A Shape Theorem for Riemannian First-Passage Percolation

Riemannian first-passage percolation (FPP) is a continuum model, with a distance function arising from a random Riemannian metric in $\R^d$. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one

No directed fractal percolation in zero area

We show that fractal (or "Mandelbrot") percolation in two dimensions produces a set containing no directed paths, when the set produced has zero area. This improves a similar result by the first author in the case of constant retention probabilities to the case of retention probabilities approaching 1

Duality and perfect probability spaces

Abstract. Given probability spaces (Xi, Ai,Pi),i =1,2,let M(P1,P2)denote the set of all probabilities on the product space with marginals P1 and P2 and let h be a measurable function on (X1 × X2, A1 ⊗A2). Continuous versions of linear programming stemming from the works of Monge (1781) and Kantorovich-Rubinˇstein (1958) for the case of compact metric spaces are concerned with the validity of the duality sup { hdP:P∈M(P1,P2)

Self-organized patterns of coexistence out of a predator-prey cellular automaton

We present a stochastic approach to modeling the dynamics of coexistence of prey and predator populations. It is assumed that the space of coexistence is explicitly subdivided in a grid of cells. Each cell can be occupied by only one individual of each species or can be empty. The system evolves in time according to a probabilistic cellular automaton composed by a set of local rules which describe interactions between species individuals and mimic the process of birth, death and predation. By performing computational simulations, we found that, depending on the values of the parameters of the model, the following states can be reached: a prey absorbing state and active states of two types. In one of them both species coexist in a stationary regime with population densities constant in time. The other kind of active state is characterized by local coupled time oscillations of prey and predator populations. We focus on the self-organized structures arising from spatio-temporal dynamics of the coexistence. We identify distinct spatial patterns of prey and predators and verify that they are intimally connected to the time coexistence behavior of the species. The occurrence of a prey percolating cluster on the spatial patterns of the active states is also examined.Comment: 19 pages, 11 figure

Site-bond representation and self-duality for totalistic probabilistic cellular automata

We study the one-dimensional two-state totalistic probabilistic cellular automata (TPCA) having an absorbing state with long-range interactions, which can be considered as a natural extension of the Domany-Kinzel model. We establish the conditions for existence of a site-bond representation and self-dual property. Moreover we present an expression of a set-to-set connectedness between two sets, a matrix expression for a condition of the self-duality, and a convergence theorem for the TPCA.Comment: 11 pages, minor corrections, journal reference adde

Spatial Scaling in Model Plant Communities

We present an analytically tractable variant of the voter model that provides a quantitatively accurate description of beta-diversity (two-point correlation function) in two tropical forests. The model exhibits novel scaling behavior that leads to links between ecological measures such as relative species abundance and the species area relationship.Comment: 10 pages, 3 figure

Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games

Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Such non-transitive relations lead to an evolution with central features represented by the `rock-paper-scissors' game, where rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity. However, population mobility is a central feature of real ecosystems: animals migrate, bacteria run and tumble. Here, we observe a critical influence of mobility on species diversity. When mobility exceeds a certain value, biodiversity is jeopardized and lost. In contrast, below this critical threshold all subpopulations coexist and an entanglement of travelling spiral waves forms in the course of temporal evolution. We establish that this phenomenon is robust, it does not depend on the details of cyclic competition or spatial environment. These findings have important implications for maintenance and evolution of ecological systems and are relevant for the formation and propagation of patterns in excitable media, such as chemical kinetics or epidemic outbreaks.Comment: Final submitted version; the printed version can be found at http://dx.doi.org/10.1038/nature06095 Supplementary movies are available at http://www.theorie.physik.uni-muenchen.de/lsfrey/images_content/movie1.AVI and http://www.theorie.physik.uni-muenchen.de/lsfrey/images_content/movie2.AV

Population Dynamics in Spatially Heterogeneous Systems with Drift: the generalized contact process

We investigate the time evolution and stationary states of a stochastic, spatially discrete, population model (contact process) with spatial heterogeneity and imposed drift (wind) in one- and two-dimensions. We consider in particular a situation in which space is divided into two regions: an oasis and a desert (low and high death rates). Carrying out computer simulations we find that the population in the (quasi) stationary state will be zero, localized, or delocalized, depending on the values of the drift and other parameters. The phase diagram is similar to that obtained by Nelson and coworkers from a deterministic, spatially continuous model of a bacterial population undergoing convection in a heterogeneous medium.Comment: 8 papes, 12 figure

