JFRabajante MS Applied Math Thesis

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    MATHEMATICAL STRATEGIES FORPROGRAMMING BIOLOGICAL CELLS

    by

    Jomar F. Rabajante

    A masters thesis submitted to the

    Institute of Mathematics

    College of Science

    University of the Philippines

    Diliman, Quezon City

    as partial fulfillment of the

    requirements for the degree of

    Master of Science in Applied Mathematics

    (Mathematics in Life and Physical Sciences)

    April 2012

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    This is to certify that this Masters Thesis entitled Mathematical Strategies for

    Programming Biological Cells, prepared and submitted by Jomar F. Rabajante

    to fulfill part of the requirements for the degree of Master of Science in Applied

    Mathematics, was successfully defended and approved on March 23, 2012.

    Cherryl O. Talaue, Ph.D.Thesis Co-Adviser

    Baltazar D. Aguda, Ph.D.Thesis Co-Adviser

    Carlene P. Arceo, Ph.D.Thesis Reader

    The Institute of Mathematics endorses the acceptance of this Masters Thesis as partial

    fulfillment of the requirements for the degree of Master of Science in Applied Mathematics

    (Mathematics in Life and Physical Sciences).

    Marian P. Roque, Ph.D.DirectorInstitute of Mathematics

    This Masters Thesis is hereby officially accepted as partial fulfillment of the requirements

    for the degree of Master of Science in Applied Mathematics (Mathematics in Life and

    Physical Sciences).

    Jose Maria P. Balmaceda, Ph.D.Dean, College of Science

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    Brief Curriculum Vitae

    09 October 1984 Born, Sta. Cruz, Laguna, Philippines

    1997-2001 Don Bosco High School, Sta. Cruz, Laguna

    2006 B.S. Applied Mathematics(Operations Research Option)University of the Philippines Los Banos

    2006-2008 Corporate Planning Assistant

    Insular Life Assurance Co. Ltd.

    2008 Professional Service Staff International Rice Research Institute

    2008-present Instructor, Mathematics DivisionInstitute of Mathematical Sciences and PhysicsUniversity of the Philippines Los Banos

    PUBLICATIONS

    Rabajante, J.F., Figueroa, R.B. Jr. and Jacildo, A.J. 2009. Modeling thearea restrict searching strategy of stingless bees, Trigona biroi, as a quasi-random walk process. Journal of Nature Studies, 8(2): 15-21.

    Esteves, R.J.P., Villadelrey, M.C. and Rabajante, J.F. 2010. Determiningthe optimal distribution of bee colony locations to avoid overpopulation us-ing mixed integer programming. Journal of Nature Studies, 9(1): 79-82.

    Castilan, M.G.D., Naanod, G.R.K., Otsuka, Y.T. and Rabajante, J.F. 2011.From Numbers to Nature. Journal of Nature Studies, 9(2)/10(1): 35-39.

    Tambaoan, R.S., Rabajante, J.F., Esteves, R.J.P. and Villadelrey, M.C. 2011.Prediction of migration path of a colony of bounded-rational species foragingon patchily distributed resources. Advanced Studies in Biology, 3(7): 333-345.

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    Table of Contents

    Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

    Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

    List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

    Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    Chapter 2. Preliminaries

    Biology of Cellular Programming . . . . . . . . . . . . . . . . . . . . 42.1 Stem cells in animals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Transcription factors and gene expression . . . . . . . . . . . . . . . . . . 82.3 Biological noise and stochastic differentiation . . . . . . . . . . . . . . . . 10

    Chapter 3. PreliminariesMathematical Models of Gene Networks . . . . . . . . . . . . . . . . 12

    3.1 The MacArthur et al. GRN . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 ODE models representing GRN dynamics . . . . . . . . . . . . . . . . . . 15

    3.2.1 Cinquin and Demongeot ODE formalism . . . . . . . . . . . . . . 163.2.2 ODE model by MacArthur et al. . . . . . . . . . . . . . . . . . . 19

    3.3 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . 20

    Chapter 4. PreliminariesAnalysis of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . 22

    4.1 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Bifurcation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Fixed point iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.4 Sylvester resultant method . . . . . . . . . . . . . . . . . . . . . . . . . . 304.5 Numerical solution to SDEs . . . . . . . . . . . . . . . . . . . . . . . . . 33

    Chapter 5. Results and Discussion

    Simplified GRN and ODE Model . . . . . . . . . . . . . . . . . . . . 355.1 Simplified MacArthur et al. model . . . . . . . . . . . . . . . . . . . . . 355.2 The generalized Cinquin-Demongeot ODE model . . . . . . . . . . . . . 385.3 Geometry of the Hill function . . . . . . . . . . . . . . . . . . . . . . . . 415.4 Positive invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.5 Existence and uniqueness of solution . . . . . . . . . . . . . . . . . . . . 52

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    Chapter 6. Results and DiscussionFinding the Equilibrium Points . . . . . . . . . . . . . . . . . . . . . 57

    6.1 Location of equilibrium points . . . . . . . . . . . . . . . . . . . . . . . . 576.2 Cardinality of equilibrium points . . . . . . . . . . . . . . . . . . . . . . 60

    6.2.1 Illustration 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2.2 Illustration 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    Chapter 7. Results and DiscussionStability of Equilibria and Bifurcation . . . . . . . . . . . . . . . . . . 73

    7.1 Stability of equilibrium points . . . . . . . . . . . . . . . . . . . . . . . . 737.2 Bifurcation of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    Chapter 8. Results and Discussion

    Introduction of Stochastic Noise . . . . . . . . . . . . . . . . . . . . . 85Chapter 9. Summary and Recommendations . . . . . . . . . . . . . . . . . . . . 100

    Appendix A. More on Equilibrium Points: Illustrations . . . . . . . . . . . . . . . 106A.1 Assume n = 2, ci = 1, cij = 1 . . . . . . . . . . . . . . . . . . . . . . . . 107

    A.1.1 Illustration 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108A.1.2 Illustration 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109A.1.3 Illustration 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111A.1.4 Illustration 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

    A.2 Assume n = 2, ci = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

    A.2.1 Illustration 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.2.2 Illustration 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115A.2.3 Illustration 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116A.2.4 Illustration 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116A.2.5 Illustration 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

    A.3 Assume n = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116A.3.1 Illustration 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116A.3.2 Illustration 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117A.3.3 Illustration 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117A.3.4 Illustration 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117A.3.5 Illustration 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

    A.3.6 Illustration 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118A.3.7 Illustration 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118A.3.8 Illustration 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

    A.4 Ad hoc geometric analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.5 Phase portrait with infinitely many equilibrium points . . . . . . . . . . 127

    Appendix B. Multivariate Fixed Point Algorithm . . . . . . . . . . . . . . . . . . 128

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    Appendix C. More on Bifurcation of Parameters:Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

    C.1 Adding gi

    > 0, Illustration 1 . . . . . . . . . . . . . . . . . . . . . . . . . 131C.2 Adding gi > 0, Illustration 2 . . . . . . . . . . . . . . . . . . . . . . . . . 132C.3 gi as a function of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

    C.3.1 As a linear function . . . . . . . . . . . . . . . . . . . . . . . . . . 134C.3.2 As an exponential function . . . . . . . . . . . . . . . . . . . . . . 137

    C.4 The effect ofij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138C.5 Bifurcation diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

    C.5.1 Illustration 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139C.5.2 Illustration 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.5.3 Illustration 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

    Appendix D.Scilab Program for Euler-Maruyama . . . . . . . . . . . . . . . . . . 147

    List of References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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    Acknowledgments

    I owe my deepest gratitude to those who made this thesis possible:

    To Dr. Baltazar D. Aguda from the National Cancer Institute, USA for providing

    the thesis topic, for imparting knowledge about models of cellular regulation, for sim-

    plifying the MacArthur et al. (2008) GRN, for giving his valuable time to answer my

    questions despite long distance communication, and for his patience, unselfish guidance

    and encouragement;To Dr. Cherryl O. Talaue for her all-out support, for spending time checking my

    proofs and editing my manuscript, for granting my requests to write recommendation

    letters, for the guidance, for the encouragement, and for always being available;

    To Dr. Carlene P. Arceo for doing the proofreading of my thesis manuscript despite

    her being on sabbatical leave, and to the members of my thesis panel for the constructive

    criticisms;

    To Mr. Mark Jayson V. Cortez and Ms. Jenny Lynn B. Carigma for checking my

    manuscript for grammatical and style errors as well as for the motivation;To the University of the Philippines Los Banos (UPLB) and to the Math Division,

    Institute of Mathematical Sciences and Physics (IMSP), UPLB for allowing me to go on

    study leave with pay;

    To Dr. Virgilio P. Sison, the Director of IMSP, for all the support and for being the

    co-maker in my DOST scholarship contract;

    To Prof. Ariel L. Babierra, the Head of the Math Division, IMSP and to Dr. Editha

    C. Jose for the invaluable suggestions, help and encouragement;

    To the Philippine Council for Industry, Energy and Emerging Technology Research

    and Development (PCIEERD), Department of Science and Technology (DOST) for the

    generous financial support; and

    To my family for the inspiration, and to El Elyon for the unwavering strength.

