H T Banks

M Doumic

C Kruse

S Prigent

H Rezaei

Keywords: Inverse problems, polyglutamine and aggregation modeling, nucleation, information content, sensitivity, Fisher matrix, uncertainty quantification, Mathematics Subject Classification: 65M32, 62P10, 64B10, 49Q12

 for nucleated polymerization in proteins, about which very little is known regarding the underlying mechanisms; thus the methodology we develop here may be of great help to experimentalists.

Introduction

As mathematical models become more complex with multiple states and many parameters to be estimated using experimental data, there is a need for critical analysis in model validation related to the reliability of parameter estimates obtained in model fitting. A recent concrete example involves previous HIV models [START_REF] Adams | Model fitting and prediction with HIV treatment interruption data[END_REF][START_REF] Banks | Modeling HIV immune response and validation with clinical data[END_REF] with 15 or more parameters to be estimated. In [START_REF] Banks | Uncertainty quantification in modeling HIV viral mechanics[END_REF], using recently developed parameter selectivity tools [START_REF] Banks | Parameter selection methods in inverse problem formulation[END_REF] based on parameter sensitivity based scores, it was shown that many parameters could not be estimated with any degree of reliability. Moreover, we found that quantifiable uncertainty varies among patients depending upon the number of treatment interruptions (perturbations of therapy). This leads to a fundamental question: how much information with respect to model validation can be expected in a given data set or collection of data sets?

Here we illustrate the use of other tools (asymptotic theories of standard error quantification using appropriate statistical models, bootstrapping, and model comparison techniques) in addition to sensitivity theory that may be used to determine the information content in data sets. We do this in the context of recent models [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF] for nucleated polymerization in proteins.

After presenting the biological context of amyloid formation, we describe the model in Section 2. In Section 3, we investigate the statistical model to be used with our noisy data. This is a necessary step in order to use the correct error model in our generalized least squares (GLS) minimization. This also reveals information on our experimental observation process. Once we have found parameters which allow a reasonable fit, we determine the confidence we may have in our estimation procedures. We do this in Section 4, using both the condition number of the covariance matrix and a sensitivity analysis. This reveals a smaller number of parameters (than those estimated in [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF]) which appear as reasonably sensitive to the data sets, whereas other do not really affect the quality of the fits to our data. To further support our sensitivity findings, we then apply a bootstrapping analysis in Section 5. We are lead to four main parameters and compare their resulting errors with the asymptotic confidence intervals of Section 4. Finally, in Section 6, we carry out model comparison tests [START_REF] Banks | Statistical methods for model comparison in parameter estimation problems for distributed systems[END_REF][START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF][START_REF] Banks | Mathematical and Experimental Modeling of Physical and Biological Processes[END_REF] as used in [START_REF] Banks | Model comparison tests to determine data information content[END_REF], and these lead us to select three well-defined parameters that can be reliability estimated out of the nine original ones estimated in [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF].

Protein Polymerization

It is now known that several neuro-degenerative disorders, including Alzheimers disease, Huntingtons disease and Prion diseases e.g., mad cow, are related to aggregations of proteins presenting an abnormal folding. These protein aggregates are called amyloids and have become a focus of modeling efforts in recent years [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF][START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF][START_REF] Xue | Systematic analysis of nucleation-dependent polymerization reveals new insights into the mechanism of amyloid self-assembly[END_REF][START_REF] Xue | Amyloid fibril length distribution quantified by atomic force microscopy single-particle image analysis[END_REF][START_REF] Xue | An imaging and systems modeling approach to fibril breakage enables prediction of amyloid behavior[END_REF]. One of the main challenges in this field is to understand the key aggregation mechanisms, both qualitatively and quantitatively. In order to test our methodology on a relatively simple case, we focus here on polyglutamine (PolyQ) containing proteins. This was also the case study chosen to illustrate the fairly general ODE-PDE model proposed in [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF]; the reason for our choice is that, as shown in [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF], the polymerization mechanisms prove to be simpler for PolyQ aggregation than for other types of proteins, e.g. PrP [START_REF] Eghiaian | Diversity in prion protein oligomerization pathways results from domain expansion as revealed by hydrogen/deuterium exchange and disulfide linkage[END_REF]. To understand data sets from experiments carried by Human Rezaei and his team at INRA, (Virologie et Immunologie Moleculaires), see [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF], we adapt the general model to this context. The data sets (DS1-DS4) of interest to us here are depicted in Figure 1 below. The data sets of interest from [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF][START_REF] Banks | Efficient numerical schemes for Nucleation-Aggregation models: Early steps[END_REF].

In [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF] and a subsequent effort in [START_REF] Banks | Efficient numerical schemes for Nucleation-Aggregation models: Early steps[END_REF], the authors sought to investigate several questions including (i) understanding the key polymerization mechanisms, (ii) how to select parameters and calibrate the model, and (iii) how to numerically approximate the model. Here we briefly summarize results related to (iii) and focus primarily on (ii).

The Model

Original ODE Model

This model we used is the same as that of [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF]. We briefly outline that model. Let (V, V * , c i ) be the concentrations of the normal monomeric proteins that we will call monomers, of the monomeric proteins presenting an abnormal configuration that we will call conformers, and of the i-polymers made of i aggregated abnormal proteins, respectively. The following comprise the fundamental dynamics modeled in [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF]:

• Monomer-conformer exchange: V k + I ⇋ k - I V * • Nucleation: V * + V * + ... + V * i 0 k N on ⇋ k N of f c i 0 • Polymerization by conformer addition: c i + V * k i on ⇋ c i+1
Other reactions like fragmentation and coalescence are negligible for the case of polyglutamine containing proteins (see [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF] for experimental justification).

The law of mass action in the deterministic framework (see [START_REF] Banks | Mathematical and Experimental Modeling of Physical and Biological Processes[END_REF][START_REF] Rubinow | Introduction to Mathematical Biology[END_REF] and the numerous references therein), translates A+B

k + I ⇋ k - I A ′ +B ′ into the ordinary differential equation d[A] dt = -k + [A][B] + k -[A] ′ [B] ′ .
Using these basic ideas we obtain the infinite system of ordinary differential equations (ODEs) studied in [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF]

dV dt = -k + I V + k - I V * , (1)
dV * dt = k + I V -k - I V * + i 0 k N of f c i 0 -V * i≥i 0 k i on c i , (2)
dc i 0 dt = k N on (V * ) i 0 -k N of f c i 0 -k i 0 on c i 0 V * , (3)
dc i dt = V * (k i-1 on c i-1 -k i on c i ), i = i 0 + 1, .... (4)
with initial conditions

V (0) = c 0 , V * (0) = 0, c i 0 (0) = c i (0) = 0
and the mass balance equation

d dt V + V * + ∞ i=i 0 ic i = 0.
The experiments of interest to us measure the total polymerized mass, i.e.,

M(t) = i≥i 0 ic i (t).

An Approximate PDE System and the Associated Forward Problem

Since very long polymers (a fibril may contain up to 10 6 monomer units) characterize amyloid formations, a PDE version of the standard model, where a continuous variable x approximates the discrete sizes i, is a reasonable approximation for large amyloid polymers. However, for small polymer sizes this curarization does not work very well. Thus we take a "hybrid approach" of leaving the ODE for smaller sizes and use the PDE for larger ones, see [START_REF] Banks | Efficient numerical schemes for Nucleation-Aggregation models: Early steps[END_REF].

We define a small parameter ε = 1 i M , and let x i = iε with i M ≫ 1 be the average polymer size defined by

i M = i≥i 0 ic i c i .
Then after definition of dimensionless quantities

c ε (t, x) = c i 1 [x i ,x i+1 ]
we may obtain a partial differential equation (PDE) to replace the infinite ODE system. Rigorous derivations of such continuous integro-PDE models may be found in [START_REF] Laurençot | From the discrete to the continuous coagulationfragmentation equations[END_REF] for coagulation-fragmentation equations, in [START_REF] Collet | The Becker-Döring system and its Lifshitz-Slyozov limit[END_REF] for the limit of the Becker-Döring system toward Lifshitz-Slyozov model, and in [START_REF] Doumic | Scaling limit of a discrete prion dynamics model[END_REF] for the growth-fragmentation "Prion Model". A formal derivation for a full model, also including nucleation, is carried out in [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF].

Let N 0 ∈ N. We then use the approximation

dV dt = -k + I V + k - I V * , dV * dt = k + I V -k - I V * + i 0 k N of f c i 0 -V * i≥i 0 k i on c i , (5)
dc i 0 dt = k N on (V * ) i 0 -k N of f c i 0 -k i 0 on c i 0 V * , (6)
dc i dt = V * (k i-1 on c i-1 -k i on c i ), i ≤ N 0 , (7)
∂ t c ε (x, t)= -V * ∂ x (k on c ε (x, t)), x ≥ N 0 , (8)
with initial conditions

V (0) = c 0 , V * (0) = 0, c i 0 (0) = c i (0) = 0, c ε (x, 0) = 0,
and the boundary condition

c ǫ (x = N 0 , t) = c N 0 (t).
Then an assumed mass balance equation becomes

d dt V + V * + N 0 i=i 0 ic i + ∞ N 0 xc ε (x) dx = 0.
In [START_REF] Banks | Efficient numerical schemes for Nucleation-Aggregation models: Early steps[END_REF] we considered requirements for a good discretization scheme including (i) it should conserve the total polymerized mass, (ii) it should be fast and most importantly, (iii) it should be accurate.

To ensure the mass conservation, we replace the ODE for V * by the mass conservation equation and obtain

dV dt = -k + I V + k - I V * , V * = c 0 -V - N 0 i=i 0 ic i - ∞ N 0 xc ε dx, dc i 0 dt = k N on (V * ) i 0 -k N of f c i 0 -k i 0 on c i 0 V * , dc i dt = V * (k i-1 on c i-1 -k i on c i ), i ≤ N 0 , ∂ t c ε (x, t) = -V * ∂ x (k on c ε (x, t)), x ≥ N 0 ,
with initial and boundary conditions as before.

We developed methodology for forward solutions in [START_REF] Banks | Efficient numerical schemes for Nucleation-Aggregation models: Early steps[END_REF]. In considering these forward solutions we first observed that the desired spatial computational domain is very large as determined by the maximum size of observed polymers, with range up to 10 6 and the peak in the distribution is at the left side of the domain of interest; for larger polymer sizes, the distribution is almost linearly decreasing.

