BACKGROUND
metabolism is one of the most complex cellular processes and a basal system for maintaining life of all organisms.
liver metabolism disorders could cause a wide range of diseases, ranging from hepatitis to hepatic tumor
<cit> .
many studies which focus on the roles of single molecule substances or single paths in regulating liver metabolism have been carried out.
for example, the interleukin receptor-associated kinase-m  has been found to negatively regulate the innate and the adaptive immune response in the liver reacting to acute insult by alcohol
<cit> ; the liver x receptors  could bind to cholesterol metabolites and regulate the cholesterol turnover
<cit> ; the metabolic changes in the glucose metabolism and the tca cycle in liver have been found to be related to diabetes progression
<cit> .
while an understanding of single molecules continues to be important, the focus is on understanding the whole metabolic network at a systems-level.
because a metabolic system is not just an assembly of metabolites, its properties cannot be fully understood merely by studying the single molecules
<cit> .
with the accumulation of the relevant data, it becomes feasible to study metabolic systems in a genome-scale.
a human metabolic model has been reconstructed based on genomic and bibliomic data
<cit> .
the reconstructed genome-scale human metabolic model has been used to study human physiology and pathology
<cit> .
based on the human metabolic model
<cit>  and a variety of different tissue-specific data, a human liver metabolic model has been derived
<cit> .
for the method for the reconstruction and analysis of metabolic models, flux balance analysis  is a mathematical approach for analyzing the flow of metabolites through a metabolic network
<cit> , which is widely used in predicting the rate of production of a bio-technologically important metabolite.
when using fba, the constructed models must satisfy the following requirements: models without gaps, electron balanced, mass balanced, etc.
while for metabolic models created by some algorithms, such as init
<cit> , they may not satisfy all the requirements.
even if the dissatisfaction exist, studies on these models could uncover novel valuable information on metabolic systems based on network analysis
<cit> .
thus, it is rewarding to study the metabolic systems from the perspective of networks.
network science is an emerging field concerned with the study of complex systems represented as networks
<cit> , which has become a powerful conceptual paradigm in the field of biology to understand biological systems at a systems-level
<cit> .
in network science, how to control a system is a central issue
<cit> .
due to the unknown architecture of a system and the dynamical rules that capture the interactions between the components, it is difficult to control the complex system
<cit> .
by fixing the weights of interactions between the components to be either  <dig> or free parameters, the structural controllability was defined and studied to show some connections between the control theory and network
<cit> .
liu et al.
have used the theory of structural controllability to many models of real networks
<cit> , and proved that by giving control signals to a minimum set of nodes , the whole network can be guided to any desired final state in finite time.
recently, structural controllability analysis has been applied to some biological networks, where interesting properties on the biological system and drug-targets have been discovered
<cit> .
it is fair to expect that there are some possible connections between the structural controllability theory and the human liver metabolic network, which could provide valuable information on the human liver metabolism, such as the discovery of essential metabolites.
abnormal states of the human liver metabolic network could lead to different metabolic diseases, such as diabetes
<cit> , obesity
<cit>  and cancers
<cit> .
sometimes, these abnormal states can be steered into normal states by different appropriate inputs: drugs, signals from environment or inside the organism, the injection of specific metabolites.
theses control inputs could lead to the changes in metabolic states  which influence the metabolic functions.
for example, the drug raltitrexed can be used in cancer chemotherapy by targeting at the metabolite thymidylate synthase
<cit> ; the injection of potassium can make the body functioning normal when the body suffers from the metabolic disorder of hypokalemia.
if an organism suffers from metabolic disorders and the metabolic network cannot be controlled with any control inputs , then the organism may develop cancer or apoptosis.
researches on the controllability of the human liver metabolic network could provide the basis for ultimately understanding liver disease mechanisms, facilitating the development of therapeutics optimized for efficacy.
in this work, we applied the structural controllability analysis to the hlmn, detecting the metabolites and reactions that play important roles in the controllability of the hlmn.
we identified driver metabolites in the hlmn, and classified the metabolites into three classes: critical, high-frequency and low-frequency driver metabolites.
among the  <dig> critical driver metabolites,  <dig> metabolites are essential, which suggests that the critical driver metabolites play important roles in the human liver metabolism.
we find that the high-frequency driver metabolites tend to participate in different metabolic pathways, which are important in regulating the whole metabolic systems.
the critical and high-frequency driver metabolites may be potential drug-targets.
moreover, we explored the other possible connections between the structural controllability theory and the hlmn.
for example, by analyzing the roles of different links of the hlmn in the robustness of controllability, we find that transport reactions and the environment are important in the human liver metabolism.
the results in this work show some connections between the structural controllability analysis and the human liver metabolism, which uncover valuable information on the human liver metabolism from a fresh perspective.
RESULTS
description of the human liver metabolic network
we used a human liver metabolic model represented by a set of metabolic reactions
<cit> , which contains  <dig> metabolites and  <dig> reactions.
the human metabolic model was generated based on mba algorithm
<cit> , which is a model-building algorithm used to derive tissue-specific metabolic models from a generic model
<cit>  by integrating a variety of tissue-specific molecular data sources, including literature-based knowledge, transcriptomic, proteomic, metabolomic and phenotypic data.
in the human liver metabolic model, each metabolite is represented in the form of a, where a is the name of a metabolite and x in the bracket  is the abbreviation of the cell compartment where the metabolite a appears .
metabolite a may appear in different cell compartments x,…,y, so there are a,…,a for the same metabolite but different cell compartments, which are counted as different metabolites.
based on the principle that a set of metabolic reactions can be translated into a network representation
<cit> , we reformulated the liver model in the following way: denoting each metabolite by a node labeled with a, and connecting two nodes by a → b if there is a chemical reaction where a is a substrate and b is a product.
the derived hlmn contains  <dig> nodes and  <dig> links .
in order to illustrate the process of reformulating the hlmn, an example with three metabolic reactions is given in figure
<dig>
for convenience and without ambiguity, we will not distinguish nodes from metabolites hereinafter when refer to the properties of the hlmn.
for example, when we say a driver node in the hlmn, we may mean a driver metabolite in the hlmn.
classification and analysis of driver metabolites
driver metabolites in the hlmn are metabolites where inputs are injected.
if the driver metabolites in a minimum driver metabolites set  are all controlled by different inputs, the hlmn can be steered from any given state to a desired state in finite time.
