Programmable double traveling waves in living liquid crystals

Domain: physics
Published: 2025-11-26
Editorial outcome: minor_revision

Abstract
Traveling patterns are ubiquitous in nature. Controlling the traveling waves in living active matter systems remains elusive due to their intrinsic chaotic nonequilibrium nature. Here, we show how to control the traveling waves in living liquid crystals (LLCs) which represent a unique class of materials where motile bacteria interact with the anisotropic environment of LCs. The concentrated bacteria form waves propagating along the predefined trajectories, showcasing a dual nature of propagation where both the bacteria and the waves themselves move in concert. The interaction between active bacterial waves and passive LCs triggers a secondary LC wave which runs after the bacterial wave with a constant phase lag, generating double traveling waves. Our experimental and theoretical investigations have unveiled the underlying mechanisms of parity and time-reversal (PT) symmetry breaking in these active-passive wave duets. The interactions between the undulating waves and the director reoriented by the waves, coupled with the symmetry broken in the designed patterns, are key to initiating wave propagation. Capitalizing on this understanding, we have successfully demonstrated the creation of single ring and even multiple rings of traveling waves with same or opposite chirality. These waves can be further programmed into letter shapes. This work opens up new avenues for the design of smart living materials and micromachines.

Introduction
Active matter is a fascinating realm of nonequilibrium systems, characterized by individual motile components that harness stored or environmental energy to generate non-conservative forces which drive the emergence of complex, coordinated motions across space and time1–10. Ubiquitous in active matter are traveling patterns and propagating waves, manifesting in diverse phenomena such as liquid-liquid phase separation11,12, metabolic synchronization waves in yeast13, and the intriguing dynamics of magnetic colloidal chiral fluids14. The recent discovery of autonomous waves in bacterial biofilms adds another layer to this complexity15,16. These biofilms exhibit not only oscillatory translations and rotations15, but also local fluctuations in cell density, culminating in synchronized spiral waves anchored by stationary cores16. Despite the unveiling of these captivating phenomena, our capacity to control the propagation of these active waves remains a formidable challenge17,18. This difficulty underscores the limitations in our understanding of how macroscopic properties in active matter systems arise from the intricate interplay of their constituent parts, despite extensive theoretical exploration1,19–24. Given the significance of mastering active traveling waves for the future control and engineering of active matter, there is a pressing need to develop delicate tools capable of manipulating these emergent waves17,18. Such advancements would not only enhance our control over these systems but also deepen our comprehension of the underlying principles governing their behavior.

Living liquid crystals (LLCs) represent a unique class of active matter, consisting of swimming bacteria embedded within passive lyotropic liquid crystals (LCs)25–27. These LCs are in the nematic phase, characterized by their alignment along a non-polar direction, represented by the director n^=−n^ 28,29. It has been shown that patterned LC directors, especially those with topological defects, can attract bacteria, leading to the formation of vortices around these defects30,31. Moreover, bacteria tend to concentrate and form polar jets within alternating bend and splay director fields32. When active LLC droplets are suspended in a passive thermotropic LC environment, their locomotion trajectories can be effectively controlled33. This controllability offers a promising platform for addressing the challenge of managing active traveling waves. In this work, we aim to leverage patterned LLCs to gain insights into the mechanisms behind the formation of traveling waves, and we will endeavor to achieve precise control and management of these dynamic waves.

Results
The LLC25 is composed of an aqueous dispersion of bacteria bacillus subtilis in chromonic nematic disodium cromoglycate (DSCG), see Supplementary Fig. 1 and methods. The LLC is confined by two photopatterned plates with a gap of 10 μm30,34. If a pattern is designed as n^=(nx, ny, nz)=(|cosψ|, sinψ, 0), where ψ= πy/L and L=100 μm, Fig. 1a, band Supplementary Fig. 2, the swimming bacteria are condensed in the splay distortions forming two unipolar streams32.

At t=0 s, swimming bacteria form two streams in opposite directions (Fig. 1c and Supplementary Movie 1). Two active traveling waves of bacteria in opposite directions are developed with time evolution, Fig. 1d, e and Supplementary Fig. 3. As the bacterial concentration crosses a critical threshold c ≈ 1.2×1015 m−3, Supplementary Fig. 3a, b, the bacterial streams experience a bending instability and develop transversal undulations, Supplementary Movie 1. The critical concentration can be calculated theoretically by the equation c= K/(αlU0L2), where K is an average Frank elastic constant of the LC, h is the cell thickness, α is a constant, l is the length of a bacterium, U0 is the hydrodynamic force dipole of the strength and L is the bacterial coherence length (period of the bacteria waves in experiment)25. In our experiment, we take K=10−12 pN, h=10 μm, α≈1, l=5 μm, U0≈1 pN μm, L=100 μm, and then we can get the critical concentration c ≈ 2×1015 m−3, which is the same order as experimental results c ≈ 1.2×1015 m−3. Undulation happens as the bacterial activity is high enough, and the activity is affected by the bacterial concentration. The fluctuation of bacterial stream increases, which disrupts the balance between bacterial orientation and LC director and thus undulation forms. The amplitude of the undulating waves increases at the early stage and finally reaches a relatively steady value, while the wavelength remains constant, Supplementary Fig. 3c, d. The temporal evolution of the undulating wave is quantified by the alignment parameter q= ⟨cos(2θ)⟩, Supplementary Fig. 3e, where θ represents the angle between the local bacterial stream and the x-axis. At t=0 s, the bacterial stream is well aligned along the x-axis, producing q≈1, Supplementary Fig. 3e. As the bending undulation evolves, the local bacterial orientations increasingly point along the y-axis, leading to a decreasing q, Supplementary Fig. 3e.

Remarkably, the undulating waves themselves migrate along the ±x direction, respectively with the bacterial propagation. Figure 1d shows that regions ① and ② in two neighboring waves come across and then separate, indicating that the two waves are traveling. The distribution of θ along the x-axis with time evolution provides another evidence of traveling waves of bacterial streams, Fig. 1e. Meanwhile, the local LC director fields are perturbed to form a similar undulating wave by the bacterial stream, following the bacterial traveling waves, Fig. 1f and Supplementary Movie 2. Compared to the average speed of swimming bacteria vb ≈ 12 μm/s, Fig. 1d, the speed of the active traveling waves is much slower with vwave ≈ 0.9 μm/s, Fig. 1i. Although the LC waves propagate in the same direction of bacterial waves, there is a phase lag between the active and passive traveling waves. As shown in Fig. 1f, at t=90 s, 120 s and 150 s, blue regions ① and ② (LC wave) always run after red regions ① and ② (bacterial wave). The interaction between the distorted director and undulating bacteria waves is the key to understand the mechanism behind the traveling waves of both bacteria and LCs.

The transition between polar jet and traveling wave of bacterial stream can be visualized by tracking a dead bacterium, Fig. 1g, h, and Supplementary Movie 3. At the initial stage of the polar jet, the dead bacterium circulates locally without translation, Fig. 1g, which represents the flow field around the bacterial stream. As the bacterial wave starts to travel, the tracer bacterium keeps circulating around the moving peak of the traveling wave, Fig. 1h, indicating that the local flow field is traveling in the same direction as the bacterial stream. We plot the temporal evolution of the wave speed at two stages (Fig. 1i), clearly showing the transition from polar jet to traveling wave state. The spatiotemporal mapping of the active traveling waves along the +x-axis is shown in Fig. 1j. Color represents the height profile of the waves, and the maximum corresponds to the peak of the waves, Fig. 1j, indicating that the waves travel from -x to +x. By applying the fast Fourier transform (FFT) to the profile data of the traveling wave, we get the power spectrum of the wave (Fig. 1k), which reflects the dispersion relation of the wave. The non-zero dispersion in the second and fourth quadrants with odd parity demonstrate the unidirectional wave traveling along the +x-axis. Both Fig. 1j, k prove that waves are unidirectional.

To gain insight into the flow field and active force, continuum simulation is conducted, Fig. 1l–m. The numerical simulation shows that there is a forward flow field with the propagating wave, and the flow field matches the active force (Fig. 1l–m and Supplementary Movie 4). The distribution of the flow field is consistent with our experiment results shown in Fig. 1h. Based on this understanding, active waves that travel in the same direction can also be rectified by using the pattern designed as n^=(nx, ny, nz)=(−cosψ, sinψ, 0), ψ= πy/L and L=100 μm (Supplementary Fig. 4 and Supplementary Movie 5). All active-passive traveling waves propagate in one direction.

To unravel the physical underpinning of active traveling waves, we model the active units as a point particle with constant self-propelled velocity v0 adhering to the Stokes drag law as a consequence of balanced hydrodynamic forces35. The self-propelled direction of the particle is defined as P, and the nematic directors are denoted as n^=(cosψ, sinψ, 0), where ψ= πy/L and L=100 μm, see Supplementary information. Figure 2a shows the traveling bacteria waves in two splay bands with opposite directions. Moving wave crests indicate the propagating waves (Fig. 2a and Supplementary Movie 6). The diagonal maximum bands distribution in the simulated space-time graph and quadrants distribution in the simulated power spectrum indicate the unidirectional waves, which are in agreement with experimental results in Figs. 1 and 2b, c.

The breaking of PT symmetry of the traveling waves is rooted in the broken symmetry of the interaction between the bacterial wave and the LC director, which is caused by the broken symmetry in the designed pattern, Fig. 2d. As shown in the first half period of Fig. 2d, the background director field is roughly represented by the red dashed lines while the bacterial stream causes the reorientation of the local director field. Due to the constant bacterial stream, the director is persistently out of equilibrium due to the surface anchoring29,33,36. The core of this interaction is a torque that the bacterial stream exerts on the LC director. To quantify this, we introduce the angle β=ψ−θ, where θ represents the angle between the local bacterial stream and the x-axis, and ψ is the nematic angle anchored on the substrates, Fig. 2f, g. This aligning torque is mathematically captured in our agent-based model for bacteria (see Supplementary Text) by the reorientation dynamics: ∂tθ=γB sin(2(ψ−θ))+Drξ(t). Here, the key term γB sin(2(ψ−θ)) creates the torque that aligns the bacterium with the local nematic director (ψ) of the LC. This is the primary mechanism for the feedback from LC to bacteria. The bacterial flows trigger a polar viscous response in the LC. The bacterial stream imposes a counter-clockwise (CCW) torque on the designed director at region ① and a clockwise (CW) torque on the director at region ②, Fig. 2d, f. The difference in designed directors between region ① and region ② produces different out-of-equilibrium director reorientation and thus different viscosities η+ ≠ η−. The difference η+ ≠ η− produces a rectified net driving force that can be estimated as Fnet ∝ (η+ − η−)vb, where vb is the speed of bacteria. Such asymmetry breaking of the driving force can be visualized numerically by mapping the spatial distribution of force, F=∇×(0,0,Tz), where Tz ∝ sin(2β) is the torque exerted from bacteria to LC, Fig. 2h, i. As indicated in Fig. 2f, h, the distribution of β and torque are not symmetric around the wave in the splay region and the induced force point rightwards, driving the wave to move rightwards. Thus, in the director field with broken symmetry, the active force caused by the swimming bacteria will push the flow field forward and thus the wave propagation.

As a comparison, the undulating waves of bacterial streams in a uniform director field are also demonstrated, in which the waves are not propagating, Supplementary Fig. 5 and Supplementary Movie 7. As shown in Fig. 2e, the bacterial stream imposes the same CCW and CW torques at regions ① and ②, which produces Fnet = 0, indicating there is no symmetry broken in the interaction between the bacterial stream and LC directors. There is no force along the x-axis and thus no traveling wave, Fig. 2g, i.

