PETER PAZMANY 

SEMMELWEIS UNIVERSITYCATHOLIC UNIVERSITY 

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework** 
Consortium leader 

PETER PAZMANY CATHOLIC UNIVERSITY 

Consortium members 

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER 

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund *** 
**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben 
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INTRODUCTION TO BIOPHYSICS 

(Bevezetés a biofizikába) 

REACTION KINETICS 

(Reakciókinetika) 
GYÖRFFY DÁNIEL, ZÁVODSZKY PÉTER 


. 
Rate equation mathematically describes thevelocities of reactions 
. 
Advanced experimental techniques exist to measure the rate of fast reactions 
. 
Reaction order of reactions can be determined by investigating the relation between initialconcentrations of reactants and the rate of reaction 
. 
Arrhenius equation reveals an exponentialdependence of rate constant on the reciprocalof temperature 

Equation of reaction 

. 
The general equation of chemical reactions is: 

..rj Aj .. .pj Ajjj 
where .jr and .jp are the stoichometrical coefficients for Aj substance as a reactant or a product, respectively 

. 
Let us consider a simple reaction: 
A.2 B . 3 C.4 D 
the velocity of the reaction is: 
v =- 

d [ A] 

dt 

=-

1

d [B ]

1

d [C ]

1

d [D ]

= 

= 

2

dt 

3

dt 

4

dt 



or generally 
1 d [ Aj ]v= 
..pj -.rj . dt 
where .p and .r are the stoichometric coefficient of
jj 
substance Aj as a product or a reactant, respectively, [ ] refers to concentration 

. 
In any reaction the rate can depend on theconcentration of reactants 
. 
This relationship is mathematically describedby the rate law 

. 
The most general form of the rate law is: 
v =k .[ Aj ]. j 
j 

where k is the rate constant and .j is the order of reaction with respect to the substance Aj 

Elementary reactions 

. 
The order of a reaction regarding a given
participant is not necessarily equal to the
stoichiometric coefficient of it 

. 
In the case of elementary reactions, however, these two measures are equal 
. 
DEF elementary reaction: reaction where theorder is equal to the stoichiometric coefficientfor each reactant and product 
. 
Complex reactions can be divided into 
elementary reactions 



hydrogen with bromine: 
H2 . g ..Br2 . g.. 2HBr .g . 

the rate law for this reaction is: 
[ H2 ][ Br2 ]3 / 2 
v=k 

[Br2 ].k' [ HBr] 

. 
An assumed mechanism of reaction is: 
– 
Initiation 
h .
Br2 .2Br· v =k1 [Br2 ] 

– 
elongation 
Br·.H2 . HBr.H· v=k 2 [Br ][H 2 ] 
H·.Br2 . HBr.Br· v=k3 [H ][Br 2] 


– 
inhibition 
H·.HBr . H2 .Br· v =k 4 [ H ][ HBr] 
– 
termination 
Br·.Br·. Br 2 v =k 5 [Br ]2 
where 
k =2 k 2 .k1 /k5 k'=k4 /k 3 


different reactants in the overall reaction are not equal to stoichiometric coefficients 
. 
On the other hand, in elementary reactions, orders are equal to stoichiometric coefficients 
. 
Order needs not to be an integer value as it can be seen in the rate law for the overall reaction 


reaction 
. 
Rate law for elementary reactions is a simplelinear differential equation 
. 
We study the concentration of reactants andproducts as they change in time 
. 
To obtain this relationship, the differential
equations of the rate law must be solved 


Example of first order reaction 
. 
Perhaps the best known first order reaction is the radioactive decay 
. 
In mathematical form, such decays are
described by 

-t /. 

I=I0 e
where I and I0 are the number of atoms of a radioactive isotope at time t and at the beginning of the reaction, respectively, and .=1/. is the lifetime, where .=k is the decay constant 

. 

The rate of decay is often characterized by the half life of reaction 
. 
DEF half life: the time needed the number of 
atoms of a radioactive isotope to drop to the
half of the initial value, in mathematical form 

I0 -.1 /2/. 

=I0 e
2 

reaction 

. 
The order of a reaction with respect to any participant can be determined by experiments 
. 
In the experiment, we measure the amount ofthe participant for which we would like to knowthe order of reaction 

www.itk.ppke.hu 


www.itk.ppke.hu 


reaction rate 
. 
Classical methods 
. 
Fast-flow methods 
. 
Stopped-flow methods 
. 
Flash photolysis 

. 
Relatively slow reactions can be monitored byclassical methods 
. 
Reactants are mixed in a reaction space 
. 
Some property which changes during the
reaction is measured 

. 
Usually some electrical property like
conductance or voltage is measured 



. 
In a normal mixer, reactants can be mixed onlyon the time scale of the reaction rate, so the obtained values are not accurate enough 
. 
In gas phase, fast flow methods are used 
. 
In these methods, reactants are injected into aflow tube and cover some distance to the detector 
. 
Concentration of reactant or some product isplotted against the distance covered. 
. 
Time scale at millisecond 


