NON-STEADY STATE DYNAMICS OF ALGAL POPULATION GROWTH: EXPERIMENTS WITH TWO CHLOROPHYTES1

The relations among dissolved phosphorus, cell quota of phosphorus, and population growth rate were determined for two Chlorophytes, Chlorella sp. and Scenedesmus quadricauda var. longispina (Chod.) G. M. Smith, in two types of non-steady state continuous culture. One of these types had relatively smooth transitions between growth under different degrees of phosphorus limitation. Under these conditions, two equations often applied to growth kinetics in steady state cultures were found to apply to non-steady state growth. Monad's equation described the relation between dissolved phosphorus concentration and population growth rate, and Droop's equation described the relation between cell quota and population growth rate. The second type of culture received phosphorus only as periodic pulses, leading to sharp changes in dissolved phosphorus, cell quota, and growth rate. A simulation model based on Droop's equation described much of the observed dynamics of cell numbers and quotas in these cultures. Droop's equation could not be convincingly fitted directly to the data, however, due to its incorrect prediction of an immediate growth response to phosphorus pulses. A third relation, predicting that saturated rates of phosphorus uptake would depend on the recent nutrient history of the cells as reflected by the cell quota, was not supported.     

Multiple resource limitation theory applied to herbivorous consumers: Liebig's minimum rule vs. interactive co-limitation

There is growing consensus that the growth of herbivorous consumers is frequently limited by more than one nutrient simultaneously. This understanding, however, is based primarily on theoretical considerations and the applicability of existing concepts of co-limitation has rarely been tested experimentally. Here, we assessed the suitability of two contrasting concepts of resource limitation, i.e. Liebig's minimum rule and the multiple limitation hypothesis, to describe nutrient-dependent growth responses of a freshwater herbivore (Daphnia magna) in a system with two potentially limiting nutrients (cholesterol and eicosapentaenoic acid). The results indicated that these essential nutrients interact, and do not strictly follow Liebig's minimum rule, which consistently overestimates growth at co-limiting conditions and thus is not applicable to describe multiple nutrient limitation of herbivorous consumers. We infer that the outcome of resource-based modelling approaches assessing herbivore population dynamics strongly depends on the applied concept of co-limitation.     

Paradox of peramorphic paedomorphosis: Heterochrony and human evolution

This paper reviews Gould's clock model for heterochronic processes and uses that model to develop simple matrix representations of growth and shape change. Matrix representations of growth and development provide a common formulation for all heterochronic processes. In particular, we show how neoteny can be diagnosed using such a matrix approach. The literature is rife with contradictory representations of how neoteny affects growth allometries and the timing of developmental events, and therefore of the role of neoteny in human evolution. Through the use of multivariate models, we explore these relationships and the internal consistency of opposing views. Gould's neoteny hypothesis for human evolution has been criticized for a number of reasons. Humans do not grow slowly. The slopes of our growth allometries show no common pattern of change vis-a-vis those of our closest relatives. Humans prolong rather than reduce rates of growth and development of body parts; the brain, for example, ceases growing later in humans than in apes, but during this prolonged period of early ontogeny, it grows at a rapid pace. This paper evaluates Gould's hypothesis and its critiques by focusing on particular questions. Does neoteny imply slow growth? Does it imply a unidirectional change in the rates of growth of traits? Under neoteny, should the brain cease growing in ancestor and descendant at the same age? Does prolongation of phases of growth and development confute neoteny? On the other hand, is paedomorphosis an inevitable consequence of prolonged growth and development? We show that, for all of these questions, the answer is no. © 1996 Wiley-Liss, Inc.     

