Stochastic actin polymerization and steady retrograde flow determine growth cone advancement

The geometry of photopolymerized topography influences neurite pathfinding by directing growth cone morphology and migration

Objective. Cochlear implants provide auditory perception to those with severe to profound sensorineural hearing loss: however, the quality of sound perceived by users does not approximate natural hearing. This limitation is due in part to the large physical gap between the stimulating electrodes and their target neurons. Therefore, directing the controlled outgrowth of processes from spiral ganglion neurons (SGNs) into close proximity to the electrode array could provide significantly increased hearing function.Approach.For this objective to be properly designed and implemented, the ability and limits of SGN neurites to be guided must first be determined. In this work, we engineer precise topographical microfeatures with angle turn challenges of various geometries to study SGN pathfinding and use live imaging to better understand how neurite growth is guided by these cues.Main Results.We find that the geometry of the angled microfeatures determines the ability of neurites to navigate the angled microfeature turns. SGN neurite pathfinding fidelity is increased by 20%-70% through minor increases in microfeature amplitude (depth) and by 25% if the angle of the patterned turn is made obtuse. Further, we see that dorsal root ganglion neuron growth cones change their morphology and migration to become more elongated within microfeatures. Our observations also indicate complexities in studying neurite turning. First, as the growth cone pathfinds in response to the various cues, the associated neurite often reorients across the angle topographical microfeatures. Additionally, neurite branching is observed in response to topographical guidance cues, most frequently when turning decisions are most uncertain.Significance.Overall, the multi-angle channel micropatterned substrate is a versatile and efficient system to assess neurite turning and pathfinding in response to topographical cues. These findings represent fundamental principles of neurite pathfinding that will be essential to consider for the design of 3D systems aiming to guide neurite growthin vivo.

Imaging actin organisation and dynamics in 3D

The actin cytoskeleton plays a critical role in cell architecture and the control of fundamental processes including cell division, migration and survival. The dynamics and organisation of F-actin have been widely studied in a breadth of cell types on classical two-dimensional (2D) surfaces. Recent advances in optical microscopy have enabled interrogation of these cytoskeletal networks in cells within three-dimensional (3D) scaffolds, tissues and in vivo. Emerging studies indicate that the dimensionality experienced by cells has a profound impact on the structure and function of the cytoskeleton, with cells in 3D environments exhibiting cytoskeletal arrangements that differ to cells in 2D environments. However, the addition of a third (and fourth, with time) dimension leads to challenges in sample preparation, imaging and analysis, necessitating additional considerations to achieve the required signal-to-noise ratio and spatial and temporal resolution. Here, we summarise the current tools for imaging actin in a 3D context and highlight examples of the importance of this in understanding cytoskeletal biology and the challenges and opportunities in this domain.

The geometry of photopolymerized topography influences neurite pathfinding by directing growth cone morphology and migration

Cochlear implants (CIs) provide auditory perception to those with profound sensorineural hearing loss: however, the quality of sound perceived by a CI user does not approximate natural hearing. This limitation is due in part to the large physical gap between the stimulating electrodes and their target neurons. Therefore, directing the controlled outgrowth of processes from spiral ganglion neurons (SGNs) into close proximity to the electrode array could provide significantly increased hearing function. For this objective to be properly designed and implemented, the ability and limits of SGN neurites to be guided must first be determined. In this work, we engineered precise topographical microfeatures with angle turn challenges of various geometries to study SGN pathfinding. Additionally, we analyze sensory neurite growth in response to topographically patterned substrates and use live imaging to better understand how neurite growth is guided by these cues. In assessing the ability of neurites to sense and turn in response to topographical cues, we find that the geometry of the angled microfeatures determines the ability of neurites to navigate the angled microfeature turns. SGN neurite pathfinding fidelity can be increased by 20-70% through minor increases in microfeature amplitude (depth) and by 25% if the angle of the patterned turn is made more obtuse. Further, by using engineered topographies and live imaging of dorsal root ganglion neurons (DRGNs), we see that DRGN growth cones change their morphology and migration to become more elongated within microfeatures. However, our observations also indicate complexities in studying neurite turning. First, as the growth cone pathfinds in response to the various cues, the associated neurite often reorients across the angle topographical microfeatures. This reorientation is likely related to the tension the neurite shaft experiences when the growth cone elongates in the microfeature around a turn. Additionally, neurite branching is observed in response to topographical guidance cues, most frequently when turning decisions are most uncertain. Overall, the multi-angle channel micropatterned substrate is a versatile and efficient system to assess SGN neurite turning and pathfinding in response to topographical cues. These findings represent fundamental principles of neurite pathfinding that will be essential to consider for the design of 3D systems aiming to guide neurite growth in vivo.

