Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Edda Klipp Systems biology 2 – Reaction kinetics Sommmersemester 2010 Humboldt-Universität.

Humboldt-Universität

Zu Berlin

Edda Klipp, Humboldt-Universität zu Berlin

Edda Klipp

Systems biology2 – Reaction kinetics

Sommmersemester 2010

Humboldt-Universität zu BerlinInstitut für BiologieTheoretische Biophysik


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Enzymes

-Proteins, often complexed with cofactors-Anorganic cofactors: metall ions-Organic cofactors (coenzymes): vitamin-derived complex groups

- remain unchanged after reaction as catalyst- have a catalytical centre- are in general highly specific - are often pH- and temperature dependent

Turnover number: 1000 /sec (100 /sec ... 10 million /sec)

Acceleration (compared to non-catalyzed reaction) by 106 to 1012 - fold

Thermodynamics: Enzymes reduce the necessary activation energy for the reaction


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Classification of enzymatic reactions

irreversible - reversible

S PS P

1 2 3 4S

0

0.2

0.4

0.6

0.8

1

V

01

23

S0

1

2

3

P

- 0.50

0.5V

01

23

S


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Number of substrates (and products)

S P

S1+S2 P

S1+S2 +S3 P1 2 3 4

S

0

0.5

1

1.5

2

2.5

V

1

S2 large (0.5)

S2 small (0)

uni

bi

ter


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Type of kinetics

v = k SLinearMass action

HyperbolicMichaelis-Menten

SigmoidalHill kinetics,Monod, Koshland

.

mKS

SVv

max

nb

nb

SK

SKY

1v

1 2 3 4S

0

0.2

0.4

0.6

0.8

1

V

“Hyperbolic” and “Sigmoidal” show saturation, “Linear” involves unlimited reaction rates.


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Deterministic kinetic modeling of biochemical reactions

Basic quantities:Concentration S : number of molecules per unit of volume Reaction rate v : concentration change per unit time

Postulat: The reaction rate v at point r in space at time t can be expressed as a unique function of the concentrations of all substances at point r at time t :

Simplifying assumptions: - spatial homogeneity (well-stirred)- autonomous systemes (not directly dependent on time)

Kinetics of Enzymatic Reactions

v(r,t) = v(S(r,t),t)

v(t) = v(S(t))


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The reaction rate is proportional to the probability of collision of reactants, This is in turn proportional to the concentration of reactants to the power of their molecularity.

(Guldberg and Waage, 19. century)

A+B 2 C

2CkBAkvvv

kk ,

eqeq

eq

BA

C

k

kq

2

Reaction rate

Rate constants

Equilibrium constant

The Mass Action Law


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Brown (1902): Mechanism for Invertase reaction (with sucrose), Which holds for one-substrate-systemes with backward reaction of effectors:

E+S ES E+P1+P2k1 k2

k-1

E – catalystS – substrateP – productki – kinetic constantcomplex

formation reversible

Michaelis, Menten (1913): rate equation under the assumptionThat second reaction will not influence the first equilibrium(Hypothesis of quasi-equilibrium)

21 kk

Briggs, Haldane (1925): more general derivation of Rate law under the assumption of a steady statefor the enzyme-substrate-complex (where )

0d

d

t

ES

ES 0

Michaelis-Menten Kinetics

complexdegradationirreversible


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E+S ES E+P1+P2k1 k2

k-1Non-linear ordinary differential equation system

ESkSEkdt

dS11

ESkkSEkdt

dES211

ESkkSEkdt

dE211

VESkdt

dP 2

- The rate of product formation is equal to the reaction rate

(1)

(2)

(3)

(4)

- The sum of equations (2) and (3) is a conservation relation for the enzyme

ESEEdt

dE

dt

dESt ,0

- The whole set of equations cannot be solved analytically. Using quasi-steady state assumption

0d

d

t

ES

Michaelis-Menten Kinetics: derivation of rate law


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E+S ES E+P1+P2k1 k2

k-1

1

21

2

kkk

S

SEkv t

Reaction rate

tEkV 2max

1

21

k

kkKm

mKS

SVv

max

Maximalvelocity

Michaelis constant

Michaelis-Menten-Rate expression S

vVmax

Vmax2

1

mK

Michaelis-Menten-Kinetics: The rate equation


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Integrated Form of MM rate law

t

S

t

PV

d

d

d

d

mKS

SV

t

S

max

d

d

SS

SKtV m ddmax

S

S

S

SmS

Smt

tS

S

SKS

S

SKtV

0 000d

dddmax

00

max ln SSS

SKtV m

Reaction rate = Product increase or substrate decrease per unit time

Integration from t0, S0 to t, S results in

and for t0 0

This is a function or . One can record a progress curve and estimate the kinetic constants

using non-linear regression.

