Optimal allocation of building blocks between nutrient uptake systems in a microbe

 A bacterial cell must distribute its molecular building blocks among various types of nutrient uptake systems. If the microbe is to maximize its average growth rate, this allocation of building blocks must be adjusted to the environmental availabilities of the various nutrients. The adjustments can be found from growth balancing considerations. We give a full proof of optimality and uniqueness of the optimal allocation regime for a simple model of microbial growth and internal stores kinetics. This proof suggests likely candidates for optimal control regimes in the case of a more realistic model. These candidate regimes differ with respect to the information that the cells control system must have access to. We pay particular attention to one of the three candidates, a feedback regime based on a cellular control system that monitors only internal reserve densities. We show that allocation converges rapidly to balanced growth under this control regime. Received: 20 November 2000 / Revised version: 7 August 2001 / Published online: 21 February 2002     

Modelling the metabolic versatility of a microbial trichome

A microbial trichome extracts nutrients from its immediate surroundings. It may also oxidize electron donors, reduce electron acceptors, and exude the ‘waste’ products of endogenous redox metabolism. Finally, it may effect light harvesting. These exchange fluxes are summed up in a generic model, which covers photoautotrophs as well as chemoheterotrophs. The focus is on endogenous metabolism and the cellular homeostasis of both reducing and phosphorylating equivalents. A novel result is the formulation of four ‘rules’, akin to the Pasteur effect, which govern the compatibility of endogenous metabolism with various assimilatory processes.     

How Microbes Can Achieve Balanced Growth in a Fluctuating Environment

A microbial colony needs several essential nutrients in order to grow. Moreover, the colony requires these nutrients in fixed combinations, which are dictated by the chemical composition of its biomass. Unfortunately, ambient availabilities of the various nutrients vary all the time. This poses the question of how microbes can achieve balanced growth.The present solution to this problem is novel in that the allocation of molecular building blocks among assimilatory machineries within the cell is regarded as dynamic. This paper shows that allocation can be adapted so as to achieve balanced growth, nearly regardless of environmental conditions. Moreover, it is shown that a feedback mechanism, which monitors internal stores, is able to achieve this allocation.     

Resource availability and the trichome defenses of tomato plants

We conducted two experiments to determine how resource availability influenced allocation by tomato (Lycopersicon esculentum) to trichomes, and how different patterns of trichome allocation by plants grown in different resource environments might then influence the behavior of tobacco hornworm (Manduca sexta) caterpillars. In the first experiment we used high and low levels of light and water, and then, using scanning electron microscopy, determined trichome densities on the leaves and stems. We sampled leaves and stems at several places throughout the plant to determine whether there were within-plant differences in allocation to trichomes. The results of the first experiment showed that resource availability influenced allocation to trichome growth. Patterns in high and low-light supported both the growth-differentiation balance hypothesis (GDBH) and the carbon-nutrient balance hypothesis (CNBH). However, the GDBH was not supported by differences among water treatments. Contrary, to predictions of the GDBH, plants with intermediate growth did not have the highest trichome densities, and plants with similar growth differed in trichome density. Possible biological and artifactual explanations are discussed. The first experiment also showed that there was within-plant variation in allocation to trichomes, and that plant resource availability may influence within-plant variation in allocation to trichomes. In the second experiment, we grew plants in high and low-light, and then monitored the behavior of tobacco hornworms on the stems of these plants in the laboratory. This experiment demonstrated that the light environment that tomato plants were grown in influenced the resting behavior of caterpillars. Furthermore, it demonstrated that both glandular and non-glandular trichomes impeded caterpillars from searching for food. Overall, this study indicated that plant resource availability can influence allocation to trichome defenses, and that these differences may affect insect herbivores.     

Evolution of dispersal distance

The problem of how often to disperse in a randomly fluctuating environment has long been investigated, primarily using patch models with uniform dispersal. Here, we consider the problem of choice of seed size for plants in a stable environment when there is a trade off between survivability and dispersal range. Ezoe (J Theor Biol 190:287–293, 1998) and Levin and Muller-Landau (Evol Ecol Res 2:409–435, 2000) approached this problem using models that were essentially deterministic, and used calculus to find optimal dispersal parameters. Here we follow Hiebeler (Theor Pop Biol 66:205–218, 2004) and use a stochastic spatial model to study the competition of different dispersal strategies. Most work on such systems is done by simulation or nonrigorous methods such as pair approximation. Here, we use machinery developed by Cox et al. (Voter model perturbations and reaction diffusion equations 2011) to rigorously and explicitly compute evolutionarily stable strategies.     

