Viscoelasticity enhances collective motion of bacteria — TEI-tagged view

Wentian LiaopersName, Igor S AransonpersName (2023date). “Viscoelasticity enhances collective motion of bacteria.” PNAS Nexustitle·j.

Object of study — collective swimming of concentrated Bacillus subtilisterm·key in synthetic and natural mucusterm·key and in a Newtonian polymer solution

Research question — how a viscoelastic, non-Newtonian medium such as mucusterm·key changes the collective motionterm·key of bacteria

Contribution — evidence that fluid elasticityterm·dom, not viscosityterm·dom, enlarges the correlation lengthterm·mth of bacterial collective motionterm·key, backed by a phenomenological model

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Viscoelasticityterm·key enhances collective motionterm·key of bacteria Abstract Bacteria form human and animal microbiota. They are the leading causes of many infections and constitute an important class of active matterterm·dom. Concentrated bacterial suspensions exhibit large-scale turbulent-like locomotion and swarming. While the collective behavior of bacteria in Newtonian fluids is relatively well understood, many fundamental questions remain open for complex fluids. Here, we report on the collective bacterial motion in a representative biological non-Newtonian viscoelastic environment exemplified by mucusterm·key. Experimentsterm·mth are performed with synthetic porcine gastric mucusterm·key, natural cow cervical mucusterm·key, and a Newtonian-like polymer solution. We have found that an increase in mucin concentration and, correspondingly, an increase in the suspension’s elasticityterm·dom monotonously increases the length scale of collective bacterial locomotion. On the contrary, this length remains practically unchanged in Newtonian polymer solution in a wide range of concentrations. The experimental observations are supported by computational modeling. Our results provide insight into how viscoelasticityterm·key affects the spatiotemporal organization of bacterial active matterterm·dom. They also expand our understanding of bacterial colonization of mucosal surfaces and the onset of antibiotic resistance due to swarming.

Mucusterm·key, a gel-like viscoelastic substance, is essential for many biological functions. Mucusterm·key lines the surfaces of cells and tissues. It is permeable to oxygen and nutrients and protects against pathogens such as bacteria, fungi, and viruses. Understanding bacterial motility in mucus-like fluids provides insights into bacteria-born infections, including sexually transmitted and gastric diseases. This work demonstrates that mucusterm·key viscoelasticityterm·key enhances bacterial organization, leading to the emergence of coherently moving bacterial groups. The results shed light on how viscoelasticityterm·key controls the spatiotemporal organization of bacterial communities and provide insight into controlling and preventing bacterial invasion of mucosal surfaces.

Introduction Bacteria are the most abundant species on Earth. They compose human and animal microbiota and are sources of many infectious diseases ( 1 ). Suspension of motile bacteriaterm·dom is an important class of active matterterm·dom: nonequilibrium systems transducing energy from the environment into mechanical motion. Concentrated bacterial suspensions often exhibit large-scale turbulent-like motion (so-called bacterial turbulence) ( 2 , 3 ). While significant knowledge is accumulated on the collective dynamics of bacterial suspensions in Newtonian liquids, complex fluids are mostly “terra incognita” ( 4 ).

Bacterial habitats are not limited to Newtonian fluids such as water. Bacteria thrive in non-Newtonian environments ( 4 ) exemplified by lyotropic liquid crystals ( 5 , 6 ), mucusterm·key ( 7 , 8 ), DNA solutions ( 9 , 10 ), blood ( 11 ), saliva ( 12 ), biofilm matrices ( 13 ), or mammalian extracellular matrices ( 14 ). Unlike the simple viscous response in a Newtonian liquid, the non-Newtonian rheology of biological viscoelastic fluid can drastically alter the bacterial collective motionterm·key ( 15 , 16 ). However, there is no consensus even on the level of individual motility of microorganisms in viscoelastic fluids. For example, some studies predicted that viscoelasticityterm·key hinders self-propulsion ( 17 , 18 ). On the contrary, other investigations suggested that the microswimmers propel faster or slower in the viscoelastic fluid but slower in a shear-thinning fluid ( 19 , 20 ). Alternatively, computational modeling ( 21 ) predicted increased bacterial swimming speed up to 60% in polymer solutions due to a nonuniform distribution of polymer molecules around a bacterium. Also, computational studies revealed that viscoelasiticity significantly affect collective motionterm·key of bacteria-like rod-shaped swimmers (pushers) but has a small effect on a suspension of algae-like swimmers (pullers) ( 15 ). Thus, examination of collective and individual dynamics of bacteria in viscoelastic fluids may produce new insights into active matterterm·dom in complex environments.

Among biological fluids, mucusterm·key, a gel-like viscoelastic substance lining the exposed organism tract, is a crucial habit of bacterial communities. The dominant glycan protein, mucin, forms a mesh network and hence endows mucusterm·key elasticityterm·dom ( 22 ). Mucusterm·key demonstrates non-Newtonian dynamics: shear-thinning, stress relaxation, yielding behavior ( 23 ), and even some liquid crystallinity ( 7 , 24 ). Most studies focus on the penetration of individual bacteria or sperm across mucusterm·key. The individual motility of bacteria such as Helibactor pylori , Escherichia coli , and Bacillus subtilisterm·key in mucusterm·key has been studied to better understand both the defensive mechanism of mucusterm·key and the invasion strategy of bacteria ( 7 , 8 , 25 ). The motility of individual sperm in cervical mucusterm·key has also been investigated to understand the mechanisms of infertility ( 26 ). However, the collective states of concentrated bacteria in mucusterm·key, which is relevant in the context of emerging antibiotic resistance ( 27 , 28 ) and colonization of mucosal surfaces in intestinal infection ( 29 ), is mostly unexplored.

