L-Cysteinium semioxalate

The title salt, C₃H₈NO₂⁺·C₂HO₄⁻, formed between L-cysteine and oxalic acid, was studied as part of a comparison of the structures and properties of pure amino acids and their cocrystals. The structure of the title salt is very different from that formed by oxalic acid and equivalent amounts of D- and L-cysteine mol­ecules. The asymmetric unit contains an L-cysteinium cation and a semioxalate anion. The oxalate anion is only singly deprotonated, in contrast with the double deprotonation in the crystal structure of bis­(DL-cysteinium) oxalate. The oxalate anion is not planar. The conformation of the L-cysteinium cation differs from that of the neutral cysteine zwitterion in the monoclinic and ortho­rhom­bic polymorphs of L-cysteine, but is similar to that of the cysteinium cation in bis­(DL-cysteinium) oxalate. The structure of the title salt can be described as a three-dimensional framework formed by ions linked by strong O—H...O and N—H...O and weak S—H...O hydrogen bonds, with channels running along the crystallographic a axis containing the bulky –CH₂SH side chains of the cysteinium cations. The cations are only linked through hydrogen bonds via semi­oxalate anions. There are no direct cation–cation inter­actions via N—H...O hydrogen bonds between the ammonium and carboxyl­ate groups, or via weaker S—H...S or S—H...O hydrogen bonds.

S1. Comment

Cysteine is the only widespread α-aminoacid that has a highly reactive sulfhydryl group (SH) in a side chain. Cysteine molecules easily form disulfide bridges, which play an important role in protein folding and in the stabilization of secondary, tertiary and quaternary structures. In biopolymers, such as proteins, the SH group can act as a proton donor in S—H···O and S—H···S interactions, or as a proton acceptor in N—H···S interactions. Studies of cysteine as an individual molecule in inert matrices (Dobrowolski et al., 2007), as a zwitterion in solutions (Li & Thomas, 1991; Li et al., 1992), in biomolecules (Kandori et al., 1998), in different crystalline polymorphs, in L- and DL-crystals and at various temperatures and pressures by spectroscopic and diffraction techniques (Harding & Long, 1968; Kerr & Ashmore, 1973; Kerr et al., 1975; Görbitz & Dalhus, 1996; Luger & Weber, 1999; Moggach et al., 2005, 2006; Kolesov et al., 2008; Minkov, Chesalov et al., 2008; Minkov, Krylov et al., 2008) provide valuable information on the conformation of the cysteine fragment in relation to the intermolecular interactions of its –CH₂SH side chain. Considering cysteine cocrystallized in a neutral (zwitterionic) form, or in an ionic form with other species, can be a valuable extension of these studies, as was shown recently for bis(DL-cysteinium) oxalate (Drebushchak et al., 2008). In the present communication, we report the structure of another oxalate of cysteine, but this time formed by only the L-enantiomer of the cation.

In the title structure, (I), the cysteine molecule is protonated, but in contrast with bis(DL-cysteinium) oxalate (Drebushchak et al., 2008), only one of the H atoms has been transferred from the oxalic acid to the weaker acid, cysteine, so that a hemioxalate ion is formed (Fig. 1). This cocrystal of L-cysteine and oxalic acid can be classified as a salt.

The conformation of cysteine is very sensitive to its crystalline environment and varies from structure to structure, being less sensitive to the degree of protonation (Fig. 2). As with all amino acids with small side chains, cysteine usually exists as two basically different conformers, gauche+ (N—C—C—S torsion angle ca 60°) and gauche- (N—C—C—S torsion angle ca -60°). As an exception, one can mention one of the two crystallographically independent molecules in the monoclinic polymorph of L-cysteine, in which the N—C—C—S torsion angle is 170.15 (7)° and the conformation is termed trans (Harding & Long, 1968; Görbitz & Dalhus, 1996). Thus, the conformation of the side chain in the orthorhombic (Kerr & Ashmore, 1973; Kerr et al., 1975) and monoclinic (Harding & Long, 1968; Görbitz & Dalhus, 1996) polymorphs of L-cysteine is gauche+ [N—C—C—S torsion angles 65.3 (2) and 74.39 (10)°, respectively; Fig. 2(b),(d)]. In the structure of (I), the cysteine cation has a gauche- conformation, which was previously only observed for L-enantiomers in racemic crystals of DL-cysteine (Luger & Weber, 1999) and bis(DL-cysteinium) oxalate (Drebushchak et al., 2008) [N—C—C—S torsion angles -62.3 (2) and -60.27 (13)°, respectively; Fig. 2(c),(f)].

