Abstract
Mechanical stresses stemming from environmental factors are a key determinant of cellular behavior and physiology. Yet, the role of selfinduced biomechanical stresses in growing bacterial colonies has remained largely unexplored. Here, we demonstrate how collective mechanical forcing plays an important role in the dynamics of the cell size of growing bacteria. We observe that the measured elongation rate of wellnourished Escherichia coli cells decreases over time, depending on the free area around each individual, and associate this behavior with the response of the growing cells to mechanical stresses. Via a cellresolved model accounting for the feedback of collective forces on individual cell growth, we quantify the effect of this mechanoresponse on the structure and composition of growing bacterial colonies, including the local environment of each cell. Finally, we predict that a mechanocrossresponse between competing bacterial strains with distinct growth rates affects their size distributions.
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Introduction
Mechanical environment is a key determinant of behavior, physiology and functions in diverse cellular systems including microorganisms and their interactions with their host cells^{1, 2}. In bacterial systems, environments impose external mechanical constraints via viscous, elastic or surface forces, with farreaching ramifications on their survival, fitness and resistance to biochemical agents, including antibiotics^{3,4}. A growing bacterial colony^{5} presents a complex biophysical setting where an interplay of extensile (due to growth) and adhesive (cell–cell and cell–substrate) interactions engender active biomechanical constraints, which evolve as the colony expands^{6,7}. Recently, it has been shown that such mechanical constraints give rise to active selfinduced stresses within bacterial and yeast colonies, the magnitude of which depends on the local position within the colony^{8,9}. Beyond a certain threshold, the response to these stresses can drive shifts in the phenotypic traits of the individual cells and trigger critical structural changes in a colony, thereby initiating biofilm formation^{8,10,11}.
At the level of a single cell, external mechanical influences on bacterial growth, including fluid flows and confinements, have been extensively investigated^{3,12,13}. Moreover, different models for the growth kinetics of individual bacteria^{14,15,16,17,18} describe cell division events as regulated toward achieving cellsize homeostasis. However, owing to the small number of cells involved in these studies, dynamical effects arising from temporal and spatial biomechanical constraints, are not fully included. Despite the crucial influence that selfinduced mechanical forces seem to have on microbial behavior and physiology, currently, we lack a cellbased mechanistic model that could capture the collective interrelations and feedback between cells that spontaneously evolve at the scale of the colony.
Motivated by the gaps in our current understanding, here we report on the phenomena arising from selfinduced collective mechanical stresses between the cells in expanding bacterial colonies. While mechanical interactions have already been identified as a key ingredient to determine the colony’s structure and shape^{19,20,21,22}, we also describe a mechanoresponse of bacterial cells that tunes the emerging distribution of cell sizes (or lengths)^{23} during the evolution of the growing colony. This mechanoresponse represents a cell’s ability to sense local surroundings, stimuli as well as the presence of other cells^{24,25} and, based on this information, adapt the growth behavior to avoid overcrowding or trigger cell death^{26,27}. A biochemical origin could lie in the inhibited signaling under mechanical stresses^{28,29}. To mimic these effects we devise an analytically accessible model founded on statistical mechanics. Inspired by the success of modeling various driven systems by means of rodlike particles^{30,31,32,33,34,35,36}, we describe the bacteria as rigid rods of variable length^{5,6,7}. Our model both explains the collective selfregulation of phenotypic bacterial traits over time^{11} and unveils that collective mechanical interactions also enable bacterial populations to dynamically tune the size (or length) of single cells.
To investigate the implications of collective stresses in a controlled setting, we perform our growth experiments on a nutrientrich agarose substrate and put forward a microscopic firstprinciples model derived from dynamical density functional theory (DDFT)^{37,38,39}, which explains the central experimental observations. Its basic ingredients are the division of a bacterium with length 2L into two shorter agents of lengthatbirth L, growth with a certain elongation rate, fluctuations of the elongation rate and a mechanoresponse to collective interactions. The celllengthdependent density^{40,41} in our DDFT describes the phenotemporal properties of the growing colony, i.e., the cells’ lengthdistribution dynamics. This allows us to gain analytic insight into the growth process, predicting the evolution towards a unique distribution of cell lengths, which reflects the stochastic nature of biological systems. A numerical evaluation further sheds light on how the length distribution depends on the spatial position. On top, we introduce a refined cellbased simulation tool, which reveals the detailed spatiotemporal aspects of the mechanoresponse to the local colony structure. Our qualitative findings are robust with respect to implementing different mechanisms for growth or cell division^{14,15,16,17,18}. Therefore, our techniques can be applied to a broad class of biological systems, providing a comprehensive understanding of collective biomechanical forces during population growth, as we exemplify by investigating a mechanocrossresponse between two competing bacterial strains.
Results
Overview
As compiled in Fig. 1, our model is designed to capture the fundamental aspects of bacterial growth (see the Methods section and Supplementary Notes 1 and 2 for more details). In our experiments, shown in Fig. 1a–h, we observe the growth of Escherichia coli at 30 °C, on a nutrientrich agarose substrate. A single cell first divides into two cells of equal length, which then continue to grow with a certain elongation rate. At later times, as the colony becomes denser, growth and cell division progress with a reduced elongation rate. This effect is significantly more pronounced for closely packed cells, whose ends are in direct contact with their neighbors, than for the loosely packed cells flanked by void regions of the colony. Overall, we observe that the average cell elongation rate is nearly halved in the grown colony, although a sufficient amount of nutrients is still available.
