1 +

Henriette Molinari
Università di Verona
Thich Nhat Hanh
(monaco zen vietnamita)
Mz -­‐My Ul#ma slide 1° lezione di Mario Piccioli – Torino 2011 The net spin polarisation along the z-axis is transferred
into a net spin polarisation along the –y axis (all spins
became phase coherent)
The net magnetic moment ⊥ to B° is called
transverse magnetisation
u  My(t) processes in the xy plane
u  Both the frequency components
along x and y are recorded by
the detection coil to define the
sign of ω
My=M0cosω0t Mx=M0sentω0t
M(t) = My + i Mx
= M0cosω0te-t/T2 +i M0senω0te-t/T2
= M0(cosω0t+isenω0t) e-t/T2
= M0eiω0te-t/T2
dwell time
Unfortunately, the information derived from NMR does not always
explicitly include structural coordinates, thereby limiting certain
nsights into the structure–dynamics–function linkage. To address this
imitation, NMR studies are often complemented with computational
imulations of protein dynamics. Conventional molecular dynamics
MD) is the most common approach whereby the three-dimensional
positions of each atom in the protein and solvent are computed over
ime using empirically determined interaction forces [33]. This high-
magnetic field denoted B0. The bulk magnetic moment for each set of
NMR-active isotopes will preferentially align with B0 along the z-axis
of the magnetic field. A weaker magnetic field temporarily applied
perpendicular to B0 rotates this bulk magnetic moment into the
transverse (x, y) plane. The bulk moment will then undergo Larmor
precession about B0 akin to a spinning top's angular momentum
precessing about the gravity field vector. The nuclei in the sample
precess at characteristic rates that differ from one another due to
NMR and the three observables
(δ, Ι, λ)
ig. 2. The free induction decay (FID) is the fundamental NMR observable and encapsulates the individual signals from each site-specific probe in the molecule. This time-dependent
ignal (left) is Fourier transformed into the frequency domain (right) for quantification of the three primary NMR observables: (1) frequency (or chemical shift δ) is the position of
he peak in the spectrum and reports on local chemical environment, (2) intensity I can be quantified by peak height or peak area and reports on populations and (3) linewidth λ is
he full peak width at half maximum height and reports on local dynamics via the relaxation rate R2 = 1/T2. Note that the more rapidly relaxing signal (B) is shorter and broader than
A) yet the total area under the peak is conserved because they are simulated with identical populations. The differences in linewidths of individual signals can be used to discern
ite-specific differences in protein dynamics.
: position of the peak, reports on
is the peak heigth/area, reports on
is the peak width at half-heigth, reports on
Chemical Shift
B loc
= B0 − Binduced
j
j
The exact resonance frequency (chemical shift) is determined
by the electronic environment of the nucleus
Lezione Piccioli-­‐Cicero Torino 2011 λ
S (ω ) = 2
λ + (ω − ω 0 )2
Lorentzian curve in absorption
λ=coherence decay constant 1
λ=
T2
Peakwidth at half heigth
(rad s-1)=2/T2=2λ
µ True T2 relaxa#on 2λ
µ B0 inhomogeneity ω0
ω (rad s-1)
1/πT2
Peakwidth at half heigth:
from each
probe in the molecule. This time-dependent
2/Tsite-specific
2 rad/s and 1/πT2 Hz
R observables: (1) frequency (or chemical shift δ) is the position of
ω/2π
ω°/2π
ight or peak area and reports on populations and (3) linewidth λ is
Rilassamento – Lezione Beringhelli Torino 2011 Inversion recovery
MZ = M0 (1 – e-­‐t/T1) Per un effettivo ritorno all’equilibrio di Mz, repetition time ≥ 5T1
Polarisation distribution after a 90° pulse
Different spins experience slightly
different magnetic fields, so that they
precess at slightly different
frequencies and get out of phase
(
) with each other
Dipolo-dipolo
Anisotropia del tensore di schermo
Quadrupolare
Accoppiamento scalare nucleare e/o elettronico
Spin-rotazionale
Per spins ½ il rilassamento è causato da campi magnetici fluttuanti al sito dello
spin nucleare, generati dai moti termici delle molecole
Man mano che la molecola si riorienta l’ampiezza e la direzione del campo
magnetico esercitato da uno spin sull’altro cambia:
Bo
J
I
θ
r
Bloc
I θ
r
J
Campi magnetici dipolari locali fluttuanti
Molecular
tumbling
Modulazione dei campi locali dipolari ad opera della rotazione molecolare
Spectral densities J(ω) indicate the probability of finding a component of the random motion of
the molecule at a particular frequency ω. A slowly tumbling molecule has more contributions at
low frequencies, a faster tumbling molecular has more contributions at higher frequencies.
