Theoretical analysis of gradient detection by growth cones

Gradients of diffusible and substrate‐bound molecules play an important role in guiding axons to appropriate targets in the developing nervous system. Although some of the molecules involved have recently been identified, little is known about the physical mechanisms by which growth cones sense gradients. This article applies the seminal Berg and Purcell (1977) model of gradient sensing to this problem. The model provides estimates for the statistical fluctuations in the measurement of concentration by a small sensing device. By assuming that gradient detection consists of the comparison of concentrations at two spatially or temporally separated points, the model therefore provides an estimate for the steepness of gradient that can be detected as a function of physiological parameters. The model makes the following specific predictions. (a) It is more likely that growth cones use a spatial rather than temporal sensing strategy. (b) Growth cone sensitivity increases with the concentration of ligand, the speed of ligand diffusion, the size of the growth cone, and the time over which it averages the gradient signal. (c) The minimum detectable gradient steepness for growth cones is roughly in the range 1–10%. (d) This value varies depending on whether a bound or freely diffusing ligand is being sensed, and on whether the sensing occurs in three or two dimensions. The model also makes predictions concerning the role of filopodia in gradient detection. © 1999 John Wiley & Sons, Inc. J Neurobiol 41: 230–241, 1999     

Growth cones turn and migrate up an immobilized gradient of the laminin IKVAV peptide

Growth cone navigation is guided by extrinsic environmental proteins, called guidance cues. Many in vitro studies have characterized growth cone turning up and down gradients of soluble guidance cues. Although previous studies have shown that axonal elongation rates can be regulated by gradients of surface-bound molecules, there are no convincing demonstrations of growth cones turning to migrate up a surface-bound gradient of an adhesive ligand or guidance cue. In order to test this mode of axonal guidance, we used a photo-immobilization technique to create grids and gradients of an adhesive laminin peptide on polystyrene culture dish surfaces. Chick embryo dorsal root ganglia (DRGs) were placed on peptide grid patterns containing surface-bound gradients of the IKVAV-containing peptide. DRG growth cones followed a path of surface-bound peptide to the middle of a perpendicularly oriented gradient with a 25% concentration difference across 30 microm. The majority of growth cones turned and migrated up the gradient, turning until they were oriented directly up the gradient. Growth cones slowed their migration when they encountered the gradient, but growth cone velocity returned to the previous rate after turning up or down the gradient. This resembles in vivo situations where growth cones slow at a choice point before changing the direction of axonal extension. Thus, these results support the hypothesis that mechanisms of axonal guidance include growth cone orientation by gradients of surface-bound adhesive molecules and guidance cues.     

A Mechanism for the Polarity Formation of Chemoreceptors at the Growth Cone Membrane for Gradient Amplification during Directional Sensing

Accurate response to external directional signals is essential for many physiological functions such as chemotaxis or axonal guidance. It relies on the detection and amplification of gradients of chemical cues, which, in eukaryotic cells, involves the asymmetric relocalization of signaling molecules. How molecular events coordinate to induce a polarity at the cell level remains however poorly understood, particularly for nerve chemotaxis. Here, we propose a model, inspired by single-molecule experiments, for the membrane dynamics of GABA chemoreceptors in nerve growth cones (GCs) during directional sensing. In our model, transient interactions between the receptors and the microtubules, coupled to GABA-induced signaling, provide a positive-feedback loop that leads to redistribution of the receptors towards the gradient source. Using numerical simulations with parameters derived from experiments, we find that the kinetics of polarization and the steady-state polarized distribution of GABA receptors are in remarkable agreement with experimental observations. Furthermore, we make predictions on the properties of the GC seen as a sensing, amplification and filtering module. In particular, the growth cone acts as a low-pass filter with a time constant ∼10 minutes determined by the Brownian diffusion of chemoreceptors in the membrane. This filtering makes the gradient amplification resistent to rapid fluctuations of the external signals, a beneficial feature to enhance the accuracy of neuronal wiring. Since the model is based on minimal assumptions on the receptor/cytoskeleton interactions, its validity extends to polarity formation beyond the case of GABA gradient sensing. Altogether, it constitutes an original positive-feedback mechanism by which cells can dynamically adapt their internal organization to external signals.     

