Determination of molecular parameters by fitting sedimentation data to finite-element solutions of the Lamm equation.

A method for fitting experimental sedimentation velocity data to finite-element solutions of various models based on the Lamm equation is presented. The method provides initial parameter estimates and guides the user in choosing an appropriate model for the analysis by preprocessing the data with the G(s) method by van Holde and Weischet. For a mixture of multiple solutes in a sample, the method returns the concentrations, the sedimentation (s) and diffusion coefficients (D), and thus the molecular weights (MW) for all solutes, provided the partial specific volumes (v) are known. For nonideal samples displaying concentration-dependent solution behavior, concentration dependency parameters for s(sigma) and D(delta) can be determined. The finite-element solution of the Lamm equation used for this study provides a numerical solution to the differential equation, and does not require empirically adjusted correction terms or any assumptions such as infinitely long cells. Consequently, experimental data from samples that neither clear the meniscus nor exhibit clearly defined plateau absorbances, as well as data from approach-to-equilibrium experiments, can be analyzed with this method with enhanced accuracy when compared to other available methods. The nonlinear least-squares fitting process was accomplished by the use of an adapted version of the "Doesn't Use Derivatives" nonlinear least-squares fitting routine. The effectiveness of the approach is illustrated with experimental data obtained from protein and DNA samples. Where applicable, results are compared to methods utilizing analytical solutions of approximated Lamm equations.     

On the applicability of ising and other statistical models to phase transition in biomembranes

The current literature is critically examined as regards to the applicability of statistical mechanics to the dipole model in order to describe phase transition in biological membranes. The numerical calculations in this context are shown to be speculative and model dependent. Implications of the X-ray diffraction studies on membranes are discussed and it is deduced that only the perpendicular orientation of the dipoles in the polar layer is consistent with the X-ray diffraction data.The author wishes to thank Drs. L. David Roper and E. Keith Hege for many helpful comments.     

DISCUSSION II ON FITTING EQUATIONS TO SENSORY DATA

In his article " On Fitting Equations to Sensory Data." Moskowitz suggests many strategies for model fitting which depart from current statistical methodology. Four areas discussed by Moskowitz are addressed here: (1) Forcing terms into a model; (2) The use of hold-out samples; (3) The use of aggregate data (averaging across people, suppressing the person-to-person variation); and (4) The use of random data as a predictor variable in a regression equation. All four of these points will be examined within this article.     

The cylindrical cell with a time-variant membrane resistance. Measuring passive parameters.

The passive electrical properties of a cable can be measured by injecting a step of current at a point and fitting the resulting potentials at several positions along the cable with analytic solutions of the cable equation. An error analysis is presented for this method (which is based on constant membrane resistance) when the membrane resistance is not constant, but increases linearly with time. The increase of rm produces a "creep" in the membrane potential at long times, as observed in cardiac, skeletal, and smooth muscle. The partial differential equation describing the time-varying cable was solved numberically for a step of current and these "data" were fit by standard constant-resistance methods. Comparing the resulting parameter values with the known true values, we suggest that a correction of the standard methods is not satisfactory for resistance changes of the kind observed; instead, the cable equation must be solved again for the particular form of rm(t). The practical implementation of a method by Adrian and Peachey for measuring the membrane capacitance and an approximate method for estimating the rate-of-change of membrane resistance are discussed in appendices.     

Comparative least-squares analysis of hemoglobin oxygen equilibrium curves

The oxygen-binding properties of hemoglobin have been studied at 600 microM protein concentration with organic phosphate, and analyzed by a series of different nonlinear least-squares analysis methods to determine whether reports of negligibly small values of the third overall Adair parameter, A3, are consequences of the data or a product of the data analysis. Data from other laboratories were analyzed as well. The single most important factor in creating a measurement that yields a small A3 is the use of equally weighted fitting in the Adair equation, while end-weighted fitting generally yields a larger A3. Endpoint extrapolation is ruled out as a major cause of abnormal A3 values. Monte Carlo simulations of the 600 microM results suggest that, if a small A3 were present, end weighting is at least as sensitive to a small A3 as equal weighting. We conclude that equally weighted fitting of the tetrameric Adair equation is unable to resolve the upper asymptote of the oxygen-binding data, resulting in an unusually small value for A3.     

Numerical method for equilibrium fluorescence polarization

A computer program for making first approximations of equation constants, and for fitting the equilibrium fluorescence polarization equation to data by successively improving the constant values is described. The techniques used in making the first approximations represent a substantial improvement over methods previously available.     

