Размерность и границы определяют механизмы конкуренции клеток в тканях

Abstract Cellular renewing active matter—assemblies of proliferating and apoptotic cells—underlies tissue homeostasis, morphogenesis, and clonal competition. Previous work in one-dimensional periodic systems identified a fitness advantage associated with rapid dead-cell clearance, an ‘opportunistic’ competition mechanism. Extending this framework, we study two-dimensional cellular aggregates and show that dimensionality modifies the interplay between competition mechanisms for clones with different clearance rates: in 2D, opportunistic and homeostatic-pressure-based competition jointly shape clonal selection, to varying degrees. We then introduce an explicit circular confinement to probe how physical boundaries modulate this interplay. While opportunistic competition persists, distinct time-dependent behaviors emerge through weakened homeostatic-pressure-based competition near boundaries. Structural analysis reveals that confinement promotes tangential alignment and spatially heterogeneous homeostatic pressure, thereby reshaping competitive outcomes at tissue edges. Partial information decomposition allows us to attribute these differing outcomes to temporally and spatially varying relative contributions of the two mechanisms. Our study connects newly discovered competition mechanisms with more realistic biological contexts, highlighting how dimensionality and spatial constraints influence tissue structures and modulate competition in heterogeneous cell populations across time and space, with implications for tumor growth dynamics and tissue development. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction The study of active matter—systems composed of units that consume energy at the microscopic scale—has opened new perspectives on the collective behavior of both living and synthetic systems [1]. While much work has focused on motile active matter [2], a particularly rich and complex class is proliferating active matter [3], in which constituents grow and divide. In biological systems such as tissues, this continual proliferation of cells is essential for functionality: enabling fluidization [4], self-organization into spatial structures [5–7] and ordering [8, 9]. Combining these with apoptosis and subsequent clearance, continuous turnover and renewal of tissue material underlie both normal homeostasis [10–12] and structural changes in morphogenetic events [13]. This dynamic turnover plays a complementary role also in competition between different cell populations [14–17]. Understanding how these characteristics of renewing active matter interact with mechanical forces and spatial organization is central to revealing the physical principles underlying living matter. In proliferating cellular populations, competition is essential for the suppression of mutated or unfit cells [18], especially in the early development of embryos. On the contrary, a competitive advantage of such cells can lead to tumors and cancer. Distinct subpopulations can compete through multiple mechanisms: adversarial competition involves active recognition of opponents via signaling pathways, followed by their elimination by ‘winner’ cells [19–21]. While coexistence is in principle possible, non-adversarial competition enables certain populations to dominate without any killing mechanism or even sensing other cell types, as demonstrated by neutral competition among stem cells, where clonal dominance emerges stochastically without direct cell-cell antagonism [17]. For example, kinetic cellular competition arises from intrinsic differences in proliferation rates, whereby faster-dividing cells progressively outnumber slower ones [19], i.e. the faster the division, the better. However, in homeostatic tissues, a high division rate is counterbalanced by increased cell death, invalidating the argument for simple kinetic advantages [22]. When growth and death depend on mechanical pressure, and the homeostatic point where they balance differs between subpopulations, homeostatic pressure can determine the outcome [14, 15]: the subpopulation with the higher homeostatic pressure becomes dominant. While some competition drivers are long-established, new directions explore the impact of adhesion and force transmission ability on tissue competition [23]. Cell removal mechanisms are necessary for homeostasis, ranging from extrusion [24–26], disintegration [27] and phagocytosis [28] to explosive bacterial lysis [29] with widely varying timescales. Moreover, tissues can exhibit heterogeneous degradation dynamics, with different subpopulations removing dead matter on distinct timescales [30]. A recent theoretical study by Pollack et al [16] established a new competition mechanism in intrinsically homeostatic systems, complementary to that of homeostatic pressure differences. Their work, based on a one-dimensional (1D) agent-based model, demonstrated that clones capable of rapidly eliminating their own dead matter increase their competitive success over those with slower degradation of dead cells. This advantage—termed the ‘opportunistic competition’ mechanism—arises because faster clearance leads to higher population turnover rates, allowing for a quicker response to space being freed. This mechanism was also found to be able to dominate over small differences in homeostatic pressure that favor