Jean-Pierre Simonin
email: jpsimonin@gmail.com

Determination of thermodynamic complexity constants and speciation for multi-complexing electrolytes within the mean spherical approximation model

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Determination of Thermodynamic Complexity Constants and Speciation for Multicomplexing Electrolytes within the Mean Spherical Approximation Model

Introduction

The subject of speciation in ionic solutions in which complexation occurs is important in various domains such as environment, toxicology (human and ecological), and sometimes in the industry. Typical multi-associating ions are multivalent metal cations, such as divalent cations Zn(II), Cd(II) or Pb(II), trivalent cations Eu(III), Am(III) or Cm(III), or tetravalent cations Th(IV) or Pu(IV). These tri-and tetravalent ions, which are mainly encountered in the nuclear industry, are very harmful to living beings. Divalent metal cations are often met in environmental pollution measurements. 1 For example, Pb(II) is a major toxic trace metal in natural waters. These cations may associate with anions such as halides, e.g., Cl - or Br -, and form various complexes MCl 2-n n in which n is the stoichiometry of the complex and 2 -n is its charge.

In environmental and toxicological studies, a key parameter is the concentration of free metal cation because, generally, this is the most toxic form of a metal in solution. [1][2][3] The composition of the medium in which the cation is present determines its degree of complexation and, consequently, the amount of free metal in this medium. For example, free zinc (II) or cadmium(II) ions may be present in their free form in the pure water of a river but, when they arrive at the mouth of the river, they meet seawater chloride anions in substantial concentration (of ∼0.6M), with which they may form complexes. Consequently the toxicity of the cation may be greatly lowered when it arrives into the sea (if however one neglects the metal cation which may be adsorbed on colloids and rock pieces in suspension and might be released from these substrates when they meet salty water).

Various programs are available for the estimation of speciation in aqueous solutions. They can be useful to researchers working in toxicological, environmental and chemical engineer-ing areas, who wish to assess the amounts of the various forms of a chemical species in a particular medium (e.g., a pollutant like Zn 2+ , Cd 2+ or Hg 2+ in natural waters). The speciation may be computed for very complicated mixtures, composed of many metal cations and complexing anions. Examples of these programs are MINEQL+, CHESS, SPECIES, MINTEQ, PHREEQC and CHEAQS, [4][5][6][7][8][9] the last four being free of charge. The determination of speciation is based on the numerical solution of the various, sometimes coupled, chemical association equilibria. Values for the thermodynamic association constants are generally taken from the literature. The computations involve formulas for the estimation of the activity coefficients of the species. Deviations from ideality arising from electrostatic interactions are computed using equations of the Debye-Hückel type with a correction, such as the Davies equation 10 in CHESS, MINTEQ, PHREEQC, or the specific interaction theory 11 (SIT) which is also available in MINTEQ for more concentrated solutions. This paper is the continuation of previous work 12 (hereafter referred as I) which was dedicated to the case of binary aqueous solutions of self-complexing electrolytes. In this work, it was found that the use of the Davies equation did not yield good results for the mean salt activity coefficient, γ S , in such solutions. First the Davies equation was found to be inaccurate in the case of strong electrolytes above 0.1M, this inaccuracy increasing dramatically with salt concentration (see Figure 1 of ref. I). Next, a thermodynamic inconsistency was observed in the case of some self-complexing electrolytes (e.g., ZnCl 2 , CdBr 2 ), in the sense that γ S exhibited a significant deviation from experimental data when literature values for the complexity constants were used in the computations. A description in terms of the mean spherical approximation (MSA) theory was proposed to replace the Davies equation.

The MSA is not an empirical model. It was derived from a statistical mechanical treatment of ions in a dielectric continuum. 13,14 It can consistently describe osmotic and activity coefficients for electrolytes up to very high concentrations (to saturation in general) when concentration-dependent solution permittivity and cation size are introduced. [15][16][START_REF] Ruas | Lanthanide salts solutions: Representation of osmotic coefficients within the binding mean spherical approximation[END_REF] In ref. I, a simplified MSA model was proposed to represent activity coefficients up to ionic strengths of about 3M (I = 1 2 ∑ k z 2 k C k ), which corresponds to concentrations of 3M for a strong 1-1 charge type salt, or of 1M for a strong 2-1 salt (ionic strength is a notion adapted to mixtures of salts of different stoichiometries). The use of this model in the speciation problem showed that some complexity constants seemed to be incorrect, as in the case of ZnCl 2 , ZnBr 2 , CdBr 2 and CdI 2 , because they led to large deviations in the mean salt activity coefficient.

Thermodynamically consistent complexity constants were obtained by minimizing the deviation of activity coefficients, computed by employing the MSA model, from the experimental ones. However these were provisional values that had to be confirmed by a closer analysis.

One of the main reasons is that these salts are generally assumed to form 4 types of complexes. Consequently one may expect many sets of complexity constants to give satisfactory representations of γ S . Nonetheless it was suggested that the case of mixtures with common anion could facilitate the determination of more constrained complexity constants because this type of system is expected to produce a higher degree of association and larger amounts of complexes of higher stoichiometry.

In the present study the case of mixtures is addressed within the MSA, as was done in The LR level is that of the experiment (the 'real world') while the MM level is that of the theoretical model (at which the solution is in osmotic equilibrium with pure water Their chloride complexes are clearly of upmost importance in geochemical studies involving seawater, brines and hydrothermal solutions 1,[START_REF] Ruaya | The stability of chlorozinc(II) complexes in hydrothermal solutions up to 350 • C. Geochim[END_REF][START_REF] Anderson | A microbeam XAFS study of aqueous chlorozinc complexing to 430 • C in fluid inclusions from the Knaumuehle granitic pegmatite, Saxonian granulite massif, Germany[END_REF] in which the chloride ion is the major anion (ca. 0.6M in seawater and up to several molar in natural brines). Zinc chloride is moreover a particularly interesting salt because of the wide range of complexity constants values that have been reported in the abundant literature about this salt, [START_REF] Sillen | Stability constants of metal-ion complexes[END_REF] and because of its peculiar, both thermodynamic and transport, properties that have been long pointed out in the literature. [START_REF] Irish | Raman study of zinc chloride solutions[END_REF][START_REF] Weingaertner | Unusual behavior of transport coefficients in aqueous solutions of zinc chloride at 25 • C[END_REF] This paper is organized as follows. The next section summarizes the experimental information available about the salt solutions studied in this work. The thermodynamic data include experimental measurements for binary and ternary solutions. Then the method used to determine the speciation in these solutions is exposed in the third section. This section includes a description of the thermodynamic MSA model. The fourth section is devoted to the presentation and discussion of the results. The last section gives some concluding remarks.

Experimental information available

Complexity constants

Hereafter we will mainly be concerned with ternary aqueous mixtures of an auto-associating salt, denoted by MCl 2 , with a strong salt, denoted by NCl p (with p = 1 or 2). The former salt is composed of the divalent metal cation M 2+ and the halide anion Cl -, which gives rise to self-complexation according to the following equilibria,

M 2+ + nCl - MCl 2-n n (β n ) (1)
in which β n is the cumulative complex formation constant (on molarity scale) corresponding to the association of the metal cation with n anions,

β n = y n y 0 (y Cl -) n C n C 0 (C Cl -) n (2)
where y and C denote activity coefficient and concentration, respectively, on molarity scale.

Subscript n denotes the number of anions in the complex MCl Let us note that polynuclear complexes may appear, but mainly in highly concentrated solutions, [START_REF] Sillen | Stability constants of metal-ion complexes[END_REF][START_REF] Irish | Raman study of zinc chloride solutions[END_REF] e.g., in solutions of CuCl 2 or ZnCl 2 (above 10M [START_REF] Irish | Raman study of zinc chloride solutions[END_REF] ), which will not be the case here.

It should be mentioned that the value of β n on molality scale may be deduced [START_REF] Robinson | Electrolyte Solutions[END_REF] from that on molarity scale by dividing the latter by d 0 n (with d 0 = 0.997045 kg L -1 the density of pure water at 25 • C). The two are therefore numerically very close for any value of n ≤ 4.

Values for the β n 's of many cations have long been reported in the literature. Many of these values have been collected in famous compilations, which include the critical tables of Sillen and Martell [START_REF] Sillen | Stability constants of metal-ion complexes[END_REF] and Smith and Martell, [START_REF] Martell | Critical Stability Constants: First Supplement[END_REF] and the IUPAC and NIST databases. 6,26 More recently, a series of critical IUPAC technical reports has been published which gives an updated account of the complexity constants of Hg(II), Cu(II), Pb(II), Cd(II) and Zn(II) [27][28][29][30][31] which all occur at trace levels in fresh water and seawater. 32 In the latter series of publications the SIT model 11 was employed to describe deviations from ideality. A look at these references shows that a large scatter generally exists in the various sets of published complexity constants. As might be expected, this is particularly so in the case of weak or relatively weak complexes like, e.g., ZnCl 2 or CuCl 2 .

Structure of the complexes

In the present work, as well as in ref. I, the structure of the complexes determines the size to be taken for these compounds in the thermodynamic calculations. In the model used here, the solute species are all modeled as (possibly charged) hard spheres. An effective diameter is assigned to these spheres, depending on the number of chloride ions and water molecules bound to the central metal cation. This diameter also depends on the type of complex, viz.

covalent or ionic.

The metal cations studied in this work are Mn 2+ , Co 2+ , Ni 2+ , Cu 2+ , Zn 2+ , Cd 2+ and Pb 2+ . It seems well established that all of these cations have an octahedral structure in their free state, 33 of formula MW 2+ 6 (with W standing for the water molecule).

