©2019 Xiang Yu ALL RIGHTS RESERVED

Xiang Yu

AND FLOW OF AMPHIPHILE-BASED COLLOIDAL PARTICLES

XIANG YU

A dissertation submitted to the

School of Graduate Studies

Rutgers, The State University of New Jersey

In partial fulfillment of the requirements

For the degree of

Doctor of Philosophy

Graduate Program in Chemical and Biochemical Engineering

Written under the direction of

Meenakshi Dutt

And approved by

______

New Brunswick, New Jersey

JANUARY, 2019 ABSTRACT OF THE DISSERTATION

Multiscale Computational Methods to Study the Formation and Flow of Amphiphile-

Based Colloidal Particles

By XIANG YU

Dissertation Director:

Meenakshi Dutt

Colloids are defined as large insoluble particles uniformly suspended in a medium including gas, liquid, and solid. Different from atoms and small molecules, one interesting property of colloids is its collective behavior: a property that nano-scaled colloids exhibit as a result of colloidal self interaction and interaction with surrounding medium. It is believed that this property plays an important role in colloidal self-organize, sedimentation, phase segregation, as well as crystallization. Hollow nanostructures, such as vesicles, due to their stability, low toxicity, intracellular communication ability, and compatibility with human tissue, has gained considerable scientific interests as a type of potential drug delivery vehicle. Due to current lack of knowledge of stability of vesicles in microfluidic devices or in blood flow, first part of this study (Chapter 2) presents how nanoparticles behave by changing architecture and relative concentrations of the molecular species of the nanoparticles under various flow conditions using a simulation technique called DPD. The second part of this study (Chapter3 and 4) focus on implantation of a new multi-scale simulation technique in investigation of membrane systems. The hybrid lattice Boltzmann- molecular dynamics method (MDLBM) is able to capture molecular details of solutes and

hydrodynamic interactions of fluid simultaneously, while saving computational time compared with traditional explicit solvent molecular dynamic methods. In Chapter 3, dynamics and mechanical properties of vesicles and bi-layers are tested and shown to be close to previous studies. In Chapter 4, current coupling scheme is modified and applied to study self-assembly process of lipids. Modified MDLBM algorithm is able to capture dynamical process of colloids, where the hydration shells are spontaneously replaced by interactions with other molecular species in solution, for example in processes encompassing aggregation or interfacial adsorption.

iii

Acknowledgment

I would like to gratefully acknowledge the School of Engineering and the

Department of Chemical and Biochemical Engineering at Rutgers, The State University of

New Jersey for providing supporting towards my graduate studies. I would like to express my gratitude to the National Science Foundation for supporting my graduate studies via award CBET-1644052.

I would like to take the opportunity to thank Extreme Science and Engineering

Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562, through allocation DMR-140125, for providing computational support for my graduate research. I would also like thank Rutgers Discovery

Informatics Institute for providing additional computational support for my graduate studies.

During my graduate studies, I have had the opportunity to attend several conferences as a part of my professional development. I would like to express my gratitude to the School of Graduate Studies at Rutgers, The State University of New Jersey for facilitating my attendance of these professional events.

Table of Contents

ABSTRACT OF THE DISSERTATION ...... ii

Acknowledgment ...... iv

List of Figures ...... vii

List of Tables ...... ix

Chapter 1 ...... 1

Introduction ...... 1

Chapter 2: Flow-induced Shape Reconfiguration, Phase Separation and Rupture of Bio- inspired Vesicles ...... 6

Introduction ...... 6

Methods ...... 8

Results and Discussion ...... 13

Conclusion ...... 32

Supporting Information ...... 35

Chapter 3: A Multi-scale Approach to Study Molecular and Interfacial Characteristics of

Vesicles ...... 38

Introduction ...... 38

Methods ...... 42

Results and Discussion ...... 51

Conclusion ...... 62

Chapter 4:Self-assembly of Amphiphilic Molecules via the Hybrid Molecular Dynamics-

Lattice Boltzmann Technique ...... 64

Introduction ...... 64

Methodology ...... 66

Results and Discussion ...... 70

Chapter 5 ...... 75

Conclusions and Future Direction ...... 75

Reference ...... 76

List of Figures

Figure 1 Representation of DPPC and PEGylated DPPC, DMPC, cholesterals and vesicles composed of relative components(using DPD coarse-grained model)...... 11

Figure 2 Morphology change and Deformation index of hairy vesicle(PEG=6) for different concentrations of hairy lipids and Re ...... 14

Figure 3 Morphology change and Deformation index of hairy vesicle(PEG=3) for different concentrations of hairy lipids and Re...... 16

Figure 4 Morphology change and Deformation index of synthetic vesicle for different concentrations of cholesterol and Re ...... 18

Figure 5: Surface stress vs Re for hairy vesicle and synthetic cell ...... 19

Figure 6 Line tension vs Re for hairy vesicle and synthetic cell...... 23

Figure 7 Shear force vs time for 10% and 50% concentration of the hairy lipids at different flow strength...... 27

Figure 8 Average shear force for different concentrations of the hairy lipids and Re, and solvent frequency outside the vesicle as a function of the distance from the outer monolayer of the vesicle...... 30

Figure 9: Images of hairy vesicle(PEG=6) with 50% hairy lipids during rupture process.

...... 31

Figure 10 Relation between force added on flow particles and mean flow velocity in the channel...... 36

Figure 11 Per-bead interaction energy between cholesterols and cholesterols for various concentrations of cholesterol in CMV...... 37

vii

Figure 12 Measurements of the shear force as a function of time, for 20% and 30% concentration of the hairy lipids...... 37

Figure 13 Representation of DPPC, PEG-DPPE hairy vesicle (using dry Martini coarse grained model) and closing of a DPPC membrane forming a stable vesicle...... 47

Figure 14 Comparison of probability of the end to end distance of PEG 19 chain between

MDLBM and explicit solvent standard Martini model...... 48

Figure 15: Scaling exponent test between radius of gyration and molecular weight for single PEG for various PEG length ...... 49

Figure 16: mean square displacement of DPPC at different temperature with and without

LB fluid ...... 54

Figure 17 Radius of gyration (Rg) (in nm) of PEG chain vs relative concentration and length...... 60

Figure 18 End to end distance(in nm) of PEG chain vs relative concentration and length.

...... 61

Figure 19 Schematic of dynamic coupling of DPPC lipid beads to LB fluid represented by 2D lattice grids ...... 69

Figure 20 Total energy of system vs time for different coupling criteria.(critical number)

...... 72

Figure 21 Time evolution of number of aggregates for a system of 218 DPPC lipids for varying coupling criteria (critical number)...... 73

viii

List of Tables

Table 1: Diffusion coefficient of DPPC lipids in membrane and vesicles, in the absence and presence of the LB fluid at both gel and fluid phase...... 56

Table 2: Area per lipid of DPPC molecules in vesicles and membranes, in the absence and presence of the LB fluid, at both gel and fluid phase...... 56

Table 3: Distance between neighboring grafting points for different PEG chain lengths and relative concentrations...... 62

Table 4: . Scaling exponent of self-assembly test for various coupling conditions ...... 74

Chapter 1

Introduction

Computer simulation creates a model based on a proved theory or developed mathematical relation or logic to study real world problems by predicting behavior of different systems. Simulation is widely used and applicable in many fields not only because it describes and explains natural phenomena but also because it is indispensable in many aspects. First, it can predict the result of an existing system which is too expensive or dangerous to run an experiment. For instance, air flow simulation is applied before manufacture of aircraft. A meltdown simulation is carried out for a nuclear reactor for safety purpose before operation. Second, Simulation is used to model a system that takes too long to observe. Numerous models have been developed to forecast population growth and climate change. Third, simulation is extremely useful to model a system that is impossible to build in real world. Quantum theory is used to predict the motions of substances at molecular or atomic level and evolution of universe.

For design and engineering of materials in microscopic level, simulation plays an important role since many processes are transient and are difficult to be captured by experimental methods. There are different simulation techniques depending on the scales of materials investigated and also multi-scale methods that combine benefits of two or more simulation techniques. To study systems at a scale from nano to micron, molecular dynamics is one of the possible ways. Molecular dynamics uses Newton’s equations of motions to determine momentum of interacting particles where intermolecular interactions are captured by various force fields. One can get microscopic details of each particle in- situ: position, velocity, and force as well as macroscopic properties of systems after

reaching equilibrium through statistical mechanics relationship. Through visualization software such as VMD, one can learn the dynamics of 3D structure of molecules of interest and compare with results from x-ray crystallography and NMR spectroscopy. Due to vast number of particles in system, molecular dynamics calculation is carried out on super computers with software that encode molecular dynamics algorithm. However, it could be expensive to investigate a large atomistic system or a small system with long timescales.

A way to overcome this issue is to build coarse-graining models which uses a one pseudo coarse grained bead to represent several actual particles. A coarse-grained model is computationally efficient, because degrees of freedom is reduced by mapping atoms into sites .Therefore, it is possible to run simulation with a larger time step, and increase speeds of computations. A coarse-grained model must retain some levels of features of atomistic models(interaction energy, structure, dynamic properties) to reproduce similar behaviors of real systems. Depending on purpose of research, degree of coarse graining scheme may vary. However, one has to be cautious in determining mapping of atoms since the advantages of coarse graining models are also their drawbacks. By increasing coarse graining degree, one sacrifices chemical details of system. A compromise way is partial coarse graining, which is considered as a multi-scale technique.

Dissipative particle dynamics(DPD) is another tool widely used to study large bio- molecules. It is considered as a mesoscale simulation method which models systems at micron scale. DPD and coarse grained molecular dynamics share many similarities. They both involve deterministic Lagrangian algorithm and they both use coarse graining scheme to reduce resolution of system. They are also very similar in system initialization and equilibration, force field calculation, neighbor list updating and many other features.

However, they differ at two main places. First, DPD uses a unique soft repulsive potential, a much simpler pairwise force field in mathematical form compared with other hard-core molecular dynamics potential. The coarse graining scheme is also much larger compared with conditional coarse grained molecular dynamics. These two factors make DPD more computational efficient than regular coarse grained MD. Second, DPD algorithm involves random force and dissipative force. Both of them together maintain the system temperature constant. Consequently, no external thermostat is needed. In Chapter 2, DPD technique is applied to investigate morphology changes of synthetic and tethered vesicle under a

Poisson flow.

An implicit solvent method is another way to save computational time. Both traditional molecular dynamics and DPD involve explicit solvent molecules in calculation.

Usually, solvents take more than 80% of simulation time. Nevertheless, properties of solvent particles are not interested. Compared with explicit solvent methods, discrete solvent beads are treated as continuum phase by approximating the mean force exerted by the external media on the solute atoms. By reducing degree of freedom, speed of simulation is increased. Despite of success of implicit method in mimicking the solvation effect by tweaking intra-molecular forces, implicit solvent methods can’t capture many features of solvents including hydrodynamic effects. Hydrodynamic interactions are long range forces induced by motions of solvent particles. Hydrodynamic interactions have been proved important in keeping correct equilibrium structure as well as dynamics of colloids including aggregation, suspension, self-assembly, and micro-phase separation.[1-

5]. For instance, hydrodynamic interaction can help copolymer system overcome barrier to

reach equilibrium during micro-phase separation. [6] It is also found that hydrodynamic effects increase diffusion of colloidal systems. [7]

Since every individual simulation method has its drawbacks (e.g. unique length and time scale), multi-scale simulation technique has been applied to study macro-molecule systems. A multi-scale approach is designed to enhance efficiency while keeping the accuracy as much as possible. For example, combined quantum mechanics and molecular dynamics are used to study reaction and dynamics simultaneously. Classical molecular dynamic and coarse grained molecular dynamic are used for different molecules or different parts of same molecule in a system. A hybrid technique of molecular dynamics with lattice Boltzmann method [8-9] is another multi-scale simulation approach. In this technique, the momentum of solute particles will be updated by molecular dynamics algorithm while the dynamics of fluid will be taken cared by lattice Boltzmann method.

