Modeling Biofilm Formation on Dynamically Reconfigurable

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Modeling Biofilm Formation on Dynamically Reconfigurable Composite Surface Ya Liu, and Anna C. Balazs Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03765 • Publication Date (Web): 02 Jan 2018 Downloaded from http://pubs.acs.org on January 3, 2018

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Modeling Biofilm Formation on Dynamically Reconfigurable Composite Surface Ya Liu and Anna C. Balazs* Chemical Engineering Department, University of Pittsburgh, 3700 O’Hara Street, Pittsburgh, PA 15261, United States Corresponding author e-mail: [email protected] Abstract We augment the dissipative particle dynamics (DPD) simulation method to model the salient features of biofilm formation. We simulate a cell as a particle containing hundreds of DPD beads and specify p, the probability of breaking the bond between the particle and surface or between the particles. At the early stages of film growth, we set p=1, allowing all bonding interactions to be reversible. Once the bound clusters reach a critical size, we investigate scenarios where p=0, so that incoming species form irreversible bonds, as well as cases where p lies in the range of 0.1 to 0.5. Using this approach, we examine the nascent biofilm development on a coating composed of a thermo-responsive gel and the embedded rigid posts. We impose a shear flow and characterize the growth rate and the morphology of the clusters on the surface at temperatures above and below Tc, the volume phase transition temperature of a gel that displays lower critical solubility temperature (LCST). At temperatures above Tc, the posts effectively inhibit the development of the nascent biofilm. For temperatures below Tc, the swelling of the gel plays the dominant role and prevents the formation of large clusters of cells. Both these antifouling mechanisms relay on physical phenomena and hence, are advantageous over chemical methods, which can lead to unwanted, deleterious effects on the environment.

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Introduction A fundamental challenge and technologically important mission is to create environmentally benign, antifouling coatings that inhibit the growth of biofilms without introducing harmful effects to the surroundings. One potential route to addressing this issue is to design surfaces that harness purely physical phenomena, rather than chemical means, to hinder the attachment of biological cells and other fouling agents. For example, researchers have used mechanical compression1 and pneumatic actuation2 to dynamically alter the surface topology and thereby expel bound particulates from the substrate. In recent studies,3 we used computational modeling to design a hybrid coating where synergistic interactions between the components produced two complementary physical routes to prevent fouling. This system integrates a thermo-responsive gel and rigid posts, which are embedded in the gel layer (see Fig. 1). The gel exhibits a lower critical solubility temperature (LCST); hence, at high temperatures, the gel collapses and exposes the posts, which provide a physical barrier to the adsorption of fouling agents. At low temperatures, the gel swells and the dynamic morphological change dislodges particulates from the surface. This hybrid coating can be augmented to yield other physical means to inhibit the settlement of particulates. For example, by anchoring chromophores (e.g. spirobenzopyran) to the gels or posts,4 light can be used to dynamically reconfigure the composite. The light-induced motion of the extended posts4 can hinder the attachment of organisms, as well as displace bound species. Hence, the gel-post composite can serve as a useful multi-functional platform that provides various physical approaches to prevent biofouling.


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Figure 1. (a) Snapshot of the initial system with the shear applied along x-axis. (b) Upper and lower panels are the top view of the arrangement of particles and rigid posts, respectively. (c) Snapshot of the system at 106 time steps for 𝛾̇ = 1.67 × 10−3, where the yellow, purple, and red particles respectively correspond to free, adhesive, and EPS particles. In the previous studies,3 we focused on a single particle (fouling agent) in the solution above the composite coating. Here, we modify our particle-based computational approach, which involves the dissipative particle dynamics (DPD) method, to simulate the incipient formation of biofilms and thereby determine the effectiveness of the coating on disrupting the development of the film on the surface. In biofilm formation, microorganisms initially form weak, reversible bonds with the surface and other nearby cells. When, however, there are a sufficient number of microorganisms that have aggregated in this manner, the cells excrete an extracellular polymer substance (EPS), which binds the particles irreversibly to each other and the surface.5,6 In other words, when a critical number of cells lie on or near the surface, they form strong inter-cellular bonds and thereby create the biofilm. To simulate the above biological behavior in a computationally efficient manner, we introduce the DPD scheme detailed in the next section. There have been a number of


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computational and numerical approaches to model the formation of biofilms.7,8 In these models, the initial substrate is typically assumed to be a flat, homogeneous surface (before the attachment of the microorganisms). An advantage of the DPD approach is that we can readily simulate interactive particles in the presence of multi-component, heterogeneous coatings in various flow fields. Hence, we can test the utility of reconfigurable, composite coatings for antifouling applications. Dissipative particle dynamics (DPD) is a particle-based computational method that provides an effective means of simulating the dynamic behavior of complex fluids and multicomponent mixtures.9–13 Unlike more atomistic molecular dynamics (MD) simulations, a DPD bead represents clusters of molecules. Similar to MD simulations, DPD captures the time evolution of a many-body system through the numerical integration of Newton’s equation of motion. An advantageous feature of DPD is that it utilizes soft repulsive interactions between the beads. Consequently, one can use a significantly larger time step Δt between successive iterations than those required by MD simulations. This, in turn, means that this approach can be used to model physical phenomena occurring at time and spatial scales many orders of magnitude greater than that captured by MD.14–16 Recently, we developed DPD simulations for thermo-responsive polymer gels,17,18 validating our model by capturing the correct volume phase transitions for lower critical solution temperature (LCST) gels. We also augmented the approach to encompass the key features of atom transfer radical polymerization (ATRP)17,18 and free radical polymerization (FRP).19 Hence, we can simulate multi-component systems involving gels, posts, particles, imposed flows, as well as chemical bonding interactions between different DPD beads in the system. In the discussion below, we first detail our new DPD scheme that captures the biofilm-