Integration of Shallow Gradients of Shh and Netrin-1 Guides Commissural Axons

During nervous system development, gradients of Sonic Hedgehog (Shh) and Netrin-1 attract growth cones of commissural axons toward the floor plate of the embryonic spinal cord. Mice defective for either Shh or Netrin-1 signaling have commissural axon guidance defects, suggesting that both Shh and Netrin-1 are required for correct axon guidance. However, how Shh and Netrin-1 collaborate to guide axons is not known. We first quantified the steepness of the Shh gradient in the spinal cord and found that it is mostly very shallow. We then developed an in vitro microfluidic guidance assay to simulate these shallow gradients. We found that axons of dissociated commissural neurons respond to steep but not shallow gradients of Shh or Netrin-1. However, when we presented axons with combined Shh and Netrin-1 gradients, they had heightened sensitivity to the guidance cues, turning in response to shallower gradients that were unable to guide axons when only one cue was present. Furthermore, these shallow gradients polarized growth cone Src-family kinase (SFK) activity only when Shh and Netrin-1 were combined, indicating that SFKs can integrate the two guidance cues. Together, our results indicate that Shh and Netrin-1 synergize to enable growth cones to sense shallow gradients in regions of the spinal cord where the steepness of a single guidance cue is insufficient to guide axons, and we identify a novel type of synergy that occurs when the steepness (and not the concentration) of a guidance cue is limiting.     

Growth cone form, behavior, and interactions in vivo: retinal axon pathfinding as a model

Studies in vitro have revealed a great deal about growth cone behaviors, especially responses to guidance molecules, both positive and negative, and the signaling systems mediating these responses. Little, however, is known about these events as they take place in vivo. With new imaging methods, growth cone behaviors can be chronicled in the complex settings of intact or semi-intact systems. With the retinal projection through the optic chiasm as a model, we examined the hypothesis previously drawn from static material that growth cone form is position-specific: growth cone form in fact reflects specific behaviors, including rate and tempo of extension, that are more or less prominent in different locales in which growth cones are situated. Other studies show that growth cones interact with cells along the pathway, both specialized nonneuronal cells and other neurons, some expressing known guidance molecules. The present challenge is to bridge dynamic imaging with electron microscopy and molecular localization, in order to link growth cone behaviors with cell and molecular interactions in the natural setting in which growth cones extend.     

Growth cone navigation in substrate-bound ephrin gradients

Graded distributions of ephrin ligands are involved in the formation of topographic maps. However, it is still poorly understood how growth cones read gradients of membrane-bound guidance molecules. We used microcontact printing to produce discontinuous gradients of substrate-bound ephrinA5. These consist of submicron-sized protein-covered spots, which vary with respect to their sizes and spacings. Growth cones of chick temporal retinal axons are able to integrate these discontinuous ephrin distributions and stop at a distinct zone in the gradient while still undergoing filopodial activity. The position of this stop zone depends on both the steepness of the gradient and on the amount of substrate-bound ephrin per unit surface area. Quantitative analysis of axon outgrowth shows that the stop reaction is controlled by a combination of the local ephrin concentration and the total amount of encountered ephrin, but cannot be attributed to one of these parameters alone.     

Modulation of growth cone morphology by substrate-bound adhesion molecules.

The growth cone, a terminal structure on developing and regenerating axons, is specialized for motility and guidance functions. In vivo the growth cone responds to environmental cues to guide the axon to its appropriate target. These cues are thought to be responsible for position-specific morphological changes in the growth cone, but the molecules that control growth cone behavior are poorly characterized. We used scanning electron microscopy to analyze the morphology of retinal ganglion cell growth cones in vitro on different adhesion molecules that axons normally encounter in vivo. L1/8D9, N-cadherin, and laminin each induced distinctive morphological characteristics in growth cones. Growth cones elaborated lamellipodial structures in response to the cell adhesion molecules L1/8D9 and N-cadherin, whereas laminin supported filopodial growth cones with small veils. On L1/8D9, the growth cones were larger and produced more filopodia. Filopodial associations between adjacent growth cones and neurites were frequent on L1/8D9 but were uncommon on laminin or N-cadherin. These results demonstrate that different adhesion molecules have profoundly different effects on growth cone morphology. This is consistent with previous reports suggesting that changes in growth cone morphology in vivo occur in response to changes in substrate composition.     

