Exponential Growth Equation Research Articles

A quantitative method for the analysis of mammalian cell proliferation in culture in terms of dividing and non-dividing cells.

The application of the exponential growth equation is the standard method employed in the quantitative analyses of mammalian cell proliferation in culture. This method is based on the implicit assumption that, within a cell population under study, all division events give rise to daughter cells that always divide. When a cell population does not adhere to this assumption, use of the exponential growth equation leads to errors in the determination of both population doubling time and cell generation time. We have derived a more general growth equation that defines cell growth in terms of the dividing fraction of daughter cells. This equation can account for population growth kinetics that derive from the generation of both dividing and non-dividing cells. As such, it provides a sensitive method for detecting non-exponential division dynamics. In addition, this equation can be used to determine when it is appropriate to use the standard exponential growth equation for the estimation of doubling time and generation time.

Systematics in bifurcations of exponential growth equations with a non-linear feedback

Bifurcation phenomena may be examined in terms of the class of non-linear differential equations of the first kind in the form x ̇ (t) = α nβx(t) − {x(t)} n+1 , where α and β are constants and n is an integer. A transformation of variables is applied taking into account the initial values x(0) = x 0. The solutions are presented in the form of dimensionless variables and in polar coordinates to visualize limit cycles. The well known logistic and pitchfork equations ( n = 1 and 2, respectively) are special cases, as is also the equation for explosive population growth ( n = 1, α and β imaginary).

Simple density-dependent matrix model for population projection

A matrix model based on a discrete time form of the logistic equation and the Leslie matrix model was developed for density-dependent population growth; the model is simpler and more easily applied than the model developed by Liu and Cohen in 1987 using a different discrete time form of the logistic equation. The new model requires no additional parameters, matrices, or mathematical functions, and it links life tables, the exponential growth equation, the logistic growth equation, and Leslie's matrix. The new model possesses the same qualitative dynamical behavior as both the model developed by Liu and Cohen and the discrete time logistic equation; it exhibits stable points, cycles, and chaos. It easily can be modified to include features developed for the logistic equation such as time lags. The model was applied to describe the growth of a white-tailed deer population introduced into a fenced reserve.

Systematics in bifurcations of exponential growth equations with a non-linear feedback

Systematics in bifurcations of exponential growth equations with a non-linear feedback

Differentiation of S. gallinarum and S. pullorum by means of growth-kinetics analysis

Bacterial growth kinetics is considered as a means of differentiation of the two serologically identical serovars, S. gallinarum and S. pullorum. Growth-kinetic parameters were calculated by fitting growth data to the exponential growth equation, using linear-regression analysis. Bacterial populations of S. gallinarum, S. pullorum and S. newport in two media (nutrient broth and selenite broth) were continuously monitored by optical-density measurements. The experiments confirmed differences in the growth of S. gallinarum and S. pullorum when cultured in the selenite broth.

Analysis and modelling of the lambda p L promoter controlled expression of a cloned thermostable α-amylase gene in Escherichia coli during batch cultivation

The p L promoter controlled expression of a Bacillus stearothermophilus gene coding for thermostable α-amylase in Escherichia coli was analyzed. Methodology was developed for modelling and controlling bioprocesses in which a protein foreign for the microbe is produced under the control of an inducible promoter. Two-stage batch cultivation was studied, the first stage at 30°C with the gene expression repressed and the second at an elevated temperature (36–42°C) in order to induce the α-amylase production. A set of 28 cultivations was carried out according to a statistical experimental design in which the pH of the first stage, the optical density of culture at the onset of induction, the temperature, and the pH of the second stage were the selected control variables. The biosystem turned out to be unstable. However, both static and kinetic models were constructed and found applicable to describe the average process behaviour. The expression temperature and pH were the most important control variables. The models suggested that derepression of the p L promoter may be dependent not only on temperature but also on pH. In the kinetic process model the exponential growth equation was applied for the first stage. During the second stage the logistic law was employed for modelling the microbial growth. According to the model the α-amylase production rate was dependent on the cell concentration and the amount of enzyme produced. The experimental and mathematical techniques proved to be useful for analyzing and modelling the cultivation, thus leading to a better understanding of the process. The kinetic techniques employed offer a possibility to model a batch process with control profiles.

Phenylphosphonate: a 31P-NMR indicator of extracellular pH and volume in the isolated perfused rabbit bladder.

