins in the control of membrane trafficking in the trans-Golgi and endosomal systems[END_REF].

Furthermore, PS itself is present in plasma membrane nanodomains, and single molecule tracking experiments revealed that these PS molecules are slow moving within the plane of the plasma membrane [31]. This raises the question of how a phospholipid may be immobilized within a membrane and whether this process could also be regulated at the tissue scale.

Temporal phospholipid patterns

The last example is perhaps the least expected as it entails PC, an archetype "structural" lipid.

Indeed, transgenic lines with imbalanced PE/PC ratio, obtained by inducible knock-down or overexpression of CTP:PHOSPHORYLETHANOLAMINE CYTIDYLYLTRANSFERASE1 (PECT1, a rate limiting enzyme in PE biosynthesis), revealed that PC levels at the shoot apex correlate with flowering time: lines with heightened PC content flower early, while plants with reduced PC levels are late flowering (Figure 4a) [START_REF] Nakamura | Arabidopsis florigen FT binds to diurnally oscillating phospholipids that accelerate flowering[END_REF][START_REF] Nakamura | Membrane Lipid Oscillation: An Emerging System of Molecular Dynamics in the Plant Membrane[END_REF]. The early flowering time phenotype observed in plants with heightened PC content is dependent upon expression of FLOWERING LOCUS T (FT), a master regulator of flowering time [START_REF] Matsoukas | Florigens and antiflorigens: a molecular genetic understanding[END_REF]. Interestingly, FT directly binds to PC but not PE in vitro [START_REF] Nakamura | Arabidopsis florigen FT binds to diurnally oscillating phospholipids that accelerate flowering[END_REF]. Furthermore, PC but not PE oscillate depending of the time of day, with specific molecular species of PC showing various temporal pattern of accumulation (Figure 4b). For example, PC species containing less unsaturated fatty acids are predominant during the day, while PC species with linolenic acid (18:3) increase during the night (Figure 4b). In vitro, PC species with linolenic acid bind poorly to FT by contrast to less unsaturated PC species [START_REF] Nakamura | Arabidopsis florigen FT binds to diurnally oscillating phospholipids that accelerate flowering[END_REF]. Transgenic plants that overexpress FATTY ACID DESATURASE3 (FAD3)

show high content in PC species with 18:3 fatty acids and are late flowering (Figure 4b) [START_REF] Nakamura | Arabidopsis florigen FT binds to diurnally oscillating phospholipids that accelerate flowering[END_REF][START_REF] Arondel | Map-based cloning of a gene controlling omega-3 fatty acid desaturation in Arabidopsis[END_REF].

Together these data suggest that FT preferentially interacts with the PC species that accumulate during the day to promote flowering (Figure 4b) [START_REF] Nakamura | Arabidopsis florigen FT binds to diurnally oscillating phospholipids that accelerate flowering[END_REF]. However, it remains unclear how PC contributes to FT activity at the molecular level and how FT discriminates the different PC species based on their degree of unsaturation.

Of note, PC is not the only phospholipid showing temporal pattern of accumulation; and some molecular species of PA or PS also cycle diurnally [START_REF] Maatta | Levels of Arabidopsis thaliana Leaf Phosphatidic Acids, Phosphatidylserines, and Most Trienoate-Containing Polar Lipid Molecular Species Increase during the Dark Period of the Diurnal Cycle[END_REF]. These cycling are not just regulated by the light/dark cycle but they are under direct circadian regulation and are abolished in clock mutants [START_REF] Kim | Interaction and Regulation Between Lipid Mediator Phosphatidic Acid and Circadian Clock Regulators[END_REF]. Interestingly, PA itself regulates the core clock components LONG ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED1 (CCA1).

PA directly binds to both LHY and CCA1, which inhibits their DNA binding capacity [START_REF] Kim | Interaction and Regulation Between Lipid Mediator Phosphatidic Acid and Circadian Clock Regulators[END_REF].

Together these results suggest that the interaction of PA with LHY/CCA1 may function in a negative feedback loop to integrate the circadian clock with lipid metabolism (Figure 4c).

Concluding remarks and future perspectives

The examples highlighted in this review show that phospholipid patterns are multiscales, from membrane nanodomains up to the organismal scale, with specific phospholipid patterns arising according to the cell type, the tissue differentiation status or the time of day. Tissue-specific control of lipid patterns is an emerging concept, but its exploration is somewhat limited by our ability to monitor their dynamics and crosstalks. By design, genetically encoded phospholipid biosensors reveal lipid subcellular localization, but not their respective accumulation [START_REF] Platre | Guidelines for the Use of Protein Domains in Acidic Phospholipid Imaging[END_REF]. As such, these biosensors are suited to detect changes in phospholipid localization, as exemplified for PS [31]. However, if the levels of phospholipids vary but not their subcellular localization, the current tools are not quantitative enough to measure these variations with spatial resolution.