"minimum" means that if signals are only input on a proper subset of s, then the hlmn cannot be guided to some final desired states in finite time.
mdmss are determined by detecting maximum matchings in the hlmn .
a maximum matching is a maximum set of links that do not share start or end nodes
<cit> .
there are different maximum matchings in a network
<cit> , which could result in different mdmss in the hlmn.
counting the number of all maximum matchings in an arbitrary network has been proven to belong to the ♯p-complete  class of problems
<cit> .
there is no currently known polynomial-time algorithm for solving a ♯p-complete problem.
the number of maximum matchings can grow exponentially with networks size, hence a network with only hundreds of nodes often leads to millions of maximum matchings.
enumeration of maximum matchings is computationally prohibitive for large networks
<cit> .
thus, the enumeration of maximum matchings in the hlmn  is hard to achieve.
classification of driver metabolites
we randomly identified  <dig> different maximum matchings  and their corresponding mdmss .
in the hlmn, a node may appear in different mdmss.
for each node v, we counted the number of mdmss that the node v appears in and then normalized the number .
the normalized values characterize the frequency f
d
of each node appearing in the  <dig> mdmss.
according to the frequency of each node, we classified the metabolites into three groups: critical driver metabolites with f
d
=  <dig>  high-frequency driver metabolites with  <dig>  ≤ f
d
<  <dig>  low-frequency driver metabolites with 0 ≤ f
d
<  <dig> .
a node with f
d
=  <dig> means that the node appears in all the mdmss.
such nodes may possess some specific properties or functions, which could provide valuable information on the hlmn.
so we classified the nodes with f
d
=  <dig> being critical driver nodes.
the reason why we chose the threshold  <dig>  to separate high-frequency driver metabolites from low-frequency driver metabolites, is that we would like to make the difference between the roles of metabolites in these two groups as big as possible .
in order to test whether the classification of metabolites based on  <dig> mdmss is reliable, we computed the frequencies of metabolites in  <dig> different families of mdmss with sizes of  <dig> , <dig> …, <dig>  the frequency of each metabolite computed based on different families of mdmss stays in a same region, where the regions are f
d
=  <dig>   <dig>  ≤ f
d
<  <dig> and 0 ≤ f
d
<  <dig>  .
in other words, the classifications of each metabolite are the same based on these different families.
hence the classification of metabolites based on  <dig> mdmss is reliable.
furthermore, we have employed an unbiased random sampling method
<cit>  to validate the results based on the  <dig> mdmss .
topological analysis of driver metabolites in the hlmn
we computed different centralities of each metabolite i in the hlmn, which include out-degree outd, in-degree ind, degree d, betweenness bc, closeness cc, in-closeness cci and out-closeness cco .
the frequency f
d
was found to decrease quickly with the in-degree  while this pattern does not hold for other centralities, which is consistent with the result in
<cit> .
for each centrality, all metabolites in the hlmn are divided into three sets of similar sizes, based on their centrality scores , in this way, seven families of sets were obtained:
fd={dl,dm,dh},
foutd={outdl,outdm,outdh},
find={indl,indm,indh},
fbc={bcl,bcm,bch},
fcc={ccl,ccm,cch},
fcci={ccil,ccim,ccih},
fcco={ccol,ccom,ccoh}, where each family contains three sets, and the subscripts l,m,h respectively represent low, medium and high.
we used set a to denote the union of metabolites from the  <dig> mdmss, and set b to denote the union set of the critical and high-frequency driver metabolites.
for each of the families
fd,foutd,find,fbc,fcc,fcci,fcco, the fractions of metabolites from set a that belong to the three sets in the family were computed ), and the fractions of metabolites from set b that belong to the three sets in the family were also computed ).
for example, for the family
fd, we computed |a ∩ d
l
|/|a|, |a ∩ d
m
|/|a|, |a ∩ d
h
|/|a|, and |b ∩ d
l
|/|b|, |b ∩ d
m
|/|b|, |b ∩ d
h
|/|b|, where |∗| denotes the size of set ∗.
comparing the results shown in figure
<dig> and figure
<dig>  we find that for each centrality, the difference between the fractions computed in set b is greater than that in set a, which means that the topological characteristic differences are bigger in the set of critical and high-frequency driver metabolites.
in-degree and in-closeness measure the susceptibility of a metabolite to be influenced by other metabolites.
higher in-degree and higher in-closeness imply that the metabolite could be more easily influenced by others.
out-closeness measures the ability of a metabolite to influence other metabolites.
higher out-closeness implies that the metabolite could influence others more easily.
the metabolites in set b tend to have low in-degree, low in-closeness, and high out-closeness.
therefore, the driver metabolites, especially the critical and high-frequency driver metabolites, tend to have strong ability to influence the states of other metabolites and weak susceptibility to be influenced by the states of other metabolites.
moreover, injecting control inputs  to critical and high-frequency driver metabolites could regulate the whole state of the hlmn, which indicates that the critical and high-frequency driver metabolites may be potential drug-targets.
for each centrality, we used chi-square test  to establish whether or not the fraction distribution in set a and set b differs from that in the whole network .
the chi-square statistic values for each centrality in set a and set b are shown in table
<dig>  while the table value for chi-square statistic is  <dig> , based on the freedom being  <dig> and the level of significance being  <dig> .
except for the cco, other chi-square statistic values are greater than the table value in set a, and the chi-square statistic values for all the centralities are greater than the table value in set b. it means that except for the cco in set a, for other centralities in set a and all the centralities in set b, the fraction distributions differ from that in the whole network.
thus, the result of the topological features of driver metabolites is of statistical significance.
d
ind
outd
bc
cc
cci
cco
properties of the critical driver metabolites
in the hlmn, we detected  <dig> critical driver metabolites .
their in-degrees are all zero, which is consistent with the result in
<cit>  and means that the  <dig> critical driver metabolites are all the start metabolites of paths .
by lin’s structural controllability theorem
<cit> , if a system is controllable, there is no inaccessible nodes .
since these start metabolites cannot be influenced by the external inputs via other metabolites, they need to be directly controlled by external inputs.
the compartment abbreviation of all critical driver metabolites are "", which means that their locations are all extracellular.
the in-degree of each critical driver metabolite is zero, which implies that each critical driver metabolite is the start metabolite of a metabolic pathway.
the categories "um", "fem", "eum" respectively represent the universal metabolites, the functional essential metabolites and the metabolites whose essentiality have not been discovered.
the  <dig> critical driver metabolites are all found to be extracellular .
by checking the biochemistry activities of the  <dig> critical driver metabolites, we find that they all participate in the transport reactions from the extracellular into the cell, which suggests that the intakes of these extracellular metabolites play important roles in the biological activities of the liver cells.
for example, appropriately increasing the intake of the critical driver metabolite gamma-tocopherol could help lower the cholesterol level, and increasing the intake of the critical driver metabolite alpha-tocopherol could decrease lipid peroxidation and hepatic stellate cells activation, which could protect liver cells and prevent liver fibrosis
<cit> .
we investigated the biological essentiality of the  <dig> critical driver metabolites.
the essentiality of a metabolite measures how important the metabolite is in the whole metabolic systems or some metabolic processes.
although a metabolite could exist in different compartments, the metabolite is recognized to be essential as long as it is found to be essential in any one of the compartments
<cit> .
based on the different essentiality of metabolites, the metabolites were classified into three groups:
• universal metabolites : some inorganic or cofactor metabolites, such as cmp and atp, which have been found to exist universally in more than 90% organisms.
the universal metabolites are usually treated as essential metabolites because most living matter cannot survive without them
<cit> .
