Bacteria-Activated Janus Particles Driven by Chemotaxis - ACS Nano

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Bacteria-Activated Janus Particles Driven by Chemotaxis Zihan Huang, Pengyu Chen, Guo-Long Zhu, Ye Yang, Ziyang Xu, and Li-Tang Yan ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.8b01842 • Publication Date (Web): 23 May 2018 Downloaded from http://pubs.acs.org on May 24, 2018

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Bacteria-Activated Janus Particles Driven by Chemotaxis Zihan Huang, Pengyu Chen, Guolong Zhu, Ye Yang, Ziyang Xu, and Li-Tang Yan* State Key Laboratory of Chemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing 100084, China

*Corresponding Author Email: [email protected]

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ABSTRACT: In the development of biocompatible nano/micro-motors for drug and cargo delivery, motile bacteria represent an excellent energy source for biomedical applications. Despite intense research of the fabrication of bacteria-based motors, how to effectively utilize the instinctive responses of bacteria to environmental stimuli in the fabrication process, particularly, chemotaxis, remains an urgent and critical issue. Here, by developing a molecular-dynamics model of bacterial chemotaxis, we present an investigation of the transport of a bacteria-activated Janus particle driven by chemotaxis. Upon increasing the stimuli intensity, we find that the transport of the Janus particle undergoes an intriguing second-order state transition: from a composite random walk, combining power-law-distributed truncated Lévy flights with Brownian jiggling, to an enhanced directional transport with size-dependent reversal of locomotion. A state diagram of Janus-particle transport depending on the stimuli intensity and particle size is presented, which allows approaches to realize controllable and predictable propulsion directions. The physical mechanism of these transport behaviors is revealed by performing a theoretical modeling based on the bacterial noise and Janus geometries. Our findings could provide a fundamental insight into the physics underlying the transports of anisotropic particles driven by microorganisms, and highlight stimulus-response techniques and asymmetrical design as a versatile strategy to possess a wide array of potential applications for future biocompatible nano/micro-devices. KEYWORDS Janus particle, bacteria, chemotaxis, second-order state transition, controllable propulsion direction


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Owing to their ability to perform selected mechanical movements, nano/micro-motors have recently drawn significant attention and been the research focuses in various fields, ranging from physics1 and chemistry2 to material science3 and nanotechnology.4 Such motor machinery with good controllability is highly desirable to develop a wide variety of integrated engineering systems5,6 and allow effective approaches for diverse applications such as medical drug and cargo delivery. 7-9 However, the majority of current motors rely on toxic fuel sources or extreme environments in their design, such as hydrogen peroxide,10 bromine11 and strongly acidic media,12 making them nearly irrelevant for biomedical applications. Bacteria, which convert surrounding chemical energy into mechanical work without the need for toxic fuels or artificial catalysts13 and have adopted symbiotic relationships with their human hosts,14 provide an excellent energy source alternative of motors for biomedical applications.2 Therefore, fabricating bacteria-activated motors and exploring the physical mechanism underlying the transport behaviors are of great significance in the development of biocompatible nano/micro-scale devices and may open doors to the realization of next-generation bio-hybrid techniques. On the other hand, as a basic but crucial biological function, the instinctive responses of living organisms to environmental stimuli, which help them to realize adaptive behaviors in changing environments, ubiquitously dominate the transport behaviors of complex biological fluids.15,16 For bacterial systems, one representative process relating to the function is the bacterial chemotaxis in response to chemical stimuli, which helps bacteria navigate to niches that are optimal for their growth and survival.17 Such bacterial process can directly control the transport directions of motile bacteria towards or away from chemicals through biasing their random walks,17 thereby possessing a wide array of applications in targeted bioremediation and delivery systems.18 However, despite intense research of the fabrication of bacteria-based motors,19-21 how to effectively utilize the stimulus responses of bacteria in the fabrication process


