Results
Difference-based deduction of initially deduced edges and the minimum equivalent gene networks
The DBRF-MEGN method consists of five processes, namely (1) difference-based deduction of initially deduced edges, (2) removal of non-essential edges from the initially deduced edges, (3) selection of the uncovered edges in main components from the non-essential edges, (4) separation of the uncovered edges in main components into independent groups, and (5) restoration of the minimum number of edges from each independent group
14
. First, we define a gene network modeled as an SDG:
Definition 1: A signed directed graph (SDG) is given by a tuple G = (V, E, f) with a set V of nodes (genes), a set E⊆V×V of directed edges, and an edge sign function f:E→{± 1}, which is an integral part of an SDG.
The first process of the DBRF-MEGN method is "difference-based deduction of initially deduced edges" (Figure 1b), which uses an assumption that is commonly made in genetics and cell biology
14
, i.e., there exists a positive (negative) regulation from gene A to gene B when the expression level of gene B in the deletion mutant of gene A is significantly lower (higher) than in the wild-type (Figure 1a). For each possible pair of genes in the profiles, the process determines whether positive (negative) regulations between those genes exist and deduces all edges consistent with both the assumption and the profiles by detecting the difference in expression levels between the wild type and deletion mutants; we call these edges initially deduced edges.
<p>Figure 1</p>An example of the deduction of MEGNs from the expression profiles of gene deletion mutants
An example of the deduction of MEGNs from the expression profiles of gene deletion mutants. (a) An assumption used in the DBRF-MEGN method. (b) Deduction of the initially deduced edges. The matrix represents a set of expression profiles and the schematic represents a set of initially deduced edges. In the matrix, A, B, ... represent expression levels of gene A, gene B, ..., and aΔ, bΔ, ... represent deletion mutants of gene A, B, ... The up (down) arrows indicate that the gene expression levels are higher (lower) in the deletion mutant than in the wild type. (c) Essential edges. Non-essential edges are gray-dotted. (d) Uncovered edges. Uncovered edges are gray-dotted and covered edges are black-dotted. (e) Exclusion of uncovered edges in peripheral components. ![]()
, ![]()
, ![]()
, and ![]()
are uncovered edges in peripheral components. The resulting four gray-dotted edges are uncovered edges in main components. (f) Independent groups of uncovered edges in main components. For each group, the minimum number of edges with which essential edges can explain all edges in the group are shown: (E, J) or (F, J) for G0, and (H, K) or (H, L) for G1. (g) Four MEGNs of the profiles. Combinations of the minimum numbers of edges of two independent groups (G0 and G1) produce all four MEGNs.
![]()
Definition 2: Let us assume the intervention experiments
have been performed for the gene set J, J ⊆
V. Let D = (d
jk
)
∈RJ×V
be a matrix such that
djk
represents the expression of gene k
after an intervention in gene j (relative to wild-type expression). From this, we deduce the graph initially deduced edges, Gide = (V,Eide,f). We assume a negative regulation of k by j if djk
> α for some suitably chosen constant α. Analogously, a positive regulation of k by j is postulated whenever djk
< β for some β (sensibly, we require β < 0 < α). Formally,
![]()
and f:Eide
→{± 1} is given by f((j,k)) = 1 if there is a positive regulation of k by j, and otherwise f((j,k)) = -1.
The thresholds α and β determine the significance of the difference in expression levels between the wild type and deletion mutants. These thresholds can be specified by various procedures such as by using fold-change or the statistical significance of the expression level
7
8
14
19
20
.
The DBRF-MEGN method deduces the most parsimonious SDGs consistent with the SDG that consists of the initially deduced edges. Before defining the most parsimonious SDGs, we need to introduce the function exp and the concept cover (Figure 2).
