Cooperative deformation of mineral and collagen in bone at the nanoscale

Gupta et al. 10.1073/pnas.0604237103.

Supporting Information

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Supporting Text
Supporting Figure 7
Supporting Figure 8
Supporting Figure 9
Supporting Figure 10
Supporting Figure 11
Supporting Figure 12
Supporting Figure 13




Supporting Figure 7

Fig. 7. Schematic of staggered model of load transfer in a two phase anisotropic composite under uniaxial tensile stress.





Supporting Figure 8

Fig. 8. (a) (Upper) Typical 2D SAXS frame of the collagen reflections from the FRELON 2000 CCD detector, with radial sector used for integration shown in dashed lines. (b) (Upper) 2D WAXD intensity maps of the apatite diffraction pattern from the Princeton Instruments detector at an angle [1/2] (2q[0002]). To allow a common visualization of the SAXS and WAXD patterns the WAXD intensity map is projected to the tangential plane of the Ewald sphere at the origin of reciprocal space (1). The remainder of the detector is masked (dark regions on the edges).

  1. Urban V, Panine P, Ponchut C, Boesecke P, Narayanan T (2003) J Appl Crystallogr 36:809-811.




Supporting Figure 9

Fig. 9. (a) Integrated profiles of SAXS intensity showing the first- and third-order peaks. (Inset) The first-order meridional collagen peak used for fitting. (b) Integrated profile of WAXD intensity showing the different lattice spacings. (Inset) The (0002) peak used for fitting.





Supporting Figure 10

Fig. 10. Simple sketch of how data binning into bins of tissue strain eT is carried out. Data points shown are schematic and are not measured data. For clarity only two sample sets are shown (black squares and white circles). Using a fixed bin width, points from the same sample in the same bin are averaged (step b). The points lying in a single tissue strain bin are averaged to give the final binned data (step c). The example shows fibril strain eF, and binning was also done for mineral strain eM, fibril strain eF, mineral strain / tissue strain eM/eT, fibril strain / tissue strain eF/eT and tissue stress s .





Supporting Figure 11

Fig. 11. The sample preparation using UV laser microdissection does not significantly damage the organic or inorganic matrix of bone samples. Raman spectra were taken of a bone sheet before (blue) and after (red) a cantilever shaped region was excised. As seen, no significant change in the spectra is observed, indicating that any changes to the bone matrix are minimal, with little damage due to heating and other factors. Inset shows the Amide I region.





Supporting Figure 12

Fig. 12. (a) Schematic of the fibrolamellar bone packet (0.10 ´ 0.15 ´ 3 mm) glued to plastic grips via a thin layer of cyanoacrylate. (b) Photograph of an actual sample, showing the grip markers at the edges of the image and the sample markers in the middle.





Supporting Figure 13

Fig. 13. Example plots showing the stress-strain curves with strain measured by the markers on the bone (dark green) and on the grips (dark red). (a) An example where the elastic tissue strain is slightly underestimated by the grip markers ((eTg/eTs) = 0.883). (b) An example where the curves coincide in the elastic regime ((eTg/eTs) = 0.983). (c) An example where the sample strain is slightly lower than the grip marker strain ((eTg/eTs) = 1.16). (d) An example where the tissue strain is underestimated by the grip markers ((eTg/eTs) = 0.766). In all cases, when the sample enters the inelastic regime the difference between the two strain measures becomes more pronounced.





Supporting Text

Staggered Model

A staggered model of load transfer at the level of mineralized fibrils has recently been described (1-4), and here we only give a brief overview of the main results. In a system such as a mineralized collagen fibril, containing a ductile (organic) phase and a stiff (inorganic) phase, we are interested in (a) the fraction of total fibril strain eF carried by the mineral particles (eM) (b) the shear strain hC in the collagen phase and (c) the effective elastic modulus of the fibril sF / eF in the elastic regime. Fig. 7 summarizes the model, and the main relations of interest are

In the Eqs. S1-S3 above, EM and EC are the elastic moduli of the mineral and collagen respectively. GC denotes the shear modulus of the collagen, and is approximated by GC »g EC, where g = 0.40. For hydrated bone tissue, collagen stiffness EC » 2 GPa, mineral stiffness EM » 100 GPa, and the mineral particle aspect ratio is taken as r » 0.1.

We note that the model can be applied to the next higher hierarchical level, where an array of (stiff) fibrils is embedded in a (ductile) extrafibrillar matrix. The equations remain the same, but the terms of interest would be fibril strain eF, tissue strain eT, macroscopic stress s T, and shear strain in the extrafibrillar matrix hE. Eq. 1 arises from combining Eq. S2 at the fibril and fibril array level to get

We use the fact that the volume fraction of the fibrils FF @ 1 in simplifying the second and third terms in Eq. S4 above. The first term Eef [equiv] (1-FF)EE is a constant depending on the extrafibrillar matrix. Taking the fibril volume fraction to be FF » 0.99 and EE » 1.0 GPa, Eef = 0.01 GPa. Recent theoretical considerations (4) have shown that hierarchical composites might be even more effective in increasing flaw resistance of materials than a one-level staggered composite (1).

SAXS and WAXD intensity profiles

Fig. 8 shows 2D SAXS and WAXD frames on bone. The collagen fibril SAXS peaks and the mineral apatite reflections are visible. Fig. 9 shows the integrated intensity profiles of the SAXS and WAXD data.

Binning procedure

Fig. 10 shows schematically how the data were binned into bins of the tissue strain eT.

