Measuring compositional and growth
properties of single cells
ARCHVES
MASSACHUSETTS
by
F
T
INSTITUTE
-ECHNOLOGY
Francisco Feij6 Delgado
JUN 27 910,3!
Licenciado, Physics Engineering,
Instituto Superior Tecnico (2005)
UBIRARIES
Submitted to the Department of Biological Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2013
© Massachusetts Institute of Technology 2013. All rights reserved.
A uth o r ....................................... ....
. ...........
Department o1 Biol
(12
Certified by.......... ................
cal Engineering
May 18, 2013
......
.............
Scott R. Manalis
Professor of Biological and Mechanical Engineering
T
e sis
Supervisor
.. V ........................
Forest M. White
Chair, Department Committee on Graduate Theses
Accepted by ................
. .........
2
Measuring compositional and growth
properties of single cells
by
Francisco Feij6 Delgado
Submitted to the Department of Biological Engineering
on May 18, 2013, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
The physical properties of a cell are manifestations of its most basic molecular and
metabolic processes. In particular, size has been a sought metric, which can be
difficult to ascertain with great resolution or for smaller organisms. The advancement
of single-cell measurement techniques and the understanding of cell-to-cell variability
have renewed the interest in size characterization. In addition, knowledge of how
individual cells grow and coordinate their growth with the cell cycle is of fundamental
interest to understanding cell development, but various approaches for describing
cellular growth patterns have often reached irreconcilable conclusions.
In this thesis, a highly sensitive microfabricated single-cell mass sensor - the
suspended microchannel resonator - is used to demonstrate cellular growth measurements by mass accumulation for several microorganism, ranging from bacterial
cells to eukaryotes and mammalian cells. From those measurements insights about
cellular growth are derived, demonstrating that larger cells grow faster than smaller
ones, consistent with exponential-like growth patterns and incompatible with linear
growth models. Subsequently, the implementation of mechanical traps as means to
optimize existing sensors is presented and the techniques are applied to the measurement of total mass, density and volume at the single-cell level. Finally, a method is
introduced to quantify cellular dry mass, dry density and water content. It is based
on weighing the same cell first in a water-based fluid and subsequently in a deuterium
oxide-based fluid, which rapidly exchanges the intracellular water content. Correlations between dry density and cellular proliferation and composition are described.
Dry density is described as a quantitative index that correlates with proliferation and
cellular chemical composition.
Thesis Supervisor: Scott R. Manalis
Title: Professor of Biological and Mechanical Engineering
3
4
Acknowledgments
I would like to start by thanking Professor Scott Manalis. Scott welcomed me into his
lab with open arms, giving me the opportunity to do this work and learn immensely.
Not only did he provide me with all the resources and toys that any aspiring scientist
may want, but his contagious enthusiasm and drive made me leave every meeting we
had with an incredible eagerness to tackle the challenges we faced.
I would like to thank my thesis committee members, Professor Jongyoon Han,
under whom I also had the pleasure to be a teaching assistant, and Professor Alan
Grossman, who from the beginning had the patience to listen to our (raw) ideas
on applying the SMR to study single-cells, contributing with much knowledge and
insight.
I'd like to thank Michel Godin, from whom I inherited my first experimental setup,
jump-starting my work at the lab. Even though I only overlapped with Mike for a few
months, he taught me a lot and was a great mentor. These last few years were spent
in the great company of the entire Manalis lab, whose members deserve my many
thanks. In particular I would like to address the ones that I most closely worked
with. I'd like to thank Wesley Weng, Selim Olcum, Sungmin Son, Andrea Bryan,
Jungchul 'Jay' Lee, Mark Stevens, Amneet Gulati and Vivian Hecht for all the help,
guidance, discussion and endless conversations; Scott Knudsen, with whom I shared
a work bench, contiguous or opposite, and all that comes along with it; Nate Cermak,
whose dedication and enthusiasm both for science and the art of building things was
a great colleague, co-author and teacher; Will Grover also taught me a lot and was
always ready to listen to and help solve any problem I'd throw at him. I also wish to
thank Alexi Goranov from the Amon lab, for enduring with my questions and Steve
Wasserman and Kristofor Payer for all there help in all lab things related.
Coming to MIT also meant coming to a new city and a new country. Such transition was an incredible one and it was made easier and more fun with the support of
my friends back home and by those who welcomed me here: my BE classmates Brian
Belmont, Jeff Wagner, Steve Goldfless, Ricardo Gonzalez, Melody Morris, Edgar
5
Sanchez, Michelle Sukup Jackson, Emily Florine, Eddie Eltoukhy, Bryan Bryson,
Karunya Srinivasan and Prabhani Atukorale, with whom I spent many hours, some
doing serious work, some having serious fun; the friends that I made here and are
too many to enumerate, as well as the people I encountered while taking part in the
Portuguese Student Association, the Portuguese-American Post-graduate Society and
the MIT Euroclub.
Finally, I'd like to thank my family whose support, encouragement and sheer
willingness to listen was, and still is, fundamental. I'd like to dedicate this thesis to
them. To my father, who never got the chance to see me reach this point, but always
knew that I would eventually become some kind of scientist. To my brother, who in
being very different from me is an integral part of who I am. To my mother, who
always gave us everything without ever asking for anything. To Elsa, with whom I
embarked in a big adventure, of which this thesis is all but a small part. Obrigado!
6
Contents
1
2
17
Introduction
1.1
Precision measurement of cellular biophysical properties.
1.2
The Suspended Microchannel Resonator
. . . . . .
17
. . .
19
1.2.1
Device description and operation
. .
. . .
19
1.2.2
Buoyant mass . . . . . . . . . . . . .
. . .
23
27
Measuring Cellular Growth with the SMR
2.1
Introduction . . . . . . . . . . . . . . . . . .
. . . . . .
27
2.2
Dynamic trap method
. . . . . . . . . . . .
. . . . . .
29
2.3
Measuring cellular growth rate . . . . . . . .
. . . . . .
32
2.3.1
Instantaneous growth rate from short trapping event s . . . . .
32
2.3.2
Long trapping events . . . . . . . . .
. . . . . .
36
2.4
A model of the experiment . . . . . . . . . .
. . . . . .
37
2.5
Materials and Methods . . . . . . . . . . . .
. . . . . .
40
2.5.1
Cell culture conditions . . . . . . . .
. . . . . .
40
2.5.2
Experimental setup and measurement conditions.
. . . . . .
42
2.5.3
Experimental errors . . . . . . . . . .
. . . . . .
44
2.5.4
Data analysis and curve fitting
. . .
. . . . . .
44
2.5.5
Model selection criteria . . . . . . . .
. . . . . .
45
2.6
Subsequent developments . . . . . . . . . . .
. . . . . .
46
2.7
Serial cantilevers
. . . . . . . . . . . . . . . . . . . . . .
48
2.8
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.9
Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
7
3
Mechanical traps for weighing single cells in different fluids
3.1
Introduction ......
3.2
Method
......
57
............................
57
........................
59
3.2.1
Position dependent error . . . . . . . . . . . . . . . . .
61
3.2.2
Windowed devices
. . . . . . . . . . . . . . . . . . . .
62
3.3
M aterials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.4.1
Measurement of polystyrene bead density . . . . . . . .
64
3.4.2
Measurement of single-cell density . . . . . . . . . . . .
65
3.4.3
Growth of captured cells . . . . . . . . . . . . . . . . .
67
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.5
4 Intracellular water exchange for measuring the dry mass, water mass
71
and changes in chemical composition of living cells
4.1
Introduction . . . . . . . . . . . . . . . . . . . .
. . . . . .
71
4.2
Measurement principle . . . . . . . . . . . . . .
. . . . . .
75
4.2.1
Distinguishing dry content from aqueous content
. . . . . .
75
4.2.2
Measurement error . . . . . . . . . . . .
. . . . . .
78
4.2.3
Necessity of single-cell measurements
. . . . . .
80
4.2.4
Evidence for complete fluid exchange
. . . . . .
81
4.3
Aqueous, non-aqueous and total cellular content
. . . . . .
84
4.4
Dry density . . . . . . . . . . . . . . . . . . . .
. . . . . .
85
4.4.1
Bacteria . . . . . . . . . . . . . . . . . .
. . . . . .
85
4.4.2
Yeast . . . . . . . . . . . . . . . . . . . .
. . . . . .
88
4.4.3
Mammalian cells . . . . . . . . . . . . .
. . . . . .
92
4.4.4
Red blood cells . . . . . . . . . . . . . .
. . . . . .
94
4.5
Discussion . . . . . . . . . . . . . . . . . . . . .
. . . . . .
95
4.6
Materials and methods . . . . . . . . . . . . . .
. . . . . .
100
4.6.1
SMR operation . . . . . . . . . . . . . .
. . . . . .
100
4.6.2
Cell culture and fixation . . . . . . . . .
. . . . . .
101
8
4.6.3
Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . 103
4.7
Measuring the density of biomolecules . . . . . . . . . . . . . . . . . .
104
4.8
Continuous dry mass measurement
. . . . . . . . . . . . . . . . . . .
107
4.9
Dual cantilever system for density measurements . . . . . . . . . . . .
109
4.9.1
Dual-cantilever bacterial SMR . . . . . . . . . . . . . . . . . .
109
4.9.2
Fluctuation minimization
. . . . . . . . . . . . . . . . . . . .
114
4.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
9
10
List of Figures
1-1
The Suspended Microchannel Resonator
. . . . . . . . . . . . . . . .
20
1-2
An SM R chip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1-3
Resonant frequency shift for a transiting particle . . . . . . . . . . . .
22
1-4
Buoyant mass of a particle . . . . . . . . . . . . . . . . . . . . . . . .
24
2-1
Trapping schematic and demonstration with polystyrene bead . . . .
30
2-2
Cell trapping and growth monitoring . . . . . . . . . . . . . . . . . .
31
2-3
Cell segregation and growth inhibition
. . . . . . . . . . . . . . . . .
32
2-4
Growth rate versus initial buoyant mass
. . . . . . . . . . . . . . . .
34
2-5
Growth rate versus initial buoyant mass at different temperatures . .
35
2-6
Long duration trapping of B. subtilis cells
. . . . . . . . . . . . . . .
37
2-7
Models for cell trapping data. . . . . . . . . . . . . . . . . . . . . . .
39
2-8
Normalized growth rates and comparison of experimental and simulated size distributions . . . . . . . . . . . . . . . . . . . . . . . . . .
41
. . . . . . . . . . . . . .
47
2-10 Serial cantilever array SMR . . . . . . . . . . . . . . . . . . . . . . .
48
. . . . . . . . . . . . . .
50
3-1
Depiction of SMRs with mechanical traps . . . . . . . . . . . . . . . .
59
3-2
Trapping schematic and demonstration with polystyrene bead . . . .
60
3-3
Position dependent error in three-channel devices. . . . . . . . . . . .
62
3-4
Windowed three-channel devices.
. . . . . . . . . . . . . . . . . . . .
63
3-5
Density and mass of polystyrene beads . . . . . . . . . . . . . . . . .
64
3-6
S. cerevisiae cells in the device trap.
. . . . . . . . . . . . . . . . . .
65
2-9
Dynamic trapping of E. coli cells in mSMR
2-11 Doubling time measurements in serial SMR
11
3-7
Density and mass of S. cerevisiae cells. . . . . . . . . . . . . . . . . .
66
3-8
Captured E. coli grown inside the SMR. . . . . . . . . . . . . . . . .
67
3-9
Growth of S. cerevisiae cells inside the cantilever. . . . . . . . . . . .
69
4-1
Buoyancy of a cell in fluids of different densities and membrane perm eabilities.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4-2
Using the SMR to measure the buoyant mass of a cell in H2 0 and D 2 0
76
4-3
Contour map of mass and density as functions of buoyant masses. . .
79
4-4
Exposure time versus calculated dry density for single bacterial cells.
82
4-5
Exposure time versus calculated dry density for single yeast cells.
. .
83
4-6
Raw data of bacterial density and mass measurements. . . . . . . . .
84
4-7
E. coli dry mass, water mass and total mass distributions.
. . . . . .
85
4-8
Dry density and dry mass of a bacterial culture. . . . . . . . . . . . .
86
4-9
Comparison of measured dry density to simulations for E. coli cells. .
87
4-10 Dry density and dry mass of a yeast culture. . . . . . . . . . . . . . .
89
4-11 Dry density distributions for budded and unbudded yeast cells. . . . .
90
. .
91
4-12 Dry density and dry mass yeast cultures treated with rapamycin.
4-13 Dry density and mass of proliferating and non-proliferating mammalian
cells. . . ... .
.........
.. .....
4-14 Dry density and mass of red blood cells.
.....
. . .....
. . . . ..
. . . . . . . . . . . . . . . .
93
94
4-15 Dry density, dry mass and dry volume of a single erythrocyte over time. 95
4-16 Error in density arising from the usage of non pure H 2 0 and D 2 0. . .
96
. . . . . . . . . . . .
105
. . . . . .
107
. . . . . . . . .
108
4-17 Densities of solutions of BSA in H 2 0 and D 2 0.
4-18 Growth curves of E. coli cultures D2 0-based LB medium.
4-19 Growing E. coli in H2 0-based and D2 0-based fluids.
4-20 Dual SMR diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4-21 Dual SMR: polystyrene beads in same fluid. . . . . . . . . . . . . . .
112
4-22 Dual SMR: polystyrene beads in different fluids. . . . . . . . . . . . . 112
4-23 Long term baseline drift in dual SMR.
. . . . . . . . . . . . . . . . . 113
4-24 Short term baseline fluctuations in dual SMR. . . . . . . . . . . . . . 113
12
4-25 Filtering the short term fluctuations with a fluidic reservoir . . . . . .
115
4-26 Simulation of short term fluctuation filtering . . . . . . . . . . . . . .
116
13
14
List of Tables
2.1
Fit parameters for linear regressions . . . . . . . . . . . . . . . . . . .
52
2.2
Fit parameters and model selection criteria for bacterial trapping data
52
2.3
Culture doubling times . . . . . . . . . . . . . . . . . . . . . . . . . .
53
2.4
Fit parameters and model selection criteria for B. subtilis long trapping
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.5
Simulation parameters used to generate Figure 2-7. . . . . . . . . . .
55
3.1
Density and population statistics of the diameter of polystyrene beads.
65
3.2
Reported measurements of yeast density. . . . . . . . . . . . . . . . .
66
4.1
Density of chemical components of cells.
. . . . . . . . . . . . . . . .
74
4.2
Approximate chemical composition of a bacterium, yeast and mam-
events.
m alian cell.
4.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
. . . . . . . . . . . . . . . .
105
Measurement of the density of a protein
15
16
Chapter 1
Introduction
"Although many biological phenomena have been discovered
and explained on the basis of qualitative analyses, new insights often
follow when they are revisited in quantitative terms."
Rob Phillips and Ron Milo, [1]
1.1
Precision measurement of cellular biophysical
properties
The knowledge of how individual cells grow and coordinate their growth with the cell
cycle is of fundamental interest to understanding cell development. The interest in
characterizing cellular size and growth is not at all new. In the 1950s, researchers were
developing techniques for measuring single-cell growth and correlate it cell size [2-4].
At that time, the methods were rudimentary and lacked the resolution to precisely
determine these quantities and parameters.
in the second half of the
2 0
th
The advent of molecular cell biology
century contributed with invaluable tools to probe the
inner workings of cellular machinery and shifted the interest away from the biophysical
characterization of cells. As technology progressed, however, renewed interest in this
field arose [5-10]
The physical properties of cells are manifestations of the most basic aspects of
17
cellular life: what is the size of a cell and what determines it, how much biomass and
water does the cell contain, how fast does it grow, or how densely are its contents organized. All of these properties are direct consequences of the biomolecular processes
occurring within the cell, from phenotypical manifestations of cell states, to the cell's
response to its environment. Yet many of these characteristics are still not explained
or connected to other aspects of the cell's molecular activity.
As advances in single-cell analysis are made, the need for physical characterization becomes greater. Size also is essential to characterize the abundance of molecular
species, thus providing context and a reference frame for the ever-growing datasets
from genomics, proteomics and other single-cell analysis methods, existent or in development. The precise characterization of size at a single-cell level will also contribute
to elucidate questions of cell-to-cell variability [11,12] -
information that is lost when
dealing with averaged population measurements. Single-cell analysis can also allow
researchers to distance themselves from methods that require cell synchronization.
Characterizing the dynamics of population growth can relay valuable information
for describing cellular growth throughout generations [13] and population homeostasis
[14,15]. In addition, the quantification of growth heterogeneity in complex samples is
of value in contexts such as microbial ecology, for profiling environmental samples [1618], or the measurement of antibiotic activity and cytotoxicity [19,20]. Furthermore,
the quantitative measurements of single-cell physical and chemical parameters are of
interest to many aspects of biological modeling [12,21].
In this thesis, the suspended microchannel resonator (SMR) single-cell mass sensor -
a highly sensitive
is used to demonstrate cellular growth measurements by
mass accumulation for several microorganism, ranging from bacterial cells to eukaryotes and mammalian cells. From those measurements insights about cellular growth
are derived, demonstrating that larger cells grow faster than smaller ones, consistent
with exponential-like growth patterns and incompatible with linear growth models
(Chapter 2). Subsequently, the implementation of mechanical traps as means to optimize existing sensors is presented and the techniques are applied to the measurement
of total mass, density and volume at the single-cell level (Chapter 3).
18
Finally, a
method is introduced to quantify cellular dry mass, dry density and water content
and the correlations of these parameters with cellular proliferation and composition
are described (Chapter 4).
The work here described is a step in the development of a multi-dimensional
technology, based on the SMR, capable of producing the above mentioned types of
information, with high precision on a single cell level, ranging a wide scope of cell
sizes and descriptive parameters.
1.2
1.2.1
The Suspended Microchannel Resonator
Device description and operation
The suspended microchannel resonator (SMR), is a microfabricated cantilever-based
mass sensor (Fig. 1-1) developed in the Manalis laboratory [22-24]. Label-free cantileverbased mass sensors had been previously described. Their measurement principle consists in determining shifts in the resonant frequency of the device induced by changes
in the mass of the resonating structure (cantilever plus the added target mass). The
SMR demonstrated that the measurement could be done by having the sample mass
located not on the cantilever itself, but within a hollow beam, while keeping the
vibrating structure in a vacuum. With such advancement, it becomes possible to
make measurements in fluidic environments without significant energy loss from fluidic dampening, also allowing for high quality factors (Q) and therefore very high
mass resolution. It is capable of weighing nanoparticles, cells ranging from bacterial
to mammalian ones [25], and sub-monolayers of adsorbed proteins with femtogram
resolution (1 Hz bandwidth) [26].
A cantilever with h height, w width and L length has a resonant frequency that
depends on its physical properties (material composition and size) according to:
k
1
f = 27r-7 m*
h
oc --
Lj2(1)
where k is the spring constant of the structure and m* the effective mass ( m* =
19
(1.1
3m
Figure 1-1: The Suspended Microchannel Resonator (SMR). Rendering depicting
a portion of the device, with a cell transiting an uncovered cantilever. Electrostatic
actuation (gold electrode) drives the cantilever, which is housed in an on-chip vacuum
chamber, and its resonant frequency is measured with an optical lever (red laser
beam). Sample reaches the resonator through two large bypass channels.
Figure 1-2: (a) An SMR chip mounted on a PCB. Scale bar is 9 mm. (b) Close-up
of the device where the microfluidic ports can be seen (red arrow), as well as the
bypass microchannels (yellow arrow). The device vacuum chamber and front optical
access window is located between the two bypasses (blue arrow). Device channel
is not distinguishable. Scale bar is 700 pm. (c) SEM image of a cantilever during
fabrication.
20
with m the mass of the structure for the first vibrational mode [27]) [23]. The mass
sensitivity for a small mass
Am
is given by:
Af
Am
1f
2m
1
wL 3
The silicon-based sensor is encapsulated in an on-chip vacuum chamber, making
the device small and easy to operate.
Microfluidic access, for sample delivery is
achieved through a clamp that holds microfluidic tubing in place with o-rings at the
interface Figure 1-2. Pressurized vials are used for controlling delivery and discarding
of sample.
The device is operated in feedback. First generation devices were driven electrostatically and cantilever deflection detected with the optical lever method [22]. More
recently piezoelectric actuation has been also been implemented [28,29], allowing for
greater drive amplitudes to be achieved, as well as a piezoresistive readout, eliminating the need for optical access, reducing the size and complexity of the measurement
apparatus and opening the possibility for easy measurement of multiple devices per
chip [28].