A two-step search and run response to gradients shapes leukocyte navigation in vivo

Migrating cells must interpret chemical gradients to guide themselves within tissues. A long-held principle is that gradients guide cells via reorientation of leading-edge protrusions. However, recent evidence indicates that protrusions can be dispensable for locomotion in some contexts, raising questions about how cells interpret endogenous gradients in vivo and whether other mechanisms are involved. Using laser wound assays in zebrafish to elicit acute endogenous gradients and quantitative analyses, we demonstrate a two-stage process for leukocyte chemotaxis in vivo: first a “search” phase, with stimulation of actin networks at the leading edge, cell deceleration, and turning. This is followed by a “run” phase, with fast actin flows, cell acceleration, and persistence. When actin dynamics are perturbed, cells fail to resolve the gradient, suggesting that pure spatial sensing of the gradient is insufficient for navigation. Our data suggest that cell contractility and actin flows provide memory for temporal sensing, while expansion of the leading edge serves to enhance gradient sampling.

Mathematical models of neuronal growth

The establishment of a functioning neuronal network is a crucial step in neural development. During this process, neurons extend neurites—axons and dendrites—to meet other neurons and interconnect. Therefore, these neurites need to migrate, grow, branch and find the correct path to their target by processing sensory cues from their environment. These processes rely on many coupled biophysical effects including elasticity, viscosity, growth, active forces, chemical signaling, adhesion and cellular transport. Mathematical models offer a direct way to test hypotheses and understand the underlying mechanisms responsible for neuron development. Here, we critically review the main models of neurite growth and morphogenesis from a mathematical viewpoint. We present different models for growth, guidance and morphogenesis, with a particular emphasis on mechanics and mechanisms, and on simple mathematical models that can be partially treated analytically.

Enhanced mechanosensing of cells in synthetic 3D matrix with controlled biophysical dynamics

3D culture of cells in designer biomaterial matrices provides a biomimetic cellular microenvironment and can yield critical insights into cellular behaviours not available from conventional 2D cultures. Hydrogels with dynamic properties, achieved by incorporating either degradable structural components or reversible dynamic crosslinks, enable efficient cell adaptation of the matrix and support associated cellular functions. Herein we demonstrate that given similar equilibrium binding constants, hydrogels containing dynamic crosslinks with a large dissociation rate constant enable cell force-induced network reorganization, which results in rapid stellate spreading, assembly, mechanosensing, and differentiation of encapsulated stem cells when compared to similar hydrogels containing dynamic crosslinks with a low dissociation rate constant. Furthermore, the static and precise conjugation of cell adhesive ligands to the hydrogel subnetwork connected by such fast-dissociating crosslinks is also required for ultra-rapid stellate spreading (within 18 h post-encapsulation) and enhanced mechanosensing of stem cells in 3D. This work reveals the correlation between microscopic cell behaviours and the molecular level binding kinetics in hydrogel networks. Our findings provide valuable guidance to the design and evaluation of supramolecular biomaterials with cell-adaptable properties for studying cells in 3D cultures.

3D culture systems can provide critical insights into cellular behaviour. Here, the authors study the binding timescale of dynamic crosslinks and the conjugation stability of cell-adhesive ligands in cell–hydrogel network interactions to evaluate the impact on stem cell behaviour, mechanosensing and differentiation.