tSS Stt

S(t)

S0

t

**

*

**

**

Henri-Michaelis-Menten-equation


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Estimation of Parameters Vmax and Km

1. Measurement of initial rates Measure initial rates for different initial concentrations , i.e. measure initial change of S.

t

SS0

V=dS/dt

2. InterpretationPlot measurement results in (S,V)-Diagram; Compare with Michaelis-Menten rate law; Estimate parameters by non-lineare regression, for example least-squares methode

V

S

*

*

*

** *

*

*

*

*

*

*

*


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Linearizations of the MM rate law

Lineweaver-Burk-Plot

maxmax

m 111

VSV

K

V

1/S

Anstieg= Km/Vmax

1/Vmax

1/V

-1/Km

Eadie-Plot

S

VKVV mmax

V

V/S

Vmax

Vmax /Km

Hanes-Plot

max

m

max V

K

V

S

V

S

S

S/V

1/VmaxAnstiegVmax

Km


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Additional aspects

RT

G

eq0

Relation to thermodynamics

Vmax is related to turnover number, kcat

Condition: completely saturated enzymes, maximal rate:

totcat E

Vk max [1/(mol*s)]

Dissoziation constante KS of theenzyme-substrate-complex: 1

1

k

kKS

[mol]


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Here: the enzyme as target of effectors

Regulation of Enzyme Activity

- Regulation of enzyme amount (Gene expression / proteine degradation)

- Action of effectors (inhibitors, activators)

- Composition of mediums (pH, ions)

- Regulation of protein activity by kinases / phosphatases / methylases....

Important mechanism for the regulation of cellularprocesses upon the adaptation to internal and external changes.


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1. Competitive inhibition: substrate and inhibitor compete for the binding place at the enzyme

E+S ES E+Pk1 k2

k-1+I

EI

k3k-3

EIESEEt

I3

3 Kk

k

EI

IE

SKI

K

SVv

Im

max

1

V

S

Vmax

1/2Vmax

Km K'm

mit Hemmung

Equilibrium for inhibitor binding

Conservation relation for the enzyme

Rate equation

Enzyme Inhibition

1/S

1/V

1/Vmax

1/Km1/K'm

i=1

i=2i=3

i=4


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Examples Competitive Inhibition

2. Acetylcholin esterasehas as substrate acetylcholin and is inhibited by Neostigmin. Note that obviously only the charged N(CH3)3

+-group is active.

3. Sulfonamide(antibiotica)block as competitive inhibitors the production of DNA, Since they are used by the enzyme instead of the vitamine precursor p-Aminobenzoesäure.

1. Succinic acid dehydrogenasehas as substrate succinic acid and is inhibited by Malonic acid.

Bernsteinsäuredehydrogenase


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2. Uncompetitive inhibition: Inhibitor binds only to the enzyme-substrate-complex

3. Non-competitive inhibition: Inhibitor binds to free and bound enzyme

E+S ES E+Pk1 k2

k-1+I

EI+Sk3

k-3ESI

I+

KI KI

IS

max

1KI

SK

SVv

Im

max

1KI

SK

SVv

E+S ES E+Pk1 k2k-1

ESI

I+

KI

Enzyme Inhibition

1/V

1/Vmax

-1/Km

-1/V´max

1/S

I=0I=3x I=2x I=1x

Anstieg=Km/Vmax

-1/K´m

Vmax

V

S

Vmax/2

Km

1/V

1/Vmax

-1/Km

-1/V´max

1/S

I=0

I=3xI=2x

I=1x

Anstieg=Km/Vmax

Anstieg=Km/V´maxV´max

V´max/2


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4. Irreversible inhibition : inhibitor binds the enzyme irreversibly, partial or complete loss of catalytic effectivity

EIIE 002max' IEkV

HICOCHSE

COCHISHE

22

22

Example: Reaction of Iod acetate with –SH groupsin cystein side chains of the reaction centre

Enzyme Inhibition, 3

1. Di-isopropyl-fluorophosphate (DFP)and other alkylphosphates bind covalently to acetylcholinesterase. This enzyme is responsible for Transmission of nerve stimuli. The organsims die of paralysis (Lähmung) of organ function.(used in military gases and insektizids)


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Allosteric Inhibition: Inhibition by a molecule that does not bind to the reaction centre. conformation change of the enzyme, Change of reaction coordinate

Enzyme Inhibition , 5

Product Inhibition : - Inhibition by the product due to allosteric inhibition

(prevents excess production)

- Reduction of the net reaction rate, due to an accumulation of product

which is substrate of the backward reaction.


