Forman-Ricci Curvature for Hypergraphs

Abstract

In contrast to graph-based models for complex networks, hypergraphs are more general structures going beyond binary relations of graphs. For graphs, statistics gauging different aspects of their structures have been devised and there is undergoing research for devising them for hypergraphs. Forman-Ricci curvature is a statistics for graphs, which is based on Riemannian geometry, and that stresses the relational character of vertices in a network through the analysis of edges rather than vertices. In spite of the different applications of this curvature, it has not yet been formulated for hypergraphs. Here we devise the Forman-Ricci curvature for directed and undirected hypergraphs, where the curvature for graphs is a particular case. We report its upper and lower bounds and the respective bounds for the graph case. The curvature quantifies the trade-off between hyperedge(arc) size and the degree of participation of hyperedge(arc) vertices in other hyperedges(arcs). We calculated the curvature for two large networks: Wikipedia vote network and Escherichia coli metabolic network. In the first case the curvature is ruled by hyperedge size, while in the second by hyperedge degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with the participation of experienced users. The curvature values of the metabolic network allowed detecting redundant and bottle neck reactions. It is found that ADP phosphorilation is the metabolic bottle neck reaction but that the reverse reaction is not that central for the metabolism.

Full text

Forman-Ricci Curvature for Hypergraphs
Wilmer Leal,1, 2, ∗Guillermo Restrepo,2, 3 Peter F. Stadler,1, 2, 3, 4, 5, 6 and J¨urgen Jost2, 6
1Bioinformatics Group, Department of Computer Science,
Universit¨at Leipzig, H¨artelstraße 16-18, 04107 Leipzig, Germany
2Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
3Interdisciplinary Center for Bioinformatics, Universit¨at Leipzig, H¨artelstraße 16-18, 04107 Leipzig, Germany
4Institute for Theoretical Chemistry, University of Vienna, W¨ahringerstraße 17, 1090 Vienna, Austria
5Facultad de Ciencias, Universidad Nacional de Colombia, KR 30-45 3, 111321, Bogot´a, Colombia
6The Santa Fe Institute, 1399 Hyde Park Rd., 87501, Santa Fe, New Mexico, USA
(Dated: November 20, 2018)
In contrast to graph-based models for complex networks, hypergraphs are more general structures
going beyond binary relations of graphs. For graphs, statistics gauging different aspects of their
structures have been devised and there is undergoing research for devising them for hypergraphs.
Forman-Ricci curvature is a statistics for graphs, which is based on Riemannian geometry, and that
stresses the relational character of vertices in a network through the analysis of edges rather than
vertices. In spite of the different applications of this curvature, it has not yet been formulated for
hypergraphs. Here we devise the Forman-Ricci curvature for directed and undirected hypergraphs,
where the curvature for graphs is a particular case. We report its upper and lower bounds and the
respective bounds for the graph case. The curvature quantifies the trade-off between hyperedge(arc)
size and the degree of participation of hyperedge(arc) vertices in other hyperedges(arcs). We calcu-
lated the curvature for two large networks: Wikipedia vote network and Escherichia coli metabolic
network. In the first case the curvature is ruled by hyperedge size, while in the second by hyperedge
degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with
the participation of experienced users. The curvature values of the metabolic network allowed de-
tecting redundant and bottle neck reactions. It is found that ADP phosphorilation is the metabolic
bottle neck reaction but that the reverse reaction is not that central for the metabolism.
I. INTRODUCTION
Hypergraphs are used to model systems whose objects
have not only binary relationships; instead, interactions
simultaneously involve multiple members [1, 2]. Exam-
ples of these systems are found in physics, biology, chem-
istry, computer science, combinatorial optimization, sci-
entometrics and several other fields [1, 3–8]. Hypergraphs
reduce to (ordinary) graphs when all relationships (hy-
peredges) are binary. Graphs have been widely used as
a mathematical model for different systems and their
mathematical properties have been extensively studied,
which include devising statistics gauging aspects of their
structures, such as vertex degree and its distributions,
clustering coefficients, betweenness centrality and more
recently Forman-Ricci curvature.
As hypergraphs are a generalization of graphs, sev-
eral of the graph statistics have been extended to hy-
pergraphs, e.g. vertex and hyperedge degrees, clustering
coefficients [3, 9] and spectral properties [10]. Most of
the commonly used quantities focus on vertices. As the
crucial structure of a graph is, however, given by the
set of its edges rather than by its vertices, we should
systematically define and evaluate quantities assigned to
the edges rather than to the vertices. In this paper we
develop the Forman-Ricci curvature for hypergraphs (di-
rected and undirected) and calculate it for networks of
∗wilmer@bioinf.uni-leipzig.de
different sizes and research fields.
II. FORMAN-RICCI CURVATURE OF
EDGES/ARCS IN GRAPHS
Recently various notions of “curvature” have been pro-
posed for graphs and other, more general, discrete struc-
tures and applied to detect various local or global prop-
erties of such structures [2, 11–19]. The name of “cur-
vature” may seem somewhat strange in this context. In
differential, and more abstractly, in Riemannian geome-
try, curvature has been found to encode and express local
and global features of smooth manifolds equipped with
metric tensors [20]. Those features themselves usually do
not depend on an underlying smooth structure, and this
has lead to abstract theories of generalized curvatures on
metric spaces. On graphs, these generalized curvatures
are particularly easy to define and to evaluate. They can
also shed considerable light on other quantities that have
been introduced in network analysis without such a clear
conceptual background as those curvatures. The simplest
among these generalized curvatures is the Ricci curvature
introduced by Forman for simplicial complexes [21]. As
graphs are one-dimensional simplicial complexes, we can
readily evaluate this curvature. As explained in detail in
Section II A, for an edge e={i, j}with vertices i, j with
degrees diand dj(the degree of a vertex is the number
of its neighbors, that is, of those other vertices that are
directly connected to it by an edge), the Forman-Ricci
curvature is simply 4−di−dj. The number 4 serves the
arXiv:1811.07825v1 [cs.DM] 19 Nov 2018

2
purpose of normalization, to make the curvature of cycle
graphs vanish. The minus signs are also conventional, to
align this curvature with the Ricci curvature of Rieman-
nian geometry. Thus, edges connecting vertices of large
degree have very negative curvature values, and the first
step in the analysis of an empirical network might con-
sist in identifying the most negatively curved edges as
the most important ones for the cohesion of the network
or for the canalization and distribution of information or
activity in the network.
Since the definition of the Forman-Ricci curvature of
an edge in an undirected graph is so clear and sim-
ple, it can be readily generalized to, for instance, di-
rected or weighted graphs, and also to structures in which
more than two elements are related. Forman himself
had introduced this curvature notion already for possibly
weighted, simplicial complexes [21]. A simplicial com-
plex is characterized by the requirement that whenever a
collection of kelements stands in relation, then this also
holds for any subcollection. This leads to mathematically
very nice properties, and simplicial complexes are basic
structures in algebraic topology, but for the modelling
and analysis of empirical data sets, we may want to re-
lax or perhaps even completely abandon that condition.
That leads us to hypergraphs, which are collections of
vertices (undirected hypergraphs) or collections endowed
with direction (directed hypergraphs). Examples of the
former are elections, where a subset of voters is an elec-
tion an the collection of elections constitutes the hyper-
graph. Chemical reactions [3, 22, 23] and particle scatter-
ings are instances of directed hypergraphs, where some
starting materials are transformed into some products.
For hypergraphs, in principle, various generalizations of
the Forman-Ricci graph curvature are possible. It is a
main contribution of this paper to identify that notion
of Forman-Ricci curvature for (un)directed hypergraphs
that is best adapted to their structure and to investigate
its properties. We also apply this to concrete empirical
hypernetworks, a social and a metabolic one.
In this section we briefly summarize the results of the
Forman-Ricci curvature for graphs and then generalize
the curvature for hypergraphs.
A. Undirected graphs
Let G= (V, E ) be a (multi)graph with vertex set V
and multiset of edges E. The Forman-Ricci curvature of
an edge e={i, j} ∈ E, as introduced in [11], is given by:
F(e) = we wi
we
+wj
we−X
el∼i
wi
√wewel−X
el∼j
wj
√wewel!(1)
where wedenotes the weight of the edge e,wiand wj
are the weights of vertices iand j, respectively. The
sums over el∼krun over all edges elincident on the
vertex kexcluding e. The curvature for the unweighted
multigraph, with vertex and edge weights set to 1, is
given by [2]
F(e)=4−di−dj(2)
where dkis the vertex degree of k. Defining D=Pk∈edk
we have
F(e) = 4 −D(3)
As a multigraph may have repeated edges, whose num-
ber is independent of the number of vertices, the bounds
for F(e) shall be expressed as a function of the known
number of edges, namely, |E|. Therefore, 2(2 − |E|)≤
F(e)≤2. The lower bound is attained when dk=|E|
for every k∈e, therefore D= 2|E|(Figure 1a). In turn
F(e) = 2, for an isolated edge e(Figure 1c). In contrast
to the multigraph case, for simple unweighted graphs, the
lower bound can be expressed as a function of the num-
ber of vertices: 2(3 − |V|)≤F(e), which is obtained for
dk=|V|−1 for every k∈e, i.e., D= 2(|V| −1) (Figure
1b). As for multigraphs, F(e) reaches its maximum value
(F(e) = 2) for an isolated edge (Figure 1c).
As seen in Figure 1, Forman-Ricci curvature quantifies
the degree of spread of the vertices in e, from maximum
spread (corresponding to min F(e)) to minimum spread
(attained when max F(e)).
min F max F
|E|=9
D=10
F=-6
|E|=1
D=2
F=2
a b c
|E|=5
D=10
F=-6
|E|=5
D=7
F=-3
|E|=4
D=5
F=-1
|E|=9
D=12
F=-8
|E|=3
D=5
F=-1
|E|=7
D=8
F=-4
|E|=2
D=3
F=1
FIG. 1. Forman-Ricci curvatures F(e) calculated for the red
edge eof the depicted undirected graphs.
B. Directed graphs
Here we are interested in an unweighted directed multi-
graph G= (V, E ), where e= (i, j)∈Eis an arc (directed
edge), and i, j ∈V. Equation 2 indicates that the curva-
ture of an edge depends on the degree of its vertices. As
in a simple directed graph the degree can be split into

3
in- and out-degree. The curvature of e= (i, j) is defined
in terms of in- and out-degrees as well [17]. There are
different possibilities for the realization of the curvature,
depending on the meaning one assigns to it. Here we
emphasize the directed spread or flow through e, i.e., fol-
lowing the direction of the arc. Therefore, we consider
the incoming arcs on i(in-degree of i, in(i)) and the out-
going arcs from j(out-degree of j, out(j)). When we
separate the curvature in (2) into the contribution 2 −di
of iand 2 −djof jand also note that the edge ecounts
for the degrees of iand j, but neither for the in-degree of
inor for the out-degree of j, then a curvature accounting
for the in-flow at i(F(→e)) and another for the out-flow
at j(F(e→)) is defined as
F(→e) = 1 −in(i)
F(e→) = 1 −out(j).(4)
Both are bounded below by 2 − |E|for in(i) = out(j) =
|E|−1, and bounded above by 1 when in(i) = out(j)=0
(Figure 2a). For the simple directed graph the lower
bound for both, in- and out-flow, is 2 − |V|, for in(i) =
out(j) = |V|−1 (Figure 2b). The upper bound is reached,
in both cases, when in(i) = out(j) = 0 (Figure 2c). The
curvature accounting for the flow through e= (i, j) is
then given by
F(→e→) = F(→e) + F(e→)
= 2 −in(i)−out(j)(5)
where 2(2 −|E|)≤F(→e→)≤2 for the multigraph case
and 2(2 − |V|)≤F(→e→)≤2 in the simple graph case.
Figure 2c shows the case where F(→e→) = 2. Some
further examples of calculations of curvatures F(→e→)
are shown in Figure 2.
If the flow-loss along eis to be considered, two addi-
tional curvatures are calculated that account for the flow
loss at i(F(←e)) and at j(F(e←)). Thus
F(←e)=1−out(i)
F(e←)=1−in(j)(6)
both bounded below by 1 − |E|, for out(i) = in(j) =
|E|, and bounded above by 0 for out(i) = in(j) = 1
(Figure 2d). For the simple directed graph we have 2 −
|V| ≤ F(←e)≤0 and 2 − |V| ≤ F(e←)≤0. Hence, the
curvature for the flow-loss along e= (i, j) is
F(←e←) = F(←e) + F(e←)
= 2 −out(i)−in(j)(7)
where 2(1 − |E|)≤F(←e←)≤0 (Figures 2a-e) holds in
the multigraph case and 2(2−|V|)≤F(→e→)≤0 in the
simple graph case. Some further examples are shown in
Figure 2.
A curvature accounting for the total flow over eis then
computed as
F(e) = F(→e→) + F(←e←) (8)
In the following section we extend the Forman-Ricci
curvature to hypergraphs.
FIG. 2. Forman-Ricci curvatures F(→e→), F(←e←), and
F(e) calculated for the red arc eof the depicted directed
graphs.
III. FORMAN-RICCI CURVATURE OF
HYPER(EDGES/ARCS) IN HYPERGRAPHS
Given a set of vertices V, a graph is a collection of
subsets (edges) of V, all of which comprise only two ele-
ments. If we call the cardinality of each subset its size,
then a graph is a collection of subsets of size two. In a
hypergraph, the size of the subsets is no longer restricted,
and subsets of any size are allowed.
A. Undirected hypergraphs
An undirected hypergraph H= (V, E) consists of a set
Vof vertices and a multiset Eof subsets of V, called
hyperedges, such that e⊆V, i.e. |e|≤|V|, for e∈E.
Some examples of hypergraphs are shown in Figure 3.
Separating the contributions of vertices iand jin
Equation 1, it can be rewritten as:
F(e) = we" wi
we−X
el∼i
wi
√wewel!+ wj
we−X
el∼j
wj
√wewel!#
(9)

