1. Introduction

2. Results and Discussion

Figure 2. Kinetics of ethanol consumption in the model. (A) – Simulated dependence of ethanol concentration on time in а 50% EBR suspension at ADH and ALDH activities equal to 30 (black line), 60 (red line), and 90 (green line) mM/h after addition of ethanol to a final concentration of 10 mM. (B) – Simulated dependence of ethanol concentration on time in the 50% EBR suspension at ADH and ALDH activities equal to 60 mM/h after addition of ethanol to a final concentration of 5 mM (black line), 10 mM (red line), and 20 mM (green line). (C) – Kinetics of NAD concentration in the model at activity of the encapsulated enzymes equal to 30 mM/h at 10 mM ethanol (black line), 60 mM/h at 10 mM ethanol (red line), and 60 mM/h at 20 mM ethanol (green line). (D) - Simulated dependence of ethanol concentration on time in the EBR suspension (hematocrit 50%) at ADH activity equal to 60 mM/h. Activity of ALDH was equal to 60 mM/h (black and red lines) or to zero (green line). Acetaldehyde concentration was calculated from the model (black and green lines) or set to zero (red line). (E) – Kinetics of acetaldehyde concentration in the model under conditions described for panel C. (F) – Experimental dependence of ethanol concentration on time obtained in [15] for the 25% suspension of human EBRs loaded with ADH (78 mM/h) and ALDH (18 mM/h), after addition of ethanol to a final concentration of 30 mM (symbols) and simulation results obtained for the same parameter values (line).

Figure 2. Kinetics of ethanol consumption in the model. (A) – Simulated dependence of ethanol concentration on time in а 50% EBR suspension at ADH and ALDH activities equal to 30 (black line), 60 (red line), and 90 (green line) mM/h after addition of ethanol to a final concentration of 10 mM. (B) – Simulated dependence of ethanol concentration on time in the 50% EBR suspension at ADH and ALDH activities equal to 60 mM/h after addition of ethanol to a final concentration of 5 mM (black line), 10 mM (red line), and 20 mM (green line). (C) – Kinetics of NAD concentration in the model at activity of the encapsulated enzymes equal to 30 mM/h at 10 mM ethanol (black line), 60 mM/h at 10 mM ethanol (red line), and 60 mM/h at 20 mM ethanol (green line). (D) - Simulated dependence of ethanol concentration on time in the EBR suspension (hematocrit 50%) at ADH activity equal to 60 mM/h. Activity of ALDH was equal to 60 mM/h (black and red lines) or to zero (green line). Acetaldehyde concentration was calculated from the model (black and green lines) or set to zero (red line). (E) – Kinetics of acetaldehyde concentration in the model under conditions described for panel C. (F) – Experimental dependence of ethanol concentration on time obtained in [15] for the 25% suspension of human EBRs loaded with ADH (78 mM/h) and ALDH (18 mM/h), after addition of ethanol to a final concentration of 30 mM (symbols) and simulation results obtained for the same parameter values (line).

Figure 3. Dependence of the steady-state ethanol consumption rate in the model on the activity of ADH and ALDH and on the ethanol concentration. (A) – The ethanol consumption rate was calculated at the ethanol concentration of 10 mM assuming that the ADH activity is equal to the ALDH activity. (B) – The ethanol consumption rate was calculated assuming that the ethanol concentration was constant and the activities of ADH and ALDH were equal to 40 mM/h. The orange, black, and cyan lines indicate stable steady states, unstable steady states with oscillation mode, and unstable steady states with metabolite accumulation respectively.

Figure 3. Dependence of the steady-state ethanol consumption rate in the model on the activity of ADH and ALDH and on the ethanol concentration. (A) – The ethanol consumption rate was calculated at the ethanol concentration of 10 mM assuming that the ADH activity is equal to the ALDH activity. (B) – The ethanol consumption rate was calculated assuming that the ethanol concentration was constant and the activities of ADH and ALDH were equal to 40 mM/h. The orange, black, and cyan lines indicate stable steady states, unstable steady states with oscillation mode, and unstable steady states with metabolite accumulation respectively.

Figure 4. Dependence of the model steady state on the activity of ADH and ALDH. The ethanol concentration was constant and equal to 10 mM. The activity of ADH was equal to the ALDH activity. Dashed lines correspond to unstable steady states.

Figure 4. Dependence of the model steady state on the activity of ADH and ALDH. The ethanol concentration was constant and equal to 10 mM. The activity of ADH was equal to the ALDH activity. Dashed lines correspond to unstable steady states.

