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By:Bahram Hemmateenejad
Complexity in Chemical Systems
• Unknown Components
• Unknown Numbers
• Unknown Amounts
Modeling Methods
• Hard modelingA predefined mathematical model is existed for the
studied chemical system (i.e. the mechanism of the reaction is known)
• Soft modelingThe mechanism of the reaction is not known
Basic Goals of MCR
1. Determining the number of components coexisted in the chemical system
2. Extracting the pure spectra of the components (qualitative analysis)
3. Extracting the concentration profiles of the components (quantitative analysis)
Evolutionary processes
• pH metric titration of acids or bases
• Complexometric titration
• Kinetic analysis
• HPLC-DAD experiments
• GC-MS experiments
• The spectrum of the reaction mixture is recorded at each stage of the process
• Data matrix (D)Nwav
Nsln
Bilinear Decomposition• If there are existed k chemical components
in the system
D = C
S
Nwav
Nsln Nsln
k Nwav
k
D = + + +
+ …. + E
Mathematical bases of MCR
• D = C S Real Decomposition• D = U V PCA Decomposition
Target factor analysis• D = U (T T-1) V
= (U T) (T-1 V) C = U T, S = T-1 V
T is a square matrix called transformation matrix
How to calculate Transformation matrix T?
Ambiguities existed in the resolved C and S
• Rotational ambiguity– There is a differene between the calculated T
and real T
• Intensity ambiguity– D = C S = (k C) (1/k S)
How to break the ambiguities (at least partially)
1. Combination of Hard models with Soft models
2. Using of local rank informations3. Implementation of some constraints
• Non-negativity• Unimodality• Closure• Selectivity• Peak Shape
MCR methods
• Non iterative methods (using local rank information)
Evolving factor analysis (EFA)
Windows factor analysis (WFA)
Subwindows factor analysis (SWFA)
• Iterative methods (using natural constrains)• Iterative target transformation factor analysis (ITTFA)
• Multivariate curve resolution-alternative least squares (MCR-ALS)
Mathematical Bases of MCR-ALS• The ALS methods uses an initial estimates
of concentration profiles (C) or pure spectra (S)
• The more convenient method is to use concentration profiles as initial estimate (C)
• D = CS • Scal = C+ D, C+ is the pseudo inverse of C
• Ccal = D S+
• Dcal = Ccal Scal Dcal D
• Lack of fit error (LOF)
(LOF) =100 ((dij-dcalij)2/dij2)1/2
• LOF in PCA (dcalij is calculated from U*V)
• LOF in ALS (dcalij is calculated from C*S)
Kinds of matrices that can by analyzed by MCR-ALS
1. Single matrix (obtained trough a single run)
2. Augmented data matrixRow-wise augmented data matrix: A single
evolutionary run is monitored by different instrumental methods. D = [D1 D2 D3]
Column-wise augmented data matrix: Different chemical systems containing common components are monitored by an instrumental method
D = [D1;D2;D3]
• Row-and column-wise augmented data matrix:
chemical systems containing common components are monitored by different instrumental method
D = [D1 D2 D3;D4 D5 D6]
Running the MCR-ALS Program
1. Building up the experimental data matrix
D (Nsoln, Nwave)
2. Estimation of the number of components in the data matrix D
PCA, FA, EFA
3. Local rank Analysis and initial estimates
EFA
4. Alterative least squares optimization
