Growth, collapse, and stalling in a mechanical model for neurite motility

A simple active fluid model unites cytokinesis, cell crawling, and axonal outgrowth

While the structural organization and molecular biology of neurons are well characterized, the physical process of axonal elongation remains elusive. The classic view posited elongation occurs through the deposition of cytoskeletal elements in the growth cone at the tip of a stationary array of microtubules. Yet, recent studies reveal axonal microtubules and docked organelles flow forward in bulk in the elongating axons of Aplysia, chick sensory, rat hippocampal, and Drosophila neurons. Noting that the morphology, molecular components, and subcellular flow patterns of growth cones strongly resemble the leading edge of migrating cells and the polar regions of dividing cells, our working hypothesis is that axonal elongation utilizes the same physical mechanisms that drive cell crawling and cell division. As a test of that hypothesis, here we take experimental data sets of sub-cellular flow patterns in cells undergoing cytokinesis, mesenchymal migration, amoeboid migration, neuronal migration, and axonal elongation. We then apply active fluid theory to develop a biophysical model that describes the different sub-cellular flow profiles across these forms of motility and how this generates cell motility under low Reynolds numbers. The modeling suggests that mechanisms for generating motion are shared across these processes, and differences arise through modifications of sub-cellular adhesion patterns and the profiles of internal force generation. Collectively, this work suggests that ameboid and mesenchymal cell crawling may have arisen from processes that first developed to support cell division, that growth cone motility and cell crawling are closely related, and that neuronal migration and axonal elongation are fundamentally similar, differing primarily in the motion and strength of adhesion under the cell body.

A simple active fluid model unites cytokinesis, cell crawling, and axonal outgrowth

Axonal outgrowth, cell crawling, and cytokinesis utilize actomyosin, microtubule-based motors, cytoskeletal dynamics, and substrate adhesions to produce traction forces and bulk cellular motion. While it has long been appreciated that growth cones resemble crawling cells and that the mechanisms that drive cytokinesis help power cell crawling, they are typically viewed as unique processes. To better understand the relationship between these modes of motility, here, we developed a unified active fluid model of cytokinesis, amoeboid migration, mesenchymal migration, neuronal migration, and axonal outgrowth in terms of cytoskeletal flow, adhesions, viscosity, and force generation. Using numerical modeling, we fit subcellular velocity profiles of the motions of cytoskeletal structures and docked organelles from previously published studies to infer underlying patterns of force generation and adhesion. Our results indicate that, during cytokinesis, there is a primary converge zone at the cleavage furrow that drives flow towards it; adhesions are symmetric across the cell, and as a result, cells are stationary. In mesenchymal, amoeboid, and neuronal migration, the site of the converge zone shifts, and differences in adhesion between the front and back of the cell drive crawling. During neuronal migration and axonal outgrowth, the primary convergence zone lies within the growth cone, which drives actin retrograde flow in the P-domain and bulk anterograde flow of the axonal shaft. They differ in that during neuronal migration, the cell body is weakly attached to the substrate and thus moves forward at the same velocity as the axon. In contrast, during axonal outgrowth, the cell body strongly adheres to the substrate and remains stationary, resulting in a decrease in flow velocity away from the growth cone. The simplicity with which cytokinesis, cell crawling, and axonal outgrowth can be modeled by varying coefficients in a simple model suggests a deep connection between them.

Mechanics of axon growth and damage: A systematic review of computational models

Normal axon development depends on the action of mechanical forces both generated within the cytoskeleton and outside the cell, but forces of large magnitude or rate cause damage instead. Computational models aid scientists in studying the role of mechanical forces in axon growth and damage. These studies use simulations to evaluate how different sources of force generation within the cytoskeleton interact with each other to regulate axon elongation and retraction. Furthermore, mathematical models can help optimize externally applied tension to promote axon growth without causing damage. Finally, scientists also use simulations of axon damage to investigate how forces are distributed among different components of the axon and how the tissue surrounding an axon influences its susceptibility to injury. In this review, we discuss how computational studies complement experimental studies in the areas of axon growth, regeneration, and damage.

Force-Regulated State Transitions of Growing Axons

Growing axons are one-dimensional active structures that are important for wiring the brain and repairing nerves. However, the biophysical mechanisms underlying the complex kinetics of growing axons remain elusive. Here, we develop a theoretical framework to recapitulate force-regulated states and their transitions in growing axons. We demonstrate a unique negative feedback mechanism that defines four distinct kinetic states in a growing axon, whose transitional boundaries depend on the interplay between cytoskeletal dynamics and axon-substrate adhesion. A phase diagram for axonal growth is formulated based on two dimensionless numbers.

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.

