8 Synchronization of Coupled Oscillators—Phase Transitions …
137
Fig. 8.3 a Our n-state model of coupled oscillators. Each oscillator moves through the n-state
cycle in the counterclockwise direction. The transition rate, k i , from state i to state i + 1 depends
on populations in states i − 1 and i + 1. b For sufficiently low values of the coupling parameter,
α, there is a homogeneous distribution over the states, i.e. p i = 1/n ∀ i. For values just above the
critical value, α c , we construct a model in which we assume that the population in a state is either
(1 + ε)/n or (1 − ε)/n, where ε is small
pump sodium and potassium ions through the membrane against the electrochemical
potential [18]. Na,K-ATPase can be present in the membrane in large concentrations. These proteins are very polar and they can be coupled through dipole-dipole
interaction. But they can also interact as they change shape in the course of the catalytic cycle and thus deform the cytoskeleton and bilayer membrane. Through these
couplings and the mechanism of Fig. 8.3, the pumps can synchronize their catalytic
cycles.
A more provocative example of the dynamics of Fig. 8.3a and Eq. (8.5) can be
found in bicycle racing. On a flat road with a smooth asphalt surface, more than 95%
of the effort goes into overcoming air resistance. Due to aerodynamic drag, bicycle
racers that ride in a group put in a smaller effort as compared to when they ride alone
[6, 9]. The larger the group, the smaller the power that has to be produced by an
individual rider. Bicycle racers thus tend to cluster in groups where they share the
burden. However, when the road goes uphill, work against gravity has to be done.
Furthermore, on a surface of cobblestones or gravel, the rolling resistance takes on
more significance. Work against gravity or rolling resistance cannot be shared and
the “racers’ coupling,” the α in our model, thus goes down. This is why it is generally
on hill climbs or on cobblestones that the peleton in a bicycle race breaks up [7].
137
Fig. 8.3 a Our n-state model of coupled oscillators. Each oscillator moves through the n-state
cycle in the counterclockwise direction. The transition rate, k i , from state i to state i + 1 depends
on populations in states i − 1 and i + 1. b For sufficiently low values of the coupling parameter,
α, there is a homogeneous distribution over the states, i.e. p i = 1/n ∀ i. For values just above the
critical value, α c , we construct a model in which we assume that the population in a state is either
(1 + ε)/n or (1 − ε)/n, where ε is small
pump sodium and potassium ions through the membrane against the electrochemical
potential [18]. Na,K-ATPase can be present in the membrane in large concentrations. These proteins are very polar and they can be coupled through dipole-dipole
interaction. But they can also interact as they change shape in the course of the catalytic cycle and thus deform the cytoskeleton and bilayer membrane. Through these
couplings and the mechanism of Fig. 8.3, the pumps can synchronize their catalytic
cycles.
A more provocative example of the dynamics of Fig. 8.3a and Eq. (8.5) can be
found in bicycle racing. On a flat road with a smooth asphalt surface, more than 95%
of the effort goes into overcoming air resistance. Due to aerodynamic drag, bicycle
racers that ride in a group put in a smaller effort as compared to when they ride alone
[6, 9]. The larger the group, the smaller the power that has to be produced by an
individual rider. Bicycle racers thus tend to cluster in groups where they share the
burden. However, when the road goes uphill, work against gravity has to be done.
Furthermore, on a surface of cobblestones or gravel, the rolling resistance takes on
more significance. Work against gravity or rolling resistance cannot be shared and
the “racers’ coupling,” the α in our model, thus goes down. This is why it is generally
on hill climbs or on cobblestones that the peleton in a bicycle race breaks up [7].