# CABeRNET: a Cytoscape app for Augmented Boolean models of gene Regulatory NETworks

###### Abstract

Background. Dynamical models of gene regulatory networks (GRNs) are highly effective in describing complex biological phenomena and processes, such as cell differentiation and cancer development. Yet, the topological and functional characterization of real GRNs is often still partial and an exhaustive picture of their functioning is missing.

Motivation. We here introduce CABeRNET, a Cytoscape app for the generation, simulation and analysis of Boolean models of GRNs, specifically focused on their augmentation when a only partial topological and functional characterization of the network is available. By generating large ensembles of networks in which user-defined entities and relations are added to the original core, CABeRNET allows to formulate hypotheses on the missing portions of real networks, as well to investigate their generic properties, in the spirit of complexity science.

Results. CABeRNET offers a series of innovative simulation and modeling functions and tools, including (but not being limited to) the dynamical characterization of the gene activation patterns ruling cell types and differentiation fates, and sophisticated robustness assessments, as in the case of gene knockouts. The integration within the widely used Cytoscape framework for the visualization and analysis of biological networks, makes CABeRNET a new essential instrument for both the bioinformatician and the computational biologist, as well as a computational support for the experimentalist. An example application concerning the analysis of an augmented T-helper cell GRN is provided.

## Introduction

Consistently with the increasing availability of big data regarding biological systems, is the need of mathematical and computational models aimed at their effective analysis and interpretation [Kit01, Kit02]. To this end, in the spirit of statistical physics and complex systems science, even simplified and abstract models have proven effective in representing complex biological phenomena, with specific focus on the emergent dynamical behaviours and the so-called generic or universal properties [Kau93, Kan06].

In the context of genomics data, one of the best examples is that provided by Boolean models of gene regulatory networks (GRNs), which have repeatedly proved fruitful in describing key properties of real systems, as well as in providing cues and hints for wet-lab experiments (see, e.g.,[Kau95, SDKZ02, SDZ02, KPST03, SVS04, KPST04, SKA05, RKYH06, SVGK07, SVG08, CQL10]). The simulation of partially characterized regulatory architectures with a Boolean approach, in particular, has recently gained attention (see, e.g., [SvHT97, AMK99, ST01, AO03, LLL04, CAES06, FNCT06, KSRL06, DB08]). A first motivation lies in the inherently “dynamical” nature of gene (de)regulation processes, and in the clear limitations of a “static” analysis capturing only a partial picture of such complex processes. For example, a structural analysis of the genomic interactions might preclude to determine the influence of a target-selective therapy on the overall GRN interplay ruling tumorigenesis.

Moreover, as compared to continuous models, such as ODE-based or stochastic models (see [KS08] for a review on GRN modeling), the Boolean abstraction allows for a clear and effective characterization of the gene activation patterns (or attractors in the terminology of dynamical systems) characterizing the different phenotypic functions, such as cell types, modes and fates, under the metaphor of “emergent collective behaviours”[Kau93, HI00, HEBYI05, FN06, RK07, HEK09, CSKJ10, FCP10, FK12, CYA13, NWV14]. In this context, phenomena such as tumorigenesis can be explained as rare emergent pattern, triggered by signals, stochastic fluctuations and biological noise (see, e.g.,[EE10, Tsi14]).

In line with this approach, we here continue on a recent research strand that has involved the development of simulation and analysis tools for dynamical Boolean GRNs (see, e.g., [ABC13]). The foundations of our theoretical framework lay in the seminal work by Stuart Kauffman on Random Boolean Networks (RBNs) [Kau69b, Kau69a] and, more recently, on the dynamical model of cell differentiation introduced in [SVB10, VBS11] and based on Noisy Random Boolean Networks (NRBNs). In this theoretical framework, cell types are associated to dynamical gene activation patterns and differentiation to cell-specific noise-resistance mechanisms, the underlying hypothesis being that such mechanisms refine along with differentiation stages (see [HCH08] and references therein). In the Background Section the main features of this approach are outlined.

