Force Dipole Distribution

For all traction force data sets, the force dipole was determined. One example of the force dipole can be found in Fig. 7.6, another in Fig. 6.6. The dipole was calculated in orientation and magnitude for each given time point apart from those images where no traction force was detected. Hence, a distinguishable dipole magnitude was only observed if an actual force existed (comp. Fig. 6.6 Bto the corresponding force curve in Fig. 7.1C).

Taking a closer look at panel B in Fig. 6.6, a very synchronised behaviour be-tween the major and minor dipole moments was observed. This behaviour was seen in most cells with the exception of highly polarised cells. Here, the minor dipole moment often remained close to 0 J or developed partially independently from the major dipole moment as seen in Fig. 7.6 B at about 500 s. Note that always the (absolute) magnitudes in dipole are shown. As the considered traction forces were contractile, the actual eigenvalues were negative.

To determine the degree of isotropy, the dipole quotient qwas calculated as de-scribed previously in Sections 5.5 and 6.2.3. An example is depicted in Fig. 6.6 C for a synchronised behaviour and in Fig. 7.6C for an unsynchronised behaviour.

Clearly, during the time interval before initial contraction at which the dipole mag-nitudes were dominated by noise in the images (synchronised case, approximately the first 400 s; unsynchronised case, about 100 s), the qchanged quickly between neighbouring time points. During the interval of distinct dipole moments, this variation decreased, leading to a more stable quotient. Only this time interval was considered during averaging. In the synchronised example, this included all data after initial contraction as indicated in panels Eand F. Contrary, the dipole quotient in Fig. 7.6 C, only showed a smaller time interval of little variance be-tween about 200 s and 700 s, after which the noise increased due to the relaxing and slightly asynchrony in behaviour of the platelet. Here, accordingly, a shorter interval for averaging was used. Interestingly, the synchronised behaviour of the majority of platelets directly indicates that the force pattern and thus the force transmitting network is established during the initial stages of spreading and does not change upon contraction within the first half an hour.

As we were first interested in whether the dipole ratio varied with the behaviour, we first divided all platelets into two groups, oscillatingvsnon-oscillating. Subse-quently, the dipole quotient qwas calculated for each platelet and assigned to its corresponding group. The accumulated averaged quotientsq are shown in panel

Contractile Behaviour on Substrata of Dierent Stiness 7.2 D. We directly saw that, on average, platelets in both groups deviated slightly from

isotropy. The median of both groups was 2.2 and 2.3 for the oscillating and non-oscillating platelets, respectively. Due to the non-normal distribution inqin each group, a Wilcoxon rank sum test was applied to test for significant differences between them. No such difference was detected, suggesting that the oscillating behaviour did not coincide with a different cytoskeletal structure.

A 16 B

Figure 7.6.: Force dipole of a contracting platelet. A Example of major (red) and minor (yellow) dipole axis on a magnitude traction force map. The length of each axis is scaled to the actual norm of the corresponding vectors. B The temporal evolution of the dipole moments in magnitude. C The dipole quotientq calculated from the dipoles shown in panel B. D Box plot for all average dipole quotientsq partitioned into two groups of platelets, oscillating and non-oscillating. The median of the groups are 2.2 and 2.3, respectively.

Additionally to studying the dipole quotient in the context of the temporal con-tractile behaviour, we also investigated the dependency ofqon the substrate stiff-ness. We previously demonstrated that the total force did not depend on the stiffness due to the small size of the platelet. If the ratio was to change with re-spect to the substrate stiffness, this would indicate that although the force was not adapted, the force transmitting system was. Calculating the dipole quotient as

be-Chapter 7 RESULTS

fore, the accumulated data is found in Fig. 7.7C. At a first glance, the differences were not large between the single stiffnesses, however, a significant difference be-tween the distributions of 19 kPa and 83 kPa as well as 29 kPa and 83 kPa was observed (p < 0.1). Thus, an increasing trend in the anisotropy was established, indicating a higher degree of polarisation on stiffer substrates than on softer.

