Boolean Approaches to Genome-Cell Interactions

1. Boolean Approaches to Genome-Cell Interactions Review, Development, Examples

2. What is it that we want to know? Signaling network discussed by Kandel, AKH 04/06/03, See: http://www.nobel.se/medicine/laureates/2000/kandel-lecture.html CREB & Memory  Silva et al, 1998, Annual Reviews Neuroscience.

3. Review: Combinatorial Logic Ref: R. Thomas & R. D’Ari, "Biological Feedback" CRC Press, Boca Raton Fl. [1990], Ch. 2.

16. Part II: Biological Feedback Thomas, R & D’Ari, R, Ch’s 9,10

17. “A simple feedback loop is positive or negative according to whether it contains an even or odd number of negative interactions. In a simple positive loop, each element exerts a positive effect on its own rate of synthesis, whereas in a simple negative loop, each element exerts a negative effect on its own rate of synthesis.”

18. Negative Loops – Metabolite Synthesis End-product inhibition End-product repression of enzyme synthesis Attenuation (The important role of consumption and growth in post-stimulus adjustment.)

19. Positive Loops: ‘Epigenetic*’ control; multiple steady states Gene X with expression dependent on the presence of its own product x. Examples: cI repressor in  lysogens NFAT in IL2 gene of T cells “…the real essence of a regulatory circuit is not whether any individual control step is positive or negative, but rather whether the feedback loops involved are positive or negative.” * “epi” means “associated with”

20. Hill Functions: F, F+, F-

21. One-element Negative loop:

22. Effective Promotion of Homeostasis* (extreme case of Hill coefficient n) In case a, k/k- > , regulation is effective; in case b, k/k- < , regulation is ineffective * coordinated response of a physiological system to any situation or stimulus tending to disturb its normal condition or function.

23. Basics • Lower-case refers to products • x=0 means "gene product absent" • x=1 means "gene product present" • Upper case refers to product generators (genes, enzymes) • X=0 means "gene off" • X=1 means "gene on" • EXAMPLE: gene X on iff z is absent and gene Y on iff z is absent and u is is present:

24. Time (cf ODE description) • If a gene has been off (X=0), then is switched on (X=1) by, say, z falling to 0, then off again as z rises – what is the time course?

25. Time (continued) • Each element has an 'on' and an 'off' time tx, tx-, not all the same. In general, each transitionshould have its own t's. • States that are inherently transient (X=1,x=0; X=0,x=1) vs those that are inherently steady (X=1,x=1; X=0,x=0).

26. Naïve Logical Description • X = 1 iff y=0, Y = 1 iff x = 1. "State table": Start with products. Fill-inrule-driven generator states. Variable states constitute a logical vector xy. Functions, XY. Then: X=1(x,y, …), Y=2(x,y,…) XY is called image of xy.

27. System evolution A stable state is defined as one in which xy and XY are equal

28. Two-element positive loop When two choices are present, the choice taken will be fixed by the time delays.

29. 'Input' variables (operational and genetic) • Operational: add a drug, change temperature. • Genetic (mutation – in gene or operator) (gene X is active iff it is genetically normal AND (product z is absent OR the operator is inactive), with gx and ox being input variables.

30. Example (already considered). Now, Xmutated in operator region and so product is thermally insensitive. Thus expression does not depend on T but product is active at T="0" and inactive at T="1".

31. Alternate representation

32. "Practical" Steady States

33. Graphical Representation Examples: Z=x; Z=xy; Z=x+y Each element a vertex; each interaction an edge. Oriented graphs. Also, as before, an oriented, signed graph:

34. A two-loop (circuit) system: A graph of interactions vs. a graph of state sequences:

35. Graphs and Matrices

36. Graphs and Matrices