桔青霉摇瓶发酵生产核酸酶P1的动力学研究

研究了桔青霉发酵生产核酸酶P1的发酵动力学特性：以Logistic方程和Luedeking—Piret方程为基础,进行最优参数估计和非线性拟合,得到了描述整个发酵过程中的茵体生长、产物合成和基质消耗的动力学模型。对实验数据与模型预测值进行比较,发现模型预测值与实验数据能较好地拟合,基本上反映了桔青霉发酵过程的动力学特征,为以后进一步研究和预测核酸酶P1发酵过程奠定了理论基础。     

Inconsistent sodium current records derived on Ranvier nodes with a commercially available potential clamp device according to Nonner

We tried to reproduce some basic implications of the Hodgkin-Huxley-Frankenhaeuser formalism by measuring sodium currents in single myelinated nerve fibres with a commercially available version of the potential clamp device according to Nonner. The following contradictory observations were made: 1. The potential dependence of the time to peak sodium currents showed a discontinuity around the sodium equilibrium potential. 2. Defining the sodium permeability PNa by the constant field equation and fitting the peak PNa-voltage relation by a sigmoid function we obtained unbelievable high values of PNa at rest. 3. Testing PNa as calculated by the constant field equation by so-called "sodium tail current" experiments we obtained instantaneous changes of PNa. Summing up, neither the kinetics of sodium currents nor the constant field concept as tested with the equipment used seem to agree satisfactorily with the standard data of sodium currents in Ranvier nodes.     

A nonlinear biphasic viscohyperelastic model for articular cartilage

Experiments on articular cartilage have shown nonlinear stress-strain curves under finite deformations as well as intrinsic viscous effects of the solid phase. The aim of this study was to propose a nonlinear biphasic viscohyperelastic model that combines the intrinsic viscous effects of the proteoglycan matrix with a nonlinear hyperelastic constitutive equation. The proposed equation satisfies objectivity and reduces for uniaxial loading to a solid type viscous model in which the actions of the springs are represented by the hyperelastic function proposed by Holmes and Mow [1990. J. Biomechanics 23, 1145-1156.]. Results of the model, that were efficiently implemented in an updated Lagrangian algorithm, were compared with experimental infinitesimal data reported by DiSilverstro and Suh [2001. J. Biomechanics 34, 519-525.] and showed acceptable fitting for the axial force (R(2)=0.991) and lateral displacement (R(2)=0.914) curves in unconfined compression as well as a good fitting of the axial indentation force curve (R(2)=0.982). In addition, the model showed an excellent fitting of finite-deformation confined compression stress relaxation data reported by Ateshian et al. [1997. J. Biomechanics 30, 1157-1164.] and Huang et al. [2005. J. Biomechanics 38, 799-809.] (R(2)=0.993 and R(2)=0.995, respectively). The constitutive equation may be used to represent the mechanical behavior of the proteoglycan matrix in a fiber reinforced model of articular cartilage.     

Study of population dynamic for a recombinant bacterium during continuous cultures: application of data filtering and smoothing

A numerical method to process experimental data concerning plasmid stability of a recombinant bacteria during continuous cultures with nonselective media is proposed here. This method differs from previous ones in that it uses the derivatve form of the state equation of the Imanaka-Aiba model for recombinant cultures. The methodology proposed here allows one to estimate values for the two model parameters without forcing them to be constant. Until now, this could not be done using classical analytical techniques because these parameters have been considered invariable because of the integration used in the evaluation of the model. These parameters are (1) the difference in the specific growth rates between plasmid-carrying cells and plasmid-free cells (deltamu), and (2) the probability of plasmid loss by plasmid-containing cells (rho(r) mu(+)). The derivative technique used here is completed by mathematical treatments involving data filtering and smoothing. The values of the two parameters are in agreement with those already publised. The current technique does not impose preconditions and permit us to further study related phenomena.     

用麦夸方法最优拟合逻辑斯谛曲线

一般对非线性逻辑斯谛生长曲线的拟合是采用首先对原方程线性化以后用线性最小二乘的方法。此方法不是最优的。本文提出用麦夸方法对曲线进行拟合,并且比较了Gause、Andrewartha、May、Pearl、Krebs、万昌秀等人提出的方法与计算结果。麦夸方法对生物实验及生态学中诸多非线性曲线的参数估计具有普遍的意义。     

A Diagram for the Description of Water Relations in Plant Cells and Organs

A diagram is described which is based on the ideal relationshipbetween osmotic potential and volume as defined by Boyle's andVan't Hoff's laws. The new graphical presentation differs fromthe related earlier one by Hammel and co-workers in the choiceof relative water content (RWC), a term depending on the volumechanges of the vacuole, as the reciprocally plotted variable.The diagram is well suited for didactic purposes and may haveadvantages for the analysis of experimental data.     