4
furthermore,
F(e) = we"X
k∈e wk
we−X
el∼k
wk
√wewel!# (10)
Since Equation 10 no longer restricts eto size two, we
present it as the Forman-Ricci curvature of the hyper-
edge e. For the unweighted hypergraph, where all vertex
weights are equal to 1, this expression simplifies to
F(e) = X
k∈e 2−dk!= 2|e| − X
k∈e
dk= 2|e| − D(11)
which is bounded below by |e|(2−|E|) when dk=|E|for
every k∈e, and bounded above by 1 when D=|e|. In
other words, the minimum curvature occurs when every
vertex in ebelongs to each hyperedge (Figures 3a,b); the
maximum is attained for an isolated hyperedge (Figure
3c).
For the particular case of simple hypergraphs, we
therefore have the lower bound 2|e|(1 −2|V|−2) when
dk=|P(V\ {k})|for every k∈e, and the upper bound
|V|, when E={V}. Note that in hypergraphs |e|≤|V|,
therefore the minimum value |e|may reach 1, unlike
graphs. In such a case, 2(1 −2|V|−2)≤F(e)≤ |V|.
Some further examples of curvature for hypergraphs are
shown in Figure 3.
FIG. 3. Forman-Ricci curvatures F(e) calculated for the blue
hyperedge eof the depicted hypergraphs.
B. Directed hypergraphs
In a directed hypergraph, each hyperedge is composed
of two subsets of vertices: the tail and the head of the
hyperedge. Formally, we say that a directed hypergraph
His the couple (V, E ) with Va set of vertices and Ea
multiset of hyperarcs. A hyperarc is a pair e= (ei, ej),
where ei⊆Vand ej⊆Vare called its tail and its head,
respectively. Figure 4 depicts some examples of diercted
hypergraphs, where the sets eiand ejare highlighted.
Starting from the definitions of curvature for an arc in
the directed graph case (Equation 4), we introduce the
curvatures F(→e) and F(e→) for a hyperarc as
F(→e) = |ei| − X
i∈ei
in(i)
F(e→) = |ej| − X
j∈ej
out(j)(12)
with bounds |ei|(1 − |E|)≤F(→e)≤ |ei|and |ej|(1 −
|E|)≤F(e→)≤ |ej|. For the simple directed hyper-
graphs, we have |ei|(1 −2|V|−1)≤F(→e)≤ |ei|and
|ej|(1 −2|V|−1)≤F(e→)≤ |ej|. With F(→e) and F(e→)
at hand, we define the curvature for the flow through
e= (ei, ej) as:
F(→e→) = F(→e) + F(e→)
=|ei|+|ej| − X
i∈ei
in(i)−X
j∈ej
out(j)(13)
with bounds (1 −|E|)(|ei|+|ej|)≤F(→e→)≤ |ei|+|ej|
in the general case and (1−2|V|)(|ei|+|ej|)≤F(→e→)≤
|ei|+|ej|for the simple directed hypergraph (Figure 4).
Note that if |e|is allowed to have its minimum value of
1, then |ek|= 1 and 2(1 − |E|)≤F(→e→)≤2. Some
examples of curvature values for directed hypergraphs are
shown in Figure 4.
The respective flow-loss curvatures are:
F(←e) = |ei| − X
i∈ei
out(i)
F(e←) = |ej| − X
j∈ej
in(j)(14)
with bounds |ei|(1 − |E|)≤F(←e)≤0 and |ej|(1 −
|E|)≤F(e←)≤0 in the general case and |ei|(1 −2|V|)≤
F(←e)≤0 and |ej|(1 −2|V|)≤F(e←)≤0 for the simple
directed hypergraphs.
Equation 14 yields the flow-loss curvature
F(←e←) = F(←e) + F(e←)
=|ei|+|ej| − X
i∈ei
out(i)−X
j∈ej
in(j)(15)
with bounds (1 − |E|)(|ei|+|ej|)≤F(←e←)≤0, which
becomes (1−2|V|−1)(|ei|+|ej|)≤F(←e←)≤0 for simple
directed hypergraphs.

5
|E|=3
F(→e)=-2
F(e→)=-1
F(←e)=0 (max)
F(e←)=0 (max)
F(→e→)=-3
F(←e←)=0
F(e)=0
a
eiej
eiej
eiej
eiej
eiej
eiej
|E|=1
F(→e)=0 (min)
F(e→)=0 (min)
F(←e)=0 (min, max)
F(e←)=0 (min, max)
F(→e→)=0 (min)
F(←e←)=0 (min, max)
F(e)=0
b
|E|=1
F(→e)=2 (max)
F(e→)=1 (max)
F(←e)=0 (min, max)
F(e←)=0 (min, max)
F(→e→)=3 (max)
F(←e←)=0 (min)
F(e)=3
c
|E|=3
F(→e)=2 (max)
F(e→)=1 (max)
F(←e)=-1
F(e←)=-2 (min)
F(→e→)=3 (max)
F(←e←)=-3
F(e)=0
d
|E|=6
F(→e)=-2
F(e→)=-1
F(←e)=-1
F(e←)=-1
F(→e→)=-3
F(←e←)=-2
F(e)=-5
e
|E|=5
F(→e)=0
F(e→)=0
F(←e)=-1
F(e←)=0 (max)
F(→e→)=0
F(←e←)=-1
F(e)=-1
f
|E|=5
F(→e)=1
F(e→)=0
F(←e)=-2
F(e←)=0 (max)
F(→e→)=1
F(←e←)=-2
F(e)=-1
g
ei=ej
FIG. 4. Forman-Ricci curvatures F(→e→), F(←e←), and
F(e) calculated for the red hyperarc e, connecting vertices
in eiwith those in ej, of the depicted hypergraphs.
In the following section we calculate the Forman-Ricci
curvature for different cases that can be modelled as hy-
pergraphs. Several applications of the Forman-Ricci cur-
vature for the graph case are found in references [2, 11–
19].
APPLICATIONS TO EMPIRICAL NETWORKS
C. Wikipedia Voting Network
Wikipedia is an encyclopedia written by volunteers. A
small part of these users are administrators, who besides
being active, regular long-term Wikipedia contributors,
have gained the general trust of the community and have
taken on technical maintenance duties. A user becomes
an administrator when a request for adminship is issued
and the Wikipedia community via a public vote decides
who to promote to administrator. Users can either sub-
mit their own requests for adminship or may be nomi-
nated by other users. Using the January 3 2008 dump
of Wikipedia page edit history [24], Leskovec et al. [25]
extracted 2,794 elections (hyperedges in our setting) and
7,066 users (vertices) participating in the elections (ei-
ther casting a vote or being voted on). We calculated
the curvature for the resulting undirected hypergraph.
Figure 5 shows the distribution of hyperedge size and of
vertex degree. The data show that many of the elections
involve a single user, although elections with 2-20 users
are also common. There are few elections with more than
100 users, the largest one including 370 users (Figure 5a).
The participation in elections is heavy-tailed distributed
(Figure 5b), with most of the users participating in a
single election and very few taking part in about a thou-
sand elections. The curvature values are mostly negative
(Figure 5c), indicating (i) the absence of elections with
unexperienced users (max F(e)6=|e|), i.e., all elections
at least include a user that takes part in at least one
other election; and (ii) for most elections the number of
elections in which users take part is greater than their
number of voting users (D > |e|in Equation 3). The
minimum curvature value (-3,112) is far from the lower
bound (−19,728,272, calculated with |e|= 7,066). This
reflects the fact that most users are experts in limited
fields only.
To have some insight about the effects of hyperedge
sizes and number of incident hyperedges on curvature, we
analyzed their distributions over the span of curvature
values (Figures 5d-f). Figure 5d shows that the more
spread the election, i.e. involving users that vote in other
elections, the larger the number of users voting. Figures
5e and f show that, in average, elections overlap with a
low number of other elections (low number of incident
elections). Thus, the curvature values are mainly ruled
by hyperedge size rather than by incident hyperedges.
D. Metabolic Network of Escherichia coli
The metabolism of Escherichia coli is one of the most
studied and best characterized among bacteria. Here we
model the metabolism K-12 (iJR904 GSM/GPR) [26] of
this bacterium as a directed hypergraph whose vertices
are the metabolites (chemical species). Each chemical
reaction is represented as a hyperarc e, whose educts
(starting materials) correspond to eiand products to ej.
There are |V|= 625 metabolites and |E|= 1,176 reac-
tions accounting for 686 non-reversible and 245 reversible
ones. These latter reactions, denoted by ei↔ejhave
been included as “forward” (ei→ej) and “backward”
(ej→ei) reactions. All curvatures (Equations 12 to 15)
and related calculations are gathered in the Supplemen-
tary Material.
As expected for chemical reactions, typically there are
not more than three educts and three products (Figure
6a). The curvature values therefore vary little in response
to hyperarc size, but rather depend more on the degree
of vertices in eiand ej. Note that these degrees result,

6
a)
100101102
100
101
102
Hyperedge degree
Frequency
b)
100101102103
100
101
102
103
Vertex degree
Frequency
c)
-30000 -25000 -20000 -15000 -10000 -5000 0
100
101
102
Frequency
d)
-30000 -25000 -20000 -15000 -10000 -5000 0
0
0.2
0.4
0.6
0.8
1
Normalized hyperedge size
e)
-30000 -25000 -20000 -15000 -10000 -5000 0
0
0.2
0.4
0.6
0.8
1
Curvature
Median degree
f)
-30000 -25000 -20000 -15000 -10000 -5000 0
0
0.2
0.4
0.6
0.8
1
Curvature
Average degree
FIG. 5. Voting Wikipedia: Distribution of a) hyperedge size
(size of elections) and b) vertex degree (participation of users
in elections). c) Histogram of curvature values with bins of
10 units. Box-plots of d) normalized hyperedge sizes, e) me-
dian, and f) average hyperedge degrees corresponding to each
curvature bin of c).
respectively, from the summation over vertex degrees of
educts and of products (Equations 12 to 15). The dis-
tribution and educts and products degrees is shown in
Figure 6. The participation of educts and products in
reactions does not yield a smooth distribution, as indi-
cated by the gaps present in Figure 6b,c. The production
of educts (Figure 6b) shows a large group of reactions
whose educts are synthesized by less than 200 reactions
and another group where they are obtained by more than
450 reactions. Likewise, there are two groups of reactions
with different levels of use of their products (Figure 6c);
one group has reactions whose products are used in less
than 100 reactions and another with more than 300 re-
actions taking their products as starting materials.
The synthesis of products and the use of educts (Fig-
ures 6d and e), shows also a discontinuous participation
of substrates in reactions. There are two groups of re-
actions according to the number of reactions synthesiz-
ing their products: one with reactions whose products
are obtained by less than 200 reactions and another by
more than 450 reactions (Figure 6d). Likewise, there are
various groups of reactions according to the use of their
educts, from some which are seldom used to some others
with about 170, 230, and more than 330 uses (Figure
6e).
a)
0.5 1 1.522.533.544.5 5 5.5 6
1
2
3
4
5
6
7
8
7
63
103
6
9
34
169
492
1
1
66
95
98
1
4
4
7
11
1
1
3
|ei|
|ej|
b)
0 100 200 300 400 500 600 700 800
100
101
102
c)
0 100 200 300 400 500 600 700 800
100
101
102
d)
0 100 200 300 400 500 600 700 800
100
101
102e)
0 100 200 300 400 500 600 700 800
100
101
102
FIG. 6. Metabolic network: a) Scatter plot of sizes of educts
(|ei|) and products (|ej|), where circle radii correspond to
log f / log 100, being fthe frequency of appearance of the cou-
ple (|ei|,|ej|) in the reactions. Numbers inside circles corre-
spond to f. Distribution of b) Pi∈eiin(i), c) Pi∈eiout(i),
d) Pj∈ejin(j), and Pj∈ejout(j).
The extent to which the educts of the reaction eare
produced from other reactions is measured by F(→e).
The more reactions lead to the educts of e, the more
negative F(→e) becomes (Figure 7a). The theoreti-
cal bounds of F(→e), assuming max |ei|= 625 are
−734,375 ≤F(→e)≤625. However, more realistic
bounds are −7,050 ≤F(→e)≤6, which results from
taking the actual max |ei|= 6 (Figure 6a). We found
that min F(→e) = −735, which is attained by four re-
actions, with four educts (all substrate abbreviations are
included in the Appendix (Table I)):
adp+h+malcoa+pi→accoa+atp+hco3
adp+h+pi+25aics→asp L+atp+5aizc
adp+dtbt+h+pi→atp+co2+dann
atp+gar+h+pi→atp+gly+pram
These reactions are those whose educts are the most
synthesized of all the metabolic reactions of E. coli (63%
of the reactions produce their educts). In three of them
atp is synthesized from adp, which shows the well-known

