Institute of Mathematical Sciences and Physics, University of the Philippines Los Baños, College, Laguna, 4031 Philippines
Correspondence: Jomar Fajardo Rabajante E-mail: [email protected] Received: March 25, 2015 Published online: April 07, 2015
Modern biology will never be the same without mathematical and computational tools. Using mind map with “epigenetics” as the root, we discuss the current advancement in the field of biomathematics for modeling cell-fate specification. In the discussions, we also present possible directions for future research.
To cite this article: Jomar Fajardo Rabajante, et al. Mathematical modeling of cell-fate specification: From simple to complex epigenetics. Stem Cell Epigenetics 2015; 2: e752. doi: 10.14800/sce.752.
In this era of information technology, the discipline of biomathematics (mathematical biology) is facing rapid development. In theoretical biology, mathematics has been used to formulate hypotheses about complex biological systems. Mathematics, computational science and bioinformatics provide tools to aid experimental and observational biologists. The data driven -omics (genomics, transcriptomics, proteomics, etc.) also benefit from mathematical techniques. The outcomes of mathematical models can guide discoveries and can help minimize cumbersome trial-and-error experiments. One of the progressive applications of biomathematics is in the field of epigenetics, particularly in modeling cell-fate specification.
In this brief review, we use mind maps to guide our discussion. Mind mapping is a novel way of writing reviews and organizing the multifaceted ideas in a fast-paced and broad field, such as epigenetics. Every year, new quantitative and qualitative investigations are published in different journals [1-104]. Visual diagrams, such as mind maps, can aid researchers to summarize the important results and explore the trending topics. We start by assessing the fundamental concepts in epigenetics, then branching out to deliberate on
REVIEW
Epigenetics
Mathematical models
Waddington's landscape
Stem cell & cell-‐fate
specification
Figure 1. The focus in this brief review is mathematical modeling in epigenetics of cell-fate specification. These four keywords (epigenetics, Waddington’s landscape, stem cell & cell-fate specification, and mathematical models) lead to multitude of search results in literatures.
new outcomes and future directions. Here we focus on mathematical modeling in epigenetics of cell-fate specification (Fig. 1).
Cell-fate specification is one of the important processes studied in epigenetics (Fig. 2). Stem cells undergo cell division to proliferate as stem cells (self-renewal) or to differentiate towards specific family/lineage of cell types (cell differentiation). The cells undergoing cell differentiation “decide” their future cell type (cell-fate specification or determination) based on the associated gene regulation and cell signaling. Stem cells are classified based on its potency, that is, the ability to differentiate to a number of types. Currently, pluripotency (ability to differentiate to almost all cells) and multipotency (ability to differentiate to many cells of related lineages) are buzzwords in this area of study. Pluripotent and multipotent stem cells are important in the formation of tissues and organs. Embryonic stem cells and induced pluripotent stem cells (iPSCs) are common examples of pluripotent cells. The umbilical cord blood, adipose tissue and bone marrow are also sources of pluripotent stem cells. We are not going to elaborate the biology of stem cells since their fundamental aspects including cell differentiation (e.g., transcription, chromatin remodeling, histone modification, DNA methylation, epigenetic clock, epigenetic memory, OCT4-SOX2- NANOG, polycomb repressive complex) are now available in standard biology books [8] as well as from various review and research articles [1-30].
For their discovery of iPSCs, Shinya Yamanaka and John Gurdon were awarded the Nobel Prize in Physiology/Medicine [101]. These iPSCs are proof that mature specialized cells can be reprogrammed back to undifferentiated states --- the process known as dedifferentiation. Several methods for acquiring iPSCs and for cellular engineering have been developed, such as transcription-factor viral transduction, footprint-free nonviral methods, reprogramming with small molecules, cell fusion, and nuclear transfer [4,6,7,11,13,21,22,25,27]. The area of cellular reprogramming has numerous successful studies but some were subjected to controversies. Scientific misconduct resulted in the retraction of published papers on stimulus-triggered acquisition of pluripotency (STAP) [102].
