Ontogenetic modulation of branch size,shape, and biomechanics produces diversity across habitats in the Bursera simaruba clade of tropical trees

Organismal size and shape inseparably interact with tissue biomechanical properties. It is therefore essential to understand how size, shape, and biomechanics interact in ontogeny to produce morphological diversity. We estimated within species branch length‐diameter allometries and reconstructed the rates of ontogenetic change along the stem in mechanical properties across the simaruba clade in the tropical tree genus Bursera, measuring 376 segments from 97 branches in nine species in neotropical dry to rain forest. In general, species with stiffer materials had longer, thinner branches, which became stiffer more quickly in ontogeny than their counterparts with more flexible materials. We found a trend from short stature and flexible tissues to tall statures and stiff tissues across an environmental gradient of increasing water availability, likely reflecting a water storage–mechanical support tradeoff. Ontogenetic variation in size, shape, and mechanics results in diversity of habits, for example, rapid length extension, sluggish diameter expansion, and flexible tissues results in a liana, as in Bursera instabilis. Even species of similar habit exhibited notable changes in tissue mechanical properties with increasing size, illustrating the inseparable relationship between organismal proportions and their tissue mechanics in the ontogeny and evolution of morphological diversity.     

Regulation of Cell Cytoskeleton and Membrane Mechanics by Electric Field: Role of Linker Proteins

Cellular mechanics is known to play an important role in the cell homeostasis including proliferation, motility, and differentiation. Significant variation in the mechanical properties between different cell types suggests that control of the cell metabolism is feasible through manipulation of the cell mechanical parameters using external physical stimuli. We investigated the electrocoupling mechanisms of cellular biomechanics modulation by an electrical stimulation in two mechanically distinct cell types—human mesenchymal stem cells and osteoblasts. Application of a 2 V/cm direct current electric field resulted in approximately a twofold decrease in the cell elasticity and depleted intracellular ATP. Reduction in the ATP level led to inhibition of the linker proteins that are known to physically couple the cell membrane and cytoskeleton. The membrane separation from the cytoskeleton was confirmed by up to a twofold increase in the membrane tether length that was extracted from the cell membrane after an electrical stimulation. In comparison to human mesenchymal stem cells, the membrane-cytoskeleton attachment in osteoblasts was much stronger but, in response to the same electrical stimulation, the membrane detachment from the cytoskeleton was found to be more pronounced. The observed effects mediated by an electric field are cell type- and serum-dependent and can potentially be used for electrically assisted cell manipulation. An in-depth understanding and control of the mechanisms to regulate cell mechanics by external physical stimulus (e.g., electric field) may have great implications for stem cell-based tissue engineering and regenerative medicine.     

Altered osteogenic commitment of human mesenchymal stem cells by ERM protein-dependent modulation of cellular biomechanics

Cellular mechanics is known to play an important role in many cellular functions including adhesion, migration, proliferation, and differentiation. Human mesenchymal stem cells (hMSCs) demonstrate unique mechanical properties distinct from fully differentiated cells. This observation suggests that the stem cell mechanics may be modulated to regulate the hMSCs' lineage commitment. Specifically, ERM (ezrin, radixin, moesin) proteins are known to mediate the membrane-cytoskeleton adhesion, cell elasticity, actin cytoskeleton organization, and therefore could serve as potential targets for modulation of the cellular mechanics. Combining silencing RNA, atomic force microscopy, and laser optical tweezers, the role of the ERM proteins involved in the regulation of stem cell biomechanics and osteogenic differentiation was quantitatively determined. Transient ERM knockdown by RNAi causes disassembly of actin stress fibers and focal adhesions, a decrease in the cell stiffness, and membrane separation from the cytoskeleton. The silencing RNA treatment not only induced mechanical changes in stem cells but impaired biochemically-directed osteogenic differentiation. The intact actin cytoskeleton and focal adhesions of hMSCs appear critical for the osteogenic induction. Thus, ERM knockdown modulates the dynamics of cell mechanical changes during hMSC differentiation and regulates the expression of tissue specific molecular markers. These findings are of particular interest for modulation of the cellular biomechanics to control hMSCs' activities and fate in tissue engineering, regenerative medicine, and other stem cell-based therapeutic applications.     

