1. 1 1.3 Systems Modelling in Cell Biology Brian Ingalls Applied Mathematics University of Waterloo Waterloo, Ontario, Canada bingalls@math.uwaterloo.ca

2. 2 1.3 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

3. 3 1.3 Models in Cell Biology Roles of modelling Abstraction for the purposes of understanding http://www.ekcsk12.org/science/regbio/bi ochemnts.html

4. 4 1.3 Models in Cell Biology Roles of modelling Abstraction for the purposes of understanding Organization of results/theories (bookkeeping) http://www.emc.maricopa.edu/faculty/fara bee/BIOBK/cendogma.gif

5. 5 1.3 Models in Cell Biology Roles of modelling Abstraction for the purposes of understanding Organization of results/theories (bookkeeping) Description of spatial or temporal relationships http://www.engin.umich.edu/dept/che/research/l inderman/Research/Images/GPCRrxn2.jpg

6. 6 1.3 Models in Cell Biology The modeller must choose an appropriate level of abstraction. Glycolysis: Focus on flow of metabolites http://oregonstate.edu/instruction/bb450/st ryer/ch16/Slide4.jpg

7. 7 1.3 Models in Cell Biology The modeller must choose an appropriate level of abstraction. Glycolysis: Focus on chemistry http://www- bioc.rice.edu/~graham/Bios302/Glycolytic_ Pathway.jpeg

8. 8 1.3 Models in Cell Biology The modeller must choose an appropriate level of abstraction. Glycolysis: Focus on regulation http://www.cm.utexas.edu/academic/cou rses/Spring2002/CH339K/Robertus/ove rheads-3/ch15_reg-glycolysis.jpg

9. 9 1.3 Dynamic Modelling Focus on how system components influence rates of change of each component in the network. Result: description of time-varying behaviour.

10. 10 1.3 Example: consider a metabolic chain which is initially inactive. Experimental perturbation: activate first enzyme in the chain. ACB System behaviour: pools of metabolites build up, system reaches steady state

11. 11 1.3 E. Coli metabolism KEGG: Kyoto Encyclopedia of Genes and Genomes (http://www.genome.ad.jp/ kegg/kegg.html ) Intuition fails when faced with a complex interconnections…

12. 12 1.3 http://www.cm.utexas.edu/academic/courses/ Spring2002/CH339K/Robertus/overheads- 3/ch15_reg-glycolysis.jpg Intuition fails when faced with a complex interconnections or feedback …

13. 13 1.3 Eric Davidson's Lab at Caltech (http://sugp.caltech.ed u/endomes/) endomesoderm specification in the sea urchin Strongylocentrotus purpuratus Intuition fails when faced with a complex interconnections or feedback, or both.

14. 14 1.3 Another complex network:

15. 15 1.3 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

16. 16 1.3 Example: irreversible isomerization AB Qualitative description: the rate v of the reaction A B increases as the concentration of A increases. v A quantitative description: mass action v = k 1 [A] (mass action)

17. 17 1.3 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 reaction rate of change of concentration

18. 18 1.3 This predictive (mechanism-based) model is far more useful than a descriptive (data-based) model The resulting differential equations can be solved to determine the time-varying system behaviour. [A] [B] time concentration

19. 19 1.3 Numerical Simulation Recursive scheme: Approximation of the Derivative: Differential Equation: Construction of Approximate Solution:....

20. 20 1.3 { Steady state behaviour: Reversible Isomerization [A] [B] time concentration AB k1k1 k2k2 } Time rate of change equals zero Algebraic solution or Long-time behaviour of simulations give Steady state (equilibrium ratio):

21. 21 1.3 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)

22. 22 1.3 Zeroth order reactions: generation of s 1 from some buffered (external) source (S). Rate: Biochemical Reactions (Metabolism and protein-protein interaction networks) or

23. 23 1.3 First order reactions: isomerization, unassisted transport, degradation/dilution, or linearization of other kinetics Rate: Biochemical Reactions (Metabolism and protein-protein interaction networks) or concentration rate [s] or

