A biological growth model using continued fraction of straight lines. Methodological aspects

To optimize the growth management of spring maize, it is essential to understand the dynamics of plant height and leaf area index (LAI) under controlled water and nitrogen supply. This study conducted two-year field experiments (2022–2023) in Karamay, Xinjiang. Three irrigation levels (75%, 100%, and 125% of Crop Evapotranspiration (ETc)) and four nitrogen application rates (0, 93, 186, and 279 kg N/ha) were set. A logistic growth model was fitted using accumulated effective temperature as the independent variable to analyze the growth and development characteristics of spring maize under various water and nitrogen conditions. The results demonstrated that the logistic models, based on relative effective accumulated temperature, had a determination coefficient (R2) of over 0.99 and a Normalized Root Mean Square Error (NRMSE) of less than 10%. Irrigation extended the rapid growth phase of plant height, whereas nitrogen application shortened the time to enter this rapid growth phase and prolonged its duration. Irrigation increased the maximum LAI growth rate and shortened and prolonged the rapid growth phase, while nitrogen extended the duration of the rapid growth phase for LAI. The W2N2 treatment, consisting of 100% ETc irrigation and 186 kg N/ha, was identified as the optimal drip irrigation water–nitrogen combination for spring maize in the study area. Under optimal water and nitrogen supply, both the maximum growth rate and the average growth rate during the rapid growth phase were higher, requiring accumulated effective temperatures of 825.16–845.74 °C·d and 856.68–890.00 °C·d, respectively, to reach these rates. The appropriate water and nitrogen supply significantly enhanced the synergistic promotion of growth and development in spring maize. This study provides a theoretical basis for the quantitative analysis of growth dynamics in summer maize using effective accumulated temperature.

BackgroundToday there is much debate about why telemedicine has stalled. Teleradiology is the only widespread telemedicine application. Other telemedicine applications appear to be promising candidates for widespread use, but they remain in the early adoption stage. The objective of this debate paper is to achieve a better understanding of the adoption of telemedicine, to assist those trying to move applications from pilot stage to routine delivery.DiscussionWe have investigated the reasons why telemedicine has stalled by focusing on two, high-level topics: 1) the process of adoption of telemedicine in comparison with other technologies; and 2) the factors involved in the widespread adoption of telemedicine. For each topic, we have formulated hypotheses. First, the advantages for users are the crucial determinant of the speed of adoption of technology in healthcare. Second, the adoption of telemedicine is similar to that of other health technologies and follows an S-shaped logistic growth curve. Third, evidence of cost-effectiveness is a necessary but not sufficient condition for the widespread adoption of telemedicine. Fourth, personal incentives for the health professionals involved in service provision are needed before the widespread adoption of telemedicine will occur.SummaryThe widespread adoption of telemedicine is a major -- and still underdeveloped -- challenge that needs to be strengthened through new research directions. We have formulated four hypotheses, which are all susceptible to experimental verification. In particular, we believe that data about the adoption of telemedicine should be collected from applications implemented on a large-scale, to test the assumption that the adoption of telemedicine follows an S-shaped growth curve. This will lead to a better understanding of the process, which will in turn accelerate the adoption of new telemedicine applications in future. Research is also required to identify suitable financial and professional incentives for potential telemedicine users and understand their importance for widespread adoption.

Probabilistic biological network growth models have been utilized for many tasks including but not limited to capturing mechanism and dynamics of biological growth activities, null model representation, capturing anomalies, etc. Well-known examples of these probabilistic models are Kronecker model, preferential attachment model, and duplication-based model. However, we should frequently keep developing new models to better fit and explain the observed network features while new networks are being observed. Additionally, it is difficult to develop a growth model each time we study a new network. In this paper, we propose BioCode, a framework to automatically discover novel biological growth models matching user-specified graph attributes in directed and undirected biological graphs. BioCode designs a basic set of instructions which are common enough to model a number of well-known biological graph growth models. We combine such instruction-wise representation with a genetic algorithm based optimization procedure to encode models for various biological networks. We mainly evaluate the performance of BioCode in discovering models for biological collaboration networks, gene regulatory networks, and protein interaction networks which features such as assortativity, clustering coefficient, degree distribution closely match with the true ones in the corresponding real biological networks. As shown by the tests on the simulated graphs, the variance of the distributions of biological networks generated by BioCode is similar to the known models' variance for these biological network types.