As individual particles or cells transit the microchannel, a shift in the resonant
frequency of the SMR is observed that corresponds to the mass change of the resonant structure. The cantilever can be operated in different vibrational modes. In
the first mode, particle mass measurement occurs at the apex of the beam, while
higher modes can quantify the mass at different positions along the the length of the
cantilever [30,31] (Figure 1-3). These higher modes have the advantage of making
the detection insensitive to flow path, unlike in first mode, in which the frequency
shift depends on whether the particle makes the turn at the apex along a more inwards or outwards trajectory. This error is termed position dependent error. In this
thesis, the majority of the measurements were made in first mode as technical and
engineering developments had not been made in order to operate the device in higher
vibrational modes. Operation in higher modes is one way to eliminate this position
dependent error, nevertheless other approaches can be taken. In Chapter 3, a design
21
Time
0
0
-0.4
0.6
W-0.8-1
Figure 1-3: Resonant frequency shift versus time for a point mass transiting along
the cantilever. First-mode peaks (left) exhibit position dependent error, while detection in second-mode (right) can eliminate the error, if measurement is performed at
anti-nodes. Solid line represents a particle transiting the cantilever along the outer
edge of the channel, while dashed line represents the inner edge trajectory.
for the SMR which includes mechanical traps is presented and one of the aims is to
specifically eliminate the position dependent error.
For a given cantilever at a given temperature, when loaded with a fluidic sample,
the baseline resonance frequency will depend on the density of the solution. Due to
the small relative size of the added mass and by controlling the driving amplitude,
the devices are operated in their linear regime. This allows for a linear calibration of
solution density using reference solutions, typically salt solutions of different concentrations prepared gravimetrically.
The mass of a transiting particle is determined by measuring the resonant frequency shift, i.e., the height of the peak depicted in Figure 1-3. A calibration factor is
obtained by measuring a monodisperse population of NIST standard size polystyrene
beads. The fact that a moving particle causes transient change in resonant frequency
allows for the determination of the baseline immediately prior to and after the measurement, and this differential measurement can be used to eliminate errors such as
a drifting baseline caused by temperature fluctuations.
22
1.2.2
Buoyant mass
If a particle transiting the cantilever were to have the same density of the surrounding
fluid, then there would be no added mass to the resonant structure of the device.
Therefore, no changes in resonant frequency would be observed 1 . In fact, what the
SMR quantifies, when using it to measure particles in suspension, is a particle's
buoyant mass, or the difference between its mass and the mass of the displaced volume
of the immersion fluid, which is defined as:
(1-3)
V(Pparticle - Pfluid)
mb
where V is the particle's mass, and
Pfluid
and
Pparticle
are the densities of the fluid
and the particle, respectively (Fig. 1-4). Buoyant mass can assume positive values
for a particle more dense than the surrounding fluid, or, in a gravitational field,
sinking -
and negative values -
for a particle less dense than the fluid, or floating.
In terms of the mass of the particle (m), the equation can be rewritten as
Mb= m(1 -
(1.4)
Pfluid
Pparticle
The buoyant mass value, by itself, constrains neither the size nor density of the
measured particle, therefore assumptions have to be made regarding one or the other
to extract meaning from the measurement. For instance, when measuring the buoyant
mass of polystyrene beads, whose density is known (p = 1.05 g-cm- 3 ), size (or mass)
can be accurately determined. Of course, buoyant mass depends on the density of
the suspension buffer, so it is not a unique value.
For biological samples, namely cells, it is worth discussing the meaning of a buoyant mass measurement. Most simply, the buoyant mass of a cell can be written as:
nbceu = rncell(1 -
Pfluid
Vcei (Pceui
-
Pfluid)
(1.5)
Pcell
Strictly speaking this might not always be true as a transiting particle could have an influence
on the device, such as disturbing the local pressure on the cantilever walls. However, such discussion
is not relevant here.
23
Cl,
C/)
m
E
C
0
Pparticle
Pfluid
Fluid density
Buoyant mass
Figure 1-4:
(mb)
of a particle (mass m and density
pparticie)
as a
function of the fluid density (Pfluid) as described by Eqs. (1.3) and (1.4). The slope
of the line is the particle's volume (V).
A cell is a phospholipid membrane-defined container of a water-based solution of
organelles, proteins, nucleic acids and many other molecules that constitute its dry
content (with a mass mdry and a bulk density Pdry). Its volume of intracellular water
is defined as Vm. Therefore, one can rewrite Eq. 1.5 as a sum of the buoyant masses
of these two different -
mbe,,
dry and wet - components:
-- mdry (1 -
Pfluid)
Pdry
+ Viw(Piw - Pfluid)
(1.6)
Since the suspension solution for a cell is typically culture medium, or a buffer solution
such as phosphate buffered saline (PBS), which have densities close to water (PPBS
1.005 g-cmfirst one -
3
_
at room temperature), the second term is considerably smaller than the
it would contribute to the total buoyant mass with about -5%
value for a cell with a dry content density of about 1.3 g-cm-
3
of the
and 75% water content
fraction. Therefore the buoyant mass of a cell suspended in a water-based solution is
analogous to looking only at the buoyant mass of the cell's non-aqueous content:
mei
mdry(1 - Pf
Pdry
24
)
(1.7)
The suspension fluid can be changed to incorporate molecules that are nonmembrane permeable and change the fluid's density. In such a case, the assumption
that the the intracellular water content has a density similar to of the surrounding
medium is no longer valid.
The essential concepts for the understanding of buoyant mass measurements with
the SMR have been introduced. The following chapters will demonstrate the application of this device to measure mass accumulation and cellular growth (Chapter 2),
to measure total mass, density and volume of single-cells in devices optimized with
mechanical traps (Chapter 3) and finally, to measure cellular dry mass, dry density
and water content (Chapter 4).
25
26
Chapter 2
Measuring Cellular Growth with
the SMR
"Growth, which is boringly continuous and
therefore harder to study, has been widely ignored"
Kim Nasmyth, Cell (via [32])
2.1
Introduction
Understanding how the rate of cell growth changes during the cell cycle and in
response to growth factors and other stimuli is of fundamental interest. Over the
decades, various approaches have been developed for describing cellular growth patterns but different studies have often reached irreconcilable conclusions, even for the
same cell types. The debate has focused on whether cells grow at a constant rate
(linear) or at a rate that is dependent on their size (exponential), although more
complex growth curves have also been suggested. The mean dry mass accumulation
of E. coli has been reported as increasing linearly [33] and cell length growth described
This chapter was done in collaboration with several other people and was published as M. Godin,
F. F. Delgado et al. "Using buoyant mass to measure the growth of single cells" Nature Methods
7:387-390, 2010. In particular, Michel Godin implemented the initial dynamic trap mechanism, and
William Grover and Sungmin Son collected data for yeast and mammalian cells, respectively.
27
as bilinear [34], bilinear and trilinear [8], and exponential [35]. The size of budding
yeast Saccharomyces cerevisiae has been reported to increase exponentially by some
approaches [10, 36], but to have a non-exponential and cell cycle dependent growth
curve by others [37]. For mammalian cells, volume measurements have shown linear
growth for rat Schwann cells [38] and exponential growth with a varying rate constant
for mouse lymphoblast cells [15]. Several factors may contribute to the discrepancies
between different growth models: i) cells are minute, irregularly shaped objects, ii)
proliferating cells increase their size only by a factor of two, so distinguishing between
different cell growth models with mathematical rigor requires highly precise measurements, iii) a wide variety of methods have been used to measure growth, including
approaches that average across populations as well as those that monitor individual cells, and iv) a cell's size includes both volume and mass, which can change at
different rates.
While both mass and volume are important parameters, mass is more fundamentally related to cell growth than is volume. Volume can change disproportionately to
mass, thereby altering a cell's density. In cells without rigid cell walls, volume can
rapidly change in response to osmotic stresses, while even in cells with cell walls, the
size of low-density intracellular vacuoles can change to alter the density of cells [39].
Fundamentally, cell growth is the creation of new biomass, the polymerization of small
molecules into the lipids, proteins and RNA that make up the membrane, cytoplasm
and organelles. But most research into cell size and growth has focused on volume,
for lack of methods to measure the mass of individual cells.
An ideal method for measuring cell growth rates would directly and continuously
monitor the mass and volume accumulation of single unperturbed cells with high
precision.
In recent years, optical microscopy has been the closest match to this
ideal [8-10], but volume determination by microscopy has lacked the precision to
conclusively distinguish between cell growth models. Potential alternatives include
using fluorescent protein reporters that are correlated with cell size [10], or using
phase microscopy to estimate dry mass during cell growth [9].
28
2.2
Dynamic trap method
The precise monitoring of single-cell growth in terms of buoyant mass is achieved by
introducing a new measurement system. Bacteria, yeast and mammalian lymphoblast
cells are demonstrated to grow at a rate that is proportional to their buoyant mass.
Buoyant mass is defined as:
mbuoyant
=
V (pceii - Pfluid) =
m(1 - Pfluid)
Pcell
(2.1)
where p is density and V is cell volume. It is dependent on the amount of biomass in
the cell, most of which is denser than water, and so is analogous to the dry mass of
the cell (see discussion in Section 1.2.2). However, this analogy is predicated in the
assumption that cellular density is mostly constant and that cellular water content
remains essentially constant.
A dynamic fluidic control system was developed, enabling the buoyant mass of
cells as small as bacteria and as large as mammalian lymphocytes to be repeatedly
measured with an SMR (see Section 1.2.1 for details in the operation of the device
and measurement principle). A feedback algorithm was implemented, which reverses
the direction of fluid flow upon detecting a cell transiting through the SMR, thereby
reintroducing the cell into the cantilever (Fig. 2-1). Continuously alternating flow
direction creates a dynamic trap that allows for consecutive buoyant mass measurements of the same cell. Since the cell fully exits the SMR prior to flow reversal, the
baseline resonant frequency is acquired after each measurement, allowing compensation for drift arising from temperature variations or accretion on the walls of the
microchannel. Dilute cultures of non-adherent cells in any desired growth medium
can be loaded directly into the system.
The dynamic trap is very stable when measuring polystyrene particles that are
less than half the size of the channel height (3-15 pm). We trapped such particles for
more than 20 hours (>32,000 measurements) (Fig. 2-1c). Sample concentration was
the main limiting factor of the trapping duration. Low concentrations (< 10 7 ml- 1)
decrease the probability of additional particles randomly drifting into the cantilever
29
a)
bC)
"18
0682
0.80
0
2
4
6
8
10 12 14 16 18 20
Time (hours)
Figure 2-1: (a) Illustration of the suspended microchannel resonator (SMR) trapping a single cell. Embedded channel cross-sections for bacteria, yeast and mammalian L1210 mouse lymphoblasts are 3 x 8 microns, 8 x 8 microns, and 15 x 20
microns, respectively. The silicon walls are opaque except in the 15 x 20 microns device, which has thinner walls. (b) Schematic of fluidics: sample is injected in parallel
through the left and right inlets (IL and IR) and collected at the left and right outlets
(OL and OR). While trapping, IL, IR and OL are kept at the same constant pressure;
variable pressure at OR applied by a computer controlled regulator determines the
direction of fluid flow in the device. (c) Trapping of a polystyrene particle. The data
show more than 32,000 consecutive buoyant mass measurements of a single 1.90pm
diameter polystyrene particle.
and becoming trapped along with the particle being measured. The maximal trapping
duration for cells was typically shorter than for polystyrene particles and was dependent on the cell type. On average, E. coli and B. subtilis could be trapped for 500
sec and 300 sec, respectively, before being lost. Yeast and L1210 mouse lymphoblast
cells could be trapped in excess of 30 minutes in a similar system as bacteria but with
larger SMR channels. When living cells were trapped, growth was observed from the
increasing amplitude of the resonant frequency peaks (Fig. 2-2). Trapped cells are in
an open system, as the suspended microchannel is in constant contact with the larger
inlet and outlet channels (Fig. 2-1a), which act as reservoirs of nutrients. Diffusion
and convection prevent local depletion of nutrients by the growing cell. Variability in
the peak amplitudes (Fig. 2-2a) limits the precision of this method and is mainly due
to the trapped cell taking different flow paths as it turns the corners at the cantilever
tip. Different flow paths, as well as increased interaction with the microchannel walls,
may also explain why cells with irregular shapes (e.g. oblong E. coli and B. subtilis)
escape the dynamic trap much more frequently than do polystyrene particles and
30
a)
0.0N
I
-0.4-0.8-1.2-
C -1.6-
0.0
0~
-0.4
-2.0-
0.8.
-2.4b.)
LL
T 1i0
-2.8I
0)
*
0
Time [(S)
I
2
g0
'S
T-
4
I
I
6
1I
12
10
8
Time (min)
E
0
0
5
10
15
20
25
30
35
40
45
50
55
Time (min)
Figure 2-2: (a) Raw data showing 400 measurements of one B. subtilis cell's buoyant
mass. The frequency shift increase with time indicates cellular growth. inset Detail
of a few peaks that show a locally stable baseline forms after each pass through the
SMR, allowing for drift compensation. (b) Several B. subtilis cells were sequentially
trapped. Each point represents the amplitude of the frequency shift, converted to
buoyant mass, as the cell transits through the cantilever. Each set of points (e.g.
from 0 to 12 minutes) is one single cell or non-segregated cells. Heavier cells have
higher growth rates.
31
a)
b)
300 -
5-
o
275 -
4-
CD 250 CD
225 -
C
E
200 -
M
175
-
150
-
125
-
3
2-
*
0
before azide treatment
after azide treatment
*
-(
1-
t
90
10
20
30
I T
040
50
3
60
Y
7y-
4
5
6
7
Initial buoyant mass (pg)
Time (min)
Figure 2-3: (a) Cellular segregation of E. coli cells growing at 23'C. A growing cell,
or possibly a cluster of two cells if division had occurred previously, segregates at t =
30 min. A brief period occurs during which both cells remain trapped and the trap
is unstable as the feedback system can only track one cell. One of the daughter cells
escapes the trap while the other remains inside the SMR. The buoyant mass ratio
at the segregation time is 2.1 ± 0.2 and the growth rate ratio is 1.67 ± 0.07. (b)
Addition of the poison sodium azide to a culture of S. cerevisiae inhibits cell growth,
demonstrating that the sequential increase in buoyant mass is due to cell growth.
round cells.
Following conversion of resonant frequency shifts, growth could be observed as
steadily increasing buoyant mass, as in a series of trapped B. subtilis cells (Fig. 22b). Occasionally, the magnitude of the frequency shifts would instantaneously drop
by a factor of two, suggesting that the trapped cell had divided into two daughter
cells, one of which had escaped the trap (Fig. 2-3a).
Adding the poison sodium
azide to a culture of S. cerevisiae continuously being loaded to the device resulted
in a greatly diminished rate of increase in buoyant mass, demonstrating that these
increases are indeed due to cell growth (Fig. 2-3b).
2.3
2.3.1
Measuring cellular growth rate
Instantaneous growth rate from short trapping events
In order to determine whether growth rate depends on size, the "instantaneous"
growth rate was measured by trapping a cell for a period much shorter than the cell's
32
own life cycle. For each trapping event, a growth rate is determined and associated
with the cell's buoyant mass at the start of the trapping event. By plotting the growth
rate versus buoyant mass, temporally localized growth rates of several individual cells
can be pieced together to determine the size dependency of growth, provided that the
measurement errors are below the natural variability. Such a plot does not necessarily
depend on knowing the position of each cell in the cell division cycle, although such
information could be valuable and may be obtainable in future devices. Cells were
sampled from exponential phase cultures of B. subtilis, E. coli, S. cerevisiae, and
L1210 mouse lymphoblasts. Growth rates for each cell were determined by performing
linear fits to the buoyant mass data from each trapping event.
Growth rates are
plotted against initial buoyant mass for B. subtilis (Fig. 2-4a), E. coli (Fig. 2-4b), E.
coli grown at low temperature -
23 'C instead of 37 'C -
(Fig. 2-5), S. cerevisiae
(Fig. 2-4c), and L1210 mouse lymphoblasts (Fig. 2-4d). A clear trend is observable
in all four cell types: heavier cells grow faster than lighter ones. The relationship
between cell size and growth rate appeared to be linear and for B. subtilis the linear
fit extrapolated close to the origin, which is suggestive of a simple exponential growth
pattern (Table 2.1).
The buoyant mass ranges displayed in Figure 2-4 clearly span over twice the lowest
values, particularly in the B. subtilis data (Fig. 2-4a). Buoyant masses that are more
than twice the smallest size in the population could potentially represent multiple
cells simultaneously entering the SMR. The larger SMR used to trap the L1210 cells
allows for optical microscope access, providing confirmation that the cells are singlets.
Both the devices used for yeast and bacteria are opaque, but the channel cross-section
of the SMR used for yeast greatly reduces the likelihood of trapping clustered cells.
However, for the bacteria, clustering is possible and some of the larger mass values are
almost certainly doublets. Note that while the exponential phase cultures of yeast
and mammalian cells were almost entirely composed of single cells when observed
under a microscope, both bacteria cultures contained 20% of non-segregated cells or
small clusters. To isolate the single cell events for bacteria, one could consider only
those events which have a buoyant mass that is less than twice the minimum buoyant
33
b)
a) 0.6. B.subtilis
E.coli
0.16
0.14
0.5.
050.12
0.4-
0.10
0.3-
0.08
0.06
. 0.2
2 0.04
-
0.
0.0
0
3Mo
0
c)
f
0.02-
200
400
0.00
0.0
0
-3
600
800
1000
60
80
11
S.cerevisae
10
6
9
5-
8
3
-
100
00
0------
120
140
160
180
200
220
Initial buoyant mass (fg)
d)
Initial buoyant mass (fg)
7
0
Mouse lymphobl st
5T. 4-
3
2-~
0
0
00
0
1,
0(9
0
7
0
8
9 .6 1,0
1,1
Initial buoyant mass (pg)
30 40
50 60
70 80
90 100 110 120 130
Initial buoyant mass (pg)
Figure 2-4: Growth rate versus initial buoyant mass. Each data point represents
a trapped cell and is plotted on the diagram according to the cell's initial buoyant
mass and the measured growth rate during the trapping period. Filled circles indicate
normal growing cells and open circles indicate fixed cells. (a) B. subtilis (Marburg
strain) from 9 cultures grown at 37 'C. (b) E. coli K12 from 11 cultures grown at 37
'C. (c) S. cerevisiae from one culture grown at 30 0 C. (d) L1210 mouse lymphoblasts
from two cultures grown at 37 'C. Curve fits are weighted linear regressions. The
growth rate errors bars for the growing cells are one standard deviation of the growth
rate measurements of the fixed cells, except in the cases when the least squares fitting parameter standard error is greater (due to particularly short trapping times).
See Materials and Methods and Table 2.1 for details on culture growth conditions,
statistical analysis and experimental errors. Figure 2-8 shows a small, but non-zero,
probability of over- or under-determining the growth rate. In light of this, the three
L1210 cells that exhibited surprisingly high growth rates (circled in red) were not
included in the linear regression.
34
0.140.12-
S0.10
0.080.06
0 .04
0
0.02-
*
0.0080
120
160
200
240
280
Initial buoyant mass (fg)
Figure 2-5: Growth rate versus initial buoyant mass at different temperatures. E.
coli cells were grown at 37 0 C (filled circles) and 23'C (open circles). Fit parameters
are reported in Table 2.1.
mass, however the presence of clustered cells can give additional information of the
growth pattern of the cells: a discontinuity of the growth rate at about twice the
value of the lowest buoyant masses would be inconsistent with exponential growth of
single bacteria (See Section 2.4, Fig. 2-7 and Table 2.2).
Although the data for all cell types are inconsistent with simple linear growth,
measurement errors and cell-to-cell variation could potentially mask growth rate
changes that would identify multiple stages of linear growth during the cell cycle.
In order to evaluate the experimental error of the growth rate determination similar
trapping experiments with fixed cells were performed. As the growth rate of fixed
cells is zero, the deviation from this value provides a measure of the experimental
error. The cell-to-cell variability in growth rates is generally greater than the error
of the method (See Section 2.5.3 and Figure 2-8 for error analysis). That is, the
deviations of cells from the fitted curves in Figure 2-4 are often not due to experimental error, but instead reflect the biological variation in an isogenic population.
Cells -
even those of the same buoyant mass, but not necessarily at the same cell
cycle position -
can exhibit different instantaneous growth rates. Previous cell cycle
35
models typically assume that all cells of a given size grow at the same rate [10, 40]
and a lack of precision in prior methodologies may have prevented growth rate variability from being observed until now. The cells grew well in our microfluidic devices
(Table 2.3), suggesting that our system does not alter normal cellular growth, and
as we currently have no information on cell cycle position, it is possible that the
measured variations reflect cell cycle-dependent changes in growth rate [10,15,37]. A
second possibility is that cell growth is variable in a manner that is independent of
cell size and cell cycle position. The source of this variability is unknown, but many
transcripts and proteins are subject to stochastic fluctuations in bacteria, yeast and
mammalian cells [41], conceivably influencing growth rate consistency. As previously
observed [3,7,8,35] we have found single cell growth to be smooth and continuous and
do not believe the observed variability occurs within a cell's lifespan (see next Section
and Fig. 2-6). Further investigations will be required to uncover the true nature of
this variability.
