Research ArticleA Boolean Network Approach to EstrogenTranscriptional Regulation

Guillermo de Anda-Jaacuteuregui1 Jesuacutes Espinal-Enr-quez12

Santiago Sandoval-Motta123 and Enrique Hernaacutendez-Lemus 12

1Computational Genomics Division National Institute of Genomic Medicine 14610 Mexico City Mexico2Centro de Ciencias de la Complejidad Universidad Nacional Autonoma de Mexico 04510 Mexico City Mexico3Research Chairs Program National Council on Science and Technology (Conacyt) 03940 Mexico City Mexico

Correspondence should be addressed to Enrique Hernandez-Lemus ehernandezinmegengobmx

Received 1 March 2019 Accepted 6 May 2019 Published 22 May 2019

Academic Editor Yongtang Shi

Copyright copy 2019 Guillermo de Anda-Jauregui et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Gene expression governs important biological processes such as the cellrsquos growth cycle and its response to environmental signalsAlterations of this complex network of transcriptional interactions often lead to unstable expression states and disease Estrogenis a sex hormone known for its roles in cell proliferation Its expression has been involved in several physiological functions suchas regulating the menstrual and reproduction cycles in women Altered expression states where estrogen levels are atypically highhave been associated with an increased incidence of breast ovarian and cervix cancer To better understand the implications ofderegulation of the estrogen and estrogen receptor regulatory networks in this work we generated a dynamical model of generegulation of the estrogen receptor transcription network based on known regulatory interactions By using an adaptation toclassical Boolean Networks dynamics we identified proliferative and antiproliferative gene expression states of the network andalso to identify key players that promote these altered states when perturbed We also modeled how pairwise gene alterations maycontribute to shifts between these two proliferative states and found that the coordinated subexpression of E2F1 and SMAD4 is themost important combination in terms of promoting proliferative states in the network

1 Introduction

11 Estrogen Regulation Estrogen is the primary humanfemale sex hormone responsible for the development of thefemale reproductive system and the emergence of secondarysexual characteristics Estrogens are primarily synthesized inthe ovaries but can also be produced in the adrenals andtestes From here these molecules are distributed to targetorgans and tissues [1 2]

To regulate biological processes estrogen binds to estro-gen receptor (ER) proteins Estrogen may readily diffusethrough the cell membrane to the cytosol and bind to ERsERs belong to either a nuclear or a membrane class Thenuclear class is comprised of two ERs alpha and beta ER-alpha is encoded by the ESR1 gene located in Chromosome6(6q251-q252) meanwhile ER-beta is encoded by the ESR2gene located in Chromosome 14 (14q232-q233) [3]

The expression of estrogen receptors is regulated throughcomplex genetic and epigenetic control mechanisms Themost extensively described regulatory mechanisms involvethe activity of transcription factor (TF) proteins [4] TheseTFs bind to regulatory regions in the genome inducingchanges (either positive or negative) in the basal transcriptionrate of the genes ultimately affecting the concentration of theencoded protein

Estrogen-bound ERs can induce changes in the biolog-ical state of a cell through two different mechanisms anongenomic and a genomic mechanism The nongenomicmechanism involves signal transduction through secondarymessengers via the Estrogen Signaling pathway and relatedcrosstalk pathways [5] The genomic mechanism of ERsinvolves its activity as a transcription factor

Estrogen-bound ERs translocate to the nucleus wherethey bind to Estrogen Response Elements located in the

HindawiComplexityVolume 2019 Article ID 8740279 10 pageshttpsdoiorg10115520198740279

2 Complexity

promoter region of estrogen-regulated genes This bind-ing can either increase or decrease their respective genetranscription rate [6] We may consider that these twophenomena are not disconnected ER-regulated genesmay beinvolved in the regulation of other TFs which in turn mayregulate ER expression forming a complex gene regulatorynetwork

Both experimental and computational studies have beenperformed to map this gene regulatory network in differentbiological contexts The description of the gene regulatorymechanisms governing the expression of ERs as well asthe genes whose expression is regulated through ERs isstill an open problem however current knowledge of thisregulatory network is enough for us to implement modelsof the gene regulatory dynamics involving estrogen andestrogen receptor genes

12 Boolean Models of Gene Regulation Boolean networkshave been increasingly used as models for simulating thedynamics of gene regulatory networksThis description treatsgenes (nodes in the network) as binary variables so genes canonly be in two possible expression states active or inactiveThe dynamical behavior of these networks has been provedto give insights into the dynamics of real cellular systemsBoolean networks have been used to model the response ofthe immune system in response to a respiratory infection[7] They have served to identify potential therapeutic tar-gets in the blood cancer T cell large granular lymphocyte[8]

Of particular interest in these models are the attractorsof the dynamical system defined as the set of recurrentstates that the networks reach after some time [9] Thedynamics of the network are simulated by determiningthe state of each element of the network at a posteriortime given the current state of its regulators The orderby which each state of each element is updated can beeither synchronous or asynchronous (see Appendix) andthus the system can be either Markovian of purely determin-istic

Since Boolean networks are discrete dynamical systemswith finite support (there are exactly 2N possible states on aBoolean network with N nodes) the evolution of the systemwill produce recurrent statesThe trajectories will fall into oneof a set of steady states or cycles called attractors The collec-tion of attractors alongwith the respective configurations thatlead to them is called the attractor landscape

The determination of the set of attractor states and theconvergence dynamics leading to those attractors constitutesthe solution to the Boolean network dynamics problem Itis important to note that these attractors are often con-sistent with what is observed experimentally in the geneexpression patterns of specific phenotypes or cell types [9ndash11]

In this paper we present a Booleanmodel for the estrogenand estrogen receptor gene regulatory network for which weanalyze its dynamics under different configurations of pres-ence absence of nodes and determine which configurationslead to a proclivity to proliferative and antiproliferative states

2 Analysis

21 Construction of the Estrogen Receptor Regulatory NetworkTo study the gene regulatory phenomenon surrounding theestrogen receptor using the Boolean formalism a suitablenetwork is required The reconstruction of such regulatorynetwork is not trivial and will often represent a fragmentof the whole set of gene regulatory interactions that couldbe involved in real organisms However taking into accountonly the first-order regulators often represents a good firstapproach model to understand how the network behavesunder different circumstances [8 12]

The problem may be approached using much formalismincluding coexpression and information-theoreticalmethods[13] literaturemining strategies [14] and even reconstructionusing the Boolean formalism itself [15] Such approachesmay become quite computationally expensive and the recon-structed network itselfmust be validated against known bibli-ographic informationWith these considerations inmind wedecided to construct a background network for our Booleandynamics based onwell-documented curated information oftranscriptional regulations available in databases

We constructed this network of gene regulation centeredon estrogen receptors 120572 and 120573 (ESR1 and ESR2) We identifyregulatory interactions previously described and validatedin the current biomedical literature For this and in orderto increase the reliability of the constructed network wemade use of two complementary databases of regulatoryinteractions RegNetwork and IPA

Our first source of regulatory interactions RegNetwork[16] is a database of transcriptional regulatory relationshipsin human This database comprises curated regulations fromvarious other databases (BioGrid FANTOM HPRD IntActJASPAR TRANSFAC TRED among others) that integratethe current understanding of regulatory interactions derivedfrom experimental and theoretical data for instance byresorting to gene expression datasets for known gene pertur-bations and ChIP-seq assays The database assigns differentconfidence levels to the listed interactions based on theinformation provided by each of its sources Furthermoreit makes a distinction between predicted and experimentallyvalidated regulatory interactions

Our second source of regulatory interactions is theIngenuity Pathway Analysis (IPA) platform (IPAcopy QIAGENRedwoodCity) A commercial software package that relies ona proprietary Ingenuity Knowledge Base [17] which containsa causal network derived from experimental observationsas well as records from manually curated biomedical litera-ture

Using these two sources we constructed our estrogenreceptor regulatory network following these steps

(1) Extract first neighbors of ESR1 and ESR2 from Reg-Network consider only interactions with experimen-tal evidence and high confidence

(2) Extract interactions between first neighbors of ESR1and ESR2 considering only interactions with exper-imental evidence and high confidence This will beNetwork 1

Complexity 3

RegNetwork

Extract ESRl and ESR2 high confidence

neighbors

Ingenuity Pathway Analysis

Extracthigh confidence

links between ESRneighbors

Generatean IPA network from

the genes found inRegNetwork

Mergenetworks keep only

intersection

FilterTo keep only

ldquoINHIBITIONrdquo and ldquoINTERACTIONrdquo links

Figure 1 Pipeline followed in this work

(3) Filter out nodes in Network 1 with 119900119906119905 minus 119889119890119892119903119890119890 = 0iteratively until there are nonodeswith 119900119906119905minus119889119890119892119903119890119890 =0 This will be Network 2

(4) Take the nodes in Network 2 and use them asthe input of an IPA analysis to generate networksbased on interactions described in IPA We merge allIPA networks and then remove nodes that were notpresent in Network 2 to generate Network 3

(5) We filter Network 3 by keeping only those links thatare described by IPA as being either rdquoINHIBITIONrdquoor rdquoACTIVATIONrdquo and we remove all nodes with119900119906119905 minus 119889119890119892119903119890119890 = 0 iteratively until there are no nodeswith 119900119906119905 minus 119889119890119892119903119890119890 = 0 This generates Network 4

(6) We generate a network with the intersection of nodesand edges of network 2 and network 4 and iterativelyremove nodes with 119900119906119905 minus 119889119890119892119903119890119890 = 0 This generatesthe final estrogen receptor regulatory network to beused in this work (Figure 1)

It is known that the general inference of this kind ofnetworks is a computationally expensive problem (NP-hard)By considering a limited search space as we just sketchedwe are able to directly delve into the phenomenologicalimplications of the studied system This is so since fixing thenetwork in compliance to established experimental facts andknownbiological information by resorting to causal inferencehas proven to be an effective way to proceed in particular incases like the estrogen pathway for which a large number ofexperimental evidence sources are available

22 Boolean Rules and Dynamics Boolean rules are the setof logical constraints that a node may have in a Boolean net-workThis rule depends on the state of regulators of the nodeand the activatoryinhibitory nature of those regulators Letus explain one dynamic rule of this signaling pathway Theestrogen receptor 1 ESR1 is a well-known gene that encodes aprotein which participates in processes such as DNAbindingactivation of transcription or sexual development This pro-tein (among other functions) binds to the androgen receptor

Table 1 ESR1 truth table

AR(119905) BRCA1(119905) STAT5A(119905) 1198641198781198771(119905 + 1)0 0 0 00 0 1 00 1 0 10 1 1 01 0 0 11 0 1 01 1 0 11 1 1 0

(AR) which in turn regulates negatively the expression ofESR1 STAT5A is another transcription factor that promotesESR1 activity At the same time BRCA1 a crucial gene inDNA damage response regulates positively the activity ofESR1

ESR1 has two positive regulators (AR and STAT5A) andone repressor (BRCA1) The dynamical state of ESR1 at time119905 + 1 will depend on the state of AR BRCA1 and STAT5A attime 119905 The logical rule of ESR1 may be written as in Table 1

In this case the single or combined action of BRCA andSTAT5A will activate the estrogen receptor (1198641198781198771 = 1) onlyif AR is not present otherwise ESR1 will acquire a valueof 0 Regarding the combined regulation of ESR1 as in therest of this dynamical system the negative regulators exert astronger influence than the positive ones This differentiatedinfluence of negative regulators has been applied in other bio-logical Boolean networks before [9ndash11] since in some cases itis not possible to know experimentally the effect of combinedaction of more than one geneprotein over a determinedmolecule It has been shown that the differentiated action ofnegative and positive regulators is often in agreementwith thebiological system