As demonstrated by means of cellresolved Langevin simulations in two dimensions, shown in Fig. 1j–q, the central experimental observations are modeled by accounting for four central ingredients in the dynamics of the bacterial length (Fig. 1i): cell division into two new individuals with lengthatbirth L when the maximal length l = 2L is reached, growth with (average) elongation rate G, fluctuations of this elongation rate with magnitude D and collective mechanical interactions generated by the same potential as the forces and torques in configurational space (see the Methods section and Supplementary Note 1 for a precise definition). Since these interactions have a weakening effect on the bacterial elongation rate, we speak of a mechanoresponse with strength quantified by the parameter S.
Sizeaveraged analytical predictions
As the main prediction of our model, we observe in Fig. 1q an effective reduction of the bacterial elongation rate over time, confirming the experimental trend in Fig. 1h. This is due to collective interactions (S > 0), which slow down the growth until the colony eventually attains a stationary state when all available space is occupied. In a first approximation, the behavior under such a mechanoresponse can be described by deriving a logistic growth equation for the total number density \(\bar{\rho }(t)\) of all cells from our general DDFT model (see the Methods section and Supplementary Note 3).
Specifying the initial value \({\bar{\rho }}_{0}:= \bar{\rho }(0)\), we find the analytic result
with \(R:= G\ln (2)/L+D{(\ln (2)/L)}^{2}\). Hence, due to the mechanoresponse, the colony does not grow indefinitely but approaches the maximal density \(\bar{\rho }(\infty )=\frac{RL}{S\ln 2}\) for t → ∞. Defining the effective elongation rate as
and inserting \(\bar{\rho }(t)\) from Eq. (1), allows us to predict analytically the behavior shown in Fig. 1q from DDFT, where G_{eff}(t) decays to zero for t → ∞. The first two terms in Eq. (2) are contributions of individual cells, while the third term emerges due to collective interactions. Cell division merely contributes to G_{eff}(t) by increasing the total cell count.
Sizeresolved DDFT
While Eqs. (1) and (2) illustrate the basic concept of a mechanoresponse, our phenotemporal DDFT, also resolving the cell size, provides deeper insight into the growth process. Thus, we now describe the growth dynamics of the bacterial colony by its density ρ(l, t) explicitly depending on the size of the bacteria, represented here by a length l (see the Methods section and Supplementary Note 2). To make analytic progress, we restrict l to the interval l ∈ [L, 2L] and, for the moment, we model cell division by the oblique boundary condition ρ(L, t) = 2ρ(2L, t). We are particularly interested in the length distribution \(h(l,t)=\rho (l,t)/\bar{\rho }(t)\) with ∫dl h(l, t) = 1, which follows from normalizing the density ρ(l, t) by the total lengthaveraged density \(\bar{\rho }(t)\). Our DDFT results compiled in Fig. 2 reveal nontrivial lengthdistribution dynamics, where the initial condition h_{0}(l) ≔ h(l, 0) is chosen as a sharp peak to represent a single bacterium of length l = 3L/2. The analytic solutions are discussed in Supplementary Note 4 and illustrated in Supplementary Fig. 1.
In the absence of both fluctuations (D = 0) and interactions (S = 0), we observe bare growth, cf. Fig. 2a, where h(l, t) ∝ 2h_{0}(l + Δl(t)) + h_{0}(l + Δl(t) − L) is periodic in time as the horizontal offset Δl(t) increases linearly from 0 to L within one period t_{G} ≔ L/G, which sets the time unit. After each generation, denoted by an integer n such that Δl(t = nt_{G}) = 0, the distribution resorts to its initial form h_{0}(l) ≔ h(l, 0) and the total density \(\bar{\rho }(nL/G)={2}^{n}{\bar{\rho }}_{0}\) has doubled.
With interactions (S > 0 but D = 0), we observe a decelerated growth, cf. Fig. 2b, where the period of h(l, t) increases from generation to generation, as the effective elongation rate in Eq. (2) decreases over time. When the density \(\bar{\rho }(\infty )=G/S\) reaches its stationary value in the course of the birth of a new generation, the length distribution is suddenly arrested. In short, the growth process comes to an end following a mechanoresponse to collective interactions.
With fluctuations (D > 0 but S = 0), we observe disperse growth, cf. Fig. 2c, where the distribution smears out and approaches the unique limit h(l, ∞) ∝ 2^{−l/L}, without inhibiting the indefinite exponential growth of \(\bar{\rho }(t)\). In short, the stochastic nature of bacterial growth results in a selfregulation of the cell length^{11}.
For extreme fluctuations (D ≫ S and \(D\gg G/{\bar{\rho }}_{0}\)), this model predicts a fluctuationdriven growth with h(l, ∞) ∝ (3L − l) (see Supplementary Note 4E).
