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Turing Patterns Inside Cells

  • Damián E. Strier,

    Affiliation Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Brussels, Belgium

  • Silvina Ponce Dawson

    To whom correspondence should be addressed. E-mail: silvina@df.uba.ar

    Affiliation Departamento de Física, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, Pabellón I, University of Buenos Aires, Buenos Aires, Argentina

Turing Patterns Inside Cells

  • Damián E. Strier, 
  • Silvina Ponce Dawson
PLOS
x

Abstract

Concentration gradients inside cells are involved in key processes such as cell division and morphogenesis. Here we show that a model of the enzymatic step catalized by phosphofructokinase (PFK), a step which is responsible for the appearance of homogeneous oscillations in the glycolytic pathway, displays Turing patterns with an intrinsic length-scale that is smaller than a typical cell size. All the parameter values are fully consistent with classic experiments on glycolytic oscillations and equal diffusion coefficients are assumed for ATP and ADP. We identify the enzyme concentration and the glycolytic flux as the possible regulators of the pattern. To the best of our knowledge, this is the first closed example of Turing pattern formation in a model of a vital step of the cell metabolism, with a built-in mechanism for changing the diffusion length of the reactants, and with parameter values that are compatible with experiments. Turing patterns inside cells could provide a check-point that combines mechanical and biochemical information to trigger events during the cell division process.

Introduction

Concentration gradients inside the cytosol are a vital piece of the cell's machinery. They underlie morphogenesis [1], and are involved in cell migration [2], cell growth [3], cell division [4], and the mechanical responses that follow fertilization, among other functions [5]. Stationary concentration gradients can spontaneously emerge from a homogeneous background in the presence of instabilities. In 1952 Alan Turing proposed a hypothetical sequential route for cell differentiation, based on a reaction-diffusion process [6]. Starting from a spatially uniform state he proved that stable inhomogeneities in “morphogen” concentration could spontaneously emerge through a diffusion-driven symmetry-breaking instability. His highly idealized model stimulated a large amount of work which eventually led to the observation of Turing patterns in open chemical reactors [7], [8]. Nevertheless, the occurrence of Turing patterns has not been unequivocally proven for a biochemical reaction in which reaction rates, concentrations and diffusion coefficients are within realistic physiological values. We do so in this paper. More specifically, we show the existence of cell-sized Turing patterns in a model of glycolysis, using realistic parameter values and equal diffusion coefficients of ATP and ADP.

The general concept behind Turing patterns involves a combination of short-range activation and large-range inhibition [9]. Several reaction-diffusion models have been proposed in the literature to explain, among others, coat patterns in mammals [9], skin patterns in fish [10] or shell patterns in mollusks [11]. However, none of these models identifies the “morphogens” involved and some of the reaction schemes lack a biological motivation. This gives almost unrestricted freedom to choose kinetic parameters and diffusion ratios. Thus, the astonishing visual similarities often observed between the predicted and the real patterns might be unrelated to the actual biological phenomena.

One of the early attempts to look for an example of a symmetry breaking mechanism in biology was due to I. Prigogine et al. [12]. They analyzed a spatially extended version of the (adiabatically reduced) 2-variable Selkov model of glycolysis, derived in [13] (see also [14], [15]). Namely, they added diffusion terms to each of the two differential equations describing the time evolution of the concentrations of ATP and ADP. It followed from their study that Turing patterns could only exist provided that ATP and ADP diffused at sufficiently unequal rates. This condition seems difficult to be met a priori, given the similarities between ATP and ADP. More recently, Hasslacher et al. [16] presented numerical simulations of Turing pattern formation in a closely related model [17] that were obtained assuming that ATP diffused 25 times faster than ADP. The authors gave a qualitative justification for their choice of diffusivities making an analogy between the role of enzymes in the glycolytic pathway and that of the immobile color indicators used for visualization in open chemical reactors [7], [8] which effectively reduce the transport rate of one of the species involved in the reactions [18]. However, the argument that applied to the latter case cannot be translated to the case of the glycolytic pathway without a deeper analysis. Firstly, enzymes affect both the dynamics of ATP and ADP. Thus, the rescaling of diffusion coefficients may occur in such a way so as to prevent the formation of patterns. Second, when the 2-variable reaction-diffusion model is obtained by the adiabatic reduction of the larger set of equations in which the enzyme dynamics is explicitly considered [19], the elimination of the fast variables (the enzymes) introduces changes of the same order of magnitude in both the diffusion and the reaction terms [20]. Thus, both changes need to be considered simultaneously in order to analyze Turing pattern formation in this setting. Finally, this theoretical discussion remains meaningless unless reasonably sized patterns can be shown to exist for realistic values of the reaction rates and of the free diffusivities.