the opposing clone. It was shown that the opportunistic advantage acts mainly near interfaces between clones, while homeostatic pressure competition is a bulk effect [14]. How the dynamics of such interfaces change with dimensionality, when new neighbor configurations are possible, and how in turn competition is affected, is an open question. In this work, we extend the findings of Pollack et al by investigating the dynamics in an epithelial-like, two-dimensional (2D) system and its effects on cellular competition. Our model incorporates different degradation modes, identical to the original study, which vary the removal process of dead cells, ranging from immediate clearance to delayed degradation. First, we characterize the interplay of opportunistic and homeostatic pressure competition in a periodic domain, and compare it to the 1D dynamics. We validate the competition dynamics found by Pollack et al in 2D for most competition cases, and find notable exceptions, where homeostatic pressure counters opportunistic competition. Pushing the applicability of the model further, we then turn to consider environmental mechanical constraints, which are invariably induced in real systems, such as for developing embryos or growing tumors [31, 32], where the balance between proliferation and removal is influenced by the surroundings [31, 33, 34]. Dynamics within tumors, which in themselves can be clonally heterogeneous [35, 36], can be affected by their healthy confining tissue, with possible implications for competition taking place within the tumor itself. Therefore, we also aim to understand the impact of a finite size and an explicit confinement on our cell populations. We model cellular aggregates enclosed within a circular boundary and study the spatial aspects imposed by the confinement to compare them to the periodic bulk-like system. We find that the competitive success of individual clones is influenced by the confining wall, where the dynamics are locally reminiscent of the 1D system. 2. Model The cell model used in this study captures multiple phenomena seen in real cells—from growth and division up to death and degradation—all in an idealized manner. The progression throughout the cell’s life and death is illustrated in figure 1(a), with representative visualizations of the cells’ states below each stage. The relative duration of the cells’ active and passive phases is not shown to scale, to reserve enough space for the sketches of different states. Each cell is a 2D dumbbell-shaped agent, consisting of two circular nodes of radius connected by an internal spring (backbone). A node diameter defines the unit length. An alive cell grows by increasing the backbone’s rest length with a base growth rate , which causes the nodes to separate. The state in the growth cycle is tracked by the growth progress . Once it reaches one, two new cells replace the mother cell’s nodes. This ensures a continuous description of division processes and involved forces [37]. Daughter cells inherit their mother cell’s backbone orientation. Two different cells interact only on contact by repulsive Hertzian forces [38], which model the elastic response of two curved surfaces and depend on the overlap in the following manner (see figure 1(b)) Within a cell, forces on nodes by the backbone follow a similar mechanical behavior, only substituting by the compression of the backbone. This results in a non-linear internal spring connecting the cell nodes, which is necessary for force continuity upon cell division, when nodes of the same cell become nodes of distinct cells interacting via the force law in equation (1). This model aims to simulate 2D dynamics, where simple nearest neighbor interactions as used in [16] are not applicable due to more complex overlap configurations compared to 1D. Therefore, an additional softness factor interpolates between interacting nodes to prevent force discontinuities, as described in [37] and in supplementary note 1. Here, we use this 2D-capable model both for 1D and 2D simulations. In a finite-size system, proliferating cells will eventually fill all available space. To reach a homeostatic state, a systematic feedback needs to be in place that adjusts cell proliferation or removal rates depending on the current state of the system. In our system, removal—which is introduced in more detail below—happens at a constant rate, while locally high densities or crowdedness can inhibit cell growth. An accessible measure for local crowdedness is the mechanical force on the cell’s backbone . To slow growth smoothly, the speed at which a cell progresses through its growth cycle is modulated by a function [16], such that with The sigmoidal form is chosen so the growth rate saturates for zero and high pressure values. Here, , the typical force in a 1D system where cells overlap by and growth is completely inhibited (). controls the steepness. The growth modulation can be seen in figure 1(c). The theoretical homeostatic pressure point is visualized by the green dot in figure 1(c). Local pressure fluctuations correct this mean-field homeostatic pressure depending on the non-linearity in equation (2), see appendix E of [16] for details. Living (active) cells have no limitation on the number of divisions in our model, but they die with a constant probability per unit time of , turning into dead (passive) cells incapable of growth and