Logarithmic transformations in ANOVA

A method is presented for choosing an additive constant c when transforming data x to y = log(x + c). The method preserves Type I error probability and power in ANOVA under the assumption that the x + c for some c are log-normally distributed. The method has advantages similar to those of rank transformations--namely, it is easy to use and is resistant to extreme observations. Since the special case c----infinity corresponds in ANOVA to y = x, the method is a useful generalization of least squares.     

Logarithmic transformation bias in allometry

It has been known for some time (DJ Finney, J. Roy. Stat. Soc. Suppl. 7:155–161, 1941) that transformation of an arithmetic data set to logarithms results in biased estimates when predicted values from a leastsquares regression are detransformed back to arithmetic units. Predicted values are estimates of the geometric mean of the dependent variable at that value of the independent variable, rather than the arithmetic mean. Since the geometric mean is always less than the arithmetic mean, detransformed predictions will underestimate the value in question. This bias affects the interpretations of allometric equations used for estimation, such as predicting fossil body mass from skeletal dimensions, and applications of allometry as a “criterion of subtraction,” in which residual variation is evaluated. A number of parametric and nonparametric corrections for transformation bias have been developed. Although this problem is relatively unexplored in mammalian morphometrics, it has received considerable attention in other disciplines that use power functions structurally identical to the allometric equation. Insights into transformation bias and the use of correction terms from economics, limnology, forestry, and hydrology are reviewed and interpreted for application to mammalian allometry. © 1993 Wiley-Liss, Inc.     

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.     

Logarithmic gap costs decrease alignment accuracy

Background  

Studies on the distribution of indel sizes have consistently found that they obey a power law. This finding has lead several scientists to propose that logarithmic gap costs, G (k) = a + c ln k, are more biologically realistic than affine gap costs, G (k) = a + bk, for sequence alignment. Since quick and efficient affine costs are currently the most popular way to globally align sequences, the goal of this paper is to determine whether logarithmic gap costs improve alignment accuracy significantly enough the merit their use over the faster affine gap costs.     

Sporulation of Yeast Harvested During Logarithmic Growth

Rapid and abundant sporulation of yeast can be obtained, with cells harvested during logarithmic growth, by employing potassium acetate rather than glucose as a carbon source.     

Active transport membrane concentration amplifiers

Three different active transport membrane configurations are suggested for achieving concentration amplification in a circulating substrate solution. The steady-state characteristics of the structures are investigated on the basis of a linear approximation of the rate of active transport. An analysis of the effects of system parameters and geometry on the concentration gradient along the flow path is presented. It is seen that the concentration gradient may be synthesized by appropriate choice of membrane arrangement, flow and physico-chemical transport parameters.     

Logarithmic timebase for stopped-flow data acquisition and analysis

A method for capturing stopped-flow and other rapid reaction records in logarithmic time, using a microcomputer, is described. Apart from the ability to record processes over several decades in time in a single experiment, the method shows distinct advantages in subsequent nonlinear regression analysis of multiexponential processes. The method is illustrated by a study of the binding of NADPH to dihydrofolate reductase and the reaction between formycin triphosphate and heavy meromyosin.     

Quantitative analysis of four EMG amplifiers

Four typical EMG amplifiers were tested quantitatively to observe the diversity and specificity of available equipment. Gain, phase, common mode rejection ratio (CMRR) and noise characteristics were measured for each device. Various gain and phase responses were observed, each best suited to specific application areas. For all amplifiers, the CMRR was shown to decrease dramatically in the presence of input impedance mismatches of more than 10 kΩ between the two electrodes. Because such impedance mismatches are common on the skin surface, these results indicate that proper skin preparation is required to maximize the noise rejection capabilities of the tested amplifiers.     

Logarithmic Sensing in Escherichia coli Bacterial Chemotaxis

We studied the response of swimming Escherichia coli (E. coli) bacteria in a comprehensive set of well-controlled chemical concentration gradients using a newly developed microfluidic device and cell tracking imaging technique. In parallel, we carried out a multi-scale theoretical modeling of bacterial chemotaxis taking into account the relevant internal signaling pathway dynamics, and predicted bacterial chemotactic responses at the cellular level. By measuring the E. coli cell density profiles across the microfluidic channel at various spatial gradients of ligand concentration grad[L] and the average ligand concentration near the peak chemotactic response region, we demonstrated unambiguously in both experiments and model simulation that the mean chemotactic drift velocity of E. coli cells increased monotonically with grad [L]/ or ∼grad(log[L])—that is E. coli cells sense the spatial gradient of the logarithmic ligand concentration. The exact range of the log-sensing regime was determined. The agreements between the experiments and the multi-scale model simulation verify the validity of the theoretical model, and revealed that the key microscopic mechanism for logarithmic sensing in bacterial chemotaxis is the adaptation kinetics, in contrast to explanations based directly on ligand occupancy.     

