Rheology of Red Blood Cells with Acoustic Force
Spectroscopy (AFS)
Xamanie Seymonson
Thesis Bachelor Project Physics and Astronomy
Study Physics & Astronomy (BSc)
Version Final version
Size 15 EC
Conducted Between 01 - 04 - 2019 & 01 - 07 - 2019
Studentnumber 11226552
Daily Supervisor Giulia Bergamaschi MSc
Supervisor Prof. Gijs J.L. Wuite
Examiner Dr. David Fokkema
University Universiteit van Amsterdam
Vrije Universiteit Amsterdam
Faculty Faculty of Science
Institute Physics of Living Systems
Abstract
Red blood cells (or erythrocytes) are vital for the overall health of our bodies; as they are the carriers of oxygen and carbon dioxide to and from every other cell in our body. While they are being pushed through our veins they undergo large deformations, therefore they have a very flexible and elastic membrane. Studies have found that due to their deformability, changes in their structure can occur during diseases which greatly influence their mechanics. In order to develop better treatments for such diseases it is crucial for us to properly understand the mechanics of healthy red blood cells. Therefore, during this project we wanted to develop a workflow to analyse the mechanics of healthy red blood cells with acoustic force spectroscopy. The acoustic force spectroscopy setup allows us to perform oscillatory stress experiments between 0.01Hz and 1Hz on multiple erythrocytes at once in a highly parallel fashion. In order to analyse the obtained data a python script was developed which is able to nicely separate the viscous and the elastic behaviours of the cells. From the data we can speculate that the elastic component did not undergo active changes within the regime we examined. However, the influence of the viscous component starts to linearly increase starting from 0.03Hz with a slope of 0.49 ± 0.06 which is nicely in the viscoelastic regime. A hypothesis for this effect is loss of linkage between the cytoskeleton and the lipid bilayer of the cellmembrane, however, more data is needed in order to confirm this hypothesis. What we have been able to show is that acoustic force spectroscopy can be used for rheology experiments of erythrocytes, and that the workflow developed during this project can be adapted for rheology experiments of other cells, nuclei or synthetic polymeric material in order to obtain a better understanding of their mechanics.
Samenvatting
Onze rode bloedcellen zijn bijzonder belangrijk voor onze lichamelijke gezondheid. Zij zijn de dragers van zuurstof en stikstof van en naar alle andere cellen en kunnen hun taak uitvoeren doordat ze gepompt worden door onze bloedvaten. Tijdens hun traject door al onze bloedvaten komen ze ook in hele nauwe plekken terecht (zoals de haarvaten) en daar worden ze erg vervormd. Dit kan omdat rode bloedcellen erg flexibel zijn en elastische celmembranen hebben. Vanwege hun flexibiliteit zijn rode bloedcellen vatbaar voor permanente vervormin-gen tijdens verschillende ziekten. Deze permanente vervorminvervormin-gen be¨ınvloeden hun werkza-amheid en het is daarom erg belangrijk om een goed begrip te hebben van hoe gezonde rode bloedcellen mechanisch in elkaar zitten voor de ontwikkeling van medische behandelingen om deze ziekten tegen te gaan. Om bij te dragen aan het begrip van hoe rode bloedcellen in elkaar zitten is er tijdens dit bachelor projekt een poging gemaakt tot het ontwikkelen van een experimentele methode waarbij de mechanica van de cellen geanalyseerd kan worden met behulp van “acoustic force spectroscopy”. Met de “acoustic force spectroscopy” techniek kon een oscillerende kracht uitgeoefend worden op meerdere bloedcellen tegelijkertijd en kon de vervorming van de cellen gemeten worden (de oscillerende krachten hadden frequenties tussen 0.01Hz en 1Hz). Verder is er voor het analyseren van de data een “python” code geschreven. Deze code kon de elastische en de vloeibare component van de vervorming van de cellen scheiden. Met deze thesis wordt aangetoond dat “acoustic force spectroscopy” gebruikt kan worden voor reologie experimenten om meer kennis te vergaren in de mechanica van rode bloedcellen en dat de ontwikkelde experimentele methode (inclusief de python code) gebruikt kan worden voor soortgelijke experimenten op andere cellen, celkernen of materialen gemaakt van polymeren.
Contents
1 Introduction 4
1.1 The importance of Red Blood Cell mechanics . . . 4
1.2 Investigating the rheological properties of cells . . . 5
1.2.1 Oscillatory rheology . . . 5
1.2.2 Acoustic Force Spectroscopy . . . 7
1.2.3 Method development for rheology on erythrocytes with AFS . . . 8
2 Methodology 9 2.1 Force calibration of the AFS setup . . . 9
2.2 Oscillatory experiments on RBCs . . . 10
2.2.1 Data analysis protocol . . . 11
3 Results 13 3.1 Force Calibration measurements . . . 13
3.2 Acoustic rheology of RBCs . . . 13
4 Discussion 15 4.1 Conclusion . . . 16
Appendices 17
A Python code developed for data analysis 17
B Supplementary figures 21
1
Introduction
1.1
The importance of Red Blood Cell mechanics
Red blood cells (RBCs) are one of the most important cells in our bodies as they are the carriers of oxygen and carbon dioxide to and from the organs and muscles; RBCs, also called erythrocytes, are thus vital for their overall health and activity. Their discovery is credited to Antoni van Leeuwenhoek in 1674 because he is the first person to give a detailed description of RBCs but red “particles” in blood had been observed before [4]. RBCs are created in the bone marrow, have a round flat shape with a concave centre and lack a nucleus. They have a diameter of about 6.2 to 6.8 µm. In figure 1a is an example of the shape of a healthy RBC.