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remains largely unexplored. This leads to an urgent and critical issue to develop valid design rules for the bacteria-activated particles driven by chemotaxis with definite physical descriptions of underlying mechanisms, which is the focus of this work. In this paper, through developing a molecular-dynamics model of bacterial chemotaxis based on the run-and-tumble dynamics and Monod-Wyman-Changeux (MWC) model,22 we present the investigation of the transport of a chemoattractant-coated Janus particle driven by chemotactic bacteria. This allows us to provide a strategy coupling stimulus-response techniques with asymmetrical design, which leads to particles with controllable propulsion directions. We find that the transport of the Janus particle undergoes a second-order state transition when increasing the stimuli intensity: from a composite random walk, combining power-lawdistributed truncated Lévy flights with Brownian jiggling, to an enhanced directional transport with size-dependent reversal of locomotion. Moreover, the state diagram of Janus-particle transport depending on the stimuli intensity and particle size is presented, which allows approaches to realize a controllable and predictable propulsion direction of the particle. Theoretical modeling based on the bacterial noise and Janus geometries is performed to reveal the physical mechanism of these transport behaviors. These results could offer a theoretical framework to gain a fundamental insight into the transports of anisotropic particles driven by microorganisms, and underscore the crucial role of asymmetrical environment in regulating the transport processes in biophysical systems. Full technical details on the developed model are described in Methods and Supporting Information I, and briefly introduced here. Bacterial motion is modeled by the run-and-tumble dynamics, and the chemotactic responses are described by the MWC model22,23. As shown in Figure 1a, we consider a suspension in a two-dimensional (2D) box24-27 L × L (L = 160 µm) with periodic boundary conditions, consisting of N bacterial cells (E. coli) with number density ϕ = N


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/ L2 and a spherical particle (radius R) where the surface concentration of the coated portion (colored in orange) is c0. Each cell is represented by a spherocylinder,28 with a diameter d0 = 1.5 µm,29 a length 2d0, and a swimming direction ei denoted by the “cap” on the cell (top inset). Both hard bodies of the bacterial cells and the immersed particle are modeled by the sum of identical force centers with short-range repulsive potentials (Figure SI.1). Runge-Kutta method is used for numerical integration with time interval δt = 10-4 s.

Figure 1. (a) The snapshot of the simulation system. Top inset: the model of bacterial cells. (b) Schematic diagram of a run-and-tumble bacterium in a chemoattractant-concentration field. The inset shows the definitions of θ and the orientation n of the particle. (c)-(e) Time evolutions of ξB (x direction) show distinct bacterial activations for particles with different geometries: (c) noncoated particle; (d) full-coated particle with c0 = 1.0 mM; (e) Janus particle with c0 = 1.0 mM. Increasing the stimuli intensity leads to a particle with a more enhanced mobility, while breaking the symmetry results in an inherent bias of bacterial noise.

Chemical stimuli are generated by the coated portion of the immersed particle, where the coating process can potentially be realized by encapsulation30 or secretion of fungus31 in


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experiments. To obtain the chemoattractant-concentration field generated by the particle, we assume that chemical spreads in the bath by classical diffusion, with a diffusion constant ε . Chemical also decays with a rate k in consequence of the enzymatic activity of bacteria. Therefore, the concentration field c obeys the reaction diffusion equation32-34 ∂ t c(r, t ) = ε∇ 2 c − kc ,

(1)

In our systems, we use ε = 500 µm2/s based on the diffusivity of MeAsp,35 a widely used chemoattractant in experiments,22,23,35 and k = 10 s-1.34 Note that the time-dependent solution of eq. 1 rapidly reaches the steady state (Figure SI.3). That is, the chemical diffuses much faster than the particle, thereby leading to a stationary concentration field at each instance (i.e., ∂tc ≡ 0).36 The validity of the model is examined by reproducing the chemotactic aggregation of bacteria (See also Supporting Information II). We turn to an immobile full-coated particle in the bacterial suspension. The concentration field is given by c(r ) = c0 Re −

k / D (r −R)

/ r (See Supporting

Information III), where r is the center-to-center distance between the cell and the particle. We specialize to the case of N = 784 bacteria with number density ϕ = 0.03 /µm2 and radius R = 5.07 µm.37,38 As shown in Figure 2a, the aggregation of bacterial cells can be clearly identified around the particle (See also Movie I), which is consistent with the experimental results (inset).39 More quantitatively, we measure the distributions of bacterial density and show the case where c0 = 1.0 mM in Figure 2b. Note that the minimal detectable concentration c* of MeAsp for E. coli is c* = 0.001 mM,40 thereby leading to a signal-detectable region (SDR) in which the cell can perform chemotactic behaviors (Figure 2b). It can be found in Figure 2c that the radii of SDR rs for a wide range of c0 show high agreements with analytic values, corroborating that the bacterial motion and chemotactic responses can be reliably mimicked by the model.


