MEASUREMENT OF HYDRODYNAMIC BOUNDARY CONDITIONS AND
VISCOSITY OF LIPID MEMBRANES
by
PHILIP ELIAS JAHL
A DISSERTATION
Presented to the Department of Physics
and the Division of Graduate Studies of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2021
DISSERTATION APPROVAL PAGE
Student: Philip Elias Jahl
Title: Measurement of Hydrodynamic Boundary Conditions and Viscosity of Lipid
Membranes
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Physics by:
Benjamin McMorran Chairperson
Raghuveer Parthasarathy Advisor
Jayson Paulose Core Member
Scott Hansen Institutional Representative
and
Andy Karduna Interim Vice Provost for Graduate Studies
Original approval signatures are on file with the University of Oregon Division of
Graduate Studies.
Degree awarded June 2021
ii
© 2021 Philip Elias Jahl
This work is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivs (United States) License.
iii
DISSERTATION ABSTRACT
Philip Elias Jahl
Doctor of Philosophy
Department of Physics
June 2021
Title: Measurement of Hydrodynamic Boundary Conditions and Viscosity of Lipid
Membranes
Cell membranes have a difficult role - they must be able to separate and
protect the interior of the cell from its environment, but they also must be able
to selectively allow desired materials in and waste out, while remaining flexible and
being able to grow. This range of functions is accomplished with a complicated
and dynamic mixture of lipids and proteins. While the proteins are responsible
for specific tasks, it is the lipids that determine the structure of the membrane
and their fluidity that allows the embedded proteins to arrange themselves.
Understanding lipid fluid properties is therefore important in order to have a full
picture of cellular dynamics.
We examine the interaction between a lipid bilayer and the surrounding
fluid in order to measure the boundary conditions. With an extremely precise
measurement of the diffusion of giant unilamellar vesicles we are able to use the
iv
Stokes-Einstein relation to determine that despite its fluidity, a lipid membrane
has boundary conditions that match those of a solid.
Next we assess the accuracy of a technique used to measure the viscosity
of a membrane using elliptical beads by performing it simultaneously with an
established technique using the motion of phase separated domains. After showing
its reliability we apply it to determine if there is any viscosity dependence on
lipid chain length, which would be impossible using the phase separated domain
method. We find that there is a small increase in viscosity with chain length.
We also examine the effect a recently discovered protein, β cell expansion
factor A, or BefA, has on membrane morphology. We find that it causes membrane
budding, resulting in clusters of small vesicles adhered to the outside of a larger
one.
This dissertation contains previously published and unpublished material.
v
CURRICULUM VITAE
NAME OF AUTHOR: Philip Elias Jahl
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, Oregon
Harvey Mudd College, Claremont, California
DEGREES AWARDED:
Doctor of Philosophy, Physics, 2021, University of Oregon
Bachelor of Science, Physics, 2016, Harvey Mudd College
AREAS OF SPECIAL INTEREST:
Membrane Physics
Image Analysis
PROFESSIONAL EXPERIENCE:
Graduate Research Assistant, Parthasarathy Lab, University of Oregon,
Eugene, Oregon, 2017-2021
Graduate Teaching Fellow, Department of Physics, University of Oregon,
Eugene, Oregon, 2016-2017
PUBLICATIONS:
Hill, J. H. & Massaquoi, M. S. & Sweeney, E. G. & Wall, E. & Jahl, P. E.
& Kallio, K. & Derrick, D. & Bell, R. & Murtaugh, L. C. & Parthasarathy,
R. & Remington, S. J. & Round, J. L. & Guillemin, K. (2021). Host and
microbe derived membrane permeabilization proteins increase β-cell mass
(under review)
Jahl, P. E & Parthasarathy, R. (2020). Lipid Bilayer Hydrodynamic Drag
Physical Review Research 2, 013132.
vi
ACKNOWLEDGEMENTS
None of the research in this dissertation would have been possible without
the support and guidance of my advisor, Dr. Raghuveer Parthasarathy. His
mentorship has shaped not just my work, but the type of scientist I want to
be. I would also like to thank the members of the lab who made the work fun
and collaborative, including Christopher Dudley, Ryan Baker, Savannah Logan,
Brandon Schlomann, Teddy Hay, Deepika Sundarraman, Julia Ngo, Jonah Sokoloff,
Vivek Ramakrishna, Noah Pettinari, Dylan Martins, and Vince Thoms. Thank
you to my thesis committee for their support. Thanks also to Dr. Emily Goers
Sweeney for her partnership in my work involving BefA.
Many thanks are also owed to my friends, my family, and my partner. In a
very real way this work would not have been possible without their tireless support
and encouragement.
vii
For my parents, who taught me to love the natural world and to examine it.
viii
TABLE OF CONTENTS
Chapter Page
I. LIPIDS, MEMBRANES, AND VESICLES . . . . . . . . . . . . . . . . 1
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Lipids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3. Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4. Vesicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
II. LIPID BILAYER HYDRODYNAMIC DRAG . . . . . . . . . . . . . . 12
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
III. COMPARISON OF MICRORHEOLOGICAL METHODS FOR
MEASURING LIPID MEMBRANE VISCOSITY . . . . . . . . . . . . 24
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2. Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3. Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4. Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . 30
3.5. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
ix
Chapter Page
3.6. Chain Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
IV. MEMBRANE INTERACTIONS OF THE β CELL EXPANSION
FACTOR A PROTEIN . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1. Microbiota and Model Organisms . . . . . . . . . . . . . . . . . . 43
4.2. β Cells and Bacteria . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3. BefA and Membrane Morphology . . . . . . . . . . . . . . . . . . 45
4.4. Continuing Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
V. CONCLUSIONS AND FUTURE DIRECTIONS . . . . . . . . . . . . . 49
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
x
LIST OF FIGURES
Figure Page
1.1. Bond line models of four example lipids, showing differences in head
group, saturation, and chain length. Images taken from Avanti
Lipids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2. A schematic of the phases of lipid membranes – from the top,
pseudocrystalline (LC), gel (Lβ), ripple (Pβ), liquid disordered (LD) . . . 6
1.3. A schematic of the liquid ordered phase achieved by adding cholesterol
when above the melting temperature (top) or below the melting
temperature (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4. Top: A schematic of a supported lipid bilayer and Bottom: a black
lipid membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5. A schematic of vesicle creation using electroformation. Lipids deposited
on ITO coated glass slides and a sucrose solution is placed between
them. The slides are connected to an alternating voltage source,
creating an oscillating electric field that encourages vesicle
formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1. (a) A schematic of the setup for light sheet fluorescence microscopy of
vesicle diffusion. The excitation laser is shown entering the sample
chamber from the left as it is focused into a thin sheet. Light emitted
by the sample is collected by the objective lens shown behind the
cuvette. (b) A typical light sheet fluorescence image of the central
plane of a 15 µm diameter polystyrene microsphere. (c)
Histogram of C for 29 particles, giving a mean ± standard error of C =
6.28 ± 0.15. (d) A typical light sheet fluorescence image of the central
plane of a fluorescein-dyed water droplet in benzyl alcohol. (e)
Histogram of C values for 25 droplets,
giving a mean ± standard error of C = 4.36 ± 0.29. In (b) and (d), the
dashed-dotted line indicates the theoretical value of C = 4.27 for water
in benzyl alcohol and the dashed and dotted lines respectively indicate
the theoretical values of C = 6 and C = 4 for the boundary conditions
of a rigid sphere and an inviscid (zero shear stress) sphere. . . . . . . . . 22
2.2. (a) A typical light sheet fluorescence image of the central plane of a
DOPC vesicle. (b) Mean-square-displacements from 10 randomly
xi
Figure Page
chosen vesicle trajectories (colored lines), along with their average
(dashed gray line). (c) Histogram of the reduced χ2 values
for each vesicle; χ2 = 1 indicates purely diffusive motion. (d) Histogram
of C for 26 vesicles giving a mean ± standard error of C = 5.92 ± 0.13.
