Complex Materials: The Tough Life of Bone

Bone was a crucial biological material for the evolution of large terrestrial organisms and is today essential for most of our daily activities and well-being. From an engineering perspective, this living material features highly desirable properties for modern load-bearing structures. It is made of abundant and environmental-friendly building blocks, which are combined into a tough and durable structure that can continuously modify itself to adapt to changes in the mechanical load imposed by the surroundings. In this review article, we compile and discuss scientific findings that allow us to understand bone as a complex system with properties that emerge from cell-mediated interactions of molecules and particles at multiple length scales. Analogous to other complex systems, such interactions lead to self-organization, hierarchical structures and adaptive behavior without the need of a central controlling unit. A rich range of physical, chemical and biological phenomena provide a framework for information to be generated and processed in this complex system. Understanding the interplay between such underlying phenomena and their emerging properties should help the diagnosis and treatment of bone-related medical conditions and might provide guidelines for the future development of more sustainable materials and engineering structures.


Introduction
From insect colonies to stock markets to the world wide web, complex systems are ubiquitous in nature, society and modern technologies [1]. In these systems, collective properties, information processing capabilities and adaptive response emerge from the interactions between a large number of individual constituents without a central controlling unit. Such features are prominent in living organisms, where complexity and diversity have been the natural outcome of billions of years of evolution. Complexity in living organisms derives from cell-mediated self-organization processes that create hierarchically structured systems capable of adapting to their environment. This holds true for a wide variety of living species from primitive E. coli bacteria to the Homo sapiens, despite the unclear role of natural selection on the increasing complexity observed [1]. While most of the organs and tissues of our body share these common attributes, bone is a remarkable example of an animal tissue where all these complex characteristics are combined with an unusual set of highly desirable material properties: low weight, stiffness and toughness.
Stiffness is the ability to carry mechanical load with minimum deformation and is the property that bone is mostly known for. Less intuitive is the strong resistance of bone against the propagation of cracks, which is referred to as fracture toughness. Stiffness and toughness provide the long-term mechanical stability * Correspondence email address: andre.studart@mat.ethz.ch required for locomotion and physical activities, whereas low weight is essential to minimize the energy needed to undertake such tasks. To fully appreciate the high mechanical stability of bone one should note that the main solid building blocks of this biological material, calcium phosphate and collagen, are intrinsically rather weak and soft, respectively. Indeed, the resistance of these mineral and organic building blocks against crack propagation is at least three-fold lower than that of the weakest engineering materials (Figure 1a, b). Mixing these weak materials in the proportions found in bone would in principle lead to a mediocre fracture toughness. Remarkably, the hierarchical structure of bone provides mechanisms to break this rule and reach toughness levels comparable to those of engineering materials. It is important to emphasize that this is achieved without sacrificing the high stiffness of the calcium phosphate mineral phase. Combining high stiffness and toughness using weak building blocks is a hallmark that differentiates biological structural materials from their engineering counterparts, which typically show a trade-off between these essential mechanical properties (Figure 1a, b). The low strength of nature's building blocks is largely compensated by their wide availability and the fact that they can be processed into stiff and tough biological materials in water near room temperature. Such mild processing conditions are highly desirable but currently unmet in most manufacturing technologies.
The impressive toughness and stiffness of bone contrast with the fascinating dynamic and living nature of this biological material. Such living character is often to the normal structure. The metacarpal bone is positioned between the "wrist" and the toes. (e) X-ray images of two distinct femoral heads showing how the architecture of the trabeculae inside the bones (dark colour) follow the principal stress lines arising from mechanical loading. The red arrows and blue lines indicate the direction of tension and compression stresses, respectively. The example on top develops higher tensile stresses compared to compressive stresses. The situation inverts in the example shown at the bottom. Graphs (a) and (b) were adapted by Dr. Florian Bouville from Espinosa et al. [5] Images (c-e) are adapted and reproduced with permission from Robling et al. [3].
illustrated by the loss of bone mass experienced by astronauts in low-gravity conditions. At the other end of the spectrum, tennis players are known to develop thicker bones in the racket-holding arm compared to the other non-dominant arm bone [2]. Lying in bed for several months is sufficient to cause significant variations in bone mass along the human body ( Figure 1c) [3]. Besides gaining or losing mass, bone is also able to adapt its internal architecture to match the stress lines developed within the tissue during mechanical loading (Figure 1d). Known as Wolff's law, this rule is a prototypical example of how structural materials in biology can dynamically adapt their structure to fit their function as opposed to the static nature of conventional engineering materials [4]. The dynamic and adaptive nature of bone results from the metabolic activity of the living cells that control the biochemistry of this tissue. These comprise primarily bone-forming osteoblasts, bone-dissolving osteoclasts and osteocytes embedded in the extracellular matrix that constitute this mineralized tissue. Together such cells are able to locally grow or dissolve bone in the regions of, respectively, high or low mechanical stresses, and to continuously renew the tissue to remove damage developed in the structure during its lifetime. As a result of these cellular processes, the average turnover of the mineral phase of our bones is as high as 10% and 20% every year for cortical and trabecular bone, respectively. The mineral turnover can be maintained at stable levels through the balanced action of osteoblasts and osteoclasts, leading to a dynamic equilibrium state known as homeostasis. In addition to damage repair, bone homeostasis is important to maintain the level of calcium in the blood stream. The interplay between the large number of osteoblasts, osteoclasts and osteocytes present in bone is at the core of the self-organization processes, hierarchical structures and adaptive response of this complex living material.
