Large-scale orientational order in bacterial colonies during inward growth

During colony growth, complex interactions regulate the bacterial orientation, leading to the formation of large-scale ordered structures, including topological defects, microdomains, and branches. These structures may benefit bacterial strains, providing invasive advantages during colonization. Active matter dynamics of growing colonies drives the emergence of these ordered structures. However, additional biomechanical factors also play a significant role during this process. Here, we show that the velocity profile of growing colonies creates strong radial orientation during inward growth when crowded populations invade a closed area. During this process, growth geometry sets virtual confinement and dictates the velocity profile. Herein, flow-induced alignment and torque balance on the rod-shaped bacteria result in a new stable orientational equilibrium in the radial direction. Our analysis revealed that the dynamics of these radially oriented structures, also known as aster defects, depend on bacterial length and can promote the survival of the longest bacteria around localized nutritional hotspots. The present results indicate a new mechanism underlying structural order and provide mechanistic insights into the dynamics of bacterial growth on complex surfaces.


INTRODUCTION
Bacterial colonization and invasion are collective phenomena. These processes are regulated through a complex interplay of physical and biological interactions in a crowded population.
Bacterial morphology, hydrodynamics, surface topology, and topography markedly alter growth mechanisms, morphology, and overall competition among bacteria [1][2][3][4][5][6]. Elucidation of the factors regulating collective bacterial growth and their competition is essential to enhance our understanding of evolutionary dynamics, bacterial infection, and the progression of inflammatory diseases.
A characteristic feature of bacterial colonization is the formation of large-scale order. Rodshaped bacteria display nematic alignment on surfaces, wherein localized stress, surface friction, and elasticity trigger the formation of ordered domains and lead to the emergence of topological defects [7][8][9][10][11] and various types of self-assembled structures, including edge fingerings [12] and vertical structures [13,14].
In particular, ± 1 2 topological defects are the typical orientational singularities observed among growing bacterial colonies and biofilms [7,9,10,15]. These topological defects have biological significance and regulate stress distribution across the structure, alter the physiology of the cells [16], and could control entire morphology; eventually, these effects trigger the formation of fruiting bodies [17] and bacterial spores in biofilms [15]. Liquid crystal theory has successfully predicted the dynamics of these defects; − 1 2 defects are stationary whereas + 1 2 defects are generally motile [18][19][20]. Another interesting structural order in bacterial colonies is anchoring, where the bacteria are tangentially oriented along the edge of the colony [2,7,8].
In this study, we assess the orientational dynamics of a crowded bacterial population competing for limited space. Unlike regular expanding colonies, if growing bacteria surround a closed area, domains of inward growth are formed. Under these conditions, entire mechanical interactions differ and lead to the formation of asters, formed as radially aligned +1 topological defects. With only a few exemptions [21,22], higher-order topological defects [23,24] are not commonly observed in extensile active matter systems, including growing bacterial colonies. These defects only appear under external modifications such as stress [25], confinement [26,27], and flow [28].
Our results also reveal that velocity profile is an important factor controlling the emergence of these radially aligned structures. Furthermore, we investigate the invasive advantages of this orientation for competing bacterial strains of different lengths.
Inward growth is commonly observed in various biological systems. During wound healing [29], cancer cell growth [30,31], and retina development [32,33], similar dynamic mechanisms are underway. Our results may provide novel mechanistic insights into these dynamics, particularly on the physical conditions for radial structural alignments during these complex growth processes.

Experimental observation of aster structures
To observe the dynamics of inward-growing bacterial colonies, we sparsely spread nonmotile Escherichia coli and Bacillus subtilis separately, on a flat agarose surface (see Materials and Methods). Time-lapse fluorescence microscopy was then performed to investigate the temporal evolution of growing colonies. With colony growth, the closed area invaded by multiple colonies was observed across the plate. Rough colony interfaces gradually converge to symmetric, relatively smooth, and enclosed circular areas. We refer to these shrinking circular regions as inward-growing bacterial domains because the growth direction is towards the center of the area. Figure 1 displays typical snapshots of the inward growth process (Fig. 1a, b, Video 1, ( Figure   1-video 1). Unlike regular expanding colonies, the bacterial orientation around these domains is generally radial. To assess the orientation, we first analyzed the radial order parameter around the center of these domains. The radial order parameter can be expressed as: where is the angular orientation with respect to x-axis and is the angular position of the bacterium i in polar coordinates about the colony center. Figure 1 d and c display the bacterial orientation and order parameter ( ) as a function of radial distance. = +1 corresponds to radial alignment and = −1 corresponds to tangential alignment. It is evident that large-scale radial order emerges across these inward growing domains (Fig. 1c, d). These structures strongly resemble +1 topological defects also known as aster structures. We also measured the velocity of the bacterial flow during this process (Fig. 1d). We found that the direction of the flow is towards the center. From these measurements, we can conclude this radial inward flow could align the bacteria in a radial direction.