In contrast, the study of complexes has often been the subject of controversy, seemingly because various techniques probe different properties of these species, and because spectra obtained using a given technique are difficult to interpret and may lead to different conclusions. This phenomenon may be observed for instance in the case of NiCl 2 which has been studied extensively in the literature. 34

In the case of zinc, it must be noticed that the monochloride complex, ZnCl + , should be at least partly an outer-sphere ionic complex (solvent-separated ion pair), not a covalent one, according to two studies using different experimental techniques. 35,36 Apparently this is also the case of the NiCl + complex for which various X-ray studies found no chloride in contact with the nickel ion. 34,37-40 All other monochloride complexes were assumed to be covalent. The complexes of higher stoichiometry were all supposed to have the same structure. Namely, the dichloro complexes were regarded as being hexa-coordinated, of formula MCl 2 W 0 4 , as was proposed for zinc. 35 The tri-and tetra-chloro complexes were taken as being tetra-coordinated, with formulas MCl 3 W -and MCl 2- 4 , again similarly to what was hypothesized in the case of zinc. 35,[41][42][START_REF] Delwaulle | Etude au moyen de l'effet Raman de la constitution des solutions d'iodure, de bromure et de chlorure de zinc dissous seuls ou en présence d'ions halogènes -Mise en évidence des molécules non ionisées ZnX 2 et des ions ZnX --4[END_REF][START_REF] Quicksall | Raman spectra of terachlorozincates and the structure of ZnCl 2-4[END_REF] The change of coordination number from 6 to 4 has been attributed to the fact that it lowers the electrostatic repulsion between the charges on the chlorides. 42

Speciation and mean coordination number of the metal cation

Some experimental data about the speciation in binary solutions of ZnCl 2 may be found in the literature. These data were obtained by using dielectric spectroscopy. 36 They included estimations of the proportions of free ions and of the first complex.

X-ray diffraction and EXAFS (Extended X-Ray Absorption Fine Structure spectroscopy)

have provided an invaluable direct information by probing the first shell about the cation.

Several studies have convincingly concluded to the absence of chloride ions in the first coordination sphere of the nickel ion. 34,37-40, [START_REF] Magini | Hydration and complex formation study on concentrated MCl2 solutions [M= Co (II), Ni (II), Cu (II)] by x-ray diffraction technique[END_REF] Other works have provided information about the presence, or absence, of Cl -ions in the first shell of most cations considered in this work, including Mn(II), [START_REF] Beagley | EXAFS studies of aqueous solutions of manganese (II) chloride and bromide: structural variations with concentration and interactions with solvent[END_REF] Co(II), [START_REF] Magini | Hydration and complex formation study on concentrated MCl2 solutions [M= Co (II), Ni (II), Cu (II)] by x-ray diffraction technique[END_REF][START_REF] Musinu | X-ray diffraction study of CoCl 2 -LiCl aqueous solutions[END_REF] Cu(II), 34,[START_REF] Magini | Hydration and complex formation study on concentrated MCl2 solutions [M= Co (II), Ni (II), Cu (II)] by x-ray diffraction technique[END_REF][START_REF] D'angelo | Structural investigation of copper(II) chloride solutions using x-ray absorption spectroscopy[END_REF] Zn(II), [START_REF] Dreier | EXAFS-study of the Zn 2+ coordination in aqueous halide solutions[END_REF] and Cd(II). 40

By denoting by p n the proportion of a complex of stoichiometry n for a complex MCl n , the mean coordination number of the metal cation with chlorides may be defined as,

N Cl ≡ 4 ∑ n=n 0 n p n (3)
with n 0 = 1 when the first complex MCl + is covalent, and n 0 = 2 when it is of ionic nature (no chlorine present in the first coordination shell).

The available experimental data related to the speciation and to N Cl (in the concentration range of this study) are summarized in Table 1. Experimental uncertainties on these measurements were either found in the reference or estimated from the apparent accuracy of the experiment. Note that in Table 1 most measurements refer to concentrations higher than 1 mol kg -1 but it was verified that the corresponding ionic strength was lower than 3M (because of association).

Thermodynamic data

Thermodynamic data are available in the literature for aqueous solutions of self-complexing salts in the case of binary solutions and ternary mixtures with strong salts in water. In this N Cl = 0.9 at 1 mol kg -1 7 %

49

N Cl = 1.6 at 2 mol kg -1 7 %
work, such data for binary and ternary solutions were considered in the calculations.

In the case of mixtures, publications mainly report osmotic coefficient values. Measurements of activity coefficients are scarce. In the present work, such values were found only for HCl in the mixture ZnCl 2 +HCl and for CdCl 2 in CdCl 2 +NaCl solution.

Values for the osmotic and mean salt activity coefficients, ϕ and γ ± , for binary solutions were retrieved from the Gamphi program developed by the NBS. 50

Determination of speciation

In this section, we expose the ingredients of the method utilized in this study to compute the speciation in ternary (and binary) mixtures.

Solution of complexation equilibria

The aqueous mixture is composed of the auto-complexing salt MCl 2 and of the strong NCl In practice the speciation in a solution S was found as follows. First a dilute mixture S 1 was considered that had the same solute mole fractions of A and B as in S, and in which the total chloride concentration was 0.1M. For this composition, the system of equations was solved analytically by neglecting deviations from ideality (ideal value y i = 1 for all species).

The same was done for a solution S 2 that was twice more concentrated than S 1 . Then, the speciations in the solutions S 3 , S 4 ,... that were 3, 4,... times more concentrated than S 1 , and were now regarded as non-ideal, were determined iteratively by a Newton-Raphson procedure in which the initial speciation was extrapolated from the speciations found in the preceding two iterations. For these solutions S 3 , S 4 ,..., the activity coefficients of the various species were computed within the MSA. The same procedure was repeated until reaching the solution S (when S k =S for some integer k). For all solutions, the salt concentrations were computed from density data (see next section and the section about densities).

Calculation of concentrations and partial molar volumes

The content of this section may be found in the Supplementary Information addendum.

MSA model for the deviations from ideality

The main features of the MSA model and formulas for the activity coefficients were presented in ref. I in the case of binary solutions. We will hereafter explain how to calculate also the osmotic coefficient and how to implement the MM-to-LR conversion in the case of mixtures.

The MSA model is derived from statistical mechanics. 14 An aqueous electrolyte solution is modeled as charged hard spheres (the ions) placed in a dielectric continuum (the water).

The various types of ions may have different diameters. The MSA model has similarities with the Debye-Hückel (DH) theory, but it better accounts for the excluded volume effects between the ions. 13,[START_REF] Blum | Theoretical Chemistry, Advances and Perspectives[END_REF] It is developed at the McMillan-Mayer level of solutions in which the solvent is in osmotic equilibrium with pure water. [START_REF] Pailthorpe | Ion-solvent interactions and the activity coefficients of real electrolyte solutions[END_REF] An important property of the MM level is that the solvent may be considered as a dielectric continuum in this framework, and the pressure of the non-ideal 'gas' of solutes coincides with the osmotic pressure of solution. [START_REF] Pailthorpe | Ion-solvent interactions and the activity coefficients of real electrolyte solutions[END_REF][START_REF] Friedman | A course in statistical mechanics[END_REF][START_REF] Stigter | Interactions in aqueous solutions. II. Osmotic pressure and osmotic coefficient of sucrose and glucose solutions[END_REF] As in ref. I, the relative permittivity of the medium was taken equal to that of water because the solutions considered here are rather dilute.

In the primitive MSA model, a simple ion is modeled as a hard sphere, whose diameter reflects the hard core interactions between ions. Consequently, the ion diameter will be larger than its crystallographic value if the ion is hydrated, [START_REF] Simonin | Real Ionic Solutions in the Mean Spherical Approximation. 1. Simple Salts in the Primitive Model[END_REF] as is the case for Li + or Mg 2+ cations 15 (see section 'Sizes assigned to the solute species' below).

The expression of the activity coefficient of an ion within this primitive MSA is composed of two terms,

= -ln x + σ i F 1 + σ 2 i F 2 + σ 3 i F 3 (5)
with σ i the diameter of i, x the volume fraction of free space (not occupied by solute parti-cles). F 1 , F 2 and F 3 are functions given elsewhere, [START_REF] Simonin | Real Ionic Solutions in the Mean Spherical Approximation. 1. Simple Salts in the Primitive Model[END_REF] and ln

y el i = -λ [ Γz 2 i 1 + Γσ i + ησ i ( 2z i -ησ 2 i 1 + Γσ i + ησ 2 i 3 )] (6)
in which λ ≡ βe 2 /4πεε 0 is twice the Bjerrum length for ion-pair formation,

β ≡ 1/k B T , k B
is Boltzmann constant, e is elementary charge, ε 0 is the permittivity of a vacuum, ε is the relative permittivity of solution (the dielectric constant of water, ε = 78.4), and Γ is the MSA screening parameter (the analog of the κ screening parameter in Debye-Hückel theory) which satisfies the equation,

Γ 2 = π λ ∑ i ρ i [ (z i -ησ 2 i )/(1 + Γσ i ) ] 2 (7)
in which ρ i is the number density of species i (ρ i = 10 3 N Av C i , with N Av standing for Avogadro's number, C i in mol L -1 and ρ i in m -3 ), and η is a quantity that cancels out when the ions are of the same size (its definition is given in refs. 15,[START_REF] Simonin | Real Ionic Solutions in the Mean Spherical Approximation. 1. Simple Salts in the Primitive Model[END_REF] ). Eq. 7 has the form Γ = f (Γ). The value of Γ may be easily determined numerically by using a simple iterative procedure, starting with the initial value, Γ 0 = κ/2 (the value of Γ at infinite dilution).

