Fatima Yousuf. A thesis submitted in conformity with the requirements for the degree of Master of Science Chemistry University of Toronto

Transcription

1 Analysis of Hybrid MD-MPC Simulations of Micelle Formation Under Neutral ph and Dynamics Under Acidic ph Using Different ph-sensitive Triblock Copolymer Structures by Fatima Yousuf A thesis submitted in conformity with the requirements for the degree of Master of Science Chemistry University of Toronto Copyright by Fatima Yousuf 2016

2 Analysis of Hybrid MD-MPC Simulations of Micelle Formation Under Neutral ph and Dynamics Under Acidic ph Using Different ph-sensitive Triblock Copolymer Structures Abstract Fatima Yousuf Master of Science Chemistry University of Toronto 2016 The solubilization of hydrophobic drug molecules in the bloodstream by micelles is a promising method for drug delivery to cancerous cells. Cancerous cells and their surrounding environment can be distinguished easily from healthy cells by their ph: acidic vs neutral respectively. This has led to the design of ph-sensitive micelles that are triggered by cancerous cells. When designing the drug or amphiphilic polymer, the drug loading and release rates are quantities that should be optimized for effective delivery. To better understand the factors affecting these quantities, ph-sensitive micelle dynamics in an acidic environment is studied computationally with molecular dynamics simulations (not experimentally due to constraints in time and length scales). By treating the solvent-solvent interactions implicitly through multiparticle collision dynamics, a coarse-grain model of polymers, drugs, and solvent is used to examine drug distribution and micelle dynamics in an acidic environment for different amphiphilic triblock copolymer structures. ii

3 Table of Contents Table of Contents... iii List of Tables...v List of Figures... vi List of Appendices... vii Chapter 1 Introduction...1 Chapter 2 Background...3 Tumors...3 Anticancer Drugs...5 Issues with Drug Delivery...6 Types of Drug Carriers...7 Micelles...8 ph-sensitive Micelles...9 Qualities of an Ideal Drug Carrier...10 Computer Simulations MD MPC ph-sensitive Micelle Response to Acidification...14 Chapter 3 Simulation Methods...15 Chapter 4 Micelle Formation under Neutral Conditions...22 Chapter 5 Micelle Dynamics under Acidic Conditions...35 Chapter 6 Discussion...42 Micelle Cross Sections...42 ph-sensitive and Drug Bead Distribution...44 clustradavg...46 (druginrad/clustrad)avg...46 iii

4 MPC Limitations...47 Chapter 7 Conclusion...49 Appendix 1: constants.h...50 Appendix 2: classes.h...55 Appendix 3: functions.h...56 Appendix 4: forces.h...58 Bibliography...59 iv

5 List of Tables Table 1: ph in different tissue and cell organelles.. 2 Table 2: Interaction energies.. 19 v

6 List of Figures Figure 1: ph-sensitive micelle drug delivery...1 Figure 2: ph-sensitive lipid structures...16 Figure 3: MPC parameters Figure 4: Micelle formation - Control graphs (1 run) 23 Figure 5: Micelle formation - Drug-Bonded graphs (1 run)..24 Figure 6: Micelle formation - Reversed + Drug-Bonded graphs (1 run)...25 Figure 7: Micelle formation - Control graphs (8 runs).. 29 Figure 8: Micelle formation - Drug-Bonded graphs (8 runs) 30 Figure 9: Micelle formation - Reversed + Drug-Bonded graphs (8 runs). 31 Figure 10: Micelle acidification - Control graphs (4 runs) Figure 11: Micelle acidification - Drug-Bonded graphs (4 runs).. 37 Figure 12: Micelle acidification - Reversed graphs (4 runs). 38 Figure 13: Micelle acidification - Reversed + Drug-Bonded graphs (4 runs) Figure 14: Before and after acidification: Cross-section of micelle (1 run).. 43 Figure 15: Before and after acidification: ph-sensitive and Drug bead distribution (1 run)...45 Figure 16: MPC limitations...47 vi

7 List of Appendices Appendix 1: constants.h Appendix 2: classes.h Appendix 3: functions.h Appendix 4: forces.h.. 58 vii

8 Chapter 1 Introduction ph-sensitive polymeric micelles used for drug delivery provide many advantages over other drug-delivery methods, such as liposomal drug carriers. These advantages include the ability to target tumor cells, increase cellular internalization, release drugs in a controlled manner, rapidly release drugs, avoid multi-drug resistance, and decrease toxicity/side effects. Compared to free drugs, encapsulated drugs do not need to have their structure and properties tuned for transportation in the bloodstream. ph-sensitive micelles have a longer circulation time than free drugs in the bloodstream due to stability and solubility, but are still able to accumulate in solid tumors and release drugs. Figure 1: ph-sensitive micelle drug delivery. The ph-sensitive micelle experiences a ph gradient in the extracellular tumor environment, and the intracellular endosomal environment. Several environmental stimuli exist for tumor cells that can be used to trigger drug release such as temperature, glucose, and salt concentration. However, ph is most commonly used. The ph is acidic in both the extracellular environment of tumor tissue and the intracellular environment of tumor cells (Table 1). The extracellular environment of the tumor tissue is acidic due to hypoxia; increased glycolysis in cancer cells, which produces lactate and protons outside the cell. Within the tumor cell, micelles can be engulfed into an endosome via endocytosis. Early endosomes have ph 4-6, which eventually become lysosomes (ph 4-5). 1

9 2 Table 1: ph in different tissue and cell organelles Tissue/cellular compartment ph Blood Stomach Duodenum Colon Early endosome Late endosome Lysosome Golgi 6.4 Tumor, extracellular While experimental measurements on micelle size, drug loading capacity, and drug release concentration do exist, there is still a lack of insight into details such as drug distribution in a micelle, how the micelle structure changes before and after acidification of the environment, the drug loading process, and the drug release process. Despite constraints in length and time scales, molecular dynamics simulations of such systems can be performed to look at drug loading, distribution, and release in various polymeric micelles. These processes will be studied using different amphiphilic triblock copolymer structures.

10 Chapter 2 Background Before getting into simulation methods and details, a background about tumors, anticancer drugs, issues with drug delivery, types of drug carriers, micelles, ph-sensitive micelles, qualities of an ideal drug carrier, and related simulation experiments will be discussed. Tumors Despite the vast amount of research that has gone into cancer and chemotherapy, there has not been a comparably great improvement in patient recovery [3]. One reason is due to the inability of anticancer drugs to exclusively target tumor tissue. This means that the drug is free to distribute itself throughout the body. If a drug is toxic to normal tissue, which many are, then this causes strong side effects in patients (e.g. bone marrow suppression, cardiac and kidney toxicity, hair loss, mucositis) [3]. Many drugs also do not last long enough in the body to even reach tumor tissue before they are expelled from the body. To try to counteract this, the dosage of the drug is usually increased to be more effective, which may worsen side effects. To summarize, the main issue with cancer treatment is the transportation of drugs to the tumor sites in an effective manner, without harming healthy tissue [7]. The obstacles limiting the effectiveness of anticancer drug delivery have been identified through the research of past decades [35]. Now the goal is to find ways to overcome them and achieve maximum drug delivery. One recent method uses nanoparticle-sized carriers, such as polymeric micelles, to deliver the drug in a systematic manner [5]. Polymer-drug nanoparticle carriers have made polymer therapeutics one of the first kinds of anticancer nanomedicine [8]. Nanoparticle carriers solubilize hydrophobic drugs by encapsulating and transporting them more efficiently through the bloodstream. Since tumor tissue has permeable blood vessels, loose junctions, and poor lymphatic drainage, the nanoparticle carriers can easily escape from the bloodstream to enter tumor tissue, but cannot easily return to the bloodstream (enhanced permeability and retention (EPR) effect) [1,5]. The EPR effect allows nanoparticle carriers to build up in tumor tissue, which increases therapeutic effectiveness and decreases side effects. Although such nanoparticle carriers may accumulate in tumor tissue, the polymer used to encapsulate the drug makes it harder to interact with the cell surface, and thus harder to enter the 3

