Torsional Behavior of Axonal Microtubule Bundles

Axonal microtubule (MT) bundles crosslinked by microtubule-associated protein (MAP) tau are responsible for vital biological functions such as maintaining mechanical integrity and shape of the axon as well as facilitating axonal transport. Breaking and twisting of MTs have been previously observed in damaged undulated axons. Such breaking and twisting of MTs is suggested to cause axonal swellings that lead to axonal degeneration, which is known as “diffuse axonal injury”. In particular, overstretching and torsion of axons can potentially damage the axonal cytoskeleton. Following our previous studies on mechanical response of axonal MT bundles under uniaxial tension and compression, this work seeks to characterize the mechanical behavior of MT bundles under pure torsion as well as a combination of torsional and tensile loads using a coarse-grained computational model. In the case of pure torsion, a competition between MAP tau tensile and MT bending energies is observed. After three turns, a transition occurs in the mechanical behavior of the bundle that is characterized by its diameter shrinkage. Furthermore, crosslink spacing is shown to considerably influence the mechanical response, with larger MAP tau spacing resulting in a higher rate of turns. Therefore, MAP tau crosslinking of MT filaments protects the bundle from excessive deformation. Simultaneous application of torsion and tension on MT bundles is shown to accelerate bundle failure, compared to pure tension experiments. MAP tau proteins fail in clusters of 10–100 elements located at the discontinuities or the ends of MT filaments. This failure occurs in a stepwise fashion, implying gradual accumulation of elastic tensile energy in crosslinks followed by rupture. Failure of large groups of interconnecting MAP tau proteins leads to detachment of MT filaments from the bundle near discontinuities. This study highlights the importance of torsional loading in axonal damage after traumatic brain injury.     

Microtubule Polymerization and Cross-Link Dynamics Explain Axonal Stiffness and Damage

Axonal damage is a critical indicator for traumatic effects of physical impact to the brain. However, the precise mechanisms of axonal damage are still unclear. Here, we establish a mechanistic and highly dynamic model of the axon to explore the evolution of damage in response to physical forces. Our axon model consists of a bundle of dynamically polymerizing and depolymerizing microtubules connected by dynamically detaching and reattaching cross-links. Although the probability of cross-link attachment depends exclusively on thermal fluctuations, the probability of detachment increases in the presence of physical forces. We systematically probe the landscape of axonal stretch and stretch rate and characterize the overall axonal force, stiffness, and damage as a direct result of the interplay between microtubule and cross-link dynamics. Our simulations reveal that slow loading is dominated by cross-link dynamics, a net reduction of cross-links, and a gradual accumulation of damage, whereas fast loading is dominated by cross-link deformations, a rapid increase in stretch, and an immediate risk of rupture. Microtubule polymerization and depolymerization decrease the overall axonal stiffness, but do not affect the evolution of damage at timescales relevant to axonal failure. Our study explains different failure mechanisms in the axon as emergent properties of microtubule polymerization, cross-link dynamics, and physical forces. We anticipate that our model will provide insight into causal relations by which molecular mechanisms determine the timeline and severity of axon damage after a physical impact to the brain.     

Geometrical nonlinear elasticity of axon under tension: A coarse-grained computational study

Axon bundles cross-linked by microtubule (MT) associate proteins and bounded by a shell skeleton are critical for normal function of neurons. Understanding effects of the complexly geometrical parameters on their mechanical properties can help gain a biomechanical perspective on the neurological functions of axons and thus brain disorders caused by the structural failure of axons. Here, the tensile mechanical properties of MT bundles cross-linked by tau proteins are investigated by systematically tuning MT length, axonal cross-section radius, and tau protein spacing in a bead-spring coarse-grained model. Our results indicate that the stress-strain curves of axons can be divided into two regimes, a nonlinear elastic regime dominated by rigid-body like inter-MT sliding, and a linear elastic regime dominated by affine deformation of both tau proteins and MTs. From the energetic analyses, first, the tau proteins dominate the mechanical performance of axons under tension. In the nonlinear regime, tau proteins undergo a rigid-body like rotating motion rather than elongating, whereas in the nonlinear elastic regime, tau proteins undergo a flexible elongating deformation along the MT axis. Second, as the average spacing between adjacent tau proteins along the MT axial direction increases from 25 to 125 nm, the Young’s modulus of axon experiences a linear decrease whereas with the average space varying from 125 to 175 nm, and later reaches a plateau value with a stable fluctuation. Third, the increment of the cross-section radius of the MT bundle leads to a decrease in Young’s modulus of axon, which is possibly attributed to the decrease in MT numbers per cross section. Overall, our research findings offer a new perspective into understanding the effects of geometrical parameters on the mechanics of MT bundles as well as serving as a theoretical basis for the development of artificial MT complexes potentially toward medical applications.     

