Goal

1. Goal Show the modeling process used by both Collins (toggle switch) and Elowitz (repressilator) to inform design of biological network necessary to encode desired dynamical behavior (bi-stability and oscillation, respectively).

2. Mathematical models predict qualitative behaviors of biological systems. Bi-stability in genetic toggle switch [1] Oscillation in genetic oscillator [2] Reversible flipping of an integrase-driven bit [3] Counting cellular events [3] . [1] Collins, Cantor 2000 [2] Elowitz 2000 [3] Us

3. Mathematics can predict qualitative behaviors of biological systems. Bi-stability in genetic toggle switch [1] Oscillation in genetic oscillator [2] Reversible flipping of an integrase-driven bit [3] Counting cellular events [3] . [1] Collins, Cantor 2000 [2] Elowitz 2000 [3] Us

4. Requirements Bi-stable[1] : holds two states Inducible switch between states [1] More than one attraction state; two stable equilibria in this case.

5. Design[1] IPTG LacI Rep2[1] Thermal induction or atC [1] Two different designs. pTAK plasmids have lacI repressor (IPTG inducible) and ptrc-2 promoter pair and PLs1con promoter with a temperature-sensitive repressor (cIts). pIKE plasmids have PLtetO-1 promoter in conjunction with the Tet repressor (tetR). pTAK plasmids switched by IPTG or thermal pulse. pIKE switched by IPTG or atC.

6. State variables[1] U: repressor 1 V: repressor 2 [1] Repressor concentration are the continuous dynamical state variables

7. Parameters ODEs Decay: Repressor degradation /dilution Repressor accumulation: cooperative repression of constitutively transcribed promoter Alpha : is the rate of protein synthesis Beta [1] : cooperativity in repression of promoter Parameters simplifications U RNA polymerase binding Open-complex formation Transcript elongation Transcript termination Repressor binding Ribosome binding Polypeptide elongation The dependence of transcription rate : Cooperativity Repression Decay rates of protein, and messenger RNA Cellular events complexity [1] The cooperativity arises from the multimerization of the repressor proteins and the cooperative binding of repressor multimers to multiple operator sites in the promoter.

8. Possible outcomes No steady state, mono stable[1], bi-stable [1] One repressor always shuts down the other

9. Question What parameter values yield bi-stability?

10. Coupled first-order ODEs Accumulation: cooperative repression of constitutively transcribed promoters Decay: Degradation/dilution of the repressors

11. Find steady state[1] 0 0 [1] Solution to both ODEs = 0, for a given set of parameter values

12. Solutions[1] 0 V U’=0 0 V’=0 U [1] Across range of U, V given parameters : alpha=1, b=y=3

13. One equilibrium (intersection) point for the system 0 V U’=0 0 V’=0 U

14. Evaluate across the parameter space to find bi-stability (> 1 intersection) Increasing cooperativity b=y=2 b=y=1 b=y=3 Alpha=1 Increasing synthesis rate Alpha=2 Alpha=3 V U

15. Bi-stable[1] when repressor expression rate and cooperativity are high b=y=2 b=y=1 b=y=3 Alpha=1 Alpha=2 Alpha=3 [1] Multiple intersections arise from sigmoidal shape, at b, y > 1, and high rate of repressor synthesis.

16. Similar[1] to what Collins shows U’=0 V V’=0 U [1] Parameters : alpha=2, b=y=3

17. Vector field shows system will move towards steady state Vector field V U [1] Parameters : alpha=2, b=y=3

18. Approaches one steady state if initial condition is high repressor 1 Initial condition: high U V Repressor level V’=0 (blue) U=2 U’=0 V=0.3 Time U U V [1] Parameters : alpha=2, b=y=3

19. Alter dynamic balance with inducer, repressor 2 maximally expressed. V V’=0 (blue) U’=0 U U V [1] Parameters : alpha=2, b=y=3

20. New initial condition for the simulation: settles into new steady state. Initial condition: high V V Repressor level V’=0 (blue) V=2 U’=0 U=0.3 Time U U V [1] Parameters : alpha=2, b=y=3