In this review article, we discuss the current understanding of the biological material bone from the perspective of complex systems. Starting at the cellular level, the self-organization processes underlying the formation of the collagen phase and its subsequent mineralization with calcium phosphate are discussed. Next, we describe the hierarchical structure that emerges from such self-organization processes and explain how structural features at different length scales of the tissue give rise to the high toughness of bone. Finally, experiments and models that have been proposed to explain bone's unique ability to adapt to its environment are reviewed in light of universal features observed in complex systems. Understanding the properties and dynamic behavior of bone requires basic knowledge from a wide range of disciplines, including physics, chemistry and biology. To help the reader navigate through these distinct disciplines, we have included tutorial panels that focus on the fundamental concepts needed to fully appreciate the complexity of this fascinating material.

Self-organization
Bone tissue is formed by an exquisite sequence of cellmediated self-organization processes that starts with the formation and assembly of collagen fibrils, and is followed by the mineralization of such fibrils with the inorganic calcium phosphate phase. Below, we describe how collagen fibrils are assembled and mineralized, highlighting the cellular processes and the supramolecular interactions involved.

The assembly of collagen fibrils
Collagens are the most abundant family of proteins present in vertebrates and the main source of tensile strength and elasticity in animal tissues. Its precursor, the procollagen α-chain, is synthesized by elongated fibroblasts (Figure 2a) as well as by highly branched and mononucleated cuboid cells called osteoblasts. Procollagens are polypeptide chains comprised of two parts that have different amino acid sequences. The central part exhibits a collagenous domain that is characterized by repeated sequences of amino acids residues in the form of Gly (glycine)-X-Y, with X and Y being typically proline (Pro) and hydroxyproline (Hyp), respectively (orange domain in Figure 2b). As the smallest amino acid, glycine plays an important role in the assembly of collagen. The repeated occurrence of this amino acid allows the α-chains to fold into highly curved left-handed helices that can intertwine with two other α-chains to eventually form right-handed triple-helical protein complexes [6]. In contrast to this central domain, the terminal parts of the α-chain consist of amino-terminal domains characterized by a sequence of amino acids that results in globular folding (green domains in Figure 2b). In this section, we will present the current understanding of the synthesis and intracellular chemical transformation of procollagen chains and their transportation, folding and secretion into the extracellular space to form the hierarchically organized fibrillar structures known as collagen fibrils.
Similarly to the production of other protein complexes in eukaryotic cells, collagen molecules are synthesized inside the cell and secreted into the extracellular space following a series of steps that are executed at specific sites within the cell. This process resembles the typical assembly line of a product in a factory [8]. The production of collagen initiates in the cell's nucleus, where the genetic information is decoded and transcripted into mRNA molecules. To carry the coding sequence further, mRNA molecules cross the nuclear membrane and reach the ribosomes that are attached onto the membranes of the endoplasmatic reticulum (ER). Ribosomes then read the coding sequence and start connecting amino acid residues to make procollagen α-chains, the synthetic precursors of the family of collagen proteins (Step 1, Figure 2b) [9]. Inside the ER, procollagen αchains are modified by the covalent addition of functional groups to specific amino acid residues that can ultimately stabilize the triple helical conformation (see tutorial box, Figure 3). First, the hydroxylation of proline and lysine residues increases the number of hydrogen bonding sites and enhances the stabilization of the triple helix conformation through non-covalent interactions (Step 2, Figure 2b). Next, some of the hydroxylysine residues are converted into glycans by addition of carbohydrate units through enzymatic glycosylation (Step 3, Figure 2b). Although the function of glycan groups in collagen still remains unknown, alterations of the glycosylation level of collagen have been reported in several bone disorders related to biomineralization, fibrillar crosslinking and collagen remodeling [10][11][12]. Following glycosylation, non-covalent interactions of amino acid residues present in the terminal propeptide domains lead to the formation of the trimeric procollagen unit. This unit is subsequently stabilized by the formation of interchain covalent disulfide bonds through oxidation of cysteine residues (Step 4, Figure 2b) [13]. The covalent association through disulfide bonds likely triggers the zipper-like folding of the procollagen domains into the triple helix conformation (Step 5, Figure 2b). Intraand interstrand supramolecular interactions within the procollagen domains are crucial for the stabilization of the triple helix conformation ( Figure 3).