The dashed and dotted lines indicate the theoretical values of C = 6
and C = 4 for the boundary conditions of a rigid sphere and an inviscid
(zero shear stress) sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1. A schematic of beads bound to a membrane. It is not obvious if the
effective radius is small, as on the left where there is a small binding
site, or large, as on the right where there is a large induced
curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2. An example image of the pole of a vesicle that has undergone phase
separation with bright liquid-disordered domains in the main
liquid-ordered phase. Imaged at room temperature, and comprised of
DPPC, DOPC,Cholesterol, DOTAP, 16:0 Biotinyl PE, and Texas Red
DHPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3. On the left, a pair of beads form a non-spherical tracer. A membrane
containing biotin binds to a neutravidin coated bead, which in turn is
bound to a biotin coated bead. On the right, a single elliptical bead
coated with neutravidin binds to the membrane, but does not form a
circular inclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4. A schematic of the device used for stretching beads, before and after
stretching, with the actual device. . . . . . . . . . . . . . . . . . . . . . 29
3.5. Translational diffusion coefficient vs. the rotational diffusion coefficient
with curves of constant viscosity based on the HPW model. Many of
the points fall outside the allowed region because the inclusions are not
rotationally symmetric. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6. Parallel diffusion coefficient vs. the rotational diffusion coefficient with
curves of constant viscosity based on the LLM model. . . . . . . . . . . 34
3.7. A histogram of the effective bead lengths L with the nominal length
shown with a dotted vertical line. There is one bead with L = 6.4 µm
which is not shown in order to provide more detail. . . . . . . . . . . . . 36
3.8. All viscosity values for the domains and beads with the mean ±
standard error plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
xii
Figure Page
3.9. A plot taken from [1] showing diffusion coefficients of lipid analogue
DiD and proteins LacY and GltT . . . . . . . . . . . . . . . . . . . . . . 38
3.10. A histogram of the effective bead lengths from all chain lengths
combined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.11. All viscosity values for the each chain length with the mean ± standard
error plotted. Fitting a line to the data has the slope 1.9 ± 1.8 × 10−10
implying a very weak, but positive affect on viscosity. . . . . . . . . . . . 41
3.12. Comparison of the viscosity at each chain length as measured with
elliptical beads (grey x’s) and viscosity calculated using lipid diffusion
values from [1] and the Saffmann-Delbrück model (red squares). All
viscosities are normalized to the 14 chain length value. Despite an
order of magnitude difference in viscosity the trend is very similar. . . . 42
4.1. An example control vesicle and multi-vesicular vesicle. Both scale bars
are 10 µm. This figure is taken from an unpublished co-authored
work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2. (a) The percentage of observed features that were multi-vesicular in the
control and 2 µM BefA. Each point is a single scan. The bars represent
the total percentage across all scans, 3% in the control and 30% in
BefA treated. (b) The concentration of features in each scan, showing
an increase with BefA exposure. This figure is taken from an
unpublished co-authored work. . . . . . . . . . . . . . . . . . . . . . . . 48
4.3. A single fluctuating vesicle, exposed to 2 µM BefA. Each image is 2
seconds apart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
xiii
CHAPTER I
LIPIDS, MEMBRANES, AND VESICLES
1.1. Introduction
The earth is home to an astounding diversity of life. It has permeated the
globe, adapting to the most extreme environments and filling the tightest niches.
This ubiquity is made possible by the four billion years of development that
has led to the complexity and variation that exists in living things. It is easy to
wonder at the millions of species, each with unique properties that allow them to
survive in disparate environments.
The complexity of living things is fascinating, but makes them a seemingly
daunting object of study for a physicist. Physics often takes a reductionist
approach, focusing not on the details of the most intricate or unique systems, but
on the uniting principles and commonalities between them. This is evident in the
hunt for fundamental particles, universal laws, or simple models. Applying the
physicist’s approach to biology is incredibly fruitful. The goal of a biophysicist is
therefore to study the fundamental aspects of living beings, because perhaps the
only things more surprising than the diversity of life are its commonalities. All
living things use DNA to store information, much of which encodes for proteins
made of the same amino acids, and whether the organism is an elephant, a
Douglas fir, or a bacterium this process occurs within a cell, one separated from
its neighbors and its environment by a lipid membrane.
These lipid membranes are one of the most important biological
materials and are the focus of my dissertation. I have focused on furthering
1
our understanding of their properties with the hope of better understanding
one of the fundamental aspects of living organisms. This understanding, while
fascinating and worthwhile in its own right, can also play an important role in
medical advances such as drug delivery – both the Moderna and Pfizer/BioNTech
vaccines for COVID-19 use lipids to encapsulate mRNA and get it inside cells.
Of particular interest in this work are the fluid properties of membranes as they
play an important role in the physical rearrangement of proteins embedded in the
membrane and therefore cellular dynamics.
This first chapter will cover some of the background of lipids and membranes
necessary to understand the work that follows. The second chapter focuses
on the hydrodynamic boundary conditions of a lipid membrane, in short, how
a membrane interacts with the fluid it exists in. I describe my experiments
characterizing the flow boundary conditions, which had never before been directly
measured. This chapter includes previously published co-authored material. I will
then discuss the fluidity of the membrane itself in chapter 3, assessing the accuracy
of a method to measure membrane viscosity before applying it to a previously
unmeasurable system. The fourth chapter discusses work done in collaboration
with Karen Guillemin’s lab examining the effect of the newly discovered bacterial
protein, BefA, on membrane topology and rigidity. This chapter contains figures
from unpublished co-authored material. The final chapter includes possible
extensions of this work and concluding remarks.
1.2. Lipids
The term lipid is a very broad one, describing a wide range of large molecules
that are amphipathic, having a hydrophilic region and a hydrophobic region.
2
These include fatty acids, sterols, detergents, and some vitamins. The lipids that
are important in this work are the lipids that self assemble into bilayers, and in
particular, phospholipids. In phospholipids, the hydrophobic part is made up of
two long hydrocarbon (acyl) chain tails that are bound to a glycerol molecule
which in turn connects them to the headgroup. This is the hydrophilic part and
contains a phosphate group, giving phospholipids their name. While the phosphate
is always present, there are many different headgroups that distinguish lipids.
Phosphatidylcholine (PC) is one of the most studied and common in membranes,
and the main headgroup used in this research. The many lipids that have PC
headgroups are distinguished by the length and saturation of their acyl chains.
[2]
Every lipid has a full chemical name that uniquely identifies it, but as these
are unwieldy it can be much clearer to use abbreviated versions. These will use the
length of the carbon chain and the number of double bonds followed by the two
letter abbreviation of the headgroup. For more information when double bonds are
present the location of the double bond is given by a ∆ position, the number of
carbons away from the headgroup that the double bond occurs. The conformation
of the carbon bonds can also be stated by including either a cis or trans. For
example, 1,2-dioleoyl-sn-glycero-3-phosphocholine can also be written as 18:1 (∆9-
Cis) PC, or even more simply DOPC. Compare this to a lipid that is identical
except there is no double bond: 1,2-distearoyl-sn-glycero-3-phosphocholine, which
can be written as 18:0 PC or DSPC. While every lipid used in this work will
have two matching acyl chains, this is by no means necessary as evidenced by 1-
palmitoyl-2-oleoyl-glycero-3-phosphocholine, or 16:0-18:1 PC, or POPC. A lipid
with identical tails to DOPC but a different headgroup is 1,2-dioleoyl-sn-glycero-
3
3-phosphoethanolamine, or 18:1 PE, or DOPE. The chemical structure of each of
these lipids is shown in Figure 1.1.
DOPE
DOPC
DSPC
POPC
FIGURE 1.1. Bond line models of four example lipids, showing differences in head
group, saturation, and chain length. Images taken from Avanti Lipids.
1.3. Membranes
Phospholipids are fairly simple molecules, but their amphipathic quality leads
to interesting and powerful behavior. When exposed to water they self assemble
into bilayers – sheets two molecules thick with the hydrophobic tails directed
inwards, protected from the water by the hydrophilic heads on the outside.
Even when only one lipid species is present, these bilayers can form several
different phases: pseudocrystalline, gel, ripple, or liquid crystal. Depending on
the chemistry of the specific lipid some phases will be unavailable, but some lipids,
like DPPC (16:0 PC) can form all of them [3]. Phase transitions can be induced
4
by changes in temperature, pressure, tension, and hydration, but because of its
comparative simplicity and more immediate applicability to most experiments, we
will only focus on temperature.
At the lowest temperature, lipids will form the pseudocrystalline LC phase
with lipids densely packed, similar to the cyrstalline structure of non-hydrated
lipids. The motion and rotation of the lipids are extremely limited in this phase.
As temperature increases, the bilayer will enter the gel phase, Lβ. In this phase,
the headgroups are less tightly packed and arrange in a distorted orthorhombic
lattice. The tails are tilted relative to the surface of the membrane. There is
slightly more lipid rotation, but it is still largely restricted. Further increasing
temperature leads to the ripple phase, Pβ. The headgroups arrange in a hexagonal
lattice and they displace vertically causing the membrane to form a “ripple”. This
allows for nearly free rotation, but they remain limited in lateral mobility. [3]
The final transition is the most significant. At high enough temperatures the
acyl chains, which were fully extended in all other phases, melt and are able to
form both cis and trans bonds. This creates the liquid crystalline (Lα) or liquid
disordered (LD) phase. The relaxed hydrocarbon chains cause the membrane
to expand laterally and thin as the lipids increase their cross sectional area and
effectively shorten their carbon chains. This greatly increases the fluidity of the
membrane, allowing for far greater lipid diffusion, orders of magnitude higher than
in the gel phase [3]. Each phase is shown in Figure 1.2.
