Biological growth and spread modeled by systems of recursions. I. Mathematical theory

We study the asymptotic behavior of solutions to a system of recursions u in+1 = Qi[mu n], i = 1, ..., k. The vector operator Q has the origin theta and a positive vector beta as fixed points and is defined for vector-valued functions bounded between theta and gamma where gamma greater than or equal to beta. In addition, Q is order-preserving, commutes with translation, and is continuous in the topology of uniform convergence on compact subsets. Let theta less than or equal to pi much less than beta, and suppose that for all pi much less than alpha much less than beta, Q(n) alpha]----beta as n----infinity. If u0 much greater than pi on a sufficiently large ball and has bounded support, then un propagates with a speed c*(xi) in the direction of the unit vector xi as n----infinity. In certain cases, c*(xi) can be calculated explicitly. The results generalize those of a scalar equation studied by Weinberger.     

A note on the nonautonomous delay Beverton-Holt model

It is well known that the periodic cycle {x(n)} of a periodically forced nonlinear difference equation is attenuant (resonant) if av(x(n)) < av(K(n))(av(x(n)) > av(K(n))),where {K ( n )} is the carrying capacity of the environment and av(t(n)) = (1/p)∑(p?1) (i=0) ti (arithmetic mean of the p-periodic cycle {t ( n )}). In this article, we extend the concept of attenuance and resonance of periodic cycles using the geometric mean for the average of a periodic cycle. We study the properties of the periodically forced nonautonomous delay Beverton-Holt model x(n+1) = r(n)x(n)/1 + (r(n?l) ? 1)x(n?k)/K(n?k), n= 0, 1, . . . , where {K ( n )} and {r ( n )} are positive p-periodic sequences; (K ( n )>0, r ( n )>1) as well as k and l are nonnegative integers. We will show that for all positive solutions {x ( n )} of the previous equation lim sup (n→∞) (∏(n?1)(i=0)xi)(1/n) ≤ ((∏(p?1)(i=0)ri)(1/p) ? 1)(∏(p?1)(i=0)(ri ? 1))(?1/p)(∏(p?1)(i=0)Ki)(1/p). In particular, in the case where {x(n)} is a p-periodic solution of the above equation (assuming that such solution exists) and r ( n )=r>1, the periodic cycle is g-attenuant, that is (∏(p?1)(i=0)x(i))(1/p)<(∏(p?1)(i=0)K(i))(p?1) Surprisingly, the obtained results show that the delays k and l do not play any role.     

Weber's theory of the kernleiter

The potential distribution about a kernleiter is determined according to Weber's method. It is shown that the distribution reduces to the solution of a telegrapher's equation when the volume of the external medium is small. The velocity of propagation as a function of the external volume is determined approximately. This involves the solution of the equation

$$\frac{{\left[ {Y_0 (k\xi )} \right]^\prime }}{{\left[ {J_0 (k\xi )} \right]^\prime }} = \frac{{\left[ {\xi ^{ - a} Y_0 (\xi )} \right]^\prime }}{{\left[ {\xi ^{ - a} J_0 (\xi )} \right]^\prime }}$$     

Modeling and analysis of a predator-prey model with disease in the prey

A system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. If the coefficient in conversing prey into predator k=k(0) is constant (independent of delay tau;, gestation period), we show that positive equilibrium is locally asymptotically stable when time delay tau; is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases. If k=k(0)e(-dtau;) (d is the death rate of predator), numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delay. A concluding discussion is then presented.     

