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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
ii
Analysis of Hybrid MD-MPC Simulations of Micelle Formation
Under Neutral pH and Dynamics Under Acidic pH Using Different
pH-Sensitive Triblock Copolymer Structures
Fatima Yousuf
Master of Science
Chemistry
University of Toronto
2016
Abstract
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.
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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 ........................................................................................................11
8.1 MD ..............................................................................................................................12
8.2 MPC ............................................................................................................................14
8.3 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
iv
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
v
List of Tables
Table 1: pH in different tissue and cell organelles……………………………………………..…2
Table 2: Interaction energies…………………………………………………..…………………19
vi
List of Figures
Figure 1: pH-sensitive micelle drug delivery……….……………………………………………..1
Figure 2: pH-sensitive lipid structures…………………………………………………………...16
Figure 3: MPC parameters…………………………………………………………………..…...19
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)………………………………...…….36
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)………………...…39
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
vii
List of Appendices
Appendix 1: constants.h……….…………………………………………..……………………..50
Appendix 2: classes.h…………………………………………………………………..………...55
Appendix 3: functions.h……………………………………………………………………..…...56
Appendix 4: forces.h……………………………………………………………………..………58
1
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).
2
Table 1: pH in different tissue and cell organelles
Tissue/cellular compartment pH
Blood 7.35-7.45
Stomach 1.0-3.0
Duodenum 4.8-8.2
Colon 7.0-7.5
Early endosome 6.0-6.5
Late endosome 5.0-6.0
Lysosome 4.5-5.0
Golgi 6.4
Tumor, extracellular 7.2-6.5
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.
3
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
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 7.2-6.6) 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].
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].
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].
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 non-
bonded 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].
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 10-1000nm 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, pH-
sensitive 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].
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,
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
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].
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, oil-
phase 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
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.
14
8.2 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.
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.
16
Figure 2: pH-sensitive lipid structures. Head bead – striped circle, pH-Sensitive bead – black
circle, Tail bead – white circle, Drug bead – white square
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 𝑡0 = √𝑚𝜎2
𝜀 where m, σ, ε are the units of mass, length, and
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 68194 beads in total, 100 lipids, and 100 drug beads. On average, the Solvent
number density in the bulk is equal to 3.96.
Bonded monomers in a lipid are bonded to one another through a spring potential
𝑉𝑏𝑜𝑛𝑑(𝑟𝑖𝑗) = −1
2𝑘𝑏𝑜𝑛𝑑𝑟∞
2𝑙𝑛 [1 − (𝑟𝑖𝑗
𝑟∞)
2
], 𝑟𝑖𝑗 ≤ 𝑟∞
where 𝑘𝑏𝑜𝑛𝑑 =20𝜀
𝜎2 and 𝑟∞ = 1.5𝜎 [14].
Next-nearest monomers in a lipid are also bonded to one another through a spring potential
𝑉𝑏𝑒𝑛𝑑(𝑟𝑖𝑗) =1
2𝑘𝑏𝑒𝑛𝑑(𝑟𝑖𝑗 − 4𝜎)
2
where 𝑘𝑏𝑒𝑛𝑑 =2.5𝜀
𝜎2 [14].
18
Head beads have a short-range Lennard-Jones repulsion to every other type of bead with a cutoff
distance of 𝑟𝑐 = 21
6𝜎. Head beads only have an attraction
𝑉𝑎𝑡𝑡(𝑟𝑖𝑗) = −𝜀𝛼𝛼′𝑐𝑜𝑠2 (0.5𝜋 [𝑟𝑖𝑗 − 𝑟𝑐
𝜔𝛼𝛼′ − 𝑟𝑐]), 𝑟𝑐 ≤ 𝑟𝑖𝑗 ≤ 𝜔𝛼𝛼′
to Solvent beads between rc and 𝜔𝛼𝛼′ = 1.65𝜎 [14].
Tail beads have a short-range Lennard-Jones repulsion to every other type of bead with a cutoff
distance of 𝑟𝑐 = 21
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 𝜔𝛼𝛼′ =
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 𝑟𝑐 = 21
6𝜎. Drug beads have an attraction to inactivated pH-Sensitive beads under
neutral pH, Tail beads, and other Drug beads between rc and 𝜔𝛼𝛼′ = 2.6𝜎.