Dedifferentiation as well as transdifferentiation (process of switching to other cell lineage) are already popular topics in biomedicine, biotechnology and bioengineering (Fig. 3). Future studies can learn from some plants and animals that undergo normal body regeneration [28,29] --- which implies that self-renewal by stem cells are vigorously effective to cure degenerative diseases and injuries. The field of stem cell research is in its golden era because of the vast research results available today. Stem cells promise extensive therapeutic applications in regenerative medicine, such as possible treatment of neurodegenerative disorders (e.g., ALS, Parkinson’s and Alzheimer’s), cardiovascular diseases, osteoarthritis, diabetes, wounds, and vision impairment [1,9,10,14,16,18,19,24]. Stem cells also become an interest in
dermatology [5]. One of the most important applications of stem cells is in cancer research, which is greatly influenced by discoveries in epigenetics [2,33,48,65,66,74,94]. Further studies on these applications are underway to ensure the safety and efficiency of stem cell therapies [1-7,9-26,30,35,48,52,53,55,58].
Waddington’s epigenetic landscape (named “creode”) is one of the cartoon diagrams that has been quantified to portray the dynamics of cell-fate specification [31-44] (Fig. 4 and 5). When a stem cell undergoes cell division, three possibilities can happen to the daughter cells: (i) both are identical to the parent cell, (ii) both are already differentiated, or (iii) one is identical to the original and the other already differentiated. Cells that undergo differentiation have several cell lineages to choose from, but the probability of their cell fates are based on some pattern formation. The creode by Waddington illustrates the paths (lineages) that a cell might take (Fig. 4). In Waddington’s model, cell differentiation is depicted by a ball descending a landscape of hills and valleys. The valleys where the ball settles without rolling further can be regarded as “attractors” that represent cell types. The hill ridges that separate the valleys portray the boundaries between cell lineages and prevent (up to some level) switching. The height of hills and depth of valleys determine the potential of cell differentiation (similar to gravitational potential).
Gene regulatory networks determine the topography of the epigenetic landscape. A pluripotent cell has high network entropy because it can differentiate towards many cell lineages [32,34]. The topography of Waddington’s landscape is said to be dynamic [32]. Changes in the topography may be part of normal processes or may entail mutations. This is the reason why epigenetic landscape has been hypothesized to have a connection with the fitness landscape in evolutionary biology [32,73]. Epigenetics is believed to affect transgenerational inheritance (transferred from parents to offspring) [103]. The dynamic topography may also dictate cells to differentiate normally, become cancer cells (due to aberration in gene regulation), or undergo apoptosis/cell death [33]. Currently, there are many variations of the landscape, such as landscape with circular paths [23,32,35]. The epigenetic landscape is not necessarily linear and one-directional because cells are plastic and reprogrammable.
The goal of mathematical modeling is to help solve emerging problems in biology and allied fields by offering quantitative techniques, analysis and solutions (Fig. 6 and Box 1). Biomathematicians aim to solve biological problems by translating biological ideas (e.g., ‘cartoon’ diagrams in biology) into the language of mathematics, then interpreting the mathematical results back to biological terms. Biological systems are too complex to be investigated as they are, where capability of current computational tools as one of the major limiting factors. Abstract mathematical models can be used to focus on a specific biological system or phenomenon. These models can then be manipulated and analyzed. For example, mathematical algorithms and in silico experiments are being used in drug discovery, especially for personalized medicine [48,65,66,71]. The logical structure in mathematics can represent physical phenomena and interacting systems in nature. In many cases, modeling and simulations can provide the necessary calculations to propose answers to the questions of biologists. Mathematical predictions and experiments propose that gene expressions can be influenced by controling the efficiency of transcription, degradation rate of proteins, amount of exogenous stimuli or inhibitor, strength and direction (from activation to repression or vice-versa) of protein-protein interaction, and kinetics of protein auto-activation [32,33,61]. However, it should be noted that mathematical models have limitations. Models are abstraction of real world phenomena, and the models are as good as the assumptions used. A useful model must capture the elements of reality with acceptable accuracy.