Age-related Decline in Forest Ecosystem Growth: An Individual-Tree, Stand-Structure Hypothesis

Forest growth is important both economically (yielding billions of dollars of annual revenues) and ecologically (with respect to ecosystem health and global carbon budgets). The growth of all forests follows a predictable general trend with age. In young forests, it accelerates as canopies develop; it then declines substantially soon after full canopy leaf area is reached. The classic explanation for the decline in growth invoked the increasing respiration costs required to sustain the larger masses of wood characteristic of older forests. Direct measurements of respiration have largely refuted this hypothesis, and recent work has focused on stand-level rates of resource supply, resource use, and growth. We developed and tested a hypothesis at the scale of individual trees (in relation to stand structure) to explain this declining stand-level rate of stem growth. According to our hypothesis, changes in stand structure allow dominant trees to sustain high rates of growth by increasing their acquisition of resources and using these resources efficiently (defined as stem growth per unit of resource used); smaller, nondominant trees grow more slowly as a result of their more limited acquisition of resources and a reduced rate of growth per unit of resource acquired. In combination, these two trends reduce overall stand growth. We tested this hypothesis by comparing growth, growth per unit of leaf area, and variation among trees within plots in two series of plantations of Eucalyptus in Brazil and by estimating individual-tree rates of growth and use of light, water, and nutrients in a plantation of Eucalyptus saligna in Hawaii. Our results supported the individual-tree hypothesis. We conclude that part of the universal age-related decline in forest growth derives from competition-related changes in stand structure and the resource-use efficiencies of individual trees. Received 19 February 2001; accepted 19 June 2001.     

Richards's equation and nonlinear mixed models applied to avian growth: why use them?

Postnatal growth is an important life‐history trait that varies widely across avian species, and several equations with a sigmoidal shape have been used to model it. Classical three‐parameter models have an inflection point fixed at a percentage of the upper asymptote which could be an unrealistic assumption generating biased fits. The Richards model emerged as an interesting alternative because it includes an extra parameter that determines the location of the inflection point which can move freely along the growth curve. Recently, nonlinear mixed models (NLMM) have been used in modeling avian growth because these models can deal with a lack of independence among data as typically occurs with multiple measurements on the same individual or on groups of related individuals. Here, we evaluated the usefulness of von Bertalanffy, Gompertz, logistic, U₄ and Richards's equations modeling chick growth in the imperial shag Phalacrocorax atriceps. We modelled growth in commonly used morphological traits, including body mass, bill length, head length and tarsus length, and compared the performance of models by using NLMM. Estimated adult size, age at maximum growth and maximum growth rates markedly differed across models. Overall, the most consistent performance in estimated adult size was obtained by the Richards model that showed deviations from mean adult size within 5%. Based on AICc values, the Richards equation was the best model for all traits analyzed. For tarsus length, both Richards and U₄ models provided indistinguishable fits because the relative inflection value estimated from the Richards model was very close to that assumed by the U₄ model. Our results highlight the bias incurred by three‐parameter models when the assumed inflection placement deviates from that derived from data. Thus, the application of the Richards equation using the NLMM framework represents a flexible and powerful tool for the analysis of avian growth.     

A theoretical approach to structuring mechanisms in the pelagic food web

In the literature there is a commonly used idealized concept of the foodweb structure in the pelagic photic zone food web, based to a large extenton size dependent relationships. An outline is here given of how theelementary size-related physical laws of diffusion and sinking, combinedwith the assumption of predators being size selective in their choice ofprey, give a theoretical foundation for this type of structure. It is shownhow such a theoretical fundament makes it possible to relate a broad specterof phenomena within one generic and consistent framework. Phenomena such asHutchinson‘s and Goldman‘s paradoxes, the influence of nutrients and watercolumn stability on the balance between microbial and classical food webs,bacterial carbon consumption, new production and export of DOC and POC tothe aphotic zone, eutrophication and diversity, can all be approached fromthis perspective. By including host-specific viruses, this approach gives ahierarchical structure to the control of diversity with nutrient contentcontrolling the maximum size of the photic zone community, size selectivityof predators regulating how the nutrient is distributed between size-groupsof osmotrophic and phagotrophic organisms, and viral host specificityregulating how the nutrients within a size group is distributed between hostgroups. I also briefly discuss how some biological strategies may besuccessful by not conforming to the normal rules of such a framework.Analyzing the behavior of these idealized systems is thus claimed tofacilitate our understanding of the behavior of complex natural food webs. This revised version was published online in July 2006 with corrections to the Cover Date.     

Network analysis of intermediary metabolism using linear optimization. I. Development of mathematical formalism.