Combining the behavior from the above special cases, the full dynamics including both fluctuations (D > 0) and interactions (S > 0) can be understood as disperse decelerated growth, cf. Fig. 2d, where the distribution h(l, t) simultaneously broadens and increases its period in the course of time. Hence, we observe in Fig. 2e that the total density \(\bar{\rho }(t)\) increases jerkily due to correlated division bursts in the young colony. Later, the mechanoresponse sets an asymptotic threshold and the prediction from Eq. (1) is recovered after the cells have selfregulated their growth behavior, approaching h(l, ∞) ∝ 2^{−l/L} due to fluctuations (other possible stationary solutions are unstable, as illustrated in Supplementary Fig. 2). Another consequence of an initially sharp length distribution is an effective elongation rate which depends on the instantaneous form of h(l, t) (see Supplementary Note 4D). As shown in Fig. 2f, the average effective elongation rate G_{eff}(t), defined in Eq. (2), decays earlier for both increasing mechanoresponse (stronger counterforce) and increasing fluctuations (faster growth). Moreover, G_{eff}(t) shows a jerky behavior for weak fluctuations, just like the total density. Comparing these observations to the approximate sizeaveraged predictions in Eqs. (1) and (2), shown as a reference in Fig. 2e, f for D = 10^{−2}GL, underlines the dynamical information hidden in the length distribution.
DDFT with position dependence
While homogeneous systems of growing bacteria, as described with ρ(l, t), can be realized in experiments^{42}, we now investigate the basic local aspects of sizeresolved colony growth using our DDFT in one spatial dimension (see the Methods section and Supplementary Note 1). Here, the density ρ(x, l, t) also depends on the position x in a onedimensional channel, resembling, e.g., cells growing in a mother machine^{43, 44}. To implement cell division, we employ a directed boundary condition, which consists of source and sink terms, representing the two daughter cells and the dividing mother cell, respectively, and prohibits flow from short to long across the boundary (see the Methods section and Supplementary Note 5).
In general, our model describes a decrease of the local cell elongation rate in the regions with high density according to the stronger local mechanoresponse. To illustrate this spatial dependence, we depict the normalized density in the early stages of the colony evolution in Fig. 3. Initially, we observe individual density peaks, each representing a cell, whose number doubles after the birth of the next generation. The effect of an increasing mechanoresponse is to slow down the overall colony growth, as predicted in Fig. 2, but here also on a local level. For example, in the bottomright snapshot of Fig. 3, the outer peaks represent cells one generation ahead of those in the center, where the local density is much higher. For a weaker mechanoresponse (top panel of Fig. 3) this effect is less pronounced and we observe that the peaks gradually merge due to growth fluctuations and spatial diffusion. Moreover, for a stronger spatial mechanorepulsion, i.e., lower substrate friction, the peaks are pushed further apart from each other (see Supplementary Fig. 3), which reduces the local strength of the mechanoresponse, such that the total cell count increases faster. Eventually, a smooth colony structure is approached which is characterized by having a higher density and a larger percentage of shorter cells in the center than in the periphery.
Cellbased simulations
To corroborate the predictions of our probabilistic DDFT and better understand the repercussions on the individual growth dynamics, we also perform cellresolved Langevin simulations in two spatial dimensions. As detailed in the Methods section, we consider rodlike bacteria that interact through Hertzian repulsion. Specifically, the Langevin equations for positions and orientations are coupled to the stochastic dynamics of the cell lengths, including a response term with respect to the same Hertzian overlap potential.
The effective elongation rate G_{eff}(t) averaged over the whole colony is shown in Fig. 1q. As in the experiment (Fig. 1h), G_{eff}(t) decreases at later times but does not exponentially approach zero since the continuously growing periphery of the colony always remains sufficiently dilute. This result is illustrated in more detail by the different snapshots compiled in Fig. 4a–d. Comparing the colony size (different scale bars) after certain times, it becomes apparent that the colony of bacteria with a stronger overall mechanoresponse (larger \(\tilde{S}\)) grows slower.
We also resolve by the color code in Fig. 4a–d the local mechanoresponse of individual cells, which is apparently stronger in the center of the colony than in the periphery. As this local quantity scales with the overall mechanoresponse, the difference between the typical forces in these two regions is more significant for larger \(\tilde{S}\). Our qualitative observations are confirmed in Fig. 4e by measuring the local elongation rate, which is steeper for larger \(\tilde{S}\) (also notice the smaller colony radius after the same amount of time) and generally increases from the colony center to the periphery, following the decrease of local density (see also the DDFT results in Fig. 3). The correlation depicted in Fig. 4f further reveals that the mechanical force on longer bacteria is typically stronger than on shorter bacteria throughout the colony evolution.
Moreover, as illustrated in Fig. 1o, p, the local force acting on a bacterium in a dense colony is smaller when it is surrounded by voids than when its ends are in close contact with its neighbors, which also feeds back on the individual elongation rate, as measured experimentally in Fig. 1f, g.
Lengthdistribution dynamics
To gain further insight into the lengthdistribution dynamics, we compare in Fig. 5 the experimental results for the mean length \(\bar{l}(t):= \int{{{{{{{\rm{d}}}}}}}}l\,l\,h(l,t)\) and the normalized variance \({{\Delta }}(t):= \int{{{{{{{\rm{d}}}}}}}}l\,{(l\bar{l}(t))}^{2}\,h(l,t)/{(\bar{l}(t))}^{2}\) to our model predictions. After early fluctuations owed to the synchronous cell cycles in the early generations, the experimental data show a clear trend that the bacteria become shorter in the dense colony, while the variance begins to plateau under the experimental growth conditions considered here. Accordingly, the moments predicted by DDFT oscillate around their stationary values with a decreasing amplitude. Recall that the version of our DDFT used here formally describes a homogeneous system with the cell lengths restricted to a fixed interval.
