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Mathematical Modelling and Scientific Computing in the Biosciences

Lecture location: HF 9904, time: Tuesdays 13:45-15:15. Computer Lab:  HF 107.
Lecturer: Dr. James Lu (Email: james.lu@oeaw.ac.at, Office: HF130)
Useful books:
●"Computational Cell Biology", C. P. Fall, E. S. Marland, J. M. Wagner, J. J. Tyson, editors. Mathematical Biology Series, 2002, Springer Verlag.
●"An Introduction to Systems Biology: Design Principles of Biological Circuits", U. Alon, Mathematical and Computational Biology Series,  2007, CRC Press.
●"Mathematical Biology I: an Introduction", J. D. Murray. Mathematical Biology Series, 2002, Springer Verlag.

Course Overview

Biological Topics/Models

● Enzyme Kinetics

        - Mass-action, Hill-Langmuir equation, Michaelis-Menton equation

● Neuron Dynamics

        - Hodgkin-Huxley model

● Cell Cycle

● Circadian Rhythm

 ◂ ▸  2 of 4

Course Overview

Mathematical Topics

● Singular Perturbation

        - Fast/slow time-scale separation, Hill equation

● Non-Dimensionalization

        - Buckingham Π-theorem

● Numerical ODE integration

        - Methods, accuracy, stability

● Dynamical Systems

        - Bifurcations: theory and numerics

● Inverse Problems

        - Parameter identification, inverse dynamical analysis

 ◂ ▸  3 of 4

Course Overview

Computational Tools

● MathSBML

    - Mathematica package for reading and analysis of models encoded in the
      Systems Biology Markup-Language (SBML) format
    - Webpage: http://sbml.org/software/mathsbml/

● MATCONT

    - MATLAB package for bifurcation analysis of dynamical systems
    - Webpage: http://www.matcont.ugent.be/

● SBML ODE Solver Library

    - C library and command line application for numeric and symbolic analysis of SBML models
    - Webpage: http://www.tbi.univie.ac.at/~raim/odeSolver/

● SBML Inverse Eigenvalue Analyzer

    - Mathematica Add-On package for exploring possibility of qualitative dynamical behaviors
       via inverse eigenvalue analysis

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Illustrative Example: Hodgkin-Huxley Model of the Squid Axon

In[1]:=

<<MathSBML.m

MathSBML Version 2.6.0.13 (21-Dec-2006) using Mathematica Version 5.2 for Linux (June 20, 2005) loaded 6-March-2007 08:08:30.064535

In[2]:=

ModelHH = SBMLRead["~/Teaching/SBMLModels/HodgkinHuxley_Squid_Axon.xml", verbose→ True, evaluateParameters→ False] ;

File Name:~/Teaching/SBMLModels/HodgkinHuxley_Squid_Axon.xml<br />SBML Level 2 Version 1

Model name: hodgkin-huxley squid-axon 1952<br />Model id: hhsa—1952

Function Definitions

----- None -----

Unit Definitions (Excluding Built-in Units)

ID MetaID Name Formula
time metaid—0000003 ··· second/1000
millisecond metaid—0000004 ··· second/1000
per—millisecond metaid—0000005 ··· 1000/second
millivolt metaid—0000006 ··· volt/1000
milliS—per—cm2 metaid—0000007 ··· (10*siemens)/metre^2
microF—per—cm2 metaid—0000008 ··· farad/(100*metre^2)
microA—per—cm2 metaid—0000009 ··· ampere/(100*metre^2)

Compartments

ID MetaID Name Dimension Size Units Derived Units Outside Constant
default ··· ··· 3 ··· volume litre ··· True
unit—compartment metaid—0000032 unit—compartment 3 ··· volume litre default True

Species

----- None -----

Global Parameters

ID MetaID Name Value Units Derived Units Constant
V metaid—0000010 transmembrane voltage -75 millivolt volt/1000 False
II metaid—0000011 applied current 0 microA—per—cm2 ampere/(100*metre^2) True
i—Na metaid—0000012 sodium current ··· microA—per—cm2 ampere/(100*metre^2) False
i—K metaid—0000013 potassium current ··· microA—per—cm2 ampere/(100*metre^2) False
i—L metaid—0000014 leakage current ··· microA—per—cm2 ampere/(100*metre^2) False
m metaid—0000015 sodium channel activation coefficient 0.05 dimensionless dimensionless False
h metaid—0000016 sodium channel inactivation coefficient 0.6 dimensionless dimensionless False
n metaid—0000017 potassium channel activation coefficient 0.325 dimensionless dimensionless False
E—R metaid—0000018 resting membrane potential -65 millivolt volt/1000 True
Cm metaid—0000019 membrane capacitance 1 microF—per—cm2 farad/(100*metre^2) True
g—Na metaid—0000020 maximum sodium channel conductance 120 milliS—per—cm2 (10*siemens)/metre^2 True
g—K metaid—0000021 maximum potassium channel conductance 36 milliS—per—cm2 (10*siemens)/metre^2 True
g—L metaid—0000022 maximum leakage conductance 0.3 milliS—per—cm2 (10*siemens)/metre^2 True
E—Na metaid—0000023 sodium equilibrium potential ··· millivolt volt/1000 False
E—K metaid—0000024 potassium equilibrium potential ··· millivolt volt/1000 False
E—L metaid—0000025 leakage equilibrium potential ··· millivolt volt/1000 False
alpha—m metaid—0000026 auxiliary alpha—m ··· per—millisecond 1000/second False
beta—m metaid—0000027 auxiliary beta—m ··· per—millisecond 1000/second False
alpha—h metaid—0000028 auxiliary alpha—h ··· per—millisecond 1000/second False
beta—h metaid—0000029 auxiliary beta—h ··· per—millisecond 1000/second False
alpha—n metaid—0000030 auxiliary alpha—n ··· per—millisecond 1000/second False
beta—n metaid—0000031 auxiliary beta—n ··· per—millisecond 1000/second False