<p>Figure 2</p>Introduction to the function "exp" and the concept "cover"
Introduction to the function "exp" and the concept "cover". (a) Initially deduced edges (Eide). (b) A set of edges (Ep). (c) Four cases of exp(a, b, c) = 1. In each case, (a, c) is explained by (a, b) and (b, c). (d) The set of edges that are covered by edges in Ep (![]()
). (A, C) and (C, D) explain (A, D); the (A, D) and (B, A) explain (B, D). Thus, (A, D) and (B, D) are covered by edges in Ep. (D, E) is not covered by edges in Ep because (D, E) cannot be explained by edges in Ep. (e) Edges that are covered by edges in Ep (black) and those that are not covered by edges in Ep (gray-dotted).
![]()
Definition 3: If, and only if, ∃ (i, j), (j, k), (i, k) | f(i, j) × f(j, k) = f(i, k), then exp(i, j, k) = 1; otherwise, exp(i, j, k) = 0.
Definition 4: Let Ep
⊆ Eide
be a set of edges. Define
![]()
and by induction
![]()
such that exp(j,i,k)=1}. Moreover, let
![]()
.
Remark: The family of edge sets on V is partially ordered by set inclusion. If E
1⊆E
2, note that by a trivial induction on r,
![]()
, and hence
![]()
. This means that the mapping
![]()
is monotonic. Let E ⊆ Eide
. By construction, an edge (j, k) from
![]()
is an element of
![]()
for suitable r,s ∈ N. This implies
![]()
. Thus
![]()
, and the mapping
![]()
is a so-called closure operation.
Lemma 1: If E
1⊆E
2,
![]()
.
Proof: The remark proves lemma 1.
Lemma 2: If
![]()
, then
![]()
.
Proof:
![]()
by monotonicity and closure of the mapping .cov.
Lemma 3: If
![]()
and
![]()
, then
![]()
.
Proof: By
![]()
by monotonicity and closure of the mapping.cov.
Now, we define the most parsimonious SDGs consistent with the expression profiles of gene deletion mutants. A most parsimonious SDG consists of the minimum number of edges that "cover" all initially deduced edges. By this definition, an edge can be redundant only when it is "explained" by two other initially deduced edges. Importantly, an edge is not redundant when it is "explained" by only three or more initially deduced edges (Figure 3a). We call the most parsimonious SDGs minimum equivalent gene networks (MEGNs).
<p>Figure 3</p>Difference between MEGN and MEG
Difference between MEGN and MEG. Deduction of the MEGN (a) and the MEG (b) from the same graph is shown. The MEGN includes the edge from A to D because no two edges explain the edge. In contrast, the MEG does not include the edge from A to D because A can reach D without using the edge from A to D (A→B→C→D). The MEGN consists only of the essential edges.
![]()
Definition 5:
![]()
(where
![]()
is the restriction of f to E
0) is a most parsimonious SDG, named a MEGN, of G = (V,Eide,f) if and only if it satisfies the following conditions: (1) E
0⊆Eide
, (2)
![]()
, (3) ∀ Ep
⊆ Eide
such that
![]()
,
![]()
. Since we keep G = (V,Eide,f) fixed for the rest of the paper, we often call G
0 simply a MEGN, without explicit reference to G.
Removal of non-essential edges from the initially deduced edges
The second process of the DBRF-MEGN method removes all non-essential edges from the initially deduced edges. The process removes all edges that are explained by two other initially deduced edges (Figure 1c). The resulting edges are called essential edges and the removed edges are called non-essential edges.
Definition 6: If there exist (i, j), (j, k), (i, k) ∈ Eide
such that exp(i, j, k) = 1, then (i, k) is called a non-essential edge. Let Enes
be the set of non-essential edges. The set Ees
of essential edges is the complement of Enes
in Eide, Ees
= Eide
\Enes
.
Essential edges and non-essential edges have the following properties.
Lemma 4: If Ep
⊆ Eide
and
![]()
, then E
p
⊇ E
es
.
Proof: Assume that there exists (i, j) ∈Ees
such that (i, j)
![]()
and (i, j) ∉ Ep
. Because (i, j)
![]()
and (i, j) ∉ Ep
, there exist (i, k), (k, j)
![]()
such that exp(i, k, j) = 1. This contradicts our assumption (i, j) ∉ Ees
.
Lemma 5: If
![]()
is a MEGN, Ees
⊆ E
0.
Proof:
![]()
, hence E0
⊇ Ees
by lemma 4.