In this paper we introduce CABeRNET, a Cytoscape [SMO03] app for the generation, import, simulation and analysis of Boolean models of GRNs. With respect to similar approaches, e.g., [LK11] or the earlier work by our group [ABC13], CABeRNET also allows to augment partially-characterized GRNs by adding user-defined entities and relations. The underlying motivation is that, despite the increasing knowledge on gene regulation in real organisms, the topological and functional characterization of real networks is stil far from being comprehensive, thus preventing to capture the complexity of the overall interplay: CABeRNET is conceived to randomly generate the missing portions of the partially-characterized GRNs, in order to obtain ensembles of augmented GRNs that share the common core and other structural and functional parameters, hence allowing to test hypotheses on the yet unknown missing portions of real networks. Besides, in line with the complex systems approach, the generic properties of such ensembles can be inferred and investigated, possibly providing an ensemble-level understanding of yet unraveled GRN phenomena. In addition, CABeRNET guides the user through various robustness analyses of the GRNs, which can be matched against genomic experimental data. Notice that CABeRNET can also generate completely random GRNs with defined structural and functional constraints, as well as simulate the dynamics of completely characterized GRNs. In the Implementation Section the main features of CABeRNET are described.

As a proof of principle, in the Results Section we present the augmentation of the T-helper cell signaling network and describe the analysis of its dynamics, with particular focus on the emergent differentiation scheme and its robustness against the knockout of specific genes. Conclusions are presented in the last Section.

## Background: the GRN model

In CABeRNET, GRNs are represented as Noisy Random Boolean Networks [VBS11]. A classical RBN is a graph composed of Boolean nodes associated with binary variables , representing the activation of a gene: if the th gene synthesizes its product (i.e., proteins or RNAs), otherwise it does not. Each node is connected to nodes which represent its regulatory inputs, i.e., those genes influencing the activation of the th gene. The way in which regulation takes place is via a node-specific Boolean function of the regulatory inputs: the value of at time is determined as . The dynamics of canonical RBNs is synchronous, deterministic and follows discrete time steps. Thus, it eventually ends up in (at least unitary) state cycles, termed attractors, which represent the emerging gene activation patterns displayed by the GRN. Remarkably, by varying the structural features of the networks, different dynamical regimes appear, ranging from ordered to disordered behaviors (see [Kau93]).

In [VBS11] Villani et al. introduced a form of stochasticity in RBNs, hence speaking of “Noisy RBNs”. NRBNs allow for the transitions among attractors as a consequence of random “flips” of ’s value, i.e., random modifications of the node’s activation value lasting a defined time span. Thus, with this approach it is possible to determine the so-called Attractors Transition Network (ATN, or matrix, ATM), i.e., a stability matrix displaying the probability of a gene activation pattern to emerge in response to a perturbation in another pattern (noise-induced transition). A noise threshold is then introduced to exclude transitions unlikely to occur for a cell in a certain differentiation stage. Therefore, multiple thresholds allow to identify distinct threshold-dependent ATNs, each one representing the stability of the GRN’s activation patterns characterizing a specific differentiation stage (e.g., toti-/multi-potents stem cells, progenitors, differentiated cells, etc). The exact mathematical definition of such patterns is given as Threshold Ergodic Sets (TESs), which: are strongly connected components in the threshold-dependent ATN, and have no outward transitions outside the components’ set. These structures are hierarchical, thus can be naturally mapped to cell types yielding to emergent differentiation trees. Such trees mimic the metaphor that less differentiated stages have less refined noise-control mechanisms, resulting hence in TES associated to lower thresholds (see, e.g., [HKG97, HLS08, FK09]). In general, the number of patterns observable in a certain cell type decreases progressively along with the differentiation stage (see Fig. 1 for a simplified representation of the model).

Notice that the approach is general, it does not refer to any specific organism, and has been shown to be suitable to describe different degrees of differentiation, i.e., from toti-/multi-potent stem cells to intermediate states, to fully differentiated cells; the stochastic differentiation process, in which a population of toti-/multi-potent cells stochastically generates progenies of distinct types; and the induced pluripotency phenomenon, according to which fully differentiated cells can revert to a pluripotent stage through the perturbation of some key genes [Yam09].

## Implementation

CABeRNET is a Java tool developed as Cytoscape’s version 3.x application; see the Availability Section for information about download and installation of the tool. CABeRNET sessions are user-defined batch computations. Parameters are specified by a step-by-step wizard and various post-simulation functions are accessible directly from the CABeRNET menu in the Cytoscape active window. The tool implements a wide range of simulation and analysis functions, which can be summarily listed as follows (see Fig. 2 for a summary of functions and parameters, and the user manual, on the plugin website, for a detailed description). Instructions on obtaining and using the code used in the paper are available on the page website. Tutorials include a package to reproduce the example application discussed in the paper (see Availability).