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    Abstract

    Mathematical Strategies for Programming Biological Cells

    Jomar F. Rabajante Co-Adviser:University of the Philippines, 2012 Cherryl O. Talaue, Ph.D.

    Co-Adviser:Baltazar D. Aguda, Ph.D.

    In this thesis, we study a phenomenological gene regulatory network (GRN) of a mes-enchymal cell differentiation system. The GRN is composed of four nodes consisting of

    pluripotency and differentiation modules. The differentiation module represents a circuit

    of transcription factors (TFs) that activate osteogenesis, chondrogenesis, and adipogen-

    esis.

    We investigate the dynamics of the GRN using Ordinary Differential Equations (ODE).

    The ODE model is based on a non-binary simultaneous decision model with autocatal-

    ysis and mutual inhibition. The simultaneous decision model can represent a cellular

    differentiation process that involves more than two possible cell lineages. We prove some

    mathematical properties of the ODE model such as positive invariance and existence-

    uniqueness of solutions. We employ geometric techniques to analyze the qualitative

    behavior of the ODE model.

    We determine the location and the maximum number of equilibrium points given

    a set of parameter values. The solutions to the ODE model always converge to a stable

    equilibrium point. Under some conditions, the solution may converge to the zero state.

    We are able to show that the system can induce multistability that may give rise to

    co-expression or to domination by some TFs.

    We illustrate cases showing how the behavior of the system changes when we vary

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    some of the parameter values. Varying the values of some parameters, such as the degra-

    dation rate and the amount of exogenous stimulus, can decrease the size of the basin of

    attraction of an undesirable equilibrium point as well as increase the size of the basin of

    attraction of a desirable equilibrium point. A sufficient change in some parameter values

    can make a trajectory of the ODE model escape an inactive or a dominated state.

    Sufficient amounts of exogenous stimuli affect the potency of cells. The introduc-

    tion of an exogenous stimulus is a possible strategy for controlling cell fate. A dominated

    TF can exceed a dominating TF by adding a corresponding exogenous stimulus. More-

    over, increasing the amount of exogenous stimulus can shutdown multistability of the

    system such that only one stable equilibrium point remains.

    We observe the case where a random noise is present in our system. We add a

    Gaussian white noise term to our ODE model making the model a system of stochastic

    DEs. Simulations reveal that it is possible for cells to switch lineages when the system is

    multistable. We are able to show that a sole attractor can regulate the effect of moderate

    stochastic noise in gene expression.

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    List of Figures

    1.1 Analysis of mesenchymal cell differentiation system. . . . . . . . . . . . 3

    2.1 Stem cell self-renewal, differentiation and programming. This diagram

    illustrates the abilities of stem cells to ploriferate through self-renewal,

    differentiate into specialized cells and reprogram towards other cell types. 5

    2.2 Priming and differentiation. Colored circles represent genes or TFs. The

    sizes of the circles determine lineage bias. Priming is represented by col-

    ored circles having equal sizes. The largest circle governs the possible

    phenotype of the cell. [70] . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    2.3 The flow of information. The blue solid lines represent general flow and

    the blue dashed lines represent special (possible) flow. The red dotted

    lines represent the impossible flow as postulated in the Central Dogma of

    Molecular Biology [41]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    2.4 C. Waddingtons epigenetic landscape creode [168]. . . . . . . . . . 11

    3.1 The coarse-graining of the differentiation module. The network in (a) issimplified into (b), where arrows indicate up-regulation (activation) while

    bars indicate down-regulation (repression). [113] . . . . . . . . . . . . . . 13

    3.2 The MacArthur et al. [113] mesenchymal gene regulatory network. Arrows

    indicate up-regulation (activation) while bars indicate down-regulation (re-

    pression). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    3.3 Gene expression or the concentration of the TFs can be represented by a

    state vector, e.g. ([X1], [X2], [X3], [X4]) [70]. For example, TFs of equal

    concentration can be represented by a vector with equal components, such

    as (2.4, 2.4, 2.4, 2.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    3.4 Hierarchic decision model and simultaneous decision model. Bars repre-

    sent repression or inhibition, while arrows represent activation. [36]. . . . 17

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    4.1 The slope ofF(X) at the equilibrium point determines the linear stability.

    Positive gradient means instability, negative gradient means stability. If

    the gradient is zero, we look at the left and right neighboring gradients.

    Refer to the Insect Outbreak Model: Spruce Budworm in [122]. . . . . . 26

    4.2 Sample bifurcation diagram showing saddle-node bifurcation. . . . . . . . 28

    4.3 An illustration of cobweb diagram. . . . . . . . . . . . . . . . . . . . . . 29

    5.1 The original MacArthur et al. [113] mesenchymal gene regulatory network. 35

    5.2 Possible paths that result in positive feedback loops. Shaded boxes denote

    that the path repeats. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    5.3 The simplified MacArthur et al. GRN . . . . . . . . . . . . . . . . . . . . 375.4 Graph of the univariate Hill function when ci = 1. . . . . . . . . . . . . . 42

    5.5 Possible graphs of the univariate Hill function when ci > 1. . . . . . . . . 43

    5.6 The graph ofY = Hi([Xi]) shrinks as the value ofKi +nj=1,j=i ij [Xj]

    cij

    increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

    5.7 The Hill curve gets steeper as the value of autocatalytic cooperativity ci

    increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

    5.8 The graph ofY = Hi([Xi]) is translated upwards by gi units. . . . . . . . 45

    5.9 The 3-dimensional curve induced by Hi([X1], [X2]) + gi and the plane in-

    duced by i[Xi], an example. . . . . . . . . . . . . . . . . . . . . . . . . . 46

    5.10 The intersections of Y = i[Xi] and Y = Hi([Xi]) + gi with varying values

    of Ki +nj=1,j=i ij[Xj]

    cij , an example. . . . . . . . . . . . . . . . . . . . 47

    5.11 The possible number of intersections of Y = i[Xi] and Y = Hi([Xi]) + gi

    where ci = 1 and gi = 0. The value of Ki +nj=1,j=i ij [Xj ]

    cij is fixed. . . 49

    5.12 The possible number of intersections of Y = i[Xi] and Y = Hi([Xi]) + gi

    where ci = 1 and gi > 0. The value of Ki +nj=1,j=i ij [Xj ]cij is fixed. . . 495.13 The possible number of intersections of Y = i[Xi] and Y = Hi([Xi]) + gi

    where ci > 1 and gi = 0. The value of Ki +nj=1,j=i ij [Xj ]

    cij is fixed. . . 50

    5.14 The possible number of intersections of Y = i[Xi] and Y = Hi([Xi]) + gi

    where ci > 1 and gi > 0. The value of Ki +nj=1,j=i ij [Xj ]

    cij is fixed. . . 50

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    5.15 Finding the univariate fixed points using cobweb diagram, an example.

    We define the fixed point as [Xi] satisfying H([Xi]) + gi = i[Xi]. . . . . . 51

    5.16 The curves are rotated making the line Y = i[Xi] as the horizontal axis.

    Positive gradient means instability, negative gradient means stability. If

    the gradient is zero, we look at the left and right neighboring gradients. . 51

    5.17 When gi = 0, [Xi] = 0 is a component of a stable equilibrium point. . . . 56

    5.18 When gj > 0, [Xj] = 0 will never be a component of an equilibrium point. 56

    6.1 Sample numerical solution in time series with the upper bound and lower

    bound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    6.2 Y =[Xi]ci

    K+[Xi]ci will never touch the point (1, 1) for 1 < ci < . . . . . . . . 70

    6.3 An example where i(Ki1/ci) > i; Y = Hi([Xi]) and Y = i[Xi] only

    intersect at the origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

    7.1 When gi = 0, ci = 1 and the decay line is tangent to the univariate Hill

    curve at the origin, then the origin is a saddle. . . . . . . . . . . . . . . . 76

    7.2 Varying the values of parameters may vary the size of the basin of at-

    traction of the lower-valued stable intersection of Y = Hi([Xi]) + gi and

    Y = i[Xi]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.3 The possible number of intersections ofY = i[Xi] and Y = Hi([Xi]) + gi

    where c > 1 and g = 0. The value of Ki +nj=1,j=i ij[Xj]

    cij is taken as a

    parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    7.4 The possible topologies when Y = Hi([Xi]) essentially lies below the decay

    line Y = i[Xi], gi = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    7.5 The origin is unstable while the points where [Xi] = Knj=1,j=i [Xj]

    are stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    7.6 Increasing the value of gi can result in an increased value of [Xi] where

    Y = Hi([Xi]) + gi and Y = i([Xi]) intersects. . . . . . . . . . . . . . . . 83

    7.7 Increasing the value of gi can result in an increased value of [Xi], and

    consequently in decreased value of [Xj] where Y = Hj([Xj]) + gj and

    Y = j([Xj]) intersects, j = i. . . . . . . . . . . . . . . . . . . . . . . . . 84

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    8.1 For Illustration 1; ODE solution and SDE realization with G(X) = 1. . . 88

