Exponential Decay Terms Research Articles

Use and elucidation of biochemical data in the prediction of the membrane separation of biocolloids

The successful design and operation of large–scale processes in the biotechnological industries requires effective methods for the prediction of process performance. Such predictive methods should be based on a knowledge of the physicochemical properties of the materials being processed. A major challenge for prediction in biological process engineering is to make use of the sophisticated three-dimensional structural information that is now available for many biological materials. The present paper develops such an approach for an important type of separation: the membrane ultrafiltration of a recombinant protein, in this case human lactoferrin. The protein is treated as a biocolloid. From the available amino acid sequence and structural data for the protein, together with the experimentally measured zeta potential, a charge–regulation model for the lactoferrin–solution interface is developed that takes into account chloride–ion binding. This surface model is then incorporated into a predictive process model for the rate of membrane ultrafiltration of proteins, which takes into account, in a sophisticated manner, the protein–protein interactions (electrostatic, London-van der Waals and entropic) that take place close to the membrane surface and which control the rate of the process. By comparison of the experimentally obtained process data with model predictions it is found that a further attractive interaction between lactoferrin molecules is present, which can be quantified through an exponential decay term. Incorporation of such an interaction in the process model allows successful prediction of the rate of ultrafiltration over a wide range of solution conditions. The existence of the further attractive interaction has been quantitatively verified by osmotic pressure measurements. Hence, the paper shows how detailed structural information may be successfully used for process prediction. It also shows how analysis of process data may itself provide quantitative information on the interactions of biocolloids. This type of approach is likely to become increasingly important as biological process engineers incorporate the latest findings in the biosciences into the development of successful industrial processes.

Fluorescence Decay of Natural Organic Matter (NOM)—Influence of Fractionation, Oxidation, and Metal Ion Complexation

Time-resolved fluorescence measurements of aquatic natural organic matter (NOM) derived from different origins were performed using the time-correlated single-photon counting technique. The obtained experimental data were analyzed with nonlinear least-squares (NLLS) algorithms. The results of a global analysis with three exponential decay terms were compared with the results obtained in a distribution analysis (exponential series method; ESM). Fulvic acid fractions from a bog lake water, from a brown coal production effluent, and from a soil seepage water as well as NOM from a municipal waste water were investigated. The influence of the emission wavelength on the NOM fluorescence decay was monitored. Furthermore, the influence of fractionation using size exclusion chromatography, of ozonation, and of metal ion complexation on the fluorescence decay of the NOM samples was investigated.

Spectral analysis of the fluorescence quenching kinetics in micelles with probe migration

The fluorescence quenching kinetics in monodisperse micelles considering intermicellar probe migration has been studied. Decay parameters obtained via eigenfunction expansion are correlated with the lifetime distribution and the rules of the exponential series method (ESM) applied to micellar kinetics. In a system where interacting micelles allow the excited probe to migrate between aggregates with a rate constant of the same order of that of intramicellar quenching, the principal eigenvalues will follow approximately the optimization constrain of the ESM method in which the inverse of the lifetimes is a linear function of the occupancy number j = 0, 1, 2, …. However, the normalized amplitudes of the discrete set of exponential decay terms are no longer Poisson distributed, although the model for quencher distribution in the equilibrium state is assumed to follow a Poisson law. This indicates that the distribution pattern recovered using the ESM cannot be uniquely correlated to the quencher occupancy statistics when probe migration occurs.