Evolving Factor Analysis(EFA)
D
FA
1f, 2f, 3f1f, 2f
FA
Forward Analysis
D
FA
1b, 2b, 3b1b, 2b
FA
Backward Analysis
-4.00
-2.00
0.00
2.00
4.00
1 3 5 7 9 11 13 15 17 19
Row Number
Lo
g e
igen
valu
es
MCR-ALS program written by Tauler • [copt,sopt,sdopt,ropt,areaopt,rtopt]=als(d,x0,nexp,
nit,tolsigma,isp,csel,ssel,vclos1,vclos2);• • Inputs:
d: data matrix (r c) Single matrix d=D
Row-wise augmented matrix d=[D1 D2 D3]Column-wise augmented matrix d=[D1;D2;D3]Row-and column-wise augmented matrix
d=[D1 D2 D3;D4;D5;D6]
• x0: Initial estimates of C or S matrices
C (r k), S (k c)
• nexp: Number of matrices forming the data set
• nit: Maximum number of iterations in the optimization step (default 50)
• tolsigma: Convergence criterion based on relative change of lack of fit error (default 0.1)
• isp: small binary matrix containing the information related to the correspondence of the components among the matrices present in data set. isp (nexp k)isp=[1 0;0 1;1 1]
• csel: a matrix with the same dimension as C indicating the selective regions in the concentration profiles
• ssel: a matrix with the same dimension as S indicating the selective regions in the spectral profiles
A B C
0 0 1
Nan Nan 1
Nan Nan Nan
Nan Nan Nan
1 Nan Nan
1 Nan 0
• vclos1 and vclos2: These input parameters are only used when we deal with certain cases of closed system (i.e. when mass balance equation can be hold for a reaction)
• vclos1 is a vector whose elements indicate the value of the total concentration at each stage of the process (for each row of C matrix)
• vclos2 is used when we have two independent mass balance equations
Outputs
• copt: matrix of resolved pure concentration profiles
• sopt: matrix of resolved pure spectra.
• sdopt: optimal percent lack of fit
• ropt: matrix of residuals obtained from the comparison of PCA reproduced data set (dpca) using the pure resolved concentration and spectra profiles.
ropt = T P’ – CS’
• areaopt: This matrix is sized as isp matrix and contains the area under the concentration profile of each component in each Di matrix. This is useful for augmented data matrices.
• rtopt: matrix providing relative quantitative information. rtopt is a matrix of area ratios between components in different matrices. The first data matrix is always taken as a reference.
An example
Protein denaturation
Protein (intermediate) Protein
(unfold) (denatured)
denaturant denaturant
Metal Complexation
• Complexation of Al3+ with Methyl thymol Blue (MTB)
Applications
Qualitative MCR-ALS
Quantitative MCR-ALS
Nifedipine 1,4-dihydro-2,6-dimethyl-4-(2-nitrophenyl)-3,5-
pyridine dicarboxilic acid dimethyl ester
– selective arterial dilator
– hypertension
– angina pectoris
– other cardiovascular disorders N
NO
COOMeMeOOC
HMeMe
2
Nifedipine is a sensitive substance
• UV light4-(2-nitrophenyl)
pyridine
• daylight 4-(2-nitrosophenyl)-
pyridine
N
NO
COOMeMeOOC
MeMe
2
N
NO
COOMeMeOOC
MeMe
0
0.5
1
1.5
2
2.5
3
225 275 325 375 425wavelength (nm)
absorb
ance
Data Analysis
• Definition of the data matrix, D (nm)– n: No. of wavelengths
– M: No. of samples
• PCA of the data D = R C– R is related to spectra of the components
– C is related to the concentration of the components
• Number of chemical components
-8
-4
0
