The dynamics of a discrete population model with threshold

The dynamics of the discrete, scalar population model x_{k + 1} = ax²_k(1 – x_k) are investigated. In addition to density dependence, which has been studied previously by many, this equation models the threshold phenomenon. Some similarities to and differences from previous models are observed. In particular, for large a values this model exhibits chaos which is restricted to a nowhere dense Cantor set of measure 0. In order to explain this, a piecewise linear simplification of the model is considered. Other models exhibiting similar dynamics are also mentioned.     

Synthesis, structures, electrochemistry and properties of dioxo-molybdenum(VI) and -tungsten(VI) complexes with novel asymmetric N2OS, and partially symmetric N2S2, NOS2N-capped tripodal ligands

A new class of asymmetric N-capped (dianionic/trianionic) tripodal proligands [H_x(Lⁿ)] (x = 2, n = 1-6; x = 3, n = 7, 8) which possess pendant arms with N₂OS, N₂S₂ or NOS₂ donor groups and with different chelate ring sizes {5,5,5} or {5,6,5} has been prepared. Treatment of these ligands with [WO₂Cl₂(dme)] (dme = 1,2-dimethoxyethane) in the presence of base (triethylamine or KOH) leads to the formation of cis-dioxotungsten(VI) complexes of the types [WO₂(Lⁿ)] (n = 1-6) and K[WO₂(Lⁿ)] (n = 7, 8). Reaction of these tetradentate ligands with [MoO₂(acac)₂] (acac = acetylacetonate) gives the corresponding Mo(VI) analogues [MoO₂(Lⁿ)] (n = 1-6) and K[MoO₂(Lⁿ)] (n = 7, 8). Moreover, a new five coordinate dioxomolybdenum(VI) complex with an NS₂ tridentate ligand [MoO₂(L⁹)] has been synthesised using similar procedure. All these compounds have been spectroscopically characterised and the molecular structures of [MoO₂(Lⁿ)] (n = 2, 6) and [WO₂(L⁶)] have been established by X-ray diffraction analysis. The electrochemistry and the catalytic activity for oxidation of allylic and benzylic alcohols of these dioxo complexes have also been investigated.     

Analysis of metabolism by mathematical programming techniques

A method is presented for analyzing metabolic interactions by procedures based on mathematical programming techniques. In the procedures described it is assumed that the organism has (through natural selection) maximized (within the constraints imposed on it by its genetic constitution) its fitness to the environment. A practical experimental procedure is described through which the constraints imposed on reaction rates can be observed and from which the metabolic objective function which, it is presumed, metabolism has optimized can be calculated. A method for testing the validity of the objective function is given. Discussion is carried out in terms of a two-dimensional example but the procedures are valid for any number of dimensions. The results of the procedures are expressed by statements of the sort: the metabolic interactions of the cell are such that Q is maximized where Q = a₁x₁ + a₂x₂ + ... + a_nx_n, where a₁, ..., a_n are constants and x₁, ..., x_n are reaction rates. Some possible uses of metabolic objective functions are given.     

Helical parameters of infinite polymer chains having a diamond-like backbone. application to homopolysaccharides and cyclic oligosaccharides.

Helical conformations of infinite polymer chains may be described by two helical parameters: n and h. General n,h maps have been produced that are independent of the monomer chemistry, except for the limitation of tetrahedral geometry. Some very simple geometrical criteria have been found to underlie the general aspects of the maps obtained. These maps are found to cover the whole range of predictable secondary structures of homopolysaccharides having the (1→2), (1→3), and (1→4) linkage-types. Moreover, a general strategy for investigating the secondary structures of (1→6)-linked homopolysaccharides is proposed. In an extension of these results, the prediction of the possibility of a stable cyclic tetrasaccharide made up of (1→6)-β-D-glucose residues, is proposed and analyzed.     

Characterization of the WEIBULL Distribution by Properties of Order Statistics

Let X_1:n, X_2:n, X_3:n…, X_n:n be the order statistics of n independent random variables with the common (absolutely continuous strictly increasing) distribution function F. The main results given in this article are:

• 1 For any fixed r and two distinct numbers s₁ and s₂ (1<r<s₁<s₂≦n) the distributions of V_i and W_i (defined in (1.11) and (1.12) are identical for i = 1,2 iff F(x) is WEIBULL (1.2).
• 2 The statistics D₁ and D₂ (as in (1.8) and (1.9)) are independent iff F(x) is WEIBULL (1.2).
• 3 The statistics U_i (1≦j≦n?1) and X_i:n (i ≦ j) are independent iff F(x) is WEIBULL (1.2).
• 4 Let X, X₁, X₂, …, X_k be random variables such that

These conditions are necessary and sufficient for F(x) to be WEIBULL .     