    8.2 For Illustration 1; ODE solution and SDE realization with G(X) = X. . 88

    8.3 For Illustration 1; ODE solution and SDE realization with G(X) = X. 898.4 For Illustration 1; ODE solution and SDE realization with G(X) = F(X). 89

    8.5 For Illustration 1; ODE solution and SDE realization using the random

    population growth model. . . . . . . . . . . . . . . . . . . . . . . . . . . 90

    8.6 For Illustration 2; ODE solution and SDE realization with G(X) = 1. . . 92

    8.7 For Illustration 2; ODE solution and SDE realization with G(X) = X. . 92

    8.8 For Illustration 2; ODE solution and SDE realization with G(X) =

    X. 93

    8.9 For Illustration 2; ODE solution and SDE realization with G(X) = F(X). 93

    8.10 For Illustration 2; ODE solution and SDE realization using the random

    population growth model. . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    8.11 For Illustration 3; ODE solution and SDE realization with G(X) = 1. . . 96

    8.12 For Illustration 3; ODE solution and SDE realization with G(X) = X. . 96

    8.13 For Illustration 3; ODE solution and SDE realization with G(X) =

    X. 97

    8.14 For Illustration 3; ODE solution and SDE realization with G(X) = F(X). 97

    8.15 For Illustration 3; ODE solution and SDE realization using the random

    population growth model. . . . . . . . . . . . . . . . . . . . . . . . . . . 988.16 Phase portrait of [X1] and [X2]. . . . . . . . . . . . . . . . . . . . . . . . 98

    8.17 Reactivating switched-off TFs by introducing random noise where G(X) = 1. 99

    9.1 The simplified MacArthur et al. GRN . . . . . . . . . . . . . . . . . . . . 100

    A.1 Intersections ofF1, F2 and zero-plane, an example. . . . . . . . . . . . . 106

    A.2 The intersection of Y = H1([X1]) + 1 and Y = 10[X1] with [X2] = 1.001

    and [X3] = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

    A.3 The intersection of Y = H2([X2]) and Y = 10[X2] with [X1] = 0.10103

    and [X3] = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

    A.4 The intersection of Y = H3([X3]) and Y = 10[X3] with [X1] = 0.10103

    and [X2] = 1.001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

    A.5 A sample phase portrait of the system with infinitely many non-isolated

    equilibrium points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

    xiii

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    C.1 Determining the adequate g1 > 0 that would give rise to a sole equilibrium

    point where [X1] > [X2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 133

    C.2 An example where without g1, [X1] = 0. . . . . . . . . . . . . . . . . . . 135

    C.3 [X1] escaped the zero state because of the introduction of g1 which is a

    decaying linear function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

    C.4 An example of shifting from a lower stable component to a higher stable

    component through adding gi(t) = it + gi(0). . . . . . . . . . . . . . . 136C.5 [X1]

    escaped the zero state because of the introduction of g1 which is a

    decaying exponential function. . . . . . . . . . . . . . . . . . . . . . . . . 137

    C.6 Parameter plot of, an example. . . . . . . . . . . . . . . . . . . . . . . 138

    C.7 Intersections ofY = i[Xi] and Y = Hi([Xi]) + gi where c > 1 and g = 0;

    and an event of bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . 139

    C.8 Saddle node bifurcation; 1 is varied. . . . . . . . . . . . . . . . . . . . . 140

    C.9 Saddle node bifurcation; K1 is varied. . . . . . . . . . . . . . . . . . . . . 141

    C.10 Saddle node bifurcation; 1 is varied. . . . . . . . . . . . . . . . . . . . . 141

    C.11 Cusp bifurcation; 1 and g1 are varied. . . . . . . . . . . . . . . . . . . . 142

    C.12 Cusp bifurcation; K1 and c are varied. . . . . . . . . . . . . . . . . . . . 142

    C.13 Cusp bifurcation; K1 and g1 are varied. . . . . . . . . . . . . . . . . . . . 143C.14 Cusp bifurcation; 1 and g1 are varied. . . . . . . . . . . . . . . . . . . . 143

    C.15 Saddle node bifurcation; 2 is varied. . . . . . . . . . . . . . . . . . . . . 144

    C.16 Saddle node bifurcation; g2 is varied. . . . . . . . . . . . . . . . . . . . . 145

    C.17 Saddle node bifurcation; 2 is varied. . . . . . . . . . . . . . . . . . . . . 146

    C.18 Saddle node bifurcation; g2 is varied. . . . . . . . . . . . . . . . . . . . . 146

    xiv

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    Chapter 1

    Introduction

    The field of Biomathematics has proven to be useful and essential for understanding

    the behavior and control of dynamic biological interactions. These interactions span a

    wide spectrum of spatio-temporal scales from interacting chemical species in a cell to

    individual organisms in a community, and from fast interactions occurring within seconds

    to those that slowly progress in years. Mathematical and in silico models enable scientists

    to generate quantitative predictions that may serve as initial input for testing biological

    hypotheses to minimize trial and error, as well as to investigate complex biological systems

    that are impractical or infeasible to study through in situ and in vitro experiments.

    One classic question that scientists want to answer is how simple cells generate com-

    plex organisms. In this study, we are interested in the analysis of gene interaction net-

    works that orchestrate the differentiation of stem cells to various cell lineages that make

    up an organism. We are also motivated by the prospects of utilizing stem cells in regen-erative medicine (such as through replenishment of damaged tissues as well as treatment

    of Parkinsons disease and diabetes) [1, 50, 107, 151, 171, 180], in revolutionizing drug

    discovery [2, 48, 136, 141, 142], and in the control of so-called cancer stem cells that had

    been hypothesized to maintain the growth of tumors [57, 65, 110, 171, 172].

    The current -omics (genomics, transcriptomics, proteomics, etc.) and systems biol-

    ogy revolution [3, 33, 61, 62, 63, 93, 96, 99, 100, 108, 133] are continually providing

    details about gene networks. The focus of this study is the mathematical analysis ofa gene network [113] involved in the differentiation of multipotent stem cells to three

    mesenchymal stromal stem cells, namely, cells that form bones (osteoblasts), cartilages

    (chondrocytes), and fats (adipocytes). This gene network shows the coupled interaction

    among stem-cell-specific transcription factors and lineage-specifying transcription factors

    induced by exogenous stimuli.

    1

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    Chapter 1. Introduction 2

    MacArthur et al. [113] proposed a model of this gene network, and we hypothesize

    that further and more substantial analytical and computational study of this model would

    reveal important insights into the control of the mesenchymal cell differentiation system.

    We refer to the process of controlling the fate of a stem cell towards a chosen lineage as

    cellular programming.

    We analyze the gene network of MacArthur et al. [113] by simplifying the network

    model while preserving the essential qualitative dynamics. In Chapter (5) of this the-

    sis, we simplify the MacArthur et al. [113] network model to highlight the essential

    components of the mesenchymal cell differentiation system and for easier analysis.

    We translate the simplified network model into a system of Ordinary Differential

    Equations (ODEs) using the Cinquin-Demongeot formalism [38]. The system of ODEs

    formulated by Cinquin-Demongeot [38] is one of the mathematical models appropriate to

    represent the dynamics depicted in the simplified MacArthur et al. [113] gene network.

    The state variables of the ODE model represent the concentration of the transcription fac-

    tors involved in gene expression. The Cinquin-Demongeot [38] ODE model can represent

    various biological interactions, such as molecular interactions during gene transcription,

    and it can represent cellular differentiation with more than two possible outcomes.

    Stability and bifurcation analyses of the ODE model are important in understanding

    the dynamics of cellular differentiation. An asymptotically stable equilibrium point is

    associated with a certain cell type. In Chapters (6) and (7), we determine the biologically

    feasible (nonnegative real-valued) coexisting stable equilibrium points of the ODE model

    for a given set of parameters. We also identify if varying the values of some parameters,

    such as those associated with the exogenous stimuli, can steer the system toward a desired

    state.

    Furthermore, in Chapter (8), we numerically investigate the robustness of the gene

    network against stochastic noise by adding a noise term to the deterministic ODEs. The

    objectives of the study are summarized in the following diagram:

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    Chapter 1. Introduction 3

    Figure 1.1: Analysis of mesenchymal cell differentiation system.

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    Chapter 2

    Preliminaries

    Biology of Cellular Programming

    2.1 Stem cells in animals

    Stem cells are very important for the development, growth and repair of tissues. These

    are cells that can undergo mitosis (cell division) and have two contrasting abilities

    ability for self-renewal and ability to differentiate into different specialized cell types.

    Self-renewal is the ability of stem cells to proliferate, that is, one or both daughter cells

    remain as stem cells after cell division. When a stem cell undergoes differentiation,

    it develops into a more mature (specialized) cell, losing its abilities to self-renew and to

    differentiate towards other cell types. In addition, scientists have shown that some cells

    can dedifferentiate and some can be transdifferentiated. Dedifferentiation means that

    a differentiated cell is transformed back to an earlier stage, while transdifferentiation

    means that a cell is programmed to switch cell lineages.

