Dependency of the Maximum Force on the Platelet Size

To answer the above question, the final spread area of each platelet determined as described in Section 6.2.1 was studied. As the platelets are produced by a budding-off process from the megakaryocytes, we expected a variation in size to be seen. Indeed, taking a look at the distribution of the final spread area of each cell as seen in Fig. 7.4A, the size varied between 17 µm² to nearly 100 µm². On average, the analysed platelets exhibited a size of about 40µm².

Next, the maximal force each platelet exerted on the substrate within the 30 min of recording time was determined. This was generally not the last time point of recording but was often reached beforehand, especially for relaxing cells. Plotting the maximal exerted total force of each plateletvs. the final spread area resulted in the scatter plot shown in Fig. 7.4A. Again, as previously noted concerning the total force development with regard to the gel elasticity, no difference was seen between the different substrate stiffnesses. At the same time, a clear increase in maximum force was seen with growing area. However, the growing variance in force with larger area, made a linear fit as done in Refs. [12, 35] infeasible.

Instead of a linear fit, the model introduced in Section 3.5 (compare also [37]) was used. Here, we modelled the platelet as a thin, circular disc coupled to an elastic substrate by a set of springs (compare Fig. 7.4B). Briefly, the platelet was characterised by its elastic modulus Ec, its Poisson’s ratio νc, its heighthc and ra-diusr_c. We further defined the substrate stiffness as E_s, its Poisson ratio ν_s and its heighth_s. The adhesion layer between both elastic layers was characterised by the adhesion layer stiffness density ^Na_L^ka, where Na denotes the number of anchor proteins,k_a their stiffness and Lthe platelet size. The combined stiffness density of the platelet and the substrate is given by Y. The contraction was lastly char-acterised by two quantities, the localisation length lL for the traction force decay in space and σ₀ as the active contractile stress of the cell. Given the previously described model using the here defined physical quantities, an expression for the

Chapter 7 RESULTS

Figure 7.4.: A The maximal reached force within 30 min of recording vs. the nal spread area.

A general increase in force is seen with increasing area of the spread platelet. B A sketch of the two-dimensional model used to t the data shown in A. In panels C and D, the graph depicted in panel A is separated into oscillating and non-oscillating, respectively. As can be seen, no dierence between the groups could be observed. The gure is partially taken from [37].

theoretical forces was derived, compare Eq. (3.41). We fitted the latter equation to the maximal forcevsarea data given the estimates for the different parameters found in Table 7.1. To find the best fit, the sum of residuals was reduced by finding the best combination of the free fitting parametersσ0 andlL. The best estimate is included in Fig. 7.4A.ⁱ For the fitting parameters, the best estimates were found to beσ₀ =162 kPa andl_L=1.8 µm. We included all data here, not separating be-tween oscillating and non-oscillating platelets as both groups did not differ from each other to warrant such a separation, compare Fig. 7.4CandD.

iThe model fitting was done by D. Probst, University of Heidelberg.

Contractile Behaviour on Substrata of Dierent Stiness 7.2 Table 7.1.: Values used for the model tting described in Section 3.5 and depicted in Fig. 7.4

A.

Physical Quantity Parameter Value

Platelet Young’s modulus [58] Ec 5 kPa Platelet Poisson ratio [121] ν_c 0.3

Platelet height [2] h_c 100 nm

Platelet size L 10 µm

Adhesion layer stiffness density ^N_L^ak2^a 0.3 nN/µm³ Substrate Young’s modulus E_s 19 - 83 kPa

Substrate height h_s 50 µm

Substrate Poisson’s ratio [64] ν_s 0.3

During the model fitting, we defined two free parameters, namely the locali-sation length l_L and the active contractile stress σ₀. The former of these can also directly be estimated from the traction force maps for each platelet. As mentioned previously, the localisation length denotes a measure on how fast an applied force on the substrate decays in space. Hence, we studied the traction force profiles in space to extract an approximation of the physical value for l_L. We already touched on the fact that a large variation between platelets was observed in terms of force and area. To compare the physical l_L to the estimated value, we thus chose platelets close to the fitted graph. An example of such cell is depicted in Fig. 7.1 with a final spread area of 21 µm² and a maximal force of 103 nN. The value for l_L was determined by first defining the start point of the traction force profiles in the centre of the cell and secondly, automatically finding the force hot spots within the traction force map. For this particular platelet, the lines at which the traction force profiles were considered are shown in the inset of Fig. 7.5A. The traction force along each line was calculated and normalised by its maximal force.

