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