### Input GRNs (generation, import and augmentation)

#### [Random network generation]

CABeRNET can randomly generate and simulate ensembles of GRNs with certain structural parameters such as: number of genes; ingoing/outgoing GRN’s topology, e.g., fixed, Erds-Rényi’s random [ER59], Barabasi-Alberts’ scale-free [BA99] or Watts-Strogatz’s small-world [WS98]; type of regulation functions (node-specific Boolean functions). Concerning the latter, these can be set by the user or randomly generated to accomodate: bias-based random functions, canalyzing [KPST04] or logical functions - expressed in the canonical AND/OR notation. Network and simulation parameters (e.g., samples size) can be defined either via an input form or a textual file.

#### [Import of characterized GRNs]

GRNs that are characterized with respect to both the topology and the regulatory functions, either fetched from public datasets via Cytoscape or loaded via input textual file, can be processed with CABeRNET.

#### [Augmentation of GRNs]

Any GRN loaded in the tool (see above) can be augmented by CABeRNET, by randomly generating a chosen number of augmented networks, in which entities and relations are added to the input network according to user-defined structural and functional parameters. Parameters for augmentation are the same that must be defined for the random generation (see above). The resulting ensemble of augmented networks will share the input topological and functional core and will differ for the randomly generated portion.

For instance, in the Results Section, the T-helper cell signaling network curated from [KSRL06] (40 genes and 51 regulatory interactions), is augmented with 160 further nodes and 349 edges according to a Erdos-Rényi random topology. Assessment of the functional effect of this augmented network is also discussed.

### GRN’s dynamics simulation

Given that the space of the possible configurations of a NRBN can be dramatically large (there are possible configurations for a network with nodes), CABeRNET can simulate the dynamics of a network by either: sampling the initial conditions to test or performing an exhaustive search (for small networks only). CABeRNET allows to investigate key statistics of the emerging attractors such as, e.g., number, length, robustness and reachability. The stability of any pattern to perturbations can be assessed either via temporary flips (with duration of 1 step) or via permanent gene knock-in/knock-out.

Following the simulation, CABeRNET can compute and display the threshold-dependent ATN (here called the TES network) for specific threshold values. Different views on such a network are available in the tool and, for instance, allow to display the genes’ configurations in a pattern, and the variation of the number of TESs alongside thresholds, as proposed in [GDP14].

### GRN-selection constrained by differentiation scheme

One might look for a GRN giving rise to a specific differentiation tree, as done e.g. in [GCDMA14]. Trees can be inputed to CABeRNET in textual format or from the Cytoscape active window; CABeRNET can select those NRBNs that display a differentiation tree structurally similar to the loaded one, where the measure of similarity is defined by the user. This feature is implemented as a batch process scanning among generated NRBNs. Notice that, usually, a single NRBN exhibits various emerging trees, according to the various possible combinations of thresholds thus this feature is the computationally most demanding in CABeRNET.

Besides, the statistically representative differentiation tree(s) of each specific network, defined as the most frequent emerging tree for different (sampled) threshold combinations, provided a specific tree depth, can be computed.

### Visualization

Each computational task is tracked by a progress bar. Once the simulation of the dynamics is completed, the powerful visualization capabilities of Cytoscape can be used to analyze the topological and dynamical properties of the networks.

In particular, with CABeRNET it is possible to visualize: the NRBNs, the attractor graph network, in which all the states of the attractors and the transitions among them are displayed, the threshold dependent ATNs and the representative differentiation tree. By clicking on a specific network, it gets visualized within Cytoscape, so that it can be further analyzed.

Different network styles have been defined and can be selected: CABeRNET network, aimed at visualizing the properties of the NRBN: the color of each node being related to the Boolean function bias and the size of each node proportional to its degree, CABeRNET attractors, for the visualization of the attractor graph network, CABeRNET TES, for the visualization of the ATN and CABeRNET collapsed TES, for the collapsed visualization of the ATN: in these last two styles, the edge size is proportional to the transition probability.

### Robustness analysis

Different kind of perturbations can be applied to a simulated network. In particular, it is possible to perform a user-defined number of temporary (i.e., flips) or permanent (i.e., knock-in/knock-out) perturbations on a chosen number of randomly selected nodes or specific nodes. The robustness analysis can be performed on single networks or on the whole ensemble of simulated GRNs.