Different from molecular dynamics or coarse-grained molecular dynamics, lattice

Boltzmann models the solvent at a much larger scale (still smaller than continuum scale).

Lattice Boltzmann method represents fluid in terms of fictive probability density streaming and colliding at uniform lattice sites. It is a dramatic reduction of the degrees of freedom compared with explicit solvent molecular dynamics. Despite the particle based nature of lattice Boltzmann method, it has been proved that property of bulk fluid such as hydrodynamic effects are maintained. [10] In Chapter 3, this coupled technique with an implicit solvent force field will be tested on a membrane colloidal system. Various dynamic and mechanical properties of membrane system will be tested and compared with experimental results. In Chapter 4, this existing coupling scheme will be modified to study self-assembly dynamics of DPPC lipids. New coupling algorithm allows dynamic

coupling of selected solute particles to LB fluid and make study dynamics of interfacial process feasible.

Chapter 2: Flow-induced Shape Reconfiguration, Phase Separation and Rupture of

Bio-inspired Vesicles

Introduction

Nanoparticles play a critical role in the encapsulation, storage and transport of therapeutic agents in a wide range of applications including drug delivery. The circulation time and bio-distribution of the nanoparticles is critically dependent upon their characteristics. The characteristics include the nanoparticle size[11-16], shape[16-20] and response to various flow conditions[21] (e.g., the Reynolds number in the human circulatory system can range from 1 in capillaries to 4000 aortas[22]). Hence, the successful delivery of therapeutic agents hinges upon the ability of the nanoparticles maintain their structural stability under diverse flow conditions. The response of the nanoparticles to flow-induced stimuli is controlled by the architecture and concentrations of the constituent molecules[23]. Therefore, suitable molecular compositions can be selected to design nanoparticles which are structurally resilient under a range of flow conditions. The selection of molecular compositions can be inspired by biological particles such as red blood cells (RBCs) and bacterial cells. RBCs possess high deformability that allows for their transit through vasculature such as lung capillaries without rupturing. [13-

25] Similarly, bacterial cells such as E. coli have attachment pili or fimbriae which help them withstand shear forces[26]. This study focuses on the response of cell- and bacteria mimetic nanoparticles to diverse flow conditions during transport in a channel. The investigation examines the role of the architecture and relative concentration of the molecular species constituting the nanoparticles on their response to various flow conditions. Of specific interest are nanoparticles with dynamic textured surfaces. Due to

the challenges in resolving physical phenomena across a range of spatiotemporal scales through experimental approaches, a mesoscopic computational technique has been adopted for the study.

Existing computational studies have investigated the behavior of different nanoparticles in microvascular flow, via techniques such as the boundary integral method,[27] multiparticle collision dynamics[28-30] and the lattice Boltzmann method.[31]

For example, elastic vertex networks have been used to model fluid vesicles to observe their shape transition in a capillary under different flow conditions[29]. In addition, particle-based models have been used to study RBCs (modeled as single component liposomes) to demonstrated different kinds of shape deformations, as well as clustering and tumbling in response to various flow rates, viscosities and confinements. [32-34] This study examines the behavior of multicomponent vesicles under Poiseuille flow in a channel and the role of specific molecular species on the response of the vesicles to different flow conditions.

In this study, we have adopted a mesoscopic simulation technique entitled Dissipative

Particle Dynamics (DPD) to examine the flow-induced response of two types of nanoparticles, namely bioinspired multicomponent and hairy vesicles. The multicomponent vesicles encompass representative amphiphilic molecular species present in biological cell membranes.[35] Whereas the hairy vesicles are composed of amphiphilic molecular species, some of which are bearing water-soluble functional groups, or tethers, to mimic the fimbriae. The selection of the tethers was determined from earlier studies which showed that tether-bearing vesicles have extended in-vivo circulation half time and enhanced biodistribution. We examine the impact of the Reynolds number and the relative

concentration of the key molecular components on the response of the two nanoparticles to the flow in a channel. We observe the vesicles to resist deformation with increasing

Reynolds number, relative concentrations of cholesterol, molecules grafted with tethers and the tether length. For sufficiently high concentrations of the molecules bearing long tethers and Reynolds number, the molecules are observed to phase separate in the outer monolayer of the vesicles due to the Marangoni effect. In addition, the vesicles are unable to preserve their structural integrity and rupture at high Reynolds number, lower concentrations of the tether-bearing molecules and tether length. The results from this study could guide the design of nanoparticles with molecular compositions optimized for desired flow-induced response characteristics for applications in drug delivery, tissue engineering and bio-sensing.[36-42]

Methods

We used a mesoscopic simulation technique entitled DPD which simultaneously resolves the molecular and continuum properties, and reproduces hydrodynamic behavior.[43-47] DPD is a mesoscopic simulation technique that uses soft-sphere coarse- grained models to capture both the molecular details of the nanoscopic building blocks and their supramolecular organization while simultaneously resolving the hydrodynamics of the system over extended time scales . In order to capture the dynamics of the soft spheres, the DPD technique integrates Newton’s equation of motion via the use of similar numerical integrators used in other deterministic particle-based simulation methods. The force acting on a soft sphere i due to its interactions with a neighboring soft sphere j has three components: a conservative force, a dissipative force and a random force, which operate within a certain cut-off distance rc from the reference particle i. These forces are pairwise

additive and yield the total force acting on particle i, which is given by Fi = ∑j≠1 Fc,ij +

Fd,ij + Fr,ij The soft spheres interact via a soft-repulsive force Fc,ij = aij(1 − rij/rc)r̂ij

d for rij < rc and Fc,ij = 0 , for rij≥rc), a dissipative force (Fd,ij = −γω (rij)(r̂ij vij)(r̂ij)

r d and a random force (F = −𝜎ω (r ) (r̂ θ ) (r̂ )where ω (푟) = (1 − 푟)2 (for r < 1), r,ij ij ij ij ij

d 2 ω (푟) = 0 (for r ≥ 1) and 𝜎 = 2훾푘퐵푇. 푎푖푗 is the maximum repulsion between spheres i and j, vij = vi − vj is the relative velocity of the two spheres, rij = ri − rj, rij = |ri − rj|, r = rij/rc, 훾 is the viscosity related parameter used in the simulations, 𝜎 is the noise

d r amplitude, θ (t) is a randomly fluctuating variable from Gaussian statistics, ω and ω ij are the separation dependent weight functions which become zero at distances greater than or equal to the cutoff distance 푟푐.The distinct molecular species, as shown in Fig. 1, are represented by established coarse-grained models.[35,48] In this study, we investigate two biologically inspired yet distinct types of vesicles: hairy vesicles (HVs) and bioinspired multicomponent vesicles (BMVs).

The HVs encompass a mixture of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine

(DPPC) and Poly Ethylene Glycol-grafted, or PEGylated DPPC, as shown in Fig. 1 (a) -

(c). The model for the PEGylated DPPC molecule (or hairy lipid) includes a hydrophilic tether (representing a PEG chain) grafted onto the head group of the lipid.[48-50] We examine the impact of two tether lengths, encompassing 3 and 6 beads, which correspond respectively to degrees of polymerization 6 and 12 for PEG molecules.[51] The BMVs are multi-component vesicles encompassing two distinct phospholipid species, DPPC and 1,2- dimyristoyl-sn-glycero-3-phosphocholine (DMPC), a glycolipid and cholesterol, as shown in Fig. 1 (d) - (g). The effective chemistry of the amphiphilic molecules and their

interactions in both systems are captured through the soft repulsive interaction parameters between the different types of beads[35,48,52]. In the cholesterol model, we introduce an additional bond between the pairs of diagonal beads (see Fig. 1 (f)) to represent the steroid ring. These additional bonds impart greater stiffness to the hydrophobic tail of cholesterol.

As a result, the fluidity of the vesicle bilayer can be controlled by varying the concentration of cholesterol.[53]

Figure 1: Image of (a) DPPC, (b) PEGylated DPPC, (c) a HV with 50% of PEGylated

DPPC and 50% of DPPC, (d) DPPC, (e) DMPC, (f) cholesterol, (g) glycolipid, and (h) a

BMV with 32.5% of DPPC, 32.5% of DMPC, 30% cholesterol and 5% glycolipid. (i) An image of a HV flowing through in a cylindrical channel. The solvent particles are not shown. The image on the left shows the channel wall to encompass particles organized on a lattice.

To model the transport of the vesicles in a cylindrical channel under Poiseuille flow conditions, we create a cuboid-shaped simulation box of dimensions 40rc*40rc*60rc. The simulation box has a periodic boundary condition in the direction of the flow (z-axis). The particle density is set at ρ = 3 so that the total number of particles is given by 288,000. The channel is represented by a cylinder spanning the box with its cylindrical axis parallel to the z direction. The diameter of the channel is set to be 32 rc, corresponding to 25nm. The particles constituting the channel wall are organized on a lattice and kept frozen during the entire duration of the simulations. The solvent particles are located both inside and outside the channel. The soft repulsive interaction parameter between the particles encompassing the channel wall and all the other particles is set to be aij = 25.[54] The number of solvent particles depends upon the composition of the vesicles so that the total number of particles in the system is conserved. Fig.1 (i) shows a HV flowing through a cylindrical channel. To drive the flow through the channel, we implement an effective pressure gradient across the channel by adding a constant body force to all solvent particles outside the vesicles.[54-58]

The flow rate can therefore be tuned by varying the body force. The soft repulsion interaction parameter between the channel wall and solvent particles imposes a no-slip boundary condition which enables an equilibrium Poiseuille flow through the channel. The

velocity at the solvent-channel interface is significantly small (factor of 10-4) compared to the magnitude of the mean flow velocity. Upon fitting the velocity to a parabolic line, we find that the transport of the solvent through the channel closely approximates Poiseuille flow. We characterize the flow via the Reynold’s number (Re) which encompasses both inertial and viscous contributions from the solvent flow. The Reynolds number is given by

UD the equation Re  zH , where  is the water density, U is mean flow velocity, D  z H is the diameter of the channel and μ is the viscosity of water. The and are mapped into real units through specific length scale (σ = 0.76 nm) and time scale (τ = 6.0 ns) obtained from a previous study on similar membrane systems. Using Re to characterize the flows will allow us to draw correspondence with biological systems involving flows of particles through confined volumes.[59,60]

We begin with an equilibrated vesicle in the cylindrical channel, surrounded by solvent

particles. The simulations are run for a time t0  1000 before a time dependent body force

t FF0 ( 1) is applied to each solvent particle in the channel. The equilibrium mean t0 flow velocity has a linear dependence on 퐹표, as shown in Fig. 10.This approach is used to

prevent the rupture of the vesicles caused by rapid solvent flow. After time tt20 2000

, the force F reaches the target value of F0 and remains constant for the remaining duration of the simulation (namely, 18,000 τ). The equilibrium mean flow velocity has a linear dependence on . Therefore, we tune the Re by varying the body force from 0

to 0.008 kTB / . We study the shape reconfiguration, phase separation and rupture of the

vesicles under different flow conditions and examine the underlying mechanisms. Four random seeds were used to test the reproducibility of the measurements for each system.

Results and Discussion

The HVs with tethers encompassing 3 and 6 beads will be referred respectively as

HV-3 and HV-6.

Figure 2 (a) Phase diagram of HV-6 for different concentrations of hairy lipids and Re.