In some cases, the stability of the lipid sensor is dependent on the quantity of this lipid within the cell. However, we do not understand the mechanistic bases for such lipid-induced stabilization/destabilization. We therefore urgently need new methods to image and quantify phospholipid accumulation and turnover with tissue and cellular resolution. Such methods could include for example the design of new quantitative lipid sensors, the use of lipid immunolocalization or mass spectrometry imaging of lipid species. The latter is particularly promising since it can not only discriminate different phospholipids but also their various unsaturation degree [START_REF] Woodfield | Spatial and Temporal Mapping of Key Lipid Species in Brassica napus Seeds[END_REF][START_REF] Lu | Spatial analysis of lipid metabolites and expressed genes reveals tissue-specific heterogeneity of lipid metabolism in high-and low-oil Brassica napus L. seeds[END_REF][START_REF] Sturtevant | Three-dimensional visualization of membrane phospholipid distributions in Arabidopsis thaliana seeds: A spatial perspective of molecular heterogeneity[END_REF][START_REF] Sturtevant | Matrix assisted laser desorption/ionization-mass spectrometry imaging (MALDI-MSI) for direct visualization of plant metabolites in situ[END_REF]. However, this technique is still limited to very abundant phospholipids, such as PC. In addition, the best resolution reported in plant samples to date is ~10 µm, which is sufficient for analyses at the organ/tissue scales but not cellular/subcellular levels [START_REF] Woodfield | Spatial and Temporal Mapping of Key Lipid Species in Brassica napus Seeds[END_REF][START_REF] Lu | Spatial analysis of lipid metabolites and expressed genes reveals tissue-specific heterogeneity of lipid metabolism in high-and low-oil Brassica napus L. seeds[END_REF][START_REF] Sturtevant | Three-dimensional visualization of membrane phospholipid distributions in Arabidopsis thaliana seeds: A spatial perspective of molecular heterogeneity[END_REF][START_REF] Sturtevant | Matrix assisted laser desorption/ionization-mass spectrometry imaging (MALDI-MSI) for direct visualization of plant metabolites in situ[END_REF].

While we begin to uncover some of the enzymes involved in lipid patterning at the tissue scale, the next challenge will be to decipher what are the downstream targets that respond to these patterns, and what are the developmental consequences. This is a particularly daunting task since phospholipids have pleiotropic targets and are therefore expected to affect the localization or activity of many effector proteins. Pinpointing relevant interactions will therefore require in depth biochemical knowledge of the protein/lipid binding interfaces in order to modulate these interactions both from the lipid but also protein sides. At the same time, such efforts will require integrative biology approaches to blend deep biochemical knowledge into their cellular and developmental contexts. Using a collection of phospholipid sensors and membrane surface charge probes, this study shows that there is an electrostatic gradient along the plant endocytic pathway that is set up by specific combinations of the anionic lipids PI4P, PA and PS.

• 14. Lee E, Vanneste S, Perez-Sancho J, Benitez-Fuente F, Strelau M, Macho AP, Botella MA, Friml J, Rosado A: Ionic stress enhances ER-PM connectivity via phosphoinositide-associated SYT1 contact site expansion in Arabidopsis. Proc Natl Acad Sci U S A 2019. Using PI(4,5)P2 biosensors, this study shows that PI(4,5)P2 content increases upon ionic stress, which in turn recruits synaptotagmins (SYTs) and remodels endoplasmic reticulum/plasma membrane contact sites.

•• 24. Gujas B, Cruz TMD, Kastanaki E, Vermeer JEM, Munnik T, Rodriguez-Villalon A:

Perturbing phosphoinositide homeostasis oppositely affects vascular differentiation in Arabidopsis thaliana roots. Development 2017, 144:3578-3589. This work, together with Ref 25, shows that the PI(4,5)P2/PI4P balance, as regulated by plasma membrane localized phosphatases and kinases, is critical for the differentiation of the root protophloem. Using various genetic background with impaired PI(4,5)P2/PI4P balance, this study establishes that both an increase or a decrease of the PI(4,5)P2 content in protophloem induce the production of undiferentiated gap cells in protophloem tracks. Furthemore, it shows that a skewed PI(4,5)P2/PI4P ratio impact vacuolar trafficking, which could in turn impact phloem diffrentiation. This work shows that PIP5K2 is not only localized at the plasma membrane but also in the nucleus. PIP5K2 is actively addressed to the nucleus by a functional nuclear localization signal, and interacts with alpha-importin. Furthermore, using immunolocalization, this study shows that both PI4P and PI(4,5)P2, which are the substrate and product of PIP5K2, respectively, are presents in the nucleus, more specicaly the nucleoplasm. Together, this work open the possibility that some of the phenotypes attributed to PI(4,5)P2 may be due to gene expression regulation rather than membrane-related events.