On the Hill plot of NMR data for titration of protein residues

The Hill plots of NMR titration data for protein residues disclose more clearly than the usual titration curves the presence of multiple weak perturbations originating from other titratable groups, and should be used whenever the conventional curve fitting is poor. For a quantitative interpretation, we derive here expressions for the Hill equation and the Hill coefficient when the titration of the observed group is perturbed by more than one titratable group. When the generalized Hill equation is fitted to the data, values of the interaction parameters between the observed group and the others are extracted provided that there are no mutual interactions between the latter groups. The method is applied to the titration data of two histidyl residues of l-arginine phosphotransferase (E.C. 2.7.3.3.) in the transition state analogue complex (enzyme-Mg²⁺-ADP-NOsk3/–l-Arg). From the Hill plots, interactions with three titratable groups are disclosed for both residues, and the fitting with the Hill equation reveals that they experience perturbations from the same three groups. Microscopic pK values are obtained for all the involved groups, indicating large changes (up to 3 pH units) upon protonation of the interacting groups. As compared to the conventional fitting procedure, the use and fitting of Hill plots yields from NMR data more information on the neighbourhood of enzyme residues and on the changes intervening therein through the steps involved in the catalysis.     

A versatile equation to describe reversible enzyme inhibition and activation kinetics: modeling beta-galactosidase and butyrylcholinesterase

Current treatments for Alzheimer's disease involve inhibiting cholinesterases. Conversely, cholinesterase stimulation may be deleterious. Homocysteine is a known risk factor for Alzheimer's and vascular diseases and its active metabolite, homocysteine thiolactone, stimulates butyrylcholinesterase. Considering the opposing effects on butyrylcholinesterase of homocysteine thiolactone and cholinesterase inhibitors, understanding how these molecules alter this enzyme may provide new insights in the management of dementia. Butyrylcholinesterase does not strictly adhere to Michaelis-Menten parameters since, at higher substrate concentrations, enzyme activation occurs. The substrate activation equation for butyrylcholinesterase does not describe the effects of inhibitors or non-substrate activators. To address this, global data fitting was used to generate a flexible equation based on Michaelis-Menten principles. This methodology was first tested to model complexities encountered in inhibition by imidazole of beta-galactosidase, an enzyme that obeys Michaelis-Menten kinetics. The resulting equation was sufficiently flexible to permit expansion for modeling activation or inhibition of butyrylcholinesterase, while accounting for substrate activation of this enzyme. This versatile equation suggests that both the inhibitor and non-substrate activator examined here have little effect on the substrate-activated form of butyrylcholinesterase. Given that butyrylcholinesterase inhibition can antagonize stimulation of this enzyme by homocysteine thiolactone, cholinesterase inhibition may have a role in treating Alzheimer and vascular diseases related to hyperhomocysteinemia.     

DNA persistence length revisited

DNA restriction fragments ranging from 79 to 789 base pairs in length have been characterized by transient electric birefringence (TEB) measurements at various temperatures between 4 and 43 degrees C. The DNA fragments do not contain runs of four or more adenine residues in a row and migrate with normal electrophoretic mobilities in polyacrylamide gels, indicating that they are not intrinsically curved or bent. The low ionic strength buffers used for the measurements contained 1 mM Tris Cl, pH 8.0, EDTA, and variable concentrations of Na(+) or Mg(2+) ions. The rotational relaxation times were obtained by fitting the TEB field-free decay signals with a nonlinear least-squared fitting program; the decay of the birefringence was monoexponential for fragments < or = 241 base pair (bp) in length and multiexponential for larger fragments. The terminal relaxation times, characteristic of the end-over-end rotation of the DNA molecules, were then used to determine the persistence length (p) and hydrodynamic radius (r) of DNA as a function of temperature and ionic strength, using several different hydrodynamic models. The specific values obtained for p and r are model dependent. The wormlike chain model of P. J. Hagerman and B. H. Zimm (Biopolymers 1981, Vol. 20, pp. 1481-1502) combined with the revised Broersma equation (J. Newman et al., Journal of Mol Biol 1997, Vol. 116, pp. 593-606) appears to be the most suitable for describing the flexibility of DNA in low ionic strength solutions. The values of p and r obtained from the global least squares fitting of this equation are independent of DNA length, and the deviations of the individual values from the average are reasonably small. The consensus r value calculated for DNA in various low ionic strength solutions containing 1 mM Tris buffer is 14.7 +/- 0.4 A at 20 degrees C. The consensus p values decrease from 814 approximately 564 A in solutions containing 1 mM Tris buffer plus 0.2-1 mM NaCl and decrease still further to 440 A in solutions containing 0.2 mM Mg(2+) ions. The persistence length exhibits a shallow maximum at 20 degrees C and decreases slowly upon either increasing or decreasing the temperature, regardless of the model used to fit the data. By contrast, the consensus values of the hydrodynamic radius are independent of temperature. The calculated persistence lengths and hydrodynamic radii are compared with other data in the literature.     

