Determination of linear fluorescence intensities from flow cytometric data accumulated with logarithmic amplifiers

Logarithmic amplifiers are useful in accumulating flow cytometric data with a large dynamic range. However, quantitative comparison of fluorescence intensities for different samples or different subpopulations within a sample is simplified by the conversion of data from log space back to linear space. A method is described in which fluorescent polystyrene spheres of differing intensities are used to construct a calibration curve for the logarithmic intensity scale. This allows calculation of relative linear intensity for each channel of the logarithmically accumulated data and determination of linear fluorescence means and coefficients of variation for comparative purposes. Fluorescent spheres of appropriate intensity may also be used as internal standards to monitor instrument and/or stain stability for samples accumulated using logarithmic amplifiers.     

Logarithmic display of auditory evoked potentials

Auditory evoked potentials (AEP) can be simultaneously recorded on-line as a succession of 11 waves, through a single input channel of a mini-computer. Since the response waves differ widely in frequency, a computing routine has been developed to display the whole response pattern in a single picture. Based upon a non-linear samples reduction of the digitized response, this routine allows a logarithmic transformation of the time axis. The method improves the identification of the AEP components and provides an objective estimate of the central auditory pathway for both neurophysiological and neuroclinical studies.     

Hyperlog-a flexible log-like transform for negative, zero, and positive valued data.

BACKGROUND: The remarkable success of cytometry over the past 30 years is largely due to its uncanny ability to display populations that vastly differ in numbers and fluorescence intensities on one scale. The log transform implemented in hardware as a log amplifier or in software normalizes signals or channels so that these populations appear as clearly discernible peaks. With the advent of multiple fluorescence cytometry, spectral crossover compensation of these signals has been necessary to properly interpret the data. Unfortunately, because compensation is a subtractive process, it can produce negative and zero valued data. The log transform is undefined for these values and, as a result, forces computer algorithms to truncate these values, creating a few problems for cytometrists. Data truncation biases displays making properly compensated data appear undercompensated; thus, enticing many operators to overcompensate their data. Also, events truncated into the first histogram channel are not normally visible with typical two-dimensional graphic displays, thus hiding a large number of events and obscuring the true proportionality of negative distributions. In addition, the log transform creates unequal binning that can dramatically distort negative population distributions. METHODS AND RESULTS: The HyperLog transform is a log-like transform that admits negative, zero, and positive values. The transform is a hybrid type of transform specifically designed for compensated data. One of its parameters allows it to smoothly transition from a logarithmic to linear type of transform that is ideal for compensated data. CONCLUSIONS: The HyperLog transform is easily implemented in computer systems and results in display systems that present compensated data in an unbiased manner.     

Application of theoretical considerations to the analysis of ELISA data

Solid-phase immunoassays such as the ELISA are in routine use in many areas of biological research. Data from these assays are analyzed in a variety of ways, frequently without taking into account the immunochemical principles of the assay. The Reference Standard Method is often used and is suitable and convenient for obtaining concentration (or activity) values from the antigen-specific ELISA or spRIA, sandwich assays, and inhibition assays. The standard curve required for this method may be obtained by simple linear regression analysis of logarithmic or logitlogarithmic transformed data obtained from titration of the reference standard. The shape of the logarithmic plot of the reference standard provides information on the performance of the assay. Examining data from multiple dilutions of the samples is essential to assure that each titrates with the same slope as does the reference standard; the analysis routine must permit this comparison to be made. ELISANALYSIS is a program for the IBM PC which was developed to perform such analyses. It is presented here as a model, with sufficient information provided for the development of similar analytical routines by interested users. This approach to ELISA data analysis is presented as an alternative to complicated empirical curve-fitting systems and simple endpoint methods, which can be immunochemically misleading or, in some cases, even invalid. The consistent use of the described routines would encourage greater uniformity in the means of data interpretation and thereby enhance our understanding of immunobiology.     