Chemical dynamic approach to synergetics and instability in biological systems

Based on Haken's theory, self-organization or synergetics is discussed using chemical dynamics to represent an autocatalytic reaction. In a simple case the changes in a self-organizing system are given by a set of two rate equations for a primary and a partial system. When these systems mutually form a feedback loop under the adiabatic condition, the rate equation of self-organization is described by a generalized Gibbs' free energy change delta U (delta x) followed by the reaction. The sign of the parameter k'3 (k0-kex; k0, kex: rate constants with or without an external stimulus) determines the instability of the coupled system in quasi-equilibrium (k'3 approximately greater than 0; k0 greater than kex). When the product exceeds the threshold (k'3 less than 0; k0 less than kex), the system transfers into a new state, or a phase transition appears. Considering the Boltzmann distribution, the transition parameter k'3 is evaluated by an average distribution of the states and the instability is discussed using the reaction velocities vqe and vqe in the quasi-equilibrium state. As an example of this model membrane excitation is discussed briefly.     

Detection and identification of transient intermediates in the reactions of tryptophan synthase with oxindolyl-L-alanine and 2,3-dihydro-L-tryptophan. Evidence for a tetrahedral (gem-diamine) intermediate

The reactions of 2,3-dihydro-L-tryptophan (DHT) and oxindolyl-L-alanine (OXA) with tryptophan synthase have been investigated by rapid-scanning stopped-flow (RSSF) spectroscopy and by the concentration dependence of rates measured by single-wavelength stopped-flow (SWSF) spectroscopy. The RSSF spectral changes for DHT and OXA show the disappearance of the internal aldimine (lambda max 412 nm), the formation and decay of intermediates absorbing less than or equal to 340 nm, and the appearance of the quinonoid (lambda max 492 and 480 nm, respectively). Rate constants determined by SWSF were either well resolved (i.e., k1[DHT], k-1 greater than k2, k-2 greater than k3, k-3) or indicative of a tightly coupled system (i.e., k1[OXA], k-1 greater than or equal to k2, k-2 greater than k3, k-3). The RSSF spectral changes and SWSF kinetic studies together with computer simulations of the kinetic time courses are consistent with a mechanism that includes formation of a bleached species. Detection of these shorter wavelength species in the reactions of OXA and DHT indicates that substrate analogues with tetrahedral geometry at C-3 induce new protein-substrate interactions that result in the accumulation of species not previously detected in the tryptophan synthase system. The bleached species with lambda max less than or equal to 340 nm are proposed as the gem-diamine intermediates.     

Some observations on the visual optics of kingfishers (Aves,Coraciformes, Alcedinidae)

Summary We studied visual optics using ophthalmoscopy in six species of coraciform birds, five species from the family Alcedinidae (kingfishers) and one from the family Meropidae (bee-eaters). All six species had large angular separations between the two foveae of one eye (angle ); angle was greater than 40° in all cases, the largest separation so far reported for any group of vertebrates. In all kingfishers, but not in the bee-eater, the plane containing the projections of both foveae in one eye (angle ) was rotated from the horizontal plane so that the projection of the monocular fovea was lower than the projection of the binocular fovea.Retinal ganglion cell isodensity maps were obtained for the sacred kingfisher (Halcyon sancta) and laughing kookaburra (Dacelo gigas). These maps were constructed in the usual way for the peripheral regions of the retina. For the high-density, multi-layered, central region of the retinal ganglion cell layer, we used a combined retinal wholemount-cross-sectional technique. It was discovered that the horizontal streak, formed by the elliptically shaped contours of retinal ganglion cell isodensity, deviated inferiorly at its nasal extent from the line of the foveae. This deviation had the same sign, and slightly greater magnitude, as the rotation (angle ) observed ophthalmoscopically when the eyes were in their primary position.Our new observations provide new insights into the functional significance of the bifoveate visual organization. In particular, the relationship between angle and angle suggests one strategy which could be used by kingfishers to maximize their visual capabilities when hunting from a perch above the substrate.     

Adsorption of monovalent cations to bilayer membranes containing negative phospholipids.