division. Those dead cells persist in the system for a time , during which they degrade mechanically by reducing the strength of their repulsive interactions (see figure 1(a), gray degradation stage). The decrease of steric forces is achieved by a scaling factor [16], which decreases monotonically from to during the degradation time according to the degradation progress (normalized time after death). The time course of this decrease is parameterized by the so-called degradation factor , such that This leads to fast softening for large positive (see figure 1(d), purple line) and slow for large negative (red), with gradual degradation for (green). In the limit , cells instantaneously lose all structure right after death or at the very end of the degradation phase, respectively. Technically, while still present in a simulation, a cell with a sufficiently low scaling factor is unable to interact with neighboring cells, therefore being effectively removed. Hence, we define the effective degradation rate as in [16] In this study, a clone or species is completely defined by its degradation factor . All other parameters are kept constant. Clones in competition with one another therefore only differ in their degradation dynamics. Videos for demonstration of the cell model are provided for clones with in supplementary videos 1-3. Simulations use the in-house developed, open-source simulation framework InPartS [39], which evolves the degrees of freedom using overdamped dynamics. The DiskCell model [37] implements core functionality of steric interactions and proliferation with important implementation details for a 2D model. This base model is expanded by the pressure-sensitive growth and the degradation aspects of the model described in [16]. Additional details on the implementation and initialization can be found in supplementary note 1. 3. Results 3.1. Dimensionality effect on competition Two possible competition mechanisms are examined in this work, which were previously established in Pollack et al [16] to be relevant in the scenario of homeostasis under pressure regulation of growth and gradual removal of dead cells: homeostatic pressure and opportunistic competition. While further mechanisms could be at play, we focus here on the interplay between these two. In the homeostatic pressure mechanism [14], the fitness of a clone is determined by the pressure at which it balances division and death. There, when two clones are placed in competition, the one that stabilizes at a higher pressure expands and presses the other one to a negative division-death imbalance. Opportunistic competition, as first described by Pollack et al, arises between clones by differences in the removal dynamics of their own dead cells, which affect the overall fraction of actively growing cells (active density). To see how dead cell removal results in a difference in active cell proportion, we recall that the homeostatic active density , i.e. the density of proliferating cells, depends on the degradation mode in the following way (see [16]) with total density and the effective degradation rate (equation (4)). Thus, clones that eliminate dead matter more rapidly maintain a larger active fraction due to the strictly monotonic increase of , which we confirm later numerically. Since free space is reclaimed mostly by active cells, a higher active cell proportion (for fast-degrading clones) translates into a fitness advantage acting at interfaces (where both clones compete for the same free space). First, we study the two competition mechanisms in a domain with periodic boundary conditions (PBC) in 1D and 2D. The 1D simulations aim to reproduce former results by Pollack et al [16] in our 2D-capable model and serve as a foundation for a meaningful comparison between 1D and 2D. The simulation domain in 1D is of length (unit length is one node diameter), and in 2D of size . To later explore how boundaries influence competition, we additionally introduce a simulation geometry with circular confinement (CC) of radius in 2D. In all scenarios, mixtures of two cell populations are initialized at time in equal proportions, only differing in their degradation mode (degradation factor ), and then evolved for time units ( generations3)—a time in which neither clone would take over the whole population. Representative snapshots illustrate the core expectation of the opportunistic mechanism: the population with faster dead matter removal (cyan, ) expands at the expense of the slower one (magenta, ), both in PBC and CC (figure 2(a)). Time series of global cell count (alive and dead cells, black) and both clones individually, averaged over realizations in PBC ( in CC), are shown in figure 2(b) (PBC as dashed lines and CC as solid lines). Clearly, the population numbers grow or shrink monotonically, validating the visual trends. The standard deviations over realizations (lighter shaded bands around the mean) grow over time. Videos that show the competition dynamics for these clones in PBC and CC are provided in supplementary videos 4 and 5, respectively. Download figure: Standard image High-resolution image In a competition scenario, the clone with a higher active density is labeled H (high), while the other is labeled L (low). To measure the fitness of a clone H or L, we count its number of cells (alive and dead cells) and use the population fraction , where is the total number