Research Article

Physics of Living Systems

Cellular organization in lab-evolved and extant multicellular species obeys a maximum entropy law

School of Physics, Georgia Institute of Technology, United States
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, United Kingdom
School of Biological Sciences, Georgia Institute of Technology, United States
Quantitative Biosciences Graduate Program, Georgia Institute of Technology, United States
Department of Molecular Biology, Umeå University, Sweden

Feb 21, 2022

https://doi.org/10.7554/eLife.72707

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Version of Record published: February 21, 2022 (This version)
Accepted: January 4, 2022
Received: August 2, 2021

1. Of interest
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Abstract
Editor's evaluation
Introduction
Results
Discussion
Materials and methods
Appendix 1
Data availability
References
Article and author information
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Abstract

The prevalence of multicellular organisms is due in part to their ability to form complex structures. How cells pack in these structures is a fundamental biophysical issue, underlying their functional properties. However, much remains unknown about how cell packing geometries arise, and how they are affected by random noise during growth - especially absent developmental programs. Here, we quantify the statistics of cellular neighborhoods of two different multicellular eukaryotes: lab-evolved ‘snowflake’ yeast and the green alga Volvox carteri. We find that despite large differences in cellular organization, the free space associated with individual cells in both organisms closely fits a modified gamma distribution, consistent with maximum entropy predictions originally developed for granular materials. This ‘entropic’ cellular packing ensures a degree of predictability despite noise, facilitating parent-offspring fidelity even in the absence of developmental regulation. Together with simulations of diverse growth morphologies, these results suggest that gamma-distributed cell neighborhood sizes are a general feature of multicellularity, arising from conserved statistics of cellular packing.

Editor's evaluation

This work uncovers a simple but far-reaching statistical principle that describes the geometry of cell packing in snowflake yeast and green algae. It draws on ideas from granular physics to offer new insight into universal rules of multicellular geometry that are otherwise easily obscured by the cell-scale idiosyncrasies of the different biological systems.

https://doi.org/10.7554/eLife.72707.sa0

Introduction

The evolution of multicellularity was transformative for life on Earth, occurring in at least 25 separate lineages (Grosberg and Strathmann, 2007). The success of multicellular organisms is due in part to their ability to assemble cells into complex, functional arrangements. Self-assembly, however, is fundamentally subject to random noise (Zeravcic and Brenner, 2014; Szavits-Nossan et al., 2014; Damavandi and Lubensky, 2019) that affects the final emergent structure (Michel and Yunker, 2019). The physiology of multicellular organisms can depend sensitively on the geometry of cellular packing (Bi et al., 2015b; Drescher et al., 2016; Jacobeen et al., 2018b; Brunet et al., 2019; Schmideder et al., 2021), and such noise may therefore have direct consequences on organismal fitness. Understanding the evolution of multicellularity, and the subsequent evolution of multicellular complexity (Bell and Mooers, 1997), requires understanding the impact of random noise on multicellular self-assembly. How do organisms accurately assemble functional multicellular components in the presence of noise?

Recent work has shown that extant multicellular organisms can either suppress (Hong et al., 2016) or leverage (Haas et al., 2018) variability in the process of reliably generating structures, and their tissues can change function based on cellular packing geometry (Bi et al., 2015a). This occurs through a coordinated developmental process involving genetic (Davidson, 2001), chemical (Sampathkumar, 2020), mechanical (Deneke and Di Talia, 2018), and bioelectric (Levin, 2004) feedbacks between interacting cells. However, even with coordinated developmental processes, noise during self-assembly results in deviations from perfectly regular structures. Further, as these developmental processes have not yet evolved in nascent multicellular organisms, it is unclear how unregulated assembly can reliably result in reproducible packing geometries and multicellular structures.

Multicellular organisms also exhibit diverse growth morphologies; for example, cells can remain attached through incomplete cytokinesis (Bonner, 1998; Grosberg and Strathmann, 2007; Knoll, 2011), they can adhere through aggregative bonds (Claessen et al., 2014), and they can assemble multicellular groups through successive cell division within a confining membrane (Angert, 2005; Herron et al., 2019). These growth morphologies can have distinct intercellular connection topologies (Yanni et al., 2020), changing how randomness is manifested. For instance, groups that grow with persistent mother-daughter bonds maintain the same intercellular connections, ‘freezing’ in place any structural randomness that arises during reproduction. In contrast, cells in aggregates can rearrange, so their final structure emerges from a combination of reproduction and intercellular interactions and noise (Delarue et al., 2016; Hartmann et al., 2019). Further, the dimensionality of multicellular groups can vary, from filaments (Herrero et al., 2016) and quasi-two-dimensional sheets (Besson and Dumais, 2011; Brunet et al., 2019) to groups that grow equally in three dimensions (Ratcliff et al., 2012; Tang et al., 2020; Butterfield, 2016). While the impact of noise on systems in thermal equilibrium is well known to depend sensitively on spatial dimensionality (Mermin and Wagner, 1966; Hohenberg, 1967; Vivek et al., 2017), no such information is yet at hand for biological development, which is intrinsically out of equilibrium. The growth morphology, connection topology, and dimensionality therefore altogether determine a multicellular architecture. Randomness resulting from many sources, such as stochastic cell division, variability in cell growth, intercellular interactions, and more, subsequently occurs as perturbations to this idealized form. It would appear that noise manifests in a unique, context-dependent manner in each of these different multicellular systems.

Here, we provide experimental evidence that, rather than being context-dependent, fluctuations in cell packing geometry instead follow a universal distribution, independent of the presence or absence of developmental regulation. We quantify the distributions of cellular space in two different types of organisms: experimentally evolved multicellular yeast (Ratcliff et al., 2012) and wild-type multicellular green algae (Goldstein, 2015). In both cases, maximum entropy considerations (Aste and Di Matteo, 2008) (see inset box) accurately predict the cell packing distribution. Building on these observations, we use computational models of diverse prescribed growth rules, mimicking extant biological morphologies, to show that cells are ubiquitously packed according to the maximum entropy principle. Detailed analysis of the case of green algae shows that correlations, that is, the lack of structural randomness, produce deviations from maximum entropy predictions, but that even a relatively small amount of randomness is sufficient to generate cellular packings that largely follow maximum entropy predictions. Next, we explore the evolutionary consequences of cell packing. We use the cell packing distribution to predict the distribution of snowflake yeast group sizes, an emergent multicellular trait that arises from cell crowding (Jacobeen et al., 2018b). Then, we use a theoretical analysis to show that the effects of fluctuations in intercellular space on the motility of green algae are small. These findings together suggest that, rather than impeding innovation, fluctuations in cell packing are highly repeatable, and may play a fundamental role in the origin and subsequent evolution of multicellular organisms.

