Simultaneous identification of growth law and estimation of its rate parameter for biological growth data: a new approach

Parameter estimation of the Solow-Swan fundamental differential equation

Background

The Solow-Swan model describes the long-term growth of the capital to labor ratio by the fundamental differential equation of Solow-Swan theory. In conventional approaches, this equation was fitted to data using additional information, such as the rates of population growth, capital depreciation, or saving. However, this was not the best possible fit.

Objectives

Using the method of least squares, what is the best possible fit of the fundamental equation to the time-series of the capital to labor ratios? Are the best-fit parameters economically sound?

Method

For the data, we used the Penn-World Table in its 2021 version and compared six countries and three definitions of the capital to labor ratio. For optimization, we used a custom-made variant of the method of simulated annealing. We also compared different optimization methods and calibrations.

Results

When comparing different methods of optimization, our custom-made tool provided reliable parameter estimates. In terms of R-squared they improved upon the parameter estimates of the conventional approach. Except for the USA, the best-fit values of the exponent were unplausible, as they suggested a too large elasticity of output. However, using a different calibration resulted in more plausible values of the best-fit exponent also for France and Pakistan, but not for Argentina and Japan.

Conclusion

Our results have shown a discrepancy between the best-fit parameters obtained from optimization and the parameter values that are deemed plausible in economy. We propose a research program to resolve this issue by investigating if suitable calibrations may generate economically plausible best-fit parameter values.

Instantaneous maturity rate: a novel and compact characterization of biological growth curve models

Modeling and analysis of biological growth curves are an age-old study area in which much effort has been dedicated to developing new growth equations. Recent efforts focus on identifying the correct model from a large number of equations. The relative growth rate (RGR), developed by Fisher (1921), has largely been used in the statistical inference of biological growth curve models. It is convenient to express growth equations using RGR, where RGR can be expressed as functions of size or time. Even though RGR is model invariant, it has limitations when it comes to identifying actual growth patterns. By proposing interval-specific rate parameters (ISRPs), Pal et al. (2018) appeared to solve this problem. The ISRP is based on the mathematical structure of the growth equations. Therefore, it is not model invariant. The current effort is to develop a measure of growth that is model invariant like RGR and shares the advantages of ISRP. We propose a new measure of growth, which we call instantaneous maturity rate (IMR). IMR is model invariant, which allows it to distinguish growth patterns more clearly than RGR. IMR is also scale-invariant and can take several forms including increasing, decreasing, constant, sigmoidal, bell-shaped, and bathtub. A wide range of possible IMR shapes makes it possible to identify different growth curves. The estimation procedure of IMR under a stochastic setup has been developed. Statistical properties of empirical IMR estimators have also been investigated in detail. In addition to extensive simulation studies, real data sets have been analyzed to prove the utility of IMR.

A generalized Gompertz growth model with applications and related birth-death processes

In this paper, we propose a flexible growth model that constitutes a suitable generalization of the well-known Gompertz model. We perform an analysis of various features of interest, including a sensitivity analysis of the initial value and the three parameters of the model. We show that the considered model provides a good fit to some real datasets concerning the growth of the number of individuals infected during the COVID-19 outbreak, and software failure data. The goodness of fit is established on the ground of the ISRP metric and the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_2$$\end{document}d2-distance. We also analyze two time-inhomogeneous stochastic processes, namely a birth-death process and a birth process, whose means are equal to the proposed growth curve. In the first case we obtain the probability of ultimate extinction, being 0 an absorbing endpoint. We also deal with a threshold crossing problem both for the proposed growth curve and the corresponding birth process. A simulation procedure for the latter process is also exploited.

Understanding gold toxicity in aerobically-grown Escherichia coli

Background

There is an emerging field to put into practice new strategies for developing molecules with antimicrobial properties. In this line, several metals and metalloids are currently being used for these purposes, although their cellular effect(s) or target(s) in a particular organism are still unknown. Here we aimed to investigate and analyze Au³⁺ toxicity through a combination of biochemical and molecular approaches.

Results

We found that Au³⁺ triggers a major oxidative unbalance in Escherichia coli, characterized by decreased intracellular thiol levels, increased superoxide concentration, as well as by an augmented production of the antioxidant enzymes superoxide dismutase and catalase. Because ROS production is, in some cases, associated with metal reduction and the concomitant generation of gold-containing nanostructures (AuNS), this possibility was evaluated in vivo and in vitro.

Conclusions

Au³⁺ is toxic for E. coli because it triggers an unbalance of the bacterium’s oxidative status. This was demonstrated by using oxidative stress dyes and antioxidant chemicals as well as gene reporters, RSH concentrations and AuNS generation.

