Arnesh Palanisamy

Gabrielle Moulin

Marco Ramaioli

Artemio Plana-Fattori

Denis Flick

Paul Menut
email: paul.menut@agroparistech.fr

Determination of the mechanical properties of gelatinized starch granule from bulk suspension characterization

Keywords: Microgel, Shear modulus, Bulk modulus, Thermal effects, Particle Elasticity

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

Introduction

In its native form, starch is found as dense granules in a variety of grains and tubers. After industrial fractionation and in some cases, chemical/thermal treatments, a large variety of starch granules with contrasted properties are obtained. These starch granules that differ in terms of textural properties, or stability against mechanical, Physico-chemical or thermal stresses, are produced for various specific applications such as enhanced oil recovery in oil drilling applications or thickening agents in food products [START_REF] Ratnayake | Starch gelatinization[END_REF][START_REF] Nawaz | Physical and chemical modifications in starch structure and reactivity[END_REF]. Global starch production is expected to reach around 156 metric tonnes by 2025(Reportlinker.com, 2020)), of this roughly 60% is used for food-based applications and 40% is expected to be used for industrial uses including oil, pharmaceutical, paper, etc.

Thermal treatments are commonly used in food processing and are usually applied to products that are highly hydrated. Applying a thermal treatment to native starch granules in the presence of excess water results in important modifications of their properties. In their native state, starch granules are solid particles, more or less spherical, in which starch macromolecules (amylopectin and amylose) are densely packed so that these granules only have a very limited water absorption capacity. However, if the temperature exceeded ∼ 70 • C, an important swelling of the particles is very rapidly observed if water is available around the granule. The influx of water is driven by the enthalpic interactions between the hydrogen bonding sites of starch and water [START_REF] Renzetti | Mechanisms controlling wheat starch gelatinization and pasting behaviour in presence of sugars and sugar replacers: Role of hydrogen bonding and plasticizer molar volume[END_REF]. This process, during which the particle can absorb more than 10 times its mass in water, is called gelatinization. During gelatinization, the volume fraction of the starch particles present in a starch suspension strongly increases, and so does the suspension viscosity.

If the volume fraction of the particles in solution remains very low, i.e. much below the random close packing volume fraction φ rcp , that is the maximum volume fraction that can be reached by solid undeformable particles packed randomly, it behaves as a hard particle suspension at low shear rates [START_REF] Einstein | Eine neue bestimmung der moleküldimensionen[END_REF][START_REF] Batchelor | The effect of brownian motion on the bulk stress in a suspension of spherical particles[END_REF]. However, at high enough starch volume fractions, these suspensions form thick pastes (gelatinization is also called starch pasting [START_REF] Chen | A dynamic model of crosslinked corn starch granules swelling during thermal processing[END_REF]) which exhibit properties that are similar to the ones of dense suspensions of soft microgel particles, with both solid-like and fluid-like properties [START_REF] Seth | Elastic properties of soft particle pastes[END_REF]. At low stresses, they behave like elastic solids and deform reversibly under stress. However, once sufficient stress is reached they begin to flow and show the properties of a fluid. The threshold stress at which they begin to flow is called the yield point. Soft particle pastes have a jammed amorphous microstructure that is the source of their rheological behaviour. At volume fractions higher than φ rcp , soft particles start to deform to accommodate more neighbouring particles [START_REF] Menut | Does size matter? elasticity of compressed suspensions of colloidal-and granular-scale microgels[END_REF][START_REF] De Aguiar | Compressive resistance of granular-scale microgels: From loose to dense packing[END_REF]. Further compression results in particle shrinkage. Therefore, their volume fraction, also called packing fraction, that is estimated from the volume occupied by the particles in the diluted state, can exceed 1 [START_REF] Seth | Elastic properties of soft particle pastes[END_REF][START_REF] Evans | Viscoelasticity of gelatinized starch dispersions[END_REF].

With the advent of particle-based simulation methods, such as Discrete Element Modelling and CFD-DEM, there is significant interest in developing simple methods to characterize material properties such as particle Young's modulus and Poisson's ratio, required as a model input. One of the most common contact models used for simulating particles is the Hertzian contact model. This model takes into account the curvature of the spherical surfaces while predicting the interaction forces on the colliding particles. According to this contact model, a repulsive force exists when the granules overlap and no force exists when they do not overlap [START_REF] Di Renzo | An improved integral non-linear model for the contact of particles in distinct element simulations[END_REF]. To predict these normal and tangential repulsive forces one needs two parameters namely, Young's modulus and Poisson's ratio.

To fully describe suspensions behaviour, lubrication forces need to be integrated, they arise due to the presence of the fluid layers in between particles. The momentum of a granule is imparted to the surrounding fluid and subsequently the other granules, hence leading to transport of momentum, even when granules are not physically in contact with each other.

One requires interstitial fluid viscosity, surface roughness, position and velocities of particles for predicting the lubrication forces. For homogeneous isotropic materials, simple relations exist between the three elastic constants are Young's modulus, the bulk modulus and the shear modulus. These relations allow calculating them all as long as two are known, as they are related by the following equations.

K = E 3(1 -2ν)
(1)

G = E 2(1 + ν) (2)
where K, G, E and ν are respectively the bulk modulus, shear modulus, Young's modulus and Poisson's ratio. These relations could be applied to describe the whole behaviour of dense suspensions, which at a macroscopic scale could be considered homogeneous and isotropic. Individual starch granules, however, are non-spherical particles, and their individual behaviour might deviate significantly from one of the homogeneous and isotropic particles. Therefore a strategy could be to determine only "averaged" properties, as it is usually done for example for their size determination, done by static light scattering assuming a spherical shape.

Techniques at the particle level exists for deciphering materials properties of individual particle material properties in literature [START_REF] Villone | Design of a microfluidic device for the measurement of the elastic modulus of deformable particles[END_REF][START_REF] Villone | Rotating tensiometer for the measurement of the elastic modulus of deformable particles[END_REF][START_REF] Abate | Measuring the elastic modulus of microgels using microdrops[END_REF][START_REF] Mohapatra | Core and surface microgel mechanics are differentially sensitive to alternative crosslinking concentrations[END_REF]. Typical techniques include atomic force microscopy(AFM) based on nano-indentation, Brillouin light scattering(BLS), microfluidics, tensionmeter. In AFM based techniques a Poisson ratio is assumed and the Young's modulus is deducted from resisting force vs displacement data [START_REF] Kaufman | Highly stiff yet elastic microcapsules incorporating cellulose nanofibrils[END_REF][START_REF] Aufderhorst-Roberts | Nanoscale mechanics of microgel particles[END_REF]. However, these techniques are not straightforward to apply to measure a particle averaged modulus for such a complex system of soft particles with a very high degree of variability in terms of size and shapes. BLS data is used to infer the Poisson's ratio [START_REF] Mohapatra | Core and surface microgel mechanics are differentially sensitive to alternative crosslinking concentrations[END_REF]. In microfluidic approaches, a particle is compressed through a tapered channel and as pressure is developed by increasing the flow rate, the particle deforms due to the pressure and penetrates further into the channel. The penetration depth is optically observed and Young's modulus is deduced [START_REF] Villone | Design of a microfluidic device for the measurement of the elastic modulus of deformable particles[END_REF]. Similarly, tensiometer and microgel based techniques make use of surface tension to generate force for deformation and optical techniques for measuring the deformation [START_REF] Abate | Measuring the elastic modulus of microgels using microdrops[END_REF][START_REF] Villone | Rotating tensiometer for the measurement of the elastic modulus of deformable particles[END_REF]. However, the applicability of these techniques is not straightforward for non-spherical, poly-disperse, complex suspensions such as starch suspensions. Typical DEM models make the spherical particle assumption, albeit the 'multi-sphere' approach and methods for specific shapes such as super-quadratics exists [START_REF] Lane | A review of discrete element method (dem) particle shapes and size distributions for lunar soil[END_REF][START_REF] Preece | Sand production modeling using superquadric discrete elements and coupling of fluid flow and particle motion[END_REF]. For non-spherical particles with no particular shape features, it is a standard practice to simulate spherical particles with averaged particle properties. One simple methodology to obtain such average particle properties is to infer them from the bulk response of the dense suspensions, which behave macroscopically as homogeneous isotropic materials, and then to determine from that an "averaged" properties at the level of the individual particles. Such use of average 'calibration techniques' are also a common feature in powder and dry granular material simulations using DEM [START_REF] He | Dem study of the mechanical strength of iron ore compacts[END_REF][START_REF] Yan | Investigating mixing and segregation using discrete element modelling (dem) in the freeman ft4 rheometer[END_REF][START_REF] Orefice | A novel framework for a rational, fully-automatised calibration routine for dem models of cohesive powders[END_REF]. This approach is also useful in 'coarse-graining' simulations where the actual system is too large and one simulates at a mesoscale with larger 'representative particles' instead of actual particle sizes to reduce the computational costs.

In this article, the average mechanical properties of a large population of starch granules are investigated from bulk rheological measurements conducted on dense suspensions. To estimate the average particle shear modulus, the suspension shear modulus was measured . Two extreme hypothesis were considered, namely the fully frictional and the friction-less scenarios, to determine an upper and a lower limit for the particle shear modulus. The likelihood of the two assumptions is critically discussed. Then, we investigate the particles bulk modulus by osmotic compression. Finally, we investigate how particles shear modulus depend on the temperature, in the range 20-90 • C.

Materials and methods

Chemically cross-linked waxy maize starch (C*Tex 06205), provided by Cargil (Baupte, France) was used for the entire study. Waxy maize starch contains more than 99 % amylopectin, and chemical cross-linking prevents the release of macromolecules during gelatinization.

True density measurement

The starch particles true density was measured using an air pycnometer (micromeritics AccuPyc 1330). We determined a true density ρ true,starch = 1.321 +/-0.002 g/ml (average value from 6 repeats).

Centrifugation

In a 2-litre mixing vessel, we introduced 15 grams of starch before the addition of 1 litre of distilled water. The mixture was heated to 90 • C while mixing a 50 rpm using an IKA eurostar 60 mixer with a 3 blade propeller of diameter 5 cm; and held at this temperature for 15 minutes to ensure complete swelling of the starch granules. After letting the suspension cool down to room temperature, it was transferred into centrifuge tubes and centrifuged (SIGMA 3-18KS centrifuge) in the required conditions of rotation speed and duration. The supernatant (excess water layer) was carefully removed and the pellet, which consisted of densely packed swollen starch granules, was recovered for rheological characterization and dry weight measurements to estimate the volume fraction. A combination of centrifugation speeds ranging from 500g to 10000g and centrifugation duration ranging from 15 minutes to 30 minutes resulted in swollen starch suspensions with volume fractions φ ranging from 0.447 to 0.552.

Osmotic compression

22.5 grams of starch were added to 1.5 l of distilled water in a mixing vessel. The mixture was gelatinized at 90 • C as described previously (2.2). After letting the suspension cool down to room temperature, it was transferred into a regenerated cellulose dialysis tube with a cut-off of 6-8 kDa (spectra/por 1, diameter 40 mm). The dialysis tube was clipped at both ends and placed for osmotic compression inside a solution of Poly-Ethylene Glycol (PEG, 20kg.mol -1 ) of known concentration. After osmotic compression, the compressed samples were recovered and subjected to rheological characterization and dry weight measurements to estimate their volume fraction. We used PEG 20kDa concentrations ranging from 0.2% to 5% (wt/wt) and compression times ranging from 2 to 15 days to produce samples with volume fractions φ ranging from 0.46 to 1.69.