31P-NMR has been used extensively to estimate intracellular pH. It also can be used to measure extracellular pH and volume when an NMR-detectable extracellular phosphorous probe is used. Phosphonic acids have been suggested as useful 31P-NMR extracellular markers. The present study was designed to assess the utility of phenylphosphonic acid (PPA) as a 31P-NMR extracellular marker in perfused smooth muscle. Rabbit bladder strips were exposed to PPA concentrations of 1-20 mM. Tension development in response to maximal carbachol challenges (10 microM) was independent of PPA concentration. Addition of PPA (6 mM) to the perfusate supplying the isolated resting rabbit bladder had no effect on 31P-NMR-detectable phosphatic compounds. PPA's resonance frequency was distinctly downfield from endogenous phosphates and demonstrated a pH-dependent chemical shift of +/- 1.12 ppm/pH unit over the range of 6.4 to 7.6 with a pK' of 7.09 at 23 degrees C. The time courses for washing PPA in and out of the resting bladder were best described by monotonic exponential growth (r = 0.972; n = 3) and decay (r = 0.972; n = 3) equations, respectively. Rate and time constants for PPA wash-in (0.039 +/- 0.004 min-1 and 25.7 +/- 2.3 minutes) and washout (0.038 +/- 0.000 min-1 and 26.3 +/- 0.0 minutes) were not significantly different. Using steady state PPA and ATP peak intensities and concentrations, an extracellular-to-intracellular ratio was calculated to be 0.31 +/- 0.03 (n = 3). These data indicate that PPA remains distributed exclusively in the extracellular spaces.(ABSTRACT TRUNCATED AT 250 WORDS)

Interspecies Variation in Mammary Gland Growth Rate: Relationship to Gestation Length

Growth of the mammary gland is measured by several indices including total wet weight, dry fat-free tissue, and deoxyribonucleic acid. The latter is a superior measure of true growth because it represents changes of cell numbers. Sufficient data have been generated to determine the relationship among species of mammals between gestation length and differences in rates of mammary growth. Exponential growth equations were estimated for eight mammalian species with gestation lengths from 16.5 d for the hamster to 280 d for the cow. The form of the most appropriate equation was Y = AeBx where Y is mammary deoxyribonucleic acid or dry fat-free tissue, x is day of gestation, e is the base of natural logs, and A and B are constants. The A term was related to body weight (W) and the B-term to gestation length (G). Resulting equations were deoxyribonucleic acid (mg) = .0547W.803 e1.98G−.98 × and dry fat-free tissue (mg) = 2.35W.779 e.719G−.77 x. First-order rate constants of mammary growth ranged in a reverse order from a high of .141 d−1 in hamsters to a low of .008 d−1 in cows; in other words, mammary deoxyribonucleic acid in hamsters doubled in 4.9 d but in the bovine it took 87 d to double.

Growth Rates and Doubling Times of Posterior Uveal Melanomas

AbstractSeventeen patients having a posterior uveal melanoma underwent enucleation following documentation of tumor growth. Duration of follow-up from diagnosis to enucleation ranged from 3 weeks to 30 months. The mean rate of enlargement in cubic tumor volume was 56.3 mm3/mo for eight spindle cell melanomas and 270.3 mm3/mo for nine mixed cell melanoma. The mean tumor doubling time calculated according to the exponential growth equation was 291.6 days for the eight spindle cell melanomas and 128.2 days for the nine mixed cell melanomas. The mean mitotic activity was 3.5 mf/40 hpf for the eight spindle cell melanomas and 5.4 mf/40 hpf for the nine mixed cell melanomas. These results suggest that mixed cell posterior uveal melanomas generally grow faster and have shorter doubling times than do spindle cell melanomas.

Chemostat and in‐situ colonization kinetics ofthermothrix thioparaon calcite and pyrite surfaces

Abstract Growth and attachment rates of Thermothrix thiopara on calcite and pyrite were quantitated in a thiosulfate‐limited chemostat and in the thermal spring where the organism is found in nature. Surface growth rates were quantitated by using the surface colonization and exponential growth equations. These two models were compared as means of determining surface growth rates. In the chemostat, T. thiopara cells colonizing calcite and pyrite surfaces grew at approximately one‐third the rate of suspended cells. However, T. thiopara attached to pyrite faster than to calcite. In the thermal spring, growth and attachment rates were equal on calcite and pyrite. It was concluded that the exponential growth equation overestimates in‐situ surface growth rates and that T. thiopara grows more slowly when colonizing mineral surfaces than when growing in suspension. Lower growth rates on surfaces may be due to a reduced cell surface area for nutrient uptake or an increased specific maintenance rate.

Evaluation of a proposed surface colonization equation usingThermothrix thiopara as a model organism

The colonization equation shown below was evaluated usingThermothrix thiopara as a model organism.[Formula: see text] where: N=number of cells on surface (cells field(-1)); A = attachment rate (cells field(-1) h(-1)); M=specific growth rate (h(-1)); t=incubation period (h).Previous studies of microbial surface colonization consider attachment and growth independently. However, the proposed colonization equation integrates the effects of simultaneous attachment and growth. Using this equation, the specific growth rate ofT. thiopara was found to be 0.38±0.3 h(-1) during in situ colonization. Estimates ofμ were independent of incubation period after 4 h (2 generations). Shorter incubations were inadequate to produce sufficient microcolonies for accurate determination of specific growth rate. Empirical data for the time course of colonization fell within the 95% confidence interval of predicted values. The attachment rate, although assumed to be constant, was found to continuously increase with time. This increase may have been an artifact due to the continuous deposition of travertine on the surface, or may indicate the need for a function to replace A in the colonization equation. Using the exponential growth equation, the progeny of cells that attach during incubation are considered to be progeny of cells that attach initially. This erroneously inflated the growth rate by 55%.

