Coordinated Development of Yeast Colonies: Quantitative Modeling of Diffusion‐Limited Growth – Part 2

A mathematical model was developed which described the growth of yeast colonies based on the assumptions that (i) these populations were built up of single cells whose proliferation was (ii) exclusively controlled by nutrient availability in the environment. The model was of a hybrid cellular automaton type and described discrete cells residing on a one‐dimensional lattice as well as on continuously distributed nutrients. Experimental results and numerical calculations were compared to elucidate under which cultivation conditions the diffusion‐limited growth (DLG) was the major construction principle in yeast colonies. Simulations were scaled to the growth of Yarrowia lipolytica and Candida boidinii colonies under carbon and nitrogen limitation. They showed that nutrient‐controlled growth of the individual cells resulted in DLG of the population. Quantitative predictions for the spatio‐temporal development of the cell‐density profile inside a growing yeast mycelium were compared to the growth characteristics of the model yeast mycelia. Only for the carbon‐limited growth of C. boidinii colonies on glucose as the limiting nutrient resource did the DLG model reproduce the cell‐density profile estimated at the end of the cultivation. Under all other cultivation conditions, strong discrepancies between calculations and experimental results were evident precluding DLG as the ruling regulatory mechanism. Thus, whether or not the development of a yeast population could be described by a DLG scenario, was strongly dependent on the particular cultivation conditions and the applied yeast species. In those cases for which the DLG hypothesis failed to explain the observed growth patterns, the underlying assumptions, i.e., the complete absence of nutrient translocation between the individual cells inside the yeast mycelia as well as the exclusively nutrient‐controlled proliferation of the cells, have to be reevaluated. The presented study demonstrated how the mathematical analysis of growth processes in yeast populations could assist the experimental identification of potential regulatory mechanisms.     

Growth of yeast colonies on solid media

Colonies on nutrient agar of the aerobic yeast Candida utilis show linear increases in diameter and height with time throughout most of the growth cycle. The concentration of glucose in the agar has a negligible effect on radial growth rate although an increase in the glucose concentration prolongs the linear radial growth phase. The rate of increase in height of the colony is proportional to the square root of the initial glucose concentration. A new model that considers both glucose diffusion and oxygen diffusion in the colony is consistent with the observed colony profiles.     

Growth rates of yeast colonies on solid media

Previous measurements of growth rates of giant yeast colonies on solid media are shown to be unreliable as they depend strongly on extraneous factors such as the proximity of other colonies and the dimensions of the apparatus used. The hitherto unexplained dependence of the growth rate on the square root of the growth limiting nutrient concentration is explained by constructing a theory based on the diffusion of nutrient towards the colony which makes use of many ideas used in the theory of flame propagation. The theory also explains why the temperature dependence of the homogeneous growth constant is different from that observed in the surface colony, and it requires the existence of a lag phase in the homogeneous culture kinetics if the velocity of propagation of the culture is to be independent of inoculum size and shape. Both phenomena are known to occur.     

Growing Yeast into Cylindrical Colonies

Microorganisms often form complex multicellular assemblies such as biofilms and colonies. Understanding the interplay between assembly expansion, metabolic yield, and nutrient diffusion within a freely growing colony remains a challenge. Most available data on microorganisms are from planktonic cultures, due to the lack of experimental tools to control the growth of multicellular assemblies. Here, we propose a method to constrain the growth of yeast colonies into simple geometric shapes such as cylinders. To this end, we designed a simple, versatile culture system to control the location of nutrient delivery below a growing colony. Under such culture conditions, yeast colonies grow vertically and only at the locations where nutrients are delivered. Colonies increase in height at a steady growth rate that is inversely proportional to the cylinder radius. We show that the vertical growth rate of cylindrical colonies is not defined by the single-cell division rate, but rather by the colony metabolic yield. This contrasts with cells in liquid culture, in which the single-cell division rate is the only parameter that defines the population growth rate. This method also provides a direct, simple method to estimate the metabolic yield of a colony. Our study further demonstrates the importance of the shape of colonies on setting their expansion. We anticipate that our approach will be a starting point for elaborate studies of the population dynamics, evolution, and ecology of microbial colonies in complex landscapes.     