For all the cell types measured in Figure 2-4, the single cell growth rates were consistent with the population doubling time of exponential phase cultures (Table 2.3).
2.3.2
Long trapping events
Occasionally B. subtilis cells would be trapped for long enough to observe a full cell
cycle (Fig. 2-6).
Optical access was not available to verify the presence of single
cells; however, for all three long-duration traps, the initial buoyant mass value was
in the lower end of the distribution of buoyant masses for the B. subtilis population
(Fig. 2-8). Therefore, it is likely that only a single cell was present for most of the
duration of each long trapping event. When fitting curves to the three long trapping
events, a simple exponential fit was a better match than linear or bilinear fits, as
verified by four different statistical tests. Results from these long-duration traps
support the conclusions of the shorter trapping events with B. subtilis (Fig. 2-4a;
see Section 2.5.4 and Table 2.4 for details of curve fitting). Importantly, analysis of
the three long trapping events and the ensemble of shorter instantaneous trapping
events yielded a consistent cellular doubling time for B. subtilis, further validating
36
a) 650 -
b)
600-70
550
'44W
( 500
->
330
E
2
CO 4-
45
6
12414
8610
Time450(
1621824
c)
0 2
5o2
4001
trapn
-din
e
Exponential
curve fit
curve fit0
E
.1
-Linear
0
2
4
6
8
10
12
14
16
18
20
22
Time (me)
3O0
210
12
1
Time (m)
Figure 2-6: (a) B. subtilis cell trapped for a period similar to the cell cycle duration.
Data is fitted to linear (red; reduced X2 = 0.00257) and exponential (blue; reduced
xain 0.00187)
2.5.3, 2.5.4 and 2.5.5 and Table
tion the factions
Sen
of the statistical analysis, additional models and model comparison.
Additional B. subtilis long trapping events.
details
(b) and (c)
the method and findings previously described.
2.4
A model of the experiment
A model was implemented to describe our experiment and to compare simulated
results for linearly and exponentially growing bacterial cells with experimental data,
taking into consideration the fact that our system may measure clustered cells.
The simulation models two different populations of growing cells according to two
different growth models, an exponential and a linear one, and measures their growth
rates at a random point in the cell cycle. The simulation requires that all cells have
specified starting buoyant mass and growth rate (both with certain Gaussian variabilities) and are allowed to grow for three generations. To simulate our experiment,
each cell's buoyant mass and growth rate are determined at a random point in its
cell cycle. These values are then blurred by simulated experimental errors attributed
to the method itself: the uncertainty in the determination of the growth rate (which
37
is inversely proportional to the duration of the trap), and the uncertainty in the cell
buoyant mass (which is related to the mass resolution of the device and the trapping
duration). The errors are assumed Gaussian distributions.
The mathematical expressions for the two growth models are, for linear growth:
m(0) =
(2.2)
m(t + 1) = m(t) + bL
and for the exponential growth model:
m(0) =
(2.3)
m(t + 1) = m(t) + bL
where mo is the starting buoyant mass, m(t) the time dependent buoyant mass, bL
the arithmetic progression difference and bE the geometric progression ratio. Cell
division is assumed to be symmetrical and occurs at a defined
tD
after which cells
may or may not segregate according to a probability PD of segregation. For the linear
growth model, bL is doubled if a cell does not segregate. In the case that a cell does
segregate, one of the daughter cells is discarded.
Simulations of the linear and exponential growth models for B. subtilis and E.
coli are shown in Figure 2-7.
The parameters used in the simulations are listed
in Table 2.5. Whenever possible, the parameters were derived from experimental
data or literature reports. The errors in the determination of the growth rate and
the determination of the cell buoyant mass are derived from the errors obtained
from the fixed cells' experimental data (see Experimental errors section); the initial
peak height are averages from the individual cell buoyant mass data (n = 100 for
B. subtilis and n = 48 for E. coli). The probability of segregation is the ratio of
single cells versus non-single cells and was estimated by counting single cells and
clustered/non-segregated cells by optical microscopy (n > 1000). The doubling time
is an average of culture doubling times determined by turbidity measurements (n = 5
for B. subtilis and n = 10 for E. coli). The cell growth parameters bL and bE are
38
0.6-
b) B.subtilis
a) B.subtilis
0.5-
Ca
0)
.904
0.40.3-
14
0.20~ ~
0.10.0 -I
250
0
0.14-
750
560
1000
250
0
750
500
1doo
d) E. coli
c) E. coli
I
0.12'0
S
I
0~~
0.10-
0
*'I
L..oo
0
C 0.08-
-0
0.06-
-
0.04-
,-
0.0275
100
125
150
175
200
225
75
100
125
150
175
200
225
Initial buoyant mass (fg)
Figure 2-7: Shown is the overlap between experimental data (red) obtained for (a)(b) B. subtilis
and (c)(d) E. coli and simulated data (grey) obtained using an exponential growth law (a)(c) and a
linear growth law (b)(d). Blue lines are weighted curve fits of the experimental data to a linear curve
(a)(c) and a piecewise constant function b), d). Error bars not presented for clarity, but are shown in
Figure 2-6. Fitting parameters and model selection criteria results are reported in Table 2.2. Initial
cell buoyant mass represents the buoyant mass value of a cell when it is introduced into the trap.
For the simulation data, these values correspond to the cell cycle time point at which the cells were
randomly selected to be measured. For B. subtilis (a)(b), the results are clearly better matched by
the exponential model, which describes both the trend and the dispersion of mass measurements.
The linear model outputs two distinguishable populations (single cells and doublets), unlike the
exponential model where the two populations overlap. Such discontinuity in the growth rate is
not observed in the experimental results. For E. coli (c)(d), however, the match is not as clear and
therefore the modelling does not allow for a definitive distinction. All the model selection criteria for
the experimental data curve fits (blue) favor the linear fit for both organisms, yielding parameters
similar to the ones used in the simulation. Integrating d = a.m + b yields m(t) = moeat - b
indicating that the above curve fits both favor the exponential growth model. For the B. subtilis
case the distinction is greater and the fit parameter b = -0.0179 ± 0.0109, is consistent with a line
intercepting the origin, suggests a simple exponential growth pattern.
39
calculated as the values needed to double the initial mass size during the doubling
time. For the exponential case, this parameter is a time constant independent of the
starting buoyant mass. The cell starting buoyant mass (i.e. weight of cell at the
beginning of the simulation and immediately following division) is a fit parameter
due to the lack of experimental values. For both bacterial strains approximately 10%
of the cells have buoyant masses below this value. The growth parameter errors are
also fit parameters and we report the minimum apparent values that fit the dispersion
of the experimental data. For E. coli, cell length variability at birth of about 35%
and coefficients of variability for linear growth rates of cell length higher than 28%
are reported in the literature [7]; the standard deviation values used for the initial
cell size distribution and growth parameters are consistent with such reporting. The
model is robust in that substantial changes in the fit parameters (± 20%) do not
significantly alter the results. Buoyant mass distributions can be obtained from the
simulated data (Fig. 2-8).
A bilinear growth model, where the rate change point might occur elsewhere than
at division, was not considered in our simulations, as one would have to introduce
other parameters in the model without any experimental cue to support them, such
as septum formation. Yet, for the B. subtilis, such a model would again predict a
discontinuity in the growth rate which was not present in the experimental data.
For B. subtilis, the trend and the dispersion of the experimental data are well
matched by an exponential growth model. For E. coli, the high variability of the
growth rate values does not allow for the identification of the best growth model.
2.5
2.5.1
Materials and Methods
Cell culture conditions
Bacillus subtilis (ATCC #6051)
and Escherichia coli (ATCC #23725) were grown
in LB (Luria-Bertani) broth (Sigma #L2542) supplemented with 0.5% BSA (Sigma
#A3059) overnight at 37 C or 23 C and then diluted 1:100 in the same heated media,
40
a)
40- 0
35
b)
L1210
Yeast
4
E.Col
0.25
-Simulated
C-
M B.subtilis
0.20
data
Early exp phase
Mid exp phase
Late exp phase
-
3025
0.15-
LZ
0.05-
20
105
-3
-2
-1
0
1
2
3
0
100
200
300
400
500 600
700
800
900 1000
Buoyant mass (fg)
Growth Rate / StDev
Figure 2-8: (a) Histogram of the growth rates of the fixed (non-growing) cells
normalized by the standard deviation of each cell type (B. subtilis: 0.012 fg/s, E.
coli: 0.004 fg/s, S. cerevisiae: 0.53pg/h, L1210: 0.24pg/h) in order to compare
across systems. Total cell count is 154. The distribution characterizes our growth
rate measurement error: approximately 68% of the cells have growth rates between
± one standard deviation, denoted by the dashed lines, which represent the size of
the error bars. (b) Experimental and simulated buoyant mass distributions. The
plots show normalized (area under the curve) buoyant mass distribution for several
exponentially growing cultures of B. subtilis that were obtained by running the SMR
in a flow-through mode, where no trapping occurs (n = 1024, 577 and 393 for early,
mid and late cultures). The simulated distribution (blue with shaded area under the
curve) is extracted from modeled data presented in Figure 2-7 (n = 1000) and fits
well to experimental results. This result doesn't show a distinction between the linear
and exponential models, as expected, but does allow for comparison between model
predictions and experimental data.
one to two hours prior to the measurement. Samples were introduced into the device
at concentrations ranging between 1 x 106
-
1
x 107
ml- 1 . Doubling times for bacteria
were determined by turbidity measurements at 560nm: E. coli: 26 t 3 min (n = 10) at
37 *C and 65 t 2 min (n = 3) at 23 C ; B. subtilis: 20 ± 1 min (n = 5). Saccharomyces
cerevisiae were grown overnight at 23 *C in YEP (yeast extract plus peptone) media
containing 2% glucose and 1 mg/mL adenine. The overnight culture was then diluted
1:250 in the same media and maintained at 30 'C during measurement. A doubling
time of 1.60 ± 0.04 hour at 30 'C (n = 6) was determined by optical absorbance at 600
nm. Yeast were introduced into the device at concentrations ranging between 1 x 105
-
1
x
106 m- 1 . L1210 mouse lymphocytes were grown in L-15 media (Invitrogen
41
#21083027)
supplemented with 10% fetal bovine serum (Invitrogen #16000-044),
0.4% glucose (Sigma #G8769) and 1% penicillin/streptomycin mix (Cellgro #MT30-002-CI) at 37 0 C. Doubling time was 12 hours at exponential growth phase and
was determined by cell concentration measured with a Coulter counter. Samples were
introduced into the device at concentrations ranging between 5 x 104 - 2 x 105 ml- 1 .
2.5.2
Experimental setup and measurement conditions
The details of the fabrication of the SMR, instrumentation for data acquisition and
software for data analysis have been previously reported [26].
Precision pressure
regulators (electronically controlled ProportionAir QPV1 and manually controlled
Omega PRG101-25) were used to control sample flow within the SMR. These regulators provided the necessary stability to precisely control the position of the target
cell within the 15 and 191 pL internal volumes of the 3 x 8 microns and 15 x 20
microns suspended microchannels. Glass vials with open-top caps and Teflon-lined
septa were used to contain the samples and collect sample waste. Nitrogen or filtered
air was used to pressurize the contents of the vials for sample introduction into the
microfluidic device and for fluidic flow control.
During an experiment, a dilute sample containing the target cells
( 105 to 107 ml
1
)
is introduced into the two bypass channels (cross-sectional size: 30 x 80 microns)
on each side of the suspended microchannel (cross-sectional size: 3 x 8 microns
for bacteria and 15 x 20 microns for L1210 cells). The pressures at all four ports
are equalized in order to limit sample flow in the bypass channels and through the
suspended microchannel. A computer-controllable pressure regulator is then used at
the outlet port of one of the bypass channels to apply a slight pressure differential
across the suspended microchannel promoting a small flow through the SMR. Custommade feedback software, implemented with National Instruments LabVIEW, then
waits for a single cell to enter the SMR. Once a mass measurement is acquired, the
software automatically adjusts the pressure to reverse the flow direction within the
suspended microchannel. The algorithm also maintains a constant flow rate ( 20
pL/sec) by monitoring the cell transit times (duration of the transient frequency
42
shifts) during each passage. The feedback software compensates for any pressure
drifts that occur over extended periods of time.
For yeast measurements, a slightly different experimental setup and 8 x 8 microns
suspended microchannels were used. A stirred cell culture at atmospheric pressure is
connected via capillary tubing to the inputs of the SMR bypass channels. The outputs of the bypass channels are connected via tubing to two waste vials. Two solenoid
valves and a vacuum regulator (SMC #ITV2090) are used to selectively depressurize the contents of one waste vial while venting the other vial to the atmosphere.
Switching the solenoid valves alternates which vial is depressurized and reverses the
direction of fluid flow in the SMR. By switching the solenoid valves immediately after
a cell passes through the SMR, the cell can be routed back and forth through the
SMR several times per second, measuring the buoyant mass of the cell with every
pass.
Fixed cells were incubated for an hour in a solution of 3.7% formaldehyde and 2%
gluteraldehyde in 100mM phosphate buffer, after being twice pelleted (3 min at 3000
rcf) and resuspended in PBS to wash away the growth media.
The SMR microchannel and fluidic system are sterilized with piranha solution (1:3
mixture of hydrogen peroxide and sulphuric acid) and thoroughly rinsed for approximately one hour with de-ionized water and growth medium prior to measurements.
Polystyrene size standard (NIST) particles (01.51 pm, Bangs Laboratories NT16N
for the suspended microchannel of 3 x 8 microns and 8 x 8 microns, 08.62 ym,
Bangs Laboratories NT25N for the 15 x 20 microns) dispersed in water are used to
calibrate the device for mass. The device is installed in a metal clamp connected to a
water circulator (Thermo NESLAB RTE7) that maintains the SMR at constant temperature, as measured using a surface-mounted thermistor connected directly onto
the device. The temperature can be adjusted quickly using a thermoelectric module
and a temperature controller (Wavelength Electronics Inc.). The pressurized sample
vials are also temperature-controlled. During an experiment, care is taken to maintain the sample temperature (both before and after entering the SMR) at the desired
temperature.
43
For population characterization, the system can be run in a flow-through mode
where cells are not trapped but measured only once.
Buoyant mass distributions
of hundreds of single measurements allow the characterization of the culture and, if
desired, the selection of a cell by its buoyant mass [42].
2.5.3
Experimental errors
The growth rate measurement errors were determined as the standard deviation of the
growth rate measurements of the fixed cells, except in the cases when the least squares
fitting parameter standard error is greater (due to particularly short trapping times;
description of curve fitting in Section 2.5.4). Figure 2-8 characterizes the distribution
of growth rate determination errors. The initial buoyant mass errors were determined
as the standard error of the Y-intersect fitting parameter but their values are too low
to be visible in the plots.
For the different cell types they range approximately
between: B. subtilis: 0.7 fg - 5.2 fg (0.3% - 0.7%), E. coli: 0.3 fg - 0.9 fg (0.3%
- 0.4%), S. cerevisiae: 4 fg - 8 fg (0.05% - 0.16%), L1210: 96 fg - 18 fg (0.05% 0.13%).
2.5.4
Data analysis and curve fitting
Curve fittings to linear models were performed in MATLAB by calculating weighted
linear least square fits. The fits to the data presented in Figure 2-3 and Figure 2-7
had the form
d
= a.m+b, where the m is the buoyant mass, and experimental errors
(es) used as fitting weights (1/c?). Fit parameters for the linear regressions and their
standard errors are presented in Tables 2.1,2.2. In addition, bacterial experimental
data were fitted to a piecewise function (Fig. 2-7) of the form
= b1 for m < mc;
dm = b2 for m > mc, where mc is the mass threshold at which
a growth rate change
occurs.
Fitting was performed as described in Baumgdrtner et al., by fitting two
constant functions to the i initial points and to the n - i last points [43]. The ith fit
with the lowest sum of the squared residuals was chosen (Table 2.2).
In order to analyze long trapping events of B. subtilis (Fig. 2-6) the buoyant mass
44
was fitted to three different models:
simple linear: m = at + b;
simple exponential: m = Aceat;
bilinear: m = ait + bi for t < t0 ; m = a 2 t + b2 for t > t0 , where to is the
time of the rate change point.
Fits were performed by least square curve fits with MATLAB's lsqcurvefit function. The fit parameters results and their standard errors are reported in Table 2.4.
2.5.5
Model selection criteria
Comparison of different curve fittings was done by calculating each of the following
model selection criteria:
Adjusted Coefficient of Determination:
R2 =
1-
1
n"-
QQ
n- p-
Z(yi - 9)
(2.4)
Reduced Chi-squared:
x
1 RSS
n -p
1-
(2.5)
Akaike information criteria:
2 p(p
+ 1)
1
AIC = nn RSS + 2 p +
n
n -p+
(2.6)
Schwarz Bayesian information criteria:
SBIC=ninn
where RSS
2
i
(,
RSS
n
+plnn
(2.7)
Yif ) 2 is the residual sum of squares, yi the ith data point, Ei
Z
the experimental error of the ith data point, yif the ith point predicted by the fitted
function,
g
the mean of the data, n the number of data points and p the number of
45
parameters of the model. Perfect linear data will have R 2 = 1; the lower the X2 , AIC
and SBIC, the better the model describes the experimental data [43]. Calculations
were performed in MATLAB.
2.6
Subsequent developments
Later developments to the dynamic trapping method were accomplished by Sungmin Son and coworkers, and were applied to further study the growth of mammalian
cells [14]. In particular, second mode actuation was implemented - vastly decreasing
the position dependent error -
and long term trapping was achieved by bringing the
cells out into the microfluidic bypasses -
providing a cleaner baseline signal, thus
increasing the signal-to-noise ratio. Unprecedented precision in cellular mass accumulation was possible, leading to the identification of a decrease in the growth rate
variability at the Gi-S phase transition, possibly indicating a growth rate threshold
for maintaining size homeostasis.
Second generation bacterial SMR devices (mSMR) were produced with first mode
sensitivities of approximately 15-30 times greater (100 and 120 pm cantilever lengths,
respectively) than the original 3 x 8 pm devices. The mSMR have channel crosssections of 3 x 3 and 3 x 5 pm. New generation readout electronics developed
by Selim Olcum and coworkers allowed for the operation of these higher frequency
devices in second vibrational mode, as described in Section 1.2.1. In second mode,
mass sensitivity is approximately 6.2 times greater than in first mode, however for
growth measurements, the greater gain comes from the fact that position dependent
error can be entirely eliminated.
Preliminary trapping data with an mSMR device actuated in second vibrational
mode is depicted in Figure 2-9. The trajectory in red is of cells that are brought out
to the device bypasses (a cell division occurs and one of the daughter cells is lost),
while the cell in blue never exits the cantilever channel. When a trapped cell remains
the entire time within the cantilever, we commonly observe a baseline mismatch
before and after the transit peak, which contributes to measurement error.
46
This
260
-I.
240
220
200
180
o
160
-MMMA
- -. Lm&h&AM
I
. 1- --WTW--W-
-----
0
T-
10
20
30
40
50
60
70
Time (mins)
Figure 2-9: Two dynamic trappings of E. coli cells (strain DH5a, grown at 23 'C in
LB medium) in an mSMR operated in second vibrational mode. For the red trapping
during which, cell division occurs, - the cells are brought into the bypasses, before
being pulled in again, contrary to the blue trap, during which the cell never exits the
cantilever channel. Green bars represent buoyant mass measurement spread for a
first mode trap of equally sized cells.
baseline mismatch is not fully understood, but is thought to arise from differences
in the densities of the fluids in the two bypasses. Bringing a cell out to the bypass
before the next measurement ensures that the cell will transit the cantilever fully
immersed in that bypass' fluid and the signal becomes cleaner, with no observable
baseline mismatch. The effect can be seen in the clearly distinguishable difference
in the spread of the buoyant mass measurements between the two different growth
trajectories:
< 1.5% of the cell's buoyant mass (red) compared to > 2.5% (blue).
The green bars represent buoyant mass measurement spread for a first mode trap of
equally sized cells which can have measurement variability that correspond to
-
15%
of the cell's buoyant mass (see Fig. 2-6).
Longer buried channels and the employment of back-syphoning - a net flow of
the fluid, when the cell is in the bypass, of opposite direction to the flow in the bypass
into which the cell accidentally or purposely is flowed into -
allow for longer duration
trappings than was previously achieved.