As shown in ref. I, the mean activity coefficient, y

(M M ) A
, of the associated salt A is given by, 12 (y

(M M ) A ) 3 = y (M M ) M 2+ (y (M M ) Cl -) 2 C M 2+ C M ( C Cl - C Cl ) 2 (8)
with C M and C Cl the total concentrations of M (C M = C A ) and of chlorine, respectively. The mean activity coefficient of added salt B is (y

(M M ) B ) 3 = y (M M ) N 2+ (y (M M )
Cl -) 2 in the case B=NCl 2 , and (y

(M M ) B ) 2 = y (M M ) N 2+ y (M M ) Cl -
if B=NCl. These later two relations for y Then, the activity coefficient of a salt S (= A or B) in a mixture is converted to the experimental LR level by using the simple approximate formula, [START_REF] Pailthorpe | Ion-solvent interactions and the activity coefficients of real electrolyte solutions[END_REF][START_REF] Simonin | Study of experimental-to-McMillan-Mayer conversion of thermodynamic excess functions[END_REF] ln y S = ln y

(M M ) S -C t V S ϕ (M M ) /ν S (9)
with V S the partial molal volume of salt S in the mixture (V S ≡ ∂V /∂m S ), ν S its stoichiometric number, and C t the total solute concentration on a complete dissociation basis,

C t ≡ 3C A + ν B C B ( 10
)
in which ν B is the total stoichiometric number of salt B (ν B = 2 for a NCl salt and ν B = 3

for NCl 2 ).

The activity coefficient on molar scale, y S , can be converted to the molal scale, γ S , by using the classic formula, 15,[START_REF] Robinson | Electrolyte Solutions[END_REF][START_REF] Simonin | Real Ionic Solutions in the Mean Spherical Approximation. 1. Simple Salts in the Primitive Model[END_REF]

γ S = y S /(V d 0 ) (11)
We now turn to the osmotic coefficient, which is first calculated at the MM level. It may be derived from the osmotic pressure which is obtained from the general relation,

P osm = ∑ i ρ i µ (M M ) i -F (M M ) (12)
in which µ

(M M ) i
is the chemical potential of i at MM level and F (M M ) is the Helmholtz energy of the gas of solute per unit volume. Eq. 12 may be transformed by expanding µ

(M M ) i
and

F (M M )
in their ideal and excess parts, which readily yields,

βP osm = ∑ i ρ i + ∑ i ρ i ln y (M M ) i -β∆F (M M ) (13)
with y i given by Eqs. 4-6, and ∆F (M M ) = ∆F HS + ∆F el in which ∆ indicates an excess quantity (over ideality). Expressions for the HS and el contributions to ∆F (M M ) may be found elsewhere (Eqs. 3 and 29 of ref. [START_REF] Simonin | Real Ionic Solutions in the Mean Spherical Approximation. 1. Simple Salts in the Primitive Model[END_REF] ). In Eq. 12, index i runs on all species present in solution, that is free ions and complexes. The first term on the r.h.s. expresses the first deviation from ideality arising from association. Indeed one has ∑ i ρ i < ρ t , in which ρ t is the sum of densities on a complete dissociation basis, ρ t ≡ 3ρ A + ν B ρ B , similarly to Eq. 10,

or ρ t = 10 3 N Av C t .
Then the MM osmotic coefficient is obtained from the classic relation,

ϕ (M M ) = βP osm /ρ t ( 14
)
We note in passing that for an ideal solution, P osm = k B T ρ t (van't Hoff relation), and therefore ϕ (M M ) = 1 from Eq. 14, as expected.

The osmotic coefficient at the experimental LR level is obtained by conversion of ϕ (M M ) as, [START_REF] Pailthorpe | Ion-solvent interactions and the activity coefficients of real electrolyte solutions[END_REF][START_REF] Simonin | Study of experimental-to-McMillan-Mayer conversion of thermodynamic excess functions[END_REF]

ϕ = ϕ (M M ) (1 -C t V ± ) (15)
with V ± the mean solute partial molar volume of solution, which is obtained from the relation, [START_REF] Simonin | Study of experimental-to-McMillan-Mayer conversion of thermodynamic excess functions[END_REF] V ± = ∂V /∂m t , in which m t ≡ 3m A + ν B m B is the total solute molality and the differentiation is done at constant solute fractions of A and B,

x A = 3m A /m t and x B = ν B m B /m t .
The analytic expression of V ± can easily be obtained by differentiation of V using Eqs. S2

and S4 (see Supplementary Information) with the use of Maple.

The effect of the MM-to-LR conversion on y

(M M ) S
(with S= A or B) in Eq. 9 and on ϕ (M M ) in Eq. 15 was examined for the various mixtures considered in this work. It was found that both are lowered upon conversion. The maximum drop is of ∼10% for y

(M M ) S
and of ∼8% for ϕ (M M ) at the highest ionic strengths encountered in this work. The effect of the conversion is therefore not negligible.

Results for the mean salt activity coefficients of A and B (given by Eq. 11), and for the osmotic coefficient (Eq. 15), may be compared with experimental measurements for these quantities because both are obtained at the same LR level. This is done below in section "Results for the β n 's".

Gibbs-Duhem relation

It was verified that the formulas for the activity and osmotic coefficients in the FORTRAN program satisfy the Gibbs-Duhem relation both at the MM and at the experimental LR level. In the case of ternary solutions it reads,

d [(3m A + ν B m B )(1 -ϕ)] + 3m A d [ln γ A ] + ν B d [ln γ B ] = 0 ( 16
)
at the LR level. A similar relation may be written down at the MM level, by replacing molalities by molar concentrations, ϕ by ϕ (M M ) , and the γ i 's by the y i 's.

Let us stress that γ

(M M ) A , γ (M M ) B (with γ (M M ) S = y (M M ) S /(V d 0 ) by which y (M M ) S

Diameters of species for MSA model

The sizes of the various species for input into the MSA model were taken as follows.

The diameters of the free cations M 2+ (denoted by σ 0 ) were obtained from simultaneous adjustments of ϕ and γ ± (the mean salt activity coefficient) for the binary solutions of perchlorate salts, M(ClO 4 ) 2 . 15 The Pauling diameter was adopted for the chloride anion, σ Cl -= 3.62 Å. 15,16 A value of 4.53 Å was used for the perchlorate anion diameter, 15 which is in excellent agreement with the 'thermochemical' value of 4.52 Å. [START_REF] Jenkins | Reappraisal of thermochemical radii for complex ions[END_REF][START_REF] Huheey | Inorganic chemistry: principles of structure and reactivity[END_REF] The sizes of the Zn 2+ • • • Cl -and Ni 2+ • • • Cl -ionic complexes were taken as being that of a sphere having the same volume as the two free ions. The diameters of the other MCl + complexes were obtained using Eq. 17 of ref. I, which should reasonably account for the effect of hydration of the metal cation. In the case of complexes of higher stoichiometry, MCl n W p 2-n with n ≥ 2, it was assumed that the molecule is not substantially further hydrated. This may be expected because the presence of negatively charged chlorides in the complex must greatly reduce the attraction of water molecules to the metal cation. Moreover these chlorides are likely to reduce the local positive electric charge on the central cation (down from +2) through electron transfer. 35 As a consequence of these phenomena, the complexes, MCl n W p 2-n for n ≥ 2, were regarded as spheres and their diameter was also approximated as being that of a sphere having the same volume as the species constituting the complex. A diameter of 2.8 Å was used for the water molecule. [START_REF] Schatzberg | Molecular diameter of water from solubility and diffusion measurements[END_REF][START_REF] Ben-Naim | The Physics and Physical Chemistry of Water[END_REF] Results and discussion

Solution densities

The results of this section are presented in the Supplementary Information addendum.

Sizes assigned to the solute species

The diameters of the free cations (N + or N 2+ , and M 2+ ) at MM level were determined by fitting simultaneously the osmotic and activity coefficients of strong electrolytes containing the cations, up to ionic strengths of 3 mol L -1 and using the MSA formulas of ref. 15 for completely dissociated salts. This limit was the upper bound adopted in ref. I. In the calculations the concentrations were obtained using the solution densities of Table S1 (see Supplementary Information).

The chloride solutions assumed to be strong electrolytes are collected in Table 2. They include HCl, and some alkali (Li + , Na + , K + ) and alkaline earth (from Mg 2+ to Ba 2+ ) chlorides. In the case of divalent cations for which the chloride salts are self-complexing in water, the perchlorate salts of these cations were assumed to be strong electrolytes. This contention is supported by experimental evidence in the case of zinc(II), 60 cadmium(II), [START_REF] Jones | A search for perchlorate complexes. Raman spectra of perchlorate solutions[END_REF][START_REF] Ohtaki | X-ray diffraction studies on the structures of cadmium iodide complexes in water and in DMSO solutions[END_REF] and cobalt(II). [START_REF] Smithson | A possible differentiation between ion-pairs and complexes[END_REF] Somewhat in contrast, Mn(ClO 4 ) 2 has been identified as being maybe slightly associated. [START_REF] Jones | A search for perchlorate complexes. Raman spectra of perchlorate solutions[END_REF] Then thermodynamic data for perchlorates were fitted within the MSA.

The results of the simultaneous fits of ϕ and γ ± , yielding values for the free cation diameters, σ 0 , are presented in Table 2. A few remarks may be made about these results. First, the σ 0 values for Zn 2+ and Cd 2+ are slightly different from those reported in ref.

I because in the latter work no MM-to-LR conversion was used for the mean salt activity coefficient.

Next, the values of σ 0 decrease when going from LiCl down to KCl in the series of alkali metals, from MgCl 2 to BaCl 2 in that of alkaline earth metals, and from Mn(ClO 4 ) 2 to Pb(ClO 4 ) 2 . As was noted in ref. 15 this outcome is very satisfactory because these decreases of σ 0 in the three series are expected at the MM level at which the ion size includes the contribution of the hydration shell. As a consequence the diameters of cations at MM level should be larger than their crystallographic diameter, or nearly equal to it if they are weakly hydrated. For this reason the size of Cl -is taken equal to its crystallographic size because this anion is thought to be nearly unhydrated. [START_REF] Bockris | Modern Electrochemistry[END_REF] Furthermore a small cation like Li + must be more heavily hydrated, and thus have a larger σ 0 than Na + and K + (Li + > Na + > K + ) in water at MM level because it carries more water molecules than Na + and K + . The same principle holds for σ 0 in the series Mg 2+ > Ca 2+ > Sr 2+ > Ba 2+ because their crystallographic sizes are in reverse order (see last column of Table 2), and in the series

Ni 2+ > Cu 2+ > Zn 2+ > Cd 2+ > Pb 2+ .
Another point worth mentioning is that the degree of hydration of a divalent cation should be higher than that of a monovalent cation of the same crystallographic diameter.

