SSNMR -- Intermediate Exchange Conformational Dynamics of Biopolymers: “A few of my favorite things”: Chemical Exchange by: Bloch-McConnell,

Stochastic Liouville and Redfield theory

Ann McDermott Winter School in Biomolecular SSNMR

Jan 10, 2018, Stowe, VT

1/9/18 2

Effects on NMR Spectra: Local, Angular Dynamics on Timescales from sec to ps

DHFR Compression: Wright et al, Chem Rev 2006

Krushelnitsky & Reichert Prog NMR 2005

1/9/18 3

€

! I LAB = ˆ I x

LAB ˆ I yLAB ˆ I z

LAB( )

Fluctuations in the Local Field Driven by Reorientation and Changes in Environment

1/9/18 4

Combined Effects of Relaxation, Coherent Evolution, and Chemical Exchange: The Bloch–McConnell

Equations

€

∂M1∂t

∂M2∂t

#

$

% %

&

'

( (

= −iΩ1 + k + R2 −k

−k −iΩ+ k + R2

#

$ %

&

' ( M1

M2

#

$ %

&

' (

€

! m (t) = Ue−DtU−1 ! m (0)

1/9/18 5

Effect of Evolution and Relaxation on Magnetization

€

∂M1∂t

∂M2∂t

#

$

% %

&

'

( (

=−iΩ1 − R2 0

0 −iΩ2 − R2

#

$ %

&

' ( M1

M2

#

$ %

&

' (

Mi ≡ Mxi + iMy

i

∂M1∂t = − iΩ1 + R2( )M1

Mi(t) = Mi(0)e− iΩ i +R2( ) t

M(t) = Mi(t)i∑

Matrix is diagonal, two separable, solvable equations.

A: Def

1/9/18 6

Chemical Exchange Intermediate Exchange Timescale

€

Keq =C2[ ]C1[ ]

=k12k21

€

C1⇔k21

k12C2

1/9/18 7

Effects of Exchange on Chemical Concentrations

€

C1⇔k21

k12C2

∂C1∂t = −k12C1 + k21C2

∂C2∂t = +k12C1 − k21C2

∂C1∂t

∂C2∂t

%

&

' '

(

)

* *

=−k12 k21k12 −k21

%

& '

(

) * C1C2

%

& '

(

) *

1/9/18 8

Effects of Chemical Exchange on Magnetization

€

∂M1∂t

∂M2∂t

#

$

% %

&

'

( (

=−k12 k21k12 −k21

#

$ %

&

' ( M1

M2

#

$ %

&

' (

1/9/18 9

Solving for Concentration over Time by Matrix Methods –Chemical Exchange Only

€

∂! c ∂t = −K! c

! c =C1C2

$

% &

'

( ) ;K =

k12 −k21−k12 k21

$

% &

'

( ) ;

! c (t) = e−Kt! c (0)D = U−1KUK = UDU−1

eKt = Ue−DtU−1

! c (t) = Ue−DtU−1! c (0)

1/9/18 10

€

K! u i = λi! u i;! u i =

ui1ui2

#

$ %

&

' (

D =λ1 00 λ2

#

$ %

&

' (

U =u11u12

u21u22

#

$ %

&

' (

K =a bc d#

$ %

&

' (

K − λI =a − λ bc d − λ

= a − λ( ) d − λ( ) − bc = 0

λ =a + d2

±a + d( )2 − 4 ad − bc( )

2

Eigenvalue Approach to Diagonalization

A: Meaning of λ

1/9/18 11

€

λ =a+ d2

±a+ d( )

2− 4 ad −bc( )2

X =a bc d$

% &

'

( ) =

0 1−1 0$

% &

'

( )

K'=a bc d$

% &

'

( ) =

k −k−k k$

% &

'

( )

K =a bc d$

% &

'

( ) =

k12 −k21−k12 k21

$

% &

'

( )

Let’s Look at Three Cases

1/9/18 12€

λ =a + d2

±a + d( )2 − 4 ad − bc( )

2

X =a bc d$

% &

'

( ) =

0 1−1 0$

% &

'

( )

λ = 0 ±0( )2 − 4 0 +1( )

2= −1

Try to Solve a Nonphysical Example

1/9/18 13

€

λ =a + d2

±a + d( )2 − 4 ad − bc( )

2

K'=a bc d$

% &

'

( ) =

k −k−k k$

% &

'

( )

λ =2k2

±2k( )2 − 4 k 2 − k 2( )

2= k ± k

D =2k 00 0

$

% &

'

( )