The qualitative experimental observations for later times are better captured by utilizing the directed boundary condition introduced for our positionresolved DDFT, as it prohibits flow from short to long (see the Methods section and Supplementary Note 5). This refined implementation allows us to effectively predict the expected decrease of \(\bar{l}(t)\) within our sizeresolved DDFT. As shown in Supplementary Figs. 4 and 5, the colony enters an additional fluctuationdriven regime after it has grown sufficiently dense, which is characterized by a gradual approach to a distinct stationary solution, where shorter bacteria balance the higher cell count (the smaller variance is due to our assumption of a restricted length interval). We also find consistent results from our cellbased simulations of a freely growing colony, where the bacteria in the dense region can effectively respond by shrinking below the lengthatbirth L (see Supplementary Fig. 6). A detailed comparison between our approaches in view of the observations in Fig. 5 can be found at the end of the Methods section.
Mechanocrossresponse of competing strains
When evolving on the same substrate, two different bacterial strains compete for the available resources, including the free space in the colony. Therefore, the growth of each individual is not only regulated by the interaction with its own kind but also with the other species. To investigate this mechanocrossresponse, we consider two interacting species A and B with different characteristic growth properties^{45}. For such a system, our multicomponent DDFT (see the Methods section) with directed boundary conditions predicts distinct length distributions in the fully grown homogeneous colony, whose general form is related analytically to the input parameters and final densities in Supplementary Note 6. We can thus use this stationary information to infer the growth properties of the two competing bacterial strains.
To understand the ultimate colony composition, we compare in Fig. 6 the stationary total densities \(\bar{\rho}_\infty^{(i)}:=\bar{\rho}^{(i)}(\infty)\) and mean lengths \(\bar{l}_\infty^{(i)}:=\bar{l}^{(i)}(\infty)\) with \(i=A,B\) for different parameters of our model. For equal strengths of the mechanoresponse (S_{A} = S_{B}, Fig. 6a), the fastergrowing strain also ends up with a larger total number of cells and thus dominates the colony. Due to the mechanocrossresponse, the other strain ends up with a larger percentage of shorter cells. Setting now, say, S_{A} > S_{B} (Fig. 6b), a dominance of species B over species A is even possible if the latter has a larger elongation rate G_{A} > G_{B}, as long as G_{B} > G_{th} exceeds an interactiondependent threshold G_{th} set by the condition \({\bar{\rho }}_{\infty }^{(A)}={\bar{\rho }}_{\infty }^{(B)}\). Slightly below this threshold, the shape of the stationary length distributions changes drastically for small variations of G_{B} ≲ G_{th}. This behavior of \({\bar{\rho }}_{\infty }^{(i)}\) and \({\bar{l}}_{\infty }^{(i)}\) is illustrated for a broader range of interaction parameters in the state diagrams in Fig. 6c, d. Moreover, as exemplified in Supplementary Fig. 7 and discussed in Supplementary Note 6, we observe for G_{B} ≳ G_{th} a dynamical crossover in the total densities, \({\bar{\rho }}^{(A)}(t)\, > \,{\bar{\rho }}^{(B)}(t)\) for t < t_{c} but \({\bar{\rho }}^{(B)}(t) \, > \,{\bar{\rho }}^{(A)}(t)\) for t > t_{c} with the crossover time t_{c}, and an adiabatic regime in which the faster growing but more mechanoresponsive species A follows a sequence of quasistationary length distributions after its total density has saturated.
The experimental investigation of such bacterial mixtures, together with a more comprehensive theoretical study that allows for a possible extinction of the dominated species, shall shed further light on the crosstalk and interfaces between distinct active growth processes in future work.
Discussion
We have introduced a DDFT model for growing bacterial colonies from which we predicted analytical length distributions and drew parallels to dynamical observations in experiments and Langevin simulations. Despite its simplicity, our model captures the basic features of collective interactions among the cells in in vitro experiments ruling out possible effects of nutrient limitation. This agreement is demonstrated by investigating the (local) reduction of the elongation rate depending on the bacterial mechanoresponse. For a mixture of competing bacterial strains, our model suggests that a mechanocrossresponse between the two species affects the (dynamical) length distributions in a nontrivial way. Our general theoretical description relates these effects to a microscopic interaction potential and, therefore, it both contributes to the understanding of collective effects in models of growing bacterial colonies and elucidates the consequences of biomechanical forces for the evolution of living samples. While a comprehensive description of biological systems surely requires the inclusion of additional effects (as discussed below), our lengthresolved tools are potentially of inherent theoretical and mathematical interest in their own right, for example regarding the interplay between cell length and topological defects.
The utility of our model can be exemplified by contemplating the onset of the monotomultilayer transition^{7,8,11} (or verticalization^{42,46,47}), i.e., when individual bacteria evade crowded regions by escaping into the third dimension to form additional layers. Such structural transitions are triggered as a consequence of the selfimposed mechanical stresses that build up in a growing bacterial colony^{7,11}. Accordingly, our model allows us to determine directly the local inplane mechanical force opposing the growth of each cell, cf. Fig. 4a–d, which can be compared to an appropriately chosen threshold. In our experiments, the transition happens after about 450 min, which roughly corresponds to t ≈ 13t_{G} for the model parameters chosen accordingly in Fig. 1q. By then the peak pressure within the colony has reached values around 10 kPa^{7}, which is consistent with the selfimposed physical pressures measured by Chu et al.^{8}. A more quantitative comparison in future work should also take into account the role of substrate friction^{27}.
