Supplementary material
Assessment of Chemical Exchange in Tryptophan-Albumin solution through
19
F 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 1.13mM 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.
19
F 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)
𝑇𝐸
kernels, exp⁡(−𝑛 ∙ 𝑇 ), profiles a set of (M-1) T2 relaxation components at N different echo times and N
2,𝑚
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).
𝑀
𝑀
2
2
∑𝑁
𝑛=1|∑𝑚=1 𝐴𝑛𝑚 𝑆𝑚 − 𝑦𝑛 | + 𝜇|∑𝑚=1 𝑆𝑚 | ,
[S1]
Given the 2 misfit defined as
𝑀
2
2
𝜒 2 = ∑𝑁
𝑛=1(∑𝑚=1 𝐴𝑛𝑚 𝑆𝑚 − 𝑦𝑛 ) /𝜎𝑛
[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:
1
𝑓(𝑥) = 𝐶1 𝑥𝜎√2𝜋 𝑒
−
𝑥
(ln ∗ )2
𝜇
2𝜎2
+ 𝑐0 , 𝑥 > 0 for single component T2 distribution,
[S3]
or
𝑓(𝑥) = 𝐶1 𝑥𝜎
1
1 √2𝜋
𝑒
−
𝑥
(ln ∗ )2
𝜇1
2𝜎1 2
+ 𝐶2 𝑥𝜎
1
2 √2𝜋
𝑒
−
𝑥
(ln ∗ )2
𝜇2
2𝜎2 2
+ 𝑐0, 𝑥 > 0 for double component T2 distribution. [S4]
The geometric mean (𝜇 ∗ or 𝜇𝑖∗ ) and multiplicative standard deviation (𝜎 ∗ = 𝑒 𝜎 or 𝜎𝑖∗ = 𝑒 𝜎𝑖 ) were
calculated from the corresponding fitting outcome.
S3
Supplementary material
13312
11264
9216
6000
5120
4608
3584
3328
2816
2304
1792
1400
960
720
492
360
240
192
Parameter setting for 19F CPMG experiments
20480
Table S1
number of
echoes
10
25
13
20
19
13.33
25
10
37
6.667
50
5
75
3.333
100
2.5
125
2
150
1.667
175
1.429
200
1.25
250
1
278
0.9
313
0.8
500
0.5
625
0.4
750
0.333
1250
0.2
CPMG (Hz)
echo spacing
(ms)
* An adjustable delay time is applied to every individual CPMG parameter set to retain the consistency in repetition time of 10 seconds
S4
Supplementary material
Figure S1
A
Decay curve
5
9
x 10
8
0.12
7
0.1
Intensity (a.u.)
Signal Intensity
T2 estimated by mono-exponential fitting
0.14
6
5
4
0.08
0.06
0.04
3
0.02
2
1
0
1000
2000
3000
4000
5000
6000
7000
8000
0
-1
10
9000 10000
0
10
1
4
10
0.14
12
0.12
10
0.1
Intensity (a.u.)
Signal Intensity
x 10
3
10
T2 estimated by mono-expoential fitting
Decay curve
5
14
10
T2 (ms)
TE (ms)
B
2
10
8
6
0.08
0.06
4
0.04
2
0.02
0
0
1000
2000
3000
4000
5000
6000
7000
8000
0
-1
10
9000 10000
0
1
10
2
10
3
10
4
10
10
T2 (ms)
TE (ms)
C
Decay curve
6
9
x 10
0.1
T2 estimated by mono-exponetial fitting
0.09
8
0.08
7
T2 estimated by one of bi-exponetial fittings
Intensity (a.u.)
Signal Intensity
0.07
6
5
4
0.05
0.04
3
0.03
2
0.02
1
0.01
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
T2 estimated by tri-exponetial fitting
0.06
0
-1
10
0
10
1
2
10
10
T2 (ms)
TE (ms)
S5
3
10
4
10
Supplementary material
5
14
x 10
12
10
Residual
8
6
4
2
0
-2
-4
-6
0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
TE (ms)
19
F 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
S6
Supplementary material
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
Fig S1A
Fig S1B
Fig S1C
Method
NNLS w/ regularization
2 (p value)
2317 (p = 0.41)
Estimated T2 (ms)
1912 (1.31)
Weight fraction
-
Mono-exponential fitting
2322 (p = 0.38)
1829 ± 14
-
NNLS w/ regularization
2370 (p = 0.16)
150 (1.52)
-
Mono-exponential fitting
2362 (p = 0.19)
132 ± 2
-
NNLS w/ regularization
2206 (p = 0.92)
237 (1.54)
1851 (1.51)
0.28
0.72
Mono-exponential fitting
6146 (p ~ 0)
1315 ± 6
-
2217 (p = 0.89)
257 ± 8
1651 ± 11
0.32
0.68
2243 (p = 0.80)
234 ± 13
1603 ± 14
0.31
0.69
2196 (p = 0.93)
211 ± 17
1023 ± 393
1944 ± 317
0.27
0.27
0.46
bi-exponential fitting
tri-exponential fitting
* For 2 goodness-of-fit test, df = 2303 in the NNLS analysis, df = 2301 in the mono-exponential fitting, df = 2299 in the biexponential 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 70:12331257 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
S7
Supplementary material
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
S8