The procedure of bringing the cell out into the bypass is only possible when the
47
Figure 2-10: Serial cantilever array - example of serial SMR devices for sequential
growth measurements with delay channels in between each device.
samples exhibit lack of, or very little mobility. When the cells are motile, they tend
to swim away from the flow path that would conduce them again into the sensor,
thus routinely lost.
2.7
Serial cantilevers
A major disadvantage of the dynamic trap method is its lack of throughput. The
device is exclusively used for a single cell at a time, making it a time-consuming
method and impractical for obtaining large data sets, which in turn makes it difficult
to identify and study population variability with significance. Profiling growth rates
within a population and correlating it with size - or other information that may arise
from further integration of the SMR -
is of usefulness, even if complete single-cell
growth trajectories are not obtainable. Serial cantilevers would provide an advantageous method to obtain this information, overcoming the throughput limitation of
the dynamic trap method, while providing a single-stream, continuous-flow mode of
operation. This makes for a more robust measurement method, which is easier for
downstream integration with other instrumentation. Such type of device was not
implemented in this thesis, however a brief comment is made for future reference.
A serial-cantilever array device consists of several sensors along the same fluidic
channel (Fig. 2-10). Between each sensor, the channel is lengthened to provide a time
delay between each buoyant mass measurements. A rectangular cross-section channel
of width w, height h < w and length L has a fluidic resistance R (kg.m-.s-1 ) [46]:
ayaL
R = wh 3
wh4
48
(2.8)
with a given by:
F
a=12 1 -
I
192h
5
ir w
rlrw1
tanh (-ii
2h
For a given pressure differential AP (Pa), the volumetric flow rate
Q
Q
(m 3 .s- 1 ) is:
= A(2.9)
R
Here a simulation of cellular growth rate measurements with a serial-cantilever
device is presented. The simulated devices have sensors with a length of 160 pm
and channel cross-section of 5 x 3 pm, while the connecting channel cross-section
is 10 x 8 pm and the distance between cantilevers is set as 10 mm. Exponentially
growing cells are modeled and their buoyant mass measured along the channel at
each sensor location. The transit times are calculated using the average flow speed
(v-
2) for a given pressure drop and the buoyant mass measurements include a
gaussian measurement error with a standard deviation of 2%. The resistance will,
of course, change with the number of devices in series as the channel is lengthened.
For all simulations n = 1000 cells are simulated and the doubling time is determined
by least squares fit of the simulated buoyant mass measurements to an exponential
curve. Simulation results presented in Figure 2-11 show median doubling times and
the spread of the data is 1 standard deviation (absolute or normalized values).
Practical implementation will have to accommodate experimental conditions that
are not described in this simulation. For instance, although flow is laminar, Poiseuille
flow has a parabolic fluid velocity profile and since the cells have finite dimensions,
may possess asymmetrical shapes, may interact with the channel walls and may be
motile, their flow trajectories will most certainly be altered in many ways. As such,
cells in a streamline may see their order changed and the transit times between devices
altered. An ideal system would have a good way to track cells, for example, by imposing low concentrations, or by optically monitoring the cell positions. Furthermore,
long channels have higher fluidic resistance, which will make cleaning procedures more
difficult in case of clogging.
49
b)
-32
1 psi
3 psi
5 psi
10 psi
2.0
.9
3
0
8
44
26
24
2
-
-
0 -
- -980
98
8642
16
4
6
Number of sensors
8
66
10
12
14
166
12
14
16
Number of sensors
d)
8,
- - - 23 min
6
0
- - - 60 min
--4 hr
0
- - - 12 hr
0.6-
4
0.4-
li
>2
0.2 -
-
-2
-0.2-
I
/
-0.4-
:P -4
I
I
II
I,
-6
-
-0.6-
-
-0.8
5,
2
4
6
8
10
12
14
16
Number of sensor s
2
4
6
8
10
Number of sensors
Figure 2-11: Serial SMR devices for growth measurement. (a) Measured cell doubling time (tD) as a function of the number of devices in series (median ± std. deviation). Numbered labels indicate the total transit time in minutes (black) and device
transit time in milliseconds (green). AP = 3 psi and simulated tD = 23 min. (b)
Doubling time for varying pressure drop conditions. Simulated tD = 23 min. Labels
are the same as in previous plot. (c) Normalized doubling time standard deviation
as a function of the number of sensors for varying cell doubling times. (d) Close-up
of the previous plot. - For all simulations n = 1000 cells.
2.8
Conclusion
The finding that growth rate is size dependent suggests that these bacteria, yeast
and mammalian cell types must actively balance their growth and division, and is
inconsistent with growth models describing linear growth. If growth and division
rates were not coordinated in cells with size-dependent growth, cell size variation in
the population would continually increase [44], which is not the case. While molec50
ular mechanisms coordinating growth and division have been described in yeast and
bacteria
[45],
such mechanisms have not yet been fully characterized in mammalian
cells.
The dynamic trapping method described in this work is envisioned to contribute to
the study of many cellular processes (e.g. growth, the cell cycle, autophagy, apoptosis,
cell differentiation) as well as cellular models of disease states. Subsequent iterations
of this system have provided more experimental power and are expected to continue to
expand in capabilities. For mammalian cells, optical access to the trapped cell allows
dynamic cellular and molecular information to be garnered from fluorescent reporters
and to be correlated in real-time with cell growth. In addition, it will be possible to
simultaneously measure the buoyant mass, volume and density of a trapped growing
cell by periodically modulating the solution density within the SMR. This technique
has since become a prime application of the SMR and it is the most precise method
for measuring cellular mass changes at the individual level.
51
2.9
Tables
Table 2.1: Fit parameters for the linear regressions presented in Figures 2-4 and
2-5.
n
Cells
B. subtilis
100
E. coli
48
E. coli (23'C)
15
S. cerevisae
36
Mouse lymphoblasts
39
R2
x2
0.8548
3.857
1
0.6469
11.814
1
0.3032
3.497
0.8092
5.557
0.6783
1.969
Parameters
a =
b=
a =
b=
a =
b=
a=
b=
a =
b=
4
6.00 x 10- t 2.6 x 10-4 s-0.0179 ± 0.0109 fg.s-1
5.62 x 10- 4 ± 6.4 x 10-4 s-0.0095 ± 0.0098 fg.s-1
1.50 x 10-4 ± 5.0 x 10-4 s-0.0048 ± 0.0085 fg.s 1
0.608 ±0.049 hr 1
-0.964 ± 0.306 pg.hr-1
0.055 ± 0.006 hr-1
-0.901 ± 0.388 pg.hr-1
1
Table 2.2: Fit parameters and model selection criteria values for linear and stepwise
function curve fits to bacterial trapping data. Fit curves are superimposed to data
presented in Figure 2-7.
Cells
Model
n
B. subtilis
Linear
100
Stepwise
Parameters
a = 6.00 x 10-4±2.6 x 10-4 S-1
b = -0.0179 ± 0.0109 fg.s-1
b1 = 0.159 ± 0.027 fg.s-1
b2 = 0.281 ± 0.010 fg.s'1
R2
x2
AIC
SBIC
0.8548
3.857
137.08
142.17
0.5217
12.701
257.38
264.94
0.6469
11.814
120.75
124.22
0.6051
120.80
233.56
238.65
me = 378 fg
E. coli
Linear
Stepwise
48
a = 5.62 x 10-4 i6.4 x 10-
4
b = -0.0095 ± 0.0098 fg.s-1
b1 = 0.056 + 0.019 fg.s-1
b2 = 0.103 ± 0.005 fg.s'1
mc = 166 fg
52
s-
1
Table 2.3: Culture doubling times obtained by standard techniques and estimated
from SMR data.
Cells
B. subtilis
E. coli
E. coli (23'C)
S. cerevisae
Mouse lymphoblasts
Doubling time from SMR data
Culture doubling time
19.3 min
20.6 min
77 min
1.1 hr*
12.6 hr
20 ±1 min
26 ± 3 min
65 ± 2 min
1.60 ± 0.04 hr
12 hr
Cellular doubling times are estimated with the assumption of simple exponential growth. The
doubling time is then td =
2, where a is the respective slope of the linear fits in Figures 2-4
and 2-5 (listed in Table 2.1). *Several factors complicate the calculation of a doubling time from
single-cell growth rate data for yeast: mother and daughter cells have unequal sizes at cell division,
mothers and daughters have different doubling times, and daughters more than double their mass
during their first cell cycle [43,47. Without information about each cell's age and position within
the cell cycle, the simple exponential fit of the experimental data represents only an approximation
of the overall growth rate, and the resulting calculated doubling time (1.1 h) has only fair agreement
with the actual bulk culture doubling time (1.6 h).
53
Table 2.4: Fit parameters and model selection criteria values for linear, bilinear and
exponential curve fits to long trapping events of B. subtilis. Data are presented in
Figure 2-6
Curve 1 (Fig. 2-4 a)
Data points: 764
Duration: 23.3 mi ns
Initial buoyant ma ss: 313.3 fg Final buoyant mass: 672.7 fg
Fit parameters:
Linear:
a = 0.228 ± 0.001 fg/s; b = 308.23 t 0.99 fg
Exponential: Ao = 325.41 ± 0.70 fg; a = 4.91 x 10- 4 ± 2.3 x 10-6 s-1
Bilinear:
ai 0.199 t 0.002 fg/s; bi = 318.90 ± 1.00;
a2
0.269 i 0.004 fg/s; b2 = 264.40 i 5.03 fg; te = 779s
Model
R2
X2
AIC
SBIC
Linear
Exponential
Bilinear
0.9785
0.9842
0.9841
183.47
134.72
135.33
3984.01
3748.07
3945.58
3993.26
3757.33
3964.08
Curve 2 (Fig. 2-4 b)
Data points: 814
Duration: 24.0 mi ns
Initial buoyant ma ss: 395.1 fg Final buoyant mass: 786.6 fg
Fit parameters:
Linear:
a = 0.228 t 0.001 fg/s; b 358.01 ± 1.16 fg
5.077 x 10~4 t 2.2 x 10-6 s-1
382.31 ± 0.80 fg; a
Exponential: Ao
Bilinear:
ai 0.229 t 0.004 fg/s; bi = 376.28 ± 1.26;
a2
0.316 ± 0.003 fg/s; b2 = 325.69 ± 3.01 fg; t, = 585s
Model
R2
X2
AIC
SBIC
Linear
Exponential
Bilinear
0.981
0.9862
0.9862
263.7
191.39
192.09
4539.9
4279.03
4509.52
4549.29
4288.42
4528.28
Curve 3 (Fig. 2-4c)
Data points: 546
Duration: 14.9 mins
Initial buoyant mass: 336.0 fg Final buoyant mass: 492.3 fg
Fit parameters:
Linear:
a = 0.185 ± 0.002 fg/s; b = 318.37 ± 1.09 fg
Exponential: Ao = 323.67 ± 0.94 fg; a = 4.64 x 10- 4 ± 5.13 x 10-6 s-1
Bilinear:
ai 0.160 ± 0.007 fg/s; bi = 324.24 ± 1.48;
a 2 0.211 ± 0.005 fg/s; b2 = 301.27 ± 3.25 fg; tc = 449s
Model
R2
X2
AIC
SBIC
Linear
Exponential
Bilinear
0.9329
0.938
0.9378
164.92
152.35
152.96
2789.6
2746.33
2768.45
2798.18
2754.91
2785.59
54
For the three curves, the exponential model scores as the one that better describes the experimental
data. Since the SMR used for the bacterial measurements is opaque, one cannot visually inspect
the cell in order to have cell cycle cues. Therefore, one cannot determine that these trapping curves
represent the entirety of a cell cycle or even that the initial buoyant mass is of a newly born cell. For
the two first curves, which span a duration consistent with the culture doubling time, the buoyant
mass essentially doubles. In addition, the initial buoyant masses are located on the lower end of the
buoyant mass spectrum. Therefore it is likely that single cells were being trapped for the majority
of the trapping events.
Table 2.5: Simulation parameters used to generate Figure 2-7.
Parameter
Value (B. subtilis)
Value (E.coli)
cell growth parameter
exponential [bE
p
n 2 + tD = 5.69 x 10-4
a 10%.y
p =mD + tD = 0.1 7
p
a
10%.yL
a
18%.y
255fg
15%.y
p
a
86fg
16%.y
linear [bL]
ln 2 +
0'
18%.y
p
doubling time
20.3 min
26.1 min
probability of segregation [PD]
85%
82%
error in determination of the
growth rate
p=0
0- = 11%
y= 0
a = 7.7%
error in determination of the
cell size
P=0
a = 0.7%
= 0
a= 0.4%
number of cells
1000
1000
[tD]
=
55
= 4.43 x 10-4
= MD + tD = 0.55
cell starting buoyant mass
(at t=0) [mo]
a
tD
56
Chapter 3
Mechanical traps for weighing
single cells in different fluids
3.1
Introduction
Early SMR implementations could provide the buoyant masses of cells in a population but could weigh each cell only once and were unable to monitor single cells
over time [48].
Later systems added fluidic controls to implement dynamic trap-
ping, during which an individual cell is repeatedly passed back and forth through the
SMR. When maintained over extended time periods, this dynamic trap can measure
the growth of individual cells in real time (see Chapter 2). But delivering chemical
stimuli to a dynamically trapped cell is challenging because viscous-dominated flow
inside the microchannel ensures that the cell moves along with the surrounding fluid.
Grover, Bryan et al. showed that by loading the SMR device with two different
fluids, single cells can be dynamically trapped within the two fluids, weighed in the
first fluid and then weighed in the second fluid [49]. This technique is well suited
This chapter was done in collaboration with several other people and was published as Y. Weng,
F. F. Delgado et al. "Mass sensors with mechanical traps for weighing single cells in different
fluids" Lab Chip 11:4174-4180, 2011. In particular Yaochung Weng implemented fluidic exchange
with concurrent growth measurements on columned SMRs (not described in this chapter, refer to
publication).
57
for measuring the density of single cells by weighing them in two fluids of different
densities. However, their technique is unsuitable for measuring the response of a
cell to chemical stimuli because the duration of exposure of the cell to the second
fluid is limited to only a few seconds, and the growth rate of a single cell cannot
be measured both pre- and post-treatment with a drug. Finally, the buoyant mass
measured by this method is subject to uncertainty caused by variations in the cells
flow path through the cantilever.
A novel strategy for overcoming these limitations is to physically trap each cell
within the SMR. The buoyant mass of the cell could then be monitored continuously while the fluid surrounding the cell is changed at will. Several designs for cell
traps were considered: methods such as standing acoustic waves [49], dielectrophoresis [50,51] and optical trapping [52,53] have their appeal in that no cell contact is necessary and that the action of capturing and releasing a cell can be readily controlled.
However, while it is possible to generate forces on the order of tens to hundreds of
piconewtons by applying acoustic or electromagnetic waves in an ordinary microfluidic channel, the multilayered geometry and complex design of the SMR device make
it difficult to implement these techniques. Mechanical structures involving cell-sized
docks preceding a constriction [54,55] and U-shaped trapping compartments [56-59]
proved to be more robust cell capturers for our application.
Mechanical trapping structures integrated with the SMR can effectively load and
unload a single cell while its buoyant mass and the density of the surrounding fluid
are continuously monitored. Two types of mechanical traps were evaluated and here
one is described, referred to as three-channel SMRs (Fig. 3-la,b), which proved to
be most suitable for single-cell density measurements. A second type, referred to
as columned SMRs (Fig. 3-1c), was used to measure cell growth before and after
exposure to a drug and is described elsewhere [60].
58
Figure 3-1:
with different
slit and (b) 8
not described
3.2
c)
b)
a)
Top perspective of SMRs with mechanical traps. Three-channel SMRs
three-channel geometries: (a) 3 x 8 pm device with a 200 nm horizontal
x 8 pum device with a vertical 2 pm wide opening. (c) Columned SMR,
here (see full publication at reference [60]).
Method
Three-channel SMRs were fabricated with a capture dock at the apex of the cantilever
(Fig. 3-la,b) and a third fluidic channel used to control flow into and out of the dock.
At the T-junction where the dock meets the third channel (channel 3), a narrow
constriction allows fluid to pass, but not cells of the appropriate size. Two versions
of the three-channel SMR with different dimensions were fabricated: one with a 3 x
8 pm channel cross-section and a 200 nm wide horizontal slit; and another with an
8 x 8 tm channel cross-section device and a 2 pm wide vertical gap. A computercontrolled fluidic system orchestrates a sequence of pressure changes that traps a cell,
measures its buoyant mass, quickly replaces the fluid around the cell, measures its
buoyant mass a second time in the new fluid, and ejects the cell.
To prime the device, fluid 1 flows from bypass channel B1 to the cantilever. Fluid
1 carries the cells to be measured (Fig. 3-2, blue). Relative pressure settings ensure
that the majority of fluid 1 travels from BI to the second bypass B2 via the SMR
to minimize the likelihood of cells being captured in the dock. After the device is
primed, the fluid velocity through the SMR is reduced, and pressure on the bypass
B3 is lowered so that a significant amount of the fluid now flows through the third
59
B
335340
-
B
-
3 35 320
B2
B1
"---
L 331720
C
331700 r
0
1
2
3
4
Time (s)
Figure 3-2: SMR. resonant frequency response is plotted versus time as the density
of a 5 pm polystyrene bead is measured. A - SMR is filled with water. B - A bead
is trapped. The short spike at the end of the step is due to the bead rounding the
corner of the wall, before entering the pocket; C - Water is replaced by D 2 0. D The particle is ejected.
channel. Thus, a transiting cell will be directed into the pocket and captured (Fig. 37 inset).
Cell capture is detected as a stepwise change in the resonant frequency
due to the change in mass inside the cantilever (if the cell sinks, mass increases and
frequency decreases; if the cell floats, mass decreases and frequency increases). Cell
immobilization eliminates position uncertainty, a source of error, which exists when
measuring samples in a flow through mode.
After a cell is trapped, the computer reverses the flow to flush the SMR with fluid
2 (Fig. 3-2, red) from bypass channel B2. Fluid 1 is completely rinsed out of the
cantilever, leaving the cell immersed in fluid 2. Prior to removing the cell from the
trap, bypass B3 is filled with fluid 2, and a constant flow is maintained. This clears
out remnants of fluid 1 that have previously exited through the third channel and is
crucial for preventing the two fluids from mixing during the ejection step. The control
system then gradually pressurizes the third channel until the cell leaves the dock.
After the cell is ejected, there is another step in the resonant frequency corresponding
to the buoyant mass of the cell in fluid 2. The gradual increase in pressure is used
to decouple the change in mass from any perturbations of resonant frequency that a
60
pressure change might induce. The entire sequence of events takes 3-5 s. The duration
of serial measurements depends on cell concentration. Smaller concentrations of cells
will increase the delay between measurements, but high concentrations will lead to
multiple cells being trapped.
With two buoyant mass measurements and two measurements of the fluid density,
the cells density can be determined. The device is calibrated with fluids of known
density, allowing the density of fluids 1 and 2 to be accurately determined. The cell
density is, therefore, determined as:
Peell
- mbuoyant2Pfluidl mbuoyant2 -
mbuoyantlPfluid2
mbuoyantl
(3.1)
Measurement error in the cells density is affected by the choice of densities of fluids
1 and 2. If the reference fluid densities are close, the buoyant mass values in both
fluids will also be close, and the measurement error will play a more significant role
in Eq. (3.1). This error was minimized by choosing reference fluids as far apart in
density as was convenient. Furthermore, care was taken to bracket the samples density
between the fluid densities (fluid 1 is less dense than the cell, and fluid 2 is more dense
than the cell). In order to calculate volume and mass, the sensitivity of the SMR,
relating buoyant mass to cantilever resonant frequency change, was determined with
NIST size standard beads as previously reported [25]. Note that since the buoyant
mass calibration factor affects all the buoyant mass terms proportionally in Eq. (3.1),
buoyant mass calibration is not actually required for measuring cell density.
3.2.1
Position dependent error
The three-channel cantilever is capable of eliminating position dependent error arising
from the different possible flow paths that a transiting cell can take. However, a
position dependent error, of lower magnitude nevertheless, might still arise in certain
conditions. In particular, if a sample is deformable, such as a cell, it can be compressed
against the dock wall by the pressure exerted on the particle by the fluid flow that
traps it. The mass spatial distribution is then altered. A simulation for the change
61
a)
2pm
b) 6%
- 0.6
0.5 C.
5%
10pm 8pm
3%
0.3
2%
F0.2
1%
0.1
0%
0
0
200 pm
3
3.5
1
1.5
2
2.5
0.5
Center of mass deviation from dock base (pm)
Figure 3-3: Position dependent error in three-channel devices. (a) Diagram and
dimensions of a three-channel device. (b) Calculated change in resonant frequency and corresponding buoyant mass in medium - for a yeast cell, whose center of mass
position changes due to cellular compression.
in buoyant mass caused by a moving center of mass of a yeast cell is depicted in
Figure 3-3b. In addition, this compression effect can lead to asymmetric frequency
changes in the loading and unloading of a cell, which can be inconvenient especially
in the case when two fluids of different densities are used.