Network’s stability is assessed via robustness analyses, by means of standard measures such as avalanches (i.e., the number of nodes whose activation pattern is different in a perturbation experiment with respect to the wild type scenario) and sensitivity (i.e., the number of perturbation experiments in which a certain node’s pattern is affected) [SVGK07, GSV11b]. The results of the analyses can be exported in csv files.

### Network analysis

CABeRNET offers a wide range of network-specific statistics. These include the distribution of the attractors’ lengths, the basins of attraction, the proportion of frozen and oscillating nodes, plus other classical network measures such as clustering coefficient, network diameter and average path length. All the statistics can be visualized and exported; further network measures are accessible via the network analysis tools included in Cytoscape.

### Outputs

All the networks and the relative topological, functional and dynamical properties can be exported as textual files, from both the Wizard and the Function menu. For instance, the complete topological and functional description of the networks can be exported so that it can be used in simulation environments external to Cytoscape such as, e.g., CHASTE [PFPB09], CompuCell3D [STB12] or the simulator described in [GCDMA14] (see [DMGA13] for a recent review on multiscale models of multicellular systems). Also, it is possible to export the complete description of all the attractor states, as well as information of their basins on attraction.

## Results

Our group has recently been focusing on the investigation of the dynamical properties of multicellular systems via multiscale simulations, with particular attention to the conditions that would favor the emergence and development of tumors. To this end, CABeRNET was recently used to generate, simulate and visualize the GRNs ruling the behaviour of an intestinal crypt in CHASTE’s multiscale simulation engine, allowing to identify conditions for cancer’s emergence and crypt’s colonization [RGC15]. In the following, we propose a further example to show some of the potential applications of CABeRNET.

#### [Augmentation of T-helper signaling network]

The signaling network of human T-helper cells was recently characterized with respect to both the topology and the regulatory functions. In [MX06] the dynamics of such network was simulated with a Boolean approach and it was shown that the attractors actually reproduce real gene activation patterns of distinctly differentiated T-helper cell types. With CABeRNET the same dynamical analysis could have been easily performed.

In the following, we present an experiment of GRN’s augmentation possible in CABeRNET. To the best of our knowledge, no experiments of this sort are possible with other RBN-based tools. The final goal is to: generate a large ensemble of random networks with the T-helper functional core, and select only those networks in which the emergent dynamical behaviour is in accordance with the hematopoietic differentiation tree, in which the T-helper cell type is supposed to be one of the leaves (see Fig. 3).

The distinct networks differ for the augmented portion, which is randomly generated with structural parameters (i.e., Erdos-Renyi random topology, average connectivity , random Boolean functions with bias ) that are classically used in similar studies (see, e.g., [SVGK07, GSV11a]). Accordingly, only certain networks will eventually display the desired emergent dynamical behavior. The underlying idea is that the matching networks might allow for the formulation of hypotheses on the missing portions of the relevant GRN ruling the overall hematopoiesis process. Besides, the characterization of the attractors could be matched with the real gene activation patterns driving the functioning of the various cell types.

In the experiment, a large number of distinct augmented networks was generated and simulated with CABeRNET: only 1 on a total of 600 augmented NRBNs actually displayed the expected dynamical behavior, i.e., the correct differentiation tree, hinting at the complexity of the tuning process driven by the evolutionary pressure that led to the topology of current GRNs.

In Fig. 3 one can see the original T-helper signaling network, originally mapped in [KSRL06], and the NRBN that was selected as correct, in which the augmented portion is highlighted. Notice that, in the augmented network, both the topology and the functions of the original core are slightly different from those of the T-helper GRN, as a consequence of adding new relations linking the new and the original portions of the net. The visualization of the network is provided via CABeRNET, by applying the suitable styles (see above).

In Fig. 3 one can notice that this specific network exhibits 8 distinct attractors (each one characterized by a length equal to 8 NRBN time steps). By pruning the ATN with increasingly larger thresholds, the TES at the higher level, including all the 8 attractors connected by noise-induced transitions and representing multi-/toti-potent cells, progressively splits in TESs enclosing an increasingly lower number of attractors, up to the 7 TESs at the lower level, which correspond to single attractors, when the threshold is equal to 1.