Based on these and other considerations discussed in [START_REF] Banks | Efficient numerical schemes for Nucleation-Aggregation models: Early steps[END_REF], the PDE was approximated by the Finite Volume Method (see [START_REF] Leveque | Finite-Volume Methods for Hyperbolic Problems[END_REF] for discussions of Upwind, Lax-Wendroff and flux limiter methods) with an adaptive mesh, refined toward the smaller polymer sizes. Furthermore, we kept the ratio between the step size and the corresponding mesh element constant, i.e., we used ∆x i

x i = q < 1 so that x i = 1 1-q x i-1 . This mesh is quasi-linear in the sense of ∆x i-1 ∆x i = 1 + O(q). The resulting Upwind and Lax-Wendroff schemes are then consistent on the progressive mesh (see [START_REF] Leveque | Finite-Volume Methods for Hyperbolic Problems[END_REF]). For further details on these schemes including examples demonstrating convergence properties, the interested reader may consult [START_REF] Banks | Efficient numerical schemes for Nucleation-Aggregation models: Early steps[END_REF].

The Inverse Problem

A major question in formulating the model for use in inverse problem scenarios consists of how to best parametrically represent the function k on for our application? Following [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF], we chose to approximate k on by a function as depicted in Figure 2. (According to our discussions between S. Prigent, H. Rezaei and J. Torrent, other choices like a Gaussian bell curve are also possible, and we discuss this later). Thus with this parametrization we have 5 more parameters k min on , k max on , x 1 , x 2 , i max in addition to the 4 basic parameters k + I , k - I , k N on , k N of f to be estimated using our data sets. Thus we seek to estimate (with acceptable quantification of uncertainties) the nine parameters k + I , k - I , k N of f , k N on , and k on (represented in parametrical form depicted above with the 5 additional unknowns k min on , k max on , x 1 , x 2 , i max ) that fit the data best! To do this we need an efficient discretization method as discussed above for the forward problem as well as a correct assumption on the measurement errors in the inverse problem.

Estimation of Parameters

We make some standard statistical assumptions (see [START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF][START_REF] Banks | Mathematical and Experimental Modeling of Physical and Biological Processes[END_REF][START_REF] Davidian | Nonlinear Models for Repeated Measurement Data[END_REF][START_REF] Seber | Nonlinear Regression[END_REF]) underlying our inverse problem formulations.

• Assume that there exists a true or nominal set of parameter θ 0 = (k - I , ..., i max )

• Let E i be iid with E(E i ) = 0 and cov(E i , E j ) = σ 2 . Let ǫ i ∈ E i .
Denote the estimated parameter for θ 0 as θ. The inverse problem is based on statistical assumptions on the observation error in the data.

If we assume an absolute error data model then data points are taken with equal importance. This is represented by observations

y i = M(t i , θ 0 ) + ǫ i . (9)
On the other hand, if one assumes some type of relative error data model then the error is proportional in some sense to the measured polymerized mass. This can be represented by observations of the form

y i = M(t i , θ 0 ) + M(t i , θ 0 ) γ ǫ i , γ ∈ (0, 1]. ( 10
)
Absolute model error formulations dictate we use Ordinary Least Squares (OLS) inverse problem [START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF][START_REF] Banks | Mathematical and Experimental Modeling of Physical and Biological Processes[END_REF] given by θ = arg min

(y i -M(t i , θ)) 2 (11)
while for relative error model one should use inverse problem formulations with Generalized Least Squares (GLS) cost functional θ = arg min

y i -M(t i , θ) M(t i , θ) γ 2 , γ ∈ (0, 1]. (12)

The Residual Plots

To obtain a correct statistical model, we used residual plots (see [START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF][START_REF] Banks | Mathematical and Experimental Modeling of Physical and Biological Processes[END_REF] for more details) with residuals given by

r i = y i -M(t i , θ) M(t i , θ) γ , γ ∈ [0, 1]
To illustrate what we are seeking for our data sets, we first used simulated relative error data (simulated data for γ = 1), then carried out the inverse problems for both a relative error cost functional (i.e., γ = 1) and an ordinary least squares cost functional (i.e., γ = 0). We then plotted the corresponding residuals vs time and also residuals vs the model values. The first plots are related to the correctness of our assumption of independency and identical distributions i.i.d. for the data whereas the second plots contain information as to the correctness of the form of our proposed statistical model.

Statistical Models of Noise

We next carried out similar inverse problems with data set (DS) 4 of our experimental data collection. We first used DS 4 on the interval t ∈ [0, 8].

Based on some earlier calculations we also chose the nucleation index i 0 = 2 for all our subsequent calculations. The residual plots given below in Figures 5 and6 suggest strongly that neither of the first attempts of assumed statistical models and corresponding cost functionals (absolute error and OLS or relative error with γ = 1 and simple GLS) are correct. Based on these initial results and the speculation that early periods of the polymerization process may be somewhat stochastic in nature, we chose to subsequently use all the data sets on the intervals [t 0 , 8] where t 0 is the first time when M(t 0 ) > 0.12 (thus 12% of the total polymerized mass). Moreover, we decided to use other values of γ between 0 and 1 to test data set 4.

We thus carried out further investigations with inverse problems for data points M(t k ) ≥ 0.12 and i 0 = 2 where we focused on the question of the most appropriate values of γ to use in a generalized least squares approach (again see [START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF] for further motivation and details). We then obtained the results with data set 4 depicted in Figure 7.

Analysis of these residuals suggest that either γ = 0.6 or γ = 0.7 might be satisfactory for use in a generalized least squares setting. Motivated by these results, we next investigated the inverse problems for each of the four experimental data sets with initial concentration c 0 = 200µmol and i 0 = 2. We carried out the optimization over all data points with M(t k ) ≥ 0.12 and used the generalized least squares method with γ = 0.6. The resulting graphics depicted in Figure 8 again suggest that γ = 0.6 is a reasonable value to use in our subsequent analysis of the polyglutamine data with regard to its information content for inverse problem estimation and parameter uncertainty quantification. We employed first the asymptotic theory for parameter uncertainty summarized in [START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF][START_REF] Banks | Mathematical and Experimental Modeling of Physical and Biological Processes[END_REF][START_REF] Davidian | Nonlinear Models for Repeated Measurement Data[END_REF] and the references therein. In the case of generalized least squares, the associated standard errors for the estimated parameters θ = (k + I , ..., i max ) (vector length κ θ = 9) are given by the following construction (for details see Chap. 3.2.5 and 3.2.6 of [START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF]):

Define the covariance matrix by the formula

SE k = Σ kk ( θ), k = 1, ..., 9,
where

Σ( θ) = σ2 (χ T ( θ)W ( θ)χ( θ)) -1 .
Here χ is the sensitivity matrix of size n × κ θ (n being the number of data points and κ θ being the number of estimated parameters) and W is defined by

W -1 ( θ) = diag(M(t 1 ; θ) 2γ , . . . , M(t n ; θ) 2γ ).
We use the approximation of the variance

σ 2 ≈ σ( θ) 2 = 1 n -κ θ n i=1 1 M(t i ; θ) 2γ (M(t i , θ) -y i ) 2 .
To obtain a finite standard error using asymptotic theory, the 9×9 matrix F = χ T ( θ)W ( θ)χ( θ) thus must be invertible. In the above problem we do indeed obtain a good fit of the curve and good residuals (for the sake of brevity, not depicted here!). However, we also found that the condition number of the matrix

F = χ T ( θ)W ( θ)χ( θ) is κ = 10 24 .
Looking more closely at the matrix F reveals a near linear dependence between certain rows, hence the large condition number. We thus quickly reach the following conclusions:

1. We obtain a set of parameters for which the model fits well, but we cannot have any reasonable confidence in them using the asymptotic theories from statistics e.g., see the references given above.

2. We suspect that it may not be possible to obtain sufficient information from our data set curves to estimate all 9 parameters with a high degree of confidence! This is based on our calculations with the corresponding Fisher matrices as well our prior knowledge in that the graphs depicted in Figure 1 are very similar to Logistic or Gompertz curves which can be quite well fit with parameterized models with only 2 or 3 carefully chosen parameters!

To assist in initial understanding of these issues, we consider the associated sensitivity matrices χ = ∂M ∂θ .

Sensitivity Analysis

For the sensitivity analysis, we follow [START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF][START_REF] Banks | Mathematical and Experimental Modeling of Physical and Biological Processes[END_REF]. Hereafter all our analysis will be carried using data set 4 and the best estimate θ obtained for the latter. We find that the model is sensitive mainly to four parameters:

k + I , k - I , k N on , k N of f
. The sensitivities for the remaining parameters are on an order of magnitude of 10 -6 or less. It also shows some sensitivity with respect to x 1 . However, the parameter x 1 appears in the model only as factor x 1 i max . The sensitivities depicted below use θ for the nine best fit GLS parameters , i.e., θ for κ θ = 9.

Sensitivity Motivated Inverse Problems

Based on the sensitivity findings depicted above, we investigated a series of inverse problems in which we attempted to estimate an increasing number of parameters beginning first with the fundamental parameters k + I and k - I . In each of these inverse problems we attempted to ascertain uncertainty bounds for the estimated parameters using both the asymptotic theory described above and a generalized least squares version of bootstrapping [START_REF] Carroll | Transformation and Weighting in Regression[END_REF][START_REF] Carroll | The effect of estimating weights in Weighted Least Squares[END_REF][START_REF] Davidian | Nonlinear Models for Univariate and Multivariate Response[END_REF][START_REF] Diciccio | Bootstrap confidence intervals[END_REF][START_REF] Efron | The Jackknife, the Bootstrap and Other Resampling Plans[END_REF].

A quick outline of the appropriate bootstrapping algorithm is given next.

Bootstrapping Algorithm: Nonconstant Variance Data

We suppose now that we are given experimental data (t 1 , y 1 ), . . . , (t n , y n ) from the underlying observation process

Y i = M(t i ; θ 0 ) + M(t i ; θ 0 ) γ E i , (13)
where i = 1, . . . , n and the E i are i.i.d. with mean zero and constant variance σ 2 0 . Then we see that E(Y i ) = M(t i ; θ 0 ) and V ar(Y i ) = σ 2 0 M 2γ (t i , θ 0 ), with associated corresponding realizations of Y i given by

y i = M(t i ; θ 0 ) + M(t i ; θ 0 ) γ ǫ i .
A standard algorithm can be used to compute the corresponding bootstrapping estimate θboot of θ 0 and its empirical distribution. We treat the general case for nonlinear dependence of the model output on the parameters θ. The algorithm is given as follows.

