Mathematical modeling for mixed culture growth of two bacterial populations with opposite substrate preferences

An unstructured mathematical model is proposed for mixed culture growth of two different bacterial species that exhibit "opposite" substrate preferences in response to the "same" environmental conditions. The model incorporates enzymatic control mechanisms such as induction, repression, and inhibition in the microorganisms as manifested in their preferential utilization of substrates and microbial interactions such as amensalism and competition. The model predicts cell mass, substrate concentrations, dissolved oxygen tension, as well as key enzyme levels. The predictions of the model are compared with experimental data for pure culture growth and for mixed culture growth on two substrates, glucose and citrate, in a batch reactor.     

Dynamics of genome rearrangement in bacterial populations

Genome structure variation has profound impacts on phenotype in organisms ranging from microbes to humans, yet little is known about how natural selection acts on genome arrangement. Pathogenic bacteria such as Yersinia pestis, which causes bubonic and pneumonic plague, often exhibit a high degree of genomic rearrangement. The recent availability of several Yersinia genomes offers an unprecedented opportunity to study the evolution of genome structure and arrangement. We introduce a set of statistical methods to study patterns of rearrangement in circular chromosomes and apply them to the Yersinia. We constructed a multiple alignment of eight Yersinia genomes using Mauve software to identify 78 conserved segments that are internally free from genome rearrangement. Based on the alignment, we applied Bayesian statistical methods to infer the phylogenetic inversion history of Yersinia. The sampling of genome arrangement reconstructions contains seven parsimonious tree topologies, each having different histories of 79 inversions. Topologies with a greater number of inversions also exist, but were sampled less frequently. The inversion phylogenies agree with results suggested by SNP patterns. We then analyzed reconstructed inversion histories to identify patterns of rearrangement. We confirm an over-representation of "symmetric inversions"-inversions with endpoints that are equally distant from the origin of chromosomal replication. Ancestral genome arrangements demonstrate moderate preference for replichore balance in Yersinia. We found that all inversions are shorter than expected under a neutral model, whereas inversions acting within a single replichore are much shorter than expected. We also found evidence for a canonical configuration of the origin and terminus of replication. Finally, breakpoint reuse analysis reveals that inversions with endpoints proximal to the origin of DNA replication are nearly three times more frequent. Our findings represent the first characterization of genome arrangement evolution in a bacterial population evolving outside laboratory conditions. Insight into the process of genomic rearrangement may further the understanding of pathogen population dynamics and selection on the architecture of circular bacterial chromosomes.     

Development of a novel continuous culture device for experimental evolution of bacterial populations

The availability of a robust and reliable continuous culture apparatus that eliminates wall growth problems would lead to many applications in the microbial field, including allowing genetically engineered strains to recover high fitness, improving biodegradation strains, and predicting likely antibiotic resistance mechanisms. We describe the design and implementation of a novel automated continuous culture machine that can be used both in time-dependent mode (similar to a chemostat) and turbidostat modes, in which wall growth is circumvented through the use of a long, variably divisible tube of growth medium. This tube can be restricted with clamps to create a mobile growth chamber region in which static portions of the tube and the associated medium are replaced together at equal rates. To functionally test the device as a tool for re-adaptation of engineered strains, we evolved a strain carrying a highly deleterious deletion of Elongation Factor P, a gene involved in translation. In 200 generations over 2 weeks of dilution cycles, the evolved strain improved in generation time by a factor of three, with no contaminations and easy manipulation.     

Neutral evolution in spatially continuous populations

We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum. The recursion allows for the interactions induced by density-dependent regulation of the population, which are inevitable in a continuous population. We give explicit series expansions for large neighbourhood size and for low mutation rates respectively and investigate the accuracy of the classical Malécot formula for these general models. When neighbourhood size is small, this formula does not give the identity even over large scales. However, for large neighbourhood size, it is an accurate approximation which summarises the local population structure in terms of three quantities: the effective dispersal rate, sigma(e); the effective population density, rho(e); and a local scale, kappa, at which local interactions become significant. The results are illustrated by simulations.     