of cells in the system. We quantify fitness typically from the perspective of clone H, and at the end of the simulation, generations. Fitness is first measured globally in the PBC systems in 1D and 2D, for all combinations of competitors with 4, as seen in figures 3(a) and (b). Each data point in the competition matrix corresponds to a specific combination of one H clone (-axis) and one L clone (-axis). Global fitness is color-coded, where blue (red) means that the H clone increases (decreases) in population fraction from the initial . In line with the findings by Pollack et al [16], most mixtures follow the prediction of the opportunistic mechanism: clones with higher active density tend to win the competition. However, in 2D, a region of combinations appears where the H clone loses, despite its higher active density, revealing that the opportunistic advantage alone cannot account for all outcomes. Neutral scenarios, where the two populations have identical degradation, show no advantage to either side, as expected. Download figure: Standard image High-resolution image To separate the contribution of the two mechanisms to fitness, we next characterize each clone in isolation, measuring active density and homeostatic pressure in homogeneous populations of only a single clone. As shown in figure 3(c), the homeostatic active density follows the prediction from equation (5) (black line) well and, importantly, recovers the strict monotonic increase with growing degradation factors . In the 1D system, the global average homeostatic pressure for clones with small is determined by the balance point between growth and death (see figure 1(c)). For higher , fluctuations in pressure increase and so does the mean pressure, as seen in figure 3(d)5. The same relative behavior is also present in 2D, only shifted towards slightly higher pressures6. To understand which traits act as the primary drivers of the competition, we examine how differences between the two clones’ properties are reflected in the final competition outcome. Specifically, we ask: when one clone has a higher active density in homeostasis or a higher homeostatic pressure, does this reliably translate into a competitive advantage? We answer this by comparing the measured fitness to the difference in active density , which reflects the opportunistic mechanism, and to the corresponding differences in homeostatic pressure, , which reflects the pressure-driven mechanism. We start by providing an intuition for how variations in active density and homeostatic pressure relate to the competition outcome, with a positive (negative) homeostatic quantity offering a first indication that the mechanism works in favor of clone H (L). To understand this visually, one can divide the data into four quadrants, where fitness above or below the initial (dashed horizontal lines in figures 3(e) and (f)) indicates which clone dominated, and each predictor’s sign determines the corresponding mechanism’s expected direction of advantage. In 1D, the picture is unambiguous: active density correlates positively with fitness. Red data points in figure 3(e) cluster predominantly in quadrant I. The range in which fitness fluctuates for neutral competition scenarios () is given as a gray band, indicating only minor noise. In contrast, homeostatic pressure does not show any strong systematic correlation with fitness. Some data points in figure 3(f) lie in quadrant II, meaning that clone H wins the competition despite having a lower homeostatic pressure. Thus, in 1D, the competition landscape in the considered parameter region is simple: the opportunistic mechanism is largely sufficient to explain the outcome of the competition. Important to note here is that and cannot be varied independently, as both depend only on the combination of and . More details are provided in supplementary note 3 B and supplementary figure 4. Having confirmed the results of Pollack et al, we now turn to the 2D PBC scenario. In contrast to the simpler behavior in 1D, increasing dimensionality enriches the competition landscape: the opportunistic mechanism is not always dominant anymore (which we already saw in figure 3(b)). Certain data points that were in quadrant I in 1D in figure 3(e) now fill quadrant IV in figure 3(g) (far beyond the noise level of neutral competition), indicating that the H population diminishes. These scenarios correspond to data points that moved from quadrant II in figure 3(f) to quadrant III in figure 3(h), where the homeostatic pressure mechanism favors the L clone. Conversely, there are still situations where the homeostatic pressure mechanism alone predicts an L advantage, yet clone H still wins. These cases are similar to the 1D dynamics, where the predictors have opposite signs as well, but the opportunistic driver is strong enough to determine the outcome (filling quadrant II in figure 3(h)). Overall, 2D competition therefore reveals a richer interplay between the mechanisms: neither active density nor pressure differences alone fully predict the dynamics, and the relative strength of the two mechanisms becomes decisive whenever their predictions diverge. This stands in marked contrast to the 1D system, where homeostatic pressure never overrides the opportunistic trend in the examined parameter range. Below in section 3.3, we carry out a more quantitative comparison of the relative contributions of these two competition drivers. 