Wind-induced stresses in cherry trees: evidence against the hypothesis of constant stress levels

We calculated the wind-induced bending moments and stresses generated in the stems of five Prunus serotina conspecifics differing in height and canopy shape and size (based on detailed measurements of stem projected area and location with respect to ground level) to test the hypothesis that wind-loads generate uniform and constant stress levels along the lengths of tree twigs, branches, and trunks. These calculations were performed using five different wind speed profiles to evaluate the relative importance of the shape of wind speed profiles versus the ’geometry’ of tree shape on stem stress distributions and magnitudes. Additionally, we evaluated the effect of absolute tree size and stem taper on wind- induced stresses by scaling the size of smaller conspecifics to the absolute height of the largest of the five trees yet retaining the original stem proportions (i.e., diameter relative to stem length) for each plant. Finally, we also determined how the factor of safety for wind-loading (i.e., the quotient of stem yield stress and wind-load stress) changed as a function of tree size (and, presumably, age). Our results indicate that wind-load stress levels (1) vary along stem length even for the same wind speed profile and the same maximum wind speed; (2) would increase to dangerous levels with increasing tree height if it were not for ontogenetic changes in stem taper and canopy shape that reduce stress intensities to manageable levels; (3) tend to be more dependent on stem taper and canopy shape and size than on the shape of the wind speed profile; and (4) the factor of safety against wind-induced mechanical failure decreases as trees get larger, but varies along the length of large trees such that preferential stem failure is likely and functionally adaptive. We thus (1) reject the hypothesis of constant wind-induced stress levels; (2) support the view that size-dependent changes in stem taper are required to maintain wind-load mechanical reliability; and (3) suggest that certain portions of mature trees are ’designed’ to fail under high winds speeds, thereby reducing drag and the bending moments and stresses experienced by trunks. Received: 24 May 1999 / Accepted: 8 October 1999     

On the economy and safety of hollow non-septate peduncles

We examined the allometry and mechanical properties of the peduncles of Hieracium pilosella and other species in the Asteraceae (H. aurantiacum, Taraxicum officinale, Tragopogon pratensi) to evaluate the hypothesis that tapered, tubular, and non-septate peduncles optimize the trade-offs among stem biomass allocation, elevating flowers and thus their wind-dispersed fruits, and the requirement for a factor of safety against mechanical failure. This hypothesis was evaluated by comparing peduncle morphometry (e.g., biomass M(s), floral biomass M(f), and length L) and mechanical properties (e.g., bending rigidity EI) for populations growing in windy and wind-sheltered sites as well as transplants between sites. Regardless of ambient wind speeds, M(s) ∝ L(4/3) and EI ∝ L(11/4) ∝ M(s)(5/3), whereas M(f)?/M(s) ∝ L(-1.15), i.e., peduncles disproportionately increase in their biomass as they increase in length, but mechanically support a disproportionately smaller floral biomass relative to their biomass. Calculations show that the tall peduncles from wind-sheltered sites have a larger fruit dispersal range and a lower factor of safety than the shorter peduncles produced in open sites. These and other observations are interpreted to indicate that tubular peduncles enhance relative fitness in terms of propagule dispersal (but not propagule number per stem) while maintaining a sufficient factor of safety against mechanical failure.     

Trubs,but no trianas: filled and empty regions of angiosperm stem length‐diameter‐mechanics space

Discrete plant habit categories such as ‘tree’, ‘shrub’, and ‘liana’ belie continuous variation in nature. To study the evolution of this continuous variation, we gathered data on stem length, diameter and tissue mechanical stiffness across a highly morphologically diverse highland xerophytic scrub on a lava flow in central Mexico. With stem allometric and mechanical data from 1216 segments from 50 species, we examined relationships between stem length–diameter proportions and tissue mechanical stiffness using linear mixed‐effects models. Rather than a series of discrete clouds in stem length–diameter–tissue stiffness space, corresponding to traditional habit categories, the plants of this xerophytic scrub formed a single continuous one. Within this cloud, self‐supporting plants had stems that became predictably longer and tissues that became stiffer for a given diameter increase, and there was no paucity of intermediates between trees and shrubs (‘trubs’). Non self‐supporting plants had a steeper stem length–diameter slope and their tissues did not increase in stiffness with stem size. The area between self‐ and non self‐supporting plants was sparsely occupied as stem size increased. We predict that this ‘empty’ space between lianas and trees is developmentally accessible but of low fitness, meaning that there should be few ‘trianas’ in nature. © 2015 The Linnean Society of London, Botanical Journal of the Linnean Society, 2015, 179 , 361–373.     

