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response_surface.py
executable file
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response_surface.py
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"""Define a response surface for methyl CCR.
The CCR rates sigma and eta, and relaxation during INEPT transfers, are all calculated from S2tc."""
#from __future__ import division # needed for python 2
import numpy as np
from numba import jit
from math import pi
# physical constants
mu0 = 4e-7 * np.pi
hbar = 1.055e-34
gH = 2.675e8
gC = 6.726e7
rCH = 1.117e-10 # Tugarinov 2004
rHH = np.sqrt(3) * rCH * np.sin(110.4*np.pi/180) # Tugarinov 2004
wC = 2*np.pi * gC/gH * 950e6 # 700 MHz
P2cosb = -1./3. # for 109 degree tetrahedral geometry
carbon_csa = 18e-6 # 18 ppm for Ile, 25 ppm for Leu/Val
# pre-factors for calculation relaxation rates
c_CHCH = 2/45. * (mu0*hbar*gC*gH / (4*pi*rCH**3))**2 * 1e-9
c_CHC = 2/5. * (mu0/(4*pi)) * P2cosb * rCH**-3 * hbar * gH * gC * wC * carbon_csa * 1e-9
c_HHHC = 1/5. * (mu0/(4*pi))**2 * rCH**-3 * rHH**-3 * hbar**2 * gH**3 * gC * 1e-9
c_HHHH = 9/20. * (mu0*hbar*gH**2 / (4*pi*rHH**3))**2 * 1e-9
# test functions
def calc_sigma(S2tc):
return c_CHCH * S2tc
def calc_eta(S2tc):
return c_CHC * S2tc
# methyl 1J(CH) scalar coupling, in Hz
J = 125.
piJ = pi * J
TAU = 0.5 / J
@jit
def calc_DELTA(S2tc):
"""Compute relaxation during INEPT transfers due to CH/CH and CH/HH dipole/dipole CCR (in macromolecular limit)."""
return np.exp(-TAU * c_HHHH * S2tc) * np.cosh(TAU * c_HHHC * S2tc);
@jit
def y(tau,phi,theta,omega):
"""Compute expected signal intensities.
y = A * exp(-lambda t) * (
Iout exp([-3 sigma - 2 eta]*t) cos([omega + 3 pi J]*t + phi)
+ Iin exp([sigma - 2/3 eta]*t) cos([omega + pi J]*t + phi)
+ Iin exp([sigma + 2/3 eta]*t) cos([omega - pi J]*t + phi)
+ Iout exp([-3 sigma + 2 eta]*t) cos([omega - 3 pi J]*t + phi)
)
Iin = 3 + 3*DELTA(S2tc)
Iout = 3 - DELTA(S2tc)
sigma = c_CHCH * tau_c
eta = c_CHC * tau_c
Args: (where M = number of time points, N = number of peaks)
t: (M x 1) array of evolution times
phi: (M x 1) array of phases (in radians)
theta: flattened (3 x N) array (with array.ravel)
theta[0,:] = amplitudes
theta[1,:] = lambda (auto-relaxation rate)
theta[2,:] = S2tc (in ns)
omega: (1 x N) array of resonance offsets, in angular units (s-1)
Returns:
y: (M x N) array of intensities
"""
t = tau.reshape((-1,1))
ph = phi.reshape((-1,1))
parameter_matrix = theta.reshape((3,-1))
A = parameter_matrix[0,:].reshape((1,-1))
lam = parameter_matrix[1,:].reshape((1,-1))
S2tc = parameter_matrix[2,:].reshape((1,-1))
# print('theta:')
# print(theta)
# print('parameter_matrix')
# print(parameter_matrix)
# print('amplitudes:')
# print(A)
# print('lambda:')
# print(lam)
# print('S2tc:')
# print(S2tc)
# print('omega:')
# print(omega)
# print('tau:')
# print(tau)
# print('t:')
# print(t)
sigma = c_CHCH * S2tc
eta = c_CHC * S2tc
DELTA = calc_DELTA(S2tc)
Iouter = 3 + 3*DELTA
Iinner = 3 - DELTA
# print('sigma')
# print(sigma)
# print('eta')
# print(eta)
# print('c_HHHC*S2tc')
# print(c_HHHC*S2tc)
# print('DELTA')
# print(DELTA)
# print('Iinner')
# print(Iinner)
# print('Iouter')
# print(Iouter)
# print( np.exp(-lam*t) )
# print(Iouter*np.exp((-3*sigma - 2*eta)*t)*np.cos((omega + 3*piJ)*t + ph))
# print(Iinner*np.exp((sigma - 2*eta/3)*t)*np.cos((omega + piJ)*t + ph))
return A * np.exp(-lam*t) * ( \
Iouter*np.exp((-3*sigma - 2*eta)*t)*np.cos((omega + 3*piJ)*t + ph) \
+ Iinner*np.exp((sigma - 2*eta/3)*t)*np.cos((omega + piJ)*t + ph) \
+ Iinner*np.exp((sigma + 2*eta/3)*t)*np.cos((omega - piJ)*t + ph) \
+ Iouter*np.exp((-3*sigma + 2*eta)*t)*np.cos((omega - 3*piJ)*t + ph))
@jit(nopython=True)
def jac(t,phi,theta,omega):
"""Compute Jacobian of response surface (at a given evolution time).