21. Mono-stability[1] Initial condition: high U Initial condition: high V [1] Single steady-state for parameters : alpha=1, b=y=3

22. To achieve bi-stability 1. Balanced and high rate of repressor synthesis 2. High co-operativity of repression 3. Induction to alters dynamic balance

23. Choose biological components (promoters / RBS / repressors) that meet these requirements!

24. Mathematics can predict qualitative behaviors of biological systems. Bi-stability in genetic toggle switch [1] Oscillation in genetic oscillator [2] Reversible flipping of an integrase-driven bit [3] Counting cellular events [3] . [1] Collins, Cantor 2000 [2] Elowitz 2000 [3] Us

25. Requirements Oscillation [1] No settling into steady state

26. Design[1]

27. State variables[1] 3 mRNA 3 repressor proteins [1] Repressor and mRNA concentration are the continuous dynamical state variables

28. Possible outcomes Steady state, or oscillation

29. Question What parameter values yield oscillation?

30. Six coupled first-order ODEs Accumulation: cooperative repression of constitutively transcribed promoters Decay: Degradation/dilution of the repressors Detailed discussion of parameters in appendix

31. Repressor logic embedded in equations The appropriate protein represses the appropriate mRNA synthesis and translation

32. Predict system behavior with respect to ODE parameters Linear algebra Prediction of parameter values that yield steady state and oscillation

33. Dynamic stability region with respect to parameters Unstable Strength of repressors Stable 1 Protein / mRNA degradation

34. Target : similar protein and mRNA degradation, minimal leakage (large drop in mRNA synthesis when repressed) Unstable Strength of repressors Stable 1 Protein / mRNA degradation

35. No leakage, high repressor expression Parameters : alpha=50, alpha0=0, beta=0.2, n=2

36. Leakage causes steady state Parameters : alpha=50, alpha0=1, beta=0.2, n=2

37. Low repressor expression causes steady state Parameters : alpha=2, alpha0=0, beta=0.2, n=2

38. Process Requirements Design Model: state variables, parameters Question: the issue model needs to resolve Collins Steady state analysis Explore parameter space Simulation Elowitz Find stability region Set parameters Simulation Understand parameter settings that encode desired dynamical behavior. Choose biological parts that adhere to parameter settings.

39. Iterate Requirements Design Model: state variables, parameters Question: the issue model needs to resolve Collins Steady state analysis Explore parameter space Simulation Elowitz Find stability region Set parameters Simulation Understand parameter settings that encode desired dynamical behavior. Choose biological parts that adhere to parameter settings.

40. Appendix

41. Parameters : mRNA model ODEs continuous dynamical state variables, (repressor concentration) Decay: Repressor degradation /dilution mRNA accumulation: cooperative repression of mRNA synthesis [5] : synthesis : leaky [2] synthesis : cooperativity of repression of promoter Parameters simplifications U RNA polymerase binding Open-complex formation Transcript elongation Transcript termination Repressor binding The dependence of transcription rate : Cooperativity Repression Decay rates of protein, and messenger RNA Cellular events complexity [1] Number of protein copies per cell n the presence of saturating amounts of repressor (owing to the `leakiness' of the promoter) [2] Here we consider only the symmetrical case in which all three repressors are identical except for their DNA-binding specificities. [3] Time is rescaled in units of the mRNA lifetime [4] mRNA concentrations are rescaled by their translation efficiency, the average number of proteins produced per mRNA molecule.

42. Parameters : Repressor protein model ODEs continuous dynamical state variables, (repressor concentration) Decay Repressor accumulation: cooperative repression of proteins produced by mRNA [5] : protein to mRNA ratio [1] : protein to mRNA decay rate ratio [1] Parameters simplifications Ribosome binding Polypeptide elongation Degradation and dilution Cellular events complexity [1] Time is rescaled in units of the mRNA lifetime [2] Protein concentrations are written in units of KM, the number of repressors necessary to half-maximally repress a promoter;