1. First obtain the estimate θ0 from the entire sample {y i } using the GLS given in [START_REF] Carroll | Transformation and Weighting in Regression[END_REF] with γ = 1. An estimate θboot can be solved for iteratively as follows.

2. Define the nonconstant variance standardized residuals si = y i -M(t ; θ0 )

M(t i ; θ0 ) γ , i = 1, 2, . . . , n.
Set m = 0.

3. Create a bootstrapping sample of size n using random sampling with replacement from the data (realizations) {s 1 ,. . . ,s n } to form a bootstrapping sample {s m 1 , . . . , s m n }.

Create bootstrapping sample points

y m i = M(t i ; θ0 ) + M(t i ; θ0 ) γ s m i ,
where i = 1,. . . ,n.

5.

Obtain a new estimate θm+1 from the bootstrapping sample {y m i } using GLS.

6. Set m = m + 1 and repeat steps 3-5 until m ≥ M where M is large (e.g., M=1000).

We then calculate the mean, standard error, and confidence intervals using the formulae

θboot = 1 M M m=1 θm , Var(θ boot ) = 1 M -1 M m=1 ( θm -θboot ) T ( θm -θboot ), ( 14
) SE k ( θboot ) = V ar(θ boot ) kk .
where θ boot denotes the bootstrapping estimator.

Estimation of two parameters

We first carried out estimation for the 2 parameters k + I and k - I . We use the GLS formulation with γ = 0.6. We fix globally (based on previous estimations with DS 4) the parameter values

k N on k N of f k min on k max on x 1
x 2 i max 4616.962 93.332 1684.381 1.5152 • 10 9 0.0626 0.859 3.542 • 10 5 and used the initial guesses for the parameters given by

k + I k - I q 0 2.1600 10.9270
We then used the bootstrapping algorithm and obtained the following means and standard errors for M = 1000 which, as reported below, compare quite well with the asymptotic theory estimates. The corresponding distributions are shown in Figures 14 and15. I . We use GLS and M=1000 runs.

k + I (boot)(GLS) k - I (boot)(GLS) k + I (asymp)(GLS) k - I (

GLS Estimation of 3 Parameters

We tried next to estimate 3 parameters. We again used the GLS formulation with γ = 0.6. Once again we fixed all the parameters describing the domain and the polymerization function k on and we also fixed either k N of f or k N on in the corresponding inverse problems. We used as initial parameter values:

k + I k - I k N
on q 0 2.1600 10.9270 4616.962

We obtained the estimated parameters together with the corresponding standard errors, variances and the condition numbers κ of the corresponding sensitivity matrices for the four data sets as reported below. The 95% confidence results based on the asymptotic theory are also depicted for DS 4 in Figure 16. To compare these asymptotic results with bootstrapping, we carried out bootstrapping with Data Set (DS) 4 for the estimation of k + I , k - I and k N on with the same initial values as above. We then obtained the following means and standard errors for a run with M = 1000, in comparison to the asymptotic Of particular interest are the values obtained for k N on and the bootstrapping standard errors for k N on which are extremely small. It should be noted that the sensitivity of the model output on k N on is also very small. Thus one might conjecture that the iterations in the bootstrapping algorithm do not change the values of k N on very much and hence one observes the extremely small SE that are produced for the bootstrapping estimates. and the initial guesses for the parameters to be estimated given by:

k + I k - I k N on SE σ 2 κ DS1 2.
k + I (boot) k - I (boot) k N on (boot) k + I (asymp) k - I (asymp) k N on (
k + I k - I k N of f
q 0 2.1600 10.9270 108.256

We obtained the estimated parameters and corresponding SE. Also in this case, we carried out bootstrapping for DS 4. The bootstrapping distributions for k + I , k - I and k N of f are found in Figures 202122. We then obtained the following means and standard errors for a run with M = 1000 in comparison to the asymptotic theory. of f . We used GLS and M=1000 runs.

k + I k - I k N of f SE σ 2 κ DS1 2.
k + I (boot) k - I (boot) k N of f (boot) k + I (asymp) k - I (asymp) k N of f (

Estimation of 4 main parameters

Following the sensitivity analysis detailed above, we tried to estimate a combination of the parameters k + I , k - I , k N on , k N of f for the parameter set with κ θ = 4. Parameters as follows were fixed from the original 9 parameter fit:

k min on k max on x 1
x 2 i max 1684 1.5 • 10 9 0.062 0.859 3.5 • 10 5

We obtained the following result for the estimation of the four parameters using the data sets 1 to 4. In all of them, the condition number of the Fischer's information matrix κ is too large to invert. This along with the sensitivity results above strongly suggests that the data sets do not contain sufficient information to estimate 4 or more parameters with any degree of certainty attached to the estimates.

k + I k - I k N on k N of f σ 2 κ DS1

Model Comparison Tests

A type of Residuals Sum of Squares (RSS) based model selection criterion [START_REF] Banks | Statistical methods for model comparison in parameter estimation problems for distributed systems[END_REF][START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF][START_REF] Banks | Mathematical and Experimental Modeling of Physical and Biological Processes[END_REF] can be used as a tool for model comparison for certain classes of models. In particular this is true for models such as those given in [START_REF] Banks | Model comparison tests to determine data information content[END_REF] in which potentially extraneous mechanisms can be eliminated from the model by a simple restriction on the underlying parameter space while the form of the mathematical model remains unchanged. In other words, this methodology can be used to compare two nested mathematical models where the parameter set Ω H θ (this notation will be defined explicitly in Section 6.1 below) for the restricted model can be identified as a linearly restricted subset of the admissible parameter set Ω θ of the unrestricted model. Indeed, the RSS based model selection criterion is a useful tool to determine whether or not certain terms in the mathematical models are important in describing the given experimental data.

Ordinary Least Squares

We now turn to the statistical model [START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF], where the measurement errors are assumed to be independent and identically distributed with zero mean and constant variance σ 2 . In addition, we assume that there exists θ 0 such that the statistical model

Y j = M(t j ; θ 0 ) + E j , j = 1, 2, . . . , n. (15)
correctly describes the observation process. In other words, ( 15) is the true model, and θ 0 is the true value of the mathematical model parameter θ.

With our assumption on measurement errors, the mathematical model parameter θ can be estimated by using the ordinary least squares method; that is, the ordinary least squares estimator of θ is obtained by solving

θ n = arg min θ∈Ω θ J n (θ; Y).
Here Y = (Y 1 , Y 2 , . . . , Y n ) T , and the cost function J n is defined as

J n (θ; Y) = 1 n n k=1 (Y k -M(t k ; θ)) 2 .
The corresponding realization θn of θ n is obtained by solving θn = arg min

θ∈Ω θ J n (θ; y),
where y is a realization of Y (that is, y = (y 1 , y 2 , . . . , y n ) T ).

As alluded to in the introduction, we might also consider a restricted version of the mathematical model in which the unknown true parameter is assumed to lie in a subset Ω H θ ⊂ Ω θ of the admissible parameter space. We assume this restriction can be written as a linear constraint, Hθ 0 = h, where H ∈ R κr×κq is a matrix having rank κ r (that is, κ r is the number of constraints imposed), and h is a known vector. Thus the restricted parameter space is Ω H θ = {θ ∈ Ω θ : Hθ = h} . Then the null and alternative hypotheses are

H 0 : θ 0 ∈ Ω H θ H A : θ 0 ∈ Ω H θ .
We may define the restricted parameter estimator as

θ n,H = arg min θ∈Ω H θ J n (θ; Y),
and the corresponding realization is denoted by θn,H . Since Ω H θ ⊂ Ω θ , it is clear that J n ( θn ; y) ≤ J n ( θn,H ; y).

This fact forms the basis for a model selection criterion based upon the residual sum of squares. Using the standard assumptions (given in detail in [START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF]), one can establish asymptotic convergence result for the test statistics (which is a function of observations and is used to determine whether or not the null hypothesis is rejected)

U n = n J n (θ n,H ; Y) -J n (θ n ; Y) J n (θ n ; Y) ,
where the corresponding realization Ûn is defined as Ûn = n J n ( θn,H ; y) -J n ( θn ; y)

J n ( θn ; y) . (16)
This asymptotic convergence result is summarized in the following theorem.

Theorem 6.1. Under assumptions detailed in [START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF][START_REF] Banks | Mathematical and Experimental Modeling of Physical and Biological Processes[END_REF] and assuming the null hypothesis H 0 is true, then U n converges in distribution (as n → ∞) to a random variable U having a chi-square distribution with κ r degrees of freedom.

The above theorem suggests that if the sample size n is sufficiently large, then U n is approximately chi-square distributed with κ r degrees of freedom. We use this fact to determine whether or not the null hypothesis H 0 is rejected. To do that, we choose a significance level α (usually chosen to be 0.05) and use χ 2 tables to obtain the corresponding threshold value τ so that P rob(U > τ ) = α. We next compute Ûn and compare it to τ . If Ûn > τ , then we reject the null hypothesis H 0 with confidence level (1α)100%; otherwise, we do not reject. We emphasize that care should be taken in stating conclusions: we either reject or do not reject H 0 at the specified level of

Generalized Least Squares

The model comparison results outlined can be extended to deal with generalized least squares problems in which measurement errors are independent with E(E k ) = 0 and V ar(E k ) = σ 2 w 2 (t k , θ), k = 1, 2, . . . , n, where w is some known real-valued function with w(t, θ) = 0 for any t. This is achieved through rescaling the observations in accordance with their variance (as discussed in [START_REF] Banks | Modeling and Inverse Problems in the Presence of Uncertainty[END_REF]) so that the resulting (transformed) observations are identically distributed as well as independent.

Results for PolyQ Aggregation Models

We then carried out a series of model comparison tests (we again used DS 4) for nested models to determine if an added parameter yields a statistically significantly improved model fit. Our null hypothesis in each case was: H 0 :

The restricted model is adequate (i.e., the fit-to-data is not significantly improved with the model containing the additional parameter as a parameter to be estimated). We obtained the following results.

1. Model with estimation of {k + I , k - I } vs. the model with estimation of {k + I , k - I , k N of f } : We find with n=699, J n ( θn H ; Y) = .0044192109, J n ( θn ; Y) = .0043709501 and Ûn = 7.7178. Thus we reject H 0 at a 99% confidence level.

Model with estimation of {k +

I , k - I } vs. the model with estimation of {k + I , k - I , k N on } : We find J n ( θn ; Y) = .0044192108 with Ûn = 7.49 × 10 -06 . Thus we don't reject H 0 at a 99% confidence level. From these and the preceding results we conclude the information content of the typical data set for the dynamics considered here will support at most 3 parameters estimated with reasonable confidence levels and these are the parameters {k + I , k - I , k N of f }.

