Houcem Gazzah
email: hgazzah@sharjah.ac.ae

Jean Pierre Delmas
email: jean-pierre.delmas@it-sudparis.eu

NONUNIFORM LINEAR ANTENNA ARRAYS FOR ENHANCED NEAR FIELD SOURCE LOCALIZATION

Keywords: Direction of arrival, range, near-field, antenna arrays, performance analysis, Cramer-Rao bound

Arrays of sensors freely located along an axis are considered in this paper that are used to localize a near-field emitting source. Using Taylor expansion and a suitable coordinate system, simple, yet rich to interpret, Cramer-Rao bounds relative to the direction and range parameters are derived. Our analysis allows in particular to unveil a family of non-uniform linear arrays with better near field estimation capabilities, compared to the well-established uniform linear arrays.

INTRODUCTION

Antenna arrays is a major field of statistical signal processing and direction finding is among the most investigated topics [START_REF] Krim | Two decades of array signal processing research[END_REF]. The complexity of the antenna array near-field propagation model explains the limited research and the scarcity of results, the existing literature being mostly concerned with antenna arrays far-field analysis and applications. A recent interest in parameter estimation of near-field sources has just emerged, focusing on the Cramer-Rao Bounds (CRB) of DOA and range estimated by means of the Uniform Linear Array (ULA) [START_REF] Begriche | Exact Cramer Rao bound for near field source localization[END_REF][START_REF] Begriche | Exact conditional and unconditional Cramer Rao bound for near field localization[END_REF] and the Uniform Circular Array (UCA) [START_REF] Delmas | CRB Analysis of nearfield source localization using uniform circular arrays[END_REF], respectively. Despite the importance and wide use of the ULA and the UCA, these results are of limited interest, especially if one thinks about the developing interest about irregular array geometries and methods to design advantageous (w.r.t. the ULA) non-uniform linear arrays [START_REF] Rubsamen | Sparse Array Design for Azimuthal Direction-of-Arrival Estimation[END_REF]. This paper addresses linear arrays with freely spaced sensors and shows that there is a room to outperform the ULA from the viewpoint of near-field source localization.

Our Taylor expansion of the CRB is based on the exact time delay formula, similarly as [START_REF] Begriche | Exact Cramer Rao bound for near field source localization[END_REF][START_REF] Begriche | Exact conditional and unconditional Cramer Rao bound for near field localization[END_REF], but uses a different coordinate system. It judiciously leads to more compact and more interpretable CRB expressions that apply to arbitrary linear arrays. For instance, we identify a class of centrosymmetric linear arrays with better near-field estimation capabilities. They include, but are not limited to, the ULA. In fact, within centro-symmetric linear arrays, we identify some geometric parameters that characterize linear (but nonuniform) arrays with better estimation than the ULA for nearfield sources, while being equivalent for far-field sources.

The paper is organized as follows. Sec. 2 presents the near-field signal model and coordinate system. Sec. 3 is dedicated to the derivation of simple and interpretable closed-form expressions of the CRB on DOA and range. Then, Sec. 4 is focused on the special class of centro-symmetric arrays and key-geometric parameters that determine their performance. This ultimately leads to non-uniform centro-symmetric arrays that achieve better accuracy than the ULA for localizing nearfield sources. A conclusion is given is Sec. 5.

SIGNAL MODEL

We consider a linear antenna array of P sensors C 1 , • • • , C P located along the x-axis, at coordinates x 1 , • • • , x P , respectively, as shown on Fig. 1. The array centroid is assumed to be at the origin O, which allows for a more compact expressions of the CRB, compared to [START_REF] Begriche | Exact Cramer Rao bound for near field source localization[END_REF][START_REF] El Korso | Conditional and unconditional Cramer Rao bounds for near-field Source localization[END_REF]. A source S emits a narrowband signal s(t) (with wavelength λ) in the direction of the antenna array. It is at a distance r from the origin O and forms an angle θ with [O, y). At time t, the source originates at sensor p the snapshot y p (t) = exp (iτ p ) s(t)+n p (t), where n p (t) represents the ambient additive noise, while τ p =2π(SO -SC p )/λ is also equal to

τ p = 2π r λ ( 1 - √ β p ) , with β p =1 -2 x p r sin(θ) + x 2 p r 2 . E y y y y y y ¨¨¨¨¨¨¨i Source θ r x C 1 C 2 O C p C P x p E
Fig. 1. Source in the near-field of the arbitrary linear array.

Based on N snapshots {y p (t)} p=1,...,P ;t=t1,...,tN , estimates of both range r and DOA θ are obtained using a variety of algorithms. Some are capable of achieving the CRB [START_REF] Gazzah | Spectral efficiency of beamforming-based parameter estimation in the single source case[END_REF]. Indeed, we adopt the stochastic CRB as our evaluation criterion, assuming the usual statistical properties about n p (t) and s(t): (i) n p (t) and s(t) are independent, (ii) {n p (t)} p=1,...,P ;t=t1,...,tN are independent, zeromean circular Gaussian distributed with variance σ2 n , (iii) {s(t)} t=t1,...,tN are assumed to be independent zero-mean circular Gaussian distributed with variance σ 2 s (the so-called unconditional or stochastic model).

TAYLOR EXPANSION OF THE CRB

To derive the stochastic CRB on θ and r, we use the general matrix expression of the stochastic CRB given for several parameters per source [START_REF] Abeida | Efficiency of subspacebased DOA estimators[END_REF] and adapted to a single source. This gives CRB(θ, r) = F -1 with

F = 2N σ 4 s σ 2 n (σ 2 n + P σ 2 s ) [ P D H D -D H aa H D ] , (1)
where a = [e iτ1 , ..., e iτP ] T is the steering vector and D the derivative

[ ∂a ∂θ , ∂a ∂r ]
. A long proof (proofs can be found in [START_REF] Gazzah | CRB based-design of linear antenna arrays for near-field source localization[END_REF]) leads to the following expression of the 2 × 2 matrix F:

2c r cos 3 (θ) [F] 1,2 = P S 3 r 3 + sin(θ) 3P S 4 -S 2 2 r 4 + o(ϵ 4 ) c r 2 cos 2 (θ) [F] 1,1 = P S 2 r 2 + 2P sin(θ) S 3 r 3 + S 4 P [ 4 sin 2 (θ) -1 ] -S 2 2 sin 2 (θ) r 4 + o(ϵ 4 ) c cos 4 (θ) [F] 2,2 = 1 4 P S 4 -S 2 2 r 4 + P S 5 -S 2 S 3 r 5 sin(θ) + P S 6 [ 23 sin 2 (θ)-3 ] -3S 2 S 4 [ 5 sin 2 (θ)-1 ] +8S 2 3 sin 2 (θ) 8r 6 +o(ϵ 6 )
where

S k = ∑ P p=1 x k p (with in particular, S 1 = 0) are purely geometric, c = λ 2 4π 2 σ 2 n (σ 2 n +P σ 2 s ) 2N σ 4 s , ϵ = 1 r max p |x p | and o(ϵ k ) sat- isfying lim ϵ→0 o(ϵ k )/ϵ k = 0. From [F] 1,2
, the DOA is decoupled from the range to the second-order in ϵ iff S 3 = 0. This class of arrays that will be studied in details in Sec. 4 will have many advantages. For the moment, we give results for arbitrary values of S 3 .

From the above, the following expressions of the CRB on the DOA and range are obtained after tedious derivations.

CRB(θ) = c P 1 S 2 - P S 2 3 P S4-S 2 2 1 + γ 1 sin(θ) r cos 2 (θ) + o(ϵ) (2) CRB(r) r 4 = 4c P S 4 -S 2 2 -P S 2 3 S2 1 + γ 2 sin(θ) r cos 4 (θ) + o(ϵ)(3)
where

γ 1 =4P S 3 S2S 2 3 -S 2 2 S4+P S 2 4 -P S3S5 (P S4-S 2 2 )(P S2S4-S 3 2 -P S 2 3 ) and γ 2 =2 3P S2S3S4+S 3 2 S3-P S 3 3 -2P S 2 2 S5 S2(P S2S4-S 3 2 -P S 2 3 )
. In these CRB expressions, terms in 1/r vanish if S 3 = 0 in (2) and if S 3 = S 5 = 0 in (3). This is not a marginal scenario as it concerns in particular the ULA. To cover these cases, we prove that, for arrays with S 3 = 0, we have

CRB(θ) = c 1 + [ 1 + ( 1 + 4P S4 P S4-S 2 2 ) sin 2 (θ) ] S4 S2 1 r 2 cos 2 (θ)P S 2 +o(ϵ 2 ).
(4) For those satisfying both S 3 = 0 and S 5 = 0, we have

CRB(r) r 4 = 4c cos 4 (θ) 1 + γ3(θ) 2(P S4-S 2 2 )r 2 P S 4 -S 2 2 + o(ϵ 2 ) ( 5
)
where

γ 3 (θ) = 18P 2 S 2 4 +2S 4 2 +3P S 2 2 S4-23P 2 S2S6 P S2 sin 2 (θ)+3P S 6 - 3S 2 S 4 .