11 4 cell. Once at the site of a cancerous tumor, some drugs have molecular targets on the surface of the cell (e.g. human epidermal growth factor receptor 2 (HER2), vascular endothelial growth factor) [27]. However, many more cytotoxic drugs may only take effect through their molecular interaction with subcellular molecules (e.g. DNA, DNA topoisomerase, tubulin) [27]. This implies that these drugs can only work if they are able to enter these cells. Thus, active-targeting by simple nanoparticle carriers is not enough to show clinical success [35]. Furthermore, if the drug can be released when nanoparticle carriers are present in tumor tissue, then this may greatly increase cytotoxicity in the tumor tissue [27]. One possibility is to use a nanoparticle carrier that is responsive to environmental stimuli [33]. As more research goes into the physiological differences between healthy tissue and tumor tissue, the design of stimuli-sensitive nanoparticle carriers for targeted drug delivery may continue to flourish [10]. Stimuli-sensitive nanoparticle carriers differ from other nanoparticle carriers in that they may interact with their environment. When this interaction occurs it should cause drug release. Using ph as the stimuli is of special interest due to the clear ph gradient between healthy (ph 7.4) and cancerous (ph ) tissue [21]. ph-triggered drug release not only increases nanoparticle carrier concentration in the leaky blood vessel step (EPR), but also increases uptake of the drug by cells [21]. Both steps incorporate a type of tumor selectivity. As more and more multi-functional nanocarriers undergo clinical studies, there is better promise to find the one that is able to conquer all barriers to drug delivery [8,35].

12 5 Anticancer Drugs Most anticancer drugs are pharmacologically effective, but not so much in clinical studies where barriers exist such as toxicity, water insolubility, lack of retention in the body (increasing the dosage to prevent this causes side effects), and improper biodistribution (causing side effects) [2,3]. A change in biodistribution of the drug must occur in order to decrease toxicity in healthy tissue and improve therapeutic efficacy. Thus, the transportation of the drug is just as important as the drug itself [4]. One way to overcome the barriers is by using a nanoparticle carrier such as a polymeric micelle [3]. The core-shell structure of polymeric micelles allows the drug to be stable within the hydrophobic core, while the shell can be designed to control the drug release rate and ensure that this rate is stable [4]. In this structure, the drug is not active and is protected [1]. Advantages of using polymeric micelles include reduced reticuloendothelial system (RES) uptake, tumor targeting, less side effects, possibility to be stimuli responsive, and equal/improved therapeutic efficacy compared to free drugs [3,4]. Various factors affect the drug loading capacity of a micelle (and thus the therapeutic efficacy), such as the process in which drugs are loaded into the micelle, the drug molecule size, and the drug molecule structure [12]. Smaller drugs diffuse into the core easier, and branched drugs are encapsulated easier than linear drugs [12]. The length of the hydrophobic block is also a factor in drug encapsulation, with a longer hydrophobic block length increasing drug encapsulation [12]. Overall, the drug encapsulation ability of polymeric micelles is still found to be very low [12]. The drug release process when triggered by ph is theorized to occur with the micelle shell swelling and creating channels, which allow the drug to escape [4]. Drug release may occur at the extracellular environment of tumor tissue or within the cancer cells through endocytosis [3].

13 6 Issues with Drug Delivery Drug efficacy is significantly dependent upon transporting the drug to the tumor site. There are two key problems that limit anticancer drug efficacy: (1) the drug not being able to make it to the tumor, and (2) the drug being denied by cancerous cells for uptake [3]. A way to overcome the nonspecific biodistribution of the drug is to encapsulate it within a nanoparticle carrier and undergo the EPR effect [3]. However, even with a nanoparticle carrier for the drug, there are still barriers to drug delivery. There is the possibility of no uptake by the cancer cells due to membrane transporters if the nanoparticle carrier is not properly designed [3]. Nanoparticle carriers that are accumulated in solid tumors still have to overcome the diverse tumor environment (improper blood supply, disordered vasculatures, diffusion-limited interstitium) and must also be able to release the drug in its active form [2,3]. Drug affinity for the nanoparticle carrier must be tuned to be able to contain the drug and release it when triggered, which is very difficult [2]. Controlling when and how the drug is released is difficult. Sometimes it is desired to have more than one type of drug or therapeutic agent (e.g. RNA and organic drugs). For cancer therapy, there is often more than one type of drug necessary for treatment, possibly up to 5 or 6. In such cases, the inside of the nanoparticle carrier should be compartmentalized to have different areas with affinity for each type of component to be released. It should also be noted that the idea of only targeting cancerous cells is not completely possible since the targets in cancer cells, however more distinct, may still exist in healthy cells [3]. The main objective for nanomedicine is the ability to both diagnose and act as therapy (theragnostics) [2]. However, a theragnostic nanoparticle that is triggered to only work under the correct disease diagnosis is still far from being achieved [2].

14 7 Types of Drug Carriers Macromolecular drug carriers that are under clinical trial or currently being used include liposomal carriers, polymeric vesicles, polymeric micelles, water-soluble polymer-drug conjugates, polymeric nanoparticles, and dendrimers [2,3]. Compared to liposomes, which interest had initially been upon, polymeric micelles have shown greater advantages in drug delivery [8]. These advantages include enhanced tumor-targeting and penetration, greater circulation in the blood due to its size being tens of nanometers, reduced toxicity (e.g. hand-foot syndrome, hypersensitivity reaction), and controlled drug release [29]. Polymeric micelles are self-assemblies of block copolymers with a special core-shell structure that can be used to carry hydrophobic compounds, metal complexes, gene and sirna, etc. [29]. These systems show positive results for drug delivery to solid tumors in both systems of nonbonded drugs and bonded drugs (to block copolymers) [8]. There is still however a need for new polymer-drug conjugates, new polymer combinations, and new stimuli-sensitive polymeric micelles [8]. For polymeric micelles, features such as particle size, stability, loading capacity, and drug release kinetics depend on the structure and physical/chemical properties of the block copolymers [29]. Tuning the structure and properties of the block copolymers in response to a stimuli can give the micelle smart functionality in order to target certain sites and release the drug, which would improve clinical results [29]. Various micelle creations have been studied in preclinical and clinical studies and have shown great promise [29]. However, the rate of drug release and safety of drug release must still be confirmed [8]. The advancement of polymeric micelles and polymer-drug conjugates are nonetheless promising fields to improve nanomedicine [8].

15 8 Micelles Polymeric micelles and polymer-drug conjugates are promising fields in nanomedicine. Amphiphilic block copolymers have the ability to directly self-assemble to create polymeric micelles in a polar solvent. Polymeric micelles have a double layer structure consisting of a hydrophilic shell enclosing a hydrophobic core when present in a hydrophilic solvent [1]. The hydrophobic core is protected from the hydrophilic environment of the solvent by this stabilizing shell-like structure. Polymeric micelles typically range in size from nm and have different biodistribution in the blood stream compared to small molecules [3]. This structure may be used for hydrophobic drug encapsulation and the hydrophilic shell may be designed to be triggered by an environmental stimuli to release the drug and also to be biocompatible [3]. The hydrophobic core of the micelle acts as a reservoir for the drug (or protein, DNA, etc.) to be transported in [1,5]. The loading and distribution of the drug in the micelle is dependent upon the drug structure, the length of the hydrophobic block length of the polymer, and the interaction between the drug and the hydrophobic block of the polymer [12]. A polymeric micelle is dynamic [31]. Micelles may be designed to behave in a certain manner by changing the chemical structure of the polymers [1]. Its stability is dependent on the polymer chemical structure, drug encapsulation, and environmental setting [31]. For instance, phsensitive micelles can be designed to be neutral under physiological conditions, but release the drug in acidic conditions [1]. Polymeric micelles as drug carriers have many advantages over the use of free drug, such as low toxicity in the body, reduced side effects, greater circulation in the blood stream due to better water solubility (avoiding phagocytic and renal clearance), enhanced tumor targeting due to adequate size for EPR, shell may be functionalized, ability to be stimuli responsive, may reduce MDR, and simple preparation [1,3,5]. The disadvantages would be that they are not suited for hydrophilic drugs [3]. Polymeric micelles are of great interest in anticancer drug delivery. They are being investigated in research and clinical studies to improve the therapeutic effectiveness of anticancer drugs and reduce side effects in patients [5,31].