Lastly, let us take a look at the actual values of q that we derived. Indepen-dent of the partition of the data we investigated, the median of the distributions was always found between 2 and 3. Pooling all data together, irrespective of the platelet’s temporal contractile behaviour or the corresponding substrate stiffness, the median was calculated to be 2.2 and the mean 2.5. We simulated 500 different point force patterns to gain an insight into what dipole ratio distributions we were to expect of a random distribution. Here, a total of 8 point forces were set around the periphery of a circle, in agreement to the traction force patterns described above. The first seven forces were all of equal magnitude, pointing towards the centre of the circle and randomly allotted. The eighth force was situated in such a way that the vectorial sum of all traction forces was 0 within the simulation. Three examples of such distributions are found in Fig. 7.7Aⁱⁱⁱ. For each of the 500 force patterns, the dipole quotient was calculated. The entire distribution in terms of q is found in Fig. 7.7 B, along with the distribution calculated from the measured data. The distributions are in good agreement with each other. The same result is seen in the box plot in Fig. 7.7C, where the simulated ratios exhibited an equiv-alent median to the distributions on the different substrate stiffnesses. Note that we only included 30 randomly sampled qvalues, in accordance to the number of platelets included in the other groups depicted here. Hence, we conclude that the slight force anisotropy seen in the platelets is equivalent to a random distribution in force along a circular circumference.

To briefly summarise, for the static experiments, we first calibrated both al-gorithms against each other, demonstrating that both approaches led to the same results in terms of total force. In the following analysis, we always used the PIV al-gorithm as it was specifically tailored to fit our problem. We showed that platelets contract along their periphery at a distinct number of spots. Platelets could be subdivided into three different categories concerning their temporal contractile behaviour; they either reached a force plateau, relaxed after initial contraction or started to oscillate after contraction with an average frequency of 13.5 mHz.

Combinations between these behaviours were observed. The different

contrac-iiiThe simulation was performed by D. Probst, University of Heidelberg.

Contractile Behaviour on Substrata of Dierent Stiness 7.3

Figure 7.7.: A Three examples of randomised point force patterns containing 8 point forces. All forces are set on the circumference of a circle and their net vector force is 0. B The dipole ratio distribution of the simulated ratios (orange) t well with the determined distributions from the experiments (blue). C The agreement between the simulation and experiments is also seen when separating the dipole quotients according to the substrate stiness. For the simulation, out of 500 dierent force patterns, 30 were randomly sampled for this gure. A slight increasing trend is observed with increasing elasticity of the substrate, where the distributions for 19 kPa and 29 kPa are signicantly dierent from that of 83 kPa (p<0.1). The gure is taken from [37].

tile developments were not reflected in the spread area evolution. The cells spread rapidly towards a final area, stabilising faster than their corresponding total forces.

Additionally, we experimentally demonstrated that the total force of the platelets is independent within the studied stiffness range. This was due to the small size of the platelets as we derived theoretically by modelling the cells as elastic, cir-cular discs coupled to the substrates. Here, we predicted that platelets were only mechano-sensitive in a stiffness range well below 10 kPa. An adaptation in the to-tal force was observed in correspondence with the spread area, where larger cells exerted a higher force than their smaller equivalents. Lastly, we showed that the contraction of platelets is slightly anisotropic. The degree of anisotropy did not change with the temporal contractile development but possessed an increasing trend with increasing substrate stiffness, suggesting a higher polarisation within the force transmitting network on stiffer substrates. The observed anisotropy, on average, was reproducible by simulating 8 random point forces along a circle’s

Chapter 7 RESULTS

Table 7.2.: The mean velocity and corresponding Reynolds number found within the measuring chamber for the dierent tested ow rates.

Flow rate [µL/h] Mean velocity [µm/s] Reynolds number

300 236 0.04

500 393 0.07

700 550 0.09

1000 786 0.13

circumference, indicating a generally random force spot distribution.