Energy transfer by chlorophyllb in detergent micelles

The concentration-dependent depolarization, concentration-dependent quenching, absorption and fluorescence spectra in solutions of chlorophyllb-containing detergent micelles with Triton X-100 were studied in a concentration range ofc = 0.4 μM–0.6mM chlorophyllb andc_d = 0.4–7.0mM Triton X-100. The concentration-dependent depolarization obeys Fo¨rster's theory of depolarization of fluorescence with a transfer distance parameterR₀ = 43 ± 2A?. The concentration-dependent quenching is described by an empirical formula for the relative fluorescence yieldη/η₀=sol1[1 + (cc_1/2)²] given by Kelly and Porter (Kelly A. R. and Porter, G. (1970) Proc. R. Soc. Lond. Ser. A. 315, 149–161). With increasing chlorophyll b concentration the red absorption band at 650 nm is shifted toward a longer wavelength and its width increases by 10 nm, the intensity of the long wave fluorescence band increases about 720 nm. The results analysed in terms of these findings lead to the conclusions that chlorophyllb molecules are (a) locally concentrated in the micelles up to the concentration range of in vivo conditions, (b) partly in an aggregated state capable for fluorescence, (c) the chlorophyllb →chlorophyllb homotransfer may be about 3–26 % of the homotransfer chlorophylla →chlorophyll-a depending on the ratio of their concentrations.     

Syntheses and characterizations of a series of silver-carboxylate polymers

Self-assembly reaction of H₃btc (1,3,5-benzenetricarboxylic acid), H₂ta (1,4-benzenedicarboxylic acid) or 1,2,4,5-benzenetetracarboxylic dianhydride with Ag(NO₃) in mixed H₂O/MeOH solution at room temperature gave rise to four novel 3D polymeric silver(I) complexes, [{Ag(H₂btc)}{Ag₂(Hbtc)}]_n (1), [Ag(ta)_1/2]_n (2), [Ag₂(btec)_1/2]_n (3) and [{Ag₃(btec)_3/4}{Ag(H₂O)₂(btec)_1/4}]_n (4) (H₄btec=1,2,4,5-benzenetetracarboxylic acid). Complex 1 crystallizes in orthorhombic space group Fddd, with a=14.877(5), b=25.926(1), c=36.377(7) Å, Z=32. Complex 2 crystallizes in monoclinic space group P2₁/c, with a=7.257(6), b=8.949(8), c=6.346(5) Å, β=111.654(3)°, Z=4. Complex 3 crystallizes in orthorhombic space group P2₁/c, with a=8.333(8), b=6.335(5), c=10.964(9) Å, Z=4, whereas complex 4 crystallizes in triclinic space group , with a=7.509(1), b=9.351(1), c=10.307(7) Å, α=69.514(2), β=84.521(2), γ=83.638(2)°, Z=2. All compounds possess 3D framework and short Ag-Ag contacts are present in 1-4. In 1, the ligand unsupported Ag-Ag interactions play an important role in the formation of the complex, which is constructed by silver chains formed by Ag-Ag interactions and carboxylate spacers.     

Structure and electrochemistry of the bridging-ligand mono-substituted diruthenium compound, [Ru2(II,III)(O2CCH3)3(admpym)(Cl)(MeOH)] (Hadmpym=2-amino-4,6-dimethylpyrimidine)

