1
Solution Dynamics and
Self-Organization
Professor William S. Price
Professor of Nanotechnology
University of Western Sydney
E-mail: [email protected]
Outline
A. Solution Dynamics and Self-Organization.
B. The Pulsed Gradient Spin Echo (PGSE) NMR method for
measuring translational diffusion.
C. Some Examples
1.
2.
3.
4.
5.
Supercooled Water
Isolated Water Molecules
Alcohol Water Systems
Drug Binding
Aggregation and Crystallization of Lysozyme
2
Solution Dynamics
Includes:
• Association → self organization and crystallisation
• Binding (e.g., drug – protein or drug -DNA)
• Phase changes
• Bio-thermodynamics (e.g., macromolecular crowding effects)
• Exchange (e.g., transmembrane)
• ….
¾ How a molecule interacts with its neighbours and
surroundings.
3
4
Types of Motion
Time taken to reorientate
by ~ 1 radian
Translational (self-)
Diffusion, D (m2s-1)
Reorientational
Correlation Time, τc (s)
Different parts of molecule may
have different reorientational
motions, but a single diffusion
coefficient characterises the whole
molecule.
Lysozyme
NMR can probe both types of motion.
5
Mw and Motion
Translational diffusion
Stokes-Einstein equation
(only holds at infinite dilution)
Species
H2O
Glycine
glucose
inorganic phosphate
creatine phosphate
sucrose
ATP
insulin
lysozyme
hemoglobin
tobacco mosaic virus
sphere
D = kT
f
T (K)
298
298
298
298
298
298
298
293
298
293
293
Reorientational motion
Mw ↑ τc↑ T2↓
f = nπη rS
friction coefficient
D (m2s-1) × 109
2.26
1.05
0.67
0.61
0.52
0.52
0.37
0.082
0.108
0.063
0.005
rs
n = 4 (slip), 6 (stick)
Mw
18
75
180
95
211
342
507
5700
14500
64500
40000000
D∝
3
1
MW
Diffusion as a Probe of Organization
Advantages:
Organization and association generally involve changes in molecular
weight and hydrodynamic properties → diffusion is a natural probe of such
phenomena.
Data analysis is facilitated by diffusion being a property of the entire
molecule (excluding exchangeable groups)
Complications:
Finite Solute Concentrations: Inter-particle collisions and interactions
(i.e., ‘obstruction effects’) also influence the measured diffusion coefficient.
Nuisance OR source of information?
6
7
PGSE NMR Diffusion Measurements
Using spatially well-defined magnetic field gradients to
spatially encode the translational motion of spins – this
includes diffusion, flow, turbulence ….
PGSE NMR measures self-diffusion not mutual
diffusion.
Also known as q-space imaging, PGSE (Pulsed
Gradient Spin-Echo), DOSY (Diffusion Ordered
Spectroscopy) or NMR diffusometry.
8
The Mechanics of NMR Diffusion Measurements
Larmor Equation
Resonance
ω = −γ B0
-1
frequency (rad s )
B0
Strength of the
static magnetic field (T)
ω
Gyromagnetic ratio (rad s-1 T-1)
If the homogeneous field (B0) is replaced by a gradient directed along
the z-axis (gz), then ω will vary spatially along the z-axis: ω ( z ) = γ g z z
spatially encode
the magnetization
+gz
apply g as pulse
of duration, δ
initial transverse
magnetization
helix
timescale of the diffusion
Δ
measurement (ms - s)
spatially decode
the magnetization
-gz
Diffusion during Δ will corrupt
the helix → less refocussing →
smaller signal.
refocussed transverse
magnetization (“echo”)
= the NMR signal (S)
9
The PGSE NMR Experiment
kept constant
Hahn spin-echo
pulse sequence
g
1.00
-9
2 -1
D = 1.41 × 10 m s
E
0.37
Diffusion
coefficient
0.14
0.0
Free Diffusion: ln ( E ) = ln ⎛⎜⎜ S ( g ) S ⎛⎜⎝ g = 0 ⎞⎟⎠ ⎞⎟⎟ = − Dγ 2 g 2δ 2 ⎛⎜⎝ Δ − δ 3⎞⎟⎠
⎝
⎠
Faster diffusion
0.5
2 2 2
1.0
9
1.5
-2
γ g δ (Δ−δ/3) × 10 (m s)
10
What About Signal Attenuation due to Relaxation?
Echo signal
S ( g ) = M exp ⎛⎜⎜ −γ 2 g 2 D ( Δ−δ 3) ⎞⎟⎟ exp ⎛⎜ − 2τ T2 ⎞⎟
⎝ 424444
0 14444444
⎝
3⎠
4244444444
3⎠ 1444
initial
magnetization
Attenuation due
to Diffusion
Attenuation due
to Relaxation
Normally, the attenuation due to relaxation is normalised out.
