14 Oscillations, Rhythms and Synchronized Time Bases …
235
One application of this Eq. (14.1) is in describing the density of individuals x t , as
a population whose growth constant is a.
Mathematical methods for exposing the complex behaviour of continuous yeast
cultures has been shown to satisfy several of the criteria [46, 59, 83, 84] for
deterministic chaos:
1. Under ‘permittistatical’ control of a continuous yeast culture, using the output
from an AC impedance measuring device to regulate growth rate by the medium
supply rate [23].
2. Stepwise decreased medium pH reveals increased complexities of trajectories
and the uncovering of a strange attractor [124].
3. Addition to cultures of a type-A monoamine oxidase inhibitor, phenelzine, gave
period 2 doubling [82, 145].
4. A long-term (3 months) culture yielding more than 36,000 points at 12 s intervals for each of 4 dissolved gases (O 2 , CO 2 , H 2 S and Ar) using an immersed
membrane inlet mass spectrometer probe indicated a low-dimensional chaotic
attractor [144].
Figure 14.3a, b shows the dissolved oxygen signal versus time. The 13 h collective
mode and bursts in circahoralian oscillatory activity are clearly visible in panel (a).
Panel (b), is an enlargement of the boxed region in panel (a), showing the 4-min
oscillations as well as the periodic re-emergence of the circahoralian rhythm. Panel (c)
shows the metabolic attractor seen in an [O 2 ], [H 2 S] projection, with points coloured
by the (baseline- corrected) CO 2 signal. The line (which is actually a plane extending
in the direction of the [CO 2 ] axis) was obtained by a simple linear regression of the
[H 2 S] versus [O 2 ] data. Panel (d) shows a section through the attractor at the level
of the plane in panel (c), which is used as simple way to de-trend the data and chose
a plane running roughly through the middle of the attractor. These short-period
oscillations were also visible in recordings from the O 2 electrode when the culture
exhibited simpler dynamics.
A metabolic attractor (the set of biochemical states visited by the culture after
decay of initial transients) of a time-series obtained for all three dissolved gases
(normalized for variations in Ar as an inert reference gas) at 12-s intervals directly
in the culture exhibited several characteristics indicative of chaotic behavior. The
attractor is shown in Fig. 14.3c, d. The capacity dimension of the attractor [28], one
measure of fractal dimension, was 2.09 ± 0.07 (95% confidence). A capacity dimension close to two suggests that this is neither a simple cycle (D = 1.0), nor a system
filling a three-dimensional (3-D) region of phase space. Although the attractor looks
like a mostly solid structure seen from the perspective of Fig. 14.3c, cuts through the
attractor show a complex structure, with regions that are nearly completely filled, and
others containing only a few points, e.g., Fig. 14.3d, which explains the dimension
that is <3. A dimension of two could be compatible with a quasiperiodic attractor.
However, an estimate of the Lyapunov exponent gave a value of 0.752 ± 0.004 h
−1 .
A positive Lyapunov exponent implies exponential divergence of nearby trajectories,
which is the signature of chaos [59]. The 3–5 min signal was also observed associated
with high respiration levels in single cells [5]. Self-similar (fractal) time structure
235
One application of this Eq. (14.1) is in describing the density of individuals x t , as
a population whose growth constant is a.
Mathematical methods for exposing the complex behaviour of continuous yeast
cultures has been shown to satisfy several of the criteria [46, 59, 83, 84] for
deterministic chaos:
1. Under ‘permittistatical’ control of a continuous yeast culture, using the output
from an AC impedance measuring device to regulate growth rate by the medium
supply rate [23].
2. Stepwise decreased medium pH reveals increased complexities of trajectories
and the uncovering of a strange attractor [124].
3. Addition to cultures of a type-A monoamine oxidase inhibitor, phenelzine, gave
period 2 doubling [82, 145].
4. A long-term (3 months) culture yielding more than 36,000 points at 12 s intervals for each of 4 dissolved gases (O 2 , CO 2 , H 2 S and Ar) using an immersed
membrane inlet mass spectrometer probe indicated a low-dimensional chaotic
attractor [144].
Figure 14.3a, b shows the dissolved oxygen signal versus time. The 13 h collective
mode and bursts in circahoralian oscillatory activity are clearly visible in panel (a).
Panel (b), is an enlargement of the boxed region in panel (a), showing the 4-min
oscillations as well as the periodic re-emergence of the circahoralian rhythm. Panel (c)
shows the metabolic attractor seen in an [O 2 ], [H 2 S] projection, with points coloured
by the (baseline- corrected) CO 2 signal. The line (which is actually a plane extending
in the direction of the [CO 2 ] axis) was obtained by a simple linear regression of the
[H 2 S] versus [O 2 ] data. Panel (d) shows a section through the attractor at the level
of the plane in panel (c), which is used as simple way to de-trend the data and chose
a plane running roughly through the middle of the attractor. These short-period
oscillations were also visible in recordings from the O 2 electrode when the culture
exhibited simpler dynamics.
A metabolic attractor (the set of biochemical states visited by the culture after
decay of initial transients) of a time-series obtained for all three dissolved gases
(normalized for variations in Ar as an inert reference gas) at 12-s intervals directly
in the culture exhibited several characteristics indicative of chaotic behavior. The
attractor is shown in Fig. 14.3c, d. The capacity dimension of the attractor [28], one
measure of fractal dimension, was 2.09 ± 0.07 (95% confidence). A capacity dimension close to two suggests that this is neither a simple cycle (D = 1.0), nor a system
filling a three-dimensional (3-D) region of phase space. Although the attractor looks
like a mostly solid structure seen from the perspective of Fig. 14.3c, cuts through the
attractor show a complex structure, with regions that are nearly completely filled, and
others containing only a few points, e.g., Fig. 14.3d, which explains the dimension
that is <3. A dimension of two could be compatible with a quasiperiodic attractor.
However, an estimate of the Lyapunov exponent gave a value of 0.752 ± 0.004 h
−1 .
A positive Lyapunov exponent implies exponential divergence of nearby trajectories,
which is the signature of chaos [59]. The 3–5 min signal was also observed associated
with high respiration levels in single cells [5]. Self-similar (fractal) time structure