Solvent beads have a short-range Lennard-Jones repulsion to every other type of non-Solvent
bead with a cutoff distance of 𝑟𝑐 = 21
6𝜎. Solvent beads only have an attraction to Head beads and
activated pH-Sensitive beads under acidic conditions between rc and 𝜔𝛼𝛼′ = 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-AT-
a) [14].
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)
𝜀ℎℎ 0.5
𝜀ℎ𝑡 1.0
𝜀ℎ𝑑 1.0
𝜀ℎ𝑠 0.05
𝜀𝑡𝑡 0.5
𝜀𝑡𝑑 0.5
𝜀𝑡𝑠 2.0
𝜀𝑑𝑑 0.5
𝜀𝑑𝑠 2.0
𝜀ℎ𝑝 1.0 0.5
𝜀𝑝𝑝 0.5 0.5
𝜀𝑝𝑡 0.5 1.0
𝜀𝑝𝑑 0.5 1.0
𝜀𝑝𝑠 2.0 0.05
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
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.
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.
23
Figure 4: Micelle formation - Control graphs (1 run)
24
Figure 5: Micelle formation - Drug-Bonded graphs (1 run)
25
Figure 6: Micelle formation - Reversed + Drug-Bonded graphs (1 run)
26
Instances where the cluster structure is not yet a stable micelle are for 1H-1P-3T from 𝑡 =
3100 − 6250𝑡0, 1H-1P(1D)-3T from 𝑡 = 2050 − 4100𝑡0, and 1H-3T-1P(1D) from times 𝑡 =
2120 − 4270𝑡0. 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
𝑉=
number of particles in system
volume of system= number density
𝛾 = �̅�𝑎3 = average number of particles in an mpc cell for a system with number density n̅
m = mass of MPC particle
𝜌 = 𝑚�̅� =𝑚𝛾
𝑎3= mean mass density
𝜂 =𝑘𝐵𝑇𝜏𝜌
2𝑚(
𝛾 + 1 − 𝑒−𝛾
𝛾 − 1 + 𝑒−𝛾) +
𝑚
12𝑎𝜏(𝛾 − 1 + 𝑒−𝛾) = shear/dynamic viscosity for MPC − AT − a
27
For the Control system, the average shear/dynamic viscosity of the MPC solvent for 𝑡 = 6,500 −
10,000𝑡0 is 1.28 (𝑚𝜀)
1
2
𝜎2⁄ . This is the time frame where the single micelle is already formed
and under equilibrium.
v =𝜂
𝜌=
𝑘𝐵𝑇𝜏
2𝑚(
𝛾 + 1 − 𝑒−𝛾
𝛾 − 1 + 𝑒−𝛾) +
𝑎2
12𝜏(
𝛾 − 1 + 𝑒−𝛾
𝛾) = kinematic viscosity for MPC − AT − a [39]
𝐷0 =𝑘𝐵𝑇𝜏
2𝑚(
𝛾 + 1 − 𝑒−𝛾
𝛾 − 1 + 𝑒−𝛾) = diffusion coefficient for MPC − AT − a [41]
𝑆𝑐 = v
𝐷0=
𝑘𝐵𝑇𝜏2𝑚 (
𝛾 + 1 − 𝑒−𝛾
𝛾 − 1 + 𝑒−𝛾) + 𝑎2
12𝜏 (𝛾 − 1 + 𝑒−𝛾
𝛾 )
𝑘𝐵𝑇𝜏2𝑚 (
𝛾 + 1 − 𝑒−𝛾
𝛾 − 1 + 𝑒−𝛾)= 1 +
𝑎2𝑚
6𝑘𝑇𝜏2[(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 1.94. 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
𝑅𝑒 = 𝑈𝐿
v=
𝑈𝐿
𝑘𝐵𝑇𝜏2𝑚 (
𝛾 + 1 − 𝑒−𝛾
𝛾 − 1 + 𝑒−𝛾) + 𝑎2
12𝜏 (𝛾 − 1 + 𝑒−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 0.265 which indicates that the micelle is under the regime
of unseparated/laminar flow. Viscous forces dominate over inertial forces. This corresponds with
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].