Conclusions and Suggested Further Efforts

For the efforts reported on above we make several conclusions. For the majority of data sets, the GLS residual plots with γ = 0.6 are random when fitted for data points M(t k ) ≥ 0.12. As conjectured earlier, this may be because the early formation of aggregates is somewhat stochastic in nature which is not well described by either the mathematical and/or statistical models. It appears that one needs special consideration of smaller polymer sizes. Indeed we suspect from additional discussions with our colleagues that perhaps the nucleation step might be dominated by a stochastic rather than deterministic process in the early stages (i.e., for small polymer sizes). This is a possible direction of further investigation.

Based on several different mathematical/statistical methodologies (sensitivities, asymptotic analysis, bootstrapping, model comparison tests), the data sets we considered do not contain sufficient information for the reliable estimation of all 9 parameters of interest. Indeed our findings suggest that at most 3 parameters can be reliably estimated with the data sets typical of those presented here, and that these parameters are {k + I , k - I , k N of f }. Recently related efforts [START_REF] Kaska Adoteye | Optimal design of non-equilibrium experiments for genetic network interrogation[END_REF] suggest that perhaps there are experimental design questions that could be addressed to collect data that might support the more sophisticated models derived in [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF], especially in order to investigate information coming from different initial concentrations. Indeed, we have considered here data sets related to experiments carried out with the same initial concentration. Adapting the previously used techniques to simultaneously or successively use all the information content in data sets carried out for different initial concentration is a challenging problem (see [START_REF] Powers | The kinetics of nucleated polymerizations at high concentrations: Amyloid fibril formation near and above the "supercritical concentration[END_REF] for a discussion of the effect of initial concentration on nucleated polymerization).

Here we conclude that at most 3 parameters {k + I , k - I , k N of f } can be reliably estimated with the data sets investigated. The two first parameters determine the balance between the normal and abnormal protein concentrations and the third represents the stability of the nucleus against the degradation into monomeric entities. These three parameters are related to the early steps of the aggregation process, and thus we conclude that the model applied to these data sets does not provide any insight into the polymerization of larger polymers. Since this is the case, there is little motivation to modify the polymerization function depicted in Figure 2 until further data collection procedures are pursued.

Figure 1 :

 1 Figure1: The data sets of interest from[START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF][START_REF] Banks | Efficient numerical schemes for Nucleation-Aggregation models: Early steps[END_REF].

Figure 2 :

 2 Figure 2: Parametric representation for k on .

Figure 3 :Figure 4 :

 34 Figure 3: Plots with simulated data: (a) Correct cost function vs. time (γ = 1); (b)Incorrect cost function vs. time (γ = 0)

Figure 5 :

 5 Figure 5: (a) M(t k ) with OLS; (b) Residuals vs Model: OLS

Figure 6 :Figure 7 :

 67 Figure 6: (a) M(t k ) with GLS, γ = 1; (b) Residuals vs Model: GLS

Figure 8 :

 8 Figure 8: Residuals for the 4 experimental data sets using γ = 0.6.

Figure 9 :Figure 10 :Figure 11 :Figure 12 :Figure 13 :

 910111213 Figure 9: (a) Sensitivity w.r.t. k - I ; (b) Sensitivity w.r.t. k + I

Figure 14 :

 14 Figure 14: Two parameters estimation (k + I , k - I ). Bootstrapping distribution for k + I . We use GLS and M=1000 runs.

5. 4 GLSFigure 15 :

 415 Figure 15: Two parameters estimation (k + I , k - I ). Bootstrapping distribution for k - I . We use GLS and M=1000 runs.

  Figure 16: Confidence Intervals

Figure 17 :

 17 Figure 17: Estimation for k + I , k - I and k N on : Bootstrapping distribution for k - I

2

 2

Figure 18 :Figure 19 :

 1819 Figure 18: Estimation for k + I , k - I and k N on : Bootstrapping distribution for k + I

5. 5

 5 GLS estimation for k + I , K - I and k N of f In another test, we fixed k N on and instead estimate k N of f (along with k + I and k - I ). We use the fixed values: 1684.381 1.5152 • 10 9 0.0626 0.859 3.542 • 10 5

Figure 20 :

 20 Figure 20: Three parameters estimation (k + I , k - I and k N of f ): Bootstrapping distribution for k + I . We used GLS and M=1000 runs.

Figure 21 :

 21 Figure 21: Three parameters estimation (k + I , k - I and k N of f ): Bootstrapping distribution for k - I . We used GLS and M=1000 runs.

Figure 22 :

 22 Figure 22: Three parameters estimation (k + I , k - I and k N of f ): Bootstrapping distribution for k Nof f . We used GLS and M=1000 runs.

3 .

 3 Model with estimation of {k + I , k - I , k N of f } vs. the model with estimation of {k + I , k - I , k N of f , k N on } : To the order of computation we find no difference in the cost functions in this case and therefore we do not reject H 0 at a confidence level of 99%.4. Model with estimation of {k +I , k - I , k N on } vs. the model with estimation of {k + I , k - I , k N on , k N of f } : We find J n ( θn ; Y) = .0043709780 with Ûn = 7.7133 and hence we reject H 0 with a confidence level of 99%.

  10 -6 3.94 • 10 11 DS4 2.16 10.91 4616.96 (0.0089, 0.0649, 45.262) 6.36 • 10 -6 7.14 • 10 10

	26 13.49 4616.96	(.012, .099, 53.925)	8.52 • 10 -6 8.89 • 10 10
	DS2 2.99 16.20 4616.96	(.021, .151, 56.691)	9.67 • 10 -6 4.37 • 10 10
	DS3 2.18 15.76 9840.31	(.011, .103, 90.466)	6.45 •

  203 12.997 99.861 (.011, .091, 1.208) 8.165 • 10 -6 4.912 • 10 7 DS2 2.893 15.474 100.019 (.019, .137, 1.279) 9.323 • 10 -6 2.486 • 10 7 DS3 2.168 15.631 41.935 (.011, .102, 0.424) 6.435 • 10 -6 9.125 • 10 6 DS4 2.181 11.090 90.536 (.009, .066, 0.936) 6.289 • 10 -6 3.043 • 10 7

  2.1431 12.4751 4616.962 108.259 8.7219 • 10 -6 6.1226 • 10 19 DS2 2.7995 14.7630 4616.957 108.4308 9.8694 • 10 -6 1.4442 • 10 19 DS3 2.180 15.757 4618.599 41.369 6.4622 • 10 -6 1.881 • 10 17 DS4 2.161 10.9278 4617.3316 93.3265 6.374 • 10 -6 2.144 • 10 18

  confidence. The table below illustrates the threshold values for χ 2 (1) with the given significance level.Similar tables can be found in any elementary statistics text or online or calculated by some software package such as Matlab, and is given here for illustrative purposes and also for use in the examples demonstrated below.

	α	τ	confidence level
	.25 1.32	75%
	.1	2.71	90%
	.05 3.84	95%
	.01 6.63	99%
	.001 10.83	99.9%

Acknowledgements

This research was supported in part (MD, CK) by the ERC Starting Grant SKIPPERAD, in part (HTB) by Grant Number NIAID R01AI071915-10 from the National Institute of Allergy and Infectious Diseases, and in part (HTB) by the Air Force Office of Scientific Research under grant number AFOSR FA9550-12-1-0188.

Information Content in Data Sets for a Nucleated-Polymerization Model


Agathe Cadène

Mathilde Fouché

Alice Rivère

Rémy Battesti

Sonia Coriani

Antonio Rizzo

Carlo Rizzo

Circular and linear magnetic birefringences in xenon at λ = 1064 nm

Keywords: numbers: 42.25.Lc, 78.20.Ls, 31.15.bw Keywords: birefringence, magneto-optical effects, coupled-cluster theory

The circular and linear magnetic birefringences corresponding to the Faraday and the Cotton-Mouton effects, respectively, have been measured in xenon at λ = 1064 nm. The experimental setup is based on time dependent magnetic fields and a high finesse Fabry-Pérot cavity. Our value of the Faraday effect is the first measurement at this wavelength. It is compared to theoretical predictions. Our uncertainty of a few percent yields an agreement at better than 1σ with the computational estimate when relativistic effects are taken into account. Concerning the Cotton-Mouton effect, our measurement, the second ever published at λ = 1064 nm, agrees at better than 1σ with theoretical predictions. We also compare our error budget with those established for other experimental published values.

I. INTRODUCTION

Magnetic birefringence corresponds to an anisotropy of the (generally complex) refractive index induced in a medium by a magnetic field. [START_REF] Barron | Molecular light scattering and optical activity[END_REF][START_REF] Rizzo | Birefringences: A Challenge for Both Theory and Experiment[END_REF] A circular birefringence arises when the magnetic field changes the angular velocity of the two eigen modes of polarization in which a linearly polarized beam is split, without deforming them. The net result is a rotation of the plane of linear polarization, a phenomenon seen also in absence of external fields in chiral samples (natural optical rotation). When the presence of the external magnetic field yields a different phase of two perpendicular components of the linear polarization vector, the net result is the appearance of an ellipticity, and we are observing an example of linear birefringence.

Two well known examples of magnetic birefringences are the Faraday and the Cotton-Mouton effects. The former corresponds to a circular birefringence induced by a longitudinal magnetic field B (aligned parallel to the direction of propagation of light). After going through the birefringent medium, the real part of the index of refraction for left circularly polarized light n -is different from that for right circularly polarized light n + . The difference ∆n

F = n --n + is proportional to B ∆n F = k F B , (1)
k F being the circular magnetic birefringence per unit magnetic field intensity. For historical reason, the Faraa) Electronic mail: carlo.rizzo@lncmi.cnrs.fr day effect is usually given in terms of the Verdet constant

V = πk F λ , (2)
where λ is the light wavelength. On the other hand, the Cotton-Mouton effect corresponds to a linear magnetic birefringence induced by a transverse magnetic field B ⊥ .

