In a fluid environment, biofilms usually form and grow into streamers attached to solid surfaces. Existing research on single streamers studied their formation and failure modes. In the experiment on biofilm growth in a microfluidic channel, we found that rings composed of bacteria and an extracellular matrix are important elements on a mesoscopic scale. In the fluid environment, the failure of these ring elements causes damage to streamers. We simulated the growth and deformation of the ring structure in the micro-channel using multi-agent simulation and fluid–structure coupling of a porous elastic body. Based on this, we simulated the biofilm evolution involving multi-ring deformation, which provides a new length scale to study the biofilm streamer dynamics in fluid environments.

  • We found that the mesoscopic element of the biofilm streamer formed in the micro-channel.

  • The failure of the ring element is the cause of damage to the streamer.

  • We used multi-agent simulation and fluid–structure interaction of a porous elastomer to simulate the deformation of a single ring element in the fluid environment and consider its growth.

In our previous research, we analyzed the growth process of Bacillus subtilis biofilms in a fluid environment, which is the minimal salt glutamate glycerol (MSgg) solution flowing through the micro-channel at a certain velocity. The biofilm formed a streamer structure in the fast flow, which is a filamentous structure composed of a few bacteria. A streamer is a viscoelastic filamentous biofilm structure, which is a special form of a biofilm in a fluid environment. We divided the streamer growth into several stages according to its characteristics, including the biofilm streamer initiation, floating stage, transverse growth stage, weak and strong attachment growth stages, failure, and drift (Zhang et al. 2022). In the further experimental study, we observed many ring structures that ran through the whole biofilm streamer growth process in the microfluidic channel. In the initial growth stage, one end of the streamer was attached to the micro-obstacle settled in the channel, and the other was free-floating. In the subsequent growth, no matter at which stage and in what form, the biofilm was composed of complex ring elements; accordingly, the biofilm streamer evolution resulted from the growth and deformation of multi-ring elements. When the biofilm was torn, the ring element transformed into a band; at this time, the biofilm streamer was made up of rings and bands. The components of the ring element included cells in different phenotypes (motile cells, matrix-producing cells, and spores) and an extracellular matrix.

In the existing experimental work, few researchers studied streamer formation at the microscopic level. Valiei et al. (2012) found that the initial viscoelastic filamentous drift band in the micro-column array is formed by extracellular polymer substance (EPS) chains and bacteria are embedded in the EPS matrix. Others studied the streamer at the macroscopic level; in microfluidic channels, the biofilm streamer structure can extend significantly with the flow, thereby spanning disconnected surfaces and favoring rapid bacterial colonization (Ghosh et al. 2021). Drescher et al. (2013) found that the sieve-like network formed by a biofilm streamer captures the flowing cells, which will cause severe catastrophes in the channel. Rusconi et al. (2010, 2011) studied the streamer pattern dependence on the microfluidic channel geometries. Biswas et al. (2016, 2017, 2018a) proposed two failure modes of different streamer structures, one growing close to the pillars of microfluidic devices and the other growing away from the wall, which were caused by different causes. Hassanpourfard et al. (2015) reported that the clogging phenomena that occur in the device are due to localized streamer failure and leakage. According to the study of Marty et al. (2012), Biswas et al. (2018b) and Drescher et al. (2013), in the low-Re regime, unlike slow-growing biofilms, the streamer structure can lead to catastrophic clogging of curved channels and filter membranes. Compared with their observation, the ring elements of the streamer in our experiment are of the size between the microscopic and macroscopic levels, which is easy and direct to be observed and used to theoretically explain the streamer evolution.

