Configurational dynamics of flexible filaments in bacterial active baths

Biopolymers with microscale length and nanoscale cross-sections subjected to active forces is a common non-equilibrium phenomenon in living creatures. It is therefore crucial to intuitively present and investigate the detailed dynamics of such flexible filaments in an active bath full of living matter. Hence, by introducing fluorescent actin filament into an active suspension of motile bacteria at different number densities in a quasi-two-dimensional chamber, we directly visualize the detailed interaction processes and find that bacteria deform a fluctuating filament by relative motion perpendicular to its principal axis of the filament and straighten a filament by parallel motions. We analyzed the evolution of bending energy in dilute and dense bacterial baths with gradual compact or coiled shapes. We successfully introduce a dimensionless number μ˜ , named as active elasto-viscous number, which governs the generic deformation of filaments in the bacterial baths by comparing the viscous force generated in the bacterial active baths to the elastic restoring force of filaments. The persistence length measuring the tangential correlation of the flexible filament is found to be proportional to μ˜ . Finally, an effective temperature of the bacterial bath is given through the relation between constant stiffness and loading forces instead of the popularly used Einstein relation. Our findings provide detailed information and specific scaling of flexible filaments interplay with active forces and in response to living and crowded environments.


Introduction
As an essential component of eukaryocyte cytoskeleton, actin filaments play crucial roles in cell shape, division, movement and mechanical stability [1][2][3][4]. Such type of flexible filaments cooperate with other biopolymers participate in complex biophysical processes from intracellular streaming to muscle contraction at macroscopic scale, which push the filament far away from their thermal equilibrium state since molecular motors continually transfer environmental energy into transformation of actin filaments [5][6][7]. Thus in recent years, remodelling of these natural processes attracts much attention to investigate different types of biopolymers in environments containing active energetic units, particularly focus on morphological dynamics and transport properties of anisotropic objects with changing stiffness under the effect of active forces.
Various numerical simulations, as well as theoretical models, are performed either by considering dry systems with short range steric interactions through passive polymer embedded in a bath of active Brownian particles [8][9][10][11], or taking into account non-negligible long range hydrodynamic interactions with active forces from environment in low-Reynolds number due to their subcellular dimensions [12]. Some common results are achieved that flexible chains showed a monotonic swell with spiral formation as with the activity increased from the surroundings and a probability distribution of end-to-end distance shifts toward larger values [8,13,14]. For stiffer polymers, peculiar nonmonotonic behavior is obtained that polymers switches between hairpin and extended states due to complex local dynamics [15]. As for an alternative dynamics of non-equilibrium behavior, actin filament can adapt versatile deformation in laminar flows, which shows interesting configuration such as buckling, snake turn and amazing three dimensional helical structure [16][17][18][19]. Although the shear flow can capture the fundamental forms of dynamical enviroments filament may experience, the real crowd intracellular confinements full of living staff possess distinct features of microscopic energy injection [20]. Rather than imposing well-controlled classical flows, complex structured flows are much more attractive to mimic the complex dynamic medium environment where actin filaments perform biological functions. There is one natural way to generate flows by swimming microbes such as bacteria, and they can coherent each other to create chaotic flows when they get together [21,22]. The active stress provided by the bacteria strikingly improve mixing within this bacterial suspension at low Reynolds number, and then enhance the transport of substance in the suspension [23][24][25][26][27][28][29][30][31].
Besides, it also stands that active forces yield a higher effective temperature of the bath and even lead to a superdiffusion of concomitant passive particles or looping kinetics of the embedded passive polymers [13,23,32]. Recently, there were experiments coming out for magnetic particle-chains showing a non-monotonic dependence of diffusion coefficient on the chain length [30,33], as well as the mean length and mean radius of gyration of chains obeying Flory's Law [34], while both the typical dimensions of cross section and bending stiffness are biased from usual biopolymers. In fact, the experimental study directly employing biopolymers is extremely challenging and lacking due to the fact that the soft polymer chains are greatly sensitive to background disturbed flow and that the visual duration is limited because of the photo bleaching issues.
Hence, we built, as an experimental model system, an active bath of motile bacteria to directly observe and characterize the behaviors of flexible filaments with varying the number density of bacteria. We confined the motile bacteria and the fluctuating filaments in a quasi two dimensional chamber made from polydimethylsiloxane(PDMS) Hele-Shaw cell to avoid imaging issue due to out of focal motions of filaments. We are therefore able to directly visualize the detailed interaction processes between active bacteria and responsive filaments. During the process, it clearly shows that bacteria deform a fluctuating filament by relative motions perpendicular to the principal axis of the filament, and straighten a filament by parallel motions. We analyzed the evolution of bending energy in dilute and dense bacteria bath with gradual compact or coiled shapes. To capture the dynamics of fluctuating filaments in response to active bath agitated by individual bacteria, we introduce a new dimensionless numberμ governing the generic deformation of filaments under bacterial active forces through force balance analyses. To extract the dimensionless number, an evaluated effective shear rate of disturbed flow caused by swimming bacteria is given with varying number density. Interestingly, we are able to define an effective temperature of the bacterial active bath, which nicely correlates the constant stiffness and loading forces instead of popular used Einstein relation. The active elasto-viscous numberμ proposed here based on local slender body theory provides a practical description for responses of soft chains when experiencing an active stimulation and being far from equilibrium.

