Introduction

At low Reynolds numbers regime, overcoming reversibility constrain to gain motility, is a challenging task which requires innovative solutions. Microorganisms utilize a variety of specialized active mechanical components to solve these difficulties by performing nonreciprocal motion^1–4. Typical examples include the rotating helical flagella and beating elastic cilia. These active mechanical systems have the ability to execute cyclic motions, thereby breaking time-reversibility constraints⁵. In addition, due to the coupling among these active microstructures, complex synchronization phenomena also known as metachronal waves could emerge^6–8. From these perspective coupling, synchronization and collective behaviors are intrinsically related and ubiquitous in nature. Such widespread dynamical processes are also biologically essential, as seen in cilia-driven mucus transport in lungs⁹, nodal flow in developing embryos¹⁰, central motor generators¹¹ and spinal cord development¹². These intricate synchronization behaviors share the same underlying dynamical principles that surprisingly align with non-Hermitian physics of active systems^13–16. All these active systems are capable of injecting and transferring energy across the system by breaking certain symmetries.

Recent advances, especially those in the field of active matter physics^17–22, have illuminated critical non-Hermitian features, such as chiral states where PT symmetry is broken. In this broken mode system could sustain the phase difference between fundamental fields or the modes which further trigger the directional energy transfer between the active compartments and generate travelling waves. Notably this non-Hermitian process is particularly driven by nonreciprocity in the coupled mechanical system^8,17,23. Although non-Hermiticity has its roots in quantum mechanics^14,24 and the fundamental principles have been broadly studied in photonics²⁵, electronics, acoustics, optomechanics^26,27, superconducting qubits²⁸, trapped ions^29,30, single-spin systems³¹, and in light-matter interactions³². This process is often achieved not only using strong nonlinearities, but also utilizing nonreciprocal interactions and coupling^33–35. On the other hand, nonreciprocity is a very common feature in biological systems. Such as hydrodynamic interactions³⁶ or prey predator³⁷ relations could easily break action reaction symmetry and results in exotic collective behaviors. Extending the basic concepts from non-Hermitian physics to biological domain holds promise for shedding light on the fundamental principles of symmetry breaking in active and, more crucially, in complex biological systems.

Various biological models with specific active micromechanical components have been studied. The most notable biological platforms are ciliated epithelial cell³⁸, bacterial carpet^39,40, starfish embryo⁴¹ social amoeba^42–44 and walking placozoa⁴⁵. Addition to flagellum or cilia, bacteria could also use nano filaments known as type IV pili to gain motility on surfaces where the rotating flagella do not work effectively. Recent studies also highlighted the importance of pili controlling the collective behaviors of bacteria^46–49 and also large-scale oscillations in the form of biofilm.

During our recent experiments, we noted striking dynamic patterns and unidirectionally propagating waves on the surface of Pseudomonas Nitroreducens (PN) bacterial biofilms. These unexpected collective activities ranging from spiral to planar wave formations, interestingly aligned with recent predictions of active matter physics^23,50. These activities are reminiscent of metachronal waves observed in various biological systems. We also observed that these collective activities are driven by the coupled pili activity of the bacteria. Recently, large-scale pili-driven activities and spiral waves were also identified in Pseudomonas aeruginosa⁴⁹. The main distinction of our observed metachronal wave lies in its localized behavior on the surface and broken left-right symmetry across the biofilm, which we refer to as the active biofilm carpet. In this study, we aim to provide a comprehensive experimental approach including biological, physical and theoretical tools to be able to understand and control these collective behaviors and also the symmetry breaking process of these biological systems.

Bacterial biofilms develop into multicellular communities characterized by interdependent biological structures. Understanding the dynamics of these dense biological formations is vital, given that their collective actions can amplify their pathogenic potential. The collective movement of pili and its variability among strains offers a crucial foundation for understanding the influence of pili activities on pathogenicity. Moreover, pili is also a primary target protein for next-generation medications and antibacterial therapies. In essence, comprehending the effect of pili dynamics on the collective behavior of biofilms can greatly influence human health by preventing severe infections.

