A Mathematical Model to Study Regulatory Properties and Dynamical Behaviour of Glycolytic Pathway Using Bifurcation Analysis

Abstract

The glycolytic pathway is an almost universal central pathway of glucose catabolism and is known to be highly conserved across various species. In some mammalian tissues and cell types (for example: erythrocytes, renal medulla, brain, and sperm), the glycolytic breakdown of glucose is the sole source of metabolic energy. In glycolysis, phosphofructokinase-1 (PFK-1) catalyzes the transfer of a phosphoryl group from ATP to fructose 6-phosphate to yield fructose 1, 6-bisphosphate. The PFK-1 reaction is essentially an irreversible reaction under cellular conditions. Additionally, it is the first “committed” step in the glycolytic pathway because glucose 6-phosphate and fructose 6-phosphate have other possible fates, but fructose 1, 6-bisphosphate is targeted for glycolysis. In this chapter, a dynamical system based on mass balance equations and S-system representation has been formulated to study the regulatory properties, rate control distributions, and dynamical behaviour of reactions in this glycolytic pathway. This representation involves several parameters. Few standard tools from the bifurcation analysis such as phase-portraits, time series plots, the Lyapunov exponents and $d_{\infty }$ plots have been employed to investigate the stability of the pathway with respect to certain chosen parameters. The study is further extended by eliminating the glycogen branch from the original pathway and by adding an external perturbation into the system.

Appendices

Appendix 1: S-System Study of Pathway

The mass-balance equation can be translated into S-system,

$$\begin{aligned} &\frac{dX_1}{dt}= \alpha _1 X_4^{g_{14}} X_6^{g_{16}} X_{11}^{g_{1,11}}- \beta _1 X_1^{h_{11}} X_2^{h_{12}} X_7^{h_{17}}, \nonumber \\ &\frac{dX_2}{dt} = \alpha _2 X_1^{g_{21}} X_2^{g_{22}} X_5^{g_{25}} X_7^{g_{27}} X_{10}^{g_{2,10}} - \beta _2 X_2^{h_{22}} X_3^{h_{23}} X_8^{h_{28}}, \nonumber \\ &\frac{dX_3}{dt} = \alpha _3 X_2^{g_{32}} X_3^{g_{33}} X_8^{g_{38}} - \beta _3 X_3^{h_{33}} X_9^{h_{39}}. \end{aligned}$$

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There is one rate constant for each pool’s production and one rate constant for its degradation in this S-system. These are $\alpha _1, \alpha _2, \alpha _3$ and $\beta _1, \beta _2, \beta _3$, respectively. For each $X_i$s, one subscript is used for each power. Power is represented by g in the production term and h in the degradation term. The parameters $\alpha _i$ and $g_{ij}$ always refer to production or gain, whereas the parameters $\beta _i$ and $h_{ij}$ always refer to degradation or loss, i.e. the greater $\alpha _i$s indicate more production and the larger $\beta _i$s imply a loss in production.

With the help of the precursor-product relationship between $X_2$ & $X_3$ and branch point constraint at $X_2$, we can reduce the number of unknowns.

Because $V_2^{-}$ and $V_3^{+}$ reflect the same process, the precursor-product relationship implies that the values of the constraints’ parameters be equal.

$$\begin{aligned} \alpha _3 = \beta _2, g_{32} = h_{22}, g_{33} = h_{23}, g_{38} = h_{28}. \end{aligned}$$

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According to the branch point restriction at $X_2$, the total of the two fluxes entering $X_2$ from $X_1$ and $X_5$, namely $V_1^{-}$ and $V_5^{-}$, must equal the influx $V_2^{+}$. The constraint on branch points can be written as

$$\begin{aligned} &V_{2}^{+} = V_{1}^{-} + V_{5}^{-}, i.e. \nonumber \\ &\alpha _{2} X_1^{g_{21}} X_2^{g_{22}} X_5^{g_{25}} X_7^{g_{27}} X_{10}^{g_{2,10}} = \beta _{1} X_1^{h_{11}} X_2^{h_{12}} X_7^{h_{17}} + \beta _{5} X_{2}^{h_{52}} X_5^{h_{55}} X_{10}^{h_{5,10}}. \end{aligned}$$

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These expressions can be understood as a power-law representation in this case. We may use the partial differentiation of the left-hand side of the equation to get the power terms $g_{21}, g_{22}, g_{25}, g_{27}$, and $g_{2,10}$.

$$\begin{aligned} &g_{27} = \frac{\partial V_{2}^{+}}{\partial X_7} \frac{X_7}{V_{2}^{+}} \nonumber \\ & = \frac{\partial (\beta _1 X_{1}^{h_{11}} X_{2}^{h_{12}} X_{7}^{h_{17}} + \beta _5 X_{2}^{h_{52}} X_{5}^{h_{55}} X_{10}^{h_{5,10}}) }{\partial X_7} \frac{X_7}{V_{2}^{+}}. \end{aligned}$$

