The storage of the system trajectory in the indirect dimension of the 2D NMR simulation shown in Fig. 1 requires 512 × 848,530 complex doubles (6.96 GB) of memory. It is clear that 3D NMR Doramapimod concentration simulations would put some strain on modern computing facilities. This would have been a difficult problem, were it not for a peculiar property of propagator semigroups – simulations can be partially run backwards, even in the presence of relaxation. The general algebraic summary is given below and a special

case of the HNCO pulse sequence is illustrated in Fig. 3. The free induction decay coming out of a 3D NMR experiment is a function of three evolution times t  1, t  2, t  3 and may be formally written as equation(6) f(t1,t2,t3)=σˆe-iLtˆˆ3Pˆˆ3e-iLtˆˆ22Mˆˆ2e-iLtˆˆ22Pˆˆ2e-iLtˆˆ12Mˆˆ1e-iLtˆˆ12Pˆˆ1ρˆ0,Lˆˆ=Hˆˆ+iRˆˆwhere

ρˆ0 is the initial density matrix, σˆ is the detection state, Lˆˆ is the background Liouvillian of the system comprising a Hamiltonian Hˆˆ and a relaxation superoperator Rˆˆ, Pˆˆn are preparation pulse and delay propagators, and Mˆˆn are propagators of refocusing pulses in the middle of evolution periods. Because semigroups are associative, the result of Eq. (6) does not depend on the partitioning of Dirac brackets. In particular, equation(7) f(t1,t2,t3)=σˆe-iLtˆˆ3Pˆˆ3e-iLtˆˆ22Mˆˆ2e-iLtˆˆ22Pˆˆ2e-iLtˆˆ12Mˆˆ1e-iLtˆˆ12Pˆˆ1ρˆ0 This transformation Stem Cells inhibitor splits a 3D NMR simulation into one forward 2D simulation from the initial state, one backward 2D simulation from the detection state and one dot product in the middle. Eq. (7) is formally equivalent to Eq. (6), but the reduction in storage requirements is considerable – for a typical protein 3D NMR experiment, instead of a dense 64 × 64 × 256 × 106 array of complex doubles (over 16 TB of data) at the end of the t3 period in Eq. (6), the arrays in Eq. (7) have dimensions

of 64 × 64 × 106 and 64 × 256 × 106 as well as better sparsity, resulting in the worst-case storage requirements of about 256 GB. As per Eq. (7), their scalar product along the last dimension returns the required 64 × 64 × 256 free induction decay. Importantly, Eq. (7) retains the parallelization opportunities and the time-memory trade-offs offered by the fact that different t1 increments may be evolved independently Florfenicol in t2 forward, and different t3 increments may be evolved independently in t2 backward. The final operation – the matrix dot product in Eq. (7) – is also intrinsically parallel. Practical testing shows that the two-sided propagation technique reduces the simulation time of 3D NMR experiments on proteins (HNCO example is given in Fig. 4) by at least an order of magnitude. Even in reduced spaces the algebraic structure of the time-domain NMR simulation problem lends itself to multiple efficiency tweaks. Sparse matrix algebra [20] is advantageous because in the Pauli basis all spin Hamiltonian matrices are guaranteed to be sparse [19].