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