When the essential edges cover all initially deduced edges, the SDG consisting of the essential edges is the only MEGN consistent with the profiles.
Theorem 1: If
![]()
, then
![]()
is the unique MEGN of G = (V, Eide, f).
Proof: By hypothesis, conditions (1) Ees
⊆ Eide
, and (2)
![]()
, of a MEGN are met. It remains to show the uniqueness and minimality of Ees
. (3) Let
![]()
be an arbitrary MEGN. Then by lemma 5, Ees
⊆ E
0, and by minimality of E
0, it follows that Ees
= E
0. The theorem is proved.
Selection of the uncovered edges in main components from the non-essential edges
The essential edges sometime fail to cover all initially deduced edges because some edges in the initially deduced edges represent direct gene regulations even when they are explained by two other edges (Figure 1d). In this case, the method restores the minimum number of non-essential edges so that the resulting edges (essential edges and the restored non-essential edges) cover all initially deduced edges. The SDG, consisting of essential edges and of the restored non-essential edges, is a MEGN. Before selecting the sets of non-essential edges to be restored, the method distinguishes non-essential edges that have a chance to be included in the MEGNs from those that do not in order to reduce the number of non-essential edges to be considered for the restoration and thus to reduce the computational cost to find non-essential edges to be restored. This third process of the DBRF-MEGN method consists of two sub-processes, namely (a) selection of uncovered edges and (b) selection of uncovered edges in main components. The resulting non-essential edges are called uncovered edges in main components, and from these edges the later processes of the DBRF-MEGN method select edges that are included in the MEGNs.
a) Selection of uncovered edges
The first sub-process distinguishes the non-essential edges that are covered by the essential edges from those that are not (Figure 1d). Those edges are called covered edges and uncovered edges, respectively.
Definition 7: Let Ecv
= (Ees
)cov\ Ees
be the set of covered edges. Let Eucv
= Eide
\( Ees
∪ Ecv
) be the set of uncovered edges. The set of initially deduced edges is thereby partitioned into three disjoint edge sets: Eide
= Ees
∪ Ecv
∪ Eucv
.
Here, we prove that the MEGNs do not include covered edges.
Lemma 6: If
![]()
is a MEGN, then Ees
⊆ E0
⊆ Ees
∪ Eucv
.
Proof: First, Ees
⊆ E0
by lemma 5. By definition 7, Ees
⊆ E0\Ecv
, hence
![]()
by monotonicity of.cov. It follows that
![]()
by lemma 2. By minimality of E
0, E0
= E0\ Ecv
, which is equivalent to E0
∩ Ecv
= Φ. By definition 7, this implies E0
⊆ Ees
∪ Eucv
, completing the proof.
b) Selection of uncovered edges in main components
The second sub-process distinguishes uncovered edges that have a chance to be included in the MEGNs from those that do not (Figure 1e; Figure 4). Those edges are called uncovered edges in main components and uncovered edges in peripheral components. The uncovered edges in peripheral components are defined as follows:
<p>Figure 4</p>Example of uncovered edges in peripheral and main components
Example of uncovered edges in peripheral and main components. (a) Initially deduced edges (Eide). (b) Essential edges (Ees). Non-essential edges are dotted. (c) Uncovered edges (Eucv). Because all non-essential edges cannot be covered by essential edges, the non-essential edges are called uncovered edges (Eucv). (d) Uncovered edges in peripheral components (gray-dotted). (B, D) and (B, E) are uncovered edges in peripheral components because (B, E) does not explain any other edges in Eucv with an edge in Eide, and (B, D) does not explain any other edges in Eucv except (B, E) with an edge in Eide. (A, C) and (B, C) are uncovered edges in main components. If edges in a MEGN cover (A, C) and (B, C), the edges also cover (B, D) and (B, E). (B, D) and (B, E) cover no edges except themselves. Thus, (B, D) and (B, E) are not included in the MEGNs.
![]()
Definition 8: Define
![]()
be the set of uncovered edges (i,j) ∈ Eucv
which cannot be used to directly explain another uncovered edge in Eucv
with the other edges (k,i) ∈ Eide
or (j,k) ∈ Eide
.