To provide a more accurate description of specific internal processes in individual cells and give a broader account of biodiversity, our model can be readily extended in various directions. Escherichia coli typically exhibits exponential growth^{11,14,18}, which can be modeled on a cellular level^{48}, and cell division occurs after adding a specific length^{15,16,17}. These processes are intrinsically related to a cell cycle^{49,50,51,52,53}. One may further include death events^{26,27,54,55,56} or describe division into multiple daughter cells^{57} and then consider more general mixtures of cells with different physiological properties^{53,58}, also allowing for phenotype switching^{59,60} or mutation^{20,61,62}. Finally, we stress that our model as employed here is formally not limited to the interpretation of mimicking mechanical interactions. An alternative application of our equations would be to conceive an interaction potential differing from that describing spatial repulsion, such that it effectively incorporates other effects limiting cell growth, such as biochemical signaling^{28,29} or nutrient depletion^{63}.
Further perspectives on the spatiotemporal implications of a heterogeneous local length distribution open up when additionally resolving positions and orientations. Both our generalized DDFT (see the Methods section) and our cellbased simulations allow us to include and compare the mechanoresponse with respect to different externally applied forces affecting the growth process on the singlecell level. For example, the bacteria can be immersed in a stiff gel^{12}, grown on rough surfaces^{64,65} or spatially confined^{26,66,67}. It is also possible to investigate the detailed colony structure^{55,68,69,70}, topological defect dynamics^{10,11,42,71,72,73}, emerging smectic order^{22,74,75,76,77,78}, growth within porous media^{79,80} or the explicit onset of threedimensional growth^{11,46,47,81,82}. Moreover, attractions through pili bonds^{83,84}, or the motility of individual bacteria^{85,86,87} can be modeled. In view of a comprehensive biological picture, one could study interactions and competition between multiple colonies or cell strains^{83,84,88,89} alongside their crosstalk with other spatially distributed agents^{90}, such as signaling molecules^{28,29} nutrients^{63,91}, antibiotics^{58,92}, parasites like bacteriophages^{93} or a secreted extracellular matrix mediating biofilm formation^{70,91}.
Finally, our DDFT equations allow for a systematic derivation^{94,95} of phasefield crystal models^{96} to recover hydrodynamic field equations^{6,71,72} and explicitly incorporate aspects related to bacterial length. Exploring the relation to active nematics^{97,98} constitutes a possible direction for future work.
Methods
General DDFT for growing bacterial colonies
We propose a dynamical density functional theory (DDFT) to model growing bacterial colonies through a timedependent density ρ(r, p, l, t), which resolves the spatial position r, orientation p and a size parameter l, which here represents the cell length. In its most general form, the DDFT reads
with the currents J_{r}(r, p, l, t), J_{p}(r, p, l, t) and J_{l}(r, p, l, t) in positional, orientational and length space, respectively. The former two terms are given in their standard DDFT form as^{37,38,39}
with the diffusion coefficients D_{r} and D_{p}, the friction coefficients γ_{r} and γ_{p} and the derivative operators ∇_{r} and \({\hat{{{{{{{{\mathcal{R}}}}}}}}}}_{{{{{{{{\bf{p}}}}}}}}}={{{{{{{\bf{p}}}}}}}}\times {{{{{{{{\boldsymbol{\nabla }}}}}}}}}_{{{{{{{{\bf{p}}}}}}}}}\) in positional and rotational space, respectively. The internal interactions between the cells and interactions with an externally imposed field are described by the excess part \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{ex}}}}}}}}}[\rho ]\) of the free energy and an external potential V_{ext}(r, p, l, t), respectively^{99}.
As the central ingredient of our model, the length current
drives the lengthdependent changes of the density, where the thermal energy k_{B}T is used as a scaling factor. The first term describes cell growth according to the growth function G(r, p, l, t) and thus drives the system out of equilibrium, while the remaining terms have a similar form as the currents in Eq. (4) but possess a slightly different interpretation. The term ∝ D is of diffusive nature and describes fluctuations of the growth function, while the terms ∝ S describe the response of the cell growth to internal and external interactions, where S is related to the substrate friction. Finally, the source and sink terms in Eq. (3) describe cell division after a length L(t) is reached. A more detailed introduction to Eq. (3) and the framework of DDFT in general can be found in Supplementary Note 1.
Overview of related approaches
In Eq. (3) we have presented the most general form of our basic model in the language of DDFT which we can, in principle, even further extend in several directions along the lines of the Discussion section. What is left to be specified are the explicit interactions between the cells. Instead of evaluating the full multidimensional DDFT, we focus in the main text on different approaches based on this model, which are further described in the remaining paragraphs of this Methods section.
Specifically, we provide in Eqs. (6), (7), and (9) a set of stochastic Langevin equations in two dimensions, which are formally equivalent to our DDFT in Eq. (3) and allow for a cellresolved investigation of the mechanoresponse. On the DDFT side, we consider various versions of Eq. (3) focusing on different aspects. First, we introduce a onedimensional version of our DDFT in Eqs. (12) and (13) to illustrate the positional dependence of the evolving length distribution. Second, we derive a homogeneous sizeresolved DDFT in Eqs. (14) and (15), which formally assumes a wellmixed system and can be analytically investigated. Third, we also generalize this sizeresolved DDFT by Eqs. (20) and (21) to a version valid for multiple bacterial species. Fourth, we demonstrate that a logistic growth equation, Eq. (22) can be recovered upon further averaging our homogeneous DDFT over the cell size. After presenting details on our experimental system, we conclude the Methods section with a discussion of how the different versions of our model are related and which experimental aspects we intend to describe.