Results

Maximum entropy

Within statistical physics, the maximum entropy principle relates randomness in low-level units (e.g. cells) to the properties of the assembly (e.g. a multicellular group). It works by enumerating all low-level configurations that conform to a set of constraints. Any particular group-level property can be generated by many different low-level configurations, but some group-level properties may correspond to more low-level configurations than others. Those that are generated by many configurations are more likely to be observed than those that correspond to relatively few configurations; in this way, the maximum entropy principle allows one to calculate the probability of observing different group properties, given a set of constraints. Multicellular groups obey a simple but universal constraint: each group has some total volume, $V$ . This volume can be divided into $N$ pieces, where $N$ is the total number of cells. Each piece is associated with a particular cell, and the $N$ pieces must sum to the total volume of the group, $V = \sum_{i} v_{i}$ , for $i = 1, 2, \dots, N$ . Using this constraint, and assuming no correlations, one can predict the most likely distribution of volumes for the $N$ pieces. This approach has been successfully used to predict the distribution of free volumes within granular materials and foams (Aste and Di Matteo, 2008; Katgert and van Hecke, 2010). Here, we use it to predict the distribution of cellular free volumes in the absence of spatial correlations in cell positions.

Consider the ensemble of all possible cellular configurations in a simple group. As first derived by Aste and Di Matteo, 2008 and Aste et al., 2007 for granular materials, the maximum entropy probability distribution $p (v)$ of cell neighborhood volumes within $V$ is the modified gamma distribution

p (v) = \frac{k^{k}}{Γ (k)} \frac{{(v - v_{c})}^{k - 1}}{{(\bar{v} - v_{c})}^{k}} \exp (- k \frac{v - v_{c}}{\bar{v} - v_{c}})

where $\bar{v}$ is the mean cell neighborhood volume, v_c is the minimum cell neighborhood volume, $Γ (k)$ is the gamma function, and $k \equiv {(\bar{v} - v_{c})}^{2} / σ_{v}^{2}$ is a shape parameter that is defined by v_c, $\bar{v}$ , and the variance of the cell neighborhood volumes, $σ_{v}^{2}$ . This distribution is expected if cell neighborhood volumes are determined independently of each other (while still conforming to the total volume constraint). In other words, volumes must be set randomly; correlations between the size of separate volumes will lead to deviations from maximum entropy predictions. If this condition holds, then maximum entropy volume distribution predictions should be valid, regardless of other geometric or structural details. For example, maximum entropy statistics hold in granular materials, despite the fact that they must obey strict force and torque balance conditions (Aste and Di Matteo, 2008; Snoeijer et al., 2004; Bi et al., 2015a). Further, the same approach applies to groups with a constraint on total area or length; this does not change the result, and $V$ can be replaced by $A$ or $L$ without other modifications. However, we would not expect this prediction to necessarily hold for confluent cell layers (i.e. packing fraction $ϕ \approx 1$ ), since in general cell size is a highly regulated process; when cells tile space, the cell neighborhood volume distribution will strongly correlate with the the cell size distribution.

In practice, we divide the total group volume or area into $N$ pieces via a Voronoi tessellation. The size of the space associated with cell $i$ includes the cell itself and the portion of intercellular space closer to its center than to the center of any other cell. As cells must have non-zero size, we therefore set v_c to be the volume of a single cell without any intercellular space (or a_c, the area of a single cell).

Experimental tests of multicellular maximum entropy predictions

To test whether different kinds of multicellular groups pack their cells according to the maximum entropy principle, we investigated cell packing in two different multicellular organisms. First, we used experimentally evolved ‘snowflake’ yeast (Ratcliff et al., 2012), a model system of undifferentiated multicellularity. Second, we used the green microalga Volvox carteri, a member of the volvocine algae that first evolved multicellularity in the Triassic (Starr, 1969; Herron et al., 2009).

Snowflake yeast

Snowflake yeast grow via incomplete cytokinesis, generating branched structures in which mother-daughter cells remain attached by permanently bonded cell walls (Figure 1A). New buds appear on ellipsoidal cells at a polar angle $⟨ θ ⟩ = 42 % \pm 23 %$ and azimuthal angle $ϕ$ that is randomly distributed ( $⟨ ϕ ⟩ = 180 % \pm 104 %$ , Figure 1—figure supplement 1). Therefore, even though snowflake yeast structure is limited to configurations in which daughter cells remain attached at their base to their mother, these new cells bud in random orientations. Each snowflake yeast cluster has a different specific configuration of cells; the ensemble of snowflake clusters will include many different configurations. We expect that this structural randomness produces predictable distributions of cellular neighborhood volumes. Conversely, if there are strong correlations in the locations of daughter cells, then we will observe deviations from maximum entropy predictions regarding the cell neighborhood volumes.

Figure 1 with 1 supplement see all

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Cell packing in two multicellular species.

(A) Cross-section of a multicellular yeast organism, which grows with persistent intercellular bonds. Scalebar is 5 m. The inset shows a smaller section, with ellipsoidal fits to individual cells along with their corresponding Voronoi polyhedra. Black overlays indicate the connection topology between yeast cells; not all connections are labeled. (B) Darkfield microscopy image of *Volvox carteri*, scalebar is $100 µ m$ . Inset: a small piece of the Voronoi-tessellated surface; black points are somatic cell positions. (C) Distributions of Voronoi polyhedron volumes as a function of cell size normalized by average size v_c for snowflake yeast. In orange is the histogram for all cells; the other three distributions correspond to different subsections of Voronoi volumes. The cells were grouped into spherical shells with radius $R$ and width $Δ R$ from the cluster center of mass. Shown are shells with edges $[0, 6.2]$ , $[6.2, 9.7]$ , and $[9.7, 20.4]$ . Black lines are maximum entropy predictions. (D) Distributions of solid angles subtended by *Volvox* somatic cells divided by a minimum solid angle $Ω_{c}$ . Solid black lines are the maximum entropy predictions. The top row shows the histogram for all cells in green and a subsection of correlated areas in gray. Bottom row illustrates the subsectioning process: blue polygon is the center of the subsectioned region. Only the Voronoi polygons, *i.e*. not the somatic cells, are shown for clarity. (**E,F**) Empirical cumulative distribution function vs entropic predictions for all distributions shown in c,d. The dashed black line represents hypothetical perfect agreement between observation and prediction.