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By doing partial differentiation, the first term is

$g_{27} = \beta _1 h_{17} X_{1}^{h_{11}} X_{2}^{h_{12}} X_{7}^{h_{17}-1} \frac{X_7}{V_{2}^{+}}$, which is equivalent to $h_{17} \frac{V_{1}^{-}}{V_{2}^{+}}$ and the second term of the equation will become zero. Thus,

$g_{27} = h_{17} \frac{V_{1}^{-}}{V_{2}^{+}}$. Similarly, we can compute the other ones i.e.

$g_{21} = h_{11} \frac{V_{1}^{-}}{V_{2}^{+}}$, $g_{25} = h_{55} \frac{V_{5}^{-}}{V_{2}^{+}}$, $g_{2,10} = h_{5,10} \frac{V_{5}^{-}}{V_{2}^{+}}$. The last one is $g_{22}$ whose derivation may be difficult because of both the terms $V_{5}^{-}$ and $V_{1}^{-}$ depend upon $X_2$.

$$\begin{aligned} &g_{22} = \frac{\partial V_{2}^{+}}{\partial X_2} \frac{X_2}{V_{2}^{+}}, \nonumber \\ & = \frac{\partial (\beta _1 X_{1}^{h_{11}} X_{2}^{h_{12}} X_{7}^{h_{17}} + \beta _5 X_{2}^{h_{52}} X_{5}^{h_{55}} X_{10}^{h_{5,10}}) }{\partial X_2} \frac{X_2}{V_{2}^{+}}, \nonumber \\ & = \beta _1 h_{12} X_{1}^{h_{11}} X_{2}^{h_{12}-1} X_{7}^{h_{17}} + \beta _5 h_{52} X_{2}^{h_{52}-1} X_{5}^{h_{55}} X_{10}^{h_{5,10}} \frac{X_7}{V_{2}^{+}}, \nonumber \\ & = h_{12} \frac{\beta _1 X_{1}^{h_{11}} X_{2}^{h_{12}} X_{7}^{h_{17}}}{V_{2}^{+}} + h_{52} \frac{\beta _5 X_{2}^{h_{52}} X_{5}^{h_{55}} X_{10}^{h_{5,10}}}{V_{2}^{+}}, \nonumber \\ & = h_{12} \frac{V_{1}^{-}}{V_{2}^{+}} + h_{52} \frac{V_{5}^{-}}{V_{2}^{+}}. \end{aligned}$$

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Appendix 2: Steady State Analysis of the Dynamical System Eq. (8)

We require equations that characterise the steady state in an explicit algebraic form for more theoretical investigations, such as a general analysis of logarithmic sensitivities. The structure and characteristics of these equations can be investigated. To find the steady state, divide each equation by its rate constant $\alpha _i$ and set the three symbolic equations to zero. The result is

$$\begin{aligned} & X_4^{g14} X_6^{g16} - \frac{\beta _1}{\alpha _1} X_1^{h_{11}} X_2^{h_{12}} X_7^{h_{17}} = 0, \nonumber \\ & X_1^{g_{21}} X_2^{g_{22}} X_5^{g_{25}} X_7^{g_{27}} X_10^{g_{2,10}} - \frac{\beta _2}{\alpha _2} X_2^{h22} X_3^{h23} X_8^{h28} = 0,\nonumber \\ & X_2^{g_{32}} X_3^{g_{33}} X_8^{g_{38}} - \frac{\beta _3}{\alpha _3} X_3^{h33} X_9^{h39} = 0. \end{aligned}$$

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Next, we define $y_i = \log X_i$ for $i = 1, 2, \ldots , 10$ and $b_i = \log \left( \frac{\beta _i}{\alpha _i}\right)$ for $i = 1, 2, 3$. Power term can be expressed as $X_i^{g_{ij}} = \exp (g_{ij} \log X_i)$, and taking log to all terms, we obtain

$$\begin{aligned} &b_1 + h_{11} y_1 + h_{12} y_2 + h_{17} y_7 - g_{14} y_4 + g_{16} y_6 = 0, \nonumber \\ &b_2 + h_{22} y_2 + h_{23}y_3 + h_{28} y_8 - g_{21} y_4 - g_{22} y_2 - g_{25} y_5 - g_{27} y_7 - g_{2,10} y_{10}, \nonumber \\ &b_3 + h_{33} y_3 + h_{39} y_9 - g_{32} y_2 - g_{33} y_3 - g_{38} y_8 = 0. \end{aligned}$$

