Re-Examination of the Sel’kov Model of Glycolysis and Its Symmetry-Breaking Instability Due to the Impact of Diffusion with Implications for Cancer Imitation Caused by the Warburg Effect

Abstract

We revisit the seminal model of glycolysis first proposed by Sel’kov more than fifty years ago. We investigate the onset of instabilities in biological systems described by the Sel’kov model in order to determine the conditions of the model parameters that lead to bifurcations. We analyze the glycolysis reaction under the circumstances when the diffusivity of both ATP and ADP reactants are taken into account. We estimate the critical value of the model’s single compact dimensionless parameter, which is responsible for the onset of reaction instability and the system’s symmetry breaking. It appears that it leads to spatial inhomogeneities of reactants, leading to the formation of standing waves instead of a homogeneous distribution of ATP molecules. The consequences of this model and its results are discussed in the context of the Warburg effect, which signifies a transition from oxidative phosphorylation to glycolysis that is correlated with the initiation and progression of cancer. Our analysis may lead to the selection of therapeutic interventions in order to prevent the symmetry-breaking phenomenon described in our work.

1. Introduction

Metabolism is a process in which transducer nutrient energy is transformed into a physiologically usable form, namely the molecule ATP (Adenosine Triphosphate). It is a molecule that is the end product of all our metabolic pathways and it has fascinated scientists since its discovery in 1929 by Karl Lohman. It was first synthesized in 1948 by Lord Alexander Todd who, for this achievement, received the Nobel Prize for Chemistry in 1957. ATP molecules store and supply energy for cellular processes. An ATP molecule consists of three building blocks: a flat purine ring system with multiple nitrogen atoms, a ribose sugar in the middle, and the three phosphate groups (Figure 1).

What makes ATP molecules the fuel for many life processes are the two outermost phosphate groups bound to the rest of ATP by so-called phosphoanhydride bonds which are high in energy and readily hydrolyzed by water. This process of ATP hydrolysis leads to ADP (Adenosine diphosphate) and one inorganic phosphate. This hydrolysis of the terminal phosphate of ATP yields about $5·{10}^4$ J per mole of usable energy, or $8.38·{10}^{-20}$ J per single ATP molecule, which is equivalent to 0.5 eV (electron-Volt).

Among many examples of its use by cells, ATP is used to make macromolecules such as proteins, sugars, and lipids. It is also used in ionic channels to create the voltage potential needed to power the nerve cells. It also fuels the motor proteins which expand and contract the muscles and provide the means of intracellular transport. Much like cars and trucks on our streets, cells move different cargoes along the cytoskeleton using dynein and kinesin motor proteins and ATP is a crucial fuel for that purpose. Amazingly, cells within the human body depend upon the hydrolysis of 100 to 150 moles (50–75 kg) of ATP per day to ensure proper functioning.

ATP is formed by the addition of two inorganic phosphate groups (${\mathrm{HPO}}_4^{2-}$) to adenosine monophosphate (AMP) through the process of phosphorylation. Since three phosphate groups on ATP carry a negative charge, a very high amount of energy is required to overcome the mutual repulsion of negative phosphates in the process of adding two phosphate groups to AMP to create ATP. Otherwise, the hydrolysis (the cleavage of the outermost phosphate by water) of ATP to ADP releases a so-called biological energy quantum of 0.5 eV.

The energy needed for chemical work in the phosphorylation of AMP is made available to the living cell by the oxidation of glucose to carbon dioxide and water with the release of associated electrostatic energy. Some of that energy is dissipated as heat, but a significant part of it is stored in other chemical bonds, primarily phosphoanhydride bonds, in ATP. The global chemical reaction for the oxidation of glucose can be written as follows:

$${\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6+6{\mathrm{O}}_2\to 6{\mathrm{CO}}_2+6{\mathrm{H}}_2\mathrm{O}+\mathrm{energy}$$

(1)

This is not an elementary process since it takes place in a series of enzymatic reactions with three main reaction sequences [1]: glycolysis, the Krebs cycle, and the electron transport. In this paper, we focus our attention specifically on the glycolysis part of the process.

Glycolysis [1] develops in three steps, starting from the phosphorylation of glucose to glucose 6-phosphate, which is then isomerized to fructose 6-phosphate. Finally, fructose 6-phosphate is phosphorylated to fructose 1,6-bisphosphate. This last reaction is catalyzed by the enzyme phosphofructokinase (PFK1). The PFK1 enzyme is allosterically inhibited by ATP. Importantly ATP is both a substrate of PFK1, binding at its catalytic site, and an allosteric inhibitor by binding at the regulatory site of PFK1.

