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Biochemical Networks
Literature:Cantor&Schimmel: Biophysical ChemistryAdam Läuger Stark : Physikalische Chemie und BiophysikVoit: Computational Analysis of Biochemical Systems
Modelling Biochemical Networks
Literature:
Voit: Computational Analysis of Biochemical SystemsAdam Läuger Stark : Physikalische Chemie und BiophysikBreckow : Biophysik
Cooperative EnzymesInhibition, RegulationKinetic RatesSynergistic SystemsParameter Estimations
Modelling of Biochemical networks
Open Systems
Source
Drain
Eq.(1)
Closed Systems
There are no Energy and mass fluxes in a closed system. The system relaxes to a steady state SY
Eq.(1) reduces to:
Be
and
hence
and
R: Boundary conditon
The steady state depends on the starting conditions! There is no possibility of regulation in a closed System!
Disturbing a closed system
(e.g. adding some kind of agents)
closed system
state variable
stat
e va
riabl
e
Kinetics
For values different to equilibrium:
and
and mass conservation
hence
with the relaxation rate
Kinetics of first order
starting rate:
steady state
Vocabluary: Order of reaction:
Reaction of 1. Order
Reaction of 2. Order
two coupled reactions
Relaxation in two phases!
General, monomolecular Reaction scheme of open systems
Reaction steps with intermediates
Solution
constant factors dumping constants
Autokatalysis, Voltera-Lotka Systems
The differential equation without feedback is :
The autocatalytic step introduces a nonlinearity
und
Extra from the lab: Crosskatlytic systems
show the AHA-Simulator
Ergebnis?
Experimental Methods for Kinetics measurement:
Time dependent measurement of concentration (Pressure, Absorbtion, electric resistance....)
Mixure experimentsContious flow, stop flow
Relaxation methods, (e.g. tempjumps)
Stopped flow method
Principle of stop-flow methods
Relaxation time spectroscopy
PNAS, November (2009)
Ingmar Schön, Hubert Krammer & Dieter Braun
Hybridisation Kinetics is different inside cells
experimental setup
quantum efficiency
illumination
periodic illuminationphase-locked relative to perturbation
0° 90° 180° 270°
collect fluorescence by slow CCD(low-pass filtering)
lock-in detection scheme
fit with transfer functionfor a first-order reaction
approach
goal: measure reaction kinetics in vivo
principle: perturbe equilibrium and analyze relaxationdetection: fluorescence resonance energy transfer (FRET)
DNA probe
RhG |5’-C AGG TTA CTA TCG TAT T C-3’
ROX |5’-C AAT ACG ATA GTA ACC T C-3’
C = L-enantiomeric cytosin
DNA probe
hybridization kinetics in a single living cell
different kinetics in subcellular compartments
dependence on concentration
calibrationbrightness of confocal image vs. DNA concentration
dependence on concentration
calibrationbrightness of confocal image vs. DNA concentration
comparison in vitro vs. in vivo
… faster Hybridization in vivo!
However 12bp probe…
… is slower: Binding with Proteins !
Molecular Crowding is no significant for short DNA
Trivial molecular crowding:excluded volume enhances local concentration, however both for 12 & 16 mer => Not found
Length dependent, specific interactions:- Catalytic speed up of Hybridization- Slowing by specific binding => Less free concentration and slower kinetics
The principle of detailed balance
In a more complex system (e.g. a cyclic system) once can think of reactions with dx/dt=0, which are thermodynamically allowed, but enable a permanent flux of material.
The equilibrium condition is valid for all subreactions of the System : „Equilibrium is way-independent“Prinziple of detailed balance(Prinziple of microscopic reversibility)
Michaelis-Menten Kinetics
Enzyme-catalysed Reactions have a Enzyme-Substrate complex as active intermediate
Katalyzes the Hydrolysis of
Ph-optimun is alcalic
Structural Properties of Alkaline Phospatase: -2 Subunits with 1 catalytic subunit each-Ser-102 binds transient covalent to the Phosphate during activity-2 Zn2+ Ions in the catalytic subunit are essential-1 Mg2+ Ion binds allosteric and increases catalytic activity-Glycoproteine-Membrane Proteinelocated on the outside of the cell membrane through covalently bound lipid
Model reaction for approvement
The Kinetics of chemical reactions
and the role of Activation energy
Enzymes lower the energy
barrier
Lower activation energies lead to higher reaction rates, since the fraction of molecules with enough energy to pass the barrier is
increased
Energy distribution of
molecules
The Kinetics of chemical reactions
and their velocity
The Kinetics of enzymatic reactions
1)The more substrate, the faster Kinetics, but...2) ..the higher occupated fraction, the more difficulties for new substrate molecules to find a new enzyme3) Maximal reaction speed is reached, as soon as all
Enzymes available are occupied by substrate
Solution of the Michaelis-Menton rate equations
To solve the equations, three assumptions are made: Creation and degeneration of complex [ES] are in a steady state, for the condition [E]<<[S]
We observe the early phase, where [P]<<[S]There is lot more substrate as is used[S]=[S
o]= constant
How educated is the assumtion of a stationary intermediate state?
