Diffusive coupling can discriminate between similar reaction mechanisms in an allosteric enzyme system
- Ronny Straube^{1}Email author and
- Ernesto M Nicola^{2}
https://doi.org/10.1186/1752-0509-4-165
© Straube and Nicola; licensee BioMed Central Ltd. 2010
Received: 9 August 2010
Accepted: 30 November 2010
Published: 30 November 2010
Abstract
Background
A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze.
Results
Motivated by recent experiments we compare two closely related mechanisms for the product activation of allosteric enzymes with respect to their ability to induce different types of reaction-diffusion waves and stationary Turing patterns. The analysis is facilitated by mapping each model to an associated complex Ginzburg-Landau equation. We show that a sequential activation mechanism, as implemented in the model of Monod, Wyman and Changeux (MWC), can generate inward rotating spiral waves which were recently observed as glycolytic activity waves in yeast extracts. In contrast, in the limiting case of a simple Hill activation, the formation of inward propagating waves is suppressed by a Turing instability. The occurrence of this unusual wave dynamics is not related to the magnitude of the enzyme cooperativity (as it is true for the occurrence of oscillations), but to the sensitivity with respect to changes of the activator concentration. Also, the MWC mechanism generates wave patterns that are more stable against long wave length perturbations.
Conclusions
This analysis demonstrates that amplitude equations, which describe the spatio-temporal dynamics near an instability, represent a valuable tool to investigate the molecular effects of reaction mechanisms on pattern formation in spatially extended systems. Using this approach we have shown that the occurrence of inward rotating spiral waves in glycolysis can be explained in terms of an MWC, but not with a Hill mechanism for the activation of the allosteric enzyme phosphofructokinase. Our results also highlight the importance of enzyme oligomerization for a possible experimental generation of Turing patterns in biological systems.
Background
The modular structure of biochemical reaction networks greatly facilitates the systematic investigation of their design principles [1, 2]. In this way it is often possible to identify small functional units called network motifs [3] which convey a particular functionality. A thorough understanding of the relationship between network design and functionality is not only important for a smart modification and regulation of existing networks, but it is also essential to design novel circuits with prescribed functionality [4, 5].
Regulatory properties of cellular networks arise from an interplay between positive and/or negative feedback reactions. These feedback reactions can be effective at the transcriptional level, at the posttranslational level or through allosteric interactions. For example, in transcriptional networks feed forward loops can act as a low pass filter [6] or as a fold-change detector [7] depending on the sign of the genetic interactions. Signal transduction cascades often utilize post-translational modifications such as phosphorylation/dephosphorylation cycles to generate ultrasensitivity [8] or bistability [9]. This behavior is advantageous for cell fate decisions where irreversible switch-like transitions are required, e.g. during maturation [10] or cell-cycle progression [11]. Metabolic enzymes are often regulated through allosteric interactions with positive and/or negative effector molecules. A classical example is the allosteric product activation of the glycolytic enzyme phospho-fructokinase (PFK) which may lead to an oscillatory behavior of the glycolytic pathway [12, 13].
So far, the relation between particular molecular reaction mechanisms and the resulting macroscopes behavior has been mainly investigated in well-mixed reaction systems where the dynamics is conveniently described by ordinary differential equations [14, 15]. However, if the enzymes in reversible modification cycles are located in different cellular compartments diffusive coupling between neighboring enzyme/substrate molecules may generate steep gradients [16] resulting in front-like wave propagation of phosphoproteins [17]. In that way spatially distributed signaling pathways may create step-like activation profiles that can affect the downstream response of the system in a threshold based manner [18]. Hence, spatial coupling can significantly alter the macroscopic behavior of biochemical reaction systems [19] and bring about new functionality to network motifs [20, 21].