Persistent and polarized global actin flow is essential for directionality during cell migration

Cell migration is hypothesized to involve a cycle of behaviours beginning with leading edge extension. However, recent evidence suggests that the leading edge may be dispensable for migration, raising the question of what actually controls cell directionality. Here, we exploit the embryonic migration of Drosophila macrophages to bridge the different temporal scales of the behaviours controlling motility. This approach reveals that edge fluctuations during random motility are not persistent and are weakly correlated with motion. In contrast, flow of the actin network behind the leading edge is highly persistent. Quantification of actin flow structure during migration reveals a stable organization and asymmetry in the cell-wide flowfield that strongly correlates with cell directionality. This organization is regulated by a gradient of actin network compression and destruction, which is controlled by myosin contraction and cofilin-mediated disassembly. It is this stable actin-flow polarity, which integrates rapid fluctuations of the leading edge, that controls inherent cellular persistence.

Heterotypic contact inhibition of locomotion can drive cell sorting between epithelial and mesenchymal cell populations

Interactions between different cell types can induce distinct contact inhibition of locomotion (CIL) responses that are hypothesised to control population-wide behaviours during embryogenesis. However, our understanding of the signals that lead to cell-type specific repulsion and the precise capacity of heterotypic CIL responses to drive emergent behaviours is lacking. Using a new model of heterotypic CIL, we show that fibrosarcoma cells, but not fibroblasts, are actively repelled by epithelial cells in culture. We show that knocking down EphB2 or ERK in fibrosarcoma cells specifically leads to disruption of the repulsion phase of CIL in response to interactions with epithelial cells. We also examine the population-wide effects when these various cell combinations are allowed to interact in culture. Unlike fibroblasts, fibrosarcoma cells completely segregate from epithelial cells and inhibiting their distinct CIL response by knocking down EphB2 or ERK family proteins also disrupts this emergent sorting behaviour. These data suggest that heterotypic CIL responses, in conjunction with processes such as differential adhesion, may aid the sorting of cell populations.

Summary: Active cell repulsion, driven by ephrin and ERK signalling, is required for fibrosarcoma cells to segregate from epithelial cells in culture.

Axon growth regulation by a bistable molecular switch

For the brain to function properly, its neurons must make the right connections during neural development. A key aspect of this process is the tight regulation of axon growth as axons navigate towards their targets. Neuronal growth cones at the tips of developing axons switch between growth and paused states during axonal pathfinding, and this switching behaviour determines the heterogeneous axon growth rates observed during brain development. The mechanisms controlling this switching behaviour, however, remain largely unknown. Here, using mathematical modelling, we predict that the molecular interaction network involved in axon growth can exhibit bistability, with one state representing a fast-growing growth cone state and the other a paused growth cone state. Owing to stochastic effects, even in an unchanging environment, model growth cones reversibly switch between growth and paused states. Our model further predicts that environmental signals could regulate axon growth rate by controlling the rates of switching between the two states. Our study presents a new conceptual understanding of growth cone switching behaviour, and suggests that axon guidance may be controlled by both cell-extrinsic factors and cell-intrinsic growth regulatory mechanisms.

The Role of Stress Fibers in the Shape Determination Mechanism of Fish Keratocytes

Crawling cells have characteristic shapes that are a function of their cell types. How their different shapes are determined is an interesting question. Fish epithelial keratocytes are an ideal material for investigating cell shape determination, because they maintain a nearly constant fan shape during their crawling locomotion. We compared the shape and related molecular mechanisms in keratocytes from different fish species to elucidate the key mechanisms that determine cell shape. Wide keratocytes from cichlids applied large traction forces at the rear due to large focal adhesions, and showed a spatially loose gradient associated with actin retrograde flow rate, whereas round keratocytes from black tetra applied low traction forces at the rear small focal adhesions and showed a spatially steep gradient of actin retrograde flow rate. Laser ablation of stress fibers (contractile fibers connected to rear focal adhesions) in wide keratocytes from cichlids increased the actin retrograde flow rate and led to slowed leading-edge extension near the ablated region. Thus, stress fibers might play an important role in the mechanism of maintaining cell shape by regulating the actin retrograde flow rate.