3.2. Influence of mechanical confinement Having seen that domain dimensionality can influence the competition outcome, the question arises whether other spatial or geometric factors of biological relevance, such as mechanical boundaries, could be decisive as well. Here, we break the homogeneity of space and introduce an explicit CC as shown in figure 2(a). We then repeat the competition and homeostasis simulations and subsequent analysis on a global scale where we confirm that observables are mostly identical to the unconfined systems, indicating that any boundary effects are small compared to the periodic system. However, since local structural changes—such as cells aligning to the wall with their long axis–are expected in confined populations, we now investigate exactly these subtleties and whether competition varies spatially. To this end, we observe the population dynamics of subpopulations in radial bins (annuli) from the center towards the confining wall (see supplementary note 2 for technical details). Indeed, we observe altered population dynamics between populations in the bulk and near the border, including some cases where the competition trend is reversed by the presence of the boundary. To illustrate this clearly, we start with such a case, specifically the competition of a clone with (gradual dead matter removal) against an clone (slow removal). In 2D PBC, the population with is a loser, despite removing dead matter faster than its opponent, as explained above. In figure 4(a), time series of population fractions are shown measured in the domain center (bulk, orange) and near the wall (cyan). Dynamics in 2D PBC (black) are shown for reference. While in bulk, the dynamics follow roughly the dynamics in PBC, the competition shows qualitatively different dynamics near the confining wall, namely the H clone having a significantly higher population fraction, even above 7, i.e. locally being a winner (at least transiently). Here we also find a timescale dependence and non-monotonic trends in the population fraction: an initial growth period in favor of the H clone (up to generations at the wall and generations in the bulk), after which it shrinks again. We expect that the downward trend in fitness of clone H persists for longer times, bringing fitness below the mark, although we have not performed simulations beyond generations. Download figure: Standard image High-resolution image However, the effect of the boundary on competition dynamics is not limited to cases where it causes non-monotonicity, as can be seen in a competition between vs (figure 4(b)), where the H clone wins as expected from the opportunistic mechanism. In this case, border and bulk dynamics are also different, although less drastically, showing a reduced population growth rate of H at the confining wall, while displaying strictly monotonic growth in all cases. This means that competition strength is reduced near the edge. The pairwise results naturally raise the question of how each clone fits into the wider hierarchy of competitors. To address this, we compare clone with every other clone across the degradation range, as shown in figure 4(c). There, we measure the fitness (population fraction at gen.) of that specific clone spatially as a function of radial distance to the colony center. The ‘sanity check’ scenario, where two populations with identical compete, is expected to show no fitness advantage at all, as validated in the figure (alternating ink blue and black line). Since globally, clone is typically a loser (with a minor advantage against ), fitness is mostly below , but it grows higher near the wall in most cases. This is in alignment with the increased population fraction at the wall seen in figure 4(a), which here corresponds to the red curve. In this specific case, the increase at the wall is even sufficient to make the clone the local winner (at the end of the simulation period), as seen from fitness values above at large . Analogously, fitness of clone in competition against all others is shown in figure 4(d) (with the sanity scenario here being the alternating purple and black line). In this case we observe the opposite behavior: a clone, winning against all other clones globally, tends to have reduced fitness at the wall against some clones. The decreased population fraction at the wall seen in figure 4(b) here corresponds to the blue curve. To understand how the confinement affects population properties and the outcome of competition, we perform a spatial analysis of the confined homogeneous system. We start by examining the effect of confinement on cell orientation. Qualitatively, this could already be seen in the snapshots on the right side of figure 2(a), but to demonstrate the effect more clearly, we show in figure 5(a) a homeostasis simulation snapshot (for a clone with ). Cells are colored by the nematic orientation of their backbone direction relative to radial direction at their position, indicating radial (red) or tangential (blue) alignment. Cells touching the wall are highly tangentially ordered, as expected from their anisotropic shapes. The inset shows a zoom-in to a representative region close to the border with dimensions . It highlights visually up to ordered layers of cells. A video of the dynamics is provided in supplementary video 6. To characterize the cell alignment