    The maturity of a stem cell is classified based on the cells potency (the cells capability

    to differentiate into various types). The three major kinds of stem cell potency are

    totipotency, pluripotency and multipotency. Figure (2.1) shows these three types of

    potencies and the differentiation process. Totipotent stem cells have the potential to

    generate all cells including extraembryonic tissues, such as the placenta, and they are the

    ancestors of all cells of an organism. A zygote is an example of a totipotent stem cell.

    Pluripotent stem cells are descendants of totipotent stem cells that have lost their

    ability to generate extraembryonic tissues but not their ability to generate all cells of the

    embryo. Examples of these stem cells are the cells of the epiblast from the inner cell mass

    of the blastocyst embryo. These stem cells can differentiate into almost all types of cells;

    specifically, they form the endoderm, mesoderm and ectoderm germ layers. Pluripotent

    4

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    Chapter 2. Preliminaries Biology of Cellular Programming 5

    Figure 2.1: Stem cell self-renewal, differentiation and programming. This diagramillustrates the abilities of stem cells to ploriferate through self-renewal, differentiate intospecialized cells and reprogram towards other cell types.

    stem cells form all cell types found in an adult organism. The stomach, intestines, liver,

    pancreas, urinary bladder, lungs and thyroid are formed from the endoderm layer; the

    central nervous system, lens of the eye, epidermis, hair, sweat glands, nails, teeth andmammary glands are formed from the ectoderm layer. The mesoderm layer connects the

    endoderm and ectoderm layers, and forms the bones, muscles, connective tissues, heart,

    blood cells, kidneys, spleen and middle layer of the skin.

    Embryonic stem (ES) cells, epiblast stem cells, embryonic germ cells (derived from

    primordial germ cells), spermatogonial male germ stem cells and induced pluripotent

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    Chapter 2. Preliminaries Biology of Cellular Programming 6

    stem cells (iPSCs) are examples of pluripotent stem cells that are cultured in vitro.

    ES cells are derived from the inner cell mass of the blastocyst embryo upon explantation

    (isolated from the normal embryo).

    Some adult stem cells, which can be somatic (related to the body) or germline (related

    to the gametes such as ovum and sperm), with embryonic stem cell-like pluripotency have

    been found by researchers under certain environments [16, 97, 103, 125, 170, 181]. Um-

    bilical cord blood, adipose tissue and bone marrow are found to be sources of pluripotent

    stem cells.

    The production of iPSCs in 2006 [109, 162] is a major breakthrough for stem cell

    research. The iPSCs are cells that are artificially reprogrammed to dedifferentiate from

    differentiated or partially differentiated cells to become pluripotent again. With only few

    ethical issues compared to embryo cloning, iPSCs can be used for possible therapeutic

    purposes such as treating degenerative diseases, repairing damaged tissues and repro-

    gramming cancer stem cells. However, there are plenty of issues on the use of iPSCs such

    as safety and efficiency. Currently, there is still no strong proof that generated iPSCs

    and natural ES cells are totally identical [158].

    Pluripotent stem cells that differentiate to specific cell lineages lose their pluripotency,

    that is, they lose their ability to generate other kinds of cells. Multipotent stem

    cells are descendants of pluripotent stem cells but are already partially differentiated

    they have the ability to self-renew yet can differentiate only to specific cell lineages.

    Multipotent stem cells are adult stem cells that are commonly considered as progenitor

    cells (cells that are in the stage between being pluripotent and fully differentiated).

    When a multipotent stem cell further differentiates, it matures to a more specialized

    cell lineage. Oligopotent and unipotent stem cells are progenitor cells that have very

    limited ability for self-renewal and are less potent. Oligopotent stem cells are descendants

    of multipotent stem cells and can only differentiate into very few cell types. Usually,

    stem cells are given special names based on the degree of potency, such as tripotent and

    bipotent depending on whether the cell can only differentiate into three and two cell fates,

    respectively. Unipotent stem cells, which are commonly called precursor cells, can only

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    Chapter 2. Preliminaries Biology of Cellular Programming 7

    differentiate into one cell type but are not the same as fully differentiated cells. Fully

    differentiated cells are at the determined terminal state, that is, they have completed

    the differentiation process, have exited the cell cycle, and have already lost the ability to

    self-renew [23, 123].

    Figure 2.2: Priming and differentiation. Colored circles represent genes or TFs. Thesizes of the circles determine lineage bias. Priming is represented by colored circles havingequal sizes. The largest circle governs the possible phenotype of the cell. [70]

    In vitro, ex vivo and in vivo programming have already been done [138, 139, 151,

    177]. The idea of programming biological cells indicates that some cells are plastic

    (i.e., some cells have the ability to change lineages). This plasticity of cells proves that

    some cells do not permanently inactivate unexpressed genes but rather retain all geneticinformation (see Figure (2.2)). Three in vitro approaches of cellular programming have

    been discussed in a review by Yamanaka [177]. These approaches are nuclear transfer,

    cell fusion and transcription-factor transduction [19, 44, 51, 58, 106, 177]. The process

    of nuclear transfer has been used to successfully clone Dolly the sheep. Transcription-

    factor transduction, commonly called direct programming, alters the expression of

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    Chapter 2. Preliminaries Biology of Cellular Programming 8

    transcription factors (TFs) by overexpression or by deletion. Overexpressing one TF

    may down-regulate other TFs that would lead to a change in the phenotype of a cell. In

    2006, Yamanaka and Takahashi [162] identified four factors OCT3/4, SOX2, c-MYC,

    and KLF4 that are enough to reprogram cells from mouse fibroblasts to become

    iPSCs (through the use of retrovirus). In 2007, Yamanaka, Takahashi and colleagues

    [161] generated iPSCs from adult human fibroblasts by the same defined factors.

    The three cellular programming approaches discussed by Yamanaka [177] have re-

    vealed common features demethylation of pluripotency gene promoters and activation

    of ES-cell-specific TFs such as OCT4, SOX2 and NANOG [113, 124, 129]. In this study,

    we only consider the TF transduction approach. To understand cellular differentiation

    and TF transduction, we need to look at gene regulatory networks. Gene regulatory

    networks (GRNs) establish the interactions of molecules and other signals for the ac-

    tivation or inhibition of genes. We consider the key pluripotency transcription factors

    OCT4, SOX2 and NANOG as the elements of the core pluripotency module in our GRN.

    For a more detailed discussion about stem cells in animals, the following references

    may be consulted [1, 12, 20, 22, 25, 34, 39, 42, 59, 74, 78, 80, 84, 103, 117, 148, 151, 159,169, 177].

    2.2 Transcription factors and gene expression

    Genes contain hereditary information and are segments of the deoxyribonucleic acid

    (DNA). Gene expression is the process in which information from a gene is used to

    synthesize functional products such as proteins. Examples of these gene products areproteins that give the cell its structure and function.

    Genes in the DNA direct protein synthesis. Transcription and translation are the two

    major processes that transform the information from nucleic acids to proteins (see Figure

    (2.3)). In the transcription process, the DNA commands the synthesis of ribonucleic

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    Chapter 2. Preliminaries Biology of Cellular Programming 9

    Figure 2.3: The flow of information. The blue solid lines represent general flow and theblue dashed lines represent special (possible) flow. The red dotted lines represent the

    impossible flow as postulated in the Central Dogma of Molecular Biology [41].

    acid (RNA) and the information is transcribed from the DNA template to the RNA. The

    RNA, specifically messenger RNA or mRNA, then carries the information to the part

    of the cell where protein synthesis will happen. In the translation process, the cell

    translates the information from the mRNA to proteins.

    During transcription, the promoter (a DNA sequence where RNA polymerase enzyme

    attaches) initiates transcription, while the terminator (also a DNA sequence) marks the

    end of transcription. However, the RNA polymerase binds to the promoter only after

    some transcription factors (TFs), a collection of proteins, are attached to the pro-

    moter.

    Gene expression is usually regulated by DNA-binding proteins (such as by TFs) at the

    transcription process, sometimes utilizing external signals. TFs play a main role in gene

    regulatory networks. A TF that binds to an enhancer (a control element) and stimulates

    transcription of a gene is called an activator; a TF that binds to a silencer (also a control

    element) and inhibits transcription of a gene is called a repressor. Hundreds of TFs

    were discovered in eukaryotes. In highly specialized cells, only a small fraction of their

    genes are activated.

    Examples of TFs are OCT4, SOX2 and NANOG as well as RUNX2, SOX9 and PPAR-

    . RUNX2, SOX9 and PPAR- stimulate formation of bone cells, cartilage cells and fat

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    Chapter 2. Preliminaries Biology of Cellular Programming 10

    cells, respectively [113].

    For a more detailed discussion about the relationship between transcription factors

    and gene expression, the following references may be consulted [24, 89, 126].

    2.3 Biological noise and stochastic differentiation

    It is believed that stochastic fluctuations in gene expression affect cell fate commitment

    in normal development and in in vitro culture of cells. The path that the cell would take

    is not absolutely deterministic but is rather affected by two kinds of noise intrinsic

    and extrinsic [128, 130, 160, 174]. Intrinsic noise is the inherent noise produced during

    biochemical processes inside the cell, while extrinsic noise is the noise produced from

    the external environment (such as from the other cells). In some cases, extrinsic noise

    dominates the intrinsic noise and influences cell-to-cell variation [174] because the internal

    environment of a cell is regulated by homeostasis.