All of these phases occur in single species lipid membranes, but additions
of other lipids can create additional phases and behaviors. The presence of
cholesterol can cause an intermediary state between the gel and liquid disordered
phases. While cholesterol itself will not form bilayers, it is nonetheless present
5
FIGURE 1.2. A schematic of the phases of lipid membranes – from the top,
pseudocrystalline (LC), gel (Lβ), ripple (Pβ), liquid disordered (LD)
6
in high concentrations in animal membranes. Cholesterol is nearly entirely
hydrophobic with only a single hydroxide group being hydrophilic. As such, it
is fully embedded in the lipid bilayer and affects the behavior of the acyl chains
above and below their melting temperature. Below the melting temperature, where
the phospholipids would normally be in the gel phase, the cholesterol increases
the spacing of the acyl chains, greatly increasing the fluidity. Above the melting
temperature, the chains appear to be more ordered [3]. This results in the liquid
ordered (LO) fluid phase with lipid diffusion slightly below that of the liquid
disordered phase [4]. This is shown in Figure 1.3.
FIGURE 1.3. A schematic of the liquid ordered phase achieved by adding
cholesterol when above the melting temperature (top) or below the melting
temperature (bottom)
These two liquid phases, ordered and disordered, can coexist, and certain
combinations of bilayer forming lipids with cholesterol will form clearly separated
domains of different phases [5]. These large, micron scale phase domains do not
occur for any single phospholipid combined with cholesterol, but will for a variety
of phospholipid pairs with cholesterol [6]. These systems are a common area of
study as a model for the somewhat controversial lipid rafts in living cells [7, 8], but
also provide a unique method in which the diffusion of the domains is measured
7
OH
OH
OH OH
OH
OH
OH
OH
OH OH
OH
OH
in order to calculate the viscosity of the membrane [9, 10]. This will be further
discussed and used in chapter 3.
1.4. Vesicles
In order to study lipid membranes, we take a reductionist approach using
purified lipids to reconstitute membranes. The complexity of a cell membrane
with proteins, cytoskeletal filaments and carbohydrates makes cellular membranes
challenging to study. Fortunately, there are many methods for creating model lipid
bilayers in vitro, allowing for control of the geometry and composition.
The different geometries each have their own advantages and uses. For
example, supported lipid bilayers (SLBs), shown in Figure 1.4(a) are formed on
a solid substrate, making them extremely stable [11] and allowing for techniques
that require proximity to a solid surface like total internal reflection fluorescence
microscopy [12]. However, this support is also its greatest limitation as it affects
the behavior of the lipids, decreasing their diffusion [13].
Another commonly used system is the black lipid membrane (BLM) [14]. In
this case, lipids dissolved in a hydrophobic solvent are painted across a circular
aperture. This results in a lipid monolayer forming on either side of the solvent,
shielding it from the water. As the solvent thins this becomes a planar bilayer that
is only supported at the edges [15], as shown in Figure 1.4(b). Because there are
large, separate aqueous solutions on either side of the bilayer, this makes it simple
to perform electrical characterization of the membrane by inserting electrodes on
either side, and it is even possible to insert ion channel proteins into the membrane
[16, 17, 18]. It is possible for the solvent used to apply the lipids to not fully leave
8
the bilayer, leaving a thin layer of solvent between the leaflets which can affect the
membrane behavior [19].
a)
b)
FIGURE 1.4. Top: A schematic of a supported lipid bilayer and Bottom: a black
lipid membrane
In order to ensure nothing altered the lipid properties, all experiments
performed in this work were performed using closed spherical bilayers, or vesicles.
Unlike the previously mentioned planar structures, the lipid bilayer of a vesicle
forms a spherical shell with fluid inside and out. This is structurally similar to
natural lipid membranes like the cell membrane or naturally produced vesicles
used for transport within a cell or between cells. They are also generally simple
to create as they will spontaneously form when dried lipids are exposed to water.
9
While there is an energy cost to bending a membrane, any edges would expose the
tails to water, so bilayers will naturally form vesicles. These can be many orders
of magnitude larger than the width of the membrane: it is easy to make vesicles
tens of microns in diameter, compared to a bilayer thickness of around 5nm. To
illustrate this difference, if a vesicle 20µm in diameter were expanded to the size of
an exercise ball, the membrane would be about as thick as a piece of paper.
While there are many methods for creating vesicles, such as various
microfluidic techniques [20, 21], or even isolating vesicles from cell cultures [22, 23],
the most common involve some variation on this spontaneous formation from
hydration. Many of the vesicles created by simply exposing dried lipids to water
will be multilamellar, with many layers inside of them. In order to achieve more
consistent unilamellar vesicles, steps can be taken after or during the hydration
process. Common techniques include sonication, placing hydrated lipids in a
sonicator [24], and extrusion, forcing hydrated lipids through membranes with
small pores [25]. Both of these result in vesicles too small for our purposes, with
diameters only getting as large as a couple hundred nanometers. Vesicles of this
size are challenging to study using traditional fluorescence microscopy as they are
smaller than the wavelength of light.
Instead we used electroformation as described by Veatch et al. [26] to
generate vesicles on the order of tens of microns. In this method, lipids of the
desired composition dissolved in chloroform are deposited onto two sheets of
indium tin oxide (ITO) coated glass and allowed to dry under vacuum. These
are sandwiched together with a silicone spacer keeping them apart and sealing
the edges. The cavity is filled with a sucrose solution and electrodes are attached
to each sheet. By varying the voltage across the ITO glass sheets we create an
10
oscillating electric field in the cavity, promoting the formation of giant unilamellar
vesicles, as shown in Figure 1.5. The details of how this process works remain
poorly understood, but its simplicity and yield have made it commonplace in
membrane biophysics research.
FIGURE 1.5. A schematic of vesicle creation using electroformation. Lipids
deposited on ITO coated glass slides and a sucrose solution is placed between
them. The slides are connected to an alternating voltage source, creating an
oscillating electric field that encourages vesicle formation.
1.5. Conclusion
In this chapter, I have attempted to provide a brief but broad background
of the chemistry and methods that underlie the work to come. I will provide
additional background specific to particular experiments in each of the subsequent
chapters. With the fundamentals of lipid chemistry, bilayer behavior, and bilayer
formation covered we can move forward to studying the physics of the interaction
of a free bilayer and the surrounding fluid environment.
11
CHAPTER II
LIPID BILAYER HYDRODYNAMIC DRAG
This chapter contains previously published co-authored material with
contributions from Raghuveer Parthasarathy. In this work I contributed to
designing the research, performing the research, analyzing the data, and writing
the paper.
2.1. Introduction
Interfaces between lipid bilayers and aqueous solutions are present in
countless environments both natural, such as at cell and organelle membranes,
and artificial, such as in suspensions of liposome-encapsulated drugs. The
hydrodynamics of bilayers and bilayer-bound objects are therefore of considerable
interest [9, 20, 27, 28, 29, 30, 31, 32]. In particular, the rheology of red blood cells
in vivo [33] and suspensions of cells and liposomes in vitro [34, 35] depends directly
on the nature of the flow boundary condition of the bilayer / water interface.
It has been widely assumed throughout work spanning many decades that this
interface is well described by the boundary condition characteristic of a rigid
solid [28, 36, 37, 38, 39, 40]. Experimental and computational studies of tracer
particles flowing inside and outside lipid vesicles are consistent with this claim
[41, 42, 43], but the complexity of the relationships between particle motions and
membrane properties prohibit straightforward readouts. However, lipid bilayers
are Newtonian fluids [9, 31, 44], and it has been speculated that their aqueous
interfaces may therefore behave more like low-shear-stress boundaries, or have
characteristics intermediate between inviscid and solid-like extremes [45]. For
12
bulk liquid droplets, intermediate shear conditions arise as internal fluid flows are
directed in the opposite direction as drag-induced surface flows. Theoretically,
therefore, an ideal spherical shell composed of an incompressible fluid would
behave similarly to a rigid solid sphere, as the fluid cannot move to the interior
space. The extent to which a real lipid vesicle behaves, in this sense, as an ideal
spherical shell is experimentally unclear. One could imagine, for example, that
differential flows between the two leaflets of the bilayer could allow relative tank-
treading motions that reduce drag from the hard-sphere value. Molecular dynamics
simulations often report finite shear at the bilayer-water interface (e.g. [29, 46]),
and also give estimates of inter-leaflet slip [46]. One could alternatively imagine
that thermally driven height fluctuations, even if too small to resolve, perceptibly
increase drag.
Remarkably, despite its fundamental importance and widespread
applicability, we are aware of almost no measurements of the hydrodynamic
boundary conditions at lipid bilayer surfaces. One study was able to use a
dynamic surface force apparatus to probe lipid bilayers supported on solid surfaces,
reporting “no-slip” boundary conditions at the bilayer-water interface, as would be
the case at a solid surface in water [47]. However, the presence of a solid support is
well known to alter membrane hydrodynamics, reducing lipid diffusion by roughly
an order of magnitude compared to free bilayers and inhibiting large-scale spatial
organization [48, 49]. To the best of our knowledge, determining the flow boundary
conditions at free lipid bilayers remains an open problem.