Allometric Effects of <Emphasis Type="Italic">Agriophyllum squarrosum</Emphasis> in Response to Soil Nutrients,Water, and Population Density in the Horqin Sandy Land of China

We monitored the allometric effects for greenhouse-grown Agriophyllum squarrosum plants in response to variations in population density and the availability of soil nutrients and water. Biomass allocations were size-dependent. The plasticity of roots, stems, leaves, and reproductive effort was “true” in response to changes in nutrient content. At a low level of soil minerals, plants allocated more resources to the development of roots and reproductive organs than to leaves, but data for stem allocations were consistent for tradeoffs between the effects of nutrients and plant size. The plasticities of leaf allocation and reproductive effort were “true” whereas those of root and stem allocations were “apparent” in response to fluctuations in soil water, being a function of plant size. Decreasing soil water content was associated with higher leaf allocation and lower reproductive effort. Except for this “apparent” plasticity of leaf allocation, none was detected with population density on biomass allocation. Architectural traits were determinants of the latter. For roots, the determining trait was the ratio of plant height to total biomass; for stems and reproduction, plant height; and for leaves, the ratio of branch numbers to plant height.     

Dynamical Allocation of Cellular Resources as an Optimal Control Problem: Novel Insights into Microbial Growth Strategies

Microbial physiology exhibits growth laws that relate the macromolecular composition of the cell to the growth rate. Recent work has shown that these empirical regularities can be derived from coarse-grained models of resource allocation. While these studies focus on steady-state growth, such conditions are rarely found in natural habitats, where microorganisms are continually challenged by environmental fluctuations. The aim of this paper is to extend the study of microbial growth strategies to dynamical environments, using a self-replicator model. We formulate dynamical growth maximization as an optimal control problem that can be solved using Pontryagin’s Maximum Principle. We compare this theoretical gold standard with different possible implementations of growth control in bacterial cells. We find that simple control strategies enabling growth-rate maximization at steady state are suboptimal for transitions from one growth regime to another, for example when shifting bacterial cells to a medium supporting a higher growth rate. A near-optimal control strategy in dynamical conditions is shown to require information on several, rather than a single physiological variable. Interestingly, this strategy has structural analogies with the regulation of ribosomal protein synthesis by ppGpp in the enterobacterium Escherichia coli. It involves sensing a mismatch between precursor and ribosome concentrations, as well as the adjustment of ribosome synthesis in a switch-like manner. Our results show how the capability of regulatory systems to integrate information about several physiological variables is critical for optimizing growth in a changing environment.     

Coupling of transcription termination to RNAi

Microbes require multiple essential elements that they acquire from the environment independently. Here we investigate how microbial stoichiometry and uptake rates depend on the conditions in which they grow. We modify a recent model of growth based on a multinutrient extension of the Droop model to allow a trade-off between ability to acquire two essential resources. In a static analysis, we show that the optimal allocation strategy is the one that results in co-limitation by both nutrients. We then add a dynamic equation to model the physiological acclimation uptake rates in changing conditions. This dynamic model predicts that the response of organismal stoichiometry to nutrient supply ratio can vary over time. The response of organismal stoichiometry and growth rate to a nutrient pulse depends on the speed at which cells adapt their uptake rates. In a variable environment, very fast or very slow acclimation may be better strategies than intermediate speed acclimation. We suggest experimental tests of the model and avenues for future model development.     

Mathematical models and simulations of bacterial growth and chemotaxis in a diffusion gradient chamber

The diffusion gradient chamber (DGC) is a novel device developed to study the response of chemotactic bacteria to combinations of nutrients and attractants [7]. Its purpose is to characterize genetic variants that occur in many biological experiments. In this paper, a mathematical model which describes the spatial distribution of a bacterial population within the DGC is developed. Mathematical analysis of the model concerning positivity and boundedness of the solutions are given. An ADI (Alternating Direction Implicit) method is constructed for finding numerical solutions of the model and carrying out computer simulations. The numerical results of the model successfully reproduced the patterns that were observed in the experiments using the DGC. Received: 3 June 1997 / Revised version: 15 August 2000 / Published online: 20 December 2000     