Logarithmic Relationship between Moisture Content and Longevity in Sesame Seeds

The relationship between seed moisture content and seed longevityin sesame (Sesamum indicum L.) in hermetic storage at 50 °Cis logarithmic. The logarithmic relationship is maintained from15 per cent down to 2 per cent – the lowest moisture contenttested — but above 15 per cent this ‘air-dry’relationship no longer holds since further increase in seedmoisture content does not reduce longevity. Tentative estimatesof constant values for the improved seed viability equationare provided, and implications for long-term storage are discussed. Sesame, Sesamum indicum L., seed storage, improved viability equation, seed moisture content, seed longevity prediction     

Logarithmic attachment to a monochromator for recording small changes in absorption

The paper describes a device consisting of a spectrophotometer (SP-4, SP-4A, SP-16), logarithmic amplifier, recording instrument and a flow-type cuvette that allows quantitative registration of biological substances during their separation on chromatographic columns.     

Chloro-oxime derivatives as novel small molecule chaperone amplifiers

Chloro-oxime derivatives were investigated as novel small molecule chaperone amplifiers. Lead optimization led to the discovery of compounds that displayed potent HSF1 activation activity, significant cytoprotection in MG-132 stress, ER stress and PolyQ stress cell models (EC₅₀ < 10 μM).     

Logarithmic expansion of LGR5+ cells in human colorectal cancer

Stem cells of the small and large intestine are marked by expression of the Wnt target gene LGR5, a leucine-rich-repeat-containing G protein-coupled receptor. Previous studies reported increased expression of LGR5 in human colorectal cancer (CRC) compared to normal tissue either by immunohistochemistry or in situ hybridization (ISH). However, as these studies were semi-quantitative they did not provide a numerical estimate of the magnitude of this effect. While we confirm that LGR5⁺ cells are exclusively located at the base of normal human small and large intestinal crypts, representing approximately 6% of total crypt cells, we show this cell population is 10-fold expanded in all grades of CRC, representing as much as 70% of the cells of tumor crypt-like structures. This expansion of the LGR5 compartment coincides with maintenance of crypt-like glandular structure (adenomas, and well and moderately differentiated adenocarcinomas), and is reduced in poorly differentiated CRC, where crypt-like glandular architecture is lost, accompanied by reduced epithelial terminal differentiation. Altogether these results indicate that LGR5⁺ cell expansion is a hallmark of CRC tumorigenesis occurring during progression to adenoma, supporting CRC as a stem cell disease with implications for CRC therapy.     

Logarithmic and Power Law Input-Output Relations in Sensory Systems with Fold-Change Detection

Two central biophysical laws describe sensory responses to input signals. One is a logarithmic relationship between input and output, and the other is a power law relationship. These laws are sometimes called the Weber-Fechner law and the Stevens power law, respectively. The two laws are found in a wide variety of human sensory systems including hearing, vision, taste, and weight perception; they also occur in the responses of cells to stimuli. However the mechanistic origin of these laws is not fully understood. To address this, we consider a class of biological circuits exhibiting a property called fold-change detection (FCD). In these circuits the response dynamics depend only on the relative change in input signal and not its absolute level, a property which applies to many physiological and cellular sensory systems. We show analytically that by changing a single parameter in the FCD circuits, both logarithmic and power-law relationships emerge; these laws are modified versions of the Weber-Fechner and Stevens laws. The parameter that determines which law is found is the steepness (effective Hill coefficient) of the effect of the internal variable on the output. This finding applies to major circuit architectures found in biological systems, including the incoherent feed-forward loop and nonlinear integral feedback loops. Therefore, if one measures the response to different fold changes in input signal and observes a logarithmic or power law, the present theory can be used to rule out certain FCD mechanisms, and to predict their cooperativity parameter. We demonstrate this approach using data from eukaryotic chemotaxis signaling.