Simultaneous nonlinear equations of growth

The simultaneous equations

$$\begin{gathered} \frac{{dx}}{{dt}} = \frac{{a_x }}{{k_x }}[k_x - x - f_x (y)] x \hfill \\ \frac{{dy}}{{dt}} = \frac{{a_y }}{{k_y }}[k_y - y - f_y (x)] y \hfill \\ \end{gathered}$$     

The basic flow equations of electrophysiology in the presence of chemical reactions: I. Development of the equations

Starting from the basic flux equation, it is possible to obtain an integral form relating the current componentsI _i at an arbitrary pointr ₂ to the distribution of mobilities and concentrationsc _i, potential forces\(\bar \mu \), and chemical productivityp _i without any restrictive assumptions such as constant mobilities, constant field, steady state, or electrical neutrality. The equation is

$$\begin{gathered} I_i (r_2 ) = G_i (r_2 )\left[ {\Delta \bar \mu _i - \int_{r_1 }^{r_2 } {z_i } FA\left( {p_i - dc_i /dt} \right)\left( {\frac{1}{{G_i (r)}}} \right)dr} \right]; \hfill \\ G_i (r) = 1/\int_{r_1 }^r {\frac{{dr}}{{z_i^2 F^2 c_i u_i }}.} \hfill \\ \end{gathered} $$     

Theory of antigen-antibody induced particulate aggreagation—I. General assumptions and basic theory

A theory of antigen-antibody induced particulate aggregation is developed by investigating the stability of model systems of particles. Conditions for the formation of large aggregates are derived by imposing the requirement that at equilibrium a statistically significant number of redundant bonds would occur in a reduced monomer-dimer model system. A relationship is obtained which predicts the fractional agglutination in the reduced dimer system as a function of the antigen, antibody and particulate concentrations: $$\frac{g}{{2f c_0 (1 - g)^{2^ - } }} = \frac{{s_1 }}{r} + \frac{{s_1 s_2 }}{{2!r^2 }} + ... + \frac{{s_1 s_2 ...s_j }}{{j!r^j }},$$ wherec ₀ is the initial concentration of monomer,f is a proximity factor,g is the fractional agglutination,s _i is the average rate of formation of theith bond from an (i?1)th bound dimer, andr is the average rate of dissociation of a single antibody-antigen bond.     

The secant condition for instability in biochemical feedback control—II. Models with upper hessenberg jacobian matrices

We consider ann-component biochemical system whose Jacobian matrixJ is of upper Hessenberg form, with principal subdiagonal elementsb ₁,b ₂, ...,b _n?1 and upper right-hand corner element ?f. The open-loop Jacobian matrixJ ₀ is formed fromJ by settingf=0. It is shown that if the characteristic roots of ?J ₀ are real and non-negative then a necessary condition for instability at a critical point (steady state) is $$\frac{{b_1 b_2 ...b_{n - 1} f}}{{\left| { - J_0 } \right|}} \geqslant (\sec \pi /n)^n $$ This condition is analyzed in terms of reaction orders. For a metabolic sequence with some reversible steps, no loss of intermediate metabolites, and competitive inhibition of the first enzyme by the last metabolite, the above necessary condition becomes $$\frac{{\beta _{N - 1} X_{n + 1} }}{{\xi _{N - 1} E_{0T} }} \geqslant (\sec \pi /N)^N $$ whereN is the number of components (metabolites, enzyme-substrate complexes, and enzyme-inhibitor complex),β _N-1 the order of the enzyme-inhibitor reaction (with respect to the inhibitor),ξ _N-1 the order of reaction for the removal of the last metabolite, andX _n+1 /E _0T the fraction of first enzyme blocked by inhibitor. It is shown that, under certain assumptions, a critical point is always stable in a single two-step enzymatic process (formation of enzyme-substrate complex, followed by conversion to product, then loss of product) with slow negative feedback by competitive product inhibition. A model is constructed showing that stable oscillations can occur in a feedback system with only two metabolic steps and negative feedback by competitive inhibition with no cooperativity. The instability is due to a slow feedback reaction and saturable removal of the second metabolite.     