CENTRO-SYMMETRIC ARRAYS

We define centro-symmetric arrays as those linear arrays with sensors disposed symmetrically w.r.t. the origin, ones with coordinates of the form ±x (1) , ±x (2) , • • •, with, possibly, a sensor at the origin. Obviously, such an array verifies S i = 0 for any odd i. The opposite is not always true. Actually we prove that 1 an array is centro-symmetric iff S i = 0 for any odd i less or equal to P . We will rather focus on the closed-form expressions ( 4)-( 5) to analyze the impact of the array geometry (in terms of the non-zero S 2 , S 4 and S 6 ) on the array estimation performance for both DOA and range, of both far-field and near-field sources.

Relations between far-field and near-field DOA performance

We start by recalling the stochastic DOA CRB in the far-field propagation model [11, rel. (5)] given for arbitrary linear arrays by

CRB FF (θ) = c cos 2 (θ)P S 2 . ( 6
)
Compared to (4), we realize that arrays for which S 3 = 0 (including centro-symmetric arrays) achieve CRB FF (θ) when the source-to-array distance tends to infinity. At the same time, θ and r estimates are decoupled, as pointed out earlier. Other advantages of centro-symmetric arrays include a larger domain of validity of our approximations as a result of the convergence in 1/r 2 compared to 1/r for non centro-symmetric arrays. More unusual is the behavior of non centro-symmetric arrays (and, in general, those with a non-zero S 3 ). Because 1 P S 4 -S 2 2 > 0, such arrays verify lim r→∞ CRB(θ) > CRB FF (θ). Now, w.r.t. [F] 1,2 , θ and r are coupled to the second-order in ϵ. More precisely, the square of [F] 1,2 and the term [F] 2,2 tend to zero with the same speed. This is in contrast to the case S 3 = 0, for which the square of [F] 1,2 tends to zero more rapidly than [F] 2,2 when r tends to ∞. Consequently, from the practical point-of-view, as far as only the DOA parameter is considered, the far-field propagation model, although approximative, may be preferable to the exact near-field model for non centro-symmetric arrays with S 3 ̸ = 0.

To illustrate the different behavior of centro-symmetric and non centro-symmetric arrays in the near-field region, we test in Fig. 2 antenna arrays of P = 4 sensors. On the one hand, the ULA with a constant inter-sensors spacing d and for which S 3 = S 5 = 0. On the other hand, the minimum hole and redundancy linear array (MHRLA) with inter-spacings d, 3d, 2d [START_REF] Meurisse | Bounds for sparse planar and volume arrays[END_REF] and for which S 3 ̸ = 0. Thanks to a larger aperture, the MHRLA exhibits a lower far-field CRB, for instance, CRB MHRLA FF (θ)/CRB ULA FF (θ) ≈ 0.22. However, due to the coupling of θ and r in the the matrix F of the MHRLA, we have lim r→∞ CRB(θ) > CRB FF (θ) for this array. Furthermore, this figure confirms that the domain of validity of our approximations is much larger for the centro-symmetric than for non centro-symmetric arrays. 2), ( 4)) ] and exact [deduced from [START_REF] Krim | Two decades of array signal processing research[END_REF]] ratios CRB(θ)/CRBFF(θ) for 4-sensors ULA and MHRLA and a source at θ = 50 • .

Near-field performance

We have favored centro-symmetric arrays verifying S 3 = 0 mainly because they do not suffer degradation in the DOA CRB when source tends to be in the antenna far-field. Among those arrays with S 3 = 0, those with identical S 2 achieve the same far-field DOA CRB. Now, out from these ones, we will be able to identify better ones, on the basis of the DOA and range (near-field) CRBs. The following rewriting of (4-5) will be helpful for this purpose ) for P ≫ 1.

CRB(θ) = c 1 S2 + 1 κ [( 1 + 4 1-κ ) sin 2 (θ) + 1 ] 1 P r 2 cos 2 (θ)P +o(ϵ 2 ) (7) CRB(r) r 4 = c 1 cos 4 (θ) 4 S 2 2 [ 1 1 κ -1 + S 2 2P r 2 × ( 18 + 3κ + 2κ 2 -23 η ) sin 2 (θ) + 3 κ 2 η -3κ (1 -κ) 2   + o(ϵ 2 ).
While S 2 determines the array far-field DOA estimation performance, κ and η are all what matters for near-field DOA and range estimation performance. In practice, only κ is important, in regard of the predominant terms in [START_REF] Gazzah | Spectral efficiency of beamforming-based parameter estimation in the single source case[END_REF] and [START_REF] Abeida | Efficiency of subspacebased DOA estimators[END_REF]. It affects [START_REF] Abeida | Efficiency of subspacebased DOA estimators[END_REF] through the increasing function 1/(1/κ -1), and affects [START_REF] Gazzah | Spectral efficiency of beamforming-based parameter estimation in the single source case[END_REF] through function f θ (κ)

f θ (κ) = 1 κ [( 1 + 4 1 -κ ) sin 2 (θ) + 1 ] ,
illustrated in Fig. 3. Considering DOA estimation, choosing κ loosely close to 1/2 ensures limited degradation in all look directions. If κ is close to (but lower than) 1/2, then it also ensures better estimation of the range parameter as well.

Comparison with the ULA

To better illustrate the impact of κ on the estimation performance of both DOA and range, we compare the 6sensors ULA (with sensors placed at ±0.1195, ±0.3586 and ±0.5976) against a non-ULA array of 6 sensors located at ±0.1281, ±0.2396 and ±0.6528. Both arrays exhibit the same S 2 = 1 (and, hence, have identical far-field DOA estimation CRBs). However, κ is equal to 0.577 for the ULA and to 0.45 for the non-ULA. In Fig. 4, we report the ratios

CRB(θ)| non-ULA CRB(θ)| ULA and CRB(r)| non-ULA CRB(r)| ULA ,
calculated using the exact CRB expressions and the approximate CRB expressions in ( 7) and ( 8). There, we can see that while we obtain similar DOA performance (with a slight degradation), the non-ULA array has better range estimation capabilities. From ( 7), all centro-symmetric linear arrays characterized by a given size P and a given value of S 2 verify

lim r→∞ CRB(θ)| non-ULA CRB(θ)| ULA = 1.
However, from ( 8), the κ-dependent function can be seen as an indicator of improvement (over the ULA) whenever it is lower than one. For instance, if P ≫ 1,

R P (κ) = lim r→∞ CRB(r)| non-ULA CRB(r)| ULA = 1 κULA -1 1 κ -1 0.5 1
R P (κ) = 4 5 1 1 κ -1 .
This ratio is illustrated in Fig. 5 for the domain2 [0.3, 0.7] of κ outside which DOA near-field performance degrades severely (as clear from Fig. 3). It can be seen from Fig. 5 that the (farfield) range CRB can be reduced by a much as 50% by antenna arrays with a κ moderately lower than that of the ULA.

CONCLUSION

Accurate, simple and interpretable closed-form expressions of the CRB for both angle and range parameters of a near-field narrow-band source have been obtained for arbitrary linear arrays using the exact expression of the time delay parameter. They show the exact geometric condition for the antenna array to have an attractive behavior in its near-field: better precision and faster convergence to the far-field DOA CRB. Such a class of centro-symmetric arrays includes, but is not restricted to the ULAs. Furthermore, it is proved that appropriately designed centro-symmetric non-ULA can largely improve the range estimates without deteriorating the DOA estimates under near-field conditions.

Fig. 2 .

 2 Fig. 2. Approximative [(from (2), (4)) ] and exact [deduced from

Fig. 3 .

 3 Fig. 3. Near-field DOA estimation performance, as expressed by f θ (κ), as function of the geometry of the centro-symmetric array, as expressed by parameter κ.

2 P 2 1 η

 221 S 6 appear as two key geometric parameters that determine the near-field accuracy of the antenna array. They also can be proved to verify the following interesting properties

Fig. 4 .

 4 Fig. 4. DOA and range CRBs of the non-ULA (κ = 0.45) normalized to that of the equivalent ULA (κ = 0.577) for θ = 40 • . Both arrays are made of P = 6 sensors and are such that S2 = 1.

Fig. 5 .

 5 Fig. 5. Centro-symmetric non-ULA versus ULA: Compared range estimation performance of far-field sources.

Proofs, detailed in[START_REF] Gazzah | CRB based-design of linear antenna arrays for near-field source localization[END_REF], about S

, S 4 and S 6 are based on the Newton-Girard Formula [10, pp. 69-74] that allows iterative calculation of S i .

In fact, extreme values of κ (i.e., 0 and 1) are achieved by impractically co-localized sensors, either at the origin, or at the same distance (and on both sides) from the origin.


Pascal Chevalier

Abdelkader Oukaci

Jean-Pierre Delmas

THIRD ORDER WIDELY NON LINEAR VOLTERRA MVDR BEAMFORMING

Keywords: Beamformer, non circular, non Gaussian, higher order, interferences, MVDR, Volterra, widely non linear, widely linear, third order, fourth order, sixth order

It is now well known that time invariant (TI) linear beamformers, such as the Capon's beamformer, are only optimal for stationary Gaussian observations whose complex envelope is necessarily second order (SO) circular. However in many applications such as in radiocommunications, most of the signals are nonGaussian and their complex envelope presents very often some SO and/or higher order (HO) non circularity properties. For this reason we propose in this paper a third order widely non linear Volterra minimum variance distortionnless response (MVDR) beamformer taking into account both the potential non Gaussian character and the potential HO non circularity (up to sixth order) of interferences. Some properties, performance and adaptive implementation of this new beamformer are presented in the paper. Illustrations shows the great interest, with respect to the existing beamformers, of this new beamformer for non Gaussian and HO non circular interferences, omnipresent in practice.