    Unregulated random fluctuations can cause negative effects to the organism. However,

    in most cases, these stochastic fluctuations are naturally regulated enough to maintain

    order [30, 111]. Stochastic fluctuations have positive effects to the system such as driving

    oscillations and inducing switching in cell fates [71, 111, 174]. The papers [113] and [176]

    discuss the importance of random noise in dedifferentiation, especially in the production

    of iPSCs.

    When a stem cell undergoes cell division, the two daughter cells may both still be

    identical to the original, may both have already been differentiated, or may have one cell

    identical to the original and the other already differentiated. Cells that would undergo

    differentiation have plenty of cell lineages to choose from, but their cell fates are based

    on some pattern formation [24]. The model creode by C. Waddington [168], as shown

    in Figure (2.4), illustrates the paths that a cell might take. In Waddingtons model, cell

    differentiation is depicted by a ball rolling down a landscape of hills and valleys. The

    parts of the valleys where the ball can stay without rolling can be regarded as attractors

    that represent cell types. GRNs determine the topography of the landscape.

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    Chapter 2. Preliminaries Biology of Cellular Programming 11

    Figure 2.4: C. Waddingtons epigenetic landscape creode [168].

    For a more detailed discussion about biological noise and stochastic differentiation,

    the following references may be consulted [9, 15, 26, 28, 30, 53, 64, 80, 81, 83, 85, 94,

    101, 105, 111, 112, 127, 131, 132, 152, 164, 176].

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    Chapter 3

    Preliminaries

    Mathematical Models of Gene Networks

    This chapter gives a review of the existing literatures on models of gene regulatory

    networks (GRN).

    Commonly, to start the mathematical analysis of GRNs, a directed graph is con-structed to visualize the interaction of the molecules involved. Various network analysis

    techniques are available to extract information from the constructed directed graph such

    as clustering algorithms and motif analysis [4, 30, 45, 68, 90]. The study of the network

    topology is important in understanding the biological system that the network represents.

    Gene regulatory systems are commonly modeled as Bayesian networks, Boolean net-

    works, generalized logical networks, Petri nets, ordinary differential equations, partial

    differential equations, chemical master equations, stochastic differential equations and

    rule-based simulations [29, 45]. The choice of mathematical model depends on the as-

    sumptions made about the nature of the GRN and on the objectives of the study.

    In this thesis, we study the directed graph constructed by MacArthur et al. [113]

    and its corresponding Ordinary Differential Equations (ODEs) formulated by Cinquin-

    Demongeot [38] and its corresponding Stochastic Differential Equations (SDEs). By using

    an ODE model, we assume that the time-dependent macroscopic dynamics of the GRN

    are continuous in both time and state space. We assume continuous dynamics because the

    process of lineage determination involves a temporal extension, that is, cells pass through

    intermediate stages [70]. We use ODEs to model the average dynamics of the GRN. ODEs

    are primarily used to represent the deterministic dynamics of phenomenological (coarse-

    grained) regulatory networks [70, 121]. In addition, we can add a random noise term to

    the ODE model to study stochasticity in cellular differentiation.

    12

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    Chapter 3. Preliminaries Mathematical Models of Gene Networks 13

    3.1 The MacArthur et al. GRN

    The MacArthur et al. [113] GRN is composed of a pluripotency module (the circuitconsisting of OCT4, SOX2, NANOG and their heterodimer and heterotrimer) and a

    differentiation module (the circuit consisting of RUNX2, SOX9 and PPAR-) [113]. The

    transcription factors RUNX2, SOX9 and PPAR- activate the formation of bone cells,

    cartilage cells and fat cells, respectively.

    Figure 3.1: The coarse-graining of the differentiation module. The network in (a) issimplified into (b), where arrows indicate up-regulation (activation) while bars indicatedown-regulation (repression). [113]

    The derivation of the core differentiation module is shown in Figure (3.1) where the

    interactions through intermediaries are consolidated to create a simplified network. The

    MacArthur et al. [113] GRN that we are going to study is shown in Figure (3.2).

    Feedback loops (which are important for the existence of homeostasis) and autoregu-

    lation (or autoactivation, which means that a molecule enhances its own expression) are

    necessary to attain pluripotency [177]. These feedback loops and autoregulation are also

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    Chapter 3. Preliminaries Mathematical Models of Gene Networks 14

    Figure 3.2: The MacArthur et al. [113] mesenchymal gene regulatory network. Arrowsindicate up-regulation (activation) while bars indicate down-regulation (repression).

    present in the MacArthur et al. GRN [113]; however, they are not enough to generate iP-

    SCs. Based on the deterministic computational analysis of MacArthur et al. [113], their

    pluripotency module cannot be reactivated once silenced, that is, it becomes resistant

    to reprogramming. However, they found that introducing stochastic noise to the systemcan reactivate the pluripotency module [113].

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    Chapter 3. Preliminaries Mathematical Models of Gene Networks 15

    3.2 ODE models representing GRN dynamics

    A state X = ([X1], [X2], . . . , [Xn]) represents a temporal stage in the cellular differentia-tion or programming process (see Figure (3.3)). We define [Xi] as a component (coordi-

    nate) of a state. A stable state (stable equilibrium point) X = ([X1], [X2], . . . , [Xn])

    represents a certain cell type, e.g., pluripotent, tripotent, bipotent, unipotent or terminal

    state.

    Figure 3.3: Gene expression or the concentration of the TFs can be represented by astate vector, e.g. ([X1], [X2], [X3], [X4]) [70]. For example, TFs of equal concentration

    can be represented by a vector with equal components, such as (2.4, 2.4, 2.4, 2.4).

    Modelers of GRN often use the function H+ (or H) which is bounded monotone

    increasing (or decreasing) with values between zero and one. Examples of such functions

    are the sigmoidal, hyperbolic and threshold piecewise-linear functions. If we use sigmoidal

    H+ and H called the Hill functions, we define

    H+([X], K , c) :=[X]c

    Kc + [X]c(3.1)

    for activation of gene expression and

    H([X], K , c) := 1 H+([X], K , c) = Kc

    Kc + [X]c(3.2)

    for repression, where the variable [X] is the concentration of the molecule involved [69,

    73, 96, 121, 144]. The parameter K is the threshold or dissociation constant and is equal

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    Chapter 3. Preliminaries Mathematical Models of Gene Networks 16

    to the value of X at which the Hill function is equal to 1/2. The parameter c is called

    the Hill constant or Hill coefficient and describes the steepness of the Hill curve. The Hill

    constant often denotes multimerization-induced cooperativity (a multimer is an assembly

    of multiple monomers or molecules) and may represent the number of cooperative binding

    sites ifc is restricted to a positive integer. However, in some cases, the Hill constant can

    be a positive real number (usually 1 < c < n where n is the number of equivalent

    cooperative binding sites) [73, 174]. If c = 1, then there is no cooperativity [38] and the

    Hill function becomes the Michaelis-Menten function which is hyperbolic. If data are

    available, we can estimate the value of c by inference.

    Various ODE models and formulations are presented in [13, 14, 27, 30, 31, 32, 43,

    47, 69, 76, 96, 115, 135, 173]. Examples of these are the neural network [166] model,

    the S-systems (power-law) [167] model, the Andrecut [7] model, the Cinquin-Demongeot

    2002 [36] model, and the Cinquin-Demongeot 2005 [38] model. The Cinquin-Demongeot

    2002 and 2005 models can represent various GRNs and are more amenable to analysis.

    3.2.1 Cinquin and Demongeot ODE formalism

    According to Waddingtons model [168], cell differentiation is similar to a ball rolling

    down a landscape of hills and valleys. The ridges of the hills can be regarded as the

    unstable equilibrium points while the parts of the valleys where the ball can stay without

    rolling further (i.e., at relative minima of the landscape) can be regarded as stable equi-

    librium points (attractors). Hence, the movement of the ball and its possible location

    after some time can be represented by dynamical systems, specifically ODEs. However,

    it should be noted that existing evidence showing the presence of attractors is limited to

    some mammalian cells [112].

    The theory that some cells can differentiate into many different cell types gives the

    idea that the model representing the dynamics of such cells may exhibit multistability

    (multiple stable equilibrium points). However, not all GRNs are reducible to binary or

    boolean hierarchic decision network (see Figure (3.4)), that is why Cinquin and Demon-

    geot formulated models that can represent cellular differentiation with more than two

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    Chapter 3. Preliminaries Mathematical Models of Gene Networks 17

    Figure 3.4: Hierarchic decision model and simultaneous decision model. Bars representrepression or inhibition, while arrows represent activation. [36].

    possible outcomes (multistability) obtained through different developmental pathways

    [3, 38, 35]. The simultaneous decision network (see Figure (3.4)) is a near approximation

    of the Waddington illustration where there are possibly many cell lineages involved.