The lag phase (L) of grape berry growth is used to determine the timing of hand sampling for yield estimation. In commercial practice, growers apply scalars to measurements of berry of cluster masses under the assumption that fruit was assessed during L, which is the short period of slowest increase in fruit mass that occurs between the first and second sigmoid curves that describe growth in fleshy fruits. To estimate L, we used an automated remote system that indirectly detects increases in vegetative and fruit mass in grapevines by monitoring the tension (T) in the main load-bearing wire of the trellis. We fitted logistic curves to the change in T (ΔT) such that the parameters could be interpreted biologically, particularly the onset of L: the asymptotic deceleration of growth. Curves fit the data well [root mean square error (RMSE) 4.2 to 14.9] in three disparate years and two vineyards. The onset of L was most sensitive to the inflection point of the first logistic curve but relatively insensitive to its shape parameter. The analytical solution of the second derivative of the first logistic curve for its minimum predicted the apparent onset of L with a range of 3 to 5 days among replicates. The roots of the third derivative allowed analytical solutions for the onset of the first rapid growth phase and L, consistently predicting the onset of L 2 to 15 days earlier than was identified by trained observers who examined ΔT curves. Remote sensing of ΔT could better time field sampling and decrease current reliance on visual and tactile assessment to identify the onset of L, thus improving yield estimation in grapes.

In this chapter we discuss the software reliability growth models (SRGMs) that describe the relationship between the number of faults removed and the number of test cases used. Firstly we describe the discrete exponential and S-shaped models in a perfect debugging environment. We also discuss flexible discrete SRGM, which can depict either exponential or S-shaped growth curves, depending upon the parameter values estimated from the past failure data. Further, we describe an SRGM for the fault removal phenomenon in a perfect debugging environment. Most testing processes are imperfect in practice, therefore we also discuss a discrete model that incorporates the impact of imperfect debugging and fault generation into software reliability growth modeling. Faults in the software are generally not of the same type, rather the faults contained in a large software may differ from each other in terms of the amount of time and skill of the removal team required to remove them. Three discrete models: the generalized Erlang model, modeling severity of faults with respect to testing time, and a model with faults of different severity incorporating logistic learning function have been discussed. A discrete model in a distributed environment is also discussed. The above discrete SRGMs assume a constant fault detection rate while testing the software under consideration. In practice, however, the fault detection rate varies because of changes in the testing skill, the system environment, and the testing strategy used to test the software. SRGMs for the fault removal phenomenon, the generalized Erlang model, and the generalized Erlang model with logistic function are discussed, incorporating the concept of change point. It is also shown how equivalent continuous models can be derived. This chapter describes the state-of-the-art in discrete modeling.

Filamentous microorganisms enable the production of a wide range of industrially relevant substances, such as enzymes or active pharmaceutical ingredients, from renewable side products and waste materials. The microorganisms' growth is characterized by the formation of complex, porous networks (mycelium) of tubular, multi-branched cells (hyphae). The mycelium is increasingly used in textiles, packaging, food and construction materials, in addition to the production of chemical substances. Overall, the mycelium's mechanical behavior is essential to many applications. In submerged cultures, spherical hyphal networks (pellets) are formed. The pellets are subjected to mechanical stress during cultivation, which can lead to structural changes affecting product titer and process conditions. To numerically investigate the mechanical behavior of pellets under normal stresses, the discrete element method (DEM) was used for the first time to simulate pellet compression. Initially, pellet structures were generated using a biological growth model and represented by a flexible fiber model. Force–displacement curves were recorded during compression to investigate the influencing factors. The effects of pellet size, fiber segment length, biological growth and DEM model parameters were studied. A strong influence of the growth parameters on the radial hyphal fraction and thus on the compression force was shown. Furthermore, the mechanical properties of the fiber joints significantly determined the pellet mechanics in the considered compression range. Overall, the simulation approach provides a novel tool for the digital investigation of stress on different mycelia, which may be used in the future to enhance mycelial structures through genetic and process engineering methods.

