Zhenyu Liao
email: zhenyu.liao@berkeley.edu

Romain Couillet
email: romain.couillet@gipsa-lab.grenoble-inp.fr

Michael W Mahoney
email: mmahoney@stat.berkeley.edu

A random matrix analysis of random Fourier features: beyond the Gaussian kernel, a precise phase transition, and the corresponding double descent

This article characterizes the exact asymptotics of random Fourier feature (RFF) regression, in the realistic setting where the number of data samples n, their dimension p, and the dimension of feature space N are all large and comparable. In this regime, the random RFF Gram matrix no longer converges to the wellknown limiting Gaussian kernel matrix (as it does when N → ∞ alone), but it still has a tractable behavior that is captured by our analysis. This analysis also provides accurate estimates of training and test regression errors for large n, p, N . Based on these estimates, a precise characterization of two qualitatively different phases of learning, including the phase transition between them, is provided; and the corresponding double descent test error curve is derived from this phase transition behavior. These results do not depend on strong assumptions on the data distribution, and they perfectly match empirical results on real-world data sets.

Introduction

For a machine learning system having N parameters, trained on a data set of size n, asymptotic analysis as used in classical statistical learning theory typically either focuses on the (statistical) population n → ∞ limit, for N fixed, or the over-parameterized N → ∞ limit, for a given n. These two settings are technically more convenient to work with, yet less practical, as they essentially assume that one of the two dimensions is negligibly small compared to the other, and this is rarely the case in practice. Indeed, with a factor of 2 or 10 more data, one typically works with a more complex model. This has been highlighted perhaps most prominently in recent work on neural network models, in which the model complexity and data size increase together. For this reason, the double asymptotic regime where n, N → ∞, with N/n → c, a constant, is a particularly interesting (and likely more realistic) limit, despite being technically more challenging [START_REF] Hyunjune | Statistical mechanics of learning from examples[END_REF][START_REF] Timothy Lh Watkin | The statistical mechanics of learning a rule[END_REF][START_REF] Haussler | Rigorous learning curve bounds from statistical mechanics[END_REF][START_REF] Engel | Statistical mechanics of learning[END_REF][START_REF] Mezard | Information, physics, and computation[END_REF][START_REF] Charles | Rethinking generalization requires revisiting old ideas: statistical mechanics approaches and complex learning behavior[END_REF][START_REF] Bahri | Statistical mechanics of deep learning[END_REF]. In particular, working in this regime allows for a finer quantitative assessment of machine learning systems, as 34th Conference on Neural Information Processing Systems (NeurIPS 2020), Vancouver, Canada. a function of their relative complexity N/n, as well as for a precise description of the under-to over-parameterized "phase transition" (that does not appear in the N → ∞ alone analysis). This transition is largely hidden in the usual style of statistical learning theory [START_REF] Vapnik | Statistical Learning Theory[END_REF], but it is well-known in the statistical mechanics approach to learning theory [START_REF] Hyunjune | Statistical mechanics of learning from examples[END_REF][START_REF] Timothy Lh Watkin | The statistical mechanics of learning a rule[END_REF][START_REF] Haussler | Rigorous learning curve bounds from statistical mechanics[END_REF][START_REF] Engel | Statistical mechanics of learning[END_REF], and empirical signatures of it have received attention recently under the name "double descent" phenomena [START_REF] Madhu S Advani | High-dimensional dynamics of generalization error in neural networks[END_REF][START_REF] Belkin | Reconciling modern machinelearning practice and the classical bias-variance trade-off[END_REF]. This article considers the asymptotics of random Fourier features [START_REF] Rahimi | Random features for large-scale kernel machines[END_REF], and more generally random feature maps, which may be viewed also as a single-hidden-layer neural network model, in this limit. More precisely, let X = [x 1 , . . . , x n ] ∈ R p×n denote the data matrix of size n with data vectors x i ∈ R p as column vectors. The random feature matrix Σ X of X is generated by pre-multiplying some random matrix W ∈ R N ×p having i.i.d. entries and then passing through some entry-wise nonlinear function σ(•), i.e., Σ X ≡ σ(WX) ∈ R N ×n . Commonly used random feature techniques such as random Fourier features (RFFs) [START_REF] Rahimi | Random features for large-scale kernel machines[END_REF] and homogeneous kernel maps [START_REF] Vedaldi | Efficient additive kernels via explicit feature maps[END_REF], however, rarely involve a single nonlinearity. The popular RFF maps are built with cosine and sine nonlinearities, so that Σ X ∈ R 2N ×n is obtained by cascading the random features of both, i.e., Σ T X ≡ [cos(WX) T , sin(WX) T ]. Note that, by combining both nonlinearities, RFFs generated from W ∈ R N ×p are of dimension 2N .

The large N asymptotics of random feature maps is closely related to their limiting kernel matrices K X . In the case of RFF, it was shown in [START_REF] Rahimi | Random features for large-scale kernel machines[END_REF] that entry-wise the Gram matrix Σ T X Σ X /N converges to the Gaussian kernel matrix K X ≡ {exp(-

x i -x j 2 /2)} n i,j=1 , as N → ∞. This follows from 1 N [Σ T X Σ X ] ij = 1 N N t=1 cos(x T i w t ) cos(w T t x j ) + sin(x T i w t ) sin(w T t x j )
, for w t independent Gaussian random vectors, so that by the strong law of large numbers, for fixed n, p, [Σ T X Σ X /N ] ij goes to its expectation (with respect to w ∼ N (0,

I p )) almost surely as N → ∞, i.e., [Σ T X Σ X /N ] ij a.s. --→ E w cos(x T i w) cos(w T x j ) + sin(x T i w) sin(w T x j ) ≡ K cos + K sin , (1) with
K cos + K sin ≡ e -1 2 ( xi 2 + xj 2 ) cosh(x T i x j ) + sinh(x T i x j ) = e -1 2 ( xi-xj 2 ) ≡ [K X ] ij . (2)
While this result holds in the N → ∞ limit, recent advances in random matrix theory [START_REF] Louart | A random matrix approach to neural networks[END_REF][START_REF] Liao | On the spectrum of random features maps of high dimensional data[END_REF] suggest that, in the more practical setting where N is not much larger than n, p and n, p, N → ∞ at the same pace, the situation is more subtle. In particular, the above entry-wise convergence remains valid, but the convergence Σ T X Σ X /N -K X → 0 no longer holds in spectral norm, due to the factor n, now large, in the norm inequality

A ∞ ≤ A ≤ n A ∞ for A ∈ R n×n and A ∞ ≡ max ij |A ij |.
This implies that, in the large n, p, N regime, the assessment of the behavior of Σ T X Σ X /N via K X may result in a spectral norm error that blows up. As a consequence, for various machine learning algorithms [START_REF] Cortes | On the impact of kernel approximation on learning accuracy[END_REF], the performance guarantee offered by the limiting Gaussian kernel is less likely to agree with empirical observations in real-world large-scale problems, when n, p are large. 1

Our Main Contributions

We consider the RFF model in the more realistic large n, p, N limit. While, in this setting, the RFF empirical Gram matrix does not converge to the Gaussian kernel matrix, we can characterize its behavior as n, p, N → ∞ and provide asymptotic performance guarantees for RFF on largescale problems. We also identify a phase transition as a function of the ratio N/n, including the corresponding double descent phenomenon. In more detail, our contributions are the following.

1.We provide a precise characterization of the asymptotics of the RFF empirical Gram matrix, in the large n, p, N limit (Theorem 1). This is accomplished by constructing a deterministic equivalent for the resolvent of the RFF Gram matrix. Based on this, the behavior of the RFF model is (asymptotically) accessible through a fixed-point equation, that can be interpreted in terms of an angle-like correction induced by the non-trivial large n, p, N limit (relative to the N → ∞ alone limit).

2.We derive the asymptotic training and test mean squared errors (MSEs) of RFF ridge regression, as a function of the ratio N/n, regularization penalty λ, training as well as test sets (Theorem 2 and 3, respectively). We identify precisely the under-to over-parameterization phase transition, as a function of the relative model complexity N/n; we prove the existence of a "singular" peak of test error at the N/n = 1/2 boundary; and we characterize the corresponding double descent behavior. Importantly, our results are valid with almost no specific assumption on the data distribution. This is a significant improvement over existing double descent analyses, which fundamentally rely on the knowledge of the data distribution (often assumed to be multivariate Gaussian for simplicity) [START_REF] Hastie | Surprises in highdimensional ridgeless least squares interpolation[END_REF][START_REF] Mei | The generalization error of random features regression: Precise asymptotics and double descent curve[END_REF].

3.We provide a detailed empirical evaluation of our theoretical results, demonstrating that the theory closely matches empirical results on a range of real-world data sets (Section 3 and Section F in the supplementary material). This includes the correction due to the large n, p, N setting, sharp transitions (as a function of N/n) in angle-like quantities, and the corresponding double descent test curves. This also includes an evaluation of the impact of training-test similarity and the effect of different data sets, thus confirming, as stated in 2., that (unlike in prior work) the phase transition and double descent hold with almost no specific assumption on the data distribution.

Related Work

Here, we provide a brief review of related previous efforts.

Random features and limiting kernels. In most RFF work [START_REF] Rahimi | Weighted sums of random kitchen sinks: Replacing minimization with randomization in learning[END_REF][START_REF] Bach | On the equivalence between kernel quadrature rules and random feature expansions[END_REF][START_REF] Haim Avron | Random fourier features for kernel ridge regression: Approximation bounds and statistical guarantees[END_REF][START_REF] Rudi | Generalization properties of learning with random features[END_REF], non-asymptotic bounds are given, on the number of random features N needed for a predefined approximation error of a given kernel matrix with fixed n, p. A more recent line of work [START_REF] Allen-Zhu | A convergence theory for deep learning via over-parameterization[END_REF][START_REF] Du | Gradient descent finds global minima of deep neural networks[END_REF][START_REF] Jacot | Neural tangent kernel: Convergence and generalization in neural networks[END_REF][START_REF] Chizat | On lazy training in differentiable programming[END_REF] has focused on the over-parameterized N → ∞ limit of large neural networks by studying the corresponding neural tangent kernels. Here, we position ourselves in the more practical regime where n, p, N are all large and comparable, and provide asymptotic performance guarantees that better fit large-scale problems.