• functional essential metabolites : the metabolites which are not um and have essential roles in some biological functions.
for example, folate is essential to numerous bodily functions, and required by the human body to synthesize, repair and methylate dna as well as to act as a cofactor in certain biological reactions
<cit> ; hyaluronan is essential for embryogenesis
<cit> ; human body requires pantothenic acid to synthesize coenzyme-a , as well as to synthesize and metabolize proteins, carbohydrates, and fats
<cit> .
• essentiality undiscovered metabolites : the metabolites whose essentiality have not been discovered.
these metabolites may be the potential essential metabolites, which demands further experimental verification.
among the  <dig> critical driver metabolites, we find that  <dig> metabolites are um;  <dig> metabolites are fem;  <dig> metabolites are eum.
therefore, among the  <dig> critical driver metabolites,  <dig> metabolites are essential, which suggests that the critical driver metabolites play important roles in human liver metabolism.
the roles of the high-frequency driver metabolites
we used simulated annealing  algorithm
<cit>  to detect modules in the hlmn.
the reason why we chose the sa algorithm is that it is a commonly used technique to detect modules, and a benchmark to validate the effectiveness of the newly developed module-detecting algorithms
<cit> .
compared with other module-detecting algorithms, such as the markov clustering method, the sa algorithm performs better in detecting modules in large scale metabolic networks and the detected modules are more biologically meaningful
<cit> , since the sa algorithm is less sensitive to noise such as experimental error or incomplete data.
according to the two parameters within-degree and the partition coefficient of each node in the modularized hlmn, the nodes were divided into seven classes: r <dig>  r <dig>  r <dig>  r <dig>  r <dig>  r <dig>  r <dig> .
since the sa algorithm is stochastic, different results of modularization could be obtained in different runs.
we have run the sa algorithm for  <dig> times.
based on the result of each run, the nodes of the hlmn were classified into the seven classes r <dig>  r <dig>  r <dig>  r <dig>  r <dig>  r <dig>  r <dig>  among the  <dig> classification results, the probability of each node being classified into each class is counted.
as shown in figure
<dig>  most nodes are always classified into a same class, which indicates that the role classification for the nodes in the hlmn based on the sa algorithm is reliable.
it has been found that the non-hubs connecting different modules are responsible for inter-module fluxes which influence the state of metabolic networks
<cit> , while the nodes with high frequency f
d
have strong ability to influence the states of other metabolites, which prompts us to think whether the nodes with high frequency f
d
tend to be non-hubs connecting different modules.
in the hlmn, more than 92% nodes are of roles r <dig> and r <dig>  which are both non-hubs and r <dig> nodes have no connection with other modules while r <dig> nodes have connections with different modules.
as shown in figure
<dig>  with the frequency threshold f
dt
increasing, the fraction of r <dig> nodes among the set of nodes with f
dt
≤ f
d
<  <dig> decreases while the fraction of r <dig> nodes increases.
the fractions of nodes with different roles fluctuate when f
dt
≥  <dig>  due to the small size of the set of nodes with f
dt
≤ f
d
<  <dig>  when f
dt
<  <dig> , the difference between the fractions of r <dig> nodes and r <dig> nodes is the biggest at around f
dt
=  <dig> .
therefore, we chose the threshold f
dt
=  <dig>  to differentiate the high-frequency driver metabolites from the low-frequency driver metabolites.
the fact that the roles of high-frequency driver metabolites tend to be r <dig>  indicates that the high-frequency driver metabolites tend to be non-hubs connecting different modules.
different modules could be mapped to different pathways
<cit> , which means that the high-frequency driver metabolites tend to participate in different metabolic pathways.
for example, the high-frequency driver metabolite cyclic adenosine monophosphate plays regulatory roles in glucose, protein and fatty metabolism pathways at the same time
<cit> .
it suggests that the high-frequency driver metabolites play important roles in human liver metabolic network.
to validate that the result of the high-frequency driver metabolites does not depend on the module detecting method sa algorithm, we used another module detecting method fast greedy
<cit>  to detect modules in the hlmn, and classify the nodes into  <dig> classes: r <dig>  r <dig>  r <dig>  r <dig>  r <dig>  r <dig>  r <dig>  with the frequency threshold f
dt
increasing, the fractions of r <dig> nodes and r <dig> nodes among the set of nodes with f
dt
≤ f
d
<  <dig> show the similar pattern as that based on the sa algorithm, which is shown in figure
<dig>  we arrived at the same conclusion that the high-frequency driver metabolites tend to be the non-hub connecting different modules.
validation for the classification and the properties of driver metabolites
the results of the properties on the driver metabolites, critical driver metabolites and high-frequency driver metabolites are all based on the  <dig> mdmss.
to validate that these results do not depend on the  <dig> mdmss, we applied an unbiased sampling method proposed by jia et al.
<cit>  to compute the frequency f
d
that each node acts as a driver node .
comparing with the results which based on the  <dig> mdmss, the set of critical driver nodes determined by this method is the same, while the set of high-frequency driver nodes determined by this method is not exactly the same, which may be caused by the randomness of sampling.
however, the following result holds for both two methods: the high-frequency driver nodes tend to be the non-hubs connecting different modules.
.
moreover, the topological analysis has been applied to the set a  and set b  detected by the method in
<cit> .
the conclusion still holds that the driver metabolites, especially the critical and high-frequency driver metabolites, tend to have strong ability to influence the states of other metabolites and weak susceptibility to be influenced by the states of other metabolites .
in conclusion, although the classification and analysis of driver metabolites are based on the  <dig> mdmss, the results on the properties of different driver metabolites do not rely on the  <dig> mdmss.
alternative classification of driver nodes and the control mode of the hlmn
a recently published paper
<cit>  has given an alternative classification of nodes based on their participation in control.
a node is critical, itermittent or redundant if it acts as a driver node in all, some or none of the minimum sets of driver nodes.
by measuring the fraction n
r
of the redundant nodes for a network with varying average degree, two distinct control modes were discovered in
<cit> .
based on the difference value of the fraction n
r
and
nrt for its transpose network , the control mode of a network can be identified: if
Δnr=nr-nrt> <dig> the network is centralized and if Δn
r
<  <dig> it is distributed.
we have applied the tools in
<cit>  to the hlmn, and find that the control mode of the hlmn is distributed.