The inhibition of PFK1 due to ATP is removed by AMP, meaning that the activity of PFK1 increases if the ratio of ATP to AMP decreases. Since the PFK1 enzyme phosphorylates fructose 6-phosphate, ATP is converted into ADP, and ADP is then converted back to ATP and AMP by the following reaction:

$$2\mathrm{ADP}\rightleftarrows \mathrm{ATP}+\mathrm{AMP}$$

(2)

This step is catalyzed by the enzyme adenylate kinase. In anaerobic respiration glycolysis, two ATP molecules are consumed, producing four new ATP molecules, two NADH (nicotinamide adenine dinucleotide H) molecules, and two pyruvates per single glucose molecule. Under certain specified conditions, the rate of glycolysis is oscillatory.

2. The Sel’kov Model

A mathematical model representing oscillatory glycolysis was originally proposed by [2]. This model has been extensively elaborated, mostly in the mathematical context [3,4,5]. The relevant simplified reaction scheme for glycolysis is as follows: Enzyme PFK1 (denoted by E) is activated or deactivated binding or unbinding with $\gamma$ molecules of ADP (denoted by $S_2$) and ATP (denoted as $S_1$), respectively. The activated form of enzyme E gives, as a product, a molecule of ADP. It is assumed that a steady supply of $S_1$ (ATP) with rate $v_1$ is provided, while the reaction product $S_2$ (ADP) is irreversibly removed by the sink with a steady rate $v_2$. This can be schematized as follows:

$$\begin{array}{c}{v}_1⟶S_1\\ v_1⟶S_1+E{S}_2^{\gamma }\underset{k_{-1}}{\overset{k_1}{\rightleftharpoons }}{S}_{1}E{S}_2^{\gamma }\stackrel{k_2}{⟶}E{S}_2^{\gamma }+S_2{v}_2⟶\\ \gamma S_2+E\underset{k_{-3}}{\overset{k_3}{\rightleftharpoons }}E{S}_2^{\gamma }\end{array}$$

(3)

$\gamma$ is the positive exponent related to the number of the molecules, $S_2$, associated with the PFK1 enzyme. Applying the law of mass action to the above kinetic scheme results in the five differential equations describing the time evolution of five ingredients of glycolytic reaction with the corresponding concentrations: $s_1=\left[S_1\right]$; $s_2=\left[S_2\right]$; $e=\left[E\right]$; $x_1=\left[E{S}_2^{\gamma }\right]$; $x_2=\left[S_{1}E{S}_2^{\gamma }\right]$,

$$\begin{array}{c}{\displaystyle \frac{d{s}_1}{dt}}=v_1-k_1{s}_1{x}_1+k_{-1}{x}_2\\ {\displaystyle \frac{d{s}_2}{dt}}=k_2{x}_2-k_3{s}_2^{\gamma }e+k_{-3}{x}_1-v_2{s}_2\\ {\displaystyle \frac{d{x}_1}{dt}}=-k_1{s}_1{x}_1+(k_{-1}+k_2)x_2+k_3{s}_2^{\gamma }e-k_{-3}{x}_1\\ {\displaystyle \frac{d{x}_2}{dt}}=k_1{s}_1{x}_1-(k_{-1}+k_2)x_2.\end{array}$$

(4)

The fifth differential equation for ${\displaystyle \frac{de}{dt}}$ is redundant because the total amount of enzyme ($e_0$) is conserved, implying a simple relation for time derivatives:

$$e+x_1+x_2=e_0;{\displaystyle \frac{de}{dt}}=-{\displaystyle \frac{d{x}_1}{dt}}-{\displaystyle \frac{d{x}_2}{dt}}.$$

(5)

It is very convenient to introduce a set of new dimensionless variables as follows:

$$\begin{array}{c}{\rho }_1=\left({\displaystyle \frac{k_1}{k_2+k_{-1}}}\right);{\rho }_2={\left({\displaystyle \frac{k_3}{k_{-3}}}\right)}^{1/\gamma };m_1={\displaystyle \frac{x_1}{e_0}};m_2={\displaystyle \frac{x_2}{e_0}};\\ \tau =\left({\displaystyle \frac{k_1{k}_2{e}_0}{k_2+k_{-1}}}\right)t,\end{array}$$