Inhibierungserscheinungen bei der alkoholischen Gärung,I. Einfluß der Ethanolkonzentration auf die spezifische Ethanolbildungsgeschwindigkeit von Saccharomyces cerevisiae

Differential values of the specific ethanol production rate \documentclass{article}\pagestyle{empty}\begin{document}$$ v_{(t)} = \frac{1}{{x_{(t)} }} \cdot \frac{{dP}}{{dt}} $$ \end{document} can be calculated exactly from experimental batch fermentation process data by use of a nonlinear regression programme. The method used is based on the fact, that the function P = f(t) can be approximated by an exponential equation. The specific ethanol production rate is calculated then from the first differential derivation of this equation using the appropriated values of actual biomass concentration. For two strains of Saccharomyces cerevisiae a linear and nonlinear kinetic pattern, respectively, was found for product formation. This result can be explained by a simple mathematical relation according to ν=ν₀ ? a . P^b,in which the exponent becomes 1 in the case of linear kinetic pattern.     

Correlation between algal species diversity index and phytoplankton community biomass in a mesotrophic lake (West Siberia, Russia)

The study deals with phytoplankton biodiversity in mesotrophic Sartlan Lake, a large natural saline water body in the south of West Siberia. Two different approaches are used: floristic and ecological cenotic. The former is good for determining and analyzing the phytoplankton species composition. The latter gives a quantitative estimation of the phytoplankton biodiversity from the equation of information theory: $ H_b = - \sum\limits_{i = 1}^n {\frac{{B_i }} {B}\log _2 \frac{{B_i }} {B}} The study deals with phytoplankton biodiversity in mesotrophic Sartlan Lake, a large natural saline water body in the south of West Siberia. Two different approaches are used: floristic and ecological cenotic. The former is good for determining and analyzing the phytoplankton species composition. The latter gives a quantitative estimation of the phytoplankton biodiversity from the equation of information theory: , where H _b is the biodiversity (bits); B _i is the population biomass of the species i; B is the entire phytoplankton community biomass (mg/l). A reliable stable negative correlation exists between the phytoplankton biomass and species diversity. Analytical equations and a diagram are given to illustrate the correlation between these values. Original Russian Text ? V.I. Ermolaev, 2009, published in Sibirskii Ekologicheskii Zhurnal, 2009, Vol. 16, No. 4, pp. 623–628.     

Energetics of the growth of Methanobacterium thermoautotrophicum and Methanococcus thermolithotrophicus on ammonium chloride and dinitrogen

Methanobacterium thermoautotrophicum was grown in continuous culture in a fermenter gassed with H₂ and CO₂ as sole carbon and energy sources, and in a medium which contained either NH₄Cl or gaseous N₂ as nitrogen source. Growth was possible with N₂. Steady states were obtained at various gas flow rates with NH₄Cl and with and the maintenance coefficient varied with the gas input and with the nitrogen source. Growth of Methanococcus thermolithotrophicus in continuous culture in a fermenter gassed with H₂, CO₂ as nitrogen, carbon and energy sources was also examined.Abbreviations molecular growth yield (g dry weight of cells per mol of CH₄ evolved) - growth rate (h^-1) - D dilution rate (h^-1) - rate (h_-1); relation of Neijssel and Tempest and of Stouthamer and Bettenhaussen - energy     

Présence dans une population congolaise deMus (Leggada) triton Th. de femelles hétérozygotes pour une délétion caractérisée par la suppression du bras court de l'un des chromosomes X métacentriques

