1. Canadian Bioinformatics Workshops
Module 3

2. Module 3

3. Systems Modelling in Cell Biology
Brian Ingalls
Applied Mathematics
University of Waterloo
Waterloo, Ontario, Canada
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4. Outline System modelling - focus on dynamics Model development
Analysis:
1) Parametric Sensitivity Analysis
2) Nonlinear Dynamics
3) Feedback regulation
Reference: System Modelling in Cellular Biology, Szallazi, Stelling and Periwal, eds
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5. Models in Cell Biology Roles of modelling
Abstraction for the purposes of understanding
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6. Models in Cell Biology Roles of modelling
Abstraction for the purposes of understanding
Organization of results/theories (bookkeeping)
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7. Models in Cell Biology Roles of modelling
Abstraction for the purposes of understanding
Organization of results/theories (bookkeeping)
Description of spatial or temporal relationships
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8. Models in Cell Biology The modeller must choose
an appropriate level of
abstraction.
Glycolysis:
Focus on flow of metabolites
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9. Models in Cell Biology The modeller must choose
an appropriate level of
abstraction.
Glycolysis:
Focus on chemistry
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10. Models in Cell Biology The modeller must choose
an appropriate level of
abstraction.
Glycolysis:
Focus on regulation
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11. Dynamic Modelling
Focus on how system components influence rates of change of each component in the network.
Result: description of time-varying behaviour.
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12. A C B Example: consider a metabolic chain which is initially inactive.
Experimental perturbation: activate first enzyme in the chain.
System behaviour: pools of metabolites build up, system reaches steady state
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13. Intuition fails when faced with a complex interconnections…
E. Coli metabolism
KEGG: Kyoto Encyclopedia of Genes and Genomes (
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14. Intuition fails when faced with a complex interconnections or feedback …
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15. Intuition fails when faced with a complex interconnections or feedback, or both.
endomesoderm specification in the sea urchin Strongylocentrotus purpuratus
Eric Davidson's Lab at Caltech (
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16. Another complex network:
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17. Outline System modelling - focus on dynamics Model development
Analysis:
1) Parametric Sensitivity Analysis
2) Nonlinear Dynamics
3) Feedback regulation
References: A Cell Biologist's Guide to Modeling and Bioinformatics, Holmes,
Systems Biology in Practice, Klipp, Herwig, Kowald, Wierling, Lehrach
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18. Example: irreversible isomerization
Qualitative description: the rate v of the reaction A B increases as the concentration of A increases.
A quantitative description: mass action
v = k1[A] (mass action)
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19. rate of change of concentration
This quantitative description of the reaction rate can be used to characterize the rates of change of the chemical species in the network:
rate of change of concentration
+/- rate of reaction
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20. The resulting differential equations can be solved to determine the time-varying system behaviour.
concentration
[A]
time
This predictive (mechanism-based) model is far more useful than a descriptive (data-based) model
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21. Numerical Simulation Differential Equation:
Approximation of the Derivative:
Recursive scheme:
Construction of Approximate Solution:
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22. } { A B Steady state behaviour: Reversible Isomerization [B] [A]k1 A B k2 } {
Time rate of change equals zero
Algebraic solution or
Long-time behaviour of simulations
give
Steady state (equilibrium ratio):
[B]
concentration
[A]
time
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23. Quantification of biochemical and genetic interactions
Foundation: Law of Mass Action
"rate of a reaction is proportional to the product of the concentrations of the reactants"
Key assumptions:
Well mixed environment (no spatial effects)
Large numbers of molecules (continuum of concentrations)
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24. Biochemical Reactions (Metabolism and protein-protein interaction networks)
Zeroth order reactions:
generation of s1 from some buffered (external) source (S).