Mechanical Effects of Dynamic Binding between Tau Proteins on Microtubules during Axonal Injury

The viscoelastic nature of axons plays a key role in their selective vulnerability to damage in traumatic brain injury (TBI). Experimental studies have shown that although axons can tolerate 100% strain under slow loading rates, even strain as small as 5% can rupture microtubules (MTs) during the fast loading velocities relevant to TBI. Here, we developed a computational model to examine rate-dependent behavior related to dynamic interactions between MTs and the MT-associated protein tau under varying strains and strain rates. In the model, inverted pairs of tau proteins can dynamically cross-link parallel MTs via the respective MT-binding domain of each tau. The model also incorporates realistic thermodynamic breaking and reformation of the bonds between the connected tau proteins as they respond to mechanical stretch. With simulated stretch of the axon, the model shows that despite the highly dynamic nature of binding and unbinding events, under fast loading rates relevant to TBI, large tensile forces can be transmitted to the MTs that can lead to mechanical rupture of the MT cylinder, in agreement with experimental observations and as inferred in human TBI. In contrast, at slow loading rates, the progressive breaking and reformation of the bonds between the tau proteins facilitate the extension of axons up to ∼100% strain without any microstructural damage. The model also predicts that under fast loading rates, individual MTs detach from MT bundles via sequential breaking of the tau-tau bonds. Finally, the model demonstrates that longer MTs are more susceptible to mechanical rupture, whereas short MTs are more prone to detachment from the MT bundle, leading to disintegration of the axonal MT ultrastructure. Notably, the predictions from the model are in excellent agreement with the findings of the recent in vitro mechanical testing of micropatterned neuronal cultures.     

Viscoelasticity of Tau Proteins Leads to Strain Rate-Dependent Breaking of?Microtubules during Axonal Stretch Injury: Predictions from a Mathematical Model

The unique viscoelastic nature of axons is thought to underlie selective vulnerability to damage during traumatic brain injury. In particular, dynamic loading of axons has been shown to mechanically break microtubules at the time of injury. However, the mechanism of this rate-dependent response has remained elusive. Here, we present a microstructural model of the axonal cytoskeleton to quantitatively elucidate the interaction between microtubules and tau proteins under mechanical loading. Mirroring the axon ultrastructure, the microtubules were arranged in staggered arrays, cross-linked by tau proteins. We found that the viscoelastic behavior specifically of tau proteins leads to mechanical breaking of microtubules at high strain rates, whereas extension of tau allows for reversible sliding of microtubules without any damage at small strain rates. Based on the stiffness and viscosity of tau proteins inferred from single-molecule force spectroscopy studies, we predict the critical strain rate for microtubule breaking to be in the range 22–44 s⁻¹, in excellent agreement with recent experiments on dynamic loading of micropatterned neuronal cultures. We also identified a characteristic length scale for load transfer that depends on microstructural properties and have derived a phase diagram in the parameter space spanned by loading rate and microtubule length that demarcates those regions where axons can be loaded and unloaded reversibly and those where axons are injured due to breaking of the microtubules.     

Viscoelasticity of Tau Proteins Leads to Strain Rate-Dependent Breaking of Microtubules during Axonal Stretch Injury: Predictions from a Mathematical Model

The unique viscoelastic nature of axons is thought to underlie selective vulnerability to damage during traumatic brain injury. In particular, dynamic loading of axons has been shown to mechanically break microtubules at the time of injury. However, the mechanism of this rate-dependent response has remained elusive. Here, we present a microstructural model of the axonal cytoskeleton to quantitatively elucidate the interaction between microtubules and tau proteins under mechanical loading. Mirroring the axon ultrastructure, the microtubules were arranged in staggered arrays, cross-linked by tau proteins. We found that the viscoelastic behavior specifically of tau proteins leads to mechanical breaking of microtubules at high strain rates, whereas extension of tau allows for reversible sliding of microtubules without any damage at small strain rates. Based on the stiffness and viscosity of tau proteins inferred from single-molecule force spectroscopy studies, we predict the critical strain rate for microtubule breaking to be in the range 22–44 s⁻¹, in excellent agreement with recent experiments on dynamic loading of micropatterned neuronal cultures. We also identified a characteristic length scale for load transfer that depends on microstructural properties and have derived a phase diagram in the parameter space spanned by loading rate and microtubule length that demarcates those regions where axons can be loaded and unloaded reversibly and those where axons are injured due to breaking of the microtubules.     