. 
It can be useful for reactions in liquid phase 

. 
The speed of mixing is fast enough not to 
distort the results 

. 
Injected reactants meet in a mix chamber
where the reaction takes place 

. 
Flow is stopped by a third piston and a switch(figure 2.3) 


Figure 2.3 



Flash photolysis 
. 
Mixing finishes before beginning of reaction 
. 
Reaction is started by a flash of light 
. 
In the case of a homogeneous ray, the
distribution of reactants will be also 
homogeneous 

. 
Time scale depends on the width of impulsebut it can reach the nanosecond interval or smaller 

Simplifying methods for
obtaining the rate law 


. 
Complex reactions can be so complicated thatthe rate law cannot be obtained purely bymathematical rearrangements 
. 
In such cases, we need to simplify theexpression describing kinetics of the reaction 
. 
Several methods are known to do this: 
– 
Steady-state approximation 
– 
Pseudo-first-order approximation 
– 
Rate limiting step approximation 


Steady state approximation 

. 
Let us consider a serial reaction 
k1 k2
A . B .C 

. 
Steady-state approximation assumes that theconcentration of the intermediate can be considered constant 
. 
The corresponding rate laws are 
d [ A] d [ B] d [C ]
=k1 [ A]=k1 [ A]-k2 [ B ]=k2 [ B ]
dt dt dt 

. 

In steady state approximation we assume 
d [ B] 
=k1 [ A]-k2 [ B ]=0
dt 
thus k1

[ B]= [ A]
k2 
and thus 
d [C ] k2·k1 
=[ A]=k1 [ A]
dt k2 


. 
Integrating the rate law gives us theconcentration of component C as a function oftime 
. 
Let us substitute [A] from the expressionabove to get 
d [C ]-k1 t 
=k1 [ A]=k1 [ A]0 e
dt 
and integrating the expression we obtain 
t -k1 t .
[C ]=[ A]0.e-k1 t dt=[ A]0 .1-e0 


approximation 

. 

Let us consider a protein with the states:unfolded (U), first intermediate (I1), second intermediate (I2) and native (N) 
For the folding of the protein the followingscheme was proposed 
k1 k 2
.

U

I 1 I 1

I 2 

. 

k 3 k4
. 

09/10/11. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 

I 1

NI 2

N 



dU d [ I 1 ]
=-k1 [U ]=k1 [U ]-.k2 .k3 .[ I 1 ]
dt dt 
d [ I 2 ] d [ N ]
=k2 [ I1 ]-k4 [ I 2 ]=k3 [ I 1 ].k4 [ I 2 ]
dt dt 
. 
Using the steady state approximation we 
assume 

d [ I 1 ] d [ I 2 ]
=k1 [U ]-.k2 .k3 .[ I 1 ]=0 =k2 [ I1 ]-k4 [ I 2 ]=0
dt dt 

. 

The concentrations of the intermediates thus 
k1
[ I1 ]= [U ]k2 .k3 
k2 k1 k2
[ I 2 ]= [ I 1 ]= [U ]k4 .k2 .k3 . k4 
. 
Substituting the concentrations of theintermediates we obtain a first order rate law for the end-product 
d [ N ] k1 k3.k1 k2 
=[U ]
dt k2.k3 


-k 1 t

[U ]=[U 0 ]e 

. 
Substituting this into the rate law for thenative state we get 
d [ N ] k1 k3.k1 k2 - k1 t -kt [U 0 ]
= e [U 0 ]=k1 e
dt k2.k3 


concentration of the native state 
t -k 1 t .
[ N ]=[U 0 ] k1 .e-k1 t =[U 0 ].1-e
0 

. 
In kinetic measurements this folding processwill resemble to a two-state folding 


folding of a protein 

Rate limiting step 


. 
Consider a serial reaction 
k1 k2
A . B .C 

. 
If one of the rate constants is far smaller than the other one, the process characterized bythat smaller rate constant is considered as the process which limits the rate of the wholereaction 
. 
Some simplification can be carried out using ofrate limiting step approximation 

. 

The rate laws for the reaction above are 
d [ A]
=-k1 [ A]
dt d [ B] 
=k1 [ A]-k2 [ B ]
dt d [C ]
=k2 [ B ]
dt 
. 
The concentration change for the substances are 
-k1 tk1 [ A]0 -k1 t -k2 t -k 2 t
[ A]=[ A]0 e [ B]= [ e -e ].[ B ]0 e .k2 .k1 .k2-k1 
[C ]=[ A]0.[ B]0.[C ]0 -[ A]-[ B] 


www.itk.ppke.hu 

[ A]0 =1 [ B]0 =0 [C ]0 =0 


. 

These complex expressions can be simplified
using of the rate limiting approximation 

. If k1<< k2 then 
-k1 t
[C ].[C ]0.[ A]0 e 
. If however k2 << k1 then 
[C ].[C ]0.[ A]0 e-k2t 
. 
If C is not present at the beginning of the
reaction then 

-k1 t -k 2 t
[C ]=[ A]0 e and [C ]=[ A]0 e respectively 


[ A]0 =1 [ B]0 =0 [C ]0 =0 k1 =10-12 k2 =10-3 



[ A]0 =1 [ B]0 =0 [C ]0 =0 k1 =10-3 k2 =10-12