Currently, Rodriguez et al. (2014) established a two-dimensional cellular automata model to simulate the growth of biofilms in a fluid environment and studied the influence of Reynolds number on the growth process, assuming that grid tiles are bacterial-sized individual units representing water, biomass, and substratum materials. Van Gestel et al. (2015) established the deformation model of a bacterial single chain by setting the rules between bacteria and simulated the ring structure at the edge of biofilms grown on the solid substrate. Others' research is carried out from a macro perspective. Towler et al. (2007) established a calculation model by using the linear viscoelastic burger constitutive relationship to solve the physical problems of fluid–structure interaction (FSI). Taherzadeh et al. (2010) used a numerical FSI framework to simulate streamer oscillation. The above theoretical works operate under the assumptions either of micro-structures like cellular components or within the continuum macroscopic framework, lacking the physical images of structures at small scales.

By choosing appropriate mesoscopic structure elements, researchers have successfully simulated the integral structure deformation. For example, Cox et al. (2018) chose the single bubble as their study element and found that dislocation is periodically reflected in the tension or compression of two independent rearrangement regions of a monodisperse foam matrix. By studying single carbon nanotubes, Magnin et al. (2021) proposed a simple theoretical model to determine the variation of the stable region of bunch carbon nanotubes based on their diameter and the number of walls. Similarly, because the scale of the ring microstructure lies between that of the micro-cell and the macro-biofilm, we can explore the deformation behavior of the ring element at the mesoscopic scale and simulate the overall growth and deformation of the biofilm streamer.

Microfluidic experiments

In this study, we adopted an experimental method similar to our previous research, including the same microfluidic device made of soft lithography, as shown in Figure 1, the same bacteria (B. subtilis), the same medium formula, and the injection speed.
Figure 1

(a) Schematic of the microfluidic device. (b) Enlargement of the channel area with a micro-obstacle. The blue arrow represents the flow direction. Dimensions of this part are D1 = 100 μm, D2 = 320 μm, W = 200 μm, and H = 20 μm.

Figure 1

(a) Schematic of the microfluidic device. (b) Enlargement of the channel area with a micro-obstacle. The blue arrow represents the flow direction. Dimensions of this part are D1 = 100 μm, D2 = 320 μm, W = 200 μm, and H = 20 μm.

Close modal

Biofilm growth

Bacillus subtilis 3610, which is one of the most studied Gram-positive bacteria and is widely used to study biofilm growth, was used in this study. Biofilms were cultivated in microfluidic channels by feeding the MSgg solution at OD600 = 0.4 with a flow rate of Q = 10 μL/h in microfluidic channels. MSgg comprised 5 mM potassium phosphate (pH 7), 100 mM MOPS (pH 7), 2 mM MgCl2, 700 μM CaCl2, 50 μM MnCl2, 50 μM FeCl2, 1 μM ZnCl2, 2 μM thiamine, 0.5% glycerol, 0.5% glutamate, 50 μg/mL tryptophan, 50 μg/mL phenylalanine, and 50 μg/mL threonine. For the complete bacterial growth experiment, refer to our previous work (Liu et al. 2022).

FSI mechanics

The shape change of a deformed object in a fluid interacts with the forces it receives, including the friction and pressure added by the fluid. That is, in addition to the force on the object being determined by its shape, the force on the object also determines the shape of the object (Vogel 2020). The growth of a biofilm streamer is usually related to the drag force of the fluid on the streamer and the viscoelastic properties of the streamer itself (Shaw et al. 2004).

In our study, the timescale of the deformation process caused by fluid drag was very short compared with the transition time required for the biofilm to show viscous flow. We ignored the viscous fluid properties of the biofilm and directly described its deformation through large deformation elastic dynamics (Shaw et al. 2004).

A biofilm is a porous body with fluid inside. The pressure gradient in the pores of the deformable biofilm leads to the slow flow represented by the linear seepage law.

We used the FSI module and porous module in COMSOL to simulate the deformation of the ring structure of the biofilm in a fluid environment due to fluid drag. The FSI module includes the fluid flow module and the solid mechanics module. The fluid module selects the peristaltic flow module because of the small Reynolds number. The FSI model we used refers to the model used by Picioreanu et al. (2018) for predicting the elastic modulus of biofilms.

