ALGORITHM FOR APPLICATION OF FOURIER ANALYSIS FOR BIORHYTHMIC BASELINES OF PHARMACODYNAMIC INDIRECT RESPONSE MODELS

The change of an indirect pharmacological response R(t) can be described by a periodic time-dependent production rate k_in (t) and a first-order loss constant k_out. If k_in(t) follows some biological rhythm (e.g., circadian), then the response R(t) also displays a periodic behavior. A new approach for describing the input function in indirect response models with biorhythmic baselines of physiologic substances is introduced. The present approach uses the baseline (placebo) response R_b(t) to recover the equation for k_in(t). Fourier analysis provides an approximate equation for R_b(t) that consists of terms (usually two or three) of the Fourier series (harmonics) that contribute most to the overall sum. The model differential equation is solved backward for k_in(t), yielding the equation involving R_b(t). A computer program was developed to perform the square L²-norm approximation technique. Fourier analysis was also performed based on nonlinear regression. Cortisol suppression after inhalation of fluticasone propionate (FP) was modeled based on the inhibition of the secretion rate k_in(t) using ADAPT II. The pharmacodynamic parameters k_out and IC₅₀ were estimated from the model equation with k_in(t) derived by the new approach. The proposed method of describing the input function needs no assumption about the behavior of k_in(t), is as efficient as methods used previously, and is more flexible in describing the baseline data than the nonlinear regression method. (Chronobiology International, 17(1), 77–93, 2000)     

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     

Stability of the analytical solution of Penna model of biological aging

There are some analytical solutions of the Penna model of biological aging; here, we discuss the approach by Coe et al. (Phys. Rev. Lett. 89, 288103, 2002), based on the concept of self-consistent solution of a master equation representing the Penna model. The equation describes transition of the population distribution at time t to next time step (t + 1). For the steady state, the population n(a, l, t) at age a and for given genome length l becomes time-independent. In this paper we discuss the stability of the analytical solution at various ranges of the model parameters—the birth rate b or mutation rate m. The map for the transition from n(a, l, t) to the next time step population distribution n(a + 1, l, t + 1) is constructed. Then the fix point (the steady state solution) brings recovery of Coe et al. results. From the analysis of the stability matrix, the Lyapunov coefficients, indicative of the stability of the solutions, are extracted. The results lead to phase diagram of the stable solutions in the space of model parameters (b, m, h), where h is the hunt rate. With increasing birth rate b, we observe critical b ₀ below which population is extinct, followed by non-zero stable single solution. Further increase in b leads to typical series of bifurcations with the cycle doubling until the chaos is reached at some b _c. Limiting cases such as those leading to the logistic model are also discussed.     

Cell-pool tryptophan phases in ergot alkaloid fermentation

Three cell-pool tryptophan phases are recorded as characteristics of the alkaloid fermentation byClaviceps paspali grown on a simple defined medium without tryptophan. Within the early phase designated “tryptophan down” the alkaloid-biosynthetic activity of the mycelium attains the maximum, protein synthesis is reduced and extracellular proteases are formed. Cell-pool tryptophan level (b) drops, tryptophan synthetase activity (c) intensifies and sums of logb+logc after different time intervals remain constant. In the subsequent “tryptophan up” phase tryptophan level (b) increases, alkaloid yield (a) becomes a function of time and reaches the top level still tolerable by tryptophan synthetase. The difference of the logb—logc is constant. The tryptophan synthetase diminishes its activity simultaneously with the alkaloid-biosynthetic activity of the mycelium. The district between the “tryptophan down” and “tryptophan up” phase is an especially promising target for the investigation of the regulation of alkaloid formation and continuous fermentation of these compounds. During the third, i.e. “tryptophan over” phase, cell-pool tryptophan accumulates and attains a concentration exerting a negative effect on the alkaloid biosynthesis.     