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surface interactions by introducing three types of particles (see Fig. 1): free (in yellow), adhesive (in purple), and EPS (in red). We incorporate the life cycle of biofilms by assigning each of these particles distinct binding properties. Using this model, we examine the growth dynamics and structural development of the biofilms under different temperatures and flow conditions. From these results, we can pinpoint the parameter space that yields effective antifouling behavior for this system. Methodology Basic components of the model Our system encompasses an array of rigid posts that are embedded in a thermoresponsive hydrogel (green beads); both the posts and the gel are anchored to a substrate (the brown beads in Fig. 1a). The gel exhibits a lower critical solution temperature (LCST), and thus swells at 𝑇 < 𝑇𝑐 and collapses at 𝑇 > 𝑇𝑐 . The non-deformable spherical particles in Fig. 1b

represent the film-forming, fouling species. The entire system is immersed in a host solution and bounded by the top and bottom walls of the simulation domain. To model the dynamic behavior of this multi-component system, we utilize dissipative particle dynamics (DPD),9–11 which is a particle-based approach used to simulate the time evolution of a many-body system that is governed by Newton’s equation of motion, 𝑚 𝑑𝐯𝑖 ⁄𝑑𝑑 =

𝐟𝑖 . Each bead i in the system experiences a force 𝐟𝑖 , which is the sum of three pairwise additive

forces: 𝐟𝑖 (𝑡) = ∑𝑗(𝐅𝑖𝑖C + 𝐅𝑖𝑖D + 𝐅𝑖𝑖R ) , where the sum is over all beads 𝑗 within a certain cutoff radius 𝑟𝑐 from bead 𝑖. The three forces are the conservative force 𝐅𝑖𝑖C , drag or dissipative force

𝐅𝑖𝑖D , and random force 𝐅𝑖𝑖R . We describe each pairwise force below.

The conservative force is a soft, repulsive force given by 𝐅𝑖𝑖C = 𝑎𝑖𝑖 (1 − 𝑟𝑖𝑖 )𝐫�𝑖𝑖 , where 𝑎𝑖𝑖

measures the maximum repulsion between beads 𝑖 and 𝑗 , 𝑟𝑖𝑖 = |𝐫𝑖 − 𝐫𝑗 |⁄𝑟𝑐 , and 𝐫�𝑖𝑖 = ACS Paragon Plus Environment

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(𝐫𝑖 − 𝐫𝑗 )⁄|𝐫𝑖 − 𝐫𝑗 |. While this soft-core force leads to a degree of overlap between neighboring

beads, this choice for the force permits the use of larger time steps than those typically used in MD simulations,11 which commonly involve hard-core potentials (e.g., the Lennard-Jones potential). The repulsive parameter 𝑎𝑖𝑖 is given in terms of 𝑘𝐵 𝑇. We choose room temperature as

the reference value and thus 𝑘𝐵 𝑇0 = 1 with 𝑇 = 25℃. The reduced temperature is introduced as

𝑇 ∗ = 𝑇⁄𝑇0 , and hence, 𝑇 ∗ = 1.077 for 𝑇 = 48℃. All the repulsive parameters are set to 25 in units of 𝑇 ∗ .

The drag force is 𝐅𝑖𝑖D = 𝛾𝜔𝐷 (𝑟𝑖𝑖 )(𝐫�𝑖𝑖 ∙ 𝐯𝑖𝑖 ), where 𝛾 is a simulation parameter related to

the viscosity arising from the interactions between the beads, 𝜔𝐷 is a weight function that goes to zero at 𝑟𝑐 , and 𝐯𝑖𝑖 = 𝐯𝑖 − 𝐯𝑗 . The random force is 𝐅𝑖𝑖R = 𝜎𝜔R (𝑟𝑖𝑖 )𝜉𝑖𝑖 𝐫�𝑖𝑖 , where 𝜉𝑖𝑖 is a zero-mean Gaussian random variable of unit variance and 𝜎 2 = 2𝑘B 𝑇𝑇 relates the amplitude of the noise to

the friction coefficient, as specified by the fluctuation-dissipation theorem.10,11 The value of 𝛾 is

chosen to ensure relatively rapid equilibration of the system’s temperature and the numerical 2

stability of the simulations for the specified time step.11 Finally, we use 𝜔D �𝑟𝑖𝑗 � = 𝜔R �𝑟𝑖𝑖 � = (1 − 𝑟𝑖𝑖 )2 for 𝑟𝑖𝑖 < 1.9,11

Each of these three pairwise forces conserves momentum locally, and thus, the DPD

simulations reproduce correct hydrodynamic behavior.9–11 The velocity-Verlet algorithm is applied to integrate the equations of motion in time. We take 𝑟𝑐 as the characteristic length scale

and 𝑘B 𝑇 as the characteristic energy scale in our simulations. The corresponding characteristic time scale is then defined as 𝜏intrinsic = �𝑚𝑟𝑐2 / 𝑘B 𝑇. The remaining simulation parameters are 𝜎 = 3 and Δ𝑡 = 0.02𝜏, with a total bead number density of 𝜌 = 3.11

We take the initial configuration of the gel to be a finite-sized tetra-functional network

with a diamond-like topology.17 The semi-flexible polymer strands are modeled as a sequence of


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30 DPD beads that are connected by harmonic bonds, with an interaction potential given by 1

𝑔

𝐸 = 2 𝐾bond (𝑟 − 𝑟0 )2 + 𝐾angle (1 + cos 𝜃) .3,20 The first term in the latter expression

characterizes the elastic energy with the elastic constant K bond and the second term represents g

the bending energy with the rigidity parameter K angle. Here, r0 is the equilibrium bond length

and 𝜃 is the bond angle between two adjacent bonds. The bond and angle potentials of the gel are set respectively at K bond = 128 and K angle = 4 to prevent bond crossing and produce a

polymer concentration comparable to the experimental results for this gel.21,22 Consequently, the total force acting on each gel bead is equal to 𝐟𝒆 + 𝐟𝒊 , where 𝐟𝒆 = −∇𝐸 and 𝐟𝒊 is the DPD

pairwise force.

