{ "cells": [ { "cell_type": "markdown", "id": "dc49fee7", "metadata": {}, "source": [ "# $T_1$ and Nuclear Overhauser Effect" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "id": "ca97a496", "metadata": {}, "source": [ "In the following, we will mimic $T_1$ relaxation and heteronuclear NOE transfer resulting from tumbling in solution. To do this, we would like to have a motion that averages the dipolar tensor to zero. A simple way to do this is to use a tetrahedral hopping motion, however, note that this only results in isotropic averaging to interactions that are initially aligned along the $z$-axis." ] }, { "cell_type": "markdown", "id": "12119164", "metadata": {}, "source": [ "## Setup" ] } , { "cell_type": "code", "execution_count": 0, "metadata": {"tags": [ "remove-cell" ]}, "outputs": [], "source": [ "# SETUP SLEEPY\n", "import sys\n", "if 'google.colab' in sys.modules:\n", " !pip install sleepy-nmr" ] }, { "cell_type": "code", "execution_count": 2, "id": "f02b5534", "metadata": {}, "outputs": [], "source": [ "import SLEEPY as sl\n", "import numpy as np\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "markdown", "id": "b7a0e7da", "metadata": {}, "source": [ "## Build the spin-system\n", "An important consideration when simulating longitudinal relaxation is that this type of relaxation is induced by interactions that are removed when using the rotating frame approximation. Therefore, we must be sure to setup our spin system in the lab frame. You can try running these calculations in the rotating frame to see that the $T_1$ relaxation vanishes.\n", "\n", "Note that adding explicit relaxation (via `L.add_relax(...)`) does not require a lab-frame calculation.\n", "\n", "We start with a 1-bond H–N dipole coupling, hopping around a tetrahedral geometry, with a correlation time of 1 ns. The tool `sl.Tools.SetupTumbling` takes `ex0` and rotates to several orientations (for higher `q`, we get more orientations), and returns a Liouvillian with different the Euler angles in exchange. The exchange matrix in the resulting Liouvillian yields an approximately monoexponential correlation function (for `q=0` or `q=1`, it is exactly monoexponential, but for higher values, it is only approximate). Note that `SetupTumbling` averages around the α- and β-angles, but not γ, so be cautious for more complex geometries." ] }, { "cell_type": "code", "execution_count": 3, "id": "4ddff027", "metadata": {}, "outputs": [], "source": [ "# By default, we get a powder average when including anisotropic terms.\n", "# so we set it explicitly to a 1-element powder average\n", "ex0=sl.ExpSys(v0H=400,Nucs=['15N','1H'],vr=0,LF=True,pwdavg='alpha0beta0')\n", "delta=sl.Tools.dipole_coupling(.102,'1H','15N')\n", "ex0.set_inter('dipole',i0=0,i1=1,delta=delta)\n", "\n", "# Set up 4-site motion\n", "L=sl.Tools.SetupTumbling(ex0,tc=1e-9,q=1) #q=1 gives just the tetrahedral orientations\n", "\n", "seq=L.Sequence(Dt=.1)" ] }, { "cell_type": "markdown", "id": "faf29262", "metadata": {}, "source": [ "## Simulate $T_1$ relaxation of $^1$H and $^{15}$N" ] }, { "cell_type": "markdown", "id": "a8080151", "metadata": {}, "source": [ "We start with both the $^{15}$N and $^1$H nuclei at thermal equilibrium (`rho0='Thermal'`) and observe decay of magnetization due to the motion. " ] }, { "cell_type": "code", "execution_count": 4, "id": "6e4bb5a5", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "rho=sl.Rho(rho0='Thermal',detect=['15Nz','1Hz'])\n", "rho.DetProp(seq,n=100)\n", "_=rho.plot(axis='s')" ] }, { "cell_type": "markdown", "id": "399a0df6", "metadata": {}, "source": [ "We observe both spins start from their thermal equilibrium and decay towards zero. The $^{15}$N magnetization is briefly enhanced by magnetization from the $^1$H spin, but then decays together with the $^1$H." ] }, { "cell_type": "markdown", "id": "d1e9f755", "metadata": {}, "source": [ "## Relaxing towards thermal equilibrium\n", "Note that exchange in simulations only can destroy magnetization, so that this approach always forces the