In contrast with the structure of bis(DL-cysteinium) oxalate, the oxalate anion in (I) is not planar; the angle between the planes of the two carbonyl groups is 38.6 (3)°. Although the planar conformation of an isolated oxalate ion is known to be energetically less advantageous than the twisted one (Dewar & Zheng, 1990), in more than 80% of the known crystal structures of metal oxalates, hemioxalates and oxalic acid itself, the oxalate fragment is planar (Boldyreva et al., 1996; Naumov et al., 1997). In the crystal structures of oxalates of amino acids, twisted oxalate ions are observed more frequently. In DL-threoninium oxalate (Nandhini et al., 2001), the value of the angle between the COO planes in the hemioxalate ion [33.8 (3)°] is very close to that in (I). In the 16 hits of structures containing an amino acid and an oxalate ion from the recent version of the Cambridge Structural Database (CSD, Version 5.29, January 2008; Allen, 2002), eight oxalate ions are planar, 16 more have an angle between the COO planes of less than 10°, seven have this angle at between 20 and 30°, and one has this angle close to 90° (Chandra et al., 1998). The total number of oxalate and hemioxalate ions included in these statistics (42) exceeds the number of crystal structures (16), since in many crystal structures there are several independent oxalate ions and hemioxalate ions.

There is no obvious correlation between the deprotonation of an oxalate ion and its twist. For example, in L-arginine hydrogen oxalate, the angle between the COO planes in the hemioxalate ion is about 10°, and in DL-arginine hydrogen oxalate it is ca 90° (Chandra et al., 1998). It is also of note that in L- and DL-arginine hydrogen oxalates, the hemioxalate ion is formed, and this fact shows that it is not obvious that a DL salt should have a completely deprotonated oxalate ion and an L salt a hemioxalate ion, as is observed for L-cysteinium oxalate (this work), bis(DL-cysteinium) oxalate (Drebushchak et al., 2008), DL-serinium oxalate dihydrate (Alagar et al., 2002), bis(glycinium) oxalate (Chitra & Choudhury, 2007) and glycinium oxalate (Subha Nandhini et al., 2001).

The different species in the crystal structure of (I) are linked by hydrogen bonds to give a three-dimensional network (Figs. 3 and 4). Each hemioxalate anion is linked to two other hemioxalate anions and five cysteinium cations. Atoms O4 and O5 of the carboxyl groups of the hemioxalate anion form a bifurcated hydrogen bond with the NH₃ group of a neighboring cysteinium cation (Fig. 3). Hydrogen bonds link cysteinium cations with hemioxalate anions, but there are no direct cation–cation interactions via N—H···O hydrogen bonds between the NH₃ and carboxylic acid groups, or via weaker S—H···S or S—H···O hydrogen bonds, in which the side groups are involved (Figs. 3 and 4). The structure is formed by ribbons of two types, each representing the arrangement of the hemioxalate anions and L-cysteinium cations within planes parallel to the ab plane. The first type of ribbons consists of infinite chains of hemioxalate ions, which are linked by the shortest hydrogen bonds in the structure [O5—H5O···O4ⁱ; symmetry code: (i) -x + 1, y+ 1/2, -z + 3/2)] and are extended along the crystallographic b axis. These chains of hemioxalate ions are linked to each other via the NH₃ and –SH groups of the cysteinium cations to form the ribbon (Fig. 3). In the second type of ribbon, L-cysteinium cations act as bridges, linking the ribbons of the first type together via hydrogen bonds formed between carboxyl groups of the hemioxalate ions and amino, –SH and –OH groups of the cysteinium cations. Thus, the structure of (I) can be described as a three-dimensional framework formed by ions linked by strong hydrogen bonds, with channels along the crystallographic a axis containing the bulky –CH₂SH side chains of the cysteinium cations (Fig. 4). These channels are very similar to those in the structure of pure orthorhombic L-cysteine (Kerr & Ashmore, 1973), which are preserved even under high hydrostatic pressure (Moggach et al., 2006).