The electrophoretic mobilities of multilamellar phosphatidylserine vesicles were measured in solutions containing monovalent cations, and the xi potentials, the electrostatic potentials at the hydrodynamic plane of shear, were calculated from the Helmholtz--Smoluchowski equation. In the presence of 0.1 M lithium, sodium, ammonium, potassium, rubidium, cesium, tetraethylammonium, and tetramethylammonium chloride, the xi potentials were -60, -62, -72, -73, -77, -80, -82, and -91 mV, respectively. Similar results were obtained with phosphatidylglycerol vesicles; different results were obtained with cardiolipin, phosphatidylinositol, and phosphatidic acid vesicles. The phosphatidylserine results are interpreted in terms of the Stern equation, a combination of the Gouy equation from the theory of the diffuse double layer, the Boltzmann relation, and the Langmuir adsorption isotherm. Evidence is presented that suggests the hydrodynamic plane of shear is 2 A from the surface of the membrane in solutions containing the alkali metal cations. With this assumption, the intrinsic association constants of the above monovalent cations with phosphatidylserine are 0.8, 0.6, 0.17, 0.15, 0.08, 0.05, 0.03, and 0 M-1, respectively. The validity of this approach was tested in two ways. First, the xi potentials of vesicles formed from mixtures of phosphatidylserine and a zwitterionic lipid, phosphatidylcholine, were measured in solutions containing different concentrations of sodium. All the data could be described by the Stern equation if the "relaxation" of the ionic atmosphere, which is predicted by classic electrostatic and hydrodynamic theory to occur at low salt concentrations and high potentials, was circumvented by using only large (diameter greater than 13 micrometers) vesicles for these measurements. Second, the fluorescent probe 2-(p-toluidinyl)naphthalene-6-sulfonate was used to estimate the potential at the surface of phosphatidylserine and phosphatidylglycerol vesicles sonicated in 0.1 M NaCl. Reasonable agreement with the predicted values of the surface potential was obtained.     

On the time course of the reversible Michaelis-Menten reaction

The methods of Padé approximants and Euler transformation are used to construct approximate solutions for the time course of the reversible Michaelis-Menten reaction. The solutions are found to describe the concentrations of the various species quite accurately throughout and beyond the transient phase. To illustrate the results, the ratio of the reverse bi-molecular rate constant to the forward bi-molecular rate constant, k2/k1, is varied from 0.1 to 5, and the initial enzyme-to-substrate concentration ratio is changed from 0.01 to 5. Only when k-2/k1 is less than one, the concentration of the intermediate complex, y(t), undergoes a maximum (steady state); for all other values of this ratio, y(t) increases monotonically with time t, to the equilibrium value, i.e. no maximum is attained. The present methods are particularly useful when the total enzyme concentration is comparable to, or greater than the initial substrate concentration, a situation commonly found under in vivo conditions.     

Theory and simulation of the time-dependent rate coefficients of diffusion-influenced reactions.

A general formalism is developed for calculating the time-dependent rate coefficient k(t) of an irreversible diffusion-influenced reaction. This formalism allows one to treat most factors that affect k(t), including rotational Brownian motion and conformational gating of reactant molecules and orientation constraint for product formation. At long times k(t) is shown to have the asymptotic expansion k(infinity)[1 + k(infinity) (pie Dt)-1/2 /4 pie D + ...], where D is the relative translational diffusion constant. An approximate analytical method for calculating k(t) is presented. This is based on the approximation that the probability density of the reactant pair in the reactive region keeps the equilibrium distribution but with a decreasing amplitude. The rate coefficient then is determined by the Green function in the absence of chemical reaction. Within the framework of this approximation, two general relations are obtained. The first relation allows the rate coefficient for an arbitrary amplitude of the reactivity to be found if the rate coefficient for one amplitude of the reactivity is known. The second relation allows the rate coefficient in the presence of conformational gating to be found from that in the absence of conformational gating. The ratio k(t)/k(0) is shown to be the survival probability of the reactant pair at time t starting from an initial distribution that is localized in the reactive region. This relation forms the basis of the calculation of k(t) through Brownian dynamics simulations. Two simulation procedures involving the propagation of nonreactive trajectories initiated only from the reactive region are described and illustrated on a model system. Both analytical and simulation results demonstrate the accuracy of the equilibrium-distribution approximation method.     