Rules

Metaid Type Formula
metaid—0000033 assignmentRule E—Na[t]==115 + E—R
metaid—0000034 assignmentRule E—K[t]==-12 + E—R
metaid—0000035 assignmentRule E—L[t]==11 + E—R
metaid—0000036 assignmentRule alpha—m[t]==(0.1*(40 + V[t]))/(1 - E^(-0.1*(40 + V[t])))
metaid—0000037 assignmentRule beta—m[t]==4*E^((-65 - V[t])/18)
metaid—0000038 assignmentRule alpha—h[t]==0.07*E^((-65 - V[t])/20)
metaid—0000039 assignmentRule beta—h[t]==(1 + E^(-0.1*(35 + V[t])))^(-1)
metaid—0000040 assignmentRule alpha—n[t]==(0.01*(55 + V[t]))/(1 - E^(-0.1*(55 + V[t])))
metaid—0000041 assignmentRule beta—n[t]==0.125*E^((-65 - V[t])/80)
metaid—0000045 rateRule m'[t]==alpha—m[t]*(1 - m[t]) - beta—m[t]*m[t]
metaid—0000046 rateRule h'[t]==alpha—h[t]*(1 - h[t]) - beta—h[t]*h[t]
metaid—0000047 rateRule n'[t]==alpha—n[t]*(1 - n[t]) - beta—n[t]*n[t]
metaid—0000042 assignmentRule i—Na[t]==g—Na*h[t]*m[t]^3*(-E—Na[t] + V[t])
metaid—0000043 assignmentRule i—K[t]==g—K*n[t]^4*(-E—K[t] + V[t])
metaid—0000044 assignmentRule i—L[t]==g—L*(-E—L[t] + V[t])
metaid—0000048 rateRule V'[t]==(II - i—K[t] - i—L[t] - i—Na[t])/Cm

Reactions

----- None -----

Events

----- None -----

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Simulating Time-Course: Hodgkin-Huxley model

In[15]:=

SolnHH = SBMLNDSolve[ModelHH, 50] ;

SBMLPlot[SolnHH, {m, h, n}, PlotRange→ All, ImageSize→ {450, 180}] ;

VPlot1 = SBMLPlot[SolnHH, {V}, PlotRange→ All, ImageSize→ {450, 180}] ;

[Graphics:HTMLFiles/index_46.gif]

[Graphics:HTMLFiles/index_47.gif]

In[6]:=

ConstantsHH = SBMLConstants/.ModelHH

Out[6]=

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Effect of Applied Current in the Hodgkin-Huxley model

In[18]:=

NewModelHH = resetParameter[ModelHH,    {II→ 5}] ;

NewSolnHH = SBMLNDSolve[NewModelHH, 50] ;

SBMLPlot[NewSolnHH, {m, h, n}, PlotRange→ All, ImageSize→ {450, 200}] ;

VPlot2 = SBMLPlot[NewSolnHH, {V}, PlotRange→ All, ImageSize→ {450, 200}] ;

[Graphics:HTMLFiles/index_59.gif]

[Graphics:HTMLFiles/index_60.gif]

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Effect of Applied Current in the Hodgkin-Huxley model

In[22]:=

NewModelHH = resetParameter[ModelHH,    {II→15}] ;

NewSolnHH = SBMLNDSolve[NewModelHH, 50] ;

SBMLPlot[NewSolnHH, {m, h, n}, PlotRange→ All, ImageSize→ {450, 200}] ;

VPlot3 = SBMLPlot[NewSolnHH, {V}, PlotRange→ All, ImageSize→ {450, 200}] ;

[Graphics:HTMLFiles/index_70.gif]

[Graphics:HTMLFiles/index_71.gif]

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Some of the Mathematical Questions Addressed in this Course

In[30]:=

Show[GraphicsArray[{VPlot1, VPlot2, VPlot3}], ImageSize→ {250 * 3, 120}] ;

[Graphics:HTMLFiles/index_78.gif]

● What solution bifurcation occurs in the transition:
    -from a stationary state to repeated, autonomous spiking?

● In the spikes, there are phases of slower variation inbetween phases of very rapid change
    -Are there multiple time-scales involved?

● What numerical methods are stable for integrating the stiff ODE:
    -Observed oscillations are 'real' and not a numerical artifact/effect?

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Some of the Biological Modelling Questions Addressed in this Course

● What dynamical behaviors can occur in gene systems:
    -bistable/hysteretic/irreversible switches
    -relaxation oscillations, bursting phenomenon, mixed-mode oscillations

● What are the motifs that occur frequently in biological networks?
    -feed-forward
    -activator-inhibitor pair
    ...

● What are the dynamical implications?

● How to design gene systems that exhibit switching behavior, and/or oscillations?

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2-Dimensional Reduced System: Phase Plane Analysis

In[31]:=

m—∞[V_] := 0.5 (1 + Tanh[(V - v1)/v2]) ;

w—∞[V_] := 0.5 (1 + Tanh[(V - v3)/v4]) ;

τ[V_] := 1/Cosh[(V - v3)/(2 v4)] ;

MorrisLecarODE//TableForm

Out[35]//TableForm=

V^′[t] == I—app - 0.5 g—Ca (1 + Tanh[(-v1 + V[t])/v2]) (-V—Ca + V[t]) - g—L (-V—L + V[t]) - g—K (-V—K + V[t]) w[t]
w^′[t] == φ Cosh[(-v3 + V[t])/(2 v4)] (0.5 (1 + Tanh[(-v3 + V[t])/v4]) - w[t])

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Morris-Lecar Model: Time-Series

In[36]:=

ODEIC = Join[MorrisLecarODE/.MorrisLecarParamRules1  ,    {w[0] == 0.01, V[0] == -10}] ;

MorrisLecarSol1 = NDSolve[   ODEIC, {V, w} , {t, 0, 200}] ; <br />

Plot[V[t]/.MorrisLecarSol1, {t, 0, 200}, PlotStyle→ {Thickness[0.01], Hue[0.7]}, ImageSize-> {400, 200}] ;

Null

[Graphics:HTMLFiles/index_106.gif]

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Morris-Lecar Model: Time-Series

