Tanya L. Leise

The

Googles success derives in large part from its PageRank algorithm, which ranks the importance of web pages according to an eigenvector of a weighted link matrix. Analysis of the PageRank formula provides a wonderful applied topic for a linear algebra course. Instructors may assign this article as a project to more advanced students or spend one or two lectures presenting the material with assigned homework from the exercises. This material also complements the discussion of Markov chains in matrix algebra. Maple and Mathematica files supporting this material can be found at www.rose-hulman.edu/~bryan.

25,000,000,000 Eigenvector: The Linear Algebra behind Google

Circadian rhythms regulate most physiological processes. Adjustments to circadian time, called phase shifts, are necessary following international travel and on a more frequent basis for individuals who work non‐traditional schedules such as rotating shifts. As the disruption that results from frequent phase shifts is deleterious to both animals and humans, we sought to better understand the kinetics of resynchronization of the mouse circadian system to one of the most disruptive phase shifts, a 6‐h phase advance. Mice bearing a luciferase reporter gene for mPer2 were subjected to a 6‐h advance of the light cycle and molecular rhythms in suprachiasmatic nuclei (SCN), thymus, spleen, lung and esophagus were measured periodically for 2 weeks following the shift. For the SCN, the master pacemaker in the brain, we employed high‐resolution imaging of the brain slice to describe the resynchronization of rhythms in single SCN neurons during adjustment to the new light cycle. We observed significant differences in shifting kinetics among mice, among organs such as the spleen and lung, and importantly among neurons in the SCN. The phase distribution among all Period2‐expressing SCN neurons widened on the day following a shift of the light cycle, which was partially due to cells in the ventral SCN exhibiting a larger initial phase shift than cells in the dorsal SCN. There was no clear delineation of ventral and dorsal regions, however, as the SCN appear to have a population of fast‐shifting cells whose anatomical distribution is organized in a ventral–dorsal gradient. Full resynchronization of the SCN and peripheral timing system, as measured by a circadian reporter gene, did not occur until after 8 days in the advanced light cycle.

Visualizing jet lag in the mouse suprachiasmatic nucleus and peripheral circadian timing system

Interactions among suprachiasmatic nucleus (SCN) neurons are required for robust circadian rhythms entrained to local time. To investigate these signaling mechanisms, we developed a functional coupling assay that uniquely captures the dynamic process by which SCN neurons interact. As a population, SCN neurons typically display synchronized rhythms with similar peak times, but will peak 6-12 hr apart after in vivo exposure to long days. Once they are removed from these conditions, SCN neurons resynchronize through a phase-dependent coupling process mediated by both vasoactive intestinal polypeptide (VIP) and GABAA signaling. Notably, GABAA signaling contributes to coupling when the SCN network is in an antiphase configuration, but opposes synchrony under steady-state conditions. Further, VIP acts together with GABAA signaling to couple the network in an antiphase configuration, but promotes synchrony under steady-state conditions by counteracting the actions of GABAA signaling. Thus, SCN neurons interact through nonredundant coupling mechanisms influenced by the state of the network.

Dynamic interactions mediated by nonredundant signaling mechanisms couple circadian clock neurons.

The mammalian pacemaker in the suprachiasmatic nucleus (SCN) contains a population of neural oscillators capable of sustaining cell-autonomous rhythms in gene expression and electrical firing. A critical question for understanding pacemaker function is how SCN oscillators are organized into a coherent tissue capable of coordinating circadian rhythms in behavior and physiology. Here we undertake a comprehensive analysis of oscillatory function across the SCN of the adult PER2::LUC mouse by developing a novel approach involving multi-position bioluminescence imaging and unbiased computational analyses. We demonstrate that there is phase heterogeneity across all three dimensions of the SCN that is intrinsically regulated and extrinsically modulated by light in a region-specific manner. By investigating the mechanistic bases of SCN phase heterogeneity, we show for the first time that phase differences are not systematically related to regional differences in period, waveform, amplitude, or brightness. Furthermore, phase differences are not related to regional differences in the expression of arginine vasopressin and vasoactive intestinal polypeptide, two key neuropeptides characterizing functionally distinct subdivisions of the SCN. The consistency of SCN spatiotemporal organization across individuals and across planes of section suggests that the precise phasing of oscillators is a robust feature of the pacemaker important for its function.

Intrinsic Regulation of Spatiotemporal Organization within the Suprachiasmatic Nucleus

Aging produces a decline in the amplitude and precision of 24 h behavioral, endocrine, and metabolic rhythms, which are regulated in mammals by a central circadian pacemaker within the suprachiasmatic nucleus (SCN) and local oscillators in peripheral tissues. Disruption of the circadian system, as experienced during transmeridian travel, can lead to adverse health consequences, particularly in the elderly. To test the hypothesis that age-related changes in the response to simulated jet lag will reflect altered circadian function, we examined re-entrainment of central and peripheral oscillators from young and old PER2::luciferase mice. As in previous studies, locomotor activity rhythms in older mice required more days to re-entrain following a shift than younger mice. At the tissue level, effects of age on baseline entrainment were evident, with older mice displaying earlier phases for the majority of peripheral oscillators studied and later phases for cells within most SCN subregions. Following a 6 h advance of the light:dark cycle, old mice displayed slower rates of re-entrainment for peripheral tissues but a larger, more rapid SCN response compared to younger mice. Thus, aging alters the circadian timing system in a manner that differentially affects the re-entrainment responses of central and peripheral circadian clocks. This pattern of results suggests that a major consequence of aging is a decrease in pacemaker amplitude, which would slow re-entrainment of peripheral oscillators and reduce SCN resistance to external perturbation.

