Hirokazu Tanimoto

Jeremy Sallé

Louise Dodin

Nicolas Minc
email: nicolas.minc@ijm.fr

Physical Forces Determining the Persistency and Centering Precision of Microtubule Asters

How large cellular structures move purposefully in a crowded and fluctuating cytoplasm remains unclear. In early embryos, microtubule asters can reach hundreds of micrometers and move at high speeds to find the geometrical center of large egg and blastomere cells, a process essential for the fidelity of cell division and development. Aster centration in these cells therefore represents a prime and ubiquitous example of directed cytoskeletal motion with acute physiological constraints on precision. We found that positional fluctuations of migrating sea urchin sperm asters were small, anisotropic, and associated with the stochasticity of dyneindependent forces moving the aster. Using in vivo magnetic tweezers to directly measure aster forces inside cells, we derived a linear aster force-velocity relationship and evidenced a springlike active mechanism stabilizing aster transverse position. The large frictional coefficient and spring constant quantitatively accounted for the amplitude and growth characteristics of athermal positional fluctuations, demonstrating that aster mechanics ensure noise-suppression to promote persistent and precise centration. These findings define generic biophysical regimes of active cytoskeletal mechanics underlying the accuracy of cell division and early embryonic development.

Microtubule (MT) asters are star-shaped cytoskeletal structures composed of MT polymers radiating from an organizing center called the centrosome. They contribute to the spatial organization of crucial functions in eukaryotic cells, ranging from cell migration to nuclear centration and mitotic spindle orientation [START_REF] Bornens | The centrosome in cells and organisms[END_REF][START_REF] Tang | Centrosome positioning in vertebrate development[END_REF][START_REF] Mitchison | Growth, interaction, and positioning of microtubule asters in extremely large vertebrate embryo cells[END_REF] . One highly conserved property of MT asters is their ability to probe the geometrical boundaries of the cell to move and position themselves in the exact cell center. This was best highlighted in seminal in vitro work reconstituting aster growth and centration in microfabricated wells of a few microns in size. In those studies, pushing forces resulting from astral MT polymerization against the chamber wall 4,5 or pulling forces provided by MT minus-end directed dynein motors attached to the wall surface [START_REF] Laan | Cortical Dynein Controls Microtubule Dynamics to Generate Pulling Forces that Position Microtubule Asters[END_REF] allowed asters to target the chamber center.

A stereotypical and ubiquitous in vivo counterpart for aster centration occurs soon after fertilization in most animal embryos [START_REF] Gilbert | Developmental Biology[END_REF] . In this context, the fertilizing sperm brings the male pronucleus and its associated centrosomal material into the side of the egg, which results in the nucleation of a "sperm aster" that continuously grows and moves to the egg center. This centration motion is critical to position the nucleus and subsequent spindle and division plane in the exact cell center. Contrary to in vitro situations, studies in systems including worm, frog, fish, and echinoderm embryos have suggested that aster centration in those cells may not primarily involve MT polymerization or cortical dynein forces [START_REF] Hamaguchi | Analysis of the Role of Astral Rays in Pronuclear Migration in Sand Dollar Eggs by the Colcemid-UV Method[END_REF][START_REF] Kimura | Computer simulations and image processing reveal length-dependent pulling force as the primary mechanism for C. elegans male pronuclear migration[END_REF][START_REF] Longoria | Cargo transport by cytoplasmic dynein can center embryonic centrosomes[END_REF][START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF][START_REF] Wuhr | A model for cleavage plane determination in early amphibian and fish embryos[END_REF] . Rather, a prominent model is that most of the forces are provided by dynein motors working along astral MTs in bulk cytoplasm [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF][START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF][START_REF] Pierre | Generic Theoretical Models to Predict Division Patterns of Cleaving Embryos[END_REF][START_REF] Wuhr | How does a millimeter-sized cell find its center?[END_REF] . Dynein motors generate plus-end directed traction forces, probably as complexes with endomembrane components such as lysosome vesicles, yolk granules [START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF] , via frictional interactions with the viscous cytoplasm. As longer MTs may associate with more dyneins, they may exert larger pulling forces on the centrosome. This length-dependent system, coupled to MT length asymmetries caused by cellular boundaries, provides a self-organization design for asters to target the cell center [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF][START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF][START_REF] Pierre | Generic Theoretical Models to Predict Division Patterns of Cleaving Embryos[END_REF] .

One outstanding physical problem implicated in aster centration in early embryos arises from the unusually large size of egg cells and early blastomeres [START_REF] Mitchison | Growth, interaction, and positioning of microtubule asters in extremely large vertebrate embryo cells[END_REF][START_REF] Wuhr | How does a millimeter-sized cell find its center?[END_REF] . These cells are typically 10-100 times larger than somatic cells [START_REF] Zhu | Finding the cell center by a balance of dynein and myosin pulling and microtubule pushing: a computational study[END_REF] or in vitro microchambers [START_REF] Laan | Cortical Dynein Controls Microtubule Dynamics to Generate Pulling Forces that Position Microtubule Asters[END_REF] , yet achieve aster centration on a timescale of only a few tens of minutes. Because of the physiological importance of aster centration in early embryos, these parameters set extreme constraints on motion persistency, speed, and centering precision. Given cytoplasmic crowding, extrinsic cellular noise, and intrinsic stochasticity of molecular elements involved [START_REF] Fulton | How crowded is the cytoplasm?[END_REF][START_REF] Brangwynne | Cytoplasmic diffusion: molecular motors mix it up[END_REF] , how moving asters may satisfy those constraints inside cells remains mysterious overall.

Here we exploited the centration of sea urchin sperm asters as a quantitative model system to derive the biophysical principles ensuring robust aster centration. By combining high-resolution tracking and direct intracellular aster force measurement, we find that aster motion is associated with large forces and small active positional fluctuations. This work demonstrates how aster mechanics may ensure noise-suppression to promote persistent and precise centration.

We first employed high-resolution microscopy (spatial resolution ~20 nm, temporal resolution ~50 ms, see supplementary information) (Fig. S1 and movie S1) to track the motion of male pronuclei attached to sperm MT asters in fertilized sea urchin eggs. We confirmed that aster speed was, on average, constant along the longitudinal centering direction (X-axis) and zero along the transverse axis (Y-axis) [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF] . Aster trajectory appeared smooth overall, but did exhibit some minor excursions away from the centering axis, which rapidly resorbed (Fig. 1, a andb).

