Chapter 8 Applications

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In chemistry – chemical reaction

Biochemical reactions: – Take place in all living organisms– Most of them involve proteins called as enzymes– Enzymes react selectively on compounds: substrates

– Biological & biochemical processes are complicated– Develop a simplifying model to understand the phenomena– Give qualitative understanding – First step to develop more realistic & complicated model

Reaction kinetics

Basic enzyme reaction: Michaelis & Menten (1913)– A substrate S reacting with an enzyme E to form a complex SE– The complex SE is converted into a product P and the enzyme

The laws of mass action: the rate of a reaction is proportional to the product of the concentrations of the reactants

1 2

1

1 1 2

,

: reaction is reversible & : reaction can go only one way

, & : constant parameters associated with the rates of reaction

k k

kS E SE SE P E

k k k

Reaction equations

Concentrations of the reactants:

Nonlinear reaction equations

Explain: The first equation for s is simply the statement that the rate of change of the concentration [S] is made up of a loss rate proportional to [S] [E] and a gain rate proportional to [SE].

[ ], [ ], [ ], [ ]s S e E c SE p P

1 1 1 1 2

1 1 2 2

, ( ) ,

( ) , ;

ds dek e s k c k e s k k c

dt dtdc dp

k e s k k c k cdt dt

Model reduction

Initial conditions:

The last equation is uncoupled

Conservation of enzyme: catalyst

Reduced system

0 0(0) , (0) , (0) 0, (0) 0s s e e c p

0 00 ( ) ( ) 0de dc

e t c t e edt dt

2 2 2

0 0 0

( ) (0) ( ') ' 0 ( ') ' ( ') 't t t

p t p k c t dt k c t dt k c t dt

1 0 1 1 1 0 1 1 2( ) , ( )ds dc

k e s k s k c k e s k s k k cdt dt

Pseudo-steady state

Pseudo-steady state solution: The reaction of complex to product is much faster than that of substrate to complex, i.e. enzyme is almost at equilibrium

The equation

1 1 2

0

1 01 1 2

1 2 1

0 ( ) 0

( 0 ) ( ) 0

dek e s k k c

dt

e e c

k e sk e c s k k c c

k k k s

1 2 01 1 2

1 2 1

k k e sdsk s e k c k c

dt k k k s

Pseudo-steady state solution

Define Michaelis constantPseudo-steady state solution

Determine rate of reaction v– Take a sequence of different initial values of– Measure the corresponding variation of s with t, – Rate of reaction– Obtain for each experiment a measurement of the initial rate

1 2

1m

k kK

k

0 2 0( )2 00( ) , 0m mK s K k e ts t

m

ds k e se s t e s e t

dt s K

0s

0( ; )s s t s2 0

m

k e sdp dsv

dt dt s K

0 2 0 2 0 0

1 1 1mK

v k e k e s

Lineweaver-Burk plot

Michaelis-Menten rate

Qualitative analysis

Nondimensionalization:

Dimensionless equation

Qualitative understanding– Steady state: u=v=0– v increases from v=0 until attains its maximum at v=u/(u+K) then decreases to v=0– u decreases monotonically from u=1 to u=0

02 1 21 0

0 0 1 0 1 0 0

( ) ( ), ( ) , ( ) , , ,

ek k ks t c tk e t u v K

s e k s k s s

( ) , ( )

(0) 1, (0) 0

du dvu u K v u u K v

d du v

Qualitative analysis

Michaelis-Menhten theory

Pseudo-steady state hypothesis: The remarkable catalytic effectiveness of enzymes is reflected in the small concentrations needed in their reactions as compared with the concentrations of the substrates.

Approximate (asymptotic) solution: – Assume

– Substitute and equate powers of – The O(1) equations:

2 70 0 0 0/ 1: typically :10 10e s e s

0 1

0 0

( ; ) ( ) , ( ; ) ( )n nn n

n n

u u v v

00 0 0 0 0 0

0 0

( ) , 0 ( )

(0) 1, (0) 0

duu u K v u u K v

du v

Michaelis-Menten theory

Solution: (nonsingular or outer solution, valid for )

Difficulty: The second equation is algebraic & does not satisfy the initial condition

10(0) 1 (0) 0

1u v

K

0 0 0 00 0 0

0 0 0

0 0

0

00 0 0

0

( )

( ) ln ( )

u (0)=1 A=1

( )u ( )+K ln ( ) , ( ) , 0.