The only contact that seems to link two neighbouring cysteine cations directly is a short C—H···O contact (Table 2). Interestingly, neighbouring thiol groups in the structure of (I) do not form any S—H···S hydrogen bonds, the minimum distance between S atoms being 4.824 (2) Å. At the same time, there are very short S···O contacts of 3.339 (2) Å and longer contacts of 3.670 (2) Å (involving O atoms from the hemioxalate ions), which can correspond to S—H···O hydrogen bonds being present in the structure, despite the disorder in the sulfhydryl group H atom, similar to the case of the orthorhombic polymorph of L-cysteine. The main difference between the hydrogen bonding in (I) and that in orthorhombic L-cysteine is that in (I) the SH group can donate its H atom to several O atoms of the COO groups. The occupancy of the position at which we have located the H atom of the SH group was estimated as 0.62, suggesting that other orientations of the SH groups in the structure are possible, and this co-existence of several types of local environments for the SH group is supported by IR spectroscopic data. For comparison, the shortest S···O distance hitherto observed in a cysteine-containing crystal structure is in orthorhombic L-cysteine at 30 K and has approximately the same value, 3.332 (1) Å (Moggach et al., 2005). In the latter case, despite this short contact, no S—H···O hydrogen bonds are formed. Instead, the ordered thiol groups are thought to form S—H···S hydrogen bonds exclusively, as deduced from X-ray diffraction data (Moggach et al., 2005). However, polarized single-crystal Raman spectroscopic data (Kolesov et al., 2008) indicate the co-existence of several cysteine conformations and local SH group environments even at 3 K, with the S—H···S contacts obviously dominating.

The types of hydrogen bond formed by the SH groups can most reliably be determined from an analysis of the vibrational spectra (Li & Thomas, 1991; Li et al., 1992; Minkov, Chesalov et al., 2008; Kolesov et al., 2008). A free thiol group in CCl₄ has an SH_str vibration at 2585±5 cm^-1 (Li et al., 1992), the SH_str vibration of this group when involved in S—H···S hydrogen bonds is observed typically in the range 2500–2550 cm^-1, and the same group forming S—H···O hydrogen bonds vibrates at higher frequencies, 2550–2585 cm^-1, indicating that the S—H···O interaction is weaker than the S—H···S one, which is opposite to what could be expected from the electronegativity of the O and S atoms, probably because the same O atom is involved simultaneously in the formation of much stronger N—H···O hydrogen bonds (Kerr et al., 1975). The SH_str vibration in the IR spectrum of (I) is split into two bands, with maxima at 2593 and 2563 cm^-1, which corresponds to one `free' thiol group (or one forming a very weak S—H···O hydrogen bond, corresponding to the long S···O contact distance in the structure) and another which forms a short S—H···O contact, or several contacts equivalent in energy. For comparison, in the structure of bis(DL-cysteinium) oxalate, there is only one SH_str band in the IR spectrum at 2576 cm^-1, in agreement with the structural data showing ordered S—H···O hydrogen bonds [with an S···O distance of 3.6200 (15) Å] and no S—H···S hydrogen bonds, despite short [3.5176 (8) Å] S···S distances in the structure (Drebushchak et al., 2008). In orthorhombic L-cysteine at ambient temperature, the thiol group is disordered between two types of contacts, S—H···S [S···S distance 3.79 (2) Å], and S—H···O [S···O distance 3.41 (2) Å] (Kerr et al., 1975), and the corresponding frequencies in the IR spectrum are 2508 and 2551 cm^-1, respectively (Minkov, Chesalov et al., 2008).

S2. Experimental

Crystals of L-cysteinium hemioxalate were obtained by slow evaporation of an aqueous–alcoholic [ethanolic?] [solvent ratio?] solution of an equimolar ratio of L-cysteine and oxalic acid dihydrate under ambient conditions. Several crystals were tested and found to be built up of several domains slightly misoriented with respect to each other.

