Global analysis of a model of plasmid-bearing,plasmid-free competition in a chemostat

A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat was proposed in a paper of Stephanopoulis and Lapidus. The model was in the form of a system of nonlinear ordinary differential equations. Such models are relevant to commercial production by genetically altered organisms in continuous culture. The analysis there was local (using index arguments). This paper provides a mathematically rigorous analysis of the global asymptotic behavior of the governing equations in the case of uninhibited specific growth rate.Research supported by the National Council of Science, Republic of ChinaResearch supported by National Science Foundation Grant, DMS-9204490Research supported by the Natural Science and Engineering Council of Canada. This author's contribution was made while on research leave visiting the Department of Ecology and Evolutionary Biology at Princeton University. She would especially like to thank Simon Levin for his guidance as well as for providing an exceptional working environment     

具有抑制剂的恒化器竞争模型的分析（英文）

考虑了具有抑制剂的两种群竞争一种微生物的恒化器模型,其中吸收函数和功能反应函数都是营养的一般单调递增函数,并且一种微生物能够分泌一种对另一种微生物起致命影响的抑制剂.利用常微分方程定性理论,首先得到了平衡点的存在条件和局部渐进稳定性;然后讨论了全局渐进稳定性,以及极限环和Hopf分支的存在性.     

Predictions of response to temperature are contingent on model choice and data quality

The equations used to account for the temperature dependence of biological processes, including growth and metabolic rates, are the foundations of our predictions of how global biogeochemistry and biogeography change in response to global climate change. We review and test the use of 12 equations used to model the temperature dependence of biological processes across the full range of their temperature response, including supra‐ and suboptimal temperatures. We focus on fitting these equations to thermal response curves for phytoplankton growth but also tested the equations on a variety of traits across a wide diversity of organisms. We found that many of the surveyed equations have comparable abilities to fit data and equally high requirements for data quality (number of test temperatures and range of response captured) but lead to different estimates of cardinal temperatures and of the biological rates at these temperatures. When these rate estimates are used for biogeographic predictions, differences between the estimates of even the best‐fitting models can exceed the global biological change predicted for a decade of global warming. As a result, studies of the biological response to global changes in temperature must make careful consideration of model selection and of the quality of the data used for parametrizing these models.     

Plasmid-bearing,plasmid-free organisms competing for two complementary nutrients in a chemostat

A model of competition for two complementary nutrients between plasmid-bearing and plasmid-free organisms in a chemostat is proposed. A rigorous mathematical analysis of the global asymptotic behavior of the model is presented. The work extends the model of competition for a single-limited nutrient studied by Stephanopoulos and Lapidus [Chem. Engng. Sci. 443 (1988) 49] and Hsu, Waltman and Wolkowicz [J. Math. Biol. 32 (1994) 731].     

Following FRET through five energy transfer steps: spectroscopic photobleaching, recovery of spectra, and a sequential mechanism of FRET.

We report the acquisition and analysis of spectrally resolved photobleaching data from a model system designed to exhibit FRET. Spectrally resolved photobleaching can be used to determine the presence of FRET in these systems and to investigate multi-step mechanisms of energy transfer. The model system was a previously described set of fluorescent beads consisting of a system of six fluorophores. In standard photobleaching experiments to determine FRET, bleaching of an acceptor molecule resulting in recovery of donor intensity or changes in photobleaching kinetics are used as indicators of FRET. Here, we use the Bateman equations to model growth and decay in a photobleaching experiment. Linked donor-acceptor growth and decay is used as an indicator of FRET. The apparatus required is relatively simple when compared to lifetime imaging systems. Several data analysis strategies, rigorous model building, global fitting procedures, and error analysis are presented. Using these procedures a five-step sequential mechanism of energy transfer was selected for these beads.     

Growth and development rates have different thermal responses

Growth and development rates are fundamental to all living organisms. In a warming world, it is important to determine how these rates will respond to increasing temperatures. It is often assumed that the thermal responses of physiological rates are coupled to metabolic rate and thus have the same temperature dependence. However, the existence of the temperature-size rule suggests that intraspecific growth and development are decoupled. Decoupling of these rates would have important consequences for individual species and ecosystems, yet this has not been tested systematically across a range of species. We conducted an analysis on growth and development rate data compiled from the literature for a well-studied group, marine pelagic copepods, and use an information-theoretic approach to test which equations best describe these rates. Growth and development rates were best characterized by models with significantly different parameters: development has stronger temperature dependence than does growth across all life stages. As such, it is incorrect to assume that these rates have the same temperature dependence. We used the best-fit models for these rates to predict changes in organism mass in response to temperature. These predictions follow a concave relationship, which complicates attempts to model the impacts of increasing global temperatures on species body size.     

