Chemical Logic Gates on Active Colloids

Abstract Recent studies have shown that active colloidal motors using enzymatic reactions for propulsion hold special promise for applications in fields ranging from biology to material science. It will be desirable to have active colloids with capability of computation so that they can act autonomously to sense their surroundings and alter their own dynamics. It is shown how small chemical networks that make use of enzymatic chemical reactions on the colloid surface can be used to construct motor‐based chemical logic gates. The basic features of coupled enzymatic reactions that are responsible for propulsion and underlie the construction and function of chemical gates are described using continuum theory and molecular simulation. Examples are given that show how colloids with specific chemical logic gates, can perform simple sensing tasks. Due to the diverse functions of different enzyme gates, operating alone or in circuits, the work presented here supports the suggestion that synthetic motors using such gates could be designed to operate in an autonomous way in order to complete complicated tasks.


I. INTRODUCTION
Extensive investigations of colloidal motors with micrometer dimensions that use various mechanisms to produce propulsion have been carried out, and review articles summarize much of this research.[1][2][3][4][5][6][7][8] Growing interest in such active agents stems from their broad potential uses as vehicles for drug delivery, cargo transport, motion-based species detection, active self-assembly, micro fluidics, medical as well as other applications.[8][9][10][11][12][13][14] In order for such micromotors to perform useful tasks effectively, their directed motion must be controlled in some way, even in the presence of strong thermal fluctuations.Chemical gradients, walls and external fields, among other means, have been used to guide their motions.[15][16][17][18][19][20][21][22][23][24][25] Rather than seeking to externally direct motor motion, it would be desirable if the motors themselves could discover ways of responding to stimuli to achieve specific goals.
Chemically-powered colloidal motors can be propelled using diffusiophoretic mechanisms [26][27][28][29][30], and in this context motors that make use of enzymatic chemical reactions on the colloid surface [31][32][33][34][35][36] are the focus of this work.Such enzyme-powered micro and nanomotors have important properties, such as biocompatibility, versatility, and fuel bioavailability, that make them attractive for applications.[35][36][37][38][39][40] Usually the chemical fuel that powers motor motion is supplied directly by the environment; however, active colloids that make use of coupled enzymatic reactions for propulsion and multi-fueled enzymatic motors have been studied in the laboratory.[41][42][43][44] Some fundamental aspects of the operation of such motors remain unexplored, including the structure of the colloidal catalytic surface and characteristics of substrate species involved in the coupled reactions, the conversion rate of fuel in surface enzyme networks, and effects related to fuel supply.
A body of earlier research has shown how DNA, RNA and protein networks can be used to construct chemical logic gates, and how circuits built from these gates could be used to carry our simple computations.[45][46][47][48][49][50][51][52][53] For example, a logic network composed of three enzymes operating in concert as four concatenated logic gates (AND/OR) was designed to process four different chemical input signals and finally produces a pH change as the output signal.[54] Also, various two-input gates built from de novo-designed proteins have been proposed recently [55], contributing to the design of programmable protein circuits.[56] Given the substantial amount of research on the construction of protein and other chemical logic gates and circuits, it should be possible to exploit this research to construct programmable enzyme-powered motors.Colloids with linear dimensions of one to a few micrometers can support tens to hundreds of thousands of enzymes on their surfaces; thus, one can construct colloids coated with several different enzymes to implement chemical logic gate functions as described above.In this connection, micromotors with a gated pH responsive DNA nanoswitch have been made and studied in the laboratory to function as on-demand payload delivery systems.[38,39] In this paper we investigate various ways in which small chemical networks that make use of enzymatic chemical reactions on the colloid surface can be used to construct motor-based chemical logic gates.In this way the motors may perform chemical computational tasks that allow them to control their dynamics by sensing the characteristics of the environment in which they move.Below we present a discussion of some basic aspects of how coupled enzyme reactions influence colloid propulsion, and provide examples of how specific chemical logic gates can be implemented to allow a colloidal motor to sense and respond to its environment in different ways, thus, performing simple tasks.

