Spin Diffusion and Cross Relaxation inCPMAS NMR of Proteins and Peptides.

Stowe Winter School, 2013

Using NMR relaxation in solid proteins to study molecular dynamics

dynamics: straightforward separation of internal from overall motion

problem: spin diffusion in uniformly enriched proteins can mask most interesting dispersion in dynamics

activation parameters: wider accessible temperature range

15CSA NΘ

NH dipoleΘ

NH

15N T1 (s) vs. sequence

05

101520253035404550

0 10 20 30 40 50 60 70 80

N CC

H D

O

N CC

H D

O

N CC

H D

O

re s id u e ire s id u e i-1 res id u e i+ 1

R D R D R D

model for 15N-15N spin diffusion in a deuterated back-exchanged protein

240 200 160 120 80 40 0 ppm

GALala, leu amidesT1 ~ 100 s

gly amineT1 ~ 0.4 s

15N—1H 15N1H3+

Zibby Fry

184 180 176 172 168 ppm60 50 40 30 20 ppm 8000 4000 0 -4000 -8000 Hz

244 K

214 K

154 K

130 K

110 K

95 K

13C 13C 1H

H2O

13C/15N glycyl-alanyl-leucine spectra vs. T

CO CH2 CH3

ala CH3

leu CH3 synthesized u-15N, u-13C/15N and also

u-13C/15N/2H

Zibby Fry

CH

δ

S1 S2

1 2SS SI I I

S z1 S z2 D D D23 23 sI I Iz x D D

H ω S ω S H H HδS dS H H

12

12

2 3δ dd δ

when |d| << δ, S1S2

flip-flops are quenched

1 2

2So

2 123S S

12 2 12

γμ P cosθ4π rd P θ

d

cos

15N115N2

1H

1H1 2S Sr

Isotropic shift difference δ truncates SSDH

15N115N2

1H

1H

1 2SS SI I I

S z1 S z2 D D D23 23 sI I Iz x D D

H ω S ω S H H HδS dS H H

1 2S Sr

12

12

2 30 d

d 0

where δeff ~ 0, flip-flops

are un-quenched

δeff ~ 0 where lines overlap

spectral overlap un-truncates SSDH

δ

S1 S2

δ

S1 S2

15N115N2

1H

1H

where δeff ~ 0, flip-flopsare un-quenched

simple theory of spin diffusion Heinrichs, Linder and HewittJCP (85), 1986, pgs. 7077-86

1 1 2 2S S S Sg 0 f ω ω f ω ω dω

overlap function

2ij ij

πR d g(0)2 spin diffusion rate

A A A A

B B B B .

z z z z1A AB AB

z z z zAB 1B AB

M M M MR R RdM M M MR R Rdt

2D spin diffusion experiment

15N

1H

CP π2

decouple

t1 t2tπ2

π2

x y

y+x-x

decoupleCP

AA BB

AA BB

2AB BA

1Δ R R21σ R R2

D Δ R R

A A A

B B B

oZ Z ZAA AB

oZ BA BB Z Z

M M MR t R tM R t R t M M

ddt

2D spin diffusion experiment

15N

1H

CP π2

decouple

t1 t2tπ2

π2

x y

y+x-x

decoupleCP

σt o o o o oA CzA 1 zA A 1 zA zB B 1 zB zA

σt o o o o oB CzB 1 zB B 1 zB zA A 1 zA zB

σtzA 1

n RΔM t ,t e cosh Dt sinh Dt M cos ω t M sinh Dt M cos ω t M MD Dn RΔM t ,t e cosh Dt sinh Dt M cos ω t M sinh Dt M cos ω t M MD D

M t ,t e co

and

o o o o oA CzA A 1 zA zB B 1 zB zA

σt o o o o oB CzB 1 zB B 1 zB zA A 1 zA zB

n RΔsh Dt sinh Dt M cos ω t M sinh Dt M cos ω t M MD Dn RΔM t ,t e cosh Dt sinh Dt M cos ω t M sinh Dt M cos ω t M MD D

