Asymmetries arising from the space-filling nature of vascular networks

Cardiovascular networks span the body by branching across many generations of vessels. The resulting structure delivers blood over long distances to supply all cells with oxygen via the relatively short-range process of diffusion at the capillary level. The structural features of the network that accomplish this density and ubiquity of capillaries are often called space-filling. There are multiple strategies to fill a space, but some strategies do not lead to biologically adaptive structures by requiring too much construction material or space, delivering resources too slowly, or using too much power to move blood through the system. We empirically measure the structure of real networks (18 humans and 1 mouse) and compare these observations with predictions of model networks that are space-filling and constrained by a few guiding biological principles. We devise a numerical method that enables the investigation of space-filling strategies and determination of which biological principles influence network structure. Optimization for only a single principle creates unrealistic networks that represent an extreme limit of the possible structures that could be observed in nature. We first study these extreme limits for two competing principles, minimal total material and minimal path lengths. We combine these two principles and enforce various thresholds for balance in the network hierarchy, which provides a novel approach that highlights the trade-offs faced by biological networks and yields predictions that better match our empirical data.Comment: 17 pages, 15 figure

Testing Foundations of Biological Scaling Theory Using Automated Measurements of Vascular Networks

Scientists have long sought to understand how vascular networks supply blood and oxygen to cells throughout the body. Recent work focuses on principles that constrain how vessel size changes through branching generations from the aorta to capillaries and uses scaling exponents to quantify these changes. Prominent scaling theories predict that combinations of these exponents explain how metabolic, growth, and other biological rates vary with body size. Nevertheless, direct measurements of individual vessel segments have been limited because existing techniques for measuring vasculature are invasive, time consuming, and technically difficult. We developed software that extracts the length, radius, and connectivity of in vivo vessels from contrast-enhanced 3D Magnetic Resonance Angiography. Using data from 20 human subjects, we calculated scaling exponents by four methods--two derived from local properties of branching junctions and two from whole-network properties. Although these methods are often used interchangeably in the literature, we do not find general agreement between these methods, particularly for vessel lengths. Measurements for length of vessels also diverge from theoretical values, but those for radius show stronger agreement. Our results demonstrate that vascular network models cannot ignore certain complexities of real vascular systems and indicate the need to discover new principles regarding vessel lengths

Variational Ansatz for PT-Symmetric Quantum Mechanics

A variational calculation of the energy levels of a class of PT-invariant quantum mechanical models described by the non-Hermitian Hamiltonian H= p^2 - (ix)^N with N positive and x complex is presented. Excellent agreement is obtained for the ground state and low lying excited state energy levels and wave functions. We use an energy functional with a three parameter class of PT-symmetric trial wave functions in obtaining our results.Comment: 9 pages -- one postscript figur

Dynamics of the chiral phase transition in the 2+1 dimensional Gross-Neveu model

The phase diagram of the Gross-Neveu (G-N) model in 2+1 dimensions as a function of chemical potential and temperature has a simple curve separating the broken symmetry and unbroken symmetry phases, with chiral symmetry being restored both at high temperature and high density. We study, in leading order in the 1/N expansion, the dynamics of the chiral phase transition for an expanding plasma of quarks in the Gross-Neveu model in 2+1 dimensions assuming boost invariant kinematics. We compare the time evolution of the order parameter (mass of the fermion) for evolutions starting in the unbroken and broken phases. The proper time evolution of the order parameter resembles previous results in the 1+1 dimensional G-N model in the same approximation. The time needed to traverse the transition is insensitive to mu.Comment: 10 pages, 3 figure

The Closest Damped Lyman Alpha System

A difficulty of studying damped Lyman alpha systems is that they are distant, so one knows little about the interstellar medium of the galaxy. Here we report upon a damped Lyman alpha system in the nearby galaxy NGC 4203, which is so close (v_helio = 1117 km/s) and bright (B_o = 11.62) that its HI disk has been mapped. The absorption lines are detected against Ton 1480, which lies only 1.9' (12 h_50 kpc) from the center of NGC 4203. Observations were obtained with the Faint Object Spectrograph on HST (G270H grating) over the 2222-3277 Angstrom region with 200 km/s resolution. Low ionization lines of Fe, Mn, and Mg were detected, leading to metallicities of -2.29, -2.4, which are typical of other damped Lyman alpha systems, but well below the stellar metallicity of this type of galaxy. Most notably, the velocity of the lines is 1160 +- 10 km/s, which is identical to the HI rotational velocity of 1170 km/s at that location in NGC 4203, supporting the view that these absorption line systems can be associated with the rotating disks of galaxies. In addition, the line widths of the Mg lines give an upper limit to the velocity dispersion of 167 km/s, to the 99% confidence level.Comment: 4 pages LaTeX, including 1 figure and 1 table, uses emulateapj.sty. Accepted for publication by Astrophysical Journal Letter

Stability of ecosystems enhanced by species-interaction constraints

Ecosystem stability is a central question both in theoretical and applied biology. Dynamical systems theory can be used to analyze how growth rates, carrying capacities, and patterns of species interactions affect the stability of an ecosystem. The response to increasing complexity has been extensively studied and the general conclusion is that there is a limit. While there is a complexity limit to stability at which global destabilisation occurs, the collapse rarely happens suddenly if a system is fully viable (no species is extinct). In fact, when complexity is successively increased, we find that the generic response is to go through multiple single-species extinctions before a global collapse. In this paper we demonstrate this finding via both numerical simulations and elaborations of theoretical predictions. We explore more biological interaction patterns, and, perhaps most importantly, we show that constrained interaction structures-a constant row sum in the interaction matrix-prevent extinctions from occurring. This makes an ecosystem more robust in terms of allowed complexity, but it also means singles-species extinctions do not precede or signal collapse-a drastically different behavior compared to the generic and commonly assumed case. We further argue that this constrained interaction structure-limiting the total interactions for each species-is biologically plausible