𝐷𝑐 =𝑘𝑇
6𝜋𝜂𝐿= diffusion coefficient of a solute particle in the fluid
𝑃𝑒 = 𝑈𝐿
𝐷𝑐=
UL
𝑘𝐵𝑇6𝜋𝜂𝐿
=6𝜋𝜂𝑈𝐿2
𝑘𝐵𝑇=
6𝜋
𝑘𝐵𝑇[𝑘𝐵𝑇𝜏𝜌
2𝑚(
𝛾 + 1 − 𝑒−𝛾
𝛾 − 1 + 𝑒−𝛾) +
𝑚
12𝑎𝜏(𝛾 − 1 + 𝑒−𝛾)] 𝑈𝐿2
The Peclet number measures the ratio of convective transport to diffusive transport [16]. For
𝑃𝑒 > 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 10-
100 [42].
29
Figure 7: Micelle formation - Control graphs (8 runs)
30
Figure 8: Micelle formation - Drug-Bonded graphs (8 runs)
31
Figure 9: Micelle formation - Reversed + Drug-Bonded graphs (8 runs)
32
The cluster number for all lipid structures exponentially decreases from a value of 20-25 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 = 2,500𝑡0 and remaining about constant at 5.75σ from 𝑡 =
5,000𝑡0 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
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 = 6,250𝑡0 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 = 625𝑡0 (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
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.
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.
36
Figure 10: Micelle acidification - Control graphs (4 runs)
37
Figure 11: Micelle acidification - Drug-Bonded graphs (4 runs)
38
Figure 12: Micelle acidification - Reversed graphs (4 runs
39
Figure 13: Micelle acidification - Reversed + Drug-Bonded graphs (4 runs)
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 = 60𝑡0, 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 5.2-6.2 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 5.2-6.2 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.
41
For all Control lipid models, the average drugInRad/clustRad exponentially decreases from 0.55-
0.75 to a steady state at 0.45-0.70 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 0.70-0.85 to a steady state at 0.50-0.70 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 0.55-0.75 to a steady state at 0.45-0.70 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 0.50-0.70 (then possibly decreases), remains at a steady
state, then decreases to 0.55-0.70 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).
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 = 200𝑡0. 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.
43
Figure 14: Before and after acidification: Cross-section of micelle (1 run)
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 = 200𝑡0.
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.
45
Figure 15: Before and after acidification: pH-Sensitive and Drug bead distribution (1 run)
46
clustRadavg
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
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
48
excessively within channels in the micelle. Formation of channels is observed in MD simulations
of this process, but with MPC, Solvent channel formation is accompanied by high Solvent
accumulation until it eventually creates infinite attraction. As a result, the acidification
simulations keep the attraction low between pH-Sensitive and Solvent beads to compensate for
using MPC. This, however, prevents channel formation, propelled Drug release, and micelle
destabilization to the extent of breaking up.
49
Chapter 7 Conclusion
In conclusion, of the various lipid structures, the one that holds the most promise in terms of
Drug release would be a Reversed + Drug-Bonded lipid with a long Tail block. Different
structures of the lipid showed different responses to the onset of an acidic environment. Half of
the lipid structures showed at least a notable growth in size due to swelling whereas the Reversed
and Reversed + Drug-Bonded lipids showed a smaller unique structure with shells of different
components. This unique structure contains an inner core of hydrophilic pH-Sensitive beads, an
outer core of hydrophobic Tail and Drug beads, and a hydrophilic outer shell with Head and pH-
sensitive beads.
MPC works well in reproducing the effects of hydrophobicity for micelle self-assembly, but it is
not appropriate to use when there are strong hydrophilic interactions since it causes excessive
clustering of Solvent beads. Endless clustering occurs because there is no repulsion between
Solvent beads, and the strong hydrophilic attraction allows too many Solvent beads in micelle
channels. Whereas previous MD studies have shown micellular break up and Drug release upon
the onset of acidification, this was not witnessed here due to limitations of MPC requiring low
hydrophilic attraction. However, information on Drug distribution within the micelle was
obtained. In the future, for broader analysis, it would be best to not use MPC for the acidification
step, and also not in a scenario where the Head-Solvent attraction is strong.
The simulation method tries to imitate acidity by instantaneously activating the pH-Sensitive
units (and breaking their respective bond to a Drug bead if any). This approach should be
analyzed for its feasibility, after which other methods of acidification can be proposed if better.
The cluster radius calculation assumes that the cluster has a spherical/micellular shape. In the
micelle formation process when some clusters have not yet reached this shape, this calculation
may not be as valuable.
