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S. Yuvan and M. Bier
fraction of the total number of oscillators pushes the phase transition to a higher value
of the coupling coefficient or, equivalently, to a smaller value of the temperature. With
this insight we expect that the critical exponent value of 1/2 will also persist if the
transition rates in Fig. 8.3a are made to vary along the cycle or if other mathematical
forms for the population dependencies of the the transition rates are tried.
There is autocatalysis or product stimulation in the system depicted in Fig. 8.3a,
i.e., the product of the j → j + 1 transition increases the rate of the j → j + 1
transition. Product stimulation is a form of positive feedback and it is commonly
the underlying driving force behind biochemical oscillations [21]. In the system in
Fig. 8.3a, the parameter α can be seen as a measure for the strength of the positive
feedback. Product stimulation is a key feature in the glycolytic oscillations that were
already mentioned in Sect. 8.1.4 [3]. The product stimulation in glycolytic oscillations is twofold. ATP binding and hydrolysis is the first step in this metabolic chain,
but subsequently ATP is produced again at several steps in the chain. Furthermore,
early in the chain energy that is harvested from the breakdown of glucose is stored
in NAD reduction: NAD
+ + H
+ + 2e
−
→ NADH. For the final step in the chain,
the conversion from acetaldehyde to ethanol, the necessary energy is derived from
the oxidation of NADH, i.e. the reverse reaction: NADH → NAD
+ + H
+ + 2e
− .
The glycolysis consists of about ten enzyme-mediated steps and because ATP and
NADH are products as well as substrates in the chain, the chain can be seen as a
cycle much like the one in Fig. 8.3a.
In a suspension of oscillating yeast cells there is a synchronized oscillation inside
every cell. The number of oscillators inside each cell, N in , is different for each
cell. The catalyzing enzymes process the substrate molecules one-by-one and it is
substrate concentrations and the number of enzymes in each cell that ultimately
determine an effective N in and a characteristic frequency for each cell.
As was mentioned before, acetaldehyde can freely permeate the membrane of
the yeast cell. The oscillations are thus coupled through the shared acetaldehyde
concentration. It is obvious that for a suspension with a small density of yeast cells,
the acetaldehyde concentration will be close to zero and not lead to any coupling.
Reference [27] models the suspension of oscillating yeast cells with a Kuramoto
model, where the shared acetaldehyde concentration provides the coupling between
the M cells. Figure 8.5a is from Ref. [27] and shows the order parameter as a function
of the density. The order parameter r is experimentally determined by simultaneously
following the fluorescing behavior of individual cells that have been fixed in their
location. The measurements are not very precise, but appear consistent with the
phase transition and the critical exponent of 1/2 that are predicted by the Kuramoto
model. A model as in Fig. 8.3a, which also predicts a critical exponent of 1/2, may be
more chemically realistic than the Kuramoto model as it explicitly includes positive
feedbacks that underlie the oscillations.
There is still a measure of controversy about the oscillations of yeast cells in a
suspension. Reference [27] reports the observation that individual cells still oscillate
even when they are in a solution that is too dilute for coupling. In Ref. [23], on
the other hand, similar measurements were described and there it was found that
cells in a diluted solution no longer oscillate. The authors of Ref. [23] describe how
S. Yuvan and M. Bier
fraction of the total number of oscillators pushes the phase transition to a higher value
of the coupling coefficient or, equivalently, to a smaller value of the temperature. With
this insight we expect that the critical exponent value of 1/2 will also persist if the
transition rates in Fig. 8.3a are made to vary along the cycle or if other mathematical
forms for the population dependencies of the the transition rates are tried.
There is autocatalysis or product stimulation in the system depicted in Fig. 8.3a,
i.e., the product of the j → j + 1 transition increases the rate of the j → j + 1
transition. Product stimulation is a form of positive feedback and it is commonly
the underlying driving force behind biochemical oscillations [21]. In the system in
Fig. 8.3a, the parameter α can be seen as a measure for the strength of the positive
feedback. Product stimulation is a key feature in the glycolytic oscillations that were
already mentioned in Sect. 8.1.4 [3]. The product stimulation in glycolytic oscillations is twofold. ATP binding and hydrolysis is the first step in this metabolic chain,
but subsequently ATP is produced again at several steps in the chain. Furthermore,
early in the chain energy that is harvested from the breakdown of glucose is stored
in NAD reduction: NAD
+ + H
+ + 2e
−
→ NADH. For the final step in the chain,
the conversion from acetaldehyde to ethanol, the necessary energy is derived from
the oxidation of NADH, i.e. the reverse reaction: NADH → NAD
+ + H
+ + 2e
− .
The glycolysis consists of about ten enzyme-mediated steps and because ATP and
NADH are products as well as substrates in the chain, the chain can be seen as a
cycle much like the one in Fig. 8.3a.
In a suspension of oscillating yeast cells there is a synchronized oscillation inside
every cell. The number of oscillators inside each cell, N in , is different for each
cell. The catalyzing enzymes process the substrate molecules one-by-one and it is
substrate concentrations and the number of enzymes in each cell that ultimately
determine an effective N in and a characteristic frequency for each cell.
As was mentioned before, acetaldehyde can freely permeate the membrane of
the yeast cell. The oscillations are thus coupled through the shared acetaldehyde
concentration. It is obvious that for a suspension with a small density of yeast cells,
the acetaldehyde concentration will be close to zero and not lead to any coupling.
Reference [27] models the suspension of oscillating yeast cells with a Kuramoto
model, where the shared acetaldehyde concentration provides the coupling between
the M cells. Figure 8.5a is from Ref. [27] and shows the order parameter as a function
of the density. The order parameter r is experimentally determined by simultaneously
following the fluorescing behavior of individual cells that have been fixed in their
location. The measurements are not very precise, but appear consistent with the
phase transition and the critical exponent of 1/2 that are predicted by the Kuramoto
model. A model as in Fig. 8.3a, which also predicts a critical exponent of 1/2, may be
more chemically realistic than the Kuramoto model as it explicitly includes positive
feedbacks that underlie the oscillations.
There is still a measure of controversy about the oscillations of yeast cells in a
suspension. Reference [27] reports the observation that individual cells still oscillate
even when they are in a solution that is too dilute for coupling. In Ref. [23], on
the other hand, similar measurements were described and there it was found that
cells in a diluted solution no longer oscillate. The authors of Ref. [23] describe how