Buckling Behavior of Individual and Bundled Microtubules

As the major structural constituent of the cytoskeleton, microtubules (MTs) serve a variety of biological functions that range from facilitating organelle transport to maintaining the mechanical integrity of the cell. Neuronal MTs exhibit a distinct configuration, hexagonally packed bundles of MT filaments, interconnected by MT-associated protein (MAP) tau. Building on our previous work on mechanical response of axonal MT bundles under uniaxial tension, this study is focused on exploring the compression scenarios. Intracellular MTs carry a large fraction of the compressive loads sensed by the cell and therefore, like any other column-like structure, are prone to substantial bending and buckling. Various biological activities, e.g., actomyosin contractility and many pathological conditions are driven or followed by bending, looping, and buckling of MT filaments. The coarse-grained model previously developed in our lab has been used to study the mechanical behavior of individual and bundled in vivo MT filaments under uniaxial compression. Both configurations show tip-localized, decaying, and short-wavelength buckling. This behavior highlights the role of the surrounding cytoplasm and MAP tau on MT buckling behavior, which allows MT filaments to bear much larger compressive forces. It is observed that MAP tau interconnections improve this effect by a factor of two. The enhanced ability of MT bundles to damp buckling waves relative to individual MT filaments, may be interpreted as a self-defense mechanism because it helps axonal MTs to endure harsher environments while maintaining their function. The results indicate that MT filaments in a bundle do not buckle simultaneously implying that the applied stress is not equally shared among the MT filaments, that is a consequence of the nonuniform distribution of MAP tau proteins along the bundle length. Furthermore, from a pathological perspective, it is observed that axonal MT bundles are more vulnerable to failure in compression than tension.     

Micromechanical analysis of brain’s diffuse axonal injury

Computational models are important tools which help researchers understand traumatic brain injury (TBI). A mechanistic multi-scale numerical approach is introduced to quantify diffuse axonal injury (DAI), the most important mechanism of TBI, induced by a mechanical insult at micro-scale regions of the white matter or voxels where fiber orientations are the same. Using the mechanical properties of a single axon with a viscoelastic constitutive relation and its functional failure in terms of electrophysiological impairment, a numerical 2D micro-level lattice method is implemented to directly analyze the percentage of injured axons in a voxel containing a bundle of axons all with the same orientation under biaxial stretches. Reference micro-injury maps are then developed with the input parameters based on the principal strain or stretch values and their direction with respect to axons, which provide the percentage of injured axons in the voxel of interest as the output. The methodology is independent of any statistical analyses of the accident data and medical reports to derive probabilistic injury risk curves for DAI. Avoiding a structurally detailed full finite element head model, this study proposes a micro-mechanical approach which considers the anatomical structure of neural axons in the white matter together with their mechanical properties using a numerical lattice method to analyze the brain’s diffuse axonal injury. This work has the potential to help develop safer prevention tools and more effective diagnosis methods for DAI.     

Polarity orientation and assembly process of microtubule bundles in nocodazole-treated, MAP2c-transfected COS cells.

Microtubule bundles reminiscent of those found in neuronal processes are formed in fibroblasts and Sf9 cells that are transfected with the microtubule-associated proteins tau, MAP2, or MAP2c. To analyze the assembly process of these bundles and its relation to the microtubule polarity, we depolymerized the bundles formed in MAP2c-transfected COS cells using nocodazole, and observed the process of assembly of microtubule bundles after removal of the drug in cells microinjected with rhodamine-labeled tubulin. Within minutes of its removal, numerous short microtubule fragments were observed throughout the cytoplasm. These short fragments were randomly oriented and were already bundled. Somewhat longer, but still short bundles, were then found in the peripheral cytoplasm. These bundles became the primordium of the larger bundles, and gradually grew in length and width. The polarity orientation of microtubules in the reformed bundle as determined by "hook" procedure using electron microscope was uniform with the plus end distal to the cell nucleus. The results suggest that some mechanism(s) exists to orient the polarity of microtubules, which are not in direct continuity with the centrosome, during the formation of large bundles. The observed process presents a useful model system for studying the organization of microtubules that are not directly associated with the centrosomes, such as those observed in axons.     