4
8
1 3 5 7 9 11 13 15No. of factors
Lo
g (
EV
)
-3
-2
-1
0
1
2
3
4
225 255 285 315 345 375 405 435
Wavelength
Sco
reScore 1
Score2
Score 3
Score Plot
0
0.5
1
1.5
2
2.5
225 275 325 375 425
Wavelength (nm)
Ab
so
rba
nc
eNifedipin (resolved)
nitroso pyridine homologue(resolved)nifedipin (experimental)
mixture
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300
Time (minute)
Fra
cti
on
of
co
mp
on
en
tsNifedipin
Nitroso pyridinehomologue
• Linear segment
CNIF = 1.181 ( 0.001) 10-4 – 4.96 (0.13) 10-9 t
r2 = 0.995
• Exponential segment
CNIF = 1.197 ( 0.003) 10-4 Exp (-6.22 ( 0.10) 10-5
t) r2 = 0.998
• Zero order 4.96 (0.13) 10-9 (mole l-1 s-1)
• First-order 6.22 ( 0.10) 10-5 (s-1)
• When iodine dissolves in a binary mixture of donating (D) and non-donating (ND) solvents, preferential solvation indicates the shape of iodine spectrum
• Nakanishi et al. (1987) studied the spectra of iodine in mixed binary solvents
• Factor analysis was used to indicate the number of component existed
• No extra works were reported
0
0.2
0.4
0.6
0.8
1
400 450 500 550 600 650
Wavelength, nm
Ab
so
rban
ce
Iodine spectra in dioxane-cyclohexane
0.00
0.40
0.80
1.20
1.60
400 450 500 550 600 650
Wavelength, nm
Ab
sorb
ance
Iodine spectra in THF-cyclohexane
Eigen-values Plot
-13
-10
-7
-4
-1
2
5
8
1 3 5 7 9Number of factors
Lo
ga
rith
m o
f e
ige
n-v
alu
e
THF
Dioxane
0.00
0.20
0.40
0.60
0.80
1.00
400.00 450.00 500.00 550.00 600.00 650.00
Wavelength (nm)
Ab
so
rba
nc
e
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.20 0.40 0.60 0.80 1.00XDioxan
Co
nc
en
tra
tio
n x
10
3
M
12 3 4
0.00
0.40
0.80
1.20
1.60
400.00 450.00 500.00 550.00 600.00 650.00
Wavelength (nm)
Ab
so
rba
nc
e
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.20 0.40 0.60 0.80 1.00XTHF
Co
nc
en
tra
tio
n x
10
3 M
1
2
3
4
Dye aggregates Dye monomer
Dye-Surfactant ion-pairing
Pre-micelle aggregate Dye partitioned in the micelle phase
Absorbance Spectra of MB
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
450 500 550 600 650 700 750 800 850
Wavelength (nm)
Abs
orba
nce
0
0.4
0.8
1.2
1.6
2
450 550 650 750 850
Wavelength (nm)
Ab
so
rba
nc
e
0.00
0.50
1.00
1.50
2.00
500 550 600 650 700 750 800
Wavelength (nm)
Ab
sorb
anve
Resolved pure spectra of the components
D
S-D
(S-D)n
D(m)
0.00
0.20
0.40
0.60
0.80
1.00
0 0.002 0.004 0.006 0.008 0.01
[SDS]
Mo
le f
rac
tio
nConcentration Profiles
D
S-D
(S-D)n
D(m)
• D + S D-S Ki = [D-S]/[D][S]
• n D-S (D-S)n Kag = [(D-S)n]/[D-S]n
• (D-S)n n D(m) Kd = [D(m)]n/[(D-S)n)
• Log Kag = log [(D-S)n] – n log [D-S]
• log [(D-S)n] = Log Kag + n log [D-S]
n = 4
log Kag = -0.058
y = 4.0249x - 0.0576
R2 = 0.9844
-2.5
-2
-1.5
-1
-0.5
-0.7 -0.5 -0.3 -0.1
log[MS]
log
[MS
(n)]
0
0.2
0.4
0.6
0.8
1
330 380 430 480 530 580
Wavelength (nm)
Ab
so
rba
nc
eInteraction of MO with CTAB
0
0.25
0.5
0.75
1
330 380 430 480 530 580
Wavelength (nm)
Ab
so
rba
nc
ePure spectra of MO Components
D
DS
(DS)n
D(m)
0
0.2
0.4
0.6
0.8
1
0 0.001 0.002 0.003 0.004 0.005
[CTAB]
Mo
le F
rac
tio
nConcentration Profiles
D(m)
(DS)n
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4
[CTAB] / [MO]
Mo
le F
rac
tio
n
D
DS
(DS)n
• D + S DS
• Ki = [DS] / [D] [S]
• CMO = 410-6 M
• [D] = 0.49 CMO
• [DS] = 0.51 CMO
• CS = 2.5 10-5 M
• [S] = CS – [DS]
Ki = 4.92 104