In order to gain insight into active-passive traveling waves, we use tensor Q to describe the orientation of LC and P to describe the orientation of bacteria, see Supplementary information. At the early stage after bacterial undulating wave forms, the orientation of LC is distorted by the movement of bacteria and the generated undulating instability is in the same phase of bacteria waves, Fig. 3a. Because of the symmetry breaking in the self-propelled polar movement of bacteria, there is a phase difference between the active bacteria wave and passive LC wave, where bacteria wave is ahead of LC wave, Fig. 3a. As a consequence of the phase difference, bacteria wave and LC wave interact with each other: bacteria rotate the LC molecule through the director-flow coupling by generating active flow, thus pushing the LC wave forward. Correspondingly, due to the anchoring of LC to bacteria, the LC wave can attract the bacterial wave and drag bacteria backward. The interaction between these two waves leads to a steady phase difference in a balanced state, thus stable double traveling waves. The effect of the bacteria on LC can be described by the equation

(∂t + v·∇)Q−S−ΓH+Fanch=0, (1)

where v·∇Q describes advection (Adv) of the nematic field by the flow and v is the fluid velocity. The tensor S=(ξA+Ω)Q+I + Q+I(ξA−Ω)−ξ(2Q+I)Tr(QW) describes alignment with the fluid flow, including flow alignment (FL, (ξA+Ω)Q+I + Q+I(ξA−Ω)) and corotation with the local vorticity (CR, ξ(2Q2+I)Tr(Q2W)), (see Supplementary text and Supplementary Table 1). ΓH and F contributions are from the free energy and external anchoring, respectively. To identify the key factor in the interaction, we analyze the contribution of different factors to the evolution of Q at the steadily traveling state, Fig. 3b. We find that elasticity and anchoring are in the same phase of Q, so do not contribute to the phase change or translational motion of the director. The spatial profiles of FL, CR and Adv are ahead of Q (LC), thus continuously increasing the phase Q, driving the director field to move forward. Since FL, CR and Adv all originate from the activity, and the phase of their total contribution is closer to that of P (bacteria), we can conclude that the LC wave is effectively attracted by the bacterial wave. In turn, the bacterial wave is attracted by the LC wave as well because of the anchoring of the LC on the bacteria. Thus, the interaction and phase difference cause the passive LC wave to travel and active-passive waves finally reach a stable traveling state. Note that the phase difference is affected by the bacteria's velocity and the anchoring strength of the LC on the bacteria. As shown in Fig. 3c, d, the simulation results of double traveling waves are in agreement with experimental results. Simulations show that active bacterial wave and passive LC wave are propagating in the same direction with a steady phase difference.

With the understanding of the mechanism, we can create swirling traveling waves with distinct chirality. For instance, a circular ring of splay band designed as n^=(nx, ny, nz)=(cosψ, sinψ, 0), where ψ=arctan(y/x)+ π(x2+y2)/L and L=100 μm is created by covering the center part during the photopatterning procedure, Supplementary Figs. 2 and 6. Double swirling traveling waves in counterclockwise (CCW) direction are formed with time evolution, Supplementary Fig. 6 and Supplementary Movie 8.

Furthermore, if a pattern with the director field is designed as ψ=arctan(y/x)+(−1)⌊(x2+y2)/L⌋ × π(x2+y2)/L, where L=100 μm, Fig. 4a, b, two polar streams of opposite chirality are formed, Fig. 4c, d and Supplementary Movie 9. Since the bacteria mainly come from outside, undulations form in the outer circle first, and the bacterial stream in the inner circle undulates later. When both of the circulating streams begin to undulate, two active waves propagate in opposite directions (Fig. 4e, f). Simulation well captures the swirling traveling waves in opposite chirality, Fig. 4g and Supplementary Movie 9.

Likewise, swirling traveling waves in the same chirality can also be realized by using the director field ψ=arctan(y/x)+ π(x2+y2)/L with L=50 μm, Fig. 4h, i. Bacterial streams form unidirectional traveling waves in the same direction (Fig. 4j, k, Supplementary Fig. 7, and Supplementary Movie 10). The experimental results are consistent with the simulation (Fig. 4l). Note that in both traveling waves with the same and opposite chirality, there are more petals (crests) in the wave for the circle with a larger radius (Fig. 4d).

Through a programmable photopatterning system, patterns can be programmed to generate desired traveling waves in arbitrary shapes37. In our experiment, we use patterns with shapes like letters “U”, “S”, “T”, and “C” (Fig. 5a, b). Hence, bacterial stream can form traveling waves following the letter shapes (Fig. 5c, d and Supplementary Movie 11). Note that in letter “T”, two different propagating bacterial streams from opposite sides of the pattern can merge into a stronger undulating stream, which then travels in a single direction. Thus, programmable traveling waves are enabled by manipulating the designed patterns.

Methods
Bacterial dispersion preparation
The LLC solution is prepared by mixing 14 wt% disodium cromoglycate (DSCG, Fig. S1) and 86 wt% terrific broth (TB) as a nematic host. TB solution serves as a growing and motility medium for B. subtilis. The bacteria grow in a sealed tube with 10 ml of TB at 37°C inside the shaking incubator at a speed of 150 rpm for 10–12 h. Then, 0.4 ml of the liquid medium with bacteria is centrifuged at 3074 × g for 2 min, and then the medium is extracted, upon which 0.1 ml of 14 wt% solution of DSCG in TB is added to the bacteria and carefully mixed with a mixer device. The dispersion of B. subtilis in DSCG is injected into the cell at 40°C on the hot stage and sealed with epoxy glue. The sample is observed under a microscope (from Soptop).

Sample preparation
Glass substrates are washed in an ultrasonic bath with detergent for 10 min. Then they are rinsed with deionized (DI) water and isopropyl alcohol (IPA) and dried in the oven at 85°C for 15 min. The glass is put in an ultraviolet (UV) ozone chamber for 10 min. An SD1 solution containing 0.2 wt% photosensitive material azo dye SD1 (Fig. S1) in N, N-dimethylformamide (DMF) solvent is spin-coated on the glass substrates at 3000 rpm for 30 s. Then the glass substrates are annealed at 90°C for 45 min to evaporate the solvent. The glass substrates are illuminated by a maskless photopatterning setup to get the predesigned pattern with photosensitive azo dye oriented perpendicular to the linearly polarized light. To protect the azo dye layer from water and anchor the LC molecules, the substrates are spin-coated with an RM257 solution at 3000 rpm for 30 s. The RM257 solution is prepared by mixing 10 wt% RM257 (Fig. S1) and 0.1 wt% photoinitiator Irgacure 651 in toluene. A UV lamp with an intensity of 1.4 mW/cm2 and wavelength of 365 nm is used to polymerize the sample for 5 min (partially fabricated by direct projection UV photolithography by Tuotuo Technology TTT-07-UVLitho-BIOS). Two substrates with mirror-symmetric patterns are assembled together, gapped by 10-μm diameter glass spheres in glue.

The maskless photopatterning setup mentioned above is based on a projector display (Fig. S2). The optical elements used in this setup are purchased from JCOPTIX, China. The pattern in Fig. 1 with n^=(nx, ny, nz)=(|cosψ|, sinψ, 0), where ψ= πy/L and L=100 μm, is generated by rectangular segments shown in Fig. S2c. The polarizer rotates at a speed of R1=18°/5 s in the clockwise sense. The segments are displayed on the y-axis at a speed of R2=10 μm/5 s. Hence, the pattern with a period of 100 μm is produced.

For a pattern designed as n^=(nx, ny, nz)=(cosψ, sinψ, 0) with a covered center part, where ψ=arctan(y/x)+ π(x2+y2)/L and L=100 μm in Fig. S6, the segment in the designed shape starts at the position at t=0 s and the rotary polarizer starts along the x-axis, Fig. S2b. The polarizer and segment rotate at the same speed of R1=R2=10°/5 s in a clockwise direction. After 180 degrees of rotation, the designed pattern is created.