Dental development in Megaladapis edwardsi (Primates, Lemuriformes): implications for understanding life history variation in subfossil lemurs

Teeth grow incrementally and preserve within them a record of that incremental growth in the form of microscopic growth lines. Studying dental development in extinct and extant primates, and its relationship to adult brain and body size as well as other life history and ecological parameters (e.g., diet, somatic growth rates, gestation length, age at weaning), holds the potential to yield unparalleled insights into the life history profiles of fossil primates. Here, we address the absolute pace of dental development in Megaladapis edwardsi, a giant extinct lemur of Madagascar. By examining the microstructure of the first and developing second molars in a juvenile individual, we establish a chronology of molar crown development for this specimen (M1 CFT = 1.04 years; M2 CFT = 1.42 years) and determine its age at death (1.39 years). Microstructural data on prenatal M1 crown formation time allow us to calculate a minimum gestation length of 0.54 years for this species. Postnatal crown and root formation data allow us to estimate its age at M1 emergence (approximately 0.9 years) and to establish a minimum age for M2 emergence (>1.39 years). Finally, using reconstructions or estimates (drawn elsewhere) of adult body mass, brain size, and diet in Megaladapis, as well as the eruption sequence of its permanent teeth, we explore the efficacy of these variables in predicting the absolute pace of dental development in this fossil species. We test competing explanations of variation in crown formation timing across the order Primates. Brain size is the best single predictor of crown formation time in primates, but other variables help to explain the variation.     

Biomechanical model of human cornea based on stromal microstructure

The optical characteristics of the human cornea depends on the mechanical balance between the intra-ocular pressure and intrinsic tissue stiffness. A wide range of ophthalmic surgical procedures alter corneal biomechanics to induce local or global curvature changes for the correction of visual acuity. Due to the large number of surgical interventions performed every day, a deeper understanding of corneal biomechanics is needed to improve the safety of these procedures and medical devices. The aim of this study is to propose a biomechanical model of the human cornea, based on stromal microstructure. The constitutive mechanical law includes collagen fiber distribution based on X-ray scattering analysis, collagen cross-linking, and fiber uncrimping. Our results showed that the proposed model reproduced inflation and extensiometry experimental data [Elsheikh et al., Curr. Eye Res., 2007; Elsheikh et al., Exp. Eye Res., 2008] successfully. The mechanical properties obtained for different age groups demonstrated an increase in collagen cross-linking for older specimens. In future work such a model could be used to simulate non-symmetric interventions, and provide better surgical planning.     

An Approach to the Biomechanics of the Masticatory Apparatus of Early Miocene (Santacrucian Age) South American Ungulates (Astrapotheria, Litopterna, and Notoungulata): Moment Arm Estimation Based on 3D Landmarks

Notoungulates, litopterns, and astrapotheres are among the most representative mammals of the early Miocene Santacrucian Age. They comprise a diversity of biological types and sizes, from small forms, comparable to rodents, to giants with no analogues in modern faunas. Traditionally, all of them have been considered herbivores; this diversity is reflected in different morphologies of the masticatory apparatus, suggesting a variety of feeding habits. The application of biomechanics to the study of fossil mammals is a good approach to test functional hypotheses. Jaws act as a lever system, with the pivot at the temporomandibular joint, with masticatory muscles providing the input force, whereas the output force is produced by the teeth on food. The moment arms of the lines of action of the muscles can be estimated to analyze relationships between bite force and bite velocity. A morphogeometric approach inspired by Vizcaíno et al. (1998) is applied to estimate muscle moment arms in a static 3D bite model based on three-dimensional landmarks and semilandmarks on crania with mandibles in occlusion. This new 3D geometric method to evaluate jaw mechanics demonstrated its reliability when applied to a control sample of extant mammals that included carnivores, herbivores, and omnivores. Our results indicate that, except for Pachyrukhos, in no Santacrucian ungulate does the masseter muscle have greater mechanical advantage than the temporalis. Among them, notoungulates have a better configuration to develop force on the molar tooth row than litopterns. This indicates a diet richer in tough plant materials for Santacrucian notoungulates (e.g., grass or even bark) than for litopterns (e.g., dicots). This is consistent with recent ecomorphological approaches applied to this fauna. Finally, the approach proposed here proves to be useful for comparing masticatory performance and it is a powerful tool to validate ecomorphological dietary hypotheses in fossil taxa.     