Self-‐renewal/proliferation
Stem cell priming
dedifferentiationdifferentiation
transdifferentiation
Figure 3. Cell-differentiation. Colored circles represent genes. The sizes of the circles determine lineage bias. Priming is represented by colored circles with equal sizes. The largest circle governs the possible phenotype of the cell. Differentiated cells lose the ability to self-renew.
The commonly used mathematical techniques in biology are statistics, probability theory, stochastic processes and simulations (e.g., Markov Chain, kinetic Monte Carlo), dynamical systems, differential equations (ordinary, delay, partial, stochastic), linear algebra, abstract algebra, graph and network theory, game theory, optimization (e.g.,
mathematical programming) and metaheuristics (e.g., genetic algorithm, neural networks), and automata. However, the techniques are not limited to this list. We can formulate hybrid models employing different mathematical techniques or we can use techniques from other branches of mathematics that are not usually employed (or not have been
X2
X1
Figure 4. Epigenetic landscape. Left: Waddington’s epigenetic landscape based on the original illustration in [31]. The cells represented by the balls roll down a terrain of hills and valleys. Right: Phase portrait of a differential equation model with a fixed set of parameter values (e.g., see Box 1) that portrays a scenario of ‘attraction’ in the epigenetic landscape.
gene regulatory network
cell lineages, probability of phenotypes, network
entropy
linear pathways, circular pathways
dynamic topography, mutations
Waddington's landscape
pattern formation, attractors
height of ridges, depth of valleys, landscape
potential
Figure 5. Mind map with “Waddington’s landscape” as root.
used yet) in biology to discover novel dynamics --- this is called the Blue Ocean strategy. For example, we can try using non-Euclidean geometries to represent the epigenetic landscape, and see if this can give us new perspectives. These different but not necessarily conflicting perspectives are common in epigenetics. For instance, the attractor in Waddington’s landscape have several mathematical
interpretations --- it can be an equilibrium point, a strange chaotic attractor, noisy attractor, or a region with cloud of states converging together [32,69,74,87,92-94,98]. Indeed, mathematical modelers need to have two important qualities: critical thinking and creativity. For general details about various mathematical models in epigenetics and gene regulation, see [45-60].
Figure 6. Visual diagram showing the basic steps of the modeling process [49,59]. Here we use the model in Box 1 as example [32,61]. Mathematical modeling is a continuous process (spiral). Models are continuously being improved to capture the real behavior of the biological system.
Mathematical modeling of decision-‐making in cell-‐fate specification
Observe and gather input datae.g., efficiency, degradation rate and basal growth rate of the gene regulatory factors;
strength of interaction among
the factors
List the simplifying and justifiable assumptionse.g., nonlinear
kinetics, continuous time and state space, deterministic dynamics
Design the modele.g., network
representation of the gene
interactions, ordinary
differential equation in Box 1
Analyze the modele.g., stability and
bifurcation analysis, numerical
simulation
Interpret the results
e.g., what is the meaning of the multistable
equilibrium points and the oscillations in terms of the epigenetic landscape?
Validate the resultse.g., compare the simulation results with exisitng data from literatures,
Intra-cellular Inter-cellular + environment No feedback system Multi-feedback system
Reductionist thinking Systems thinking
want to include more realistic assumptions in a model to improve its robustness in understanding the full mechanisms of cell-fate specification and its associated diseases, without sacrificing reliability, usefulness and validity of the model results. Table 1 shows the directions that future researches may take. Researchers, especially systems and synthetic biologists, have already started extending simple models to become more multifaceted. Still, there are more that can be explored in the realm of mathematical modeling that could help us uncover novel dynamics. For example, the usual ordinary differential equation (ODE) models can be extended to include stochasticity, delay and spatial aspects. One may use stochastic DE, delay DE and partial DE, but one may use intensive Monte Carlo individual/agent-based simulations. Moreover, perturbation and sensitivity analyses should always be done to check the robustness of the model and of the tool for analysis against instabilities and noise. Bifurcation analysis can also be performed to examine how the behavior of the biological system (e.g., gene regulation) changes with parameter values.