Data acquisition and analysis
We use an advanced upright polarized light microscope from Soptop with a 20× magnification objective. Optical microscopy images are captured by a 9 MP USB 3.0 C-mount microscope camera from Soptop (resolution 4096 × 2160 pixels). The video sequences are recorded at 10 and 20 Hz rate. The sequences of images are analyzed by ImageJ2-Fiji. The velocity map is done by particle image velocimetry (PIV) MATLAB package. By enhancing the contrast of the image, we can get the number of bacteria by using analyzing particles in ImageJ-Fiji. We then divided the number of bacteria in a stripe by the stripe volume V=HW h to calculate the concentration, where H=263.7 μm is the height of the stripe, W=701.6 μm is the width of the image along the x-axis and h=10 μm is the cell thickness. Polar angle φ and the angle θ between bacteria and flow direction are calculated by tracking the bacteria trajectories and recording the position data of bacteria at the given moment. With two neighboring positions (x1, y1) and (x2, y2), θ is calculated by θ=arctan((y2−y1)/(x2−x1)). The velocity of the traveling wave is calculated by tracking several peaks of the wave at different moments and calculating the average. To plot the space-time graph, we extract the x and y (height profile h) coordinates of the bacteria wave at different moments and plot the map using the software Tecplot 360. We carry a fast Fourier transform (FFT) on the h(x,t) data and square the absolute value of this FFT to get a power spectrum in frequency ω, and wavenumber k.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_2": "The manuscript by Wu and co-workers presents studies on an interesting system composed of a water based lyotropic liquid crystalline base fluid doped by living bacteria. The dynamics of the active matter system of motile microorganisms can be controlled by the orientation of the anisotropic liquid crystal medium. The authors identified undulated travelling waves of the active bacteria, which can be controlled by the orientational field of the liquid crystal medium prescribed by photopatterning. A fascinating result is that the passive liquid crystal medium interacts with the wave of active microbes, which perturbs the orientational field and makes a secondary wave in the passive but orientationally elastic medium. The manuscript presents thorough experimental analysis of the phenomenon, which is supported by theoretical calculations and simulations. After starting with simple orientation patterns with initially straight movement of the bacteria, the authors demonstrate the ability of controlling bacterial motion in more complicated structures including circles and even letters showing to possible future application of the system. In my opinion, the paper is well written, the results are interesting for the broad readership of Nature Communications. The findings are expected to have high impact; therefore I suggest the publication after a revision.\nMy questions and comments are the following:\n1. The authors write that the instability resulting in the undulating bacterial trajectories is related to the concentration of bacteria. Can a critical concentration be calculated theoretically? I think there should be more discussion to shed light on the reason for undulation.\n2. On page 4, line 83-84 we can read: “bacterial streams experience a bending instability and develop longitudinal undulations”. Aren’t they transversal?\n3. Please indicate unit or definition for omega and k in Fig. 1k. It would be beneficial to add more physical explanation and insight into the usefulness of displaying such dispersion relations. Please add more discussion and explain the meaning of Figure 1j and 1k in relation with Fig 2b and 2c.\n4. In the part “Mechanism of active traveling waves” the hand-waving explanation of the instability is too vague; a better description of the arguments is needed. The origins of the different torques playing role have to be clarified as well as the calculation of the net force (or torque?). Fig. 2d and 2e must be improved for easier understanding. The authors subsequently discuss with torques and forces, that should also be improved to avoid confusion.\n5. Repetition of equation 1 must be eliminated. First it appears on page 10 then on page 19.", "reviewer_3": "Key Results:\nThe paper investigates the behavior of swimming bacteria within liquid crystal (LC) environments, focusing on the controllability of traveling waves through the design of the director force field using advanced photopatterning techniques. The key findings provide a deep understanding of the interplay between the externally imposed nematic director field and the bacterial motion following an undulating instability. This interplay results in the emergence of a double traveling wave, marked by an \"active\" bacterial wave followed by a \"passive\" wave of the nematic components of the LC.\nThe basic mechanism is as follows: initially, the nematic field of the LC aligns with the externally imposed photopattern containing splay distortions. The bacteria move along these splay distortions, forming a jet that captures more and more bacteria. As the bacterial concentration in the jet increases, the linear bacterial stream undergoes an undulating instability, forming a traveling wave along the same splay distortion. This bacterial wave then \"pushes\" the nematic molecules in the LC, causing them to reorient in a traveling wave pattern that moves at the same speed as the bacterial wave but with a constant phase lag.\nThe authors explain the observed double-wave mechanism in terms of surface anchoring, changes in viscosity due to bacterial motion, and torque imbalances on the nematic field at different regions of the bacterial wave caused by the splay distortion of the externally imposed nematic director. These findings are substantiated with clear experimental data and matching computational simulations. Furthermore, the authors leverage the precise control of the director field design granted by the photopatterning setup to generate double traveling waves with varied complexities, including simple patterns, spirals with different chiralities, and letters.\n-----\nValidity:\nThe data presented in the paper appear technically sound, obtained using appropriate experimental, theoretical, and computational techniques. The interpretation of results is generally robust, with the authors providing sufficient evidence for their claims, particularly in demonstrating the impact of controlled nematic director fields on bacterial behavior. However, the explanation of the basic interaction generating the feedback mechanism from bacteria to LC could use more clarity concerning the terms in the mathematical model.\nThe authors successfully explain the formation of the double wave and the phase lag in terms of torques due to forces between bacteria and nematic molecules. However, it is not immediately clear where these forces are accounted for in their equations. The authors mention that bacteria rotate LC molecules through director-flow coupling by generating active flow, thus pushing the LC wave forward, and that the LC wave is effectively attracted by the bacterial wave. They provide a helpful diagram (Figure 3b) comparing contributions to the LC wave from bacterial interaction, including flow alignment, corotation, and advection. However, the only clearly identifiable term dependent on bacterial position and alignment in their mathematical model is the active part of the stress tensor in the Navier-Stokes equation. Although the authors state that their model is based on one derived by Genkin et al., they significantly alter the model topology by employing an agent-based model instead of continuous PDEs. Presenting the complete employed model in its entirety would benefit the readability and clarity of the theoretical part without requiring constant references to other publications.\n-----\nSignificance:\nThe conclusions drawn from the study have substantial potential significance for the fields of active nematics and living liquid crystals. The ability to control bacterial movement using externally imposed nematic director fields could open new avenues for applications in bioengineering, materials science, and directed bacterial transport. The clear and comprehensive explanation of double traveling waves under controlled conditions represents a novel finding compared to previous works in the field. It might be beneficial to highlight the conceptual progress offered by this manuscript in relation to the Nature Physics paper by Turiv et al., given the similarity in the theoretical model, with the exception of the agent-based swimmers, and the analogous experimental setup.\n-----\nData and Methodology:\nWhile I do not possess the expertise necessary to thoroughly evaluate the experimental setup, the overall quality of the data and presentation is high, with detailed figures, clear supplementary videos, and comprehensive supplementary information. The techniques used to analyze the data, including dead bacterium path tracing to infer the force field, FFT to identify the power spectrum and dispersion relations of the instability, and schematic visualization of the nematic director field, are appropriate and comprehensive. Here are a few minor suggestions:\n1) In Supplementary Figure 3a, it is unclear which bacterial concentration is plotted. Is it the number of bacteria in the entire system or the number of bacteria along the bacterial stream over the splay distortion? How was this number measured?\n2) Lines 82-84 reference a critical bacterial concentration at which the bacterial stream experiences a bending instability, shown in Figure 3a. Contrarily to a similar figure from Turiv et al., the slope in Supplementary Figure 3a is not as pronounced relative to fluctuations. This raises the question of whether the interpolated number of bacteria at the time of undulation is a critical instability threshold or wether we are simply observing a transient state towards asymptotic traveling wave behavior already after the wave instability critical threshold.\n3) What algorithm or procedure did the authors use to calculate theta vs x in Figure 1e? Did they average the angle of each bacterium in the stream region, or did they identify the stream region with imaging techniques and calculate the stream's angle with respect to the x-axis?\n4) In the captions of Figures 1g and 1h, it would be clearer if the authors mentioned that the tracked bacterium is dead, as stated in the main text.\n5) The values of ⟨v⟩ in Figure 1i before the wave instability seem to be zero with no fluctuations, which is surprising. How were these values calculated, and why is there no fluctuation? Is this an artifact of the algorithm used by the authors?\n6) In the space-time plot Figure 1j, it is unclear what the color legend of the heat map represents (number of bacteria? Some normalized density signal?). Additionally, it is not clear what segment of space the x-axis refers to. I suggest identifying the y coordinate in some of the other figures that the space-time map pertains to.\n7) In the captions of Figure 1k, specify that omega refers to the wave frequency, given that omega is also used for vorticity in Figures 1l and 1m. The authors mention what omega and k are in the materials and methods section, but clarifying this in the captions would be beneficial.\n8) Regarding lines 120-122, when I tried plotting the rectifying pattern defined as is, it was reflected along the y-axis compared to Supplementary Figure 4 and movie 5. The pattern shown by the authors has n_x=−cos(ψ), rather than n_x=cos⁡(ψ).\n9) The nematic director mentioned in lines 143-145 is unclear. The authors seem to refer to the general definition of a 2D nematic director where ψ is intended as the nematic angle. However, since ψ was previously defined as π*y/L, its interpretation is confusing. Additionally, earlier context identified the x component of the nematic director as ∣cos⁡(ψ)∣, not cos(ψ). I suggest clarifying what this nematic director refers to and whether ψ retains its previous definition or has a different one.\n10) The term \"symmetry broken\" in lines 148 and 149 is unusual. Consider changing it to \"broken symmetry.\"\n-----\nAnalytical Approach:\nThe hybrid PDE/agent-based model employed is innovative, cleverly devised, and suitable for representing the experimental system. However, understanding the model in its entirety is challenging due to its presentation. There is no single place where all model equations are clearly laid out, all terms explained, and all symbols defined. Instead, readers must cross-reference multiple papers (references [26, 32]), definitions within the main section of the manuscript, and separate sections of the materials and methods. In addition to that, some sections appear inconsistent:\n1) In equation (7) (line 393), the external force term is missing, contrary to equation (1) (lines 348 and 191).\n2) Regarding the position of the bacteria, what equation was actually used, equation (3) or equation (4)? In line 357, the authors state, \"For each bacterium, its equation of motion in the x direction could be [equation (3)].\" This hypothetical phrasing is confusing. Is equation (3) actually used in the model, or is it a simpler reference equation?\n3) The explanation of the agent-based model equations in lines 353-362 is confusing. The statistical calculations connecting the agent-based model to the continuum model (lines 360-362) are written in a cryptic way. Referencing the programming language used for simulation code (line 362) doesn't help clarify the employed simulation algorithm.\nGiven the novel theoretical model topology and the impressive reproduction of experimental results, a much more comprehensive and clear theory and simulation section is needed. The current information makes reproducing the simulation results challenging, if not impossible. I suggest the authors remove the \"Theoretical model of numerical simulations for bacteria wave\" section from the appendix and write a more comprehensive \"theoretical model and numerical simulations\" section in the Supporting Information, laying out the full model and outlining the hybrid simulation algorithm. Additionally, I recommend the simulation script be made available for review as supplementary material.