Growing yeast into cylindrical colonies

A mathematical model for nutrient-limited uniaxial growth of a compressible tissue

We consider the uniaxial growth of a tissue or colony of cells, where a nutrient (or some other chemical) required for cell proliferation is supplied at one end, and is consumed by the cells. An example would be the growth of a cylindrical yeast colony in the experiments described by Vulin et al. (2014). We develop a reaction-diffusion model of this scenario which couples nutrient concentration and cell density on a growing domain. A novel element of our model is that the tissue is assumed to be compressible. We define replicative regions, where cells have sufficient nutrient to proliferate, and quiescent regions, where the nutrient level is insufficient for this to occur. We also define pathlines, which allow us to track individual cell paths within the tissue. We begin our investigation of the model by considering an incompressible tissue where cell density is constant before exploring the solution space of the full compressible model. In a large part of the parameter space, the incompressible and compressible models give qualitatively similar results for both the nutrient concentration and cell pathlines, with the key distinction being the variation in density in the compressible case. In particular, the replicative region is located at the base of the tissue, where nutrient is supplied, and nutrient concentration decreases monotonically with distance from the nutrient source. However, for a highly-compressible tissue with small nutrient consumption rate, we observe a counter-intuitive scenario where the nutrient concentration is not necessarily monotonically decreasing, and there can be two replicative regions. For parameter values given in the paper by Vulin et al. (2014), the incompressible model slightly overestimates the colony length compared to experimental observations; this suggests the colony may be somewhat compressible. Both incompressible and compressible models predict that, for these parameter values, cell proliferation is ultimately confined to a small region close to the colony base.

MIL-CELL: a tool for multi-scale simulation of yeast replication and prion transmission

The single-celled baker's yeast, Saccharomyces cerevisiae, can sustain a number of amyloid-based prions, the three most prominent examples being [URE3], [PSI+], and [PIN+]. In the laboratory, haploid S. cerevisiae cells of a single mating type can acquire an amyloid prion in one of two ways (i) spontaneous nucleation of the prion within the yeast cell, and (ii) receipt via mother-to-daughter transmission during the cell division cycle. Similarly, prions can be lost due to (i) dissolution of the prion amyloid by its breakage into non-amyloid monomeric units, or (ii) preferential donation/retention of prions between the mother and daughter during cell division. Here we present a computational tool (Monitoring Induction and Loss of prions in Cells; MIL-CELL) for modelling these four general processes using a multiscale approach describing both spatial and kinetic aspects of the yeast life cycle and the amyloid-prion behavior. We describe the workings of the model, assumptions upon which it is based and some interesting simulation results pertaining to the wave-like spread of the epigenetic prion elements through the yeast population. MIL-CELL is provided as a stand-alone GUI executable program for free download with the paper. MIL-CELL is equipped with a relational database allowing all simulated properties to be searched, collated and graphed. Its ability to incorporate variation in heritable properties means MIL-CELL is also capable of simulating loss of the isogenic nature of a cell population over time. The capability to monitor both chronological and reproductive age also makes MIL-CELL potentially useful in studies of cell aging.

Metabolic heterogeneity and cross-feeding within isogenic yeast populations captured by DILAC

Genetically identical cells are known to differ in many physiological parameters such as growth rate and drug tolerance. Metabolic specialization is believed to be a cause of such phenotypic heterogeneity, but detection of metabolically divergent subpopulations remains technically challenging. We developed a proteomics-based technology, termed differential isotope labelling by amino acids (DILAC), that can detect producer and consumer subpopulations of a particular amino acid within an isogenic cell population by monitoring peptides with multiple occurrences of the amino acid. We reveal that young, morphologically undifferentiated yeast colonies contain subpopulations of lysine producers and consumers that emerge due to nutrient gradients. Deconvoluting their proteomes using DILAC, we find evidence for in situ cross-feeding where rapidly growing cells ferment and provide the more slowly growing, respiring cells with ethanol. Finally, by combining DILAC with fluorescence-activated cell sorting, we show that the metabolic subpopulations diverge phenotypically, as exemplified by a different tolerance to the antifungal drug amphotericin B. Overall, DILAC captures previously unnoticed metabolic heterogeneity and provides experimental evidence for the role of metabolic specialization and cross-feeding interactions as a source of phenotypic heterogeneity in isogenic cell populations.