In this paper we overcome the drawbacks of these previous works by showing that the full 5-variable Selkov model describing the PFK-controlled steps of the glycolytic pathway supports Turing patterns for realistic reaction rates and equal diffusivities of ATP and ADP. We show that Turing structures of subcellular size (10 µm) may be found by increasing the glycolytic flux and the enzyme concentration while keeping fixed the set of kinetic constants that give good agreement with the classic experimental results on homogeneous oscillations in yeast extracts [21], [22], [23], [24], [14]. The emergence of these patterns can be traced back to the differential interactions of ATP and ADP with PFK and its complexes. We then conclude that the key enzymatic step responsible for glycolytic oscillations may also provide a robust mechanism for the formation of steady state inhomogeneities in the concentration of ATP and ADP at the cellular and supracellular level.

Results

The 5-variable Selkov model reads [13]:(1)where E is the free enzyme (PFK), which can form the complexes, ES2γ and S1ES2γ; the substrate, S1 (ATP) is supplied by an external source at the rate, ν1, and the product, S2γ (ADP) is removed by a first order reaction at rate ν2[S2]. From scheme (1) it is clear that the enzyme is inactive unless it has γ product molecules bound, forming the complex ES2γ. As done in [13], we set γ = 2 all throughout this paper. It is this activating step of the allosteric enzyme which provides the mechanism responsible for the instabilities. We assume that ATP and ADP diffuse while the enzyme and its complexes are immobile, due to the larger mass. We also assume that, initially, all concentrations are spatially uniform. Using mass action kinetics, the reaction-diffusion equations describing this model can be written, in dimensionless form, as:(2)where the dimensionless concentrations are:(3)with e0, the total concentration of enzyme which remains constant and uniform throughout the evolution. The other quantities are(4)The dimensionless space and time coordinates are obtained by dividing the dimensional ones by an arbitrary length-scale, L, and(5)respectively. Assuming that the diffusion coefficients of ATP (D1) and ADP (D2) are equal (D≡D1 = D2), we take L≡10(DT)1/2. Using the diffusion coefficients in vivo, D∼150 µm2 s−1 [25], we obtain L = 122 µm (T s−1)½. while the dimensionless diffusion coefficients are d1 = d2 = 0.01. The relatively small amount of enzyme used in experiments implies that ε≪1. For this reason, a quasi-steady state approximation of Eqs. (2) was analyzed in [13] considering, furthermore, spatially uniform concentrations.

We now constrain the values of the various parameters of Eqs. (2) according to previous estimations and measurements [13], [15], [26], [21], [22], [23]. Under the experimental conditions of [22], in which experiments are done using yeast extracts, the values of ν1 and ν2 at which the oscillations start are ν1*∼5.8 µMs−1 and ν2*∼0.04 s−1 [15]. The enzyme is very diluted in these experiments, with e0* between 3 and 10 µM, the average concentrations of ATP and ADP are [S1*] = 630 µM and [S2*] = 150 µM, respectively, while the period of the oscillations, T, is between 3 and 5 min [15]. It has been shown unequivocally that the transition to oscillatory behavior in glycolysis is due to a Hopf bifurcation [24]. This means that the transition can be generically encountered by varying only one parameter. Equations (2) have a single homogeneous fixed point solution that depends on several parameters, which indeed undergoes a Hopf bifurcation. Thus, there are various ways by which this bifurcation can be reached. Among them, we choose the set of parameter values that are compatible with the observed frequency of oscillations and ATP and ADP concentrations. In particular, we find that at η = η* = 0.15, ν = ν*≈0.0041, ε = ε* = 10−6, α = 15, K1 = 1500, K3 = 1, a Hopf bifurcation occurs for the set of Eqs. (2) in the spatially homogeneous case. Using the definitions of these dimensionless quantities in terms of dimensional ones and the experimentally determined values, ν1*∼5.8 µMs−1 and ν2*∼0.04 s−1, we obtain that the total amount of enzyme at the Hopf bifurcation is e0 = e0*∼7.9 µM and that the various reaction rates satisfy(6)These numbers imply that, at the Hopf bifurcation, the stationary homogeneous solution of Eqs. (2) occurs at [S1*]∼150 µM and [S2*]∼145 µM, and that the dimensional period of the oscillations is T*∼2.7 min, which agrees with the experimentally determined values. We show the oscillatory behavior of [ATP] and [ADP] close to this bifurcation in Fig. 1 (a).