As the collagenous protein complexes are much larger (300 nm) than the transportation vesicles (60-80 nm) typically produced by cells, special membrane-bound compartments called vesicular tubular clusters are assembled to transport the collagen complex from the ER to the Golgi apparatus (Step 6, Figure 2b). Once the complex molecule enters the Golgi apparatus, the glycosylated residues of the propeptide domain are further modified with chains containing multiple carbohydrates units (Step 7, Figure 2b). Chemical modifications mediated by enzymes present in the Golgi apparatus seem to act as a signal that instruct the cell's transportation system to deliver the procollagen molecules to their final destination within the cell [14]. This final intracellular transportation process is carried out by membranebound compartments known as fibripositors. Next to the plasma membrane, enzymatic cleavage of the terminal propeptides chains results in the formation of tropocollagen triple helices (Step 8, Figure 2b) that eventually self-organize into collagen fibrils (fibrillogenesis, Step 9, Figure 2b). Finally, the membranes of the fibripositor and the cell fuse to secrete the initial fragments of collagen fibrils into the extracellular space (Step 10, Figure 2b). Fibrillogenesis continues in the extracellular space (Step 11, Figure 2b) where each tropocollagen triple helix is offset along the longitudinal axis by a distance (D) of 67 nm, to yield a periodic crossstriated structure with well-defined gap (0.6D) and overlap (0.4D) zones within the assembled collagen fibril ( Figure 3). The fibrils show average diameter ranging typically from 10 to 500 nm and length that can reach up to a few millimeters in size ( Figure 2c). The mechanical strength and stiffness of such fibrils are afterwards enhanced through a biomineralization process.

Biomineralization
Biomineralization in bone tissues is still a topic of debate in the scientific community. This process is regulated by osteoblasts ( Figure 4a) and occurs after the non-mineralized collagen matrix is deposited on the bone surface as osteoid seams. In healthy bones, the time between osteoid formation and the onset of the mineralization process, known as mineralization lag time, typically ranges between 2 and 3 weeks [21]. In the last decades, several models have been proposed to explain the mineralization of collagen fibrils in bone through the formation of the crystalline calcium phosphate phase hydroxyapatite. Some of these models include (i) cell-independent processes, in which noncollagenous proteins interact with the gap zones in collagen fibrils, activate the capture of ions and trigger the crystallization of hydroxyapatite [22]; (ii) cellcontrolled processes, in which vesicles that are secreted by mineralizing cells accumulate ions in the extracellular space, deliver their contents to the collagen fibrils, and induce hydroxyapatite precipitation [23]; and (iii) Figure 3: Tutorial on the supramolecular chemistry of collagen fibrils [15][16][17][18][19]. All images except 3.2c are adapted and reproduced with permission from Zhu et al. and Hahn [19,20]. Image 3.2.c was originally published in the Journal of Biological Chemistry [18] ©. cell-controlled processes, in which amorphous calcium phosphate is transported via intracellular vesicles and later undergoes phase transformation into hydroxyapatite particles within the collagen fibrils [24]. Recently, strong experimental evidences have been shown to support the phase transformation model. According to this model, intracellular vesicles containing amorphous droplets of calcium-based mineral precursor are produced within the mitochondria of osteoblasts ( Figure 4a). In this section, we will discuss the experimental observations that support this biomineralization model of bone tissue.
Mitochondria are membrane-bound organelles responsible for the storage of chemical energy in the form of adenosine triphosphate (ATP) molecules inside eukaryotic cells. In these organelles, calcium ions are internally accumulated to balance the electric potential that is created by the proton gradient used to power the enzyme ATP-synthase [27]. Such calcium ions form insoluble complexes in the presence of proteins and phosphate ions, which are released into the cytosol as droplets inside membrane-bound vesicles (Figures 4a,  b) [28][29][30][31][32]. These intracellular vesicles are transported through the cytosol, secreted into the extracellular space and end their journey by delivering their content onto non-mineralized collagen fibrils located at the mineralization front of the bone tissue (Figure 4c, e). Once released, the amorphous calcium phosphate from the droplets binds to specific sites located at the end of the tropocollagen triple helices, diffuses and solidifies within the gap zones of the collagen fibrils [33,34]. Eventually, the amorphous precursor undergoes a phase transformation into oriented platelet-like hydroxyapatite crystals (Figure 4c, f).
Although classical nucleation theory has been used to explain the crystallization of hydroxyapatite particles in bone tissue [35], the phase transformation model described above is in good agreement with several experimental observations [24,30,31,36]. Biomineralization via phase transformation of an intermediate amorphous phase reduces the flow of water during crystallization, enables pre-shaping of the amorphous phase into defined and unusual geometries, eases transportation of precursors to their final destinations, and enhances the crystallization rates up to two orders of magnitude compared to classical nucleation and growth mechanisms [37,38]. The remarkably faster crystallization achieved by phase transformation results from the reduced activation energy for the formation of the metastable amorphous phase (∆G * 1 , continuous black line in Figure 4d) in comparison to the energy barrier (∆G * n , dashed red line in Figure 4d) involved in the nucleation and growth of ion clusters in the classical mechanism [34]. This suggests that the biomineralization process occurs via pathways that facilitate the crystallization of reinforcing mineral platelets within the collagen fibrils.