INTRODUCTION

Conventional time invariant (TI) linear beamformers, such as the Capon's beamformer [START_REF] Capon | Multidimensional maximum likelihood processing of a large aperture seismic array[END_REF], are only optimal for stationary Gaussian observations whose complex envelope is necessarily second order (SO) circular. However in many applications such as in radiocommunications, most of the signals are nonGaussian and their complex envelope presents very often some SO and/or higher order (HO) non circularity properties [START_REF] Picinbono | On circularity[END_REF]. For example a M -PSK (Phase Shift Keying with M states) signal is a nonGaussian signal whose complex envelope is noncircular at an order q such that q ≥ M . For this reason, a third order Volterra generalized sidelobe canceller (GSC) [START_REF] Griffiths | An alternative approach to linearly constrained adaptive beamforming[END_REF] structure has been proposed in [START_REF] Souloumiac | Improvement in nonGaussian jammers rejection with a non linear spatial filter[END_REF] for a few years to improve the performance of the Capon's beamformer in nonGaussian context. However this Volterra beamformer does not exploit the potential noncircularity of the interferences. To overcome this drawback, a widely linear [START_REF] Picinbono | Widely linear estimation with complex data[END_REF] MVDR beamformer has been proposed recently in [START_REF] Chevalier | Widely linear MVDR beamformers for the reception of an unknown signal corrupted by noncircular interferences[END_REF] to improve the performance of the Capon's beamformer in SO noncircular context. However this beamformer does not exploit the potential nonGaussian character of the interferences. In order to keep the advantages of these two beamformers, we propose in this paper a so-called third order widely nonlinear Volterra MVDR beamformer which takes into account both the potential nonGaussian character and the potential HO noncircularity (up to sixth order) of interferences. Some properties, performance and adaptive implementation of this new beamformer are presented in the paper. Illustrations show the great interest of this new beamformer with respect to the existing ones for nonGaussian and HO noncircular interferences, omnipresent in practice. Note that widely nonlinear Volterra complex filters have already been used in [START_REF] Chevalier | Complex transversal Volterra filters optimal for detection and estimation[END_REF] and [START_REF] Chevalier | Complex Linear-Quadratic Systems for Detection and Array Processing[END_REF] but for mean square estimation and detection respectively.

The paper is organized as follows. Section 2 introduces hypotheses and formulates the problem. Section 3 introduces the new third order widely nonlinear Volterra beamformer. Section 4 presents some properties and computes the output signal to interference plus noise ratio (SINR) at the output of this new beamformer. Finally section 5 compares the performance of the proposed beamformer to those of the existing ones for both BPSK and QPSK interferences.

HYPOTHESES AND PROBLEM FORMULATION

Hypotheses

Let us consider an array of N narrowband sensors. Each sensor is assumed to receive a signal of interest (SOI) corrupted by a total noise potentially composed of interference and background noise. Under these assumptions, the vector x t of complex amplitudes of the signals at the output of these sensors can be written as follows

x t = s t s + n t , ( 1
)
where s t and s correspond to the complex envelope, assumed zero-mean, and the steering vector, assumed perfectly known and such that its first component is equal to one, of the SOI respectively. The vector n t is the total noise vector, assumed to be zero-mean, potentially nonGaussian and noncircular and independent of the SOI s t .

Problem formulation

The problem addressed in this paper is to estimate the unknown signal s t from the observation x t . Naturally, the best estimate y t of s t according to the minimum mean square error (MMSE) is the conditional expectation y t = E(s t /x t ). Note that for respectively circular or noncircular mutually Gaussian distributions of (s t , x t ), this conditional expectation becomes linear or widely linear [START_REF] Picinbono | Widely linear estimation with complex data[END_REF]. But for nonGaussian distribution of (s t , x t ), the derivation of this conditional expectation needs this distribution which is unknown in practice.

For this reason, we try in this paper, to find an approximation of this conditional expectation through the analysis of a particular class of nonlinear beamformer corresponding to what we call in this paper complex TI widely nonlinear Volterra beamformers, introduced for the first time in [START_REF] Chevalier | Complex transversal Volterra filters optimal for detection and estimation[END_REF]. The general input/output relation of a full M -th order complex TI widely nonlinear Volterra beamformer is defined by

y t = M ∑ m=1 m ∑ q=0 w H m,q (x ⊗q t ⊗ x * ⊗(m-q) t ) (2)
where (w m,q ) 0≤q≤m,1≤m≤M is a N m × 1 TI complex filter and ⊗ denotes the Kronecker product. ( 2) defines a widely linear filter [START_REF] Picinbono | Widely linear estimation with complex data[END_REF] for M = 1 and a full complex linear-quadratic filter [START_REF] Chevalier | Complex Linear-Quadratic Systems for Detection and Array Processing[END_REF] for M = 2. Note that the best M th-order complex approximation corresponds to the orthogonal projection of s t on the subspace generated by [START_REF] Picinbono | On circularity[END_REF]. Consequently the term

w H m,q (x ⊗q t ⊗ x * ⊗(m-q) t ) brings information only if w H m,q (x ⊗q t ⊗ x * ⊗(m-q) t
) is correlated with s t , which requires that the (m + 1)th-order moment E(s q t s * (m+1-q) t

) is not zero. In practice, the distribution of s t is often symmetric and consequently the odd order moments of s t are zero. To take advantage of this M thorder complex approximation for arbitrary distribution, we only consider complex TI Volterra filters such that M is odd, containing only polynomial terms of odd order.

TI VOLTERRA MVDR BEAMFORMING

To simplify the notations, we only consider in the following that M = 3 for which

y t = w H 1,1 x t 0 + w H 1,0 x * t 1 + w H 3,0 (x * t ⊗ x * t ⊗ x * t ) 2 + w H 3,1 (x t ⊗ x * t ⊗ x * t ) 3 + w H 3,2 (x t ⊗ x t ⊗ x * t ) 4 + w H 3,3 (x t ⊗ x t ⊗ x t ) 5 ( 3
) def = w H x 13,t , (4)
where

w def = [w T 1,1 , w T 1,0 , w T 3,0 , w T 3,1 , w T 3,2 , w T 3,3 ] T and x 13,t def = [x T t , x H t , (x * t ⊗x * t ⊗x * t ) T , (x t ⊗x * t ⊗x * t ) T , (x t ⊗x t ⊗x * t ) T , (x t ⊗ x t ⊗ x t ) T ]
T is the augmented observation [START_REF] Comon | Path-wise wide-sense polynomial receiver for UMTS communications[END_REF]. Analyzing the different terms of (3), it is possible to show that all the terms convey information for a rectilinear interference such as an ASK or a BPSK interference whereas only terms 0 and 4 convey information for a circular interference. For QPSK interferences, which is SO circular but fourth-order noncircular, terms 0, 2 and 4 convey information. The third order Volterra beamformer associated only with the terms 0 and 4 is a so called linear-cubic beamformer, used in [START_REF] Souloumiac | Improvement in nonGaussian jammers rejection with a non linear spatial filter[END_REF]. When at least one of the terms between 1, 2, 3, 5 is taken into account in addition to terms 0 and 4, the third order beamformer is so called widely nonlinear. In this case as long as the terms 1, 2, 3, 5 are not all taken into account, the beamformer is a partial third order widely nonlinear Volterra beamformer. It becomes a full third order widely nonlinear Volterra beamformer when both 1, 2, 3 and 5 terms are taken into account in addition to 0 and 4 terms.

To introduce, our Volterra MVDR beamformer, we focus now only on the particular partial structure

y t = w H 1,1 x t + w H 3,2 (x t ⊗ x t ⊗ x * t )
, because the other partial and full structures follow the same derivation. This beamformer is a linearly constrained minimum variance beamformer [START_REF] Frost | An algorithm for linearly constrained adaptive array processing[END_REF] which uses only s as a prior knowledge. It corresponds to the filter w that minimizes its output power

w H R x,13 w (5)
with R x,13 def = E( x 13,t x H 13,t ) under multiple linear constraints. Apart from the constraint w H 1,1 s = 1 to preserve the SOI, s t , we must elaborate a little bit to consider the third-order term

w H 3,2 [(s t s + n t ) ⊗ (s t s + n t ) ⊗ (s t s + n t ) * ] (6)
because the total noise n t is random. A solution consists in decomposing n t on a deterministic orthogonal basis (s, u 1 , ...u N -1 ) ∈ C N because s is known:

n t = α 0,t s + ∑ N -1 i=1 α i,t u i .
Hence, excluding the terms u i ⊗ u j ⊗ u * k of (6) which will be considered in the minimization of the output power [START_REF] Picinbono | Widely linear estimation with complex data[END_REF], a solution to avoid correlated terms with s t in y t is to impose the 1 + 3(N -1) + 3(N -1) 2 = N 3 -(N -1) 3 following constraints:

w H 3,2 (s ⊗ s ⊗ s * ) = 0 w H 3,2 (u i ⊗ s ⊗ s * ) = 0, w H 3,2 (s ⊗ u i ⊗ s * ) = 0 w H 3,2 (s ⊗ s ⊗ u * i ) = 0, w H 3,2 (u i ⊗ u j ⊗ s * ) = 0 w H 3,2 (s ⊗ u i ⊗ u * j ) = 0, w H 3,2 (u i ⊗ s ⊗ u * j )
= 0 where i, j ∈ {1, .., N -1}. Putting all the constraints in a matrix form, we obtain the following constraint:

C H w = f (7) with C = [ s O N ×[N 3 -(N -1) 3 ] 0 N 3 ×1 A ]
, where the columns of A contain the previous vectors s ⊗ s ⊗ s * , ...,

u i ⊗ s ⊗ u * j and f def = (1, 0 T N 3 -(N -1) 3 ) T .
We note that R x,13 is singular due to the redundancies in the augmented observation x 13,t . Consequently, the filter w that minimizes w H R x,13 w under constraint (7) cannot be directly derived. To solve this constrained minimization problem, the redundancies of x 13,t have to be withdrawn to give the new augmented observation x ′ 13,t with covariance R ′

x,13

which is now not singular. And the new filter w ′ associated with the new constraint matrix C ′ would be given by

w ′ = R ′ -1 x,13 C ′ [C ′ H R ′ -1
x,13 C ′ ] -1 f . Naturally all this presentation and those of the next section extends straightforwardly to arbitrary other partial or full structure. For example, for the full structure, C has dimension

(2N + 4N 3 ) × (2 + 4(N 3 -(N -1) 3 ).