magnetization towards zero. We may add a correction term in SLEEPY, that will cause the magnetization to approach thermal equilibrium rather than zero. This is included by running `L.add_relax('DynamicThermal')`. Then, we can start the magnetization at zero (`rho0='zero'`) and allow both spins to recover." ] }, { "cell_type": "code", "execution_count": 5, "id": "0e9a1112", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "L.add_relax('DynamicThermal')\n", "rho=sl.Rho(rho0='zero',detect=['15Nz','1Hz'])\n", "rho.DetProp(seq,n=100)\n", "ax=rho.plot(axis='s')\n", "ax.set_xlim(ax.get_xlim())\n", "ax.plot(ax.get_xlim(),ex0.Peq[0]*np.ones(2),linestyle=':',color='black')\n", "_=ax.plot(ax.get_xlim(),ex0.Peq[1]*np.ones(2),linestyle='--',color='grey')" ] }, { "cell_type": "markdown", "id": "f92760f7", "metadata": {}, "source": [ "`'DynamicThermal'` works by calculating the full Liouvillian (referred to here as $\\hat{\\hat{L}}_0$) and equilibrium density matrix, $\\hat{\\rho}_{eq}$. Then, a correction term is calculated, such that:\n", "\n", "$$\n", "\\hat{\\rho}_{corr}=-\\hat{\\hat{L}}_0\\cdot\\hat{\\rho}_{eq}\n", "$$\n", "\n", "Then, if $\\hat{\\rho}=\\hat{\\rho}_{eq}$, the sum of the product of the Liouvillian and the density matrix plus the correction term is zero:\n", "\n", "$$\n", "\\frac{d}{dt}\\hat{\\rho}=\\hat{\\hat{L}}_0\\cdot\\hat{\\rho}+\\hat{\\rho}_{corr}\n", "$$\n", "\n", "The correction term is inserted into the Liouvillian such that it only interacts with the identity in the density matrix. Remember, the density matrix is extended to a vector in Liouville space. However, take for example, a 2-spin-1/2 system. The 4x4 identity matrix is stretched into a 16 element vector, with elements at the 0th, 5th, 10th, and 15th position. In this system, we have four states in exchange, so that yields elements at positions: 0,5,10,15,16,21,26,31,32,37,42,47,48,53,58,63. We can see how this works in the plot below, where the columns corresponding to the listed positions are each occupied by the product $\\hat{\\hat{L}}_0\\cdot\\hat{\\rho}_{eq}$" ] }, { "cell_type": "code", "execution_count": 6, "id": "53f9db63", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "_=L.plot('Lrelax')" ] }, { "cell_type": "markdown", "id": "681c95d9", "metadata": {}, "source": [ "Then, it is important to note: this correction will result in polarization on, for example, the $\\hat{S}_z$ term, by continually reducing the population of $\\hat{S}_\\beta$ and increasing the population of $\\hat{S}_\\alpha$, whereas the rest of the Liouvillian is continuously try to equilibrate these populations. However, we also expect asymmetric transfer between, for example, the $\\hat{I}^+\\hat{S}_\\alpha$ and $\\hat{I}^+\\hat{S}_\\beta$ populations. This is what gives rise to the [contact shift](../Chapter5/Ch5_ContactShift.ipynb) occuring due to fast relaxing electrons. However, both the $\\hat{I}^+\\hat{S}_\\alpha$ and $\\hat{I}^+\\hat{S}_\\beta$ terms are zero under thermal conditions, and therefore are not included in the \"DynamicThermal\" correction term, as we can see above where we note that the correction is calculated from the density matrix at thermal equilibrium, $\\hat{\\rho}_{eq}$. We will later see that we can correctly produce contact and pseudo-contact shifts using Lindblad formalism,$^1$ but this is obtained with a different set of settings which we will use later for paramagnetic effects.\n", "\n", "[1] C. Bengs, M. Levitt. [*J. Magn. Reson.*](https://doi.org/10.1016/j.jmr.2019.106645). **2020**, 310,106645." ] }, { "cell_type": "markdown", "id": "fb1fbcb2", "metadata": {}, "source": [ "We note that the 'DynamicThermal' option is susceptible to numerical error due to the small size of the correction compared to the large exchange rates possible. Please see [$T_1$ tests](Ch2_T1_limits.ipynb) for more details." ] }, { "cell_type": "markdown", "id": "df55f373", "metadata": {}, "source": [ "## Obtaining the NOE enhancement" ] }, { "cell_type": "markdown", "id": "8aa0b6ba", "metadata": {}, "source": [ "Finally, we may observe how saturating one spin can lead to a change in the magnetization on the second spin via the Nuclear Overhauser Effect.