(
)
2 2
exp
−
γ
g D( Δ−δ 3) exp( − 2τ T2 )
⎛
⎞
S(g)
=
= exp ⎜⎜ −γ 2 g 2 D ⎛⎜ Δ − δ 3⎞⎟ ⎟⎟
E=
⎝
⎠⎠
S ( 0)
⎝
exp( − 2τ T2 )
Echo Signal Attenuation
Unfortunately, this ‘elegant’ solution is not general - it only holds for a
single component.
A Summary of the Characteristics of PGSE
“Rule of Thumb” Mw ↑ D ↓
PGSE can measure diffusion in the range of 10-9 – 10-14 m2s-1.
H2O
solid polymer
If the mean square displacement during the experiment (Δ ~ 10 ms – 1 s)
is such that a sufficient population of spins make contact with the
boundaries ⇒ complicated non-exponential attenuation profiles.
Experimentally this corresponds to barriers with characteristic distances, R
100 μm.
At finite concentrations the diffusing species obstruct each other ⇒ D ↓
Differential relaxation weighting is a problem in polydisperse systems.
The combination of these effects can complicate data interpretation
11
12
SUPERCOOLED WATER
What is the nature of the water ↔ ice transition?
Self-nucleation temperature is about 231 K.
Metastable ‘Grey area’ from 273 to 231 K.
PGSE is one of the few applicable techniques.
Sample for Supercooled Water
Diffusion Measurements
Using a small sample volume makes it possible
to approach the self-nucleation temperature
5 mm NMR tube
0.13 mm i.d.
sealed capillaries
Water
(~ 0.5 μl, length ~ 8 mm)
Diffusion measured along the capillary.
Methanol
(for temperature calibration)
13
14
Supercooled 1H2O Diffusion
-9
Self-nucleation
2 -1
D = 2.30 × 10 m s
errors in D are smaller
than the symbol size
Supercooled
3.0
-1
EA → 44.4 kJ mole
2.0
10
ln(D × 10 ) m s
2 -1
2.5
Our measurements
Gillen et al (1972)
T = 25 ºC
1.5
Non-Arrhenius behaviour
T = -35.4 ºC
1.0
T = 0 ºC
-10
D = 1.58 × 10
0.5
3.3
3.4
3.5
3.6
3.7
3.8
3.9
-1
J. Phys. Chem. A (1999) 103, 448
1000/T (K )
2 -1
ms
4.0
4.1
4.2
4.3
15
Fractional Power Law (FPL) and VogelTamman-Fulcher (VTF) Relations
FPL equation (sharp transition from water to ice)
γ
⎛
⎞
T
1
2
D = D0 T ⎜ −1⎟
⎜T
⎟
⎝ S
⎠
TS: low temperature limit
(singularity)
D0, γ : fitting parameters
VTF equation (smooth transition)
D = D0 exp ⎪⎧⎨⎪−B / ⎛⎜T −T0 ⎞⎟ ⎪⎫⎬⎪
⎩
⎝
⎠⎭
T0: related to the glass
transition temperature
D0, B : fitting parameters
16
Modeling 1H2O Diffusion Using the
FPL and VTF Relations
VTF
Supercooled
D0 = 4.00 ± 0.87 × 10-8 m2s-1
B = 371 ± 45 Κ
T0 = 169.7 ± 6.1 K
2 -1
ln(D) (m s )
-20
VTF
-22
T = 0 ºC
FPL
3.4
3.6
3.8
4.0
-1
1000/T (K )
4.2
FPL
D0 = 7.66 ± 0.24 × 10-10 m2s-1
TS = 219.2 ± 2.6 Κ
γ = 1.74 ± 0.10
17
Apparent Activation Energy of 1H2O
Diffusion
Singularity at TS= 219.2 K
70
FPL
50
-1
EA (kJ mol )
60
40
30
VTF
20
10
0
220
230
240
250
260
T (K)
270
280
290
300
18
ISOLATED-WATER MOLECULES
Anomalous behaviour of liquid water arises from hydrogen bond
network.
Water dissolved in a hydrophobic solvent
‘isolated’ water molecules.
No hydrogen bonding between water molecules.
Studied the diffusion (17O PGSE) and 17O longitudinal relaxation
(T1) of H217O dissolved in nitromethane.
‘Non-standard’, low γ nuclei are now becoming accessible to PGSE
measurement with the increasingly larger applied gradients available.
