J. theor. Biol, (1975) 51, 51-58
Do Mineral Crystals StifFen Bone by Straitjacketing its Collagen? C. W. MCCUTCHEN Laboratory of Experimental Pathology, National Institute of Arthritis, Metabolism and Digestive Diseases, National Institute of Health, Bethesda, Maryland 20014, U.S.A. (Received 8 February
1974, and in revisedform
21 June 1974)
Theories that assume that the collagen and the mineral of bone have the same properties in bone as they do when they are by themselves have difficulty in accounting for bone’s tensile stiffness. The collagen fibers are long but stretchy: the apatite crystals are stiff enough, but too short to be effective tension carriers. In bone the mineral is so finely divided that each collagen molecule may have a mineral crystal next to it. It is proposed that collagen is the prime tension carrier in bone, and that via short molecular struts the mineral crystals prevent the naturally kinky collagen molecules from straightening under tensile stress, which greatly increases the collagen’s tensile stiffness.
1. The Paradox The major constituents of bone, present in about equal proportions by volume, are impure apatite, a stiff, brittle mineral, and collagen, a strong, tough, fibrous natural polymer (Currey, 1964). The tensile stiffness (Young’s) modulus of bone apatite has not been measured. That of synthetic hydroxyapatite is 1-06x lo6 kg/cm’, of human bone 1.82 x 10’ kg/cm’ and of collagen 1.22 x IO4 kg/cm’ (Welch, 1970). The quantity referred to is stiffness, the force per unit area divided by the fractional change in length caused by the force. This is distinguished from tensile strength, the force per unit area required to break the material. The collagen is in the form of fibrils of indefinite length, made by joining end-to-end three-stranded molecules that are 3000 A long, and 15 A in diameter (Ramachandran & Kartha, 1955; Yannas, 1972). The apatite crystals, 400 A long, more or less, lie within the collagen fibrils and are aligned parallel with them (Welch, 1970; Currey, 1969). Their other dimensions are perhaps 45 by 45 A, but the precise shape of their cross section is unknown (Currey, 1969; Eanes, 1973). 51
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Several models have been proposed to explain how the collagen and apatite co-operate to give bone its observed tensile stiffness. Because the collagen stretches unbroken for long distances, it could carry tensile loads with no help from the mineral. But the tensile stiffness modulus would be that of collagen times the fraction of the bone’s volume that the collagen occupies, or about one thirtieth the stiffness of bone. Under tension the apatite crystals would separate from each other as shown in Fig. l(a) and not affect the stiffness modulus. The apatite crystals are several times stiffer than bone, and might account for its compressive stiffness, because endwise compression could push the crystals against each other end-to-end as in Fig. l(b). Were the bone prestressed, as Knese (1958) suggests, its collagen in tension and its mineral in compression, the apatite crystals would be held clamped together as shown in Fig. l(c), even when the bone was in tension. The compressive stiffness of the columns of mineral could thus also account for bone’s stiffness in tension. But prestressing uses up so much of the ultimate strength of the apatite that bone would be weaker in compression than it is (Currey, 1964). The apatite crystals could carry tension if they were attached to each other end-to-end. There might be non-mineralized bridges between the ends of adjacent crystals, but the joint between bridge and crystal should be as strong as the weaker of the two, perhaps not easily achieved between materials with different structures. Though this arrangement would explain bone’s behavior up to its elastic limit at 0.76 % elongation, it does not explain why bone can be stretched inelastically by a further 3 % before it breaks (Burstein, Reilly & Frankel, 1973). In the absence of end-to-end bridges neighboring columns would have to be arranaged so the joints between the crystals were not opposite each other. The tension could then get around the joints by zig-zagging back and forth from column to column, passing from one to the other as shear stress in the intervening layer of collagen as proposed by Currey (1969) [see Fig. l(d)]. One cannot prove that the collagen molecules have not the needed shear strength parallel to their long axes for them to serve as glue. But their structure seems wrong. The shear stress can be analyzed into a pure tension and a pure compression at right angles to each other, and both at 45” to the faces of the crystals. To transmit the stresses the strength members of the collagen should run diagonally across the gap from the lateral face of one crystal to that of another. Instead they run parallel to the crystal faces. Furthermore, if this is how stress is carried why were the mineral crystals made so short? Longer crystals would impose less shear stress on the collagen-glue. Why not make them thicker as well? This would increase the strength of bone by reducing the fraction of its volume that is devoted to the glue layers.