Numerical simulation of bacterial orientation during inward growth
To clarify the impact of flow-induced alignment and differences in orientation between inwardgrowing and regular expanding colonies, we simulated 2D bacterial growth using a hard rod model. We used the open-source simulation code GRO [34] which provides a fast platform to observe bacterial growth (see Materials and Methods). To determine the morphology of the inward growing domain, we initially distributed bacteria in a random orientation. With growth, bacteria form small colonies, which eventually fuse into a growing annulus (Fig. 1e). To visualize the large-scale order, we color-coded bacteria on the basis of their radial orientation, with red representing radial orientation, and blue representing tangential orientation around the center of the hole (Fig. 1e, Video 2). These simulation results captured the experimentally observed radial order across the colony.
However, regular expanding colonies only formed microdomains with random local orientations ( Fig. 1g, Figure 1-video 2). Regular expanding colonies represent the outward growth initiating from a single bacterium displayed in Figure 1g. Based on these simulations, the primary difference between regular expanding and inward-growing colonies is the sudden change in the direction of the surface drag force which depends on the velocity (Fig. 1f, h). In inward-growing colonies, this force flips its sign at a critical radius where the local radial velocity of the colony vanishes.
To further quantify the effects of the critical radius, we determined the stress distribution and radial velocity profile , in growing colonies. Figure 2 summarizes the comparison and time evolution of these parameters. We first focused on radial and azimuthal stress profiles. We noted that the stresses (| | and | |) (Figure 2-figure supplement 1) are maximum around the critical radius during inward growth (Fig. 2a, b). The stress profiles initially show the quadratic form which is particularly dictated by the radial velocity profile [13].
Then we observed that, as colonies grew, only inward-growing colonies developed substantial radial order ( ) (Fig. 2c, d). Furthermore, radial velocity profiles significantly differed between regular expanding and inward-growing colonies. In contrast with regular expanding colonies, which have a linear radial velocity profile, inward-growing colonies developed radially nonlinear velocity, which vanishes at the critical radius ( Based on these results, the velocity profile appears to be the key physical parameter regulating the flow-induced alignment and the formation of the radial order.

Velocity profile and radial alignment
To better understand the association between the velocity profile on radial alignment, we first focused on the development of a minimum theoretical model based on active nematics. The theory of active nematics and liquid crystal physics provides a robust framework for understanding the dynamics of bacterial orientation. The primary characteristic of expanding colonies is the constant growth rate of the colony structure. The incompressibility criteria in 2D results in a linear relation between bacterial growth rate and radial velocity profile of the colony, where ( ) is the local growth rate and Λ is the exponential bacterial growth rate. These coefficients are related to incompressible expanding bacterial colonies.
This relation was previously referred to as a Hubble-like constant owing to its similarity to the expansion of the universe [8]. We considered the same approximations to obtain insights into bacterial orientation during inward growth. First, we used the assumption that without molecular field and convection terms, the time evolution of the orientational angle is simply regulated as follows (see Materials and Methods): where is the angular position of the bacteria in polar coordinates, and is the flow alignment parameter. Furthermore, g(r) is the local growth rate of the colony and its spatial derivative ′ (r) regulating the stability of the bacterial orientation. The constant growth rate observed in regular expanding colonies does not provide any orientational preference, = 0. However, this condition significantly differs during inward growth, wherein the local growth rate can be expressed as follows: Our assumption of a constant critical radius ( Supplementary Fig. 1b) indicates that the spatial derivative of the growth rate is positive everywhere across the colony, ′( ) > 0, suggesting the possibility of a stable state with = . The stable radial orientation stimulates aster formation, being referred to as a +1 topological defect. This finding is significant because ′( ) ≠ 0 is generally possible in compressible structures and also only around leading edges of growing colonies due to sudden drop [8]. Although bacterial colonies are not compressible, inward growth and the shrinking hole structure alter the overall velocity profile and lead to an essential local growth rate.
The radial orientation is stable throughout the colony and not only below the critical radius. To