The cell membrane is made up of the lipid bilayer, transmembrane proteins and the cytoskeleton [1]. The lipid bilayer is the outer surface of the membrane and is lined with proteins. Spectrin is a cytoskeletal protein underneath the lipid bilayer which is organized as a meshwork Interestingly, this meshwork of spectrin not only maintains the shape of the cell but also plays a key role in the flexibility of RBCs [23]. In figure 1b a schematic view of the membrane is given. Here is clearly shown the lipid bilayer lined with proteins which are also linked to the meshwork of spectrin underneath.
The overall function of the cellmembrane is to separate the extracellular medium from the cytoplasm within the cell. Furthermore, the proteins which line the membrane control which molecules or ions get in and out of the cell. Erythocytes contain, in this manner, a protein called Hemoglobin which has an iron atom that binds oxygen in our lungs and transports it to every cell in the body. Hemoglobin can also carry the waste product of oxygen, carbon dioxide, from the cells to our lungs so we can breathe it out [15]. In figure 1c is shown where oxygen binds to the iron atom of hemoglobin.
(a) RBCs in blood vessel [24] (b) The cell membrane [18] (c) Hemoglobin molecule [6]
Figure 1: Red blood cells structure. (a) An artist impression of healthy RBCs in a blood vessel. The round flat shape is clearly shown with the concave centre. A front (or on top) view is given of the cell in the centre of the image. (b) A schematic depiction of the cell membrane is shown. The outer membrane is made up of the lipid bilayer which is lined with proteins. Underneath their is a meshwork of spectrin which the proteins are also linked to. This meshwork functions as the cytoskeleton maintaining the shape and flexibility of the RBCs. (c) A schematic depiction of the protein hemoglobin in RBCs. It shows where oxygen binds to the iron atom of hemoglobin.
In order for RBCs to fulfil their tasks they are transported throughout our body in blood vessels. While they are being pushed through our veins they endure large deformations especially when passing through microcapillaries [16]. Due to their deformability changes in their structure and mechanics may occur during diseases such as anemia [12], diabetes [9], malaria [20, 19] and cancer [8]. These changes can greatly affect their function which is why it is important to analyse the rheological properties of RBCs.
1.2
Investigating the rheological properties of cells
Similarly to other biological and soft matter systems, RBCs are viscoelastic. This means that RBCs have properties between solids (the purely elastic regime) and between liquids (the purely viscous regime); and their mechanical properties are time-dependent. Viscoelastic materials are characterised by their response to strain, showing typical behaviours such as stress-relaxation (figure 2a), creep (figure 2b) and hysteresis (figure 2c).
(a) Stress-relaxation (b) Creep response (c) Hysteresis energy loss
Figure 2: Characteristic responses of viscoelastic materials. (a) Given a constant strain the stress-relaxation response occurs. (b) Given a constant stress a creep response occurs. (c) The coloured area shows the amount of energy dissipated in the system during applied stress and strain. Figures taken from [5].
As a matter of fact, stress-relaxation will occur when a constant strain is generated in the structure for a finite amount of time. This causes the deformation to remain constant while the stress in the system decreases over time. Interestingly, a creep response occurs during an almost similar process; subjecting the system to a constant stress for a finite amount of time. However, this response indicates that the material deforms continuously showing an increasing strain until it reaches a plateau where the strain remains constant. Furthermore, the stress can also be plotted against the strain giving a hysteresis curve which is an indication for the energy dissipated in the system. Another way of measuring the rheological response of materials to strain are oscillatory stress experiments, often performed in the soft matter field.
1.2.1 Oscillatory rheology
In oscillatory stress experiments, given an input stress of the form
σ(t) = σ0sin ωt, (1)
where σ is the input signal, σ0the amplitude and ω the frequency, a viscoelastic material returns
the resulting strain (or displacement) of the form
ε(t) = ε0sin ωt − δ, (2)
where ε is the displacement through time, ε0the amplitude of the displacement and δ the phase.
The phase indicates the time difference it takes for the material to respond to the stress (or deformation). For a purely elastic material, the stress and the strain are in phase so the response to deformation if immediate (δ = 0). However, for a purely viscous material the strain lags stress by 90◦(or δ = π
2). Viscoelastic materials have behaviours somewhere in-between and thus we can
expect to find a strain signal which will have a phase of 0 <δ <π
2 (see figure 3a). The ratio of
stress to strain resulting from such oscillatory experiments can be represented by a complex shear modulus G∗ given by
G∗= G0+ iG00, (3)
where G0 and G00 are the storage and loss modulus respectively. These moduli are depended on the phase shift and the amplitude of the stress and the strain signals in the form
G0= σ0 ε0 cos δ, G00=σ0 ε0 sin δ. (4)
(a) Purely viscous and elastic delays versus viscoelastic delays (b) Rheometer measuring shear modulus of cells [10]
Figure 3: Typical elastic, viscous and viscoelastic responses. (a) An example is shown for the expected response of a purely elastic (completely in phase), purely viscous (90◦phase shift) and viscoelastic material (phase shift between 0◦ and 90◦). (b) A schematic depiction of a rheometer that can be used to study these viscous and elastic components of viscoelastic materials such as cells.