Fluid flow outside the biofilm: for the fluid region outside the streamer region, we used the momentum equation and mass conservation equation of the incompressible Navier–Stokes equation.
formula
(1)
formula
(2)

The condition assumed in the simulation is that the distance between the inlet and outlet of the microflow pipeline and the experimental process is far enough, and the flow field generated when the fluid enters and exits the pipeline will not affect the area involved in the experiment. Therefore, the laminar flow with the average velocity given in the experiment is set at the inlet in this paper. The pressure set at the outlet is constant p = 0. The fluid does not slip at the boundary where there is no bacterial biofilm growth.

Fluid flow in the streamer: due to the slow flow caused by the pressure gradient in the pores inside the streamer domain, Darcy's law with fluid mass conservation is set in the streamer domain.
formula
(3)
formula
(4)

The pore fluid pressure gradient derived from the porous deformation structure is associated with the permeability of the bacterial biofilm and the flow velocity of the bacterial biofilm domain, and the flow velocity in the two domains is the same due to the continuity of fluid inside and outside the boundary between the bacterial biofilm domain and the fluid domain (fluid–solid coupling boundary). The continuity of the fluid makes the pressure on both sides of the fluid–structure coupling boundary the same pb = pf.

Streamer mechanics: inside the streamer, we use the Navier equation of equilibrium solid to change the deformation of the biofilm matrix. This equation can ignore the deformation outside the two-dimensional plane.
formula
(5)
formula
(6)

Bacterial biofilms exhibit elastic deformation under short-term stress, while it exhibits viscous flow under long-term stress. However, the single step of deformation caused by drag during the deformation of the bacterial biofilm strip structure studied in this paper is much shorter than the transition time from elastic deformation to viscous flow of the bacterial biofilm, which is 20 min. Therefore, we ignored the viscous flow of the bacterial biofilm and only considers its elastic behavior.

At the same time, the fluid pressure inside the bacterial biofilm needs to balance with the pressure in the fluid domain so the porous structure of the bacterial biofilm cannot be ignored. Therefore, the stress in the bacterial biofilm should include the porous elastic and isotropic single-line parts. Therefore, Equation (7) is substituted into Equation (6):
formula
(7)
In our experiment, we injected the MSgg solution containing B. subtilis into a microfluidic channel at a constant flow rate of 10 μL/h. The biofilm appeared at the leftmost end of the micro-column within 5 min, and then bacteria proliferated and differentiated rapidly, which made the biofilm larger (Liu et al. 2022). Also, we observed many similar structures containing numbers of bands and rings in the growth of the B. subtilis biofilm streamer, as shown in Figure 2(a)–2(l) , which are snapshots of the streamer at different time points after streamer initiation. At 3 h, the biofilm showed the ring structure as the cube oblique view, as shown in Figure 2(a). After 10 min, the shape of the oblique view of the cube became the structure of the outer ring, wrapping the inner ring under the drag of the fluid, as shown in Figure 2(b). At 4 h and 25 min into the streamer growth process, cell proliferation induced more rings, including bigger rings; interactions between these rings generated ring deformation, accumulation, and motion, as shown in Figure 2(c). At 11 h and 5 min, the biofilm band came up due to the fluid flow tearing the streamer, as shown in Figure 2(d). The fluid flow continuously tore the streamer until 16 h and 5 min, and the remaining part of the streamer crawled and grew in a band structure, as shown in Figure 2(e). Another newly formed streamer underwent the crawling growth process at 17 h and 40 min, as shown in Figure 2(f). After 1 day and 17 h growth, the streamer formed a dense structure that strongly adhered to the obstacle and the wall; the band formed at the dense structure peripheral, as shown in Figure 2(g). After 3 days and 17 h, two obvious ring structures with half the width of the channel appeared downstream of the newly formed biofilm, as shown in Figure 2(h). More and more rings appeared with the streamer growth; the growth time was 3 days and 20 h, as shown in Figure 2(i). With fluid force getting larger and larger, the dense structure of the streamer was destroyed eventually, only a piece of the dense structure remained, and a new biofilm in the ring structure grew around this piece, as shown in Figure 2(j); the growth time was 5 days and 13 h. At 6 days and 2 h, foam structure with more rings formed, as shown in Figure 2(k). After the dense structure totally disappeared, the streamer started a new growth cycle at 8 days and 3 h, as shown in Figure 2(l). We noticed that the bacteria are preferentially accumulated in the bottom part of figures. Since the experimental micro-channel was placed horizontally, we did not discuss the influence of gravity and considered it to be the randomness of bacterial attachment and growth.
Figure 2