II. CHEMICAL GATES AND MOTORS WITH COUPLED ENZYMATIC REACTIONS
Coupled enzymatic reactions involving glucose oxidase and catalase enzymes coated on a Janus surface have been studied experimentally [42][43][44].In this system the reaction of glucose (Glc) in the presence of O 2 catalysed by the enzyme glucose oxidase (GO x ) yields gluconic acid (GlcA) and not a desirable environmental species in biochemical applications it is advantageous to supply fuel directly on the catalytic surface where it is used, instead of the more conventional global fuel supply; thus, the chemical kinetics of systems with locally supplied fuel merit detailed study.
The properties of colloidal motors that support chemical logic gates involve such coupled enzymatic reactions on the colloid surface, and some of these reactions may be used to implement to sensing tasks, while others are responsible for motor motion.Indeed, Fig. 1 shows how the reaction catalysed by GO x can be mapped onto an AND gate.The inputs to the gate are x 1 = Glc and x 2 = O 2 , while the target output is hydrogen peroxide, y 1 = H 2 O 2 .The GlcA product is not monitored.The reaction models an AND gate since both substrates are required for product formation.[57] FIG. 1.A small protein network that simulates an AND gate whose output is a species that is fuel for a second enzyme.The glucose and oxygen substrates correspond to the inputs x1 and x2 to the AND gate while hydrogen peroxide is the desired output y1.The product y2 = GlcA is not monitored.The hydrogen peroxide is then processed by the catalase enzyme to produce O2 and H2O.
A simple version of such coupled chemical reactions on the colloid surface can serve to illustrate the roles of various factors, such as reaction rates, manner of fuel supply, and enzyme distributions on the colloid dynamics.For a spherical colloid with radius R we suppose that on the upper hemisphere (H u ) a fraction f u 1 of surface area is covered by enzyme E 1 , randomly distributed on that hemisphere, while the remaining fraction f u 2 = 1 − f u 1 is covered by enzyme E 2 (see sketch in fig.2(a)).On the lower hemispherical cap (H ) a fraction f 1 is covered by enzyme E 1 , while on the remainder of the hemispherical surface the fraction f 0 = 1−f 1 sites are inactive enzymes E 0 or empty sites.
Enzyme 1 catalyzes the reaction where species with a superscript * are assumed to be held constant by reservoirs or are in excess.The fixed concentration c * R1 of R * 1 is incorporated in the rate constant k 1 and κ 1 = k 1 /(4πR 2 ) is the rate constant per unit surface area.(If c * R1 is in excess or above a threshold the reaction (1) is controlled by the concentration of A. In this case, one input is always on, and the reaction can act effectively as a buffer gate that produces species I if A is an input.)Enzyme 2 catalyzes the reaction, with κ 2 = k 2 /(4πR 2 ).In these reactions we see that the product I produced on the motor surface by reaction (1) serves as the fuel for the motor reaction (2).In addition, we assume that the species B and I are degraded in the bulk phase and ultimately produce A. (Alternatively, one can suppose that B and I are removed from the system by catabolic reactions and A is supplied by reservoirs.)Thus, the reactions B This scheme could also model reactions catalysed by other enzymes; for example, the enzyme choline oxidase (CHO) that catalyzes the reaction of choline (Ch) and O 2 to give betaine aldehyde (Be) and Instead of a single catalytic reaction, the reaction network using both Eqs. ( 1) and (2) also can model an AND gate using the coupled reactions of two enzymes such as GO x and horseradish peroxidase (HRP).The H 2 O 2 from the GO x catalysis is then processed by HRP in the reaction, 1 above) concentration fixed, one can take as inputs x 1 = Glc and x 2 = ABTS, where ABTS is 2,2'-azinobis(3-ethylbenzothiazoline-6-sulfonic acid) and the output of the AND gate is ABTS ox , the oxidized form of ABTS.[51] We use continuum theory and molecular simulation to investigate the properties of coupled enzyme kinetics on colloids based on Eqs.(1) and (2).