23 Boolean Model and Attractor Landscape Analysis Oursimplified estrogen receptor regulatory network consists of14 nodes (Figure 2) In this dynamic model each node can

4 Complexity

RARA

E2F1JUN

SMAD4

SP1STAT5A

ESR2 AR

MYC CREBBP

ESR1 HIF1ATP53

BRCA1

Figure 2 The estrogen receptor regulatory network It was constructed from curated transcriptional interactions found in RegNetwork andIPA Gene nodes include ESR1 and ESR2 the estrogen receptors It has 25 interactions representing transcriptional regulation among thesegenes Inhibitory regulation is represented in red activatory regulation is depicted in black

acquire a set of discrete values that correspond to its possibleexpression levels Due to the lack of experimental data onthe kinetic constants for each interaction of the networkwe constructed a model that focuses on the functionalstate of expression of each component rather than on theirexact concentrations These levels of expression are modeledthrough discrete variables that take a finite number of valuesSince all the elements of the network are considered binaryour network has fourteen binary nodes giving a total of Ω =214 possible dynamical states for the network

At every time step of the dynamics the expression levelof all components of the network is updated simultaneouslyaccording to

120590119899 (119905 + 1) = 119865119899 (1205901119899 (119905) 1205902119899 (119905) 120590119896119899 (119905)) (1)

where 120590119899(119905) represents the state of the 119899119905ℎ element of thenetwork at time 119905 Here 1205901119899 1205902119899 120590119896119899 are the 119896119899 regulatorsof 120590119899 and 119865119899(sdot) is a discrete function stated as a logicalrule that explicitly states the corresponding expression levelof 120590119899 given the current expression levels of its regulators119865119899(sdot) is constructed according to experimental evidenceregarding the regulatory interactions (activator or inhibitor)for each node (see methods for references) All the functions119865119899(sdot) for the estrogen network are listed in SupplementaryInformation 1

Since there is a finite number of possible dynamical con-figurations for the entire network (Ω) starting the dynamicsfrom any of these configurations and successively iterating (1)for each node will make the network traverse through a seriesof states until a periodic pattern of activity is reached Thisperiodic set of states is known as an attractor

Several attractors might exist for a given network andseveral initial configurations may lead to the same attractorFor a fixed set of logical functions 119865119899() the particularattractor the network falls into depends entirely on the initialcondition the network starts from Each attractor has a basinof attraction defined by all of the initial conditions that lead to

that particular attractorThese attractors can be thought of asstable patterns of activity of real biological systems as has beenshown previously [8 9] In this case the attractors reachedwill represent the proliferative or antiproliferative state in cellsunder the transcription signals triggered by estrogen

Despite these parallelisms a direct comparison of anattractor to real expression levels might not be so straight-forward Cyclic attractors (attractors composed by severalstates) are very common whereas experimental gene expres-sion is often presented as a single value Additionally geneexpression measurements are commonly taken from cellpopulations which makes the final measurement an averageof single measurements taken over a time window For thisreason we have used a modification of the classic BooleanNetwork approach where we average the gene expressionin the whole attractor basin to end up with a single levelof expression for each gene This approach has been usedpreviously to accurately simulate the behavior of small generegulatory networks [12]

Here the state of each element of the network is repre-sented by its average expression over a time lapse The lengthof the window (L) where values will be averaged correspondsto the length of the attractor reached For instance let usassume a cyclic attractor consisting in four network states isreached For simplicity let us assume the states are

State (1) [1]1[0]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14State (2) [0]1[0]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14State (3) [1]1[1]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14State (4) [0]1[1]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14

Now we want the averaged state of [119899119900119889119890]1 whichis represented by the first digit of the network state

Complexity 5

([1] [0] [1] [0])Then the corresponding expression level for1198991199001198891198901 for this attractor will be (1 + 0 + 1 + 0)4 = 05 where4 is the size of the attractor

Since a network can have more than one attractor we willend up with an expression level for each of these attractorsIn order to account for each of their basins of attraction andend up with a single value for the expression level of a genewe have incorporated a weighted average using the entire setof attractors (N) for each network We define the averageexpression level of 120590119899 as

120590119899 = 119873sum119886=1

120596119886(sum119871119886120591

120590119899 (120591)119871119886 ) (2)

where N represents all the different attractors so theexternal sum is carried out over all the existing attractorsnumbered 1 to N The parameter 120596119886 is the fraction of initialconditions that lead to attractor ldquoardquo over the total possibleconditions that is the size of the basin of the 119886119905ℎ attractoroverΩThe internal sum is carried out over all the120590119899(120591) statesof the 119886119905ℎ attractor of size 119871119886 This means that the final levelof expression of the gene 120590119899 will be the sum of the averagedexpression levels of 120590119899 in each attractor with each attractoraverage weighted by its corresponding basin of attraction

This modification apart from allowing an easier compar-ison between the model and experimental data resemblesthe way in which experimental data is gathered for geneexpression where traditionally measurements of the level ofexpression represent the population average as cells in thepopulation may be at different stages of a stable pattern ofgene expression

It is important to note that mutations in our model (ega deletionmalfunction of a gene) are represented by keepingthe value of the deleted node equal to zero throughout allthe dynamics In the case of gene overexpression we kept thevalue of the overexpressed gene equal to its maximum stateover the whole simulation

24 Perturbation Analysis To simulate altered physiologicalstates we explored the dynamics of the network if the state ofgenes is perturbed

(i) We simulated the overexpression of a gene by settingit to an ON state at the beginning of the simulationand keeping it that way throughout the simulationregardless of the state of its regulators

(ii) We simulated the knockout of a gene by setting itto an OFF state at the beginning of the simulationand keeping it that way throughout the simulationregardless of the state of its regulators

The simulation of the overexpression of a gene mayrepresent either an increase in the activity or concentrationof said gene in a phenotype and may also be used as a modelof the activity of an agonist drug Similarly the simulation ofthe knockout of a gene may also be used as a model of theactivity of an antagonist drug

We analyzed the full set of single overexpression andknockout perturbations for all genes in the network We

also analyzed the full set of 2-hit perturbations for all genesin the network including all overexpressionoverexpressionoverexpressionknockout and knockoutknockout pairs Thishas been conducted successfully in other biological systemsby members of our group such as the calcium-dependentsignaling pathway of the spermatozoa of the sea urchin Spurpuratus in its searching for the egg [10]

25 Proliferation Index We used the Boolean dynamics toquantify biological features in a particular phenotype In thiswork we focused on identifying whether a given phenotypemay tend to be proliferative or antiproliferative To do thiswe considered whether the expression of each gene may beinvolved in processes that are proliferative or antiprolifera-tive beyond their role as regulators in the network

We constructed a Proliferation Index (PI) in which weconsider for a given phenotype the average state of eachgene throughout the attractor landscapes associated to thephenotype

119875119868 = sum ⟨119875⟩ minus sum ⟨119860119875⟩⟨sum ⟨119875⟩ + sum ⟨119860119875⟩⟩ (3)

Where 119875 are proliferative genes and 119860119875 are anti-proliferative genes and ⟨sdot⟩ are the appropriate ensembleaverages In order to assess whether a gene was con-sidered proliferative or anti-proliferative we performed asystematic analysis of the literature using a combina-tion of Pubmed httpswwwncbinlmnihgovpubmed theGene database [18] and the Genetics Home Referencehttpsghrnlmnihgov In Supplementary Information 2 weprovide the bibliographic evidence used to asign a prolifer-ative or anti-proliferative value to each gene in the networkOur index should not be confusedwith the proliferative index(or growth fraction) which is used in the clinical setting

It is worth to mention that the Proliferation Index (PI)defined here is the result of averaging the state value ofnodes during the attractor period The PI is a measure thatintegrates the attractor landscape in terms of the prolifera-tiveantiproliferative phenotype

3 Results

31 The Estrogen Transcriptional Network By following ourconstruction methodology we are able to recover an estrogenreceptor regulatory network composed of 14 nodes and 25directed and signed interactions Four of these interactionsare inhibitory while the rest correspond to activation Avisualization of this network may be found in Figure 2

The network dynamics of this network is depicted inFigure 3 where each dynamical state is represented as a col-ored point and the transition between two consecutive statesis represented as a straight line As previously mentionedattractors of the network dynamicsmay be punctual or cyclicIn the figure we observe both cases

32 Effects of Perturbations on Proliferation Based on Net-work Dynamics Through the use of well-curated biologi-cal knowledge along with Boolean network dynamics we

Complexity 3

RegNetwork

Extract ESRl and ESR2 high confidence

neighbors

Ingenuity Pathway Analysis

Extracthigh confidence

links between ESRneighbors

Generatean IPA network from

the genes found inRegNetwork

Mergenetworks keep only

intersection

FilterTo keep only

ldquoINHIBITIONrdquo and ldquoINTERACTIONrdquo links

Figure 1 Pipeline followed in this work

(3) Filter out nodes in Network 1 with 119900119906119905 minus 119889119890119892119903119890119890 = 0iteratively until there are nonodeswith 119900119906119905minus119889119890119892119903119890119890 =0 This will be Network 2

(4) Take the nodes in Network 2 and use them asthe input of an IPA analysis to generate networksbased on interactions described in IPA We merge allIPA networks and then remove nodes that were notpresent in Network 2 to generate Network 3

(5) We filter Network 3 by keeping only those links thatare described by IPA as being either rdquoINHIBITIONrdquoor rdquoACTIVATIONrdquo and we remove all nodes with119900119906119905 minus 119889119890119892119903119890119890 = 0 iteratively until there are no nodeswith 119900119906119905 minus 119889119890119892119903119890119890 = 0 This generates Network 4

(6) We generate a network with the intersection of nodesand edges of network 2 and network 4 and iterativelyremove nodes with 119900119906119905 minus 119889119890119892119903119890119890 = 0 This generatesthe final estrogen receptor regulatory network to beused in this work (Figure 1)

It is known that the general inference of this kind ofnetworks is a computationally expensive problem (NP-hard)By considering a limited search space as we just sketchedwe are able to directly delve into the phenomenologicalimplications of the studied system This is so since fixing thenetwork in compliance to established experimental facts andknownbiological information by resorting to causal inferencehas proven to be an effective way to proceed in particular incases like the estrogen pathway for which a large number ofexperimental evidence sources are available

22 Boolean Rules and Dynamics Boolean rules are the setof logical constraints that a node may have in a Boolean net-workThis rule depends on the state of regulators of the nodeand the activatoryinhibitory nature of those regulators Letus explain one dynamic rule of this signaling pathway Theestrogen receptor 1 ESR1 is a well-known gene that encodes aprotein which participates in processes such as DNAbindingactivation of transcription or sexual development This pro-tein (among other functions) binds to the androgen receptor

Table 1 ESR1 truth table

AR(119905) BRCA1(119905) STAT5A(119905) 1198641198781198771(119905 + 1)0 0 0 00 0 1 00 1 0 10 1 1 01 0 0 11 0 1 01 1 0 11 1 1 0

(AR) which in turn regulates negatively the expression ofESR1 STAT5A is another transcription factor that promotesESR1 activity At the same time BRCA1 a crucial gene inDNA damage response regulates positively the activity ofESR1

