# Quantifying the Biophysical Impact of Budding Cell Division on the Spatial Organization of Growing Yeast Colonies

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Nutrient-Rich Growth: Budding Division Impacts Local Colony Organization in Simulated Yeast Colonies

#### 2.1.1. Budding Does Not Impact Large-Scale Colony Growth or Structure (Expanse or Sparsity)

#### 2.1.2. Budding Does Not Change Global Age and Spatial Structure but Impacts Local Connectivity

#### 2.1.3. Budding Division Maintains Closeness between Mothers and Daughters after Physical Separation

#### 2.1.4. Budding Division Promotes Subcolony Connectivity

**subcolony**to be the subset of all cells whose common ancestor is the same immediate daughter of the founder cell (Section 4.2.2 and Figure 17). Note that a colony has as many subcolonies as immediate daughters of the founding cell. We then analyze how well each of these subcolonies is connected in terms of the spatial layout of the colony. To do this, we considered colonies at the final time point (24 h) and compared the number of connected components of the first five subcolonies between budding and non-budding division conditions. We found that the average number of connected components was significantly lower for budding colonies (Figure 6A). This demonstrates that budding division acts as a mechanism to increase spatial adjacency within subcolonies as well as impact overall subcolony connectivity.

#### 2.2. Nutrient-Limited Growth: Differential Growth Rates Impact Global Organization of Yeast Colonies

#### 2.2.1. Nutrient Limitation Slows Colony Growth but Does Not Change Large-Scale Colony Structure

#### 2.2.2. Nutrient Limitation Creates Age-Structured Colonies by Promoting Birth at the Colony Boundary

#### 2.2.3. Nutrient-Limited Growth Promotes Colony Connectivity

#### 2.2.4. Nutrient Limitation Further Promotes Subcolony Connectivity

#### 2.2.5. Nutrient Limitation Changes Global Colony Organization by Driving Variation in Subcolony Sizes

## 3. Discussion

## 4. Materials And Methods

#### 4.1. Computational Model

#### 4.1.1. Cell-Cell Interaction and Spatial Arrangement Of Cells

#### 4.1.2. Budding Cell Division

#### 4.1.3. Cell Growth and Cell Cycle Length

**Cell Progress**$(CP\in [0,1\left]\right)$ is used to track the progress of individual cells through the $G1$ and $G2$ phases. In our model, the progress of cell i at time t is given by:

#### 4.1.4. Nutrient-Limited Growth

#### 4.1.5. Simulation Run Time

#### 4.2. Colony Metrics

#### 4.2.1. Colony Shape Metrics

**colony expanse**quantifies how large the colony is, while the

**colony sparsity**quantifies how circular the colony is. Both depend on the center of mass of the colony (Figure 15). Let $N\left(t\right)$ be the number of cells in the colony at time t, each of which has position ${\overrightarrow{x}}_{i}\left(t\right)=({x}_{i}\left(t\right),{y}_{i}\left(t\right))$ and radius ${R}_{i}\left(t\right)$. Since we assume all cells are of the same type, the mass of each cell, ${m}_{i}\left(t\right)$, is proportional to the area of each cell with the same constant. As such, the center of mass of the colony at time t is given by the 2D point, $\overrightarrow{C}\left(t\right)$, defined by:

#### 4.2.2. Colony Organization Metrics

**colony connectivity**, is the fraction of mother-daughter edges that are also in the intersection graph:

**subcolony**to be the subset of all cells whose common ancestor is an immediate daughter of the founder (Figure 17). We index the daughters of the founder cell by ${d}_{1},{d}_{2},\cdots {d}_{F}$ where ${d}_{k}$ denotes the k-th daughter of the founder cell. Note, every cell in the colony belongs to one of the subcolonies founded by an immediate daughter of the founder cell. Thus, for each ${c}_{i}\in V\backslash \left\{{c}_{1}\right\}$, we define $\mathrm{Sub}\left({c}_{i}\right)$ to be the subcolony that cell i belongs to. Moreover, each daughter of the founder is considered to be the founder of its own colony (i.e. a subcolony of the original colony) denoted $\mathrm{Sub}\left({d}_{k}\right)$ and the total number of subcolonies is equal to F, the total number of immediate daughters of the founder cell.

**subcolony graph**${G}_{\mathrm{sub}}({V}_{\mathrm{sub}},{E}_{\mathrm{sub}})$, where

**connected components**of ${G}_{\mathrm{sub},{d}_{k}}$ for each subcolony. A connected component is defined to be any maximal subgraph ${G}_{connect}\subseteq {G}_{\mathrm{sub},{d}_{k}}$ such that any two vertices in ${G}_{connect}$ are connected by a path and not connected to any other vertices in ${G}_{\mathrm{sub}}$. The total number of connected components for the ${k}^{th}$ subcolony is the total number of maximal subgraphs that partition the ${k}^{th}$ subcolony graph ${G}_{\mathrm{sub},{d}_{k}}$.

#### 4.3. Statistical Analysis

`statannot`package in python [88]. In addition, we performed Kaplan-Meier survival analysis and generated Kaplan-Meier survival curves using the lifelines library in python [89]. The survival function defines the probability that a death event (i.e. loss of a mother-daughter edge in ${G}_{S}$ or a given subcolony splitting into more than 15 connected components) has not occurred yet at time t, or equivalently, the probability of surviving past time t [90]. We then used the log-rank test available in the lifelines library to compare the survival curves between budding and non-budding colonies in all cases (Figure 5D, Figure 6D, Figure 9D and Figure 10D). Finally, we computed the Kolmogorov-Smirnov statistic using the SciPy library in python [91] to compare the probability distributions of birth location between nutrient-rich and nutrient-limited colonies.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ABM | Agent-Based Model |

