Dynamical phase transition of a one-dimensional transport process including death

Abstract

Motivated by biological aspects related to fungus growth, we consider the competition of growth and corrosion. We study a modification of the totally asymmetric exclusion process, including the probabilities of injection ␣ and death of the last particle ␦. The system presents a phase transition at ␦ c ͑␣͒, where the average position of the last particle ͗L͘ grows as ͱ t. For ␦ Ͼ ␦ c , a nonequilibrium stationary state exists while for ␦ Ͻ ␦ c the asymptotic state presents a low density and max current phases. We discuss the scaling of the density and current profiles for parallel and sequential updates. DOI: 10.1103/PhysRevE.81.042101 PACS number͑s͒: 05.40.Ϫa, 87.10.Hk, 87.10.Rt Fungi are eukaryotic organisms that include microorganisms such as yeasts and molds, as well as the familiar mushrooms. The growth of a fungus is formed by the combination of the apical growth and the branching process which leads to the development of a mycelium. Most of them grow as hyphae, which are a cylindrical thread-like structures of 5 -10 m in diameter and up to several centimeters in length ͓1͔. The apical growth is a quasi-one-dimensional process that extends the hypha by transport of material from the seed to the front tip. At this later position, enzymes are released into the environment, where the new wall material is synthesized. The rate of extension, in a favorable environment, can be extremely rapid, up to 40 m per minute. Many biophysical models ͓2-6͔ describing the growth of fungal colonies and/or single hypha have been studied. In this context, efforts focused on the nonequilibrium properties of a modification of the totally asymmetric exclusion process ͑TASEP͒, by considering a distinct dynamics of one of the two boundary sites ͓7-11͔.An important aspect not taken into account yet, is that fungi have the ability to grow in a wide range of habitats, including extreme environments ͓12-14͔ and survive intense UV/cosmic radiation during space travel ͓15͔. Since the wall of the tip is usually structurally weak ͓1͔, in such a situation the extension rate can be slowed down and as the hypha is progressively aging, it may break down or be broken by other organisms ͓1͔. Our analysis is focused on the theoretical description of the above-mentioned growing process in competition with a corrosive environment. The theoretical model used is based on a simple modification of the TASEP and captures the general behavior induced by the two competing processes.Proposed in 1968 to study the motion of ribosomes along mRNA ͓16,17͔, numerous modifications of the TASEP were introduced including multilane systems and multi species transport ͓18-22͔, Langmuir dynamics ͓23-25͔, extended particles ͓26,27͔ as well as systems with finite resources ͓28,29͔. In a general framework, such models are considered as toy models of transport phenomena in order to better understands physics far from equilibrium.In the first part of this Brief Report we define the model and the system parameters. After a careful definition of the parallel update dynami...

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