F_ij = dy_j / dq_i
Args: (where N = number of peaks)
t (float): evolution time
phi (float): phase (in radians)
theta: flattened (3 x N) array (with array.ravel)
theta[0,:] = amplitudes
theta[1,:] = lambda (auto-relaxation rate)
theta[2,:] = S2tc (in ns)
omega: (1 x N) array of resonance offsets, in angular units (s-1)
Returns:
F: (3N x N) array with jacobian matrix
"""
N = omega.size
F = np.zeros((3*N,N), dtype=np.float64)
parameter_matrix = theta.reshape((3,-1))
# loop over spins:
for n in range(N):
A = parameter_matrix[0,n]
lam = parameter_matrix[1,n]
S2tc = parameter_matrix[2,n]
sigma = c_CHCH * S2tc
eta = c_CHC * S2tc
DELTA = calc_DELTA(S2tc)
Iouter = 3 + 3*DELTA
Iinner = 3 - DELTA
w = omega[0,n]
exp_lam = np.exp(-lam*t)
x1 = np.exp((-3*sigma - 2*eta)*t) * np.cos((w + 3*piJ)*t + phi)
x2 = np.exp((sigma - 2*eta/3)*t) * np.cos((w + piJ)*t + phi)
x3 = np.exp((sigma + 2*eta/3)*t) * np.cos((w - piJ)*t + phi)
x4 = np.exp((-3*sigma + 2*eta)*t) * np.cos((w - 3*piJ)*t + phi)
# dy/dA:
F[3*n,n] = exp_lam * ( \
Iouter*x1 \
+ Iinner*x2 \
+ Iinner*x3 \
+ Iouter*x4)
# dy/dlambda:
F[3*n+1,n] = -t * A * exp_lam * ( \
Iouter*x1 \
+ Iinner*x2 \
+ Iinner*x3 \
+ Iouter*x4)
# dy/d(S2tc) # TODO check CHCH and HHHH are correct!
F[3*n+2,n] = A * exp_lam * ( \
np.exp((-3*sigma - 2*eta)*t) * \
((3+3*DELTA)*(-2*eta-3*sigma)*t - 3*TAU*DELTA*c_HHHH + 3*np.exp(-TAU*c_HHHH*S2tc)*TAU*c_HHHC*np.sinh(TAU*c_HHHC*S2tc)) * \
np.cos((w + 3*piJ)*t + phi) \
+ np.exp((sigma - 2*eta/3)*t) * \
((3-DELTA)*(-2/3*eta+sigma)*t + TAU*DELTA*c_HHHH - np.exp(-TAU*c_HHHH*S2tc)*TAU*c_HHHC*np.sinh(TAU*c_HHHC*S2tc)) * \
np.cos((w + piJ)*t + phi) \
+ np.exp((sigma + 2*eta/3)*t) * \
((3-DELTA)*(2/3*eta+sigma)*t + TAU*DELTA*c_HHHH - np.exp(-TAU*c_HHHH*S2tc)*TAU*c_HHHC*np.sinh(TAU*c_HHHC*S2tc)) * \
np.cos((w - piJ)*t + phi) \
+ np.exp((-3*sigma + 2*eta)*t) * \
((3+3*DELTA)*(2*eta-3*sigma)*t - 3*TAU*DELTA*c_HHHH + 3*np.exp(-TAU*c_HHHH*S2tc)*TAU*c_HHHC*np.sinh(TAU*c_HHHC*S2tc)) * \
np.cos((w - 3*piJ)*t + phi) )
return F