Analysis of metabolic networks using linear optimization theory allows one to quantify and understand the limitations imposed on the cell by its metabolic stoichiometry, and to understand how the flux through each pathway influences the overall behavior of metabolism. A stoichiometric matrix accounting for the major pathways involved in energy and mass transformations in the cell was used in our analysis. The auxiliary parameters of linear optimization, the so-called shadow prices, identify the intermediates and cofactors that cause the growth to be limited on each nutrient. This formalism was used to examine how well the cell balances its needs for carbon, nitrogen, and energy during growth on different substrates. The relative values of glucose and glutamine as nutrients were compared by varying the ratio of rates of glucose to glutamine uptakes, and calculating the maximum growth rate. The optimum value of this ratio is between 2-7, similar to experimentally observed ratios. The theoretical maximum growth rate was calculated for growth on each amino acid, and the amino acids catabolized directly to glutamate were found to be the optimal nutrients. The importance of each reaction in the network can be examined both by selectively limiting the flux through the reaction, and by the value of the reduced cost for that reaction. Some reactions, such as malic enzyme and glutamate dehydrogenase, may be inhibited or deleted with little or no adverse effect on the calculated cell growth rate.     

The many growth rates and elasticities of populations in random environments

Despite considerable interest in the dynamics of populations subject to temporally varying environments, alternate population growth rates and their sensitivities remain incompletely understood. For a Markovian environment, we compare and contrast the meanings of the stochastic growth rate (lambdaS), the growth rate of average population (lambdaM), the growth rate for average transition rates (lambdaA), and the growth rate of an aggregate represented by a megamatrix (shown here to equal lambdaM). We distinguish these growth rates by the averages that define them. We illustrate our results using data on an understory shrub in a hurricane-disturbed landscape, employing a range of hurricane frequencies. We demonstrate important differences among growth rates: lambdaS lambdaM. We show that stochastic elasticity, ESij, and megamatrix elasticity, EMij, describe a complex perturbation of both means and variances of rates by the same proportion. Megamatrix elasticities respond slightly and stochastic elasticities respond strongly to changing the frequency of disturbance in the habitat (in our example, the frequency of hurricanes). The elasticity EAij of lambdaA does not predict changes in the other elasticities. Because ES, although commonly utilized, is difficult to interpret, we introduce elasticities with a more direct interpretation: ESmu for perturbations of means and ESsigma for variances. We argue that a fundamental tool for studying selection pressures in varying environments is the response of growth rate to vital rates in all habitat states.     

Equivalence of regression curves sharing common parameters

In clinical trials, the comparison of two different populations is a common problem. Nonlinear (parametric) regression models are commonly used to describe the relationship between covariates, such as concentration or dose, and a response variable in the two groups. In some situations, it is reasonable to assume some model parameters to be the same, for instance, the placebo effect or the maximum treatment effect. In this paper, we develop a (parametric) bootstrap test to establish the similarity of two regression curves sharing some common parameters. We show by theoretical arguments and by means of a simulation study that the new test controls its significance level and achieves a reasonable power. Moreover, it is demonstrated that under the assumption of common parameters, a considerably more powerful test can be constructed compared with the test that does not use this assumption. Finally, we illustrate the potential applications of the new methodology by a clinical trial example.     

Merging resource availability with isotope mixing models: the role of neutral interaction assumptions

Background

Bayesian mixing models have allowed for the inclusion of uncertainty and prior information in the analysis of trophic interactions using stable isotopes. Formulating prior distributions is relatively straightforward when incorporating dietary data. However, the use of data that are related, but not directly proportional, to diet (such as prey availability data) is often problematic because such information is not necessarily predictive of diet, and the information required to build a reliable prior distribution for all prey species is often unavailable. Omitting prey availability data impacts the estimation of a predator''s diet and introduces the strong assumption of consumer ultrageneralism (where all prey are consumed in equal proportions), particularly when multiple prey have similar isotope values.

Methodology

We develop a procedure to incorporate prey availability data into Bayesian mixing models conditional on the similarity of isotope values between two prey. If a pair of prey have similar isotope values (resulting in highly uncertain mixing model results), our model increases the weight of availability data in estimating the contribution of prey to a predator''s diet. We test the utility of this method in an intertidal community against independently measured feeding rates.