Rate: or or
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25. Biochemical Reactions (Metabolism and protein-protein interaction networks)
First order reactions:
isomerization, unassisted transport, degradation/dilution, or linearization of other kinetics
Rate: or or
concentration
rate
[s]
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26. Biochemical Reactions (Metabolism and protein-protein interaction networks)
Second order reactions:
Binding/association events
Rate:
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27. Biochemical Reactions (Metabolism and protein-protein interaction networks)
Michaelis-Menten Kinetics:
Enzyme-catalysed reaction (metabolic, signal transduction, active transport)
Rate: or
concentration
rate
[s]
Vmax
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28. Biochemical Reactions (Metabolism and protein-protein interaction networks)
Hill-type Kinetics:
Catalysis by cooperative enzyme or lumped description of multi-step process
Rate: or k
rate
[s]
concentration
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29. Biochemical Reactions (Metabolism and protein-protein interaction networks)
Allosteric inhibition:
Inhibitor I binds enzyme E and reduces its catalytic activity
Rate:
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30. Genetic Circuits unregulated mRNA transcription: zeroth order
translation
transcription
protein
mRNA
gene
unregulated mRNA transcription: zeroth order
unregulated protein translation: first order in mRNA concentration
degradation/dilution: first order
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31. Genetic Circuits Regulated mRNA transcription: Hill type kinetics
transcription factor P
transcription or
mRNA
gene
Regulated mRNA transcription: Hill type kinetics
Activation:
Inhibition:
multimerization of P
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32. Example: autoinhibitory gene circuit: trp operon
enzyme
mRNA
gene
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33. Species: R R* trp enzyme mRNA gene m (mRNA) R (Repressor)
e (enzyme) R* (Active Repressor)
T (trp)
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34. R k2 R* c1
k-2
trp k1 
enzyme
mRNA d2
gene d1
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35. Lab exercise open "trp.ode" in XPPAUT
check the differential equations (Eqns tab) and parameter values (Param tab)
Run the simulation (Initialconditions -> Go)
Check the output (Data tab)
Add curves to the plot (Graphic stuff -> Add curves -> variable name on y-axis, color=1-9)
Change view by Window/Zoom -> Zoom in (draw box) OR Fit
Note the overshoot in m and e
Explore the effect of changing the available pool of repressor by changing the initial value of R in the ICs tab (Initial conditions)
References:
(XPPAUT) Simulating, Analyzing and Animating Dynamical Systems, Ermentrout
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36. Other Software Packages
Gepasi (
E-Cell (
Cellerator (
JWS online (
SBML: common markup language for biochemical and genetic models (
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37. Example: protein-protein interaction network
Modelling the initiation of DNA replication in the eukaryotic cell cycle
Joint work with B. Duncker, B. McConkey, Dept. of Biology, University of Waterloo
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38. The cell cycle in budding yeast
Initiation of replication
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39. Model focusing on construction of pre-replicative machinery
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40. Detailed model indicating cyclic behaviour of pre-replicative machinery
mcms
cdt1
SCF
cdc6 p
mcms
cdt1
clb5
cdc6
mcms
cdc6
swi5
cdc6
cdt1
orc
cdc6
orc
orc
mcms
cdt1