## References

- DiSalvo, S.; Serio, T.R. Insights into prion biology: Integrating a protein misfolding pathway with its cellular environment. Prion
**2011**, 5, 76–83. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Frazer, C.; Hernday, A.D.; Bennett, R.J. Monitoring Phenotypic Switching in Candida albicans and the Use of Next-Gen Fluorescence Reporters. Curr. Protoc. Microbiol.
**2019**, 53, e76. [Google Scholar] [CrossRef] [PubMed] - Klaips, C.L.; Hochstrasser, M.L.; Langlois, C.R.; Serio, T.R. Spatial quality control bypasses cell-based limitations on proteostasis to promote prion curing. eLife
**2014**, 3, e04288. [Google Scholar] [CrossRef] [PubMed] - Liebman, S.W.; Chernoff, Y.O. Prions in yeast. Genetics
**2012**, 191, 1041–1072. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Giometto, A.; Nelson, D.R.; Murray, A.W. Physical interactions reduce the power of natural selection in growing yeast colonies. Proc. Natl. Acad. Sci. USA
**2018**, 115, 11448–11453. [Google Scholar] [CrossRef] [Green Version] - Ben-Jacob, E.; Schochet, O.; Tenenbaum, A.; Cohen, I.; Czirok, A.; Vicsek, T. Generic modelling of cooperative growth patterns in bacterial colonies. Nature
**1994**, 368, 46–49. [Google Scholar] [CrossRef] - Shapiro, J.A. The significances of bacterial colony patterns. Bioessays
**1995**, 17, 597–607. [Google Scholar] [CrossRef] - Mitri, S.; Clarke, E.; Foster, K.R. Resource limitation drives spatial organization in microbial groups. ISME J.
**2016**, 10, 1471–1482. [Google Scholar] [CrossRef] [Green Version] - Hallatschek, O.; Hersen, P.; Ramanathan, S.; Nelson, D.R. Genetic drift at expanding frontiers promotes gene segregation. Proc. Natl. Acad. Sci. USA
**2007**, 104, 19926–19930. [Google Scholar] [CrossRef] [Green Version] - Johnson, C.R.; Boerlijst, M.C. Selection at the level of the community: The importance of spatial structure. Trends Ecol. Evol.
**2002**, 17, 83–90. [Google Scholar] [CrossRef] - Hallatschek, O.; Nelson, D.R. Life at the front of an expanding population. Evol. Int. J. Org. Evol.
**2010**, 64, 193–206. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Alber, M.S.; Jiang, Y.; Kiskowski, M.A. Lattice gas cellular automation model for rippling and aggregation in myxobacteria. Phys. Nonlinear Phenom.
**2004**, 191, 343–358. [Google Scholar] [CrossRef] [Green Version] - Amiri, A.; Harvey, C.; Buchmann, A.; Christley, S.; Shrout, J.D.; Aranson, I.S.; Alber, M. Reversals and collisions optimize protein exchange in bacterial swarms. Phys. Rev. E
**2017**, 95, 032408. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Qin, B.; Fei, C.; Bridges, A.A.; Mashruwala, A.A.; Stone, H.A.; Wingreen, N.S.; Bassler, B.L. Cell position fates and collective fountain flow in bacterial biofilms revealed by light-sheet microscopy. Science
**2020**, 369, 71–77. [Google Scholar] [CrossRef] [PubMed] - Noble, S.M.; Gianetti, B.A.; Witchley, J.N. Candida albicans cell-type switching and functional plasticity in the mammalian host. Nat. Rev. Microbiol.
**2017**, 15, 96. [Google Scholar] [CrossRef] [Green Version] - Miller, M.G.; Johnson, A.D. White-opaque switching in Candida albicans is controlled by mating-type locus homeodomain proteins and allows efficient mating. Cell
**2002**, 110, 293–302. [Google Scholar] [CrossRef] [Green Version] - Lohse, M.B.; Johnson, A.D. White–opaque switching in Candida albicans. Curr. Opin. Microbiol.
**2009**, 12, 650–654. [Google Scholar] [CrossRef] [Green Version] - Lee, P.S.; Greenwell, P.W.; Dominska, M.; Gawel, M.; Hamilton, M.; Petes, T.D. A fine-structure map of spontaneous mitotic crossovers in the yeast Saccharomyces cerevisiae. PLoS Genet.
**2009**, 5, e1000410. [Google Scholar] [CrossRef] [Green Version] - Krafzig, D.; Klawonn, F.; Gutz, H. Theoretical analysis of the effects of mitotic crossover in large yeast populations. Yeast
**1993**, 9, 1093–1098. [Google Scholar] [CrossRef] - Ramírez-Zavala, B.; Reuß, O.; Park, Y.N.; Ohlsen, K.; Morschhäuser, J. Environmental induction of white–opaque switching in Candida albicans. PLoS Pathog.
**2008**, 4, e1000089. [Google Scholar] [CrossRef] [Green Version] - Xie, J.; Tao, L.; Nobile, C.J.; Tong, Y.; Guan, G.; Sun, Y.; Cao, C.; Hernday, A.D.; Johnson, A.D.; Zhang, L.; et al. White-opaque switching in natural MTL a/α isolates of Candida albicans: Evolutionary implications for roles in host adaptation, pathogenesis, and sex. PLoS Biol.
**2013**, 11, e1001525. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Magno, R.; Grieneisen, V.A.; Marée, A.F. The biophysical nature of cells: Potential cell behaviours revealed by analytical and computational studies of cell surface mechanics. BMC Biophys.
**2015**, 8, 8. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cullen, P.J.; Sprague, G.F. The regulation of filamentous growth in yeast. Genetics
**2012**, 190, 23–49. [Google Scholar] [CrossRef] [PubMed] - Kron, S.J.; Styles, C.A.; Fink, G.R. Symmetric cell division in pseudohyphae of the yeast Saccharomyces cerevisiae. Mol. Biol. Cell
**1994**, 5, 1003–1022. [Google Scholar] [CrossRef] [PubMed] - Drubin, D.G.; Nelson, W.J. Origins of cell polarity. Cell
**1996**, 84, 335–344. [Google Scholar] [CrossRef] [Green Version] - Ni, L.; Snyder, M. A genomic study of the bipolar bud site selection pattern in Saccharomyces cerevisiae. Mol. Biol. Cell
**2001**, 12, 2147–2170. [Google Scholar] [CrossRef] [Green Version] - Chant, J.; Mischke, M.; Mitchell, E.; Herskowitz, I.; Pringle, J.R. Role of Bud3p in producing the axial budding pattern of yeast. J. Cell Biol.
**1995**, 129, 767–778. [Google Scholar] [CrossRef] [Green Version] - Byers, B. Cytology of the yeast life cycle. In The Molecular Biology of The Yeast Saccharomyces: Life Cycle and Inheritance; Springer: Berlin/Heidelberg, Germany, 1981; pp. 59–96. [Google Scholar]
- Nadell, C.D.; Foster, K.R.; Xavier, J.B. Emergence of spatial structure in cell groups and the evolution of cooperation. PLoS Comput. Biol.