Figure 1—source data 1 Experimental data files enumerating the cell centers positions for each organism sampled. The folders are subdivided into those on snowflake yeast (from SEM studies) and Volvox (from lightsheet studies). Each subdirectory contains an explanatory README file.: https://cdn.elifesciences.org/articles/72707/elife-72707-fig1-data1-v1.zip
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To determine the distribution of cell neighborhood volumes, we first must measure the position of every cell in a cluster. It is difficult to image individual cells within snowflake yeast clusters due to excessive light scattering. Instead, we used a serial block face scanning electron microscope equipped with a microtome to scan and shave thin ( $50 nm$ ) layers off a resin block with embedded yeast clusters with stained cytoplasms. This process allowed us to determine the 3D structure of $N = 20$ snowflake yeast clusters and locate cell centers with nanometer precision.

We define the group volume as the smallest convex hull that surrounds all cells in the cluster and computed the 3D Voronoi tessellation of cell centers within that (Figure 1A). The distribution of cellular Voronoi volumes closely matched the predicted k-gamma distribution (Figure 1C, $k = 2.88$ ). This agreement is quantified via ‘P-P plots’ of the empirical cumulative distribution function (CDF) plotted against the predicted k-gamma CDF. We find a root-mean-square residual $r_{R M S} = \sqrt{⟨ {(F (v) - F_{i})}^{2} ⟩} = 0.02$ , where $F_{i}$ is the empirical CDF and $F (v)$ is the predicted k-gamma CDF.

The influence of the convex hull on these results was investigated by using an alternative procedure in which the Voronoi volumes were binned into shells centered at the cluster’s center of mass (Figure 1C and E). We binned cells into shells with shell edges of $[0, 6.2]$ , $[6.2, 9.7]$ , and $[9.7, 20.4]$ away from the center of mass. We found that the distribution of Voronoi volumes within each shell matched the predicted k-gamma distribution, with $r_{R M S} = {0.037, 0.020, 0.014}$ , $k = {3.45, 3.08, 4.63}$ in the shells shown in Figure 1C(ii-iv).

Volvocine algae

To test if cell neighborhood volumes in extant multicellular organisms are consistent with maximum entropy cell packing predictions, we examined cell packing within the green microalgae Volvox carteri. Development in V. carteri, which evolved over millions of years, is highly regulated, occurring through a stereotyped morphological progression (Kirk, 2005). V. carteri embryos arise as a spherical cellular monolayer from palintomic cell divisions with incomplete cytokinesis, which leaves the cells attached via cytoplasmic bridges. These bridges disappear when ECM is secreted by the cells, filling the entire sphere, and eventually moving the cells apart. The approximately 1000 somatic cells remain embedded on the surface of a translucent sphere of extracellular matrix (Figure 1B). While six-fold coordination is the most frequent local arrangement of somatic cells, the fact that the cells are embedded in a surface with spherical topology requires there to be ‘defects’ with differing coordination number (e.g. 5,7), and these are found interspersed around the spheroid. Thus, despite their developmental regulation, somatic cells exhibit a degree of disorder with respect to coordination number. From a physics perspective, the local hexatic order in the somatic cell arrangement is low (see Materials and methods).

To determine the distribution of Volvox cell neighborhood sizes, we imaged somatic cells using their chlorophyll autofluorescence in a light sheet microscope. Since the somatic cells are arranged around a surface embedded in 3D space, we constructed a 2D Voronoi tessellation of somatic cells on the surface. Each organism imaged had a different size, and therefore had a different mean Voronoi area $⟨ A ⟩$ . To compare distributions across organisms, we removed the systematic area differences by recording the solid angle $Ω_{i} = 4 π A_{i} / S$ subtended by each somatic cell, where $S = \sum_{i} A_{i}$ is the total surface area of the organism. We found that the k-gamma distribution largely matched the distribution of solid angles (Figure 1D, $k = 2.40$ , $r_{R M S} = 0.04$ ). However, there are systematic deviations between the data and maximum entropy predictions (Figure 1F).

We next investigated if maximum entropy predictions are more accurate within subregions with similar mean solid angles; specifically we examine regions whose mean is $⟨ Ω ⟩ = 0.0185 \pm 0.0003$ , obtained across six organisms. The distribution of Voronoi solid angles within these subregions closely follows the k-gamma distribution (Figure 1F, $k = 10.66$ , $r_{R M S} = 0.01$ ). This observation suggests that while there are systematically correlated subregions of cells, within these subregions cells are largely arranged randomly. Thus, the organization of Volvox carteri somatic cells is consistent with maximum entropy predictions.

Simulations of different growth morphologies

We next used simulations to investigate the impact on cell packing of four different growth morphologies: growth via incomplete cell division (cf. snowflake yeast), cells distributed on a spherical surface (cf. Volvox), aggregation, and palintomy. The goal of these studies was to determine if morphological details and constraints impact entropic packing using simplified models that capture the essential features of the growth and behavior of these varied organisms. These simulations captured a couple basic properties of multicellularity: organisms may grow in both two and three dimensions, and they may assemble with two different classes of bonds: bonds that are reformable, and bonds that are not reformable. Nonreformable, or ‘fixed’ bonds (like those formed from incomplete cytokinesis) are common in plants, algaes, and fungi; on the other hand, reformable bonds (such as those formed via sticky proteins) are common in animal tissues and cellular aggregates, like biofilms and slime molds. Finally, palintomy, or growth confined inside a maternal membrane, is also common from algae to animals.

First, we performed geometric simulations of multicellular groups that grow via incomplete cell division; these simulations were inspired by previous simulations of snowflake yeast (Jacobeen et al., 2018a; Jacobeen et al., 2018b). Daughter cells bud from mother cells with experimentally determined polar angle and random azimuthal angle, and remain attached to mother cells with rigid bonds. We ran 9, 100 simulations starting from a single cell, each of which underwent seven generations of division, and calculated the Voronoi tessellation of the final structure from each simulation. The distribution of Voronoi volumes closely matched the k-gamma distribution across four orders of magnitude (Figure 2A, $k = 2.26$ , $r_{R M S} = 0.007$ ).

Figure 2 with 3 supplements see all

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Entropic packing is a general feature of simple multicellularity.