Besides their prominent role in the assembly of collagen fibrils (Figure 3), non-covalent molecular interactions have also been proposed to explain the controlled crystallization of platelet-like hydroxyapatite particles in bone tissues. Although the current spatial and temporal resolutions of in vitro and in vivo experiments are not sufficient to fully elucidate the interplay between the supramolecular assembly of collagen and the phase transformation of the mineral phase [39], functional groups present in the tropocollagen triple helix seem to work in synergy with inhibitors of hydroxyapatite nucleation so as to actively control the process of mineralization. Additionally, charged amino acids located at the end of the tropocollagen triple helices may attract electrostatically the amorphous calcium phosphate droplets and favor their infiltration into the collagen fibrils. These charged amino acids residues can also serve as heterogeneous nucleation sites to induce the phase transformation of amorphous calcium phosphate precursor into parallel arrays of oriented hydroxyapatite crystals [40]. The elaborate biological and chemical processes behind the synthesis of mineralized collagen fibrils well illustrate the complex nature of the bone formation process at its smallest length scales. Beyond the scale of individual fibrils, bone features a sophisticated hierarchical architecture that finds no counterparts among structural engineering materials.

Structural features across length scales
The assembly of collagen molecules and hydroxyapatite nanocrystals into mineralized fibrils is the result of a self-organization process driven by individual bone cells. The collective action of such cells eventually gives rise to an exquisite hierarchical architecture that spans from the nano-to the centimeter scale and makes bone a lightweight material with unusually high stiffness and fracture toughness (Figure 5a) [41]. To understand how these mechanical properties emerge and to appreciate the complexity of this biological material, we describe the main structural features of bone from the largest to progressively smaller length scales.
At the macroscopic scale (∼10 0 cm), bone displays a core-shell cylindrical morphology comprising a dense outer shell known as cortical bone and a highly porous inner core referred to as trabecular bone [42]. Trabecular bone contains about 20 vol% of mineralized tissue, the rest being occupied by marrow and fat. The mineralized tissue in the core forms a load-bearing continuous open network that provides shock resistance and transfers mechanical loads to cortical bone. Following a mechanical design often used in engineering structures subjected to bending, the low fraction of mineralized tissue in trabecular bone enables weight reduction without compromising the mechanical performance of the overall structure. Indeed, 80% of the overall bone mass is present in cortical bone, where the highest mechanical stresses are applied [42].
To cope with the high stresses developed during running and jumping, cortical bone is organized in a hierarchical architecture that displays high strength and fracture resistance. At the scale of hundreds of micrometers, the main structural elements of cortical bone are the osteons. These are concentric lamellar structures featuring a central channel that hosts blood vessels and nerves, which is known as the Haversian canal [41]. In cortical bone, the osteons reinforce bone in a similar way to fibers in engineering composite materials. Efficient reinforcement is achieved due to the orientation of the osteons along the direction of the highest mechanical stresses within cortical bone. The fiber-like osteons are embedded in the extraosteonal bone matrix and surrounded by 1 to 5 µm-thick cement lines [44,45]. Because of their softer nature and low mineral content, cement lines play a key role in the high fracture resistance of bone, as discussed later in the text [46].
At smaller length scales, each osteon is composed of concentric lamellae that are reinforced by arrays of mineralized collagen fibrils. The orientation of individual collagen fibrils varies within each osteonal lamella, leading to sublayers of fibrils aligned in specific directions [43,47]. Several models have been proposed to describe the fibrillar alignment patterns across the different sublayers in the osteonal lamellar tissue [48][49][50][51]. The recent identification of twisted and oscillating arrangements within the same osteon suggests that the fibrils follow complex and mixed orientation patterns in the osteonal lamellar tissue. With a diameter of ∼100 nm and length up to ∼10 µm, a fibril is selfassembled from collagen molecules of approximately 1.5 nm thickness and 300 nm length. In the selfassembly process, hydroxyapatite nanoplatelets grow in the gaps between the collagen molecules, as explained in the previous section. At the nanoscale, the collagen fibrils are composed of staggered arrays of tropocollagen triple helices that are held together by supramolecular interactions [52].
Our current understanding of the hierarchical architecture of bone has relied on characterization techniques that enable the analysis of structural features spanning across multiple length scales. Synchrotron-based X-ray diffraction methods (Wide-Angle X-ray Diffraction and Small-Angle X-ray Scattering), electron microscopy (Scanning and Transmission Electron Microscopy) and vibrational spectroscopy approaches are some of the techniques that have been utilized for this purpose [53]. To illustrate how these tools can shed light on the intricate architecture of biological materials, we briefly describe the use of polarized Raman spectroscopy for the analysis of the orientation of collagen fibrils within the lamellae and sub-lamellae of osteons (Figure 5b) [43].