Results

Emergence of metachronal waves on active biofilm carpets

Bacterial biofilms are commonly used in various experiments particularly to investigate host-pathogen interactions. During our recent screen of biofilm library for nematode C.elegans⁵¹, we observed that Pseudomonas Nitroreducens (PN, Materials and Methods) bacterial biofilms exhibit rhythmic activities approximately 10 hours post-inoculation (Figure1a-b). In their firing phase, faint propagating pairs of spiral wave patterns emerged on the biofilm’s surface. These waves have a periodicity of around 2-7 minutes (Figure1c). We also determined that the visibility of these waves improves dramatically with oblique contrast and polarized light-based microscopy (Figure1a-b, Supplementary Video1-2). Later on, these waves were very dynamic and converged to various combinations of spiral, target and planar waves. The most striking feature of these waves is their broken left-right symmetries. They, often unidirectionally propagate from right to left (edge to the center) or radially in inward direction across the colony (Figure1a-b). Furthermore, the contrast of the wave’s switches between the dark to bright sharp lines depending on the direction of propagation. This suggest that the front kink of these waves scatters the light asymmetrically (Supplementary Figure 1). We observed that these waves predominantly propagate on the biofilm’s surface rather than its bulk region (Figure1e). This critical feature is particularly revealed by the nature of our optical imaging systems which is very sensitive to the surface topography. To validate this observation, we employed cytoplasmic GFP for imaging. Yet, no waves were detected in the green fluorescence channel (Supplementary Figure 2). Similarly, when we labeled extracellular DNA with a dye, the waves remained unobserved in the fluorescence signal (Supplementary Figure 2). Moreover, close-up imaging (100X) highlighted the bacterial displacement together with clear optical dark or bright contrast on the surface (Supplementary Video3). Imaging growing biofilm starting from a single bacterium clarifies that oscillations first starts from dense regions and gradually propagate through the entire colony (Supplementary Video4). Further motivated by recent studies^48,49, we also tested the Pseudomonas aeruginosa strain PA14 under the same conditions. Similarly, PA14 exhibited weak but comparable wave patterns, though within a shorter time window (Supplementary Figure 3).

Figure 1.

To delve deeper into these phenomena, our next focus was on the necessary media, given that the original experiments were conducted on C. elegans using Nematode Growth Media (NGM) plates (see Materials and Methods). Interestingly, we found that these activities were exclusive to NGM. Substituting NGM with a standard LB plate did not reproduce the same dynamics; instead, the bacteria grow rapidly. Yet, upon omitting yeast extract and replacing bactotryptone with bactopeptone, a major nutritional component, from the LB media PN resumed its activity (Supplementary Figure 5). These findings underscore the significance of nutrient restriction as a primary factor. Remarkably, these observations are unexpected, but become evident due the use of NGM plate, which is not common in microbiology experiments.

Although PA14 generates similar waves we observed that PAO1, a close relative of PA14, failed to produce waves on both NGM and LB plates (Supplementary Figure 4). Worth noting, PAO1 is also an important human pathogen causing Cystic Fibrosis (CF). Finally, through this comprehensive screening effort, we successfully identified a group of bacteria, offering the potential to examine various factors, from genetics to physical domain, influencing the emergence of these spiral waves. Both PA01 and PA14 possess robust genetic toolkits, while PN strain display a clear and prolonged active carpet state for detailed physical investigations. The other critical difference between all these biofilms is the finger formation around the leading edge. Unlike PAO1, PN and PA14 form very strong instabilities leading unstable extensions (Figure1f).

We then shifted our attention to the genetic distinctions between PN, PA14 and PAO1. Initially, we first explored the contribution of bacterial pili and flagella as an active mechanical structure. We found that PA14:ΔPilB and PA14:ΔPilA mutations eliminated the waves but not PA14:ΔFilK (Supplementary Figure 3). This suggests that activity is exclusively originating from pili dynamics and the original dynamics is similar to the recent work⁴⁹, but activities are localized to the surface. We then explored the potential of hyper piliation, which might amplify the coupling of pili on the biofilm surfaces. Yet, the PAO1 ΔPilH mutant, known for its polarized and hyper piliation condition^46,47, did not manifest the waves but provided only small fluctuations on the surface (Supplementary Figure 4). Then, we pinpointed that a significant variation was in the pili subunit groups⁵². While PA14 possesses group 3 Pilin, PAO1 has group 2 Pilin. It’s essential to highlight that all these bacteria have type IV pili, though they might be categorized into separate groups. Currently, our hypothesis posits that diverse pili types might exhibit varied physical characteristics such as elasticity or rigidity. Our understanding of PN at the genetic level remains limited but it shares strong similarity with PA. Nonetheless, the various genome sequences of PN are accessible. Our analysis of their pili related genomic region revealed a resemblance to the group 4 and 5 pili (Supplementary Figure 6). All these firing pili groups 3 and 4, 5 have additional accessory genes in the genome and they may contribute to the expression or folding of these pilins. Together with the unstable edge fingering (Figure1f), the comparison of PN and PA strains support the idea that group of pili and particularly pili expression play a critical role in driving waves on these bacterial biofilms.