The relative length scale in the figure does not represent the real size of the vesicles. The flow is in -z direction. (b) Deformation index of HV-6 as a function of the concentration of hairy lipids and Re. All the measurements were performed using four random seeds. a. Flow-induced shape reconfiguration of HVs and BMVs

The shape of the HVs and BMVs are observed to undergo significant reconfiguration for a range of Re. Fig 2 shows the characterization of the shape of HV-6 for different

concentrations of the hairy lipids and Re. For small values of the mean flow velocity (U z

-4 < 0.1 rc / τ) and Re (Re < 3.5 × 10 ), we do not find significant changes in the shape of the HVs. These results are observed for all concentrations of the hairy lipids. The HVs are observed to possess an approximately spherical shape when no flow is applied. As Re increases, the shape of the vesicles transitions from a sphere to a bullet with a slightly tapered head towards the direction of the flow. We note that at Re = 2.2 × 10-3, for a vesicle with 10% molar concentration of the hairy lipids, the shape of the vesicle undergoes a transition from a symmetric to an asymmetric bullet-like vesicle. This transition is significantly distinct from vesicles under the same flow conditions but with higher concentration of the hairy lipids. For higher concentrations of hairy lipids and Re, the hairy lipids are observed to phase separate on the surface of the HVs. This behavior will be discussed in the next section.

To understand the role of the molecular composition and flow conditions on the deformation of the HVs, we perform quantitative measurements of the shape reconfiguration. The bullet-like shapes are approximately axially symmetric and therefore can be characterized through the Deformation Index (DI).[61] The DI is defined as

X DI 1 , where X and Y are respectively the long and short principal axes of the Y vesicles excluding the tethers. We measure the DI of the HV-6, for a range of concentrations of the hairy lipids (10% - 50%) and Re (0, 3.5 × 10-4, 7 × 10-4, 1 × 10-3, 1.4

× 10-3, 1.8 × 10-3, 2.2 × 10-3 and 2.6 × 10-4), as shown in Fig. 2(b). We would like to note that for solvent flows with Re > 2.6 × 10-4, the vesicles were observed to rupture under steady-state flow conditions and yield bicelles. Whereas for Re = 2.6 × 10-4, the HV-6 with lower concentration of hairy lipids (10 % - 30%) are observed to rupture. HV-6 with higher concentration of hairy lipids maintain their structural integrity and bullet-like shapes. Since

DI measurement on compositions and flow conditions which yield bicelles cannot be compared with corresponding measurements for unruptured vesicles, those cases are excluded from Fig 2.

The shape transition of HV-6 from a sphere to a bullet is characterized by an increase of the DI, with Re > 3.5 × 10-4. At Re = 1.8 × 10-3, HV-6 with 10% and 20% concentrations of hairy lipids under go another transition from a bullet-like to a parachute- like shape which leads to rupture. This transition is captured by the sharp increase in DI.

The DI measurements are found to support the phase diagram of HV-6 for different concentrations of hairy lipids and flow conditions. It is also observed that HV-6 maintains axial symmetry under different flow conditions, in comparison to HV-3 and BMV. The axial symmetry could be promoted by the spatial reorganization of the hairy lipids to enhance the resistance against the high flow-induced shear.

Figure 3: (a) Phase diagram for the shapes of the HV-3 at different concentrations of the hairy lipids and Re. The relative length scale in the figure does not represent the real size of the vesicles. (b) Measurement of the deformation index of the HV-3 as a function of the concentration of hairy lipids and Re. All the measurements were performed using four random seeds.

We examined the effect of tether length on the shape of the HVs under diverse flow conditions, as shown in Fig 3. Similar to the results for HV-6, significant reconfiguration of the shape of the HV-3 is observed for Re > 7 × 10-4. For 10% concentration of hairy lipids and Re > 1.8 × 10-3, the vesicle undergoes a transition from a bullet-like to an unstable worm-like shape. However, for higher concentrations of the hairy lipids, the DI increases with Re but the shape of the HV-3 remains stable. The presence of 20% concentration of hairy lipids stabilizes the shape of HV-3 under strong perturbative flows

(that is, Re  1.8 × 10-3). However, it is unclear whether higher concentrations of hairy lipids would reduce the shape reconfiguration of the HV-3 for high Re. We note that all the HV-3 rupture at Re = 2.6 × 10-3.

We examined the flow-induced shape reconfiguration of BMVs for different concentrations of cholesterol, as shown in Fig. 4. The shape of the BMVs are found to be impacted by the flow conditions. Also, BMVs with higher concentration of cholesterol are observed to resist shape reconfiguration for higher values of Re. We surmise that the presence of cholesterol in the bilayer increases its rigidity and enables the bilayer to resist shear-induced deformations caused by the solvent flow. Based on the DI measurements, a

BMV encompassing 10% concentration of cholesterol could have 50% change in its shape compared with a BMV with 50% concentration of cholesterol at the highest Re.

Figure 4: (a) Phase diagram for the shapes of BMVs for different concentrations of cholesterol and Re. The relative length scale in the figure does not represent the real size of the vesicles. (b) Measurements of the deformation index of BMV as a function of the concentration of cholesterol and Re. All the measurements were performed using four random seeds.

Figure 5: Measurements of the surface stress as a function of Re for (a) HV-6, (b) HV-3 and (c) BMV. The measurements for the HVs have varied the concentration of the hairy lipids. Similarly, the measurement of the BMVs have varied the concentration of cholesterol. All the measurements were performed using four random seeds.

The role of the concentration of molecular components on the deformation underlying the shape reconfiguration of the vesicles can be understood through the surface stress. The surface stress is defined as the reversible work needed per unit area to deform a surface. In this study, we use the surface stress to evaluate the ability of a vesicle surface to resist shear-induced deformation under flow. We evaluated the surface stress through

1  F y y the following equation:      .[62] Where Ac is the surface area 2ANc   

of the vesicle, N is the total number of particles involved in this calculation, F is the non- bonded force between particle  and  , and y is the position vector of a particle. The surface area of each vesicle is evaluated through Delaunay triangulation of a three dimensional point set of the lipid head beads. The surface stress is averaged over all the lipid head beads and measured as a function of the Re, as shown in Fig. 5. We observe a decrease in the surface stress for both BMVs and HVs with Re. This result indicates that the vesicles are less resistant to deformation at higher Re. This observation agrees well with the measurements of DI as it suggests the vesicles tend to deform at a higher rate for larger values of Re. The shape deformations of HVs depends on the concentration of hairy lipids and tether length, as shown in Figs. 2 and 3. Hence, it becomes important to understand the effect of tether length and concentration on the surface stress. This knowledge can link the resistance of HVs to flow-induced deformations and molecular compositions.

Higher concentration of the hairy lipids does not appear to have a significant effect on the deformation resistance of the HVs. The surface stress is significantly lowered for shorter tether length, for 10% concentration of the hairy lipids. This effect becomes more

pronounced at high Re. This result explains the sudden change in the deformation of the vesicle as characterized by the DI at the corresponding flow conditions. These results are consistent with the phase diagrams shown in Figs. 2 and 3, where large deformations are observed for HV-3 with 10% concentration of hairy lipids.

For the BMVs, the concentration of the cholesterol is observed to influence the surface stress of the vesicles as demonstrated by the different levels of deformation, especially at high Re. The surface stress increases with the concentration of cholesterol, with the vesicles undergoing smaller deformations at certain values of the Re. These results are supported by experimental studies which reported that cholesterol-encompassing liposome deformed less than their pure lipid counterparts under microstreaming flow.[63]

As the concentration of cholesterol in the bilayer increases, the bilayer will be more rigid.

Hence, more work will be required to deform the surface of the vesicle. This will result in increasing surface stress and less shear-induced deformations in the bilayer. b. Flow-induced phase separation in HVs

The morphological transition of the HV-6 from a spherical to bullet-like shape is accompanied by a phase separation of the hairy lipids on the surface of the vesicle (see Fig.

2). An examination of the internal cavity of a HV-6 shows the hairy lipids to be uniformly distributed in the inner monolayer. The PEGylated lipids in the outer monolayer are exposed to the solvent flow in the channel whereas those in the inner monolayer are not.

We surmise that the phase separation is promoted by the solvent particles impinging on the tethers emanating from the outer monolayer of the HV-6. The solvent particles in the center of the channel move faster than those closer to the channel wall. Averaged over a long time scale, the vesicle under steady state moves as a single entity with particles possessing

uniform velocity. This uniform velocity is approximately the mean flow velocity.

Therefore, the velocity gradient of the flow acts as an effective pulling shear force in the direction of the flow and as a drag shear force in a direction opposite to the flow, sweeping the hairy lipids to the front end and the rear part of the vesicle. This phenomenon is known as the Marangoni effect which was earlier used to explain the influence of surfactants on the rising velocity of a gas bubble in solution.[64] Another study reported gathering of surfactants molecules in the liquid-liquid interface on a deformed liquid droplet in shear flow.[65] Our observation demonstrates stable surface phase separation due to the

Marangoni effect in Poiseuille flow. In addition, the impact of the velocity gradient (in terms of the effective pulling and drag forces) on the vesicle increases with the length and concentration of the tethers. Therefore, the degree of phase separation in HV-6 increases with the concentration of the hairy lipids. Hence, the phase separation on the outer monolayer of the HV-6s is induced by the solvent flow, and not promoted by the phase behavior of the surfactants. We note that HV-3 and BMV do not demonstrate flow-induced phase separation.

Figure 6: Measurement of the line tension as a function of Re for (a) HV-6, (b) HV-3 and (c) BMV.

All the measurements were performed using particle trajectories generated from simulations with four different random seeds.

We examine the correlation between the degree of phase segregation and Re by measuring the interfacial line tension under different flow conditions, as shown in Fig.6.

The measurements are performed on equilibrium configurations of the HVs and BMVs, for a range of concentrations of hairy lipids and cholesterol. The line tension of an interface between two segregated phases can be measured through the excess free energy per unit contact length along the interface. We estimate the line tension λ of the domain boundary

1 through the following equation  UUUAA+ BB AB / lmo for a binary system 2

(e.g., HV).[66] Similarly, the line tension for a quaternary system (e.g., BMV) can be written as a superposition of six binary terms and is given by

3 ()(/UUUUAA  BB  CC  DD  UUUUUUl AB  AC  AD  BC  BD  CD mo . Uij are the 2 pair interaction energies between two like (i = j) or unlike components (i  j), and ɭmo is given by 1.1rc which is the lateral size of the lipid molecules determined from the area per lipid.[66]

Our results show that under equilibrium conditions, the line tension decreases with increasing concentration of the hairy lipids in the HVs, as shown in Fig. 6. This result is in good agreement with an earlier study which demonstrated that the increasing excluded volume of the tethers reduces the interactions between the hairy lipids.38 For BMVs under equilibrium conditions, the line tension is observed to increase with the concentration of cholesterol. We believe that this result arises from the formation and growth of cholesterol-

enriched domains which increases the interaction between the cholesterol molecules, as illustrated in Fig. 11. We find the line tension for both BMV and HV-3 to remain independent of Re. Whereas for HV-6, the line tension is influenced by both Re and the concentration of the hairy lipids. These results show that phase segregation does not occur in HV-3, which agrees with the phase diagram in Fig.3. For HV-6 with hairy lipid concentrations higher than 10%, we observe the line tension to rapidly increase for Re >

3.5 × 10-4. This trend continues until the line tension reaches a value where the phase segregated state is unable to be sustained due to the perturbative effect of the solvent

-3 -3 particles at a critical Re (namely, Rec). For higher Re (Re ≥ 2.6 × 10 and Re ≥ 3.0 × 10 for respectively 10% - 30% and 40% - 50% concentration of hairy lipids), the line tension drops and the vesicle ruptures. The relation between the line tension and Re demonstrates that the phase segregation on the outer monolayer of HV-6 is induced by flow. In addition, the degree of the phase segregation is dependent on Re. We also conclude that Rec tends to increase with the concentration of the hairy lipids. A possible explanation for this trend could be that solvent particles with increasing momentum and kinetic energy are required to disrupt or prevent the phase separation of higher concentrations of hairy lipids. Hence, the HV-6 are able to increasingly resist flows that reduce the degree of phase separation on the surface of the vesicle with higher concentrations of the hairy lipids. It is worthwhile to point out that for hairy lipids concentration of 10%, we do not observe a significant variation in the line tension for different values of the Re in comparison to the other

-3 concentrations. The value of Rec for this systems is 1.0 × 10 , which is the smallest among all the concentrations of hairy lipids. c. Role of solvent on shape reconfiguration and phase separation in HVs

To understand the role of the solvent on the shape reconfiguration and phase separation, we examine the shear force acting on the vesicles. The shear force is defined as the force between solvent and tether beads in the direction of flow (that is, the z direction). The shear force on the HV is measured during the transient and equilibrium phases of the flow. These measurements were performed for Fo = 0.0004, 0.0008, 0.002, 0.003, 0.004, 0.005, and

-4 -4 -4 -3 - 0.006 kTB / which corresponds to Re = 1.5 × 10 , 3 × 10 , 7 × 10 , 1 × 10 , 1.4 × 10

3, 1.8 × 10-3 and 2.2 × 10-3. Fig. 7 shows the shear force on the tethers for 10% and 50%

concentration of hairy lipids, for both HVs at Fo = 0.0004, 0.002, and 0.005 kTB / . Fig.

