# Neural Differentiation Dynamics Controlled by Multiple Feedback Loops in a Comprehensive Molecular Interaction Network

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Construction of a Neuronal Differentiation Network

#### 2.2. Contraction of the Network

#### 2.3. Mathematical Model Construction

#### 2.4. Simulation and Analysis

## 3. Results

#### 3.1. Signaling Network of Neuronal Differentiation

#### 3.2. Mathematical Model of the Core Network

#### 3.3. Simulation of the Oscillatory Dynamics

^{−1}) and minimum (264 h

^{−1}) Kcat values were acquired from 27 records for mammals in BioNumbers; all Kcat values in the model were within this range. Detailed records are provided as an Excel file in the Supplementary Materials. These data indicated that the oscillatory state could be established under physiologically relevant conditions. The model could simulate the HES1 and ASCL1 oscillation within the 2.5 h period reported by Imayoshi et al. [14] in the undifferentiated state as the basal condition (Figure 5A and Figure A7).

#### 3.4. Model Validation

#### 3.5. Model Analysis

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Supporting Figures and Tables

**Figure A1.**Cascade-contracted network. To find the feedback loop, the whole signaling network was contracted by cascade contraction.

**Figure A2.**Diagram of the HES1 negative self-feedback loop model without delay. The model consists of an HES1 negative self-feedback loop with three nodes without delay. The red edge represents positive regulation. The blue edge represents negative regulation.

**Figure A3.**Simulation result of the HES1-loop model. The model could simulate the oscillatory state with a period of almost 2 hours. The following parameters were used: k3 = 0.00129, k3r = 0.0232, kg2 = 4.25, nmre1s9 = 5, kSmre1s9 = 0.00228, kg1 = 3.24, ks3d = 0.0258, and ks2d = 0.0355.

**Figure A4.**Diagram of the positive-feedback loop between PI3K and aPKC_PAR3_PAR6. (

**A**) The positive-feedback loop in the whole signaling network. (

**B**) The contracted positive-feedback loop in the toy model. Red edges represent positive regulations. Background colors show correspondence relationships between before and after contraction.

**Figure A5.**Diagram of the negative-feedback loop between PTEN and GSK3B. (

**A**) The negative-feedback loop in the whole signaling network. (

**B**) The contracted negative-feedback loop in the toy model. Red edges represent positive regulations. Blue edges represent negative regulations. Background color shows correspondence relationship between before and after contraction.

**Figure A6.**Diagram of negative-feedback loop between beta-catenin and HES1. (

**A**) The negative-feedback loop in the whole signaling network. (

**B**) The contracted negative-feedback loop in the toy model. Red edges represent positive regulations. Blue edges represent negative regulations. Background colors show correspondence relationships between before and after contraction.