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Figure 2. (a) Reproduction of chemotactic aggregation. Inset: photomicrograph showing attraction of E. coli bacteria to chemoattractant.39 (b) Plot of the bacterial density distribution where c0 = 1.0 mM. The blue circle is the analytic SDR. (c) The radius of SDR rs versus c0. The red line represents the analytic values determined by c(rs) = c*. RESULTS AND DISCUSSION Effects of Chemotactic Bacterial Activation We begin by examining the transport of full-coated particles with homogeneous chemical stimuli to individually explore the effects of activation given by chemotactic bacteria. We also specialize to the case with ϕ = 0.03 /µm2 and R = 5.07 µm, and systematically change the coating concentration c0. Average over 20 independent runs are performed for each c0. Figure 2a shows the mean square displacements (MSDs) 〈∆r 2 (t )〉 = 〈| x (t ) − x (0) |2 〉 of particle for a wide range of c0, where x(t) is the position vector of the particle and 〈L〉 denotes the ensemble average. It can be found that the particle motion is short-time superdiffusive and long-time Brownian, with the effective diffusivity Deff defined as Deff = lim〈∆r 2 (t )〉 / 4t increases monotonically with c0 (inset t →∞

of Figure 3a). Moreover, the 2D non-Gaussian parameter α 2 (t ) = 〈∆r 4 (t )〉 / 2〈∆r 2 (t )〉 2 − 1 is also calculated to characterize the heterogeneity of transport dynamics. 41 The dramatic departure of

α 2 (t ) from 0 at short times shows the emergence of non-Gaussian transport behaviors (Figure


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3b). To gain a physical insight into the mechanism of such anomalous transports, we use the Langevin formalism to describe the motion of particle, written as

x& (t ) = ξ B (t ) + 2 D ξT (t ) ,

(2)

where D is the intrinsic diffusivity of the particle. ξ B denotes the perturbations exerted by bacteria (i.e., bacterial noise) and ξT is the Gaussian white noise describing the interaction with solvent molecules. We measure the bacterial noise ξ B in the lab frame, and calculated the autocorrelation function 〈 ξ B (t ) ⋅ ξ B (0)〉 . As shown in Figure 3c, 〈 ξ B (t ) ⋅ ξ B (0)〉 are found to follow an exponential decay as 〈 ξ B (t ) ⋅ ξ B (0)〉 = 4 I e −t /τ / τ , where I and τ are the noise intensity and correlation time of bacterial noise respectively, and both increase with the increasing c0. In such cases, the theoretical MSD reads

〈∆r 2 (t )〉 = 4 Dt + 4 I (t + τ e − t /τ − τ )

(3)

based on eq. 2, which is consistent with the simulation results, i.e., short-time superdiffusive ( 〈∆r 2 (t )〉 = 4 It 2 / τ + 4 Dt for t > τ). In addition, the effective diffusivity Deff = lim〈∆r 2 (t )〉 / 4t = I + D increases with I , where I t →∞

dramatically increases when raising the stimuli intensity. That is, increasing the intensity of chemical stimuli directly leads to a particle with a more enhanced mobility.