12 shows similar measurements for the remaining concentrations of the hairy lipids and Fo.

Figure 7:Measurements of the shear force as a function of time, for 10% (left) and 50%

(right) concentration of the hairy lipids, at Fo = 0.0004, 0.002 and 0.005 kTB / .

For low Re or applied force (~ 0.0004 kTB / ), the shear force on the tethers is relatively constant. As the Re increases, the fluctuations in the shear force captures the response of the tethers on HV-6 to the impinging solvent (see Figs. 7 (c), (d)). Yet the shear force on the tethers is insufficient to induce a large degree of phase separation. In addition,

the larger spacing between the hairy lipids and conformational configurations of the longer tethers introduces greater variability in the number of solvent beads trapped between the tethers. This results in greater fluctuation in the shear force for 10% concentration of hairy lipids due to the larger separation between neighboring tethers. When Re increases further, the hairy lipids completely phase separate on the surface of the HV-6 (see Fig. 2). For 10% concentration of hairy lipids, the time evolution of the shear force shows the phase separation to minimize the opposition to the flow of the solvent beads and reduce the shear force on HV-6 (see Fig. 7 (e)). Hence, the shape reconfiguration and phase separation processes preserve the structural integrity of HV-6 under flow. However, for 50% concentration of hairy lipids, their complete phase separation is accompanied by the shear force attaining an equilibrium value higher than its initial value (see Fig. 7(f)). We surmise that after the hairy lipids have redistributed to the extreme ends of the vesicle (see Fig. 2), the concentration of hairy lipids is sufficiently high that the tethers continue to encounter a large number of impinging solvent beads. We would like to point out that HVs with shorter tethers are determined to be significantly less responsive to increasing Re.

Comparison of the shear force for different tether lengths, concentrations and Re can be obtained by comparing the average equilibrium shear force, as shown in Fig. 8 (a). For

HVs with shorter tethers, the shear force increases with the concentration of hairy lipids but remains primarily unresponsive to changes in Re. One possible hypothesis for this observation is that there are fewer solvent beads interacting with the shorter tethers than there are with the longer tethers.

We test this hypothesis by computing the solvent frequency which is the number of solvent particles at a given distance from the outer monolayer of a HV divided by the total

number of solvent beads (see Fig. 8 (b)). The solvent frequency between distances 0 to

0.899 rc (length of short tether) is the same for both tether lengths. For larger distances, the long tether continues to be exposed to the solvent, as captured by the solvent frequency.

Hence, the response of HV-6 to variations in Re is due to the solvent beads trapped between the longer tethers at a distance ranging from 0.899 rc to 1.838 rc (namely, the tether length).

These results support our proposed hypothesis of more solvents beads interacting with longer tethers.

The shear force on the tethers of HV-6 at 10% and 20% concentration of the hairy lipids increases with Re but is not accompanied by any phase separation. For Re > 1.0 ×

10-3, the shear force begins to decrease. The higher values of Re are accompanied by movement of the hairy lipids towards the extremities of the vesicle to reduce opposition of the tethers to the solvent flow. For higher concentrations of hairy lipids, the shear forces are observed to increase until Re = Rec before attaining an equilibrium value. This behavior at higher Re is attributed to the limited space at the extremities of the vesicle which is unable to accommodate all the hairy lipids. Whereas phase separation is observed to occur at higher Re and concentrations of the hairy lipids, some of these lipids will drift away from the extremities and oppose the solvent flow. We also note that for increasing concentrations of hairy lipids at a given Re, the shear force on the tethers in HV-6 increases due to higher number of the solvent beads impinging upon the tethers and occupying the space between the tethers. With increasing concentration, the shear forces reach a maximum at 30% concentration of the hairy lipids before decreasing due to the limited volume between the tethers available for the solvent beads to occupy. These results support the hypothesis that phase segregation in the outer monolayer of the vesicle is flow induced.

We would like to note that at 30% concentration of hairy lipids, the shear force increases after it reaches a minimum at Re=32. At Re=32 the hairy lipids reorganize themselves to the extremities of the vesicles so as to minimize the opposition of the tethers to the solvent flow. When Re increases to 40, the hairy lipids are unable to move to the extremities of the vesicles. This causes the tethers to oppose the flow of the solvent, thereby increasing the shear force.

Figure 8: Measurements of the (a) average shear force experienced by HV-3 and HV-6 for different concentrations of the hairy lipids and Re, and (b) the normalized solvent frequency outside the vesicle as a function of the distance from the outer monolayer of the vesicle.

d. Flow-induced rupture of HVs and BMVs

Figure 9: Images of HV-6 with 50% hairy lipids during rupture process: time (a) t’ = 0 τ,

(b) t’ = 200 τ, (c) t’ = 260 τ, (d) t’ = 360 τ.

The vesicles are unable to maintain their structural integrity beyond a specific Re.

We observe BMV and HV-3 to rupture at Re ≥ 2.6 × 10-3, for all concentrations of cholesterol and hairy lipids respectively. Similarly, HV-6 with hairy lipids at concentrations given by 10 - 30% and 40 - 50% rupture respectively at Re = 2.8 × 10-3 and

Re = 3.0 × 10-3. These observations indicate that the concentration of cholesterol does not have a significant effect on preserving the integrity of the vesicles under flow. However, longer tethers and higher concentrations of hairy lipids enhance the structural integrity of the vesicles at a higher Re.

The rupture mechanism is induced by the spatial reorganization of the molecular components in the outer monolayer, as shown in Fig. 9. Due to the strong flow-induced shear force, a few molecules are pushed out from the bilayer and form a tail. These molecules can disaggregate from the vesicle and re-aggregate to form a micelle (Fig. 9 (a)).

As the vesicle continues to lose molecules, the molecular population in the outer monolayer reduces. The molecules in the inner monolayer reorganize to shield the hydrophobic region of the bilayer from exposure to the solvent. This process leads to the formation of a transient pore in the bilayer, as shown in Fig. 9 (b). The size of the transient pore increases due to the extensional shear applied on the bilayer by the flow, fully exposing the aqueous interior region of the vesicle and eventually leading to its rupture. The mechanism underlying the rupture process is similar for all the vesicles.

Distortion of the vesicles due to flow-induced extensional shear could also induce their rupture. The molecules in the bilayer will lose their tight packing as the vesicle elongates and deviates from a spherical shape. This process will increase the exposure of the hydrophobic tails of the molecules to the aqueous environment. The molecules will reorient their head groups to prevent energetically unfavorable interactions between the hydrophobic tails and the solvent, which would cause the formation of transient holes, and eventually rupture the vesicle. This mechanism is pertinent for BMVs, where we do not observe disaggregation of the molecules from the vesicle and their re-aggregation to form micelles.

Conclusion

Via DPD, we examined the transport of cell-mimicking and bacteria-mimicking vesicles in a cylindrical channel under Poiseuille flow conditions. Our results demonstrate the concentration of cholesterol, hairy lipids, the length of the tethers and Re to influence the shape, phase separation and structural integrity of the vesicles.

The concentration of cholesterol is observed to impact the mechanical stiffness of the bilayer. Hence, higher concentrations of cholesterol impart greater resistance to shear- induced deformations. Cholesterol-rich domain formation is observed for higher concentrations. However, no large-scale phase separation of cholesterol is observed. The structural integrity of the BMV is determined to be independent of the concentration of cholesterol.

The tether length is found to impact the interactions between the solvent and tether beads. Increasing the tether length results in greater excluded volume which in turn traps solvent beads at the bilayer solvent interface, and shields it from rapidly moving solvent beads. The population of the solvent beads occupying the region between the bilayer- solvent interface and the length of the short tethers is the same for both tether lengths.

However, the short tethers are unable to trap the solvent beads between the tethers and protect the bilayer-solvent interface from flow-induced shear. Hence, the vesicle becomes increasingly resistant to flow-induced deformation with tether length. This effect is enhanced for increasing tether concentrations. Similarly, the structural stability of the vesicle increases with tether length and concentration. In addition, the shape of the vesicles is observed to remain symmetric along the channel cylindrical axis for higher tether length and concentrations.

The velocity gradient in the channel applies an effective pulling and dragging force on the tethers, thereby sweeping the hairy lipids to the extremities of the vesicle. This effect is enhanced by high Re, tether length and concentration, and results in the phase separation of the hairy lipids on the surface of the vesicle due to the Marangoni effect in Poiseuille

flow. The phase separation of the hairy lipids prevents the opposition to the flow of the solvent, thereby aiding the preservation of the structural integrity of the vesicle.

The predictions from our investigation can be tested via some experimental techniques. Techniques such as electron microscopy can be used to visualize lipid vesicles at nanometer resolutions[11] . The electron microscopy technique can be combined with immuno-gold staining to investigate specific molecules on the vesicle surface, and hence be used to test some of our predictions[12] . Since the vesicles are flowing in the fluid in our study, a better alternative method to test our predictions is flow cytometry. The sensitivity of this method has been recently improved to enable the detection of biological particles whose diameters are less than 100 nm.3 For example, in a recent investigation

[14], a miniaturized nano flow cytometer was used to analyze single vesicles with diameters in the range of 100 nm. This investigation demonstrated fluorescent labelling of single vesicles and multi-color detection of specific molecules on the surface of vesicles within this size range. Although the vesicles in our study are smaller than this range (30-

40 nm), we would expect similar observations as reported for the size range of 100 nm as the vesicles would approximately be in same scale. To test our predictions, we can label one of our lipid species (such as the PEGylated lipids). Using this approach, we can track the locations of the PEG groups as a function of the flow rate.

The results from this study can potentially aid the design of drug delivery vehicles which can sustain transport through confined volumes such as blood capillaries, or microfluidic devices in tissue engineering for enhanced in vivo transport efficiency. For example, our results provide insight on the organization of PEG groups on the surface of the vesicle as function of the concentration and length of the PEG chains, and the flow rate

of the solution. We observed phase separation of the PEG chains at certain concentrations of the PEG chains and flow rates. The phase separation results in the formation of non-

PEG bearing patches on the vesicle surface, which can enable the binding of these vesicles to immune cell surfaces. Thus, the design parameters can be selected based on the blood flow rate in the targeted region of the body. In addition, the shape and elastic properties of the drug delivery vehicles can influence their circulation time and biodistribution profile.[15, 16] Our results can provide insight on suitable cholesterol concentrations for

BMVs or PEG concentrations for HVs at various flow rates to obtain a specific shape for the vesicles. In addition, it has been shown that deformable particles with shapes and elastic properties similar to red blood cells have longer circulation time and improved biodistribution profile.[15] Thus, although our vesicles do not have underlying cytoskeleton encompassing spectrin, they can be designed to mimic the shape and elastic properties of red blood cells.

Supporting Information

Figure 10: Relation between force added on flow particles and mean flow velocity in the channel. Error bars are too small to be visible.

Figure 11: Per-bead interaction energy between cholesterols and cholesterols for various concentrations of cholesterol in CMV

Figure 12:Measurements of the shear force as a function of time, for 20% (left) and 30%

(right) concentration of the hairy lipids, at Fo = 0.0004, 0.002 and 0.005 kTB / .

