^{1}

^{2}

^{*}

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

The authors have declared that no competing interests exist.

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 (

Concentration gradients inside the cytosol are a vital piece of the cell's machinery. They underlie morphogenesis

The general concept behind Turing patterns involves a combination of short-range activation and large-range inhibition

One of the early attempts to look for an example of a symmetry breaking mechanism in biology was due to I. Prigogine

In this paper we overcome the drawbacks of these previous works by showing that the full 5-variable Selkov model describing the

The 5-variable Selkov model reads _{2}^{γ} and _{1}_{2}^{γ}; the substrate, _{1} (_{1}, and the product, _{2}^{γ} (_{2}[_{2}]. From scheme (1) it is clear that the enzyme is inactive unless it has γ product molecules bound, forming the complex _{2}^{γ}. As done in _{0}, the total concentration of enzyme which remains constant and uniform throughout the evolution. The other quantities are_{1}) and _{2}) are equal (_{1} = _{2}), we take ^{1/2}. Using the diffusion coefficients ^{2} s^{−1} ^{−1})^{½}. while the dimensionless diffusion coefficients are _{1} = _{2} = 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

We now constrain the values of the various parameters of Eqs. (2) according to previous estimations and measurements _{1} and ν_{2} at which the oscillations start are ν_{1}*∼5.8 µMs^{−1} and ν_{2}*∼0.04 s^{−1} _{0}* between 3 and 10 µM, the average concentrations of _{1}*] = 630 µM and [_{2}*] = 150 µM, respectively, while the period of the oscillations, ^{−6}, α = 15, _{1} = 1500, _{3} = 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, ν_{1}*∼5.8 µMs^{−1} and ν_{2}*∼0.04 s^{−1}, we obtain that the total amount of enzyme at the Hopf bifurcation is e_{0} = e_{0}*∼7.9 µM and that the various reaction rates satisfy_{1}*]∼150 µM and [_{2}*]∼145 µM, and that the dimensional period of the oscillations is

(a) Glycolytic oscillations in _{1} (solid curve) and _{2} (dashed curve) for η = 0.15, ν = 0.00345, ε = 10^{−6}, α = 15, _{1} = 1500, _{3} = 1. (b) Linear growth rate of the unstable modes as a function of the square of the wavenumber, _{1} = 1500, _{3} = 1, and _{1} = _{2} = 0.01. Inset: Evolution of [_{1}] 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

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, _{1} and _{3} 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 (_{1} as ν_{2} also gets larger, ^{−4}). We show in _{c}, when η and ν are varied in the shaded region of _{2}, 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 _{1} = 1500 and the previous rate constant estimates (Eqs. (6)) obtained at the Hopf bifurcation, which we assume remain fixed, we conclude that _{0}≥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 _{0} decreases, leading, in turn, to the disappearance of the patterns.

We finally integrate numerically Eqs. (2) in a square domain of side _{1} after 10 min have passed since the initial situation, at which the concentrations are given by the homogeneous stationary solution ([_{1}] = 2.4 mM and [_{2}] = 71 µM), with 10% added noise. The typical size of the pattern agrees with the critical wavelength of the linear stability analysis, l_{c} ≈12 µm. The simulation shows that Eqs. (2) support stable Turing patterns with equal diffusion coefficients of

Stationary pattern in [_{1} = 1500, _{3} = 1, and _{1} = _{2} = 0.01.

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

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

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.