Using the concept of isomorphism, we provide a quantitative description of the thermodynamic characteristics of the Logistic, Bertalanffy, and Gompertz biological growth models. With the help of the entropy expressions derived from the isomorphic probabilistic models, we show that the Logistic, Bertalanffy, and Gompertz growth models have distinct thermodynamic characteristics, though they have similar sigmoid trajectories in the time domain. The entropic analysis further reveals that the Gompertz model corresponds to a completely open system, in which cells receive adequate nutrition and competition among cells can be neglected. We have also applied the present entropic analysis to the modeling of tumor growth. Our results suggest that the entropic expression provides a theoretical justification for using the Gompertz model for describing the development of tumor cell populations. The entropic analysis may play an important role in studying the mechanisms of different biological growth processes.

We investigate the rapid growth phase of supermassive black holes (BHs) within the hydrodynamical cosmological \eagle simulation. This non-linear phase of BH growth occurs within $\sim$$L_{*}$ galaxies, embedded between two regulatory states of the galaxy host: in sub $L_{*}$ galaxies efficient stellar feedback regulates the gas inflow onto the galaxy and significantly reduces the growth of the central BH, while in galaxies more massive than $L_{*}$ efficient AGN feedback regulates the gas inflow onto the galaxy and curbs further non-linear BH growth. We find evolving critical galaxy and halo mass scales at which rapid BH growth begins. Galaxies in the low-redshift Universe transition into the rapid BH growth phase in haloes that are approximately an order of magnitude more massive than their high-redshift counterparts (\M{200} $\approx 10^{12.4}$~\Msol at $z \approx 0$ decreasing to \M{200} $\approx 10^{11.2}$~\Msol at $z \approx 6$). Instead, BHs enter the rapid growth phase at a fixed critical halo virial temperature ($T_{\mathrm{vir}} \approx 10^{5.6}$~K). We additionally show that major galaxy--galaxy interactions ($\mu \geq \frac{1}{4}$, where $\mu$ is the stellar mass ratio) play a substantial role in triggering the rapid growth phase of BHs in the low-redshift Universe, whilst potentially having a lower influence at high redshift. Approximately 40\% of BHs that initiate the rapid BH growth phase at $z \approx 0$ do so within $\pm 0.5$ dynamical times of a major galaxy--galaxy merger, a fourfold increase above what is expected from the background merger rate. We find that minor mergers ($\frac{1}{10} \leq \mu < \frac{1}{4}$) have a substantially lower influence in triggering the rapid growth phase at all epochs.

The hyper-geometric distribution is used to estimate the number of initial faults residual in software at the beginning of the test-and-debug phase. The hyper-geometric distribution growth model (HGD model) is well suited to making estimates for the observed growth curves of the accumulated number of detected faults. The advantage of the proposed model is the applicability to all kinds of observed data. By application of a single model, exponential growth curves as well as S-shaped growth curves can be estimated. The precise formulation of the HGD model is presented. The exact relationship of this model to the NHPP Goel-Okumoto growth model and the delayed S-shaped growth model is shown. With the introduction of a variable fault detection rate, the goodness of fit of the estimated growth curve to the growth curve of real observed faults is increased significantly. Different examples of the applicability of the model to real observed data are presented. >