VOLTERRA GSC STRUCTURE

Equivalent Volterra GSC structure

It can be easily verified that our Volterra MVDR beamformer has this equivalent simpler to implement GSC structure [START_REF] Griffiths | An alternative approach to linearly constrained adaptive beamforming[END_REF] w H proposed recently in [START_REF] Chevalier | Widely linear MVDR beamformers for the reception of an unknown signal corrupted by noncircular interferences[END_REF] and to Capon beamformer [START_REF] Capon | Multidimensional maximum likelihood processing of a large aperture seismic array[END_REF], both implemented by SMI algorithms. Although less fast than the MVDR 1 and Capon beamformers, we see that the partially and fully structured Volterra MVDR beamformers practically converge from a number of 50 and 100 snapshots respectively. Furthermore, in this scenario the partially and fully structured Volterra MVDR beamformers largely outperform the MVDR 1 and Capon beamformers. Naturally there is no gain for |α j,s | = 0 (s and j orthogonal, total rejection of the interference) and |α j,s | = 1 (s and j colinear, no rejection of interference).

For a BPSK interference, term (1) of eq.( 3) conveys more information than term (4) because the nonGaussianity of BPSK is relatively weak (kurtosis of -2) with respect to its SO noncircularity that is maximum (rectilinear). Consequently the MVDR 1 beamformer [START_REF] Chevalier | Widely linear MVDR beamformers for the reception of an unknown signal corrupted by noncircular interferences[END_REF] that uses only term (1) outperforms the partial third order Volterra beamformer. In contrast, all the terms of eq.( 3) convey information and thus the total third order Volterra beamformer largely outperforms the partially structured one. Furthermore, we note that the gain can be very large (over 8dB) in this scenario.

For a QPSK interference, the MVDR 1 beamformer [6] does not improve the Capon beamformer because term (1) of eq.( 3) does not convey any information for SO circular interference. In contrast, for the partially [resp., totally] structured MVDR Volterra beamformer, term (4) [resp., terms (2) and ( 4)] of eq.( 3), convey information for fourth-order noncircular interference. Consequently, these two MVDR Volterra beamformers outperform the Capon beamformer. Furthermore, we note that the gain can be large (about 2dB and 6dB) in this scenario.

In these two experiments, we see the interest to take into account, both nonGaussianity and noncircularity of the interference through our full third order widely nonlinear Volterra beamformer.

Fig. 2

 2 Fig.2 Estimated SINR at the output of different beamformers.In the second experiment, we are interested by the gain in SINR with respect to the Capon algorithm in the steady state, as a function of the spatial correlation |α j,s | defined by α j,s = j H s |j||s| = |α j,s |e iϕj,s , The interference is a BPSK (a) or QPSK (b) modulation with SNR = 20dB and INR = 20dB. Naturally there is no gain for |α j,s | = 0 (s and j orthogonal, total rejection of the interference) and |α j,s | = 1 (s and j colinear, no rejection of interference).For a BPSK interference, term (1) of eq.(3) conveys more information than term (4) because the nonGaussianity of BPSK is relatively weak (kurtosis of -2) with respect to its SO noncircularity that is maximum (rectilinear). Consequently the MVDR 1 beamformer[START_REF] Chevalier | Widely linear MVDR beamformers for the reception of an unknown signal corrupted by noncircular interferences[END_REF] that uses only term (1) outperforms the partial third order Volterra beamformer. In contrast, all the terms of eq.(3) convey information and thus the total third order Volterra beamformer largely outperforms the partially structured one. Furthermore, we note that the gain can be very large (over 8dB) in this scenario.For a QPSK interference, the MVDR 1 beamformer [6] does not improve the Capon beamformer because term (1) of eq.(3) does not convey any information for SO circular interference. In contrast, for the partially [resp., totally] structured MVDR Volterra beamformer, term (4) [resp., terms (2) and (4)] of eq.(3), convey information for fourth-order noncircular interference. Consequently, these two MVDR Volterra beamformers outperform the Capon beamformer. Furthermore, we note that the gain can be large (about 2dB and 6dB) in this scenario.In these two experiments, we see the interest to take into account, both nonGaussianity and noncircularity of the interference through our full third order widely nonlinear Volterra beamformer.

Fig. 3

 3 Fig.3 Gain of Volterra MVDR beamformers as a function of the spatial correlation.


Dr Maria T Proetto

Pr Kelsey N Alexander

Dr Mohand Melaimi
email: mmelaimi@ucsd.edu

Guy Bertrand
email: gbertrand@ucsd.edu

Nathan C Gianneschi
email: nathan.gianneschi@northwestern.edu.edu

Pr N Gianneschi

Pr G Bertrand

Cyclic (Alkyl)(Amino)Carbene (CAAC

Keywords: Antitumoral Agents, CAACs, Cancer, Transmission Electron Microscopy, TrxR

Introduction

Many civilizations, starting with ancient China, employed gold containing concoctions, long touted for their healing powers in the fight against disease. 1 In the late 19th century, scientific study began with the examination of gold(I) complexes against pulmonary tuberculosis, 2 and then later against rheumatoid arthritis. 3 The latter became a successful endeavor with a clear demonstration of the beneficial effect of gold-thiolate drugs. Introduced in the late 1970's, Auranofin (Scheme 1, A) proved very promising and was approved for clinical use in 1985. 3a The antitumor activity of Auranofin was reported for the first time in 1979. 4 Since that time, numerous gold(I) complexes supported by phosphorous-, nitrogen-or carbon-based ligands have been investigated. 5 Although it is still not fully understood, the proposed mechanism is associated with the ability of Auranofin and other gold-based drugs to bind a selenocysteine moiety located in the C-terminal active site of the Thioredoxin Reductase (TrxR) enzyme, 6 which is involved in controlling Reactive Oxygen Species (ROS) homeosta-sis. 7 It should be noted that the 2',3',4',6'-tetra-O-acetyl--D-gluco-pyranosyl-1'thiolate (Glc) moiety was selected due to the over-expression of the corresponding glucose transporters on cancerous cells, resulting in a higher uptake of the gold complex compared to versions lacking this anionic carbohydrate ligand. 8 Despite its potency, Auranofin suffers from significant side effects due to its undesired binding to cysteine residues on serum proteins and intracellular proteins. 9 A step forward in the design of gold(I) drugs was achieved with the replacement of the phosphine ligand by N-Heterocyclic Carbenes (NHCs) B, [START_REF]For thematic issues and books on NHCs[END_REF]11 which form stronger ligand-metal bonds. Similarly to Auranofin, neutral complexes of type Au(B)Cl mainly exhibit cytotoxicity to their ability to inhibit TrxR overexpressed in cancer cells. Alternatively, cationic bis-carbene complexes of type Au(B) 2+ X - show higher stability under physiological conditions than neutral complexes and are characterized by a better ability to navigate through membrane with elevated potential, to accumulate in the mitochondria, to interact with membranal vital systems and also to induce deregulation and/or alterations of elements essential for cell life. 5d-f More recently, another family of carbenes, namely cyclic (alkyl)(amino) carbenes (CAACs) [START_REF] Lavallo | For the synthesis of CAACs[END_REF][START_REF] Soleilhavoup | For reviews on CAACs see[END_REF] C have found many applications, due to their peculiar steric and electronic properties. In-deed, when compared to NHCs, the replacement of one nitrogen atom (-attractor and -donor) by a quaternary carbon (solely -donor) results in a higher HOMO (carbene lone pair) and lower LUMO (formal empty orbital) (Schame 1). 14 Therefore, CAACs are simultaneously amongst the most basic and the most -acidic singlet carbenes. The advantages of this class of carbene over NHCs are best exemplified by success in the isolation of active catalytic intermediates, 15 efficient and robust pre-catalysts, 16 and complexes with unusual configuration and/or oxidation states. 17 In all cases the stronger carbene-metal bond is characteristic of these CAAC bound complexes with respect to NHC analogs. There-fore, there is reason to believe that CAACbound gold complexes would exhibit lower affinity for cellular proteins compared to desired biological targets, leading to more specific drug-target interactions and decreased side effects. Recently, different CAAC-bound gold, silver and copper complexes have been reported as potent cytotoxic agents. Further, their mechanism of action appears to be metal dependent. 18 Herein, we report a detailed investigation on the in vitro antitumoral activity of a set of gold(I) complexes (vide infra) all bearing CAAC ancillary ligands for better comparison between them, namely [Au(CAACEt) 2 ] + TfO -1, Au(CAACEt)Cl 2 17d and Au(CAACEt)Glc 3. We have also evaluated the ability of these complexes to inhibit TrxR, measured their affinity for bovine serum albumin (BSA) and provide evidence for the mechanism of cell death.