while in
<cit> , the control modes of the three involved metabolic networks cannot be identified, which is caused by the incompleteness of the metabolic networks, whose average degrees are in the ‘pre-bifurcation’ region .
with more information on these metabolic networks being uncovered, the average degrees increase and result in identifiable control modes.
for example, the e. coli metabolic network
<cit>  studied in
<cit>  was assembled in  <dig>  and its control mode cannot be identified; however, when we applied the tools to the e. coli metabolic network ijo1366
<cit> , which was assemble in  <dig>  we can find that the control mode of network ijo <dig> is centralized.
it is not easy to figure out the reason why the control mode of the human liver metabolic network is distributed and the e. coli metabolic network ijo <dig> is centralized, due to the incompleteness of these two networks, whose control mode may alter with the increase of the network scale.
the role of reactions in the robustness of the controllability in the hlmn
reaction failures could happen in metabolic systems, and different reaction failures have different impacts on the robustness of the metabolic function.
robustness characters the ability of metabolic systems to behavior normally under reaction failures.
some reaction failures would break the cellular homeostasis, resulting in an anti-proliferative effect
<cit>  or apoptosis
<cit> , while some almostly have no influence on the cellular functions
<cit> .
in what follows, we focus on the impacts of different reaction failures on the robustness  of the controllability in the hlmn.
based on different impacts on the robustness of controllability caused by links absence, the links have been classified into three categories
<cit> : "critical" if its absence causes the minimum number of driver nodes increased so as to maintain full control; "redundant" if it can be removed without affecting the current set of driver nodes; "ordinary" if it is neither critical nor redundant.
from the fractions of critical, ordinary and redundant links in the hlmn, which are shown in figure
<dig>  we can find that few links are critical and most links are ordinary, whose absence may change the current set of driver nodes, but the network could still be controlled with the same number of driver nodes.
in the human liver metabolism, there are only a few reactions represented by critical links, which provides an explanation to why human liver metabolism could function well under many different circumstances.
in the human liver metabolic model
<cit> , the reactions have been classified into three classes: core high reactions for these reactions included in human-curated tissue-specific pathways, which are essential in the human liver metabolism; core moderate reactions for these reactions testified by molecule data; non-core reactions for the other, most of which are not associated with genes in the model and 50% are transport reactions.
we computed the fractions of links representing the core high, core moderate and non-core reactions among the set of critical, ordinary and redundant links and the set of all links in the hlmn, which are shown in figure
<dig>  comparing with the fractions among the sets of the ordinary, redundant links and the whole link set in the hlmn, the fraction of links representing core high reactions are the lowest and the fraction of non-core reaction links are the highest in the set of critical links, which indicates that the reactions represented by critical links tend to be the non-core reactions.
transport reactions transfer metabolites across compartments, many of them transfer metabolites from the environment into the cell.
the fraction of transport reaction links among the set of critical links is  <dig> %, while that among the whole link set in the hlmn is  <dig> %.
moreover, we computed the fraction of links representing transport reactions which transfer metabolites from the environment into the cell among the set of critical links and that among the whole link set in the hlmn, which are  <dig> % and  <dig> %, respectively.
these comparisons indicate that transport reactions and the environment are important in influencing the robustness of controllability of the hlmn.
the metabolites carried in by transport reactions could activate a series metabolic reactions in human liver cells, which could change the state of the liver metabolism and influence the controllability of the hlmn.
validation for the result that the reactions represented by critical links tend to be the non-core reactions
we used chi-square test  to test whether or not the differences between the fractions of the core high, core moderate and non-core reaction links among the whole network and those among the set of critical links are out of chance.
the observed data are the number of core high, core moderate and non-core reaction links among each of the sets of the critical links, which are  <dig>   <dig> and  <dig> respectively.
the expected percentages are the fractions of core-high, core moderate and non-core reaction links among the whole network, which are  <dig> % and  <dig> % and  <dig> % respectively.
the chi-square statistic value was computed based on the chi-square formula , which is  <dig> .
with the freedom degree being  <dig> and the significance level being  <dig> , the table value for chi-square statistic is  <dig> .
the chi-square statistic value is bigger than the table value, so there is a significant difference between the fractions among the set of critical links and those among the whole network, which means that the reactions represented by critical links tend to be the non-core reactions.
CONCLUSIONS
in this study, we have detected the driver metabolites in the hlmn and classified the metabolites into three classes: critical, high-frequency and low-frequency driver metabolites.
among the  <dig> critical driver metabolites,  <dig> metabolites are essential, which suggests that the critical driver metabolites play important roles in the human liver metabolism.
moreover, the compartments where the critical driver metabolites appear are all extracellular.
it is consistent with our knowledge that the substances imported from the environment play important roles in steering the behavior of the whole metabolic network.
the liver metabolic system could be regulated by controlling the intakes of the critical driver metabolites.
for example, the increase of the intake of the critical driver metabolite alpha-tocopherol could decrease lipid peroxidation and hepatic stellate cells activation, so as to protect liver cells and prevent liver fibrosis
<cit> .
we find that the high-frequency driver metabolites tend to participate in different metabolic pathways, which are important in regulating the whole metabolic systems.
for example, the high-frequency driver metabolite cyclic adenosine monophosphate, which acts as a second messenger in many biological processes, plays important regulatory roles in glucose, protein and fatty metabolism pathways at the same time
<cit> .
in addition, the states of the critical and high-frequency driver metabolites have strong ability in steering the state of the whole hlmn, indicating that the critical and high-frequency driver metabolites may be potential drug-targets.
by analyzing the roles of different links in the robustness of controllability, we find that transport reactions and the environment are important in the robustness of controllability in the hlmn under reactions failures.
the metabolites carried in by transport reactions could activate a series metabolic reactions in human liver cells, leading to changes in the state of liver metabolism.
moreover, we have explored some other possible connections between the structural controllability theory and the hlmn.
based on the structural controllability theory, two key concepts control centrality
<cit>  and control mode
<cit>  have been proposed.
the control centrality of a node measures the number of nodes that can be independently controlled by controlling this node alone.
we attempted to reveal the possible connections between the control centrality and the actual importance of a metabolite in the hlmn, but we find there is no such connection .
we also applied the tools in
<cit>  to give an alternative node classification in the hlmn based on node’s participation in control, and find that the control mode of the hlmn is distributed.
it is not easy to figure out the reason why the hlmn is distributed, due to the incompleteness of this network, whose control mode may alter with the increase of the network scale.
in summary, we find that the driver metabolites have essential biological functions, and the metabolites connecting different pathways play crucial roles in the controllability of the hlmn.
the crucial role of extracellular metabolites and the transport reactions highlight the importance of the environment in the health of human liver metabolism.
the work presented here raises a number of questions.
for example, what properties do the low-frequency driver metabolites have?
how can we quantify the influence of each driver metabolite on the state of hlmn?
answers to these questions could further provide theoretical foundation for designing experiments of regulating the human liver metabolism.