In[50]:=

ODEIC = Join[MorrisLecarODE/.MorrisLecarParamRules2  ,    {w[0] == 0.01, V[0] == -10}] ;

MorrisLecarSol2 = NDSolve[   ODEIC, {V, w} , {t, 0, 200}] ; <br />

Plot[V[t]/.MorrisLecarSol2, {t, 0, 200}, PlotStyle→ {Thickness[0.01], Hue[0.7]}, ImageSize-> {400, 200}] ;

[Graphics:HTMLFiles/index_116.gif]

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Null Clines

In[28]:=

MorrisLecarNullClines = MorrisLecarODE/.{V^′[t] → 0, w^′[t] → 0}//TableForm

Out[28]//TableForm=

0 == I—app - 0.5 g—Ca (1 + Tanh[(-v1 + V[t])/v2]) (-V—Ca + V[t]) - g—L (-V—L + V[t]) - g—K (-V—K + V[t]) w[t]
0 == φ Cosh[(-v3 + V[t])/(2 v4)] (0.5 (1 + Tanh[(-v3 + V[t])/v4]) - w[t])

In[29]:=

MorrisLecarNullClines/.MorrisLecarParamRules2//TableForm

Out[29]//TableForm=

0 == 150 - 2.2 (1 + Tanh[1/18 (1.2 + V[t])]) (-120 + V[t]) - 2 (69 + V[t]) - 8 (84 + V[t]) w[t]
0 == 0.04 Cosh[1/60 (-2 + V[t])] (0.5 (1 + Tanh[1/30 (-2 + V[t])]) - w[t])

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Null Clines and Phase Plane Plot:  Low Applied Current

In[30]:=

<<DiffEqs`DEGraphics`

VF1 = Map[ #[[2]] &, MorrisLecarODE/.MorrisLecarParamRules1]/.{V[t] -> V, w[t] ->w} ;

Null

[Graphics:HTMLFiles/index_138.gif]

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Null Clines and Phase Plane Plot:  High Applied Current

In[34]:=

VF2 = Map[ #[[2]] &, MorrisLecarODE/.MorrisLecarParamRules2]/.{V[t] -> V, w[t] ->w} ;

Null

[Graphics:HTMLFiles/index_148.gif]

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Solution Continuation and Bifurcation Detection

● The system becomes unstable as I—app is increased.

● The structural stability is lost as an eigenvalue pair crosses the imaginary axis from negative half-plane.

● To locate the value of I—app were lost-of-stability occurs, we can:
    -continue the equilibrium solution with respect to I—app
    -detect change in stability using test function for Hopf bifurcation.

In[26]:=

MCBifPlot = Import["~/Teaching/NoteBooks/MATCONT_FILES/MorrisLecar_Bif.jpg"] ;

Show[MCBifPlot, ImageSize→ {400, 300}] ;

[Graphics:HTMLFiles/index_156.gif]

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Stiff ODE System

In[54]:=

Plot[Evaluate[{y[x], z[x]}/.sol], {x, 0, 1}, PlotRange→ All, PlotStyle→ {{Thickness[0.01], Hue[0.7]}, {Thickness[0.01], Hue[0.3]}}, ImageSize→ {500, 250}]

Null

[Graphics:HTMLFiles/index_165.gif]

Out[55]=

-Graphics -

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Stiff ODE System : Stability of Numerical Methods

In[39]:=

Plot[Evaluate[{y[x], z[x]}/.sol], {x, 0, 1}, PlotRange→ All, PlotStyle→ {{Thickness[0.01], Hue[0.7]}, {Thickness[0.01], Hue[0.3]}}, ImageSize→ {300, 150}]

[Graphics:HTMLFiles/index_174.gif]

Out[40]=

-Graphics -

In[41]:=

Plot[Evaluate[{y[x], z[x]}/.sol], {x, 0, 1}, PlotStyle→ {{Thickness[0.01], Hue[0.7]}, {Thickness[0.01], Hue[0.3]}}, ImageSize→ {300, 150}]

[Graphics:HTMLFiles/index_178.gif]

Out[42]=

-Graphics -

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Cell Cycle Model: Chen et al, 2004

ModelCellCycle = SBMLRead["~/Teaching/SBMLModels/Chen2004_CellCycle.xml", verbose→ True] ;

SolnCellCycle = SBMLNDSolve[ModelCellCycle, 1000] ;

File Name:~/Teaching/SBMLModels/Chen2004_CellCycle.xml<br />SBML Level 2 Version 1

Model name: Chen2004—CellCycle<br />Model id: Model2<br />Model metaid: metaid—0000002

Function Definitions

Unit Definitions (Excluding Built-in Units)

ID MetaID Name Formula
time metaid—0000366 min 60*second

Compartments

ID MetaID Name Dimension Size Units Derived Units Outside Constant
cell—0 metaid—0000171 cell 3 1 volume litre ··· True