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FIG. 1. (a) If the collagen carried tension with no help from the apatite crystals, bone would be far stretchier than it is, because collagen has a low stiffness modulus. Pulling endwise on the bone would stretch the collagen and cause the apatite crystals to separate from each other. (b) In compression the apatite crystals can carry stress and provide high stiffness, because the joints between crystals are no bar to compressive stress. (c) If the collagen were pretensioned so as to hold the apatite crystals clamped together tensile loads would be carried by the collagen, yet most of the stiffness would be provided by the apatite. But prestressing leaves only a fraction of the ultimate compressive strength of the apatite available for load carrying. (d) If most of the tensile load is to be carried by the mineral crystals then either they must be very strongly attached end-to-end, or the joints in adjacent columns of crystals must be staggered as shown here, and the collagen used as glue to take the stress zig-zagging from column to column so it does not have to cross the joints. The scale of the drawing is given by the apatite crystals, which are 4.00 A long.
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The above theories treat bone as analogous to steel-reinforced concrete. If we know its structure we can predict the behavior of reinforced concrete from the properties that each of its constituents shows when it is by itself. Attempts to predict this way the properties of bone all seem to have one flaw or another. Alexander (1968) advanced a different idea, here quoted in toto. “Filled rubbers perhaps offer a better analogy to bone, though they are neither as strong nor as stiff. They are mixtures of rubber with fine particles of another material which is usually carbon black. Carbon black is fine soot. Its particles are roughly spherical and in some of the grades used for reinforcing rubber are about the size of the crystals in bone. Rubber filled with carbon black is used for making tyres. “The rubber seems to become very dimly attached to the carbon black, and the attachments prevent the molecules from straightening as much as they otherwise would, when the rubber is stretched. This is particularly so since the carbon particles tend to join together in short chains (Holliday, 1966). Hence, Young’s modulus is higher than for pure rubber. The tensile strength is also higher. Rubber containing 50% carbon black can be as much as 16 times as strong as pure rubber (Bueche, 1958).” The presence of carbon black modifies the properties of the rubber, and the composite is stiffer and stronger than it would be if the rubber were not modified. In what follows, the analogous effect of the apatite crystals on the collagen of bone appears as a natural consequence of the mutual arrangement of the collagen fibers and the apatite crystals, and so far as theory can ever verify theory, seems to justify Alexander’s suggestion. 2. The Straitjacket
Theory
For one constituent of&bone to modify the properties of the other the two must be closely intermingled, and they are. The apatite crystals are so small that, if their cross section is square and they are oriented parallel to each other and spaced the same distance apart, the distance between their faces is only 18.6 A. This is space enough for one, or at most two layers of collagen molecules. If each collagen molecule is in contact with one or maybe two apatite crystals we may expect its properties to be much modified by the association. The mineral crystals, on the other hand, are many lattice spaces thick, and so should behave like bulk material. I suggest that collagen is the prime tension carrier in bone, and that bone’s stiffness modulus is much greater than that of isolated collagen because intimate association with the mineral raises the collagen’s tensile stiffness. The extensibility of tendon collagen must result from straightening of kinks in its three backbones rather than
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stretching of the backbones’ bonds. The high Young’s moduli of most crystalline materials shows that bonds are very stiff in pure extension. Not all kinks need be flexible, but some of them must be so to account for collagen’s extensibility. The flexibility can result from either bending or rotation of bonds. Yannas (1972) says that on average there is a hinge every 9 A along the collagen backbone. Suppose that at every point where a collagen backbone has a flexible kink it is connected to the surface of a mineral crystal by bonds or by struts formed of side chains. There is a side chain for every 2446 A of axial distance along the minor helix (Ramachandran & Kartha, 1955), so even ignoring hydrogen bonding there are enough potential struts to prevent bending at the hinges. The bonds and struts would hold the molecule in the kinky shape it has when unstressed. Bending or rotation of the backbone bonds would be prevented. Stretching the molecule would lengthen the bonds of its backbones, and need far greater force than stretching an unbraced molecule whose backbones were free to straighten to the full extent permitted by their one per 86 A twist around each other. At most two braces in different directions at each bend are needed to prevent the backbones from straightening under tension. The braces can all be perpendicular to the long axes of the molecules and crystals, but it is not essential that they be so. Figure 2 shows how a single backbone could be braced to one or two apatite crystals. Bracing three backbones need not require three times the
FIG. 2. Bracing the flexible kinks in the collagen molecules to the stiff apatite crystals would prevent the molecules from straightening under load, and greatly increase their tensile stiffness. Only one of the three backbones of each molecule is shown, and the drawing probably underestimates the number of flexible kinks. If the apatite crystals have square cross sections 45 by 45 A the spaces between them are 18.6 A wide and most collagen molecules might be braced to two apatite crystals as shown at the top. Were the crystal cross sections 45 by 150 A the cracks between them would be 30 A thick. The collagen molecules would lie between them in a double layer, perhaps staggered, and each molecule could be braced directly to only one crystal as shown at the bottom. Intermolecular and intramolecular struts no doubt are present, but are not shown.