ESR1 has two positive regulators (AR and STAT5A) andone repressor (BRCA1) The dynamical state of ESR1 at time119905 + 1 will depend on the state of AR BRCA1 and STAT5A attime 119905 The logical rule of ESR1 may be written as in Table 1

In this case the single or combined action of BRCA andSTAT5A will activate the estrogen receptor (1198641198781198771 = 1) onlyif AR is not present otherwise ESR1 will acquire a valueof 0 Regarding the combined regulation of ESR1 as in therest of this dynamical system the negative regulators exert astronger influence than the positive ones This differentiatedinfluence of negative regulators has been applied in other bio-logical Boolean networks before [9ndash11] since in some cases itis not possible to know experimentally the effect of combinedaction of more than one geneprotein over a determinedmolecule It has been shown that the differentiated action ofnegative and positive regulators is often in agreementwith thebiological system

23 Boolean Model and Attractor Landscape Analysis Oursimplified estrogen receptor regulatory network consists of14 nodes (Figure 2) In this dynamic model each node can

4 Complexity

RARA

E2F1JUN

SMAD4

SP1STAT5A

ESR2 AR

MYC CREBBP

ESR1 HIF1ATP53

BRCA1

Figure 2 The estrogen receptor regulatory network It was constructed from curated transcriptional interactions found in RegNetwork andIPA Gene nodes include ESR1 and ESR2 the estrogen receptors It has 25 interactions representing transcriptional regulation among thesegenes Inhibitory regulation is represented in red activatory regulation is depicted in black

acquire a set of discrete values that correspond to its possibleexpression levels Due to the lack of experimental data onthe kinetic constants for each interaction of the networkwe constructed a model that focuses on the functionalstate of expression of each component rather than on theirexact concentrations These levels of expression are modeledthrough discrete variables that take a finite number of valuesSince all the elements of the network are considered binaryour network has fourteen binary nodes giving a total of Ω =214 possible dynamical states for the network

At every time step of the dynamics the expression levelof all components of the network is updated simultaneouslyaccording to

120590119899 (119905 + 1) = 119865119899 (1205901119899 (119905) 1205902119899 (119905) 120590119896119899 (119905)) (1)

where 120590119899(119905) represents the state of the 119899119905ℎ element of thenetwork at time 119905 Here 1205901119899 1205902119899 120590119896119899 are the 119896119899 regulatorsof 120590119899 and 119865119899(sdot) is a discrete function stated as a logicalrule that explicitly states the corresponding expression levelof 120590119899 given the current expression levels of its regulators119865119899(sdot) is constructed according to experimental evidenceregarding the regulatory interactions (activator or inhibitor)for each node (see methods for references) All the functions119865119899(sdot) for the estrogen network are listed in SupplementaryInformation 1

Since there is a finite number of possible dynamical con-figurations for the entire network (Ω) starting the dynamicsfrom any of these configurations and successively iterating (1)for each node will make the network traverse through a seriesof states until a periodic pattern of activity is reached Thisperiodic set of states is known as an attractor

Several attractors might exist for a given network andseveral initial configurations may lead to the same attractorFor a fixed set of logical functions 119865119899() the particularattractor the network falls into depends entirely on the initialcondition the network starts from Each attractor has a basinof attraction defined by all of the initial conditions that lead to

that particular attractorThese attractors can be thought of asstable patterns of activity of real biological systems as has beenshown previously [8 9] In this case the attractors reachedwill represent the proliferative or antiproliferative state in cellsunder the transcription signals triggered by estrogen

Despite these parallelisms a direct comparison of anattractor to real expression levels might not be so straight-forward Cyclic attractors (attractors composed by severalstates) are very common whereas experimental gene expres-sion is often presented as a single value Additionally geneexpression measurements are commonly taken from cellpopulations which makes the final measurement an averageof single measurements taken over a time window For thisreason we have used a modification of the classic BooleanNetwork approach where we average the gene expressionin the whole attractor basin to end up with a single levelof expression for each gene This approach has been usedpreviously to accurately simulate the behavior of small generegulatory networks [12]

Here the state of each element of the network is repre-sented by its average expression over a time lapse The lengthof the window (L) where values will be averaged correspondsto the length of the attractor reached For instance let usassume a cyclic attractor consisting in four network states isreached For simplicity let us assume the states are

State (1) [1]1[0]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14State (2) [0]1[0]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14State (3) [1]1[1]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14State (4) [0]1[1]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14

Now we want the averaged state of [119899119900119889119890]1 whichis represented by the first digit of the network state

Complexity 5

([1] [0] [1] [0])Then the corresponding expression level for1198991199001198891198901 for this attractor will be (1 + 0 + 1 + 0)4 = 05 where4 is the size of the attractor

Since a network can have more than one attractor we willend up with an expression level for each of these attractorsIn order to account for each of their basins of attraction andend up with a single value for the expression level of a genewe have incorporated a weighted average using the entire setof attractors (N) for each network We define the averageexpression level of 120590119899 as

120590119899 = 119873sum119886=1

120596119886(sum119871119886120591

120590119899 (120591)119871119886 ) (2)

where N represents all the different attractors so theexternal sum is carried out over all the existing attractorsnumbered 1 to N The parameter 120596119886 is the fraction of initialconditions that lead to attractor ldquoardquo over the total possibleconditions that is the size of the basin of the 119886119905ℎ attractoroverΩThe internal sum is carried out over all the120590119899(120591) statesof the 119886119905ℎ attractor of size 119871119886 This means that the final levelof expression of the gene 120590119899 will be the sum of the averagedexpression levels of 120590119899 in each attractor with each attractoraverage weighted by its corresponding basin of attraction

This modification apart from allowing an easier compar-ison between the model and experimental data resemblesthe way in which experimental data is gathered for geneexpression where traditionally measurements of the level ofexpression represent the population average as cells in thepopulation may be at different stages of a stable pattern ofgene expression

It is important to note that mutations in our model (ega deletionmalfunction of a gene) are represented by keepingthe value of the deleted node equal to zero throughout allthe dynamics In the case of gene overexpression we kept thevalue of the overexpressed gene equal to its maximum stateover the whole simulation

24 Perturbation Analysis To simulate altered physiologicalstates we explored the dynamics of the network if the state ofgenes is perturbed

(i) We simulated the overexpression of a gene by settingit to an ON state at the beginning of the simulationand keeping it that way throughout the simulationregardless of the state of its regulators

(ii) We simulated the knockout of a gene by setting itto an OFF state at the beginning of the simulationand keeping it that way throughout the simulationregardless of the state of its regulators

The simulation of the overexpression of a gene mayrepresent either an increase in the activity or concentrationof said gene in a phenotype and may also be used as a modelof the activity of an agonist drug Similarly the simulation ofthe knockout of a gene may also be used as a model of theactivity of an antagonist drug

We analyzed the full set of single overexpression andknockout perturbations for all genes in the network We

also analyzed the full set of 2-hit perturbations for all genesin the network including all overexpressionoverexpressionoverexpressionknockout and knockoutknockout pairs Thishas been conducted successfully in other biological systemsby members of our group such as the calcium-dependentsignaling pathway of the spermatozoa of the sea urchin Spurpuratus in its searching for the egg [10]

25 Proliferation Index We used the Boolean dynamics toquantify biological features in a particular phenotype In thiswork we focused on identifying whether a given phenotypemay tend to be proliferative or antiproliferative To do thiswe considered whether the expression of each gene may beinvolved in processes that are proliferative or antiprolifera-tive beyond their role as regulators in the network

We constructed a Proliferation Index (PI) in which weconsider for a given phenotype the average state of eachgene throughout the attractor landscapes associated to thephenotype

119875119868 = sum ⟨119875⟩ minus sum ⟨119860119875⟩⟨sum ⟨119875⟩ + sum ⟨119860119875⟩⟩ (3)

Where 119875 are proliferative genes and 119860119875 are anti-proliferative genes and ⟨sdot⟩ are the appropriate ensembleaverages In order to assess whether a gene was con-sidered proliferative or anti-proliferative we performed asystematic analysis of the literature using a combina-tion of Pubmed httpswwwncbinlmnihgovpubmed theGene database [18] and the Genetics Home Referencehttpsghrnlmnihgov In Supplementary Information 2 weprovide the bibliographic evidence used to asign a prolifer-ative or anti-proliferative value to each gene in the networkOur index should not be confusedwith the proliferative index(or growth fraction) which is used in the clinical setting

It is worth to mention that the Proliferation Index (PI)defined here is the result of averaging the state value ofnodes during the attractor period The PI is a measure thatintegrates the attractor landscape in terms of the prolifera-tiveantiproliferative phenotype

3 Results

31 The Estrogen Transcriptional Network By following ourconstruction methodology we are able to recover an estrogenreceptor regulatory network composed of 14 nodes and 25directed and signed interactions Four of these interactionsare inhibitory while the rest correspond to activation Avisualization of this network may be found in Figure 2

The network dynamics of this network is depicted inFigure 3 where each dynamical state is represented as a col-ored point and the transition between two consecutive statesis represented as a straight line As previously mentionedattractors of the network dynamicsmay be punctual or cyclicIn the figure we observe both cases

32 Effects of Perturbations on Proliferation Based on Net-work Dynamics Through the use of well-curated biologi-cal knowledge along with Boolean network dynamics we

4 Complexity

RARA

E2F1JUN

SMAD4

SP1STAT5A

ESR2 AR

MYC CREBBP

ESR1 HIF1ATP53

BRCA1

Figure 2 The estrogen receptor regulatory network It was constructed from curated transcriptional interactions found in RegNetwork andIPA Gene nodes include ESR1 and ESR2 the estrogen receptors It has 25 interactions representing transcriptional regulation among thesegenes Inhibitory regulation is represented in red activatory regulation is depicted in black

acquire a set of discrete values that correspond to its possibleexpression levels Due to the lack of experimental data onthe kinetic constants for each interaction of the networkwe constructed a model that focuses on the functionalstate of expression of each component rather than on theirexact concentrations These levels of expression are modeledthrough discrete variables that take a finite number of valuesSince all the elements of the network are considered binaryour network has fourteen binary nodes giving a total of Ω =214 possible dynamical states for the network

At every time step of the dynamics the expression levelof all components of the network is updated simultaneouslyaccording to

120590119899 (119905 + 1) = 119865119899 (1205901119899 (119905) 1205902119899 (119905) 120590119896119899 (119905)) (1)

where 120590119899(119905) represents the state of the 119899119905ℎ element of thenetwork at time 119905 Here 1205901119899 1205902119899 120590119896119899 are the 119896119899 regulatorsof 120590119899 and 119865119899(sdot) is a discrete function stated as a logicalrule that explicitly states the corresponding expression levelof 120590119899 given the current expression levels of its regulators119865119899(sdot) is constructed according to experimental evidenceregarding the regulatory interactions (activator or inhibitor)for each node (see methods for references) All the functions119865119899(sdot) for the estrogen network are listed in SupplementaryInformation 1

Since there is a finite number of possible dynamical con-figurations for the entire network (Ω) starting the dynamicsfrom any of these configurations and successively iterating (1)for each node will make the network traverse through a seriesof states until a periodic pattern of activity is reached Thisperiodic set of states is known as an attractor

Several attractors might exist for a given network andseveral initial configurations may lead to the same attractorFor a fixed set of logical functions 119865119899() the particularattractor the network falls into depends entirely on the initialcondition the network starts from Each attractor has a basinof attraction defined by all of the initial conditions that lead to

that particular attractorThese attractors can be thought of asstable patterns of activity of real biological systems as has beenshown previously [8 9] In this case the attractors reachedwill represent the proliferative or antiproliferative state in cellsunder the transcription signals triggered by estrogen

