Bell's Equation Research Articles

Accumulation of Hydrogen Bonds and van der Waals Interactions Determines Force Response between Two Parallel Cellulose Chains: Steered Molecular Dynamics Simulations.

We investigated the response forces between two parallel cellulose chains during the shearing and tearing processes by using steered molecular dynamics simulations. It was found that there are two logarithmic dependencies between response force and pulling speed in shearing processes but only one in tearing, according to Bell's equation by fitting the f-ln v curve. The mechanism is that there are 2-fold interactions determining the force response between two parallel cellulose chains resisting chain separation during a shearing process. Our results indicate that hydrogen bonds dominate the interchain interactions in the fast pull mode (FPM) for shearing, while van der Waals interactions dominate in the slow pull mode (SPM). For tearing, the one-by-one breaking of hydrogen bonds and van der Waals interactions plays a main role.

Investigating the Effect of Tyrosine Kinase Inhibitors on the Interaction between Human Serum Albumin by Atomic Force Microscopy

It is important for elucidating the regulation mechanism of life activities, as well as for the prevention, diagnosis, and drug design of diseases, to study protein–protein interactions (PPIs). Here, we investigated the interactions of human serum albumin (HSA) in the presence of tyrosine kinase inhibitors (TKIs: imatinib, nilotinib, dasatinib, bosutinib, and ponatinib) using atomic force microscopy (AFM). The distribution of rupture events including the specific interaction force Fi and the non-specific interaction force F0 between HSA pairs was analyzed. Based on the force measurements, Fi and F0 between HSA pairs in the control experiment were calculated to be 47 ± 1.5 and 116.1 ± 1.3 pN. However, Fi was significantly decreased in TKIs, while F0 was slightly decreased. By measuring the rupture forces at various loading rates and according to the Bell equation, the kinetic parameters of the complexes were investigated in greater detail. Molecular docking was used as a complementary means by which to explore the force of this effect. The whole measurements indicated that TKIs influenced PPIs in a variety of ways, among which hydrogen bonding and hydrophobic interactions were the most important. In conclusion, these outcomes give us a better insight into the mechanisms of PPIs when there are exogenous compounds present as well as in different liquid environments.

Simplified Model to Approach the Theoretical Clear Sky Solar PV Generation Curve through a Gaussian Approximation

Radiation forecast is the milestone of the solar energy industry, making possible the existence of the whole market. Solar radiation models allow scientists and engineers to predict the behaviour of a PV system to perform technical and economic analysis. Despite the existence of numerous models, most of them are highly complex or require massive amounts of data, limiting solar energy start-ups. As a result, a state-of-the-art review was performed and based on it this study proposed a simple model that allows emerging solar companies to create preliminary analysis for their clients. The proposed model calculates the instant power of the envelope curve of PV generation, based on the Gaussian bell equation, by using the daily specific energy and a deviation proportional to the sun hours of the geographical information as parameters. With accurate meteorological data, results showed an acceptable performance, with a more straightforward implementation, when compared against those reported in the literature.

Refined modelling of the single-mode cylindrical Richtmyer–Meshkov instability

Abstract

Measurement of a Richtmyer-Meshkov Instability at an Air-SF_{6} Interface in a Semiannular Shock Tube.

We report the first measurements of the perturbation amplitude in the converging Richtmyer-Meshkov instability in a semiannular shock tube. At early stages, the amplitude growth agrees well with the impulsive model considering the geometrical convergence effect. A quick decrease of the growth rate at late time, even to be negative, before the reshock is observed for the first time. The reduction of the growth rate is ascribed to the Rayleigh-Taylor stabilization caused by the interface deceleration motion only presented in the converging circumstance. By reasonably evaluating the Rayleigh-Taylor stabilization, a modified model based on the Bell equation is proposed, which well predicts the perturbation growth in a converging geometry from early to late stages before the reshock. It is also found that the flow compressibility is significant in the converging Richtmyer-Meshkov instability.