Here, we examine the collective motionterm·key of concentrated bacteria Bacillus subtilisterm·key in synthetic porcine gastric mucusterm·key and natural cow cervical mucusterm·key. We contrasted the collective behavior in viscoelastic mucusterm·key with essentially Newtonian polyvinylpyrrolidone (PVP360term·key) polymer solution at a wide range of concentrations. We have found that the velocity correlation lengthterm·mth of bacterial collective locomotion increases monotonously with the increase of mucin concentration in both synthetic mucusterm·key and natural mucusterm·key. The elevation of mucin concentration increases both storage and loss modules which is consistent with a previous observation ( 23 ). On the contrary, the correlation lengthterm·mth of bacterial motion in the PVP360term·key solution remained practically unchanged regardless of PVP360term·key concentration, consistent with the correlation lengthterm·mth predictions for Newtonian fluid ( 30 ). The study implies that the spatial correlation of bacterial active matterterm·dom is primarily affected by fluid elasticityterm·dom rather than viscosityterm·dom. The experimental results are supported by the computational analysis of phenomenological model for bacterial collective motionterm·key in viscoelastic fluids. The result sheds light on manipulating active matterterm·dom by modifying the elasticityterm·dom of the suspending fluid.

Results Experimentsterm·mth were performed with motile aerobic bacteria Bacillus subtilisterm·key , strain DK400. The bacteria were grown in a Terrific Broth (TB) medium (Sigma T5574) and then centrifuged to obtain a condensed bacterial pellet. The pellet was diluted in fresh TB/synthetic mucusterm·key/natural mucusterm·key/PVP360term·key solutions to reach concentrations of the order 1 − 3 × 10 10 cells/cm 3 . Then, the bacterial suspension was pipetted into a free-standing film to ensure sufficient oxygen supply for the respiration of bacteria, Fig. 1 A. A similar setup was used for the experimentsterm·mth on bacterial locomotion in Newtonian fluids ( 30 ). The bright-field or fluorescent images were recorded at the rate of about 33 frames per second and then processed with PIVlab ( 31 ) and custom MATLAB script.

Instant flow patterns of bacterial collective motionterm·key in mucusterm·key and PVP360term·key. A) Schematics of the experimental setup. A free-standing film containing bacterial suspension in mucusterm·key is stretched between 4 movable wires. B) A schematic representation of motile bacteria (blue bodies with flagella) swimming in tunnels formed by mucin polymers (black tubes). The blue (brighter in grey-scale image representation) arrow indicates the swimming direction of mobile bacteria. The trapped bacteria are shown in red. C–E) Select frames illustrating instant flow patterns at the bottom of the film for different concentrations of mucin/PVP360term·key; black arrows depict the direction and magnitude of bacterial flow, and colors show the vorticity w of bacterial flow. C) Instant flow pattern in 50 mg/mL mucin solution. D) Instant flow pattern in 200 mg/mL mucin solution. E) Instant flow pattern in 125 mg/mL PVP360term·key solution. The scale bar in all figures is 80 μ m.

Synthetic mucusterm·key, fabricated by mixing purified porcine stomach mucin powder (M1778 Sigma-Aldridge) with TB, was used in the experimentterm·mth. The mucin solution’s most profound feature is the deformable polymer network which endows mucusterm·key with viscoelasticityterm·key, Fig. 1 B. Synthetic mucin solution preserves most of the rheological properties of fresh mucusterm·key that is hard to acquire consistently in large quantities ( Supplementary Fig. S1 ). Natural mucusterm·key was collected directly from the cervical cow tract and diluted with broth immediately. Two dilutions with mucin concentrations of 0.9375 and 1.875 mg/mL were used for experimentsterm·mth in the inset to Fig. 2 A; see Methods for determining mucin concentration. For simplification, all mucin solutions mentioned in the following text represents synthetic mucusterm·key unless specified explicitly. Control experimentsterm·mth were conducted with a Newtonian suspension of PVP360term·key Sigma-Aldridge. The measurements were performed in a wide range of mucin and PVP360term·key concentrations.

Properties of bacterial collective motionterm·key in mucusterm·key and PVP360term·key. A) Velocity correlation lengthterm·mth L for different mucin/PVP360term·key concentrations. The correlation lengthterm·mth is normalized on the correlation lengthterm·mth L 0 ≈ 80 − 100 μ m in pure TB. Inset: Correlation lengthterm·mth L for different natural mucusterm·key concentrations. B) The root mean squareterm·mth (rmsterm·mth) velocity of collective motionterm·key V for different mucin/PVP360term·key concentrations. The velocity is normalized on velocity V 0 ≈ 43 μ m/s in pure TB. The rmsterm·mth velocity is compared with the normalized correlation velocity ( r / r 0 , r 0 = L 0 / τ 0 ) defined by normalized correlation lengthterm·mth divided by normalized correlation time. C) Velocity correlation time τ for different mucin/PVP360term·key concentrations. The correlation time is normalized on the correlation time τ 0 ≈ 3 s in pure TB. D) The standard deviation of the vorticity probability distribution σ and the mean vorticity absolute value w ¯ for different mucin/PVP360term·key concentrations. Solid blue squares and red circles (brighter in grey-scale image representation) represent PVP360term·key and mucin concentrations, respectively. In all panels, the symbols depict the measured values, lines are the guides for the eye. Error bars are the standard deviations.