Atomic force microscopy reveals important differences in axonal resistance to injury

Axonal degeneration after traumatic brain injury and nerve compression is considered a common underlying cause of temporary as well as permanent disability. Because a proper functioning of neural network requires phase coherence of all components, even subtle changes in circuitry may lead to network failure. However, it is still not possible to determine which axons will recover or degenerate after injury. Several groups have studied the pressure threshold for axonal injury within a nerve, but difficulty accessing the injured region; insufficient imaging methods and the extremely small dimensions involved have prevented the evaluation of the response of individual axons to injury. We combined microfluidics with atomic force microscopy and in vivo imaging to estimate the threshold force required to 1), uncouple axonal transport without impairing axonal survival, and 2), compromise axonal survival in both individual and bundled axons. We found that rat hippocampal axons completely recover axonal transport with no detectable axonal loss when compressed with pressures up to 65 ± 30 Pa for 10 min, while dorsal root ganglia axons can resist to pressures up to 540 ± 220 Pa. We investigated the reasons for the differential susceptibility of hippocampal and DRG axons to mechanical injury and estimated the elasticity of live axons. We found that dorsal root ganglia axons have a 20% lower elastic modulus than hippocampal axons. Our results emphasize the importance of the integrity of the axonal cytoskeleton in deciding the axonal fate after damage and open up new avenues to improve injury diagnosis and to identify ways to protect axons.     

Association of actin filaments with axonal microtubule tracts

Axoplasmic organelles move on actin as well as microtubules in vitro and axons contain a large amount of actin, but little is known about the organization and distribution of actin filaments within the axon. Here we undertake to define the relationship of the microtubule bundles typically found in axons to actin filaments by applying three microscopic techniques: laser-scanning confocal microscopy of immuno-labeled squid axoplasm; electronmicroscopy of conventionally prepared thin sections; and electronmicroscopy of touch preparations-a thin layer of axoplasm transferred to a specimen grid and negatively stained. Light microscopy shows that longitudinal actin filaments are abundant and usually coincide with longitudinal microtubule bundles. Electron microscopy shows that microfilaments are interwoven with the longitudinal bundles of microtubules. These bundles maintain their integrity when neurofilaments are extracted. Some, though not all microfilaments decorate with the S1 fragment of myosin, and some also act as nucleation sites for polymerization of exogenous actin, and hence are definitively identified as actin filaments. These actin filaments range in minimum length from 0.5 to 1.5 µm with some at least as long as 3.5 µm. We conclude that the microtubule-based tracks for fast organelle transport also include actin filaments. These actin filaments are sufficiently long and abundant to be ancillary or supportive of fast transport along microtubules within bundles, or to extend transport outside of the bundle. These actin filaments could also be essential for maintaining the structural integrity of the microtubule bundles.     

Changes in the Pattern of Organization of Microtubules During Microspore Formation in Rice (Oryza sativa L.)

Changes in the pattern of organization of microtubules in the developing microspore of rice ( Oryza sativa L. ) have been followed using immunofluorescence confocal microscopy. At the microsporocyte stage of development the cell possessed a network of highly branched and thickened microtubule bundles. In the central cytoplasm numerous bundles mn circumferentially around the nucleus. From the circumferentially distributed microtubule bundle network some microtubule bundles radiated towards the conical region of the cell. The microsporocyte after Meiosis Ⅰ became a dyad. In the dyad cell microtubule bundles emanated radially from the nucleus. In the cortex of the dyad cell some of the microtubule bundles became randomly oriented. The dyad then underwent Meiosis Ⅱ to become tetrad. Microtubule bundles in the tetrad cell radiated from the nucleus. No randomly oriented microtubule bundles were present in the cortical region of the tetrad cell. Mter- wards the four cells that made up the tetrad dissociated from each other and each became a microspore. At the early stage of the microspore development most of the microtubule bundles were randomly distributed. Later, some of the microtubules converged towards a bud-like cytoplasmic protrusion. This bud-like protrusion later developed into a germ pore (or pollen pore). At the late stage of microspore formation, microtubule bundles became thinner and reticulately oriented to form a tightly knitted network.     