Spectral density
• The FT of g(t) is called the spectral density func0on, J(ω), and since g(t) is a decaying exponen#al, J(ω) is a Lorentzian curve: J(ω)
J(ω) =
2 τc
1 + ω2τc2
ωo
ωo * τc >> 1 (large molecules)
ωo * τc ≈ 1
ωo * τc << 1 (small molecules)
log(ω)
• The spectral density function J(ω) tells us the power available from the lattice (i.e.,
from molecular motions) to bring about relaxation, as a function of molecular
tumbling rate.
J(0)=2τc
Fluttuazioni di Bloc lungo z danno
contributi a frequenza 0
For spherical molecules: 3
4πa η
τr =
3kT
Stokes-­‐Einstein law tr
T = temperature, a = radius of the protein h = viscosity of the solvent Longitudinal and transverse magne#za#on B
Net Magnetic
Moment
Thermal
Equilibrium
Thermal equilibrium Longitudinal magne1za1on •  The difference in populations between two state
has physical significance
•  The difference in spin state populations
indicates net longitudinal spin polarisation,
i.e. magnetisation of the sample in the direction
of the field
B
Net Polarization
Transverse magne1za1on B
•  Coherence requires the
existence of spins which
have transverse
polarisations partially
aligned. The phase of the
coherences indicates the
direction of the transverse
spin polarisation in the xy
plane
B
Net
Polarization
z
x
No Coherence
y
z
y
x
Coherence
Effetto NOE
• 
• 
• 
• 
L’effetto NOE si origina dalle interazioni dipolari che si stabiliscono tra i
nuclei di una molecola.
Un nucleo, dotato di momento magnetico di spin non nullo, in un campo
magnetico statico, può essere considerato come un dipolo fluttuante, che
influenza il rilassamento magnetico dei dipoli nucleari circostanti.
L’interazione nucleare dipolare si trasmette attraverso lo spazio,
diversamente dall’interazione nucleare scalare che viene propagata per
mezzo degli elettroni dei legame.
Due dipoli magnetici, vicini nello spazio, scambiano di energia
⎛ µ ⎞⎛ µ ⎞
A
Bdipole
= ⎜ 0 ⎟⎜ 3 ⎟ 3 cos2 ϑ − 1
⎝ 4π ⎠⎝ r ⎠
(
)
•  è u n e ff e W o q u a d r a # c o dell’energia di interazione dipolare, quindi dipende da 1/r6 •  Dipende da τc • L’effetto nucleare Overhauser è un processo
incoerente nel quale due spin nucleari
“cross-rilassano”.
Un singolo spin rilassa
tramite i meccanismi T 1 (rilassamento
longitudinale o spin-lattice) o T2 (rilassamento
trasversale).
Two spin energy diagram
-1/2
W1S
W2IS
0
βα
IS
ββ
W1I
0
αβ
W0IS
W1S
W1I
αα
+1/2
Intensity of I: p(αα)-­‐p(βα)+p(αβ)-­‐p(ββ)=1 Intensity of S: p(αα)-­‐p(αβ)+p(βα)-­‐p(ββ)=1 WIS is saturated • W0IS and W2IS become the only way I can relax The W2 increases the population
difference for I spin
c) N+Δ/2-(N-Δ)=3/2Δ NOE > 0
N-Δ
The W0 process decreases the
population for I spin
d) N - (N-Δ/2)= Δ/2 NOE < 0
N-Δ/2
Effetto NOE
N+Δ/2
N-Δ/2
N+Δ
N
N
N+Δ/2
Ogni volta che sono operativi i
meccanismi W2 e W0 il rilassamento
dello spin I è influenzato dalla
presenza dello spin J.