19
DH
2
17 O
PGSE and Relaxation Measurements
EA (kJ mol-1)
10.0 ± 0.3
17O
T1 → reorientational correlation time, τc
-26.8
DNitromethane 9.7 ± 0.2
7.7 ± 0.1
ƒ D depends on inertial
and solvent effects
ƒ τc dominated by
inertial effects
τc H
-19.0
17
2
2 -1
2
17 O
ln(D/[m s ])
τc H
-27.0
O
-19.5
DH
17
2
-20.0
3.2
J. Chem. Phys. (2000) 113, 3686
O
DNitromethane
-20.5
3.4
3.6
1000/(T/K)
3.8
4.0
-27.2
-27.4
-27.6
ln(τc /s)
17O
20
Correlation between Reorientational Correlation Time
and Diffusion in the Hydrodynamic Continuum Model
Stokes-Einstein (SE) Equation
1.7
0.6
τc H
O
1.4
1.2
1.1
0.4
0.3
1.5
2.0
2.5
1.0
3.0
SEDStick
SEStick
0.2
H2O Stokes radius, rs = 1.21 Å
SE: lit. η → SE → D
SED: τc → SED → SE → D
All simulations
overestimate D
1.3
0.5
1/(D/[109 m2 s-1])
4π rS3η
τc =
3 kT
17
2
n = 4 (slip), 6 (stick)
Stokes-Einstein-Debye (SED)
Equation
1.5
τc /ps
kT
D = nπη
rS
1.6
0.1
DH
SEDSlip
SESlip
1.5
17
2
2.0
2.5
6
10 (η /[Pa s])/(T/K)
O
3.0
21
ALCOHOL-WATER
Alcohols are amphiphilic → complicated solution
chemistry.
Methanol, Ethanol and tert-Butanol differ only in
the size of the alkyl group.
“Primordial” lipids - models for micelle assembly.
Diffusion in the Methanol-Water
System at 298 K
2.4
Equimolar protons
1.8
1.5
-9
2 -1
D (× 10 m s )
2.1
1.2
Methanol
OH
H2O
0.9
0.6
H2O-OH
Maximum solution
structuring
H2O-calculated
0.3
0.0
0.2
0.4
0.6
xA
0.8 1.0
Alcohol mole
fraction
22
23
Diffusion in the Ethanol-Water
System at 298 K
2.4
-9
2 -1
D (× 10 m s )
2.1
1.8
Ethanol
OH
H2O
1.5
H2O-OH
1.2
0.9
0.6
0.3
wine
whisky
0.0
0.0
0.2
0.4
0.6
xA
0.8
1.0
24
Diffusion in the tert-Butanol-Water
System at 298 K
25
Butanol
OH
H2O
H2O-OH
H2O-calculated
15
D (× 10
-10
2 -1
ms )
20
10
5
0.0
0.2
0.4
0.6
xA
0.8
1.0
25
Arrhenius Activation Energy for
Diffusion
45
40
t-Butanol
Calculated from
diffusion measured at
273, 285 and 298 K
-1
EA, EW (kJ mol )
35
30
Ethanol
25
Pure H2O
20
15
10
Methanol
Solid = Alkyl
Hollow = OH
0.0
0.2
0.4
0.6
xA
0.8
1.0
26
Stokes-Einstein Analysis of the
Methanol System at 298 K
5
2.00
1.75
Equimolar protons
xA=1
rA , rW
4
Methanol
H2O
1.25
3
1.00
2
0.75
0.50
1
Polynomial interpolation of
literature viscosity data
0.25
0.00
0.0
0.2
0.4 0.6
xA
0.8
1.0
0
Bulk viscosity
η (mPa s)
xA= 0
Stokes radii normalized
to the pure solvent values 1.50
27
Stokes-Einstein Analysis of the
Ethanol System at 298 K
5
2.00
Local maximum in
the Stokes radius of
the ethanol
1.75
Ethanol
H2O
1.50
4
xA=1
rA , rW
3
1.00
2
0.75
0.50
1
0.25
0.00
0.0
0.2
0.4
0.6
xA
0.8
1.0
0
η (m Pa s)
xA= 0
1.25
Stokes-Einstein Analysis of the
tert-Butanol System at 298 K
2.00
5
1.75
butanol
self-association
4
1.50
xA=1
rA , rW
3
1.00
0.75
tert-Butanol
H2O
2
0.50
0.25
0.00
1
0.0
0.2
0.4
0.6
xA
0.8
1.0
The data is consistent with small butanol
clusters (e.g., tetramer, pentamer).