    In 2002, Cinquin and Demongeot proposed an ODE model representing the simulta-

    neous decision network [36]. In 2005, they proposed another ODE model representing

    the simultaneous decision network but with autocatalysis (autoactivation) [38]. Both the

    Cinquin-Demongeot models are based on the simultaneous decision graph where there is

    mutual inhibition. All elements in the Cinquin-Demongeot models are symmetric, that

    is, each node has the same relationship with all other nodes, and all equations in the

    system of ODEs have equal parameter values.

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    Chapter 3. Preliminaries Mathematical Models of Gene Networks 18

    Equations (3.3) and (3.4) are the Cinquin-Demongeot ODE models without autocatal-

    ysis (2002 version, [36]) and with autocatalysis (2005 version, [38]), respectively. Let us

    suppose we have n antagonistic transcription factors. The state variable [Xi] represents

    the concentration of the corresponding TF protein such that the TF expression is subject

    to a first-order degradation (exponential decay). The parameters c, and g represent the

    relative speed of transcription (or strength of the unrepressed TF expression relative to

    the first-order degradation), cooperativity and leak, respectively. The parameter g is a

    basal expression of the corresponding TF and a constant production term that enhances

    the value of [Xi], which is possibly affected by an exogenous stimulus. For simplification,

    only the transcription regulation process is considered in [38]. The models are assumed

    to be intracellular and cell-autonomous (i.e., we only consider processes inside a single

    cell without the influence of other cells).

    Without autocatalysis :d[Xi]

    dt=

    1 +

    nj=1,j=i

    [Xj]c

    [Xi], i = 1, 2, . . . , n (3.3)

    With autocatalysis :d[Xi]

    dt=

    [Xi]c

    1 +nj=1

    [Xj]c

    [Xi] + g, i = 1, 2, . . . , n (3.4)

    The terms

    1 +nj=1,j=i

    [Xj]c

    and [Xi]c

    1 +nj=1

    [Xj]c

    (3.5)

    are Hill-like functions. In this study, we only consider Cinquin-Demongeot (2005 version)

    model (3.4) because autocatalysis is a common property of cell fate-determining factors

    known as master switches [38].

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    Chapter 3. Preliminaries Mathematical Models of Gene Networks 19

    In [38], Cinquin and Demongeot observed that their model (with autocatalysis) can

    show the priming behavior of stem cells (i.e., genes are equally expressed) as well as

    the up-regulation of one gene and down-regulation of the others. They also proved that

    multistability of their model where g = 0 is manipulable by changing the value of c

    (cooperativity); however, manipulating the level of cooperativity is of minimal biological

    relevance. Also, their model is more sensitive to stochastic noise when the equilibrium

    points are near each other.

    3.2.2 ODE model by MacArthur et al.

    MacArthur et al. [113] proposed an ODE model (Equations (3.6) and (3.7)) to rep-

    resent their GRN (refer to Figure (3.2)). Let [Pi] be the concentration of the TF

    protein in the pluripotency module, specifically, [P1] := [OCT4], [P2] := [SOX2] and

    [P3] := [NANOG]. Also, let [Li] be the concentration of the TF protein in the differen-

    tiation module where [L1] := [RUNX2], [L2] := [SOX9] and [L3] := [PPAR]. Theparameter si represents the effect of the growth factors stimulating the differentiation

    towards the i-th cell lineage, specifically, s1 := [RA + BM P4], s2 := [RA + T GF] and

    s3 := [RA +Insulin]. In mouse ES cells, RUNX2 is stimulated by retinoic acid (RA) andBMP4; SOX9 by RA and TGF-; and PPAR- by RA and Insulin. The derivation of the

    ODE model and the interpretation of the parameters are discussed in the supplementary

    materials of [113].

    d[Pi]

    dt= k1i[P1][P2](1+[P3])

    (1+k0

    j sj)(1+[P1][P2](1+[P3])+kPL

    j [Lj]) b[Pi] (3.6)

    d[Li]dt

    = k2(si+k3

    j=i sj)[Li]2m

    1+kLC1 [P1][P2]+kLC2 [P1][P2][P3]+[Li]2+kLL(si+k3

    j=i sj)

    j=i[Lj ]

    2 b[Li] (3.7)

    However, this system of coupled ODEs is difficult to study using analytic techniques.

    MacArthur et al. [113] simply conducted numerical simulations to investigate the behav-

    ior of the system. They tried to analytically analyze the system but only for a specific

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    Chapter 3. Preliminaries Mathematical Models of Gene Networks 20

    case where Pi = 0, i = 1, 2, 3, that is, when the pluripotency module is switched-off.

    The ODE model (3.8) that they analyzed when the pluripotency module is switched-off

    follows the Cinquin-Demongeot [38] formalism with c = 2, that is,

    d[Li]

    dt=

    [Li]2

    1 + [Li]2 + aj=i

    [Lj]2

    b[Li], i = 1, 2, 3 (3.8)

    MacArthur et al. [113] analytically proved that the three cell types (tripotent, bipo-

    tent and terminal states) are simultaneously stable for some parameter values in ( 3.8).

    However, as the effect of an exogenous stimulus is increased above some threshold value,

    the tripotent state becomes unstable leaving only two stable cell types (bipotent and

    terminal state). If the effect of the exogenous stimulus is further increased, the bipotent

    state also becomes unstable leaving the terminal state as the sole stable cell type. In

    addition, MacArthur et al. [113] showed that dedifferentiation is not possible without

    the aid of stochastic noise.

    3.3 Stochastic Differential Equations

    A time-dependent Gaussian white noise term can be added to the ODE model to inves-

    tigate the effect of random fluctuations in gene expression. This Gaussian white noise

    term combines and averages multiple heterogeneous sources of temporal noise. Equations

    (3.10) to (3.13) show some of the different SDE models [71, 72, 113, 174] of the form

    dX = F(X)dt + G(X)dW (3.9)

    that we use in this study. We employ different G(X) to observe the various effects of

    the added Gaussian white noise term. We let F(X) be the right-hand side of our ODE

    equations, be a diagonal matrix of parameters representing the amplitude of noise, and

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    Chapter 3. Preliminaries Mathematical Models of Gene Networks 21

    W be a Brownian motion (Wiener process). If the genes in a cell are isogenic (essentially

    identical) then we can suppose the diagonal entries of the matrix are all equal.

    dX = F(X)dt + dW (3.10)

    dX = F(X)dt + XdW (3.11)

    dX = F(X)dt +

    XdW (3.12)

    dX = F(X)dt + F(X)dW (3.13)

    Notice that in Equations (3.11) and (3.12), the noise term is affected by the value

    of X. As the concentration X increases, the effect of the noise term also increases.

    Whereas, in Equations (3.13), the noise term is affected by the value of F(X), that is,

    as the deterministic change in the concentration X with respect to timedXdt = F(X)

    increases, the effect of the noise term also increases. In Equation (3.10), the noise term

    is not dependent on any variable.

    For a more detailed discussion about various modeling techniques, the following ref-

    erences may be consulted [6, 11, 18, 21, 46, 52, 55, 60, 66, 67, 75, 77, 79, 87, 88, 92, 118,

    137, 140, 143, 149, 153, 154, 163, 165, 175, 179, 182].

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    Chapter 4

    Preliminaries

    Analysis of Nonlinear Systems

    This chapter gives a brief discussion of the theoretical background on the qualitative

    analysis of coupled nonlinear dynamical systems.

    Consider autonomous system of ODEs

    d[Xi]

    dt= Fi([X1], [X2], . . . , [Xn]), i = 1, 2, . . . , n , (4.1)

    with initial condition [Xi](0) := [Xi]0 i. We assume that t 0 and Fi : B Rn,i = 1, 2, . . . , n where B Rn. If we have a nonautonoumous system of ODEs, d[Xi]dt =Fi([X1], [X2], . . . , [Xn], t), i = 1, 2, . . . , n, then we convert it to an autonomous system by

    defining t := [Xn+1

    ] and d[Xn+1]dt

    = 1 [134].

    For simplicity, let F := (Fi, i = 1, 2, . . . , n), X := ([Xi], i = 1, 2, . . . , n) and X0 :=

    ([Xi]0, i = 1, 2, . . . , n).

    For an ODE model to be useful, it is necessary that it has a solution. Existence of a

    unique solution for a given initial condition is important to effectively predict the behavior

    of our system. Moreover, we are assured that the solution curves of an autonomous system

    do not intersect with each other when existence and uniqueness conditions hold [56].

    Suppose X(t) is a differentiable function. The solution to (4.1) satisfies the following

    integral equation:

    X(t) = X0 +

    t0

    F(X())d . (4.2)

    22

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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 23

    The following are theorems that guarantee local existence and uniqueness of solutions

    to ODEs:

    Theorem 4.1 Existence theorem(Peano, Cauchy). Consider the autonomous system

    (4.1). Suppose thatF is continuous onB. Then the system has a solution (not necessarily

    unique) on [0, ] for sufficiently small > 0 given any X0 B.

    Theorem 4.2 Local existence-uniqueness theorem (Picard, Lindelorf, Lipschitz,

    Cauchy). Consider the autonomous system (4.1). Suppose that F is locally Lipschitz

    continuous on B, that is, F satisfies the following condition: For each point X0 B

    there is an -neighborhood of X0 (denoted as B(X0) where B(X0) B) and a positiveconstant m0 such that |F(X) F(Y)| m0 |X Y| X, Y B(X0). Then the systemhas exactly one solution on [0, ] for sufficiently small > 0 given any X0 B.