4.64 104
Quinone reduction
• In the presence of proton source
• Q + e Q- (1)
• Q- + HB QH + B- (2)
• QH + e QH- (3)
• QH- +HB QH2 + B- (4)
Our data set
• Vis. Spectra of 0.1 mM solution of 9,10-anthraquinone at different applied potential in DMF solution
• Optically transparent thin layer electrode
(OTTLE)
The experiment was conducted in Arak University
0
0.5
1
1.5
2
380 430 480 530 580 630 680
Wavelength (nm)
Ab
sorb
ance
C
Table 1: Result of factor analysis of spectroelectrochemical data
No. of
factors
Log (eigenvalues) % Eigenvalue Cumulative % of eigenvalue
1 7.2847 85.9782 85.9782
2 5.0597 9.2918 95.2700
3 4.3647 4.6372 99.9072
4 -0.0273 0.0574 99.9645
5 -0.6388 0.0311 99.9957
6 -3.8141 0.0013 99.9970
7 -4.1098 0.0010 99.9980
8 -4.3311 0.0008 99.9987
9 -4.7288 0.0005 99.9992
10 -5.2691 0.0003 99.9996
11 -5.4931 0.0002 99.9998
EFA Plot
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
1.00 3.00 5.00 7.00 9.00 11.00
Row Nnumber
log
(ei
gen
valu
e)
Pure spectra
1) AQ-o 2) AQH- 3) AQ2-
0.00
0.50
1.00
1.50
2.00
380 430 480 530 580 630 680 730
Wavelength (nm)
Ab
sorb
ance
1
3
2
Concentration Profiles
0.00
0.20
0.40
0.60
0.80
1.00
-2.00-1.80-1.60-1.40-1.20Potential (V)
Fra
ctio
n o
f co
mp
on
ents 1 2
3
• Conversion of AQ-o to AQH-
• AQ-o + H+ AQH-
• E = E - (0.0592/n) log ([AQH-]/[AQ-o][H+])
• E = E - (0.0592/n) log(1/[H+])
- (0.0592/n) log ([AQH-]/[AQ-o])
R2 = 0.996 Slope = 0.0594 intercept = -1.37
-1.45
-1.40
-1.35
-1.30
-0.80 -0.30 0.20 0.70 1.20
log ([AQH-]/[AQo- ])
Po
ten
tial
(V
)
MCR-ALS of polarographic data applied to the study of the copper-binding ability
of tannic acid
Structures of tannic acid (TA) (a) and condensed tannin (b)
R. Tauler et al Anal. Chim. Acta 424 (2000) 203–209
DPP obtained for the system Cu(II) + TA during the titration of a 1× 10- 5 mol l-1 Cu(II) solution with TA in the presence of 0.01 mol l-1 KNO3 and 0.01 mol l-1 acetate buffer (pH = 5.0). The thick line denotes the polarogram measured for the metal ions in the absence of TA.
Cu+2
I = CV + E
Cu+2 + TA
Singular value decomposition (SVD) for the data repre-sented
Concentration profiles (a, c, e) and normalised pure voltammograms (b, d, f), in arbitrary units, obtained in the MCR-ALS decomposition of the data matrix of Fig. 2 according to different assumptions: three components with selectivity, non-negativity and unimodality constrains (a, b) (lof 8.1%); four components with selectivity, non-negativity and unimodality (c, d) (lof 4.4%) or four components with selectivity, non-negativity and signal shape (e, f) (lof 6.5%)
Study of the interaction equilibria between the ploynucleotide
poly (inosinic)-poly(cytidilic) acid and Ethidium bromide by
means of coupled spectrometric techniques
R. Tauler et al. Anal. Chim. Acta 424 (2000) 105-114
poly(I)-poly(C)
Ethidium bromide (EtBr)(3,8-diamino-5-ethyl-6-phenylphenantridinium)
Activator of in vivo the interferon biosynthesis
Fluorescent dye
poly(I)-poly(C) concentration constant
EtBr concentration constant
37 oC, neutral pH, KH2PO4 0.021 M, Na2HPO4 0.029 M, and NaCl 0.15 M, Itotal=0.26 M
TechniquesMolecular absorption
Fluorscence
Circular dicroism (CD)
MethodsContinous variation
Mole-ratio
Experimental conditions
300 400 500 6000
0.5
1
Abs
orba
nce
(a.u
.)