Tensile Forces Originating from Cancer Spheroids Facilitate Tumor Invasion

The mechanical properties of tumors and the tumor environment provide important information for the progression and characterization of cancer. Tumors are surrounded by an extracellular matrix (ECM) dominated by collagen I. The geometrical and mechanical properties of the ECM play an important role for the initial step in the formation of metastasis, presented by the migration of malignant cells towards new settlements as well as the vascular and lymphatic system. The extent of this cell invasion into the ECM is a key medical marker for cancer prognosis. In vivo studies reveal an increased stiffness and different architecture of tumor tissue when compared to its healthy counterparts. The observed parallel collagen organization on the tumor border and radial arrangement at the invasion zone has raised the question about the mechanisms organizing these structures. Here we study the effect of contractile forces originated from model tumor spheroids embedded in a biomimetic collagen I matrix. We show that contractile forces act immediately after seeding and deform the ECM, thus leading to tensile radial forces within the matrix. Relaxation of this tension via cutting the collagen does reduce invasion, showing a mechanical relation between the tensile state of the ECM and invasion. In turn, these results suggest that tensile forces in the ECM facilitate invasion. Furthermore, simultaneous contraction of the ECM and tumor growth leads to the condensation and reorientation of the collagen at the spheroid’s surface. We propose a tension-based model to explain the collagen organization and the onset of invasion by forces originating from the tumor.

Confinement and low adhesion induce fast amoeboid migration of slow mesenchymal cells

The mesenchymal-amoeboid transition (MAT) was proposed as a mechanism for cancer cells to adapt their migration mode to their environment. While the molecular pathways involved in this transition are well documented, the role of the microenvironment in the MAT is still poorly understood. Here, we investigated how confinement and adhesion affect this transition. We report that, in the absence of focal adhesions and under conditions of confinement, mesenchymal cells can spontaneously switch to a fast amoeboid migration phenotype. We identified two main types of fast migration--one involving a local protrusion and a second involving a myosin-II-dependent mechanical instability of the cell cortex that leads to a global cortical flow. Interestingly, transformed cells are more prone to adopt this fast migration mode. Finally, we propose a generic model that explains migration transitions and predicts a phase diagram of migration phenotypes based on three main control parameters: confinement, adhesion, and contractility.

Inter-cellular forces orchestrate contact inhibition of locomotion

Contact inhibition of locomotion (CIL) is a multifaceted process that causes many cell types to repel each other upon collision. During development, this seemingly uncoordinated reaction is a critical driver of cellular dispersion within embryonic tissues. Here, we show that Drosophila hemocytes require a precisely orchestrated CIL response for their developmental dispersal. Hemocyte collision and subsequent repulsion involves a stereotyped sequence of kinematic stages that are modulated by global changes in cytoskeletal dynamics. Tracking actin retrograde flow within hemocytes in vivo reveals synchronous reorganization of colliding actin networks through engagement of an inter-cellular adhesion. This inter-cellular actin-clutch leads to a subsequent build-up in lamellar tension, triggering the development of a transient stress fiber, which orchestrates cellular repulsion. Our findings reveal that the physical coupling of the flowing actin networks during CIL acts as a mechanotransducer, allowing cells to haptically sense each other and coordinate their behaviors.

Quantification of collagen contraction in three-dimensional cell culture

Many different cell types including fibroblasts, smooth muscle cells, endothelial cells, and cancer cells exert traction forces on the fibrous components of the extracellular matrix. This can be observed as matrix contraction both macro- and microscopically in three-dimensional (3D) tissues models such as collagen type I gels. The quantification of local contraction at the micron scale, including its directionality and speed, in correlation with other parameters such as cell invasion, local protein or gene expression, can provide useful information to study wound healing, organism development, and cancer metastasis. In this article, we present a set of tools to quantify the flow dynamics of collagen contraction, induced by cells migrating out of a multicellular cancer spheroid into a three-dimensional (3D) collagen matrix. We adapted a pseudo-speckle technique that can be applied to bright-field and fluorescent microscopy time series. The image analysis presented here is based on an in-house written software developed in the Matlab (Mathworks) programming environment. The analysis program is freely available from GitHub following the link: http://dx.doi.org/10.5281/zenodo.10116. This tool provides an automatized technique to measure collagen contraction that can be utilized in different 3D cellular systems.