It is important to note that this complex shear modulus G∗ is used in shear oscillatory stress experiments in which parallel surfaces of material being experimented with slide past each other. Thus, from equation 4 we can obtain a shear modulus G∗ which is the trigonometrical vectorial sum of the storage and loss moduli G0 and G00.
Analogous to the complex shear modulus we can also define the complex elastic (or compres-sion) modulus E∗ for oscillatory experiments in which a stress is directly applied to a viscous substance and the deformation is recorded. This complex elastic modulus is also dependent on the storage and loss moduli (E0 and E00respectively), which are in turn dependent on the phase shift and the stress and strain amplitudes. Furthermore, we can also obtain a loss tangent (tan δ) which is the ratio between E0 and E00. These equations are as follows
E∗= E0+ iE00, E0= σ0 ε0 cos δ, E00=σ0 ε0 sin δ, tan δ = E 00 E0. (5)
Interestingly, the shear and elastic modulus are connected through Poisson’s ratio as
E∗= 2(1 + ν)G∗, (6)
which means that cells will show similar behaviours during shear or strain oscillatory experiments while values may vary.
Usually, to perform rheological experiments in the soft matter field, devices such as rheometers are used. These allow to directly apply an oscillatory stress in between two plates measuring the shear responses given in equation 3 and 4 [10]; a schematic depiction of the setup of such experiments is given in figure 3b. However, these are bulk studies which focus on the average response of cells at once, limiting the accuracy of these experiments. To study more precisely the mechanics of cells (and macromolecules such as DNA and proteins) many single cell and single molecule techniques have been developed over the years such as magnetic tweezers, optical tweezers, atomic force spectroscopy and micropipette aspiration. With these techniques single cells can be probed using a wide range of controlled forces from piconewtons to nanonewtons and their response can be recorded and analysed. Oscillatory stress test have also been performed
on, for example, nuclei using atomic force microscopy (AFM) [13] and on F-actin solution using magnetic beads [25]. Typically, these single molecule techniques can only probe one cell at a time which limits how much data can be obtained for statistics [21]. In order to solve these problems a new method was developed relatively recently (2014), namely acoustic force spectroscopy, which combines the accuracy of single-molecule techniques with multiplexing.
1.2.2 Acoustic Force Spectroscopy
Acoustic Force spectroscopy (AFS) is a relatively new method which uses acoustic forces to probe cells (or molecules) that are tethered between a glass surface and a microsphere (bead). With this technique multiple cells can be symmetrically probed during one measurement, giving an advantage in data collecting for statistics. Furthermore, this technique makes it possible to use highly controllable forces in the piconewton range and thus giving an advantage to more accurate probing.
In the AFS setup a piezo element is used through which an oscillating voltage is sent to a microfluidics (or flow cell). This oscillating voltage creates a resonant planar acoustic standing wave over the flow cell. A bead in the flow cell will experience a vertical (z-direction) acoustic radiation force Fradgiven by
Frad= −V ∇ 1 − κ? 4 κmp 2− ρ ?− 1 2ρ?+ 1ρmν 2 , (7)
with V the volume of the bead, p the acoustic pressure and ν the velocity of the acoustic wave. ρ? and κ?are respectively the density and compressibility ratio between the particle and the medium [21]. The acoustic force can push the beads upwards to the acoustic pressure node. An LED makes imaging with a 40× air-spaced objective and a digital camera possible and the x, y and z-displacement of the beads can be tracked with previously developed software. The algorithm for determining the z-displacement of the beads is determined by analysis of the diffraction pattern. A schematic depiction of the setup and the flow cell is given in figure 4.
(a) Schematic depiction of the AFS setup
(b) Schematic depiction of the flow cell zoomed in
Figure 4: The AFS setup. (a) A schematic view of the AFS setup is shown. The sample in the fluid chamber of the flow cell can be imaged by use of an inferted microscope. (b) A depiction of a typical measurement of DNA tethers. The piezo element is the driving force behind the acoustic standing wave created in the fluid chamber. Due to the acoustic force created by this wave the beads connected to the DNA tethers are pulled up stretching the DNA. Figures taken from [3]
1.2.3 Method development for rheology on erythrocytes with AFS
The AFS setup, as previously described, has already been used to study the mechanics of DNA by performing force-extension experiments [21]. Recently, it has also been used to study properties of RBCs by performing force-clamp experiments while introducing changes in the cell mechanics via chemical additions [22]. During these experiments highly controlled forces were used in the piconewton range; which are also ideal for performing oscillatory stress tests in order to stay in the linear regime. This gives us the opportunity to understand the viscous and elastic properties of healthy RBCs separately as we can untangle them from equation 5. Furthermore, oscillatory stress tests on human blood have been performed by using other methods [11, 2] but these experiments have not yet been done on single RBCs. The goal during this project is to show that AFS can be used to study the viscoelastic properties of RBCs by performing oscillatory stress tests on single cells. Furthermore, we want to develop an analysis workflow to study the obtained data from the oscillatory experiments and understand the viscous and elastic properties of healthy RBCs.