(6)

which leads to a transformed system of Equation (4):

$$\begin{array}{c}{\displaystyle \frac{d{\rho }_1}{d\tau }}=\nu -\left({\displaystyle \frac{k_2+k_{-1}}{k_2}}\right)m_1{\rho }_1+\left({\displaystyle \frac{k_{-1}}{k_2}}\right)m_2\\ {\displaystyle \frac{d{\rho }_2}{d\tau }}=\beta \left[m_2-\left({\displaystyle \frac{k_{-3}}{k_2}}\right)(1-m_1-m_2){\rho }_2^{\gamma }+\left({\displaystyle \frac{k_{-3}}{k_2}}\right)m_1\right]-\eta {\rho }_2\\ \epsilon {\displaystyle \frac{d{m}_1}{d\tau }}=m_2-m_1{\rho }_1+\left({\displaystyle \frac{k_{-3}}{k_2+k_{-1}}}\right)\left[(1-m_1-m_2){\rho }_2^{\gamma }-m_1\right]\\ \epsilon {\displaystyle \frac{d{m}_2}{d\tau }}=m_1{\rho }_1-m_2.\end{array}$$

(7)

Here, the new dimensionless parameters are introduced as follows:

$$\begin{array}{c}\nu ={\displaystyle \frac{v_1}{k_2{e}_0}};\beta ={\displaystyle \frac{k_2+k_{-1}}{k_1}}{\left({\displaystyle \frac{k_3}{k_{-3}}}\right)}^{1/\gamma };\\ \eta ={\displaystyle \frac{(k_2+k_{-1})v_2}{k_1{k}_2{e}_0}};\epsilon ={\displaystyle \frac{e_0{k}_1{k}_2}{{\left(k_2+k_{-1}\right)}^2}}.\end{array}$$

(8)

It is reasonable to consider parameter $\epsilon$ to be much smaller than unity, meaning that both $m_1$ and $m_2$ change remarkably faster than ${\rho }_1$ and ${\rho }_2$ since they are catalytic effects. Thus, these fast variables can be safely set to be equal to their quasi-equilibrium values, $\epsilon {\displaystyle \frac{d{m}_1}{d\tau }}=\epsilon {\displaystyle \frac{d{m}_2}{d\tau }}=0$, yielding

$$\begin{array}{c}{m}_2-m_1{\rho }_1+\left({\displaystyle \frac{k_{-3}}{k_2+k_{-1}}}\right)\left[\left(1-m_1-m_2\right){\rho }_2^{\gamma }-m_1\right]=0\\ m_1{\rho }_1-m_2=0,\end{array}$$

(9)

so that solutions of Equation (9) are expressed as

$$m_1={\displaystyle \frac{{\rho }_2^{\gamma }}{{\rho }_2^{\gamma }\left({\rho }_1+1\right)+1}}$$

(10)

$$m_2={\displaystyle \frac{{\rho }_1{\rho }_2^{\gamma }}{{\rho }_2^{\gamma }\left({\rho }_1+1\right)+1}}.$$

(11)

Inserting $m_1({\rho }_1,{\rho }_2)$ and $m_2({\rho }_1,{\rho }_2)$ from Equations (10) and (11) into Equation (7), we obtain

$${\displaystyle \frac{d{\rho }_1}{d\tau }}=\nu -f({\rho }_1,{\rho }_2)=F({\rho }_1,{\rho }_2)$$

(12)

$${\displaystyle \frac{d{\rho }_2}{d\tau }}=\beta f({\rho }_1,{\rho }_2)-\eta {\rho }_2=G({\rho }_1,{\rho }_2)$$

(13)

where we use a new denotation,

$$f({\rho }_1,{\rho }_2)=m_2({\rho }_1,{\rho }_2)$$

(14)

The relative steady-state reaction for Equation (12) has the following form:

$${\displaystyle \frac{d{\rho }_1}{d\tau }}=0;\nu =f({\rho }_1,{\rho }_2)$$

(15)

In the following, we will specify the exponent $\gamma$ to be exactly two, so that it gives

$$f({\rho }_1,{\rho }_2)={\displaystyle \frac{{\rho }_1{\rho }_2^2}{{\rho }_2^2({\rho }_1+1)+1}}$$

(16)

The function Equation (15) is presented as a 3D diagram in Figure 2.