Species

ID Name Compartment initialType Value Units of the Species Derived Units of the Species B.C Constant Charge
BCK2—1 BCK2 cell—0 ··· ··· substance mole False False ···
BUB2—2 BUB2 cell—0 initialAmount 0.2 substance mole False False ···
BUD—3 BUD cell—0 initialAmount 0.008473 substance mole False False ···
C2—4 C2 cell—0 initialAmount 0.238404 substance mole False False ···
C2P—5 C2P cell—0 initialAmount 0.024034 substance mole False False ···
C5—6 C5 cell—0 initialAmount 0.070081 substance mole False False ···
C5P—7 C5P cell—0 initialAmount 0.006878 substance mole False False ···
CDC14—8 CDC14 cell—0 initialAmount 0.468344 substance mole False False ···
CDC14T—9 CDC14T cell—0 initialAmount 2 substance mole False False ···
CDC15—10 CDC15 cell—0 initialAmount 0.656533 substance mole False False ···
CDC15i—11 CDC15i cell—0 initialAmount 0.343466 substance mole False False ···
CDC20—12 CDC20 cell—0 initialAmount 0.444296 substance mole False False ···
CDC20i—13 CDC20i cell—0 initialAmount 1.472044 substance mole False False ···
CDC6—14 CDC6 cell—0 initialAmount 0.10758 substance mole False False ···
CDC6P—15 CDC6P cell—0 initialAmount 0.015486 substance mole False False ···
CDC6T—16 CDC6T cell—0 ··· ··· substance mole False False ···
CDH1—17 CDH1 cell—0 initialAmount 0.930499 substance mole False False ···
CDH1i—18 CDH1i cell—0 initialAmount 0.0695 substance mole False False ···
CKIT—19 CKIT cell—0 ··· ··· substance mole False False ···
CLB2—20 CLB2 cell—0 initialAmount 0.1469227 substance mole False False ···
CLB2T—21 CLB2T cell—0 initialAmount 0.17 substance mole False False ···
CLB5—22 CLB5 cell—0 initialAmount 0.0518014 substance mole False False ···
CLB5T—23 CLB5T cell—0 initialAmount 0.12 substance mole False False ···
CLN2—24 CLN2 cell—0 initialAmount 0.0652511 substance mole False False ···
CLN3—25 CLN3 cell—0 ··· ··· substance mole False False ···
ESP1—27 ESP1 cell—0 initialAmount 0.301313 substance mole False False ···
F2—29 F2 cell—0 initialAmount 0.236058 substance mole False False ···
F2P—30 F2P cell—0 initialAmount 0.0273938 substance mole False False ···
F5—31 F5 cell—0 initialAmount 0.00007240000000000001 substance mole False False ···
F5P—32 F5P cell—0 initialAmount 0.00007910000000000001 substance mole False False ···
IE—33 IE cell—0 initialAmount 0.8985 substance mole False False ···
IEP—34 IEP cell—0 initialAmount 0.1015 substance mole False False ···
LTE1—35 LTE1 cell—0 initialAmount 0.1 substance mole False False ···
MAD2—36 MAD2 cell—0 initialAmount 0.01 substance mole False False ···
MASS—37 MASS cell—0 initialAmount 1.206019 substance mole False False ···
MCM1—38 MCM1 cell—0 ··· ··· substance mole False False ···
NET1—40 NET1 cell—0 initialAmount 0.018645 substance mole False False ···
NET1P—41 NET1P cell—0 initialAmount 0.970271 substance mole False False ···
NET1T—42 NET1T cell—0 initialAmount 2.8 substance mole False False ···
ORI—43 ORI cell—0 initialAmount 0.000909 substance mole False False ···
PDS1—44 PDS1 cell—0 initialAmount 0.025612 substance mole False False ···
PE—45 PE cell—0 initialAmount 0.7 substance mole False False ···
PPX—46 PPX cell—0 initialAmount 0.123179 substance mole False False ···
RENT—47 RENT cell—0 initialAmount 1.04954 substance mole False False ···
RENTP—48 RENTP cell—0 initialAmount 0.6 substance mole False False ···
SBF—49 SBF cell—0 ··· ··· substance mole False False ···
SIC1—50 SIC1 cell—0 initialAmount 0.0228776 substance mole False False ···
SIC1P—51 SIC1P cell—0 initialAmount 0.00641 substance mole False False ···
SIC1T—52 SIC1T cell—0 ··· ··· substance mole False False ···
SPN—53 SPN cell—0 initialAmount 0.03 substance mole False False ···
SWI5—54 SWI5 cell—0 initialAmount 0.95 substance mole False False ···
SWI5P—55 SWI5P cell—0 initialAmount 0.02 substance mole False False ···
TEM1GDP—56 TEM1GDP cell—0 initialAmount 0.1 substance mole False False ···
TEM1GTP—57 TEM1GTP cell—0 initialAmount 0.9 substance mole False False ···