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Substrate Excess Inhibition

E+S ES

ESS

E+PKS k2

KI

+S

mS KKES

SE

IK

ESS

SES

ESSESEEt

Im

max2

1K

SSK

SVESkV

vopt

vopt

KI=Km

KI=100 Km

v

Vmax

S

0.5Vmax

Km

Imopt KKS

Im

maxopt

21 KK

VV

CO2-

CO2-

CH2

CH2 CH2 CH2CO2- CO2-

CO2-CO2- CH2CH2

Example: Succinic acid dehydrogenase

Binding a further substrate molecule to ES-complex Enzyme-Substrate-Complex ESS,Which does not transforms to reaction products. Reversible inhibition, if one molecule dissociates.

Equilibrium assumptions

Enzyme conservation

Reaction rate

Optimum


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ActivationIncrease of the rate by

- Change of substrate binding

- Acceleration of product formation

E+S ES+A

EA+S

+A

EAS

E+P

EA+P

KS

KS

KAKA

k

k

,

,

,

EASkESkV

Enzyme Activation

SESkESkV

SESSEESEEt

ES

SEK S

A

ASS K

KKK

SE

SEK A

SES

SESK A

AA

SS

Att

K

S

K

K

S

K

K

SEkEk

V

1

Vmax

v

S

,

Example Substrate activation:

Substrats S acts as activator A. Reaction rate = Product formation rate

Enzyme conservation

Quasi-equilibrium condition

Reaction rate


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Activation and Inhibition for Mass Action Kinetics

A

PS ASkv

AK

ASkv 1+

I

PSI

KSkv I

IKI

Skv

1

1-

compulsory additional


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Ligand: compound that binds to enzyme / protein

Here: Binding of ligands to monomeric und oligomeric proteins.

several ligand binding sites at a protein:Possibility of interactions between these sites during binding This phenomenon is called cooperativity

Positive/negative cooperativity: Binding of a ligand molecule increases/reduces the affinity of the protein for further ligands.

Homotrope/heterotrope cooperativity :Binding of a ligand molecule affects binding of further molecules Of the same/ other ligands.

Ligand Binding and Cooperativity


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Case of 1 binding site:Binding of S (Ligand) to E (Protein) ESSE

Binding constante SE

ESKb

Definition: Fractional Saturation subunits ofnumber total

ligands bound that subunits, ofnumber Y

Fractional saturation for 1 subunit 10

SK

SK

EES

ES

E

ESY

b

b

Y

S

Plot of Y versus S is hyperbolic

YE

ES

V

V

0max

Fractional Saturation


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Positive, homotrope cooperativitySimplest case: dimeric protein - two similar ligand binding sites- Binding of first ligand increases affinity to second ligand

22schnell

2

2langsam

2

SMSSM

SMSM

SMSM 222

2222

22

2

22

SMSMM

SM

M

SMY

tot

Assumption: Binding of S increases affinityM2S reacts with S as soon as it is formed

Fractional saturation:

Complete cooperativity (each subunit is either empty or completey saturated) 222 SM2SM

22

22

SM

SMKb

2

2

222

22

2

22

1 SK

SK

SMM

SM

M

SMY

b

b

tot

M = monomere Untereinheit, M2 = Dimer

Binding constante

Hill-Kinetik

Fractional saturation:


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For complete homotrope cooperativity of a protein with n subunits holds:This is a form of the Hill equation n

b

nb

SK

SKY

1

nn 22 OHbOHb

Hemoglobin: sigmoid bindung curve of oxygen against oxygen partial pressure

Hill (1909): Interaction between binding sites - positive cooperativity

Known: hem binds oxygen molecules

Unknown: number of subunits per protein

Assumption: complete cooperativity - experimental Hill coefficient h=2.8

•Four Binding sites per hemoglobin molecule •No complete cooperativity •High oxygen partial pressure in lungs: good binding of oxygen to Hb

•Low oxygen partial pressure in body – easy delivery of O2

Y

S

Hill Kinetics


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Monod-Wyman-Changeux model for enzymes with sigmoidal kinetics

Model assumptions (J.Mol.Biol.(1965),12,88) : i) Enzyme consists of several identical subunits (SU)ii) each SU can assume one of two conformations (active = R or inactive = T)iii) all SU of an enzyme have the same conformation Conformation change for all SU at the same time (concerted transition).

