A proposal for robust temperature compensation of circadian rhythms
Christian I. Hong*,†Emery D. Conrad‡, and
John J. Tyson*,§
Departments of *Biological Sciences and
‡Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24060; and
†Department of Genetics, Dartmouth Medical School, Hanover, NH 03755
PNAS January 23, 2007 vol. 104 no. 4 1195-1200 [Open Access]Abstract
The
internal circadian rhythms of cells and organisms coordinate their
physiological properties to the prevailing 24-h cycle of light and dark
on earth. The mechanisms generating circadian rhythms have four
defining characteristics: they oscillate endogenously with period close
to 24 h, entrain to external signals, suffer phase shifts by aberrant
pulses of light or temperature, and compensate for changes in
temperature over a range of 10°C or more. Most theoretical descriptions
of circadian rhythms propose that the underlying mechanism generates a
stable limit cycle oscillation (in constant darkness or dim light),
because limit cycles quite naturally possess the first three defining
properties of circadian rhythms. On the other hand, the period of a
limit cycle oscillator is typically very sensitive to kinetic rate
constants, which increase markedly with temperature. Temperature
compensation is therefore not a general property of limit cycle
oscillations but must be imposed by some delicate balance of
temperature dependent effects. However, “delicate balances” are
unlikely to be robust to mutations. On the other hand, if circadian
rhythms arise from a mechanism that concentrates sensitivity into a few
rate constants, then the “balancing act” is likely to be more robust
and evolvable. We propose a switch-like mechanism for circadian rhythms
that concentrates period sensitivity in just two parameters, by forcing
the system to alternate between a stable steady state and a stable
limit cycle.
Since the breakthrough discovery of the period (per) gene by Konopka and Benzer in 1971 (1),
molecular biologists have identified many new circadian rhythm genes
and have uncovered a complex network of interacting feedback loops
which comprise the control system. In the consensus view, an endogenous
daily rhythm is created by a negative feedback loop whereby the PERIOD
(PER) protein inhibits its own expression by interfering with
transcription factors (2, 3). This mechanism has been studied in great detail theoretically by many authors (4–12).
The time-delayed negative feedback loop generates limit cycle
oscillations with many properties characteristic of physiological daily
rhythms, except for one: the autonomous circadian period (T) is quite insensitive to variations of the kinetic constants, a property that is not characteristic of generic limit cycle
oscillators. This insensitivity shows up in two ways: (i) T varies little among individual organisms even though individuals show considerable genetic and/or proteomic variability that
translates into variations of kinetic parameters, and (ii) T is temperature compensated, even though kinetic constants are strongly temperature dependent. Physiologically, this robustness
of the period (T
≈ 24 h despite genetic variability and environmental fluctuations) is
essential to circadian physiology. If the autonomous period of the
clock drifts too far from 24 h, then the circadian rhythm would not
reliably entrain to the external 24 h light/dark Zeitgeber.
(i) Consider first the tight distribution of T [24 ± 1 h, coefficient of variation (CV) = 4%] across populations of fruit flies (13, 14). In general, the period of a limit cycle oscillator depends on many rate constants (ki), and the variability of T as a result of rate constant variations is given by 
Now, the relative variability of rate constants between individuals is
likely to be very large (e.g., 50% in the case of heterozygosity for a
loss-of-function mutation), and the variability in one parameter is
surely independent of the variability in another parameter. So, the
only way that CV can be small in the face of arbitrary, large Δki values is for T somehow to be independent of most rate constants in the mechanism.
(ii) Temperature compensation leads us to the same conclusion. Rate constants depend on temperature (θ) according to Arrhenius'
law, ki = αie
−Eirθ, where R is the universal gas constant, αi determines the value of ki at θ = 298 K, and Ei is the activation energy of the ith reaction. Ruoff and colleagues (15) pointed out that a limit cycle oscillator would be temperature compensated if, according to the chain rule, 
This sum is a balance of positive and negative terms (because ∂T/∂ki is positive for some i
and negative for others), and it can always be set close to zero by
choosing a suitable set of activation energies. The proposal of Ruoff
and colleagues is a reasonable and popular explanation of temperature
compensation (16–19).
If Ruoff's balance hypothesis is correct, we would expect that most
mutations of circadian rhythm genes (which change kinetic constants and
activation energies in the underlying control system) are likely to
disrupt this balance and, therefore, to exhibit failures in temperature
compensation of the circadian rhythm. As expected, there are mutants
with defective temperature compensation in both Neurospora crassa and Drosophila melanogaster (Table 1). However, more intriguingly, geneticists (13, 14, 20–23) have identified many circadian rhythm mutations (aberrant period) that leave temperature compensation intact; indeed, 60–70%
of the representative mutants in Table 1 maintain temperature compensation. In order for temperature compensation to survive in the face of a variety of mutations
at many different places in the mechanism, many terms in the balance equation must be negligible, i.e., ∂T/∂ki · kiEi ≅ 0 for many i. It is unlikely that Ei ≅ 0 for many kinetic constants, leading us to the conclusion that ∂T/∂ki ≅ 0 for many ki
values. Hence, the mechanism of circadian rhythms must somehow generate
a 24-h period independently of the precise values of many of the rate
constants in the mechanism. Based on the fact that the majority of
circadian rhythm mutants maintain temperature compensation, we suggest
that temperature compensation is not the result of a delicate balance
of opposing influences of all of the rate constants in the mechanism,
but rather that temperature compensation is embedded directly in the
molecular machinery, analogous to the way perfect adaptation appears to
be embedded in the mechanism of bacterial chemotaxis (24).
The cell division cycle (25)
has a similar kind of compensation embedded in its control mechanism:
it is a periodic process governed by a complex regulatory network, but
the period of the cell cycle is completely independent of the rate
constants in the underlying network. For unicellular organisms, like
yeast, the period of the cell cycle is always equal to the mass
doubling time of the culture, Td = ln(2)/μg, where μg
= specific growth rate, which is a function of the nutritional value of
the growth medium, not the rate constants for the kinase and
phosphatase reactions that dominate the control network. This
physiological property of “balanced growth and division,” which is
essential to the reproduction of unicellular organisms, is a
consequence of the control mechanism, which triggers cell division
(2-fold reduction of cell mass) every time growth increases cell mass
by 2-fold (26). A similar “resetting mechanism” for the circadian rhythm might be responsible for minimizing the number of rate constants
that strongly influence its period.
If ∂T/∂ki ≅ 0 for most ki valuess, then the robustness of circadian period is readily understood. (i) CV = ΔT/T is small because most terms in the sum, 
are zero; and (ii) ∂T/∂θ ≅ 0 because most terms in the sum, 
are also zero. Only mutations in specific genes i, for which ∂T/∂ki ≠ 0, will have strong effects on circadian period and temperature compensation. Identifying these mutations may lead us to
that part of the mechanism responsible for robustness of circadian period.
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