16 9 ph-sensitive Micelles The use of simple micelles to target tumor tissue has not shown the desired efficacy in cancer therapy. New strategies need to be in effect in order to release as much of the drug from micelles to tumor tissue. The biological environment of tumor tissue may be used to tweak responsiveness of nanocarriers. That is, micelles can be designed to be sensitive to certain stimuli such as ph, temperature, hypoxia, light, salt concentration, and/or redox potential [10,33]. The added feature of stimuli-responsiveness may enhance the therapeutic efficacy of anticancer drugs through improved drug release. The stimulus which is plainly obvious is the ph difference between healthy tissue and tumor tissue [6,24,33]. The micelle can be designed to be stable under physiological ph, then be susceptible to instability under the weakly acidic extracellular environment of tumor tissue and/or under the more acidic endosomal compartments within tumor cells. Polymeric micelles can be altered to be ph-sensitive by having reversible protonation/deprotonation units in the polymers (hydrolysis) or an acid-liable bond between the polymeric units and the anticancer drug (dissociation) [24]. Under the acidic ph trigger, whether it be in the extracellular tissue or intracellular lysosomes/endosomes, such ph-sensitive micelles should change structure and release the encapsulated drugs. In the first scenario, drug release would be after the EPR effect when ph-sensitive nanoparticle carriers build up in the extracellular tissue [24]. The second scenario goes one step further with the ph-sensitive nanoparticle carriers being taken up by cancerous cells in the tumor tissue through endocytosis [24]. In vitro studies have been performed with ph-sensitive micelles designed to be stable under ph 7.4 (physiological environment/healthy tissue) and unstable under ph 5.0 (endosome inside of cancerous cell) [20]. For cancer cells that could take in the drug, the uptake of ph-sensitive micelles was about the same as for free drug. For cancer cells that could not take in the drug, the ph-sensitive micelles were mostly taken in, while the free drug was not. The ph-sensitive micelles killed the cancerous cells efficiently and also showed no toxicity to the healthy cells. The advantages of ph-sensitive micelles for drug delivery are numerous. They have relatively rapid drug release at the desired tumor site, greater cellular uptake for intracellular ph-targeting, decreased multidrug resistance for intracellular ph-targeting, reduced toxicity to healthy cells,

17 10 reduced side effects, and again better tumor targeting [24]. It is no surprise that these nanoparticle carriers are of special interest and will continue to be. Qualities of an Ideal Drug Carrier In the last century, focus has been upon the development of new drugs to improve medicine [3]. Now, the focus is upon creating an ideal drug carrier. A key quality of an ideal drug carrier is that it should be able to safely and precisely transport the right quantity of drug to solid tumors [2]. In order to do this, the drug must be protected by the carrier to slow down the degradation of the drug, enhance the drug targeting, control biodistribution by preventing accumulation in healthy cells, reduce drug toxicity to healthy cells, and control the release of the drug naturally or through external stimuli. Drug carriers should also be designed with knowledge of the correlation between physicochemical properties of the carrier and the carrier s behavior in the body. Lastly, since the drug should not be released too early, the attraction between the drug and the carrier should be tuned to prevent this [3]. Polymeric micelles with a core-shell structure can transport drugs from the blood, across the hematoencephalic barrier, and into the central nervous system [4]. If the shell material is chosen to reduce RES uptake and thereby increase circulation time in the body, then the carrier may avoid being targeted by the immune system. The physical and chemical structure of the shell may also be designed to target certain cells, tissue, or a specific location in the body. Ideally, the shell should be biodegradable to release the drug in response to environmental changes. The shell may also be used to control the rate of drug release to ensure stable release. Currently, PEG is almost always used as the shell component for amphiphilic copolymer micelles, however, there is evidence that proteins bind to the micelle surface to destabilize them [31]. This means that either there needs to be new hydrophilic components, or that micelles should initially be incubated in certain proteins before IV injection in order to alter biological response. For polymeric micelles that are intended to change under a physical stimulus, the aim is release the drug under the stimulus or to add stress to the cancer cell [3]. For polymeric micelles that are intended to change under a chemical stimulus, the aim is for the stimuli to change the micelle from amphiphilic to just hydrophilic and destabilize the micelle, thereby releasing the drug. ph is often used as a chemical stimulus because tumors have a slightly lower and acidic ph compared to normal tissue. If the micelle has polymers with protonatable groups, then they may

18 11 become protonated in tumor tissue, which would ideally break down the micelle and release its drug content. Dissociation is an alternative ph-stimulus approach where the drug is bonded to the polymer and released under at the trigger ph. For instance, drugs may be bonded to core segments of the micelle through acid-labile linkers that are stable under physiological conditions (ph 7.4), but cleavable under acidic intracellular conditions in endosomes and lysosomes (ph 5-6) [2]. Drug carriers should be biocompatible and biodegradable, greater than 10 nm to avoid renal clearance and less than 5 μm to allow cell uptake, and have a positively charged surface to interact with negatively charged components of cancer cell membranes to allow cell uptake if extracellular drug delivery is desired [3]. Future shell-core structures should be designed with all of these qualities in mind, and combine useful features into one delivery system [4]. Computer Simulations Computer simulations may be used to give insight into the dynamics and structure of real systems, without having to give quantitatively exact results to experimental results [9]. This is not to say that quantitative results cannot be extracted from simulations to explain experimental results. For example, the tilt transition, tilt angle, and direction have been correctly predicted for monolayers with respect to experimental data [19]. At the atomistic or molecular level, important processes otherwise not understood experimentally may be explained by computer simulations [32]. The exact molecular detail in simulations is usually not replicated to allow for greater time scales in a short amount of real time [19]. A coarse-grain model of lipids and drugs in solvent can be used with elastically-bound beads representing the lipids, single beads representing the drug, and single beads representing the solvent [14]. In fact, the best insight into self-assembly and drug solubilisation has come from coarse-grain models [19]. The processes of drug loading and drug release from micelles are not clear, but are important to understand to achieve better drug delivery [32]. Computer simulations may be used to capture these processes and observe drug distribution. Simulations have provided useful insight into research relating surfactant structure, dynamics, and rheology to surfactant self-assembly, micelles, amphiphilic monolayers, bilayers, and oil solubilisation [19]. Micelle deformation and drug release, for instance, has been observed under a swelling mechanism from simulations [4].