Neuritic growth rate described by modeling microtubule dynamics

A model is developed to describe neuronal elongation as a result of the polymerization of microtubules and elastic stretching of the neurites by force produced by the growth cone. The model for a single segment with a single growth cone revealed a constant elongation rate, while the concentration of tubulin in the soma rises, and the concentration of tubulin becomes constant in the growth cone. Extending the model to a neurite with a single branch point and two growth cones revealed the same results. When the assembly or the disassembly rate of microtubules is unequal in both growth cones, transient retraction of one of the terminal segments occurs, which results in complete retraction of the segment when the difference in (dis)assembly rate between the two growth cones is large enough. When the model is applied to large trees, a maximal sustainable number of terminal segments as a function of the production rate of tubulin appears. Mechanisms to stop outgrowth are discussed in relation to the establishment of synaptical contacts between cells.     

Response of retinal ganglion cell axons to striped linear gradients of repellent guidance molecules

Although molecular gradients have long been postulated to play a role in the development of topographic projections in the nervous system, relatively little is known about how axons evaluate gradients. Do growth cones respond to concentration or to slope? Do they react suddenly or gradually? Is there adaptation? In the developing retinotectal system, temporal retinal ganglion cell axons have previously been shown to avoid repellent cell-surface activities distributed in gradients across the optic tectum. We confronted temporal retinal axons with precisely formed striped linear gradients of repellent tectal membranes and of two candidate repellent molecules, ephrin-A2 and -A5. Axons entered gradient stripes independently of their slope and extended unhindered in the uphill direction until they suddenly avoided an apparent threshold concentration of repellent material that was independent of slope. This critical concentration was similar in both linear and nonlinear gradients, and hence independent of gradient shape. When gradients of identical slope were formed on different basal levels of repellent material, axons grew uphill for a fixed increment of concentration, possibly measured from the lowest point of the gradient, rather than up to a fixed absolute concentration. The speed of growth cones was not affected by repellent unstriped gradients below the critical concentration level. Similar results were found with membranes from cell lines stably transfected with either ephrin-A5 or ephrin-A2, two previously identified growth cone repellent cell-surface proteins. These data suggest that growth cones or axons can integrate guidance information over large distances, probably by a combined memory and adaptation mechanism. © 1998 John Wiley & Sons, Inc. J Neurobiol 37: 541–562, 1998     

Mechanisms of growth cone guidance and motility in the developing grasshopper embryo

During neuronal pathfinding in vivo, growth cones must reorient their direction of migration in response to extracellular guidance cues. The developing grasshopper limb bud has proved to be a model system in which to examine mechanisms of growth cone guidance and motility in vivo. In this review we examine the contributions of adhesion and multiple guidance cues (semaphorins 1 and 2) in directing a growth cone steering event. Recent observations have suggested that the tibial pioneer growth cones are not directed via mechanisms of differential adhesivity. We present a model of growth cone steering that suggests a combination of adhesive and guidance receptors are important for a correct steering event and that guidance molecules may be important regulators of adhesive interactions with the actin cytoskeleton.     

A stochastic model of neuronal growth cone guidance regulated by multiple sensors

Neuronal growth cones migrate directionally under the control of axon guidance molecules, thereby forming synapses in the developing brain. The signal transduction system by which a growth cone detects surrounding guidance molecules, analyzes the detected signals, and then determines the overall behavior remains undetermined. In this study, we describe a novel stochastic model of this behavior that utilizes multiple sensors on filopodia to respond to guidance molecules. Overall growth cone behavior is determined by using only the concentration gradients of guidance molecules in the immediate vicinity of each sensor. The detected signal at each sensor, which is treated as a vector quantity, is sent to the growth cone center and then integrated to determine axonal growth in the next step by means of a simple vector operation. We compared the results of computer simulations of axonal growth with observations of actual axonal growth from co-culture experiments using olfactory bulb and septum. The probabilistic distributions of axonal growth generated by the computer simulation were consistent with those obtained from the culture experiments, indicating that our model accurately simulates growth cone behavior. We believe that this model will be useful for elucidating the as yet unknown mechanisms responsible for axonal growth in vivo.     