 metabolism is one of the most complex cellular processes and a basal system for maintaining life of all organisms. liver metabolism disorders could cause a wide range of diseases, ranging from hepatitis to hepatic tumor. many studies which focus on the roles of single molecule substances or single paths in regulating liver metabolism have been carried out. for example, the interleukin receptor-associated kinase-m has been found to negatively regulate the innate and the adaptive immune response in the liver reacting to acute insult by alcohol; the liver x receptors could bind to cholesterol metabolites and regulate the cholesterol turnover; the metabolic changes in the glucose metabolism and the tca cycle in liver have been found to be related to diabetes progression. while an understanding of single molecules continues to be important, the focus is on understanding the whole metabolic network at a systems-level. because a metabolic system is not just an assembly of metabolites, its properties cannot be fully understood merely by studying the single molecules. with the accumulation of the relevant data, it becomes feasible to study metabolic systems in a genome-scale. a human metabolic model has been reconstructed based on genomic and bibliomic data. the reconstructed genome-scale human metabolic model has been used to study human physiology and pathology. based on the human metabolic model and a variety of different tissue-specific data, a human liver metabolic model has been derived. for the method for the reconstruction and analysis of metabolic models, flux balance analysis is a mathematical approach for analyzing the flow of metabolites through a metabolic network, which is widely used in predicting the rate of production of a bio-technologically important metabolite. when using fba, the constructed models must satisfy the following requirements: models without gaps, electron balanced, mass balanced, etc. while for metabolic models created by some algorithms, such as init, they may not satisfy all the requirements. even if the dissatisfaction exist, studies on these models could uncover novel valuable information on metabolic systems based on network analysis. thus, it is rewarding to study the metabolic systems from the perspective of networks. network science is an emerging field concerned with the study of complex systems represented as networks, which has become a powerful conceptual paradigm in the field of biology to understand biological systems at a systems-level. in network science, how to control a system is a central issue. due to the unknown architecture of a system and the dynamical rules that capture the interactions between the components, it is difficult to control the complex system. by fixing the weights of interactions between the components to be either or free parameters, the structural controllability was defined and studied to show some connections between the control theory and network. liu et al. have used the theory of structural controllability to many models of real networks, and proved that by giving control signals to a minimum set of nodes, the whole network can be guided to any desired final state in finite time. recently, structural controllability analysis has been applied to some biological networks, where interesting properties on the biological system and drug-targets have been discovered. it is fair to expect that there are some possible connections between the structural controllability theory and the human liver metabolic network, which could provide valuable information on the human liver metabolism, such as the discovery of essential metabolites. abnormal states of the human liver metabolic network could lead to different metabolic diseases, such as diabetes, obesity and cancers. sometimes, these abnormal states can be steered into normal states by different appropriate inputs: drugs, signals from environment or inside the organism, the injection of specific metabolites. theses control inputs could lead to the changes in metabolic states which influence the metabolic functions. for example, the drug raltitrexed can be used in cancer chemotherapy by targeting at the metabolite thymidylate synthase; the injection of potassium can make the body functioning normal when the body suffers from the metabolic disorder of hypokalemia. if an organism suffers from metabolic disorders and the metabolic network cannot be controlled with any control inputs, then the organism may develop cancer or apoptosis. researches on the controllability of the human liver metabolic network could provide the basis for ultimately understanding liver disease mechanisms, facilitating the development of therapeutics optimized for efficacy. in this work, we applied the structural controllability analysis to the hlmn, detecting the metabolites and reactions that play important roles in the controllability of the hlmn. we identified driver metabolites in the hlmn, and classified the metabolites into three classes: critical, high-frequency and low-frequency driver metabolites. among the critical driver metabolites, metabolites are essential, which suggests that the critical driver metabolites play important roles in the human liver metabolism. we find that the high-frequency driver metabolites tend to participate in different metabolic pathways, which are important in regulating the whole metabolic systems. the critical and high-frequency driver metabolites may be potential drug-targets. moreover, we explored the other possible connections between the structural controllability theory and the hlmn. for example, by analyzing the roles of different links of the hlmn in the robustness of controllability, we find that transport reactions and the environment are important in the human liver metabolism. the results in this work show some connections between the structural controllability analysis and the human liver metabolism, which uncover valuable information on the human liver metabolism from a fresh perspective.  description of the human liver metabolic network we used a human liver metabolic model represented by a set of metabolic reactions, which contains metabolites and reactions. the human metabolic model was generated based on mba algorithm, which is a model-building algorithm used to derive tissue-specific metabolic models from a generic model by integrating a variety of tissue-specific molecular data sources, including literature-based knowledge, transcriptomic, proteomic, metabolomic and phenotypic data. in the human liver metabolic model, each metabolite is represented in the form of a, where a is the name of a metabolite and x in the bracket is the abbreviation of the cell compartment where the metabolite a appears. metabolite a may appear in different cell compartments x,…,y, so there are a,…,a for the same metabolite but different cell compartments, which are counted as different metabolites. based on the principle that a set of metabolic reactions can be translated into a network representation, we reformulated the liver model in the following way: denoting each metabolite by a node labeled with a, and connecting two nodes by a → b if there is a chemical reaction where a is a substrate and b is a product. the derived hlmn contains nodes and links. in order to illustrate the process of reformulating the hlmn, an example with three metabolic reactions is given in figure for convenience and without ambiguity, we will not distinguish nodes from metabolites hereinafter when refer to the properties of the hlmn. for example, when we say a driver node in the hlmn, we may mean a driver metabolite in the hlmn. classification and analysis of driver metabolites driver metabolites in the hlmn are metabolites where inputs are injected. if the driver metabolites in a minimum driver metabolites set are all controlled by different inputs, the hlmn can be steered from any given state to a desired state in finite time. "minimum" means that if signals are only input on a proper subset of s, then the hlmn cannot be guided to some final desired states in finite time. mdmss are determined by detecting maximum matchings in the hlmn. a maximum matching is a maximum set of links that do not share start or end nodes. there are different maximum matchings in a network, which could result in different mdmss in the hlmn. counting the number of all maximum matchings in an arbitrary network has been proven to belong to the ♯p-complete class of problems. there is no currently known polynomial-time algorithm for solving a ♯p-complete problem. the number of maximum matchings can grow exponentially with networks size, hence a network with only hundreds of nodes often leads to millions of maximum matchings. enumeration of maximum matchings is computationally prohibitive for large networks. thus, the enumeration of maximum matchings in the hlmn is hard to achieve. classification of driver metabolites we randomly identified different maximum matchings and their corresponding mdmss. in the hlmn, a node may appear in different mdmss. for each node v, we counted the number of