In principle, the interfacial boundary condition of an object can be
determined from measurements of its Brownian Motion. For a sphere of radius
13
R, the diffusion coefficient D is given by the the Stokes-Einstein relation
kBT
D = , (2.1)
CπηR
where kB is Boltzmann’s constant, T is the temperature, η is the external fluid
viscosity, and C is a dimensionless constant that characterizes the boundary
condition. It is well known that C = 6 corresponds to the limiting case of a rigid
sphere, and C = 4 corresponds to an interface with no shear stress, as is the case
for a liquid sphere of zero viscosity. More generally, for a sphere of viscosity ηinternal
in an external liquid of viscosity ηexternal, with
ηinternal
λ = , (2.2)
ηexternal
the boundary constant is [50, 51, 52]:
3λ+ 2
C = 4 (2.3)
2λ+ 2
We note that determining C by measuring D does not provide insight into
local flows that may occur at length-scales much smaller than the vesicle size or
time-scales much shorter than attainable measurement times. Rather, it provides a
characterization of the average, effective hydrodynamic properties of the vesicle.
In practice, determining C by measuring D is non-trivial due to potential
hydrodynamic influence from nearby surfaces such as container walls, and the
requirement of high-precision determination of the object’s positions and radius.
We surmount these challenges by applying light sheet fluorescence
microscopy [53, 54, 55, 56] together with fast, accurate tracking techniques [57]
14
to characterize the diffusive motion of spherical phospholipid vesicles. Light sheet
fluorescence microscopy provides optical sectioning of three-dimensional samples,
enabling the imaging of vesicles hundreds of microns (tens of vesicle diameters)
away from the walls of the imaging chamber [54]. We verified our methodology
by characterizing the diffusive motion of solid microspheres in an aqueous medium
and water droplets in benzyl alcohol, as detailed below, which gives C = 6.28±0.15
and C = 4.36 ± 0.28, respectively, consistent with theoretical expectations. We are
therefore able to accurately measure the entire range of boundary behaviors. We
then characterized lipid vesicles composed primarily of the common phospholipid
DOPC (1,2-dioleoyl-sn-glycero-3-phosphocholine), determining that C = 5.92±0.13
(stat.) ± 0.16 (syst.). This establishes that the bilayer/water interface is well
described by the interfacial shear stress condition of a solid object.
2.2. Methods
We performed light sheet fluorescence microscopy using a home-built
instrument that closely follows the design of Keller et al. [56] and is described
in detail in [55]. In brief, excitation light was provided by a 488 nm laser with
an output power of 50 mW, which was scanned by a galvanometer mirror and
focused by an objective lens to form a sheet in the sample chamber. The minimum
thickness of the sheet was ≈ 3 µm, extending over a lateral extent (Rayleigh
length) of ≈ 100 µm. Images were captured through a 40× 1.0 NA Plan-apo
objective lens (Zeiss) perpendicular to the excitation plane and recorded with a
5.5 Mpixel sCMOS camera (pco.edge, Cooke Corp.). A schematic of the setup is
shown in Figure 2.1(a).
15
Suspensions of either lipid vesicles or polystyrene microspheres in 0.1 Molar
sucrose, or deionized water droplets in benzyl alcohol, were placed in a square
cross-section glass cuvette (Starna Cells, part number 3-3.45-SOG-3), which was
mounted to a movable translation stage and inserted in the light sheet microscopy
chamber. The distance between imaged objects and the cuvette walls was several
hundred micrometers.
For beads, droplets, and vesicles the optical plane intersecting the sphere
center (the “equatorial” plane) was readily evident due to a lack of out-of-focus
light outside the bright ring or disk, due to the few micron sheet thickness. This is
illustrated in Supplemental Movie 1, which shows a three-dimensional scan through
a lipid vesicle.
To assess the accuracy of our methods, we examined diffusion of objects with
well known flow boundary conditions: solid polystyrene microspheres and deionized
water droplets suspended in benzyl alcohol.
The microspheres were FITC-labeled polystyrene beads of nominal diameter
15.45 ± 0.70 µm (mean ± standard deviation; Bangs Laboratory, part number
FSDG009). Light sheet fluorescence images were captured for durations of 15
seconds at 33.33 frames per second. A typical image is shown in Figure 2.1(b).
For the first image, we determine the particle center using the radial symmetry-
based algorithm described in [57], which provides rapid localization with accuracy
close to theoretical limits. In brief, the center is calculated as the point that
minimizes total distance to lines derived from intensity gradients throughout the
image. The original algorithm was modified to only weight intensity gradients
from the vicinity of the bead, limiting the effects of noise outside the particle,
such as fluorescent debris or light from other beads. Each remaining image was
16
cross correlated with the previous image and the original radial symmetry-based
algorithm was applied to the cross-correlation to determine the shift between each
frame. From the particle positions, we determine the diffusion coefficient D using
the covariance based estimator described by Vestergaard et. al. [58]. Not only
does this provide greater accuracy than, for example, linear fits of mean-squared-
displacements to time intervals, but it also provides robust estimates of localization
accuracy and goodness-of-fit to a random walk model [58] that we make use of
in assessing vesicle data below. We experimentally determined the radius of each
bead by Hough transformation following edge detection [59], which produces from
each input image a series of output images corresponding to each possible radius
candidate, with the true object radius giving a bright, compact transform image.
Across all beads, this gives an average diameter of 15.06 ± 0.41 µm (mean ±
standard deviation), consistent with the nominal value of 15.45 ± 0.70 µm. Using
the measured radius, the literature value for the viscosity of deionized water, and
the ambient temperature T = 293 K, we use Eq. 2.1 to determine C for each
microsphere. We show the histogram of C values for N = 29 microspheres in
Figure 2.1(b). The mean ± standard error is C = 6.28 ± 0.15, slightly higher than
but consistent with the expected C = 6 for solid particles. (Using the nominal
rather than the measured microsphere radius gives C = 6.12, matching the
theoretical value within uncertainties. The microsphere edges are less well-defined
than those of the vesicles; in the latter case, we note explicitly the uncertainty in
R below.)
The assessment of liquid droplets is similar. The droplets were deionized
water dyed with 75 mg/ml fluorescein, suspended in benzyl alcohol, formed into
an emulsion by vigorous shaking. Fluorescein has very low solubility in water, but
17
nonetheless preferentially labels the aqueous phase. Benzyl alcohol was chosen
because its density, 1.045 g/ml, is similar to that of water, limiting gravity-induced
drift. The viscosity of benzyl alcohol is η = 6.29 × 10−3 Pa·s [60], and so the
expected C = 4.27. The droplets were imaged for 60 seconds at 8.33 frames per
second. A typical image is shown in Figure 2.1(c). Using the same procedures
described above for center positions and radii, we determined C for each water
droplet. Figure 2.1(d) shows the histogram of C values for N = 25 droplets.
The mean ± standard error is C = 4.36 ± 0.29, consistent with the expected
value. Notably, C increased with time for these droplets, likely due to adsorption
of fluorescein to the boundary as the poorly soluble dye lowers the oil/water
interfacial energy. With sufficient adsorption, the droplet properties are those of an
oil/fluorescein rather than an oil/water interface and may reflect possible ordering
of a fluorescein monolayer or pure monolayer hydrodynamics. The C value stated
was determined from data within 20 minutes of emulsion preparation.
The lipid vesicles we examined were composed of 94% DOPC (1,2-dioleoyl-
sn-glycero-3-phosphocholine), and 6% NBD-PE (1,2-dipalmitoyl-sn-glyercero-3-
phophoethanolamine-N-(7-nitro-2-1,3-benzoxadiazol-4-yl) (Avanti Polar Lipids).
Phosphatidylcholine lipids are a major constituent of cellular membranes. The
headgroup-conjugated NBD probe has an isotropic orientation relative to the
lipid bilayer plane (unlike, for example, probes such as Texas Red [61]), and so
provides vesicle images of symmetric fluorescence under polarized laser excitation.
The vesicles had radii between 3.1 and 25.7 µm with a mean of 12.2 µm and a
standard deviation of 4.0 µm. Vesicles were imaged for 15 seconds at 33.33 frames
per second. The vesicles were created by electroformation, as in [26]. In brief, the
desired lipids were dried on glass slides with an indium tin oxide coating, hydrated
18
with a 0.1 M sucrose solution, and subjected to an oscillating electric field to
stimulate vesicle formation. Vesicles were added to a sample cuvette containing
0.1 M sucrose so that the interior and exterior of the vesicles would be matched
in density and osmolarity. We use the literature value for the viscosity of 0.1 M
sucrose, η = 1.095× 10−3 Pa·s [62], in our analyses. Low levels of drift were present
in the experiments, possibly due to convection and imperfect density matching.