Kinetic modeling of sporulation and product formation in stationary phase by Bacillus coagulans RK-02 vis-à-vis other Bacilli

A logistic kinetic model was derived and validated to characterize the dynamics of a sporogenous bacterium in stationary phase with respect to sporulation and product formation. The kinetic constants as determined using this model are particularly important for describing intrinsic properties of a sporogenous bacterial culture in stationary phase. Non-linear curve fitting of the experimental data into the mathematical model showed very good correlation with the predicted values for sporulation and lipase production by Bacillus coagulans RK-02 culture in minimal media. Model fitting of literature data of sporulation and product (protease and amylase) formation in the stationary phase by some other Bacilli and comparison of the results of model fitting with those of Bacillus coagulans helped validate the significance and robustness of the developed kinetic model.     

Simulation of citric acid production by rotating disk contactor

A simple model was presented to describe the time courses of citric acid production by a rotating disc contactor (RDC) using Aspergillus niger. The model is expressed by Monod-type cell growth, Luedeking-Piret-type citric acid production rate equations, and the diffusion equation for oxygen in the biofilm. The model contains five parameters which were determined by the nonlinear least squares method by fitting the numerical solution to the experimental data. In solving the equations, the cell density of the biofilm was estimated from the value of cellular mass per unit of biofilm area using an empirical equation. The experimental time courses in citric acid production period were well simulated with this model. The relation between the specific biofilm surface area and the rate of citric acid production was also explained by the simulation using the average values of five parameters of twelve runs. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 689-696, 1997.     

A versatile equation to describe reversible enzyme inhibition and activation kinetics: Modeling β-galactosidase and butyrylcholinesterase

Current treatments for Alzheimer's disease involve inhibiting cholinesterases. Conversely, cholinesterase stimulation may be deleterious. Homocysteine is a known risk factor for Alzheimer's and vascular diseases and its active metabolite, homocysteine thiolactone, stimulates butyrylcholinesterase. Considering the opposing effects on butyrylcholinesterase of homocysteine thiolactone and cholinesterase inhibitors, understanding how these molecules alter this enzyme may provide new insights in the management of dementia. Butyrylcholinesterase does not strictly adhere to Michaelis–Menten parameters since, at higher substrate concentrations, enzyme activation occurs. The substrate activation equation for butyrylcholinesterase does not describe the effects of inhibitors or non-substrate activators. To address this, global data fitting was used to generate a flexible equation based on Michaelis–Menten principles. This methodology was first tested to model complexities encountered in inhibition by imidazole of β-galactosidase, an enzyme that obeys Michaelis–Menten kinetics. The resulting equation was sufficiently flexible to permit expansion for modeling activation or inhibition of butyrylcholinesterase, while accounting for substrate activation of this enzyme. This versatile equation suggests that both the inhibitor and non-substrate activator examined here have little effect on the substrate-activated form of butyrylcholinesterase. Given that butyrylcholinesterase inhibition can antagonize stimulation of this enzyme by homocysteine thiolactone, cholinesterase inhibition may have a role in treating Alzheimer and vascular diseases related to hyperhomocysteinemia.     

Moment arms and musculotendon lengths estimation for a three-dimensional lower-limb model

This paper presents a set of polynomial expressions that can be used as regression equations to estimate length and three-dimensional moment arms of 43 lower-limb musculotendon actuators. These equations allow one to find, at a low computational cost, the musculotendon geometric parameters required for numerical simulation of large musculoskeletal models. Nominal values for these biomechanical parameters were established using a public-domain musculoskeletal model of the lower limb (IEEE Trans. Biomed. Eng. 37 (1990) 757). To fit these nominal values, regression equations with different levels of complexity were generated, based on the number of generalized coordinates of the joints spanned by each musculotendon actuator. Least squares fitting was used to identify regression equation coefficients. The goodness of the fit and confidence intervals were assessed, and the best fitting equations selected.     

Data plotting of warfarin binding to human serum albumin

The binding of warfarin to human serum albumin was studied by equilibrium dialysis at pH 7.4 in a 67 mM sodium phosphate buffer at 37 degrees C. The equilibrium data were analysed using a computer program for curve fitting. The analysis was made fitting the data to equations for one, two and three classes of binding sites with one, two and three sites at the primary binding site (n(1)=1, 2 or 3). The data fitting was acceptable for two and three classes of binding sites but the best fit was obtained with the equation for two classes of binding sites, allowing us to define the binding by a model with two independent classes of binding sites on the serum albumin molecule.