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total length of bracing struts, because some points may be braced to points on other backbones that are in turn braced to the apatite crystals. In Fig. 1 the struts are represented as additions to the collagen molecule. In fact, they probably consist largely of side groups like those already present on tendon collagen. Each crystal will experience a complex distribution of lateral pulls, pushes, and shears from the braces linking it to the collagen, the magnitude of each being the backbone tension times twice the sine of half the angle of the bend in the collagen backbone where the brace is attached. If the flexible kinks are large in a number the angle at each can be small, and the braces much weaker than the backbone. The stiff crystals, three times thicker than the collagen molecules, should be deformed very little by these relatively small forces. As well as increasing the stiffness of the collagen molecules straitjacketing ought to increase their strength. A bond bent or rotated out of its equilibrium shape is part way up the slope of its potential well, and should fail under less tension than a bond that is neither bent nor twisted. The braces between the mineral and collagen will not be absolutely intextensible, and neither are the bonds of the collagen backbones. The collagen molecules, even when braced, will extend a bit under load, straining the braces in shear at each end of the mineral crystals. The crystals are stiff and hardly stretch at all. If the crystals were too long the end braces would be sheared off long before the collagen backbones broke. The shortness of the crystals is necessary if they are to serve as splints for bracing the collagen molecules.
Human bone can be stretched 0.76% and recover completely (Burstein, Reilly & Frankel, 1973). As the crystals are 400 A long this means that the braces at each end are sheared by l-52 A minus a small correction for the extension of the crystal, which seems a reasonable value for the greatest shear they can survive without breaking. It is between -& and 5 of the likely lengths of the braces, and makes the latter lean by up to 42” relative to their unstrained orientation. The result will be to pull the collagen and the apatite closer together by a factor of 2. If this movement is sterically hindered the braces will be stretched as well as sheared. The shear stress at the ends of each crystal passes some tensile stress from the collagen into the crystal and back to the collagen. Now much stress is thus routed depends on how stiff the braces and their junctions to the collagen and apatite are in bending, and on their stiffness in extension if they lie out of the plane at right angles to the axes of the collagen molecules. It seems likely that the length of the apatite crystals is such that if the bone is overstressed the braces start breaking a bit before the collagen backbones do. The first braces to break should be those near the ends of the crystals, because these are sheared, as well as being pulled or pushed by their bracing
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function. Breaking of the braces allows the collagen backbones to straighten and the bone to stretch, and might account for at least part of the inelastic stretching of bone before it breaks. Straining the bone beyond its elastic limit, yet not enough to break it, should tear loose the collagen from the ends of each crystal but leave it attached in a region around the middle. The detached lengths of collagen should behave with increased compliance, hence overstrained bone should still behave elastically, but with a reduced stiffness modulus, from its initial length up to the greatest previous strain. Straining it more should detach additional lengths of collagen, further reducing the tensile stiffness modulus and increasing the maximum elastic strain. Figure 2 of Burstein et al. (1973) shows that, except for one thing, this happens. Neglected