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An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation pp 115–138Cite as

Markov State and Diffusive Stochastic Models in Electron Spin Resonance

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Part of the book series: Advances in Experimental Medicine and Biology ((AEMB,volume 797))

Abstract

Electron spin resonance (ESR) spectra of biological macromolecules reflect a wide range of dynamical molecular motions. However, because an electron spin is strongly coupled to its environment, the quantal degrees of freedom must be propagated for hundreds of nanoseconds to calculate spectra with a reasonable resolution of detail. Furthermore, a large number of independent “samples” are necessary for a reliable estimate of the ESR spectrum. For this reason, a direct calculation from molecular dynamics (MD) simulations is inefficient and wasteful route. As a practical alternative, we present a methodology in which stochastic are first constructed from MD simulations and then used to calculate ESR spectra. Discrete Markov state models (MSMs) offer a natural representation of the jump-like isomerization dynamics of a spin label attached to a protein through a flexible linker. A pedagogical introduction to the second half of the formalism—accounting for the coupling between the molecular and the spin dynamics—is also provided. The chapter concludes with a successful application of the methodology to multi-frequency ESR spectroscopy of spin-labeled T4 Lysozyme.

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Notes

  1. 1.

    Although not rigorously correct, the assumption that the classical dynamics is completely uninfluenced by the states of the quantum system is typically an excellent approximation for room-temperature magnetic resonance. One minor inconvenience is that the equilibrium population of the states of the spin system corresponds to an infinite temperature. This, however, affects only the longitudinal magnetization but not the transverse magnetization whose evolution is calculated.

  2. 2.

    A time-dependent or constant rotation matrix is associated with each successive transformation in a motional model like (10.5). The matrix for the net transformation from L to N, to be employed in (10.4), is obtained as the product of the successive rotation matrices: R LN(t)=R LM R MS(t)R SN.

  3. 3.

    Splitting the molecular motion according to (10.7) assumes that the overall molecular tumbling and the motion of the spin label with respect to the global molecular frame are independent [23]. Clearly, this approximation may break in some cases, e.g., when an internal structural rearrangement changes the overall structure—and hence the rotational diffusion tensor—of the whole molecule. Nevertheless, in many instances with spin-labeled biomacromolecules the approximation of decoupled global and internal motions is well justified.

  4. 4.

    In protein crystal structures the side chain of cysteine is very rarely seen to adopt a conformation with χ 1≈+60∘ when located on α helices since this places the cysteine sulfur atom in unfavorable steric contact with the backbone atoms of the helix.

  5. 5.

    The observation that, considering the multiplicity of its dihedral angles, the R1 side chain can adopt 108 different rotameric states motivated the choice of number of microstates.

  6. 6.

    Since the absolute value of the cw-ESR measurement depends on instrumental factors and is not relevant for our purposes, proportionality constants relating the magnetizations and the respective spin operators have been neglected in (10.16).

  7. 7.

    These are

    $$\begin{aligned} \hat{S}_0&= \left [\begin{array}{c@{\quad}c}1 & 0 \\ 0 & 1 \end{array} \right ] , \qquad \hat{S}_+= \left [\begin{array}{c@{\quad}c}0 & 1 \\ 0 & 0 \end{array} \right ] ,\\ \hat{S}_-&= \left [\begin{array}{c@{\quad}c}0 & 0 \\ 1 & 0 \end{array} \right ] ,\qquad \hat{S}_z=\frac{1}{2} \left [\begin{array}{c@{\quad}c}1 & 0 \\ 0 & -1 \end{array} \right ] . \end{aligned} $$
  8. 8.

    In principle, the submatrix ρ 0 in (10.17) contains the 3×3 or 2×2 identity matrix I 0 along its main diagonal. However, the part proportional to the identity matrix is neither affected by the relaxation or the coherent evolution nor does it affect the evolution of the rest of the density matrix. Hence, ρ 0 can be taken as traceless.

  9. 9.

    As mentioned above, the justification lies in the fact that the time scale T 1—on which ρ z and ρ 0 build up—depends on motions at the time scale of the Larmor precession and is much longer than the time scale T 2—on which ρ + decays—dominated by slow motions. The high field approximation automatically excludes the possibility to account for the contribution of T 1 processes to T 2 relaxation using Eq. (10.22).

  10. 10.

    The numerical advantages associated with working in the rotating frame are apparent from (10.27), where the transverse magnetization M +(t) consists of a rapidly oscillating “carrier” wave whose amplitude is modulated by the slowly changing “signal” \(M_{+}^{\prime}(t)\). Thus, following M +(t) numerically would require an integration time step sufficient to resolve the fast oscillations on the time scale of the Larmor precession (cf. Table 10.3). In contrast, calculating the slowly changing \(M_{+}^{\prime}(t)\) numerically allows us to take time steps larger by several orders of magnitude.

  11. 11.

    Note that the same propagator matrix acts on both sides of \({\rho}_{+}^{\prime}\) in this equation, which is different from the propagation of the density matrix ρ in the full Hilbert space. The source of the difference lies in replacing the commutator in (10.19) by an anticommutator in (10.22).

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Authors and Affiliations

  1. Faculty of Engineering and Natural Sciences, Sabancı University, Istanbul, Turkey

    Deniz Sezer

  2. Department of Biochemistry and Molecular Biology, The University of Chicago, Chicago, USA

    Benoît Roux

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  1. Deniz Sezer
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  2. Benoît Roux
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Correspondence to Deniz Sezer .

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Editors and Affiliations

  1. University of California, Berkeley, California, USA

    Gregory R. Bowman

  2. Department of Chemistry, Stanford University, Stanford, California, USA

    Vijay S. Pande

  3. Freie Universität Berlin, Berlin, Germany

    Frank Noé

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Sezer, D., Roux, B. (2014). Markov State and Diffusive Stochastic Models in Electron Spin Resonance. In: Bowman, G., Pande, V., Noé, F. (eds) An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation. Advances in Experimental Medicine and Biology, vol 797. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7606-7_10

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