The reaction between S (polymer) and D (dye) can be described by a three step series mechanism as

Appendix VI

129

APPENDIX VI

Derivation of relaxation times for a three step series mechanism

The reaction between S (polymer) and D (dye) can be described by a three step series mechanism as

(VI.1)

where S and D denote respectively the free sites of the nucleotide and the free dye, and D,S, DS

and DS

different bound forms.

Such a system can theoretically display three different relaxation effects, each one being characterised by its relaxation time λ.

The (VI.1) process is composed of three thermodynamically independent reactions

D + S ' D,S (VI.2)

D,S ' DS

(VI.3)

DS

' DS

(VI.4)

The three reciprocal relaxation times 1/τ

= λ

are related to the equilibrium concentrations and to the rate constants of the three steps trough the determinantal equation (Maggini et al, 1997)

(VI.5)

where g

= g

and the g

coefficients are related to the equilibrium concentrations by the equations (Castellan, 1963)

S]

[D, 1 [D]

1 [S]

g

= 1 + + (VI.6)

r₁g₁₁- λ_k r₁g₁₂ r₁g₁₃ r₂g₂₁ r₂g₂₂ - λ_k r₂g₂₃ r₃g₃₁ r₃g₃₂ r₃g₃₃ - λ_k

= 0

S + D k

D,S k

DS

DS

k

k

k

k

Relaxation times equations

130

S]

[D,

g

= − 1 (VI.7)

0

g

= (VI.8)

] [DS

1 S]

[D, g 1

I

22

= + (VI.9)

] [DS g 1

I

23

= − (VI.10)

] [DS

1 ]

[DS g 1

II I

33

= + _(VI.11)

where [S], [D], [D,S], [DS

] e [DS

] are the equilibrium concentrations of the S, D, D,S, DS

e DS

species respectively.

Eq. (VI.5) is a cubic equation with respect to λ which leads to quite complicate expression for λ

, λ

and λ

.

Nevertheless, the problem can be greatly simplified if the three kinetic effects take place in very different time ranges, that is λ

>> λ

>> λ

. If this is the case, like it happens indeed for the systems here investigated, three distinct expressions can be used for each relaxation time deriven from the general expression (Castellan, 1963)

1 k

k k k

D D r

−

=

λ (VI.12)

where D

(D stays for Determinant) are given by

D

= 1 (VI.13)

D

= g

(VI.14)

(VI.15)

D₂ =

g₁₁ g₁₂ g₂₁ g₂₂

Appendix VI

131 (VI.16)

whereas r

are the exchange rate of the reactions (VI.2), (VI.3) and (VI.4).

r

= k

[D][S] = k

[D,S] (VI.17)

r

= k

[D,S] = k

[DS

] (VI.18)

r

= k

[DS

] = k

[DS

] (VI.19)

We can, then, obtain the final expressions for the reciprocal relaxation times

1 1

11 1 0

1 1

r g k ([D] [S]) k

D

r D = = + +

=

=

1

1

λ /τ (VI.20)

2 1

2 1 11

2 2 12 22 2 11

21 12 22 2 11 1

2 2

k

[S]) ([D]

K 1

[S]) ([D]

k K g

r g g g r

g g g r g D

r D +

+ +

= +

−

− =

=

=

=

2

1

λ /τ (VI.21)

(where K

= k

/k

)

3 2

1 3 2 1 21 12 22 11

21 13 23 11 32 33 3 2 3 3 3

3

k

]) S [ ] D )([

K 1 ( K 1

]) S [ ] D ([

k ) K 1 ( K g

g g g

) g g g g g ( g D r r D

1 +

+ +

+

+

= +

⎭ ⎬

⎫

⎩ ⎨

⎧

−

− −

=

= τ

=

λ (VI.22)

(where K

= k

/k

).

The above relationships have been derived for a mechanism of ordinary reactions.

It has been demonstrated (Jovin and Striker, 1977) that when one of the reactant is a linear polymer the expression for g

should be modified in

S]

[D, 1 [D]

1 [S]

(r) '

g₁₁ =−f + +

(VI.23)

D₃ =

g₁₁ g₁₂ g₁₃ g₂₁ g₂₂ g₂₃ g₃₁ g₃₂ g₃₃

Relaxation times equations

132

Under the circumstances, taking into account that [S] = C

f(r), the quantity expressing the sum of the reactant concentrations in eqs. (VI.20), (VI.21), (VI.22) should be replaced by the function

F(C) = C

f(r)-f ’(r)[D] (VI.24)

F(C) is therefore a function of polymer concentration C

and of free dye concentration [D] (Jovin and Striker, 1977).

A plot of 1/τ

vs. F(C) is a straight line whose slope and intercept yield k

and k

respectively.

A plot of 1/τ

vs. F(C) is a curve following eq. (VI.21), whose interpolation yields the K

, k

and k

parameters.

In an analogous way interpolation of a 1/τ

vs. F(C) plot on eq. (VI.22) yields the values of k

, k

and K

(1+K

).

The first step of eq. (VI.1) is generally too fast to be recorded with the techniques used in present studies. Therefore, it might sometimes not appear explicitly in the observed relaxation effects.

In such cases a two-step mechanism could be proposed on a first approximation

(VI.25)

Nevertheless, either because one finds that λ

tends to a level off as F(C) increases, or because the k

value results to be too slow compared to diffusion controlled reaction steps, it turns out that the first step of scheme (VI.25) is not a simple one but encloses the first and the second steps of scheme (VI.1), so that the model that correctly depicts the dynamics of the system is (VI.1), even if only two kinetic effects are observed.

S + D k

DS

DS

k

k

k