It is obvious that this function exhibits saturation for (${\rho }_1\to \infty$; ${\rho }_2\to \infty$) as being bounded by 1. This implies that for $\nu >1$ the solution of Equation (12) blows up, which is unphysical. Hence, it makes sense to consider the restriction

$$0<\nu <1.$$

(17)

It is of interest to find the steady state for Equations (12) and (13). Taking

$${\displaystyle \frac{d{\rho }_1}{d\tau }}=0\mathrm{and}{\displaystyle \frac{d{\rho }_2}{d\tau }}=0,$$

(18)

and solving for ${\rho }_1$ and ${\rho }_2$, we obtain the unique steady state

$${\rho }_{10}={\displaystyle \frac{\nu (1+{\rho }_{20}^2)}{(1-\nu ){\rho }_{20}^2}};{\rho }_{20}={\displaystyle \frac{\beta }{\eta }}\nu$$

(19)

The stability of this steady state can be analyzed by linearizing governing Equations (12) and (13) about the steady-state Equation (19) and solving the eigenvalues of the thus linearized system. We will perform this procedure for a simplified version later given by the Equations (23) and (24). The nullclines of the vector field flow are given by the following equations:

$$\begin{array}{ccc}{\rho }_1={\displaystyle \frac{\nu }{1-\nu }}\left(1+{\displaystyle \frac{1}{{\rho }_2^2}}\right),\hfill & \mathrm{for}\hfill & {\displaystyle \frac{d{\rho }_1}{d\tau }}=0\hfill \\ {\rho }_1={\displaystyle \frac{1+{\rho }_2^2}{{\rho }_2\left(\frac{\beta }{\eta }-{\rho }_2\right)}},\hfill & \mathrm{for}\hfill & {\displaystyle \frac{d{\rho }_2}{d\tau }}=0\hfill \end{array}$$

(20)

The first curve in Equation (20) is monotonically decreasing hyperbola (dotted line in Figure 3), while the second nullcline in Equation (20) has the minimum and two vertical asymptotes (Figure 3, dashed line).

The crossing point of these nullclines is the steady state with coordinates given by Equation (19). Here, we restrict the consideration for the experimentally observed fact that the self-sustained oscillations of concentrations ${\rho }_1$ and ${\rho }_2$ in glycolysis appear at a very low rate of ATP influx reaction. This means that the inequality

$$\nu \ll 1$$

(21)

safely holds. Under this condition, Equation (12) becomes simplified as

$$\nu \approx {\rho }_1{\rho }_2^2.$$

(22)

Taking this fact into account, the system of equations, Equations (12) and (13), can be rearranged as follows:

$${\displaystyle \frac{dX}{d\theta }}=1-X{Y}^2,$$

(23)

$${\displaystyle \frac{dY}{d\theta }}=\alpha X{Y}^2-\alpha Y,$$

(24)

where the new variables X, Y, $\theta$ and the single compact parameter $\alpha$ are expressed in this way:

$$X=k_2{e}_0\left({\displaystyle \frac{k_3}{k_{-3}}}\right)\left({\displaystyle \frac{v_1}{v_2^2}}\right){\rho }_1;Y=\left({\displaystyle \frac{k_{-3}}{k_3}}\right)\left({\displaystyle \frac{v_2}{v_1}}\right){\rho }_2,$$

(25)

$$\theta =\left({\displaystyle \frac{k_3}{k_{-3}}}\right){\displaystyle \frac{v_1^2}{v_2^2}}\tau$$

(26)

$$\alpha =\left({\displaystyle \frac{k_{-1}+k_2}{k_1{k}_2{e}_0}}\right)\left({\displaystyle \frac{k_{-3}}{k_3}}\right){\displaystyle \frac{v_2^3}{v_1^2}}$$

(27)

The cubic autocatalytic step expressed by the terms $X{Y}^2$ lies at the heart of the simplest model for oscillatory and other complex nonlinear behavior in closed systems and in a reaction coupled with diffusion, which will be demonstrated in the next section. The above system of Equations (23) and (24) provides the minimally adequate model of a product (ADP)-activated and substrate (ATP)-inhibited reaction. In a finite part of the $(X,Y)$ vector field phase plane, the system has just one fixed point representing the equilibrium state:

$$X_0=Y_0=1.$$

(28)

This means that a homogeneous steady state of the Sel’kov model is achieved when the concentration of ATP and ADP are equal and constant in space and time. The neighborhood of this fixed point system can be linearized by the expansion

$$\begin{array}{c}X=X_0+x;{\displaystyle \frac{x}{X_0}}\ll 1\\ Y=Y_0+y;{\displaystyle \frac{y}{Y_0}}\ll 1\end{array}$$

(29)

The linearization is given in Appendix A. The characteristic equation for the linearized system now reads

$$\left|\begin{array}{cc}-1-\lambda & -2\\ \alpha & \alpha -\lambda \end{array}\right|=0$$

(30)

This gives the square quadratic equation for eigenvalues as follows:

$${\lambda }^2-(\alpha -1)\lambda +\alpha =0,$$

(31)

with the roots expressed as

$${\lambda }_{\pm }={\displaystyle \frac{\alpha -1}{2}}\left[1\pm \sqrt{1-{\displaystyle \frac{4\alpha }{{(\alpha -1)}^2}}}\right]$$

(32)

and represented in Figure 4 as a function of $\alpha$.