Understanding reproductive allometry in turtles: A slippery “slope”

Measures of reproductive output in turtles are generally positively correlated with female body size. However, a full understanding of reproductive allometry in turtles requires logarithmic transformation of reproductive and body size variables prior to regression analyses. This allows for slope comparisons with expected linear or cubic relationships for linear to linear and linear to volumetric variables, respectively. We compiled scaling data using this approach from published and unpublished turtle studies (46 populations of 25 species from eight families) to quantify patterns among taxa. Our results suggest that for log–log comparisons of clutch size, egg width, egg mass, clutch mass, and pelvic aperture width to shell length, all scale hypoallometrically despite theoretical predictions of isometry. Clutch size generally scaled at ~1.7 to 2.0 (compared to an isometric expectation of 3.0), egg width at ~0.5 (compared to an expectation of 1.0), egg mass at ~1.1 to 1.3 (3.0), clutch mass at ~2.5 to 2.8 (3.0), and pelvic aperture width at 0.8–0.9 (1.0). We also found preliminary evidence that scaling may differ across years and clutches even in the same population, as well as across populations of the same species. Future investigators should aspire to collect data on all these reproductive parameters and to report log–log allometric analyses to test our preliminary conclusions regarding reproductive allometry in turtles.     

Enhanced Vitreous Imaging in Healthy Eyes Using Swept Source Optical Coherence Tomography

Purpose

To describe enhanced vitreous imaging for visualization of anatomic features and microstructures within the posterior vitreous and vitreoretinal interface in healthy eyes using swept-source optical coherence tomography (SS-OCT). The study hypothesis was that long-wavelength, high-speed, volumetric SS-OCT with software registration motion correction and vitreous window display or high-dynamic-range (HDR) display improves detection sensitivity of posterior vitreous and vitreoretinal features compared to standard OCT logarithmic scale display.

Design

Observational prospective cross-sectional study.

Methods

Multiple wide-field three-dimensional SS-OCT scans (500×500A-scans over 12×12 mm²) were obtained using a prototype instrument in 22 eyes of 22 healthy volunteers. A registration motion-correction algorithm was applied to compensate motion and generate a single volumetric dataset. Each volumetric dataset was displayed in three forms: (1) standard logarithmic scale display, enhanced vitreous imaging using (2) vitreous window display and (3) HDR display. Each dataset was reviewed independently by three readers to identify features of the posterior vitreous and vitreoretinal interface. Detection sensitivities for these features were measured for each display method.

Results

Features observed included the bursa premacularis (BPM), area of Martegiani, Cloquet''s/BPM septum, Bergmeister papilla, posterior cortical vitreous (hyaloid) detachment, papillomacular hyaloid detachment, hyaloid attachment to retinal vessel(s), and granular opacities within vitreous cortex, Cloquet''s canal, and BPM. The detection sensitivity for these features was 75.0% (95%CI: 67.8%–81.1%) using standard logarithmic scale display, 80.6% (95%CI: 73.8%–86.0%) using HDR display, and 91.9% (95%CI: 86.6%–95.2%) using vitreous window display.

Conclusions

SS-OCT provides non-invasive, volumetric and measurable in vivo visualization of the anatomic microstructural features of the posterior vitreous and vitreoretinal interface. The vitreous window display provides the highest sensitivity for posterior vitreous and vitreoretinal interface analysis when compared to HDR and standard OCT logarithmic scale display. Enhanced vitreous imaging with SS-OCT may help assess the natural history and treatment response in vitreoretinal interface diseases.     

Measurement scale in maximum entropy models of species abundance

The consistency of the species abundance distribution across diverse communities has attracted widespread attention. In this paper, I argue that the consistency of pattern arises because diverse ecological mechanisms share a common symmetry with regard to measurement scale. By symmetry, I mean that different ecological processes preserve the same measure of information and lose all other information in the aggregation of various perturbations. I frame these explanations of symmetry, measurement, and aggregation in terms of a recently developed extension to the theory of maximum entropy. I show that the natural measurement scale for the species abundance distribution is log-linear: the information in observations at small population sizes scales logarithmically and, as population size increases, the scaling of information grades from logarithmic to linear. Such log-linear scaling leads naturally to a gamma distribution for species abundance, which matches well with the observed patterns. Much of the variation between samples can be explained by the magnitude at which the measurement scale grades from logarithmic to linear. This measurement approach can be applied to the similar problem of allelic diversity in population genetics and to a wide variety of other patterns in biology.     

Data transformations for improved display and fitting of single-channel dwell time histograms.

A.L. Blatz and K.L. Magleby (1986a. J. Physiol. [Lond.]. 378:141-174) have demonstrated the usefulness of plotting histograms with a logarithmic time axis to display the distributions of dwell times from recordings of single ionic channels. We derive here the probability density function (pdf) corresponding to logarithmically binned histograms. Plotted on a logarithmic time scale the pdf is a peaked function with an invariant width; this and other properties of the pdf, coupled with a variance-stabilizing (square root) transformation for the ordinate, greatly simplify the interpretation and manual fitting of distributions containing multiple exponential components. We have also examined the statistical errors in estimation, by the maximum-likelihood method, of kinetic parameters from logarithmically binned data. Using binned data greatly accelerates the fitting procedure and introduces significant errors only for bins spaced more widely than 8-16 per decade.     