**2010**, 6, e1000716. [Google Scholar] [CrossRef] [Green Version] - Tam, A.; Green, J.E.F.; Balasuriya, S.; Tek, E.L.; Gardner, J.M.; Sundstrom, J.F.; Jiranek, V.; Binder, B.J. Nutrient-limited growth with non-linear cell diffusion as a mechanism for floral pattern formation in yeast biofilms. J. Theor. Biol.
**2018**, 448, 122–141. [Google Scholar] [CrossRef] [Green Version] - Gontar, A.; Bottema, M.J.; Binder, B.J.; Tronnolone, H. Characterizing the shape patterns of dimorphic yeast pseudohyphae. R. Soc. Open Sci.
**2018**, 5, 180820. [Google Scholar] [CrossRef] [Green Version] - Tronnolone, H.; Gardner, J.M.; Sundstrom, J.F.; Jiranek, V.; Oliver, S.G.; Binder, B.J. Quantifying the dominant growth mechanisms of dimorphic yeast using a lattice-based model. J. R. Soc. Interface
**2017**, 14, 20170314. [Google Scholar] [CrossRef] [PubMed] - Broach, J.R. Nutritional control of growth and development in yeast. Genetics
**2012**, 192, 73–105. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Merchant, S.S.; Helmann, J.D. Elemental economy: Microbial strategies for optimizing growth in the face of nutrient limitation. In Advances in Microbial Physiology; Elsevier: Amsterdam, The Netherlands, 2012; Volume 60, pp. 91–210. [Google Scholar]
- Plocek, V.; Váchová, L.; Št’ovíček, V.; Palková, Z. Cell Distribution within Yeast Colonies and Colony Biofilms: How Structure Develops. Int. J. Mol. Sci.
**2020**, 21, 3873. [Google Scholar] [CrossRef] [PubMed] - Kayser, J.; Schreck, C.F.; Yu, Q.; Gralka, M.; Hallatschek, O. Emergence of evolutionary driving forces in pattern-forming microbial populations. Philos. Trans. R. Soc. Biol. Sci.
**2018**, 373, 20170106. [Google Scholar] [CrossRef] [Green Version] - Smith, W.P.; Davit, Y.; Osborne, J.M.; Kim, W.; Foster, K.R.; Pitt-Francis, J.M. Cell morphology drives spatial patterning in microbial communities. Proc. Natl. Acad. Sci. USA
**2017**, 114, E280–E286. [Google Scholar] [CrossRef] [Green Version] - Van Liedekerke, P.; Palm, M.; Jagiella, N.; Drasdo, D. Simulating tissue mechanics with agent-based models: Concepts, perspectives and some novel results. Comput. Part. Mech.
**2015**, 2, 401–444. [Google Scholar] [CrossRef] [Green Version] - Glen, C.M.; Kemp, M.L.; Voit, E.O. Agent-based modeling of morphogenetic systems: Advantages and challenges. PLOS Comput. Biol.
**2019**, 15, e1006577. [Google Scholar] [CrossRef] [Green Version] - Gorochowski, T.E. Agent-based modelling in synthetic biology. Essays Biochem.
**2016**, 60, 325–336. [Google Scholar] - Jönsson, H.; Levchenko, A. An explicit spatial model of yeast microcolony growth. Multiscale Model. Simul.
**2005**, 3, 346–361. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.; Lo, W.C.; Chou, C.S. A modeling study of budding yeast colony formation and its relationship to budding pattern and aging. PLoS Comput. Biol.
**2017**, 13, e1005843. [Google Scholar] [CrossRef] [Green Version] - Aprianti, D.; Haryanto, F.; Purqon, A.; Khotimah, S.; Viridi, S. Study of budding yeast colony formation and its characterizations by using circular granular cell. J. Phys. Conf. Ser.
**2016**, 694, 012079. [Google Scholar] [CrossRef] - Aprianti, D.; Khotimah, S.; Viridi, S. Budding yeast colony growth study based on circular granular cell. J. Phys. Conf. Ser.
**2016**, 739, 012026. [Google Scholar] [CrossRef] - Aji, D.P.P.; Aprianti, D.; Viridi, S. Stochastic Simulation of Yeast Cells and Its Colony Growth by Using Circular Granular Model for Cases of Growth and Birth Probabilities Depends on Position. J. Phys. Conf. Ser.
**2019**, 1245, 012010. [Google Scholar] - Purnama, F.A.; Meiriska, W.; Aji, D.P.P.; Aprianti, D.; Viridi, S. Network Analysis of Saccharomyces cerevisiae. J. Phys. Conf. Ser.
**2019**, 1245, 012081. [Google Scholar] [CrossRef] - Meiriska, W.; Purnama, F.; Aji, D.; Aprianti, D.; Viridi, S. Network Analysis of Saccharomyces Cerevisiae Colony: Relation between Spatial Position and Generation. J. Phys. Conf. Ser.
**2019**, 1245, 012006. [Google Scholar] [CrossRef] - Drasdo, D.; Loeffler, M. Individual-based models to growth and folding in one-layered tissues: Intestinal crypts and early development. Nonlinear Anal.-Theory Methods Appl.
**2001**, 47, 245–256. [Google Scholar] [CrossRef] - Drasdo, D.; Forgacs, G. Modeling the interplay of generic and genetic mechanisms in cleavage, blastulation, and gastrulation. Dev. Dyn. Off. Publ. Am. Assoc. Anat.
**2000**, 219, 182–191. [Google Scholar] [CrossRef] - Drasdo, D.; Höhme, S. A single-cell-based model of tumor growth in vitro: Monolayers and spheroids. Phys. Biol.
**2005**, 2, 133. [Google Scholar] [CrossRef] - Drasdo, D.; Hoehme, S.; Block, M. On the role of physics in the growth and pattern formation of multi-cellular systems: What can we learn from individual-cell based models? J. Stat. Phys.
**2007**, 128, 287. [Google Scholar] [CrossRef] - Hornung, R.; Grünberger, A.; Westerwalbesloh, C.; Kohlheyer, D.; Gompper, G.; Elgeti, J. Quantitative modelling of nutrient-limited growth of bacterial colonies in microfluidic cultivation. J. R. Soc. Interface
**2018**, 15. [Google Scholar] [CrossRef] [Green Version] - Warren, M.R.; Sun, H.; Yan, Y.; Cremer, J.; Li, B.; Hwa, T. Spatiotemporal establishment of dense bacterial colonies growing on hard agar. Elife
**2019**, 8, e41093. [Google Scholar] [CrossRef] [PubMed] - De Virgilio, C. The essence of yeast quiescence. FEMS Microbiol. Rev.
**2012**, 36, 306–339. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Herskowitz, I. Life cycle of the budding yeast Saccharomyces cerevisiae. Microbiol. Rev.
**1988**, 52, 536. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Minois, N.; Frajnt, M.; Wilson, C.; Vaupel, J.W. Advances in measuring lifespan in the yeast Saccharomyces cerevisiae. Proc. Natl. Acad. Sci. USA
**2005**, 102, 402–406. [Google Scholar] [CrossRef] [Green Version] - Sheu, Y.J.; Barral, Y.; Snyder, M. Polarized growth controls cell shape and bipolar bud site selection in Saccharomyces cerevisiae. Mol. Cell. Biol.
**2000**, 20, 5235–5247. [Google Scholar] [CrossRef] [Green Version] - Mable, B. Ploidy evolution in the yeast Saccharomyces cerevisiae: A test of the nutrient limitation hypothesis. J. Evol. Biol.
**2001**, 14, 157–170. [Google Scholar] [CrossRef] [Green Version] - Serio, T.R.; The University of Chicago, Chicago, IL, USA. Personal Communication, 2020.
- Binder, B.J.; Sundstrom, J.F.; Gardner, J.M.; Jiranek, V.; Oliver, S.G. Quantifying two-dimensional filamentous and invasive growth spatial patterns in yeast colonies. PLoS Comput. Biol.