was apparent “friction” caused, perhaps, by new, weak bonds replacing the broken ones between the collagen and the crystals. The bone returns only about 3 of the way toward its original length after inelastic straining, and when pulled again has approximately its original stiffness up to about 0.4 of its initial yield stress. Above that, it seems that the new bonds have broken, and the stiffness drops to a lower and quite constant value that holds until the strain reaches the greatest previous strain. Here the stressstrain curves flatten, with the stress rising more slowly as inelastic straining recommences and, presumably, further lengths of collagen become detached. Below this knuckle the slope of each curve is such that, extrapolated backward, the curves all pass remarkably close to the zero stress-zero strain point, just as they would if the freed regions of collagen were acting as springs whose unstressed length corresponded to the initial length of the bone, as the theory says they ought. In terms of this extrapolation, inelastically strained bone has a memory of its length before it was stretched. In describing the tensile-mineral, collagen-glue model of bone Currey (1969) suggests that the mineral might raise the tensile modulus of collagen as much as five times. But a factor of 30, or somewhat less if part of the stress is routed through the apatite crystals, is needed to reinstate collagen as a major carrier of tensile stress in bone. Currey did not reinstate it, nor did he give a mechanism for the stiffening effect. The straitjacket mechanism and the tensile-mineral collagen-glue mechanism might operate simultaneously in the same piece of bone. The second would be ruled out if the end-to-end joints in neighboring columns of crystals were found not to be staggered, if they were as shown in Fig. l(a) rather than Fig. l(d). There would be no overlap between crystals, so the tensile stress could not be passed from one to another as shear in the collagen. I thank Leon Sokoloff, Walter Stewart and Ned Feder for helpful discussions.
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REFERENCES Animal Mechanics, pp. 118-119. London:
Sidgwick & Jackson. BUECHE, F. (1958). J. Polymer Sci. 33, 259. BURSTEIN, A. H., REILLY, D. T. & FRANKEL, V. H. (1973). In Perspectives in Biomedical Engineering (R. M. Kenedi, ed.) pp. 131-134. London: MacMillan. SURREY, J. D. (1964). BiorheoZogy 2,l. CURREY, J. D. (1969). J. Biomech. 2,477. hNES, E. D. (1973). In Biological Mineralization (I. Zipkin, ed.) pp. 243-247. New York: John Wiley. HOLLIDAY, L. (1966). (Ed.) Composite Materials. Amsterdam: Elsevier. KNESE. K.-H. (1958). Knockenstraktur als Verbundbau. Stuttgart: Thieme. RAMA&ANDR~N, G: N. & KARTHA, G. (1955). Nature, LonEi 176, 593. WELCH, D. 0. (1970). Recent Adv. engng Sci. 5, 245. YANNAS, I. V. (1972). In Biomedical Physics and Biomaterials Science (H. E. Stanley, ed.) pp. 43-45. Cambridge, Massachusetts: M.I.T. press. ALEXANDER,
R.
McN.
(1968).
Do Mineral Crystals StifFen Bone by Straitjacketing its Collagen? C. W. MCCUTCHEN Laboratory of Experimental Pathology, National Institute of Arthritis, Metabolism and Digestive Diseases, National Institute of Health, Bethesda, Maryland 20014, U.S.A. (Received 8 February
1974, and in revisedform
21 June 1974)
Theories that assume that the collagen and the mineral of bone have the same properties in bone as they do when they are by themselves have difficulty in accounting for bone’s tensile stiffness. The collagen fibers are long but stretchy: the apatite crystals are stiff enough, but too short to be effective tension carriers. In bone the mineral is so finely divided that each collagen molecule may have a mineral crystal next to it. It is proposed that collagen is the prime tension carrier in bone, and that via short molecular struts the mineral crystals prevent the naturally kinky collagen molecules from straightening under tensile stress, which greatly increases the collagen’s tensile stiffness.