Despite these parallelisms a direct comparison of anattractor to real expression levels might not be so straight-forward Cyclic attractors (attractors composed by severalstates) are very common whereas experimental gene expres-sion is often presented as a single value Additionally geneexpression measurements are commonly taken from cellpopulations which makes the final measurement an averageof single measurements taken over a time window For thisreason we have used a modification of the classic BooleanNetwork approach where we average the gene expressionin the whole attractor basin to end up with a single levelof expression for each gene This approach has been usedpreviously to accurately simulate the behavior of small generegulatory networks [12]

Here the state of each element of the network is repre-sented by its average expression over a time lapse The lengthof the window (L) where values will be averaged correspondsto the length of the attractor reached For instance let usassume a cyclic attractor consisting in four network states isreached For simplicity let us assume the states are

State (1) [1]1[0]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14State (2) [0]1[0]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14State (3) [1]1[1]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14State (4) [0]1[1]2[0]3[0]4[0]5[0]6[0]7[0]8[0]9[0]10[0]11[0]12[0]13[0]14

Now we want the averaged state of [119899119900119889119890]1 whichis represented by the first digit of the network state

Complexity 5

([1] [0] [1] [0])Then the corresponding expression level for1198991199001198891198901 for this attractor will be (1 + 0 + 1 + 0)4 = 05 where4 is the size of the attractor

Since a network can have more than one attractor we willend up with an expression level for each of these attractorsIn order to account for each of their basins of attraction andend up with a single value for the expression level of a genewe have incorporated a weighted average using the entire setof attractors (N) for each network We define the averageexpression level of 120590119899 as

120590119899 = 119873sum119886=1

120596119886(sum119871119886120591

120590119899 (120591)119871119886 ) (2)

where N represents all the different attractors so theexternal sum is carried out over all the existing attractorsnumbered 1 to N The parameter 120596119886 is the fraction of initialconditions that lead to attractor ldquoardquo over the total possibleconditions that is the size of the basin of the 119886119905ℎ attractoroverΩThe internal sum is carried out over all the120590119899(120591) statesof the 119886119905ℎ attractor of size 119871119886 This means that the final levelof expression of the gene 120590119899 will be the sum of the averagedexpression levels of 120590119899 in each attractor with each attractoraverage weighted by its corresponding basin of attraction

This modification apart from allowing an easier compar-ison between the model and experimental data resemblesthe way in which experimental data is gathered for geneexpression where traditionally measurements of the level ofexpression represent the population average as cells in thepopulation may be at different stages of a stable pattern ofgene expression

It is important to note that mutations in our model (ega deletionmalfunction of a gene) are represented by keepingthe value of the deleted node equal to zero throughout allthe dynamics In the case of gene overexpression we kept thevalue of the overexpressed gene equal to its maximum stateover the whole simulation

24 Perturbation Analysis To simulate altered physiologicalstates we explored the dynamics of the network if the state ofgenes is perturbed

(i) We simulated the overexpression of a gene by settingit to an ON state at the beginning of the simulationand keeping it that way throughout the simulationregardless of the state of its regulators

(ii) We simulated the knockout of a gene by setting itto an OFF state at the beginning of the simulationand keeping it that way throughout the simulationregardless of the state of its regulators

The simulation of the overexpression of a gene mayrepresent either an increase in the activity or concentrationof said gene in a phenotype and may also be used as a modelof the activity of an agonist drug Similarly the simulation ofthe knockout of a gene may also be used as a model of theactivity of an antagonist drug

We analyzed the full set of single overexpression andknockout perturbations for all genes in the network We

also analyzed the full set of 2-hit perturbations for all genesin the network including all overexpressionoverexpressionoverexpressionknockout and knockoutknockout pairs Thishas been conducted successfully in other biological systemsby members of our group such as the calcium-dependentsignaling pathway of the spermatozoa of the sea urchin Spurpuratus in its searching for the egg [10]

25 Proliferation Index We used the Boolean dynamics toquantify biological features in a particular phenotype In thiswork we focused on identifying whether a given phenotypemay tend to be proliferative or antiproliferative To do thiswe considered whether the expression of each gene may beinvolved in processes that are proliferative or antiprolifera-tive beyond their role as regulators in the network

We constructed a Proliferation Index (PI) in which weconsider for a given phenotype the average state of eachgene throughout the attractor landscapes associated to thephenotype

119875119868 = sum ⟨119875⟩ minus sum ⟨119860119875⟩⟨sum ⟨119875⟩ + sum ⟨119860119875⟩⟩ (3)

Where 119875 are proliferative genes and 119860119875 are anti-proliferative genes and ⟨sdot⟩ are the appropriate ensembleaverages In order to assess whether a gene was con-sidered proliferative or anti-proliferative we performed asystematic analysis of the literature using a combina-tion of Pubmed httpswwwncbinlmnihgovpubmed theGene database [18] and the Genetics Home Referencehttpsghrnlmnihgov In Supplementary Information 2 weprovide the bibliographic evidence used to asign a prolifer-ative or anti-proliferative value to each gene in the networkOur index should not be confusedwith the proliferative index(or growth fraction) which is used in the clinical setting

It is worth to mention that the Proliferation Index (PI)defined here is the result of averaging the state value ofnodes during the attractor period The PI is a measure thatintegrates the attractor landscape in terms of the prolifera-tiveantiproliferative phenotype

3 Results

31 The Estrogen Transcriptional Network By following ourconstruction methodology we are able to recover an estrogenreceptor regulatory network composed of 14 nodes and 25directed and signed interactions Four of these interactionsare inhibitory while the rest correspond to activation Avisualization of this network may be found in Figure 2

The network dynamics of this network is depicted inFigure 3 where each dynamical state is represented as a col-ored point and the transition between two consecutive statesis represented as a straight line As previously mentionedattractors of the network dynamicsmay be punctual or cyclicIn the figure we observe both cases

32 Effects of Perturbations on Proliferation Based on Net-work Dynamics Through the use of well-curated biologi-cal knowledge along with Boolean network dynamics we

Complexity 5

([1] [0] [1] [0])Then the corresponding expression level for1198991199001198891198901 for this attractor will be (1 + 0 + 1 + 0)4 = 05 where4 is the size of the attractor

Since a network can have more than one attractor we willend up with an expression level for each of these attractorsIn order to account for each of their basins of attraction andend up with a single value for the expression level of a genewe have incorporated a weighted average using the entire setof attractors (N) for each network We define the averageexpression level of 120590119899 as

120590119899 = 119873sum119886=1

120596119886(sum119871119886120591

120590119899 (120591)119871119886 ) (2)

where N represents all the different attractors so theexternal sum is carried out over all the existing attractorsnumbered 1 to N The parameter 120596119886 is the fraction of initialconditions that lead to attractor ldquoardquo over the total possibleconditions that is the size of the basin of the 119886119905ℎ attractoroverΩThe internal sum is carried out over all the120590119899(120591) statesof the 119886119905ℎ attractor of size 119871119886 This means that the final levelof expression of the gene 120590119899 will be the sum of the averagedexpression levels of 120590119899 in each attractor with each attractoraverage weighted by its corresponding basin of attraction

This modification apart from allowing an easier compar-ison between the model and experimental data resemblesthe way in which experimental data is gathered for geneexpression where traditionally measurements of the level ofexpression represent the population average as cells in thepopulation may be at different stages of a stable pattern ofgene expression

It is important to note that mutations in our model (ega deletionmalfunction of a gene) are represented by keepingthe value of the deleted node equal to zero throughout allthe dynamics In the case of gene overexpression we kept thevalue of the overexpressed gene equal to its maximum stateover the whole simulation

24 Perturbation Analysis To simulate altered physiologicalstates we explored the dynamics of the network if the state ofgenes is perturbed

(i) We simulated the overexpression of a gene by settingit to an ON state at the beginning of the simulationand keeping it that way throughout the simulationregardless of the state of its regulators

(ii) We simulated the knockout of a gene by setting itto an OFF state at the beginning of the simulationand keeping it that way throughout the simulationregardless of the state of its regulators

The simulation of the overexpression of a gene mayrepresent either an increase in the activity or concentrationof said gene in a phenotype and may also be used as a modelof the activity of an agonist drug Similarly the simulation ofthe knockout of a gene may also be used as a model of theactivity of an antagonist drug

We analyzed the full set of single overexpression andknockout perturbations for all genes in the network We

also analyzed the full set of 2-hit perturbations for all genesin the network including all overexpressionoverexpressionoverexpressionknockout and knockoutknockout pairs Thishas been conducted successfully in other biological systemsby members of our group such as the calcium-dependentsignaling pathway of the spermatozoa of the sea urchin Spurpuratus in its searching for the egg [10]

25 Proliferation Index We used the Boolean dynamics toquantify biological features in a particular phenotype In thiswork we focused on identifying whether a given phenotypemay tend to be proliferative or antiproliferative To do thiswe considered whether the expression of each gene may beinvolved in processes that are proliferative or antiprolifera-tive beyond their role as regulators in the network

We constructed a Proliferation Index (PI) in which weconsider for a given phenotype the average state of eachgene throughout the attractor landscapes associated to thephenotype

119875119868 = sum ⟨119875⟩ minus sum ⟨119860119875⟩⟨sum ⟨119875⟩ + sum ⟨119860119875⟩⟩ (3)

Where 119875 are proliferative genes and 119860119875 are anti-proliferative genes and ⟨sdot⟩ are the appropriate ensembleaverages In order to assess whether a gene was con-sidered proliferative or anti-proliferative we performed asystematic analysis of the literature using a combina-tion of Pubmed httpswwwncbinlmnihgovpubmed theGene database [18] and the Genetics Home Referencehttpsghrnlmnihgov In Supplementary Information 2 weprovide the bibliographic evidence used to asign a prolifer-ative or anti-proliferative value to each gene in the networkOur index should not be confusedwith the proliferative index(or growth fraction) which is used in the clinical setting

It is worth to mention that the Proliferation Index (PI)defined here is the result of averaging the state value ofnodes during the attractor period The PI is a measure thatintegrates the attractor landscape in terms of the prolifera-tiveantiproliferative phenotype

3 Results

31 The Estrogen Transcriptional Network By following ourconstruction methodology we are able to recover an estrogenreceptor regulatory network composed of 14 nodes and 25directed and signed interactions Four of these interactionsare inhibitory while the rest correspond to activation Avisualization of this network may be found in Figure 2

The network dynamics of this network is depicted inFigure 3 where each dynamical state is represented as a col-ored point and the transition between two consecutive statesis represented as a straight line As previously mentionedattractors of the network dynamicsmay be punctual or cyclicIn the figure we observe both cases

32 Effects of Perturbations on Proliferation Based on Net-work Dynamics Through the use of well-curated biologi-cal knowledge along with Boolean network dynamics we

6 Complexity

Figure 3 Excerpt of the attractor landscape Fan-like represen-tation of a set of 4 attractors of the attractor landscape of theestrogenestrogen receptor network Each point represents a state ofthe network Connections represent temporal succession betweenstates with the outward points representing preceding states (seeblack arrow) Same colored fans represent a collection of states thatlead to the same future state of the network (eg red dashed circle)A cyclic attractor of size = 2 is also represented (see blue and purplecircular arrows)The length of the connections is inconsequential asall time steps between states are fixed

developed a model that may elucidate the contribution ofgene perturbations to an observable phenotypic trait Wefocused on the proliferative state that is achieved throughgene perturbation This could reflect the changes in cellgrowth observed in diseases such as cancer but it also canbe used to model the effects that an external perturbation(such as a pharmacological intervention) may have in thephenotype