The ligand substitution reaction of Ru₂(O₂CCH₃)₄Cl with 2-amino-4,6-dimethylpyrimidine (Hadmpym) under gentle refluxing conditions in methanol led to the formation of a bridging-ligand mono-substituted compound, [Ru₂(O₂CCH₃)₃(admpym)(Cl)(MeOH)] (1). Compound 1 crystallized in monoclinic space group P2₁/n (no. 14) with a=8.3074(8) Å, b=12.3722(8) Å, c=18.913(1) Å, β=95.559(3)°, V=1934.8(3) ų, and Z=4. Temperature dependence of the magnetic susceptibility of 1 revealed it to be in a spin ground state S=3/2 arising from the electronic configuration of σ²π⁴δ²(δ*π*)³. Compound 1 undergoes three metal-centered redox reactions in electrochemistry: E_1/2 ^(ox)=+0.72 V (I_a/I_c<1, ΔE_p=0.17 V); E_1/2 ^(1,red)=−0.65 V (I_a/I_c≈1, ΔE_p=0.10 V); and E_1/2 ^(2,red)=−1.80 V (I_a/I_c?1, ΔE_p=0.16 V). Then, the redox species produced by electrolysis were characterized by spectroscopic studies.     

Preparation and structure of [Ru2(O2CMe)4]2[Fe(CN)5NO] and magnetically ordered Hx[Ru2(O2CMe)4]3−x[Cr(CN)5NO] possessing interpenetrating lattices

Reaction of [Ru₂(O₂CMe)₄]Cl with K₃[Cr(CN)₅NO] in water forms H_x[Ru^II/III₂(O₂CMe)₄]_3−x-[Cr(CN)₅NO]·zH₂O (x = 0.2) that magnetically orders at 4.0 K and possesses an interpenetrating body centered cubic [a = 13.2509(2) Å] structure with random locations of the bridging nitrosyl ligands, and x/3 vacant cation sites. Similarly, the aqueous reaction of [Ru₂(O₂CMe)₄]Cl with Na₂[Fe(CN)₅NO] forms paramagnetic [Ru₂(O₂CMe)₄]₂[Fe(CN)₅NO]·H₂O, which has a similar tetragonal interpenetrating structure [a = 13.0186(1) Å, c = 13.0699(2) Å] where the NO ligands are presumably nonbridging and 1/3 of the expected cation sites are unoccupied. The presence of uncoordinated NO sites in addition to missing neighboring [Ru₂(O₂CMe)₄]⁺ units, results in significant vacancies (or holes) in the lattice.     

Quantifying the trans influence of triphenylarsine. Crystal and molecular structures of cis-[PtCl2(SMe2)(AsPh3)] and cis-[PtCl2(AsPh3)2]·CHCl3

Reaction between a mixture of cis-trans-[PtCl₂(SMe₂)₂] and 1 equiv. AsPh₃ in chloroform gives cis-[PtCl₂(SMe₂)(AsPh₃)] crystallizing in P2₁/n with a=10.397(2), b=14.876(3), c=13.956(3) Å, β=90.86(3)° and Z=4. Selected geometrical parameters are PtAs 2.3531(10), PtS 2.262(2), PtCl (trans to S) 2.301(2), PtCl (trans to As) 2.328(2) Å and SPtAs 88.85(6), SPtCl(2) 90.77(8), AsPtCl(1) 91.07(6) and ClPtCl 89.42(7)°. cis-[PtCl₂(AsPh₃)₂]·CHCl₃ crystallizes in P2₁/c with a=20.557(4), b=9.5951(19), c=20.147(4) Å, β=96.77(3)° and Z=4. Selected geometrical parameters are PtAs(1) 2.3599(9), PtAs(2) 2.3770(9), PtCl(1) (trans to As(1)) 2.3515(18), PtCl(2) (trans to As(2)) 2.3251(18) Å and AsPtAs 97.87(3), As(1)PtCl(2) 88.67(5), As(2)PtCl(1) 84.30(5) and ClPtCl 89.32(7)°. By comparison with related structures from the literature the following trans influence series was established PMe₂Ph>PPh₃>AsPh₃≈SbPh₃>Me₂SO≈SMe₂≈SPh₂>NH₃≈olefin>Cl⁻>MeCN.     