Microbial genome evolution: sources of variability

Comparative genome analyses of close relatives have yielded exciting insight into the sources of microbial genome variability with respect to gene content, gene order and evolution of genes with unknown functions. The genomes of free-living bacteria often carry phages and repetitive sequences that mediate genomic rearrangements in contrast to the small genomes of obligate host-associated bacteria. This suggests that genomic stability correlates with the genomic content of repeated sequences and movable genetic elements, and thereby with bacterial lifestyle. Genes with unknown functions present in a single species tend to be shorter than conserved, functional genes, indicating that the fraction of unique genes in microbial genomes has been overestimated.     

Mathematical modeling and analysis of glucose and glutamine utilization and regulation in cultures of continuous mammalian cells

A number of factors have been shown to affect the metabolism of glucose and glutamine in mammalian cells and their mechanisms have been partially elucidated. Despite these efforts, a quantitative knowledge of the significance of these factors, the regulation of glucose and glutamine utilization, and particularly the interactions of these two nutrients is still lacking. Controversies exist in the literature. To clarify some of these controversies, mathematical models are proposed in this work which enable to separate and identify the effects of individual factors. Experimental data from five cell lines obtained in batch, fed-batch, and continuous cultures, both under steady-state and transient conditions, were used to verify the model formulations. The resulting kinetic models successfully describe all these cultures. According to the models, the specific consumption rate of glucose (Q(Glc)) of continuous animal cells under normal culture conditions can be expressed as a sum of three parts: a part owing to cell growth; a part owing to glucose excess; and a part owing to glutamine regulation. The specific consumption rate of glutamine (q(Glc)7) can be expressed as a sum of only two parts: a part owing to cell growth; and a part owing to glutamine excess. Using the kinetic models the interaction and regulation of glucose and glutamine utilizations are quantitatively analyzed. The results indicate that, whereas q(Glc) is affected by glutamine, q(Gln) appears to be not or less significantly affected by glucose. It is also shown that the relative utilizations of glucose and glutamine by anabolism and catabolism are mainly affected by the residual concentrations of the respective compounds and are less sensitive to growth rate and the nature of growth limitation.(c) 1995 John Wiley & Sons, Inc.     

Phenotypic-dependent variability and the emergence of tolerance in bacterial populations

Ecological and evolutionary dynamics have been historically regarded as unfolding at broadly separated timescales. However, these two types of processes are nowadays well-documented to intersperse much more tightly than traditionally assumed, especially in communities of microorganisms. Advancing the development of mathematical and computational approaches to shed novel light onto eco-evolutionary problems is a challenge of utmost relevance. With this motivation in mind, here we scrutinize recent experimental results showing evidence of rapid evolution of tolerance by lag in bacterial populations that are periodically exposed to antibiotic stress in laboratory conditions. In particular, the distribution of single-cell lag times—i.e., the times that individual bacteria from the community remain in a dormant state to cope with stress—evolves its average value to approximately fit the antibiotic-exposure time. Moreover, the distribution develops right-skewed heavy tails, revealing the presence of individuals with anomalously large lag times. Here, we develop a parsimonious individual-based model mimicking the actual demographic processes of the experimental setup. Individuals are characterized by a single phenotypic trait: their intrinsic lag time, which is transmitted with variation to the progeny. The model—in a version in which the amplitude of phenotypic variations grows with the parent’s lag time—is able to reproduce quite well the key empirical observations. Furthermore, we develop a general mathematical framework allowing us to describe with good accuracy the properties of the stochastic model by means of a macroscopic equation, which generalizes the Crow-Kimura equation in population genetics. Even if the model does not account for all the biological mechanisms (e.g., genetic changes) in a detailed way—i.e., it is a phenomenological one—it sheds light onto the eco-evolutionary dynamics of the problem and can be helpful to design strategies to hinder the emergence of tolerance in bacterial communities. From a broader perspective, this work represents a benchmark for the mathematical framework designed to tackle much more general eco-evolutionary problems, thus paving the road to further research avenues.     