L’intensità della transizione dello spin
I è alterata da un cambio delle
popolazioni dello spin J
Origin of NOE -1/2
0
-1/4
ββ
0
βα
-1/4
αβ
+1/2
αα
ββ
+1/4
βα
αβ
IS
+1/4
αα
• W1S are not possible (we have the same populations in these levels)
• W1I are not happening (we have not affected the equilibrium for this
spin)
• W0IS and W2IS become the only way I can relax
Origin of NOE • If W2IS is the preponderant source of S relaxa#on, the I popula#on will be increased (posi1ve NOE) • If W0IS is the preponderant source of S relaxa#on, the I popula#on will be decreased (nega1ve NOE) The Solomon equation
W2IS - W0IS
η = γI / γS *
2 * W1S + W2IS + W0IS
The defini1on of NOE Frac1onal varia1on of signal intensity when another signal is perturbed Transient (equivalent to I N OESY exps) J Steady state I J Reference Irradiated Difference ηJ →I
ηI → J
σ IJ
=
ρI
σ IJ
=
ρJ
ηJ →I (max) = ηI →J (max) =
⎛ R + D ⎞
= ⎜
⎟
⎝ R − D ⎠
−
( R−D)
2D
⎛ R + D ⎞
− ⎜
⎟
⎝ R − D ⎠
−
( R+ D)
2D
1
R = (ρ I + ρ J ) D = 1 (ρ I − ρ J )2 + σ IJ 2
2
4
Theore1cal basis laid by Overhauser, Phys. Rev. 91, 476 (1953) [
1
2
]
Dependence of NOE from τr
The equa1ons of NOE For two spins I and J dipolarly coupled the longitudinal transfer of magne#za#on is described by σ IJ
−ρ t
(
η IJ =
1− e )
ρI
I
k
ρI = 6
rIJ
⎛
⎞
τ
3τ c
6τ c
c
⎜
⎟
+
+
2
2
2
2
2
2
⎜ 1 + (ω − ω ) τ
1 + ω I τ c 1 + (ω I + ω J ) τ c ⎟⎠
I
J
c
⎝
k
σ IJ = 6
rIJ
⎛
⎞
6τ c
τ
c
⎜
⎟
⎜ 1 + (ω + ω )2τ 2 − 1 + (ω − ω )2τ 2 ⎟
I
J
c
I
J
c ⎠
⎝
2
2
2
2
µ
2

γ
γ
⎛ 0 ⎞
I
J I ( I + 1)
k = ⎜ ⎟
15
⎝ 4π ⎠
B0 J I θ
rI,J Transi#on Rates and Correla#on Time (
µ0
K=
W0IS = 0.1 K2 J(ωI - ωS)
W2IS = 0.6 K2 J(ωI + ωS)
W/K2
10
rIS3
-8
W2 dominates
10
4π )γ I γ S
W0 dominates
-9
W2IS(300MHz) J(ω) =
10
-10
10
-11
W2IS(700MHz) 2 τc
1+
W0IS ω2τc2
10
-12
0,01
0,1
1
τc [ns]
10
2D Spectroscopy •  Introduc#on of a second pulse and of a variable delay t1
t1 is systema#cally increased Slide Cicero Torino 2011 After Fourier transform along t2
t1
t2
t2
F1
ω2
F2
F1
Evoluzione per effetto dell’accoppiamento: spettri di correlazione scalare
y x Spettri in antifase per nuclei accoppiati
Se non ho accoppiamento
dopo t1=1/2J:
Antiphase magnetisation represents the coherence of one spin (red vectors)
linked to the population of the coupling partner (blue vectors) and is denoted
as the product of a transverse magnetisation with a z-magnetisation (2I1xI2z)
2 1 One vector results from spin 1
bound to spin 2 in α-state, while
the second vector corresponds to
spin 1 bound to spin 2 in the βstate
Coherence refers to the
presence of phase relationship
between excited spins
in-phase, coherence -I1y, causes a normal
doublet, where both peaks have the same sign
and phase
antiphase magnetisation, coherence 2I1yI2z
results in a characteristic doublet with the two
peaks having opposite signs.