    Theorem (4.2) can be extended to a global case stated as:

    Theorem 4.3 Global existence-uniqueness theorem. If there is a positive constant

    m such that |F(X) F(Y)| m |X Y| X, Y B (i.e., F is globally Lipschitzcontinuous on B) then the system has exactly one solution defined for all t R forany X0 B.

    If all the partial derivatives Fi[Xj ] i, j = 1, 2, . . . , n are continuous on B (i.e., F C1(B)) then F is locally Lipschitz continuous on B. If the absolute value of these partial

    derivatives are also bounded for all X B then F is globally Lipschitz continuous onB. The global condition says that if the growth of F with respect to X is at most linear

    then we have a global solution. If F satisfies the local Lipschitz condition but not the

    global Lipschitz condition, then it is possible that after some finite time t, the solution

    will blow-up.

    We define a point X = ([X1], [X2], . . . , [Xn]) as a state of the system, and the collec-

    tion of these states is called the state space. The solution curve of the system starting

    from a fixed initial condition is called a trajectory or orbit. The collection of trajectories

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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 24

    given any initial condition is called the flow of the differential equation and is denoted by

    (X0). The concept of the flow of the differential equation indicates the dependence of

    the system on initial conditions. The flow of the differential equation can be represented

    geometrically in the phase space Rn using a phase portrait. There exists a corresponding

    vector defined by the ODE that is tangent to each point in every trajectory; and the

    collection of all tangent vectors of the system is a vector field. A vector field is often

    helpful in visualizing the phase portrait of the system. Moreover, various methods are

    also available to numerically solve the system (4.1) such as the Euler and Runge-Kutta

    4 methods.

    4.1 Stability analysis

    In nonlinear analysis of systems, it is important to find points where our system is at rest

    and determine whether these points are stable or unstable. In modeling cellular differ-

    entiation, an asymptotically stable equilibrium point, which is an attractor, is associated

    with a certain cell type. For any initial condition in a neighborhood of the attractor, the

    trajectories tend towards the attractor even if slightly perturbed.

    Definition 4.1 Equilibrium point. The point X := ([X1], [X2]

    , . . . , [Xn]) Rn is

    said to be an equilibrium point (also called as critical point, stationary point or steady

    state) of the system (4.1) if and only if F(X) = 0.

    Finding the equilibrium points corresponds to solving for the real-valued solutions to

    the system of equations F(X) = 0. It is possible that this system of equations has a

    unique solution, several solutions, a continuum of solutions, or no solution.

    In order to describe the local behavior of the system (4.1) near a specific equilibrium

    point X, we linearize the system by getting the Jacobian matrix JF(X), defined as

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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 25

    JF(X) =

    F1[X1]

    F1[X2]

    F1[Xn]

    F2[X1]

    F2[X2]

    F2[Xn]

    ......

    . . ....

    Fn[X1]

    Fn[X2]

    Fn[Xn]

    (4.3)

    and then evaluating JF(X). If none of the eigenvalues of the matrix JF(X) has zero

    real part then X is called a hyperbolic equilibrium point. In this chapter, we focus

    the discussion on hyperbolic equilibrium points; but for details about nonhyperbolic

    equilibrium points, refer to [134].

    We use the eigenvalues of JF(X) to determine the stability of equilibrium points.

    Definition 4.2 Asymptotically stable and unstable equilibrium points. The

    equilibrium point X is asymptotically stable when the solutions near X converge to X

    as t . The equilibrium point X is unstable when some or all solutions near Xtend away from X as t .

    Theorem 4.4 Stability of equilibrium points. If all the eigenvalues ofJF(X) have

    negative real parts then X is an asymptotically stable equilibrium point. If at least one

    of the eigenvalues of JF(X) has a positive real part then X is an unstable equilibrium

    point.

    For simplicity, we will call an asymptotically stable equilibrium point, stable. There

    are various tests for determining the stability of an equilibrium point such as by using

    Theorem (4.4), or by using geometric analysis as shown in Figure (4.1). In addition, we

    define X as a saddle if it is an unstable equilibrium point but JF(X) has at least

    one eigenvalue with negative real part. For further details regarding the local behavior

    of nonlinear systems in the neighborhood of an equilibrium point, refer to the Stable

    Manifold Theorem and the Hartman-Grobman Theorem [134].

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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 26

    Figure 4.1: The slope ofF(X) at the equilibrium point determines the linear stability.Positive gradient means instability, negative gradient means stability. If the gradient iszero, we look at the left and right neighboring gradients. Refer to the Insect OutbreakModel: Spruce Budworm in [122].

    It is also useful to determine the set of initial conditions X0 with trajectories con-

    verging to a specific stable equilibrium point X. We call this set of initial conditions

    the domain or basin of attraction of X, denoted by

    X := X0 : limt(X0) = X

    . (4.4)

    In addition, a set B B is called positively invariant with respect to the flow (X0)if for any X0 B, (X0) B for all t 0, that is, the flow of the ODE remains in B.

    There are other types of attractors, such as -limit cycles and strange attractors

    [56]. A limit cycle is a periodic orbit (a closed trajectory which is not an equilibrium

    point) that is isolated. An asymptotically stable limit cycle is called an -limit cycle.

    Strange attractors usually occur when the dynamics of the system is chaotic. Moreover,

    under some conditions, a trajectory may be contained in a non-attracting but neutrally

    stable center (see [56] for discussion about centers). However, the extensive numerical

    simulations by MacArthur et al. [113] suggest that their ODE model (Equations (3.6)

    and (3.7)) does not have oscillators (periodic orbit) and strange trajectories. Cinquin and

    Demongeot [38] also claim that the solutions to their model (refer to Equations (3.4))

    always tend towards an equilibrium and never oscillate [38].

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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 27

    The existence of a center, -limit cycle or strange attractor that would result to

    recurring changes in phenotype is abnormal for a natural fully differentiated cell. Limit

    cycles are associated with the concept of continuous cell proliferation (self-renewal) where

    there are recurring biochemical states during cell division cycles [82]. However, cell

    division is beyond the scope of this thesis.

    Various theorems are available for checking the possible existence or non-existence of

    limit cycles (although most are for two-dimensional planar systems only). The Poincare-

    Bendixson Theorem for planar systems [134] states that ifF C1(B) and a trajectoryremains in a compact region ofB whose -limit set (e.g. attracting set) does not contain

    any equilibrium point, then the trajectory approaches a periodic orbit. Furthermore, if

    F C1(B) and a trajectory remains in a compact region of B as well as if there areonly a finite number of equilibrium points, then the -limit set of any trajectory of the

    planar system can be one of three types an equilibrium point, a periodic orbit or a

    compound separatrix cycle.

    Some researches have shown the effect of the presence of positive or negative feedback

    loops in GRNs such as possible multistability (existence of multiple stable equilibrium

    points) and existence of oscillations [8, 37, 45, 104, 119, 155]. It is also important to notethat a strange (chaotic) attractor will not exist for n < 3 [56].

    4.2 Bifurcation analysis

    The behavior of the solutions of system (4.1) depends not only on the initial conditions

    but also on the values of the parameters. The parameters of the model may be associated

    with real-world quantities that can be manipulated to control the solutions. Varying the

    value of a parameter (or parameters) may result in dramatic changes in the qualitative

    nature of the solutions, such as a change in the number of equilibrium points or a change

    in the stability. Here, we now let F be a function of the state variables X and of the

    parameter matrix (i.e., F(X, )). We define the values of the parameters where such

    dramatic change occurs as bifurcation value, denoted by . If we simultaneously vary

    the values of p number of parameters then we have a p-parameter bifurcation.

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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 28

    Ifp-parameter bifurcation is sufficient for a bifurcation type to occur then we classify

    the bifurcation type as codimension p. Examples of codimension one bifurcation type

    are saddle-node (fold), supercritical Poincare-Andronov-Hopf and subcritical Poincare-

    Andronov-Hopf bifurcations. Transcritical, supercritical pitchfork and subcritical pitch-

    fork bifurcations are also often regarded as codimension one. Cusp bifurcation is of

    codimension two.

    Figure 4.2: Sample bifurcation diagram showing saddle-node bifurcation.

    In a local bifurcation, the equilibrium point X is nonhyperbolic at the bifurcation

    value. For n 2, if JF(X) has a pair of purely imaginary eigenvalues and no othereigenvalues with zero real part at the bifurcation value then under some assumptions a

    Hopf bifurcation may occur and a limit cycle might arise from X. We can visualize the

    bifurcation of equilibria using a bifurcation diagram. For further details about bifurcation

    theory, refer to [86, 102, 134]. There are softwares available for numerical bifurcation

    analysis such as Oscill8 [40] which uses AUTO (http://indy.cs.concordia.ca/auto/ ).

