Illia Horenko & Wilhelm Huisinga Einführungsvortrag zum Seminar Modellierung dynamischer Prozesse in der Zellbiologie: Deterministischer Zugang Freie Universität Berlin, 08. May 2003
Illia Horenko & Wilhelm Huisinga Einführungsvortrag zum Seminar Modellierung dynamischer Prozesse in der Zellbiologie: Deterministischer Zugang Freie Universität.
Mar 26, 2015
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Illia Horenko &Wilhelm Huisinga
Einführungsvortrag zum Seminar
Modellierung dynamischer Prozesse in der Zellbiologie: Deterministischer
Zugang
Freie Universität Berlin, 08. May 2003
Mat
h. M
odel
lieru
ng in
der
Zel
lbio
logi
eContents
• Homogeneous deterministic model– Derivation of reaction equation– Matlab-application (Michelis-Menten)
• Heterogeneous deterministic model– Derivation of reaction-diffusion equation– Numerical realisation (Michaelis-Menten)
Mat
h. M
odel
lieru
ng in
der
Zel
lbio
logi
eMass Action Law
Bimolecular reaction:
ESSEk
k
1
2
Model assumptions: M, V constant; distribution of particles is homogeneous
0
0
)()(
)()(
SESS
EESE
NtNtN
NtNtN
Mat
h. M
odel
lieru
ng in
der
Zel
lbio
logi
eMass Action Law
Change of particles number due to the reactions:
ttNkttNtNkttN ESSEES )()()()( 21
)())())((()( 200
1 tNktNNtNNktNdt
dESESSESEES
0t :
Mat
h. M
odel
lieru
ng in
der
Zel
lbio
logi
eODE’s: Solution
Linear autonomous ODE:
00 )( yty
Ayydt
d
Analytical solution:
00 ))(exp()( yttAty
Numerical methods
Implicit(ode15s)
Rational representation:
))((
))(())(exp(
0
00 ttAR
ttAQttA
n
nn ttAn
ttA )(!1
))(exp( 00
Polynomial representation: Explicit(ode23, ode45)
where
n
n AAAA ...
Mat
h. M
odel
lieru
ng in
der
Zel
lbio
logi
eODE’s: Matlab Numerics
Bimolecular kinetics:
ESSEk
k
1
2
Timesteps are adapted to absolute error tolerance AbsTol → odeset(…)
Mat
h. M
odel
lieru
ng in
der
Zel
lbio
logi
eODE’s: Stiffness
Michaelis-Menten enzyme kinetics:
PEESSEk
k
1
2
Models with very differrent timescales are “stiff“ → implicit integrators such as ode15s are recommended
Mat
h. M
odel
lieru
ng in
der
Zel
lbio
logi
e Efficient Modelling of heterogeneity
compartment models (partially heterogeneous): (1) well-stirred within compartment (ODE,MP) & (2) interaction between compartment (consistent coupling)
photo: http://genome-www.stanford.edu/Saccharomyces/yeast_images.shtml
Note: the more complex the model, the more parameters it needs!
reaction-diffusion models (fully heterogeneous) (a) concentration in time and space (deterministic PDE) or (b) 3d-molecular positions (random walk and reaction probabilities)
Mat
h. M
odel
lieru
ng in
der
Zel
lbio
logi
eHeterogeneous deterministic Model
Bimolecular reaction:
ESSEk
k
1
2
Model assumptions: M, V constant; distribution of particles is inhomogeneous (dependent on coordinate R)
Change of concentration i due to the reactions and diffusion:
diffusion
idiff
reactions
lil
li ttRcttRctRckttRc ),(),(),(),(
Mat
h. M
odel
lieru
ng in
der
Zel
lbio
logi
e
),(),(),()(),( tRctRcktRcRDdivtRct i
llliiRii
Species i is described through concentration and diffusionconstant
Modelling heterogeneity (PDE)
),( tRci)(RDi
Modell reduction: use spherical symmetry
),(),(),(2
),(),( 2
2
trctrcktrcrr
Dtrc
rDtrc
t il
lliii
iii
Mat
h. M
odel
lieru
ng in
der
Zel
lbio
logi
ePDE: Numerics
When is large w. r. to reaction constatnts => homogenous modelling is sufficient!
11
11 ),(),(),(
jj
jijijiR RR
tRctRctRc
1.Spatial discretisation 2.Time discretisation
System of ODEs for concentrationsat grid points
Finite differences:
Method of lines
)(RDi
Mat
h. M
odel
lieru
ng in
der
Zel
lbio
logi
ePDE: Michaelis-Menten
S, ES and P are fixed on the membrane (D=0)E diffuse freely from nucleus to membrane
Related Documents