19 12 Computer simulations may shed some light on questions pertaining to micelle stability [31]. If the hydrophobic core and drug interact, then how is it released and how does drug release affect micelle stability? If drug loading is a limiting factor in drug delivery, then how can it be improved and how would that affect micelle stability? 8.1 MD Molecular dynamics (MD) simulations have been used for various surfactant, oil, and water experiments. These include the self-assembly of micelles, solubilisation of oil by micelles, oil diffusion into the core of a micelle, and micelle collision [9,12,18,22]. A lot of insight has been gained from such simulations, such as the fact that that hydrogen-bonding is not necessary for micelle formation, micelle dynamics and morphology depend heavily on surfactant structure, oilphase oil is transferred to micelles through three different processes (1. Dissolution to the solvent before being encapsulated by micelle, 2. Exchange to micelle through a soft collision, and 3. Surfactants adsorbing onto the oil-phase and extracting oil to micelle), for micelle aggregation between two micelles there are three steps (1. Molecular contact, 2. Neck formation, and 3. Neck growth) followed by drug exchange between micelles, there are two rate-limiting steps during micelle aggregation (1. Breaking the water film between two micelles, and 2. Creation of a pore in both micelles), increased Head group repulsion makes aggregation more difficult, Head group length makes aggregation more difficult due to increased steric repulsion, and drug presence helps micelle formation by pulling surfactants. Pertaining to the self-assembly of soluble amphiphiles in aqueous solution, this has been examined experimentally and modelled theoretically for some decades now. At low amphiphile concentration, amphiphiles dissolve in aqueous solution as single molecules [38]. As the concentration increases to the critical micelle concertation (CMC), the amphiphiles then begin to self-assemble into micelles. The CMC is an inflection point in amphiphile concentration where the physicochemical properties such as surface tension are a function of concentration [37]. Above the CMC, the single amphiphile molecule concentration in solution is about constant since newly added amphiphiles go towards micelle formation. Micelle formation is driven by amphiphile Tail attraction (hydrophobic and Van der Waals) and limited by the amphiphile Head repulsion (electrostatic and steric/entropic). The resulting shape of the micelle depends on these interactions. Micelle formation is a spontaneous process driven by the increase in entropy created

20 13 when the hydrophobic Tail of the amphiphiles is removed from water and disturbs the ordered structure of water in that region. The free energy minimum state obtained by this process encourages micelle formation to occur. For typical amphiphiles (single Head and single Tail), the next phase to be formed as the concentration increases is one of three types [38]. Amphiphiles with large Head group area per molecule, which form spherical micelles up until the second CMC, then form a discontinuous cubic phase consisting of discrete micelles in a cubic-like lattice at greater concentration. Amphiphiles with smaller Head group area per molecule, which form sphere-like micelles that change shape to cylinders at the second CMC, then form organised cylinders in a hexagonal pattern at greater concentration. Amphiphiles with the smallest Head group area per molecule, which form micelles in the shape of flat bilayers at the second CMC, then form a lamellar phase consisting of stacked uniformly spaced bilayers at greater concentration. Any other component in the solution besides the solvent and the amphiphile may have an effect on surfactant self-assembly at the CMC and second CMC [38]. This is because they may change interactions between amphiphiles and introduce competing interactions with amphiphiles or solvent molecules. Molecular dynamics has been used for simple models of amphiphilic molecules and water, which have shown self-assembly of the amphiphiles to form micelles [9,13]. In simulations, the structure and shape formed by self-assembling amphiphiles depends on the structure and interactions of the amphiphile. This self-assembly is even possible without the need to incorporate Hydrogen-bonding forces. In one simulation experiment, the initial conditions consisted of the amphiphiles, drug, and water in a homogeneous mixture [25]. As time progressed, the amphiphiles aggregated together to form clusters with drugs adsorbed to the surface of these clusters. The clusters continued to aggregate get bigger until they reached a single stable micelle with the drugs engulfed within. Thus, amphiphilic self-assembly is easily possible through molecular dynamics simulation.

21 MPC Multiparticle collision dynamics (MPC) or Stochastic Rotation Dynamics (SRD) is a simulation method at the mesoscale for fluid flow [11]. This method involves switching between streaming and collision steps in an ensemble of solvent point particles. The collisions are represented by dividing the solvent particles to be in certain collision cells where mass, momentum, and energy are conserved in each. MPC allows complete hydrodynamic interactions and thermal fluctuations. MPC of mesoscopic particles may reproduce the right hydrodynamics of solvent fluids at the macroscopic scale. Most applications of MPC algorithm are studies of equilibrium dynamics and flow properties of colloids, polymers, and vesicles in solvent. Recent applications are to study colloid and particle dynamics, behaviour of vesicles and cells in hydrodynamic flow, and dynamics of viscoelastic fluids. For more complicated systems where thermal fluctuations are key, MPC will become more and more useful. For instance, interactions of colloids, polymers, and membranes with the mesoscale solvent can all be treated on the same basis. A big advantage of this algorithm is that it easily allows one to model dynamics of constituents in the solvent with a hybrid MD-MPC basis. This hybrid approach still gives quantitatively correct results with theoretical predictions, and other simulation methods. By modeling the solvent using MPC, the forces between solvent-solvent particles do not need to computed, which they would otherwise have to be in MD/DPD. The simulations are faster as a result. 8.3 ph-sensitive Micelle Response to Acidification An interesting process that has been witnessed in various MD/DPD acidification simulations of ph-sensitive drug-loaded micelles is the swelling of the micelle [4,22,25]. The first step is micelle swelling, followed by drug release as the micelle demicellizes, and/or finally the micelle breaking apart into free polymer units. One such simulation has shown several interesting results including that the hydrophilic block length affects drug release no matter what the drug distribution is, the length of the ph-sensitive block has a great effect on drug release, when drugs distribute near the ph-sensitive block then the effect of acidification is greater, and the hydrophobic block length affects drug release differently depending on drug distribution [28]. MD and MPC may provide a powerful and efficient tool for drug and polymer design.

22 15 Chapter 3 Simulation Methods The various lipid structures that are used are shown below in Figure 2. The lipids consist of a hydrophilic Head block, ph-sensitive block, hydrophobic Tail block, and/or a hydrophobic Drug component. The ph-sensitive bead behaves as any other Tail bead under neutral conditions, but under acidic conditions it behaves as any other Head bead. These beads can be used to represent groups of monomers in an actual polymer. The first class of lipid structures, (1) Control, are chains that go in the order Head ph-sensitive Tail, the second class, (2) Drug-Bonded, is the same except with a Drug bonded to the ph- Sensitive bead, the third class, (3) Reversed, is the same as (1) except the ph-sensitive block is at the terminal end of the chain, and the fourth class, (4) Reversed + Drug-Bonded, is the same as (3) except with a Drug bonded to the ph-sensitive block. Systematically going through the physical differences between these structures and the micelle dynamics may shed some new light and assist with polymer design for ph-sensitive polymeric micelles.

23 16 Figure 2: ph-sensitive lipid structures. Head bead striped circle, ph-sensitive bead black circle, Tail bead white circle, Drug bead white square

24 17 The simulation for each lipid structure has two main steps: (1) the formation of a stable micelle from random initial coordinates of all system components, and (2) the subsequent response of the micelle when the ph-sensitive units are activated. The unit of time in MD units is t 0 = mσ2 where m, σ, ε are the units of mass, length, and ε energy respectively in MD units. The MD timestep size is dt = 0.005t0 and the MPC timestep size is τ = 40dt = 0.2t0. The time allocated for micellization is 10,000t0 and for acidification is 200t0. The mass of the Head, ph-sensitive, Tail, and Drug beads are all the same at unity, while the mass for each Solvent bead is 0.5m. The temperature of the system is initialized at kt/ε = 1.0. The domain is a 26σ edge length cube with periodic boundary conditions. It is divided into MD cells of edge length 2.6σ and MPC cells of edge length σ. Regardless of the lipid structure, the system has beads in total, 100 lipids, and 100 drug beads. On average, the Solvent number density in the bulk is equal to Bonded monomers in a lipid are bonded to one another through a spring potential V bond (r ij ) = 1 2 k bondr 2 ln [1 ( r 2 ij ) ], r r ij r where k bond = 20ε σ 2 and r = 1.5σ [14]. Next-nearest monomers in a lipid are also bonded to one another through a spring potential V bend (r ij ) = 1 2 k bend(r ij 4σ) 2 where k bend = 2.5ε σ 2 [14].