To quantitatively examine the stochastic fluctuations around the mean motion, we detrended aster trajectory by subtracting its local velocity and computed the residual displacements, X and Y, as a function of lag time, t (eq. S2) [START_REF] Winkler | Fluctuation Analysis of Centrosomes Reveals a Cortical Function of Kinesin-1[END_REF] . The probability distribution functions (PDFs) of X and Y were nearly similar for t < 30 sec, and were well described by Gaussian functions (Fig. 1c). For t > 30 sec, while the PDF of Y kept a near-constant shape, the PDF of X appeared to deviate from a Gaussian and had a non-zero mean, which could reflect more complex behaviors such as higher order slow changes in the aster mean speed.

We characterized the statistical properties of aster fluctuations by plotting the second order moment of X and Y (mean-squared residual displacement, MSrD) as a function of t (Fig. 1d). Both MSrDs were flat at t < 1 sec, due to measurement noise. They then grew linearly above the measurement noise with a slope close to 1 for 1 < t < 30 sec, suggesting diffusive dynamics with similar diffusion coefficients along both axes: 𝐷 𝑥 = 1.7*10 -3 m 2 /sec and 𝐷 𝑦 = 1.8*10 -3 m 2 /sec. These results indicate the existence of uniform random forces which cause asters to fluctuate in a diffusive manner.

The two MSrDs had different behaviors at t > 30 sec. While the longitudinal fluctuations kept growing, the transverse fluctuations saturated, likely reflecting a positional feedback that stabilizes aster trajectory transversely (Fig. 1d). Accordingly, these transverse fluctuations were well described by a random walk under spring-like restoration forces (Fig. 1d, inset) [START_REF] Pecreaux | The Mitotic Spindle in the One-Cell C. elegans Embryo Is Positioned with High Precision and Stability[END_REF] , so that MSrD 𝑦 (𝛿𝑡) = 2𝜏𝐷 𝑦 (1 -𝑒 -𝛿𝑡/𝜏 ), (Eq. 1)

with a saturation amplitude 2𝜏𝐷 𝑦 = 0.17m 2 and saturation timescale 𝜏 = 46 sec. This corresponded to the typical size and restoration time of excursion events away from the centering axis. The saturation amplitude allowed for estimating a mean deviation distance of the aster from the centering axis of √ 2𝜏𝐷 𝑦 = 0.41 µm, which was typically less than 1% of the cell radius, demonstrating a remarkable centration precision. Thus, aster fluctuations are small and anisotropic with characteristics determined by a balance between random forces and viscous dampening, and additional spring-like feedback in the transverse axis.

To discern if the observed fluctuations reflected thermal noise or active processes, we manipulated cytoskeletal components using specific chemical inhibitors (Fig. 1, d and e and Fig. S3-4). To separately characterize the diffusive fluctuations from the effects of positional feedback, we computed a fluctuation amplitude defined as MSrD at t=5 sec, where asters exhibit purely diffusive behavior. Strikingly, addition of 100 M of Ciliobrevin D, which inhibits dynein activity and halts aster motion [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF] , decreased positional fluctuation amplitude by almost an order of magnitude. Thus dynein force-generation events which drive aster centration may also add active noise to this motion because of their stochastic nature. Actin depolymerization with 20 µM Latrunculin B affected cell cortex and cell shape 11 , but did not affect aster motion or fluctuation amplitude. This suggests that dynein drives aster fluctuations within the bulk cytoplasm not from the cortex, and by associating with cytoplasmic elements independent of actin [START_REF] Pecreaux | The Mitotic Spindle in the One-Cell C. elegans Embryo Is Positioned with High Precision and Stability[END_REF][START_REF] Almonacid | Active diffusion positions the nucleus in mouse oocytes[END_REF] . Finally, treatment with 20 µM Nocodazole to depolymerize MTs also halted aster motion, but caused the sperm nucleus to fluctuate more than in controls, suggesting that MTs may contribute to a large fraction of aster viscous drag. Importantly, in Nocodazole and Ciliobrevin D treatments, longitudinal and transverse MSrDs both grew diffusively for the entire timescale without saturation (Fig. 1d and fig. S4). This indicates that MTs and dynein contribute to the transverse spring-like feedback.

To understand how those kinetic properties may emerge from the mechanical properties of moving asters, we set out to directly measure the physical forces of asters inside cells. We modified a magnetic tweezer strategy recently used to measure forces for mitotic spindle maintenance in C. elegans [START_REF] Garzon-Coral | A force-generating machinery maintains the spindle at the cell center during mitosis[END_REF] to be able to apply larger forces of several hundreds of pN to moving asters in arbitrary directions. These modifications rested on the injection of magnetic beads with highly persistent MT minus-end targeting activity, which rapidly aggregated and bound to the aster center upon fertilization in a MT-and dynein-dependent manner [START_REF] Hamaguchi | Microinjected polystyrene beads move along astral rays in sand dollar eggs[END_REF] . Application of large calibrated forces was achieved by bringing a sharpened steel piece connected to a magnet to a controlled distance from the internalized beads (Fig. 2a and figs. S5 to S9) (see supplementary information).

Application of external longitudinal forces against aster-centering motion caused aster speed to decrease in a dose-dependent manner (Fig. 2, b-d, Fig. S10 and movie S2). In these experiments, we focused on a short timescale response by computing aster speed within typically ~ 30 sec after force application, to minimize long-term aster adaptive responses. In Fig. 2b-d, we first applied a 260 pN force, which dropped aster longitudinal speed Vx by almost a factor 2 without altering Vy. This force was subsequently increased to 570 pN which further decreased Vx to a negative value, thus reverting aster motion. After the force was released, the aster restored a centering velocity close to its original value, suggesting that external forces did not grossly perturb aster organization. Conversely, applying external forces along the centering direction caused asters to accelerate (Fig. 2, e-g, Fig. S11 and movie S3). In Fig. 2e-g, we applied a 670 pN force in the positive X direction which increased aster speed by nearly 2-fold.

Systematic repetition of these measurements allowed derivation of an aster force-velocity relationship for a wide range of external forces, from +1500 pN to -700 pN (a positive force corresponds to a rear pull). Consequent changes in longitudinal aster speed Vx varied from -0.13 to 0.2 m/sec and collapsed into a single linear curve (Fig. 2h and fig. S12). These results indicate that aster motion is governed by a simple linear friction law, so that

𝑉 𝑥 = 1 𝛾 (𝐹 𝑎𝑠𝑡𝑒𝑟 -𝐹 𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 ). (Eq. 2)
Importantly, this linear relationship holds for external forces applied along and against aster centering motion, suggesting that contributions from compressive MT forces at the aster rear, close to the cortex, may be negligible in these experiments. Using those results, we determined an aster stall force which is equal to the aster endogenous force of 𝐹 𝑎𝑠𝑡𝑒𝑟 = 580 +/-21 pN, and a frictional coefficient 𝛾 of 8400 +/-280 pN*sec/m (+/-indicates the standard error in fitting parameters unless specified). Detached bead aggregates with a similar size to the male pronucleus moved much faster than asters under the same forces, indicating that most of this friction may be associated with MTs in the aster (fig. S6). These results demonstrate that the centering motion of sperm asters obeys a simple linear friction law involving large self-propelling forces and drag.