Metabolic regulation in bacterial continuous cultures: II

The transient behavior of a continuous culture of Klebsiella pneumoniae with mixed feed of glucose and xylose arising from step-up and step-down in dilution rates and from a feed-switching experiment is presented. he organism gradually switches from simultaneous utilization of the substrates at low growth rates to preferred utilization of the faster substrate (i.e, supporting a higher growth rate) at high dilution rates. The metabolic lags following a step increase in dilution rate and a significant accumulation of the slower substrate during the transient period result from the effects of metabolic regulation. The cybernetic modeling approach that successfully described the foregoing situations with single-substrate feeds is employed to describe mixed substrate behavior. The parameters in the mixed-substrate (glucose and xylose) model are the same as those in the single-substrate models with the singular exception of the rate constant for the xylose growth enzyme synthesis. The reason for this discrepancy is discussed in detail. It appears that the constitutive rate of enzyme synthesis for growth on a given substrate may be related to the past history of the organism in regard to whether or not the organism has been exposed to the particular substrate. Thus, the results further demonstrate the ability of the framework to effectively describe metabolic regulation in batch, fedbatch, and continuous microbial cultures.     

An algebraic view of bacterial genome evolution

Rearrangements of bacterial chromosomes can be studied mathematically at several levels, most prominently at a local, or sequence level, as well as at a topological level. The biological changes involved locally are inversions, deletions, and transpositions, while topologically they are knotting and catenation. These two modelling approaches share some surprising algebraic features related to braid groups and Coxeter groups. The structural approach that is at the core of algebra has long found applications in sciences such as physics and analytical chemistry, but only in a small number of ways so far in biology. And yet there are examples where an algebraic viewpoint may capture a deeper structure behind biological phenomena. This article discusses a family of biological problems in bacterial genome evolution for which this may be the case, and raises the prospect that the tools developed by algebraists over the last century might provide insight to this area of evolutionary biology.     

Mathematical modeling of mortality dynamics of mammalian populations exposed to radiation

A mathematical model is developed which describes the dynamics of radiation-induced mortality of a non-homogeneous (in radiosensitivity) mammalian population. It relates statistical biometric functions with statistical and dynamic characteristics of a critical system in organism of specimens composing this population. The model involves two types of distributions, the normal and the log-normal, of population specimens with respect to the radiosensitivity of the critical system cells. This approach suggests a new pathway in developing the methods of radiation risk assessment.     

Growth and production modeling in hybridoma continuous cultures

Several experimental data on continuous cultures of hybridoma cells show that monoclonal antibody productivity is a decreasing function of dilution rate. It has been suggested that this unusual behavior may be due to the arrest of a fraction of cycling cells at a critical point of Phase G(1). Although this hypothesis has been recently investigated by using population balance models, mathematical analysis has been performed without accounting for the dynamics of the arrested cells properly. In this article, a more general and accurate approach is presented and new specific assumptions are introduced to characterize the arrest and the later progress through the cycle. Two different models (stochastic and deterministic) and two different critical points for the arrest (at the beginning and at the end of G(1)) are considered. The cell cycle parameters are estimated so that data predicted by the model fit those reported in the literature. In particular, the fraction of arrested cells, the cell arrest probability, and the mean cell generation time are computed as functions of the dilution rate. Results so far obtained predict that there is an optimal value of dilution rate for maximizing specific production rate of monoclonal antibody. (c) 1993 John Wiley & Sons, Inc.     

Bayesian modeling of recombination events in bacterial populations

Background  

We consider the discovery of recombinant segments jointly with their origins within multilocus DNA sequences from bacteria representing heterogeneous populations of fairly closely related species. The currently available methods for recombination detection capable of probabilistic characterization of uncertainty have a limited applicability in practice as the number of strains in a data set increases.     