Evaluation of an Automated Colony Counter

An automated colony counter was found to readily detect surface and subsurface bacterial colonies of 0.3-mm size or greater with a high degree of precision. On a logarithmic scale, counting efficiency consistently ranged from 89 to 95% of corresponding manual count determinations for plates containing up to 1,000 colonies. In routine application, however, automated plate counts up to approximately 400 colonies were selected as a more practical range for operation. The automated counter was easily interfaced with an automated data acquisition system.     

On rethinking allometry

Analysis of sets of intra-species and inter-species allometric relationships reveals that the inter-species data generally fit an exponential model better than a linear model. The intra-species data seem equally suited to either model. Skewness of the data and the effect of logarithmic transformations on correlation coefficients are examined in the light of these findings. Inter-species data are approximately lognormally distributed and logarithmic transformations are necessary to produce linear relationships. As a consequence, correlation coefficients usually increase after logarithmic transformation of inter-species data.     

Software sensor design considering oscillating conditions as present in industrial scale fed‐batch cultivations

Investigations of inhomogeneous dynamics in industrial‐scale bioreactors can be realized in laboratory simulators. Such studies will be improved by on line observation of the growth of microorganisms and their product synthesis at oscillating substrate availability which represents the conditions in industrial‐scale fed‐batch cultivations. A method for the kinetic monitoring of such processes, supported by on line measurements accessible in industrial practice, is proposed. It consists of a software sensor (SS) system composed of a cascade structure. Process kinetics are simulated in models with a structure including time‐varying yield coefficients. SSs for measured variable kinetics have classical structures. The SS design of unmeasured variables is realized after a linear transformation using a logarithmic function. For these software sensors, a tuning procedure is proposed, at which an arbitrary choice of one tuning parameter value that guarantees stability of the monitoring system allows the calculation of the optimal values of six parameters. The effectiveness of the proposed monitoring approach is demonstrated with experimental data from a glucose‐limited fed‐batch process of Bacillus subtilis in a laboratory two‐compartment scale down reactor which tries to mimic the conditions present in industrial scale nutrient‐limited fed‐batch cultivations. Biotechnol. Bioeng. 2013; 110: 1945–1955. © 2013 Wiley Periodicals, Inc.     

Gamma generalized linear models for pharmacokinetic data

Summary .   This article considers the modeling of single-dose pharmacokinetic data. Traditionally, so-called compartmental models have been used to analyze such data. Unfortunately, the mean function of such models are sums of exponentials for which inference and computation may not be straightforward. We present an alternative to these models based on generalized linear models, for which desirable statistical properties exist, with a logarithmic link and gamma distribution. The latter has a constant coefficient of variation, which is often appropriate for pharmacokinetic data. Inference is convenient from either a likelihood or a Bayesian perspective. We consider models for both single and multiple individuals, the latter via generalized linear mixed models. For single individuals, Bayesian computation may be carried out with recourse to simulation. We describe a rejection algorithm that, unlike Markov chain Monte Carlo, produces independent samples from the posterior and allows straightforward calculation of Bayes factors for model comparison. We also illustrate how prior distributions may be specified in terms of model-free pharmacokinetic parameters of interest. The methods are applied to data from 12 individuals following administration of the antiasthmatic agent theophylline.     

Univariate Dose-Response Models in Case-Control Studies

For modelling dose-response relationships in case-control studies the multiplicative logistic regression model, assuming the relative risk to be an exponential function of the dose, is widely known. If the relative risk is assumed to be a linear function of the dose, several authors (see e.g. BERRY (1980)) have proposed an additive (linear) model. This model has a better fit with the data if such a linear relation holds. Confidence limits for the relative risk derived from the information matrix, however, appear to be rather inaccurate. Therefore, use of the ‘standard’ logistic model in two different ways was studied: extension with a quadratic term or a logarithmic transformation of the dose. By applying the methods both to an empirical data set and in a simulation experiment, it is shown that appropriate transformation (often logarithmic) of the dosage and then applying the ‘standard’ logistic model is an useful approach if a linear dose-response relationship holds.     