Finally, in some conditions, it was observed that the sample could mildly adhere
to the dock walls, which would lead to it rolling along the wall before being fully
ejected. This can contribute to a position dependent error caused by the fact that
the ejection resonant frequency step is referenced to a position distinct of the capture
resonant frequency step.
3.2.2
Windowed devices
Devices with etched and subsequently oxidized regions of the cantilever lid were tested.
The oxidized silicon is transparent to visible light allowing for better imaging of the
samples at the dock as well as fluorescence readout (Fig. 3-4). Most windows revealed
integrity under normal usage conditions, however on several occasions breakdown was
observed. These devices exhibited higher noise levels than windowless ones and were
not used on the experiments described below. Furthermore, without piezoresistive
readout, fluorescence microscopy at wavelengths close to the one of the readout laser
are compromised, requiring the laser to be turned off. Imaging elsewhere along the
device microchannel, outside of the cantilever region 62
without surface deformation
Figure 3-4: Windowed three-channel devices. Silicon oxide window at the tip of
the cantilever makes the device lid locally transparent to visible light. Fluorescence
measurements can be made while the sample is immobilized in the dock.
from caused by the vibration -
3.3
should be considered in future iterations.
Materials
Buffers - Yeast was grown and measured in yeast extract plus peptone medium supplemented with 2% glucose and 1 mg-ml
Bertani (LB) broth (Sigma-Aldrich L2542).
adenine (YEPD) and bacteria in LuriaFor the secondary fluid for the den-
sity measurements we used Milli-Q ultrapure water, deuterium oxide (Sigma-Aldrich
151882) and a 1:9 dilution of 10 x PBS (Omnipur 6505) in high-density Percoll (SigmaAldrich P4937, modified as reported by Grover et al. [49])
Cells - Saccharomyces cerevisiae cells (strain A2587) were grown in YEPD at 30
'C with agitation and measured about 2.5 hours after the culture had been started,
prior to beginning of exponential growth phase. Escherichia coli (ATCC 25922) were
grown in LB overnight at 37 'C then diluted 1:100 in 1 h before the measurement.
63
a) 4
b) 80
78
*
4.4d
*
76
4.;
4.
0-
*
74
s*eo
00
3.
72
J% )"O
.
.
83.
*
*
0
0
0
6
0
0000
S
66
3.
3
1.047
1.048
0
1.049
1.050
1.051
1.052
0
720
*
1.053
64
1.047
1.048
em
1.049
0
C.
1.050
1.051
1.052
1.053
Density (g-cm~)
Density (g-cm-3)
Figure 3-5: Density and mass of polystyrene size standard beads measured in two
different three-channel devices. Mean densities: (a) 1.9 pm particles p = 1.0497 ±
0.0010 g-cm-3, CV = 0.09%, n = 231 (b) 5.003 pm particles p = 1.0491 ± 0.0008
g-cm-3, CV = 0.08%, n = 247
Polystyrene Beads - The beads used in the measurements and calibrations were
the size standards from Bangs Labs NT17N (1.9 pm) and Thermo Scientific 4205A
(5 pm).
Results
3.4
Measurement of polystyrene bead density
3.4.1
The technique was first applied to polystyrene beads, which were measured in water
(p = 0.9983 g-cm- 3 ) and deuterium oxide (p = 1.1046 g-cm- 3 ). All experiments were
performed at 23.3 'C. The measurements, shown in Table 3.1 and Figure 3-5 for
1.9 tm and 5 pm beads, were carried out in the 3 x 8 pm and 8 x 8 pm devices
respectively. The results, p1.9 = 1.0497 ± 0.0010 g-cm- 3 and p5 = 1.0491 ± 0.0008
g-cm-
3
(mean ± standard deviation), matched the reported density of polystyrene
(~1.05 g-cm- 3 ). In addition, the population statistics of the calculated diameters
were determined by assuming the volume of a sphere in Eq. 3.1.
Both samples
showed lower standard deviations and coefficients of variation than the ones reported
by the manufacturers (Table 3.1).
64
Table 3.1: Density and population statistics of the diameter of polystyrene beads.
Diameter was calculated from Eq. 3.1 assuming the volume of a sphere. Mean
values (*) of diameter are the same by definition as they were used as buoyant mass
calibration, therefore are merely indicative.
Diameter
Density
Sample
3
[g.cm- ]
1.9 pm
(n = 231)
5 pm
(n = 247)
3.4.2
Mean
St. Dev
CV (%)
Mean
St. Dev
CV (%)
[pm]
1.0497
1
x10-
3
0.09%
1.0491
8 x10-4
0.08%
measured
datasheet
1.9*
0.03
1.40%
1.9*
0.03
1.60%
5.003*
0.04
0.80%
5.003*
0.05
1.00%
Measurement of single-cell density
The method was also used to measure the density of S. cerevisiae cells with an 8 x
8 pm SMR (Fig. 3-6). The results obtained by consecutive measurements of the cells
in their medium (p = 1.0182 g.cm-3) and PBS:Percoll (p = 1.1667 g-cm-3) are shown
in Figure 3-7. An average cell density p = 1.1042 ± 0.0057 g-cm- 3, CV = 0.59%,
n = 244 (totaling 2 runs) was determined. This value is in accordance with single-cell
yeast density measurements obtained through other methods (Table 3.2). The two
experimental runs were measured from two samples taken from the same culture, one
hour apart. The measured densities were p = 1.1049
0.0068 g-cm- 3, CV = 0.62%
Figure 3-6: Micrographs of two differently sized, non-budding S. cerevisiae cells
immobilized in the trap. Scale bar: 10 pm.
65
240
220
200
180
160
00 0
140
120
~~46
100
60
a
0
s
-75
.6020 - 45
-. 1.045 ,1.0 60
1.09
0
J1
.0
0
- 90
40
0
a.,
0.
80
S
1.055.
1.10
0
0
0
I
1.11
1.12
Density (g.cm~f)
Figure 3-7: Density and mass of S. cerevisiae cells. Mean density p = 1.1042
t 0.0066 g-cm-3, CV = 0.59%, n = 244. inset plot Density and mass of 5 pm
polystyrene beads measured in the same conditions as the cells. Scales are the same
as the main plot. The biological variability as determined by the spread of cell
measurements is less than the instrument variability as predicted by the measurement
and dispersion of bead samples.
for the earlier one (n = 132) and p = 1.1033 t 0.0061 g cm-3, CV = 0.56% for the
later sample (n = 112). The results for calculated mass are m = 95.08 ± 46.30 pg,
CV = 48.7% and volume V = 78.1 ± 35.3 pm 3 , CV = 45.2% and V = 95.9 ± 47.5
pm3 , CV = 49.6%, respectively. Volume was also measured with a Coulter Counter
(Beckman Coulter, Multisizer 4), V = 78.5 ± 49.2 pm3 , CV = 62.6% and V = 91.45
t 58.5 pm 3 , CV = 64.0%, respectively (n = 20,000 cells).
Table 3.2: Reported measurements of yeast density.
Reference
Density
Method
[g.cm-3]
Reu et al. [61]
Aiba et al. [62]
Haddad et al. [63]
Baldwin et al. [64]
Bryan al. [48]
current result
1.0952 ± 0.011
1.090 t 0.0112
1.087 t 0.026
1.1126 t 0.010
1.1029 ± 0.0026
1.1042 i 0.0066
66
Anton Parr DMA 45 densitometer
settling velocity
settling velocity
density gradient centrifugation
SMR + Coulter Counter
SMR
3.4.3
Growth of captured cells
1023-
1024
1028
1029
0
10
20
30
40
50
60
70
80
90
100
Time (min)
Figure 3-8: E. coli grown inside the SMR, while immobilized at the pocket of
a three-channel device. Cells enter at time 0, and baseline is recovered after cell
ejection at ~95 min. No noticeable growth is observed during the first -20 min.
Raw data (maroon), exponential curve fit (green), apparent baseline drift (blue).
Attempts were made to observe cellular growth at the tip, with a cell immobilized
in the pocket.
recorded.
The measurement is passive and only the cantilever frequency is
Since several factors can influence resonant frequency, such as changes
in density of the fluid or variations in temperature, these sources of noise have to
be negligible in order for the measure to be successful, unless a second cantilever
or density tags are used as reference. These solutions, however, are not trivial to
implement.
Nevertheless, once the cell is parked and adequate fluidic conditions
achieved, the system is immune from pressure fluctuations. If the noise sources are
not completely eliminated this method is unlikely is less useful to follow individual
cell growth trajectories. However, it can be useful to detect growth, or absence of it,
in a rapid manner, with low cell concentration and with controlled exposure to drugs.
The experiment was conducted by loading the cells through one bypass and when
captured at the pocket, the fluid flow was reversed and clean media was injected from
the other bypass. This ensured that no other cell would enter the system and disrupt
67
the experiment. A slight pressure drop across the dock is needed to avoid the cell
drifting away. For more than 15 attempts, growth of E. coli cells was only observed
in one case, in which, out of curiosity and lack of results for single cells, more than
one cell was loaded into the pocket. The result is shown in Figure 3-8. Although a
doubling time of about 23 minutes is determined, consistent with culture doubling
time, there is a lag phase of about 24 minutes where no growth seems to occur. This
is somewhat puzzling as after this lag phase, the cells appear to grow normally.
Similar attempt was made with S. cerevisiae cells, without success, either for
singularly trapped cells or for small clusters (2-5 cells). In order to identify which
factor or factors were conditioning cellular growth, the experiment was ran with
the device not being actuated and growth inspected optically constant subjection to vibration acceleration, turned off -
in order eliminate
with the laser and microscope light
to eliminate possible local heating or radiation exposure -
and with
conditioned media - in case specific growth factors were need for a single cell to grow
optimally. Nevertheless, no growth was observed. Possibly, there are uncontrollable
flow conditions or surface properties that are constraining cellular growth.
Yet, as with E. coli, there was a puzzling case when a cluster of cells did grow
in a device left loaded overnight. As the time-lapse images depicted in Figure 3-9
demonstrate, there is accumulation of biomass inside the cantilever: the cells identified
in red and green are pushed towards the top bypass entrance (reservoir of fresh media),
while the cell labeled in green remains essentially immobile.
This indicates that
cells are growing and dividing throughout the cantilever, in particular at the tip,
and that the growing cells push each other along the free room in the channel. It
should be noted that the cells in the lower channel are more compact, due to the
fact that this is the channel connected to the lower bypass (reservoir of media with
actively growing cells), and therefore became crowded faster, leaving less room to be
freely occupied by the growing cells. Also, since the cell labelled in green is almost
immobile (even moving slightly towards the lower bypass entrance, at the last image
of the squence) the movement indicated by the displacement of the red and blue cells
cannot be an effect of net fluid flow. In fact the fluid flow is set up in the opposite
68
I
Figure 3-9: Growth of S. cerevisiae cells inside the cantilever (base). Time series
demonstrating that mass accumulation occurs within the device as no cells are entering from the culture bypass side (green cell, essentially immobile) and cells are being
pushed towards the clean media bypass (red and blue cells).
69
direction, to ensure only clean media is driven into the device. In this case, the
device was being actuated and the resonant frequency read with the optical lever;
microscope illumination was also present and no conditioned media was used. This
is not consistent with any of the hypothesis for lack of growth of individual or small
clusters of cells and did not shed light on what was constraining their growth in the
previous attempts.
3.5
Conclusion
The results demonstrate that the described mechanical traps can be used to immobilize cells inside the device and that effective fluidic buffer exchange can be accomplished. The method can successfully and accurately be used to determine the density,
mass, and volume of single cells to the extent that osmotic shock can be avoided or
minimized and that the density of the cell being measured can be bracketed by the
appropriate solution densities. More than one cell can be trapped at the same time if
a cluster of cells enter simultaneously or if the flow reversal time is not short enough
to prevent an additional cell from being captured. For density measurements, the
capture of multiple cells will result in a measured average, which will mask the variability amongst those cells. However, these events can be detected optically or by
size signatures and can be rejected by data analysis.
Further attempts were made to achieve the same measurement on bacterial cells
(E. coli) with the 3 x8 pm SMR. However, although single bacterium can be trapped
within the pocket, ejection proved to be difficult without large pressure differentials
due to cell adhesion to the walls. Further developments such as bacteria-resistant
surfaces [65] are still required to successfully measure the density of single bacteria.
70
Chapter 4
Intracellular water exchange for
measuring the dry mass, water
mass and changes in chemical
composition of living cells
4.1
Introduction
The dry and wet content of the cell as well as its overall chemical composition are
tightly regulated in a wide range of cellular processes. Bacteria and yeast increase
their ribosomal RNA content to achieve faster growth rates [66-69], the wet and dry
content of yeast can change disproportionately during the cell cycle [70-72] and the
water content of mammalian cells is reduced following apoptosis [73].
Despite the
fundamental significance of these physical parameters, the techniques for measuring
This chapter was done in collaboration with several other people and was submitted for publication as F. F. Delgado, N. Cermak et al. "Intracellular water exchange for measuring single cell
dry mass and changes in chemical composition", PLoS ONE, 2013. In particular, Nathan Cermak
contributed to the error analysis of the measurement and gathered the bacterial data and Vivian
Hecht gathered mammalian cell data.
71
a)
water content
whole cell
41F -
f
Pfluid
M I
buoyancy
t
i
b)
biomass
C)
J.,
U)
CO
0
4-0
density
Intracellular water
buoyant mass
dry
I
dry density
mas..--
C
0
total/
U
ci)
E
0
/
~a)
C)
m3
C:
/
a
I-
total
mass
Fluid density
Figure 4-1: Buoyancy of a cell in fluids of different densities and membrane permeabilities. (a) In an H2 0 or D2 0 based fluid (1 or 3), the cell sinks as a result of the dry
content's density being higher than the surrounding fluid. In a dense impermeable
fluid (2), the buoyancy of the cell's water content dominates and the cell floats. (b)
The pairing of the different buoyant mass measurements allows the determination of
different biophysical parameters of the cell as shown in the plot (not to scale).
them directly, particularly in living cells, are limited. Dry and wet mass are typically
obtained by weighing a population before and after baking to remove the intracellular water [74]. Although dry mass can be measured in living cells by quantitative
phase microscopy [75], the conversion factor between refractive index and dry mass
concentration must be known. While this factor is similar for most globular proteins
72
(typically varying by less than 5%), it can vary by almost 20% for carbohydrates
or lipids [75]. Approaches based on vibrational spectroscopy can provide chemical
composition of living cells [76], but do not reveal the dry and wet mass.
To address these limitations, an approach was developed that exploits the high
water permeability of cellular membranes for obtaining the water mass, dry mass, and
an index of chemical composition for living cells (Fig. 4-1). When a cell is weighed in
fluids of distinct densities -
an H2 0-based and a deuterium oxide-based (D2 0) fluid
the aqueous portion of the cell is neutrally buoyant in both measurements since
intracellular H2 0 is rapidly replaced by D2 0 upon immersion in D2 0. The paired
weighings (Fig. 4-la,b, blue and red) therefore offer direct quantification of the cell's
dry mass and its non-aqueous volume, which allows the determination of a parameter
termed dry density [77, 78] -
the density of the cell's dry material (Fig. 4-1b). If
instead the first measurement is made in an impermeable fluid as dense as D2 0, the
intracellular H 2 0 buoys up the cell. Upon immersing the cell in D 2 0, the intracellular
H2 0 is replaced by D2 0, and the aqueous portion of the cell no longer contributes
to its buoyancy. The differential between these two measurements (Fig. 4-la,b, green
and red) yields the intracellular water mass, as it excludes the dry material whose
buoyant mass is identical in both cases.
Here, this approach is validated and used to measure the dry mass and dry density
of various cell types, from microbes to mammalian cells. Dry density is related to
the chemical composition of cells: it is an average of the densities of the different
components of the cell's biomass (RNA, proteins, lipids, etc.) (Table 4.1) weighted
by their relative amounts (Table 4.2). It is different from dry mass density, which
refers to the concentration of cellular dry mass, i.e. dry mass per unit cell volume. In
contrast to total cell density or dry mass density, dry density is independent of the
cell's water content, making the measurement invariant to water uptake or expulsion
due to osmotic pressures. Dry density is also size independent, whenever the relative
chemical composition remains unchanged.
73
Table 4.1: Density of chemical components of cells.
Density
References
[g-cm-3]
DNA
1.4-2.0
RNA
2
Protein 1.22-1.43
[79,80]
[79]
[79,81]
Table 4.2: Approximate chemical composition of a bacterium, yeast and mammalian
cell.
E. coli
% total weight
Water
% dry weight
S. cerevisiae Mammalian Cell
70
80
70
DNA
RNA
Proteins
Lipids
3
20
50-55
7-9
0.1-0.6
6-12
35-60
4-10
1
4
60
13
References
[82-84]
[85-89]
[84]
74
Dry density is shown to increase between stationary and exponential phases in
E. coli and S. cerevisiae, as might have been expected due to known changes in
RNA/protein ratio, since RNA is denser than most cellular components. Further
changes are observed in dry density of mammalian cells that are manifestations of
their different states: healthy proliferating mouse embryonic fibroblasts, FL5.12 cells
and L1210 lymphocytic leukemia cells all show higher dry density values than confluent fibroblasts, nutrient-starved FL5.12 cells and cycloheximide-treated L1210 cells,
respectively, even though in some cases their dry mass distributions do not undergo
noticeable alterations. These examples suggest that dry density may be used to determine the bulk cellular composition that is necessary for proliferation.
4.2
Measurement principle
This work builds upon a previously published method for measuring a particle's total
density, mass and volume. Like Grover et al. [49] a suspended microchannel resonator
(SMR) is used to determine a single particle's buoyant mass, defined as
mb = V(pcea - Pfluid) = m(1 - Pf)luid
(4.1)
Pcell
where V is the volume, m is the mass and pcel is the density of the particle immersed in
a fluid of density Pfluid (Fig. 4-2). One buoyant mass measurement does not constrain
either the volume or the mass of a particle, but with two sequential buoyant mass
measurements in fluids of differing densities, it is possible to solve for the particle's
mass and volume (Fig. 4-1b).
4.2.1
Distinguishing dry content from aqueous content
This method is altered by rendering the intracellular water content of a cell neutrally
buoyant in both buoyant mass measurements, allowing the paired measurements to
isolate the physical properties of the dry content alone. As described in Section 1.2.2,
this is formalized by decomposing a cell's buoyant mass into two parts - the buoyant
75
Figure 4-2: Using the SMR to measure the buoyant mass of a cell in H2 0 and
D2 0. The measurement starts with the cantilever filled with H2 0 (blue, box 1). The
density of the red fluid is determined from the baseline resonance frequency of the
cantilever. When a cell passes through the cantilever (box 2), the buoyant mass of the
cell in water is measured as a transient change in resonant frequency. The direction
of fluid flow is then reversed, and the resonance frequency of the cantilever changes
as the cantilever fills with D2 0, a fluid of greater density (red, box 3). The buoyant
mass of the cell in D2 0 is measured as the cell transits the cantilever a second time
(box 4). From these four measurements of fluid density and cell buoyant mass, the
absolute mass, volume, and density of the cell's dry content are calculated (adapted
from Grover et al. [49]).
76
mass of the dry material and the buoyant mass of the intracellular water:
mbflud
=
mdry(1
Pfluid) + Viw(piw -
-
Pfluid)
(4.2)
PcelI
where mdry and Pdry are the mass and density of the cell's dry content, or biomass,
and Viw, pi, are the volume and the density of the exchangeable water content,
respectively. Assuming that the cell is measured first in pure H2 0 and secondly in
pure D2 0, and that the intracellular H2 0 molecules are all replaced by D2 0 molecules,
in each measurement the buoyant mass of the exchanged volume (the latter term in
Equation (4.2)) is zero. The two cases yield:
mH2O
(43)
Pdry
SmbDOn
= mdry(1 -PD
2
)
and we can solve for the dry mass:
mdry = mb1P2 - mb 2 Pl
P2 - P1
(4.4)
dry volume:
Vdry =
1,b-
P2 -
mb 2
P1
(4.5)
and dry density (Fig. 4-1b):
Pdry
=
mb 1 P2 - mb 2 Pl
mb, - mb
2
(4.6)
Additionally, the method can be easily modified to determine the cell's water
content, owing to the rapid exchange of H2 0 by D2 0. A cell is first weighed in a
dense, non-cell permeable fluid such as OptiPrep (iodixanol in H2 0) and then weighed
in D2 0. If the fluids' densities are adjusted to match, the contribution to the cell's
buoyant mass of the dry content (first term in Equation (4.2)) is identical in both
77
fluids.