S3. Refinement

The selected crystal was found to be a twin with two large domains (two orientation matrices, where 58 and 39% of the reflections from the whole data set could be indexed and integrated). A small number of reflections could not be indexed with either of the two orientation matrices; this happens quite often with the Stoe IPDS diffractometer. The twin law for transforming hkl(1) to hkl(2) is: 0.99986 -0.00105 -0.00068; 0.00231 0.99886 0.05342; 0.00179 -0.07266 0.99791. Structure solution and refinement were carried out using data corresponding solely to the major domain (data from the minor domain and overlaps were subtracted from the whole data set). This has led to the omission of some 13% of the unique reflections from the data set. Nonetheless, the data:parameter ratio has remained high. Attempts to refine the structure by including the overlapping reflections and correcting their intensities for the contribution from the minor domain gave worse results.

Methine and methylene H atoms were placed in geometrically calculated positions and constrained to ride on their parent atoms, with C—H = 0.98 and 0.97 Å, respectively. The H atoms of the NH₃ group were also constrained to an ideal geometry, with N—H = 0.89 Å, but were allowed to rotate freely about the N—C bond. The positions of the sulfhydryl and hydroxyl H atoms were found from difference Fourier maps and were refined freely. For all H atoms, U_iso(H) was set to 1.2U_eq(parent atom). In order to locate the H atom of the SH group, a dummy SH₃ group was introduced. The occupancies of several possible H-atom positions were refined and eventually estimated as 0.6, 0.2 and 0.05 from the difference Fourier maps using PLATON (Spek, 2003). The location suggested in the present CIF corresponds to site occupancy 0.6, and the H atom at this position can form two S—H···O contacts, a longer and a shorter one, which agrees with the spectroscopic data.

Computing details

Data collection: X-AREA (Stoe & Cie, 2007); cell refinement: X-AREA (Stoe & Cie, 2007); data reduction: X-RED (Stoe & Cie, 2007); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: Mercury (Macrae et al., 2006); software used to prepare material for publication: publCIF (Westrip, 2008).

Figures

	Fig. 1. The asymmetric unit of (I), with the atom-labelling scheme. Displacement ellipsoids are drawn at the 50% probability level and H atoms are shown as spheres of arbitrary radii.
	Fig. 2. The conformation of L-cysteine in different crystalline environments. (a) L-cysteinium oxalate. (b) orthorhombic L-cysteine. (c) bis(DL-cysteinium oxalate. (d) and (f) Two crystallographically independent molecules in monoclinic L-cysteine. (e) DL-cysteine.
	Fig. 3. Hydrogen bonding (dashed lines) between the hemioxalate anions and L-cysteinium cations of (I). [Symmetry codes: (i) -x + 1, y - 1/2, -z + 3/2; (ii) -x, y - 1/2, -z + 3/2; (iii) x - 1, y, z.]
	Fig. 4. A fragment of the crystal structure of (I), projected onto the bc plane. Hydrogen bonds are shown as dashed lines.

L-Cysteinium hemioxalate

Crystal data

C3H8NO2S^+·C2HO4^−	F(000) = 440
Mr = 211.20	Dx = 1.602 Mg m^−3
Orthorhombic, P212121	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: P 2ac 2ab	Cell parameters from 6447 reflections
a = 7.0529 (11) Å	θ = 2.6–29.6°
b = 10.2407 (12) Å	µ = 0.37 mm^−1
c = 12.1199 (15) Å	T = 295 K
V = 875.4 (2) Å^3	Prism, colourless
Z = 4	0.46 × 0.38 × 0.22 mm

Data collection

Stoe IPDS 2 diffractometer	2052 independent reflections
Radiation source: sealed X-ray tube, 12 x 0.4 mm long-fine focus	1727 reflections with I > 2σ(I)
Plane graphite monochromator	Rint = 0.036
Detector resolution: 6.67 pixels mm^-1	θmax = 29.2°, θmin = 2.6°
rotation method scans	h = −9→8
Absorption correction: numerical (X-SHAPE; Stoe & Cie, 2003)	k = −14→14
Tmin = 0.834, Tmax = 0.926	l = −16→14
6447 measured reflections

