Cell cycle control at the first restriction point and its effect on tissue growth

Cell cycle is controlled at two restriction points, R ₁ and R ₂. At both points the cell will commit apoptosis if it detects irreparable damage. But at R ₁ an undamaged cell also decides whether to proceed to the S phase or go into a quiescent mode, depending on the environmental conditions (e.g., overpopulation, hypoxia). We consider the effect of this decision at the population level in a spherical tissue {r < R(t)}. We prove that if the cells have full control at R ₁, they can manipulate the size of R(t) to ensure that 0 < c ≤ R(t) ≤ C < ∞; simulations further show that R(t) can be made nearly stationary. In the absence of such control, R(t) will either increase to ∞ or decrease to 0. The mathematical model and analysis involve a system of PDEs in {r < R(t)}.     

Cross-Correlation Functions for a Neuronal Model

Cross-correlation functions, R_XY(t,τ), are obtained for a neuron model which is characterized by constant threshold θ, by resetting to resting level after an output, and by membrane potential U(t) which results from linear summation of excitatory postsynaptic potentials h(t). The results show that: (1) Near time lag τ = 0, R_XY(t,τ) = f_U [θ-h(τ), t + τ] {h′(τ) + E_U [u′(t + τ)]} for positive values of this quantity, where f_U(u,t) is the probability density function of U(t) and E_U [u′(t + τ)] is the mean value function of U′(t + τ). (2) Minima may appear in R_XY(t,τ) for a neuron subjected only to excitation. (3) For large τ, R_XY(t,τ) is given approximately by the convolution of the input autocorrelation function with the functional of point (1). (4) R_XY(t,τ) is a biased estimator of the shape of h(t), generally over-estimating both its time to peak and its rise time.     

Design and synthesis of glucose-templated proline-lysine chimera: polyfunctional amino acid chimera with high prolyl cis amide rotamer population

We describe the synthesis of two glucose-templated proline-lysine chimeras (GlcTProLysCs) that differ in the stereochemistry of the hydroxymethyl substituent at the C-5′ position of the pyrrolidine ring. The key synthetic steps involve C-glycosylation of an exocyclic glucose-based epoxide with allyltributylstannane, which affords functionalized C-ketosides containing an α-hydroxy ester moiety; introduction of an amino group at C-2 through stereoselective reductive amination; and regioselective installation of the azide group at C-6 on the glucose scaffold. Incorporation of these chimeras into the model peptides Ac-GlcTProLysC-NHMe and Ac-GlcTProLysC-OMe demonstrates that the stereochemistry of the hydroxymethyl substituent at the C-5′ position has a profound effect on the equilibrium constant of prolyl amide cis/trans isomerization. The equilibrium constant K_c/t for the peptide mimic Ac-GlcTProLysC-NHMe with C-5′(R) stereochemistry was determined to be 3.03 ± 0.04, while the K_t/c for the C-5′(S) diastereoisomer was 0.56 ± 0.04 in D₂O. Temperature coefficient experiments indicate that the origin of these effects is derived from two critical hydrogen bonds involving the C-5′ hydroxymethyl substituent: one to the N-terminal amide carbonyl group, and the other to the primary amino group in the glucose moiety.     

Reactivity of platina-β-diketones towards chelating nitrogen and sulfur donors: formation of acyl(hydrido)platinum(IV) and acyl(chloro)platinum(II) complexes

