The percentage of overlap (p) between two vectors, V1 and V2 was calculated using their dot product:

where V1 and V2 are the norms of vectors V1 and V2.

In this study, we calculated the percentage of overlap (or projection) between the modes of a structure that we call A and the difference coordinates vector between structures A and B. So V1=qmA, where qmA is mode m of structure A, with m∈[7, 206], and V2=ΔR=RB-RA, where RA and RB are the coordinates of structures A and B, respectively. Vector ΔR may also be written AB. Prior to the projection, structure B is superimposed on structure A using Cα atoms. The projection is also done considering only Cα atoms in both vectors, qm and ΔR, in order to avoid misleading directions due to sidechains. The higher the percentage of overlap between these two vectors (qm and ΔR), the most straightforward is the transition from A to B. Here A refers to one of the two energy-minimized structures (E1.Mg2+ and E1.Mg2+:SLN) and B to one of the two chosen crystal structures (1VFP for state E1.2Ca2+ and 3W5C for state E2). For the E1.2Ca2+ state, 1VFP was preferred to the other structures, 3AR2, 1T5S, and 3TLM, despite its slightly lower resolution in some cases, because 1T5S and 3TLM lack the disulfide bridge that stabilizes the protein, and the ATP analog in 3AR2 chelates Ca2+ instead of Mg2+ (see Supplementary Table 1). For the E2 state, the structure of 3W5C, although lacking an ATP analog, was preferred to 2DQS, because it is free of exogenous molecules like thapsigargin. 2DQS, which has the ATP analog, also contains thapsigargin to stabilize it and disrupt the communication between the transmembrane domain and the cytosolic headpiece of the ATPase (Picard et al., 2006; Montigny et al., 2007). However, despite these differences, the two E2 structures, 3W5C and 2DQS, are very similar, with an RMSD between all their Cα atoms of only 0.6 Å.

We also calculated the percentage of overlaps between all the modes calculated from one structure (B) over one chosen mode calculated from another structure (A), in which case, V1=qmA and V2=qnB, where m = 54 when A = E1.Mg2+:SLN and n∈[7, 206] for B = E1.Mg2+, or m = 40 when A = E1.Mg2+ and n∈[7, 206] for B = E1.Mg2+:SLN (see subsection The P-Domain Plays a Central Role in the Transition Toward the E2 State in the Results).

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