Polarized Raman spectroscopy is a non-destructive characterization method that has been applied to obtain structural and chemical information about the collagen and mineral phases of bone. The technique relies on the inelastic scattering of light resulting from interactions with the probed material. Such scattering leads to a frequency shift in the Raman signal that can be assigned to a vibrational mode of a specific chemical group. Since the intensity of Raman scattering is influenced by the orientation of the incident polarized light relative to the vibrating molecular units, this method provides an effective means to quantify molecular alignment in anisotropic materials. In the case of bone, the amide groups of the collagen molecules generate an anisotropic Raman active vibrational band (∼1660 cm −1 ) that has been used to map the orientation of the mineralized collagen fibrils across a cross section of an osteon. The gradually changing orientation of mineralized fibrils within an osteon lamella is evident from the strong dependence of the Raman intensity signal on the polarization of the incident laser ( Figure 5c) [43,54]. Orientation maps obtained with this analysis reveal that the concentric lamellae within the osteon is composed of sublayers displaying collagen fibrils aligned in a specific direction. As the orientation of the collagen fibrils changes between distinct sublayers, this analysis confirmed the coexistence of twisted and oscillating plywood patterns in the fibrillar alignment within the same osteonal lamellar tissue [43]. Such a plywood-like arrangement of fibrils is expected to improve the resistance of bone against fracture by enabling the arrest and twisting of cracks at the length scale of the osteon lamellae.

Strength and fracture toughness
The high fracture resistance of cortical bone arises from reinforcing and toughening mechanisms operating at multiple length scales. As in engineering materials, fracture in bone is dictated by the interactions of a propagating crack with the structure of the material. While several structural features may provide resistance against crack propagation in a material [55,56] we focus on two important reinforcing and toughening mechanisms that contribute to the high strength and fracture toughness of cortical bone (Figure 6a), namely (1) multiscale cooperative deformation and (2) crack deflection and twisting. By operating in a complementary fashion at multiple length scales, these mechanisms highlight the importance of the hierarchical structure for the mechanical properties of bone. A more detailed analysis of these toughening mechanisms is essential to understand how bone can exceed the mechanical properties expected from its relatively soft collagen and weak hydroxyapatite building blocks.
The cooperative deformation of fibril arrays, individual mineralized collagen fibrils and mineral platelets at progressively smaller length scales has been suggested to be an important contribution for bone's high strength [55]. This has been demonstrated by performing in situ synchrotron-based X-ray scattering measurements on bone samples subjected to external mechanical forces (Figure 6a). The central questions in such experiment were: how are the applied external forces and strains distributed inside such a material containing stiff and soft phases at different length scales? Which of the structural elements present at multiple length scales in bone contributes to the global stiffness and strength of the material? To answer these questions, the strain developed within the mechanically loaded material at increasing length scales was experimentally measured. Wide-Angle X-ray Diffraction (WAXD) was used to quantify the strain at the scale of the mineral platelets, whereas the strain at the coarser level of mineralized collagen fibrils was measured by Small-Angle X-ray Scattering (SAXS). The diffraction and scattering data were complemented by the global strain at the tissue level measured directly on the macroscopic sample.
The in situ synchrotron measurements on bone samples show that both mineralized collagen fibrils and mineral platelets are strained upon the application of increasing external tensile forces. When normalized with respect to the tissue strain, the strain level on these two structural elements was found to remain constant during the initial elastic deformation of the tissue, before decreasing monotonically once the material starts to deform plastically (yielding). These results support the idea that global stresses are transmitted throughout the hierarchical structure of bone by shear forces in the extrafibrillar matrix that are translated in tensile forces in the mineralized collagen fibrils, which in turn develop shear forces in the collagen matrix that are converted into tensile stresses in the mineral platelets (Figure 6b). This coordinated stress transfer mechanism across hierarchical levels makes it possible for all the structural elements to contribute to the overall stiffness of bone, while maintaining the local strain levels below the critical limit that would cause failure [55].
While cooperative deformation is an important mechanism to enhance bone's stiffness and strength, the resistance of bone against fracture arises mostly from the deflection and twisting of cracks at the cement line between osteons [41]. Cracks propagating in the transverse direction perpendicular to the orientation of the osteon are strongly deflected at the cement line due to the weak nature of this interfacial layer. Such deflection moves the crack tip away from the plane of maximum tensile stress, strongly reducing local stress concentration and thus requiring higher loads to propagate the fracture further [57]. Because the cement lines form a weak interface also along the circumferential direction around the osteons, in-plane crack deflection is accompanied by out-of-plane twisting of the crack. This twisting effect reduces further the stress concentration at the crack tip and leads to significant toughening [57]. The deflection and twisting of transverse cracks at cement lines of cortical bone undergoing fracture have been confirmed by scanning electron microscopy and in situ synchrotron X-ray computed tomography (Figure 6c, d) [57].