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Figure 1. Behaviors predicted by the 5-variable Selkov model for different parameter values.

(a) Glycolytic oscillations in S1 (solid curve) and S2 (dashed curve) for η = 0.15, ν = 0.00345, ε = 10−6, α = 15, K1 = 1500, K3 = 1. (b) Linear growth rate of the unstable modes as a function of the square of the wavenumber, k for η = 1.215, ν = 0.03, ε = 0.0003, α = 15, K1 = 1500, K3 = 1, and d1 = d2 = 0.01. Inset: Evolution of [S1] in the spatially homogeneous case for the same parameter values. (c) Turing space (shadowed domain) as a function of the (dimensionless) input and output rates of ATP (ν) and ADP (η), for the same parameter values as in (b). (d) Predicted value of the wave-length of the most unstable mode at each point in the Turing space of (c).

https://doi.org/10.1371/journal.pone.0001053.g001

Given these previous estimates, we explore the behavior of Eqs. (2), in the spatially extended case, varying the parameters ν, η, and ε and keeping the purely kinetic constants, α, K1 and K3 fixed at the previously mentioned values. We obtain the set of parameter values for which the homogeneous stationary solution of Eqs. (1) is unstable against spatially inhomogeneous perturbations (i.e., the Turing space) by analyzing the dispersion relation of the linearized evolution equations. We show in Fig. 1(b) the (dimensionless) linear growth rate, q, of the unstable modes as a function of the square of the (dimensionless) wavenumber, k, for ν = 0.03, η = 1.215, ε  = 0.0003. There is a bounded band of unstable modes for k≠0, while for k = 0 (see the inset) the fixed point is stable. We show in Fig. 1(c) the Turing space on the (ν,η) plane for ε = 0.0003. As it may be observed, patterns may exist for larger values of ν1 as ν2 also gets larger, i.e., as the glycolytic flux increases. A similar picture holds on the (ε,η) plane (data not shown): no pattern is possible at low enzyme concentration (if ε≤10−4). We show in Fig. 1(d) the wave-length of the most unstable mode, lc, when η and ν are varied in the shaded region of Fig. 1(c). We then conclude that the characteristic size gets smaller as the rate of product removal, ν2, becomes larger. We observe that, for the parameter values considered, the length-scale is always less than 23 µm, so that it can fit inside a typical cell. Using the definition of ε with the value of K1 = 1500 and the previous rate constant estimates (Eqs. (6)) obtained at the Hopf bifurcation, which we assume remain fixed, we conclude that e0≥800 µM for the Turing instability to occur. As we discuss later, the patterns arise due to the effective rescaling of the diffusion coefficients of ADP and ATP that the enzyme produces. This rescaling gets smeared out as e0 decreases, leading, in turn, to the disappearance of the patterns.

We finally integrate numerically Eqs. (2) in a square domain of side 8L = 84.5 µm with periodic boundary conditions, using a finite-difference scheme on a 150×150 square grid. Fig. 2 shows the resulting pattern in S1 after 10 min have passed since the initial situation, at which the concentrations are given by the homogeneous stationary solution ([S1] = 2.4 mM and [S2] = 71 µM), with 10% added noise. The typical size of the pattern agrees with the critical wavelength of the linear stability analysis, lc ≈12 µm. The simulation shows that Eqs. (2) support stable Turing patterns with equal diffusion coefficients of ATP and ADP. The existence of these patterns can be understood in terms of the reduced set of equations obtained in the quasi-steady state approximation [19], [20], [27]. The effective diffusion coefficients of ATP and ADP that are obtained in this approximation are different due to the different interaction of ATP and ADP with the enzymes that belong to the pathway. In this scheme, ATP cannot bind to the immobile activated complex unless γ = 2 ADP molecules are already bound. Thus, the effective diffusivity of ADP –which plays the role of the activator of the step– is reduced by a larger amount than that of ATP.