Fluctuation analyses in the transverse axis supported the existence of a spring-like feedback mechanism stabilizing aster position around the centering axis. To characterize this feedback, we applied magnetic forces perpendicular to the motion direction (Fig. 3,a and b, fig. S13 and movie S4). In Fig. 3a-b, we applied a 470 pN force in the Y positive direction for 140 sec. The external force did not affect aster motion along the X-axis, but caused a continuous drift in Y which eventually saturated ~7 µm away from the X-axis. Remarkably, after force cessation the Y position restored to its original value within tens of seconds (Fig. 3b). Computing the maximum Y-displacement at saturation as a function of various applied forces yielded a linear forcedisplacement curve (Fig. 3c). These results directly demonstrate the existence of a linear spring stabilizing aster position around the centration axis, with a spring constant of = 59 +/-2.8 pN/m. The stiffness of this spring is ~4 times higher than in C. elegans [START_REF] Garzon-Coral | A force-generating machinery maintains the spindle at the cell center during mitosis[END_REF] , plausibly revealing different force-generation mechanisms. Accordingly, the transverse speed Vy following force application was comparable to the changes in Vx in the longitudinal force experiments, ruling out a major contribution of MT compressive forces to this transverse feedback. In addition, it has been shown that MT laser severing along the transverse axis in this system yields aster motion away from the site of ablation [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF] . These data support that this centering spring is mostly associated with MT pulling forces.

These transverse force experiments are consistent with a Kelvin-Voigt model, in which an elastic spring and viscous dashpot are connected in parallel [START_REF] Garzon-Coral | A force-generating machinery maintains the spindle at the cell center during mitosis[END_REF] (Fig. 4). This model predicts that the mean-squared displacement driven by internal random forces should saturate in an exponential manner, as observed in our fluctuation analysis, with a timescale equal to the relaxation timescale upon displacement by an external load. Accordingly, quantification of the recovery dynamics after force cessation revealed a restoring kinetic well described by an exponential relaxation with a single characteristic timescale,  = 72 +/-18 sec (+/-indicates standard deviation), independent of the initial Y-offset (Fig. 3d). This timescale was close to the saturation timescale observed in the transverse fluctuations ( = 46 sec), supporting the consistency between our passive and active characterizations of the transverse positional feedback. Given that fluctuation saturation depended on MTs and dynein, these results suggest that the positional feedback maintaining the aster around the centering axis relies on dynein pulling forces on MTs.

These findings may be consistent with the length-dependent mechanism proposed to drive aster centration in sea urchins and other embryos [START_REF] Hamaguchi | Analysis of the Role of Astral Rays in Pronuclear Migration in Sand Dollar Eggs by the Colcemid-UV Method[END_REF][START_REF] Kimura | Computer simulations and image processing reveal length-dependent pulling force as the primary mechanism for C. elegans male pronuclear migration[END_REF][START_REF] Longoria | Cargo transport by cytoplasmic dynein can center embryonic centrosomes[END_REF][START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF][START_REF] Wuhr | A model for cleavage plane determination in early amphibian and fish embryos[END_REF][START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF][START_REF] Pierre | Generic Theoretical Models to Predict Division Patterns of Cleaving Embryos[END_REF][START_REF] Kimura | Intracellular organelles mediate cytoplasmic pulling force for centrosome centration in the Caenorhabditis elegans early embryo[END_REF][START_REF] Shinar | A model of cytoplasmically driven microtubule-based motion in the single-celled Caenorhabditis elegans embryo[END_REF][START_REF] Barbosa | Dynactin binding to tyrosinated microtubules promotes centrosome centration in C. elegans by enhancing dynein-mediated organelle transport[END_REF] . This system has the properties of an 'active spring' with respect to cell geometry: a displacement away from the cell center yields a length imbalance on the two sides of the aster and creates a dynein-dependent restoration force proportional to the displacement [START_REF] Hamaguchi | Analysis of the Role of Astral Rays in Pronuclear Migration in Sand Dollar Eggs by the Colcemid-UV Method[END_REF] . During aster centration, this spring is expected to function only along the transverse axis, because MTs reach the cortex along this axis, while front MTs do not reach to the opposite side until the very end of centration (Fig. 4) [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF] . Using the simplest linear length-dependency for MT forces, 𝐹 𝑀𝑇 = 𝛼𝐿 𝑀𝑇 , we can relate the spring constant 𝜅 to the length-dependency factor, so that𝛼 = 𝜅 2 ⁄ . This analysis indicates that ~30 pN forces are generated per 1 m-depth region of the aster surface (corresponding to a volume of 5*10 3 m 3 for an aster radius of 20 m). This suggests a lower bound of 100-200 dynein motors involved in moving and positioning these asters, much higher than in previous indirect estimates [START_REF] Minc | Influence of cell geometry on division-plane positioning[END_REF][START_REF] Grill | The distribution of active force generators controls mitotic spindle position[END_REF] .

Aster mechanical properties along the two different axes appeared to be largely consistent. For asters to move to the cell center, front MTs should be longer than those at the back, because their growth is not restricted by the cortex. We recently estimated a difference in length between front and rear MTs of about 𝛿𝐿 ~ 10 µm [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF] . This would correspond to a net force of 𝐹 𝑎𝑠𝑡𝑒𝑟 = 𝛼𝛿𝐿 ~ 300 pN; smaller yet close to the direct longitudinal force measurement calculated above. Furthermore, the relaxation timescale in the transverse force experiments allows for defining a frictional coefficient given by γ = κ • τ = 4200 pN*sec/m. This value is comparable to but twice smaller than that obtained from direct measurements in the longitudinal axis. Because the time-window used to compute the frictional coefficient in the transverse axis (>100 sec) is larger than in the longitudinal axis (~30 sec), we envision that the differences in γ could reflect some shape changes of the aster, which could modify the effective drag over this longer timescale. Together, these findings indicate that the viscous dampening in the transverse position is essentially the same as the drag associated with aster longitudinal centering motion, and that length-dependent dynein-MT forces may account for both transverse feedback and net directed force.