$^2$\n", "\n", "Here, we have to use a special tool in SLEEPY to apply an oscillating RF field in the lab frame. Keep in mind, longitudinal relaxation is driven by terms in the Hamiltonian that are dropped when we go to the rotating frame, so a lab frame calculation is required. However, in the lab frame, the applied RF fields are no longer time-independent and their time-dependence needs to be explicitly simulated. SLEEPY uses several tricks for doing this. First, we use a two-step square wave to produce the correct frequency, and scale it to produce the correct amplitude at center frequency band. The resulting propagator is then brought into its eigenbasis and propagated either to one step in a rotor period, or here just to the desired length of the propagator. It should be noted that this approach does indeed produce sidebands to the RF, but we do not have any indication that this negatively impacts the simulations. \n", "\n", "[2] A.W. Overhauser. [*Phys. Rev.*](https://doi.org/10.1103/PhysRev.92.411), **1953**, 92, 411-415." ] }, { "cell_type": "code", "execution_count": 7, "id": "6b2e92c4", "metadata": {}, "outputs": [], "source": [ "L.kex=sl.Tools.fourSite_sym(tc=1e-9)\n", "\n", "seq=L.Sequence(Dt=.1).add_channel('1H',v1=500)\n", "#The lfrf object creates a propagator with a fixed\n", "#field strength on one channel, \n", "#and length equal to the input sequence length (0.1 s)\n", "lfrf=sl.LFrf(seq)" ] }, { "cell_type": "markdown", "id": "4feb4b8a", "metadata": {}, "source": [ "Below, we plot the square wave used to produce RF-irradiation in the lab frame." ] }, { "cell_type": "code", "execution_count": 8, "id": "d3d2ce1c", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "_=lfrf.seq.plot()" ] }, { "cell_type": "markdown", "id": "7bf346f9", "metadata": {}, "source": [ "First, we get the propagator for the lab-frame RF. Then, we apply it to a system at thermal equilibrium, to observe it evolve away from thermal equilibrium due to the RF irradation saturating the $^1$H." ] }, { "cell_type": "code", "execution_count": 9, "id": "94081468", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "U=lfrf.U(progress=False)\n", "\n", "rho=sl.Rho('Thermal',['15Nz','1Hz'])\n", "rho.DetProp(U,n=100)\n", "_=rho.plot(axis='s')" ] }, { "cell_type": "code", "execution_count": 10, "id": "1e6448d7", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Enhancement: -2.1\n" ] } ], "source": [ "print(f'Enhancement: {rho.I[0][-1].real/ex0.Peq[0]:.1f}')" ] }, { "cell_type": "markdown", "id": "4845f98f", "metadata": {}, "source": [ "Then, the $^1$H magnetization is saturated, and the $^{15}$N signal switches from negative to positive and is enhanced by just over a factor of 2.\n", "\n", "We finally calculate the correlation time dependence of the NOE enhancement." ] }, { "cell_type": "code", "execution_count": 11, "id": "d6a88990", "metadata": {}, "outputs": [], "source": [ "seq=L.Sequence(Dt=.1).add_channel('1H',v1=5000)\n", "rho=sl.Rho('Thermal','15Nz')\n", "\n", "tc0=np.logspace(-10,-7.5,50)\n", "for tc in tc0:\n", " rho.reset()\n", " L.kex=sl.Tools.fourSite_sym(tc=tc)\n", " lfrf=sl.LFrf(seq)\n", " U=lfrf.U(progress=False)\n", " (U**np.inf*rho)()" ] }, { "cell_type": "code", "execution_count": 12, "id": "38cba459", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "ax=plt.subplots()[1]\n", "ax.semilogx(tc0,rho.I[0].real/ex0.Peq[0])\n", "ax.set_ylim([-4.5,1])\n", "ax.set_xlabel(r'$\\tau_c$ / s')\n", "_=ax.set_ylabel(r'$\\varepsilon$')" ] }, { "cell_type": "markdown", "id": "ab052a07", "metadata": {}, "source": [ "\n", "We then get the expected enhancement values for an H–N heteronuclear NOE. A little noise appears at short correlation times where the `DynamicThermal` method is less numerically stable. Limits to this method are discussed in detail in [T$_1$tests](Ch2_T1_limits.ipynb)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 5 }