The PDF of the article https://academic.oup.com/femsre/article/23/1/13/575607 is downloaded from the link given in the above URL. Quoting from the PDF we have:
4H2 + CO2 → CH4 + 2H2O (1)
...
Eq. 1 is the sum of Eq. 3a, Eq. 4a, Eqs. 5-7, Eq. 8a, Eq. 8b and Eqs. 9-12.
We check the validity of the model below.
2H2 → 4e- + 4H+ (3a)
2H2 + 2F420 → 2F420H2 (4a)
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)
Let us cancel the terms that appear in LHS and RHS.
2H2 → 4e- + 4H+ (3a)
The RHS has 4e- + 4H+. We find 2e- + 2H+ on the LHS of (5) and (12). We cancel these to get the equations (3a)1; (5a)1; and (12)1
2H2 + 2F420 → 2F420H2 (4a)
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)
======================
In the following equations the cancelled portions are shown in bold.
After cancelling 4e- + 4H+ we get
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)1
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)1
2F420H2 appears in the RHS of (4a) and once in the LHS of (8a) and (9). We cancel these and get (4a)1; (8a)1; and (9)1.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)1
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)1
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)1
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)1
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)1
formyl_MF in RHS of (5)1. It occurs in the LHS of (6). We cancel these and get (5)2 and (6)1.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)1
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)2
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)1
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)1
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)1
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)1
H2O that appears on the RHS of (5)2 does not appear in the LHS. 5_formyl_H4MPT appears on the RHS of (6)1 and the LHS of (7). MF that appears on the RHS of (6)1 appears on the LHS of (5)2. We delete these and get (5)3; (6)2: and (7)1
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)1
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)3
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)2
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)1
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)1
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)1
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)1
5,10_methenyl_H4MPT+ appears on the RHS of (7)1 and the LHS of (8a)1. The H2O that appears on the RHS of (7)1 does not appear in any LHS. We make the changes and gt (7)2 and (8a)2.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)1
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)3
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)2
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)2
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)2
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)1
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)1
5,10_methylene_H4MPT appears on the RHS of (8a)2 and LHS of (9)1. Cancelling the appearances, we get (8a)3 and (9)2.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)1
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)3
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)2
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)2
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)3
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)2
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)1
F420 appears once in the RHS of (8a)3 and (9)2. It appears with a multiplier of 2 in the LHS of (4a)1. Cancelling these appearances, we get (8a)4; (9)3; and (4a)2.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)2
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)3
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)2
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)2
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)4
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)3
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)1
H4MPT occurs in the LHS of (6)2 and the RHS of (10). Cancelling both the appearances, we get (6)3 and (10)1.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)2
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)3
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)3
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)2
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)4
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)3
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)1
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)1
H+ occurs in the LHS of (7)2 and the RHS of (8a)4. Removing from both sides, we get (7)3 and (8a)5.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)2
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)3
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)3
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)3
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)5
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)3
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)1
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)1
5_methyl_H4MPT is found in the RHS of (9)3 and LHS of (10)1. Simplifying we get (9)4 and (10)2.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)2
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)3
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)3
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)3
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)5
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)4
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)2
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)1
We find HS_CoM in the LHS of (10)2 and (12)1. Removing both occurrences, we get (10)3 and (12)2.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)2
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)3
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)3
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)3
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)5
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)4
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)3
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)2
CH3_S_CoM is found in RHS of (10)3 and LHS of (11). Removing both occurrences, we get (10)4 and (11)1.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)2
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)3
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)3
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)3
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)5
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)4
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)4
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)1
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)2
HS_CoB appears in LHS of (11)1 and RHS of (12)2. On simplification, we get (11)2 and (12)3.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)2
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)3
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)3
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)3
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)5
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)4
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)4
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)2
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)3
CoM_S_S_CoB is found in the RHS of (11)2 and LHS of (12)3. In the next step, we get (11)3 and (12)4.
2H2 → 4e- + 4H+ (3a)1
2H2 + 2F420 → 2F420H2 (4a)2
CO2 + MF + 2e- + 2H+ → formyl_MF + H2O (5)3
formyl_MF + H4MPT → 5_formyl_H4MPT + MF (6)3
5_formyl_H4MPT + H+ → 5,10_methenyl_H4MPT+ + H2O (7)3
5,10_methenyl_H4MPT+ + F420H2 → 5,10_methylene_H4MPT + F420 + H+ (8a)5
5,10_methenyl_H4MPT+ + H2 → 5,10_methylene_H4MPT + H+ (8b)
5,10_methylene_H4MPT + F420H2 → 5_methyl_H4MPT + F420 (9)4
5_methyl_H4MPT + HS_CoM → CH3_S_CoM + H4MPT (10)4
CH3_S_CoM + HS_CoB → CH4 + CoM_S_S_CoB (11)3
CoM_S_S_CoB + 2e- + 2H+ → HS_CoM + HS_CoB (12)4
No more simplifications exist. (3a)1 has 2H2 in LHS, (4a)2 has 2H2 in LHS, (5)3 has CO2 in LHS, (5)3 has H2O in the RHS, (7)3 has H2O in the RHS, (8b) does not belong in this set, and (11)3 has CH4 in the RHS.
Collecting the terms in the LHS we get 2H2 + 2H2 + CO2. Likewise collecting the terms in the RHS we get H2O + H2O + CH4.
In conclusion, CO2 + 4H2 → CH4 + 2H2O.
We could not verify equations (5) and (7) which have H2O in the RHS because the structure of the cofactors are not found in the paper. The structure of cofactor MF alone is given in the paper. In the absence of the structure of the cofactors formyl_MF, 5_formyl_H4MPT, and 5,10_methenyl_H4MPT+ we are forced to believe in the correctness of the model.