Does population ecology have general laws?

There is a widespread opinion among ecologists that ecology lacks general laws. In this paper the author argues that this opinion is mistaken. Taking the case of population dynamics, the author points out that there are several very general law-like propositions that provide the theoretical basis for most population dynamics models that were developed to address specific issues. Some of these foundational principles, like the law of exponential growth, are logically very similar to certain law of physics (Newton's law of intertia, for example, is almost a direct analogue of exponential growth). The author discusses two other principles (population self-limitation and resource-consumer oscillations), as well as the more elementary postulates that underlie them. None of the "laws" that the author proposes for population ecology are new. Collectively ecologists have been using these general principles in guiding development of their models and experiments since the days of Lotka, Volterra, and Gause.     

Does population ecology have general laws?

There is a widespread opinion among ecologists that ecology lacks general laws. In this paper I argue that this opinion is mistaken. Taking the case of population dynamics, I point out that there are several very general law-like propositions that provide the theoretical basis for most population dynamics models that were developed to address specific issues. Some of these foundational principles, like the law of exponential growth, are logically very similar to certain laws of physics (Newton's law of inertia, for example, is almost a direct analogue of exponential growth). I discuss two other principles (population self-limitation and resource-consumer oscillations), as well as the more elementary postulates that underlie them. None of the "laws" that I propose for population ecology are new. Collectively ecologists have been using these general principles in guiding development of their models and experiments since the days of Lotka, Volterra, and Gause.     

Generic Darwinian selection in catalytic protocell assemblies

To satisfy the minimal requirements for life, an information carrying molecular structure must be able to convert resources into building blocks and also be able to adapt to or modify its environment to enhance its own proliferation. Furthermore, new copies of itself must have variable fitness such that evolution is possible. In practical terms, a minimal protocell should be characterized by a strong coupling between its metabolism and genetic subsystem, which is made possible by the container. There is still no general agreement on how such a complex system might have been naturally selected for in a prebiotic environment. However, the historical details are not important for our investigations as they are related to assembling and evolution of protocells in the laboratory. Here, we study three different minimal protocell models of increasing complexity, all of them incorporating the coupling between a 'genetic template', a container and, eventually, a toy metabolism. We show that for any local growth law associated with template self-replication, the overall temporal evolution of all protocell's components follows an exponential growth (efficient or uninhibited autocatalysis). Thus, such a system attains exponential growth through coordinated catalytic growth of its component subsystems, independent of the replication efficiency of the involved subsystems. As exponential growth implies the survival of the fittest in a competitive environment, these results suggest that protocell assemblies could be efficient vehicles in terms of evolving through Darwinian selection.     

Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.     

THE KINETICS OF SENESCENCE

The course of decline of vitality with age due to the process of senescence, when not complicated by the process of growth, follows a simple exponential law; that is the degree of vitality or of senescence (defining vitality as the reciprocal of senescence) at any moment is, regardless of age, a constant percentage of the degree of vitality or senescence of the preceding moment. This exponential law is the same as the law of monomolecular change in chemistry. During the actively growing period of life the index of vitality rises, due to the process of growth and the course of vitality in the case when the growing period is included in the vitality curve, follows a rising and falling course. This rising and falling course may often be represented by an equation containing two exponential terms which is practically the equation used to represent the course of accumulation and disappearance of a substance as the result of two simultaneous consecutive monomolecular chemical reactions.     

Growth of Ca-D-malate crystals in a bioreactor

To develop a bioreactor for solid-to-solid conversions, the conversion of solid Ca-maleate to solid Ca-D-malate by permeabilized Pseudomonas pseudoalcaligenes was studied. In a bioreactor seeded with product (Ca-D-malate) crystals, growth of Ca-D-malate crystals is the last step in the solid-to-solid conversion and is described here. Crystal growth is described as a transport process followed by surface processes. In contrast to the linear rate law obeyed by the transport process, the surface processes of a crystal-growth process can also obey a parabolic or exponential rate law. Growth of Ca-D-malate crystals from a supersaturated aqueous solution was found to be surface-controlled and obeyed an exponential rate law. Based on this rate law, a kinetic model was developed which describes the decrease in supersaturation due to Ca-D-malate crystal growth as a function of the constituent ions, Ca(2+) and D-malate(2-). The kinetic parameters depended on temperature, but, as expected (surface-controlled), they were hardly affected by the stirring speed.     

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

Scientific formalizations of the notion of growth and measurement of the rate of growth in living organisms are age-old problems. The most frequently used metric, “Average Relative Growth Rate” is invariant under the choice of the underlying growth model. Theoretically, the estimated rate parameter and relative growth rate remain constant for all mutually exclusive and exhaustive time intervals if the underlying law is exponential but not for other common growth laws (e.g., logistic, Gompertz, power, general logistic). We propose a new growth metric specific to a particular growth law and show that it is capable of identifying the underlying growth model. The metric remains constant over different time intervals if the underlying law is true, while the extent of its variation reflects the departure of the assumed model from the true one. We propose a new estimator of the relative growth rate, which is more sensitive to the true underlying model than the existing one. The advantage of using this is that it can detect crucial intervals where the growth process is erratic and unusual. It may help experimental scientists to study more closely the effect of the parameters responsible for the growth of the organism/population under study.     

