Eosin fluorescence changes during Rb+ occlusion in the Na+/K(+)-ATPase

We used suspensions of partially purified Na(+)/K(+)-ATPase from pig kidney to compare the effects of Rb(+), as a K(+) congener, on the time course and on the equilibrium values of eosin fluorescence and of Rb(+) occlusion. Both sets of data were collected under identical conditions in the same enzyme preparations. The incubation media lacked ATP so that all changes led to an equilibrium distribution between enzyme conformers with and without bound eosin and with and without bound or occluded Rb(+). Results showed that as Rb(+) concentration was increased, the equilibrium value of fluorescence decreased and occlusion increased along rectangular hyperbolas with similar half-maximal values. The time courses of attainment of equilibrium showed an initial phase which was so quick as to fall below the time resolution of our rapid-mixing apparatus. This phase was followed by the sum of at least two exponential functions of time. In the case of fluorescence the fast exponential term accounted for a larger fraction of the time course than in the case of occlusion. Comparison between experimental and simulated results suggests that fluorescence changes express a process that is coupled to Rb(+) occlusion but that is completed before occlusion reaches equilibrium.     

An evaluation of fitting methods for the sum of two hyperbolas: application to uptake studies

Analysis of data in terms of the sum of two rectangular hyperbolas is frequently required in solute uptake studies. Four methods for such analysis have been compared. Three are based on least-squares fitting whereas the fourth (partition method I) is an extension of a single hyperbola fitting procedure based on non-parametric statistics. The four methods were tested using data sets which had been generated with two primary types of random, normal error in the dependent variable: one of constant error variance and the other of constant coefficient of variation. The methods were tested on further data sets which were obtained by incorporating single 10% bias errors at different positions in the original two sets. Partition method I consistently gave good estimates for the four parameters defining the double hyperbola and was highly insensitive to the bias errors. The least-squares procedures performed well under conditions satisfying the least-squares assumptions regarding error distribution, but frequently gave poor estimates when these assumptions did not hold. Our conclusion is that in view of the errors inherent in many solute uptake experiments it would usually be preferable to analyse data by a method such as partition method I rather than to rely on a least-squares procedure.     

A general multivariate approach to linear modeling in human genetics.

The general linear structural equation model is applied to problems in human genetics where there may be more than one measured phenotype per individual. A modeling convention, termed conditional associations, is developed to extend the general linear model so that it can handle the unique problems in human genetic models that arise from the pairing up of individuals or families under assortment between mates and the assortative placement of adoptees. Formulas are presented to generate expected covariance matrices for assortment or assortative placement on many variables simultaneously. It is demonstrated that all linear models in human genetics can be reduced in form to two fundamental equations. An algorithm is presented that will allow the application of these two equations to linear modeling in human genetics.     

Calculating thermodynamic data for transitions of any molecularity from equilibrium melting curves

In this paper, we derive the general forms of the equations required to extract thermodynamic data from equilibrium transition curves on oligomeric and polymeric nucleic acids of any molecularity. Significantly, since the equations and protocols are general, they also can be used to characterize thermodynamically equilibrium processes in systems other than nucleic acids. We briefly review how the reduced forms of the general equations have been used by many investigators to evaluate mono- and bimolecular transitions, and then explain how these equations can be generalized to calculate thermodynamic parameters from common experimental observables for transitions of higher molecularities. We emphasize the strengths and weaknesses of each method of data analysis so that investigators can select the approach most appropriate for their experimental circumstances. We also describe how to analyze calorimetric heat capacity curves and noncalorimetric differentiated melting curves so as to extract both model-independent and model-dependent thermodynamic data for transitions of any molecularity. The general equations and methods of analysis described in this paper should be of particular interest to laboratories that currently are investigating association and dissociation processes in nucleic acids that exhibit molecularities greater than two.     

Generalized finite element solution to one-dimensional flux problems

A finite element numerical solution to the general one-dimensional flow equation is derived in a form that provides a convenient and general means to simulate a wide variety of one-dimensional flow techniques of interest to biological scientists, e.g., ultracentrifugation, electrophoresis, chromatography, etc. Diverse physical models defined in terms of column geometry, solute interactions, and the dependence of transport parameters on column position, time, or concentrations of one or more solutes, can be accommodated. A particularly useful aspect of the formulation is that a wide variety of boundary conditions can be simply applied to the end result, without rederivation of the solution for each new case. The numerical solution is expressed as matrix equations that are sufficiently general so that incorporation of particular models can be effected by substitution of appropriate quantities into the final result.     