orc
mcms
cdc45
Nuclear
export
orc
orc
cdt1
cdc45
mcms
cdc45
mcms
orc
cdc45
elongation
Nuclear
export
clb2
clb5
and
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41. rate of change of concentration
Detailed model indicating cyclic behaviour of pre-replicative machinery
mcms
cdt1
SCF
cdc6 p
mcms
cdt1
clb5
cdc6
mcms
cdc6
swi5
cdc6
cdt1
orc
cdc6
orc
RC2
RC3
orc
mcms
cdt1
RC4
orc
RC1
RC7
rate of formation
RC5
mcms
cdc45
Nuclear
export
orc
orc
cdt1
cdc45
RC6
mcms
cdc45
mcms
orc
cdc45
rate of degradation
elongation
Nuclear
export
rate of change of concentration
clb2
clb5
and
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42. Detailed model indicating cyclic behaviour of pre-replicative machinery
mcms
cdt1
SCF
cdc6 p
mcms
cdt1
clb5
cdc6
mcms
cdc6
swi5
cdc6
cdt1
orc
cdc6
orc
orc
mcms
cdt1
orc
mcms
cdc45
Nuclear
export
orc
orc
cdt1
cdc45
mcms
cdc45
mcms
orc
cdc45
elongation
Nuclear
export
clb2
clb5
and
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43. Complete Dynamic Model
# Budding yeast DNA replication model
#pre-rc formation (pre)
dCDC6/dt= kt6*SWI5+kpre1'*RC2+kinit1*RC3-(kpre1*RC1*CDC6+Vpre1*CDC6/(Jpre1+CDC6)+kinit1'*CDC6*RC4)
dRC2/dt=kpre1*RC1*CDC6+kpre2'*RC3-(kpre2*MCMSCDT1*RC2+kpre1'*RC2)
dMCMSCDT1/dt= kpre3*MCMS*CDT1+kpre2'*RC3-(kpre3'*MCMSCDT1+kpre2*MCMSCDT1*RC2)
#initiation (init)
dRC3/dt=kinit1'*CDC6*RC4+kpre2*MCMSCDT1*RC2-(kinit1*RC3+kpre1'*RC3)
dRC4/dt=kinit3'*CDT1*RC5+kinit1*RC3-(kinit3*RC4+kinit1'*CDC6*RC4)
#nuclear export of cdt1
dCDT1P/dt = Vinit4*CDT1/(Jinit4+CDT1)+kinit6'*CDT1PC-(kinit6*CDT1P+Vinit4rev*CDT1P/(Jinit4'+CDT1P))
dCDT1PC/dt = kinit6*CDT1P-kinit6'*CDT1PC
#S-phase (elon)
dRC5/dt=kelon1'*RC6+kinit3*RC4-(kinit3'*CDT1*RC5+kelon1*(kb2*CLB2T+kb5*CLB5T)*RC5*CDC45)
#post-replicative complex (post)
dRC6/dt=kpost1'*MCMS*RC7+kelon1*(kb2*CLB2T+kb5*CLB5T)*RC5*CDC45-(kelon1'*RC6+kpost1*RC6)
dRC7/dt=kpost2'*(kb2*CLB2T+kb5*CLB5T)*CDC45*RC1+kpost1*RC6-(kpost2*RC7+kpost1'*MCMS*RC7)
#nuclear export of MCMS
dMCMSP/dt = Vpost3*MCMS/(Jpost3+MCMS)+kpost4'*MCMSPC-(kpost4*MCMSP+Vpost3rev*MCMSP/(Jpost3'+MCMSP))
dMCMSPC/dt = kpost4*MCMSP-kpost4'*MCMSPC
#conservation of mass equations
#RC's, cdt1, mcms, cdc45 are all conserved. cdc6 is not conserved
RC1 = RCT-RC2-RC3-RC4-RC5-RC6-RC7
MCMS = MCMT-MCMSCDT1-MCMSP-MCMSPC-RC3-RC4-RC5-RC6
CDT1 = CDT1T-MCMSCDT1-CDT1P-CDT1PC-RC3-RC4
CDC45 = CDC45T-RC6-RC7
#rate functions
Vpre1=kb2*CLB2T+kb5*CLB5T
Vinit1=kb2*CLB2T+kb5*CLB5T
Vinit4=kb2*CLB2T+kb5*CLB5T
Vinit4rev=1
Vpost3=kb2p*CLB2T+kb5p*CLB5T
Vpost3rev=1
#parameters
param kt6=0.12
#pre
param kpre1=1,kpre1'=0.1,kpre2=1,kpre2'=0.1,kpre3=10,kpre3'=1
param Jpre1=1
#init
param kinit1=5,kinit1'=5,kinit3=1,kinit3'=1,kinit6=1,kinit6'=0.01
param Jinit4=1,Jinit4'=1
#elong
param kelon1=1,kelon1'=1
#post
param kpost1=0.01,kpost1'=0.01,kpost2=1,kpost2'=1,kpost4=1,kpost4'=0.01
param Jpost4=1,Jpost3=1,Jpost3'=1
#weights for clb2 and clb5 concentrations
param kb2=10,kb5=10,kb2p=1,kb5p=1
#total concentrations
param MCMT=7,CDT1T=5,CDC45T=1,RCT=1
#signals: needs modifying
SWI5=cos(t/16)*heav(cos(t/16))
CLB2T=-1.2*sin(t/16)*heav(-sin(t/16))
CLB5T=0.5*sin(t/16-35/16)*heav(sin(t/16-35/16))
@Maxstore=100000,bound=300
@Meth=Stiff, total=201, xplot=t, yplot=CDC6, xlo=0, xhi=505, ylo=0, yhi=5
#additional variables to plot
aux SWI5=SWI5
aux MCMS=MCMS
aux CDT1=CDT1
aux CDC45=CDC45
aux RC1=RC1
aux CLB2T=CLB2T
aux CLB5T=CLB5T
#initial conditions. Simulates beginning of G1 of the cell cycle.
init CDC6=0.5
init CDT1P=0.025,CDT1PC=4.8
init MCMSCDT1=0.1,MCMSP=0.03,MCMSPC=5.5
init RC2=0.1,RC3=0.001,RC4=0.0005,RC5=0.3,RC6=0,RC7=0
done
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44. Results: Simulation
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45. Where do the numbers (model parameters) come from?