600 700 8000
0.1
0.2
Flu
or.
int.
(a.
u.)
300 400 500 600-0.2
0
0.2
CD
(a.
u.)
300 400 500 6000
0.5
1
1.5
Abs
orba
nce
(a.u
.)
600 700 8000
0.5
1
Flu
or.
int.
(a.
u.)
300 400 500 600
-1
0
1
CD
(a.
u.)
300 400 500 6000
0.2
0.4
Wavelength (nm)
Abs
orba
nce
(a.u
.)
600 700 8000
0.1
0.2
0.3
Wavelength (nm)
Flu
or.
int.
(a.
u.)
300 400 500 600
-0.2
0
0.2
Wavelength (nm)
CD
(a.
u.)
300 400 500 6000
0.5
1
Abs
orba
nce
(a.u
.)
600 700 8000
0.1
0.2
Flu
or.
int.
(a.
u.)
300 400 500 600-0.2
0
0.2
CD
(a.
u.)
300 400 500 6000
0.5
1
1.5
Abs
orba
nce
(a.u
.)
600 700 8000
0.5
1
Flu
or.
int.
(a.
u.)
300 400 500 600
-1
0
1
CD
(a.
u.)
300 400 500 6000
0.2
0.4
Wavelength (nm)
Abs
orba
nce
(a.u
.)
600 700 8000
0.1
0.2
0.3
Wavelength (nm)
Flu
or.
int.
(a.
u.)
300 400 500 600
-0.2
0
0.2
Wavelength (nm)
CD
(a.
u.)
DUV-Visvar Dfluor
var DDCvar
DUV-VisEt Dfluor
Et DDCEt
DUV-Vispoly Dfluor
poly DDCpoly
Data matrices arrangement: (a) analysis of a single spectroscopic data matrix; (b) simultaneous analysis of several spectroscopic data matrices corresponding to different spectroscopic techniques and different experiments.
250 300 350 400 450 500 550 6000
1
2
3
4
5
6
7x 10
4
Wavelength (nm)
Abs
ortiv
ity
550 600 650 700 750 800 8500
1
2
3
4
5
6
7
8
x 104
Wavelength (nm)
Fluo
resc
ence
(a.
u.)
220 240 260 280 300 320 340 360 380 400-4
-2
0
2
4
6
8x 10
4
Wavelength (nm)
CD
(a.
u.)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
x 10-5
Xpoly
Con
cent
ratio
n (M
)
0 1 2 3 4 5 60
0.5
1
1.5
2
2.5x 10
-5
r poly:dye
Con
cent
ratio
n (M
)
0.75 0.8 0.85 0.9 0.95 10
0.5
1
1.5
2
2.5
3x 10
-5
Xpoly
Con
cent
ratio
n (M
)
Cvar SUV-Vis Sfluor SCD
CEt
Cpol
y
poly(I)-poly(C)
EtBr
poly(I)-poly(C)-Et
Poly(I)-poly(C) + EtBr EtBr poly complex
Kapp = [EtBr poly complex] /[Poly(I)-poly(C)
EtBr]RESULTSThe intercalation sites occur every 2-3 base pairs and the value for the log Kapp was 4.6 0.1 M-1
R. Tauler, R. Gargallo, M. Vives and
A. Izquierdo- Ridorsa
Chemometrics and Intelligent Lab
Systems, 1998
Study of conformational equilibria
of polynucleotides
Poly(adenylic)-poly(uridylic) acid system
Melting dataA
bso
rban
ce
Wavelength (nm) Temperatu
re (°
C)
Melting data recorded at = 260 nm
(univariate data analysis)
Temperature (°C)
Ab
sorb
an
ce
Melting Curve
Melting recorded at = 280 nm
Temperature (°C)
Ab
sorb
an
ce
Melting Curve
Poly(A)-poly(U) system. Two different melting experiments
ALS recovered concentration profiles
poly(A)-poly(U)-poly(U) ts
poly(A)-poly(U) ds
poly(U) rc
poly(A) rc
poly(A) cs
Rel
ativ
e co
ncen
trat
ion
Temperature (°C)
ALS recorded pure spectra