Universal foliage-stem scaling across environments and species in dicot trees: plasticity, biomechanics and Corner's Rules

Trees range from small-leaved, intricately branched species with slender stems to large-leaved, coarsely branched ones with thick stems. We suggest a mechanism for this pattern, known as Corner's Rules, based on universal scaling. We show similar crown area–stem diameter scaling between trunks and branches, environments, and species spanning a wide range of leaf size and stem biomechanics. If crown and stem maintain metabolically driven proportionality, but similar amounts of photosynthates are produced per unit crown area, then the greater leaf spacing in large-leaved species requires lower density stem tissue and, meeting mechanical needs, thicker stems. Congruent with this scenario, we show a negative relationship between leaf size and stem Young's modulus. Corner's Rules emerge from these mutual adjustments, which suggest that adaptive studies cannot consider any of these features independently. The constancy of scaling despite environmental challenges identifies this trait constellation as a crucial axis of plant diversification.     

Safety and streamlining of woody shoots in wind: an empirical study across 39 species in tropical Australia

? Wind is a key mechanical stress for woody plants, so how do shoot traits affect performance in wind? ? We used a vehicle mounted apparatus to measure drag, streamlining and mechanical safety in 127 vertical lead-shoots, 1.2 m long, across 39 species in tropical Australia. ? Shoot dimensions and stem tissue properties were closely coupled so that shoots with low stem specific gravity or larger projected area had thicker stems. Thicker stems provide larger second moment of area (I), which increased shoot safety and bending stiffness but impeded shoot reconfiguration in strong winds, including frontal area reduction. Nonetheless, increasing I also improved streamlining. Streamlining was unrelated to traits except I. Stem tissue material properties only had small effects. Higher modulus of rupture increased shoot safety and higher Young's modulus impeded shoot reconfiguration. ? We found no conflict between bending stiffness and streamlining for woody shoots. Stiffness might help streamlining by increasing damping and stability, thereby reducing flagging in wind. Tissue-level traits did influence shoot-level mechanical safety and behaviour, but shoot geometry was much more important. Variable shoot and stem traits, which all influenced shoot biomechanics, were integrated in shoots to yield a relatively narrow range of outcomes in wind.     

Stem biomechanics of three columnar cacti from the Sonoran Desert

The allometric relationship of stem length L with respect to mean stem diameter D was determined for 80 shoots of each of three columnar cactus species (Stenocereus thurberi, Lophocereus schottii, and S. gummosus) to determine whether this relationship accords with that predicted by each of three contending models purporting to describe the mechanical architecture of vertical shoots (i.e., geometric, stress, and elastic similitude, which predict L proportional to D(alpha), with alpha = 1/1, 1/2, and 2/3, respectively). In addition, anatomical, physical, and biomechanical stem properties were measured to determine how the stems of these three species maintain their elastic stability as they increase in size. Reduced major axis regression of L with respect to D showed that alpha = 2.82 ± 0.14 for S. thurberi, 2.32 ± 0.19 for L. schottii, and 4.21 ± 0.31 for S. gummosus. Thus, the scaling exponents for the allometry of L differed significantly from that predicted by each of the three biomechanical models. In contrast, these exponents were similar to that for the allometry previously reported for saguaro. Analyses of biomechanical data derived from bending tests performed on 30 stems selected from each of the three species indicated that the bulk stem tissue stiffness was roughly proportional to L2, while stem flexural rigidity (i.e., the ability to resist a bending force) scaled roughly as L3. Stem length was significantly and positively correlated with the volume fraction of wood, while regression analysis of the pooled data from the three species (i.e., 90 stems) indicated that bulk tissue stiffness scaled roughly as the 5/3-power of the volume fraction of wood in stems. These data were interpreted to indicate that wood served as the major stiffening agent in stems and that this tissue accumulates at a sufficient rate to afford unusually high scaling exponents tot stem length with respect to stem diameter (i.e., disproportionately large increments of stem length with respect to increments in stem diameter). Nevertheless, the safety factor against the elastic failure of stems (computed on the basis of the critical buckling height divided by actual stem length) decreased with increasing stem size tot each species, even though each species maintained an average safety factor equal to two. We speculate that the apparent upper limit to plant height calculated for each species may serve as a biomechanical mechanism for vegetative propagation and the establishment of dense plant colonies by means of extreme stem flexure and ultimate breakage, especially for S. gummosus.     