We therefore subtracted the best-fit linear trajectory (i.e. constant velocity) from
each vesicle trajectory, and used only the horizontal component of trajectories.
Rare frame-to-frame displacements more than three standard deviations from the
mean (< 0.5% of frames) appeared to indicate large-scale instrument vibrations,
and trajectories were analyzed piecewise around such points.
We assessed the accuracy of center- and radius-finding algorithms for vesicle
images by applying them to simulated images of bright rings with a range of radii
and signal-to-noise ratios (SNRs), mimicking the form of the vesicle images, as
in Ref. [54] and similar to the ring images in Ref. [57]. In brief, a high-resolution
image of a thin annulus was convolved with the detection point-spread function
(PSF), pixelated, and subjected to Poisson-distributed noise. We used the
theoretical PSF of the emission wavelength and numerical aperture, which has a
full-width at half-maximum of 0.19 µ m. Because of the occasional presence of
lipid matter in and around the vesicle edge, the radial center-finding algorithm is
weighted with a hyperbolic tangent function centered around the ring of the vesicle
so as to only use the intensity gradient from the edge of the vesicle. From analysis
of simulated images, the radial-symmetry-based localization gives an estimated
localization error of ≈ 3 nm. Independently, the localization error estimated from
vesicle trajectories by the method of Vestergaard et al.[58] is ≈ 10 nm. The vesicle
19
radius is determined by Hough transformation, as in the bead and droplet image
analysis. The estimated uncertainty in R from the standard deviation of simulated
images at the appropriate SNR is ≈ 0.005 µm. More significant, however, is the
standard deviation in R over the course of an image series, due for example to
changes in position relative to the sheet plane. This is ≈ 0.03 µm, which is small
compared to the typical 10 µm vesicle radii and which contributes negligibly to
the overall uncertainty in C. From vesicle position data, we calculated D and C as
described above.
2.3. Results
Light sheet fluorescence microscopy provides clear images of lipid vesicles.
A typical example is shown in Figure 2.2(a). Assessment of images as described
above provides vesicle positions, radii, diffusion coefficients (D), and the flow
boundary constant (C), along with their uncertainties. We provide the complete
set of positions, radii, diffusion coefficients, and boundary coefficient values for
every microsphere, water droplet, and lipid vesicle examined as Supplemental File.
In Figure 2.2(b) we show ten randomly chosen examples of vesicle mean-
squared-displacement(MSD) (colored lines) as a function of lag time together with
their average (dashed gray line). The linearity of the MSD curve is indicative of
free diffusion. A more robust assessment of the Brownian character of vesicle
motion comes from the goodness of fit calculation provided by the covariance
based estimator of D, which gives a reduced χ2 value whose value should be ≈ 1
for a model of pure diffusion. A histogram of the measured χ2 values is shown in
Figure 2.2(c), and is consistent with simple Brownian diffusion.
20
As described in Methods, we use measurements of vesicle radii and diffusion
coefficients to determine the flow boundary condition constant C. The histogram
of C for N = 26 vesicles is shown in Figure 2.2(d). The mean ± standard error
is C = 5.92 ± 0.13. To estimate possible systematic error, we assume that the
polystyrene microspheres for which we calculated C (see Methods) are ideal hard
spheres. The standard deviation of the beads’ C value would therefore be the
spread inherent in our methodology. In order to account for this uncertainty we
can compute a systematic standard error by dividing the standard deviation of the
beads’ C by the square root of the number of vesicles. This gives us a constant for
vesicles of C = 5.92 ± 0.13 (stat.) ± 0.16 (syst.).
Our measurements show that at least over micrometer length scales and
millisecond-to-second timescales, lipid bilayers are well described by the shear
stress boundary conditions that characterize solid surfaces in fluid. Perhaps
reassuringly, the standard assumption that is ubiquitous in treatments of liposome
hydrodynamics is well supported.
Our method illustrates that a conceptually simple imaging-based approach
can provide precision measurements of microscale fluid properties. We expect
that this can be extended to, for example, fluctuating membranes driven by
either temperature [54, 63] or active forces [64] to investigate couplings between
topography and drag. Put differently, the precision with which the value C
= 6 can be determined, even if one views the value itself as an unsurprising
confirmation of expectations, illuminates the precision that future studies of
more exotic membrane systems can achieve. Finally, we note that the methods
presented here will also be applicable in non-Newtonian fluids, for which detailed
understanding of microscale rheology continues to be an active area of study.
21
(a)
(b) (c)
5 μm
(d) (e)
15 μm
5 μm
FIGURE 2.1. (a) A schematic of the setup for light sheet fluorescence microscopy
of vesicle diffusion. The excitation laser is shown entering the sample chamber
from the left as it is focused into a thin sheet. Light emitted by the sample is
collected by the objective lens shown behind the cuvette. (b) A typical light
sheet fluorescence image of the central plane of a 15 µm diameter polystyrene
microsphere. (c) Histogram of C for 29 particles, giving a mean ± standard error
of C = 6.28 ± 0.15. (d) A typical light sheet fluorescence image of the central plane
of a fluorescein-dyed water droplet in benzyl alcohol. (e) Histogram of C values for
25 droplets, giving a mean ± standard error of C = 4.36± 0.29. In (b) and (d), the
dashed-dotted line indicates the theoretical value of C = 4.27 for water in benzyl
alcohol and the dashed and dotted lines respectively indicate the theoretical values
of C = 6 and C = 4 for the boundary conditions of a rigid sphere and an inviscid
(zero shear stress) sphere. 22
(a) (b)
10 uμm
(c) (d)
(d)
FIGURE 2.2. (a) A typical light sheet fluorescence image of the central plane
of a DOPC vesicle. (b) Mean-square-displacements from 10 randomly chosen
vesicle trajectories (colored lines), along with their average (dashed gray line).
(c) Histogram of the reduced χ2 values for each vesicle; χ2 = 1 indicates purely
diffusive motion. (d) Histogram of C for 26 vesicles giving a mean ± standard
error of C = 5.92 ± 0.13. The dashed and dotted lines indicate the theoretical
values of C = 6 and C = 4 for the boundary conditions of a rigid sphere and an
inviscid (zero shear stress) sphere.
23
CHAPTER III
COMPARISON OF MICRORHEOLOGICAL METHODS FOR
MEASURING LIPID MEMBRANE VISCOSITY
3.1. Introduction
Cellular function is heavily dependent on the fluidity of lipid membranes,
allowing proteins, macromolecules, lipid domains, and the lipids to self-organize
into assemblies [9, 10, 65, 66, 67, 68]. The viscosity of the membrane sets the
timescale at which these dynamics occur and so an accurate understanding is
required to understand and predict their behavior. But despite measurements
of lipid and protein diffusion being made regularly, it is not simple to use these
to determine the viscosity. Part of the difficulty stems from the effect that the
diffusing object has on the membrane. The effective radius of a protein or other
tracer particle embedded in or attached to the membrane is not clear as it could be
defined by a small binding site or by a large induced curvature, as shown in Figure
3.1.
FIGURE 3.1. A schematic of beads bound to a membrane. It is not obvious if the
effective radius is small, as on the left where there is a small binding site, or large,
as on the right where there is a large induced curvature.
24
Methods exist to address this issue. As in [9, 31, 32, 41, 44] phase separated
domains can be used as tracers, as shown in Figure 3.2. Because these regions
of distinct lipid phase are a part of the membrane itself rather than affixed to it,
the radius of the domain can be taken as the effective radius. Alternatively, as in
[30], elliptical tracer particles can be used. While the effective radius is unknown,
by using both translational and rotational diffusion coefficients the viscosity, η,
and radius can both be determined, as detailed below. This technique has the
advantage of being applicable to a wide range of lipid systems, only requiring a
low concentration of lipids that the particles are able to bind to, while using phase
separated domains requires multiple lipid species and specific compositions that
undergo phase separation.
