1
1. Experimental data used in the calculations
TableS1Dependence of mole fraction CO on reaction time for 11 different samples of coke – data from [26]
time / mole fraction of carbon monooxide, [CO]number of coke sample
h / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11
0.28 / 0.24 / 0.24 / 0.12 / 0.23 / 0.17 / 0.19 / 0.18 / 0.17 / 0.25 / 0.18 / 0.15
0.56 / 0.43 / 0.45 / 0.30 / 0.50 / 0.35 / 0.40 / 0.45 / 0.46 / 0.48 / 0.39 / 0.41
0.84 / 0.47 / 0.47 / 0.38 / 0.52 / 0.40 / 0.46 / 0.49 / 0.49 / 0.51 / 0.46 / 0.48
1.12 / 0.46 / 0.50 / 0.37 / 0.49 / 0.37 / 0.44 / 0.46 / 0.46 / 0.49 / 0.44 / 0.46
1.40 / 0.45 / 0.44 / 0.36 / 0.45 / 0.35 / 0.42 / 0.44 / 0.43 / 0.45 / 0.41 / 0.44
1.68 / 0.43 / 0.42 / 0.35 / 0.42 / 0.34 / 0.39 / 0.40 / 0.40 / 0.42 / 0.38 / 0.39
1.96 / 0.41 / 0.40 / 0.33 / 0.39 / 0.32 / 0.37 / 0.37 / 0.37 / 0.38 / 0.33 / 0.37
2.24 / 0.38 / 0.38 / 0.31 / 0.36 / 0.29 / 0.34 / 0.34 / 0.35 / 0.34 / 0.32 / 0.35
2.52 / 0.36 / 0.35 / 0.29 / 0.33 / 0.26 / 0.31 / 0.30 / 0.33 / 0.32 / 0.30 / 0.33
2.80 / 0.32 / 0.33 / 0.26 / 0.31 / 0.25 / 0.29 / 0.29 / 0.30 / 0.28 / 0.28 / 0.31
3.08 / 0.31 / 0.30 / 0.25 / 0.29 / 0.23 / 0.27 / 0.26 / 0.28 / 0.24 / 0.26 / 0.30
3.36 / 0.29 / 0.28 / 0.23 / 0.25 / 0.21 / 0.23 / 0.23 / 0.26 / 0.22 / 0.24 / 0.28
3.64 / 0.26 / 0.25 / 0.20 / 0.21 / 0.19 / 0.20 / 0.21 / 0.24 / 0.20 / 0.22 / 0.27
3.92 / 0.23 / 0.23 / 0.19 / 0.20 / 0.17 / 0.18 / 0.20 / 0.20 / 0.18 / 0.19 / 0.25
4.20 / 0.20 / 0.21 / 0.17 / 0.18 / 0.16 / 0.16 / 0.18 / 0.20 / 0.16 / 0.17 / 0.23
TableS2Dependence of mole fraction CO on reaction time and absolute temperature for 2 selected samples of coke – data from [26]
time / mole fraction of carbon monooxide, [CO]temperature of reaction, K
h / blast furnace coke (No 3 in Table S1) / domestic coke (No 9 in Table S1)
1173 / 1223 / 1273 / 1323 / 1373 / 1423 / 1173 / 1223 / 1273 / 1323 / 1373 / 1423
0.28 / 0.03 / 0.06 / 0.12 / 0.20 / 0.34 / 0.42 / 0.07 / 0.16 / 0.25 / 0.26 / 0.53 / 0.60
0.56 / 0.09 / 0.19 / 0.30 / 0.48 / 0.61 / 0.76 / 0.20 / 0.36 / 0.48 / 0.62 / 0.74 / 0.85
0.84 / 0.10 / 0.22 / 0.38 / 0.58 / 0.67 / 0.82 / 0.24 / 0.38 / 0.51 / 0.68 / 0.78 / 0.86
1.12 / 0.09 / 0.22 / 0.37 / 0.54 / 0.62 / 0.79 / 0.23 / 0.37 / 0.49 / 0.67 / 0.75 / 0.78
1.40 / 0.08 / 0.21 / 0.36 / 0.52 / 0.56 / 0.73 / 0.22 / 0.36 / 0.45 / 0.60 / 0.63 / 0.70
1.68 / 0.07 / 0.20 / 0.35 / 0.50 / 0.51 / 0.60 / 0.22 / 0.34 / 0.42 / 0.54 / 0.55 / 0.61
1.96 / 0.07 / 0.20 / 0.33 / 0.46 / 0.47 / 0.52 / 0.22 / 0.33 / 0.38 / 0.47 / 0.48 / 0.52
2.24 / 0.06 / 0.18 / 0.31 / 0.43 / 0.42 / 0.44 / 0.21 / 0.31 / 0.34 / 0.44 / 0.41 / 0.42
2.52 / 0.06 / 0.18 / 0.29 / 0.33 / 0.37 / 0.36 / 0.20 / 0.29 / 0.32 / 0.38 / 0.34 / 0.34
2.80 / 0.05 / 0.16 / 0.26 / 0.35 / 0.34 / 0.30 / 0.20 / 0.27 / 0.28 / 0.34 / 0.31 / 0.28
3.08 / 0.05 / 0.16 / 0.25 / 0.31 / 0.30 / 0.24 / 0.20 / 0.26 / 0.24 / 0.30 / 0.26 / 0.22
3.36 / 0.04 / 0.15 / 0.23 / 0.26 / 0.26 / 0.20 / 0.18 / 0.24 / 0.22 / 0.26 / 0.22 / 0.18
3.64 / 0.04 / 0.12 / 0.20 / 0.23 / 0.23 / 0.15 / 0.18 / 0.22 / 0.20 / 0.23 / 0.18 / 0.14
3.92 / 0.04 / 0.09 / 0.19 / 0.21 / 0.20 / 0.11 / 0.18 / 0.20 / 0.18 / 0.22 / 0.16 / 0.10
4.20 / 0.03 / 0.06 / 0.17 / 0.19 / 0.17 / 0.08 / 0.17 / 0.20 / 0.16 / 0.20 / 0.13 / 0.08
2. System of linear equations 1st-order according to [S1]
A system of equations for a reaction of disappearing substrate A(τ) via an intermediate product B(τ) leading to a final product C(τ) is analyzed in a dissertation [S1, pp.8-9] according to the scheme:
(S1)
(S2)
.(S3)
From equations (S1) and (S2) for the initial condition A(τ=0) =1, one can obtain a solution for the intermediate product B:
.(S4)
The system of equations (S1) to (S3) and the solution (S4) is characteristic for consecutive reactions with an intermediate product B(τ), and exhibits a maximum.
Expanding on the considerations presented in [S1], equation (S4) can be substituted into equation (S3) and the condition C(τ=0) =0 can be applied, leading to an expression for the final reaction product:
.(S5)
For ,, i.e., the contribution of carbon monoxide reaches a value of [CO] = 1, because function (S5) is constantly increasing and exhibits no maximum.
3. Analogies with Michaelis-Menten enzyme kinetic model
The proposed equation (18) refers to processes involving intermediate products and active complexes. In the case of enzymatic reactions, the Michaelis-Menten (M-M) scheme:
.(S6)
A similar situation is observed. The reaction of the substrate S with the enzyme E leads through the intermediate product ES. For such defined scheme kinetic equations are presented in the form [S2-S4]:
(S7)
(S8)
(S9)
.(S10)
Paths of the M-M reaction are often simplified by assuming . Using the steady-state requirement: equation (S10) can be solved to the following relation:
(S11)
which implies that in the full cycle of consumption and regeneration of the enzyme (turnover time) and the concentration of the intermediate product , which is incompatible with the available experimental results – Fig. S1.
Participation of the intermediate product ES during the reaction exhibits a maximum value. This fact can be shown in two ways:
a) starting from the balance of intermediate product, it can be demonstrated that . As relation time exhibits a minimum value and enzyme E or E' regenerates after turnover time [S3], i.e., ,
b) only the intermediate product [ES] is considered, and single-molecule approach presented in [S3] and [S6] (note 31 on pages 1881-1882) can be used. This approach, together with the additional assumption k-2=0, simplifies system of equations (S7)-(S10) to form:
(S12)
.(S13)
Relating (S12) and (S13) for the initial assumptions after Laplace transform takes the form:
(S14)
.(S15)
After dissolution and inverse transformation, the following expected relationship can be obtained:
(S16)
(S17)
.(S18)
This expression predicts that time exhibits a minimum value.
Fig.S1 Relation of enzyme concentration [E] and intermediate product [ES] vs time [S5]
References for Supplementary Materials
S1.Rydlewski JP. On the maximum likelihood estimator in the generalized nonlinear regression model. Ph. Thesis. Faculty of Mathematics and Computer Since Jagiellonian University. Kraków; 2009. (in Polish).
S2.Qian H. A Guide for Michaelis-Menten Enzyme Kinetic Models, National Simulation Resource, University of Washington, Seattle, /xsim_modeldocs/OLD/Software/DEMO/MICMEN/index.html. Accessed Jul 2014.
S3.Molski A. Single-molecule Michaelis-Menten kinetics: Effect of substrate fluctuations. Chemical Physics. 2008;352:276-80.
S4.Molski A. Are single-enzyme kinetics of lipase B from Canadian antarctica governed by a stretch exponential? Chemical Physics Letters. 2006;428:196-9.
S5.Wolfram Demonstrations Project - Michaelis-Menten Enzyme Kinetics and the Steady-State Approximation, Accessed Jul 2014.
S6.Lu HP, Xun L, Xie S. Single-Molecule Enzymatic Dynamics. Science. 1998;282:1877-82.