Intrinsic activity in cells and the brain

Motile cells such as bacteria, amoebae, and fibroblasts display a continual level of energy-consuming reactions involving the cytoskeleton and signal pathways, regardless of whether or not they are actually migrating. I draw parallels between these "silent signals" and the intrinsic activity of the human brain, especially that associated with the brain stem. In both cases, it can be argued that the organism continually rehearses possible future actions, so it can act quickly and accurately when suitable cues are received from the environment.

Active mechanics and dynamics of cell spreading on elastic substrates

The spreading area of cells has been shown to play a central role in the determination of cell fate and tissue morphogenesis; however, a clear understanding of how spread cell area is determined is still lacking. The observation that cell area and force generally increase with substrate rigidity suggests that cell area is dictated mechanically, by means of a force-balance between the cell and the substrate. A simple mechanical model, corroborated by experimental measurements of cell area and force is presented to analyze the temporal force balance between the cell and the substrate during spreading. The cell is modeled as a thin elastic disc that is actively pulled by lamellipodia protrusions at the cell front. The essential molecular mechanisms of the motor activity at the cell front, including, actin polymerization, adhesion kinetics, and the actin retrograde flow, are accounted for and used to predict the dynamics of cell spreading on elastic substrates; simple, closed-form expressions for the evolution of cell size and force are derived. Time-resolved, traction force microscopy, combined with measurements of cell area are performed to investigate the simultaneous variations of cell size and force. We find that cell area and force increase simultaneously during spreading but the force develops with an apparent delay relative to the increase in cell area. We demonstrate that this may reflect the strain-stiffening property of the cytoskeleton. We further demonstrate that the radial cell force is a concave function of spreading speed and that this may reflect the strengthening of cell-substrate adhesions during spreading.

Different modes of growth cone collapse in NG 108-15 cells

In the fundamental process of neuronal path-finding, a growth cone at the tip of every neurite detects and follows multiple guidance cues regulating outgrowth and initiating directional changes. While the main focus of research lies on the cytoskeletal dynamics underlying growth cone advancement, we investigated collapse and retraction mechanisms in NG108-15 growth cones transiently transfected with mCherry-LifeAct and pCS2+/EMTB-3XGFP for filamentous actin and microtubules, respectively. Using fluorescence time lapse microscopy we could identify two distinct modes of growth cone collapse leading either to neurite retraction or to a controlled halt of neurite extension. In the latter case, lateral movement and folding of actin bundles (filopodia) confine microtubule extension and limit microtubule-based expansion processes without the necessity of a constantly engaged actin turnover machinery. We term this previously unreported second type fold collapse and suggest that it marks an intermediate-term mode of growth regulation closing the gap between full retraction and small scale fluctuations.

Optical fiber light source directs neurite growth

Guiding the growth of a neurite by directing ~800 nm laser light to the leading edge of the neurite's growing region can be accomplished by controlling the position and direction in three dimensional space of a tapered optical fiber through which the light is projected. We control the position, angle and power of the laser beam to direct the growth of actin accumulations in neurites which affects their mobility.