2
Methodology
2.1
Force calibration of the AFS setup
The experiments on the RBCs are done by making use of the AFS setup discussed in the previous section. Before the measurements on RBCs can be carried out we need to find a conversion factor between the acoustic force in the flow cell and the voltage placed over the piezo element. This force calibration is done by applying a voltage to the piezo element of the flow cell which creates an acoustic force inside the flow cell and kicks the beads up to the node. The displacement of the beads in the z-direction is of interest and this is tracked with software which analyses the change in the diffraction pattern of the beads. When the beads are at the node the voltage is turned off and they can fall back down to the surface due to gravity. In figure 5a an interpretation of this process is given. The beads start at the bottom of the flow cell, then a voltage is driven through the piezo element generating an acoustic planar standing wave which kicks the beads up with the force given in equation 7. However, this is not the only force that will be acting on the beads. As they are suspended in a solution, they will also be subjected to gravity (FGrav), the buoyancy
force (FBuoyancy) and the Stokes’ drag force (Fdrag) [7]. When the beads are at the node these
forces balance each other out in the force balance equation given by
FGrav− FBuoyancy+ Fdrag− Frad= 0. (8)
From this the algorithm finds the force felt by the beads at a given voltage. Finally, the voltage is turned off and the beads are allowed to float back to the surface. Figure 5b shows an example of a trace for a calibration experiment in which a bead goes to the node and floats back down to the surface.
(a) Depiction of the beads movement during force cal-ibration
(b) Bead step experiment [7]
Figure 5: Force calibration. (a) A schematic figure of the flow cell during the force calibration is show. The bottom surface of the flow cell is incubated with a 0.5% BSA - 0.5% pluronics mixture. A voltage difference is placed over the piezo element which creates an acoustic force in the flow cell kicking up the beads. When the beads are at the node the voltage is switched off allowing them to float back down to the surface due to gravity. (b) An example is given of a calibration experiment. The red trace depicts the voltage put over the piezo element. The black trace shows the displacement of the bead; going up as the voltage is added and floating back down when the voltage is turned off.
The force needed to send the beads up to the node is recorded per voltage for a number of beads and the average force is plotted for each voltage. On the data an allometric curve
y = axb (9)
is fitted. Here a is the conversion factor needed and b = 2 because the force and the voltage scale quadratically.
Beads and Flow cell preparation. For all experiments “shooting” beads are used. These are beads which are not coated with biochemical substances such as, for example streptavidin or concanavalin A. The beads were made of silicon dioxide (SiO2) and had a size of 6.59 ± 0.16 µm.
In order to prepare the beads for the experiment they were first washed with phosphate-buffered saline (PBS) in a centrifuge by spinning them down for 2 minutes with a speed of 500rcf at 23◦C. They are then resuspended in ringer’s buffer, which is a solution that has a similar salt and glucose concentration as the human body and helps to maintain a constant pH value [17]. The flow cell is incubated for a minimum of 30 minutes with a 0.5% BSA - 0.5% pluronics mixture to prevent the beads from sticking to the surface. Before flushing the beads into the flow cell some ringer’s buffer is flushed in to remove the excess BSA-pluronics.
2.2
Oscillatory experiments on RBCs
In order to perform oscillatory experiments on RBCs they need to be attached to the surface of the flow cell. The RBCs are collected right before the experiments from one healthy donor by finger pricking and diluted in Ringer’s buffer to keep ATP production active. Furthermore, the beads need to be attached to the cells so they can be probed.
Beads and Flow cell preparation. To attach the cells to the surface, this is incubated with 0.1 mg/ml of poly-L-lysine (PLL) for a minimum of 30 minutes. After the flow cell passivation, Ringer’s buffer is flushed in to remove the excess PLL and the RBCs are flushed in so they can spread onto the surface. In order for the beads to bind to the cells they are activated with 3% HCl for 10 minutes. They are then incubated for 30 mins at 4◦C with 1 mg/ml concanavalin A (con-A). This is a binding protein that makes the beads stick to the RBCs through carbohydrate binding. The beads are then diluted in ringer buffer to a concentration of approximately 3% w/v and flushed into the flow cell so they can bind with the RBCs. A typical field of view (FOV) with beads on cells is shown in figures 6a and 6b. Figures 6c and 6d give an impression of how the beads pull on the cells.
(a) Beads on RBCs (b) Beads on RBCs zoomed in
(c) Beads stuck on RBCs [22] (d) Beads pulling on RBCs [22]
Figure 6: Beads stuck on RBCs. (a) The beads on Red blood cells along with beads stuck to the surface are tracked during measurement. The beads that are stuck to the surface are needed to account for the average drift of the beads in the fluid. (b) Shows a zoomed in field of view for beads on RBCs. (c) The RBCs are stuck to the PLL on the surface and beads are stuck on RBCs as to the PLL on the surface between the RBCs. (d) The acoustic force pulls the beads up to the node which in turn pull on the RBCs or on the PLL on the surface.
Experimental assay. Once the sample is set we can perform oscillatory stress tests on the cells. A voltage peak-to-peak amplitude, which corresponds to a force pulling on the beads, is modulated at a specified frequency causing the acoustic force to oscillate at the set amplitude in a sinusoidal manner. The frequency range used for these experiments was between 0.01 Hz and 1 Hz. The oscillating z-displacement of the beads is recorded and, using the force calibration, the oscillating voltage is converted into force. These two signals are of interest, more precisely the phase shift between the input force and output cellular response.