The resulting emergent differentiation tree perfectly matches that of hematopoietic cells (taken from [LIYT13], see Fig. 3), which is characterized by a multi-potent progenitor (MPP; antecedent hematopoietic stem cells, HSCs, are not shown in the scheme), with the potential to differentiate into two lineages, i.e., common myeloid progenitor (CMP) and common lymphoid progenitor (CLP). CMP further divide into megakaryocyte-erythroid progenitor (MEP) and granulocyte/monocyte progenitor (GMP), finally committing to mature blood cells including erythrocytes (EC), megakaryocyte (MK), monocyte (M) and granulocytes, i.e. neutrophils, eosinophils, basophils (N/E/B). Conversely, CLP further differentiate into B-cell progenitors (B PROG) and T-cell and natural killer cell progenitors (T/NK PROG), with a final commitment to mature B cells (B), T cells (T) and NK cells (NK).

Hence, it is possible to hypothesize that this specifically selected NRBN might present topological and functional properties ensuring the correct emergent differentiation scheme. We remark that, in case augmented networks were more likely to display a matching emergent tree, one may exploit CABeRNET to perform ensemble-level analyses on the matching set, aimed at the formulation of hypotheses on the generic properties of real networks.

A robustness analysis on the matching NRBN was also performed. By simulating selective single knockouts of the genes in the original T-helper core, we can assess the distinctive relevance in maintaining the correct differentiation scheme. In this example, we performed 40 single knockout experiments (KO), by forcing the specific Boolean function of each gene to inactivation (i.e., 0 output for any regulatory input), and we tried to match the resulting differentiation trees with that of hematopoietic cells. Remarkably, in 35 cases (), the KO experiment resulted in a mismatching tree, hinting at the role of those specific genes in the interplay leading to the emergence of the hematopoietic tree. We measured the similarity among the hematopoietic tree, , and a tree resulting from a KO experiment as follows:

where and are the maximum depth and the maximum number of a node’s children in both and . Function returns the number of nodes at level with children in tree ; thus, this quantity measures the structural level-by-level similarity of two trees by assessing the number of parents with children, per level. Since we focus on differentiation trees, this can be interpreted as a measure of the ability of a certain cell type, a progenitor, to differentiate in a set of distinct subtypes.

In Fig. 3 we show values of in our experiments; the lower the value the closer is to . Values of range around 9, with a maximum of 17 and minimum 0; in 8 cases, the value lower than 5 suggests a close similarity between the emergent and the hematopoietic tree. Besides, the dynamics turned out to be completely insensitive (i.e., ) to the KO of 5 specific genes, i.e., CRE, Ca, PLCg-a, Fos and Gads, which, accordingly, might be not relevant in the differentiation process. Clearly, further investigations are needed to corroborate this hypothesis.

## Conclusions

In this work we introduced CABeRNET – a new Cytoscape app for the generation, simulation and analysis of augmented Boolean models of gene regulatory networks – and described some of its key functionalities, as well as an example application to real GRN data.

CABeRNET is the final result of a long-time effort aimed at bridging different fields and disciplines, such as computer science, statistics and complex systems science, for the effective study of complex biological systems. The numerous modeling and simulation functionalities, the various effective analysis tools and the fine integration within the widely used Cytoscape framework, might settle the ground for CABeRNET becoming a powerful instrument for bioinformaticians and computational biologists, especially in providing a computational support for experimentalists.

In particular, CABeRNET can provide an essential tool to effectively investigate key and still partially undeciphered biological phenomena, such as, e.g., gene regulation, cell differentiation and tumorigenesis, with particular focus on the properties of dynamical gene activation patterns and their relation with biological noise.

### Availability and requirements

Project name:
CABeRNET: a Cytoscape app for the generation and the Analysis of Boolean models of gene Regulatory NETworks

Version:
1.0

Plugin website:
http://bimib.disco.unimib.it/index.php/CABERNET

Operating systems:
platform independent

Software requirement:
Cytoscape 3.x (http://www.cytoscape.org/)

Programming language:
Java

License:
BSD-like license (see website)

## Acknowledgements

This project was partially supported by the ASTIL program, project “RetroNet”, grant n. 12-4-5148000-40; U.A 053, and by NEDD Project [ID14546A Rif SAL-7] Fondo Accordi Istituzionali 2009.