A sample of 12Mus (Leggada) triton Th. from the region of Bukavu (Democratic Republic of Congo) contains 5 ♂♂ and 7 ♀♀. 2N=32. All the autosomes are acrocentric. The sex-chromosomes of the ♂ are of the typeX—Y, theX beeing a big submetacentric (I.C.=0,4). Three ♀♀ possess two metacentricX, as expected. By four ♀♀, there is only one typicalX whose partner is acrocentric and as long as the long arm of a normalX. ThisX must have been arisen through the deletion of the short arm and is calledX _ddc. The statistical analysis of the sample is compatible with this pattern:

$$\begin{array}{*{20}c} { \circ \circ } \\ { + + } \\ \end{array} \begin{array}{*{20}c} {X---X = 4/9} \\ {X---X_{dc} = 4/9} \\ {X_{dc} ---X_{dc} = 1/9} \\ \end{array} \begin{array}{*{20}c} { \nearrow \nearrow } \\ { \circ \circ } \\ \end{array} \begin{array}{*{20}c} {X---Y = 2/3} \\ {X_{dc} ---Y = 1/3} \\ \end{array} $$     

Interaction-induced electric (hyper)polarizability in the dihydrogen–neon pair: basis set and electron correlation effects

We investigated the interaction (hyper)polarizability of neon–dihydrogen pairs by performing high-level ab initio calculations with atom/molecule-specific, purpose-oriented Gaussian basis sets. We obtained interaction-induced electric properties at the SCF, MP2, and CCSD levels of theory. At the CCSD level, for the T-shaped configuration, around the respective potential minimum of 6.437 a₀, the interaction-induced mean first hyperpolarizability varies for 5?<? R/a₀?<?10 as

$$ \left[{\overline{\beta}}_{\mathrm{int}}(R)\hbox{-} {\overline{\beta}}_{\mathrm{int}}\left({R}_{\mathrm{e}}\right)\right]/{e}^3{a_0}^3{E_{\mathrm{h}}}^{-2}=-0.91\left(R\hbox{-} {R}_{\mathrm{e}}\right)+0.50{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^2\hbox{--} 0.13{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^3+0.01{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^4. $$

Again, at the CCSD level, but for the L-shaped configuration around the respective potential minimum of 6.572 a₀, this property varies for 5?<? R/a₀?<?10 as

$$ \left[{\overline{\beta}}_{\mathrm{int}}(R)\hbox{-} {\overline{\beta}}_{\mathrm{int}}\left({R}_{\mathrm{e}}\right)\right]/{e}^3{a_0}^3{E_{\mathrm{h}}}^{-2}=-1.33\left(R\hbox{-} {R}_{\mathrm{e}}\right)+0.75{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^2-0.20{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^3+0.02{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^4. $$

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Graphical Abstract Interaction-induced mean dipole polarizability (\( \overline{a} \)) for the T-shaped configuration of H₂–Ne calculated at the SCF, MP2, and CCSD levels of theory

     

A hall dynamo effect driven by two-fluid tearing instability

A quasi-linear prediction of the two-fluid dynamo effect is analyzed with the use of tearing eigenfunctions obtained for force-free equilibrium. In the range of parameters of practical interest, the basic shear Alfvén mode is decoupled from fast compressional Alfvén and slow magneto-acoustic modes. Kinetic Alfvén modification of the shear Alfvén wave drives an instability with a growth rate ∝δ^1/3ρ _s ^2/3 , where δ is the electron skin depth and ρ_s is the ion-sound gyroradius. A net dynamo effect parallel to the magnetic field is calculated at ρ _s ?δ for large values of the stability factor \(\Delta '\rho _s^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \delta ^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \gg 1\). The dynamo effect caused by the j×B Hall term dominates the contribution from the v×B term (the alpha effect) by a factor ∝(ρ_s/δ)² in the narrow electron layer, while in the broader ion layer these contributions are comparable. The results are compared with the case of a strong guiding field where ρ _s ?δ and the tearing instability is described by resistive MHD.     