Rodriguez-Villalon

• 13 .

 13 Platre MP, Noack LC, Doumane M, Bayle V, Simon MLA, Maneta-Peyret L, Fouillen L, Stanislas T, Armengot L, Pejchar P, et al.: A Combinatorial Lipid Code Shapes the Electrostatic Landscape of Plant Endomembranes. Dev Cell 2018, 45:465-480 e411.

  A, Gujas B, van Wijk R, Munnik T, Hardtke CS: Primary root protophloem differentiation requires balanced phosphatidylinositol-4,5biphosphate levels and systemically affects root branching. Development 2015, 142:1437-1446. • 26. Gerth K, Lin F, Daamen F, Menzel W, Heinrich F, Heilmann M: Arabidopsis phosphatidylinositol 4-phosphate 5-kinase 2 contains a functional nuclear localization sequence and interacts with alpha-importins. Plant J 2017, 92:862-878.

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: PI(4,5)P2 patterning in meristematic tissues and impact of imbalanced PI(4,5)P2/PI4P ratio on protophloem differentiation. (a) Schematic representation of the shoot apical meristem organization (top) and PI(4,5)P2 accumulation pattern together with the predicted tensile stress pattern (bottom). (b) Schematic representation of protophloem differentiation in Arabidopsis root meristem and phenotype associated with experimental variations in the PI(4,5)P2/PI4P balance. Root schematic was modified from B. Peret: https://figshare.com/articles/Primary_and_lateral_root_ai/5143987

Figure 3 :Figure 4 :

 34 Figure 3: Impact of PS variations on ROP6-mediated auxin signaling. (a) Variation of PS content at the plasma membrane during root differentiation. (b) Stabilization of ROP6 into PScontaining nanodomains upon auxin treatment and (c) impact of experimental variations of total PS content on ROP6 nanoclustering and downstream auxin signaling. Root schematic was modified from B. Peret: https://figshare.com/articles/Primary_and_lateral_root_ai/5143987

  The main rate-limiting enzymes involved in phospholipid interconversion are indicated (enzymes in red are discussed in this review). Abbreviations: DAG, diacylglycerol; PA, phosphatidic acid; PC, phosphatidylcholine; PE, phosphatidylethanolamine; PG, phosphatidylglycerol; PS, phosphatidylserine; PI, phosphatidylinositol; PI3P, phosphatidylinositol-3-phosphate; PI4P, phosphatidylinositol-4-phosphate; PI5P, phosphatidylinositol-5-phosphate; PI(3,5)P2,

	phosphatidylinositol-(3,5)-bisphosphate; PI(4,5)P2, phosphatidylinositol-(4,5)-bisphosphate;
	LPAAT, LYSOPHOSPHATIDIC ACID ACYLTRANSFERASE; PAP, PA PHOSPHATASE;
	AAPT, AMINOALCOHOL AMINOPHOSPHOTRANSFERASE; DGK, DAG KINASE;
	PECT,	CYTIDINE	TRIPHOSPHATE:PHOSPHORYLETHANOLAMINE
	CYTIDYLYLTRANSFERASE; PSS, PS SYNTHASE; CDS, CYTIDINE DIPHOSPHATE-
	DAG SYNTHASE; CMP-PA, CYTIDINE MONOPHOSPHATE PA; PIS, PI SYNTHASE;
	PI4K, PI4-KINASE; PIP5K, PI4P 5-KINASE; PLC, PHOSPHOLIPASE C; PLD,
	PHOSPHOLIPASE D.

Acknowledgements.

We thank the L. Armengot, P. Moreau, O. Hamant, M. Platre and T. Stanislas for comments on the manuscript, Y.J. is funded by the French National Research Agency ANR caLIPSO (ANR-18-CE13-0025-02) and ANR STAYING-TIGHT (ANR-18-CE13-0016-02), L.C. is funded by a PhD fellowships from the French Ministry of Higher Education.