mdmss that the node v appears in and then normalized the number. the normalized values characterize the frequency f d of each node appearing in the mdmss. according to the frequency of each node, we classified the metabolites into three groups: critical driver metabolites with f d  =  high-frequency driver metabolites with  ≤ f d  <  low-frequency driver metabolites with 0 ≤ f d  < . a node with f d  =  means that the node appears in all the mdmss. such nodes may possess some specific properties or functions, which could provide valuable information on the hlmn. so we classified the nodes with f d  =  being critical driver nodes. the reason why we chose the threshold to separate high-frequency driver metabolites from low-frequency driver metabolites, is that we would like to make the difference between the roles of metabolites in these two groups as big as possible. in order to test whether the classification of metabolites based on mdmss is reliable, we computed the frequencies of metabolites in different families of mdmss with sizes of, …, the frequency of each metabolite computed based on different families of mdmss stays in a same region, where the regions are f d  =   ≤ f d  <  and 0 ≤ f d  < . in other words, the classifications of each metabolite are the same based on these different families. hence the classification of metabolites based on mdmss is reliable. furthermore, we have employed an unbiased random sampling method to validate the results based on the mdmss. topological analysis of driver metabolites in the hlmn we computed different centralities of each metabolite i in the hlmn, which include out-degree outd, in-degree ind, degree d, betweenness bc, closeness cc, in-closeness cci and out-closeness cco. the frequency f d was found to decrease quickly with the in-degree while this pattern does not hold for other centralities, which is consistent with the result in. for each centrality, all metabolites in the hlmn are divided into three sets of similar sizes, based on their centrality scores, in this way, seven families of sets were obtained: fd={dl,dm,dh}, foutd={outdl,outdm,outdh}, find={indl,indm,indh}, fbc={bcl,bcm,bch}, fcc={ccl,ccm,cch}, fcci={ccil,ccim,ccih}, fcco={ccol,ccom,ccoh}, where each family contains three sets, and the subscripts l,m,h respectively represent low, medium and high. we used set a to denote the union of metabolites from the mdmss, and set b to denote the union set of the critical and high-frequency driver metabolites. for each of the families fd,foutd,find,fbc,fcc,fcci,fcco, the fractions of metabolites from set a that belong to the three sets in the family were computed ), and the fractions of metabolites from set b that belong to the three sets in the family were also computed ). for example, for the family fd, we computed |a ∩ d l |/|a|, |a ∩ d m |/|a|, |a ∩ d h |/|a|, and |b ∩ d l |/|b|, |b ∩ d m |/|b|, |b ∩ d h |/|b|, where |∗| denotes the size of set ∗. comparing the results shown in figure and figure we find that for each centrality, the difference between the fractions computed in set b is greater than that in set a, which means that the topological characteristic differences are bigger in the set of critical and high-frequency driver metabolites. in-degree and in-closeness measure the susceptibility of a metabolite to be influenced by other metabolites. higher in-degree and higher in-closeness imply that the metabolite could be more easily influenced by others. out-closeness measures the ability of a metabolite to influence other metabolites. higher out-closeness implies that the metabolite could influence others more easily. the metabolites in set b tend to have low in-degree, low in-closeness, and high out-closeness. therefore, the driver metabolites, especially the critical and high-frequency driver metabolites, tend to have strong ability to influence the states of other metabolites and weak susceptibility to be influenced by the states of other metabolites. moreover, injecting control inputs to critical and high-frequency driver metabolites could regulate the whole state of the hlmn, which indicates that the critical and high-frequency driver metabolites may be potential drug-targets. for each centrality, we used chi-square test to establish whether or not the fraction distribution in set a and set b differs from that in the whole network. the chi-square statistic values for each centrality in set a and set b are shown in table while the table value for chi-square statistic is, based on the freedom being and the level of significance being. except for the cco, other chi-square statistic values are greater than the table value in set a, and the chi-square statistic values for all the centralities are greater than the table value in set b. it means that except for the cco in set a, for other centralities in set a and all the centralities in set b, the fraction distributions differ from that in the whole network. thus, the result of the topological features of driver metabolites is of statistical significance. d ind outd bc cc cci cco properties of the critical driver metabolites in the hlmn, we detected critical driver metabolites. their in-degrees are all zero, which is consistent with the result in and means that the critical driver metabolites are all the start metabolites of paths. by lin’s structural controllability theorem, if a system is controllable, there is no inaccessible nodes. since these start metabolites cannot be influenced by the external inputs via other metabolites, they need to be directly controlled by external inputs. the compartment abbreviation of all critical driver metabolites are "", which means that their locations are all extracellular. the in-degree of each critical driver metabolite is zero, which implies that each critical driver metabolite is the start metabolite of a metabolic pathway. the categories "um", "fem", "eum" respectively represent the universal metabolites, the functional essential metabolites and the metabolites whose essentiality have not been discovered. the critical driver metabolites are all found to be extracellular. by checking the biochemistry activities of the critical driver metabolites, we find that they all participate in the transport reactions from the extracellular into the cell, which suggests that the intakes of these extracellular metabolites play important roles in the biological activities of the liver cells. for example, appropriately increasing the intake of the critical driver metabolite gamma-tocopherol could help lower the cholesterol level, and increasing the intake of the critical driver metabolite alpha-tocopherol could decrease lipid peroxidation and hepatic stellate cells activation, which could protect liver cells and prevent liver fibrosis. we investigated the biological essentiality of the critical driver metabolites. the essentiality of a metabolite measures how important the metabolite is in the whole metabolic systems or some metabolic processes. although a metabolite could exist in different compartments, the metabolite is recognized to be essential as long as it is found to be essential in any one of the compartments. based on the different essentiality of metabolites, the metabolites were classified into three groups: • universal metabolites: some inorganic or cofactor metabolites, such as cmp and atp, which have been found to exist universally in more than 90% organisms. the universal metabolites are usually treated as essential metabolites because most living matter cannot survive without them. • functional essential metabolites: the metabolites which are not um and have essential roles in some biological functions. for example, folate is essential to numerous bodily functions, and required by the human body to synthesize, repair and methylate dna as well as to act as a cofactor in certain biological reactions; hyaluronan is essential for embryogenesis; human body requires pantothenic acid to synthesize coenzyme-a, as well as to synthesize and metabolize proteins, carbohydrates, and fats. • essentiality undiscovered metabolites: the metabolites whose essentiality have not been discovered. these metabolites may be the potential essential metabolites, which demands further experimental verification. among the critical driver metabolites, we find that metabolites are um; metabolites are fem; metabolites are eum. therefore, among the critical driver metabolites, metabolites are essential, which suggests that the critical driver metabolites play important roles in human liver metabolism. the roles of the high-frequency driver metabolites we used simulated annealing algorithm to detect modules in the hlmn. the reason why we chose the sa algorithm is that it is a commonly used technique to detect modules, and a benchmark to validate the effectiveness of the newly developed module-detecting algorithms. compared with other module-detecting algorithms, such as the markov clustering method, the sa algorithm performs better in detecting modules in large scale metabolic networks and the detected modules are more biologically meaningful, since