Ideally, from characterizations of individual interactions, e.g. enzymological data. (but still issues with conditions, cell types, etc.)
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46. Where do the numbers (model parameters) come from?
More often, parameters are inferred by fitting model behaviour to experimental measures of system behaviour
optimization algorithm (simulated annealing, genetic algorithm,...)
p1 = 3.4
p2 = 13.6
p3 = 0.7
...
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47. Purposes and implications of dynamic modelling:
Analysis:
Testing for fidelity: a model is a falsifiable manifestation of a hypothesis.
In silico experiments: behaviour of the model can suggest (predict?) behaviour of the system
Parametric sensitivity analysis - hypothesis generation, study of influence/function of system components
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48. Purposes and implications of dynamic modelling:
Design:
Results of in silico experiments can suggest experimental design
Model-based design: metabolic engineering, rational drug design, “synthetic biology”
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49. Outline System modelling - focus on dynamics Model development
Analysis:
1) Parametric Sensitivity Analysis
2) Nonlinear Dynamics
3) Feedback regulation
References: (local) Parametric Sensitivity in Chemical Systems, Varma, Morbidelli, Wu, (global) Sensitivity Analysis in Practice, Saltelli, Tarantola, Campolongo, Ratto
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50. Parametric Sensitivity Analysis: Example
reaction kinetics:
steady state:
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51. local sensitivity analysis:
steady state:
local sensitivity analysis:
effect of perturbation/ intervention:
relative sensitivity:
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52. steady state: sensitivity analysis: vector notation
implicit differentiation
steady state:
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53. Sensitivity Analysis: General Computation
model:
steady state:
differentiate:
absolute sensitivity:
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54. complete sensitivity analysis:
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55. Parametric Sensitivity Analysis
Parameters
1. Enzyme activity levels
2. Kinetics constants
3. Decay rates
4. Boundary conditions
Variables
1. Concentrations
2. Pathway fluxes
3. Dynamic response
4. Growth rate
Parametric sensitivity analysis investigates the relationship between the variables and parameters in a biochemical network.
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56. Application:
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57. sensitivity of flux J to enzyme activities:
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58. sensitivity of flux J to enzyme activities:
Summation Theorem of Metabolic Control Analysis: conservation law for sensitivities
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59. Applications of Sensitivity Analysis
Predicting the effect of interventions
Drug development
Trypanosome metabolism. Bakker et al., 1999,J. Biol. Chem
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60. Applications of Sensitivity Analysis
Predicting the effect of interventions
Drug development
Medicine
Tumour growth and thiamine, Comin-Anduix et al., 2001, Eur. J. Biochem.
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61. Applications of Sensitivity Analysis
Predicting the effect of interventions
Drug development
Medicine
Metabolic engineering
Diacetyl production in Lactococcus lactis, Hoefnagel et al. 2002, Microbiology
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62. Applications of Sensitivity Analysis
Predicting the effect of interventions
Drug development
Medicine
Metabolic engineering
Model construction and analysis
Identifying key variables
NF-B pathway. Ihekwaba et al., 2004, IEE Sys. Biol.
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63. Applications of Sensitivity Analysis
Predicting the effect of interventions
Drug development
Medicine
Metabolic engineering
Model construction and analysis
Identifying key variables
Model calibration
Identifiability. Zak et al. 2003, Genome. Res.
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64. Lab exercise: Parametric Sensitivity of the trp operon model
Return to trp.ode in XPPAUT
Consider the effect of small (10%) parameter changes on the steady state value of Trp (make changes under parameter tab: hit default button to return to nominal values)
Is Trpss more sensitive to X (substrate for trp production), kcat (enzymatic activity of trp production) or c1 (rate of Trp consumption)?
Consider a large parameter change: increase  (rate of mRNA transcription) to 150.