Deformation responses of a physically cross-linked high molecular weight elastin-like protein polymer

Recombinant protein polymers were synthesized and examined under various loading conditions to assess the mechanical stability and deformation responses of physically cross-linked, hydrated, protein polymer networks designed as triblock copolymers with central elastomeric and flanking plastic-like blocks. Uniaxial stress-strain properties, creep and stress relaxation behavior, as well as the effect of various mechanical preconditioning protocols on these responses were characterized. Significantly, we demonstrate for the first time that ABA triblock protein copolymers when redesigned with substantially larger endblock segments can withstand significantly greater loads. Furthermore, the presence of three distinct phases of deformation behavior was revealed upon subjecting physically cross-linked protein networks to step and cyclic loading protocols in which the magnitude of the imposed stress was incrementally increased over time. We speculate that these phases correspond to the stretch of polypeptide bonds, the conformational changes of polypeptide chains, and the disruption of physical cross-links. The capacity to select a genetically engineered protein polymer that is suitable for its intended application requires an appreciation of its viscoelastic characteristics and the capacity of both molecular structure and conditioning protocols to influence these properties.     

Interplay between Brownian motion and cross-linking controls bundling dynamics in actin networks

Morphology changes in cross-linked actin networks are important in cell motility, division, and cargo transport. Here, we study the transition from a weakly cross-linked network of actin filaments to a heavily cross-linked network of actin bundles through microscopic Brownian dynamics simulations. We show that this transition occurs in two stages: first, a composite bundle network of small and highly aligned bundles evolves from cross-linking of individual filaments and, second, small bundles coalesce into the clustered bundle state. We demonstrate that Brownian motion speeds up the first stage of this process at a faster rate than the second. We quantify the time to reach the composite bundle state and show that it strongly increases as the mesh size increases only when the concentration of cross-links is small and that it remains roughly constant if we decrease the relative ratio of cross-linkers as we increase the actin concentration. Finally, we examine the dependence of the bundling timescale on filament length, finding that shorter filaments bundle faster because they diffuse faster.     

Mechanics of microtubule bundles in pillar cells from the inner ear.

The mechanical properties of cross-linked microtubule bundles were measured from outer pillar cells isolated from the mammalian inner ear. Measurements were made using a three-point bending test and were incorporated into a mathematical model designed to distinguish between the stiffness contributions from microtubules and their cross-linking proteins. Outer pillar cells were composed of 1000-3000 parallel bundled microtubules in a square array that was interdigitated and cross-linked with actin filaments. The average midpoint bending stiffness of intact cells was 7 x 10(-4) N/m. After removal of both the actin filaments and cross-links with detergent in the presence of DNase I, the square array was disrupted and the stiffness decreased by a factor of 4, to 1.7 x 10(-4) N/m. The bending modulus for individual microtubules was calculated to be 7 x 10(-23) Nm2, and the Young's modulus for these 15 protofilament microtubules was 2 x 10(9) Pa. The shear modulus between microtubules in intact cells was calculated to be 10(3) Pa. It was concluded that cross-linking proteins provided shear resistance between microtubules, which resulted in a fourfold increase in stiffness. The model can be used to estimate the mechanical properties of cross-linked microtubule bundles in cells from which direct measurements are not available.     

Spindle pole mechanics studied in mitotic asters: dynamic distribution of spindle forces through compliant linkages

During cell division, chromosomes must faithfully segregate to maintain genome integrity, and this dynamic mechanical process is driven by the macromolecular machinery of the mitotic spindle. However, little is known about spindle mechanics. For example, spindle microtubules are organized by numerous cross-linking proteins yet the mechanical properties of those cross-links remain unexplored. To examine the mechanical properties of microtubule cross-links we applied optical trapping to mitotic asters that form in mammalian mitotic extracts. These asters are foci of microtubules, motors, and microtubule-associated proteins that reflect many of the functional properties of spindle poles and represent centrosome-independent spindle-pole analogs. We observed bidirectional motor-driven microtubule movements, showing that microtubule linkages within asters are remarkably compliant (mean stiffness 0.025 pN/nm) and mediated by only a handful of cross-links. Depleting the motor Eg5 reduced this stiffness, indicating that Eg5 contributes to the mechanical properties of microtubule asters in a manner consistent with its localization to spindle poles in cells. We propose that compliant linkages among microtubules provide a mechanical architecture capable of accommodating microtubule movements and distributing force among microtubules without loss of pole integrity—a mechanical paradigm that may be important throughout the spindle.     