The platina-β-diketone [Pt₂{(COMe)₂H}₂(μ-Cl)₂] (1) was found to react with chelating N,N-ligands 2(R^′NCR)C₅H₄N (R/R^′=Ph/OH, H/Ph, Me/Ph) to form acyl(hydrido)platinum(IV) complexes [Pt(COMe)₂Cl(H){2-(R^′NCR)C₅H₄N}] (R/R^′=Ph/OH 2a; H/Ph 2b; Me/Ph (2c)). Reactions of complex 1 with chelating S,S- and N,S-donors (RS-CH₂-CH₂-SR, 2-(RSCH₂)C₅H₄N, R=Et, Ph, t-Bu) afforded acyl(chloro)platinum(II) complexes [Pt(COMe)Cl(RSCH₂CH₂SR)] (R=Et, 3a; Ph, 3b; t-Bu, 3c) and [Pt(COMe)Cl{2-(RSCH₂)C₅H₄N}] (R=Et, 4a; Ph, 4b; t-Bu, 4c), respectively. All complexes were fully characterized by microanalysis, IR and NMR (¹H, ¹³C) spectroscopy. Furthermore, molecular structures of complexes 3b and 4b were determined by single-crystal X-ray diffraction analyses revealing close to square-planar configuration. In complex 4b the acetyl ligand is trans to pyridine N atom (configuration index SP-4-2). The reactions are discussed in terms of consecutive oxidative addition and reductive elimination reactions.     

Correlation between R/S enantiomer ratio of lansoprazole and CYP2C19 activity after single oral and enteral administration

The purpose of this study was to investigate whether CYP2C19 activity can be estimated from plasma concentrations of lansoprazole enantiomers 4 h (C_4h) after single administration by oral and enteral routes. Sixty‐nine subjects, 22 homozygous extensive metabolizers (homEMs), 32 heterozygous EMs (hetEMs), and 15 poor metabolizers (PMs), participated in the study. After a single oral or enteral dose of racemic lansoprazole (30 mg), plasma concentrations of lansoprazole enantiomers were measured 4 h postdose. The R/S ratio of lansoprazole at 4 h differed significantly among the three groups (P < 0.0001) regardless of the administration route. The R/S ratio of lansoprazole in CYP2C19 PMs ranged from 3.0 to 13.7, whereas in homEMs and hetEMs the ratio ranged from 8.6 to 90 and 2.1 to 122, respectively. The relationship between (S)‐lansoprazole concentration and R/S ratio of lansoprazole at C_4h is given by the following formula: log₁₀ [R/S ratio] = 2.2 – 0.64 × log₁₀ [C_4h of (S)‐lansoprazole] (r = 0.867, P < 0.0001). Thus, phenotyping CYP2C19 using the R/S enantiomer ratio of lansoprazole seems unlikely. However, to obtain a pharmacological effect similar to that in CYP2C19 PMs, we can presume that lansoprazole has a sufficient effect in the patient with an R/S enantiomer ratio at 4 h ≤ 13.70 and (S)‐lansoprazole concentration at 4 h ≥ 50 ng/ml. Chirality 2010. © 2009 Wiley‐Liss, Inc.     

Chiroptical properties of diastereomeric five-coordinate Pt(II)–olefin complexes. Molecular structure of [PtCl2{(S,S)-(E)-CNCHCHCN}(R,R)-{C6H5CH(CH3)N(CH3)CH2}2]·C6H6

The chiroptical properties of five-coordinate diastereomeric complexes of general formula [PtCl₂(R,R)-{C₆H₅CH(CH₃)N(CH₃)CH₂}₂{olefin}], with olefin ligands having electron-withdrawing substituents, have been investigated. The sign of CD bands in the 28 000–30 000 cm⁻¹ region appears to be correlated to the absolute configuration of the prochiral coordinated alkene. Single-crystal X-ray diffraction structure determination has been performed on the single diastereomer [PtCl₂(E-but-2-enedinitrile)(R,R)-{C₆H₅CH(CH₃)N(CH₃)CH₂}₂]· C₆H₆. The compound crystallizes in the monoclinic space group C2 with a = 17.842(2), b = 8.466(1), c = 10.464(1) Å, β = 109.34(1)°, Z = 2. The number of observed reflections was 1943 and the final R and R_w values were 0.020 and 0.028 respectively. Trigonal-bipyramidal geometry is observed around the Pt atom, with the two Cl atoms in axial positions. The unsaturated ligand lies in the equatorial plane disclosing S,S absolute configuration.     