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By solving the system of linear equations (32), we will get two solution sets. One is with respect to the parameters $h_{55}$, $h_{52}$, $h_{33}$, $h_{5,10}$, $\alpha _i$ and $\beta _i$ for $i = 1, 2, 3$. While putting the value of the parameters from Sect.  2, Table 1. The value of the variables $y_i$ will be $\log (X_i)$, all $X_i$ are from Sect. 2, Table 2. Corresponding solutions set for this system Eq. (32) is

$$\begin{aligned} &h_{33} = h_{33}, h_{52} = h_{52}, h_{55} = h_{55}, h_{5,10} = h_{5,10}, \nonumber \\ &\alpha _1 = 0.8333333333e-2 \exp (-1.530000000 y_1 + 0.5900000000 y_2 + 1.519706161) \beta _1, \nonumber \\ & \alpha _2 = \exp (0.5129289136 h_{5,10} + 0.5954920276 h_{55} - 0.3700000000 y_2 h_{52} + 0.9486000000 y_1 \nonumber \\ & - 3.604200000 y_2 + 3.060000000 y_3 - 2.625549624) \beta _2, \nonumber \\ & \alpha _3 = \alpha _3, \beta _1 = \beta _1, \beta _2 = \beta _2, \nonumber \\ &\beta _3 = 0.7352941176e-2 \exp (h_{33} y_3 - 3.970000000 y_2 + 3.060000000 y_3 + 1.050821625)\alpha _3. \end{aligned}$$

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The second solution set is generated only for $Y_1, Y_2, Y_3$, in which only $\alpha _i$, $\beta _i$ and $h_{33}, h_{55}, h_{52}$ and $h_{5,10}$ parameters are unknown. Solution set is

$$\begin{aligned} &Y_1=- \frac{1}{(46250. h_{52} h_{33} + (4.04800e+05) h_{33} + (1.41525e+05) h_{52} - (2.79837e+05))}((2.042483660e+06) \nonumber \\ &((-1.405361185e+10) h_{33} h_{55} - (1.210512236e+10) h_{33} h_{5,10} + (1.441680000e+11) \nonumber \\ &\log \frac{\alpha _1}{\beta _1} h_{33} + (4.836322660e+10) h_{52} h_{33} - (2.355910293e+11) + (4.528800000e+10) h_{52} \log \frac{\alpha _1}{\beta _1}\nonumber \\ & + (1.480000000e+10) \log \frac{\alpha _1}{\beta _1} h_{33} h_{52} - (4.477392000e+10) \log \frac{\alpha _1}{\beta _1} + (7.221600000e+10) \log \frac{\alpha _3}{\beta _3} + \nonumber \\ & (7.221600000e+10) \log \frac{\alpha _2}{\beta _2} + (5.330730828e+11) h_{33} + (2.360000000e+10) h_{33} \log \frac{\alpha _2}{\beta _2} \nonumber \\ & + (1.479914734e+11) h_{52} - (4.300405227e+10) h_{55} - (3.704167443e+10) h_{5,10})), \end{aligned}$$

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$$\begin{aligned} &Y_2 = - \frac{1}{(46250 h_{52} h_{33} + (4.04800e+05) h_{33} + (1.41525e+05) h_{52} - (2.79837e+05))}((3.125000000e+04) \nonumber \\ &((-2.381968110e+08) h_{33} h_{55} - (2.051715654e+08) h_{33} h_{5,10} + (1.860630674e+09) h_{33} + \nonumber \\ & (4.00000000e+08) h_{33} \log \frac{(\alpha _2)}{\beta _2} - (7.288822418e+08) h_{55} - (6.2782499033+08) h_{5,10} + \nonumber \\ &(2.48000000e+08) \log \frac{\alpha _1}{\beta _1} h_{33} + (1.224000000e+09) \log \frac{\alpha _2}{\beta _2} + (9.666459509 e+08) + \nonumber \\ & (7.58880000e+08) \log \frac{\alpha _1}{\beta _1} + (1.224000000e+09) \log \frac{\alpha _3}{\beta _3})), \end{aligned}$$

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$$\begin{aligned} &Y_3 = \frac{1}{(46250 h_{52} h_{33} + (4.04800e+05) h_{33} + (1.41525e+05) h_{52} - (2.79837e+05))} (0.6250000000e-5 \nonumber \\ &((2.857756613e+10) h_{52} + (4.728206699e+10) h_{55} + (4.072655574e+10) h_{5,10} - (7.400000000e+09) \nonumber \\ &h_{52} \log \frac{(\alpha _3}{\beta _3} - (1.192119721e+11) - (7.940000000e+10) \log \frac{\alpha _2}{\beta _2} - (4.922800000e+10) \nonumber \\ & \log \frac{\alpha _1}{\beta _1} - (6.476800000e+10) \log \frac{\alpha _3}{\beta _3})). \end{aligned}$$

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