Exemplary Model Description Draft 4

Description

We are modeling the transcription of target genes regulating the self-renewal or differentiation of stem cells based on the effects of outside signals that increase or decrease the concentrations of transcription factors OCT4, SOX2, and NANOG and the complexes they form. Four differential equations are used to model the changing concentrations of the three transcription factors and the OCT4-SOX2 heterodimer that is formed, which are affected by each of the other factors as well as signals A₊ and B_-. The transcription of the target genes is then described by one of two similar models: a coherent model involving one differential equation to describe the switch between self-renewal and differentiation and an incoherent model involving two differential equations, one describing the self-renewal target gene product concentration and the other the differentiation target gene product concentration.

Assumptions

This model is based on many assumption about how the different transcription factors and signals regulate each other. It is previously known that NANOG, the OCT4-SOX2 heterodimer, and the OCT4-SOX2-NANOG complex positively regulate OCT4, SOX2, and NANOG. OCT4-SOX2 and NANOG regulate the target gene transcription, in which the OCT4-SOX2 heterodimer must first bind to the DNA and then recruit NANOG in order for transcription to occur efficiently. Therefore, the combined effects of OCT4, SOX2, and NANOG regulate transcription.

Additionally, it is assumed that the signals A₊ and B_- can accurately model all possible combinations of signals, since A₊ activates OCT4 and SOX2 and B_- negatively regulates NANOG. It is also assumed that the signals and transcription factors do not directly interact, since the binding region of the signals is far from the transcription factor binding sites.

Finally, there are two models created based on a coherent and incoherent system. The coherent model involves a feedforward loop where OCT4-SOX2 is an activator for transcription of the stem cell and differentiation genes and NANOG is a weak repressor for the stem cell genes and a stronger repressor for the differentiation genes, such that the loop only responds to persistent signals. The incoherent model involves a loop with OCT4-SOX2 and NANOG acting as activators for the stem cell genes and a repressors for the differentiation genes, which allows the system to strongly react to short signals.

Equations

Transcription Factors

These four equations govern the concentration of the four transcription factors, OCT4, SOX2, NANOG, and OCT4-SOX2. The equations for OCT4, SOX2, and NANOG are based on the Shea-Ackers rate equation model, such that part of the equation is based on a ratio between the transcription rate of the factor based on outside factors over the degradation rate. These equations also include terms that model the effect of constant degradation rates and formation and dissociation of the OCT4-SOX2 complex from OCT4 and SOX2. The equation for OCT4-SOX2 is based on the formation, dissociation, and degradation rates of OCT4-SOX2 based on constants multiplied by the concentration of the complex. Each term is explained in a table following the equation.

OCT4

$\frac{d[O]}{dt} = \frac{\eta_1 + a_1 [A_+ ] + a_2 [OS] +a_2 [OS]}{1 + \eta_2 + b_1 [A_+ ] + b_2 [OS] + b_3 [OS] [N]} - \gamma_1 [O] - k_{1c} [O] [S] - k_{2c} [OS]$

Table 1

$\eta_1$	$a_1 [A_+ ]$	$a_2 [OS]$	$a_3 [OS] [N]$	$1 + \eta_2$	$b_1 [A_+ ]$	$b_2 [OS]$	$b_3 [OS] [N]$	$\gamma_1 [O]$	$k_{1c} [O] [S]$	$k_{2c} [OS]$
Basal Transcription Rate	Transcription Rate from A+	Transcription Rate from OCT4-SOX2	Transcription Rate from OCT4-SOX2-NANOG Complex	Basal Degradation Rate	Degradation Rate from A+	Degradation Rate from OCT4-SOX2	Degradation Rate from OCT4-SOX2-NANOG Complex	Constant Degradation of OCT4	Formation Rate of OCT4-SOX2	Dissociation Rate of OCT4-SOX2

SOX2

$\frac{d[S]}{dt} = \frac{\eta_3 + c_1 [A_+ ] + c_2 [OS] + c_3 [OS] [N]}{1 + \eta_4 + d_1 [A_+ ] + d_2 [OS] + d_3 [OS] [N]} - \gamma_2 [O] - k_{1c} [O] [S] - k_{2c} [OS]$