    4.3 Fixed point iteration

    Definition 4.3 Fixed point. The point X is a fixed point of the real-valued function

    Q if Q(X) = X.

    http://indy.cs.concordia.ca/auto/http://indy.cs.concordia.ca/auto/
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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 29

    We use fixed point iteration (FPI) to find approximate stable equilibrium points of

    the Cinquin-Demongeot [38] model. If X is a stable equilibrium point then for initial

    conditions X0 sufficiently close to X (where X0 = X), the sequences generated by FPIwill converge to X (i.e., is locally convergent). If X0 = X, we can either have a stable

    or unstable equilibrium point.

    Algorithm 1 Fixed point iterationSuppose Q is continuous on the region B.Input initial guess X(0) := X0 B and acceptable tolerance error R.While

    X(i+1) X(i)

    > do X(i+1) := Q(X(i)).

    If X(i+1) X(i) is satisfied then X(i+1) is the approximate fixed point.

    Figure 4.3: An illustration of cobweb diagram.

    The geometric illustration of FPI is called a cobweb diagram as illustrated in Figure

    (4.3).

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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 30

    4.4 Sylvester resultant method

    To find the equilibrium points, we can rewrite the Cinquin-Demongeot ODE model wherethe exponent is a positive integer as a system of polynomial equations. Assume F(X) = 0

    can be written as a polynomial system P(X) = 0. The topic of solving multivariate

    nonlinear polynomial systems is still in its development stage. However, there are already

    various available algebraic and geometric methods for solving P(X) = 0 such as Newton-

    like methods, homotopic solvers, subdivision methods, algebraic solvers using Grobner

    basis, and geometric solvers using resultant construction [120]. In resultant construction,

    we treat the problem of solving P(X) = 0 as a problem of finding intersections of curves.

    All Pi(X) should have no common factor of degree greater than zero so that P(X)

    has a finite number of complex solutions. The following Bezout Theorem gives a bound

    on the number of complex solutions including the multiplicities.

    Theorem 4.5 Bezout theorem. Consider real-valued polynomialsP1, P2, . . . , P n where

    Pi has degree degi. Suppose all the polynomials have no common factor of degree greater

    than zero (i.e., they are collectively relatively prime). Then the number of isolated

    complex solutions to the system P1(X) = P2(X) = . . . = Pn(X) = 0 is at most

    (deg1)(deg2) . . . (degn).

    The method of using the Sylvester resultant is a classical algorithm in Algebraic

    Geometry used to find the complex solutions of a system of two polynomial equations in

    two variables. It can also be used for solving a polynomial system of n equations with

    n variables where n > 2, by repeated application of the algorithm. The idea of usingSylvester resultants for solving multivariate polynomial systems is to eliminate all except

    for one variable. There are other resultant construction methods for solving multivariate

    polynomial systems with n > 2 such as the Dixon resultant, Macaulay resultant and

    U-resultant methods, but we will only focus on the Sylvester resultant. The algorithm

    for using Sylvester resultants is illustrated in the following paragraphs.

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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 31

    Consider two polynomials P1([X1], [X2]) and P2([X1], [X2]). We eliminate [X1] by

    constructing the Sylvester matrix associated to the two polynomials with [X1] as the

    variable (i.e., we take [X2] as fixed parameter). The size of the Sylvester matrix is

    (deg1 + deg2) (deg1 + deg2) where deg1 and deg2 are the degrees of the polynomial P1and P2 in the variable [X1], respectively.

    We give an example to show how to construct a Sylvester matrix. Let us suppose

    P1([X1], [X2]) = 2[X1]3 + 4[X1]

    2[X2] + 7[X1][X2]2 + 10[X2]

    3 + 8 (4.5)

    P2([X1], [X2]) = 5[X1]2 + 2[X1][X2] + [X2]

    2 + 6. (4.6)

    Since the degree ofP1 in terms of [X1] is 3 and the degree of P2 in terms of [X1] is 2, then

    the size of the Sylvester matrix (with [X1] as variable) is 5 5. The Sylvester matrix ofP1 and P2 with [X1] as variable is

    2 4[X2] 7[X2]2 10[X2]

    3 + 8 0

    0 2 4[X2] 7[X2]2 10[X2]

    3 + 8

    5 2[X2] [X2]2 + 6 0 0

    0 5 2[X2] [X2]2 + 6 0

    0 0 5 2[X2] [X2]2 + 6

    . (4.7)

    The first row of the Sylvester matrix contains the coefficients of [X1]3, [X1]

    2, [X1]1 and

    [X1]0 in P1. We shift each element of the first row one column to the right to form the

    second row. The third row contains the coefficients of [X1]2, [X1]

    1 and [X1]0 in P2. We

    shift each element of the third row one column to the right to form the fourth row. We

    again shift each element of the fourth row one column to the right to form the fifth row.

    Generally, we continue the process of shifting each element of the previous row to form

    the next row until the coefficient of [X1]0 reaches the last column. All cells of the matrix

    without entries coming from the coefficients of the polynomials are assigned the value

    zero.

    We use the determinant of the Sylvester matrix to find the intersection of P1 and P2.

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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 32

    Definition 4.4 Sylvester resultant. We call the determinant of the Sylvester matrix

    of P1 and P2 in [X1] (where [X2] is a fixed parameter) the Sylvester resultant, denoted by

    res(P1, P2; [X1]).

    Theorem 4.6 Zeroes of the Sylvester resultant. The values whereres(P1, P2; [X1]) =

    0 are the complex values of [X2] where P1([X1], [X2]) = P2([X1], [X2]) = 0.

    We denote the complex values of [X2] where P1([X1], [X2]) = P2([X1], [X2]) = 0 by

    [X2]. To find [X

    1], we solve the univariate system P

    1([X

    1],[X

    2]) = P

    2([X

    1],[X

    2]) = 0

    for all possible values of[X2].

    The following theorem can be used to determine if P1 and P2 either do not intersect,

    or intersect at infinitely many points.

    Theorem 4.7 None and infinitely many solutions. res(P1, P2; [X1]) is nonzero

    for any [X2] if and only if P1([X1], [X2]) = P2([X1], [X2]) = 0 has no complex solutions.

    Furthermore, the following statements are equivalent:

    1. res(P1, P2; [X1]) is identically zero (i.e., zero for any values of [X2]).

    2. P1 and P2 have a common factor of degree greater than zero.

    3. P1 = P2 = 0 has infinitely many complex solutions.

    We can extend the Sylvester resultant method to a multivariate case, say with threepolynomials P1([X1], [X2], [X3]), P2([X1], [X2], [X3]) and P3([X1], [X2], [X3]), by getting

    R1 = res(P1, P2; [X1]) and R2 = res(P2, P3; [X1]). Notice that R1 and R2 are both in

    terms of [X2] and [X3]. We then get R3 = res(R1, R2; [X2]) which is in terms of [X3].

    We solve the univariate polynomial equation R3 = 0 by using available solvers to obtain

    [X3]. After this, we find [X2]

    by substituting [X3] in R1 and R2 and solve R1 =

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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 33

    R2 = 0. We then find [X1] by solving P1([X1],[X2]

    ,[X3]) = P2([X1],[X2]

    ,[X3]) =

    P3([X1],[X2],[X3]

    ) = 0.

    For a more detailed discussion on solving systems of multivariate polynomial equa-

    tions, the following references may be consulted [17, 49, 98, 156, 157, 178].

    4.5 Numerical solution to SDEs

    The solutions to ODEs are functions, while the solutions to SDEs are stochastic processes.

    We define a continuous-time stochastic process X as a set of random variables X(t)where the index variable t 0 takes a continuous set of values. The index variable t mayrepresent time.

    Suppose we have an SDE model of the form dX = F(X)dt + G(X)dW where W

    is a stochastic process called Brownian motion (Wiener process). The differential dW of

    W is called white noise. Brownian motion is the continuous version of random walk

    and has the following properties:

    1. For each t, the random variable W(t) is normally distributed with mean zero and

    variance t.

    2. For each ti < ti+1, the normal random variable W(ti) = W(ti+1) W(ti) is in-dependent of the random variables W(tj), 0 j ti (i.e., W has independentincrements).

    3. Brownian motion W can be represented by continuous paths (but is not differen-

    tiable).

    Suppose W(t0) = 0. We can simulate a Brownian motion using computers by dis-

    cretizing time as 0 = t0 < t1 < . . . and choosing a random number that would represent

    W(ti1) from the normal distribution N(0, ti ti1) =

    ti ti1N(0, 1). This impliesthat we obtain W(ti) by multiplying

    ti ti1 by a standard normal random number and

    then adding the product to W(ti1).

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    Chapter 4. Preliminaries Analysis of Nonlinear Systems 34

    The solution to an SDE model has different realizations because it is based on

    random numbers. We can approximate a realization of the solution by using numerical

    solvers such as the Euler-Maruyama and Milstein methods. In this thesis, we use the

    Euler-Maruyama method. The Euler-Maruyama method is similar to the Euler method

    for ODEs.

    Algorithm 2 Euler-Maruyama methodDiscretize the time as 0 < t1 < t2 < .. . < tend.Suppose Yti is the approximate solution to X(ti).Input initial condition Xt0. Let Yt0 := Xt0.For i = 0, 1, 2 . . . , e n d

    1 do

    W(ti) = ti+1 tirandN(0,1), whe