Refinement

Refinement on F^2	Secondary atom site location: difference Fourier map
Least-squares matrix: full	Hydrogen site location: inferred from neighbouring sites
R[F^2 > 2σ(F^2)] = 0.034	H atoms treated by a mixture of independent and constrained refinement
wR(F^2) = 0.072	w = 1/[σ^2^(Fo^2^) + (0.0408P)^2^] where P = (Fo^2^ + 2Fc^2^)/3
S = 1.00	(Δ/σ)max < 0.001
2052 reflections	Δρmax = 0.19 e Å^−3
128 parameters	Δρmin = −0.16 e Å^−3
0 restraints	Absolute structure: Flack (1983), with 854 Friedel pairs
Primary atom site location: structure-invariant direct methods	Absolute structure parameter: −0.02 (9)

Special details

Refinement. Refinement of F^2^ against ALL reflections. The weighted R-factor wR and goodness of fit S are based on F^2^, conventional R-factors R are based on F, with F set to zero for negative F^2^. The threshold expression of F^2^ > σ(F^2^) is used only for calculating R-factors(gt) etc. and is not relevant to the choice of reflections for refinement. R-factors based on F^2^ are statistically about twice as large as those based on F, and R- factors based on ALL data will be even larger.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
N1	0.1853 (2)	0.62449 (14)	0.86481 (12)	0.0284 (3)
H1N	0.2663	0.5840	0.8201	0.034*
H2N	0.2168	0.7084	0.8699	0.034*
H3N	0.0684	0.6176	0.8378	0.034*
C1	0.3893 (3)	0.58949 (16)	1.02322 (15)	0.0315 (4)
C2	0.1930 (3)	0.56325 (17)	0.97653 (15)	0.0277 (4)
H2	0.1001	0.6075	1.0237	0.033*
C3	0.1436 (3)	0.41916 (16)	0.97346 (17)	0.0345 (4)
H31	0.2284	0.3752	0.9226	0.041*
H32	0.1637	0.3820	1.0462	0.041*
O1	0.4283 (2)	0.51598 (15)	1.10817 (14)	0.0496 (4)
H1O	0.545 (5)	0.529 (3)	1.133 (2)	0.059*
O2	0.4936 (2)	0.67027 (16)	0.98526 (13)	0.0495 (4)
S1	−0.09886 (10)	0.38890 (6)	0.93173 (6)	0.05446 (19)
H1S	−0.075 (5)	0.346 (3)	0.862 (2)	0.065*
C4	0.6203 (3)	0.48966 (15)	0.72255 (14)	0.0250 (3)
C5	0.6677 (2)	0.63579 (16)	0.73432 (14)	0.0252 (3)
O3	0.7364 (2)	0.41845 (11)	0.67468 (11)	0.0362 (3)
O4	0.4653 (2)	0.45372 (12)	0.76298 (13)	0.0368 (3)
O5	0.5167 (2)	0.70732 (13)	0.72027 (11)	0.0322 (3)
H5O	0.532 (4)	0.789 (3)	0.7288 (18)	0.039*
O6	0.8256 (2)	0.67562 (13)	0.75251 (13)	0.0420 (3)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
C1	0.0300 (9)	0.0256 (9)	0.0389 (9)	0.0009 (7)	−0.0048 (8)	0.0002 (7)
C2	0.0235 (9)	0.0252 (9)	0.0344 (9)	0.0012 (6)	−0.0004 (7)	0.0009 (6)
C3	0.0359 (11)	0.0259 (10)	0.0417 (10)	−0.0025 (7)	0.0019 (8)	0.0023 (7)
C4	0.0228 (9)	0.0226 (8)	0.0297 (9)	−0.0009 (6)	−0.0006 (7)	0.0004 (6)
C5	0.0225 (9)	0.0224 (8)	0.0308 (9)	−0.0002 (6)	0.0004 (6)	0.0018 (6)
O1	0.0359 (11)	0.0548 (10)	0.0579 (10)	−0.0118 (7)	−0.0189 (7)	0.0228 (7)
O2	0.0420 (10)	0.0474 (8)	0.0591 (10)	−0.0200 (7)	−0.0172 (7)	0.0162 (7)
O3	0.0348 (8)	0.0252 (7)	0.0486 (8)	−0.0004 (5)	0.0126 (6)	−0.0071 (5)
O4	0.0271 (8)	0.0212 (6)	0.0622 (9)	−0.0022 (5)	0.0114 (6)	−0.0024 (5)
O5	0.0268 (8)	0.0186 (6)	0.0512 (8)	0.0004 (5)	−0.0021 (6)	−0.0008 (5)
O6	0.0261 (7)	0.0292 (7)	0.0707 (10)	−0.0047 (5)	−0.0080 (6)	0.0032 (6)
N1	0.0224 (8)	0.0243 (7)	0.0385 (8)	−0.0013 (5)	−0.0021 (6)	0.0018 (6)
S1	0.0454 (4)	0.0550 (4)	0.0631 (4)	−0.0224 (3)	−0.0133 (3)	0.0076 (3)