Continuum model:
The velocity of the motor can be computed using the formula for the diffusiophoretic propulsion velocity [26][27][28][29][30] to give [58] where the overline denotes an average over the surface of the colloid, and Λ αα = R+rc R dr (r − R) e −βVαc − e −βV α c accounts for the interactions of solute molecules of type α with the colloid through repulsive potentials V α with finite range r c .The velocity is directed along the unit vector û that points from the H to H u hemispheres as shown in Fig. 2 (a).The numerical values of the motor velocity can be found by substituting the solutions of the reaction-diffusion equations, subject to boundary conditions on the motor surface, into Eq.( 3).This calculation is given in Appendix A. Among other factors, the propulsion velocity depends on the rate constants k 1 and k 2 , and the fractional coverage.
By contrast, if the fuel I is supplied directly from the fluid phase, as is usually done for diffusiophoretic motors, we have The full expression is also given in Appendix A.
Comparisons of Eqs. ( 3) and ( 4) allow one to quantify the differences between fuel supply at the motor surface and supply from the boundaries.Not only do two rate constants, k 1 and k 2 , appear when fuel is supplied locally, but the bulk phase reactions and boundary conditions couple all three concentrations, c A , c B and c I .Hence, surface fuel production leads to a rich structure that could be exploited control motor motion.
Simulation: Simulation of the motor dynamics is based on a coarse-grained microscopic description of the entire system.The surface of the colloid is covered by spherical beads that act as coarse-grained groups of enzymes.The E 1 and E 2 enzyme groups are uniformly and randomly distributed on the colloid surface with fractions as described above and sketched in Fig. 2 (a).The reactions on the surface of the colloid take place on the chemically active enzymatic sites.The strengths of the repulsive colloid-solute potentials are gauged by α energy parameters, and for these simulations The evolution of the system is carried out using hybrid molecular dynamics-multiparticle collision dynamics [59][60][61][62] Further information on the simulation model and parameters are given in Appendix B.
Concentration fields and motor velocity: From Eq. ( 3) one can see that the concentration gradient fields in the colloid vicinity play an important part in the diffusiophoretic mechanism, and, for the sequential reaction mechanism of interest here, the results of simulations and continuum calculations allow one to assess the factors giving rise to the spatial structure of these fields.Specifically, the gradient fields ∂ θ c B (R, θ) and ∂ θ c I (R, θ) enter the expression for the V u in Eqs (3) and (A20).

FIG. 2. (a)
A colloidal Janus motor with radius R = 4.0 and 1000 coarse-grained enzymatic sites on its surface.The fractions of E1 sites (blue) on the two hemispheres are f u 1 = f 1 = 1/2, so that E1 and E2 each occupy 250 sites on the Hu cap, while there are 250 E1 and 250 E0 sites (grey) on the H cap. The motor axis is defined by the unit vector û pointing from the center-of-mass of the colloid to that of the Hu cap containing E2 sites (red).The energy parameters are H caps meet.One also observes the depletion of locallysupplied I on the upper hemisphere.The pronounced asymmetry of these fields around the colloid indicates that local fuel can provide effective self-propulsion.The overall structure of the concentration fields of all three species, α = A, I, B near the H u cap is seen in the plots of the radial concentrations, cα (r), in Fig. 2  (c).These radial concentrations were constructed from averages of the species density fields only over the angles corresponding to the H u cap.The plots show that I is efficiently converted to product B in the motor vicinity, while species A remains in high concentration farther from the colloid.Thus, the colloid is able to function as a motor without high concentrations of the undesirable fuel I in the bulk phase, an important feature for biological applications where undesirable fuels such as H 2 O 2 are required.Furthermore, the consumption characteristics of the intermediates can have an influence on the design of motor chemical gates based on coupled enzyme networks.
To examine more directly how the locally produced fuel I is converted to product B molecules, we take f 1 = 0 since then I is produced only on the H u cap that also has E 2 enzymes.The ratio of the average number of product B molecules, NB , to the average number fuel I molecules, NI , resulting from reactions on the H u cap defines the conversion ratio, CR = NB / NI .This ratio depends on the fractional occupancies of the E 1 and E 2 enzymes on the H u cap, N u E2 /N u E1 = f u 2 /f u 1 , since the probability of species I to undergo a surface reaction to B before escaping to the bulk depends on this ratio.This dependence is shown in Fig. 2 (d).For N u E2 /N u E1 < 1, CR increases rapidly and then reaches a plateau value CR ≈ 0.75.Since the amount of supplied fuel decreases as N u E2 /N u E1 increases, this suggests that a fractional coverage of f u 1 ≈ 1/2 may be optimal.Results for the propulsion velocity of the motor are given in Fig. 3.The component of the average motor velocity along û may be computed in the simulation from V u = V (t) • û , where V (t) is the instantaneous velocity of the center-of-mass of the colloid and the angular brackets denote an average over time and different realizations of the dynamics.The simulation values are compared to the continuum results using Eqs.( 3) and (A20) in Fig. 3 (a) for different values of k 1 and fixed k 2 .In the simulation k 1 is varied by changing the reaction probability p R 1 on E 1 with p R 2 = 1.0 for E 2 .In the continuum model these rate coefficients are estimated from the kinetic theory expression in Appendix B. (The abscissa values in the plots are the kinetic theory rate coefficients.)The increase in V u with k 1 is expected since more fuel is supplied locally to the E 2 enzymatic reaction that powers propulsion; however, the deviation from linear increase can be attributed to the inability of E 2 to process the fuel quickly enough before I escapes to the bulk phase.Both simulation and continuum theory are in qualitative agreement and show this effect.The velocity V u also depends on fractional occupancy of E 1 and E 2 on the H u cap, f u 2 /f u 1 = N u E2 /N u E1 as shown in Fig. 3 (b).Here, in contrast to Fig. 2 (d), the H cap occupancy is still f 1 = f 0 = 1/2.The rapid increase of V u with N u E2 /N u E1 reflects the corresponding increase in local fuel supply (seen Fig. 2 (d)) on increased numbers of E 2 enzymes, while the decrease for larger values of N u E2 /N u E1 is due to the saturation of CR with decreased numbers of E 1 enzymes.These results indicate that choice of the optimal fractional occupancy of E 1 and E 2 gives rise to the most powerful propulsion, another feature that could be exploited when designing motors with coupled enzyme kinetics.
The above results are compared to those where fuel I is directly supplied from the bulk in Fig. 3 (c).The colloid now has a fraction f u 2 = 1/2 of E 2 sites on the H u cap, while all other sites on the colloid are chemically inactive E 0 sites.Only the reaction rate k 2 enters this calculation (see reaction (A2) and Eq.(A30) in Appendix A) .The plots are similar to those in panel (a), although the velocity is somewhat larger.This is evident in Fig. 3 (d) where the velocity probability distribution functions p(V u ) are plotted.The peaks lie at V u = 0.0065 and V u = 0.0084 for local and global fuel supply, respectively, and one also sees that thermal fluctuations are strong.Lastly, for global fuel supply simulation and continuum theory are in quantitative agreement, as shown in fig.3(c).For local fuel supply, the continuum model is not able to capture the full reactive dynamics in the boundary layer since the reaction rates it uses are averages over the entire hemisphere.