The metabolic rates associated with resting, and with the performance of agonistic, submissive and digging behaviours in the cichlid fish Neolamprologus pulcher (Pisces: Cichlidae)

We measured the metabolic rates as a direct estimate of energy expenditure of individual Neolamprologus pulcher, a cooperatively breeding cichlid fish, when resting and when performing agonistic, submissive or digging behaviours in a respirometer. Standard and routine metabolic rates increased linearly with body mass (range 0.9–8.4 g) when plotted on a doubly logarithmic scale (linear regression equations: standard metabolic rate: log individual oxygen consumption rate = 0.65 + 0.86 log body mass; routine metabolic rate: log individual oxygen consumption rate = 0.75 + 0.86 log body mass). Routine metabolic rates were, on average, 30% higher than standard metabolic rates. Submissive and agonistic behaviours raised routine metabolic rates by factors of 3.3 and 3.9, respectively. Digging resulted in a 6.1-fold increase of routine metabolic rates. Differences in metabolic rates between active and resting rates were statistically significant. However, those between the three behaviours were not. Mean opercular beat frequencies correlated significantly with routine metabolic rates and with metabolic rates when performing specific behaviours, which offers methodological prospects for field measurements. In N. pulcher, the high energy expenditure for submissive behaviour may indicate that this is a reliable signal. The considerable energy expenditure involved in territory defence suggests that these costs should be considered in addition to risk in cost-benefit analyses. This is the first study in which the energy expenditures of specific social and territory maintenance behaviours of individual fish were measured directly by respirometry and within the usual social setting of the fish. Accepted: 20 February 1998     

Non-linear analysis of GeneChip arrays

The application of microarray hybridization theory to Affymetrix GeneChip data has been a recent focus for data analysts. It has been shown that the hyperbolic Langmuir isotherm captures the shape of the signal response to concentration of Affymetrix GeneChips. We demonstrate that existing linear fit methods for extracting gene expression measures are not well adapted for the effect of saturation resulting from surface adsorption processes. In contrast to the most popular methods, we fit background and concentration parameters within a single global fitting routine instead of estimating the background before obtaining gene expression measures. We describe a non-linear multi-chip model of the perfect match signal that effectively allows for the separation of specific and non-specific components of the microarray signal and avoids saturation bias in the high-intensity range. Multimodel inference, incorporated within the fitting routine, allows a quantitative selection of the model that best describes the observed data. The performance of this method is evaluated on publicly available datasets, and comparisons to popular algorithms are presented.     

Do rats represent time logarithmically or linearly?

This study examined two possible psychophysical time scales, a logarithmic representation of time with constant variability and a linear representation of time with scalar variability. Twenty-four rats were tested on a modified temporal bisection procedure in which multiple responses could occur in a 10-s time window after the termination of a stimulus. The number of “long” or “short” responses was used as a measure of the rats’ certainty of the duration of the presented interval. Data were analyzed with signal detection theory, and straight lines were fit to the zROC curves with the assumptions of a logarithmic representation of time with constant variability and a linear representation of time with scalar variability. The logarithmic representation of time with constant variability provided a better fit to the data than the linear representation of time with scalar variability.     

Proteomics in hematologic malignancies

Basic science research in hematology has been determining the functions of gene products using classical approaches that typically involve studying one or a few genes at a time. Proteomics, defined as the study of protein properties on a large scale, provides tools to globally analyze malignant hematologic cells. A major challenge in cancer therapy is the identification of drugs that kill tumor cells while preserving normal cells. Differential display via proteomics enables analysis of direct as well as side-effects of drugs at a molecular level. Proteomics also allows a better understanding of cell signaling pathways involved during apoptosis in hematologic cells. Storing the information in a 2D electrophoresis database enhances the efficiency of proteome research on malignant cells. Finally, the work needed to be carried out on proteomic analysis prior to routine clinical adoption is discussed, and the necessity for multi-institutional collaborations is emphasized.     

Proteomics in hematologic malignancies

Basic science research in hematology has been determining the functions of gene products using classical approaches that typically involve studying one or a few genes at a time. Proteomics, defined as the study of protein properties on a large scale, provides tools to globally analyze malignant hematologic cells. A major challenge in cancer therapy is the identification of drugs that kill tumor cells while preserving normal cells. Differential display via proteomics enables analysis of direct as well as side-effects of drugs at a molecular level. Proteomics also allows a better understanding of cell signaling pathways involved during apoptosis in hematologic cells. Storing the information in a 2D electrophoresis database enhances the efficiency of proteome research on malignant cells. Finally, the work needed to be carried out on proteomic analysis prior to routine clinical adoption is discussed, and the necessity for multi-institutional collaborations is emphasized.