The growth of pure and simple microbial competitors in a moving distributed medium.

The dynamics of pure and simple competition between two microbial species are examined for the case of interaction arising in a distributed and nonstagnant environment. The environment is modeled as a tubular reactor. It is shown that for relatively small values of the dispersion coefficient (i.e., for small, but nonzero, backmixing of the medium), the two competing populations can coexist in a stable steady state. It has been assumed that the species grow uninhibited and that if there are maintenance requirements they are satisfied from endogenous sources. From numerical studies it has been found that a necessary condition for coexistence is that the net specific growth rate curves of the two competitors cross each other at a positive value of the concentration of the rate-limiting substrate. The model equations have been numerically solved by using the methods of orthogonal and spline collocation.     

Flux balancing of light and nutrients in a biofilm photobioreactor for maximizing photosynthetic productivity

This article reports a combined experimental and numerical study on the efficient operation of Porous Substrate Bioreactors. A comprehensive model integrating light transport, mass transport, and algal growth kinetics was used to understand the productivity of photosynthetic biofilms in response to delivery rates of photons and nutrients. The reactor under consideration was an evaporation driven Porous Substrate Bioreactor (PSBR) cultivating the cyanobacteria Anabaena variabilis as a biofilm on a porous substrate which delivers water and nutrients to the cells. In an unoptimized experimental case, this reactor was operated with a photosynthetic efficiency of 2.3%, competitive with conventional photobioreactors. Moreover, through a scaling analysis, the location at which the phosphate delivery rate decreased the growth rate to half of its uninhibited value was predicted as a function of microorganism and bioreactor properties. The numerical model along with the flux balancing techniques presented herein can serve as tools for designing and selecting operating parameters of biofilm based cultivation systems for maximum productivity. © 2014 American Institute of Chemical Engineers Biotechnol. Prog., 30:348–359, 2014     

Measurement and mathematical modelling of competition between fast- and slow-growing ordinary heterotrophic organisms in low and high substrate-loaded systems

Ordinary heterotrophic organisms (OHO) of an activated sludge wastewater treatment system showed an atypical growth behaviour when they are inoculated to batch aerobic growth tests with a high substrate-loaded condition. For example, the OHO maximum specific growth rates on readily biodegradable substrates (μ _H) increased with a high ratio of substrate concentration to OHO active biomass concentration (So/Xo), although they were assumed to be constant in a conventional microbial growth kinetic model with a single OHO population group. We, therefore, set a hypothesis in that the change of OHO maximum specific growth rates in the batch test condition is caused by turnover of fast-growing OHO population against slow-growing OHO population. And, a competitive microbial growth kinetic model of the fast- and slow-growing OHO population groups was developed and validated with model-data fitting analysis for the batch test results. The competitive microbial growth kinetic model of process selection, rather than that of kinetic selection, was capable of simulating microbial growth kinetics in high substrate-loaded dynamic systems (i.e., batch tests) and low substrate-loaded steady-state systems (i.e., continuously operated wastewater treatment systems), better than the conventional non-competitive growth kinetic model.     

ETA: Robust software for determination of cell specific rates from extracellular time courses

Accurate quantification of cell specific rates and their uncertainties is of critical importance for assessing metabolic phenotypes of cultured cells. We applied two different methods of regression and error analysis to estimate specific metabolic rates from time‐course measurements obtained in exponentially growing cell cultures. Using simulated data sets to compute specific rates of growth, glucose uptake, and lactate excretion, we found that Gaussian error propagation from prime variables to the final calculated rates was the most accurate method for estimating parameter uncertainty. We incorporated this method into a MATLAB‐based software package called Extracellular Time‐Course Analysis (ETA), which automates the analysis workflow required to (i) compute cell specific metabolic rates and their uncertainties; (ii) test the goodness‐of‐fit of the experimental data to the regression model; and (iii) rapidly compare the results across multiple experiments. ETA was used to estimate the uptake or excretion rate of glucose, lactate, and 18 different amino acids in a B‐cell model of c‐Myc‐driven cancer. We found that P493‐6 cells with High Myc expression increased their specific uptake of glutamine, arginine, serine, lysine, and branched‐chain amino acids by two‐ to threefold in comparison to low Myc cells, but exhibited only modest increases in glucose uptake and lactate excretion. By making the ETA software package freely available to the scientific community, we expect that it will become an important tool for rigorous estimation of specific rates required for metabolic flux analysis and other quantitative metabolic studies. Biotechnol. Bioeng. 2013; 110: 1748–1758. © 2013 Wiley Periodicals, Inc.     