On Stress-Induced Tribochemical Reaction Rates

A number of recent experiments have demonstrated that normal and shear stresses are able to accelerate the rates of chemical reactions of adsorbates on surfaces, yielding an approximately exponential increase in reaction rate with force. It seems that such mechanochemical reactions play a central role in tribochemistry and the consequent formation of anti-wear or friction-reducing films from lubricant additives. It is thus appropriate to summarize the theories that can be used to analyse such tribochemical data. The models have a strong parallel with those used to analyse sliding friction, the Prandtl/Tomlinson model. It is found that the force-dependent activation barrier for chemical reactions varies as $$E_{\text{act}} (F) = E_{\text{act}} + AF + BF^{2}$$ . The linear variation in activation barrier with force, the so-called Bell equation, and the quadratic dependence, the so-called Hammond effect, are analysed using model reaction energy profiles, which reveal a dependence of the parameters A and B on the shape of the energy profile. The influence of contact pressure and sliding velocity is also discussed. The above models assume that the normal and shear stresses are coplanar with the thermal reaction coordinate for the tribochemical processes, and the effects of this not being the case are also discussed. Finally, the existing results of tribochemical reactions of oxygen-functionalized graphene and zinc dialkyl dithiophosphates are also analysed using these models.

Elasticity of the transition state for oligonucleotide hybridization

Despite its fundamental importance in cellular processes and abundant use in biotechnology, we lack a detailed understanding of the kinetics of nucleic acid hybridization. In particular, the identity of the transition state, which determines the kinetics of the two-state reaction, remains poorly characterized. Here, we used optical tweezers with single-molecule fluorescence to observe directly the binding and unbinding of short oligonucleotides (7–12 nt) to a complementary strand held under constant force. Binding and unbinding rate constants measured across a wide range of forces (1.5–20 pN) deviate from the exponential force dependence expected from Bell's equation. Using a generalized force dependence model, we determined the elastic behavior of the transition state, which we find to be similar to that of the pure single-stranded state. Our results indicate that the transition state for hybridization is visited before the strands form any significant amount of native base pairs. Such a transition state supports a model in which the rate-limiting step of the hybridization reaction is the alignment of the two strands prior to base pairing.

The energy dependent Pál–Bell equation and the influence of the neutron energy on the survival probability in a supercritical medium

The survival probability of a neutron injected into a supercritical fissile medium is studied with respect to the energy dependence of the incoming neutron. We assume a point model but allow the energy dependence to be included through a general energy exchange model. We have studied the effect of slowing down on the survival probability by means of the Pál–Bell equation and the Goertzel-Greuling kernel, the separable kernel, the Hansen–Roach dataset and a WIMS generated dataset for a homogenised reactor to approximate the slowing down process. The Goertzel-Greuling model is known to be exact for A=1 and reverts to age theory for large mass ratios. It is also accurate for all intermediate mass numbers, except possibly when strong resonances are present. The separable kernel is a simple model of energy exchange, corresponding to the thermalisation of neutrons in a single collision, which ignores the slowing down process but provides a simple result allowing both ends of the reactor spectrum to be included. The Hansen–Roach data set is obtained from a realistic slowing down model in a fast system and the thermal system is modelled by homogenised reactor data, generated using WIMS, and is typical of the material found in a PWR. Using a scattering cross section which is an arbitrary function of energy, and capture and fission cross sections which are proportional to 1/v, we find that the survival probability is energy independent insofar as it depends only on the values of the ratio of the fission and capture cross sections. For non-1/v cross sections there is an energy dependence which we discuss below. The formalism developed is robust enough for studies to be made of the influence of resonance cross sections and inelastic scattering on survival probability.

Optimal Loading of Molecular Bonds

A corollary of the Bell equation of bond rupture is that the bond transfers a maximal impulse during its lifetime when the applied force equals k(B)T/x*. It is proposed that the designs of biological systems converge toward loading bonds with this optimal force to minimize the need for self-healing and to optimize energy transfer.

Duration-frequency relationships of heavy rainfall in Santa Catarina, Brazil

The purpose of this study was to adjust equations that establish relationships between rainfall events with different duration and data from weather stations in the state of Santa Catarina, Brazil. In this study, the relationships between different duration heavy rainfalls from 13 weather stations of Santa Catarina were analyzed. From series of maximum annual rainfalls, and using the Gumbel-Chow distribution, the maximum rainfall for durations between 5 min and 24 h were estimated considering return periods from 2 to 100 years. The data fit to the Gumbel-Chow model was verified by the Kolmogorov-Smirnov test at 5 % significance. The coefficients of Bell's equation were adjusted to estimate the relationship between rainfall duration t (min) and the return period T (y) in relation to the maximum rainfall with a duration of 1 hour and a 10 year return period. Likewise, the coefficients of Bell's equation were adjusted based on the maximum rainfall with a duration of 1 day and a 10 year return period. The results showed that these relationships are viable to estimate short-duration rainfall events at locations where there are no rainfall records.

