Inferring the Topology of Gene Regulatory Networks: An Algebraic Approach to Reverse Engineering - Mathematical Concepts and Methods in Modern Biology - page 86

Example 3.6. Consider a 3-node system having the following states in Z 5 :

200
431
314
043

which we interpret as

s 1 = 2 0 → t 1 = 4 1 ,

, 0 , , 3 ,

s 2 = 4 1 → t 2 = 3 4 ,

, 3 , , 1 ,

s 3 = 3 4 → t 3 = 0 3 .

, 1 , , 4 ,

The corresponding ideals are
I ( s 1 ) = x 1 − 2 , x 2 , x 3 ,

I ( s 2 ) = x 1 − 4 , x 2 − 3 , x 3 − 1 ,

.
To compute the intersection I of the ideals, we can use the computer algebra system
Macaulay 2 [ 26 ] with the following code:

I ( s 3 ) = x 1 − 3 , x 2 − 1 , x 3 − 4

R=ZZ/5[x1,x2,x3]
I1=ideal(x1-2,x2,x3)
I2=ideal(x1-4,x2-3,x3-1)
I3=ideal(x1-3,x2-1,x3-4)
I=intersect{I1,I2,I3} .
Then I is computed as

2 x 3 + x 3 + − x 1 + 2 x 2 + x 3 + 2 , 2 x 2 x 3 − x 3 , − x 2 x 3 − x 2 + x 3 ,

x 2 + 2 x 2 − x 2 x 3 + 2 x 2 + 2 x 3 , 2 x 2 x 3 + x 2 , x 1 x 2 + x 1 x 3 + x 2 + x 3 ,

2 − 2 x 1 x 2 + 2 x 1 x 3 − x 1 − x 2 + x 3 + .

Note that negative coefficients can be written as positive numbers: − 1 ≡ 4mod5 ,

− 2 3 mod 5, etc.

To find the function f 1 (

≡

x 1 , x 2 , x 3 ) for node x 1 such that

f 1 2

0 =

, 0 , 4 ,

f 1 4 1 =

, 3 , 3 , (3.1)

f 1 3 4 = , 1 , 0 ,

we compute the r polynomials:
r 1 x 1 , x 3 = 2 4 3 x 1 − 4 2 3 3 x 1 − 3 = 3 x 1 + x 2 , − − 4 x 1 + 1 ,

r 2 x 1 , x 3 = 4 2 3 x 1 − 2 4 3 3 x 1 − 3 =

3 x 1 + x 2 , − − 3 ,

r 3 x 1 , x 3 = 3 2 3 x 1 − 2 3 4 3 x 1 − 4 =

4 x 1 + x 2 , − − x 1 + 2 .