Improved prediction of the collagen fiber architecture in the aortic heart valve

Living tissues show an adaptive response to mechanical loading by changing their internal structure and morphology. Understanding this response is essential for successful tissue engineering of load-bearing structures, such as the aortic valve. In this study, mechanically induced remodeling of the collagen architecture in the aortic valve was investigated. It was hypothesized that, in uniaxially loaded regions, the fibers aligned with the tensile principal stretch direction. For biaxial loading conditions, on the other hand, it was assumed that the collagen fibers aligned with directions situated between the principal stretch directions. This hypothesis has already been applied successfully to study collagen remodeling in arteries. The predicted fiber architecture represented a branching network and resembled the macroscopically visible collagen bundles in the native leaflet. In addition, the complex biaxial mechanical behavior of the native valve could be simulated qualitatively with the predicted fiber directions. The results of the present model might be used to gain further insight into the response of tissue engineered constructs during mechanical conditioning.     

Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling

At autopsy, 13 nonstenotic human left anterior descending coronary arteries [71.5 +/- 7.3 (mean +/- SD) yr old] were harvested, and related anamnesis was documented. Preconditioned prepared strips (n = 78) of segments from the midregion of the left anterior descending coronary artery from the individual layers in axial and circumferential directions were subjected to cyclic quasi-static uniaxial tension tests, and ultimate tensile stresses and stretches were documented. The ratio of outer diameter to total wall thickness was 0.189 +/- 0.014; ratios of adventitia, media, and intima thickness to total wall thickness were 0.4 +/- 0.03, 0.36 +/- 0.03, and 0.27 +/- 0.02, respectively; axial in situ stretch of 1.044 +/- 0.06 decreased with age. Stress-stretch responses for the individual tissues showed pronounced mechanical heterogeneity. The intima is the stiffest layer over the whole deformation domain, whereas the media in the longitudinal direction is the softest. All specimens exhibited small hysteresis and anisotropic and strong nonlinear behavior in both loading directions. The media and intima showed similar ultimate tensile stresses, which are on average three times smaller than ultimate tensile stresses in the adventitia (1,430 +/- 604 kPa circumferential and 1,300 +/- 692 kPa longitudinal). The ultimate tensile stretches are similar for all tissue layers. A recently proposed constitutive model was extended and used to represent the deformation behavior for each tissue type over the entire loading range. The study showed the need to model nonstenotic human coronary arteries with nonatherosclerotic intimal thickening as a composite structure composed of three solid mechanically relevant layers with different mechanical properties. The intima showed significant thickness, load-bearing capacity, and mechanical strength compared with the media and adventitia.     

F actin bundles in Drosophila bristles. I. Two filament cross-links are involved in bundling

Transverse sections though Drosophila bristles reveal 7-11 nearly round, plasma membrane-associated bundles of actin filaments. These filaments are hexagonally packed and in a longitudinal section they show a 12-nm periodicity in both the 1.1 and 1.0 views. From earlier studies this periodicity is attributable to cross-links and indicates that the filaments are maximally cross-linked, singed mutants also have 7-11 bundles, but the bundles are smaller, flattened, and the filaments within the bundles are randomly packed (not hexagonal); no periodicity can be detected in longitudinal sections. Another mutant, forked (f36a), also has 7-11 bundles but even though the bundles are very small, the filaments within them are hexagonally packed and display a 12-nm periodicity in longitudinal section. The singed-forked double mutant lacks filament bundles. Thus there are at least two species of cross-links between adjacent actin filaments. Hints of why two species of cross-links are necessary can be gleaned by studying bristle formation. Bristles sprout with only microtubules within them. A little later in development actin filaments appear. At early stages the filaments in the bundles are randomly packed. Later the filaments in the bundles become hexagonally packed and maximally cross-linked. We consider that the forked proteins may be necessary early in development to tie the filaments together in a bundle so that they can be subsequently zippered together by fascin (the singed gene product).