Aging differentially affects the re-entrainment response of central and peripheral circadian oscillators

Biological oscillators naturally exhibit stochastic fluctuations in period and amplitude due to the random nature of molecular reactions. Accurately measuring the precision of noisy oscillators and the heterogeneity in period and strength of rhythmicity across a population of cells requires single-cell recordings of sufficient length to fully represent the variability of oscillations. We found persistent, independent circadian oscillations of clock gene expression in 6-week-long bioluminescence recordings of 80 primary fibroblast cells dissociated from PER2::LUC mice and kept in vitro for 6 months. Due to the stochastic nature of rhythmicity, the proportion of cells appearing rhythmic increases with the length of interval examined, with 100% of cells found to be rhythmic when using 3-week windows. Mean period and amplitude are remarkably stable throughout the 6-week recordings, with precision improving over time. For individual cells, precision of period and amplitude are correlated with cell size and rhythm amplitude, but not with period, and period exhibits much less cycle-to-cycle variability (CV 7.3%) than does amplitude (CV 37%). The time series are long enough to distinguish stochastic fluctuations within each cell from differences among cells, and we conclude that the cells do exhibit significant heterogeneity in period and strength of rhythmicity, which we measure using a novel statistical metric. Furthermore, stochastic modeling suggests that these single-cell clocks operate near a Hopf bifurcation, such that intrinsic noise enhances the oscillations by minimizing period variability and sustaining amplitude.

Persistent Cell-Autonomous Circadian Oscillations in Fibroblasts Revealed by Six-Week Single-Cell Imaging of PER2::LUC Bioluminescence

Analysis of circadian oscillations that exhibit variability in period or amplitude can be accomplished through wavelet transforms. Wavelet-based methods can also be used quite effectively to remove trend and noise from time series and to assess the strength of rhythms in different frequency bands, for example, ultradian versus circadian components in an activity record. In this article, we describe how to apply discrete and continuous wavelet transforms to time series of circadian rhythms, illustrated with novel analyses of 2 case studies involving mouse wheel-running activity and oscillations in PER2::LUC bioluminescence from SCN explants.

Wavelet-Based Time Series Analysis of Circadian Rhythms

This article offers an accessible but rigorous and essentially self-contained account of some of the central ideas in compressed sensing, aimed at nonspecialists and undergraduates who have had linear algebra and some probability. The basic premise is first illustrated by considering the problem of detecting a few defective items in a large set. We then build up the mathematical framework of compressed sensing to show how combining efficient sampling methods with elementary ideas from linear algebra and a bit of approximation theory, optimization, and probability allows the estimation of unknown quantities with far less sampling of data than traditional methods.

Making Do with Less: An Introduction to Compressed Sensing

Circadian neural circuits generate near 24-hr physiological rhythms that can be entrained by light to coordinate animal physiology with daily solar cycles. To examine how a circadian circuit reorganizes its activity in response to light, we imaged period (per) clock gene cycling for up to 6 days at single-neuron resolution in whole-brain explant cultures prepared from per-luciferase transgenic flies. We compared cultures subjected to a phase-advancing light pulse (LP) to cultures maintained in darkness (DD). In DD, individual neuronal oscillators in all circadian subgroups are initially well synchronized but then show monotonic decrease in oscillator rhythm amplitude and synchrony with time. The small ventral lateral neurons (s-LNvs) and dorsal lateral neurons (LNds) exhibit this decrease at a slower relative rate. In contrast, the LP evokes a rapid loss of oscillator synchrony between and within most circadian neuronal subgroups, followed by gradual phase retuning of whole-circuit oscillator synchrony. The LNds maintain high rhythmic amplitude and synchrony following the LP along with the most rapid coherent phase advance. Immunocytochemical analysis of PER shows that these dynamics in DD and LP are recapitulated in vivo. Anatomically distinct circadian neuronal subgroups vary in their response to the LP, showing differences in the degree and kinetics of their loss, recovery and/or strengthening of synchrony, and rhythmicity. Transient desynchrony appears to be an integral feature of light response of the Drosophila multicellular circadian clock. Individual oscillators in different neuronal subgroups of the circadian circuit show distinct kinetic signatures of light response and phase retuning.

Light Evokes Rapid Circadian Network Oscillator Desynchrony Followed by Gradual Phase Retuning of Synchrony

A variety of methods to determine phase markers and period length from experimental data sets have traditionally assumed a rhythm of fixed period and amplitude. But most biological oscillations exhibit fluctuations in both period and amplitude, leading to the recent interest in the application of wavelet transforms that can measure how rhythms vary over time. Here we examine how wavelet-based methods can be extended to the analysis of conventional actograms, including the detection of onsets in circadian activity and temperature rhythms of rodents.