Global Parameters

Rules

Reactions, contexts suppressed

ID MetaID Name Fast Reaction Reactants
ID MetaID		 Products
ID MetaID		 Modifiers
ID MetaID		 Parameters
ID=val MetaID		 Formula
(Substance/Volume)
Growth—225 metaid—0000262 Growth False Ø →  MASS—37 ···
MASS—37 ···
··· ··· Mass—Action—0—223[MASS—37[t]*mu—39[t]]
SynthesisofCLN—226 metaid—0000263 Synthesis of CLN2 False Ø →  CLN2—24 ···
CLN2—24 ···
SBF—49 ···
MASS—37 ···
··· Mass—Action—0—223[0.15*MASS—37[t]*SBF—49[t]]
DegradationofCLN—227 metaid—0000264 Degradation of CLN2 False CLN2—24 →  Ø
CLN2—24 ···
··· ··· ··· Mass—Action—1—222[0.12, CLN2—24[t]]
SynthesisofCLB—228 metaid—0000265 Synthesis of CLB2 False Ø →  CLB2—20 ···
CLB2—20 ···
MCM1—38 ···
MASS—37 ···
··· Mass—Action—0—223[MASS—37[t]*(0.001 + 0.04*MCM1—38[t])]
DegradationofCLB—229 metaid—0000266 Degradation of CLB2 False CLB2—20 →  Ø
CLB2—20 ···
··· ··· ··· Mass—Action—1—222[Vdb2—63[t], CLB2—20[t]]
SynthesisofCLB—230 metaid—0000267 Synthesis of CLB5 False Ø →  CLB5—22 ···
CLB5—22 ···
SBF—49 ···
MASS—37 ···
··· Mass—Action—0—223[MASS—37[t]*(0.0008 + 0.005*SBF—49[t])]
DegradationofCLB—231 metaid—0000268 Degradation of CLB5 False CLB5—22 →  Ø
CLB5—22 ···
··· ··· ··· Mass—Action—1—222[Vdb5—64[t], CLB5—22[t]]
SynthesisofSIC—232 metaid—0000269 Synthesis of SIC1 False Ø →  SIC1—50 ···
SIC1—50 ···
SWI5—54 ···
··· Mass—Action—0—223[0.012 + 0.12*SWI5—54[t]]
PhosphorylationofSIC—233 metaid—0000270 Phosphorylation of SIC1 False SIC1—50 →  SIC1P—51
SIC1—50 ···
SIC1P—51 ···
··· ··· Mass—Action—1—222[Vkpc1—69[t], SIC1—50[t]]
DephosphorylationofSIC—234 metaid—0000271 Dephosphorylation of SIC1 False SIC1P—51 →  SIC1—50
SIC1P—51 ···
SIC1—50 ···
··· ··· Mass—Action—1—222[Vppc1—72[t], SIC1P—51[t]]
FastDegradationofSICP—235 metaid—0000272 Fast Degradation of SIC1P False SIC1P—51 →  Ø
SIC1P—51 ···
··· ··· ··· Mass—Action—1—222[1, SIC1P—51[t]]
AssocofCLBandSIC—236 metaid—0000273 Assoc. of CLB2 and SIC1 False CLB2—20 + SIC1—50 →  C2—4
CLB2—20 ···
SIC1—50 ···
C2—4 ···
··· ··· Mass—Action—2—221[50, CLB2—20[t], SIC1—50[t]]
DissocofCLBSICcomplex—237 metaid—0000274 Dissoc. of CLB2/SIC1 complex False C2—4 →  CLB2—20 + SIC1—50
C2—4 ···
CLB2—20 ···
SIC1—50 ···
··· ··· Mass—Action—1—222[0.05, C2—4[t]]
AssocofCLBandSIC—238 metaid—0000275 Assoc. of CLB5 and SIC1 False CLB5—22 + SIC1—50 →  C5—6
CLB5—22 ···
SIC1—50 ···
C5—6 ···
··· ··· Mass—Action—2—221[50, CLB5—22[t], SIC1—50[t]]
DissocofCLBSIC—239 metaid—0000276 Dissoc. of CLB5/SIC1 False C5—6 →  CLB5—22 + SIC1—50
C5—6 ···
CLB5—22 ···
SIC1—50 ···
··· ··· Mass—Action—1—222[0.06, C5—6[t]]
PhosphorylationofC—240 metaid—0000277 Phosphorylation of C2 False C2—4 →  C2P—5
C2—4 ···
C2P—5 ···
··· ··· Mass—Action—1—222[Vkpc1—69[t], C2—4[t]]
DephosphorylationofCP—241 metaid—0000278 Dephosphorylation of C2P False C2P—5 →  C2—4
C2P—5 ···
C2—4 ···
··· ··· Mass—Action—1—222[Vppc1—72[t], C2P—5[t]]
—242 metaid—0000279 Phosphorylation of C5 False C5—6 →  C5P—7
C5—6 ···
C5P—7 ···
··· ··· Mass—Action—1—222[Vkpc1—69[t], C5—6[t]]
—243 metaid—0000280 Dephosphorylation of C5P False C5P—7 →  C5—6
C5P—7 ···
C5—6 ···
··· ··· Mass—Action—1—222[Vppc1—72[t], C5P—7[t]]
DegradationofCLBinC—244 metaid—0000281 Degradation of CLB2 in C2 False C2—4 →  SIC1—50
C2—4 ···
SIC1—50 ···
··· ··· Mass—Action—1—222[Vdb2—63[t], C2—4[t]]
—245 metaid—0000282 Degradation of CLB5 in C5 False C5—6 →  SIC1—50
C5—6 ···
SIC1—50 ···
··· ··· Mass—Action—1—222[Vdb5—64[t], C5—6[t]]
DegradationofSICinCP—246 metaid—0000283 Degradation of SIC1 in C2P False C2P—5 →  CLB2—20
C2P—5 ···
CLB2—20 ···
··· ··· Mass—Action—1—222[1, C2P—5[t]]
—247 metaid—0000284 Degradation of SIC1P in C5P False C5P—7 →  CLB5—22
C5P—7 ···
CLB5—22 ···
··· ··· Mass—Action—1—222[1, C5P—7[t]]
DegradationofCLBinCP—248 metaid—0000285 Degradation of CLB2 in C2P False C2P—5 →  SIC1P—51
C2P—5 ···
SIC1P—51 ···
··· ··· Mass—Action—1—222[Vdb2—63[t], C2P—5[t]]
—249 metaid—0000286 Degradation of CLB5 in C5P False C5P—7 →  SIC1P—51
C5P—7 ···
SIC1P—51 ···
··· ··· Mass—Action—1—222[Vdb5—64[t], C5P—7[t]]
CDCanotherCKIlikeSIC—250 metaid—0000287 CDC6 synthesis False Ø →  CDC6—14 ···
CDC6—14 ···
SWI5—54 ···
SBF—49 ···
··· Mass—Action—0—223[0.024 + 0.004*SBF—49[t] + 0.12*SWI5—54[t]]
—251 metaid—0000288 Phosphorylation of CDC6 False CDC6—14 →  CDC6P—15
CDC6—14 ···
CDC6P—15 ···
··· ··· Mass—Action—1—222[Vkpf6—70[t], CDC6—14[t]]
—252 metaid—0000289 Dephosphorylation of CDC6 False CDC6P—15 →  CDC6—14
CDC6P—15 ···
CDC6—14 ···
··· ··· Mass—Action—1—222[Vppf6—73[t], CDC6P—15[t]]
—253 metaid—0000290 Degradation of CDC6P False CDC6P—15 →  Ø
CDC6P—15 ···
··· ··· ··· Mass—Action—1—222[1, CDC6P—15[t]]
—254 metaid—0000291 CLB2/CDC6 complex formation False CDC6—14 + CLB2—20 →  F2—29
CLB2—20 ···
CDC6—14 ···
F2—29 ···
··· ··· Mass—Action—2—221[15, CLB2—20[t], CDC6—14[t]]