III. MOTORS WITH CHEMICAL GATES THAT SENSE THEIR ENVIRONMENTS
We now show how other chemical logic gates can be used by motors to sense certain aspects of their environments and respond to these chemical signals by carrying out specific tasks.The examples are simple and intended to be illustrative, although the tasks the colloids perform are not very challenging.

INH gate:
The first task is to prevent a colloid from being captured by a source.We suppose that the system contains a colloid (C) and a fixed spherical source (S) of a chemical species I.The colloid responds to the gradient field of this species through a diffusiophoretic mechanism and may be attracted to the source.We want the motor to sense the magnitude of this concentration field and respond to it in a way that prevents capture by the source.We do this by implementing an inhibitor (IHN) gate on the colloid.
Protein logic gates can be constructed using the properties of inhibitors that prevent certain reactions from taking place.For this purpose we consider the reaction scheme in Fig. 4 that represents an IHN gate.This gate is modeled on earlier work for a NOT gate that employs inhibitor molecules.[45] In this scheme an enzyme (E) catalyzes the reaction R * Since the catalyst E is uniformly distributed over the entire surface of the colloid, in the absence of the source sphere and with a uniform supply of fuel A, the net propulsion force is zero by symmetry.When the S sphere is present and the colloid enters its vicinity it will experience a depletion of A fuel on its face that points towards the source.The subsequent reaction A E → B converts A around the colloid to B. Since B > I the diffusiophoretic force will cause the colloid to move towards the S sphere and be captured by it, as shown in the inset in Fig. 5