We simulated four different growth morphologies: (A) Tree-like groups formed with rigid, permanent bonds between cells, (B) surface-attached cells located on a spherical surface, (C) aggregates formed with attractive ‘sticky’ interactions, and (D) groups formed by rapid cell division within a maternal membrane. In all subfigures, left panel shows the predicted and observed probability distributions, and right panel plots the observed cumulative distribution vs. the expected cumulative distribution. Histogram bars represent measured Voronoi volume distribution in simulations, and black solid line represents the maximum entropy prediction. Maximum entropy predictions accurately described the distribution of cellular volumes/areas, despite their varying mechanisms of group formation ( $r_{R M S} \leq 0.01$ ).

Figure 2—source data 1 Simulation data (enumerating the cell center positions) for the six classes of numerical studies; aggregation, apoptosis, polydispersity, snowflake yeast growth, tree-like growth, and Volvox growth. Each subdirectory contains an explanatory README file.: https://cdn.elifesciences.org/articles/72707/elife-72707-fig2-data1-v1.zip
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Inspired by Volvox, we simulated cells distributed across the two-dimensional surface of a sphere through a random Poisson point process. We completed 10 simulations, each with 1000 cells, and computed the distribution of solid angles subtended by Voronoi cells. As shown in Figure 2B, the distribution of Voronoi solid angles is consistent with maximum entropy predictions ( $k = 9.29$ , $r_{R M S} = 0.009$ ).

Next, we simulated organisms that stick together via reformable cell-cell adhesions, a mechanism of group formation that is common in biofilms and extant aggregative multicellular life (Claessen et al., 2014) (i.e. Dictylostelium and Myxococcus; Figure 2C). In these simulations, multicellular aggregates were grown from a single cell. Seven generations of cell division occurred, in which new cells appear on the surface of existing cells at random positions, and steric interactions force cells to separate after division and occupy space. Aggregative bonds were modeled through harmonic interactions of the cell centers. The observed Voronoi volume distributions were consistent with maximum entropy predictions ( $k = 7.84$ and $r_{R M S} = 0.007$ ).

Finally, we modeled cells undergoing palintomic division within a maternal cell wall, as is common in green algae (Lürling and Van Donk, 1997; Boraas et al., 1998; Ratcliff et al., 2013; Fisher et al., 2016; Herron et al., 2019), and is reminiscent of both baeocyte production in Stanieria bacteria (Angert, 2005) and neoproterozoic fossils of early multicellularity (Xiao et al., 1998; Figure 2D). The details of these simulations remained similar to the simulations of aggregative multicellularity, with the important difference being that instead of harmonic interactions between cell centers enforcing groups to stay together, cells interacted with a spherical maternal wall acting as a corral. The Voronoi volume distributions for these simulations were also consistent with maximum entropy predictions ( $k = 15.16$ and $r_{R M S} = 0.013$ ).

Taken together, the results of these simulations suggest that a broad distribution of cell neighborhood sizes is a general feature of multicellular growth morphologies. In particular, when cell locations are random under these rules, cell neighborhood size distributions closely follow the k-gamma distribution.

The role of spatial correlations

While we have shown that the distribution of cell neighborhood volumes closely follows the k-gamma distribution in two very different organisms, we have also seen that in some cases maximum entropy predictions are more accurate in sub-sections of an organism than across its entirety. For instance, in Volvox we observed that $r_{R M S}$ is much smaller within subregions with similar mean solid angles than across the whole organism. This observation suggests that correlations exist in the arrangement of V. carteri somatic cells. Such a correlation is not wholly unexpected; V. carteri have a well-known anterior-posterior polarity which they use to orient and swim toward light sources (Drescher et al., 2010b). This polarity may affect the somatic cell packing distribution in different subregions of the organism, leading to deviations from maximum entropy predictions.

The spatial correlations in the cellular areas in Volvox were studied first by plotting a 3D heatmap of Voronoi solid angle sizes (Figure 3A). It is apparent that extended spatial regions have well-defined and non-random mean Voronoi solid angles. We quantified this feature by calculating the spatial correlation function $C (Q)$ of the solid angle

C (Q) = \frac{⟨ (Ω - ⟨ Ω ⟩) Y_{Q} ⟩}{σ_{Ω} σ_{Y_{Q}}},

where $Y_{Q} = J {(Q)}^{- 1} \sum_{j} (Ω_{j} - ⟨ Ω ⟩)$ is the average deviation of the solid angle of a given polygon’s neighbors at a neighbor distance $Q$ from the mean. Here, the number of neighbors is $J (Q)$ , a function of $Q$ , which enumerates the network distance from the polygon of interest (i.e. $Q = 1$ calls the nearest neighbors, of which there are $J (1)$ , $Q = 2$ calls the next nearest neighbors, of which there are $J (2)$ , and so on). The standard deviation of the solid angle across the population is $σ_{Ω}$ , and $σ_{Y_{Q}}$ is the standard deviation of $Y_{Q}$ across the population. We find that Volvox Voronoi solid angles are positively correlated at distances as large as $Q = 10$ (Figure 3A). This analysis suggests that there are systematic differences in Volvox group structure in different spatial regions. We therefore should expect to observe deviations from the k-gamma distribution, which was derived under the assumption that there are no correlations in the division of space among cells.

Figure 3

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Correlations lead to deviations from maximum entropy predictions in *Volvox carteri*.

(A) Correlation function of Voronoi polygon areas vs. network neighbor distance $Q$ . Green circles represent all experimental *Volvox* data. Lines indicate the same correlation function calculated in subsections of size $Q_{0} = {6, 10, 14, 18}$ . Inset: visualization of spatial correlations of solid angle; one *Volvox’s* Voronoi tessellation is displayed with a three-color heatmap corresponding to polygons with areas smaller than (light gray), within (gray) and larger (black) than one standard deviation of the mean. Scale bar is $200 µ m$ . (B) PP plots for the observed vs predicted cumulative distribution function. In green is the *Volvox* distribution for all cells before corrections for correlations. A selection of differently-sized subsections is also plotted, corresponding to sizes $Q_{0} = {6, 10, 14, 18}$ . Arrow indicates direction of increasing Q₀ value. (C) Root-mean-square residual deviation from maximum entropy predictions as a function of subsection size, as a function of nearest neighbor number Q₀. As the subsection size increases (including more and more uncorrelated Voronoi areas), the deviation from predictions first decreases until $Q_{0} = 6$ , then increases.