**2015**, 11, e1004070. [Google Scholar] [CrossRef] [Green Version] - Binder, B.J.; Simpson, M.J. Cell density and cell size dynamics during in vitro tissue growth experiments: Implications for mathematical models of collective cell behaviour. Appl. Math. Model.
**2016**, 40, 3438–3446. [Google Scholar] [CrossRef] - Ruiz-Riquelme, A.; Lau, H.H.; Stuart, E.; Goczi, A.N.; Wang, Z.; Schmitt-Ulms, G.; Watts, J.C. Prion-like propagation of β-amyloid aggregates in the absence of APP overexpression. Acta Neuropathol. Commun.
**2018**, 6, 26. [Google Scholar] [CrossRef] - Halfmann, R.; Jarosz, D.F.; Jones, S.K.; Chang, A.; Lancaster, A.K.; Lindquist, S. Prions are a common mechanism for phenotypic inheritance in wild yeasts. Nature
**2012**, 482, 363–368. [Google Scholar] [CrossRef] [Green Version] - Weinberg, R.P.; Koledova, V.V.; Shin, H.; Park, J.H.; Tan, Y.A.; Sinskey, A.J.; Sambanthamurthi, R.; Rha, C. Oil palm phenolics inhibit the in vitro aggregation of β-amyloid peptide into oligomeric complexes. Int. J. Alzheimer’S Dis.
**2018**, 2018. [Google Scholar] - Esler, W.P.; Stimson, E.R.; Jennings, J.M.; Vinters, H.V.; Ghilardi, J.R.; Lee, J.P.; Mantyh, P.W.; Maggio, J.E. Alzheimer’s disease amyloid propagation by a template-dependent dock-lock mechanism. Biochemistry
**2000**, 39, 6288–6295. [Google Scholar] [CrossRef] [PubMed] - Brugger, S.D.; Baumberger, C.; Jost, M.; Jenni, W.; Brugger, U.; Mühlemann, K. Automated counting of bacterial colony forming units on agar plates. PLoS ONE
**2012**, 7, e33695. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bewes, J.; Suchowerska, N.; McKenzie, D. Automated cell colony counting and analysis using the circular Hough image transform algorithm (CHiTA). Phys. Med. Biol.
**2008**, 53, 5991. [Google Scholar] [CrossRef] - Ferrari, A.; Lombardi, S.; Signoroni, A. Bacterial colony counting with convolutional neural networks in digital microbiology imaging. Pattern Recognit.
**2017**, 61, 629–640. [Google Scholar] [CrossRef] - Scherz, R.; Shinder, V.; Engelberg, D. Anatomical analysis of Saccharomyces cerevisiaestalk-like structures reveals spatial organization and cell specialization. J. Bacteriol.
**2001**, 183, 5402–5413. [Google Scholar] [CrossRef] [Green Version] - Nguyen, B.; Upadhyaya, A.; van Oudenaarden, A.; Brenner, M.P. Elastic instability in growing yeast colonies. Biophys. J.
**2004**, 86, 2740–2747. [Google Scholar] [CrossRef] [Green Version] - Reynolds, T.B.; Fink, G.R. Bakers’ yeast, a model for fungal biofilm formation. Science
**2001**, 291, 878–881. [Google Scholar] [CrossRef] - Brückner, S.; Mösch, H.U. Choosing the right lifestyle: Adhesion and development in Saccharomyces cerevisiae. FEMS Microbiol. Rev.
**2012**, 36, 25–58. [Google Scholar] [CrossRef] [Green Version] - Dranginis, A.M.; Rauceo, J.M.; Coronado, J.E.; Lipke, P.N. A biochemical guide to yeast adhesins: Glycoproteins for social and antisocial occasions. Microbiol. Mol. Biol. Rev.
**2007**, 71, 282–294. [Google Scholar] [CrossRef] [Green Version] - Smith, A.E.; Zhang, Z.; Thomas, C.R.; Moxham, K.E.; Middelberg, A.P. The mechanical properties of Saccharomyces cerevisiae. Proc. Natl. Acad. Sci. USA
**2000**, 97, 9871–9874. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Stenson, J.D.; Hartley, P.; Wang, C.; Thomas, C.R. Determining the mechanical properties of yeast cell walls. Biotechnol. Prog.
**2011**, 27, 505–512. [Google Scholar] [CrossRef] [PubMed] - Hoehme, S.; Drasdo, D. A cell-based simulation software for multi-cellular systems. Bioinformatics
**2010**, 26, 2641–2642. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Byrne, H.; Drasdo, D. Individual-based and continuum models of growing cell populations: A comparison. J. Math. Biol.
**2009**, 58, 657. [Google Scholar] [CrossRef] - Farhadifar, R.; Röper, J.C.; Aigouy, B.; Eaton, S.; Jülicher, F. The influence of cell mechanics, cell-cell interactions, and proliferation on epithelial packing. Curr. Biol.
**2007**, 17, 2095–2104. [Google Scholar] [CrossRef] [Green Version] - Kursawe, J.; Brodskiy, P.A.; Zartman, J.J.; Baker, R.E.; Fletcher, A.G. Capabilities and Limitations of Tissue Size Control through Passive Mechanical Forces. PLoS Comput. Biol.
**2015**, 11, e1004679. [Google Scholar] [CrossRef] [Green Version] - Newman, T.J. Modeling multicellular systems using subcellular elements. Math. Biosci. Eng.
**2005**, 2, 613–624. [Google Scholar] [CrossRef] - Brewer, B.J.; Chlebowicz-Sledziewska, E.; Fangman, W.L. Cell cycle phases in the unequal mother/daughter cell cycles of Saccharomyces cerevisiae. Mol. Cell. Biol.
**1984**, 4, 2529–2531. [Google Scholar] [CrossRef] [Green Version] - Di Talia, S.; Skotheim, J.M.; Bean, J.M.; Siggia, E.D.; Cross, F.R. The effects of molecular noise and size control on variability in the budding yeast cell cycle. Nature
**2007**, 448, 947–951. [Google Scholar] [CrossRef] - Váchová, L.; Palková, Z. How structured yeast multicellular communities live, age and die? FEMS Yeast Res.
**2018**, 18, foy033. [Google Scholar] [CrossRef] - Milani, M.; Batani, D.; Bortolotto, F.; Botto, C.; Baroni, G.; Cozzi, S.; Masini, A.; Ferraro, L.; Previdi, F.; Ballerini, M.; et al. Differential Two Colour X-ray Radiobiology of Membrane/Cytoplasm Yeast Cells: TMR Large-Scale Facilities Access Programme; NASA: Washington, DC, USA, 1998.
- Finch, A.M.; Wilson, R.C.; Hancock, E.R. Matching delaunay graphs. Pattern Recognit.
**1997**, 30, 123–140. [Google Scholar] [CrossRef] [Green Version] - Lee, D.T.; Schachter, B.J. Two algorithms for constructing a Delaunay triangulation. Int. J. Comput. Inf. Sci.
**1980**, 9, 219–242. [Google Scholar] [CrossRef] - Weatherill, N.P.; Hassan, O. Efficient three-dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints. Int. J. Numer. Methods Eng.
**1994**, 37, 2005–2039. [Google Scholar] [CrossRef] - Weier, M.H. Wal-Mart Chooses Neoview Data Warehouse. 2007. Available online: http://www.informationweek.com/news/201202317 (accessed on 2 April 2020).
- Davidson-Pilon, C.; Kalderstam, J.; Jacobson, N.; Kuhn, B.; Zivich, P.; Williamson, M.; Abdeali, J.K.; Datta, D.; Fiore-Gartland, A.; Parij, A.; et al. CamDavidsonPilon/lifelines: v0.24.16; Zenodo: Genève, Switzerland, 2020. [Google Scholar] [CrossRef]
- Efron, B. Logistic regression, survival analysis, and the Kaplan-Meier curve. J. Am. Stat. Assoc.
**1988**, 83, 414–425. [Google Scholar] [CrossRef] - Virtanen, P.; Gommers, R.; Oliphant, T.E.; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nat. Methods
**2020**, 17, 261–272. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Spatial Phenotypes are the Consequence of Processes at Different Scales. (