2D spin diffusion experiment

15N

1H

CP π2

decouple

t1 t2tπ2

π2

x y

y+x-x

decoupleCP

o o ozA zB z A Bassuming M M M and n n 1

zA 1 σt CA 1 B 1o

z

zB 1 σt CB 1 A 1o

z

M t ,t RΔe cosh Dt sinh Dt cos ω t sinh Dt cos ω tD DMM t ,t RΔe cosh Dt sinh Dt cos ω t sinh Dt cos ω tD DM

2D spin diffusion experiment

15N

1H

CP π2

decouple

t1 t2tπ2

π2

x y

y+x-x

decoupleCP

σt

σtC

σt

σtC

ΔAA t e cosh Dt sinh DtDRAB t e sinh DtD

ΔBB t e cosh Dt sinh DtDRBA t e sinh DtD

Cif R Δ

(σ D)t

σ D t

AA t BB t AB t BA t eAA 0 BB 0

AA t BB t AB t BA t eAA 0 BB 0

auto and cross peak intensities

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 20 40 60 80

peak

inte

nsity

mixing time (s)

15N-15N 2D spin diffusion measurements on u-15N GAL

A B

X

BBAA

ABBA

15N—1H15N1H3

+

BB

AA

AB

BA

15N @ 80 MHz, νr = 18kHz1t / T

total(AA BB AB BA) M e

SD2t / Ttotal(AA BB AB BA) M e

0

1000

2000

3000

4000

5000

6000

0 5 10 15

peak

inte

nsity

mixing time (s)

15N-15N 2D spin diffusion measurements on u-15N GAL

A B

X

BBAA

ABBA

BB

AA

AB

BAXX

XAXX

XA

15N @ 80 MHz, νr = 7 kHz

SD2t / Ttotal(AA BB AB BA) M e

1t / Ttotal(AA BB AB BA) M e

15N—1H15N1H3

+

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 20 40 60 80

7 kHz11.5 kHz15 kHz18 kHz

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 20 40 60 80

7 kHz11.5 kHz15 kHz18 kHz

ln(A

A+BB

+AB+

BB)

time (s)

T1 decay TSD decay

time (s)

TSD and T1 depend on MAS rate

ln(A

A+BB

-AB-

BB)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 20 40 60 80

7 kHz11.5 kHz15 kHz18 kHz

R C (

s-1)

2π·103 /ωr (s)

1/TSD vs. 1/ωr TSD decay

time (s)

TSD and T1 depend on MAS rate

from initial slope

ln(A

A+BB

-AB-

BB)

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 2 4 6 8

7 kHz11.5 kHz15 kHz18 kHz

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 20 40 60 80

7 kHz11.5 kHz15 kHz18 kHz

ln(A

A+BB

+AB+

BB)

TSD decay TSD decay

time (s)

TSD decay looks like classic diffusion barrier limited kinetics

ln(A

A+BB

-AB-

BB)

1/ 2time (s )Root-time decay has same form as for nuclei

relaxed by randomly distributed paramagnetic impurities

Due to distribution of N-N vectors in rotor frame

15N spectra of gly-ala-leu 15N/13C/2H crystallized from D2O/H2O

leu NH(D)ala NH(D)

gly NH3+

(D)

138 136 134 132 130 128 126 124 122 120 118 116 ppm 34 33 32 31 30 29 28 27 26 ppm