As in our previous studies,3 the finite-sized network contains cross-links, which are beads with a connectivity of four, and dangling ends, which are located on the surface of the lattice and have a connectivity of less than four. Here, the gel contains 12,096 beads and consists of 396 strands, 175 cross-links, and 41 dangling ends. This network is periodic in the lateral (x and y) directions. To properly model the thermo-responsive behavior of this gel, we relate the repulsive parameter between a polymer and solvent bead, 𝑎ps , to the Flory-Huggins parameter

characterizing the polymer-solvent interaction, 𝜒ps , as follows: 𝑎ps = 𝑎 + 𝑘𝐵 𝑇𝜒ps ⁄0.306.11 In

studies of thermo-responsive gels (e.g., poly(N-isopropylacrylamide) (PNIPAAm)), it is typically assumed that 𝜒ps depends on temperature and polymer concentration.23–26 Hence, we

assume that 𝜒ps (𝑇, 𝜑) = 𝜒1 (𝑇) + 𝜒2 𝜑p ,17 where 𝜑p is the polymer volume fraction in the gel

and 𝜒1 (𝑇) = (𝛿ℎ − 𝑇𝑇𝑇)⁄𝑘𝐵 𝑇 , with 𝛿ℎ and 𝛿𝛿 being the respective changes in enthalpy and entropy.23,27 Note that 𝜑p is calculated as 𝜑p = 𝜌̅p ⁄𝜌̅gel where 𝜌̅p is the time-averaged polymer


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number density and 𝜌̅gel is the time-averaged total number density of the gel including the

polymer and solvent beads. Here, we set 𝑎 = 25 and take 𝛿ℎ = −14.331 × 10−14 erg , 𝛿𝛿 =

−5.452 × 10−16 ergK −1 and 𝜒2 = 0.596 to produce a continuous volume transition between

𝑇 = 30℃ and 35℃.11,28–31 With this choice of 𝑎ps , we reproduce the experimentally observed

temperature-induced volume phase transitions of PNIPAAm gels.17,23,32

The gel layer is attached via an adhesive interaction to the substrate (brown beads in Fig. 1a). The effective attraction between the gel and substrate is modeled by setting the interaction parameter between the beads in the gel and bottom wall, 𝑎gw , at 𝑎gw = 𝑎ps − 8 so that the gel

remains anchored to the wall in all our simulations.

The non-deformable spherical particle in Fig. 1b is constructed from 325 DPD beads that are dispersed on two spherical layers with an outer layer radius of 2.5 and interlayer spacing of 0.5. Each spherical layer is modeled by geodesic grids generated by subdividing an icosahedron (Fig. 1b).33In this way, we construct a particle with a well-defined smooth surface, with an outer diameter of 5.0 particles (yellow beads). These particles are initially dispersed in the bulk solution and represent the free fouling species. The total force and torque acting on each particle is computed as the sum of the forces and torques on its constituent DPD beads. The corresponding number density of the particle shell is 10.1, which is sufficiently high to prevent the penetration of the gel and post beads into the sphere and does not induce an unrealistic depletion force between the solvent beads (a behavior that can occur in particle-based simulation methods such as DPD). The posts are modeled as rigid polymers with a fixed bond length of one, and each post is 28 units in length. The 5 × 5 array of posts is anchored onto the substrate with an inter-post

spacing of 4.92, as shown in Fig. 1a. Each post is capped with a head that contains 10 DPD


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beads arranged in a regular structure that is based on the diamond cubic morphology (Fig. 1b). The lattice spacing in the head of the post is taken as one, which ensures that the head density is close to the system density of 𝜌 = 3. These head groups prevent the penetration of the posts into the surface of the non-deformable particles even under high shear rates. The separation between the posts is less than the diameter of the particles and hence, these structures effectively prevent the particles from reaching the substrate. Modeling the interactions between biofilm particles The particles in the bulk solution interact with each other through the excluded-volume repulsion and are initially dispersed in the upper channel as shown in Fig. 1a. The structural evolution of biofilms involves complex processes that, nonetheless, can be categorized into two main events: biofilm expansion or biofilm reduction.34,35 Biofilm expansion includes cell attachment through deposition or adsorption, cell growth and division, and the secretion of extracellular polymeric substance (EPS), which helps maintain the structural integrity of the biofilm.5 On the other hand, biofilm reduction includes erosion (the removal of single cells or small cell clusters from the biofilm), cell death, and sloughing (the removal of a large number of microorganisms in a single event) due to liquid-biofilm hydrodynamic interactions.5,35 Herein, we present a simple phenomenological model to characterize the main features of biofilm development by introducing three types of particles: yellow particles representing free cells, adhesive purple particles representing attached cells, and red particles representing the EPS, as shown in Fig. 1c. In the discussion below, we demonstrate that by utilizing these three different types of particles, we can capture the salient and essential dynamics of biofilm development. In the simulations, in the early stages of biofilm development, the free yellow particles can come into contact with the posts. These particles can bind to the post by forming a harmonic


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bond with post beads that lie within a cutoff range 𝑅𝑐𝑐𝑐 . There is no binding interaction if this

separation is greater than 𝑅𝑐𝑐𝑐 . In this way, we model the respective deposition and desorption processes illustrated in Fig. 2a. a)

Deposition

Desorption b)

Adsorption

Desorption

c)

EPS Production

Growth d)

Desorption

e) Detachment


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Figure 2. Schematic of the biofilm-growth model. (a) A free particle (in yellow) attaches/detaches to a post through deposition/desorption process; the attached adhesive particle is indicated in purple. (b) A free particle attaches/detaches to an attached particle through adsorption/desorption process. (c) EPS particle is produced if the attached particle is surrounded by more than four neighboring particles. (d) A free particle attaches/detaches to an EPS particle through growth/desorption process and forms an attached particle. (e) The detachment of biofilm after the applied shear breaks the binding to the post. Once detached from the post, the biofilm breaks into free particles.