quantitatively, we examine collective nematic order in the polar coordinate system as a function of radial distance from the center, as shown in figure 5(b). Each cell’s orientation contributes towards this order parameter with a weight , which is the area fraction of that cell in the considered region (see supplementary note 2). Consequently, the order parameter is the averaged weighted orientation in a set of cells, where or means perfect radial and tangential alignment, respectively, and zero means complete disorder. The visually clear tangential order seen in the snapshot is confirmed near the boundary and also propagates into the bulk for multiple cell diameters, as discussed in supplementary note 3. The orientation is shown only for the region near the boundary (), since the bulk region is fully disordered with , as expected. Download figure: Standard image High-resolution image Having characterized how the mechanical confinement shapes population structure, we now turn to examine its effect on the active density and pressure, being the principal drivers of competition. In figure 5(c), the spatial profile of the active density is essentially flat, with no considerable dependence on position. For different clones, this homogeneous level varies monotonically with the degradation factors , matching the trend in figure 3(c). In contrast, the homeostatic pressure shows a complex spatial profile (figure 5(d)), which can strongly influence competition outcomes. Notably, the difference in between clones diminishes towards the boundary, as shown in supplementary note 3. 3.3. Interplay of competition mechanisms The strong variation of homeostatic pressure shown above suggests that the interplay between altered pressure and active density near the wall jointly shapes the local fitness landscape. To move from a descriptive view towards a more quantitative statement about the relative importance of both competition drivers, we make use of an information-theoretical concept: the partial information decomposition (PID) [40, 41]. PID provides intuition by separating how multiple variables contribute to a target quantity. Specifically, it identifies the unique information that each source carries about an outcome8. In our context, PID therefore offers a way to evaluate how much of the fitness outcome is uniquely attributed to either active density or homeostatic pressure, and therefore the influence of the two mechanisms. We compute PID across all clone-pair combinations simultaneously, treating as a time-dependent dataset, which additionally allows us to resolve how the relative contributions of the two mechanisms evolve during the simulations. To establish a baseline for interpreting the confined case, we first analyze PID in the periodic system. Here, we focus on unique contributions. The full analysis of PID, including synergistic and redundant information, is reported in supplementary figure 6. First, we summarize the results for the periodic system. As we see for 1D in figure 6(a), the unique contribution from active density differences grows (green), while the information from homeostatic pressure differences is near zero (orange). This is consistent with the interpretation that in 1D, opportunistic competition alone decides about the winning clone. In 2D (figure 6(b)), the information is similar to 1D on short timescales, while eventually, opportunistic contributions decrease and pressure becomes important again, indicating their joint importance at the simulation end, as discussed earlier. This timescale separation is also found in individual population fraction time series, similar to figure 2(b), where non-monotonicities can be observed, favoring the H clone on short and the L clone on long timescales (see supplementary figure 7). Download figure: Standard image High-resolution image Applying the PID analysis to the confined cell populations now puts us in a position to develop a hypothesis on the relation between the spatial order evident in the confined homeostasis populations and the observed change in fitness near the boundary. Since the homeostatic pressure shows a complex behavior along the radial coordinate due to layering effects of the cells, while fitness gradients are mostly monotonic (figures 4(c) and (d)), we therefore analyze PID separately in the bulk and at the wall. The PID signals in the colony center closely match the 2D PBC baseline: both active density and homeostatic pressure contributions show similar magnitudes and temporal trends (compare figures 6(b) and (d)). This indicates that competition mechanisms in the bulk of confined colonies operate essentially as in the periodic system. This agreement is expected, since the population in the center is not influenced by the outer wall and exhibits the same disordered structure and pressure levels as in the setting without mechanical boundaries, as shown in figure 5 (and supplementary figure 3 for further insights). However, near the population’s edge, the unique information is reminiscent of the 1D case, with homeostatic pressure giving almost no unique contribution to the competition outcome while active density and therefore opportunistic dynamics locally decide about winner and loser (figure 6(c)). We hypothesize that the boundary region behaves as a quasi-1D system: strong alignment and layering make opportunistic, interface-driven competition dominant at early times, similar to true 1D dynamics. However, unlike a strictly 1D interface, the boundary layer exchanges cells with the bulk (see supplementary figure 8), and this slow radial mixing eventually restores the influence of homeostatic pressure. The mixing between layers happens on the same timescale as the shift in competition dynamics, possibly explaining the cases in which the initial opportunistic dominance gives way to pressure-driven fitness differences at later times, as also observed in scenarios such as figure 4(a). 