Morphology and Size of Bacterial Colonies Control Anoxic Microenvironment Formation in Porous Media

Bacterial metabolisms using electron acceptors other than oxygen (e.g., methanogenesis and fermentation) largely contribute to element cycling and natural contaminant attenuation/mobilization, even in well-oxygenated porous environments, such as shallow aquifers. This paradox is commonly explained by the occurrence of small-scale anoxic microenvironments generated by the coupling of bacterial respiration and dissolved oxygen (O₂) transport by pore water. Such microenvironments allow facultative anaerobic bacteria to proliferate in oxic environments. Microenvironment dynamics are still poorly understood due to the challenge of directly observing biomass and O₂ distributions at the microscale within an opaque sediment matrix. To overcome these limitations, we integrated a microfluidic device with transparent O₂ planar optical sensors to measure the temporal behavior of dissolved O₂ concentrations and biomass distributions with time-lapse videomicroscopy. Our results reveal that bacterial colony morphology, which is highly variable in flowing porous systems, controls the formation of anoxic microenvironments. We rationalize our observations through a colony-scale Damköhler number comparing dissolved O₂ diffusion and a bacterial O₂ uptake rate. Our Damköhler number enables us to predict the pore space fraction occupied by anoxic microenvironments in our system for a given bacterial organization.

A Model for Cell Proliferation in a Developing Organism

In mathematical biology, there is a great deal of interest in producing continuum models by scaling discrete agent-based models governed by local stochastic rules. We discuss a particular example of this approach: a model for the proliferation of neural crest cells that can help us understand the development of Hirschprung's disease, a potentially-fatal condition in which the enteric nervous system of a new-born child does not extend all the way through the intestine and colon. Our starting point is a discrete-state, continuous-time Markov chain model proposed by Hywood et al. (2013a) for the location of the neural crest cells that make up the enteric nervous system. Hywood et al. (2013a) scaled their model to derive an approximate second order partial differential equation describing how the limiting expected number of neural crest cells evolve in space and time. In contrast, we exploit the relationship between the above-mentioned Markov chain model and the well-known Yule-Furry process to derive the exact form of the scaled version of the process. Furthermore, we provide expressions for other features of the domain agent occupancy process, such as the variance of the marginal occupancy at a particular site, the distribution of the number of agents that are yet to reach a given site and a stochastic description of the process itself.

Modelling uniaxial non-uniform yeast colony growth: Comparing an agent-based model and continuum approximations

Biological experiments have shown that yeast can be restricted to grow in a uniaxial direction, vertically upwards from an agar plate to form a colony. The growth occurs as a consequence of cell proliferation driven by a nutrient supply at the base of the colony, and the height of the colony has been observed to increase linearly with time. Within the colony the nutrient concentration is non-constant and yeast cells throughout the colony will therefore not have equal access to nutrient, resulting in non-uniform growth. In this work, an agent based model is developed to predict the microscopic spatial distribution of labelled cells within the colony when the probability of cell proliferation can vary in space and time. We also describe a method for determining the average trajectories or pathlines of labelled cells within a colony growing in a uniaxial direction, enabling us to connect the microscopic and macroscopic behaviours of the system. We present results for six cases, which involve different assumptions for the presence or absence of a quiescent region (where no cell proliferation occurs), the size of the proliferative region, and the spatial variation of proliferation rates within the proliferative region. These six cases are designed to provide qualitative insight into likely growth scenarios whilst remaining amenable to analysis. We compare our macroscopic results to experimental observations of uniaxial colony growth for two cases where only a fixed number of cells at the base of the colony can proliferate. The model predicts that the height of the colony will increase linearly with time in both these cases, which is consistent with experimental observations. However, our model shows how different functional forms for the spatial dependence of the proliferation rate can be distinguished by tracking the pathlines of cells at different positions in the colony. More generally, our methodology can be applied to other biological systems exhibiting uniaxial growth, providing a framework for classifying or determining regions of uniform and non-uniform growth.