Permanence of some ecological systems with several predator and one prey species

The stability criterion used in the following is permanence. Permanence means that all trajectories starting in the interior are ultimately bounded away from the boundary and that this bound is independent of the initial values. Hence sufficiently small fluctuations cannot lead to extinction of any species. In the following we deal with one-prey, two-predator resp. one-prey, three-predator systems and a one-prey, two-predator, one-top-predator system with three trophic levels. It turns out that the characterization of permanence for such models described by Lotka-Volterra dynamics is rather simple and elegant.     

Qualitative permanence of Lotka–Volterra equations

In this paper, we consider permanence of Lotka-Volterra equations. We investigate the sign structure of the interaction matrix that guarantees the permanence of a Lotka-Volterra equation whenever it has a positive equilibrium point. An interaction matrix with this property is said to be qualitatively permanent. Our results provide both necessary and sufficient conditions for qualitative permanence.     

Shifts in frequency tuning of electroreceptors in androgen-treated mormyrid fish

Summary Several species of mormyrid electric fish have a sex difference in the pulse waveform of their electric organ discharge (EOD). Field studies in Gabon, West Africa have shown for one such species,Brienomyrus brachyistius (triphasic), that the sexually mature male EOD differs in shape and is nearly twice the duration of the EODs of females and juveniles. Fourier analysis reveals that differences in EOD duration correlate with those in the EOD power spectrum which has a peak at 0.3 kHz in males and 1.3 kHz in females and juveniles. We find a corresponding sex difference in the frequency tuning of at least one class of electroreceptors known as Knollenorgans. The average best or characteristic frequency of Knollenorgans is lower in males compared to females and juveniles. This correlates with a lower peak in the power spectrum of the male's pulse. When females are treated with gonadal androgens, their EODs increase 2–3 fold in duration, and the power spectra of their pulses are correspondingly lowered to match that of mature males. The average best frequency of Knollenorgans decreases by nearly 1 kHz which matches the downward shift of their EOD's power spectrum.For a second species ofBrienomyrus (sp. 2) which is commercially imported from Nigeria, we have not detected a sex difference in the power spectrum or duration of the EOD. The power spectrum peaks at about 4.2 kHz in males, females, and juveniles. Androgens, however, do cause a coincident downward shift in the average peak of the EOD power spectrum (from 4.2 to 1.3 kHz) and the average best frequency of Knollenorgans (from 2.3 to 1.4 kHz).Specimens ofBrienomyrus (sp. 2) that have been electrically silenced by surgical means are tuned, on the average, only 0.2 kHz higher than control animals. Silenced animals that have been treated with androgens are tuned, on the average, 0.2 kHz below controls. The results suggest that electroreceptor tuning is only partially modifiable during androgen treatment if the electroreceptors arenot being stimulated by an external electrical stimulus, i.e. the animal's own EOD. Since androgen treatment has a dramatic effect on receptor tuningonly in intact fish, it seems likely that retuning isnot due to a direct action of androgens on receptors, but rather due to the action of the principal electrical stimulus upon the receptors, i.e. the EOD. The implications of such results for the development of species and sex differences in electro-receptor tuning is discussed.     

The effect of dispersal on single-species nonautonomous dispersal models with delays

In this paper, single-species nonautonomous dispersal models with delays are considered. An interesting result on the effect of dispersal for persistence and extinction is obtained. That is, if the species is persistent in a patch then it is also persistent in all other patches; if the species is permanent in a patch then it is also permanent in all other patches; if the species is extinct in a patch then it is also extinct in all other patches. Furthermore, some new sufficient conditions for the permanence and extinction of the species in a patch are established. The existence of positive periodic solutions is obtained in the periodic case by employing Teng and Chen's results on the existence of positive periodic solutions for functional differential equations. Received: 26 June 2000 / Revised version: 6 October 2000 / Published online: 10 April 2001     