25 18 Head beads have a short-range Lennard-Jones repulsion to every other type of bead with a cutoff distance of r c = 2 1 6σ. Head beads only have an attraction V att (r ij ) = ε αα cos 2 (0.5π [ r ij r c ω αα r c ]), r c r ij ω αα to Solvent beads between rc and ω αα = 1.65σ [14]. Tail beads have a short-range Lennard-Jones repulsion to every other type of bead with a cutoff distance of r c = 2 1 6σ. Tail beads in a lipid are attracted to Tail beads in another lipid, to inactivated ph-sensitive beads under neutral ph, and to Drug beads between rc and ω αα = 2.6σ. Under neutral ph, ph-sensitive beads behave as any other Tail bead, except with the added spring-like bond to a Drug bead for Drug-Bonded systems. Under acidic conditions, ph- Sensitive beads behave as any other Head bead. Drug beads have a short-range Lennard-Jones repulsion to every other type of bead with a cutoff distance of r c = 2 1 6σ. Drug beads have an attraction to inactivated ph-sensitive beads under neutral ph, Tail beads, and other Drug beads between rc and ω αα = 2.6σ. Solvent beads have a short-range Lennard-Jones repulsion to every other type of non-solvent bead with a cutoff distance of r c = 2 1 6σ. Solvent beads only have an attraction to Head beads and activated ph-sensitive beads under acidic conditions between rc and ω αα = 1.65σ. Since most of the system consists of Solvent beads, most of the computational time would have gone into the interactions between Solvent beads. To avoid this, explicit Solvent-Solvent interactions are omitted by using the constant temperature version of multiparticle collision dynamics (MPC-ATa) [14].

26 19 Figure 3: MPC parameters The total potential energy of the system at a given time can summed up below: V = VHH + VHP + VHT + VHD + VHS + VPP + VPT + VPD + VPS + VTT + VTD + VTS + VDD + VDS The interaction energies used for the Lennard-Jones repulsions and the attraction forces are summarized in Table 2. Table 2: Interaction energies Interaction Energy Neutral Conditions (ε) Acidic Conditions (ε) ε hh 0.5 ε ht 1.0 ε hd 1.0 ε hs 0.05 ε tt 0.5 ε td 0.5 ε ts 2.0 ε dd 0.5 ε ds 2.0 ε hp ε pp ε pt ε pd ε ps

27 20 Each MD cell has a particle list that is updated at each timestep. Every bead, except for Solvent beads, has a neighbor list of beads that are within a radius of 2.6σ. The neighbor list only counts particle lists from directly neighboring cells. The acceleration of each bead is calculated from its neighbor list. The Verlet algorithm is used to compute the position and velocity of each particle for every timestep [34]. In each simulation the mean free path of the Solvent is less than the MPC cell size, σ, with the average Solvent speed being ~ 2(ε/m) 1/2. To prevent the same Solvent beads colliding over and over with each other for many timesteps, grid-shifting is implemented before the MPC collision step [16]. The initial velocity of each bead is Gaussian distributed with zero mean and variance kt/m [14]. The initial positions for the lipid beads are randomly chosen such that the domain is divided into fcc units and the lipid monomers are randomly placed at the fcc sites. Afterwards, the Drug beads are placed randomly at unoccupied fcc sites. Then finally, the Solvent beads are placed randomly anywhere in the domain such that there is a padding of space around the previously positioned non-solvent beads. The cluster analysis at each time step counts the number of lipid clusters, identifies which cluster a lipid belong to if it does at all, calculates average cluster radius from first Head monomer positions, counts number of encapsulated Drugs, identifies which cluster an encapsulated Drug belongs to and calculates the average ratio of encapsulated Drug radius to corresponding cluster radius (centre calculated from first Head monomer positions). A cluster is defined as two or more lipids that have any monomers, excluding the first Head monomer, which are within a distance of 2.6σ. The calculation for the cluster radius uses the sum of positions of outermost Head monomers of the lipids and assumes a there is a micellular/spherical structure. If a cluster has not yet formed a micellular structure, then the significance of the cluster radius diminishes. Drugs are reported to be encapsulated if they are within a distance of 2.6σ to a monomer of a lipid (excluding the first Head monomer) that is part of a cluster. A drawback of this approach is that if a Drug bead is outside of a cluster, but within a close enough distance, then it will be reported as encapsulated when it is in fact not. In the future, a possible way to avoid this is by counting the nearest neighbors within a small radial distance of the Drug bead and ensuring that the number of Solvent beads is at a minimum. Under neutral conditions when the Drug is still

28 21 bonded in the Drug-Bonded systems, the Drug is reported as encapsulated regardless because it is bonded to a lipid and not considered free. Once a single stable micelle forms after t = 10,000t0, the final positions and velocities are used as initial conditions in the acidification step. Acidification of the environment is modeled by instantaneously making the ph-sensitive beads behave as Head beads. That is, they will no longer be hydrophobic and will become hydrophilic. In addition, in the case of Drug-Bonded systems, the bond between a ph-sensitive bead and a Drug bead instantaneously breaks. The code is written in C++ and utilizes header files constants.h, classes.h, functions.h, forces.h, and vecmat3.h [43]. The latter is a header file that allows powerful matrix and vector operations. More details on these files are explained in Appendices 1, 2, 3, and 4.

29 22 Chapter 4 Micelle Formation under Neutral Conditions In this chapter, the micelle formation process is looked at starting from random initial coordinates of system constituents to an end result of one or two micelles in equilibrium. The loading capability of polymeric micelles is just one of the many valuable quantities that may be analysed under this process since it is a measure of the number of drugs a micelle can hold. Measured quantities during the micelle formation process include the cluster number (clustnum), average cluster radius (clustradavg), average number of lipids per cluster (clustlipnumavg), number of encapsulated drugs (druginnum), average number of encapsulated drugs per cluster (clustdruginnumavg), and the average ratio of encapsulated drug radius to corresponding cluster radius (druginrad/clustrad)avg where the radius starts from the centre of the lipid cluster. Since the ph-sensitive beads are treated as any other Tail bead in the neutral ph system, the results for the Control system can apply for the Reversed lipid structures in a neutral environment. For each lipid structure, micelle formation simulations were performed eight times to average results. With the initial conditions and interactions employed, aggregation of lipids and Drug beads naturally occurs. The results pertaining to one run should be explained first before analyzing the results after an average of eight runs. The results for average cluster radius were compared to the visualization of the simulation. Due to the maximum range of rc = 2.6σ to determine if two lipid chains are part of the same cluster, two clusters which have not yet aggregated into a single stable micellular structure, but that are sufficiently close enough, can be classified as one cluster in the cluster analysis. This is reflected in the average cluster radius graph (Fig. 4-6) by the unsteady jumps and dips which precede the more constant value in which the cluster adopts a stable single micellular form. This is because the cluster analysis uses the sum of the positions of the outermost Head monomer beads of the lipids to calculate the cluster centre and radius. That is, it assumes a micellular structure.

30 Figure 4: Micelle formation - Control graphs (1 run) 23

31 Figure 5: Micelle formation - Drug-Bonded graphs (1 run) 24

32 Figure 6: Micelle formation - Reversed + Drug-Bonded graphs (1 run) 25

33 26 Instances where the cluster structure is not yet a stable micelle are for 1H-1P-3T from t = t 0, 1H-1P(1D)-3T from t = t 0, and 1H-3T-1P(1D) from times t = t 0. This is reflected in the graph for average cluster radius by the more erratic jumps and dips since the cluster radius calculation assumes a stable micellular structure. As a result, the value for average druginrad/clustrad is also erratic in this time interval. The average cluster radius results for 2H-1P-3T show that the clusters swiftly assume a micellular form upon fusion from two to one clusters because the average cluster radius value undergoes a quick sharp jump from one steady state to another. For the Control system, the descending order for reaching one final stable micelle the quickest goes (3H)1H-1P-3T, 1H-1P-3T(3T), 1H-1P-4T, 1H-1P-3T, 2H-1P-3T. For the Drug-Bonded system, the descending order for reaching one final stable micelle the quickest goes 2H-1P(1D)-3T, 1H-1P(1D)-3T, (3H)1H-1P(1D)-3T, and finally 1H- 1P(1D)-4T and 1H-1P(1D)-3T(3T). For the Reversed + Drug-Bonded system, the descending order for reaching one final cluster number the quickest goes (3H)1H-3T-1P(1D), 2H-3T- 1P(1D), 1H-3T(2T)-1P(1D), 1H-4T-1P(1D), and 1H-3T-1P(1D). Dimensionless numbers that characterize the system (Schmidt, Reynolds, and Peclet) were also measured at each timestep starting when the system had formed one micelle under equilibrium. The single micellular structure may be treated as a single solute particle for these measurements. a = MPC cell length n = N V = number of particles in system volume of system = number density γ = n a 3 = average number of particles in an mpc cell for a system with number density n m = mass of MPC particle ρ = mn = mγ a 3 η = k BTτρ 2m = mean mass density + 1 e γ (γ γ 1 + e γ) + m 12aτ (γ 1 + e γ ) = shear/dynamic viscosity for MPC AT a