Table 2

$\eta_3$	$c_1 [A_+ ]$	$c_2 [OS]$	$c_3 [OS] [N]$	$1 + \eta_4$	$d_1 [A_+ ]$	$d_2 [OS]$	$d_3 [OS] [N]$	$\gamma_2 [S]$	$k_{1c} [O] [S]$	$k_{2c} [OS]$
Basal Transcription Rate	Transcription Rate from A+	Transcription Rate from OCT4-SOX2	Transcription Rate from OCT4-SOX2-NANOG Complex	Basal Degradation Rate	Degradation Rate from A+	Degradation Rate from OCT4-SOX2	Degradation Rate from OCT4-SOX2-NANOG Complex	Constant Degradation of SOX2	Formation Rate of OCT4-SOX2	Dissociation Rate of OCT4-SOX2

NANOG

$\frac{d[N]}{dt} = \frac{\eta_5 + e_1 [OS] + e_2 [OS] [N]}{1 + \eta_6 + f_1 [OS] + f_2 [OS] [N] + f_3 [B_- ]} - \gamma_3 [N]$

Table 3

$\eta_5$	$e_1 [OS]$	$e_2 [OS] [N]$	$1 + \eta_6$	$f_1 [OS]$	$f_2 [OS] [N]$	$f_3 [B_- ]$	$\gamma_3 [N]$
Basal Transcription Rate	Transcription Rate from OCT4-SOX2	Transcription Rate from OCT4-SOX2-NANOG Complex	Basal Degradation Rate	Degradation Rate from OCT4-SOX2	Degradation Rate from OCT4-SOX2-NANOG Complex	Degradation Rate from B-	Constant Degradation of NANOG

OCT4-SOX2

$\frac{d[OS]}{dt} = k_{1c} [O] [S] - k_{2c} [OS] - k_{3c} [OS]$

Table 4

$k_{1c} [O] [S]$	$k_{2c} [OS]$	$k_{3c} [OS]$
Formation Rate of OCT4-SOX2	Dissociation Rate of OCT4-SOX2	Degradation Rate of OCT4-SOX2

Target Genes

There are two sets of equations that are used to describe the concentration of the target gene product: the coherent model, consisting of an integrated equation for the self-renewal and differentiation genes, and the incoherent model, using two separate equations. The coherent model is based on a rate equation model similar to equations above, and the concentration is affected by OCT4-SOX2 and OCT4-SOX2-NANOG, with a degradation rate directly related to the concentration of the product. The equations specifically governing the self-renewal or differentiation genes for the incoherent model are very similar to the coherent model, but, unlike in the coherent model, OCT4-SOX2-NANOG positively affects the transcription of the self-renewal target genes and OCT4-SOX2 only has a negative effect on the differentiation target genes. Each term is further explained in the table following each equation.

Integrated Target Genes

$\frac{d[TG]}{dt} = \frac{\eta_7 + g_1 [OS]}{1 + \eta_8 + h_1 [OS] + h_2 [OS] [N]} - \gamma_4 [TG]$

Table 5

$\eta_7$	$g_1 [OS]$	$1 + \eta_8$	$h_1 [OS]$	$h_2 [OS] [N]$	$\gamma_4 [TG]$
Basal Transcription Rate	Transcription Rate from OCT4-SOX2	Basal Degradation Rate	Degradation Rate from OCT4-SOX2	Degradation Rate from OCT4-SOX2-NANOG Complex	Constant Degradation of Target Genes

Self-Renewal Target Genes

$\frac{d[TG_{self-renewal}]}{dt} = \frac{\eta_9 + m_1 [OS] + m_2 [OS] [N]}{1 + \eta_{10} + n_1 [OS] + n_2 [OS] [N]} - \gamma_5 [TG_{self-renewal}]$

Table 6

$\eta_9$	$m_1 [OS]$	$m_2 [OS] [N]$	$1 + \eta_{10}$	$n_1 [OS]$	$n_2 [OS] [N]$	$\gamma_5 [TG_{self-renewal}]$
Basal Transcription Rate	Transcription Rate from OCT4-SOX2	Transcription Rate from OCT4-SOX2-NANOG Complex	Basal Degradation Rate	Degradation Rate from OCT4-SOX2	Degradation Rate from OCT4-SOX2-NANOG Complex	Constant Degradation of Self-Renewal Target Genes

Differentiation Target Genes

$\frac{d[TG_{differentiation}]}{dt} = \frac{\eta_{11}}{1 + \eta_{12} + q_1 [OS] + q_2 [OS] [N]} - \gamma_6 [TG_{differentiation}]$

