Robustness of Intracellular Oscillators and Bistable Switches

1. Robustness of Intracellular Oscillators and Bistable Switches Elling W. Jacobsen Automatic Control Lab School of Electrical Engineering KTH

2. Outline • The role of autonomous oscillations and multistability in cellular biology – some examples • Models and uncertainty / perturbations • parametric vs structural perturbations • Quantifying robustness and identifying fragilities using distance to structural instability • Applications • two minimal gene regulatory networks • metabolism of activated white blood cells • mammalian circadian clock • B cell differentiation • Conclusions

3. Circadian Clocks • provides periodic control of biological activity • autonomous oscillations in mammalian SCN generated by a network of interacting genes

4. The Segmentation Clock • Formation of somites in vertebrate segmentation are driven by an autonomous gene regulatory oscillator

5. Oscillatory Calcium Signaling • Amplitude and frequency modulated Ca2+ oscillatory signals controls a wide variety of intra- and intercellular processes

6. Metabolic Oscillations in Activated Neutrophils • Neutrophils make up first line of defense against bacterial infections • oscillatory production of lethal chemicals. Hypothesis: provide peaks that kill bacteria without harming the cell itself

7. The lac operon and phage lambda switches • lac genes turned on when glucose low and lactose available • positive feedback loop • first experimental evidence of intracellular bistability (1957) • Phage ¸ virus infecting a cell switches between Lytic and Lysogenic states through a bistable switch • A double negative feedback generates switch

8. Apoptosis – Programmed Cell Death • The death decision in programmed cell death corresponds to a bistable switch • ~50 billion cells/day undergo apoptosis in a healthy adult human

9. Cell Cycle Control • Gene regulatory networks provide bistable switches that control transitions between the various phases of the cell cycle

10. Blood Stem Cell Differentiation • Antigens stimulate differentiation of B cells into antibody secreting cells through a bistable switch

11. Intracellular Oscillators and Switches • Autonomous oscillators and multi-/bistable switches are frequently observedin the cell; biology utilizes strongly nonlinear phenomena • An important aim of systems biology is to determine the architecture of the underlying biochemical networks • Modeling: typically bottom-up, ODEs • Robustness analysis important for • model (in)validation • determining essential/non-essential components and interactions • identifying fragilities • ultimately elucidating the principles behind biological robustness

12. A minimal gene oscillator Bifurcation diagram

13. A minimal gene switch Bifurcation diagram State space

14. Robustness Analysis • How robust are these behaviors? • quantitative properties • persistence of behavior • Perturbations? • reflecting real biological uncertainty • model uncertainty; parametric, structural

15. Perturbations / Uncertainty • For analysis of biological robustness, perturbations should reflect true biological uncertainty • external disturbances (environment) • internal perturbations (gene mutations) • internal noise (stochastic fluctuations) • For modeling purposes, perturbations should • reflect model uncertainty (uncertain kinetic models, uncertain components, …) • provide mechanistic insight

16. Parametric vs Structural Perturbations • Structural uncertainty: • ¢ represents a dynamic perturbation of the direct interactions between biochemical components • allows for uncertain strength/dynamics of interactions, uncertain number of nodes and edges • motivated by the fact that involved components usually highly uncertain, standardized description of interactions (reaction kinetics). • Parametric uncertainty: • “standard” uncertainty description in biological models • mainly motivated by the fact that parameters are fitted to experimental data R Pc Pn

17. Structural Robustness Analysis • Consider persistence of qualitative behavior )determinesmallest perturbation that makes system structurally unstable • Assume nominal model possesses a steady-state and consider perturbations that translatesit to a bifurcation point • sustained oscillations: translate steady-state to Hopf point • bistability: translate steady-state to saddle-node • Linear problem: write network on feedback form and apply Small Gain Theorem R Pc Pn

18. Network on Feedback Form • Taylor expansion • Jacobian A determines structural stability • 2nd and 3rd order terms B and C determines type of bifurcation, e.g., sub- or supercritical • Write the linear part on feedback form where is stable

19. Perturbing the Network • The smallest distance to a structurally unstable steady-state is given by the robustness radius • Where and ¢ in general will be a structured matrix • The corresponding dynamic perturbation

20. Perturbing the activity of all network nodes • ¢ is a diagonal n £ n matrix • The robustness radius quantifies overall robustness of network behavior

21. Perturbing the activity of single nodes • ¢ is a scalar • provides information on the role of single components

22. Perturbing individual edges • ¢ij is a scalar • provides information on the role of specific interactions

23. Comments • All perturbations are relative • Complex perturbations ¢) tight bounds for can in general be computed (for moderaten) • There may exist smaller perturbations that induce bifurcations away from considered steady-state. • Thus, can in principle only identify non-robust features