R – Conc. active conformationR0 – R- Conc. without bound substrate

R1 – R- Conc. with 1 bound substrate

T – Conc. of inactive conformationT0 – Conc. without bound substrates

R - active T - inactive

0

0

R

TL

L

Conformation equilibrium

Allosteric constant


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Monod-Wyman-Changeux model

n = 4 subunits

Binding constante for substrate S to one SU: KR or KT

(Assumption: Binding only to active form, S +

KR

00 TR KK ,

For each enzyme there are the following possible bound states:

R0 - Concentration of R without substrate binding,

R1 - Conc. of R with 1 bound molecule of S

R2 - Conc. of R with 2 bound molecules of S



1 possibility

4 possibilities

6 possibilities.

4 possibilities

1 possibility

General: Possibilities of substrate binding for Ri !!

!

iin

n

i

n


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RKSR

R4

0

1

SRKR R 01 4

RKSR

R

4

6

1

2

202 6 SKRR R

RKSR

R

6

4

2

3

303 4 SKRR R

RKSR

R

4

1

3

4

404 SKRR R 0

00 SKRR R

It holds:

iRi SKR

i

nR 0

General:

n

i

iR

n

ii SK

i

nRRR

00

0 n

RSKRR 10

YTRn

iRn

ii

1

TSKRn

SKRi

ni

Yn

R

n

i

iR

10

10

00

10

1

1

TSKR

SKSKRY

nR

nRR

Sum of all active states:

with binomic Formula:

Fractional saturation

Replacement of R and Ri

T exists only as T0


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TSKR

SKSKRY

nR

nRR

1

1

0

10

It follows

10

1

00

1

10

10

100

1

1

!!1

!1

:1 !1-!

!1

!1-!

!1

!!

!

nRR

jR

n

jR

iR

n

iR

iR

n

i

iR

n

i

iR

n

i

SKSKnR

SKjjn

nSKnR

jiSKiin

nSKnR

SKiin

nnR

SKiin

niRSKR

i

ni

inin

i

n bai

nba

1

LSK

SK

SK

SKY

nR

nR

R

R

1

1

1

nR

R

R

SK

LSK

SKY

11

1

1

YEkV t tEkV max

nR

R

R

SK

LSK

SKVV

11

1

1max

Reaction rate

Michaelis-Menten-Term

"Regulatory Term"


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0 2 4 6 8 100

0.2

0.4

0.6

0.8

1 0 102 103

104

activation

inhibition

S

v

For S∞ : Monod-Kinetics approaches Michaelis-Menten-Kinetics

small S: regulatory term important depending on L L = 0: MM-Kinetics L >> 0: sigmoidal curve, shifted to right.

Explanation of the action of activators and inhibitors: - Activators bind to active conformation- Inhibitors bind to inactive conformation -Shift of equilibrium to R or T

A R

I T

AI KK , Bindungskonstanten n

A

nI

AK

IKLL

1

1


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F6 ATP FDP ADPPFKP

Example: Phosphofructokinase: experimentaly well studied system

Activators: Inhibitors: DPG, ATPTypical value for

AMP, NH K4+,

L 104


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E+S ES E+Pk1 k2

k-1 k-2

Derivation of rate equation for steady state

0dtESd VdtdP

Relation between equilibrium constant q and kinetic constants of elementary steps 21

21

kk

kkq

121

2

221

1 1

k

P

kk

k

kkk

SkPSq

EV t

mPmS

mP

rück

mS

vor

K

P

K

S

PK

VS

K

V

V

1

maxmax

mSrück

mPvor

KV

KVq

max

max

Reaction rate

Kinetics of Reversible Reactions


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E+S ES E+Pk1 k2

k-1 k-2

Relation to phenomenological quantities

S very high, P=0 P very high, S=0 Half-maximal forward rate Half-maximal backward rate

vorttt VkE

Sk

kSkE

kkSk

kSkEV max2

1

21

211

21

rücktt

t VkEPk

kPkE

Pkkk

kPkEV max1

2

21

221

21

mSvor K

k

kkSVV

1

21max

2

1

mPKk

kkPVV

2

21rückmax

2

1

For S and P very small holds

This resembles Mass action kinetics

(Also called linear kinetics).

1 mPmS KPKS PkSkPK

VS

K

VV

mP

rück

mS

vor

maxmax

Kinetics of Reversible Reactions


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Several activated complexes


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Methode of King and AltmanEmpirical methode to derive steady-state rate equations for reactions, Which are catalyzed by an enzyme (no interaction between enzymes!)