—255 metaid—0000292 CLB2/CDC6 dissociation False F2—29 →  CDC6—14 + CLB2—20
F2—29 ···
CLB2—20 ···
CDC6—14 ···
··· ··· Mass—Action—1—222[0.5, F2—29[t]]
—256 metaid—0000293 CLB5/CDC6 complex formation False CDC6—14 + CLB5—22 →  F5—31
CLB5—22 ···
CDC6—14 ···
F5—31 ···
··· ··· Mass—Action—2—221[0.01, CLB5—22[t], CDC6—14[t]]
—257 metaid—0000294 CLB5/CDC6 dissociation False F5—31 →  CDC6—14 + CLB5—22
F5—31 ···
CLB5—22 ···
CDC6—14 ···
··· ··· Mass—Action—1—222[0.01, F5—31[t]]
—258 metaid—0000295 F2 phosphorylation False F2—29 →  F2P—30
F2—29 ···
F2P—30 ···
··· ··· Mass—Action—1—222[Vkpf6—70[t], F2—29[t]]
—259 metaid—0000296 F2P dephosphorylation False F2P—30 →  F2—29
F2P—30 ···
F2—29 ···
··· ··· Mass—Action—1—222[Vppf6—73[t], F2P—30[t]]
—260 metaid—0000297 F5 phosphorylation False F5—31 →  F5P—32
F5—31 ···
F5P—32 ···
··· ··· Mass—Action—1—222[Vkpf6—70[t], F5—31[t]]
—261 metaid—0000298 F5P dephosphorylation False F5P—32 →  F5—31
F5P—32 ···
F5—31 ···
··· ··· Mass—Action—1—222[Vppf6—73[t], F5P—32[t]]
—262 metaid—0000299 CLB2 degradation in F2 False F2—29 →  CDC6—14
F2—29 ···
CDC6—14 ···
··· ··· Mass—Action—1—222[Vdb2—63[t], F2—29[t]]
—263 metaid—0000300 CLB5 degradation in F5 False F5—31 →  CDC6—14
F5—31 ···
CDC6—14 ···
··· ··· Mass—Action—1—222[Vdb5—64[t], F5—31[t]]
—264 metaid—0000301 CDC6 degradation in F2P False F2P—30 →  CLB2—20
F2P—30 ···
CLB2—20 ···
··· ··· Mass—Action—1—222[1, F2P—30[t]]
—265 metaid—0000302 CDC6 degradation in F5P False F5P—32 →  CLB5—22
F5P—32 ···
CLB5—22 ···
··· ··· Mass—Action—1—222[1, F5P—32[t]]
—266 metaid—0000303 CLB2 degradation in F2P False F2P—30 →  CDC6P—15
F2P—30 ···
CDC6P—15 ···
··· ··· Mass—Action—1—222[Vdb2—63[t], F2P—30[t]]
—267 metaid—0000304 CLB5 degradation in F5P False F5P—32 →  CDC6P—15
F5P—32 ···
CDC6P—15 ···
··· ··· Mass—Action—1—222[Vdb5—64[t], F5P—32[t]]
SynthesisofSWI—268 metaid—0000305 Synthesis of SWI5 False Ø →  SWI5—54 ···
SWI5—54 ···
MCM1—38 ···
··· Mass—Action—0—223[0.005 + 0.08*MCM1—38[t]]
DegradationofSWI—269 metaid—0000306 Degradation of SWI5 False SWI5—54 →  Ø
SWI5—54 ···
··· ··· ··· Mass—Action—1—222[0.08, SWI5—54[t]]
DegradationofSWIP—270 metaid—0000307 Degradation of SWI5P False SWI5P—55 →  Ø
SWI5P—55 ···
··· ··· ··· Mass—Action—1—222[0.08, SWI5P—55[t]]
ActivationofSWI—271 metaid—0000308 Activation of SWI5 False SWI5P—55 →  SWI5—54
SWI5P—55 ···
SWI5—54 ···
CDC14—8 ···
··· Mass—Action—1—222[2*CDC14—8[t], SWI5P—55[t]]
InactivationofSWI—272 metaid—0000309 Inactivation of SWI5 False SWI5—54 →  SWI5P—55
SWI5—54 ···
SWI5P—55 ···
CLB2—20 ···
··· Mass—Action—1—222[0.05*CLB2—20[t], SWI5—54[t]]
ActivationofIEP—273 metaid—0000310 Activation of IEP False IE—33 →  IEP—34
IE—33 ···
IEP—34 ···
··· ··· MichaelisMenten—220[Vaiep—59[t], 0.1, 1, IE—33[t]]
Inactivation—274—IEP metaid—0000311 Inactivation False IEP—34 →  IE—33
IEP—34 ···
IE—33 ···
··· ··· MichaelisMenten—220[1, 0.1, 0.15, IEP—34[t]]
SynthesisofinactiveCDC—275 metaid—0000312 Synthesis of inactive CDC20 False Ø →  CDC20i—13 ···
CDC20i—13 ···
MCM1—38 ···
··· Mass—Action—0—223[0.006 + 0.6*MCM1—38[t]]
DegradationofinactiveCDC—276 metaid—0000313 Degradation of inactiveCDC20 False CDC20i—13 →  Ø
CDC20i—13 ···
··· ··· ··· Mass—Action—1—222[0.3, CDC20i—13[t]]
DegradationofactiveCDC—277 metaid—0000314 Degradation of active CDC20 False CDC20—12 →  Ø
CDC20—12 ···
··· ··· ··· Mass—Action—1—222[0.3, CDC20—12[t]]
ActivationofCDC—278 metaid—0000315 Activation of CDC20 False CDC20i—13 →  CDC20—12
CDC20i—13 ···
CDC20—12 ···
IEP—34 ···
··· Mass—Action—1—222[0.05 + 0.2*IEP—34[t], CDC20i—13[t]]
Inactivation—274—CDC20 metaid—0000316 Inactivation False CDC20—12 →  CDC20i—13
CDC20—12 ···
CDC20i—13 ···
MAD2—36 ···
··· Mass—Action—1—222[MAD2—36[t], CDC20—12[t]]
—279 metaid—0000317 CDH1 synthesis False Ø →  CDH1—17 ···
CDH1—17 ···
··· ··· Mass—Action—0—223[0.01]
—280 metaid—0000318 CDH1 degradation False CDH1—17 →  Ø
CDH1—17 ···
··· ··· ··· Mass—Action—1—222[0.01, CDH1—17[t]]
—281 metaid—0000319 CDH1i degradation False CDH1i—18 →  Ø
CDH1i—18 ···
··· ··· ··· Mass—Action—1—222[0.01, CDH1i—18[t]]
Activation—282 metaid—0000320 CDH1i activation False CDH1i—18 →  CDH1—17
CDH1i—18 ···
CDH1—17 ···
··· ··· MichaelisMenten—220[Vacdh—58[t], 0.03, 1, CDH1i—18[t]]
Inactivation—274—CDH1 metaid—0000321 Inactivation False CDH1—17 →  CDH1i—18
CDH1—17 ···
CDH1i—18 ···
··· ··· MichaelisMenten—220[Vicdh—67[t], 0.03, 1, CDH1—17[t]]
—283 metaid—0000322 CDC14 synthesis False Ø →  CDC14—8 ···
CDC14—8 ···
··· ··· Mass—Action—0—223[0.2]
—284 metaid—0000323 CDC14 degradation False CDC14—8 →  Ø
CDC14—8 ···
··· ··· ··· Mass—Action—1—222[0.1, CDC14—8[t]]
AssocwithNETtoformRENT—285 metaid—0000324 Assoc. with NET1 to form RENT False CDC14—8 + NET1—40 →  RENT—47
CDC14—8 ···
NET1—40 ···
RENT—47 ···
··· ··· Mass—Action—2—221[200, CDC14—8[t], NET1—40[t]]
