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Systems Modelling in Cell Biology

Systems Modelling in Cell Biology. Brian Ingalls Applied Mathematics University of Waterloo Waterloo, Ontario, Canada bingalls@math.uwaterloo.ca. Outline. System modelling - focus on dynamics Model development Analysis: 1) Parametric Sensitivity Analysis

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Systems Modelling in Cell Biology

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

  2. 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. Models in Cell Biology Roles of modelling • Abstraction for the purposes of understanding http://www.ekcsk12.org/science/regbio/biochemnts.html

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

  5. 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/linderman/Research/Images/GPCRrxn2.jpg

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

  7. 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. Models in Cell Biology The modeller must choose an appropriate level of abstraction. Glycolysis: Focus on regulation http://www.cm.utexas.edu/academic/courses/Spring2002/CH339K/Robertus/overheads-3/ch15_reg-glycolysis.jpg

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

  10. A B C 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

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

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

  13. 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 (http://sugp.caltech.edu/endomes/)

  14. Another complex network:

  15. 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. Example: irreversible isomerization v A B 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)

  17. rate of change of concentration +/- rate of reaction This quantitative description of the reaction rate can be used to characterize the rates of change of the chemical species in the network:

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

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

  20. Steady state behaviour: Reversible Isomerization 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

  21. 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. Biochemical Reactions (Metabolism and protein-protein interaction networks) Zeroth order reactions: generation of s1 from some buffered (external) source (S). Rate: or or

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

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

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

  26. 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 concentration [s]

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

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

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

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

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

  32. R k2 R* c1 k-2 trp k1  enzyme mRNA d2 d1 gene

  33. 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. 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. 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. The cell cycle in budding yeast Initiation of replication http://www.schoolscience.co.uk/content/4/biology/sgm/images/yeast.jpg http://www.bmb.psu.edu/courses/biotc489/notes/cycle.jpg

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

  38. mcms cdt1 swi5 cdc6 elongation clb2 and clb5 Detailed model indicating cyclic behaviour of pre-replicative machinery mcms cdt1 SCF cdc6 p clb5 cdc6 mcms cdc6 cdt1 orc cdc6 orc mcms cdt1 orc orc mcms cdc45 Nuclear export orc orc cdt1 cdc45 mcms cdc45 mcms orc cdc45 Nuclear export

  39. mcms cdt1 swi5 cdc6 elongation clb2 and clb5 Detailed model indicating cyclic behaviour of pre-replicative machinery mcms cdt1 SCF cdc6 p clb5 cdc6 mcms cdc6 cdt1 orc cdc6 orc RC2 RC3 mcms cdt1 orc 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 Nuclear export rate of change of concentration

  40. mcms cdt1 swi5 cdc6 elongation clb2 and clb5 Detailed model indicating cyclic behaviour of pre-replicative machinery mcms cdt1 SCF cdc6 p clb5 cdc6 mcms cdc6 cdt1 orc cdc6 orc mcms cdt1 orc orc mcms cdc45 Nuclear export orc orc cdt1 cdc45 mcms cdc45 mcms orc cdc45 Nuclear export

  41. 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

  42. Results: Simulation

  43. 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. 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 ...

  45. 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. 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. 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. Parametric Sensitivity Analysis:Example reaction kinetics: steady state:

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

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

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