Finally, future work in this field should look at how these lipid structures respond when the
interactions and affinities are varied for the Head, pH-Sensitive, Tail, and Drug beads. After all,
the structure of lipids is not the only factor affecting the resultant micelle structure, Drug
distribution, and possible Drug release.
50
Appendix 1: constants.h
int drug_bonded……………………………………..………...1 if drug is bonded to lipid, 0 if not
int drug_released……………………..1 if drug-lipid bond is broken in acidification step, 0 if not
int rev_lip……………………………………………………...1 if lipid is of reverse type, 0 if not
double m = 1.0………………………………………………...…………………….MD mass unit
double sigma = 1.0…………………….................................................................MD distance unit
double epsilon = 1.0…………………………………………………………….....MD energy unit
double t0 = sqrt(m*sigma*sigma*(1.0/epsilon))………………………………….....MD time unit
double dt = 0.005*t0…………………………………………………….............MD timestep size
int tau = 40……………………………………………..MPC timestep factor to MD timestep size
int equilibstepTot = 2000000…………………………Number of timesteps for micelle formation
int acidicstepTot = 40000…………………...…......Number of timesteps for micelle acidification
double mh = m…………………………………………………………….………Head bead mass
double mp = m………………………………………..…………..………pH-Sensitive bead mass
double mt = m………………………………………………………………….…...Tail bead mass
double md = m……………………………………………………………….........Drug bead mass
double ms = 0.5*m………………………………………………………….…..Solvent bead mass
double kT = 1.0*epsilon………………………………………………………………Temperature
double rc = pow(2.0, 1.0/6.0)*sigma…………………………….....Max distance for LJ repulsion
51
double rinf = 1.5*sigma……………………...………….........Max monomer bond length in lipid
double rclust = 2.6*sigma..........Max distance between lipid monomers to be part of same cluster
double whs = 1.65*sigma – rc…………...………..1.65*sigma is max distance for H-S attraction
double wtt = 2.6*sigma – rc………………………...2.6*sigma is max distance for T-T attraction
double wtd = 2.6*sigma - rc…………..……….…...2.6*sigma is max distance for T-D attraction
double wdd = 2.6*sigma - rc……………………….2.6*sigma is max distance for D-D attraction
double wpp = 2.6*sigma - rc……….…......2.6*sigma is max distance for P-P attraction (Neutral)
double wpt = 2.6*sigma - rc………….....2.6*sigma is max distance for P- T attraction (Neutral)
double wpd = 2.6*sigma - rc…………......2.6*sigma is max distance for P-D attraction (Neutral)
double wps = 1.65*sigma – rc...…….........1.65*sigma is max distance for P-S attraction (Acidic)
double ehh = 0.5*epsilon…….............................................Interaction energy between H-H beads
double eht = epsilon…….....................................................Interaction energy between H-T beads
double ehd = epsilon……....................................................Interaction energy between H-D beads
double ehs = 0.05*epsilon……............................................Interaction energy between H-S beads
double ett = 0.5*epsilon……................................................Interaction energy between T-T beads
double etd = 0.5*epsilon……..............................................Interaction energy between T-D beads
double ets = 2.0*epsilon……...............................................Interaction energy between T-S beads
double edd = 0.5*epsilon…….............................................Interaction energy between D-D beads
double eds = 2.0*epsilon……..............................................Interaction energy between D-S beads
double ehp……......Interaction energy between H-P beads (Neutral: epsilon, Acidic: 0.5*epsilon)
52
double epp = 0.5*epsilon……………………………..........Interaction energy between P-P beads
double ept……Interaction energy between P-T beads (Neutral: 0.5*epsilon, Acidic: 1.0*epsilon)
double epd.......Interaction energy between P-D beads (Neutral: 0.5*epsilon, Acidic: 1.0*epsilon)
double eps......Interaction energy between P-S beads (Neutral: 2.0*epsilon, Acidic: 0.05*epsilon)
double kbond = 20.0*epsilon*(1.0/(sigma*sigma))………………Monomer bond spring constant
double kbend = 2.5*epsilon*(1.0/(sigma*sigma))……......Next-nearest monomer spring constant
double cellsize = 2.6*sigma…………………………………………………………..MD cell size
int xind = 10………………………………………x,y,z-direction MD cell indices go from 0 to 9.