DissocfromRENT—286 metaid—0000325 Dissoc. from RENT False RENT—47 →  CDC14—8 + NET1—40
RENT—47 ···
NET1—40 ···
CDC14—8 ···
··· ··· Mass—Action—1—222[1, RENT—47[t]]
AssocwithNETPtoformRENTP—287 metaid—0000326 Assoc with NET1P to form RENTP False CDC14—8 + NET1P—41 →  RENTP—48
CDC14—8 ···
NET1P—41 ···
RENTP—48 ···
··· ··· Mass—Action—2—221[1, CDC14—8[t], NET1P—41[t]]
DissocfromRENP—288 metaid—0000327 Dissoc. from RENP False RENTP—48 →  CDC14—8 + NET1P—41
RENTP—48 ···
CDC14—8 ···
NET1P—41 ···
··· ··· Mass—Action—1—222[2, RENTP—48[t]]
—289 metaid—0000328 Net1 synthesis False Ø →  NET1—40 ···
NET1—40 ···
··· ··· Mass—Action—0—223[0.084]
—290 metaid—0000329 Net1 degradation False NET1—40 →  Ø
NET1—40 ···
··· ··· ··· Mass—Action—1—222[0.03, NET1—40[t]]
—291 metaid—0000330 Net1P degradation False NET1P—41 →  Ø
NET1P—41 ···
··· ··· ··· Mass—Action—1—222[0.03, NET1P—41[t]]
NETphosphorylation—292 metaid—0000331 NET1 phosphorylation False NET1—40 →  NET1P—41
NET1—40 ···
NET1P—41 ···
··· ··· Mass—Action—1—222[Vkpnet—71[t], NET1—40[t]]
dephosphorylation—293—NET1P metaid—0000332 dephosphorylation False NET1P—41 →  NET1—40
NET1P—41 ···
NET1—40 ···
··· ··· Mass—Action—1—222[Vppnet—74[t], NET1P—41[t]]
RENTphosphorylation—294 metaid—0000333 RENT phosphorylation False RENT—47 →  RENTP—48
RENT—47 ···
RENTP—48 ···
··· ··· Mass—Action—1—222[Vkpnet—71[t], RENT—47[t]]
dephosphorylation—293—RENTP metaid—0000334 dephosphorylation False RENTP—48 →  RENT—47
RENTP—48 ···
RENT—47 ···
··· ··· Mass—Action—1—222[Vppnet—74[t], RENTP—48[t]]
DegradationofNETinRENT—295 metaid—0000335 Degradation of NET1 in RENT False RENT—47 →  CDC14—8
RENT—47 ···
CDC14—8 ···
··· ··· Mass—Action—1—222[0.03, RENT—47[t]]
DegradationofNETPinRENTP—296 metaid—0000336 Degradation of NET1P in RENTP False RENTP—48 →  CDC14—8
RENTP—48 ···
CDC14—8 ···
··· ··· Mass—Action—1—222[0.03, RENTP—48[t]]
DegradationofCDCinRENT—297 metaid—0000337 Degradation of CDC14 in RENT False RENT—47 →  NET1—40
RENT—47 ···
NET1—40 ···
··· ··· Mass—Action—1—222[0.1, RENT—47[t]]
DegradationofCDCinRENTP—298 metaid—0000338 Degradation of CDC14 in RENTP False RENTP—48 →  NET1P—41
RENTP—48 ···
NET1P—41 ···
··· ··· Mass—Action—1—222[0.1, RENTP—48[t]]
TEMactivation—299 metaid—0000339 TEM1 activation False TEM1GDP—56 →  TEM1GTP—57
TEM1GDP—56 ···
TEM1GTP—57 ···
LTE1—35 ···
··· MichaelisMenten—220[LTE1—35[t], 0.1, 1, TEM1GDP—56[t]]
inactivation—300—TEM1GTP metaid—0000340 inactivation False TEM1GTP—57 →  TEM1GDP—56
TEM1GTP—57 ···
TEM1GDP—56 ···
BUB2—2 ···
··· MichaelisMenten—220[BUB2—2[t], 0.1, 1, TEM1GTP—57[t]]
CDCactivation—301 metaid—0000341 CDC15 activation False CDC15i—11 →  CDC15—10
CDC15i—11 ···
CDC15—10 ···
TEM1GDP—56 ···
TEM1GTP—57 ···
CDC14—8 ···
··· Mass—Action—1—222[0.001*CDC14—8[t] + 0.002*TEM1GDP—56[t] + TEM1GTP—57[t], CDC15i—11[t]]
inactivation—300—CDC15 metaid—0000342 inactivation False CDC15—10 →  CDC15i—11
CDC15—10 ···
CDC15i—11 ···
··· ··· Mass—Action—1—222[0.5, CDC15—10[t]]
PPXsynthesis—302 metaid—0000343 PPX synthesis False Ø →  PPX—46 ···
PPX—46 ···
··· ··· Mass—Action—0—223[0.1]
degradation—303—PPX metaid—0000344 degradation False PPX—46 →  Ø
PPX—46 ···
··· ··· ··· Mass—Action—1—222[Vdppx—66[t], PPX—46[t]]
PDSsynthesis—304 metaid—0000345 PDS1 synthesis False Ø →  PDS1—44 ···
PDS1—44 ···
SBF—49 ···
MCM1—38 ···
··· Mass—Action—0—223[0.055*MCM1—38[t] + 0.03*SBF—49[t]]
degradation—303—PDS1 metaid—0000346 degradation False PDS1—44 →  Ø
PDS1—44 ···
··· ··· ··· Mass—Action—1—222[Vdpds—65[t], PDS1—44[t]]
DegradationofPDSinPE—305 metaid—0000347 Degradation of PDS1 in PE False PE—45 →  ESP1—27
PE—45 ···
ESP1—27 ···
··· ··· Mass—Action—1—222[Vdpds—65[t], PE—45[t]]
AssocwithESPtoformPE—306 metaid—0000348 Assoc. with ESP1 to form PE False ESP1—27 + PDS1—44 →  PE—45
PDS1—44 ···
ESP1—27 ···
PE—45 ···
··· ··· Mass—Action—2—221[50, PDS1—44[t], ESP1—27[t]]
DissofromPE—307 metaid—0000349 Disso. from PE False PE—45 →  ESP1—27 + PDS1—44
PE—45 ···
PDS1—44 ···
ESP1—27 ···
··· ··· Mass—Action—1—222[0.5, PE—45[t]]
—308 metaid—0000350 DNA synthesis False Ø →  ORI—43 ···
ORI—43 ···
CLB5—22 ···
CLB2—20 ···
··· Mass—Action—0—223[2*(0.45*CLB2—20[t] + 0.9*CLB5—22[t])]
—309 metaid—0000351 Negative regulation of DNA synthesis False ORI—43 →  Ø
ORI—43 ···
··· ··· ··· Mass—Action—1—222[0.06, ORI—43[t]]
—310 metaid—0000352 Budding False Ø →  BUD—3 ···
BUD—3 ···
CLN2—24 ···
CLN3—25 ···
CLB5—22 ···
··· Mass—Action—0—223[0.2*(CLB5—22[t] + 0.25*CLN2—24[t] + 0.05*CLN3—25[t])]
—311 metaid—0000353 Negative regulation of Cell budding False BUD—3 →  Ø
BUD—3 ···
··· ··· ··· Mass—Action—1—222[0.06, BUD—3[t]]
—312 metaid—0000354 Spindle formation False Ø →  SPN—53 ···
SPN—53 ···
CLB2—20 ···
··· Mass—Action—0—223[(0.1*CLB2—20[t])/(0.14 + CLB2—20[t])]
—313 metaid—0000355 Spindle disassembly False SPN—53 →  Ø
SPN—53 ···
··· ··· ··· Mass—Action—1—222[0.06, SPN—53[t]]