7
central metabolic role of atp [27, 28].
max F(→e) = 1 corresponds to a single reac-
tion: cyan+tsul→h+so3+tcynt, where only one of
its two educts is a product of a single reaction:
atp+h2o+tsul→adp+h+pi+tsul.
Figure 7a shows that the most frequent curvature value
is 0 (for 73 reactions), i.e. 6% of the reactions have a
trade-off between the number of educts and the num-
ber of reactions producing them; most of the remaining
reactions have more ways to produce their educts than
the number of educts. It is also found that there are
almost no reactions with curvatures between -200 and
-450, indicating that educts of reactions are mainly ob-
tained either by less than 200 reactions (less than 17% of
the reactions) or by 450 to 600 reactions (38 to 51% of
the reactions). This is a consequence of the heavy-tailed
in-degree distribution of substrates [27].
Figure 7b shows the curvature values F(e→), which
quantify the extent to which products of reactions are
used in further reactions as educts. By taking max |ej|=
8 (Figure 6a) this curvature takes values −9,400 ≤
F(e→)≤8. The actual min F(e→) = −729, for
adp+h+pi→atp+h+h2o, i.e., this is the reaction whose
three products are most used in other reactions as start-
ing materials (used in 62% of the reactions). In contrast,
there are four reactions with max F(e→) = 1:
agpe EC +pg EC →apg EC+g3pe
agpc EC +pg EC →apg EC+g3pc
agpg EC +pg EC →apg EC+g3pg
udpgal →udpgalfur
Hence, for those three reactions with two products, these
substrates are only used in a further reaction as educts,
while udpgalfur is not further used, i.e. it is a metabolic
“dead-end” [26]. As most of the reactions (96%) have
negative values of F(e→), this indicates the efficient use
of reaction products [28], which can be divided into two
regimes. For about half of the reactions their products
are used in no more than 9% of the reactions and about
40% of the reactions have products that are used in more
than a quarter of the reactions. This is a consequence
of the heavy-tailed distribution, this time, of the out-
degrees of the substrates [27].
F(→e) showed that for most of the reactions their
educts are produced by other reactions and F(e→) that
the products are used in several other reactions. The
question that arises whether those popular educts are
connected through reactions with the popular products
is positively answered by F(→e→), which takes negative
values for most of the reactions. The min F(→e→) =
−1,463 corresponds to adp+h+pi→atp+h+h2o. Hence,
this is the reaction whose educts are most synthesized by
other reactions and whose products are the most used
as educts in other reactions. It is the bottleneck of the
E. coli metabolism. Other reactions of this sort, with
F(→e→)<−1,000 (Figure 7e), are:
adp+h+pi→atp+h+h2o
h+o2+q8h2→h+h2o+q8
h+o2+q8h2→h+h2o+q8
h+no3+q8h2→h+h2o+no2+q8
h+mql8+no3→h+h2o+mqn8+no2
Having analyzed the metabolism following the direc-
tion of educts to products in reactions, we now proceed
to study the curvature in the backward direction, which
quantifies to which extent a reaction is just one of the
many connecting popular educts with popular products.
We start by analyzing F(←e) that shows to which ex-
tent educts of a reaction participate in other reactions.
The theoretical bounds are −734,375 ≤F(←e)≤0 and
we found that F(←e) takes values in between -729 and
0; the minimum is attained by atp+h+h2o→adp+h+pi,
indicating that atp in an acidic aqueous medium is the
most often used starting material. max F(←e) occurs for
51 reactions, whose involved 56 educts are only used in
those 51 reactions, i.e. they are very specialized educts
for very particular metabolic reactions. The distribution
of F(←e) values shows that for half of the reactions, their
educts participate in less than 9% of the reactions, while
for the rest, their educts take part in more than 15% of
the reactions.
F(e←) shows to which extent products of a reaction
are synthesized by other reactions. The theoreti-
cal bounds are given by max |ej|= 8, leading to
−9,400 ≤F(e←)≤0. The actual values range from
-788 to 0. The minimum is reached by reaction:
dxyl5p+nad+phthr→co2+h+h2o+nadh+pdx5p+pi,
i.e. this set of products is the most synthesized by E. coli
metabolism, which is expected, for the likelihood of a
set of substances to be synthesized scales with the size of
the set. This reaction with six products is one of the few
where more than the frequent one to four products are
synthesized (Figure 6a). Moreover, among the products,
co2,h,h2o,nadh, and pi are often products of other
reactions.
max F(e←) = 0 is attained by 29 reactions, all of them
leading to a single product, except for three reactions,
each one with two products. Thus, those 32 products are
of little synthetic relevance for the metabolism. The dis-
tribution of curvature values shows that there are three
kinds of reactions whose products are synthesized by dif-
ferent number of reactions. For 60% of the reactions their
products are synthesized by less than 200 reactions (17%
of the reactions) and for the rest of the reactions by more
than 450 reactions (38% of the reactions).
Curvatures F(←e) and F(e←) showed that half of the
educts are often used and 40% of the products are of-
ten synthesized, which indicates that it is very likely to
find alternative ways to link educts with products of ex-
isting reactions, as found in [27–29]. A measure of this
degree of redundancy of a reaction or of its replaceability
is given by F(←e←), which indicates to which extent a
reaction connects popular educts with popular products.
The more negative the curvature, the more redundant or
likely replaceable the reaction is.
By analyzing F(←e←) distribution (Figure 7f) it is
seen an ample spectrum of curvatures, with almost no
gaps, indicating different degrees of redundancy for the

8
metabolic reactions. min F(←e←) = −1,463 corresponds
to the hydrolysis of ATP, i.e., atp+h+h2o→adp+h+pi,
indicating, e.g., that the dephosphorilation of atp to adp
can be achieved by many other reactions (12% of the re-
actions). max F(←e←) = 0 occurs for the following eight
reactions, which are unique as they are the only way to
connect their educts with their products:
mmcoa R→mmcoa S
5mdr1p→5mdru1p
gdpddman→gdpofuc
adphep D,D→adphep L,D
dhnpt→gcald+6hmhpt
glu1sa→5aop
prfp→prlp
pran→2cpr5p
a)
−800 −700 −600 −500 −400 −300 −200 −100 0
100
101
b)
−800 −700 −600 −500 −400 −300 −200 −100 0
100
101
c)
−800 −700 −600 −500 −400 −300 −200 −100 0
100
101
d)
−800 −700 −600 −500 −400 −300 −200 −100 0
100
101
e)
−1,600−1,400−1,200−1,000 −800 −600 −400 −200 0
100
101
f)
−1,600−1,400−1,200−1,000 −800 −600 −400 −200 0
100
101
1
FIG. 7. Metabolic network: Histograms of curvature values
for a) F(→e), b) F(e→), c) F(←e), d) F(e←), e) F(→e→),
and f) F(←e←) with bins of 500 units.
CONCLUSIONS AND OUTLOOK
The Forman-Ricci curvature emphasizes the impor-
tance of the relational character of (hyper)edges, thereby
providing a view of the network structure that comple-
ments traditional vertex-centered descriptors. It also em-
beds the characterization in a formal mathematical the-
ory, namely Riemannian geometry.
The results here reported include a brief review of
Forman-Ricci curvature for (un)directed graphs and gen-
eralize the curvature to both undirected and directed hy-
pergraphs. Graph curvatures used in previous studies
thus become particular cases of the curvature for hyper-
graphs [2, 11–19]. We determined the upper and lower
bounds for Forman-Ricci curvature for graphs and hy-
pergraphs, which so far had not been studied.
The curvatures here presented aim at quantifying the
trade-off between hyperedge(arc) size and the degree of
participation of vertices members of the hyperedge(arc)
in other hyperedges(arcs). For undirected hypergraphs,
the curvature takes negative values when the degree of
vertices of the hyperedge is more significant than the size
of the hyperedge. For directed hypergraphs we devised
four curvatures that gauge different aspects of hyperarcs.
F(e→) quantifies the trade-off between the size of the hy-
perarc tail and the input of its vertices, F(→e) do so for
the size of the hyperarc head and the output of its ver-
tices; while F(←e) and F(e←) consider the size of the tail
and the output of their vertices, and the size of the head
and the input of their vertices, respectively. These cur-
vatures are combined into F(→e→) and F(←e←), which
account for the flow through hyperarc eand for its re-
dundancy, respectively.
The Forman-Ricci curvature for hypergraphs intro-
duced here differs from the alternative construction pro-
posed in [30]. There, hyperedges are interpreted as sim-
plices. Here, we focus entirely on hyperedges, their
sizes, and the degrees of vertices, thereby avoiding the
re-interpretation of hyperedges as “higher-dimensional”
objects and implicitly introducing additional structures,
like boundaries of simplices, that are not part of the orig-
inal data. For the particular case of directed hypergraphs
we disentangled the curvature in the four aforementioned
informative measures that allow a detailed exploration
of the hypergraph structure. Moreover, we applied these
curvatures to the analysis of two large networks, one of
social and the other of chemical interactions.
The analysis of Wikipedia vote network exemplified
the Forman-Ricci curvature of undirected hypergraphs,
where elections constituted hyperedges and users/voters
vertices. We found that curvature is mostly ruled by
hyperedge size rather than by hyperedge degree. Like-
wise, the more users involved in elections, the more the
presence of experienced users. In a traditional graph set-
ting [24, 25], with users as vertices and votes as couple
of users, conclusions such as the previous one on elec-
tions cannot be drawn. This shows the richness of hy-
pergraphs and their curvatures, which for this particular
case allowed the definition of a hyperedge as an election.
Forman-Ricci curvature for directed hypergraphs was
computed over the metabolic network of E. coli, which
traditionally has been analyzed through a graph setting
[27–29] and which has shed light on the important role
of several substrates for the metabolic stability. In our
approach, rather than focusing on substrates, we did on
reactions, which were characterized as hyperarcs connect-
ing sets of educts with sets of products.

9
In contrast to the Wikipedia vote network, we found
that curvature values for the metabolic network were
ruled by the degree of hyperarcs, i.e. of in- and out-
degrees of tails and heads of hyperarcs. This is a chem-
ical consequence, for it is unlikely that several educts
collide simultaneously to give place to a product. In fact
reactions where more than five educts participate in a
single-step reaction are scarce [31].
We emphasize that the strong dependence of hyperarc
curvature is on the summation of the degrees of the vertex
belonging to the hyperarc, which is different from the
traditional degree of isolated vertices.
With curvature results at hand we defined “bottle
neck” reactions as those few reactions whose educts are
readily available (obtained from several reactions) and
whose products are often used as educts. They are char-
acterized by having very negative F(→e→) values. For
E. coli this reactions is: adp+h+pi→atp+h+h2o. Bot-
tle neck reactions can be considered as assortative ones,
for they transform popular products into popular educts.
Curvature values also allowed detecting redundant re-
actions (“one of the crow reactions”), which can be easily
replaced by others. The suitable curvature for detecting
such reactions is F(←e←), whose most negative values
correspond to reactions where popular sets of educts are
connected to popular sets of products. For E. coli, this
reaction is atp+h+h2o→adp+h+pi. Thus, adp phos-
phorilation is the metabolic bottle neck reaction but the
reverse reaction is not that central for the metabolism.
Our results show that E. coli metabolic network makes
use of a wealth amount of the products of its reactions
to start other reactions. This contrast with the historical
trend in wet-lab chemistry reactions, where most of the
products are seldom used in further reactions [32]. As the
historical study was conducted over single substances,
rather than over educts and products, further work on
the curvature of wet-lab chemical reactions needs to be
done to determine whether the behaviour found for E.
coli is also a trend of chemical reactions, in general.
The curvatures here presented, as indicated in Equa-
tion 1 and as used in [2, 11–19], can be weighted. In
the recent sketch of curvature for hypergraphs [30], the
weights are calculated based on the volume of the sim-
plex associated to the hyperedge. Weights, however, can
also be based on meta information of the network, e.g.
user’s seniority in the Wikipedia example or stoichiomet-
ric coefficients in the metabolic network. This and other
weighting schemes need to be explored in future stud-
ies on the curvature of hypergraphs, which our approach
allows.
ACKNOWLEDGMENTS
WL was supported by a PhD scholarship from
the German Academic Exchange Service (DAAD):
Forschungsstipendien-Promotionen in Deutschland,
2017/2018 (Bewerbung 57299294).
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11
TABLE I. Partial list of substrates of the E. coli metabolic
network.
Abbreviation Chemical name
25aics (S)-2-[5-Amino-1-(5-phospho-D-
ribosyl)imidazole-4-carboxamido]succinate
5aizc 5-amino-1-(5-phospho-D-ribosyl)imidazole-4-
carboxylate
5aop 5-Amino-4-oxopentanoate
2cpr5p 1-(2-Carboxyphenylamino)-1-deoxy-D-
ribulose
5-phosphate
6hmhpt 6-hydroxymethyl dihydropterin
5mdr1p 5-Methylthio-5-deoxy-D-ribose 1-phosphate
5mdru1p 5-Methylthio-5-deoxy-D-ribulose 1-phosphate
accoa Acetyl-CoA
adp Adenosine diphosphate
adphep D,D ADP-D-glycero-D-manno-heptose
adphep L,D ADP-L-glycero-D-manno-heptose
asp L L-Aspartate
atp Adenosine triphosphate
co2 Carbon dioxyde
cyan Hydrogen cyanide
dann 7,8-Diaminononanoate
dhnpt Dihydroneopterin
dtbt Dethiobiotin
dxyl5p 1-deoxy-D-xylulose 5-phosphate
tsul Thiosulfate
gar N1-(5-Phospho-D-ribosyl)glycinamide
gcald Glycolaldehyde
gdpddman GDP-4-dehydro-6-deoxy-D-mannose
gdpofuc GDP-4-oxo-L-fucose
glu1sa L-Glutamate 1-semialdehyde
gly Glycine
hH+
h2o Water
hco3 Bicarbonate
malcoa Malonyl CoA C24H33N7O19P3S
mmcoa R (R)-Methylmalonyl-CoA
mmcoa S (S)-Methylmalonyl-CoA
mql8 Menaquinol 8
mqn8 Menaquinone 8
nad Nicotinamide adenine dinucleotide
nadh Nicotinamide adenine dinucleotide - reduced
no2 Nitrite
no3 Nitrate
o2 Molecular oxygen
pdx5p Pyridoxine 5’-phosphate
phthr O-Phospho-4-hydroxy-L-threonine
pi Phosphate
pram 5-Phospho-beta-D-ribosylamine
pran N-(5-Phospho-D-ribosyl)anthranilate
prfp 1-(5-Phosphoribosyl)-5-[(5-
phosphoribosylamino)methylideneamino]imidazole-
4-carboxamide
prlp 5-[(5-phospho-1-deoxyribulos-1-
ylamino)methylideneamino]-1-(5-
phosphoribosyl)imidazole-4-carboxamide
q8 Ubiquinone-8
q8h2 Ubiquinol-8
so3 Sulfite
tcynt Thiocyanate
tsul Thiosulfate