Now, on the basis of Equation (30) we determine the characteristic values of parameter $\alpha$ which follow from the following equations:

$$\begin{array}{ccc}\alpha -1=0& \Rightarrow & {\alpha }_0=1\\ {(\alpha -1)}^2-4\alpha =0& \Rightarrow & {\alpha }_{\pm }=3\pm 2\sqrt{2}.\end{array}$$

(33)

According to these critical numbers, one obtains the characteristic intervals for stability features of the system of Equations (23) and (24) with the corresponding boundaries, respectively:

$$\begin{array}{cc}\left(1\right) \mathrm{a}\mathrm{stable}\mathrm{node}\mathrm{at}\hfill & 0<\alpha <3-2\sqrt{2}\hfill \\ \left(2\right) \mathrm{a}\mathrm{stable}\mathrm{focus}\mathrm{at}\hfill & 3-2\sqrt{2}<\alpha <1\hfill \\ \left(3\right)\mathrm{an}\mathrm{unstable}\mathrm{focus}\mathrm{at}\hfill & 1<\alpha <3+2\sqrt{2}\hfill \\ \left(4\right)\mathrm{an}\mathrm{unstable}\mathrm{node}\mathrm{at}\hfill & 3+2\sqrt{2}<\alpha <\infty .\hfill \end{array}$$

(34)

Let us illustrate an unstable focus taking $\alpha =3$. Then, Equation (32) gives

$${\alpha }_{\pm }=1\pm i\sqrt{2};i=\sqrt{-1},$$

(35)

implying perturbations of the form

$$\begin{array}{c}x=x_{0}exp\left(\tau \right)\left[cos\left(\sqrt{2}\tau \right)+isin\left(\sqrt{2}\tau \right)\right]\hfill \\ y=y_{0}exp\left(\tau \right)\left[cos\left(\sqrt{2}\tau \right)-isin\left(\sqrt{2}\tau \right)\right].\hfill \end{array}$$

(36)

These perturbations apparently blow up due the exponential dependence on time. Different mathematical aspects of the Sel’kov model are examined in depth by many authors [3,4,5].

3. The Sel’kov Model as a Spatial Dissipative Structure

Some five decades ago Nobel prize winner Ilya Prigogine [6,7,8] proposed the concept of dissipative structures as the dynamical bases of non-equilibrium self-organization in biological systems. There are at least two types of biochemical instabilities.

The first is generated by homogeneous perturbation where the system goes from a homogeneous steady state to another homogeneous state. The second type is provided by space-time-dependent inhomogeneous perturbations where diffusion acts as an essential factor of symmetry-breaking events.

Diffusion can provide a positive contribution to the excess entropy production in the reaction–diffusion system, leading to the stabilization of the steady state. Otherwise, it increases the manifold perturbations compatible with macroscopic evolution equations. If this effect is dominant, the symmetry-breaking instabilities are possible [9].

In this respect, we consider glycolysis as a spatio-temporal dissipative structure based on diffusion and having the form of sustained oscillations, as described in the previous section. We now extend the Sel’kov system of Equations (23) and (24) and we take into account the diffusion of both reactants X and Y. To simplify, we assume a one-dimensional medium represented by the dimensionless length $\xi$, concerning the diffusion of molecules ATP and ADP, within the following glycolytic reaction:

$${\displaystyle \frac{dX}{d\theta }}=1-X{Y}^2+D_x{\displaystyle \frac{{\partial }^{2}X}{\partial {\xi }^2}},$$

(37)

$${\displaystyle \frac{dY}{d\theta }}=\alpha X{Y}^2-\alpha Y+D_y{\displaystyle \frac{{\partial }^{2}Y}{\partial {\xi }^2}}$$

(38)

where the dimensionless diffusion coefficients are $D_x$ and $D_y$.