Chapter 3: A Multi-scale Approach to Study Molecular and Interfacial

Characteristics of Vesicles

Introduction

Colloidal suspensions are critical for a wide range of technological applications due to their intricate interfacial properties. The surface chemistry and profile of these particles dictate their stability, response to external stimuli and self-organization in suspensions. The complexity of the interfacial characteristics of colloids arises from the combination of physical interactions and processes spanning large spatiotemporal scales which impact the structural and dynamical properties of these systems. Hence, the multiscale characteristics of these colloidal particles are impacted by the molecular scale, long range electrostatic, hydrodynamic interactions and their interplay. A fundamental understanding of the links between the molecular composition, the interfacial and collective properties of colloidal particles requires consideration of their multiscale characteristics. Whereas experimental approaches can provide insight into those links, limitations in their spatiotemporal resolution gives rise to the need for suitable computational approaches.

A single computational approach which can address physical interactions at disparate spatiotemporal scales is well suited to identify the predominant factors and mechanisms responsible for the characteristics of these particles. Of specific interest is the ability to determine molecular and interfacial characteristics of colloidal particles using such an approach. In this study, we use a computational approach which captures the particle dynamics and the hydrodynamic interactions in the system. We focus on two types of colloidal particles whose molecular components have different levels of sustained interactions with the solvent. One of the colloids encompassed amphiphiles with small

hydrophilic groups (compared to the hydrophobic groups). The other colloid was composed of two types of amphiphilic molecules with one of the molecules bearing a long hydrophilic group which extended into the solvent.

Potential particle dynamics-based approaches such as Multi-particle Collision

Dynamics [67-71] and Dissipative Particle Dynamics [70,72-76] are suitable for simultaneously resolving particle dynamics while capturing the hydrodynamics of colloidal systems. [7,77-78] However, these approaches are typically restricted to values of the Schmidt number (i.e., the ratio of the diffusive momentum transfer rate to the diffusive mass transfer rate) which are considerably lower than that corresponding to water at standard temperature and pressure, [9, 79] although efforts have been made to address this gap. [80] In addition, the soft repulsive interactions in Dissipative Particle Dynamics are unable to accurately capture the structure of molecules within colloidal particles. [81]

Another approach [82] has examined the integration of hydrodynamic forces into intra and inter-molecular interactions. Yet it is not clear if this approach can yield the long-tail behavior characteristic of hydrodynamic interactions.

We use a hybrid approach interfacing a continuum fluid dynamics technique known as the lattice Boltzmann method (LBM) [10, 83] to the classical Molecular Dynamics (MD) method. This approach has been developed to simultaneously capture deterministic particle dynamics, hydrodynamic interactions and thermal fluctuations, [9, 79] and used to examine colloidal suspensions and flow. [72, 73] The hybrid approach is suitable for addressing collective phenomena in colloids where the particles continuously maintain an interface with the solvent. The particle dynamics is resolved by the MD simulation technique. [84-

87] The fluid and its dynamical properties is modeled by the LBM [10, 83] which yields a

hydrodynamic description provided by the Navier-Stokes equation that is valid down to very short spatiotemporal scales. Hence, the solution to the Navier-Stokes equation for the flow field around a particle is accurate for distances greater than a few lattice spacings. [9,

79] Similarly, an all-atom MD [88] representation will yield hydrodynamic behavior for distances greater than a few water molecule diameters. The two approaches are equivalent as long as they produce the same long-range flow field. [9, 79] The two techniques are coupled to enable the particles and fluid to exchange momentum and introduce hydrodynamic interactions between the particles. Hence, this approach enables the resolution of the impact of both long range hydrodynamic and electrostatic forces on the particle dynamics.

There exist multiple implementations of the hybrid Molecular Dynamics-Lattice

Boltzmann method (MDLBM). [9,79,88-91] An earlier implementation [92] by Ladd modeled a colloid by point particles on its surface which interacted with the fluid through a set of bounce-back rules. Thermal fluctuations were introduced into this implementation to study Brownian motion. [93] An alternate implementation [9, 79] by Dunweg used a frictional force proportional to the difference in the local velocities of the particle and fluid, and modeled large particles by representing their surface via a collection of smaller particles. Thermal fluctuations were introduced via a Langevin thermostat acting on the

MD particles and fluctuating LB fluid. [89] This approach has been extended to examine the dynamics of polyelectrolytes and charged colloids with long range electrostatic interactions. [79] A different implementation [88,90,91] by Denniston uses a force coupling method which removes the need for an external Langevin thermostat acting on the particles, and accounts for thermal fluctuations. Existing methods have been used to

study the dynamics of single, large colloidal particles [9,79,88,90,93] or macromolecules

[79,94,95] in solution. However, these methods have not been extended to study the multiscale structural and dynamical characteristics of colloidal particles or emulsions.

We demonstrate the feasibility of a MDLBM implementation which includes aspects of the earlier methods to capture the multiscale structural and dynamical characteristics of colloids. We study the impact of hydrodynamics on the molecular characteristics of pure phospholipid vesicles and two component hairy vesicles encompassing DPPC (dipalmitoylphosphatidylcholine) and Poly ethylene glycol (PEG)- grafted DPPC (or, hairy lipids). These two vesicles were selected to be simplified representations of colloidal particles. The amphiphilic molecules in the vesicles had different degrees of sustained interactions with the solvent. The head groups of the phospholipid molecules and the PEG chains grafted to the lipids maintain their interactions with the fluid through the duration of the study. The PEG chains were able to adopt extended conformations in the solvent as compared to the lipid head groups. The different degrees of sustained interactions of the molecules with the solvent could potentially impact their organization and dynamics in the vesicle bilayer along with their conformation in the solvent. The pair and non-pair interactions between the particles were represented by a coarse grained implicit solvent model which is based upon the Dry MARTINI model. [96]

As a part of the study, we investigate the molecular properties of phospholipids and

PEGylated lipids in liposomes and hairy vesicles, respectively. We find our results for the molecular characteristics of DPPC in a vesicle bilayer to be good agreement with corresponding results from experiments and other simulation studies. In addition, the characterization results for the PEG chains on the surface of the hairy vesicles and their

trends, as a function of relative concentration and length of the PEG chains, are in agreement with existing theoretical and experimental studies. Our results validate the use of the hybrid MDLBM technique in conjunction with a coarse-grained implicit solvent force field in modeling colloidal particles. Our findings can be extended to apply this technique to probe multiscale characteristics of the interactions between colloids in suspensions under different flow conditions, and their relation to molecular properties.

Methods

Hybrid Molecular Dynamics-Lattice Boltzmann Method

The dynamics of the particles is resolved via the MD simulation technique using the canonical ensemble with a Langevin thermostat. The dynamics of the fluid is resolved through LBM with thermal fluctuations using the implementation introduced by Denniston

[88,90,91]. The hydrodynamic interactions impact the particle dynamics by coupling LBM with the MD technique. At each time step, the momenta of the fluid and the particles are coupled through their microscopic relations at the fluid-solid boundary but resolved and independently updated by the respective methods.

In the LBM introduced by Denniston, [90] a large spherical colloid is modeled through its surface which encompasses a network of evenly distributed node particles. The node particles enable the interaction between the fluid and the colloid, and are distributed to nearby lattice sites. Mesh effects are prevented by setting the number of node particles sufficiently large so that the distance between each of the node particles is smaller than the lattice spacing ∆푥. The mass of the colloids is the sum of the mass of the individual cage nodes. However, discretizing the surface of individual cage nodes (which encompass a

coarse-grained model of the lipid molecules) into a network of node particles is computationally expensive. Given the large number of hydrophilic lipid beads which will interface with the fluid, the computational cost for calculating all the intermolecular interaction between the node particles of two neighboring coarse-grained beads will be extremely high. In addition, coarse grained lipid beads are essentially point particles with no surface area, which makes it unnecessary to discretize them. These factors remove the need to discretize the surface of each coarse-grained bead.

In this study, the lattice Boltzmann fluid is coupled directly to the center of mass of each hydrophilic bead by interpolating the mass and velocity of individual coarse- grained beads to the nearby fluid lattice sites. The treatment of hydrophilic coarse-grained beads as point particles follows the essence of Dunweg’s implementation of LBM, [9] where the mass of the particle is distributed using linear interpolation. In this study, the immersed boundary method [97] is applied to interpolate point particles to the local fluid lattice sites. This approach has been demonstrated by several investigations to be more effective in distributing singular force densities, thereby reducing mesh effects for point particle-like objects. [91, 98] Similar to Dunweg’s implementation, the coupling between the LB fluid and the MD particle is modeled through a friction force that is proportional to the velocity difference between the center of mass of the MD particle and LB fluid:

퐹푥푦 = 훾(푉푦 − 푈푓)휉푥푦

퐹푦 = −훾(푉푦 − 푈푓)

Where 퐹푥푦 represents the local friction force applied on the MD bead y by a lattice site x,

훾 is coupling parameter, 푉푦 is the velocity of the MD bead y, 푈푓 is velocity of LB fluid

interpolated to the particle position based on local lattice sites, and 휉푥푦 is interpolation factor so that ∑x 휉푥푦 = 1. According to Newton’s third law, the total friction exerted by the MD bead y is reflected back and given by 퐹푦. The coupling parameter 훾 is modeled by elastic collision process between the MD point particle and the LB fluid mass:

2푚푓푚푛 1 훾 = ( ) 푚푓 + 푚푛 ∆푡푐표푙푙푖푠푖표푛

where 푚푛 is the mass of a lipid head bead in contact with fluid, and 푚푓 is the mass of the fluid colliding with each hydrophilic bead. Since we treat MD particles as point particles, the fluid mass at each interpolation position is calculated as 𝜌∆푥3, where 𝜌 is bulk fluid density (set as the density of water at the corresponding temperature) and ∆푥 is lattice space

(set as 10 nm). In this study, the mass of all MD beads were set as 72 g/mol except the PEG bead whose mass is 45 g/mol.

Interaction Parameters between MD Particles

The effective chemistry between the MD particles is captured by the pair and non- pair interactions. The forces and potential energies associated with pair, bond, angle and dihedral interactions will require an implicit solvent model. Existing solvent-free models for lipids have included the chemistry through force matching between all atom and coarse- grained systems; [99-105] Newton inversion method; [99-105] long-range attractive interactions of the hydrocarbon tails, [99-106] hard-sphere and square well potentials; [99-

105] analytical potentials combined with force matching potentials, [99-105] and Lennard-

Jones (LJ) potentials for non-bonded interactions between the beads (known as the Dry

MARTINI implicit solvent coarse-grained force field). [96] The Dry MARTINI model

follows from the standard MARTINI coarse-grained scheme [106] of grouping approximately four heavy atoms into one bead, and will be adopted due to its advantages over the other mentioned models. These advantages include the simplistic form of the interaction potentials and the large range of molecules currently parameterized under the

MARTINI model. [106] In the model, each bead is classified based on its charge, polarity, and ability to form hydrogen bonds. The non-bonded, non-electrostatic pair interactions are modeled by the shifted truncated 12-6 LJ potential with the potential cutoff distance rc set at 2.5σ for all bead pairs (σ = 0.47 nm). The non-bonded, electrostatic pair interactions are modeled by the shifted Coulombic potential with σ = 0.62 nm. The bond and angle interactions are modeled respectively by the weak harmonic and cosine harmonic potentials where the dihedral is modeled through the CHARMM potential. [96,107,108]

The colloids of interest are vesicles encompassing one or two types of amphiphiles:

DPPC and PEG-grafted DPPC molecules, or hairy lipids, as shown in Figure 13. The interaction parameters for DPPC and pegylated lipid was obtained from earlier studies.

[96,107,108] Non-bonded interactions of the hairy lipids using the standard MARTINI model were modified to compensate for the loss of the solvent. In an earlier study, [107-

108] the PEG-PEG interaction was treated as a SNda-SNda interaction whereas the PEG-

DPPC interaction was treated as N0-x where x represents the different bead types in the

DPPC molecule. However, using the Dry MARTINI model, the PEG beads were observed to embed themselves into the hydrophobic region of the vesicle bilayer. In addition, with increase in the PEG concentration, multiple PEG chains were observed to coil together.