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Second-Order State Transition: from Composite Random Walk to Enhanced Directional Transport Having established the effects of chemotactic bacterial activation, we now turn to the design of motors driven by chemotactic bacteria. As shown in Figure 1e, breaking the spatial symmetry of chemical stimuli can result in an inherent bias of ξ B , making the particle with asymmetrical geometry a potential candidate to realize directional transports through the persistent drifts exerted by bacteria. To demonstrate this point, the transport process of the Janus (half-coated) particle is investigated, as Janus particles are the simplest and most elementary asymmetrical particles.42-48 The concentration field of Janus particle is given by ∞

c(r ,θ ) = c0 ∑ n =0 an kn (λ r ) Pn (cos θ ) where Pn(t) and kn(t) (n = 0, 1, 2, 3, …) are the Legendre polynomial and modified spherical Bessel function of the second kind, respectively. The definition of θ is delineated in the inset of Figure 1b. an =

2n + 1 1 Pn (t )dt determined by the 2kn (λ R ) ∫0

boundary condition where λ = k / ε (See Supporting Information III). We focus on the system where ϕ = 0.03 /µm2 as well, and systematically change the surface concentration c0. The rotation of Janus particles is nearly free and regardless of c0, and the characteristic time of particle rotation is long enough to cover up the time scale in our systems (See Supporting Information IV). That is, the transports of Janus particles are mainly determined by the bacterial noise. Intriguingly, unlike the full-coated particle which always performs long-time Brownian transports, two distinct transport patterns are identified for Janus particles when increasing c0. Representative trajectories of such two patterns when R = 5.07 µm are shown in Figure 4a for c0 = 0.04 (main view) and 1.0mM (bottom inset) respectively. At low coating concentration (e.g., c0 = 0.04 mM), the trajectory appears random to the eye. However, differing from conventional Brownian transport where each step is statistically identical, the


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Janus particle alternatively performs persistent walks and local jiggling, which are schematically indicated by arrows and dashed circles respectively. To reveal the inner nature of this unusual randomness, a wavelet-based method49 is employed to rapidly separate the persistent “active” runs from random “passive” jiggling (See also Methods). The active portions of tracked positions (partially shown as grey dots) detected by the wavelet analysis are highlighted by grey circles in Figure 4a. To uncover the underlying differences between these active and passive steps, we denote the length of active or passive segment as “step size”, l, and measure the corresponding probability distributions p(l) based on over 100 independent trajectories. As shown in Figure 4b, the size of passive step is exponentially distributed, indicating that statistics of local jiggling obeys the descriptions of classic Brownian random walks.50 For active steps, a power-law tail with slope µ = -2 can be identified. The inset of Figure 4b highlights the exponential and powerlaw distributions, which are both mathematically determined by the maximum likelihood estimation51 (See Supporting Information V). Such power-law tail suggests that the active steps follow the landscape of truncated Lévy flights, where the distribution of step size is heavy-tailed with slope µ satisfying -3 < µ < -1.52 Therefore, the transport of Janus particle at low coating concentration is identified to be a composite random walk (CRW) combining power-law-tail distributed truncated Lévy flights with Brownian jiggling. However, such CRW doesn't hold for the high coating concentration (e.g., c0 = 1.0 mM), where the motion of particle maintains directional persistence as delineated in the bottom inset of Figure 4a. Such reinforcing directionality can be quantitatively characterized by the distribution of turning angles between consecutive steps, which are measured by an error-radius analysis.52 As shown in Figure 4c, the uniformity of turning angles for c0 = 0.04 mM (top) implies that no direction is preferred for turns in CRW, while the biased distribution for c0 = 1.0 mM (bottom)


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shows that the corresponding transport is actually directional. Such directionality of transport has also been confirmed to be persistent for very long time (Figure SI.5). Therefore, a state transition for Janus particle can be identified, from a composite random walk to an enhanced directional transport (EDT) when increasing the intensity of chemical stimuli. Such directional transport also corroborates that the asymmetrical design could be a valid approach to fabricate bacteriaactivated particles with directed propulsion.