Experimental details
The bacteria used in our experiments have an elliptical body with a few microns in length and around 0.5 µm in radius of cross section, with multiple helical flagellum of few microns, rotating with high frequency and resulting in a typical swimming speed around 20 µm s −1 , as shown in figure 1(A). The actin filaments used here have a length of 10−40 µm with a thin diameter d about 8 nm. Thus a 60X objective with a depth of focus around 2 µm is used to observe the interaction between bacteria and filaments. To confine the movement of bacteria and the deformation of flexible filaments in focal plane, a PDMS Hele-Shaw cell with a height of 3 µm and a width of 200 µm is opted to use. To avoid the collapse of the channel with such a large aspect ratio, a special processing process is needed by following the [35] and the height of the channel is confirmed by thin layer scan with a confocal microscopy. The glass cover slip is spin coated with a thin layer of PDMS with a thickness around 5 µm to have a symmetric top and bottom boundaries and the PDMS channel is soaked with 7 mg ml −1 bovine serum protein for 15 min before experiments to prevent the adhesion of actin filaments to PDMS channel. Both bacteria and filaments are fluorescent labelled and the interaction between them can be directly recorded and showed in figure 1

Protocol of actin filaments
Non-fluorescent G-actin (Cytoskeleton, Inc.) is mixed in G-Buffer (10 mM Tris-HCl, pH = 7.8, 0.2 mM ATP, 0.1 mM CaCl 2 , 1 mM DTT) to obtain a solution of G-actin with a concentration of 48 µM. Concentrated G-actin is placed into F-buffer (10 mM Tris-HCl, pH = 7.8, 0.2 mM ATP, 0.2 mM CaCl 2 , 1 mM DTT, 1 mM MgCl 2 , 100 mM KCl, 0.2 mM EGTA, and 0.145 mM DABCO) at a final concentration of 1∼3 µM. At the same time, Alexa488-fluorescent phalloidin in the same molarity as G-actin is added to visualize and stabilize filaments. After 1 h of polymerization in the dark at room temperature, concentrated F-actin can be stored at 4 • C for experiments in the following weeks.