34 27 For the Control system, the average shear/dynamic viscosity of the MPC solvent for t = 6,500 10,000t 0 is 1.28 (mε)1 2. This is the time frame where the single micelle is already formed and under equilibrium. v = η ρ = k BTτ 2m D 0 = k BTτ 2m S c = v D 0 = σ e γ a2 (γ γ 1 + e γ) + 12τ 1 + e γ (γ ) = kinematic viscosity for MPC AT a [39] γ + 1 e γ (γ γ 1 + e γ) = diffusion coefficient for MPC AT a [41] k B Tτ + 1 e γ 2m (γ γ 1 + e γ ) + a2 12τ k B Tτ + 1 e γ 2m (γ γ 1 + e γ ) 1 + e γ (γ γ ) = 1 + a2 m 6kTτ 2 [(γ 1 + e γ ) 2 γ(γ + 1 e γ ) ] The Schmidt number is the ratio of the rate of diffusive momentum transfer to the rate of diffusive mass transfer [16]. The average value of the Schmidt number for the single micelle in MPC solvent was found to be This indicates that the micelle is in a particle regime and the MPC solvent behaves more gas-like than liquid-like. U = characteristic velocity (velocity of centre of micelle) L = characteristic length (radius of micelle) v = kinematic viscosity R e = UL v = UL k B Tτ + 1 e γ 2m (γ γ 1 + e γ ) + a2 12τ 1 + e γ (γ γ ) The Reynolds number measures the relative importance of inertial and viscous forces in the system [16]. For large-scale turbulent flow the Reynolds number is large and inertia dominates. For small particle motion in dense fluids inertial effects are unimportant and the Reynolds number is small. Low Reynolds numbers result from the small sizes and low velocities of the particles, in combination with the fact that they move in a medium with relatively high viscosity. The Reynolds number was found to be which indicates that the micelle is under the regime of unseparated/laminar flow. Viscous forces dominate over inertial forces. This corresponds with

35 28 the fact that the majority of microfluidic devices developed to date employ low Reynold s number flows and rely on the dominance of viscous forces over inertial forces [40]. D c = kt = diffusion coefficient of a solute particle in the fluid 6πηL P e = UL D c = UL k B T 6πηL = 6πηUL2 k B T = 6π k B T [k BTτρ + 1 e γ (γ 2m γ 1 + e γ) + m 12aτ (γ 1 + e γ )] UL 2 The Peclet number measures the ratio of convective transport to diffusive transport [16]. For P e > 1 the particle will move convectively over distances greater than its size. The average Peclet number was found to be 21.4 which indicates that convection dominates over diffusion. The Peclet number for micelles obtained from experiment is usually somewhere in the range [42].

36 Figure 7: Micelle formation - Control graphs (8 runs) 29

37 Figure 8: Micelle formation - Drug-Bonded graphs (8 runs) 30

38 Figure 9: Micelle formation - Reversed + Drug-Bonded graphs (8 runs) 31

39 32 The cluster number for all lipid structures exponentially decreases from a value of to 1-2. The various lipid models start off from random initial positions then eventually aggregate into one or two cluster(s) by the end. They reach this state at around the same time as one another. For the Control system, before the cluster number reaches a steady state between one and two, 2H-1P-3T and 1H-1P-3T are the slowest at aggregating, followed by 1H-1P-4T. The branched lipids are the fastest at aggregating. For the Drug-Bonded system, before the cluster number reaches a steady state between one and two, (3H)1H-1P(1D)-3T, 1H-1P(1D)-3T, 1H-1P(1D)-4T, and 2H-1P(1D)-3T have about the same aggregation rate and are the slowest. 1H-1P(1D)-3T(3T) is the fastest at aggregating. For the Reversed + Drug-Bonded system, before the cluster number reaches a steady state between one and two, 1H-3T-1P(1D) is the slowest at aggregating, followed by 2H-3T-1P(1D) and 1H-3T(2T)-1P(1D), and then 1H-4T-1P(1D). (3H)1H-3T- 1P(1D) is the fastest at aggregating. The average cluster radius exponentially increases from 1-3σ to 5-7σ for all Control lipid models, from 1-3σ to 5-6σ for all Drug-Bonded lipid models, and from 1-3σ to 4-7σ for all Reversed + DrugBonded lipids models. As the cluster number decreases, the increase in average cluster radius also decreases (eventually to zero if the simulations can run long enough to reach clustnum = 1). The system with (3H)1H-1P-3T has the average cluster radius exponentially increasing then decreasing at about t = 2,500t 0 and remaining about constant at 5.75σ from t = 5,000t 0 and onwards. A similar graph shape can be seen for 1H-1P(1D)-3T, 1H-1P(1D)- 3T(3T), 1H-3T-1P(1D), and 1H-4T-1P(1D). This unsteady peak, before dropping to the steadier/constant average cluster radius may be due to cluster(s) not yet adopting a stable micellar form. 1H-1P-4T, 1H-1P-3T(3T), (3H)1H-3T-1P(1D) have an average cluster radius that exponentially increases, decreases, then reaches a steadier (albeit not constant) state. 2H-1P-3T and 2H-1P(1D)-3T has an average cluster radius that exponentially increases then reaches a steadier (albeit not constant) steadier state. 1H-1P-3T, 1H-1P(1D)-4T, (3H)1H-1P(1D)-3T, 2H- 3T-1P(1D), and 1H-3T(2T)-1P(1D) have an average cluster radius that exponentially increases then reaches a constant value. For the Control system, the final average cluster radius in descending order is 1H-1P-3T(3T), 1H-1P-4T, (3H)1H-1P-3T, 2H-1P-3T, and 1H-1P-3T. For the Drug-Bonded system, the final average cluster radius in descending order is 2H-1P(1D)-3T, (3H)1H-1P(1D)-3T, 1H-1P(1D)-3T(3T), 1H-1P(1D)-4T, and 1H-1P(1D)-3T. For the Reversed