To summarize, the total interaction between a free cell and the posts is expressed as:

𝐸𝑏𝑏

1 𝑗 𝑗 𝑘𝑏𝑏 � (�𝐑𝑖𝑏 − 𝐑 𝑝 � − 𝑟0 )2 , �𝐑𝑖𝑏 − 𝐑 𝑝 � < 𝑅𝑐𝑐𝑐 = �2 𝑖∈𝑏,𝑗∈𝑝

0, �𝐑𝑖𝑏 −

𝑗 𝐑𝑝 �

≥ 𝑅𝑐𝑐𝑐

(1)

𝑗

where 𝐑𝑖𝑏 is the ith beads on the free particle, 𝐑 𝑝 is the jth bead on the posts, and 𝑘𝑏𝑏 is the

spring constant. Here 𝑅𝑐𝑐𝑐 = 1 is equal to the cutoff range for the DPD pairwise repulsion, and 𝑟0 = 1.

Once attached to the posts, the cells are indicated in purple. These attached purple cells

are adhesive; namely, they can bind reversibly to other cells that lie within the distance 𝑅𝑐𝑐𝑐 , as

illustrated in Fig. 2b. If a free yellow cell becomes bound to an attached purple cell, it too becomes a purple unit. As a simplification, the relevant binding interaction is also set to 𝐸𝑏𝑏 . Alternatively, if the free cell does not bind to the anchored cell, it can continue to diffuse freely in the solution and continues to be labeled as “yellow”. To model the stability of the colony once it has reached a certain size and the role of the extracellular polymeric substance, we introduce red beads that represent an EPS particle. In particular, if an attached particle is bound to at least four neighbors, its label is changed from


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“purple” to “red” (see Fig. 2c). The interaction between a non-bonded pair containing a red EPS particle is specified as: 𝐸𝐸𝐸𝐸 = 𝑗

1 𝑘 2 𝐸𝐸𝐸

�

𝑖∈𝐸𝐸𝐸,𝑗∈𝑝

𝑗

��𝐑𝑖𝐸𝐸𝐸 − 𝐑 𝑏 � − 𝑟0 �

2

(2)

𝑗

if �𝐑𝑖𝐸𝐸𝐸 − 𝐑 𝑏 � < 𝑅𝑐𝑐𝑐 . Here 𝐑𝑖𝐸𝐸𝐸 is the ith bead on the EPS particle, 𝐑 𝑏 is the jth beads on the

neighboring particle within the cutoff range 𝑅𝑐𝑐𝑐 , and 𝑘𝐸𝐸𝐸 = 𝑘𝑏𝑏 . The formed harmonic 𝑗

bonds described by Eq. 2 are breakable when �R𝑖𝐸𝐸𝐸 − R 𝑏 � ≥ 𝑅𝑐𝑐𝑐 with a bond breaking probability 𝑝. In the study, we fix 𝑘𝐸𝐸𝐸 = 𝑘𝑏𝑏 = 0.05, which ensures negligible energy

fluctuations occurring during the bond breakage and formation. Note that if 𝑝 = 0, the

desorption process shown in Fig. 2d is restricted and the attachment to the EPS particle is irreversible. On the other hand, the behavior of an EPS particle is identical to that of an attached purple particle if 𝑝 = 1. Unless specified otherwise, we set 𝑝 = 0 in the studies described below.

Shear is applied to the system by moving the upper wall of the simulation box along the x

direction at specified velocities. The shear rate 𝛾̇ takes the values 1.67 × 10−4, 1.67 × 10−3, and

8.33 × 10−3 , which correspond respectively to the velocities of the upper wall of 0.01, 0.1, and

0.5 in dimensionless units. The force from the imposed shear stretches the bonds between the particles and the posts, as well as between the particles themselves. The reversible bonds are broken if the bond length exceeds the specified equilibrium length. We note that the bond forming and breaking is checked every 20 time steps. Independent simulations show that the results are insensitive to this value if the checks are done at time steps in the range of [5,100].

Due to this imposed shear, the biofilm can undergo the reduction process as shown in

Fig. 2e, where a cluster of cells is removed from the posts by the flowing fluid. The cells return to the bulk fluid particles since the strong shear flow breaks the reversible bonds between the


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cluster and the post. In the simulation, after this detachment, the cells recover the free-particle form. Notice in this case, all the bonds between particles including the EPS particles are removed. Finally, the simulation box is 24.6 × 24.6 × 60 units in size and is filled with 89603

solvent beads, maintaining the total density of the system at 𝜌sys = 3 . Eight independent

simulations are carried out for 𝑡 = 5 × 106 time steps for each parameter set. (In the ensuing

discussion, t is specified in simulation time steps, which can be related to physical units of time as discussed below.) Comparison of simulation parameters to physical values We can relate the dimensionless simulation parameters to physical values through the value of the collective diffusion coefficient of the polymer network. If we assume that each solvent bead represents 10 water molecules,36,37 then a DPD solvent bead occupies a volume of 3

3

300Å since a water molecule (of mass density 1 g/cm3) has a volume ≈ 30Å . The total bead number density in our system is 𝜌sys = 3, and using 𝜌sys = 3𝑟𝑐−3 and the mass density of water,

we obtain the unit of length 𝑟𝑐 = 0.97nm and the characteristic mass m = 180Da. By matching

the mass density of a polymer bead in the simulation to the mass density of amorphous PNIPAAm (1.1 g⁄cm3 ), we find that a polymer bead represents 1.6 PNIPAAm monomers.