1. The Paradox The major constituents of bone, present in about equal proportions by volume, are impure apatite, a stiff, brittle mineral, and collagen, a strong, tough, fibrous natural polymer (Currey, 1964). The tensile stiffness (Young’s) modulus of bone apatite has not been measured. That of synthetic hydroxyapatite is 1-06x lo6 kg/cm’, of human bone 1.82 x 10’ kg/cm’ and of collagen 1.22 x IO4 kg/cm’ (Welch, 1970). The quantity referred to is stiffness, the force per unit area divided by the fractional change in length caused by the force. This is distinguished from tensile strength, the force per unit area required to break the material. The collagen is in the form of fibrils of indefinite length, made by joining end-to-end three-stranded molecules that are 3000 A long, and 15 A in diameter (Ramachandran & Kartha, 1955; Yannas, 1972). The apatite crystals, 400 A long, more or less, lie within the collagen fibrils and are aligned parallel with them (Welch, 1970; Currey, 1969). Their other dimensions are perhaps 45 by 45 A, but the precise shape of their cross section is unknown (Currey, 1969; Eanes, 1973). 51
52
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W.
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Several models have been proposed to explain how the collagen and apatite co-operate to give bone its observed tensile stiffness. Because the collagen stretches unbroken for long distances, it could carry tensile loads with no help from the mineral. But the tensile stiffness modulus would be that of collagen times the fraction of the bone’s volume that the collagen occupies, or about one thirtieth the stiffness of bone. Under tension the apatite crystals would separate from each other as shown in Fig. l(a) and not affect the stiffness modulus. The apatite crystals are several times stiffer than bone, and might account for its compressive stiffness, because endwise compression could push the crystals against each other end-to-end as in Fig. l(b). Were the bone prestressed, as Knese (1958) suggests, its collagen in tension and its mineral in compression, the apatite crystals would be held clamped together as shown in Fig. l(c), even when the bone was in tension. The compressive stiffness of the columns of mineral could thus also account for bone’s stiffness in tension. But prestressing uses up so much of the ultimate strength of the apatite that bone would be weaker in compression than it is (Currey, 1964). The apatite crystals could carry tension if they were attached to each other end-to-end. There might be non-mineralized bridges between the ends of adjacent crystals, but the joint between bridge and crystal should be as strong as the weaker of the two, perhaps not easily achieved between materials with different structures. Though this arrangement would explain bone’s behavior up to its elastic limit at 0.76 % elongation, it does not explain why bone can be stretched inelastically by a further 3 % before it breaks (Burstein, Reilly & Frankel, 1973). In the absence of end-to-end bridges neighboring columns would have to be arranaged so the joints between the crystals were not opposite each other. The tension could then get around the joints by zig-zagging back and forth from column to column, passing from one to the other as shear stress in the intervening layer of collagen as proposed by Currey (1969) [see Fig. l(d)]. One cannot prove that the collagen molecules have not the needed shear strength parallel to their long axes for them to serve as glue. But their structure seems wrong. The shear stress can be analyzed into a pure tension and a pure compression at right angles to each other, and both at 45” to the faces of the crystals. To transmit the stresses the strength members of the collagen should run diagonally across the gap from the lateral face of one crystal to that of another. Instead they run parallel to the crystal faces. Furthermore, if this is how stress is carried why were the mineral crystals made so short? Longer crystals would impose less shear stress on the collagen-glue. Why not make them thicker as well? This would increase the strength of bone by reducing the fraction of its volume that is devoted to the glue layers.
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FIG. 1. (a) If the collagen carried tension with no help from the apatite crystals, bone would be far stretchier than it is, because collagen has a low stiffness modulus. Pulling endwise on the bone would stretch the collagen and cause the apatite crystals to separate from each other. (b) In compression the apatite crystals can carry stress and provide high stiffness, because the joints between crystals are no bar to compressive stress. (c) If the collagen were pretensioned so as to hold the apatite crystals clamped together tensile loads would be carried by the collagen, yet most of the stiffness would be provided by the apatite. But prestressing leaves only a fraction of the ultimate compressive strength of the apatite available for load carrying. (d) If most of the tensile load is to be carried by the mineral crystals then either they must be very strongly attached end-to-end, or the joints in adjacent columns of crystals must be staggered as shown here, and the collagen used as glue to take the stress zig-zagging from column to column so it does not have to cross the joints. The scale of the drawing is given by the apatite crystals, which are 4.00 A long.