Fatigue at the Molecular Scale

The influence of force on the stability of inter- and intramolecular bonds has been studied in great detail. Experiments, e.g. with AFMs or the Biomembrane Force Probe, have shown that the relationship between the lifetime of biomolecules and applied forces is well described by the Bell equation. However, rather than experiencing a constant (or continuously increasing) force, some biomolecules are subject to repeated force cycles. For instance, myosin motors in muscles experience millions of force cycles over their lifetime. At the macroscopic level, repeated stress cycles can lead to ‘fatigue failure’. Fatigue failure originates in permanent micro-damages and occurs at subcritical stress levels, provided that the number of stress cycles is large enough. Fatigue is a major cause of failure in machinery; however there is no comprehensive theory of fatigue. To design machines, engineers use heuristic methods such as the Palmgren-Miner rule or S-N diagrams. Here, we investigate for the first time a molecular theory of fatigue failure. We aim to provide a theoretical basis for the heuristic methods engineers use to avoid fatigue failure and draw conclusions about the design of molecular machines such as kinesin or myosin motors.

Kinetics of Myosin-I-Membrane Detachment Under Load

Myo1c is a single-headed motor that links cell membranes to the underlying actin cytoskeleton. Actin binding occurs via the motor domain, while the tail domain interacts with phosphatidylinositol 4,5-bisphosphate (PI(4,5)P2) through a putative pleckstrin homology (PH) domain. In this study, we measured the strength and lifetime of the myo1c attachment to spherical supported lipid bilayers (SSL) composed of 1 μm diameter silica or polystyrene beads coated with various mole fractions of PI(4,5)P2, dioleoyl-phosphatidylcholine (DOPC), and dioleoyl-phosphatidylserine (DOPS). The SSLs trapped by a laser beam were brought into contact with immobilized spherical (2 μm) silica pedestals sparsely labeled with the myo1c tail domain. Upon repeated contact and retraction cycles, binding events and subsequent rupture forces were measured. The most probable rupture forces (< 10 pN) from membranes containing 2% PI(4,5)P2 were found to be largely independent of the loading rate (pN/s). Nevertheless, the frequency of single myo1c molecules interacting with membranes containing 2% PI(4,5)P2 - 98% DOPC was considerably higher than with membranes that contained 20 - 80% DOPS. Lifetime measurements of myo1c attachment to 2% PI(4,5)P2 - 98% DOPC membranes under constant pulling forces (1 - 3 pN) were fitted to Bell's equation and the extrapolated duration of the bond at zero force was found to be ∼ 25 ms. Although phosphoinositide binding is crucial for the proper cellular targeting of myo1c, our results suggest that it is unlikely that this connection is a suitable anchor for force generation.

Relações entre precipitações intensas de diferentes durações ocorridas no município de Urussanga, SC

Neste trabalho se analisaram os pluviogramas do período de 1981 a 2004 da estação meteorológica de Urussanga, SC (Latitude 28° 31' S, longitude 49° 19' W, altitude 49 m). A partir das séries de chuvas máximas anuais foram estimadas, por meio da distribuição de Gumbel-Chow, as chuvas máximas com durações variando de 5 min a 24 h, e também para a duração de 1 dia considerando-se os períodos de retorno de 2, 5, 10, 20, 25, 50 e 100 anos. A aderência dos dados ao modelo de Gumbel-Chow foi comprovada através do teste de Kolmogorov-Smirnov, com nível de significância de 5%. A razão entre a precipitação máxima de 24 h e a precipitação máxima de 1 dia variou de 1,17 para o período de retorno de 2 anos até 1,11 para o período de retorno de 100 anos, com média de 1,13. Ajustaram-se os coeficientes da equação proposta por Bell a fim de estimar a relação entre a chuva com duração t min e período de retorno T anos, em função da chuva máxima com período de retorno de 10 anos e duração de 1 h. Do mesmo modo, se ajustaram também os coeficientes da equação de Bell cuja base foi a chuva máxima com período de retorno de 10 anos e duração de 1 dia.