Transglucosidic reactions of the Aspergillus niger family 3 beta-glucosidase: qualitative and quantitative analyses and evidence that the transglucosidic rate is independent of pH

The hydrolytic and transglucosidic reactions of the Aspergillus niger Family 3 beta-glucosidase were characterized. Michaelis-Menten plots of the rates of aglycone formation were normal (hyperbolic) at low [substrate]. However, at high [substrate] the rates decreased at pH below approximately 5.5 but increased at pH above approximately 5.5. Each decrease or increase took the form of a second hyperbola adjoining the first. Thin layer chromatography, gas-liquid chromatography, and NMR analyses indicated that the substrates became transglucosidic acceptors when present at high concentrations. When pNPGlc and cellobiose reacted as acceptors, the C6 hydroxyl of the non-reducing substrate component reacted to form beta-D-glucopyranosyl-(1-6)-beta-D-glucopyranosyl-p-nitrophenol and beta-D-glucopyranosyl-(1-6)-beta-D-glucopyranosyl-(1-4)-D-glucopyranose, respectively. The acceptor action accounted for the second adjoining hyperbolas. Rate equations were derived for the production of the aglycone and the transglucosidic intermediate, and these equations described the data very well. Hydrolytic Vmax {Vmax(h)}, hydrolytic Km {Km(h)}, transglucosidic Vmax {Vmax(t)}, and transglucosidic Km {Km(t)} values were obtained by non-linear regression analysis using these equations. Vmax(h) pH profiles were bell shaped with optima between pH 4 and 4.5 but the Vmax(t) values did not change substantially between pH 3 and 7. These differences in the pH profiles explain the decreasing and increasing adjoining hyperbolas since Vmax(t) is lower than Vmax(h) at pH less than approximately 5.5 but higher than Vmax(h) at pH greater than approximately 5.5. The reason for these pH effects is that the value of the hydrolytic rate constant (k3) decreases while the value of the transglucosidic rate constant (k4) does not change between pH 3 and 7. The study also showed that gentiobiose forms by an intermolecular reaction of the C6 hydroxyl of Glc rather than an intramolecular reaction and that an equatorial orientation of the C2 hydroxyl, the presence of a C6 primary hydroxyl and beta-linkages with oligosaccharide acceptors are important for acceptor reactivity.     

A mathematical contribution to the study of orientation of organisms

Organisms orient themselves to a stimulus by two general methods. One method is by directed orientation (taxis); the other is by undirected locomotory reaction (kinesis). An equation, and the methods for finding the necessary parameters of this equation, is derived for the distribution of organisms within a container, with the following limitations: (1) the organisms have no accommodation, (2) they are always active, and (3) the stimulus changes slowly with position. Necessary modifications of the equation are then derived, so that the last two limitations may be eliminated. The equation cannot be solved excatly because of its complexity; hence an approximation method must be used. This method is discussed, an approximate solution is found, and a time constant for equilibrium to be established is derived. Applications tovarious experiments in the literature are then made with fairly satisfactory results. A new interpretation of the theory of klino-kinesis with accommodation is found upon application of the equations developed to experimental work. Further limitations and uses of these equations are then discussed. This work was done while the author was Public Health Service Research Fellow of The National Institute of Mental Halth, Federal Security Agency.     

Modelling the maximum voluntary joint torque/angular velocity relationship in human movement

The force exerted by a muscle is a function of the activation level and the maximum (tetanic) muscle force. In "maximum" voluntary knee extensions muscle activation is lower for eccentric muscle velocities than for concentric velocities. The aim of this study was to model this "differential activation" in order to calculate the maximum voluntary knee extensor torque as a function of knee angular velocity. Torque data were collected on two subjects during maximal eccentric-concentric knee extensions using an isovelocity dynamometer with crank angular velocities ranging from 50 to 450 degrees s(-1). The theoretical tetanic torque/angular velocity relationship was modelled using a four parameter function comprising two rectangular hyperbolas while the activation/angular velocity relationship was modelled using a three parameter function that rose from submaximal activation for eccentric velocities to full activation for high concentric velocities. The product of these two functions gave a seven parameter function which was fitted to the joint torque/angular velocity data, giving unbiased root mean square differences of 1.9% and 3.3% of the maximum torques achieved. Differential activation accounts for the non-hyperbolic behaviour of the torque/angular velocity data for low concentric velocities. The maximum voluntary knee extensor torque that can be exerted may be modelled accurately as the product of functions defining the maximum torque and the maximum voluntary activation level. Failure to include differential activation considerations when modelling maximal movements will lead to errors in the estimation of joint torque in the eccentric phase and low velocity concentric phase.     