Step response of pressure-sensitive paints

An experimental and analytical characterization of the response of three typical pressure-sensitive paint (PSP) formulations to a step change in pressure is presented. The three PSP formulations tested have all been previously used in wind-tunnel experiments and have been shown to have acceptable aerodynamic surface properties and mechanical durability. The time response of the PSP coatings is dependent on the diffusion of oxygen through the PSP binder material with typical time constants ranging from 0.05 to 2.5 s. The PSP response is best characterized by a two-term exponential decay function representing a complex first-order dynamic system. In contrast, a standard pressure transducer responds as a second-order dynamic system. The two-exponential curvefit provides a convenient way to compare the time response of different PSP formulations. A mass diffusion model of the PSP response is also presented that is capable of predicting the time response of PSP layers. Nomenclature Aj = constant in mass diffusion solution, Eq. (7) a = binder layer thickness B = constant in two-exponential curvefit, Eq. (10) Ci, C2 = constants in two-exponential curvefit, Eq. (10) Dm = mass diffusivity, Eq. (5) /i, /2 = functional relationships / = light intensity N, n = mass concentration in the binder layer P = pressure R = calibration constant, Eq. (1) S = calibration constant, Eq. (1) x = distant above the substrate e}, e2 = exponential decay terms, Eq. (10) X = wavelength A./ = eigenvalues in mass diffusion solution, Eq. (7) a = Henry's constant, Eq. (2) TI , T2 = time constants, Eq. (10) r = time constant, Eq. (8) Subscripts

State Space Modelling of Contaminant Transport Processes by Extending Discrete Vortex Simulations-1. Horizontal Dispersion.

In order to diagnose and predict the contaminant transport process in large scale geosystems using a nonlinear filtering strategy the state space model has been derived from simulating horizontal contaminant dispersion by extending the discrete vortex method. The dispersion area S and the dispersion rate S have been taken into account for evolving the state space model of the horizontal contaminant transport process. Eventually, S/S has been found to be represented by a function composed of exponential decay terms, which is transformed to the autonomous state equation and serves as the state space model accompanied with the observation equation. The state equation is extended to involve a noise process corresponding to disturbances such as viscous diffusion and inhomogeneity of the vortex strength as observed in real systems. The parameters involved in the state equation are adaptively estimated from the current state observations using a non-linear filtering algorithm. The state space model has been verified by a series of numerical demonstrations carried out using the discrete vortex method.

On the dynamics of high Rydberg states of large molecules

In this paper we explore the level structure, the optical excitation modes and the dynamics of a mixed Stark manifold of very high Rydberg states (with principal quantum numbers n=80–250) of large molecules, e.g., 1,4 diaza bicyclo [2,2,2] octane (DABCO) and bis (benzene) chromium (BBC) [U. Even, R. D. Levine, and R. Bersohn, J. Phys. Chem. 98, 3472 (1994)] and of autoionizing Rydbergs of atoms [F. Merkt, J. Chem. Phys. 100, 2623 (1994)], interrogated by time-resolved zero-electron kinetic energy (ZEKE) spectroscopy. We pursue the formal analogy between the level structure, accessibility and decay of very high Rydbergs in an external weak (F≂0.1–1 V cm−1) electric field and intramolecular (interstate and intrastate) relaxation in a bound molecular level structure. The onset n=nM of the strong mixing (in an external field F and in the field exerted by static ions) of a doorway state, which is characterized by a low azimuthal quantum number l, a finite quantum defect δ, and a total nonradiative width Γs≂Γ0/n3, with the inactive high l manifold is specified by nM≂80.6δ1/5(F/V cm−1)−1/5. At n≥nM the level structure and dynamics are characterized by the product γρ, where ρ is the density of states and γ=ΓsD(n) is the average decay width of the eigenstates, with the dilution factor D(n)≊n−2 for (lml) mixing and D(n)≂n−1 for (l) mixing, whereupon γρ=(Γ0/4δR)(nM/n)5, being independent of D(n). The sparse level structure is realized for γρ≪1, while the dense level structure prevails for γρ≳1, resulting in two limiting situations; (a) a dense limit for n≥nM and a sparse limit for n≫nM, and (b) a sparse limit for all n≥nM. The experimental information currently available on the decay dynamics of molecular (DABCO and BBC) and atomic (Ar) Rydbergs for n≥nM corresponds to case (b). The time-resolved dynamics was characterized in terms of the excited state total population probability P(t) and the population probability I(t) of the doorway state. P(t), which is interrogated by time-resolved ZEKE spectroscopy, will exhibit for both the sparse and dense level structures and for all excitation conditions a superposition of exponential temporal decay terms with an average lifetime of ∼ℏ/γ. I(t) can be used to interrogate coherence effects, which in case (b) are manifested in quantum beats, while case (a) corresponds to a giant resonance with a molecular time characterized by the reciprocal energetic spread of the Stark manifold. The experimental data for the onset of strong mixing and for the diluted lifetimes [ℏ/ΓsD(n) with D(n)∼n−1] of the high Rydbergs (n∼100–200) of BBC and of DABCO are in accord with the predictions of the theory for the limit of strong (l) mixing. While strong mixing is realized for F̄=Fn5/3.4×109δ≳1, we expect that for the weak mixing regime (F̄<1) the dynamics of ultrahigh Rydbergs will be characterized by two distinct (∼ns and ∼μs) time scales. Finally, we emphasize the universality of the model, which provides a unified description of the level structure and dynamics of high Rydbergs of molecules and of autoionizing atoms.

