Quantifying the effects of cell death and agar density on yeast colony biofilms using an extensional-flow mathematical model

By combining experimental measurements with a thin-film extensional-flow mathematical model, this study demonstrates that increasing agar density inhibits nutrient uptake and enhances biofilm-substratum adhesion, with the latter identified as the most consistent factor driving changes in *Saccharomyces cerevisiae* colony biofilm growth.

The Gist

Imagine a tiny, microscopic city being built by yeast cells on a plate of jelly. This isn't just a flat pancake; it's a living, breathing "biofilm" that grows, spreads, and changes shape depending on how firm or soft the jelly (agar) underneath it is.

This paper is like a detective story where scientists used experiments (watching the yeast grow) and math (building a virtual simulation) to figure out exactly how the "jelly's" stiffness changes the yeast's behavior.

Here is the breakdown of their findings using simple analogies:

1. The Setup: The Yeast City and the Jelly Floor

The scientists grew a specific type of yeast (Saccharomyces cerevisiae, the kind used for bread and beer) on square plates filled with agar. Think of the agar as the floor of the city.

• They made four different types of floors: some were soft and squishy (low agar density), and some were firm and stiff (high agar density).
• They watched the yeast spread out over 21 days, taking photos and even cutting the colonies in half to see how thick they got.

2. The Mystery: Why does the shape change?

When the yeast grew on the soft floor, it spread out wide and flat, like a puddle of water on a smooth table.
When the yeast grew on the stiff floor, it didn't spread out as much. Instead, it got thicker and taller, like a pile of sand that refuses to slide away.

The scientists wanted to know: Why? Is it because the yeast eats less? Does it die more? Or is it just stuck to the floor?

3. The Math Model: The "Slippery vs. Sticky" Simulation

To solve this, the researchers built a mathematical model. Imagine this as a video game simulation where they could tweak the rules of physics to see what matched the real yeast.

They tested three main theories:

1. The "Hungry Yeast" Theory: Maybe the stiff jelly blocks the yeast from eating nutrients?
2. The "Dead Yeast" Theory: Maybe the yeast dies faster on stiff jelly?
3. The "Sticky Feet" Theory: Maybe the yeast gets stuck to the stiff jelly and can't slide as easily?

4. The Big Discovery: It's All About the "Sticky Feet"

After running the numbers, the scientists found the answer: The "Sticky Feet" theory was the winner.

• On Soft Jelly (Low Density): The yeast has slippery feet. It can slide around easily, spreading out horizontally like a runner on a smooth track. It doesn't get stuck, so it grows wide.
• On Stiff Jelly (High Density): The yeast has sticky feet. The stiff surface acts like double-sided tape. The yeast tries to slide, but the friction holds it back. Because it can't slide forward easily, the only place for the new cells to go is up. This is why the colony gets thicker and taller on stiff floors.

5. The Other Clues (What didn't matter as much)

• Eating Habits: The yeast did eat slightly less on the stiff jelly, but this wasn't the main reason for the shape change.
• Death Rate: The yeast didn't die significantly more on the stiff jelly. The "dead zone" in the middle of the colony was similar regardless of the floor type.
• Growth Speed: The yeast grew at roughly the same speed in terms of making new cells; the difference was just where that growth went (sideways vs. upwards).

6. Why Does This Matter?

This isn't just about bread yeast.

• Medical Relevance: Fungi (like yeast) can form biofilms on medical devices (like catheters) inside hospitals. Understanding how they stick to different surfaces helps doctors figure out how to stop them from spreading.
• Universal Rule: The scientists found that bacteria behave similarly. Whether it's a tiny bug or a yeast cell, if the surface is rough or sticky, the colony stops sliding and starts piling up.

The Takeaway

Think of the yeast colony as a crowd of people trying to leave a room.

• If the floor is ice (soft agar), everyone slides out quickly, spreading wide across the room.
• If the floor is mud (stiff agar), everyone gets stuck. They can't slide out, so they just keep piling up on top of each other, getting taller and thicker.

The paper proves that the texture of the surface is the most important factor in deciding whether a microbial city spreads out or builds a skyscraper.

Technical Summary

1. Problem Statement

The study addresses the gap in understanding how agar density and cell death influence the growth dynamics of Saccharomyces cerevisiae (baker's yeast) colony biofilms. While previous mathematical models (specifically by Tam et al., 2019) successfully described yeast biofilm expansion via sliding motility, they had two significant limitations:

1. They largely neglected cell death, which is a critical factor in harsh environments and releases nutrients for the colony.
2. They were restricted to low-density agar (0.3%), failing to account for how varying substrate stiffness (agar density) alters growth mechanics.

Recent studies on bacterial colonies suggest that higher substrate density reduces nutrient uptake and increases cell-substrate friction, promoting vertical over horizontal growth. However, these mechanisms remain unexplored in fungal colonies. The authors aim to quantify these effects by combining experimental data with an enhanced mathematical model.

2. Methodology

Experimental Approach

• Organism: A diploid prototrophic strain of S. cerevisiae (Σ1278b).
• Setup: Rectangular streaks of yeast were inoculated onto YPD agar plates with four different densities: 0.6%, 0.8%, 1.2%, and 2.0%.
• Measurements:
  • Horizontal Expansion: Colony width was tracked over 21 days using macroscopic imaging.
  • Cell Viability: Samples were stained with Phloxine B (a dye indicating dead cells) and analyzed via flow cytometry and microscopy to determine the ratio of living to dead cells at the center, middle, and edge of the colony.
  • Aspect Ratio: Cross-sectional microscopy was used to measure colony thickness. For low-density agar, where cutting distorted the shape, aspect ratios were estimated using cell-count data and scaling from the high-density measurements.
• Replicates: A total of 15 experimental replicates were analyzed.