Figure 3—source data 1 Used in this figure is specifically Volvox cartesian cell center position measured from lightsheet microscopy.: https://cdn.elifesciences.org/articles/72707/elife-72707-fig3-data1-v1.zip
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A natural question is whether maximum entropy predictions are more accurate within correlated subregions of an organism. We measured the Voronoi distribution in subregions with similar mean solid angles across six organisms and, for each subregion, a central node and its neighbors up to Q₀ were identified. We varied Q₀ from 3 (corresponding to, on average, $38 \pm 2$ cells in the subregion) to $Q_{0} = 20$ ( $928 \pm 10$ cells on average in the subregion) to measure the Voronoi solid angle distributions in subregions of different sizes. Spatial correlations were weaker within smaller domains (Figure 3A), and deviations from maximum entropy predictions smaller as well (Figure 3B) with the minimum $r_{R M S}$ at $Q_{0} = 6$ (Figure 3C, average of $133 \pm 5$ cells). These observations suggest that while there are systematic correlations between subregions, cell neighborhood sizes are largely randomly distributed within subregions.

The crucial role of randomness

How much randomness is necessary for the k-gamma distribution to predict cell neighborhood size distributions? Our analysis of the solid angle distribution of Volvox cells demonstrates that maximum entropy principle predictions are relatively accurate ( $r_{R M S} = 0.04$ ) even in the presence of some spatial correlations. However, it is unclear how much randomness is necessary for the k-gamma distribution to be predictive. Developmental processes are known to incur random effects (Hong et al., 2016; Haas et al., 2018), but, at the same time, fluctuations must be sufficiently suppressed for complex structures to assemble. This dilemma is augmented because there are few methods for measuring the degree of randomness in developmental processes. Therefore, it is difficult to disentangle exactly which reproducible qualities and traits stem from regulated processes, and which ones stem from random processes. In this section, we address this gap by assessing the stability of maximum entropy packing distributions to correlative perturbations (Figure 4). We do so by simulating three different sources of correlations: (i) size polydispersity, (ii) precisely defined growth patterns, and (iii) coordinated cellular apoptosis, each of which is a common multicellular developmental process. In each scenario, we varied the relative strength of correlations and noise, and monitored how closely the cell neighborhood size distributions agreed with the k-gamma distribution via P-P plots and the $r_{R M S}$ . We do not attempt to exactly recapitulate naturally-occurring levels of randomness strength; rather, we investigate the limits of how well maximum entropy predictions can measure the impact of correlations.

Figure 4

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Introducing correlations and structure can break the maximum entropy distribution.

In A-C are PP plots of the observed vs. predicted cumulative distribution function for three different simulations. The colors correspond to increasing levels of noisiness in the simulations, from red (strongest correlations/determinism) to blue (strongest noise). The dashed black line in each represents $y = x$ , or exact predictive efficacy. (A) Aggregative groups with bimodal size polydispersity; noise is introduced by varying the probability that small cells reproduce into small or large cells, and vice versa. (B) Tree-like groups with persistent intercellular bonds that grow according to a growth plan modified by noise in cell placement. (C) Surface-bound groups with programmed cell death events that may be localized or randomly dispersed. (D) The root mean square deviation from predicted values for each simulation case. Circles are aggregative simulations from A, triangles are tree-like simulations from B, and squares are surface-bound simulations from C. Note: the discrete points in (B) arise because absent randomness in the cell locations the exact same cellular structure is achieved by every simulation. Therefore, the observed distribution of Voronoi volumes is noticeably discrete for this case. Upon adding randomness, the cellular structure is altered between successive simulations, and the distribution of Voronoi volumes becomes less discrete.

Figure 4—source data 1 This figure was generated from the data provided through simulations of aggregative growth with size polydispersity, tree-like growth with noise in cell position, and growth on a spherical surface with planned apoptosis.: https://cdn.elifesciences.org/articles/72707/elife-72707-fig4-data1-v1.zip
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The impact of heritable size polydispersity was investigated by simulating aggregative groups consisting of large and small cells. All simulations were seeded with one small cell and one large cell. We then varied the probability $ξ$ that a new cell is the same size as its mother from 0.5 to 1.0. When $ξ = 1$ , cells always produce offspring with the same radius; for $ξ = 0.5$ it is equally likely that a small cell produces a small or large daughter (and vice versa for large cells). Therefore, groups with $ξ = 1$ have correlated regions of cell size, but the degree of correlations decreases with decreasing $ξ$ . While groups with $ξ = 1$ deviate significantly from the k-gamma distribution ( $r_{R M S} = 0.09$ ), we observed that even a small amount of randomness results in excellent agreement between simulated groups and the k-gamma distribution (in order from $ξ = 1$ to $ξ = 0.5$ , $r_{R M S} = {0.09, 0.04, 0.03, 0.03, 0.02}$ ).

Next we investigated groups with varying amounts of noise on top of defined growth patterns. In these simulations, new cells bud in precise positions; the first daughter at the position ( $θ = 0 \pm η$ , $ϕ = 0 \pm η$ in spherical coordinates), the second at $θ = 90 \pm η$ , $ϕ = 0 \pm η$ , the third at $θ = 90 \pm η$ , $ϕ = 180 \pm η$ , etc., where the noise is uniformly distributed with zero mean and width $η$ . For $η = 0$ (no noise), the distribution of Voronoi volumes was discontinuous, since cells could only access a finite number of local configurations. As expected, as $η$ increases ( $η = {0, 5, 30, 60, 90}$ ), $r_{R M S}$ decreases ( $r_{R M S} = {0.07, 0.05, 0.02, 0.02, 0.02}$ ).

Finally, we investigated groups with localized and random cell death. In these simulations, 50 cells were confined to the surface of a sphere of unit radius following the protocol described above. One cell is randomly selected to die. Centered at this cell, a spherical region of radius $R$ is defined, and then 10 cells in this region were randomly selected to die (and disappear, thereby not contributing to the Voronoi tessellation). For small $R$ , cell death is highly localized, and thus spatially correlated. As $R$ increases, cell death events become less localized, and therefore more random. We find that highly correlated cell death resulted in large deviations from maximum entropy predictions. Conversely, as $R$ increases dead cells become less localized, the observed distribution becomes more accurately described by the k-gamma distribution; as $R$ increases from $R = {0.75, 1, 1.25, 1.5, 1.75, 2}$ , we find $r_{R M S} = {0.10, 0.07, 0.03, 0.02, 0.01}$ .

In summary, absent randomness, spatial correlations lead to large deviations from the k-gamma distribution. Yet, with even a small amount of randomness, the k-gamma distribution holds significant predictive power. These simulations suggest that maximum entropy predictions are likely to be robust against even moderate correlations.