Emergent complexity of the cytoskeleton: from single filaments to tissue

Despite their overwhelming complexity, living cells display a high degree of internal mechanical and functional organization which can largely be attributed to the intracellular biopolymer scaffold, the cytoskeleton. Being a very complex system far from thermodynamic equilibrium, the cytoskeleton's ability to organize is at the same time challenging and fascinating. The extensive amounts of frequently interacting cellular building blocks and their inherent multifunctionality permits highly adaptive behavior and obstructs a purely reductionist approach. Nevertheless (and despite the field's relative novelty), the physics approach has already proved to be extremely successful in revealing very fundamental concepts of cytoskeleton organization and behavior. This review aims at introducing the physics of the cytoskeleton ranging from single biopolymer filaments to multicellular organisms. Throughout this wide range of phenomena, the focus is set on the intertwined nature of the different physical scales (levels of complexity) that give rise to numerous emergent properties by means of self-organization or self-assembly.

Membrane tension, myosin force, and actin turnover maintain actin treadmill in the nerve growth cone

A growth cone is a motile structure at the tips of axons that is driven by the actin network and guides axon extension. Low actin adhesion to the substrate creates a stationary actin treadmill that allows leading-edge protrusion when adhesion increases in response to guidance cues. We use experimental measurements in the Aplysia bag growth cone to develop and constrain a simple mechanical model of the actin treadmill. We show that actin retrograde flow is primarily generated by myosin contractile forces, but when myosin is inhibited, leading-edge membrane tension increases and drives the flow. By comparing predictions of the model with previous experimental measurements, we demonstrate that lamellipodial and filopodial filament breaking contribute equally to the resistance to the flow. The fully constrained model clarifies the role of actin turnover in the mechanical balance driving the actin treadmill and reproduces the recent experimental observation that inhibition of actin depolymerization causes retrograde flow to slow exponentially with time. We estimate forces in the actin treadmill, and we demonstrate that measured G-actin distributions are consistent with the existence of a forward-directed fluid flow that transports G-actin to the leading edge.

Strength in the periphery: growth cone biomechanics and substrate rigidity response in peripheral and central nervous system neurons

There is now considerable evidence of the importance of mechanical cues in neuronal development and regeneration. Motivated by the difference in the mechanical properties of the tissue environment between the peripheral (PNS) and central (CNS) nervous systems, we compare substrate-stiffness-dependent outgrowth and traction forces from PNS (dorsal root ganglion (DRG)) and CNS (hippocampal) neurons. We show that neurites from DRG neurons display maximal outgrowth on substrates with a Young's modulus of ∼1000 Pa, whereas hippocampal neurite outgrowth is independent of substrate stiffness. Using traction force microscopy, we also find a substantial difference in growth cone traction force generation, with DRG growth cones exerting severalfold larger forces compared with hippocampal growth cones. The traction forces generated by DRG and hippocampal growth cones both increase with increasing stiffness, and DRG growth cones growing on substrates with a Young's modulus of 1000 Pa strengthen considerably after 18-30 h. Finally, we find that retrograde actin flow is almost three times faster in hippocampal growth cones than in DRG. Moreover, the density of paxillin puncta is significantly lower in hippocampal growth cones, suggesting that stronger substrate coupling of the DRG cytoskeleton is responsible for the remarkable difference in traction force generation. These findings reveal a differential adaptation of cytoskeletal dynamics to substrate stiffness in growth cones of different neuronal types, and highlight the potential importance of the mechanical properties of the cellular environment for neuronal navigation during embryonic development and nerve regeneration.

Rac1 drives melanoblast organization during mouse development by orchestrating pseudopod- driven motility and cell-cycle progression

During embryogenesis, melanoblasts proliferate and migrate ventrally through the developing dermis and epidermis as single cells. Targeted deletion of Rac1 in melanoblasts during embryogenesis causes defects in migration, cell-cycle progression, and cytokinesis. Rac1 null cells migrate markedly less efficiently, but surprisingly, global steering, crossing the dermal/epidermal junction, and homing to hair follicles occur normally. Melanoblasts navigate in the epidermis using two classes of protrusion: short stubs and long pseudopods. Short stubs are distinct from blebs and are driven by actin assembly but are independent of Rac1, Arp2/3 complex, myosin, or microtubules. Rac1 positively regulates the frequency of initiation of long pseudopods, which promote migration speed and directional plasticity. Scar/WAVE and Arp2/3 complex drive actin assembly for long pseudopod extension, which also depends on microtubule dynamics. Myosin contractility balances the extension of long pseudopods by effecting retraction and allowing force generation for movement through the complex 3D epidermal environment.