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The above theories treat bone as analogous to steel-reinforced concrete. If we know its structure we can predict the behavior of reinforced concrete from the properties that each of its constituents shows when it is by itself. Attempts to predict this way the properties of bone all seem to have one flaw or another. Alexander (1968) advanced a different idea, here quoted in toto. “Filled rubbers perhaps offer a better analogy to bone, though they are neither as strong nor as stiff. They are mixtures of rubber with fine particles of another material which is usually carbon black. Carbon black is fine soot. Its particles are roughly spherical and in some of the grades used for reinforcing rubber are about the size of the crystals in bone. Rubber filled with carbon black is used for making tyres. “The rubber seems to become very dimly attached to the carbon black, and the attachments prevent the molecules from straightening as much as they otherwise would, when the rubber is stretched. This is particularly so since the carbon particles tend to join together in short chains (Holliday, 1966). Hence, Young’s modulus is higher than for pure rubber. The tensile strength is also higher. Rubber containing 50% carbon black can be as much as 16 times as strong as pure rubber (Bueche, 1958).” The presence of carbon black modifies the properties of the rubber, and the composite is stiffer and stronger than it would be if the rubber were not modified. In what follows, the analogous effect of the apatite crystals on the collagen of bone appears as a natural consequence of the mutual arrangement of the collagen fibers and the apatite crystals, and so far as theory can ever verify theory, seems to justify Alexander’s suggestion. 2. The Straitjacket
Theory
For one constituent of&bone to modify the properties of the other the two must be closely intermingled, and they are. The apatite crystals are so small that, if their cross section is square and they are oriented parallel to each other and spaced the same distance apart, the distance between their faces is only 18.6 A. This is space enough for one, or at most two layers of collagen molecules. If each collagen molecule is in contact with one or maybe two apatite crystals we may expect its properties to be much modified by the association. The mineral crystals, on the other hand, are many lattice spaces thick, and so should behave like bulk material. I suggest that collagen is the prime tension carrier in bone, and that bone’s stiffness modulus is much greater than that of isolated collagen because intimate association with the mineral raises the collagen’s tensile stiffness. The extensibility of tendon collagen must result from straightening of kinks in its three backbones rather than
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stretching of the backbones’ bonds. The high Young’s moduli of most crystalline materials shows that bonds are very stiff in pure extension. Not all kinks need be flexible, but some of them must be so to account for collagen’s extensibility. The flexibility can result from either bending or rotation of bonds. Yannas (1972) says that on average there is a hinge every 9 A along the collagen backbone. Suppose that at every point where a collagen backbone has a flexible kink it is connected to the surface of a mineral crystal by bonds or by struts formed of side chains. There is a side chain for every 2446 A of axial distance along the minor helix (Ramachandran & Kartha, 1955), so even ignoring hydrogen bonding there are enough potential struts to prevent bending at the hinges. The bonds and struts would hold the molecule in the kinky shape it has when unstressed. Bending or rotation of the backbone bonds would be prevented. Stretching the molecule would lengthen the bonds of its backbones, and need far greater force than stretching an unbraced molecule whose backbones were free to straighten to the full extent permitted by their one per 86 A twist around each other. At most two braces in different directions at each bend are needed to prevent the backbones from straightening under tension. The braces can all be perpendicular to the long axes of the molecules and crystals, but it is not essential that they be so. Figure 2 shows how a single backbone could be braced to one or two apatite crystals. Bracing three backbones need not require three times the
FIG. 2. Bracing the flexible kinks in the collagen molecules to the stiff apatite crystals would prevent the molecules from straightening under load, and greatly increase their tensile stiffness. Only one of the three backbones of each molecule is shown, and the drawing probably underestimates the number of flexible kinks. If the apatite crystals have square cross sections 45 by 45 A the spaces between them are 18.6 A wide and most collagen molecules might be braced to two apatite crystals as shown at the top. Were the crystal cross sections 45 by 150 A the cracks between them would be 30 A thick. The collagen molecules would lie between them in a double layer, perhaps staggered, and each molecule could be braced directly to only one crystal as shown at the bottom. Intermolecular and intramolecular struts no doubt are present, but are not shown.
