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MBW:Dissipative Structures for an Allosteric Model

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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 12:1302-1315. [1]

Executive Summary

This 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.


The 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.


  • There have been many studies on oscillations in enzyme systems. In enzymatic systems the chain of reactions is activated and/or inhibited by its substrates and/or products. Oscillations occur due to this type of regulation i.e. positive or negative feedback loops and they are characterized by a shorter period in enzyme reactions.
  • Sustained oscillations are seen in the glycolytic system due to positive feedback and a model for this is presented below along with experimental results from various papers researching the glycolytic system in yeast. Most other models currently being investigated focus on end-product inhibition of the enzyme rather than activation.[2][3]
  • It is theorized that the purpose for oscillations could be to regulate the concentrations of the metabolites and possibly produce specific spacially differentiated patterns. There have been studies on systems that have a diffusing component in which the system can develop spacial and temporal order. Prigogine describes this order as a dissipative structure. The Zhabotinsky reaction, which is a good example of a chemical clock, confirms the connection between the spacial and temporal patterns.[4][5]
  • The model presented below is an extension of the global models presented by Higgins and Sel'kov for glycolytic ocsillations. The physical significance of the parameters in terms of effectors that were studied include activation by the product, the influence of the substrate, and cooperativity. Diffusion is also considered for the diffusion of the metabolites away from the enzyme.

Biological Background

  • Dissapative structure

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.

  • Allosteric Proteins

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 Model

The following model is used to explain the experimental facts found by the author.

  • v_{1}=the rate of the supplied substrate
  • R= the active form of the enzyme dimer
  • R_{0}= the active form of the enzyme not bound to the ligand
  • T= the inactive form of the enzyme dimer
  • T_{0}= the inactive form of the enzyme not bound to the ligand
  • k_{s}=proportionality factor of the product leaving the system proportional to its concentration
  • a_{1},d_{1}=kinetic constants for binding and dissociation (respectively) of the substrate for the active forms
  • a_{2},d_{2}=kinetic constants for binding and dissociation (respectively) of the product for the active forms
  • a_{3},d_{3}=kinetic constants for binding and dissociation (respectively) of the substrate for the inactive form
  • k_{1}=kinetic constant related to the conversion of R_{0} to T_{0}
  • k_{2}=kinetic constant related to the conversion of T_{0} to R_{0}
  • k=kinetic constant related to the irreversible chemical reaction
  • A_{2}=concentration of the product
  • A_{3}=concentration of the substrate

Model I


A simple model shown in figure 1 can be constructed by:

  (1)  {\frac  {dA_{3}}{dt}}=v_{1}-2a_{3}A_{3}T_{0}+(d_{3}-a_{3}A_{3})T_{1}+2d_{3}T_{2}-2a_{1}A_{3}(R_{0}+R_{{01}}+R_{{02}})+

With the following constraints

  1. The substrate supply rate v_{1} is constant.
  2. The enzyme is a dimer that can exist in an active and inactive form, R and T respectively, and are converted from R_{0} to V_{0} and vice-versa.
  3. The substrate binds to both the active an inactive forms, while the product only binds to the active form.
  4. The active forms of the enzyme R carrying the substrate decompose irreversibly to the product.
  5. The product leaves the system at rate k_{s}.

The active and inactive forms of the enzyme follow a conservation relation such that

D_{0}=\displaystyle \sum {R}+\displaystyle \sum {T}

where both sums account for all the enzymatic forms in their respective state.

Model II

The author then normalizes the concentrations of the product and substrate by

\alpha ={\frac  {A_{3}}{K_{{A_{3}(R)}}}}, \gamma ={\frac  {A_{2}}{K_{{A_{2}(R)}}}}

where K_{{A_{3}(R)}}={\frac  {d_{1}}{a_{1}}} and K_{{A_{2}(R)}}={\frac  {d_{2}}{a_{2}}} are the equilibrium dissociation constants for the active forms of A_{3} and A_{2}. Now, for simplicity, the author lets a_{1}=a_{2}=a and d_{1}=d_{2}=d, and defines a change of variables as the following:

\sigma _{1}={\frac  {v_{1}}{d}},\sigma _{2}={\frac  {k_{s}}{a}},\epsilon ={\frac  {k}{d}}

With these above conditions, and taking into account the diffusion of the product and intermediates while neglecting the diffusion of R and T (from assuming earlier they are conserved) the system can be described by the following differential equations.

