Boolean Approaches to Genome-Cell Interactions
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Boolean Approaches to Genome-Cell Interactions
Review, Development, Examples
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CREB & Memory Silva et al, 1998, Annual Reviews Neuroscience.
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
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Review: Combinatorial Logic
Ref: R. Thomas & R. D’Ari, "Biological Feedback" CRC Press, Boca Raton Fl. [1990], Ch. 2.
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Part II: Biological Feedback
Thomas, R & D’Ari, R, Ch’s 9,10
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+ + -
?++
“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.”
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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.)
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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
* “epi” means “associated with”
“…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.”
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Y
X
0.00
0.25
0.50
0.75
1.00
0.00 1.00 2.00
Hill Functions: F, F+, F-
4
4 4
4
4 4
All of these can be written with the non-dimensional variable, /
, , , = 1 = 1= 1
x
Kx
x
x
x
x
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One-element Negative loop:
( ) ( )dx
H x kF x k xdt
( ) Since the RHS
is positive, monotonically decreasing
function of x, it can intersect the line
of unit slope only once. There is just
one steady state.
kx F xk
Steady State :
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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.
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Basics
Lower-case refers to productsx=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:
X = z; Y = zu
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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?
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Time (continued)
Each element has an 'on' and an 'off' time tx, tx
-, not all the same. In general, each transition should 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).
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Naïve Logical Description
X = 1 iff y=0, Y = 1 iff x = 1. "State table":
x y
+
-X = y; Y = x
x y X Y
0 0 1 0
0 1 0 0
1 1 0 1
1 0 1 1
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.
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System evolution
x y X Y
0 0 1 0
0 1 0 0
1 1 0 1
1 0 1 1
yy_ t
00 10__
11_
00 10
01
__
_11_
x
y t
t
y_
01
_ x_
x_
t x
A stable state is defined as one in which xy and XY are equal
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Two-element positive loop
x y
-
-x y X Y
0 0 1 1
0 1
1 1 0
0 1
0
1 0 1 0
When two choices are present, the choice taken will be fixed by the time delays.
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'Input' variables (operational and genetic)
Operational: add a drug, change temperature.Genetic (mutation – in gene or operator)
For x xX = z X = g (z + o )one can write
(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.
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Example
X = y; Y = x
X = y + o
Y = x + T
01 01
1
00 01 11 10
00 11 11 11 11
01 11 11
11 10 01 00
10 11 11
normal
operator,
low T
1
1
10 0
xXY o ,T
xy
(already considered). Now, X mutated 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".
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Alternate representation
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"Practical" Steady States
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Graphical Representation
z
+
+AND
x
y
z
+
+
x
y
OR
z+x
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:
x y
-
-
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A two-loop (circuit) system:
A graph of interactions vs. a graph of state sequences:
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Graphs and Matrices
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Graphs and Matrices