太白山红桦种群结构与空间分析

 为了揭示太白山红桦(Betula albo-sinensis)种群的维持机制, 结合径级结构、静态生命表和存活曲线, 对太白山红桦种群结构进行了研究; 并应用单变量、双变量K函数对不同径级立木与残干的空间格局、空间关联性进行了多尺度分析。研究显示, 太白山红桦种群在幼苗幼树阶段死亡率较高, 进入中龄期后死亡率有所降低, 随着年龄增长, 死亡率逐渐增加, 直至年龄极限。其存活曲线基本接近Deevey I型(曲线凸型)。其个体死亡方式主要为掘根和折干(由大雪所致), 能够为种群更新提供潜在空间。除海拔2 250 m立木整体格局呈集群分布外, 该地区红桦立木与残干在不同尺度下的整体格局均为随机分布。不同发育阶段的个体均呈集群分布, 表现为斑块状同龄群。在红桦更新所需的最适林窗大小尺度上, 红桦立木与残干的空间关联性为显著正相关。以上结果表明: 太白山红桦种群具有同龄集群生长的特性, 种群由时空上不连续的局部斑块同龄个体组成, 其更新有赖于以掘根及折干方式死亡个体所形成的林窗斑块, 这种以局部林窗斑块更新维持种群整体稳定的特性, 可能是其长期稳定存在的重要机制。     

Using optical full-field measurement based on digital image correlation to measure strain on a tree subjected to mechanical load

Key message

The results indicate the usability of DIC technique in tree biomechanics is advantageous compared to standard procedure since it reveals the induced strain in full-field way, accurately and noninvasively.

Abstract

The goal of this study was to analyze and measure the displacement and strain response behavior of the tree (Juglans regia L.) subjected to a mechanical load. The analysis addresses issues of tree biomechanics that call for a high-quality deformation data measured on trees’ surfaces in a noninvasive way. For these purposes, the 3D digital image correlation (3D-DIC) was proposed and tested against standard extensometers. The measured tree was subjected to a bending by a pulling test to induce strain on the tree stem. The DIC technique successfully provided strain field on a tree bark despite its highly complicated geometry. Vertical averaging of the strain field obtained by the DIC revealed that a tree stem behaves according to beam theory exhibiting compression and tension parts. The absolute values of strain measured by both techniques agree with each other in order of magnitude, DIC returning lower values (approximately 21.1 and 40.8 % for compression and tension part, respectively). The results reveal necessity for future study of stress/strain transfer from xylem to bark. The sensitivity analysis shows that the computed strain highly depends on subset size used in the DIC computation. Based on the projection error and strain, the optimal subset size was found out to be between 21 and 31 pixels. The measurement proved that the DIC method can be successfully used in tree biomechanics. In general, our results and the character of DIC technique that allows testing at different scales depending on imaging methods indicate its big potential in plant biomechanics.     

干细胞发育的生物力学调控

生物力学是采用力学方法对生物系统的结构和功能进行的研究,与生物化学信号一起是调节胚胎发育、干细胞发育分化和组织器官形成的重要因素。近年来,随着学科交叉的深入,生物力学因素越来越受到研究者的重视。目前的研究表明：在心血管和造血系统,血流产生的流体剪切力对于血管内皮和造血细胞的发育分化至关重要;此外,对于广泛研究的间充质干细胞,由细胞外基质物理特性诱导的细胞张力对于干细胞功能及其向不同子代细胞的分化也扮演了重要的角色;而在肝脏等上皮组织来源的器官,也有研究提示生物力学因素,如基质弹性等在疾病的发生发展过程中起到了不可忽视的作用。总之,在干细胞发育分化过程中,生物力学调控与生物化学信号通路怎样协同发挥作用将成为今后研究的重点。     

Mechanical adaptations of cleavers (Galium aparine)