The field induces a difference between the real parts of the refraction index for light polarized parallel with respect to that polarized perpendicular to the magnetic field. The difference ∆n CM = n -n ⊥ is proportional to the square of the magnetic field

∆n CM = k CM B 2 ⊥ , (3)
with k CM the linear magnetic birefringence per square unit magnetic field intensity. For the Cotton-Mouton effect, k CM has two contributions, the first one due to the distortion of the electronic structure while the second one corresponds to a partial orientation of the molecules. When working in the conditions of constant volume, the orientational contribution is proportional to the inverse of the temperature T , and it usually dominates, often hiding the first temperature independent contribution. For axial molecules, for examples, k CM is given by the expression 3

k CM = πN A V m 4πǫ 0 ∆η + 2 15k B T ∆α∆χ . (4)
Above N A is the Avogadro constant, V m the molar volume, k B the Boltzmann constant, ǫ 0 the electric constant, ∆η the frequency dependent hypermagnetizability anisotropy, ∆α the optical electric dipole polarizability anisotropy, and ∆χ the magnetic susceptibility anisotropy. For spherical molecules or for atoms, such as xenon, however, the temperature dependent contribution vanishes. Measurements on noble gases, for example, allow to focus on the hypermagnetizability anisotropy ∆η term. On the other hand, since the Langevin-type orientational term vanishes, the magnetic birefringence is much lower than the one observed in non spherical molecules. From an experimental point of view, measurements on such gases require a very sensitive apparatus, with a ∆n CM of the order of 10 -16 for helium and 10 -14 for xenon at one atmosphere and with a magnetic field of one Tesla. In comparison, ∆n F is typically 10 5 bigger.

The computational determination of the Verdet constant and of the Cotton-Mouton effect requires the farfrom-trivial calculation of higher-order response functions, [START_REF] Rizzo | Birefringences: A Challenge for Both Theory and Experiment[END_REF][START_REF] Rizzo | The Cotton-Mouton effect in gases : experiment and theory[END_REF] and it has often served as test bed for the validation of new electronic structure methods. For atoms, in order to obtain accurate results one must properly account for the appropriate description of one-electron (basis set), N-electron (correlation) and relativistic effects. As far as correlation is concerned, coupled cluster (CC) methods are nowadays among the most accurate tools in electronic structure theory. [START_REF] Helgaker | Recent advances in wave function-based methods of molecular-property calculations[END_REF][START_REF] Christiansen | Response functions from Fourier component variational perturbation theory applied to a time-averaged quasienergy[END_REF] Both birefringences treated here, and in particular the Cotton-Mouton effect, require a good description of the outer valence space of the system at hand, and therefore the presence of diffuse functions in the one-electron basis set is mandatory. [START_REF] Rizzo | Birefringences: A Challenge for Both Theory and Experiment[END_REF][START_REF] Helgaker | Recent advances in wave function-based methods of molecular-property calculations[END_REF] Whereas for light atoms relativistic corrections are minor, their importance increases and they become significant for heavier atoms. For example, Ekström et al [START_REF] Ekström | Four-component Hartree-Fock calculations of magnetic-field induced circular birefringence-Faraday effect-in noble gases and dihalogens[END_REF] have calculated that for helium the relativistic effects add -0.03% to the non-relativistic Verdet value. For xenon, the heaviest non-radioactive noble atom, relativistic corrections add 3 to 4%, depending on the chosen wavelength. In this case, relativistic effects cannot be ignored in accurate calculations.

In this article, we report both measurements and calculations of Faraday and Cotton-Mouton effects at λ = 1064 nm. We perform the first measurement of the Faraday effect of xenon at this wavelength, and our estimate bears an uncertainty of a few percent. Concerning the Cotton-Mouton effect, our measurement, the second ever published at λ = 1064 nm, agrees at better than 1σ with theoretical predictions and we also compare our error budget with those established for other experimental published values. Our theoretical predictions, that can be considered of state-of-the-art quality, were obtained at the coupled cluster singles and doubles (CCSD) [START_REF] Purvis | A full coupled cluster singles and doubles model: The inclusion of disconnected triples[END_REF][START_REF] Koch | The integral-direct coupled cluster singles and doubles model[END_REF][START_REF] Hättig | Frequency-dependent first hyperpolarizabilities using coupled cluster quadratic response theory[END_REF] and coupled cluster singles, doubles and approximate triples (CC3) 10-13 levels of theory, and they include estimates of relativistic effects. For both effects, our theoretical predictions are within 1σ of our experimental data.

II. EXPERIMENTAL SETUP

A. Principle of the measurement

Experimentally, we determine the Faraday and the Cotton-Mouton effects by measuring, respectively, the rotation induced by a longitudinal magnetic field and the ellipticity induced by a transverse magnetic field on an incident linear polarization. For small angles, the induced rotation θ F depends on the circular birefringence as follows

θ F = π L B λ ∆n F , (5)
where L B is the length of the magnetic field region. The induced ellipticity ψ CM is related to the linear birefringence by the formula:

ψ CM = π L B λ ∆n CM sin 2θ P , (6)
where θ P is the angle between the light polarization and the magnetic field.

B. General setup

The apparatus has already been described in detail elsewhere. [START_REF] Battesti | The BMV experiment: a novel apparatus to study the propagation of light in a transverse magnetic field[END_REF][START_REF] Cadène | Faraday and Cotton-Mouton effects of helium at λ=1064 nm[END_REF] Briefly, light comes from a Nd:YAG laser at λ = 1064 nm (see Fig. 1). It is linearly polarized by a first polarizer P, before going through a transverse or a longitudinal magnetic field. The polarization is then analyzed by a second polarizer A, crossed at maximum extinction compared to P. The beam polarized parallel to the incident beam, reflected by the polarizer A as the ordinary ray, is collected by the photodiode Ph t . Its power is denoted by I t . The beam polarized perpendicular to the incident beam (power I e ), corresponding to the extraordinary ray that passes through the polarizer A, is collected by the low noise and high gain photodiode Ph e . This setup has been designed to measure the linear magnetic birefringence of vacuum [START_REF] Cadène | Vacuum magnetic linear birefringence using pulsed fields : status of the bmv experiment[END_REF] and its sensitivity allows to perform precise measurements on gases. [START_REF] Cadène | Faraday and Cotton-Mouton effects of helium at λ=1064 nm[END_REF][START_REF] Berceau | Magnetic linear birefringence measurements using pulsed fields[END_REF] All the optical components from A to P are placed in an ultrahigh-vacuum chamber. To perform birefringence measurement on gases, we fill the vacuum chamber with a high-purity gas. For this particular measurement, we have used a bottle of xenon with a global purity higher than 99.998 %.

C. Fabry-Pérot cavity

Magnetic birefringence measurements on dilute gases are difficult, especially at low pressure, because one has to detect very small variations of light polarization. To increase the measured signal, one needs high magnetic fields. One also needs an as large as possible path length in the field L B (cf. Eqs ( 5) and ( 6)). To this end, optical cavities are used to trap light in the magnetic field region and therefore enhance the signal to be measured.

As shown in Fig. 1, the cavity is formed by two mirrors M 1 and M 2 , placed at both sides of the magnetic field region. The laser frequency is locked to the cavity resonance frequency, using the Pound-Drever-Hall technique. [START_REF] Pound | Electronic Frequency Stabilization of Microwave Oscillators[END_REF] The electro-optic modulator generates 10 MHz sidebands and the signal reflected by the cavity is detected by the photodiode Ph r . The laser frequency is adjusted with the acousto-optic modulator, the piezoelectric and the Peltier elements of the laser.

This cavity increases the distance traveled by light in the magnetic field by a factor 2F/π, where F is the cavity finesse. Therefore, the rotation induced by the longitudinal magnetic field becomes

Θ F (t) = 2F π θ F (t), (7)
with θ F the rotation acquired without any cavity. In the same way, the ellipticity induced by the transverse magnetic field becomes

Ψ CM (t) = 2F π ψ CM (t), (8)
with ψ CM denoting the ellipticity acquired without any cavity. The cavity finesse is inferred from the measurement of the photon lifetime τ inside the cavity 19

F = 2π∆ FSR τ, (9)
with ∆ FSR the cavity free spectral range. For the Faraday effect, the cavity finesse was about F = 475 000. For the Cotton-Mouton effect, two sets of mirrors were used with a respective finesse of about 400 000 and 480 000.

D. Raw signals

We measure the circular and the linear magnetic birefringence by measuring the ratio I e /I t I e (t)

I t,f (t) = σ 2 + [Γ + Ψ CM (t)] 2 + [ǫ + Θ F (t)] 2 . ( 10
)
As said previously, I e (I t ) corresponds to the power of light polarized perpendicular (parallel) to the incident beam. The subscript f indicates that we need to take into account the cavity filtering, as explained in details in previous papers. [START_REF] Cadène | Faraday and Cotton-Mouton effects of helium at λ=1064 nm[END_REF][START_REF] Berceau | Dynamical behaviour of birefringent Fabry-Prot cavities[END_REF] The term σ 2 corresponds to the extinction ratio of polarizers P and A, Γ is the total static ellipticity due to the cavity mirrors and ǫ is the static angle between the major axis of the elliptical polarization and the incident polarization. The extinction ratio and the static birefringence are measured before each magnetic pulse. The static angle ǫ can be estimated but its value is not needed for the analysis.

III. CIRCULAR MAGNETIC BIREFRINGENCE A. Magnetic field

The magnetic field is generated by a solenoid previously used for Faraday effect measurement in helium. [START_REF] Cadène | Faraday and Cotton-Mouton effects of helium at λ=1064 nm[END_REF] Its characteristics have already been explained in details. [START_REF] Cadène | Faraday and Cotton-Mouton effects of helium at λ=1064 nm[END_REF] Here we just briefly recall its main features. It generates a longitudinal magnetic field with an equivalent length L B = (0.308 ± 0.006) m at 1σ. This magnetic field is modulated at the frequency ν = 18 Hz: B = B ,0 sin(2πνt + φ). The rotation of the polarization due to the Faraday effect is thus given by

Θ F = Θ 0 sin(2πνt + φ), (11) with
Θ 0 = 2F π V B ,0 L B . (12)
B. Data analysis Expanding Eq. ( 10), the raw signal becomes

I e (t) I t,f (t) = σ 2 + Γ 2 + ǫ 2 + 2ǫΘ F (t) + Θ 2 F (t). ( 13
)
This gives three main frequency components: a DC signal, a signal at the frequency ν, and a signal at the double frequency 2ν. To measure the Verdet constant, we use the amplitude of the signal at 2ν 15

A 2ν = Θ 2 0 2 1 + 2ν νc 2 , (14)
where ν c = 1/4πτ is the cavity cutoff frequency, introduced to take into account the cavity filtering. [START_REF] Berceau | Dynamical behaviour of birefringent Fabry-Prot cavities[END_REF] A 2ν is measured for different magnetic field amplitudes, from 0 to about 50×10 -3 T. The whole is fitted by K V B 2 ,0 . The Verdet constant finally depends on the measured experimental parameters as follows

V (T, P ) = K V 2 1 + (8πτ ν) 2 1/4 2τ ∆ FSR L B , (15)
where T and P are respectively the temperature and pressure of the gas.