On a new graphical method in the theory of multiple fixations and its application to the wurmser theory of isohemagglutination

The fundamental equation of the theory of multiple fixations without interaction (MFWI) is

$$\frac{1}{r} = \frac{1}{m} + \frac{1}{{mKA}}$$     

Biological similarity relative to allometric quantities

It is an empirical finding that an allometric quantity with dimensional exponents α, β and γ relative to mass, length, and time, respectively, has a value for its allometric exponentb satisfying the relation

$$\tfrac{1}{3}(3\alpha + \beta + {\gamma \mathord{\left/ {\vphantom {\gamma 2}} \right. \kern-\nulldelimiterspace} 2}) \leqslant b \leqslant \tfrac{1}{3}(3\alpha + \beta + \gamma ).$$

A theoretical derivation is given of this double inequality using only the fact of constant density and the plausible assumption that metabolic rate is a dominant allometric quantity.

     

Carbonic anhydrases in the mouse harderian gland

The harderian gland is located within the orbit of the eye of most terrestrial vertebrates. It is especially noticeable in rodents, in which it synthesises lipids, porphyrins, and indoles. Various functions have been ascribed to the harderian gland, such as lubrication of the eyes, a site of immune response, and a source of growth factors. Carbonic anhydrases (CAs) are zinc-containing metalloenzymes that catalyse the reaction \( {\text{CO}}_{2} + {\text{H}}_{2} {\text{O}} \Leftrightarrow {\text{H}}^{ + } + {\text{HCO}}_{3}^{ - } \). They are involved in the adjustment of pH in the secretions of different glands. Thirteen enzymatically active isozymes have been described in the mammalian α-CA family. Here, we first investigated the mRNA expression of all 13 active CAs in the mouse harderian gland by quantitative real-time PCR. Nine CA mRNAs were detectable in the gland. Car5b and Car13 showed the highest signals. Car4, Car6, and Car12 showed moderate expression levels, whereas Car2, Car3, Car7, and Car15 mRNAs were barely within the detection limits. Immunohistochemical staining was performed to study the expression of Car2, Car4, Car5b, Car12, and Car13 at the protein level. The epithelial cells were intensively stained for CAVB, whereas only weak signal was detected for CAXIII. Positive signals for CAIV and CAXII were observed in the capillary endothelial cells and the basolateral plasma membrane of the epithelial cells, respectively. This study provides an expression profile of all CAs in the mouse harderian gland. These results should improve our understanding of the distribution of CA isozymes and their potential roles in the function of harderian gland. The high expression of mitochondrial CAVB at both mRNA and protein levels suggests a role in lipid synthesis, a key physiological process of the harderian gland.     

An improved Four-Russians method and sparsified Four-Russians algorithm for RNA folding

Background

The basic RNA secondary structure prediction problem or single sequence folding problem (SSF) was solved 35 years ago by a now well-known \(O(n^3)\)-time dynamic programming method. Recently three methodologies—Valiant, Four-Russians, and Sparsification—have been applied to speedup RNA secondary structure prediction. The sparsification method exploits two properties of the input: the number of subsequence Z with the endpoints belonging to the optimal folding set and the maximum number base-pairs L. These sparsity properties satisfy \(0 \le L \le n / 2\) and \(n \le Z \le n^2 / 2\), and the method reduces the algorithmic running time to O(LZ). While the Four-Russians method utilizes tabling partial results.