Measurements of viscosity of phase separated vesicles have been performed
by several groups and returned consistent values. For example, work done in our
lab found η = 0.75 ± 0.15 × 10−9 Pa · s · m for the DOPC enriched liquid-
disordered phase of phase separated vesicles [31], in close agreement with other
labs’ measurements [41, 44]. However, this is more than an order of magnitude
smaller than the viscosity measured using non-spherical tracer particles for nearly
pure DOPC membranes, η = 15.9 ± 2.3 × 10−9 Pa · s · m, the only study to date
using non-spherical tracers. This is a large discrepancy, even when considering the
other lipids present in the LD phase. It is most likely caused by residual solvent
left between the leaflets of the black lipid membrane. It should be noted that the
non-spherical tracers used in [31] were pairs of beads, one coated with neutravidin
and the other with biotin. The lipid membrane included biotinylated lipids so that
only the neutravidin coated bead would bind to the membrane. When imaging
a pair of bound beads would appear as a single extended body enabling the
25
10μm
FIGURE 3.2. An example image of the pole of a vesicle that has undergone
phase separation with bright liquid-disordered domains in the main liquid-ordered
phase. Imaged at room temperature, and comprised of DPPC, DOPC,Cholesterol,
DOTAP, 16:0 Biotinyl PE, and Texas Red DHPE
measurement of the rotational motion while behaving as a rotationally symmetric
circular inclusion. In order to simplify the binding and preparation process, we use
elliptical beads as the non-spherical tracers. These are not necessarily rotationally
symmetric circular inclusions and so will require a different analysis, described
below. A schematic of both methods is shown in Figure 3.3.
To resolve the discrepancy and assess the accuracy of the determination
of membrane viscosity using non-spherical tracers, we apply the method on
phase separated vesicles, measuring viscosity from elliptical beads and domains
simultaneously, the first direct comparison of the methods.
26
FIGURE 3.3. On the left, a pair of beads form a non-spherical tracer. A
membrane containing biotin binds to a neutravidin coated bead, which in turn
is bound to a biotin coated bead. On the right, a single elliptical bead coated with
neutravidin binds to the membrane, but does not form a circular inclusion.
3.2. Experimental Methods
A lipid concentration to form phase separated vesicles was chosen based
on [6], but with modifications made in order ensure that beads bind. We used
35.5% DPPC (1,2-dipalmitoyl-sn-glycero-3-phospocholine), 15.5% DOPC
(1,2-dioleoyl-sn-glycero-3-phosphocholine), 40% Cholesterol, 8% DOTAP
(1,2-dioleoyl-3-trimethylammonium-propane (chloride salt)), 0.5% 16:0
Biotinyl PE (1,2-dipalmitoyl-sn-glycero-3-phosphoethanolamine-N-(biotinyl)
(sodium salt)), and 0.5% Texas Red DHPE (1,2-Dihexadecanoyl-sn-glycero-3-
Phosphoethanolamine, Triethylammonium Salt). The Texas Red DHPE was
purchased from ThermoFisher Scientific and all other lipids were purchased from
Avanti Polar Lipids. This concentration of lipids will produce liquid-disordered
(LD) domains in a liquid-ordered (LO) main phase. The fluorescently tagged Texas
Red DHPE preferentially partitions into the LD phase resulting in bright domains.
The streptavidin coated beads strongly bind to the Biotinylated PE lipids. All
vesicles were created using electroformation, as described in [26]. The lipids of
27
interest were deposited on indium tin oxide coated glass slides and dried before
hydrating in 0.1M sucrose in an oscillating electric field.
Elliptical polystyrene beads were formed based on the method described
in [69]. In short, 1 µm spherical polystyrene beads (Invitrogen catalog number
F8776) were placed in a solution of 6.2% polyvinyl alcohol and 2.5% glycerol
by mass in deionized water which was allowed to evaporate, leaving the beads
embedded in a thin, flexible sheet. This sheet was placed in a mechanical
stretching device as shown in Figure 3.4. But before extending the device, the
film was submerged overnight in toluene, which dissolves the polystyrene beads
without affecting the sheet. The sheet is then stretched and removed from the
toluene allowing the particles to solidify in the shape of the now deformed cavities.
Once the beads have solidified in the stretched film, the film can be dissolved in
water and the beads are isolated with a centrifuge. Prior to use, the beads are
soaked overnight in a 0.01 g/ml streptavidin solution in order to ensure binding to
the biotinylated lipids in the vesicles. The beads are centrifuged and removed from
the streptavidin solution and sonicated to break up any clumps. They are added to
vesicles suspended in sucrose an hour before imaging to allow time for binding.
All images were recorded at 20 frames per second on a Nikon TE2000
inverted fluorescence microscope with a 60× oil immersion objective and a
Hamamatsu Orca CCD camera at room temperature. A Cairn OptoSplit was used
to simultaneously record two color channels, one for the domains and one for the
beads.
28
FIGURE 3.4. A schematic of the device used for stretching beads, before and after
stretching, with the actual device.
3.3. Image Analysis
The analysis of phase separated domains follows the method described in
[31]. Domains were identified with intensity thresholding and their centers were
found by fitting Gaussian profiles using maximum likelihood estimation. The
boundaries of domains were found with a bilateral filter and the area of each
domain was used to determine the radius. Domain positions across frames were
assigned to tracks by performing a nearest neighbor linking, only keeping time
series for domains that were tracked continuously for at least 100 frames to ensure
sufficient statistics to accurately calculate their diffusion coefficients.
29
Beads were detected by identifying the brightest single object in each frame
and the centers were found using the radial symmetry-based algorithm described
in [57], a rapid localization method with accuracy close to theoretical limits. The
center is the point that minimizes the total distance from lines along intensity
gradients throughout the image. The angle of the bead is found using the center
as the starting point for fitting an ellipse to the image by determining the ellipsoid
that matches the covariance matrix of the intensity.
The accuracy of this localization method was determined by applying it to
simulated images of bright ellipses that mimic the size and signal-to-noise ratio of
our data. A high-resolution image of an ellipse was convolved with the detection
point-spread function based on the emission wavelength and numerical aperture,
pixelated, and subjected to Poisson distributed noise. The center is found with
a variance of ≈35 nm, well below the pixel size of 110 nm. The variance of the
bead’s angle was ≈0.25◦.
Diffusion coefficients for domains and beads were determined using the
covariance based method described by Vestergaard et al. [58]. This method
provides greater accuracy than linear fits of mean-square displacements and
provides estimates of localization accuracy and goodness of fit to a random walk.
3.4. Mathematical Analysis
The membranes formed by lipids can be many orders of magnitude larger
than the thickness of the membrane, which is on the order of 5nm. This allows
the membrane to be treated as a two dimensional fluid where the individual lipid
molecules freely flow and diffuse within the membrane, but are confined to the
thin sheet. This forms the basis of the models we use of membrane fluidity. In
30
1975 Saffman and Delbrück put forward a model for the diffusion of a cylinder in
a two dimensional fluid bounded on either side by a bulk fluid [70]. They found
that by accounting for the behavior of the membrane and the surrounding fluid the
translational diffusion is given by
( )
kBT η
DT = log − γ (3.1)
4πη ηwa
and the rotational diffusion is given by
kBT
DR = (3.2)
4πηa2
where η is the two dimensional viscosity of the membrane, ηw is the viscosity of
the surrounding fluid, a is the radius of the cylinder, and γ is Euler’s constant.
However, this method can only be applied only when the useful dimensionless
quantity  = 2ηwa  1, i.e. systems where the membrane inclusion is small and the
η
viscosity is large. This holds in lipid membranes for nano-scale objects, but will
fail for micrometer scale lipid domains or particles. Hughes, Pailthorpe, and White
(HPW) extended this model to arbitrary  [71]. The HPW model is substantially
more complicated and cannot be expressed in simple closed-form equations. Petrov
and Schwille provide a simple analytical approximation that fits the full HPW
model across all  [44]:
[ ( ) ( ) ( )]
kBT 2 4 
2 2
DT = [ln ( )− γ(+ − ln4πη ) π) 2  ]−1 (3.3)3
1 −  2ln + β(, bT1, bT2, cT1, cT2)
π 
31
is the translational diffusion, where
cT1
bT1
β(, bT1, bT2, cT1, cT2) = (3.4)
1 + c b 2T2 T
and bT1 = 2.74819, bT2 = 0.51465, cT1 = 0.73761, and cT2 = 0.52119. The
rotational diffusion is given by
2 [ ](2η ) 3 −1w
D = 2
4
R + + β(, bR1, bR2, cR1, cR2) (3.5)
4πη3 3π
where β is the same function as in equation 3.4, but the constant values are given
by bR1 = 2.91587, bR2 = 0.68319, cR1 = 0.31943, and cR2 = 0.60737.
The full HPW model and the Petrov-Schwille approximation have been
shown to apply to phase separated domains [31, 44]. It should be noted that
this model is for the diffusion of a circular inclusion in the membrane. As such,
it is not clear that it can be applied to the motion of the elliptical beads. An
elliptical bead may induce a roughly circular curvature or be bound to the
membrane in only a small region, but there is no guarantee it will bind in a
spherically symmetric way. If we plot curves of constant viscosity in rotational
and translational diffusion coefficient space, as in Figure 3.5, we see that the
low viscosities asymptote. Several of the measured translational and rotational
diffusion coefficients of the elliptical beads fall beyond this asymptote in a
disallowed region that does not correspond to any viscosity. In order to describe
the motion of our elliptical beads we use the work of Levine, Liverpool, and
MacKintosh (LLM) on the motion of extended bodies in membranes [72]. They
give the rotational diffusion
1 kBT
DR = (3.6)
cR(λ) 4πηL2
32
FIGURE 3.5. Translational diffusion coefficient vs. the rotational diffusion
coefficient with curves of constant viscosity based on the HPW model. Many of the
points fall outside the allowed region because the inclusions are not rotationally
symmetric.
where L is the length of the rod and cR is a function that depends on λ =
2ηwL .
