chapter 8 applications in physics in biology in chemistry in engineering in political sciences in...
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Chapter 8 Applications
In physicsIn biologyIn chemistryIn engineeringIn political sciencesIn social sciencesIn business
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