Nemato-hydrodynamics and continuum modeling
Thereafter, we investigated whether the same defects were obtained through the continuum nemato-hydrodynamics equations of growing active matter [35][36][37]. Due to coarse graining over specific physical details, the continuum model could provide generality of our observation. The model is based on continuity, Navier-Stokes equations, and dynamics of the order parameter tensor (see Materials and Methods). The coupled differential equations governing the primary material fields density , , and velocity can be expressed as follows: Here, is the material derivative, and the stress tensor is given as Here, ( ) represents the active stress originating from the extensile nature of bacterial growth. and are the traceless strain rate and vorticity, respectively (see Materials and

Inward growing domains in multi-layered colonies
Growing bacterial colonies on elastic substrates generally form multi-layered structures. We  Figure 4 shows the prototypical FEM simulation outcome from inwardgrowing colonies. As expected, accumulated stress triggers verticalization and multi-layer formation around the critical radius of the colony (Fig. 4a, b and Video 4). However, a bacterial monolayer was observed only around the inner and outer leading edges of the colony. The formation of a monolayer region around growing colonies has been investigated in great detail [4]. We found that these monolayers could also result in planar radial alignment (Video 5). The width of the monolayer was approximately r =90±30μ (Fig. 4c). This width defines the size of the aster structures observed herein.

Inward growing domains in monolayer colonies
So far, we experimentally studied naturally emerged inward growing domains on agar surfaces. These domains are randomly formed across the plate. Due to random seeding of bacteria, outward growing edges of multiple colonies merge and form multilayered structures.
In these experiments particularly, the confinement is defined by crowded multilayered environments. Thus, observing critical radius, outer growing edge and detailed velocity profiles are not possible around these dense regions. Our simulations showed that the initial annulus shape could overcome these limitations. We asked whether we could induce similar annulus structures by patterning the initial distribution of bacteria to observe both inward and outward growing domains. We first tried to imprint bacteria on an agarose surface using soft PDMS molds. However, the wet surface and capillary effect quickly disturbed the initial bacterial patterns defined by the mold. Then we preferred non-contact lithographic techniques for patterning. Using a photomask ( Figure 5-figure supplement 1), we exposed randomly distributed bacteria with blue light to define an initial growth geometry by killing the remaining part of the pattern (see Materials and Methods). Figure 5a shows the time evolution of growing bacteria starting from annulus-shaped initial distribution. We observed that on a regular agar surface, again multi-layer formation dominates the overall colony morphology. Only very narrow monolayer regions are observable around the inner and outer edges of the colony.
We then focus our attention on how to eliminate this multilayering process. A simple glass or PDMS confinement cannot eliminate this multi-layering ( Figure 4-figure supplement 1).
Previous studies showed that attractive biochemical interactions between bacteria and surface could generate additional strong friction force [41].

Biological significance
Finally, to assess the biological significance of radial alignment, we investigated whether these structures potentially affect the competition among bacteria during inward growth. In general, near the leading edge of a bacterial colony, competition strongly depends on physical parameters. The most prominent example is a genetic drift based on random fluctuations [43,44]. This phenomenon could be altered through steric interactions among the cells, which can potentially alter the evolutionary dynamics of competing bacteria [45]. Although bacterial orientation is generally tangential at the expanding colony edge, radial bacterial alignment potentially contributes to inward growth. We hypothesized that longer rod-shaped bacteria potentially have an advantage owing to the torque balance. The basic premise is that the torque depends on the length of the bacteria, resulting in rapid radial alignment. Radial alignment further leads to lane formation and promotes an invasive advantage to the longest one, which could be beneficial in terms of approaching nutritional hotspots localized around the defect core more effectively.
To assess this competition, we initially simulated the growth dynamics of a mixed population with different division lengths from the same random initial distribution on a circle ( Figure   6-figure supplement 1). This is the most challenging condition to test the impact of the length difference on the bacterial alignment. Although the initial distribution of the bacteria is random, long bacteria can develop a higher radial order during inward growth (Fig. 6a-f). This radial order gradually allows the longest bacteria to approach the center of the defect more effectively (Fig. 6g). The bacterial growth is local, and it can create strong segregation within the colonies. The impact of the length could be more significant in segregated colonies. To visualize the difference, we initially segregated the bacterial strains with different lengths around the edge of the colony. Similar segregation can be commonly observed around the edge of the colony owing to random fluctuations. These segregations can also occur during inward growth ( Figure 6-figure supplement 2). Instead of expanding segments owing to perimeter inflation, we observed shrinking segments owing to the deflation of the hole geometry. Computationally, the advantage of radial alignment was more evident in segregated bacterial colonies (Fig. 6h). Interestingly, in a monolayer colony, radial alignment promotes the invasion of both the center and the leading outer edge of the colony through the longest bacteria (Fig. 6g). After the complete invasion of the center, the radial lanes buckle ( Figure   6-figure supplement 3, Video 6, 7). However, experimental verification of this competition remains challenging. Although precise regulation of the aspect ratio of bacterial morphology is well established [46], cell length can still not be independently tuned without perturbing other essential physiological properties, including growth rate and the biofilm-forming potential of the bacteria.