Mucusterm·key microrheological properties are summarized in Discussions and Supplementary Fig. S1 . See also Methods for describing the technique. Overall, mucusterm·key exhibits viscoelastic behavior with the effective elastic constant E monotonously increasing with the increase in the mucin concentration. On the contrary, the PVP360term·key solution shows a negligible elastic response in the entire range of the polymer concentrations. A similar monotonous increase was also observed in macrorheological measurements; see Supplementary Fig. S1 . However, the results have a significant scatter due to the difficulty of uniformly loading of the sample at high mucin concentrations.

To examine the effect of viscoelasticityterm·key on collective behavior, we recorded the image sequences of bacterial motion in viscoelastic mucusterm·key and viscous Newtonian PVP360term·key solutions. We determined the velocity field from the particle-image velocimetryterm·mth (PIVterm·mth) using the PIVlab software. Select results for the velocity fields in synthetic mucusterm·key and PVP360term·key solutions are shown in Fig. 1 C–E. Visual comparison of the velocity field snapshots Fig. 1 C and D (mucin) and Fig. 1 E (PVP360term·key) indicates a drastically different response of the bacterial collective motionterm·key on the increase in the mucin/PVP360term·key concentration. In the concentrated mucin solution (Fig. 1 D), the spatial scale of collective motionterm·key increases by a factor of 2–3 compared to the dilute case (Fig. 1 C). In the PVP360term·key solution, the correlation lengthterm·mth remains practically unchanged regardless of the polymer concentration (Fig. 1 E). See also Supplementary Videos 1–4 .

Further characterization of collective and individual bacterial dynamics is presented in Figs. 2 and 3 . First, we determined the correlations length L dependencies on concentration of mucin and PVP360term·key; the main results are shown in Fig. 2 A. From the experimental velocity fields, we extracted the structure factors and the radial spatial correlation functions K ( r ) . The corresponding velocity correlation lengthterm·mth L was obtained by fitting the correlation function K ( r ) to an exponential function K ( r ) ∼ exp ( − r / L ) + const . The correlation lengthterm·mth (about 100 μ m) at zero mucin concentration (pure TB) is one order magnitude higher than a typical bacteria size (about 0.8 μ m in diameter and 5–7 μ m in length), confirming an emergence of collective behavior. The change of the mucin content in synthetic mucusterm·key from 0 to 200 mg/mL increases the correlation lengthterm·mth L by a factor of two or three, from 80–100 to 250 μ m. At this concentration range, the effective (macroscopic) viscosityterm·dom of mucin solution changes by two orders of magnitude associated with the increase in elasticityterm·dom at a much higher rate as shown in Fig. 4 A.

Characterization of individual bacterial dynamics. A) Trajectories of individual bacteria in pure TB. B) Trajectories of individual bacteria in 100 mg/mL mucin solution. C) The rotational diffusion coefficient of individual bacteria in mucin/PVP360term·key solutions at different concentrations where the solid blue line with solid squares and solid red line (brighter in grey-scale image representation) with solid circles represents measured value in PVP360term·key and mucin solutions, respectively. The blue dashed line with empty squares and the red dashed line with empty circles represent expected values for thermal rotational diffusion in PVP360term·key and mucin solution, respectively. D) Individual bacterial speeds (v) scaled by the speed in pure TB ( v 0 ) in mucin/PVP360term·key solutions at different concentrations where the solid blue line with solid squares and solid red line with solid circles represents PVP360term·key and mucin solution, respectively. Individual bacterial speeds from Ref. ( 34 ) in PVP360term·key polymer solutions are shown with empty black triangles. In all panels, the symbols depict the measured values; lines are the guides for the eye. Error bars are the standard deviations.

Dependence of collective motionterm·key on the elastic properties of mucusterm·key. A) Microrheological measurements of mucin/PVP360term·key solutions at different concentrations. Red circles (brighter in grey-scale image representation) correspond to the effective viscosityterm·dom η (solid line) and elasticityterm·dom E (dashed line) of mucin solutions. Blue solid line with solid squares corresponds to the effective viscosityterm·dom η of PVP360term·key solutions. PVP360term·key is effectively viscous Newtonian fluid so no detectable elasticityterm·dom is observed. The viscosityterm·dom is defined as G ″ ( ω ) / ω at ω = 1 Hz, where G ″ ( ω ) is the loss modulus at the measurement frequency ω. The elasticityterm·dom E is defined as G ′ ( ω ) at ω = 5 Hz, where G ′ ( ω ) is the storage modulus. B) Elasticityterm·dom E determines the correlation lengthterm·mth L . The red solid circles shows the dependence of the correlation lengthterm·mth L in mucin solution. The black solid circles shows the dependence of the correlation lengthterm·mth L for different natural mucusterm·key concentrations. The symbols depict the measured values. Red dashed line is the theoretical prediction, Eq. 7 , L ≈ L 0 ( 1 + κ 1 E ) / ( 1 + κ 2 E ) with E = G ′ taken at ω = 5 Hz. κ 1 , 2 are fitting parameters. Inset: The dependence of the correlation lengthterm·mth L vs. mucusterm·key elasticityterm·dom in linear scale.