η (mPa s)
xA= 0
1.25
28
29
Alcohol-Water Summary
At low xA the alcohol molecules associate due to
hydrophobic hydration. The alcohol molecules sit at
the centre of hydration shells.
As xA increases there comes a point where there are
insufficient H2O to form the shells.
The three alcohols have quite different properties.
tert-Butanol has the strongest hydrophobic hydration
but the weakest H-bonding.
Ideally, we would like to study the very low xA.
30
DRUG BINDING
Diffusion is a very powerful method for screening
drugs and characterizing their binding properties.
31
The Practicalities of Diffusion-based Binding Assays
Wish to determine:
i. Is there any binding.
ii. Dissociation constants.
iii. Decide if there are one or more
classes of binding constants.
To do so it is necessary to:
i. Accurately measure D of the drug over large concentration
ranges (esp. v. low concentrations) – ideally in non-deuterated
samples (i.e., in 1H2O not 2H2O).
ii. Remove the signals of the receptor.
iii. Consider the effects of NMR relaxation on the measurement.
The Basis of Diffusion-based Binding Assays
Db = D of protein (v. slow)
(~ unchanged by drug binding)
Df = D of drug (fast)
If the exchange is fast on the NMR timescale, the observed drug
diffusion coefficient D, will be the population weighted average of the
diffusion coefficients: D = Pb Db + PfDf
Can use a diffusion filter to detect binding or a more detailed
analysis to determine the dissociation constant (Kd).
The use of NMR diffusion measurements to study drug binding
is sometimes referred to as “Affinity NMR”.
General ref.: Encyclopedia of Nuclear Magnetic Resonance (2002) 9, 364-374.
32
33
Effects of Ligand Relaxation
The bound and free states of the drug have different relaxation rates.
dSf
Sf S b Sf
2
= − ( γδ g ) Df Sf − + −
τ f τ b T2f
dt
dS b
S b Sf S b
2
= − ( γδ g ) Db S b − + −
τ b τ f T2b
dt
Ignoring
Population weighted average
relaxation effects
diffusion coefficient
(
S = exp − ( γδ g ) [ Pf Df + Pb Db ] Δ
2
)
Df T2f
Kd
T2b
Db
Kärger et al Adv. Magn. Reson. (1988) 12, 1-89.
Relaxation affects the population weighting:
Ignoring it in the data analysis will result in incorrect answers.
Using it provides and additional source of information.
Drug
PGSE-WATERGATE
A Hahn-based sequence is preferable to a Stimulated Echo-based
PGSE sequence due to: (1) better removal of the protein resonances due
to relaxation, (2) larger drug signal and (3) no complications from crossrelaxation effects.
Echo signal
S ( g ) = M exp ⎛⎜⎜ −γ 2 g 2 D ( Δ−δ 3) ⎞⎟⎟ exp ⎛⎜ − 2τ T2 ⎞⎟
initial
magnetization
⎝ 424444
0 14444444
⎝
3⎠
4244444444
3⎠ 1444
Attenuation due
to Diffusion
Attenuation due
to Relaxation
34
35
Salicylate binding to BSA
80 mM salicylate
0.5 mM BSA
90% 1H2O 298 K
7.0
7.0
6.8
Two Site Model:
D = (1 − Pb ) Df + Pb Db
Pb = α − α − β
2
Kd
PL
P+L
6.5
6.6
6.4
CL + nCP + K d )
(
α=
nC
β= P
CL
Magn. Reson. Chem. (2002) 40, 391
2CL
6.0
0.01
6.2
6.0
5.8
5.6
0.00
0.02
Kd = 0.030 M
n = 33
0.01
0.02
0.03
CP/CL
0.04
0.05
36
Using Q-Switching to counter Radiation Damping
The advantages of Q-switching (i.e., the rf circuitry is effectively
disconnected) during acquisition are well-known.
Less well-known are the advantages of switching during the sequence.
90% 1H2O 9% 2H2O 1% Ethanol at 500 MHz
Radiation damping has negligible
effects on the (small) ethanol signals.
Magn. Reson. Chem. (2002) 40, S128.
Q-switching
Using the PGSE-Q-Switch Sequence
90% 1H2O at 500 MHz
Δ = 30 ms, δ = 2 ms
Normal PGSE spectra – very distorted
(High-Q is the normal spectrometer condition)
PGSE-Q-Switch spectra
g
With Q-switching we have complete
freedom to move the gradient pulses
within the sequence and can work at
any sample volume.
37
38
PROTEIN ASSOCIATION
Involved in normal physiology and in disease (e.g., cataracts,
Alzheimer’s disease).
Aggregation is the initial step in the crystallisation process.
Proteins are both colloids and polymers.