Random matrix theory. From a random matrix theory perspective, nonlinear Gram matrices of the type Σ T X Σ X have recently received an unprecedented research interests, due to their close connection to neural networks [START_REF] Pennington | Nonlinear random matrix theory for deep learning[END_REF][START_REF] Pennington | Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice[END_REF][START_REF] Benigni | Eigenvalue distribution of nonlinear models of random matrices[END_REF][START_REF] Pastur | On random matrices arising in deep neural networks[END_REF], with a particular focus on the associated eigenvalue distribution.

Here we propose a deterministic equivalent [START_REF] Couillet | Random matrix methods for wireless communications[END_REF][START_REF] Hachem | Deterministic equivalents for certain functionals of large random matrices[END_REF] analysis for the resolvent matrix that provides access, not only to the eigenvalue distribution, but also to the regression error of central interest in this article. While most existing deterministic equivalent analyses are performed on linear models, here we focus on the nonlinear RFF model. From a technical perspective, the most relevant work is [START_REF] Louart | A random matrix approach to neural networks[END_REF][START_REF] Mei | The generalization error of random features regression: Precise asymptotics and double descent curve[END_REF]. We improve their results by considering generic data model on the popular RFF model.

Statistical mechanics of learning.

A long history of connections between statistical mechanics and machine learning models (such as neural networks) exists, including a range of techniques to establish generalization bounds [START_REF] Hyunjune | Statistical mechanics of learning from examples[END_REF][START_REF] Timothy Lh Watkin | The statistical mechanics of learning a rule[END_REF][START_REF] Haussler | Rigorous learning curve bounds from statistical mechanics[END_REF][START_REF] Engel | Statistical mechanics of learning[END_REF], and recently there has been renewed interest [START_REF] Charles | Rethinking generalization requires revisiting old ideas: statistical mechanics approaches and complex learning behavior[END_REF][START_REF] Charles | Traditional and heavy tailed self regularization in neural network models[END_REF][START_REF] Charles | Statistical mechanics methods for discovering knowledge from modern production quality neural networks[END_REF][START_REF] Charles | Heavy-tailed Universality predicts trends in test accuracies for very large pre-trained deep neural networks[END_REF][START_REF] Bahri | Statistical mechanics of deep learning[END_REF]. Their relevance to our results lies in the use of the thermodynamic limit (akin to the large n, p, N limit), rather than the classical limits more commonly used in statistical learning theory, where uniform convergence bounds and related techniques can be applied.

Double descent in large-scale learning systems. The large n, N asymptotics of statistical models has received considerable research interests in the machine learning community [START_REF] Pennington | The emergence of spectral universality in deep networks[END_REF][START_REF] Hastie | Surprises in highdimensional ridgeless least squares interpolation[END_REF], resulting in a (somehow) counterintuitive phenomenon referred to as the "double descent." Instead of focusing on different "phases of learning" [START_REF] Hyunjune | Statistical mechanics of learning from examples[END_REF][START_REF] Timothy Lh Watkin | The statistical mechanics of learning a rule[END_REF][START_REF] Haussler | Rigorous learning curve bounds from statistical mechanics[END_REF][START_REF] Engel | Statistical mechanics of learning[END_REF][START_REF] Charles | Rethinking generalization requires revisiting old ideas: statistical mechanics approaches and complex learning behavior[END_REF], the "double descent" phenomenon focuses on an empirical manifestation of the phase boundary and refers to the empirical observations of the test error curve as a function of the model complexity, which differs from the usual textbook description of the bias-variance tradeoff [START_REF] Madhu S Advani | High-dimensional dynamics of generalization error in neural networks[END_REF][START_REF] Liao | The dynamics of learning: A random matrix approach[END_REF][START_REF] Belkin | Reconciling modern machinelearning practice and the classical bias-variance trade-off[END_REF][START_REF] Friedman | The elements of statistical learning[END_REF]. Theoretical investigation into this phenomenon mainly focuses on various regression models [START_REF] Dobriban | High-dimensional asymptotics of prediction: Ridge regression and classification[END_REF][START_REF] Peter L Bartlett | Benign overfitting in linear regression[END_REF][START_REF] Deng | A model of double descent for high-dimensional binary linear classification[END_REF][START_REF] Liang | Just interpolate: Kernel "ridgeless" regression can generalize[END_REF][START_REF] Hastie | Surprises in highdimensional ridgeless least squares interpolation[END_REF][START_REF] Mei | The generalization error of random features regression: Precise asymptotics and double descent curve[END_REF]. In most cases, quite specific (and rather strong) assumptions are imposed on the input data distribution. In this respect, our work extends the analysis in [START_REF] Mei | The generalization error of random features regression: Precise asymptotics and double descent curve[END_REF] to handle the RFF model and its phase structure on real-world data sets.

Notations and Organization of the Paper

Throughout this article, we follow the convention of denoting scalars by lowercase, vectors by lowercase boldface, and matrices by uppercase boldface letters. In addition, the notation (•) T denotes the transpose operator; the norm • is the Euclidean norm for vectors and the spectral or operator norm for matrices; and a.s.

--→ stands for almost sure convergence of random variables.

Our main results on the asymptotic training and test MSEs of RFF ridge regression are presented in Section 2, with proofs deferred to the Appendix. In Section 3, we provide detailed empirical evaluations of our main results, as well as discussions on the corresponding phase transition behavior and the double descent test curve. Concluding remarks are placed in Section 4. For more detailed discussions and empirical evaluations, we refer the readers to an extended version of this article [START_REF] Liao | A random matrix analysis of random fourier features: beyond the gaussian kernel, a precise phase transition, and the corresponding double descent[END_REF].

Main Technical Results

In this section, we present our main theoretical results. To investigate the large n, p, N asymptotics of the RFF model, we shall technically position ourselves under the following assumption. Assumption 1. As n → ∞, we have

1. 0 < lim inf n min{ p n , N n } ≤ lim sup n max{ p n , N n } < ∞;
or, practically speaking, the ratios p/n and N/n are only moderately large or moderately small. 2. lim sup n X < ∞ and lim sup n y ∞ < ∞, i.e., they are normalized with respect to n.

Under Assumption 1, we consider the RFF regression model. For training data X ∈ R p×n of size n, the associated random Fourier features, Σ X ∈ R 2N ×n , are obtained by computing WX ∈ R N ×n , for standard Gaussian random matrix W ∈ R N ×p , and then applying entry-wise cosine and sine nonlinearities on WX, i.e., Σ T X = [cos(WX) T , sin(WX) T ] with W ij ∼ N (0, 1). Given this setup, the RFF ridge regressor β ∈ R 2N is given by, for λ ≥ 0,

β ≡ 1 n Σ X 1 n Σ T X Σ X + λI n -1 y • 1 2N >n + 1 n Σ X Σ T X + λI 2N -1 1 n Σ X y • 1 2N <n . (3)
The two forms of β in (3) are equivalent for any λ > 0 and minimize the (ridge-regularized) squared loss 1 n y -Σ T X β 2 + λ β 2 on the training set (X, y). Our objective is to characterize the large n, p, N asymptotics of both the training MSE, E train , and the test MSE, E test , defined as

E train = 1 n y -Σ T X β 2 , E test = 1 n ŷ -Σ T Xβ 2 , (4)
with Σ T X ≡ [cos(W X) T , sin(W X) T ] ∈ R n×2N on a test set ( X, ŷ) of size n.

Asymptotic Deterministic Equivalent

To start, we observe that the training MSE, E train , in (4), can be written as

E train = λ 2 n Q(λ)y 2 = -λ 2
n y T ∂Q(λ)y/∂λ, which depends on the quadratic form y T Q(λ)y of

Q(λ) ≡ 1 n Σ T X Σ X + λI n -1 ∈ R n×n , (5)
the so-called resolvent of 1 n Σ T X Σ X (also denoted Q when there is no ambiguity) with λ > 0. In order to assess the asymptotic training MSE, it thus suffices to find a deterministic equivalent for Q(λ) (i.e., a deterministic matrix that captures the asymptotic behavior of the latter). One possibility is the expectation

E W [Q(λ)]. Informally, if the training MSE E train (that is random due to random W)
is "close to" some deterministic quantity Ētrain , in the large n, p, N limit, then Ētrain must have the same limit as

E W [E train ] = -λ 2 n ∂y T E W [Q(λ)]y/∂λ for n, p, N → ∞.
However, E W [Q] involves integration (with no closed-form due to the matrix inverse), and it is not a convenient quantity with which to work. Our objective is to find an asymptotic "alternative" for

E W [Q] that is (i) close to E W [Q]
in the large n, p, N → ∞ limit and (ii) numerically more accessible.