To obtain the correct characteristic time scale, we relate the collective diffusion coefficient of the

polymer network in the simulations, 𝐷0sim = 1.74 × 10−2 nm2 ⁄𝜏DPD obtained from the swelling kinetics exp

of the gel, to the experimental value 𝐷0

= 2 × 10−11 m2 ⁄s.38 We thus obtain the following physical

values for the simulation parameters: 𝜏DPD = 0.87ns , the simulation box size is 23.9 × 23.9 ×

58.2𝑛𝑛, the fiber equilibrium length is 27.2nm, and the particle diameter is 9.7nm. The maximum

value of the applied shear rate 8.33 × 10−3 corresponds to 𝛾̇ = 9.6 × 106 /𝑠 .39,40Additionally, ACS Paragon Plus Environment

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𝐾bond = 128 and 𝐾angle = 4 correspond to the respective values of 0.56 N⁄m and 4𝑘𝐵 𝑇.33

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Results and Discussion

In the ensuing discussion, we first analyze the effects of the shear flow on the growth dynamics and architecture of biofilms at temperatures above the polymer volume phase transition point, where the gel collapses below the posts. We then increase the number of posts anchored to the substrate and examine how this variation impacts the biofilm structure. We vary the bond breaking probability in the system to examine how the reversibility of the bonds affects structure formation in the presence of the exposed posts. Finally, we cool the system to induce the swelling of the LCST gel and thereby analyze the contribution of the gel to the antifouling properties of the composite. Hydrodynamic interactions between the fouling species and the surrounding fluid can influence both the initial contact between the individual species and the posts, and the ultimate detachment of the biofilm. Hence, the growth and architecture of the biofilm are significantly affected by the imposed flow. In the following studies, we vary the shear rate, 𝛾̇ , in the simulation box to determine how this variable affects the behavior of 25 free particles (drawn in yellow) that are initially randomly dispersed in the bulk solution above the gel-post composite (Fig. 1a). The temperature of the system is fixed at T = 48℃ and thus, the LCST gel collapses

below the height of the posts. Under these conditions, we can focus on the role that the posts play in the dynamic behavior of the film. After the temperature is fixed, the entire system is relaxed for 104 time steps, and the following shear rates are applied along the x-direction: 𝛾̇ = 1.67 × 10−4, 1.67 × 10−3, and 8.33 × 10−3 .

Figure 3 shows snapshots of the system at different time steps at the lowest value of shear

considered here 𝛾̇ = 1.67 × 10−4. At the early stages, the motion of the free particles is controlled


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by a combination of diffusion and convection. If a motile particle comes within the interaction range of the rigid posts, harmonic bonds are formed between the particle and post. If, in the next update, all these bonds are not broken by the imposed shear, then the particle becomes attached to the posts and is converted into a purple particle. If these particle-post bonds are ruptured by the shear flow, the particle desorbs and migrates into the bulk fluid. Figure 3a reveals that at 𝑡 = 1.2 × 105, six purple particles are attached to the posts and effectively nucleate the formation

of the biofilm.

Figure 3. Snapshots of the system for the shear rate 𝛾̇ = 1.67 × 10−4 at (a) 𝑡 = 1.2 × 105 , (b) 𝑡 = 9 × 105 , (c) 𝑡 = 1.6 × 106 , and (d) 𝑡 = 3 × 106 .

This nascent film will absorb more approaching species and continue to grow, as shown in Fig. 3b. In particular, if a free particle comes within the interaction range of a bound particle, it forms bonds with the bound unit and thus contributes to the biofilm growth. Figure 3b reveals that at 𝑡 = 9 × 105 , bound particles are stacked on the initially adsorbed units and form multiple layers. When a given particle is bound to at least four neighbors and this cluster of particles is


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connected to a post, that particle is tagged as a “red” particle. We consider the red particle to be part of the matrix that is formed by extracellular polymeric substances (EPS) and refer to it as an “EPS particle”. The EPS is vital for binding adherent cells to the growing film; 5,6 we mimic the functionality of the EPS in our simulations by stipulating that the red particles can form permanent, irreversible bonds (regardless of the magnitude of the imposed shear) with neighboring units and thus, enhance the stability of the cluster. At 𝑡 = 1.6 × 106 (Fig. 3c), five

EPS particles are associated with a cluster and promote the biofilm growth.

Eventually, at 𝑡 = 3 × 106 , all the free particles in the solution are absorbed and form a

biofilm with 14 EPS particles and 11 purple particles. At this stage, the biofilm structure is highly stable, with small fluctuations under the shear rate of 𝛾̇ = 1.67 × 10−4.

To quantify the rate of growth of the adsorbed layer, we monitor the temporal evolution

of the total number of particles in the biofilm, 𝑁𝑎 , by averaging over ten independent simulations that were carried out for 5 × 106 time steps. As shown in Fig. 4, the black, red, and green curves

correspond to the respective low ( 𝛾̇ = 1.67 × 10−4 ), intermediate ( 𝛾̇ = 1.67 × 10−3 ), and high

(𝛾̇ = 8.33 × 10−3) shear rates. The shading about each curve represents the standard deviations

calculated from the ten simulations. At these shear rates, 𝑁𝑎 grows from zero to the saturation value 25 when all cell particles are attached to the substrate (Fig. 4a). As the shear rate is

increased, however, the growth rate increases and the corresponding fluctuations decrease. We extract the characteristic time for the growth with the aid of a fitting function 𝐹(𝑡) = 𝑎 − 𝑏exp(−𝑡/𝑡0 ),

(3)

as shown in Fig. 4a, and obtain the following values for 𝑡0 as the shear rate is increased:𝑡0 = 1.0 × 106 ± 4.3 × 103 , 7.4 × 105 ± 4.2 × 103 , and 5.5 × 105 ± 2.4 × 103 . As described in the Supplemental Information, the characteristic time for the particles on a flat substrate (without the