## References

- [ABC13] M. Antoniotti, G.D. Bader, G. Caravagna, S. Crippa, A. Graudenzi, and G. Mauri. GESTODifferent: a Cytoscape plugin for the generation and the identification of gene regulatory networks describing a stochastic cell differentiation process. Bioinformatics, 29(4):513–14, 2013.
- [AMK99] Tatsuya Akutsu, Satoru Miyano, Satoru Kuhara, et al. Identification of genetic networks from a small number of gene expression patterns under the boolean network model. In Pacific Symposium on Biocomputing, volume 4, pages 17–28. Citeseer, 1999.
- [AO03] R. Albert and H.G Othmer. The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in drosophila melanogaster. J. Theor. Biol., 223:1–18, 2003.
- [BA99] A.-L. Barabasi and R. Albert. Emergence of scaling in random networks. Science, 286:509–512, 1999.
- [CAES06] Alvaro Chaos, Max Aldana, Carlos Espinosa-Soto, Berenice García Ponce de León, Adriana Garay Arroyo, and Elena R Alvarez-Buylla. From genes to flower patterns and evolution: dynamic models of gene regulatory networks. Journal of Plant Growth Regulation, 25(4):278–289, 2006.
- [CQL10] Daizhan Cheng, Hongsheng Qi, and Zhiqiang Li. Analysis and control of Boolean networks: a semi-tensor product approach. Springer Science & Business Media, 2010.
- [CSKJ10] M.A. Canham, A.A. Sharov, M.S.H. Ko, and Brickmanm J.M. Functional heterogeneity of embryonic stem cells revealed through translational amplification of an early endodermal transcript. PLoS Biol., 8(5):e1000379, 2010.
- [CYA13] Wei-Yi Cheng, Tai-Hsien Ou Yang, and Dimitris Anastassiou. Biomolecular events in cancer revealed by attractor metagenes. PLoS computational biology, 9(2):e1002920, 2013.
- [DB08] Maria I Davidich and Stefan Bornholdt. Boolean network model predicts cell cycle sequence of fission yeast. PloS one, 3(2):e1672, 2008.
- [DMGA13] G. De Matteis, A. Graudenzi, and M. Antoniotti. A review of spatial computational models for multi-cellular systems, with regard to intestinal crypts and colorectal cancer development. Journal of Mathematical Biology, 66(7):1409–1462, 2013.
- [EE10] A. Eldar and M.B Elowitz. Functional roles for noise in genetic circuits. Nature, 467:167–173, 2010.
- [ER59] P. Erdős and A. Rényi. On random graphs. Publicationes Mathematicae, 6:290–297, 1959.
- [FCP10] Nadia Felli, Luciano Cianetti, Elvira Pelosi, Alessandra Carè, Chang G Liu, George A Calin, Simona Rossi, Cesare Peschle, Giovanna Marziali, and Alessandro Giuliani. Hematopoietic differentiation: a coordinated dynamical process towards attractor stable states. BMC systems biology, 4(1):85, 2010.
- [FK09] C. Furusawa and K. Kaneko. Chaotic expression dynamics implies pluripotency: when theory and experiment meet. Biol. Direct., 4, 2009.
- [FK12] Chikara Furusawa and Kunihiko Kaneko. A dynamical-systems view of stem cell biology. Science, 338(6104):215–217, 2012.
- [FN06] G. Forgacs and S.A. Newman. Biological physics of the developing embryo. Cambridge University Press, 2006.
- [FNCT06] Adrien Fauré, Aurélien Naldi, Claudine Chaouiya, and Denis Thieffry. Dynamical analysis of a generic boolean model for the control of the mammalian cell cycle. Bioinformatics, 22(14):e124–e131, 2006.
- [GCDMA14] A. Graudenzi, G. Caravagna, G. De Matteis, and M. Antoniotti. Investigating the relation between stochastic differentiation, homeostasis and clonal expansion in intestinal crypts via multiscale modeling. PLoS ONE, 9(5):e97272, 2014.
- [GDP14] Alex Graudenzi, Chiara Damiani, Andrea Paroni, Alessandro Filisetti, Marco Villani, Roberto Serra, and Marco Antoniotti. Investigating the role of network topology and dynamical regimes on the dynamics of a cell differentiation model. In Advances in Artificial Life and Evolutionary Computation, pages 151–168. Springer, 2014.
