1. Mathematical Modelling for Synthetic Biology aGEM Workshop on Mathematical Modelling July 22, 2012, Lethbridge, AB Brian Ingalls Department of Applied Mathematics University of Waterloo Waterloo, ON

2. Workshop Outline Introduction to mathematical modelling of biochemical reaction networks Modelling of gene regulatory networks Lab I: simulation of kinetic models Tools for model analysis Lab II: model-based design of gene regulatory networks

3. Models in Science and Engineering Models are abstractions of reality

4. Models in Science and Engineering Models are abstractions of reality Models can be physical Ball-and-stick model of molecular structure http://mariovalle.name/ChemViz/representations/index.htmlhttp://srxawordonhealth.com/2012/04/ Mouse model of obesity

5. Models in Science and Engineering Models are abstractions of reality Models can be physical or conceptual http://www.nature.com/scitable/topicpage/gpcr-14047471 Interaction diagram model of G-protein signalling Kinetic model of bacterial chemotaxis signalling pathway

6. Models in Science and Engineering Models are abstractions of reality Models can be physical or conceptual Mathematical models are mechanistic (based on physico-chemical laws) and predictive (allow inferences beyond the data used for their construction)

7. How are mathematical models used in molecular biology? Models summarize data Models allow of falsification of hypotheses Models allow exploration of system behaviour (in silico experiments) Model-based design allows easy exploration of design space

8. Model Construction

9. Modelling Chemical Reaction Networks Chemical reaction: Rate constant: Law of mass action:

10. Using derivatives to describe rates of change Ex: decay reaction: rate of change of [A] at time t rate of reaction at time t differential equation model

11. Solution of the differential equation model Model simulation = in silico experiment

12. Model Simulation

13. Numerical simulation of differential equation models Approximate derivative by a difference quotient Rearrange to yield an update rule:

14. Implemented in MATLAB, XPPAUT, Copasi, Mathematica, Maple, … Tutorials in notes for XPPAUT (freeware, simulation and analysis of differential equation models) MATLAB (licensed, general-use computational software) Repeated application of the update rule (starting from known initial concentration):

15. Example network model networ k simulation model

16. Model Analysis

17. Separation of time-scales Every model is formulated around a specific time-scale  Processes occurring on a slower time-scale are treated as frozen in time  Processes occurring on a faster time-scale are presumed to occur instantaneously

18. Treating rapid processes Rapid equilibrium approximation: presume at all times. Quasi-steady-state approximation: presume [A] is in steady-state with respect to [B] at all times.

19. Sensitivity analysis Measure sensitivity of steady-state species concentrations to changes in model parameters

20. Applications of Sensitivity analysis Identification of optimal drug targets (steps with high sensitivities) Bakker et al. 1999, ‘What controls glycolysis in bloodstream form Trypanosoma brucei? JBC 274.

21. Applications of Sensitivity analysis Interpretation of regulation schemes: role of negative feedback

22. Biochemical Kinetics

23. Saturation: Michaelis-Menten kinetics Rates of enzyme-catalysed reactions exhibit saturation: Michaelis-Menten kinetics

24. Cooperativity: Hill function kinetics Processes involving multiple interacting components can exhibit sigmoidal activity (e.g. cooperative binding of O 2 to hemoglobin) Hill function (empirical fit)

25. Gene Regulatory Networks

26. Modelling constitutive gene expression mRNA dynamics: protein dynamics:

27. Modelling constitutive gene expression If mRNA dynamics are fast (compared to protein dynamics): Treat mRNA in ‘quasi-steady-state’ Reduced model only describes protein concentration :

28. Regulated gene expression Constitutive expression Repressed expression

29. Regulated gene expression Constitutive expression Activated expression

30. Modelling regulated expression

32. Regulation by multiple transcription factors Distribution of states: If A=B=P, with cooperative binding:

33. Natural Gene Regulatory Networks

34. Autoregulating genes Autorepressor: (regulation enhances robustness and response timing) Autoactivator: (bistable ON/OFF switching behaviour)

35. Gene switch: lac operon Gene autoactivates in response to lactose

36. Gene switch: lysis/lysogeny decision in phage lambda cro cI Double negative feedback locks in one of two states

37. Oscillatory gene network: the Goodwin oscillator Delayed negative feedback leads to sustained oscillations

38. Oscillatory gene network: circadian rhythm generator Delayed negative feedback leads to sustained oscillations Model: Goldbeter, 1996

39. Developmental gene networks Endomesoderm specification in purple sea urchin (Davidson, Bolouri et al.) Segmentation in Drosophila

40. Engineered Gene Circuits

41. The Collins Toggle Switch Gardner, Cantor, and Collins, Nature, 2000 Double repression locks in one of two possible states. Inducers allow transitions

42. Collins toggle switch: implementation

43. The Repressilator Elowitz and Leibler, Nature, 2000 Three-step repression ring generates delayed negative feedback: sustained oscillations

44. Repressilator: Implementation progeny single cell

45. Improved oscillator design: relaxation oscillator Interplay of positive and negative feedback lead to robust sustained rise-and-crash oscillations

46. Stricker/Hasty oscillator Stricker et al., Nature, 2008 Interplay of positive and negative feedback lead to robust sustained oscillations

47. Stricker/Hasty oscillator implementation

48. Lab I Goals: 1)Simulate a differential equation model in XPPAUT 2)Determine sensitivity coefficients for a simple network model 3)Explore system behaviour in models of gene regulatory networks: the Collins toggle switch, the repressilator, the Hasty/Stricker oscillator