Growing Yeast into Cylindrical Colonies

Microorganisms often form complex multicellular assemblies such as biofilms and colonies. Understanding the interplay between assembly expansion, metabolic yield, and nutrient diffusion within a freely growing colony remains a challenge. Most available data on microorganisms are from planktonic cultures, due to the lack of experimental tools to control the growth of multicellular assemblies. Here, we propose a method to constrain the growth of yeast colonies into simple geometric shapes such as cylinders. To this end, we designed a simple, versatile culture system to control the location of nutrient delivery below a growing colony. Under such culture conditions, yeast colonies grow vertically and only at the locations where nutrients are delivered. Colonies increase in height at a steady growth rate that is inversely proportional to the cylinder radius. We show that the vertical growth rate of cylindrical colonies is not defined by the single-cell division rate, but rather by the colony metabolic yield. This contrasts with cells in liquid culture, in which the single-cell division rate is the only parameter that defines the population growth rate. This method also provides a direct, simple method to estimate the metabolic yield of a colony. Our study further demonstrates the importance of the shape of colonies on setting their expansion. We anticipate that our approach will be a starting point for elaborate studies of the population dynamics, evolution, and ecology of microbial colonies in complex landscapes.     

Synchronous plasma membrane electrochemical potential oscillations during yeast colony development and aging

Microorganisms that survive in natural environments form organized multicellular communities, biofilms and colonies with specific properties. During stress and nutrient limitation, slow growing and senescent cells in such communities retain vital processes by maintaining plasma membrane integrity and retaining the ability to generate transmembrane electrochemical gradients. We report the use of a Saccharomyces cerevisiae colonial model to show that population growth in a multicellular community depends on nutrient diffusion and that resting cells start to accumulate from the beginning of the second acidic phase of colony development. Despite differentiation of colony members, synchronous transmembrane potential oscillation was detected in the organized colony. The electrochemical membrane potential periodically oscillated at frequencies between those for circadian to infradian rhythms during colony aging and transiently decreased at time points previously linked with rebuilding of yeast metabolism. Despite extensive decreases in the intracellular ATP concentration and in the amount and activity of the plasma membrane proton pump during nutrient limited growth and colony aging, the transmembrane electrochemical potential appeared to be maintained above a level critical for population survival.     

Uptake of Glucose and Phosphorus by Growing Colonies of Fusarium oxysporum as Quantified by Image Analysis

Olsson, S. 1994. Uptake of glucose and phosphorus by growing colonies of Fusarium oxysporum as quantified by image analysis. Experimental Mycology 18, 33-47. The simplest of all heterogeneous environments for fungal colony growth is the petri dish with an agar medium. As the colony grows there will be a depression of nutrient concentrations under the colony caused by the uptake of nutrients by the growing colony. Image analysis methods have been developed for measuring medium concentrations of glucose and phosphorus with simultaneous biomass density determinations in agar systems. Maps of the concentrations in the agar medium under the colony and of colony biomass density were produced. A new method for weighing fungal colonies grown on agar is also presented. For Fusarium oxysporum phosphorus and glucose uptake from the medium was the same irrespective of the C/mineral ratios in the medium within the measured range of ratios. Even the concentration profiles of the nutrients under the colony were the same irrespective of nutrient ratios. Distribution of biomass density was affected by differences in glucose concentrations, being highest at the colony margin at the lower concentrations. The results indicate that the fungal colony is able to take up nutrients at the margin in excess of the local needs.     

Colony pattering ofAspergillus oryzae on agar media

To clarity the effects of nutrient concentration and diffusion on the pattern formation of fungal colonies, the colony patterning ofAspergillus oryzae at various nutrient and agar levels was studied experimentally and was summarized in a colony morphology diagram. Roles of the nutrient content and the relaxation of nutrient distribution on the colony patterning were discussed based on a computer model of the mycelial growth. The colony morphology changed from compact to ramified as the nutrient and agar levels were lowered. No clear boundary was found between these two morphologies. The deterioration of substrate around the growing colony was detected when the morphic switching from homogeneous into splitting patterns emerged in the growth of ramified colonies. In the mycelial growth model, dense compact colonies developed at low growth rates and high nutrient influx into the colonized area. Under low nutrient levels, splitting colonies appeared at high growth rates as compared with the nutrient influx.     

Applying dimorphic yeasts as model organisms to study mycelial growth: part 2. Use of mathematical simulations to identify different construction principles in yeast colonies

The dimorphic yeasts Candida boidinii and Yarrowia lipolytica were applied as model organisms to study mycelial growth. A mathematical model of hybrid cellular automaton type was developed to analyze the impact of different biological assumptions on the predicted development of filamentous yeast colonies. The one-dimensional model described discrete cells and continuous distribution of nutrients. The simulation algorithm accounted for proliferation of cells, diffusion of nutrient, as well as biomass decay and recycling inside the mycelium. Simulations reproduced the spatio-temporal development of C. boidinii colonies when a diffusion-limited growth algorithm based on the growth of pseudohyphal cells was applied. Development of Y. lipolytica colonies could only be reproduced when proliferation was restricted to the colony boundary, and cell decay and biomass recycling were incorporated into the model. The results suggested that cytoplasm, which served as the secondary nutrient resource, had to be translocated inside the hyphal network.     