Similarly, the correlation lengthterm·mth in natural cow cervical mucusterm·key increases with mucin concentration and reaches 144 μ m at a mucin concentration of around 1.875 mg/mL as shown in the inset to Fig. 2 A. However, reproducible experimentsterm·mth with natural mucusterm·key are challenging. When mucin concentration is above 1.875 mg/mL, the natural mucusterm·key network prevents the homogeneous mixing of bacterial pellets in mucusterm·key without breaking the innate mucusterm·key network. As a result, the experimentsterm·mth with natural mucusterm·key at a mucin concentration higher than 1.875 mg/mL are very sensitive to the preparation details and therefore there are only two data points for natural mucusterm·key in the inset to Fig. 2 A. An example of bacterial collective motionterm·key in natural mucusterm·key at a concentration higher than 1.875 mg/mL (3.75 mg/mL) is shown in Supplementary Video 5 with a spatial scale of ∼200 μ m, which is close to that in 200 mg/mL synthetic mucusterm·key. Overall, the correlation lengthterm·mth L dependencies on mucin concentration in natural mucusterm·key exhibit a similar monotonous increase. Still, they could be different in values because the restored mucusterm·key network from synthetic mucusterm·key has smaller viscoelasticityterm·key compared with fresh, natural mucusterm·key at the same mucin concentration ( 32 ).

Compared to mucusterm·key, increasing the PVP concentration does not affect the spatial scale of collective motionterm·key, Fig. 2 A. The main impact of PVP360term·key is slowing down the dynamics. Across the whole PVP360term·key concentration range, the effective (macroscopic) viscosityterm·dom of suspension changes by two orders of magnitude. However, the corresponding elasticityterm·dom is negligible compared to mucusterm·key (see Discussions and Fig. 4 A). The results imply that fluid elasticityterm·dom enhances correlation lengthterm·mth while fluid viscosityterm·dom mainly slows down the dynamics.

Figure 2 B displays the root mean squareterm·mth (rmsterm·mth) velocity of collective motionterm·key V . At 0 mg/mL (pure TB), the rmsterm·mth velocity of collective motionterm·key is about 43 μ m/s, consistent with the previous study ( 30 ). The rmsterm·mth velocity of individual bacteria in TB with a viscosityterm·dom of approximately 1 cp is about 20 μ m/s. The doubled swimming speed indicates the emergence of collective motionterm·key. The rmsterm·mth speed of bacterial flow continues to increase after 0 mg/mL and peaks at 50 mg/mL for mucusterm·key and 25 mg/mL for PVP360term·key solution, with a value of around 77 μ m/s, and 53 μ m/s, respectively.

After the peak, the speed in both fluids gradually decreases and reaches a plateau at 30 μ m/s for mucin solution and 21 μ m/s for PVP360term·key solution. In all concentrations, the speed of collective motionterm·key in PVP360term·key is always lower than in mucin solution. It results from the higher viscosityterm·dom of PVP360term·key solution in the same concentration, see Discussions. A plausible explanation for the velocity increase is the suppression of the velocity fluctuations due to polymer stretching and reorientation by swimming bacteria ( 33 ). At higher concentrations (over 50 mg/mL), the increased suspension viscosityterm·dom slows the bacteria down. However, compared to viscoelastic mucusterm·key, the swimming speed enhancement in viscous but otherwise Newtonian polymer liquid is modest. For comparison, we plot in Fig. 2 B the correlation velocity defined r = L / τ . The correlation velocity r exhibits a similar nonmonotonic trend as the rmsterm·mth velocity V , although the peak is less pronounced.

We further examined the temporal velocity correlation time τ dependencies on concentration of mucin and PVP360term·key. To determine τ , each image was partitioned into 127 × 127 subdomains. The correlation time was extrapolated by fitting the velocity autocorrelation function at each subdomain into the exponential dependence (as for spatial correlations). The correlation time of all subdomains was averaged to obtain the mean value. Figure 2 C presents correlation time τ as a function of concentration. Unlike the correlation lengthterm·mth L dependencies, the correlation times increase with the increase of solid fraction for both mucusterm·key and PVP360term·key solution at most concentrations (>25 mg/mL). This observation complies with the increase of effective viscosityterm·dom of the suspension with the increase of solid fraction and, corresponding, overall slowdown of the motion ( 30 ). However, mucusterm·key and PVP360term·key solutions show a minor decrease in the correlation time for small concentrations, about 25 mg/mL. This decrease in the correlation time is likely related to the increase in the velocity of collective motionterm·key.