In reaction-diffusion systems spiral shaped concentration waves and stationary Turing patterns are among the most fascinating spatiotemporal structures. While spiral waves can occur in systems with excitable and oscillatory reaction dynamics [22, 23] Turing patterns typically emerge in activator-inhibitor systems with long range inhibition [24, 25]. Here, we investigate the effect of different mechanisms of product activation on the generation of such reaction-diffusion patterns in an enzymatic reaction system centered around the PFK which is a central part of the glycolytic pathway. Under well-stirred conditions this system exhibits oscillatory behavior in both cell free extracts [26, 27] and in living cells [28], and diffusive coupling was shown to generate waves of glycolytic activity in yeast extracts [29–31]. Recently, we have observed a novel type of spiral wave behavior in that system [32]. By increasing the overall protein concentration of the extract a transition from outward to inward rotating spiral waves (also known as anti-spirals) was induced. While outward propagating waves have been observed in several biological systems [29, 33, 34] inward rotating spiral waves were, so far, only observed in purely chemical systems [35, 36].
Although we could reproduce the inward propagating waves in numerical simulations with the Goldbeter model [32] the underlying molecular mechanism for their generation is still unclear. The simulations have shown that the negative feedback on the PFK activity, as provided by its substrate ATP, is not required to generate anti-waves. Therefore, we focus here on the allosteric activation of the PFK by its product ADP. Specifically, we address the question whether the symmetry model of Monod, Wyman and Changeux (MWC) [37], as employed in the Goldbeter model, is necessary to generate inward propagating waves or whether a more simple Hill kinetics, as it was used by Sel'kov [12] to model the PFK activation, is suffcient. Since the regulatory properties of the PFK play a key role for the emergence of oscillatory behavior in glycolysis [27, 38], in particular in yeast extracts [39], these simple models have been remarkably successful in describing general aspects of glycolytic oscillations [12, 40].
whose solution types are well known [42]. Here, the complex amplitude A(x, t) describes slow spatio-temporal modulations around the unstable steady state. At the level of the CGLE the details of the molecular reaction mechanism are encoded in the dependence of the two real parameters c_{1} = c_{1}(D, p) and c_{3} = c_{3}(p) on the original system parameters p and D in Eq. 1. To find the mapping between the two sets of parameters is tedious, but straight-forward [41] (see Methods). The transition between inward and outward propagating waves is marked by the curve c_{1} - c_{3} = 0 [43–45] where the region c_{1} - c_{3}> 0 corresponds to inward propagating waves in Eq. 1.
By explicitly calculating the two CGLE coefficients c_{1} and c_{3} we show that inward propagating waves can arise in the Goldbeter model due to the sequential binding of product molecules to the allosteric enzyme as implied in the MWC mechanism. In contrast, in the limit of an in finitely large binding a finity, as implicitly assumed in the Sel'kov model, the formation of inward propagating waves is sup-pressed by a Turing instability. We also find a relation between enzyme cooperativity and the occurrence of inward propagating waves. However, it is not the absolute magnitude of the cooperativity which is important here (as it is for the occurrence of oscillations [40]), but the sensitivity of the co-operativity with respect to changes in the activator concentration. Finally, we observe that the sequential activation mechanism has a stabilizing effect on the wave dynamics. Together, this shows that in the presence of diffusive coupling the particular choice of a molecular mechanism can have a significant impact on the type and the stability of spatio-temporal patterns even though the dynamics under well-mixed conditions is qualitatively the same.
Model Definitions
Sel'kov Model
where ϕ_{ S } denotes the fractional saturation. Substrate (α = ATP/K_{ M } ) and product $(\gamma =ADP/{K}_{P}^{app})$ concentrations are measured in terms of the Michaelis-Menten constant ${K}_{M}=(k+{k}_{S}^{-})/{k}_{S}^{+}$ and the apparent dissociation constant for product binding ${K}_{P}^{app}={({k}_{P}^{-}/{k}_{P}^{+})}^{1/n}$. The other parameters are given by $\nu ={\nu}_{i}/{K}_{M}{k}_{d},\sigma =k{e}_{0}/{K}_{M}{k}_{d}$ and $q={K}_{M}/{K}_{P}^{app}$. Time is measured in units of 1/k_{ d } and e_{0} denotes the total enzyme concentration.
