Circadian rhythm, the 24-hour cycle that governs our waking and sleeping, has been a familiar concept since the beginning of mankind. Yet only recently have researchers uncovered the biological mechanism for creating this cycle. We now know that a complex relationship between a small number of genes governs the human circadian cycle under the influence of the suprachiasmatic nucleus (SCN) — an area of the brain behind the eyes. The SCN governs peripheral clocks that regulate a panoply of processes, including hormonal cycles and even mitochondrial energy regulation.
Intriguing work in several labs has suggested the presence of intrinsic rhythms in cells that are not governed by the SCN and indeed are not 24-hour circadian in nature. While published data hinted at the ubiquity of these rhythms, a new mathematical approach was necessary to uncover them.
Figure i represents RNA as detected by microarray via two probes. This raw data suggest oscillations. When the eigenvalue pencil is applied, each probe can be separated and the oscillations become clear. Data courtesy of .
The central dogma of genetics is that DNA makes RNA, which in turn makes protein. Work in recent decades has modified the dogma (with some heresy thrown in), and now we know that the process is regulated at several points. Complex feedback loops in cells generate autonomous rhythms of RNA (transcription) and protein (translation) production. If we consider the RNA outputs of oscillating genes over time as signals, we can profitably apply signal processing mathematics to this type of analysis.
We used a novel signal processing method that employs the eigenvalue decomposition of a pencil of matrices. This approach is ideally suited for transcription examination, in that it is not biased to finding periodicity. Additionally, it can identify independent oscillations by virtue of the angles between signals. If the angle approaches 90\(^\circ\), we conclude that the oscillations are orthogonal. In this sense, orthogonality means that further computations will not affect an oscillation once it has been determined. In other words, the fundamental oscillations are independent of each other. This is a critical point, as previous work has suggested that when researchers discover a rhythm other than the 24-hour circadian rhythm they can consider it a harmonic. The eigenvalue pencil paved the way for biological experiments proving that the 12-hour rhythm in cells is autonomous, not dependent on circadian cycles [1, 4].
The eigenvalue pencil is a new arrow in the figurative quiver of the biologist. Other applications of this technique to natural sciences include the study of turbulence in wind farms . Taken together, this combination of digital signal processing and fundamental biology emphasizes the universality of science, called “musica universalis” .
Acknowledgments: Athanasios C. Antoulas acknowledges funding from National Science Foundation, CCF-1320866; the German Science Foundation, AN-693/1-1; and the Max-Planck Institute for the Dynamics of Complex Technical Systems. Clifford C. Dacso acknowledges funding from the Center for the Advancement of Science in Space, GA-2014-136; the National Science Foundation, CISE-11703170; the National Science Foundation, NeTS: 1801865; the Brockman Medical Research Foundation; the National Institutes of Health, 1PO1DK113954-01A1; Philip J. Carroll, Jr. Professorship; the Joyce Family Foundation; and Sonya and William Carpenter.
 Antoulas, AC, Zhu, B., Zhang, Q., York, B., O’Malley, B.W., & Dacso, C.C. (2018). A novel mathematical method for disclosing oscillations in gene transcription: A comparative study. PLoSOne, 13(9), e0198503.
 Khodkar, M.A., Antoulas, A.C., & Hassanzadeh, P. (2018). Data-driven Spatio-temporal Prediction of High-dimensional Geophysical Turbulence using Koopman Operator Approximation Preprint, arXiv:1812.09438.
 Zhu, B., Dacso, C.C., & O’Malley, B.W. (2018). Unveiling ‘Musica Universalis’ of the Cell: A Brief History of Biological 12-Hour Rhythms. J. Endo. Soc., 2(7), 727-752.
 Zhu, B., Zhang, Q., Pan, Y., Mace, E.M., York, B., Antoulas, A.C., Dacso, C.C., & O’Malley, B.W. (2017). A Cell-Autonomous Mammalian 12 hr Clock Coordinates Metabolic and Stress Rhythms. Cell Metab., 25(6), 1305-1319.