Figure 2 D shows the vorticity magnitude dependencies. We use two complementary definitions of vorticity magnitudes σ , w ¯ to provide a more comprehensive characterization of bacterial collective dynamics. To extract σ and w ¯ from the sequence of experimental images, we calculated the vorticity w from the velocity field. Vorticity magnitude w ¯ is the mean absolute value of vorticity w . The magnitude σ is defined as the standard deviation of the vorticity probability distribution P ( w ) approximated by the Gaussian law P ( w ) ∼ exp ( − w 2 / 2 σ 2 ) ; see Supplementary Figs. S2–S5 . As one sees from Fig. 2 D, different definitions σ , w ¯ exhibit similar trends. It implies that the probability distribution P ( w ) is close to the Gaussian one. The rmsterm·mth velocity V and the vorticity magnitudes σ and w ¯ exhibit similar nonmonotonic trends vs. concentration of mucin or PVP360term·key, Fig. 2 B and D: initial increase followed by a gradual decrease.

We investigated the movements of individual bacteria in synthetic mucusterm·key and PVP360term·key solutions to obtain additional insights into of the correlation lengthterm·mth dependence in a viscoelastic fluid. The experimentsterm·mth on the individual bacterial dynamics are performed on the glass slide. Due to the interaction between bacteria and solid surface, the bacteria move on circular orbits in Newtonian fluids ( 35 ), Fig. 3 A. However, in polymer solutions, the trajectories can straighten due to a viscoelastic lift force, Fig. 3 B ( 36 ). This viscoelastic lift force does not necessarily affect the bacterial collective dynamics studied in a surface-free standing film. Also, as the solid fraction of mucin solution increases with concentration, the relative fraction of trapped or slowly moving bacteria increases, as displayed by the decreased number of trajectories presented in Fig. 3 B, also see Supplementary Videos 6–8 . The short trajectories less than 20 μ m are the results of impurity in mucin solution and bacteria trapped in the mucin mesh. They have been filtered out before the calculation on velocity and rotational diffusion coefficient.

We examined individual bacteria’s rotational diffusion D R . It was calculated by fitting the mean square angular displacementterm·mth (MSADterm·mth) ⟨ Δ θ 2 ⟩ of each tracked trajectory into the linear law ⟨ Δ θ 2 ⟩ = 2 D R t . The MSADsterm·mth of tracked trajectories in mucin/PVP360term·key solution are shown in Supplementary Figs. S6–S8 . The black dashed line represents the average, respectively. The rotational diffusion, see Fig. 3 C, shows no apparent dependence on the mucin concentration (solid red line) until ∼ 50 mg/mL, after which it decreases with viscosityterm·dom. However, D R gradually decreases in PVP360term·key solutions with the concentration increase in the whole range (solid blue line). Estimates for the thermal rotational diffusion D R ′ (red and blue dashed lines) are shown in Fig. 3 C to highlight the differences in rotational diffusion in mucin/PVP360term·key solutions. The estimate considers the medium to be homogeneous and isotropic across all concentrations in mucin and PVP360term·key solutions. Thus D R ′ should decrease inversely with the effective viscosityterm·dom η as the red and blue dashed lines in Fig. 3 C. However, the mucin solution is not homogenous neither isotropic due to the polymer network, Fig. 1 B. Inside the pore space of the mesh network, the bacteria can swim as they do in the solvent, giving a similar rotational diffusion as in broth. Therefore, the tracked bacteria produce a rotational diffusion in mucin solution with a much weaker concentration dependence compared with PVP360term·key solution. In the PVP360term·key solution, where the medium is more homogeneous and isotropic for bacteria, the rotational diffusions decrease inversely with the effective viscosityterm·dom, as has been reported by previous studies ( 37 ).

The speed of individual bacteria also shows different concentration dependencies in mucin and PVP360term·key solutions. In mucin solutions, the motile bacteria have a similar speed as in TB until ∼100 mg/mL and decreases then. The difference hints at the specific mucin solution structure. A cow cervical mucusterm·key shows a mesh network with an average pore size of around 2 μ m ( 7 ). Inside the mesh network, the fluid viscosityterm·dom is similar to or only slightly higher than that of water. Therefore, the motion of bacteria inside the mucusterm·key pores is similar to that in water. As the concentration of mucin increases, the mesh pore size decreases, which will trap more bacteria. Hence, a smaller number of motile bacteria is observed. However, the speed of motile bacteria is still the same since the viscosityterm·dom inside the pore does not change. In PVP360term·key solutions, the speed initially increases until 20 mg/mL and decreases afterward. The initial increase was expected to be the influence of the small molecular weight impurity from PVP360term·key solutions ( 34 ). After 20 mg/mL, the speed decreases as the viscosityterm·dom increases.