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Figure 2. Turing pattern obtained with the 5-variable Selkov model in two space dimensions.

Stationary pattern in [ATP] achieved after 10 min from an initial condition randomly distributed around the Turing-unstable fixed point. White corresponds to [ATP] = 2.47 mM and black to 1.1 mM. The simulation was done for η = 1.3, ν = 0.0175, ε = 0.0005, α = 15, K1 = 1500, K3 = 1, and d1 = d2 = 0.01.

https://doi.org/10.1371/journal.pone.0001053.g002

Discussion

Conlusions

In this paper we have provided the first closed example of Turing pattern formation in a model of a vital piece of a cell's real biochemistry, with a built-in mechanism for the change of the morphogens diffusion length, and with parameter values that are compatible with experiments. Our results suggest that the pattern of enzyme regulation that gives rise to the glycolytic oscillations may also provide the basis for the formation of stationary spatial structures both at the cellular and supracellular level. The model we have used is highly idealized and cannot account for certain observations. However, we think that some of its basic dynamical features should be common to those of more sophisticated models [22], [28] and of the real system. In particular, it does reproduce the oscillations that are observed for dilute enzyme concentrations [22], [23], and has helped the finding of rotating spirals in vitro [29]. There are studies that show that the Turing and Hopf bifurcations are intimately related in systems with immobile species [30]. Thus, we expect all these models to share the existence of a Turing bifurcation at large enough (immobile) enzyme concentration and a Hopf bifurcation at lower ones. Our study has shown that the interactions involved in the PFK catalyzed step of the glycolytic pathway change the “effective” diffusion coefficients of ATP and ADP in the necessary direction for Turing pattern formation. These results could be tested in the type of open reactors that have especially been conceived for the investigation of spatio-temporal dynamics in glycolysis [31].

We have also found that the patterns can fit inside a typical cell and that the time it takes for the patterns to form is relatively short (of the order of minutes). The formation of Turing patterns in this biochemical pathway could then be related to organizing centers in eukaryotic cells, playing a role during cell division [4]. The fact that the Turing patterns have an intrinsic length-scale implies that there could be zero, one or several spots of high [ATP] inside the cell, depending on the relationship between the cell and the pattern sizes (see Fig. 2). In particular, there are two properties which imply that the number of high [ATP] spots could change during the cell division cycle. First, the pattern size decreases as the strength of the glycolitic flux increase. This flux must increase during interphase for biosynthesis and growth. Thus, the pattern size might become small enough to fit inside the cell only after the glycolytic flux has increased sufficiently. The change of the cell size also acts in the same direction, allowing more “room” for more high [ATP] spots to fit as the cell grows during interphase. In this way, the change in the number of high [ATP] spots as the cell grows could provide a check-point that combines mechanical information (cell size) and biochemistry to trigger the subsequent chain of events in the cell division process. If a key stage of the replicating machine of every eukaryotic cell strongly relies on a metabolic pathway, it would be unlikely that such pathway had evolved after the appearance of the first eukaryotic cell. The pathway should be old and highly conserved. The glycolytic pathway shares both properties.

Acknowledgments

This paper is dedicated to B. Hasslacher who recently passed away and whose pioneering ideas were a key factor for the development of the present work. We also acknowledge useful conversations with G. B. Mindlin, J.E. Pearson and A. Goldbeter.

Author Contributions

Conceived and designed the experiments: SP. Performed the experiments: DS. Analyzed the data: DS. Wrote the paper: SP DS.

