Growth produces coordination trade-offs in Trichoplax adhaerens, an animal lacking a central nervous system

The ciliated cells of T. adhaerens, each of which is able to beat in variable directions (13), allows the animal to traverse its environment at around 5 μm per second. We developed a high-resolution automated tracking microscopy system that allows us to observe the ventral surface of T. adhaerens individuals during locomotion, being able to record both the displacement of the individual animal and the relative movement of constitutive tissues within the animal (Fig. 2A). We observed the intercellular dynamics involved in locomotion for 58 animals varying in linear size by nearly an order of magnitude, from c. 200 µm to 2000 µm in diameter, quantifying the internal deformations using optical flow (Fig. 2B). For T. adhaerens, the thin body plan of the animal produces a granular texture on the ventral epithelium when illuminated from above, marking the cells of the tissue. The ventral ciliated cells of T. adhaerens have a diameter of c. 2 micrometers while our recordings are roughly at c. 1.2 micrometers per pixel. While this is sufficient to visualize individual cells, it does not allow for the direct observation of cilia. We therefore used optical flow to track the tissue movement by subdividing this texture into regions and tracking the texture movement within each region. We chose the size of these regions to approximate the size of an individual fiber cell diameter or a 10 × 10 cluster of ciliated cells.

Individual animals exhibited a variety of behavioral states throughout the period of recording, ranging from highly ordered (polarized) locomotion, to highly ordered rotating motion, to disordered movement. We quantify the degree of collective order in the resultant vector fields through order parameters that represent polarization (Eq. 1), rotation/vorticity (Eq. 2), and expansion/contraction (Eq. 3) (equations from ref. (16) and (17)). In these equations, N is the number of regions/cells that we are tracking, ${\widehat{r}}_{\mathit{ic}}$ is the unit vector from the animal’s center of mass to the region, and $\widehat{v_i}$ is the unit vector of the region’s velocity. We thereby produced the instantaneous collective order measures that represented the movement of the animal’s cells for each half-second of locomotion.T. adhaerens dynamically moves through different states of collective movement throughout its locomotion, shifting from polarized, to rotating, to disordered movement. We therefore represented the overall collective order exhibited by an animal through a unified order measure $O(t)=\sqrt{P{(t)}^2+R{(t)}^2+\mathrm{\Delta }{(t)}^2}$, which we refer to as the collective order. This value is bound from 0, which represents completely disordered motion, to 1, which represents a pure affine transformation. By considering the mean collective order exhibited by an animal over time, we find that larger animals are significantly less ordered than smaller ones (Fig. 2F; linear fit: t = −5.73, p <  5 × 10⁻⁷; adj.R² = 0.30). This suggests that larger animals have greater difficulty establishing a consensus of locomotion across their body plans, resulting in the animal’s characteristic shape changes (18).

This observed decrease in collective order with animal size poses an important question: Is this trade-off between size and collective order inevitable in such decentralized animals, or can it be avoided under the right conditions? For instance, larger T. adhaerens individuals may become disordered because the coupling interactions between cells (mechanical or otherwise) may be insufficiently strong to propagate movement information beyond a certain spatial range, causing cells on one side of the animal to be unaware of changes in movement on the other. Strengthening these interactions could make it easier to maintain collective order, but would such a change make the animal unresponsive to perturbations and new information from the environment?

In order to explore these possibilities, we developed a simple yet generic model of T. adhaerens as a distributed, collective system, where we can systematically tune model parameters to identify the regime that best recapitulates our experimental observations. Our simulated representation consists of a mesh of self-propelled particles (SPPs) with elastic interactions (Methods and Fig. 3A). Similar elastic interaction models have been used previously to simulate T. adhaerens locomotion (13), though these prior models relied on an alignment interaction rule to generate coordinated locomotion. However, it is unclear whether and how individual cells can sense orientations of neighboring cells, which is implicitly assumed by an explicit alignment interaction. We also note that the cilia-driven propulsion of Placozoa takes place on a substrate, which is from a physics point very different from collective ciliary swimming like in Volvox carteri (8) since the substrate greatly reduces the importance of long-range hydrodynamical couplings (19). We therefore developed a model—inspired by another model for collective cell migration (20)—that is able to generate order even in the absence of such a rule. In our model, each particle uses only information provided by elastic forces— whose strength is governed by a spring constant k—and coordination emerges from coupling of the self-propelled direction of the particles to the elastic force they experience (see also ref. (21) and (22) for related models). Our model is similar to another recently produced model of Placozoa tissue dynamics (23), though while this other model more explicitly accounts for cilia oscillations, their analysis is performed in the limit of weak coupling between the propulsion orientation and cilia oscillations. It therefore provides a detailed discussion for the emergence of long-ranged correlations in the ordered state (see also ref. (22)) but did not fully explore the parameter space that results in the transition from ordered or disordered movement.

