Hill Equation Research Articles

Quantifying Drug Combination Synergy along Potency and Efficacy Axes.

Two goals motivate treating diseases with drug combinations: reduce off-target toxicity by minimizing doses (synergistic potency) and improve outcomes by escalating effect (synergistic efficacy). Established drug synergy frameworks obscure such distinction, failing to harness the potential of modern chemical libraries. We therefore developed multi-dimensional synergy of combinations (MuSyC), a formalism based on a generalized, multi-dimensional Hill equation, which decouples synergistic potency and efficacy. In mutant-EGFR-driven lung cancer, MuSyC reveals that combining a mutant-EGFR inhibitor with inhibitors of other kinases may result only in synergistic potency, whereas synergistic efficacy can be achieved by co-targeting mutant-EGFR and epigenetic regulation or microtubule polymerization. In mutant-BRAF melanoma, MuSyC determines whether a molecular correlate of BRAFi insensitivity alters a BRAF inhibitor's potency, efficacy, or both. These findings showcase MuSyC's potential to transform the enterprise of drug-combination screens by precisely guiding translation of combinations toward dose reduction, improved efficacy, or both.

Normalized shear strength of trapezoidal corrugated steel webs: Improved modeling and uncertainty propagation

This paper presents a comprehensive analysis of mathematical models for estimating the shear strength of trapezoidal-corrugated steel webs. The existing database of experimental data on trapezoidal corrugated steel web beams and girders is updated by adding more test results, yielding a total of 144 tests. The three-parameter model, previously developed by the authors based on Richards equation is reduced to a two-parameter improved version based on Hill equation. Both models were found to have, essentially, the same performance when compared against test results of the updated database. Based on the Akaike Information Criterion, the new model was retained. The performance of the improved and previously published models is reviewed and tested against the updated database of test results. It was found that the improved model performed better than the existing models in many aspects. An uncertainty analysis was performed on the proposed model considering both, geometric and material parameters as random variables. Bootstrapping and the Point Estimate Method (PEM) were employed to propagate uncertainty and construct confidence intervals for the normalized shear strength. It was found that, overall, the coefficient of variation of the normalized shear strength increases with the slenderness ratio.

Galoisian approach to complex oscillation theory of some Hill equations

We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponent of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. These are solutions obtainable by suitable differential field extension constructions. In particular, we establish a full correspondence between solutions of non-oscillatory type and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation whose potential is a combination of four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of a biconfluent Heun equation. We further show that these Liouvillian solutions exhibit novel single and double orthogonality, and satisfy Fredholm integral equations over suitable integration regions in $\mathbb{C}$ that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lamé and Whittaker-Hill equations, discovered by Whittaker and Ince almost a century ago.

New insights into the evolution of solid and liquid phases during hydrothermal carbonization of lignocellulosic biomasses

This paper is the first illustration of an innovative procedure for studying the dynamics of condensed phases in hydrothermal carbonization of lignocellulosic waste. A novel technique, based on a particular implementation of van Krevelen plots, is shown to be capable of monitoring the compositions precisely during this process. Two fundamental laws describe the system evolution: double-exponential decay and a modified form of the Hill equation. The modeling was tested using batch experiments (200 °C, up to 2 h) performed with three different lignocellulosic materials (silver fir, beech wood, and olive pruning). A high goodness of fit (R2 as high as 0.999) supports the approach and heuristically indicates the way for further research. The method allows for prediction of the time course of solid yield, heating value, atomic ratios, and other relevant process variables. The technique also provides valuable and easy-to-obtain information to address the process optimization of industrial plants.