Long-range signaling in growing neurons after local elevation of cyclic AMP-dependent activity

Cyclic AMP-dependent activity at the growth cone or the soma of cultured Xenopus spinal neurons was elevated by local extracellular perfusion of the neuron with culture medium containing 8-bromoadenosine 3',5'-cyclic monophosphate (8-br-cAMP) or forskolin. During local perfusion of one of the growth cones of multipolar neurons with these drugs, the perfused growth cone showed further extension, while the distant, unperfused growth cones were inhibited in their growth. Local perfusion of the growth cone with culture medium or local perfusion with 8-br-cAMP at a cell-free region 100 microns away from the growth cone did not produce any effect on the extension of the growth cone. Reduced extension of all growth cones was observed when the perfusion with 8-br-cAMP was restricted to the soma. The distant inhibitory effect does not depend on the growth of the perfused growth cone since local coperfusion of the growth cone with 8-br-cAMP and colchicine inhibited growth on both perfused and unperfused growth cones, while local perfusion with colchicine alone inhibited only the perfused growth cone. The distant inhibitory effect was abolished when the perfusion of 8-br-cAMP was carried out together with kinase inhibitor H- 8, suggesting the involvement of cAMP-dependent protein kinase and/or its downstream factors in the long-range inhibitory signaling. Uniform exposure of the entire neuron to bath-applied 8-br-cAMP, however, led to enhanced growth activity at all growth cones. Thus, local elevation of cAMP-dependent activity produces long-range and opposite effects on distant parts of the neuron, and a cytosolic gradient of second messengers may produce effects distinctly different from those following uniform global elevation of the messenger, leading to differential growth regulation at different regions of the same neuron.     

Gradients and growth cone guidance of grasshopper neurons.

The generation of a functional nervous system is dependent on precise pathfinding of axons during development. This pathfinding is directed by the distribution of local and long-range guidance cues, the latter of which are believed to be distributed in gradients. Gradients of guidance cues have been associated with growth cone function for over a hundred years. However, little is known about the mechanisms used by growth cones to respond to these gradients, in part owing to the lack of identifiable gradients in vivo. In the developing grasshopper limb, two gradients of the semaphorin Sema-2a are necessary for correct neuronal pathfinding in vivo. The gradients are found in regions where growth cones make critical steering decisions. Observations of different growth cone behaviors associated with these gradients have provided some insights into how growth cones respond to them. Growth cones appear to respond more faithfully to changes in concentration, rather than absolute levels, of Sema-2a expression, whereas the absolute levels may regulate growth cone size.     

The enrichment of a neuronal growth cone collapsing activity from embryonic chick brain

We have devised a simple bioassay for the identification of molecules that inhibit growth cone motility. Chick dorsal root ganglion (DRG) growth cones extending on laminin collapse when exposed to a suspension of embryonic brain membranes. Detergent-solubilized membranes from which the detergent has been removed collapse DRG growth cones extending on either laminin or chick L1. Collapse occurs over a time course of minutes and is fully reversible. Solubilized liver, primary fibroblast, or RN22 schwannoma cell membranes do not collapse DRG or retinal growth cones. Solubilized PC12 membranes cause retinal but not DRG growth cones to collapse. The collapsing activity from embryonic brain is heat-labile, is trypsin-sensitive, and behaves as a macromolecule on a sizing column. It can be enriched 100-fold by chromatography on heparin and hydroxylapatite. These results are consistent with the idea that growth cone motility is inhibited by specific membrane-associated proteins in the developing nervous system.     

Time course and Ca(2+) dependence of sensitivity modulation in cyclic GMP-gated currents of intact cone photoreceptors