Conclusions

Our results indicate that our weighting procedure increases the accuracy by which consumer diets can be inferred in situations where multiple prey have similar isotope values. This suggests that the exchange of formalism for predictive power is merited, particularly when the relationship between prey availability and a predator''s diet cannot be assumed for all species in a system.     

A critical test of the two prevailing theories of plant response to nutrient availability

Whereas the "law of the minimum" (LM) states that plant growth is limited by a single resource at any one time, the "multiple limitation hypothesis" (MLH) proposes that optimum plant behavior results from balancing resource costs and benefits so that all resources limit plant growth simultaneously. We tested the hypothesis that neither the LM nor the MLH account for plant responses to all mineral nutrients. Fronds of the aquatic plant Lemna minor were grown in nutrient solutions with increasing levels of four nutrients: nitrogen, phosphorus, potassium, and magnesium. Neither LM or MLH adequately predicted plant responses to all of these nutrients: 23 of the 60 responses analyzed were classified as belonging to the LM; 20 cases were classified as undefined; and 17 cases as MLH. The type of response strongly depended on the specific pair of nutrients considered. The validity of the MLH model would depend on the accompanying resource limiting plant growth and on the severity of the stress. We propose that a "nutrient-specific" analysis, considering the biology of each mineral nutrient rather than grouping plant resources as a whole, is more appropriate than general models in understanding plant responses to nutrient availability.     

Effects of models of rate evolution on estimation of divergence dates with special reference to the metazoan 18S ribosomal RNA phylogeny

The molecular clock, i.e., constancy of the rate of evolution over time, is commonly assumed in estimating divergence dates. However, this assumption is often violated and has drastic effects on date estimation. Recently, a number of attempts have been made to relax the clock assumption. One approach is to use maximum likelihood, which assigns rates to branches and allows the estimation of both rates and times. An alternative is the Bayes approach, which models the change of the rate over time. A number of models of rate change have been proposed. We have extended and evaluated models of rate evolution, i.e., the lognormal and its recent variant, along with the gamma, the exponential, and the Ornstein-Uhlenbeck processes. These models were first applied to a small hominoid data set, where an empirical Bayes approach was used to estimate the hyperparameters that measure the amount of rate variation. Estimation of divergence times was sensitive to these hyperparameters, especially when the assumed model is close to the clock assumption. The rate and date estimates varied little from model to model, although the posterior Bayes factor indicated the Ornstein-Uhlenbeck process outperformed the other models. To demonstrate the importance of allowing for rate change across lineages, this general approach was used to analyze a larger data set consisting of the 18S ribosomal RNA gene of 39 metazoan species. We obtained date estimates consistent with paleontological records, the deepest split within the group being about 560 million years ago. Estimates of the rates were in accordance with the Cambrian explosion hypothesis and suggested some more recent lineage-specific bursts of evolution.     

Species synchrony and its drivers: neutral and nonneutral community dynamics in fluctuating environments

Independent species fluctuations are commonly used as a null hypothesis to test the role of competition and niche differences between species in community stability. This hypothesis, however, is unrealistic because it ignores the forces that contribute to synchronization of population dynamics. Here we present a mechanistic neutral model that describes the dynamics of a community of equivalent species under the joint influence of density dependence, environmental forcing, and demographic stochasticity. We also introduce a new standardized measure of species synchrony in multispecies communities. We show that the per capita population growth rates of equivalent species are strongly synchronized, especially when endogenous population dynamics are cyclic or chaotic, while their long-term fluctuations in population sizes are desynchronized by ecological drift. We then generalize our model to nonneutral dynamics by incorporating temporal and nontemporal forms of niche differentiation. Niche differentiation consistently decreases the synchrony of species per capita population growth rates, while its effects on the synchrony of population sizes are more complex. Comparing the observed synchrony of species per capita population growth rates with that predicted by the neutral model potentially provides a simple test of deterministic asynchrony in a community.     

