On the lag phase and initial decline of microbial growth curves

The lag phase is generally thought to be a period during which the cells adjust to a new environment before the onset of exponential growth. Characterizing the lag phase in microbial growth curves has importance in food sciences, environmental sciences, bioremediation and in understanding basic cellular processes. The goal of this work is to extend the analysis of cell growth curves and to better estimate the duration of the lag phase. A non-autonomous model is presented that includes actively duplicating cells and two subclasses of non-duplicating cells. The growth curves depend on the growth and death rate of these three subpopulations and on the initial proportion of each. A deterministic and a stochastic model are both developed and give the same results. A notable feature of the model is the decline of cells during the early stage of the growth curve, and the range of parameters when this decline occurs is identified. A limited growth model is also presented that accounts for the lag, exponential growth and stationary phase of microbial growth curves.     

Modeling and estimating bacterial lag phase

A branching process model of a bacterial population with initial lag phase is developed. Approximations are established in order to facilitate parameter estimation. The validity of approximations and estimation procedures is tested with simulated data.     

Individual-based modelling of angiogenesis inside three-dimensional porous biomaterials

This paper presents a simulation modelling framework to study the growth of blood vessels and cells through a porous tissue engineering scaffold. The model simulates the migration of capillaries and the formation of a vascular network through a single pore of a tissue engineering scaffold when it is embedded in living tissue. The model also describes how the flow of blood through the network changes as growth proceeds. Results are given for how the different strategies of seeding the pore with cells affects the extent of vascularisation. Also simulations are made to compare results where the values of different model parameters are varied such as the pore dimensions, the density of endothelial cells seeded into the pore, and the release rate of growth factor from the scaffold into the pore. The modelling framework described in this paper is useful for exploring experimental strategies for producing well-vascularised tissue engineered constructs, and is therefore potentially important to the field of regenerative medicine.     

A rule-based approach to the modelling of bacterial ecosystems

This paper presents an approach to ecological/evolutionary modelling that is inspired by natural bacterial ecosystems and bacterial evolution. An individual-based artificial ecosystem model is proposed, which is designed to explore the evolvability of adaptive behavioural strategies in artificial bacteria represented by rule-based learning classifier systems. The proposed ecosystem model consists of a n-dimensional environmental grid, which can contain different types of artificial resources in arbitrary arrangements. The resources provide the energy that is necessary for the organisms to sustain life, and can trigger different types of behaviour in the organisms, such as movement towards nutrients and away from toxic substances, growth, and the controlled release of signalling resources. The balance between energy and material is modelled carefully to ensure that the ecosystem is dissipative. Those organisms that are able to efficiently exploit the available resources gradually accumulate enough energy to reproduce (by binary fission) and generate copies of themselves in the environment. Organisms are also able to produce their own resources, which can potentially be used as markers to send signals to other organisms (a behaviour known as quorum sensing). The complex relationships between stimuli and actions in the organisms are stochastically altered by means of mutations, thus enabling the organisms to adapt to their environment and maximise their lifespan and reproductive success. In this paper, the proposed bacterial ecosystem model is defined formally and its structure is discussed in detail. This is followed by results from simulation experiments that illustrate the model's operation and how it can be used in evolutionary modelling/computing scenarios.     

Individual-based modelling of Pinus sylvestris invasion after grazing abandonment in the French Massif Central

In the Chaîne des Puys, a mid-elevation volcanic mountain of the French Massif Central, Scots pine proves to be an invasive species colonizing abandoned lawns or heathlands, and forms in a few years monospecific natural forests. Most of the abandonment occurred 30 to 40 years ago and this process has now stopped. Thus, we lack data on the very first phase of tree colonization. We anticipate that a simulation tool could bring an appreciable help in (i) rebuilding the entire colonization process – including the initial phase – of pine settlement and (ii) answering questions about the origin of the narrow and unimodal distributions of age of pine stands we observed. In addition, such a simulator could help managers to forecast extension of Scots pine and to predict growth and evolution of present secondary forests. A spatially explicit individual-based model is presented. The model takes into account both space and time and includes growth of trees, seed production and seed dispersal, death and competition between individuals. The influence of the initial parameters are analyzed and elements of validation given. The model was then used to predict tree settlement and stand establishment using the initial conditions from a natural stand studied in the field whose characteristics before abandonment were known (number and age-distribution of trees, location of mother trees, time of abandonment). Three simulations were achieved by using the same initial conditions but following different scenarios for the recruitment process. The scenario of a fluctuating resistance of the resident vegetation (that controls the susceptibility of the environment to tree establishment) seemed as one of the most probable to explain the actual stand characteristics. We thus concluded that dynamic models could be improved by taking into account the resistance of the vegetation to colonization as a fluctuating parameter instead of a static and permanent attribute.     