Shape of population interfaces as an indicator of mutational instability in coexisting cell populations

Cellular populations such as avascular tumors and microbial biofilms may 'invade' or grow into surrounding populations. The invading population is often comprised of a heterogeneous mixture of cells with varying growth rates. The population may also exhibit mutational instabilities, such as a heavy deleterious mutation load in a cancerous growth. We study the dynamics of a heterogeneous, mutating population competing with a surrounding homogeneous population, as one might find in a cancerous invasion of healthy tissue. We find that the shape of the population interface serves as an indicator for the evolutionary dynamics within the heterogeneous population. In particular, invasion front undulations become enhanced when the invading population is near a mutational meltdown transition or when the surrounding 'bystander' population is barely able to reinvade the mutating population. We characterize these interface undulations and the effective fitness of the heterogeneous population in one- and two-dimensional systems.

Gas production reveals the metabolism of immobilized Chlorella vulgaris during different trophic modes

The metabolism of heterotrophic and mixotrophic cultivation modes of Chlorella vulgaris, a potential source of biofuel and CO₂ mitigation, was studied in immobilized cultures. The gas concentration (O₂ and CO₂) was measured thanks to an original device manufactured using 3D printing. The biomass was monitored by 3D imaging and image processing. Net O₂ and CO₂ sources were obtained by a balance equation considering a calibrated leakage and the dissolved gas. Combined experimental and theoretical gas yields (mass of gas per mass of biomass), the photosynthesis proportion of mixotrophic colony was determined. Its increase with light intensity is not linear. Therefore, the highest light intensity (104μmol∙m^-2∙s^-1) revealed the limit of photosynthesis potential in the growth of mixotrophic colony. In the presence of light, the colony adopts a cylindrical shape instead of a spherical cap. This study proposed mechanisms of synergy inside the colony for heterotrophic and mixotrophic modes.

Observing Nutrient Gradients, Gene Expression and Growth Variation Using the "Yeast Machine" Microfluidic Device

The natural environment of microbial cells like bacteria and yeast is often a complex community in which growth and internal organization reflect morphogenetic processes and interactions that are dependent on spatial position and time. While most of research is performed in simple homogeneous environments (e.g., bulk liquid cultures), which cannot capture full spatiotemporal community dynamics, studying biofilms or colonies is complex and usually does not give access to the spatiotemporal dynamics at single cell level. Here, we detail a protocol for generation of a microfluidic device, the "yeast machine", with arrays of long monolayers of yeast colonies to advance the global understanding of how intercellular metabolic interactions affect the internal structure of colonies within defined and customizable spatial dimensions. With Saccharomyces cerevisiae as a model yeast system we used the "yeast machine" to demonstrate the emergence of glucose gradients by following expression of fluorescently labelled hexose transporters. We further quantified the expression spatial patterns with intra-colony growth rates and expression of other genes regulated by glucose availability. In addition to this, we showed that gradients of amino acids also form within a colony, potentially opening similar approaches to study spatiotemporal formation of gradients of many other nutrients and metabolic waste products. This approach could be used in the future to decipher the interplay between long-range metabolic interactions, cellular development, and morphogenesis in other same species or more complex multi-species systems at single-cell resolution and timescales relevant to ecology and evolution.