Modelling of propagating spiral waves

To get more intuition about the dynamics of the waves and emergence of the broken symmetry across the biofilm, we next focused on the mathematical modelling. From an active matter perspective, spiral waves can emerge in non-equilibrium and excitable media^53–55 as well as in coupled oscillatory systems with delays⁵⁶. Despite these systems exhibiting similar wave patterns, the internal mechanisms that trigger the waves vary significantly. Recent studies particularly on the concept of nonreciprocity provides comprehensive approach^23,50. It is important to note that coupling between active mechanical units could introduce nonreciprocity into the system due to local symmetry-breaking processes. Inspired from the previous studies on the dynamics of cilia^17,57, and recent robotic system⁵⁸ we proposed that coupled active pili system could also show similar limit cycles oscillations defined by bacterial displacement (U) and the polarization of the bacterial pili (P) (Figure1e). To study these phenomena, we first utilized a minimal phenomenological phase field model that includes a nonreciprocal coupling term. The oscillatory dynamics of the model are driven by oscillatory extension and retraction processes. The Kuramoto based model^23,56,59, which effectively captures the phase dynamics of coupled oscillators, is extended here by incorporating a delay term (α) that disrupts the odd symmetry—representing action and reaction symmetry between interacting oscillators—and enriches the system’s dynamic repertoire. This modified model is described by the equation:

Here, ∅ represents the local phase of individual pili oscillations (The phase of parameter space defined by U and P, Figure 1e). We assume interactions only among nearest neighbors, where w₀represents the uniform intrinsic oscillation frequency of an isolated oscillator. The term α specifically break odd interaction symmetry between pili, facilitating the formation of travelling waves. Note that traveling waves indicate this broken PT symmetry between these fields. It’s important to emphasize that the detailed biophysical mechanisms behind this nonreciprocity remain unclear. Several possibilities, such as force relaxation dynamics or hydrodynamics interactions, could also result in similar nonreciprocal behavior⁶⁰. Our phenomenological model accurately captures the principal features of these phenomena. While reciprocally coupled oscillators (α = 0) tend to achieve a globally synchronized state (Figure 2a), nonreciprocity favors the formation of traveling waves (Figure 2b-c, Supplementary Video5). The excitability term (b) shapes the pulsative nature of the waves (illustrated in Figure 2d-e) by amplifying the effects of specific phases within the coupling mechanism^56,61,62.

Figure 2.

Next, we turned our focus to continuum simulations to capture the essential properties of these dynamic processes (Supplementary Video6). The recently developed active solid model provides a broad and robust framework for studying collective behaviors of interacting active components, spanning from robotic systems to human crowds^58,63. The core idea of this model is based on two nonreciprocally coupled fields: displacement (U), representing bacterial displacement within the biofilm lattice, and polarization (P), representing the orientation of pili, which exert active forces whose direction can be modulated by the displacement field (Figure1d). The dynamics of the active biofilm lattice can be described by coupled equations, where F_e denotes elastic force and D_r represents polarization relaxation. The underlying rationale of these equations is that retracted pili exert active forces on bacteria, which are elastically coupled to the biofilm structure. Local biofilm displacement can guide the growth direction of pili extension, influenced by local deformation. Importantly, since pili are anchored to the substrate, their orientation dynamically adjusts based on extension processes. Our simulations demonstrate that the active solid model successfully captures critical dynamical phenomena, such as the formation of spiral and planar waves (Figure 2 f-g). As anticipated, fields U and P develop a characteristic phase difference, indicative of limit cycle oscillations. Moreover, analysis of vector configurations around defect cores reveals directional reversals of displacement U on opposite sides of defects, highlighting the fundamental mechanism underlying defect formation, where U diminishes, and the phase (∅) becomes undefined.

