Modelling the emergence of spiral colony morphology in the yeast Magnusiomyces magnusii

1. Problem Statement

The study addresses the challenge of understanding the macroscopic morphological patterns formed by yeast colonies, specifically a novel, highly ordered spiral morphology observed in an isolate of the hyphae-forming yeast Magnusiomyces magnusii.

• Context: Yeasts exhibit diverse growth modes (biofilms, pseudohyphal, hyphal) influenced by genetics, nutrients, and environmental stress. While Saccharomyces cerevisiae is a well-studied model, M. magnusii (bioprospected from a Tasmanian cider gum tree) displays a unique "galaxy-like" spiral pattern under high-density, nutrient-limited laboratory conditions.
• Gap: The microscale cellular mechanisms driving this specific spiral formation were unknown. Specifically, the angle between successive hyphal segments, which theoretically dictates the curvature of the spiral, had not been quantified for this species.
• Goal: To develop a mathematical framework to simulate this growth, infer the underlying cellular parameters from experimental images, and quantify the specific biological rules (e.g., extension angles) that generate the spiral pattern.

2. Methodology

The authors employed a combination of experimental biology, off-lattice agent-based modeling (ABM), and likelihood-free Bayesian inference.

A. Experimental Setup

• Organism: An in-house isolate of M. magnusii.
• Conditions: Cultured on Chemically Defined Sap (CDS) and Yeast Nitrogen Base (YNB) media.
• Variables: Tested under low and high colony densities. The spiral morphology was observed specifically under high-density conditions (4 × 10³ cells mL⁻¹), suggesting it is an adaptation to nutrient competition.
• Imaging: Macroscopic colony images were captured and processed into binary masks using the Tool for Analysis of the Morphology of Microbial Colonies (TAMMiCol).

B. Mathematical Modeling (Off-Lattice Agent-Based Model)

The authors adapted a 2D off-lattice ABM to simulate hyphal growth.

• Agents: The colony is represented by rod-like agents (ellipses approximated as dodecagons) with a fixed aspect ratio (3:1), representing hyphal segments.
• Growth Rules:
  • Regular Hyphae: Can extend from four sites on their sides.
  • Filament-Forming Hyphae: Extend only from the distal end at a prescribed acute angle ($\theta_p$) relative to the previous segment. This angle is the primary driver of the spiral.
  • Volume Exclusion: Agents cannot overlap; growth is aborted if a proposed position intersects an existing agent.
• Key Parameters ($\theta$):
  • $n^*$: Threshold proportion of colony size triggering the switch from regular to filament-forming growth (nutrient stress proxy).
  • $p_a$: Probability of selecting a regular vs. filament-forming hypha to extend.
  • $p_d$: Probability that a regular hypha produces a filament-forming daughter.
  • $p_b$: Probability that a filament-forming hypha produces a regular daughter (affecting filament thickness).
  • $\gamma$: Probability of switching to a different filament-forming agent (forking control).
  • $\theta_p$: The angle of extension between successive filament segments.

C. Parameter Inference (Sequential Neural Likelihood - SNLE)

Since the model is stochastic and high-dimensional, a closed-form likelihood function is unavailable.

• Approach: Likelihood-free Bayesian inference using Sequential Neural Likelihood (SNLE).
• Process:
  1. A Gaussian Mixture Density Network (MDN) was trained as a surrogate likelihood model using 1,000 simulations per round.
  2. Four rounds of training were performed to refine the posterior distribution.
  3. Summary Statistics: Five spatial metrics were used to compare simulations to experiments: Maximum Radius ($R_{max}$), Mean Radius ($R_{mean}$), Filamentous Area ($I_{AF}$), Branch Count ($I_B$), and Compactness ($I_C$).
  4. Optimization: The golden section search was used to match the simulated colony area to the experimental image.

3. Key Contributions

1. Quantification of Microscale Mechanics: The study successfully quantified the angle between successive hyphal segments ($\theta_p$) for M. magnusii, a parameter previously unknown.
2. Novel Modeling Framework: Adapted an off-lattice ABM to specifically handle hyphal extension and branching, distinguishing between regular hyphae and filament-forming agents.
3. Robust Inference Pipeline: Demonstrated the efficacy of SNLE in inferring biological parameters from macroscopic colony images where microscopic agent behavior is unobservable.
4. Multi-Morphology Capability: Showed that the same model framework, with different parameter sets, can reproduce not only the spiral morphology but also hexagonal (Eden-like) and straight filamentous morphologies observed in M. magnusii under different conditions.

4. Results

• Inferred Parameters:
  • Spiral Angle ($\theta_p$): The posterior distribution yielded a mean angle of 2.3° (95% credible interval: [1.1°, 3.6°]). This small, constant angle is sufficient to generate the observed large-scale spiral patterns.
  • Filament Initiation ($n^*$): Strongly identified with a mode of 0.76, indicating that filamentation (and thus spiral formation) begins only when the colony reaches ~76% of its maximum capacity, supporting the hypothesis that it is a stress response to nutrient limitation.
  • Filament Formation Probability ($p_d$): Inferred to be very small (< 0.1), consistent with the observation of long, distinct filaments rather than short, dense clusters.
  • Weakly Identified Parameters: Parameters $p_a$, $p_b$, and $\gamma$ were less constrained by the data, suggesting that the summary statistics used did not fully capture filament width or forking complexity.
• Model Validation:
  • The posterior predictive distributions for colony size (radius) and compactness closely matched the experimental data.
  • Visual simulations using inferred parameters successfully reproduced the "galaxy-like" spiral pattern seen in the experiments (Figure 9).
• Morphological Plasticity:
  • By varying $n^*$, the model reproduced the transition from rounded/hexagonal colonies (low stress, $n^*=1$) to spiral/straight filaments (high stress, $n^* < 1$).
  • Varying $p_b$ and $\gamma$ successfully simulated differences in filament thickness, matching experimental observations of straight filaments in other conditions.

5. Significance

• Biological Insight: The research confirms that the spiral morphology in M. magnusii is an adaptive response to high-density nutrient competition, mediated by a specific, quantifiable cellular rule (a ~2.3° extension angle).
• Methodological Advancement: It provides a template for "bioprospecting" new yeast strains. Instead of purely descriptive biology, researchers can now use computational modeling to reverse-engineer the cellular mechanisms of new, uncharacterized strains based on their macroscopic growth patterns.
• Future Applications: The framework is extensible to 3D to study invasive growth (penetration into agar), which is critical for understanding pathogenicity in clinical yeast infections. The open-source code provided facilitates further research into yeast colony dynamics and parameter inference.

In summary, this paper bridges the gap between microscopic cellular behavior and macroscopic colony patterns, using advanced statistical inference to reveal that a tiny, consistent angular bias in hyphal growth is the fundamental driver of the complex spiral architecture in M. magnusii.