The toughening effect induced by crack deflection and twisting can be quantified by measuring the resistance imposed by the material against crack propagation as the crack grows larger into the microstructure. The resistance of the material against crack growth is often expressed in terms of the critical stress intensity factor at the crack tip, K IR (see tutorial panel, Figure 7). Measurements of K IR as a function of crack extension, known as the R-curve, are used to quantify the evolution of the fracture toughness of the material during crack propagation. Experimental R-curves show that the fracture toughness of cortical bone increases by a factor of five or more as a crack penetrates 0.5 mm in the transverse direction (Figure 6e). The maximum K IR value of 25 MPa·m 1/2 achieved rivals that of the toughest engineering ceramics, despite the soft and weak nature of the collagen and hydroxyapatite building blocks. For sake of comparison, the strong glass cover of a smart phone displays fracture toughness lower than 1 MPa·m 1/2 and no crack growth resistance (rising R-curve), whereas tough zirconia ceramics used in dental applications show a relatively low K IR of approximately 6 MPa·m 1/2 . It is important to note that the presence of large heterogeneities lowers bone's fracture strength compared to these other structural materials ( Figure 7) and that the highly anisotropic architecture of bone makes its fracture toughness very dependent on the crack propagation direction relative to the orientation of osteons. Indeed, the K IR value measured for cracks propagating in the longitudinal direction show a relatively minor increase with crack extension, reaching levels five times lower than those achieved in the transverse configuration. This explains why it is easier to split than to break bone. Since transverse cracks are susceptible to the highest stress concentration under the bending loads developed in physiological conditions, the orientation of osteons in bone is designed to provide high toughness in this particular crack propagation direction. Such a match between loading conditions and material architecture illustrates how the skeletal tissue is structurally designed and adapted to best fulfill its biological function.

Adaptation
The self-organization processes underlying bone formation and the hierarchical architecture responsible for its high fracture toughness strongly rely on the metabolic activity and concerted action of the living cells present in this biological material. Such cells are the key drivers of the modeling and remodeling processes that take place in the skeletal tissue.
Bone modeling typically involves the deposition of new tissue in the periosteal (external) surface of the bone to increase the global rigidity of the structure and the removal of tissue from under-stressed regions in the internal part of bone to minimize weight. This modeling process occurs predominantly during growth and development of the skeleton. In remodeling processes, bone cells work together to replace potentially damaged    [59]. (c) Predictions of the mechanostat model and comparison with experimental data for bone under disuse followed by remobilization [59]. (d) Schematics illustrating the cellular interactions involved in the control of bone resorption via the metabolic activity of osteoclasts [3]. Stromal cells (blue) enhance or reduce the activity of osteoclasts (green) through the expression of the ligands RANKL or OPG, respectively. Graphs (a-c) were adapted and reproduced with permission from Turner [59]. Drawing shown in (d) was adapted and reproduced with permission from Robling et al. [3]. regions within the skeleton by new flawless tissue, thus providing resilience and longevity to the structure.
From an engineering perspective, the process of bone modeling is a remarkable feat. It is analogous to the task of permanently reconstructing and growing the primary structure of a skyscraper using minimal material resources while the building is already loaded with people working in offices at full capacity. How do our bones manage to accomplish such a challenging task? Research on bone modeling suggests that this is achieved by constantly monitoring the local strain within the entire structure and using this information to decide where existing bone should be removed and where new tissue must be deposited. The fact that this occurs through the autonomous action of a large quantity of individual cells without control from a central decision making unit (brain) demonstrates the complexity underlying the adaptive behavior and modeling process of bone.

Bone as a mechanostat
In the 1960s, bone modeling was proposed by Frost to operate as a closed loop control system [59][60][61]. A wellknown example of such a system is the thermostat used in the kitchen at home to control the temperature of an oven. In this so-called mechanostat model, a negative feedback control system ensures that the level of local mechanical strain in the bone is maintained between two physiologically relevant setpoint values (Figure 8a). The upper setpoint (ε u ) is defined by the maximum local strain that can be tolerated by bone without undergoing long-term failure, which typically falls within the range 1.5 − 2.5 × 10 −3 [59]. The value defining the lower setpoint (ε l ) has been suggested to lie between 0.05 and 0.2 × 10 −3 [59], which is probably a strain level that indicates a sufficient load-bearing contribution of the local tissue to the overall rigidity of the bone. Using this mechanostat model, the adaptive behavior of bone can be described using the language and terms typically employed for dynamic control systems (Figures 8b, c). In a negative feedback closed loop configuration, such a system may comprise a sensor (S) to measure the parameter of interest, a regulator (R) to process the sensing signal, and an effector (E) to keep the parameter of interest within the pre-defined setpoints.
