Mathematical Modelling for Synthetic Biology

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 Mouse model of obesity http://mariovalle.name/ChemViz/representations/index.html http://srxawordonhealth.com/2012/04/

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

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 silicoexperiments) • 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: differential equation model rate of reaction at time t rate of change of [A] at time t

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

12. Model Simulation

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

14. Repeated application of the update rule (starting from known initial concentration): 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)

15. Example network model network 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 Trypanosomabrucei? 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 sigmoidalactivity (e.g. cooperative binding of O2 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 Repressed expression Constitutive expression

29. Regulated gene expression Activated expression Constitutive expression

30. Modelling regulated expression

31. 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 cI cro 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, Bolouriet 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 single cell progeny

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

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

47. Stricker/Hasty oscillator implementation

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

49. Lab I • 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 • Open XPPAUT with file Lab2.ode • 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. • 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 filesLab_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