Discussions Our experimental study revealed a distinct effect of viscoelasticityterm·key: while the medium viscosityterm·dom results in the overall slowdown of collective motionterm·key, the elasticityterm·dom increases the correlation lengthterm·mth of collective motionterm·key. By examining individual bacterial dynamics, one may come to the conclusion that an increase in the correlation lengthterm·mth is an outcome of the straightening of bacterial trajectories ( 33 , 37 ). However, this issue is more subtle, and the curvature reduction of individual bacteria trajectories does not necessarily translates into the increase of the collective motionterm·key scale, as shown by our study, see Fig. 1 C–E. Straightening of the bacterial trajectories could be an interplay of multiple factors: suppression of run-and-tumble behavior, bundling of multiple flagella in a single tight bundle due to polymer interaction, effect of the lift force near solid surface, or alignment of bacterial bodies with the polymer network. In our situation, the rise in the correlation lengthterm·mth is due to the stretching and realigning of mucusterm·key elastic polymer network by bacteria, a phenomenon similar to that in liquid crystals ( 5 , 6 ). However, we do not anticipate the onset of long-range liquid crystalline order for the mucin concentrations used in our experimentsterm·mth. Here, the competition between the mucusterm·key network’s elastic deformations and the bacteria’s hydrodynamic torques increases spatial correlation lengthterm·mth L .

The collective bacterial flow has a scale of 80–250 μ m. It exceeds the mucin network’s mesh size (2 μ m). The recoil during the mesh network distortion can effectively slow the bacterial flow. Besides increasing viscosityterm·dom, this slowdown may explain the increased temporal correlation in mucin solution with increased concentrations. The Deborah numberterm·dom (Determ·dom) describes the effect of viscoelastic relaxation of flow patterns. It is defined as Determ·dom = λ a / λ b , where λ a is the fluid relaxation time and λ b is the observation time of fluid flow. Depending on the concentration, the characteristic relaxation time of mucin solution is around λ a ∼ 1 − 10 s. The observation time of fluid flow λ b associated with the collective bacterial flow is l c / U c , where l c is the length of a bacterium and U c is the bacterial flow speed. Thus, λ b is ∼0.7–3 s and the Deborah numberterm·dom Determ·dom ∼ 0.6 − 14 . Therefore, flow elasticityterm·dom can be essential in the onset of collective motionterm·key.

We develop a description of bacterial collective motionterm·key in a viscoelastic fluid. It consists of a phenomenological equation for bacterial dynamics ( 38–40 ) coupled to the Johnson–Segelman model of polymer viscoelasticityterm·key ( 41 , 42 ). The dynamics of incompressible bacterial liquid is governed by a 2D complex Swift–Hohenberg-type equation for the effective velocity v in thin-film geometry (incorporating bacterial and solvent velocity): complemented by the incompressibility condition ∇ ⋅ v = 0 . Here, T is the stress tensor, the viscous friction term α v is due to surface integration ( 43 ), and the nonlinear term ζ | v | 2 v is needed for saturation of the instability. The “effective” stress tensor T is of the form: Here, p is the isotropic pressure, I is the 2D identity matrix, Σ is the polymer stress and D is the velocity gradient tensor Here, the superscript T implies the transpose matrix. Terms β D , γ ∇ 2 D include, on the phenomenological level, the active dipolar stresses produced by swimming bacteria, as well as viscous and collision stresses. According to Ref. ( 30 ), the increase in viscosityterm·dom is absorbed in the model coefficients α , β and γ in Eq. 2 and does not change the scale of collective motionterm·key. It implies that the effect of increased polymer viscosityterm·dom, as in PVP360term·key, should modify the coefficients proportionally to a common factor: ( α ′ , β ′ , γ ′ ) = ( 1 + g μ ) ( α , β , γ ) , g is phenomenological parameter, μ is the polymer viscosityterm·dom. The phenomenological factor ϕ ( τ ) characterizes the polymer nonviscous (i.e. elastic) response. Here, τ is the polymer relaxation time. Correspondingly, ϕ → 0 for τ → 0 .

The polymer stress Σ is described by the Johnson–Segelman equation: Here, Σ ▽ is the notation for the Gordon-Schowalter time derivative (upper convective derivative), τ is the polymer relaxation time, E = μ / τ is effective polymer elasticityterm·dom, and D p is the polymer thermal diffusion coefficient ( 41 , 42 ). For simplicity, we assume polymer deformation small and neglect the polymer nonlinearity, reducing the Gordon–Schowalter upper convective derivative to partial time derivative: Σ ▽ = ∂ t Σ . We also neglect the diffusion term D p ∇ 2 D ; the coefficient D p is of the order of thermal diffusivity of polymer molecules and is important only for very large deformations ( 41 ). The resulting model captures the bacterial collective behavior in Newtonian (for τ = 0 ) and viscoelastic ( τ ≠ 0 ) fluids. It is also significantly simpler than the model for viscoelastic liquid crystals ( 44 , 45 ).

Thus, resulting equations describing the bacterial collective motionterm·key in viscoelastic fluid assume the form In the absence of the polymer stress Σ , Eq. 1 exhibits a short-wave instability for β < 0 with the optimal wavenumber k 0 = | β | / 2 γ ( 38 ). The instability results in the onset of collective motionterm·key closely resembling bacterial turbulence observed experimentally ( 30 , 38 , 46 ). The examination of Eqs. 5 , 6 confirmed that the characteristic scale of bacterial collective motionterm·key increases with the increase of polymer stress contribution proportional to μ ϕ ( τ ) ), see Fig. 5 and Supplementary Video 9 . Equations 5 , 6 also predict the dependence of the characteristic wavelength of the bacterial collective motionterm·key k ~ as the following k ~ 2 = ( | β | ( 1 + ( g − ϕ / | β | ) τ E ) / 2 ( 1 + g τ E ) ) . Here, we used that μ = τ E . In the following, we assume g > ϕ / | β | . Then, the increase in the polymer stress will decrease the optimal instability wavenumber and increase the correlation lengthterm·mth. Correspondingly, the characteristic length L vs. polymer stress behaves as where L 0 is the characteristic length in the solvent, and κ 1 > κ 2 > 0 are fit parameters. The dependence agrees with the experimental data in Fig. 4 .