For the estimation of bulk modulus, we used the same protocol as described above, however, to ensure a constant osmotic pressure (that could decrease with time due to water diffusion from the sample to the PEG bath), the PEG bath was renewed 3 times. To determine the osmotic pressure corresponding to a given PEG concentration, we used the phenomenological one-parameter equation of state proposed by [START_REF] Cohen | A phenomenological one-parameter equation of state for osmotic pressures of peg and other neutral flexible polymers in good solvents[END_REF], which correctly described the behaviour of PEG (and others neutral flexible polymers in good solvents). This equation, based on the sum of a van 't Hoff and a des Cloizeaux terms, include a fitted parameter α, also called the "crossover index", which is polymer size-dependent and for which we used the value proposed by [START_REF] Li | Equation of state of peg/peo in good solvent. comparison between a one-parameter eos and experiments[END_REF] for PEG 20kDa. In that case, the osmotic compression lasted around 17 days to ensure equilibrium without any degradation or granule softening.

Limited water swelling

After the addition of a known quantity of starch (ranging from 5 to 25 grams) into 100ml of water in a mixing vessel, the suspension was heated up to 90 • C and held at this temperature for 15 min under constant mixing. The sample was then cooled down to room temperature and recovered for rheological characterization and dry weight measurements.

This resulted in starch suspension with volume fractions φ ranging from 0.42 to 1.90.

Dry weight measurement and volume fraction determination

Dry weight measurements were conducted at 100 • C in an oven (Chopin technologies).

Prior measurements showed that after 24h at 100 • C the sample had reached a constant weight, therefore all samples were let in the oven for 24h before dry weight measurement.

Samples were introduced in an aluminium foil previously weighted, and the weight of the sample was determined before (M 1 ) and after (M 2 ) drying so that the sample dry starch content could be determined as S dry = M 2 /M 1 . Therefore, the starch concentration in the suspension, C s (in g of starch per 100g of water) is given by C s = 100 *

S dry 1-S dry

The corresponding equivalent volume fraction of unswollen starch granules, φ ug , is given by

φ ug = C s ρ true,starch * 1 100 + Cs ρ true,starch (3)
With ρ true,starch the true density of unswollen starch previously determined assuming density for water ρ water = 1000 kg/m 3 .

To estimate the volume fraction of swollen granules, we used data from previous microscopic experiments showing that, on average, the starch granule radius (the geometric mean of the Feret diameters) increases 2.35 times when gelatinized in excess of water [START_REF] Palanisamy | Kinetic modelling of individual starch granules swelling[END_REF][START_REF] Deslandes | Estimation of individual starch granule swelling under hydro-thermal treatment[END_REF]. Thus, assuming that the granules are spherical and the total volume is conserved (volume gained by the starch during swelling is lost by water), one can estimate the volume of the swollen starch granules divided by the volume of the suspension: φ, which is also known as the volume fraction. Please note that based on the definition here volume fraction is notional and thus can exceed 1. Similar definitions have been used extensively in the literature for various other suspensions as well [START_REF] Mattsson | Soft colloids make strong glasses[END_REF]. φ = (2.35 3 ) * φ ug (4)

Rheology

The rheological measurements were all performed in a Couette cell using an Anton Paar The effect of temperature on the rheological properties was also evaluated. A rejuvenation and ageing step was performed prior to the first measurement, in order to limit measurements error resulting from the sample history [START_REF] Cloitre | Rheological aging and rejuvenation in microgel pastes[END_REF]). Rejuvenation For cooling the sample from 90 • C to 20 • C, a ramp rate of 0.4 • C per minute was used, and sample properties were again measured once thermal equilibrium was reached.

Confocal laser scanning microscopy

Gelatinized samples obtained by the limited water swelling method and characterized by a particle volume fraction close to 0.4 were observed using a confocal laser scanning microscope (CLSM) (Leica TCS SP8, Germany). 0.5 ml of starch solution was stained using 50 µL of Congo Red solution (1% in water, Sigma-Aldrich). Two 250 µm thick spacers were used to place the stained solutions on the microscope slides. The solutions were excited with a laser at 488 nm and the emitted fluorescence was detected in the range 497 -717 nm.

The laser was used at 85% of the maximum intensity. Samples were observed with a 40X

water objective. The total number of pixels in each picture is 3224 x 3224 and each pixel correspond to 0.09 µm. Thus, the length and width of each picture corresponds to 290.62 µm. A total of 192 scans were performed using 0.422 µm step for a total sample height of 80.67 µm to generate a stack of images. Each of these 192 images was manually analyzed to count the number of neighbours of some representative granules.

Bulk modulus measurements through osmotic compression

To measure the bulk modulus of the suspension using osmotic compression, one first needs to establish the time taken for the sample to equilibrate osmotically with the surrounding medium. To do so, we followed the dry weight evolution during osmotic compression, at two different PEG concentrations, and determined the corresponding volume fraction (Table 1).

After 17 days, the volume fraction tend to reach a plateau. Above 17 days, the increase in volume fraction is very limited, but we observed by rheology a decrease in the sample shear modulus, that could result from starch degradation and granule softening, thus only samples obtained after 17 days will be considered for analysis.

Results and Discussion

Suspension rheology and estimation of the particle shear modulus

We observed that starting from φ ∼ 0.4, suspensions exhibit a finite value of elastic modulus. By contrast to what is observed in colloidal systems, which are liquid till the random closed packing, suspensions of granular particles can exhibit a finite modulus at volumes fractions much lower, due to the formation of a self-supporting stress-bearing network. The minimum volume fraction at which this network can form is called the random loose packing, its value depends on the shape but also the frictional properties of the particles [START_REF] Silbert | Jamming of frictional spheres and random loose packing[END_REF][START_REF] Dong | Role of interparticle forces in the formation of random loose packing[END_REF]. Starting from the random loose packing, the shear modulus (measured at f = 1Hz and γ = 0.01) of dense suspensions of swollen starch granules strongly increases with volume fraction, as illustrated in figure 1. Whatever the method used to prepare dense suspensions, namely centrifugation, osmotic compression or limited water swelling, the data superimposed. Whatever the history associated with each procedure, the final stage that is reached is reproducible in terms of rheological characterization and the granule volume fraction is the key parameter that controls the elastic behaviour of the paste. Even in the case of limited water swelling, for which granule never reached their full swelling capacity due to the limited amount of water (for φ > 1), the properties of the suspension are identical to the one of fully swollen starch granules that were partially dehydrated by compression.

Two regimes are observed concerning the elastic modulus dependency toward volume fraction. First, from above the random loose packing to φ ≈ 0.7, the shear modulus strongly increases with the volume fraction. Then, a transition zone exists for 0.7 < φ < 1. For volume fraction φ > 1, the dependence of shear modulus on volume fraction is much weaker.

Insert in Figure 1 shows the log-log plot which shows the existence of the two regimes and the power law dependence of shear modulus with the notional volume fraction of granules.

Fitting a power law to these regimes resulted in the following equation

G ′ =      1600φ 3.15 0.4 ≤ φ ≤ 0.7 550φ 0.53 1 ≤ φ
A similar increase in shear modulus with notional volume fraction have been reported

by [START_REF] Evans | Concentration dependence of the linear elastic behaviour of model microgel dispersions[END_REF]; [START_REF] Abdulmola | Effect of xanthan on the small-deformation rheology of crosslinked and uncrosslinked waxy maize starch[END_REF]; [START_REF] Rodriguez-Hernandez | Rheology-structure properties of waxy maize starch-gellan mixtures[END_REF]. The existence of two regimes of shear modulus with volume fraction for swollen starch suspensions is reported in [START_REF] Evans | Viscoelasticity of gelatinized starch dispersions[END_REF]. Also, for comparison [START_REF] Evans | Viscoelasticity of gelatinized starch dispersions[END_REF] For friction-less systems, [START_REF] Evans | Concentration dependence of the linear elastic behaviour of model microgel dispersions[END_REF] derived an analytical equation that links the suspension shear modulus with the individual granule shear modulus and the average number of neighbouring granules. This model assumes a Hertzian potential (contact model) between granules. The equation, valid for φ rcp < φ < 1, is as follows:

G ′ susp = φ rcp nG ′ p 5π(1 -ν)   φ φ rcp 1 3 1 - φ φ rcp -1 3 1 2 - 8 3 φ φ rcp 2 3 1 - φ φ rcp -1 3 3 2   (5)
where G ′ susp is the suspension shear modulus, φ rcp is the volume fraction at random close packing and n is the average number of neighbouring granules respectively.

Given a granule size distribution one can estimate the φ rcp based on the relationship proposed by [START_REF] Farr | Close packing density of polydisperse hard spheres[END_REF] as previously done in Shewan and Stokes (2015a,b); [START_REF] Shewan | Viscoelasticity of non-colloidal hydrogel particle suspensions at the liquid-solid transition[END_REF]. In the system investigated, the size distribution of Feret diameters of swollen starch granules has been previously determined, it is characterized by a normal distribution with mean = 33.1 µm and standard deviation = 10.7 µm. With this value, φ rcp = 0.68 was determined . The average number of neighbours for each granule, n, was estimated using confocal microscopy. This method was previously used for PMMA spheres by [START_REF] Van Der Vaart | Rheology of concentrated soft and hard-sphere suspensions[END_REF] and also allowed to measure the pair correlation function. In fact, they predicted suspension shear moduli at high frequency with Young's modulus obtained from atomic force microscopy and pair correlation function from confocal microscopy.

In figure 1 we show the fit of equation 5 for the truncated data in the volume fraction range of φ rcp < φ < 1. As we can see from equation 5, G p depends on the assumption of the average coordination number n of each granule. To determine n, confocal images were acquired on suspensions and the coordination number of 10 different granules was manually counted for each of them, based on the contacts that were observed in different layers. 2017). As a result, the increase in coordination number with volume fraction might be limited, especially if friction is present in the system. From the image acquired by confocal microscopy, it is evident that the starch granules are far from being spherical. It should be noted that if the granule seems hollow, this is likely due to the fact that dye (congo red) is adsorbed only on the surface of the granule. In figure 2 we tracked one selected granule from its base (top-left image) to its top (bottom-right image). For this selected granule 5 contacts were observed with neighbouring granules (highlighted as blue boxes on the pictures). This

analysis was carried out to analyze ten different granules, for which we found coordination numbers varying from 5 (min) to 8 (max).

Therefore, with n = (5,8) and a Poisson's ratio n = 0.5, values obtained for the particle modulus were between G p ≈ 5300 Pa and G p ≈ 3300 Pa.

However, as seen in the confocal images, the particles are far from being spherical and have hair-like protrusions on the surface. This suggests that friction might be present in the system, which is coherent with the measure of a finite shear modulus at low volume fraction, the hallmark of the system above random loose packing concentration. Assuming a fully frictional case, G p ≈ G suspension at φ around 1, and we obtain G p ≈ 550 Pa.