Analysis of protein distribution in budding yeast

Flow cytometry is a fast and sensitive method that allows monitoring of different cellular parameters on large samples of a population. Protein distributons give relevant information on growth dynamics, since they are related to the age distribution and depend on the law of growth of the population and the law of protein accumulation during the cell cycle. We analyzed protein distributions to evaluate alternative growth models for the budding yeast Saccharomyces cerevisiae and to monitor the changes in population dynamics that result from environmental modifications; such an analysis could potentially give parameters useful in the control of biotechnological processes. Theoretical protein distributions (taking into account the unequal division of yeast cells and the exponential law of protein accumulation during a cell cycle) quantitatively fit experimental distributions, once appropriate variability sources are introduced. Best fits are obtained when the protein threshold required for bud emergence increases at each new generation of parent cells.     

Taylor’s power law of fluctuation scaling and the growth-rate theorem

Taylor’s law (TL), a widely verified empirical relationship in ecology, states that the variance of population density is approximately a power-law function of mean density. The growth-rate theorem (GR) states that, in a subdivided population, the rate of change of the overall growth rate is proportional to the variance of the subpopulations’ growth rates. We show that continuous-time exponential change implies GR at every time and, asymptotically for large time, TL with power-law exponent 2. We also show why diverse population-dynamic models predict TL in the limit of large time by identifying simple features these models share: If the mean population density and the variance of population density are (exactly or asymptotically) non-constant exponential functions of a parameter (e.g., time), then the variance of density is (exactly or asymptotically) a power-law function of mean density.     

THE RATE OF GROWTH OF THE DAIRY COW : V. EXTRAUTERINE GROWTH IN LINEAR DIMENSIONS.

Barring fluctuations due to the cyclic phenomena, the extrauterine course of growth in linear dimensions and in weight of the dairy cow follows an exponential law having the same form as the law representing the course of monomolecular change in chemistry. This suggests the interpretation that the general course of growth is limited by a monomolecular chemical process, and that the cyclic phenomena are due to subsidiary processes in the fundamentally exponential course of growth. The fact that growth follows or tends to follow an exponential course may be stated more simply as follows: if the unit of time is taken sufficiently large so that fluctuations due to the cyclic phenomena are balanced or eliminated, then the amount of growth made during the given unit of time at any age tends to be a constant percentage of the growth made during the preceding unit of time. Thus, the growth in height at withers made during any year is about 34 per cent of the growth made during the preceding year. Similarly the growth in weight made during any year is about 56 per cent of the growth in weight made during the preceding year. This is in accordance with expectations if it is assumed that each animal begins life with a definite endowment of limiting substance necessary for the process of growth, and that this endowment is used up at a constant rate (or percentage) of itself.     

Emergence of protocellular growth laws

Template-directed replication is known to obey a parabolic growth law due to product inhibition (Sievers & Von Kiedrowski 1994 Nature 369, 221; Lee et al. 1996 Nature 382, 525; Varga & Szathmáry 1997 Bull. Math. Biol. 59, 1145). We investigate a template-directed replication with a coupled template catalysed lipid aggregate production as a model of a minimal protocell and show analytically that the autocatalytic template-container feedback ensures balanced exponential replication kinetics; both the genes and the container grow exponentially with the same exponent. The parabolic gene replication does not limit the protocellular growth, and a detailed stoichiometric control of the individual protocell components is not necessary to ensure a balanced gene-container growth as conjectured by various authors (Gánti 2004 Chemoton theory). Our analysis also suggests that the exponential growth of most modern biological systems emerges from the inherent spatial quality of the container replication process as we show analytically how the internal gene and metabolic kinetics determine the cell population's generation time and not the growth law (Burdett & Kirkwood 1983 J. Theor. Biol. 103, 11-20; Novak et al. 1998 Biophys. Chem. 72, 185-200; Tyson et al. 2003 Curr. Opin. Cell Biol. 15, 221-231). Previous extensive replication reaction kinetic studies have mainly focused on template replication and have not included a coupling to metabolic container dynamics (Stadler et al. 2000 Bull. Math. Biol. 62, 1061-1086; Stadler & Stadler 2003 Adv. Comp. Syst. 6, 47). The reported results extend these investigations. Finally, the coordinated exponential gene-container growth law stemming from catalysis is an encouraging circumstance for the many experimental groups currently engaged in assembling self-replicating minimal artificial cells (Szostak 2001 et al. Nature 409, 387-390; Pohorille & Deamer 2002 Trends Biotech. 20 123-128; Rasmussen et al. 2004 Science 303, 963-965; Szathma ry 2005 Nature 433, 469-470; Luisi et al. 2006 Naturwissenschaften 93, 1-13).     