Computational modeling results. A) No polymer stress, ϕ = 0 . B) Effect of polymer stress, τ = 0.4 , ϕ = 1 . Other parameters of the model: domain size 50 × 50 dimensionless units, β = − 1.05 , α = 0.0303 , γ = 0.303 , ζ = 0.01 , μ = 1.3 , g = 0.5 . The vorticity w is shown in pseudocolors, the flow velocity is displayed by arrows.

Conclusion We have shown that the viscoelasticityterm·key of suspending media profoundly affects the collective behavior of bacterial active matterterm·dom. Namely, the elasticityterm·dom of the solution increases spatial correlation lengthterm·mth, whereas the viscosityterm·dom mainly results in motion slowing down and, in turn, increasing the correlation time. This statement is supported by the correlation measurements in the viscoelastic mucusterm·key solution and contrasted with the study of essentially Newtonian PVP360term·key polymer suspension. We anticipate an onset of a global vortex when the correlation lengthterm·mth becomes of the order of system size, similar to ( 10 , 47 ). However, whether this global vortex exhibits spontaneous direction reversal, as in Ref. ( 10 ) could depend on subtle properties of the viscoelastic medium, e.g. the ratio between the relaxation time of the medium and the correlation time of the bacterial locomotion, nonlinearity of the stress–strain relation, etc.

The increased correlation lengthterm·mth L in mucusterm·key with elasticityterm·dom E is the consequence of the reorientation of the mucin network by collectively swimming bacteria. The elastic deformations of mucin network “flattens” small-scale bacterial collective motionterm·key and lead to the increased size of bacterial vortices and streams. A similar effect occurs in liquid crystals where the correlation lengthterm·mth increases with the elastic constant ( 5 , 6 ). A corresponding model for mucusterm·key requires a set of highly nonlinear equations coupling the local orientation and polymer extension and therefore is challenging. Here, we adopted a simplified framework where the alignment effect described by the polymer’s elastic response factor ϕ ( τ ) q in Eq. 5 . In this approach, the dependence results from pure physical interaction between swimmers and viscoelastic medium; therefore, neither bacteria- nor media-specific. We anticipate that our results provide insights into a broad class of active matterterm·dom composed of viscoelastic suspending medium (liquid crystals, polymers, biological fluids) and self-propelled agents, both biological (bacteria) and synthetic. The synthetic realizations could include Janus microswimmers energized by chemical reactions, light, or ultrasound ( 48 ). Our results also provide insights into bacteria behavior on mucosal surfaces with possible implications for human and animal health. The enhancement of large-scale collective motionterm·key of bacteria by mucusterm·key viscoelasticityterm·key may result in accelerated bacteria penetration and, in turn, faster bacteria colonization of internal tissues.

Methods Bacteria growth and general experimental procedure Experimentsterm·mth were performed with a DK400 strain of Bacillus subtilisterm·key . These bacteria have a diameter of 0.8 μ m and a body length of 5–7 μ m. They were grown on an LB agar plate at room temperature for 2–3 days until scattered colonies were observed. Then, a single colony was inoculated to Terrific Broth fluid (Sigma T5574) in sealed centrifuged tubes under microaerobic conditions. Bacteria grown under microaerobic conditions show enhanced oxygen deprivation resistance and motility at high bacterial concentrations. The bacterial suspension was centrifuged at the end of the log phase corresponding to an optical density of 0.6 . The bacterial pellet was mixed with fresh broth, mucin solutions, or PVP360term·key solutions with different concentrations to produce a bacteria/broth, bacteria/mucin or bacteria/PVP360term·key mixture. Five microliter of the mixture was pipetted into a 0.25 cm × 0.25 cm square free-standing film. A 10× magnification objective was used to record a 10 s, 30 fps video for each mixture. Each video was processed by a MATLAB PIVterm·mth script ( 31 ). The velocity field obtained from the PIVterm·mth was used to generate the temporal velocity autocorrelation and spatial velocity correlation functions. Correlation time and length were extrapolated by fitting each correlation function into each exponentially decaying function. For the experimentterm·mth on individual bacteria movements, one part of the bacterial culture at the log phase was mixed with nine parts of different solutions (mucin or PVP 360) before imaging. The bacteria concentration during imaging (about 10 7 cells/cm 3 ) is much lower than the concentration threshold for the onset of collective motionterm·key ( 10 10 cells/cm 3 ).

Synthetic mucusterm·key solution and PVP360term·key solution fabrication Partially purified mucin powder from the porcine stomach (M1778 Sigma-Aldrich) or PVP360term·key powder (PVP360term·key Sigma-Aldrich) was mixed with Terrific Broth (T9179 Sigma-Aldrich) in different ratios to fabricate mucusterm·key/PVP360term·key solutions in different concentrations. Fabricated mucusterm·key/PVP360term·key solution was stored in 4 ∘ C before the experimentsterm·mth with bacteria and rheological measurements.