Physiological studies on the effect of Ca2+ on the duration of the lag phase of Saccharomyces cerevisiae

Abstract Cell multiplication and growth of Saccharomyces cereviseae were followed in 2-ml test tubes containing Wickerham's synthetic medium or very dilute synthetic media supplemented in various ways. The ability of the cell cultures to leave the lag phase and enter the exponential phase of growth was investigated. Multiplication was assessed by microscopical observation. The results showed great differences in times required for the cultures to leave the lag phases and begin multiplication. In Wickerham's medium, all cultures grew well 6 h after inoculation. In the dilute medium, several days elapsed before all the cultures grew. These cultures went into exponential growth with approximately first order kinetics. In the unsupplemented medium, the 'half-lives' in the lag phase were about 28 h. Addition of either Ca²⁺ or Ca²⁺ plus A23187 (calcimycin) reduced the half-lives to 10 and 6 h, respectively. The doubling times in the exponential phases of growth were not shortened by these additions. We suggest that Ca²⁺ plays a crucial role as a signal to switch on the mode of cell proliferation in S. cerevisiae .     

Connection between stochastic and deterministic modelling of microbial growth

We present in this paper various links between individual and population cell growth. Deterministic models of the lag and subsequent growth of a bacterial population and their connection with stochastic models for the lag and subsequent generation times of individual cells are analysed. We derived the individual lag time distribution inherent in population growth models, which shows that the Baranyi model allows a wide range of shapes for individual lag time distribution. We demonstrate that individual cell lag time distributions cannot be retrieved from population growth data. We also present the results of our investigation on the effect of the mean and variance of the individual lag time and the initial cell number on the mean and variance of the population lag time. These relationships are analysed theoretically, and their consequence for predictive microbiology research is discussed.     

Analysis of bacterial populations in a grassland soil according to rates of development on solid media

Abstract The process of colony formation by bacteria from grassland soil sampled in April, July and September was simulated by a colony-forming curve (CFC). The CFC was a super-imposition of several component curves (cCFC) given theoretically by the first order reaction (FOR) model [3,6]. The pattern of FOR model curves was not influenced by the time of sampling and four cCFCs were always recognized during an incubation period of 160 h. It was considered that the CFC describes an inherent property of the bacterial population of the field. Bacterial isolates were obtained from colonies produced in each of four cCFCs on agar plates. Isolates corresponding to one cCFC were classified as one group. The bacterial isolates were characterized by morphological and physiological tests and subsequently clustered. Few oligotrophic bacteria were obtained among bacteria which produced visible colonies within 63 h of incubation time. On the other hand, approx. 50% of bacteria which produced v colonies after 63 h were oligotrophic bacteria. The time required for the appearance of the first colony, t _r of the FOR model, was very similar in the isolates belonging to one group. A close linear relationship was observed between t _r value and doubling time of isolates.     

Optimization of quorum quenching mediated bacterial attenuation of Solanum torvum root extract by response surface modelling through Box-Behnken approach

The present study was intended to optimize the quorum sensing inhibitory action of Solanum torvum root extract against Chromobacterium violaceum. Factors such as bacterial density, frequency of administration and concentration of extract were analysed. Plant samples were collected from Thrissur District, Kerala, India. Response surface modelling of factors by Box-Behnken approach was employed for optimizing quorum quenching activity of extract. The adequacy of mathematical model was verified by ANOVA and Cook’s distance table. Results revealed that quorum quenching property of Solanum torvum root extract is highly influenced by variables studied whereas maximum activity was found during administration of 300?µg/ml extract thrice in a day. It was also understood that extract does not possess any bactericidal activity wherein it only silence its quorum sensing mediated functions. This observations can be further used in quorum quenching studies.     