Controlling the transition between the waveforms on active biofilm carpet

Extensive parameter testing has revealed that our models not only capture the emergence of spiral waves but also predicts transitions into target and plane wave solutions which we commonly observe in grooving biofilms. During the transition to target waves, two oppositely spinning spiral waves with topological charges of +1 and −1 merge, forming a symmetrically expanding target wave. Subsequently, fast oscillating plane waves progressively dominate the entire simulation domain (Figure3 a-c, Supplementary Video5).

Motivated by our simulation results, we hypothesize that our active carpet system could also facilitate similar transitions between different forms of traveling waves^64,65, offering a means to externally control the dynamics of the system and may explain the details of broken symmetries. We observed that adding a water droplet to the biofilm surface led to the transformation of spiral waves into target and planar waves (Figure 3d-f). To recover the spiral waves, we increased the surface temperature by gently heating the system. As expected, with the temperature rise, the leading edge of the planar waves first became noisy, and multiple target waves randomly appeared then finally converge to spiral waves (Supplementary Video7). These findings are in interesting similarity with recent theoretical predictions in different nonreciprocal systems^50,64.

Figure 3.

Additionally, by incorporating the water-soluble polymer polyethylene glycol (PEG) into the biofilm, we managed to not only alter wave propagation patterns but also direct the movement of inward growing target waves. Intriguingly, the addition of small drop of PEG modified the intrinsic oscillation period of the pili dynamics, creating a radial gradient profile largely due to the slow evaporation rate and uneven deposition of PEG on the surface. This inward wave guidance is particularly controlled by frequency gradient (Figure 4a-d). The center has slow oscillations than the edge. This gradient profile is the critical feature to understand the basics of broken symmetry observed in the circular biofilms. In naturally growing colony oscillation frequency is also varying (Figure 4a-b). We hypnotize that this variation is age dependent where the center of the colony is slowing down due to the nutrition or oxygen depletion process. Before delving deeper into this interesting symmetry-breaking process, we investigated how the aging of the biofilm affected the oscillation dynamics. This is because symmetry breaking occurs at late stage where the aging is critical. To do so, we first measured the oscillation periods as the uniform colony aged. We observed that the periods of these spiral waves continuously increased (Figure 5a-b). These results suggest that the intrinsic oscillation dynamics of the Pili decreased, likely due to depleted nutrients or oxygen, during first 24 hours.

Figure 4.

Figure 5:

Further, we examined the growing biofilm starting from a thin inoculation strip, we reproducibly achieved a broken left-right symmetry for plane waves across the colony (Supplementary Video8). The waves propagated from the edges toward the center. This asymmetry became particularly pronounced a few days after inoculation, during which the colony edges continued to grow while the center remained stationary (Figure 6a, b). To finally confirm this observation, we numerically solve the coupled oscillators under varying frequency. Similarly, we found similar broken left-right symmetry where the planar waves propagating towards the slower region of the biofilm center (Figure 6c, d, Supplementary Video9).

Figure 6.

Controlling left-right symmetry breaking

As a final step we focused on how to control the symmetry breaking and dynamics of metachronal waves. We noticed that temperature adjustments also reversibly effect the oscillatory behavior of the waves which increased the oscillation frequency (Figure 7a, b). This suggest that temperature could be also used as a complementary technique to dynamically guide the wave propagation by controlling the frequency gradients. To do so we embedded metallic pipes in the bulk agar plate and tuned the local temperatures (Supplementary Figure 7). Setting different temperatures in these pipes creates temperature gradient (∇T∼10°C/cm). This gradient profile converted spiral waves quickly to planar waves, propagating from warmer (31°C) to cooler (21°C) areas. Similarly, when we switched the temperature profiles wave propagation also reversed (Figure 7c-d). This result indicates that spatial temperature variations can effectively dictate wave propagation directions. These comprehensive observations confirm that the developed frequency gradients of intrinsic Pili oscillations is the critical feature breaking symmetry and controlling the propagation directions of the waves.