An S-shaped reliability growth curve is used to describe a reliability growth trend with a lower rate of debugging and growth at the early stage and a higher rate later on as better fixes are found and successfully implemented. In some situations, it is unreasonable to set a high growth rate of improvement during the early stages of development for complex systems because of the difficulty of identifying the sources and causes of the failures, hence requiring the use of S-shaped reliability growth curves. One of the models now in use is the Gompertz model. From several data sets, it was seen that, in general, reliability growth data with an S-shaped trend could not be described well by the conventional Gompertz model, because this model has a fixed value of reliability at its inflection point. Therefore, only a small fraction of reliability growth data sets, following an S-shaped reliability growth curve, could be fitted to it. The modified Gompertz model that is presented in this paper overcomes this shortcoming by using a fourth parameter, which shifts it vertically, thus accommodating S-shaped growth data sets. The resulting model is much more flexible than it's predecessor, especially for fitting data with S-shaped trends. >

Sigmoidal kinetic curves have been reported for a number of cooperative phenomena in nature. These curves may be fit by purely mathematical functions that, however, do not correspond to any physical model. The 1997 Finke–Watzky (F–W) two-step model of slow, continuous nucleation (A → B, rate constant k1) and fast, autocatalytic growth (A + B → 2B, rate constant k2) provides both a physical model and a mathematical solution. As a result, the F–W two-step kinetic model has been successfully applied to a large number of cooperative phenomena throughout nature that display sigmoidal kinetic curves. Herein, we derive formulas for the first, second, and third derivatives of the concentration of product versus time, [B]t, expressed in terms of the F–W parameters k1, k2, and the initial concentration of monomer, [A]0. Mathematical expressions are then derived for key empirical parameters in sigmoidal curves, including the induction period and (maximum) slope, which are then examined under the limit k1 ≪ k2[A]0 wh...

Biochemical diagnostic procedures, such as protein binding, rely on biomolecular interactions as its diagnostic modality, and as a consequence, their calibration curves are more complex. In addition, sigmoidal curves are often seen in these tests. In the event of asymmetry, a logistic (5PL) curve, or a logistic (4PL) curve, may be the best way to represent a distinctive sigmoidal relationship. It is possible that the linearization of an otherwise nonlinear connection by log transformation may result in a disruption of the error structure of the curve, and that this will have the opposite effect of decreasing or even eliminating error in the relationship. Previously, a surface plasmon resonance biosensor for the detection of squamous cell carcinoma antigen (SCCa) using nanoparticle technology was developed. However, based on the calibration curve, it conforms to the majority on sigmoidal shape curve for antibody-type sensing system. The resultant curve showed a sigmoidal calibration curve but was not modelled according to any of the sigmoidal models available. The LOD value obtained through the 4PL modelling exercise based on the classical method was 0.255 pM (95% confidence interval of 0.167 to 0.379) while the pooled standard deviation (PSD) method yielded an LOD value of 0.035 pM (95% confidence interval of 0.011 to 0.067), which indicates that the PSD method was superior.