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65. Outline System modelling - focus on dynamics Model development
Analysis:
1) Parametric Sensitivity Analysis
2) Nonlinear Dynamics
3) Feedback regulation
Reference: Nonlinear Dynamics and Chaos, Strogatz
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66. Nonlinear Dynamics
Phase plane Analysis
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67. Phaseplane Analysis Time Course ([si] vs. time)
Phase Plane ([s1] vs. [s2])
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68. Direction Field
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69. Nullclines: Turning points
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70. Nullclines: Turning points
s1 nullcline: s1 not changing
s2 nullcline: s2 not changing
intersection of nullclines: neither s1 nor s2 changing: steady state
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71. Stability Long time (asymptotic) behaviour of these systems is either
convergence to a steady state or;
periodic oscillation (convergence to a limit cycle)
Other behaviours (divergence, chaos) are rare and may indicate poor model construction.
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72. Y <<X Stability steady state independent of initial conditions
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73. Y <<X Bistability Nullclines intersect once: one steady state
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74. Bistability
Y = X
steady state depends on initial conditions
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75. Bistability
Y = X
Nullclines intersect three times: three steady states
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76. Bistability
Y = X
Nullclines intersect three times: three steady states
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77. Bistability
Y = X
Middle steady state repels trajectories
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78. Stability A steady state is
stable if nearby trajectories converge to it
unstable if nearby trajectories diverge from it
Linearization: a direct test for stability (details in notes)
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79. Oscillatory behaviour
q=2
single stable steady state
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80. Oscillatory behaviour
q=3
single unstable steady state and limit cycle
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81. Bifurcations
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82. Bifurcations
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83. Lab Exercise:Bistability
open bimet.ode in XPPAUT
Under Viewaxis -> 2D select s1 for the X-axis and s2 for the Y-axis (i.e. the phaseplane) Set Xmin=Ymin=0, Xmax=Ymax=0.025.
Choose Initialconditions -> mIce and select a number of initial conditions to explore the system behaviour
Select Nullclines -> New to see the nullclines.
Erase the plot and consider values of the parameter Prod in the range between 4 and 2. How does the system behaviour change as the symmetry is lost?
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84. Lab Exercise: Oscillations
open glycosc.ode in XPPAUT
Under Viewaxis -> 2D select s1 for the X-axis and s2 for the Y-axis (i.e. the phaseplane) Set Xmin=Ymin=0, Xmax=Ymax=4.
Choose Initialconditions -> mIce and select a number of initial conditions to explore the system behaviour
Select Nullclines -> New to see the nullclines.
Erase the plot and change the parameter q to 3. How is the behaviour different? You may want to use Window/zoom -> Zoom In to examine the behaviour near the steady state.
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85. Outline System modelling - focus on dynamics Model development
Analysis:
1) Parametric Sensitivity Analysis
2) Nonlinear Dynamics
3) Feedback Regulation
Reference: Feedback Systems: An introduction for Scientists and Engineers, Astrom and Murray
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86. Feedback regulation
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87. Simplest Feedback Strategy: (proportional) negative feedback
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88. An alternative: Integral Control
Proportional Control:
Steady state:
Proportional-Integral Control:
cover 2
Steady state:
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89. Primer on Integral Feedback Control
Time integral of system error is fed back.
Ensures that steady-state error approaches zero despite changes in the input or in the system parameters.
Ubiquitous in complex engineered systems.
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90. Lab Exercise: Metabolic Regulation
Goal: regulate s2 against variation in the substrate S S s1 s2 s3
proportional feedback
integral feedback
open metreg.ode in XPPAUT
Run the simulation (Initialconditions -> Go). Note the effect of the perturbation in S at time 5.
Open the Parameters Tab p1=gain on proportional feedback p2=gain on integral feedback
Change p1 to 1 and rerun. Note the effect of the regulation
Double the gain (to p1=2) and rerun. Note the level of improvement in the response
Change p1 back to 1 and add an integral feedback of equal strength (p2=1) to arrive at the same total gain as in the previous case. Note the vast improvement in steady state behaviour.
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91. Conclusions
Dynamic mathematical modelling is a valuable tools for exploring the behaviour of biochemical and genetic networks
Ordinary Differential Equations (ODEs) provide a powerful and accessible framework for the development of dynamics models
Local parametric sensitivity analysis allows the investigation of how system behaviour depends on system features (through parameters)
Nonlinear Dynamics addresses non-intuitive system behaviour such as bistability and oscillations
Control Theory provides tools for addressing regulation of system behaviour
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