(2) {\begin{cases}{\frac  {\partial \alpha }{\partial t}}=a[\sigma _{1}-{\frac  {[{\frac  {2D_{0}\epsilon }{(\epsilon +1)}}]\alpha (1+\gamma )^{2}[1+{\frac  {\alpha }{(\epsilon +1)}}]}{L(1+\alpha c)^{2}+(1+\gamma )^{2}[1+{\frac  {\alpha }{(\epsilon +1)}}]^{2}}}]+{\mathcal  {D}}_{\alpha }{\frac  {\partial ^{2}\alpha }{\partial r^{2}}}\\{\frac  {\partial \gamma }{\partial t}}=a[{\frac  {[{\frac  {2D_{0}\epsilon }{(\epsilon +1)}}]\alpha (1+\gamma )^{2}[1+{\frac  {\alpha }{(\epsilon +1)}}]}{L(1+\alpha c)^{2}+(1+\gamma )^{2}[1+{\frac  {\alpha }{(\epsilon +1)}}]^{2}}}-\sigma _{2}\gamma ]+{\mathcal  {D}}_{\gamma }{\frac  {\partial ^{2}\gamma }{\partial r^{2}}}\\{\bar  {R}}={\frac  {(1+\gamma )^{2}[1+{\frac  {\alpha }{(\epsilon +1)}}]^{2}}{L(1+\alpha c)^{2}+(1+\gamma )^{2}[1+{\frac  {\alpha }{(\epsilon +1)}}]^{2}}}\end{cases}}

In system (2), {\mathcal  {D}}_{\alpha } and {\mathcal  {D}}_{\gamma } are the diffusion coefficients of the substrate A_{3} and the productA_{2}, respectively. c is the nonexclusive binding coefficient, meaning it can bind to the active and inactive states of the enzyme, and is defined as c={\frac  {K_{{A_{3}(R)}}}{K_{{A_{2}(R)}}}}. This gives the degree of activation or inhibition by the substrate. L is the allosteric constant, and is equal to the ratio of the active and inactive forms of the enzyme in the absence of the ligand, or L={\frac  {T_{0}}{R_{0}}}, and expresses the coopertivity of the enzyme. Positive cooperativity occurs when binding of the substrate molecule increases the affinity of the other active sites, where negative cooperativity decreases the affinity.

The equations for {\frac  {\partial \alpha }{\partial t}} and {\frac  {\partial \gamma }{\partial t}} give the change in substrate and product concentrations, respectively, while {\bar  {R}} relates the state of the enzyme(active or inactive) to the product and substrate concentrations and is equivalent to

{\bar  {R}}={\frac  {\displaystyle \sum {R}}{\displaystyle \sum {R}+\displaystyle \sum {T}}}

Dissipative structures

Looking at system (2), the author finds two separate steady states, (\alpha _{{01}},\gamma _{0}) and (\alpha _{{02}},\gamma _{0}) where,

\gamma _{0}=\sigma _{1}/\sigma _{2}

\alpha _{0}={\frac  {[({\frac  {D_{0}\epsilon }{(\epsilon +1)}}\Gamma ^{2}-\sigma _{1}[Lc+{\frac  {\Gamma ^{2}}{(\epsilon +1)}}]\pm \Gamma {\sqrt  {\delta }}}{[\sigma _{1}(Lc^{2}+{\frac  {\Gamma ^{2}}{(\epsilon +1)^{2}}}]-{\frac  {2D_{0}\epsilon \Gamma ^{2}}{(\epsilon +1)^{2}}}}}

with \alpha _{{01}} corresponding to the added \Gamma {\sqrt  {\delta }} term, and \alpha _{{02}} corresponding to the subtracted \Gamma {\sqrt  {\delta }} term, and

\Gamma =(1+{\frac  {\sigma _{1}}{\sigma _{2}}})

\delta =2\sigma _{1}L[{\frac  {D_{0}\epsilon }{(\epsilon +1)}}][{\frac  {1}{(\epsilon +1)}}-c]+[{\frac  {D_{0}\epsilon \Gamma }{(\epsilon +1)}}]^{2}-\sigma _{1}^{2}L[{\frac  {1}{(\epsilon +1)}}-c]^{2}

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.

\delta X=xe^{{\omega i+{\frac  {ir}{\lambda }}}}

Which plugged back into the evolution equations around the steady state, we get the following dispersion equation

(3)0=\omega ^{2}+\omega [aC(A-B)+\sigma _{2}a+{\frac  {({\mathcal  {D_{\alpha }}}+{\mathcal  {D_{\gamma }}})}{\lambda ^{2}}}]+[a^{2}\sigma _{2}CA+aC({\frac  {A{\mathcal  {D_{\gamma }}}}{\lambda ^{2}}}-{\frac  {B{\mathcal  {D_{\alpha }}}}{\lambda ^{2}}})+{\frac  {\sigma _{2}a{\mathcal  {D_{\alpha }}}}{\lambda ^{2}}}+{\frac  {{\mathcal  {D_{\alpha }}}{\mathcal  {D_{\gamma }}}}{\lambda ^{4}}}]