⇒ need to consider size and electrostatic interactions
39
Aggregation of Lysozyme
Lysozyme - forms a complicated polydisperse system.
+ ++
++
+ ++
++
+ ++
++
+ ++
++
+ ++
++
+ ++
++
+ ++
++
+ ++
++
+ ++
++
+ ++
++
+ ++
++
reasonably
globular
+
+ ++ + +
+
+
++
+
+
+ ++ +
++ ++
concentration
Its aggregation state is sensitive to the solution environment (e.g., salt
concentration, pH).
The spectra of the different oligomeric states overlap.
40
Theoretical
Calculation of
Protein Diffusion
How to Analyse Polydisperse
Protein PGSE Data ?
Monomer/
Oligomer
Hydrodynamics
Aggregate
Distribution
PGSE Measurement
of Protein Diffusion
Crowding/
Obstruction
Effects
Ensemble
Averaging
Ensemble
Averaging
Experimental
Diffusion
Coefficient
Theoretical
Diffusion
Coefficient
Aggregation
Model
41
PGSE of a Polydisperse System
Assume slow exchange between the different oligomeric states w.r.t. Δ.
Multiexponential
Echo signal decay
Number concentration
of the ith oligomer
S ( g ) ∝ ∑ Mwin exp ⎛⎜ −bDi ⎞⎟ exp ⎛⎜⎜ − 2τ T2,i ⎞⎟⎟
i
Sum over the all
oligomeric states
i
⎝ 2444
⎝ 244443⎠
14243 1444
3⎠ 14444
Magnetisation Diffusion
1st Fudge: Neglect relaxation and normalise
⎛
⎞
Mw
n
exp
−
bD
⎜
⎟⎟
∑
⎜
i
i
i
⎝
⎠
S
g
(
)
i
E= ⎛ ⎞=
S ⎜⎝ 0 ⎟⎠
∑ Mwi ni
i
Relaxation
b = γ 2 g 2δ 2 ⎛⎜⎜ Δ−δ
⎝
3
⎞
⎟
⎟
⎠
Probably OK for small oligomers as most
of the signal is from the side-chains whose
motion is reasonably independent of the
overall reorientation rate.
But even highly aggregated protein solutions
give single exponential echo decay.
⇓
Some sort of microscopic population weighted ensemble averaging
of the oligomeric diffusion coefficients to give a single average D (i.e., D ).
42
Ensemble Averaging of the Diffusion
Coefficients
2nd Fudge: expand the exponential
Neglect higher terms
⎛
⎜
⎜
⎜
⎜
⎜
⎜⎜
⎝
2
ln E =−b D + b2 D − D2
W 2
W
W
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
⎞
⎟
⎟
⎟
⎟
⎟
⎟⎟
⎠
weight-averaged
diffusion coefficient
D
W
=
∑ Mwi ni Di
i
∑ Mwi ni
i
PGSE gives ~
D
C
D ).
(actually
W
W
Effects of crowding are
inherently included
43
Modelling the Oligomer Hydrodynamics
Monomers and higher oligomers have complex shapes.
Models (numerical and analytical) do exist.
However, given the quality of the diffusion data and lack of
knowledge of the oligomeric shapes ⇒ assume spherical shapes.
+ ++
++
i=2
+ ++
++
+ ++
++
+ ++
+
+ +++
+ + ++
+
++
+ ++
++
+ ++
++
+ ++
++
i=4
+
+ ++ + +
++ ++
+
+ ++ + +
++ ++
?
+ ++
++
+ ++
++
+ ++
++
+ ++
++
i=1
i=3
+
+ ++ + +
++ ++
+
+ ++ + +
+
+
++
Infinite dilution
D = kT
6πηrS
0
rS
0
D
1
D =3
0
i
i
Diffusion coefficient of i-mer
sphere of
equivalent
volume
44
Importance of Charge Effects I
Isoelectric point ~ pH 11 → net positive charge at normal pH
1.4
308 K
1.2
298 K
1.0
293 K
C
10
2 -1
<D>W × 10 (m s )
Bead model
Approximation 1.6
(theoretical estimate)
1.5 mM Lysozyme 0 M NaCl
288 K
283 K
0.8
0.6
3
4
effective size of the lysozyme decreases as the
pH increases → less obstruction
5
6
pH
7
8
9
45
Importance of Charge Effects II
1.5 mM Lysozyme 0.15 M NaCl
308 K
1.4
1.2
298 K
1.0
293 K
C
10
2 -1
<D>W × 10 (m s )
1.6
288 K
283 K
0.8
0.6
3
4
5
6
pH
7
8
9
Competition between electrostatic screening (D ↑) and aggregation (D ↓).