In the following theorem (proved in Appendix B), we introduce an asymptotic equivalent for

E W [Q].
Instead of being directly related to the Gaussian kernel K X = K cos + K sin as suggested by [START_REF] Allen-Zhu | A convergence theory for deep learning via over-parameterization[END_REF] in the large-N -only limit, it depends on the two components K cos , K sin in a more involved manner. Theorem 1 (Asymptotic equivalent for E W [Q]). Under Assumption 1, for Q defined in (5) and λ > 0, we have, as

n → ∞ E W [Q] -Q → 0 for Q ≡ N n ( Kcos 1+δcos + Ksin 1+δsin ) + λI n -1 , K cos ≡ K cos (X, X), K sin ≡ K sin (X, X) ∈ R n×n and K cos (X, X ) ij = e - x i 2 + x j 2 2 cosh(x T i x j ), K sin (X, X ) ij = e - x i 2 + x j 2 2 sinh(x T i x j ), (6)
where (δ cos , δ sin ) is the unique positive solution to

δ cos = 1 n tr(K cos Q), δ sin = 1 n tr(K sin Q). (7)
Remark 1 (Correction to large-N behavior). Taking N/n → ∞, one has δ cos → 0, δ sin → 0 so that

Kcos 1+δcos + Ksin 1+δsin → K cos + K sin = K X and Q n N K -1
X , for λ > 0, in accordance with the classical large-N -only prediction. In this sense, the pair (δ cos , δ sin ) introduced in Theorem 1 accounts for the "correction" due to the non-trivial n/N , as opposed to the N → ∞ alone analysis. Also, when the number of features N is large (i.e., as N/n → ∞), the regularization effect of λ flattens out and Q behaves like (a scaled version of) the inverse Gaussian kernel matrix K -1 X (that is well-defined if x 1 . . . , x n are all distinct, see [START_REF] Scholkopf | Learning with kernels: support vector machines, regularization, optimization, and beyond[END_REF]Theorem 2.18]).

Remark 2 (Geometric interpretation). Since Q shares the same eigenspace with Kcos 1+δcos + Ksin 1+δsin , one can geometrically interpret (δ cos , δ sin ) as a sort of "angle" between the eigenspaces of K cos , K sin and that of Kcos 1+δcos + Ksin 1+δsin . For fixed n, as N → ∞, one has

1 N N t=1 cos(X T w t ) cos(w T t X) → K cos , 1 N N t=1
sin(X T w t ) sin(w T t X) → K sin , the eigenspaces of which are "orthogonal" to each other, so that δ cos , δ sin → 0. On the other hand, as N, n → ∞, the eigenspaces of K cos and K sin "intersect" with each other, captured by the non-trivial (δ cos , δ sin ).

Asymptotic Training Performance

Theorem 1 provides an asymptotically more tractable approximation of E W [Q]. Together with some additional concentration arguments (e.g., from [START_REF] Louart | A random matrix approach to neural networks[END_REF]Theorem 2]), this permits us to provide a complete description of the limiting behavior of the random bilinear form a T Qb, for a, b ∈ R n of bounded Euclidean norms, in such a way that a T Qba T Qb a.s.

--→ 0, as n, p, N → ∞. This, together with the fact that E train = λ 2 n y T Q(λ) 2 y = -λ 2 n y T ∂Q(λ)y/∂λ, leads to the following result on the asymptotic training error, the proof of which is given in Appendix C. --→ 0, Ētrain =

λ 2 n Qy 2 + N n λ 2 n 2 tr( QKcos Q) (1+δcos) 2 tr( QKsin Q) (1+δsin) 2 Ω y T QK cos Qy y T QK sin Qy
for Q defined in Theorem 1 and

Ω -1 ≡ I 2 - N n 1 n tr( QKcos QKcos) (1+δcos) 2 1 n tr( QKcos QKsin) (1+δsin) 2 1 n tr( QKcos QKsin) (1+δcos) 2 1 n tr( QKsin QKsin) (1+δsin) 2 . ( 8
)
One can show that (i) for a given n and λ > 0, Ētrain decreases as the model size N increases; and

(ii) for a given ratio N/n, Ētrain increases as the regularization penalty λ grows large, as expected.

Asymptotic Test Performance

Theorem 2 holds without any restriction on the training set, (X, y), except for Assumption 1, since only the randomness of W is involved, and thus one can simply treat (X, y) as known in this result. This is no longer the case for the test error. Intuitively, the test data X cannot be chosen arbitrarily, and one must ensure that the test data "behave" statistically like the training data, in a "well-controlled" manner, so that the test MSE is asymptotically deterministic and bounded as n, n, p, N → ∞. Following this intuition, we work under the following assumption.

Assumption 2 (Data as concentrated random vectors [START_REF] Louart | Concentration of measure and large random matrices with an application to sample covariance matrices[END_REF]). The training data x i ∈ R p , i ∈ {1, . . . , n}, are independently drawn (non-necessarily uniformly) from one of K > 0 distribution classes2 µ 1 , . . . , µ K . There exist constants C, η, q > 0 such that for any x i ∼ µ k , k ∈ {1, . . . , K} and any 1-Lipschitz function f : R p → R, we have

P (|f (x i ) -E[f (x i )]| ≥ t) ≤ Ce -(t/η) q , t ≥ 0. (9)
The test data xi ∼ µ k , i ∈ {1, . . . , n} are mutually independent, but may depend on training data X and

E[σ(WX) -σ(W X)] = O( √ n) for σ ∈ {cos, sin}.
To facilitate the discussion of the phase transition and the double descent, we do not assume independence between training data and test data (but we do assume independence between different columns within X and X). In this respect, Assumption 2 is weaker than the classical i.i.d. assumption, and it permits us to illustrate the impact of training-test similarity on the model performance (Section 3.3).

A first example of concentrated random vectors satisfying ( 9) is the random Gaussian vector N (0, I p ) [START_REF] Ledoux | The concentration of measure phenomenon[END_REF]. Moreover, since the concentration property in ( 9) is stable over Lipschitz transformations [START_REF] Louart | Concentration of measure and large random matrices with an application to sample covariance matrices[END_REF], it holds, for any 1-Lipschitz mapping g : R d → R p and z ∼ N (0, I d ), that g(z) also satisfies [START_REF] Chizat | On lazy training in differentiable programming[END_REF]. In this respect, Assumption 2, although seemingly quite restrictive, represents a large family of "generative models", including notably the "fake images" generated by modern generative adversarial networks (GANs) that are, by construction, Lipschitz transformations of large random Gaussian vectors [START_REF] Goodfellow | Generative adversarial nets[END_REF][START_REF] Seddik | Random matrix theory proves that deep learning representations of GAN-data behave as gaussian mixtures[END_REF]. As such, from a practical consideration, Assumption 2 provides a more realistic and flexible statistical model for real-world data.

With Assumption 2, we have the following result on the asymptotic test error, proved in Section D. Theorem 3 (Asymptotic test performance). Under Assumptions 1 and 2, we have, for test MSE E test defined in (4) and test data ( X, ŷ)

satisfying lim sup n X < ∞, lim sup n ŷ ∞ < ∞ with n/n ∈ (0, ∞) that, as n → ∞ E test -Ētest a.s. --→ 0, Ētest = 1 n ŷ - N n Φ Qy 2 + N 2 n 2 n Θcos (1+δcos) 2 Θsin (1+δsin) 2 Ω y T QK cos Qy y T QK sin Qy
for Ω defined in [START_REF] Benigni | Eigenvalue distribution of nonlinear models of random matrices[END_REF],

Θ σ = 1 N tr K σ ( X, X) + N n 1 n tr Q ΦT Φ QK σ - 2 n tr Q ΦT K σ ( X, X), σ ∈ {cos, sin}, (10)
and Φ ≡ Kcos 1+δcos + Ksin 1+δsin , Φ ≡ Kcos( X,X) 1+δcos + Ksin( X,X) 1+δsin , with K cos ( X, X), K sin ( X, X) ∈ R n×n and K cos ( X, X), K sin ( X, X) ∈ R n×n defined as in [START_REF] Peter L Bartlett | Benign overfitting in linear regression[END_REF].

Taking ( X, ŷ) = (X, y), one gets Ētest = Ētrain , as expected. From this perspective, Theorem 3 can be seen as an extension of Theorem 2, with the "interaction" between training and test data (i.e., training-versus-test K σ ( X, X) and test-versus-test K σ ( X, X) interaction matrices) summarized in the scalar parameter Θ σ defined in [START_REF] Cortes | On the impact of kernel approximation on learning accuracy[END_REF], for σ ∈ {cos, sin}.

Empirical Evaluations and Practical Implications

In this section, we provide a detailed empirical evaluation, including a discussion of the behavior of the fixed-point equation in Theorem 1, and its consequences in Theorem 2 and Theorem 3. In particular, we describe the behavior of the pair (δ cos , δ sin ) that characterizes the necessary correction in the large n, p, N regime, as a function of the regularization λ and the ratio N/n. This explains: (i) the mismatch between empirical regression errors from the Gaussian kernel prediction (Figure 1); and (ii) the behavior of (δ cos , δ sin ) as a function of N/n, which clearly indicates two phases of learning (Figure 3) and the corresponding double descent test error curves (Figure 4).

Correction due to the Large n, p, N Regime

The RFF Gram matrix Σ T X Σ X /N is not close to the classical Gaussian kernel matrix K X in the large n, p, N regime; and, as a consequence, its resolvent Q, as well the training and test MSE, E train and E test (that are functions of Q), behave quite differently from the Gaussian kernel predictions. As already discussed in Remark 1 after Theorem 1, for λ > 0, the pair (δ cos , δ sin ) characterizes the correction when considering n, p, N all large, compared to the large-N -only asymptotic behavior:

δ cos = 1 n tr K cos Q, δ sin = 1 n tr K sin Q, Q = N n K cos 1 + δ cos + K sin 1 + δ sin + λI n -1 . (11)
To start, Figure 1 compares the training MSEs of RFF ridge regression to the predictions from Gaussian kernel regression and to the predictions from our Theorem 2, on the popular MNIST data set [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF]. Observe that there is a huge gap between empirical training errors and the Gaussian kernel predictions, especially when N/n < 1, while our theory consistently fits empirical observations almost perfectly.