Estimation of particle bulk modulus

To estimate the starch granule bulk modulus, we used osmotic compression experiments as previously done by de Aguiar et al. ( 2017) on model microgels. In general terms, the isotropic compressibility of material of volume V under a pressure P is measured by its bulk elastic modulus K = -V dP dV . In this study, we do not measure directly the volume of the sample, but instead, we determined φ, which is directly proportional to the sample volume through the relation φ = n p V d /V , where n p is the particle's number in the sample, V d is the average volume of a fully swollen starch granule in the diluted state, and V the volume of the entire suspension. The bulk modulus of a gelatinized starch paste, compressed under an osmotic pressure π, can then be written as K = -(1/φ) dπ d(1/φ) , or in a more simple way

K = φ dπ in dφ (6)
where π, φ are the applied osmotic pressure and the starch particle volume fraction in the suspension respectively. In our experiments, the osmotic pressure is directly linked to the PEG content in the water bath, so that osmotic pressures ranging from about 18 kPa to about 19 kPa were experimentally reached ( modulus at φ = 1 is roughly equal to 0.8×particle bulk modulus, and that the ratio of the suspension bulk modulus to the particle bulk modulus is independent of volume fraction for φ > 1. From the linear fit in figure 3, it appears that the bulk modulus of the suspension is about 8.4 kPa at φ = 1. Therefore, an estimate for our starch particle bulk modulus is K particle ≈ 10.5 kPa. Using this estimate for bulk modulus and shear modulus of the suspension at φ = 1, we obtain a Poisson's ratio of 0.47, which is close to the maximum value of 0.5. By contrast, a Poisson's ratio of 0.33 is obtained if it is calculated using the friction-less spherical particle assumption. Poisson's ratio for hydrogels reported in literature ranged from 0.38 to 0.5 [START_REF] Mohapatra | Core and surface microgel mechanics are differentially sensitive to alternative crosslinking concentrations[END_REF][START_REF] Boon | Swelling of micro-hydrogels with a crosslinker gradient[END_REF]. This methodology to characterize the modulus of particles is complementary to the techniques such as nano-indentation, AFM,etc [START_REF] Kaufman | Highly stiff yet elastic microcapsules incorporating cellulose nanofibrils[END_REF][START_REF] Aufderhorst-Roberts | Nanoscale mechanics of microgel particles[END_REF].

Such existing alternative techniques are not straightforward to apply to measure a particle averaged modulus for such a complex system of soft particles with a very high degree of variability in terms of size and shapes.

Suspension material properties with temperature

Conclusion

This study focused on the rheological properties of fully gelatinized chemically modified waxy maize starch. We characterized suspension shear modulus and suspension bulk modulus at different particles volume fractions, which was reached thanks to three different and independent techniques: centrifugation of a diluted suspension, osmotic compression of a diluted suspension, and controlled swelling of the granules with a limited amount of water. These different techniques allowed to reach a large range of volume fractions but also gave coherent results showing the results are not technique-dependant. The shear and bulk moduli were determined by rheometry and osmotic compression, respectively. From these macroscopic measurements, we estimated the mechanical properties of an "averaged" swollen granule. We first discussed the equation previously used by [START_REF] Evans | Concentration dependence of the linear elastic behaviour of model microgel dispersions[END_REF], which can be used to estimate the particle shear modulus from the suspension shear modulus, and required an estimate of the average number of neighbouring granules. This was roughly estimated manually using the layered scans of confocal microscopy. Besides [START_REF] Evans | Concentration dependence of the linear elastic behaviour of model microgel dispersions[END_REF] equation is only valid for perfectly spherical friction-less spheres, which is rarely the case in naturally occurring systems such as starch suspensions, this approach could give an upper bound for the particle shear modulus which was estimated to be around 4400 Pa for this starch. For the lower bound, we considered frictional interactions so that the particle shear modulus is nearly identical to the suspension shear modulus at a volume fraction close to 1. From this assumption, an estimate of particle modulus of 550 Pa was obtained. It seems reasonable, notably in view of the 'roughness' of the observed granules to consider rather the second hypothesis. Therefore, assuming frictional interactions, the bulk modulus of the particle was estimated to be around 10.5 kPa. Interestingly, the suspension shear modulus, measured at different volume fractions, were found to be independent of temperature, which strongly suggests that starch granules mechanical properties are temperature independent in the range of temperature investigated here. Finally, in a first approach, for this kind of starch, in a fully swollen state, Young's modulus and Poisson's ratio of the granules are approximately 1.8 kPa and 0.47 respectively. These estimates could find use in research using modelling and simulation techniques such as DEM and CFD-DEM which in turn help in designing process equipment, process intensification, optimization and designing model-based process control. This methodology can be applied to estimate properties of similar microgel systems and of different grades of chemically modified waxy maize starches.

Acknowledgments

This research has been carried out with funding from European Union as part of EU RESEARCH FRAMEWORK PROGRAMME: H2020 / Marie Sk lodowska-Curie Actions ITN MATHEGRAM [813202].

  figure 1. The frequency and loading effects were negligible compared to the volume effects.

  consists of shearing the sample at shear stress greater than the yield stress thus allowing the flow of the sample. Here, the sample was sheared at a shear rate γ = 50 s -1 . The following ageing step consists in holding the sample at rest for 45 minutes. Sedimentation experiments showed the absence of settling in the sample during all the duration of the experiments, which can be attributed to the large volume fraction of the samples (above random loose packing), combined with a low density difference between the swollen starch granules and the surrounding water. Rheological properties were measured at 20 • C; 40 • C, 60 • C, 80 • C and finally 90 • C, the ramp rate during heating being 2 • C per minute. At each temperature, the rheological properties were measured at constant frequency f = 1 Hz and constant strain γ= 0.01 for 120 seconds with an acquisition rate of 1 point per second.

Figure 1 :

 1 Figure 1: Suspension shear modulus as a function of volume fraction. Triangles[△], Squares[ ], circles[ ] represent samples prepared via centrifugation, limited water swelling and osmotic compression respectively(γ =0.01, f = 1 Hz). The solid dotted line represents the polynomial fit of overall experimental data (G suspension = -352φ 2 + 1229φ -330). The purple continuous line represents the power-law fit for volume fractions φ < φ rcp , Green dashed line represents the best fit from the Evans and Lips (1990) model for 0.8 < φ < 1 , Pink continuous line represents the power law fit for volume fractions φ > 1. Insert: The same figure in the log-log plot. Right-side plot: Dependence on frequency: volume fraction of three samples prepared via centrifugation, limited water swelling and osmotic compression correspond to 0.4, 0.91, 1.31.

Figure 2

 2 Figure 2 shows the confocal cross-section image of the starch granule suspension at φ = 0.4, the maximum fraction that could be reached for which each granule could still be easily distinguished from its neighbours. Here, φ < φ rcp and has n could depend on φ(van Hecke, 2009), the value determined at this volume fraction might differ from the one at φ = 1, that should be used to apply the Evans and Lips model. Experimental studies have shown that once particles are in contact, further compression first result in a decrease in the void fraction de Aguiar et al. (2018) and local deformation at the contact point de Aguiar et al. (2017). As a result, the increase in coordination number with volume fraction might be

Figure 2 :

 2 Figure 2: Confocal stack images of starch granule suspension. Blue Boxes ( ) correspond to the contact of the traced granule with the neighbouring granule.

Using equation 6 Figure 3 :

 63 Figure 3: Suspension bulk modulus as function of volume fraction. Asterisks[ * ] correspond to experimental data.

Figure 4 :

 4 Figure 4: Storage modulus at different temperatures. Asterisks[*], circles[ ], boxes[ ] and triangles[△] correspond to volume fractions φ of 0.42, 0.63, 0.83 and 1.03 respectively. Error bars correspond to one standard deviation. The dashed line is the mean value of all the measurements across temperatures.

Table 1 :

 1 Evolution of the sample volume fraction during osmotic compression, at two different PEG concentrations

	time [days] φ-3% PEG20K [wt/wt] φ-4% PEG20K [wt/wt]
	4	0.76	0.99
	9	1.20	1.60
	14	1.39	1.67
	17	1.45	1.69
	21	1.46	1.75
	24	1.47	1.75

Table 2

 2

	)

Table 2 :

 2 Equilibrium osmotic pressure and volume fractions


Sebastian Schöps

Innocent Niyonzima

Markus Clemens

Parallel-In-Time Simulation of Eddy Current Problems Using Parareal

Keywords: Parallel-in-time, eddy currents, time stepping, finite element

In this contribution the usage of the Parareal method is proposed for the time-parallel solution of the eddy current problem. The method is adapted to the particular challenges of the problem that are related to the differential algebraic character due to non-conducting regions. It is shown how the necessary modification can be automatically incorporated by using a suitable time stepping method. The paper closes with a first demonstration of a simulation of a realistic four-pole induction machine model using Parareal.

I. INTRODUCTION

T HE numerical simulation of the eddy current problem in time domain is computationally expensive due to implicit time stepping. This is particularly challenging if long time periods have to be considered as for example when the start-up of an electrical machine is simulated, possibly with surrounding circuitry [START_REF] Tsukerman | Finite element differential-algebraic systems for eddy current problems[END_REF]. Most implementations, as in GetDP [START_REF] Geuzaine | GetDP: a general finite-element solver for the de Rham complex[END_REF], use loworder time stepping schemes as for example the θ-method, [START_REF] Hairer | Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems[END_REF]. On the other hand, higher order time integration methods, e.g. [START_REF] Nicolet | Implicit Runge-Kutta methods for transient magnetic field computation[END_REF], [START_REF] Benderskaya | Embedded Runge-Kutta methods for field-circuit coupled problems with switching elements[END_REF], have been proposed but are rarely used in practice. Recently, explicit methods gained interest as computational hardware architectures seem to favor those algorithms [START_REF] Außerhofer | Discontinuous Galerkin finite elements in time domain eddy-current problems[END_REF], [START_REF] Schöps | Higher order half-explicit time integration of eddy current problems using domain substructuring[END_REF], [START_REF] Dutiné | Explicit time integration of transient eddy current problems[END_REF]. Another approach is time domain parallelization [START_REF] Gander | 50 Years of Time Parallel Time Integration[END_REF]. For example the reformulation of the time stepping process as one big system of equations has been proposed in [START_REF] Takahashi | Time-domain parallel finite-element method for fast magnetic field analysis of induction motors[END_REF]. However, it requires to rewrite the time-stepping code.

In this contribution the usage of the (non-intrusive) Parareal method [START_REF] Lions | A parareal in time discretization of PDEs[END_REF], [START_REF] Gander | Nonlinear Convergence Analysis for the Parareal Algorithm[END_REF] is proposed and its application to a real world electrical engineering problem, i.e., an electrical machine, is shown. Furthermore, the method is adapted to the particular challenges of space discretized eddy current problems related to the differential algebraic character of the equation. The Parareal method has already been applied to wide range of problems in mathematics and in physics. These problems include for example linear and nonlinear parabolic problems [START_REF] Liu | A parareal waveform relaxation algorithm for semi-linear parabolic partial differential equations[END_REF], molecular dynamics [START_REF] Baffico | Parallelin-time molecular-dynamics simulations[END_REF], stochastic differential equations [START_REF] Engblom | Parallel in time simulation of multiscale stochastic chemical kinetics[END_REF], Navier Stokes [START_REF] Trindade | Parallel-in-time simulation of the unsteady Navier-Stokes equations for incompressible flow[END_REF], quantum control [START_REF] Maday | Monotonic parareal control for quantum systems[END_REF].