46
Importance of Charge Effects III
Electrostatic screening complete
above 0.5 M NaCl
1.5 mM Lysozyme 0.5 M NaCl
1.4
308 K
1.2
298 K
293 K
288 K
1.0
C
10
2 -1
<D>W × 10 (m s )
1.6
0.8
283 K
0.6
3
4
5
6
7
8
9
pH
Conditions favourable for aggregation (D ↓) – a DLVO problem.
+
+
+
higher
oligomers
47
Two Models for the Effects of Obstruction
on Diffusion in Lysozyme Solution
0
D1 in 0.5 M NaCl pH 4.6 at 298 K
Current models for
obstruction
do not include:
Non-interacting
1.15
2 -1
1.10
10
D (10 × m s )
1. Aggregation effects
2. Electrostatics
1.20
1.05
decrease in D due
only to obstruction
(Han and Herzfeld, Biophys. J.
1993, 65, 1155-1161)
1.00
(Tokuyama and Oppenheim,
Phys. Rev. E 1994, R16-R19)
0.95
0
1
2
3
4
5
Concentration of Lysozyme (mM)
6
48
The Theoretical Diffusion Coefficient
Combining all the steps and convenient fudges:
Includes concentration effects
(i.e., obstruction)
C
D
W
= ∑αi
i
D10 i
Sum over the different
oligomeric states
Mole fraction
(from aggregation model)
−
1
3
⎛
⎜
⎜⎜
⎝
⎞
⎟
⎟⎟
⎠
fC C = D
W
⎛
⎜
⎜⎜
⎝
fC C
⎞
⎟
⎟⎟
⎠
Simplistic obstruction correction
The inclusion of ensemble averaging makes this equation identical to that
obtained using the fast exchange but without ensemble averaging.
49
Lysozyme Aggregation
1.3
0
Dmonomer
C
10
2 -1
<D>W × 10 (m s )
1.2
corrected
Dmonomer
1.1
1.0
Simulations:
1. Monomer-Dimer
Tokuyama model
0.9
corrected
Disodesmic
0.8
0.5 M NaCl, pH 4.6 at 298 K
0.7
0.0
Crowding
1.0
Aggregation
Kd = 313 ± 555
(corrected)
2. Isodesmic
Ke = 1056 ± 173 Μ-1
(uncorrected)
Ke = 118 ± 12 Μ-1
(corrected)
2.0
3.0
4.0
5.0
6.0
Concentration of lysozyme (mM)
Analysis is wildly wrong if
obstruction is not accounted for.
J. Am. Chem. Soc. (1999) 121, 11503
What Happens at Higher Protein
Concentrations ?
With reasonably small aggregates neglecting relaxation
weighting is ‘reasonable’.
However, with more polydisperse systems it must be
considered.
−1 ⎛ ⎞
D
= ∑αi D10 i 3 f C ⎜⎜⎜C ⎟⎟⎟ exp(− 2τ T2,i ) =
⎝
⎠ 144444244444
3
W ,R i
C
D
W ,R
⎛
⎜
⎜⎜
⎝
fC C
⎞
⎟
⎟⎟
⎠
Relaxation
It has the effect of a high molecular weight filter with a
very broad cut-off.
50
51
What the PGSE experiment will ‘see’
Supersaturated
protein solution
increasing
association
All species are NMR invisible
larger mean
free paths
partially
NMR invisible
NMR invisible
The PGSE experiment provides information
on the smaller aggregates still in solution
52
Probing the Time-Dependence of Aggregation
As aggregation proceeds the larger aggregates
(~ ‘solid’ phase) become ‘NMR invisible’ and
PGSE measures only those still in solution.
The protein remaining in solution diffuses faster
due to less obstruction.
Gradient strength permitting, the relaxation
weighting can be ‘tuned’ to different Mw ranges.
⇒ PGSE provides a means to studying the timedependence of aggregation.
Time-Dependence of Lysozyme Aggregation I
obstruction corrected monomer
diffusion coefficients 1.2
Very suitable conditions for aggregation
pH 6, 0.5 M NaCl at 298 K
C = 3, 5, 6, 7 mM
D1
C=1.5 mM
D~1
1.0
Lysozyme
Concentration
3 mM
5 mM
6 mM
7 mM
0.9
0.8
C
10
2 -1
<D>W,R (10 × m s )
1.1
0
~D1
sigmoidal
time dependence
0.7
0.6
Of the aggregating
species still in solution
0
D
Biophys. J. (2001) 80, 1585.