Models are not the ultimate goal of biomathematics research. Both simple and sophisticated models have similar goals --- to solve biological problems using mathematics. We also need to answer several assessment questions during the modeling process, such as
Is my goal to describe only the biological systems or to make predictions?;
Do I need to use mechanistic models or phenomenological models?; and
Am I modeling correlations or causality?
Moreover, most of the time, a single model is insufficient to
describe a real and complex biological phenomenon. Several models may be employed in different modules of the research project. For example, in network pharmacology [65,66,71], we use bioinformatics algorithms to search pool of data for the factors that define a disease. Then, we use different models to formulate and optimize drugs to target those factors and reduce side effects.
Close collaboration of biomathematicians (applied mathematicians, physicists, computational chemists and bioengineers) with biologists are important in making successful exploration of epigenetics. We need to relate interdisciplinary studies at the molecular level to the dynamics at the cellular level, and the cellular level to the development of tissues and organs --- just like the goal of physicists in uniting the general theory of relativity to quantum theory. For example, we need to answer the question: How can we relate Waddington’s landscape to the spatial pattern formation of tissues and of the whole organism? Likewise, future studies can focus from a generalist model to a specialized model, such as models describing the epigenetics of certain organs (e.g., brain, skin, heart), of certain species (e.g., microorganisms, animals, plants), or of certain diseases (e.g., diabetes, Parkinson’s disease, cancer). However, unless the outcome of a mathematical model is confirmed experimentally, it will stay as a hypothesis.
This brief review is clearly not enough to present all the significant breakthroughs in epigenetics of cell-fate specification. However, the advantage of mind mapping is that the maps can be extended to include more ideas not mentioned in this review. The definition of “epigenetics” has evolved in recent years. While working definition is important, we cannot be restricted by it because epigenetics is evolving and progressing [3,100]. There are several groups that are at the forefront of mathematical modeling and cell-fate specification research, such as the group of Sui Huang [104]. Additional references about mathematical modeling are listed here [61-99].
Author contributions
All authors are involved in drafting the manuscript and in revising it. JFR is the lead author. The authors declare that they have no conflicting interests.
References
1. Bruin JE, Saber N, Braun N, Fox JK, Mojibian M, Asadi A, et al. Treating diet-induced diabetes and obesity with human embryonic stem cell-derived pancreatic progenitor cells and antidiabetic drugs. Stem Cell Reports 2015; 4:1-16.
2. Baxter E, Windloch K, Gannon F, Lee JS. Epigenetic regulation in
Table 1. Possible directions for future researches: from simple to complex models
3. Benitah SA, Bracken A, Dou Y, Huangfu D, Ivanova N, Koseki H, et al. Stem cell epigenetics: looking forward. Cell Stem Cell 2014; 14:706-209.
4. David L, Polo JM. Phases of reprogramming. Stem Cell Res 2014; 12(3):754-761.
5. Ogliari KS, Brum DE, Marinowic D, Loth F. Stem cells in dermatology. An Bras Dermatol 2014; 89:286-291.
6. Tonge PD, Corso AJ, Monetti C, Hussein SM, Puri MC, Michael IP, et al. Divergent reprogramming routes lead to alternative stem-cell states. Nature 2014; 516:192-197.
7. Buganim Y, Faddah DA, Jaenisch R. Mechanisms and models of somatic cell reprogramming. Nat Rev Genet 2013; 14:427-439.
8. Reece JB, Urry LA, Cain ML, Wasserman SA, Minorsky PV, Jackson RB. Campbell Biology. 10th Edition. San Francisco: Pearson Benjamin Cummings; 2013.
9. Eguizabal C, Montserrat N, Veiga A, Belmonte JCI. Dedifferentiation, transdifferentiation, and reprogramming: future directions in regenerative medicine. Semin Reprod Med 2013; 31:82-94.