Dynamics of biomass partitioning in two competing meadow plant species

The optimal allocation theory predicts that growth is allocated between the shoot and the roots so that the uptake of the most limiting resource is increased. Allocation is dynamic due to resource depletion, interaction with competitors, and the allometry of growth. We assessed the effects of intra- and inter-specific competition on growth and resource allocation of the meadow species Ranunculus acris and Agrostis capillaris, grown in environments with high (+) or low (−) availability of light (L) and nutrients (N). We took samples twice a week over the 7 weeks experiment, to follow the changes in root-to-shoot ratios in plants of different sizes, and carried out a larger scale harvest at the end of the experiment. Of all the tested factors, availability of nutrients had the largest effect on the growth rate and shoot-to-root allocation in both species, although both competition and light had significant effects as well. The highest root-to-shoot ratios were measured from the L+N− treatment, and the lowest from the L−N+ treatment, as predicted by the optimal allocation theory. Competition changed resource allocation, but not always toward acquiring the resource that is most limiting to growth. We thus conclude that the greatest variation in shoot-to-root allocation was due to the resource availability and the effects of competition were small, probably due to low density of plants in the experiment.     

Environmental Heterogeneity and Population Differentiation in Plasticity to Drought in <Emphasis Type="Italic">Convolvulus Chilensis</Emphasis> (Convolvulaceae)

Plant populations may show differentiation in phenotypic plasticity, and theory predicts that greater levels of environmental heterogeneity should select for higher magnitudes of phenotypic plasticity. We evaluated phenotypic responses to reduced soil moisture in plants of Convolvulus chilensis grown in a greenhouse from seeds collected in three natural populations that differ in environmental heterogeneity (precipitation regime). Among several morphological and ecophysiological traits evaluated, only four traits showed differentiation among populations in plasticity to soil moisture: leaf area, leaf shape, leaf area ratio (LAR), and foliar trichome density. In all of these traits plasticity to drought was greatest in plants from the population with the highest interannual variation in precipitation. We further tested the adaptive nature of these plastic responses by evaluating the relationship between phenotypic traits and total biomass, as a proxy for plant fitness, in the low water environment. Foliar trichome density appears to be the only trait that shows adaptive patterns of plasticity to drought. Plants from populations showing plasticity had higher trichome density when growing in soils with reduced moisture, and foliar trichome density was positively associated with total biomass. Co-ordinating editor: F. Stuefer     

Indeterminate growth is selected by a trade-off between high fecundity and risk avoidance in stochastic environments

 We developed a stage-structured model to describe optimal energy allocation among growth, reproduction, and survival. Our model includes stochastic fluctuations in survival rate at age 0 but constant survival rate at older ages. Many mammals and birds cease to grow after maturity (i.e., determinate growth), whereas organisms in a number of other taxa grow beyond maturation (i.e., indeterminate growth). We discuss the conditions under which each of the following strategies is optimal: (I) semelparity, (II) iteroparity with determinate growth, and (III) iteroparity with indeterminate growth. Our model demonstrates that iteroparity with indeterminate growth is selected for when a nonlinear relationship exists between weight and energy production; this strategy is also often selected for in stochastic environments, even with a linear relationship between weight and energy production. The optimal strategy in stochastic environments is to maximize the long-term population growth rate, which does not correspond with maximization of total fecundity. The optimal life history is determined by a balance between spreading a risk and increasing the number of offspring. Our model suggests that optimal life history strategy depends on the magnitude of environmental fluctuations, the advantage of investing in growth, the cost of survival, and the nonlinearity between weight and energy production. Received: February 20, 2002 / Accepted: September 20, 2002 Acknowledgments We thank Drs. Y. Matsumiya, K. Morita, K. Shirakihara, and Y. Watanabe for encouragement and helpful advice. We also thank the responsible editor and anonymous reviewers for helpful comments. This work was supported by a Japan Society for the Promotion of Science grant to H.M. Correspondence to:Y. Katsukawa     

Periodic solutions: a robust numerical method for an S-I-R model of epidemics

 We describe and analyze a numerical method for an S-I-R type epidemic model. We prove that it is unconditionally convergent and that solutions it produces share many qualitative and quantitative properties of the solution of the differential problem being approximated. Finally, we establish explicit sufficient conditions for the unique endemic steady state of the system to be unstable and we use our numerical algorithm to approximate the solution in such cases and discover that it can be periodic, just as suggested by the instability of the endemic steady state. Received: 1 September 1995 / Revised version: 30 April 1997     