Growth cones as soft and weak force generators

Many biochemical processes in the growth cone finally target its biomechanical properties, such as stiffness and force generation, and thus permit and control growth cone movement. Despite the immense progress in our understanding of biochemical processes regulating neuronal growth, growth cone biomechanics remains poorly understood. Here, we combine different experimental approaches to measure the structural and mechanical properties of a growth cone and to simultaneously determine its actin dynamics and traction force generation. Using fundamental physical relations, we exploited these measurements to determine the internal forces generated by the actin cytoskeleton in the lamellipodium. We found that, at timescales longer than the viscoelastic relaxation time of τ = 8.5 ± 0.5 sec, growth cones show liquid-like characteristics, whereas at shorter time scales they behaved elastically with a surprisingly low elastic modulus of E = 106 ± 21 Pa. Considering the growth cone's mechanical properties and retrograde actin flow, we determined the internal stress to be on the order of 30 pN per μm(2). Traction force measurements confirmed these values. Hence, our results indicate that growth cones are particularly soft and weak structures that may be very sensitive to the mechanical properties of their environment.

Engineered in vitro/in silico models to examine neurite target preference

Research on spinal cord injury (SCI) repair focuses on developing mechanisms to allow neurites to grow past an injury site. In this article, we observe that numerous divergent paths (i.e., spinal roots) are present along the spinal column, and hence guidance strategies must be devised to ensure that regrowing neurites reach viable targets. Therefore, we have engineered an in vitro micropatterned model in which cultured E7 dorsal root ganglia (DRG) explants may enter alternate pathways (?roots?) along a branching micropattern. Alongside this in vitro model, we have developed an in silico simulation that we validate by comparison with independent experiments. We find in both in silico and in vitro models that the probability of a neurite entering a given root decreases exponentially with respect to the number of roots away from the DRG; consequently, the likelihood of neurites reaching a distant root can be vanishingly small. This result represents a starting point for future strategies to optimize the likelihood that neurites will reach appropriate targets in the regenerating nervous system, and provides a new computational tool to evaluate the feasibility and expected success of neurite guidance in complex geometries.

Leading-edge-gel coupling in lamellipodium motion

We present a model for actin-based motility that combines the dynamics of the semiflexible region at the leading edge of the lamellipodium with actomyosin gel properties in the bulk described by the theory of active polar gels. We calculate the velocity of the lamellipodium determined by the interaction of the gel and adhesion with forces in the semiflexible region. The stationary concave force-velocity relation of the model reproduces experimental results. We suggest that it is determined by retrograde flow at small forces and gel formation and retrograde flow at large ones. The variety of dynamic regimes of the semiflexible region reproducing experimentally observed morphodynamics is conserved when we couple the leading edge to the gel.

Actin dynamics: from nanoscale to microscale

The dynamic nature of actin in cells manifests itself constantly. Polymerization near the cell edge is balanced by depolymerization in the interior, externally induced actin polymerization is followed by depolymerization, and spontaneous oscillations of actin at the cell periphery are frequently seen. I discuss how mathematical modeling relates quantitative measures of actin dynamics to the rates of underlying molecular level processes. The dynamic properties addressed include the rate of actin assembly at the leading edge of a moving cell, the disassembly rates of intracellular actin networks, the polymerization time course in externally stimulated cells, and spontaneous spatiotemporal patterns formed by actin. Although several aspects of actin assembly have been clarified by increasingly sophisticated models, our understanding of rapid actin disassembly is limited, and the origins of nonmonotonic features in externally stimulated actin polymerization remain unclear. Theory has generated several concrete, testable hypotheses for the origins of spontaneous actin waves and cell-edge oscillations. The development and use of more biomimetic systems applicable to the geometry of a cell will be key to obtaining a quantitative understanding of actin dynamics in cells.