H

H H

D

D D

short CP

long CP

138 134 130 126 122 118 ppm 34 32 30 28 26 ppm

large secondary isotopic chemical shifts

120

121

122

123

124

125

126

127

128

129

131 130 129 128 127 126 125 124 123 122 121 120 ppm

AHAD

BHBD

Amide-Amide spin diffusion is 1H driven

2D 15N-15N30 s spin exchangespectrumfor 50/50

GAL

AH-BHcrosspeaks

120

121

122

123

124

125

126

127

128

129

131 130 129 128 127 126 125 124 123 122 121 120 ppm

AHAD

BHBD

Amide-Amide spin diffusion is 1H driven

2D 15N-15N30 s spin exchangespectrumfor 50/50

GAL

AH-BDcrosspeaks

120

121

122

123

124

125

126

127

128

129

131 130 129 128 127 126 125 124 123 122 121 120 ppm

AHAD

BHBD

Amide-Amide spin diffusion is 1H driven

2D 15N-15N30 s spin exchangespectrumfor 50/50

GAL

AD-BHcrosspeaks

120

121

122

123

124

125

126

127

128

129

131 130 129 128 127 126 125 124 123 122 121 120 ppm

AHAD

BHBD

Amide-Amide spin diffusion is 1H driven

2D 15N-15N 30 s spin exchangespectrumfor 50/50

GAL

No AD-BDcrosspeaks

x

x

N CC

H D

O

N CC

H D

O

N CC

H D

O

re s id u e ire s id u e i-1 res id u e i+ 1

R D R D R D

15N-15N spin diffusion should be detected in the 15N T1 relaxation

240 200 160 120 80 40 0 ppm

GALala, leu amidesT1 ~ 100 s

gly amineT1 ~ 0.4 s

15N—1H 15N1H3+

15N T1 independent of deuterium exchange

T1 amine = 0.4 s T1 amide = 800 s as r ∞

Amide 15N T1 MAS independent if gly-15N is removed

T1 amine = 0.4 s T1 amide = 800 s as r ∞

о ala in 2-15N GALо leu in 2-15N GAL

Nuclear Cross-Relaxation Induced by Specimen RotationE.R. Andrew et al, Physics Letters (4) 1963, pg 99-100

31P spectrum of PCl5

6PCl4PCl

1H T1 and methyls - Slichter, Douglass ......

13C T1 and methyls – White, Law ......

2H T1 with relaxation sinks as CD3 – Gan, Wimperis.....

Relaxation by spin diffusion to relaxation sinks

2 4 0 2 0 0 1 6 0 1 2 0 8 0 4 0 0 p p m

CSA enabled 15N-15N spin diffusion

apparent R1 fast MAS ~ R1 NH dipolar

8000 Hz

amide amine

overlap ~ constant at slow

MAS rates

R iso3R

overlap decreases as 1 once ω Δωω

2.2 kHz

5.0 kHz

12 kHz

spin diffusion under MAS

Kubo and McDowell

rimω t12 2 12 12 m

m 1, 2d P cosθ (t) d A ed

Fourier components of SSDH

Suter and Ernst (32) 1985, 5608-27

2m m Ry r

m 1, 2SD2

Ry r Ry r Ry r Ry r

1 d A A K (mω )Td K (ω ) K ( ω ) K (2ω ) K ( 2ω )15

RyK (ω)zero quantum lineshape

powder sum

240 200 160 120 80 40 0 ppm

amine isotropic

zero quantum amplitudes are approximately the overlap integral for the amine line with the amide

sidebands

26

o2 4 2

N

1ij

μ γ1 πT 2 160π

ij ij ij

ij

d f fr T1 amine = 0.4 s T1 amide = 800 s as r ∞

0100200300400500600700800900

0 5 10 15 20 25 30

H dipolarD dipolarCSA onlyexperimental

scaled overlap function fij

vr (kHz)

TSD (s)

0100200300400500600700800900

0 5 10 15 20 25 30

CSA onlyH dipolarD dipolarexperimental

vr (kHz)

Amide T1 T1 (s)