the sa algorithm is less sensitive to noise such as experimental error or incomplete data. according to the two parameters within-degree and the partition coefficient of each node in the modularized hlmn, the nodes were divided into seven classes: r r r r r r r. since the sa algorithm is stochastic, different results of modularization could be obtained in different runs. we have run the sa algorithm for times. based on the result of each run, the nodes of the hlmn were classified into the seven classes r r r r r r r among the classification results, the probability of each node being classified into each class is counted. as shown in figure most nodes are always classified into a same class, which indicates that the role classification for the nodes in the hlmn based on the sa algorithm is reliable. it has been found that the non-hubs connecting different modules are responsible for inter-module fluxes which influence the state of metabolic networks, while the nodes with high frequency f d have strong ability to influence the states of other metabolites, which prompts us to think whether the nodes with high frequency f d tend to be non-hubs connecting different modules. in the hlmn, more than 92% nodes are of roles r and r which are both non-hubs and r nodes have no connection with other modules while r nodes have connections with different modules. as shown in figure with the frequency threshold f dt increasing, the fraction of r nodes among the set of nodes with f dt  ≤ f d  <  decreases while the fraction of r nodes increases. the fractions of nodes with different roles fluctuate when f dt  ≥  due to the small size of the set of nodes with f dt  ≤ f d  <  when f dt  < , the difference between the fractions of r nodes and r nodes is the biggest at around f dt  = . therefore, we chose the threshold f dt  =  to differentiate the high-frequency driver metabolites from the low-frequency driver metabolites. the fact that the roles of high-frequency driver metabolites tend to be r indicates that the high-frequency driver metabolites tend to be non-hubs connecting different modules. different modules could be mapped to different pathways, which means that the high-frequency driver metabolites tend to participate in different metabolic pathways. for example, the high-frequency driver metabolite cyclic adenosine monophosphate plays regulatory roles in glucose, protein and fatty metabolism pathways at the same time. it suggests that the high-frequency driver metabolites play important roles in human liver metabolic network. to validate that the result of the high-frequency driver metabolites does not depend on the module detecting method sa algorithm, we used another module detecting method fast greedy to detect modules in the hlmn, and classify the nodes into classes: r r r r r r r with the frequency threshold f dt increasing, the fractions of r nodes and r nodes among the set of nodes with f dt  ≤ f d  <  show the similar pattern as that based on the sa algorithm, which is shown in figure we arrived at the same conclusion that the high-frequency driver metabolites tend to be the non-hub connecting different modules. validation for the classification and the properties of driver metabolites the results of the properties on the driver metabolites, critical driver metabolites and high-frequency driver metabolites are all based on the mdmss. to validate that these results do not depend on the mdmss, we applied an unbiased sampling method proposed by jia et al. to compute the frequency f d that each node acts as a driver node. comparing with the results which based on the mdmss, the set of critical driver nodes determined by this method is the same, while the set of high-frequency driver nodes determined by this method is not exactly the same, which may be caused by the randomness of sampling. however, the following result holds for both two methods: the high-frequency driver nodes tend to be the non-hubs connecting different modules.. moreover, the topological analysis has been applied to the set a and set b detected by the method in. the conclusion still holds that the driver metabolites, especially the critical and high-frequency driver metabolites, tend to have strong ability to influence the states of other metabolites and weak susceptibility to be influenced by the states of other metabolites. in conclusion, although the classification and analysis of driver metabolites are based on the mdmss, the results on the properties of different driver metabolites do not rely on the mdmss. alternative classification of driver nodes and the control mode of the hlmn a recently published paper has given an alternative classification of nodes based on their participation in control. a node is critical, itermittent or redundant if it acts as a driver node in all, some or none of the minimum sets of driver nodes. by measuring the fraction n r of the redundant nodes for a network with varying average degree, two distinct control modes were discovered in. based on the difference value of the fraction n r and nrt for its transpose network, the control mode of a network can be identified: if Δnr=nr-nrt> the network is centralized and if Δn r  <  it is distributed. we have applied the tools in to the hlmn, and find that the control mode of the hlmn is distributed. while in, the control modes of the three involved metabolic networks cannot be identified, which is caused by the incompleteness of the metabolic networks, whose average degrees are in the ‘pre-bifurcation’ region. with more information on these metabolic networks being uncovered, the average degrees increase and result in identifiable control modes. for example, the e. coli metabolic network studied in was assembled in and its control mode cannot be identified; however, when we applied the tools to the e. coli metabolic network ijo1366, which was assemble in we can find that the control mode of network ijo is centralized. it is not easy to figure out the reason why the control mode of the human liver metabolic network is distributed and the e. coli metabolic network ijo is centralized, due to the incompleteness of these two networks, whose control mode may alter with the increase of the network scale. the role of reactions in the robustness of the controllability in the hlmn reaction failures could happen in metabolic systems, and different reaction failures have different impacts on the robustness of the metabolic function. robustness characters the ability of metabolic systems to behavior normally under reaction failures. some reaction failures would break the cellular homeostasis, resulting in an anti-proliferative effect or apoptosis, while some almostly have no influence on the cellular functions. in what follows, we focus on the impacts of different reaction failures on the robustness of the controllability in the hlmn. based on different impacts on the robustness of controllability caused by links absence, the links have been classified into three categories: "critical" if its absence causes the minimum number of driver nodes increased so as to maintain full control; "redundant" if it can be removed without affecting the current set of driver nodes; "ordinary" if it is neither critical nor redundant. from the fractions of critical, ordinary and redundant links in the hlmn, which are shown in figure we can find that few links are critical and most links are ordinary, whose absence may change the current set of driver nodes, but the network could still be controlled with the same number of driver nodes. in the human liver metabolism, there are only a few reactions represented by critical links, which provides an explanation to why human liver metabolism could function well under many different circumstances. in the human liver metabolic model, the reactions have been classified into three classes: core high reactions for these reactions included in human-curated tissue-specific pathways, which are essential in the human liver metabolism; core moderate reactions for these reactions testified by molecule data; non-core reactions for the other, most of which are not associated with genes in the model and 50% are transport reactions. we computed the fractions of links representing the core high, core moderate and non-core reactions among the set of critical, ordinary and redundant links and the set of all links in the hlmn, which are shown in figure comparing with the fractions among the sets of the ordinary, redundant links and the whole link set in the hlmn, the fraction of links representing core high reactions are the lowest and the fraction of non-core reaction links are the highest in the set of critical links, which indicates that the reactions represented by critical links tend to be the non-core reactions. transport reactions transfer metabolites across compartments, many of them transfer metabolites from the environment into the cell. the fraction of transport reaction links among the set of critical links is %, while that among the whole link set in the hlmn is %. moreover, we computed the fraction of links