A new framework for growth curve fitting based on the von Bertalanffy Growth Function

All organisms grow. Numerous growth functions have been applied to a wide taxonomic range of organisms, yet some of these models have poor fits to empirical data and lack of flexibility in capturing variation in growth rate. We propose a new VBGF framework that broadens the applicability and increases flexibility of fitting growth curves. This framework offers a curve-fitting procedure for five parameterisations of the VBGF: these allow for different body-size scaling exponents for anabolism (biosynthesis potential), besides the commonly assumed 2/3 power scaling, and allow for supra-exponential growth, which is at times observed. This procedure is applied to twelve species of diverse aquatic invertebrates, including both pelagic and benthic organisms. We reveal widespread variation in the body-size scaling of biosynthesis potential and consequently growth rate, ranging from isomorphic to supra-exponential growth. This curve-fitting methodology offers improved growth predictions and applies the VBGF to a wider range of taxa that exhibit variation in the scaling of biosynthesis potential. Applying this framework results in reliable growth predictions that are important for assessing individual growth, population production and ecosystem functioning, including in the assessment of sustainability of fisheries and aquaculture.

A Novel Unification Method to Characterize a Broad Class of Growth Curve Models Using Relative Growth Rate

Growth curve models serve as the mathematical framework for the quantitative studies of growth in many areas of applied science. The evolution of novel growth curves can be categorized in two notable directions, namely generalization and unification. In case of generalization, a modeler starts with a simple mathematical form to describe the behavior of the data and increases the complexity of the equation by incorporating more parameters to obtain a more flexible shape. The unification refers to the process of obtaining a compact representation of a large number of growth equations. An enormous number of growth equations are made available in the literature by means of the generalization of existing growth laws. However, the unification of growth equations has received relatively less attention from the researchers. Two significant unification functions are available in the literature, namely the Box-Cox transformation by Garcia (For Biometry Model Inf Sci 1:63-68, 2005) and generalized logarithmic and exponential functions by Martinez et al. (Phys A 387:5679-5687, 2008; Phys A 388:2922-2930, 2009). Existing unification approaches are found to have limited applications if the growth equation is characterized by the relative growth rate (RGR). RGR has immense practical value in biological growth curve analysis, which has been amplified by the construction of size and time covariate models, in which; RGR is represented either as a function of size or time or both. The present study offers a unification function for the RGR growth curves. The proposed function combines a broad class of the growth curves and possesses a greater generality than the existing unification functions. We also propose the notion of generalized RGR, which is capable of making interrelations among the unifying functions. Our proposed method is expected to enhance the generality of software and may aid in choosing an optimal model from a set of competitor growth equations.

Evolution of model specific relative growth rate: Its genesis and performance over Fisher's growth rates

Growth curve models play an instrumental role to quantify the growth of biological processes and have immense practical applications across disciplines. In the modelling approach, the absolute growth rate and relative growth rate (RGR) are two most commonly used measures of growth rates. RGR is empirically estimated by Fisher (1921) assuming exponential growth between two consecutive time points and remains invariant under any choice of the underlying growth model. In this article, we propose a new measure of RGR, called modified RGR, which is sensitive to the choice of underlying growth law. The mathematical form of the growth equations are utilized to develop the formula for model dependent growth rates and can be easily computed for commonly used growth models. We compare the efficiency of Fisher's measure of RGR and modified RGR to infer the true growth profile. To achieve this, we develop a goodness of fit testing procedure using Gompertz model as a test bed. The relative efficiency of the two rate measures is compared by generating power curves of the goodness of fit testing procedure. The asymptotic distributions of the associated test statistics are elaborately studied under Gompertz set up. The simulation experiment shows that the proposed formula has better discriminatory power than the existing one in identifying the true profile. The claim is also verified using existing real data set on fish growth. An algorithm for the model selection mechanism is also proposed based on the modified RGR and is generalized for some commonly used other growth models. The proposed methodology may serve as a valuable tool in growth studies in different research areas.

A simple approximation of moments of the quasi-equilibrium distribution of an extended stochastic theta-logistic model with non-integer powers

The stochastic versions of the logistic and extended logistic growth models are applied successfully to explain many real-life population dynamics and share a central body of literature in stochastic modeling of ecological systems. To understand the randomness in the population dynamics of the underlying processes completely, it is important to have a clear idea about the quasi-equilibrium distribution and its moments. Bartlett et al. (1960) took a pioneering attempt for estimating the moments of the quasi-equilibrium distribution of the stochastic logistic model. Matis and Kiffe (1996) obtain a set of more accurate and elegant approximations for the mean, variance and skewness of the quasi-equilibrium distribution of the same model using cumulant truncation method. The method is extended for stochastic power law logistic family by the same and several other authors (Nasell, 2003; Singh and Hespanha, 2007). Cumulant truncation and some alternative methods e.g. saddle point approximation, derivative matching approach can be applied if the powers involved in the extended logistic set up are integers, although plenty of evidence is available for non-integer powers in many practical situations (Sibly et al., 2005). In this paper, we develop a set of new approximations for mean, variance and skewness of the quasi-equilibrium distribution under more general family of growth curves, which is applicable for both integer and non-integer powers. The deterministic counterpart of this family of models captures both monotonic and non-monotonic behavior of the per capita growth rate, of which theta-logistic is a special case. The approximations accurately estimate the first three order moments of the quasi-equilibrium distribution. The proposed method is illustrated with simulated data and real data from global population dynamics database.