Next, from [START_REF] Couillet | Random matrix methods for wireless communications[END_REF] we know that both δ cos and δ sin are decreasing functions of λ. (See Lemma 7 in Appendix E for a proof of this fact.) Figure 2 shows that: (i) over a range of different N/n, both δ cos and δ sin decrease monotonically as λ increases; (ii) the behavior for N/n < 1, which is decreasing from an initial value of δ 1, is very different from the behavior for N/n 1, where an initially flat region is observed for small values of λ and we have δ < 1 for all values of λ; and (iii) the impact of regularization λ becomes less significant as the ratio N/n becomes large. This is in accordance with the limiting behavior of

Q n N K -1
X in Remark 1 that is independent of λ as N/n → ∞. Note also that, while δ cos and δ sin can be geometrically interpreted as a sort of weighted "angle" between different kernel matrices (as in Remark 2), and therefore one might expect to have δ ∈ [0, 1], this is not the case for the leftmost plot of Figure 1 with N/n = 1/4. There, for small values of λ (say λ 0.1), both δ cos and δ sin scale like λ -1 , while they are observed to saturate to a fixed O(1) value for N/n = 1, 4, 16. This corresponds to two different phases of learning in the ridgeless λ → 0 limit, as discussed in the following section.

Phase Transition and Corresponding Double Descent

Both δ cos and δ sin in [START_REF] Couillet | Random matrix methods for wireless communications[END_REF] are decreasing functions of N , as depicted in Figure 3. (See Lemma 6 in Appendix E for a proof.) More importantly, Figure 3 also illustrates that δ cos and δ sin exhibit qualitatively different behavior, depending on the ratio N/n. For λ not too small (λ = 1 or 10), both δ cos and δ sin decrease smoothly, as N/n grows large. However, for λ relatively small (λ = 10 -3 and 10 -7 ), we observe a "phase transition" on two sides of the interpolation threshold 2N = n. (Note that the scale of the y-axis is different in different subfigures.) More precisely, in the leftmost plot with λ = 10 -7 , δ cos and δ sin "jump" from order O(1) (when 2N > n) to much higher values of the same order of λ -1 (when 2N < n). A similar behavior is also observed for λ = 10 -3 .

This phase transition can be theoretically justified by considering the ridgeless λ → 0 limit in Theorem 1. First note that, for λ = 0 and 2N < n, the (random) resolvent Q(λ = 0) in ( 5) is simply undefined, as it involves inverting a singular matrix Σ T X Σ X ∈ R n×n that is of rank at most 2N < n. As a consequence, we expect to see both Q and Q scale like λ -1 as λ → 0 for 2N < n, while for 2N > n this is no longer the case. As a consequence, we have the following two phases:

1.Under-parameterized with 2N < n. Here, Q is not well-defined (indeed Q scales like λ -1 ) and one must consider instead the properly scaled λδ cos , λδ sin and λ Q as λ → 0.

2.Over-parameterized with 2N > n, where one can take λ → 0 in [START_REF] Couillet | Random matrix methods for wireless communications[END_REF] to get δ cos , δ sin and Q. Remark 3 (Double descent test error curves). On account of the above two phases, it is not surprising to observe a "singular" behavior at 2N = n , when no regularization is applied. Here, we consider the (asymptotic) test MSE in Theorem 3 in the ridgeless λ → 0 limit and focus on the situation where the test data X is sufficiently different from the training data X (see more discussions on this point in Section 3.3 below). Then, the two-by-two matrix Ω defined in [START_REF] Benigni | Eigenvalue distribution of nonlinear models of random matrices[END_REF] diverges to infinity at 2N = n as λ → 0. (Indeed, the determinant det(Ω -1 ) scales as λ, per Lemma 5 in Appendix E.) As a consequence, we have Ētest → ∞ as N/n → 1/2, resulting in a sharp deterioration in the test performance around the interpolation threshold 2N = n. It is also interesting to note that, while Ω also appears in Ētrain , we still obtain (asymptotically) zero training MSE at 2N = n, despite the divergence of Ω as λ → 0, essentially due to the prefactor λ 2 in Ētrain .

Figure 4 depicts the empirical and theoretical test MSEs with different λ. In particular, for λ = 10 -7 and λ = 10 -3 , a double-descent-type behavior is observed, with a singularity at 2N = n, while for larger values of λ (λ = 1 and 10), a smoother and monotonically decreasing test error curve is observed, as a function of N/n, in accordance with the observations in [START_REF] Mei | The generalization error of random features regression: Precise asymptotics and double descent curve[END_REF] on Gaussian data.

Remark 4 (Double descent as a consequence of phase transition). While the double descent phenomenon has received considerable attention recently, our analysis makes it clear that in this model (and presumably many others) it is a natural consequence of the phase transition between two qualitatively different phases of learning [START_REF] Charles | Rethinking generalization requires revisiting old ideas: statistical mechanics approaches and complex learning behavior[END_REF].

Impact of Training-test Similarity

We see that the (asymptotic) test error behaves entirely differently, depending on whether the test data X is "close to" the training data X or not. For X = X, one has Ētest = Ētrain that decreases monotonically as N grows large; while for X sufficiently different from X (in the associated kernel space in the sense that K σ (X, X) is sufficiently different from K σ ( X, X) for σ ∈ {cos, sin}), Ētest diverges at 2N = n and establishes a double descent behavior. To have a more quantitative assessment of the impact of training-test similarity on the RFF model performance, we consider here the special case ŷ = y. Since in the ridgeless λ → 0 limit, Ω scales as λ -1 at 2N = n (Remark 3), one must then have Θ σ ∝ λ to "compensate" so that Ētest does not diverge at 2N = n as λ → 0. A first example is the case where the test data is a small perturbation of the training data. In Figure 5, the test data are generated by adding Gaussian white noise of variance σ 2 to the training data, i.e., xi = x i + σε i [START_REF] Deng | A model of double descent for high-dimensional binary linear classification[END_REF] for independent ε i ∼ N (0, I p /p). In Figure 5, we observe that (i) below the threshold σ 2 = λ, the test error coincides with the training error and both are relatively small for 2N = n; and (ii) as soon as σ 2 > λ, the test error diverges from the training error and grows large (but linearly in σ 2 ) as the noise level increases. Note also from the two rightmost plots of Figure 5 that the training-to-test "transition" at σ 2 λ is sharp only for relatively small values of λ, as predicted by our theory.

Conclusion

We have established a precise description of the resolvent of RFF Gram matrices, and provided asymptotic training and test performance guarantees for RFF ridge regression, in the n, p, N → ∞ limit. We have also discussed the under-and over-parameterized regimes, where the resolvent behaves dramatically differently. These observations involve only mild regularity assumptions on the data, yielding phase transition behavior and double descent test error curves for RFF regression that closely match experiments on real-world data. Extended to a (technically more involved) multi-layer setting in the more realistic large n, p, N regime as in [START_REF] Fan | Spectra of the conjugate kernel and neural tangent kernel for linear-width neural networks[END_REF], our analysis may shed new light on the theoretical understanding of modern deep neural nets, beyond the large-N alone neural tangent kernel limit.

Broader Impact

In this article, we provide theoretical assessment of the popular random Fourier features (RFFs), in the practical setting where n, p, N are all large and comparable. Asymptotic performance guarantees are provided for RFF ridge regression in this n, p, N → ∞ limit, as an important positive impact of this work on the development of more reliable large-scale machine learning systems. The theoretical framework developed in this article presents fair and non-offensive societal consequence.

Theorem 2 (

 2 Asymptotic training performance). Under Assumption 1, for a given training set (X, y) and training MSE, E train defined in (4), as n → ∞ E train -Ētrain a.s.

N/n = 2 λFigure 1 :N/n = 16 λFigure 2 :

 21162 Figure 1: Training MSEs of RFF ridge regression on MNIST data (class 3 versus 7), as a function of regression penalty λ, for p = 784, n = 1 000, N = 250, 500, 1 000, 2 000. Empirical results displayed in blue circles; Gaussian kernel predictions (assuming N → ∞ alone) in black dashed lines; and Theorems 2 in red solid lines. Results obtained by averaging over 30 runs.

Figure 3 :

 3 Figure 3: Behavior of (δ cos , δ sin ) on MNIST data (class 3 versus 7), as a function of N/n, p = 784, n = 1 000, λ = 10 -7 , 10 -3 , 1, 10. The black dashed line is the interpolation threshold 2N = n.

Figure 4 :

 4 Figure 4: Empirical (blue crosses) and theoretical (red dashed lines) test errors of RFF regression as a function of the ratio N/n, on MNIST data set (class 3 versus 7), for p = 784, n = 500, λ = 10 -7 , 10 -3 , 1, 10. The black dashed line is the interpolation threshold 2N = n.

10 -10 10 -7 10 - 4 10 - 5 10 - 3 10 - 1 λ = 10 - 7 Noise variance σ 2 MSE 10 - 5 10 -3 10 - 1 10 - 2 10 - 1 10 0 λ = 10 - 3 Noise variance σ 2 10 -5 10 - 10 Noise variance σ 2

 1045311072105121103210102

Figure 5 :

 5 Figure 5: Empirical training (red crosses) and test (blue circles) errors of RFF ridge regression on MNIST data (class 3 versus 7), as a function of the noise level σ 2 , for N = 512, p = 784, n = n = 1 024 = 2N , λ = 10 -7 , 10 -3 , 1, 10. Results obtained by averaging over 30 runs.

For readers not familiar with the impact of spectral norm error in learning, or with the random matrix theory techniques that we will use in our analysis, such as resolvent analysis and the use of deterministic equivalents, see Appendix A for a warm-up discussion.

K ≥ 2 is included to cover multi-class classification problems; and K should remain fixed as n, p → ∞.

Acknowledgments. We would like to acknowledge the UC Berkeley CLTC, ARO, IARPA, NSF, and ONR for providing partial support of this work. Our conclusions do not necessarily reflect the position or the policy of our sponsors, and no official endorsement should be inferred. Couillet's work is partially supported by MIAI at University Grenoble-Alpes (ANR-19-P3IA-0003).