321 The Effects of Single Perturbations on the ProliferativePhenotype In Figure 4 we present the result of the pertur-bation of single genes in terms of the Proliferation Index (PI)compared to the PI value for the wild-type (WT) phenotype

A total of 28 perturbations were performed which maybe seen in Figure 4(a) Overall 15 of these perturbationsinduce PI value higher than the one for the WT phenotype(119875119868 = minus00761) whereas 13 lead to a reduction of the PIvalue with respect to the WT The maximum PI value isachieved through the knockout of E2F1 (119875119868 = 03029) whilethe minimum PI value is achieved through the knockout ofSTAT5A (119875119868 = minus05600)

We may observe that the effects of gene overexpressionand knockout are different in terms of the PI In Figures4(b) and 4(c) we may observe the PI for overexpressionsand knockouts separately In the case of overexpressions(Figure 4(b)) the PI values are less spread ranging fromminus03670 to 00335with 9 perturbations increasing the PI withrespect to theWTand 5decreasing it In the case of knockouts(Figure 4(c)) these cover a broader range including the afore-mentioned overallmaximum (E2F1 knockout) andminimum

(STAT5A knockout) 4 perturbations increase the PI withrespect to WT and 10 decrease it

In Figure 4(d) we present the genes in the network ina scatterplot where the x-axis represents PI when the geneis knocked-out and the y-axis represents PI when the geneis overexpressed We trace four quadrants with respect toPI for the WT phenotype We may observe that all fourantiproliferative genes are placed in the lower right quadrantindicating that their knockouts lead to more proliferationwhile their overexpression leads to less proliferation

322 Effects of Two-Hit Perturbations on the ProliferativePhenotype In Figure 5 we present the results of the simul-taneous perturbation of two genes of the Estrogen ReceptorRegulatory Network in terms of the Proliferative Indexas heatmaps In Figure 5(a) we present the result of thesimultaneous overexpression of two genes In Figure 5(b)we show the effect of the combined overexpression of a gene(shown in the rows of the heatmap) and the knockdown ofanother gene (shown in the columns of the heatmap) Finallyin Figure 5(c) we show the effect of double gene knockouts

For each type of two-hit perturbation we may find amaximum and minimum PI value In the case of the doubleoverexpression the minimum PI value is achieved with theoverexpression of TP53 and AR (119875119868 = minus05313) while themaximum PI value is achieved with the overexpression ofESR1 and AR (119875119868 = 01876) For overexpressionknockoutcombinations the minimum PI value results from the over-expression of TP53 and the knockout of STAT5A(119875119868 =minus08871) while the maximum PI value comes from overex-pressing ESR1 and knocking out SMAD4 (119875119868 = 04719) Inthe case of double knockouts knocking out both STAT5Aand JUN leade to the minimum PI value (119875119868 = minus10196)while the double knockout of SMAD4 and E2F1 generates themaximum PI value (119875119868 = 08033)

Through the double perturbation of genes it is possibleto reach more extreme changes in PI than by targeting asingle gene alone For instance the lowest PI value obtained(119875119868 = minus10196 from the double knockout of STAT5A andJUN) is much lower than the lowest PI obtained from a singlegene perturbation (119875119868 = minus05600 from the single knockoutof STAT5A) Similarly the highest PI value obtained (119875119868 =08033 from the double knockout of SMAD4 and E2F1)is higher than the highest PI value obtained from singleperturbations (119875119868 = 03029 from the knockout of E2F1)Importantly and similar to what was observed in singleperturbations the most extreme changes in PI come fromknockout perturbations

4 Discussion

We have shown that with the Boolean approach it is possibleto perturb the dynamical state of the estrogen transcriptionalnetwork and observe single or multitarget perturbations Asit is expected the effect of altering one or more elementsin the network dynamics will be different in terms of theProliferation Index

In the upper left quadrant of the scatterplot inFigure 4(d) representing single gene perturbations we

Complexity 7

STAT5A_OFFESR1_OFFJUN_OFFTP53_ON

ESR2_OFFAR_ON

SP1_OFFRARA_OFFBRCA1_ON

E2F1_ONMYC_OFF

HIF1A_OFFCREBBP_OFF

AR_OFFSMAD4_ON

WT_WTSP1_ON

JUN_ONMYC_ON

HIF1A_ONCREBBP_ON

RARA_ONSTAT5A_ON

ESR2_ONESR1_ON

TP53_OFFBRCA1_OFFSMAD4_OFF

E2F1_OFF

minus04

minus02

00

02

04Proliferation Index

(a)

TP53_ON

AR_ON

BRCA1_ON

E2F1_ON

SMAD4_ON

WT_WT

SP1_ON

JUN_ON

MYC_ON

HIF1A_ON

CREBBP_ON

RARA_ON

STAT5A_ON

ESR2_ON

ESR1_ON

minus04

minus02

00

02

04Proliferation Index

(b)

STAT5A_OFF

ESR1_OFF

JUN_OFF

ESR2_OFF

SP1_OFF

RARA_OFF

MYC_OFF

HIF1A_OFF

CREBBP_OFF

AR_OFF

WT_WT

TP53_OFF

BRCA1_OFF

SMAD4_OFF

E2F1_OFF

minus04

minus02

00

02

04Proliferation Index

(c)

AR

BRCA1

CREBBP

E2F1

ESR1

JUN

RARA

SMAD4

SP1

STAT5A

TP53

WT

minus03

minus02

minus01

00

minus06 0200minus02minus04Knockminusout Proliferation Index

Ove

rexp

ress

ion

Prol

ifera

tion

Inde

xANTI

PRO

Wild_Type

Proliferation indices

ESR2

(d)

Figure 4 Proliferation indexes for single gene perturbations Each column shows the PI values after a perturbation (a) shows the set ofoverexpression and knockouts (bc) represent overexpression and knockout separately (d) is a scatterplot showing the PI values of all genesafter overexpressing and knocking out the genes

may find proliferative genes (STAT5A ESR2 MYC JUNetc) meaning the overexpression of these genes lead to moreproliferation while their knockout leads to less proliferationAn interesting finding is the curious case of AR This isthe only gene in the network that is located in the lowerleft quadrant indicating that both its overexpression andknockout lead to a decrease of the proliferative index withrespect to the wild type

The E2F1 gene is a well known tumor suppressor geneIt participates in both control of cell cycle and cell deathprocesses It has been observed experimentally that lowerexpression values of E2F1 gene are frequent in malignanttumors in breast cancer [19] As in our network dynamicsthe highest 119875119868 value was obtained by knocking out E2F1gene which is in agreement with the experimental resultsAnalogously as STAT5A being one of the main activators ofESR1 and ESR2 its inhibition decreases substantially 119875119868

By observing Figure 5 representing two-hit perturba-tions it is evident that each type of perturbation generatesdifferent clustering patterns It may be seen that in the caseof double overexpressions we observe a more homogeneousdistribution of the PI values In the case of the overexpres-sionknockout combinations the PI patterns tend to be moredominated by the knockout genes (as the pattern observedis of vertical stripes) Finally in the case of double knockoutswemay findwell defined clusters that are related to the doubleknockout of antiproliferative genes proliferative genes or thecombination of a proliferative and antiproliferative gene

It is worth noting that the lowest PI value results fromthe concerted action of the overexpression of a keystonetumor suppressor (TP53) and the concomitant knockout of aproproliferative gene (STAT5A) whose single knockout leadsto the lowest individual PI value The aforementioned resultsmay have important implications in different scenarios suchas cancer where drug combinationsmay have deep impact inclinical outcomes

Interestingly the resulting network after causal infer-ence contains Master Regulators such as P53 E2F SMAD4STAT5A AR ESR1 MYC FOS or JUN It is well knownthat these Master Regulators determine the cell phenotype inhealth and disease and its deregulation may have profoundimplications in cases such as cancer [20]

PI was constructed acknowledging the pro- and antipro-liferative activity of said regulators The relevance of havingMaster Regulators in our network is that the ldquofine tuningrdquo ofthem would imply the switch to a proliferative state or a cellcycle arrest

The Boolean approach used here has several advantagessuch as the fast and direct set of results that are obtainedby a relatively simple model It is not necessary to knowthe reaction rates or other biological parameters that areoften difficult to obtain experimentally Perturbation analysisis also easy to obtain and interpret Another advantage isthe possibility to perturb more than one molecule in silicoand analyze results in terms of transient times attractorlandscapes or basins of attraction

8 Complexity

ESR1

JUN

TP53

ESR2

AR

SP1

BRCA1

MYC

HIF1A

CREBBP

RARA

STAT5A

SMAD4

E2F1

015

000

minus015

minus030

minus045

TP53

STAT

5ASP1

SMA

D4

RARA

MYCJU

N

HIF

1A

ESR2

ESR1

E2F1

CREB

BP

BRCA

1

AR

(a) Overexpressedoverexpressed

025

000

minus025

minus050

minus075

ESR1

JUN

TP53

ESR2

AR

SP1

BRCA1

MYC

HIF1A

CREBBP

RARA

STAT5A

SMAD4

E2F1

TP53

STAT

5ASP1

SMA

D4

RARA

MYCJU

N

HIF

1A

ESR2

ESR1

E2F1

CREB

BP

BRCA

1

AR

(b) Overexpressedknockout

04

08

00

minus04

minus08

ESR1

JUN

TP53

ESR2

AR

SP1

BRCA1

MYC

HIF1A

CREBBP

RARA

STAT5A

SMAD4

E2F1

TP53

STAT

5ASP1

SMA

D4

RARA

MYCJU

N

HIF

1A

ESR2

ESR1

E2F1

CREB

BP

BRCA

1

AR

(c) Knockoutknockout

Figure 5 Proliferation indexes for two-hit perturbations The three heatmaps show the PI value for perturbation of a couple of genes in thenetwork

However these kinds of models also present some issuesthat must be taken into account to have a better interpre-tation the model only uses two discrete states loosing thefine-tuning of studying the system as a continuous modelTime evolution is also discrete but it is widely known thatbiological molecules have a particular time for reactionDespite the fact that the Boolean model uses a discrete timeevolution this does not significantly differ from an attractorlandscape obtained by a nonsynchronous update evolutiondynamics

All of these caveats obviously have influence on the inter-pretation but after a careful construction of the dynamicalrules of the network the results of the Boolean dynamics area good generator of hypotheses andmay be used as a first stepin the searching for experimental corroborations

5 Conclusions

In this work we have demonstrated in silico that altering thedynamical state of key biomolecules of the proliferative estro-gen regulatory networks is possible to shift the dynamicalstate from a proproliferative towards an antiproliferative oneand vice versaTheProliferation Index presented here despitebeing similar to other indexes used in cancer-related Booleannetworks [21] provides elements of analysis and suggestspossible experimental approaches in terms of altering theestrogen-dependent cell proliferation

These kinds of approaches may be useful to test the usageof different drugs with a known or unknown effects andevaluate the final outcome searching for a more personalizedmedicine