\n-----\nReferences:\nThe manuscript references previous literature appropriately, including works by Genkin et al., highlighting the transition from spontaneous alignment methods to externally imposed director fields. The references are relevant and well-integrated, providing a solid foundation for the study. However, a minor omission was found: in reference 30, the journal name (Science) is missing."}, "author_response": "We greatly appreciate all the reviewers for evaluating our manuscript of NCOMMS-25-40685-A, and providing valuable comments. All reviewers’ comments and suggestions are quite helpful for us to substantially improve the quality and readability of this manuscript.\nWe have carefully revised our manuscript and Supplementary Information by taking the respected reviewers’ comments as appropriate into account. Our point-by-point response to the reviewers’ comments and the changes made in our revised manuscript with highlighted yellow color are included as follows. We hope the new version will be found more convincing and meet the journal’s criteria for publication.\nResponse to reviewers\nReviewer #2\nThe manuscript by Wu and co-workers presents studies on an interesting system composed of a water based lyotropic liquid crystalline base fluid doped by living bacteria. The dynamics of the active matter system of motile microorganisms can be controlled by the orientation of the anisotropic liquid crystal medium. The authors identified undulated travelling waves of the active bacteria, which can be controlled by the orientational field of the liquid crystal medium prescribed by photopatterning. A fascinating result is that the passive liquid crystal medium interacts with the wave of active microbes, which perturbs the orientational field and makes a secondary wave in the passive but orientationally elastic medium. The manuscript presents thorough experimental analysis of the phenomenon, which is supported by theoretical calculations and simulations. After starting with simple orientation patterns with initially straight movement of the bacteria, the authors demonstrate the ability of controlling bacterial motion in more complicated structures including circles and even letters showing to possible future application of the system. In my opinion, the paper is well written, the results are interesting for the broad readership of Nature Communications. The findings are expected to have high impact; therefore I suggest the publication after a revision.\nResponse:\nWe thank the Reviewer for suggesting the publication of our work. We greatly appreciate the respected reviewer’s positive comments on the interest and novelty of our work. According to the valuable suggestions, we have made changes throughout the revised manuscript.\nQ2.1 The authors write that the instability resulting in the undulating bacterial trajectories is related to the concentration of bacteria. Can a critical concentration be calculated theoretically? I think there should be more discussion to shed light on the reason for undulation.\nA2.1 Response:\nWe sincerely thank the reviewer for their constructive comments. According to the paper (see Ref. Proceedings of the National Academy of Sciences 111, 1265-1270 (2014) (Ref. 25)), the critical concentration can be calculated theoretically by the equation c = K / (alpha * l * U_0 * L), where K is an average Frank elastic constant of the LC, h is the cell thickness, alpha is a constant, l is the length of a bacterium, U_0 is hydrodynamic force dipole of the strength and L is the bacterial coherence length (period of the bacteria waves in experiment). In our experiment, we take K = 10^-12pN, h = 10μm, alpha~1, l = 5μm, U_0 ~1pN μm, L = 100μm , and then we can get the critical concentration c~2*10^13m^-3, which is the same order as experimental results c = 1.2*10^13m^-3. Undulation happens as the bacteria activity is high enough and activity is affected by bacteria concentration. The fluctuation of bacterial stream increases, which disrupt the balance between bacterial orientation and LC director and thus undulation forms. We have added more description of undulation in the revised manuscript.\nOriginal Text in Original Manuscript:\nLines 82-84, page 4: “As the bacterial concentration crosses a critical threshold c = 1.2*10^13m^-3, Supplementary Fig. 3a, the bacterial streams experience a bending instability and develop longitudinal undulations, Supplementary Movie 1.”\nRevised Text in Revised Manuscript:\nLines 82-93, page 4: “As the bacterial concentration crosses a critical threshold c = 1.2*10^13m^-3, Supplementary Fig. 3a, the bacterial streams experience a bending instability and develop longitudinal undulations, Supplementary Movie 1. The critical concentration can be calculated theoretically by the equation c = K / (alpha * l * U_0 * L), where K is an average Frank elastic constant of the LC, h is the cell thickness, alpha is a constant, l is the length of a bacterium, U_0 is hydrodynamic force dipole of the strength and L is the bacterial coherence length (period of the bacteria waves in experiment)25. In our experiment, we take K = 10^-12pN, h = 10μm, alpha~1, l = 5μm, U_0 ~1pN μm, L = 100μm, and then we can get the critical concentration c~2*10^13m^-3, which is the same order as experimental results c = 1.2*10^13m^-3. Undulation happens as the bacteria activity is high enough and activity is affected by bacteria concentration. The fluctuation of bacterial stream increases, which disrupt the balance between bacterial orientation and LC director and thus undulation forms.”\nQ2.2 On page 4, line 83-84 we can read: “bacterial streams experience a bending instability and develop longitudinal undulations”. Aren’t they transversal?\nA2.2 Response:\nWe sincerely thank the reviewer for this valuable comment. We totally agree with the comment that undulations are transversal and change the description in the revised manuscript.\nOriginal Text in Original Manuscript:\nLines 83-84, page 4: “the bacterial streams experience a bending instability and develop longitudinal undulations”\nRevised Text in Revised Manuscript:\nLines 83-84, page 4: “the bacterial streams experience a bending instability and develop transversal undulations”\nQ2.3 Please indicate unit or definition for omega and k in Fig. 1k. It would be beneficial to add more physical explanation and insight into the usefulness of displaying such dispersion relations. Please add more discussion and explain the meaning of Figure 1j and 1k in relation with Fig 2b and 2c.\nA2.3 Response:\nWe thank the reviewer for the valuable comment. ω refers to normalized frequency with the unit s^-1 and k refers to normalized wavenumber with the unit μm^-1. According to the reviewers' comments, we add the definition and unit of omega and k in Fig. 1k and change the labels in Fig 2b-c to be consistent with Fig 1j-k.\nThe spatiotemporal mapping of the active traveling waves along the +x axis is shown in Fig. 1j. Color represents the height profile of the waves and the maximum corresponds to the peak of the waves, Fig. 1j, indicating that the waves travel from -x to +x. By applying Fast Fourier Transform (FFT) to the profile data of traveling wave, we get the power spectrum of the wave (Fig. 1k), which reflects the dispersion relation of the wave. The non-zero dispersion in the second and forth quadrants with odd parity demonstrate the unidirectional wave traveling along the +x axis. Both Fig. 1j and 1k prove that waves are unidirectional.\nFig. 2b-c are the simulated space-time graph and FFT power spectrum of the bacterial traveling wave. The diagonal maximum bands distribution in simulated space-time graph and quadrants distribution in simulated power spectrum indicate the unidirectional waves, which are in agreement with experimental results in Figure 1.\nOriginal figures in Original Manuscript:\nLines 136-137, page 7: Figure 1. ……(k) Power spectrum of the bacterial wave by FFT.\nLines 170-172, page 9: Figure 2. ……(b) Simulated space-time graph of active traveling wave along the +x-axis. (c) Corresponding simulated power spectrum of the traveling wave.\nRevised figures in Revised Manuscript:\nLines 149-150, page 8: Figure 1. ……(k) Power spectrum of the bacterial wave by FFT. ω and k refer to normalized frequency and wavenumber, respectively.\nLines 192-194, page 10: Figure 2. ……(b) Simulated space-time graph of active traveling wave along the +x-axis. (c) Corresponding simulated power spectrum of the traveling wave. ω and k refer to normalized frequency and wavenumber, respectively.\nOriginal Text in Original Manuscript:\nLines 111-115, page 5: “The spatiotemporal mapping of the active traveling waves along the +x axis is shown in Fig. 1j. By applying Fast Fourier Transform (FFT) to the profile data of traveling wave, we get the power spectrum of the wave (Fig. 1k), which reflects the dispersion relation of the wave. The non-zero dispersion in the second and forth quadrants demonstrate the unidirectional wave traveling along the +x axis.”\nRevised Text in Revised Manuscript:\nLines 119-126, page 6: “The spatiotemporal mapping of the active traveling waves along the +x axis is shown in Fig. 1j. Color represents the height profile of the waves and the maximum corresponds to the peak of the waves, Fig. 1j, indicating that the waves travel from -x to +x. By applying Fast Fourier Transform (FFT) to the profile data of traveling wave, we get the power spectrum of the wave (Fig. 1k), which reflects the dispersion relation of the wave. The non-zero dispersion in the second and forth quadrants with odd parity demonstrate the unidirectional wave traveling along the +x axis. Both Fig. 1j and 1k prove that waves are unidirectional.”\nOriginal Text in Original Manuscript:\nLines 146-147, page 7: “The space-time graph and power spectrum are in agreement with the experimental results, Fig. 2b-c.”\nRevised Text in Revised Manuscript:\nLines 161-163, page 8: “The diagonal maximum bands distribution in simulated space-time graph and quadrants distribution in simulated power spectrum indicate the unidirectional waves, which are in agreement with experimental results in Figure 1, Fig.2b-c.”\nQ2.4 In the part “Mechanism of active traveling waves” the hand-waving explanation of the instability is too vague; a better description of the arguments is needed. The origins of the different torques playing role have to be clarified as well as the calculation of the net force (or torque?). Fig. 2d and 2e must be improved for easier understanding. The authors subsequently discuss with torques and forces, that should also be improved to avoid confusion.\nA2.4 Response:\nWe thank the reviewer for the valuable comment. As suggested by the reviewer, we add four plots in Figure 2 to have a better understanding of the torque and force. To visualize the distribution of torque around the wave, we first calculated the angle between bacteria and LC director beta = theta - psi, where theta represents the angle between the local bacterial stream and the x-axis and psi is the nematic angle anchored on the substrates, Fig. 2f-g. The bacterial flows trigger a polar viscous response in the LC. The bacterial stream imposes a counterclockwise(CCW) torque on the designed director at region ① and a clockwise (CW) torque on the director at region ②, Fig. 2d and 2f.\nThe difference of designed directors between region ① and region ② produces different out-of-equilibrium director reorientation and thus different viscosities eta_+ = eta_- . The difference eta_+ = eta_- produces a rectified net driving force that can be estimated as F_driving proportional to (eta_+ - eta_-)v, where v is the speed of bacteria. Such symmetry broken of driving force can be visualized numerically by mapping the spatial distribution of force, F = nabla*(0,0,T), where T proportional to sin(2*beta) is the torque exerted from bacteria to LC, Fig. 2h-i. As indicated in Fig. 2f and 2h, the distribution of beta and torque are not symmetric around the wave in splay region and the induced force point rightwards, driving the wave to move rightwards. Thus, in director field with broken symmetry, active force caused by the swimming bacteria will push the flow field forward and thus the wave propagation.\nAs a comparison, the undulating waves of bacterial streams in a uniform director field are also demonstrated, in which the waves are not propagating, Supplementary Fig. 5 and Supplementary Movie 7. As shown in Fig. 2e, the bacterial stream imposes the same CCW and CW torques at regions ① and ②, which produces F_driving = 0, indicating there is no symmetry broken of the interaction between the bacterial stream and LC directors. There is no force along x axis and thus no traveling wave, Fig. 2g and 2i.\nNewly Added Figures:\nLines 200-205, page 11: Figure 2. ……(f) Spatial distribution of the angle beta (color bar) between bacteria and LC director in splay director field. Solid black line is the center line of the bacterial wave in splay region. Brown arrows represent the orientation of bacteria with angle theta. Black short bars are director field with angle psi. (g) Spatial distribution of the angle beta (color bar) in uniform director field. (h) Spatial distribution of the torque in splay region. Brown and black arrows represent the bacteria orientation and the force induced by the torque, respectively. (i) Spatial distribution of the torque and force in uniform director field.\nRevised Text in Revised Manuscript:\nLines 168-189, page 9-10: “Due to the constant bacterial stream, the director is persistently out of equilibrium due to the surface anchoring29,33,36. To visualize the distribution of torque around the wave, we first calculated the angle between bacteria and LC director beta = theta - psi, where theta represents the angle between the local bacterial stream and the x-axis and psi is the nematic angle anchored on the substrates, Fig. 2f-g. The bacterial flows trigger a polar viscous response in the LC. The bacterial stream imposes a counterclockwise (CCW) torque on the designed director at region ① and a clockwise (CW) torque on the director at region ②, Fig. 2d and 2f. The difference of designed directors between region ① and region ② produces different out-of-equilibrium director reorientation and thus different viscosities eta_+ = eta_- . The difference eta_+ = eta_- produces a rectified net driving force that can be estimated as F_driving proportional to (eta_+ - eta_-)v, where v is the speed of bacteria. Such symmetry broken of driving force can be visualized numerically by mapping the spatial distribution of force, F = nabla*(0,0,T), where T proportional to sin(2*beta) is the torque exerted from bacteria to LC, Fig. 2h-i. As indicated in Fig. 2f and 2h, the distribution of beta and torque are not symmetric around the wave in splay region and the induced force point rightwards, driving the wave to move rightwards. Thus, in director field with broken symmetry, active force caused by the swimming bacteria will push the flow field forward and thus the wave propagation.