This paper is structured as follows: after this introduction, Section II discusses the modeling and the differential algebraic character of the system. Section III introduces the Parareal algorithm and its adaption. Finally, the numerical results are presented in Section IV. Conclusions are given in Section V.

Γ = ∂Ω Ω s Ω Ω σ

II. MODELING AND DISCRETIZATION

When disregarding displacement currents, and introducing the magnetic vector potential A as unknown, one obtains the eddy current problem in A -formulation

σ∂ t A + ∇ × (ν∇ × A) = J s (t) (1)
on the domain Ω × I and I := (t 0 , t end ], see Fig. 1. The problem is well posed when supplying a gauge condition, suitable boundary conditions, e.g. Dirichlet n × A| Γ = 0 where Γ = ∂Ω and an initial value A( r, t 0 ) = A 0 ( r) with r ∈ Ω. The material is described by the conductivity σ and nonlinear reluctivity ν; the current density J s = k χ s,k i k is given by strandedconductor winding functions χ s,k , which homogeneously distribute the currents i k .

A. Discretization by Finite Elements

Equation ( 1) is reformulated in the following weak form:

find A ∈ H 0 (curl, Ω) such that Ω w • σ∂ t A + ∇ × w • (ν∇ × A) dΩ = Ω w • J s dΩ Figure 2
. Mesh view of the four-pole induction machine model "im 3kw" from the GetDP library [START_REF] Geuzaine | GetDP: a general finite-element solver for the de Rham complex[END_REF] as described in [START_REF] Gyselinck | Multi-slice FE modeling of electrical machines with skewed slots-the skew discretization error[END_REF].

for all w ∈ H 0 (curl, Ω). Discretization by a finite set of edge elements [START_REF] Monk | Finite Element Methods for Maxwell's Equations[END_REF] A( x, t) ≈ n i=1 w i ( x) a i (t) yields for the induction machine "im 3kw", cf. Fig. 2, the following system of differential algebraic equations (DAEs)

M σ d t a(t) + K ν a(t), θ(t) a(t) = j s (t) (2)
where a(t) ∈ R n is the vector of (line-integrated) magnetic vector potentials, M σ ∈ R n×n denotes the (singular) mass matrix representing the conductivities and j s (t) ∈ R n describes the discretized source current density. Finally, K ν (a, θ) ∈ R n×n is the curl-curl matrix which depends on rotor angle and flux. Movements are considered by the moving band approach [START_REF] Ferreira Da Luz | Analysis of a permanent magnet generator with dual formulations using periodicity conditions and moving band[END_REF] determined by the mechanical equation

d t θ(t) = ω(t) and Id t ω(t) + κθ(t) = T a(t) (3)
with initial values θ(t 0 ) = θ 0 and ω(t 0 ) = ω 0 , where ω is the angular velocity, I the inertia, κ the torsion coefficient and T defines the mechanical excitation given by the magnetic field. Let us address in the following the current driven coupled problem ( 2) and (3) as

Md t u(t) + K u(t))u(t) = f (t) (4)
with unknown u = [a , θ, ω] and the obvious definitions for M, K and f .

B. Differential algebraic equations

The solution of the DAE ( 1) is straightforward since the system is an index-1 DAE [START_REF] Bartel | Structural analysis of electrical circuits including magnetoquasistatic devices[END_REF], [START_REF] Nicolet | Implicit Runge-Kutta methods for transient magnetic field computation[END_REF]. It can be treated with standard techniques, while higher index problems are increasingly more difficult to solve [START_REF] Hairer | Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems[END_REF].

Let M + be the Moore-Penrose pseudo inverse of M such that P = M + M and Q = I -P denote projectors decomposing the vector potential u i = u(t i ) at each time instance into its differential and algebraic components, respectively

u i = Pu i,σ + Qu i,0 .
When solving (2) for given currents i i = i(t i ), only initial conditions for the differential components u 0,σ may be prescribed. The algebraic components u 0,0 must be consistently determined by solving the constraint

Q K ν Qu 0,0 = Q X s i 0 -Q K ν Pu 0,σ .
(5) However, when using the implicit Euler method to solve an initial value problem with inconsistent data, i.e, u 0,σ and u 0,0 do not fulfill ( 5), a projection is automatically carried out: the time stepping instruction for t i to t i+1 = t i + δt

T 0 T 1 T 2 T 3 T 4 T 5 a i (t)
1 δt M + K u i+1 u i+1 = f i+1 + 1 δt Mu i (6)
ignores inconsistent algebraic components after the first step due to the term Mu i = MPu i = Mu i,σ . This is generally not the case for higher index DAEs and other time-stepping schemes, see e.g. [START_REF] Bartel | Structural analysis of electrical circuits including magnetoquasistatic devices[END_REF].

The implicit Euler method is a numerical implementation of the solution operator F :

I × I × R n → R n such that u i = F(t i , t 0 , u 0 )
which propagates u 0 through time. Let us define another coarse propagator denoted by G : I×I×R n → R n of lower precision, e.g., implicit Euler with a time step ∆t δt.

III. THE PARAREAL METHOD

Let us split the total time interval into smaller intervals I j := (T j-1 , T j ] with t 0 = T 0 < T 1 < . . . < T N = t end according to the number of CPUs N available. On each interval, one defines the equation ( 4) with initial value U j-1 := u(T j-1 ), final value U j := u(T j ). Continuity at the interfaces T j is established by matching conditions, see Fig. 3 H

(U) :=            U 0 -u 0 = 0, U 1 -F(T 1 , T 0 , U 0 ) = 0, . . . U N -1 -F(T N -2 , T N -1 , U N -2 ) = 0. (7)
In other words, the problem of matching can be considered as the unknown of a nonlinear equation H :

R N •n → R N •n in the variable U = [U 0 , ..., U j , ..., U N -1 ].

A. Interpretation as Newton's Method

The system (7) can be solved by Newton's method. It reads using the superscript (k) to account for the iterations

∂H ∂U U (k-1) (U (k) -U (k-1) ) = -H(U (k-1) ). ( 8
)
Algorithm 1: Parareal as proposed in [START_REF] Lions | A parareal in time discretization of PDEs[END_REF].

1 init: U (k)
0 ← u 0 (for all k) and ū(0) j , ũ(0) j ← 0 (for all j); 2 set counter: k ← 1; The j-th row of the Jacobian matrix ∂H/∂U has only two entries in columns j -1 and j. It is given by

3 while k ≤ 2 or max j U (k) j -U (k-1) j > tol do 4 for j ← 1 to N do 5 solve coarse: ū(k) j ← G(T j , T j-1 , U (k)
∂H j ∂U • = 0, . . . 0, - ∂F ∂U T j , T j-1 , • , I, 0 . . . , 0 (9)
where I denotes the identity of dimension n. After rearranging the terms, equation ( 8) is equivalent to the explicit update formula

U (k) j = F(T j , T j-1 , U (k-1) j-1 ) (10)
+ ∂F ∂U T j , T j-1 , U (k-1) j-1 (U (k) j-1 -U (k-1) j-1 ),
in which the linearization is approximated by the difference

≈ F(T j , T j-1 , U (k-1) j-1 ) (11)
+ G(T j , T j-1 , U (k) j-1 ) -G(T j , T j-1 , U (k-1) j-1
) which is a Quasi-Newton method as proposed e.g. in [START_REF] Gander | Nonlinear Convergence Analysis for the Parareal Algorithm[END_REF].

Due to the splitting of the time interval, one may take advantage of the parallel architecture of modern computers to speed up the time integration similar to multiple shooting methods [START_REF] Lions | A parareal in time discretization of PDEs[END_REF]. The pseudo code of the resulting Parareal algorithm is shown in Alg. 1.

B. Discussion of the Algorithm

The algorithm solves two kinds of problems in a nested loop until convergence is reached: a cheap problem defined on a coarse time and possibly spatial grid is solved sequentially (line 5, Alg. 1) to propagate missing initial conditions and highfidelity problems are solved in parallel on the intervals I j (line 9, Alg. 1). For both problems the implicit Euler method (6) can chosen, or alternatively a higher order method.

The solution of the cheap problem at T j is denoted by

ūj = G(T j , T j-1 , U j-1 ) (12)
which is computed by propagating the initial value U j-1 from T j-1 to T j by coarsely discretizing (2) in time, i.e., using large time steps ∆t. The solution of the high-fidelity problem is obtained by solving (2) with initial condition U j-1 using fine discretizations, i.e., small δt. This allows to rewrite the update equation [START_REF] Lions | A parareal in time discretization of PDEs[END_REF] in Alg. 1 (line 6) as

ũj = F(T j , T j-1 , U j-1 ) (13)
U (k) j = ũ(k-1) j + ū(k) j - ū(k-1) j .
It can be shown, [12, Theorem 1], that Alg. 1 yields the correct solution until time T k after k iterations, so the correct solution is obtained after at most N iterations. This implies that Parareal does not take longer than the sequential time stepping procedure using the fine solver F from (13) if neglecting the computational costs of the coarse solution operator G. However, in this case Parareal requires up to N -times more CPU time due to its parallel processing.

C. Discussion of the Algebraic Equations

For the eddy current problem line 6 of Alg. 1 must be adapted to reflect the differential algebraic character, i.e.

PU (k) j = Pũ (k-1) j + Pū (k) j -Pū (k-1) j
with a subsequent solve of (5) to obtain a consistent Qu (k) j . However, when using Implicit Euler as shown in [START_REF] Außerhofer | Discontinuous Galerkin finite elements in time domain eddy-current problems[END_REF], this step is automatically taken care of. Similarly, the norm in line 3 should be adapted to only account for differential components, e.g. by considering a projection or the eddy current losses.

IV. NUMERICAL EXAMPLE

Parareal is particularly interesting for problems with multitone solutions, e.g. due to slotting as present in the machine "im 3kw" as visible in Fig. 4. This was also observed in molecular-dynamics [START_REF] Baffico | Parallelin-time molecular-dynamics simulations[END_REF]. However, speed-ups have also been observed for less favorable problems [START_REF] Gander | Nonlinear Convergence Analysis for the Parareal Algorithm[END_REF].

The algorithm was implemented in GNU Octave [START_REF] Eaton | The GNU Octave 4.0 Reference Manual 1/2: Free Your Numbers[END_REF]. GetDP is used for the simulation of the 2D model with 8308 degrees of freedom [START_REF] Geuzaine | GetDP: a general finite-element solver for the de Rham complex[END_REF], [START_REF] Gyselinck | Multi-slice FE modeling of electrical machines with skewed slots-the skew discretization error[END_REF]. They are executed on an Intel Xeon cluster with 80 × 2.00GHz cores, i.e., 8×E7-8850 and 1TB DDR3 memory.

A. Sequential Time Steppers

As sequential time stepper the implicit Euler scheme is applied for 10 electrical periods, i.e., I = (0, 0.2] s. The time grid of this simulation is refined from δt = 10 -3 s and δt = 10 -4 s to δt = 10 -5 s. The coarsest sequential simulation takes approx. 15min, while the ones with finer grids correspondingly more time, i.e., 2h and 20h, respectively. Each time step requires ca. 0.3s for matrix reassembly, Newton and solving the systems.