double mpc_cellsize = sigma………………………………………………………..MPC cell size
int mpc_xind = 26…………………...……….…x,y,z-direction MPC cell indices go from 0 to 25
double length = xind*cellsize……………………………………………….System cubic box size
double lpspacing = 0.65*sigma………….Initial conditions lattice position spacing for D beads
int lpdim = 40…………………………..Initial conditions lattice positions per edge for D beads
int lpsites = 4*pow(lpdim, 3)……….........Initial conditions total lattice position sites for D beads
double lpspacing_lip = 1.3*sigma…….……...Initial conditions lattice position spacing for lipids
int lpdim_lip = 20…………………….……..Initial conditions lattice positions per edge for lipids
int lpsites_lip = 4*pow(lpdim_lip, 3)…….....Initial conditions total lattice position sites for lipids
int headmonNum…………………………………………..…...Number of H monomers in a lipid
int pHsemonNum…………………………………………….…Number of P monomers in a lipid
int tailmonNum…………………………………………….…...Number of T monomers in a lipid
53
int monNum……………………………………………………….Number of monomers in a lipid
int branchNum[monNum]……….…...Array of number of branches a monomer in lipid chain has
int branchmonNum[monNum][monBranchNumMax]……………………………………………..
……………………………………………………Array of number of monomers each branch has
int prevbranchmonTot[monNum][monBranchNumMax]……………………………….…………
……………...Array of total number of previous branch monomers before a main chain monomer
vecmat3::Vector<int> branchtypemonTot…………………………................................................
………………………….Total number of branch monomers in a lipid for each lipid particle type
int lipidbranchmonTot = branchtypemonTot(0) + branchtypemonTot(1) + branchtypemonTot(2)
……………………………………………………..….Total number of branch monomers in lipid
int lipTot = 100………………………………...……………..……...Total number of lipid chains
int branchTot = lipidbranchmonTot*lipTot……………….…..Total number of branch monomers
int drugTot = 100……………….……………………...………………...Total number of D beads
int solvTot = 68194 - (monNum*lipTot + branchTot + drugTot)…….....Total number of S beads
int partTot = (monNum*lipTot) + branchTot + drugTot + solvTot…….…Total number of beads
int step………………………………………………………………………..…...Current timestep
int clustNum……………………………………………………………………Number of clusters
int drugIn………………………………………………...…...…Number of encapsulated D beads
int drugOut………………………………………………………………..Number of free D beads
int MPCcellNum_with_no_nonsolv…………....……..Number of MPC cells without non-S beads
54
double TIME……………………………………………………………………………..…...Time
double ekTot………………………………………………..………Total kinetic energy of system
double uTot……………………………………………………………..…...Total potential energy
double pressTot……………………………………………………………………...Total pressure
double eTot……………………………………………………………………………Total energy
Vector lin_momTot……………………………………………………......Total linear momentum
double clustLipNum_avg……………………………….…...Average number of lipids per cluster
double clustRad_avg………………………………………………………..Average cluster radius
double micRad_avg……………………………………………………......Average micelle radius
double clustDrugNum_avg……………………………..…...Average number of drugs per cluster
double drugInRad_div_clustRad_avg………………………………………………………………
……...Average ratio of encapsulated D radius to cluster radius with origin being centre of cluster
double solvNum_avg_MPCcell_with_no_nonsolv………………………………………………...
………………………………………….......Average number of S beads per MPC cell in the bulk
55
Appendix 2: classes.h
class Bead…………………………………………………………………………………………..
Class Bead is used for all beads in the system and has members for particle type, monomer, lipid,
cluster, position, velocity, acceleration, MD cell, MPC cell, grid shift position, and random MPC
velocity.
class Cell……………………………………………………………………………………………
Class Cell is used for the tridimensional cell array and this class has one member for particle list.
class MPC_Cell…………………………………………………………………………………….
Class MPC_Cell is used for the tridimensional mpc_cell array and this class members for
average solvent velocity, random solvent velocity sum, number of beads, number of solvent
beads, and whether or not it has a non-Solvent bead.
class Cluster………………………………………………………………………………………...
class Cluster is used for the lipid clusters formed during micelle formation. It has members for
indexing which cluster it is, the number of lipids it contains, the number of Drugs it contains, a
list of positions of the first Head monomer for each lipid in the cluster, the centre position of the
cluster, and the radius of the cluster.