The pcsA gene from Streptomyces diastaticus var. 108 encodes a polyene carboxamide synthase with broad substrate specificity for polyene amides biosynthesis

Two structurally related polyene macrolides are produced by Streptomyces diastaticus var. 108: rimocidin (3a) and CE-108 (2a). Both bioactive metabolites are biosynthesized from the same pathway through type I polyketide synthases by choosing a starter unit either acetate or butyrate, resulting in 2a or 3a formation, respectively. Two additional polyene amides, CE-108B (2b) and rimocidin B (3b), are also produced “in vivo” when this strain was genetically modified by transformation with engineered SCP2*-derived vectors carrying the ermE gene. The two polyene amides, 2b and 3b, showed improved pharmacological properties, and are generated by a tailoring activity involved in the conversion of the exocyclic carboxylic group of 2a and 3a into their amide derivatives. The improvement on some biological properties of the resulting polyenes, compared with that of the parental compounds, encourages our interest for isolating the tailoring gene responsible for the polyene carboxamide biosynthesis, aimed to use it as tool for generating new bioactive compounds. In this work, we describe the isolation from S. diastaticus var. 108 the corresponding gene, pcsA, encoding a polyene carboxamide synthase, belonging to the Class II glutamine amidotransferases and responsible for “in vivo” and “in vitro” formation of CE-108B (2b) and rimocidin B (3b). The fermentation broth from S. diastaticus var. 108 engineered with the appropriate pcsA gene construction, showed the polyene amides to be the major bioactive compounds.     

Measurement of the adhesive strength of a gingival cell line to agar

Summary A new technique is described for measuring the adhesive strength of a gingival cell line to an agar substratum by the modification of the original “blister” test for adhesives. A cell monolayer was developed on a Petri dish with a hole in the center of the growing surface, overlayed with agar, and the system pressurized to debond the cells from the agar surface. Pressure changes were measured by a capacitance pressure transducer the output of which was measured by a strip-chart recorder. The modulus (E) of the agar overlay was determined and used in the calculation of the adhesive-bond strength (γ_a). The (γ_a) yield for the gingival cell line (cell-agar debond) was 48.8 ergs per cm², and for the control (no cells) (agar-polystyrene debond) was 30.0 ergs per cm². This research was supported by National Institute of Dental Research Grand DE 03983-02     

Optimizing selection for function-valued traits

We consider a function-valued trait z(t) whose pre-selection distribution is Gaussian, and a fitness function W that models optimizing selection, subject to certain natural assumptions. We show that the post-selection distribution of z(t) is also Gaussian, compute the selection differential, and derive an equation that expresses the selection gradient in terms of the parameters of W and of the pre-selection distribution. We make no assumptions on the nature of the “time” parameter t.      

Theory of biological similarities,nondimensional parameters and invariant numbers

In previous papers (1955–1957) a theory of biological similarity was established, assuming that the limits are the mechanical and the electrodynamical similarity criteria. The range of this theory lies between the coefficient of the time exponent (γ) for mechanical (0.5γ) and electrodynamical (1.0γ) similarities, being the mode 0.93γ. Moreover, for certain functions this restricted theoretical range should be extended to the hydrodynamical similarity criterion (2γ), so that the dimensionless numbers commonly used in Physics (Reynolds, Froude, Weber, etc.) can be included within the total range (0.5–2γ) of biological similarities. From dimensional analysis of physiological, functions it was possible to obtain, by means of dimensional and solution matrices, a group of “nondimensional numbers” by applying Buckingham's Pi-theorem. Nevertheless, only if a single similarity criterion was applied, the residual weight exponent was exactly zero; in all other instances the weight exponent was not zero, due to the existence of a range for biological similarities and to the statistical meaning of exponent (b) of the allometric equations. From the similarity criteria “invariant numbers” can be obtained, by means of which it is possible to establish correlations between numerous morphological and physiological characteristics of a particular system (circulation, respiration, metabolism, etc.).     

Correlations between the psychological peculiarities of an individual and the efficacy of a single neurofeedback session (by the EEG characteristics)

Modifications of different EEG rhythms induced by a single neurofeedback session (by the EEG characteristics) directed toward an increase in the ratio of the spectral powers (SPs) of the α vs θ oscillations were compared with the psychological characteristics of the tested subjects (the group included 30 persons). A generally accepted neurofeedback technique was used; the intensity of acoustic white noise served as the feedback signal. EEG potentials were recorded from the C3 and C4 leads. Psychological testing was carried out using Eysenck’s (EPQ), Rusalov’s (OST), and (16 PF) questionnaires. The directions of changes in the SPs of EEG frequency components were found to significantly correlate with some individuality-related peculiarities of the tested subjects. The SP of the δ rhythm correlated with the EPQ scale “neuroticism,” OST scale “social plasticity,” and 16 PF factors H (“parmia”), I (“premsia”), and Q₃ (“self-control of behavior”). The SP of the θ component demonstrated correlations with the OST scales “ergisity,” “plasticity,” and “social temp” and with 16 PF factors M (“autia”), Q₄ (“frustration”), and Q1 (“exvia”). The SP of the α rhythm correlated with 16 PF factors Q₃ (“self-control of behavior”), G (“strength of superEgo”), O (“hypothymia”), L (“protension”), and N (“shrewdness”). The SP of the β rhythm correlated with the OST scale “emotionality,” while that of the γ rhythm showed correlations with the 16 PF indices L (“protension”) and M (“autia”). Changes in the ratio of the α vs θ SPs correlated with the EPQ scale “neuroticism.” Thus, our data confirm the statement that a high individual variability of the results of a single (first in the series) neurofeedback session is to a great extent related to peculiarities of the individual psychological pattern of the subject. Neirofiziologiya/Neurophysiology, Vol. 38, No. 3, pp. 239–247, May–June, 2006.     

