Modelling of biochemical systems

Vangelis Simeonidis

Manchester Centre for Integrative Systems Biology

Biochemical networks

Course Objectives

Know how to:

• BUILD a model of a biochemical network

• SIMULATE a model of a biochemical network

• ANALYSE a model of a biochemical network

Motivation: The cycle of knowledge

OBSERVATIONSChemistry/Biology/

Genetics ExperimentsKNOWLEDGE

Structured Databases

Model

Synthesis/Induction

Hypothesis/Deduction

Modelling Inspiration: The Virtual Heart Story

BioEssays 24:1155–1163, 2002

•Noble, 1960: First computer model of heart

•DiFrancesco and Noble, 1985: Transporters and pumps

•Noble, 2006: The virtual heart

The Future of Biochemical Modelling

Personalised organ scale or human scale models

“Let me check your model…”

“Will this drug help me?”

Patient Doctor

Types of Biochemical Networks

• Metabolic networks for energy and synthesis

• Signalling networks - cells response to their environment

• Gene expression networks

• PK/PD (Pharmacokinetic/Pharmacodynamic) models

DNA

RNA

Enzymes

Metabolites

Chemical Kinetics

Example:

[A][B]kν ff

kf

kb

A + B C+ D

Forward reaction:

Backward reaction: [C][D]kν bb Mass action kinetics:

At Equilibrium forward andbackward rates balance:

eqb

f

bfbf

Kk

k

[A][B]

[C][D]

[C][D]k[A][B]kνν

Equilibriumconstant

Enzyme Kinetics

• Enzymes are proteins that catalyse biochemical reactions

• Specific enzymes for specific reactions: alcohol dehydrogenase

• Enzymes reduce the activation energy of the reaction enabling it to run faster

Reaction Coordinate

Gibbs free energy (G)

Reactants

Products

Ecat

EEcat<< E

Enzyme Kinetics

ADH1CH3CH2OH + NAD+ CH3CHO + NADH + H+

ADH2

• Example: Fermentation by yeast

(EtOH) (AcAld)

Alcohol dehydrogenase

Byproduct of glycolysis

Michaelis-Menten Kinetics

Substrate Concentration [S](mmol l-1)

Rate of reaction (mmol l-1 min-1)

Vmax = kcat [E]

Km

0.5 Vmax

Vmax [S]

Km + [S]

Low sensitivity to [S]

High sensitivity to [S]

S PE

Rate of reaction Rate of consumption of substrate S = Rate of production of product P

[S]K

[S]V

dt

d[P]

dt

d[S]

m

max

Ordinary differential equation

Michaelis-Menten Kinetics - Derivation

ESkESkdt

Sd

dt

Pd1-1

21-

1

00

21-

10

kkSk

1

EEE

kk

ESkEEESE

21-

121-1 kk

ESkES0ESkESkESk

dt

ESd

But total enzyme is constant:

Substituting for [E]:

Now we assume [ES] is approximately constant:

21-

21

21-

1-1 kk

kkES

kk

k1kES

S

kkk

SEk

kk

kk

kkSk

1

ES

1

21-

02

21-

21

21-

1

0

SK

SV

m

max

1

21-m

02max

k

kkK

EkV

Substituting for [ES]:

Rate of reaction:

where:

S + E ES E + Pk1

k-1

k2

Variables vs. Parameters

• For a biochemical network: variables = species concentrations

• Integration of differential eqns gives concentration profiles over time

maxmax

m

m

max

V

1

S

1

V

K

ν

1

SK

SVν

• Parameters found by experiment– e.g. Lineweaver-Burke plot for Michaelis-Menten equation

time

Concentration

Species X

Species Y

Slope =ν

1

S

1

X

X

X

X

X

X

max

m

V

K

mK

1

Differential Equations for Biochemical Networks

S + E ES E + Pk1

k-1

k2

SK

SV

dt

dS

m

max

• …are converted into ordinary differential equations (ODEs):

• Chemical equations governing interactions & transformations…

(Association, Dissociation, Catalysis)

Rate of change ofconcentration with time

Biochemical Network Structure

• Mathematical representation as a graph of connected nodes

SPECIES nodes

REACTION nodes

connected by FLUXES

Stoichiometric Matrix

2

1

1

1

1

1

0

0

1

1

1

0

1

1

1

P

SE

E

S

• Back to simple example S + E ES E + Pk1

k-1

k2

S

E

ES P

2

211

211

11

dt

Pddt

ESddt

Eddt

Sd

Stoichiometric matrix

vector ofreaction rates

General systems of Differential Equations

m21n21ii k,...,k,k,X,...,X,Xνν

2

1

1

1

1

1

0

0

1

1

1

0

1

1

1

P

SE

E

S

NνX

Stoichiometricmatrix

vector of timederivative of speciesconcentrations (system

variables)

Each reaction rate is a function of the species concentrations and some parameters

vector of reaction rates

• Previous example:

• Generally:

parametersvariables

SK

SV

m

max

e.g. Michaelis-Menten

Data required to solve an ODE Model

Stoichiometry linking the species and reactions (network structure)

Functional form of the reaction rate equations

The values of the rate constants in these equations

The initial values of all species concentrations

The time horizon

Solving (i.e. Integrating) ODE Models

• Use an ODE solver (Matlab, Mathematica, Silicon Cell, etc.)