1
SUPPLEMENTARY MATERIAL
Curvature values for chemical reactions from
the metabolism of Escherichia coli K-12 (iJR904
GSM/GPR).
Reaction F(→e)F(e→)F(→e→)F(←e)F(e←)F(←e←)
ALATA_L→-24 -42 -66 -30 -77 -107
ALATA_L←-77 -30 -107 -42 -24 -66
ALAR→-7 -5 -12 -10 -4 -14
ALAR←-4 -10 -14 -5 -7 -12
ASNN -90 -24 -114 -236 -41 -277
ASNS2 -78 -340 -418 -196 -558 -754
ASNS1 -131 -368 -499 -434 -592 -1026
ASPT -7 -17 -24 -17 -44 -61
ASPTA→-24 -32 -56 -37 -40 -77
ASPTA←-40 -37 -77 -32 -24 -56
VPAMT -8 -17 -25 -13 -47 -60
DAAD -91 -21 -112 -245 -84 -329
ALARi -7 -5 -12 -10 -4 -14
FFSD -86 -6 -92 -233 -8 -241
A5PISO→-3 -1 -4 -3 0 -3
A5PISO←0 -3 -3 -1 -3 -4
MME→-1 -1 -2 0 0 0
MME←0 0 0 -1 -1 -2
MICITD -86 0 -86 -233 -1 -234
ALCD19→-498 -55 -553 -364 -45 -409
ALCD19←-45 -364 -409 -55 -498 -553
LCADi -130 -366 -496 -287 -499 -786
TGBPA→-1 -16 -17 0 -21 -21
TGBPA←-21 0 -21 -16 -1 -17
LCAD→-130 -366 -496 -287 -499 -786
LCAD←-499 -287 -786 -366 -130 -496
ALDD2x -132 -365 -497 -287 -508 -795
ARAI→-3 -1 -4 -3 0 -3
ARAI←0 -3 -3 -1 -3 -4
RBK_L1 -37 -358 -395 -173 -590 -763
RBP4E→-2 -3 -5 0 -4 -4
RBP4E←-4 0 -4 -3 -2 -5
ACACCT -14 -3 -17 -18 -14 -32
BUTCT -14 -3 -17 -18 -12 -30
AB6PGH -85 -3 -88 -233 -8 -241
PMANM→0 -2 -2 -1 -3 -4
PMANM←-3 -1 -4 -2 0 -2
PPM2→-5 -1 -6 -5 0 -5
PPM2←0 -5 -5 -1 -5 -6
PPM→-5 -3 -8 -5 -8 -13
PPM←-8 -5 -13 -3 -5 -8
DRPA 0 -13 -13 -1 -17 -18
GALCTND -1 -233 -234 -2 -86 -88
DDPGALA→-1 -23 -24 0 -55 -55
DDPGALA←-55 0 -55 -23 -1 -24
DDGALK -37 -358 -395 -172 -589 -761
DHAPT -6 -21 -27 -23 -52 -75
FAO4 -128 -362 -490 -290 -505 -795
ALDD19x -127 -362 -489 -283 -496 -779
FRUK -37 -359 -396 -172 -590 -762
FCLPA→-1 -11 -12 0 -12 -12
FCLPA←-12 0 -12 -11 -1 -12
FCI→-2 -1 -3 -2 0 -2
FCI←0 -2 -2 -1 -2 -3
FCLK -37 -358 -395 -173 -589 -762
LCAR→-499 -52 -551 -366 -43 -409
LCAR←-43 -366 -409 -52 -499 -551
UDPG4E→-3 -2 -5 -5 -1 -6
UDPG4E←-1 -5 -6 -2 -3 -5
GALKr→-41 -359 -400 -174 -589 -763
GALKr←-589 -174 -763 -359 -41 -400
UGLT→-4 -11 -15 -6 -7 -13
UGLT←-7 -6 -13 -11 -4 -15
GALU→-456 -12 -468 -339 -75 -414
GALU←-75 -339 -414 -12 -456 -468
GALCTD -1 -233 -234 -2 -87 -89
GLTPD→-42 -363 -405 -50 -496 -546
GLTPD←-496 -50 -546 -363 -42 -405
GLYCTO2 -8 -6 -14 -17 -18 -35
GLYCTO3 -11 -8 -19 -13 -13 -26
GLYCTO4 -10 -7 -17 -9 -8 -17
GLYCDx -45 -365 -410 -55 -499 -554
GLYK -41 -365 -406 -177 -597 -774
PGLYCP -85 -40 -125 -233 -151 -384
G3PD2→-53 -374 -427 -25 -475 -500
G3PD2←-475 -25 -500 -374 -53 -427
GLCRAL -1 -15 -16 0 -45 -45
DHPPD -41 -362 -403 -50 -497 -547
DHCIND -41 -362 -403 -50 -497 -547
PPPNDO -501 -50 -551 -384 -41 -425
CINNDO -499 -50 -549 -382 -41 -423
HPYRI→0 -1 -1 -2 -2 -4
HPYRI←-2 -2 -4 -1 0 -1
IDOND→-499 -52 -551 -364 -46 -410
IDOND←-46 -364 -410 -52 -499 -551
IDOND2 -469 -20 -489 -369 -49 -418
GNK -42 -359 -401 -175 -589 -764
5DGLCNR→-469 -21 -490 -369 -49 -418
5DGLCNR←-49 -369 -418 -21 -469 -490
DDGLK -40 -358 -398 -174 -589 -763
LACZ -87 -5 -92 -235 -20 -255
MLTP1→-150 -12 -162 -40 -9 -49
MLTP1←-9 -40 -49 -12 -150 -162
MLTP2→-150 -13 -163 -40 -10 -50
MLTP2←-10 -40 -50 -13 -150 -163
MLTP3→-147 -13 -160 -37 -10 -47
MLTP3←-10 -37 -47 -13 -147 -160
AMALT1 -2 -6 -8 -7 -19 -26
AMALT2 -4 -7 -11 -8 -20 -28
AMALT3 -5 -7 -12 -9 -20 -29
AMALT4 -5 -4 -9 -9 -17 -26
MLTG1 -87 -8 -95 -235 -17 -252
MLTG2 -89 -5 -94 -236 -17 -253
MLTG3 -90 -6 -96 -237 -19 -256
MLTG4 -90 -7 -97 -237 -20 -257
MLTG5 -87 -7 -94 -234 -20 -254
MAN6PI→-3 -8 -11 -2 -9 -11
MAN6PI←-9 -2 -11 -8 -3 -11
GALS3 -86 -5 -91 -234 -20 -254
3HCINNMH -501 -283 -784 -384 -128 -512
3HPPPNH -501 -283 -784 -384 -128 -512
DHCINDO -5 0 -5 -20 0 -20
HPPPNDO -5 0 -5 -20 0 -20
HKNDDH -86 -336 -422 -233 -471 -704
HKNTDH -86 -334 -420 -233 -460 -693
OP4ENH -87 0 -87 -233 0 -233
HOPNTAL 0 -18 -18 0 -48 -48
ACALDi -69 -378 -447 -75 -509 -584
M1PD→-42 -370 -412 -50 -505 -555
M1PD←-505 -50 -555 -370 -42 -412
arXiv:1811.07825v1 [cs.DM] 19 Nov 2018

2
AGDC -87 -4 -91 -233 -15 -248
G6PDA -89 -15 -104 -234 -43 -277
ACNML 0 -15 -15 -1 -43 -44
TRE6PS -11 -327 -338 -9 -456 -465
TRE6PP -87 -38 -125 -234 -146 -380
PACCOAL -60 -12 -72 -193 -105 -298
PFK_2 -37 -358 -395 -173 -589 -762
PGMT→-6 -4 -10 -9 -8 -17
PGMT←-8 -9 -17 -4 -6 -10
MCITL2→-1 -26 -27 0 -64 -64
MCITL2←-64 0 -64 -26 -1 -27
MCITS -94 -345 -439 -239 -472 -711
MCITD 0 -233 -233 0 -86 -86
ACCOAL -61 -72 -133 -194 -287 -481
PTA2 -148 -21 -169 -38 -24 -62
RBK -37 -361 -398 -173 -596 -769
RMI→-1 -1 -2 -1 0 -1
RMI←0 -1 -1 -1 -1 -2
RMK -37 -358 -395 -173 -589 -762
RMPA→-1 -11 -12 0 -12 -12
RMPA←-12 0 -12 -11 -1 -12
MMM2 -2 0 -2 -4 -1 -5
KG6PDC -448 -8 -456 -324 -58 -382
X5PL3E 0 0 0 0 -2 -2
SBTPD→-42 -370 -412 -50 -505 -555
SBTPD←-505 -50 -555 -370 -42 -412
TAUDO -21 -343 -364 -41 -531 -572
PPAKr→-140 -173 -313 -34 -38 -72
PPAKr←-38 -34 -72 -173 -140 -313
OBTFL -23 -7 -30 -22 -12 -34
TREHe -86 -3 -89 -235 -16 -251
TRE6PH -87 -7 -94 -234 -24 -258
TREH -86 -3 -89 -235 -16 -251
TARTD -1 -237 -238 -2 -92 -94
PEAMNO -89 -8 -97 -253 -38 -291
ALTRH 0 -235 -235 -1 -89 -90
TAGURr→-41 -363 -404 -51 -497 -548
TAGURr←-497 -51 -548 -363 -41 -404
GUI1→-2 -1 -3 -2 -1 -3
GUI1←-1 -2 -3 -1 -2 -3
GUI2→-2 -1 -3 -2 -1 -3
GUI2←-1 -2 -3 -1 -2 -3
MNNH 0 -235 -235 -1 -89 -90
MANAO→-41 -363 -404 -51 -497 -548
MANAO←-497 -51 -548 -363 -41 -404
XYLI2i 0 -3 -3 -2 -16 -18
XYLI1→-2 -1 -3 -2 0 -2
XYLI1←0 -2 -2 -1 -2 -3
XYLK -37 -361 -398 -173 -592 -765
DKGLCNR1 -465 -19 -484 -369 -44 -413
HPYRRx -496 -50 -546 -364 -43 -407
HPYRRy -466 -18 -484 -369 -46 -415
GLCRD -1 -233 -234 -2 -87 -89
MMCD -449 -10 -459 -325 -60 -385
PPCSCT -23 -5 -28 -14 -3 -17
DKGLCNR2y -465 -20 -485 -369 -47 -416
2DGLCNRx -495 -53 -548 -363 -46 -409
2DGLCNRy -465 -21 -486 -368 -49 -417
DKGLCNR2x -495 -52 -547 -364 -44 -408
2DGULRx -496 -52 -548 -363 -46 -409
2DGULRy -466 -20 -486 -368 -49 -417
DOGULNR -495 -49 -544 -362 -41 -403
ICL -1 -15 -16 -2 -25 -27
MALS -103 -352 -455 -252 -476 -728
ME2 -48 -65 -113 -25 -118 -143
PPCK -43 -62 -105 -176 -200 -376
PPA -158 -360 -518 -240 -595 -835
PPC -147 -364 -511 -261 -601 -862
ME1 -45 -60 -105 -57 -148 -205
MDRPD 0 -234 -234 -1 -86 -87
DKMPPD2 -86 -365 -451 -234 -606 -840
PTRCTA -23 -28 -51 -26 -34 -60
SSALx -127 -374 -501 -284 -517 -801
ABUTD -127 -365 -492 -283 -500 -783
MTRK -37 -358 -395 -172 -589 -761
MTRI→-1 -1 -2 0 0 0
MTRI←0 0 0 -1 -1 -2
DKMPPD -90 -365 -455 -254 -606 -860
G5SADs -2 -558 -560 0 -536 -536
UNK3 -35 -21 -56 -28 -19 -47
ACGS -47 -345 -392 -44 -472 -516
ACGK -37 -34 -71 -172 -140 -312
AGPR→-191 -367 -558 -56 -467 -523
AGPR←-467 -56 -523 -367 -191 -558
ACOTA→-17 -30 -47 -21 -35 -56
ACOTA←-35 -21 -56 -30 -17 -47
NACODA -87 -3 -90 -235 -14 -249
ACODA -86 -10 -96 -234 -18 -252
ARGSS -44 -337 -381 -190 -555 -745
ARGSL→-1 -15 -16 0 -13 -13
ARGSL←-13 0 -13 -15 -1 -16
OCBT→-8 -361 -369 -8 -595 -603
OCBT←-595 -8 -603 -361 -8 -369
AST -5 -345 -350 -9 -472 -481
SOTA -17 -28 -45 -20 -34 -54
SADH -535 -15 -550 -557 -92 -649
SGSAD -127 -362 -489 -283 -496 -779
SGDS -86 -40 -126 -233 -55 -288
CBPS -125 -423 -548 -421 -770 -1191
SSALy -130 -379 -509 -252 -487 -739
ABTA -21 -29 -50 -23 -34 -57
GSPMDA -86 -5 -91 -233 -5 -238
GSPMDS -42 -394 -436 -177 -734 -911
MTAN -86 -5 -91 -233 -6 -239
G5SD -466 -54 -520 -367 -192 -559
GLU5K -71 -34 -105 -200 -139 -339
P5CR -467 -22 -489 -368 -48 -416
P5CD -128 -390 -518 -284 -530 -814
PROD2 -5 -325 -330 -11 -457 -468
ARGDC -452 -8 -460 -329 -58 -387
AGMT -86 -7 -93 -233 -9 -242
ORNDC -455 -14 -469 -331 -64 -395
ADMDCr→-451 -9 -460 -331 -58 -389
ADMDCr←-58 -331 -389 -9 -451 -460
SPMS -6 -327 -333 -7 -451 -458
SPMDAT1 -15 -344 -359 -19 -472 -491
SPMDAT2 -15 -344 -359 -19 -472 -491
ORNTA -23 -28 -51 -27 -36 -63
PPTGS 0 -323 -323 0 -451 -451
UDCPDP -88 -361 -449 -233 -595 -828
KDOPP -86 -36 -122 -233 -146 -379
PEPT_EC→-461 -2 -463 -330 -1 -331
PEPT_EC←-1 -330 -331 -2 -461 -463
PAPA_EC -89 -37 -126 -234 -147 -381
ECAP_EC 0 -323 -323 0 -451 -451
AACPS1 -60 -15 -75 -181 -106 -287
AACPS2 -59 -13 -72 -179 -106 -285
AACPS3 -60 -13 -73 -181 -106 -287