Mathematical interpretation of dose-response curves

The dose-response of an individual organism can be described by a step functions if the organism survives when the dose is below a certain lethal level and dies when this level is exceeded. If, in a population of organism, the lethal dose for an individual has a unimodal distribution, the latter's properties will determine the shape of the population's response in the following manner. If the distribution is symmetric the dose-response curve has a symmetric sigmoid shape when plotted on linear coordinates. The location of the inflection point and the curve's slope around it are determined by the distribution's mode and variance. When the distribution is skewed, the dose-response curve has an asymmetric sigmoid shape which becomes reminiscent of an exponential decay when the distribution is strongly skewed to the right. The population's dose-response curve can be constructed by integration of the step changes over the distribution range. The step function representing the dose-response of an individual organism can be approximate by a Fermi function, and the distribution of an lethal doses can be represented by the Weibull distribution function. When the two functions are combined, the resulting dose-response of the populationS(X)), which is the fraction of survivors after exposure to a doseX, is given by:S(X)=∫ ₀ ¹ [1/{+exp{(X-X _c (φ))/a _i ]}]dω whereX _c (ω)={(1/b)[-ln(1-ω)]}^(1/n),n andb being the constants of the Weibull distribution anda _i an arbitrarily small number, i.e.a _i ≪[X−X _c (ϕ)], whose actual magnitude is of little significance. This model can be used to determine the underlying distributions of experimental dose-response relationship. It was applied to published survival data of microorganisms exposed to pulsed electric field, X-ray radiation and ozone to show that the different observed shapes of the dose-response curve, and shifts between them, can be expressed in terms of the correponding distribution parameters, namely the mode, variance and skewness.     

The smallest value of the axon density for which ‘ignition’ can occur in a random net

As shown by A. Rapoport (1952), when a very brief stimulation or “instantaneous input” is applied to a random net, the subsequent events are determined by the parameters of the net as follows: If the axon densitya is sufficiently large and the fraction γ of the neurons initially stimulated exceeds a certain value γ₁ (theover-all threshold of the net for instantaneous stimulation), excitation will spread through the net until a steady state is reached in which a fraction γ₂ ⩾ γ₁ of the neurons is firing (“ignition phenomenon”). If γ < γ₁ the activity in the net dies out. However, if the axon density is too small, the activity will ultimately die out, no matter how large the fraction of initially stimulated neurons. Thus there exists a limiting valueA of the axon density below which the net cannot “ignite”. ThisA is a function ofh, theindividual threshold of the neurons constituting the net (we assume hereh≥2, since forh=1 the situation is essentially different). Geometrically γ₁ and γ₂ are determined as the two intersection points of a straight line with a sigmoid curve. Whena<A the two curves do not intersect and fora=A they are tangent. In this paper the “tangency case” is investigated and the general features of the functionA(h) are determined. It is shown thatA increases monotonically withh (as one would expect). For all values ofh>1 we haveA(h)>h, but the fractionA(h)/h and the derivativedA(h)/dh approach unity ash increases. An analytical expression of the functionA(h) valid for very large values ofh is derived.     