Figure 5. Steady oscillations of the metabolite concentrations and enzymatic reaction rates in the model. Oscillations of the metabolite concentrations in the upper part of glycolysis (A), the lower part of glycolysis (B, C), adenine nucleotides (D), NAD, NADH, and acetaldehyde (E), and enzymatic reaction rates (F). Calculations were made at the constant ethanol concentration of 10 mM and at the ADH and ALDH activities equal to 102.5 mM/h.

Figure 5. Steady oscillations of the metabolite concentrations and enzymatic reaction rates in the model. Oscillations of the metabolite concentrations in the upper part of glycolysis (A), the lower part of glycolysis (B, C), adenine nucleotides (D), NAD, NADH, and acetaldehyde (E), and enzymatic reaction rates (F). Calculations were made at the constant ethanol concentration of 10 mM and at the ADH and ALDH activities equal to 102.5 mM/h.

Figure 6. Dependence of oscillation parameters in the model on the ADH and ALDH activity. The dependence of [FDP] oscillation amplitude (A) and period (B) on the ADH and ALDH activity. The red dotted lines show a switch of the model between fast low-amplitude and slow high-amplitude oscillations. Vertical black dashed lines indicate a border of the oscillation mode at high activity of the encapsulated enzymes. The ethanol concentration was constant and equal to 10 mM. The activity of ADH was equal to the ALDH activity.

Figure 6. Dependence of oscillation parameters in the model on the ADH and ALDH activity. The dependence of [FDP] oscillation amplitude (A) and period (B) on the ADH and ALDH activity. The red dotted lines show a switch of the model between fast low-amplitude and slow high-amplitude oscillations. Vertical black dashed lines indicate a border of the oscillation mode at high activity of the encapsulated enzymes. The ethanol concentration was constant and equal to 10 mM. The activity of ADH was equal to the ALDH activity.

Figure 7. Kinetics of [FDP] (A) and [ATP] (B) in the model at high ADH and ALDH activity. Calculations were performed at the constant ethanol concentration of 10 mM and ADH and ALDH activity equal to 120 (black line), 150 (red line), 180 (green line), and 240 (blue line) mM/h.

Figure 7. Kinetics of [FDP] (A) and [ATP] (B) in the model at high ADH and ALDH activity. Calculations were performed at the constant ethanol concentration of 10 mM and ADH and ALDH activity equal to 120 (black line), 150 (red line), 180 (green line), and 240 (blue line) mM/h.

Figure 8. Time shifts between principal metabolite concentrations and reaction rates during steady oscillations in the model. (A) – Comparison of ATP concentration with GAP concentration multiplied by 3 and AMP concentration multiplied by 10. (B) – Comparison of doubled PFK reaction rate and GAPDH reaction rate. Calculations were carried out at the constant ethanol concentration of 10 mM and ADH and ALDH activities of 102.5 mM/h.

Figure 8. Time shifts between principal metabolite concentrations and reaction rates during steady oscillations in the model. (A) – Comparison of ATP concentration with GAP concentration multiplied by 3 and AMP concentration multiplied by 10. (B) – Comparison of doubled PFK reaction rate and GAPDH reaction rate. Calculations were carried out at the constant ethanol concentration of 10 mM and ADH and ALDH activities of 102.5 mM/h.

3. Mathematical model

Table 2. Differential and algebraic equations describing the kinetics of metabolite concentrations in the model of erythrocyte-bioreactors prepared for ethanol consumption.

Table 3. Model equations for the rates of enzymatic reactions.

Table 3. Model equations for the rates of enzymatic reactions.

Hexokinase [21] $a_{HK}=12$ mM/h, $K_{HK}^1=1$ $mM$ $,K_{HK}^2=5.5\cdot {10}^{-3}$ $mM$

Glucose-6-phosphate isomerase [21] $a_{GPI}=360$ mM/h, $K_{GPI}^1=3$ $mM$ $,K_{GPI}^2=0.3$ $mM$, $K_{GPI}^3=0.2$ $mM$

Phosphofructokinase [21] $a_{PFK}=380$ mM/h$,K_{PFK}^1=0.1$ $mM$ $,K_{PFK}^2=2$ $mM$ $,K_{PFK}^3=0.01$ $mM$ $,K_{PFK}^4=0.195$ $mM$ $,K_{PFK}^5=3.7\cdot {10}^{-4}$ $mM$