24. Minimal Gene Oscillator – overall robustness • , i.e., a 2% change in the component activities sufficient to translate steady-state into HB point

25. Minimal Gene Oscillator – perturbing single nodes and edges • Perturbing single components: • Perturbing single interactions:

26. Minimal Gene Oscillator - impact of perturbing an edge on limit cycle • Impact of perturbing effect of transcription factor on gene activity R Pc Pn

27. Minimal Gene Oscillator – impact on bifurcation diagram

28. Minimal gene switch – perturbing effect of gene A on gene B

29. But the main use is of course for ….. more complex networks

30. Application to Modeling of Metabolic Oscillations in Activated Neutrophils Olsen et al, BioPhysJ • two compartments: cytosol and phagosome • model: 7 metabolites in cytosol, 9 metabolites in phagosome • 14 known chemical reactions, 5 transport eqs • lumped model with 14 ODEs • 25 parameters, fitted to experimental data

31. Neutrophil Network • The 14 state model corresponds to a highly connected network • predicts experimentally observed period and amplitudes • robust to parameter variations up to 20% (Olsenet al, BioPhysJ)

32. Spatial Waves – extending the model • The Olsen model assumes perfectly mixed cell environment, i.e., no spatial gradients • Experimental observations: oscillations correspond to spatiotemporal traveling waves • Cedersund (2008): oscillations in Olsen model disappear even for very small spatial gradients (large diffusion rates) ) appears unrobust to structural perturbations

33. Robust Instability of Steady-state From Nyquist: poor robustness margins ) some small perturbation of network will make underlying steady-state structurally unstable

34. Perturbing all nodes Overall robustness: can only tolerate up to 0.2% simultaneous change in all component activities

35. Perturbing Individual Components Perturbing the activity of metabolite 1 (per3+) or 4 (H2O2) by approx 1% completely removes oscillatory behavior

36. Perturbing Single Edges The most significant fragilities involve interactions between metabolites 1, 2 and 4

37. Perturbing the Nonlinear Model • Most severe fragility involves effect of metabolite 1 on metabolite 2 • Can be fitted to a time-delay • Add delay to corresponding reaction kinetics of original model

38. Effect of Delay on Robust Instability of Steady-State Nyquist Locus

39. Effect of Delay on Limit Cycle Behavior • Effect of delay µ in reaction R2 on bifurcation behavior of full model • linear analysis yields good prediction of nonlinear behavior • the added delay should be compared to oscillation period T>20 s

40. Robustification • Robustness analysis reveals unrobust features of model and identifies main fragilities • Robustification: optimize robustness by adding generic perturbations to network edges ) provides experimentally testable hypotheses on plausible model modificiations Neutrophil model: increase robustness significantly through small modifications of kinetics for reactions involving metabolites 2 and 4 (Nenchev and Jacobsen, to appear)

41. MammalianCircadian Clock – non-essential interactions Leloup and Goldbeter (PNAS, 100, 12) • What is purpose of the apparent excessive complexity? • Common hypothesis: improved robustness

42. Perturbing Individual Nodes • Nodes with robustness radius can be removed without inducing a qualitative change in network behavior ) non-essential • 11 out of 16 components in LeloupGoldbeter model non-essential

43. Perturbing Individual Edges • 29 out of 34 edges (interactions) non-essential • Only interactions related to Per gene appear essential for oscillatory behavior

44. A single loop generates the oscillations • Reduced model with 5 states predicts full model oscillations well • Most of the network apparently not required for generating circadian oscillations

45. What about robustness? • Robustness almost unaffected by non-essential nodes • Purpose of network complexity?

46. Bistable Switch in B Cell Differentiation • Antigens stimulate differentiation of B cells into antibody secreting cells through a bistable switch • Robustness for perturbation of all nodes

47. Bistable Switch in B Cell Differentiation • Perturbing individual components • all components except APIp (10), TA (11) and TAA (12) appear essential

48. Bistable Switch in B Cell Differentation • Perturbing single edges • most interactions needed to provide bistability

49. Perturbing the nonlinear model • Perturbing effect of Blimp1 mRNA on Blimp1 protein by 30%

50. Summary • Nonlinear phenomena frequently utilized to create critical functionality in cellular biology • Analyzing the robustness of bistable switches and autonomous oscillators important for modeling and elucidating the mechanisms of the underlying biochemical networks • We considered robustness in terms of persistence of behavior and proposed to determine smallest perturbation that makes a steady-state structurally unstable (bifurcation) • By strategically adding perturbations to nodes and edges fragilities can be identified and key mechanisms unravelled