A new series of iodocarbonyl ruthenium (II) complexes with unsymmetrical phosphine-phosphine sulfide ligands of the type Ph2P(CH2)nP(S)Ph2, n = 1-4: Isolation and structural investigation

Hexa-coordinated chelate complex cis-[Ru(CO)₂I₂(P∩S)] (1a) {P∩S = η²-(P,S)-coordinated} and penta-coordinated non-chelate complexes cis-[Ru(CO)₂I₂(P∼S)] (1b-d) {P∼S = η¹-(P)-coordinated} are produced by the reaction of polymeric [Ru(CO)₂I₂]_n with equimolar quantity of the ligands Ph₂P(CH₂)_nP(S)Ph₂ {n = 1(a), 2(b), 3(c), 4(d)} in dichloromethane at room temperature. The bidentate nature of the ligand a in the complex 1a leads to the formation of five-membered chelate ring which confers extra stability to the complex. On the other hand, 1:2 (Ru:L) molar ratio reaction affords the hexa-coordinated non-chelate complexes cis,cis,trans-[Ru(CO)₂I₂(P∼S)₂] (2a-d) irrespective of the ligands. All the complexes show two equally intense terminal ν(CO) bands in the range 2028-2103 cm⁻¹. The ν(PS) band of complex 1a occurs 23 cm⁻¹ lower region compared to the corresponding free ligand suggesting chelation via metal-sulfur bond formation. X-ray crystallography reveals that the Ru(II) atom occupies the center of a slightly distorted octahedral geometry. The complexes have also been characterized by elemental analysis, ¹H, ¹³C and ³¹P NMR spectroscopy.     

Chemical implantation of Group 4 cations on silica via cyclopentadienyl- and N,N-dialkylcarbamato derivatives

Chemical implantation of Group 4 cations [Ti(III), Ti(IV), Zr(IV), Hf(IV)] has been carried out under mild conditions by the reaction of polycyclopentadienyl- (MCp_n; M = Ti, n = 3, 4; M = Zr, Hf, n = 4), mixed cyclopentadienyl/N,N-dialkylcarbamato (ML_x(O₂CNEt₂)_y; M = Ti, L = Cp, C₅Me₅ (Cp*), x = 2, y = 1; M = Hf, L = Cp, x = 1, y = 3), and N,N-dialkylcarbamato (M(O₂CNR₂)_n, M = Ti, n = 3, R = ⁱPr; M = Ti, Hf, n = 4, R = Et; M = Zr, n = 4, R = ⁱPr) derivatives, with the silanol groups of amorphous silica. Cyclopentadiene/pentamethylcyclopentadiene and/or carbon dioxide and the secondary amine are released in the process. The amount of implanted cations depends on the metal and on the ligands, the pentamethylcyclopentadienyl complex being less reactive than the unsubstituted congener. The starting complexes and the final products have been characterized by EPR or by ¹³C CP-MAS NMR spectroscopy.     

Three coordination modes of the pentadentate ligand 2,6-diacetylpyridinedisemicarbazone

The pentadentate ligand 2,6-diacetylpyridinedisemicarbazone, DAPSC, reacts with Cr(NO₃)₃·9H₂O and forms two kinds of complexes. At pH=3, the ligand is singly-deprotonated and crystals of [Cr- (DAPSCH)(H₂O)₂](NO₃)₂·H₂O (Ia) are obtained. Evaporation of a solution at pH=0, yields crystals of [Cr(DAPSC)(H₂O)₂](NO₃)₃·2H₂O (II) in which the ligand is fully protonated. The reaction of DAPSC with UO₂(O₂CCH₃)₂ in methanol, followed by crystallization of the product from DMSO yields crystals of [UO₂(DAPSC2H)(H₂O)]·2DMSO (III) in which the ligand is fully deprotonated. Compound Ia is monoclinic, space group P2₁/n with a=11.746(1), b=14.752(2), c=11.866(1) Å,β=105.53(2)°, V= 1981(1) ų and Z=4. Compound II is monoclinic, space group, P2₁/n with a=38.000(3), b= 14.939(2), c=8.233(1) Å, β=96.12(2)°, V= 4647(1) Å and Z=8. Compound III is monoclinic, space group P2₁/n with a=18.048(2), b=15.207(2), c=8.842(1) Å,β=97.72(2)°, V=2405(1) ų and Z=4. The structures were refined using 2084, 4169 and 2516 reflections to R values of 4.4%, 7.8% and 4.8% respectively.     