Figure 7:

Discussion

Synchronization is a ubiquitous phenomenon in biological systems. A classic hallmark signature of this dynamical process is global synchronization, where all components of the system tend to merge in phase, as observed in fireflies, clocks and metronomes. Hydrodynamics or intrinsic activities of the interactions, however, introduce nonreciprocity into these tightly coupled systems, fundamentally altering the synchronization process. This action reaction asymmetry drives the emergence of large-scale propagating waves, often emerge as spiral waves in two dimensions. Remarkably, diverse systems from proteins in cell membrane to social amoebae exhibit similar dynamics, highlighting the special importance of these waves in biological systems particularly due to their transport capabilities.

From a symmetry perspective, nonreciprocity is the key feature in understanding these phenomena. Recent advancements in active matter physics have bridged various seemingly unrelated fields around this concept. Particularly, non-Hermitian physics, including PT-symmetry, provides a beneficial macroscopic metric for understanding a system’s response through the classical formalism borrowed from quantum physics and optics. Despite the diverse nature of the dynamic processes and the varying types of interactions—from predator-prey dynamics, specific chemical reactions, to elasticity of active metamaterials—the common feature across these systems is to maintain phase difference between the fundamental fields that support limit cycle oscillations enabling collective motility and transport capabilities to the interacting active system.

We have observed that bacterial biofilm surfaces also known as active carpet exhibit similar dynamics due to coupled and cyclic pili extension and retraction activities. Although these nanoscale activities make the biophysical mechanisms complex and demanding to elucidate, the fundamental characteristic of the process remains consistent: the cyclic motion of bacterial pili forms limit cycle oscillations in phase space to disrupt local or action-reaction symmetry among the interacting components.

Using a phenomenological model, we have concentrated on identifying critical parameters that control the system dynamics and synchronization. We have also developed methods to manipulate and control the collective dynamics and propagation of metachronal waves by adjusting these critical parameters. From the perspective of active matter, the controllability of densely coupled active mechanical components is crucial, especially at low Reynolds numbers where controlling turbulence and managing long-range transport is vital. Biological systems can effectively address these challenges and precisely control active and programable transport⁶⁶. A deeper understanding of the underlying principles and critical symmetries governing this controllability could provide new insights into the complexities of biological systems. We speculate that these waves could promote active diffusion of oxygen and DNA. Further biophysical investigations are needed to delve deeper into these fascinating collective behaviors and its biological significance. Furthermore, the dynamical similarities between non-Hermitian physics and PT symmetry and the metachronal waves due to nonreciprocity are remarkable. Finally, we also hypothesize that the concepts developed in no-Hermitian physics could bring new perspective to dissect the complexities of these collective behaviors. Finally, our findings also raise several questions, particularly regarding the biological significance of these waves in the physiology of the biofilm. Rhythmic activities^67,68 play a critical role in bacterial communities. We hypothesize that this synchronization may facilitate rhythmic behavior across the biofilm or enable expression and the transport of specific biomolecules under stressful conditions. Further studies are needed to elucidate the mechanisms underlying this emerging phenomenon.

Materials and Methods

Analysis of Pilin Proteins and Accessory Genes

Genetic analyses were conducted using the complete genome sequence of Pseudomonas nitroreducens strain L4 chromosome⁶⁹ (NCBI Reference Sequence: NZ_CP120376.1). The reference genomic region between pilB and tRNA-thr was specifically compared.

Imaging System

Time-lapse imaging was performed using multiple microscopy techniques, selected for their capability to visualize surface topography effectively. We observed that oblique and polarized illumination techniques significantly enhanced the visibility of surface waves. During the evaporation process, the roughness of the biofilm significantly varies. During the evaporation process, the roughness of the biofilm varies significantly. We observed that oblique illumination is more effective on rough surfaces, whereas polarized imaging provides clearer results on flat surfaces. Oblique imaging was primarily conducted using Zeiss Smartzoom digital microscopes and Nikon Stereo SMZ18 microscopes equipped with contrast illumination. Additional polarized imaging utilized a Zeiss Axio Imager M2m. Successive images were captured at intervals of 10 seconds. Fluorescence imaging for GFP was conducted using 488 nm blue-light excitation with GFP emission filters, and DNA staining was visualized using RFP emission filters and imaged with an Andor EMCCD camera. The typical optical magnification was 10X.