These observations demonstrated that the PEG-PEG and PEG-DPPC interactions in the

Dry MARTINI model were highly enthalpically favorable. Intuitively, the lack of the

solvent would remove the effects of hydration and hydrogen bonding on the behavior of the solute beads (that is, PEG and hydrophilic lipid head beads), and therefore increase the favorable enthalpic interactions. This difficulty was overcome by treating the PEG-DPPC interaction as a P2-DPPC interaction and the PEG-PEG interaction as a SP2-SP2 interaction. The bead types P2 and SP2 are from the Martini force field. [106] This change maintains the same interaction energy level between the PEG and DPPC hydrophilic groups (Choline Qa, Phosphate Q0, and glycerol Na), but decreases the interaction between the PEG beads and the hydrophobic beads of DPPC, namely C1.

The implicit solvent model of PEG and lattice Boltzmann fluid as a good solvent are validated through measurements of the probability of the end to end distance of the

PEG chain. [109] A membrane composed of 814 DPPC lipids with a PEG 19 chain

(molecular weight ~ 838) grafted to it was simulated for approximately 100 ns. Multiple trajectories (namely, 500) were used to calculate the frequency of the end to end distance.

Results from MDLBM with the coarse grained implicit solvent model and MD simulations using the standard MARTINI model were compared (see Figure 14). Correspondence in the results from MDLBM with the coarse grained implicit solvent model and the standard

MARTINI model validates the former. [109] In addition, the radius of gyration of single

PEG chains of different lengths were computed and examined as a function of the corresponding molecular weight, as shown in Figure 15. The scaling exponents relating the radius of gyration to the molecular weight were determined, and found to be in good agreement with those from an earlier study [107] using the standard MARTINI model. This result validates the potential for the PEG chain.

Figure 13: Coarse grained model of (a) DPPC, PEG-DPPE with PEG chain length of (b)

6, (c) 12, (d) 28, and (e) 45. The red, blue, and purple beads represent the head group bead types Q0, Qa and Na, respectively. The green beads represent the hydrophobic tail beads (type C1). The cyan and brown beads represent types SP2 and N0 beads, respectively. (f) Closing of a DPPC membrane encompassing 814 DPPC molecules to form a stable vesicle. (g) Hairy vesicle with PEG 12 at a concentration of 10%.

0.16

0.14

0.12

0.1

0.08 wet mdlbm Probability 0.06

0.04

0.02

0 0 1 2 3 4 5 6 Distance (A)

Figure 14: Probability of the end to end distance of PEG 19 chain. The PEG chain was grafted to a membrane composed of 814 DPPC lipids. The membrane was equilibrated at

336 K until the area per lipid corresponded with values reported in literature. [110]

1.6

1.4

1.2

0.8

0.6 log log (nm) Rg

0.4

0.2

0 2 2.5 3 3.5 4 log (MW)

Figure 15: A plot of the log of the radius of gyration (Rg) (in nm) versus log of the molecular weight (MW) for single PEG chains of length 10, 19, 28, 37, 45, 77, 91, 113,

136 and 159. The measurement was performed using a process similar to that reported by an earlier study. [108]

System Setup and Simulation Details

The open source community-based MD package entitled LAMMPS [111] was used for this study. The two colloids of interest were: a pure DPPC lipid vesicle and a hairy vesicle encompassing DPPC and hairy lipids. The lipid vesicle was created through the spontaneous fusion of the edges of a pre-assembled membrane consisting of 814 DPPC lipids. The membrane was placed in a simulation box whose lateral area in the x-y plane was larger than the total area of a leaftlet of the membrane. A DPPC lipid membrane was equilibrated in a cubic box of dimensions 40 nm using the canonical ensemble for 1500 ns with a time step of 30 fs. Temperature of the system was maintained at 400 K using the

Langevin thermostat. Prior studies have demonstrated using the Langevin thermostat on

MD beads coupled to the LB fluid lattice reproduces the correct hydrodynamic behavior and particle dynamics [79, 89]. Unless specially stated, the damping rate of the Langevin thermostat is set to 2000 fs, which maintains the desired temperature [96]. This temperature was higher than 314 K, the transition temperature of DPPC, [112] to ensure that the bending energy of the membrane was lower than its interfacial energy. This allowed the membrane to minimize its interfacial energy by bending and fusing its edges to form a vesicle, as shown in Figure 13. This approach prevents the buildup of large asymmetric stresses across the bilayer leaflets which render the vesicle unstable. This simulation was performed in the absence of the LB fluid. After the vesicle was formed, it was ‘immersed’ into the

LB fluid by coupling the hydrophilic head beads of the lipids to the fluid lattice. The lipid head beads were coupled to the LB fluid as these beads were exposed to the LB fluid for the entire duration of the simulations. The time step was maintained at 10 fs to ensure 1:1 ratio of the lattice space to the timestep.

Some of the measurements required to be performed on a DPPC membrane. For this purpose, a membrane encompassing 3265 DPPC lipids was preassembled. The membrane was equilibrated using the MD technique in the canonical ensemble with a

Langevin thermostat (310 K) and a Berendsen barostat (1 bar) for 15 ns at a time step of

30 fs. The dimension of the simulation box was fixed along z-direction so that membrane can only expand along the x- and y- directions with the modulus set to 3333 atm

(compressibility 3 ∗ 10−4atm−1) and the damping rate given by 1000 fs. The equilibrated membrane was immersed in a LB fluid with the lipid hydrophilic head groups coupled to the fluid lattice, and was equilibrated for another 10 ns at a time step of 10 fs. The simulation box volume was kept constant to maintain the lattice space fixed. Therefore, the canonical ensemble was used to equilibrate the membrane using MDLBM. The vesicle and membrane systems simulated using MD and MDLBM were characterized when the bilayer was in the gel and fluid phases. The gel and fluid phases of a DPPC membrane correspond respectively to temperatures of 310 K and 336 K.

The hairy vesicle composed of DPPC and hairy lipids was preassembled using 3265 amphiphilic molecules. The relative concentration of the hairy lipids was varied from 2.2% to 20%. In addition, the length of the PEG chains was varied (that is, the number of beads encompassing the PEG chains was set to 6, 12, 28 and 45). Each system was equilibrated using MDLBM in the canonical ensemble at a temperature of 336 K using the Langevin

thermostat for 50 ns with a time step of 10 fs. Each system was simulated for an additional

10 ns to perform measurements of the system properties. The lipid head groups and the

PEG beads were coupled to the LB fluid during the MDLBM simulations. Three independent particle trajectories were used for the characterization of each vesicle.

Results and Discussion

The organization and collective dynamics of the lipid molecules will be impacted by the interfacial characteristics of the molecules in the vesicle. The dynamics and packing of DPPC molecules in a pure phospholipid vesicle was determined using MDLBM. These molecular properties were characterized through the diffusion coefficient and the area per lipid. The use of the centers of mass of the lipids for measurements of the diffusion coefficient through the mean square displacement yields larger diffusion coefficient as well as nonlinear mean square displacement curves in the first few nanoseconds. This observation has been attributed to the fact that the dynamics of the lipid tails is faster than the lipid head groups. [113] This is known as the tail-wagging effect. [113] To prevent the tail-wagging effect, the mean square displacement of 200 head beads of the DPPC lipid molecules were measured for 90 ns. Measurements from time 15 ns to 75 ns were used to determine the diffusion coefficient. To circumvent the impact of the bilayer curvature, the diffusion coefficient of the phospholipids in the outer and inner leaflets of the vesicle bilayer are measured separately. In addition, the diffusion coefficient of the lipids in a

DPPC membrane was measured with the goal of comparing the results with corresponding findings in a vesicle. Since the vesicle and the membrane were free to move during the span of the simulation, the mean square displacements of the center of mass of the vesicles and membranes were subtracted from the measurements of those for the lipids. To improve

the statistical accuracy of the measurement, the mean square displacement of the lipids at time τ was estimated to be the average of the difference between the mean square displacements (MSD) at 푡푖푚푒 푠푡푒푝 = 푡표 and 푠푡푒푝 = 푡표 + 휏.

푁푡 푁 1 1 푀푆퐷(휏) = ∑ ∑|푅(푛 + 휏) − 푅(푛)|2 푁푡 푁 푛=0 푖=1

Where 푅 is the displacement of a particle at a certain time. 푁푡 is total number of intervals the mean square displacements are averaged over, and N represents the number of lipids used for the calculation. A similar approach was used in earlier studies. [114,115] Figure

3-4 (a to d) shows the mean square displacements of the DPPC lipids (for 휏 ranging from

0 to 2000000 iterations) at 310 K and 336 K, both with and without the LB fluid.

Figure 3-4 shows the lipids in the outer monolayer to move faster than those in the inner monolayer. This observation is based upon the packing of the molecules; lipids in the outer monolayer tend to pack less tightly than those in the inner monolayer. Earlier studies [116] have demonstrated that whereas the overall tension in the bilayer of the vesicle is zero, the tension in the inner leaflet of a vesicle is negative, while the outer leaflet has a positive tension. The lipid molecules will pack with higher or lower density in monolayers with negative or positive tension, respectively. The density of the molecular packing will impact the mobility of the molecules. A molecule in a monolayer that is under negative tension will encounter many more “obstacles” or neighboring molecules (as compared to a monolayer under positive tension) as it diffuses in the monolayer. Hence, molecules in a monolayer that is under negative tension will have lower diffusion coefficient than molecules present in a monolayer under positive tension. Our observation agrees with earlier studies. [117-119]. Furthermore, a curved bilayer has been known to

have a lower lipid packing density than a planar bilayer. [120] Therefore, the dynamics of the lipids in membranes will be slower than those in either leaflet of the vesicle bilayer as demonstrated by our measurements. This result shows the dynamics of the lipids at a given temperature to not only be related to its molecular chemistry, but also to its environment.

To understand the impact of the LB fluid on the dynamics of the lipid molecules, the diffusion coefficients of the lipids was calculated using the slope of the interpolation of the mean square displacements:

1 1 1 D = ∗ ∗ 2d k timestep

Where the variable d is 2 and 3 respectively for membranes and vesicles. Table 3-1 summarizes the diffusion coefficient of DPPC lipids in a membrane and vesicle at 310K and 336K, with and without the LB fluid.

800 1200 outer outer 700 inner inner 1000 600 mem mem

) ) 500 800

)

2 2

400 600 MSD(A

300 MSD(A 400 200 100 200 0 0 0 1000000 2000000 0 1000000 2000000 Iterations Iterations

(a) (b)

500 1200 outer outer 450 inner 400 inner 1000 mem 350 mem

800

)

) 2

300 2

250 600 MSD(A 200 MSD(A 150 400 100 200 50 0 0 0 1000000 2000000 0 1000000 2000000 Iterations Iterations

Figure 16: (a) The mean square displacements (MSD) at 336K with LB fluid; (b) MSD at 336K without LB fluid; (c) MSD at 310K with LB fluid, and (d) MSD at 310K without

LB fluid. Blue and red line indicates measurements performed on the outer and inner monolayer of vesicle, respectively; the green line represents measurements performed on the membrane. Number of lipids used in all three measurements is same.

The measurements summarized in Table 1 demonstrate the lipids to diffuse faster in the presence of the LB fluid. We surmise that a particle immersed in the LB fluid passes its momentum to nearby particles by creating a flow field in the fluid. The force from the flow field produced by the original particle will accelerate the neighboring particles. The discrepancy between the results from experiments and our simulations arises as the experiments typically use giant unilamellar vesicles for the measurements. These colloids are sufficiently large that the curvature of the vesicle could be ignored. However, the simulations examined a vesicle of dimension 15 nm which is significantly smaller than what is typically used in experimental studies. Therefore, the experimental measurements

of the diffusion coefficient are akin to studying the dynamics of lipids on a membrane.