24. 24 1.3 Second order reactions: Binding/association events Rate: Biochemical Reactions (Metabolism and protein-protein interaction networks)

25. 25 1.3 Michaelis-Menten Kinetics: Enzyme-catalysed reaction (metabolic, signal transduction, active transport) Rate: Biochemical Reactions (Metabolism and protein-protein interaction networks) or concentration rate [s] V max

26. 26 1.3 Hill-type Kinetics: Catalysis by cooperative enzyme or lumped description of multi- step process Rate: Biochemical Reactions (Metabolism and protein-protein interaction networks) or concentration rate [s] k

27. 27 1.3 Allosteric inhibition: Inhibitor I binds enzyme E and reduces its catalytic activity Rate: Biochemical Reactions (Metabolism and protein-protein interaction networks)

28. 28 1.3 Genetic Circuits unregulated mRNA transcription: zeroth order unregulated protein translation: first order in mRNA concentration degradation/dilution: first order mRNA protein transcription translation gene

29. 29 1.3 Genetic Circuits Regulated mRNA transcription: Hill type kinetics Activation: Inhibition: mRNA transcription gene transcription factor P or multimerization of P

30. 30 1.3 Example: autoinhibitory gene circuit: trp operon mRNA enzyme gene trp R R*R*

31. 31 1.3 mRNA enzyme gene trp R Species: m (mRNA)R (Repressor) e (enzyme)R * (Active Repressor) T (trp) R*R*

32. 32 1.3 mRNA enzyme  k1k1 gene d2d2 d1d1 trp R R*R* c1c1 k -2 k2k2

33. 33 1.3 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

34. 34 1.3 Other Software Packages Gepasi (http://www.gepasi.org/) E-Cell (http://www.e-cell.org/ecell/) Cellerator (http://www.cellerator.org/) JWS online (http://jjj.biochem.sun.ac.za/index.html) SBML: common markup language for biochemical and genetic models (http://sbml.org/Main_Page)

35. 35 1.3 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

36. 36 1.3 http://www.bmb.psu.edu/courses/b iotc489/notes/cycle.jpg http://www.schools cience.co.uk/conten t/4/biology/sgm/ima ges/yeast.jpg The cell cycle in budding yeast Initiation of replication

37. 37 1.3 http://www.science.uwaterloo.ca/biology/people/faculty/duncker/duncker.html Model focusing on construction of pre- replicative machinery

38. 38 1.3 orc cdc6 mcms cdt1 mcmscdt1 cdc6 mcms cdt1 orc cdc6 cdt1 orc cdt1 mcms orc mcms cdc45 orc mcms cdc45 orc cdc45 mcms elongation Nuclear export Nuclear export cdc6 swi5 cdc6 p clb2clb5 and SCF Detailed model indicating cyclic behaviour of pre- replicative machinery clb5

39. 39 1.3 orc cdc6 mcms cdt1 mcmscdt1 cdc6 mcms cdt1 orc cdc6 cdt1 orc cdt1 mcms orc mcms cdc45 orc mcms cdc45 orc cdc45 mcms elongation Nuclear export Nuclear export cdc6 swi5 cdc6 p clb2clb5 and RC1 RC2RC3 RC4 RC5 RC6 RC7 SCF clb5 Detailed model indicating cyclic behaviour of pre- replicative machinery rate of change of concentration rate of formation rate of degradation

40. 40 1.3 orc cdc6 mcms cdt1 mcmscdt1 cdc6 mcms cdt1 orc cdc6 cdt1 orc cdt1 mcms orc mcms cdc45 orc mcms cdc45 orc cdc45 mcms elongation Nuclear export Nuclear export cdc6 swi5 cdc6 p clb2clb5 and SCF Detailed model indicating cyclic behaviour of pre- replicative machinery clb5

41. 41 1.3 # 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 Complete Dynamic Model

42. 42 1.3 Results: Simulation

43. 43 1.3 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.)

44. 44 1.3 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,...) p 1 = 3.4 p 2 = 13.6 p 3 = 0.7...