49. Lab I 1)Simulate a differential equation model in XPPAUT a) Open XPPAUT with file Lab1.ode or generate your own file, content is simply: par k=1 x’=-k*x init x=1 done b) Select Initialconds|Go (I|G). Resize the window with Window/zoom|Fit (W|F). Then choose Initialconds|New (I|N) and run a simulation with initial value of x set to 0.5. c) Open the param window, change the value of k to 1. Re-run your simulations from x(0)=1 and x(0)=0.5. How has the behaviour changed? 2) Determine sensitivity coefficients for a simple network model a)Open XPPAUT with file Lab2.ode b)Select Initialconds|Go (I|G). Use the Data window to view all four species concentrations. Select Graphic stuff|Add curve (G|A) to add additional time-series to the plot. c)Open the param window. Explore the effect of changing the parameter values. Consider the sensitivities of the steady state of [A] with respect to (i) k1; (ii) k2; and (ii) k3. 3 ) Explore system behaviour in models of gene regulatory networks: the Collins toggle switch, the repressilator, the Hasty/Stricker oscillator a) Open XPPAUT with one of the files Lab_toggle.ode, Lab_repressilator.ode, or Lab_hastyosc.ode b) Select Initialconds|Go (I|G). Verify the system’s desired behaviour: oscillations or bistability (for bistability, modify the initial conditions). c) Explore the effect of modifying the model parameters (param window) on the desired behaviour

50. Tools for analysis of dynamic mathematical models

51. The phase plane

52. Example network: Time series: species concentrations plotted against time Phase portrait: species concentrations plotted against one another The phase plane

53. Time series: multiple simulations (range of initial concentrations) Phase portrait: multiple simulations reveal system behaviour

54. The phase plane Direction field Nullclines (turning points) Intersection of nullclines: steady state.

55. Stability

56. Example: symmetric antagonistic network: Case I: Non-symmetric inhibition strengths: unique long-time (steady-state) behaviour

57. Stability Example: symmetric antagonistic network: Case II: Symmetric inhibition strengths: two potential long-time (steady-state) behaviours

58. Stability Nullclines intersect three times. Intermediate steady- state is unstable, other two are stable.

59. Stability A bistable system

60. Stability monostability bistability saddle point

61. Linearized Stability Analysis Evaluate Jacobian at steady state: Model: Determine eigenvalues of Jacobian: If all eigenvalues have negative real part, then the steady state is stable

62. Oscillations

63. Oscillatory behavior Example: autocatalytic pathway: Case I: weak autocatalysis: damped oscillations (settling to steady state)

64. Oscillatory behavior Example: autocatalytic pathway: Case II: strong autocatalysis: sustained (limit cycle) oscillations Limit cycle

65. Bifurcation analysis

66. Bifurcation diagrams The location of steady states depends on model parameters Plot of steady-state concentration against parameter value: continuation diagram

67. Bifurcation diagram: bistability Parameter values at which the number or stability of steady states change are called bifurcation points nullclines vary with parameter values Bistable network

68. Bifurcation diagram: oscillations Autocatalytic network Damped (transient) oscillation persistent (limit cycle) oscillation

69. Model-based design

70. The Collins Toggle Switch repression Network: Goal: bistability

71. Model: Model-based design analysis Conclusions: need strong expression, highly nonlinear repression

72. The Repressilator Network: Goal: sustained oscillations

73. Conclusions: need low leak, strong non-linearity, short-lived proteins/long-lived mRNA Model: Model-based design analysis

74. Alternative Modelling Frameworks

75. Alternative modelling frameworks Stochastic modelling Compartmental modelling Boolean modelling Spatial Modelling (PDEs)

76. Lab II Goal: Model-based design analysis of the Collins toggle switch and the Hasty/Stricker oscillator

77. Lab II 1)Model-based design analysis of the Collins toggle switch a)Open XPPAUT with Lab_toggle.ode. b)Generate a phase plane (Viewaxis|2D -- select P1 and P2 for the axes, set the window to [0,4]x[0,4]). Select Initialconds|mIce (I|I) and generate multiple trajectories c)Generate the nullclines: Nullclines|New (N|N) d)Open the param window and change b to b=1. Reconstruct trajectories and nullclines. (Erase the previous phase portrait.) How has the behaviour changed? e)Explore a range of values for each parameter. What changes improve the robustness of the bistability? Which changes eliminate it? What design choices can you recommend? 2 ) Model-based design analysis of the Hasty/Stricker oscillator relaxation a)Open XPPAUT with Lab_hastyosc.ode b)Repeat (b) and (c) above. (Set the window to X,Y [0,4]x[0,4]). c)Open the param window and change alpha to alpha=5. Reconstruct trajectories and nullclines. (Erase the previous phase portrait.) How has the behaviour changed? d)Explore a range of values for each parameter. What changes eliminate the oscillations? What design choices can you recommend? 3 ) Stability analysis Determine the robustness of steady-state stability by determining the eigenvalues of the system Jacobian: select Sing. pts.|Mouse, click near an equilibrium, print eigenvalues (negative real part signifies stability) 4) Bifurcation analysis Generate a bifurcation diagram. Set parameters to their default values. Run a trajectory to steady state. Hit (I|L) a few times to ensure steady state has been reached. Open AUTO: F|A. Set the parameter of interest to Par1 in the Parameter window. Set the window size in the Axes window. Set the parameter value range (Par Min and Par Max) in the Numerics window. Tap Run.