Hence, our measurements of the diffusion coefficient for the phospholipids in membranes in the fluid and gel phase are in agreement with corresponding experimental and computational studies. [121, 122]

Further insight into the molecular packing of the lipids can be obtained through the measurements of the area per lipid. Table 2 summarizes the area per lipid of membranes and vesicles, in the absence and presence of the LB fluid, at temperatures of 310 K and 336

K. The measurements for the membrane and vesicle in the gel phase is observed to be in good agreement with experimental values. [110, 123-124] In the fluid phase, in the absence of the LB fluid the measurement of the area per lipid is higher than that reported by experimental studies. Our measurements also support the understanding that the lipids are more tightly packed in the inner monolayer of the vesicle than the outer monolayer. Our results match those from previous studies which have shown that area per lipid and diffusion coefficient of lipids are intercorrelated [125-128]. Furthermore, we demonstrate that coupling the polar lipid groups to the LB fluid does not affect the packing of the molecules in both monolayers.

Planar- Outerlayer Innerlayer Literature membrane (m2/s) (m2/s) [58, 59] (m2/s) no fluid 2.6722E-11 8.20444E-12 4.625E-12 ~2.10E-12 310K LB fluid 3.656E-11 1.12517E-11 6.303E-12 no fluid 2.8707E-11 1.05917E-11 2.220E-11 ~2.25E-11 336K LB fluid 6.0796E-11 2.06467E-11 1.089E-11

Table 1: Diffusion coefficient of DPPC lipids in membrane and vesicles, in the absence and presence of the LB fluid. The measurements were performed at temperature corresponding to the gel (310 K) and fluid phase (336 K) of a DPPC bilayer.

Temperature 336K 310K No LB fluid planar- 62.9±0.2 (Å2) 64.3±0.2 membrane LB fluid (Å2) 64.4±0.2 62.5±0.2 OM: 94.6±0.5 OM: 92.1±0.5 No LB fluid (Å2) IM: 68.0±2 IM: 64.0 ±1.0 vesicle OM: 94.2±0.4 OM: 91.8±0.5 LB fluid (Å2) IM: 67.0 ±2 IM 64.0 ±1.0 Experimental (Å2) [47, 60, 61] ~ 67 ~ 63 Table 2: Area per lipid of DPPC molecules in vesicles and membranes, in the absence and presence of the LB fluid, at temperatures of 310 K and 336 K.

The interfacial properties of a colloid encompassing amphiphilic molecules bearing solvophilic chains can be determined by characterizing the molecules. The characteristics of molecules in vesicles with extended conformations in the solvent was investigated through a hairy vesicle. The hairy vesicle encompassed a binary mixture of phospholipid

DPPC and PEGylated phospholipids PEG-DPPE, which were present in different relative concentrations. In addition, PEG chains of different molecular weights or lengths were also studied (that is, 6, 12, 28 and 45). We examined the impact of the length and relative concentration of the PEG chains on its conformational characteristics. We performed the measurements of these characteristics using MDLBM. Our investigations provide insight on the conformational characteristics of the PEG chains and their comparison with existing theory.

The conformational configuration of the PEG chains can be characterized via the radius of gyration. The radius of gyration is measured for each PEG chain in a given bilayer leaflet and averaged over all the chains in the leaflet. Figure 17 shows the measurements of the radius of gyration for different PEG chain length and relative concentrations in the outer leaflet of the hairy vesicle bilayer. The radius of gyration (Rg) and molecular weight

(MW) are related through the following scaling relation Rg ~ MWa. The scaling exponent a is determined to lie in the range of 0.61 to 0.63. The value for exponent a agrees with the corresponding value for a free PEG chain in a LB fluid, as shown in Figure 15. This result implies that the scaling relation between the radius of gyration and the molecular weight of a PEG chain is independent of its grafting density, length and curvature of the grafting surface.

As expected, the radius of gyration is observed to increase with the length of the

PEG chain. However, the relation between RG and PEG concentration is non-trivial. For short tethered systems(PEG 6 and PEG 12), there is no significant difference between radius of gyration for different PEG concentrations. This is reflected with large overlap of error bars. As PEG length increases, increase of radius of gyration with respect to concentration is more obvious. For PEG length 45, it is apparent that no overlap exists between different data points. In addition, the increase in the radius of gyration becomes more pronounced with the relative concentration of the PEG chains. Changing of radius of gyration versus tether length/concentration can be explained by morphology of PEG. For low grafting densities or relative concentrations, the PEG chains do not interact with each other laterally and adopt ‘mushroom-like’ conformations. For high grafting densities, the

PEG chains avoid interacting with each other laterally by adopting increasingly ‘stretched-

out’, or ‘brush-like’ conformations [108, 129-130]. Since radius of gyration is a measurement of excluded volume, a mushroom-like PEG will have smaller RG comparing with brush-like PEG chain of same length. For PEG 6, 12 and 24, overlap of RG at 2% and 10% indicates a mix of mushroom and brush like PEG chains. We can say that changing of variation of radius of gyration is a result of percentage of mushroom or brush- like PEG.

To determine whether the PEG chains transition from a mushroom to a brush regime, we determine the end-to-end distances for each system in the presence of the LB fluid (as shown in Figure 18. The end-to-end distance is given by the distance between the center of mass of the last bead of the PEG chain and its grafting point on the lipid molecule.

The end-to-end distance is observed to increase with the PEG chain length and relative concentration. We observe a rapid increase of the end-to-end distance for PEG 28 and 45 at around 10% concentration of the PEG chains. This behavior is indicative of a change in the morphology of the PEG chains. To determine whether the PEG chains transition from a mushroom to a brush conformation, the scaling of the end-to-end distance with the degree of polymerization (N) and the distance between the grafting points of the neighboring PEG chains (D) is measured. D is an average of the distances between each grafting point and its nearest neighbor on the outer monolayer of the vesicle bilayer. Earlier studies [129,130] have shown the PEG chain extension length from a grafting point on a

3/5 surface to be given by the Flory Radius 푅푓 = a N for a chain which adopts a mushroom conformation. The monomer length is given by a which is 0.35 nm. We have found that at low concentration (< 5%) of PEG chains, the scaling relation for the PEG chains was in good agreement with the relation corresponding to the mushroom conformation. This result

indicates that majority of the PEG chains are adopting a mushroom conformation. As the relative concentration of the PEG chains increase, the end-to-end distance of the PEG chains deviates from that corresponding to the mushroom conformation. The relation between the neighboring grafting point distance D and 푅푓 predict the morphology of the

PEG chains. [129-131] When D < 푅푓, the grafted PEG chains are in the brush regime. As shown in Table 3, the shortest PEG chain (that is PEG 6) is still in the mushroom regime when its concentration reaches 20%. However, for PEG 12, the mushroom to brush transition occurs between the relative concentrations of 10% and 20%. For PEG 28 and 45, the mushroom to brush transition occurs between the relative concentrations of 2.5% and

10%. Hence, our results show the mushroom to brush transition to occur at lower relative concentrations with increasing PEG chain lengths, which is in agreement with earlier studies. [129]

An earlier study [132] using a similar force field in conjunction with MD simulations qualitatively captured the mushroom to brush transition with increasing concentration of PEG 5000 chains, each grafted to amphiphiles in a bilayer. This finding was validated by experimental investigations [132] and supports our results. However, this study did not couple the solvophilic beads of the amphiphiles to the LB fluid. Another study [133] employed a similar computational approach (but without the coupling to the

LB fluid) to study the chain conformations of PEG 5000 grafted to amphiphilic molecules which were present in different concentrations in a lipid bilayer. The measurements of the chain conformations [133] were unable to capture differences in the values for the end-to- end distance and radius of gyration of the PEG chains for different concentrations of the

PEG chains. These results demonstrate that the momentum exchanged between the

solvophilic beads and the discreet fluid is essential for capturing the conformations of the

PEG chains.

0.442 0.682 0.4415 0.681 0.68 0.441 0.679 0.4405 0.678 0.677 0.44 0.676

0.4395 (nm) Rg Rg (nm) Rg 0.675 0.439 0.674 0.673 0.4385 0.672 0.438 0.671 0 0.1 0.2 0.3 0 0.1 0.2 0.3 Relative Concentration Relative Concentration

(A) (B)

1.145 1.55 1.14 1.54 1.53 1.135 1.52 1.13 1.51

1.125 1.5 Rg (nm) Rg Rg (nm) Rg 1.12 1.49 1.48 1.115 1.47 1.11 1.46 1.105 1.45 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 Relative Concentration Relative Concentration

Figure 17: Radius of gyration (Rg) (in nm) of PEG chain as a function of the relative concentration and length (A = PEG 6, B = PEG 12, C = PEG 28 and D = PEG 45). The relative concentration is defined as the fraction of PEGylated lipids divided by the total number of amphiphiles in the outer monolayer of vesicle.

1.27 1.78 1.26 1.25 1.76 1.24 1.74 1.23 1.72 1.22 1.7 1.21 1.68 1.2 1.66

End to endEndto distance (nm) 1.19

1.18 endEndto distance (nm) 1.64 1.17 1.62 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 Relative Concentration Relative Concentration

(A) (B)

3 4.2 4.1 2.95 4 2.9 3.9 2.85 3.8 2.8

3.7 End to end(nm)to End distance End to endEndto distance (nm) 2.75 3.6

2.7 3.5 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 Relative Concentration Relative Concentration

Figure 18: End to end distance of PEG chain as a function of the outer monolayer PEG chain relative concentration and length (A=PEG 6, B=PEG 12, C=PEG 28 and D= PEG

45).

PEG chain length 6 12 28 45

Rf (nm) 1.026 1.554 2.584 3.436

Relative concentration D (nm)

0.022 3.85 3.66 4.04 4.32

0.1 2.01 1.94 2.05 2.04

0.2 1.41 1.51 1.53 1.47

Table 3: Distance between neighboring grafting points for different PEG chain lengths and relative concentrations.

Conclusion

We have implemented a hybrid MDLBM technique and examined it feasibility to capture molecular and interfacial characteristics of colloidal particles. For simplicity, we focused on two vesicles: one encompassing DPPC and the other composed of DPPC and

PEG-DPPE. The emphasis of this study was the determination of the molecular characteristics of these vesicles using the hybrid technique. Specifically, we focused on the characteristics of DPPC in the bilayer and the extended conformations of the PEG chains grafted to the vesicle in the solvent.

We measured the diffusion coefficient and area per lipid of DPPC at two temperatures corresponding to the gel and fluid phases of a DPPC bilayer. Our results were in good agreement with corresponding experimental and computational approaches. In addition, our determination of the PEG chain conformational characteristics and trends

with PEG chain length and concentration supported earlier theoretical and experimental studies. We find the hybrid MDLBM technique to be suitable for capturing molecular and interfacial characteristics of colloidal particles.

In the future, the application of the hybrid technique can be extended to examine the interactions between colloids in a suspension, or the effect of flow on interfaces of colloidal particles, and their relation to molecular characteristics. Hence, the hybrid technique has the potential to resolve multiscale interfacial, structural and dynamical characteristics of colloidal particles.

Chapter 4:Self-assembly of Amphiphilic Molecules via the Hybrid Molecular

Dynamics-Lattice Boltzmann Technique

Introduction

Nanoscale colloids are defined as assemblies of small bonded molecules including lipids, amino acids, nucleic acids, copolymers, and other amphiphilic molecules. Self- assembly is a spontaneous process during which disordered building blocks form an ordered assembly as a result of non-covalent intramolecular interactions. Mechanism of molecular self-assembly is of great interest and has been intensely studied because of its omnipresence and significance in biological systems as well as its potential in chemical synthesis of nanostructures for medical purposes. [134-139] In particular, micelles, nanotubes, liposome and other hollow nanostructures composed of amphiphilic molecules are popular for usages as drug delivery vehicles due to their stability, low toxicity and high compatibility with human tissues. [140-143] Transition from micelles to vesicles or micellar self-assembly usually happens at a time scale of milliseconds or longer. [144-146]

However, self-assembly at initial stage (eg: forming micelles or other transient) often occurs at a smaller length and time scale and might be difficult to be captured by traditional experimental method. [145-146] Computational simulation may provide a way to study this process. There has been numerous computational studies focusing on self assembly of lipids in aqueous solution. [146-153] Fujiwara et al [147]carried out molecular dynamic simulation to study micelle formation. Huang et al[144] used Lattice dynamic Monte

Carlo method to investigate vesicle formation from block copolymer A1B3. Noguchi et al

[149] and Yamamoto [147] et al have applied Brownian dynamics and dissipative particle dynamics to study formation of vesicles.