The frictional coefficient of MT asters is remarkably high, 60 times larger than the reported value for static mitotic spindles in C. elegans [START_REF] Garzon-Coral | A force-generating machinery maintains the spindle at the cell center during mitosis[END_REF] . The linear relationship between aster force and velocity suggests a viscous interaction between the aster-dynein complex and the cytoplasm. Based on the measured cytoplasmic viscosity of fertilized sea urchin eggs [START_REF] Hiramoto | Mechanical properties of the protoplasm of the sea urchin egg. II. Fertilized egg[END_REF] , this friction would amount to that of an object with a hydrodynamic radius of ~440 m, typically ~20 times larger than the aster physical radius. Such large friction may not be readily explained by internal structures of the aster. Given the high density of astral MTs, their association with endomembranes such as the endoplasmic reticulum [START_REF] Wuhr | How does a millimeter-sized cell find its center?[END_REF] , an aster may be viewed as a sphere with low permeability [START_REF] Nazockdast | Cytoplasmic flows as signatures for the mechanics of mitotic positioning[END_REF][START_REF] Nazockdast | A fast platform for simulating semi-flexible fiber suspensions applied to cell mechanics[END_REF] . The frictional coefficient of asters is thus expected be close to that of a non-permeable counterpart, and much smaller than the sum of the drag of individual MTs. One possible source for such a large frictional coefficient is the confinement set by the cell boundary: the hydrodynamic interaction between the aster and cell boundaries could significantly reduce the hydrodynamic mobility of the aster. In support of this, recent quantitative hydrodynamic simulations suggest that even a moderate cell-confinement (aster/cell size ~ 0.5) can lead to a 10-30 times increase of aster frictional coefficient [START_REF] Nazockdast | A fast platform for simulating semi-flexible fiber suspensions applied to cell mechanics[END_REF] . Our experimental results may thus highlight the overlooked physical effect of cell-confinement on the mobility of intracellular structures.

Finally, we propose a simple model which explains how measured mechanical properties may account for aster positional fluctuations. Aster motion kinetics can be represented by one lengthscale defined by dividing the diffusion coefficient by the mean speed, 𝐷 𝑥 /𝑉 𝑥 = 20 nm. This small length-scale reflects the high persistency of aster motion. To illustrate how this characteristic length-scale may emerge from aster mechanics, we introduced a simple model for aster longitudinal motion (Fig. 4 and Supporting text). In this model, a single force-generation event created by a moving dynein on a MT causes a fixed aster displacement, 𝛿 𝑑 , towards or away from the cell center. We assume that force-generation obeys a first-order reaction, (i.e. it is limited by either the binding or the activation of dynein), with a reaction rate 𝐾 𝑓 (𝐾 𝑟 ) proportional to the front (rear) aster radius 𝐿 𝑓 (𝐿 𝑟 ). Using Poisson statistics [START_REF] Svoboda | Fluctuation analysis of motor protein movement and single enzyme kinetics[END_REF] , we can express 𝛿 𝑑 as a function of 𝐷 𝑥 /𝑉 𝑥 as (see supporting information):

𝛿 𝑑 = 2𝐷 𝑥 𝑉 𝑥 𝐾 𝑓 -𝐾 𝑟 𝐾 𝑓 +𝐾 𝑟 .
(Eq. 3).

Assuming force balance between the aster and a single dynein-cargo complex, and using typical cargo vesicle parameters (radius r ~ 0.5 m and run-length l ~ 5 m 11 ) and aster shape asymmetry 𝐿 𝑟 𝐿 𝑓 ⁄ ~ 0.8 [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF] , this model predicts an aster hydrodynamic radius as R = (𝑙 • 𝑟) 𝛿 𝑑 ⁄ ~620 m, which is comparable to our direct measurement of ~440 m. This result demonstrates how the large aster frictional coefficient may suppress the motion error caused by dynein active fluctuations.

In conclusion, based on the comprehensive characterization of aster kinetics and forces, we find that aster centration in large embryos is extremely precise, significantly more so than in wellcontrolled in vitro aster centration assays [START_REF] Holy | Assembly and positioning of microtubule asters in microfabricated chambers[END_REF][START_REF] Faivre-Moskalenko | Dynamics of microtubule asters in microfabricated chambers: the role of catastrophes[END_REF][START_REF] Laan | Cortical Dynein Controls Microtubule Dynamics to Generate Pulling Forces that Position Microtubule Asters[END_REF] , and that this precision is achieved by unusually large endogenous forces and friction. Based on the measured frictional coefficient, we estimate that the cell dedicates an energy of at least γ𝑉 2 ~ 1000 ATP molecules/sec for centering MT asters. This large frictional coefficient enables asters to take an ensemble average of stochastic dynein force generation events and ensures motion persistency. The transverse feedback further suppresses fluctuations and maintains aster trajectory along a precise axis related to cell geometry. As demonstrated by external force applications, this active feedback can bring back the aster along its centering trajectory even after accidental deviations larger than those caused by intrinsic dynein fluctuations. Most fertilizing embryos are associated with large rotational flows, shape changes and other cytoplasmic re-organization [START_REF] Gilbert | Developmental Biology[END_REF] . Our results suggest that sperm MT asters may be equipped with near-optimal physical designs to stabilize their motion in such an unfavorable environment to rapidly and precisely target the geometrical center of the large egg cell.

Fig. 1 .

 1 Fig. 1. Fluctuation analysis of centering MT asters. (a and b) High-resolution tracking of sperm MT asters during the constant speed centration phase. The representative 3 min trajectory corresponds to 3600 time points. Aster XY position was defined using X as the centering axis. (c) PDFs of residual displacements along X and Y for different t. Bold lines are best-fit Gaussian distributions. (d) X and Y MSrDs for control and Ciliobrevin D-treated samples plotted as a function of t in log scale. The broken line indicates a slope of 1. Inset: Transverse MSrD (blue curve) of controls superimposed with a fit of the model from Eq. 1 (broken line), plotted on a linear scale. The absolute magnitude of the residual error between the fit and the data is also depicted (green curve). (e) Contributions of cytoskeletal components to aster positional fluctuations. The fluctuation amplitude was characterized by computing the MSrD at t=5 sec.

Fig. 2 .

 2 Fig. 2. Force-velocity relationships of MT asters. (a) Aggregates of injected magnetic beads were targeted to the aster center to directly apply magnetic forces to centering asters. (b-g) External magnetic forces were applied to asters either against (b-d) or along (e-g) the centering direction. The 1D kymographs (c and f) and time-evolution of aster XY positions (d and g) show how applied forces consistently change aster longitudinal speed. (h) Aster longitudinal speed Vx plotted as a

Fig. 3 .

 3 Fig. 3. Direct demonstration of a transverse feedback stabilizing asters along their centering (a and b) External forces were applied orthogonally to the centering direction along the Y-axis for 140 sec. The applied force causes a drift towards the magnet tip. (c) Aster Y saturation (shown in b) plotted as a function of external force amplitude. N=10 measurements from 10 cells. The spring constant κ=59 pN/m was determined from the slope of the linear fit (red broken line). (d) Recovery dynamics of aster Y position. Aster Y position after force cessation was plotted as a function of time. N=10 measurements from 10 cells. Solid lines indicate best fits with exponential function (eq. S6), yielding a mean recovery timescale τ = 72 +/-18 sec. Inset: Recovery timescale plotted as a function of Y saturation. The correlation analysis indicates that there is no correlation between the two variables.