0
2
0
2
0
1
0
1
In-phase coherence of A along y
Anti-phase coherence of A along y
with respect to X
Spectrum of A COSY
ν1 ν2 t1
t2
I1
spin 1 H1z/I1z
spin 2 H2zI2z
I2
I1
H1y
I1y
90°x
x
y
y
x
H1xH2z
2I1xI2z
t1
x
H1zH2x
2I1zI2x
90°
y
x
x
Anti-phase magnetization
evolving at ΩI
y
t1
2IxSz
90ºy
-2IzSx
t2
y
Anti-phase magnetization
evolving at ΩS
Slide di Cicero, Torino 2011 Slide di Cicero, Torino 2011 gives rise to an anti-phase absorption multiplet on spin 2. Using the
relationship sin B sin A = 21 {− cos( B + A) + cos( B − A)} the modulation in t1
can be expanded
sin πJ12 t 1 sin Ω 1t =
J12
F1
F2
ew of the crosset from a COSY
The circles are
indicate the twodouble absorption
lustrated below;
represent positive
open
represent
nsity.
1
2
{− cos(Ω t
1 1
}
+ πJ 12 t1 ) + cos(Ω 1t1 − πJ 12 t )
Two peaks in F1, at Ω1 ± πJ12, are expected; these are just the two lines of
the spin 1 doublet. Note that the two peaks have opposite signs – that is
they are anti-phase in F1. In addition, since these are cosine modulated we
expect the absorption lineshape (see section 3.2). The form of the crosspeak multiplet can be predicted by "multiplying together" the F1 and F2
multiplets, just as was done for the diagonal-peak multiplet. The result is
shown opposite. This characteristic pattern of positive and negative peaks
that constitutes the cross-peak is know as an anti-phase square array.
The second π/2 pulse is replaced by a train
of strong π pulses spanning the mixing
interval τm
shifts, and coherence transfer between spins is
At B0 Δδ(Hz) >> J(Hz) the effects on the energy of the system arising
d by them.
systemareismuch
said to
be first-order:
fromThe
couplings
smaller
than those due to chemical shifts
H = H + HJ + … with H >> HJ + …
Hδ is called the Halmitonian and
represents the energy of the
system
If the magnetisation is spin-locked in the xy plane by a composite pulse
d the Hamiltonian, and represents the energy of
the system.
gs change if the system is spin-locked. Since we
ss removed  (B1 << Bo) but not couplings, we
have that HJ >> Hd.
The frequencies of all the
transitions of the system are
proportional to BSL
coherence transfer occurs due to scalar coupling
Strong coupling condition
In-phase transfer magnetisation under strong coupling in TOCSY
z z Spin I1 x z x y x y y z z z Spin I2 x y x y x y z z z Spin I3 x y x y x y Ambiguity in a COSY spectrum
Diagonal
X’
X
M,M’
A’
A
A
A’
M,M’
X
X’
If we only could detect spin-spin couplings bejond 3J… TOCSY spectra!
Diagonal
X’
X
M,M’
A’
A
A
A’
M,M’
X
X’
t2 t1 τ
τ
DQF-COSY t1 t1: frequency labelling
t1 τ
t2 τ: cross-relaxation between close nuclei can take place t2 NOESY
NOESY 90x 90x tm t1 90
x t2 aquisition
90
90
t1
90
tm
t2
NOESY
“inversion”
-­‐Izcos(ΩIt1) σIS(small mol)
-­‐Izcos(ΩIt1) Iycos(ΩIt1) Iycos(ΩIt1)cos (ΩIt2) Diagonal peak σIS(large mol)
Szcos(ΩIt1) -­‐Szcos(ΩIt1) -­‐Sycos(ΩIt1) Sycos(ΩIt1) -­‐Sycos(ΩIt1)cos (ΩSt2) Cross-­‐peak Sycos(ΩIt1)cos (ΩSt2) Cross-­‐peak z Peak amplitude Long τc slow mo#on z Rcross > 0 Spin I1 x y y Cross-­‐relaxa#on z z Spin I2 x autorelaxa#on tm
y cross tm /s autorelaxa#on x diagonal x y z Peak amplitude Small τc fast mo#on z Rcross < 0 Spin I1 tm /s autorelaxa#on x x y y Cross-­‐relaxa#on z z Spin I2 x autorelaxa#on tm
y diagonal x y cross The power of the NOESY experiment is that the intensity of an NOE
peak will be related to the nuclear separation.