4. Discussion Cell competition in growing tissues can arise through several mechanisms, including adversarial interactions and homeostatic competition driven by mechanical feedback [14]. A key factor identified in earlier work is the rate at which dead cells are cleared: faster clearance enables a larger population of active growing cells and higher likelihood to fill newly freed space leading to a competitive advantage, even when division and death rates are identical. This ‘opportunistic’ mechanism was demonstrated in a 1D model by Pollack et al [16]. Here we extend this picture to 2D aggregates and show that dimensionality and confinement substantially reshape the competitive landscape. In 1D systems without mechanical boundaries, our results recover the prediction: faster-degrading clones reach higher homeostatic density and dominate through the opportunistic mechanism. In 2D, however, pressure again plays a stronger role on long time scales, sometimes shifting competition outcomes toward the clone favored by homeostatic pressure differences, in this case the L clone. The origin of this 2D-specific behavior remains open and may relate to the more complex spatial arrangements possible in higher dimensions. Introducing a rigid circular boundary adds another layer of geometric structure, since competitive advantages become spatially dependent. Globally losing clones can perform significantly better near the confining wall, sometimes even becoming local winners, while globally dominant clones can lose part of their advantage in these regions. Beyond our findings, we expect cells of higher aspect ratio to show much stronger alignment effects with the wall as seen in other settings [8, 9] and micro-domains in bulk [6, 7]. How such micro-domain dynamics would be affected by cell removal is an open question, but a stronger alignment with the confining wall could result in even stronger spatial fitness heterogeneities, which could be an interesting direction of future research. However, it is already interesting to see that the minimal anisotropy (aspect ratio ) and minimal alignment can already alter the fitness of a clone locally to such an extent. The PID analysis provides a quantitative view of how the two competition mechanisms shape fitness across geometries. In periodic systems, PID revealed a clear dimensional dependence: in 1D, competition is governed almost exclusively by opportunistic dynamics, whereas in 2D, the relative influence of pressure increases over time. Under mechanical confinement, PID showed that these contributions become spatially dependent: the colony center behaves like a 2D bulk environment, while the boundary region exhibits quasi-1D characteristics with a pronounced early-time dominance of opportunistic competition. Yet, slow radial mixing at the wall eventually restores the role of pressure, demonstrating that the balance between the two mechanisms is not static but time-dependent and shaped by emergent cell flows. Together, the PID results highlight that neither mechanism is intrinsically dominant; rather, their relative importance emerges from the interplay between dimensionality, interface availability, and confinement-driven spatial order. This framework thus offers a mechanistic lens for predicting how tissue architecture and microenvironmental constraints modulate competitive outcomes. An interesting possibility is that the observed non-monotonicities and time-dependent effects we observe (in population fraction as well as PID) might be related to a coarsening of the system: Although initially well mixed, small clone patches vanish stochastically while bigger clusters grow, even without competition (see supplementary note 3 and supplementary figure 5). Such coarsening dynamics might interact with the competition mechanisms: since opportunistic competition acts primarily at the interface between populations (appendix D in [16]), could the shrinking of these interfaces during coarsening reduce its overall influence? On the other hand, homeostatic pressure competition is a bulk effect [14], seemingly independent of interfaces, so it might remain largely unaffected. Similar coarsening occurs in 1D as well, although to a lesser extent (supplementary figure 5), and without an obvious shift in the balance between the competition mechanisms. However, a comparison of this coarsening for 1D and 2D is not straightforward, since topological effects of dimensionality, including the number of possible neighbors and the opportunities for rearrangements are not negligible. For example, in 1D a cell has only two neighbors and if one dies, half of its possible movement directions open up, while in 2D, this percentage drops. While the fraction of dead neighbors should be identical on average, small-number effects due to the low number of direct neighbors might differ between different