Accuracy of Optimized Chemical-exchange Parameters Derived by Fitting CPMG R 2 Dispersion Profiles when R 2 0a ≠ R 2 0b

The transverse relaxation rate, R₂, measured as a function of the effective field (R₂ dispersion) using a Carr-Purcell-Meiboom-Gill (CPMG) pulse train, is well suited to detect conformational exchange in proteins. The dispersion data are commonly fitted by a two-site (sites a and b) exchange model with four parameters: the relative population, p_a, the difference in chemical shifts of the two sites, δω, the correlation time for exchange, τ_ex, and the intrinsic relaxation rate (i.e., transverse relaxation rate in the absence of chemical exchange), R₂⁰. Although the intrinsic relaxation rates of the two sites, R₂^0a and R₂^0b, can differ, they are normally assumed to be the same (i.e., R₂^0a = R₂^0b = R₂⁰) when fitting dispersion data. The purpose of this investigation is to determine the magnitudes of the errors in the optimized exchange parameters that are introduced by the assumption that R₂^0a = R₂^0b. In order to accomplish this goal, we first generated synthetic constant-time CPMG R₂ dispersion data assuming two-site exchange with R₂^0a ≠ R₂^0b, and then fitted the synthetic data assuming two-site exchange with R₂⁰ = R₂^0a = R₂^0b. Although all the synthetic data generated assuming R₂^0a ≠ R₂^0b were well fitted (assuming R₂^0a = R₂^0b), the optimized values of p_a and τ _ex differed from their true values, whereas the optimized values of δω values did not. A theoretical analysis using the Carver–Richards equation explains these results, and yields simple, general equations for estimating the magnitudes of the errors in the optimized parameters, as a function of ( R₂^0a − R₂^0b).     

Kinematics of nonstationary locomotion in the centipede <Emphasis Type="Italic">Scolopendra cingulata</Emphasis>: A single instant change of speed

The spatiotemporal parameters of leg movement in Scolopendra instantly changing the locomotion speed from V ₁ to V ₂ ≠ V ₁ are investigated. It is shown that (i) the principle of “in trail” leg placement is kept upon a change of speed; (ii) the continuity of the metachronal coordination is not disturbed; but (iii) for some time after changing speed the set of ipsilateral legs comprises (a) head-proximal legs creating a new trackway of steps with pace ℓ₂, and (b) distal legs still using the old trackway with pace ℓ₁. The two groups differ in kinematic parameters. Group (a) works in the stationary mode corresponding to speed V ₂, while group (b) works in a transitory mode. Consecutively, with a metachronal wave propagating backwards along the body but immobile relative to the ground, more and more legs from group (b) switch to the new stationary mode.     

Porcine IgG: structure,genetics, and evolution

Eleven genomic porcine Cγ gene sequences are described that represent six putative subclasses that appear to have originated by gene duplication and exon shuffle. The genes previously described as encoding porcine IgG1 and IgG3 were shown to be the IgG1^a and IgG1^b allelic variants of the IGHG1 gene, IgG2a and IgG2b are allelic variants of the IGHG2 gene, while “new” IgG3 is monomorphic, has an extended hinge, is structurally unique, and appears to encode the most evolutionarily conserved porcine IgG. IgG5^b differs most from its putative allele, and its C_H1 domain shares sequence homology with the C_H1 of IgG3. Four animals were identified that lacked either IgG4 or IgG6. Alternative splice variants were also recovered, some lacking the C_H1 domain and potentially encoding heavy chain only antibodies. Potentially, swine can transcribe >20 different Cγ chains. A comparison of mammalian Cγ gene sequences revealed that IgG diversified into subclasses after speciation. Thus, the effector functions for the IgG subclasses of each species should not be extrapolated from “same name subclasses” in other species. Sequence analysis identified motifs likely to interact with Fcγ receptors, FcRn, protein A, protein G, and C1q. These revealed IgG3 to be most likely to activate complement and bind FcγRs. All except IgG5^a and IgG6^a should bind to FcγRs, while all except IgG6^a and the putative IgG5 subclass proteins should bind well to porcine FcRn, protein A, and protein G. An erratum to this article can be found at     

Theoretical evaluation of safety of dives completed by non-stop ascent to surface

Our model of decompression sickness determines the cumulative probability of developing symptoms of this illness by the exponential equation whose index is the integral cumulative risk function of all body tissue lesions by bubbles, F _cum(t) = ΣF _n(t). The underwater dives may be considered as practically safe in the context of this model when the function F _cum(t) during its growth will not exceed some small value F _cum-max = ΣF _n-max. Using hypothetical values of parameters of tissues and functions F _n(t), we calculated the curves depth-duration for practically safe non-stop dives with air and with mixtures of oxygen with helium, neon and argon. Doing so, we obtained the distributions of values F _n-max in regard to the values of inert gas washout half-times from tissues which show that the tissues experienced the largest risks of bubble lesions are different for dives with different duration. The comparison of the indicated curves shows that the short-term dives with air are less dangerous and the long-term dives are more dangerous than the dives with helium-oxygen mixture. At the same time, the least risk of bubble lesions of tissues arises at dives with neon-oxygen mixture and the greatest risk arises at dives with argon-oxygen mixture.     