The load dependence of rate constants

As experimental techniques in biophysics have progressed at the single molecule level, there has been considerable interest in understanding how external mechanical influences (such as load) affect chemical reactions. The majority of biophysical studies investigating load-dependent kinetics use an equation where the rate constant exponentially depends on force, which is sometimes called Bell's equation. This equation requires the determination of two parameters that describe the potential energy-strain function: k(0), which is the reaction rate in the absence of load, and x(c), which is the difference in strain between the reactant and transition states. However, there have been theoretical studies based on Kramers' theory suggesting that the rate constant should have load-dependent pre-exponential terms and nonlinear load-dependent terms in the exponential. Kramers' theory requires an exact knowledge of the potential energy-strain function, which is in general not known for an experimental system. Here, we derive a general approximation of Kramers' theory where the potential energy-strain function is described by five parameters, which can, for small loads, be reduced to four-, three-, and finally to two parameters (Bell's equation). We then use an idealized physical system to validate our approximations to Kramers' theory and show how they can predict parameters of interest (such as k(0) and x(c)) better than Bell's equation. Finally, we show previously published experimental data that are not well fitted by Bell's equation but are adequately fitted by these more exact equations.

The Saunders-Bell Analysis of Tunnel Effects in Reactions with Kinetic Isotope Effects

The primary kinetic isotope effects of deuterium were investigated in 22 hydrogen and deuterium transfer reactions, including enzymatic and nonenzymatic hydride transfer reactions, elimination reactions, and reactions catalyzed by enzymes lipooxygenase, amine dehydrogenase, and methylmalonyl-CoA mutase. In each case, the Saunders-Bell analysis was applied to calculate the tunnel effects and the corresponding thermodynamic parameters. The Saunders-Bell analysis was effective in 14 cases out of 22. A high degree of correlation was found between the barrier factor, the tunnel factor, and the entropy factor among all reactions studied. From this, a general relationship between the three factors was derived, based on the Saunders-Bell analysis of the Bell equation; the Saunders-Bell analysis is valid within certain limits of the barrier factor. This general relationship is universally valid for all hydrogen/deuterium transfer reactions in nature with moderate tunneling, when the Saunders-Bell analysis applies.

Active earth pressure in cohesive soils with an inclined ground surface

An analytical solution is developed to determine the active lateral earth pressure distribution on a retaining structure when it consists of a cohesive backfill (internal friction angle ϕ &gt; 0, cohesion c &gt; 0) with an inclined ground surface. The solution derived encompasses both Bell's equation (for cohesive or cohesionless backfill with a horizontal ground surface) and Rankine's solution (for cohesionless backfill with an inclined ground surface). The orientation of the failure surface is also determined. Results indicate that, unlike the soil-wall scenarios of Bell and Rankine where the failure planes are parallel with a fixed orientation independent of the overburden pressure, for sloping cohesive backfill (ϕ &gt; 0, c &gt; 0) the slope of the failure surface is a function of the overburden pressure and becomes shallower with depth, thus forming a curvilinear failure surface. The solution developed can also be used to check the sustainability of a slope. The analytical solution can be programmed conveniently in a computer.Key words: retaining structure, active earth pressure, cohesive backfill.

An investigation of the quantum chemical description of the ethylenic double bond in reactions. I. The electrophilic addition of hydrochloric acid to ethylene

The bimolecular reaction mechanism for adding hydrochloric acid to ethylene is studied within a number of contemporary quantum chemical models. The transition state structure and energy is examined in detail, and high-level calculations support a picture of an intimate association of a chloride anion and a bridged ethyl cation, with some covalent bonding retained between chlorine and hydrogen. A tunneling correction of 1 kcal/mol in the reaction barrier is obtained by the Bell equation. The methods employed include configuration interaction (CI) and gradient-corrected density functional theory (DFTG), with a range of basis sets. High-end CI and DFT methods perform equally well with respect to enthalpies of reaction and activation, when used in connection with large bases. However, when bases of double-ζ plus polarization quality are used, an unfortunate accumulation of errors makes DFTG inferior. This is in contrast to the excellent values for a number of ground state properties of relevance to the reaction barrier which are obtained at this level of theory. A recently proposed extrapolation procedure for CI energies (PCI-80) is shown to fail for the electron affinity of chlorine, leading to an exaggerated estimate of the barrier for the title reaction. The single-particle description of the π bond in ethylene converges very slowly with the number of basis functions, and this affects the reaction enthalpy as computed by DFT and CI methods to a similar degree.