1. Conservation of total enzyme amount:

n

iitotal EXE

1

2. Relative concentration of each enzyme speciesis equal to ratio of two sums of terms,where every term Tij is the product of n-1 rate constantsand the related concentrations.

n

i jji

jji

total

i

T

T

E

EX

1,

,

3. Every term Tij contains the rate constants (times substrate conc.),which are associated with the steps leading individually or sequentially to EX i .The sum of all possible combinations (j) are the numerator, the sum of all numerators for all EXi is the denominator.

EXi - freies Enzym

4. The reaction is:

n

i jji

jjnn

jjnn

tnnnn

T

TPkTk

EEXPkEXkdt

dPv

1

1

1

,

,,


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King-Altman for 3-Step reaction mechanism

E ES

EP

Sk1

k-1 k-2k2k3

Pk-3

1. Conservation of total enzyme amount: : EPESEEtotal

2., 3. Listing of all possibilities of n-1 = 2 lines leading to each enzyme species:

For Ek-1

k3

k-1

k-2 k3 k2

322113

3

1

kkkkkkTj

jE

,

For ESSk1

k3

Sk1

k-2 Pk-3 k-2

322131

3

1

PkkkSkkSkTj

jES ,

For EPSk1

k2 Pk-3 k2

322131

3

1

PkkkSkPkkTj

jEP ,

k-1

Pk-3

21232321322113

3

1

3

1

3

1

kkkPkkkkSkkkkkkkTTTNennerj

jEPj

jESj

jE ,,,

21232321322113

32132133

kkkPkkkkSkkkkkkk

kkPkkkSkEEPkEPkv total4. Reaction rate:


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E EA EAB EPQ EQ EAk1 Bk2 k3 k4

k-1 k-2 Pk-3 Qk-4

Ordered bi-bi-Mechanismus (Example: Kreatinkinase)

Further typical Mechanisms

iBmBiA

mAmP

iAmQ

eqr

f

mAiAmBiP

eq

f

KB

PKK

BKKQ

KA

PKKV

VBKKAK

KP

AB

KPQ

ABV

v

1111


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E EA EAB EPQ EQ EAk1 Bk2 k3 k4

k-1 k-2 Pk-3 Qk-4

Ordered bi-bi-Mechanism (Example: Kreatinkinase)

Ping-Pong-Mechanism (Example : Transaminase, Nukleosid-Diphosphokinase)

E EA FB EQ EAk1 k2 Bk3 k4

k-1 k-3Pk-2 Qk-4FP F

Random bi-uni-Mechanism (Example : an Aldolase-Type)

E

EA

EAB

EB

E

Ak1 Bk2

Bk3 Ak4

k-1 k-2

k-4k-3

k5

Pk-5

Further typical Mechanisms


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Unbranched Reaction Chain

EXn

EX1

1k~

1k~

EX2

EX2

EXn-1

nk~

nk~

2k~

2k~

iii kRk ~

n

i i

i

k

kq

1~

~~

n

i

ji

ir r

rn

j i

n

i i

total

k

k

kk

qEv

1

11

1 11

11

1

~

~

~~

~

Apparent rate constants

Apparent equilibrium constants

General rate law

Holds for all sequential reaction mechanisms


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Example


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Convenience Kinetics(actually a generalized random kinetics….)

-200 -100 100 200

-400

-200

200

400

-200 -100 100 200

-400

-200

200

400

-400 -200 200 400

-400

-200

200

400

Convenience Kinetics

Ord

ere

d K

ineti

cs

r=0.946 r=0.975 r=0.983

Ordered KineticsPin

g-p

ong K

ineti

cs

Convenie

nce

Kin

eti

cs

Ping-pong Kinetics


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Other types of kinetics: S-SystemsIntroduced by M. Savageau, 1976 („synergistic systems“)

mn

j

hji

mn

j

gji

iii

jiji XX

VVX

11

,,

For i = 1...n n independent variablesm dependent variables

Steady state:

mn

jjiii

mn

jjiii

mn

j

hji

mn

j

gji

mn

j

hji

mn

j

gji

i

LogXhLogLogXgLog

XLogXLog

XX

X

jiji

jiji

11

11

11

0

0

,,

,,

,,

Xi

Xj2Xj3

Xj4

Xj5Xj1 Vi+ Vi

-

g, h – positive or negative, usually no integers


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Other types of kinetics: Lin-Log Kinetics

Sef Heijnen and others

0000lnln1

c

c

S

S

E

E

v

v

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Edda Klipp Systems biology 2 – Reaction kinetics Sommmersemester 2010 Humboldt-Universität.

Documents

reaction slide

rate law reaction rate

rate equation slide

products s p s

concentration s

s results

derivation of rate law

maximal rate