- [GSV11a] A. Graudenzi, R. Serra, M. Villani, C. Damiani, A. Colacci, and S.A. Kauffman. Dynamical properties of a boolean model of gene regulatory network with memory. Journal of Computational Biology, 18(10):1291–1303, 2011.
- [GSV11b] Alex Graudenzi, Roberto Serra, Marco Villani, Annamaria Colacci, and Stuart A Kauffman. Robustness analysis of a boolean model of gene regulatory network with memory. Journal of Computational Biology, 18(4):559–577, 2011.
- [HCH08] M. Hoffman, H. H. Chang, S. Huang, D. E. Ingber, M. Loeffler, and J. Galle. Noise driven stem cell and progenitor population dynamics. PLoS ONE, 3:e2922, 2008.
- [HEBYI05] Sui Huang, Gabriel Eichler, Yaneer Bar-Yam, and Donald E Ingber. Cell fates as high-dimensional attractor states of a complex gene regulatory network. Physical review letters, 94(12):128701, 2005.
- [HEK09] S. Huang, I. Ernberg, and S.A. Kauffman. Cancer attractors: a systems view of tumors from a gene network dynamics and developmental perspective. Semin Cell Dev Biol., 20(7):869–76, 2009.
- [HI00] S. Huang and D. E. Ingber. Shape-dependent control of cell growth, differentiation, and apoptosis: switching between attractors in cell regulatory networks. Experimental Cell Research, 261(1):91–103, 2000.
- [HKG97] M. Hu, D. Krause, M. Greaves, S. Sharkis, M. Dexter, C. Heyworth, and T. Enver. Multilineage gene expression precedes commitment in the hemopoietic system. Genes Dev., 11:774–785, 1997.
- [HLS08] K. Hayashi, S. M. Lopes, and M. A. Surani. Dynamic equilibrium and heterogeneity of mouse pluripotent stem cells with distinct functional and epigenetic states. Cell Stem Cell, 3:391–440, 2008.
- [Kan06] K. Kaneko. Life: An Introduction to Complex Systems Biology. Springer Berlin Heidelberg, 2006.
- [Kau69a] S.A. Kauffman. Homeostasis and differentiation in random genetic control networks. Nature, 224:177, 1969.
- [Kau69b] S.A. Kauffman. Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol., 22:437–467, 1969.
- [Kau93] S.A. Kauffman. The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, 1993.
- [Kau95] S.A. Kauffman. At home in the universe. Oxford University Press, 1995.
- [Kit01] H. Kitano. Foundations of Systems Biology. MIT Press, 2001.
- [Kit02] H. Kitano. Computational systems biology. Nature, pages 206–210, 2002.
- [KPST03] S.A. Kauffman, C. Peterson, B. Samuelsson, and C. Troein. Random boolean network models and the yeast transcriptional network. Proc. Natl Acad. Sci. USA, 100:14796–14799, 2003.
- [KPST04] S.A. Kauffman, C. Peterson, B. Samuelsson, and C. Troein. Genetic networks with canalyzing boolean rules are always stable. Proc. Natl Acad. Sci. USA, 101:17102–7, 2004.
- [KS08] Guy Karlebach and Ron Shamir. Modelling and analysis of gene regulatory networks. Nature Reviews Molecular Cell Biology, 9(10):770–780, 2008.
- [KSRL06] Steffen Klamt, Julio Saez-Rodriguez, Jonathan A Lindquist, Luca Simeoni, and Ernst D Gilles. A methodology for the structural and functional analysis of signaling and regulatory networks. BMC bioinformatics, 7(1):56, 2006.
- [LIYT13] Wai Feng Lim, Tomoko Inoue-Yokoo, Keai Sinn Tan, Mei I Lai, and Daisuke Sugiyama. Hematopoietic cell differentiation from embryonic and induced pluripotent stem cells. Stem Cell Res Ther, 4(3):71, 2013.
- [LK11] Duc-Hau Le and Yung-Keun Kwon. NetDS: a Cytoscape plugin to analyze the robustness of dynamics and feedforward/feedback loop structures of biological networks. Bioinformatics, 27(19):2767–2768, 2011.
- [LLL04] Fangting Li, Tao Long, Ying Lu, Qi Ouyang, and Chao Tang. The yeast cell-cycle network is robustly designed. Proceedings of the National Academy of Sciences of the United States of America, 101(14):4781–4786, 2004.
- [MX06] Luis Mendoza and Ioannis Xenarios. A method for the generation of standardized qualitative dynamical systems of regulatory networks. Theoretical Biology and Medical Modelling, 3:13, 2006.
- [NWV14] Svetoslav Nikolov, Olaf Wolkenhauer, and Julio Vera. Tumors as chaotic attractors. Molecular BioSystems, 10(2):172–179, 2014.