References

  1. 1. Lawrence PA (1992) The Making of a Fly: the genetics of animal design. Boston: Blackwell Scientific Publications. PA Lawrence1992The Making of a Fly: the genetics of animal designBostonBlackwell Scientific Publications
  2. 2. Xu H-T, Yuan X-B, Guan C-B, Duan S, Wu C-P, et al. (2004) Calcium signaling in chemorepellant Slit2-dependent regulation of neuronal migration. Proc Natl Acad Sci USA 101: 4296–4301.H-T XuX-B YuanC-B GuanS. DuanC-P Wu2004Calcium signaling in chemorepellant Slit2-dependent regulation of neuronal migration.Proc Natl Acad Sci USA10142964301
  3. 3. Hepler PK, Vidali L, Cheung AY (2001) Polarized cell growth in higher plants. Annu Rev Cell Dev Biol 17: 159–187.PK HeplerL. VidaliAY Cheung2001Polarized cell growth in higher plants.Annu Rev Cell Dev Biol17159187
  4. 4. Sharp DJ, Rogers GC, Scholey JM (2000) Microtubule motors in mitosis. Nature 407: 41–47.DJ SharpGC RogersJM Scholey2000Microtubule motors in mitosis.Nature4074147
  5. 5. Clapham DE (1995) Replenishing the stores. Nature 375: 634–635.DE Clapham1995Replenishing the stores.Nature375634635
  6. 6. Turing AM (1952) The chemical basis of morphogenesis. Phil Trans Roy Soc B 237: 37–72.AM Turing1952The chemical basis of morphogenesis.Phil Trans Roy Soc B2373772
  7. 7. Castets V, Dulos E, Boissonade J, De Kepper P (1990) Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys Rev Lett 64: 2953–2956.V. CastetsE. DulosJ. BoissonadeP. De Kepper1990Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern.Phys Rev Lett6429532956
  8. 8. Ouyang Q, Swinney H (1991) Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352: 610–612.Q. OuyangH. Swinney1991Transition from a uniform state to hexagonal and striped Turing patterns.Nature352610612
  9. 9. Murray JD (1989) Mathematical Biology. 12 York: Springer. JD Murray1989Mathematical Biology12 YorkSpringer
  10. 10. Kondo S, Asai R (1995) A Reaction-Diffusion Wave on the Skin of the Marine Angelfish Pomacanthus. Nature 376: 765–767.S. KondoR. Asai1995A Reaction-Diffusion Wave on the Skin of the Marine Angelfish Pomacanthus.Nature376765767
  11. 11. Meinhardt H (1999) On pattern and growth. In: Chaplain MAJ, Singh GD, McLachlan JC, editors. On Growth and Form: Spatio-temporal Pattern Formation in Biology. London: John Wiley & Sons. pp. 129–148.H. Meinhardt1999On pattern and growth.MAJ ChaplainGD SinghJC McLachlanOn Growth and Form: Spatio-temporal Pattern Formation in BiologyLondonJohn Wiley & Sons129148
  12. 12. Prigogine I, Lefever R, Goldbeter A, Herschkowitz-Kaufman M (1969) Symmetry Breaking Instabilities in Biological Systems. Nature 223: 913–916.I. PrigogineR. LefeverA. GoldbeterM. Herschkowitz-Kaufman1969Symmetry Breaking Instabilities in Biological Systems.Nature223913916
  13. 13. Selkov EE (1968) On the Mechanism of Single-Frequency Self-Oscillations in Glycolysis. I. A Simple Kinetic Model. Eur J Biochem 4: 79–86.EE Selkov1968On the Mechanism of Single-Frequency Self-Oscillations in Glycolysis. I. A Simple Kinetic Model.Eur J Biochem47986
  14. 14. Goldbeter A (1997) Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour. Cambridge: Cambridge University Press. A. Goldbeter1997Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic BehaviourCambridgeCambridge University Press
  15. 15. Selkov EE (1968) Mol Biol 2: 252–266 (English Translation).EE Selkov1968Mol Biol2252266 (English Translation)
  16. 16. Hasslacher B, Kapral R, Lawniczak A (1993) Molecular Turing structures in the biochemistry of the cell. Chaos 3: 7–13.B. HasslacherR. KapralA. Lawniczak1993Molecular Turing structures in the biochemistry of the cell.Chaos3713
