Axonal growth and guidance
Brain function depends critically on precisely specified patterns of wiring between neurons, and failures of wiring can compromise normal function. This wiring develops during early life as axons grow and navigate to find their appropriate targets, often over long distances. Axonal navigation is guided by spatial patterns of molecular cues in the developing brain. Over the past two decades major advances have been made in identifying the genes and molecules that comprise these molecular cues (O'Donnell et al, 2009), and mutations in these genes are now known to be associated with a variety of brain disorders (Stoekli, 2012). In addition, appropriate regeneration of axons is important for recovery of function after injury, and it is thus critical to understand how to promote regrowth of axons after damage to the adult nervous system (Harel & Strittmatter, 2006).
Axonal navigation is guided by the growth cone at the axonal tip (Gordon-Weeks, 2005; Lowery & Van Vactor, 2009), and also sometimes by selective branching from the axon shaft. Molecular cues in the environment are transduced by receptors on the growth cone and shaft, and converted into changes in the cytoskeleton that generate movement. These molecular cues often take the form of concentration gradients, so that many examples of axon guidance can be seen as a form of chemotaxis. One particularly common type of axon guidance is the generation of topographic maps, whereby nearby neurons in the input structure map to nearby neurons in the target structure.
The phenomena of axon growth and guidance provide a rich target for computational modelling. Here we briefly review some key aspects of growth cone biology, and how these have been addressed by theoretical models. For other reviews see e.g. Maskery and Shinbrot (2005), Graham and van Ooyen (2006), Simpson et al (2009).
The growth cone
The growth cone consists of central and peripheral domains (for a review from a physics perspective see Franze and Guck (2010)). The former contains microtubules extending from the axon shaft and a network of actin. The peripheral domain contains actin filaments and is characterised by finger-like filopodia joined by veil-like lamelipodia (Fig. 1). Growth cones are highly dynamic, with growth and reorganisation of actin filaments and microtubules constantly reshaping their structure on timescales of less than a minute (Kalil and Dent, 2005). Filopodia often appear to be actively searching their environment for guidance cues, and an individual filopodium can lead the growth cone in a specific direction. Forward movement of the growth cone is driven by myosin-based molecular motors via a `clutch' mechanism. Filamentous actin flows retrogradely from peripheral to central regions, but if these filaments are coupled to the substrate (via e.g. integrin receptors) then the retrograde flow pushes the growth cone in the opposite (forward) direction (Suter and Forscher, 2000). The growth cone steers by increasing actin polymerisation on one side and decreasing it on the other. Theoretical models have considered actin and microtubule dynamics both in growth cones (Kiddie et al, 2005; Graham et al, 2006) and in motile cells more generally (Mogilner, 2009), filopodial dynamics (Buettner et al, 1994), and potential roles for filopodia in steering growth cones (Goodhill et al, 2004). For a detailed review see Mortimer et al (2012).
Families of guidance molecules
Until the mid-1990s the molecules that guide axons in vivo were almost completely unknown. However since then over 100 have now been identified (Dickson, 2002), some of which have been associated with cognitive disorders as diverse as autism spectrum disorders, dyslexia and dementia (Stoekli, 2012). These molecules are organised into several families, the most prominent of which are the netrins, slits, semaphorins, and ephrins. The netrins, signalling through DCC and Unc5 receptors, were the first to be identified and play a role in many guidance events in vivo. They are particularly important for guiding commissural axons to the midline, via a concentration gradient established by diffusion from midline structures. However netrins can be both attractive and repulsive. Slits, signalling through robo receptors, are also involved in midline guidance, but primarily in a repulsive capacity. They can also promote axon branching. Semaphorins, signalling through plexin and neuropilin receptors, can act as both attactive and repulsive cues in both diffusible and contact-mediated forms. Semaphorins are a large family that play many diverse roles. Ephrins, signalling through eph receptors, are critical for establishing topographic maps (discussed further below), and act in a primarily repulsive fashion.
Signal transduction in growth cones
Binding of guidance receptors on the surface of the growth cone leads to directed motion through a complex sequence of signalling events that is only partially understood. Calcium transients are related to overall growth cone motility, while calcium gradients across the growth cone are important for determining direction of growth (Zheng and Poo, 2007). For many guidance cues the cyclic nucleotides cAMP and cGMP also play a critical role in determining guidance direction: a high ratio of cAMP to cGMP inside the growth cone tends to promote attraction while the opposite promotes repulsion, though there is also an interaction with calcium levels such that the opposite can sometimes apply. Two important outputs of the calcium / cyclic nucleotide signalling network are CaMKII and CaN, where CaMKII tends to promote attraction while CaN tends to promote repulsion. A mathematical model recently unified these phenomena, by borrowing a theoretical framework from the analysis of the signalling events underlying the switch betwen LTP and LTD (Forbes et al, 2012). The growth cone was modeled as two compartments (up-gradient and down-gradient), and guidance was assumed to occur towards the compartment with the higher ratio of CaMKII to CaN, as determined by the different calcium levels in each compartment due to the external gradient. Besides explaining previous data, this model made clear that calcium and cyclic nucleotide levels interact in a subtle way to determine the direction of guidance.