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To reveal the origin of these intriguing transport behaviors and understand the physical mechanisms of the state transition, we turn to the distributions of bacterial noise. As shown in


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Figure 5a, differing from the bacterial noise with an inherent bias (non-zero mean) at high c0 (Figure 1e), ξ B is still long-time vanishing at low c0, making the trajectories overall random to the eye. In particular, in the magnified view of ξ B (Figure 5b), athermal jumps of noise with dramatic departures from 0, which result in the active Lévy flights,53 can be clearly identified. However, the bias of chemical stimuli at low c0 is not sufficient to maintain such departures. Hence, these jumps can be dissipated by the following fluctuations and then become intermittently distributed, thereby making the particle motion alternatively persistent and random. However, when c0 is high, the bias of stimuli is able to keep these jumps steady, leading to a non-vanishing mean (Figure 1e) that gives rise to a long-time directional persistence. Therefore, as shown in Figure 5c, we calculate the autocorrelation functions of bacterial noise

〈 ξ B (t ) ⋅ ξ B (0)〉 for various c0 to uncover the fundamental change of particle transports when undergoing the CRW-to-EDT transition. As expected, over sufficiently long times (See Figure SI.6 for the evolutions of 〈 ξ B (t ) ⋅ ξ B (0)〉 covering up to 800 s), 〈 ξ B (t ) ⋅ ξ B (0)〉 decays to zero for low c0 due to the dissipation, and demonstrates a non-zero steady-state value (denoted as V) for high c0 in consequence of the steady athermal jumps. Further, the dependence of V on c0 is given in Figure 5d, showing clearly a critical point at c0 = 0.1mM (See the determination of critical point in Supporting Information VII). Note that the dependence of V on c0 (or log10c0) is continuous-like, indicating such transition may be analogous to the continuous (second-order) phase transition. A central feature of second-order phase transition is that, physical quantities show power-law dependence with characteristic critical exponents near the critical point.54 Hence, to probe this idea in our system, we look for an analogous critical behavior near the critical point, expressed as V − Vc ∝ ( P − Pc ) β where P = log10c0 and β is the critical exponent. Vc


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and Pc are corresponding values at the critical point. As shown in Figure 5e, V - Vc versus P - Pc indeed demonstrates a power-law relation with a critical exponent β = 1.5. Moreover, since the critical exponent is usually system-independent,55 we systematically change the bacterial density ϕ to examine the universality of such second-order transitions. As expected, power-law relation can still be identified, with β remaining 1.5 regardless of the bacterial density. Therefore, the transition indeed conforms to the scenario of the second-order phase transition.

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portion is corresponding to the trajectory highlighted by the blue rectangle in Figure 4a. (b) The magnified view of the blue portion in (a). (c) Time evolutions of 〈 ξ B (t ) ⋅ ξ B (0)〉 for various c0. The long-time steady value is denoted as V. (d) The plot of V versus c0. The black dashed line denotes the critical point. The insets depict the transport patterns of the particle. (e) The log-log plot of V − Vc versus P − Pc showing the power-law dependence for various bacterial densities ϕ.


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Size-Dependent Reversal of Locomotion

Furthermore, as the mobility of particle is usually influenced by its size,37 we examine the transport of particles with different R. As shown in Figure 6, for the particle with R = 6.97 µm, despite that the particle transports at c0 = 0.05 and 1.0 mM are both in the EDT states, the bacterial densities around the particle demonstrate distinct distributions. At low c0 (e.g., 0.05 mM), chemotactic aggregation occurs on the coated side (Figure 6a). That is, the perturbations exerted by bacteria mainly locate near the coated side, leading to a propulsion direction that is along the particle orientation n. However, such asymmetrical distribution of bacterial density counter-intuitively reverses (Figure 6b) at high c0 (e. g., 1.0 mM), thereby making the propulsion direction against the orientation.

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0.05 mM. (b) c0 = 1.0 mM. Asymmetrical distributions of bacterial density lead to a bias of bacterial noise, which determines the propulsion direction of particle. The blue lines denote the analytic SDRs, showing high agreements with the density distributions.