Analitical theory Flouqet-Bloch waves for gyrotropic magnetophotonic krystals

The relevance of the problem considered in the work is determined by the widespread use of magnetophotonic crystals in various devices of the terahertz microwave and optical ranges. The key is the analytical solution of the third boundary value problem for the Hill equation with mixed Cauchy boundary conditions. This approach made it possible to explicitly find new solutions for electromagnetic fields in the crystal layers and dispersion characteristics for TE and TM waves, which is important for the development of new multifunction devices in the terahertz range.. The purpose of the work is to develop an analytical theory of Floquet-Bloch waves for gyrotropic magnetophotonic crystals with a transverse magnetic field. Materials and methods. Magnetophotonic crystals consist of gyrotropic (gyroelectric or gyromagnetic materials) two-layer structures over a period, the parameters of which vary from the magnitude of the applied magnetic field. Analytical methods for solving the Hill equation through fundamental solutions of the third boundary value problem. Results. The fundamental solutions of the Hill equation are determined in an analytical form. Analytical expressions for the dispersion characteristics of TE and TM Floquet-Bloch waves are found. The existence of bulk and surface waves in the transmission zones of a magnetophotonic crystal is established. The existence of an extraordinary surface wave with an atypical field distribution in the crystal layers for positive effective electric or magnetic permeability is shown. Conclusions. The proposed new approach for determining the solutions of the Hill equation based on the fundamental solutions of the third boundary-value problem made it possible to obtain in an analytical form the dispersion characteristics and fields of controlled gyromagnetic magnetophotonic crystals for TE and TM Floquet-Bloch waves. This will make it relatively easy to calculate various devices based on controlled Bragg structures.

Bohr equation and the lost allosteric Bohr effects in symmetry.

Bohr, Hasselbalch and Krogh demonstrated a group of sigmoid curves under various carbon dioxide contents in 1904. Hill fitted these curves in 1910 with Hill equation without the physical meanings of Hill coefficient and dissociation constant. In 1965, Monod-Wyman-Changeux model (MWC) popularized the word “allostery” with 81 words of symmetry to define an orthosteric nature of cooperativity in a single and symmetric sigmoid curve. Paradoxically the MWC model didn’t quantify the homotropic Hill coefficient and confusingly described the symmetry of sigmoid shapes with three allosteric variables. A heterotropic Bohr equation, by clarifying the biophysical symmetry in allostery, suggests the solution of allosteric coefficients with only one Bohr variable. We reveal that the mathematical need of a fictional monomer by MWC model justify a symmetric logistic curve with a parabolic kernel of dissociation constant to model the 1904 sigmoid curves. The logistic-derived Bohr equation and its half-saturated P50 equation successfully used the embedded P50 values in the 1904 sigmoidal curves to quantify their hyperbolic conformational shifts and Hill coefficients (n) pending for a century. Both are the logarithmic functions of carbon dioxide. This truly quantitative Bohr equation digitizes the allosteric regulation of the orthosteric affinity by precisely cloning the original group of dissociation/association curves published in 1904. The Bohr equation honestly suggests that nature should have chosen the allosteric Bohr effects to modify hemoglobin to cope with the swift dynamic of gas exchange. The discovery of the Bohr function in Bohr equation challenges the feasibility of the orthosteric cooperativity of hemoglobin.

Fatiguing stimulation increases curvature of the force-velocity relationship in isolated fast-twitch and slow-twitch rat muscles.

In skeletal muscles, the ability to generate power is reduced during fatigue. In isolated muscles, maximal power can be calculated from the force-velocity relationship. This relationship is well described by the Hill equation, which contains three parameters: (1) maximal isometric force, (2) maximum contraction velocity and (3) curvature. Here, we investigated the hypothesis that a fatigue-induced loss of power is associated with changes in curvature of the force-velocity curve in slow-twitch muscles but not in fast-twitch muscles during the development of fatigue. Isolated rat soleus (slow-twitch) and extensor digitorum longus (EDL; fast-twitch) muscles were incubated in Krebs-Ringer solution at 30°C and stimulated electrically at 60 Hz (soleus) and 150 Hz (EDL) to perform a series of concentric contractions to fatigue. Force-velocity data were fitted to the Hill equation, and curvature was determined as the ratio of the curve parameters a/F0 (inversely related to curvature). At the end of the fatiguing protocol, maximal power decreased by 58±5% in the soleus and 69±4% in the EDL compared with initial values in non-fatigued muscles. At the end of the fatiguing sequence, curvature increased as judged from the decrease in a/F0 by 81±20% in the soleus and by 31±12% in the EDL. However, during the initial phases of fatiguing stimulation, we observed a small decrease in curvature in the EDL, but not in the soleus, which may be a result of post-activation potentiation. In conclusion, fatigue-induced loss of power is strongly associated with an increased curvature of the force-velocity relationship, particularly in slow-twitch muscles.