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posts and gel) at the lowest shear rate is 𝑡0 = 7.9 × 105 ± 2.5 × 103 . The latter value is smaller than the characteristic time at the comparable shear when the system encompasses the exposed

posts. The smaller value of t0 leads to a higher value of the growth rate F(t) in the case of the flat substrate and thus, indicates the utility of the posts in preventing the biofilm growth. (a)

Na

(b) spike

Figure 4. (a) Time evolution of 𝑁𝑎 for the shear rates 𝛾̇ = 1.67 × 10−4 (black), 1.67 × 10−3 (red), and 8.33 × 10−3 (green). The solid smooth curve and shading under each curve are the corresponding fitting by Eq. 2 and standard deviation, respectively. The inset is snapshot of the system at 𝑡 = 2 × 106 time steps for 𝛾̇ = 1.67 × 10−4 . The circled cluster composing all purple ACS Paragon Plus Environment

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and red particles corresponds to 𝑁𝑎 . The height is defined as the z-component of the distance between the top and bottom beads in the cluster. (b) Time evolution of the height. The spikes correspond to the slough of small cluster of biofilm particles.

The higher shear rate leads to more frequent particle-post contact, as well as a greater possibility of the particle-post bond rupture. The competition between the two factors leads to the distinct dynamics. In the parameter range we considered, the particle-post contacts dominate the overall growth and thus as the shear rate is increased, the biofilm develops at a faster rate. The particle-post bond rupture is more pronounced in the early stage than in the late stage of growth and leads to larger fluctuations in the measurement of 𝑁𝑎 .

We also monitored the temporal evolution of the biofilm height, which is measured as the

z-component of the distance from the bottom beads to the top beads of the biofilm particles (see the inset in Fig. 4a). In all three cases, the height increases and ultimately saturates, as shown in Fig. 4b. At early times, the plots display large functions, especially for the high shear rate 𝛾̇ = 8.33 × 10−3 . These fluctuations are due to the detachment of the cell clusters from the

biofilm (as illustrated in Fig. 2e). The weaker shear leads to the greater height, indicating that the structure of the biofilm is sensitive to the imposed flow. To characterize the architecture of the biofilm, we calculate the relative shape anisotropy, 𝜅 2 ,41 and introduce the parameter 𝜅𝑧 to measure the directional anisotropy with respect to the z2

2

axis. Here 𝜅 2 = ��𝜆𝑥 − 𝜆𝑦 � + (𝜆𝑥 − 𝜆𝑧 )2 + �𝜆𝑦 − 𝜆𝑧 � � /(2𝑅𝑔4 ) and 𝜅𝑧 = 𝜆𝑧 /(𝜆𝑥 + 𝜆𝑦 + 𝜆𝑧 ) , 1

𝑁𝑎 𝑖 𝑖 where 𝜆𝑥 , 𝜆𝑦 , and 𝜆𝑧 are the eigenvalues of the gyration tensor 𝑆𝑚𝑚 = 𝑁 ∑𝑖=1 𝑟𝑚 𝑟𝑛 . As above, 𝑎

N a is the total number of biofilm particles and 𝐫 𝑖 is the position vector of the center of the ith

particle with respect to the center of mass of the entire biofilm. The radius of gyration, 𝑅𝑔 , is


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calculated as 𝑅𝑔 = �𝜆𝑥 + 𝜆𝑦 + 𝜆𝑧 . For an ideal sphere, 𝜅 2 = 0 and 𝜅𝑧 = 1/3. For an ideal rod, 𝜅 2 = 1 , and 𝜅𝑧 = 1 if the rod aligns along the z direction and 𝜅𝑧 = 0 if it aligns in x-y plane.41

To illustrate how the parameters the 𝜅 2 and 𝜅𝑧 characterize structure, we consider a

spheroidal ellipsoid described by

𝑥2

𝑎2

+

𝑦 2 +𝑧 2 𝑏2

= 1, with the aspect ratio 𝜀 = 𝑎/𝑏. Figure 5 shows

𝜅 2 as a function of 𝜀 for the corresponding spheroids. Note that the parameter 𝜅 2 alone is insufficient to capture the overall structural anisotropy. For instance, 𝜅 2 = 0.2 corresponds to

two possible configurations: one protrudes along the z-axis with 𝜅𝑧 = 0.48 and the other expands like a pancake in x-y plane with 𝜅𝑧 = 0.12. Hence, we also need to specify 𝜅𝑧 to accurately describe the structure of the object, such as the biofilm.

Figure 5. Asymmetric parameters 𝜅 2 and 𝜅𝑧 as a function of aspect ratio for spheroids. I and II are two configurations with the same 𝜅 2 = 0.2 .As noted above, the EPS particle forms permanent bonds with other approaching particles, and thus acts like the central core in the development of the biofilm. Here, we focus on characterizing this EPS core, which contains only red particles, as shown in Fig. 6a.


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(a)

Shear (b)

(c)

Figure 6. (a) Typical conformations of the EPS core of the biofilm as the shear increases. (b) and (c) are the temporal evolutions of 𝜅 2 and 𝜅𝑧 , respectively, for shear rates 𝛾̇ = 1.67 × 10−4 ( black), 1.67 × 10−3 (red), and 8.33 × 10−3 (green).


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As the shear rate is increased, three typical structures are observed. At the low shear rate, the core extends preferentially in the direction along z-axis. At the intermediate shear rate, the core displays a more symmetric, globular conformation. At the high shear rate, the core resembles a chain that is stretched in the x-y plane. Figures 6b and 6c show the respective temporal evolution of 𝜅 2 and 𝜅𝑧 for the different shear rates considered here; these quantitative measurements of the height are consistent with the respective graphical images. For the low (black curve) and high (green curve) shear rates, the

have large values since both structures

are highly anisotropic. The large value of 𝜅𝑧 in the case of the weak shear rate indicates that the structure develops vertically against the flow, while for the strong shear, the structure aligns horizontally (small 𝜅𝑧 ) since the vertical development is suppressed by the imposed flow. In the

moderate shear, 𝜅 2 and 𝜅𝑧 are small since the biofilm grows both vertically and horizontally and the structure is more isotropic.