Image Processing

To enhance wave visibility, successive images taken at 10-second intervals were subtracted, and the resulting false-colored difference images were superimposed onto the original ones. The time-dependent responses of surface pulses were analyzed using ImageJ software.

Bacterial growth and Preparation

The Pseudomonas nitroreducens (PN) bacterial strain used in this study was initially isolated from environmental samples and identified through 16S rRNA sequencing. Verification was performed through comparison with the corresponding strain obtained from DSMZ. For optical imaging, PN bacteria were grown overnight in LB broth at 21 °C on a shaker, and subsequently, 100 µl drops of bacterial suspension were placed onto 10-cm, 1-day-old, 1.4% solid agar, NGM plates. To ensure slow evaporation of excess moisture, humidity was maintained at approximately 80%, and plate lids were partially left open. Nematode growth medium (NGM) plates were prepared following standard protocols.

Numerical Simulation

Numerical simulations based on the Kuramoto model were performed using custom-developed MATLAB code with finite difference algorithms. The original code for the active solid model was obtained from prior literature and modified for improved visualization. Through extensive parameter exploration, we determined that large lattice structures with fixed boundary conditions effectively reproduced spiral and planar wave dynamics.

Data availability

The critical experimental data generated or analyzed during this study are provided as supporting video files. We did not generate additional data sets.

Software availability

All the codes used in the study will be available online.

Supplementary Figures and Tables

Supplementary Figure 1.

Supplementary Figure 2.

Supplementary Figure 3.

Supplementary Figure 4.

Supplementary Figure 5.

Supplementary Figure 6.

Supplementary Figure 7.

Table 1:

Acknowledgements

This work was supported by Airforce Office of Scientific Research (AFOSR) under Grant No: FA9550-22-1-0431 with Program Manager Ali Sayir and Tübitak Project No: 121C159. S.K.O acknowledges the support from Air AFOSR Multidisciplinary University Research Initiative (MURI) Award No. FA9550-21-1-0202. We thank Zeiss, Turkey division and Kenan Doğru for digital microscopy installation and training.

Additional information

Author contributions

B.A. and A.Ko. initiated the project and performed the experiments. B.A., Y.I.Y., and M.B. Y.G. developed and executed numerical modeling. B.K. conducted 16S DNA sequencing and GFP labeling of PN. N.G. performed genome sequence analysis. E.T.G. and A.Ü. developed and conducted imaging for the temperature gradient control system. S.K.Ö. and C.K. conceptualized non-Hermitian and PT-symmetry principles. A.Ko. and B.A. drafted the manuscript, with contributions from all authors toward the final manuscript.

Additional files

Video 1. Time-lapse imaging of active Pseudomonas nitroreducens biofilm surfaces generating metachronal waves. Global left-right symmetry is broken, and merging spiral waves at the edge drive unidirectional planar waves toward the biofilm center. The wave wavelength is approximately 100 µm. Associated with Figure 1a.

Video 2. Time-lapse imaging of circularly symmetric active biofilm surfaces generating inward-propagating metachronal waves. Associated with Figure 1b.

Video 3. High magnification (100X) time-lapse imaging of biofilm surfaces illustrating bacterial displacement and optical contrast changes.

Video 4. Time-lapse imaging of biofilm growth initiated from a single bacterium. After growth cessation, dense regions begin firing first, and waves dominate the biofilm surface.

Video 5. Numerical simulation using the nonreciprocal Kuramoto model. Colors indicate oscillation phases. Spiral waves merge to form target waves, eventually converging into planar waves. Associated with Figure 2c.

Video 6. Numerical simulation based on the active solid model. Emergence of spiral and planar waves within the lattice. Associated with Figure 2e.

Video 7. Time-lapse imaging of controlling the transition between waves to target and spiral by manipulating the moisture of biofilm surface. Associated with Figure 3d-g

Video 8. Time-lapse imaging of a bacterial biofilm with broken left-right symmetry. Associated with Figure 6.

Video 9. Time-lapse imaging of controlling propagating planar waves under a varying temperature gradient. Associated with Figure 7.

Video 10. Numerical simulation of propagating planar waves influenced by a frequency gradient. Associated with Figure 7c.