Cellbased Langevin simulations
In the particlebased approach to modeling growing bacterial colonies, the cells are considered as rigid rods. Their positions r_{i}, orientations θ_{i} and lengths l_{i} (of their long axis) evolve in time according to coupled Langevin equations. Here i = 1, …, N is the cell index, where the total number N(t) of bacteria may increase after each time step due to cell division. The short axis of each rod is kept fixed with length d_{0}.
The position r_{i} of rod i evolves according to
where γ is the friction coefficient and F_{ij} are steric forces stemming from the interactions with other rods. Further, the orientation of the rod is measured by the angle θ_{i} with respect to the xaxis in a Cartesian coordinate system. The dynamics of the angles are given by
where r_{ij} = r_{i} − r_{j} is the distance vector between particles i and j and e_{z} is the vector perpendicular to the rods’ plane of motion. The forces between rods are calculated by a Hertzian repulsion
where h_{ij} is the overlap of rods i and j, F_{0} is the strength of the force, and n_{ij} is the vector normal to the closest point of contact of the particles.
In the same spirit, we now allow the length of a rod i to evolve as
with the constant elongation rate G, a white noise ξ_{i} of unit variance accounting for fluctuations of magnitude D and mechanical interactions mediated by the overlap between particles h_{ij}. The parameter \(\tilde{S}\) quantifies the strength of this mechanoresponse and takes the role of an inverse friction coefficient. Here, we have absorbed the other parameters from Eq. (8), such that \(\tilde{S}\) is formally different from S in Eq. (19) given the different nature of interactions considered. When the length l_{i} of a cell exceeds the value 2L after a certain time step, it is reset to l_{i} = L, where L is the lengthatbirth. Then, a second cell with a new particle label and the same length L is introduced and the total cell count N is increased accordingly. In the course of this cell division, the positions of the two daughter cells are shifted from the rod’s original position by L/2 along the direction of the rod axis.
In the main text, the repulsion strength is fixed as F_{0} = 10^{6}G/(γL) and the length of the short axis of each rod is d_{0} = L/8. The data presented in our plots are averaged over five simulation runs.
DDFT in one spatial dimension
To illustrate the spatial evolution of the length distribution in DDFT, we consider a system in one spatial dimension described by the density ρ(x, l, t). As a minimal model, we take the growth function G as a constant elongation rate^{6,7}, focus on a freely growing colony in the absence of an external potential V_{ext} = 0 and employ a softrepulsive pair interaction
in the form of Gaussian cores^{100} whose width follows from the lengths l and \({l}^{{\prime} }\) of two interacting bacteria. Such a potential is conveniently included through the meanfield functional
where the integrals over x and \({x}^{{\prime} }\) run over the full onedimensional space, while l and \({l}^{{\prime} }\) are restricted under the present assumptions to the interval [L, 2L]. More general interactions could also be incorporated through appropriate alternative freeenergy functionals^{99,101,102,103}.
With the above choices, we rewrite the length current from Eq. (5) as
The onedimensional version of Eq. (3) then reads
where a detailed derivation is given in Supplementary Note 1D.
Although the cell length l can, in principle, take any positive value, we restrict it here to the fixed interval l ∈ [L, 2L] with constant lengthatbirth L and consider a directed boundary condition (see Supplementary Note 5 for a detailed description and also Eq. (16) below). At l = L, this corresponds to a specific noflux boundary condition in Eq. (12) imposed for the fluctuations (the term ∝ D) and to enforcing a vanishing density ρ(x, l = L, t) = 0 if the driftlike terms (those ∝ G and ∝ S) result in a negative contribution to the current. At l = 2L, we use an absorbing boundary.
Homogeneous sizeresolved DDFT
For the sizeresolved calculations presented here, we employ a version of Eq. (3) for the dynamics of ρ(l, t) after averaging over positional and orientational coordinates, while assuming again that the external potential V_{ext}(l, t) vanishes (or does not depend on the cell length) and taking a constant elongation rate G. The resulting DDFT reads
where the length current J_{l}(l, t) is given by
The third term of J_{l}(l, t) is derived assuming a meanfield expression of \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{ex}}}}}}}}}[\rho ]\), Eq. (11), for soft interactions in the form of Gaussian cores, Eq. (10). Hence, this sizeresolved DDFT is fully consistent with Eqs. (13) and (12) in one spatial dimension. The expression in Eq. (15) can also be used in any spatial dimension after absorbing a trivial dimensional scaling factor into S.
The spatially homogeneous nature of our sizeresolved DDFT allows us to consider two types of boundary conditions for l ∈ [L, 2L]. First, when working with Eq. (14), we use the directed boundary condition (compare Supplementary Note 5A)
for the current in Eq. (15). Second, a convenient alternative is to incorporate cell division through the oblique boundary condition
assuming that individual cell growth is homeostatic, while rewriting Eq. (15) as
which yields the DDFT equation
This approximation allows for a detailed analytic understanding of the lengthdistribution dynamics, where the effective elongation rate G_{eff}(t) in Eq. (2) can be conveniently defined from the term on the righthand side. A more precise specification of G_{eff}(t), a full derivation of Eq. (15), details on the role of the different boundary conditions, analytic analysis and further results are provided in Supplementary Notes 2, 4, and 5.