Solitary wave and spatially locked solitary pattern in a chemical reaction system

We investigate the behavior of a one-dimensional two component dynamical system. The dynamical equations are obtained by extracting an essence out of equations which describe the behavior of a biochemical reaction catalyzed by an allosteric protein. The obtained dynamical equations are similar to van der Pol equations. The dynamical equations are solved numerically. In the continuous system, a solitary wave is found to occur in certain ranges of the parameter space. The condition of occurrence of the solitary wave is investigated. The solitary wave can be induced by various initial perturbations, including rectangular ones with space-wise length longer than a certain critical value. The property of the solitary wave is similar to that of the impulses in nervous systems. In the discrete system, a spatially locked solitary pattern is found to occur in certain ranges of the parameter space.     

Use of the biological stimulus in determining parameters of drug action, and its relationship to the drug effect: a contribution to the theory of drug action

Drug action is considered a function of the biological stimulus, which in turn is proportional to the receptor occupancy; but the nature of the functional relationship is unknown, as is also the magnitude of the drug-receptor affinity and of the efficacy (the proportionality constant of the stimulus-occupancy relationship). Methods exist to determine the receptor affinity of (a) a drug of high efficacy provided it is possible to block part of the receptor irreversibly, or (b) a drug of low efficacy provided it can be observed together with a drug of high efficacy acting on the same receptor. The former method is generally accepted, the latter is shown to rest on a questionable simplification. An improvement of method (b) is presented which avoids all simplifying assumptions: making use of the affinity constant of the high-efficacy drug determined by method (a), and of the thesis that identical biological stimuli correspond to identical effects, it is shown that the equations of theoretical pharmacology yield the affinity constant of the low-efficacy drug. A more general version of the method allows to calculate the affinity constants of any two drugs acting on the same receptor, without previous knowledge of anything but the dose-response curves of the two drugs. In both cases, the relative efficacies of the two drugs can also be calculated and so can the stimuli corresponding to each drug concentration, so that it is possible to find empirically the nature of the functional relationship between the calculated stimulus and the observed drug effect.     

Generalization of the Monod-Wyman-Changeux model for the case of multisubstrate reactions]

A general equation is derived for the rate of multisubstrate reaction catalyzed by oligomeric enzyme E(R, T) liable to concerted transitions Ro in equilibrium To or Ro in equilibrium 2To. It is shown that with some assumptions about the enzymes the rate equations can be constructed from the rates of corresponding reactions catalyzed by a single active site. These single active site rate equations are known for the majority of catalysis mechanisms, otherwise they can be easily deduced. As an example the rate equation is derived for the reaction S1 + S2 + S3 in equilibrium S4 + S5 catalyzed by an oligomeric enzyme according to the ordered ter-bi mechanism.     

Erratum to: Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.     

Observations on programs to estimate the parameters of enzyme kinetics

This paper gives a critical account of the two major programs so far published specifically to estimate the parameters of enzyme kinetics. The dangers of submitting data to the programs without proper checks are discussed, and a screening test is described to identifty sets of data which may not be best fitted by a rectangular hyperbola.     

Evolution in response to social selection: the importance of interactive effects of traits on fitness

Social interactions have a powerful effect on the evolutionary process. Recent attempts to synthesize models of social selection with equations for indirect genetic effects (McGlothlin et al. 2010) provide a broad theoretical base from which to study selection and evolutionary response in the context of social interactions. However, this framework concludes that social selection will lead to evolution only if the traits carried by social partners are nonrandomly associated. I suggest this conclusion is incomplete, and that traits that do not covary between social partners can nevertheless lead to evolution via interactive effects on fitness. Such effects occur when there are functional interactions between traits, and as an example I use the interplay in water striders (Gerridae) between grasping appendages carried by males and spines by females. Functional interactive effects between traits can be incorporated into both the equations for social selection and the general model of social evolution proposed by McGlothlin et al. These expanded equations would accommodate adaptive coevolution in social interactions, integrate the quantitative genetic approach to social evolution with game theoretical approaches, and stimulate some new questions about the process of social evolution.     

A confidence interval for the unknown value of the regressor with given expectation of the regressand in model I of linear regression

In the case of model I of linear regression there is derived a confidence interval for that x_o where the “true line” will reach a given value y_o. The interval can be given by the intersections between the line y = y_o and the hyperbolas providing pointwise confidence intervals of the expectations of y.     