BA 1B1B

1 R RT

T1 computed from amine overlap with r, 2r sidebands of amide

T1 amine = 0.4 s

T1 amide = 800 sas r ∞

26

o2 4 2

N

1ij

μ γ1 πT 2 160π

ij ij ij

ij

d f fr

amide linewidth

~2500 Hz

15N

1H

increase overlap to decrease T1

π2

π2

1 rω ω decouple

trelax

15N

1H

increase overlap to decrease T1

π2

π2

1 rω ω decouple

trelax

trelax (s)0.0 0.1 0.2 0.5 1 2 5 10 50 100 200 400 1000

ωr/2π = 7.0 kHz

15N

1H

increase overlap to decrease T1

π2

π2

1 rω ω decouple

trelax

trelax (s)0.0 0.1 0.2 0.5 1 2 5 10 50 100 400 1000 3000

ωr/2π = 20.0 kHz

MAS and anisotropic relaxation

H Nm m

WI

WI

W2

Wo

WS

WS

β

r oω t γ

Bo

Mθ15N

1H

I S0I S

1S r 2 2I S

2

I S0I S

1S 2 2I S

2

WτW f(β,ω t γ) and g(τ,ω) 1 ω τ

W

WτW average over t, β g(τ,ω) 1 ω τ

W

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 t

powder sum decay

decay with average T1

longest T1 component

shortest T1 component

T1

T1

T1

◦

time

11/ Te

Model like 13C T1 relaxation of a methyl group under MAS

i) internal rotation dominantii) temperature dependence of known

2

CH

Θ 69.1 f or C H vector S 0.93Θ 90 f or H H vector r = 1.106 Å

I I I I I S I S I SI I I 1 2 S 0 1I 2

SS SS I S I S I SSS S 1 2 I 0 1S 2

I S I SI S I 2 0

I S I SSI S 2 0

R 2(n 1)(W W ) n (W 2W W )R 2(n 1)(W W ) n (W 2W W )R n (W W )R n (W W )

I I

S S S

oZ Z ZII I I S

oZ Z ZSI SS

M M MR t R tM M MR t R t

ddt

θ

ΘBo

1H13C

D. Torchia and A. Szabo JMR 49, (1982) pg 107

Ottiger and Bax, JACS 121

(1999) pg 4690

AE / RT1 2 oτ 4τ τ e

τ

dipolar T1 for a 15N-1H pair freely diffusing on surface of a cone

2 12Cone

11 24

1

Θ f rom S cosΘ 1 cosΘ1τ τ f romT

2 2 22I S

1 I s61 I SMAS

42 I s

21 S

42 S

21 I S

42 I S

0.075γ γ1 {g(τ,ω ω )sin 2ΘT rg(τ ,ω ω )sin Θ3g(τ,ω )sin 2Θ3g(τ ,ω )sin Θ6g(τ,ω ω )sin Θ6g(τ ,ω ω )sin Θ}

15N

1H1H

Θ

15CSA NΘ

NH dipoleΘ

NH

-6-30369

12151821

-17 -15 -13 -11 -9 -7 -5 -3 -1

NOEamide R1amine R1

θ

Θ Bo1H

15N

15N

1H1H

Θamide in“fast” motion

limit

τc ~ 10-11s

amine in“slow” motion

limit

τc ~ 10-7s

log(τc)

R 1 (s

-1) o

r NOE

13C1H 1H

1H

13C T1 vs. MAS rate for 13C, 15N enriched GAL

150 ppm

13C O

R1 enhancementsfor fast vs. slow MAS

rates could be used tomeasure CH3 to CO rCC

or… compare natural abundance with a single 13CH3

330 s

330 s

170 s 67 s

70 s

8 s

13 12SD 1 C 1 C

1 1 1T T T

6 calcsd sdT (s) 1/ r T (s)

L(CO) 88 0.00026 183A(CO) 8.4 0.0044G(CO) 83 0.00045 81

or… compare natural abundance with a single 13CH3

303 s

21 s

111 s

2 s

13 12SD 1 C 1 C

1 1 1T T T

6 calcsd sdT (s) 1/ r T (s)

L(Ca) 175 0.0003 123A(Ca) 1.8 0.084 0.5

but some R1 enhancements are not so simple to explain

17 s

0.2 s

6 calcsd sdT (s) 1/ r T (s)

L(Cg) 5.2 0.0003 122L(Cd1) 2.1 0.00048L(Cd2) 2.2 0.00039A(Cb) 0.7 0.00007

0.3 s

0.4 s

0.6 s4 s 0.5 s

0.8 s

2ij ij

sd

1 πR d g(0)T 2

Yale University• Van Phan• Zibby Fry• Suvrajit Sengupta• Victoria Mooney• Lacey Ketzner• Shan Kuang• Josh Hernandez• Chris Bennett• Hannah Fuson• Catalina Espinosa• Josh Karli

coworkers:

Funding: NSF Experimental Physical Chemistry

Agilent Foundation