Parent-offspring fidelity via maximum entropy packing

So far, we have shown that randomness in cellular packing leads to highly predictable packing statistics. Here we show that maximum entropy statistics can directly impact the emergence of a highly heritable multicellular trait, organism size.

Prior work has shown that the size of snowflake yeast at fragmentation is remarkably heritable – higher, in fact, than the traits of most clonally reproducing animals (Ratcliff et al., 2015). The size to which snowflake yeast grow depends strongly on the aspect ratio of its constituent cells; more elongated cells allow the growth of larger clusters before strain from cellular packing causes group fragmentation (Jacobeen et al., 2018a; Jacobeen et al., 2018b). Recently, experiments with engineered yeast showed that this emergent multicellular trait, group size, was in fact more heritable than the underlying cellular trait upon which it was based (cellular aspect ratio), despite the fact that the mutations engineered in this system only affected cellular aspect ratio directly (Zamani-Dahaj et al., 2021). Simulations of multicellular chemotaxis observed a similar effect (Colizzi et al., 2020). While at first glance this may seem surprising, we show below that the high heritability of snowflake yeast group size arises from the direct dependence of size on the robust maximum entropy distribution of volume within groups.

Before addressing how fracture impacts the distribution of cluster sizes by impacting the number of cells within a group, we first must address fluctuations in size among clusters with the same number of cells. Given a number of cells $N$ in the cluster, variation in cell packing fraction results in variation of the total volume. The arguments given above for predicting the distribution of individual cell volumes also applies to the distribution of total volume (Aste and Di Matteo, 2008); the distribution of total volume for clusters with the same number of cells should follow the k-gamma distribution. To generate enough clusters with identical $N$ to test this prediction, we used simulations. We generated 3000 snowflake yeast clusters, each with 100 cells, and measured their total volumes. The distribution of volumes is consistent ( $r_{R M S} = 0.0043$ , $k = 23.0$ ) with the k-gamma distribution as shown in Figure 5A. Further, these fluctuations in size are small compared to the differences in size gained via reproduction of cells or lost via fracture.

Figure 5

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Maximum entropy cell packing generates a consistent and predictable life cycle in snowflake yeast.

(A) Distribution of total cluster volume for 3000 simulated snowflake clusters, each with $N = 100$ cells. The total volume is divided by the minimum possible volume $N v_{c}$ . The k-gamma distribution (black line, $k = 23.0, r_{R M S} = 0.0043$ ) provides a good description of the data. (B) Distribution of all experimental Voronoi volumes (black circles) and the maximum entropy prediction (black line). The vertical black dashed line is the critical Voronoi volume $v^{*} = 2.02$ predicted from simulations. Orange filled region integrates up to $p^{*}$ the probability that any one cell occupies a volume less than $v^{*}$ . Insets: sequential brightfield microscope images of one yeast cluster undergoing group fragmentation. White arrowhead indicates location of fracture point. The images measure $150 µ m$ across from top to bottom. (C), Experimentally measured yeast cluster size distribution (solid black line) along with the prediction from weakest link theory (orange line). Gray region represents $1 σ$ confidence bounds on the measured distribution from estimating the number of cells in a group. (D), Coefficient of variation in group radius ( $σ / \bar{R}$ ) for snowflake groups (S) and flocculating groups (F). Data from Pentz et al., 2020.

Figure 5—source data 1 This figure was generated from a combination of the simulation data and experimental data provided with Figures 1 and 2, as well as epxerimental size measurements included here from a particle multisizer, or Coulter Counter.: https://cdn.elifesciences.org/articles/72707/elife-72707-fig5-data1-v1.zip
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To predict the group size distribution, we consider the probability of fragmentation via a weakest-link model of fracture. As the location of new cells is random (see Figure 1—figure supplement 1), each new cell has a chance of causing intercellular bond fracture. It was previously observed that bonds only break if cells are highly confined, that is they have smaller Voronoi volumes; otherwise flexible cellular branches simply bend (Jacobeen et al., 2018b). We model fracture as occurring when a cell’s Voronoi volume is below a critical value denoted by $v^{*}$ (Figure 5B) such that its motion is completely restricted. We measure $v^{*}$ from simulations that determine the maximum local packing density for groups with same cell size and shape distributions as seen in experiments (see Materials and methods). The probability that a particular cell is confined to a Voronoi volume $v \leq v^{*}$ is the integral

p^{*} = \int_{v_{c}}^{v^{*}} p (v) d v .

As each cell in a cluster of $N$ cells independently has probability $p^{*}$ of having $v \leq v^{*}$ (and thus causing fracture), the probability of a cluster with $N$ cells not fragmenting is

P (N) = {(1 - p^{*})}^{N}

As we do not model the fate of products of fragmentation (i.e. the size of the separate pieces post-fracture), we expect the weakest link model to be more accurate for larger clusters than it is for smaller clusters.

We measured group size for approximately $10, 000$ snowflake clusters, all descendants of a single isolate, using a particle multisizer, and found strong agreement between the experimentally observed cluster size distribution and the weakest-link prediction (the coefficient of determination is $r^{2} = 0.97$ for $\log (C o u n t s)$ vs $N$ ) (Figure 5C). Hence, the predictable statistics of entropic cell packing guides the distribution of group size among offspring of a single isolate.

For context, we compared the distribution of group size in snowflake yeast to that of flocculating yeast, which forms multicellular groups via aggregation. The multicellular size of flocculating yeast depends on the rate of collisions with other cells and groups of cells. The growth rate of aggregates is thus typically proportional to their size, as larger aggregates are more likely to contact more cells (Pentz et al., 2020). In fact, the maximum size of a flocculating yeast aggregate is bounded by the duration of aggregation, an extrinsic parameter, while the minimum size can be a single cell (Stratford, 1992). Using data from Pentz et al., 2020, we compared the group size distributions of snowflake yeast and flocculating yeast grown in the same environmental conditions. We find that flocculating yeast groups exhibit a much larger coefficient of variation in size compared to snowflake yeast groups (Pentz et al., 2020; Figure 5D). These results demonstrate that randomly assembled groups can exhibit more reproducible group traits than groups assembled with correlations.