τ [ms] 1.14 2.0 × 10 -2 KV [rad T -1 ] 1.07 3 × 10 -3 3.2 × 10 -2 ∆ FSR [MHz] 65.996 3 × 10 -4 LB [m] 0.308 1.9 × 10 -2 V × 10 5 1.66 1.8 × 10 -2 2.5 × 10 -2 [radT -1 m -1 ]

C. Measurement and error budget

The A-and B-type uncertainties associated to the measurement of V are detailed in Tab. I. [START_REF] Cadène | Faraday and Cotton-Mouton effects of helium at λ=1064 nm[END_REF][START_REF] Berceau | Magnetic linear birefringence measurements using pulsed fields[END_REF] They are given at 1σ (coverage factor k = 1). The A-type uncertainty is dominated by the photon lifetime uncertainty. The main contributions of the B-type uncertainty comes from the uncertainty of the magnetic length and of the fit constant K V which includes the B-type uncertainty of the magnetic field and of the photodiodes conversion factor [START_REF] Berceau | Magnetic linear birefringence measurements using pulsed fields[END_REF] .

We have measured the Verdet constant in xenon at T = (294 ± 1) K and for 5 pressures from 1.01 × 10 -3 to 5.01 × 10 -3 atm. In this range of pressure, xenon can be considered as an ideal gas and the Verdet constant is thus proportional to the pressure. Data are fitted by a linear equation:

V (T, P ) = V n P, (16)
giving a normalized Verdet constant (P = 1 atm) at λ = 1064 nm and T = (294 ± 1) K

V n = (3.31 ± 0.09) × 10 -3 atm -1 rad T -1 m -1 . ( 17
)
The uncertainty is given at 1σ and is detailed in Tab. II. With a scale law on the gas density, this corresponds to a normalized Verdet constant at T = 273.15 K of

V N = (3.56 ± 0.10) × 10 -3 atm -1 rad T -1 m -1 . ( 18
)
Using Eq. ( 2), we can also give the normalized Faraday constant at T = 273.15 K

k N F = (1.21 ± 0.03) × 10 -9 atm -1 T -1 . ( 19
)
IV. LINEAR MAGNETIC BIREFRINGENCE

A. Magnetic field

The transverse magnetic field B ⊥ is generated by an X-Coil, specially designed by the High Magnetic Field

V × 10 5 1.66 1.8 × 10 -2 2.5 × 10 -2 [radT -1 m -1 ] P × 10 3 5 2 × 10 -3 [atm] linear fit 3.31 1.5 × 10 -2 ×10 3 [atm -1 rad T -1 m -1 ] V n × 10 3 3.31 1.5 × 10 -2 2.5 × 10 -2 [atm -1 rad T -1 m -1 ]
National Laboratory (LNCMI-Toulouse, France) for the measurement of the vacuum magnetic birefringence. This coil has been presented and discussed in great details in several previous papers. [START_REF] Battesti | The BMV experiment: a novel apparatus to study the propagation of light in a transverse magnetic field[END_REF][START_REF] Batut | A Transportable Pulsed Magnet System for Fundamental Investigations in Quantum Electrodynamics and Particle Physics[END_REF] Very briefly, the magnet delivers a pulsed magnetic field over an equivalent length L B of 0.137 m. The total duration of the pulse is about 10 ms with a maximum reached within 2 ms. For the present measurements, a maximum magnetic field of 3 T has been used. Finally, the high-voltage connections can be remotely switched to reverse the direction of the field. Thus we can set B ⊥ parallel or antiparallel to the x direction, as shown in Fig. 1.

B. Data analysis

The data analysis follows the one described for the Cotton-Mouton effect measurement in helium. [START_REF] Cadène | Faraday and Cotton-Mouton effects of helium at λ=1064 nm[END_REF] We will however detail the main steps, since a slightly different method was used in the present case.

To extract the ellipticity Ψ CM (t) from Eq. ( 10), we calculate the following Y (t) function

Y (t) = Ie(t) I t,f (t) -I DC 2|Γ| = γΨ CM (t) + Ψ 2 CM (t) 2|Γ| + γ |ǫ|Θ F (t) 2|Γ| + Θ 2 F (t) 2|Γ| , (20)
where γ stands for the sign of Γ. I DC is the static signal measured just before the application of the magnetic field. The absolute value of the static ellipticity |Γ| is also measured before each pulse.

Two parameters are adjustable in the experiment: the sign γ of the static ellipticity Γ and the direction of the transverse magnetic field. We acquire signals for both signs of Γ and both directions of B ⊥ : parallel to x is denoted as > 0 and antiparallel is denoted as < 0. This gives four data series: (Γ > 0, B ⊥ > 0), (Γ > 0, B ⊥ < 0), (Γ < 0, B ⊥ < 0) and (Γ < 0, B ⊥ > 0).

For each series, signals calculated with Eq. ( 20) are averaged and denoted as Y >> , Y >< , Y << and Y <> . The first subscript corresponds to Γ > 0 or < 0 while the second one corresponds to B ⊥ parallel or antiparallel to x.. This average function can be written in a more general form than the one of Eq. (20). It is the sum of different effects with different symmetries, denoted as s

Y >> = +Ψ + 1 2 1 Γ >> s ++ + 1 Γ >> s --+ 1 2 1 Γ >> s +-, Y >< = +Ψ + 1 2 1 Γ >< s ++ + 1 Γ >< s --+ 1 2 1 Γ >< s +-, Y << = -Ψ + 1 2 1 Γ << s ++ + 1 Γ << s --+ 1 2 1 Γ << s +-, Y <> = -Ψ + 1 2 1 Γ <> s ++ + 1 Γ <> s --+ 1 2 1 Γ <> s +-. (21)
The first subscript in s corresponds to the symmetry with respect to the sign of Γ and the second one to the symmetry with respect to the direction of B ⊥ . The subscript + indicates an even parity while the subscript -indicates odd parity. The ratio < 1/Γ > is the average of 1/|Γ| measured during corresponding series. The terms Ψ 2 CM and Θ 2 F are included in s ++ , γ|ǫ|Θ F are included in s --, and s +-corresponds to a spurious signal with an odd parity towards the direction of B ⊥ and an even parity with respect to the sign of Γ. The ellipticity γΨ CM corresponds to s -+ .

From this set of four equations with four unknown quantities (Ψ CM , s ++ , s --and s +-), we extract Ψ CM (t), which is fitted by αB 2 ⊥,f . The cavity filtering should again be taken into account, as indicated by the subscript f. [START_REF] Cadène | Faraday and Cotton-Mouton effects of helium at λ=1064 nm[END_REF][START_REF] Berceau | Dynamical behaviour of birefringent Fabry-Prot cavities[END_REF] The Cotton-Mouton constant k CM finally depends on the measured experimental parameters as follows:

k CM (T, P ) = α 4πτ ∆ FSR λ L B 1 sin 2θ P . (22)

C. Measurement and error budget

The A-and B-type uncertainties associated to the measurement of k CM are detailed in Tab. III and are given at 1σ. The B-type uncertainties have been evaluated previously and detailed in Ref. 17. They essentially come from the length of the magnetic field L B and the fit constant α.

We have measured the Cotton-Mouton constant in xenon at T = (293 ± 1) K and for nine pressures from 3 × 10 -3 to 8 × 10 -3 atm. The data as a function of the

[T -2 atm -1 ] k n CM 2.41 1.5 × 10 -1 3.1 × 10 -2 ×10 14 [T -2 atm -1 ]
pressure are fitted by a linear equation, and we obtain for the value of the Cotton-Mouton constant at P = 1 atm

k n CM = (2.41 ± 0.37) × 10 -14 T -2 atm -1 . (23)
The uncertainty given at 1σ is detailed in Tab. IV. The dominant uncertainty comes from the linear fit of the Cotton-Mouton constant versus pressure (A-type). The value of k n CM normalized at 273.15 K is calculated with a scale law on the gas density

k N CM = (2.59 ± 0.40) × 10 -14 T -2 atm -1 . (24)
V. OUR CALCULATIONS

The Verdet constant and the Cotton-Mouton birefringence were computed within Coupled Cluster response theory, [START_REF] Helgaker | Recent advances in wave function-based methods of molecular-property calculations[END_REF][START_REF] Christiansen | Response functions from Fourier component variational perturbation theory applied to a time-averaged quasienergy[END_REF] at the CCSD 7-9 and CC3 10-13 levels of approximation. Specifically, the Verdet constant was obtained from the following frequency-dependent quadratic response function 4,21-23

V (ω) = Cω µ x ; µ y , L z ω,0 , (25)
with C = N e 8meǫ0c0 = 0.912742 × 10 -7 in atomic units, N the number density (N = P kBT for ideal gases), e the elementary charge, m e the electron mass, c 0 the speed of light in vacuo, ω/2π the frequency of the probing light, and µ x,y and L z are Cartesian components of the electric dipole, and angular momentum operators, respectively. The hypermagnetizability anisotropy ∆η entering the Cotton-Mouton birefringence in Eq. ( 4) (the only term contributing for atoms) is given by the combination of a quadratic and a cubic response functions

3 ∆η = - 1 4 µ x ; µ x , L z , L z ω,ω,0 - 1 4 µ x ; µ x , Θ xx ω,0 (26)
≡ ∆η p + ∆η d .

with Θ xx the Cartesian component of the traceless quadrupole operator. At the CC3 level, calculations were performed at three different wavelengths, namely 1064, 632.8 and 514.5 nm. At the CCSD level, we computed the dispersion coefficients, as done in our previous study, [START_REF] Coriani | The electric-field-gradientinduced birefringence of Helium, Neon, Argon, and SF 6[END_REF] i.e., for the Verdet constant

V (2n) = 2nS(-2n -2); (27)
V (ω) = C ∞ n=1 ω 2n V (2n); (28)
(ω) = ∞ n=0 ω 2n ∆η(2n). (29) ∆η
Above, S(k) is the Cauchy moment

S(k) = m =0 2ω k+1 m0 0 | µ z | m m | µ z | 0 ( 31
)
with ω m0 indicating the excitation energy from the ground state 0 to the excited state(s) m, and B(2n) is the dispersion coefficient introduced when expanding, for frequencies below the lowest excitation energy, the electric dipole-electric dipole-electric quadrupole quadratic response function B x,x,xx (-ω; ω, 0) = µ x ; µ x , Θ xx ω,0 in a convergent power series in the circular frequency ω

B x,x,xx (-ω; ω, 0) = ∞ n=0 ω 2n B(2n) (32)
For further details on how the above Cauchy moments and dispersion coefficients of the given quadratic response function are computed within coupled cluster response theory, the reader should refer to Refs. 24-26.