Applying dimorphic yeasts as model organisms to study mycelial growth: Part 1. Experimental investigation of the spatio-temporal development of filamentous yeast colonies

Colony development of the dimorphic yeasts Yarrowia lipolytica and Candida boidinii on solid agar substrates under glucose limitation served as a model system for mycelial development of higher filamentous fungi. Strong differences were observed in the behaviour of both yeasts: C. boidinii colonies reached a final colony extension which was small compared to the size of the growth field. They formed cell-density profiles which steeply declined along the colony radius and no biomass decay processes could be detected. The stop of colony extension coincided with the depletion of glucose from the growth substrate. These findings supported the hypothesis that glucose-limited C. boidinii colonies can be regarded as populations of single cells which grow according to a diffusion-limited growth mechanism. Y. lipolytica colonies continued to extend after the depletion of the primary nutrient resource, glucose, until the populations covered the entire growth field which was accomplished by utilization of mycelial biomass.     

A microfluidic chemostat for experiments with bacterial and yeast cells

Bacteria and yeast frequently exist as populations capable of reaching extremely high cell densities. With conventional culturing techniques, however, cell proliferation and ultimate density are limited by depletion of nutrients and accumulation of metabolites in the medium. Here we describe design and operation of microfabricated elastomer chips, in which chemostatic conditions are maintained for bacterial and yeast colonies growing in an array of shallow microscopic chambers. Walls of the chambers are impassable for the cells, but allow diffusion of chemicals. Thus, the chemical contents of the chambers are maintained virtually identical to those of the nearby channels with continuous flowthrough of a dynamically defined medium. We demonstrate growth of cell cultures to densely packed ensembles that proceeds exponentially in a temperature-dependent fashion, and we use the devices to monitor colony growth from a single cell and to analyze the cell response to an exogenously added autoinducer.     

Diversity of the growth patterns of Bacillus subtilis colonies on agar plates

Abstract A Bacillus subtilis strain showed a variety of colony growth patterns on agar plates. The bacterium grew to a fractal colony through the diffusion-limited aggregation process, a round colony reminiscent of the Eden model, a colony with a straight and densely branched structure similar to the dence branching, morphology, a colony spreading without any openings, and a colony with concentric rings, on plates with various agar and nutrient concentrations. The microstructures of these colonies were also characteristic and dynamic. The patterns of these bacterial colonies were thought to grow in relation to the diffusion of nutrient in the agar plate.     

The effects of growth dynamics upon pH gradient formation within and around subsurface colonies of Salmonella typhimurium

pH measurements made in and around submerged colonies of Salmonella typhimurium grown within a model gelatin gel system using pH-sensitive micro- and macroelectrodes indicated some pH heterogeneity occurring in and around the bacterial colony. Inoculation density, initial pH and glucose concentration were all found to influence colony diameter and metabolism of Salmonella colonies. Colony growth in the presence of glucose, at pH 7.0 with an inoculation density of 1 cell ml^-1 led to a pH fall of 1–2 pH units after 2 d. At pH 5.0, with glucose, colony growth rates were much slower than at pH 7.0, and the pH change varied by less than one pH unit often becoming alkaline. In the absence of glucose, only small pH changes were observed within the medium, although growth rates were similar to those in glucose-containing media. At the higher inoculation density ( ca 1000 cells ml^-1), isolated pH changes were not observed. Morphological changes, such as the production of annular rings, were noted in stationary phase colonies as was alkali production in colonies. These results are discussed in relation to observations with surface colonies.     

Two continuum models for the spreading of myxobacteria swarms

We analyze the phenomenon of spreading of a Myxococcus xanthus bacterial colony on plates coated with nutrient. The bacteria spread by gliding on the surface. In the first few hours, cell growth is irrelevant to colony spread. In this case, bacteria spread through peninsular protrusions from the edge of the initial colony. We analyze the diffusion through the narrowing reticulum of cells on the surface mathematically and derive formulae for the spreading rates. On the time scale of tens of hours, effective diffusion of the bacteria, combined with cell division and growth, causes a constant linear increase in the colony's radius. Mathematical analysis and numerical solution of reaction-diffusion equations describing the bacterial and nutrient dynamics demonstrate that, in this regime, the spreading rate is proportional to the square root of both the effective diffusion coefficient and the nutrient concentration. The model predictions agree with the data on spreading rate dependence on the type of gliding motility.     

