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An extension of the totally asymmetric exclusion process, which incorporates a dynamically extending lattice is explored. Although originally inspired as a model for filamentous fungal growth, here the dynamically extending exclusion process (DEEP) is studied in its own right, as a nontrivial addition to the class of nonequilibrium exclusion process models. Here we discuss various mean-field approximation schemes and elucidate the steady state behaviour of the model and its associated phase diagram. Of particular note is that the dynamics of the extending lattice leads to a new region in the phase diagram in which a shock discontinuity in the density travels forward with a velocity that is lower than the velocity of the tip of the lattice. Thus in this region the shock recedes from both boundaries.

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Abstract. We consider a large class of two-lane driven diffusive systems in contact with reservoirs at their boundaries and develop a stability analysis as a method to derive the phase diagrams of such systems. We illustrate the method by deriving phase diagrams for the asymmetric exclusion process coupled to various second lanes: a diffusive lane; an asymmetric exclusion process with advection in the same direction as the first lane, and an asymmetric exclusion process with advection in the opposite direction. The competing currents on the two lanes naturally lead to a very rich phenomenology and we find a variety of phase diagrams. It is shown that the stability analysis is equivalent to an 'extremal current principle' for the total current in the two lanes. We also point to classes of models where both the stability analysis and the extremal current principle fail.arXiv:1103.4677v1 [cond-mat.stat-mech]

We propose a simple model for mass transport within a fungal hypha and its subsequent growth. Inspired by the role of microtubule-transported vesicles, we embody the internal dynamics of mass inside a hypha with mutually excluding particles progressing stochastically along a growing onedimensional lattice. The connection between long range transport of materials for growth, and the resulting extension of the hyphal tip has not previously been addressed in the modelling literature. We derive and analyse mean-field equations for the model and present a phase diagram of its steady state behaviour, which we compare to simulations. We discuss our results in the context of the filamentous fungus, Neurospora crassa. PACS numbers: 87.10.+e, 87.16.Ac, 87.16.Ka , Biologically, fungi are distinct from both plants and animals. In addition to their intrinsic interest, they impact immensely on human affairs and on the ecosystem [1].Key to the evolutionary success of fungi, is their unique mode of growth. Filamentous fungi grow by the polarised extension of thread-like hyphae, which make up the body, or mycelium, of a fungus. Except for branching (which initiates new hyphae) the site of growth is localised to a single region at the tip of each elongating hypha.There are many theoretical models for the growth of fungal colonies and of single hyphae (reviewed in [2,3]). Most models of single hypha growth concentrate on biomechanics [4,5]. Of more interest for us here is the "vesicle supply centre" (VSC) model [6,7], in which raw materials for growth are packaged in secretory vesicles and distributed to the hyphal surface from a single "supply centre" (often identified with an organelle complex known as the Spitzenkörper, or apical body [8,9]) situated within the growing tip. This model is capable of predicting the shape of hyphal tips; but the speed of growth (equivalent to the speed of the VSC) is an input parameter. Moreover, all transport processes are subsumed into a single rate of vesicle supply at the VSC. We are aware of just one model that takes explicit account of transport along the growing hypha [10]. A major interest of this early work, however, was the initiation of branching; these authors did not relate vesicle transport to growth velocity. This latter issue remains poorly understood.In this work, we propose a simple one-dimensional model which makes an explicit connection between the long-distance transport of building materials along a hypha and the resulting extension as they are delivered to its apical site of growth.It is a highly idealised model, encompassing many complicated biological processes (many of which are still poorly understood) with two key parameters: the rate at which vesicles enter the system and the efficiency with which they extend the length of the hypha. We demonstrate that by altering these rates, steady states can be attained whereby the hypha is extending at a constant speed while being supplied with materials far behind the tip. Our model has features in common with [10]. Like [10], we...