MWC Model
Based on experimental evidence Goldbeter proposed an alternative approach to describe the allosteric regulation of the PFK [13] which utilizes the Monod-Wyman-Changeux mechanism [37]. Here, the free form of the oligomeric enzyme performs concerted transitions between a catalytically active (R_{00}) and a catalytically inactive (T_{00}) conformation where the allosteric constant L = k^{+}/k^{ - } defines the equilibrium between both conformations in the absence of any ligands (Figure 1B). The enzyme is activated by sequential binding of product molecules with dissociation constant ${K}_{P}=({k}_{P}^{-}/{k}_{P}^{+})$ for each binding step. Hence, there are n + 1 active enzyme forms R_{0m}to which substrate molecules can bind to form n + 1 enzyme-substrate complexes R_{1m}. Each complex can release product molecules at the specific rate k.
The parameters have the same meaning as in Eqs. 3 if the apparent dissociation constant ${K}_{P}^{app}$ is replaced by the true dissociation constant K_{ P } .
Compared with the original Goldbeter model we have neglected the cooperativity with respect to substrate binding and the inhibitory effect of ATP on the PFK activity (as suggested by numerical simulations [32]). With these simplifications we treat the Sel'kov and the Goldbeter model on an equal footing which allows for a direct comparison between their PFK activation mechanisms. Since our model retains the MWC mechanism for PFK activation as an essential part we shall call it the MWC model. We also remark that the PFK actually exhibits sigmoidal behavior with respect to its second substrate fructose-6-phosphate while it does not show any co-operativity with respect to ATP [46, 47]. Hence, the simplifying assumption of a hyperbolic dependence of the PFK activity on ATP as a substrate seems to be reasonable.
Diffusion and Unified Description
The simplest way to incorporate diffusive coupling between the PFK effectors ATP and ADP is to add 'diffusion terms' in Eqs. 3 with constant (effective) diffusion coefficients. Thereby, we neglect complications arising from allosteric interactions between the PFK effectors and the enzyme which may lead to cross-diffusion terms (non-diagonal elements in the diffusion matrix D) and a dependence of the effective diffusion coefficients on the effector concentrations [48, 49]. For a recent review of the effects of cross-diffusion on pattern formation see Ref. [50].
where the parameter δ ≡ D_{ATP}/D_{ADP} denotes the ratio between the effective diffusion coefficients of inhibitor and activator. Length scales are measured in units of the activator diffusion length given by (D_{ADP}/k_{ d } )^{1/2} = (D_{ATP}/δk_{ d } )^{1/2}.
Eqs. 6 and 7 will be analyzed near a supercritical Hopf bifurcation where the dynamics is well de-scribed by the CGLE in Eq. 2. We are particularly interested in the type and the stability of the emerging patterns as we change from a sequential activation mechanism (L_{ M } = L > 1, ε_{ M } = 1) to a Hill-like activation mechanism (L_{ S } ≡ 1, ε_{ S } = 0). Note that the Hill mechanism leads to a factor γ^{ n } in Eq. 7 while the sequential mechanism produces a factor (1 + γ) ^{ n } . The latter results from the (binomial) summation over the intermediate enzyme states R_{ l0 } , ..., R_{ ln } (l = 0, 1) (see Methods).
Transition from the MWC to the Sel'kov Model
Given the structural similarity between the Sel'kov and the MWC model it will be beneficial to investigate the relation between the two models in more detail. In particular, we expect that the MWC mechanism reduces to that of the Sel'kov model as the affinity for subsequent product binding steps increases (i.e. K_{ P } decreases) such that the product activation becomes more and more cooperative.