-
mbf
mbf
2
mdry(1
Pf luid )
Pdry
= mdry(1
Pfluid)
MdrY
+ MH 2 o0
Pfluid)
PH2
(47)
Therefore the differential measurement allows for the determination of the mass and
volume of the cell's water content, since the value is simply the buoyant mass of the
intracellular water when weighed in the non-cell permeable fluid.
mnbf ludl
mH
2o =
-
Mbfluid 2 _
lbin1
fuid
(4.8)
PH 2 0
4.2.2
Measurement error
As can be seen in Eqs. (4.4) and (4.6), mass and volume are linear combinations of
the two buoyant mass measurements. Density, however is not. Figure 4-3 shows the
contour lines for both mass and density obtained from two buoyant mass measurements in H2 0 and D2 0. Density is monotonically encoded in the angle of the two
buoyant mass measurements, however this function is far from linear. For particles
in the fourth quadrant (consisting of particles which sink in one fluid and float in the
other), the transform between buoyant masses and density is relatively linear. However when the particle density is far beyond the density of either fluid (in the first or
third quadrants), as is the case of the measurements here described, the gradient is
extremely steep and so small errors in buoyant mass generate large errors in density.
Perturbations to cellular growth is well known and [90-92]
To understand the error sources in our measurements, we first consider only the
case of errors in buoyant mass estimation. We take these to be predominantly additive
errors and estimate their magnitude by making repeated measurements on a single
cell.
In calculating mass and volume, since they are linear combinations of buoyant
masses (Eqs. (4.4) and (4.6)), the errors are also transformed linearly. Hence, for a
particle measured in H2 0 (p _ 1.0 g-cm- 3 ) and D2 0 (p _ 1.1 g-cm- 3 ) with buoyant
mass errors with standard deviation oI
the standard deviation of the resulting
mass estimate is 14 .8 7 umb. For a particle measured in fluids of density pi and P2, the
78
a)
Contours of constant density
b)
C)
Contours of constant mass
E
-1.0
NIL,
-0.s
0.0
0.6
0
1.0
135
90
46
180
-0.5
-1,0
0.0
1,0
0.8
Buoyant mass in H20
Angle (degrees)
Buoyant mass in H20
Figure 4-3: (a) Contour map of density as a function of two buoyant mass measurements. (b) In polar coordinates, the angle can be shown to map directly to density.
(c) Contour map showing cell mass as a function of two buoyant masses. This function is linear, with a gradient oriented to the lower right (higher buoyant mass in
H20, lower buoyant mass in D20).
standard deviation of rfn is:
oU
mb
(4-9)
rbi
m P2 - mb2 P1
Mb1 - mnb2
(4.10)
= V2
P2
2
1
-P1
Now we turn to the density estimator (Eq. (4.6)):
p-
If we again assume that each buoyant mass measurement includes a random error ei
(for the measurement made in fluid i), then we can rewrite the above by substituting
in mbi = m(1 - P') + Ei:
p
+ P2Ci - Pi2
m(P2 - PI) + 6i - E2
m(p2 - PI)
+ PW1 - Pi62
m(P2 - Pi) + E1 - E2
Pm(p2 - Pi)
(4.11)
(.2
Here we see that the variance of the density estimator will depend on the true mass and
density. We turn to Monte Carlo simulations to understand how a
joint distribution
over mass and density is affected by errors in buoyant mass. In particular, we take the
true
joint distribution to be constrained to only one possible density and then observe
79
how the addition of noise to the buoyant mass measurements affects the observed joint
distribution (see more details in Section 4.6.3).
4.2.3
Necessity of single-cell measurements
While previous works by coworkers and others [42,93] have used population means
to estimate mean particle density, this method will not work for estimating dry density. The method used in these publications amounts to first measuring hundreds to
thousands of particles in fluid 1, followed by measuring a similar number in fluid 2.
The mean buoyant mass of each population is calculated, yielding P, and p 2 , which
are then used as mb, and mb2 in Equation (4.6). However, this measure relies on a
reasonable degree of certainty in
si and A2, which
in turn depends on several factors:
" the actual dispersion of the population masses, om
" the sample sizes used, ni and n2
"
the magnitude of the error in a single measurement, o-
We are interested in the mean of the buoyant mass distribution in fluid i. Since
the standard deviation of mass measurements in fluid i is given by:
o62
- 6)2
P +
F-m
=o(1
(4.13)
and we assume measurements are independent and identically distributed, the standard error of 42 is:
ot(1
SEg =
-
a.)2+UE
P
n
(414
(4.14)
For very monodisperse particles (as were typically measured in previous work [42,93]),
this error is dominated by o-, which is typically quite small. In the case of populations
of cells, populations are often very heterogeneous (CV ~ 33% for typical E. coli
samples), and so om dominates to such a degree that to achieve high precision in
42
requires tens to hundreds of thousands of cells, a number currently beyond the
throughput of the SMR within a several-hour experiment.
80
Making repeated single-cell measurements provides a much more accurate estimate
of the true dry density since it avoids the problem of the population variance each cell is measured in two fluids individually, and those measurements are paired
together, allowing density determination without requiring population parameters.
4.2.4
Evidence for complete fluid exchange
If the intracellular H2 0 molecules were not being mostly replaced by D2 0 molecules,
then one would expect to measure a lower density -
in the limit of no exchange
occurring at all, then the dry density would be equivalent to the total density of
the particle and total density values for cells are known to be well below 1.3 g-cm-3
[49,94]. While the exchange of 100% of the intracellular water at the time of the the
second measurement cannot be fully ascertained, a statistically significant correlation
between time spent in D2 0 and dry density for E. coli was not observed (Fig. 4-4).
In yeast, of four replicate experiments, only once was a statistically significant
correlation between dry density and time spent in D2 0 observed, however the correlation explained only 5% of the variance in dry density, and suggested that the dry
density was changing by 0.003 g-cm- 3 .s-
1
(Fig. 4-5). This suggests that at a bare
minimum, 2-3 seconds after immersing a cell in D2 0 that the exchange process has
reached an asymptote, which we think is likely to be near complete water exchange.
This is consistent with previous findings [66].
81
1.60
ef= 1.55
1.60-
-
R2= 0.06
-
1.50 -
1.55 -
I
.0
1.60R2 =0.01
1.50
-
I
a
0
1.45 -
1.45-
1.40 -
1.40
1.35
P=0.014
1.30
-
I
2
I
4
I
6
I
8
1.50
-
R2 =0.01
1.45-
1.35
-
1.30
-
P=0.11
P =0.096
1.30-
I I
10 12
2
Exposure time (a)
1.60
-
1.40-
-
1.35
-
1.55
4
6
8
10 12
I
I
2
4
Exposure time (a)
1.60
-
I
8
I
I I
8 10 12
Exposure time (a)
1.60
-
-
2
R2= 0.02
1.55 -
1.55
1.50 -
R2 =0.01
*
-
SR2 =0.05
1.55 -
1.50 -
1.80 -
1.45-
1.45-
1.40-
1.40-
4b
of0
0,
CD
I
1.45V
1.40-
C
1.35
*
*
1.35 -
-
P =0.9
1.30
-
I
2
I
4
I
6
I
8
I I
10 12
1.35
-
1.30
-
P =0.84
1.30 -
2
Exposure time (a)
P=0.016
I I I I I
4 6 8 10 12
2
Exposure time (a)
2
R2= 0.00
2
R2= 0.03
1.55
1.60
-
f.. 1.55
-
1.50
-
1.50
Cn 1.50
1.45
1.45
1.45-
1.40
1.40
1.40-
1.35
1.30
0)
1.35
P =0.44
I
I
I
I
2
4
6
8
I
I
10 12
Exposure time (a)
4
6
8
10 12
Exposure time (a)
1.60
1.60
1.55
JS
RR22=0.01
1.35P =0.036
1.30
P=0.19
1,30-
2
4
6
8
10 12
Exposure time (a)
2
4
6
8
10 12
Exposure time (a)
Figure 4-4: Time between measurements (exposure time) versus calculated dry
density for single cells in each of nine analyses of E. coli samples (2-3 technical
replicates for each of 4 samples). Assuming the cell was nearly immediately immersed
in D2 2 0 after the first measurement, this should be a good approximation of time
spent in D2 0. Line shows ordinary least squares fits, which agreed well with robust
fits (Huber weights). Correlations are all statistically insignificant at a = 0.05 (a =
0.006 for each test, using Bonferroni correction). P-values are given for slope being
non-zero using one-sided t-test.
82
1.8 -
1.8 1.7 -
2
1.6 -
1.6 g
f
1.5 -o
C
2
R =0.05
1.7 -
R =0.00
,
1.4 -
L
1.3 -
1.5 1.4 1.3 -
P= 0.34
1.2 -
I
I
I
I
2
4
6
8
P = 2.4e-05
1.2 -
10
10
2
Exposure time (s)
1.6 -
I
C
1.3 -
1.6 -
ae
-I
~0
Rer-
1.5 1.4 1.3 -
P =0.033
-
10
R2 .0.0
1.7 -
1.4 -
1.2
8
1.8 R2 .0.01
1.7 -
1.5 -
6
Exposure time (s)
1.8 -
0)
4
2
2
P 0.46
I
I
I
I
4
6
8
10
1.2 2
Exposure time (s)
4
6
8
10
Exposure time (s)
Figure 4-5: Time between measurements (exposure time) versus calculated dry
density for single S. cerevisiae cells in four experiments. Line shows ordinary least
squares fits, which never account for more than 5% of the total variance. Because
these experiments were done three-channel devices, much more precise control over
exposure time could be achieved, and this parameter was deliberately varied, yielding
the discrete times seen above. Only one experiment showed a statistically significant
correlation (a = 0.05/4 = 0.0125 using Bonferroni correction). P-values are given for
slope being non-zero using one-sided t-test.
83
Aqueous, non-aqueous and total cellular content
4.3
As an initial test of the method, the water content, dry content and total content of
individual cells from a sample of early stationary E. coli were separately determined.
Since the measurement time typically exceeded several doublings of the culture, cells
were fixed to ensure synchrony. The single-cell water mass distribution was determined by sequentially measuring the cells in OptiPrep:PBS (p = 1.101 g-cm- 3 ) fol_
lowed by D2 0:PBS (p = 1.101 g-cm
3 ),
as exemplified in Figure 4-6a. The median
water content in these cells was 516 ± 12 fg. Then cells were sequentially measured
in H2 0:PBS (p = 1.005 g-cm-3) and D2 0:PBS to obtain the dry mass distribution
(exemplified in Figure 4-6b), yielding a median value of 203 ± 5 fg. Finally, we measured the total mass distribution by the method of Grover et al. [49] and the median
value was 727 ± 15 fg. To ensure that the osmotic pressure experienced by the cells
was equal in both fluids of each measurement, phosphate buffered saline (PBS) was
added to the all the solutions in order to match their osmolarity.
1000.
a)
b)
1200-
1150.
920.
298
298.5
299
290.5
300
300.5
301
301.5
609.5 70
70.5
71
71.5
72
72.5
73
73.5
74
Time (s)
Time (s)
Figure 4-6: Raw data of representative bacterial density and mass measurements.
(a) Total density and mass by consecutive buoyant mass determinations in PBS:H 2 0
and Percoll:PBS. The cell floats while immersed in the second fluid. (b) Dry density and dry mass by consecutive buoyant mass determinations in PBS:H 2 0 and
PBS:D 2 0. The cell sinks while immersed in the second fluid. - Shaded areas denote
fluidic exchange and data acquisition mixdown value change (step change in baseline frequency). Drift in the second fluid baseline arises from presence of residual
quantities of the first fluid (typically < 1%). Due to higher viscosities than aqueous
solutions, Percoll and Optiprep containing fluids exhibit less clean transitions.
84
1.0
U Dry mass (n = 124)
0.8 -
* Water mass (n = 112)
U Total cell mass (n = 284)
0.6
00.4
Q) 0.2
0.00
500
1000
1500
2000
Mass (fg)
Figure 4-7: Kernel density estimates of probability densities for dry mass, water
mass and total mass of a sample of fixed stationary-phase E. coli. Functions were
resealed so that their maxima were one. Solid bars represent sample medians.
The results presented above demonstrate that the method is self-consistent, as the
water content of the cells plus the dry mass (sum of median values equals 719 ±
13 fg) accounts for the total mass value (Fig. 4-7c). This suggests the median early
stationary E. coli cell is roughly 28% dry material by mass and 20% by volume.
4.4
4.4.1
Dry density
Bacteria
Whether and how bacterial dry density and dry mass change with culture growth
phases was investigated by growing E. coli cells and analyzing fixed samples of the
culture at four time points -
stationary, early exponential (after dilution into new
culture), late exponential and a second stationary point (Fig. 4-8). Each fixed sample was analyzed two to three times over several days to verify that the results were
consistent. Dry mass was found to have increased in early exponential phase, then
rapidly decreased upon entry into stationary phase, which has been reported previously [70,82]. Dry density exhibited a similar trend, initially increasing when stationary E. coli were diluted into fresh medium and entered exponential growth. As the
culture progressed towards late exponential phase, the dry density decreased and by
stationary phase had returned to the same value as the previous stationary culture.
85
a)
c)
2
0
0.02
2
1.55.
-
10
0
20
30
40
E
Time (hours)
b)
stationary
a eauly log
E late lg
1.50o
a*stationary 2
n=96
1.400.6-
n=93
0.4-
n=138
1.36 -
n=153
n=105
n=100 =~
n=80
0.21.301
so
100
200
50
1000
stationary
2000
early log
late log
stationary 2
Dry mass (fg)
Figure 4-8: Dry density and dry mass of a bacterial culture (a) Growth curves of
E. coli cultures. A culture was grown for 24 hours, diluted 1000-fold, and allowed
to grow again for 24 hours. Samples from the cultures were fixed for analysis at the
colored time-points. Solid line is the fit to logistic growth model. (b) The dry density
of the culture by sampling time point. Technical replicates of these fixed samples
show that the changes in density are reproducible and not attributable to instrument
error. (c) Probability distributions of dry mass, rescaled so that the modal mass had
a density of one. Lines of the same color are technical replicates, measured several
days apart.
The presence of a subpopulation of cells in the early exponential sample was noted,
exhibiting masses and dry densities characteristic of stationary cells (left shoulder on
blue distributions in Figure 4-8b, see also Figure 4-9).
Compared to the variation in total density reported previously for E. coli [95] and
other microorganisms such as yeast [60], the single-cell dry density measurements here
reported were much more variable. To investigate the source of this variation, propagation of errors in buoyant mass measurement to dry density estimates was considered
and simulated error distributions for our experimental conditions (see Sections 4.2.2
and 4.6.3). For E. coli measurements, the simulated and observed distributions match
quite well (Fig. 4-9), suggesting that the majority of the observed heterogeneity arises
from buoyant mass measurement error. Thus, it is likely that the true dry density
86
variation is lower than what we observe.
stationary
stationary
stationary
0
0
0
I0.
0
-
I i
i
i
1.40
1.50
i0
0
0
i
0
1.30
CO
1.60
1.40
1.30
Dry density (g -m~-)
1.40
1.50
Dry density (g-cm)
Dry density (g -cm-)
early log
late log
early log
I
1.30
1.80
1.50
1.60
LO
0
0
I
0
0
-- -
--
1.30
1.40
0
1.50
1.30
1.60
3
Dry density (g -cm~ )
1.50
1.40
1.30
1.80
1.50
1.80
Dry density (g-cm~)
Dry density (g-cm~)
stationary 2
stationary 2
late log
1.40
0
-
~
I
0
O
I
I
0
I
0
\
0
~
-" S I i
i
i
0
0
1.30
1.40
1.50
Dry density (g-cm~)
1.60
-
1.30
|
I'
-
,-
1.40
1.50
Dry density (g -cm -)
Figure 4-9: Comparison of measured
mass measurement errors propagating
samples. Dashed lines show expected
have the same density and that density
line).
0O
-
1.80
1.30
1.40
1.50
1.80
Dry density (g -cm~)
data (solid lines) to simulations of buoyant
through the density calculation for E. coli
dry density distributions assuming all cells
is the median observed dry density (vertical
87
4.4.2
Yeast
Subsequently, it was investigated whether these patterns of changes were unique to
bacteria, or if they might also be found in eukaryotic cells. As with E. coli, a culture of
yeast cells was grown for a 24h growth cycle, taking samples throughout the culture's
growth phases for fixation and quantification of their dry mass and dry density (Fig. 410). The measurements were repeated both with technical and biological replicates
showing consistency amongst the measurements and trends (Fig. 4-10e).
Single-cell distributions are shown for dry density (Fig. 4-10b) and dry mass (Fig. 410c). The first time point during growth, 3h after the dilution, shows a concurrent
increase in cell dry mass and dry density, when cells are growing at their fastest
growth rate and actively dividing. The gradual decrease in dry density accompanies
the slowing speed of culture growth as the culture approaches saturation. In contrast
to the E. coli results, the computed error distributions are less variable than what we
observe (Fig. 4-10b), showing true density heterogeneity and possibly distinct subpopulations. Additionally, cells were concurrently annotated as budded or unbudded
by brightfield microscopy. In early stationary phase cultures, the dry density distribution of budded cells showed higher median values than of unbudded cells, but in
exponential phase, the dry densities were not significantly different. (Fig. 4-11).
Growth perturbation
The perturbation of cell growth was experimented by incubating a growing culture
with 1 pM rapamycin (a TOR inhibitor), either at the moment of inoculation (Ohculture) or three hours post-inoculation (3h-culture). The growth of the 3h-culture
was delayed, while the Oh-culture exhibited essentially no change in optical density
(Fig. 4-12, a). In both cases, dry mass does not increase significantly as compared
to the controls (Fig. 4-12, c). However, with the 3h-culture, dry density seems to
follow a trend similar to the control during the observed time, with an increase and
subsequent decrease in dry density, while the Oh-culture demonstrates a continuously
increasing dry density, with no growth observed in the bulk culture.
88
a)
b)
10
c)
Oh
n=191
0.5
0.5
100
I
0
0/
1.3
10'
1.5
1.6
01
0
20
40
60
80
100
20
40
60
80
100
0
20
40
60
80
100
0
20
40
60
80
100
40
60
80
100
3h
n=162
10-2
25
0.5
Culture time (h)
d)
1.4
0.5
I
1.60
2 1.3
C?
E
1.4
1.5
1.6
0
0
1.56
8h
n=211
1.52
--------------
0
1.48
0.5.
0.5
0
1.3
1.4
1.5
1.6
0
1.44
0
e)
20
40
60
80
Dry mass (p
15h
n=186
1.54
0.5
0.5
C.)1.52
1
1.4
0
1.3
1.50
1.5
1.6
0
1.48
C
C
1.461
2
1.44
3
4
.
0
5
10
15
Culture time (h)
20
25
24h
n=194
0.5
0.5
0'
0
1.3
1.4
1.5
Dry density (g-cm-3)
1.6
0
20
Dry mass (pg)
Figure 4-10: Dry density and dry mass of a yeast culture (a) Growth curve of a
culture started from a 1000-fold dilution of a recently-saturated culture (time Oh).
(b) Distributions of dry densities for the time points indicated in (a). Distributions
expected due purely to measurement error (see text) are shown as black dashed lines.
(c) Dry mass distributions for the same time points. (d) Single-cell data for time
point 8h. Solid lines is median dry density and dashed lines are 99% bounds on the
expected dry densities if all cells actually had the median dry density, given known
measurement error (see Methods). (e) Dry density distribution medians for several
replicates: curves 1 and 2 are technical replicates and 2-4 are biological replicates;
curve 3 is for data show in (b).
89
15
N
unbudded
0 budded
Oh
n=1 49
n=39
10
P=0.0006
5
0
1.35
1.30
1.40
1.45
1.50
1.55
1.60
20
3h
n=51
n=102
15
P=0.1020
10
5
0
1.35
1.30
1.45
1.50
1.55
1.60
8h
n=95
n=97
15
0
1.40
10
P=0.6371
Z,
:5
0
ai-
0
1.30
1.35
1.40
1.45
1.50
15h
n=73
n=98
15
10
1.55
1.60
P=0.4213
5
0
1.30
1.35
1.40
1.50
1.55
1.60
1.30
1.35
1.50
1.45
1.40
Dry density (g -cm-3)
1.55
1.60
1.45
14
12
10
8
6
4
2
0
Figure 4-11: Dry density distributions for budded and unbudded yeast cells, by
timepoint. In early stationary phase cultures (t = Oh, 24h), the dry density distribution of budded cells showed higher median values than of unbudded cells, but in
exponential phase, the dry densities were not significantly different. P-values are for
two-sided Mann-Whitney U tests.