Soybeans and radish leaves contain only one of the sulfonium diastereoisomers of S-adenosylmethionine

Using a liquid chromatography method that separates the two sulfonium diastereoisomers of adenosylmethionine, we have found that immature soybeans, soybean callus culture, radish leaves, yeast and rat liver contain only the (S)-sulfonium form of S-adenosylmethionine. Our findings contradict the suggestion by Stolowitz and Minch that 10–20% of naturally-occurring adenosylmethionine may have the (R)-configuration at the sulfonium pole. Absence of the (R)-sulfonium isomer of adenosylmethionine in biological materials indicates that the (R)-sulfonium form of adenosylmethionine present in commercial adenosylmethionine samples is an artifact of the isolation procedure. Our method of measuring the isomers of adenosylmethionine enabled us to readily determine the rate of racemization and hydrolysis of adenosylmethionine. Our rate constants for racemization (K_r) and hydrolysis (K_h) were 2.4 × 10^?6 sec^?1 and 12.3 × 10-^?6 sec^?1, respectively; values which are noticeably different from those of Wu and co-workers which were obtained with a more complicated method (K_r = 8 × 10^?1 sec^?1; K_h = 6 × 10^?6 sec^?1). We believe the absence of the (R)-isomer in vivo is best explained by stabilization of the (S)-isomer as suggested by Wu et al. Although the tissues we have analysed contained the (S)-sulfonium form of adenosylmethionine exclusively, when ethionine-resistant soybean cell lines were given ethionine, they accumulated both sulfonium diastereoisomers of adenosylethionine.     

DYNAMICS OF ARBUSCULE DEVELOPMENT AND DEGENERATION IN A ZEA MAYS MYCORRHIZA

A quantitative light and electron microscope study of developing and degenerating mycorrhizal arbuscules of Glomus fasciculatum in Zea mays was carried out in order to estimate three parameters during the colonization cycle. These were: 1) V_v(f,c), the fraction of the host cell volume occupied by a volume of fungus; 2) V_v(cy,c), the fraction of the host cell volume occupied by host cytoplasm; 3) S_v(pr,c), the surface-area-to-volume ratio of the host protoplast to the whole host cell. Uninfected cortical cells had an S_v(pr,c) of 0.13 μm²/μm³. As the fungus penetrates the cell wall, the protoplast invaginates, causing a decrease in protoplast volume and an increase in protoplast S_v. The S_v(pr,c) of a cell containing a mature arbuscule is 1.275 μm²/μm³. Because of the shrinkage of the protoplast, the S_v of the protoplast to its own volume rather than the original cell volume is 2.55 μm²/μm³, or almost a 20-fold increase. Total cell size is unaffected. When the arbuscule is mature, the fungus occupies 42% of the cell, with 24% as 1-μm-diam branches, and 18% as trunk. Arbuscular branch formation progresses at a linear rate and is the most important factor in causing the increased host S_v. The correlation coefficient for V_v(br,c) the volume fraction for arbuscular branches, vs. Sv(pr,c) is r = 0.932 (P < 0.001). Degeneration of the arbuscule is marked by a rapid decrease in branches, host S_v, and host cytoplasm. The trunk develops and degenerates at a slower rate than the branches.     

Synthesis and coordination chemistry of tripodal dialkyl[1,2-bis(diethylcarbamoyl)ethyl]phosphonates with lanthanide nitrates