Raman spectroscopy on bone samples

To check whether the UV laser microdissection procedure damages the bone matrix, we used Raman spectroscopy on the bone samples, with a confocal Raman microscope (CRM200, WITec, Ulm, Germany) with a diode pumped green laser excitation (532 nm). Fig. 11 shows Raman spectra on a bone sheet before (blue) and after (red) a cantilever shaped section was cut out using the UV laser microdissector. As can be seen, no significant change in the spectra is observed after laser microdissection.

Machine Compliance Correction

When calculating the tissue strain from two markers placed beyond the point of bonding of the sample to the substrate via a thin cyanoacrylate glue layer as shown in Fig. 12a (5), errors in strain measurement arising due to the glue - sample bonding must be considered. We did this by both experimentally and with a simple calculation.

Experimentally, tensile stretch - to - failure experiments were carried out in the lab under the same conditions as during the in situ synchrotron measurements, but with the addition that strain was measured (a) from markers on the bone itself and (b) from markers on the grips, as used in the synchrotron measurements.

Gentle brushing of ink lines onto the bone tissue with a fine single hair from a paintbrush was used to put optical markers on the single fibrolamellar bone packets, without damaging the sample (Fig. 12b). By tracking the distance between (a) the marker pair on the bone tissue and (b) the marker pair on the grips and plotting both curves on the same graph, we can quantify the difference between the two methods. We denote the strain measured from the markers just beyond the bone ends as "grip strain" eTg and the strain measured by markers on the bone tissue itself as "sample strain" eTs. 4 example data sets are shown in Fig. 13.

We carried out a series of tensile stretch to failure experiments on single fibrolamellar bone packets, at the same velocity (1 mm·s-1) used for the in - situ synchrotron experiments. The bone samples were taken from bovine bone of the same age, sex, bone type (femur), location in the tissue (periosteum) and geometry as used for the synchrotron experiments. In total, n = 38 samples were tested, and for each data set, the stress, grip strain, and sample strain was measured. The effective modulus using the grip strain (ETg) and sample strain (ETs) was calculated by linear regressions in the elastic regime. The average ratio ( eTg / eTs) of the grip strain to the sample strain was calculated in the elastic regime for each sample.

The distribution of (eTg / eTs) values was tested for normality using the Kolmogorov - Smirnov test and passed (P = 0.104; SigmaStat for Windows version 3.11, SyStat Software). The mean value is 0.843 ± 0.031 (error: standard error of mean). The fact that (eTg /eTs) is on average <1 implies that the grip strain is actually a slight underestimate of the sample strain and not an overestimate. The mean value of the ratio of the moduli (ETs / ETg) is the same (0.841 ± 0.030) within statistical error, as expected (the grip moduli are larger since the grip strain values are smaller). Hence, to correct for the tissue strain values measured using the grip markers in the paper, the tissue strain has to be multiplied by a factor of 1/0.843 = 1.19.

The reason for the grip strain values being smaller than the actual sample strain values is clear when we consider how the sample is fixed to the grips (Fig. 12). Two cellophane foils enclose the sample gauge length in a drop of water. At the ends of the sample, it is glued via a thin layer of cyanoacrylate glue to the grip (Fig. 12a). These ends of the tissue are drier than the central gauge length, because they are not in water and are bonded at the surface to the dry glue layer. The effective stiffness of the entire length of the sample is thus slightly elevated relative to the stiffness of the main gauge length. Grip strain values (eTg) give information on the strain in the entire length (including the bonded ends) and hence are smaller than the sample strain (eTs) values. An estimate of the shear deformation of the thin cyanoacrylate glue layer between the sample and the grip shows that it does not deform significantly (< 1 mm) at the loads (< 250 grams) used in our tests.

Hence the grip strain provides a slight underestimate of the tissue strain, and should be multiplied by a factor of 1.19 to correct for the difference.

Compliance correction via aluminium foils: We also made compliance test samples of layered aluminium foils with successively greater number of layers (1, 2, 5, 10, and 20) which we glued in the same way as the bone samples to the plastic grips and stretched to 80% of the maximum rating of the 250 gram load cell (200 grams; 80% to avoid damage to the load cell). The extension was measured again with the black markers placed just beyond the glue bond to the foil samples. The limiting value of the effective sample stiffness was about 2.38 N/m m. Since the maximum force reached in the experiments was typically 0.8 - 1.0 N, this result means less than 1 mm of deformation occurs in the region between the sample bond and grip markers.

1. Gao HJ, Ji BH, Jager IL, Arzt E, Fratzl P (2003) Proc Natl Acad Sci USA 100:5597-5600.

2. Jager I, Fratzl P (2000) Biophys J 79:1737-1746.

3. Gupta HS, Schratter S, Tesch W, Roschger P, Berzlanovich A, Schoeberl T, Klaushofer K, Fratzl P (2005) J Struct Biol 149:138-148.

4. Gao HJ (2006) Int J Fracture 138:101-137.

5. Burgert I, Fruhmann K, Keckes J, Fratzl P, Stanzl-Tschegg SE (2003) Holzforschung 57:661-664.

6. Urban V, Panine P, Ponchut C, Boesecke P, Narayanan T (2003) J Appl Crystallogr 36:809-811.

This Article

  1. Published online before print November 9, 2006, doi: 10.1073/pnas.0604237103 PNAS November 21, 2006 vol. 103 no. 47 17741-17746
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