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total length of bracing struts, because some points may be braced to points on other backbones that are in turn braced to the apatite crystals. In Fig. 1 the struts are represented as additions to the collagen molecule. In fact, they probably consist largely of side groups like those already present on tendon collagen. Each crystal will experience a complex distribution of lateral pulls, pushes, and shears from the braces linking it to the collagen, the magnitude of each being the backbone tension times twice the sine of half the angle of the bend in the collagen backbone where the brace is attached. If the flexible kinks are large in a number the angle at each can be small, and the braces much weaker than the backbone. The stiff crystals, three times thicker than the collagen molecules, should be deformed very little by these relatively small forces. As well as increasing the stiffness of the collagen molecules straitjacketing ought to increase their strength. A bond bent or rotated out of its equilibrium shape is part way up the slope of its potential well, and should fail under less tension than a bond that is neither bent nor twisted. The braces between the mineral and collagen will not be absolutely intextensible, and neither are the bonds of the collagen backbones. The collagen molecules, even when braced, will extend a bit under load, straining the braces in shear at each end of the mineral crystals. The crystals are stiff and hardly stretch at all. If the crystals were too long the end braces would be sheared off long before the collagen backbones broke. The shortness of the crystals is necessary if they are to serve as splints for bracing the collagen molecules.
Human bone can be stretched 0.76% and recover completely (Burstein, Reilly & Frankel, 1973). As the crystals are 400 A long this means that the braces at each end are sheared by l-52 A minus a small correction for the extension of the crystal, which seems a reasonable value for the greatest shear they can survive without breaking. It is between -& and 5 of the likely lengths of the braces, and makes the latter lean by up to 42” relative to their unstrained orientation. The result will be to pull the collagen and the apatite closer together by a factor of 2. If this movement is sterically hindered the braces will be stretched as well as sheared. The shear stress at the ends of each crystal passes some tensile stress from the collagen into the crystal and back to the collagen. Now much stress is thus routed depends on how stiff the braces and their junctions to the collagen and apatite are in bending, and on their stiffness in extension if they lie out of the plane at right angles to the axes of the collagen molecules. It seems likely that the length of the apatite crystals is such that if the bone is overstressed the braces start breaking a bit before the collagen backbones do. The first braces to break should be those near the ends of the crystals, because these are sheared, as well as being pulled or pushed by their bracing
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function. Breaking of the braces allows the collagen backbones to straighten and the bone to stretch, and might account for at least part of the inelastic stretching of bone before it breaks. Straining the bone beyond its elastic limit, yet not enough to break it, should tear loose the collagen from the ends of each crystal but leave it attached in a region around the middle. The detached lengths of collagen should behave with increased compliance, hence overstrained bone should still behave elastically, but with a reduced stiffness modulus, from its initial length up to the greatest previous strain. Straining it more should detach additional lengths of collagen, further reducing the tensile stiffness modulus and increasing the maximum elastic strain. Figure 2 of Burstein et al. (1973) shows that, except for one thing, this happens. Neglected was apparent “friction” caused, perhaps, by new, weak bonds replacing the broken ones between the collagen and the crystals. The bone returns only about 3 of the way toward its original length after inelastic straining, and when pulled again has approximately its original stiffness up to about 0.4 of its initial yield stress. Above that, it seems that the new bonds have broken, and the stiffness drops to a lower and quite constant value that holds until the strain reaches the greatest previous strain. Here the stressstrain curves flatten, with the stress rising more slowly as inelastic straining recommences and, presumably, further lengths of collagen become detached. Below this knuckle the slope of each curve is such that, extrapolated backward, the curves all pass remarkably close to the zero stress-zero strain point, just as they would if the freed regions of collagen were acting as springs whose unstressed length corresponded to the initial length of the bone, as the theory says they ought. In terms of this extrapolation, inelastically strained bone has a memory of its length before it was stretched. In describing the tensile-mineral, collagen-glue model of bone Currey (1969) suggests that the mineral might raise the tensile modulus of collagen as much as five times. But a factor of 30, or somewhat less if part of the stress is routed through the apatite crystals, is needed to reinstate collagen as a major carrier of tensile stress in bone. Currey did not reinstate it, nor did he give a mechanism for the stiffening effect. The straitjacket mechanism and the tensile-mineral collagen-glue mechanism might operate simultaneously in the same piece of bone. The second would be ruled out if the end-to-end joints in neighboring columns of crystals were found not to be staggered, if they were as shown in Fig. l(a) rather than Fig. l(d). There would be no overlap between crystals, so the tensile stress could not be passed from one to another as shear in the collagen. I thank Leon Sokoloff, Walter Stewart and Ned Feder for helpful discussions.
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