Strong NOE crosspeaks - 2.5 Å
Weak NOE crosspeaks - 2.5-3.5 Å
Extending the mixing time will permit the observation of nuclei
separated by 5Å – Not all spin systems will give a detectable peak
though. The absence of a peak does not preclude close approach.
Similarly a weaker crosspeak does not always prove a larger
internuclear distance.
Therefore tend to be cautious and define distance ranges.
Strong (1.8-2.5Å), medium (1.8-3.5Å), weak (1.8-5.0Å).
Since this works through space we can use the NOE to connect spin
systems that we assigned with the COSY and TOCSY spectra.
There is a choice as to whether to observe 13C/15N or 1H when recording a
shift correlation spectrum.
It is very advantageous from the sensitivity point of view to record 1H:
ü  1H magnetization is larger than that of 13C because there is a larger
separation between the spin energy levels giving, by the Boltzmann
distribution, a greater population difference.
ü  A given magnetization induces a larger voltage in the coil the higher the
NMR frequency becomes.
Sensitivity of NMR spectroscopy
S/N ~ N
exc
3/2
det
B03/2 NS T21/2
S/N
signal-to-noise
N
number of spins
sample concentration
exc
gyromagnetic ratio of excited spins
isotope labeling
det
gyromagnetic ratio of detected spins
B0
magnet “size”
static magnetic field
(e.g. 14.1 Tesla or 600 MHz for 1H)
number of scans
measurement time
T2
transverse relaxation ~1/(line width)
molecular weight
signal intensity
NS
line width
~ 1/( T2)
broad
nal)
narrow
Fast relaxation
Slow relaxation
Broad lines
Narrow lines
largemolecule
protein
Large
small protein
Small
molecule
fast
slow
Spin echoes in homonuclear spin systems
Chemical shift
J coupling
βx
Δ
180: Refocusing pulse as the
evolution during the first delay Δ is
undone by the second Δ
Ix Ix αx
Δ
The evolution of the coupling
is not affected by the
sequence
Ix Ix cos 2πJt + 2IySz sin 2πJt Spin echoes in heteronuclear spin systems
X Chemical shift
J coupling
1H Δ
Δ
X Ix Ix cos ω0(2Δ) Iy sin ω0(2Δ) Ix Ix refocussed
1H Δ
Δ
X Offset of both spin is refocused
J coupling evolves for time 2Δ
INEPT: Insensi1ve Nuclei Enhancement by Polarisa1on Transfer INEPT block X: 90x 90 180x 90y tD 1H: 90x(H) x 180 x 1/4JX-­‐H y tD x 180x(H) y Selective inversion of
one of the proton
transition
x 1/4JX-­‐H y z 90y(H) y 180x(X) 90x(X) -­‐ KIIy x INEPT
Heteronuclear polarization transfer
1H 13C 2ΔC -ΔH+ΔC αCβH 13C • •
-ΔH-­‐ΔC 4 2 2ΔH 1H 3,4 2ΔH ΔH-­‐ΔC 1H αCαH 1,2 βCβH • • • • •
• • • • •
1 ΔH+ΔC • • • •
• • • •
β α 1,3 2,4 C H 13C 3 2ΔC I
S 2ΔH/2ΔC ∝ γH/γC ≈ 4 Here the population differences between the energy levels
reflect that we have a 1 to 4 ratio between 13C and 1H due to
the differences in the gyromagnetic rations.