Hence, ϕ_{ ε } interpolates between ϕ_{ M } and ϕ_{ S } since ϕ_{1 ≡}ϕ_{ M } and as ε → 0 (the binding a finity in-creases) ϕ_{ ε } approaches ϕ_{ S } provided that the product L_{ M } ε^{ n } converges to L_{ S } = 1. However, this means that in the MWC model the enzyme cooperativity, as measured by the allosteric constant L_{ M } , has to become increasingly large which is in agreement with the idea that the product activation becomes more cooperative as we change from the MWC to the Sel'kov mechanism.
This relation between the MWC and the Sel'kov model will be helpful when we analyze how the type and the stability of the spatio-temporal patterns changes as we change the PFK activation mechanism from the MWC to the Hill type.
Results
The diffusive coupling of locally oscillatory reactions as in Eqs. 6 and 7 can generate different types of reaction-diffusion wave patterns which can be broadly classified into outward and inward propagating waves [45]. Near a supercritical Hopf bifurcation the transition between these wave types occurs for c_{1} -c_{3} = 0 (Eq. 2). Depending on the initial and/or boundary conditions these waves may appear in the form of circular or spiral shaped waves.
More complex dynamic behavior can occur near a Benjamin-Feir instability which is indicated by the condition 1 + c_{1}c_{3}< 0 [42]. In this bifurcation plane wave solutions become unstable against long wave length perturbations which may result in the occurrence of spatio-temporal chaos.
Finally, when the spatial scale separation δ becomes sufficiently large the oscillatory instability may be suppressed and stationary Turing patterns can emerge. The transition between wave dynamics and stationary patterns is indicated by the codimension-two Turing-Hopf bifurcation.
To compare the spatio-temporal dynamics of the Sel'kov and the MWC model we have calculated the two CGLE coefficients (c_{1} and c_{3}) and the Turing-Hopf curve for Eqs. 6 and 7 as a function of the systems parameters L_{ i } , ε_{ i } , ν, q, n and δ (see Methods). Note that σ has been eliminated by the requirement for the system to be near the Hopf bifurcation.
Sequential vs. Hill-like Activation Mechanism
As the affinity for the sequential binding of product molecules increases (ε decreases) the stability region of inward propagating waves (dark shaded area) decreases (Figure 2A and 2B). At ε = 0.1 the transition curve between outward and inward propagating waves (c_{1} - c_{3} = 0, dashed line) has crossed the Turing-Hopf bifurcation line (Figure 2C). Thus, for ε ≤ 0.1 the transition to inward propagating waves occurs in the non-oscillatory regime where wave behavior (shaded area) is suppressed in favor of stationary Turing patterns. This shows that the inward propagating waves, as predicted by the CGLE, are not necessarily observable at the level of the original reaction-diffusion system (Eqs. 6 and 10). Since the binding of subsequent product molecules becomes more cooperative as ε → 0 the occurrence of inward propagating waves in the MWC model seems to be related to the sequential activation of the PFK (Figure 1B) which exhibits less cooperativity because the binding a finity (1/εK_{ p } ) is finite for ε = 1.
Wave Stability and Numerical Simulations
Inward Propagating Waves and Enzyme Cooperatively
Positive (negative) cooperativity corresponds to values n_{ H } > 1 (n_{ H } < 1) while n_{ H } = 1 indicates no cooperativity.
To perform the transition from the MWC to the Hill mechanism when L is not necessarily large we introduce in Eq. 8 an effective allosteric constant as L_{ eff } ≡ L_{ M } ε^{ n } = ε (L - 1) + 1. This definition ensures that L_{ eff } has the correct limiting behavior as required by Eq. 9, i.e. L_{ eff } = L_{ M } = L for ε = 1 and L_{ eff } → L_{ S } = 1 as ε → 0. Note that the true allosteric constant L_{ M } increases as 1/ε^{ n } when ε → 0.