Solve the Eigenvalue Problem for Symmetric Two Site Chemical Exchange

1/9/18 14

€

K! u i = λi! u i

k −k−k k$

% &

'

( ) ! u 1 = 2k! u 1

ku11 − ku12 = 2ku11;−ku11 + ku12 = 2ku12;u11 = −u12k −k−k k$

% &

'

( ) ! u 2 = 0k! u 2

ku21 − ku22 = 0ku21;−ku21 + ku22 = 0ku22;u21 = u22

U =u11u12

u21u22

$

% &

'

( ) =

1 1−1 1$

% &

'

( )

U =a bc d$

% &

'

( ) ;U−1 =

1U

d −b−c a$

% &

'

( ) ;U−1 =

121 −11 1$

% &

'

( )

Solve for the Transformation Matrices U and U-1

1/9/18 15

€

U =a bc d"

# $

%

& '

U−1 =1U

d −b−c a"

# $

%

& '

Homework: Confirm that this expression yields the inverse generally

Homework: Confirm in this specific case that this is the correct transformation matrix

€

K =k −k−k k#

$ %

&

' (

U =1 1−1 1#

$ %

&

' (

Homework: Write equations for the concentration over time and check their validity

1/9/18 16€

λ =a + d2

±a + d( )2 − 4 ad − bc( )

2

K =a bc d$

% &

'

( ) =

−k12 k21k12 −k21

$

% &

'

( )

λ =−k12 − k21

2±

−k12 − k21( )2 − 4 k12k21 − k12k21( )2

λ = −k ± kk ≡ k12 + k21

Arbitrary Two-Site Exchange Can Also be Solved

See Bernasconi 1976 “Relaxation Kinetics” Academic Press

1/9/18 17

Note that K is not symmetric; is it always diagonalizable for larger kinetic schemes (3-component or N-component kinetic system)?

Connected with assumption of microscopic reversibility, a similarity transformation U, below, can be used to symmetrize K (to form K’), assuring us that it is always diagonalizable.

Exercise: Show that if U is defined as below, it renders the matrix of rate constants symmetric. A real symmetric matrix is always diagonalizable.

€

Cnknm = CmkmnUij = P iδij

K'ij = KijK ji = Pj PiKij

1/9/18 18

Combined Effects of Relaxation, Coherent Evolution, and

Chemical Exchange

€

∂M1∂t

∂M2∂t

#

$

% %

&

'

( (

= −iΩ1 + k + R2 −k

−k −iΩ+ k + R2

#

$ %

&

' ( M1

M2

#

$ %

&

' (

€

! m (t) = Ue−DtU−1 ! m (0)

1/9/18 19

K+R+ iL =iΩ+ R2 + k −k

−k −iΩ+ R2 + k

#

$

%%

&

'

((

λ = R2 + k( )± k2 −Ω2

U = a bc d

#

$%

&

'(;U−1

ij =1

ad − bcd −b−c a

#

$%

&

'(

a = i(Ω+ Ω2 − k2 )

b = i(Ω− Ω2 − k2 )c = d = k

M (t) =M1(t)+M2 (t) =a+ c( ) d − b( )ad − bc

e−λ1t +b+ d( ) a− c( )ad − bc

e−λ2t

Question: what is the meaning of Eigenvalues, the Imaginary vs. Real part

General Questions

1/9/18 20

Meaning of lambda values? Real part? Imaginary part?Meaning of the coefficients in front of the exponent?

Analogy with Single Spin, no Exchange

M (t) =M1(t)+M2 (t) =a+ c( ) d − b( )ad − bc

e−λ1t +b+ d( ) a− c( )ad − bc

e−λ2t

λ = R2 + k( )± k2 −Ω2

Mi (t) =Mi (0)e− iΩi+R2( )t

Questions about Fast Limit

1/9/18 21

What is the frequency when k>>Ω? What is/re the number of lines when k>>Ω?What is/are the linewidths when k>>Ω?

M (t) =M1(t)+M2 (t) =a+ c( ) d − b( )ad − bc

e−λ1t +b+ d( ) a− c( )ad − bc

e−λ2t

λ = R2 + k( )± k2 −Ω2

λ ~ R2;λ ~ R2 + 2k

Questions about Slow Limit

1/9/18 22

What is the number of lines when k<<Ω?What is/are the frequencies when k<<Ω? What is/are the linewidths when k<<Ω?