3D-QSAR study for DNA cleavage proteins with a potential anti-tumor ATCUN-like motif

Genomics projects have elucidated several genes that encode protein sequences. Subsequently, the advent of the proteomics age has enabled the synthesis and 3D structure determination for these protein sequences. Some of these proteins incorporate metal atoms but it is often not known whether they are metal-binding proteins and the nature of the biological activity is not understood. Consequently, the development of methods to predict metal-mediated biological activity of proteins from the 3D structure of metal-unbound proteins is a goal of major importance. More specifically, the amino terminal Cu(II)- and Ni(II)-binding (ATCUN) motif is a small metal-binding site found in the N-terminus of many naturally occurring proteins. The ATCUN motif participates in DNA cleavage and has anti-tumor activity. In this study, we calculated average 3D electrostatic potentials (xi(k)) for 265 different proteins including 133 potential ATCUN anti-tumor proteins. We also calculated xi(k) values for the total protein or for the following specific protein regions: the core, inner, middle, and outer orbits. A linear discriminant analysis model was subsequently developed to assign proteins into two groups called ATCUN DNA-cleavage proteins and non-active proteins. The best model found was: ATCUN=1.15.xi(1)(inner)+2.18.xi(5)(middle)+27.57.xi(0)(outer)-27.57.xi(0)(total)+0.09. The model correctly classified 182 out of 197 (91.4%) and 61 out of 66 (92.4%) proteins in training and external predicting series', respectively. Finally, desirability analysis was used to predict the values for the electrostatic potential in one single region and the combined values in two regions that are desirable for ATCUN-like proteins. To the best of our knowledge, the present work is the first study in which desirability analysis has been used in protein quantitative-structure-activity-relationship (QSAR).     

On the inhibition of camel retina acetylcholinesterase activity by cycloheximide in vitro

The kinetic parameters of inhibition of camel retinal acetylcholinesterase (AChE) activity by cycloheximide (CH) were investigated. For the control system, the Michaelis–Menten constant (K _m)for the hydrolysis of acetylthiocholine iodide was found to be 0.076 mmol/L and the V _max was 0.547 mol/min per mg protein. In contrast, these parameters were decreased in the CH-treated systems. Dixon and Lineweaver–Burk plots, and their secondary replots, indicated that the inhibition was of the linear mixed type, which seems to be a combination of partial competitive and pure noncompetitive inhibition. The values of K_i(slope) and K _I(intercept) were estimated to be 3.50 and 5.68 mmol/L, respectively. K _i was greater than K_i, indicating that CH has a greater binding affinity for the peripheral site than the active site.     

Adaptive omnivory and species coexistence in tri-trophic food webs

The commonness of omnivory in natural communities is puzzling, because simple dynamic models of tri-trophic systems with omnivory are prone to species extinction. In particular, the intermediate consumer is frequently excluded by the omnivore at high levels of enrichment. It has been suggested that adaptive foraging by the omnivore may facilitate coexistence, because the intermediate consumer should persist more easily if it is occasionally dropped from the omnivore's diet. We explore theoretically how species permanence in tri-trophic systems is affected if the omnivore forages adaptively according to the "diet rule", i.e., feeds on the less profitable of its two prey species only if the more profitable one is sufficiently rare. We show that, compared to systems where omnivory is fixed, adaptive omnivory may indeed facilitate 3-species persistence. Counter to intuition, however, facilitation of 3-species coexistence requires that the intermediate consumer is a more profitable prey than the basal resource. Consequently, adaptive omnivory does not facilitate persistence of the intermediate consumer but enlarges the persistence region of the omnivore towards parameter space where a fixed omnivore would be excluded by the intermediate consumer. Overall, the positive effect of adaptive omnivory on 3-species persistence is, however, small. Generally, whether omnivory is fixed or adaptive, 3-species permanence is most likely when profitability (=conversion efficiency into omnivores) is low for basal resources and high for intermediate consumers.