From macroscopic measurements to microscopic mechanisms of protein aggregation

The ability to relate bulk experimental measurements of amyloid formation to the microscopic assembly processes that underlie protein aggregation is critical in order to achieve a quantitative understanding of this complex phenomenon. In this review, we focus on the insights from classical and modern theories of linear growth phenomena and discuss how theory allows the roles of growth and nucleation processes to be defined through the analysis of experimental in vitro time courses of amyloid formation. Moreover, we discuss the specific signatures in the time course of the reactions that correspond to the actions of primary and secondary nucleation processes, and outline strategies for identifying and characterising the nature of the dominant process responsible in each case for the generation of new aggregates. We highlight the power of a global analysis of experimental time courses acquired under different conditions, and discuss how such an analysis allows a rigorous connection to be established between the macroscopic measurements and the rates of the individual microscopic processes that underlie the phenomenon of amyloid formation.     

Global dynamics of a selection model for the growth of a population with genotypic fertility differences

A population growth model is considered for a one locus two allele problem with selection based entirely on fertility differences. A local stability analysis is carried out for the critical points — which include possible polymorphic states — of the resulting nonlinear differential equations. The methods of dynamical systems theory are applied to obtain limiting genotypic proportions for every initial state. Thus the results are global and there are no periodic solutions.Research for this paper was partially supported by the National Science and Engineering Research Council of Canada Grant NSERC A-8130Research for this paper was partially supported by the National Science and Engineering Research Council of Canada Grant NSERC A-4823Research supported by NSF Grant MCS 7901069. A portion of the work was carried out while the author was a Visiting Professor at the University of Utah, Salt Lake City, Utah     

Mathematical analysis of a delayed stage-structured predator–prey model with impulsive diffusion between two predators territories

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete, but in reality, many species diffuse only during a single period, and diffusion often occurs in regular pulses. Further, in forest habitats, the highest-level predator species are restricted to a specific territory, but prey can impulsively move between territories. Therefore, in this paper, we consider a delayed stage-structured predator–prey model with impulsively diffusive prey between two patches; in the model, patches represent the territories of two different predator populations. Here, we analytically obtain the global attractivity condition of predator-extinction periodic solutions for the system by using the concepts of Hui and Chen (2005); a numerical simulation is also included to illustrate this result. Further, we establish permanence conditions for the coexistence of the species using the theory of impulsive delayed differential equations. Finally, we explore the possibilities of the permanence of the system by using the growth rates of immature predators and the impulse period as critical parameters, and we also obtain the parameters’ threshold limits using numerical experimentation.     

A unified statistical model for functional mapping of environment-dependent genetic expression and genotype x environment interactions for ontogenetic development

The effects of quantitative trait loci (QTL) on phenotypic development may depend on the environment (QTL x environment interaction), other QTL (genetic epistasis), or both. In this article, we present a new statistical model for characterizing specific QTL that display environment-dependent genetic expressions and genotype x environment interactions for developmental trajectories. Our model was derived within the maximum-likelihood-based mixture model framework, incorporated by biologically meaningful growth equations and environment-dependent genetic effects of QTL, and implemented with the EM algorithm. With this model, we can characterize the dynamic patterns of genetic effects of QTL governing growth curves and estimate the global effect of the underlying QTL during the course of growth and development. In a real example with rice, our model has successfully detected several QTL that produce differences in their genetic expression between two contrasting environments. These detected QTL cause significant genotype x environment interactions for some fundamental aspects of growth trajectories. The model provides the basis for deciphering the genetic architecture of trait expression adjusted to different biotic and abiotic environments and genetic relationships for growth rates and the timing of life-history events for any organism.     