Results and Discussion

First, we reacted free carbene CAACEt with a half-equivalent of Au(THT)Cl in tetrahydrofuran. Subsequent anion exchange by treating the cationic complex with lithium trifluoromethanesulfonate in methylene chloride afforded the homoleptic salt 1 in 91% yield (Scheme 2, top). The neutral complex 2 is prepared by a two-step sequence in which CAACEt is first reacted with Au(PPh3)Ph to substitute the PPh3 ligand. Then protolysis of the resulting Au(CAACEt)Ph with dry HCl in benzene allowed for the isolation of the desired complex 2 in an overall yield of 87% (Scheme 2 bottom center). Note that both compounds 1 and 2 can be synthesized by bypassing the isolation of free carbene CAACEt. In the former case, addition of one equivalent of KHMDS to the mixture of the cyclic iminium CAACEtH + TfO -and Au(THT)Cl also led to complex 1, albeit with overall lower yields of 62%. Similarly, addition of one equivalent of KHMDS to the mixture of the cyclic iminium CAACEtH + TfO - and Au(PPh3)Ph and further reaction with HCl afforded complex 2 with an 82% yield. Finally, reaction of complex 2 with the gluco-pyranosyl-1-thiole in the presence of triethylamine in methylene chloride led to the expected CAAC-bound Auranofin analog 3 in good yield (94%, Scheme 2, bottom right). Each of these complexes were found to be indefinitely air and moisture stable, and were fully characterized by NMR, HR-MS and X-ray diffraction of single crystals (Figure 1). As for any CAAC supported divalent gold(I) complex, 19 we expected a linear geometry at the metal center. Analysis of the solid-state structures 20 confirmed this spatial arrangement with C-Au-C/X angles of 170.56 o (13), 178.05 o (7) and 176.47 o (10) for compounds 1, 2 and 3, respectively. The slightly bent geometry in the homoleptic complex 1 is unexpected with regards to previously reported [Au(CAAC)2] + fragments in which the carbene-Au-carbene angle is found to be 180 o . However, this bent geometry is commonly observed in bis-CAAC complexes of Ni(0), 21a Pd(0) 21b and Pt(0), 21b which are isolobal to Au(I), and exhibit carbene-metal-carbene angles of 169 o ± 4 o . More importantly, in complex 1, the trifluoromethanesulfonate anion is located more than 5 Å away from the gold atom and therefore does not engage in any interaction with the metal center. Finally, carbene-gold bond lengths remain relatively constant (1: 2.036 (3) Å, 2: 1.975 (3) Å, 3: 2.004 (4) Å). Note that the minor shortening of the carbene bond length for the goldchloride complex has already been observed for analogous compounds. With complexes 1-3 and the CAACEtH + TfO -salt in hand, we tested their cytotoxicity against three different human cancer cell lines: HeLa (cervical cancer), A549 (lung cancer), HT1080 (fibrosar-coma) and Caov-3 (ovarian cancer). A cell titer blue assay was used to determine the extent of the inhibitory effect of the com-plexes on cancer cell growth. Cytotoxic activities were demon-strated to be compound and cell line dependent: against HeLa cells, activities decreased following the order 1~2>A>3>> CAACEtH + TfO -. whereas, against A549 and HT1080, the order 1>A>3~2>> CAACEtH + TfO -was observed. Finally, the activities followed 1>A>3>2>> CAACEtH + TfO -when exposed to Caov-3. Overall, Auranofin A, Au(CAACEt)Cl 2 and Au(CAACEt)Glc 3 triggered death on all cell lines with IC50 values in the micromolar range (Table 1). Interestingly, [Au(CAACEt) 2 ] + TfO -1 proved to be more cytotoxic than the others, with IC50 values lower by 1 to 2 orders of magnitude across all cell lines. This trend has previously been observed for NHC supported gold(I) complexes and is linked to the ability of delocalized lipophilic cations to pass through hydrophobic barriers and mitochondrial membranes to reach their targets in the mitochondria. 22 Auranofin (A) was observed to be slightly more active than the CAACEt-analog 3 (Au(CAACEt)Glc) in all cell lines. Remarkably, compound 2 exhibited higher cytotoxicity than Auranofin only on HeLa cells, suggesting a cell dependent mechanism of action. Furthermore, the ligand precursor CAACEtH + OTf -was found to be at least 5-to 1000-fold less toxic than the complexes against this set of cancer cell lines. The significantly lower cytotoxicity of CAACEtH + OTf -, clearly demonstrates the gold center involvement in the antitumor activity. In general, the gold complexes bearing CAACEt as a mono-or bis-carrier ligand were demonstrably more active than those reported with bulkier CAAC-ligands,18 suggesting that steric parameters and the hydrophilicity/lipophilicity balance are key features impacting activity. Thus, as part of an investigation on the physiochemical properties of the CAAC-gold complexes, we determined their lipophilicity using the octanol-water partition protocol. Based on our experimental results, the most lipophilic compound was Au(CAACEt)Cl 2, followed by Au(CAACEt)Glc 3. As expected because of its positively charged nature [Au(CAACEt) 2 ] + TfO -1 showed the lowest lipophilicity value (Table 2). Indeed, it has been reported that the lipophilicity of compounds can be finely tuned to increase their ability to cross cellular membranes and accumulate in mitochondria, therefor modifying their toxicity. 6,23 To examine the impact of lipophilicity on cytotoxicity, we studied the cellular uptake of [Au(CAACEt) 2 ] + TfO -1, Au(CAACEt)Cl 2 and Au(CAACEt)Glc 3 in Caov-3 at 1 µM concentration after 6 h incubation. Our results showed a 10-fold greater uptake for the cationic complex [Au(CAACEt) 2 ] + TfO -1 when compared to the neutral complexes Au(CAACEt)Cl 2 and Au(CAACEt)Glc 3. Interestingly, complex Au(CAACEt)Cl 2, which is the most lipophilic compound, showed the lowest uptake and cytotoxicity values on Caov-3 cells. Among these CAAC-complexes, lower lipophilicity is associated with higher cellular uptake and higher cytotoxicity.

Based on the accepted mode of action for Auranofin A and other gold complexes, we opted to explore affinity towards TrxR, a known target for gold complexes. In parallel, we investigated non-specific interactions between gold complexes and serum proteins, using bovine serum albumin as a model. Compounds A, 1, 2 and 3 were incubated with TrxR and bovine serum albumin and results are summarized in Table 2. Compound LogP Uptake (ng Au/mg protein) [a] TrxR (IC50, nM) [b] BSA binding (%) [c] Auranofin A n. [a] Uptake in Caov-3 cells after 6 h incubation with 1 µM of the respective Aucomplex. [b] IC50 values represent the concentration that caused 50% cell death reported. [c] BSA binding was estimated as the percentage of free drug recovered in the presence and absence of protein. [d] not determined .

Under our protocol and in accordance with previously published literature, 24 Auranofin A showed the highest TrxR inhibitory effect at low nanomolar concentrations. Complex 2 showed increased activity over 3, with both exhibiting TrxR-IC50 values in the micromolar ranges. Note that previously reported, analogous NHC and CAAC complexes show TrxR inhibition at similar concentration ranges. 25 Interestingly, despite displaying the highest cytotoxicity against all cancer cell lines studied here, the cationic complex [Au(CAACEt) 2 ] + TfO -1 did not inhibit TrxR up to its solubility limit (100 µM). Moreover, as expected, CAACEtH + TfO -bearing no gold metal center, was demonstrated to have no affinity for TrxR up to its solubility limit.