It was previously demonstrated in other SPP models that such collective systems can undergo a phase transition from disordered to ordered locomotion by tuning a variety of parameters governing interparticle interactions (16, 24). In our model, we show that by varying only the spring constant k, and keeping all other parameters constant, it is possible to control the degree of order in the simulated animals, and thereby cause the animal to transition from a disordered to ordered locomotion (Fig. 3C), with the continuous phase transition occurring near the critical point k_c ≈ 3.7 (red highlight). We confirm this is the critical point by measuring the susceptibility χ_o of the system as in ref. (25) (Fig. 3D), a parameter which is known to diverge (maximize) at criticality (26–28). Our results also recapitulate a well-known phenomenon of finite-size systems, namely that the control parameter value that maximizes susceptibility changes with animal size.

The phase transition in our system reveals several relationships between the interaction strength k, system size N, and collective order O. First, collective order decreases with system size only in the critical and the disordered regimes. Second, collective order decreases more quickly in response to weakening k for larger system sizes. Thirdly, the narrow range of collective order we observe in the actual T. adhaerens (Fig. 2C) is reproduced only in simulations in the critical regime (red highlight). These combined observations suggest that T. adhaerens exists in this special parameter region, which could place Placozoa in a family of other biological collective systems that have been used as evidence for the criticality hypothesis (29).

Determining exactly which simulated regime best represents T. adhaerens requires investigating several other known statistical properties and phenomena. First, we investigated the rate at which collective order decreases as we change our control parameter k, as it is known that this phase transition will be steeper in larger collective systems (24, 30). Our simulations reveal this pattern as well in the critical regime (Fig. 3C), but while it is trivial to adjust the control parameter k in our simulations, we cannot perform a similar manipulation in T. adhaerens. However, we can assume that corresponding control parameters in T. adhaerens must undergo some natural dynamic variation due to various noise sources intrinsic to living systems. We therefore define a proxy noise value ${\eta }^{\ast }=\frac{⟨\Vert {\mathbf{u}}^2\Vert ⟩}{⟨\Vert {\mathbf{v}}^2\Vert ⟩}$, where v represents the full velocity vector of a given particle or cell, and u represents the velocity fluctuation vector, which is the velocity that remains once the collective movement components have been subtracted (Materials and Methods). Where noise is maximal (η^* = 1), the velocity fluctuation represents the entirety of the cell velocity, and there is no collective movement component. This quantity can be measured both in T. adhaerens and our simulations and is found to be strongly inversely correlated with the control parameter k (SI Appendix). By measuring the instantaneous η^* and O at multiple time points for animals (Fig. 4 A, inset) and simulations (Fig. 4D, inset) of varying sizes, we are able to characterize the steepness of the phase transition, defined as O′(η^*), and show that it is steeper (more negative) with increasing organism size in both T. adhaerens (Fig. 4A) and simulations (Fig. 4D).

Another important measure of coordination in collective movement is the spatial correlation of the velocity fluctuations between individual cells/particles, a method proposed by Cavagna et al. (31) that indicates the range at which cells influence each other’s movement. In this measure (Eq. 4), i and j are two particles or cells separated by a distance r_ij, and their respective velocity fluctuations are u_i and u_j. We normalize the correlations at all pair-wise distances by a constant c_o such that C(0)=1, with the range bound between 1 (perfect correlation) and −1 (perfect anticorrelation). We use the summations and Kronecker delta function δ(r − r_ij) to determine the mean correlation for any given distance r and thereby characterize the correlation function C(r) across the entire animal body (Fig. 4B).

We characterized the observed correlations by the velocity correlation length λ_v, with C(λ_v)=0. The general expectation for interacting particle systems with local coupling is that λ_v has a finite, constant value that is independent of system size and determined by the relative strength of the coupling between particles. However, we find that λ_v scales linearly with animal size (Fig. 4B; β = 0.26, p <  10⁻¹⁶, adj.R² = 0.96) across the entire size distribution of T. adhaerens, a phenomenon that has been noted in other biological collectives and that has been argued to be an indicator of criticality (17, 31–33). By measuring this quantity in simulated systems, we find that linear scaling occurs both in critical and ordered regimes, while simulations in the disordered regime exhibit sublinear scaling (Fig. 4E). This result is consistent with the behavior of simulated elastic sheets of SPPs described in other studies (22).

In addition to the correlation length, we can further describe the correlation profile by its integral χ measured from r = 0 to λ_v (17, 34). We find that this quantity increases sublinearly with animal diameter in T. adhaerens (Fig. 4C), which is visually apparent if one rescales each animal’s correlation profile C(r) by λ_v to produce C(r/λ) (Fig. 4C, inset). This is in contrast to other biological collectives like bacterial swarms or bird flocks where both λ_v and χ increased linearly with size, such that the correlation profiles could be collapsed onto a scale-invariant function (31, 33, 35). By contrast, in T. adhaerens, we measure a roughly five-fold increase in χ over a corresponding seven-fold increase in linear size, a ratio that we observe for our simulations in the critical regime (k ≈ 3.5; Fig. 4F, black highlight).