Differential Equations, contexts suppressed

Events

ID MetaID Name Trigger Delay TimeUnits Event
Assignment		 Assignment
MetaID
event—1 metaid—0000359 ··· CLB2—20[t] + CLB5—22[t] - KEZ2—172[t] < 0 0 time
ORI—43[t]->0
···
event—2 metaid—0000360 ··· -1 + ORI—43[t] > 0 0 time
MAD2—36[t]->mad2h—214
BUB2—2[t]->bub2h—76
···
···
event—3 metaid—0000361 ··· -1 + SPN—53[t] > 0 0 time
MAD2—36[t]->mad2l—215[t]
LTE1—35[t]->lte1h—212[t]
BUB2—2[t]->bub2l—77[t]
···
···
···
event—4 metaid—0000362 ··· CLB2—20[t] - KEZ—171[t] < 0 0 time
MASS—37[t]->F—28[t]*MASS—37[t]
LTE1—35[t]->lte1l—213[t]
BUD—3[t]->0
SPN—53[t]->0
···
···
···
···

In[23]:=

SBMLPlot[SolnCellCycle, {CDC14—8,   BUD—3, SWI5—54, CLN2—24}, ImageSize→ {450, 220}] ;

[Graphics:HTMLFiles/index_199.gif]

 ◂ ▸  FormBox[StyleBox[RowBox[{CounterBox[SlideShowNavigationBar], of , CounterBox[SlideShowNavigationBar, {None, SlideShowHeader, -1}]}], SR], StandardForm]

Circadian Clock Example: Leloup and Goldbeter, 1999

● Circadian (approximately daily) clocks underly 24 hour sleep-wake cycle in many organisms

● Leloup and Goldbeter model: interlocked negative and positive regulation of Period, Timeless genes

In[66]:=

visualizeSBMLModel["~/Teaching/SBMLModels/CircClock_LG99.xml"] ;

[Graphics:HTMLFiles/index_206.gif]

 ◂ ▸  FormBox[StyleBox[RowBox[{CounterBox[SlideShowNavigationBar], of , CounterBox[SlideShowNavigationBar, {None, SlideShowHeader, -1}]}], SR], StandardForm]

Circadian Clock Example: Leloup and Goldbeter, 1999

● Dynamical property: robust, temperature-compensated ~24 hr oscillation period

In[61]:=

ModelCircClock = SBMLRead["~/Teaching/SBMLModels/CircClock_LG99.xml", verbose→ False] ;

SolnCircClock = SBMLNDSolve[ModelCircClock, 150] ;

In[67]:=

SBMLPlot[SolnCircClock, {CC, Cn}, PlotRange→ {{40, 150}, {0, 2.2}}, ImageSize→ {600, 300}] ;

[Graphics:HTMLFiles/index_215.gif]

Created by Mathematica  (March 6, 2007) Valid XHTML 1.1!