  17. 17. Richter PH, Rehmus P, Ross J (1981) Control and dissipation in oscillatory chemical engines. Prog Theor Phys 66: 385–405.PH RichterP. RehmusJ. Ross1981Control and dissipation in oscillatory chemical engines.Prog Theor Phys66385405
  18. 18. Lengyel I, Epstein I (1991) Modeling of Turing Structures in the Chlorite—Iodide—Malonic Acid—Starch Reaction System. Science 251: 650–652.I. LengyelI. Epstein1991Modeling of Turing Structures in the Chlorite—Iodide—Malonic Acid—Starch Reaction System.Science251650652
  19. 19. Strier DE, Ponce Dawson S (2000) Rescaling of diffusion coefficients in two-timescale chemical systems. J Chem Phys 112: 825–834.DE StrierS. Ponce Dawson2000Rescaling of diffusion coefficients in two-timescale chemical systems.J Chem Phys112825834
  20. 20. Strier DE, Ponce Dawson S (2004) Role of complexing agents in the appearance of Turing patterns. Phys Rev E 69: 066207 1–10.DE StrierS. Ponce Dawson2004Role of complexing agents in the appearance of Turing patterns.Phys Rev E69066207 110
  21. 21. Chance B, Ghosh AK, Pye EK, Hess B, editors. (1974) Biological and Biochemical Oscillators. 12 York: Academic Press. B. ChanceAK GhoshEK PyeB. Hess1974Biological and Biochemical Oscillators12 YorkAcademic Press
  22. 22. Hess B, Boiteaux A (1971) Oscillatory phenomena in biochemistry. Annu Rev Biochem 40: 237–258.B. HessA. Boiteaux1971Oscillatory phenomena in biochemistry.Annu Rev Biochem40237258
  23. 23. Boiteux A, Goldbeter A, Hess B (1975) Control of Oscillating Glycolysis of Yeast by Stochastic, Periodic, and Steady Source of Substrate: A Model and Experimental Study. Proc Natl Acad Sci USA 72: 3829–3833.A. BoiteuxA. GoldbeterB. Hess1975Control of Oscillating Glycolysis of Yeast by Stochastic, Periodic, and Steady Source of Substrate: A Model and Experimental Study.Proc Natl Acad Sci USA7238293833
  24. 24. Dano S, Sørensen PG, Hynne F (1999) Sustained oscillations in living cells. Nature 402: 320–322.S. DanoPG SørensenF. Hynne1999Sustained oscillations in living cells.Nature402320322
  25. 25. Bezrukov SM, Vodyanoy I, Parsegian VA (1994) Counting polymers moving through a single ion channel. Nature 370: 279–281.SM BezrukovI. VodyanoyVA Parsegian1994Counting polymers moving through a single ion channel.Nature370279281
  26. 26. Pye EK, Chance B (1966) Proc Nat Acad Sci USA 55: 888–894.EK PyeB. Chance1966Proc Nat Acad Sci USA55888894
  27. 27. Strier DE (2002) PhD thesis, Universidad de Buenos Aires. DE Strier2002PhD thesis, Universidad de Buenos Aires.
  28. 28. Goldbeter A, Lefever R (1972) Dissipative structures for an allosteric model. Biophys J 12: 1302–1315.A. GoldbeterR. Lefever1972Dissipative structures for an allosteric model.Biophys J1213021315
  29. 29. Mair T, Muller SC (1996) Traveling NADH and proton waves during oscillatory glycolysis in vitro. J Biol Chem 271: 627–630.T. MairSC Muller1996Traveling NADH and proton waves during oscillatory glycolysis in vitro.J Biol Chem271627630
  30. 30. Pearson JE, Bruno W (1992) Pattern Formation in an N+Q Component Reaction-Diffusion System. Chaos 2: 513–524.JE PearsonW. Bruno1992Pattern Formation in an N+Q Component Reaction-Diffusion System.Chaos2513524
  31. 31. Bagyan S, Mair T, Dulos E, Boissonade J, DeKepper P, et al. (2005) Glycolytic oscillations and waves in an open spatial reactor: Impact of feed-back regulation of phosphofructokinase. Biophys Chem 116: S. 67–76.S. BagyanT. MairE. DulosJ. BoissonadeP. DeKepper2005Glycolytic oscillations and waves in an open spatial reactor: Impact of feed-back regulation of phosphofructokinase.Biophys Chem116S. 6776