Through our investigation, we were able to characterize the effect of size on coordination in T. adhaerens and identify the parameter regime in simulations that best replicated the empirical observations. We find that animals become increasingly disordered in their locomotion as they increase in size, a phenomenon that occurs in our simulations in both the critical and low-order regimes. When we investigated the statistical properties of the velocity fluctuations, we found that T. adhaerens exhibits a linear scaling of the correlation length and sublinear scaling of the correlation integral with animal diameter. When compared with our simulations, both of these scaling phenomena are best recapitulated by simulations tuned to either the critical or ordered regimes. The combination of our observations in T. adhaerens therefore occur only when the model is tuned to a critical regime, suggesting that a similar tuning may be responsible for the phenomena in the organism.

The capacity of a collective system to coordinate as it increases in size is essential for all decentralized systems, including the first multicellular animals such as T. adhaerens. By finding that order decreases as T. adhaerens increases in size, we identified a trade-off in early decentralized organisms between growth and coordination. We used simulation modeling to replicate this phenomenon and showed through a variety of statistical observations that only simulations tuned to a critical point can replicate the entirety of our observations in T. adhaerens. This work provides support for the so-called “criticality hypothesis” that biological systems are tuned to phase transitions that optimize collective information processing (29, 36), and it was shown that being at this regime of instability enables collective agility and responsiveness (37). Our simulation also indicates that this trade-off between size and collective order—by occurring even at the critical regime—suggests that this coordination challenge is inescapable as such decentralized organisms grow in size. This raises questions and interesting hypotheses for why more complex coordination structures and body plans evolved in later-diverged multicellular animals.

It is interesting to understand the limitations to coordination in T. adhaerens by contrasting them with other biological collectives, such as starling flocks (31) or fish schools (32). These animal groups often grow to tremendous scales and yet can still achieve highly ordered and coordinated behaviors. By comparing cells in T. adhaerens to animals in groups, we can identify multiple restrictions on the former that place them at a distinct disadvantage in coordinating at scale. One such restriction is that in T. adhaerens, the interactions among the constitutive cells are only local, typically in a Voronoi network of nearest neighbors. Comparatively, animal groups can coordinate over long ranges with other sensory modalities such as over a visual network (38, 39). Animals in groups can also quickly switch neighbors at relatively fast timescales compared to cells in T. adhaerens, that are physically much more constrained. These additional constraints may explain why such animal groups have scale-invariant correlation profiles that collapse on a single generic function irrespective of size, while in T. adhaerens, correlations decay much faster over distance as a proportion of the animal size. These limitations of simple multicellular life may compound each other and explain why large animals lacking a nervous system are typically sessile.

Our study has several limitations that provide promising expansions for future work. One limitation is our simplified representation of T. adhaerens as a cellular sheet, abstracting its true but functionally unknown anatomy. In particular, it is unclear whether the internal fiber cells in T. adhaerens, whose function and network structure is unknown, is involved in coordination. It was previously considered that these cells are neural or muscular precursors (40), though this was dismissed by more recent anatomical studies (41). however, there is still too little known about their network structure or process lengths to discount them having some role in coordination. In the absence of contradictory evidence, we have assumed that the number of such cells increases in proportion to the animal’s size and that their network topology is largely lattice-like. T. adhaerens also possesses specialized cells that bind to neurotransmitter-like molecules (41, 42) and were reported to arrest the beating of nearby cilia during feeding (43). More recent assays have shown that glycine can act as a chemoattractant for T. adhaerens (44), which provides a promising avenue for the impact of size on coordination in tracking gradients and contrasting this with similar studies on animal groups (45).

One important implication of the detrimental impact of size on collective order and coordination is that the maximal size of T. adhaerens could be limited by an inability to coordinate. It has recently been demonstrated that breakdown in coordinated locomotion in T. adhaerens causes microfractures in the animal’s epithelium and that this plays an important role in asexual fission (23). If poor coordination at larger sizes increases the frequency of these microfractures and thereby generates one of the most characteristic behaviors of T. adhaerens, it would both reveal this as an emergent phenomenon (as opposed to one that is developmentally or genetically regulated) and provide an interesting parallel to size mediation in animal groups (17, 46).

Our approach extends the knowledge of collective systems to provide insight into the effect of size on coordination in the earliest multicellular animals. We view this as part of a larger effort to reveal not only how system-wide coordination can be achieved spontaneously (47, 48) but also to identify the limitations of such decentralized decision-making. The detrimental effect of size on coordination in such decentralized animals reveals a limitation that such organization poses for coordination and puts forward a hypothesis for the evolution of hierarchical structures as animals were faced with the evolutionary pressure to grow to bigger (49).

This work was supported in part by the US NSF through the Center for the Physics of Biological Function PHY-1734030 (T.G.); the NIH Grant R01GM097275 (T.G.); the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project number RO 4766/2-1 (P.R.); the DFG under Germany’s Excellence Strategy-EXC 2002/1-390523135 (P.R.); the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy-EXC https://www.sciencedirect.com/science/article/pii/S0169534722002889#gts0020 2117–422037984 (I.D.C.); the NSERC Julie Payette PGS M Fellowship (M.D.); the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 860949 (I.D.C.); and Office of Naval Research N00014-19-1-2556 (I.D.C.). We thank Corina Tarnita, Allyson Sgro, Simon LeBlanc, and Duong Nguyen for provided expertise.