Lemma 7:
![]()
.
Proof: By definition 8, the edges in
![]()
cannot explain another uncovered edges in Eucv
. Therefore, the edges in
![]()
can be explained by the edges in
![]()
. The lemma is proved.
Definition 9: Following the definition 8, define
![]()
which cannot be used to directly explain another uncovered edge in
![]()
with the other edges (k,i) ∈ Eide
or (j,k) ∈ Eide
}. Let
![]()
be the set of uncovered edges in peripheral components. Let
![]()
be the set of uncovered edges in main components. The set of initially deduced edges is thereby partitioned into four disjoint edge sets:
![]()
.
In the following, we prove that the MEGNs do not include uncovered edges in peripheral components. First, we prove that uncovered edges in peripheral components have the following properties.
Lemma 8:
![]()
.
Proof: We prove lemma 8 by mathematical induction. (1) By lemma 7, lemma 8 is true when r = 0. By definitions 8 and 9,
![]()
, hence
![]()
. By lemmas 2 and 7,
![]()
. Thus, lemma 8 is true when r = 1. (2) Assume that lemma 8 is true when r = m. This means that we assume that
![]()
(2a). By definition 9,
![]()
(2b). Because
![]()
and (2b),
![]()
(2c). Because (2a), (2c) and lemma 3,
![]()
(2d). Because (2b) and (2d),
![]()
. Thus, lemma 8 is true when r = m +1, if it is true when r = m. By (1) and (2), lemma 8 is true.
Lemma 9:
![]()
.
Proof: By lemma 8,
![]()
. Because
![]()
, lemma 9 is true.
Now we prove that the MEGNs do not include uncovered edges in peripheral components.
Lemma 10: If
![]()
is a MEGN,
![]()
.
Proof: Assume that there exists
![]()
. Because of lemma 5 and definition 7,
![]()
, hence
![]()
by lemma 6. By the assumption
![]()
and definition 8,
![]()
, hence
![]()
. By lemmas 2 and 9,
![]()
. This contradicts our assumption that
![]()
is a MEGN. Therefore,
![]()
. By definition 9 and lemma 6, this implies
![]()
, completing the proof.
Separation of the uncovered edges in main components into independent groups and restoration of the minimum number of edges from each independent group
The fourth process of the DBRF-MEGN method separates uncovered edges in main components into "independent groups" so that edges to be restored can be deduced independently for each group (Figure 1f; Figure 5). For each group, the fifth process of the DBRF-MEGN method deduces the minimum number of edges with which essential edges can cover all edges in the group. All sets of such edges are deduced for each group. The essential edges and any possible combination of these sets from each group generate a MEGN of the profiles (Figure 1g).
<p>Figure 5</p>An example of independent groups
An example of independent groups. (a) Initially deduced edges (Eide). (b) Essential edges (Ees). Uncovered edges in main components (![]()
) are dotted. (c) Independent groups of uncovered edges in main components. Uncovered edges in main components are separated into two independent groups G0 (![]()
) and G1 (![]()
). Edges in one group do not explain those in other group. ![]()
consists of ![]()
and ![]()
. ![]()
consists of ![]()
, ![]()
, ![]()
, ![]()
, and ![]()
.
![]()
The independent groups are generated so that the edges in one group do not cover those in other groups.
Definition 10: Define
![]()
be a set of an edge
![]()
, and by induction
![]()
such that exp(i, j, k) = 1 or
![]()
such that exp(k, i, j) = 1 or
![]()
such that exp(i, k, j) = 1 or
![]()
such that exp(i, k, j) = 1}. Let
![]()
be the set of edges in an independent group. Let
![]()
, where
![]()
is a set of an edge
![]()
and by induction
![]()
such that exp(i, j, k) = 1 or
![]()
such that exp(k, i, j) = 1 or
![]()
such that exp(i, k, j) = 1 or
![]()
such that exp(i, k, j) = 1}. Then,
![]()
.
The essential edges and a combination of sets of the minimum number of edges for each independent group generate a MEGN of the profiles.