Relativistic effects were approximately accounted for by employing relativistic effective core potentials (ECPs), [START_REF] Dolg | Effective core potentials[END_REF] and specifically pseudo-potentials (PP). "Small core" effective pseudo-potentials were used to describe the 28 inner electrons (that is, the [Ar]3d 10 core), whereas the remaining 26 valence electrons were correlated as in standard non-relativistic calculations. The basis sets used were constructed starting from the singly augmented aug cc pvxz pp (x=t,q) sets of Peterson et al. [START_REF] Peterson | Systematically convergent basis sets with relativistic pseudopotentials. II. Small-core pseudopotentials and correlation consistent basis sets for the post-d group 16-18 elements[END_REF] Since single augmentation is usually not sufficient to ensure converged results, at least for the Cotton-Mouton birefringence. Additional sets of diffuse functions were added by applying an even-tempered generation formula commonly used for this purpose to the orbital functions describing the valence electrons, while retaining the pseudo-potential of the original set. The resulting sets are labeled d-aug and t-aug, for double and triple augmentation, respectively.

Where pseudo-potentials parametrically account for relativistic effects on the innermost orbitals, other relativistic effects (e.g. higher-order and picture change effects, spin-orbit coupling) could play a significant role. [START_REF] Saue | Spin-interactions and the non-relativistic limit of electrodynamics[END_REF][START_REF] Saue | Relativistic Hamiltonians for Chemistry: A Primer[END_REF] When dealing with valence properties like electric hyperpolarizabilities, the higher-order relativistic effects and picture change effects (for the dipole operator and also the electron-electron interaction) are expected to be not so important. Also, spin-orbit coupling should be quite weak. Both the Faraday and Cotton Mouton birefringences, however, involve the magnetic dipole operator. In general relativistic effects on magnetic properties can be more significant and more difficult in terms of picture change (the operators look different in relativistic and non-relativistic theory and this may require a correction of the property operator that one uses as a perturbation). [START_REF] Saue | Spin-interactions and the non-relativistic limit of electrodynamics[END_REF][START_REF] Saue | Relativistic Hamiltonians for Chemistry: A Primer[END_REF] Nonetheless, also given that the most stringent requirement in terms of basis set convergence is the inclusion of diffuse functions as in the case of the electric hyperpolarisability, it is reasonable to assume that both properties are essentially valence properties, for which picture change effects are typically small, and we reckon therefore that the use of (PP)ECPs can be considered accurate enough.

The results obtained in the x=q basis sets are summarized in Tab. V and Tab. VI, for CCSD and CC3, respectively.

All calculations were performed with the Dalton code. We can compare our value of the normalized Verdet constant to other published values. The most extensive experimental compilation of Verdet constants has been reported by Ingersoll and Liebenberg in 1956, for several gases including xenon 32 for wavelengths ranging from 363.5 to 987.5 nm, with a total uncertainty of about 1 %. These values are plotted in Fig. 2.

No datum has ever been reported for λ = 1064 nm. Nevertheless, we can extrapolate its value from the points of Fig. 2, by fitting the data with a function of form V = A/λ 2 + B/λ 4 (solid curve in Fig. 2). [START_REF] Ingersoll | Faraday Effect in Gases and Vapors. II[END_REF][START_REF] Rosenfeld | Zur theorie des faradayeffekts[END_REF] A supplementary systematic uncertainty should also be added, since the authors measured the ratio between Faraday effects in xenon and in distilled water, and rescaled their measurements with accepted values for water. [START_REF] Ingersoll | Faraday Effect in Gases and Vapors. II[END_REF][START_REF] Rosenfeld | Zur theorie des faradayeffekts[END_REF] . Thus it does not correspond to absolute measurements of the Faraday effect, contrary to ours. At λ = 1064 nm and T = 273.15 K we obtain V N = (3.46 ± 0.04) × 10 -3 atm -1 rad.T -1 m -1 . The 1σ uncertainty includes the one given by the fit. This value is compatible with our experimental value (Eq. ( 18)), represented as the open circle in Fig. 2 and as the straight and dashed lines in Fig. 3. Points : theoretical predictions (both ours and from the literature). See text and Tab. VII for the references.

Theory

We can also compare our experimental value with theoretical predictions (both ours and from the literature), plotted in Fig. 3 and summarized in Tab. VII at 1 atm, 273.15 K and with the gas number density of an ideal gas. To convert from theoretical results given in atomic units into the units used experimentally, we exploited the relation:

V (atm -1 rad.T -1 m -1 ) = V (a.u.) × 8.039617 × 10 4 .
(33) Our experimental value is compatible within 1σ with both our "best" coupled cluster results (t-aug cc pvqz pp basis) and the theoretical prediction of Ekström et al, [START_REF] Ekström | Four-component Hartree-Fock calculations of magnetic-field induced circular birefringence-Faraday effect-in noble gases and dihalogens[END_REF] within 2σ with the estimate of Ikäläinen et al, [START_REF] Ikäläinen | Fully relativistic calcu-lations of faraday and nuclear spin-induced optical rotation in xenon[END_REF] and within 3σ with that of Savukov. [START_REF] Savukov | Particle-hole configuration-interaction polarizabilities and Verdet constants of noble-gas atoms[END_REF] The uncertainty of a few percent obtained on our experimental value allows to comment on the agreement with theoretical predictions as a function of the theoretical approximation or model. Savukov [START_REF] Savukov | Particle-hole configuration-interaction polarizabilities and Verdet constants of noble-gas atoms[END_REF] has used a relativistic particle-hole configuration interaction (CI) method. He does not give a value at 1064 nm, but the latter can be interpolated, as done with the previous experimental data of Ingersoll and Liebenberg, [START_REF] Ingersoll | Faraday Effect in Gases and Vapors. II[END_REF] obtaining the value of Tab. VII, with an uncertainty given by the fit. The agreement between theory and experiment is only within 3σ, even if relativistic effects are taken into account. Ekström et al 6 have used the nonrelativistic time-dependent Hartree-Fock (TDHF in Fig. 3) and the relativistic time-dependent Dirac-Hartree-Fock (TDDHF in Fig. 3). There is clearly a better agreement (better than 1σ), between their calculations and our experimental value when relativistic effects are taken into account. Finally, Ikäläinen et al 34 have used the nonrelativistic Hartree-Fock method (NR in Fig. 3), the exact two-component method (X2C in Fig. 3), and the fully relativistic four-component method (DHF in Fig. 3). The same authors also report (in the supporting information file) a non relativistic CCSD result (NR-CCSD in Fig. 3).

While their uncorrelated results confirm that relativistic effects should be taken into account to improve agreement with experiment, their non-relativistic CCSD result highlights how the inclusion of correlation effects is equally important. Also worth noticing is the rather poor performance of the BLYP and B3LYP functionals, which overestimate the value of the Verdet constant in both non-relativistic and relativistic calculations. This also applies for the BHandHLYP functional in the relativistic calculations, whereas the non-relativistic BHandHLYP value is still within 1σ of our experimental result (See Only a few measurements of the Cotton-Mouton effect in xenon have been discussed in the literature. There is one at λ = 514.5 nm by Carusotto et al, [START_REF] Carusotto | Measurement of the magnetic birefringence of noble gases[END_REF] one at λ = 632.8 nm by Hüttner (reported as a private communication by Bishop et al ), [START_REF] Bishop | Hypermagnetizability anisotropy (Cotton-Mouton effect) for the rare gases and methane[END_REF] and finally one at λ = 1064 nm by Bregant et al. [START_REF] Bregant | Measurement of the cotton-mouton effect in krypton and xenon at 1064 nm with the pvlas apparatus[END_REF][START_REF] Bregant | Erratum to "Measurement of the Cotton-Mouton effect in krypton and xenon at 1064 nm with the PVLAS apparatus[END_REF] Our experimental value, referring to λ = 1064 nm is compatible within 1σ with the datum of Refs. 38,39. The set of results is shown in Tab. VIII and plotted as a function of the wavelength in Fig. 4.

Our measurement has an uncertainty of about 15%. This value, which is larger than that of the other reported values, especially those given for wavelengths of 514.5 nm and 632.8 nm, was established via a complete error budget. Note that no information is available on the setup, the number of pressures, the error budget and the evaluation of the uncertainty for the value reported at λ = 632.8 nm by Bishop et al 37 as a private communication of Hüttner. The value reported at λ = 514.5 nm by Carusotto et al [START_REF] Carusotto | Measurement of the magnetic birefringence of noble gases[END_REF] was measured only at 1 atm, and by comparing the observed magnetic birefringence with that of nitrogen under the same experimental conditions, therefore taking as a reference, free of uncertainty, the Cotton-Mouton constant of nitrogen. It is safe to say therefore that the uncertainty associated to their datum might be underestimated. Finally, the value reported by Bregant et al [START_REF] Bregant | Measurement of the cotton-mouton effect in krypton and xenon at 1064 nm with the pvlas apparatus[END_REF][START_REF] Bregant | Erratum to "Measurement of the Cotton-Mouton effect in krypton and xenon at 1064 nm with the PVLAS apparatus[END_REF] at λ = 1064 nm corresponds to the weighted average between measurements at two different pressures (9 pressures for our measurement) and the uncertainty is similar to ours.

Theory

The Cotton-Mouton constant k CM is linked to ∆η by the relationship 3 k CM (atm -1 T -2 ) = 6.18381 × 10 -14 T ∆η (a.u.). (34) Only one theoretical prediction has been published sofar for the Cotton-Mouton effect in xenon. [START_REF] Bishop | Calculation of electromagnetic properties of the noble gases[END_REF] The calculation of Bishop and Cybulski was performed at the self-consistent-field (SCF) level of approximation, and it yielded static hypermagnetizability anisotropy ∆η. As stated by the authors, relativistic effects were not taken into account, even though the authors expected them to play a substantial role. Our experimental value agrees with that theoretical prediction within 1σ.