Modeling nonhomogeneous Markov processes via time transformation

Summary .   Longitudinal studies are a powerful tool for characterizing the course of chronic disease. These studies are usually carried out with subjects observed at periodic visits giving rise to panel data. Under this observation scheme the exact times of disease state transitions and sequence of disease states visited are unknown and Markov process models are often used to describe disease progression. Most applications of Markov process models rely on the assumption of time homogeneity, that is, that the transition rates are constant over time. This assumption is not satisfied when transition rates depend on time from the process origin. However, limited statistical tools are available for dealing with nonhomogeneity. We propose models in which the time scale of a nonhomogeneous Markov process is transformed to an operational time scale on which the process is homogeneous. We develop a method for jointly estimating the time transformation and the transition intensity matrix for the time transformed homogeneous process. We assess maximum likelihood estimation using the Fisher scoring algorithm via simulation studies and compare performance of our method to homogeneous and piecewise homogeneous models. We apply our methodology to a study of delirium progression in a cohort of stem cell transplantation recipients and show that our method identifies temporal trends in delirium incidence and recovery.     

Metabolic capacity of Bacillus subtilis for the production of purine nucleosides, riboflavin, and folic acid

We developed a stoichiometric model of Bacillus subtilis metabolism for quantitative analysis of theoretical growth and biochemicals production capacity. This work concentrated on biochemicals that are derived from the purine biosynthesis pathway; inosine, guanosine, riboflavin, and folic acid. These are examples of commercially relevant biochemicals for which Bacillus species are commonly used production hosts. Two previously unrecognized, but highly desirable properties of good producers of purine pathway-related biochemicals have been identified for optimally engineered product biosynthesis; high capacity for reoxidation of NADPH and high bioenergetic efficiency. Reoxidation of NADPH, through the transhydrogenase or otherwise, appears to be particularly important for growth on glucose, as deduced from the corresponding optimal carbon flux distribution. The importance of cellular energetics on optimal performance was quantitatively assessed by including a bioenergetic efficiency parameter as an unrestricted, ATP dissipating flux in the simulations. An estimate for the bioenergetic efficiency was generated by fitting the model to experimentally determined growth yields. The results show that the maximum theoretical yields of all products studied are limited by pathway stoichiometry at high bioenergetic efficiencies. Simulations with the estimated bioenergetic efficiency of B. subtilis, growing under glucose-limiting conditions, indicate that the yield of these biochemicals is primarily limited by energy and thus is very sensitive to the process conditions. The maximum yields that can reasonably be expected with B. subtilis on glucose were estimated to be 0.343, 0.160, and 0.161 (mol product/mol glucose) for purine nucleosides, riboflavin, and folic acid, respectively. Potential strategies for improving these maximum yields are discussed.     

Comparing relative rates of pollen and seed gene flow in the island model using nuclear and organelle measures of population structure

We describe a method for comparing nuclear and organelle population differentiation (F(ST)) in seed plants to test the hypothesis that pollen and seed gene flow rates are equal. Wright's infinite island model is used, with arbitrary levels of self-fertilization and biparental organelle inheritance. The comparison can also be applied to gene flow in animals. Since effective population sizes are smaller for organelle genomes than for nuclear genomes and organelles are often uniparentally inherited, organelle F(ST) is expected to be higher at equilibrium than nuclear F(ST) even if pollen and seed gene flow rates are equal. To reject the null hypothesis of equal seed and pollen gene flow rates, nuclear and organelle F(ST)'s must differ significantly from their expected values under this hypothesis. Finite island model simulations indicate that infinite island model expectations are not greatly biased by finite numbers of populations (>/=100 subpopulations). The power to distinguish dissimilar rates of pollen and seed gene flow depends on confidence intervals for fixation index estimates, which shrink as more subpopulations and loci are sampled. Using data from the tropical tree Corythophora alta, we rejected the null hypothesis that seed and pollen gene flow rates are equal but cannot reject the alternative hypothesis that pollen gene flow is 200 times greater than seed gene flow.     