Aldolase [21] $a_{ALD}=76$ mM/h$,K_{ALD}^1=2\cdot {10}^{-4}$ $mM$ $,K_{ALD}^2=1.2\cdot {10}^{-5}$ $m{M}^2$ $,K_{ALD}^3=0.01$ $mM$ $,K_{ALD}^4=0.032$ $mM$ $,K_{ALD}^5=2.1\cdot {10}^{-3}$ $mM$ $,K_{ALD}^6=2$ $mM$ $,K_{ALD}^7=0.065$ $mM$

Triosephosphate isomerase [21,28] $a_{TPI}=19522$ mM/h, $K_{TPI}^1=0.82$ $mM$, $K_{TPI}^2=0.45$ $mM$, $K_{TPI}^3=0.43$ $mM$

Glyceraldehyde phosphate dehydrogenase [21] $V_{GAPDH}=a_{GAPDH}\frac{\left(\left[GAP\right]\left[NAD\right]\left[P_i\right]-\left[\mathrm{1,3}DPG\right]\left[NADH\right]/K_{GAPDH}^4\right)/K_{GAPDH}^1{K}_{GAPDH}^2{K}_{GAPDH}^3}{1.29\left(1+\frac{\left[GAP\right]}{K_{GAPDH}^1}+\frac{\left[\mathrm{1,3}DPG\right]}{K_{GAPDH}^5}\right)\left(1+\frac{\left[NAD\right]}{K_{GAPDH}^2}+\frac{\left[NADH\right]}{K_{GAPDH}^6}\right)}$ $a_{GAPD}=690$ mM/h, $K_{GAPDH}^1=0.13$ $mM$, $K_{GAPDH}^2=0.13$ $mM$, $K_{GAPDH}^3=3.4$ $mM$, $K_{GAPDH}^4=0.136$ ${mM}^{-1}$, $K_{GAPDH}^5=0.013$ $mM$, $K_{GAPDH}^6=2\cdot {10}^{-3}$ mM

Phosphoglycerate kinase [21,28] $a_{PGK}=2115$ mM/h, $K_{PGK}^1=2.2\cdot {10}^{-3}$ $mM$, $K_{PGK}^2=0.14$ $mM$, $K_{PGK}^3=380$ $mM$, $K_{PGK}^4=0.3$ $mM$, $K_{PGK}^5=0.27$ $mM$, $K_{PGK}^6=1.4$ $mM$, $K_{PGK}^7=0.4$ $mM$

Diphosphoglycerate mutase [21] $a_{DPGM}=3892$ mM/h, $K_{DPGM}^1=0.04$ $mM$, $K_{DPGM}^2=0.013$ $mM$

Diphosphoglycerate phosphatase [21] $a_{DPGP}=0.65$ mM/h, $K_{DPGP}^1=0.02$ $mM$, $K_{DPGD}^2=6\cdot {10}^{-3}$ mM

Phosphoglycerate mutase [21] $a_{PGM}=1100$ mM/h$,K_{PGM}^1=0.27$ $mM$ $,K_{PGM}^2=0.24$ $,K_{PGM}^3=0.02$ $mM$

Enolase [21,28] $a_{ENO}=120$ mM/h, $K_{ENO}^1=0.056$ $mM$, $K_{ENO}^2=6.7$, $K_{ENO}^3=2\cdot 10^{-3}$ $mM$

Pyruvate kinase [21] $a_{PK}=120$ mM/h, $K_{PK}^1=0.05$ $mM$, $K_{PK}^2=0.43$ $mM$, $K_{PK}^3=0.35$ $mM$

Lactate dehydrogenase [21] $V_{LDH}=a_{LDH}\frac{([PYR][NADH]-[LAC][NAD]/K_{LDH}^3)/K_{LDH}^1{K}_{LDH}^2}{1+\frac{[PYR]}{K_{LDH}^1}+\frac{[NADH]K_{LDH}^4+[PYR][NADH]+[LAC][NADH]K_{LDH}^4/K_{LDH}^5}{K_{LDH}^1{K}_{LDH}^2}+\frac{C}{K_{LDH}^5{K}_{LDH}^6}}$ $C=K_{LDH}^7[NAD]+K_{LDH}^6[LAC]+[NAD][LAC]+K_{LDH}^7[PYR][NAD]/K_{LDH}^1$ $a_{LDH}=550$ mM/h, $K_{LDH}^1=0.022$ $mM$, $K_{LDH}^2=7\cdot {10}^{-3}$ $mM$, $K_{LDH}^3=426$, $K_{LDH}^4=0.14$ $mM$, $K_{LDH}^5=380$ $mM$, $K_{LDH}^6=0.1$ $mM$, $K_{LDH}^7=170$ $mM$