We determined the Ca(2+) dependence and time course of the modulation of ligand sensitivity in cGMP-gated currents of intact cone photoreceptors. In electro-permeabilized single cones isolated from striped bass, we measured outer segment current amplitude as a function of cGMP or 8Br-cGMP concentrations in the presence of various Ca(2+) levels. The dependence of current amplitude on nucleotide concentration is well described by the Hill function with values of K(1/2), the ligand concentration that half-saturates current, that, in turn, depend on Ca(2+). K(1/2) increases as Ca(2+) rises, and this dependence is well described by a modified Michaelis-Menten function, indicating that modulation arises from the interaction of Ca(2+) with a single site without apparent cooperativity. (Ca)K(m), the Michaelis-Menten constant for Ca(2+) concentration is 857 +/- 68 nM for cGMP and 863 +/- 51 for 8Br-cGMP. In single cones under whole-cell voltage clamp, we simultaneously measured changes in membrane current and outer segment free Ca(2+) caused by sudden Ca(2+) sequestration attained by uncaging diazo-2. In the presence of constant 8Br-cGMP, 15 micro, Ca(2+) concentration decrease was complete within 50 ms and membrane conductance was enhanced 2.33 +/- 0.95-fold with a mean time to peak of 1.25 +/- 0.23 s. We developed a model that assumes channel modulation is a pseudo-first-order process kinetically limited by free Ca(2+). Based on the experimentally measured changes in Ca(2+) concentration, model simulations match experimental data well by assigning the pseudo-first-order time constant a mean value of 0.40 +/- 0.14 s. Thus, Ca(2+)-dependent ligand modulation occurs over the concentration range of the normal, dark-adapted cone. Its time course suggests that its functional effects are important in the recovery of the cone photoresponse to a flash of light and during the response to steps of light, when cones adapt.     

B‐type Eph receptors and ephrins induce growth cone collapse through distinct intracellular pathways

Forward and reverse signaling mediated by EphB tyrosine kinase receptors and their transmembrane ephrin‐B ligands play important roles in axon pathfinding, yet little is known about the intracellular pathways involved. Here we have used growth cones from the ventral (EphB receptor‐bearing) and dorsal (ephrin‐B‐bearing) embryonic Xenopus retina to investigate the signaling mechanisms in both forward and reverse directions. We report that unclustered, but not clustered, EphB2 ectodomains trigger fast (5–10 min) transient collapse responses in growth cones. This collapse response is mediated by low levels of intracellular cyclic GMP and requires proteasome function. In contrast, clustered, but not unclustered, ephrin‐B1 ectodomains cause slow (30–60 min) growth cone collapse that depends on high cGMP levels and is insensitive to inhibition of the proteasomal pathway. Upon receptor‐ligand binding, endocytosis occurs in the reverse direction (EphB2‐Fc into dorsal retinal growth cones), but not the forward direction, and is also sensitive to proteasomal inhibition. Endocytosis is functionally important because blocking of EphB2 internalization inhibits growth cone collapse. Our data reveal that distinct signaling mechanisms exist for B‐type Eph/ephrin‐mediated growth cone guidance and suggest that endocytosis provides a fast mechanism for switching off signaling in the reverse direction. © 2003 Wiley Periodicals, Inc. J Neurobiol 57: 323–336, 2003     

Modeling the role of myosin 1c in neuronal growth cone turning

We addressed the mechanical basis for how embryonic chick dorsal root ganglion growth cones turn on a uniform substrate of laminin-1. Turning is significantly correlated with lamellipodial area but not with filopodial length. We assessed the lamellipodial contribution to turning by asymmetric micro-CALI of myosin isoforms that causes localized lamellipodial expansion (myosin 1c) or filopodial retraction (myosin V). Episodes of asymmetric micro-CALI of myosin 1c (or myosin 1c and V together) caused significant turning of the growth cone. In contrast, repeated micro-CALI of myosin V or irradiation without added antibody did not turn growth cones. These findings argue that lamellipodia and not filopodia are necessary for growth cone turning. To model the role of myosin 1c on growth cone turning, we fitted the measured trajectories from asymmetric micro-CALI of myosin 1c-treated and untreated growth cones to the persistent random walk model. The first parameter in this equation, root-mean-square speed, is indistinguishable between the two data sets whereas the second parameter, the persistence of motion, is significantly increased (2.5-fold) as a result of asymmetric inactivation of myosin 1c by micro-CALI. This analysis demonstrates that growth cone turning results from an increase in the persistence of directional motion rather than a change in speed. Taken together, our results suggest that myosin 1c is a molecular correlate for directional persistence underlying growth cone motility.     