DDFT for multiple bacterial species
It is in general straightforward to generalize a given DDFT model to mixtures of κ species with different properties by adding an additional species label i = 1, …, κ and consider the individual evolution equations for ρ_{i}, which are coupled via their collective interactions^{39}. For the purpose of the present study, we generalize the sizeresolved DDFT from Eq. (14) to
with the currents
In these equations, we assumed for simplicity that all species have the same lengthatbirth L and the same magnitude D of growth fluctuations. Moreover, we consider κ = 2 different species A and B and define the elongation rates G_{A} ≔ G_{1} and G_{B} ≔ G_{2}, as well as the strengths S_{A} ≔ S_{11} and S_{B} ≔ S_{22} of the intraspecies mechanoresponse, where we assume S_{12} = S_{21} = (S_{A} + S_{B})/2 for the crossinteractions (see Supplementary Note 6 for further discussion).
Sizeaveraged logistic growth
The phenotemporal description of ρ(l, t) in Eq. (19) represents an averaged model after integrating out positions and orientations of a more general DDFT for ρ(r, p, l, t). In turn, if one is only interested in the increase of the density \(\bar{\rho }(t)\) (or number of cells), we can show upon further averaging out the dependence of the cell size (see Supplementary Note 3) that our model is consistent with the logistic growth equation
widely used to describe (spaceresolved) population dynamics^{104,105,106,107}. From our Eq. (19) we identify here the overall growth rate \(R:= G\ln (2)/L+D{(\ln (2)/L)}^{2}\). Solving Eq. (22) with the initial condition \({\bar{\rho }}_{0}:= \bar{\rho }(0)\), we find an analytic expression for the time evolution of the total density \(\bar{\rho }(t)\), as given by Eq. (1). Note that generalized growth equations^{108,109} can also be derived within our framework from different microscopic interactions.
Experiments on growing bacterial colonies
We use nonmotile strains of E. Coli bacteria, NCM3722 deltamotA, growing at 30 °C on a millimeterthick agarose matrix. The experimental time scales are short enough to ensure that the growing bacterial monolayers remain nutrientreplete and do not undergo physiological changes throughout the duration of the experiments. Nutrientlimitation, if any, will impact all cells irrespective of their location in the colony. This setup ensures that collective (mechanical) stresses constitute the main cause of (locally) limited cell growth reported in Fig. 1a–h.
All experiments have been performed for a minimum of three distinct biological replicates. Cells were grown and monitored using standard protocols and control experiments^{11}. The growth of a single bacterium (or two initial cells, in some cases) into colonies was imaged while maintaining the growth temperature of 30 °C within the microscope environment. Single bacteria acting as monoclonal nucleating sites expand horizontally on the nutrientrich agarose layers. Initially, the colony expanded in two dimensions as a bacterial monolayer over multiple generations, subsequently penetrating into the third dimension.
We visualize the colony growth over the entire period using timelapse phasecontrast microscopy. For the current work, we focus primarily on the horizontal spreading of the colony and analyze the data till the transition to the multilayer structure sets off. Images were acquired using a Hamamatsu ORCAFlash Camera (1 μm = 10.55 pixels) that was coupled to an inverted microscope (Olympus CellSense LSIXplore). We use a 60X oil objective and, in some cases, 100X oil objectives to zoom into specific regions of the growing colonies. Overall, this gave a minimum resolution of 0.11 μm.
Each experiment lasted typically 15–18 h, allowing us to capture the monotomultilayer configurations and the structure and dynamics of multilayer colonies. Prior to image acquisition, multiple locations on the agarose surface (where a single bacterium or up to two cells were present) were identified and recorded, allowing us to additionally extract technical replicates from the same sample. The microscope was automated to scan these prerecorded coordinates and to capture the images of the gradually increasing colonies after every five minutes while maintaining the focus across all the colonies captured.
We extracted the cell dimensions (width and length), position (centroid) and orientation of each bacterium from the phasecontrast images using the combination of opensource packages of Ilastik^{110} and ImageJ as well as MATLAB (MathWorks). The combination of phase contrast and timelapse imaging allowed us to quantify phenotypic traits at the resolution of individual cells and thereby extract the reported statistics after image analyses while ensuring that the cells do not tilt out of the plane^{46,47} (see Supplementary Fig. 8). A detailed description of cell culturing, fabrication and imaging of cell monolayers can be found in Supplementary Note 7. The obtained celllength statistics are shown in Supplementary Tables 1–3 and provided as Supplementary Data 1–3.
Comparison of different approaches
Our cellbased Langevin model essentially describes the same physics of growing bacterial colonies as our general DDFT. However, the former conveniently ignores positional and rotational diffusion terms, which are typically required in DDFT. Apart from this minor difference, Eqs. (6) and (7) are conceptually equivalent to the first and second terms in Eq. (3), respectively (the sole difference being the way interactions are chosen and implemented, as discussed below). The dynamical description of cell size is the heart of our model and the four ingredients illustrated in Fig. 1i are accounted for in all our approaches. While the implementation of cell division is straightforward in the Langevin model, our DDFT requires the translation to a slightly different directed boundary condition (16). The description of growth and fluctuations in the Langevin model is stochastically equivalent to our DDFT when we compare the first two terms in Eq. (9) to those of the probability current in Eqs. (5), (12) and (15), or to those on the righthandside of Eq. (19). Finally, despite the different appearance of the third term in these equations, the treatment of collective interactions is also conceptually equivalent: they are derived from the same expression as in configurational space (steric forces F_{ij} in the Langevin model and an excess free energy \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{ex}}}}}}}}}\) in DDFT), which is the defining feature of the mechanoresponse.