In the case of bone, pre-osteoblastic cells in the marrow and osteocytes embedded in the mineralized tissue have been suggested to act as local mechanical sensors [3]. The role of effectors that change the rigidity of bone by favoring either mineralization or resorption processes are predominantly played by osteoblasts and osteoclasts, respectively. During operation of the mechanostat, the local strain (ε) is the parameter of interest that is constantly monitored by the sensing cells and compared with the lower and upper set point values (ε l and ε u ). If the actual strain falls below ε l , the balance between the activity of osteoblasts and osteoclasts is shifted towards local mineral resorption. Instead, if ε is higher than the upper set point (ε u ), the balance is changed to promote more bone formation. This negative feedback control mechanism allows the strain to be maintained within a desired interval (ε l ≤ ε ≤ ε u ) that ensures mechanical stability with minimum weight. Although the mechanostat model does not consider local effects and does not specify the underlying mechanisms that drive the system, it has been successfully used to describe the overall changes in bone rigidity in response to changes in the mechanical loading conditions (Figure 8c). Such a model has also been used to interpret the effect of non-mechanical agents, such as hormones or pharmaceuticals, on bone rigidity and stability. These non-mechanical agents presumably shift the balance between osteoblast and osteoclast activity by changing the set points of the mechanostat [3].
One of the mechanisms that contributes to the observed homeostatic control is the formation of new osteoclasts induced by stromal-derived cells present in the bone marrow (Figure 8d). This process, known as osteoclastogenesis, is regulated by several biomolecules that enable signal processing between cells through specific ligand-receptor interactions. Osteoclastogenesis is presumably favored under low mechanical loading conditions due to the release of RANK ligands (RANKL) by stress-sensitive stromal-derived cells. Such ligands bind to RANK receptors present on the surface of osteoclasts, promoting the metabolic activity of these cells and thus bone resorption. In contrast, higher mechanical stresses cause stromal-derived cells to preferentially secrete OPG, which is another biomolecule that works as a dummy receptor for RANKL. Binding of OPG to RANKL prevents the RANK receptors on the osteoclasts from interacting with the resorptionpromoting ligand. This eventually leads to the programmed death (apoptosis) of osteoclasts and therefore a reduction in bone resorption. Such biochemical signaling process provides a molecular mechanism for the effect of mechanical stress on changes in bone rigidity. In the mechanostat model, the stromal-derived cells play the role of the sensor that monitors the local strain and of the regulator that generates an amplified signal to trigger bone resorption, whereas osteoclasts are the effectors that release chemicals close to the bone surface to dissolve its mineral phase and promote resorption.

Osteocyte network and bone remodeling
While the adaptive behavior and disease conditions of bone can be described in terms of changes in the balance between the metabolic activity of osteoblasts and osteoclasts, osteocytes have been shown to also play a major role in bone dynamics, particularly for homeostasis and the long-term mechanical stability of the tissue. By forming an extensive interconnected network throughout the bone tissue, osteocytes are about 20 times more numerous than osteoblasts and osteoclasts. The openings that host the main body of the osteocytes in the mineralized bone tissue are known as lacunae. Lacunae are connected via numerous small channels called canaliculi, which host long cytoplasmic extensions known as osteocyte processes. This network architecture and the abundance of this cell type is related to the tissue morphogenesis. Osteocytes originate from osteoblasts that are entrapped within the mineralized tissue at the end of the bone formation process. Recent research suggests that the network of osteocytes functions as a sensor that constantly monitors the local strain distribution throughout the tissue. Sensing is hypothesized to be associated with fluid flow in the canaliculi, which might enable mechanotransduction via the build-up of electrical charge or deformation of transmembrane proteins (Figures 9 and 10) [3]. Such a fluid-mediated strain amplification process was put forward to explain the ability of osteocytes to sense strain, after it became clear that the elastic strains developed within the solid phase of bone tissue were too low to induce cellular response.
The sensing capabilities of the osteocyte network is crucial for the remodeling process of bone. Remodeling occurs through the coordinated action of osteoblasts, osteoclasts and other bone cells inside a large agglomerate of cells known as the Basic Multicellular Unit (BMU). Such multicellular units are present on the surface of trabecular bone or embedded within the structure of cortical bone. In cortical bone, the BMU is analogous to a boring machine that excavates bone to remove damaged sites and replace them with new flawless tissue. BMUs are presumably recruited by osteocytes when the local strain level in the tissue exceeds the physiologically relevant set point (Figure 9a). Alternatively, it has also been proposed that the recruitment of BMUs is triggered by cell death caused by micro-cracks that interrupt cell processes of osteocytes. Operating in parallel along the longitudinal axis of the bone, these remarkable cell agglomerates are responsible for the formation of osteons, which are their main structural signature. To move within the tissue and enable bone remodeling, an individual BMU is equipped with osteoclasts at the front and osteoblasts at the rear. This configuration allows osteoclasts to resorb bone in the flawed areas detected by the osteocytes and prompts osteoblasts to lay down new defect-free bone tissue as replacement. All the information required for this orchestrated action to take place is processed and signaled through intricate biochemical pathways, many of which are not yet fully understood ( Figure 10).  (3) and osteoblasts (4) that form the BMU are indicated in the cartoon. Adapted and reproduced with permission from Studart [65]. (b) The osteocyte network shown by a combination of optical microscopy, confocal microscopy and a computer-generated graphical representation. Adapted and reproduced with permission from Weinkamer et al. [63]. (c) Selected area of the network depicting (in red) nodes that form "highways" for fast nutrient and signal transmission throughout the tissue. Adapted and reproduced with permission from Weinkamer et al. [63]. (d) Expected gain in transmission speed for distinct types of bone depending on the ratio between the transport velocity in the network relative to the bone matrix. Adapted and reproduced with permission from Kollmannsberger et al. [62]. (e) Dependence of the small-worldness index S on the network size (N) for the osteocyte lacunacanalicular system (colored dots) compared to other real-world networks (black and grey). Adapted and reproduced with permission from Kollmannsberger et al. [62].