Definition 11: Let
![]()
be the set of edges in ith independent group that satisfies (1)
![]()
, (2)
![]()
, and (3)
![]()
such that
![]()
,
![]()
.
We prove that the essential edges and a combination of sets of the minimum number of edges for each independent group generate a MEGN of the profiles as follows:
Lemma 11: If there exist (i, j) ∈
![]()
, (i, k), (k, j) ∈ Eide
such that exp(i, k, j) = 1, then {(i, k), (k, j)} ∩ Eucv
⊆
![]()
.
Proof: By definition 10, lemma 11 is true.
Lemma 12:
![]()
.
Proof: By definitions 7 and 11,
![]()
. Because
![]()
,
![]()
. By lemmas 2 and 8,
![]()
. Therefore,
![]()
.
Theorem 2:
![]()
is a MEGN.
Proof: (1)
![]()
by the condition of theorem 2.(2) By lemma 12,
![]()
. (3) By lemmas 4, 11 and definition 11, ∀ Ep
⊆ Eide
such that
![]()
,
![]()
. Because
![]()
and lemma 2, ∀ Ep
⊆ Eide
such that
![]()
,
![]()
. The theorem is proved.
Remark: When there exist more than one solution of the minimum number of edges for independent groups, the SDGs each of which consists of the essential edges and a possible combination of sets of the solutions for each independent group are MEGNs because these SDGs must satisfy the conditions in definition 5.
Algorithms of the DBRF-MEGN method
We are concerned with algorithms that are computationally efficient for deducing MEGNs from expression profiles of single-gene deletion mutants. We list these in a form easily translatable into a computer program.
(A1) Algorithm for deducing initially deduced edges
double
d[n][n]: gene expression profiles
int
t[n][n]
void
dbrf()
int
i, j;
for
i = 1 to
n
do
for
j = 1 to
n
do
if
d[i][j] < β & i ≠ j
then
t[i][j]: = +1;
else if
d[i][j] > α & i ≠ j
then
t[i][j]: = -1;
else
t[i][j]: = 0;
The matrix d[n][n] represents the gene expression profiles. Each entry d[i][j] represents the log-ratio of the expression of gene j in gene i deletion mutants to that in the wild-type. The non-zero entries of the resulting matrix t[n][n] represent the initially deduced edges. If an entry t[i][j] is +1 or-1, it represents a positive or negative edge from gene i to gene j, respectively. The number of complete iterations is bounded by n2
.
(A2) Algorithm for distinguishing the essential edges from the non-essential edges
int
t[n][n]: initially deduced edges
void
ess_noness()
int
i, j, k;
for
j = 1 to
n
do
for
i = 1 to
n
do
if
t[i][j] ≠ 0 then
for
k = 1 to
n
do
if
t[j][k] ≠ 0 & t[i][k] ≠ 0 & t[i][k] = t[i][j]× t[j][k] then
check(t[i][k]);
The checked entries of the matrix t[n][n] represent non-essential edges. The unchecked non-zero entries of the resulted matrix t[n][n] represent essential edges. We created this algorithm by modifying Warshall's algorithm
21
. The number of complete iterations is bounded by n3
.
(A3.1) Algorithm for distinguishing uncovered edges from covered edges
int
t[n][n]: initially deduced edges
int
e[n][n]: essential edges
void
covered_edge()
int
i, j, k;
bool
finished;
finished : = false;
while
finished = false
do
finished : = true;
for
i = 1 to
n
do
for
j = 1 to
n
do
if
e[i][j] ≠ 0 then
for
k = 1 to
n
do
if
e[j][k] ≠ 0 & t[i][k] ≠ 0 & t[i][k] = e[i][j] × e[j][k] then
e[i][k]: = t[i][k];
check(e[i][k]);
finished : = false;
The checked entries of the matrix e[n][n] represent covered edges. The non-zero entries of the matrix t[n][n] that differ from the non-zero entries of the resulted matrix e[n][n] represent uncovered edges. This algorithm iterates over the while loop to find edges in Enes
that can be covered by the essential edges. Thus, the number of iterations is bounded by
![]()
.