This principle is verified in Table 2 as may be seen for example in the case of Li + , Mg 2+ and Zn 2+ , or in the case of Na + and Ca 2+ , that have comparable crystallographic sizes, and σ 0 is significantly higher for the divalent cations than for the monovalent one.

It is also observed in Table 2 that the AARD of fit of ϕ is generally appreciably lower than that of γ ± , with average values of 1.4% for ϕ vs. 2.5% for γ ± . It is noticed that the AARD's are a bit higher in the case of Mn(ClO 4 ) 2 . However, it should be noticed that for this electrolyte the possibility of a small degree of association was mentioned in a study using Raman spectroscopy. [START_REF] Jones | A search for perchlorate complexes. Raman spectra of perchlorate solutions[END_REF] This fact may be the reason for the higher AARD's of fit found in the case of Mn(ClO 4 ) 2 , if it is regarded as a strong salt.

A last point is the fact that σ 0 for K + (∼ 2.5 Å) is a bit smaller than the crystallographic diameter. This shortcoming seems to be due to the fact that the MSA model used here is simplified. This did not occur in the more refined model of ref. 15 in which the cation size and the solution permittivity were taken as being concentration dependent. In this latter version of the model, the K + diameter (at infinite dilution) was 3.45 Å and it remained larger than σ cryst at all concentrations. However it would not be possible to develop a MSA model at the same level as in ref. 15 for the present topic of mixtures containing self-associating electrolytes, because one would have to introduce dependencies of the two cation diameters (and of the solution permittivity) with the composition of solution. Such a model would be cumbersome and would introduce too many unknown parameters. Therefore one is compelled to use a simplified MSA model, which necessarily may have some shortcoming. So one has to admit that the MSA diameter of K + is small, but since it allows a good description of ϕ and γ ± in the case of KCl solution it will be accepted that this diameter value may be used in the case of mixtures in the present framework.

Then the sizes for the complexes were assessed following the method described in the section about the diameters of species. The values of the σ n 's are collected in Table 3. In the case of NiCl 2 only the value of σ 1 is given in the table because no complex of higher stoichiometry had been found in two independent experimental studies. [START_REF] Moore | Anion exchange studies. IV. Cobalt and nickel in hydrochloric acid solutions[END_REF][START_REF] Herber | Anion-exchange Studies. III. Ni(II) in Aqueous Hydrochloric Acid and Lithium Chloride[END_REF]

Results for the β n 's

In this section, wide intervals of β n values were explored for each self-complexing salt, and possibly each added salt. Initially, the lower and upper bounds were taken as 1/100 and 100 times, respectively, typical values found in the literature. For every set of β n 's the speciation, the mean coordination number, N Cl , and the thermodynamic quantities were computed according to the methods depicted in the preceding sections.

A set of values was stored when the AARD on the computed value of ϕ or γ S (in the case S= HCl or CdCl 2 , see Table 4 or 5) for the various compositions was lower than some imposed limit. In view of the results of Table 2, for most salts this uncertainty was estimated to be of the order of 1.4% on ϕ and 2.5% on γ S , which are the mean deviations of these quantities in this table. Thus, (typically a few thousands of) sets of β n values that gave an AARD <1.4% for ϕ or <2.5% for γ S were recorded. Two exceptions were the case of NiCl 2 for which the upper limit was raised to 2% for ϕ because the lowest AARD was found to be 1.5% as shown below (see Table 4), and that of PbCl 2 for which the limit was reduced to 0.9% because the lowest AARD was 0.36% for ϕ (see Table 4).

Let us underline that, in all calculations, thermodynamic measurements concerning pure binary solutions were also included in the data [START_REF] Goldberg | Program Gamphi for Calculating Activity and Osmotic Coefficients of Aqueous Electrolyte Solutions at 298[END_REF] (in addition to data for ternary solutions).

However, in the case of solutions of Mn(II), Cu(II) and Zn(II) chlorides, for which experimental speciation-related information is available (Table 1) a set of β n 's had to satisfy the additional condition that the calculated p 0 and p 1 , and/or N Cl values were in agreement with experiment, within experimental uncertainty. If not, the set of β n 's was discarded. This first analysis determined sets of 'possible' complexity constants. In the case of CoCl 2 and ZnCl 2 for which data for two different mixtures are available, the same ranges of β n values were explored.

After this scan of β n 's was accomplished, the set that gave the best consistency with experimental thermodynamic data, ϕ or γ S , was determined. Actually, in order to assess this 'optimum' set accurately, the intervals of search were subsequently sliced in smaller ones of equal width (on log scale), step by step, until this width reached a value of 0.01. A scan typically lasted between several hours and two days in the first calculations, and a few minutes in the last stage of the search.

The results for the 'optimum' sets of complexity constants are presented in Table 4 (decimal logarithms of the β n 's). It is seen in this table that the number of ϕ or γ S data was large for all mixtures, ranging from 23 to 136. The corresponding results for the speciation and for N Cl are shown in Table S2 of the Supplementary Information addendum (see also Table 1).

In a second type of calculation, a statistical analysis of the values of the possible β n 's recorded in the first stage was performed. The mean value and the related uncertainties (root mean square deviations) and AARD's (on ϕ or γ S ) were computed. They are shown in Table 5. It is seen that the AARD's for each mixture are still quite acceptable after averaging the β n values. After having determined values for the β n 's we now examine the results from the MSA model in the case of pure binary solutions. So, the osmotic coefficient, ϕ, or the mean salt activity coefficient, γ ± , for the binary solutions obtained using the mean β n values of Table 5 or 6 (see below) are shown in Figures 12345678(in which the size of the symbols is about the experimental uncertainty of ∼1%).

These results are now discussed for each salt.

Case of MnCl 2

Ref. [START_REF] Sillen | Stability constants of metal-ion complexes[END_REF] gives a value for the first complexity constant only, namely log 10 β 1 ∼ 0. It was found in a PhD thesis of 1947. This work does not seem to have been published. No value for the thermodynamic complexity constants (or, equivalently, the constants at infinite dilution) can be found in ref. [START_REF] Martell | Critical Stability Constants: First Supplement[END_REF] Figure 1 shows that the MSA model describes very well the osmotic coefficient of MnCl 2 solution up to 1.5 mol kg -1 , although some deviation is observed at low concentrations (around 0.1 mol kg -1 ) which may be due in part to a known inaccuracy of the MSA in that region. 13 The deviation is ∼ 2% at 1.75 mol kg -1 , and ∼ 4% at 2 mol kg -1 . These increasing discrepancies are very likely due to the fact that the assumptions of constant solution permittivity and constant ionic diameters become unrealistic (and unsuitable) when the concentration is increased. 15,[START_REF] Simonin | Real Ionic Solutions in the Mean Spherical Approximation. 1. Simple Salts in the Primitive Model[END_REF] However the overall representation of ϕ is satisfactory with an AARD of 1% with the β n 's of Table 5.

Case of CoCl 2

Refs. [START_REF] Sillen | Stability constants of metal-ion complexes[END_REF] and 25 give log 10 β 1 = -1.1, and log 10 β 2 = -1.28 and -3.95, respectively.

Two different sets of values for β 1 and β 2 are given in By combining the results for β 1 and β 2 for the two added salts one gets the final values shown in Table 6. With these values the AARD on ϕ remains satisfactory. Although some accuracy is inevitably lost upon averaging, they do not lead to a bad description of the osmotic coefficient. 1.3 % ZnCl 2 0.12 ± 0.10 0.17 ± 0.12 0.14 ± 0.07 -1.4 ± 0.4 1.5 % a Mean AARD for the two different mixtures (see Table 5); b β n = 0.

The MSA result for ϕ in the case of the binary solution is in particularly good agreement with the experimental data as may be seen in Figure 2, although a notable discrepancy occurs around 0.1 mol kg -1 (see previous section for MnCl 2 ).

Case of NiCl 2

Refs. [START_REF] Sillen | Stability constants of metal-ion complexes[END_REF] and 25 report no value for the thermodynamic complexity constants (i.e. β n 's at infinite dilution).

In the present study, only the first complexity constant β 1 was taken into account in the case of Ni(II), in conformity with experimental observations. 64,65 An ionic complex was assumed for NiCl + because no chloride had been found in the first coordination shell of Ni. 34,[37][38][39][40] The value found here, log 10 β 1 = 0.31 ± 0.07, should be reliable because this only unknown parameter was regressed in the calculation.

The plot of ϕ in Figure 3 reveals that, overall, the MSA model satisfactorily describes the experimental data, although some discrepancy begins to appear at the highest concentrations and a deviation is observed around 0.1 mol kg -1 (see section for the case of MnCl 2 ).

The deviation is ∼ 4.7% at 1.5 mol kg -1 .

0.8 0.9

Case of CuCl 2

Values for log 10 β 1 ranging from 0.05 to 0.4 may be found in the critical tables. For the other constants we get β 3 = 0 and log 10 β 4 = -3.7 ± 0.5. The uncertainty on the value of β 4 might actually be larger because very little amounts of the fourth complex were found in the calculation (maximum value of p 4 ∼ 0.03).

The result from the MSA model for ϕ in the case of pure binary CuCl 2 solution with the

β n values of

Case of ZnCl 2

Thermodynamic data for two mixtures containing ZnCl 2 were retrieved from the literature.

Different quantities were measured for these two different mixtures, namely activity coefficient values γ HCl for added HCl, and osmotic coefficient values ϕ for added BaCl 2 .

