1. Using Logical Circuits to Analyze and Model Genetic Networks Leon Glass Isadore Rosenfeld Chair in Cardiology, McGill University

2. Introduction to logical models – history and application Mathematics and applications – evolution of electronic circuits and the inverse problem

3. Puzzle: Is there a simple way to think about genetic networks? If no, we are in trouble. If yes, then might ideas developed using logical networks be relevant?

4. “From molecular to modular cell biology” Hartwell, Hopfield, Leibler, Murray (Nature, 1999) “The next generation of students should learn how to look for amplifiers and logic circuits, as well as to describe and look for molecules and genes.”

5. Logical Network Models Neural Networks – A logical calculus of the ideas immanent in nervous activity. McCulloch and Pitts (1943). See also Kling & Szekely, Cowan, Sejnowski, Grossberg, Hopfield and many others Genetic Networks – Teleonomic mechanisms in cellular metabolism, growth and differentiation. Jacob and Monod (1961). See also Sugita, Kauffman, Thomas, Bray and many others

6. Kling and Szekely, Kybernetik, 1968

7. Logical Models - Positive Many superb papers identify logical functions as key controllers in biological systems and have developed models based on this concept.

8. (Bull. Math. Biol. 1995)

9. (Science, 1998)

10. (J. Theor. Biol. 1998)

11. Logical Models - Negative Logical models are not well known to experimental biologists. Typical models often consist of complex networks without an analytical context. If logical models really worked, people would use them.

12. Logical Models - Positive Some experimental systems show clear evidence of discrete genetic expression patterns in time and space.

13. Slide from John Reinitz

14. Logical Models - Negative Some data does not show any obvious evidence of the operation of discrete expression levels (Circ Res 2007)

15. Logical Models - Positive Biobricks Website, MIT – Many parts are based on logical models. Synthetic biology competition.

16. Logical Models - Negative (Science, 2002)

17. Logical Models - Positive Beautiful mathematical formulation for analyzing such networks (I will describe this in a minute).

18. Logical Models- Negative Logical formulations are not easily derived from mass action kinetics unless one has special features such as cooperativity or cascades to achieve threshold-like control. Many analytic problems may arise due to important factors such as time delays, stochasticity, spatial structure that have not yet been carefully addressed.

19. Synthetic Biology Uses Ideas from Logical Models Toggle switch Inhibitory loops (repressilator)

20. Construction of a genetic toggle switch in Escherichia coli Gardner, Cantor & Collins (2000)

21. Construction of the plasmid Gardner, Cantor & Collins (2000)

22. Two stable steady states Gardner, Cantor & Collins (2000)

23. gene A mRNA A protein A PC gene B mRNA B protein B PA gene C mRNA C protein C PB TetR LacI cI Elowitz and Leibler, 2000 A synthetic oscillatory network of transcriptional regulators

24. amp R SC101 origin P L tetO1 cI-lite P R lacI-lite P L lacO1 tetR-lite ColE1 kan R P L tetO1 gfp-aav RepressilatorReporter TetR GFP Plasmids TetR LacI cI Elowitz and Leibler, 2000

25. 60140250300390450550600 Fluorescence (a.u.) time (min) Observation in Individual Cells GFP Fluorescence Bright-Field Elowitz and Leibler, 2000

26. Problem: How can we develop mathematical models that represent the dynamics in real networks?

27. A Boolean Switching Network X i is either 1 or 0 B i is a Boolean function Random boolean networks as gene models (Kauffman, 1969)

28. A differential equation Glass, Kauffman, Pasternack, 1970s

29. Rationale for the equation A method was needed to relate the qualitative properties of networks (connectivity, interactions) to the qualitative properties of the dynamics The equations allow detailed mathematical analysis. Discrete math problems (classification), nonlinear dynamics (proof of limit cycles and chaos in high dimensions)

30. The Repressilator

31. The Hypercube Representation

32. The Hypercube Representation for Dynamics (N genes) 2 N vertices – each vertex represents an orthant of phase space N x 2 N-1 edges – each edge represents a transition between neighboring orthants For networks with no self-input, there is a corresponding directed N-cube in which each edge is oriented in a unique orientation

33. Fixed Points A vertex that only has in arrows represents a stable fixed point. It is robust under changes in parameter values

34. Cyclic Attractors Any attracting cycle on the hypercube corresponds to either a stable limit cycle or a “stable focus” in the differential equation (Glass and Pasternack, 1978)

35. Evolving Rare Dynamics Long cycle Chaotic dynamics - increased complexity using topological entropy as a measure of complexity

36. The number of different networks in N dimensions Glass, 1975; Edwards and Glass, 2000

37. An Evolvable Circuit (J. Mason, J. Collins, P. Linsay, LG, Chaos, 2004)

38. Why study electronic circuits? It is real It leads us to think about issues in real circuits – i.e. not all decay rates will be equal Circuits could be useful

39. The Hybrid Analog-Digital Circuit

40. Circuit Elements

41. Distribution of Cycle Lengths in Electronic Circuit (300 random circuits with stable oscillations) Choose a target period of 80 ms

42. Sample Evolutionary Run

43. Optimal Mutation Rate - Data Each trial starts with oscillating network 25 Trials at each mutation rate for 250 generations Mutation rates of 2.5%, 5%, 10%, 20%, 100%

44. Prediction of Optimal Mutation Rate Compares favorably with experimentally determined value of ~5-10%

46. The Inverse Problem. Compute the number of logical states needed to determine connectivity diagram Perkins, Hallett, Glass (2004)

47. Compute the number of switches needed to determine the entire network

48. Gene expression in Drosophila Perkins, Jaeger, Reinitz, Glass PLOS Computational Biology 2006

52. ( Perkins, Jaeger, Reinitz, Glass, PLOS Computational Biology 2006 )

54. Proposed network for gene control

55. Comparison with different models

56. Some Important Ideas About Logical Network Models They do not require discrete time or states Logical networks can be embedded in differential equations (that’s the main idea of this talk) Qualitative features of networks are often preserved by changing step function control to sigmoidal function control Neural network models are a subclass of the differential equations I described

57. Mathematical Models of Neural and Gene Networks are Closely Related

58. Properties of Networks Based on Logical Structure “Extremal” stable fixed points Limit cycles associated with cyclic attractors (stability and uniqueness) Necessary conditions for limit cycles and chaos Analysis of chaos in some networks Upper limit on topological entropy

59. Conclusions Logical models do provide a rich class of models appropriate for many real biological systems The limitations of this class of models is not known

60. Thanks Stuart Kauffman, Joel Pasternack, Rod Edwards, Jonathan Mason, Paul Linsay, James Collins, Ted Perkins, Yogi Jaeger, John Reintiz. NSERC, MITACS