Protocol of bacteria culture
Two types of bacteria, fluorescent labeled Escherichia coli (strain AW405) and non-fluorescent Bacillus subtilis (strain BS6633), are used in the experiments. The bacteria stored at −80 • C are transferred into the Luria-Bertani broth (LB solution, with 1.0% (w/v) Tryptone, 0.5% (w/v) Yeast and 1.0% (w/v) NaCl in deionized water) in a shaker with 200 rpm for overnight growth at 30 • C. Thereafter, the bacterial suspension at middle exponential phase is diluted into a fresh LB solution with a ratio of 1:100 for further growth until bacteria concentration reaches an optical density (OD) ∼0.6 under the light with a wavelength of λ = 600 nm. To keep bacteria motile but not dividable, the diluted bacteria suspension is centrifuged at 3000 rpm for 5 min and redispersed into a motility buffer (1M sodium lactate, 100 mM EDTA, 1 mM L-methionine, and 0.1 M of potassium phosphate buffer, pH = 7.0). Note that, we wash the bacteria suspension at least three times with deionized water in order to remove secretion residuals. To stain the bacterial body and flagella, the bacterial suspension is added into 5% sodium bicarbonate (1 M, pH = 7.8), with 2% Fluor TM 488 (Thermo Fisher Alexa) in DMSO (dimethyl sulfoxide, 99.9%, Sigma-Aldrich), under shaking at 200 rpm for 30 min in dark environment at room temperature. Finally, the bacterial suspension is adjusted to the target volume fractions 0.01%∼2.5%. The volume fraction of bacteria is calculated based on the approximation that number density of bacteria is n 0 ≈ 3 × 10 8 cells ml −1 corresponding to 0.1% in volume fraction with OD ∼1.

Morphological dynamics of individual filaments in bacterial bath
We first perform experiments with bacteria of fluorescent labelled body and flagella as well as fluorescent actin filaments to directly present the interacting processes in a quasi two-dimensional chamber in case of dilute bacteria so that the interaction between bacteria can be neglected. Figure 2 shows the relative positions and configurations of bacteria and filaments during 4 s. As shown in figure 2(A), a bacterium is approaching to a filament and deforming the filament with a local high curvature, followed by a tumble event and then swimming away perpendicularly from the side. During time steps from 2.2 s to 2.3 s, bacteria may have a steric interaction with the filament through its body and flagella, since the local shear rate induced by fast rotating flagella in the close region can reach 10 4 s −1 at maximum [36]. In time t ∼ 2.4 s, bacteria stir the medium and affect the filament with hydrodynamic interaction. Since both bacteria and filaments are subjected to fluidic environment at low Reynolds number due to micro-scale dimensions, the dynamics of flow field can be described through Stokes equations by neglecting inertia effect. The bacterium pushes the fluid outward at the head and tail, thus named a pusher swimmer [37]. The pusher bacteria exert backward trust force on fluid by rotating the flagella and an opposite equal forward force at head accounting for a force-free approximation, which is modelled as a force dipole [21]. One of the basic solutions, flow field caused by a force dipole can be approximated as disturbed flow generated by a swimming bacterium with an elliptical body and rotating helical flagella at the leading order. Differently, when the bacteria swim towards the filament but parallel to its axis, the local extensional part of the disturbed flow straighten the filament, as shown in figure 2 To avoid the confusion of fluorescence from bacteria, experiments are then carried out by fluorescent labelling actin filaments merely. Moreover, the dynamical behaviors of the actin filaments in an active bath filled with dense bacteria swimmers need to exclude the signal noise from the shining bacteria with fluorescence. So that, the dyed actin filaments are imaged in this active bath with embedded bacteria, which are motile but not fluorescently stained. The typical configurations under bacterial active forces with increasing number density of bacteria are showed in figure 3. As a reference, the configurations of a filament in equilibrium with thermal fluctuations are showed in figure 3(A). As number density increase, the shapes of filaments gets frizzier both on average or most coiled state as shown in figures 3(B) and (C). The hydrodynamical interaction among dense bacteria generates the active flows, which serves as the active bath and gives rise the violent fluctuation of actin filament compared to thermal relaxation in equilibrium heat bath. This strong coupling flows induced by great number of force dipoles, provide the active disturbed flow field and transfer the injected energy to the surfing actin filament.