3
AACPS4 -59 -14 -73 -179 -106 -285
AACPS5 -59 -13 -72 -179 -106 -285
ALAALAr→-41 -395 -436 -177 -734 -911
ALAALAr←-734 -177 -911 -395 -41 -436
DAGK_EC -38 -359 -397 -173 -591 -764
ETHAAL 0 -11 -11 0 -39 -39
GDMANE 0 0 0 0 0 0
GOFUCR -466 -17 -483 -367 -44 -411
GALUi -456 -12 -468 -339 -75 -414
UDPGALM -1 1 0 -2 0 -2
GF6PTA -10 -29 -39 -20 -37 -57
UAGDP -450 -11 -461 -330 -73 -403
G1PACT -13 -345 -358 -17 -472 -489
GPDDA1 -87 -333 -420 -233 -461 -694
GPDDA2 -87 -331 -418 -233 -458 -691
GPDDA3 -85 -340 -425 -233 -465 -698
GPDDA4 -87 -336 -423 -233 -462 -695
GPDDA5 -85 -330 -415 -233 -459 -692
GMAND 0 -233 -233 0 -86 -86
S7PI -1 0 -1 -2 0 -2
GMHEPPA -86 -36 -122 -233 -146 -379
KDOPS -89 -36 -125 -254 -146 -400
KDOCT2 -2 -9 -11 -6 -72 -78
MOAT 0 -329 -329 -2 -460 -462
MOAT2 0 -330 -330 -2 -460 -462
UAGAAT→-2 -8 -10 -5 -17 -22
UAGAAT←-17 -5 -22 -8 -2 -10
LPADSS 0 -326 -326 -1 -455 -456
UHGADA -86 -3 -89 -234 -12 -246
U23GAAT -1 -332 -333 -1 -466 -467
TDSK -37 -358 -395 -172 -588 -760
EDTXS1 0 -7 -7 -2 -17 -19
EDTXS3 -1 -7 -8 -2 -17 -19
MAN1PT2 -456 -36 -492 -329 -146 -475
PAPPT3 0 -1 -1 -1 -7 -8
PGAMT→0 -1 -1 -1 -3 -4
PGAMT←-3 -1 -4 -1 0 -1
EDTXS4 -1 -6 -7 -2 -17 -19
EDTXS2 -1 -7 -8 -2 -17 -19
UAGCVT -4 -36 -40 -24 -146 -170
UAPGR -466 -18 -484 -367 -44 -411
UAMAS -44 -394 -438 -182 -734 -916
UAMAGS -37 -394 -431 -173 -734 -907
UAAGDS -38 -394 -432 -175 -734 -909
UGMDDS -37 -394 -431 -173 -734 -907
UAGPT3 -1 -326 -327 -4 -455 -459
GLUR→0 -28 -28 -1 -34 -35
GLUR←-34 -1 -35 -28 0 -28
PLIPA3 -85 -329 -414 -233 -476 -709
PLIPA1 -87 -329 -416 -237 -476 -713
PLIPA2 -87 -329 -416 -234 -476 -710
LPLIPA1 -86 -328 -414 -234 -477 -711
LPLIPA2 -86 -328 -414 -234 -477 -711
LPLIPA3 -86 -328 -414 -234 -477 -711
LPLIPA4 -1 1 0 -5 -3 -8
LPLIPA5 -1 1 0 -5 -3 -8
LPLIPA6 -1 1 0 -5 -3 -8
AGMHE 0 0 0 0 0 0
GMHEPAT -486 -7 -493 -496 -72 -568
GMHEPK -37 -358 -395 -172 -588 -760
LPSSYN_EC -3 -366 -369 -8 -607 -615
G1PTT -455 -7 -462 -335 -72 -407
TDPGDH 0 -234 -234 0 -86 -86
TDPDRE 0 0 0 -1 0 -1
TDPDRR -466 -17 -483 -367 -44 -411
MI1PP -85 -35 -120 -233 -147 -380
UDPGD -130 -361 -491 -288 -496 -784
USHD -86 -325 -411 -234 -456 -690
ACGAMT -1 -1 -2 -5 -7 -12
UAG2Ei -1 0 -1 -4 0 -4
UACMAMO -127 -362 -489 -283 -496 -779
TDPADGAT -13 -345 -358 -16 -472 -488
TDPAGTA -34 -20 -54 -29 -17 -46
AADDGT 0 -325 -325 0 -451 -451
ACMAMUT 0 -326 -326 0 -455 -455
ACONT→-2 -2 -4 -2 -1 -3
ACONT←-1 -2 -3 -2 -2 -4
CITL -2 -7 -9 -2 -18 -20
FRD2 -19 -21 -40 -15 -27 -42
FRD3 -14 -17 -31 -14 -26 -40
FUM→-96 -7 -103 -243 -4 -247
FUM←-4 -243 -247 -7 -96 -103
CS -105 -347 -452 -253 -474 -727
ICDHyr→-45 -71 -116 -20 -92 -112
ICDHyr←-92 -20 -112 -71 -45 -116
TEST_AKGDH -81 -50 -131 -91 -107 -198
MDH→-45 -366 -411 -57 -502 -559
MDH←-502 -57 -559 -366 -45 -411
MDH2 -7 -7 -14 -20 -20 -40
MDH3 -10 -9 -19 -16 -15 -31
SUCD1i -22 -10 -32 -19 -17 -36
SUCOAS→-81 -74 -155 -205 -287 -492
SUCOAS←-287 -205 -492 -74 -81 -155
PMDPHT -86 -36 -122 -233 -147 -380
DNMPPA -86 -36 -122 -233 -146 -379
NADPPPS -130 -86 -216 -251 -187 -438
NMNN -88 -327 -415 -237 -458 -695
NADDPe -127 -334 -461 -283 -484 -767
PPCDC -449 -8 -457 -324 -58 -382
DPCOAK -37 -379 -416 -172 -611 -783
HYPOE -88 -36 -124 -234 -146 -380
PYDXPP -88 -36 -124 -236 -146 -382
PDXPP -88 -36 -124 -234 -146 -380
NMNDA -88 -8 -96 -237 -36 -273
TMKr→-38 -359 -397 -173 -590 -763
TMKr←-590 -173 -763 -359 -38 -397
PPNCL2 -4 -336 -340 -12 -532 -544
ACPS1 -22 -331 -353 -21 -467 -488
AMAOTr→-3 -1 -4 -8 -1 -9
AMAOTr←-1 -8 -9 -1 -3 -4
BTS2→-3 -334 -337 -7 -458 -465
BTS2←-458 -7 -465 -334 -3 -337
DBTSr→-96 -395 -491 -181 -735 -916
DBTSr←-735 -181 -916 -395 -96 -491
AOXSr→-456 -30 -486 -334 -82 -416
AOXSr←-82 -334 -416 -30 -456 -486
BSORx -495 -283 -778 -363 -129 -492
BSORy -465 -251 -716 -368 -132 -500
CBIAT→-123 -44 -167 -405 -218 -623
CBIAT←-218 -405 -623 -44 -123 -167
CBLAT→-124 -43 -167 -406 -219 -625
CBLAT←-219 -406 -625 -43 -124 -167
PNTK -38 -358 -396 -173 -588 -761
PTPATi -486 -7 -493 -496 -72 -568
ADOCBLS 0 -326 -326 0 -455 -455
NNDMBRT -1 -325 -326 -1 -451 -452
ADOCBIK -37 -358 -395 -173 -588 -761
ACBIPGT -450 -7 -457 -335 -72 -407

4
HEMEOS -86 -6 -92 -235 -72 -307
SHCHD2 -41 -362 -403 -50 -496 -546
SHCHF 0 -323 -323 -2 -449 -451
DXPRIi -467 -18 -485 -369 -44 -413
DXPS -504 -10 -514 -347 -59 -406
DHBD→-42 -363 -405 -50 -497 -547
DHBD←-497 -50 -547 -363 -42 -405
ICHORT -86 -14 -100 -234 -44 -278
ENTCS 0 -329 -329 -2 -482 -484
DHBSr→-38 -8 -46 -173 -72 -245
DHBSr←-72 -173 -245 -8 -38 -46
SERASr→-493 -8 -501 -505 -72 -577
SERASr←-72 -505 -577 -8 -493 -501
E4PD→-129 -363 -492 -286 -497 -783
E4PD←-497 -286 -783 -363 -129 -492
DHFR→-468 -22 -490 -367 -49 -416
DHFR←-49 -367 -416 -22 -468 -490
DHNPA2 0 0 0 0 0 0
DHFS -71 -70 -141 -200 -287 -487
GTPCI -87 -5 -92 -244 -10 -254
HPPK2 -37 -330 -367 -172 -482 -654
DHPS2 0 -331 -331 0 -521 -521
MECDPDH -449 -234 -683 -324 -86 -410
GLUTRS -71 -13 -84 -200 -105 -305
ALATA_D2 -6 -15 -21 -8 -45 -53
ALATA_L2 -9 -15 -24 -13 -45 -58
GTHOr→-466 -20 -486 -367 -47 -414
GTHOr←-47 -367 -414 -20 -466 -486
GLUCYS -73 -394 -467 -206 -734 -940
GTHS -42 -396 -438 -177 -737 -914
GLUTRR -466 -18 -484 -367 -44 -411
PPBNGS 0 -557 -557 0 -535 -535
HMBS -86 -7 -93 -233 -34 -267
UPP3S 0 -234 -234 0 -86 -86
UPPDC1 -449 -8 -457 -325 -58 -383
CPPPGO -453 -241 -694 -344 -144 -488
PPPGO -4 -233 -237 -20 -86 -106
FCLT 0 -324 -324 -2 -449 -451
G1SATi 0 0 0 0 0 0
UPP3MT -2 -325 -327 -8 -455 -463
IPDDIi 0 0 0 -4 -1 -5
DMATT -1 -7 -8 -4 -72 -76
GRTT 0 -9 -9 -4 -72 -76
OCTDPS 0 -8 -8 -6 -72 -78
MEPCT -451 -7 -458 -330 -72 -402
CDPMEK -37 -358 -395 -172 -588 -760
MECDPS 0 -5 -5 0 -11 -11
NADDP -127 -334 -461 -283 -484 -767
DMPPS -496 -283 -779 -363 -128 -491
IPDPS -496 -287 -783 -363 -127 -490
DHNAOT 0 -344 -344 -1 -584 -585
NPHS 0 -21 -21 0 -23 -23
SUCBZS 0 -233 -233 0 -86 -86
OXGDC2 -467 -8 -475 -345 -58 -403
SHCHCS2 0 -15 -15 -1 -44 -45
SUCBZL -60 -13 -73 -193 -105 -298
ICHORSi 0 -1 -1 -4 0 -4
AMMQT8_2 -7 -334 -341 -12 -461 -473
QULNS -12 -593 -605 -7 -681 -688
ASPO3 -10 -3 -13 -30 -17 -47
ASPO4 -13 -5 -18 -26 -12 -38
ASPO5 -17 -12 -29 -27 -24 -51
ASPO6 -11 -1 -12 -37 -7 -44
NNDPR -450 -16 -466 -335 -132 -467
NNAT -488 -7 -495 -497 -72 -569
NMNAT -488 -57 -545 -500 -113 -613
NADS1 -71 -387 -458 -179 -595 -774
DNTPPA -86 -331 -417 -233 -521 -754
ADCS -1 -28 -29 -16 -34 -50
ADCL 0 -338 -338 0 -492 -492
MOHMT -89 -4 -93 -239 -5 -244
PANTS -37 -338 -375 -172 -555 -727
ASP1DC -456 -8 -464 -341 -58 -399
DPR -466 -18 -484 -367 -44 -411
PDX5PS -42 -640 -682 -54 -788 -842
PERD→-42 -363 -405 -51 -497 -548
PERD←-497 -51 -548 -363 -42 -405
PDX5PO→-6 -4 -10 -21 -6 -27
PDX5PO←-6 -21 -27 -4 -6 -10
PYAM5PO -92 -11 -103 -254 -40 -294
PYDXNK -37 -359 -396 -172 -590 -762
PYDAMK -37 -359 -396 -172 -590 -762
PYDXK -37 -361 -398 -172 -590 -762
RZ5PP -86 -36 -122 -233 -146 -379
NNAM -87 -8 -95 -233 -36 -269
NAMNPP -126 -78 -204 -417 -359 -776
GTPCII2 -87 -336 -423 -244 -531 -775
DB4PS -3 -329 -332 -3 -459 -462
APRAUR -466 -18 -484 -367 -44 -411
DHPPDA2 -535 -7 -542 -557 -34 -591
RBFSa -1 -269 -270 0 -232 -232
RBFK -37 -358 -395 -172 -588 -760
FMNAT -486 -14 -500 -496 -73 -569
RBFSb 0 0 0 0 -1 -1
OHPBAT→-35 -22 -57 -29 -17 -46
OHPBAT←-17 -29 -46 -22 -35 -57
AMPMS -86 -329 -415 -234 -460 -694
PMPK -38 -34 -72 -172 -139 -311
HMPK1 -36 -358 -394 -172 -589 -761
TMPPP -450 -8 -458 -324 -74 -398
TMPKr→-39 -35 -74 -173 -140 -313
TMPKr←-140 -173 -313 -35 -39 -74
HETZK -36 -358 -394 -172 -589 -761
THZPSN -42 -587 -629 -183 -706 -889
4HTHRS -86 -35 -121 -235 -146 -381
HBZOPT 0 -7 -7 -1 -72 -73
OPHHX -4 0 -4 -20 0 -20
CHRPL 0 -14 -14 -4 -43 -47
OPHBDC -449 -8 -457 -324 -58 -382
OMBZLM -2 -325 -327 -7 -455 -462
OMMBLHX -4 0 -4 -20 0 -20
OHPHM -2 -325 -327 -7 -455 -462
DMQMT -2 -328 -330 -7 -469 -476
OMPHHX -4 0 -4 -20 0 -20
UDCPDPS 0 -7 -7 -6 -74 -80
DXYLK -36 -360 -396 -172 -589 -761
NADK -78 -376 -454 -222 -632 -854
BPNT -87 -42 -129 -233 -179 -412
ADSK -37 -358 -395 -172 -588 -760
SADT2 -124 -47 -171 -417 -225 -642
SERAT→-20 -22 -42 -25 -23 -48
SERAT←-23 -25 -48 -22 -20 -42
PAPSR 0 -324 -324 -8 -461 -469
SULR→-131 -367 -498 -252 -469 -721
SULR←-469 -252 -721 -367 -131 -498
CYSS -1 -333 -334 -2 -463 -465
TRPAS1 -88 -22 -110 -239 -78 -317
GCALDD -127 -366 -493 -283 -501 -784