**A**) Cells transition between different phenotypic states due to genetic mutations or epigenetic determinants. For example, alternative conformations of the prion protein in S. cerevisiae can function as epigenetic determinants of transmissible phenotypes. (

**B**) Daughter cells inherit their phenotype from their mother. In some cases, inefficient transmission of different intracellular constituents (i.e., prion aggregates) can lead to loss of phenotype. (

**C**) Individual cell behaviors impact the propagation, loss and spatial arrangement of phenotypes within the colony. In this paper we investigate the impact of budding division in S. cerevisiae on overall shape, size and spatial organization of cells. During budding division, the new daughter cell forms as a bud on the mother cell and remains attached until it reaches a mature size and they physically separate. (

**D**) The outcome of processes at the molecular, subcellular and cellular scales lead to different morphological traits such as sector-like regions in S. cerevisiae colonies where all cells have lost the prion phenotype.

**Figure 2.**Simulated Yeast Colonies in Nutrient-Rich Conditions and Corresponding Lineage Relationships. We compared overall growth, shape and spatial organization between budding (Middle) and non-budding (Bottom) colonies. Colonies are depicted at three time points (

**A**) 12 h (∼100 cells); (

**B**) 18 h (∼1300 cells) and (

**C**) 24 h (∼15,000 cells). In addition to the physical layout of cells, we analyzed two different networks associated with our colonies, the lineage graph (${G}_{L}$, Top) and the spatial graph (${G}_{S}$, see Section 4.2.2). (Top row) The lineage graph (${G}_{L}$) represents mother-daughter relationships and does not consider cell position in space. As such, the lineage graph is the same for the two different colonies depicted below (budding and non-budding). Cells in colonies and edges in the lineage graph are colored according to the unique subcolony each cell belongs to, where a

**subcolony**is the subset of all cells whose common ancestor is the same immediate daughter of the founder cell: Founder (red), Subcolony 1 (maroon), Subcolony 2 (blue), Subcolony 3 (dark green), Subcolony 4 (light green), Subcolony 5 (lavender), Subcolony 6 (purple), Subcolony 7 (dark orange), Subcolony 8 (gold), Subcolony 9 (yellow), Subcolony 10 (rust), Subcolony 11 (magenta), Subcolony 12 (light pink) and Subcolony 13 (grey). Lineage graphs display the first five subcolonies only. (See Section 4.2.2 for details.)

**Figure 3.**Population Growth, Expanse and Sparsity of Simulated Yeast Colonies in Nutrient-Rich Conditions. (

**A**) Colony growth is exponential with doubling time ∼105 min (inset). Bar plots represent average number of cells in each colony across all 50 simulations for both budding and non-budding colonies under nutrient-rich conditions calculated at 4.5 h intervals; (

**B**) Colony expanse increases over time for both budding and non-budding colonies; (

**C**) Colony sparsity decreases to 1 (implying that the colony is becoming more circular) as the size of the colony increases over time for both budding and non-budding colonies. (See Section 2.1.1 for details.)

**Figure 4.**Age and Spatial Organization of Cells in Nutrient-Rich Colonies. (

**A**) Empirical probability density function for cell ages in synthetic colonies after 24 h of colony growth. We observe that the probability that a given cell is ≤ 8 h old is 0.9388; (

**B**) Empirical probability density function for the normalized distance from the colony center of mass to the birth location of cells born in the last hour of colony growth (${D}_{norm}$). We observe that the probability that ${D}_{norm}$ is ≥ 0.5 is 0.6844; (

**C**) Cell age (h) versus cell distance from the colony center of mass for budding (red) and non-budding (blue) colonies at the 24 h time point. In both (

**C**,

**D**), average values were determined for different age groups using a sliding window with a fixed window size of 20 min. We conclude that cell distance from the colony center of mass is not impacted by cell age or budding division since 95% confidence intervals overlap; (

**D**) Cell age (h) versus cell distance from its mother for budding (red) and non-budding (blue) colonies. The difference between budding and non-budding colonies is given for age groups 2,4,6,8. We conclude that this difference is significant since 95% confidence intervals do not overlap. (See Section 2.1.2 for details).