The specific choice and implementation of the interaction terms is what distinguishes our two computational approaches. As we are interested in the basic phenomenology resulting from our model we have chosen to work with expressions that are standard in each case. Hence, we implement our Langevin simulations with the established Hertzian overlap function (8)^{5,6,7} and equip our DDFT with a Gaussian softrepulsive potential (10) which can be treated in the meanfield way (11). To corroborate the general compatibility of both approaches, let us note that, for a wellmixed system, the DDFT equation from Eq. (19) is stochastically equivalent to the Langevin model
where N = ∑_{i}1 is the current number of particles and V is the total volume of the system, thus \(\bar{\rho }=N/V\). As shown in Supplementary Fig. 4, the simulations of Eq. (23) with l_{i} ∈ [L, 2L] yield practically the same length distributions as DDFT with the directed boundary condition.
In view of the most accurate description of our experiments, a specific interaction potential would have to be measured for interacting cells and implemented in our mechanical terms. However, our qualitative observations are largely independent of such a choice, as long as the assumed interaction is sufficiently repulsive. The parameters entering our model equations are empirical and specific to the experimental nonequilibrium systems of interest. In particular, the strength S (or \(\tilde{S}\)) of the mechanoresponse is a measure of how strongly the growth behavior of a cell is actually affected by a mechanical stimulus—just like friction with the substrate determines the extent of the spatial drift induced by a repulsive force. If desired, other growthlimiting effects that are not of mechanical origin (such as nutrient depletion) can be effectively described by an appropriate adaption of this parameter to experimental measurements.
Although it is not the focus of the present work, we stress that the onset of cell division can also be affected by different biological or mechanical mechanisms. Hence, to better represent the real bacterial system, our basic model could be finetuned by allowing for a timedependent lengthofbirth L(t) in future work. For example, individual Escherichia coli cells grow according to the adder model and divide after having grown by a certain length^{15,16,17}. More specifically, taking a closer look at our experimental data, we find that the periodicity of the oscillations in Fig. 5 decreases in the course of the colony evolution, i.e., on average, a cell divides every 33–36 min in the dilute case and every 24–27 min in the dense case. As the elongation rate, averaged over the colony, also decreases over time, this observation is accompanied by a reduction of the maximum length an individual cell reaches before the division event, i.e., the lengthatbirth decreases from generation to generation. In addition, the lengthatbirth depends on a cell’s local position in relation to its neighbors in the growing colony. Hence, we conclude that the observed decrease of the mean cell length in Fig. 5 is consistent with our current model of a mechanoresponse depending on the local density (even in the simple form with a constant lengthatbirth L) and that the individual growth kinetics play only a minor role. In an extended model, the lengthatbirth should thus also depend on the density, which could be modeled by similar terms as used for the length current in Eq. (5).
Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
Data availability
Data that support the plots within this paper and other findings of this study are available from the manuscript and the accompanying Supplementary Notes. All experimental data are provided as Supplementary Data 1–3. Animations of the data shown in Fig. 2a–d are provided as Supplementary Movies 1–4. Animations of the data shown in Fig. 3 are provided as Supplementary Movies 5–7. Any additional detail can be obtained from the corresponding authors upon reasonable request.
Code availability
Relevant software descriptions, opensource packages and commercial packages for image analysis are available in the text. Additional software used in this study is available from the corresponding authors upon reasonable request.
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Acknowledgements
The authors would like to thank Michael te Vrugt, Nicola Pellicciotta, Marco Mazza, Simon Schnyder, and Jens Elgeti for stimulating discussions and Marcel Funk for his contributions to implementing the sizeresolved Langevin simulations (23). R.W. and H.L. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) through the SPP 2265, under grant numbers WI 5527/11 (R.W.) and LO 418/251 (H.L.). A.S. thanks the Institute for Advanced Studies, University of Luxembourg (AUDACITY Grant: IAS20/CAMEOS) and the Luxembourg National Research Fund’s ATTRACT Investigator Grant (Grant no. A17/MS/11572821/MBRACE) and CORE Grant (C19/MS/13719464/TOPOFLUME/Sengupta) for supporting this work.
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R.W. and F.J.S. conceived research. R.W. designed DDFT model and performed analytics. G.H.P.N. performed numerical DDFT. F.J.S. designed and performed simulations. A.S. designed and performed experiments. H.L. supervised the research. All authors discussed and interpreted the results. R.W. and A.S. wrote the paper, with inputs from all authors.
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A.S. is a Guest Editor of the Collection ‘Interactive Active Matter: crosstalk and interfaces between distinct active systems’ for Communications Physics, but was not involved in the editorial review of, or the decision to publish this article. The other authors declare no competing interests.
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Wittmann, R., Nguyen, G.H.P., Löwen, H. et al. Collective mechanoresponse dynamically tunes cellsize distributions in growing bacterial colonies. Commun Phys 6, 331 (2023). https://doi.org/10.1038/s4200502301449w
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DOI: https://doi.org/10.1038/s4200502301449w
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