Discussion
Radially aligned structures can be considered as a +1 aster defect. These are ubiquitous topological structures observed in biological [47][48][49][50] or synthetic [51,52]  topological defects. This study shows that stable radially aligned, aster structures can also emerge during inward growth. In particular, we report the critical role of the colony velocity profile during this process, which depends on numerous factors. Although the bacterial growth rate is constant throughout the colony, growth geometry, confinement, or boundary conditions can alter the velocity profile.
Together, these biomechanical interactions change the bacterial orientation and stability, thus generating ordered structures. Different types of ordered structures have been observed in bacterial biofilms [53] and 3D colonies [4]. Furthermore, we believe that the velocity profile of growing structures on flat surfaces plays a significant role in bacterial alignment. Future studies are required to investigate the contribution of these effects.
We should emphasize that inward growing bacterial colonies and wrinkling thin circular sheets have geometric similarities [54]. In these elastic circular objects, under axisymmetric tensile load, azimuthal stress (hoop stress, ) show transition from tensile to compressive profile which eventually creates radial wrinkling pattern below critical radius. However, unlike elastic objects, growing bacterial colonies can only develop compressive stress due to negligible attractive force between bacteria. Experimental measurement of internal stress could provide more details, but it remains challenging. We noticed that the packing fraction of the bacteria shows a correlated profile (Figure 2-figure supplement 2). However, particularly for aligned bacteria, it is still very difficult to extract this information. In the future, new molecular probes could be useful for the experimental measurement of accumulated stress in the bacterial colonies [55,56].

Materials and Methods
Bacterial preparation and growth conditions. Bacterial cultures (BAK47 and BAK51) were grown in Luria-Bertani (LB) broth at 37 o C on a shaker. An overnight culture was diluted 100x and grown for 8 h. The culture was diluted 10000x, and 10 µl of culture was seeded on an LB agarose plate. These isolated bacteria on plates were grown at 21 o C for 12 h and then imaged.
Strains used in experiments are described in Table 1. In B. Subtilis bacterial strains, the flagellaproducing gene (hag) was mutated to eliminate the swimming-induced motion. The background strain TMN1138 was obtained from R. Losick Lab.

Microscopy imaging. Fluorescence time-lapse imaging was performed using a Nikon inverted
and Stereo SMZ18 microscopes. Images were obtained using a Andor EMCCD camera. Time intervals between successive images were set to 5-10 minutes.

2D hard-rod simulations of a growing colony.
We used the open-source simulation program GRO based on a hard-rod model. The code is available from https://depts.washington.edu/soslab/gro/. We modified the original code to be able to change the initial bacterial position and to extract the orientation of the bacteria.

Bacterial patterning.
We used photolithographic techniques to define the initial distribution of the bacteria by killing with structured blue light illumination. We used 15 min exposure under 5mW/mm 2 480 nm uniform light beam. We think the killing mechanism is mainly based on the local drying process.
The geometry was defined by chromium photomask. The mask (Supplementary Fig. 8) was fabricated by using Heidelberg DWL 66+ Direct Writing Lithography System and developed with chromium etchant. We have tested different annulus-shaped patterns by tuning the inner and outer radius. Due to light diffraction, the final exposed pattern depends on the spacing between the PC filter and a mask. Our optimized pattern has 400 m inner and 800 m outer radius.