To further understand TrxR inhibition by the gold complexes, we chose to study the binding of gold complexes 1-3 with dodecapeptide D ox (Ac-SGGDILQSGCUG-NH2, Scheme 3) 6b which is analogous to the peptide sequence found in TrxR that is the specific target of gold complexes. Note that in its oxidized form, D ox corresponds to the non-active form of the TrxR enzyme. Peptide D ox was prepared according to literature procedures and was converted into D red H2 (Ac-SGGDILQSGCHUHG-NH) upon treatment with tris(2carboxyethyl)phosphine (TCEP). Under its reduced form, D red H2, as well as analogous deprotonated forms of the reduced peptide (D red H -and D red.2-), correspond to the active forms of the TrxR. To investigate the interactions between the gold complexes and peptides of type D, we incubated freshly reduced peptide D red H2/D red H -/D red.2-with gold complexes 1-3 and analyzed the results by mass spectrometry. As expected, complexes 2 and 3 form anionic adducts [D red -Au(CAACEt)] -in which the reduced dianionic peptide D red.2-is bound to the [CAAC-Au] + cationic fragment resulting from the formal substitution of the anionic ligand (X= Cl -for 2 or X= Glc -for 3). Interestingly, incubation of the cationic complex 1 under the same conditions as above, did not allow for observation of the corresponding adduct [B red -Au(CAACEt)2] -; thus, explaining the lack of inhibition of TrxR detailed in Table 2. As stated, one of our goals was to overcome concerns regarding the undesired affinity of Auranofin to albumin. Here, all complexes showed reduced binding compared to Auranofin A (See Table 2). Moreover, compared to similar NHC supported gold complexes, 26 the use of CAAC ligands lead to decreased non-specific binding from close to 100% 27 to 55%. We believe the lower affinity of CAAC-supported complexes 1, 2 and 3 to BSA compared to A and NHC analogs complexes is a result of the stronger ligand-metal bond, and thus the greater stability of complexes 1, 2 and 3. Several important pieces of information arise from these data. First, it appears that the cytotoxicity of complex 1 results without targeting the TrxR enzyme, hence causing cell death through a mechanism different than for complexes A, 2 and 3. Also, even though the lack of interaction between [Au(CAACEt)2] + TfO -1 and the selenocysteine moiety is not surprising, it does suggest that the CAAC ligand is not labile under these conditions, and that the fragment [Au(CAACEt)2] + retains its structural integrity. Again, this stability is a clearly the result of the stronger carbene-metal bond strength compared to other supporting ligands. 28 Finally, while the Auranofin A inhibition of TrxR, in the nanomolar range, is much stronger than complexes 2 and 3, the higher propensity of A to undergo non-selective binding to BSA [START_REF]Non-specific binding of Auranofin A to HSA[END_REF] partially accounts for similar cytotoxicity observed for complexes A, 2 and 3. Indeed, in contrast to A, complexes 2 and 3 inhibit TrxR in the micromolar range but show lower non-specific binding. In other words, taken together, the results from Table 1 and 2 confirm the high efficiency at which CAAC-bound complexes reach their biological target and may be reflected less by lower affinity, non-specific interactions with other, undesired, biomolecular targets. This increased selectivity might also result in improved bioavailability and bio-distribution profiles which are important parameters for experimental drugs at the developmental stage. After establishing these intrinsic features, we decided to investigate the modes of cell death (necrosis vs. apoptosis) induced by each compound, A, 1, 2 and 3. We used a standard protocol in which HeLa cells were incubated with our set of gold compounds (10 µM) and their effects were studied in a timedependent manner. Thus, at each selected time-point, cells were stained with FITC-Annexin V and Propidium Iodide (PI). Annexin V is a protein that binds to phosphatidylserine, a lipid constituent of plasma membrane which is normally restricted to the inner membrane surface but becomes exposed in the outer leaflet when cells undergo apoptosis. PI is used as a marker of cell membrane integrity and combined with Annexin V it becomes a tool to distinguish necrotic from apoptotic cells using flow cytometry. [START_REF] Crowley | [END_REF] Analysis at various selected times, show that complexes A, 2 and 3 induce apoptosis at early time-points, after which cells probably undergo secondary necrosis resulting in an increase of necrotic cells with respect to cells undergoing apoptosis. Among this series of compounds, the glucopyranosyl containing complex 3 stands out as the most potent cell death promoter, with almost 80% of the total population of cells undergoing either apoptosis or necro-sis. Interestingly, when exposed to [Au(CAACEt)2] + TfO -1 for a duration of up to 6 h, a negligible amount of cells undergo either apoptosis or necrosis. These results, as well as the previously discussed lack of TrxR inhibition exhibited by [Au(CAACEt)2] + TfO -1, supports the hypothesis that its cell death mechanism involves a different pathway than the gold complexes bearing only one CAAC ligand. To further understand these results, we studied the ability of these compounds to induce depolarization and loss of mitochondrial trans-membrane potential (m), which is closely related to apoptotic cellular pathways. 31 To test this, we employed a lipophilic cationic dye, 5,5',6,6'-tetrachloro-1,1',3,3'tetraethylbenzimidazolylcarbocya-nine iodide (JC-1), which is known to enter mitochondria and arrange in red fluorescent Jaggregates on cells with high m. Upon mitochondrial membrane depolarization JC-1 remains in its green fluorescent monomeric form. Flow cytometry plots were obtained (FL1, green vs. FL2, red) for HeLa cells pretreated with Auranofin A, [Au(CAACEt)2] + TfO -1, Au(CAACEt)Cl 2, Au(CAACEt)Glc 3 or dimethylformamide (DMF, solvent control) and subsequently stained with JC-1 (Figure S7). A healthy cell population was gated on an untreated cell sample (P1, double positive) and compared to the plot from the cells incubated with DMF or the gold complexes. Accordingly, a significant decrease of red fluorescent aggregates in cells exposed to Auranofin A, Au(CAACEt)Cl 2 and Au(CAACEt)Glc 3 can be observed. Once more, the latter display the largest effect, while [Au(CAACEt) 2 ] + TfO-1, showed only minimal decrease on Jaggregates or mitochondrial depolarization, suggesting a different mechanism of action. 3F-I). Lastly, TEM micro-graphs of the control experiment, with cells exposed to only DMF (Figure 3E and3J) were obtained to ascertain the effect of each gold complex. We consider normal cellular morphologies as rounded with homogenous cellular and nuclear membranes, each containing a nucleolus, chromatin, in addition to well-preserved cytoplasmic organelles. Auranofin A (Figure 3A and3F) is well known to promote apoptosis on a wide variety of cancer cells. 33 Under our experimental protocol, we could observe clear signs of apoptosis such as nuclear fragmentation or karyorrhexis (Figure 3A, yellow arrows), and swollen cytoplasmic organelles and vacuoles (Figure 3F, red and green arrows respectively). 32b,34 Au(CAACEt)Cl 2 (Figure 3C and 3H) pretreated cells show morphological signs of early apoptosis such as condensation of chromatin at the nuclear mem-brane (Figure 3C, yellow arrow) and preserved mitochondria (Figure 3H, red arrows). Cells exposed to Au(CAACEt)Glc 3 (Figure 3D and 3I) show the highest degree of pathological morphological features with cells transitioning from apoptosis to secondary necrosis such as swelling of mitochondria and disruption of plasma membrane (Figure 3I, red and green arrows respectively). In contrast, cells exposed to [Au(CAACEt) 2 ] + TfO -1, (Figure 3B and 3G) show well rounded nuclei and preserved cytoplasmic organelles (Figure 3G, red arrows). Some degree of chromatin fragmentation and cytoplasmic vacuoles were observed (Figure 3B, yellow and green arrows respectively), which may suggest a population of cells undergoing early apoptosis.

Conclusion

In conclusion, we have demonstrated the effect of a CAAC ligand as a gold ligand in anticancer cell studies. As previously reported with NHC supported complexes, mono-and bis-CAAC gold com-plexes possess different cellular targets. 7b Both monoand bis-CAAC supported gold centers benefit from the peculiar coordination environment and intrinsic electronic properties provided by the CAAC ligand. As a result, this new family of cytotoxic agents demonstrated significantly reduced non-specific binding toward non-tumor associated proteins such as albumin, while still retaining high affinity to the target TrxR (Au(CAACEt)Cl 2 and Au(CAACEt)Glc 3). Moreover, morphological and biochemical changes suggest that Au(CAACEt)X complexes 2 and 3 induce cell death through a mechanism indicative of apoptosis followed by secondary necrosis. [Au(CAACEt) 2 ] + TfO -1, showed the highest cytotoxic activity with an onset of activity at longer time-points than the mono-ligated complexes and no affinity to TrxR. Therefore, other mechanisms of action need to be considered, such as the induction of Mitochondrial Membrane Permeabilization, 35 or binding to telomeric DNA Gquadruplexes. 36 Such mechanisms of action unrelated to TrxR inhibition are currently under investigation for [Au(CAACEt) 2 ] + TfO -1. Altogether this investigation reports on the applicability of CAAC-supported gold complexes with application as antitumor agents and reveals their biological targets which may lead to lower undesired side-effects for gold metalbased therapeutics.

Scheme 2 .

 2 Scheme 2. Syntheses of complexes 1 (top), 2 (bottom center) and 3 (bottom right).

19

 19

Figure 1 .

 1 Figure 1. Solid-state structures of complexes 1, 2 and 3. Ortep views are shown with ellipsoids at 50% probability. H atoms, solvents are omitted for clarity.

Scheme 3 .

 3 Scheme 3. Dodecapeptide D, analogous to TrxR active sequence, in its oxidized (D ox ) and reduced (D red H2) forms.

Figure 2 .

 2 Figure 2. Population of apoptotic vs. necrotic HeLa cells after a 30 min to 6 h exposure to complexes A, 1, 2 and 3. Full bars represent % cells undergoing apoptosis, while gridded bars represent cells under-going necrosis. The height of the bars represents the % of cells undergoing apoptosis and necrosis.

Figure 3 .