To investigate the stability of the steady-state $(X_0,Y_0)=(1,1)$, we first consider the dispersion equation within the framework of a linear analysis. We start with spatio-temporal perturbation of state in the one-dimensional case as the s-direction, of total length L, as follows:

$$\begin{array}{c}X=X_0+xexp(\omega \theta +i\xi /\lambda )\\ Y=Y_0+yexp(\omega \theta +i\xi /\lambda )\end{array};\xi =2\pi {\displaystyle \frac{s}{L}}$$

(39)

where $X_0$ and $Y_0$ are known steady-state values and $\omega$ and $\lambda$ are dimensionless frequency and wavelength, respectively.

The inequalities

$${\displaystyle \frac{x}{X_0}}\ll 1;{\displaystyle \frac{y}{Y_0}}\ll 1$$

(40)

are again of basic importance for the validity of the used approach.

Inserting expressions of Equation (39) into the Sel’kov model, Equations (37) and (38), we are able to obtain the algebraic system of linearized equations as follows:

$$(\omega +1+{\Lambda }_x)x=-2y;{\Lambda }_x={\displaystyle \frac{D_x}{{\lambda }^2}}$$

(41)

$$(\omega -\alpha +{\Lambda }_y)=\alpha x;{\Lambda }_y={\displaystyle \frac{D_y}{{\lambda }^2}}$$

(42)

The straightforward evaluation of Equations (41) and (42) is performed in Appendix B.

Using substitutions of x from Equation (41) to Equation (42) and canceling y, we obtain the dispersion equation

$${\omega }^2-(\alpha -1-{\Lambda }_x-{\Lambda }_y)\omega -\left[\alpha ({\Lambda }_x-1)-{\Lambda }_y({\Lambda }_x+1)\right]=0$$

(43)

According to the Prigogine scenario [6,7,8], we consider the critical condition for the onset of dissipative instability which is established by zero frequency ($\omega =0$). This implies, based on Equation (43), the $\omega$ free term to be equal to zero:

$$\alpha ({\Lambda }_x-1)-{\Lambda }_y({\Lambda }_x+1)=0,$$

(44)

or, explicitly,

$$\alpha ={\displaystyle \frac{{\Lambda }_y({\Lambda }_x+1)}{{\Lambda }_x-1}}=D_y{\displaystyle \frac{(D_x+{\lambda }^2)}{{\lambda }^2(D_x-{\lambda }^2)}}.$$

(45)

This is an even function of wavelength $\lambda$ with two vertical asymptotes $\lambda =0$ and ${\lambda }^2=D_x$. The shape of the function represented by Equation (45) is given in Figure 5.

The function has a minimum which defines the critical ${\lambda }_c$ and critical compacted parameter of the Sel’kov model ${\alpha }_c$. Thus, the first derivative with respect to $\lambda$ should be equal to zero:

$${\displaystyle \frac{d\alpha }{d\lambda }}=D_y{\displaystyle \frac{2{\lambda }^3(D_x-{\lambda }^2)-({\lambda }^2+D_x)(2D_x\lambda -4{\lambda }^3)}{{\lambda }^4{(D_x-{\lambda }^2)}^2}}=0$$

(46)

This gives the equation for critical wavelength:

$${\lambda }_c^4+2D_x{\lambda }_c^2-D_x^2,$$

(47)

$${\lambda }_c^2={\displaystyle \frac{1}{2}}\left[-2D_x\pm \sqrt{4D_x^2+4D_x^2}\right]$$

(48)

Just a positive solution makes physical sense:

$${\lambda }_c^2=(\sqrt{2}-1)D_x.$$

(49)

Interestingly, the critical squared wavelength depends only on the diffusion coefficient of ATP molecules.

Inserting Equation (49) into the expression in Equation (45), one obtains the critical value of the Sel’kov fundamental compacted parameter $\alpha$:

$${\alpha }_c=\left({\displaystyle \frac{\sqrt{2}}{3\sqrt{2}-4}}\right)\left({\displaystyle \frac{D_y}{D_x}}\right)=5.83\left({\displaystyle \frac{D_y}{D_x}}\right)$$

(50)

For ATP and ADP, $D_x\simeq D_y=D=0.35·{10}^{-9} {\mathrm{m}}^2/\mathrm{s}$ [10]. This yields

$${\alpha }_c=5.83$$

(51)

If we compare this number with the intervals in Equation (34), we see that it pertains to a segment of unstable nodes reflecting the circumstance that the system is indeed in the regime of dissipative instability.