Fig. 4 .

 4 Fig. 4. Aster mechanics ensure persistent and precise aster centration. The large frictional coefficient of asters suppresses active fluctuation along the longitudinal axis. This process was analyzed using a simple Poisson model, in which a single dynein-force generation event causes an aster step motion (see main and supporting text). Along the transverse axis, fluctuations are further suppressed by a dynein-dependent feedback mechanism, which stabilizes the centering direction with respect to cell geometry.

Acknowledgements:

The authors acknowledge M. Coppey and J. Azimzadeh for technical support, and S. Dmitrieff, T. Strick, M. Piel, M. Thery and K. Laband for careful reading of the manuscript. This research was supported by the CNRS and grants from the "Mairie de Paris emergence" program, the FRM "amorçage" grant AJE20130426890 and the European Research Council (CoG Forcaster N° 647073).


C Champenois

M Jacquey

S Lepoutre

M Büchner

G Trénec

J Vigué

Index of refraction of gases for matter waves: effect of the motion of the gas particles on the calculation of the index

Two different formulae relating the index of refraction n of gases for atom waves to the scattering amplitude have been published. We show here that these two formulae are not consistent with the definition of the total scattering cross-section while the formula developed by one of us (C.C.) in her thesis is in agreement with this standard knowledge. We discuss this result, in particular in the neutron case for which such an index was first introduced. We finally evaluate the index of refraction as a function of well known quantities and we discuss the order of magnitude of the ratio of (n -1)/nt, where nt is the gas density.

I. INTRODUCTION

To describe the propagation of matter waves through a medium, it is possible to use an index of refraction, as commonly done for light. This idea was first introduced around 1940 for the propagation of neutrons through matter (see the review papers by L. L. Foldy [1] and by M. Lax [2]).

Atom interferometry has permitted to study the propagation of an atom wave through a dilute gas and the first measurements of the index of refraction of gases for sodium waves were made in 1993 at MIT by J. Schmiedmayer et al. [START_REF] Schmiedmayer | Lecture notes in Physics[END_REF][START_REF] Schmiedmayer | [END_REF] who measured the attenuation and the phase shift of the transmitted wave. Further works in the same laboratory have led to the observation of glory oscillations of the index as a function of the sodium atom velocity [START_REF] Schmiedmayer | Atom interferometry[END_REF][START_REF] Hammond | [END_REF]7]. More recently, our group has measured the index of refraction of several gases for lithium waves [8].

Whatever is the nature of the wave and of the medium, the index of refraction describes the modification of the propagation of an incident wave due to the waves scattered in the forward direction by the particles of the medium: the scattered waves interfere with the incident wave and modify its phase and its amplitude. The modification of the phase induces a modification of the wave velocity, described by the real part of the index of refraction, while the modification of the amplitude is described by its imaginary part.

In practice, the index of refraction is proportional to the complex forward scattering amplitude [START_REF] Landau | Quantum mechanics Pergamon Press[END_REF][START_REF] Joachain | Quantum collision theory[END_REF]. The imaginary part of this amplitude is related to the total cross section, which is traditionally measured by beam scattering experiments, whereas its real part can be measured only by atom interferometry. This amplitude exhibits resonances, for a collision energy comparable to the potential well depth, and glory oscillations, for larger energies. Glory oscillations are the quantum consequence of the existence of an undeflected classical trajectory due to the compensation of attractive and repulsive forces [START_REF] Pauly | Elastic Cross sections Section I: spherical potentials[END_REF].

For atom waves, the forward scattering amplitude can be calculated if the interaction potential between a particle of the wave and a particle of the target gas is known. Several papers [START_REF] Vigué | [END_REF]13,14,15,16,17,18,19,20] have discussed the theory of the index of refraction for atom waves and their equations have been used to compare calculated values of the index of refraction [13,14,15,16,17,21,22,23] with experimental results. A detailed comparison is possible only if the thermal motion of the target gas is not neglected and we show here that the formulae [15,16] introduced to take into account this motion disagree with collision physics. We propose a formula developed by one of us (C. C.) in her thesis [17]: this formula agrees with standard results of collision physics as well as with a recent calculation based on quantum Boltzmann equation due to K. Hornberger [24].

In the present paper, we first recall the previous results concerning the index of refraction. Then, we explain why these formulae are not in agreement with the Beer-Lambert law and we extract from this discussion a new formula giving the imaginary part of the index of refraction, which is generalized to the real part of the index of refraction. We then discuss the differences between the old and new formulae. We verify that our formula is in agreement with the results concerning the index of refraction for slow neutrons. Finally, we compare the order of magnitude of the index of refraction of gases for light and atomic waves. In the cases where the index of refrac-tion for atom waves has been measured, the values of the ratio (n -1)/n t , n t being the target gas density, are very close to the value of this ratio for the index of refraction for ordinary gases for visible light. This similarity is in fact a coincidence, without any physical meaning.

II. ORIGIN OF THE PROBLEM

When one compares light waves and atom waves interacting with an atomic (or molecular) target gas, there is a large difference, which is precisely related to the motion of the target particles.

In the case of light, the photon velocity, almost equal to c in dilute matter, is considerably larger than the target particle velocity. At the same time, the photon momentum is usually considerably smaller than the target particle momentum. The atom motion induces Doppler effect and the photon momentum induces atom recoil. Because the velocity of light is usually so much larger than the one of the target particles, these two effects have small consequences on the index of refraction, if we except the frequency range close to a sharp resonance line. This simple idea remains true, even if the target particles move rapidly in the laboratory, because one can always study the problem in their rest frame.

In the case of atomic waves, the velocity of an atom of the wave is usually comparable to the velocity of a target particle and, at the same time, the scattering properties depend rapidly on the relative velocity. Moreover, the momentum of an atom of the wave and the one of a target particle are usually of comparable magnitudes. In practice, it is absolutely necessary to take into account the target particle motion to make a realistic calculation of the index of refraction.

III. NOTATIONS

In the laboratory frame, a plane wave of wavevector k p describes the propagation in vacuum of a projectile p of mass m p and velocity v p :

hk p = m p v p (1)
The projectile can be any massive particle described by quantum mechanics, a neutron, an electron, an atom or a molecule. This wave propagates through a gas of density n t made of target particles t of mass m t and velocity v t , the target particles being also described by quantum mechanics. The wavevector k r describes the relative motion of the projectile p in the center of mass frame for a collision with a target particle t: k r = µv r /h, where v r = |v p -v t | is the relative velocity and µ = m p m t /(m p +m t ) is the reduced mass. The present notations will be used to write all previous formulae in order to facilitate comparison.