 3 Figure 3. TEM images of cells pretreated with Auranofin A (A, F), [Au(CAACEt)2] + TfO -1 (B, G), Au(CAACEt)Cl 2 (C, H), Au(CAACEt)Glc 3 (D, I) at a 10 µM concentration and DMF as a vehicle control (E, J). Top panels show low magnification images (x2,900, A-E, scale bars represent 2 µm) and bottom panels show high magnification images (x6,800, F-J, scale bars represent 1 µm). Specific features of cell morphologies are shown with arrows (further discussion in the main text).Apoptosis and necrosis are the result of different biochemical events, which can be monitored as cellular morphological changes.32 Owing to its high resolution and the possibility of introducing high-contrast staining agents to the samples, Transmission Electron Microscopy (TEM) is widely used to study cellular ultra-structures. Therefore, HeLa cells were incubated with DMF solutions of complexes Auranofin A, [Au(CAACEt) 2 ] + TfO -1, Au(CAACEt)Cl 2 and Au(CAACEt)Glc 3 (10 µM) plus a control voided of any gold complex for 30 min and processed for TEM imaging. Representative TEM micrographs of cells exposed to gold complexes are shown in low magnification (x2, 900, Figure3A-D) and high magnification (x6, 800, Figure3F-I). Lastly, TEM micro-graphs of the control experiment, with cells exposed to only DMF (Figure3E and 3J) were obtained to ascertain the effect of each gold complex. We consider normal cellular morphologies as rounded with homogenous cellular and nuclear membranes, each containing a nucleolus, chromatin, in addition to well-preserved cytoplasmic organelles. Auranofin A (Figure3A and 3F) is well known to promote apoptosis on a wide variety of cancer cells.33 Under our experimental protocol, we could observe clear signs of apoptosis such as nuclear fragmentation or karyorrhexis (Figure3A, yellow arrows), and swollen cytoplasmic organelles and vacuoles (Figure3F, red and green arrows respectively). 32b,34 Au(CAACEt)Cl 2 (Figure 3C and 3H) pretreated cells show morphological signs of early apoptosis such as condensation of chromatin at the nuclear mem-brane (Figure 3C, yellow arrow) and preserved mitochondria (Figure 3H, red arrows). Cells exposed to Au(CAACEt)Glc 3 (Figure 3D and 3I) show the highest degree of pathological morphological features with cells transitioning from apoptosis to secondary necrosis such as swelling of mitochondria and disruption of plasma membrane (Figure 3I, red and green arrows respectively). In contrast, cells exposed to [Au(CAACEt) 2 ] + TfO -1, (Figure 3B and 3G) show well rounded nuclei and preserved cytoplasmic organelles (Figure

Table 1 .

 1 Cytotoxicity (IC50) against cancer cell lines (in µM)[a]

	Compound	HeLa	A549	HT1080	Caov-3
	Auranofin A	1.7 ± 0.2	3.7 ± 0.6	1.4 ± 0.9	0.9 ± 0.3
	[Au(CAACEt)2] + OTf -1	0.3 ± 0.2	0.07 ± 0.06 0.14 ± 0.04	0.3 ± 0.2
	Au(CAACEt)Cl 2	0.6 ± 0.2	4.5 ± 0.7	4.4 ± 1.3	3.9 ± 0.8
	Au(CAACEt)Glc 3	2.7 ± 0.1	6.6 ± 2.5	3.1 ± 1.8	1.9 ± 0.4
	CAACEtH + TfO -	60 ± 13	76 ± 34	26 ± 16	> 50

[a] IC50 values represent the concentration that caused 50% cell death reported. These values were estimated by interpolation in the sigmoidal dose response fitted curve and are the results of at least three independent experiments

Table 2 .

 2 . LogP values, Cellular Uptake, TrxR Inhibition and % BSA binding

Acknowledgements

Thanks are due to the US National Science Foundation (CHE-1954380). M. T. P. thanks the UCSD Cancer Researchers in Nanotechnology (CRIN) for a postdoctoral fellowship and the mentorship of Dr A. Kummel within that program.


Apel

; Cardoso De Oliveira

; Pacheco De Oliveira

After several years in the Tanzanian courts, without any favourable resolutions, four pastoralist Maasai villages from Loliondo area decided to file a case against the government at the East African Court of Justice (EACJ) in 2017, now motivated by new evictions that happened that year. Filing a lawsuit collectively in this judicial level is unprecedented in the history of the Tanzanian pastoralist and hunter-gatherer peoples.

In this article, I will be focusing on two specific situations related to what has been called "the Loliondo case" that took place during the preparatory meetings at the Pan-African Lawyers Union (PALU) headquarters and the hearings at the EACJ 1 , in May and June 2019. In both events, the encounters between lawyers and Maasai villagers (who were the witnesses) highlighted incongruities of translations, showing the existence of different "communities of communication" 2 . Such communities are organised in these social Aline C. Rabelo. "Haki hupewi. Haki udai" [Right is not Granted; You Have to Claim it] November 6, 2019 2 situations under "asymmetrical interdependent relations"3 and the effects of the powers issued from these asymmetries help us to better understand the ways in which hierarchy is established in interethnic dialogues. On the other hand, we will also note that the role of the Maasai activists -being "indigenous brokers" who are themselves lawyers or highly educated people and, therefore, master the codes of the different communities of communication -reveals the capacity of making negotiations even under (and through) the logics of hegemonic knowledge.

I will start, however, by giving a brief contextualisation about the reasons behind this litigation process. Connected to these reasons, a few aspects of the activism carried out by the pastoralist (or "indigenous") NGOs will be introduced in the next section in order to provide complementary elements about the pastoralists' struggles for land rights in the United Republic of Tanzania.

The overlaps

"Haki hupewi. Haki udai." I heard this statement once, in a meeting organised by four pastoralist NGOs who wanted to gather updated information on pastoralists' condition in north-eastern Tanzania with regard to land conflicts and evictions that seemed predictable. A Maasai lawyer spoke about the importance of taking action: those pastoralists villagers, who were invited to participate in the meeting, should not expect to take their rights as granted. They should fight for them.

"Tusikubali tena!"4 [We shall not agree again!]; a Maasai villager resonated, referring to what happened to the Serengeti plains. In 1959, Maasai elders were impelled to sign an agreement, through a "manufactured consent"5 , in order to relocate their people to Ngorongoro and Loliondo areas, once Serengeti was being declared a national park. Currently, both Ngorongoro and Loliondo are under the status of national protected (or conservation) areas, which has been increasingly imposing more and more limitations for the pastoralist who live there. There have been attempts from sectors of the government at transforming Game Controlled Areas, such as Loliondo, into Game Reserves, another type of conservation area within which human settlements and grazing activities are not allowed. That was the main reason for the arrangement of that meeting with Maasai villagers. Although not having always the same interests neither the same priorities (often, they are even contradictory), these pastoralist NGOs circumstantially work together. Despite -and because of -the friction6 , they produce movement, that is, they take action to fight for land rights.

In practice, evictions and other types of human rights violations have been happening even within Game Controlled Areas (GCA). To be more precise, most of the land conflicts Aline C. Rabelo. "Haki hupewi. Haki udai" [Right is not Granted; You Have to Claim it] November 6, 2019 3 in the centre-north Tanzania take place mainly within GCA. The main cause for that is an overlap of land categories in Tanzanian land laws and therefore the different uses made out from these lands: many GCA overlap completely with registered village lands. And, as Nelson has noted, "the main use of wildlife in Game Controlled Areas is as tourist hunting concessions" granted by the Tanzanian government "without the permission or involvement of the landholder" -in this cases, the registered villages. "The result is that tourist hunting is conducted extensively on community lands (…) [and] may conflict with livestock grazing or forest products collection or other local economic activities. " (2005:8) "Overlap of GCA's and village lands in Loliondo and Sale Divisions, Ngorongoro District. GCA demarcated by dotted line, solid lines are village boundaries." (Nelson, 2005:7) It is important to note that land tenure conflicts have taken place in Tanzania since colonial times, but they have been recently reinforced by contradictory laws and inadequate implementation of the National Land Policy (1995). There is a lack, for instance, of precise definitions about what a Game Controlled Area may be, and a huge confusion created around the allowance of human settlement and grazing activity within it, a contradiction mainly triggered by the enactment of the Wildlife Conservation Act (2009). This act established restrictions for GCAs that are not in accordance with other still valid laws. Such overlap favours land grabbing in the name of "conservation". In fact, conservation areas destined for safaris and other tourism purposes have become a profitable business for the country's economy and foreign "investors". As shown on the map, Loliondo is a game controlled area overlapping with several registered villages. A part of Loliondo Game Controlled Area is set aside as a "hunting block" -the "hunting blocks" are areas destined for concessions, where international millionaire companies can sign agreements with the Tanzanian government and use the land for leisure hunting purposes, as private game parks. In 1992, the Dubai royal family was granted the Loliondo block, through its safari company Ortelo Business Corporation (OBC), an agreement that lasted for 25 years and provoked mass evictions (among other human rights violations) six times.

Although pastoralist NGOs have been to the national courts several times, it has not been effective so far, since the actor responsible for implementation is the perpetrator itself, that is, the Tanzanian government. That is why four Loliondo villages decided to file a lawsuit at the East African Court of Justice in 2017. Pastoralist NGOs support the case's costs via the engagement of international partners, while a current "independent activist" from Loliondo -John 7 -is connecting the different actors (pastoralist NGOs, Maasai witnesses and lawyers).

"The Loliondo case" became worldwide known mainly as a result of Avaaz campaigns 8 , which denounced threats and forced evictions of the Maasai of Loliondo villages. As John has once noted, it became a "global scandal", receiving the support of several organisations and groups from all over the world. It also highlighted the participation of the Maasai in the global indigenous movement, through the solidarity manifested towards them from indigenous groups of other continents. The preparatory meetings at Palu At the end of May 2019, twenty Maasai witnesses from Loliondo came to Arusha in order to be heard by the PALU's lawyers, who took the case on a pro bono basis. The meetings were carried out during 3 entire days, in order to file affidavits for the cross-examination. There was not so much time left to register the documents for the upcoming hearings at the court.