M (t) =M1(t)+M2 (t) =a+ c( ) d − b( )ad − bc

e−λ1t +b+ d( ) a− c( )ad − bc

e−λ2t

λ = R2 + k( )± k2 −Ω2

λ ~ R2 + k( )+ iΩ;λ ~ R2 + k( )− iΩ

Differential Equation Describing Precession, Chemical Exchange, and Relaxation

€

∂M1∂t

∂M2∂t

#

$

% %

&

'

( (

=−iΩ1 − k12 − R2 k21

k12 −iΩ2 − k21 − R2

#

$ %

&

' ( M1

M2

#

$ %

&

' (

€

Keq =C2[ ]C1[ ]

=k12k21

€

C1⇔k21

k12C2

€

S fast (ω) ≈2R2

R22 + ω( )2

Sslow(ω) =k + R2

R2 + k( )2 + Ω−ω( )2+

k + R2R2 + k( )2 + Ω+ω( )2

1/9/18 24

Amplitudes in More Detail: Slow Exchange: k<<Ω

€

M(t) = M1(t) + M2(t) =a + c( ) d − b( )ad − bc

e−λ1t +b + d( ) a − c( )ad − bc

e−λ2t

λ1,2 = R2 + k( ) ± k 2 −Ω2 ≈ R2 + k ± iΩ

a = i(Ω+ Ω2 − k 2 ) ≈ 2iΩ

b = i(Ω− Ω2 − k 2 ) ≈ 0c = d = k

M(t) = M1(t) + M2(t) = e−λ1t + e−λ2t = e− iΩte−(R2 +k )t + eiΩte−(R2 +k )t

S(ω) =k + R2

R2 + k( )2 + Ω−ω( )2+

k + R2R2 + k( )2 + Ω+ω( )2

A: Meaning of Ampl Re vs Im

1/9/18 25

Amplitudes in More Detail: Fast Limit: k>>Ω

€

M(t) = M1(t) + M2(t) =a + c( ) d − b( )ad − bc

e−λ1t +b + d( ) a − c( )ad − bc

e−λ2t

λ1,2 = R2 + k( ) ± k 2 = R2 + 2k( ),R2a = i(Ω+ Ω2 − k 2 ) ≈ −k

b = i(Ω− Ω2 − k 2 ) ≈ kc = d = k

M(t) = M1(t) + M2(t) ≈ 0e−λ1t + 2e−λ2t ≈ 2e−R2t

S(ω) ≈ 2R2R2

2 + ω( )2

1/9/18 26

Slow Exchange Limit Contributions to Linewidth

€

K +R+ iL =iΩ+ R2 + k −k

−k −iΩ+ R2 + k$

% &

'

( )

λ = R2 + k( ) ± k 2 −Ω2

k <<Ω

λ ≈ R2 + k ± iΩ

1/9/18 27

Fast Limit Contributions to Linewidth

K +R + iL =iΩ+ R2 + k −k

−k −iΩ+ R2 + k

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

λ = R2 + k( )± k 2 −Ω2

k >>Ωλ ≈ R2 + 2k( ), R2( )x <<1; 1+ x ≈1+ x

2+ ...

λ ≈ R2 + 2k( ), R2 +Ω2

2k( )

1/9/18 28

Exchange Broadening Interference of Conformational Exchange with Shift Evolution

•  Slow Limit scales with k or 1/τc •  Fast Limit scales with 1/k or τc •  Coalescence k~Δωiso /sqrt(2) •  The ability to probe this broadening allows

one to document rare events (poorly populated species) and study the rates of processes. In contrast to R1 measurements, its sensitive to timescales from high nanosecond to mid-millisecond, and therefore to important functional events in binding recognition folding and catalysis.

1/9/18 29Sharon Rozovsky et al, JMB 2001

Triosephosphate Isomerase: Lid Opening ~ Turnover

1/9/18 30

General Solution: Numerical Simulations of SLE

€

∂! m ∂t = − iL + R + K( ) ! m

∂! ρ ∂t = − iL + R + K( ) ! ρ

L = H,ρ[ ]! ρ (t0 + t) = e− iL(t )+R+K( ) ! ρ (t0)

Numerical integration of the (linearized, Taylors expansion) equations, evaluation of derivatives Spinevolution (Vershtort and Griffin JMR 2006) …Asymmetric N site hop model subsequently added. SPINACH (Kuprov)

1/9/18 31

e.g. Deuterium Powder Relaxation, Exchange: Wittebort Olejniczak and Griffin 1987,

1/9/18 32

CSA Tensor Orientation Change

See Solum Zilm Michl Grant 1983

Analysis according to above 2 site hop theory– but in place of two environments/isotropic shifts, the two alternative “sites” correspond to two angles of the truncated CSA , keeping time averaged terms that commute with B0. Each crystal orientation analyzed separately and then powder pattern constructed.