With the variables A,B, and C set as

A=L(1+\gamma _{0})[\alpha _{0}^{2}c({\frac  {2}{(\epsilon +1)}}-c)+{\frac  {2\alpha _{0}}{(\epsilon +1)}}+1]+(1+\gamma _{0})^{3}[1+{\frac  {\alpha _{0}}{(\epsilon +1)}}]^{2}

B=2\alpha _{0}L[1+{\frac  {\alpha _{0}}{(\epsilon +1)}}](1+\alpha _{0}c)^{2}

C=2[{\frac  {D_{0}\epsilon }{(\epsilon +1)}}]{\frac  {(1+\gamma _{0})}{[L(1+\alpha _{0}c)^{2}+(1+\gamma _{0})^{2}[1+{\frac  {\alpha _{0}}{(\epsilon +1)}}]^{2}]^{2}}}

Now by neglecting the effects caused by diffusion, equation (3) becomes

(4) 0=\omega ^{2}+\omega [aC(A-B)+\sigma _{2}a]+a^{2}\sigma _{2}CA

which represents dispersion without diffusion. The system is also subject to the following instability relations.

C(A-B)+\sigma _{2}<0 and A>0

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

The author then looks at the inhomogeneous case by first letting

\theta ={\frac  {{\mathcal  {D_{\gamma }}}}{{\mathcal  {D_{\alpha }}}}}

Then, in order for a spacial dissipative structure to occur, the following condition must be satisfied

{\frac  {\theta {\mathcal  {D_{\alpha }}}^{2}}{\lambda ^{4}}}+a({\frac  {{\mathcal  {D_{\alpha }}}}{\lambda ^{2}}})[C(\theta A-B)+\sigma _{2}]+a^{2}\sigma _{2}CA<0

Numerical Methods

Spacial Case

The numerical study suggests that the only steady state physically acceptable is (\alpha _{{02}},\gamma _{0}) 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 \theta \geq 0.1 and therefore these conclusions can be applied to all cases where the the product molecule is similar to the subtrate molecule.

Temporal Case

The stability of the allosteric model for the temporal case was studied with respect to the most important parameters \sigma _{1},\epsilon ,c and L because the conditions obtained for instability could not be described by one parameter. The stability diagrams for L-c,L-\epsilon , and , L-\sigma _{1} are shown below. The conditions for which there occurs an unstable focus were evaluated around the steady states (\alpha _{{01}},\gamma _{0}), and (\alpha _{{02}},\gamma _{0}) The stationary states are stable in domain I. In domain II there are limit cycles. Domain III there is no physical steady state. NumericalJZ.jpg NumericalJZ2.jpg

Conclusions from temporal case

  1. Instabilities occur for large values of L, the allosteric constant (L>10^{4}). (see Figs. 3a, 3b, 4c, 5a, 5b)
  2. For (0\leq c<1) there is a greater range for oscillations when c decreases. In the presense of substrate inhibition the system can still become unstable but region II is reduced (see Figure 1 and 3). As inhibition increases the oscillations disappear and there are only two region, I and III. Goldberg et al. explains “in physical terms this means that the substrate accumulates in such a way that the enzyme cannot follow any longer” [1](see Figure 2 c). This can occur when \sigma _{1},c or L are too large or when \epsilon is too small (Fig 4c). Physically this occurs when the inactive form (T) increases meaning that the active form concentration decreases for large values of c and L.
  3. The values for \sigma _{1} and \epsilon for which instabilities occur are in an acceptable physiological range. The values were calculated and compared with experimental data and were found to occur at physical values. When there is substrate activation \epsilon has a wide range for instabilities to occur.
  4. A domain of oscillations exists (Region II) when more of the enzyme is in the T form. When \epsilon takes on small values oscillations can occur under the active R form.

Glycolytic Oscillations

Phosphofructokinase 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 (fructose-diphosphate). It is also inhibited by excess ATP, its first substrate, and activated by F-6-P (fructose-6-phosphate), 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 F-6-P 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 \alpha represents the substate ATP and \gamma 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, \sigma _{1}=10^{{-8}} 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, c. 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

  1. 1.0 1.1 1.2 Goldbeter A. and R. Lefever. 1972. Dissipative structures for an allosteric model. Biophysical Journal 12:1302-1315.
  2. Walter, C. 1969. J. Theor. Biol. 23:23.
  3. Morales, M. and D. McKay. 1967. Biophys J. 7:621.
  4. Busse, H. 1969. J. Phys. Chem. 73:750
  5. Herschkowitz-Kaufman, M. 1970. C. R. H. Acad. Sci. Ser. C. 270:1049.
  8. Betz, A. and E. Sel'kov. 1969. FEBS (Fed. Eur. Biochem. Soc.) Lett. 3:5
  9. Hess, B., A. Boiteux, and J. Kruger. 1969. Adv. Enzyme Regul. 7:149.