5
MTHFC→-87 -3 -90 -234 -2 -236
MTHFC←-2 -234 -236 -3 -87 -90
MTHFD→-46 -368 -414 -21 -467 -488
MTHFD←-467 -21 -488 -368 -46 -414
GLYCL -51 -56 -107 -59 -141 -200
MTHFR2 -498 -50 -548 -365 -41 -406
FTHFD -88 -333 -421 -236 -464 -700
GLUDC -483 -11 -494 -352 -62 -414
GLUDy→-164 -394 -558 -279 -517 -796
GLUDy←-517 -279 -796 -394 -164 -558
GLNS -105 -406 -511 -207 -735 -942
GLUSy -484 -46 -530 -399 -78 -477
GLUN -87 -35 -122 -245 -68 -313
SERD_D -1 -21 -22 -2 -77 -79
GHMT2 -12 -241 -253 -13 -93 -106
GLYATi -23 -21 -44 -21 -18 -39
PGCD -43 -362 -405 -52 -496 -548
PSP_L -86 -45 -131 -233 -153 -386
PSERT -34 -20 -54 -28 -17 -45
SERD_L -7 -21 -28 -9 -77 -86
THRD -46 -362 -408 -56 -496 -552
PDH -107 -62 -169 -85 -118 -203
G1PP -92 -39 -131 -242 -162 -404
ENO→-1 -253 -254 -1 -89 -90
ENO←-89 -1 -90 -253 -1 -254
FBA→-2 -16 -18 -1 -21 -22
FBA←-21 -1 -22 -16 -2 -18
FBP -88 -44 -132 -234 -155 -389
F6PA→-9 -12 -21 -8 -15 -23
F6PA←-15 -8 -23 -12 -9 -21
GAPD→-199 -363 -562 -95 -497 -592
GAPD←-497 -95 -592 -363 -199 -562
GLCS1 0 -358 -358 0 -588 -588
GLGC -492 -7 -499 -505 -72 -577
GLCP -146 -9 -155 -36 -6 -42
HEX1 -53 -362 -415 -175 -596 -771
PGM→-1 -2 -3 -1 -2 -3
PGM←-2 -1 -3 -2 -1 -3
PFK -46 -359 -405 -180 -590 -770
PGI→-8 -8 -16 -4 -9 -13
PGI←-9 -4 -13 -8 -8 -16
PGK→-39 -35 -74 -174 -140 -314
PGK←-140 -174 -314 -35 -39 -74
PPS -166 -386 -552 -419 -631 -1050
PYK -591 -186 -777 -378 -80 -458
TPI→-9 -9 -18 -7 -12 -19
TPI←-12 -7 -19 -9 -9 -18
TRSAR -498 -50 -548 -363 -43 -406
GLXCL -453 -9 -462 -327 -60 -387
GLYCK -39 -360 -399 -172 -590 -762
GLYCLTDy -470 -22 -492 -370 -49 -419
GLYCLTDx -500 -54 -554 -365 -46 -411
PRMICIi 0 0 0 0 0 0
IGPDH 0 -233 -233 0 -86 -86
HISTP -86 -36 -122 -233 -146 -379
HSTPT -34 -20 -54 -28 -17 -45
HISTD -127 -364 -491 -283 -499 -782
IG3PS -1 -353 -354 -12 -485 -497
ATPPRT -38 -7 -45 -183 -72 -255
PRATPP -86 -331 -417 -233 -521 -754
PRAMPC -86 0 -86 -233 0 -233
PRPPS→-45 -341 -386 -175 -483 -658
PRPPS←-483 -175 -658 -341 -45 -386
ACCOACr→-51 -395 -446 -192 -735 -927
ACCOACr←-735 -192 -927 -395 -51 -446
ACACT1r→-13 -21 -34 -16 -25 -41
ACACT1r←-25 -16 -41 -21 -13 -34
CDAPPA_EC -88 -330 -418 -236 -463 -699
DASYN_EC→-454 -10 -464 -331 -74 -405
DASYN_EC←-74 -331 -405 -10 -454 -464
CLPNS_EC→-1 -5 -6 -4 -4 -8
CLPNS_EC←-4 -4 -8 -5 -1 -6
C140SN -467 -268 -735 -382 -206 -588
C120SN -467 -267 -734 -382 -205 -587
MACPD -449 -9 -458 -334 -59 -393
KAS14 -450 -20 -470 -335 -76 -411
MCOATA→-18 -31 -49 -8 -23 -31
MCOATA←-23 -8 -31 -31 -18 -49
C160SN -467 -266 -733 -382 -206 -588
KAS16 -466 -34 -500 -378 -120 -498
C181SN -467 -266 -733 -382 -206 -588
C141SN -467 -266 -733 -382 -206 -588
KAS15 -462 -34 -496 -350 -82 -432
ACOATA→-30 -22 -52 -23 -24 -47
ACOATA←-24 -23 -47 -22 -30 -52
C161SN -467 -267 -734 -382 -206 -588
FAO2 -194 -391 -585 -485 -621 -1106
FAO1 -194 -391 -585 -485 -621 -1106
FAO3 -188 -391 -579 -484 -621 -1105
PGPP_EC -86 -40 -126 -234 -147 -381
PGSA_EC→-11 -330 -341 -10 -460 -470
PGSA_EC←-460 -10 -470 -330 -11 -341
PASYN_EC -14 -8 -22 -10 -20 -30
PSD_EC -449 -9 -458 -325 -59 -384
PSSA_EC→-9 -330 -339 -12 -460 -472
PSSA_EC←-460 -12 -472 -330 -9 -339
AHC→-92 -7 -99 -234 -7 -241
AHC←-7 -234 -241 -7 -92 -99
DHPTDC 0 -232 -232 0 -86 -86
RHCCE 0 -1 -1 0 -2 -2
HSST -2 -21 -23 -6 -23 -29
SHSL1 -2 -336 -338 -6 -470 -476
CYSTL -86 -22 -108 -233 -79 -312
METS -2 -5 -7 -1 -7 -8
METAT -125 -50 -175 -406 -220 -626
AHCYSNS -92 -5 -97 -234 -6 -240
LGTHL -3 0 -3 -2 0 -2
GLYOX -86 -329 -415 -233 -455 -688
MGSA -9 -36 -45 -7 -146 -153
UGLYCH -535 -18 -553 -557 -96 -653
ALLTN -87 -324 -411 -235 -449 -684
ALLTAH -86 -1 -87 -233 -3 -236
CYNTAH -450 -15 -465 -329 -92 -421
CMPN -97 -4 -101 -238 -9 -247
ADA -540 -11 -551 -563 -39 -602
DADA -537 -8 -545 -559 -36 -595
ADNK1 -42 -364 -406 -178 -621 -799
ADK1→-70 -34 -104 -178 -139 -317
ADK1←-139 -178 -317 -34 -70 -104
DADK→-38 -35 -73 -173 -141 -314
DADK←-141 -173 -314 -35 -38 -73
ADK4→-33 -34 -67 -6 -139 -145
ADK4←-139 -6 -145 -34 -33 -67
ADK3→-34 -38 -72 -17 -146 -163
ADK3←-146 -17 -163 -38 -34 -72
AMPN -119 -8 -127 -239 -14 -253
AP4AH -85 -358 -443 -233 -588 -821
GP4GH -85 -328 -413 -233 -456 -689

6
AP5AH -85 -530 -615 -233 -625 -858
ADPT -7 -13 -20 -16 -105 -121
CYTD -538 -11 -549 -561 -39 -600
DCYTD -536 -9 -545 -558 -37 -595
CYTK2→-38 -35 -73 -173 -141 -314
CYTK2←-141 -173 -314 -35 -38 -73
CYTK1→-48 -36 -84 -177 -141 -318
CYTK1←-141 -177 -318 -36 -48 -84
UMPK→-44 -36 -80 -173 -145 -318
UMPK←-145 -173 -318 -36 -44 -80
CSND -536 -12 -548 -558 -39 -597
ADNCYC -37 -6 -43 -172 -72 -244
DCTPD -536 -8 -544 -559 -36 -595
TMDPP→-150 -5 -155 -40 -5 -45
TMDPP←-5 -40 -45 -5 -150 -155
DURIPP→-149 -10 -159 -38 -10 -48
DURIPP←-10 -38 -48 -10 -149 -159
NTPTP1 -87 0 -87 -235 -4 -239
NTPTP2 -87 -1 -88 -244 -4 -248
DUTPDP -88 -333 -421 -234 -523 -757
GK1→-42 -38 -80 -174 -146 -320
GK1←-146 -174 -320 -38 -42 -80
DGK1→-38 -35 -73 -173 -141 -314
DGK1←-141 -173 -314 -35 -38 -73
XPPT -5 -8 -13 -15 -73 -88
HXPRT -5 -10 -15 -15 -75 -90
GUAPRT -5 -9 -14 -17 -77 -94
INSK -42 -361 -403 -176 -591 -767
GSNK -40 -360 -400 -174 -593 -767
NTPP3 -87 -332 -419 -235 -522 -757
NTPP4 -88 -336 -424 -239 -532 -771
NTPP5 -87 -332 -419 -234 -522 -756
NTPP6 -123 -337 -460 -405 -554 -959
NTPP7 -86 -332 -418 -235 -524 -759
NTPP8 -87 -332 -419 -239 -528 -767
NTPP1 -87 -332 -419 -235 -522 -757
NTPP2 -87 -333 -420 -244 -526 -770
NDPK1→-44 -45 -89 -176 -140 -316
NDPK1←-140 -176 -316 -45 -44 -89
NDPK2→-43 -40 -83 -174 -140 -314
NDPK2←-140 -174 -314 -40 -43 -83
NDPK3→-39 -40 -79 -174 -141 -315
NDPK3←-141 -174 -315 -40 -39 -79
NDPK5→-39 -36 -75 -173 -140 -313
NDPK5←-140 -173 -313 -36 -39 -75
NDPK6→-39 -35 -74 -173 -141 -314
NDPK6←-141 -173 -314 -35 -39 -74
NDPK7→-39 -36 -75 -173 -140 -313
NDPK7←-140 -173 -313 -36 -39 -75
NDPK8→-39 -35 -74 -173 -140 -313
NDPK8←-140 -173 -313 -35 -39 -74
NDPK4→-39 -36 -75 -173 -139 -312
NDPK4←-139 -173 -312 -36 -39 -75
RNDR1 -139 -234 -373 -42 -96 -138
RNDR2 -7 -234 -241 -12 -96 -108
RNDR4 -6 -234 -240 -10 -96 -106
RNDR3 -2 -234 -236 -10 -96 -106
RNTR1 -37 -234 -271 -180 -95 -275
RNTR2 -1 -235 -236 -19 -95 -114
RNTR3 -2 -235 -237 -14 -95 -109
RNTR4 -1 -234 -235 -14 -96 -110
URIDK2r→-39 -35 -74 -174 -141 -315
URIDK2r←-141 -174 -315 -35 -39 -74
DURIK1 -40 -360 -400 -174 -590 -764
TMDK1 -41 -359 -400 -176 -591 -767
TMDS -4 -1 -5 -5 -5 -10
DTMPK→-40 -35 -75 -173 -141 -314
DTMPK←-141 -173 -314 -35 -40 -75
URIK2 -6 -329 -335 -15 -463 -478
CYTDK2 -4 -333 -337 -15 -467 -482
PYNP2r→-151 -10 -161 -40 -10 -50
PYNP2r←-10 -40 -50 -10 -151 -161
UPPRT -6 -8 -14 -16 -79 -95
NTD1 -88 -38 -126 -235 -149 -384
NTD5 -89 -40 -129 -234 -150 -384
NTD6 -87 -38 -125 -234 -148 -382
NTD8 -87 -37 -124 -234 -149 -383
NTD3 -87 -37 -124 -234 -147 -381
NTD4 -97 -40 -137 -238 -149 -387
NTD7 -119 -42 -161 -239 -151 -390
NTD9 -91 -38 -129 -235 -149 -384
NTD11 -89 -40 -129 -236 -151 -387
NTD10 -87 -38 -125 -234 -149 -383
NTD2 -93 -40 -133 -234 -151 -385
PUNP6→-148 -9 -157 -37 -9 -46
PUNP6←-9 -37 -46 -9 -148 -157
PUNP5→-151 -9 -160 -40 -9 -49
PUNP5←-9 -40 -49 -9 -151 -160
PUNP2→-148 -10 -158 -38 -11 -49
PUNP2←-11 -38 -49 -10 -148 -158
PUNP4→-149 -11 -160 -37 -9 -46
PUNP4←-9 -37 -46 -11 -149 -160
PUNP1→-151 -10 -161 -42 -11 -53
PUNP1←-11 -42 -53 -10 -151 -161
PUNP3→-149 -11 -160 -38 -9 -47
PUNP3←-9 -38 -47 -11 -149 -160
PUNP7→-149 -9 -158 -38 -9 -47
PUNP7←-9 -38 -47 -9 -149 -158
GUAD -539 -11 -550 -563 -38 -601
ADD -541 -11 -552 -562 -38 -600
L-LACD2 -6 -17 -23 -17 -57 -74
L-LACD3 -9 -19 -28 -13 -52 -65
ATPS4r→-734 -729 -1463 -394 -572 -966
ATPS4r←-572 -394 -966 -729 -734 -1463
CRNBTCT→-2 -3 -5 -2 -3 -5
CRNBTCT←-3 -2 -5 -3 -2 -5
CRNCBCT→-3 -2 -5 -3 -2 -5
CRNCBCT←-2 -3 -5 -2 -3 -5
CRNCDH→-2 -234 -236 -2 -87 -89
CRNCDH←-87 -2 -89 -234 -2 -236
CYTBD -467 -570 -1037 -347 -538 -885
CYTBO3 -467 -570 -1037 -347 -538 -885
LDH_D2 -6 -17 -23 -16 -57 -73
DMSOR1 -8 -241 -249 -8 -95 -103
DMSOR2 -3 -237 -240 -7 -94 -101
TMAOR1 -457 -241 -698 -332 -95 -427
TMAOR2 -452 -237 -689 -331 -94 -425
FDH2 -462 -335 -797 -342 -521 -863
FDH3 -465 -337 -802 -338 -516 -854
GLCDe -105 -330 -435 -249 -468 -717
G3PD7 -14 -11 -25 -12 -13 -25
G3PD6 -15 -12 -27 -16 -18 -34
G3PD5 -12 -10 -22 -20 -23 -43
HYD3 -454 -328 -782 -331 -453 -784
HYD1 -452 -327 -779 -339 -463 -802
HYD2 -455 -329 -784 -335 -458 -793
NO3R1 -463 -573 -1036 -329 -542 -871
NO3R2 -458 -569 -1027 -331 -545 -876