A mathematical model to study the effects of drugs administration on tumor growth dynamics

A mathematical model for describing the cancer growth dynamics in response to anticancer agents administration in xenograft models is discussed. The model consists of a system of ordinary differential equations involving five parameters (three for describing the untreated growth and two for describing the drug action). Tumor growth in untreated animals is modelled by an exponential growth followed by a linear growth. In treated animals, tumor growth rate is decreased by an additional factor proportional to both drug concentration and proliferating cells. The mathematical analysis conducted in this paper highlights several interesting properties of this tumor growth model. It suggests also effective strategies to design in vivo experiments in animals with potential saving of time and resources. For example, the drug concentration threshold for the tumor eradication, the delay between drug administration and tumor regression, and a time index that measures the efficacy of a treatment are derived and discussed. The model has already been employed in several drug discovery projects. Its application on a data set coming from one of these projects is discussed in this paper.     

Introduction to the modelling of venereal disease

Summary The continually rising trend in the incidence of venereal diseases, especially gonorrhoea, in a large number of countries, both developed and developing is causing considerable public health concern. There is a disquieting volume of human suffering involved, as well as large economic losses in treatment and hospitalization. The present paper reviews the existing state of development in the mathematical modelling of the relevant disease dynamics. The criss-cross nature of the infections, which in heterosexual contacts switch between the male and female populations, together with the nonlinear form of the rate of spread normally occurring in infectious diseases, leads to special types of simultaneous nonlinear differential equations.The simplest deterministic models available entail threshold phenomena connecting the maintenance of endemic states to the contact-rates, the personto-person infection-rates, and the removal-rates. A few stochastic results are also available.Special attention is given to the aspects of nonhomogeneous mixing, analysis of contact-rates, infection without immunity, allowance for asymptomatic infection, the recognition of many different classes of infected individuals, and the problems of public health forecasting and control. In some cases transient solutions of the equations can be used to forecast future trends in disease incidence, depending on appropriate assumptions about alternative public health interventions.It is concluded that further mathematical work should be concentrated on relatively simple models comprising no more than three or four district epidemiological groups for each sex. There should be both (i) more intense mathematical investigations, and (ii) new attempts to assimilate the models directly to public health venereal disease control.     

Coupled modelling of tumour angiogenesis, tumour growth and blood perfusion

We propose a mathematical modelling system to investigate the dynamic process of tumour cell proliferation, death and tumour angiogenesis by fully coupling the vessel growth, tumour growth and blood perfusion. Tumour growth and angiogenesis are coupled by the chemical microenvironment and the cell-matrix interaction. The haemodynamic calculation is carried out on the updated vasculature. The domains of intravascular, transcapillary and interstitial fluid flow were coupled in the model to provide a comprehensive solution of blood perfusion variables. An estimation of vessel collapse is made according to the wall shear stress criterion to provide feedback on vasculature remodelling. The simulation can show the process of tumour angiogenesis and the spatial distribution of tumour cells for periods of up to 24 days. It can show the major features of tumour and tumour microvasculature during the period such as the formation of a large necrotic core in the tumour centre with few functional vessels passing through, and a well circulated tumour periphery regions in which the microvascular density is high and associated with more aggressive proliferating cells of the growing tumour which are all consistent with physiological observations. The study also demonstrated that the simulation results are not dependent on the initial tumour and networks, which further confirms the application of the coupled model feedback mechanisms. The model enables us to examine the interactions between angiogenesis and tumour growth, and to study the dynamic response of a solid tumour to the changes in the microenvironment. This simulation framework can be a foundation for further applications such as drug delivery and anti-angiogenic therapies.     

Flow-induced permeation of non-occlusive blood clots: an MRI study and modelling

The success of clot thrombolysis very much depends on efficient clot permeation with blood plasma carrying the thrombolytic agent. In this paper clot permeation was studied by dynamic magnetic resonance imaging (MRI) on artificial non-occlusive blood clots inserted in an artificial circulation system filled with blood plasma to which an MRI contrast agent was added. The MRI results revealed that clot permeation is much faster and more efficient at the entrance of the flow channel across the clot. Clot permeation with fluid was simulated numerically as well. The simulation was based on numerical solution of Navier-Stokes equations for the flow in the channel and within the clot. The clot was considered as a porous material with known permeability and porosity. Based on the calculated velocity profiles, concentration profiles of fluid in the clot were modelled. These agreed well with the MRI results. The presented model of clot permeation with fluid may also serve as a useful extension to numerical modelling of dissolution of non-occlusive blood clots during thrombolytic therapy.     