Our computed coupled cluster results, both CCSD and CC3, in the largest (t-aug cc pvqz pp) basis sets for the three wavelengths at which experimental results are given in Tab. VIII. Both the CCSD and CC3 values at 1064 nm are well within 1σ of our experimental measurement, and just outside 1σ of the result by Bregant et al. [START_REF] Bregant | Measurement of the cotton-mouton effect in krypton and xenon at 1064 nm with the pvlas apparatus[END_REF][START_REF] Bregant | Erratum to "Measurement of the Cotton-Mouton effect in krypton and xenon at 1064 nm with the PVLAS apparatus[END_REF] At 632.8 nm the agreement of our CC3 value with the experimental result of Hüttner 37 is just outside 3σ. At 514.5 nm our computed values fall well outside 3σ of the estimate of Carusotto et al. [START_REF] Carusotto | Measurement of the magnetic birefringence of noble gases[END_REF] This apparently confirms that the error associated to this measured value might be underestimated.

VII. CONCLUSION

We have carried out a thorough analysis of the Faraday (circular) and Cotton Mouton (linear) birefringences of xenon, at a wavelength of 1064 nm. The study involves both an experimental segment, exploiting the capabilities of a state-of-the-art optical setup, and a computational element, where sophisticated wavefunction structure and optical response models (and with an estimate of the effect of relativity) were employed.

Our experimental estimate for the normalized Verdet constant of xenon at a temperature of 273.15 K and λ=1064 nm, V N = (3.56±0.10) ×10 -3 atm -1 rad T -1 m -1 , is very well reproduced by our theoretical approach, which yields a value (V N = 3.52 ×10 -3 atm -1 rad T -1 m -1 using the CC3 approximation) within 1σ of the measured datum.

With respect to the Cotton Mouton effect, at T =273.15 K and λ=1064 nm, experiment yields a normalized constant k N CM = (2.59±0.40) ×10 -14 atm -1 T -2 , whereas we compute (again with our most sophisticated model, CC3) a value of k N CM = 2.78 ×10 -14 atm -1 T -2 , therefore within 1σ of experiment.

FIG. 1 .

 1 FIG. 1. Experimental setup. EOM = electro-optic modulator; AOM = acousto-optic modulator; PDH = Pound-Drever-Hall; Ph = photodiode; P = polarizer; A = analyzer. See text for more details.

  + 1)(2n + 2)S(-2n -4) + B(2n)];

  FIG. 2.△: Experimental values of xenon normalized Verdet constant at T = 273.15 K reported by Ingersoll and Liebenberg[START_REF] Ingersoll | Faraday Effect in Gases and Vapors. II[END_REF] for wavelength from 363 nm to 987.5 nm. These values are fitted by the law A/λ 2 + B/λ 4 (solid line). •: Our experimental value at T = 273.15 K.

FIG. 3 .

 3 FIG. 3. Normalized Verdet constant of xenon at T = 273.15 K at λ = 1064 m. Solid line: our experimental mean value. Dashed lines: our experimental value with 1σ uncertainty. Points : theoretical predictions (both ours and from the literature). See text and Tab. VII for the references.

FIG. 4 .

 4 FIG. 4. Reported values of Cotton-Mouton constant of xenon for λ ranging from 514.5 nm to 1064 nm and with 1σ uncertainty. Experimental values: black triangle: Carusotto et al, 36 open triangle: Hüttner (private communication by Bishop et al.), 37 black diamond: Bregant et al. 38,39 , open diamond: this work. Theoretical predictions: dashed line: SCF method for λ = ∞ by Bishop, 40 open circle: this work, CCSD, black circle: this work, CC3

TABLE I .

 I Parameters and their respective relative A-and B-type uncertainties at 1σ that have to be measured to infer the value of the Verdet constant V . Typical values are given at P = 5 × 10 -3 atm.

	Parameter Typical	Relative	Relative
		A-type	B-type
	value	uncertainty	uncertainty

TABLE II .

 II Parameters and their respective relative A-and B-type uncertainties at 1σ that have to be measured to infer the value of the normalized Verdet constant V n . The uncertainty given by the linear fit takes into account the A-type uncertainty of V .

	Parameter Typical	Relative	Relative
		A-type	B-type
	value	uncertainty	uncertainty

TABLE III .

 III Parameters that have to be measured to infer the value of the Cotton-Mouton constant kCM and their respective relative A-and B-type uncertainties at 1σ. Typical values are given at P = 8 × 10 -3 atm.

	Parameter Typical	Relative	Relative
			A-type	B-type
		value	uncertainty	uncertainty
	τ [ms]	1.14	2.0 × 10 -2
	α × 10 5 [T -2 ] 2.82	2.8 × 10 -4	2.2 × 10 -2
	∆ FSR [MHz] 65.996		3 × 10 -4
	LB [m]	0.137		2.2 × 10 -2
	λ [nm]	1064.0		< 5 × 10 -4
	sin 2θP	1.0000		9 × 10 -4
	kCM	2.31	2.0 × 10 -2	3.1 × 10 -2
	×10 16 [T -2 ]

TABLE IV .

 IV Parameters and their respective relative A-and B-type uncertainties at 1σ that have to be measured to infer the value of the normalized Cotton-Mouton constant k n CM .

	Parameter Typical	Relative	Relative
			A-type	B-type
		value	uncertainty	uncertainty
	kCM	2.31	2.0 × 10 -2	3.1 × 10 -2
	×10 16 [T -2 ]
	P × 10 3	5		2 × 10 -3
	[atm]
	linear fit	2.41	1.5 × 10 -1
	×10 14

TABLE V .

 V Dispersion coefficients of the Verdet and Cotton-Mouton response functions at the CCSD level of theory (atomic units).

	n	B(2n)	S(-2n -4)	S(-2n -2)	V (2n)	∆η(2n)
				aug cc pvqz pp
	0	-654.89471	126.50595			100.47070
	1	-8903.3825	763.59899	126.50595	253.01190	-64.951345
	2	-92298.251	5369.0486	763.59899	3054.3960	-17193.302
	3	-860869.9	41692.560	5369.0486	32214.292	-368478.36
	4			41692.560	333540.48
				d-aug cc pvqz pp
	0	-739.15630	126.97174			121.30323
	1	-9822.9127	774.87190	126.97174	253.94348	131.11247
	2	-106369.19	5553.2321	774.87190	3099.4876	-15056.943
	3	-1074975.1	44095.369	5553.2321	33319.393	-348591.39
	4			44095.369	352762.95
				t-aug cc pvqz pp
	0	-748.34187	126.91927			123.62583
	1	-9940.7218	774.47234	126.91927	253.83854	161.76343
	2	-107513.63	5551.3771	774.47234	3097.8894	-14756.921
	3	-1084127.2	44088.280	5551.3771	33308.263	-346204.12
	4			44088.280	352706.24

TABLE VI .

 VI CC3 values of the response function components (in atomic units) involved in the Verdet and Cotton-Mouton birefringences. The Verdet constant V N (ω) is given in atm -1 rad. T -1 m -1 and the Cotton-Mouton constant k N CM is in T -2 .atm -1 at 273.15 K.

	λ[nm]	µx; µy, Lz ω,0	V N (ω) × 10 3	µx; µx, Θxx ω,0	µx; µx, Lz, Lz ω,ω,0	∆η	k N CM × 10 14
				aug cc pvqz pp
	1064	11.1587	3.505	-668.242	272.564	98.9195	2.239
	632.8	19.5823	10.35	-700.706	308.069	98.1593	2.222
	514.5	24.9438	16.22	-728.260	339.617	97.1607	2.200
				d-aug cc pvqz pp
	1064	11.2155	3.522	-755.936	274.099	120.459	2.727
	632.8	19.6927	10.40	-791.994	310.285	120.427	2.726
	514.5	25.0963	16.32	-822.705	342.514	120.048	2.718
				t-aug cc pvqz pp
	1064	11.2127	3.521	-765.680	274.031	122.912	2.782
	632.8	19.6878	10.40	-802.186	310.210	122.994	2.784
	514.5	25.0901	16.31	-833.274	342.435	122.710	2.778
	VI. RESULTS AND DISCUSSION
	A. Faraday effect
	1. Experiments

TABLE VII .

 VII Experimental and theoretical values of the normalized Verdet constant at T = 273.15 K, λ = 1064 nm, with uncertainties at 1σ.

	Ref.	V N × 10 3	Remarks
		(atm -1 rad.
		T -1 m -1 )
	Experiment
	Ingersoll et al 32	3.46 ± 0.04	Interpolated with
			A/λ 2 + 2B/λ 4 .
			Scaled to water.
	This work	3.56 ± 0.10
	Theory
	Savukov 35	3.86 ± 0.01	Interpolated in this work
			with A/λ 2 + B/λ 4 .
	Ekström et al 6	3.35	TDHF
	Ekström et al 6	3.46	TDDHF
	Ikäläinen et al 34	3.34	NR
	Ikäläinen et al 34	3.48	X2C
	Ikäläinen et al 34	3.46	DHF
	Ikäläinen et al 34	3.52	NR-CCSD
	This work	3.49	CCSD/t-aug cc pvqz pp
	This work	3.52	CC3/t-aug cc pvqz pp

  Table S5 of the Supporting Information file of Ref. 34).

TABLE VIII .

 VIII Experimental (uncertainties of 1σ) and theoretical values of the Cotton-Mouton constant of xenon at T = 273.15 K.

	Ref.		λ (nm)	k N CM × 10 14
				(T -2 .atm -1 )
	Experiment
	Carusotto et al 36		514.5	(2.29 ± 0.10)
	Hüttner 37		632.8	(2.41 ± 0.12)
	Bregant et al 38,39		1064	(3.02 ± 0.27)
	This work		1064	(2.59 ± 0.40)
	Theory
	Bishop et al 40		∞	2.665
	This work,		514.5	2.803
	CCSD/t-aug cc pvqz pp
	This work,		632.8	2.808
	CCSD/t-aug cc pvqz pp
	This work,		1064	2.804
	CCSD/t-aug cc pvqz pp
	This	work,	514.5	2.778
	CC3/t-aug cc pvqz pp
	This	work,	632.8	2.784
	CC3/t-aug cc pvqz pp
	This	work,	1064	2.782
	CC3/t-aug cc pvqz pp
	B. Cotton Mouton Effect
	1. Experiments

ACKNOWLEDGMENTS

We thank all the members of the BMV collaboration, and in particular J. Béard, J. Billette, P. Frings, B. Griffe, J. Mauchain, M. Nardone, J.-P. Nicolin and G. Rikken for strong support. We are also indebted to the whole technical staff of LNCMI. Sonia Coriani acknowledges useful discussions with Lucas Visscher and Trond Saue. We acknowledge the support of the Fondation pour la recherche IXCORE and the Agence National de la Recherche (Grant No. ANR-14-CE32-0006).