IV. INDEX FOR FIXED SCATTERING CENTERS

When a plane wave of wavevector k p enters in a medium made of target particles t, its propagation is modified, with a different wavevector k p,m given by:

k p,m = nk p ( 2
)
where n is the index of refraction. If the medium is described by a random distribution of fixed scattering centers, the index of refraction n is given by:

n = 1 + 2πn t f (k p ) k 2 p ( 3
)
where f (k p ) is the forward scattering amplitude of the wave scattered by one scattering center. f (k p ) has the dimension of a length and, as usual, the index n is a dimensionless quantity. A general derivation of this formula is found in the paper of L. Foldy [1] (see also the review of M. Lax [2] and references therein). Equation (3) is the first order term of an expansion in powers of the gas density n t and it is a good approximation if the following conditions are fulfilled:

• the mean distance n -1/3 t between nearest scattering centers must be considerably larger than the projectile de Broglie wavelength

λ p = 2π/k p , i.e. n -1/3 t λ p ≪ 1.
• the mean distance n -1/3 t between scattering centers must be considerably larger than the range of the interaction potential. Unfortunately, as realistic atom-atom interaction potentials V (r) vanish only for an infinite distance r, the range has not a clear definition for such potentials.

• the density n t of scattering centers is low enough so that the mean field correction is negligible i.e.

(n -1) ≪ 1. This last condition, which involves a condition on k p and f (k p ), depends on the interaction potential and collision energy.

Practically, for thermal atoms waves and a target gas near room temperature, with relative velocities of the order of 10 3 m/s and target densities up to n t ≈ 10 19 m -3 used in the experiments [START_REF] Schmiedmayer | Lecture notes in Physics[END_REF][START_REF] Schmiedmayer | [END_REF][START_REF] Schmiedmayer | Atom interferometry[END_REF][START_REF] Hammond | [END_REF]7,8], the mean interatomic distance n -1/3 t is larger than 5 × 10 -7 m, the index of refraction is of the order of |n -1| < ∼ 10 -10 and these three conditions are well fulfilled.

In the early days of atom interferometry, the target particles were treated as fixed scattering centers and the index was given by equation (3): this was the case of the first paper [START_REF] Schmiedmayer | Lecture notes in Physics[END_REF] dealing with the index of refraction for atom waves, written by J. Schmiedmayer et al. in 1993 and of the paper written by one of us (J.V.) in 1994 [START_REF] Vigué | [END_REF]. As soon as experimental values [START_REF] Schmiedmayer | [END_REF] of the index of refraction of gases for atomic waves became available in 1995, it appeared necessary to take into account the target gas thermal motion. The next section presents the equations used by different groups.

V. FORMULAE GIVING THE INDEX OF A GAS AT THERMAL EQUILIBRIUM

Three research groups have been involved in the calculation of the index of refraction of gas for atomic waves and have worked on the proper way to take into account the motion of the targets. For convenience and clarity, the formulae used by each group are presented separately.

A. Publications of D. Pritchard and co-workers

After a first paper where targets were considered at rest [START_REF] Schmiedmayer | Lecture notes in Physics[END_REF], this research group described in 1995 a set of measurements of the index of refraction of gases for sodium waves [START_REF] Schmiedmayer | [END_REF]. To interpret their experiment, these authors use the following equation:

n = 1 + 2πn t f (k r ) k p k r (4)
where the thermal average is not explicitly discussed. The same equation is also used in the review paper [START_REF] Schmiedmayer | Atom interferometry[END_REF] written by this group in 1997. A thermal average is taken into account in the paper by T. D. Hammond et al. [START_REF] Hammond | [END_REF] published in 1997, with an index of refraction given by:

n = 1 + 2π n t k p f (k r ) k r (5)
where the brackets mean the average over the velocity distribution of the target particles.

B. Publications of A. Dalgarno and co-workers

Equation (5) appears for the first time in the paper of R. C. Forrey et al. [15], written in 1995 and published in 1996. In their paper, R. C. Forrey et al. [15] calculate the distribution of the relative wavevector k r and, for sake of completeness, we reproduce this calculation in Appendix A. The group of A. Dalgarno has published a series of papers on the index of refraction for atom waves [20,23] or for electron waves [19] and this work was continued by R. C. Forrey and co-workers [22].

In 2002, R. C. Forrey et al. published a paper [18] entitled 'On the statistical averaging procedure for the refractive index of matter waves'. In this paper, they do not give a complete derivation of their formula but they state that a key step is the Lorentz invariance of the ratio

f (k r )/k r = f (k p )/k p .

C. Publications of our research group in Toulouse

In 1995, E. Audouard et al. published a calculation of the index of refraction of argon gas for sodium waves [13]. This work was the first one by our group taking into account the effect of thermal averaging. We have made an error in the algebra which was corrected in [14]. The derivation of the thermal average formula was given in a following paper by C. Champenois et al. [16]. Our calculation was based on the Fizeau effect: this effect is a frame dragging effect well known for light, which has also been studied in the case of neutron matter waves [25,26,27]. We had written:

k p,m = k p + (n CM -1)k r (6)
with the center of mass index of refraction n CM given by:

n CM = 1 + 2πn t f (k r ) k 2 r ( 7
)
As above, the brackets stand for the average over the velocity distribution of the target particles. We will not recall the final result [16] corresponding to a Boltzmann distribution of the target gas, because this result is not correct. After discussion with the group of A. Dalgarno in 1998, whose results are presented above, we were convinced that our formula was wrong, in particular because our description of the Fizeau effect is not correct. In her thesis [17], Caroline Champenois derived a new formula (see equations V.78 and V.81):

n = 1 + 2πn t m p + m t m t f (k r ) k 2 p ( 8
)
As the derivation of this formula was very involved, we will not reproduce it here but, in the next section, we use a simpler argument to convince the reader that this formula is the right one.

VI. DISAGREEMENT WITH CLASSIC RESULTS AND A NEW FORMULA

In this part, we show that neither equation ( 5) proposed by R. C. Forrey et al. nor our equations ( 6) and ( 7) are in agreement with well accepted atomic collision results. We first introduce the total scattering cross section and relate it to the imaginary part of the index of refraction.