Mathematical Modeling

The authors developed a thin-film extensional-flow continuum model based on lubrication theory, extending the work of Tam et al. (2019).

• Two-Phase System: The biofilm is modeled as a mixture of two phases:
  1. Living phase ($\phi_n$): Proliferating cells consuming nutrients.
  2. Inactive phase ($\phi_m$): Dead cells and extracellular matrix (ECM).
• Governing Equations: The model incorporates mass and momentum conservation equations, including:
  • Biomass production driven by nutrient consumption (first-order kinetics).
  • Cell death occurring at a rate proportional to the living cell volume fraction.
  • Nutrient diffusion in both the agar substratum and the biofilm.
  • Slip condition: A generalized slip boundary condition at the biofilm-agar interface ($\mu \frac{\partial u}{\partial z} = \lambda u$), where $\lambda$ represents the biofilm-substratum adhesion strength.
• Nondimensionalization: The system was scaled using the thin-film approximation ($\epsilon \ll 1$), leading to a set of 1D partial differential equations describing the evolution of colony width, thickness, and composition.

Parameter Estimation

• Five unknown dimensionless parameters were estimated by fitting the model to the experimental data using numerical optimization (Differential Evolution followed by LBFGS):
  1. $\Psi_n$: Biomass-production rate.
  2. $\Psi_d$: Cell-death rate.
  3. $Q^*$: Nutrient-uptake coefficient.
  4. $\Upsilon$: Nutrient-consumption rate.
  5. $\lambda^*$: Biofilm-substratum adhesion strength (slip parameter).
• Validation: Robustness was tested via sensitivity analysis (parameter-pair heat maps) and synthetic data simulations (generating 50 synthetic experiments per agar density based on experimental variance).

3. Key Results

Parameter Trends vs. Agar Density

• Adhesion Strength ($\lambda^*$): This parameter showed the most significant and consistent increase with agar density.
  • On low-density agar (0.6%), $\lambda^*$ was small, suggesting near-perfect slip (consistent with previous models).
  • On high-density agar (2.0%), $\lambda^*$ increased substantially, indicating strong friction/adhesion.
  • Physical Consequence: Increased adhesion inhibits horizontal expansion, forcing the colony to grow vertically, which explains the observed increase in aspect ratio (thickness) on denser agar.
• Nutrient Uptake ($Q^*$): The nutrient uptake rate decreased as agar density increased. This aligns with findings in bacterial colonies, suggesting that denser matrices restrict nutrient diffusion to the biofilm interface.
• Biomass Production ($\Psi_n$) & Cell Death ($\Psi_d$): These rates showed minimal variability across different agar densities. This suggests that yeast cells do not actively alter their intrinsic proliferation or death rates in response to substrate stiffness; rather, the growth morphology changes due to mechanical constraints and nutrient availability.
• Nutrient Consumption ($\Upsilon$): Showed a decreasing trend with density but was less robustly identified than adhesion strength.

Model Fit and Sensitivity

• The model successfully reproduced the experimental expansion curves, cell viability distributions, and aspect ratios for all four agar densities.
• Sensitivity Analysis: The adhesion parameter ($\lambda^*$) and biomass production rate ($\Psi_n$) were the most "well-identified" parameters (narrow confidence intervals). In contrast, nutrient uptake ($Q^*$) was harder to identify due to its coupling with consumption rates, though the trend of decreasing uptake with density remained consistent.
• Synthetic Data: Simulations confirmed that the trend of increasing adhesion with agar density is robust against biological variability, whereas trends in nutrient uptake showed slightly higher variability.

4. Significance and Contributions

1. Mechanistic Insight into Fungal Biofilms: The study provides the first quantitative evidence that substrate stiffness (agar density) is a primary driver of S. cerevisiae colony morphology, acting primarily through increased biofilm-substratum adhesion rather than changes in cellular proliferation rates.
2. Model Advancement: The extension of the thin-film model to include cell death and generalized slip conditions allows for a more accurate representation of biofilm dynamics in varying environments. It validates the "sliding motility" hypothesis while quantifying the frictional resistance imposed by the substrate.
3. Cross-Species Consistency: The findings align with recent bacterial studies (e.g., E. coli), suggesting a universal biophysical principle where high-density substrates increase friction and reduce nutrient uptake, shifting growth from horizontal spreading to vertical stacking.
4. Clinical and Industrial Relevance: Understanding how substrate properties dictate biofilm shape and cell death is crucial for controlling fungal infections (e.g., on medical devices) and optimizing industrial fermentation processes where biofilm formation is a factor.

5. Limitations and Future Work

• The model assumes the agar is rigid; future work could incorporate agar deformation (dehydration or mechanical stress).
• Osmotic swelling, known to affect bacterial biofilms, was neglected but could be a factor in yeast.
• The model currently focuses on accidental cell death; the role of regulated cell death (which can create necrotic cores) requires further investigation.
• Future models could be extended to multispecies biofilms and investigate the formation of interfacial instabilities (wrinkles) observed at later growth stages.