Ring elements involved in different biofilm growth stages in a fluid environment.

Figure 2

Ring elements involved in different biofilm growth stages in a fluid environment.

Close modal

Simulation of the bacterial single ring

From the above images, we can see that the ring structure runs through the whole streamer growth process. To illustrate the ring growth and deformation, we chose a period of growth involving the whole ring evolution cycle, as shown in Figure 3. There was a ring with two ends fixed to the obstacle. Initially, the ring was sharply tilted to the left, as shown in Figure 3(a); then, the ring gradually changed from b to f due to the combined effects of fluid flow and its own growth.
Figure 3

Single ring growth and deformation process; the time interval is 15 min.

Figure 3

Single ring growth and deformation process; the time interval is 15 min.

Close modal

The deformation of the ring structure in a fluid environment is a combination of two deformation processes caused by fluid drag and its own growth. We divided the coupling of these two processes into two parts, namely, growth deformation and deformation caused by fluid drag, to simulate the deformation process of the biofilm ring structure in the microfluidic channel.

Growth deformation simulation

At the edge of the biofilm cultured on solid medium, Van Gestel et al. (2015) found a ring structure on the edge of the B. subtilis biofilm. In their study, they likened the bundle structure, resembling Van Gogh's painting “Starry Night,” to a “Van Gogh bundle,” and they divided the growth process of the Van Gogh bundle into three processes: growth, division, and rotation. We adopted this growth mechanism in our simulations, as shown in Figure 4.
Figure 4

Growth and deformation rules of the Van Gogh bundle.

Figure 4

Growth and deformation rules of the Van Gogh bundle.

Close modal
Cells grow as end-to-end chains. There are three main processes: (A) growth, the cells elongate in length and squeeze with the cells in the growth direction to produce rotation; (B) division, when a cell grows to a certain length, it divides into two cells of equal length; and (C) turning, we take three cells as a group, give one cell a rotation based on the random angle of normal distribution, and make the other two cells complete the movement. We determined whether rotation occurs by calculating the increase or decrease of energy. If the energy decreases, rotation occurs. The energy calculation formula is as follows:
formula
(8)
These three processes are not required to be carried out in sequence, and turning is also carried out before growing to the division length. Different processes are also carried out on different cells at the same time. We simulated several initial cases in our experiment by using the rules of Van Gogh's bundle, and the simulation results are shown in Figure 5.
Figure 5

Three types of single ring growth deformations observed in experiments (a, c, e), and the corresponding simulation results (b, d, f); the time intervals in (a, c, and e) are 10, 15, and 20 min, respectively.

Figure 5

Three types of single ring growth deformations observed in experiments (a, c, e), and the corresponding simulation results (b, d, f); the time intervals in (a, c, and e) are 10, 15, and 20 min, respectively.

Close modal

In the first case, the linear slender streamer expanded and bent, as shown in Figure 5(a) and 5(b) (experiment and simulation, respectively). In the second case, the ring structure of the biofilm exhibited a cross shape, as shown in Figure 5(c) and 5(d) (experiment and simulation, respectively). The biofilm in the microfluidic channel is not suitable to be simply regarded as a two-dimensional structure in all cases. Due to the thickness of the microfluidic channel, there is a height when the bacteria and their secreted matrix adhere to the micro-obstacle, although the streamer of the biofilm is generated at half the height. However, in the later stage of growth, a large number of band structures of biofilm occupy the whole height; even a single ring can be located in different planes, which probably causes ring cross. In the third case, the folded ring expanded, as shown in Figure 5(e) and 5(f) (experiment and simulation, respectively).

