Model description and definitions

We define the gene regulatory network as in Section 14.1.4. We further assume that it satisfies the following conditions:

1.

When the boolean function f_v assigned to v has k inputs, k input lines (directed edges) come from k distinct nodes u₁,...,u_k other then v.

2.

For each i = 1,...,k there exists an input $(a_1,..,a_k) \in \{0,1\}^{k}$ with $f_v( a_1,...,a_k) \neq f_v( a_1,..,\bar{a_i},...,a_k)$ where $\bar{a_i}$ is a complement bit of a_i.

3.

A node v with no inputs has a constant value (0 or 1).

  

Figure: Example of gene regulatory network with 16 genes ( $\oplus$ means "activation" and $\ominus$ means "deactivation" of the gene). Gene F is activated by gene A and is also inactivated by gene L ( $f_F(A,L) = l(A) \wedge \neg l(L)$). For gene D, it expresses if its all predecessors C,F,X1,X2 express (AND - node).

  

Figure 14.10: Gene expressions by disruption and overexpression from the gene regulatory network Fig. 14.9 (0 - the gene is not expressed , 1 - the gene is expressed).

For a gene v, a disruption of v forces v to be inactive and a overexpression of v forces v to be active. Let x₁,...,x_p,y₁,...,y_q be mutually distinct genes of G. An experiment with gene overexpressions x₁,...,x_p and gene disruptions y₁,...,y_q is denoted by $e = \langle x_1,...,x_p$ , $\neg y_1,...,\neg y_q \rangle$. The cost of e is defined by the number p+q. Three cases of gene expression conditions (normal, disruption of A, overexpression of gene B ) are presented in Fig. 14.10. Let us define the nodes with unique values given experiment e :

We now define different types of states of gene regulatory network G:

1.

A global state of G is a mapping $\psi : V \rightarrow\{0,1\}$. The global states of the genes need not be consistent with the gene regulation rules.

2.

The global state $\psi$ of G is stable under experiment $e = \langle x_1,...,x_p$ , $\neg y_1,...,\neg y_q \rangle$ if $\psi(x_i) = 1$ (i = 1,...,p) , $\psi(y_j)= 0$ (j = 1,...,q) and it is consistent with all gene regulation rules, i.e., for each node v with inputs u₁,...,u_k , $\psi(v) = f_v(\psi(u_1),...,\psi(u_k))$. Otherwise, it is called unstable.

3.

The global state $\psi$ of G is observed global state under experiment $e = \langle x_1,...,x_p$ , $\neg y_1,...,\neg y_q \rangle$ if it satisfies all gene regulation rules for invariant nodes.

4.

The observed global state $\psi$ of G is native global state when no perturbations are made $(e = \langle \rangle)$.

We shall now prove the upper and lower bounds for the number of experiments required for identifying a gene regulatory network with n genes, depending on the in-degree constraint and acyclicity. The Table 14.1 summarizes the results. Computationally, all algorithms for the results with polynomial experiments in Table 14.1 run in polynomial time.

 

Table 14.1: Bounds on number of experiments needed for reconstruction (n - number of genes, D - maximum in-degree)

	Constraints	Lower bounds	Upper bounds
	None	$\Omega(2^{n-1})$	O(2n-1)
	In-degree $\leq D$	$\Omega(n^D)$	O(n2D)
	In-degree $\leq D$ All genes are AND-nodes (OR-nodes)	$\Omega(n^D)$	O(nD+1)
	In-degree $\leq D$ Acyclic	$\Omega(n^D)$	O(nD)
	In-degree $\leq 2$ All genes are AND-nodes (OR-nodes). No inactivation edges.	$\Omega(n^2)$	O(n2)