P Thérond

A Boutron

E Lebigot
email: elise.lebigot@aphp.fr

M Hully

L Amazit

P Gaignard

T Michel

M Rio

M Lombès

M P Golinelli-Cohen

Expanding the phenotype of mitochondrial disease: Novel pathogenic variant in ISCA1 leading to instability of the iron-sulfur cluster in the protein

Keywords: MMDS ISCA1 Iron-sulfur cluster PDH Lipoic acid PAGE, Blue-native polyacrylamide gel electrophoresis, CSF, Cerebro-spinal fluid, ISC, Iron-sulfur cluster, α-KGDHc, Alpha-ketoglutarate dehydrogenase, MMDS, Multiple mitochondrial dysfunction syndrome, NGS, Next-generation sequencing, OCR, Oxygen consumption rate, PBS, Phosphate-buffered saline, PDHc, Pyruvate dehydrogenase complex, RC, Respiratory chain, WT, Wild-type

published or not. The documents may come

Expanding the phenotype of mitochondrial disease: Novel pathogenic variant in ISCA1 leading to instability of the iron-sulfur cluster in the protein E Lebigot, M Hully, L Amazit, P Gaignard, T Michel, M Rio, Marc Lombès,

Introduction

In mammals, the mitochondrial machinery for iron-sulfur cluster (ISC) biogenesis requires at least 18 proteins [START_REF] Braymer | Iron-sulfur cluster biogenesis and trafficking in mitochondria[END_REF]. Three major steps are necessary for the formation of the ISC and its delivery to recipient proteins. The first step is the de novo assembly of a [2Fe-2S] cluster on the scaffold protein ISCU using inorganic iron and sulfur ions. Then, the cluster is transferred to a first temporary recipient protein (GLRX5) and secondarily to mitochondrial Fe-S-assembling proteins [START_REF] Beilschmidt | Mammalian Fe-S cluster biogenesis and its implication in disease[END_REF]. Maturation of [2Fe-2S]-containing proteins can directly proceed from GLRX5, while maturation of the [4Fe-4S]containing proteins requires supplementary steps with the biosynthesis of [4Fe-4S] cluster and its delivery to mitochondrial [4Fe-4S] proteins. Five proteins, ISCA1, ISCA2, BOLA3, NFU1 and IBA57, are involved in these last steps but the precise role of each of these proteins is not yet completely understood. Several in vitro studies suggested that ISCA1 and ISCA2 are essential for the biosynthesis of [4Fe-4S] clusters from [2Fe-2S]-containing GLRX5 [START_REF] Brancaccio | Formation of [4Fe-4S] clusters in the mitochondrial iron-sulfur cluster assembly machinery[END_REF][START_REF] Brancaccio | 4Fe-4S] cluster assembly in mitochondria and its impairment by copper[END_REF]. NFU1 and BOLA3 seem to be dedicated to [4Fe-4S] cluster delivery to client apoproteins including LIAS and SDHB [START_REF] Cameron | Mutations in iron-sulfur cluster scaffold genes NFU1 and BOLA3 cause a fatal deficiency of multiple respiratory chain and 2-oxoacid dehydrogenase enzymes[END_REF][START_REF] Melber | Role of Nfu1 and Bol3 in iron-sulfur cluster transfer to mitochondrial clients[END_REF]. IBA57 could also take part in [4Fe-4S] cluster biogenesis or [4Fe-4S] cluster transfer to recipient apoproteins [START_REF] Sheftel | The human mitochondrial ISCA1, ISCA2, and IBA57 proteins are required for [4Fe-4S] protein maturation[END_REF][START_REF] Ajit Bolar | Mutation of the iron-sulfur cluster assembly gene IBA57 causes severe myopathy and encephalopathy[END_REF][START_REF] Gourdoupis | IBA57 recruits ISCA2 to form a [2Fe-2S] cluster-mediated complex[END_REF]. Mutations in genes encoding for NFU1, BOLA3, IBA57, ISCA2 and ISCA1 lead to multiple mitochondrial dysfunction syndrome (MMDS) types 1 to 5, respectively. In 2017, the first four patients from two Indian families with MMDS type 5 were described and associated to the homozygous p.(Glu87Lys) pathogenic variant in ISCA1 [START_REF] Shukla | Homozygous p. (Glu87Lys) variant in ISCA1 is associated with a multiple mitochondrial dysfunctions syndrome[END_REF]. Then, a similar fifth case was reported [START_REF] Shukla | Report of the third family with multiple mitochondrial dysfunctions syndrome 5 caused by the founder variant p.(Glu87Lys) in ISCA1[END_REF]. Finally, an Italian patient carrying a novel homozygous pathogenic variant p.(Val10Gly) in ISCA1 was described with extended biochemical investigation [START_REF] Torraco | ISCA1 mutation in a patient with infantile-onset leukodystrophy causes defects in mitochondrial [4Fe-4S] proteins[END_REF].

Herein, we report a young boy from Egyptian origin carrying a novel pathogenic variant in ISCA1 leading to the yet undescribed p. (Tyr101Cys) mutation in ISCA1. A tetrapyramidal spasticity episode and skills regression were noted at 20 months of age associated with cavitating leukodystrophy on brain imaging. At 4 years of age, he recovered walking with easy fatigability but language was severely delayed. Investigation in fibroblasts from the patient showed that lipoic acid synthesis was impaired and activities of complex I and II of respiratory chain were also decreased. Thus, ISCA1 deficiency leads to dysfunction of mitochondrial [4Fe-4S]-containing proteins either involved in lipoic acid synthesis (LIAS) or essential for complexes I and II activities. After bacterial overexpression, purification of human ISCA1 showed that the p.(Tyr101Cys) mutation leads to an increased instability of its Fe-S cluster. This report shows similarities with well documented other MMDS types 1 to 4 but reveals that clinical feature of MMDS type 5 is wide ranging from early onset and fatal encephalopathy to more moderate symptomatology with predominant spasticity and neurodevelopmental delay. Thereafter, patients with pathogenic variant in ISCA1, ISCA2 or IBA57 will be respectively named ISCA1, ISCA2 or IBA57 patients to facilitate the reading of the manuscript.

Materials and methods

Cell culture

Individuals without mitochondrial disease were referred as control subjects. Cultured skin fibroblasts were grown in Ham's F-10 medium (Eurobio, Les Ulis, France) supplemented with antibiotics (100 IU/mL penicillin and 100 μg/mL streptomycin), 50 mg/L uridine and 12% fetal bovine serum (Life Technologies, Carlsbad, CA, USA) for fibroblasts and 20% fetal bovine serum for myoblasts. Cells were cultured at 37 °C in a humidified incubator with 5% CO 2 . Cultured myoblasts from the ISCA1 patient were obtained from Biobank of Cochin Hospital (Paris). Clinical, biochemical and brain imaging data were collected retrospectively. Informed consent prior to skin biopsy was obtained from parents of patients included in this study.

Biochemical studies

PDHc activity in fibroblasts was carried out as described [START_REF] Imbard | Molecular characterization of 82 patients with pyruvate dehydrogenase complex deficiency. Structural implications of novel amino acid substitutions in E1 protein[END_REF]. Polarographic or spectrophotometric assays of mitochondrial respiratory chain (RC) complexes, α-KGDHc and citrate synthase activities were measured according to standard procedures [START_REF] Rustin | Biochemical and molecular investigations in respiratory chain deficiencies[END_REF]. Complex I activity assay was performed on mitochondrial enriched fractions prepared from permeabilized cells with digitonin as previously described [START_REF] Chretien | Assay of mitochondrial respiratory chain complex I in human lymphocytes and cultured skin fibroblasts[END_REF]. RC complex activity was normalized to citrate synthase activity.

Immunoblot and BN-PAGE analysis

Western blot analysis was performed on mitochondria-enriched fractions or on total cell lysates. About 30 μg of protein extracts were separated by SDS-PAGE and transferred to PVDF membranes, blocked in PBS-Tween containing 5% non-fat milk and then incubated overnight with primary antibodies. Following antibodies directed against lipoic acid (LA) (Abcam, ab58724), ISCA1 (ThermoFisher, PA5-60121), SDHB (Abcam, ab14714), Rieske protein (Abcam, ab14746), ferrochelatase (kindly provided by Dr. H. Puccio, Strasbourg, France), mt-aconitase (kindly provided by Dr. R. B. Franklin, University of Maryland, Baltimore, USA), GRIM19 (Abcam, ab10240), GAPDH (Sigma-Aldrich, G9545) and prohibitin (Novusbio, NBP2-37563) were used. Secondary peroxidase-conjugated anti-mouse (Sigma-Aldrich, A4416) and antirabbit (Sigma-Aldrich, NA934V) antibodies were used. Detection was achieved using the ECL chemiluminescence kit (GE Healthcare) and G-Box (Syngene) associated with GeneSNAP software. Blue-native PAGE (BN-PAGE) analysis was carried out on mitochondria-enriched fractions from fibroblasts [START_REF] Hescot | Mitotane alters mitochondrial respiratory chain activity by inducing cytochrome c oxidase defect in human adrenocortical cells[END_REF]. Immunoblotting was performed using antibodies raised against the complex I subunit GRIM19 (Abcam, ab10240), the 70 kDa subunit of complex II (SDHA, Abcam, ab14715), the complex III subunit core2 (UQCRC2, Abcam, ab14745) and the MTCO1 subunit of complex IV (Abcam, ab110270).