1/9/18 33

Intermediate Exchange Powder Lineshapes

dtk

dttP 1)( =

H H

1/9/18 34

Chemical Exchange During MAS

δ  is the tensor width : σzz- σiso

η  is the asymmetry : (σxx - σyy) / δRZ(α)RY’(β)RZ’’(γ) transforms to the Rotor Fixed Frame at t=0

from the Laboratory Fixed Frame

€

ω t( ) =ω iso + δ C1 cos γ +ωrt( ) + C2 cos 2γ + 2ωrt( ) + S1 sin γ +ωrt( ) + S2 cos 2γ + 2ωrt( )[ ]

C1 = −22sin 2β( ) 1+

η3cos 2α( )

)

* +

,

- .

C2 = −12sin2 β − η

31+ cos2 β( )cos 2α( )

)

* +

,

- .

S1 = −η 23sin β( )sin 2α( )

S2 = −η3cos β( )sin 2α( )

1/9/18 35

Exchange during MAS: Linewidths f(ωr)

Figure 2 A Schmidt and S Vega J Chem Phys 1987 Linewidths for a system Undergoing 2 site flips: Floquet Treatment vs. Perturbation Approx.

Suwelack Rothwell and Waugh ‘80 €

Rex ≈δ 2

151+

η2

3%

& '

(

) *

τ c1+ 4ωr

2τ c2( )

+2τ c

1+ωr2τ c

2( )-

.

/ /

0

1

2 2

1/9/18 36

Chemical Exchange, Tensor Reorientation Interferes with MAS

See Frydman et al 1990 and Schmidt et al 1986

1/9/18 37

MAS Lineshapes: Sensitivity to Near Intermediate Exchange Conditions

Kristensen et al JMR

Characterizing the Exchange Linewidth

1/9/18 38

Sometimes Rex can be extracted from the spectrum….. but more practically, with other contributions to linewidths, and irregular lineshapes, Rex can be obtained from the decay of echo-refocused magnetization. Either an echo (single pi pulse), a CPMG train, or a spin lock refocuses the chemical shift, to allow an accurate measurement of relaxation, and provides experimentally controllable variables, such as a field strength, frequency. Exchange processes will cause failure to refocus, decay of the echo with increasing delay (in addition to R2

o).

y

y

R1ρ : Relaxation During SpinlockDEC

20

15

10

5

0

intensity

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5kHz

2.0

1.5

1.0

0.5

0.0

mag

netiz

atio

n

80006000400020000time (msec)

Inte

nsity

Time (ms) 8000Freq. (kHz)

4000ms

0ms

near resonant, in phase, spin lock, variable strength and length

Measurements Reflect Sum of R2o and Rex

If R2 is known or predicted, Rex can be extracted an analyzed. They are of comparable order of magnitude in many cases.

1/9/18 40

R1ρ = R1cos2θ + R2°sin2θ + Rexsin2 θRex : Chemical Exchange modulates an interaction (e.g. Isotropic Shift)θ  : Effective Field AngleR1, R2° : in the absence of exchange

Separating R2 from Rex Refocusing time (field strength for T1rho, interpulse pulse frequency for CPMG) can be a tool to discern the rate constant, if they are of comparable magnitude to the rate constant! For example, if τc~1 ms, and the echo delay is 10 ms, magnetization will fail to recover. With a 100us delay echo (done 100 times) the magnetization will largely recover (Carr Purcell Meiboom Gill sequence Rev. Sci. Instrum. 29, 688 (1958);) will succeed. A rare chemical event would causes dephasing only during the 100us echo element it occurs in, but would not accumulate over the whole 10 ms period.

1/9/18 41

CPMG

T1rho

Lab Frame vs. Rotating Frame Exchange Relaxation �Fast Limit (Redfield) of Asymmetric 2 Site Hop

1/9/18 42

Rex = p1p2ΔΩ2 τ c1+ωe

2τ 2c( )

Rex = p1p2τ cΩ2

High applied field asymptote of Rex is zero, obtaining a R2

o estimate.

Free Procession

During Spin Lock (T1rho)

1/9/18 43

€

A = e iH ot( ) Ae −iH ot( )

€

H = H0 +H1

€

∂σ∂t

= −i H,σ[ ]

€

∂σ ∂t

= −i H 1,σ [ ]

Relaxation Due to Rapid Angular Fluctuations

H0 is the average (constant). H1 fluctuates and averages to 0.

Move to a frame where the action of H0 is the invisible (overbar). How do the H1 fluctuations, which averages to 0, affect the observables?