- [PFPB09] Joe Pitt-Francis, Pras Pathmanathan, Miguel O. Bernabeu, Rafel Bordas, Jonathan Cooper, Alexander G. Fletcher, Gary R. Mirams, Philip Murray, James M. Osborne, Alex Walter, S. Jon Chapman, Alan Garny, Ingeborg M.M. van Leeuwen, Philip K. Maini, Blanca RodrÌguez, Sarah L. Waters, Jonathan P. Whiteley, Helen M. Byrne, and David J. Gavaghan. Chaste: A test-driven approach to software development for biological modelling. Computer Physics Communications, 180(12):2452 – 2471, 2009.
- [RGC15] S. Rubinacci, A. Graudenzi, G. Caravagna, J. Osborne, J. Pitt-Francis, G. Mauri, and M. Antoniotti. Cognac: a chaste plugin for the multiscale simulation of gene regulatory networks driving the spatial dynamics of tissues and cancer. Cancer Informatics, submitted, 2015.
- [RK07] A.S. Ribeiro and S.A. Kauffman. Noisy attractors and ergodic sets in models of gene regulatory networks. J. Theor. Biol., 247:743–755, 2007.
- [RKYH06] P. Ramo, Y. Kesseli, and O Yli-Harja. Perturbation avalanches and criticality in gene regulatory networks. J. Theor. Biol., 242:164–170, 2006.
- [SDKZ02] Ilya Shmulevich, Edward R Dougherty, Seungchan Kim, and Wei Zhang. Probabilistic boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics, 18(2):261–274, 2002.
- [SDZ02] Ilya Shmulevich, Edward R Dougherty, and Wei Zhang. From boolean to probabilistic boolean networks as models of genetic regulatory networks. Proceedings of the IEEE, 90(11):1778–1792, 2002.
- [SKA05] I. Shmulevich, S.A. Kauffman, and M. Aldana. Eukaryotic cells are dynamically ordered or critical but not chaotic. Proc. Natl Acad. Sci. USA, 102:13439–44, 2005.
- [SMO03] P. Shannon, A. Markiel, O. Ozier, N. S. Baliga, J. T. Wang, D. Ramadge, N. Amin, B. Schwikowski, and T. Ideker. Cytoscape: a software environment for integrated models of biomolecular interaction networks. Genome Res., 13(11):2498–504, 2003.
- [ST01] L. Sanchez and D. Thieffry. A logical analysis of the drosophila gap-gene system. J. Theor. Biol., 211:115–141, 2001.
- [STB12] M. Swat, G. L. Thomas, J. M. Belmonte, A. Shirinifard, D. Hmeljak, and J. A. Glazier. Multi-Scale Modeling of Tissues Using CompuCell3D. COmputational Methods in Cell Biology – Methods in Cell Biology, 110:325–366, 2012.
- [SVB10] R. Serra, M. Villani, A. Barbieri, S.A. Kauffman, and A. Colacci. On the dynamics of random boolean networks subject to noise: attractors, ergodic sets and cell types. J. Theor. Biol., 265:185–193, 2010.
- [SVG08] Roberto Serra, Marco Villani, Alex Graudenzi, Annamaria Colacci, Stuart A Kauffman, et al. The simulation of gene knock-out in scale-free random boolean models of genetic networks. Networks and heterogeneous media, 3(2):333, 2008.
- [SVGK07] R. Serra, M. Villani, A. Graudenzi, and S.A. Kauffman. Why a simple model of genetic regulatory networks describes the distribution of avalanches in gene expression data. J. Theor. Biol., 249:449–460, 2007.
- [SvHT97] L. Sanchez, J. van Helden, and D. Thieffry. Establishement of the dorso-ventral pattern during embryonic development of drosophila melanogasater: a logical analysis. J. Theor. Biol., 189:377–389, 1997.
- [SVS04] R. Serra, M. Villani, and A. Semeria. Genetic network models and statistical properties of gene expression data in knock-out experiments. J. Theor. Biol., 227:149–157, 2004.
- [Tsi14] Lev S Tsimring. Noise in biology. Reports on Progress in Physics, 77(2):026601, 2014.
- [VBS11] M. Villani, A. Barbieri, and R. Serra. A dynamical model of genetic networks for cell differentiation. PLoS ONE, 6(3):e17703. doi:10.1371/journal.pone.0017703, 2011.
- [WS98] Duncan J Watts and Steven H Strogatz. Collective dynamics of ‘small-world’networks. nature, 393(6684):440–442, 1998.
- [Yam09] H. Yamanaka. Elite and stochastic models for induced pluripotent stem cell generation. Nature, 460:49–52, 2009.