A model of microbial survival after exposure to pulsed electric fields

AbstractSurvival curves of microorganisms exposed to pulsed electric fields have a characteristic sigmoid shape when plotted in linear coordinates. They can be described by the phenomenological model S(V, n) ‐ 100/(1 + exp(V ‐ Vc(n)) /a((n))) where S(V, n) is the percent surviving organisms, V the field intensity, n the number of pulses, Vc(n) critical field intensity corresponding to 50% survival and a(n) a constant representing the curve's steepness (about 90% of the loss in number occurs within Vc ± 3a). Testing the model with published data showed an excellent fit. Moreover, both Vc(n) and a(n) could be described in terms of a single exponential decay term, indicative of the increased lethality of the field as the number of pulses increases. These enabled the creation of three dimensional plots of the survival‐field intensity‐number of pulses relationships, with linear or semi‐logarithmic scales, from which similarity and differences in resistance can be revealed at a glance. The tabulated model's constants can also be used for quantitative comparison of the survival curves of different organisms. WhenV ≫ Vc the model is reduced to log[S(V, n)/100] = ‐V/a(n), that is it entails the same type of semi‐logarithmic relationship between survival and field intensity, a relationship that is traditionally used to present survival curves.

Kinetic properties of skeletal-muscle-like high-threshold calcium currents in a non-fusing muscle cell line.

Macroscopic kinetics and single-channel properties of skeletal-muscle-type calcium currents were studied in the non-fusing, clonal muscle cell line, BC3H1. Slowly activating, dihydropyridine(DHP)-sensitive currents, associated with T-tubular DHP receptors and ion channels, could be isolated from rapidly activating, DHP-resistant currents. Description of macroscopic current activation kinetics required only a brief delay term (tau o <1 ms), two ascending exponential terms with voltage-dependent time constants (2 < tau 1 > 20 ms and 10 < tau 2 < 200 ms), and a single exponential decay term (0.5 < tau 3 < or = 5 s). Steady-state activation voltage dependence required description by two Boltzmann distribution terms with V 1/2 and slope factors differing by 20 mV and 3.5- to 4-fold respectively. These two distributions were correlated with the steady-state voltage dependence of the two ascending kinetic terms described by tau 1 and tau 2 respectively. Rundown of the DHP-sensitive slow current was correlated with a negative shift in the voltage dependence of current decay (tau 3). Three conductance levels (4.5 pS, 8 pS and 12 pS) were detected in single-channel records, two of which (the 8-pS and 12-pS events) were prolonged by BayK8644 and thus associated with DHP-sensitive single-channel events. Description of single-channel open time distributions required a minimum of two exponential terms (2.5 +/- 0.9 ms and 10.3 +/- 5.4 ms at -10 mV). Slow transitions among closed states results in biexponential latency-to-first-event distributions (47 +/- 12 ms and 470 +/- 123 ms at -10 mV).