Generating Second Order Expressions

1/9/18 44

€

∂σ ∂t

= − H 1 t( ), H 1 t'( ), σ t '( )−σ eq( )[ ][ ]

0

t

∫ dt '

€

∂σ ∂t

= −i H 1,σ [ ]

€

σ t( ) =σ 0( ) − i H 1 t'( ),σ t '( )[ ]

0

t

∫ dt '

Lab Frame vs. Rotating Frame Exchange Relaxation �Redfield of Asymmetric 2 Site Hop Intermediate

1/9/18 45

Rex = p1p2ΔΩ2 τ c1+ωe

2τ 2c( )

Rex = p1p2τ cΩ2

At high applied field it tends (slowly) to zeroà obtain R2 “control”Caveat– strong fields needed, heating

Free Procession

During Spin Lock T1rho

Determining kex from Rex

•  A unique solution for ΔΩ p1 p2 and kex = kf +kr based on Rex – if ΔΩ and p1 p2

are available from from other data, and assume a simple kinetic model.

•  In addition, variation of the spin lock strength ωe is potentially useful (dispersion curves). When its weak, the full effect is expected. If it is high enough, the exchange contribution is removed.

1/9/18 46

Rex = p1p2ΔΩ2 τ c1+ωe

2τ 2c( )

Rex = p1p2ΔΩ2τ cIn lab frame

In rotating frame

ωe >>1/ τ c22

1/9/18 47

Rates, s-1

50015003000750012500

Loria et al Accounts Chem Res. 2008

CPMG

T1rho

Dispersion Curves : A Constraint for Identifying the rates

“Frame Transformation” of Spectral Density Centered on “CS” Conditions ω1= nωr

Random Fluctuation Model - Approximate Initial Relaxation due to CSA during a Spinlock and MAS for Fast Limit

Farès, Qian, and Davis, J Chem Phys 2005

Experimental Analysis Involving Dipolar Mechanisms:Methyl Rotors: Akasaka Ganapathy MacDowell Naito J Chem Phys 1983Experimental DMS: Krushelnitsky et al SSNMR 2002

Redfield and Model Free expressions for R1ρ During MAS: 1/τc >> ωr , ω1> δ

R2 expressions: Range of Validity

1/9/18 50

R1ρ = R1cos2θ + R2°sin2θ + Rexsin2 θ Rex : Due to chemical exchange θ  : Effective Field Angle R2° : in the absence of exchange- knowledge needed to

extract Rex ImentionedthatthefluctuationismoreeffectiveforR2andR1ρwhenitsslower.ButR2~Δω2(τc)blowsupwhenthecorrelationtimeistoolong.

Unphysical!R2(τc)=Δω2(τc2)<<1,soτc<Δω

Valid Application Range for R1ρ expressions

τc << T1ρ and R1ρτc = Δω2 (τc2/(1+ω2 τc

2) <<1 (where ω=ω1+/-nωρ)

àFast processes

ω2τc2<<1R1ρτc ~ Δω2 (τc2/(1+ω2 τc

2)<<1 so(τc.Δω)2<<1τc<10-5

àSlowerprocessesweakperturbingfieldω2τc2>>1,Δω2(τc2/(1+ω2τc2)<<1soΔω2(τc2/ω2τc2)<<1 Δω2 << ω2 Δω ∼ 104 and ω1+/- nωρ ∼ 106

Assuming Δω ∼ 104-105 ω1+/-nωρ∼ 104-106

1/9/18 51

Redfield expressions Indicate Novel FeaturesCompared to Solution

MAS R1ρ Experiments

R1ρ = R1cos2θ + R2°sin2θ + Rexsin2 θ Rex : Due to chemical exchange θ  : Effective Field Angle R1, R2° : in the absence of exchange

R1r

ho, s

-1

Figure2.(A)SimulationsofDMSR1ρrelaxation,showingthedependenceofR1ρonthetimescaleofdynamics:kex=1Hz(green),1kHz(blue),30kHz(red),infinitelyfast(yellow).(B)Expansionofintermediateexchangesimulationsintheregionof20-40kHzappliedspin-lockpower.(inset)StructureandmotionalmodeofDMS.

Spinlock Field, kHz

10 30 50 30 40

k= 1, 1000, 30,000, >10^10 s-1

Caution Regarding Interpretation

1/9/18 54Vanderhart and Garroway J Chem Phys 1979

Global Analysiskex(T, ωo, ω1, ωr)

R2(T, ωo, ω1, ωr)

I(t, T, ωo, ω1, ωr)

Prior 2H lineshape and relaxation Analysis: Brown Vold & Hoatson ’96This Analysis by 13C T1ρ (C. Quinn & McDermott ’09) T2 ~ 10 - 60 s-1