The first important remark in the case of ZnCl 2 is that attempts of calculations with n 0 = 2, i.e. assuming that ZnCl + is an outer-sphere ion pair, were unsuccessful. No β n values could be found that gave AARD's of fit below the limits imposed (< 1.4% for ϕ or < 2.5% for γ HCl together with satisfactory p 0 , p 1 and N Cl values). The origin of this unexpected result is difficult to ascertain. Various possibilities may be mentioned, such as the first ZnCl + complex being (totally or partly) an inner-sphere complex instead of an outer-sphere ionic complex, inaccuracies in the values of p 0 , p 1 or N Cl , or the existence of an additional phenomenon of unknown nature. Let us notice that the latter possibility is not quite unlikely in view of the many peculiarities exhibited by zinc chloride solutions. [START_REF] Irish | Raman study of zinc chloride solutions[END_REF][START_REF] Weingaertner | Unusual behavior of transport coefficients in aqueous solutions of zinc chloride at 25 • C[END_REF] In the face of these uncertainties, it was assumed that the monochloro complex is not ionic but covalent (inner-sphere ZnCl + complex). The value n 0 = 1 was taken in eq. 3 for the two added salts HCl and BaCl 2 , and the two conditions on the value of p 1 (see Table 1)

were relaxed because the estimation of p 1 included the hypothesis of an ionic complex. The diameter of the first complex was now σ 1 = 6.104 Å.

The results from this procedure are given in Table 5. One notices in this table that the confidence intervals for the two sets of β n values in Table 5 do overlap. Combination of the results for the β n 's for the two added salts leads to values given in Table 6.

Values for the complexity constants may be found in the critical tables [START_REF] Sillen | Stability constants of metal-ion complexes[END_REF][START_REF] Martell | Critical Stability Constants: First Supplement[END_REF] (mostly taken from the work of Marcus The results from this work (Table 5) are compatible with literature values but at the lower end of their confidence intervals as may be seen in Figure 5. [START_REF] Sillen | Stability constants of metal-ion complexes[END_REF][START_REF] Martell | Critical Stability Constants: First Supplement[END_REF]31 We note that, in the present computations, appreciable quantities of the third and fourth complexes are formed when the added salt concentration is much larger than that of ZnCl 2 .

The maximum proportions of these complexes in mixtures with HCl and BaCl It is noticed in Figure 6 that the result for ϕ with set 1 is very good below 0.8 mol kg -1 .

Above this concentration it deviates from the experimental data, gradually to the downside.

The deviation is ∼ -4.7% at 1.5 mol kg -1 .

However it is much better than the plot corresponding to set 2 (the dashed line in Figure 6). The result found using set 2 follows experimental data up to a low concentration of ∼0.05 mol kg -1 . The discrepancy from the experimental data at 1.5 mol kg 4 and5) for β 1 and β 2 are in keeping with the literature values. Our value for β 3 , log 10 β 3 ∼ 2.9, is a bit larger.

Here we find, β 4 = 0, that is no complex of order 4. As pointed out in ref. 30 the constant proposed by Smith and Martell is quite uncertain.

As already mentioned in ref. I, the first complex is rather strong (rather high complexity constant β 1 ). Consequently the behavior of the binary solution at low concentration is governed by the effect of this first complex, and there is therefore very little uncertainty on the value of β 1 whatever the model used to determine it (as long as the model is equivalent to DH theory at low concentration). This is also true for the second complex.

Figure 7 shows that the MSA result reproduces very well the experimental values for γ ± in pure binary solution with the β n 's of Table 5

Case of PbCl 2

The values for the constants reported in the critical tables [START_REF] Sillen | Stability constants of metal-ion complexes[END_REF][START_REF] Martell | Critical Stability Constants: First Supplement[END_REF] are in the following ranges: 1.1-1.62, 1.78-2.44, 1.4-1.86 and 0.81-1.4, for n = 1, . . . , 4 respectively. More recent studies 29,[START_REF] Luo | Stability constants for the formation of lead chloride complexes as a function of temperature and ionic strength[END_REF] report log 10 β 1 = 1.5, log 10 β 2 ∼ 2 and 2.1, log 10 β 3 ∼ 2, and no fourth complex, β 4 = 0 as found previously. [START_REF] Hefter | Stability constants for the lead (II)-halide systems[END_REF] The values found here (see Table 5), log 10 β 1 ∼ 1.5 and log 10 β 2 ∼ 1.9, and β 4 = 0 are in keeping with these latter values. The value of the first complexity constant is likely to be accurate because the first complex is rather strong. At the low maximum concentration of 0.039 mol kg -1 (which corresponds to saturation), the high calculated proportion of the first complex, p max 1 ∼ 0.37, guarantees that the model is sensitive to β 1 . In contrast, for the second complex at saturation we find p 2 ∼ 0.033, which is likely insufficient to ensure a good accuracy in the determination of β 2 .

Our value for the third complexity constant (log 10 β 3 ∼ 3.3) is larger than the literature values. However, since p 3 ∼ 0.046 at saturation, the actual uncertainty on β 3 must be rather high.

The low solubility of PbCl 2 in water at 25 • C cannot allow us to assess these constants accurately. Data for mixtures with a chloride salt would be needed to be able to reach the constants β 2 and β 3 reliably. Such data seem to be unavailable at present time.

It is seen in Figure 8 that the osmotic coefficient, ϕ, computed from the MSA model with the β n values of Table 5 is in very good accord with the experimental data.

Sensitivity of results to the diameters of the complexes

The sensitivity of the above results for the β n 's w.r.t. the sizes of the complexes was examined by varying their diameters by ±5%. The variation was made for one complex at a time.

This modification had nearly no effect in the case of CdCl 2 or PbCl 2 because the maximum ionic strength was rather low for the first salt (0.8M) and very low for the second (0.08M).

It was reduced in the case of the mixture ZnCl 2 +HCl.

This was not so for the other salts where the ionic strength was of 2.4M or 2.8M. This is because the hard sphere parts of the osmotic and activity coefficients vary rapidly with the size of the species (see, e.g., Eq. 5). At the highest ionic strengths it was observed that the values of log 10 β n varied by ∼ ± 0.1 on average when the diameter of a complex was individually varied by 5%, with variations in the range of 0.02 to 0.18.

Conclusion

Values for the complexity constants of the divalent cations Mn 2+ , Co 2+ , Ni 2+ , Cu 2+ , Zn 2+ , Cd 2+ , and Pb 2+ , with the chloride anion have been determined from thermodynamic and speciation data. The thermodynamic data consisted of values for the osmotic coefficient ϕ or the mean activity coefficient γ S , for binary solutions of these self-complexing salts and for mixtures with strong salts in water. The speciation-related data included proportions of free cation and first complex, and mean metal-chlorine coordination number.

The MSA model was employed to describe the thermodynamic properties of these ionic solutions. The simplest version of this model has been used in which the solute species have a fixed diameter and the permittivity of solution is that of pure water. The diameters of the species were assessed from their structure. The study was limited to ionic strengths below 3M. In this framework the complexity constants are the only unknown parameters.

In order to be able to consider higher ionic strengths, a natural extension of the present model would be to introduce concentration dependencies for the solution permittivity and for the diameters of the ionic species, as has been done before. 15,16,[START_REF] Triolo | Simple electrolytes in the mean spherical approximation. 2. Study of a refined model[END_REF] The main difficulty in the present work is that many ionic species are present. This would therefore introduce many unknown parameters (at least 5 already for a binary solution), which would make the model hardly tractable in practice. Nevertheless it might be proposed to use empirical formulas for the parameters. This possibility should be examined in future work.

In the present study, the thermodynamic properties and speciation data (when available) could be described consistently for all salts, except in the case of ZnCl 2 for which no satisfactory result could be found when the first complex was assumed to be ionic. Alternatively the ZnCl + pair was supposed to be an inner-sphere covalent complex, in which case sets of β n values could be determined.

The exception of ZnCl 2 in this work reminds one of the peculiarities that have long been observed experimentally when measuring the thermodynamic and the transport properties of this electrolyte. [START_REF] Irish | Raman study of zinc chloride solutions[END_REF][START_REF] Weingaertner | Unusual behavior of transport coefficients in aqueous solutions of zinc chloride at 25 • C[END_REF] It is the view of the present author that the structure of ZnCl 2 aqueous solutions is still not well known. It is not unlikely that some unknown phenomenon is at work in these solutions, that remains to be discovered.

  ref. I. However, in contrast with ref. I, since mostly osmotic coefficient data (ϕ) are available in the literature for these mixtures, it was thought preferable to handle thermodynamicallyconsistent osmotic and ionic activity coefficients (this was not the case in ref. I, in which only mean salt activity coefficients were optimized). This constraint requires the use of the so-called McMillan-Mayer-to-Lewis-Randall (MM-to-LR) conversion of these quantities.

1- 1

 1 salt, or NCl 2 2-1 salt. For convenience we will hereafter denote by A the self-complexing salt MCl 2 and by B the strong salt NCl or NCl 2 .For a given solution composition and a given set of β n values for n going from 1 to 4, the speciation (viz. the set of C n values) was found by employing a Newton-Raphson method in 6 dimensions. The 6 equations to solve were the 4 equations for the equilibria (Eq. 2 for n=1,...,4) in which the activity coefficients, y n , were computed according to the MSA model, plus the 2 relations for the conservation of total M and Cl (the concentration of N + or N 2+ is not modified by association). In the Newton-Raphson algorithm, the equations were written as a function of the variables {ln C n } (not the C n 's) in order to avoid the appearance of negative values of the C n 's in the iterations. The inverse of the 6×6 Jacobian matrix was calculated analytically, and thereafter translated directly into FORTRAN, by using Maple (the symbolic calculation software). Equilibration was performed at the McMillan-Mayer level.

  the result that the chemical potentials of A and B are equal to the sum of the chemical potentials of the corresponding free ions (see pp. 9724-9725 of ref. I).

  is converted to molality scale), and ϕ (M M ) do not satisfy the Gibbs-Duhem relation. MM-to-LR conversion (Eqs. 9 and 15) is required to obtain quantities on molality scale that indeed satisfy this fundamental relation. Fulfilment of Eq. 16 was examined numerically for various arbitrary values of the β n 's.The differentials (d [• • • ]) were computed in the vicinity of 2 particular molalities, e.g. m A = 1 mol kg -1 and m B = 2 mol kg -1 . The Gibbs-Duhem relation was deemed to be verified when the sum of the 3 terms in Eq. 16 was much smaller than any of the 3 terms. This was found to be the case for any values of m A and m B , with a typical accuracy of the order of 10 -8 or 10 -9 in a double precision FORTRAN program.The numerical fulfilment of Eq. 16 is a stringent condition that ensures, with a high level of confidence, that all thermodynamic quantities (ϕ (M M ) , ϕ, the y i 's and the γ i 's) are calculated correctly in the program. It was observed that even a slight modification in a single formula of the FORTRAN program resulted in a clear unfulfillment of the Gibbs-Duhem relation.