Molecular analysis

Genomic DNA was extracted from fibroblasts by standard methods on automated nucleic acid extraction system QIAsymphony (Qiagen) and molecular analysis was performed using next-generation sequencing (NGS). The entire coding exons ( ± 50 bp) of a panel of 117 genes related to mitochondrial diseases, including pyruvate crossing abnormalities and mitochondrial cytopathies were analyzed. This panel was designed using the Agilent Sure Design web application (Agilent). Preparation of genomic DNA for NGS analysis and bioinformatics workflow were performed as previously described [START_REF] Bouali | New MCM8 mutation associated with premature ovarian insufficiency and chromosomal instability in a highly consanguineous Tunisian family[END_REF]. High Throughput Sequencing was performed on an Illumina MiSeq instrument. Sanger sequencing was performed after PCR amplification with specific primers to validate the identified ISCA1 mutation and to analyze parental DNAs. The integrated software ALAMUT V.2.7 (Interactive Biosoftware; http:// www.interactive-biosoftware.com including Align GVGD, SIFT and MutationTaster) and Polyphen2 software were used for in silico analysis of DNA variant.

Mitochondrial network study

Skin fibroblasts were seeded at subconfluence on a glass coverslip and incubated for 40 min in darkness at 37 °C with 0.1 µM MitoTracker Red CMXRos (Invitrogen, M7512) in Ham's F-10 medium. Cells were rinsed with PBS and then fixed in 4% paraformaldehyde in PBS. Lastly, cells were incubated for 1 min with 1 µg/mL Hoechst (Invitrogen, H3570) in PBS. Fluorescent cells were observed with an Olympus Provis AX70 (Rungis, France), and images were acquired with Qcapture Pro version 5.1 (Q Imaging Inc., Surrey, British Columbia, Canada) using an Evolution VF Monochrome camera (Media Cybernetics Inc., Bethesda, MD).

Expression and purification of mature ISCA1

The mature form of human ISCA1 (NCBI Q9BUE6, residues 24-129) and the p.Tyr101Cys mutated form were expressed in E. coli using the pET28a vector (GenScript). The N-term His-tagged proteins were expressed using Rosetta(DE3)pLys cells in an 2xTY medium with 50 mg/L kanamycin at 37 °C. At OD 600 = 0.5, 25 μM FeCl 3 was added and the temperature was decreased to 18 °C. Forty minutes later, 0.2 mM IPTG was added to the medium. Cells were cultured at 18 °C for 20 h and then harvested by centrifugation for purification steps. Bacterial pellets were re-suspended in 50 mL of Lysis Buffer (50 mM Tris-HCl pH 8, 500 mM NaCl, 5 mM imidazole, 0.1% triton X100, 1 mM DTT) containing an antiprotease cocktail (P2714; Sigma-Aldrich). After sonication and ultracentrifugation (1h at70,000xg), the cleared lysate was applied to a 5 mL HiTrap chelating HP column (GE Healthcare) on an Äkta FPLC (GE Healthcare) pre-equilibrated with Lysis Buffer. Proteins were eluted using a 0-60% linear gradient between Lysis buffer and Elution Buffer (50 mM Tris-HCl pH 8, 500 mM NaCl, 500 mM imidazole). Fractions containing ISCA1 were pooled and concentrated using an Amicon Ultra-15 centrifugal filter unit (Millipore) with a 5 kDa cutoff. Protein purity was assessed to be > 99% using SDS-PAGE. Protein concentrations were measured using Bradford assay with bovine serum albumin as standard [START_REF] Bradford | A rapid and sensitive method for the quantitation of microgram quantities of protein utilizing the principle of protein-dye binding[END_REF]. Protein aliquots were frozen in liquid nitrogen. UV-visible absorption spectra were recorded between 240 and 900 nm with a Cary 100 (Agilent) spectrophotometer.

Iron quantification

Iron content in purified protein was quantified using a bathophenanthroline colorimetric assay adapted from [START_REF] Fish | Rapid colorimetric micromethod for the quantitation of complexed iron in biological samples[END_REF]. For total iron quantification of a 38.6 µL protein sample, 2.9 µL of perchloric acid 54% was added. After 30 min at room temperature, 20.6 µL of bathophenanthroline sulfonate 1.7 mg/mL, 5.1 µL of sodium ascorbate 76 mg/mL and 2.9 µL of saturated ammonium acetate solution were added and the reaction was completed with water up to 100 µL. After 30 min at room temperature, the absorbance at 535 nm was measured. Mohr's salt solution was used as standard.

Results

Case report

The boy, sixth child from consanguineous parents native from Egypt, was born at 31 weeks of pregnancy by caesarean delivery due to retro-placental hematoma. He was able to walk at 18 months of age and to pronounce some syllables at about 20 months. At this age, an episode of motor deficiency of the left arm occurred. Neurological examination showed tetrapyramidal syndrome with spasticity of the limbs, and no neuropathy. After this episode, regression of skills was noted with loss of independent walking and language. At 30 months of age, he partially recovered and was able to walk few steps with help. The left arm also partially recovered but remained weak. At 4 years and a half, he presented with spastic diplegia enabling him walking despite frequent falls and fatigability. He was then able to say isolated few words. The brain MRI (bMRI) performed at 20 months of age showed leukodystrophy without calcifications on cerebral scan. At 2 years old, electroneuromyogram showed no peripheral neuropathy. BMRI revealed cavitating leukodystrophy with diffuse lesions of periventricular supra tentorial white matter with hyperintensity on T2 ponderation images and with hypointensity on T1 ponderation sequences, sparing U fibers. Apparent diffusion coefficient was decreased. Vacuolization was found in supratentorial white matter, not in the corpus callosum. A mild ventriculomegaly and a thin corpus callosum were observed. No cortical malformation was detected. 4 years of age, bMRI remained otherwise stable (Fig. 1, panels A, B, C and D) and brain spectroscopy detected a lactate peak (Fig. 1, panel E). Lactate was high in blood (3.2 mmol/L; reference values (RV): < 2.0 mmol/L) with elevated pyruvate (0.23 mmol/L; RV: < 0.15 mmol/L) but lactate/pyruvate ratio was in the normal range (13.9; RV: 12.0-18.0). All others biological investigations were unremarkable including extensive metabolic screening for lysosomal and peroxysomal diseases as well as for interferonopathies. Pyruvate dehydrogenase (PDH) deficiency and mitochondrial cytopathy were also suspected to cause the clinical feature of this patient.

Biochemical analyses

Oxygen consumption rate (OCR), PDH activity and respiratory chain complexes activity assays were performed in primary cultured skin fibroblasts of the patient when he was 30 months old. Results are presented in Table 1.

These results showed a moderate decrease of PDHc activity and a marked decreased OCR with succinate, suggesting dysfunction of complex II. It was confirmed by a slightly decreased activity of complex II and decreased complex II/citrate synthase activities ratio without any observable effect of the other RC complex activities compared to reference values. These observations led us to analyze DNA of the patient by NGS analysis on our panel of nuclear genes dedicated to mitochondrial diseases.

Molecular results

Applying filters for variations with low allelic frequency (< 0.03 in 1000 genome DATABASE), the analysis of NGS data sets revealed a homozygous variation (c.302A > G; p.Tyr101Cys) located in exon 4 of ISCA1 (NM_030940.4) (Fig. 2,part A). Seventy other variants were identified including 3 variants predicted as deleterious in genes ACAD9, SDHD and COQ6 at the heterozygous state. Mutations in these genes are related to mitochondrial disease with recessive transmission and no other gene variation (intragenic deletion or insertion) was detected in these three genes. Thus, the homozygous ISCA1 mutation remained the only candidate for the observed phenotype and the biochemical data. We validated this variant by family analysis showing that each parent of the patient carried the ISCA1 mutation at the heterozygous state. This ISCA1 variation was never reported before (in 1000 genome, dbSNP and EXAC databases) and was predicted as a deleterious variant by in silico analysis using prediction softwares be sensitive to oxygen, previous anaerobic purification of mouse ISCA1 leads to protein precipitation [START_REF] Beilschmidt | ISCA1 is essential for mitochondrial Fe4S4 biogenesis in vivo[END_REF]. Then, we purified the overexpressed protein under aerobic conditions and obtained a colored WT protein. The UV-visible spectra of the as-purified ISCA1 is characterized by absorption bands at 320 and 420 nm and shoulders at 460 and 550 nm (Fig. 3, panel A). These results are compatible with the presence of a [2Fe-2S] cluster in the aerobically purified WT form of ISCA1. Then, we expressed the Tyr101Cys mutated form of human ISCA1 and purified it using similar conditions than those used for the WT form. We obtained a colorless protein. The absence of cluster was confirmed by the UV-visible absorption spectrum, which does not present any significant absorption in the visible part (Fig. 3, panel A). Chemical analysis confirmed that the as-purified Tyr101Cys mutated form contains only low amounts of iron compared to the WT form (Fig. 3, panel B). Thus, aerobic purifications of overexpressed mature ISCA1 led to a [2Fe-2S] cluster-containing WT form while Tyr101Cys form lacks its Fe-S cluster.

Western blot

To further confirm and better understand the molecular consequences of the ISCA1 Tyr101Cys mutation, we extended the biochemical investigation. We first looked at ISCA1 protein level in the patient fibroblasts compared to fibroblasts of patients with either deficiency in ISCA2 or in IBA57 (P12 and P10 respectively, described in our previously published study [START_REF] Lebigot | Impact of mutations within the [Fe-S] cluster or the lipoic acid biosynthesis pathways on mitochondrial protein expression profiles in fibroblasts from patients[END_REF]). Immunoblot analysis revealed a statistical lower amount of ISCA1 protein in ISCA1 patient fibroblasts compared to the control fibroblasts (Fig. 4B). ISCA1 protein amount in ISCA2 or IBA57 patient was similar to those of controls.