(b).
To avoid capture of colloid by the source, an INH gate is constructed on the colloid.Species I acts as the inhibitor.If the local concentration of I around a enzyme bead, c I , exceeds a pre-set threshold value, c T I , we suppose the reaction A E → B is inhibited.If the value is either lower or higher than the threshold, the output of the gate is defined to be either 0 or 1, respectively.Therefore, when the colloid moves toward the source, the catalytic activities of enzymes facing the source are suppressed if c I > c T I .In this circumstance the local concentration of A is greater than that on the face where the A → B reaction takes place and, since A > B > I , a diffusiophoretic force pushes the colloid away from the source.lower thresholds the colloid tends to remain at large distances from the source with quite sharp probability distributions, while for a large threshold it is able explore regions closer to the colloid with a broad distribution, without collapsing to a bound pair.
In Fig. 6 we consider a system with a colloid and four source spheres.The gate threshold is set to have an intermediate value, c T I = 0.3.One sees that the colloid can move among the array of sources without touching them and, in fact, is trapped in a small region between the sources as a result of the action of the IHN gate.
OR gate: An OR gate is shown in Fig. 7 (a).This figure also shows an enzymatic implementation of such a gate that has been discussed in the literature.[51] The single enzyme acetylcholinesterase (AcCHE) can catalyse the decomposition of both acetylcholine (AcCH) and butyrycholine (BuCh) to give the common product choline (Ch), so that it mimics the action of an OR gate.
The task using the OR gate is variation on that for the INH gate above but shows how a gate can be used to change the propulsion direction of a Janus colloid through a chemotactic effect in order to be captured by a source.We again have a source sphere that produces I by consuming A. However, the colloid is Janus particle where one hemisphere H u is randomly covered by enzymes E 1 and E 2 with equal probability 1/2, while H is randomly covered by E 1 and inactive E 0 enzymes, also with probability 1/2, as shown in fig.8 B takes place on the E 2 sites that are uniformly distributed on the colloid (see Fig. 8 (a)) simulating an OR gate.Now B will also be preferentially produced on the face of the colloid that points towards the source, since species I has higher concentration near to the source (see Eq. ( 5)).In general this face will not point in the same direction as vector û.As a result the colloid diffusiophoretic force lies in a direction determined by the combined production of B due to both catalytic reactions.This biases the propulsion direction towards the source, and leads to capture of the colloid.Figure 8 (b) shows several stochastic trajectories that lead to capture by the source through this OR gate mechanism.

IV. DISCUSSION
The results in this paper showed how small enzyme networks on colloidal motors could be used to construct a number of different chemical logic gates, and how these gates can enable the active colloids to perform simple tasks.The simulation results and continuum models provide detailed information on the way surface enzyme networks function to produce the inhomogeneous concentration fields that play an important part in how chemical gates function, how fuel is consumed in the network, and The examples of sensing tasks presented above, while simple, show how an active colloid with chemical gates can use propulsion and sensing to change its dynamics to achieve a goal.Small enzyme networks of the sort shown in these examples can be used to construct all the logic gates, and strategies for constructing such gates have been described recently.[52] The results lay the foundation for further research.The experiments on micromotors with a gated pH responsive DNA nanoswitch mentioned earlier are laboratory examples of how such gated active colloids could be used in applications.[38,39] There have been other studies that consider how active motion can be combined with logic functions or rules to yield complex dynamics; for example, model gates have been used in the context of active microfluidics [63], run-and-tumble particles with simple rules for gates [64], and the use of chemical signals to communicate among self-propelled particles [65].In the work presented here, model protein networks were used to build the chemical gates, and the simulations of gate operations were carried out at a coarse-grained microscopic level that takes into account the full reactive dynamics and fluid flows that underlie the propulsion mechanism and gate operations of the colloids, along with their interactions with their environments.
Proteins generally have a high specificity for substrate molecules enabling them to sense their temporally and spatially varying environments.Thus, the proteins on the surfaces of the colloids, either in chemical gates or acting singly, can be used to enable diverse sensing capability.Circuits built from these gates [47,51], either on single colloids or collections of colloids, should be able to carry out computations needed to accomplish complicated tasks.The work presented here could be extended to treat these more complex and interesting situations, and may serve to interpret or suggest future experimental work on active colloids that sense and autonomously respond to their environments.
The N E =1000 surface beads with radius σ = 1.0 are coarse-grained enzymatic sites.The colloid is contained in a cubic simulation box with size V = L x ×L y ×L z containing reactive particles.Periodic boundary conditions are applied.
For simulation where the colloid is confined to the center plane in a simulation box with a slab geometry, five beads are selected: one at the center-of-mass of the colloid, and others on a circle with r = 3.5 in the x-y plane at z = L z /2.They interact with the walls at z = 0 and z = L z through a 9-3 LJ potential, V 93 LJ (r) = w [(σ w /r) 9 − (σ w /r) 3 ], where w and σ w are the wall energy and distance parameters, respectively.The interaction between the fluid particles and the source sphere are 6-12 repulsive LJ potentials with = 0.1.
Reactive events that convert substrate to product take place on the enzyme beads E ν (ν labels different enzymes) with probabilities p R ν .[67] For the continuum calculation, the reaction rate constant for the colloid can be estimated from the kinetic theory expression, and the reduced mass is µ cs = (M c m)/(M c + m).
The system is evolved using hybrid molecular dynamics-multiparticle collision (MPC) dynamics.[59][60][61][62] Simulation results are reported in dimensionless units based on energy , mass m, and cell length a 0 .The time is in units of t(ma 2 0 / ) 1/2 → t, distance parameter r/a 0 → r, and temperature k B T / → T .The system temperature T is 1  6 .The average number of fluid particles per cell is c 0 = 10.2.The mass of the colloid is given by M c = 4  3 πR 3 c 0 , so that the colloid is approximately neutrally buoyant.The MPC rotation angle is φ = π The values of α are given in the text.Grid shifting is employed to ensure Galilean invariance.[68] take place in the fluid phase with bulk phase rate constants k B b and k I b , and maintain the system in a nonequilibrium state.
Plots of c B (5, θ) and c I (5, θ) versus θ are shown in Fig. 2 (b) for a radius value r = 5, somewhat outside the boundary region at r c = 4.125 where the potentials act.The c B and c I profiles are typical of Janus colloids where the largest gradient is in the vicinity of the equator where the H u and 1.0 and B = 0.1.(b) The intermediate fuel species I and product B concentrations cα(θ) versus θ at r=5.0.(c) The Hu-cap radial concentrations cα(r) for three substrate species, A (triangles), I (squares), and B (circles).(d) The dependence of conversion ratio CR on N