Trifunctional dialkyl [1,2-bis(diethylcarbamoyl)- ethyl] phosphonates, (RO)₂P(O)CH[C(O)N(C₂H₅)₂]- [CH₂C(O)N(C₂H₅)₂] R  CH₃, C₂H₅, i-C₃H₇, n-C₆H₁₃ were prepared from the respective sodium salts, Na[(RO)₂P(O)CHC(O)N(C₂H₅)₂] and N,N- diethylchloroacetamide, and they were characterized by elemental analysis, mass, infrared and NMR spectroscopy. The molecular structure of (i-C₃H₇O)₂- P(O)CH[C(O)N(C₂H₅)₂][CH₂C(O)N(C₂H₅)₂] was determined by single crystal X-ray diffraction analysis and found to crystallize in the monoclinic space group P2₁/c with a=15.589(6), b=9.783(4), c= 16.283(7) Å, β = 110.90(3)°, Z = 4 and V= 2320(2) ų. The structure was solved by direct methods and blocked least-squares refinement converged with Rf = 5.7% and R_wF= 4.4% on 2266 unique data with F>4σ(F). Important bond distances include PO 1.459(3) Å, CHCO 1.228(3) Å and CHCH₂CO 1.223(3) Å. The coordination chemistry of the ligand with several lanthanides was examined, and the structure of the complex Gd(NO₃)₃{[(i-C₃H₇O)₂P(O)CH[C(O)N(C₂H₅)₂][CH₂C(O)N(C₂H₅)₂]}₂·H₂O was determined. The complex crystallized in the monoclinic space group P2₁/n with a = 13.524(5), b = 22.033(4), c = 19.604(4) Å β = 106.22(2)°, Z = 4 and V= 5609(3) ų. The structure was solved by heavy atom techniques and blocked least-squares refinement converged with R_F = 5.9% and R_wF = 4.1% on 5275 reflections with F > 4σ(F). Both trifunctional ligands were found to bond to Gd(III) through only the phosphoryl oxygen atoms. The remainder of the Gd coordination sphere was composed of three bidentate nitrate oxygen atoms and an oxygen bonded water molecule. Several important bond distances include GdO(phosphoryl)_av = 2.343(5) Å, GdO(nitrate)_av = 2.475(7) Å, GdO(water) = 2.354(5) Å, PO(phosphoryl)_av = 1.467(6) Å, CHCO_av = 1.242(10) Å and CHCH₂CO_av = 1.209(11) Å.     

Synthesis and structure of a novel molybdenum-iron-sulfur cluster with Mo2Fe2 core and all-disulfide chelate ligands, [Mo2Fe2(μ3-S)4(S2CNEt2)5] CH3CN

The cluster compound [Mo₂Fe₂(μ₃-S)₄- (S₂CNEt₂)₅]CH₃CN has been prepared from the reaction system containing (NH₄)₂MoS₄, FeCl₃, NaS₂CNEt₂, PhSH and NaOCH₃. The crystal and molecular structure have been determined by the low temperature X-ray diffraction technique. The compound crystallizes in space group P2₁/c of the monoclinic system with a = 19.397(7), b = 10.891(7), c = 24.302(8) Å, β = 108.95(2)° and Z = 4. With use of 2647 reflections (I)>2.5σ(I)) the structure was refined to R(R_w) = 0.045(0.036). The cluster Mo₂Fe₂S₄(S₂CNEt₂)₅ has a cubane-like skeleton [Mo₂Fe₂S₄]⁵⁺. Each metal atom is coordinated by three μ₃-S atoms and a disulfide chelate terminal ligand. The fifth S₂CNEt₂ group as a bridging ligand coordinates to two Mo atoms. In a molecule of the compound, the two Mo atoms are equivalent but the two Fe atoms are unequivalent.     

Improvements of enzyme activity and enantioselectivity in lipase-catalyzed alcoholysis of (R,S)-azolides