Multicellular motility is robust to cellular area heterogeneity

One of the issues arising from the existence of the broad distribution of somatic cell areas in Volvox is the extent to which colony motility is affected by that heterogeneity. Each of the somatic cells at the surface of a Volvox colony has two flagella that beat at $\sim 30$ Hz, in planes that are primarily oriented in the anterior-posterior (AP) direction but with a slight lateral tilt that makes each colony spin around its AP axis. A longstanding focus in biological fluid mechanics of multicellular flagellates has been to understand the connection between the beating of the carpet of flagella that cover their surface and their self-propulsion. Measurements of the flow fields around micropipette-held (Short et al., 2006) and freely-swimming colonies (Drescher et al., 2010a) have shown that despite the discreteness of the flagella, the flow is remarkably smooth, albeit often displaying metachronal waves (Brumley et al., 2015), long-wavelength phase modulations of the beating pattern. However, the effect of heterogeneity in cell area on motility is as yet unknown; naively, we might expect that heterogeneities in flagella packing may lower the motility of the organism in comparison to a highly regular arrangement. Here, we show in fact that Volvox motility is largely unaffected by fluctuations in cell area, indicating that area heterogeneity may not be detrimental to the organism’s swimming fitness. The absence of such a barrier may represent a scenario where trait optimization can be achieved without first evolving sophisticated developmental genes.

A heuristic explanation for the smoothness of the flows can be developed by noting first that the flow arising from each flagellum, beating close to the no-slip surface of the colony, will fall off only as an inverse power of distance $r$ from the flagellum. Thus, the superposition of the flows from many flagella will be sensitive to contributions from distant neighbors (Boselli et al., 2021) and will tend to wash out local variations in flagellar actuation. This argument can be made quantitative using two different models for the motility of such flagellates. The first is the ‘squirmer’ model (Lighthill, 1952), in which the flagellate is characterized by a tangential ‘slip’ velocity $u (θ)$ on the surface, which can be thought of as corresponding to the mean motion of the flagella tips. Here, $θ \in [0, π]$ is the polar angle with respect to the AP axis. In this approach the details of the fluid velocity profile below the tips are not resolved, and in particular the no-slip condition at the surface of the ECM is ignored. In the second approach (Ishikawa et al., 2020), which builds on earlier work (Short et al., 2006) that specified a force density at the colony surface instead of a slip velocity, there is a specified force density applied at some small distance above the no-slip colony surface, and the flow field below that locus is resolved. This approach, termed the ‘shear stress, no-slip’ model, captures the very large viscous dissipation that occurs in the region between the ECM and the locus of forcing. Within either of these two approaches above the effects of area inhomogeneities can be investigated by coarse-graining the flagella dynamics; either the local slip velocity $u (θ)$ or the local tangential force density $f (θ)$ has noise.

In the squirmer model, the swimming speed $U$ is (Lauga, 2020)

U = \frac{1}{2} \int_{0}^{π} d θ \sin θ u_{θ} (θ) V_{1} (θ),

where

V_{n} (θ) = \frac{2}{n (n + 1)} P_{n}^{^{'}} (\cos θ) \sin θ,

$P_{n}$ is the Legendre polynomial, and the prime indicates differentiation with respect to its argument. If we represent the effects of area inhomogeneities as noise in the slip velocity, then it is most natural to use $V_{n}$ as the basis functions for the tangential slip velocity, expressed as

u_{θ} (θ) = \sum_{n = 1}^{\infty} u_{n} V_{n} (θ),

where $V_{n} (0) = V_{n} (π) = 0$ , guaranteeing that the slip velocity vanishes at the anterior and posterior poles (Short et al., 2006). Accurate experimental measurements of the azimuthal velocity field of Volvox (Drescher et al., 2010b) show that it is well-captured by that lowest mode, leading to a modest anterior-posterior asymmetry. From the orthogonality relation for the $V_{n}$ ,

\int d θ \sin θ V_{1} (θ) V_{n} (θ) = \frac{2 n (n + 1)}{2 n + 1} δ_{1 n},

we see immediately that the contributions from all modes $n > 1$ vanish identically, and thus the swimming speed is given identically by the amplitude of the lowest mode $V_{1} (θ) = \sin θ$ ,

U = \frac{2}{3} u_{1} .

Thus, within the squirmer model, motility is essentially insensitive to area inhomogeneities. This result does not preclude effects of those higher modes, only that such effects will be on quantities other than the swimming speed, such as the nutrient uptake rate (Magar et al., 2003).

In the shear-stress, no-slip model, the velocity field in the region between the colony radius $R$ and the radius $R (1 + ϵ)$ at which the shear stress is applied is solved separately from that for $r > R (1 + ϵ)$ and the two flow fields are matched at $R (1 + ϵ)$ through boundary conditions of continuity in velocity and normal stress and the specified discontinuity in shear stress. Analogously to the expansion of the slip velocity in the squirmer model (7), noise in that discontinuity can be expressed by assuming that the coarse-grained shear force applied by the flagella has spatial variations, and can be expanded in the form

f_{θ} (θ) = \sum_{n = 1}^{\infty} f_{n} V_{n} (θ) .

The swimming speed again depends only on the lowest-order mode in this expansion,

U = \frac{2 ϵ R}{3 μ} f_{1},

and we again have insensitivity of $U$ to inhomogeneities in the area per somatic cell.

Discussion

In this paper, we demonstrated that universal cellular packing geometries are an inevitable consequence of noisy multicellular assembly. We measured the distribution of Voronoi polytope sizes in both nascent and extant multicellular organisms, and showed that they are consistent with the k-gamma distribution, which arises via maximum entropy considerations. Using simulations, we demonstrated that k-gamma distributions arise in many different growth morphologies, and do so requiring only a relatively small amount of structural randomness. Further, we showed that the distribution of cell neighborhood sizes can be used to distinguish the effects of randomness from the effects of developmental patterning. Finally, we demonstrated that consistent packing statistics can lead to highly reproducible, and thus heritable, multicellular traits, such as group size in snowflake yeast. Altogether, these results indicate that entropic cell packing is a general organizing feature of multicellularity, applying to multicellular organisms with varying growth morphologies, connection topologies, and dimensionalities.

One of the strengths of the packing-based maximum entropy framework employed here is its simplicity. We have demonstrated that the distribution of cell neighborhood sizes can be predicted with high accuracy, in many different multicellular morphologies, from only the first two moments of the distribution. Deviations from maximum entropy predictions therefore encode important information about additional correlations that can arise via a variety of sources, such as developmental regulation or interactions with the environment. These additional correlations could be explored via, for example, higher order structures to the maximum entropy model (Schneidman et al., 2003). Relatedly, it would be interesting to extend our analysis to include topological information, which has recently been applied to theoretical and experimental systems (Skinner et al., 2021; Jeckel et al., 2021). Developing methods to incorporate these higher-order effects may prove especially useful when assessing the importance of random fluctuations in organisms with structures more complex than those studied here.