\nAs a comparison, the undulating waves of bacterial streams in a uniform director field are also demonstrated, in which the waves are not propagating, Supplementary Fig. 5 and Supplementary Movie 7. As shown in Fig. 2e, the bacterial stream imposes the same CCW and CW torques at regions ① and ②, which produces F_driving = 0, indicating there is no symmetry broken of the interaction between the bacterial stream and LC directors. There is no force along x axis and thus no traveling wave, Fig. 2g and 2i.”\nQ2.5 Repetition of equation 1 must be eliminated. First it appears on page 10 then on page 19.\nA2.5 Response:\nWe thank the reviewer for the valuable comment. As suggested by the reviewers, the repeated equation has been removed.\nReviewer #3\nThe paper investigates the behavior of swimming bacteria within liquid crystal (LC) environments, focusing on the controllability of traveling waves through the design of the director force field using advanced photopatterning techniques. The key findings provide a deep understanding of the interplay between the externally imposed nematic director field and the bacterial motion following an undulating instability. This interplay results in the emergence of a double traveling wave, marked by an \"active\" bacterial wave followed by a \"passive\" wave of the nematic components of the LC.\nThe basic mechanism is as follows: initially, the nematic field of the LC aligns with the externally imposed photopattern containing splay distortions. The bacteria move along these splay distortions, forming a jet that captures more and more bacteria. As the bacterial concentration in the jet increases, the linear bacterial stream undergoes an undulating instability, forming a traveling wave along the same splay distortion. This bacterial wave then \"pushes\" the nematic molecules in the LC, causing them to reorient in a traveling wave pattern that moves at the same speed as the bacterial wave but with a constant phase lag.\nThe authors explain the observed double-wave mechanism in terms of surface anchoring, changes in viscosity due to bacterial motion, and torque imbalances on the nematic field at different regions of the bacterial wave caused by the splay distortion of the externally imposed nematic director. These findings are substantiated with clear experimental data and matching computational simulations. Furthermore, the authors leverage the precise control of the director field design granted by the photopatterning setup to generate double traveling waves with varied complexities, including simple patterns, spirals with different chiralities, and letters.\nResponse:\nWe greatly appreciate the respected reviewer’s positive comments on our work.\nQ3.1 The data presented in the paper appear technically sound, obtained using appropriate experimental, theoretical, and computational techniques. The interpretation of results is generally robust, with the authors providing sufficient evidence for their claims, particularly in demonstrating the impact of controlled nematic director fields on bacterial behavior. However, the explanation of the basic interaction generating the feedback mechanism from bacteria to LC could use more clarity concerning the terms in the mathematical model.\nA3.1 Response:\nWe thank the reviewer for this insightful comment. The feedback mechanism from bacteria to the liquid crystal (LC) is primarily mediated through the active stress term (sigma_act) in the momentum balance equation (Eqn. 6 in our manuscript). This term, sigma_act = Lambda*c*P, is the crucial coupling point where bacterial properties directly influence the LC dynamics. Here, c is the bacterial concentration, Lambda is active constant and P is the orientation field, both interpolated from the agent-based particle model onto the continuum grid using Gaussian kernels (Eqns. 8 & 9).\nThe physical interpretation is that each swimming bacterium acts as a microscopic force dipole, exerting pushes and pulls on the surrounding fluid. The collective effect of these dipoles, summed via sigma_act, generates an active flow (v). This flow then enters the Beris-Edwards equation (Eqn. 1) through the term S = (xi*A+Omega)*(Q+I/2)+(Q+I/2)*(xi*A-Omega)-xi*(2*Q+I)Tr(Q*W), which accounts for the coupling between the flow field (v), its derivatives (rate-of-strain tensor A and vorticity tensor Omega), and the nematic director field (n). This director-flow coupling is the mechanism that reorients the LC, \"pushing the LC wave forward\" as mentioned in the manuscript.\nIn summary, the pathway is:\nBacteria (Agents) → Interpolated Fields (c, P) → Active Stress (sigma_act) → Active Flow (v) → Director-Flow Coupling (S) → LC Director Reorientation (∂Q/∂t)\nNewly Added Text (Supplementary information, pages 2-4):\nAgent-based particle model for bacteria. We used an agent-based particle model to describe the motion of swimming bacteria in passive liquid crystals. Each of the bacteria is represented by its position r_i and a polarization p_i which describe the self-purpose direction of bacteria. And we used an over-damped equation to describe the evolution of the position r_i:\n∂r_i / ∂t = v_0 * p_i + gamma*F_ij + v, (2)\nwhere the v_0 is self-propose velocity of bacteria, gamma*F_ij describes the reaction between neighboring bacteria, and v is the fluid velocity to describe the advection of bacteria. We saw our system as a quasi-2D system so the use a linear cut-off force F_ij(r_ij) to describe the soft excluded volume interactions where r_ij means the distance between two bacteria.\nF_ij = k*(r_ij - r_c) / r_ij (r_ij < r_c) (3)\n0 (r_ij >= r_c)\nwhere r_c means the cut-off distance of our reaction between neighboring bacteria.\nBacterium tends to align with the local orientation of liquid crystals, p_i = (cos(theta),sin(theta)), as shown in ref.1,2, the form leads to the evolution of the angle of bacteria as follows:\n∂theta / ∂t = gamma_B * sin(2*(psi-theta)) + D_r * xi(t), (4)\nHere the bacterium tends to align with the local orientation of the nematic director n_hat = (cos(psi),sin(psi)) from the continuum model) at a rate gamma_B, and to reorient randomly driven by a stochastic white noise xi(t) with D_r.\nContinuum model for liquid crystals. Just like previous works1,3,4, we used the Beris-Edwards (BE) equations for the background nematic liquid crystals coupled with the linear momentum balance equation for the flow field. The first equation is in terms of a tensorial order parameter Q5,6 (equation 1).\nThe tensor Q is defined Q = q(n*n - I/2) where q is the order parameter representing the nematic ordering and n is a unit vector representing the nematic orientation. The tensor S = (xi*A+Omega)*(Q+I/2)+(Q+I/2)*(xi*A-Omega)-xi*(2*Q+I)Tr(Q*W) describes alignment with the fluid flow. We keep flow-aligning parameter xi = 0.3 in this study. The next term H describes relaxation to minimize the Landau-de Gennes free energy:\nF_LdG = integral (-a/2 * Tr(Q^2) + b/4 * Tr(Q^2)^2 + L/2 * |nabla*Q|^2) dr. (5)\nas H = delta*F_LdG / delta*Q. The parameters a and b are used to describe the mean order parameter of tensor Q or the temperature of liquid crystals systems. In our system we set a = 1, b = 2, so the equilibrium state is a nematic phase with q = sqrt(2a/b) The final term F_anchoring quantifies the anchoring energy, which characterizes the interaction between the liquid crystal orientation and the patterned substrate. Besides, the over-damped stokes equation to describe the flow field is in the form of:\nnabla*(sigma_a + sigma_s + sigma_visc + sigma_active - p*I) - gamma_v * v = 0 (6)\nnabla*v = 0 (7)\nThe equation 6 represents the linear momentum balance equation, commonly employed to describe the dynamics of overdamped systems. The terms sigma_a and sigma_s denote the anisotropic and isotropic components of the total stress tensor, respectively. Additionally, sigma_visc characterizes the viscous stress. The final stress tensor, sigma_active = Lambda*c*P, is the sole term in the Landau–de Gennes equation specifically associated with the active bacterial system. Here, c is the bacterial concentration, Lambda is active constant and P is the orientation field The last term gamma_v * v describe the friction from the substrate.\nTo convert particle configurations into c and P, we adopt a particle-to-grid interpolation scheme based on Gaussian spreading functions. Specifically, the contribution from each particle decays Gaussianly with distance, such that field values at position x are\nc(x) = sum G_sigma(|x_i - x|) (8)\nP(x) = sum_i (p_i*p_i - I/2) * G_sigma(|x_i - x|) / sum_i G_sigma(|x_i - x|) (9)\nwhere x_i and p_i are the position and orientation of the i-th particle, and G_sigma(r) = 1/(2*pi*sigma^2) * exp(-r^2 / (2*sigma^2)) is the Gaussian interpolation kernel. Details of the numerical implementation and code can be found in the Supplemental Material.\nTo make the model dimensionless, we rescale the length by the characteristic nematic liquid crystals length l_0 = sqrt(K/a), time by the orientational relaxation time tau = 1/(Gamma*a), and energy by the free energy of liquid crystals per unit length epsilon_0 = sqrt(a*K). Here Gamma is the relaxation rate. The main model parameters are summarized in Table I. The continuum fields are initialized with small-amplitude random perturbations near the zero state, and the particles are randomly seeded with uniformly distributed positions and orientations throughout the simulation box.\nQ3.2 The authors successfully explain the formation of the double wave and the phase lag in terms of torques due to forces between bacteria and nematic molecules. However, it is not immediately clear where these forces are accounted for in their equations. The authors mention that bacteria rotate LC molecules through director-flow coupling by generating active flow, thus pushing the LC wave forward, and that the LC wave is effectively attracted by the bacterial wave. They provide a helpful diagram (Figure 3b) comparing contributions to the LC wave from bacterial interaction, including flow alignment, corotation, and advection. However, the only clearly identifiable term dependent on bacterial position and alignment in their mathematical model is the active part of the stress tensor in the Navier-Stokes equation. Although the authors state that their model is based on one derived by Genkin et al., they significantly alter the model topology by employing an agent-based model instead of continuous PDEs. Presenting the complete employed model in its entirety would benefit the readability and clarity of the theoretical part without requiring constant references to other publications.\nA3.2 Response:\nWe agree with the reviewer that presenting a self-contained model description improves clarity. While our approach is inspired by Genkin et al., the hybrid agent-continuum nature of our model represents a significant modification. Below, we provide a consolidated summary of the complete model as implemented in our study.\nOur complete coupled model consists of two parts:\n1. Agent-Based Model for Bacteria:\nMotion:\n∂r_i / ∂t = v_0 * p_i + gamma*F_ij + v,\nwhere F_ij is a short-range repulsive force (Eqn. 3) to prevent overcrowding, and v is the fluid velocity field that advects the bacteria.\nOrientation:\n∂theta / ∂t = gamma_B * sin(2*(psi-theta)) + D_r * xi(t),\nThis equation governs the reorientation of bacteria. The key term gamma_B*sin(2*(psi-theta)) creates the torque that aligns the bacterium with the local nematic director (psi) of the LC. This is the primary mechanism for the feedback from LC to bacteria.\n2. Continuum Model for Liquid Crystals and Flow:\nNematic Order (Beris-Edwards Eqn.):\n(∂_t + v*nabla)Q - S - Gamma*H + F_anchoring = 0, (1)\nS = (xi*A+Omega)*(Q+I/2)+(Q+I/2)*(xi*A-Omega)-xi*(2*Q+I)Tr(Q*W) handles the coupling of the director to the flow field H = delta*F_LdG / delta*Q governs the relaxation of the LC based on the Landau-de Gennes free energy (Eqn. 5) and the anchoring potential.\nFlow Field (Overdamped Stokes Eqns.):\nnabla*(sigma_a + sigma_s + sigma_visc + sigma_active - p*I) - gamma_v*v = 0\nnabla * v = 0\nThe meaning of the terms is:\nsigma_a, sigma_s: LC elastic stresses; sigma_visc: viscous stress; -pI: isotropic pressure; sigma_active = Lambda*c*P: Active stress (The key term from bacteria)\nCoupling Scheme:\nThe agent-based and continuum models are bidirectionally coupled:\nBacteria → LC: Bacterial positions and orientations are converted into continuous fields c and P via Gaussian interpolation (Eqns. 8, 9). These fields source the active stress sigma_active in the momentum equation, affecting the flow field to generate flow that distorts the LC. According to equation 1, Q is affected by advection of the nematic field by the flow (v*nabla*Q), flow alignment ((xi*A+Omega)*(Q+I/2)+(Q+I/2)*(xi*A-Omega)), corotation with the local vorticity xi*(2*Q+I)Tr(Q*W), free energy Gamma*H and external anchoring F_anchoring. All of the factors contribute influencing the LC. The description of these terms has been updated in the manuscript.\nLC → Bacteria: The local LC director angle psi enters the agent's orientation dynamics (Eqn. 4), exerting a torque that aligns the bacteria. The fluid velocity v advects the bacteria (Eqn. 2).\nThe description about the simulation part have been updated, please see A3.1 Response.\nOriginal Text in Original Manuscript:\nLines 190-194, page 9: “The effect of the bacteria on LC can be described by the equation\n(∂_t + u*nabla)Q - S - Gamma*H + F_anchoring = 0, (1)\nwhere u*nabla*Q describes advection (Adv) of the nematic field by the flow and u is the flow field. S characterizes the reorientation of nematic director by the flow gradient, including flow alignment (FL) and corotation with the local vorticity (CR).”