More Efficient Strategies for Special Cases

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More Efficient Strategies for Special Cases

In this section we consider a case where the network consists of AND and/or OR nodes. In this case we assume that any AND(resp. OR) node c is inactive (resp. active) if at least one literal appearing in the boolean function assigned to c is forced to 0 ( resp. 1) by disruption or overexpression of the gene corresponding to the literal. The above assumption is biologically reasonable even the network contains inconsistent nodes.
$\begin{theorem} A gene regulatory network which consists of AND and/or OR nodes... ...in-degree D can be identified by $O(n^{D+1})$\space experiments. \end{theorem}$

$\begin{proof}% latex2html id marker 259 Here we only show strategy for a network... ...rty holds and $O(n^{3})$\space experiments are sufficient in total. \end{proof}$
Next, we consider an acyclic case for which we can obtain an optimal bound.
$\begin{definition}{A set of nodes $\{x_1,x_2,...,x_k\}$\space has $influence$\sp... ...ce activates $y$\space and $e_2$\space inactivates $y$ . } \end{definition}$

$\begin{definition}{A set of nodes $\{x_1,x_2,...,x_k\}$\space has $influence$\sp... ...,...,x_k\}$\space has influence on at least one of $y_i$ . } \end{definition}$

$\begin{definition}{A set of nodes $\{x_1,x_2,...,x_k\}$\space has {\em strong in... ...1$\space differs from $e_2$\space only on a single $x_i$ . } \end{definition}$
The above definitions are invalid if the network is unstable (i.e., has an inconsistent node) or has multiple stable states. But, in what follows, the network can not have inconsistent nodes except ones that are disrupted or overexpressed. Moreover, for stable networks, we make a biologically reasonable assumption that a set of nodes $\{x_1,x_2,...,x_k\}$ does not have influence on a node to which there is no direct path from x₁ or x₂ or ... x_k.
$\begin{theorem} An acyclic gene regulatory network of maximum in-degree D can be identified by $\Theta(n^{D})$ experiments. \end{theorem}$

$\begin{proof}% latex2html id marker 279 The lower bound directly follows from Pr... ...the network by $O(n^{2})$\space experiments with maximum cost 2. \end{proof}$
For cyclic networks with in-degree, 2 experiments of cost 2 do not suffice. It is possible to identify such network in some cases in O(n^D) experiments. The strategy is based on detection of strongly connected components.

Proposition Suppose each set U of nodes does not have influence on any node to which there is no path to U. Then strongly connected components can be identified by O(n^D) experiments of maximum cost D for a gene regulatory network of bounded in-degree D.

$\begin{proof}We prove the proposition only for $D=2$ , where those for the other... ...ted components of $G(E,V)$\space by computing those of $G'(E',V)$ . \end{proof}$

**Figure:** Example of construction E' from E (Prop. 14.9)
InfTo(A)={(B,C),(B,D),(B,E),(C,D),(C,E),(D,E),
InfTo(B)={(A,C),(A,D),(A,E),(C,D),(C,E),(D,E)},
InfTo(C)={},
InfTo(D)={(B,C),(B,A),(B,E),(C,A),(C,E),(A,E)},
InfTo(E)={(B,C),(B,D),(B,A),(C,D),(C,A),(D,A)}
$\scalebox{0.8}{\includegraphics {lec14_fig/lect14_4.ps}}$

Next: Heuristic Strategy for the Up: Identification of Gene Regulatory Previous: Bounded In-degree Case With

Peer Itsik
2001-03-04