Results

In this paper, we explore three different algorithmic speedups. We first expand the reformulate the single sequence folding Four-Russians \(\Theta \left(\frac{n^3}{\log ^2 n}\right)\)-time algorithm, to utilize an on-demand lookup table. Second, we create a framework that combines the fastest Sparsification and new fastest on-demand Four-Russians methods. This combined method has worst-case running time of \(O(\tilde{L}\tilde{Z})\), where \(\frac{{L}}{\log n} \le \tilde{L}\le min\left({L},\frac{n}{\log n}\right)\) and \(\frac{{Z}}{\log n}\le \tilde{Z} \le min\left({Z},\frac{n^2}{\log n}\right)\). Third we update the Four-Russians formulation to achieve an on-demand \(O( n^2/ \log ^2n )\)-time parallel algorithm. This then leads to an asymptotic speedup of \(O(\tilde{L}\tilde{Z_j})\) where \(\frac{{Z_j}}{\log n}\le \tilde{Z_j} \le min\left({Z_j},\frac{n}{\log n}\right)\) and \(Z_j\) the number of subsequence with the endpoint j belonging to the optimal folding set.

Conclusions

The on-demand formulation not only removes all extraneous computation and allows us to incorporate more realistic scoring schemes, but leads us to take advantage of the sparsity properties. Through asymptotic analysis and empirical testing on the base-pair maximization variant and a more biologically informative scoring scheme, we show that this Sparse Four-Russians framework is able to achieve a speedup on every problem instance, that is asymptotically never worse, and empirically better than achieved by the minimum of the two methods alone.

     

Mathematical modeling of mixing in a horizontal rotating tubular bioreactor: “Simple flow” model

The construction of the horizontal rotating tubular bioreactor (HRTB) represents a combination of a thin-layer bioreactor and a biodisc reactor. The bioreactor was made of a plastic tube whose interior was divided by the O-ring shaped partition walls. For the investigation of mixing properties in HRTB the temperature step method was applied. The temperature change in the bioreactor as a response to a temperature step in the inlet flow was monitored by six Pt-100 sensors (t ₉₀ response time 0.08 s and resolution 0.002 °C) which were connected with an interface unit and personal computer. Mixing properties of the bioreactor were modeled using the modified tank in series concept which divided the bioreactor into ideally mixed compartments. A mathematical mixing model with simple flow was developed according to the physical model of the compartments network and corresponding heat balances. Numerical integration of an established set of differential equations was done by the Runge-Kutt-Fehlberg method. The final mathematical model with simple flow contained four adjustable parameters (N1,Ni, F _cr andF _p ) and five fixed parameters.List of Symbols A _u m² inner surface of bioreactor's wall - A _ui m² i-th part of inner surface of bioreactor's wall - A _v m² outlet surface of bioreactor's wall - A _vi m² i-th part of outlet surface of bioreactor's wall - C _p kJ kg^–1 K^–1 heat capacity of liquid - C _pr kJ kg^–1 K^–1 heat capacity of bioreactor's wall - D h^–1 dilution rate - E °C °C^–1 h^–1 error of mathematical model - F _cr dm³s^–1 circulation flow in the model - F _p dm³ s^–1 back flow in the model - F _t dm³s^–1 inlet flow in the bioreactor - I °C intensity of temperature step, the difference in temperature between the temperature of the inlet liquid flow and the temperature of liquid in the bioreactor before the temperature step - K1 Wm^–2K^–1 heat transfer coefficient between the liquid and bioreactor's wall - K2 Wm^–2K^–1 heat transfer coefficient between the bioreactor's wall and air - m _s kg mass of bioreactor's wall - L m length of bioreactor - L _k m wetted perimeter of bioreactor - n min^–1 rotational speed of bioreactor - n _s number of temperature sensors - N1 number of cascades - Ni number of compartments inside the cascade - Nu Nusselt number - Pr Prandtl number - r _u m inner diameter of bioreactor - r _v m outside diameter of bioreactor - Re Reynolds number - s(t) step function - t s time - T °C temperature - T _c °C calculated temperature - T _m °C measured temperature - T _N1,Ni °C temperature of liquid in a defined compartment inside cascade - T _N1,S °C temperature of defined part of bioreactor's wall - T _S °C temperature of bioreactor's wall - T _v °C temperature of liquid in bioreactor - T _z °C temperature of surrounding air - V _t dm³ volume of liquid in the bioreactor Greek Symbols kJm^–1s^–1 K^–1 thermal conductivity of liquid in the bioreactor - kgm^–3 density of liquid in the bioreactor - m²s^–1 kinematic viscosity of liquid in the bioreactor Matrix Coefficient B - C - D - E B+C+D - G1 - G2 - G3 - A _ui - A _vi - Q ₁ - Q ₂ - Q ₃      