Theory for Durotactic Axon Guidance

During the development of the nervous system, neurons extend bundles of axons that grow and meet other neurons to form the neuronal network. Robust guidance mechanisms are needed for these bundles to migrate and reach their functional target. Directional information depends on external cues such as chemical or mechanical gradients. Unlike chemotaxis that has been extensively studied, the role and mechanism of durotaxis, the directed response to variations in substrate rigidity, remain unclear. We model bundle migration and guidance by rigidity gradients by using the theory of morphoelastic rods. We show that, at a rigidity interface, the motion of axon bundles follows a simple behavior analogous to optic ray theory and obeys Snell's law for refraction and reflection. We use this powerful analogy to demonstrate that axons can be guided by the equivalent of optical lenses and fibers created by regions of different stiffnesses.

Integrating Chemistry and Mechanics: The Forces Driving Axon Growth

The brain is our most complex organ. During development, neurons extend axons, which may grow over long distances along well-defined pathways to connect to distant targets. Our current understanding of axon pathfinding is largely based on chemical signaling by attractive and repulsive guidance cues. These cues instruct motile growth cones, the leading tips of growing axons, where to turn and where to stop. However, it is not chemical signals that cause motion-motion is driven by forces. Yet our current understanding of the mechanical regulation of axon growth is very limited. In this review, I discuss the origin of the cellular forces controlling axon growth and pathfinding, and how mechanical signals encountered by growing axons may be integrated with chemical signals. This mechanochemical cross talk is an important but often overlooked aspect of cell motility that has major implications for many physiological and pathological processes involving neuronal growth.

Mechanical Regulation of Neurite Polarization and Growth: A Computational Study

The densely packed microtubule (MT) array found in neuronal cell projections (neurites) serves two fundamental functions simultaneously: it provides a mechanically stable track for molecular motor-based transport and produces forces that drive neurite growth. The local pattern of MT polarity along the neurite shaft has been found to differ between axons and dendrites. In axons, the neurons' dominating long projections, roughly 90% of the MTs orient with their rapidly growing plus end away from the cell body, whereas in vertebrate dendrites, their orientations are locally mixed. Molecular motors are known to be responsible for cytoskeletal ordering and force generation, but their collective function in the dense MT cytoskeleton of neurites remains elusive. We here hypothesized that both the polarity pattern of MTs along the neurite shaft and the shaft's global extension are simultaneously driven by molecular motor forces and should thus be regulated by the mechanical load acting on the MT array as a whole. To investigate this, we simulated cylindrical bundles of MTs that are cross-linked and powered by molecular motors by iteratively solving a set of force-balance equations. The bundles were subjected to a fixed load arising from actively generated tension in the actomyosin cortex enveloping the MTs. The magnitude of the load and the level of motor-induced connectivity between the MTs have been varied systematically. With an increasing load and decreasing motor-induced connectivity between MTs, the bundles became wider in cross section and extended more slowly, and the local MT orientational order was reduced. These results reveal two, to our knowledge, novel mechanical factors that may underlie the distinctive development of the MT cytoskeleton in axons and dendrites: the cross-linking level of MTs by motors and the load acting on this cytoskeleton during growth.

The influence of device handles in single-molecule experiments

We deduce a fully analytical model to predict the artifacts of the device handles in single molecule force spectroscopy experiments. As we show, neglecting the handle stiffness can lead to crucial overestimation or underestimation of the stability properties and unfolding thresholds of multistable molecules.

Growth and remodelling of living tissues: perspectives, challenges and opportunities

One of the most remarkable differences between classical engineering materials and living matter is the ability of the latter to grow and remodel in response to diverse stimuli. The mechanical behaviour of living matter is governed not only by an elastic or viscoelastic response to loading on short time scales up to several minutes, but also by often crucial growth and remodelling responses on time scales from hours to months. Phenomena of growth and remodelling play important roles, for example during morphogenesis in early life as well as in homeostasis and pathogenesis in adult tissues, which often adapt to changes in their chemo-mechanical environment as a result of ageing, diseases, injury or surgical intervention. Mechano-regulated growth and remodelling are observed in various soft tissues, ranging from tendons and arteries to the eye and brain, but also in bone, lower organisms and plants. Understanding and predicting growth and remodelling of living systems is one of the most important challenges in biomechanics and mechanobiology. This article reviews the current state of growth and remodelling as it applies primarily to soft tissues, and provides a perspective on critical challenges and future directions.