Aggregation of ferrihaems. Dimerization and protolytic equilibria of protoferrihaem and deuteroferrihaem in aqueous solution

1. The absorption spectra of deutero- and proto-ferrihaem in aqueous solution at 25°C show marked changes with concentration and pH in the Soret band region. Quantitative studies of these phenomena imply that they are associated with ferrihaem dimerization and with protolytic equilibria involving monomeric (M) and dimeric (D) ferrihaem species according to the scheme: [Formula: see text] 2. For deuteroferrihaem we obtain K=1.9×10⁻², pK_a(M)=7.1, pK_a(D)=7.4. Protoferrihaem has a much higher dimerization constant, K=4.5 and pK_a(D)=7.5 (pK_a(M) is not accessible). 3. Possible structural relationships between monomeric and dimeric ferrihaem species in solution are discussed in relation to recent work on the oxo-bridged nature of crystalline ferrihaem dimers.     

Dynamical consequences of harvest in discrete age-structured population models

The role of harvest in discrete age-structured one-population models has been explored. Considering a few age classes only, together with the overcompensatory Ricker recruitment function, we show that harvest acts as a weak destabilizing effect in case of small values of the year-to-year survival probability P and as a strong stabilizing effect whenever the survival probability approaches unity. In the latter case, assuming n=2 age classes, we find that harvest may transfer a population from the chaotic regime to a state where the equilibrium point (x₁*, x₂*) becomes stable. However, as the number of age classes increases (which acts as a stabilizing effect in non-exploited models), we find that harvest acts more and more destabilizing, in fact, when the number of age classes has been increased to n=10, our finding is that in case of large values of the survival probabilities, harvest may transfer a population from a state where the equilibrium is stable to the chaotic regime, thus exactly the opposite of what was found in case of n=2. On the other hand, if we replace the Ricker relation with the generalized Beverton and Holt recruitment function with abruptness parameter larger than 2, several of the conclusions derived above are changed. For example, when n is large and the survival probabilities exceed a certain threshold, the equilibrium will always be stable.Revised version: 18 September 2003     

Structural change of supramolecular coordination polymers of itaconic acid and 1,10-phenanthroline along lanthanide series

Five new supramolecular lanthanide coordination polymers with three different structures, {[La₂(IA)₃(phen)₂] · 2H₂O}_n (1), {[Ln(IA)_1.5(phen)] · xH₂O}_n [x = 1, Ln = Eu (2); x = 0.25, Ln = Dy (3)], and [Ln(IA)_1.5(phen)]_n [Ln = Er (4); Yb (5)], were prepared by hydro- and solvothermal reactions of lanthanide chlorides with itaconic acid (H₂IA) and 1,10-phenanthroline (phen), and structurally characterized by single crystal X-ray diffraction. 1 Comprises 1-D double-chains that are further assembled to a 3-D supramolecular structure via hydrogen bonds and π-π stacks between phen molecules. 2 and 3 have 2-D infinite networks which are further constructed to form 3-D supramolecular architectures with 1-D channels by π-π aromatic interactions. 4 and 5 have 2-D layer structures consisting of three types of rings which are further architectured to form 3-D supramolecular structures by C-H?O hydrogen bonds. The H₂IA ligands are all completely deprotonated and exhibit tetra-, penta-, and hexadentate coordination modes in the titled complexes. The high-resolution emission spectrum of 2 shows only one Eu³⁺ ion site in 2, which is in agreement with the result of X-ray diffraction. And the magnetic property and the thermal stability of 2 were also investigated.     

Steady-state kinetics of electron transfer through cytochrome chain of uncoupled submitochondrial particles. II. Influence of pH on kinetics of electron transfer

pH Dependences of steady-state kinetic parameters of cytochrome chains of submitochondrial particles have been studied. It has been shown that the lifetimes of activated states (τ) of the pairs of cytochromes b → c₁ and a → a₃ have different pH dependences; those for the c₁ → c and c → a cytochrome pairs being similar. The rate constants for the non-activated state of the respiratory chains decreased for the b → c₁ pair and increased for the a → a₃ pair when the pH value was increased.The values of pK calculated from these dependences for the pairs b → c₁ and a → a₃ were 7.2 and 8.9, respectively. It has been supposed that the ratio of activated to non-activated electron carriers may be controlled by the local pH value in the mitochondrial membrane, the latter being dependent upon the rate of electron transfer. The kinetic model based on this assumption allows one to explain the experimental dependences on pH of the rate constants for cytochromes b → c, and a → a₃.The values of the diffusion rate constants for H⁺ and OH^? ions in the mitochondrial membrane estimated from these kinetic data obtained in this study weree 10⁴–10⁵ s^?1 and 10²–10³ s^?1, respectively.     