**Figure 5.**Colony Connectivity in Nutrient-Rich Colonies. Comparison of colony connectivity between budding (red) and non-budding (blue) colonies at 12, 18 and 24 h. We compared connectivity for all cells in (

**A**), as well as cells in two distinct phases: during the time when cells are attached for budding colonies in (

**B**) and after separation in (

**C**). (See Section 4.2.2 for how we define connectivity.) (

**A**) We observe a statistically significant difference in colony connectivity at 18 h ($p=1.423\times {10}^{-27}$) and 24 h ($p=2.008\times {10}^{-99}$). (

**B**) We observe a rapid decrease in connectivity between both 12–18 and 18–24 h in non-budding colonies. This decrease leads to a statistically significant difference between budding and non-budding colonies at 12 h ($p=3.387\times {10}^{-02}$), 18 h ($p=6.521\times {10}^{-34}$) and 24 h ($p=2.867\times {10}^{-103}$). (

**C**) We observe a statistically significant difference in connectivity at 18 h ($p=3.684\times {10}^{-19}$) and 24 h ($p=7.615\times {10}^{-89}$). (p-values for connectivity were computed using independent t-tests. p-value annotation is as follows. *: $1.0\times {10}^{-2}$ p $\le 5.0\times {10}^{-2}$, **: $1.0\times {10}^{-3}$ p $\le 1.0\times {10}^{-2}$, ***: $1.0\times {10}^{-4}$ p $\le 1.0\times {10}^{-3}$, ****: p $\le 1.0\times {10}^{-4}$.) (

**D**) Kaplan-Meier survival curves for the edge connecting mother-daughter cell pairs in ${G}_{S}$ for budding (red) and non-budding (blue) colonies. The y-axis is the probability that a given mother-daughter edge will remain in ${G}_{S}$ for longer than t hours after separation, where time is on the x-axis. We observe that the survival curves are different between the two groups ($p=7.217\times {10}^{-25}$), indicating that the probability that a mother-daughter edge remains in ${G}_{S}$ for longer than t hours is greater for budding colonies. (The p-value comparing survival curves was calculated using a log-rank test as described in Section 4.3.) (See Section 2.1.3 for details.)

**Figure 6.**Subcolony Structure and Organization in Nutrient-Rich Colonies. (

**A**) Comparison of the number of connected components for the first five subcolonies between budding (red) and non-budding (blue) colonies at the 24 h time point. We observe a statistically significant difference in the number of connected components for subcolony 1 ($p=2.885\times {10}^{-10}$), subcolony 2 ($p=1.139\times {10}^{-10}$), subcolony 3 ($p=3.075\times {10}^{-04}$), subcolony 4 ($p=1.539\times {10}^{-06}$) and subcolony 5 ($p=1.959\times {10}^{-03}$). (

**B**) Comparison of time from creation until each of the first five subcolonies splits into 15 connected components. We observe a statistically significant difference between budding (red) and non-budding (blue) colonies for subcolony 1 ($p=1.140\times {10}^{-04}$), subcolony 2 ($p=4.727\times {10}^{-04}$), and subcolony 3 ($p=1.581\times {10}^{-02}$). (

**C**) Comparison of the number of cells in a connected component with less than 10 cells. Note that overall very few cells are in small connected components; however, we observe a statistically significant difference between budding (red) and non-budding (blue) colonies at 12 h ($p=7.408\times {10}^{-40}$), 18 h ($p=7.896\times {10}^{-19}$) and 24 h ($p=1.233\times {10}^{-59}$). (p-values in (

**A**–

**C**) were computed using independent t-tests. p-value annotation is as follows. *: $1.0\times {10}^{-2}$ p $\le 5.0\times {10}^{-2}$, **: $1.0\times {10}^{-3}$ p $\le 1.0\times {10}^{-2}$, ***: $1.0\times {10}^{-4}$ p $\le 1.0\times {10}^{-3}$, ****: p $\le 1.0\times {10}^{-4}$.). (

**D**) Kaplan-Meier survival curves for the length of time a subcolony is made up of less than 15 connected components for budding (red) and non-budding (blue) colonies. The y-axis is the probability that a subcolony consists of less than 15 connected components for longer than t hours, where time is on the x-axis. We observe that the survival curves are different between the budding and non-budding colonies ($p=1.165\times {10}^{-06}$), indicating that budding promotes subcolony connectivity. (The p-value comparing survival curves was calculated using a log-rank test as described in Section 4.3. In addition, 95% confidence intervals for the survival function are shown. See Section 2.1.4 for details.)

**Figure 7.**Population Growth, Expanse and Sparsity of Simulated Yeast Colonies in Nutrient-Limited Conditions. (

**A**) Colony growth is exponential with doubling time ∼123 min (inset). Bar plots represent average number of cells in each colony across all 50 simulations for both budding and non-budding colonies under nutrient-limited conditions calculated at 4.5 h intervals; (

**B**) Colony expanse increases over time for both budding and non-budding colonies; (

**C**) Colony sparsity decreases to 1 as the size of the colony increases over time for both budding and non-budding colonies. (See Section 2.2.1 for details.)

**Figure 8.**Age and Spatial Organization of Cells in Nutrient-Limited Colonies. (

**A**) Empirical probability density function for cell ages in synthetic colonies after 28 h of colony growth. We observe that the probability that a given cell is ≤ 12 h old is 0.9454; (

**B**) Empirical probability density function for the normalized distance from the colony center of mass to the birth location of cells born in the last hour of colony growth (${D}_{norm}$). We observe that the probability that ${D}_{norm}$ is ≥ 0.5 is 0.889; (

**C**) Cell age (h) versus cell distance from the colony center of mass for budding (red) and non-budding (blue) colonies at the 28 h time point. In both (

**C**,

**D**) average values were determined for different age groups using a sliding window with a fixed window size of 20 min. We conclude that distance from the colony center of mass at 28 h is strongly correlated with age. However, we see that cell distance from the colony center of mass is not impacted by budding division since 95% confidence intervals overlap; (

**D**) Cell age (h) versus cell distance from its mother for budding (red) and non-budding (blue) colonies. The difference between budding and non-budding colonies is given for ages 2, 4, 6, 8, 10 and 12. We conclude that this difference is significant since 95% confidence intervals do not overlap. Moreover, since nutrient-limited colonies are smaller than nutrient-rich colonies, this difference becomes even larger when it is considered relative to the size of the colony. (See Section 2.2.2 for details.)