B. Parareal

The Parareal implementation uses OpenMP parallelized calls of GetDP. The implicit Euler method is used with time step sizes δt = 10 -5 s and ∆t = 10 -3 s for the fine and coarse problem, respectively. Fig. 5 shows the errors in comparison with the sequential reference simulation at δt = 10 -5 s. For the Parareal simulation N = 40 cores have been used. After k = 4 iterations, a relative l 2 accuracy of 10 -2 has been obtained. The sudden increase of the error after approx. T 4 can be explained by the convergence of the Parareal algorithm: it is expected to reproduce the sequential time-stepper's solution for t < T k (k = 4). The computation has a potential speed-up of N/k = 10 with respect to the reference simulation when neglecting communication costs and the coarse grid solution. The speedup is obtained since the effective length of the time interval was reduced by a factor of N = 40, but iterated k = 4 times. The parallelization allows to obtain errors below 1% within the same time that a sequential simulation needs to get errors of 100%. However, due to suboptimal implementation the actual speed-up was only ca. 3 times.

V. CONCLUSION

This paper proposes the usage of Parareal for eddy current problems and discussed necessary modification due to nonconducting regions in the computational domain. A first implementation shows quick convergence of the iterative algorithm and promises a high speed-up. Future research will investigate the optimal choice of the coarse propagator.

Figure 1 .

 1 Figure 1. Sketch of the computational domain of an eddy current problem

Figure 3 .

 3 Figure 3. Jumps at the interfaces are compensated by Newton's method; coarse solution G (blue), correction via gradient (red) and fine solution F (green)

Figure 4 .

 4 Figure 4. Time stepping dynamics of the discrete magnetic vector potential

1 Figure 5 .

 15 Figure 5. Relative l 2 error w.r.t. classical backward Euler (δt = 10 -5 ); dashed vertical lines show the 40 intervals of size h = 0.05 s on which the problem is solved in parallel in the case of Parareal (green)

ACKNOWLEDGEMENT

This work was supported by DFG grants SCHO 1562/1-1 and CL 143/11-1, the Excellence Initiative of German Federal and State Governments and the Graduate School for CE at TU Darmstadt. The authors would like to thank R.V. Sabariego and N. Marsic for the fruitful discussions on GetDP.


Jean-Antoine Girault
email: jean-antoine.girault@inserm.fr

Angus C Nairn

DARPP-32 40 years later

Keywords: dopamine, striatum, phosphorylation, protein kinase, protein phosphatase, cAMP, drugs of abuse, L-DOPA-induced dyskinesia, schizophrenia

published or not. The documents may come

Introduction

DARPP-32 stands for dopamine-and cAMP-regulated phosphoprotein with an apparent Mr of 32 000 on polyacrylamide gel electrophoresis. Now, it is also known as PPP1R1B, phosphoprotein phosphatase 1 regulatory subunit 1B, based on its major biochemical property, which is to inhibit phosphoprotein phosphatase 1 (PPP1, previously known as PP1) when DARPP-32 is phosphorylated on Thr-34 (see below). DARPP-32 is arguably one of the most prototypical proteins discovered in Paul Greengard's laboratory and certainly one of the most iconic (its name appears in more than 1200 Pubmed references and 236 000 hits in Google). Since its discovery forty years ago, it has been the subject of intense study and has been key to understanding the complexity of signal transduction in neurons, especially in striatal projection neurons in which DARPP-32 is particularly abundant. Three related proteins with sequence similarities to DARPP-32 are found in vertebrates encoded by the PPP1R1A, PPP1R1B, and PPP1R1C

genes, but only the first two are highly expressed in neurons. PPP1R1B/DARPP-32 has a number of unique features that distinguish it from the two other paralogues. This review summarizes what we have discovered about DARPP-32 over the past 40 years, focusing on the contributions of the laboratories of Paul Greengard and his close collaborators and placing them in the perspective of current knowledge of neuronal signaling mechanisms.

The discovery of DARPP-32

In the '70s, Greengard and his colleagues identified mechanisms by which neurotransmitters exert many of their effects, through second messenger systems and activation of protein kinases [START_REF] Greengard | Phosphorylated proteins as physiological effectors[END_REF].

In doing so they extended and generalized findings initially done in the study of glycogen metabolism in the liver and skeletal muscle [START_REF] Krebs | Protein phosphorylation and cellular regulation, I. Nobel lectures[END_REF]. These studies provided a basis for elucidating the molecular mechanisms of complex synaptic properties such as neuromodulation and synaptic plasticity. In the course of this work several novel second messenger-regulated protein kinases were identified and their actions investigated in neurons. In the late '70s and early '80s a major question was the identity of the substrates of these protein kinases, the "third messengers" that were ultimately responsible for the action of neurotransmitters and other intercellular messengers [START_REF] Nestler | Protein phosphorylation in the brain[END_REF]. In an attempt to identify these substrates and to determine whether they were the same in different cells, new approaches were explored to compare potential substrates for various protein kinases in different brain regions. Notably, this was more than ten years before the word proteome was coined and tools became available to explore it. [START_REF] Walaas | Regional distribution of calcium-and cyclic adenosine 3':5'-monophosphate-regulated protein phosphorylation systems in mammalian brain. II. Soluble systems[END_REF] published 2 papers describing many proteins whose phosphorylation was enhanced by activation of diverse second messengers (Walaas, Nairn, & Greengard, 1983a[START_REF] Walaas | Regional distribution of calcium-and cyclic adenosine 3':5'-monophosphate-regulated protein phosphorylation systems in mammalian brain. I. Particulate systems[END_REF].

A few of these proteins attracted particular attention because they were detected in only one or a few of the brain regions investigated. Among these, a protein in the "soluble fraction" phosphorylated in response to cAMP stimulation was rapidly scrutinized because of its enrichment in the striatum, a region densely innervated by dopamine, which was known from previous work to act, at least in part, through stimulation of cAMP production [START_REF] Kebabian | Dopamine-sensitive adenyl cyclase: possible role in synaptic transmission[END_REF]. Walaas and colleagues showed the phosphorylation of this protein was increased by cAMP and dopamine in striatal slices and named it "DARPP-32" for that reason [START_REF] Walaas | A dopamine-and cyclic AMP-regulated phosphoprotein enriched in dopamine-innervated brain regions[END_REF]. Attempts to purify DARPP-32 were rapidly successful and the first information obtained about its properties showed it resembled a molecule known as protein phosphatase 1 inhibitor-1 (I1) [START_REF] Hemmings | DARPP-32, a dopamine-and adenosine 3':5'-monophosphate-regulated phosphoprotein enriched in dopamine-innervated brain regions. II Purification and characterization of the phosphoprotein from bovine caudate nucleus[END_REF]. I1 had been discovered in the lab of Philip Cohen in Dundee, a leader in the description of phosphatases, and shown to inhibit one of them, PP1. Sensitivity to phosphorylated I1 was in fact one of the key operational criteria to classify protein phosphatases, at a time when their sequences were not known [START_REF] Ingebritsen | The protein phosphatases involved in cellular regulation. 1 Classification and substrate specificities[END_REF]. Hugh Hemmings Jr., the graduate student who had purified DARPP-32 in Paul Greengard's laboratory, went for a few weeks to Philip Cohen's laboratory and showed that DARPP-32 was a very potent PP1 inhibitor when it was phosphorylated by cAMP-dependent protein kinase (PKA) [START_REF] Hemmings | DARPP-32, a dopamine-regulated neuronal phosphoprotein, is a potent inhibitor of protein phosphatase-1[END_REF]. These discoveries were major steps in the early understanding of how dopamine could act in the striatum. It opened up a broad perspective for understanding of the mechanisms of neuromodulation by dopamine since phosphatase inhibition was potentially capable to have numerous modulatory effects on target cells by reinforcing the effects of diverse signaling pathways, including, but not limited to, cAMP.

Detailed immunohistofluorescence studies showed that DARPP-32 was highly enriched in all striatal projection neurons (SPNs, a.k.a. medium-size spiny neurons or MSNs), as well as in a number of other cells in dopaminoceptive regions such as the cerebral cortex and beyond, for instance in Purkinje cells in the cerebellum or in non-neuronal cells in the hypothalamus [START_REF] Ouimet | DARPP-32, a dopamine-and adenosine 3':5'-monophosphate-regulated phosphoprotein enriched in dopamine-innervated brain regions. III Immunocytochemical localization[END_REF]. Further studies of DARPP-32 revealed that it had a wider distribution and more complex properties and functions than initially thought. First, DARPP-32 is expressed in many neurons and a few other cell types including some that do not receive dopamine innervation. For example DARPP-32 is present at high levels in choroid plexus where it is regulated by serotonin [START_REF] Snyder | Phosphorylation of DARPP-32 and protein phosphatase inhibitor-1 in rat choroid plexus: regulation by factors other than dopamine[END_REF]. Outside of the CNS, DARPP-32 is found, for example, in kidney [START_REF] Aperia | The significance of L-amino acid decarboxylase and DARPP-32 in the kidney[END_REF] or parathyroid (Meister, Askergren, Tunevall, Hemmings, & Greengard, 1991). Even in the main target of dopamine neurons, the striatum, it was apparent that the SPNs are divided in two populations, striatonigral and striatopallidal, with only the first being enriched in D1 dopamine receptors which activate adenylyl cyclase [START_REF] Gerfen | D 1 and D 2 dopamine receptor-regulated gene expression of striatonigral and striatopallidal neurons[END_REF]. Therefore the link between DARPP-32 and dopamine was not obligatory and turned out to be more complex than originally thought [START_REF] Svenningsson | DARPP-32: an integrator of neurotransmission[END_REF]. Part of this complexity is related to the multiple mechanisms of regulation of DARPP-32 by phosphorylation which we briefly discuss [for in-depth reviews see [START_REF] Svenningsson | DARPP-32: an integrator of neurotransmission[END_REF]Walaas, Hemmings, Greengard, & Nairn, 2011;[START_REF] Yger | DARPP-32, Jack of All Trades[END_REF]].

Regulation of DARPP-32 by phosphorylation of Thr34

DARPP-32 was discovered as a protein phosphorylated by PKA with a notable distribution in brain regions that generally overlapped with dopamine innervation [START_REF] Walaas | A dopamine-and cyclic AMP-regulated phosphoprotein enriched in dopamine-innervated brain regions[END_REF]. The first biochemical studies showed that DARPP-32 is a protein with little or no stable 3-D structure, making it resistant to acidic pH and heat [START_REF] Hemmings | DARPP-32, a dopamine-and adenosine 3':5'-monophosphate-regulated phosphoprotein enriched in dopamine-innervated brain regions. II Purification and characterization of the phosphoprotein from bovine caudate nucleus[END_REF], corresponding to a typical intrinsically unstructured protein [START_REF] Dancheck | Detailed structural characterization of unbound protein phosphatase 1 inhibitors[END_REF]. The key sequences for PPP1 inhibition include the phosphorylated Thr34 and a conserved motif found in many proteins interacting with PPP1 (Lys-Ile-Gln-Phe or KIQF) [START_REF] Kwon | Characterization of the interaction between DARPP-32 and protein phosphatase 1 (PP-1): DARPP-32 peptides antagonize the interaction of PP-1 with binding proteins[END_REF]. It is likely that the combination of these two binding sites is responsible for the very low IC50 of pThr34-DARPP-32 on PPP1 activity [START_REF] Desdouits | Mechanism of inhibition of protein phosphatase 1 by DARPP-32: studies with recombinant DARPP-32 and synthetic peptides[END_REF][START_REF] Kwon | Characterization of the interaction between DARPP-32 and protein phosphatase 1 (PP-1): DARPP-32 peptides antagonize the interaction of PP-1 with binding proteins[END_REF].