Natural cow cervical mucusterm·key Mucusterm·key samples were collected from Holstein dairy heifers during estrus before artificial insemination. All procedures involving animals are reviewed and approved by the Pennsylvania State University Institutional Animal Care and Use Committee (protocol #200346584) and comply with the Guide for the Care and Use of Agricultural Animals in Agricultural Research and Teaching.

The collected mucusterm·key was then stored in 50 mL conical tubes and kept at 4 ∘ C until further use. The mucin content (7.5 mg/mL) of natural mucusterm·key was estimated by the weight of lyophilized natural mucusterm·key (70 mg) divided by the volume of natural mucusterm·key (9.3 mL) while ignoring the weight of other small proteins, lipids, and DNA. Natural mucusterm·key was sonicated for 10 min with broth in different ratios and placed in the ice bath for 2 h before experimentsterm·mth. The sonication helps to dilute natural mucusterm·key while preventing significant damage to the mucin polymer network. The final mucin concentrations of natural mucusterm·key dilutions used in the inset of Fig. 2 A are 0.9375 and 1.875 mg/mL. Experimentsterm·mth with natural mucusterm·key dilution at a concentration higher than 1.875 mg/mL are unreliable due to the difficulty in preparing homogeneous bacteria/mucusterm·key mixture. An example of bacterial collective motionterm·key in natural mucusterm·key dilution at around 3.75 mg/mL is shown in Supplementary Video 5 illustrating a scale of collective movement at roughly 200 μ m.

Bulk rheology We observed that mucusterm·key oscillation frequency measurements appear to be unreliable due to issues with loading samples, especially for high mucin concentration. Namely, the elastic mucusterm·key drop did not fill the rheometer probe uniformly. It leads to a scatter of the experimental data. Nevertheless, the rheometry data show a gradual increase of both storage and elastic moduli with the increase of mucin concentration, Supplementary Fig. S1E and F .

On the contrary, viscous polymer PVP360term·key solutions show no elasticityterm·dom as no network is formed. Therefore, instead of using oscillation frequency measurement to probe both viscous and storage modulus as we do with mucusterm·key solutions, a flow sweep measurement was used to characterize the viscosityterm·dom in the PVP360term·key solutions at different concentrations. Flow sweep measurement was performed on PVP360term·key solution for small strain values ( Supplementary Fig. 1F ). Supplementary Fig. S1B and F gives an apparent increase in the average viscosityterm·dom, which is similar to Ref. ( 49 ).

Microrheology The viscoelasticityterm·key of mucin solutions was also examined by particle tracking microrheology ( 50 ). Briefly, the subdiffusion of micron-sized particles (3 μ m in diameter) was tracked and extrapolated to obtain the viscous and storage moduli. Supplementary Fig. 1A and B summarizes the frequency-dependent storage (triangles) and viscous (circles) moduli. Overall, both storage and viscous moduli increase as the mucin concentration increases. In Supplementary Fig. S1C and D , the ratio of the viscous modulus to the elastic modulus tan ( δ ) is presented. Below 150 mg/mL, the viscous modulus dominates ( tan ( δ ) > 1 ) at all frequencies. At high mucin concentrations (150 mg/mL and above), tan ( δ ) is close to or smaller than 1, indicating a nontrivial elasticityterm·dom. To probe the microrheology of mucin solutions, we tracked the random, diffusive behavior of micro-sized polystyrene latex particles characterized by the mean square displacement (MSD) q , where q = ⟨ | r ( t ) − r 0 | 2 ⟩ , r , r 0 are the current and initial positions of the tracer particle, correspondingly. The MSD data of each concentration was converted into frequency-dependent elastic and viscous moduli using a modified algebraic form of the generalized Stokes–Einstein equations: where Here, a is the particle radius, k B is the Boltzmann constant, and Γ is the gamma function. α ( ω ) and β ( ω ) are the first- and second-order logarithmic time derivatives of the MSD data, respectively, while α ′ ( ω ) and β ′ ( ω ) are the local first- and second-order logarithmic derivatives of G ( ω ) , respectively ( 50 ).

Thermal rotational diffusion coefficient An estimate for the (thermal) rotational diffusion coefficient (blue and red dashed lines in Fig. 3 C) can be found from the Einstein–Stokes relation for an ellipsoid: D R ′ = 3 k B T ( log ( 2 b / a ) − 1 / 2 ) / 8 π η b 3 . Here, a , b are the major and minor axis of the ellipsoid-shaped bacteria cell body, η is the effective viscosityterm·dom of the suspension, k B is the Boltzmann constant, and T is the temperature.

Numerical implementation To solve Eqs. 5 , 6 , the following technique was used. First, we introduced a stream function φ and applied curl operation to Eq. 5 to exclude pressure p and satisfy the incomprehensibility condition. The resulting equation for vorticity w = − ∇ 2 φ was solved by the quasispectral method in periodic boundary conditions ( 40 ). Typically, we used 1024 × 1024 grid points in the domain of the order of 100 × 100 dimensionless units.

Supplementary Material Click here for additional data file.