7
NADH5 -499 -53 -552 -375 -55 -430
NADH9 -501 -54 -555 -367 -45 -412
NADH10 -502 -55 -557 -371 -50 -421
NTRIR2x -500 -290 -790 -365 -161 -526
NADH6 -499 -377 -876 -375 -504 -879
NADH7 -502 -379 -881 -371 -499 -870
NADH8 -501 -378 -879 -367 -494 -861
THD2 -540 -417 -957 -380 -507 -887
POX -132 -14 -146 -260 -84 -344
SUCD4→-10 -10 -20 -13 -15 -28
SUCD4←-15 -13 -28 -10 -10 -20
TEST_NADTRHD -58 -56 -114 -93 -91 -184
TMAOR1e -457 -241 -698 -332 -95 -427
TMAOR2e -452 -237 -689 -331 -94 -425
DMSOR1e -8 -241 -249 -8 -95 -103
DMSOR2e -3 -237 -240 -7 -94 -101
TRDR -474 -26 -500 -367 -44 -411
PGL -86 -325 -411 -234 -450 -684
EDA -1 -23 -24 0 -55 -55
EDD -1 -233 -234 -1 -87 -88
GND -45 -54 -99 -19 -78 -97
RPE→-3 -3 -6 -3 -4 -7
RPE←-4 -3 -7 -3 -3 -6
RPI→-8 -3 -11 -3 -3 -6
RPI←-3 -3 -6 -3 -8 -11
TALA→-13 -11 -24 -11 -11 -22
TALA←-11 -11 -22 -11 -13 -24
TKT1→-12 -11 -23 -6 -13 -19
TKT1←-13 -6 -19 -11 -12 -23
TKT2→-6 -17 -23 -6 -21 -27
TKT2←-21 -6 -27 -17 -6 -23
G6PDH2r→-52 -368 -420 -22 -466 -488
G6PDH2r←-466 -22 -488 -368 -52 -420
GMPS2 -125 -367 -492 -418 -593 -1011
IMPD -130 -363 -493 -286 -497 -783
GMPR -471 -28 -499 -369 -81 -450
ADSS -11 -364 -375 -31 -603 -634
ADSL2r→-1 -11 -12 -1 -12 -13
ADSL2r←-12 -1 -13 -11 -1 -12
ADSL1r→-1 -16 -17 0 -43 -43
ADSL1r←-43 0 -43 -16 -1 -17
PRASCS→-45 -395 -440 -190 -735 -925
PRASCS←-735 -190 -925 -395 -45 -440
PRAGSr→-43 -396 -439 -177 -735 -912
PRAGSr←-735 -177 -912 -396 -43 -439
AIRC3→-1 0 -1 -1 -1 -2
AIRC3←-1 -1 -2 0 -1 -1
GLUPRT -88 -35 -123 -256 -107 -363
AICART→-4 -5 -9 -4 -6 -10
AICART←-6 -4 -10 -5 -4 -9
IMPC→-89 -1 -90 -236 -1 -237
IMPC←-1 -236 -237 -1 -89 -90
AIRC2 -38 -394 -432 -177 -735 -912
PRFGS -125 -422 -547 -418 -768 -1186
PRAIS -37 -395 -432 -172 -734 -906
GARFT→-3 -329 -332 -5 -455 -460
GARFT←-455 -5 -460 -329 -3 -332
GART -48 -395 -443 -179 -735 -914
ASPCT -9 -360 -369 -18 -596 -614
DHORTS→-86 -324 -410 -235 -450 -685
DHORTS←-450 -235 -685 -324 -86 -410
DHORD2 -3 -3 -6 -15 -16 -31
DHORD5 -6 -5 -11 -11 -11 -22
ORPT→-72 -11 -83 -8 -3 -11
ORPT←-3 -8 -11 -11 -72 -83
OMPDC -449 -9 -458 -325 -65 -390
CTPS2 -125 -428 -553 -423 -770 -1193
CBMK -129 -359 -488 -187 -590 -777
AMANAPE -1 0 -1 0 -1 -1
AMANK -37 -358 -395 -173 -589 -762
ALLTNt2r→-450 -326 -776 -326 -450 -776
ALLTNt2r←-450 -326 -776 -326 -450 -776
ARGORNt7→-9 -12 -21 -12 -9 -21
ARGORNt7←-9 -12 -21 -12 -9 -21
ACACt2→-450 -326 -776 -326 -450 -776
ACACt2←-450 -326 -776 -326 -450 -776
BUTt2r→-450 -326 -776 -326 -450 -776
BUTt2r←-450 -326 -776 -326 -450 -776
CYNTt2 -449 -325 -774 -325 -449 -774
GALCTt2r→-450 -326 -776 -326 -450 -776
GALCTt2r←-450 -326 -776 -326 -450 -776
GLCRt2r→-450 -326 -776 -326 -450 -776
GLCRt2r←-450 -326 -776 -326 -450 -776
PPPNt2r→-450 -326 -776 -326 -450 -776
PPPNt2r←-450 -326 -776 -326 -450 -776
HPPPNt2r→-450 -326 -776 -326 -450 -776
HPPPNt2r←-450 -326 -776 -326 -450 -776
HCINNMt2r→-450 -326 -776 -326 -450 -776
HCINNMt2r←-450 -326 -776 -326 -450 -776
GLUABUTt7→-38 -31 -69 -31 -38 -69
GLUABUTt7←-38 -31 -69 -31 -38 -69
ALAt2r→-456 -334 -790 -334 -456 -790
ALAt2r←-456 -334 -790 -334 -456 -790
URAt2r→-454 -329 -783 -329 -454 -783
URAt2r←-454 -329 -783 -329 -454 -783
GLYBt2r→-453 -326 -779 -326 -453 -779
GLYBt2r←-453 -326 -779 -326 -453 -779
CHLabc -126 -396 -522 -407 -737 -1144
GLYBabc -127 -396 -523 -407 -738 -1145
TARTRt7→-22 -14 -36 -14 -22 -36
TARTRt7←-22 -14 -36 -14 -22 -36
SUCCabc -144 -406 -550 -417 -755 -1172
GUAt2 -453 -330 -783 -330 -453 -783
XANt2 -453 -328 -781 -328 -453 -781
ACKr→-49 -35 -84 -175 -140 -315
ACKr←-140 -175 -315 -35 -49 -84
ACS -72 -29 -101 -196 -118 -314
ADHEr→-509 -73 -582 -378 -66 -444
ADHEr←-66 -378 -444 -73 -509 -582
LDH_D→-44 -376 -420 -53 -539 -592
LDH_D←-539 -53 -592 -376 -44 -420
FHL -459 -10 -469 -329 -58 -387
PTAr→-159 -22 -181 -52 -24 -76
PTAr←-24 -52 -76 -22 -159 -181
PFL -66 -21 -87 -35 -23 -58
SDPTA→-17 -28 -45 -21 -35 -56
SDPTA←-35 -21 -56 -28 -17 -45
ASAD→-191 -368 -559 -56 -467 -523
ASAD←-467 -56 -523 -368 -191 -559
DHDPS -44 -557 -601 -16 -535 -551
DHDPRy -466 -18 -484 -367 -44 -411
THDPS -88 -21 -109 -237 -24 -261
SDPDS -86 -12 -98 -234 -22 -256
DAPE→-1 -3 -4 0 -1 -1
DAPE←-1 0 -1 -3 -1 -4
LYSDC -453 -8 -461 -328 -59 -387
THRAr→-5 -9 -14 -6 -10 -16
THRAr←-10 -6 -16 -9 -5 -14

8
DAPDC -450 -12 -462 -327 -62 -389
HSDy→-44 -369 -413 -20 -467 -487
HSDy←-467 -20 -487 -369 -44 -413
ASPK→-44 -35 -79 -189 -140 -329
ASPK←-140 -189 -329 -35 -44 -79
HSK -37 -358 -395 -174 -588 -762
THRS -86 -42 -128 -233 -151 -384
12PPDt→-2 -2 -4 -2 -2 -4
12PPDt←-2 -2 -4 -2 -2 -4
NMNt7 -88 -327 -415 -237 -458 -695
ACALDt→-5 -4 -9 -4 -5 -9
ACALDt←-5 -4 -9 -4 -5 -9
GUAt→-4 -6 -10 -6 -4 -10
GUAt←-4 -6 -10 -6 -4 -10
HYXNt→-4 -4 -8 -4 -4 -8
HYXNt←-4 -4 -8 -4 -4 -8
XANt→-4 -4 -8 -4 -4 -8
XANt←-4 -4 -8 -4 -4 -8
NACUP -2 -1 -3 -1 -2 -3
ASNabc -127 -397 -524 -408 -738 -1146
ASNt2r→-453 -327 -780 -327 -453 -780
ASNt2r←-453 -327 -780 -327 -453 -780
DAPabc -124 -397 -521 -408 -735 -1143
CYSabc -125 -400 -525 -411 -736 -1147
ACt2r→-461 -327 -788 -327 -461 -788
ACt2r←-461 -327 -788 -327 -461 -788
ETOHt2r→-451 -326 -777 -326 -451 -777
ETOHt2r←-451 -326 -777 -326 -451 -777
PYRt2r→-492 -338 -830 -338 -492 -830
PYRt2r←-492 -338 -830 -338 -492 -830
O2t→-4 -20 -24 -20 -4 -24
O2t←-4 -20 -24 -20 -4 -24
CO2t→-58 -8 -66 -8 -58 -66
CO2t←-58 -8 -66 -8 -58 -66
H2Ot→-86 -233 -319 -233 -86 -319
H2Ot←-86 -233 -319 -233 -86 -319
DHAt→-3 -3 -6 -3 -3 -6
DHAt←-3 -3 -6 -3 -3 -6
NH3t→-34 -7 -41 -7 -34 -41
NH3t←-34 -7 -41 -7 -34 -41
ARBt2r→-452 -327 -779 -327 -452 -779
ARBt2r←-452 -327 -779 -327 -452 -779
ARBabc -126 -397 -523 -408 -737 -1145
HISt2r→-452 -326 -778 -326 -452 -778
HISt2r←-452 -326 -778 -326 -452 -778
PHEt2r→-451 -326 -777 -326 -451 -777
PHEt2r←-451 -326 -777 -326 -451 -777
LEUt2r→-452 -326 -778 -326 -452 -778
LEUt2r←-452 -326 -778 -326 -452 -778
VALt2r→-453 -327 -780 -327 -453 -780
VALt2r←-453 -327 -780 -327 -453 -780
ILEt2r→-452 -327 -779 -327 -452 -779
ILEt2r←-452 -327 -779 -327 -452 -779
CBL1abc -124 -395 -519 -406 -735 -1141
CADVt -454 -328 -782 -328 -454 -782
CRNt7 -3 -3 -6 -3 -3 -6
NAt3_1→-457 -332 -789 -332 -457 -789
NAt3_1←-457 -332 -789 -332 -457 -789
CITt7 -23 -14 -37 -14 -23 -37
CSNt2 -450 -325 -775 -325 -450 -775
MALTpts -4 -13 -17 -25 -43 -68
ACGApts -2 -14 -16 -20 -44 -64
DALAt2r→-453 -329 -782 -329 -453 -782
DALAt2r←-453 -329 -782 -329 -453 -782
DSERt2r→-450 -326 -776 -326 -450 -776
DSERt2r←-450 -326 -776 -326 -450 -776
GLYt2r→-454 -329 -783 -329 -454 -783
GLYt2r←-454 -329 -783 -329 -454 -783
SULabc -123 -395 -518 -406 -734 -1140
ASPt2_2 -456 -341 -797 -341 -456 -797
FUMt2_2 -459 -334 -793 -334 -459 -793
MALt2_2 -453 -331 -784 -331 -453 -784
SUCCt2_2 -470 -336 -806 -336 -470 -806
ASPt2_3 -456 -341 -797 -341 -456 -797
MALt2_3 -453 -331 -784 -331 -453 -784
SUCCt2_3 -470 -336 -806 -336 -470 -806
SUCCt2b -470 -336 -806 -336 -470 -806
FUMt2_3 -459 -334 -793 -334 -459 -793
SUCFUMt→-31 -22 -53 -22 -31 -53
SUCFUMt←-31 -22 -53 -22 -31 -53
GALCTNt2r→-450 -326 -776 -326 -450 -776
GALCTNt2r←-450 -326 -776 -326 -450 -776
GALURt2r→-451 -326 -777 -326 -451 -777
GALURt2r←-451 -326 -777 -326 -451 -777
GLCURt2r→-451 -326 -777 -326 -451 -777
GLCURt2r←-451 -326 -777 -326 -451 -777
OCDCAt2 -449 -325 -774 -325 -449 -774
HDCAt2 -455 -326 -781 -326 -455 -781
TTDCAt2 -455 -326 -781 -326 -455 -781
FE2abc -123 -396 -519 -407 -734 -1141
FORt→-10 -5 -15 -5 -10 -15
FORt←-10 -5 -15 -5 -10 -15
FUCt→-451 -326 -777 -326 -451 -777
FUCt←-451 -326 -777 -326 -451 -777
ABUTt2 -453 -327 -780 -327 -453 -780
GALt2 -453 -326 -779 -326 -453 -779
GLCt2 -465 -327 -792 -327 -465 -792
GALTpts -2 -14 -16 -20 -44 -64
GLNabc -124 -406 -530 -417 -735 -1152
GLYCt→-4 -5 -9 -5 -4 -9
GLYCt←-4 -5 -9 -5 -4 -9
GLYALDt→-2 -2 -4 -2 -2 -4
GLYALDt←-2 -2 -4 -2 -2 -4
UREAt→-3 -1 -4 -1 -3 -4
UREAt←-3 -1 -4 -1 -3 -4
GLYC3Pt6 -155 -43 -198 -43 -155 -198
ASPabc -130 -411 -541 -422 -741 -1163
GLUabc -157 -422 -579 -433 -768 -1201
ASPt2 -456 -341 -797 -341 -456 -797
GLUt2r→-483 -352 -835 -352 -483 -835
GLUt2r←-483 -352 -835 -352 -483 -835
GLUt4 -42 -36 -78 -36 -42 -78
ORNabc -129 -401 -530 -412 -740 -1152
ARGabc -126 -399 -525 -410 -737 -1147
HISabc -126 -396 -522 -407 -737 -1144
LYSabc -127 -398 -525 -409 -738 -1147
IDONt2r→-454 -326 -780 -326 -454 -780
IDONt2r←-454 -326 -780 -326 -454 -780
GLCNt2r→-454 -327 -781 -327 -454 -781
GLCNt2r←-454 -327 -781 -327 -454 -781
DDGLCNt2r→-452 -326 -778 -326 -452 -778
DDGLCNt2r←-452 -326 -778 -326 -452 -778
Kabc -125 -396 -521 -407 -736 -1143
AKGt2r→-466 -344 -810 -344 -466 -810
AKGt2r←-466 -344 -810 -344 -466 -810
LCTSt→-450 -326 -776 -326 -450 -776
LCTSt←-450 -326 -776 -326 -450 -776
ILEabc -126 -397 -523 -408 -737 -1145

9
THRabc -128 -400 -528 -411 -739 -1150
ALAabc -130 -404 -534 -415 -741 -1156
VALabc -127 -397 -524 -408 -738 -1146
LEUabc -126 -396 -522 -407 -737 -1144
D-LACt2→-452 -327 -779 -327 -452 -779
D-LACt2←-452 -327 -779 -327 -452 -779
GLYCLTt2r→-454 -328 -782 -328 -454 -782
GLYCLTt2r←-454 -328 -782 -328 -454 -782
L-LACt2r→-452 -328 -780 -328 -452 -780
L-LACt2r←-452 -328 -780 -328 -452 -780
LYSt2r→-453 -328 -781 -328 -453 -781
LYSt2r←-453 -328 -781 -328 -453 -781
MALTPTabc -127 -398 -525 -409 -738 -1147
MALTabc -124 -399 -523 -410 -735 -1145
MALTTTRabc -126 -397 -523 -408 -737 -1145
MALTHXabc -127 -398 -525 -409 -738 -1147
MALTTRabc -124 -396 -520 -407 -735 -1142
FRUpts2 -3 -22 -25 -22 -52 -74
MANpts -2 -16 -18 -20 -46 -66
GAMpts -2 -15 -17 -20 -46 -66
MELIBt2 -449 -325 -774 -325 -449 -774
METabc -125 -395 -520 -406 -736 -1142
METDabc -123 -394 -517 -405 -734 -1139
GALabc -127 -396 -523 -407 -738 -1145
INDOLEt2r→-452 -327 -779 -327 -452 -779
INDOLEt2r←-452 -327 -779 -327 -452 -779
ACNAMt2 -449 -325 -774 -325 -449 -774
NO3t7 -4 -5 -9 -5 -4 -9
NO2t2r→-453 -327 -780 -327 -453 -780
NO2t2r←-453 -327 -780 -327 -453 -780
NAt3_2 -457 -332 -789 -332 -457 -789
NAt3_1.5 -457 -332 -789 -332 -457 -789
GSNt2 -452 -326 -778 -326 -452 -778
DGSNt2 -452 -325 -777 -325 -452 -777
INSt2 -454 -328 -782 -328 -454 -782
DINSt2 -451 -325 -776 -325 -451 -776
ADNt2 -454 -330 -784 -330 -454 -784
URIt2 -454 -328 -782 -328 -454 -782
CYTDt2 -452 -328 -780 -328 -452 -780
DCYTt2 -450 -325 -775 -325 -450 -775
DURIt2 -452 -326 -778 -326 -452 -778
DADNt2 -451 -326 -777 -326 -451 -777
THMDt2 -453 -328 -781 -328 -453 -781
PNTOt4 -9 -9 -18 -9 -9 -18
PIt2r→-595 -360 -955 -360 -595 -955
PIt2r←-595 -360 -955 -360 -595 -955
NMNP -2 -4 -6 -4 -2 -6
PTRCabc -129 -400 -529 -411 -740 -1151
SPMDabc -125 -397 -522 -408 -736 -1144
PTRCORNt7→-12 -13 -25 -13 -12 -25
PTRCORNt7←-12 -13 -25 -13 -12 -25
PTRCt2r→-455 -330 -785 -330 -455 -785
PTRCt2r←-455 -330 -785 -330 -455 -785
PROt2r→-453 -328 -781 -328 -453 -781
PROt2r←-453 -328 -781 -328 -453 -781
PROabc -127 -398 -525 -409 -738 -1147
PIabc -269 -394 -663 -441 -734 -1175
ACMANApts -3 -14 -17 -21 -44 -65
MNLpts -2 -14 -16 -20 -44 -64
SUCpts -2 -14 -16 -20 -43 -63
GLCpts -19 -18 -37 -23 -51 -74
FRUpts -3 -14 -17 -22 -43 -65
TREpts -3 -15 -18 -22 -44 -66
PROt4 -12 -12 -24 -12 -12 -24
RMNt -450 -325 -775 -325 -450 -775
TSULabc -123 -395 -518 -406 -734 -1140
SERt2r→-456 -333 -789 -333 -456 -789
SERt2r←-456 -333 -789 -333 -456 -789
THMabc -124 -395 -519 -406 -735 -1141
SBTpts -2 -14 -16 -20 -44 -64
SERt4 -15 -17 -32 -17 -15 -32
THRt4 -13 -14 -27 -14 -13 -27
TAURabc -123 -395 -518 -406 -734 -1140
THRt2r→-454 -330 -784 -330 -454 -784
THRt2r←-454 -330 -784 -330 -454 -784
TRPt2r→-453 -326 -779 -326 -453 -779
TRPt2r←-453 -326 -779 -326 -453 -779
Kt2r→-451 -326 -777 -326 -451 -777
Kt2r←-451 -326 -777 -326 -451 -777
TYRt2r→-451 -327 -778 -327 -451 -778
TYRt2r←-451 -327 -778 -327 -451 -778
GLYC3Pabc -132 -401 -533 -412 -743 -1155
MAN6Pt6_2 -149 -38 -187 -38 -149 -187
G6Pt6_2 -154 -40 -194 -40 -154 -194
FUCPt6_2 -146 -36 -182 -36 -146 -182
URAt2 -454 -329 -783 -329 -454 -783
XTSNt2r→-452 -326 -778 -326 -452 -778
XTSNt2r←-452 -326 -778 -326 -452 -778
INSt2r→-454 -328 -782 -328 -454 -782
INSt2r←-454 -328 -782 -328 -454 -782
ADNt2r→-454 -330 -784 -330 -454 -784
ADNt2r←-454 -330 -784 -330 -454 -784
CYTDt2r→-452 -328 -780 -328 -452 -780
CYTDt2r←-452 -328 -780 -328 -452 -780
THMDt2r→-453 -328 -781 -328 -453 -781
THMDt2r←-453 -328 -781 -328 -453 -781
URIt2r→-454 -328 -782 -328 -454 -782
URIt2r←-454 -328 -782 -328 -454 -782
XYLt2 -451 -326 -777 -326 -451 -777
XYLabc -125 -396 -521 -407 -736 -1143
CHLt2r→-452 -326 -778 -326 -452 -778
CHLt2r←-452 -326 -778 -326 -452 -778
ADEt2r→-455 -329 -784 -329 -455 -784
ADEt2r←-455 -329 -784 -329 -455 -784
RIBabc -123 -395 -518 -406 -734 -1140
PSCVT→-4 -37 -41 -20 -146 -166
PSCVT←-146 -20 -166 -37 -4 -41
DHQS 0 -36 -36 0 -147 -147
CHORS 0 -40 -40 -1 -146 -147
DHQD→-1 -234 -235 0 -87 -87
DHQD←-87 0 -87 -234 -1 -235
SHK3Dr→-467 -19 -486 -368 -44 -412
SHK3Dr←-44 -368 -412 -19 -467 -486
DDPA -91 -36 -127 -256 -146 -402
SHKK -37 -358 -395 -173 -589 -762
PPNDH -449 -241 -690 -325 -145 -470
CHORM 0 -1 -1 -4 0 -4
TRPAS2→-90 -24 -114 -235 -80 -315
TRPAS2←-80 -235 -315 -24 -90 -114
TRPS2 -10 -235 -245 -12 -90 -102
TRPS3 0 -12 -12 -1 -15 -16
TRPS1 -7 -244 -251 -10 -102 -112
PRAIi 0 0 0 0 0 0
IGPS -449 -242 -691 -324 -144 -468
ANS -1 -366 -367 -16 -526 -542
ANPRT -1 -7 -8 -11 -72 -83
PPND -41 -46 -87 -51 -106 -157
TYRTA→-19 -28 -47 -23 -35 -58

10
TYRTA←-35 -23 -58 -28 -19 -47
PHETA1→-19 -28 -47 -22 -35 -57
PHETA1←-35 -22 -57 -28 -19 -47
ATPM -123 -394 -517 -405 -734 -1139
BETALDHx -126 -364 -490 -284 -500 -784
BETALDHy -129 -369 -498 -252 -470 -722
HCO3E→-144 -328 -472 -241 -450 -691
HCO3E←-450 -241 -691 -328 -144 -472
CYANST 1 -323 -322 -1 -452 -453
CAT -4 -253 -257 -1 -90 -91
SELNPS -122 -41 -163 -405 -179 -584
SPODM -448 -21 -469 -324 -8 -332
ACONMT -1 0 -1 -7 -6 -13
THRD_L -5 -8 -13 -6 -34 -40
ACHBS -492 -8 -500 -339 -58 -397
ACLS -492 -8 -500 -338 -58 -396
KARA2i -466 -18 -484 -367 -44 -411
KARA1i -466 -18 -484 -367 -44 -411
DHAD2 0 -233 -233 0 -87 -87
DHAD1 0 -236 -236 0 -87 -87
ILETA→-20 -28 -48 -23 -35 -58
ILETA←-35 -23 -58 -28 -20 -48
VALTA→-21 -31 -52 -23 -35 -58
VALTA←-35 -23 -58 -31 -21 -52
LEUTAi -34 -22 -56 -28 -20 -48
IPPS -100 -345 -445 -252 -473 -725
IPMD -41 -362 -403 -51 -496 -547
OMCDC -449 -8 -457 -324 -58 -382
IPPMIa→0 -234 -234 -1 -87 -88
IPPMIa←-87 -1 -88 -234 0 -234
IPPMIb→-87 0 -87 -234 -1 -235
IPPMIb←-1 -234 -235 0 -87 -87