Trehalose and glycogen accumulation is related to the duration of the G1 phase of Saccharomyces cerevisiae

Several factors may control trehalose and glycogen synthesis, like the glucose flux, the growth rate, the intracellular glucose-6-phosphate level and the glucose concentration in the medium. Here, the possible relation of these putative inducers to reserve carbohydrate accumulation was studied under well-defined growth conditions in nitrogen-limited continuous cultures. We showed that the amounts of accumulated trehalose and glycogen were regulated by the growth rate imposed on the culture, whereas other implicated inducers did not exhibit a correlation with reserve carbohydrate accumulation. Trehalose accumulation was induced at a dilution rate (D)相似文献   

Modelling the ontogeny of ectotherms exhibiting indeterminate growth

Numerous growth functions exist to describe the ontogeny of animals. Such functions (e.g., von Bertalanffy's equation, thermal-unit growth coefficient) are currently applied to ectotherms even though they fail to provide analytical expressions that adapt to a wide range of fluctuating temperatures. The underlying mechanisms responsible for the ontogeny of ectotherms exhibiting indeterminate growth have not yet been summarised in terms of a simple but meaningful mathematical equation. Here, a growth function is developed, with parameters having physical or biological interpretation that accommodates indeterminate growth under fluctuating temperatures assuming the latter vary seasonally. The equation is derived as a special case of von Bertalanffy's equation providing realistic growth trajectories throughout the ontogeny of several groups of ectotherms (R²>0.90). The results suggest that the effect of temperature on growth trajectory supersedes that of reproduction in an environment with fluctuating temperature. Furthermore, values of the allometric weight exponent (0<b<0.75) indicate that the rules of body surface and body weight do not apply under certain circumstances. Finally, the growth function circumvents problems associated with models based on thermodynamic and chemical kinetic principles (e.g., inability to predict growth of organisms in which ontogeny exceeds 3 months) and on rule of thermal summation (e.g., reliable only in a certain range of temperature). The growth function can handle a wide range of temperature fluctuations, encompass life stages and apply to key organisms in ecology, fisheries and agriculture.     

Mathematical modelling of liver regeneration after intoxication with CCl(4)

Liver regeneration is a complex process, having evolved to protect animals from the consequences of liver loss caused by food toxins. In this study, we established a mathematical spatial-temporal model of the liver lobule regenerating after CCl(4) intoxication. The aim of modelling the regeneration process by matching experimental observations with those from a mathematical model is to gain a better understanding of the process and to recognize which parameters are relevant for specific phenomena. In order to set up a realistic minimal model, we first reconstructed a schematised liver lobule after determination of: (i) the mean number of hepatocytes between the central vein and the periphery of the lobule, (ii) the mean size of the hepatocytes and (iii) the mean number of hepatocyte columns in the inner, midzonal and peripheral ring of the lobule. In a next step, we determined the time course of cell death and BrdU incorporation after intoxication of male Sprague Dawley rats with CCl(4), thereby differentiating between inner, midzonal and peripheral hepatocytes. These parameters were used to construct a model. The basic unit of this model is the individual cell. The detailed behaviour of the cells is studied, controlled by the model parameters: (1) probability of cell division at defined positions of the lobule at a given time, (2) "coordinated cell orientation", i.e., the ability of the cells to align during the regeneration process into columns towards the central vein of a liver lobule, (3) cell cycle duration, (4) the migration activity and (5) the polarity of the hepatocytes resulting in polar cell-cell adhesion between them. In a schematised lobule, the model shows that CCl(4) initially induced cell death of a pericentral ring of hepatocytes, followed by a wave of proliferation that starts in the surviving hepatocytes next to the inner ring of dead cells and continues to the peripheral hepatocytes, finally restoring the characteristic micro-architecture of the lobule in a 7-day process. This model was used to systematically analyze the influence of parameters 1-5. Interestingly, coordinated cell orientation and cell polarity were identified to be the most critical parameters. Elimination led to destruction of the characteristic micro-architecture of the lobule and to a high degree of disorder characterized by hexagonal cell structures. Our model suggests that the ability of hepatocytes to realign after cell division by a process of coordinated cell orientation (model parameter 2) in combination with cell polarity (model parameter 5) may be at least as critical as hepatocyte proliferation (model parameter 1) itself.