Discussion
Beginning in the 1960s glycolytic oscillations have become one of the best studied biochemical oscillators both in cell-free extracts [26, 38, 53] and in living cells [28, 54]. Later it was found in studies with yeast cell populations that glycolytic oscillations represent a collective phenomenon. The oscillations in individual cells are synchronized through the exchange of metabolic intermediates such as acetaldehyde [55] or glucose [56]. At low cell densities the oscillations at the population level disappear (synchronously in all cells) indicating a quorum sensing mechanism [57]. Synchronized behavior was also observed in cell-free extracts where diffusive coupling of glycolytic enzymes can generate waves of glycolytic activity [30, 31]. However, clear experimental evidence for metabolic waves in living cells remains scarce [58] although mathematical modeling supports the feasability of such waves [59].
Recently, a novel type of wave dynamics, called inward rotating spiral waves, has been observed in cell-free yeast extracts [32]. Such wave behavior has, so far, only been observed in purely chemical systems [35, 36]. Here, we have investigated the molecular mechanism underlying the generation of such anti-spiral waves in simple glycolytic model systems which focus on the allosteric activation of the glycolytic enzyme phosphofructokinase (PFK). We have shown that in the Goldbeter model inward rotating spiral waves can arise due to the sequential activation of the PFK implied in the Monod-Wyman-Changeux mechanism [13, 37]. In the limit of an in finitely large binding a finity where the PFK activation is described by a Hill function, as in Sel'kov model [12], the capability to generate inward propagating waves is lost. This suggests that the MWC mechanism, as in Figure 1B, can not be further implied. On the other hand, as we have shown earlier [32] the capability to generate anti-waves is retained by the Goldbeter model where the cooperativity with respect to substrate binding and the allosteric inhibition by ATP are additionally taken into account. Hence, the MWC model can be regarded as a 'core' mechanism for the generation of inward propagating waves for allosteric enzyme systems with product activation.
For well-mixed reaction systems a simple Hill function is often employed to model cooperative behavior in a 'generic' way. Near the onset of oscillations choosing a Hill kinetics instead of a more complex activating function, as in the MWC model, does not lead to a qualitative change in the dynamics under well-stirred conditions. However, as we have shown, the choice of the activating function can significantly change the type and the stability of dynamic patterns in the presence of diffusive coupling. For example, the appearance of a Benjamin-Feir instability in the Sel'kov model indicates the occurrence of spatio-temporal chaos which is mostly absent in the MWC model (Figure 2, 3 and 4). This suggests that the intermediate enzyme forms in the MWC model, which are only partially saturated with product molecules, can stabilize the system dynamics against long wave length perturbations.
Sel'kov and Goldbeter have shown that for the PFK mediated reaction to become oscillatory a sufficiently strong positive enzyme cooperativity is required [12, 60]. However, as far as oscillations are concerned the detailed shape of the Hill coefficient curve (Eq. 11) is not important. Consequently, they occur on the ascending branch of the cooperativity curve (where dn_{ H }/dγ_{ s } > 0) as well as on the descending branch (where dn_{ H } /dγ_{ s } < 0) as long as n_{ H } > 1 (Figure 4A, B, C and 4D). Interestingly, the occurrence of inward propagating waves does not seem to depend on the magnitude of the enzyme cooperativity, but on its sensitivity with respect to changes in the activator concentration. Our simulations show that the formation of inward propagating waves correlates with a positive sensitivity (dn_{ H } /dγ_{ s } > 0) which indicates that for the pattern forming aspects of allosteric enzyme systems more subtle enzyme properties play a role than they do for the occurrence of oscillations.
