«Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt ...»
Moreover, it can be shown analytically that the distribution of nearest obstacle distances in a random line networks follows a Rayleigh distribution even when the network is anisotropic (SI).
8: Rayleigh distribution of nearest obstacle distances. (A) The distribution of nearest obstacle distances in simulated line networks is determined for 3 different line densities by computing the Euclidean distance to the nearest solid phase pixel (inset). Regardless of line density, the resulting probability density calculated for 105 test points randomly placed in the liquid phase (dots) follows a Rayleigh distribution (solid lines). (B) Probability densities of nearest obstacle distances calculated for an isotropic line network (black) and two anisotropic networks with cut-off angles of 40° and 55°. In all cases, the probability distribution of the nearest obstacle distance follows a Rayleigh distribution (solid lines).
albeit with a right shifted peak and a broader distribution compared to the isotropic network (Fig. 3.1.8 B), as expected. Therefore, the characteristic mean value rmean of the Rayleigh distribution fitted to the p(rnod) distribution of nearest obstacle distances can be taken as a measure of the average pore size of the network imaged with either CRM or CFM.
This independence of the Rayleigh distribution from fiber anisotropy offers the opportunity to reconstruct the unbiased network pore size from CRM and SHG images and to correct for the blind spot effect. It can be analytically shown that rmean,unbiased of an isotropic network and rmean,biased of the same network but with invisible fibers above a cut-off angle are related by a factor that is simply the square root of the fraction of the visible fibers (SI). For a cut-off angle of 51°, this correction factor is √cos(51°) = 0.793.
The correction factor is constant for different network densities and only depends on the
optical cut-off angle θcut of the imaging system:
(2) √ At first glance, an over-estimation of the pore size by 25% from uncorrected confocal reflection images may not seem dramatic. However, several recent reports have demonstrated that cell migration in a porous environment critically depends on pore size, with a surprisingly sharp cut-off below which cells cannot migrate [23, 24]. A 25% change in pore size around a mean diameter of 2 µm, for instance, was associated with a more than 10-fold change in transmigration efficiency .
Comparison with the covering radius transform
To compare the pore sizes obtained using the nearest obstacle distance method with the pore sizes obtained using the established covering radius transform (CRT) method , the scaling factor between the two measures was numerically determined for simulated isotropic networks (SI). The average pore radius given by the covering radius transform, rCRT, is by a factor of f = 1.82 larger than rmean,unbiased obtained with the nearest obstacle distance method.
32 III RESULTS AND DISCUSSION(3) √ For random isotropic networks, the factor f relating the pore sizes obtained with the two methods is independent of the imaging setup and the cut-off angle.
Application to collagen data To test whether these analytical and numerical results also hold for real biopolymer networks, we applied our method to a set of collagen gels polymerized at different monomer concentrations imaged with the 20x dip in objective (NA=1.0). For each gel, we obtained a CRM and CFM data set and measured the distributions of nearest obstacle distances. For all concentrations imaged, the rnod distributions followed Rayleigh distributions for both CRM and CFM. As expected, the CRM data showed consistently larger pores than the CFM data (Fig. 3.1.9 A-D). With decreasing collagen concentrations, these distributions broadened and the maxima shifted to larger pore sizes. The quality of the Rayleigh fit was very high, with correlation coefficients between r2=0.987 for 2.4 mg/ml collagen gels and r2=0.997 for 0.3 mg/ml collagen gels.
The CRM data of fibrin gels and the SHG data of collagen gels also followed a Rayleigh distribution, thus further confirming that the nearest obstacle distance of random biopolymer networks is Rayleigh distributed, irrespective of imaging mode or protein composition.
We then computed rmean,biased by fitting a Rayleigh distribution (Eq. 1) to the anisotropic CRM data, and predicted the unbiased distribution of nearest obstacle distances for an isotropic network according to Eq. (2). The Rayleigh distribution predicted from the CRM data and the measured distribution from CFM data show excellent agreement for all collagen concentrations (Fig. 3.1.9 A-D), confirming the validity of our method.