(A3.2) Algorithm for finding uncovered edges in peripheral components
int
t[n][n]: initially deduced edges
int
u[n][n]: uncovered edges
void
peripheral_uncovered()
int
i, j, k;
bool
flag, sflag;
flag : = false;
while
flag = false
do
flag : = true;
for
j = 1 to
n
do
for
i = 1 to
n
do
if
u[i][j] ≠ 0 then
sflag : = false;
for
k = 1 to
n
do
if
t[j][k] ≠ 0 & u[i][k] ≠ 0 & t[i][k] = t[i][j] × t[j][k] then
sflag : = true;
if
t[k][i] ≠ 0 & u[k][j] ≠ 0 & t[k][j] = t[k][i] × t[i][j] then
sflag : = true;
if
sflag = false
then
check(u[i][j]);
flag : = false;
rm_checked_edge(); // set all checked entries to 0
The entries of the resulted matrix u[n][n] that have been changed from +1 or -1 to 0 represent uncovered edges in peripheral components. The non-zero entries of the resulted matrix u[n][n] represent uncovered edges in main components. This algorithm iterates over the while loop to find edges in Eucv
that are to be included in
![]()
. Thus, the number of complete iterations is bounded by
![]()
.
(A4.1) Algorithm for dividing uncovered edges in main components (
![]()
) into independent groups
int
t[n][n]: initially deduced edges
int
e[n][n]: uncovered edges in main components
ig
indgrp : independent group
list < edge > el : edge list
list < ig > igl : independent group list
void
independent_group()
int
i, j;
for
i = 1 to
n
do
for
j = 1 to
n
do
if
e[i][j] ≠0 then
el.clear();
el.append(eij
);
append_group(i, j);
indgrp.init();
indgrp.set_el(el); // store edge list el in indgrp
igl.append(indgrp); // indgrp : an independent group
void
append_group(int
i, int
j)
int
x;
for
x = 1 to
n
do
if
t[i][x] ≠0 & t[x][j] ≠0 then
if
t[i][j] = t[i][x] × t[x][j] then
if
e[i][x] ≠0 then
el.append(eix
);
e[i][x]: = 0;
append_group(i, x);
if
e[x][j] ≠0 then
el.append(exj
);
e[x][j]: = 0;
append_group(x, j);
if
t[x][i] ≠0 & t[x][j] ≠0 then
if
t[x][j] = t[x][i] × t[i][j] then
if
e[x][j] ≠0 then
el.append(exj
);
e[x][j]: = 0;
append_group(x, j);
if
t[j][x] ≠0 & t[i][x] ≠0 then
if
t[i][x] = t[i][j] × t[j][x] then
if
e[i][x] ≠0 then
el.append(eix
);
e[i][x]: = 0;
append_group(i, x);
The number of complete iterations of independent_group() is bounded by n
2. The number of complete iterations of append_group(int, int) is bounded
![]()
. Thus, the number of complete iterations is bounded by
![]()
.
(A4.2) Algorithm for finding all sets of minimum number of edges to be restored in each independent group
int
e[n][n]: essential edges and uncovered edges in peripheral components
ig
indgrp : independent group
list < ig > igl : independent group list
list < edge > el, tmp_el : edge list
list < edge list > combi_el : combination of edge list
void
find_min_ig()
int
i, num_edge;
for
i = 1 to igl.size() do
combi_el.clear(); el.clear();
indgrp ← igl.get_ig(i); // copy the ith independent group from igl
el ← indgrp.get_el();
for
num_edge = 1 to
el.size() do
add_edge(num_edge, 1);
if (combi_el.size() > 0) then
break;
set_min_combi_el(i, combi_el); // store combi_el in the ith independent group
void
add_edge(int
num_edge, int
start)
int
j;
if
start + num_edge - 1 > el.size() then
return;
for
j = start
to
el.size() do
set_edge(j); // set the entry of e[n][n] corresponding to the jth edge
// in el to +1 or -1 according to the sign of the edge
check(el.get_edge(j)); // check the jth edge in el
if
num_edge > 1 then
add_edge(num_edge - 1, j + 1);
else
if (confirm() = true) then
tmp_el.clear();
set_tmp_el(); // append all checked edges to tmp_el
combi_el.append(tmp_el); // tmp_el : a set of the minimum number of
// edges to be restored
reset_edge(j); // set the entry of e[n][n] corresponding to the jth edge
// in el to 0
uncheck(el.get_edge(j)); // uncheck the jth edge in el
bool
confirm(): when resulting edges e[n][n] can covered all edges in the group, return true.