Synthesis, electrochemistry, protonation and X-ray analysis of meso-aryl substituted open-chain pentapyrroles

Five meso-tetraaryl open-chain pentapyrroles were synthesized and characterized as to their electrochemistry and protonation reactions in nonaqueous media. The investigated compounds are represented as (Ar)4PPyH3 where Ar [Formula: see text],[Formula: see text]-F2Ph, [Formula: see text]-BrPh, Ph, [Formula: see text],[Formula: see text],[Formula: see text]-(OMe)3Ph or [Formula: see text]-MePh and were characterized by UV-vis and 1H NMR spectroscopy, mass spectrometry and electrochemistry. Cyclic voltammetry was used to measure redox potentials, while protonation involving the conversion of (Ar)4PPyH3 to [(Ar)4PPyH5][Formula: see text] was monitored by UV-vis absorption spectroscopy. Equilibrium constants for proton addition were calculated using the Hill equation. One of the pentapyrroles was also structurally characterized. The electrochemical data, protonation constants and crystal structure were then compared with data for previously examined pentapyrroles and analyzed as a function of the solvent properties and nature of substituents on the meso-phenyl rings of the macrocycle.

Broadside Radiation in Leaky-Wave Antenna Using Multiperiodic Width-Modulated Microstrip Lines

An analytical-numerical method in a multitone modulated dielectric medium is presented for the transverse magnetic mode of which the wave equation is in the form of the Hill equation. In order to suppress the second stopband, dispersion characteristics are analytically considered in terms of “stability chart” and “dispersion diagram.” Having suppressed the second stopband in the multitone modulation of a dielectric medium, an effective value of the dielectric constant is realized by width-modulated microstrip lines in a grounded dielectric substrate. Then, a new planar leaky-wave antenna is designed and fabricated based on multitone periodicity for broadside radiation at 14 GHz. The fabricated antenna is able to scan through the broadside without any drop in the gain near the broadside. Simulation and measurement results are in a good agreement with theoretical results.

Inhibitory effects of rubratoxin A, a potent inhibitor of protein phosphatase 2, on the Ca2+-dependent contraction of skinned carotid artery from guinea pig.

Rubratoxin A, a potent inhibitor of PP2A, is known to suppress smooth muscle contraction. The inhibitory role of PP2A in smooth muscle contraction is still unclear. In order to clarify the regulatory mechanisms of PP2A on vascular smooth muscle contractility, we examined the effects of rubratoxin A on the Ca2+-induced contraction of β-escin skinned carotid artery preparations from guinea pigs. Rubratoxin A at 1 µM and 10 µM significantly inhibited skinned carotid artery contraction at any Ca2+ concentration. The data fitting to the Hill equation in [Ca2+]-contraction relationship indicated that rubratoxin A decreased Fmax-Ca2+ and increased [Ca2+]50, indices of Ca2+ sensitivity for the force and myosin-actin interaction, respectively. These results suggest that PP2A inhibition causes downregulation of the myosin light chain phosphorylation and direct interference with myosin-actin interaction.

Beaded fiber composites—Stiffness and strength modeling

We present a theoretical analysis of the elastic stresses in a composite reinforced with beaded fibers by extending the classic Cox shear lag theory. The motivation for reinforcing a composite with beaded fibers is to improve both strength and toughness, two often conflicting properties. It is found that owing to their geometry beads intermittently placed on a fiber enhance fiber anchoring in the matrix, and can potentially dissipate energy by deforming the matrix during failure. The composite stiffness is shown to improve compared to a composite with beadless fibers, particularly when the beads are large and stiffer than the surrounding matrix. The stress profiles in the fiber, bead, matrix and along their respective interfaces incur periodic perturbations induced by the beads, modeled by Hill equation. For given elastic constants and bead geometry, these profiles reveal the weakest link loci in the structure, and consequently determine the composite strength and failure mode. A finite element analysis is presented that confirms our results. The bead-fiber and bead-matrix interfaces may be tuned by choice of materials and coatings to achieve desired mechanical properties.