η
In this case, λ is a dimensionless variable analogous to the  used in the SD and
HPW models. Because the translational diffusion of an elliptical object will
depend on its orientation, Levine provides expressions for the diffusion along the
axes parallel and perpendicular to the extended body’s long axis. These are
1 kBT
D⊥ = (3.7)
c⊥(λ) 4πη
1 kBT
D‖ = (3.8)
c‖(λ) 4πη
33
The functions that define the coefficients in each of these equations are taken from
[72], but they are not provided explicitly. We fit a curve to the plots in order to
solve for viscosity and effective bead length analytically. These fits were compared
to values provided by the authors to ensure accuracy. Our data falls into the
allowed regime of this theory, shown in Figure 3.6.
FIGURE 3.6. Parallel diffusion coefficient vs. the rotational diffusion coefficient
with curves of constant viscosity based on the LLM model.
3.5. Results
We first assess the viscosity of phase separated vesicles to allow a comparison
of the methods. The viscosity was calculated from the diffusion of phase separated
domains using their translational diffusion, radius as determined from images,
34
and the Petrov-Schwille approximation of the HPW model. Each vesicle had
multiple domains, ranging from 2 to 18, which were fit to a single viscosity value
by calculating translational diffusion coefficients for a wide range of viscosity
values and minimizing the mean square distance from the measured translational
diffusion coefficients. For N = 13 vesicles this resulted in an average ± standard
error of ηdomain = 3.0±0.4×10−9 Pa ·s ·m. Uncertainties for individual vesicles were
about 30% of the value, but decreased for vesicles with more tracked domains.
Beads were analyzed using both the rotational diffusion coefficient and the
parallel diffusion coefficient with the LLM equations used to numerically solve
for both the viscosity and the effective length of the beads L. It is possible to use
either the parallel diffusion coefficient, the perpendicular diffusion coefficient, or to
use their ratio. The difference between the parallel and perpendicular diffusion
coefficients, both in our data and the theoretical expectation, is small and so
attempting to use the ratio is dominated by noise. We chose to use the parallel
diffusion coefficient as it was less noisy than the perpendicular. When beads were
added in high concentrations issues arose with beads clumping together and so
we used vesicles that had a single bound bead. For N = 11 vesicles the average
± standard error was ηbead = 3.6 ± 0.4 × 10−9 Pa · s · m. The L values had an
average ± standard error of 3.1 ± 0.3 µm. This average matches the observed
length of the beads, but it is important to fit L rather than assuming that it is
the observed length of the bead because the effective length can vary greatly from
much smaller than the bead, due to a small binding site, to much larger than the
bead, due to an induced curvature in the membrane. Forcing the effective length
to be equal to the average value results in a higher viscosity value with a larger
spread: 4.6 ± 0.7 × 10−9 Pa · s · m. Furthermore, two of the vesicles used were
35
bound to “double” beads – beads that became stuck together end to end, forming
one long bead. One had a calculated length of L = 3.2 µm, much smaller than the
measured length of 5.3 µm, but the other had a calculated length of L = 6.4 µm,
slightly larger than the measured length of 4.6 µm, and well outside the range of
the “single” beads. In Figure 3.7 the histogram of L values excludes this double
bead to provide more detail of the other values.
FIGURE 3.7. A histogram of the effective bead lengths L with the nominal length
shown with a dotted vertical line. There is one bead with L = 6.4 µm which is not
shown in order to provide more detail.
The close agreement of these viscosities, within their uncertainties, as
well as both of their agreement with previously measured values for this phase
36
FIGURE 3.8. All viscosity values for the domains and beads with the mean ±
standard error plotted.
(η = 3.9 ± 0.4 × 10−9 Pa · s · m from [31] and similar in [9, 41, 44]), suggests
that viscosity determination using elliptical beads provides accurate measures of
membrane viscosity, allowing us to apply the method to other systems.
3.6. Chain Length
The simplest lipid membrane is one composed of a single type of lipid. The
dependence of its viscosity on the chain length of the lipids is of course impossible
to study using phase separated domains as it would require phase separation into
37
different compositions to occur. Using elliptical beads to measure the viscosity,
this becomes a feasible experiment.
Previously, the effect of lipid chain length on membrane diffusion has been
studied. In general, as chain length increases there is a slight decrease in diffusion
coefficient [73, 74, 75] as shown in Figure 3.9 taken from [1], but this does not
necessarily imply an increase in viscosity.
FIGURE 3.9. A plot taken from [1] showing diffusion coefficients of lipid analogue
DiD and proteins LacY and GltT
We examined monounsaturated PC lipids with chain lengths of 14, 16,
18, and 20 carbons. These were 1,2-dimyristoleoyl-sn-glycero-3-phosphocholine,
38
1,2-dipalmitoleoyl-sn-glycero-3-phosphocholine, 1,2-dioleoyl-sn-glycero-3-
phosphocholine, and 1,2-dieicosenoyl-sn-glycero-3-phosphocholine respectively,
but we will refer to them for simplicity solely by their chain lengths. The double
bonds of 14, 16, and 18 all occur at the ∆9 position whereas the double bond of 20
occurs at the ∆11 position. We expect this difference to have no significant effect
on the behavior.
Vesicle composition was 91% PC lipid, 8% DOTAP, 0.5% 16:0 Biotinyl PE,
and 0.5% Texas Red DHPE. They were formed and analyzed in the same manner
as the phase separated vesicles. The bead lengths across chain lengths had an
average ± standard error of 3.7 ± 0.2 µm, slightly larger than the nominal value
of 3.1 µm. The L values also had a larger spread than the phase separated data,
ranging from 2 µm to 6.5 µm, as shown in the histogram in Figure 3.10. The
mean ± standard error for each chain length is
η14 = 2.3 ± 0.6 × 10−9 Pa · s · m
η −916 = 2.5 ± 0.4 × 10 Pa · s · m
η −918 = 3.6 ± 0.8 × 10 Pa · s · m
η20 = 3.2 ± 0.9 × 10−9 Pa · s · m
If we fit a line to these values, as shown in Figure 3.11, it has a slope of 1.9 ± 1.8 ×
10−10, implying that while very weak, increasing chain length does have a positive
effect on viscosity. Interestingly, if we compare this to the lipid diffusion values
taken from [1], finding the viscosity using the Saffmann-Delbrück model, we find
that the trend is very similar, shown in Figure 3.12. This similarity is surprising
because the microscale viscosity experienced by the lipids themselves is quite
39
FIGURE 3.10. A histogram of the effective bead lengths from all chain lengths
combined.
different from the macroscale viscosity of larger objects in the membrane. The
values we find are an order of magnitude smaller - ranging from 1.2×10−10 Pa ·s ·m
to 2.5×10−10 Pa·s·m. So despite the large difference in values, increasing the chain
length of lipids appears to have a similar effect on both macroscale and microscale
viscosity.
3.7. Conclusion
This work allows us to assert that elliptical beads can provide an accurate
measure of the viscosity of a lipid membrane. This opens the door to measuring
40
FIGURE 3.11. All viscosity values for the each chain length with the mean ±
standard error plotted. Fitting a line to the data has the slope 1.9 ± 1.8 × 10−10
implying a very weak, but positive affect on viscosity.
viscosity of a wide range of systems that previously would have been unreachable.
To demonstrate this, we considered the effect of lipid chain length, which we
found to be weakly positive. The importance of lipid viscosity and the scarcity of
direct measurements make this a rich area of study. Without the reliance on phase
separation, viscosity measurements are possible with any bilayer forming lipids at
nearly pure concentrations and any combination of lipids. It is also a conceptually
simple extension to study the effect of various proteins on lipid viscosity.
41
FIGURE 3.12. Comparison of the viscosity at each chain length as measured with
elliptical beads (grey x’s) and viscosity calculated using lipid diffusion values from
[1] and the Saffmann-Delbrück model (red squares). All viscosities are normalized
to the 14 chain length value. Despite an order of magnitude difference in viscosity
the trend is very similar.
42
CHAPTER IV
MEMBRANE INTERACTIONS OF THE β CELL EXPANSION
FACTOR A PROTEIN
This chapter contains unpublished co-authored material with contributions
from Raghuveer Parthasarathy, Emily Sweeney, Karen Guillemin, Jennifer Hill,
Michelle Sconce Massaquoi, Elena Wall, Karen Kallio, Daniel Derrick, Rickesha
Bell, L. Charles Murtaugh, S. James Remington, and June Round. In this work, I
contributed to designing the research, performing the research, analyzing the data,
and writing the paper.