10. Harding J, Roberts RM, Mirochnitchenko O. Large animal models for stem cell therapy. Stem Cell Res Ther 2013; 4:23.
11. Malik N, Rao MS. A review of the methods for human iPSC derivation. Methods Mol Biol 2013; 997:23-33.
12. Feil R, Fraga MF. Epigenetics and the environment: emerging patterns and implications. Nat Rev Genet 2012; 13:97-109.
13. Yamanaka S. Induced pluripotent stem cells: past, present, and future. Cell Stem Cell 2012; 10:678-684.
14. Gupta PK, Das AK, Chullikana A, Majumdar AS. Mesenchymal stem cells for cartilage repair in osteoarthritis. Stem Cell Res Ther 2012; 3:25.
15. Balazsi G, van Oudenaarden A, Collins JJ. Cellular decision making and biological noise: from microbes to mammals. Cell 2011; 144:910-925.
16. Fortier LA, Travis AJ. Stem cells in veterinary medicine. Stem Cell Res Ther 2011; 2:9.
17. Zhou JX, Huang, S. Understanding gene circuits at cell-fate branch points for rational cell reprogramming. Trends Genet 2011; 27:55-62.
18. Christen B, Robles V, Raya M, Paramanov I, Belmonte JCI. Regeneration and reprogramming compared. BMC Biology 2010; 8:5.
19. Dantuma E, Merchant S, Sugaya K. Stem cells for the treatment of neurodegenerative diseases. Stem Cell Res Ther 2010; 1:37.
20. Hanna JH, Saha K, Jaenisch R. Pluripotency and cellular reprogramming: facts, hypotheses, unresolved issues. Cell 2010; 143:508-525.
21. Pera MF, Tam PPL. Extrinsic regulation of pluripotent stem cells. Nature 2010; 465:713-720.
23. Enver T, Pera M, Peterson C, Andrews PW. Stem cell states, fates and the rules of attraction. Cell Stem Cell 2009; 4:387-397.
24. Nadig RR. Stem cell therapy - hype or hope? A review. J Conserv Dent 2009; 12:131-138.
25. Yamanaka S. Elite and stochastic models for induced pluripotent stem cell generation. Nature 2009; 460:49-52.
26. Magnus T, Liu Y, Parker GC, Rao MS. Stem cell myths. Phil Trans R Soc B 2008; 363:9-22.
27. Takahashi K, Yamanaka S. Induction of pluripotent stem cells from mouse embryonic and adult fibroblast cultures by defined factors. Cell 2006; 126:663-676.
28. Stocum DL. Amphibian Regeneration and Stem Cells. In Regeneration: Stem Cells and Beyond. Current Topics in Microbiology and Immunology Volume 280. 1st Edition. Edited by Heber-Katz E. Berlin/Heidelberg: Springer; 2004:1-70.
29. Giles KL. Dedifferentiation and regeneration in bryophytes: A selective review. New Zeal J Bot 1971; 9:689-694.
30. The National Academies. Understanding Stem Cells: An Overview of the Science and Issues from the National Academies [http://nas-sites.org/stemcells]
31. Waddington CH. The Strategy of the Genes. London: George Allen & Unwin; 1957.
32. Rabajante JF, Babierra AL. Branching and oscillations in the epigenetic landscape of cell-fate determination. Prog Biophys Mol Biol 2015; doi: 10.1016/j.pbiomolbio.2015.01.006.
33. Li C, Wang J. Quantifying the underlying landscape and paths of cancer. J R Soc Interface 2014; 11:20140774.
34. Banerji CR, Miranda-Saavedra D, Severini S, Widschwendter M, Enver T, Zhou JX, et al. Cellular network entropy as the energy potential in Waddington’s differentiation landscape. Sci. Rep. 2013; 3:3039.
35. Ladewig J, Koch P, Brustle O. Leveling Waddington: the emergence of direct programming and the loss of cell fate hierarchies. Nat Rev Mol Cell Biol 2013; 14:225-236.
36. Li C, Wang J. Quantifying Waddington landscapes and paths of non-adiabatic cell fate decisions for differentiation, reprogramming and transdifferentiation. J R Soc Interface 2013; 10:20130787.