Mum, why do you keep on growing?Impacts of environmental variability on optimal growth and reproduction allocation strategies of annual plants

In their 1990 paper Optimal reproductive efforts and the timing of reproduction of annual plants in randomly varying environments, Amir and Cohen considered stochastic environments consisting of i.i.d. sequences in an optimal allocation discrete-time model. We suppose here that the sequence of environmental factors is more generally described by a Markov chain. Moreover, we discuss the connection between the time interval of the discrete-time dynamic model and the ability of the plant to rebuild completely its vegetative body (from reserves). We formulate a stochastic optimization problem covering the so-called linear and logarithmic fitness (corresponding to variation within and between years), which yields optimal strategies. For ``linear maximizers', we analyse how optimal strategies depend upon the environmental variability type: constant, random stationary, random i.i.d., random monotonous. We provide general patterns in terms of targets and thresholds, including both determinate and indeterminate growth. We also provide a partial result on the comparison between ``linear maximizers' and ``log maximizers'. Numerical simulations are provided, allowing to give a hint at the effect of different mathematical assumptions.     

Increased forest carbon storage with increased atmospheric CO2 despite nitrogen limitation: a game‐theoretic allocation model for trees in competition for nitrogen and light

Changes in resource availability often cause competitively driven changes in tree allocation to foliage, wood, and fine roots, either via plastic changes within individuals or through turnover of individuals with differing strategies. Here, we investigate how optimally competitive tree allocation should change in response to elevated atmospheric CO₂ along a gradient of nitrogen and light availability, together with how those changes should affect carbon storage in living biomass. We present a physiologically‐based forest model that includes the primary functions of wood and nitrogen. From a tree's perspective, wood is an offensive and defensive weapon used against neighbors in competition for light. From a biogeochemical perspective, wood is the primary living reservoir of stored carbon. Nitrogen constitutes a tree's photosynthetic machinery and the support systems for that machinery, and its limited availability thus reduces a tree's ability to fix carbon. This model has been previously successful in predicting allocation to foliage, wood, and fine roots along natural productivity gradients. Using game theory, we solve the model for competitively optimal foliage, wood, and fine root allocation strategies for trees in competition for nitrogen and light as a function of CO₂ and nitrogen mineralization rate. Instead of down‐regulating under nitrogen limitation, carbon storage under elevated CO₂ relative to carbon storage at ambient CO₂ is approximately independent of the nitrogen mineralization rate. This surprising prediction is a consequence of both increased competition for nitrogen driving increased fine root biomass and increased competition for light driving increased allocation to wood under elevated CO₂.     

Reconstructing parameters of the FitzHugh–Nagumo system from boundary potential measurements

We consider distributed parameter identification problems for the FitzHugh–Nagumo model of electrocardiology. The model describes the evolution of electrical potentials in heart tissues. The mathematical problem is to reconstruct physical parameters of the system through partial knowledge of its solutions on the boundary of the domain. We present a parallel algorithm of Newton–Krylov type that combines Newton’s method for numerical optimization with Krylov subspace solvers for the resulting Karush–Kuhn–Tucker system. We show by numerical simulations that parameter reconstruction can be performed from measurements taken on the boundary of the domain only. We discuss the effects of various model parameters on the quality of reconstructions. Action Editor: David Terman     

Within-individual plasticity explains age-related decrease in stress response in a short-lived bird

A crucial problem for every organism is how to allocate energy between competing life-history components. The optimal allocation decision is often state-dependent and mediated by hormones. Here, we investigated how age, a major state variable affects individuals'' hormonal response to a standardized stressor: a trait that may reflect allocation between self-maintenance and reproduction. We caught free-living house sparrows and measured their hormonal (corticosterone) response to capture stress in consecutive years. Using a long-term ringing dataset, we determined the age of the birds, and we partitioned the variation into within- and among-individual age components to investigate the effects of plasticity versus selection or gene flow, respectively, on the stress response. We found large among-individual variation in the birds'' hormone profiles, but overall, birds responded less strongly to capture stress as they grew older. These results suggest that stress responsiveness is a plastic trait that may vary within individuals in an adaptive manner, and natural selection may act on the reaction norms producing optimal phenotypic response in the actual environment and life-history stage.     