Most of them involve using coarse grained modeling for molecules, meaning that a pseudo atom represents multiple real atom particles. By using a coarse-grained force field, degree of freedom is lowered and it becomes feasible to study large systems with longer times. For highly coarse grained system like DPD, structure of molecules in colloids cannot be captured accurately. Despite of coarse graining effect, computation could still be expensive due to large number of solvent atoms in the system. Cook et al [153] has developed a solvent free model to investigate self-assembly of bilayer membranes.

However, removing explicit solvents will also eliminate hydrodynamic effect which is also an important factor in self-assembly of lipid systems. Noguchi and Gompper[7] found that hydrodynamic interactions speed up self-assembly process and determines morphology of vesicle formed. However, including explicit solvent molecules in the simulation could be computationally expensive even though a coarse-grained force field is used.

A compensating simulation method is a hybrid approach that couples lattice

Boltzmann method, a technique solves fluid dynamic numerically, with molecular dynamic that solves particle’s dynamic simultaneously. This method includes hydrodynamic interactions without involving explicit solvent beads in the system. This hybrid technique has been used to simulate soft matter systems including polymers [9], pre-assembled bilayer systems[154], and even composite colloids.[8] Despite success of current MDLB algorithm in investigating static colloidal systems, its drawback in studying self-assembly of amphiphilic colloidal systems can’t be ignored. Since self assembly is a dynamic process, individual building block may or may not contact with LB fluid during timescale of simulation. In this study, we modify an existing MDLB coupling scheme[8] so that individual MD particle will dynamically coupled to lattice Boltzmann fluid.

Methodology

(a) Lattice Boltzmann method

Lattice Boltzmann method solves fluid dynamics governed by Navier–Stokes equation through modeling the streaming and colliding process of fictive particles on 3D lattice grids with lattice space ∆x. The algorithm uses finite difference method to solve a velocity discretized version of the linearized Boltzmann equation:

∆푡 푓(푥 + 푒 ∆푡, 푡 + ∆푡) = 푓 (푥, 푡) − (푓 (푥, 푡) − 푓푒푞(푥, 푡)) (1) 푖 푖 휏 푖 푖

Where 푓푖(푥, 푡) represents particle partial distribution function that conserves mass and momentum of local fluid in-situ.

푁

𝜌 = ∑ 푓푖 (2) 푖=1

푁

𝜌푣 = ∑ 푓푖푒푖 (3) 푖=1

Density and momentum are particle velocity moments of 푓푖. The second term of right hand side of equation (1) is BGK collision factor which relaxes the fluid toward

푒푞 equilibrium distribution, 푓푖 which is determined based on the weight factors 푤푖,

푎 푎 Normalization factor 푁 , equilibrium momentum of velocity vectors 푒푖, 푀푒푞. Local

푒푞 equilibrium 푓푖 is set to ensure that Navier-stoke macroscopic equations are recovered.

18 푒푞 푎 푎 푎 푓푖 = 푤푖 ∑ 푚푖 푀푒푞푁 (4) 푎=0 Relaxation term 휏 determines how fast system reaches equilibrium and is related with fluid viscosity. 푒푖, represents velocity vector at each lattice site. We use a D3Q19 (3

dimensional 19 velocity vector) model in which, ei, velocity vectors at each lattice site are set to 푒푖=(0,0,0), (±1,0,0), (0,±1,0), (0,0,±1), (±1, ±1, 0), (±1, 0,±1), (0, ±1, ±1).

(b) Molecular dynamics

Particle dynamics is modeled by coarse-grained molecular dynamics in which positions and velocities of MD particles are updated by velocity verlet algorithm. Pair and non-pair potentials of MD beads are captured by dry Martini force field in previous investigation.[96,154]

Dynamics of fluid is captured by lattice Boltzmann algorithm. Both molecular dynamics and lattice Boltzmann simulation will be carried out through open source parallel program LAMMPS. As it is discussed in previous publications [8,154], since size of individual MD bead is smaller than lattice space ∆x, it is not necessary to avoid mesh effects by discretization of the particle’s surface. Therefore, each MD bead is treated as a point particle and directly coupled to lattice Boltzmann fluid. Since coupling of MD with

LB fluid occurs at lattice site, we distribute mass and velocity of MD particles to nearby lattice site through Peskin interpolation. As demonstrated in previous studies[8], a point

MD particle is coupled to nearby lattice sites by applying a frictional force proportional to velocity difference between the particle V , determined from molecular dynamics equation and fluid interpolated at the particle position, 푈푓, derived from equation (1), (2) and (3).

퐹 = 훾(푉 − 푈푓) (5)

Where 훾 is the friction coefficient determined from elastic collision process between the

MD point particle and the LB fluid mass, F is sum of force acting on MD bead. Meanwhile,

a force of same magnitude and opposite direction is exerted back to fluid due to Newton’s third law.

퐹푓 = −훾(푉 − 푈푓) (6)

Modification of current MDLB algorithm is carried out in the step of calculation frictional force acting on the MD bead. When a MD bead is not exposed to LB fluid, frictional force

F and 퐹푓 are both set to 0. The condition to judge whether MD beads contact with fluid is described as following: When the number of MD particles N fall with the force cutoff of a particular MD bead i is larger than a critical number Nc, then 퐹 = 퐹푓 = 0 for particle i.

(d) Pseudo code of implementation

Modification of current MDLB algorithm is carried out on a user defined package USR-

LB on an open source parallel program: LAMMPS. The file modified is fix_lb_fluid.cpp.

1. Calculate partial distribution function 푓푖(푥, 푡) for all mesh grids

2. Based on equation (2) and (3), implement fluid velocity at each mesh grid

3. Calculate fluid velocity 푈푓 on MD bead i through interpolation of nearby lattice

sites.

4. Calculate friction force exerted on fluid and MD bead i

If N(i)>Nc

퐹(푖) = 퐹푓 = 0

Else

퐹 = 훾(푉(푖) − 푈푓)

퐹푓 = −훾(푉(푖) − 푈푓)

END

5. Repeat step 3 and 4 for all MD beads in system.

6. Update partial distribution function 푓푖 from equation (1).

Figure 19: Schematic of coupling of DPPC lipid beads to LB fluid represented by 2D lattice grids. 8 beads fall within interaction range (force field cut-off) of orange MD bead.

In case CN=5, then LB interaction of orange bead is shut off.

(e) Force Field

The Dry Martini model [96] is an implicit solvent model and modified from the standard Martini coarse-grained scheme. However, without explicit solvent molecules,

long-range hydrodynamic interactions are not captured, and therefore raising the necessity of coupling MD beads to LB fluid.

(f) Simulation Details

We randomly put 216 DPPC molecules in an 80*80*80 퐴3 cubic box with periodic boundaries and let the system equilibrated under NVT ensemble with a timestep of 10 fs.

Lattice space ∆x is set as 10 A to maintain dx/dt=1. Temperature of system is maintained at 310K using Langevin thermostat. Each DPPC lipid bead is composed of 4 hydrophilic head beads and 8 hydrophobic tail beads. All twelve beads are coupled to LB fluid using the modified MDLBM coupling scheme with different critical number Nc (5,10,15,20, and

25). We manually set skin distance of neighbor list to 0.05A to store the neighbors of each

MD bead. A small skin distance ensures that neighbors of each MD bead are approximately those within the interaction range. We invoke conditional LB coupling to MD beads though modified LAMMPS command fix lb/fluid/dynamic/coupling in which we add two parameters compared with original version: group name of MD beads that will be dynamically coupled to LB fluid and critical number Nc. We repeat our simulations using

5 different seeds for accuracy of results. We are interested to see whether modified

MDLBM algorithm is able to capture dynamics of self-assembly process.

Results and Discussion

Hydrophobic interactions between tail beads of DPPC lipids drive the lipids to form aggregates that protect hydrophobic parts from explicit solvents. In this study, an aggregate is defined as a cluster that includes more than two lipid molecules whose hydrophobic beads fall within the interaction range of each other. During simulation, small clusters

diffuse and merge with each other and form larger aggregates. Due to scope of this research, we cease simulation until aggregation visible.

(a)

(b) (c)

(d) (e)

(f) (g)

Figure 20: (a)Total energy of system versus time for different Nc. Energy will still drop until a single aggregate is formed. (b-g) Different stages of self-assembly process for CN=5.

(b) t=0ns (c) t=2ns (d) t=4ns (e) t=6ns (f) t=8ns (g) t=9ns. As time goes by, self-assembly process slows down.

Dynamics of self-assembly is captured by tracking numbers of clusters as simulation goes by. To track cluster numbers, we used a LAMMPS built-in command compute aggregate/atom cut-off to assign DPPC molecules that fall within the cut-off distance of each other the same cluster ID. Then, the number of different cluster ID is same as number of clusters. Since cluster size increases during simulation, we increase the cut- off in the command to avoid underestimate of cluster number. We measure the scaling exponent of following relation N(t) = C ∗ ta . where N(t) is number of clusters, C is constant and a is scaling exponent.

Figure 21: Time evolution of number of aggregates for a system of 218 DPPC lipids for varying critical number Nc.

Critical

number 5 10 15 20 25

average -0.8 -0.9 -0.9 -0.89 -0.94

standard

deviation 0.1 0.1 0.1 0.09 0.05

Table 4: Scaling exponent a for different critical number Nc. Our value is close to previous experimental and simulation results. [155-156]

Our results show that scaling exponent of lipid self-assembly is not affected by different critical numbers. In addition, the time when system reaches equilibrium is also same for all scenarios. It indicates that dynamic of self-assembly of DPPC lipids is a spontaneous macroscopic process and not determined by microscopic level details.

Conclusion

We studied self-assembly of DPPC lipid systems based on an existing algorithm that couples molecular dynamics approach with lattice Boltzmann method. We modified previous algorithm so that each individual MD bead is dynamically coupled to LB fluid in- situ based on number of its neighbors that fall within the force field cut-off range. This dynamical coupling scheme has been proved to capture dynamics of self-assembly of lipids with correct scaling exponent comparing with previous experimental and simulation investigation. We also found that number of interacting neighbors of an MD bead has no effects on dynamics of self-assembly process.

Chapter 5

Conclusions and Future Direction

Due to important functions of membranes in chemical and biological fields, they have been studied thoroughly both experimentally and theoretically. In Chapter 2, we investigated morphology change of synthetic and hairy vesicle under a Poisson flow in a cylindrical tube through DPD method. The effects of flow strength, PEG concentration,

PEG length and cholesterol concentration were tested. In Chapter 3 and 4, we used hybrid lattice Boltzmann-molecular dynamics technique to investigate membrane system composed of DPPC lipids. We found this muti-scale simulation technique able to capture mechanical and dynamic property ofmembrane systems while maintain hydrodynamic effects of fluid at both gel and fluid phase. In Chapter 4, Existing coupling scheme between lattice Boltzmann method and molecular dynamics was modified. MD particles were coupled dynamically to LB fluid based upon number of its neighbors fall within force cut off range.

We found this new coupling algorithm accurate in capturing dynamics of self- assembly of DPPC lipids into micelles. This new technique can be applied to interfacial characterizations of surfactants. For instance, dynamic coupling could be used to investigate self-organization process of amphiphilic surfactants where hydrophobic parts of solutes not always contact with hydrophilic fluid and therefore dynamically coupled to

LB fluid.

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