We next predicted the covering radius transform pore size, rCRT, for different collagen gels from the fit of the Rayleigh distribution to the confocal reflection data according to Eq. (3). The predicted rCRT was then compared to the directly measured rCRT from the confocal fluorescence images. We found a close agreement between predicted and directly measured pore sizes for the different collagen concentrations (Fig. 3.1.9 E), confirming the validity of Eq. (3) for converting the different pore size measures.
III RESULTS AND DISCUSSION 33Figure 3.1.
9: Blind spot correction. (A) From the binarized reflection data, the nearest obstacle distance is determined (green squares) and fitted with a Rayleigh distribution (grey line). Rescaling this distribution with a correction factor according to Equ.(2) yields a prediction for the distribution of nearest obstacle distance for CFM data, as indicated by the red arrows. The distribution of the nearest obstacle distances calculated from a binarized fluorescent data set obtained for the same collagen sample (blue squares) is correctly predicted by the re-scaled Rayleigh curve (this curve is not a fit to the measurements). (B-D) Repeating this procedure for different collagen concentrations from 0.3 mg/ml to 2.4 mg/ml gave excellent agreement between predicted and measured distributions. (E) Average pore size rmean, biased for different collagen concentrations calculated from reflection data (black line), blind-spot corrected (rmean,unbiased, blue dashed line) with Equ. (2), and converted to the corresponding mean CRT values (rCRT, red dashed line) with Equ.(3). All predicted values are in good agreement with the measured data from fluorescent images (solid lines). The error bars indicate the standard deviation between different fields of view (n=8) of the same samples.
Relationship between pore size and protein concentration
Our data show that collagen gels polymerized at higher monomer concentrations have smaller pore sizes, consistent with previous findings [16, 23] (Fig. 3.1.3 and Tab.3.1.2).
Moreover, by analyzing the relationship between protein concentration and pore size for both, collagen and fibrin, it is possible to address the question whether higher monomer
34 III RESULTS AND DISCUSSIONconcentrations are stoichiometrically incorporated into the fibers, and whether the incorporated monomers contribute to increased fiber length or instead to increased fiber thickness.
We can take advantage of the fact that the scaling factor rmean of the Rayleigh function
If protein monomers are stoichiometrically incorporated into fibers, and if they contribute only to fiber lengthening and not to fiber thickening, we expect that the pore size increases with concentration cm according to rmean ~cm-1/2. The pore sizes determined for collagen polymerized at concentrations between 0.6 and 2.4 mg/ml and for fibrin polymerized at concentrations between 0.125 and 8 mg/ml closely follow this prediction (Fig. 3.1.10). However, for very high fibrin concentrations and accordingly small pore sizes, the relationship deviates from the expected square root dependency (Fig. 3.1.10, grey area), suggesting that pore sizes below 1 µm cannot be reliably resolved.
Collagen polymerization is known to be strongly influenced by the source of the collagen (36) but also by pH and temperature [15, 16, 75, 76], which differ between protocols from different laboratories. Previous studies  have reported a stronger relationships between collagen monomer concentration and pore size, according to rmean ~cm-1. Our gels were polymerized at 37° degrees and with a pH of 10, however, which leads to a faster polymerization  compared to the more widely used pH of 7 or lower [16, 17, 76], and therefore may have contributed to these differences.
Furthermore, it has been suggested that fluorescent labeling of collagen might affect its polymerization behavior . Even though we do not see systematic pore size differences between fluorescently labeled and unlabeled collagen, our CRM-based method avoids this issue altogether.
III RESULTS AND DISCUSSION 35
-0.5 4 Collagen Pore size [µm] Fibrin
-0.5 0.5 1 2 4 8 Concentration [mg/ml] Figure 3.1.