Genetic control of trichome branch number in Arabidopsis: the roles of the FURCA loci

We are using trichome (hair) morphogenesis as a model to study how plant cell shape is controlled. During a screen for new mutations that affect trichome branch initiation in Arabidopsis, we identified seven new mutants that show a reduction in trichome branch number from three branches to two. These mutations were named furca, after the Latin word for two-pronged fork. These seven recessive mutations were placed into four complementation groups that define four new genes: FURCA1, FURCA2, FURCA3 and FURCA4. The trichome branch number phenotype indicates that the FURCA genes encode positive regulators of trichome branch initiation. Analysis of double mutants suggests that primary and secondary branch initiation events are not genetically distinct, but rely on the levels of partially redundant groups of regulators of trichome branch initiation. Based on the analysis of both epistatic and additive genetic interactions between the FURCA genes and other genes that control trichome branch number, we propose a model that explains how these genes interact to control trichome branch initiation. This model successfully predicts the phenotypes of all the single and double mutants examined and suggests points of control of the trichome branch pathway.     

King eiders use an income strategy for egg production: a case study for incorporating individual dietary variation into nutrient allocation research

The use of stored nutrients for reproduction represents an important component of life-history variation. Recent studies from several species have used stable isotopes to estimate the reliance on stored body reserves in reproduction. Such approaches rely on population-level dietary endpoints to characterize stored reserves (“capital”) and current diet (“income”). Individual variation in diet choice has so far not been incorporated in such approaches, but is crucial for assessing variation in nutrient allocation strategies. We investigated nutrient allocation to egg production in a large-bodied sea duck in northern Alaska, the king eider (Somateria spectabilis). We first used Bayesian isotopic mixing models to quantify at the population level the amount of endogenous carbon and nitrogen invested into egg proteins based on carbon and nitrogen isotope ratios. We then defined the isotopic signature of the current diet of every nesting female based on isotope ratios of eggshell membranes, because diets varied isotopically among individual king eiders on breeding grounds. We used these individual-based dietary isotope signals to characterize nutrient allocation for each female in the study population. At the population level, the Bayesian and the individual-based approaches yielded identical results, and showed that king eiders used an income strategy for the synthesis of egg proteins. The majority of the carbon and nitrogen in albumen (C: 86 ± 18%, N: 99 ± 1%) and the nitrogen in lipid-free yolk (90 ± 15%) were derived from food consumed on breeding grounds. Carbon in lipid-free yolk derived evenly from endogenous sources and current diet (exogenous C: 54 ± 24%), but source contribution was highly variable among individual females. These results suggest that even large-bodied birds traditionally viewed as capital breeders use exogenous nutrients for reproduction. We recommend that investigations of nutrient allocation should incorporate individual variation into mixing models to reveal intraspecific variation in reproductive strategies.     

Constrained proteome allocation affects coexistence in models of competitive microbial communities

Microbial communities are ubiquitous and play crucial roles in many natural processes. Despite their importance for the environment, industry and human health, there are still many aspects of microbial community dynamics that we do not understand quantitatively. Recent experiments have shown that the structure and composition of microbial communities are intertwined with the metabolism of the species that inhabit them, suggesting that properties at the intracellular level such as the allocation of cellular proteomic resources must be taken into account when describing microbial communities with a population dynamics approach. In this work, we reconsider one of the theoretical frameworks most commonly used to model population dynamics in competitive ecosystems, MacArthur’s consumer-resource model, in light of experimental evidence showing how proteome allocation affects microbial growth. This new framework allows us to describe community dynamics at an intermediate level of complexity between classical consumer-resource models and biochemical models of microbial metabolism, accounting for temporally-varying proteome allocation subject to constraints on growth and protein synthesis in the presence of multiple resources, while preserving analytical insight into the dynamics of the system. We first show with a simple experiment that proteome allocation needs to be accounted for to properly understand the dynamics of even the simplest microbial community, i.e. two bacterial strains competing for one common resource. Then, we study our consumer-proteome-resource model analytically and numerically to determine the conditions that allow multiple species to coexist in systems with arbitrary numbers of species and resources.Subject terms: Biodiversity, Microbial ecology, Microbial ecology, Bacterial physiology