Boolean Approaches to Genome-Cell Interactions

Review, Development, Examples

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

Review: Combinatorial Logic

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

Part II: Biological Feedback

Thomas, R & D’Ari, R, Ch’s 9,10

+ + -

?++

“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.”

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.)

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.”

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

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 :

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.

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

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?

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).

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.

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

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.

'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.

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".

Alternate representation

"Practical" Steady States

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

-

-

A two-loop (circuit) system:

A graph of interactions vs. a graph of state sequences:

Graphs and Matrices

Graphs and Matrices