\nRevised Text in Revised Manuscript:\nLines 218-223, pages 11-12: “The effect of the bacteria on LC can be described by the equation\n(∂_t + v*nabla)Q - S - Gamma*H + F_anchoring = 0, (1)\nwhere v*nabla*Q describes advection (Adv) of the nematic field by the flow and here v is the fluid velocity. The tensor S = (xi*A+Omega)*(Q+I/2)+(Q+I/2)*(xi*A-Omega)-xi*(2*Q+I)Tr(Q*W) describes alignment with the fluid flow, including flow alignment (FL, (xi*A+Omega)*(Q+I/2)+(Q+I/2)*(xi*A-Omega)) and corotation with the local vorticity (CR, xi*(2*Q+I)Tr(Q*W)), see Supplementary information.”\nQ3.3 The conclusions drawn from the study have substantial potential significance for the fields of active nematics and living liquid crystals. The ability to control bacterial movement using externally imposed nematic director fields could open new avenues for applications in bioengineering, materials science, and directed bacterial transport. The clear and comprehensive explanation of double traveling waves under controlled conditions represents a novel finding compared to previous works in the field. It might be beneficial to highlight the conceptual progress offered by this manuscript in relation to the Nature Physics paper by Turiv et al., given the similarity in the theoretical model, with the exception of the agent-based swimmers, and the analogous experimental setup.\nA3.3 Response:\nWe appreciate the reviewer's insightful questions. In the Nature Physics paper by Turiv et al., they demonstrated that the bacteria move along the splay distortions, forming a jet that captures more and more bacteria in the splay regions. As the concentration of bacteria exceeds certain threshold, undulating instability is formed. They further demonstrated the application of polar jets in cargo transport.\nInspired by this work, we make a substantial step forward. We discovered that as the bacterial concentration in the jet increases, the linear bacterial stream undergoes an undulating instability, forming a traveling wave along the same splay distortion. This bacterial wave then \"pushes\" the nematic molecules in the LC, causing them to reorient in a traveling wave pattern that moves at the same speed as the bacterial wave but with a constant phase lag. We explained the observed double-wave mechanism in terms of surface anchoring, changes in viscosity due to bacterial motion, and torque imbalances on the nematic field at different regions of the bacterial wave caused by the splay distortion of the externally imposed nematic director. The interplay between the externally imposed nematic director field and the bacterial motion results in the emergence of a double traveling wave, marked by an \"active\" bacterial wave followed by a \"passive\" wave of the nematic components of the LC. Finally, we leveraged the precise control of the director field design granted by the photopatterning setup to generate double traveling waves with varied complexities, including simple patterns, spirals with different chirality, and letters.\nWe have highlighted the following text in the main text on Page 18.\nIn this work, the orientational order of LCs and the interactions between the active units—namely, the bacteria—and the out-of-equilibrium reorientation of LC molecules introduce new dynamic effects. The nonequilibrium interaction within the system is responsible for triggering active traveling waves within the bacterial streams in the background LCs. The active traveling waves can further actuate the LCs to travel, leading to the formation of a unique traveling pattern of active-passive traveling waves with a steady phase difference.\nQ3.4 While I do not possess the expertise necessary to thoroughly evaluate the experimental setup, the overall quality of the data and presentation is high, with detailed figures, clear supplementary videos, and comprehensive supplementary information. The techniques used to analyze the data, including dead bacterium path tracing to infer the force field, FFT to identify the power spectrum and dispersion relations of the instability, and schematic visualization of the nematic director field, are appropriate and comprehensive. In Supplementary Figure 3a, it is unclear which bacterial concentration is plotted. Is it the number of bacteria in the entire system or the number of bacteria along the bacterial stream over the splay distortion? How was this number measured?\nA3.4 Response:\nWe appreciate the reviewer's valuable comments. We calculate the bacterial concentration by using the number of bacteria in the field of view for one bacterial stream. To visualize the area, we newly add a picture of the region named Supplementary Figure 3a in the supplementary and previous Supplementary Figure 3a become Supplementary Figure 3b. By enhancing the contrast of the image, we can get the number of bacteria by using analyzing particles in ImageJ-Fiji. We then divided the number of bacteria in a stripe by the stripe volume V = HWh to calculate the concentration, where H=263.7μm is the height of the stripe, W = 701.6μm is the width of the image along the x axis and h=10μm is cell thickness. We have added the details in the revised manuscript.\nOriginal Figures in Original Supplementary:\nSupplementary Fig. 3. Active traveling wave in opposite directions in Fig. 1. (a) Temporal evolution of bacterial concentration. Dashed line shows the onset of undulation. (b) Temporal evolution of instability amplitude of two waves. (c) Temporal evolution of instability wavelength. (d) Temporal evolution of the alignment parameter q. (e) Map of bacterial velocities. The length of arrows and the color scale linearly with the speed of bacteria. Scale bars: 100 μm. Error bars: standard deviations.\nRevised Figures in Revised Supplementary: (on the pages 3-4)\nSupplementary Fig. 3. Active traveling wave in opposite directions in Fig. 1. (a) The area to count bacterial number. White dots represent bacteria. W and H represents the width and height of the given area. (b) Temporal evolution of bacterial concentration. Dashed line shows the onset of undulation. (c) Temporal evolution of instability amplitude of two waves. (d) Temporal evolution of instability wavelength. (e) Temporal evolution of the alignment parameter q. (f) Map of bacterial velocities. The length of arrows and the color scale linearly with the speed of bacteria. Scale bars: 100 μm. Error bars: standard deviations.\nOriginal Text in Original Manuscript:\nLines 335-337, page 18: “By enhancing the contrast of the image, we can then get the number of bacteria using analyzing particles in ImageJ-Fiji and then calculate the bacterial concentration in the given region.”\nRevised Text in Revised Manuscript:\nLines 365-369, page 20: “By enhancing the contrast of the image, we can get the number of bacteria by using analyzing particles in ImageJ-Fiji. We then divided the number of bacteria in a stripe by the stripe volume V = HWh to calculate the concentration, where H=263.7μm is the height of the stripe, W = 701.6μm is the width of the image along the x axis and h=10μm is cell thickness.”\nQ3.5 Lines 82-84 reference a critical bacterial concentration at which the bacterial stream experiences a bending instability, shown in Figure 3a. Contrarily to a similar figure from Turiv et al., the slope in Supplementary Figure 3a is not as pronounced relative to fluctuations. This raises the question of whether the interpolated number of bacteria at the time of undulation is a critical instability threshold or wether we are simply observing a transient state towards asymptotic traveling wave behavior already after the wave instability critical threshold.\nA3.5 Response:\nWe thank the reviewer for the insightful questions. The reason for the inconspicuous slope relative to fluctuations compared to the figure from Turiv et al. is that we use different methods to calculate the concentration. In the paper by Turiv et al., the concentration of bacteria in an undulating jet was calculated as the total number of bacteria n in the jet divided by the volume of the jet. The volume of the jet changes with developing jet and thus concentration fluctuates. But in our case, we use a fixed region containing bacterial streams with a fixed volume to calculate bacterial concentration. The slope in Supplementary Figure 3a is not as pronounced relative to fluctuations. By observing the transition towards undulating wave, we get the beginning time of undulation.\nQ3.6 What algorithm or procedure did the authors use to calculate theta vs x in Figure 1e? Did they average the angle of each bacterium in the stream region, or did they identify the stream region with imaging techniques and calculate the stream's angle with respect to the x-axis?\nA3.6 Response:\nWe appreciate the reviewer's insightful questions. theta represents the angle between the local bacterial stream and the x-axis. We identify the bacterial stream like Figure 1d and then measure the position of stream. With two neighboring positions (x1, y1) and (x2, y2), theta at the first position can be calculated by theta = tan^-1((y2-y1)/(x2-x1)). Then theta along x-axis can be calculated. We have added the details in the revised manuscript.\nOriginal Text in Original Manuscript:\nLines 337-339, page 18: “Polar angle phi and the angle theta between bacteria and flow direction are calculated by tracking the bacteria trajectories and recording the position data of bacteria at the given moment. Angles are calculated by adjacent data.”\nRevised Text in Revised Manuscript:\nLines 369-372, page 20-21: “Polar angle phi and the angle theta between bacteria and flow direction are calculated by tracking the bacteria trajectories and recording the position data of bacteria at the given moment. With two neighboring positions (x1, y1) and (x2, y2), theta at the first position can be calculated by theta = tan^-1((y2-y1)/(x2-x1)). Then theta along x-axis can be calculated.”\nQ3.7 In the captions of Figures 1g and 1h, it would be clearer if the authors mentioned that the tracked bacterium is dead, as stated in the main text.\nA3.7 Response:\nWe thank the reviewer for the valuable comment. As suggested by the reviewers, the figure captions in Figure 1g and 1h now include the information of tracked dead bacterium, with revisions highlighted in yellow.\nRevised figure captions in Revised Manuscript:\nLines 143-146, page 8: Figure 1. ……(g) Trajectory of the tracked dead bacterium before wave propagates. Black line represents the bacterial stream. White straight arrow indicates the traveling direction and curved arrow represents rotation direction. (h) Trajectory of the tracked dead bacterium at the traveling wave stage.\nQ3.8 The values of ⟨v⟩ in Figure 1i before the wave instability seem to be zero with no fluctuations, which is surprising. How were these values calculated, and why is there no fluctuation? Is this an artifact of the algorithm used by the authors?\nA3.8 Response:\nWe appreciate the reviewer's insightful questions. It is an artifact of the algorithm because the wave is not obvious. We remeasure the velocity at several moments before wave instability and revise Figure 1i.\nOriginal figures in Original Manuscript: Revised figure in Revised Manuscript:\nLines 146-147, page 8: Figure 1. ……(i) Temporal evolution of average speed of the traveling waves.\nQ3.9 In the space-time plot Figure 1j, it is unclear what the color legend of the heat map represents (number of bacteria? Some normalized density signal?). Additionally, it is not clear what segment of space the x-axis refers to. I suggest identifying the y coordinate in some of the other figures that the space-time map pertains to.\nA3.9 Response:\nWe appreciate the reviewer's valuable comment. The color legend of the heat map represents the height profile of the wave h(x, t), which equals to y coordinate of the wave. We have added the definition of h and correspondence of x-axis in Figure 1f and the description in figure caption (highlighted yellow). We change the labels of the coordinate in Fig. 1k and Fig. 2b-c that the space-time map pertains to.\nOriginal figures in Original Manuscript:\nLines 135-137, page 7: Figure 1. ……(j) Space-time graph of the traveling wave. (k) Power spectrum of the bacterial wave by FFT.\nLines 170-172, page 9: Figure 2. ……(b) Simulated space-time graph of active traveling wave along the +x-axis. (c) Corresponding simulated power spectrum of the traveling wave.\nRevised figures and figure captions in Revised Manuscript:\nLines 143-150, page 8: Figure 1. ……(f) ……. The bottom red vertical arrow indicates the h in (j). (j) Space-time graph of the traveling wave. h(x, t) represents the height profile of the wave (y coordinate). The definition of h and correspondence of x-axis are indicated in (f). (k) Power spectrum of the bacterial wave by FFT. ω and k refer to normalized frequency and wavenumber, respectively.\nLines 192-194, page 10: Figure 2. ……(b) Simulated space-time graph of active traveling wave along the +x-axis. (c) Corresponding simulated power spectrum of the traveling wave. ω and k refer to normalized frequency and wavenumber, respectively.\nWe have also highlighted the following text from Methods for the Reviewer’s convenience (Lines 372-377, page 21):\nThe velocity of the traveling wave is measured by tracking several peaks of the wave at different moments and calculating the average. To plot the space-time graph, we extract the x and y (height profile h) coordinates of the bacteria wave at different moments and plot the map by software Tecplot360. We carry a fast Fourier transform (FFT) on the h(x, t) data and square the absolute value of this FFT to get a power spectrum in frequency ω, and wavenumber k.\nQ3.10 In the captions of Figure 1k, specify that omega refers to the wave frequency, given that omega is also used for vorticity in Figures 1l and 1m. The authors mention what omega and k are in the materials and methods section, but clarifying this in the captions would be beneficial.