FIG. 3 .
FIG. 3. (a) VelocityVu as a function of the reaction rate constant k1 for fixed k2 for the colloid described in Fig.2 (a).The squares and circles are calculated from simulation and theory, respectively.(b) The dependence of Vu on the ratioN u E 2 /N u E 1 .(c)The velocity Vu from simulation (squares) and theory (circles) as a function of k2 when fuel I is supplied from the bulk.(d) Velocity probability distribution P (Vu) versus Vu for local (squares) and global (circles) fuel supply.The data is averaged from ten independent realizations.

1 +FIG. 4 .FIG. 5 .
FIG. 4.An enzymatic reaction that simulates an inhibitor INH gate.The enzyme E catalyzes the reaction of substrates A and R * 1 to form products B and P * 1 .The open circle with a bar signifies that species In is an inhibitor for the reaction.The substrate A and inhibitor In correspond to the inputs x1 = In and x2 = A to the INH gate, while the output is y1 = B.

Figure 5 (FIG. 6 .
FIG.6.The concentration profile of species I and trajectories of the colloid obtained from five independent realizations of the dynamics in a system with size Lx = Ly = 80 and Lz = 14 containing four source spheres with c T I = 0.3.

E1BE2B. 1 FIG. 7 .E2B
FIG. 7.An enzymatic reaction that simulates an OR gate.The enzyme acetylcholinesterase (AcCHE) catalyzes the decomposition of both acetylcholine (AcCH) and butyrycholine (BuCh) to the product choline (Ch).The inputs to the OR gate are x1 = AcCh and x2 = BuCh and the output is y1 = Ch.

FIG. 8 .
FIG. 8. (a) Janus colloid with equal numbers, N u E1 = N u E2 = 250, of E1 and E2 enzymes randomly distributed on the upper Hu hemisphere.Likewise, equal numbers, N E1 = N E0 = 250, of E1 and E0 enzymes are randomly distributed on the lower hemisphere H .Both substrates A and I can be converted to B by E1 and E2, respectively.Enzymes E0 are chemically inactive.The interaction parameters are chosen to be A = I = 2.0 and B = 0.1.(b) Three examples of the evolution of the distance (RSC ) between the colloid and the source.

2
and collision time interval is τ M P C = 0.5.The velocity Verlet algorithm is used to integrate the Newton's equation of motion with τ M D = 0.005.The Schmidt number is S c = 1.39.Other parameters: harmonic spring force constant k s =60, wall w = 5.0, σ w = L z /2, fluid rate constants k B b = k I b = 0.001, common diffusion coefficient D = 0.097, viscosity from MPC expression η = 1.35.