Inversion of one transition
ΔH+ΔC 13C • • • • •
αCβH • • • • •
-­‐ΔH-­‐ΔC 4 2 βCβH • • • •
• • • •
β α
C H • •
1 -­‐ΔH+ΔC 13C 3 ΔH-­‐ΔC 1,2 I
1,3 2,4 S Poiché 2ΔH =4x (2ΔC) -­‐3 I 3,4 1H 2ΔH -2ΔH 1H αCαH 2,4 +5 1,3 HSQC
τ=1/4J
τ=1/2J
Decoupling can be applied 15N J J 1 2 3 1H Undesired effects of 1H chemical shift evolution
A mixture of HxNz and HyNz is produced
MQ
This part of
magnetisation
is lost
The transfer efficiency of 1H to the heteronucleus has been decreased due to 1H
chemical shift evolution
A 1H 180° pulse at the centre of the 1/2J period will refocus 1H
chemical shift but also leads to refocussing of 1JNH evolution
The effect of J-coupling has been refocused and the magnetisation cannot
be transferred to 15N
The introduction of two 180° pulsed at the centre of the (1/2J)
period refocuses the effect of 1H chemical shift, while retaining the
effect of JXH coupling
1/4J
1/4J
x Nx magnetisation has been created from 1H magnetisation
KiHy
KiSx
Decoupling can be applied
decoupling sequence applied to 15N
A 180° pulse applied to one of two
J-coupled nuclei at the midpoint of
a delay refocuses the effect of J
coupling during that delay
15N 1 2 3 1H Sensitivity enhanced HSQC
Chemical shift evolution generates MQ components which could be recovered.
The procedure is to add a second reverse INEPT period. During this period the NORmal
pathway magnetisation is stored along the z axis while the MQ magnetisation is recovered by
turning it into single quantum magnetisation. In order to separately detect the NOR and REC
components, the 90° 15N pulse following t1 is applied with phases +x and –x in alternate scans.
This results in a sign change of NOR magnetisation (-Hy becomes +Hy), but no change in the
REC magnetisation which is in the NxHz state at the time of the pulse. The signal from these
two scans is alternatively subtracted and added to select for the NOR or REC (-Hy, Hx). The
chemical shift info encoded in the split of the magnetisation into the two pathways can be
extracted. These two pathways provide quadrature detection.
Heteronuclear experiments: HMQC
180° 90° 1H-12C/1H-14N
=1/2J 90° 90-t-180-t-FID
chemical shift refocussing
90° 1H-13C/15N
15N/
t=1/2J before the 90°15N pulse
-I1y
I1xS2z
I1y
-I1xS2z
-II1y 1y
Antiphase terms evolve into observables
Measuring 90° 15N/13C pulse recording 1H: 90°(1H)-1/2J- 0, 90 or 180° (15N)-FID
x
HxNz (HyNz)
HxNz does not imply a net magnetisation of 1H along x or of 15N along z. It denotes
that the state of 1H along x is correlated with the state of 15N along z
no signal
90°
HxNx (HyNx)
180°
-HxNx (HyNx)
multiple-quantum coherence
Terms including 1H chemical shift evolution are however refocussed
Terms including 1H chemical shift evolution are however refocussed
901H-­‐1/2J-­‐ -­‐1801H-­‐ -­‐90°15N-­‐ -­‐t1/2-­‐ t1/2 -­‐90°15N-­‐ Why using a ten pulse sequence HSQC, rather then the
simpler HMQC?
HMQC: during t1 –NyHx and other MQ terms relax
HSQC: during t1 –NyHz and other anti-phase terms relax
Hx relaxation is due to 1H T2 while Hz relaxation is due to 1H
T1 and T2 ≤ T1
In HMQC the amide 1H is transverse during t1, is subject to 3JHNHα coupling to Hα proton. This coupling is not refocussed by the 1H 180° pulse, as this pulse affect both the HN and Hα spins. Thus the value of 3JHNHα is added to 15N line width. The image cannot be displayed. Your computer may not have enough memory to open
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