Synthesis and characterization of amine complexes of the cyclopentadienyliron dicarbonyl complex cation, [Cp(CO)2Fe]

The organometallic Lewis acid, [CpFe(CO)₂]⁺ (Cp = η⁵-C₅H₅) reacts with excess dry diethyl ether at low temperatures to form the labile complex [CpFe(CO)₂(Et₂O)]⁺[BF₄]⁻ (1) which is stable at low temperatures and has been fully characterized. Complex 1 in turn reacts with 1-aminoalkanes and α,ω-diaminoalkanes to form new complexes of the type [CpFe(CO)₂NH₂(CH₂)_nCH₃]BF₄ (n = 2-6) (2) and [{CpFe(CO)₂}₂μ-(NH₂(CH₂)_nNH₂)](BF₄)₂ (n = 2-4) (3), respectively. These complexes have been fully characterized and the mass spectral patterns of complexes 2 are reported. The structures of compounds 2a (n = 2) and 2b (n = 3) have been confirmed by single crystal X-ray crystallography. The single crystal X-ray diffraction data show that complex 2a, [CpFe(CO)₂NH₂(CH₂)₂CH₃]BF₄, crystallizes in a triclinic space group while 2b, [CpFe(CO)₂NH₂(CH₂)₃CH₃]BF₄, crystallizes in an orthorhombic Pca2₁ space group with two crystallographically independent molecular cations in the asymmetric unit. Furthermore, the reaction of 1 with 1-alkenes gives the η²-alkene complexes in high yield.     

Synthesis and characterization of isopropylamine complexes of lanthanide(II) diiodides: Molecular structure of TmI2(PrNH2)4 and EuI2(PrNH2)4

It was found that the lanthanide diiodides LnI₂ (1) (Ln = Nd, Sm, Eu, Dy, Tm, Yb) are dissolved in isopropylamine (IPA) without redox transformations. Stability of the formed solutions decreases in a row Eu ≈ Yb > Sm > Tm > Dy > Nd. Removing of a solvent in vacuum leaves complexes LnI₂(IPA)_x (2) (Nd, x = 5; Sm, Eu, Dy, Tm, Yb, x = 4) as crystalline colored solids. Stability of 2-Nd,Dy,Tm is higher than that of known THF or DME coordinated salts. Divalent state of metal in the products is confirmed by data of UV-Vis spectroscopy, magnetic measurements and their chemical behavior. Structure of 2-Eu and 2-Tm was established by X-ray diffraction analysis. Oxidation of 2-Nd,Dy in IPA affords amine-amides (PrⁱNH)Ln(IPA)_y (3) (Nd, y = 4; Dy, x = 3). n-Propylamine also dissolves the iodides 1-Sm,Eu,Dy,Tm,Yb but stability of the solutions is significantly lower. 1-Nd vigorously reacts with PrⁿNH₂ even at −30 °C which hampers the formation of the solution.     

Purification and characterisation of 1F-fructosyltransferase from the roots of asparagus (asparagus officinalis L.)

A fructosyltransferase that transfers a terminal d-fructosyl group from a (2→1)-β-linked fructosaccharide to HO-1 of another d-fructosyl group has been purified from an extract of asparagus roots by successive chromatography with DEAE-cellulose, octyl-Sepharose, Sephadex G-200, and raffinose-coupled Sepharose 6B. The disc-electrophoretically homogeneous enzyme was free from β-d-fructofuranosidase, sucrose:sucrose 1-fructosyltransferase, and 6^G-frutosyltransferase activity, and catalysed the d-fructosyl transfer from 1-kestose more rapidly to saccharides of the neokestose series [1^F(1-β-d-fructofuranosyl)_m-6^G(1-β-d-fructofuranosyl)_nsucrose] than to those of the 1-kestose series [1^F(1-β-d-fructofuranosyl)_nsucrose]. The enzyme was tentatively termed 1^F-fructosyltransferase. The general properties of the enzyme were as follows: mol. wt., ~64,000; optimum pH, ~5.0; stable at pH 5.0–5.5 at 45° for 20 min; stable at 30–45° for 10 min; inhibited by Hg²⁺, p-chloromercuribenzoate, and Ag⁺.