Control of the specific growth rate of Bacillus subtilis for the production of biosurfactant lipopeptides in bioreactors with foam overflow

This paper formulates a feeding law for a bioprocess dedicated to the production of an antibiotic surfactant using Bacillus subtilis. The specificity of the process relies on the use of the surface active property of the product to extract it by foaming. The control law is designed to maintain a constant specific biomass growth rate while taking into account the particularity of the process. This law can be regarded as a generalization of the conventional exponential feeding strategy and is generic enough to encompass the case of continuous processes with partial recycling. Conventional exponential feeding strategies indeed fail to account for the loss of biomass induced by the foaming. Previous experiments have provided a model of the process and values for its parameters. From this information, a feeding rate law was computed using the feeding strategy proposed in this paper and applied to an experimental culture. This experiment allows discussion of the modeling of the biomass extraction method used in this study. The results on the estimated specific growth rate highlight the complete agreement between the expected and experimental features. Further process optimization studies can now be performed on the basis of the constant specific biomass growth rate.     

Simple growth laws and selection consequences

It is commonly thought that various types of population growth can be satisfactorily modelled as deviations from an inherently exponential (malthusian) growth law. Consideration of kinetic results from research on the origin of life, laser physics and more-conventional population dynamics makes it clear, however, that in certain cases the simplest and mechanistically most satisfactory assumption is either a basic subexponential or a hyperbolic growth law. Although these simple growth laws cannot be used instead of more-complicated models of density-dependent population growth when exact quantities are important, the insight gained by thinking them over can be substantial. Ideas about species packing, for example, await reconsideration.     

Relation of DNA replication and cell division frequency in a population of microorganisms to the cell growth rate

DNA replication and the frequency of cell division were studied in a microbial population in relation to the rate of cell growth. The relationship is based on the law of cell biomass linear increase during the cell cycle and on the exponential law of mean cell mass increase, and depends on the specific rate of population growth. The cell mass in the initiation of DNA replication is correlated with the number of initiation points basing on the Cooper-Helmstetter theory of DNA replication and taking account of the linear growth of mass in one cell. Possible applications of these relationships are discussed.     

A general expression for sequential DNA-fluorescence histograms

A general expression for time sequences of DNA-fluorescence histograms from flow microfluorometry is given in this paper. Such expression is given in terms of the law of DNA growth along S phase, the flux of cells into S, and the efflux out of M. Special conditions of growth (e.g. the exponential steady-state growth), and the case of blocks in S phase are also analyzed. Some simulations of the expressions obtained are presented.     

Stochastic kinetics of reproduction of virions inside a cell

We analyze intracellular viral kinetics in the framework of the model incorporating viral genome replication, mRNA synthesis and degradation, protein synthesis and degradation, capsid assembly, and virion release from a cell. Due to the existence of the critical concentration of viral capsid proteins and other features of reproduction of virions inside a cell, the kinetics is demonstrated to exhibit three distinct initial stages. Specifically, (i) the exponential growth of the viral genome, mRNA and protein concentrations is followed by (ii) the transient stage to (iii) the steady-state regime. The formation of mature virions starts during the transient stage. Comparison of the kinetics, obtained by using the mass-action law and Monte Carlo (MC) technique, indicates that they are nearly identical during the initial exponential growth of the viral intermediates and also during the steady-state stage. The transition from the initial stage to the steady-state regime occurs however somewhat faster in the determenistic case even if the steady-state populations of virions and genomes are appreciable (e.g., about 250 and 500, respectively).     

Statistical Basis for Predicting Technological Progress

Forecasting technological progress is of great interest to engineers, policy makers, and private investors. Several models have been proposed for predicting technological improvement, but how well do these models perform? An early hypothesis made by Theodore Wright in 1936 is that cost decreases as a power law of cumulative production. An alternative hypothesis is Moore''s law, which can be generalized to say that technologies improve exponentially with time. Other alternatives were proposed by Goddard, Sinclair et al., and Nordhaus. These hypotheses have not previously been rigorously tested. Using a new database on the cost and production of 62 different technologies, which is the most expansive of its kind, we test the ability of six different postulated laws to predict future costs. Our approach involves hindcasting and developing a statistical model to rank the performance of the postulated laws. Wright''s law produces the best forecasts, but Moore''s law is not far behind. We discover a previously unobserved regularity that production tends to increase exponentially. A combination of an exponential decrease in cost and an exponential increase in production would make Moore''s law and Wright''s law indistinguishable, as originally pointed out by Sahal. We show for the first time that these regularities are observed in data to such a degree that the performance of these two laws is nearly the same. Our results show that technological progress is forecastable, with the square root of the logarithmic error growing linearly with the forecasting horizon at a typical rate of 2.5% per year. These results have implications for theories of technological change, and assessments of candidate technologies and policies for climate change mitigation.