• Solver output is the concentration profiles over time

time

Concentration

Species X

Species Y

• Compare model output to measured concentration profiles

• If all profiles become flat then the system reaches ‘steady state’

Steady States

• At steady state all the fluxes are in balance

• Total production of each species = total consumption

• Example: Synthesis and degradation of mRNA and proteins

deg

syn

SS k

kmRNA

DNA

mRNA

Protein

RNA PolymerasesNucleotides

RibosomesAmino Acids

RNAases

Proteases

mRNAkkdt

mRNAd degsyn synk

degk

Stability of Dynamic Systems

• If system returns to steady state after small perturbations then– Steady state is STABLE

dt

dxUnstable steady state

Stablesteady state

Unstable steady state

x

Kinetic modelling

226 16

26 16

6 62 2

26 16 ( )0

6

6 ( )6 ( )6 ( ) 1

1 1 1PFK PFK PFK AK PFKF bP F bP AMP ATP

PFK PFK PFKF bP F bP AMP

PFK RR PFK PFK PFK PFK

F P ATP ATP F PPFK

C F bP C F P t C Keq Ci

K K KPFK PFKATP F P

g F P tF P tg Vm F P t

Km Km Km Kmv

LKm Km

2

26 16 6 6

2 2

2 22 26 ( )16 ( ) 6 ( )26

1

1 1 1 1

PFKATP

PFK PFKATP ATP

AKR

PFK PFK PFK PFK PFK PFK PFK PFKF bP F bP AMP ATP F P ATP ATP F P

C

Ki Km

g F P tF P t Keq F P tF bPK K K Ki K Km Km Km

Teusink et al. glycolysis model (Eur J

Biochem 267:5313, 2000)

aims to characterize fully the mechanics of each enzymatic reaction

2 2 2( ) 4 ( ) 2 ( ) 8 ( ) ( ) 4 ( )

2 8

AK AK AKAXP AXP AXP AXP

AK

P t Keq P t P t Keq P t P t Keq P t

Keq

2 2 22 ( ) 8 ( ) ( ) 4 ( )

1 4

AK AKAXP AXP AXP AXP

AK

P t Keq P t P t Keq P t

Keq

( ) GLK GLTind GLC tv v

dt

6 ( )2

..............................................................

GLK GLYCOGEN PGI TREHALOSEd G P tv v v v

dt

Kinetic modelling

Teusink et al. glycolysis model (Eur J Biochem 267:5313, 2000)

aims to characterize fully the mechanics of each enzymatic reaction

full detail

costly; time-consuming

unknown mechanics

Metabolic Control Analysis (MCA)

• MCA can be applied to both metabolic and signalling pathways

• MCA = sensitivity analysis– how does a small change in parameter X effect model output Y?

• studies the relative control exerted by each step (enzyme) on the system's variables (fluxes and metabolite concentrations)

• apply perturbation and measure the effect on the variable of interest after the system has settled to a new steady state

• Resources: • http://bip.cnrs-mrs.fr/bip10/mcafaq.htm• http://dbkgroup.org/mca_home.htm

Metabolic Control Analysis (MCA)

• control coefficient: relative measure of how much a perturbation affects a system variable

• e.g. Flux control coefficients CeJ for metabolic networks

– CeJ= % change in flux J due to a 1% change in level of enzyme e

– If 1% increase in enzyme gives 5% increase in flux then CeJ = 5

– If 1% increase in enzyme gives 0.4% increase in flux then CeJ = 0.4

MCA: The Summation Theorem

• for a given flux the sum of its flux-control coefficients of all steps in the system is equal to unity

• For concentration control coefficients:

– increases in some of the flux-control coefficients imply decreases in the others so that the total remains the same

– control coefficients are global properties

– in metabolic systems, control is a systemic property, dependent on all of its steps

The “rate limiting step”

• enzymes often after a branch point and catalysing irreversible reactions (with very high equilibrium constants)

• The assumption was that such enzymes operate at a lower velocity than others, and so they “control” the pathway

• But in MCI, the summation theorem applies:

• experimental studies have revealed that a large increase of enzyme is not accompanied by equivalent increases in pathway flux

• increasing the amount of the “rate-limiting” enzyme, its control over the pathway flux would decrease until it eventually approached 0

Control is shared between all enzymes in different proportions

What about kinetics data?

• properties of each enzyme are measured using a sensitivity, known as the elasticity coefficient

• defined as the ratio of relative change in local rate to the relative change in one parameter (eg the concentration of an effector)

• NOT systemic properties

• Connectivity theorem:

Case study 1: Yeast Glycolysis

Silicon Cell website http://jjj.biochem.sun.ac.za/database/index.html

• Find steady state flux through:– Glucose Transporter (GLT)– Glycogen (GLYCO)– Trehalose (Treha)– Phosphofructokinase (PFK)

• How long does it take system to reach steady state?

• What is the effect of decreasing extracellular glucose level from 50 mM to 2 mM on flux through ADH (flux of ethanol)?

Teusink et al. glycolysis model (Eur J Biochem 267:5313, 2000)