Na+-K+-АТPase [21] $a_{NaKATP}=0.045$ mM/h, $\left[{Na}^{+}\right]=10mM$ [30]

All other ATPases are presented by the following equation [21] $a_{ATP}=1.6$ mM/h, $K_{ATP}=1$ $mM$

Alсohol dehydrogenase [26,27] $V_{ADH}={}_{ADH}\frac{\left[NAD\right]\left[ETH\right]-\frac{\left[ACALD\right]\left[NADH\right]}{K_{ADH}^{eq}}}{D_{ADH}}$ $D_{ADH}=K_{ADH}^2{K}_{ADH}^3+K_{ADH}^3\left[NAD\right]+K_{ADH}^1\left[ETH\right]+\left[NAD\right]\left[ETH\right]+\frac{K_{ADH}^2{K}_{ADH}^3}{K_{ADH}^8}\left[NADH\right]+\frac{K_{ADH}^2{K}_{ADH}^3{K}_{ADH}^7}{K_{ADH}^5{K}_{ADH}^8}\left[ACALD\right]+\frac{K_{ADH}^2{K}_{ADH}^3}{K_{ADH}^5{K}_{ADH}^8}\left[ACALD\right]\left[NADH\right]+\frac{K_{ADH}^3{K}_{ADH}^7}{K_{ADH}^5{K}_{ADH}^8}\left[ACALD\right]\left[NAD\right]+\frac{K_{ADH}^1}{K_{ADH}^8}\left[ETH\right]\left[NADH\right]+\frac{\left[ACALD\right]\left[ETH\right]\left[NAD\right]}{K_{ADH}^6}+\frac{K_{ADH}^2{K}_{ADH}^3}{K_{ADH}^4{K}_{ADH}^5{K}_{ADH}^8}\left[ACALD\right]\left[ETH\right]\left[NADH\right]$ $K_{ADH}^1=0.074mM,K_{ADH}^2=0.61mM,K_{ADH}^3=13mM,K_{ADH}^4=0.43mM,$ $K_{ADH}^5=0.78mM,K_{ADH}^6=0.67mM,K_{ADH}^7=0.11mM,K_{ADH}^8=0.018mM$ $K_{ADH}^{eq}=2\bullet {10}^{-4}$ ${}_{ADH}$- ADH activity – this is a varied parameter in the model.

Acetaldehyde dehydrogenase [20] $V_{ALDH}={}_{ALDH}\frac{\left[NAD\right]\left[ACALD\right]}{\left(\left[NAD\right]\left[ACALD\right]+K_{ALDH}^3\left[NAD\right]+K_{ALDH}^1\left[ACALD\right]+K_{ALDH}^2{K}_{ALDH}^3\right)\left(1+\frac{\left[NADH\right]}{K_{ALDH}^4}\right)}$ $K_{ALDH}^1=0.02mM,K_{ALDH}^2=0.07mM,K_{ALDH}^3=0.009mM,K_{ALDH}^4=0.1mM$ ${\alpha }_{ALDH}$ – ALDH activity - this is a varied parameter in the model

Pyruvate transport [25] $V_{trPYR}=A_{trPYR}\frac{{\left[PYR\right]}_{ext}-\left[PYR\right]}{{\left[PYR\right]}_{ext}+\left[PYR\right]+\left(1+\frac{\left[LAC\right]+{\left[LAC\right]}_{ext}}{K_I^{trPYR}}\right)K_m^{trPYR}}$ $A_{trPYR}=120mM/h$; $K_m^{trPYR}=1.9mM$; $K_I^{trPYR}=11mM$

Lactate transport [25] $V_{trLAC}=A_{trLAC}\frac{{\left[LAC\right]}_{ext}-\left[LAC\right]}{{\left[LAC\right]}_{ext}+\left[LAC\right]+\left(1+\frac{\left[PYR\right]+{\left[PYR\right]}_{ext}}{K_I^{trLAC}}\right)K_m^{trLAC}}$ $A_{trLAC}=120mM/h$; $K_m^{trLAC}=9mM$; $K_I^{trLAC}=1.6mM$

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