Deformation caused by drag force

In this paper, the FSI module and the porous module in COMSOL were used to simulate the deformation of the ring structure in the single micro-obstacle channel. Because the channel was completely symmetrical both above and below and the deformation of the ring structure only occurs on one side of the channel, we simulated half of the channel. The fine circular area was taken as the streamer domain, and the two ends of the ring were fixed to the micro-obstacle. The simulation results are shown in Figure 6. In this process, the ring structure moved to the downstream direction and flattened, which is consistent with the phenomenon observed in Figure 3(d)–3(f).
Figure 6

Ring deformation caused by fluid drag.

Figure 6

Ring deformation caused by fluid drag.

Close modal

Coupling deformation simulation

The specific process of deformation alternates between growth deformation and deformation caused by fluid drag. First, the single ring is divided into pieces according to the bacterial size, and the divided bacterial angle is input into the NetLogo program for growth deformation simulation. After the growth simulation, the shape of the result is introduced into the single micro-obstacle channel of COMSOL for FSI simulation to redefine the liquid environment and solid domain; then, the simulated ring structure undergoes dividing again. We repeated the division and growth simulation in NetLogo and deformation simulation in COMSOL alternately. The finite-element mesh was created according to the subdomain and boundary size, and as the length and shape of the ring elements changes, the number of finite elements also varied between approximately 7,000 and 9,000 elements. The coupling simulations are shown in Figure 7. The simulation results are consistent with the phenomena observed in Figure 3(a)–3(f), which shows that this model is reasonable.
Figure 7

(a) Initial state of the ring element. (b–i) Subsequent deformation under the growth and fluid drag.

Figure 7

(a) Initial state of the ring element. (b–i) Subsequent deformation under the growth and fluid drag.

Close modal

Deformation of several multi-ring structures

In the process of streamer growth in a microfluidic channel fluid environment, a very independent single ring structure is very rare; instead, the whole biofilm is always made up of a large number of rings and bands, whose deformation represents the streamer evolution, as shown in Figure 8.
Figure 8

Three groups of multi-ring structure evolution in different growth periods; the time intervals in (a–c) are 10, 5, and 15 min, respectively.

Figure 8

Three groups of multi-ring structure evolution in different growth periods; the time intervals in (a–c) are 10, 5, and 15 min, respectively.

Close modal
We used the FSI simulation to study the deformation process of the multi-ring structure. To simplify the model, we evenly distributed the multi-ring structure in rows and set the contact part between the ring structure and the bottom of the channel as a fixed boundary. The initial situation and steady-state results of the simulation are shown in Figure 9(a) and 9(b). This shows that the displacement of the upper ring structure is large, and the lower ring structure has large deformation due to the superposition of the resistance of each layer. We found that the newly grown biofilm was thinner. In the simulation of the multi-ring structure, the difference in thickness affects the change in Young's modulus. The higher the layer, the smaller the Young modulus of the ring. The simulation results after changing the Young's modulus distribution are shown in Figure 9(c). After we modified the Young's modulus of each row of the ring structure, the deformation and displacement of the upper ring increased. According to the simulation results, the original circular ring structure deformed and folded under the action of fluid drag. If the deformation continued in this trend, it led to dense slender bands, as shown in Figure 9(d), which was observed in our experiment.
Figure 9

Simulation of uniform multi-ring structure deformation (a–c), and the following deformation of multi-ring structure after further folding and extrusion was observed in experiment.

Figure 9

Simulation of uniform multi-ring structure deformation (a–c), and the following deformation of multi-ring structure after further folding and extrusion was observed in experiment.