90
a)
104
<0
0
101
0
5
-
10
15
20
10
15
20
10
15
20
b)
1.52
1.5
.......
..
Z
1.48
1.46
1.44
(U
1.42
1.4
1
C)
40
35
30
E
25
20
15
10 -
0
5
Culture time (h)
0
A- no drug
D
B- no drug
A - RAP@Oh
A - RAP@3h
3
-
B- RAP@Oh
B - RAP@3h
Figure 4-12: Dry density and dry mass yeast cultures treated with rapamycin.
(a) Growth of two sets of biological replicates of yeast cells treated with rapamycin
at inoculation, 3h post-inoculation, or untreated. Dry density (b) and dry mass (c)
distribution medians. While the 3h-culture dry density seems to follow a trend similar
to the control during the observed time, the Oh-culture demonstrates a continuously
increasing dry density, with no growth observed in the bulk culture. The discrepancy
in dry density values between the two cultures is unaccounted for.
91
4.4.3
Mammalian cells
Finally, changes in dry density that occur when varying mammalian cells are subjected
to similar changes in growth conditions were measured. Four cell types were chosen
mouse embryonic fibroblasts, (MEFs), L1210 mouse lymphocytic leukemia cells,
FL5.12 mouse prolymphocytic cells, and CD8 T cells from an OT-1 transgenic mouse
and their proliferative states manipulated. For MEFs, cells were grown either to
70% or 100% confluency. L1210 cells were treated with cycloheximide and measured
before and 24 hours after treatment. FL5.12 cells were measured before and 20 hours
after being placed in media lacking interleukin 3 (IL-3). Finally, naYve OT-1 CD8 T
cells were activated with an ovalbumin peptide and measured before and 96 hours
after activation. All measurements were performed without cell fixation.
With the exception of the activated OT-1 cells, proliferating cells appeared to have
higher dry densities than their non-proliferating counterparts (Fig. 4-13). Moreover,
both MEF cells grown to confluency (Fig. 4-13a) and L1210 cells treated with cycloheximide (Fig. 4-13b) did not show a substantial decrease in dry mass relative to
steady state populations. FL5.12 cells, however, decreased both in dry density and
dry mass when starved of IL-3 (Fig. 4-13c). Interestingly, primary OT-1 T CD8 cells,
which are quiescent and non-proliferating (naive), had higher dry density prior to
activation than following activation (Fig. 4-13d). Of these four mammalian cell lines,
only for the naive OT-1 T cells is the variation nearly completely accounted for by
measurement error, suggesting non-negligible biological variation in the other populations. Additionally, for all the cells except the OT-1 cells, the observed variation in
dry density increased upon interfering with proliferation.
92
a) 1.40
b)
MEF
L1210
E
o 1.35
1.30
200
0
600
400
800
1000
50
100
150
Dry mass (pg)
C) 1.6 -
200
250
300
350
Dry mass (pg)
d)
FL5.12
OT-1 CD8+ T
1.50 1
0
0
1.5 '
1.45
B
E
C
E
B
'
1.4
-
0
0
'
.0.
0
0
~4
0
*0
0
*S
-9--
1.3
1.35
with IL-3
without IL-3
1.2
0
100
200
300
1.30
400
0
Dry mass (pg)
50
100
150
200
250
300
350
Dry mass (pg)
Figure 4-13: Dry density and mass of proliferating and non-proliferating mammalian cells. Solid lines are median dry densities and dashed lines are 99% bounds
on the expected dry densities if all cells actually had the median dry density, given
known measurement error. (a) Confluent and proliferating (75% confluency) mouse
embryonic fibroblasts. (b) Cycloheximide-treated and proliferating L1210 cells. (c)
IL-3-depleted and proliferating FL5.12 cells. (d) Naive and activated OT-1 T cells.
93
a)
40
1345b)
32*
38
1.340
E1.335
4
~
1.330
38
T
42
1.325
30
1.320
o
e3
36
*
1.315
15
20
25
28
*
30
35
40
45
50
26
28
30
32
34
36
38
40
Dry mass (pg)
Dry mass (pg)
Figure 4-14: Dry density and mass of red blood cells (a) Single-cell dry density and
dry mass of two different human erythrocyte samples. Different shapes are technical
replicates. Solid lines are median dry densities and dashed lines are 99% bounds
on the expected dry densities if all cells actually had the median dry density, given
known measurement error. inset Population mean values from four patient samples.
Error bars are standard deviation of the population. (b) Comparison to hemoglobin
mass per cell determined with an Advia instrument. Dashed line indicates y = x and
solid line is total least squares fit.
4.4.4
Red blood cells
Human erythrocytes are a unique sample for this method because they are deformable
enough that we can flow them through sensitive 3 x 5 pm channel devices designed
for bacteria. No cell lysis was observed, consistent with reports of unimpeded flow
of red blood cells through 3 pm diameter pores [96]. As a result, there is essentially
no error caused by variability in cell transit flow paths, and because they are 40 to
160 times larger than bacteria, the signal-to-noise ratio is higher than for any other
sample. From four different human samples, we find that erythrocytes have extremely
narrow dry density distributions (median sample standard deviation of 0.0024 g-cm- 3,
maximum 0.0051 g-cm- 3) and the measurements are highly reproducible (Fig. 4-14a).
The narrowness of the dry density distributions allow us to distinguish differences in
dry density amongst different populations that may or may not have distinct dry mass
distributions. We also compared the dry mass of the red blood cells, with the mean
hemoglobin content quantified by the FDA-approved Siemens ADVIA instrument
(Fig. 4-14b). More than 1000 consecutive measurements of the same cell, during a
94
2.0
140
9 1.5
120
100
Cell dry density
10
. Cell dry mass
21.0
'
- ' ''
1.0 -ot",
. Cell dry volume
"" ~ '
2
0
E
c80
E
'6
40
0.5
20
0
5
0
10
15
20
0
25
Time (min)
Figure 4-15: Dry density dry mass, and dry volume of a single erythrocyte over
time. The dry content parameters of a red blood cell exhibit no change over 25
minutes of monitoring; during this time the intracellular water content of the cell
was swapped out (H2 0 to D2 0 and back) over one thousand times (measurement by
William Grover).
period of about 25 minutes, did not exhibit variation in the dry mass or dry density
(Fig. 4-15).
4.5
Discussion
This thesis introduces a non-optical technique for quantifying the dry mass, water
content and dry density of either living or fixed cells that does not require any assumptions about the cell's composition. However, it does rely on two key assumptions.
First, to avoid osmotic perturbations that might damage or lyse a cell, the measurements are not made in pure H2 0 and D2 0, but in isotonic solutions. Therefore the
assumption that the intracellular water volume is exactly neutrally buoyant in the
immersion fluid is an approximation, as there will be a difference between the densities of the intracellular water (or deuterium oxide) and of the fluids in which the
cell is immersed. However, assuming the cell volume is 80% water, this error will
be small (< 0.04 g-cm-
3
- see Fig. 4-16). Knowing the exact fraction of water con-
tent would allow us to correct for this effect. Second, it is assumed that complete
95
p=1.3, water fraction = 0.8
p =1.3, water fraction = 0.7
J
-
1.102 --
1.102
E
a
1.100
2 1.100 -
1.098
$
1.098 -
1.096
1.096
1.094
1.094
0.998
0.998 1.000
1.000 1.002 1.004 1.006 1.008
1.102
1.102 -r
E
1.100
S
1.100
1.098
1.098
.2
$
1.096
0.998 1.000 1.002 1.004 1.006
1.096
0.998
1.008
1.000 1.002 1.004 1.006 1.008
density
(g -cm-)
H20 solution
density
(g -cm)
HO solution
p = 1.5, water fraction
p = 1.5, water fraction = 0.7
0.8
- 1.102
E
1.102
a
2 1.100
0
4
-
1.094
1.094
-
1.008
p = 1.4, water fraction = 0.8
p =1.4, water fraction = 0.7
a
1.002 1.004 1.006
H20 solution
density(g- cm~3)
H20 solution
density
(g -cm~)
1.100
1.098
1.098
1.096
1.096
1.094
1.094
0.998
1.000 1.002 1.004 1.006 1.008
0.998
1.000 1.002 1.004 1.006 1.008
H20 solution
density(g-cm-)
H20 solution
density
(g-cm-3)
Figure 4-16: Contour plots of dry density estimates when the buoyant mass measurements aren't made in pure H2 0 or pure D2 0. Intracellular water fractions are in
fraction of total volume. Dashed line shows equal departure (in density) from pure
fluids. Pure H2 0 and 9:1 (v/v) D20:H2 0 densities are the red dot in the lower left
corner of each figure, at which point the dry density is calculated correctly. As salts
(or other impermeable components) are added to the fluid, it becomes more dense
and the intracellular water is no longer neutrally buoyant. This introduces systematic
error into the dry density measurement, which depends on how much of the cell is
water. The measurements we've made using 1x PBS in both fluids are shown as
black dots.
96
exchange of intracellular water occurs. This is justified, as observed correlations between dry density and the time a cell spends immersed in D2 0 are at most very weak
and typically statistically insignificant (Section 4.2.4 and Figs. 4-4 and 4-5). Indeed,
previous measurements of water permeation and diffusion across the membrane have
demonstrated the almost instantaneous nature of the event [66].
The presented dry mass results are consistent with previous reported measurements of the described single-cell or bulk methods. For instance, two TEM-based
studies found median E. coli dry masses of 489 fg and 710 fg for exponentially growing cells and 179 fg and 180 fg for stationary ones [70, 72].
In this work, median
values for dry mass of 725 fg and 179 fg, respectively, are reported, pooling technical
replicates shown in Figure 4-8. Budding yeast dry mass content per cell is not widely
reported, but the results obtained are in line with the values reported by Mitchison [71].
Furthermore, the results for the human erythrocytes agree well with the
hemoglobin content concurrently quantified by the FDA-approved Siemens ADVIA
instrument (Fig. 4-14b), or by QPM [97]. It should be noted that, even though the
ADVIA measures only hemoglobin content, this protein has been shown to account
for more than 95% of the cell's dry mass [98,99]. By comparison to the results herein
reported, the mean hemoglobin content determined by the ADVIA accounts for 97.7
± 1.3% of the total dry mass content.
A unique aspect of this measurement is the concurrent determination of dry density. The few mentions of this parameter in the literature are seemingly limited to
measurements of wet and dry spores [77, 78, 100] or an application in an H2 0/D 2 0
density gradient [101]. However, none of these reports connect dry density to chemical composition of the dry content. The results herein presented suggest that dry
density is a direct manifestation of the changes in chemical composition of a cell's
biomass and we demonstrate that the parameter can be measured for single cells.
While related to total cell density, dry density is independent of the intracellular water content and should not be perturbed by uptake or expulsion of water. In contrast,
since the majority of a cell's volume is composed of water, total density is likely to
be much more indicative of changes in cell water content.
97
For E. coli [67,82,1021 and yeast [68,69], the RNA/protein ratio has been extensively correlated with growth rate: faster growing cells have an increased proportion
of RNA relative to protein. These growth-rate-dependent changes in chemical composition are observed directly as changes in dry density, since the average density of
proteins is lower than of RNA (Table 4.2). Faster growing cells - in early exponential
phase -
have a higher dry density, consistent with higher RNA/protein ratio, and as
growth rate diminishes, so does dry density. Furthermore, the annotation of budded
and unbudded yeast populations also reveals that at early saturation time points,
budded cells tend to have higher dry densities than unbudded cells. However, as the
culture enters exponential phase, the unbudded cells no longer possess a distinctive
dry density profile. It is speculated that the dry density variation results from heterogeneous proliferative states within a well-mixed culture - cells with higher dry
densities are those that are currently or were more recently proliferating. The treatment of yeast cultures with rapamycin, where dry density is shown to increase, even
though dry mass does not change considerably and bulk growth is essentially nonexistent, is consistent with reports that rapamycin treated yeast exhibit a G1 growth
arrest and accumulate storage carbohydrates such as glycogen [103]. The density of
glycogen is reported being greater than of proteins (~1.3 -
>1.6 g.cm-3) and, if so,
the changes in dry density are consistent with the synthesis of a denser biomolecule
without net mass accumulation [79,104].
The results with mammalian cells demonstrate that dry density changes when
growth is perturbed. Both MEF and FL5.12 cells decrease their dry density as they
transition into unfavorable growth conditions characterized by culture crowding or
nutrient deprivation from IL-3 depletion. The decrease in MEF dry density can be
compared to the results reported by Short et al. [105] for fibroblast derived M1 cells,
in which the relative amounts of RNA and protein content decrease and lipid content
increases for overcrowded non-proliferating cells when compared to proliferating ones.
Changes in dry density are not necessarily correlated with dry mass. L1210 cells
treated with a lethal dose of cycloheximide - which blocks protein synthesis - show
a decrease in dry density even though the overall dry mass distribution does not show
98
a substantial decrease. This suggests that during this period the cell has no notable
net mass exchange with its environment, but the inner constituents of the cell are
undergoing substantial biochemical alterations. As fewer proteins are synthesized
during this time, degradation is likely the primary force lowering the relative protein
content [106]. This increases the relative contribution of lower density components,
such as lipids, thereby decreasing the overall dry density. The alteration of dry density
in these situations suggests that this parameter can be indicative of cell proliferative
state.
If proliferating cells amongst a steady state population have different dry
densities, dry density could be complementary to proliferation detection assays such
as Ki-67 labeling.
It was of interest to determine whether changing mammalian cell growth rate by
activating growth from a natural quiescent state would result in a change in dry
density. In their naive state, CD8 T cells are quiescent and only begin proliferating
following antigen stimulation. Comparison of naive and activated OT-1 CD8 T cells
show that T cell activation is accompanied by dramatic changes in both dry density
and dry mass, suggesting, as in previous results, that as cells alter their proliferative
state, changes in their chemical composition occur.
It is notable that naive CD8
T cells freshly isolated from mice show high dry density but very low dry mass.
Stimulation of these cells into proliferation is associated with an expected increase in
dry mass as well as a change in dry density to similar values of cultured L1210 cells.
The increase in dry mass is consistent with the growth of the cells as they undergo
proliferation. However, the high dry density of nafve CD8 T cells was unexpected, in
light of the observation that the dry density decreases when mammalian cells undergo
growth arrest due to stress or depletion of nutrition. Naive CD8 T cells are compact
with little cytoplasm -
consistent with their low dry mass value -
and their high
dry density may reflect the lack of cytoplasm and organelles, which are rich in lipids.
As a result, the proportion of nucleic acids and proteins is higher for naive T cells
than actively dividing T cells.
Finally, it is demonstrated that the concept of rapid exchange of intracellular water
can be used to quantify a cell's water content. Although currently it is only possible
99
to measure an average or modal intracellular water fraction, the ability to weigh single
cells in three different fluids will allow all of the described quantities to be determined
simultaneously.
4.6
4.6.1
Materials and methods
SMR operation
Three types of SMRs were used to perform the measurements. For bacteria and red
blood cells, the procedure is identical to that of Grover et al. [49] but smaller 3 x
3 x 100 Am and 3 x 5 x 120 pm (channel height x width x length) cantilevers
were used. Budding yeast cells were measured as previously described for single-cell
density by Weng et al. [60], with an 8 x 8 pm, 210 pm-long three-channel device.
Finally, the larger mammalian cells were measured with a 15 x 20 Am, 450 pm-long
channel SMR with the Grover et al. method, however the device was being operated
in the second vibrational mode [30].
To measure dry mass and dry density, the cells are weighed twice, first in a waterbased solution (1x PBS in H2 0), secondly in a deuterium oxide-based solution (1x
PBS in 9:1 D 2 0:H 2 0) (Fig. Si). The actuation of the cantilever in the second vibrational mode increases sensitivity and decreases error by eliminating the flow path
dependency of the buoyant mass measurement. The three-channel devices also eliminate this error, as described elsewhere [30]. However, technical constraints prevent
us from using either of these methods with the smaller bacterial-sized devices.
After the first weighing, the cell is immersed in the second fluid and the two
measurements can be done several seconds apart. The fluidic exchange occurs on a
faster time-scale. While it is difficult to make the two measurements faster in less
than several seconds using regular cantilevers, with the three-channel devices the fluid
environment can be switched from H2 0 to D2 0 very rapidly (250 ms).
To measure water content in single E. coli, cells were initially immersed in a solution
of roughly 18% OptiPrep (w/v), 0.9x PBS in H2 0. Because it is essential that the
100
fluid densities match precisely, this solution density was manually adjusted with a few
drops of water or 60% OptiPrep to match the density of 1 x PBS in 9:1 D2 0:H 2 0. Cell
buoyant masses were then sequentially measured in the OptiPrep:PBS:H 2 0 solution,
followed by the PBS:D 2 0 solution.
SMR buoyant mass measurements were calibrated using polystyrene particles of
varying sizes (depending on SMR type) from Duke Scientific and from Bangs Labs.
Fluid density measurements were calibrated with NaCl standard solutions. All measurements were done at 22-23 0 C.
4.6.2
Cell culture and fixation
Escherichia coli - cells (ATCC 23725) were grown on Luria Broth (LB) agar plates
from frozen stock, and single colonies were transferred into 35 mL liquid cultures
(LB) and grown for 24 hours at 37 'C with vigorous shaking. Two cultures were
grown to verify similar growth behavior by optical density at 600nm. After 24 hours,
several milliliters from one culture were fixed in 2% paraformaldehyde and 2.5%
glutaraldehyde for 1 hour at 4 C. Some of the same culture was also used to inoculate
a 35 mL culture at a 1000-fold dilution. Cells were then taken at OD600 0.17 and
1.17 and fixed (220 minutes and 340 minutes, respectively), and then a final sample
was fixed at 24 hours (OD600 ~3.3).
Saccharomyces cerevisiae -
haploid cells (702 W303) were grown in YEPD
medium at 30 C, well-shaken. For suspended culture growth experiments cells were
started from a plated culture and grown for 24h. At that point an aliquot was sampled
and a new culture was started with a 1000-fold dilution. Subsequent aliquots were
sampled at the times described in the text. Each sample was spun down, suspended
in PBS, sonicated and fixed in 4% paraformaldehyde overnight.
Red blood cells -
Four human erythrocytes samples were collected in EDTA from
subjects under a research protocol approved by the Partners Healthcare Institutional
Review Board. Samples were diluted in PBS prior to each dry measurement. "Advia"
hemoglobin mass measurements were performed on a Siemens Advia 2120 instrument.
L1210 -
cells were grown at 37 'C in L-15 media supplemented with 0.4 % (w/v)
101
glucose, 10% (v/v) fetal bovine serum (FBS), 100 IU penicillin and 100 pg-mL-1
streptomycin. For cycloheximide treatment, 5 pL of a 10 mg.mL-' cycloheximide in
DMSO stock solution were added to a 5 mL culture (7.5 x 105 cells.mL- 1 ) of L1210
cells. Treated cells were maintained in an incubator at 37 'C for 24 hours prior to
measurement. Before loading the sample into the SMR, cells were washed twice in
PBS by spinning down for 5 minutes each time at 100 RCF. The concentration of the
cell sample was adjusted to 5 x 105 cells-mL- 1 .
FL5.12 -
cells were grown at 37 'C in RPMI media supplemented with 10% (v/v)
FBS, 100 IU penicillin, 100 pg-mL-
1
streptomycin and 0.02 pg-mL-
1
IL-3. For FL-
5.12 starvation, a confluent culture of FL5.12 cells (106/mL) was washed three times
before culturing for 20h in RPMI media lacking IL-3. Before measurement in the
SMR, cells were washed twice with PBS as with the L1210 cells.
Mouse endothelial fibroblasts -
cells were grown at 37 'C in DMEM media
supplemented with 10% (v/v) FBS, 100 IU penicillin and 100 pg.mL-
1
streptomycin.
Cells were trypsinized and measured at 70% confluency (106 cells on a 25 cm 2 flask)
or overconfluency (2 x 106 cells.mL- 1 ). Cells were washed twice with PBS before
loading into the SMR.
OT1 CD8 T cells -
Lymph nodes were harvested from OT1-ragl-/- mice, ground
and filtered using a 70 pm nylon cell strainer. To activate T cells, OT-1 cells were
stimulated with 2 pg.mL-
1
OVA257-264 peptide (SIINFEKL) at 37 'C for 24 hours
in RPMI media supplemented with 10% (v/v) FBS, 100 IU penicillin, 100 pg-mL-1
streptomycin, 2 mM L-glutamine, 1 mM sodium pyruvate, 55 pM 2-mercaptoethanol
and 100 pM nonessential amino acids, followed with culture for another 4 days in
the presence of 50 IU IL-2. Cells were washed twice with PBS before measurements.