Mapping the stiffness-temperature-moisture relationship of solid biomaterials at and around their glass transition

Mechanical changes in biomaterials at and around their glass transition are key factors in their functionality and/or stability. They are described in terms of a relationship betwen a relative stiffness R(T,M) defined as the ratio between a modulus or storage modulus at a temperature T, and moisture M, and its magnitude in the glassy state. The relationship, in turn, is expressed by the model R(T,M) = 1/[1+exp [{T-Tc,(M))/a(M)]} where Tc(M) is a critical temperature identifying the transition temperature range and a(M) a constant representing the relationship's slope. The proposed model correctly accounts for the downward concavity of the stiffness vs temperature relationship at the transition onset. Published data on biosolids indicate that Tc(M) can be described by a single exponential decay term, and so most probably also a(M). Incorporation of these terms into the model enables the creation of realistic three-dimensional maps of the relative stiffness-temperature-moisture relationship at and in the neighborhood of the glass transition region. In principle, the same method can also be used to describe the effect of plasticizers other than water if their influence on Tcand the steepness parameter can be formulated as an algebraic expression.

Shear-wave attenuation and site response in Guerrero, Mexico

Spectra from moderate earthquakes are used to investigate shear-wave attenuation and site response for accelerographs located on rock outcrops in Guerrero, Mexico. A least-squares method is used to model the strong-motion spectra assuming an ω−2 source and an exponential decay term, e −πκ f , to describe the high-frequency spectral shape. The spectral decay variable, κ, associated with attenuation, is parameterized as the sum of a distance dependent component, κ ˜ ( r ) , and a site term, κ( S ), which takes a different value for each site, S . Frequency-dependent site response is determined by comparing observed spectra to model event spectra corrected for κ and geometrical spreading. Averaging these residuals over several earthquakes results in an estimate of site response that is generally free of source and path effects. The distance dependence of the spectral decay parameter as expressed by κ ˜ ( r ) is very weak in the Guerrero region in contrast to that in southern California (Anderson, 1991b). This difference may be due to greater shear strength of the top 20 km of crust in the Guerrero subduction zone. Near-site attenuation, represented by κ( S ), is greater in Guerrero compared to the stations in the Anza array of southern California. No obvious correlation with site geology was found. A combination of a deeper weathered layer of crystalline rock and varying subsurface lithology may be responsible for the greater mean values of κ( S ) in Guerrero. The frequency-dependent site functions show significant amplification and deamplification effects for the hard-rock sites. No direct correlation with local topography is observed to explain these effects. The site functions obtained here by constraining the source shape and geometrical spreading closely resemble those derived for Guerrero stations by Castro et al. (1990) using much less restrictive assumptions about the source and propagation effects.

Analytical method for the compensation of eddy-current effects induced by pulsed magnetic field gradients in NMR systems

Most NMR localization techniques use pulsed magnetic field gradients which, however, induce multiexponentially decaying eddy currents that distort images and spectra. This work describes a comprehensive strategy to measure and to exactly compensate for the induced gradient and the shift in B 0 field which are produced and also to compensate for other terms like cross-talk or nonlinear terms. The time dependence of the gradient, and, if desired, of the B 0 shift and other terms, is measured from FID signals with a method that distinguishes these components. The signals are obtained from a small sample successively at two or more positions by driving the gradient coil with a step function current pulse. A multiexponential fit through the measured temporal behavior of the gradient determines the amplitudes and time constants of the various exponential decay terms. A model allows calculation of the exact shape and parameters of the current pulse required to compensate for the eddy-current effects. This also turns out to be a multiexponential function, with, however, time constants differing from those of the eddy currents. They would only be the same if the shaping did not itself produce eddy currents. After compensation of the gradient component is achieved, the ΔB 0( t) component is measured and similarly corrected, as can also be the other terms. The procedure is suited for automation and should avoid long and tedious adjustments by trial and error of the compensation, including that with shielded gradients.