Thr34 is phosphorylated by PKA and also by cGMP-dependent protein kinase, in vitro (Hemmings, Nairn, & Greengard, 1984) and in cells [START_REF] Snyder | Phosphorylation of DARPP-32 and protein phosphatase inhibitor-1 in rat choroid plexus: regulation by factors other than dopamine[END_REF][START_REF] Tsou | Nitric oxide/cGMP pathway stimulates phosphorylation of DARPP-32, a dopamine-and cAMP-regulated phosphoprotein, in the substantia nigra[END_REF]. It is dephosphorylated by protein phosphatases 2A (PP2A, now PPP2) (Hemmings, Greengard, et al., 1984) and 2B (PP2B, now PPP3, a.k.a. calcineurin), a Ca 2+ -activated phosphatase [START_REF] King | Mammalian brain phosphoproteins as substrates for calcineurin[END_REF]. This latter property provides, among others, an interesting cross-talk between cAMP and stimuli that raise intracellular free Ca 2+ , including glutamate acting on NMDA receptors which can have opposing effects on DARPP-32 Thr34 phosphorylation [START_REF] Halpain | Activation of NMDA receptors induces dephosphorylation of DARPP-32 in rat striatal slices[END_REF]. Over the years a large number of extracellular signals increasing cAMP, or sometimes cGMP, levels have been shown to enhance DARPP-32 Thr34 phosphorylation [START_REF] Svenningsson | DARPP-32: an integrator of neurotransmission[END_REF]. Conversely various stimuli increasing intracellular free Ca 2+ trigger its dephosphorylation, thereby finely tuning PPP1 activity [START_REF] Nishi | Role of calcineurin and protein phosphatase-2A in the regulation of DARPP-32 dephosphorylation in neostriatal neurons[END_REF]. Importantly the regulation and function of Thr34 in DARPP-32 is similar to that of the homologous threonine residue in I1 (PPP1R1A), which is also abundant in the striatum but has a much broader distribution throughout brain [START_REF] Hemmings | The distribution of protein phosphatase inhibitor-1 in brain and peripheral tissues of various species: comparison with DARPP-32[END_REF]. This raises the question of why DARPP-32 was selected throughout evolution to exhibit its unique expression pattern. The answer may come from additional properties of DARPP-32, which were discovered through the thorough investigation of its other phosphorylation sites (see Figure 1).

[Insert Fig. 1 about here]

Regulation of DARPP-32 by phosphorylation of multiple residues

When labeled with 32 P in living striatal slices, DARPP-32 is mostly phosphorylated on serine [START_REF] Girault | Phosphorylation of DARPP-32, a dopamine-and cAMP-regulated phosphoprotein, by casein kinase II[END_REF][START_REF] Halpain | Activation of NMDA receptors induces dephosphorylation of DARPP-32 in rat striatal slices[END_REF]. Study of these site(s) first led to the discovery that DARPP-32 is a substrate for "casein kinase II", now referred to as protein kinase CK2, which is broadly expressed and highly active in the brain (Girault, Hemmings, Zorn, Gustafson, & Greengard, 1990). Mutant mice lacking CK2 in D1-expressing neurons have a strong phenotype including novelty-induced hyperlocomotion and exploratory behavior as well as defective motor control and motor learning, presumably resulting from enhanced D1 signaling [START_REF] Rebholz | Selective knockout of the casein kinase 2 in d1 medium spiny neurons controls dopaminergic function[END_REF]. CK2 phosphorylates Ser45 and Ser97 (in mouse sequence numbering, Ser102 in rat) [START_REF] Girault | Phosphorylation of DARPP-32, a dopamine-and cAMP-regulated phosphoprotein, by casein kinase II[END_REF]. Although phosphorylation at these site(s) appears to facilitate DARPP-32 phosphorylation by PKA [START_REF] Girault | Phosphorylation of DARPP-32, a dopamine-and cAMP-regulated phosphoprotein, by casein kinase II[END_REF], the phosphorylation of Thr34 was increased in CK2 KO mice [START_REF] Rebholz | Selective knockout of the casein kinase 2 in d1 medium spiny neurons controls dopaminergic function[END_REF], suggesting that this direct effect was overcome by indirect mechanisms in vivo. In agreement with this hypothesis, CK2 has a strong negative effect on G s and G olf [START_REF] Rebholz | CK2 negatively regulates Galphas signaling[END_REF], the latter being the G protein that activates adenylyl cyclase in SPNs [START_REF] Corvol | Galpha(olf) is necessary for coupling D1 and A2a receptors to adenylyl cyclase in the striatum[END_REF].

Importantly, phosphorylation of Ser97 regulates intracellular localization of DARPP-32, facilitating its nuclear export [START_REF] Stipanovich | A phosphatase cascade by which rewarding stimuli control nucleosomal response[END_REF]. Dephosphorylation of Ser97 is catalyzed by PPP2 [START_REF] Girault | Phosphorylation of DARPP-32, a dopamine-and cAMP-regulated phosphoprotein, by casein kinase II[END_REF]. This dephosphorylation can be stimulated by phosphorylation of PPP2 regulatory subunit B56 (PPP2R5D), triggering nuclear accumulation of pThr34-DARPP-32, thus facilitating inhibition of nuclear PPP1 and enhancing phosphorylation of histone H3 [START_REF] Stipanovich | A phosphatase cascade by which rewarding stimuli control nucleosomal response[END_REF]. Conversely, dephosphorylation of Ser97 by PPP2 associated with a Ca 2+ -sensitive regulatory subunit, PR72 (PPP2R3A) in response to glutamate stimulation favors nuclear accumulation of dephospho-Thr34-DARPP-32 and reduces PKA-dependent DARPP-32 signaling [START_REF] Nishi | Glutamate Counteracts Dopamine/PKA Signaling via Dephosphorylation of DARPP-32 Ser-97 and Alteration of Its Cytonuclear Distribution[END_REF]. DARPP-32 is also phosphorylated by casein kinase I (protein kinase CK1) on Ser130 (in mouse sequence, Ser137 in rat sequence) [START_REF] Desdouits | Phosphorylation of DARPP-32, a dopamine-and cAMP-regulated phosphoprotein, by casein kinase I in vitro and in vivo[END_REF]. This phosphorylation takes place at the C-terminus of a long stretch of acidic residue present in DARPP-32 from many species. It partly corrects the aberrant mobility of DARPP-32 in SDS/PAGE, which has a mobility of a 30-32-kDa protein whereas its real molecular mass is about 22 kDa, an observation which led to the identification of this phosphorylation site [START_REF] Desdouits | Phosphorylation of DARPP-32, a dopamine-and cAMP-regulated phosphoprotein, by casein kinase I in vitro and in vivo[END_REF]. Phosphorylation of DARPP-32 by CK1 renders phosphoThr34 resistant to the action of PPP3 [START_REF] Desdouits | Dopamine-and cAMP-regulated phosphoprotein DARPP-32: Phosphorylation of Ser-137 by casein kinase I inhibits dephosphorylation of Thr-34 by calcineurin[END_REF] and is enhanced by stimulation of metabotropic glutamate receptors [START_REF] Liu | Regulation of cyclin-dependent kinase 5 and casein kinase 1 by metabotropic glutamate receptors[END_REF]. PhosphoSer130 is dephosphorylated by PPP2 and, especially in vivo, by PP2C (phosphoprotein phosphatase requiring Mg 2+ /Mn 2+ , PPM) [START_REF] Desdouits | Dephosphorylation of Ser-137 in DARPP-32 by protein phosphatases 2A and 2C: different roles in vitro and in striatonigral neurons[END_REF].

Finally, DARPP-32 Thr75 is a substrate of Cdk5 and when DARPP-32 is phosphorylated on that site it becomes an inhibitor of PKA catalytic subunit [START_REF] Bibb | Phosphorylation of DARPP-32 by Cdk5 modulates dopamine signalling in neurons[END_REF]. Thus, DARPP-32 can switch from mediating cAMP-induced PPP1 inhibition to inhibiting PKA in response to phosphorylation by Cdk5. The switch-like property is enhanced by the fact that PKA can activate PPP2 associated with B56 (PPP2R5D), which dephosphorylates Thr-75 and amplifies cAMP signaling [START_REF] Nishi | Amplification of dopaminergic signaling by a positive feedback loop[END_REF].

Recent work has identified protein partners of DARPP-32 and emphasizes the importance of localized regulation of PKA activity by DARPP-32. DARPP-32 can interact with adducins [START_REF] Engmann | DARPP-32 interaction with adducin may mediate rapid environmental effects on striatal neurons[END_REF], which are important linkers between the actin/spectrin cytoskeleton and the plasma membrane.

Phospho-Thr75-DARPP-32 facilitates phosphorylation of -adducin at Ser713 through inhibition of a cAMP-dependent protein kinase/phosphatase-2A cascade thereby facilitating the depolymerization of the cortical cytoskeleton [START_REF] Engmann | DARPP-32 interaction with adducin may mediate rapid environmental effects on striatal neurons[END_REF]. Behavioral evidence suggests that this regulatory mechanism is implicated in the effects of enriched environment on the properties of SPNs and their responsiveness to cocaine [START_REF] Engmann | DARPP-32 interaction with adducin may mediate rapid environmental effects on striatal neurons[END_REF]. Dysregulation of this mechanism may be implicated in the depressive phenotype of a mouse model of Huntington's disease [START_REF] Brito | Cyclin-Dependent Kinase 5 Dysfunction Contributes to Depressive-like Behaviors in Huntington's Disease by Altering the DARPP-32 Phosphorylation Status in the Nucleus Accumbens[END_REF]. Thr75 phosphorylation is enhanced by caffeine and may contribute to its psychostimulatory effects [START_REF] Lindskog | Involvement of DARPP-32 phosphorylation in the stimulant action of caffeine[END_REF]. Thr75 is also implicated in the interaction of DARPP-32 with the CLOCK protein, which prevents the binding of CLOCK to BMAL and its transcriptional effects, suggesting a possible link with caffeine-induced effects on mood [START_REF] Trautmann | A regulatory pathway linking caffeine action, mood and the diurnal clock[END_REF].

These regulations by multiple phosphorylation sites distinguish DARPP-32 from the other related inhibitors of PPP1, PPP1R1A and PPP1R1C. Experimental evidence suggests these sites are implicated in the fine regulation of DARPP-32 function. However, the resulting imbrication of these various regulatory mechanisms and the multiple potential positive and negative feedback interactions make it challenging to make simple predictions about their contribution to regulatory processes. Therefore it has been of great interest to use the power of computer modeling to dissect the potential regulation and roles of DARPP-32 in a more integrated context.

