A Multiscale Computational Model of Chemotactic Axon Guidance

Introduction

The ability of cells of responding to chemical signals present in the environment is of utmost importance for life, for example to recognize peers or locating food sources. Chemical cues also serve to mark pathways, which lead cells to a target (attractive cues) as well as repel them from selected regions (repulsive cues). Pathfinding by chemical cues is a key mechanism in the embryo, where sets of cells have to organize and reach specific areas to form the different body tissues. Cells crawl along the concentration gradient, towards (or away from) the direction of increasing diffusible chemical signal, moving from the peripheries to the source. This phenomenon is known as chemotaxis and its discovery dates back to the 18th century, allowed by the invention of the microscopy. An interesting example of chemotaxis is found in the developing nervous system, where axons, long and slender projections of nerve cells, find the targets they will innervate navigating in the extracellular environment through a chemotactic guidance mechanism (see, e.g., Tessier-Lavigne & Goodman, 1996; Mueller, 1999; Song & Poo, 2001). Detection and transduction of navigational cues in chemotactic axon guidance is mediated by the growth cone (GC), a highly dynamic structure located at the axon tip (see, e.g., Guan & Rao, 2003 and refer to Figure 1). From the microscopic point of view, directional sensing is initiated by differential binding with the extracellular ligand (the chemical cue) of the specialized receptors located on the opposite sides of the GC membrane. In order to respond to the very shallow ligand gradients observed in nature, the GC must optimize concentration measurements, overcoming the surrounding noise. Several mathematical models investigate this concept. In the seminal work of Berg & Purcell (1977), each receptor is considered as a “measuring device” which provides an estimation of the local ligand concentration based on its average time of permanence in the binding state during a certain period. In Mortimer et al. (2009b), it is shown that, if -in addition- the number of unbound-to-bound transitions is also signalled by the receptor, a more precise measure of the ligand concentration is yielded. When coming to consider the complete pool of receptors present on the GC, a strategy to weight the whole set of binding measurements should also be envisaged. In Mortimer et al. (2009a), it is shown that the optimal measuring strategy gives to each receptor a weight proportional to its distance from the geometrical center of the GC. In the present work, we propose a modification of this latter concept, introducing a weighting strategy depending on the distance of the single receptor from the center of mass of the receptor pool, a quantity dynamically varying according to the activity level of each single receptor (as will be defined more thoroughly in the following). The biophysical fact which motivates this hypothesis is the recent finding of Bouzigues et al. (2007a, 2007b) that, in presence of an attractive gradient of the diffusible cue GABA, ligand-bound GC receptors undergo two fundamental types of motion on the membrane: the first kind of motion is free diffusion, which is present even under an uniform external field, while the second kind of motion is a biased drift toward the side facing the attractive ligand source. This latter motion is driven by the physical interaction of bound receptors with the GC microtubules, which serve as conveyor belts (Saxton, 1994; Saxton & Jacobson, 1997). The overall effect of this mechanism is the establishment of an autocatalytic loop: bias in receptor localization induces, via internal polarization of molecules, preferential growth of the microtubules toward the leading edge of the GC and this, in turn, enhances convey of receptors on that same side (Bouzigues et al., 2007b). Once a weighting strategy for the receptor measurements is established, one should model the subsequent internal polarization chain leading to motion. Mathematical models in this context most often do not enter into the details of the extremely complex biochemical signalling cascade, but rather adopt phenomenological simplified descriptions that provide a “black box” information of the functional behaviour of the system. A first class of approaches (see, e.g., Buettner et al., 1994; Maskery & Shinbrot, 2005) is based on persistent random walk models. The GC trajectory is typically described by a system of ordinary differential equations accounting for a deterministic velocity field and random “kicks” arising from stochastic terms, macroscopically representing fluctuations in gradient sensing and signal transduction. Evolutions of these models are presented in Hentschel & van Ooyen (2000) and further in Aletti & Causin (2008), where the GC trajectory is described by more sophisticated stochastic partial differential systems of equations, including diffusion and inertia contributions. A second class of models are investigated in Aeschlimann & Tettoni (2001); Goodhill & Urbach (1999); Goodhill et al. (2004); Xu et al. (2005), where there is the attempt of introducing a description of the intracellular chain. Namely, the probability of finding a transmembrane receptor at a certain angular position on the GC is supposed to be linked to some significant intracellular parameter, for example the local concentration of ionic calcium in Aeschlimann & Tettoni (2001). We also refer to the mathematical models presented in a series of works by some of the Authors of this Chapter: in particular, we refer to Aletti et al. (2008a), where a novel modelling of ligand-receptor binding was proposed, introducing Markov chains to describe the state of receptors; to Causin & Facchetti (2009), where a detailed analysis of the internal polarization chain triggered by the receptor redistribution was carried out, performing an analysis of the amplification steps occurring in the chain; to Aletti & Causin (2008b), where the random walk models were used to study axon trajectories and to infer the internal organization of the GC.