Table 7

$\eta_{11}$	$1 + \eta_{12}$	$q_1 [OS]$	$q_2 [OS] [N]$	$\gamma_6 [TG_{differentiation}]$
Basal Transcription Rate	Basal Degradation Rate	Degradation Rate from OCT4-SOX2	Degradation Rate from OCT4-SOX2-NANOG Complex	Constant Degradation of Differentiation Target Genes

Method

This model is replicated in Mathematica and is used to create a series of steady-state diagrams and bifurcation plots. This is done by setting each of the equations equal to zero and using Mathematica to solve the system for lines of equilibria. These lines are then plotted by varying inputs, including A₊, B₊, and OCT4-SOX2, or various parameters, thus creating steady-state diagrams and bifurcation plots. These diagrams show the bistable behavior of the system and how varying parameter values changes whether or not the system is capable of bistability.

State Variables and Parameters

All state variables, parameters, and constants are listed in the two tables below, along with descriptions and typical initial value or range of values.

Table 8: State Variables

Variable	Description	Initial Value
[O]	Concentration of OCT4	O0
[S]	Concentration of SOX2	S0
[N]	Concentration of NANOG	N0
[OS]	Concentration of OCT4-SOX2	0
TG	Concentration of the Target Gene Product	0

Table 9: Parameters and Constants

Parameter	Description	Range of Values
[A+]	Concentration of Signal A+	0 - 600
[B-]	Concentration of Signal B-	0 - 120
k1c	Formation Rate of OCT4-SOX2 Complex	0 - 10
k2c	Dissociation Rate of OCT4-SOX2 Complex	0 - 10
k3c	Degradation Rate of OCT4-SOX2 Complex	0 - 10
a1, a2, a3	Binding Strength between OCT4 and A+, OCT4-SOX2, and OCT4-SOX2-NANOG, respectively	0 - 10
b1, b2, b3	Repression Strength between OCT4 and A+, OCT4-SOX2, and OCT4-SOX2-NANOG, respectively	0.00055 - 0.011
c1, c2, c3	Binding Strength between SOX2 and A+, OCT4-SOX2, and OCT4-SOX2-NANOG, respectively	0 - 1
d1, d2, d3	Repression Strength between SOX2 and A+, OCT4-SOX2, and OCT4-SOX2-NANOG, respectively	0.00055 - 0.011
e1, e2	Binding Strength between NANOG and OCT4-SOX2 and OCT4-SOX2-NANOG, respectively	0 - 10
f1, f2, f3	Repression Strength between NANOG and OCT4-SOX2, OCT4-SOX2-NANOG, and B-, respectively	0 - 0.05
g1	Binding Strength between the Target Genes and OCT4-SOX2	0.1 - 6
h1, h2	Repression Strength between the Target Genes and OCT4-SOX2 and OCT4-SOX2-NANOG	0 - 0.5
m1, m2	Activation Strength between the Target Genes and OCT4-SOX2 and OCT4-SOX2-NANOG for Self-Renewal	0.1
n1, n2	Repression Strength between the Target Genes and OCT4-SOX2 and OCT4-SOX2-NANOG for Self-Renewal	0.001 - 0.01
q1, q2	Repression Strength between the Target Genes and OCT4-SOX2 and OCT4-SOX2-NANOG for Differentiation	0.001 - 0.01
η1	Basal Transcription Rate of OCT4	0 - 10
η2	Basal Degradation Rate of OCT4	10-7
η3	Basal Transcription Rate of SOX2	0.0001
η4	Basal Degradation Rate of SOX2	10-7
η5	Basal Transcription Rate of NANOG	0 - 10
η6	Basal Degradation Rate of NANOG	10-7
η7	Basal Transcription Rate of the Target Gene Product	10-4; 10-5
η8	Basal Degradation Rate of the Target Gene Product	10-7
η9	Basal Transcription Rate of the Target Gene Product for Self-Renewal	0.0001
η10	Basal Degradation Rate of the Target Gene Product for Self-Renewal	10-7
η11	Basal Transcription Rate of the Target Gene Product for Differentiation	1
η12	Basal Degradation Rate of the Target Gene Product for Differentiation	0.001
γ1	Degradation Constant of OCT4	0 - 5
γ2	Degradation Constant of SOX2	1
γ3	Degradation Constant of NANOG	0 - 2
γ4	Degradation Constant of the Target Gene Product	0.01
γ5	Degradation Constant of the Target Gene Product for Self-Renewal	0.05
γ6	Degradation Constant of the Target Gene Product for Dissociation	0.01