Figure 2 :

 2 Figure 2: Result for the osmotic coefficient, ϕ, of CoCl 2 as a function of molality computed with the β n values ofTable 6. ( ⊙ ) Experimental data; 50 solid line: result from MSA model.

Figure 5 :

 5 Figure 5: Values for the β n 's in the case of ZnCl 2 for n = 1,2,3: (⊙) Values found in this work in the case of mixtures with HCl, and ( ) in the case of BaCl 2 added salt; (•) Literature values.[START_REF] Sillen | Stability constants of metal-ion complexes[END_REF][START_REF] Martell | Critical Stability Constants: First Supplement[END_REF]31

25

 25 Figure 6: Result for the osmotic coefficient, ϕ, of ZnCl 2 solution as a function of molality. ( ⊙ ) Experimental data; 50 solid line: result from MSA model with the β n 's of Table6; dashed line: result from MSA model computed with the β n values of Smith and Martell.[START_REF] Martell | Critical Stability Constants: First Supplement[END_REF]

and β 3

 3 are very close to these latter values (no value was proposed for β 4 because of the high uncertainty on constant). The values found in this work (Tables

Figure 8 :

 8 Figure 8: Result for the osmotic coefficient, ϕ, of PbCl 2 as a function of molality with the β n values of Table 5. ( ⊙ ) Experimental data; 50 solid line: result from MSA model.

  [START_REF] Pailthorpe | Ion-solvent interactions and the activity coefficients of real electrolyte solutions[END_REF]

  2-n n . The subscript n = 0 is used to designate the free metal ion M 2+ . So, C 0 ≡ C(M 2+ ) and C n ≡ C(MCl 2-n n ). For most metal cations considered in this work, experiment has suggested a maximum value of n = 4.

Table 1 :

 1 Experimental information on the speciation and on N Cl .

	Salt	Ref.	Experimental data	Uncertainty
	MnCl 2	46	N Cl = 0.4 at 1 M	50 %
	CuCl 2	48	N Cl = 0.8 for 0.1 M CuCl 2 + 0.8 M NaCl	50 %
		48	N Cl = 1 for 0.1 M CuCl 2 + 2.8 M NaCl	25 %
	ZnCl 2	36	p 0 = 0.51 at 1.62 M	20 %
		36	p 0 = 0.46 at 2.08 M	20 %
		36	p 1 = 0.14 at 1.47 M	20 %
		36	p 1 = 0.12 at 1.89 M	20 %
		49

Table 2 :

 2 Results for the regression of cation diameters, σ 0 , in the simultaneous fits of ϕ and γ ± for strong binary electrolytes (with σ Cl -= 3.62 Å and σ ClO -

	4 = 4.53 Å).

Table 3 :

 3 Values of the diameters σ n (in Å) taken for the complexes (with n the number of chlorine atoms in the complex), with σ Cl -=3.62 Å.

	Salt	σ 1	σ 2	σ 3	σ 4
	MnCl 2 6.523 5.482 5.162 5.352
	CoCl 2 6.360 5.478 5.158 5.348
	NiCl 2 6.618	-	-	-
	CuCl 2 6.202 5.513 5.180 5.364
	ZnCl 2 6.418 5.495 5.177 5.366
	CdCl 2 5.751 5.549 5.237 5.422
	PbCl 2 5.134 5.658 5.343 5.516

Table 4 :

 4 'Optimum' values for the complex formation constants, log 10 β n .

	c	β 2	c	β 3	c	β 4	c

Salt

Added salt N a Max.

I b AARD of fit Ref.

β 1 a Number of data points; b Maximum ionic strength; c Decimal logarithm; d β n = 0; e Measurement of γ HCl ; f Measurement of γ CdCl 2 ; g Binary solution.

Table 5 :

 5 Mean values for the complexity constants, log 10 β n . Binary solution; b β n = 0; c Measurement of γ HCl ; d Measurement of γ CdCl 2 .

	Salt	Added salt	log 10 β 1	log 10 β 2	log 10 β 3	log 10 β 4	AARD
	MnCl 2 + NaCl	0.46 ± 0.04 -0.36 ± 0.06	-b	-b	1 %
	CoCl 2 + NaCl	0.29 ± 0.06 -0.67 ± 0.12	-b	-b	1.1 %
		+ KCl	0.17 ± 0.09 -0.35 ± 0.09	-b	-b	1.4 %
	NiCl 2	+ NaCl	0.31 ± 0.07	-b	-b	-b	1.5 %
	CuCl 2 + NaCl	0.43 ± 0.02 0.16 ± 0.02	-b	-3.7 ± 0.5 0.8 %
	ZnCl 2	+ HCl c	0.10 ± 0.09 0.17 ± 0.12 0.14 ± 0.06 -1.4 ± 0.5 1.8 %
		+ BaCl 2	0.20 ± 0.08 0.15 ± 0.11 0.12 ± 0.08 -1.5 ± 0.4 1.1 %
	CdCl 2 + NaCl d	1.94 ± 0.02 2.55 ± 0.06 2.88 ± 0.08	-b	2 %
	PbCl 2	a	1.50 ± 0.03 1.90 ± 0.06	3.3 ± 0.2	-b	0.4 %

a

Table 5

 5 Figure 1: Result for the osmotic coefficient, ϕ, of MnCl 2 as a function of molality computed with the β n values ofTable 5. ( ⊙ ) Experimental data; 50 solid line: result from MSA model.

	1.3
	1.2
	1.1
	φ
	1
	0.9
	0.8	0	0.5	1	1.5	2
				m (mol/kg)

for the two added salts NaCl and KCl, for which experimental data were obtained by two different groups. The two sets are in quite reasonable agreement.

Table 6 :

 6 Final values for the complexity constants of CoCl 2 and ZnCl 2 .

	Salt	log 10 β 1	log 10 β 2	log 10 β 3	log 10 β 4	AARD a
	CoCl 2 0.23 ± 0.10 -0.52 ± 0.19	-b	-b

  Figure 3: Result for the osmotic coefficient, ϕ, of NiCl 2 as a function of molality computed with the β n values ofTable 5. ( ⊙ ) Experimental data; 50 solid line: result from MSA model.

	1.3
	1.2
	1.1
	φ
	1
	0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6
				m (mol/kg)

  [START_REF] Sillen | Stability constants of metal-ion complexes[END_REF][START_REF] Martell | Critical Stability Constants: First Supplement[END_REF] A more recent work of Powell et al.28 proposed the value log 10 β 1 = 0.83±0.09. The value found here, 0.43 ± 0.02, is at the high end of the range of values from the first source and below that of the second one.As regards the other constants only a value for β 2 was given in the same work: 28 log 10 β 2 = 0.6±0.3. No value for the other constants is available in the literature. Here we obtained log 10 β 2 = 0.16 ± 0.02, which is lower than that of Powell et al. but of the same order of magnitude.

Table 5 in

 5 Figure 4 is good up to 1.5 mol kg -1 . Above, it deviates more and more from experiment (see section for MnCl 2 ). The maximum deviation is of ∼ 2% at 2 mol kg -1 .Figure 4: Result for the osmotic coefficient, ϕ, of CuCl 2 as a function of molality computed with the β n values of Table 5. ( ⊙ ) Experimental data; 50 solid line: result from MSA model.

	1.1
	1
	φ
	0.9
	0.8
	0	0.5	1	1.5	2
			m (mol/kg)

  [START_REF] Marcus | Anion exchange of metal complexes. VIII. The effect of the secondary cation. The zinc-chloride system[END_REF] from which uncertainties may be estimated as mentioned in ref. I) and from a recent work.31 From these sources we get: log 10 β 1 ∼ 0.4±0.2, log 10 β 2 ∼ 0.6±0.3, log 10 β 3 ∼ 0.5 ± 0.5, and log 10 β 4 ∼ 0.2 ± 0.2. The last value is however highly uncertain as pointed out by Powell et al.31 and should not be retained according to this last reference.

  2 were p max HCl is often in good agreement with the experimental measurement. Consequently the result for ϕ or γ HCl is sensitive to the values of β 3 and β 4 when the amount of chloride is much larger than that of Zn(II).The MSA result with the β n 's of Table6(denoted as set 1) and that using Smith and Martell values 25 (set 2) are plotted in Figure6. The latter values from Smith and Martell are: log 10 β n = 0.46, 0.60, 0.50 and 0.20 for n = 1,. . .,4, respectively.

		3	∼
	0.31 and 0.49, and p max 4	∼ 0.23 and 0.34, respectively. Moreover in these cases the calculated
	ϕ or γ

  As was found in ref. I, these results suggest that the classic β n values give thermodynamic quantities (ϕ or γ S ) that are not consistent with experimental data in the case of ZnCl 2The case of cadmium chloride is simpler than the previous one because it gives rise to stronger complexes. The recommended values from ref.[START_REF] Martell | Critical Stability Constants: First Supplement[END_REF] are log 10 β 1 = 1.98 ± 0.03, log 10 β 2 = 2.6 ± 0.1, log 10 β 3 = 2.4 ± 0.1 and log 10 β 4 = 1.7. The values of Powell et al. 30 for β 1 , β 2Figure 6: Result for the osmotic coefficient, ϕ, of ZnCl 2 solution as a function of molality. ( ⊙ ) Experimental data; 50 solid line: result from MSA model with the β n 's of Table 6; dashed line: result from MSA model computed with the β n values of Smith and Martell.

