By:
Bahram Hemmateenejad
Complexity in Chemical Systems
• Unknown Components
• Unknown Numbers
• Unknown Amounts
Modeling Methods
• Hard modeling
A predefined mathematical model is existed for the
studied chemical system (i.e. the mechanism of
the reaction is known)
• Soft modeling
The 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)
Nsln
Nwav
Bilinear Decomposition
• If there are existed k chemical components
in the system
Nwav
Nwav
k
S
D
Nsln
=
Nsln
C
k
D=
+
+
+
…. + E
+
Mathematical bases of MCR
• D= CS
• D=UV
Real Decomposition
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 informations
3. 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 matrix
Row-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)
Forward Analysis
D
FA
1f, 2f
FA
1f, 2f, 3f
Backward Analysis
D
FA
1b, 2b
FA
1b, 2b, 3b
4.00
Log eigenvalues
2.00
0.00
-2.00
-4.00
1
3
5
7
9
11
Row Number
13
15
17
19
MCR-ALS program written by Tauler
• [copt,sopt,sdopt,ropt,areaopt,rtopt]=als(d,x0,nex
p,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
(unfold)
denaturant
(intermediate)
denaturant
Protein
(denatured)
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)-3,5pyridine dicarboxilic acid dimethyl ester
–
–
–
–
selective arterial dilator
hypertension
angina pectoris
other cardiovascular disorders
NO2
COOMe
MeOOC
Me
N
H
Me
Nifedipine is a sensitive substance
• UV light
• daylight
4-(2-nitrophenyl)
pyridine
4-(2-nitrosophenyl)pyridine
NO2
COOMe
MeOOC
Me
N
Me
NO
COOMe
MeOOC
Me
N
Me
3
2.5
absorbance
2
1.5
1
0.5
0
225
275
325
wavelength (nm)
375
425
Data Analysis
• Definition of the data matrix, D (nm)
– n: No. of wavelengths
– M: No. of samples
• PCA of the data
D=RC
– R is related to spectra of the components
– C is related to the concentration of the components
• Number of chemical components
Log (EV)
8
4
0
-4
-8
1
3
5
7
9
11
No. of factors
13
15
Score Plot
4
Score 1
Score2
Score 3
3
Score
2
1
0
-1
-2
-3
225
255
285
315
345
Wavelength
375
405
435
2.5
Nifedipin (resolved)
nitroso pyridine homologue
(resolved)
nifedipin (experimental)
Absorbance
2
mixture
1.5
1
0.5
0
225
275
325
Wavelength (nm)
375
425
1.2
Nifedipin
Fraction of components
1
Nitroso pyridine
homologue
0.8
0.6
0.4
0.2
0
0
50
100
150
Time (minute)
200
250
300
• 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
• First-order
4.96 (0.13)  10-9 (mole l-1 s-1)
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
Iodine spectra in dioxane-cyclohexane
1
Absorbance
0.8
0.6
0.4
0.2
0
400
450
500
550
Wavelength, nm
600
650
Iodine spectra in THF-cyclohexane
1.60
Absorbance
1.20
0.80
0.40
0.00
400
450
500
550
Wavelength, nm
600
650
Eigen-values Plot
Logarithm of eigen-value
8
5
THF
Dioxane
2
-1
-4
-7
-10
-13
1
3
5
Number of factors
7
9
Absorbance
1.00
0.80
0.60
0.40
0.20
0.00
400.00
450.00
500.00
550.00
600.00
650.00
1.00
Concentration x 10
M
3
Wavelength (nm)
2
0.80
3
4
1
0.60
0.40
0.20
0.00
0.00
0.20
0.40
XDioxan
0.60
0.80
1.00
1.20
0.80
0.40
0.00
400.00
450.00
500.00
550.00
600.00
650.00
Wavelength (nm)
Concentration x 10 M
1.00
3
Absorbance
1.60
4
0.80
3
1
0.60
2
0.40
0.20
0.00
0.00
0.20
0.40
XTHF
0.60
0.80
1.00
Dye aggregates
Dye monomer
Dye-Surfactant ion-pairing
Pre-micelle aggregate
Dye partitioned in the
micelle phase
Absorbance Spectra of MB
1.8
Absorbance
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
450
500
550
600
650
700
Wavelength (nm)
750
800
850
2
Absorbance
1.6
1.2
0.8
0.4
0
450
550
650
Wavelength (nm)
750
850
Resolved pure spectra of the
components
2.00
Absorbanve
D(m)
1.50
D
1.00
(S-D)n
0.50
S-D
0.00
500
550
600
650
700
Wavelength (nm)
750
800
Concentration Profiles
1.00
S-D
Mole fraction
0.80
D(m)
(S-D)n
0.60
0.40
D
0.20
0.00
0
0.002
0.004
0.006
[SDS]
0.008
0.01
• 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]
-0.5
log[MS(n)]
-1
y = 4.0249x - 0.0576
R2 = 0.9844
-1.5
-2
-2.5
-0.7
-0.5
-0.3
log[MS]
n=4
log Kag = -0.058
-0.1
Interaction of MO with CTAB
1
Absorbance
0.8
0.6
0.4
0.2
0
330
380
430
480
Wavelength (nm)
530
580
Pure spectra of MO Components
Absorbance
1
0.75
D(m)
D
0.5
DS
0.25
(DS)n
0
330
380
430
480
Wavelength (nm)
530