37. Ferrell JE Jr. Bistability, bifurcations, and Waddington’s epigenetic landscape. Curr Biol 2012; 22:R458-R466.
38. Li M, Liu G-H, Belmonte JCI. Navigating the epigenetic landscape of pluripotent stem cells. Nat Rev Mol Cell Biol 2012; 13:524-535.
39. Pujadas E, Feinberg AP. Regulated noise in the epigenetic landscape of development and disease. Cell 2012; 148:1123-1131.
40. Takahashi K. Cellular reprogramming - lowering gravity on Waddington’s epigenetic landscape. J Cell Sci 2012; 125:2553-2560.
41. Bhattacharya S, Zhang Q, Andersen ME. A deterministic map of Waddington’s epigenetic landscape for cell fate specification. BMC Syst Biol 2011; 5:85.
42. Huang S. The molecular and mathematical basis of Waddington’s epigenetic landscape: A framework for post-Darwinian biology? Bioessays 2011; 34:149-157.
43. Wang J, Zhang K, Xu L, Wang E. Quantifying the Waddington landscape and biological paths for development and
differentiation. Proc Natl Acad Sci USA 2011; 108(20):8257-8262.
44. Wang J, Xu L, Wang E, Huang S. The potential landscape of genetic circuits imposes the arrow of time in stem cell differentiation. Biophys J 2010; 99:29-39.
45. Wang P, Song C, Zhang H, Wu Z, Tian XJ, Xing J. Epigenetic state network approach for describing cell phenotypic transitions. Interface Focus 2014; 4:20130068.
46. Wolkenhauer O. Why model? Front Physiol. 2014; 5:21.
47. Bernot B, Comet JP, Richard A, Chaves M, Gouze JL, Dayan F. Modeling and Analysis of Gene Regulatory Networks. In Modeling and Computational Biology and Biomedicine: A Multidisciplinary Endeavor. Volume 28. 1st Edition. Edited by Cazals F, Kornprobst P. Heidelberg: Springer-Verlag; 2013:47-80.
48. Creixell P, Schoof EM, Erler JT, Linding R. Navigating cancer network attractors for tumor-specific therapy. Nat Biotechnol 2012; 30:842-848.
49. Voit EO. A First Course in Systems Biology. New York: Garland Science; 2012.
50. Huang S. Systems biology of stem cells: three useful perspectives to help overcome the paradigm of linear pathways. Phil Trans R Soc B 2011; 366:2247-2259.
51. Radde N. The role of feedback mechanisms in biological network models. Asian J Control 2011; 13:597-610.
52. Lim SJ, Tan TW, Tong JC. Computational Epigenetics: the new scientific paradigm. Bioinformation 2010; 4:331-337.
53. Liu E, Lauffenburger D (Eds). Systems Biomedicine: Concepts and Perspectives. Massachusetts: Academic Press; 2009.
54. Shmulevich I, Aitchison JD. Deterministic and Stochastic Models of Genetic Regulatory Networks. In Methods in Enzymology. Volume 467. 1st Edition. Edited by Johnson ML, Brand L. Amsterdam: Elsevier BV; 2009:335-356.
55. McArthur BD, Ma’ayan A, Lemischka IR. Systems biology of stem cell fate and cellular reprogramming. Nat Rev Mol Cell Biol 2009; 10:672-681.
56. Tracik G, Bialek W. Cell Biology: Networks, Regulation and Pathways. In Encyclopedia of Complexity and Systems Science. Edited by Meyers RA. New York: Springer; 2009:719-741.
57. Aguda BD, Friedman A. Models of Cellular Regulation. New York: Oxford Univ. Press; 2008.
58. Bock C, Lengauer T. Computational epigenetics. Bioinformatics 2008; 24:1-10.
59. Voit EO, Qi Z, Miller GW. Steps of modeling complex biological systems. Pharmacopsychiatry 2008; 41(Suppl 1):S78- S84
60. Schlitt T, Brazma A. Current approaches to gene regulatory network modelling. BMC Bioinformatics 2007; 8(Suppl 6):S9.