Geometric parameters (Å, º)

O1—C1	1.305 (2)	C2—N1	1.493 (2)
O1—H1O	0.89 (3)	C2—C3	1.517 (2)
O5—C5	1.304 (2)	C2—H2	0.9800
O5—H5O	0.85 (3)	C5—O6	1.206 (2)
O2—C1	1.199 (2)	C5—C4	1.540 (2)
S1—C3	1.810 (2)	C3—H31	0.9700
S1—H1S	0.97 (3)	C3—H32	0.9700
O4—C4	1.253 (2)	N1—H1N	0.8900
O3—C4	1.241 (2)	N1—H2N	0.8900
C1—C2	1.520 (3)	N1—H3N	0.8900
C1—O1—H1O	112.0 (18)	O3—C4—O4	125.88 (15)
C5—O5—H5O	115.7 (17)	O3—C4—C5	118.10 (16)
C3—S1—H1S	99 (2)	O4—C4—C5	116.02 (15)
O2—C1—O1	124.86 (19)	C2—C3—S1	113.00 (13)
O2—C1—C2	122.56 (17)	C2—C3—H32	109.0
O1—C1—C2	112.58 (16)	S1—C3—H32	109.0
N1—C2—C3	112.22 (15)	C2—C3—H31	109.0
N1—C2—C1	107.24 (14)	S1—C3—H31	109.0
C3—C2—C1	113.00 (15)	H32—C3—H31	107.8
N1—C2—H2	108.1	C2—N1—H2N	109.5
C2—C3—H32	108.1	C2—N1—H3N	109.5
C1—C2—H2	108.1	H2N—N1—H3N	109.5
O6—C5—O5	126.02 (16)	C2—N1—H1N	109.5
O6—C5—C4	123.09 (16)	H2N—N1—H1N	109.5
O5—C5—C4	110.88 (15)	H3N—N1—H1N	109.5
O2—C1—C2—N1	15.7 (2)	O5—C5—C4—O3	141.00 (17)
O1—C1—C2—N1	−164.98 (16)	O6—C5—C4—O4	142.04 (19)
O2—C1—C2—C3	139.8 (2)	O5—C5—C4—O4	−39.0 (2)
O1—C1—C2—C3	−40.8 (2)	N1—C2—C3—S1	−64.76 (18)
O6—C5—C4—O3	−38.0 (3)	C1—C2—C3—S1	173.84 (13)

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
N1—H1N···O4	0.89	2.06	2.912 (2)	161
N1—H1N···O5	0.89	2.49	3.041 (2)	121
N1—H2N···O3^i	0.89	2.24	3.0977 (19)	161
N1—H3N···O6^ii	0.89	2.09	2.926 (2)	157
O1—H1O···O3^iii	0.89 (3)	1.71 (3)	2.587 (2)	170 (3)
O5—H5O···O4^i	0.85 (3)	1.69 (3)	2.5346 (18)	172 (2)
O5—H5O···O3^i	0.85 (3)	2.59 (2)	3.0796 (19)	118 (2)
S1—H1S···O3^ii	0.97 (3)	2.73 (3)	3.3387 (16)	121 (2)
S1—H1S···O6^iv	0.97 (3)	2.84 (3)	3.6702 (17)	144 (2)
C2—H2···O2^v	0.98	2.40	3.105 (2)	128

Symmetry codes: (i) −x+1, y+1/2, −z+3/2; (ii) x−1, y, z; (iii) −x+3/2, −y+1, z+1/2; (iv) −x+1, y−1/2, −z+3/2; (v) x−1/2, −y+3/2, −z+2.