The above simulations results show qualitative agreement with experimental observations. Single-celled microorganisms such as Haloferax volcanii develop into structured biofilms in static liquids.42 When the biofilms reach maturation, their morphology resembles a multiplelayer tower, similar as our observation in Fig. 6a.42 When the biofilms are grown under flow, experiments show the biofilm structures at maturity depend on the fluid velocity.5 For example, under the flow velocity of 0.03ms-1, the developed morphology can be isotropic, with a mushroom-like structure.5 Under the high-shear flow with the velocity of 1ms-1, the biofilms attain a highly anisotropic ripple structure.5,43These experimental observations are consistent with the respective simulation results at intermediate and high shear. The rigid posts can delay the formation of the clusters (such as those shown in Fig. 6), and thus, can suppress the production of EPS, which is secreted by cells when the colony reaches


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a certain size. In this context, the separation between the posts can play a significant role. To investigate this hypothesis, we examine the behavior of the particles in the presence of a 6 × 6 array of posts, which are uniformly anchored on the substrate. Figure 7 shows the temporal evolution of 𝑁𝑎 . Fitting the data to Eq. 3, we obtain the characteristic time scale of 𝑡0 = 3.1 ×

105 ± 9.6 × 102 for the growth at the low shear 𝛾̇ = 1.67 × 10−4.

Figure 7. Time evolution of 𝑁𝑎 , 𝜅 2 , and 𝜅𝑧 under the weak shear rates 𝛾̇ = 1.67 × 10−4 with an array of 6 × 6 posts. The solid smooth curve and shading under each curve are the corresponding fitting by Eq. 2 and standard deviation, respectively. The insets are the snapshots of the system at 𝑡 = 3.4 × 104 (left) and 𝑡 = 1.49 × 106 (right). The latter value of 𝑡0 is a factor of three faster than that for the 5 × 5 array of posts. Since the

inter-post spacing is now decreased to 4.17, the attached particles can easily bind neighboring particles in solution. Consequently, the probability (and transition rate) of a bound particle becoming an EPS particle is increased and development of the biofilm is accelerated. The large values of 𝜅 2 and 𝜅𝑧 suggests that the biofilm develops vertically with high anisotropy; this is

consistent with the conformation shown in the insets. At 𝑡 = 3.4 × 104, four biofilm particles are ACS Paragon Plus Environment

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present and at 𝑡 = 1.49 × 106 , all the particles have become attached to the posts and

accumulated vertically.

In all the above studies, we imposed the most stringent constraint on the nature of the binding interactions between the EPS particles and potential binding partners. In particular, we assumed that the EPS particles from permanent bonds with the incoming species. With this assumption, we could probe the effectiveness of the posts in the “worse-case scenario”. It is, however, useful to consider the behavior of the system when these red particles can form breakable bonds, where the probability of breaking is nonetheless relatively small. Within the framework of our computational model, we can vary the probability of bond breaking, p, between the EPS particles and the neighboring species. In particular, we carried out simulations for the range 𝑝 ∈ [0.1,0.5]; the latter simulations are comparable to scenario in Fig. 8, where 𝑝 = 0.1.

Figure 8 shows the typical temporal evolution of the height at 𝑇 = 48℃ for the case of

strong shear 𝛾̇ = 8.33 × 10−3 . At 𝑡 = 7.4 × 105 , multiple EPS particles are attached on the posts

and the biofilm develops into a vertical structure with a height of 22.7. Since the particle binding is no longer permanent, partial biofilm clusters detach from the colony. At 𝑡 = 1.9 × 106, only one EPS remains in the biofilm under the influence of the strong shear, and the height of the biofilm is 13.2. The height evolution has large fluctuations, which indicate the frequent attachment and detachment of particles. At 𝑡 = 5 × 106 , the height is 7.7 with one EPS particle left in the layer. As noted above, this behavior is typical for the range of bond breaking probabilities 𝑝 ∈ [0.1,0.5].


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I

Height of two layers II III

I

II

III

Figure 8 . Temporal evolution of the height of the biofilm where the probability to break the EPS bond is 𝑝 = 0.1 under the strong shear 𝛾̇ = 8.33 × 10−3. I, II, and III are the corresponding snapshots of the system at 𝑡 = 7.4 × 105 , 1.9 × 106 , and 5 × 106 , respectively. The above results provide significant insight into a factor controlling the film formation. Specifically, the simulations indicate that even a small probability of bond breaking between the red and the other particles significantly affects the development of the film. In the case where


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p=0.1, the posts are effective at inhibiting the formation of a cluster of red particles at the surface. Hence, while the results of the simulation are relatively insensitive to the choice of p in the range [0.1,0.5], the introduction of bond-breaking with respect to the EPS particle does lead to significantly different results compared to the case of p=0. In other words, while the binding of the cells to the surface and to each other remains reversible, the posts can play a vital role in deterring fouling. This behavior can occur in the presence of the EPS even if the rate constant for bond breaking becomes smaller, as long as the bonds remain reversible. In essence, the results for p in the range [0.1,0.5] indicate that the posts are highly effective at inhibiting strong binding to the surface in the early stages of biofilm development (where the behavior of the cells is dominated by reversible binding interactions). On the other hand, if the binding becomes irreversible (for example, as may be the case in the later stages of development),44 then the post still serve a vital function. In this regime, they retard the rate of growth of the biofilm. In the above discussion, we harnessed the LCST behavior of the gel to expose the posts at the relatively high temperature of T = 48℃ and thus, used the narrow, rigid pillars to prevent biofilm formation on the substrate. The expansion of the underlying gel can also play a role in preventing fouling. To investigate the effect of gel swelling on the biofilm growth, we prepare the system as depicted in Fig. 1a and cool it to T = 25℃, focusing on the case of weak shear,

𝛾̇ = 1.67 × 10−4 . Here again, we set p=0. As shown in Fig. 9a, at 𝑡 = 2 × 105 , the gel swells

above the tips of the posts. The excluded volume repulsion between the swelling gel and the free particles hinders the deposition process (Fig. 2a); only two particles are attached to the surface in Fig. 9a. At 1.2 × 106 time steps, the biofilm contains 10 purple particles, and is reduced to nine purple particles at 1.22 × 106 time steps. At 𝑡 = 5 × 106 , the layer does not contain any EPS (red) ACS Paragon Plus Environment

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particles and 15 particles remain in the bulk solution. The latter scenario is significantly different from the p=0 cases shown above, as all particles attach to the posts when the gel is collapsed.