61. Rabajante JF, Talaue CO. Equilibrium switching and mathematical properties of nonlinear interaction networks with concurrent antagonism and self-stimulation. Chaos Soliton Fract 2015; 73:166-182.
63. Bogdan P, Deasy BM, Gharaibeh B, Roehrs T, Marculescu R. Heterogeneous structure of stem cells dynamics: statistical models and quantitative predictions. Sci Rep 2014; 4:4826.
64. Chung KM, Kolling FW 4th, Gajdosik MD, Burger S, Russell AC, Nelson CE. Single Cell Analysis Reveals the Stochastic Phase of Reprogramming to Pluripotency Is an Ordered Probabilistic Process. PLoS ONE 2014; 9: e95304.
65. El Baroudi M, La Sala D, Cinti C, Capobianco E. Pathway landscapes and epigenetic regulation in breast cancer and melanoma cell lines. Theor Biol Med Model 2014; 11:S8.
66. Huether R, Dong L, Chen X, Wu G, Parker M, Wei L, et al. The landscape of somatic mutations in epigenetic regulators across 1,000 pediatric cancer genomes. Nat Commun 2014; 5:3630.
67. Lang AH, Li H, Collins JJ, Mehta P. Epigenetic Landscapes Explain Partially Reprogrammed Cells and Identify Key Reprogramming Genes. PLoS Comput Biol 2014; 10:e1003734.
68. Morris R, Sancho-Martinez I, Sharpee TO, Izpisua Belmonte JC. Mathematical approaches to modeling development and reprogramming. Proc Natl Acad Sci USA 2014; 111:5076-5082.
69. Nikolov S, Wolkenhauer O, Vera J. Tumors as chaotic attractors. Mol Biosyst 2014; 10:172-179.
70. Przybilla J, Rohlf T, Loeffler M, Galle J. Understanding epigenetic changes in aging stem cells - a computational model approach. Aging Cell 2014; 13:320-328.
71. Shi SH, Cai YP, Cai XJ, Zheng XY, Cao DS, Ye FQ, et al. A Network Pharmacology Approach to Understanding the Mechanisms of Action of Traditional Medicine: Bushenhuoxue Formula for Treatment of Chronic Kidney Disease. PLoS ONE 2014; 9:e89123.
72. Verd B, Crombach A, Jaeger J. Classification of transient behaviours in a time-dependent toggle switch model. BMC Syst Biol 2014; 8:43.
73. Foster DV, Rorick MM, Gesell T, Feeney LM, Foster JG. Dynamic landscapes: A model of context and contingency in evolution. J Theor Biol 2013; 334:162-172.
74. Huang S, Kauffman S. How to escape the cancer attractor: Rationale and limitations of multi-target drugs. Semin Cancer Biol 2013; 23:270-278.
75. Sasai M, Kawabata Y, Makishi K, Itoh K, Terada TP. Time scales in epigenetic dynamics and phenotypic heterogeneity of embryonic stem cells. PLoS Comput Biol 2013; 9:e1003380.
76. Teles J, Pina C, Eden P, Ohlsson M, Enver T, Peterson C. Transcriptional regulation of lineage commitment - a stochastic model of cell fate decisions. PLoS Comput Biol 2013; 9:e1003197.
77. Wu M, Su RQ, Li X, Ellis T, Lai YC, Wang X. Engineering of regulated stochastic cell fate determination. Proc Natl Acad Sci USA 2013; 110:10610-10615.
78. Flöttmann M, Scharp T, Klipp E. A stochastic model of epigenetic dynamics in somatic cell reprogramming. Front Physiol 2012; 3:216.
79. Kim KH, Sauro HM. Adjusting phenotype by noise control. PLoS Comput Biol 2012; 8:e1002344.
80. Nene NR, Zaikin A. Interplay between path and speed in decision making by high-dimensional stochastic gene regulatory networks.
