jfrabajante ms applied math thesis
TRANSCRIPT
-
8/2/2019 JFRabajante MS Applied Math Thesis
1/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
2/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
3/179
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.
iii
-
8/2/2019 JFRabajante MS Applied Math Thesis
4/179
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
iv
-
8/2/2019 JFRabajante MS Applied Math Thesis
5/179
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
v
-
8/2/2019 JFRabajante MS Applied Math Thesis
6/179
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
vi
-
8/2/2019 JFRabajante MS Applied Math Thesis
7/179
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.
vii
-
8/2/2019 JFRabajante MS Applied Math Thesis
8/179
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
viii
-
8/2/2019 JFRabajante MS Applied Math Thesis
9/179
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.
ix
-
8/2/2019 JFRabajante MS Applied Math Thesis
10/179
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
x
-
8/2/2019 JFRabajante MS Applied Math Thesis
11/179
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
xi
-
8/2/2019 JFRabajante MS Applied Math Thesis
12/179
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
xii
-
8/2/2019 JFRabajante MS Applied Math Thesis
13/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
14/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
15/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
16/179
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:
-
8/2/2019 JFRabajante MS Applied Math Thesis
17/179
Chapter 1. Introduction 3
Figure 1.1: Analysis of mesenchymal cell differentiation system.
-
8/2/2019 JFRabajante MS Applied Math Thesis
18/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
19/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
20/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
21/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
22/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
23/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
24/179
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.
-
8/2/2019 JFRabajante MS Applied Math Thesis
25/179
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].
-
8/2/2019 JFRabajante MS Applied Math Thesis
26/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
27/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
28/179
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].
-
8/2/2019 JFRabajante MS Applied Math Thesis
29/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
30/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
31/179
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.
-
8/2/2019 JFRabajante MS Applied Math Thesis
32/179
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].
-
8/2/2019 JFRabajante MS Applied Math Thesis
33/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
34/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
35/179
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].
-
8/2/2019 JFRabajante MS Applied Math Thesis
36/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
37/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
38/179
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
-
8/2/2019 JFRabajante MS Applied Math Thesis
39/179
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].
-
8/2/2019 JFRabajante MS Applied Math Thesis
40/179
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].
-
8/2/2019 JFRabajante MS Applied Math Thesis
41/179
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.
-
8/2/2019 JFRabajante MS Applied Math Thesis
42/179
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/ -
8/2/2019 JFRabajante MS Applied Math Thesis
43/179
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).
-
8/2/2019 JFRabajante MS Applied Math Thesis
44/179
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.
-
8/2/2019 JFRabajante MS Applied Math Thesis
45/179
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.
-
8/2/2019 JFRabajante MS Applied Math Thesis
46/179
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 =
-
8/2/2019 JFRabajante MS Applied Math Thesis
47/179
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).
-
8/2/2019 JFRabajante MS Applied Math Thesis
48/179
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