		1
		0.95
		0.9
		0.85
	φ	0.8
		0.75
		0.7
		0.65
		0.6
		0.55	0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6
						m (mol/kg)
	solution.
	Case of CdCl 2

-1 reaches -23%. This deviation is similar to that found in ref. I for the mean salt activity coefficient when deviations from ideality were computed using the (inaccurate) Davies equation or the SIT model and β n sets of values taken from the literature.

  . At 1.5 mol kg -1 the mean activity coefficient of CdCl 2 is very low (γ ± ∼ 0.05) because of the strong association. Figure 7: Result for the mean salt activity coefficient, γ ± (log-scale), of CdCl 2 as a function of molality computed with the β n values ofTable 5. ( ⊙ ) Experimental data; 50 solid line: result from MSA model.

		0.9
		0.5
		0.4
		0.3
	γ ±	0.2
		0.1
		0.05
		0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6
					m (mol/kg)
						29

Table S2 :

 S2 Results for the densities of ternary aqueous electrolyte mixtures (using Eq. S2). C i , summed on all ions (on a complete dissociation basis). results in the case of a self-associating salt mixed with a strong salt. Nevertheless, this latter accuracy should be sufficient in this work in which densities are used to compute concentrations.

	Mixture		Max. I a Accuracy of fit Ref.
			(mol L -1 )
	CaCl 2 + NaCl		11.2	0.05 %	15
	CaCl 2 + KCl			6.5	0.06 %	15
	ZnCl 2 + KCl a Maximum ionic strength: I = 1 2	∑	Z 2 i	9.8	0.7 %	16

Results for the proportion of free cation, p 0 , and for N Cl

Table S3 :

 S3 Experimental (exp) and calculated (cal ) values for p 0 and p 1 , and the mean coordination number, N Cl , for the β n values of Table6.

	Salt	Concentration	p exp 0	p cal 0	N exp Cl	N cal Cl
	MnCl 2	1 M			0.4	0.6
	CuCl 2 0.1 M CuCl 2 + 0.8 M NaCl			0.8 0.41
		0.1 M CuCl 2 + 2.8 M NaCl			1	1.2
	ZnCl 2	1.62 M	0.51 0.45
		2.08 M	0.46 0.40
		1 mol kg -1			0.9 0.88
		2 mol kg -1			1.6 1.42

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI:...

Calculation of concentrations and partial molar volumes; Results for solution densities;

Results for the proportion of free cation, p 0 , and for N Cl .

(26) Smith, R.; Martell

Abstract Graphic

-osmotic coefficient E-mail: jpsimonin@gmail.com

Self-complexing electrolyte mixture

MSA thermodynamic model

Calculation of concentrations and partial molar volumes

Concentrations and partial molar volumes of salts were derived from an expression for the density of solution. In ref. I, the density of binary solutions of a salt S was represented by the formula,

in which d 0 is the density of pure water, m S the molality of salt S, and d 1S and d 2S are two salt-dependent parameters.

In the present work, it was assumed to have the form,

The form of Eq. S2 was inspired from an equation in terms of concentrations. 1 The concentration, C i , of any species i can be found from its molality m i as,

in which V is the specific volume of solution, i.e. the volume of solution per kg of water in the solution. One has,

with M S the molar mass of salt S (in kg) and d the density given by Eq. S2.

The partial molar volume of salt S (= A or B) in a mixture, V S = ∂V /∂m S , can be obtained by analytic differentiation of Eq. S4 with d given by Eq. S2. The software Maple was used to handle this type of symbolic calculation throughout, and to translate the formulas directly into FORTRAN.

Results for solution densities

Experimental densities of binary solutions at 25 • C were fitted using Eq. S1. The results for the parameters d 1S and d 2S are given in Table S1. and infinite dilution partial molar volume data for Pb 2+ and ClO - 4 ions. 13,14 Then, the accuracy of Eq. S2 to describe the densities of ternary solutions of chloride salts at 25 • C was examined in the case of strong and associating electrolyte mixtures. Such systems are not plentiful but data for three mixtures were found in the literature. In this comparison with experimental values no new parameter was introduced. The results are shown in Table S2.

These results may suggest that Eq. S2 gives accurate results for the estimation of ternary solution densities of strong electrolytes up to high concentrations, and probably less accurate


Dr Dennis Kurzbach
email: kurzbach@ens.fr

Estel Canet

Prof Andrea G R Flamm

Aditya Jhajharia

Prof Emmanuelle M M G Weber

Robert Konrat

Geoffrey Bodenhausen

Departementdec Bodenhausen

Himie

Konrat

Investigation of Intrinsically Disordered Proteins through Exchange with Hyperpolarized Water

Keywords: dynamic nuclear polarization, hyperpolarized water, intrinsically disordered proteins, N MR spectroscopy, solvent exchange

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Intrinsically disordered proteins (IDPs) constitute ac lass of physiologically active proteins that lack stable secondary and tertiary structures. [1] Theh ighly dynamic nature of these proteins endows them with an inherent ability to interact simultaneously with multiple binding partners in the cellular environment. [2] IDPs thus frequently constitute important hubs in protein interaction networks.T he malfunction of IDPs often leads to severe diseases associated with deregulation of the cellular machinery. [3] Since intrinsically disordered proteins or regions constitute approximately 30 %o f the human proteome, [4] understanding their functionality and structure-function relationships is clearly desirable,especially with aview to developing novel therapeutic strategies for the treatment and prevention of diseases related to IDP malfunction. [5] Thei nherent dynamics and flexibility of IDPs explain their astonishingly rich interactome.N uclear magnetic resonance (NMR) spectroscopy has developed into akey method for the study of IDPs. [6] Multidimensional techniques and high-field NMR spectrometers give access to residue-resolved dynamics of IDPs at ambient temperature and low pH. [START_REF] Zawadzka-Kazimierczuk | [END_REF] However,u nder physiological conditions,t he spectra typi-cally suffer from strong signal overlap and broadening,t hus leading to peak intensities below the detection threshold. [START_REF] Rule | Fundamentals of Protein NMR Spectroscopy[END_REF] To the best of our knowledge,only one study has been published so far that employs 1 H-15 Ncorrelation spectra of IDPs under physiological conditions at pH 7.4 and 37 8 8C. [START_REF] Theillet | [END_REF] These limitations in the study of IDPs can be overcome by using hyperpolarized HDO [10] produced through dissolution dynamic nuclear polarization (D-DNP). [11] Because exchange between hyperpolarized HDO and amide H N protons only enhances al imited set of solvent-exposed residues under near-physiological conditions, [START_REF] Rule | Fundamentals of Protein NMR Spectroscopy[END_REF] the 1 H-15 Ncorrelation spectra are sparse and can be assigned despite the poor dispersion of their chemical shifts. [12] Such sparse spectra are reminiscent of "spectral dilution" techniques that rely on 13 Cl abelling of selected amino acids. [13] Felli and co-workers have advocated the use of 13 Cd irect detection techniques that allow the observation of IDPs under physiological conditions in 13 C-15 N correlation spectra but do not show any proton signals,which are the object of our work. [14] Osteopontin (OPN,Figure 1a)isacancer-and metastasisassociated IDP that can interact with heparin, an analogue of the natural ligand hyaluronic acid (HA). [15] By using hyperpolarized HDO to selectively observe residues that undergo fast exchange with the solvent, it is possible to follow the structural and dynamical adaptions of OPN induced by binding to heparin at pH 7.4 and 37 8 8C.

To hyperpolarize water, we employ dissolution DNP (D-DNP). [16] Av olume of 100 mLo fa20 mm solution of 4hydroxy-2,2,6,6-tetramethylpiperidin-1-oxyl (TEMPOL) in H 2 O, mixed 1:1(v/v) with glycerol as aglass-forming solvent, is frozen at 1.2 Ki nam agnetic field of 6.7 T. Microwave irradiation near 188.2 GHz on resonance with the paramagnetic centers leads to hyperpolarization P( 1 H) = 89 %i n the glassy solid. [10a] The2 00 mLh yperpolarized solid are rapidly dissolved by ab urst of 5mL( 25-fold dilution) preheated D 2 O( 180 8 8C, 10.5 bar) and transported in approximately 6sthrough am agnetic tunnel to as uitably buffered protein solution that is waiting in a5mm NMR sample tube in ah igh-resolution 800 MHz NMR spectrometer.T he protein solution was adjusted to pH 8b efore dissolution, so that anear-physiological pH 7.5 is reached after dilution. The temperature of the sample was approximately 308 K( 35 8 8C) after mixing.O ur approach is summarized in Figure 1b. [17] Themost exposed amide (H N )protons of the protein rapidly exchange with the hyperpolarized HDO (after 40-fold dilution of H 2 Oi nD 2 O, most solvent protons are found in the form of HDO), while non-exposed residues exchange slower.A fter ad elay of 8s to allow turbulence to settle, ab and-selective optimized-flip-angle short-transient hetero-nuclear multiple-quantum correlation ( 15 N-1 HS OFAST-HMQC) spectrum [18] was recorded in 19.2 s(see the Supporting Information).

Frydman and co-workers [19] have reported similar experiments,a lthough their 1 H-15 Nc orrelation spectra do not feature any resolved resonances of individual amino acids that would be amenable to assignment. By contrast, we show that it is possible to monitor the folding and unfolding of individual residues of an IDP induced by the presence of ligands like heparin under near-physiological conditions. After dissolution, the TEMPOL concentration drops below 0.25 mm.T he lifetime of the HDO proton hyperpolarization is T 1 ( 1 H) = 29 s, which is close to the value of HDO without TEMPOL.

Thea verage relaxation time T 1 ( 1 H N )o famide protons in OPN is on the order of 1s , [20] but the proton polarization is replenished by exchange as long as the HDO (which is approximately in 100-fold excess with respect to the protein) remains hyperpolarized. Theproton signals must be detected before the polarization becomes too weak.