Since the glycolytic model systems in Eqs. 6 and 7 are of the substrate-depletion type [14] it is not surprising that both models predict the occurrence of stationary Turing patterns if the spatial scale separation between inhibitor and activator dynamics becomes sufficiently large [49]. What is surprising is the fact that this transition already occurs for comparably small values of δ = 2, ..., 4 if the number n of enzyme subunits is sufficiently large (Figure 2A and 2D). This strong dependence on the enzyme cooperativity has been largely neglected in earlier work [61, 62] which mostly focused on the case n = 2 (corresponding to muscle PFK). However, in yeast the PFK is an octamer (n = 8) for which Turing pat-terns are predicted to occur for δ > 4 in the MWC model and for δ > 2 in the Sel'kov model. The necessary spatial scale separation could be generated, for example, through preferential allosteric interactions of the PFK effectors with immobilized enzymes [32, 63], including the PFK itself. Although Turing patterns can be systematically generated only in chemical systems yet [25] our results suggest that high oligomeric enzyme systems are promising candidates to generate such patterns also in properly designed biochemical reaction-diffusion systems.
Conclusions
In well-mixed reaction systems the systematic investigation of molecular reaction mechanisms has led to considerable insights into the design principles for the generation of a specific type of dynamic behavior such as bistability or oscillations [14, 15, 64]. Here, we have expanded this approach to the case of spatially extended systems. Specifically, we have demonstrated that amplitude equations are a valuable tool to investigate how the occurrence of particular spatio-temporal patterns depends on the details of the underlying molecular reaction mechanism in the presence of diffusive coupling. In that way we could provide a molecular explanation for the occurrence of inward rotating spiral waves as they were recently observed in glycolysis in cell-free yeast extracts. Our results support the view that in yeast the allosteric enzyme phosphofructokinase is activated by a Monod-Wyman-Changeux and not by a Hill mechanism. They also highlight the importance of the number of enzyme subunits for a possible experimental generation of Turing patterns in biological systems.
Methods
Derivation of the rate law for the PFK in the MWC model
is the fractional saturation function which measures the number of occupied substrate binding sites relative to the total number of substrate binding sites. In quasi-steady state the active enzyme states R_{ lm } can expressed in terms of binding constants and substrate/product concentrations. For example, for the case n = 2 shown in Figure 1B we have R_{01} = 2R_{00}ADP/K_{ S } and R_{02} = R_{01}ADP/ 2K_{ S } = R_{00} (ADP/K_{ S } )^{2}. Hence, the summation over the intermediate enzyme states in ${\sum}_{l=0}^{2}{R}_{0l}}={(1+ADP/{K}_{S})}^{2}{R}_{00$ produces a binomial series. Similarly, we obtain ${\sum}_{l=0}^{2}{R}_{1l}}={(1+ADP/{K}_{S})}^{2}{R}_{00}(ATP/{K}_{M})$ such that ϕ as defined in Eq. 12 reproduces Eq. 5 by taking into account that L = T_{00}/R_{00}.
Calculation of the Hopf and the Turing Instability
where the Sel'kov model is characterized by ε_{ S } = 0 and L_{ S } = 1 while the Monod-Wyman-Changeux model is obtained for ε_{ M } = 1 and L_{ M } = L > 1. In Eqs. 13 the vector p = (ν, σ , q, n, L_{ i } , ε_{ i } ) collectively denotes the kinetic parameters appearing in the functions f_{α} and fγ.
Steady States
For α_{ s } > 0 to be positive we require that σ > ν.
Turing and Hopf bifurcation thresholds
Instabilities occur if there exists a k_{ c } for which Re (λ(k_{ c } )) > 0. The type of instability depends on whether this occurs for k_{ c } = 0 (Hopf bifurcation) or for k_{ c } ≠ 0 (corresponding to a Turing bifurcation if, in addition, Im (λ(k_{ c } )) = 0).
Turing bifurcation
The critical wave number k_{ c } of the most unstable mode in the Turing bifurcation is determined by da_{0}/dk = 0, and the corresponding parameter set is implicitly given by a_{0}(k_{ c } ) = 0.