Tension- and Adhesion-Regulated Retraction of Injured Axons

Damage-induced retraction of axons during traumatic brain injury is believed to play a key role in the disintegration of the neural network and to eventually lead to severe symptoms such as permanent memory loss and emotional disturbances. However, fundamental questions such as how axon retraction progresses and what physical factors govern this process still remain unclear. Here, we report a combined experimental and modeling study to address these questions. Specifically, a sharp atomic force microscope probe was used to transect axons and trigger their retraction in a precisely controlled manner. Interestingly, we showed that the retracting motion of a well-developed axon can be arrested by strong cell-substrate attachment. However, axon retraction was found to be retriggered if a second transection was conducted, albeit with a lower shrinking amplitude. Furthermore, disruption of the actin cytoskeleton or cell-substrate adhesion significantly altered the retracting dynamics of injured axons. Finally, a mathematical model was developed to explain the observed injury response of neural cells in which the retracting motion was assumed to be driven by the pre-tension in the axon and progress against neuron-substrate adhesion as well as the viscous resistance of the cell. Using realistic parameters, model predictions were found to be in good agreement with our observations under a variety of experimental conditions. By revealing the essential physics behind traumatic axon retraction, findings here could provide insights on the development of treatment strategies for axonal injury as well as its possible interplay with other neurodegenerative diseases.

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.

An Integrated Cytoskeletal Model of Neurite Outgrowth

Neurite outgrowth underlies the wiring of the nervous system during development and regeneration. Despite a significant body of research, the underlying cytoskeletal mechanics of growth and guidance are not fully understood, and the relative contributions of individual cytoskeletal processes to neurite growth are controversial. Here, we review the structural organization and biophysical properties of neurons to make a semi-quantitative comparison of the relative contributions of different processes to neurite growth. From this, we develop the idea that neurons are active fluids, which generate strong contractile forces in the growth cone and weaker contractile forces along the axon. As a result of subcellular gradients in forces and material properties, actin flows rapidly rearward in the growth cone periphery, and microtubules flow forward in bulk along the axon. With this framework, an integrated model of neurite outgrowth is proposed that hopefully will guide new approaches to stimulate neuronal growth.

Piconewton Mechanical Forces Promote Neurite Growth

Investigations over half a century have indicated that mechanical forces induce neurite growth, with neurites elongating at a rate of 0.1-0.3 μm h-1 pN-1 when mechanical force exceeds a threshold, with this being identified as 400-1000 pN for neurites of PC12 cells. In this article, we demonstrate that neurite elongation of PC12 cells proceeds at the same previously identified rate on application of mechanical tension of ∼1 pN, which is significantly lower than the force generated in vivo by axons and growth cones. This observation raises the possibility that mechanical tension may act as an endogenous signal used by neurons for promoting neurite elongation.

Modeling the Axon as an Active Partner with the Growth Cone in Axonal Elongation

Forces generated by the growth cone are vital for the proper development of the axon and thus brain function. Although recent experiments show that forces are generated along the axon, it is unknown whether the axon plays a direct role in controlling growth cone advance. Here, we use analytic and finite element modeling of microtubule dynamics and the activity of the molecular motors myosin and dynein to investigate mechanical force balance along the length of the axon and its effects on axonal outgrowth. Our modeling indicates that the paradoxical effects of stabilizing microtubules and the consequences of microtubule disassembly on axonal outgrowth can be explained by changes in the passive and active mechanical properties of axons. Our findings suggest that a full understanding of growth cone motility requires a consideration of the mechanical contributions of the axon. Our study not only has potential applications during neurodevelopment but might also help identify strategies to manipulate and promote axonal regrowth to treat neurodegeneration.

Are Homeostatic States Stable? Dynamical Stability in Morphoelasticity

Biological growth is often driven by mechanical cues, such as changes in external pressure or tensile loading. Moreover, it is well known that many living tissues actively maintain a preferred level of mechanical internal stress, called the mechanical homeostasis. The tissue-level feedback mechanism by which changes in the local mechanical stresses affect growth is called a growth law within the theory of morphoelasticity, a theory for understanding the coupling between mechanics and geometry in growing and evolving biological materials. This coupling between growth and mechanics occurs naturally in macroscopic tubular structures, which are common in biology (e.g., arteries, plant stems, airways). We study a continuous tubular system with spatially heterogeneous residual stress via a novel discretization approach which allows us to obtain precise results about the stability of equilibrium states of the homeostasis-driven growing dynamical system. This method allows us to show explicitly that the stability of the homeostatic state depends nontrivially on the anisotropy of the growth response. The key role of anisotropy may provide a foundation for experimental testing of homeostasis-driven growth laws.

Post-buckling behaviour of a growing elastic rod

We consider mechanically-induced pattern formation within the framework of a growing, planar, elastic rod attached to an elastic foundation. Through a combination of weakly nonlinear analysis and numerical methods, we identify how the shape and type of buckling (super- or subcritical) depend on material parameters, and a complex phase-space of transition from super- to subcritical is uncovered. We then examine the effect of heterogeneity on buckling and post-buckling behaviour, in the context of a heterogeneous substrate adhesion, elastic stiffness, or growth. We show how the same functional form of heterogeneity in different properties is manifest in a vastly differing post-buckled shape. Finally, a fourth form of heterogeneity, an imperfect foundation, is incorporated and shown to have a more dramatic impact on the buckling instability, a difference that can be qualitatively understood via the weakly nonlinear analysis.

