Signal processing in cellular clocks

Abstract

Many biochemical events within a cell need to be timed properly to occur at specific times of day, after other events have happened within the cell or in response to environmental signals. The cellular biochemical feedback loops that time these events have already received much recent attention in the experimental and modeling communities. Here, we show how ideas from signal processing can be applied to understand the function of these clocks. Consider two signals from the network sðtÞ and rðtÞ, either two variables of a model or two experimentally measured time courses. We show how sðtÞ can be decomposed into two parts, the first being a function of rðtÞ, and the second the derivative of a function of rðtÞ. Geometric principles are then derived that can be used to understand when oscillations appear in biochemical feedback loops, the period of these oscillations, and their time course. Specific examples of this theory are provided that show how certain networks are prone or not prone to oscillate, how individual biochemical processes affect the period, and how oscillations in one chemical species can be deduced from oscillations in other parts of the network.biochemical clocks | circadian rhythms | gene networks | biological time M any biological systems perform specific functions at specific times (1). On a 24-h (circadian) timescale, intracellular circadian clocks trigger biological events to occur at specific times of the day. Faster, noncircadian clocks properly time many other events, such as those that occur in development, in cell division, and in metabolism (2-7). Our knowledge of both of these types of clocks has grown tremendously in the past two decades with new experimental techniques, a growing library of mathematical models (8, 9), and even the building of synthetic cellular clocks (10, 11).However, in each of these cellular clocks, many genes and proteins work together in complex ways to produce oscillations. A new challenge has arisen in incorporating all these recent results into a mathematical theory that can be used along with computation and experimentation (12) to better understand the system's behavior. One possibility, recently advocated by Sontag for biological clocks (13), is to use ideas from signal processing and Hilbert space. Although this approach has been around for over 30 y (2), it has received limited attention with respect to biological clocks, probably because the biochemistry of clocks is often nonlinear (14), and most techniques consider linear systems (15). Here, we use these mathematical ideas to determine how to relate different oscillating elements in a genetic network.In this current study, we draw upon this approach to show that oscillating signals in nonlinear biochemical clocks obey geometric properties. We apply these properties to three questions of wide study: (i) When are oscillations possible in biochemical feedback loops? (ii) How do individual components of the biochemical feedback loops determine the period? (iii) How does one element of the feedback loop in...

Engineered temperature compensation in a synthetic genetic clock

“…Experimental evidence suggests that the thermal stability of the protein and the temperature dependent splicing of its mRNA play an important role in temperature compensation (45,46). Simple mathematical models based on Goodwin's oscillator (45,48) have already shown that these mechanisms can impart temperature compensation. Similarly in mammalian circadian clocks, the Per1 gene provides negative feedback and has been implicated in temperature compensation (49).…”

Protein sequestration versus Hill‐type repression in circadian clock models

“…Similar to the HT models [91][92][93] Hill exponent is critical for HT models to generate rhythms ( Fig. 2b), a large number of phosphorylation sites at the activator (WCC) is required in Neurospora.…”

“…In particular, for both models, it has been shown that the logarithmic sensitivity should be greater than 8 at the steady state (see Appendix for detailed analysis) [34,59,[91][92][93]. Importantly, conditions to achieve such high logarithmic sensitivity differ depending on the repression mechanisms (Figs.…”

“…For instance, the required Hill exponent decreases as more intermediate steps are included to generate time delay in the NFL (Eq. 1), which is known as the secant condition (see Appendix for details) [91][92][93][94]. The Michaelis-Menten type of repressor clearance also reduces the necessary Hill exponent as it can serve as an additional source of non-linearity [35,[95][96][97][98].…”

Regulation of oscillation dynamics in biochemical systems with dual negative feedback loops

“…Symmetry among elements within a feedback loop was predicted to increase the occurrence likelihood of oscillations [35]. It is, therefore, interesting to study how the degradation rates affect systems dynamics in our dual-feedback context.…”