Impact of spatial heterogeneity on a predator-prey system dynamics

This paper deals with the study of a predator-prey model in a patchy environment. Prey individuals moves on two patches, one is a refuge and the second one contains predator individuals. The movements are assumed to be faster than growth and predator-prey interaction processes. Each patch is assumed to be homogeneous. The spatial heterogeneity is obtained by assuming that the demographic parameters (growth rates, predation rates and mortality rates) depend on the patches. On the predation patch, we use a Lotka-Volterra model. Since the movements are faster that the other processes, we may assume that the frequency of prey and predators become constant and we would get a global predator-prey model, which is shown to be a Lotka-Volterra one. However, this simplified model at the population level does not match the dynamics obtained with the complete initial model. We explain this phenomenom and we continue the analysis in order to give a two-dimensional predator-prey model that gives the same dynamics as that provided by the complete initial one. We use this simplified model to study the impact of spatial heterogeneity and movements on the system stability. This analysis shows that there is a globally asymptotically stable equilibrium in the positive quadrant, i.e. the spatial heterogeneity stabilizes the equilibrium.     

The effects of nutrient limitation and growth rate in the chemostat on the immunogenicity of Vibrio cholerae

The possibility was examined of using the chemostat to produce cholera vaccine of immunogenicity similar or superior to that of conventional vaccine produced from organisms grown on nutrient agar. The immunogenicity in mice of vaccines prepared from organisms grown under carbon, nitrogen, magnesium and phosphate limitation each at low and high growth rates was inferior to that of conventional vaccine. It was concluded that under the conditions used in this study the chemostat could not be used to produce cholera vaccine of acceptable immunogenicity. The sensitivity of organisms grown under the same conditions in the chemostat to agglutination by specific agglutinating antiserum showed phenotypic variation and under carbon limitation was growth rate related, although the increased sensitivity to agglutination of carbon-limited organisms at high growth rates was not paralleled by increased ability of these organisms to induce agglutinating antibodies.     

Computer programs for modeling mammalian cell batch and fed-batch cultures using logistic equations

A MATLAB^® toolbox was developed for applying the logistic modeling approach to mammalian cell batch and fed-batch cultures. The programs in the toolbox encompass sensitivity analyses and simulations of the logistic equations in addition to cell specific rate estimation. The toolbox was first used to generate time courses of the sensitivity equations for characterizing the relationship between the logistic variable and the model parameters. Subsequently, the toolbox was used to describe CHO cell data from batch and fed-batch mammalian cell cultures. Cell density, product, glucose, lactate, glutamine, and ammonia data were analyzed for the batch culture while fed-batch analysis included cell density and product concentration. In all instances, experimental data were well described by the logistic equations and the resulting specific rate profiles were representative of the underlying cell physiology. The 6-variable batch culture data set was also used to compare the logistic specific rates with those from polynomial fitting and discrete derivative methods. The polynomial specific rates grossly misrepresented cell behavior in the initial and final stages of culture while those based on discrete derivatives had high variability due to computational artifacts. The utility of logistic specific rates to guide process development activities was demonstrated using specific protein productivity versus growth rate trajectories for the 3 cultures examined in this study. Overall, the computer programs developed in this study enable rapid and robust analysis of data from mammalian cell batch and fed-batch cultures which can help process development and metabolic flux estimation.     

Multiscale modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures

This work was motivated by the problems of analysing detailed 3D models of vascular districts with complex anatomy. It suggests an approach to prescribing realistic boundary conditions to use in order to obtain information on local as well as global haemodynamics. A method was developed which simultaneously solves Navier-Stokes equations for local information and a non-linear system of ordinary differential equations for global information. This is based on the principle that an anatomically detailed 3D model of a cardiovascular district can be achieved by using the finite element method. In turn the finite element method requires a specific boundary condition set. The approach outlined in this work is to include the system of ordinary differential equations in the boundary condition set. Such a multiscale approach was first applied to two controls: (i) a 3D model of a straight tube in a simple hydraulic network and (ii) a 3D model of a straight coronary vessel in a lumped-parameter model of the cardiovascular system. The results obtained are very close to the solutions available for the pipe geometry. This paper also presents preliminary results from the application of the methodology to a particular haemodynamic problem: namely the fluid dynamics of a systemic-to-pulmonary shunt in paediatric cardiac surgery.