dimensions and have non-trivial effects, which also are worth further investigation. This raises the question whether dimensionality plays a more intricate role than anticipated, or whether other mechanisms compensate for the reduced coarsening in lower dimensions. Our results point to several directions for future investigation. First, extending the present framework to three-dimensional tissues would provide a more complete picture of how dimensionality and spatial constraints shape competition. A 3D geometry would capture architectural features relevant to many in vivo contexts, bridging from epithelia [17] to solid tumors [35]. Second, a soft or deformable confinement, such as a boundary composed of passive cells from a surrounding tissue [18, 22], would likely produce weaker spatial order than rigid walls but may still generate spatially heterogeneous fitness patterns. Studying how soft or dynamically remodeled confinement influences the local contributions of pressure and active density would help determine whether the phenomena identified here persist under more physiological conditions. Third, the dependence on the specific form of the growth modulation in equation (2) could be probed. We know that the non-linear form connects local pressure fluctuations to an increased homeostatic pressure (appendix E of [16]), which is the reason why pressure changes with and explains the elevated pressure values for high as seen in figure 3(c). Investigating how pressure-based competition mechanisms are affected by varied growth inhibition might be an interesting next step towards a better understanding of the interplay with removal-based competition. Similarly, opportunistic competition is directly affected by the function scaling steric repulsion during degradation, equation (3). Its specific form determines effective degradation rates and therefore the active density. Finally, cell motility offers a natural route to reintroduce mixing [42] within otherwise structured or coarsened colonies. Active rearrangements can restore or enhance interface-rich configurations, suggesting that motility may shift competition toward the opportunistic mechanism even in settings where static architecture suppresses it. Together, these extensions would broaden the applicability of the information-based approach introduced here and help clarify how real tissues balance mechanical constraints, spatial organization, and competitive interactions. Acknowledgments We thank Jonas Isensee and Lukas Hupe for their technical support, development, and maintenance on software and the simulation framework InPartS, and Christina Goss and David Beine for early contributions towards the model implementation. We thank Stefan Klumpp, Timo Betz, Ulrich Parlitz, Torben Sunkel and Tobias Büchner for valuable discussions, feedback, and suggestions. We thank Ramin Golestanian and the Department of Living Matter Physics at MPI-DS for their support. We acknowledge support from the Max Planck Society as well as the Max Planck School Matter to Life, which is jointly funded by the Federal Ministry of Education and Research (BMBF) of Germany and the Max Planck Society. We acknowledge support also from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 449750155 – RTG 2756. Data availability statement The InPartS framework [39] is publicly available at www.inparts.org/. The model presented here is published within the InPartSBiome package. Example simulation and analysis scripts are contained in the supporting data linked below. Generated and analyzed data can be reproduced with the code or can be provided on reasonable request. The data that support the findings of this study are openly available at the following URL/DOI: https://hdl.handle.net/21.11101/0000-0008-00F2-6 [43]. Supplementary Material available at https://doi.org/10.1088/1367-2630/ae7ea6/data1. Supplementary Video 1 available at https://doi.org/10.1088/1367-2630/ae7ea6/data2. Supplementary Video 2 available at https://doi.org/10.1088/1367-2630/ae7ea6/data3. Supplementary Video 3 available at https://doi.org/10.1088/1367-2630/ae7ea6/data4. Supplementary Video 4 available at https://doi.org/10.1088/1367-2630/ae7ea6/data5. Supplementary Video 5 available at https://doi.org/10.1088/1367-2630/ae7ea6/data6. Supplementary Video 6 available at https://doi.org/10.1088/1367-2630/ae7ea6/data7. Supplementary Video 7 available at https://doi.org/10.1088/1367-2630/ae7ea6/data8. Conflict of interest The authors declare no competing interests. Footnotes - 3A generation is the average doubling time, which is set by balanced growth matching the death rate, . - 4These values sample from slow to fast degradation, within the limits explored by Pollack et al—expanded range including extreme see supplementary note 3 and supplementary figure 2. - 5The link between pressure fluctuations and homeostatic pressure was previously established in appendix E of [16]. - 6Here, the average cell’s backbone force serves as a proxy for homeostatic pressure. However, we confirmed that it shows identical behavior to the virial pressure calculated from stress tensor contributions (see supplementary note 3). - 7The fitness value of is the barrier between winning and losing of clone H, as highlighted by dashed horizontal lines. - 8Additionally, it provides information shared redundantly, and the information that emerges only from their joint action.