Figure 2a shows aSOFAST-HMQC using hyperpolarized HDO (red) obtained in less than 30 scompared with aregular SOFAST-HMQC spectrum (black) of the same sample obtained in thermal equilibrium over 18 h. Note that the peaks of the regular SOFAST-HMQC are very broad, as expected at pH 7.5 and 35 8 8C. [START_REF] Rule | Fundamentals of Protein NMR Spectroscopy[END_REF] IDPs generally give rise to poorly dispersed peaks,and the signals are further broadened by fast proton exchange under our experimental conditions. [21] Relaxation of the amide protons (H N )ofOPN by TEMPOL is negligible. [22] In the red hyperpolarized spectrum in Figure 2a,weonly see enhanced signals for the most labile H N protons,which can exchange rapidly with HDO. [START_REF] Rule | Fundamentals of Protein NMR Spectroscopy[END_REF] Thee xchange rates are estimated to be between 30-140 s À1 at 35 8 8C.

Thep robability that ah yperpolarized proton exchanges with OPN is at best 1/40 owing to the dilution of hyperpolarized protons with non-polarized protons and deuterons. Thus,f or an exchange rate of 30 s À1 and ar ecovery delay of 0.3 s( which was optimized empirically for our experiments), the probability to detect ahyperpolarized H N amide proton is only about 25 %. This probability rises for exchange rates near 140 s À1 .T hus,o nly rapidly exchanging protons have as ufficiently high probability to be detected, as displayed in Figure 2a.

Only residues that can exchange efficiently with the solvent on this time scale will be visible in our experiments, while residues that exchange more slowly will not be observed. Thes pectrum appears as if observed through ahyperpolarized exchange filter that suppresses 1 H-15 Ncross peaks for protons that slowly exchange with the solvent. Additionally,m any peaks appear in the hyperpolarized SOFAST HMQC that cannot be observed through other methods owing to weak signal intensity at pH 7.5 and 35 8 8C. Thus,e xchange with the hyperpolarized HDO boosts the magnetization of fast exchanging residues above the detection limit.

In Figure 2b,a ne xpansion of the serine and threonine region is shown. Our method not only has the advantage that the signals are amplified, but also that the enhancement is selective.T he signals can therefore be assigned to their respective residues,which is not possible in aconventional 2D correlation spectrum because of extensive signal overlap (Figure 2b,b lack). Figure 2c shows the "bulk" signal of our hyperpolarized exchange filtered spectrum. Figure 2d and Figure S1 in the Supporting Information show overlays of OPN spectra, either enhanced by hyperpolarized HDO at pH 7.5 and 35 8 8C, or detected in thermal equilibrium at pH 6.5 and 25 8 8C, where the NMR signals are strongest in the absence of hyperpolarization. [15b] In high-resolution spectra obtained without hyperpolarization, many signals disappear when going from NMR-optimized to near-physiological conditions, that is,f rom pH 6.5 and 25 8 8Ct o7 .5 and 35 8 8C. These signals can be recovered through exchange with hyperpolarized water. Thus,afurther advantage of our D-DNP technique is the observation of signals that are broadened beyond detection in conventional methods,a lthough the resolution in hyperpolarized HMQC spectra suffers from rapid accumulation (< 30 s) after dissolution. Notwithstanding significant signal overlap,s ome peaks can still be assigned unambiguously.E ven the N-terminal amino acids can be observed, which is normally impossible owing to fast solvent exchange.H ere,h owever,f or the very same reason, they become observable.Our method is therefore ideal for studying systems that cannot be studied otherwise owing to excessively fast exchange-a frequent obstacle is turned into an advantage in this method. Note that the highest enhancement factors that were achieved for OPN were about 40 for the overall spectrum and 58 for the most intense peaks with respect to thermal equilibrium. Figure 2e shows ac omparison between the signal intensities of OPN in ah yperpolarized exchangefiltered spectrum and ac onventional spectrum in thermal equilibrium. Frydman and co-workers reported enhancement factors up to 330 at pH 7and 60 8 8Cfor denatured Tr ypsin (247 amino acids,l ike OPN). Thee nhancement factors should be compared with caution since the experimental conditions were quite different. Frydman and co-workers used am uch faster injection system (1-3 s) and employed radical-separation techniques.D espite these differences,w em ay conclude that while we aim for physiological conditions that should be favorable for system-oriented studies,s tronger signal enhancements may be obtained at other temperatures and pH. Furthermore,ithas to be taken into account that in OPN, the compacted core features reduced exchange rates so that many cross peaks are not enhanced by hyperpolarized HDO,w hile in entirely denatured proteins like that of Frydman and co-workers,a ll residues can be enhanced homogeneously.

In the complex of OPN bound to heparin, the binding site [15b, 20] is located between residues 145-165 of OPN (Figure 1a). Between residues 100 and 145, ac ompensatory site is located that masks the binding site in the apo-state (i.e., in the absence of heparin), but is exposed to the solvent in the holo-state (i.e., when OPN is bound to heparin). [20] This feature,w hich compensates for entropy loss,i sr etained at near-physiological pH and temperature.

Figure 3a shows the Gly region of the hyperpolarized exchange filtered (red, 30 s) and normal (gray,o btained by time-averaging for 18 h) SOFAST HMQC spectra in the presence of heparin. Clearly, hyperpolarization allows the assignment of signals that are invisible in the normal spectrum owing to strong signal broadening. Most importantly,t he glycine residues are not observable in the absence of heparin. Figure 3b shows how the H N exchange rates are affected when heparin binds to OPN. This leads to distinct effects in the glycine region. For exchange rates that increase upon binding,w eo bserve that new cross-peaks emerge (green arrows in Figure 3d). In contrast, the only glycine that was observable in the apo-form becomes weaker owing to ad ecreased exchange rate in the holo-form (red arrow in Figure 3d). All visible glycine residues lie in the region between residues 100 and 180. They are more exposed to the solvent in the holo-form than in the apo-form. [20] This exposure accelerates exchange with hyperpolarized HDO,t hereby rendering them amenable to detection. Only one glycine signal disappears upon heparin binding.I ti sl ocated within the binding interface between residues 140-165, which display reduced exchange rates.

Figure 3c shows chemical-shift perturbations along the primary sequence of OPN observed by hyperpolarized exchange spectroscopy.These chemical shifts are in excellent agreement with earlier studies and highlight both the heparin binding site and the compensatory site. [20,23] Hyperpolarized HDO thus provides ar eliable method to monitor conformational changes in IDPs.

Figure 1 .

 1 Figure 1. a) Sketch of the functionala rchitecture of quail osteopontin (OPN). Full-length OPN consists of 267 amino acids. The metastasisassociated construct studied here comprises the 220 C-terminal residues (47-267) of the primary sequence. The heparin interaction epitope is located between residues 100 and 165. The heparin binding site comprises residues 145-165. The compensatory residues 100-145 become more mobile when bound to heparin. b) Amixture of H 2 O and glycerol containing 10 mm TEMPOL is hyperpolarized at 1.2 Ka nd 6.7 T, dissolved with a20-fold excess of superheated D 2 O, and transferred to an 18.8 T(800 MHz) spectrometerw here it is mixed with asolution of OPN that is waiting in the NMR tube (ca. pH 7.5 and 308 Kafter mixing.) Strong signals in the 1 H-15 Ncorrelation spectra arise for rapidly exchanging protons (H N )inOPN, while slowly exchanging residues remain below the detection threshold.

Figure 2 .

 2 Figure 2. a) Red: 1 H-15 Ncorrelation spectrum of OPN selectivelye nhanced by hyperpolarized HDO. Black:Aregular spectrum of the same sample after return to thermal equilibrium. b) Region showing resonances of serines selectively enhanced by hyperpolarized HDO. c) The bulk signal still allows the assignmento fsome cross peaks. d) Overlay of ah yperpolarized exchange filtered SOFAST HMQC at pH 7.5 and 35 8 8C(red), ahigh resolution HSQC with arecycling time of 1sand 256 complex increments in the indirect dimension at pH 7.5 and 35 8 8C( black), and the same HSQC with arecycling time of 1sat pH 6.5 and 25 8 8C(blue). Differencesi nchemical shifts between the red and the blue spectra stem from the different experimental conditions. Forthe overlay of the entire spectrum, see the Supporting Information.e )Comparison of the projection of the full 1 H-15 Nspectra of OPN onto the proton dimension,d etected with hyperpolarized water and SOFAST HMQC (red, see Figure 2c)inasingle accumulationa nd the exact same experimentd etectedw ith 256 accumulations(black) after return to thermal equilibriumofthe sample after dissolution.Inb oth cases the recycling delay was set to 0.3 s. The two projections are scaled by the ratio of the number of spectra averaged (hyperpolarized/ thermalized = 1:256) and the applied receiver gain (hyperpolarized/thermalized = 128:912). The signal enhancement due to hyperpolarized HDO was 40.

Figure 3 .

 3 Figure 3. a) Expanded Ser/Thr region of the regular correlation spectrum of apo-OPN (partly folded in the absence of heparin;gray) and the spectrum of apo-OPN boosted by hyperpolarized HDO (colored). b) Differential exchange rates Dk ex = k ex (holo)Àk ex (apo). The green arrows indicate Gly residues that only appear in 2D correlation spectra of the holo-state of OPN. The red arrow indicates the only glycine resonancec an be observed by means of hyperpolarized HDO in the apo-form. Its signal becomes weaker upon heparin binding. c) Chemical-shift perturbation (CSP) of resolved amino acid upon the binding of heparin to OPN.

Acknowledgments

We thank Dr. Pavel Kader ˇµvek, Dr. Daniel Abergel, and Dr. Phillippe Pelupessy for their support, and Bruker BioSpin for generously providing the D-DNP prototype.T his work was supported by the French CNRS,t he European Research Council (ERC contract "Dilute para-water"), and the Austrian Science Foundation (FWF), P26317.