Reviewed by: Growth: From Microorganisms to Megacities by Vaclav Smil Ruth Schwartz Cowan (bio) Growth: From Microorganisms to Megacities By Vaclav Smil. Cambridge, MA: MIT Press, 2019. Pp. 634. Growth: From Microorganisms to Megacities By Vaclav Smil. Cambridge, MA: MIT Press, 2019. Pp.634. Remember that kid in third grade? The one who memorized all the rivers in Africa, then made you feel like a jerk because she knew so much more than you? Or maybe that kid was you? Or me? Thankfully, we both grew out of it. Vaclav Smil did not. Smil, distinguished professor of environmental science emeritus at the University of Manitoba, is the author of 37 previous single-authored books, all focused on some aspect of pre-pandemic global environmental, economic, and social crises. Growth has a similar focus, but you won't find that out until you have plowed your way through 448 poorly organized pages containing dozens and dozens of graphs about the growth patterns of—the subtitle actually minimizes the scope of the enterprise—everything from microorganisms to megacities, plus empires and civilizations. In his preface, Smil tells us that he intends to examine growth quantitatively, looking for patterns in the data, particularly "S-shaped (sigmoid) growth curves" (p. xxi). He's a bit cagey, however, about whether all the data and S-shaped curves he intends to display will, or even should, lead to secure forecasts of future growth, or lack thereof. He cautions against "any simplistic embrace of even the best statistical fits for long-term forecasting," then in the next paragraph declares that "parts of the book are helpfully predictive" (p. xxiii). The pages of Growth are riddled with similarly ambivalent and protective statements about the ultimate meaning of the S-shaped growth patterns he describes. In excruciating detail. But before we get to that detail, a sixty-nine-page introductory chapter, as its title suggests, is intended to elucidate "Trajectories: or common patterns of growth." Unfortunately, much of the chapter will only be intelligible to readers who understand a "perfect power-law function (approximating the form f(x)=ax-k where a and k are constant)" (p. 60). A reader unfamiliar with the nuances of advanced statistics may well cower in frightful admiration, an effect (and affect) which strikes me as the grownup version of what that kid in third grade was hoping to achieve. Four substantive chapters, occupying 448 pages, follow the introduction: Nature, Energies, Artifacts, and Populations, Societies, Economies. Each chapter is divided into subsections with no rhyme and little reason; each subsection is filled with a miscellany of data analyses and charts. None ends with a concluding generalization that makes sense out of the very different growth patterns discussed. Historians of technology who might want to use Growth as a handy reference [End Page 916] work about growth patterns on such diverse subjects as the maximum thrust of jet engines or the total length of Chinese highways will need to be exceedingly careful. One graph, presumably based on data that I happen to know something about (adoption rates of household appliances in the U.S., 1900–2005, p. 294) is multiply fraudulent. Leaving aside the fact that there is no reliable national data for the period 1900–1960, the graph Smil publishes cannot be found in the source he cites. That source (for which Smil gives both the wrong date and wrong URL) contains data (and a graph) about something else altogether, and only the years 1973–2006. The mistakes just keep coming. P. Taylor et al. "Luxury or necessity?" is at www.pewresearch.org/wp-content/uploads/sites/3/2010/10/Luxury.pdf. One can compare the URL publication date in Growth (p. 605). Colleagues, you have been warned! Growth concludes with a sixty-page chapter "What Comes After Growth: or demise and continuity" and a five-page "Coda." Here Smil reveals that the growth trajectory he really cares about is the one for "high energy civilization." Unfortunately, after treating us to extended critiques of both pro-growth and limits-to-growth economists, as well as arguments against the concepts of sustainable growth and de-materialization, Smil concludes that everyone who has ever modeled the...

The postnatal growth of rats involves a developmental phase (0 to approximately 3 weeks), a rapid growth phase ( approximately 3 to approximately 10 weeks), and a slower maturation phase ( approximately 10 weeks+). In this study, we investigated the age-related changes in excitation-contraction (E-C) coupling characteristics of mammalian skeletal muscle, during rapid growth (4-10 weeks) and maturation (10-21 weeks) phases, using single, mechanically skinned fibres from rat extensor digitorum longus (EDL) muscle. Fibres from rats aged 4 and 8 weeks produced lower maximum T-system depolarization-induced force responses and fewer T-system depolarization-induced force responses to 75% run-down than those produced by fibres from rats aged 10 weeks and older. The sensitivity of the contractile apparatus to Ca(2+) in fibres from 4-week rats was significantly higher than that in fibres from 10-week rats; however, the maximum Ca(2+)-activated force per skinned fibre cross-sectional area (specific force) developed by fibres from 4-week rats was on average approximately 44% lower than the values obtained for all the other age groups. In agreement with the age difference in specific force, the MHC content of EDL muscles from 4-week rats was approximately 29% lower than that of 10-week rats. Thus, mechanically skinned fibres from rats undergoing rapid growth are less responsive to T-system depolarization and maximal Ca(2+) activation than fibres from rats at the later stage of maturation or adult rats. These results suggest that during the rapid growth phase in rats, the structure and function of elements involved in E-C coupling in fast-twitch skeletal muscle continue to undergo significant changes.

A unified approach to the Richards-model family for use in growth analyses: Why we need only two model forms