Inward Rotating Spiral Waves in Glycolysis

Abstract

We report on the first observation of inward rotating spiral waves (antispirals) in a biochemical reaction-diffusion system. Experiments are performed with extracts from yeast cells in an open spatial reactor. By increasing the protein concentration of the extract we observe a transition from outward to inward propagating waves of glycolytic activity. Numerical simulations with an allosteric model for the phosphofructokinase can reproduce these inward propagating waves over a wide range of parameters if the octameric structure of yeast phosphofructokinase is taken into account.

Spiral-shaped concentration waves are common patterns in biological reaction-diffusion systems (1–3) which occur in both excitable and oscillatory media (4). Recently, it was shown (5–7) that oscillatory systems near a supercritical Hopf bifurcation may also support a new type of spiral wave pattern called inward rotating spiral wave or antispiral. In contrast to normal spiral waves the phase fronts of antispirals propagate toward the spiral core. The group velocity is positive for both antispirals and normal spirals, and consequently local perturbations of the wave propagate always away from the spiral core. Despite the fact that antispirals should be quite common in oscillatory media they were only observed in two chemical reaction-diffusion systems (8,9).

We report on the generation of inward propagating NADH waves in an enzyme reaction system centered on the allosteric enzyme phosphofructokinase (PFK). Glycolysis is the central metabolic pathway in virtually all organisms, and the allosteric feedback regulation of the PFK is known to be essential for the oscillatory behavior of that pathway (10–12). We have previously shown that diffusive coupling generates waves of glycolytic activity in yeast extracts (3,12,13). Here, we induce a transition from outward to inward propagating waves by increasing the protein concentration of the extract from 25 mg/ml to 60 mg/ml. In numerical simulations with the allosteric enzyme model proposed by Goldbeter (14) the antiwaves can be reproduced if the high oligomeric structure of yeast PFK is taken into account.

Experiments are performed in an open spatial reactor as described in (12). Briefly, the yeast extract, which contains the glycolytic enzymes, is fixed in a thin layer of 1.65% agarose gel. The gel is separated from a continous stirred tank reactor (CSTR) (Volume flow: 6.2 ml/h, stirring rate 500 rpm) by a cellulose triacetate membrane (MW-cut off 10 kD; Sartorius) such that the large enzymes are kept in the gel compartment whereas the metabolites and cofactors are continuously refreshed by diffusive exchange with the CSTR. Glycolytic reactions are kept in an oscillatory state by properly adjusting the inflow streams of the CSTR. The spatio-temporal dynamics of glycolytic activity were monitored via NADH fluorescence changes in the yeast extract (12).

In the range between 25 mg/ml–34 mg/ml protein concentration only outward propagating NADH waves, mostly in the form of circular-shaped waves (target patterns), are observed (Figs. 1 A and C). As the protein concentration is increased inward propagating NADH waves appeared in the range of concentrations between 40 mg/ml–60 mg/ml in eight independent experiments. In the range between 40 mg/ml–50 mg/ml, we always observed several coexisting antispirals with a wavelength larger than the system size (Movie S1 in the Supporting Material). Such a behavior is generically expected to occur close to a supercritical Hopf bifurcation or near the transition between spiral waves and antispirals where the wavelength diverges (7). At 59 mg/ml we obtained a single antispiral with a wavelength comparable to the system size (Movie S2). It was coexisting with an inward propagating circular wave (antitarget) for ∼4hs (Fig. 1 B). During this period the spiral core exhibited a slow upward drift (Fig. 1 D) until it was pushed out of the observation area.

Figure 1.

Transition from outward to inward propagating NADH waves as a function of the protein concentration of the extract. (A) Snapshot of an outward propagating target pattern. Local oscillation period (measured at the white dot): 12.6 min. (B) Snapshot of an inward rotating spiral wave and anti-target (AT). Arrows indicate the direction of wave propagation. (C and D) Space-time plots taken along the dashed lines in (A) and (B). White arrow heads in (B) and (D) mark the location of the spiral core. Scale bars: (A and B) 2mm, (C) and (D) horizontal − 2mm, vertical − 10 min. (E) Power spectrum of local oscillations close to the spiral core (1) and in the bulk (2) as indicated in (B). (F) Negative dispersion dω/dk < 0 measured from a one-dimensional space-time plot along the white line with bar ends shown in (B). Error bars are standard errors obtained from the measurement of seven subsequent wave fronts.

Antispiral waves are characterized by two distinct properties (5,7,8): a negative dispersion relation and an oscillation frequency ω_AS that is smaller than that of spatially homogeneous bulk oscillations ω_bulk. To confirm the latter property (ω_AS < ω_bulk) we measured the local oscillation frequency of the NADH fluorescence close to the spiral core (Fig. 1 B, region 1) and in the lower left corner (Fig. 1 B, region 2) where bulk oscillations arrived from a region outside of the observation area. The corresponding spectra in Fig. 1 E show that the local oscillation frequency close to the spiral core (dashed line) is smaller by ∼10% as compared to that of the bulk oscillations (solid line). This is in qualitative agreement with theoretical predictions using the complex Ginzburg-Landau equation (5,7).

