Browsing by Subject "dynamics"

Now showing 1 - 6 of 6

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
Extraction of state-to-state reactive scattering attributes from wave packet in reactant Jacobi coordinates
(2010) Sun, Zhigang; Guo, Hua; Zhang, Dong H
The S-matrix for a scattering system provides the most detailed information about the dynamics. In this work, we discuss the calculation of S-matrix elements for the A+BC -> AB+C, AC+B type reaction. Two methods for extracting S-matrix elements from a single wave packet in reactant Jacobi coordinates are reviewed and compared. Both methods are capable of extracting the state-to-state attributes for both product channels from a single wave packet propagation. It is shown through the examples of H+HD, Cl+H-2, and H+HCl reactions that such reactant coordinate based methods are easy to implement, numerically efficient, and accurate. Additional efficiency can be gained by the use of a L-shaped grid with two-dimensional fast Fourier transform. (C) 2010 American Institute of Physics. [doi:10.1063/1.3328109]
Open Access
Solow's Harrod: Transforming Cyclical Dynamics into a Model of Long-Run Growth
(Economic Research Initiatives at Duke (ERID), 2013-03-27) Halsmayer, V; Hoover, KD
Modern growth theory derives mostly from Robert Solow’s “A Contribution to the Theory of Economic Growth” (1956). Solow’s own interpretation locates the origins of his “Contribution” in his view that the growth model of Roy Harrod implied a tendency toward progressive collapse of the economy. He formulates his view in terms of Harrod’s invoking a fixed-coefficients production function. We challenge Solow’s reading of Harrod’s “Essay in Dynamic Theory,” arguing that Harrod’s object in providing a “dynamic” theory had little to do with the problem of long-run growth as Solow understood it, but instead addressed medium-run fluctuations, the “inherent instability” of economies. It was an attempt to isolate conditions under which the economy might tend to run below potential. In making this argument, Harrod does not appeal to a fixed-coefficients production function – or to any production function at all, as that term is understood by Solow. Solow interpreted Harrod’s “Essay” in the light of a particular culture of understanding grounded in the practice of formal modeling that emerged in economics in the post-World War II period. The fate of Harrod’s analysis is a case study in the difficulties in communicating across distinct interpretive communities and of the potential for losing content and insights in the process. From Harrod’s English Keynesian point of view, Solow’s interpretation arose out of a culture of misunderstanding, and his objects – particularly, of trying to account for a tendency.
Open Access
The dynamics of successive induction in larval zebrafish.
(J Exp Anal Behav, 2010-09) Staddon, JER; MacPhail, RC; Padilla, S
Charles Sherrington identified the properties of the synapse by purely behavioral means-the study of reflexes-more than 100 years ago. They were subsequently confirmed neurophysiologically. Studying reflex interaction, he also showed that activating one reflex often facilitates another, antagonistic one: successive induction, which has since been demonstrated in a wide range of species, from aphids to locusts to dogs and humans. We show a particularly orderly example in zebrafish (Danio rerio) larvae; the behavior (locomotion) of larvae is low in dark and intermediate in light, but low in light and substantially higher in dark when dark followed light. A quantitative model of a simple dynamic process is described that readily captures the behavior pattern and the effects of a number of manipulations of lighting conditions.
Open Access
Transient anomalous diffusion of tracer particles in soft matter
(2009) McKinley, Scott A; Yao, Lingxing; Forest, M Gregory
This paper is motivated by experiments in which time series of tracer particles in viscoelastic liquids are recorded using advanced microscopy. The experiments seek to infer either viscoelastic properties of the sample [Mason and Weitz, Phys. Rev. Lett. 74, 1250-1253 (1995)] or diffusive properties of the specific tracer particle in the host medium [Suh et al., Adv. Drug Delivery Rev. 57, 63-78 (2005); Matsui et al., Proc. Natl. Acad. Sci. U. S. A. 103, 18131-18136 (2006); Lai et al., Proc. Natl. Acad. Sci. U. S. A. 104, 1482-1487 (2007); Fricks et al., SIAM J. Appl. Math. 69, 1277-1308 (2009)]. Our focus is the latter. Experimentalists often fit data to transient anomalous diffusion: a sub-diffusive power law scaling (t(v), with 0 < v < 1) of mean-squared displacement (MSD) over a finite time interval, with longtime viscous scaling (t(1)). The time scales of sub-diffusion and exponents v are observed to vary with particle size, particle surface chemistry, and viscoelastic properties of the host material. Until now, explicit models for transient sub-diffusive MSD scaling behavior [Doi and Edwards, The Theory of Polymer Physics (Oxford University Press, New York, 1986); Kremer and Grest, J. Chem. Phys. 92, 5057-5086 (1990); Rubinstein and Colby, Polymer Physics (Oxford University Press, New York, 2003)] are limited to precisely three exponents: monomer diffusion in Rouse chain melts (t(1/2)), in Zimm chain solutions (t(2/3)), and in reptating chains (t(1/4)). In this paper, we present an explicit parametrized family of stochastic processes (generalized Langevin equations with prescribed memory kernels) and derive their closed-form solutions which (1) span the complete range of transient sub-diffusive behavior and (2) possess the flexibility to tune both the time window of sub-diffusive scaling and the power law exponent v. These results establish a robust family of sub-diffusive models, for which the inverse problem of parameter inference from experimental data [Fricks et al., SIAM J. Appl. Math. 69, 1277-1308 (2009)] remains to be developed. (C) 2009 The Society of Rheology. [DOI: 10.1122/1.3238546]