81. Andrecut M, Halley JD, Winkler DA, Huang S. A general model for binary cell fate decision gene circuits with degeneracy: indeterminacy and switch behavior in the absence of cooperativity. PLoS ONE 2011; 6:e19358.
82. Suzuki N, Furusawa C, Kaneko K. Oscillatory protein expression dynamics endows stem cells with robust differentiation potential. PLoS ONE 2011; 6:e27232.
83. Nordon RE, Ko KH, Odell R, Schroeder T. Multi-type branching models to describe cell differentiation programs. J Theor Biol 2011; 277:7-18.
84. Zhang L, Zheng Y, Li D, Zhong Y. Self-organizing map of gene regulatory networks for cell phenotypes during reprogramming. Comput Biol Chem 2011; 35:211-217.
85. Gonze D. Coupling oscillations and switches in genetic networks. BioSystems 2010; 99:60-69.
86. Graham TGW, Tabei SMA, Dinner AR, Rebay I. Modeling bistable cell-fate choices in the Drosophila eye: qualitative and quantitative perspectives. Development 2010; 137:2265-78.
87. Huang S. Cell Lineage Determination in State Space: A Systems View Brings Flexibility to Dogmatic Canonical Rules. PLoS Biol 2010; 8:e1000380.
88. Ito Y, Uchida K. Formulas for intrinsic noise evaluation in oscillatory genetic networks. J Theor Biol 2010; 267:223-234.
89. Micheelsen MA, Mitarai N, Sneppen K, Dodd IB. Theory for stability and regulation of epigenetic landscapes. Phys Biol 2010; 7:026010.
90. Zhdanov VP. Periodic perturbation of the bistable kinetics of gene expression. Phys A 2010; 390:57-64.
91. Foster DV, Foster JG, Huang S, Kauffman SA. A model of sequential branching in hierarchical cell fate determination. J Theor Biol 2009; 260:589-597.
92. Furusawa C, Kaneko K. Chaotic expression dynamics implies pluripotency: when theory and experiment meet. Biol Direct 2009 4:17.
93. Huang S. Reprogramming cell fates: reconciling rarity with robustness. Bioessays 2009; 31:546-560.
94. Huang S, Ernberg I, Kauffman S. Cancer attractors: A systems view of tumors from a gene network dynamics and developmental perspective. Semin Cell Dev Biol 2009; 20:869-876.
95. Alvarez-Buylla ER, Chaos A, Aldana M, Benitez M, Cortes-Poza Y, Espinosa-Soto C, et al. Floral Morphogenesis: Stochastic Explorations of a Gene Network Epigenetic Landscape. PLoS ONE 2008; 3:e3626.
96. Andrecut M, Cloud D, Kauffman SA. Monte Carlo simulation of a simple gene network yields new evolutionary insights. J Theor Biol 2008; 250:468-474.
97. MacArthur BD, Please CP, Oreffo ROC. Stochasticity and the molecular mechanisms of induced pluripotency. PLoS ONE 2008; 3:e3086.
98. Ribeiro A, Kauffman SA. Noisy attractors and ergodic sets in models of gene regulatory networks. J Theor Biol 2007; 247:743-755.
99. Huang S, Eichler G, Bar-Yam Y, Ingber DE. Cell fates as high-dimensional attractor states of a complex gene regulatory network. Phys Rev Lett 2005; 94:128701.
100. Ledford H. Language: Disputed definitions. Nature 2008; 455:1023-1028.
101. The Nobel Prize in Physiology or Medicine 2012 Press Release [http://www.nobelprize.org/nobel_prizes/medicine/laureates/2012/press.html]
102. The Japan Times: Haruko Obokata [http://www.japantimes.co.jp/tag/haruko-obokata]
103. Hughes V. Epigenetics: The sins of the father. The roots of inheritance may extend beyond the genome, but the mechanisms remain a puzzle [http://www.nature.com/news/epigenetics-the-sins-of-the-father-1.14816]
104. Institute for Systems Biology, Sui Huang group [https://www.systemsbiology.org/sui-huang-group]