Reductive dehalogenation of carbon tetrachloride by Escherichia coli K-12

The formation of radicals from carbon tetrachloride (CT) is often invoked to explain the product distribution resulting from its transformation. Radicals formed by reduction of CT presumably react with constituents of the surrounding milieu to give the observed product distribution. The patterns of transformation observed in this work were consistent with such a hypothesis. In cultures of Escherichia coli K-12, the pathways and rates of CT transformation were dependent on the electron acceptor condition of the media. Use of oxygen and nitrate as electron acceptors generally prevented CT metabolism. At low oxygen levels (approximately 1%), however, transformation of [14C]CT to 14CO2 and attachment to cell material did occur, in accord with reports of CT fate in mammalian cell cultures. Under fumarate-respiring conditions, [14C]CT was recovered as 14CO2, chloroform, and a nonvolatile fraction. In contrast, fermenting conditions resulted in more chloroform, more cell-bound 14C, and almost no 14CO2. Rates of transformation of CT were faster under fermenting conditions than under fumarate-respiring conditions. Transformation rates also decreased over time, suggesting the gradual exhaustion of transformation activity. This loss was modeled with a simple exponential decay term.

Viscoelasticity

We study the behaviour in viscoelastic fluids of the high-frequency viscosity factor C(n(ωm)). This factor depends upon the number n(ωm) of exponential decay terms needed to describe the observed relaxation behaviour in a frequency range 0 to ωm. Quite generally n(ωm) increases with increasing ωm and decreasing temperature (T). C can be interpreted as the ratio η(0)fast/η(0) where η(0) is the zero frequency viscosity, and η(0)fast is that component of the viscosity that is associated with rapidly decaying relaxation processes, i.e. with processes that decay in times shorter than ωm -1. As T decreases, more relaxation processes become slow thereby accounting for a decrease in C; at sufficiently low T, the only fast processes are the very fast (molecular) ones that vary little with T and are called the ‘bare processes’. Thus at low T, C is small and varies as η-1. At high T, 1 — C can be associated with the shear-orientation parameter R, and at low T the quantity Cη(0)k 2/ρ can be associated with the k-dependent part of the shear wave relaxation frequency.

Hot-electron zero-field mobility and diffusion in rare-gas moderators.

An efficient discrete-ordinate method of solution of the time-dependent Boltzmann equation is employed in the calculation of the zero-field electron mobility and diffusion coefficients for hot-electron thermalization in rare-gas moderators. The discrete-ordinate method is modified to permit a rescaling of the quadrature points. This procedure is somewhat analogous to the two-temperature-moment methods employed in the theoretical analysis of electron swarms. The time-dependent transport coefficients are given as a sum of exponential decay terms characterized by the discrete eigenvalues of the Lorentz-Fokker-Planck operator for elastic electron-atom collisions. For argon, krypton, and xenon, the time dependence is strongly influenced by the Ramsauer-Townsend minimum and leads to maxima in the transient mobility and diffusion coefficient. Helium and neon with hard-sphere-like cross sections exhibit transient mobilities which initially are below the thermal zero-field mobility and then increase to the thermal mobilities as the electron distribution approaches equilibrium. The transient mobility for cross sections with Ramsauer minima are sufficiently sensitive to the details of the cross sections such that it may be feasible to distinguish between different cross sections experimentally. The calculations also indicate that the transient mobility is insensitive to the initial distribution function. A nonequilibrium phenomenon not previously recognized is the possibility of a negative transient mobility which occurs provided that the momentum-transfer cross section increases sufficiently rapidly with energy.

Theory of Longitudinal Relaxation of an AX Spin System under Double Resonance. I. Density Matrix Formulation of Saturation Recovery

Concrete equations of motion describing longitudinal relaxation of an AX system under irradiation of a selective rf field are worked out on the basis of Bloch's density matrix theory. These equations are composed of thermal relaxation and coherent motion of the stirred spin, which determine dynamical characteristics of the spin system. Explicit equations of motion are obtained also for the magnetizations of another spin observed in the saturation-recovery experiment under double resonance. When the stirring field is not too strong to cause any appreciable splitting of the observed line, the recovery of the observed line can be described in terms of several exponential decay terms together with a pair of oscillating terms corresponding to the coherent nutation.