Complexity 9

Appendix

Boolean Networks as Dynamical Systems

This brief appendix provides some definitions of dynamicalsystems and Boolean networks included for the sake of com-pletenessThe study of the dynamical evolution of networkedsystems has been gaining importance and recognition inthe physicsappliedmathematicscomplex systemscomputersciences literature This is so since a wide variety of non-trivial phenomena has been characterized as arising of thedynamic evolution of interdependent agents Features likecooperation spreading and synchronization dynamics onnetworks have been characterized For instance the workof Wang and coworkers [22] presents an application ofnovel centrality measures to account for modified diffusion(spreading) on complex networks while information sharingand cooperation have been characterized in the works of theChengyi group [23 24]

Boolean networks in particular are a class of (determin-istic or stochastic) sequential dynamical systems Booleannetworks usually consist on a (finite) set of Boolean logicvariables governed by a set of finitary functions of the formF B119896 997888rarr B where B = 0 1 is a binary logic or Booleandomain (eg an algebra of logical truth values) (it is possibleto build Boolean domains with more than two logical statesThe formalism extension to these cases is straightforward)and 119896 is the arity (number of arguments or Cartesian productdimension) a nonnegative integer ABoolean function is thusa propositional formula in 119896 variables which takes a series ofinputs from a subset of the Boolean variables and as an outputproduces the state of the corresponding variable The set ofBoolean functions determines the connectivity of the set ofvariables that become the nodes of a network whose topologyis given by the combination of Boolean functions for all thevariables [25 26]

For the Boolean network dynamics the state of thenetwork at a given time 119905 + 1 is determined via the evaluationof each of the variablesrsquo function on the state of the networkat a previous time 119905 This may be done on a synchronous (allnodesrsquo states updated at once) or asynchronous (hierarchicalupdating given the position of a given node in the network)way Depending upon updating procedures the systemrsquosdynamicsmay beMarkovian or non-Markovian (often finite-Markovian) [26 27]

Given the fact that Boolean networks are discrete dynam-ical systems with finite support (there are exactly 2119873 possiblestates on a classicalmdashie 2-statemdashBoolean network with Nnodes) the evolution of the system will produce recurrentstatesThe trajectories will fall into one of a set of steady statesor cycles called attractors The set of attractors of a dynamicalsystem is called the attractor landscape The determinationof the set of attractor states and the convergence dynamicsleading to those attractors constitutes the solution to theBoolean network dynamics problem [27]

The Boolean networks studied here belong to a classof deterministic dynamical systems Such systems may berepresented by a set of differential equations describing thedynamical evolution in phase space Deterministic Boolean

networksmay also be represented as a discrete dynamical sys-tem (a map) that when iterated reproduces the full dynamicsof the network including the set of attractors This was theway we proceeded here Since iterated maps and differentialequations are two equivalent representations of the evolutionof a dynamical system [28] our approach does not loose anygenerality

Data Availability

All relevant data has been included in the supplementarymaterials

Conflicts of Interest

The authors have no conflict of interest to declare

Authorsrsquo Contributions

Guillermo de Anda-Jauregui Jesus Espinal-Enrıquez andSantiago Sandoval-Motta contributed equally to this work

Acknowledgments

The research leading to these results has received fundingfrom Consejo Nacional de Ciencia y Tecnologıa (grantnumber 2855442016 Ciencia Basica and 21152017 Fron-teras de la Ciencia (Jesus Espinal-Enrıquez)) as well asfederal funding from the National Institute of GenomicMedicine (EnriqueHernandez-Lemus) EnriqueHernandez-Lemus also acknowledges support from the 2016 MarcosMoshinsky Research Chair in the Physical Sciences JesusEspinal-Enrıquez acknowledges support from FundacionMiguel Aleman in Health Research Santiago Sandoval-Motta acknowledges support from the program CatedrasCONACYT The funders had no role in the design of thisresearch

Supplementary Materials

Supplementary 1 Supplementary Material 1 Regulatoryfunctions of the estrogen transcriptional networks Eachfile contains the regulatory function for all those genes inthe network including the regulatory genes as well as thediscrete value of the target gene after taking into account thevalue of its regulatorsSupplementary 2 Supplementary Material 2 Bibliographicevidence associated with the proliferative and antiprolifera-tive nature of the genes in the network

References

[1] W E Stumpf ldquoNuclear concentration of 3H-estradiol in targettissues Dry-mount autoradiography of vagina oviduct ovarytestis mammary tumor liver and adrenalrdquo Endocrinology vol85 no 1 pp 31ndash37 1969

[2] J Cui Y Shen and R Li ldquoEstrogen synthesis and signalingpathways during aging from periphery to brainrdquo Trends inMolecular Medicine vol 19 no 3 pp 197ndash209 2013

Complexity 7

STAT5A_OFFESR1_OFFJUN_OFFTP53_ON

ESR2_OFFAR_ON

SP1_OFFRARA_OFFBRCA1_ON

E2F1_ONMYC_OFF

HIF1A_OFFCREBBP_OFF

AR_OFFSMAD4_ON

WT_WTSP1_ON

JUN_ONMYC_ON

HIF1A_ONCREBBP_ON

RARA_ONSTAT5A_ON

ESR2_ONESR1_ON

TP53_OFFBRCA1_OFFSMAD4_OFF

E2F1_OFF

minus04

minus02

00

02

04Proliferation Index

(a)

TP53_ON

AR_ON

BRCA1_ON

E2F1_ON

SMAD4_ON

WT_WT

SP1_ON

JUN_ON

MYC_ON

HIF1A_ON

CREBBP_ON

RARA_ON

STAT5A_ON

ESR2_ON

ESR1_ON

minus04

minus02

00

02

04Proliferation Index

(b)

STAT5A_OFF

ESR1_OFF

JUN_OFF

ESR2_OFF

SP1_OFF

RARA_OFF

MYC_OFF

HIF1A_OFF

CREBBP_OFF

AR_OFF

WT_WT

TP53_OFF

BRCA1_OFF

SMAD4_OFF

E2F1_OFF

minus04

minus02

00

02

04Proliferation Index

(c)

AR

BRCA1

CREBBP

E2F1

ESR1

JUN

RARA

SMAD4

SP1

STAT5A

TP53

WT

minus03

minus02

minus01

00

minus06 0200minus02minus04Knockminusout Proliferation Index

Ove

rexp

ress

ion

Prol

ifera

tion

Inde

xANTI

PRO

Wild_Type

Proliferation indices

ESR2

(d)

Figure 4 Proliferation indexes for single gene perturbations Each column shows the PI values after a perturbation (a) shows the set ofoverexpression and knockouts (bc) represent overexpression and knockout separately (d) is a scatterplot showing the PI values of all genesafter overexpressing and knocking out the genes

may find proliferative genes (STAT5A ESR2 MYC JUNetc) meaning the overexpression of these genes lead to moreproliferation while their knockout leads to less proliferationAn interesting finding is the curious case of AR This isthe only gene in the network that is located in the lowerleft quadrant indicating that both its overexpression andknockout lead to a decrease of the proliferative index withrespect to the wild type

The E2F1 gene is a well known tumor suppressor geneIt participates in both control of cell cycle and cell deathprocesses It has been observed experimentally that lowerexpression values of E2F1 gene are frequent in malignanttumors in breast cancer [19] As in our network dynamicsthe highest 119875119868 value was obtained by knocking out E2F1gene which is in agreement with the experimental resultsAnalogously as STAT5A being one of the main activators ofESR1 and ESR2 its inhibition decreases substantially 119875119868

By observing Figure 5 representing two-hit perturba-tions it is evident that each type of perturbation generatesdifferent clustering patterns It may be seen that in the caseof double overexpressions we observe a more homogeneousdistribution of the PI values In the case of the overexpres-sionknockout combinations the PI patterns tend to be moredominated by the knockout genes (as the pattern observedis of vertical stripes) Finally in the case of double knockoutswemay findwell defined clusters that are related to the doubleknockout of antiproliferative genes proliferative genes or thecombination of a proliferative and antiproliferative gene

It is worth noting that the lowest PI value results fromthe concerted action of the overexpression of a keystonetumor suppressor (TP53) and the concomitant knockout of aproproliferative gene (STAT5A) whose single knockout leadsto the lowest individual PI value The aforementioned resultsmay have important implications in different scenarios suchas cancer where drug combinationsmay have deep impact inclinical outcomes

Interestingly the resulting network after causal infer-ence contains Master Regulators such as P53 E2F SMAD4STAT5A AR ESR1 MYC FOS or JUN It is well knownthat these Master Regulators determine the cell phenotype inhealth and disease and its deregulation may have profoundimplications in cases such as cancer [20]

PI was constructed acknowledging the pro- and antipro-liferative activity of said regulators The relevance of havingMaster Regulators in our network is that the ldquofine tuningrdquo ofthem would imply the switch to a proliferative state or a cellcycle arrest

The Boolean approach used here has several advantagessuch as the fast and direct set of results that are obtainedby a relatively simple model It is not necessary to knowthe reaction rates or other biological parameters that areoften difficult to obtain experimentally Perturbation analysisis also easy to obtain and interpret Another advantage isthe possibility to perturb more than one molecule in silicoand analyze results in terms of transient times attractorlandscapes or basins of attraction

8 Complexity

ESR1

JUN

TP53

ESR2

AR

SP1

BRCA1

MYC

HIF1A

CREBBP

RARA

STAT5A

SMAD4

E2F1

015

000

minus015

minus030

minus045

TP53

STAT

5ASP1

SMA

D4

RARA

MYCJU

N

HIF

1A

ESR2

ESR1

E2F1

CREB

BP

BRCA

1

AR

(a) Overexpressedoverexpressed

025

000

minus025

minus050

minus075

ESR1

JUN

TP53

ESR2

AR

SP1

BRCA1

MYC

HIF1A

CREBBP

RARA

STAT5A

SMAD4

E2F1

TP53

STAT

5ASP1

SMA

D4

RARA

MYCJU

N

HIF

1A

ESR2

ESR1

E2F1

CREB

BP

BRCA

1

AR

(b) Overexpressedknockout

04

08

00

minus04

minus08

ESR1

JUN

TP53

ESR2

AR

SP1

BRCA1

MYC

HIF1A

CREBBP

RARA

STAT5A

SMAD4

E2F1

TP53

STAT

5ASP1

SMA

D4

RARA

MYCJU

N

HIF

1A

ESR2

ESR1

E2F1

CREB

BP

BRCA

1

AR

(c) Knockoutknockout

Figure 5 Proliferation indexes for two-hit perturbations The three heatmaps show the PI value for perturbation of a couple of genes in thenetwork

However these kinds of models also present some issuesthat must be taken into account to have a better interpre-tation the model only uses two discrete states loosing thefine-tuning of studying the system as a continuous modelTime evolution is also discrete but it is widely known thatbiological molecules have a particular time for reactionDespite the fact that the Boolean model uses a discrete timeevolution this does not significantly differ from an attractorlandscape obtained by a nonsynchronous update evolutiondynamics

All of these caveats obviously have influence on the inter-pretation but after a careful construction of the dynamicalrules of the network the results of the Boolean dynamics area good generator of hypotheses andmay be used as a first stepin the searching for experimental corroborations

5 Conclusions

In this work we have demonstrated in silico that altering thedynamical state of key biomolecules of the proliferative estro-gen regulatory networks is possible to shift the dynamicalstate from a proproliferative towards an antiproliferative oneand vice versaTheProliferation Index presented here despitebeing similar to other indexes used in cancer-related Booleannetworks [21] provides elements of analysis and suggestspossible experimental approaches in terms of altering theestrogen-dependent cell proliferation