representing transport reactions which transfer metabolites from the environment into the cell among the set of critical links and that among the whole link set in the hlmn, which are % and %, respectively. these comparisons indicate that transport reactions and the environment are important in influencing the robustness of controllability of the hlmn. the metabolites carried in by transport reactions could activate a series metabolic reactions in human liver cells, which could change the state of the liver metabolism and influence the controllability of the hlmn. validation for the result that the reactions represented by critical links tend to be the non-core reactions we used chi-square test to test whether or not the differences between the fractions of the core high, core moderate and non-core reaction links among the whole network and those among the set of critical links are out of chance. the observed data are the number of core high, core moderate and non-core reaction links among each of the sets of the critical links, which are and respectively. the expected percentages are the fractions of core-high, core moderate and non-core reaction links among the whole network, which are % and % and % respectively. the chi-square statistic value was computed based on the chi-square formula, which is. with the freedom degree being and the significance level being, the table value for chi-square statistic is. the chi-square statistic value is bigger than the table value, so there is a significant difference between the fractions among the set of critical links and those among the whole network, which means that the reactions represented by critical links tend to be the non-core reactions.  in this study, we have detected the driver metabolites in the hlmn and classified the metabolites into three classes: critical, high-frequency and low-frequency driver metabolites. among the critical driver metabolites, metabolites are essential, which suggests that the critical driver metabolites play important roles in the human liver metabolism. moreover, the compartments where the critical driver metabolites appear are all extracellular. it is consistent with our knowledge that the substances imported from the environment play important roles in steering the behavior of the whole metabolic network. the liver metabolic system could be regulated by controlling the intakes of the critical driver metabolites. for example, the increase of the intake of the critical driver metabolite alpha-tocopherol could decrease lipid peroxidation and hepatic stellate cells activation, so as to protect liver cells and prevent liver fibrosis. we find that the high-frequency driver metabolites tend to participate in different metabolic pathways, which are important in regulating the whole metabolic systems. for example, the high-frequency driver metabolite cyclic adenosine monophosphate, which acts as a second messenger in many biological processes, plays important regulatory roles in glucose, protein and fatty metabolism pathways at the same time. in addition, the states of the critical and high-frequency driver metabolites have strong ability in steering the state of the whole hlmn, indicating that the critical and high-frequency driver metabolites may be potential drug-targets. by analyzing the roles of different links in the robustness of controllability, we find that transport reactions and the environment are important in the robustness of controllability in the hlmn under reactions failures. the metabolites carried in by transport reactions could activate a series metabolic reactions in human liver cells, leading to changes in the state of liver metabolism. moreover, we have explored some other possible connections between the structural controllability theory and the hlmn. based on the structural controllability theory, two key concepts control centrality and control mode have been proposed. the control centrality of a node measures the number of nodes that can be independently controlled by controlling this node alone. we attempted to reveal the possible connections between the control centrality and the actual importance of a metabolite in the hlmn, but we find there is no such connection. we also applied the tools in to give an alternative node classification in the hlmn based on node’s participation in control, and find that the control mode of the hlmn is distributed. it is not easy to figure out the reason why the hlmn is distributed, due to the incompleteness of this network, whose control mode may alter with the increase of the network scale. in summary, we find that the driver metabolites have essential biological functions, and the metabolites connecting different pathways play crucial roles in the controllability of the hlmn. the crucial role of extracellular metabolites and the transport reactions highlight the importance of the environment in the health of human liver metabolism. the work presented here raises a number of questions. for example, what properties do the low-frequency driver metabolites have? how can we quantify the influence of each driver metabolite on the state of hlmn? answers to these questions could further provide theoretical foundation for designing experiments of regulating the human liver metabolism. 
the accumulation in relevant data makes it feasible to derive a large-scale human liver metabolic network and to discover important biological principles or drug-targets based on network analysis. the exploration on the connections between structural controllability theory and the hlmn could be used to uncover valuable information on the human liver metabolism from a fresh perspective. moreover, we explored some other possible connections between the structural controllability theory and the hlmn, and find that transport reactions and the environment play important roles in the human liver metabolism. there are interesting connections between the structural controllability theory and the human liver metabolism: driver metabolites have essential biological functions; the crucial role of extracellular metabolites and transport reactions in controlling the hlmn highlights the importance of the environment in the health of human liver metabolism. some studies have shown that interesting biological phenomenon and drug-targets could be discovered by applying structural controllability analysis to biological networks. we applied structural controllability analysis to the hlmn and detected driver metabolites. abnormal states in human liver metabolism are major causes of human liver diseases ranging from hepatitis to hepatic tumor. in addition, the metabolites were classified into three classes: critical, high-frequency and low-frequency driver metabolites. the driver metabolites tend to have strong ability to influence the states of other metabolites and weak susceptibility to be influenced by the states of others. among the identified critical driver metabolites, metabolites were found to be essential; the high-frequency driver metabolites tend to participate in different metabolic pathways, which are important in regulating the whole metabolic systems.
 abnormal states in human liver metabolism are major causes of human liver diseases ranging from hepatitis to hepatic tumor. the accumulation in relevant data makes it feasible to derive a large-scale human liver metabolic network and to discover important biological principles or drug-targets based on network analysis. some studies have shown that interesting biological phenomenon and drug-targets could be discovered by applying structural controllability analysis to biological networks. the exploration on the connections between structural controllability theory and the hlmn could be used to uncover valuable information on the human liver metabolism from a fresh perspective.  we applied structural controllability analysis to the hlmn and detected driver metabolites. the driver metabolites tend to have strong ability to influence the states of other metabolites and weak susceptibility to be influenced by the states of others. in addition, the metabolites were classified into three classes: critical, high-frequency and low-frequency driver metabolites. among the identified critical driver metabolites, metabolites were found to be essential; the high-frequency driver metabolites tend to participate in different metabolic pathways, which are important in regulating the whole metabolic systems. moreover, we explored some other possible connections between the structural controllability theory and the hlmn, and find that transport reactions and the environment play important roles in the human liver metabolism.  there are interesting connections between the structural controllability theory and the human liver metabolism: driver metabolites have essential biological functions; the crucial role of extracellular metabolites and transport reactions in controlling the hlmn highlights the importance of the environment in the health of human liver metabolism. human liver metabolic networkcontrollabilitydriver metabolite 