All normalised traction forces were then averaged and fitted to the function T(r)

For the platelet depicted in Fig. 7.5A, a value of 1.3µm was found, in good agree-ment with the value estimated from the fit to all data. We thus concluded that the estimates from the theoretical model very well describe the actual measurements.

In the following, we used the value for l_L as derived from the model. The rea-son for this choice was the variation found for thel_Lvalues between the platelets

Chapter 7 RESULTS

close and farther away the fitted graph. As we knew that the graph was a good representation of all data, we continued with this value.

Figure 7.5.: A The localisation length can be determined from the traction force maps as shown here. For the traction force map depicted in the inset, the connection lines between the cell centre and the force maxima are determined (red lines). Along these lines, the relative traction force is calculated (dashed lines) and from the averaged curve (bold), lLis determined via tting (black) according to Eq. 7.1. For this particular cell, the total force was 103 nN, the area 21µm and the localisation length 1.3µm. B From the model and the estimation of the localisation length, the dependency of the total force on the substrate stiness can be estimated. This is depicted here for dierent cell heights and adhesion layer stiness density. The bold lines denote a stiness density of 0.3 nN/µm³ and the dashed lines 1 nN/µm³. The images are taken from Hanke et al. [37].

Let us briefly go back to the model fit in Fig.7.4 A. Here, we observe that no point of saturation at larger areas was reached. From Eq. (3.42), assuming that r_c l_L, we derived that a saturation in force was reached such that F_theo = 2πhcrcσ0 described its asymptotic maximum value. Given the set values of the cell and substrate characteristics as well as the estimates for the free fitting param-eters, a saturation was calculated to be found at a platelet radius of approximately 30 µm, corresponding to an area of 2800 µm², a size two orders of magnitude larger than the real platelet size. Continuing along this line, we derived in Sec-tion 3.5 that the total force depends on cell properties E_ch_c and on the adhesion layer stiffness density ^N_L^a2^ka. We previously observed that no influence of the sub-strate stiffness on the total force was measurable. This was further explored by considering the definition of l_Las given in Eq. (3.36) and the relation between the force and the localisation length. For different heights of the cell itself as well as different adhesion layer stiffness densities, the approximations shown in Fig. 7.5 B. were derived. In our specific case of platelets, we know from literature that the

Contractile Behaviour on Substrata of Dierent Stiness 7.2 actual height was 100 nm or below. Within this regime, the stiffness density did

not influence the result significantly. Most interestingly, we noted that a saturation in the force was detected below a substrate stiffness of 10 kPa, suggesting that all stiffnesses higher than 10 kPa result in a maximal force response in the platelet.

In other words, given the physical properties of the platelets, especially their size, and our data, we were able to show that platelets are unable to sense a difference in the stiffness range studied here in terms of total force.ⁱⁱ

Until now, we saw that platelets contracted near their periphery, exhibited three distinct behaviours or a combination of these and did not adapt to the stiffness of their surrounding in the studied range of elasticity. The exerted maximum force, however, was dependent on the size of the cells. Using the model of a contractile circular disc with an elastic coupling to the substrate, it became clear that the size of the platelets played an important role in the elasticity insensitivity. In particular, from our data, we predicted that due to their small height, a saturation in total force would not be detected for reasonable platelet sizes and an insensitivity exists to substrates of stiffnesses over 10 kPa. However, that still does not explain the different behaviours observed. Hence, another physical measure was explored as to explain the different behaviours.

As we could already see in Fig. 7.4, panel C and D, neither the final spread area nor the maximal total force was correlated with the individual contractile behaviour of each platelet. Could the behaviour then correspond to an internal structure? As was shown previously in Ref. [25, 48, 90], the actin cytoskeleton may adapt several different shapes such as a pointy ellipse with dense end points or a triangular shape. Let us assume that the major transfer of force from the cytoskeleton to the substrate is done at the pointed ends of the inner actin struc-ture. Then a triangular actin network generally would yield to a more isotropic contraction pattern compared to an elliptical structure. In particular, a triangular actin network does then correlated to,e.g., three distinct hot spots in the traction force distribution as seen above. Can we explain the observed oscillations by a different inner actin architecture? To study this possible aspect, the force dipole was calculated as described in Sections 5.5.1 and 6.2.3. To determine the deviation from the isotropic contraction, the dipole quotientq was determined as given in Section 5.5.2 using the error estimation via variance calculation approach.

iiThe calculations depicted in Fig. 7.5 B were conducted by A. Zemel, Hebrew University of Jerusalem.

Chapter 7 RESULTS