With Candida antarctica lipase B (CALB)-catalyzed alcoholysis of (R,S)-naproxenyl 1,2,4-triazolide at the optimal conditions (i.e. anhydrous MTBE as the solvent, and methanol as the acyl acceptor at 45 °C) as the model system, the enzyme enantioselectivity in terms of V_R/V_S = 105.8 and specific activity for the fast-reacting (R)-azolide V_R/(E_t) = 0.979 mmol/(h g) were greatly improved in comparison with V_R/V_S = 8.0 and V_R/(E_t) = 0.113 mmol/(h g) of using (R,S)-naproxenyl 2,2,2-trifluoroethyl ester as the substrate. The resolution strategy was successfully extended to other (R,S)-profenyl 1,2,4-triazolides and lipases from Candida rugosa (Lipase MY) and Carica papaya (CPL) having opposite enantioselectivity to CALB. Moreover, the kinetic constants were estimated, compared with those obtained via hydrolysis, and employed for modeling time-course conversions of (R,S)-naproxenyl 1,2,4-triazolide in anhydrous MTBE. The advantages of easy substrate preparation, high enzyme reactivity and enantioselectivity, as well as easy product separation from the remaining substrate via reactive extraction demonstrate merits of using (R,S)-azolides but not the corresponding esters for the alcoholytic resolution.     

The interaction of cytosolic epoxide hydrolase with chiral epoxides

• 1.1. The kinetic parameters of the cytosolic epoxide hydrolase were examined with two sets of spectrophotometric substrates. The (2S,3S)- and (2R,3R)-enantiomers of 4-nitrophenyl trans-2,3-epoxy-3-phenylpropyl carbonate had a K_mof 33 and 68 μm and a V_max of 16 and 27 μmol/min/mg, respectively. With the (2S,3S)- and (2R,3R)- enantiomers of 4-nitrophenyl trans-2,3-epoxy-3-(4-nitrophenyl)propyl carbonate, cytosolic epoxide hydrolase had a K_M of 8.0 and 15 μM and a V_max of 7.8 and 5.0 μmol/min/mg, respectively.
• 2.2. Glycidyl 4-nitrobenzoate had the lowest I₅₀ of the compounds tested in the glycidyl 4-nitrobenzoate series (I₅₀= 140 μM). The I₅₀ of the (2R)-enantiomer was 3.7-fold higher. The inhibitor with the lowest i₅₀ in the glycidol series, and the lowest I₅₀ of any compound tested, was (2S,3S)-3-(4-nitrophenyl)glycidol (I₅₀ = 13.0μM). It also showed the greatest difference in I₅₀ between the enantiomers (330-fold).
• 3.3. All enantiomers of glycidyl 4-nitrobenzoates and _trans-3-phenylglycidols gave differential inhibition of cytosolic epoxide hydrolase. However, neither the (S,S)-/(S)- or (R,R)-/(R)-enantiomer always had the lower I₅₀.
• 4.4. Addition of one or more methyl groups to either enantiomer of glycidyl 4-nitrobenzoate resulted in increased I₅₀. However, addition of a methyl group to C₂ of either enantiomer of 3-phenylglycidol resulted in a decreased I₅₀. Finally, when the hydroxyl group of trans-3-(4-nitrophenyl)glycidol was esterified the I₅₀ of the (2S,3S)- but not the (2R,3R)-enantiomer increased.

     

Analysis of a mathematical model for the growth of tumors

 In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown function r=s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r=R ₀ (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf _t→∞ s(t)>0, i.e. once engendered, tumors persist in time. Indeed, we further show that (ii) If c is sufficiently small then s(t)→R ₀ exponentially fast as t→∞, i.e. the steady state solution is globally asymptotically stable. Further, (iii) If c is not “sufficiently small” but is smaller than some constant γ determined explicitly by the parameters of the problem, then lim sup _t→∞ s(t)<∞; if however c is “somewhat” larger than γ then generally s(t) does not remain bounded and, in fact, s(t)→∞ exponentially fast as t→∞. Received: 25 February 1998 / Revised version: 30 April 1998     

Influence of glucose-templated proline mimetics on the β-turn conformation of the peptide fragment Ac-Leu-d-Phe-Pro-Val-NMe2 found in Gramicidin S