These kinds of approaches may be useful to test the usageof different drugs with a known or unknown effects andevaluate the final outcome searching for a more personalizedmedicine

Complexity 9

Appendix

Boolean Networks as Dynamical Systems

This brief appendix provides some definitions of dynamicalsystems and Boolean networks included for the sake of com-pletenessThe study of the dynamical evolution of networkedsystems has been gaining importance and recognition inthe physicsappliedmathematicscomplex systemscomputersciences literature This is so since a wide variety of non-trivial phenomena has been characterized as arising of thedynamic evolution of interdependent agents Features likecooperation spreading and synchronization dynamics onnetworks have been characterized For instance the workof Wang and coworkers [22] presents an application ofnovel centrality measures to account for modified diffusion(spreading) on complex networks while information sharingand cooperation have been characterized in the works of theChengyi group [23 24]

Boolean networks in particular are a class of (determin-istic or stochastic) sequential dynamical systems Booleannetworks usually consist on a (finite) set of Boolean logicvariables governed by a set of finitary functions of the formF B119896 997888rarr B where B = 0 1 is a binary logic or Booleandomain (eg an algebra of logical truth values) (it is possibleto build Boolean domains with more than two logical statesThe formalism extension to these cases is straightforward)and 119896 is the arity (number of arguments or Cartesian productdimension) a nonnegative integer ABoolean function is thusa propositional formula in 119896 variables which takes a series ofinputs from a subset of the Boolean variables and as an outputproduces the state of the corresponding variable The set ofBoolean functions determines the connectivity of the set ofvariables that become the nodes of a network whose topologyis given by the combination of Boolean functions for all thevariables [25 26]

For the Boolean network dynamics the state of thenetwork at a given time 119905 + 1 is determined via the evaluationof each of the variablesrsquo function on the state of the networkat a previous time 119905 This may be done on a synchronous (allnodesrsquo states updated at once) or asynchronous (hierarchicalupdating given the position of a given node in the network)way Depending upon updating procedures the systemrsquosdynamicsmay beMarkovian or non-Markovian (often finite-Markovian) [26 27]

Given the fact that Boolean networks are discrete dynam-ical systems with finite support (there are exactly 2119873 possiblestates on a classicalmdashie 2-statemdashBoolean network with Nnodes) the evolution of the system will produce recurrentstatesThe trajectories will fall into one of a set of steady statesor cycles called attractors The set of attractors of a dynamicalsystem is called the attractor landscape The determinationof the set of attractor states and the convergence dynamicsleading to those attractors constitutes the solution to theBoolean network dynamics problem [27]

The Boolean networks studied here belong to a classof deterministic dynamical systems Such systems may berepresented by a set of differential equations describing thedynamical evolution in phase space Deterministic Boolean

networksmay also be represented as a discrete dynamical sys-tem (a map) that when iterated reproduces the full dynamicsof the network including the set of attractors This was theway we proceeded here Since iterated maps and differentialequations are two equivalent representations of the evolutionof a dynamical system [28] our approach does not loose anygenerality

Data Availability

All relevant data has been included in the supplementarymaterials

Conflicts of Interest

The authors have no conflict of interest to declare

Authorsrsquo Contributions

Guillermo de Anda-Jauregui Jesus Espinal-Enrıquez andSantiago Sandoval-Motta contributed equally to this work

Acknowledgments

The research leading to these results has received fundingfrom Consejo Nacional de Ciencia y Tecnologıa (grantnumber 2855442016 Ciencia Basica and 21152017 Fron-teras de la Ciencia (Jesus Espinal-Enrıquez)) as well asfederal funding from the National Institute of GenomicMedicine (EnriqueHernandez-Lemus) EnriqueHernandez-Lemus also acknowledges support from the 2016 MarcosMoshinsky Research Chair in the Physical Sciences JesusEspinal-Enrıquez acknowledges support from FundacionMiguel Aleman in Health Research Santiago Sandoval-Motta acknowledges support from the program CatedrasCONACYT The funders had no role in the design of thisresearch

Supplementary Materials

Supplementary 1 Supplementary Material 1 Regulatoryfunctions of the estrogen transcriptional networks Eachfile contains the regulatory function for all those genes inthe network including the regulatory genes as well as thediscrete value of the target gene after taking into account thevalue of its regulatorsSupplementary 2 Supplementary Material 2 Bibliographicevidence associated with the proliferative and antiprolifera-tive nature of the genes in the network

References

[1] W E Stumpf ldquoNuclear concentration of 3H-estradiol in targettissues Dry-mount autoradiography of vagina oviduct ovarytestis mammary tumor liver and adrenalrdquo Endocrinology vol85 no 1 pp 31ndash37 1969

[2] J Cui Y Shen and R Li ldquoEstrogen synthesis and signalingpathways during aging from periphery to brainrdquo Trends inMolecular Medicine vol 19 no 3 pp 197ndash209 2013

8 Complexity

ESR1

JUN

TP53

ESR2

AR

SP1

BRCA1

MYC

HIF1A

CREBBP

RARA

STAT5A

SMAD4

E2F1

015

000

minus015

minus030

minus045

TP53

STAT

5ASP1

SMA

D4

RARA

MYCJU

N

HIF

1A

ESR2

ESR1

E2F1

CREB

BP

BRCA

1

AR

(a) Overexpressedoverexpressed

025

000

minus025

minus050

minus075

ESR1

JUN

TP53

ESR2

AR

SP1

BRCA1

MYC

HIF1A

CREBBP

RARA

STAT5A

SMAD4

E2F1

TP53

STAT

5ASP1

SMA

D4

RARA

MYCJU

N

HIF

1A

ESR2

ESR1

E2F1

CREB

BP

BRCA

1

AR

(b) Overexpressedknockout

04

08

00

minus04

minus08

ESR1

JUN

TP53

ESR2

AR

SP1

BRCA1

MYC

HIF1A

CREBBP

RARA

STAT5A

SMAD4

E2F1

TP53

STAT

5ASP1

SMA

D4

RARA

MYCJU

N

HIF

1A

ESR2

ESR1

E2F1

CREB

BP

BRCA

1

AR

(c) Knockoutknockout

Figure 5 Proliferation indexes for two-hit perturbations The three heatmaps show the PI value for perturbation of a couple of genes in thenetwork

However these kinds of models also present some issuesthat must be taken into account to have a better interpre-tation the model only uses two discrete states loosing thefine-tuning of studying the system as a continuous modelTime evolution is also discrete but it is widely known thatbiological molecules have a particular time for reactionDespite the fact that the Boolean model uses a discrete timeevolution this does not significantly differ from an attractorlandscape obtained by a nonsynchronous update evolutiondynamics

All of these caveats obviously have influence on the inter-pretation but after a careful construction of the dynamicalrules of the network the results of the Boolean dynamics area good generator of hypotheses andmay be used as a first stepin the searching for experimental corroborations

5 Conclusions

In this work we have demonstrated in silico that altering thedynamical state of key biomolecules of the proliferative estro-gen regulatory networks is possible to shift the dynamicalstate from a proproliferative towards an antiproliferative oneand vice versaTheProliferation Index presented here despitebeing similar to other indexes used in cancer-related Booleannetworks [21] provides elements of analysis and suggestspossible experimental approaches in terms of altering theestrogen-dependent cell proliferation

These kinds of approaches may be useful to test the usageof different drugs with a known or unknown effects andevaluate the final outcome searching for a more personalizedmedicine

Complexity 9

Appendix

Boolean Networks as Dynamical Systems

This brief appendix provides some definitions of dynamicalsystems and Boolean networks included for the sake of com-pletenessThe study of the dynamical evolution of networkedsystems has been gaining importance and recognition inthe physicsappliedmathematicscomplex systemscomputersciences literature This is so since a wide variety of non-trivial phenomena has been characterized as arising of thedynamic evolution of interdependent agents Features likecooperation spreading and synchronization dynamics onnetworks have been characterized For instance the workof Wang and coworkers [22] presents an application ofnovel centrality measures to account for modified diffusion(spreading) on complex networks while information sharingand cooperation have been characterized in the works of theChengyi group [23 24]

Boolean networks in particular are a class of (determin-istic or stochastic) sequential dynamical systems Booleannetworks usually consist on a (finite) set of Boolean logicvariables governed by a set of finitary functions of the formF B119896 997888rarr B where B = 0 1 is a binary logic or Booleandomain (eg an algebra of logical truth values) (it is possibleto build Boolean domains with more than two logical statesThe formalism extension to these cases is straightforward)and 119896 is the arity (number of arguments or Cartesian productdimension) a nonnegative integer ABoolean function is thusa propositional formula in 119896 variables which takes a series ofinputs from a subset of the Boolean variables and as an outputproduces the state of the corresponding variable The set ofBoolean functions determines the connectivity of the set ofvariables that become the nodes of a network whose topologyis given by the combination of Boolean functions for all thevariables [25 26]

For the Boolean network dynamics the state of thenetwork at a given time 119905 + 1 is determined via the evaluationof each of the variablesrsquo function on the state of the networkat a previous time 119905 This may be done on a synchronous (allnodesrsquo states updated at once) or asynchronous (hierarchicalupdating given the position of a given node in the network)way Depending upon updating procedures the systemrsquosdynamicsmay beMarkovian or non-Markovian (often finite-Markovian) [26 27]

Given the fact that Boolean networks are discrete dynam-ical systems with finite support (there are exactly 2119873 possiblestates on a classicalmdashie 2-statemdashBoolean network with Nnodes) the evolution of the system will produce recurrentstatesThe trajectories will fall into one of a set of steady statesor cycles called attractors The set of attractors of a dynamicalsystem is called the attractor landscape The determinationof the set of attractor states and the convergence dynamicsleading to those attractors constitutes the solution to theBoolean network dynamics problem [27]

The Boolean networks studied here belong to a classof deterministic dynamical systems Such systems may berepresented by a set of differential equations describing thedynamical evolution in phase space Deterministic Boolean

networksmay also be represented as a discrete dynamical sys-tem (a map) that when iterated reproduces the full dynamicsof the network including the set of attractors This was theway we proceeded here Since iterated maps and differentialequations are two equivalent representations of the evolutionof a dynamical system [28] our approach does not loose anygenerality

Data Availability

All relevant data has been included in the supplementarymaterials

Conflicts of Interest

The authors have no conflict of interest to declare

Authorsrsquo Contributions

Guillermo de Anda-Jauregui Jesus Espinal-Enrıquez andSantiago Sandoval-Motta contributed equally to this work

Acknowledgments

The research leading to these results has received fundingfrom Consejo Nacional de Ciencia y Tecnologıa (grantnumber 2855442016 Ciencia Basica and 21152017 Fron-teras de la Ciencia (Jesus Espinal-Enrıquez)) as well asfederal funding from the National Institute of GenomicMedicine (EnriqueHernandez-Lemus) EnriqueHernandez-Lemus also acknowledges support from the 2016 MarcosMoshinsky Research Chair in the Physical Sciences JesusEspinal-Enrıquez acknowledges support from FundacionMiguel Aleman in Health Research Santiago Sandoval-Motta acknowledges support from the program CatedrasCONACYT The funders had no role in the design of thisresearch

Supplementary Materials

Supplementary 1 Supplementary Material 1 Regulatoryfunctions of the estrogen transcriptional networks Eachfile contains the regulatory function for all those genes inthe network including the regulatory genes as well as thediscrete value of the target gene after taking into account thevalue of its regulatorsSupplementary 2 Supplementary Material 2 Bibliographicevidence associated with the proliferative and antiprolifera-tive nature of the genes in the network