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For a more quantitative description, we define v p = 〈ξ B cos ϕ 〉 as the propulsion velocity of Janus particle, where φ is the included angle between the orientation n and instantaneous ξ B . The dependence of vp on c0 where R = 6.97 µm is shown in Figure 7a. It can be found that the sign of vp changes from positive to negative at c0 = 0.28 mM, corroborating that the locomotion of particle indeed reverses from along to against its orientation upon increasing c0, as denoted by the insets. This indicates that the propulsion direction of particle can be controlled by the stimuli intensity and particle size. For the sake of pinpointing the origin of such reversion, we quantify the bias of bacterial noise which results in the enhanced directional transport (See more details of this theoretical modeling in Supporting Information VIII). We denote Φ c and Φ nc as the net influx of bacteria into the asymmetric SDR on the coated and non-coated sides respectively. Thus, the influx difference ζ = Φ c − Φ nc can be utilized to characterize the bias level ϑ. For the reason that both Φ c and Φ nc depend on the geometry of SDR, we use two characteristic lengths, i.e., rc satisfying c(rc, 0) = c* and rnc satisfying c(rnc, 0) = c* (Figure 7b), to calculate Φ c and Φ nc . As the transition is continuous-like, the bias level ϑ can be estimated by ζ using a first-order Taylor series expansion, given by

ϑ = κ 0 + κ1ζ + O(ζ 2 ) ,

(4)

where κ0 is the intrinsic bias induced by the particle geometry and κ1 is the factor measuring the effective influx which collides with the tracer. In such a way, the CRW-to-EDT transition occurs when ϑ exceeds the maximum bias that can finally be dissipated, while the reversal of locomotion emerges once ϑ < 0. Thus, based on our theoretical analysis (See Supporting Information IV), the theoretical boundaries for the CRW-to-EDT transition and reversal of locomotion read rc R = χ c , rnc / R = χ nc ,

(5)


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respectively. As shown in Figures 7c, rc and rnc at critical points for different R show high agreements with eq. 5. The constants χc and χnc are obtained in Figure SI.9. Moreover, a state diagram of vp on c0 - R space is given in Figure 7d, which is highly consistent with the theoretical

(a) 3

EDT+

10

(d)

EDT-

1

9

EDT+

0

vp 8 (µm/s)

-1 -2

(b)

c0 (mM) 0.01

0.1

Φc

40 20 0

-40

Φ nc -40 -20 0

20 40

x (µm)

32 24 16

30 20 10 0

1

7

10

6 4

rc

rnc

-20

(c) rc (µm)

-3

rnc (µm)

vp (µm/s)

2

R (µm)

predictions, allowing approaches to realize controllable and predictable propulsion directions.

y (µm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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6

8

10

CRW 0 3 6 9 12

R (µm)

0.01

EDT0.1

c0 (mM)

1

5 4

Figure 7. (a) The dependence of vp on c0 for particle with R = 6.97µm. The dashed line denotes

the point where the reversal of locomotion occurs. (b) Schematic diagram of the modeling. The red and blue solid lines are the SDRs for c0 = 0.05 and 1.0mM respectively. (c) The dependences of rc and rnc (at critical points) on R. The green and orange lines denote rc R = χ c and rrc / R = χ nc respectively. (d) State diagram of particle transport on c0 - R space. The squares and triangles denote the critical points. The solid lines are the theoretical predictions determined by eq. 5. CONCLUSION

In summary, by developing a molecular-dynamics model of bacterial chemotaxis based on the run-and-tumble dynamics and Monod-Wyman-Changeux model, we present an investigation of the transport of a bacteria-activated Janus particle driven by the chemotaxis. This


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allows us to provide an effective strategy coupling stimulus-response techniques with asymmetrical design, which leads to particles with controllable propulsion directions. Upon increasing the stimuli intensity, a second-order state transition of Janus-particle transport is identified upon increasing the stimuli intensity: from a composite random walk, combining power-law-distributed truncated Lévy flights with Brownian jiggling, to an enhanced directional transport with size-dependent reversal of locomotion. The state diagram of Janus-particle transport depending on the stimuli intensity and particle size is presented, which allows approaches to realize controllable and predictable propulsion directions of particles. The physical mechanisms of these transport behaviors are revealed by performing a theoretical modeling based on the bacterial noise and Janus geometries. Furthermore, the developed model of bacterial chemotaxis can potentially be applied in other studies of bacterial systems. Our findings constitute a well-defined bacteria-activated motor system, which could provide a significant advance in the design of future biocompatible nano/micro-devices and suggest an approach to achieve efficient targeted delivery to a specific lesion based on the stimulus-response techniques.