The synthesis of tetrapeptide-based β-turn mimetics containing spirocyclic glucose-templated 3-hydroxyproline hybrids Glc3′(S)-5′(R)(CH₂OH)HypH and Glc3′(S)-5′(S)(CH₂OH)HypH as proline mimetics is presented. NMR-based conformational analysis of Ac-Leu-d-Phe-[Glc3′(S)-5′(R)(CH₂OH)HypH]-Val-NMe₂ and Ac-Leu-d-Phe-[Glc3′(S)-5′(S)(CH₂OH)HypH]-Val-NMe₂ demonstrates the presence of β-turn conformations. Different turn structures were observed by changing the stereochemistry at 5′-position of Glc3′(S)-5′(R)(CH₂OH)HypH. The major prolyl amide cis isomer of glucose-protected tetrapeptide Ac-Leu-d-Phe-[Glc(MOM)₄3′(S)-5′(R)(CH₂OMOM)HypH]-Val-NMe₂11 and glucose unprotected Ac-Leu-d-Phe-[Glc3′(S)-5′(R)(CH₂OH)HypH]-Val-NMe₂13 forms a type VI β-turn conformation. In contrast, the major prolyl amide trans rotamer of tetrapeptide Ac-Leu-d-Phe-[Glc(MOM)₄3′(S)-5′(S)(CH₂OMOM)HypH]-Val-NMe₂12 conserves a similar β-turn conformation as the Gramicidin S-based peptide fragment Ac-Leu-d-Phe-Pro-Val-NMe₂16.     

2(S),3(S)-3-hydroxy-4-methyleneglutamic Acid from Gleditsia caspica

A new natural product, 2(S),3(S)-3-hydroxy-4-methyleneglutamic acid (G3) has been isolated from seeds of Gleditsia caspica. The structure has been established by chemical and spectroscopic methods. Catalytic reduction of G3 yields 2(S),4(S)-4-methylglutamic acid and a new amino acid, 2(S),3(S),4(S)-3-hydroxy-4-methylglutamic acid. Ozonolysis of G3 followed by oxidation gives 2(S),3(R)-3-hydroxyaspartic acid. The S- (or l-) configurations at C₂ in G3 and in 2(S),3(S),4(S)-3-hydroxy-4-methyglutamic acid and the S-configurations at C₃ for G3 and 2(S),3(S),4(S)-3-hydroxy-4-methylglutamic acid and at C₄ for 2(S),3(S),4(S)-3-hydroxy-4-methylglutamic acid are inferred from the configurations at C₂ in 2(_S),4(_S)-4-methylglutamic acid and at C₂ and C₃ in 2(S),3(R)-3-hydroxyaspartic acid. The seeds also contain appreciable quantities of 2(S),3(S),4(R)-3-hydroxy-4-methylglutami c acid (G1) and 2(S),4(R)-4-methylglutamic acid.     

Chirality and biosynthesis of lilac compounds in Actinidia arguta flowers

Biosynthesis of lilac compounds in ‘Hortgem Tahi’ kiwifruit (Actinidia arguta) flowers was investigated by treating inflorescences with d₅-linalool. The incorporation of the deuterium label into 8-hydroxylinalool, 8-oxolinalool, the lilac aldehydes, alcohols, and alcohol epoxides was followed by GC-MS and enantioselective GC-MS. Both (R)- and (S)-linalool were produced naturally by the flowers, but 8-hydroxylinalool, 8-oxolinalool, and the lilac aldehydes and alcohols occurred predominantly as the (S) and 5′(S)-diastereoisomers, respectively. The enantioselective step in the biosynthesis of the lilac aldehydes and alcohols was concluded to be the oxidation of linalool to (S)-8-hydroxylinalool. In contrast, the lilac alcohol epoxides had a 5′(R):(S) ratio, the same as for linalool, which suggests that either these compounds are not synthesised from the 5′(S)-configured lilac aldehydes and alcohols, or that if indeed they are, then it is by an enantioselective step that favours utilisation of the 5′(R)-configured compounds.     