The Rho GTPases, particularly Rac, RhoA and Cdc42, are important molecules linking the level of the molecules discussed to the actin cytoskeleton, and have been widely implicated in cell motility more generally. Attractive cues activate Rac and Cdc42, promoting actin polymerization and growth cone extension, while repulsive cues activate Rho, promote actin-myosin contraction and growth cone retraction. A number of theoretical models have addressed the spatio-temporal dynamics of the Rac/RhoA/Cdc42 signalling network and their role in guidance (Sakumura et al, 2005; Jilkine et al, 2007). Spatial gradients of adhesion complex assembly and disassembly are also important for guidance (Tojima et al, 2011). More generally mathematical analyses at this level can tie into broader efforts to understand the critical motifs underlying signaling networks that allow robust molecular computation at the level of single cells.
Noise in axon guidance
Detection of concentration gradients by small sensing devices such as growth cones is fundamentally limited by noise: random Brownian fluctuations in the number of molecules present in the external gradient and the downstream signaling pathways, and the stochastic nature of receptor binding. Chemotaxis is therefore an example of decision-making in the face of unreliable data, and can be analysed theoretically using the type of ideal observer approaches that have proved useful for understanding the constraints governing other types of sensory perception (Kording, 2007). Mortimer et al (2009) applied a Bayesian approach to calculate the growth cone's optimal strategy for extracting gradient direction from noisy receptor binding measurements. Testing the predictions experimentally revealed a good fit of the data to the chemotactic sensitivity curve predicted by the model. Such approaches have also recently applied to understanding the constraints on chemotaxis for other cellular-scale systems such as leukocytes and dictyostelium (Levine and Rappel, 2013).
Axonal branching and fasciculation
Axonal branching also plays an important role in the formation of connections (Bilimoria and Bonni, 2013). This can occur via growth cone splitting of the primary axon, or by interstitial branching from the shaft of the axon. When axons reach their targets terminal arborisation often occurs, to form connections with many nearby neurons. In contrast interstitial branching has shown to be particularly important for the formation of retinotectal maps in mammals (see below). Many of the same families of molecules that are important in guiding axons also play a role in promoting or repressing branching, though much remains to be understood about the molecular details of this process. Branching can also be controlled by activity-dependent processes (Ruthazer et al, 2003). Theoretical work has addressed axonal branching via phenomenological approaches designed to capture the statistics of the resulting tree structures (e.g. Van Pelt et al, 1997), and mechanistic models considering the roles of signalling molecules and cytosketal remodelling (e.g. Hely et al, 2001; for reviews see Graham and van Ooyen (2006) and Van Ooyen (2011)). Fasciculation refers to the tendency for neighboring axons to stick to each other as they grow, mediated by cell adhesion molecules. This allows, for instance, later-growing axons to follow the track laid down by pioneer axons. Processes of fasciculation and defasciculation have been addressed theoretically in models such as Hentschel and Van Ooyen (1999).
Topographic map formation
A particularly important system for understanding axon guidance events in vivo is the formation of topographic maps. These occur throughout the brain, with a paradigm model system being the mapping from the retina to the tectum / superior colliculus. To explain how these maps form Sperry (1963) proposed the idea of chemospecificity: retinal axons are guided to their targets in the tectum by the matching of molecular labels whose expression is graded across the retina with corresponding labels whose expression is graded across the tectum. The identities of some of the key labels involved were subsequently discovered to be ephrin ligands binding to their receptors the ephs (Fig 3; Mclaughlin and O'Leary, 2005). The experimental data regarding the formation of retinotectal maps is particularly rich, including both modern genetic manipulations of the labels involved and earlier surgical manipulations which investigated how the mapping adapted to, for instance, the loss of half the retina or half the tectum. Explaining these latter results requires additional mechanisms (e.g. competition) beyond strict label matching. Theoretical models have contributed to our understanding of these data from several perspectives (Goodhill and Xu, 2005), ranging from the basic constraints on label-matching (Prestige and Willshaw, 1975) to how multiple constraints interact to determine the final mapping (Koulakov and Tsigankov, 2004; Willshaw, 2006; Simpson and Goodhill, 2011). For a recent comparison of models see Hjorth et al (2013).
Some future directions
While the last 20 years has seen a revolution in our understanding of the molecules involved in axon guidance, there is still much work to be done in understanding these phenomena more quantitatively. Despite recent attempts (Rosoff et al, 2004; von Philipsborn et al, 2006) a major limitation in this regard remains the difficulty of producing precisely known and controllable concentration distributions of guidance molecules. A promising new approach is the development of new assays based on microfluidics technologies (e.g. Morel et al (2012)). Although there are challenges to be overcome due to the relative delicacy of neuronal growth cones compared to other cell types, microfluidics is poised to deliver a new generation of insights into the mechanisms of axon guidance. Some important questions include understanding better the signal transduction mechanisms that convert an external cue into directed movement, and how multiple different cues interact to determine guidance decisions in vivo.
Another important emerging area is understanding the role mechanical forces play in axon guidance (Franze, 2013). Growth cones can sense mechanical tension, and their growth is affected by substrate stiffness. Recent advances in technologies for controlling substrate stiffness, and measuring the forces involved, will make it possible to directly assess the degree to which mechanical forces work together with molecular cues to shape brain wiring in vivo.
In summary, as this article has hopefully made apparent, theoretical/computational models have made an important contribution to our understanding of axon guidance. This contribution is very likely to increase in the future, due to the necessity for teasing apart multiple influences on axon growth and guidance from complex combinations of molecular and mechanical cues.
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