4.1. Microbiota and Model Organisms
In this chapter, I will discuss work done with the Guillemin lab involving a
bacterial protein they discovered that affects membranes. While I will focus on
the work I have performed, understanding the motivation and importance of the
work requires further biological background about the microbiota and the way it is
studied.
Since their discovery, microbes have largely been portrayed as villains –
bringers of disease and death. But this ignores the millions of bacteria that call
us home, living on and in our bodies. And this is not just a peaceful coexistence,
as our understanding of the human microbiota has increased it has become clear
that it plays a crucial role in our health [76, 77].
This has led to greatly increasing interest in microbiota, but it is difficult to
study. Much of the work on the gut microbiome has been done using fecal samples
[78, 79] which provide a snapshot of the relative abundance of bacteria present
43
at a particular time. This cannot provide insight into the temporal dynamics or
spatial structure of the bacteria. And while understanding how the entire bacterial
community affects overall health is important, the vastly complicated system
makes it difficult to learn about specific interactions and behaviors.
Much as I have used model membranes to study aspects of complicated
biological systems, model organisms can be used to study the microbiota. A
successful model organism will be easy to breed and maintain while retaining
a genetic similarity to humans. Each commonly used model organism has
trade offs: C. elegans and Drosophilia melanogaster reproduce quickly, are
very well understood genetically, and have been used in microbiota studies
[80, 81, 82, 83, 84]. However, both are far simpler than humans. On the other
side of the spectrum, mice are much more similar to humans, but are expensive to
house and slow to reproduce and still depend mainly on fecal samples or dissection
[85].
Zebrafish occupy a middle ground. They are vertebrates with genetic
similarities to humans [86] while still reproducing quickly and in large numbers
[87]. They can also be raised germ-free, totally devoid of any bacteria [88], and are
largely transparent while in their larval stage, making it possible to take detailed
images of them and their microbiota. This makes them a uniquely powerful model
organism for this type of work and has led to many interesting discoveries [89].
4.2. β Cells and Bacteria
β cells are insulin producing cells found in the pancreas. Zebrafish hatch with
a small number of β cells and in the days following they undergo an expansion
[90]. This is very similar to the development of β cells in humans, although the
44
expansion lasts until age 5 [91]. In both zebrafish and humans this expansion
occurs concurrently with the colonization of the gut with bacteria and it has been
shown that type 1 diabetes, in which a loss of β cell function results in abnormal
glucose homeostasis, is associated with, and in fact preceded by, decreased
microbiota diversity [92, 93, 94].
In their 2016 paper, Jennifer Hill et al. [95] compared the number of β cells
in germ free (GF) fish with conventional (CV) fish, fish that are exposed to a
wide array of bacteria. The number of β cells in CV fish increased steadily over
time, but GF fish did not undergo β cell expansion and had elevated levels of
circulating glucose, implying that they were unable to correctly regulate glucose
levels. They then examined fish that were inoculated with a single species of
bacteria and found that, depending on the type of bacteria used, the β cells would
develop as in the CV fish or remain stagnant as in the GF fish. Having pinpointed
which bacteria were driving the β cell expansion, they were able to isolate exactly
what was behind this: a protein that they called β cell expansion factor A, or
BefA. Purified BefA on its own will cause β cell expansion and mutant bacteria
do not produce BefA but are otherwise identical will not cause β cell expansion,
meaning that BefA is both necessary and sufficient for proper β cell development.
Interestingly, they were also able to find homologous proteins in bacterial species
found in human microbiota that will also cause β cell expansion in zebrafish.
4.3. BefA and Membrane Morphology
Once the protein was identified, it was important to understand how
it was interacting with the zebrafish to cause these changes. Sequencing the
protein revealed that it contained a SYLF domain. These domains are not well
45
understood, but have been shown to be associated with membranes [96, 97, 98,
99, 100]. This, as well as the observation of invagination in cells exposed to BefA,
inspired an examination of BefA’s interaction with lipid membranes.
Simple vesicles of 99.5% 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC)
and 0.5% Texas Red 1,2 dihexadecanoyl-sn-glycero-3-phosphoethanolamin (Texas
Red DHPE), a fluorescently tagged lipid, were observed in the absence and
presence of 0.2 µM BefA. While it is not uncommon to have imperfect vesicles,
those treated with BefA were frequently observed to form clusters of small vesicles
adhered to the outside, which we refer to as multi-vesicular vesicles, as shown in
Figure 4.1. In order to quantify this, we used a light sheet microscope, described
in Chapter 2, to take large 3D scans of vesicles suspended in a cuvette. We
then detected bright objects and categorized them based on their morphology.
We found that only 3% of the control were multi-vesicular vesicles, while 30%
of the BefA treated vesicles were. Each point in Figure 4.2(a) represents the
concentration of multi-vesicular vesicles for a single scan. Figure 4.2(b) shows the
total number of features, both vesicles and multi-vesicular vesicles, in the control
and BefA treated experiments. The doubling in number of features/nanoliter
suggests that multi-vesicular vesicles are not formed by separate vesicles binding
to each other, but by budding and fragmentation of larger vesicles.
4.4. Continuing Work
In our search of BefA treated vesicles, we also noticed that some of the
vesicles were undergoing thermally driven fluctuations, as shown in Figure 4.3.
While fluctuations can occur in untreated vesicles, they appeared far more
frequently after exposure to BefA. This behavior offers a potential insight into
46
FIGURE 4.1. An example control vesicle and multi-vesicular vesicle. Both scale
bars are 10 µm. This figure is taken from an unpublished co-authored work.
how BefA is causing the membrane budding. Fluctuations have been used to
measure the bending modulus of membranes [54]. By measuring these fluctuations
for untreated vesicles and at several concentrations of BefA we could understand
the effect of BefA on the rigidity. We have begun this work, but it will largely be
completed in the future.
47
(a) (b)
FIGURE 4.2. (a) The percentage of observed features that were multi-vesicular
in the control and 2 µM BefA. Each point is a single scan. The bars represent
the total percentage across all scans, 3% in the control and 30% in BefA treated.
(b) The concentration of features in each scan, showing an increase with BefA
exposure. This figure is taken from an unpublished co-authored work.
10 μm
FIGURE 4.3. A single fluctuating vesicle, exposed to 2 µM BefA. Each image is 2
seconds apart.
48
CHAPTER V
CONCLUSIONS AND FUTURE DIRECTIONS
The work presented in the previous three chapters represent the completed
work of several years. While seeking answers to three different questions, each
increase our understanding of the properties and behavior of lipid bilayers and
inspire further exploration.
In particular, the ability to determine membrane viscosity from elliptical
tracer particles opens the door to many systems. With the newfound capability
to measure viscosity of nearly single species membranes, it would be beneficial to
do this for the most commonly used lipids. The fact that this work remains to be
done emphasizes the difficulty in measuring membrane viscosity. As the viscosities
of more lipid species are determined, dependencies on various biophysical
properties will emerge, as we observed for chain length. It would be illuminating
to study the viscosity dependence on headgroup, acyl chain saturation, and
temperature.
Beyond these fundamentals, the effect of various membrane-interacting
proteins on viscosity could be studied. For example, this method could be applied
to the BefA protein discussed in chapter 4. While the morphological changes
caused by BefA could make this experimentally challenging, it is a conceptually
simple extension from the previous work.
Between the viscosity and the already begun measurements of bending
modulus dependence on BefA, there is much to be done to understand how the
protein interacts with lipids. But the work done to characterize the morphology of
large scans of vesicles could also have a very different application. Sorting through
49
detected objects to classify into vesicles and multi-vesicular vesicles resulted in a
large collection of cropped videos of objects typical of a bulk solution of vesicles,
each with a label. This type of data set is the start of the most important part of
any neural network: the training data.
Any neural network is only as good as the information it is given, and
creating and curating the labeled data to train the network can be a very time
consuming step. With this prepared labeled data it would be possible to train a
classifier that could sort potential vesicles without any human input, turning an
extremely tedious analysis into something fully automated.
With more effort, this network could even be applied during data collection.
While being able to count and sort large number of vesicles would be occasionally
useful, it is more common to be interested in videos of a single vesicle. The
ability to identify a vesicle paired with the speed of neural networks could make
automated data collection possible. A scanning “search” state could feed images
to the neural network until an object is identified as a vesicle. The location of the
vesicle would be delivered to the stage control and a video would be taken. If the
network were already functioning, this extension would be only focused on getting
the hardware and software to communicate.
The work described in this dissertation has covered a range of topics and
can be continued in many directions. The theme uniting everything is an effort
to understand fundamental biophysics of lipid bilayers. This encompasses their
boundary conditions with surrounding fluids, methods for measuring their
viscosity, and their interaction with proteins. There is ample room for this work
to continue and I am excited to watch it develop.
50
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