A model of photosystem II for the analysis of fast fluorescence rise in plant leaves

The polyphasic patterns of fluorescence induction rise in pea leaves in vivo and after the treatment with ionophores have been studied using a Plant Efficiency Analyzer. To analyze in detail photosystem II (PS II) electron transfer processes, an extended PS II model was applied, which included the sums of exponential functions to specify explicitly the light-driven formation of the transmembrane electric potential (ΔΨ(t)) as well as pH in the lumen (pH_L(t)) and stroma (pH_S(t)). PS II model parameters and numerical coefficients in ΔΨ(t), pH_L(t), and pH_S(t) were evaluated to fit fluorescence induction data for different experimental conditions: leaf in vivo or after ionophore treatment at low or high light intensity. The model imitated changes in the pattern of fluorescence induction rise due to the elimination of transmembrane potential in the presence of ionophores, when ΔΨ = 0 and pH_L(t), pH_S(t) changed to small extent relative to control values in vivo, with maximum ΔΨ(t) ∼ 90 mV and ΔΨ(t) ∼ 40 mV for the stationary state at ΔpH ≅ 1.8. As the light intensity was increased from 300 to 1200 μmol m⁻² s⁻¹, the heat dissipation rate constants increased threefold for nonradiative recombination of P680⁺Phe⁻ and by ∼30% for P680⁺Q_A⁻. The parameters ΔΨ, pH_S and pH_L were analyzed as factors of PS II redox state populations and fluorescence yield. The kinetic mechanism of fluorescence quenching is discussed, which is related with light-induced lumen acidification, when ⁺Q_A⁻ and P680⁺ recombination probability increases to regulate the Q_A reduction.     

Scattering spectrum properties and their relationship to biogeochemical parameters: a case study in Taihu Lake

The scattering spectrum properties of highly turbid and eutrophic inland case 2 water from Taihu Lake were studied during three cruises from 2006 to 2007. The scattering [b _p(λ)] and backscattering [b _bp(λ)] coefficients and the backscattering probability (B) for Taihu Lake were found to show a clear spectral dependence, and this dependence was well simulated by a power-law function. This dependence, however, became weak when algae dominated the sample points. The mean values of the power-law index for b _p(λ), v, in Oct 2006, Mar 2007 and Nov 2007 were −0.6712, −0.8129 and −0.7600, respectively. To interpret the spectral characteristics and mechanisms of b _p(λ) and b _bp(λ), water samples were collected simultaneously for the biogeochemical characterization of suspended particles. The average values of the specific scattering coefficients for total suspended matter, inorganic suspended matter (ISPM) and organic suspended matter (OSPM) were 0.634 (550 nm), 1.057 (532 nm), and 0.396 g m⁻² (532 nm), respectively. The power-law index of b _bp(λ) (Y) was significantly related to ISPM/OSPM and b _bp(532 nm), but only weakly related to the particle size distribution index. The mean (spatial and wavelength) values of B in Oct 2006, Mar 2007, and Nov 2007 were 0.0108, 0.0138, and 0.0125, respectively. B decreases with increasing ISPM concentration because of the large contribution of ISPM to b _b(λ) and the strong restraint on b _bp(λ) caused by the multi-scattering effect under high-turbidity conditions.     

Development of optical criteria to discriminate various types of highly turbid lake waters

Our aim was to refine the optical classification of turbid waters in order to improve the performance of water color algorithms. Bio-optical measurements and sampling of optically active substances were performed in highly turbid lakes Taihu, Chaohu, and Dianchi, and in Three Gorges reservoir in China. Based on strong correlations between trough depths of remote sensing reflectance (R _rs(λ)) near 680 nm (denoted as TD680) and the ratios of inorganic suspended matter (ISM) to total suspended matter (TSM) concentrations, an empirical model was developed for water classification. In the 400–900 nm spectral range, different correlations between R _rs(λ), TSM and chlorophyll a (Chla) concentrations indicate discrepancies among the following ISM/TSM ranges: ISM/TSM ≤ 0.5, 0.5 < ISM/TSM < 0.8, and ISM/TSM ≥ 0.8. Corresponding findings support an important conclusion that only high ISM/TSM ratios, usually above 0.5, and using the more sensitive and stable near infrared spectral range (730–820 nm), can assure good performances of the TSM remote sensing algorithms. Meanwhile, the particulate absorption a _p(λ) and scattering b _p(λ) were strongly influenced by different ranges of ISM/TSM ratios. Typically the a _p(675) peaks became more and more vague as ISM/TSM increased, and the b _p(λ) values first decreased and then increased with a marked inflexion at ISM/TSM ≈ 0.5. The TD680 threshold values were derived to discriminate three types of turbid waters, i.e., Type 1 (TD680 ≥ 0.0082 sr⁻¹), Type 2 (0.0082 sr⁻¹ > TD680 > 0 sr⁻¹), and Type 3 (TD680 ≤ 0 sr⁻¹). This study provides a promising tool for identifying various types of highly turbid waters, and is significant for the development of class-based algorithms of water color remote sensing.     