Numeric simulation with 5 intermediate states
Enzyme Kinetics in quasi-stationary regime
The concentration of enzymes [E] is limited by the altogether amount of
Enzyme [E0], hence [E]=[E
0]-[E
S].That leads to:
Together with
it follows the solution
Michalelis-Menton-constantMichalelis-Menton-Rate Maximal rate
Michalelis-Menton-constantMichalelis-Menton-Rate Maximal rate
Enzyme Kinetics in quasi-stationary regieme
Experimental measurement of maximal reaction speed
Plotting the reaction speed against the substrate concentration reveals a saturation curve that converges against an upper Limit
The half-maximum-speed is reached, if half of all Enzymes are occupied with substrate.
Reaction speed, affinity and Michaelis-Menden-Constant
The better a substate binds to an enzyme, the lower is the necessary concentration of substrate to bind half of all enzyme molecules
Reaction speed, affinity and Michaelis-Menden-Constant
The better a substate binds to an enzyme, the lower is the necessary concentration of substrate to bind half of all enzyme molecules
Kd is the substrate-concentration
at which the hal-maximum speed is reached.
Kd is therefore a concentration
with the dimesions of mol/L Kd
The Lineweaver-Burk Diagramjust another way of plotting...
In a MichaeDiagram it is kind of hard to measure V
max
precisely.Therefore it ilis-Menten Diagram it s also not possible to measure V
max/2 and the K
d-
value precisely
The Lineweaver-Burk Diagramjust another way of plotting...
In a Lineweaver-Burk diagram the reciprocal 1/V and 1/S is plotted instead of V and S. Then the intersection with the x-axis has the value of -1/K
d
Biological Regulation via Enzyme inhibition
Effects of noncompetitive inhibition
Non-competitive inhibition
Competitive Inhibition
Inhibitor competes withsubstrate for binding toenzymeExample 1: most drugsExample 2: Product inhibition
Problem :Die kompetitive Hemmung hat unzureichende Regeleigenschaften
Multiside binding and the effect of Cooperativity
„Switch-off function“
Sollwert
rate is dependent on inhibitory substance
An enzyme with more then one binding site for an inhibitor allowas a sharp regulation
Cooperativity of allosteric Enzymes
Michaelis-Menten-Kinetiks(n=1)
Hill equation
The Hill-coefficient is extracted from experimental data by a logarithmic plot of v(s) (Hill Plot)
Das Operon-Regelsystem nach Monod:Beispiel allosterischer Kontrolle
G: Genprodukt (z.B. Enzym, das Bildung von P aus Substrat St katalysiert)
Für die Komplexbildung von Produkt P mit Konzentration yP und dem regulatorische Gen R wird eine kooperative Rückkopplung angesetzt
Autokatalysis, Voltera-Lotka Systems
The differential equation without feedback is :
The autocatalytic step introduces a nonlinearity
und
Predator-Prey-System
In nonlinear systems more then one steady state can appear. Diskriminative Switch
Formation of inactive complexes between two partners
Relationships (Shiraishi-Savageau, 1992)
Homogeneous3D reactions-> pos. integers
Kinetic orders= weighted averages ofmore elementary ko´s (Alves-Savageau, 2000)
Modelling of biochemical networks
Quelle: Stelling, Curr.Op.MicroBio 2004
Metabolische Netzwerke
Metabolische Netzwerke sind durch eine Netzwerktopologie (pathway) und biochemische Ratengleichungen beschrieben.
S-Systeme : einfache nichtlineare Näherung mit numerischen Vorteilen
Elementare Fluss Moden Analyse : Stoichiometrisches Fliessgleichgewicht
Computergestützte Analyse
Metabolic Netzwworks
Metabolic networks are described with help of an network topology (pathways and biochemical rate constants
S-Systeme : simple, nonlinear approximation with numeric benefits
Elementar flux modes : Stoichiometric Balance of fluxes
Computonal Analysis
S-Systemes
• Produktansatz for influx and outflux Vi+ and Vi
-.
dXi/dt = V+-V-=αi ∏j=1n+m Xj
gij - βi ∏j=1n+m Xj
hij
αi and βi : rate constans
- gij and hij : cinetic exponents
– Xi : concentrations of all the metabolites that are involved in the network
Why do S-Systems work ?
Educated assumtion :
* Analytic Steady-State-solution * Maths is simple* Arbitrary Systems of Differential equations can be translated into S-Systems.* Parameter estimation is possible
Benefits:
Most biochemical Systems are in a quasistatic state, the dynamics of sytem regulation is slow compared with the dynamics of the system itself. S-Systems are a extension around a steady stateBiochemical Systems are robust. Functionality is more or less independent of concentration
aus Torres: Pathway Analysis
Order of Kinetics from experimental data
Stociometric Matrix:flux analysis