A. The Beer-Lambert law

In nonrelativistic mechanics, the total scattering cross section σ(v r ) is related to the number of collisions dN coll occurring during a time dτ in a volume dV between a projectile p and a target t:

dN coll dτ dV = n p n t σ(v r )v r (9)
From this equation, we deduce the mean number of collisions dN coll,p /dτ encountered by a projectile p per unit time:

dN coll,p dτ = n t σ(v r )v r (10)
We now consider a beam of projectiles p crossing a slab of target gas, with the velocity v p perpendicular to the slab. A slab of thickness dL is crossed by a projectile p in a time dτ = dL/v p and the mean number of collisions for a projectile is given by:

dN coll,p = n t σ(v r ) v r v p dL (11)
From this equation, one can deduce the transmission T of the slab i.e. the fraction of the incoming flux which has crossed the slab without any collision. For a finite thickness L, the transmission T is obtained by a straightforward integration:

T = exp -n t σ(v r ) v r v p L (12)
This equation is the Beer-Lambert law, usually written with an effective cross section σ ef f (v p ) = σ(v r )v r /v p :

T = exp [-n t σ ef f (v p )L] (13)
If the target velocity is spread, with a normalized distribution P (v t ) (i.e. verifying P (v t )d 3 v t = 1), we must replace the effective cross section σ ef f (v p ) in equation ( 13) by its average σ ef f (v p ) given by:

σ ef f (v p ) = P (v t )σ(v r ) v r v p d 3 v t . (14)

B. Wave-like description of the attenuation of a beam by a slab

We are going to calculate the transmission T of the beam through the same slab of length L, using the wave point of view. Let ψ inc be the incident wave and ψ trans the transmitted wave given by: ψ trans = e i(n-1)kpL ψ inc = te iϕ ψ inc (15) with t = exp [-Im(n -1)kL] and ϕ = Re(n-1)kL. The transmission in intensity is T = t 2 , which depends solely on the imaginary part of (n -1):

T = exp [-2Im(n -1)k p L] (16)
C. Consequences for the index of refraction

ASs the attenuation of the beam calculated in the two formalisms must be the same, the imaginary part of the index of refraction is related to the effective cross section by the following equation:

Im(n -1) = n t σ ef f (v p ) 2k p (17)
The total cross section is related by the optical theorem to forward scattering amplitude [START_REF] Joachain | Quantum collision theory[END_REF]:

σ(v r ) = 4π Im(f (k r )) k r (18)
Using this relation and equations ( 14) and ( 17), we obtain a formula giving the imaginary part of the index of refraction in agrement with Beer-Lambert law:

Im(n -1) = 2πn t m p + m t m t Im(f (k r )) k 2 p ( 19
)
where the brackets mean the average over the target velocity distribution P (v t ). Once we have an expression for the imaginary part of (n -1), we get the real part by a simple generalization:

n = 1 + 2πn t m p + m t m t f (k r ) k 2 p ( 20
)
This equation agrees with the result derived by C. Champenois in her thesis [17]. It agrees also with a result recently obtained by K. Hornberger using the formalism of the quantum Boltzmann equation [24]. We want to point out that equation ( 20) can be applied with any type of velocity distributions, as could be produced by a gas flow or a molecular beam, and not only with a Maxwell-Boltzmann distribution corresponding to thermal equilibrium.

D. Comparison of the new and previous formulae

The new formula, equation (20), differs from equation (5) established by R. C. Forrey et al. [15,18]:

-the denominator is k r k p in equation ( 5) and k 2 p in equation (20); -the mass ratio (m p + m t )/m t present in equation ( 20) is absent from equation ( 5).

-from equation ( 5), we deduce an effective cross section given by:

σ ef f (v p ) = P (v t )σ(v r )dv t (21)
This expression of σ ef f (v p ) differs from equation ( 14) and in the v p ≪ α limit, where α is the thermal velocity defined in Appendix A, the effective cross section given by equation ( 21) is independent of the projectile velocity v p , while the correct behavior given by equation ( 14) is σ ef f (v p ) ∝ 1/v p . This is a well known result, recognized in everyday's life: we run under the rain to get less wet! However, in the opposite limit when the target gas temperature vanishes, v t = 0 so that v r = v p and k r = µv p , it is easy to verify that equation (20) and equation ( 5) are then equivalent. This equivalence suggests that, even when the temperature does not vanish, the index calculated by these two formulae will not differ strongly as long as α ≪ v p .

VII. DISCUSSION OF THE NEUTRON CASE

It is also interesting to apply equation (20) to the well known case of neutron waves. Neutrons are scattered only by nuclei, if we except the case of ferromagnetic materials in which the magnetic interactions of the neutron spin cannot be neglected. In the low energy domain where the formalism of the index of refraction is useful, the neutron-nucleus scattering process is almost always dominated by s-wave scattering [28,[START_REF] Rauch | Neutron interferometry[END_REF].

The index of refraction is frequently calculated as the consequence of an effective potential which is related to the scattering length of the neutron-nucleus interaction potential. However, there are few papers devoted to the theoretical relation between scattering theory and the index of refraction for neutrons, in which the motion of the nuclei is taken into account. This is the case of the papers by B. A. Lippmann and J. Schwinger [START_REF] Lippmann | [END_REF] and also of a brief note by D. Kleinman and G. Snow [31], who state: "This derivation of the index clearly shows that there is no Doppler effect due to the motion of the nuclei, because the λ in the formula is the neutron wavelength relative to the boundary of the slab".

Obviously, the neutron case differs from the case of atom waves only by the fact that s-wave (l = 0) scattering dominates the forward scattering amplitude, which, in this case, is given by:

f (k r ) = exp (iδ 0 ) sin δ 0 k r ≈ -a(1 -ik r a) (22)
where δ 0 is the s-wave phase shift, a is the scattering length defined by a = -lim (tan δ 0 /k r ) when k r → 0. If we keep only the leading term in equation ( 22), f (k r ) ≈ -a, our formula (20), is equivalent to the result of D. Kleinman and G. Snow [31]:

n = 1 -2πn t m p + m t m t a k 2 p (23)
As k r is absent from the result, there is no Doppler effect on the index of refraction. However, if we take into account the first order term in k r in equation ( 22), the forward scattering amplitude has a non-vanishing imaginary part, Im (f (k r )) ≈ k r a 2 . As this imaginary part is linear in k r , the imaginary part of the index of refraction is sensitive to the motion of the target particles but this imaginary part, which is very small, is usually ignored. The k r -dependence of the imaginary part of the index of refraction has no practical consequences but this remark proves that the absence of Doppler effect on the index of refraction of matter for neutrons is a very special property valid only for the real part of the index of refraction in the s-wave limit.

VIII. COMPARISON OF THE ORDER OF MAGNITUDE OF THE INDEX OF REFRACTION OF GASES FOR LIGHT AND MATTER WAVES

Up to now, we have not discussed the numerical value of the index of refraction of gases for matter waves. A somewhat surprising feature is that the index of refraction of gases for matter waves [START_REF] Schmiedmayer | [END_REF]8] and the index of refraction of transparent gases for light has similar values, when the gas density n t is the same. In this section, we calculate the value of the (n -1)/n t ratio from first principles in both cases and we compare the results, in order to understand the origin of this similarity.