The complexity of bone is not limited to the multiple biochemical pathways and associated chemical networks involved in the remodeling process. Remarkably, experimental and modeling studies have shown that the network of interconnected osteocytes also exhibits distinctive features of complex systems [62][63][64]. One of these features involves the notion of a smallworld network. Taking nodes interconnected by links as an abstract model of a network, the small-world characteristic is observed when any two nodes of a large network are separated by only a small number of links [1]. This small-world feature has been found in a wide range of self-organized complex systems, including neuron networks in the brain, the world wide web and ecological networks [1]. The small-world nature of the osteocyte network is manifested by the presence of a relatively small number of central cells that work as hubs to which many canaliculi are connected (Figure 9b, c).
By enabling the connection of distant nodes via a small number of links, small-world networks facilitate communication while minimizing the energy and costs associated with information processing. In bone, this allows for the supply of nutrients to osteocytes and the transmission of mechano-transduced signals to BMUs with minimum energy input [64]. The expected gain in transmission speed through the osteocyte network has been estimated for two types of skeletal tissues: the fibrolamellar bone of sheep and the woven bone of mouse. The more ordered structure of fibrolamellar bone leads to a stronger small-world character, which is expected to accelerate the transportation of nutrients by a factor of 10 compared to the bone matrix without a network. This estimate assumes that diffusion via the canaliculae is 100 times faster than in the bone matrix (v network /v matrix = 100). Likewise, the less organized woven architecture leads to a 6-fold increase in transport speed relative to the matrix (Figure 9d). Compared to other complex networks, the osteocyte network displays a small-world character that lies in between that of the world wide web and the neural network of the model worm C. elegans (Figure 9e). This network connectivity correlates well with the intermediate size of the osteocyte network in comparison to these other examples.
The network of osteocytes is expected to also show another universal feature of small-world networks, namely the trade-off between resilience and vulnerability. On the one hand, the network is resilient against the loss of random nodes with low interconnectivity, which constitute the vast majority in the network. On the other hand, the reliance of the network on a small quantity of essential hubs increases its vulnerability to targeted attacks. The possible effect of this trade-off on the longterm mechanical stability of bone and its behavior under the influence of medication or disease is an open question Figure 10: Tutorial on bone mechanotransduction [3,[66][67][68][69]. Images are adapted and reproduced with permission from Robling et al. [3] and Papachroni et al. [67]. of direct relevance for the understanding of this complex material and for the possible identification of better protocols for the medical diagnosis and treatment of injured bone.

Conclusions
Bone is a fascinating material that combines the living nature of a biological tissue with mechanical properties that are quite unusual in view of its weak and soft chemical constituents. Based on the way information is processed during growth and maintenance of the tissue, bone can be understood as a complex system in which properties emerge from the interactions of individual cells rather than the top-down control by a central processing unit. This leads to universal features that are also found in other complex systems, such as self-organization, hierarchy and adaptive behavior. Selforganization is driven by the activities of living cells, which explore supramolecular, covalent and ionic bonds to assemble collagen molecules and hydroxyapatite platelets in remarkably ordered mineralized fibrils. The arrangement of such fibrils at multiple length scales results in hierarchical structures that are simultaneously stiff and fracture resistant, two antagonistic properties that are seldom found in engineering materials. Through the concerted action of living cells, the stiff and tough structure of bone is able to grow and adapt itself to mechanical stimuli from the environment. From an engineering perspective, this set of properties makes bone quite a remarkable material that serve as an inspiration for the development of future lightweight, sustainable and durable load-bearing structures. The recent finding that osteocytes interconnected throughout bone display network properties analogous to those found in neural networks shows how much there is still to be explored in this fascinating material. Further understanding of bone as a complex system will require expertise from multiple disciplines, which makes this material also an exciting platform for collaborative teaching and research.

Author Contributions
T.M and R.L have equally contributed to this manuscript and are therefore jointly first authors. All authors contributed to the conceptualization, drafting, discussion and revision of the manuscript. R.L. compiled information about collagen synthesis and bone mineralization models, discussed the underlying self-organization process and drafted the tutorial panel describing the supramolecular chemistry of collagen. T.M. reviewed the hierarchical structure of bone and the spectroscopic characterization of osteons, revisited the mechanical properties of bone and drafted the tutorial on fracture mechanics of materials. A.R.S. worked on the overall structure of the manuscript, reviewed the adaptive behavior and the remodeling processes of bone, and discussed the properties of the osteocyte network. A.K. wrote the tutorial panel about mechanotransduction and biochemical pathways in bone, and critically assessed the cellular mechanisms underlying bone remodeling.