Browsing by Subject "mathematics, applied"

Now showing 1 - 4 of 4

Open Access
Bifurcation Analysis of Gene Regulatory Circuits Subject to Copy Number Variation
(2010) Mileyko, Yuriy; Weitz, Joshua S
Gene regulatory networks are comprised of many small gene circuits. Understanding expression dynamics of gene circuits for broad ranges of parameter space may provide insight into the behavior of larger regulatory networks as well as facilitate the use of circuits as autonomous units performing specific regulatory tasks. In this paper, we consider three common gene circuits and investigate the dependence of gene expression dynamics on the circuit copy number. In particular, we perform a detailed bifurcation analysis of the circuits' corresponding nonlinear gene regulatory models restricted to protein-only dynamics. Employing a geometric approach to bifurcation theory, we are able to derive closed form expressions for conditions which guarantee existence of saddle-node bifurcations caused by variation in the circuit copy number or copy number concentration. This result shows that the drastic effect of copy number variation on equilibrium behavior of gene circuits is highly robust to variation in other parameters in the circuits. We discuss a possibility of extending the current results to higher dimensional models which incorporate more details of the gene regulatory process.
Open Access
Chaos for cardiac arrhythmias through a one-dimensional modulation equation for alternans
(2010) Dai, Shu; Schaeffer, David G
Instabilities in cardiac dynamics have been widely investigated in recent years. One facet of this work has studied chaotic behavior, especially possible correlations with fatal arrhythmias. Previously chaotic behavior was observed in various models, specifically in the breakup of spiral and scroll waves. In this paper we study cardiac dynamics and find spatiotemporal chaotic behavior through the Echebarria-Karma modulation equation for alternans in one dimension. Although extreme parameter values are required to produce chaos in this model, it seems significant mathematically that chaos may occur by a different mechanism from previous observations. (C) 2010 American Institute of Physics. [doi:10.1063/1.3456058]
Open Access
Computational and Analytic Perspectives on the Drift Paradox
(2010) Pasour, VB; Ellner, SP
The fact that many small aquatic and marine organisms manage to persist in their native environments in the presence of constant advection into unfavorable habitat is known as the "drift paradox." Although advection may determine large scale biological patterns, individual behavior such as predation or vertical/horizontal migration can dominate at smaller scales. Using both computational and analytical methods to model flow in an idealized channel, we explore the extent to which biological processes can counteract physical drivers. In particular, we investigate how different zooplankton migration behaviors affect biological retention time under a variety of flow regimes and whether a combination of physical/biological regimes exists that can resolve the drift paradox, i.e., allow the zooplankton to avoid washout for time periods much greater than the hydrologic retention time. The computational model is a three-dimensional semi-implicit hydrodynamic model which is coupled with an individual-based model for zooplankton behavior, while the analytical model is a simple partial differential equation containing both advective and behavioral components. The only behavior exhibited by the zooplankton is diel vertical migration. Our studies show that the interaction of zooplankton behavior and exchange flow can significantly influence zooplankton residence time. For a channel without vegetation, the analytical methods give biological residence times that vary by at most a day from the computational results.
Open Access
SMALL-SIZE epsilon-NETS FOR AXIS-PARALLEL RECTANGLES AND BOXES
(2010) Aronov, Boris; Ezra, Esther; Sharir, Micha
We show the existence of epsilon-nets of size O (1/epsilon log log 1/epsilon) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane and "fat" triangular ranges and for point sets in R-3 and axis-parallel boxes; these are the first known nontrivial bounds for these range spaces. Our technique also yields improved bounds on the size of epsilon-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of epsilon-nets of size O (1/epsilon log log log 1/epsilon) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Bronnimann and Goodrich or of Even, Rawitz, and Shahar, we obtain improved approximation factors (computable in expected polynomial time by a randomized algorithm) for the HITTING SET or the SET COVER problems associated with the corresponding range spaces.