Exponential growth and allometric equations

Earlier the simple exponential growth law has been used when attempts have been made to explain theoretically the existence of allometric equations in biology (Bertalanffy, 1968; Giinther, 1975). We show here that the exponential growth law in which the proportion coefficient also depends on time is a better approximation. We consider especially the metabolic rate and the body mass of man during the growth process.

Evaluation of a proposed surface colonization equation usingThermothrix thiopara as a model organism

The colonization equation shown below was evaluated usingThermothrix thiopara as a model organism.[Formula: see text] where: N=number of cells on surface (cells field(-1)); A = attachment rate (cells field(-1) h(-1)); M=specific growth rate (h(-1)); t=incubation period (h).Previous studies of microbial surface colonization consider attachment and growth independently. However, the proposed colonization equation integrates the effects of simultaneous attachment and growth. Using this equation, the specific growth rate ofT. thiopara was found to be 0.38±0.3 h(-1) during in situ colonization. Estimates ofμ were independent of incubation period after 4 h (2 generations). Shorter incubations were inadequate to produce sufficient microcolonies for accurate determination of specific growth rate. Empirical data for the time course of colonization fell within the 95% confidence interval of predicted values. The attachment rate, although assumed to be constant, was found to continuously increase with time. This increase may have been an artifact due to the continuous deposition of travertine on the surface, or may indicate the need for a function to replace A in the colonization equation. Using the exponential growth equation, the progeny of cells that attach during incubation are considered to be progeny of cells that attach initially. This erroneously inflated the growth rate by 55%.

Growth and Gas Exchange of Embryonic Sea Turtles (Chelonia, Caretta)

The increase in mass of embryonic sea turtles (Chelonia mydas, Caretta caretta) with incubation time can be described by a decaying exponential growth equation and appears sigmoidal in pattern. The coefficients required to fit the growth equation to the embryonic data vary among and within species depending on the mass and incubation times of the sea turtle population. When groups of eggs are incubated in artificial nests where gas exchange can be manipulated, rates of growth and hatching success are related to nest gas exchange. Maximal rates of growth (incubation - 60 days) and hatching success occur in a respiratory environment quite similar to that measured in natural nests. As movement of respiratory gases between the nest and the atmosphere is limited, embryonic rates of growth and egg hatching success are reduced.

Quantitation of microbial growth on surfaces

An equation describing the initial phases of microbial surface colonization is presented. Simultaneous microbial attachment and growth are considered as the primary components of colonization. A table is given that permits determination of growth rate from the density and distribution of cells present on surfaces after incubation in situ. Other methods used to calculate microbial growth rate on surfaces are evaluated. The new procedure is more accurate and less time consuming than those used previously. Published data on microbial surface colonization more closely follow the proposed colonization equation than the exponential growth equation, which overestimates the growth rate.

Analysis of the Exponential Growth Equation1

Solutions for growth equations that estimate dry matter from photosynthetic, respiratory, and photosynthetic area variables were tested by using field data for cotton (G. hirsutum L. var. ‘Deltapine Smoothleaf’). The equations based on assumptions of exponential leaf area expansion and exponential extraction of light by leaves fitted the data test. A leaf area expansion function that depends upon morphogenetic variables should improve the fit.

An Approach to the Calculation of the Allowable Specific Activities in Marine Fishes

A mathematical expression is given by Isaacs et al. for calculating the specific activity in fish using an assumed exponential growth equation. In the present work, experimentally verified limited growth cubic equation proposed by Von Bertalanffy is used in place of the exponential growth equation of Isaacs'. Further, assuming an age distribution function of the type n = alpha t e−at which is found to be applicable for the actual fish catches along the Indian coast, an expression is derived for the weighted average specific activity in fishes in an actual catch. There is a good agreement between the values calculated from the model proposed here and Isaacs' expression for nuclides having short physical and/or biological half lives. However, for nuclides having long physical and biological half lives (e.g. 45Ca, 55Fe and 90Sr), the values from the present model are significantly higher than Isaacs' limiting values and approach Isaacs' limiting values only at infinite age of the fish.

Weight of the skeleton during postnatal development.

AbstractThe weight of all bones and the length of humeri, radii, femora and tibiae have been determined in a series of 150 dry, fat‐free skeletons from American Whites and Negroes of both sexes, ranging in age from 23 days to 22 years. Six skeletons were eliminated from the series because of evidence of previous illness. A comparison of the lengths of femur plus tibia of this series with the mean statures of a large series of living children at given ages indicates similarity in the growth patterns. Statistical analyses of the data show that the skeletal weight cannot be estimated reliably from age by cither an exponential growth equation or by a logistic function. The weight of the skeleton, however, is related to the lengths of the measured limb bones by allometric equations, and such equations involving each of the four bones are presented for estimation of skeletal weight in the living. Although the standard errors of estimate of the equations based on lengths of each of these four bones differ very little, the radius is recommended over the other three because it is more readily accessible in the living for a roentgenogram and its shadow on the film shows least distortion.