Influence of steric hindrance in molecules of substituted 2-phenyl-1,1,3,3-tetramethylguanidines on the kinetics and isotope effects of their proton transfer reactions with 4-nitrophenylnitromethane in acetonitrile solvent

Rate constants have been determined for the proton and deuteron transfer reactions of 4-nitrophenylnitromethane, NO2C6H4CH2NO2, with three phenyl substituted derivatives of 2-phenyl-1,1,3,3-tetramethylguanidine, X—C6H4—N=C[N(CH3)2]2, where X = 4-Cl, 2-Cl, and 2-Br, in the temperature range 278–308 K in acetonitrile solvent. In the concentration ranges used, the reaction to form an ion-paired product is first order in acid and first order in base. The reaction was followed spectrophotometrically using the stopped-flow technique. The reaction is fastest for the 4-Cl substituted base and slowest for the 2-Br analogue. The observed deuterium kinetic isotope effects show a clear dependence on the position and the size of the substituent in the base molecule. Kinetic isotope effects at 25 °C, kH/kD = 9.7 ± 0.3, 12.5 ± 0.3, and 15.6 ± 0.3, were observed for 4-Cl, 2-Cl, and 2-Br substituted bases respectively. The appropriate ratios of Arrhenius preexponential factors, AH/AD = 0.98 ± 0.10, 0.87 ± 0.10, and 0.77 ± 0.10, may indicate increasing importance of tunnelling of the proton when the substituent is moved from the para to the ortho position or when the size of ortho substituent increases. Values of tunnelling correction factors were calculated using Bell's equation by means of the Caldor and Saunders method.

Kinetic hydrogen isotope effects in the reaction between 2,4,6-trinitrotoluene and 1-ethylpiperidine in acetonitrile. The effect of pressure

The effect of pressure on the kinetic isotope ratio kH/kD and tunnelling factor Q has been examined for proton and deutron transfer from 2,4,6-trinitrotoluene (TNT) to 1-ethylpiperidine in acetonitrile. The reaction rate has been measured in the pressure range 1–1000 bar and in the temperature range 15–35 °C by means of a high-pressure stopped-flow method. Even at 1000 bar, the kinetic isotope ratio kH/kD(16.4), the activation energy difference Ea(D)–Ea(H)(10.0 kJ mol–1) and the ratio of Arrhenius pre-exponential factor A(D)/A(H)(4.0) are large, suggesting a considerable tunnelling contribution. However, the tunnelling factor Q(H), calculated according to Bell's equation, hardly changes with pressure (being 2.39 at 1 bar and 2.26 at 1000 bar). Tunnelling phenomena and related pressure effects are discussed in relation to the curvature of potential-energy barriers. The present results lead us to conclude that large changes of kH/kD with pressure do not always confirm the occurrence of tunnelling.

Density, food habits, and economic importance of raccoons (Procyon lotor) in Quebec agrosystems

During two years of trapping, in a sweet and field corn growing area in the Eastern Townships (P.Q.), population levels of raccoons based on a Lincoln index and on a Bell's equation varied from 0.17 to 2.90 individuals per hectare of corn. In 1978, corn, chokecherries, and raspberries were the main food sources. The following year spring foods, mainly invertebrates, were replaced during the summer by chokecherries, corn, hawthorn drupes, and apples. Corn and apples constituted nearly all of the food ingested in the fall of 1979. Corn crops are damaged yearly through being a staple in the diet of raccoons. Monetary losses to sweet corn were estimated at $61.03/ha in 1978 and $20.56/hain 1979. Field corn losses were very low, less than 1% of total crop biomass for both years. Damage levels varied from 10.61 kg/ha to 98.49 kg/ha implying monetary losses of $0.19/ha and $1.60/ha. Comparing the 1979 Canadian market value of $41.00 for a raccoon hide with the maximum crop loss of $4.42 per raccoon indicates that management of the species could be beneficial.