45. 45 1.3 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

46. 46 1.3 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”

47. 47 1.3 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

48. 48 1.3 Parametric Sensitivity Analysis: Example reaction kinetics: steady state:

49. 49 1.3 steady state: local sensitivity analysis: effect of perturbation/ intervention: relative sensitivity:

50. 50 1.3 steady state: sensitivity analysis: vector notation implicit differentiation

51. 51 1.3 Sensitivity Analysis: General Computation model: steady state: differentiate: absolute sensitivity:

52. 52 1.3 complete sensitivity analysis:

53. 53 1.3 Parametric Sensitivity Analysis Parametric sensitivity analysis investigates the relationship between the variables and parameters in a biochemical network. Variables 1. Concentrations 2. Pathway fluxes 3. Dynamic response 4. Growth rate 5..... Parameters 1. Enzyme activity levels 2. Kinetics constants 3. Decay rates 4. Boundary conditions 5.....

54. 54 1.3 Application:

55. 55 1.3 sensitivity of flux J to enzyme activities:

56. 56 1.3 sensitivity of flux J to enzyme activities: Summation Theorem of Metabolic Control Analysis: conservation law for sensitivities

57. 57 1.3 Applications of Sensitivity Analysis Trypanosome metabolism. Bakker et al., 1999,J. Biol. Chem Predicting the effect of interventions Drug development

58. 58 1.3 Applications of Sensitivity Analysis Predicting the effect of interventions Drug development Medicine Tumour growth and thiamine, Comin-Anduix et al., 2001, Eur. J. Biochem.

59. 59 1.3 Applications of Sensitivity Analysis Predicting the effect of interventions Drug development Medicine Metabolic engineering Diacetyl production in Lactococcus lactis, Hoefnagel et al. 2002, Microbiology

60. 60 1.3 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.

61. 61 1.3 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.

62. 62 1.3 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 Trp ss more sensitive to X (substrate for trp production), k cat (enzymatic activity of trp production) or c 1 (rate of Trp consumption)? Consider a large parameter change: increase  (rate of mRNA transcription) to 150.

63. 63 1.3 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

64. 64 1.3 Nonlinear Dynamics Phase plane Analysis

65. 65 1.3 Phaseplane Analysis Time Course ([s i ] vs. time)Phase Plane ([s 1 ] vs. [s 2 ])

66. 66 1.3 Direction Field

67. 67 1.3 Nullclines: Turning points

68. 68 1.3 Nullclines: Turning points s 1 nullcline: s 1 not changing s 2 nullcline: s 2 not changing intersection of nullclines: neither s 1 nor s 2 changing: steady state

69. 69 1.3 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.

70. 70 1.3 Stability Y <<X steady state independent of initial conditions

71. 71 1.3 Bistability Y <<X Nullclines intersect once: one steady state

72. 72 1.3 Bistability Y = X steady state depends on initial conditions

73. 73 1.3 Bistability Y = X Nullclines intersect three times: three steady states

74. 74 1.3 Bistability Y = X Nullclines intersect three times: three steady states

75. 75 1.3 Bistability Y = X Middle steady state repels trajectories

76. 76 1.3 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)

77. 77 1.3 Oscillatory behaviour q=2 single stable steady state

78. 78 1.3 Oscillatory behaviour q=3 single unstable steady state and limit cycle

79. 79 1.3 Bifurcations

80. 80 1.3 Bifurcations

81. 81 1.3 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?

82. 82 1.3 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.

83. 83 1.3 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

84. 84 1.3 Feedback regulation

85. 85 1.3 Simplest Feedback Strategy: (proportional) negative feedback http://www.cofc.edu/poolel/trp_operon.jpg

86. 86 1.3 An alternative: Integral Control Proportional Control: Steady state: Proportional-Integral Control:

87. 87 1.3 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.

88. 88 1.3 Lab Exercise: Metabolic Regulation 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. Goal: regulate s 2 against variation in the substrate S Ss1s1 s2s2 s3s3 proportional feedback integral feedback

89. 89 1.3 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