2.2.1 Data analysis protocol
Both signals were of a sinusoidal form, where the signal of the force had the form of equation 1 which was smooth (because this is our digital input) and the signal of the displacements of the beads had the form of equation 2 which also had noise that needed to be filtered out. In order to analyse the data and filter out the noise a python script was developed. In this script a file for each frequency containing rows for the time, force and average distance (response of the beads) could be loaded. The program separates the data in arrays which can be used for the data analysis process. The array for the force is used to determine the frequency, phase and offset. The amplitude is determined by taking the highest and the lowest number in the average distance array, subtracting them and dividing by 2. These values are used as guesses for a least-square sinefitting of the cellular response, as follows
A sin(2πf t − δ) + B (10)
which filters out the noise. Here A is the amplitude of the average distance, f is the frequency, t the time, δ the phase and B is the offset. The fit nicely capture the peaks of the response in 3 orders of magnitude. A typical trace for the displacement and the sine fit to filter out the noise is given in figure 7 for data of 0.01Hz, 0.1Hz and 1Hz.
(a) Sine fit of 0.01Hz (b) Sine fit of 0.1Hz
(c) Sine fit of 1Hz
Figure 7: Typical sine fit. The blue line in the graphs shows a typical trace of the beads during the oscillation experiments for frequencies 0.01Hz, 0.1Hz and 1Hz. The red line is the obtained sine fit and this fit is used for the rest of the data analysis.
(a) Force and displacement traces (b) Time delay
Figure 8: Force and displacement traces. (a) A typical trace is shown of the applied force and the response from the cells. These traces are analysed by the developed python program; the program uses the peaks to find the phase shift which is used to find the viscous and elastic components and the loss tangent as described in section 2.2.1. The traces shown have an input frequency of 0.1Hz. (b) A zoomed in view of the traces for the applied force and the response is given. Here the phase shift is clearly visible.
The program then determines the peaks of the sine fit and the force signal and can from these peaks determine the time delay ∆t and phase shift ∆δ between the force and the displacement peaks; a typical trace of the force and displacement with detected peaks and phase shift is shown in figure 8a and figure 8b respectively. As given in equation 5 the viscous and elastic contributions can be obtained from the data. We can determine the ratio between the viscous (E00) and elastic (E0) contributions from the tangent of the phase shift, the loss tangent given by
tan(∆δ) = E
00
E0. (11)
The viscous and elastic contribution are both multiplied by a factor called the young’s mod-ulus, which measures a materials ability to withstand deformations. Within the program an apparent young’s modulus (just young’s modulus from now on), E∗, can be determined by using an approximation
E∗= AF 3πaAD
, (12)
where AF is the amplitude of the force signal, a is the radius of the bead and ADis the amplitude
from the sine fit on the average distance signal and thus obtain the separate contributions of E00 and E0
E00= E∗sin(∆δ),
E0= E∗cos(∆δ). (13)
3
Results
3.1
Force Calibration measurements
In order to calibrate the force as described in section 2.1 a voltage difference in the range of 0.5V and 1.2V is placed over the piezo element which kicks the beads up to the node. The maximum force needed is analysed by the algorithm from equation 8. A typical trace of a calibration experiment is given in figure 9a; where a voltage is placed over the piezo for a finite amount of time and the trajectory of the bead going to the node and back to the surface is plotted. The force needed per voltage is plotted in figure 9b and the curve fit is used to obtain the conversation factor a. Approximately 15 ± 3 beads were recorded for every voltage (detailed statistics are given in appendix C table 1). For the SiO2 beads used in Ringer’s buffer a = 13.06 ± 0.29.
(a) Bead step trace (b) The force calibration results
Figure 9: Force calibration. (a) A voltage difference is placed over the piezo element for a finite amount of time and the displacement of the bead going to the node en back to the surface is recorded. (b) Voltage differences between 0.5V and 1.2V where placed over the piezo and the forces recorded. The conversion factor obtained from the data is 13.06 ± 0.29.
3.2
Acoustic rheology of RBCs
The data of the response of the cells was obtained on multiple days. In the time delay between the response and the input signal there was a day to day variability which was accounted for by taking the standard error of the mean for the results of the loss tangent, the young’s modulus, the viscous component and the elastic component. These results were plotted versus the frequency. The measurements were done on 12 days and per frequency approximately 64 ± 11 beads were tracked (detailed statistics are shown in appendix C table 2). In figure 10a the loss tangent is plotted for the frequencies measured. There are two regions; at low frequency the loss tangent starts off with a plateau and starts to increase at 0.03Hz. A linear fit through the data from 0.03Hz to 1Hz has been done to find the power law where the two extrema are 0 and 1; here 1 indicates a purely viscous material and 0 a purely elastic material. The slope is 0.33 ± 0.03 indicating that the cells are nicely in the viscoelastic regime, with a predominant elastic contribution.
The loss tangent can also be factorised into its viscous component (E00) and its elastic com-ponent (E0) given in equation 13. These values are multiplied by the young’s modulus which is obtained by equation 12. The viscous and elastic components are plotted together in figure 10b. The viscous component seems to have a similar shape as the loss tangent and the data between 0.3Hz and 1Hz has been analysed with a linear fit. The slope of the linear fit is 0.49 ± 0.06, which is nicely in the viscoelastic regime. Given the error bar range this slope could be even steeper
in-dicating that the cells become more viscous. Furthermore, the prefactor of the viscous and elastic contributions, the young’s modulus, is plotted in figure 10c. From the young’s modulus a complex viscosity (η∗) could be obtained by dividing the young’s modulus values by the frequency. These values are plotted for the frequency in figure 10d. Interestingly, the complex viscosity seems to have a very clear linear trend downward.
(a) Loss tangent (b) Viscous(E00) and elastic (E0) components
(c) Young’s modulus (d) reduced viscosity (η∗)
Figure 10: Viscoelastic mechanics of RBCs. (a) The loss tangent has two regions; a plateau from 0.01Hz to 0.03Hz and goes up from 0.03Hz to 1Hz. The lineair fit has a slope of 0.33 ± 0.03. (b) The elastic component is seemingly constant and dominates the viscous component. The viscous component has a similar shape as the loss tangent with a plateau from 0.01Hz to 0.03Hz and rising from 0.03 to 1Hz. The linear fit has a slope of 0.49 ± 0.06. (c) The young’s modulus is seemingly constant for each measured frequency. (d) The reduced viscosity, obtained by dividing the young’s modulus by the frequency, seems to have a very clear linear trend downwards.
4
Discussion
Comparing figure 10a and 10b there seems to be a resemblance between the shape of the loss tangent and E00. Given that E0 is seemingly constant, that means that the viscous component is the determining factor for the slope of the loss tangent. Our loss tangent values are nicely in line with literature, such as measurements for fibroblast cells within a frequency range of 0.01Hz and 30Hz [10]. However, our results do indicate that RBCs are softer in comparison to fibroblast, as they found a seemingly constant loss tangent between 0.25 and 0.35 (see appendix B figure 12) while we found a loss tangent in a wider range, specifically between 0.084±0.019 and 0.328±0.014. This data does make sense because RBCs do not have a nucleus and thus are more flexible than fibroblast cells. The increase of the loss tangent indicates that the response of the RBCs becomes increasingly “fluid” as the frequency increases. Furthermore, the elastic (G0) and viscous (G00) components they found both increase parallel to each other in a linear fashion as the frequency increases (see appendix B figure 12). Interestingly, our elastic component E0 is constant over the frequency range used, while our viscous E00 component does increase in a linear fashion. Also important to note is that the experiments on the fibroblast cells were oscillatory shear stress tests and thus the shear moduli are computed. The relation between the shear moduli and elastic moduli is given in equaction 6, which explains why our values are not in the same range as [10]. Another explanation could be the prefactor used for the elastic and viscous component, namely the young’s modulus depicted in figure 10c. An approximation was used to determine the values of the young’s modulus given in equation 12. However, this approximation was adapted from [25] where a bead was completely embedded in viscoelastic material as opposed to the beads used for this experiment which are on top of viscoelastic material, thus this approximation could have an influence on the values obtain causing them not to be in line with [10]. However, the constant behaviour of the elastic component can be an indication that probing cells at these forces and frequencies (which are, in comparison to other research, on the low side) doesn’t actively probe a change in elastic behaviour of the cells. Interestingly, for the RBCs the elastic behaviour would mainly be due to the cytoskeleton, i.e. the meshwork of spectrin. The fairly constant elastic component thus hints that the spectrin is not actively deformed in this frequency region. Furthermore, the viscous component would mainly be due to the outer side of the cell; the lipid bilayer.
Between 0.01Hz and 0.03Hz the viscous component is seemingly constant. Interestingly, start-ing from 0.03Hz the viscous component starts to rise linearly which indicates that as the frequency increases the cells become more deformable or “fluid”. That the cells become more “fluid” starting from 0.03Hz can be due to loss of linkage between the lipid bilayer and the spectrin meshwork as the beads pull on the cells. This effect would gradually dominate the membrane fluctuations which the increasing viscous component possibly hints to. Furthermore, this loss of linkage could be possible because the beads used where activated with con-A, which is a lectin that binds to the glycoproteins of lipid bilayer; some of these glycoproteins are directly connected to spectrin which makes it easier to pop out linkage with the lipid bilayer. Important to note is that PLL, which was used to incubate the surface, causes the cells to spread out over the surface and thus slightly stretching them horizontally while the beads stretch them vertically during measurement. As the frequency increases this could cause the bond between the glycoproteins and the spectrin to snap giving way to a more viscous response of the cells. Furthermore, examining the plot of the com-plex viscosity (figure 10d) it has a clear linear trend downwards, which shows that with increasing frequency, the resistance of the cells to deformation decreases (they “flow” easier or become more “fluid”). In order to confirm this, further measurements can be done using a different binding protein for the beads, which does not make a direct connection to glycoproteins and therefore to spectrin or using no binding proteins. Also, another binding agent could be used to coat the surface which does not cause the erythrocytes to spread out as PLL does but instead allows them to retain their original concave disk shape. A possibility could be using fibronectin instead as it was recently observed in the lab that when fibronectin was used to incubate the surface that the RBCs retained their shape (unpublished data). Furthermore, tests could be done with addition of chemicals that act on the lipid bilayer or the spectrin, in order to understand which structural components contribute to the viscous and elastic behaviour.
Examining the first three points for E00in figure 10b they seem to give way to a dip in the data instead of a plateau. This could be possible as very similar behaviour was found for oscillatory experiments on F-actin solutions in the frequency range of 10−3Hz to 105Hz [14]. In the range of 0.01Hz and 1Hz our results closely match the shape of the graph for the storage and loss moduli versus the frequency [14]; our E0 and their G0 having a plateau shape, and our E00 and their G00 seemingly having a dip at low frequency. However, as they performed shear oscillatory stress test the values are not in line (the graph of [14] is given in appendix B figure 11 for comparison). This dip was ruled out for the data shown here, because the viscous and elastic component combined for the loss tangent do give a plateau for these points. The first point is also higher for both the viscous and elastic component which could be due to a systematic error. Furthermore, there is no data obtained at lower frequencies to confirm that the viscous component plays a more significant role in that region in comparison to 0.01Hz to 0.03Hz. Therefore, it would be beneficial to obtain data at even lower frequencies. With this data it is not possible to say with certainty which structural component contributes to the viscous behaviour to become more present as the frequency increases starting from 0.03Hz, thus more measurements are needed.
4.1
Conclusion
During this project the goal was to develop an analysis workflow to understand the viscous and elastic contributions of healthy erythrocytes’ mechanics with the AFS setup. In order to probe the RBCs mechanics the AFS setup allowed for oscillatory experiments to be performed on multiple cells at once in parallel in the linear regime. The frequencies used during these experiments were between 0.01Hz and 1Hz, and the data showed that for this frequency range the elastic behaviour was not actively probed. Therefore, a speculation that can be made is that this data hints that the spectrin network underneath the lipid bilayer did not undergo large deformations in this regime. Furthermore, the data shows that there is an increase in viscous behaviour starting from 0.03Hz; this could be due to breakage of linkage between proteins in the lipid bilayer to the spectrin meshwork underneath giving way to a more “fluid” cellmembrane. This hypothesis could be a possibility because the beads used for probing were coated with con-A which binds to the glycoproteins in the lipid bilayer, some of which are directly connected to spectrin. However, more data is needed in order to solidify this hypothesis. With our current data we cannot say with certainty which structural components of the erythrocytes are responsible to the viscous and elastic behaviours of their mechanics. However, what we have been able to show is that AFS can be used for these rheology experiments on cells and that the algorithm developed for data analysis can give us an insight into the cell mechanics. This analysis workflow can also be adapted for other cell types, nuclei or synthetic polymeric material.
Appendices
A
Python code developed for data analysis
# automated data analysis for oscillatory experiments with beads on cells # author: Xamanie Seymonson
# with help from Giulia Bergamaschi # necessary libraries
import scipy import numpy as np
import matplotlib.pyplot as plt from scipy.optimize import curve_fit from scipy.signal import find_peaks from scipy.optimize import leastsq # function reads in csv file
# generates time, force and average distance arrays def read_csv(file): data_file = open(file, ’r’) row_number = 0 time_list = [] force_list = [] distance_list = [] for row in data_file:
row_number += 1 if row_number >= 3: cut_data = row.split(’,’) time_list.append(float(cut_data[0])) force_list.append(float(cut_data[1])) distance_list.append(float(cut_data[-1])) time = np.array(time_list) force = np.array(force_list) distance = np.array(distance_list) print ("File results: ", file) return time, force, distance # LOAD IN CSV FILE HERE
data = read_csv(’example.csv’)
def peaks(time, array):
peaks , _ = find_peaks(array, height = 0.01, distance = 1) peak_list = [] peaktime_list = [] for j in peaks: peak = array[j] peak_time = time[j] peak_list.append(peak) peaktime_list.append(peak_time) peak_array = np.array(peak_list) peaktime_array = np.array(peaktime_list) return peaktime_array, peak_array
# generate peak height and peak time arrays for the input force_peaktime_peaks = peaks(data[0], data[1])
# plots the peaks of the input for check
plt.plot(data[0], data[1], ’b-’, label = ’input’)
plt.plot(force_peaktime_peaks[0], force_peaktime_peaks[1], ’gx’, label = ’peaks’) plt.xlabel(’time (s)’)
plt.ylabel(’force (pN)’) plt.legend()
plt.show()
# function finds the frequency and period for sinusoidal traces def freq_finder(time_array): delta_peak_time_list = [] frequency_list = [] counter = -1 for i in range(len(time_array) - 1): counter += 1
delta_peak_time = np.subtract(time_array[counter + 1], time_array[counter]) delta_peak_time_list.append(delta_peak_time)
frequency = 1/delta_peak_time frequency_list.append(frequency)
avg_period = sum(delta_peak_time_list)/len(delta_peak_time_list) avg_frequency = sum(frequency_list)/len(frequency_list)
return avg_period, avg_frequency
# generate the average frequency and phase of the input
avg_freq_delta_time_input = freq_finder(force_peaktime_peaks[0]) avg_freq_input = avg_freq_delta_time_input[1]
# function fits a general sine to data
def fitting_function(time, signal, guess_amp, guess_freq, guess_phase, guess_mean): optimize_func = lambda x: x[0]*np.sin(2 * np.pi * x[1]*time - x[2]) + x[3] - signal
est_amp, est_freq, est_phase, est_mean = leastsq(optimize_func, [guess_amp, guess_freq, guess_phase, guess_mean])[0] data_fit = est_amp*np.sin(2 * np.pi * est_freq*time- est_phase) + est_mean
# print optimized variables for check print (’fit amp:’, est_amp)
print (’fit freq:’, est_freq) print (’fit phase:’, est_phase) print (’fit offset:’, est_mean) return est_freq, data_fit, est_amp # define the guess variables for the fit
guess_amp = 0.5 * (data[2].max() - data[2].min()) guess_freq = avg_freq_input
guess_phase = avg_phase_input guess_mean = np.mean(data[2]) # print guess variables for check print (’guess amp:’, guess_amp) print (’guess freq:’, guess_freq) print (’guess phase:’, guess_phase) print (’guess offset:’, guess_mean)
# call on the fit with the guess variables and the response trace
fit_signal = fitting_function(data[0], data[2], guess_amp, guess_freq, guess_phase, guess_mean) # plot the response and the fit for check
plt.plot(data[0], data[2], ’b-’, markersize = 0.75, label = ’raw output response’) plt.plot(data[0], fit_signal[1], ’r-’, label = ’sine fit of the response’)
plt.xlabel(’time(s)’)
plt.ylabel(’displacement (um)’) plt.legend(loc = ’upper right’) plt.show()
# generate peak height and peak time arrays for the response response_peaktime_peaks = peaks(data[0], fit_signal[1]) # from micrometer to nanometer for the response
distance_in_nm = fit_signal[1] * 10**3
distance_peaks_in_nm = response_peaktime_peaks[1] * 10**3 # plot the input, response and peaks of both
plt.plot(data[0], data[1], ’b-’, label = ’force’)
plt.plot(force_peaktime_peaks[0], force_peaktime_peaks[1], ’rx’)
plt.plot(data[0], distance_in_nm, ’g-’, label = ’sine fit of response’)
plt.plot(response_peaktime_peaks[0], distance_peaks_in_nm, ’rx’, label = ’peaks’) plt.xlabel(’time(s)’)
plt.ylabel(’displacement (nm) and force(pN)’) plt.legend(loc = ’lower right’)
plt.show()
# function finds the time difference between the input and response peaks # and finds the average phase difference and the loss tangent
def delta_timepeaks_intput_response(input_peaktime, response_peaktime, frequency): delta_peaktimes_array = abs(np.subtract(response_peaktime, input_peaktime)) avg_delta_peaktimes = np.sum(delta_peaktimes_array)/len(delta_peaktimes_array) avg_delta_phase = 2 * np.pi * avg_delta_peaktimes * frequency
tangent_phase = np.tan(avg_delta_phase)
delta_phase_array = 2 * np.pi * delta_peaktimes_array * frequency standard_error_delta_phase = np.std(delta_phase_array)
return delta_peaktimes_array, avg_delta_peaktimes, avg_delta_phase, tangent_phase, standard_error_delta_phase # generates the average phase and time difference and the loss tangent
final_result = delta_timepeaks_intput_response(force_peaktime_peaks[0], response_peaktime_peaks[0], fit_signal[0]) # amplitude of the force from piconewton to newton
# amplitude of the distance from micrometer to meter
force_amp = 0.5 * (data[1].max() - data[1].min()) * 10**-12 response_amp = abs(fit_signal[2]) * 10**-6
# calculates the young’s modulus (E*) and its components in newton per meter squared young_modulus = force_amp / ( 6 * np.pi * 0.5 * 6.59 * 10**(-6) * response_amp) elastic_component = young_modulus * np.cos(final_result[2])
viscous_component = young_modulus * np.sin(final_result[2]) # prints all final results
print (’delta times:’, final_result[0])
print (’average delta time:’, final_result[1]) print (’average delta phase:’, final_result[2])
print (’standard error delta phase:’, final_result[4]) print (’loss tangent:’, final_result[3])
print (’young modulus:’, young_modulus)
print (’elastic component:’, elastic_component) print (’viscous component:’, viscous_component)
B
Supplementary figures
Figure 11: F-actin shear storage and loss moduli. The solid symbols are the storage (G0) moduli and the open symbols are the loss (G00) moduli for 2 different trials. At low frequency (between 0.001Hz and 10Hz) the G0dominates with a plateau, similar to our data (between 0.01Hz and 1Hz) for the elastic component (E0). Interestingly, here there is a clear dip for G00at the low frequency. Our data also seems to have this dip for E00, however, more data needs to be obtained to confirm this. In this graph G0 and G00are given in dynes/cm2 which is relates to 10−1
N/m2
Figure 12: Fibroblast shear storage G0 and loss G00moduli. The plot shows the relation between G0 and G00versus the frequency, as the frequency increases there is a linear increase of both in a parallel fashion. Also, their ratio G00/G0 which is the loss tangent (tan δ) is given which is constant within this frequency range.
C
Statistics
Table 1: Force calibration statistics. The data collected for the force calibration is spread over 4 days. For each input voltage the force needed to send the beads up to the node was recorded for a number of beads. The average force of these beads was plotted versus the voltage to find the conversion factor discussed in section 2.1 and 3.1.
Voltage (V) Number of beads
0.5 7 0.6 12 0.7 19 0.8 23 0.9 20 1 23 1.1 10 1.2 9
Table 2: Oscillation experiments statistics. The data for the oscillatory experiments on the RBCs was collected over 12 days. On each day multiple data sets where used due to switching to different fields of view. The recorded displacement of the beads in separate datasets were averaged and these averaged results for datasets where brought together in the end results for each frequency. These results are discussed in section 3.2
Frequency (Hz) Number of beads
0.01 35 0.02 48 0.03 42 0.04 48 0.05 54 0.06 47 0.07 47 0.08 47 0.09 47 0.01 70 0.02 55 0.03 43 0.05 137 1 179
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