Photosynthesis in Stands of Green Peppers. An Application of Empirical and Mechanistic Models to Controlled-environment Data

The photosynthetic response of stands of green peppers to lightand CO² is examined by means of various mathematical models.Several single leaf response equations are considered, fromthe simple and ubiquitous rectangular hyperbola, to more sophisticatedforms incorporating photorespiration and the oxygen effect.By making the usual assumptions about stand structure and lightpenetration, mechanistic crop response equations arc derivedfor each of the different leaf response equations. First, asa purely empirical approach, the leaf response equations areapplied directly to crop data, and it is concluded that therectangular hyperbola with a term for dark respiration [eqn(2)] gives an adequate summary of crop response to light andCO₂. Second, four mechanistic equations of crop response areapplied to the data, and, although the results are equivocal,it is suggested that the simpler crop equations [eqns (15) and(17)] are satisfactory at the present time, and it is not yetpossible to detect the results of photorespiration and the oxygeneffect directly in the crop data.     

医用直线加速器矩形野散射因子计算方法研究

目的：医用直线加速器矩形野散射因子计算方法加以改进。方法：（1）实际测量矩形野的散射因子;（2）利用Kim的经验公式和面积-周长比公式分别计算理论上的矩形野的散射因子,并与矩形野的测量结果进行比较。结果：采用该方法计算医用加速器散射因子的最大误差在0.5%以下,而采用面积-周长比原理确定的等效方边长,最大误差达到3%左右。结论：采用Kim计算方法算出的等效边长,然后再根据矩形野测量出的数据计算等效射野的Scp结果比传统的面积周长比算法更能减小计算误差,并且精度也比较高。     

Hand breakaway strength model-effects of glove use and handle shapes on a person's hand strength to hold onto handles to prevent fall from elevation

This study developed biomechanical models for hand breakaway strength that account for not only grip force but also hand-handle frictional coupling in generation of breakaway strength. Specifically, models for predicting breakaway strength for two commonly-used handle shapes (circular and rectangular handles) and varying coefficients of friction (COF) between the hand and handle were proposed. The models predict that (i) breakaway strength increases with increasing COF and (ii) a circular handle with a 50.8 mm-diameter results in greater mean breakaway strength than a handle with a rectangular cross-section of 38.1 by 38.1 mm for COFs greater than 0.42. To test these model predictions, breakaway strengths of thirteen healthy young adults were measured for three frequently-encountered COF conditions (represented by three glove types of polyester (COF=0.32), bare hand (COF=0.50), and latex (COF=0.74) against an aluminum handle) and for the two handle shapes. Consistent with the model predictions, mean breakaway strength increased with increasing COF and was greater for the circular than rectangular handle for COFs of 0.50 and 0.74. Examination of breakaway strength normalized to body weight reveals that modification of COF and handle shapes could influence whether one can hold his/her body using the hands or not (thus must fall), highlighting the importance of considering these parameters for fall prevention. The biomechanical models developed herein have the potential to be applied to general handle shapes and COF conditions. These models can be used to optimize handle design to maximize breakaway strength and minimize injuries due to falls from ladders or scaffolds.     

Using Communication Tools to Facilitate Regulatory Review

Submittals under CERCLA, RCRA, and related environmental regulatory programs often require voluminous data and narrative, supported by multiple appendices of supplemental data. Often lost in the encyclopedic maze of documentation is the key message of the respondent. Also lost is the fact that every document, however data-driven or technical, has a message that should be readable and understandable to both the technical expert and the general reader alike. This is especially true for documents that will undergo public scrutiny at some phase. The regulatory review process can be facilitated if the respondents submitting the documentation present a clear and understandable message that is first understood by the respondents and subsequently by the intended audience. Technical communicators can play an important role in assuring that even a highly complex technical document filled with assumptions, qualifiers, analysis, and quantification retains a clear and understandable message. Simple tools that we may take for granted can be employed throughout the writing process. These tools can range from titles, outlines, and team message slogans to editorial style sheets and review checklists. These tools, used as message reminders, can assist members of a documentation team to stay en pointe, to keep the message in focus while the document is being prepared. Properly applied, these tools translate into a clear message for readers, despite the legitimate and real concerns of sophisticated analysis and scientific inquiry. A clear message, in turn, can expedite the regulatory review process by helping to minimize extraneous comments and excessive requests for clarification from regulatory agencies.