\nA3.10 Response:\nWe thank the reviewer for the valuable suggestion. As suggested by the reviewer, the figure captions in Figure 1k now include the meaning of omega and k, with revisions highlighted in yellow. In addition, we change the ω for vorticity in Figure 1l and 1m to Ω.\nOriginal figures in Original Manuscript:\nRevised figure captions in Revised Manuscript:\nLines 149-152, page 8: Figure 1. ……(k) Power spectrum of the bacterial wave by FFT. ω and k refer to normalized frequency and wavenumber, respectively. (l) Simulated flow field around the wave. Black arrows indicate flow field velocity. Green line and arrow indicate bacterial wave and traveling direction, respectively. Color bar: normalized vorticity Ω. (m) Simulated active force indicated by black arrows.\nQ3.11 Regarding lines 120-122, when I tried plotting the rectifying pattern defined as is, it was reflected along the y-axis compared to Supplementary Figure 4 and movie 5. The pattern shown by the authors has n_x=−cos(ψ), rather than n_x=cos⁡(ψ).\nA3.11 Response:\nWe thank the reviewer for the valuable comment. As suggested by the reviewers， c the pattern shown in Supplementary Figure 4 and movie 5 should have n_x = -cos(psi). We have revised the following text:\nOriginal Text in Original Manuscript:\nLines 120-121, page 6: “active waves travel in the same direction can also be rectified by using the pattern designed as n_hat = (n_x, n_y, n_z) = (cos(psi), sin(psi), 0), psi = pi*y/L and L = 100 μm”\nRevised Text in Revised Manuscript:\nLines 131-132, page 6: “active waves travel in the same direction can also be rectified by using the pattern designed as n_hat = (n_x, n_y, n_z) = (-cos(psi), sin(psi), 0), psi = pi*y/L and L = 100 μm”\nQ3.12 The nematic director mentioned in lines 143-145 is unclear. The authors seem to refer to the general definition of a 2D nematic director where ψ is intended as the nematic angle. However, since ψ was previously defined as π*y/L, its interpretation is confusing. Additionally, earlier context identified the x component of the nematic director as ∣cos(ψ)∣, not cos(ψ). I suggest clarifying what this nematic director refers to and whether ψ retains its previous definition or has a different one.\nA3.12 Response:\nWe thank the reviewer for the valuable comment. Here n_hat and psi are the same as previous definition |cos(psi)| and in lines 77-78, page 4. We revise the text in the manuscript.\nOriginal Text in Original Manuscript:\nLines 143-144, page 7: “The swimming direction of the particle is defined as p_hat and the nematic directors are denoted as n_hat = (cos(psi), sin(psi))”\nRevised Text in Revised Manuscript:\nLines 157-159, page 8: “The self-propose direction of the particle is defined as p_hat and the nematic directors are denoted as n_hat = (|cos(psi)|, sin(psi), 0), where psi = pi*y/L and L = 100 μm”\nQ3.13 The term \"symmetry broken\" in lines 148 and 149 is unusual. Consider changing it to \"broken symmetry.\"\nA3.13 Response:\nWe thank the reviewer for the valuable comment. We have revised the following text:\nOriginal Text in Original Manuscript:\nLines 148-149, page 7: “The breaking of PT symmetry of the traveling waves is rooted in the symmetry broken of the interaction between the bacterial wave and the LC director, which is caused by the symmetry broken”\nRevised Text in Revised Manuscript:\nLines 164-165, page 8: “The breaking of PT symmetry of the traveling waves is rooted in the broken symmetry of the interaction between the bacterial wave and the LC director, which is caused by the broken symmetry”\nQ3.14 In equation (7) (line 393), the external force term is missing, contrary to equation (1) (lines 348 and 191).\nA3.14 Response:\nWe thank the reviewer for the valuable comment. The conflict comes from the different simulation methods. Now we revised this part and use equation (1) (line 219) to replace previous equation (7).\nQ3.15 Regarding the position of the bacteria, what equation was actually used, equation (3) or equation (4)? In line 357, the authors state, \"For each bacterium, its equation of motion in the x direction could be [equation (3)].\" This hypothetical phrasing is confusing. Is equation (3) actually used in the model, or is it a simpler reference equation?\nA3.15 Response:\nWe thank the reviewer for the valuable comment. The conflict comes from the different simulation methods. Now we revised this part and use equation (2) (∂r_i / ∂t = v_0 * p_i + gamma*F_ij + v) (previous equation 4) to calculate the position of bacteria.\nQ3.16 The explanation of the agent-based model equations in lines 353-362 is confusing. The statistical calculations connecting the agent-based model to the continuum model (lines 360-362) are written in a cryptic way. Referencing the programming language used for simulation code (line 362) doesn't help clarify the employed simulation algorithm.\nA3.16 Response:\nWe thank the reviewer for the valuable comment. As suggested by the reviewers, the simulation method part has been totally revised and moved to Supplementary information: (on the pages 2-4)\nAgent-based particle model for bacteria. We used an agent-based particle model to describe the motion of swimming bacteria in passive liquid crystals. Each of the bacteria is represented by its position r_i and a polarization p_i which describe the self-purpose direction of bacteria. And we used an over-damped equation to describe the evolution of the position r_i:\n∂r_i / ∂t = v_0 * p_i + gamma*F_ij + v, (2)\nwhere the v_0 is self-propose velocity of bacteria, gamma*F_ij describes the reaction between neighboring bacteria, and v is the fluid velocity to describe the advection of bacteria. We saw our system as a quasi-2D system so the use a linear cut-off force F_ij(r_ij) to describe the soft excluded volume interactions where r_ij means the distance between two bacteria.\nF_ij = k*(r_ij - r_c) / r_ij (r_ij < r_c) (3)\n0 (r_ij >= r_c)\nwhere r_c means the cut-off distance of our reaction between neighboring bacteria.\nBacterium tends to align with the local orientation of liquid crystals, p_i = (cos(theta),sin(theta)), as shown in ref.1,2, the form leads to the evolution of the angle of bacteria as follows:\n∂theta / ∂t = gamma_B * sin(2*(psi-theta)) + D_r * xi(t), (4)\nHere the bacterium tends to align with the local orientation of the nematic director n_hat = (cos(psi),sin(psi)) from the continuum model) at a rate gamma_B, and to reorient randomly driven by a stochastic white noise xi(t) with D_r.\nContinuum model for liquid crystals. Just like previous works1,3,4, we used the Beris-Edwards (BE) equations for the background nematic liquid crystals coupled with the linear momentum balance equation for the flow field. The first equation is in terms of a tensorial order parameter Q5,6 (equation 1).\nThe tensor Q is defined Q = q(n*n - I/2) where q is the order parameter representing the nematic ordering and n is a unit vector representing the nematic orientation. The tensor S = (xi*A+Omega)*(Q+I/2)+(Q+I/2)*(xi*A-Omega)-xi*(2*Q+I)Tr(Q*W) describes alignment with the fluid flow. We keep flow-aligning parameter xi = 0.3 in this study. The next term H describes relaxation to minimize the Landau-de Gennes free energy:\nF_LdG = integral (-a/2 * Tr(Q^2) + b/4 * Tr(Q^2)^2 + L/2 * |nabla*Q|^2) dr. (5)\nas H = delta*F_LdG / delta*Q. The parameters a and b are used to describe the mean order parameter of tensor Q or the temperature of liquid crystals systems. In our system we set a = 1, b = 2, so the equilibrium state is a nematic phase with q = sqrt(2a/b) The final term F_anchoring quantifies the anchoring energy, which characterizes the interaction between the liquid crystal orientation and the patterned substrate. Besides, the over-damped stokes equation to describe the flow field is in the form of:\nnabla*(sigma_a + sigma_s + sigma_visc + sigma_active - p*I) - gamma_v * v = 0 (6)\nnabla*v = 0 (7)\nThe equation 6 represents the linear momentum balance equation, commonly employed to describe the dynamics of overdamped systems. The terms sigma_a and sigma_s denote the anisotropic and isotropic components of the total stress tensor, respectively. Additionally, sigma_visc characterizes the viscous stress. The final stress tensor, sigma_active = Lambda*c*P, is the sole term in the Landau–de Gennes equation specifically associated with the active bacterial system. Here, c is the bacterial concentration, Lambda is active constant and P is the orientation field The last term gamma_v * v describe the friction from the substrate.\nTo convert particle configurations into c and P, we adopt a particle-to-grid interpolation scheme based on Gaussian spreading functions. Specifically, the contribution from each particle decays Gaussianly with distance, such that field values at position x are\nc(x) = sum G_sigma(|x_i - x|) (8)\nP(x) = sum_i (p_i*p_i - I/2) * G_sigma(|x_i - x|) / sum_i G_sigma(|x_i - x|) (9)\nwhere x_i and p_i are the position and orientation of the i-th particle, and G_sigma(r) = 1/(2*pi*sigma^2) * exp(-r^2 / (2*sigma^2)) is the Gaussian interpolation kernel. Details of the numerical implementation and code can be found in the Supplemental Material.\nTo make the model dimensionless, we rescale the length by the characteristic nematic liquid crystals length l_0 = sqrt(K/a), time by the orientational relaxation time tau = 1/(Gamma*a), and energy by the free energy of liquid crystals per unit length epsilon_0 = sqrt(a*K). Here Gamma is the relaxation rate. The main model parameters are summarized in Table I. The continuum fields are initialized with small-amplitude random perturbations near the zero state, and the particles are randomly seeded with uniformly distributed positions and orientations throughout the simulation box.\nQ3.17 The hybrid PDE/agent-based model employed is innovative, cleverly devised, and suitable for representing the experimental system. However, understanding the model in its entirety is challenging due to its presentation. There is no single place where all model equations are clearly laid out, all terms explained, and all symbols defined. Instead, readers must cross-reference multiple papers (references [26, 32]), definitions within the main section of the manuscript, and separate sections of the materials and methods. Given the novel theoretical model topology and the impressive reproduction of experimental results, a much more comprehensive and clear theory and simulation section is needed. The current information makes reproducing the simulation results challenging, if not impossible. I suggest the authors remove the \"Theoretical model of numerical simulations for bacteria wave\" section from the appendix and write a more comprehensive \"theoretical model and numerical simulations\" section in the Supporting Information, laying out the full model and outlining the hybrid simulation algorithm. Additionally, I recommend the simulation script be made available for review as supplementary material.\nA3.17 Response:\nWe thank the reviewer for the valuable comment. As suggested by the reviewer, the simulation method part has been revised and a new \"theoretical model and numerical simulations\" section has been added in the Supporting Information, also see A3.16 Response. The main model parameters are summarized in Table I:\nTable I. Main model parameters\nParameter Value Description\nLambda -15 - -25 Active constant\nK 1 Elastic constant\na 1 Landau–de Gennes coefficient\nb 2 Landau–de Gennes coefficient\neta 1 Isotropic viscosity\nzeta 0.48 Aspect ratio parameter\nGamma 1 Relaxation rate\ngamma_B 1-3 Orientation relaxation rate\nv_0 10-20 Bacterial velocity\nQ3.18 The manuscript references previous literature appropriately, including works by Genkin et al., highlighting the transition from spontaneous alignment methods to externally imposed director fields. The references are relevant and well-integrated, providing a solid foundation for the study. However, a minor omission was found: in reference 30, the journal name (Science) is missing.\nA3.18 Response:\nWe thank the reviewer for the valuable comment. We have added the journal name of reference 30.\nRevised reference in Revised Manuscript:\nLines 456-457, page 23: “Peng, C., Turiv, T., Guo, Y., Wei, Q.-H. & Lavrentovich, O. D. Command of active matter by topological defects and patterns. Science 354, 882-885 (2016).”\nWe sincerely appreciate the editors and reviewers for their time in evaluating our manuscript (NCOMMS-25-40685A) and providing valuable comments. Their suggestions have significantly improved both the academic quality and readability of this work. We have revised the manuscript according to both the editorial requests in the attached Author Checklist and the reviewers’ comments. All modifications in the revised manuscript are highlighted in yellow as follows:\n1. Respond to Reviewer 2's comments and include a discussion on the ODE regarding the orientation of the bacteria in the mains.\nResponse:\nNewly Added Text: (on page 7)\nThe core of this interaction is a torque that the bacterial stream exerts on the LC director. To quantify this, we introduce the angle beta = psi - theta, where theta represents the angle between the local bacterial stream and the x-axis, and psi is the nematic angle anchored on the substrates, Fig. 2f-g. This aligning torque is mathematically captured in our agent-based model for bacteria (see Supplementary Text) by the reorientation dynamics: ∂theta / ∂t = gamma_B * sin(2*(psi-theta)) + D_r * xi(t) . Here, the key term gamma_B * sin(2*(psi-theta)) creates the torque that aligns the bacterium with the local nematic director (psi) of the LC. This is the primary mechanism for the feedback from LC to bacteria.\n2. We ask all authors to submit their code in a public repository. Please refer to the checklist for details.\nResponse:\nWe have deposited the computer codes for simulations and data processing in the Figshare repository.\n(Figshare private link: https://figshare.com/s/37530bcc8e111cd77135, Figshare DOI: https://doi.org/10.6084/m9.figshare.30490277 ).\nResponse to reviewers\nReviewer #2\nThe manuscript has been sufficiently improved compared to its previous state; therefore, I recommend its acceptance for publication. All questions are properly answered and accordingly, proper modifications have been done in the manuscript.