Hopf bifurcation
Turing-Hopf codimension-2 bifurcation
In general, oscillations are observed for σ ≤ σ_{ H } if σ_{ H } < σ_{ T } while Turing patterns emerge for σ ≤ σ_{ T } if σ_{ T } < σ_{ H } and σ_{ T } is the smallest (real) root of Eq. 17. However, if both of these codimension-1 bifurcations occur simultaneously (σ_{ H } = σ_{ T } ) a Turing-Hopf codimension two bifurcation takes place. An implicit expression for this bifurcation curve is obtained by using the explicit representation for σ_{ H } (Eq. 18) in the expression for T (δ, p) = 0 (Eq. 17). Note that near the Turing-Hopf bifurcation curve it can be difficult to predict whether wave or Turing patterns are observed since both can be simultaneously stable. Alternatively, mixed mode patterns can appear near a Turing-Hopf bifurcation [66].
Calculation of the CGLE coefficients c_{1} and c_{3}
Here, A(x, t) is a complex amplitude describing slow spatio-temporal modulations around the (spatially homogeneous) unstable steady state of Eqs. 13 while c_{1} and c_{3} are real coefficients. In general, c_{3} = c_{3}(p) only depends on the reaction mechanism through the kinetic parameters while c_{1} = c_{1}(p,δ ) additionally depends on the ratio of the diffusion coefficients δ = D_{α}/D_{ γ } .
To determine the borderline between inward and outward propagating waves, given by c_{1} - c_{3} = 0, we will calculate the two CGLE coefficients c_{1} and c_{3} as a function of the original system parameters following the approach in Ref. [41].
Calculation of c_{1}
Calculation of c_{3}
where Id ≡ diag(1, 1) denotes the 2 × 2 identity matrix. The matrix L_{0} and the eigenvalue λ_{0} have been defined in Section.
Explicit expression for c_{1} and c_{3} in the limit of low glycolytic flux
Note, that for the Sel'kov case, where ε_{ i } ≡ ε_{ S } = 0, the coefficients c_{1} and c_{3} become independent of the parameter combination νq corresponding to the steady state value of the activator concentration.
Numerical Simulations
Parameters for the Numerical Simulations shown in Figure 3.
model | Sel'kov | MWC | ||||
---|---|---|---|---|---|---|
panel | A | B | C | D | E | F |
system parameters | ||||||
δ | 1 | 1.5 | 3 | 1 | 3 | 5 |
σ | 1770 | 1770 | 1770 | 50 | 60 | 60 |
α_{ s } (Eq. 14) | 0.073 | 0.073 | 0.073 | 0.4 | 0.34 | 0.34 |
γ_{ s } (Eq. 14) | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |
integrator parameters | ||||||
spatial step: dx = dy | 3 | 3 | 0.3 | 3 | 10 | 0.3 |
time step: dt | 0.05 | 0.05 | 3.10^{-3} | 0.05 | 0.1 | 3.10^{-3} |
side length l_{ s } | 528 | 431 | 30.5 | 528 | 10^{3} | 23.6 |
where r is a random number equally distributed in the interval [0,1] and x, y = 1, ...,N. A similar expression was used for γ.
To generate Figure 3A and 3B we have chosen k = 1/70 while Figure 3D and 3E were generated with k= 2/3.
denotes the diffusion length of the activator, dx is the spatial step size of the discretization and N = 176 is the number of grid points which we kept fixed for all simulations. As a result, the physical dimensions of each panel in Figure 3 are different since the simulations were done for different values of the spatial scale separation δ = D_{ ATP } /D_{ ADP } and dx (cf. Table 1). Once specific values for the ATP diffusion co-efficient D_{ ATP } and the product consumption rate k_{ d } are provided the physical side length is determined by (D_{ ATP } /k_{ d } )^{1/2}l_{ s } where l_{ s } = dx · N/δ^{1/2} denotes the dimensionless side length.
Declarations
Acknowledgements
RS acknowledges financial support from the Ministry of Education of Saxony-Anhalt within the Research Center 'Dynamic Systems'.
Authors’ Affiliations
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