Cyclic and high molecular weight chiral binaphthoxy-phosphazene polymers carrying Br or trimethylsilyl-acetylene groups

The new chiral and functionalized cyclic binaphthoxyphosphazenes R,R,R-[N₃P₃(O₂C₂₀H₁₀Br₂)₃] (R-1), R,R,R-[N₃P₃(O₂C₂₀H₁₀(CCSiMe₃)₂)₃] (R-2), and the high molecular weight linear polymers R/S-[NP(O₂C₂₀H₁₀Br₂)]_n (R/S-3), R-[NP(O₂C₂₀H₁₀Br₂)]_n (R-3), and R-{NP[O₂C₂₀H₁₀(CCSiMe₃)₂]}_n, (R-4), with M_w on the order of 10⁶ and very high T_g, have been synthesized and characterized by IR and NMR spectroscopy. The optically active polymer (R-3) was configurationally stable below 300 °C, but at higher temperatures an atropisomerization process took place that became faster near the glass transition temperature (ca. 350 °C).     

The rate of convergence of a generalized stable population

In an age-structured population that grows exponentially, each age groupP _i(t) at periodt is asymptotically equivalent tox ₀ ^t for some positive number x₀. In this paper we show that the speed at which the ith age group reaches its exponential state of equilibrium can be measured by the rate at which the ratio v_i(t)=P_i(t)/p_i(t–1) converges tox ₀. The age specific rate of convergence is determined by considering a quantityr satisfyingv _i(t)-x ₀ ¦ r ^t whent is large;R _i=Infr (over all initial populations,r satisfying the above inequality) is the R-factor used in numerical analysis to measure the rate at which the sequencev _i (t) converges tox ₀;S _i =- In R_i is then defined as the rate of convergence to stability of the ith age group. The case of constant net maternity rates is studied in detail; in this contextS ₀ is compared to the population entropyH, which was proposed by Tuljapurkar (1982) as a measure of the rate of convergence to stability.     

Complexes with asymmetric tetraamine ligands. IV. Cobalt(III) complexes containing two chiral ligands. Structure of Λ-cis-β2-(SS)-[S-alaninato-2S, 5R, 9S-trimethyltriencobalt(III)] perchlorate hydrate

Complexes of cobalt(III) with two optically active ligands have been prepared and characterized. One of the ligands was an amino acid (S-alanine or S-isoleucine) and the other was a methyl-substituted derivative of triethylenetetramine (trien), either 2S,5R,9S-Me₃trien or 2S,5S,9S-Me₃trien. The amino acid complexes were prepared from the cis-α or cis-β dichloro complexes of Co(III) with the appropriate trien derivative. The crystal and molecular structure of one of these complexes, with S-alanine and 2S,5R, 9S-Me₃trien, is reported.Since the cobalt(III) complex contains two chiral ligands of known absolute configuration, the overall configuration of the complex could be assigned unambiguously as the Λ isomer. Although one of the three chelate rings of the chiral tetraamine deviates from ideal symmetric skew geometry, all three methyl groups are in equatorial positions. Both secondary nitrogen atoms have S configuration, and the methyl group on the central chelate ring of the tetraamine is adjacent to the ‘flat’ rather than the ‘apical’ secondary N. The alanine anion is coordinated with its nitrogen trans to a secondary nitrogen and its oxygen trans to a primary nitrogen of the tetraamine, to give the β₂ geometric isomer. Both perchlorate anions are disordered.Crystallographic data for the title complex are as follows: C₁₂H₃₀O₁₁N₅Cl₂Co, M_r=550.23, tetragonal, P4₃2₁2, Z=8, a=9.166(5), c=53.51(4) Å, V=4495.4 ų, D_m=1.63(1), D_c=1.657 g cm⁻³, CuKα, λ=1.54178 Å, μ(CuKα)=90.89 cm⁻¹, F(000)=2288, room temperature, final R=[Σ(6F_o| −|F_c|)/Σ|F_o|]=O.089 for 2620 unique, observed reflections.