John would make translations between Maa (the Maasai's mother tongue) and Kiswahili (the Tanzanian official language) when requested. This would mostly happen when women were interviewed, as most female pastoralists have not been to school and therefore do not speak Kiswahili. The lawyers, therefore, preferred calling first those witnesses who could speak the language -in this case, mostly the men -to avoid using the translator. However, on the third day, the lawyers realised that they had interviewed only 12 villagers (out of 20), and because they would not have enough time to finish interviewing the others and also because they evaluated that most of the affidavits were quite similar, they made the decision to call the remaining group and ask if they wanted to add to what had already stated. The result was a torrent of new information coming from the women. They are, actually, the most affected by the evictions: they are the ones who build the houses and carry out the day-to-day activities with their children in the bomas [complex of houses and livestock], while the men are mostly away, grazing cattle or travelling. The Tanzanian lawyers were surprised: they were not aware of that. The women were also the ones who saw the houses burning down and had to run away with only the clothes on their backs. All this means that the burning of the bomas is not only a material loss to pastoralist women; it is the loss of their very daily lives.

While translating, John regretted that those women had not been listened to one by one. "Wanawake Wamaasai wanateseka sana" [Maasai women suffer a lot], he said. He was aware of the reality back in the village. After that collective talk, John tried to convince the main attorney that it was primordial to listen individually to at least two of those women (in order to prepare their affidavits and make them apt to be called by the judges). One was an elder woman, who had a strong feminist discourse and much more to say; the other, a young mother whose child became disabled after being heavily beaten by the same guards who burnt their bomas. The lawyer then chose the younger one, finding it more strategic to be used at the court. The other women's testimonies could be all concentrated in just one affidavit. John also insisted on some categories and themes that should be emphasised by the lawyers, such as vulnerability, genocide, culture and discrimination (as Maasai in a country of a Bantu majority). The lawyer then refused, by saying that the judges would probably not understand, that those were contentious issues, and that they had already enough proofs.

"We should take this opportunity to emphasise our condition as indigenous, Aline.", John told me. "Because if we win this case, it is going to be a reference to many other indigenous Aline C. Rabelo. "Haki hupewi. Haki udai" [Right is not Granted; You Have to Claim it] November 6, 2019 6 peoples… But these Swahili 10 lawyers don't understand that, because they are also from the main society; they are from the mainstream.", he said.

At the East African court of justice

At the beginning of the hearings, the applicants' lawyers, from PALU, were told that they could choose the witnesses, but preferably those who could speak Kiswahili so that the court clerk could easily translate their testimonies into English. An "ordinary" villager, although being able to speak the national language, found himself in trouble.

"How old are you? In what year were you born?", the government's attorney asks. "I think I am 40… I am not sure about the year I was born.", the villager replies. "So you don't remember your age… Where were you when the houses were supposedly burning down?" "I was in the village." "So did you see the houses burning?" "No, I didn't because when I knew about the fire, I run away." "So you mean that you were NOT in the village." "Yes, I was there in the village, but a bit far from the bomas" "On your affidavit, you are confirming that all the bomas in the village were burnt. How can you say so if you weren't in the village?", the attorney says, with a sarcastic smile. "Yes! The whole village was burnt! Wait, I know you want to confuse me, but I was in the village!" "How far from the houses in flame? How many meters or kilometres?", the attorney insists. "I don't know. I just know I was a bit far, but not too much." "So you cannot confirm where you were exactly. And what was then the date it happened?" "The date…? The day was… I cannot remember the date…", the villager says. "My Lady, my Lords, how can we trust in what this person is saying if he does not remember anything?!" Many Maasai families, especially those whose parents have never been to school, are not used to counting days neither to registering dates. As John has noted, the Maasai calendar is based on the moon's light and their years are counted according to the rains period.

The Maasai activists who attended the session (some of them lawyers) also realised that the language became a problem, as different meanings of the word "village" created misunderstandings. According to them, the word "village" [kijiji] in Kiswahili is a more defined area than what it is in Maa. In Maa, village can mean a much broader area; it can actually even represent "the whole world".

It was clear that there were gaps between the different ways of conceiving time and space, and that the government's attorney was taking advantage of a dominant intelligibility and literacy in order to forge contradiction from the witness.

Due to these and other gaps related to incongruities of translation -not just linguistic, but also cosmological -, the activists told the PALU attorneys about the urgent need to let 10 The (Wa)Swahili are only one among more than 130 ethnic groups in Tanzania, being from Bantu and Arab origins and mainly settled on the coast. Since the national language is Kiswahili (the Swahili's language) and over 90% of the Tanzanian population is of Bantu origin, the Maasai, who are Nilotic speakers, see themselves as a minority and marginalised group. They call the rest of the Tanzanian population "The Swahilis" as a metonymy. 7 the witnesses speak and express themselves freely in their mother tongue. A Maasai professional translator was then requested to take over the translation. That move made it possible to call the selected Maasai woman (the disabled child's mother) to give her own testimony. I was also told that the PALU lawyers could afterwards file new documents addressing the cultural specificities of the Maasai in order to clarify what was the object of confusion.

Conclusion

The preparatory meetings at PALU and the hearing sessions at the EACJ are highly formal mechanisms of interaction 11 , where the different actors are in interdependent relations. They need each other as a condition for the existence of those social situations. Although in interdependence, they occupy asymmetric positions, mainly due to the inequality of power which with they are vested. 12 Institutionalised rituals in urban areas often take a (hegemonic) language (in both figurative and literal senses) as granted 13 , not considering a priori the fact that the participants do not come all from the same "community of communication". There are also social situations, as we have seen, where the multiple sorts of asymmetries are deliberately used in someone's favour. However, what I want to highlight here is more than the boundaries of the "communities of communication"; instead, I want to shed some light on their porosity aspect and the possibilities of negotiation even under hierarchical interethnic dialogues.

Apel's concept of "communities of communication", as an analytical tool, seems to be useful for the sake of situating the actors in clusters according to their different social strategies, which combine shared values and interests 14 . I would add that the contingent aspect can be central in the formation of these communities of communication, revealing the possibility of different arrangements depending on the situations. They must be, therefore, seen as fluid, not as rigid clusters. The concept also allows us to better understand the movements of the indigenous activists, as they can belong to both communities of communication (of lawyers and Maasai villagers) and are circumstantially in transit between them. Eventually, we could see these Maasai brokers as effective translators as they master the codes and "language games" of both communities. In fact, they are as much in a position of transit between the different communities of communication as they are circumstantially constituting their own community of Maasai activists.

We can then observe that indigenous brokers can play an important role in institutionalised ways of governing, not just because they understand the gaps created by hegemonic and ruling knowledge, but because they represent themselves a nuance in these apparent "radical" asymmetries. Through these social situations, we were able to see how these actors negotiate spaces where hegemonic powers would tend to impose themselves. In such cases, the asymmetric interdependent relations should be seen less as arrangements 11 Gagné, 2018. 12 De l'Estoile, 2015 and Gagné, 2018. 13 De l'Estoile, 2015. 14 Pacheco de Oliveira, 2014.

  Aline C. Rabelo. "Haki hupewi. Haki udai" [Right is not Granted; You Have to Claim it] November 6, 2019 4

7

  Indigenous youth of Indonesia show their support to the Maasai of Loliondo. Source: "Serge Marti, LifeMosaic. Barisan Pemuda Adat Nusantara (Indonesian Archipelago Indigenous Youth Front). Photo taken August 25, 2016." 9 Here anonymised for security reasons. 8 Avaaz is a "online activist network", i.e., a website destined for the creation of online petitions which are widely disseminated all around the world. See https://www.avaaz.org/page/en/about/. 9 Laltaika & Askew, 2018:16. Available at https://www.un.org/development/desa/indigenouspeoples/wpcontent/uploads/sites/19/2018/01/Laltaika-and-Askew_UN-paper_rev3.pdf. Aline C. Rabelo. "Haki hupewi. Haki udai" [Right is not Granted; You Have to Claim it] November 6, 2019 5

  Aline C. Rabelo. "Haki hupewi. Haki udai" [Right is not Granted; You Have to Claim it] November 6, 2019

[START_REF] De L'estoile | Observer en réunion[END_REF][START_REF] Gagné | Ethnographie de l'État aux confins de la République. Le dispositif de l'audience et la justice foraine en Polynésie française[END_REF]

The language used during the workshop was mostly Kiswahili, the national language of Tanzania, despite the fact that 85% of the participants were Maasai (and some of they could not understand Kiswahili).

[START_REF] Jim | Power and Force in Tanzania Civil Society: The story of Barabaig NGOs in the Hanang Community Development Project[END_REF]

[START_REF] Anna | Friction. An ethnography of global connection[END_REF]

November 6, 2019 8 guided by the differences of power, but rather, as highlighting the interdependence aspect as the pivot for negotiation and less dualistic determinisms.

If rights are not granted and it is then needed to fight for them, it is not against the hegemonic apparatus that this fight takes place, but rather, through it. Claiming rights includes participation in asymmetrical interdependent relations as well as to find ways to take action despite -and within -the gaps.