10: Concentration dependence of the average pore size of fibrin (red) and collagen (black) gels. Average pore size rmean,unbiased reconstructed from 3D CRM data vs.
protein concentration. The error bars indicate the standard deviation of the mean from different collagen gels (n≥3) and different fields of view (n≥5 for each gel). Both data sets are in good agreement with a power law with an exponent of -0.5 (gey dashed line), as expected for stoichiometrically polymerizing networks where monomer addition contributes to fiber lengthening but not to fiber thickening. The grey area illustrates the range (r 1 µm) where the pore sizes cannot be reliably measured.
Collagen Pore size [µm], control Pore size [µm], glutaraldehyde concentration
2.4 mg/ml Table 3.1.2: Average pore size dependence on collagen concentration and glutaraldehyde treatment. Average pore size rmean,unbiased reconstructed from 3D CRM data. With increasing concentration, the pore size of the network decreases. Glutaraldehyde treatment does not change the average pore size. The error bars indicate the standard deviation of the mean from different collagen gels (n≥3) and different fields of view (n≥5 for each gel).
36 III RESULTS AND DISCUSSION
Relationship between pore size and polymerization temperature
As mentioned before the pore size of a self-assembling biopolymer network is crucially dependent on the surrounding conditions during the polymerization [16, 41, 79]. As such, the pH of the collagen solution and the polymerization temperature play a significant role. To characterize the influence of the polymerization temperature on our collagen gels, we varied the polymerization temperature in the incubator. All other conditions (pH of the solution, CO2 level and humidity) remained constant. However, due to the properties of our incubators, we could not set the temperature below 22°C.
The results showed that the average pore size was decreasing with an increase in temperature, which could be explained by a faster polymerization. The effect was independent of the collagen monomer concentration used in the experiments (Fig 3.1.11). For 1.2 mg/ml collagen and 2.4 mg/ml collagen gels, the decrease showed even a similar trend. However, there is no stoichiometric theory how exactly the polymerization temperature influences the network assembling. Therefore we desisted from fitting any curve to the measured data. Moreover, due to the fact that we could not pre-estimate the pore size when the polymerization temperature was changed we did not pursue an experiment where also cells would be seeded in or on those collagen gels.
1.2 mg/ml 2.4 mg/ml Pore Size [µm]
Temperature [°C] Figure 3.1.
11: Polymerization temperature dependence of the average pore size of collagen gels. Average pore size rmean,unbiased of a 1.2 mg/ml gel (blue stars) and a 2.4 mg/ml gel (red circles); reconstructed from 3D CRM data vs. the temperature during polymerization. The error bars indicate the standard deviation of the mean from different collagen gels (n≥4) and different fields of view (n≥5 for each gel). Both data sets showed a decrease in pore size with increasing polymerization temperature. However, any mathematical function could be fitted to the curves
III RESULTS AND DISCUSSION 37and because no theory of dependence known so far, we resigned on fitting a function to the curves.
Relationship between pore size and a chemical cross-linker We have shown that the pore size of reconstituted collagen networks depend on the collagen monomer concentration and polymerization conditions including temperature.
Others have shown, that the pore size also depends on the pH [15, 16, 19, 51, 78].
However, the pore size of a reconstituted collagen gel should not change if the gel is treated with a cross-linker. To test this hypothesis, we polymerized eight dishes of 1.2 mg/ml collagen gels and treated four of them with the cross-linker glutaraldehyde. We used a 0.2% solution of glutaraldehyde in PBS for the cross-linking procedure, which already saturates the free binding sites on the collagen fibrils [80, 81]. After the treatment we obtained the pore size of all four of these gels. As expected we did not see a difference between the pore sizes of the treated and the untreated gels. We repeated the experiment with 3 more sample sets on different days. For all gels, regardless on which day they were produced, we did see no difference in the resulting pore size (Fig.
3.1.12, A). Repeating this experiment on different samples prepared on different days showed highly reproducible results, with a variation in pore sizes of ± 0.15 µm (Fig.
3.1.12, A, black line). Note that if a new batch of collagen solution was used the resulting pore size can vary up to ± 2 µm 1. Therefore, if a series of experiments is planned with collagen gels, enough collagen solution from the same batch should be stored.