Pectin hydrolysis in cashew apple juice by Aspergillus aculeatus URM4953 polygalacturonase covalently-immobilized on calcium alginate beads: A kinetic and thermodynamic study

The kinetics and thermodynamics of pectin hydrolysis in cashew apple juice by polygalacturonase (PG) from Aspergillus aculeatus URM4953 covalently-immobilized on calcium alginate beads were investigated. Immobilized-PG activity in cashew apple juice was the highest at 20 °C, showing a maximum hydrolysis rate of 58.2 mg/mL·min, a catalytic constant of 166.2 s−1 and an affinity constant of 113.0 mg/mL. Since the enzyme exhibited an allosteric behavior, the hydrolysis rate was modeled, with excellent accuracy, by the Hill Equation as function of pectin concentration. The Hill coefficient increased from 3 to 5 with increasing temperature from 20 to 50 °C, evidencing a positive cooperativity mechanism. The reaction activation energy and the standard enthalpy variation of enzyme unfolding were 80.3 and 16.6 kJ/mol, respectively. Consistently with the kinetic results, PG-catalyzed pectin hydrolysis proceeded with maximum spontaneity at 20 °C, showing activation Gibbs free energy, enthalpy and entropy of 59.3 kJ/mol, 77.9 kJ/mol and 63.4 J/mol·K, respectively. Immobilized PG was successful in the hydrolysis of cashew apple juice pectin, requiring a low temperature to act optimally.

Rheological studies on nanocrystalline cellulose/alginate suspensions

Rheological behavior of mixtures containing nanocrystalline cellulose (CNC) and alginate was studied. A shear-thinning behavior was observed in all samples, which was progressively enhanced as a function of CNC and alginate concentrations. Electrostatic interactions between anionic alginate chains and anionic CNC rods promote physical gelation at small CNC concentrations (4wt%), resulting in soft gels with pronounced shear-thinning and solid-like behavior. Next, hydrogels were obtained using the internal gelation method, and the crosslinking kinetics were monitored by time sweep tests. The experimental data were well described by a modified Hill equation. Fourier transform infrared spectroscopy (FTIR) was performed in order to analyze the hydrogen bonding interactions that occur between CNC rods and alginate chains, and the alginate ionic crosslinking.

Simulação do efeito sedativo de doses altas de detomidina em equinos

The objective of the study was to simulate the sedative effect of the administration of detomidine in doses of 80 ug/kg by intravenous and intramuscular route in equines. A pharmacokinetic/pharmacodynamic (PK/PD) model was performed with the sigmoid curve of plasma concentration, obtained by Mama and collaborators, depending on the effect on the head position by means of Hill equation. A sedative effect simulation of 80 ug/kg intravenous (IV) and intramuscular (IM) doses by Hill's equation was then performed. Higher and prolonged plasmatic concentrations were obtained after administrations of 80 ug/kg, when compared to administrations of 30 ug/kg resulting in higher and longer effect. The simulation of the effect of detomidine after administration of 80 ug/kg IV determined effects in the head height higher than 90%, 75% and 50% up to 2, 4 and 6 hours after administration. Meanwhile, the simulation of the effect of detomidine after the administration of 80 ug/kg IM determined effects in head height higher than 90%, 75% and 50% up to 3, 4 and 6 hours after administration. The present simulation showed that the effects of the position of the head after the administration of 80 ug/kg of detomidine via IV and IM in equines are superior and more prolonged when compared to the administration of 30 ug/kg. Therefore, in vivo studies should be performed in order to verify the adverse effects of these dosages in equines.

Propagation of optical waves in planar periodically inhomogeneous chiral structures

The analytic solution of the problem of propagation of eigen modes in planar periodically inhomogeneous optical waveguides based on alternating dielectric and chiral (or with optical activity) layers is examined. The method is based on the solution of the Hill equations for the electric field strengths of modes with circular polarizations. As a result, a dispersion equation for the eigen modes of a planar periodically inhomogeneous chiral-dielectric waveguide was obtained and its numerical solution was performed. As a result of the analysis of numerical results, it is shown that the structure under study can operate in different modes: propagation of modes with right and left circular polarizations, propagation of a mode with only one type of circular polarization and the optical key mode.

