Supplementary material

Assessment of Chemical Exchange in Tryptophan-Albumin solution through 19F Multicomponent Transverse Relaxation Dispersion Analysis

Ping-Chang Lin

Department of Radiology, College of Medicine

Howard University, Washington, DC 20060, USA

*to whom the correspondence should be addressed

pingchang.lin@howard.edu

S1

Supplementary material

Supporting information

Sample preparation. 6-fluoro-DL-tryptophan (6F-TRP) (Gold Biotechnology, Inc., St. Louise, MO)

was dissolved in 0.05M HCl at room temperature to make a 45-mM stock solution. Bovine serum

albumin (BSA) (Amresco, LLC, Solon, OH) was then dissolved in the 6F-TRP stock to prepare a 1.13-

mM BSA/ 45-mM 6F-TRP complex solution for characterization of chemical exchange process between

the free and BSA-bound states of 6F-TRP. To mimic a condition that macromolecules cannot penetrate

cellular barriers such as cytoplasmic membranes, the semi-permeable dialysis membrane of 8-10 kD

MWCO was implemented to separate the BSA-6F-TRP complex solution from the 6F-TRP solution.

All the samples were separately placed in 5-mm NMR tubes for 19F transverse relaxation experiments.

NMR measurements. 19F NMR experiments were acquired at a 9.4T Bruker Avance NMR

spectrometer (Bruker Biospin, GmbH, Rheinstetten, Germany) equipped with a 5-mm 1H/ multinuclear

broadband probe at 20 ± 1 °C. 19F T2 relaxation data were collected using a spectroscopic CPMG pulse

sequence for each sample prepared. The CPMG experiments were acquired with a sufficient number of

transients (either 128 or 256), repetition time of 10 s, and echo-spacing τCPMG varying from 0.2 to 25 ms

in 19 steps, referring to the number of echoes from 20480 down to 192, respectively (detailed in Table

S1). Signals acquired at individual echoes in each CPMG sequence setting were then collected to

generate a T2 decay curve for the NNLS analysis (Figures 2 and S1). For the 19F relaxation data

acquired, the signal-to-noise ratio ranged from 16 to 442.

Fitting of multicomponent T2 relaxation data. The NNLS method has been adopted for

multicomponent T2 relaxation analysis (Reiter et al. 2009). By incorporating regulations into the linear

equation system of describing multi-exponential decays, the NNLS approach is easily changed to

construct a continuous spectrum (Graham et al. 1996; Reiter et al. 2009; Whittall and MacKay 1989).

Briefly, a set of linear equations, yn=Anm Sm, illustrate the multiple T2 exponential decays in discrete

form through using M-1 relaxation components over N echoes generated by the CPMG pulse train. As

described in the text, the vector yn includes N echo amplitudes; the matrix Anm, consisting of N x (M-1)

kernels, exp (−n ∙ TET 2 ,m

), profiles a set of (M-1) T2 relaxation components at N different echo times and N

x 1 elements of value 1 in the M-th column for baseline offset adjustment; and the array Sm consists of

M unknown amplitudes associated with the M-1 T2 components and a baseline offset in this linear

equation system. The NNLS approach makes no a priori assumptions about the number of relaxation

components present. A minimum energy constraint, i.e. a Tikhonov regularization of second kind in our

study, is imposed into the function to lessen the impact of noise on the curve fitting and to permit

S2

Supplementary material

generation of a continuous T2 distribution (Equation S1) (Graham et al. 1996; Reiter et al. 2009; Whittall

and MacKay 1989).

∑n=1

N |∑m=1

M

Anm Sm− yn|2

+μ|∑m=1

M

Sm|2

, [S1]

Given the 2 misfit defined as

χ2=∑n=1

N

(∑m=1

M

Anm Sm− yn)2

/σ n2 [S2]

, which is the sum of variances of the prediction errors divided by the standard deviation of yn, a

nonnegative set of Sm was obtained by performing regularization of NNLS fits. An appropriate value of

the regularizer μ was selected for an optimal condition that the 2 misfit value from the regularized fit

was 100.5% of the non-regularized 2 based on the strategy of “least squared-based constraints”, which

evenly regularizes all datasets on a percentage basis across the study (Graham et al. 1996; Reiter et al.

2009). The T2 distribution, which was constructed by the T2 values and the associated component

fractions Sm, was interpreted in terms of matrix composition. The T2 distribution consisting of one or

two T2 relaxation components was then fitted using a 4- or 7-parameter lognormal model. All fitting

routines were implemented using MATLAB (MathWorks, Natick, MA, USA).

Lognormal curve fitting. The T2 distributions resulting from the NNLS fits were further fitted into the

probability density function of a lognormal distribution:

f ( x )=C11

xσ √2 πe

−(ln xμ¿ )

2

2σ2

+c0, x>0 for single component T2 distribution, [S3]

or

f ( x )=C11

x σ1 √2πe

−(ln xμ1

¿ )2

2 σ12

+C21

x σ2 √2 πe

−(ln xμ2

¿ )2

2 σ 22

+c0, x>0 for double component T2 distribution. [S4]

The geometric mean (μ¿ or μi¿) and multiplicative standard deviation (σ ¿=eσ or σ i

¿=eσ i) were calculated

from the corresponding fitting outcome.

S3

Supplementary material

echo spacing (m

s)

C

PMG (H

z)

number of

echoes

0.2

1250

20480

0.333

750

13312

0.4

625

11264

0.5

500

9216

0.8

313

6000

0.9

278

5120

1 250

4608

1.25

200

3584

1.429

175

3328

1.667

150

2816

2 125

2304

2.5

100

1792

3.333

75

1400

5 50

960

6.667

37

720

10 25

492

13.33

19

360

20 13

240

25 10

192

S4

Table S1 Parameter setting for

* An adjustable delay time is applied to every individual CPMG parameter set to retain the consistency in repetition time of 10 seconds

10-1

100

101

102

103

104

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

T2 (ms)

Inte

nsity

(a.u

.)