The number of complete iterations is bounded by
![]()
, where G is the number of independent groups, Rj
is the number of edges in the jth independent group, nj
is the number of genes in the jth independent group, and mj
is the number of edges to be restored in the jth independent group.
(A5) Algorithm for deducing all MEGNs by making all possible combinations of sets of the minimum number of edges for each independent group
int
e[n][n]: essential edges
int
megn : the number of MEGNs
list < ig > igl : independent group list
list < edge list > tmp_combi_el : combination of edge list
list < edge > tmp_el : edge list
void
megn()
int
i;
i : = 1; megn : = 0;
sub_megn(i);
if
megn = 0 then
e[n][n]: MEGN // e[n][n] represents the MEGN when
![]()
void
sub_megn(int
i)
int
x, y, count;
if
i > igl.size() then
return;
tmp_combi_el ← get_min_combi_el(i); // copy combi_el of the ith independent
// group
for
y = 1 to
tmp_combi_el.size() do
tmp_el ← tmp_combi_el.get_el(y); // copy the yth edge list of tmp_combi_el
set_edges(tmp_el); // set the entries of e[n][n] corresponding to the edges in
// tmp_el to +1 or -1 according to the signs of the edges
if
i = igl.size() then
megn++;
e[n][n]: MEGN // e[n][n] represents a MEGN when
![]()
else
sub_megn(i + 1)
reset_edges(tmp_el); // set the entries of e[n][n] corresponding to the edges in
// tmp_el to 0
The number of complete iterations is bounded by
![]()
, where Sj
is the number of sets of minimum number of edges to be restored for the jth independent group.
Discussion
We have described in detail the algorithm of the DBRF-MEGN method and have proved that the algorithm provides all of the exact solutions of the most parsimonious gene networks consistent with expression profiles of gene deletion mutants. The resulting gene networks, called MEGNs, are the most parsimonious SDGs consistent with an SDG that consists of the initially deduced edges. In graph theory, many algorithms have been developed for deducing the most parsimonious unsigned directed graphs consistent with a given unsigned directed graph; these graphs are called minimum equivalent graphs (MEGs)
22
23
24
25
. MEGN is not just an "SDG version" of MEG, as is explained below. Although both MEGN and MEG are the most parsimonious graphs of a given graph, the parsimoniousness of the graph is defined differently between these graphs. MEGN consists of the minimum number of edges that cover all edges of a given graph (initially deduced edges), whereas MEG consists of the minimum number of edges that retain the reachability of a given graph
22
. MEGNs use the cover instead of the reachability because a MEGN is a prediction of a gene network consisting only of direct gene regulations
14
. When positive regulations from gene A to gene B, from gene B to gene C, from gene C to gene D, and from gene A to gene D are detected and regulation from gene A to gene C is not detected, the regulation from gene A to gene D is likely to be a direct regulation instead of an indirect regulation as a result of the other three regulations (Figure 3a). The use of cover makes MEGNs include edges representing such likely direct regulations (Figure 3a). In contrast, the MEGs, using reachability, do not include those edges (Figure 3b). Therefore, the DBRF-MEGN method, which deduces MEGNs, is fundamentally different from algorithms that deduce MEGs or algorithms for transitive reduction of SDG
16
17
18
.
The selection of uncovered edges in main components (the third process) and the generation of independent groups (the fourth process) make the DBRF-MEGN method applicable to large-scale gene expression profiles. Without these processes, the computational cost for finding all sets of non-essential edges to be included in the MEGNs is
![]()
where n is the number of genes and m is the number of non-essential edges to be included in a MEGN. This computation is impractical for large-scale gene expression profiles because
![]()
increases rapidly as
![]()
or m increase. The selection of uncovered edges in main components reduces the computational cost to
![]()
and the generation of independent groups further reduces it to
![]()
, where t is the number of independent groups, nj
is the number of genes in the jth independent group, and mj
is the number of edges in the jth independent group to be included in a MEGN.