Energy stored in filaments with varying activity
The stretching and recoiling of filaments indicate that release and storing of extra elastic energy are transferred into filaments from the motile active particles. Especially in dense bacterial suspension, filaments do not have enough time to relax from strikingly deformed shape back to their equilibrium states. We then    figure 4(B) shows that the overall unit length energy is slightly larger than that in equilibrium, where some small peaks occur and span a typical duration of few seconds. As the volume fraction continue increase, as shown in figures 4(C) and (D), there are multiple higher peaks, revealing stronger effect of bacterial action on filaments. Figure 5 shows that as the concentration of bacteria increase, both the ensemble average bending energy per unit length and its standard deviation increase, which are calculated based on several movies lasting 30 s at each concentration. To calculate the average and standard deviation of number density, tens of independent measurements are made in each case of OD. As is commonly understood, passive molecules in a classical thermal bath supply the activation energy necessary to agitate the filaments embedded within it, but they also generate a drag-resistant force. As a result, activation and drag dissipation of the filament stem from the same source and define the equilibrium state of the flexible filament. Whereas, the active bath on the other hand, increase the degree of deformation higher up to one order of magnitude when the active swimmer gets dense. More specifically, the filaments extract the energy from the active bath and store this elastic energy by shape fluctuation and the vital deformation, showing a dependence on the number density of the active constituents in a linear way.

Activity induced soften of flexible filaments
Then the question arises here is that how to understand the energy transfer from active forces to the filaments deformation with versatile configurations. Can we describe the activity from the bath as the active flows disturbed by the motile microswimmers with additive contribution when they get dense. From hydrodynamic interaction point of view, the dynamics of a flexible filament in bacterial active bath can be described by the force balance equation along the arc length of the filament based on local slender body theory as follows, where, r t and v a is the motion of a filament and the velocity field of active bath respectively with the subscript stands for partial differential. µ is the viscosity of the medium. D = I + r s r T s arises from drag anisotropy of slender body and c = ln (ϵ 2 /e) is a logarithm correction of drag coefficient refer to aspect ratio ϵ = L/d, e is the nature exponential. f is the elastic restoring force of the deformed filament. To uncover the physical components needed in the dynamics, the equation (2) needs to be dimensionless. More importantly, it is tricky that the background flow v a from bacterial disturbance is not known. We need to find a way to calculate v a , which however is complicated particularly as function of number density of the bacteria. The usual way to have the resultant flows generated by interacting swimmers, is to use linear properties of Stokes equation and count on contributions of each bacteria with the far-filed model of force dipoles. It is possible to simplify this calculation of background flow that we can derive the flow field by experimental measurement of velocity of individual bacteria. Fortunately, the mean kinetic energy from the stirring bacteria is able to measure the activity of the active bath, thus provide the information of the background flow for filaments. To nondimensionalize equation (2), we could numerically solve this equation with the help of velocity measurement of bacteria. The filament shows complex morphology in response flows and thermal agitation, which can be decomposed into different fluctuation modes, therefore, a typical relaxation time at the slowest is proportional to µL 4 /B with a bending rigidity of B [17,19]. The length of filament L is used as the natural characteristic length. Then, the active flow is normalized by Lγ, whereγ is defined as the characteristic shear rate to measure the strength of background flow. Then we have, whereμ called active elasto-viscous number asμ = (8π µL 4 )/(Bcτ ). In Stokes flow, A dimensionless number µ = 8π µγL 4 /Bc has been derived, which is called elasto-viscous number in previous studying on filament dynamics [17,38]. Therefore, a similar active elasto-viscous number due to the active force of swimming bacteria is proposed asμ = 8πµγ eff L 4 /Bc. It is a comparison of two dominant forces at play, the viscous forces acted by fluid on filaments ∼ µγ eff L 2 and the elastic restoring forces of curved filaments ∼B/L 2 . This number can also be regarded as the a comparison of the typical time scale of slowest relaxation time of a filament and the typical time scale of the background disturbed flow. Here, in the case of filaments in bacterial active bath, the viscous force is generated by the disturbed flow of swimming bacteria with a format of extensile force dipole. Since the filament is deformed by the drag force due to velocity gradient of background flow, it is essential to estimate an effective shear rateγ eff filaments experienced on average in the different active baths with increasing bacteria number density. Experimentally, we increase the number density to enhance the disturbed flow field which is highly fluctuating. Consider a system of a square with side length of l s , thus the typical length of each bacterium occupied on average is given by √ l 2 s /N, where N is the population of bacteria. Suppose the bacteria have a mean speed of v b , the velocity gradient caused by swimming bacteria is then evaluated to be v b / √ l 2 s /N. Taking into account the individual variation of bacteria velocity, the typical shear rate of bacterial active system with different population N can be given byγ eff = , v i is the average speed of ith bacterium considered in the system. Meanwhile, a typical time scale τ = 1/γ eff is obtained, characterizing the time scale of bacteria swimming across a giving distance on average. As shown in figure 6, as the number density increase, the effective shear rate of the system increases in the order of 1. It turns out that the effective shear rate generated by interacting motile bacteria is linearly dependent on the bacterial number density. Also interestingly, two types of bacteria are used here to confirm the typical shear rate of the system, and the same tendency is achieved.
A flexible filament tends to coil under the instantaneous collisions by solvent molecules, which is counter-acted by the rigidity of the filament. In such competence, persistence length ℓ p is defined to quantify the flexural strength under the thermal agitation in equilibrium. In case of two-dimensional system, the persistence length ℓ p is the exponential decay coefficient of the correlation function of the tangential vectors along the arc length as the following equation, Here, angle brackets represent ensemble average. However, under the active forces of bacteria, the filaments are deformed and tend to be more coiled shapes with higher bending energy comparing to thermal fluctuations. It is expected that the persistence length will decrease when the bath gets more active, i.e. bath with high number density of bacteria. Hence, we calculated the effective persistence length of the active system with varying active elasto-viscous number. As shown in figure 7(A), asμ increase, the scaled persistence lengthl p by persistence length in equilibrium decreases. To calculate the average and standard deviation of effective persistence length, thousands of configurations from tens of independent experiments are considered. In equilibrium, a smaller persistence length calculated from much more curved and coiled shapes indicates softer of the filaments and weaker ability to resist thermal forces. It has been accepted that the bacteria bath acts as a strikingly energetic surrounding to enhance the diffusion of the object embedded, and can be equivalently considered as a thermal bath but with high temperature [23]. In fact, the flexural rigidity of filaments keeps constant, while active forces produced by swimming bacteria are much greater. We can use an analogy of active flow to thermal forces at higher temperature, which can be derived through T eff = B/(k B ℓ p ). As shown in figure 7(B), as the active contribution of bacteria rises, the effective temperature increases up to an order of 10 2 ∼ 10 3 K.