\nResponse:\nWe sincerely thank the reviewer’s valuable comments and positive endorsement. We are pleased that the revisions have addressed all concerns satisfactorily. We fully support the reviewer’s recommendation for acceptance and are available for any post-review requirements.\nReviewer #3\nI would like to thank the authors for thoroughly and comprehensively addressing the questions and points I raised. I found their response satisfactory, and the modifications they made to the manuscript and supplementary information address all the issues I highlighted in my previous review. I especially appreciated the reformulation of the theoretical model and simulation algorithm section, which is now clear, comprehensive, and easy to follow for anyone who might want to reproduce their results. I do have a couple of minor points which are not critical for publication but could still enhance the value of this manuscript for readers. However, these suggested revisions are minor and do not impact my overall recommendation, which is to publish the manuscript in the current form.\nResponse:\nWe gratefully acknowledge the reviewer's insightful suggestions and favorable evaluation of our revisions. We have carefully revised our manuscript accordingly following the reviewer’s suggestions.\n- Suggested minor revision 1:\nIn their response to my question 3.2, the authors point to the ODE for the orientation of the bacteria and mention:\n\"This equation governs the reorientation of bacteria. The key term gamma_Bsin(2(psi-theta)) creates the torque that aligns the bacterium with the local nematic director (psi) of the LC. This is the primary mechanism for the feedback from LC to bacteria.\"\nI believe this is crucial information that should not be confined to the rebuttal letter but should be explicitly included in the manuscript's discussion of the torque. The authors have significantly improved their explanations of the torque arguments central to the mechanism of active traveling waves, as suggested by reviewer 2 (question and answer 2.4 to reviewer 2). The aforementioned sentence should be included in that section to clarify the origin of the torque in the mathematical model used by the authors.\nResponse:\nWe gratefully acknowledge the reviewer's insightful suggestions and favorable evaluation of our revisions. We have added a discussion about on the ODE regarding the orientation of the bacteria to the main text.\nNewly Added Text: (on page 7)\nThe core of this interaction is a torque that the bacterial stream exerts on the LC director. To quantify this, we introduce the angle beta = psi - theta, where theta represents the angle between the local bacterial stream and the x-axis, and psi is the nematic angle anchored on the substrates, Fig. 2f-g. This aligning torque is mathematically captured in our agent-based model for bacteria (see Supplementary Text) by the reorientation dynamics: ∂theta / ∂t = gamma_B * sin(2*(psi-theta)) + D_r * xi(t) . Here, the key term gamma_B * sin(2*(psi-theta)) creates the torque that aligns the bacterium with the local nematic director (psi) of the LC. This is the primary mechanism for the feedback from LC to bacteria.\n- Suggested minor revision 2:\nThe clear and detailed explanation of the mathematical model implemented by the authors indirectly fulfills the requirement for reproducibility of the simulations. I had previously suggested that the authors share their simulation code if possible. At lines 76-77 of the newly added \"Theoretical model and numerical simulations\" in the Supplementary Material, the authors state:\n\"Details of the numerical implementation and code can be found in the Supplemental Material.\"\nIt is unclear if this sentence indicates an intention to share their simulation code or refers to another document in the supplementary material package. As far as I can see, the only section that provides information on the numerical implementation is the aforementioned section, and there is no actual code in the supplementary material beyond the theoretical explanation of the algorithm. The only code added to the supplementary material is for generating figures 2 and 4, which does not include any simulations.\nWhile adding the simulation code would significantly enhance the reproducibility of the authors' analyses and results, if the authors decide not to share their code, they should remove the aforementioned sentence.\nResponse:\nWe gratefully acknowledge the reviewer's insightful suggestions and favorable evaluation of our revisions. We have deposited the computer codes for simulations and data processing in the Figshare repository.\n(Figshare private link: https://figshare.com/s/37530bcc8e111cd77135, Figshare DOI: https://doi.org/10.6084/m9.figshare.30490277 ).\n- Suggested minor revision 3:\nRegarding the code shared by the authors to generate figures 2 and 4, the code would greatly benefit from in-line commenting. Additionally, running the code in Matlab as it is currently written does not work because it references local hard drive paths (e.g., ['Z:\\zeyang\\bacteria',path,'\\thp.out']) that are inaccessible to the reader. I suggest the authors rewrite these input/output operations using local relative paths and ensure that the code works immediately upon downloading on a different computer.\nResponse:\nWe gratefully acknowledge the reviewer's insightful suggestions and favorable evaluation of our revisions. Following the reviewer's suggestion, we have rewritten these input/output operations to use local relative paths, ensuring that the code runs immediately upon download on different computers. We have deposited the computer codes for simulations and data processing in the Figshare repository.\n(Figshare private link: https://figshare.com/s/37530bcc8e111cd77135, Figshare DOI: https://doi.org/10.6084/m9.figshare.30490277 )."}, {"round": 2, "reviewer_comments": {"reviewer_2": "The manuscript has been sufficiently improved compared to its previous state; therefore, I recommend its acceptance for publication. All questions are properly answered and accordingly, proper modifications have been done in the manuscript.", "reviewer_3": "I would like to thank the authors for thoroughly and comprehensively addressing the questions and points I raised. I found their response satisfactory, and the modifications they made to the manuscript and supplementary information address all the issues I highlighted in my previous review. I especially appreciated the reformulation of the theoretical model and simulation algorithm section, which is now clear, comprehensive, and easy to follow for anyone who might want to reproduce their results. I do have a couple of minor points which are not critical for publication but could still enhance the value of this manuscript for readers. However, these suggested revisions are minor and do not impact my overall recommendation, which is to publish the manuscript in the current form.\n- Suggested minor revision 1:\nIn their response to my question 3.2, the authors point to the ODE for the orientation of the bacteria and mention:\n\"This equation governs the reorientation of bacteria. The key term gamma_Bsin(2(psi-theta)) creates the torque that aligns the bacterium with the local nematic director (psi) of the LC. This is the primary mechanism for the feedback from LC to bacteria.\"\nI believe this is crucial information that should not be confined to the rebuttal letter but should be explicitly included in the manuscript's discussion of the torque. The authors have significantly improved their explanations of the torque arguments central to the mechanism of active traveling waves, as suggested by reviewer 2 (question and answer 2.4 to reviewer 2). The aforementioned sentence should be included in that section to clarify the origin of the torque in the mathematical model used by the authors.\n- Suggested minor revision 2:\nThe clear and detailed explanation of the mathematical model implemented by the authors indirectly fulfills the requirement for reproducibility of the simulations. I had previously suggested that the authors share their simulation code if possible. At lines 76-77 of the newly added \"Theoretical model and numerical simulations\" in the Supplementary Material, the authors state:\n\"Details of the numerical implementation and code can be found in the Supplemental Material.\"\nIt is unclear if this sentence indicates an intention to share their simulation code or refers to another document in the supplementary material package. As far as I can see, the only section that provides information on the numerical implementation is the aforementioned section, and there is no actual code in the supplementary material beyond the theoretical explanation of the algorithm. The only code added to the supplementary material is for generating figures 2 and 4, which does not include any simulations.\nWhile adding the simulation code would significantly enhance the reproducibility of the authors' analyses and results, if the authors decide not to share their code, they should remove the aforementioned sentence.\n- Suggested minor revision 3:\nRegarding the code shared by the authors to generate figures 2 and 4, the code would greatly benefit from in-line commenting. Additionally, running the code in Matlab as it is currently written does not work because it references local hard drive paths (e.g., ['Z:\\zeyang\\bacteria',path,'\\thp.out']) that are inaccessible to the reader. I suggest the authors rewrite these input/output operations using local relative paths and ensure that the code works immediately upon downloading on a different computer."}, "author_response": "We gratefully acknowledge the reviewer's insightful suggestions and favorable evaluation of our revisions. We have carefully revised our manuscript accordingly following the reviewer’s suggestions.\n- Suggested minor revision 1:\nIn their response to my question 3.2, the authors point to the ODE for the orientation of the bacteria and mention:\n\"This equation governs the reorientation of bacteria. The key term gamma_Bsin(2(psi-theta)) creates the torque that aligns the bacterium with the local nematic director (psi) of the LC. This is the primary mechanism for the feedback from LC to bacteria.\"\nI believe this is crucial information that should not be confined to the rebuttal letter but should be explicitly included in the manuscript's discussion of the torque. The authors have significantly improved their explanations of the torque arguments central to the mechanism of active traveling waves, as suggested by reviewer 2 (question and answer 2.4 to reviewer 2). The aforementioned sentence should be included in that section to clarify the origin of the torque in the mathematical model used by the authors.\nResponse:\nWe gratefully acknowledge the reviewer's insightful suggestions and favorable evaluation of our revisions. We have added a discussion about on the ODE regarding the orientation of the bacteria to the main text.\nNewly Added Text: (on page 7)\nThe core of this interaction is a torque that the bacterial stream exerts on the LC director. To quantify this, we introduce the angle beta = psi - theta, where theta represents the angle between the local bacterial stream and the x-axis, and psi is the nematic angle anchored on the substrates, Fig. 2f-g. This aligning torque is mathematically captured in our agent-based model for bacteria (see Supplementary Text) by the reorientation dynamics: ∂theta / ∂t = gamma_B * sin(2*(psi-theta)) + D_r * xi(t) . Here, the key term gamma_B * sin(2*(psi-theta)) creates the torque that aligns the bacterium with the local nematic director (psi) of the LC. This is the primary mechanism for the feedback from LC to bacteria.\n- Suggested minor revision 2:\nThe clear and detailed explanation of the mathematical model implemented by the authors indirectly fulfills the requirement for reproducibility of the simulations. I had previously suggested that the authors share their simulation code if possible. At lines 76-77 of the newly added \"Theoretical model and numerical simulations\" in the Supplementary Material, the authors state:\n\"Details of the numerical implementation and code can be found in the Supplemental Material.\"\nIt is unclear if this sentence indicates an intention to share their simulation code or refers to another document in the supplementary material package. As far as I can see, the only section that provides information on the numerical implementation is the aforementioned section, and there is no actual code in the supplementary material beyond the theoretical explanation of the algorithm. The only code added to the supplementary material is for generating figures 2 and 4, which does not include any simulations.\nWhile adding the simulation code would significantly enhance the reproducibility of the authors' analyses and results, if the authors decide not to share their code, they should remove the aforementioned sentence.\nResponse:\nWe gratefully acknowledge the reviewer's insightful suggestions and favorable evaluation of our revisions. We have deposited the computer codes for simulations and data processing in the Figshare repository.\n(Figshare private link: https://figshare.com/s/37530bcc8e111cd77135, Figshare DOI: https://doi.org/10.6084/m9.figshare.30490277 ).\n- Suggested minor revision 3:\nRegarding the code shared by the authors to generate figures 2 and 4, the code would greatly benefit from in-line commenting. Additionally, running the code in Matlab as it is currently written does not work because it references local hard drive paths (e.g., ['Z:\\zeyang\\bacteria',path,'\\thp.out']) that are inaccessible to the reader. I suggest the authors rewrite these input/output operations using local relative paths and ensure that the code works immediately upon downloading on a different computer.\nResponse:\nWe gratefully acknowledge the reviewer's insightful suggestions and favorable evaluation of our revisions. Following the reviewer's suggestion, we have rewritten these input/output operations to use local relative paths, ensuring that the code runs immediately upon download on different computers. We have deposited the computer codes for simulations and data processing in the Figshare repository.\n(Figshare private link: https://figshare.com/s/37530bcc8e111cd77135, Figshare DOI: https://doi.org/10.6084/m9.figshare.30490277 )."}]

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