Computational modeling of DARPP-32 phosphorylation and activity

The positioning of DARPP-32 at the crossroad of signaling by important neurotransmitters with a strong importance in the control of basal ganglia functions and in several pathological conditions has attracted the attention of several computationally-oriented groups. A first attempt showed that DARPP-32 is a robust signal integrator, whose response depends on the delay between cAMP and Ca 2+ signals [START_REF] Fernandez | DARPP-32 is a robust integrator of dopamine and glutamate signals[END_REF], providing a mechanism for tuning physiological responses to the timing of inputs to SPNs. Moreover, a non-intuitive convergent synergism of cAMP-and Ca 2+ -dependent pathways on DARPP-32 phosphorylation was disclosed by a different study [START_REF] Lindskog | Transient calcium and dopamine increase PKA activity and DARPP-32 phosphorylation[END_REF]. A collaborative effort between Paul Greengard's lab and his son, who is mathematician, used an approach of dimensionality reduction to provide evidence that the network in which DARPP-32 participates has particularly robust features in the presence of noise [START_REF] Barbano | A mathematical tool for exploring the dynamics of biological networks[END_REF]. A model including additional steps identified a positive feedback loop consisting of PKA, PPP2, and pThr-75 DARPP-32 serving as a major switch for inducing LTD and LTP [START_REF] Nakano | A kinetic model of dopamine-and calciumdependent striatal synaptic plasticity[END_REF]. Calcium input modulates this loop through the PPP3-CK1-Cdk5-Thr75 pathway and PPP2, whereas Ca 2+ and dopamine input activate the loop via PKA. Interestingly this positive feedback loop displayed robust bistable responses. A recent study also provided experimental and computational evidence for a switch-like PKA response in the nucleus of SPNs that requires a positive feedback through a phosphatase inhibitor like DARPP-32 [START_REF] Yapo | Switch-like PKA responses in the nucleus of striatal neurons[END_REF].

A different approach investigating DARPP-32 as an index of signaling from synapses to the nucleus, revealed the importance of dendrite diameter in the cellular responses [START_REF] Li | Dendritic geometry shapes neuronal cAMP signalling to the nucleus[END_REF]. Model predictions of nuclear DARPP-32 phosphorylation as a readout of cAMP signaling pointed to the particular efficiency of synapses on secondary dendrites to activate nuclear PKA, which was confirmed by live imaging [START_REF] Li | Dendritic geometry shapes neuronal cAMP signalling to the nucleus[END_REF]. Taken together these modelling approaches have shown that the complex regulatory steps in which DARPP-32 plays a central role have important properties including robustness to noise and bistability. They also led to the discovery of the importance of dendritic tree geometry on synapse-to-nucleus signaling. It is important to emphasize that the regulation of protein phosphatases by DARPP-32 is only one aspect of regulatory signaling networks in SPNs which also include several other regulatory proteins and phosphatase inhibitors (see paper by Cunningham and Nairn in this issue) that shape the particular responses of SPNs.

Role of DARPP-32 in the physiology of the basal ganglia

Since DARPP-32 is a phosphorylation-controlled phosphatase/kinase inhibitor, with the complex regulations emphasized above, it has been difficult to ascribe to it a simple function. Depending on its state of phosphorylation, its intracellular localization, and its interaction with partners, it can modulate a variety of targets. To try and address the functional implications of this signaling regulations, mutant mice lacking DARPP-32 either completely [START_REF] Fienberg | DARPP-32: Regulator of the efficacy of dopaminergic neurotransmission[END_REF] or in specific cell types [START_REF] Bateup | Distinct subclasses of medium spiny neurons differentially regulate striatal motor behaviors[END_REF] have been used. In addition, the consequences of point mutation of the various phosphorylation sites have been investigated in some detail [START_REF] Svenningsson | Diverse psychotomimetics act through a common signaling pathway[END_REF]. The first conclusion from these studies was that, in laboratory housing conditions, mice do perfectly well without DARPP-32 [START_REF] Fienberg | DARPP-32: Regulator of the efficacy of dopaminergic neurotransmission[END_REF]. They breed and reproduce as their wild type littermates. However, the consequences of the absence of the protein become apparent when mice are challenged. The initial studies revealed that a variety of direct or indirect responses to dopamine or other neurotransmitters are altered [START_REF] Fienberg | DARPP-32: Regulator of the efficacy of dopaminergic neurotransmission[END_REF][START_REF] Hiroi | Neuronal and behavioural abnormalities in striatal function in DARPP-32-mutant mice[END_REF]. In most instances these effects are not completely lost but require stronger stimuli to occur; in other words the dose-response curves are shifted to the right. However, in particular experimental conditions, the responses can be lost. For example it is striking that in striatal slices, in the absence of DARPP-32, specific forms of long-term potentiation and depression (LTP and LTD) at corticostriatal synapses disappeared and were restored by bath application of a cell-permeant phosphatase inhibitor [START_REF] Calabresi | Dopamine and cAMP-regulated phosphoprotein 32 kDa controls both striatal long-term depression and long-term potentiation, opposing forms of synaptic plasticity[END_REF]. An elegant study compared the consequences of a conditional loss of DARPP-32 in either of the two populations of SPNs [START_REF] Bateup | Distinct subclasses of medium spiny neurons differentially regulate striatal motor behaviors[END_REF]. The loss of DARPP-32 in striatonigral neurons decreased basal and cocaine-induced locomotion. Conversely, the loss of DARPP-32 in striatopallidal neurons produced a robust increase in locomotor activity and strongly reduced cataleptic response to the antipsychotic drug haloperidol. Point mutation of the main phosphorylation sites in knock-in mice [START_REF] Svenningsson | Diverse psychotomimetics act through a common signaling pathway[END_REF] allowed a dissection of their contribution in vivo and was helpful to address the role of DARPP-32 in pathology.

What is the role of DARPP-32 in the brain in physiological conditions? It is probably over simplistic to look for a straightforward correlation between a signaling molecule with complex regulatory functions and a "simple" functional output at the behavioral level. Several studies, however, provide evidence for an implication of DARPP-32 in the processing of natural rewards. In rats it was shown that the phosphorylation pattern of DARPP-32 undergoes specific modifications in the nucleus accumbens in response to exposure to food reward, in line with the important role of dopamine in this context (see [START_REF] Scheggi | DARPP-32 in the orchestration of responses to positive natural stimuli[END_REF] for a review). Mutation of DARPP-32 does not grossly impair food intake and DARPP-32 KO mice learn to nose-poke for food as wild type controls [START_REF] Heyser | DARPP-32 knockout mice exhibit impaired reversal learning in a discriminated operant task[END_REF]. However, these mice lacking DARPP-32 are impaired when the rewarded hole is reversed [START_REF] Heyser | DARPP-32 knockout mice exhibit impaired reversal learning in a discriminated operant task[END_REF] and motivation for food reward is decreased in Ser97Ala mutant mice [START_REF] Stipanovich | A phosphatase cascade by which rewarding stimuli control nucleosomal response[END_REF], providing evidence of the implication of this molecule in tuning food related behaviors. Evidence also suggests a role of DARPP-32 in some aspects of sexual behavior both in female [START_REF] Mani | Requirement for DARPP-32 in progesterone-facilitated sexual receptivity in female rats and mice[END_REF][START_REF] Meredith | Matingrelated stimulation induces phosphorylation of dopamine-and cyclic AMP-regulated phosphoprotein-32 in progestin receptor-containing areas in the female rat brain[END_REF] and male [START_REF] Mchenry | Dopamine D1 receptors and phosphorylation of dopamine-and cyclic AMP-regulated phosphoprotein-32 in the medial preoptic area are involved in experience-induced enhancement of male sexual behavior in rats[END_REF] rodents. These effects appear to indicate a role of DARPP-32 in hypothalamic nuclei, and, at least in females, its contribution to actions of progesterone. These results suggest that DARPP-32 is implicated at various levels in the correct processing of natural rewards and their consequences on behavior, possibly improving complex adaptive responses.

DARPP-32 in pathological conditions

The possible role of DARPP-32 various pathological conditions has been investigated both in humans, using post-mortem tissues and genetic approaches, and in animal models. Notably, besides its role in neurons, DARPP-32 and its shorter splice variant that lacks the N-terminal moiety including Thr34 and termed "truncated DARPP-32" (tDarpp), have been implicated in several cancers (see [START_REF] Avanes | Darpp-32 and t-Darpp protein products of PPP1R1B: Old dogs with new tricks[END_REF] for a recent review). These important findings are not discussed here and we focus on neurological and psychiatric conditions.

Drugs of abuse and addiction

Given the enrichment of DARPP-32 in SPNs, which are the major target of dopamine, and since drugs of abuse all share the ability to increase extracellular dopamine in the striatum, particularly in its ventral component, the nucleus accumbens (Di Chiara, 1999), it is not surprising that many drugs of abuse act via DARPP-32. Many drugs of abuse alter DARPP-32 phosphorylation and their effects are modified in DARPP-32 KO mice (see [START_REF] Nairn | The role of DARPP-32 in the actions of drugs of abuse[END_REF] for a review). DARPP-32 participates in the signaling mechanisms that promote ERK phosphorylation in response to cocaine [START_REF] Valjent | Regulation of a protein phosphatase cascade allows convergent dopamine and glutamate signals to activate ERK in the striatum[END_REF], an important step in the long-term responses of striatal neurons [START_REF] Girault | ERK2: a logical AND gate critical for druginduced plasticity?[END_REF].

The use of point mutant mice pinpointed the importance of diverse components of DARPP-32 signaling.

Mice bearing the Thr34Ala point mutation display a decreased locomotor sensitization in response to cocaine [START_REF] Valjent | Regulation of a protein phosphatase cascade allows convergent dopamine and glutamate signals to activate ERK in the striatum[END_REF] or morphine [START_REF] Valjent | Mechanisms of locomotor sensitization to drugs of abuse in a two-injection protocol[END_REF] administration in a 2-injection protocol.

Cocaine-induced conditioned place preference was also decreased in DARPP-32 KO [START_REF] Zachariou | Reduction of cocaine place preference in mice lacking the protein phosphatase 1 inhibitors DARPP 32 or Inhibitor 1[END_REF] and Thr34Ala mutant mice [START_REF] Zachariou | Phosphorylation of DARPP-32 at Threonine-34 is required for cocaine action[END_REF]. Acquisition of stable cocaine selfadministration was delayed in Thr34Ala mice but both Thr34Ala and Ser130Ala DARPP-32 mutant mice self-administered more cocaine than their respective wild-type controls [START_REF] Zhang | Cocaine self-administration in mice is inversely related to phosphorylation at Thr34 (protein kinase A site) and Ser130 (kinase CK1 site) of DARPP-32[END_REF]. Interestingly it appears that acute injection and chronic self-administration of cocaine have different effects on Thr34 phosphorylation, which is increased in the first case [START_REF] Bateup | Cell type-specific regulation of DARPP-32 phosphorylation by psychostimulant and antipsychotic drugs[END_REF][START_REF] Nishi | Amplification of dopaminergic signaling by a positive feedback loop[END_REF] and decreased in the second [START_REF] Zhang | Cocaine self-administration in mice is inversely related to phosphorylation at Thr34 (protein kinase A site) and Ser130 (kinase CK1 site) of DARPP-32[END_REF]. Accordingly, induction of Cdk5 by chronic cocaine and the resulting phosphorylation of DARPP-32 at Thr75, may act as a negative feedback homeostatic mechanism with respect to the behavioral effects of cocaine [START_REF] Bibb | Effects of chronic exposure to cocaine are regulated by the neuronal protein Cdk5[END_REF]. In conclusion, although DARPP-32 is not necessary for the appearance of addiction-like behavior, it plays a dual modulatory role in animal models of cocaine addiction, with positive and negative effects on different aspects.