Standard and fractional Ornstein-Uhlenbeck process on a growing domain

We study normal diffusive and subdiffusive processes in a harmonic potential (Ornstein-Uhlenbeck process) on a uniformly growing or contracting domain. Our starting point is a recently derived fractional Fokker-Planck equation, which covers both the case of Brownian diffusion and the case of a subdiffusive continuous-time random walk (CTRW). We find a high sensitivity of the random walk properties to the details of the domain growth rate, which gives rise to a variety of regimes with extremely different behaviors. At the origin of this rich phenomenology is the fact that the walkers still move while they wait to jump, since they are dragged by the deterministic drift arising from the domain growth. Thus, the increasingly long waiting times associated with the aging of the subdiffusive CTRW imply that, in the time interval between two consecutive jumps, the walkers might travel over much longer distances than in the normal diffusive case. This gives rise to seemingly counterintuitive effects. For example, on a static domain, both Brownian diffusion and subdiffusive CTRWs yield a stationary particle distribution with finite width when a harmonic potential is at play, thus indicating a confinement of the diffusing particle. However, for a sufficiently fast growing or contracting domain, this qualitative behavior breaks down, and differences between the Brownian case and the subdiffusive case are found. In the case of Brownian particles, a sufficiently fast exponential domain growth is needed to break the confinement induced by the harmonic force; in contrast, for subdiffusive particles such a breakdown may already take place for a sufficiently fast power-law domain growth. Our analytic and numerical results for both types of diffusion are fully confirmed by random walk simulations.

A microfluidic device for inferring metabolic landscapes in yeast monolayer colonies

Microbial colonies are fascinating structures in which growth and internal organization reflect complex morphogenetic processes. Here, we generated a microfluidics device with arrays of long monolayer yeast colonies to further global understanding of how intercellular metabolic interactions affect the internal structure of colonies within defined boundary conditions. We observed the emergence of stable glucose gradients using fluorescently labeled hexose transporters and quantified the spatial correlations with intra-colony growth rates and expression of other genes regulated by glucose availability. These landscapes depended on the external glucose concentration as well as secondary gradients, for example amino acid availability. This work demonstrates the regulatory genetic networks governing cellular physiological adaptation are the key to internal structuration of cellular assemblies. This approach could be used in the future to decipher the interplay between long-range metabolic interactions, cellular development and morphogenesis in more complex systems.

The Cost of Being Big: Local Competition, Importance of Dispersal, and Experimental Evolution of Reversal to Unicellularity

Multicellularity provides multiple benefits. Nonetheless, unicellularity is ubiquitous, and there have been multiple cases of evolutionary reversal to a unicellular organization. In this article, we explore some of the costs of multicellularity as well as the possibility and dynamics of evolutionary reversals to unicellularity. We hypothesize that recently evolved multicellular organisms would face a high cost of increased competition for local resources in spatially structured environments because of larger size and increased cell densities. To test this hypothesis we conducted competition assays, computer simulations, and selection experiments using isolates of Saccharomyces cerevisiae that recently evolved multicellularity. In well-mixed environments, multicellular isolates had lower growth rates relative to their unicellular ancestor because of limitations of space and resource acquisition. In structured environments with localized resources, cells in both multicellular and unicellular isolates grew at a similar rate. Despite similar growth, higher local density of cells in multicellular groups led to increased competition and higher fitness costs in spatially structured environments. In structured environments all of the multicellular isolates rapidly evolved a predominantly unicellular life cycle, while in well-mixed environments reversal was more gradual. Taken together, these results suggest that a lack of dispersal, leading to higher local competition, might have been one of the main constraints in the evolution of early multicellular forms.