Comparison of two methods for stress relaxation data presentation of solid foods

Published exponential relaxation equations, derived from Maxwellian models, were used to generate data for linear representation in the form ofP(0) ·t/(P(0) —P(t)) =k 1 +k 2t whereP(t) is the decaying parameter (force, stress or modulus),P(0) its initial value (att = 0) andk 1 andk 2 constants. The computer plots indicated that the fit of this normalized and linearized form was excellent for equations containing at least three exponential decay terms. The fit was not as good for some of the two-term exponential equations mainly due to the lack of accurate account for the initial stage of the relaxation process. In all the cases, however, the linear representation could clearly reveal the general rheological character of the analysed materials in terms of the relative degree of solidity.

Kinetics of unfolding and folding of horse heart ferricytochrome c with urea.

Kinetic data for the reversible folding and unfolding by urea of horse heart ferricytochrome c, in 0.05 M phosphate + 0.25 M Na2SO4 buffer, pH 7.0, in the region of the main denaturation transition, 4-9 M, are reported. Stopped flow technique and absorptivity at 695, 528, and 361 nm as the monitoring probes were used. The decay profiles in the region of the transition 6-7.2 M urea are adequately described by a rate law with two exponential decay terms, but a rate law with only a single term is found to be applicable at the lower and higher limits of the transition. The apparent rate constant for the fast phase exhibits urea dependence with a minimum value at about 6.5 M urea, while the apparent rate constant of the slow phase is found to be independent of urea and has a value of 0.04 +/- 0.02 s-1. The assignment of the two apparent rate constants to the respective steps and the characterization of the processes involved were carried out through correlation of the kinetic data to the results from equilibrium studies for urea denaturation ( Myer, Y. P., MacDonald, L. H., Verma, B. C., and Pande, A. (1980) Biochemistry 19, 199-207). A mechanism X1 in equilibrium X2 in equilibrium D, where the first step is the urea-dependent unfolding and folding, and the second, an apparent urea-independent process involving possibly reorganization of the unfolded form X2, has been proposed to account for the above findings. The X2 in equilibrium D process is further considered in light of various possibilities: the incorrect folding of the unfolded form, the alteration of heme iron coordination, and the cis-trans isomerization of proline.

Modeling the Ames test

Despite the value and widespread use of the Ames test, little attention has been focused on standardizing quantitative methods of analyzing these data. In this paper, a realistic and statistically tractable model is developed for the evaluation of Ames-type data. The model assumes revertant colony formation at a any dose follows a Poisson process, while the mean number of revertants per plate is a nonlinear functions of up to 4 parameters. An exponential decay term can be included in the model to adjust for toxicity. The resultant system of nonlinear equations is solved using a modified Gauss-Newton iterative scheme to obtain maximum likehood estimates of the model parameters. Significance of the key parameters is tested by fitting reduced models and using likelihood ratio tests. The model's performance is demonstrated on data from organic extracts of various environmental contaminants. Among the advantages of the proposed model are (1) no data is discarded in the parameter estimation process, (2) no arbitrary constants need to be added to zero counts or doses, and (3) no mathematical transformation of the data is required.

Infinite elements

AbstractA series of shape functions analogous to Lagrange polynomials, but including an exponential decay term, are suggested as appropriate for elements which extend to infinity. The shape functions are applied to a number of examples, mainly of viscous flow, where the domain extends to infinity. Both explicitly integrated and numerically integrated elements are described. The technique is applicable to many problems where the domain is supposed to extend to infinity.

Fluorescence depolarization in a model system: carbonic anhydrase-dansyl sulphonamide

The model system, carbonic anhydrase to which dansyl sulphonamide is specifically adsorbed, was chosen to test the fundamentals of the fluorescence depolarisation method. Fluorescence decay was shown to be satisfactorily represented by a single exponential decay term and the derived lifetime did not seem to vary significantly with mean excitation or emission wavelength. Depolarisation measurements were quite reproducible and those made with varying temperature were consistent with those performed with varying solution viscosity, thus excluding localised dye rotation. The calculated relaxation time, which was independent of wavelength of excitation or emission, was in conformity with a rigid spherical molecule of molecular weight 29 000 with 30% hydration.