The effect of random noise has been an important area of research in developmental biology (Tsimring, 2014; Lander, 2011). During development, cellular growth, reproduction, differentiation, and patterning combine to form a multicellular organism. Random noise introduced at any stage in this process can result in phenotypic variability, which may affect an organism’s fitness (Waddington, 1957). But while some multicellular traits exhibit high variability, others are tightly conserved, leading to a wide body of research addressing the origin of mechanisms underlying robustness and stability, and the nature of feedback mechanisms that must be present to manage the large number of stochastic fluctuations in gene expression and growth (Gregor et al., 2007; Haas et al., 2018; Hong et al., 2016; Sampathkumar, 2020; Deneke and Di Talia, 2018). In this context, our results demonstrate that random noise can itself lead to highly reproducible multicellular traits such as the cell packing distribution.

Our observation that heritable properties can arise from random processes is reminiscent of the reproducible structures and phenomena generated by random noise in a wide range of physical (Shinbrot and Muzzio, 2001; Manoharan, 2015) and biological systems (Tsimring, 2014; Lander, 2011). While it may be surprising that the distribution of free space in snowflake yeast and Volvox follow the same k-gamma distributions despite the many differences between these organisms, this universality actually extends beyond multicellular organisms to non-living materials, such as those seen in granular materials and foams (Katgert and van Hecke, 2010; Varadan and Solomon, 2002; Aste and Di Matteo, 2008). This broad universality likely arises due to the simple requirements for application of the maximum entropy principle to packing; specifically, there must be a total volume, individual volumes cannot overlap, and volumes must be determined independently (subject to the total volume constraint). It is thus important to note that entropic packing is not necessarily adaptive; it can readily emerge as a consequence of random cellular reproduction or interactions. While entropic packing statistics may produce advantages in some cases, they could be neutral or detrimental in others.

An example of one possible advantage granted by entropic packing is the parent-offspring fidelity that arises from its ensemble statistics. Since both parents and their offspring are assembled through similar noisy processes, they achieve similar cell packing distributions. This statistical similarity therefore details at least one heritable multicellular trait that does not rely on genetically regulated multicellular development. Other multicellular traits that build on the cell packing distribution are similarly affected by this emergent process and could become heritable as well. Such parent-offspring heredity could play a crucial role in the evolutionary transition to multicellularity, providing a mechanism for nascent multicellular organisms to participate in the evolutionary process without first having to possess genetically regulated development. Over time, developmental innovation may arise via multicellular adaptation, modifying or replacing entropic cell packing as a mechanism of multicellular heredity. Consistent with this hypothesis, maximum entropy retains considerable predictive power in extant multicellular organisms such as Volvox, animal embryos (Alsous et al., 2018), and epithelial tissue monolayers (Atia et al., 2018), each of which have canalized development. There may be other examples of organisms with highly regulated development which pack cells according to maximum entropy predictions, and future work could address cell packing in, for example, animal embryos, brain tissue, and more. Finally, as fragmentation is a common mode of multicellular reproduction (Larson et al., 2020; Prakash et al., 2021; Angert, 2005; Keim et al., 2004; Koyama et al., 1977), fracture driven by maximum entropy packing statistics may be relevant to organisms other than snowflake yeast.

The broad distributions in cellular volumes we have found in two very different types of organisms, with two very different modes of reproduction and growth, suggest that noise in developmental geometry may be an inevitable consequence of almost any microscopic mechanism. In this sense, they may be just as unavoidable in biological contexts as thermal fluctuations are in systems that obey the rules of equilibrium statistical physics. As an example, we recall the ‘flicker phenomenon’ of erythrocytes, in which the red blood cell membrane exhibits stochastic motions around its equilibrium biconcave discoid shape. Thought for many years to be a consequence of specific biochemical processes associated with living systems, flickering was eventually shown by quantitative video microscopy (Brochard and Lennon, 1975) to be consistent with equilibrium thermal fluctuations of elastic biomembranes immersed in water. This was later confirmed by similar studies of shape fluctuations exhibited by large lipid vesicles (Schneider et al., 1984). The generalization of these considerations to homeostatic tissues with cell division, rearrangements and apoptosis has also been considered (Risler et al., 2015; Kalziqi et al., 2018). While such membrane systems may differ greatly in the specific values of their elastic modulus (and, indeed, of their microscopic membrane constituents), the viscosity of the surrounding fluid, and their physical size, the space-time correlation function of fluctuations about the equilibrium shape adopts a universal form in appropriately rescaled length and frequency variables.

These results on equilibrium fluctuations provide a conceptual precedent for the results reported here. A central issue that then arises from our results is how to connect any given stochastic biochemical growth process defined at the microscopic level to the more macroscopic probability distribution function observed for cellular volumes. Mathematically, this is the same question that arises in the theory of random walks, wherein a Langevin equation defined at the microscopic level leads, through suitable averaging, to a Fokker-Planck equation for the probability distribution function of displacements. Can the same procedure be implemented for growth laws?

K-gamma ( $μ, k$ )	$μ = \bar{V}$	$k = {(\bar{V} - v_{c})}^{2} / s^{2}$
Normal ( $μ, σ$ )	$μ = \bar{V}$	$σ = s$
Log-normal ( $μ, σ$ )	$μ = \exp (\bar{V} + s^{2} / 2)$	$σ = (\exp (s^{2}) - 1) \exp (2 \bar{V} + s^{2})$
Beta prime ( $α, β$ )	$α = \bar{V} (\bar{V} + {\bar{V}}^{2} + s^{2}) / s^{2}$	$β = (\bar{V} + {\bar{V}}^{2} + 2 s^{2}) / s^{2}$

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Cell packing in two multicellular species.

Figure 1—source data 1

Entropic packing is a general feature of simple multicellularity.

Figure 2—source data 1

Correlations lead to deviations from maximum entropy predictions in Volvox carteri.

Figure 3—source data 1

Introducing correlations and structure can break the maximum entropy distribution.

Figure 4—source data 1

Maximum entropy cell packing generates a consistent and predictable life cycle in snowflake yeast.

Figure 5—source data 1

Model parameters for distribution comparisons.

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Thomas C Day

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Stephanie S Höhn

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Seyed A Zamani-Dahaj

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David Yanni

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Anthony Burnetti

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Jennifer Pentz

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Aurelia R Honerkamp-Smith

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Hugo Wioland

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Hannah R Sleath

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William C Ratcliff

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Raymond E Goldstein

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Peter J Yunker

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