**Figure 9.**Colony Connectivity in Nutrient-Limited Colonies. Comparison of colony connectivity between budding and non-budding colonies in nutrient-limited conditions at 12 h, 18 h, 24 h and 28 h. We compared connectivity for all cells in (

**A**), as well as cells in two distinct phases: during the time when cells are attached for budding colonies in (

**B**) and after separation in (

**C**). (See Section 4.2.2 for how we define connectivity.) (

**A**) We observe a statistically significant difference in colony connectivity at 12 h ($p=3.602\times {10}^{-04}$), 18 h ($p=8.921\times {10}^{-29}$), 24 h ($p=1.094\times {10}^{-79}$) and 28 h ($p=3.389\times {10}^{-110}$). (

**B**) We observe a rapid decrease in connectivity for non-budding colonies between 12–18, 18–24, and 24–28 h. This decrease leads to a statistically significant difference between budding and non-budding colonies at 12 h ($p=6.834\times {10}^{-05}$), 18 h ($p=4.670\times {10}^{-40}$), 24 h ($p=1.050\times {10}^{-92}$) and 28 h ($p=2.658\times {10}^{-122}$). (

**C**) We observe a statistically significant difference in connectivity at 18 h ($p=1.144\times {10}^{-15}$), 24 h ($p=1.888\times {10}^{-58}$) and 28 h ($p=2.140\times {10}^{-83}$). (p-values for connectivity were computed using independent t-tests. p-value annotation is as follows. *: $1.0\times {10}^{-2}$ p $\le 5.0\times {10}^{-2}$, **: $1.0\times {10}^{-3}$ p $\le 1.0\times {10}^{-2}$, ***: $1.0\times {10}^{-4}$ p $\le 1.0\times {10}^{-3}$, ****: p $\le 1.0\times {10}^{-4}$.); (

**D**) Kaplan-Meier survival curves for the edge connecting mother-daughter cell pairs in ${G}_{S}$ for budding (red) and non-budding (blue) colonies. The y-axis is the probability that a given mother-daughter edge will remain in ${G}_{S}$ for longer than t hours after separation, where time is on the x-axis. We observe that the survival curves are different between the two groups ($p=1.314\times {10}^{-75}$) indicating that the probability that a mother-daughter edge remains in the spatial graph for longer than t hours is greater for budding colonies. (The p-value comparing survival curves was calculated using a log-rank test as described in Section 4.3. See Section 2.2.3 for details.)

**Figure 10.**Subcolony Structure and Organization in Nutrient-Limited Colonies. (

**A**) Comparison of the number of connected components for the first five subcolonies between budding (red) and non-budding (blue) colonies at the 28 h time point. We observe a statistically significant difference in the number of connected components for subcolony 1 ($p=3.39\times {10}^{-10}$), subcolony 2 ($p=4.454\times {10}^{-17}$), subcolony 3 ($p=8.537\times {10}^{-11}$), subcolony 4 ($p=2.884\times {10}^{-11}$) and subcolony 5 ($p=3.783\times {10}^{-06}$). (

**B**) Comparison of time until each of the first five subcolonies splits into 5 connected components. We observe a statistically significant difference between budding (red) and non-budding (blue) colonies for subcolony 1 ($p=8.967\times {10}^{-05}$), subcolony 2 ($p=6.246\times {10}^{-08}$), subcolony 3 ($p=8.636\times {10}^{-05}$), and subcolony 4 ($p=9.154\times {10}^{-06}$). (

**C**) Comparison of the number of cells in a connected component with less than 10 cells. Note that nutrient-limited growth results in a large decrease in the total number of cells in small components at the 24 h time point compared to nutrient-rich growth. However, we still observe a statistically significant difference between budding (red) and non-budding (blue) colonies at 18 h ($p=3.537\times {10}^{-02}$), 24 h ($p=9.040\times {10}^{-07}$), and 28 h ($p=5.410\times {10}^{-18}$). (

**D**) Kaplan-Meier survival curves for the length of time a subcolony is made up of less than 5 connected components for budding (red) and non-budding (blue) colonies. The y-axis is the probability that a subcolony consists of less than 5 connected components for longer than t hours, where time is on the x-axis. We observed that the survival curves are different between the budding and non-budding colonies ($p=2.428\times {10}^{-23}$) indicating that the probability that a given subcolony remains connected (i.e., less than 5 connected components) for longer than t hours is greater for budding colonies. (The p-value comparing survival curves was calculated using a log-rank test as described in Section 4.3. In addition, 95% confidence intervals for the survival function are shown. See Section 2.2.4 for details.)

**Figure 11.**Nutrient Limitation Drives Spatial Organization of Cells. We compared growth and overall spatial organization between four different types of colonies (budding/non-budding, nutrient-rich/nutrient-limited). Cells in colonies are colored according to the unique subcolony each cell belongs to: Subcolony 1 (dark green), Subcolony 2 (blue), Subcolony 3 (cyan), Subcolony 4 (teal), Subcolony 5 (light green), Subcolony 6 (yellow-green), Subcolony 7 (yellow), Subcolony 8 (gold), Subcolony 9 (orange), Subcolony 10 (red), Subcolony 11 (magenta), Subcolony 12 (purple) and Subcolony 13 (pink). Typical simulation output from budding and non-budding colonies grown in nutrient-limited conditions: 12 hours (

**A**), 18 hours (

**B**), 24 hours (

**C**), 28 hours (

**D**). (

**E**) Nutrient limitation results in more variation in the percentage of the population contained in each subcolony (Subcolony 1 ($p=2.779\times {10}^{-20}$), Subcolony 2 ($p=6.143\times {10}^{-25}$), Subcolony 3 ($p=4.636\times {10}^{-23}$), Subcolony 4 ($p=6.574\times {10}^{-32}$), and Subcolony 5 ($p=2.985\times {10}^{-09}$)). (p-values were calculated using the Levene test for equal variances. p-value annotation is as follows. *: $1.0\times {10}^{-2}$ p $\le 5.0\times {10}^{-2}$, **: $1.0\times {10}^{-3}$ p $\le 1.0\times {10}^{-2}$, ***: $1.0\times {10}^{-4}$ p $\le 1.0\times {10}^{-3}$, ****: p $\le 1.0\times {10}^{-4}$.) (

**F**) Birth Location of cells born within the last hour of colony growth changes significantly between nutrient-rich (red) and nutrient-limited (cyan) conditions ($p<1.0\times {10}^{-32}$). (p-value was calculated using the Kolmogorov-Smirnov statistic in python. See Section 2.2.5 for details.)

**Figure 12.**Biophysical Model of Cell-Cell Interactions. We simulate budding (

**A**) and non-budding (

**B**) colonies using a 2D center-based modeling approach where cells interact through different potentials. (

**A**) For budding colonies, the mechanical interactions of all cell pairs (${E}^{CC}$) are governed by a combination of repulsive and attractive interactions using a modified Hertz-model described in Equation (1). We use a linear spring to model the additional adhesive force between mother cells and new daughter cells during the budding phase (${E}^{MB}$) as in Equation (4); (

**B**) For non-budding colonies, the mechanical interactions are similar to that of cells in budding colonies except that the adhesive force between mother cells and new daughter cells during the budding phase (${E}^{MB}$) is neglected (see Section 4.1.1 for details).