1389. Pharmacokinetic/Pharmacodynamic (PK/PD) Evaluation of a Novel Aminomethylcycline Antibiotic, KBP-7072, in the Neutropenic Murine Pneumonia Model Against S. aureus (SA) and S. pneumoniae (SPN)

BackgroundKBP-7072 is a novel aminomethylcycline antibiotic with broad-spectrum activity that includes organisms with drug-resistance to β-lactams and tetracyclines. We examined the PK/PD relationship between KBP-7072 drug exposures and treatment effect using a neutropenic murine pneumonia model against a diverse group of SA and SPN.MethodsFive SAs (three MRSAs) and six SPNs (three PCNs NS, two TetR) strains were used. MICs were determined by CLSI Methods. Plasma and ELF PK was determined after SC dosing (range 1–256 mg/kg). Lung burden was assessed by CFU counts at the beginning and end of therapy (24 hours). Infected mice were treated with KBP-7072 by SC route: SA dose range 0.25–64 mg/kg/6 hours, SPN dose range 0.06–16 mg/kg/6 hours. The Emax Hill equation was used to model the dose–response data to the PK/PD index AUC/MIC. The magnitude of the PK/PD index (plasma free and ELF total concentrations) associated with net stasis, 1- and 2-log kill were determined in the pneumonia model for all strains.ResultsSA MICs were 0.25 mg/L for all isolates and SPN MICs were 0.008–0.016 mg/L. Plasma PK of KBP-7072 included: Cmax 0.12–25.2 mg/L, AUC0-∞ 1.1–234 mg hour/L, T1/2 3.2–4.6 h. ELF PK by urea correction methods included: Cmax 0.06–13.3 mg/L, AUC0-∞ 0.4–95 mg hour/L, T1/2 3.1–4 hours. ELF penetration based on free plasma drug concentrations (77.5% bound) ranged from 82 to 238%. AUC was linear over the dose range (R2 = 0.99). Potent dose-dependent cidal activity (3–5 log kill) was observed against all strains. AUC/MIC was a robust predictor of efficacy (SA R2 = 0.89, SPN R2 0.80). Median static, 1- and 2-log kill AUC/MIC values are shown in the table.GroupStasis Plasma fAUC/MICStasis ELF AUC/MIC1-Log Kill Plasma fAUC/MIC1-Log Kill ELF AUC/MIC2-Log Kill Plasma fAUC/MIC2-Log Kill ELF AUC/MICSA0.971.722.484.415.817.51SPN1.121.993.686.5413.0623.22ConclusionKBP-7072 demonstrated potent in vivo efficacy against SA and SPN, including strains with elevated minocycline MIC and β-lactam resistance, in the neutropenic murine pneumonia model. A 3–5 log kill was observed and AUC/MIC was strongly associated with efficacy. The AUC/MIC target for net stasis was comparable between SA and SPN at a plasma fAUC/MIC target of ~1 and ELF AUC/MIC target ~2. Cidal targets were similarly very low. All targets were numerically lower than comparative tetracyclines. These results should prove useful for clinical dosing regimen optimization.Disclosures A. J. Lepak, KBP Biosciences: Research Contractor, Research support. Q. Liu, KBP Biosciences: Employee, Salary. P. Wang, KBP Biosciences: Employee, Salary. Y. Wang, KBP Biosciences: Employee, Salary. J. C. Bader, KBP Biosciences: Research Contractor, Research support. P. G. Ambrose, KBP Biosciences: Research Contractor, Research support. D. R. Andes, KBP Biosciences: Research Contractor, Research support.

Perturbation approach to the Hill equation with a slowly varying parameter with application to the inflationary cosmology

We develop a general perturbative approach to solving the Hill equation with a slowly varying parameter based on the Floquet theory and asymptotic expansions in the vicinity of the exact solutions with a “frozen” parameter. Equations of this type describe parametric resonance in a wide class of physical systems being under the influence of slowly varying factors. In particular, such equations describe the parametric instability of the fluctuations of the inflaton scalar field oscillating near a minimum of the effective potential in an expanding universe. We give a general procedure for constructing asymptotic solutions of the Hill equation and write out explicit formulas of the zero-order and first-order approximations. As an example, we consider the ϕ2–ϕ4 inflaton potential and construct the approximate solutions of the corresponding Lamé equation with the energy density of the oscillating scalar background as a slowly varying parameter. The obtained solutions are found to be in good agreement with the results of direct numerical integration. Based on these, we find the shape and characteristic size of a single scalar field fluctuation.