580
Concentration Profiles
D(m)
Mole Fraction
1
0.8
0.6
0.4
(DS)n
0.2
0
0
0.001
0.002
0.003
[CTAB]
0.004
0.005
1
D
(DS)n
Mole Fraction
0.8
0.6
0.4
DS
0.2
0
0
1
2
[CTAB] / [MO]
3
4
• D+S
DS
• Ki = [DS] / [D] [S]
• CMO = 410-6 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Q- + HB  QH + BQH + e  QHQH- +HB  QH2 + B-
(1)
(2)
(3)
(4)
Our data set
• Vis. Spectra of 0.1 mM solution of 9,10anthraquinone at different applied potential
in DMF solution
• Optically transparent thin layer electrode
(OTTLE)
The experiment was conducted in Arak
University
C
Absorbance
2
1.5
1
0.5
0
380
430
480
530
580
Wavelength (nm)
630
680
Table 1: Result of factor analysis of spectroelectrochemical data
No. of
Log (eigenvalues)
% Eigenvalue
Cumulative % of eigenvalue
factors
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
log (eigenvalue)
3.00
2.00
1.00
0.00
-1.00
-2.00
-3.00
1.00
3.00
5.00
7.00
Row Nnumber
9.00
11.00
Pure spectra
Absorbance
2.00
2
3
1.50
1
1.00
0.50
0.00
380
430
480
530
580
630
680
Wavelength (nm)
1) AQ-o
2) AQH-
3) AQ2-
730
Concentration Profiles
Fraction of components
1.00
2
1
0.80
3
0.60
0.40
0.20
0.00
-1.20
-1.40
-1.60
Potential (V)
-1.80
-2.00
• 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])
Potential (V)
-1.30
-1.35
-1.40
-1.45
-0.80
-0.30
0.20
0.70
1.20
log ([AQH-]/[AQo- ])
R2 = 0.996
Slope = 0.0594
intercept = -1.37
MCR-ALS of polarographic data applied
to the study of the copper-binding ability
of tannic acid
R. Tauler et al Anal. Chim. Acta 424 (2000) 203–209
Structures of tannic acid (TA) (a) and condensed tannin (b)
Cu+2 + TA
Cu+2
I = CV + E
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.
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
Activator of in vivo the interferon
biosynthesis
Fluorescent dye
Ethidium bromide (EtBr)
poly(I)-poly(C)
(3,8-diamino-5-ethyl-6-
phenylphenantridinium)
Techniques
Methods
Molecular absorption
Fluorscence
Circular dicroism (CD)
Continous variation
Mole-ratio
poly(I)-poly(C) concentration constant
EtBr concentration constant
Experimental conditions
37 oC, neutral pH, KH2PO4 0.021 M, Na2HPO4 0.029 M,
and NaCl 0.15 M, Itotal=0.26 M
0
300
500
DUV-VisEt
1
0.5
0
0.4
0.2
0
600
Fluor. int. (a.u.)
1.5
400
300
400
500
DUV-Vispoly
600
300 400 500 600
Wavelength (nm)
CD (a.u.)
0.2
0.1
0
DDCvar
600
700
0.2
0
-0.2
800
DfluorEt
1
CD (a.u.)
0.5
Dfluorvar
0.5
300 400
500 600
300 400
500 600
DDCEt
1
0
-1
0
600
700
800
Dfluorpoly
DDCpoly
0.3
CD (a.u.)
Fluor. int. (a.u.)
1
Fluor. int. (a.u.)
Absorbance (a.u.)
Absorbance (a.u.)
Absorbance (a.u.)
DUV-Visvar
0.2
0.1
0.2
0
-0.2
0
600
700
800
Wavelength (nm)
300 400 500 600
Wavelength (nm)
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.
Cvar
SUV-Vis
-5
Sfluor
4
x 10
7
SCD
4
x 10
x 10
4
8
8
x 10
2
6
7
5
6
4
3
4
5
CD (a.u.)
1
Fluorescence (a.u.)
Absortivity
Concentration (M)
6
4
2
3
0
2
2
1
0
0
0.1
0.2
0.3
0.4
0.5
Xpoly
0.6
0.7
0.8
0.9
1
0
CEt
250
300
350
400
450
Wavelength (nm)
500
550
600
0
550
600
650
700
Wavelength (nm)
750
800
850
-4
220
240
260
280
300
320
Wavelength (nm)
340
360
380
poly(I)-poly(C)
-5
2.5
-2
1
x 10
EtBr
1.5
1
poly(I)-poly(C)-Et
0.5
0
0
1
2
3
r poly:dye
4
5
6
Poly(I)-poly(C) + EtBr  EtBr poly complex
Cpol
-5
3
x 10
Kapp = [EtBr poly complex]
/[Poly(I)-poly(C) EtBr]
y
2.5
2
Concentration (M)
Concentration (M)
2
1.5
RESULTS
1
0.5
0
0.75
0.8
0.85
0.9
Xpoly
0.95
1
The intercalation sites occur every 2-3 base pairs and
the value for the log Kapp was 4.6  0.1 M-1
400
Study of conformational equilibria
of polynucleotides
R. Tauler, R. Gargallo, M. Vives and
A. Izquierdo- Ridorsa
Chemometrics and Intelligent Lab
Systems, 1998
Poly(adenylic)-poly(uridylic) acid system
Absorbance
Melting data
Melting data recorded at  = 260 nm
Absorbance
(univariate data analysis)
Temperature (°C)
Melting Curve
Absorbance
Melting recorded at  = 280 nm
Temperature (°C)
Melting Curve
Poly(A)-poly(U) system. Two different melting experiments
Relative concentration
ALS recovered concentration profiles
poly(A)-poly(U) ds
poly(U) rc
poly(A) rc
poly(A)-poly(U)-poly(U) ts
poly(A) cs
Temperature (°C)
ALS recorded pure spectra