Figure 9 . Snapshots of the system at (a) 𝑡 = 2 × 105 , (b) 𝑡 = 1.2 × 106 , (c) 𝑡 = 1.22 × 106 , and (d) 𝑡 = 5 × 106 for the weak shear 𝛾̇ = 1.67 × 10−4 when the LCST gel swells at T = 25℃. For the moderate (𝛾̇ = 1.67 × 10−3) and strong (𝛾̇ = 8.33 × 10−3 ) shear rates, we again

observe the absence of EPS particles at late times. The efficient inhibition of the biofilm

development at T = 25℃ occurs because the swelling gel fills the space between attached particles and effectively inhibits further particle-particle contacts; this prevents the accumulation

of particles into the biofilm colony and the appearance of EPS particles. Without the EPS particles, the biofilm is fragile and the particles are easily removed under the applied shear. We note that the in the above simulations, the value of 𝑎𝑖𝑖 is the same for the particle-

particle and particle-gel interactions ( 𝑎𝑖𝑖 = 25 ) and hence, the interactions are dominated by ACS Paragon Plus Environment

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steric repulsion. The behavior of the system at T = 25℃ would depend on the relative value of

the gel-particle interactions and the gel would be less active at inhibiting biofilm formation if the binding of the particle to the gel were enthalpically favorable. Conclusions We extended the DPD approach in a number of distinct ways. First, we augmented the method to allow us to specify a particle type (yellow, purple or red in this scheme) and permit different bonding interactions based on the types of particles that were involved in the pairings. We also modified the approach so that we can update the particle type once it underwent the bond formation. Additionally, we updated the particle type if bonds were broken and a particle changed state (between red and purple) or became a free particle (yellow). With these changes, we could specify whether a particle was weekly or strongly bound to the surface layer, or if it was still an unbound species. In this way, we could simulate the nascent stages of biofilm formation through the DPD simulations. This constitutes an important advance because the DPD simulations also allow us to examine the behavior of substrates formed from hybrid materials or displaying complex architectures, and hence, we can now examine how the materials properties and structural features of the substrate contribute to the antifouling behavior of the system. Using this augmented approach, we examined the initial stages of biofilm development on composite substrates formed from rigid posts embedded in a thermo-responsive gel. With the gel collapsed at T=480C and the posts exposed, we analyzed the growth rate and structural development of the film at three different shear rates. In accordance with experimental observations, we found that the morphology of the developing layer depended on the flow rate of the surrounding fluid. Moreover, the specific structures observed in the simulations showed qualitative agreement with experiments on biofilm formation in the presence of flow fields.


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Importantly, we showed that the exposed posts decreased the growth rate of the biofilm relative to that on a bare, flat surface (see SI). The above studies were performed for the situation where the binding between the EPS particles and the other neighboring sites was taken to be irreversible. When the probability of bond breaking was set to a finite value between 0.1 and 0.5, the EPS particles and neighboring sites formed reversible bonds. In this case, the posts were highly effective at inhibiting the formation of large EPS clusters. The latter results may be particularly applicable to the early stages of biofilm development. Notably, approaches that are effective at inhibiting the early stages of film formation are particularly valuable since they prevent the buildup of the layer. We then considered the behavior of the system when the temperature was lowered to T=250C. Namely, at the latter temperature the simulations revealed that the gel swells and occupies the region between the posts. Hence, approaching particles experience a steric repulsion from the swollen gel. The combination of the steric repulsion from the neutral gel (aij=25) and a high shear rate acted in concert to inhibit the nucleation of a strongly bound biofilm layer. The simulations indicate the hybrid system provides an effective platform at inhibiting the fouling of the surfaces, where each component of the composite can play an important role in this application. In particular, the system physically inhibits the settlement of organisms both above and below the volume phase transition temperature of the LCST PNIPAAm gel. At temperatures above the volume phase transition temperature, the posts effectively inhibit the development of the nascent biofilm. If the temperature dips below this value (T~33.50C), the swelling of the gel now plays the dominant role and prevents the formation of large clusters of cells. Both mechanisms relay on physical phenomena, rather than chemical additives.


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Approaches that rely purely on physical means to prevent fouling are highly advantageous over chemical methods, which can lead to unwanted, deleterious effects on the environment. Acknowledgements ACB gratefully acknowledges financial support from the ONR and helpful conversations with Ms. Cathy Zhang, Mr. Stefan Kolle and Prof. Joanna Aizenberg

Supporting Information Available: Growth dynamics of biofilm on a flat substrate This material is available free of charge via the Internet at https://na01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fpubs.acs.org&data=01%7C 01%7Cbalazs%40pitt.edu%7C4a154674798f491c197208d5375fc68c%7C9ef9f489e0a04eeb87c c3a526112fd0d%7C1&sdata=syPgidIHXoUThDXa81SqNdEyk2jsJzy%2BgXoIdlLH7Is%3D&r eserved=0.

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For Table of Contents Use Only

Gel swells

Shear rate

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Gel collapses


Modeling Biofilm Formation on Dynamically Reconfigurable

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