In addition, we performed some analysis to evaluate how ISCA1 deficiency could affect maturation of ISC protein in the patient fibroblasts. We first looked at the level of lipoylation of the E2 subunit of PDHc and of α-KGDHc to evaluate the enzymatic activity of LIAS which is dependent on the insertion of its [4Fe-4S] cluster [START_REF] Landgraf | Radical S-adenosylmethionine enzymes in human health and disease[END_REF] (Fig. 5A). Furthermore, we examined the levels of several mitochondrial [2Fe-2S]-and [4Fe-4S]-containing proteins (Fig. 5B andC). Finally, several other subunits of RC complexes were tested. We found an alteration of the lipoylation of PDH-E2 and a more severe decrease for α-KGDH-E2, likely secondary to a LIAS dysfunction. We also observed a marked decrease in SDHB expression in the patient fibroblasts. Other proteins of complex II and of complex I, which do not contain ISC (SDHA and GRIM19, respectively), and proteins carrying a [2Fe-2S] cluster (Rieske protein of complex III and ferrochelatase) were not affected by the ISCA1 deficiency.

RC and TCA functionality in fibroblasts and myoblasts

We further studied the consequences of ISCA1 deficiency on RC complexes and TCA enzymes activities in fibroblasts. As shown in Table 1, RC complexes and α-KGDHc activities were only weakly affected by ISCA1 deficiency, despite lipoylation defect (Fig. 5A) and decreased level of SDHB protein (Fig. 5C). We explored these enzymatic activities in myoblasts to evaluate if this cellular type would be more affected than fibroblasts (Fig. 6). Complex I activity was decreased in fibroblasts but unfortunately, limitation in the amount of cellular material did not allow us to measure complex I activity in myoblasts. Both results in myoblasts and in fibroblasts showed decreased complex II activity while others enzymatic activities were near values obtained in control cells.

BN-PAGE analysis was performed using fibroblasts to assess the RC complex assembly (Fig. 7). This experiment underlined the effects on the integrity of complexes I and II and revealed additional effects on complex IV.

Our investigations showed several mitochondrial damages in ISCA1 deficient fibroblasts including RC complexes and lipoic acid metabolism dysfunction. As mitochondrial network morphology can vary from hyperfused to fragmented morphology in accordance with OXPHOS activities or nutrient status [START_REF] Wai | Mitochondrial dynamics and metabolic regulation[END_REF], we questioned whether mitochondrial network morphology could be affected in fibroblasts of patient with ISCA1 deficiency. We observed ISCA1 patient cells after incubation with MitoTracker Red CMXRos® which stains functional mitochondria and found similar tubular mitochondrial network with homogenous cellular distribution in patient and in control fibroblasts (Fig. 8).

Discussion

Pathogenic variants in genes encoding for proteins involved in the late step of the mitochondrial ISC machinery lead to various types of MMDS pathology. MMDS type 5, caused by mutations in ISCA1, was recently described in 5 patients from three Indian families [START_REF] Shukla | Homozygous p. (Glu87Lys) variant in ISCA1 is associated with a multiple mitochondrial dysfunctions syndrome[END_REF][START_REF] Shukla | Report of the third family with multiple mitochondrial dysfunctions syndrome 5 caused by the founder variant p.(Glu87Lys) in ISCA1[END_REF] and one Italian patient [START_REF] Torraco | ISCA1 mutation in a patient with infantile-onset leukodystrophy causes defects in mitochondrial [4Fe-4S] proteins[END_REF]. MMDS type 5 leads to early-onset, fatal encephalopathy and/or severe spasticity and seizures [START_REF] Shukla | Homozygous p. (Glu87Lys) variant in ISCA1 is associated with a multiple mitochondrial dysfunctions syndrome[END_REF][START_REF] Shukla | Report of the third family with multiple mitochondrial dysfunctions syndrome 5 caused by the founder variant p.(Glu87Lys) in ISCA1[END_REF][START_REF] Torraco | ISCA1 mutation in a patient with infantile-onset leukodystrophy causes defects in mitochondrial [4Fe-4S] proteins[END_REF] and resembles MMDS types 3 and 4 due to pathogenic variants in IBA57 and ISCA2, respectively [START_REF] Ajit Bolar | Mutation of the iron-sulfur cluster assembly gene IBA57 causes severe myopathy and encephalopathy[END_REF][START_REF] Lebigot | Impact of mutations within the [Fe-S] cluster or the lipoic acid biosynthesis pathways on mitochondrial protein expression profiles in fibroblasts from patients[END_REF][START_REF] Lossos | Fe/S protein assembly gene IBA57 mutation causes hereditary spastic paraplegia[END_REF][START_REF] Debray | Mutation of the iron-sulfur cluster assembly gene IBA57 causes fatal infantile leukodystrophy[END_REF][START_REF] Al-Hassnan | ISCA2 mutation causes infantile neurodegenerative mitochondrial disorder[END_REF]. Extensive leukodystrophy on brain MRI and ophthalmic symptoms such as optic atrophy or nystagmus were described in most of the patients with MMDS type 3, 4 or 5. The present article describes a new patient with ISCA1 deficiency. This patient from Egyptian origin presented with a psychomotor regression with transient loss of gait at 20 months of age. He progressively recovered but, at 4 years of age, he remained impaired with spastic diplegia leading to recurrent falls while walking associated with fatigability and severe language deficiency. Thus, his clinical symptoms, despite less severe, were similar to the Italian patient ones. As for this latter and others patients with MMDS (de Souza et al., 2018), bMRI of the French patient revealed abnormalities. Cavitating leukodystrophy, ventriculomegaly and thin corpus callosum were observed similar to the described patients with MMDS type 5 especially the Italian one, but with less impairment. All the previously reported ISCA1 patients died between 11 months and 11 years of age. Currently, the patient that we described here is 6 years old and keep improving skills. MMDS 3 and 4 also led to variable clinical symptomatology and evolution. Thus, IBA57 or ISCA2 patients display variable clinical features closed to those observed in ISCA1 patients with progressive neurological regression and 4S]-containing proteins (mt-aconitase and SDHB) and proteins involved in complex I (GRIM19) or complex II (SDHA). Samples of patient were loaded twice. serves as loading control for samples obtained after mitochondrial enrichment and GAPDH serves as loading control for total cell lysate samples. For each antibody, all the samples were analyzed on the same gel; however, unnecessary lanes were removed and replaced by black lines. leukodystrophy or severe encephalopathy with early onset and sometimes early death [START_REF] Ajit Bolar | Mutation of the iron-sulfur cluster assembly gene IBA57 causes severe myopathy and encephalopathy[END_REF][START_REF] Lossos | Fe/S protein assembly gene IBA57 mutation causes hereditary spastic paraplegia[END_REF][START_REF] Toldo | Neonatal mitochondrial leukoencephalopathy with brain and spinal involvement and high lactate: expanding the phenotype of ISCA2 gene mutations[END_REF][START_REF] Debray | Mutation of the iron-sulfur cluster assembly gene IBA57 causes fatal infantile leukodystrophy[END_REF][START_REF] Al-Hassnan | ISCA2 mutation causes infantile neurodegenerative mitochondrial disorder[END_REF][START_REF] Ishiyama | IBA57 mutations abrogate iron-sulfur cluster assembly leading to cavitating leukoencephalopathy[END_REF][START_REF] Torraco | Novel mutations in IBA57 are associated with leukodystrophy and variable clinical phenotypes[END_REF][START_REF] Hamanaka | Expanding the phenotype of IBA57 mutations: related leukodystrophy can remain asymptomatic[END_REF][START_REF] Liu | Phenotypic spectrum of mutations in IBA57, a candidate gene for cavitating leukoencephalopathy[END_REF][START_REF] Alfadhel | Further delineation of the phenotypic spectrum of ISCA2 defect: a report of ten new cases[END_REF][START_REF] Alaimo | Loss-of-function mutations in ISCA2 disrupt 4Fe-4S cluster machinery and cause a fatal leukodystrophy with hyperglycinemia and mtDNA depletion[END_REF]. Among all reported patients, no optic atrophy was reported for ISCA1 patients while it was frequent in IBA57 or ISCA2 patients [START_REF] Al-Hassnan | ISCA2 mutation causes infantile neurodegenerative mitochondrial disorder[END_REF][START_REF] Hamanaka | Expanding the phenotype of IBA57 mutations: related leukodystrophy can remain asymptomatic[END_REF]. Only nystagmus and visual impairment were present in the Italian ISCA1 patient [START_REF] Torraco | ISCA1 mutation in a patient with infantile-onset leukodystrophy causes defects in mitochondrial [4Fe-4S] proteins[END_REF].

