Exact NMR simulation of protein-size spin systems using tensor train formalism

We introduce a new method, based on alternating optimization, for compact representation of spin Hamiltonians and solution of linear systems of algebraic equations in the tensor train format. We demonstrate the method's utility by simulating, without approximations, a ${}^{15}\mathrm{N}$ NMR spectrum of ubiquitin—a protein containing several hundred interacting nuclear spins. Existing simulation algorithms for the spin system and the NMR experiment in question either require significant approximations or scale exponentially with the spin system size. We compare the proposed method to the Spinach package that uses heuristic restricted state space techniques to achieve polynomial complexity scaling. When the spin system topology is close to a linear chain (e.g., for the backbone of a protein), the tensor train representation is more compact and can be computed faster than the sparse representation using restricted state spaces.

Figure 1

Human ubiquitin protein (PDB code 1D3Z): 76 amino acids, 563 magnetic nuclei in the extended backbone (H, N, C, CA, CB, HA, HB)

Figure 2

Performance comparison for binary and AMEn tensor train summation with relative accuracy parameter $\varepsilon ={10}^{-12}$ during the construction of the NMR spin Hamiltonian. Top: human ubiquitin backbone (H, N, C, CA, HA); bottom: human ubiquitin extended backbone (H, N, C, CA, HA, CB, HB). Here $H_0$ refers to the isotropic part of the Hamiltonian and $Q_{k,n}$ to the irreducible spherical components of the anisotropic part.

Figure 3

Amino group region of the pulse-acquire ${}^{15}\mathrm{N}$ NMR spectrum of human ubiquitin. Top: spectrum computed by AMEn algorithm [37] with accuracy parameter $\varepsilon ={10}^{-6}$ is compared to the results obtained by the restricted state space (RSS) approximation [4] with basis containing local spin correlations of orders up to 5 and 6. Bottom: accurate RSS computation is used as a reference $O_{☆}\left(\omega \right)$ and compared to the spectra $O\left(\omega \right)$ computed by AMEn and DMRG [35, 36], both using the accuracy parameter $\varepsilon ={10}^{-3}.$ Right subgraph: convergence of AMEn and DMRG methods at two points of the frequency domain (dashed lines: an off-peak point at 100 ppm, solid lines: a peak at 122 ppm).

Figure 4

Theoretical (left) and experimental (right) ${}^1\mathrm{H}$-${}^{15}\mathrm{N}$ HSQC spectra of ${}^{15}\mathrm{N}$-,${}^{13}\mathrm{C}$-labeled human ubiquitin. Top: proton decoupling switched off in the indirect dimension and nitrogen decoupling switched off in the direct dimension to demonstrate accurate quantum mechanical treatment of spin-spin coupling by the simulation. Bottom: additionally, carbon decoupling is switched off in the indirect dimension. Signal groups marked A–F in the theoretical spectrum are not visible in the experimental data due to partial deuteration and slow conformational exchange of the corresponding amino acid residues.