Movement disorders

The treatment of Parkinsonian patients with L-DOPA corresponds to another condition in which there is a strong pharmacological stimulation of dopamine signaling in SPNs. Converging evidence shows that this intermittent stimulation, beyond its initial symptomatic efficacy, contributes to the appearance of the abnormal movements triggered by L-DOPA that complicate treatment of Parkinson's disease after several years. These movements are designated as L-DOPA-induced dyskinesia or LID. Intermittent L-DOPA treatment alternates with periods of lack of dopamine stimulation, a situation that presents some similarity with what is observed in addiction when the self-administration of drugs triggers irregular and intense stimulation of dopaminergic neurotransmission. Although it striatal levels are not altered in Parkinsonian patients [START_REF] Girault | Striatal phosphoproteins in Parkinson disease and progressive supranuclear palsy[END_REF][START_REF] Raisman-Vozari | Lack of change in striatal DARPP-32 levels following nigrostriatal dopaminergic lesions in animals and in parkinsonian syndromes in man[END_REF], DARPP-32 contributes to the generation of LID. In rats with lesioned dopamine neurons that display LID following L-DOPA treatment, the lack of synaptic depotentiation is accompanied by high levels of pThr34 DARPP-32 [START_REF] Picconi | Loss of bidirectional striatal synaptic plasticity in L-DOPA-induced dyskinesia[END_REF]. LID are attenuated in DARPP-32 KO mice as well as following treatment with a compound that prevents activation of the ERK pathway [START_REF] Santini | Critical involvement of cAMP/DARPP-32 and extracellular signal-regulated protein kinase signaling in L-DOPA-induced dyskinesia[END_REF]. Thr34Ala mutation diminishes LID and ERK activation [START_REF] Santini | Dopamine-and cAMP-regulated phosphoprotein of 32-kDa (DARPP-32)-dependent activation of extracellular signal-regulated kinase (ERK) and mammalian target of rapamycin complex 1 (mTORC1) signaling in experimental parkinsonism[END_REF] suggesting some similarities in the consequences of L-DOPA treatment and cocaine administration [START_REF] Valjent | Regulation of a protein phosphatase cascade allows convergent dopamine and glutamate signals to activate ERK in the striatum[END_REF]. In the case of L-DOPA, however, ERK activation is not completely dependent on DARPP-32 [START_REF] Alcacer | Galpha(olf) mutation allows parsing the role of cAMP-dependent and extracellular signal-regulated kinase-dependent signaling in L-3,4-dihydroxyphenylalanine-induced dyskinesia[END_REF][START_REF] Gerfen | Differences between dorsal and ventral striatum in Drd1a dopamine receptor coupling of dopamine-and cAMP-regulated phosphoprotein-32 to activation of extracellular signal-regulated kinase[END_REF]. Strikingly, conditional KO of DARPP-32 in striatonigral neurons abolishes LID [START_REF] Bateup | Distinct subclasses of medium spiny neurons differentially regulate striatal motor behaviors[END_REF]. These results indicate that DARPP-32 is an important component of the signaling pathways that result in LID, and be involved in the induction or in the expression of these abnormal movements or in both.

Psychiatric disorders

Because of DARPP-32's role in mediating dopamine actions and the implication of dopamine in schizophrenia, several studies have been carried out in animal models and in humans to address the possible involvement of DARPP-32 in this disease or group of diseases. In mice various psychotomimetic drugs change DARPP-32 phosphorylation and their effects are altered by selective mutation of DARPP-32 phosphorylation sites [START_REF] Svenningsson | Diverse psychotomimetics act through a common signaling pathway[END_REF]. These results supported the role of DARPP-32 in the signaling mechanisms activated by these drugs, but evidence for a possible role in human patients required the investigation of human brain samples. A first post-mortem study found that DARPP-32 protein levels were decreased in the prefrontal cortex of schizophrenic patients as compared with matched controls [START_REF] Albert | Evidence for decreased DARPP-32 in the prefrontal cortex of patients with schizophrenia[END_REF]. Similar observations were then reported in patients with schizophrenia or bipolar disorder [START_REF] Ishikawa | Immunohistochemical and immunoblot analysis of Dopamine and cyclic AMP-regulated phosphoprotein, relative molecular mass 32,000 (DARPP-32) in the prefrontal cortex of subjects with schizophrenia and bipolar disorder[END_REF], in schizophrenic patients dead from suicide [START_REF] Feldcamp | Reduced prefrontal cortex DARPP-32 mRNA in completed suicide victims with schizophrenia[END_REF] and in schizophrenic patients [START_REF] Kunii | Differential protein expression of DARPP-32 versus Calcineurin in the prefrontal cortex and nucleus accumbens in schizophrenia and bipolar disorder[END_REF]. A frequent 7-single nucleotide polymorphism (7-SNP) haplotype in DARPP-32 gene (PPP1R1B) predicting the expression of its isoforms in postmortem human brain, was associated with enhanced performance on cognitive tests depending on frontostriatal function [START_REF] Meyer-Lindenberg | Genetic evidence implicating DARPP-32 in human frontostriatal structure, function, and cognition[END_REF]. Although preliminary evidence suggested that this haplotype could be associated with risk of schizophrenia [START_REF] Meyer-Lindenberg | Genetic evidence implicating DARPP-32 in human frontostriatal structure, function, and cognition[END_REF], the association of PPP1R1B locus with increased risk for major psychiatric was not found in other small scale studies or in major GWAS. However, relationships between genetic variants and expression levels have been confirmed. In a study including a large number of cases, tDarpp was found to be increased in the dorsolateral prefrontal cortex of patients with schizophrenia and bipolar disorder, and was strongly associated with PPP1R1B genotypes [START_REF] Kunii | Revisiting DARPP-32 in postmortem human brain: changes in schizophrenia and bipolar disorder and genetic associations with t-DARPP-32 expression[END_REF]. Some evidence even suggested that genetic variants associated with higher tDarpp expression predict worse cognitive performance [START_REF] Kunii | Revisiting DARPP-32 in postmortem human brain: changes in schizophrenia and bipolar disorder and genetic associations with t-DARPP-32 expression[END_REF]. Taken together these studies support the existence of a decrease in DARPP-32 associated with an increased expression of tDarpp in the prefrontal cortex of patients with schizophrenia and possibly bipolar disorder. Since tDarpp may oppose cAMPregulated functions of DARPP-32, this combination suggests an overall decreased function of DARPP-32 in these conditions. How these observations are related with the disease mechanisms remains to be determined. One obvious link could be through the consequences of chronic treatment with antipsychotics, but, in rats, prolonged administration of dopamine antagonists or antipsychotics does not alter DARPP-32 or tDARPP levels [START_REF] Grebb | Chronic treatment of rats with SCH-23390 or raclopride does not affect the concentrations of DARPP-32 or its mRNA in dopamineinnervated brain regions[END_REF][START_REF] Kunii | Revisiting DARPP-32 in postmortem human brain: changes in schizophrenia and bipolar disorder and genetic associations with t-DARPP-32 expression[END_REF]. It is therefore possible that the observed changes reflect primary cellular alterations in patients which warrant further exploration. Given the reported alterations of DARPP-32 in both bipolar disorder and schizophrenia and the heterogeneity of the latter condition, it is possible that they are more pronounced in specific subgroups of patients. Whether this corresponds to particular clinical features or response to treatment is an intriguing question. However, the current impossibility to non-invasively evaluate DARPP-32 levels in living subjects makes it very difficult to address.

Conclusion

Throughout his academic career Paul Greengard provided major discoveries related to the mechanisms of action of neurotransmitters in general, but especially to the details of dopamine signaling. After the initial discovery that dopamine activated the production of cAMP in the striatum in the early '70s, he pioneered the elucidation of the mechanisms by which cAMP acts in striatal neurons. This demonstrated the paramount importance of intracellular protein phosphorylation pathways, an idea that Greengard put forward early on. In-depth exploration of the regulation and properties of DARPP-32 in striatal neurons led to the discovery of a variety of mechanisms whose importance, or even existence, in brain was previously not known. They the role of several protein kinases such as CK1, CK2, and Cdk5 as well as the importance and intricacy in neurons of the regulation of major protein phosphatases, PPP1, PPP2, PPP3, and PPM. Given that DARPP-32 is a regulatory protein with multiple functions at the crossroads of several signaling pathways, it is not surprising that its role cannot be restricted to a simple functional output. The use of computer modeling revealed that it provides robustness to intracellular signaling pathways and endows them with a bistable activity, switching between states of strong and weak cAMP signaling. As a result we can hypothesize that DARPP-32 enhances the contrast in the responses of the striatal network to stimulation, and increases the overall signal to noise ratio. A characteristic of the regulatory networks in which DARPP-32 participates is their high degree of redundancy, which makes them difficult to fully evaluate in vivo, since altering any single component has only relatively mild consequences. However the properties of these multiple intricate signaling networks, including their redundancy, are likely to be essential for the fine and adapted control of action and movement as well as its adaptation over time through reward-controlled procedural learning that characterizes the integrated function of striatal projection neurons. The main amino acid residues are indicated with the kinases that phosphorylate them (green) and the phosphatases that remove phosphates from these residues (red). Some additional features are indicated: KIQF, consensus motif (Lys-Ile-Gln-Phe) that binds PPP1 and contributes to its inhibition with phosphoThr34, the nuclear export sequence (NES) and the acidic stretch characteristic of DARPP-32. Other abbreviations: CK1, casein kinase 1, CK2, casein kinase 2, Cdk5, cyclin-dependent protein kinase 5, PKA, cAMP-dependent protein kinase, PKG, cGMP-dependent protein kinase, PPP1, protein phosphatase 1 (a.k.a. PP1), PPP2, protein phosphatase 2A (PP2A), PPP3, calcineurin (a.k.a. protein phosphatase 2B, PP2B, calcineurin), PPM, phosphoprotein phosphatase requiring Mg 2+ /Mn 2+ (a.k.a. PP2C).

Figure legend Figure 1 :

 legend1 Figure legend

Acknowledgements

The authors dedicate this review the memory of Paul Greengard who was a visionary scientist, an extraordinary mentor, and a wonderful friend. Research in the laboratory of JAG is currently supported by Inserm and Sorbonne Université and grants from Agence Nationale de la Recherche (ANR-16-CE16-0018, ANR-19-CE16-0020), Laboratory of Excellence Bio-Psy (Investissements d'Avenir, ANR-11-IDEX-0004-02), and Fondation pour la Recherche Médicale. Research in the laboratory of ACN is supported by the National Institutes of Health (NIH) (AG047270, AG062306, AG066508, DA018343) and the State of Connecticut Department of Mental Health and Addiction Services.