56
Appendix 3: functions.h
void Assign_Type_Lipid_Monomer_Indices………………………………………………………
This function labels each system bead with a particle type (integers 1 to 5), each lipid particle
with a lipid index (1 to 100), and each lipid monomer with an index (1 to monNum).
void Initial_Velocities………………………………………………………………………………
This function randomly sets the initial velocities of all the system beads based on a normal
distribution with mean zero and variance kT/m. The random number generator is the Mersenne
Twister from the Boost C++ library.
void KineticEnergy_LinearMomentum……………………………………………………………..
This function calculated the total kinetic energy and total linear momentum of the system at a
given timestep,
void Lipid_Initial_Positions………………………………………………………………………..
This function sets the position of the lipid monomers randomly on an fcc cubic lattice that the
domain is divided into. It also sets the Drug position randomly if the Drug bead is bonded to the
lipid.
void NonLipid_Initial_Positions……………………………………………………………………
This function randomly sets the initial position of the Drug beads if they are not bonded to the
lipid such that there is a padding of space around lipid molecules and previously placed drug
beads. It then sets the Solvent beads randomly everywhere else under the guideline that there is a
padding of space around lipid molecules and Drug beads.
void Cluster_Drug_Analysis………………………………………………………………………
This function analyses if there are lipid clusters, how many, the radius, the number of lipids in
them, the number of Drugs in them, and the relative distance of the encapsulated Drug beads
away from the core of the respective cluster. The lipids are set in an array and each lipid is
looked at one after the other. First, the first lipid is checked to see if any other lipid is within
57
rclust distance of it. If so, those lipids are labelled as part of the same cluster. Afterwards, those
lipids that were labelled then undergo the same process themselves against unlabeled lipids. This
process continues until the lipid does not have any other lipid within rclust distance of it. This is
repeated going along the lipid array for unlabeled lipids. In the end, a list of clusters is formed
with their respective labels, lipids, Drugs, centre, and radius.
void Particle_Lists………………………………………………………………………………….
This function goes through the array of beads in the system and uses the position of the bead to
add it to the particle list for the corresponding MD cell. This particle list is later used for the
force calculation on each relevant system bead.
void MPC_Particle_Lists…………………………………………………………………………...
This function goes through the array of Solvent beads in the system and uses the position of the
bead to add it to the particle list for the corresponding MPC cell. This MPC particle list is later
used for the calculation of the new velocities for the solvent particles at the “collision” step.
void Grid_Shift_Assign_vrand……………………………………………………………………...
This functions shifts the Solvent beads by a random amount chosen from a uniform distribution
from -0.5*mpc_cellsize to 0.5*mpc_cellsize.
void Calc_vmean……………………………………………………………………………………
This function calculates the average velocity of the Solvent in each MPC cell.
void Update_Solvent_Velocities……………………………………………………………………
This function updates the velocities of the Solvent beads based on the constant temperature
version of MPC (Anderson-Thermostat).
void Return_Positions………………………………………………………………………………
This function returns the solvent beads to their original positions before the grid shift.
58
Appendix 4: forces.h
void Acceleration_uTot_FreeDrug_Neutral………………………………………………………..
For the system with the Drug not bonded to the lipid and undergoing micelle formation, this
function calculates the acceleration of each bead (excluding Solvent beads not interacting with
non-Solvent beads). It is invoked at each timestep. It first looks at the lipids and calculates the
net force from within the lipid on each monomer from LJ repulsion, bonding, and bending. It
then goes on to calculate the net force on each bead that is not a Solvent bead. Each of those
bead has its neighbor list formed by neighboring cell particle lists within a radial cutoff distance
of the MD cell size. If the particle is within this distance and comes after the current particle in
the particle array then it added to that particle’s neighbor list. The force between these two
particles is determined and added to both particles.
void Acceleration_uTot_BondedDrug_Neutral…………………………………………………….
Similar to function Acceleration_uTot_FreeDrug_Neutral, but for Drug-bonded system.
void Acceleration_uTot_FreeDrug_Acidic………………………………………………………
Similar to function Acceleration_uTot_FreeDrug_Neutral, but for acidic system. The pH-
Sensitive beads instantaneously behave as Head beads
void Acceleration_uTot_BondedDrug_Acidic……………………………………………………
Similar to function Acceleration_uTot_BondedDrug_Neutral, but for acidic system. The pH-
Sensitive beads instantaneously behave as Head beads and P-D bonds are instantaneously
broken.
59
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