粗纹举腹蚁蚁巢解剖及其数学建模

为了弄清与排泄蜜露的昆虫有密切关系的粗纹举腹蚁Crematogaster macaoensis Wheeler的生物学特性,于2009年8月至2010年7月在云南省墨江县雅邑乡紫胶园采集并调查了42个粗纹举腹蚁蚁巢。粗纹举腹蚁蚁巢类型为层纸巢,灰黑色,筑于紫胶虫寄主植物树干的中上部,由干枯树叶、杂草、碎屑和蚁分泌物粘结而成;巢内疏松,蚁道纵横交错;单蚁后制,工蚁单型;平均每巢的蚁后、繁殖雌蚁、雄蚁和工蚁的数量分别为(1.00±0.00)头,(488.75±242.59)头,(3 096.86±2 923.75)头和(40 839.07±8 597.98)头;卵、幼虫和蛹的数量分别为(9 936.60±2 448.62)粒,(3 214.83±781.71)头和(4 582.00±656.07)个。粗纹举腹蚁蚁巢体积和每巢工蚁数量的计算公式分别为:V=0.365×(43×πabc)1.056和WN=135.236×V0.660(式中V代表蚁巢体积,a、b、c分别代表蚁巢的长半径、短半径和极半径,WN代表工蚁数量)。这两个公式为监测粗纹举腹蚁的种群数量变化提供了一种手段。     

Structure,diversity, and catabolic potentialities of aerobic heterotrophic bacterial populations associated with continuous cultures of natural marine phytoplankton

Variations of structure (probable taxonomic generic groups; ecological profiles), diversity (Shannon index, ¯H), and average catabolic potentialities (strain's average exoenzyme equipment, EAI; average carbonaceous compound utilization, UAI) of bacterial populations during two experimental phytoplankton blooms are described and show a certain overall unity. Oligotrophic conditions are characterized by high diversity levels (¯H from 3.60 to 4) and moderate catabolic potentialities (EAI and UAI close to 40%). During phytoplankton exponential growth phase bacteria show an EAI stability, but there is an increase of UAI with maximal values at the beginning of chlorophyll plateau (52–57%) and higher values of diversity (¯H greater than 4). Phytoplankton mortalities appear to cause an EAI increase and a decrease of both UAI and ¯H (1.50 to 2). Vibrio-like organisms seem to be closely related to this period.In spite of these similar patterns, many differences appear between both experiments from a taxonomic point of view, the autumnal population being more diversified than the spring one.The results obtained show the value of simultaneous analysis of these different aspects and of this ecological methodology allowing spatial or temporal comparisons.     

Parallels in genome evolution in mitochondria and bacterial symbionts

Mitochondria, the energy-producing organelles of the eukaryotic cell, originate from an endosymbiotic alpha-proteobacterium. These organelles are believed to have arisen only once in evolutionary history, but despite their common ancestry, mitochondrial DNAs vary extensively throughout eukaryotes in genome architecture and gene content. New insights into early mitochondrial genome evolution come from the investigation of primitive mitochondriate eukaryotes, as well as the comparison between mitochondria and intracellular bacterial symbionts.     

Mathematical modeling of evolution of horizontally transferred genes

We describe a stochastic birth-and-death model of evolution of horizontally transferred genes in microbial populations. The model is a generalization of the stochastic model described by Berg and Kurland and includes five parameters: the rate of mutational inactivation, selection coefficient, invasion rate (i.e., rate of arrival of a novel sequence from outside of the recipient population), within-population horizontal transmission ("infection") rate, and population size. The model of Berg and Kurland included four parameters, namely, mutational inactivation, selection coefficient, population size, and "infection." However, the effect of "infection" was disregarded in the interpretation of the results, and the overall conclusion was that horizontally acquired sequences can be fixed in a population only when they confer a substantial selective advantage onto the recipient and therefore are subject to strong positive selection. Analysis of the present model in different domains of parameter values shows that, as long as the rate of within-population horizontal transmission is comparable to the mutational inactivation rate and there is even a low rate of invasion, horizontally acquired sequences can be fixed in the population or at least persist for a long time in a substantial fraction of individuals in the population even when they are neutral or slightly deleterious. The available biological data strongly suggest that intense within-population and even between-populations gene flows are realistic for at least some prokaryotic species and environments. Therefore, our modeling results are compatible with the notion of a pivotal role of horizontal gene transfer in the evolution of prokaryotes.