J. Schmiedmayer et al. [START_REF] Schmiedmayer | [END_REF] give the values of the real and imaginary parts of (n-1) of several gases at T = 300 K for sodium matter waves with a velocity v p = 1000 m/s. All the values of Re(n -1) or of Im(n -1) are in the range (0.14 -2.49) × 10 -10 for a gas pressure equal to 1 mTorr at T = 300 K. This pressure corresponds to a gas density n t ≈ 3.2 × 10 19 m -3 from which we find that the ratio Re(n -1)/n t of Im(n -1)/n t is in the range (0.4 -8) × 10 -30 m 3 . Our results for lithium waves [8] are somewhat larger, in the range 1 -2 × 10 -29 m 3 .

It is well known that the index of refraction of air for visible light is almost purely real with its value given by (n -1) ≈ 2.8 × 10 -4 at ordinary pressure and a temperature of 288 K, with a density n t ≈ 2.55×10 25 m -3 . From these values, we calculate a ratio (n-1)/n t ≈ 1.1×10 -29 m 3 for air. Similar values will be obtained for other transparent gases.

The ratio (n -1)/n t has comparable values for light or matter waves, although the only common feature is that they involve an index of refraction. We now evaluate these two ratio from first principles.

In optics, the index of refraction of an atomic or molecular gas is dominated by the electric dipole transitions, with its value given by:

n(ω) = [1 + 4πn t α(ω)] 1/2 ≈ 1 + 2πn t α(ω) (24)
where n t is the gas density, α(ω) the atomic or molecular electric polarizability for an angular frequency ω. In the visible region of the spectrum, where ω is usually smaller than the angular frequency of the main resonance transitions, α(ω) is close to its static value α(0). We now use atomic units, with α(0) = a 3 0 α au (0), where a 0 is the Bohr radius (a 0 ≈ 0.529 × 10 -10 m) and we get:

n(ω) -1 n t ≈ 2πa 3 0 α ua (0) (25)
Tabulated values of α au for atoms [32] vary from 1.4 a.u.

for Helium up to 450 a.u. for Francium and the polarizability of small molecules has similar values (for instance, nitrogen dimer N 2 has a polarizability α ua (0) ≈ 11.9 [33]). For matter waves, the index of refraction is given by equation (20). The imaginary part Im(n-1) is related to the total cross-section σ(v r ) and, for a purely attractive -C 6 /r -6 potential, there is a closed form expression of σ(v r ) [START_REF] Landau | Quantum mechanics Pergamon Press[END_REF]:

σ(v r ) = 8.08 C 6 hv r 2/5 (26)
The 8.08 factor is the numerical value of a complicated expression involving gamma function. This result is valid in an intermediate range of energy, with many partial waves contributing to the scattering amplitude [START_REF] Pauly | Elastic Cross sections Section I: spherical potentials[END_REF]. This result does not explain the glory oscillations, which exists only when the potential is attractive at long range and repulsive at short range, but this result gives the correct value of the cross-section averaged over the glory oscillations. As we want only an order of magnitude, we will make some simplifications, by neglecting the thermal motion of the target particle and by assuming that m p ≪ m t , so that we can replace v r by v p . We thus get an expression of the imaginary part of (n -1)/n t :

Im(n -1) n t = 4.04 k p C 6 hv p 2/5 (27)
For a purely attractive -C 6 /r -6 potential, the real and imaginary parts have similar values: Schmiedmayer et al. [START_REF] Schmiedmayer | [END_REF] calculated the ratio ρ = Re(n -1)/Im(n -1) and found that it is constant and equal to ρ = 0.7265 (see also [16]). We can now express Im(n -1)/n t , using atomic units for C 6 = C 6au m e c 2 α 2 f s a 6 0 (m e is the electron mass and α f s ≈ 1/137.037 is the fine-structure constant). We get:

Im(n -1) n t = 4.12 × 10 -3 a 3 0 m e m p c v p 7/5 (C 6ua ) 2/5
(28) Tabulated values of the atom-atom C 6au coefficient span the range from 1.47 for Helium up to ca. 7260 a.u. for Cesium [33,34] (we quote these references which provide a large set of C 6 values). The comparison of equation ( 25) and (28) proves that, the index of refraction of gases for light and matter waves have very different expressions and, if they have comparable values as in the cases discussed above, this must be considered as a pure coincidence. The index for matter waves has a rapid velocity dependence in v -7/5 p in a large velocity range and it would be considerably larger for lower projectile velocities, provided that the target gas velocity can still be neglected.

IX. CONCLUSION

In this paper, we have shown that the equations proposed by R. C. Forrey et al. [15,18] and by our group [16] to take into account the motion of the gas particles in the calculation of the index of refraction for matter waves are not consistent with the traditional definition of a cross-section. Following the result obtained by C. Champenois in her thesis [17], we propose a new formula for the index of refraction:

n = 1 + 2πn t m p + m t m t f (k r ) k 2 p ( 29
)
This formula is consistent with classic results of collision theory and it is also in agreement with a result recently obtained by K. Hornberger using the formalism of the quantum Boltzmann equation [24]. We have used this formula when comparing our measurement of the index of refraction of gases for lithium waves with theoretical values [8]. Equation ( 29) agrees also with the formula giving the index of refraction of matter for neutrons. Finally, in the limiting case where the thermal velocity α of the gas is considerably smaller than the projectile velocity v p , equation [START_REF] Rauch | Neutron interferometry[END_REF] and equation ( 5) proposed by R. C. Forrey et al. [15,18] are equivalent but they differ considerably in the opposite limit v p ≪ α.

X. ACKNOWLEDGEMENTS XI. APPENDIX A

We calculate the distribution P (v r ) of the relative velocity v r , starting from a normalized Maxwell-Boltzmann distribution of the target velocity v t :

P MB (v t )d 3 v t = 1 π 3/2 α 3 exp -v 2 t /α 2 d 3 v t (30)
with α = 2k B T /m t , k B being the Boltzmann constant and T the temperature. Using v t = v pv r , we can write: (31) We take the z axis along to the projectile velocity v p , we use spherical coordinates for v r , v r being its modulus, θ its angle with the z-axis θ and and ϕ its azimuth. Integration over ϕ and θ is easy and we get:

P (v r )d 3 v r = 1 π 3/2 α 3 exp -(v p -v r ) 2 /α 2 d 3 v r
P (v r )dv r = 2v r π 1/2 αv p exp - v 2 p + v 2

  v r )dv r = 1.

We thank K. Hornberger for the communication of his recent results prior to publication and for a very stimulating discussion. We also thank A. Dalgarno for inviting C. Champenois for one month in 1998 and for the interaction between our research groups at that time.