References

[1] W E Stumpf ldquoNuclear concentration of 3H-estradiol in targettissues Dry-mount autoradiography of vagina oviduct ovarytestis mammary tumor liver and adrenalrdquo Endocrinology vol85 no 1 pp 31ndash37 1969

[2] J Cui Y Shen and R Li ldquoEstrogen synthesis and signalingpathways during aging from periphery to brainrdquo Trends inMolecular Medicine vol 19 no 3 pp 197ndash209 2013

Complexity 9

Appendix

Boolean Networks as Dynamical Systems

This brief appendix provides some definitions of dynamicalsystems and Boolean networks included for the sake of com-pletenessThe study of the dynamical evolution of networkedsystems has been gaining importance and recognition inthe physicsappliedmathematicscomplex systemscomputersciences literature This is so since a wide variety of non-trivial phenomena has been characterized as arising of thedynamic evolution of interdependent agents Features likecooperation spreading and synchronization dynamics onnetworks have been characterized For instance the workof Wang and coworkers [22] presents an application ofnovel centrality measures to account for modified diffusion(spreading) on complex networks while information sharingand cooperation have been characterized in the works of theChengyi group [23 24]

Boolean networks in particular are a class of (determin-istic or stochastic) sequential dynamical systems Booleannetworks usually consist on a (finite) set of Boolean logicvariables governed by a set of finitary functions of the formF B119896 997888rarr B where B = 0 1 is a binary logic or Booleandomain (eg an algebra of logical truth values) (it is possibleto build Boolean domains with more than two logical statesThe formalism extension to these cases is straightforward)and 119896 is the arity (number of arguments or Cartesian productdimension) a nonnegative integer ABoolean function is thusa propositional formula in 119896 variables which takes a series ofinputs from a subset of the Boolean variables and as an outputproduces the state of the corresponding variable The set ofBoolean functions determines the connectivity of the set ofvariables that become the nodes of a network whose topologyis given by the combination of Boolean functions for all thevariables [25 26]

For the Boolean network dynamics the state of thenetwork at a given time 119905 + 1 is determined via the evaluationof each of the variablesrsquo function on the state of the networkat a previous time 119905 This may be done on a synchronous (allnodesrsquo states updated at once) or asynchronous (hierarchicalupdating given the position of a given node in the network)way Depending upon updating procedures the systemrsquosdynamicsmay beMarkovian or non-Markovian (often finite-Markovian) [26 27]

Given the fact that Boolean networks are discrete dynam-ical systems with finite support (there are exactly 2119873 possiblestates on a classicalmdashie 2-statemdashBoolean network with Nnodes) the evolution of the system will produce recurrentstatesThe trajectories will fall into one of a set of steady statesor cycles called attractors The set of attractors of a dynamicalsystem is called the attractor landscape The determinationof the set of attractor states and the convergence dynamicsleading to those attractors constitutes the solution to theBoolean network dynamics problem [27]

The Boolean networks studied here belong to a classof deterministic dynamical systems Such systems may berepresented by a set of differential equations describing thedynamical evolution in phase space Deterministic Boolean

networksmay also be represented as a discrete dynamical sys-tem (a map) that when iterated reproduces the full dynamicsof the network including the set of attractors This was theway we proceeded here Since iterated maps and differentialequations are two equivalent representations of the evolutionof a dynamical system [28] our approach does not loose anygenerality

Data Availability

All relevant data has been included in the supplementarymaterials

Conflicts of Interest

The authors have no conflict of interest to declare

Authorsrsquo Contributions

Guillermo de Anda-Jauregui Jesus Espinal-Enrıquez andSantiago Sandoval-Motta contributed equally to this work

Acknowledgments

The research leading to these results has received fundingfrom Consejo Nacional de Ciencia y Tecnologıa (grantnumber 2855442016 Ciencia Basica and 21152017 Fron-teras de la Ciencia (Jesus Espinal-Enrıquez)) as well asfederal funding from the National Institute of GenomicMedicine (EnriqueHernandez-Lemus) EnriqueHernandez-Lemus also acknowledges support from the 2016 MarcosMoshinsky Research Chair in the Physical Sciences JesusEspinal-Enrıquez acknowledges support from FundacionMiguel Aleman in Health Research Santiago Sandoval-Motta acknowledges support from the program CatedrasCONACYT The funders had no role in the design of thisresearch

Supplementary Materials

Supplementary 1 Supplementary Material 1 Regulatoryfunctions of the estrogen transcriptional networks Eachfile contains the regulatory function for all those genes inthe network including the regulatory genes as well as thediscrete value of the target gene after taking into account thevalue of its regulatorsSupplementary 2 Supplementary Material 2 Bibliographicevidence associated with the proliferative and antiprolifera-tive nature of the genes in the network

References

[1] W E Stumpf ldquoNuclear concentration of 3H-estradiol in targettissues Dry-mount autoradiography of vagina oviduct ovarytestis mammary tumor liver and adrenalrdquo Endocrinology vol85 no 1 pp 31ndash37 1969

[2] J Cui Y Shen and R Li ldquoEstrogen synthesis and signalingpathways during aging from periphery to brainrdquo Trends inMolecular Medicine vol 19 no 3 pp 197ndash209 2013

10 Complexity

[3] F Pedeutour B J Quade S Weremowicz P Dal Cin S Aliand C C Morton ldquoLocalization and expression of the humanestrogen receptor beta gene in uterine leiomyomatardquo GenesChromosomes and Cancer vol 23 no 4 pp 361ndash366 1998

[4] L Giacinti P P Claudio M Lopez and A Giordano ldquoEpi-genetic information and estrogen receptor alpha expression inbreast cancerrdquoThe Oncologist vol 11 no 1 pp 1ndash8 2006

[5] G DeAnda-Jauregui R AMejıa-Pedroza J Espinal-Enrıquezand E Hernandez-Lemus ldquoCrosstalk events in the estrogensignaling pathwaymay affect tamoxifen efficacy in breast cancermolecular subtypesrdquoComputational Biology andChemistry vol59 pp 42ndash54 2015

[6] P Ascenzi A Bocedi and M Marino ldquoStructure-functionrelationship of estrogen receptor 120572 and 120573 Impact on humanhealthrdquo Molecular Aspects of Medicine vol 27 no 4 pp 299ndash402 2006

[7] J Thakar M Pilione G Kirimanjeswara E T Harvill andR Albert ldquoModeling systems-level regulation of host immuneresponsesrdquo PLoS Computational Biology vol 3 no 6 Article IDe109 2007

[8] A Saadatpour R-S Wang A Liao et al ldquoDynamical andstructural analysis of a t cell survival network identifies novelcandidate therapeutic targets for large granular lymphocyteleukemiardquo PLoS Computational Biology vol 7 no 11 Article IDe1002267 2011

[9] J Espinal M Aldana A Guerrero C Wood A Darszon andGMartınez-Mekler ldquoDiscrete dynamics model for the speract-activated Ca 2+ signaling network relevant to sperm motilityrdquoPLoS ONE vol 6 no 8 Article ID e22619 2011

[10] J Espinal-Enrıquez A Darszon A Guerrero and GMartınez-Mekler ldquoIn Silico determination of the effect of multi-targetdrugs on calcium dynamics signaling network underlying seaurchin spermatozoa motilityrdquo PLoS ONE vol 9 no 8 ArticleID e104451 2014

[11] J Espinal-Enrıquez D A Priego-Espinosa A Darszon CBeltran andGMartınez-Mekler ldquoNetworkmodel predicts thatCatSper is themainCa2+ channel in the regulation of sea urchinsperm motilityrdquo Scientific Reports vol 7 no 1 article no 42362017

[12] S Perez-Landero S Sandoval-Motta C Martınez-Anaya et alldquoComplex regulation of Hsf1-Skn7 activities by the catalyticsubunits of PKA in Saccharomyces cerevisiae Experimentaland computational evidencesrdquo BMC Systems Biology vol 9 no1 article no 42 2015

[13] S Barbosa B Niebel SWolf KMauch and R Takors ldquoA guideto gene regulatory network inference for obtaining predictivesolutions Underlying assumptions and fundamental biologicaland data constraintsrdquo BioSystems vol 174 pp 37ndash48 2018

[14] B A McGregor S Eid A E Rumora et al ldquoConserved tran-scriptional signatures in human andmurine diabetic peripheralneuropathyrdquo Scientific Reports vol 8 no 1 2018

[15] S Barman and Y-K Kwon ldquoA novel mutual information-based Boolean network inference method from time-seriesgene expression datardquo PLoS ONE vol 12 no 2 Article IDe0171097 2017

[16] Z-P Liu C Wu H Miao and H Wu ldquoRegNetwork Anintegrated database of transcriptional and post-transcriptionalregulatory networks in human and mouserdquoDatabase vol 2015pp 1ndash12 2015

[17] A Kramer J Green J Pollard and S Tugendreich ldquoCausalanalysis approaches in ingenuity pathway analysisrdquo Bioinfor-matics vol 30 no 4 pp 523ndash530 2014

[18] N A OrsquoLeary M W Wright J R Brister et al ldquoReferencesequence (RefSeq) database at NCBI Current status taxonomicexpansion and functional annotationrdquo Nucleic Acids Researchvol 44 no 1 pp D733ndashD745 2016

[19] D Worku F Jouhra G W Jiang N Patani R F Newbold andK Mokbel ldquoEvidence of a tumour suppressive function of E2F1gene in human breast cancerrdquo Anticancer Reseach vol 28 no 4B pp 2135ndash2139 2008

[20] H Tovar R Garcıa-Herrera J Espinal-Enrıquez and EHernandez-Lemus ldquoTranscriptional master regulator analysisin breast cancer genetic networksrdquo Computational Biology andChemistry vol 59 pp 67ndash77 2015

[21] J Espinal-Enriquez R A Meja-Pedroza and E Hernndez-Lemus ldquoA Boolean network model for invasive thyroid carci-nomardquo in Proceedings of the Artificial Life Conference 2016 pp570ndash577 Cancun Mexico July 2016

[22] J Wang C Li and C Xia ldquoImproved centrality indicatorsto characterize the nodal spreading capability in complexnetworksrdquo Applied Mathematics and Computation vol 334 pp388ndash400 2018

[23] C Xia X Li Z Wang and M Perc ldquoDoubly effects ofinformation sharing on interdependent network reciprocityrdquoNew Journal of Physics vol 20 no 7 Article ID 075005 2018

[24] C Chen Y Hu and L Li ldquoNRP1 is targeted by miR-130aand miR-130b and is associated with multidrug resistance inepithelial ovarian cancer based on integrated gene networkanalysisrdquoMolecular Medicine Reports vol 13 no 1 pp 188ndash1962016

[25] M Leone A Pagnani G Parisi and O Zagordi ldquoFinite sizecorrections to random Boolean networksrdquo Journal of StatisticalMechanics Theory and Experiment no 12 Article ID P120122006

[26] B Derrida and Y Pomeau ldquoRandom networks of automata Asimple annealed approximationrdquo EPL (Europhysics Letters) vol1 no 2 pp 45ndash49 1986

[27] U Bastolla and G Parisi ldquoThe modular structure of Kauffmannetworksrdquo Physica D Nonlinear Phenomena vol 115 no 3-4pp 219ndash233 1998

[28] M W Hirsch R L Devaney and S Smale Differential Equa-tions Dynamical Systems and Linear Algebra vol 6 AcademicPress New York NY USA 1974

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom