For the interpretation of our data, we model the contracting cell on the elastic substrate. In particular, we are interested in the interaction between the cell and the stiffness of the substrate. Here, different models exist, varying in complexity.
For our specific problem, we employ the model introduced by Edwardset al.[26].
A sketch of the model can be found in Fig. 3.4. Unless declared otherwise, the same variable notation holds as used in Section 3.1. The entire derivation for our specific problem of a contractile blood platelet,i.e. the deviations from the original works by Edwardset al.[26], can be found in Hanke et al.[37].i
The original model [26] was derived to analytically solve the problem of a cell layer contracting on a set of micro-pillars. Here, the micro-pillars are described as a set of springs with a given spring constant or stiffness. We, however, study a single platelet contracting on an elastic substrate. Thus, to mathematically describe the interactions between cell and substrate, we define an elastic string stiffness density
iI am indepted to D. Probst, U. Schwarz (University of Heidelberg) and A. Zemel (Hebrew Uni-versity of Jerusalem) for providing theses derivations.
Chapter 3 THEORY
Figure 3.4.: A schematic representation of the model described in Hanke et al. [37]. The platelets is reduced to a circular disc (orange) of radius rc (red) and heighthc. The platelet is characterised further by the elastic modulus Ec and Poisson's ratioνc. The active stress of the platelet during contraction is given byσ0. The combined stiness density of the anchor proteins and substrate is given asY (black), a function of the single stiness components ka (protein) andES (substrate). As the problem is axially symmetric, the independent variable is denoted by r.
Yto reflect both the stiffness of the substrate as well as the anchor proteins between cell body and substrate. Additionally, instead of studying a cell layer, our object is a single platelet approximated by a contractile disc situated on the springs.
Next, let us define a number of physical properties of the blood platelet. First, for the elastic platelet, we define the elastic modulusEcas well as a Poisson’s ratio νc. Furthermore, as we approximate our platelet as a contractile disc, we set the radius torcand the height tohc. Lastly, the active stress of the platelet is set toσ0. The force balance is then given by
∇σ = Y
hcu. (3.32)
We again assume small displacements within the deformed layer as done previ-ously in Section 3.1, which allows us to linearise the strain. Then, the single stress’
components take the form We can further simplify Eq. (3.33) if one assumes that the active stress within the platelet is constant at each point, hence, its derivative in space is 0. The force balance is now given by
Modelling a Contractile Cell 3.5 We then define a localisation lengthlLsuch that
lL= s
hcEc
Y(1−ν2c). (3.35)
This length describes the decay of a point force as seen by the deformation of the substrate. If we now decouple the assembled spring constantY into its two components, namely the stiffness of the substrate as well as the protein bonds as previously demonstrated in Ref. [5], we re-write Eq. (3.35) as
lL= withNa/L2 being the ligand density andka its stiffness,Esthe elastic modulus of the substrate, νs its Poisson’s ratio andhs its height. We combine the properties of the adhesion region,i.e. the layer of the ligands, by setting NLa2ka as the so called adhesion layer stiffness density.
Let us now take a look at the displacement we observe in the substrate. As our cell is modelled as a circular disc of isotropic contraction, in mathematical terms, the problem is axially symmetric and the strain tensor is given in spherical coordinates as
withubeing the displacement as before. Note that due to the condition of isotropy, the angle does not play a role in this equation. We further have the boundary conditionu(r=0) =0 and
with I0 and I1 being the modified Bessel functions of first kind. The total force is given by
F =
Z
Y|u|dx. (3.40)
Chapter 3 THEORY
Evaluating this integral, we obtain for the theoretical total force the expression Ftheo= π2rchcσ0
I1(rlc
L)L0(rlc
L)−I0(rlc
L)L1(rlc
L) I0(rlc
L)−(1−νc)·lrL
c ·I1(rlc
L). (3.41)
Here, Ln(x) denotes the modified Struve function. From this relation, we can determine the behaviour of the total force in its limiting cases. The first limit is the case of soft substrates. As L0(0) = L1(0) = 0 and the localisation length increases towards infinity, the total force goes to 0. On stiff substrates, however, the localisation length decreases to its minimal value which in turn increases the total force to its maximal value. Note that the minimal localisation length is given by the expression lL,min =
qhcEcL2
Naka . Hence, lL and thus the maximal theoretical total force is governed by two entities, one stemming from the cell properties (Echc) and one stemming from the adhesion layer (NLa2ka).
Lastly, let us find a relationship between the total force and the cell radius. We approximate the Bessel functions given in Eq. (3.41) forx >>nby the expression In(x)≈ √ex
2πx. This yields for Ftheo(rc)that Ftheo
2πrc ≈ σ0hc 1+ 1
2−νc lL
rc +O lL
rc 2!
. (3.42)
From here, we can directly see that, ifrclL, it follows that 2πrFtheo
chc =σ0.
4
Materials and Methods
In the present chapter, all methods used for the experiments are summarised.
In Section 4.1, the isolation of platelets from concentrates is described, followed by the fabrication of the PAA gels used in all experiments in Section 4.2. The assembly of the different microfluidic devices is outlined in Section 4.3. Finally, the recording of the data is explained in Section 4.4. Note that Sections 4.1, 4.2.1 and 4.4.2 are part of Hankeet al.[37] and Sections 4.2.2, 4.3 and 4.4.3 are part of Ref. [36].
All buffers and solutions mentioned in this chapter are listed in Table 4.1 along with their chemical composition. All commercial chemicals and tools used are found in Table 4.3 along with the corresponding supplier.
Chapter 4 MATERIALS AND METHODS
4.1 Platelet Isolation From Plasma Concentrates
The plasma isolation was done according to Ref. [107]. The platelets were isolated from plasma concentrates donated to the University Medical Center Göttingen but expired for clinical use. Thus, platelets were used 4 to 6 days after donation. All work described below was performed in a clean bench. Before starting the isola-tion, at least 30 min in advance, the tubes containing PSG (Pipes Saline Glucose) and HT buffer (Hepes-Tyrode buffer) were pre-heated in the incubator, slightly opened. This was done to ensure that the buffers exhibited similar conditions as found in vivo, namely 5 % CO2 content and a temperature of 37◦ C. When not used, the tubes were always stored in the incubator during the isolation process.
From the blood concentrates, the cap on the outlet was removed. The bag was still closed at this point as an additional, self-sealing rubber stopper was incorpo-rated into the bag itself. A 2 mL syringe was connected to a needle of size 1 and carefully inserted into the rubber stopper without perforating the bag. Slowly, the syringe was filled, detached from the needle and discarded including the 2 mL of blood plasma. A 5 mL syringe was subsequently attached to the needle and filled with 4 mL of plasma. The syringe was again detached and the content transferred into a 15 mL reaction tube by slowly letting it run down the tube’s wall. 90µL of PSG was mixed with 10µL of PGE1 (Prostaglandin E1) and added into the 15 mL tube. Mixing it shortly by swirling it gently by hand, the tube was put into the centrifuge kept at 21◦C. A counterbalance was inserted, the centrifuge was closed and set to 480gfor 20 min.
After centrifuging, a pellet consisting of platelets and red blood cells were found in the bottom of the tube. The supernatant was carefully removed and 4 mL of PSG was added. The cells were now brought back into suspension by gently pipetting the fluid up and down until no clumps of cells were observed anymore.
It is important to note that the re-suspension was done very gently as to not stress the platelets unnecessarily.
Subsequently, 10µL PGE1was mixed with 90µL PSG and added to the platelet suspension. After swirling the tube, it was inserted into the centrifuge and put to the previous settings. In total, the cells including the PGE1 were centrifuged and re-suspended in 4 mL PSG three times. During the centrifugation, in a separate reaction tube, BSA (bovine serum albumin) was added to the HT buffer to a final concentration of 5 mg/mL. After the last centrifugation step, the cells were re-suspended in 1 mL of HT-BSA buffer until the solution was a homogeneous, milky
Platelet Isolation From Plasma Concentrates 4.1
height of liquid column
height of platelet pellet clay surface centrifugation
Figure 4.1.: A sketch of the process of cell counting. A capillary is lled with a suspension of platelets (yellow). The uid upper surface should not exceed the red line at the top of the capillary. Leakage is prevented by sealing the capillary with clay at the bottom (brown). After centrifugation, the platelets are found directly above the clay. By aligning the capillary to a reader chart, the height from the upper clay surface to the upper boundary of the platelets yields the concentration of cells per millilitre.
mixture. 90µL of the suspension was transferred into a small reaction tube while the larger tube was fastened into a smaller table rotator and kept in motion with 12 rpm. The cells on the rotator were later used for the experiments. To avoid aggregation due to stagnation of the solution, they were kept in constant motion.
The small amount of platelet suspension in the smaller tube was then used to determine the concentration of cells. The fluid was filled into a haematocrit capillary using capillary forces (Fig. 4.1). On the capillary, a small red ring was displayed around its circumference to mark the maximum fluid level which not to exceed. The filled capillary was sealed at the lower end using modelling clay, wrapped in a piece of low-lint tissue wipe (KimTech tissue, see Table 4.3) to avoid breakage and inserted into a 15 mL tube. The tube was put into the centrifuge and rotated at 1000g at 21◦ for 10 min. The cells inside the capillary were then pelleted directly above the clay and, using a haematocrit reader chart, the total concentration of cells was determined. Typically, the concentration amounted to 4·109 cells/mL to 6·109 cells/mL. For the experiments, a final concentration of 2·107cells/mL was used.
For the static experiments, the platelets were stained with a membrane dye, CellMask Deep Red dye, to identify the time point of attachment during recording.
When staining the platelets, 0.5µL of dye was mixed in 999.5µL of HT-BSA buffer to a final concentration of 2.5µg/mL until the solution’s colour was uniform. The platelets were then diluted to their final concentration in the dye containing buffer.
Chapter 4 MATERIALS AND METHODS
All following steps had to be made quickly as the platelets tended to take up the dye into the cytoplasma, thus reducing the time in which reliable detection is possible. In particular, between the first staining step and the start of recording, on average, 20 min elapsed. The tube with the stained platelets was incubated for 5 min at 37◦ C. Meanwhile, 10 µL PGE1 was mixed with 90 µL HT-BSA buffer.
25 µL of the diluted PGE1 was pipetted into the cell suspension before the tube was transferred into the centrifuge pre-set to 480g for 5 min. The supernatant was removed and the platelets re-suspended in 1 mL HT-BSA buffer for 8 min to 10 min. Note that the cell pellet was not visible anymore at this stage. The stained platelets were then directly used for recording.
Additionally to the above steps, two further aliquots had to be prepared. One aliquot of some millilitres of HT-BSA buffer solution was prepared for the later washing of the substrates. Furthermore, thrombin solution was prepared with a concentration of 40 u/mL by mixing 25 µL of the thrombin stock as denoted in Table 4.1 with 225µL HT-BSA solution.