![]()
and mj
are usually far smaller than
![]()
and m. Because of these reductions of the computational cost, the DBRF-MEGN method successfully deduced MEGNs from sets of large-scale gene expression profiles
14
[see Additional file 2, Table S1; Additional file 3]. Although there is no guarantee that the method will deduce MEGNs from any given expression profiles in an acceptable time, the method would most probably deduce MEGNs from most sets of expression profiles in an acceptable time.
<p>Additional file 2</p>
Supporting text for the applicability of the DBRF-MEGN method to the large-scale expression profiles and the sensitivity of the DBRF-MEGN method to the noise of the expression profiles.
Click here for file
<p>Additional file 3</p>
The MEGNs deduced from large-scale gene expression profiles.
Click here for file
Because MEGNs are deduced from initially deduced edges, the accuracy of MEGNs depends on that of initially deduced edges. The primary source for the inaccuracy in initially deduced edges is the noise of the expression profiles. Importantly, the number of false-positive edges in MEGN depends more on that of falsely-detected edges than that of falsely-missed edges in initially deduced edges; the number of false-negative edges in MEGN depends more on that of falsely-missed edges than that of falsely-detected edges in initially deduced edges [see Additional file 2, Table S2; Additional file 2, Figure S1]. These dependencies suggest the following guideline for the thresholds α and β (Definition 2): when the number of false-positive edges is more important than that of false-negative edges in MEGN, α (β) should be a little higher (lower) than the optimal value; in contrast, when the number of false-negative edges is more important than that of false-positive edges in MEGN, α (β) should be a little lower (higher) than the optimal value.
The DBRF-MEGN method is applicable not only to gene expression profiles of deletion mutants but also to those of gene overexpressions and conditional knock-downs/knock-outs
26
27
28
. We cannot obtain gene expression profiles of deletion mutants for essential genes. Thus, the method cannot deduce gene networks including essential genes when we use gene expression profiles of deletion mutants. A possible solution for this problem is to use the expression profiles of gene overexpressions or conditional knock-downs/knock-outs. Applications of the DBRF-MEGN method to those profiles will deduce gene regulations that cannot be deduced from gene expression profiles of gene deletion mutants.
A limitation of the DBRF-MEGN method is its inability to deduce (1) self-regulation of genes, and (2) combinatorial gene regulations such as regulation in which the expression of gene A is down-regulated only when both gene B and gene C are inactive. Self-regulation could be deduced by using chromatin immunoprecipitation
29
. Combinatorial gene regulations could be deduced by using the expression profiles of multiple gene deletion mutants
30
. Synthetic genetic arrays can systematically construct a collection of double-gene deletion mutants
31
. A combination of the DBRF-MEGN method and the above techniques would provide more accurate information about gene networks.
When the DBRF-MEGN method is applied to gene expression profiles measured by using DNA microarray, each of the deduced edges represents regulation of one gene's mRNA level by another gene's activity. Therefore, the deduced MEGNs do not include edges that represent post-transcriptional gene regulations although they play major roles in the cell. However, because the algorithm of the DBRF-MEGN method is based on logic that is most commonly used in genetics and cell biology to infer gene networks from small-scale experiments, we can predict post-transcriptional modulators of transcriptional activity from those MEGNs. We predicted total 72 transcriptional regulators and 232 post-transcriptional modulators of 18 transcriptional regulators from the MEGNs deduced from a set of gene expression profiles for 265 Saccharomyces cerevisiae genes
14
. The DBRF-MEGN method is applicable not only to gene expression profiles measured by using DNA microarray but also to those measured by using other technologies such as 2D-PAGE-MS
32
and protein chips
33
. MEGNs deduced from those non-DNA microarray expression profiles will include edges that represent post-transcriptional gene regulations in the cell.