On the Hydrodynamic Equilibrium of a Rod in a Lattice Fluid

We model the behavior of a big (Brazil) nut in a medium of smaller nuts with a stochastic asymmetric simple exclusion dynamics of a polymer-monomer lattice system. The polymer or `rod' can move up or down in an external negative field, occupying N horizontal lattice sites where the monomers cannot enter. The monomers (at most one per site) or `fluid particles' are moving symmetrically in the horizontal plane and asymmetrically in the vertical direction, also with a negative field. For a fixed position of the rod, this lattice fluid is in equilibrium with a vertical height profile reversible for the monomers' motion. Upon `shaking' (speeding up the monomers) the motion of the `rod' dynamically decouples from that of the monomers resulting in a reversible random walk for the rod around an average height proportional to log N.Comment: 19 pages, 2 figure

Metabolic engineering of Saccharomyces cerevisiae to produce a reduced viscosity oil from lignocellulose

Citation: Tran, T. N. T., Breuer, R. J., Narasimhan, R. A., Parreiras, L. S., Zhang, Y. P., Sato, T. K., & Durrett, T. P. (2017). Metabolic engineering of Saccharomyces cerevisiae to produce a reduced viscosity oil from lignocellulose. Biotechnology for Biofuels, 10, 12. doi:10.1186/s13068-017-0751-yBackground: Acetyl-triacylglycerols (acetyl-TAGs) are unusual triacylglycerol (TAG) molecules that contain an sn-3 acetate group. Compared to typical triacylglycerol molecules (here referred to as long chain TAGs; lcTAGs), acetyl-TAGs possess reduced viscosity and improved cold temperature properties, which may allow direct use as a drop-in diesel fuel. Their different chemical and physical properties also make acetyl-TAGs useful for other applications such as lubricants and plasticizers. Acetyl-TAGs can be synthesized by EaDAcT, a diacylglycerol acetyltransferase enzyme originally isolated from Euonymus alatus (Burning Bush). The heterologous expression of EaDAcT in different organisms, including Saccharomyces cerevisiae, resulted in the accumulation of acetyl-TAGs in storage lipids. Microbial conversion of lignocellulose into acetyl-TAGs could allow biorefinery production of versatile molecules for biofuel and bioproducts. Results: In order to produce acetyl-TAGs from abundant lignocellulose feedstocks, we expressed EaDAcT in S. cerevisiae previously engineered to utilize xylose as a carbon source. The resulting strains were capable of producing acetyl-TAGs when grown on different media. The highest levels of acetyl-TAG production were observed with growth on synthetic lab media containing glucose or xylose. Importantly, acetyl-TAGs were also synthesized by this strain in ammonia fiber expansion (AFEX)-pretreated corn stover hydrolysate (ACSH) at higher volumetric titers than previously published strains. The deletion of the four endogenous enzymes known to contribute to lcTAG production increased the proportion of acetyl-TAGs in the total storage lipids beyond that in existing strains, which will make purification of these useful lipids easier. Surprisingly, the strains containing the four deletions were still capable of synthesizing lcTAG, suggesting that the particular strain used in this study possesses additional undetermined diacylglycerol acyltransferase activity. Additionally, the carbon source used for growth influenced the accumulation of these residual lcTAGs, with higher levels in strains cultured on xylose containing media. Conclusion: Our results demonstrate that S. cerevisiae can be metabolically engineered to produce acetyl-TAGs when grown on different carbon sources, including hydrolysate derived from lignocellulose. Deletion of four endogenous acyltransferases enabled a higher purity of acetyl-TAGs to be achieved, but lcTAGs were still synthesized. Longer incubation times also decreased the levels of acetyl-TAGs produced. Therefore, additional work is needed to further manipulate acetyl-TAG production in this strain of S. cerevisiae, including the identification of other TAG biosynthetic and lipolytic enzymes and a better understanding of the regulation of the synthesis and degradation of storage lipids