( )

u du u uv u u K

u K d u K u K

u K u A

uu A v

u K

0 (1)o

Michaelis-Menten theory

The solution is not a uniformly valid approximation for all The original problem is a singular perturbation problem since the second equation is multiplied by a small parameterThe assumption is not valid near Initial layer existsIntroduce the transformations

New equations

0

0 1 ( )

dvO

d

0

0

, ( ; ) ( ; ), ( ; ) ( ; )u U v V

( ) , ( ) ,

(0) 1, (0) 0

dU dVU U K V U U K V

d dU V

Michaelis-Menten theory

Assume

O(1) equations

The solutions (singular or inner solution, valid for )

0 0

( ; ) ( ) , ( ; ) ( )n nn n

n n

U U V V

0 00 0 0

0 0

0, ( )

1, 0

dU dVU U K V

d dU V

10 0( ) 1, ( ) (1 ) (1 exp[ (1 ) ])U V K K

0 1

Michaelis-Menten theory

Matching: – The limit of the outer solution when – The limit of the inner solution when

Initial (or boundary) layer:

let 0, for a fixed 0 1

0

0 0 0 00

1lim [ ( ), ( )] [1, ] lim [ ( ), ( )]

1U V u v

K

1 100| 1

dVdV

d d

Michaelis-Menten theory

Singular perturbation, a systemic way– Outer solution in the form of a regular series expansion

– Inner solution expansion

00 0 0 0 0 0

011 0 0 1 1 0 0 1

(1) : ( ) , 0 ( )

O( ): ( 1) ( ) , (1 ) ( )

duO u u K v u u K v

ddvdu

u v u K v u v u K vd d

0 00 0 0

1 10 0 0 0 1 0 1

(1) : 0, ( )

( ) : ( ) , (1 ) ( )

dU dVO U U K V

d ddU dV

O U V K V V U V K Vd d

Michaelis-Menten theory

– Initial conditions:

– Thus the singular solutions are determined completely

– Outer solutions

– Matching of the inner and outer solutions

0 0

0 10

00

1 (0; ) (0) (0) 1, (0) 0

0 (0; ) (0) (0) 0

nn n

n

nn n

n

U U U U

V V V

10 0( ) 1, ( ) (1 ) (1 exp[ (1 ) ])U V K K

0 0 0 00

lim [ ( ), ( )] lim [ ( ), ( )]U V u v

00 0 0

0

( )u ( )+K ln ( ) , ( ) , 0

( )

with A the constant of integration

uu A v

u K

Michaelis-Menten theory

Uniformly expansion

0 0 00

00 0 00

0

0 0 0

lim ( ) 1 lim ( ) (0)

(0)1lim ( ) lim ( ) (0)

1 (0)

(0) 1 (0) ln (0) 1

U u u

uV v v

K u K

u u K u A A

0 0 0

10 0

00 0

0

( ; ) ( ) ( ); ( ) ln ( ) 1

( ) ( ); ( ) (1 ) 1 exp( (1 ) , 0 1;

( ; )( )

( ) ( ); ( ) , 0( )

u u O u K u

V O V K K

vu

v O vu K

Michaelis-Menten theory

Michaelis-Menten theory

Uniformly matched asymptotic expansion: inner+outer-middle

Explain– Rapid change in substrate-enzyme takes place in dimensionless time– Very short, in many experimental cases, singular solutions is not observed– The reaction for the complex is essentially in a steady state – The v-reaction is so fast it is more or less in equilibrium at all times– This is Michaelis and Menten’s pseudo-steady state hypothesis

0 0 0 0

0 0

1 0

0

( ; ) ( ) 1 1 ( ) ( ) ( ); ( ) ln ( ) 1

1( ; ) ( ) ( ) ( )

1( ) 1

(1 ) 1 exp( (1 ) ( ); 0( ) 1

u u O u O u K u

v V v OK

uK K O

u K K

( )O

Other chemical reactions

Cooperative phenomena

– Enzyme has more than one binding site for substrate molecules – An enzyme + a substrate is called as cooperative: if a single enzyme molecule,

after binding a substrate molecule at one site can then bind another substrate molecule at another site.

– Example: enzyme molecule E binds a substrate molecule S to form a single bound substrate-enzyme complex C1. C1 can break to form a product P and enzyme E & combine with another substrate molecule S to form a dual bound substrate-enzyme complex C2. C2 breaks down to form the product P and single bound complex C1.

Autocatalysis, Activation & Inhibition: systems with feedback controls

31 2 4

1 3

1 1 2 1,kk k k

k kS E C E P S C C P C

1 2

1

2 ,k k

kA X X B X C

Other models

Biological oscillators & switches: Feedback control

Reaction diffusion, Chemotaxis