Spatially Constrained Growth Enhances Conversional Meltdown

Cells that mutate or commit to a specialized function (differentiate) often undergo conversions that are effectively irreversible. Slowed growth of converted cells can act as a form of selection, balancing unidirectional conversion to maintain both cell types at a steady-state ratio. However, when one-way conversion is insufficiently counterbalanced by selection, the original cell type will ultimately be lost, often with negative impacts on the population's overall fitness. The critical balance between selection and conversion needed for preservation of unconverted cells and the steady-state ratio between cell types depends on the spatial circumstances under which cells proliferate. We present experimental data on a yeast strain engineered to undergo irreversible conversion: this synthetic system permits cell-type-specific fluorescent labeling and exogenous variation of the relative growth and conversion rates. We find that populations confined to grow on a flat agar surface are more susceptible than their well-mixed counterparts to fitness loss via a conversion-induced "meltdown." We then present analytical predictions for growth in several biologically relevant geometries-well-mixed liquid media, radially expanding two-dimensional colonies, and linear fronts in two dimensions-by employing analogies to the directed-percolation transition from nonequilibrium statistical physics. These simplified theories are consistent with the experimental results.

Development of a process for generating three-dimensional microbial patterns amenable for engineering use

We describe in detail a process for generating three-dimensional patterns of microbes on an optimum substrate in such a way that the patterns are amenable for engineering applications.

Survival probabilities at spherical frontiers

Motivated by tumor growth and spatial population genetics, we study the interplay between evolutionary and spatial dynamics at the surfaces of three-dimensional, spherical range expansions. We consider range expansion radii that grow with an arbitrary power-law in time: R(t) = R0(1 + t/t(∗))Θ, where Θ is a growth exponent, R0 is the initial radius, and t(∗) is a characteristic time for the growth, to be affected by the inflating geometry. We vary the parameters t(∗) and Θ to capture a variety of possible growth regimes. Guided by recent results for two-dimensional inflating range expansions, we identify key dimensionless parameters that describe the survival probability of a mutant cell with a small selective advantage arising at the population frontier. Using analytical techniques, we calculate this probability for arbitrary Θ. We compare our results to simulations of linearly inflating expansions (Θ = 1 spherical Fisher-Kolmogorov-Petrovsky-Piscunov waves) and treadmilling populations (Θ = 0, with cells in the interior removed by apoptosis or a similar process). We find that mutations at linearly inflating fronts have survival probabilities enhanced by factors of 100 or more relative to mutations at treadmilling population frontiers. We also discuss the special properties of "marginally inflating" (Θ = 1/2) expansions.

A simple mathematical model that describes the growth of the area and the number of total and viable cells in yeast colonies

We propose a model, based on the Gompertz equation, to describe the growth of yeasts colonies on agar medium. This model presents several advantages: (i) one equation describes the colony growth, which previously needed two separate ones (linear increase of radius and of the squared radius); (ii) a similar equation can be applied to total and viable cells, colony area or colony radius, because the number of total cells in mature colonies is proportional to their area; and (iii) its parameters estimate the cell yield, the cell concentration that triggers growth limitation and the effect of this limitation on the specific growth rate. To elaborate the model, area, total and viable cells of 600 colonies of Saccharomyces cerevisiae, Debaryomyces fabryi, Zygosaccharomyces rouxii and Rhodotorula glutinis have been measured. With low inocula, viable cells showed an initial short exponential phase when colonies were not visible. This phase was shortened with higher inocula. In visible or mature colonies, cell growth displayed Gompertz-type kinetics. It was concluded that the cells growth in colonies is similar to liquid cultures only during the first hours, the rest of the time they grow, with near-zero specific growth rates, at least for 3 weeks.

SIGNIFICANCE AND IMPACT OF THE STUDY

Mathematical models used to predict microbial growth are based on liquid cultures data. Models describing growth on solid surfaces, highlighting the differences with liquids cultures, are scarce. In this work, we have demonstrated that a single Gompertz equation describes accurately the increase of the yeast colonies, up to the point where they reach their maximum size. The model can be used to quantify the differences in growth kinetics between solid and liquid media. Moreover, as all its parameters have biological meaning, it could be used to build secondary models predicting yeast growth on solid surfaces under several environmental conditions.