The novel ISCA1 homozygous variant (p.( Tyr101Cys)), the third to be described, that we identified, led to a reduction in the level of ISCA1 protein in fibroblasts. This observation suggests that mutated protein could be degraded due to abnormal protein folding even though we did not explore this hypothesis further. Overexpression and purification of WT ISCA1 and of the mutated form confirmed that mutation of Tyr101, a residue close to the Fe-S cluster, leads to cluster destabilization. Two other mutations were described previously. The p.(Val10Gly) mutation, involving a residue from the mitochondrial targeting peptide, led to decrease of the mitochondrial import of the protein [START_REF] Torraco | ISCA1 mutation in a patient with infantile-onset leukodystrophy causes defects in mitochondrial [4Fe-4S] proteins[END_REF] inducing a large decrease of ISCA1 protein level in fibroblasts. Concerning the other known mutation in ISCA1, p.(Gln87Lys), no functional analysis was performed to understand the mechanism of the pathogenicity of this mutation. In all the described patients, dysfunction of ISCA1 also causes abnormal respiratory chain and lipoylated protein functionalities. In fibroblasts and/or myoblasts of our studied patient, activities of complexes I and II were affected probably by defect of maturation of their Fe-S cluster containing subunits such as SDHB for complex II. BN-PAGE analysis showed an altered assembly of these two complexes but also of complex IV. Activities of PDH and α-KGDH complexes were moderately decreased in fibroblasts or myoblasts secondary to limited lipoylation defect. The effect was clearly less severe than for the Italian patient [START_REF] Torraco | ISCA1 mutation in a patient with infantile-onset leukodystrophy causes defects in mitochondrial [4Fe-4S] proteins[END_REF]. Surprisingly, deleterious biochemical consequences of the mutation were not more pronounced in myoblasts than in fibroblasts. As in several MMDS [START_REF] Ajit Bolar | Mutation of the iron-sulfur cluster assembly gene IBA57 causes severe myopathy and encephalopathy[END_REF][START_REF] Lebigot | Impact of mutations within the [Fe-S] cluster or the lipoic acid biosynthesis pathways on mitochondrial protein expression profiles in fibroblasts from patients[END_REF][START_REF] Ahting | FeS cluster biogenesis defect in a patient with mutations in ISCA2[END_REF][START_REF] Nishioka | An infant case of diffuse cerebrospinal lesions and cardiomyopathy caused by a BOLA3 mutation[END_REF], complex IV is also affected in MMDS type 5 either with decrease of total complex IV assembly (present report) or decrease of a complex IV subunit with unknown mechanism [START_REF] Torraco | ISCA1 mutation in a patient with infantile-onset leukodystrophy causes defects in mitochondrial [4Fe-4S] proteins[END_REF]. Importantly, all the results show that the ISCA1 Tyr101Cys mutant functionally impaired mitochondrial [4Fe-4S] proteins assembly and was causative of the observed clinical features. Despite dysfunctions of many RC complexes, we observed that the mitochondrial network did not show fragmented morphology in fibroblasts. The repartition of mitochondria is homogeneous in cytoplasm compared to results obtained in fibroblasts of one ISCA2 patient [START_REF] Alaimo | Loss-of-function mutations in ISCA2 disrupt 4Fe-4S cluster machinery and cause a fatal leukodystrophy with hyperglycinemia and mtDNA depletion[END_REF]. Two studies suggested that complete ISCA1 deficiency was unsustainable in mammals [START_REF] Beilschmidt | ISCA1 is essential for mitochondrial Fe4S4 biogenesis in vivo[END_REF][START_REF] Yang | Knockout of ISCA 1 causes early embryonic death in rats[END_REF] and would be lethal at the embryo state in rats [START_REF] Yang | Knockout of ISCA 1 causes early embryonic death in rats[END_REF]. Even if ISCA1 was severely decreased in the patient fibroblasts, we hypothesized that residual ISCA1 functionality must be present in tissues others than fibroblasts for the French ISCA1 patient who presented with moderate clinical phenotype.

Fibroblasts Myoblasts

Fig. 6. Activities of RC complexes and TCA enzymes (citrate synthase and alpha-KGDHc) in fibroblasts and myoblasts of the ISCA1 patient . Activities were compared to those of control cells (fibroblasts and myoblasts respectively) measured in same conditions, in total cell lysates or in mitochondrial enriched fractions (complex I only). Citrate synthase activity was used as an indication for mitochondrial content. Data represent mean ± SEM of 2 independent experiments except for complex I activity (only one experiment in fibroblasts and no analysis in myoblasts; ND: not done) Fig. 7. Blue-native PAGE analysis of respiratory chain complexes. BN-PAGE analysis of mitochondrial enriched fibroblasts of the ISCA1 patient and control was performed using antibodies against respiratory chain complex subunits without [Fe-S] cluster (complex I: GRIM19, complex II: SDHA, complex III: UQCRC2, complex IV: MTCO1). Results for both samples are issued from the same experiment that needed two different gels represented in two blocks (unnecessary lanes were removed and replaced by black lines).

ISCA1 is one of the proteins involved in the late step of maturation of ISC containing protein but its functionality inside this machinery is not clearly known as yet. Since 2004, mitochondrial localization of ISCA1 was proven and its role in Fe-S cluster biogenesis was proposed [START_REF] Cózar-Castellano | hIscA: a protein implicated in the biogenesis of ironsulfur clusters[END_REF]. Further studies in HeLa cells suggested that ISCA1 collaborates with ISCA2 and IBA57 in the late step of the maturation of mitochondrial Fe-S proteins [START_REF] Sheftel | The human mitochondrial ISCA1, ISCA2, and IBA57 proteins are required for [4Fe-4S] protein maturation[END_REF]. Recently, in vitro studies demonstrated that the biogenesis of [4Fe-4S] cluster from two [2Fe-2S] clusters required ISCA1 and ISCA2 proteins [START_REF] Brancaccio | Formation of [4Fe-4S] clusters in the mitochondrial iron-sulfur cluster assembly machinery[END_REF][START_REF] Brancaccio | 4Fe-4S] cluster assembly in mitochondria and its impairment by copper[END_REF]. Other analyses suggested that homodimeric ISCA1 could be sufficient to form a [4Fe-4S] cluster from two [2Fe-2S] clusters [START_REF] Beilschmidt | ISCA1 is essential for mitochondrial Fe4S4 biogenesis in vivo[END_REF]. Finally, IBA57 and ISCA2 may also collaborate with ISCA1 for [4Fe-4S] cluster biogenesis [START_REF] Gourdoupis | IBA57 recruits ISCA2 to form a [2Fe-2S] cluster-mediated complex[END_REF]. Interestingly, a study revealed that ISCA2 protein was decreased in fibroblasts of an ISCA1 patient [START_REF] Torraco | ISCA1 mutation in a patient with infantile-onset leukodystrophy causes defects in mitochondrial [4Fe-4S] proteins[END_REF] while ISCA1 protein level was not decreased in fibroblasts of an ISCA2 patient (this study). These results suggest that ISCA2 required ISCA1 to be stabilized but ISCA1 is not destabilized by low level of ISCA2 in the same tissue. ISCA1 and ISCA2 are able to interact with each other but are not redundant in their functions in ISC machinery. With recent in vitro studies and multiple descriptions of MMDS type 5, it appears that ISCA1 protein is an essential protein for maturation of ISC-containing protein. ISCA1 deficiency leads to biochemical and clinical consequences similar to those reported in IBA57 or ISCA2 deficiency, all involved in the late steps of mitochondrial [4Fe-4S] cluster biogenesis.

In conclusion, we report a patient with spastic diplegia and severe language delay who carries a novel pathogenic variant in ISCA1. This Tyr101Cys variant clearly destabilizes the Fe-S cluster of the ISCA1 protein and reduces stability and functionality of this latter. Mitochondrial [4Fe-4S]-containing proteins such as subunits of RC complexes or proteins involved in lipoic acid synthesis are therefore affected by ISCA1 dysfunction, leading to MMDS type 5.

  Fig. 1. Images of brain MRI and brain spectroscopy of the ISCA1 patient. Cavitating leukodystrophy with diffuse lesions of periventricular supratentorial white matter with hyperintensity on T2 (A) and flair (B) and with hypointensity on T1 (C). (D) Vacuolization was located in supratentorial white matter. (E) Lactate peak on brain spectroscopy is pointed out by a yellow arrow.

Fig. 2 .

 2 Fig. 2. Localization of the novel variant in ISCA1 and conservation of residue 101 in ISCA1 protein in different orthologues. (A) Sanger validation of the variant in exon 4 of ISCA1 in the patient and his parents. Other variants reported in ISCA1 are localized in exon 4 (10) and in exon 1 (12).(B) Alignment of ISCA1 protein from different species with CLUSTAL O(1.2.4) multiple sequence alignment from NCBI orthologues sequences. Residue Tyr (Y) number 101 in Homo sapiens, noted with red star, is conserved in all represented species except Danio i.e. zebrafish.

Fig. 3 .Fig. 4 .

 34 Fig.3. UV-visible absorption spectra and iron content analysis of purified WT and Tyr101Cys mutant. (A) UV-visible spectra of the WT (black line) and Tyr101Cys (red dotted line) forms of purified ISCA1 at 40 µM. (B) Iron contains of purified ISCA1 forms. Results were expressed as the ratio of the iron and the protein concentrations. Data represent mean ± SEM of 3 independent experiments. Statistical analysis: ttest (significance: **p < 0.01).

Fig. 5 .

 5 Fig. 5. Protein levels of mitochondrial Fe-S proteins in fibroblasts of patient with ISCA1 pathogenic variant. (A) Lipoylated proteins were detected using antibody against lipoic acid. (B) Levels of mitochondrial [2Fe-2S] proteins (Rieske and ferrochelatase). (C) Levels of [4Fe-4S]-containing proteins (mt-aconitase and SDHB) and proteins involved in complex I (GRIM19) or complex II (SDHA). Samples of patient were loaded twice. serves as loading control for samples obtained after mitochondrial enrichment and GAPDH serves as loading control for total cell lysate samples. For each antibody, all the samples were analyzed on the same gel; however, unnecessary lanes were removed and replaced by black lines.

Fig. 8 .

 8 Fig. 8. Mitochondrial network in ISCA1 patient fibroblasts. Mitochondria of patient fibroblasts were labelled with the Mitotracker probe (A) and compared to control fibroblasts (B). Hoechst was used to counterstain the nuclei as indicated in the figure (right panel). The inset shows a magnification of the selected mitochondrial subcellular region (white square). Scale bar represents 10 µm.

Table 1

 1 Oxygen consumption rate (OCR), α-KGDHc-, PDHc-and respiratory chain complex activity assays performed on primary cultured skin fibroblasts of the ISCA1 patient.Citrate synthase activity served as mitochondrial content control. Values in bold indicate results outside of the normal value range.

	Result	Reference values

E.Lebigot, et al. Mitochondrion 52 (2020) 75-82

Acknowledgements

We thank the AMMI association (Association contre les Maladies Mitochondriales) for its financial contribution to this work. We thank Allison CARDOSO and Yannick SANDRE for their technical help. We are grateful for the excellent expertise from Dr Abdel SLAMA during this work.