20
C
W,R
40
60
⎛
⎜
⎜
⎝
⎞
⎟
0 ⎟⎠
C
80 100 120 140 160 180
t (h)
Dt
=
W,R
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1+ e
− D ⎛⎜ t∞ ⎞⎟
⎝
t −t
sigm
⎞
⎟
⎟
⎟
⎟
⎟
⎠
⎠
tS
midpoint of inflection
C
W,R
+ D ⎛⎜ t∞ ⎞⎟
⎝
time scaling
⎠
C
W,R
53
54
Time-Dependence of Lysozyme Aggregation II
Recast using the Stokes-Einstein Equation.
C
Of the aggregating
species still in solution
<Mw>R (Lysozyme Monomer)
0
Crystal Formation
1
2
3
Tetramer
4
5
Lysozyme
Concentration
3 mM
5 mM
6 mM
7 mM
6
0
20
40
60
80 100 120 140 160 180
t (h)
It has been suggested that the critical nucleus is a
tetramer and that the growth unit is an octamer.
55
Time-Dependence of Lysozyme Aggregation III
When the sample tube was removed at the end of the experiment
lysozyme crystals were visible.
Initial solution = 5 mM Lysozyme
Lysozyme crystals are
invisible to PGSE NMR
Shigemi
NMR tube
Using PGSE to Probe the TimeDependence of Aggregation
Lysozyme
D ( t0 )
Concentration
(× 10
-10
C
W
2 -1
ms )
D ( t∞ )
(× 10
-10
C
tsigm
Slope at inflection
(h)
(× 10-17 m2s-2)
Result
W
2 -1
ms )
3 mM
0.89 ± 0.00
1.05 ± 0.00
66.5 ± 0.8
9.4
Small no. large crystals
5 mM
0.72 ± 0.01
1.03 ± 0.00
68.9 ± 1.2
14.8
Small no. large crystals
6 mM
0.65 ± 0.01
1.04 ± 0.00
23.9 ± 0.9
27.4
Many small crystals
7 mM
0.61 ± 0.01
1.03 ± 0.00
41.4 ± 0.9
27.0
Many small crystals
small no. of critical nuclei
long induction period
and large crystals
large no. of critical nuclei
short induction period
and small crystals
56
57
RELATED POSTERS
‰ Fröba et al (350) “Mutual Diffusion Coefficients in Fluids by Dynamics Light Scattering”.
‰ Gädke and Nestle (392) “Apparent Longitudinal Relaxation of Mobile Spins in Thin,
Periodically Excited States”.
‰ Lang et al (356) “Molecular Motions of Calix[4]Arene and Thiacalix[4]Arene in Solution
Studied by NMR Relaxation”.
‰ Marek et al (358) “ESRI Study of Diffusion Processes in Poly(2-Hydroxyethyl Methacrylate)
Gels and Concentrated Solutions”.
‰ Fernandez et al (444) “New Options for Measuring Diffusion in Zeolites by MAS PFG NMR”.
‰ Schönfleder et al (470) “NMR Studies of Diffusion and Pore Size Distribution on Water
Containing Aquifer Rocks and Construction Materials”.
‰ Grinberg (564) “Ultraslow Molecular Dynamics of Organized Fluids: NMR Experiments and
Monte-Carlo simulations”.
‰ Roland et al (584) “Influence of Phase Transitions on the Mobility of Organic Pollutants in
Synthetic and Natural Polymers”.
‰ Sagidullin et al (588) “Water Diffusion through Assymetric Polymer Membranes and
Polyelectrolyte Multilayers”.
58
Acknowledgements
NSW State Government BioFirst Award.
Peter Stilbs, Royal Institute of Technology, Sweden.
Olle Söderman, University of Lund, Sweden.
Fredrik Elwinger, GE Healthcare, Sweden.
Yoji Arata, Water Research Institute, Japan.
Hiroyuki Ide, Ajinomoto, Japan.
Fumihiko Tsuchiya, Applied Biosystems Japan.
Cécile Vigouroux, Royal Institute of Technology,
Sweden.
Markus Wälchli, Bruker Biospin, Japan.
59
Water Suppression
One of the best suppression techniques is the WATERGATE sequence.
Selective π pulse (e.g., a binomial pulse) that
gives a null on the solvent resonance
Small gradient pulses
all other
resonances
WATERGATE
solvent
The WATERGATE sequence resembles a PGSE sequence except
diffusion effects (i.e., signal loss) are minimized.
60
61
WATERGATE
Residual 1H2O peak
Excitation ‘null’
(the width depends
on the selective pulse)
Very flat baseline
(good for integration)
1
Lysozyme (10 mM in 90% H2O)
12
10
8
6
4
ppm
2
0
-2
Suppression of the water peak by a factor of at least 10000.