Graphical Approach to Compare Concentration Constants of Hill and Michaelis-Menten Equations

The aim of present study was to describe the graphical technique how to go from Hill concentration constant to Michaelis constant. To compare enzymatic processes, the kinetics of which is subjected to different regularities, it is possible to use constants that characterize catalytic activity (Vmax) and concentration constants that are the substrate concentration at which the rate of the enzymatic process is equal to a half of maximum permissible rate. Concentration constants are S0.5 for Hill equation and Km for Michaelis-Menton equation. The graphical approach was proposed in order to go from concentration constant of Hill equation to Michaelis concentration of the process that could be characterized by the same catalytic activity (the same values of minimum and maximum rates) similar to that observed in the process described by Hill equation.

Evaluation of aromatic hydrocarbon decomposition catalyzed by the dioxygenase system and substitution of ferredoxin and ferredoxin reductase.

In this study, the catalytic activity and kinetic characteristics of the aromatic hydrocarbon dioxygenase system and the possibility of substituting its ferredoxin and ferredoxin reductase components were evaluated. The genes encoding toluene dioxygenase and toluene dihydrodiol dehydrogenase were cloned from Pseudomonas putida F1, and the corresponding enzymes were overexpressed and purified to homogeneity. Oxidative hydroxylation of toluene to cis-toluene dihydrodiol was catalyzed by toluene dioxygenase, and its subsequent dehydrogenation to 3-methylcatechol was catalyzed by toluene dihydrodiol dehydrogenase. The specific activity of the dioxygenase was 2.82U/mg-protein, which is highly remarkable compared with the values obtained in previous researches conducted with crude extracts or insoluble forms of enzymes. Kinetic parameters, as characterized by the Hill equation, were vmax = 497.2μM/min, KM = 542.4μM, and nH = 2.2, suggesting that toluene dioxygenase has at least three cooperative binding sites for toluene. In addition, the use of alternative ferredoxins and reductases was examined. Ferredoxin cloned from CYP153 could transfer electrons to the iron sulfur protein component of toluene dioxygenase. The ferredoxin could be reduced by ferredoxin, rubredoxin, and putidaredoxin reductases of CYP153, alkane-1 monooxygenase, and camphor 5-monooxygenase, respectively. The results provide useful information regarding the effective enzymatic biotreatment of hazardous aromatic hydrocarbon contaminants.

Soup with or without meatballs: Impact of nutritional factors on the MIC, kill-rates and growth-rates

BackgroundThe Minimum Inhibitory Concentration (MIC) is a reference value for susceptibility testing of bacteria. However, the MIC is a net result of growth and killing after a certain duration of exposure under standardized favourable in vitro conditions. Killing and growth characteristics of a drug may yield more information on its activity and help to explain discrepancies between efficacy observed in vitro and in vivo. MethodsThe MIC of meropenem was determined for P. aeruginosa ATCC 27853 both by microdilution and the E-test in dilutions of Mueller Hinton (MH) broth from 100% to 1%. Time-kill curves were obtained for twofold dilutions of meropenem. Growth rates and kill rates at each concentration and dilution were obtained by linear regression. The Hill equation was fit to the kill rates vs concentrations. ResultsGrowths rates decreased log linearly from 0.63/h at 100% to 0.29/h at 6% MH. Over the 100–6% MH dilution range, there was a log-linear decrease of the MIC of meropenem of both the E-test and microdilution. The EC50s decreased from 0.29 mg/L to 0.07 mg/L, which is in agreement with the MIC results. There was a log-linear relationship between MIC and EC50 for the various dilutions MH. ConclusionsThe availability of nutritional factors is related to the MIC, and a lower availability is related to both a lower growth rate and higher kill rate. Since nutritional factors are less abundantly available in vivo as compared to in vitro, this should be taken into account when translating in vitro to in vivo pharmacodynamics.