Surface colony growth in a controlled nutrient environment. 1 The exponential law.

An apparatus designed to study the growth rates of surface colonies in constant conditions, i.e. not affected by nutrient diffusion as in a closed Petri dish, is described. In contrast to classical experiments in closed systems, an exponential growth of colony radius is obtained for a period of more than 72 h. Nutrient concentration gradients are shown to be eliminated by analytical techniques.     

pH gradients through colonies of Bacillus cereus and the surrounding agar.

pH-sensitive microelectrodes, constructed with a tip diameter of about 4 microns, were deployed through 24 h and 48 h colonies of Bacillus cereus incubated on CYS medium (Casamino acids, yeast extract, salts), with and without glucose. Measurements of pH were used to construct pH profiles through the colony and the surrounding agar. pH gradients could be detected for at least 800 microns into the agar beneath a 24 h colony, and to approximately 10 mm horizontally away from the edge of the colony. In older colonies, the lateral gradient extended for over 20 mm. The pH of the underlying agar was increased by up to 1.45 pH units after 48 h growth without glucose. When colonies were grown with glucose, a significant area of acidification was observed within the colony in addition to a zone of alkalinization present at its periphery. Acidification was thought to be due to the anaerobic fermentation of glucose producing organic acids whilst alkalinization was due to the aerobic oxidation of amino acids releasing ammonia.     

Cluster growth by mitosis

A stochastic mitosis growth model is introduced. The colony enlarge by random mitosis of a mother cell which is replaced by two daughter cells. Unlike growth models based on a grid, the present model is isotropic. Using simulation of colonies of 10,000 cells, it is found that the density is constant and the boundary is fractal.     

Growth model and metabolic activity of brewing yeast biofilm on the surface of spent grains: a biocatalyst for continuous beer fermentation

In the continuous systems, such as continuous beer fermentation, immobilized cells are kept inside the bioreactor for long periods of time. Thus an important factor in the design and performance of the immobilized yeast reactor is immobilized cell viability and physiology. Both the decreasing specific glucose consumption rate (q(im)) and intracellular redox potential of the cells immobilized to spent grains during continuous cultivation in bubble-column reactor implied alterations in cell physiology. It was hypothesized that the changes of the physiological state of the immobilized brewing yeast were due to the aging process to which the immobilized yeast are exposed in the continuous reactor. The amount of an actively growing fraction (X(im)act) of the total immobilized biomass (X(im)) was subsequently estimated at approximately X(im)act = 0.12 g(IB) g(C)(-1) (IB = dry immobilized biomass, C = dry carrier). A mathematical model of the immobilized yeast biofilm growth on the surface of spent grain particles based on cell deposition (cell-to-carrier adhesion and cell-to-cell attachment), immobilized cell growth, and immobilized biomass detachment (cell outgrowth, biofilm abrasion) was formulated. The concept of the active fraction of immobilized biomass (X(im)act) and the maximum attainable biomass load (X(im)max) was included into the model. Since the average biofilm thickness was estimated at ca. 10 microm, the limitation of the diffusion of substrates inside the yeast biofilm could be neglected. The model successfully predicted the dynamics of the immobilized cell growth, maximum biomass load, free cell growth, and glucose consumption under constant hydrodynamic conditions in a bubble-column reactor. Good agreement between model simulations and experimental data was achieved.     

ARRESTED GROWTH OF CHINESE HAMSTER OVARY (CHO) CELL COLONIES ON AGAR

Chinese hamster ovary (CHO) cells plated on agar form two classes of colonies; those which increase continuously in diameter and those which become arrested in outward growth. All colonies continue to increase in mass and thickness as demonstrated by computer-assisted analysis of time sequence photographs of several thousand colonies and by examination of histological sections. Colonies which shift from a monolayer to a mounded morphology at fairly large colony diameter (?1 mm) continue to increase in diameter. Colonies in which mounding occurs at smaller colony diameter (?1 mm) cease to increase in diameter but continue to increase in thickness, as demonstrated by histological examination and by computer-assisted analysis. Rapid cell division occurs at the edge of all colony classes, as shown by the distribution of mitotic figures. In arrested colonies these dividing cells must move toward the colony core to compensate for dying cells. Necrotic cells are found as a discrete zone at the air-colony interface in all cases for CHO cell colonies growing on agar. As such, necrosis is probably due to limitations in nutrient diffusion upward from the agar rather than oxygen diffusion downward.