**Figure 13.**Selecting a Bud Site. The choice of the next bud location for a cell depends on whether it is a mother cell (left) or a new daughter cell (right). (Left) For mother cells, the next bud location will be chosen either adjacent to the previous bud scar with probability $0.5$ (

**A**) or opposite to the previous bud scar with probability $0.5$ (

**B**). In the case when the location of the new bud site overlaps with a previous bud site (

**A**), we adjust the location of the new bud site by increments of ${10}^{\circ}$ in either the clockwise (probability $0.5$) or counterclockwise (probability $0.5$) direction until we arrive at a location with no previous bud scar. (Right) For new daughter cells the next bud location (red) is chosen opposite to the previous birth scar (black) with probability 1 (

**C**). New daughter cells that have successfully completed a full cell cycle are considered mother cells for the remainder of the simulation. (See Section 4.1.3.)

**Figure 14.**Cell Cycle Length. (Left): The $G1$ phase for mother cells is approximately 15 min. Since mother cells have already reached their adult size, the $G1$ phase serves as a waiting period before the mother cell enters $G2$ and forms a bud. When the mother cell enters $G2$, the new daughter cell forms as a bud and stays attached for ∼75 min as it grows. After ∼75 min, the mother and new daughter physically separate resulting in two unevenly sized cells. At this time, the mother cell enters $G1$ and begins a new cell cycle. (Right:) The new daughter cell continues to grow until it reaches its adult size (∼45 min) and forms its own bud. Under nutrient-limited conditions, the length of the $G1$ and $G2$ phases are increased for both mother and daughter cells (see Section 4.1.4 for details).

**Figure 15.**Colony Sparsity and Expanse. We first compute the colony center of mass (represented as a green square) using Equation (13). Then we determine the radius (length of the blue line) of the smallest circle which surrounds the entire colony centered at the center of mass (shown in red) using Equation (14). The colony sparsity is then computed using Equation (16) with the result of Equation (14), the area of the circle, and the total area of the cells using Equation (15). (Left): The space within the circle is more dense, thus covering more area within the circle with radius equal to the colony expanse, resulting in a small colony sparsity. (Right): The space that cells occupy within the circle is less dense, resulting in a higher colony sparsity.

**Figure 16.**Colony Spatial Graphs. The vertex set for all three colony graphs is the same (all cell centers). The founder cell is designated in black. The edge set differs depending on the relationships between cells; (

**A**): The edge set for the spatial graph ${G}_{S}$ (blue edges) are those induced by the the Delaunay triangulation applied to the cell centers (Equation (17)); (

**B**): The edges for the lineage graph ${G}_{L}$ (red edges) correspond to mother-daughter pairs (Equation (18)); (

**C**): The edge set for the intersection graph ${G}_{I}$ (purple edges) include those edges that belong to the two previous edge sets (Equation (19)).

**Figure 17.**Constructing Subcolony Graphs. Subcolony graphs are constructed from partitions of the spatial graph ${G}_{S}$ according to the following procedure. (

**A**): We generate the the same spatial graph (${G}_{S}$) using the Delaunay triangulation (Equation (17)). (The founder cell is indicated in black.) (

**B**): We define a subcolony as a subset of cells in the lineage graph consisting of a daughter of the founder cell along with all of its descendants (Equation (21)). Edges are colored based one which subcolony each cell belongs to (Equation (22)). We then remove edges from the spatial connecting cells from different subcolonies (dotted black edges). (

**C**): Removing these edges results in the subcolony graph ${G}_{\mathrm{sub}}$, a set of subgraphs of ${G}_{S}$ that we index by daughter cells: ${G}_{\mathrm{sub},{d}_{k}}$. These graphs preserve the spatial relationship between cells within the same subcolony (Equation (25)).

**Table 1.**Parameter Values Used in ABM. Descriptions of the biophysical and biological processes corresponding to these variables are detailed in Section 4.

Parameter | Symbol | Value | Units | Meaning | Reference |
---|---|---|---|---|---|

Poisson ratio | $\sigma $ | 0.3 | Incompressibility of yeast cells | [38,50,74] | |

Young’s Modulus | E | 1000 | $kPa$ | Mechanical property of yeast cell walls | [38,50,74] |

Receptor Surface Density | $\varphi $ | 10E15 | ${\mathrm{m}}^{-2}$ | Density of surface adhesion molecules in the contact area | [38,50] |

Single Bond Binding Energy | ${W}_{s}$ | 25 ${k}_{B}T$ | [38,50] | ||

${E}^{MB}$ Linear Spring Constant | ${K}_{\mathrm{bud}}$ | 25 | $nN/\mathsf{\mu}$m | Attachment of bud on mother cell | calibrated |

Damping Coefficient | $\eta $ | 2.5 | $Ns/\mathsf{\mu}{\mathrm{m}}^{2}$ | Viscosity of the growth media | [38,50] |

Average Length of $G2$ phase | $G{2}_{\mathrm{avg}}$ | 75 | min | [82,83] | |

Average Length of $G1$ phase (new daughters) | $G{1}_{{\mathrm{avg}}_{\mathrm{daughter}}}$ | 120 | min | [82,83] | |

Average Length of $G1$ phase (mothers) | $G{1}_{{\mathrm{avg}}_{\mathrm{mother}}}$ | 15 | min | [82,83] | |

Average Mature Radius Size | ${R}_{\mathrm{avg}}$ | 2.58 | $\mathsf{\mu}$m | [84] | |

Carrying Capacity | ${M}_{j,\mathrm{max}}$ | $18\pi {R}_{\mathrm{avg}}^{2}$ | $\mathsf{\mu}$${\mathrm{m}}^{2}$ | Maximal possible biomass for each subdomain | calibrated |

Subdomain Size | ${D}_{i}\left(t\right)$ | 25 | $\mathsf{\mu}$${\mathrm{m}}^{2}$ | Area of each subdomain | calibrated |

Rate of Maximum Cell Cycle Adjustment | r | 0.003 | Controls the amount cell cycle is adjusted at each timestep | calibrated | |

Timestep | $\mathsf{\Delta}t$ | 0.00144 | min | calibrated |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Banwarth-Kuhn, M.; Collignon, J.; Sindi, S.
Quantifying the Biophysical Impact of Budding Cell Division on the Spatial Organization of Growing Yeast Colonies. *Appl. Sci.* **2020**, *10*, 5780.
https://doi.org/10.3390/app10175780

**AMA Style**

Banwarth-Kuhn M, Collignon J, Sindi S.
Quantifying the Biophysical Impact of Budding Cell Division on the Spatial Organization of Growing Yeast Colonies. *Applied Sciences*. 2020; 10(17):5780.
https://doi.org/10.3390/app10175780

**Chicago/Turabian Style**

Banwarth-Kuhn, Mikahl, Jordan Collignon, and Suzanne Sindi.
2020. "Quantifying the Biophysical Impact of Budding Cell Division on the Spatial Organization of Growing Yeast Colonies" *Applied Sciences* 10, no. 17: 5780.
https://doi.org/10.3390/app10175780