Axonal guidance by an avoidance mechanism

1. On a substrate consisting of alternating lanes of anterior and posterior tectal membranes, temporal retinal axons have a strong tendency to grow on the lanes of anterior membranes and to avoid the lanes of posterior membranes. 2. Temporal axons do extend neurites on posterior material, and in equivalent numbers and lengths to that of anterior membranes if the substrate consists of pure anterior or posterior membranes. 3. Inactivation of posterior membranes by heat or the enzyme phosphatidylinositol-specific phospholipase C (PI-PLC) abolishes their ability to induce the avoidance reaction of temporal axons. It is concluded that the posterior membranes contain a repulsive component for temporal retinal axons. 4. Growth cones growing on anterior membranes, which encounter posterior membranes at the strip boundary, in general do not become reduced in their growth rate. 5. These results are most easily explained by a "gradient-reading model" similar to chemotaxis where the steering of a growth cone is independent on the growth rate. 6. According to the model, a gradient of a guiding component outside the growth cone is transformed into an internal gradient which gives the growth cone its directionality. 7. Other models like growth inhibition cannot be ruled out but need at least two additional assumptions like habituation for growth on the putative posterior inhibitory substrate and a strong local restriction of the inhibitory effect within the growth cone which contacts the posterior material.     

Local calcium changes regulate the length of growth cone filopodia

Previous studies have demonstrated that the free intracellular calcium concentration ([Ca(2+)](i)) in growth cones can act as an important regulator of growth cone behavior. Here we investigated whether there is a spatial and temporal correlation between [Ca(2+)](i) and one particular aspect of growth cone behavior, namely the regulation of growth cone filopodia. Calcium was released from the caged compound NP-EGTA (o-nitrophenyl EGTA tetrapotassium salt) to simulate a signaling event in the form of a transient increase in [Ca(2+)](i). In three different experimental paradigms, we released calcium either globally (within an entire growth cone), regionally (within a small area of the lamellipodium), or locally (within a single filopodium). We demonstrate that global photolysis of NP-EGTA in growth cones caused a transient increase in [Ca(2+)](i) throughout the growth cone and elicited subsequent filopodial elongation that was restricted to the stimulated growth cone. Pharmacological blockage of either calmodulin or the Ca(2+)-dependent phosphatase, calcineurin, inhibited the effect of uncaging calcium, suggesting that these enzymes are acting downstream of calcium. Regional uncaging of calcium in the lamellipodium caused a regional increase in [Ca(2+)](i), but induced filopodial elongation on the entire growth cone. Elevation of [Ca(2+)](i) locally within an individual filopodium resulted in the elongation of only the stimulated filopodium. These findings suggest that the effect of an elevation of [Ca(2+)](i) on filopodial behavior depends on the spatial distribution of the calcium signal. In particular, calcium signals within filopodia can cause filopodial length changes that are likely a first step towards directed filopodial steering events seen during pathfinding in vivo.     

Activation of ADF/cofilin mediates attractive growth cone turning toward nerve growth factor and netrin‐1

Proper neural circuitry requires that growth cones, motile tips of extending axons, respond to molecular guidance cues expressed in the developing organism. However, it is unclear how guidance cues modify the cytoskeleton to guide growth cone pathfinding. Here, we show acute treatment with two attractive guidance cues, nerve growth factor (NGF) and netrin‐1, for embryonic dorsal root ganglion and temporal retinal neurons, respectively, results in increased growth cone membrane protrusion, actin polymerization, and filamentous actin (F‐actin). ADF/cofilin (AC) family proteins facilitate F‐actin dynamics, and we found the inactive phosphorylated form of AC is decreased in NGF‐ or netrin‐1‐treated growth cones. Directly increasing AC activity mimics addition of NGF or netrin‐1 to increase growth cone protrusion and F‐actin levels. Extracellular gradients of NGF, netrin‐1, and a cell‐permeable AC elicit attractive growth cone turning and increased F‐actin barbed ends, F‐actin accumulation, and active AC in growth cone regions proximal to the gradient source. Reducing AC activity blunts turning responses to NGF and netrin. Our results suggest that gradients of NGF and netrin‐1 locally activate AC to promote actin polymerization and subsequent growth cone turning toward the side containing higher AC activity. © 2010 Wiley Periodicals, Inc. Develop Neurobiol 70: 565–588, 2010