T2 estimated by one of bi-exponetial fittings

T2 estimated by mono-exponetial fitting

T2 estimated by tri-exponetial fitting

Supplementary material

Figure S1

S5

A

B

C

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100001

2

3

4

5

6

7

8

9x 10

5

TE (ms)

Sign

al In

tens

ityDecay curve

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

2

4

6

8

10

12

14x 10

5 Decay curve

TE (ms)

Sign

al In

tens

ity

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7

8

9x 10

6

TE (ms)

Sign

al In

tens

ity

Decay curve

10-1

100

101

102

103

104

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

T2 (ms)

Inte

nsity

(a.u

.)

T2 estimated by mono-expoential fitting

10-1

100

101

102

103

104

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

T2 (ms)

Inte

nsity

(a.u

.)

T2 estimated by mono-exponential fitting

Supplementary material

19F T2 decay curves and the corresponding fitting analyses in (A) the 6F-Trp solution, (B) the BSA-6F-

Trp complex solution, and (C) the two-compartmental 6F-Trp system. The echo spacing present in the

CPMG pulse sequence was τCPMG=¿2 ms for all the solution systems. In the TE vs. signal intensity plot

for each solution system, green circles represent T2 decay signals acquired at the respective echoes; blue

fitting curve exhibits the NNLS fit of the T2 decay data; and red curve represents the mono-exponential

fit without restraints. Additional two T2 decay analyses through use of bi- and tri-exponential fittings

without regularization are included in (C), which are barely distinct from the fitting curve genereated

from the NNLS analysis. As expected, the arithmetic means of T2 in the mono-exponential fits were

close to the geometric means of the T2 dispersions yielded from the NNLS analysis in (A) and (B), but

not in (C) (Table S2). On the other hand, the variances of fits in the bi- and tri-exponential analyses were

compatible with that in the NNLS analysis in (C), evidenced in the 2 statistics and in the residual plot

(blue circle: residuals of the NNLS fit; red circle: residuals of the mono-exponential fit; purple cross:

residuals of the bi-exponential fit; and green triangle: residuals of the tri-exponential fit). Regarding the

bi-exponential and tri-exponential analyses, the residual plot in (C) does not show any substantial

difference between the fits although the number of exponential components and their decay rates differ

quiet significantly. In fact, the variances of fits cannot be evaluated with an F-test because the residuals

of the fits are not independent of each other, which is due to the consequence of non-orthogonality of

exponentials (Istratov and Vyvenko 1999; Johnson 2008). Therefore, there is no compelling evidence

supporting either the bi-exponential or tri-exponential model in the two-compartmental 6F-Trp system.

In addition, the T2 curve fitting outcomes are shown in the T2 vs. intensity plots of (A), (B) and (C),

respectively, with the discrete T2 values, resulting from the mono-, bi- or tri-exponential fitting,

presented by the line segments, accompanied with the corresponding T2 distributions resulting from the

NNLS analysis. For all the curve fittings, the 2 statistics were applied to goodness-of -fit tests, with the

calculated 2 shown in Table S2.

Table S2 Fitting analysis of T2 relaxation curves acquired at τCPMG=¿2 ms

Method 2 (p value) Estimated T2 (ms) Weight fraction

Fig S1A

NNLS w/ regularization 2317 (p = 0.41) 1912 (1.31) -

Mono-exponential fitting 2322 (p = 0.38) 1829 ± 14 -

Fig S1B

NNLS w/ regularization 2370 (p = 0.16) 150 (1.52) -

Mono-exponential fitting 2362 (p = 0.19) 132 ± 2 -

Fig S1C NNLS w/ regularization 2206 (p = 0.92) 237 (1.54) 0.28

S6

Supplementary material

1851 (1.51) 0.72

Mono-exponential fitting 6146 (p ~ 0) 1315 ± 6 -

bi-exponential fitting

2217 (p = 0.89)257 ± 8

1651 ± 110.320.68

2243 (p = 0.80)234 ± 131603 ± 14

0.310.69

tri-exponential fitting 2196 (p = 0.93)211 ± 17

1023 ± 3931944 ± 317

0.270.270.46

* For 2 goodness-of-fit test, df = 2303 in the NNLS analysis, df = 2301 in the mono-exponential fitting, df = 2299 in the bi-exponential fitting, and df = 2297 in the tri-exponential fitting.

** Estimated T2 is presented as geometric mean (multiplicative standard deviation) for the NNLS analysis and as arithmetic mean ± standard deviation for the mono-, bi- or tri-exponential fitting.

References

Graham SJ, Stanchev PL, Bronskill MJ (1996) Criteria for analysis of multicomponent tissue T2 relaxation data Magn Reson Med 35:370-378

Istratov AA, Vyvenko OF (1999) Exponential analysis in physical phenomena Rev Sci Instrum 70:1233-1257 doi:10.1063/1.1149581

Johnson ML (2008) Nonlinear least-squares fitting methods Method Cell Biol 84:781-805 doi: 10.1016/S0091-679x(07)84024-6

Reiter DA, Lin PC, Fishbein KW, Spencer RG (2009) Multicomponent T-2 Relaxation Analysis in Cartilage Magnetic Resonance in Medicine 61:803-809 doi:10.1002/mrm.21926

Whittall KP, MacKay AL (1989) Quantitative interpretation of NMR relaxation data Journal of Magnetic Resonance (1969) 84:134-152 doi:10.1016/0022-2364(89)90011-5

S7