Suppression of the rayleigh-taylor instability of a low-density imploding liner by a longitudinal magnetic field

The possibility of suppressing the Rayleigh-Taylor instability in a low-density plasma, Π=ω _pi ² Δ²/c²?1 (where Δ is the thickness of the current-carrying slab), is investigated for the case in which the electron currents are much higher than the ion currents. The suppression of this instability in an imploding cylindrical liner by an axial external magnetic field \(B_{0z} \) is considered. It is shown that, for the instability to be suppressed, the external magnetic field \(B_{0z} \) should be stronger than the magnetic field B_0θ of the current flowing through the liner.     

Theoretical study of AlH(2+)n (n=1-7) dications

The structures and stability of 1–7 dications were calculated at the ab initio MP2/aug-cc-pVTZ level of theory. The dications AlH²⁺ 1 and 2 were characterized to be unstable thermodynamically. However, these and the stable dications, 3–7 have considerable kinetic barriers for deprotonation. Each of the structures 3–7 contains one or more two-electron three-center (2e–3c) bonds. Aluminum atoms of these dications carry most of the positive charges, as indicated by NBO charge calculations.Dedicated to Professor Dr. Paul von Ragué Schleyer on the occasion of his 75th birthday     

Concentration equations for the diffusion of a reversibly reacting substance in gel-like media

The concentration of a diffusible substanceA(x, t) in a semi-infinite geometry is studied for the set of reversible reactionsA+B _i ?C _i ;i=1...n, whereB _i andC _i are assumed to be associated with non-diffusible biological structures. Assuming chemical equilibrium prevails throughout for each reaction, it is shown that a single uncoupled partial differential equation is sufficient to specifyA(x, t) and indirectlyB _i (x, t) andC _i (x, t) as well: $$\left[ {1 + \sum\limits_i {\frac{{K_i \beta _i }}{{\left( {1 + K_i A} \right)^2 }}} } \right]\frac{{\partial A}}{{\partial t}} = D_A \frac{{\partial ^2 A}}{{\partial x^2 }}$$ whereK _i is the chemical equilibrium constant of theith reaction, β₁ is concentration of binding sites of theith species (i.e.B _i+C _i) andD _A is the usual diffusion constant forA. Numerical solutions for boundary conditions amenable to the Boltzman transformation are presented and the range of parameters established over which the uniqueness and convergence of the solutions can be proven.     

Sampling bias in estimating Design II variance components with S1 families

Summary The use of several S₁ individuals to represent an S₀ individual permits the use of a Design II mating scheme for plants with only one pistillate flower per plant. Estimates of additive (V _A ) and dominance (V _D ) variance from this mating scheme will be biased upwards, when a small number (10) of individuals of each S₁ line are used. This bias can be computed, and the additive and dominance estimates can be corrected. Of particular interest is the observation that the additive genetic variance contributes to bias in estimates of V _D . When S₀ plants are non inbred and their selfedprogeny (S₁ lines) are used to represent them in developing families for use in the Design II, where m₁ is the number of individuals used to represent an S₁ line in developing half sib-families and m₂ is the number of individuals used to represent the S₁ line in making up full sib-families. For example, in a 3×3 Design II, with about 10 individuals used to represent each S₁ line in each cross, m₂ = 10 and m₁ = 30. When m₁ = m₂ = 1, and Joint contribution from Department of Agronomy, University of Nebraska 68583, and the S. S. Cameron Laboratory, Werribee, Victoria 3030, Australia. Published as paper No. 7395, Journal Series