Conclusion
In this work, we built an experimental model system by combining fluorescent bacteria and actin filaments to investigate the dynamics of flexible filaments in bacterial active bath from dilute to dense cases. The detailed interaction courses are vividly demonstrated that filaments are deformed or straighten by swimming bacteria respect to different relative positions. Quantitatively, we calculated the bending energy of filaments under dilute and dense bacterial forces comparing to it in equilibrium and find a positive relation between bacteria number density and average bending energy per unit length. The dynamics of filaments under active forces are then qualitatively analyzed from a mechanical point of view through local slender body theory. To characterize the activity strength of background flow generated by a group of bacterial swimmers, an effective shear rate of the bacterial bath is naturally related to the number density of bacteria, and thus used as a determined parameter to regulate velocity differences along filament on average. Luckily, an analogy to previous defined elasto-viscous number [39],μ here comparing viscous force generated in bacterial active bath to elastic restoring force of deformed filament, is derived through dimension analysis as the govern parameter of the flexible filaments in response to bacterial bath with active forces and can be named as active elasto-viscous number. As active elasto-viscous numberμ increases, the persistence length linearly decreases, meaning an activity induce soften effect in normal temperature. In an equivalent way, the response of the filaments is regarded as in a thermal bath with effective temperature in the order of 10 3 K. Apart from that, it is also interesting to investigate the transport properties of these anisotropic and flexible filament under bacterial active forces which is an ongoing work. Our observations and quantitative analyses provide detailed information and specific scaling of intermediate filaments interplay with active forces and help to understand better the mechanical functions of those biopolymers associated with basic living processes.

Data availability statement
The data cannot be made publicly available upon publication because the cost of preparing, depositing and hosting the data would be prohibitive within the terms of this research project. The data that support the findings of this study are available upon reasonable request from the authors.