40 33 + Drug-Bonded system, the final average cluster radius in descending order is (3H)1H-3T- 1P(1D), 2H-3T-1P(1D), 1H-4T-1P(1D), 1H-3T(2T)-1P(1D), and 1H-3T-1P(1D). The average size of the clusters in terms of lipid number increases almost linearly before settling to a constant value for all lipid models. For the Control system, the increase is slower for 1H-1P- 3T and 2H-1P-3T compared to 1H-1P-4T, (3H)1H-1P-3T and 1H-1P-3T(3T). 1H-1P-3T reaches the maximum lipid number size of 100 first, followed by 1H-1P-3T(3T) and 1H-1P-4T. (3H)1H- 1P-3T does not quite reach 100 and is followed by 2H-1P-3T at around 80 lipids. For the Drug- Bonded system, the increase is initially the fastest for 1H-1P(1D)-3T(3T), which then becomes the slowest from t = 6,250t 0 onwards. (3H)1H-1P(1D)-3T reaches the maximum lipid number size of 100 first, followed by 2H-1P(1D)-3T, and 1H-1P(1D)-3T. 1H-1P(1D)-4T reaches only about 95, and 1H-1P(1D)-3T(3T) is lower at about 89 lipids. For the Reversed + Drug-Bonded system, the increase is the fastest for (3H)1H-3T-1P(1D) and 2H-3T-1P(1D). (3H)1H-3T-1P(1D) and 2H-3T-1P(1D) reach the maximum lipid number size of 100 first, followed by 1H-4T- 1P(1D). 1H-3T(2T)-1P(1D) reaches only about 93, and 1H-3T-1P(1D) is lower at about 81. For the Control system, the number of encapsulated Drug beads exponentially increases to the constant and maximal value of 100 for all lipid structures. The shape of the plots for the different lipid structures are about the same. The average number of encapsulated Drug beads per cluster is the same as the clustlipnumavg,. For the Drug-Bonded and Reversed + Drug-Bonded Systems, the number of encapsulated Drugs is constant at 100 because all bonded Drugs are not considered to be free. Therefore, the average number of encapsulated Drug beads per cluster is the same as the clustlipnumavg because every lipid chain has one Drug bead bonded to it. For the Control system, the average druginrad/clustrad decreases exponentially (as average cluster radius increases exponentially) to eventually reach a steady state. As the cluster number decreases, the decrease in average druginrad/clustrad also decreases (eventually to zero if the simulations can run long enough to all reach clustnum = 1). In the end, the descending order of average druginrad/clustrad is 1H-1P-4T and 1H-1P-3T(3T), 1H-1P-3T and 2H-1P-3T, and (3H)1H-1P-3T. The micelle with lipid structure (3H)1H-1P-3T has the Drugs closest to the core relative to the cluster radius. For the Drug-Bonded system, the average druginrad/clustrad initially decreases until about t = 625t 0 (clustnum ~ 8 and clustlipnumavg ~ 10) for all lipid structures then jumps back up to a steady state for the remainder of the simulation. In the end, the

41 34 descending order of average druginrad/clustrad is 1H-1P(1D)-3T(3T), 1H-1P(1D)-4T, 1H- 1P(1D)-3T, 2H-1P(1D)-3T, and (3H)1H-1P(1D)-3T. The micelle with lipid structure (3H)1H- 1P(1D)-3T has the Drugs closest to the core relative to its cluster radius. For the Reversed + Drug-Bonded system, the average druginrad/clustrad decreases to a steady state for all lipid structures. In the end, the descending order of average druginrad/clustrad is 1H-3T(2T) - 1P(1D), 1H-4T-1P(1D) and 1H-3T-1P(1D), and finally 2H-1P(1D)-3T and (3H)1H-3T-1P(1D). The micelles with lipid structures 2H-1P(1D)-3T and (3H)1H-1P(1D)-3T have the Drugs closest to the core relative to their cluster radius.

42 35 Chapter 5 Micelle Dynamics under Acidic Conditions Acidic conditions upon the stable micelle are modelled by instantaneously activating all the ph- Sensitive beads and breaking any bond to a Drug bead. Activation for the ph-sensitive beads means they no longer behave as hydrophobic Tail beads and begin to behave as hydrophilic Head beads. The importance of this is seeing how the Drug distribution within the micelle is affected by activated ph-sensitive beads. If one wants micelles that can efficiently release their drugs when triggered, then this process needs to be better understood. Not all of the eight simulations for each lipid structure in the previous micelle formation step ended with one cluster. The lowest amount of runs ending with one cluster for a lipid structure was four. Accordingly, only four runs ending with one cluster for each lipid structure will be used as initial conditions in the acidification step. For all lipid structures, the average cluster number stays constant at one because only micelle formations leading to one cluster in the end were chosen for the acidification step and the micelle never breaks apart. Since the micelle never breaks apart, the average lipid number per cluster also remains unchanged at 100. Micelle break up is not witnessed because under MPC, ph- Sensitive beads cannot be set to be significantly attracted to the Solvent without an unrealistic amount of Solvent beads accumulating in micelle pores. For all lipids structures, the average number of encapsulated Drug beads stays constant at 100 during the acidification step since the micelle neither breaks apart to release the Drug beads nor launches the Drug beads out. Likewise, the average encapsulated Drug number per cluster stays constant at 100.

43 Figure 10: Micelle acidification - Control graphs (4 runs) 36

44 Figure 11: Micelle acidification - Drug-Bonded graphs (4 runs) 37

45 Figure 12: Micelle acidification - Reversed graphs (4 runs 38

46 Figure 13: Micelle acidification - Reversed + Drug-Bonded graphs (4 runs) 39

47 40 For all Control lipid models, the average cluster radius is about constant, but at different values between 5.2σ and 6.2σ. The beginning of the acidification step comes with the average cluster radius quickly increasing then staying about constant. The overall average cluster radius size in descending order goes 1H-1P-3T(3T) and (3H)1H-1P-3T, 2H-1P-3T, 1H-1P-4T, and finally the lowest with 1H-1P-3T. The average cluster radius for the systems with branched lipids are about the same and the average cluster radius for the systems with elongated lipids are about the same. The average cluster radius for (3H)1H-1P-3T is about constant throughout the simulation, but shows a sharp peak towards the end of simulation, as does 2H-1P-3T at about t = 60t 0, and 1H- 1P-3T(3T) near the beginning. For all Drug-Bonded lipid models, the average cluster radius is about constant, but at different values between 5.2σ and 6.2σ. The beginning of the acidification step comes with the average cluster radius quickly increasing, then afterwards staying about constant. The overall average cluster radius size in descending order goes (3H)1H-1P(1D)-3T and 1H-1P(1D)-3T(3T), 2H-1P(1D)-3T, 1H-1P(1D)-4T, and finally the lowest with 1H-1P(1D)- 3T. The average cluster radius for the systems with branched lipids are about the same and the average cluster radius for the systems with elongated lipids are about the same. For all Reversed lipid models, the average cluster radius decreases exponentially from σ, then either continues to decrease, or stays at a steady state between 4.8σ and 6.2σ. The overall average cluster radius size in descending order goes (3H)1H-3T-1P, 1H-4T(2T)-1P, 2H-3T-1P, 1H-4T- 1P, and finally the lowest with 1H-3T-1P. The average cluster radius for the systems with branched lipids are about the same. 2H-3T-1P and (3H)1H-3T-1P are about constant throughout the rest of the simulation, while the other lipid structures show a slow decrease. For all Reversed + Drug-Bonded lipid models, the average cluster radius decreases exponentially from σ then stays at a steady state between 4.8σ and 6.2σ. The overall average cluster radius size in descending order goes (3H)1H-3T-1P(1D), 1H-3T(2T)-1P(1D) and 2H-3T-1P(1D), 1H-4T- 1P(1D), and finally the lowest with 1H-3T-1P(1D). The average cluster radius of 2H-3T-1P(1D) and 1H-3T(2T)-1P(1D) are about the same.