Evoking the expression for $\alpha$, Equation (24), we can equate it with Equation (50) introducing the critical reaction rates $v_{1c}$ and $v_{2c}$, so that we have the equation

$$\left[{\displaystyle \frac{(k_{-1}+k_2)k_{-3}}{k_1{k}_2{k}_3{e}_0}}\left({\displaystyle \frac{v_{2c}^3}{v_{1c}^2}}\right)\right]=\left({\displaystyle \frac{\sqrt{2}}{3\sqrt{2}-4}}\right)\left({\displaystyle \frac{D_y}{D_x}}\right).$$

(52)

Solving for $v_{2c}$, we finally obtain the critical value of the sink rate for the onset of reaction instability:

$$v_{2c}={\left[\left({\displaystyle \frac{3\sqrt{2}-4}{\sqrt{2}}}\right)\left({\displaystyle \frac{D_y}{D_x}}\right)\left({\displaystyle \frac{k_1{k}_2{k}_3{e}_0}{(k_{-1}+k_2)k_{-3}}}\right)\right]}^{1/3}·v_{1c}^{2/3}$$

(53)

The sink rate is proportional to the cube root of the product of forward reaction rate constants inversely proportional to the backward rate constants. If we take that $D_x=D_y$, the above expression is simplified to the sublinear function of input rate $v_{1c}$:

$$v_{2c}=\kappa v_{1c}^{2/3};\kappa ={\left[5.83{\displaystyle \frac{k_1{k}_2{k}_3{e}_0}{(k_{-1}+k_2)k_{-3}}}\right]}^{1/3}$$

(54)

If we return to the function $\alpha$, Equation (45), (Figure 5), we can notice that for $\alpha >{\alpha }_c$ there exists two possible values of $\lambda$ for dissipative instability. Thus, ${\alpha }_c$ represents a kind of pitchfork bifurcation point separating the excitability of shorter and longer wavelengths, respectively. It is plausible that longer wavelengths are more probable.

This fact can also be viewed as a symmetry-breaking effect. The disturbance which has the form of Equation (39), taken with $\omega =0$, reduces to a spatially-distributed standing wave of the shape

$$x=Acos(\xi /\lambda );y=Bcos(\xi /\lambda ).$$

(55)

The maximal wavelength is limited by the dimension of a reaction dish (the diameter of a red blood cell, for example):

$${\xi }_{max}=2\pi {\displaystyle \frac{s_{max}}{L}}.$$

(56)

Taking $s_{max}=L$, it follows that

$${\xi }_{max}=2\pi .$$

(57)

Let us impose the boundary conditions in the form of a zero gradient as follows:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial x}{\partial \xi }}=0\mathrm{for}& \xi =0;\mathrm{and}& {\displaystyle \frac{\partial x}{\partial \xi }}=0\mathrm{for}{\xi }_{max}=2\pi ,\hfill \end{array}$$

(58)

This yields

$${\displaystyle \frac{\partial }{\partial \xi }}cos(\xi /\lambda )={\displaystyle \frac{1}{\lambda }}sin(\xi /\lambda )=0.$$

(59)

At ${\xi }_{max}=2\pi$, we have

$$sin(2\pi /\lambda )=0;$$

(60)

which leads to

$$\lambda ={\displaystyle \frac{2}{n}};n=1,2,3,\dots$$

(61)

So, the maximal wavelength is

$$\begin{array}{ccc}\hfill \lambda =2& \mathrm{for}& n=1,\hfill \end{array}$$

(62)

See Figure 6a. The next mode is given by $\lambda =1$ for $n=2$ (Figure 6a,b). Smaller wavelengths give higher local gradients of the reactant concentration, thus enhancing the strength of diffusion, which tends to obliterate imposed perturbation. Otherwise, the mode with the largest wavelength ($\lambda =2$) is the most favorable for a sustained symmetry-breaking instability within the system under consideration.

In Figure 6a, the concentration of ATP is increased in the left half of the “tube” where the glucose source is located, and in Figure 6b, the concentration elevation is evenly distributed around the source and the drain of the reaction.

The amplitudes are exaggerated for better visibility. The dot line represents the steady-state level of concentration.

4. Summary

In this paper, we first briefly elaborated on the importance of ATP as the energy currency in living organisms. Then, we re-examined the Sel’kov model of glycolysis and focused our attention on the compact format, Equations (23) and (24), which contains just one concise dimensionless parameter $\alpha$ (Equation (26)). The main contribution of the paper is the extension of the Sel’kov model to take into account the diffusivity of ATP and ADP molecules as the main reactants participating in this model of glycolysis.

