Time scale of glycation in collagen of bovine pericardium-derived bio-tissues

Many pathological side-effects of diabetes and aging are directly linked to non-enzymatic glycation of collagen as an abundant, multi-purpose tissue in human beings. We describe a model that quantifies glycation and its structural effects as a function of sugar level (glucose and ribose) and time, based on SAXS and WAXS data.

1 These authors contributed equally to this work.

Number of sugar molecules involved in the glycation
To calculate more quantitatively the number of sugar molecules involved in the glycation, per staggered repetition unit, we computed the relative electron density values by means of the following relation (1): Here, the absolute electron density values of the collagen ρ c and the solution ρ s are expressed in terms of # el./ Å 3 . The factor 5 is because the gap region, (1-σ)×d M , contains one gap and four' overlapping sections; this region is alternated to the denser zones, σ×d M , made of five overlapping sections, without any gap, as schematically shown in Figure 1.
To determine the absolute electron density values for the case under study, we have reported all the relevant physical properties, both for collagen and for the ribose and glucose in Table S1. Using theoretical interpolation formulae, the volume of the solution of 40 g of sugar and 1 L of water could be approximated with 1.014±0.001 L for all sugars used here (2). Measures of volume's variations of 1 mL of water, with added 39.9±0.1 mg of ribose and 40.4±0.1 mg of glucose have IUCrJ (2022). 9, https://doi.org/10.1107/S2052252521010344 Supporting information, sup-2 given, respectively, 1018±1 µL and 1010±1 µL. Thus, the volumes of the solutions of 1 L of water and 40 g either of ribose or glucose, are 1.018±0.001 L, 1.010±0.001 L, respectively. Table S1. Physical properties of ribose and collagen needed to calculate the electron density seen by X-rays. N A is the Avogadro Number. In particular, the absolute electron density values of the last column are obtained from those reported in the second-last, multiplying them by the Avogadro Number N A and converting the volume in Å 3 . In turn, the second-last column's values are determined by the number of moles/volume (sixth column) times Ne (the number of electrons/molecules in the fourth column). For ribose, we obtain: To estimate how many sugar molecules are involved in the glycation process, we need to evaluate the variation of the volumes V and V of the gap and overlap zones in the collagen structure, respectively. This variation can be estimated by approximating this volume with a cylinder of height 5×d M and diameter d E , as schematically shown in Figure 1, and by imposing that the relative electron density's changes must be equal to the values previously determined:

Mass
Here the subscripts "i" and "f" denote initial (day=0) and final (days=90) values. ∆ is the number of electrons needed to explain the changes in the relative electron density due to sugar molecules involved in the glycation. Eq. (S4) can be readily derived by Eq. (S2) and it is, in fact, a generalization for ∆ρ f ., obtained after the incubation of collagen in the sugar solution.
We have determined the difference of the electron density in the overlapping region, averaged on the five nanofibrils, with respect to the average value in the gap region, averaged on four nanofibrils only. Therefore, denoted with N the average number of sugar molecules involved in the glycation process in the whole collagen staggered repetition unit, we can put in Eq. (S5) ∆e≅N × N e , obtaining: we finally have For σ = σ = σ, Eq. (S8) becomes: plotted in Figure 8.

In-plane packing areal density of glycated collagen molecules
Given the sugar concentration m/V, we want to evaluate how many sugar molecules are contained in a cylindrical volume of solution comparable to the volume of a collagen staggered repetition unit, i.e., of diameter d E =1.5 nm and length d M =5×65.5 nm: Here  Table 1), to which corresponds, based on Eq. (S10), To quantify this aspect, Figure S1 shows a schematic drawing of a standard in-plane packing, with the collagen molecules aligned along the three principal planes of molecular packing (5) but allowing enough space to accommodate ribose molecules (green circles). The circles have been drawn with a size proportional to the actual ratio of molecules. In other words, the ratio of the red and green circles is equal to 1.51/0.54=0.28. When we put in contact these circles, we impose a planar geometrical constraint. If we put in contact red circles (collagen molecules) and green circles (ribose molecules) we found that the rectangular blue box of Figure 9, which contains five aligned IUCrJ (2022). 9, https://doi.org/10.1107/S2052252521010344 Supporting information, sup-5 collagen molecules, has a size which is 5.5 that of the red circle, 10% larger than the size of 5 collagen in-contact molecules. This simple geometrical model, of circles in contact, leads to a 10% larger space between collagen molecules (red circles), to allow the presence of green circles (ribose molecules). This lateral spacing variation is comparable to the increase in the measured lateral distance between nanofibrils, experimentally found from WAXS (Table 1)  To be more quantitative, for an arbitrary lateral packing of cylinders, we will denote with w the fraction of space between cylinders, complementary to the in-plane packing areal density. After 90 =0.32±0.05 (S11) in agreement with the previous estimate, based on the geometrical model sketched in Figure 9, since 1-0.68=0.32.
The maximum (2D hexagonal lattice) density packing of cylinders, in contact with each other, would fill about 91% of the available space. It is known that collagen molecules in plane are not characterized by an optimal spacing in a quasi-hexagonal array (5), thus showing smaller packing areal density than the maximum possible value. We found 68% after ribose-glycation. However, the squared relative variation of d E equatorial spacing varies linearly as a function of the incubation time (see Figure 9). Given the linear relation deduced by Figure 9, the fraction of space between collagen molecules after 90 days of incubation in a ribose solution of 40 mg/mL, is about 26% larger than the initial value at t=0 days which, consequently, is = (0)=0.255±0.04, leading to a native value of the lateral areal packing of collagen molecules -without glycation -of about 1-=0.745±0.04. This value implies a 2D native density packing of about 75%, confirming the not-optimal in-plane spacing of collagen molecules (5).
The fraction of space between cylinders after 90 days, in presence of glucose, for the geometrical model shown in Figure S1, should be only 4% larger than the initial value, at 0 days, giving Therefore, from the linear fit of Figure 9, we can deduce: Another fraction of the cylindrical volume, that can be associated to a collagen staggered repetition unit, available for sugar molecules, is the gap region. It is equal to about 1/10≅(1-σ)/5. Thus, the total effective volume available for sugar molecules will be: (S15) IUCrJ (2022). 9, https://doi.org/10.1107/S2052252521010344 Supporting information, sup-7

Density variations along the axis of collagen molecules
Also the squared relative variation of d M meridional period varies linearly as a function of the incubation time. The fitting line of the ribose data, shown in Figure 10, has the following slope: ∆d M =−3.1×10 -6 ±0.15×10 -6 days -1 . Thus ∆d M is two order of magnitude smaller than ∆d E .
Consequently, even at a concentration of 40 mg/mL, any variation on the glucose data is not after 90 days of incubation, from Eq. (S17) it follows that we have a variation of the , ( ) period of only 0.3%, i.e., of about 0.18 nm, too close to the experimental error of ±0.1 nm to be well evidenced by experiments, as shown in Figure 10.
Also the variation of the meridional period contributes to the total variation of the effective fraction of volume available for sugar molecules, between collagen molecules. Thus, taking into account also the density variations along the collagen molecule axis, Eqs. (S14) and (S15) can be generalized as follows: denoted in the main text as ( ), meaning Eq. (S18) for ribose and (S19) for glucose.