62
PGSE-WATERGATE
A Hahn-based sequence is preferable to a Stimulated Echo-based
PGSE sequence due to: (1) better removal of the protein resonances due
to relaxation, (2) larger drug signal and (3) no complications from crossrelaxation effects.
Echo signal
S ( g ) = M exp ⎛⎜⎜ −γ 2 g 2 D ( Δ−δ 3) ⎞⎟⎟ exp ⎛⎜ − 2τ T2 ⎞⎟
initial
magnetization
⎝ 424444
0 14444444
⎝
3⎠
4244444444
3⎠ 1444
Attenuation due
to Diffusion
Attenuation due
to Relaxation
PGSE Spectra of Salicylate in BSA Solution
63
The binding of salicylate to albumin governs its transport and tissue distribution.
80 mM salicylate
0.5 mM BSA
90% 1H2O 298 K
500 MHz
Δ = 120 ms, δ = 2 ms
Magn. Reson. Chem. (2002) 40, 391.
The protein background is negligible.
Unless the solvent suppression is excellent, the three proton resonances
appear to give different diffusion coefficients.
Problems with NMR of Strong Signals
64
‘Strong’ samples present particular difficulties to NMR measurements
due to the induction of radiation damping effects.
Radiation damping is in effect a feedback loop between the sample
magnetization and the rf coil/circuitry.
Static field strength
Radiation damping
⎡ μ0γ 3 h 2 B0 c A
⎤
1
=⎢
RRD =
χ water ⎥ηQ
rate constant
TRD ⎣
8kT
⎦
Often: TRD << T1 !
Filling factor
‘Quality’ factor
of rf coil
Gets worse in better spectrometers and more sensitive probes (e.g.,
cryoprobes). It affects both T1 and T2 (and therefore lineshape) and has
very deleterious effects on pulse sequences – esp. PGSE sequences.
Reducing B0, η or Q is not a solution when trying to detect small
resonances.
Radiation Damping Effects on Relaxation
Magnetization
recovery after an inversion pulse at 300 MHz
M
0
Mz (normalized)
100
Radiation
damping
T1RD~ 0.4 s
Protein
0
Water
without radiation damping
T1 ~ 3.7 s
(NB The relaxation of the protein is unaffected by
radiation damping due to its much smaller signal)
-M0
-100
0
5
10
Time after inversion (s)
The non-linear behaviour of the water makes pulse sequence design
very difficult.
65
Radiation Damping and the PGSE Sequence
Radiation damping effects are ~ negligible when the magnetization
is gradient encoded (the vector sum of the net magnetization is zero).
Susceptible to radiation damping
The effects of radiation damping during t1 are constant but change
during t2 in a complicated manner on the gradient parameters (δ, g, Δ).
66
67
Mild Radiation Damping Effects on PGSE
Small 1H2O sample at 300 MHz
0
Effects of RD
Disastrous
ln(E)
-1
-2
Significant
-3
-4
0.0
0.5
1.0
1.5
2 2 2
2.0
9
-2
γ g δ (Δ-δ/3) (× 10 m s)
J. Magn. Reson. (2001) 150, 49.
2.5
Negligible
Unfortunately, the best sequence
is generally impracticable.
68
Using Q-Switching
The advantages of Q-switching (i.e., the rf circuitry is effectively
disconnected) during acquisition are well-known.
Less well-known are the advantages of switching during the sequence.
90% 1H2O 9% 2H2O 1% Ethanol at 500 MHz
Radiation damping has negligible
effects on the (small) ethanol signals.
The initial part of the signal is free
of radiation damping effects.
Magn. Reson. Chem. (2002) 40, S128.
Q-switching
Using the PGSE-Q-Switch Sequence
90% 1H2O at 500 MHz
Δ = 30 ms, δ = 2 ms
Normal PGSE spectra – very distorted
(High-Q is the normal spectrometer condition)
PGSE-Q-Switch spectra
g
With Q-switching we have complete
freedom to move the gradient pulses
within the sequence and can work at
any sample volume.
69
70
71
Diffusive Diffraction
(between parallel PLANES)
1
T = 318.4 K
δ = 2 ms Δ = 2 s
0.1
The position of the ‘diffractive’ minima
are related to the separation, R.
1
E
0.01
Restricted
Free
9
10
20
2 2
Free
diffusion
gz
2 -1
D = 3.6 × 10 m s
0
two separate
peaks in the
NMR spectrum
2
30
40
9
-2
γ g δ (Δ-δ/3) (10 m s)
50
H2O
R ~ 100 μm
Restricted
diffusion
(not to scale)
Glass