Physical Biology of Axonal Damage

Excessive physical impacts to the head have direct implications on the structural integrity at the axonal level. Increasing evidence suggests that tau, an intrinsically disordered protein that stabilizes axonal microtubules, plays a critical role in the physical biology of axonal injury. However, the precise mechanisms of axonal damage remain incompletely understood. Here we propose a biophysical model of the axon to correlate the dynamic behavior of individual tau proteins under external physical forces to the evolution of axonal damage. To propagate damage across the scales, we adopt a consistent three-step strategy: First, we characterize the axonal response to external stretches and stretch rates for varying tau crosslink bond strengths using a discrete axonal damage model. Then, for each combination of stretch rates and bond strengths, we average the axonal force-stretch response of n = 10 discrete simulations, from which we derive and calibrate a homogenized constitutive model. Finally, we embed this homogenized model into a continuum axonal damage model of [1-d]-type in which d is a scalar damage parameter that is driven by the axonal stretch and stretch rate. We demonstrate that axonal damage emerges naturally from the interplay of physical forces and biological crosslinking. Our study reveals an emergent feature of the crosslink dynamics: With increasing loading rate, the axonal failure stretch increases, but axonal damage evolves earlier in time. For a wide range of physical stretch rates, from 0.1 to 10 /s, and biological bond strengths, from 1 to 100 pN, our model predicts a relatively narrow window of critical damage stretch thresholds, from 1.01 to 1.30, which agrees well with experimental observations. Our biophysical damage model can help explain the development and progression of axonal damage across the scales and will provide useful guidelines to identify critical damage level thresholds in response to excessive physical forces.

Mechanical stress as a regulator of cell motility

The motility of a cell can be triggered or inhibited not only by an applied force but also by a mechanically neutral force couple. This type of loading, represented by an applied stress and commonly interpreted as either squeezing or stretching, can originate from extrinsic interaction of a cell with its neighbors. To quantify the effect of applied stresses on cell motility we use an analytically transparent one-dimensional model accounting for active myosin contraction and induced actin turnover. We show that stretching can polarize static cells and initiate cell motility while squeezing can symmetrize and arrest moving cells. We show further that sufficiently strong squeezing can lead to the loss of cell integrity. The overall behavior of the system depends on the two dimensionless parameters characterizing internal driving (chemical activity) and external loading (applied stress). We construct a phase diagram in this parameter space distinguishing between static, motile, and collapsed states. The obtained results are relevant for the mechanical understanding of contact inhibition and the epithelial-to-mesenchymal transition.

Neurite elongation is highly correlated with bulk forward translocation of microtubules

During the development of the nervous system and regeneration following injury, microtubules (MTs) are required for neurite elongation. Whether this elongation occurs primarily through tubulin assembly at the tip of the axon, the transport of individual MTs, or because MTs translocate forward in bulk is unclear. Using fluorescent speckle microscopy (FSM), differential interference contrast (DIC), and phase contrast microscopy, we tracked the movement of MTs, phase dense material, and docked mitochondria in chick sensory and Aplysia bag cell neurons growing rapidly on physiological substrates. In all cases, we find that MTs and other neuritic components move forward in bulk at a rate that on average matches the velocity of neurite elongation. To better understand whether and why MT assembly is required for bulk translocation, we disrupted it with nocodazole. We found this blocked the forward bulk advance of material along the neurite and was paired with a transient increase in axonal tension. This indicates that disruption of MT dynamics interferes with neurite outgrowth, not by disrupting the net assembly of MTs at the growth cone, but rather because it alters the balance of forces that power the bulk forward translocation of MTs.

Modeling molecular mechanisms in the axon

Axons are living systems that display highly dynamic changes in stiffness, viscosity, and internal stress. However, the mechanistic origin of these phenomenological properties remains elusive. Here we establish a computational mechanics model that interprets cellular-level characteristics as emergent properties from molecular-level events. We create an axon model of discrete microtubules, which are connected to neighboring microtubules via discrete crosslinking mechanisms that obey a set of simple rules. We explore two types of mechanisms: passive and active crosslinking. Our passive and active simulations suggest that the stiffness and viscosity of the axon increase linearly with the crosslink density, and that both are highly sensitive to the crosslink detachment and reattachment times. Our model explains how active crosslinking with dynein motors generates internal stresses and actively drives axon elongation. We anticipate that our model will allow us to probe a wide variety of molecular phenomena-both in isolation and in interaction-to explore emergent cellular-level features under physiological and pathological conditions.