A dynamic interaction process between KaiA and KaiC is critical to
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1 A dynamic interaction process between KaiA and KaiC is critical to the cyanobacterial circadian oscillator Pei Dong 1,2, Ying Fan 2, Jianqiang Sun 3, Mengting Lv 2, Ming Yi 4, Xiao Tan 1,2, Sen Liu 1,2,* 1 Hubei Key Laboratory of Tumor Microenvironment and Immunotherapy, China Three Gorges University, Yichang , China 2 College of Medical Science, China Three Gorges University, Yichang , China 3 School of Statistics, Shandong Institute of Business and Technology, Yantai, , China 4 Department of Physics, College of Sciences, Huazhong Agricultural University, Wuhan , China * Corresponding author: senliu.ctgu@gmail.com
2 Supplementary Text S1. The analysis of the KaiA-KaiC binding equilibrium In this section we will prove that K Dfit, which is the apparent value of the real equilibrium constants of two steps of the KaiA-KaiC binding processes, should be smaller than the apparent values corresponding to K Dapp1 and K Dapp2 in the following part. For simplicity, A stands for KaiA, and C stands for KaiC (C1 and C2 are two different conformational statuses). According to the binding processes shown above and the mass action principle, we obtain that: d[ac! ] = ka dt! A C kd! AC! + k! AC! k! AC! (1) d AC! = ka dt! A C kd! AC! + k! AC! k! AC! (2) Moreover, we suppose that all steps above are rapid and pre-equilibrium processes compared to the phosphorylation processes. Thus, we have d[ac! ] = ka dt! A C kd! AC! + k! AC! k! AC! = 0 (3) d AC! = ka dt! A C kd! AC! + k! AC! k! AC! = 0 (4) Combining (3) with (4), we may conclude that (ka! + ka! ) A C = kd! AC! + kd! AC! (5) Suppose that [AC!"! ] is the total concentration of complex AC, i.e. [AC!"! ] = AC! + AC! (6) Thus, there is a number γ (0 < γ < 1) such that AC! = γ[ac!"! ] (7) Combining (5), (6), and (7), we have (ka! + ka! ) A C = γkd! + 1 γ kd! [AC!"! ] (8) (ka! + ka! ) A C = kd! + 1 γ γ kd! [AC! ] (9) γ (ka! + ka! ) A C = 1 γ kd! + kd! [AC! ] (10) Denoting K Dapp1 = [A][C]/[AC 1 ], K Dapp2 = [A][C]/[AC 2 ], and from (8), (9), (10), we obtain K!"#$ =!! [!"!"! ] = γkd! + 1 γ kd! ka! + ka!, (11)
3 A C K!"#$! = [AC! ] = γkd! + 1 γ kd! γ(ka! + ka! ) A C K!"#$! = [AC! ], (12) = γkd! + 1 γ kd! (1 γ)(ka! + ka! ), (13) According to (11), (12) and (13), it is obvious that K!"#$ = γ K!"##$ = 1 γ K!"##$ (14) Supplementary Figure S1. The sequence analyses of the clock proteins. (A) The sequence alignment of 65 KaiA sequences aligned in MEGA v5.05. The alignments were rendered with STRAP. The secondary structure elements were shown according to the protein structure of Synechococcus elongates PCC (B) The sequence alignment of 65 KaiB sequences aligned in MEGA v5.05. (C) The sequence alignment of 65 KaiC sequences aligned in MEGA v5.05. (D) The phylogenetic tree generated from concatenated KaiA/KaiB/KaiC sequences. (E) The calculated Relative Evolution Rates (RERs) of KaiA residues. (F) The calculated RERs of KaiC residues.
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25 (D) (E) Coil Sheet Helix Connecting Line 3 Relative Evolution Rate (A. U.) Residue (KaiA, S. e. PCC 7942)
26 (F) 4 Coil Sheet Helix Connecting Line Relative Evolution Rate (A. U.) CIIABD 1.5 Inter Domain Linker Residue (KaiC, S. e. PCC 7942)
27 Supplementary Figure S2. The molecular dynamics simulation of the apo structure of KaiA C-terminal domains (1Q6B). (A) The backbone RMSD variation of the whole C-terminal domains compared to the apo structure (1Q6B) and the holo structure (1SV1). (B) The backbone RMSD variation of the helices 9 and 10 compared to the apo structure (1Q6B) and the holo structure (1SV1) respectively. Compared to the simulation of the holo structure described in the main text, the conformational change of the apo structure had a lower oscillation frequency between these two conformations. (A) (B) Supplementary Video S1. The snapshots of the molecular dynamics simulation of the apo structure of KaiA C-terminal domains. The apo structure (1R6B, in cyan) and the holo structure (1SV1, in magenta) are shown for comparison. Only 20 ns simulations were shown. Supplementary Video S2. The snapshots of the molecular dynamics simulation of the holo structure of KaiA C-terminal domains. The apo structure (1R6B, in cyan) and the holo structure (1SV1, in magenta) are shown for comparison. Only 20 ns simulations were shown.
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