On the existence of MHn species with M=Al, Ga and n=4, 5, 6. Computational study of structures, stabilities and bonding

Based on second-order perturbation theory (MP2) predictions with large 6-311++G(3df, 3pd) basis set we have reviewed the possible structures and stabilities of a series of neutral MH_n(M=Al, Ga; n=4, 5, 6) species. For AlH₄ and AlH₅, our results confirm the previous theoretical findings, which indicate the dihydrogen C_s complexes (²A′) AlH₂(H₂) and (¹A′) AlH₃(H₂), respectively, as the lowest energy isomers. We found, similarly, C_s (²A′) GaH₂(H₂) and (¹A′) GaH₃(H₂) van der Waals complexes as the most stable species of the gallium analogues GaH₄ and GaH₅. The calculated H₂ dissociation energies (D_e) for AlH₂(H₂) and AlH₃(H₂) are of the order 1.8–2.5 kcalmol¹, whereas this range of values for GaH₂(H₂) and GaH₃(H₂) is 1.4–1.8 kcalmol¹ . Symmetry-adapted perturbation theory (SAPT) was used to analyze the interaction energies of these dihydrogen complexes (for n=5) to determine why the Ga species show a smaller binding energy than the Al species. The SAPT partitioning of the interaction energy showed significant differences between AlH₃(H₂) and GaH₃(H₂), resulting from the much stronger “hydride” character of the aluminum species. The experimental observation of AlH₂(H₂) and AlH₃(H₂), and likely GaH₃(H₂), via low-temperature matrix isolation has been reported recently by Pullumbi et al. and Andrews et al., supporting the theoretical predictions. For n=6, we found the degenerate C₂(²A) and C_s(²A′) MH₂(H₂)₂ “double H₂” type van der Waals complexes as the lowest energy species for both M=Al and Ga.Electronic Supplementary Material Supplementary material is available for this article at     

Genetic control of aluminium tolerance in rye (Secale cereale L.)

 Aluminium (Al) tolerance in roots of two cultivars (“Ailés” and “JNK”) and two inbred lines (“Riodeva” and “Pool”) of rye was studied using intact roots immersed in a nutrient solution at a controlled pH and temperature. Both the cultivars and the inbred lines analysed showed high Al tolerance, this character being under multigenic control. The inbred line “Riodeva” was sensitive (non-telerant) at a concentration of 150 μM, whereas the “Ailes” cultivar showed the highest level of Al tolerance at this concentration. The segregation of aluminium-tolerance genes and several isozyme loci in different F₁s, F₂s and backcrosses between plants of “Ailés” and “Riodeva” were also studied. The segregation ratios obtained for aluminium tolerance in the F₂s analysed were 3 : 1 and 15 : 1 (tolerant : non-tolerant) while in backcrosses they were 1 : 1 and 3 : 1. These results indicated that Al tolerance is controlled by, at least, two major dominant and independent loci in rye (Alt1 and Alt3). Linkage analyses carried out between Al-tolerance genes and several isozyme loci revealed that the Alt1 locus was linked to the aconitase-1 (Aco1), nicotinamide adenine dinucleotide dehydrogenase-2 (Ndh2), esterase-6 (Est6) and esterase-8 (Est8) loci, located on chromosome arm 6RL. The order obtained was Alt1-Aco1-Ndh2-Est6-Est8. The Alt3 locus was not linked to the Lap1, Aco1 and Ndh2 loci, located on chromosome arms, 6RS, 6RL and 6RL respectively. Therefore, the Alt3 locus is probably on a different chromosome. Received: 18 March 1997 / Accepted: 21 March 1997