The dispersion relation describes how the oscillation frequency of a wave changes with its wave number. For a spiral wave, it has to be measured in the far field where the wave front is approximately planar. Hence, we generated a one-dimensional space-time plot from a transversal intersection with the antispiral wave front in a region far away from the spiral core (cf. Fig. 1 B). By means of an edge-detection algorithm the wave front profiles were extracted and locally fitted by smoothing splines. From the slope of the wave front profiles we measured how the local phase velocity changes with the local oscillation period. From these two quantities we computed the local angular frequency ω and the local wave number k. The resulting dispersion curve has a negative slope (cf. Fig. 1 F) as expected for an antispiral (7).

To investigate the generic conditions for the occurrence of inward propagating waves in glycolysis we conducted numerical simulations using the classical Goldbeter model (14). Instead of describing the whole glycolytic pathway it focuses on the allosteric regulation of the PFK. It is assumed that the PFK comprises of n subunits to which ATP can either bind as a substrate or as an allosteric inhibitor while ADP acts as an allosteric activator of the PFK in that model. In dimensionless units it can be written as (see Supporting Material)

$$\frac{\partial }{\partial t}\alpha =\delta {\nabla }_{\mathbf{x}}^2\alpha +\nu -\sigma {\varphi }_n\left(\alpha ,\gamma \right)$$ (1)

$$\frac{\partial }{\partial t}\gamma ={\nabla }_{\mathbf{x}}^2\gamma +q\sigma {\varphi }_n\left(\alpha ,\gamma \right)-\gamma$$ (2)

where α ∝ ATP denotes the inhibitor and γ ∝ ADP the activator. The function

$${\varphi }_n\left(\alpha ,\gamma \right)\equiv \frac{a\alpha {\left(1+a\alpha \right)}^{n-1}{\left(1+\gamma \right)}^n}{L{\left(1+c\alpha \right)}^n+{\left(1+a\alpha \right)}^n{\left(1+\gamma \right)}^n}$$ (3)

describes the allosteric regulation of the PFK. ν and σ are proportional to the substrate influx and the PFK concentration, respectively. The parameter $\delta \equiv D_{ATP}^{eff}/D_{ADP}^{eff}$ describes a spatial scale separation. It is given by the ratio between the effective diffusion coefficients of the inhibitor and the activator. The remaining parameters c, L, a, and q were fixed at values which are compatible with experiments in well-mixed extracts (see Supporting Material) whereas σ and ν were constraint by the requirement to generate oscillatory behavior (Fig. 2 A).

Figure 2.

Results from numerical simulations with the Goldbeter model. (A) Phase diagram for the occurrence of spiral waves and antispirals as a function of the normalized PFK concentration σ and the normalized substrate influx rate ν (HB denotes a supercritical Hopf bifurcation). (B) Phase diagram for the occurrence of spiral waves, antispirals and Turing patterns as a function of the number of PFK subunits n and the spatial scale separation δ. For each n the value of ν was adjusted as ν = 0.95ν_H, ₂(n) where ν_H, ₂(n) is the larger of the two Hopf bifurcation points shown in A. (C)-(E) Snapshots of two dimensional simulations of anti-spirals for increasing values of δ for ν = 0.1 and σ/L = 7·10⁻³ (see Movie S3). Arrows mark the direction of wave propagation. For visual comparison with Fig. 1B the experimental observation area is indicated by a dashed line. Scale bars: 5 mm. Other parameters: L = 10⁶, q = 5, c = 0, a = 1.

In extensive numerical simulations we found that the number of PFK subunits n as well as the spatial scale separation δ have a significant impact on the formation of inward propagating waves (Fig. 2 B). Interestingly, for δ ≤ 5 antiwaves could not be generated for a dimeric PFK (n = 2) as it was used in the original Goldbeter model (14). However, when we took into account the octameric structure of yeast PFK (n = 8) (15) antiwaves readily appeared for a moderate spatial scale separation of δ ≥ 1.5 (Fig. 2 A and B). At larger values of δ antiwaves turn into stationary Turing patterns.

It is likely that in our experiments a small spatial scale separation emerges from the reversible interactions between the PFK, which is largely immobilized in the agarose gel, and its allosteric effectors (16). Since glycolytic enzymes account for at least 65% of the total amount of soluble yeast protein (17) increasing the total protein concentration of the extract (as we have done) should increase the PFK concentration in a proportional manner. A simple estimate shows that δ should increase with increasing PFK concentrations in the extract (see Supporting Material) and, therefore, favor the formation of inward propagating waves. This view is supported by the fact that the wavelength of the simulated antispirals decreased as δ was increased (Fig. 2 C–E). A similar scenario was also observed in the experiments where the wavelength of the antispiral waves decreased as the protein concentration of the extract was increased.

In agreement with theoretical predictions (5,6) the antiwaves arise close to the supercritical Hopf bifurcation, but could also be generated inside the oscillatory region (Fig. 2 A). Since the Goldbeter model is based on the quite general Monod-Wyman-Changeux mechanism (18) to describe the interaction of allosteric enzymes with their effectors, it can be expected that inward propagating waves may also occur in other diffusively coupled allosteric enzyme systems to which the Monod-Wyman-Changeux mechanism is applicable.

Supporting Material

Additional text and three movies are available at http://www.biophysj.org/biophysj/supplemental/S0006-3495(10)00483-2.