The Malthusian parameter of ascents: What prevents the exponential increase of one’s ancestors?

The reason that the indefinite exponential increase in the number of one’s ancestors does not take place is found in the law of sibling interference, which can be expressed by the following simple equation:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\begin{matrix}{\mathit{N}}_{{\mathit{n}}} \enskip & \\ {\mathit{{\blacksquare}}} \enskip & \\ {\mathit{ASZ}} \enskip & \end{matrix} {\mathrm{\hspace{.167em}{\times}\hspace{.167em}2\hspace{.167em}=\hspace{.167em}}}{\mathit{N_{n+1},}}\end{equation*}\end{document} where Nn is the number of ancestors in the nth generation, ASZ is the average sibling size of these ancestors, and Nn+1 is the number of ancestors in the next older generation (n + 1). Accordingly, the exponential increase in the number of one’s ancestors is an initial anomaly that occurs while ASZ remains at 1. Once ASZ begins to exceed 1, the rate of increase in the number of ancestors is progressively curtailed, falling further and further behind the exponential increase rate. Eventually, ASZ reaches 2, and at that point, the number of ancestors stops increasing for two generations. These two generations, named AN SA and AN SA + 1, are the most critical in the ancestry, for one’s ancestors at that point come to represent all the progeny-produced adults of the entire ancestral population. Thereafter, the fate of one’s ancestors becomes the fate of the entire population. If the population to which one belongs is a successful, slowly expanding one, the number of ancestors would slowly decline as you move toward the remote past. This is because ABZ would exceed 2. Only when ABZ is less than 2 would the number of ancestors increase beyond the AN SA and AN SA + 1 generations. Since the above is an indication of a failing population on the way to extinction, there had to be the previous AN SA involving a far greater number of individuals for such a population. Simulations indicated that for a member of a continuously successful population, the AN SA ancestors might have numbered as many as 5.2 million, the AN SA generation being the 28th generation in the past. However, because of the law of increasingly irrelevant remote ancestors, only a very small fraction of the AN SA ancestors would have left genetic traces in the genome of each descendant of today.

Interlocked feedback loops contribute to the robustness of the Neurospora circadian clock

Interlocked feedback loops may represent a common feature among the regulatory systems controlling circadian rhythms. The Neurospora circadian feedback loops involve white collar-1 (wc-1), wc-2, and frequency (frq) genes. We show that WC-1 and WC-2 proteins activate the transcription of frq gene, whereas FRQ protein plays dual roles: repressing its own transcription, probably by interacting with the WC-1/WC-2 complex, and activating the expression of both WC proteins. Thus, they form two interlocked feedback loops: one negative and one positive. We establish the physiological significance of the interlocked positive feedback loops by showing that the levels of WC-1 and WC-2 determine the robustness and stability of the clock. Our data demonstrate that with WC-1 being the limiting factor in the WC-1/WC-2 complex, the greater the levels of WC-1 and WC-2, the higher the level of the FRQ oscillation and the more robust the overt rhythms. Our data also show that, despite considerable changes in the levels of WC-1, WC-2, and FRQ, the period of the clock has been limited to a small range, suggesting that the interlocked circadian feedback loops are also important for determining the circadian period length of the clock.