In that respect, we evaluated the critical value of parameter $\alpha$ as the function of the diffusion coefficients. This critical value indicates the onset of a symmetry-breaking instability in the extended Sel’kov model. It exhibits a sublinear proportionality between the rate of ATP influx in the reaction scheme and the ADP forward sink from the reaction (Equation (55)). This instability leads to the redistribution of reactant concentrations following the resulting standing wave pattern (Figure 6). The most favorable standing wave should be one with the maximal wavelength ($\lambda =2$) limited by the dimension of the reaction dish. We intuitively associate diffusion with a smoothing and homogenizing influence that eliminates chemical gradients leading to a uniform spatial distribution. Here, it appears that diffusion acts in an opposite manner promoting gradient formation and fostering non-uniform pattern-like forms of reactant concentrations.

5. Discussion and Conclusions

Our results provide quantitative conditions for the symmetry-breaking transition in the production of ATP from uniform to non-uniform spatial distribution. In an earlier publication [11] researching an integrated multidisciplinary model describing the initiation of cancer and the Warburg hypothesis [12], a theoretical model supported by experimental data showed that a characteristic of cancer cells, namely the Warburg shift from oxidative phosphorylation to glycolysis, is well described by the symmetry-breaking effect where the non-uniform distribution of reactants and products of glycolytic reactions is associated with a cancer phenotype as opposed to a uniform distribution present in the normal cellular phenotype. These wave-like patterns include glucose, pH and NADH and have consequences on mitochondrial activity (their decreased involvement in cellular bioenergetics), cytoskeletal dynamics and cell survival. The present paper provides a mathematical demonstration of how this symmetry effect is caused by parametric changes in the Sel’kov model of glycolysis.

The pathological state of the cell with a preferential use of glycolysis over oxidative phosphorylation may start by an imbalance of glucose external to the cell, a gradual disconnect between ATP and ADP concentrations inside the cell or, finally, by internal defects causing metabolic enzyme redistribution processes in the cytoplasm. This type of imbalance is essentially creating a chemical potential difference between extra- and intracellular compartments of the cell, producing stress on the cell. This chemical gradient may lead to conditions favoring inhomogeneous solutions to the enzymatic reactions further causing mitochondrial destabilization, which may trigger a symmetry-breaking effect as a non-equilibrium phase transition. The above analysis of conditions for the Warburg effect initiation, or the transition from aerobic glycolysis to anaerobic glycolysis, suggests possible avenues for the treatment and prevention of cancer. As shown in Equations (50)–(53), imbalances in the diffusion coefficients as well as their consequences in terms of the reaction rates of the system in Equation (3) set up the conditions for the bifurcation, leading to the onset of inhomogeneous solutions.

We, therefore, hypothesize that excess glucose or metabolic enzyme reaction rate abnormalities, through a symmetry-breaking phenomenon, may cause a cell to prefer to process this energy source using substrate glycolysis. Continued excess substrate glycolysis will cause phase transitions to disrupt the mitochondria. When a cell then passes through mitosis, the chance of mitotic failure is increased. All this suggests that a low-glycemic diet would lower the incidence of cancer, and may suggest a mechanism why metformin, which lowers blood glucose levels, is associated with improved outcomes in diabetic cancer patients [13,14] and reduced risk of pancreatic cancer [15]. This, then, suggests targeting cells that have made the glycolytic switch. For instance, the work of [16] and colleagues [17,18,19] has used 3-bromopyruvate to inhibit glyceraldehyde 3-phosphate dehydrogenase (GAPDH), which effectively inhibits glycolysis [20]. In addition, 3-bromopyruvate may force, via Le Chatelier’s principle, some reverse reactions to essentially deprive the cancer cell of substrate-created ATP. We further hypothesize that a Br derivative of 3-phosphoglycerate would similarly facilitate a reverse reaction to deprive a cancer cell of ATP.

Finally, we wish to comment on the importance of the Sel’kov model. The introduction and importance of Selkov’s cubic autocatalytic chemical reaction model given by Equations (23) and (24) applies far beyond glycolysis itself. It has been extended to many other issues including technological attempts towards the implementation of different pattern formation examples in the context of Turing ideas. In this respect, we have examined the importance of the diffusivity of reactants, which has a profound meaning. For example, the control of diffusivity can be achieved by microwave heating of either an activator or an inhibitor in a given reaction. It should also be stated that while the Sel’kov model is a vast oversimplification of the complexity of the many coupled biochemical reactions occurring in cells during the production of ATP via glycolysis, it captures the essential features that lead to the emergence of limit cycles under specific parameter conditions. It has been used as a “toy” model for these processes and can provide useful information about the sensitivity of the system to parameter changes and environmental conditions. As such, it plays a useful role in our understanding of these complex biological systems without the need to resort to massive numerical simulations.