The concept of multiple-nutrient-limited growth of microorganisms and its application in biotechnological processes

The "law of the minimum" (Liebig's law) states that usually one nutrient restricts the maximum quantity of biomass that can be produced within a system, whereas all other nutrients are in excess. This general rule has been applied also to the growth of microorganisms, e.g., by adjusting the relative concentrations of the individual nutrients in growth media such that one of them, in the case of heterotrophic microbes, usually the carbon source, determines the maximum cell density that can be obtained in a culture. However, experimental data demonstrated that growth of microbial cultures can be limited simultaneously by two or more nutrients. These authors reported that during growth of bacteria and yeasts at a constant dilution rate in the chemostat, three distinct growth regimes were recognised as a function of the C:N ratio in the inflowing medium: (1) a clearly carbon-limited regime with the nitrogen source in excess, (2) a transition ("double-nutrient-limited") growth regime where both the carbon and the nitrogen source were below the detection limit, and (3) a clearly nitrogen-limited growth regime with the carbon source in excess. Subsequent calculations suggested that the extension and position of this double-nutrient-limited zone should be strongly dependent on the imposed growth rate: Whereas it is very narrow at high growth rates it should become very broad during slow growth. This pattern as a function of growth rate has now been confirmed for a number of different organisms. In industrial processes, microbial growth is always in some way controlled by the limited availability of nutrients, and limitation of specific nutrients is frequently used to force microbial cultures into a productive physiological state. This article will discuss what the consequences of multiple-nutrient-limited growth are for industrial processes and how the concept might be applied. Specific examples will be given that demonstrate the advantages and the potential of multiple nutrient-limited growth conditions for industrial production processes.     

Universality of thermodynamic constants governing biological growth rates

Background

Mathematical models exist that quantify the effect of temperature on poikilotherm growth rate. One family of such models assumes a single rate-limiting ‘master reaction’ using terms describing the temperature-dependent denaturation of the reaction''s enzyme. We consider whether such a model can describe growth in each domain of life.

Methodology/Principal Findings

A new model based on this assumption and using a hierarchical Bayesian approach fits simultaneously 95 data sets for temperature-related growth rates of diverse microorganisms from all three domains of life, Bacteria, Archaea and Eukarya. Remarkably, the model produces credible estimates of fundamental thermodynamic parameters describing protein thermal stability predicted over 20 years ago.

Conclusions/Significance

The analysis lends support to the concept of universal thermodynamic limits to microbial growth rate dictated by protein thermal stability that in turn govern biological rates. This suggests that the thermal stability of proteins is a unifying property in the evolution and adaptation of life on earth. The fundamental nature of this conclusion has importance for many fields of study including microbiology, protein chemistry, thermal biology, and ecological theory including, for example, the influence of the vast microbial biomass and activity in the biosphere that is poorly described in current climate models.     

The dynamic evolution of the power exponent in a universal growth model of tumors

We have previously reported that a universal growth law, as proposed by West and collaborators for all living organisms, appears to be able to describe also the growth of tumors in vivo after an initial exponential growth phase. In contrast to the assumption of a fixed power exponent p (assumed by West et al. to be equal to 3/4), we propose in this paper a dynamic evolution of p, using experimental data from the cancer literature. In analogy with results obtained by applying scaling laws to the study of fragmentation of solids, the dynamic behaviour of p is related to the evolution of the fractal topology of neoplastic vascular systems. Our model might be applied for diagnostic purposes to mark the emergence of an efficient neo-angiogenetic structure if the results of our in silico experiments are confirmed by clinical observations.     

Zebrafish Offers New Perspective on Developmental Role of TOR Signaling

Genetic studies on the molecular basis of growth control have converged on the target of rapamycin (TOR) pathway as a key regulator. When stimulated by nutrients (i.e. amino acids) or growth factors (i.e. insulin), TOR activates protein synthesis and other anabolic pathways to promote cell growth. Our knowledge of TOR's function in vivo is still rudimentary, particularly in the setting of vertebrate development. An important question is whether TOR functions as a constitutive regulator of growth in all cell types, or as a stage- and organ-specific regulator. Recently we employed the zebrafish as a vertebrate model system to study the developmental role of TOR signaling. We found that TOR signaling was required for a discrete step prior to epithelial differentiation. The results support the view that different organs may be reliant on TOR activity to differing degrees. In the case of the zebrafish, the digestive tract exhibits the greatest sensitivity to rapamycin, which may reflect its reliance on TOR signaling for normal growth. We suggest the hypothesis that TOR signaling may regulate the size of the intestine's absorptive surface area in response to systemic nutrient demand.