METHODS Mechanical Interactions and Run-and-Tumble Dynamics

Both hard bodies of the bacterial cells and the motor are modeled by the sum of identical force centers with short-range repulsive potentials chosen as f(r) = Ar/r14 (See Supporting Information I for more details). To mimic the run-and-tumble motion of bacteria, a constant drift velocity v0 along ei is added on the cell in the running state to create a straight swim, while a random torque is exerted during the tumbling state to randomize the direction of next run. At


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each time step, the running cell has a probability Γδt to switch in a tumbling state, and a probability Ξδt to switch back, where δt is the time interval. Realistic physical value suitable for motile E. coli cells can be obtained as d0 = 1.5 µm, v0 = 30 µm/s, Ξ = 10 s-1, Γ = 1 s-1 (without chemotactic responses).29

Monod-Wyman-Changeux Model

For chemotactic responses, we use average kinase activity pon and the average methylation level m of the methyl-accepting chemotaxis protein receptors to quantitatively describe the chemotaxis pathway of E. coli. The kinase activity controls the run-and-tumble motion of the cell by regulating the flagellar motor's probability Γ for counter-clockwise (run) or clockwise (tumble). In our simulations, the activity pon is given by the Monod-Wyman-Changeux (MWC) two-state model.22,23

1 e − Fon pon = − Fon = , − Foff e +e 1 + eF

(6)

where F = Fon − Foff . Here Fon/off is the free energy of the MWC cluster to be on/off as a whole. For a cluster composed of Nr receptors, the total free-energy difference F = N r f m , where fm is the individual free-energy difference between the receptor on and off states and reads

 1 + c / K off f m = ε (m) + ln  on  1+ c / K

 . 

(7)

Here ε (m) is the methylation-level-dependent free energy difference, and is taken to be linear in m as ε (m) = α (m0 − m) . The kinetics of the methylation level follows

dm / dt = k R (1 − pon ) − k B pon ,


(8)

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ACS Nano

where kR (kB) is the rate of methylation (de-methylation) for the inactive (active) receptors. The flagellar motor's probability Γ can be given as

Γ = Γ 0 ( pon / p0 ) H ,

(9)

where Γ0 = 1 s-1 is the probability without chemotactic response, p0 is the kinase activity at steady-state, and H is the Hill coefficient of the motor response function. In our model, we use Nr = 6, Koff = 0.02 mM, Kon = 0.5 mM, α = 1.7, m0 = 1, kR = kB = 0.1 s-1 (leading to p0 = 0.5), and H = 10 for MeAsp binding by Tar receptors. 22,23

Wavelet-Based Method The separation of active steps and passive steps from the particle trajectory is realized using the wavelet-based method.49 Briefly, such an analysis method comprises following three steps. Firstly, choose an appropriate wavelet. The wavelet is a weighting function which is used in calculating the local integral values of time series over different scales. Wavelets can take different forms and thereby can acquire different integral values. Typically, “Haar wavelet” is commonly used in the wavelet analysis. Secondly, perform the wavelet transform. The wavelet transform is mathematically defined as a local integral. During the wavelet transform, the timedependent data can be transformed into a time- and scale-dependent representation of the original data. Finally, determine the scale and threshold. We must set a scale on which a threshold is used to decide what differences are large enough to matter. This threshold serves as the decision criterion for classifying the type of dynamics and identifying dynamical heterogeneity.


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In our systems, we calculate the Haar wavelet coefficients at a scale of 500 frames for all trajectories. At this scale, the wavelet transform coefficients can be clearly distinguished. The threshold was set according to the “universal thresholding”.56

ASSOCIATE CONTENT Supporting Information The details of simulation model and theoretical analysis, and additional figures and movies. This material is available free of charge on the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Authors * Email: [email protected]

Notes The authors declare no competing financial interest.

ACKONWLEDGEMENT We are thankful for helpful discussions with Bing Miao. We acknowledge financial support from National Natural Science Foundation of China (Grant Nos. 21422403, 51273105, 51633003, 21174080). L.-T. Y. acknowledges financial support from Ministry of Science and Technology of China (Grant No. 2016YFA0202500).

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