Measurements of naive CD8 T cells were carried out immediately after harvesting from
mice. All experiments with mice were performed in accordance with the institutional
guidelines.
102
4.6.3
Statistical analysis
To estimate uncertainty in dry density and dry mass measurements, the uncertainty
in buoyant mass measurements is first estimated and then the measurement error
propagation through the density and mass calculations is simulated. Buoyant mass
uncertainty is estimated from the discrepancy between two sequential measurements
of a cell in the same fluid, as the two measurements are expected to be identical. Two
sequential measurements are made from approximately 100 cells, and for each cell,
the difference between the two measurements is calculated. As the difference in each
pair of measurements is the difference of two presumed independent errors, rescaling
the distribution of differences by V2_ yields approximately the distribution of errors
that might occur on a single buoyant mass measurement.
For dry mass, the standard error in an estimate is approximately 15 times greater
than the error in a single buoyant mass measurement (see analysis in Section 4.2.2).
However, dry density is a non-linear function of the two buoyant mass measurements
and so we simulate the effect of buoyant mass errors on a population with no dry
density variability. One begins by assuming all particles have a density equal to the
median observed dry density and a mass distribution equal to our observed dry mass
distribution. Subsequently, 10,000+ hypothetical particle masses are sampled from
the observed real dry mass distribution and the buoyant masses calculated for those
particles in each of the two fluids used for the experiment. Afterwards, errors from
the measured error distribution are sampled, then this random noise is added to each
buoyant mass measurement, and the dry density for each noisy' pair of buoyant mass
measurements is calculated. Although no dry density heterogeneity went into this
calculation, the resulting dry density measurements have a non-zero variation due to
buoyant mass errors, and we qualitatively compare these distributions to our observed
dry density distributions.
103
4.7
Measuring the density of biomolecules
The dry density of the cell, as previously described, corresponds to the bulk density
of it's non-aqueous content. Simply put, it is the ratio between the total mass of the
proteins, nucleic acids, lipids and other components of the cell, and the total volume
they occupy. More descriptively, the latter is the volume of water that these molecules
displace.
In order to model biochemical compositions of cells, the density of these elemental
components of the dry mass is of relevance. Furthermore this parameter is useful in
applications such as macromolecular structure determination in X-ray single crystal
crystallography or analytical ultra centrifugation [81]. However, these values are not
immediately accessible (Table 4.1) and, when determined experimentally through
ultracentrifugation, they can value considerably depending on which gradient solution
the molecules were suspended in and their own composition [80]. Different theoretical
calculations have show disagreements and have yet to fully explain the experimentally
determined values [81,107].
Here, a similar measurement as described above for cells is described for proteins
(or any other water soluble molecule). In this case, it is not an intracellular aqueous
component that is being exchanged, but the fluid surrounding the molecules.
protein solution with some mass concentration
(" ,
VH2 0
"ein)will have the density:
p = aPprotein + (1 - a)Pfluid
where a
A
(4.15)
*Hroteinis the volume fraction occupied by the protein and Pfluid
Vprotein+VH 2 O
pH2 0. By preparing a D2 0-based solution of the same concentration and measuring
the density of both solutions pi and P2 (H20 and D2 0-based, respectively), the density
of the protein can be calculated as:
PD 20
protein
P2-PD
-
1
P1 -PH
104
2
0
PH2OPPH 2O
P2-PD 2 0
2
0
(4.16)
Table 4.3: Measurement of the density of a protein. Density, mass and volume of
BSA in solutions of different concentrations. Each solution density measurement is
a reported as mean ± absolute error for 3 prepared solutions. Calculated errors are
obtained by propagation of errors
Concentration
[mg-m-1]
PH20
[g.cm- 3]
PD 2 0
[g.cm-3]
Pprotein
[g-cm~ 3 ]
0
0.997340.4x10- 4
1.0119O.5x10~4
1.0168±4.Oxi- 4
1.0212±4.7x10-4
1.1036+0.6x10- 4
1.1132+0.9 x 104
1.1163+1.4x10~4
1.1194+1.6x10-4
-
-
-
1.3071+0.0071
1.3029+0.0137
1.3111+0.0138
49.45±1.61
67.85+4.75
82.30+5.73
64.63+1.21
88.41+3.57
107.91+4.28
60
80
100
Vprotein
[ml] per ml
mprotein
[mg] per ml
Concentration indicates the concentration of the prepared solutions, 3 water-based and 3 deuterium
oxide-based. PH2 O and PD2 o are the densities of the water- and deuterium oxide-based solutions,
respectively. pprotein, Vprotein and mprotein are the density, mass and volume of the protein. The
two latter quantities are per 1 ml of solvent.
a)
b)
1.025
1.020
1.125
u.2
E
1-015
9
0
1.016
U
1.115
-1.005s
1.105
1.000
O.986
0
20
40
s0
80
100
120
1.10040
1
Prepared solution concentration (mg-m- )
-
20
40
----
60
80
100
120
Prepared solution concentration (ng-min')
Figure 4-17: Densities of (a) water-based and (b) deuterium oxide-based bovine
serum albumin solutions of different concentrations. Detailed values and protein
density calculations are in Table 4.3
Three different H2 0- and D20-based 100 mg-ml'
solutions of bovine serum albu-
min (BSA) were prepared for each solvent and subsequently diluted. Their densities
were measured at 23 *C and the results are reported in Figure 4-17 and Table 4.3. A
mean value of p,,tezn
-
1.3070 t 0.0138 g-cm 3 is reported.
This measurement has the shortcoming that preparing solutions of the same concentration with great precision is error-prone, in particular if the sample is limited
in quantity. In fact, the measurement can be done without a second fluid, by using
dilutions with known constraints. Still an assumption in solution preparation has to
be made, which is that two identical volumes can be measured with low enough error.
105
If the same volume of water (VH2 o) is added to the solution of density pi (Eq. (4.15)),
then the resulting solution has a volume of water of 2 x (1 - a), in terms of the initial
volume fractions, which now should be normalized by a total volume of:
a + 2 x (1 - a) = 2 - a
This second solution, therefore, has a density of:
a
2(1 - a)
+
P2 P22 - - a poem
ot
2 - a
PH2 0
The density of the protein is then given by:
Pprotein =
(4.17)
-1)
P2P1 + (P2 - 2p1)pH 2o
2
P2 - P1 - PH2O
and the volume fraction:
2P2
-
P2
P1
-
-
PH 2O
PH 2O
Therefore the mass of protein used is:
mprotein
Pprotein
a
(4.18)
VH 2O
For this measurement, the initial mass of protein needs not to be known, although
care should be taken to ensure that there's enough dissolved protein to change the
overall solution density beyond the instrument's limit of detection -
the higher the
concentration, the better.
The execution of this method has, however, a solvable yet impractical implementation issue. After the preparation of the first solution, the total volume should not be
altered, so that the volume fractions are preserved for when the dilution with VH2 O occurs, in the preparation of the second solution. This poses a problem as some volume
of the first solution needs to be consumed for the density measurement. A potential
way to solve this is to prepare the first solution with 2 x VH 0 and then divide this
2
volume in half. This has to be done gravimetrically, as the total volume of protein
106
and solvent is now unknown and will be another potential source of experimental
error.
4.8
Continuous dry mass measurement
As demonstrated in Figure 4-15 with an erythrocyte, continuous measurements of
dry density and dry mass can be achieved by sequentially immersing the cell in H2 0and D2 0-based solutions. Here, the growth of E. coli cells (strain DH5a, grown at 23
one H2 0-based and the other D2 0-based
'C) in two preparations of LB medium -
is shown, so as to demonstrate the direct measurement of the dry mass of a live
growing cell. Dynamic trapping with two different fluids is implemented and the
device is actuated in second mode, as described in Section 2.6. In this experiment, a
cell, which is growing initially in a water-based medium, is brought into the dynamic
trap and from that moment on is exposed to both the water-based and the deuterium
oxide-based fluids roughly the same period of time.
Even though cellular growth has been reported at high D2 0 concentrations [90-92],
growth in LB medium prepared solely with D2 0 was not observed in dynamic trap
00
50%
75%
90%
100%
V
O6
0
200
400
600
800
1000
Time (min)
Figure 4-18: Growth curves of E. coli cultures in different concentrations of D2 0based LB medium at 37 0 C
107
250-
I
CL
2002
E
Time (mins)
Figure 4-19: Growing E. coli cells in H20-based and 50% D20-based LB medium
at 23 'C. (a) Each color represents a different cell and each shows two distinct sets of
data points (top and bottom), which are the consecutive buoyant mass measurements
in the different fluids. Some trajectories exhibit clear growth rate change after an
initial faster growth phase (1,3,9). Lines are ordinary least square fits. (b) Computed
dry mass for each cell. Individual points represent calculated dry mass from each
pair of buoyant mass data points and the green lines are their linear regression curve,
while the red lines represent the calculation based on the buoyant mass linear fits.
(c) Dry density for each cell, computed as in (b).
experiments. Bulk cultures, nevertheless, did exhibit growth (Fig. 4-18). A second
LB preparation with 50% D20/50% H20-based was used and growth was observed
(Fig. 4-19).
As expected, larger cells grew faster than smaller ones, but on a few
occasions some cells did not grow at all, with buoyant mass loss, a characteristic sim-
ilar to what was observed with 100% D 2 0-based medium (data not shown). Many
cells did exhibit a sudden drop in growth rate after an initial period of faster growth
(Fig. 4-19, curves 1,3,9). It is known that exposure to D20 is a stress factor that induces a response similar to bacterial adaptation to oxidative and osmotic stresses [92]
and that growth in medium containing D20 not only is perturbed, but requires adaptation [90-92].
The observed changes in growth rate might be due to such effects.
Entire cell cycle growth trajectories were not acquired and a exhaustive characterization of the growth under these conditions was not performed. For the purpose of
demonstrating single-cell dry mass accumulation it suffices, however further studies
should be made to fully ascertain the growth characteristics of cells in these alternating conditions.
108
The dry mass and dry density values for each cell, measured throughout the
trapped period, are shown in Figure 4-19. These were computed either by using
consecutive buoyant mass pairs, or by using linear least square fits of the buoyant
mass vales. For trapping events ranging larger fractions of the cell cycle, the ideal
analysis should segment these periods in order to account for deviations from pure
linear changes. The fact that the second fluid is now only 50% D2 0 causes its density
and the density difference between the two fluids -
to be lower than in the exper-
iments reported earlier in this Chapter. As mentioned in Section 4.2.2, this causes
the dry density measurement to be even more sensitive to errors in buoyant mass.
Even though there are multiple buoyant mass measurements for each cell, dry density
is still highly susceptible to errors in the paired buoyant mass measurements, most
evident in the shorter trapping events (curves 4,5). In this case, for these trapping
durations and these growth rates, the trends described by the least square fits are
only significant for the dry masses in some of the longest cases (curves 1, 2 and 7 F-test at 5% significance).
4.9
4.9.1
Dual cantilever system for density measurements
Dual-cantilever bacterial SMR
Dual-cantilever devices for mammalian single-cell density measurements have been
previously reported [108]. They consist of two SMR in series with a fluidic mixing
structure in between them. A cell's buoyant mass is first measured in one device,
while immersed in fluid 1. Subsequently, before reaching the second cantilever, a
fluidic junction delivers a second fluid to the fluid stream. Unlike the previously
described methods for measuring density (Sections 3.2 and 4.2), complete or almost
complete immersion in a second fluid does not occur, but the second fluid (fluid 2)
will be a mixture of fluid 1 and the delivered fluid. As these devices operate mostly in
laminar flow regimes (Reynolds number < 1), mixing occurs essentially by diffusion.
The transit time - and therefore mixing time 109
between the two devices is increased
Figure 4-20: Detail of dual SMR device. In blue are the sample bypass channels,
in green the main microfluidic channel that goes through both cantilevers (width is
5 pm in first device and 8 pm in second) and orange is the channel for delivery of
the second fluid. Serpentine region (-
10 pm long) increases transit time between
devices.
with a lengthened channel (-
10 pm long).
Serial dual-cantilever devices provide a continuous stream mode of operation, which
increases the throughput of the measurement and eases subsequent steps for downstream integration with other instruments and sorting mechanisms.
Furthermore,
operation should be simpler and require less monitoring, as current methods necessitate constant supervision.
This thesis presents a preliminary analysis of bacterial sized dual-cantilever devices
(Fig. 4-20). The delivery of the second fluid increases the net flow and the transit
speed of the cells in the second device. High speeds can be problematic for data
acquisition, therefore this imposes a limitation on the fluid mixing ratio. Too mitigate
this effect, the width of the channel is increased after the junction (from 5 pm to 8
pam). These devices can be operated in second mode and thus the widening of the
channel does not contribute to increasing position dependent error.
Mixing by diffusion occurs perpendicularly to the axis of the channels. For dry
110
density and dry mass measurements the two fluids are water and deuterium oxide; the
diffusion coefficient is approximately D ~ 2 x 10-9 m 2 - S-1 [109]. The characteristic
diffusion time across half of the channel width is, therefore,
tD =
L
~1 Ims, which
is considerably lower than the several second transit between the two sensors.
Preliminary data was gathered for polystyrene beads.Figure 4-21 shows five 1.6 pum
diameter polystyrene beads transiting both devices, immersed in water in both cases.
The difference in baseline noise and peak sizes is due to the fact that the devices are of
different lengths (120 pm and 160 pm, respectively) and therefore have different noise
characteristics and mass sensitivities. In Figure 4-21 the beads are now in different
fluids: the first cantilever has only H2 0 and the second has a mixture of H2 0 and
D2 0. The beads now float in the second fluid, since its density is greater than of
polystyrene. The peak widths reveal different transit velocities in the two devices the second one is greater due to the greater flux. Importantly, the baseline in the
second device (Fig. 4-21, red) shows a significantly far less stable baseline, which has
implications in the the buoyant mass measurement precision.
Firstly, consider the long term drift displayed in Figure 4-23. During a period of
about 15 minutes, the baseline of the first device does not change more than ~ 5 Hz
(<
0.5 x 10~5 g-cm- 3 ). However, the second device shows a drift of in the device's
resonant frequency of more than 700 Hz (equivalent to 1.078 g-cm-,
or 78%, D2 0
to 1.081 g-cm-3, or 81% D2 0). This is possible due to a monotonic change in the
pressure conditions, which lead to a slowly changing mixing ratio. It can be caused,
for example, by differing changes in height of the sample volumes in the containers,
but is not problematic as the density of the mixed fluid can be continuously monitored.
The first device resonant frequency is mostly insensitive to pressure instabilities,
providing the fluid flow direction is not reversed. However, fluctuations in pressure
have great influence downstream of the fluidic junction and therefore on the second
device signal. This is due to the fact that any imbalanced pressure variation will
result in a change of the fluidic mixing ratio, which translates into the short term
random drift observable in Figure 4-24. Such fluctuations make the determination of
the baseline more error prone and degrade the overall signal-to-noise ratio. At the
111
Fa-
2
Time (s)
212427
-
~
112
L15
'w
'
y
~
32
212
25-
21
21M
210
22
21
0
2:3
214
24
242
21
21I24
Time (in)
Figure 4-21: Five 01.6 pm polystyrene beads transiting both devices, immersed in
the same fluid. Differences in baseline noise and peak sizes are due to different sized
devices having different noise characteristics and mass sensitivities - 120 pum (blue)
and 160 pm (red) long.
2102
2
1
(
-Li
27
7
271.
2794
2000
2001
212
2003
M20
2.
"
2:08
.
2007
H20
21
241
2412
2M13
2424
2015
248
(a)
+ D20
Tio (a)
Figure 4-22: Two 01.6 pm polystyrene beads transiting both devices. The first
one has only H 2 0 (blue) and the second has a mixture of H2 0 and D 2 0 (red). Peak
widths reveal different transit velocities (scales are equal). Baseline fluctuations in
second device are due to mixing instability (see text).
112
2.103
10'
1.7
0300
-c
-
1.078 g - cm4
8200
S
78% D20
2
2.1025
2.10200
C
2 500
2.101 .
50D
1.081 g.cm- 3
-
2.1005
5600
81% D20
2400
2500
2600
2700
2800
2800
3100
3000
3200
3300
3400
240
2500
2800
2700
lime (s)
2800
2900
3000
3100
3200
3300
3400
Time (s)
Figure 4-23: Long term baseline drift. The first device does not display a significant
change in baseline resonant frequency during a period of approximately 15 mins (blue).
However, the second device exhibits a slow change in mixing ratio with time (red).
X10,
2.1024
21022
-
2.1022.101821018
2 1014
21012
2101
21006 210082.1004
2000
5670
2610
220
2840
2830
2800
2850
2670
-ime
(s)
can be caused by density variations
$850-Fluctuations
of < 0.00005 g -cm0580
-
57902010
2820
2830
2640
2850
2688
2870
2880
280
lime (s)
Figure 4-24: Short term baseline fluctuations in the second cantilever. During a
period of about one minute, the baseline of the first device is essentially flat (blue).
However, the second cantilever exhibits a random-like short term fluctuation (red).
At that location they account for variations in density of less than 5 x 10-5 gcm-3.
113
location of the second device, these fluctuations account for variations in density of
less than 5 x 10-5 g-cm- 3 (some dilution would have occurred along the channel, so
at the junction, it is higher). These fluctuations are not happening across the width
of the channel, which, as mentioned above, would have enough time to dissipate,
but occur lengthwise: pressure imbalances as a function of time translate into bulk
density variations as a function of position along the fluid stream -
the device is
also a pressure gauge. These fluctuations are large enough not to be evened out by
diffusion.
4.9.2
Fluctuation minimization
Dual cantilevers in series provide the ability to greatly increase the throughput of
density measurements, while reducing the need for constant monitoring of the experiments. Whilst the usage of second mode actuation eliminates the position dependent
error, the short term fluctuations described above cripple the implementation of the
device. As such, it the device can be altered to include an element that would filter
out these fluctuations.
A reservoir along the channel in between the two devices, such as depicted in
Figure 4-25, provides a time delay and increases the fluid divergence across the
channel, increasing the mixing by diffusion capacity along the length of the channel.
It is analog to an inductor in a electric circuit, filtering high frequency components.
A two-dimensional simulation of the effect of a 25 pm-diameter cylindrical reservoir
on a 4 pm-wide channel reservoir is shown in Figure 4-26. Time dependent fluctuations of the mixing ratio at the beginning of the channel propagate along it (blue),
however the presence of the reservoir greatly reduces their amplitude (red, yellow).
114
Time-0.25 Sice: Nea fraction (1)
TImne-2.25 Slice: Mass fraction (1)
A 1.0359
A 1.036
Figure 4-25: Filtering out the short term fluctuations with a fluidic reservoir. A
cylindrical reservoir (25 pm diameter, 20 pm height) is positioned after the x -junction
of the 4 x 4 pm square cross-section channels.
115
0.67
0.66
t= 5 ms
-
0.65
0.64
-
0.63
0.62
t
=
17 ms
----------------0
I
-
0.670.66-
t = 20 ms
0
-
K>
0.65
0.64
0.63
n aliI
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Length (pm)
Figure 4-26: Two-dimensional simulation of the effect on a 4 pm-wide channel of
a 25 pm-diameter cylindrical reservoir. Time-dependent pressure fluctuations at the
beginning of the channel (L = 0) are shown as a varying mixing ratio, propagating
along the channel length. The blue line represents a device with no reservoir - at
t = 0 ms the channel is filled with one fluid and at t = 20 ms the system is in
stationary state. Amplitude fluctuations are greatly decreased after the reservoirs,
located at different distances from the x-junction, indicated by the green dashed lines
(120 pm red and 450 pm yellow).
116
4.10
Conclusion
This thesis introduces a non-optical method for the determination of the mass and volume of the bulk aqueous and non-aqueous content of single living cells. It is achieved
by the measurement of a cell's buoyant mass in two fluids of different densities and
by completely replacing the intracellular water with deuterium oxide, thus allowing
the isolation of the dry content of the cell.
The method further allows for the determination of the bulk density of the cell's
dry content, which is correlated with changes in the chemical composition of the cell.
These changes occur as natural consequences of cellular growth or due to deliberate
perturbation. Further developments will increase the precision in the measurement
of buoyant mass, increasing the resolution in dry density. Also, the ability to weigh
single cells in three different fluids will allow these quantities to be determined simultaneously with the total cellular mass and water content fraction.
The combined dry mass and dry density characterization of cells provides a useful set of parameters for cellular discrimination as it describes cellular size, while
providing a quantitative index of chemical composition.
117
118
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