Close modal
We also observed ring failure in the deformation of the multi-ring structure, as shown in Figure 10(a) and 10(b). Different from the preceding continuous deformation, as shown in Figure 9(d), we observed that a multi-ring structure changed greatly between the two adjacent frames with an interval of 5 min during the deformation, and the rings in red color suddenly shifted along the flow direction when the blue ring claps, as shown in Figure 10(a) and 10(b).
Figure 10

(a, b) Ring structure (red) deformation process with the failure of one of the rings (blue). (c, d) Simulation about the multi-ring structure deformation removing the supporting ring. (e, f) Simulation about the multi-ring structure deformation with the supporting ring.

Figure 10

(a, b) Ring structure (red) deformation process with the failure of one of the rings (blue). (c, d) Simulation about the multi-ring structure deformation removing the supporting ring. (e, f) Simulation about the multi-ring structure deformation with the supporting ring.

Close modal

In this multi-ring structure, the blue ring is believed to play a supporting role so that the whole multi-ring structure does not have large deformation under the fluid drag. When the adhesion between the supporting ring and the channel wall becomes weak, resulting in the ring falling off, or when the supporting ring itself breaks, the rest of the multi-ring structure undergoes large deformation under fluid drag.

To validate whether ring failure is the reason causing the large deformation in Figure 10(a) and 10(b), we simulated two deformation situations through the finite-element method. In one case, the simulation is about the multi-ring structure deformation after removing the supporting ring, as shown in Figure 10(c) and 10(d), which is consistent with the experiment observation in Figure 10(a) and 10(b). In the other case, the simulation is about the multi-ring structure deformation with the supporting ring, as shown in Figure 10(e) and 10(f). As expected, the deformation is largely decreased.

In the multi-ring structure, due to the uneven growth of the streamer and fluid drag, the failure occurs not only on the supporting ring but also on other rings in the multi-ring structure. The sudden deformation caused by the fracture or failure of one ring in the support ring or other rings can be simulated by FSI.

Biofilms in a fluid environment usually begin to grow in the form of a streamer, and in the subsequent growth, their growth is achieved by the combination of growth and deformation of the ring structure. Valiei et al. (2012) found that the streamers did not form a permanent structure, and the fluid stress played an unstable role in the formation of streamers. The formation of streamers may be precursors to mature microbial structures found in porous media. Our study also provided a theoretical basis for the removal of biofilms. People can prevent biofilm growth by changing the flow and destroying the stability of biofilm microstructure.

In this study, first, we used multi-agent simulation software NetLogo and multi-physical field simulation software COMSOL to simulate the ring element deformation under fluid flow considering the cell proliferation. Then, we used COMSOL to simulate the deformation of multi-ring structure in the flow field. The conclusions are summarized as follows:

  • (1) We found the mesoscopic element of the biofilm streamer formed in the micro-channel, which is the ring element composed of cells and an extracellular matrix. The ring structure runs through the whole streamer growth, and the streamer evolution is the growth and deformation of the multi-ring structure.

  • (2) The failure of the ring element is the cause of the damage to the streamer. There are two types of streamer damage: one is a large deformation in the streamer due to the failure of the supporting ring element, and the other is the tearing and falling off of the streamer due to the failure of one or more rings among the ring structure.

  • (3) To verify how the ring element contributes to the streamer dynamics, we used multi-agent simulation and FSI of the porous elastomer to simulate the deformation of a single ring element in the fluid environment and considering its growth, as well as the multi-ring structure with and without the supporting ring failure under the fluid flow.

In this paper, we simulated the growth and deformation of biofilm ring elements in two dimensions. However, the simulation of the failure of the multi-ring structure of the biofilm was not perfect. In the future, we should explore the effect of microstructure on the stability of biofilm at the spatial scale.

The authors thank Professor David A. Weitz and Professor Shmuel Rubinstein from Harvard University for their experimental support.

This work was supported by the National Natural Science Foundation of China (Grant numbers 11972074, 11772047, 11620101001, and 12372321).

All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by XW, ZZ, YT, CT, JZ, FD, SL, and DZ. The first draft of the manuscript was written by JZ, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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