48 41 For all Control lipid models, the average druginrad/clustrad exponentially decreases from to a steady state at for all lipid structures. The initial average druginrad/clustrad in descending order goes 1H-1P-4T and 1H-1P-3T(3T), 1H-1P-3T, 2H-1P-3T, and finally the lowest with (3H)1H-1P-3T. The final average druginrad/clustrad is in a similar order, but 1H- 1P-3T(3T) is greater than 1H-1P-4T. The average druginrad/clustrad for 1H-1P-3T(3T) and 1H-1P-4T are close to each other. Some lipid models show troughs where peaks are shown in the average cluster radius plot. For all Drug-Bonded lipid models, the average druginrad/clustrad exponentially decreases from to a steady state at for all lipid structures. The initial average druginrad/clustrad in descending order goes 1H-1P(1D)-3T(3T), 1H-1P(1D)-3T, 1H-1P(1D)-4T, 2H-1P(1D)-3T, and finally the lowest with (3H)1H-1P(1D)-3T. The final average druginrad/clustrad is in a similar order, but 1H-1P(1D)-4T is greater than 1H-1P(1D)- 3T. The average druginrad/clustrad for 2H-1P(1D)-3T and (3H)1H-1P(1D)-3T are close to each other. For all Reversed lipid models, the average druginrad/clustrad linearly decreases from to a steady state at for all lipid structures. The initial average druginrad/clustrad in descending order goes 1H-4T-1P and 1H-4T(2T)-1P, 1H-3T-1P, 2H-3T- 1P, and finally the lowest with (3H)1H-3T-1P. The final average druginrad/clustrad is in a similar order, but 1H-4T-1P is similar to 1H-4T(2T)-1P. The average druginrad/clustrad for 1H-3T-1P, 1H-4T-1P, and 1H-4T(2T)-1P are close to each other. 1H-3T-1P initially shows a clear increase before decreasing. For all Reversed + Drug-Bonded lipid models, the average druginrad/clustrad increases from (then possibly decreases), remains at a steady state, then decreases to for all lipid structures. The overall average druginrad/clustrad in descending order goes 1H-3T-1P(1D), 1H-4T-1P(1D) and 1H-3T(2T)-1P(1D), 2H-3T-1P(1D), and finally the lowest with (3H)1H-3T-1P(1D). The average druginrad/clustrad for 1H-3T- 1P(1D), 1H-4T-1P(1D), and 1H-3T(2T)-1P(1D) are close to each other and the average druginrad/clustrad for 2H-3T-1P(1D) and (3H)1H-3T-1P(1D) are close to each other. The greatest and quickest jump in average druginrad/clustrad occurs for 1H-4T-1P(1D).

49 42 Chapter 6 Discussion In this chapter, differences in micelle and Drug distribution after acidification are discussed. From the results of the previous chapter, a design for the ideal amphiphilic triblock copolymer is also hypothesized. Finally, limitations from the use of MPC in the acidification step are presented. Micelle Cross Sections In Figure 14, cross sections of the micelle are shown first under neutral conditions at t = 0, then at the end of the acidification simulation at t = 200t 0. It s worth noting that the structure of the micelle still maintains a sphere-like shape after acidification. Visual Molecular Dynamics is used to visualize the system [44]. For the Control micelles, acidification causes the micelle to expand. The Head beads extend outwards towards the Solvent and the Drug beads move closer to the core where there are less lipid chains. This is especially apparent for (3H)1H-1P-3T. For the Drug-Bonded micelles, acidification causes the Drug beads to migrate away from the shell and towards the core of the micelle. The Head beads also extend outward towards the Solvent, allowing channels for Solvent beads to enter, which is clear for 2H-1P(1D)-3T. For 1H- 1P(1D)-3T, the Drug beads seem more sprawled around the micelle than just at the core. For the Reversed and Reversed + Drug-Bonded micelles, acidification causes some lipid chains to turn out to the surface of the micelle since the terminal monomer becomes hydrophilic under acidic conditions. For the lipids that do not turn out, they form a hydrophilic core in the centre of the micelle. This creates two hydrophilic barriers for the Drug bead: the shell (Head beads and hydrophilic ph-sensitive beads) and the core (hydrophilic ph-sensitive beads). As a result, Drug beads settle in the shell of space containing the core and surrounded by the shell.

50 Figure 14: Before and after acidification: Cross-section of micelle (1 run) 43

51 44 ph-sensitive and Drug Bead Distribution In Figure 15, distributions of the ph-sensitive beads and Drug beads are shown. First under neutral conditions at t = 0, then at the end of the acidification simulation at t = 200t 0. For the Control lipid structures, ph-sensitive beads move more outward as a response to acidification, while the Drug beads move closer to the core. The Drug beads in the (3H)1H-1P- 3T system move more inward collectively. In the 1H-1P-3T(3T) system, there is a clear dense core of Drug beads surrounded by a less dense shell of Drug beads. For the Drug-Bonded system, the ph-sensitive beads move more outward as a response to acidification, while the Drug beads move closer to the core. This condensing to the core is especially clear for the (3H)1H-1P(1D)-3T lipid model. Drug beads in the 1H-1P(1D)-3T(3T) lipid model end up more dispersed throughout the micelle compared to other lipid models. For the Reversed and Reversed + Drug-Bonded system, the ph-sensitive beads either stay near the core of the micelle or expand outwards. The Drug beads settle in between the outer shell and core of ph-sensitive beads. The density and quantity of the ph-sensitive core varies between different lipid structures.

52 Figure 15: Before and after acidification: ph-sensitive and Drug bead distribution (1 run) 45

53 46 clustrad avg For the systems with Control and Drug-Bonded lipids, the onset of acidification causes the average cluster radius to sharply and quickly increase, then be more or less steady for the rest of the simulation. That is, the micelle swells quickly in response to the newly acidic environment, then stays at a steady size. An increase in lipid chain length by a monomer unit increases the average cluster radius. Branched lipids increase the average cluster radius more compared to a single monomer unit increase. An increase in chain length by a Head monomer unit increases the average cluster radius more compared to a Tail monomer unit. A branched Head monomer has about the same average cluster radius as a branched Tail monomer. For the systems with Reversed and Reversed + Drug-Bonded lipids, the average cluster radius exponentially decreases and by less as time goes on until it reaches a steady state. That is, the micelle shrinks in size, then stays at a steady size. An increase in lipid chain length by a monomer unit increases the average cluster radius. An increase in chain length by a Head monomer unit increases the average cluster radius more compared to a Tail monomer unit. Branched lipids make the average cluster radius greater compared to their unbranched counterparts. (druginrad/clustrad) avg The Control and Drug-Bonded micelles have the average druginrad/clustrad decreasing exponentially and by less as time goes on. The Drug beads move closer towards the core of the micelle in response to the micelle swelling due to the acidic environment. Swelling leaves space for more Solvent beads, which repel the Drug beads away. The average druginrad/clustrad is smaller for an increase in lipid chain length by a Head monomer unit compared to an increase by a Tail monomer unit. A branched Head pushes the Drugs closer to the core than an increase in chain length by a Head monomer. A branched Tail allows the Drug to be farther from the core than an increase in chain length by a Tail monomer. The Reversed micelles have the average druginrad/clustrad either increasing then staying constant (and/or decreasing afterwards), or just decreasing linearly. In the first scenario, the Drug beads are pushed away from the core of the micelle and do not go back because of their repulsion

54 47 to the newly hydrophilic core. The average druginrad/clustrad stays about the same when a Tail monomer unit is added or if the Tail is branched. It becomes lower when a Head monomer unit is added or if the Head is branched (particularly for the latter). The Reversed + Drug-bonded micelles show that a longer Tail can cause the Drug beads to be shot away from the core by more than 5% of the micelle radius. Therefore, a proposed structure to efficiently shoot Drug beads outwards is one in which the ph-sensitive bead is at the terminal end of the lipid chain, the Drug beads bonded to the ph-sensitive beads, and the hydrophobic Tail being longer. The hydrophilic shell must be less of a repulsive barrier than the hydrophilic ph-sensitive monomers at the core. In this scenario, if just the right proportion of ph-sensitive units is at the terminal end of the lipid chain, then it can perhaps shoot the Drug beads out of the micelle without them being stopped by the hydrophilic shell. A system with a branched Head, would not be suitable to push the Drug beads away from the core to the extent that they can exit the micelle. MPC Limitations Figure 16: MPC limitations. Solvent bead accumulation in channels within the micelle (acidic environment). This is a result of strong ph-sensitive bead attraction to Solvent beads. One of the noticeable observations in the acidification step is that the micelle never breaks up for any lipid structure, while it does in previous MD studies. This is due to the low attraction (0.05ε) between ph-sensitive and Solvent beads. The attraction is chosen to be weak because although MPC reproduces similar results as explicit MD for micelle formation, it fails in the acidification step if ph-sensitive beads are too attracted to Solvent beads. The Solvent beads accumulate