BACKGROUND AND AIMS: Cleavers (Galium aparine) is a fast-growing herbaceous annual with a semi-self-supporting, scrambling-ascending growth habit. Mature plants often use upright species for support. It is common in hedgerows and on waste ground. This study aims to characterize the mechanical behaviour of the stem and roots of cleavers and relate this to the arrangement of structural tissue, the net microfibrillar orientations in the cell walls, and plant growth habit. METHODS: The morphology and mechanics of mature cleavers was investigated using plants grown in pots and ones collected from the grounds at the University of Lincoln, Lincoln, UK. Tensile tests were carried out on the stem and the basal section of the first-order lateral roots. The net orientation of cellulose microfibrils in the cell walls was investigated using polarized light microscopy. KEY RESULTS: Results show that the basal regions of the stem and first-order lateral roots were highly extensible. Breaking strains of 24 +/- 7% were recorded for the stem base and 28 +/- 6% for the roots. Anatomical observations showed that the lower stem (base + 100 mm) was circular in cross-section with a solid central core of vascular tissue, whereas further up the stem the transverse section showed a typical four-angled shape with a ring-like arrangement of vascular tissue and sclerenchyma bundles in the corners. The net orientation of wall microfibrils in the secondary xylem diverges from the longitudinal by between 8 and 9 degrees . CONCLUSIONS: The basal region of the stem of cleavers is highly extensible, but the mechanism by which the stem is able to withstand such high breaking strains is unclear; reorientation of the cellulose fibrils in the stem along the axis of loading is not thought to be responsible.     

Stem and wood allometric relationships in Cacteae (Cactaceae)

Allometric relationships in organisms are considered a universal phenomenon. A positive scaling has been reported between stem size and cellular size of tracheary elements in wood of different vascular plants, but few studies have been carried out in slow-growing succulent plants. The aim of this study was to evaluate if a relationship exists between size, growth form and wood cell size among individual species of Cacteae. Forty-four species belonging to 16 genera of the tribe Cacteae with differing growth forms and sizes were studied. When analyzing plant size, we found a positive allometric scaling and the larger-sized species showing a higher percentage of succulent tissue and less accumulation of wood tissue. The positive scaling found between plant size (height and diameter) and vessel elements and fiber length support the universality of the allometric relationship proposed for other vascular plants with non-succulent stems. Notably, wide-band tracheids do not scale with plant size or growth form. Succulence associated with narrow vessel elements with distinctive helical secondary walls and wide-band tracheids suggest they are the key adaptations to tolerate drought and provide support to the stems of most taxa in Cacteae. Fibers do not have the primary role of giving mechanical support; therefore, we assume the scarce fibers in clusters represent reaction wood that, along with the fundamental tissue, maintains the vertical position and shape of those species growing in rocky cracks. Our results with species having short succulent stems support the universal theory of positive allometric scaling of vascular plants.     

Age, growth and reproduction of the Caspian roach (Rutilus rutilus caspicus) in the Anzali and Gomishan wetlands, North Iran

The hypothesis of environmentally related life history variations between two Rutilus rutilus caspicus populations was investigated by comparing Gomishan and Anzali populations along the southern shores of the Caspian Sea. The roach growth rate in the Anzali wetland was higher than in the Gomishan wetland. There were significant differences in growth patterns between males and females from both wetlands, however the sex ratios were not significantly different. Fecundity or reproductive investment (Gonadosomatic index – GSI) of the Gomishan roach was significantly more than the Anzali roach. In both areas the absolute fecundity was significantly positive correlated to age and total length.     

Computational simulation of fracture healing: influence of interfragmentary movement on the callus growth

Bone fractures heal through a complex process involving several cellular events. This healing process can serve to study factors that control tissue growth and differentiation from mesenchymal stem cells. The mechanical environment at the fracture site is one of the factors influencing the healing process and controls size and differentiation patterns in the newly formed tissue. Mathematical models can be useful to unravel the complex relation between mechanical environment and tissue formation. In this study, we present a mathematical model that predicts tissue growth and differentiation patterns from local mechanical signals. Our aim was to investigate whether mechanical stimuli, through their influence on stem cell proliferation and chondrocyte hypertrophy, predict characteristic features of callus size and geometry. We found that the model predicted several geometric features of fracture calluses. For instance, callus size was predicted to increase with increasing movement. Also, increases in size were predicted to occur through increase in callus diameter but not callus length. These features agree with experimental observations. In addition, spatial and temporal tissue differentiation patterns were in qualitative agreement with well-known experimental results. We therefore conclude that local mechanical signals can probably explain the shape and size of fracture calluses.