Growing monolayer colonies on low friction surfaces.
In order to minimize surface friction, we replaced the agarose surface with a filter membrane.
We have tested several membrane filters, including nylon, polycarbonate (PC), polyethersulfone, cellulose acetate. Only white PC filters with 0.4 m pore size supported stable and large-scale monolayer colony formation. We noted that the brown PC filter has similar low surface friction; however, it has strong light absorption and does not allow noncontact lithography due to heavy condensation on the photomask.

SEM imaging.
A PC filter paper with a pore size of 0.4 µm was placed on LB agar surface.
After seeding the 10000x diluted bacteria on the filter paper, bacteria were grown on the paper for 12 h at 21 °C. Then the filter paper was peeled off from the surface, and the colonies were fixed using paraformaldehyde and left to dry. Fixed colonies were coated with 20nm gold and imaged using a Zeiss Ultra Plus Field Emission Electron Microscope.

Calculating stress distribution in a growing colony
The stress inside the colony can be calculated from the virial expansion  [40]. Analogous to [15], the bacteria were modeled as an isotropic, linearly elastic continuum whose initial stress-free shapes were spherocylinders. The bacteria were assumed to maintain a uniform circular cross-section with radius = 0.5µm, a mass density of 1gcm In order to simulate inward growing biofilm structure, we used our previous biofilm model [15] and the same parameters. 8 identical replica of small biofilm structures are circularly distributed to form an initial annulus shape. We used fracture strain (0.3) to relax the extreme bending condition by triggering filament division.

Radial Velocity Profile.
To calculate radial velocity profile, , during inward growth, we assume there is a critical radius , where is equal to zero. For < and the initial domain size equals to : Similarly, the time derivative is = 2 (12) Which results in the same equation. Consider that for lower than , velocity will be negative (inward direction) and for greater than velocity will be positive (outward direction).

Continuum
and we defined scalar order parameter: The molecular field can be obtained starting from the Landau-de Gennes free energy density given as: Approximation for growth-induced alignment. The following approximation and equations are received from Dell'Arciprete et.all [8]. These approximations were used to explain the tangential alignment of bacteria at the edge of growing colonies. The equation of motion for 2D nematodynamics without any free energy and no spatial variation of can be written as If we assume From this formulation we can conclude that : Where is the Cartesian component of the position vector Where = . Now this tensor is symmetric, with calculating and = 0 putting it in (Eq. 17) above we get: With writing using : In polar coordinates ( , ) right hand side of (Eq. 21) is: If we combine (Eq.23) and (Eq.24): Multiply first equation (Eq.25) by − (2 ) and second equation (Eq.26) by (2 ) and sum them up: Thus if ′ > 0 , equation above has a stable equilibrium for = (aster).

Data Availability
All data supporting the findings in this study are available from the corresponding authors on request.

Code Availability
All custom codes used in this study are available from the corresponding authors on request.

Competing interests
Authors declare no competing interests.      The video is associated with Figure 1g.                                The longest bacteria can invade the center more effectively. After a complete invasion of the center, the radial lanes buckle.

Supplementary Video Captions:
Video 1-Movie 1-Radial alignment during inward growth and emergence of asters. The video shows the fluorescence image of GFP labeled E.coli (BAK 55) during inward growth.

Video 2-2D
Simulation of a bacterial colony during inward growth. This video is associated with Figure 1f.
Video 3-Continuum simulation of 2D colony growth during inward growth. The scalar order parameter is overlapped with the director field pattern. This video is associated with Figure 3b.
Video 4-3D FEM simulation of colony growth during inward growth. Color represents vertical displacement. This video is associated with Figure 4b.
Video 5-3D FEM simulation of colony growth during inward growth. Color represents the radial order parameter. This video is associated with Figure 4a.
Video 6-2D simulation results of competing bacterial strains with identical division lengths. RFP and GFP labeled bacteria are initially segregated. Color represents the bacterial type. This video is associated with Figure 6h.
Video 7-2D simulation results of competing bacterial strains with different division lengths. RFP and GFP labeled bacteria are initially segregated. Color represents the bacterial type. This video is associated with Figure 6h.