
MBW:Dissipative Structures for an Allosteric ModelFrom MathBio(Redirected from Dissipative Structures for an Allosteric Model)
The following paper was reviewed by JaeAnn Dwulet and Zach Vaughan. Goldbeter A. and R. Lefever. 1972. Dissipative structures for an allosteric model. Biophysical Journal 12:13021315. ^{[1]} ContentsExecutive SummaryThis paper presents an allosteric model that describes oscillations in a monosubstrate enzyme reaction. In this model, the oscillations occur as a result of the regulation of the enzyme pathway by its metabolites. The model presented in this paper describes a system that is activated by its product. The system can form dissipative structures independent of time or space and these structures are characterized by their organization when the system is thermodynamically far from equilibirum. Two models are presented to represent both types of instabilites, in time or in space. The temporal case is studied numerically and the results suggest that there exists a region which consists of limit cycle behavior. Sustained oscillations are favored by substrate activation. Oscillations can still occur with substrate inhibition but the range decreases. The conditions for the spacial instabilities always occur before instabilities involving diffusion. The model is applied to the phosphofructokinase pathway in glycolysis and is responsible for glycolitic oscillations. The phosphofructokinase pathway is activated by one of its products ADP and therefor can be described accurately with this model. ContextThe biological phenonmenon under consideration is the ability of allosteric proteins, specifically enzymes, to form organized structures far from equilibrium. These structures are called dissipative structures. The paper includes discussion about an application to glycolytic oscillations in which the phosphofructokinase pathway is discussed. The positive feedback loop or the activation by one of enzyme's products, ADP, provides the allosteric character that fits the model presented. History
Biological Background
Ilya Prigogine used the phrase “dissipative structures” to describe the organization of structures far from thermodynamic equilibrium. These structures follow specific nonlinear kinetics. Dissipative structures develop in open systems, systems that are in contact with their surroundings allowing energy exchange to occur.^{[6]} In other words, these structures “in time or in space can only maintain themselves beyond a minimal level of energy dissipation.”^{[1]} Dissipative structures disappear if the energy exchange stops.
Goldbeter et al. describe the conditions needed for the occurance of dissipative structures for an allosteric model. The allosteric model used is the “well assessed” MWC model presented by Monod, Wyman and Changeux. This model describes the allosteric transitions for the regulation of proteins such as enzymes. The theory suggests that some enzymes can exist in different states in the absense of a regulator. These types of proteins are called allosteric proteins. An allosteric enzyme can exist in two different conformational states the ‘R’/relaxed state or the ‘T’/tense state. The R state has a higher affinity for the ligand than the T state and therefor the R state is also known as the active state whereas the T state is known an the inactive state. In allosteric enzymes, the ligand can inhibit or activate the activity of the enzyme. Goldbeter et al. consider a model in which an allosteric enzyme is activated by its product. ^{[7]} The ModelThe following model is used to explain the experimental facts found by the author.
Model I
A simple model shown in figure 1 can be constructed by: (1) With the following constraints
where both sums account for all the enzymatic forms in their respective state. Model IIThe author then normalizes the concentrations of the product and substrate by
The equations for and give the change in substrate and product concentrations, respectively, while relates the state of the enzyme(active or inactive) to the product and substrate concentrations and is equivalent to
Dissipative structuresLooking at system (2), the author finds two separate steady states, and where,
with corresponding to the added term, and corresponding to the subtracted term, and
To obtain the dispersion equation, the author looks at how system (2) responds to infinitesimal perturbations. He finds that as the equations become linear with time and space independent coefficients, the system gives a solution of the following form.
Which plugged back into the evolution equations around the steady state, we get the following dispersion equation
(4) which represents dispersion without diffusion. The system is also subject to the following instability relations. and The following phase plane diagram in Figure 2 (a) shows an unstable focus point the evolves to a stable limit cycle which is shown as the darker line. These sustained oscillations can be seen in Figure 2 (b) with both the product and substrate as functions of time. Fig. 2
Then, in order for a spacial dissipative structure to occur, the following condition must be satisfied Numerical MethodsSpacial CaseThe numerical study suggests that the only steady state physically acceptable is and A is always positive. This means that in the homogeneous case instabilities of the stationary state will always occur before an instability occuring from diffusion. The numerical study shows that this is true for values of and therefore these conclusions can be applied to all cases where the the product molecule is similar to the subtrate molecule. Temporal CaseThe stability of the allosteric model for the temporal case was studied with respect to the most important parameters and because the conditions obtained for instability could not be described by one parameter. The stability diagrams for , and , are shown below. The conditions for which there occurs an unstable focus were evaluated around the steady states and The stationary states are stable in domain I. In domain II there are limit cycles. Domain III there is no physical steady state. Conclusions from temporal case
Glycolytic OscillationsPhosphofructokinase is the enzyme partly responsible for glycolytic oscillations. The phosphofructokinase pathway is regulated in a particular way that causes these periodicities. Phosphofructokinase is an allosteric enzyme in yeast and is activated by its two products ADP via AMP and FDP (fructosediphosphate). It is also inhibited by excess ATP, its first substrate, and activated by F6P (fructose6phosphate), its second substrate. The self oscillating behavior of the glycolytic system in yeast have been well studied and experimental results are presented here. In a paper by Betz and Sel'kov ^{[8]} results show that the regulation is different under the conditions that produce oscillations in cell free extract with all effectors present. In this case ATP is ineffective in inhibiting phosphofructokinase while AMP and F6P are effective activators. Product activation is essential for sustained oscillations. The allosteric model I described above can be used to model phosphofructokinase because ADP acts as an activator of the enzyme. Therefore represents the substate ATP and represents the product ADP. The experimental observations agree with the model's prediction for sustained oscillations (see Fig 2b). When the system enters a region of oscillations, the period is accurately described by the calculation for period described by the model. Hess et al. ^{[9]} has also obtained experimental results that verify the model. Hess provided experimental evidence that the glycolytic system remained stable as long as the rate of substrate supply was very low, mM. When this rate increased oscillations occured. Increasing even more led to a shortening of the period of oscillations and then the system reached a new stationary state which can be described by Region III of the above model. Betz et al. experiments also suggest that sustained oscillations are favored when the substrate is activated. This effect can only be described by an allosteric model in which the substrate depends exclusively on the binding coefficent, . The experimental evidence agrees very well with the allosteric models in this paper. These results justified the choice of values used in the stability diagrams. In this way the values used were physiologically acceptable and were confirmed experimentally. References and Suggested Reading