**Figure A10.**Diagram of the collapsed model of the positive-feedback loop between aPKC_PAR3_PAR6 and PI3K.

**Figure A11.**Diagram of the collapsed model of the negative-feedback loop between beta-catenin and HES1.

Node Name | Reason | Integrated To (Identifier in the Toy Model) |
---|---|---|

NEUROG2 | Cascade (whole signaling network) | Rho_kinase |

MLC | Cascade (whole signaling network) | Rho_kinase |

RhoA | Cascade (whole signaling network) | Rho_kinase |

RAP1B | Cascade (whole signaling network) | PIP3 |

CDC42_GEF | Cascade (whole signaling network) | PIP3 |

Cofilin | Cascade (whole signaling network) | PAK |

LIMK | Cascade (whole signaling network) | PAK |

Stathmin | Cascade (whole signaling network) | PAK |

N_WASP | Cascade (whole signaling network) | CDC42 |

MRCK | Cascade (whole signaling network) | CDC42 |

KLC | Cascade (whole signaling network) | GSK3B |

APC | Cascade (whole signaling network) | GSK3B |

b-catenin | Cascade (whole signaling network) | GSK3B |

mTOR | Cascade (whole signaling network) | PIP3 |

RHEB | Cascade (whole signaling network) | PIP3 |

PDK1 | Cascade (whole signaling network) | PIP3 |

ILK | Cascade (whole signaling network) | PIP3 |

AKT | Cascade (whole signaling network) | PIP3 |

L1 | Cascade (whole signaling network) | RAS |

CREB | Cascade (whole signaling network) | RAS |

MAPKAP_K1 | Cascade (whole signaling network) | RAS |

MAPK | Cascade (whole signaling network) | RAS |

MEK | Cascade (whole signaling network) | RAS |

RAF | Cascade (whole signaling network) | RAS |

MARK2 | Cascade (whole signaling network) | aPKC_PAR3_PAR6 |

Arp2/3 | Feedback loop extraction | - |

IQGAP3 | Feedback loop extraction | - |

PAK | Feedback loop extraction | - |

p35/CDK5 | Feedback loop extraction | - |

SRA1_WAVE1 | Feedback loop extraction | - |

MAP1B | Feedback loop extraction | - |

Tau | Feedback loop extraction | - |

CRMP-2 | Feedback loop extraction | - |

RAS | Parameterization | - |

Rho_kinase | Cascade (core network) | PTEN_ca (s22) |

TIAM1/2 | Cascade (core network) | PI3K_ca (s20) |

RAC1 | Cascade (core network) | PI3K_ca (s20) |

PIP3 | Cascade (core network) | PIP_ca (s16) |

GSK3B | Cascade (core network) | GSK3B_ca (s24) |

PTEN | Cascade (core network) | PTEN_ca (s22) |

aPKC_PAR3_PAR6 | Cascade (core network) | aPKC_ca (s18) |

Equation No. | Differential equations |
---|---|

1 | $\frac{d\left[HES1\right]}{dt}=k{g}_{2}\xb7\left[mHES1\right]-{k}_{s{3}_{d}}\xb7\left[HES1\right]-2\xb7{k}_{3}\xb7{\left[HES1\right]}^{2}+2\xb7{k}_{3r}\xb7\left[dime{r}_{HES1}\right]$ |

2 | $\frac{d\left[mHES1\right]}{dt}=k{g}_{1}\xb7\left(1-\frac{{\left[dime{r}_{HES1}\right]}^{n{m}_{re{1}_{s9}}}}{{\left[dime{r}_{HES1}\right]}^{n{m}_{re{1}_{s9}}}+kS{m}_{re{1}_{s9}}^{n{m}_{re{1}_{s9}}}}\right)-{k}_{s{2}_{d}}\left[mHES1\right]$ |

3 | $\frac{d\left[dime{r}_{HES1}\right]}{dt}={k}_{3}\xb7{\left[HES1\right]}^{2}-{k}_{3r}\xb7\left[dime{r}_{HES1}\right]$ |

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**Figure 2.**Signaling network of neural stem cell differentiation based on publicly available data. (

**A**) The entire signaling network. (

**B**) Core signaling network: feedback loops extracted from the entire network. Rectangular nodes are generic proteins. Oval nodes are small molecules. The arrowhead node is a receptor. Dashed-line nodes are active forms. Bold-line nodes are neuronal markers. Node color codes: yellow, receptor; red, transcription factor; orange, enzyme; blue, molecule related to a function of a mature neuron; green, other.

**Figure 3.**Feedback loops and contraction. (

**A**) Examples of positive and negative feedback loops. (

**B**) Example of contraction.

**Figure 4.**Diagram of the toy model. The model consists of multiple feedback loops extracted from the core signaling network. The red edge is a component of a positive feedback loop. Blue edges are components of negative feedback loops. Node color codes: yellow, receptor; red, transcription factor; white, contracted node.

**Figure 5.**Simulation of the toy model. (

**A**) Basic condition with a 2.5 h period. (

**B**) Negative and (

**C**) positive perturbation of the NOTCH concentration at 50 h. (

**D**) Bifurcation analysis of the HES1 and ASCL1 concentration dependence on the NOTCH concentration. Background colors show cell differentiation state; green, neuron; yellow, astrocyte.

**Figure 6.**Simulation of HES1 and ASCL1 concentrations under the Id2 gene knockdown or overexpression conditions. (

**A**) Id2 gene knockdown with kSm_Id = 0.1. (

**B**) Id2 gene overexpression with kSm_Id = 10. Background colors show cell differentiation state; green, neuron; yellow, astrocyte.

**Figure 7.**Bifurcation analysis of the HES1 and ASCL1 concentration dependence on the NOTCH concentration when each feedback loop is collapsed. (

**A**) Collapsed HES1 self-feedback loop. (

**B**) Collapsed negative-feedback loop between GSK3B and PTEN. (

**C**) Collapsed positive-feedback loop between aPKC_PAR3_PAR6 and PI3K. (

**D**) Collapsed negative-feedback loop between beta-catenin and HES1. Background colors show cell differentiation state; green, neuron; yellow, astrocyte.

Equation No. | Equation |
---|---|

1 | $\frac{d\left[HES1\right]}{dt}=k{g}_{2}\xb7\left[mHES1\right]-{k}_{s{3}_{d}}\xb7\left[HES1\right]-2\xb7{k}_{3}\xb7{\left[HES1\right]}^{2}+2\xb7{k}_{3r}\xb7\left[dime{r}_{HES1}\right]$ |

2 | $\begin{array}{ll}\frac{d\left[mHES1\right]}{dt}=& \frac{k{g}_{1}\xb7{\left[NOTCH\right]}^{n{p}_{re{2}_{s11}}}}{{\left[NOTCH\right]}^{n{p}_{re{2}_{s11}}}+kS{p}_{re{2}_{s11}}^{n{p}_{re{2}_{s11}}}}\\ & \left(1-\frac{{\left[dime{r}_{HES1}\right]}^{n{p}_{re{2}_{s9}}}}{{\left[dime{r}_{HES1}\right]}^{n{p}_{re{2}_{s9}}}+kS{m}_{re{2}_{s9}}^{n{p}_{re{2}_{s9}}}}\right)\\ & \left(1-\frac{{\left[GSK3{B}_{ca}\right]}^{n{p}_{re{2}_{s24}}}}{{\left[GSK3{B}_{ca}\right]}^{n{p}_{re{2}_{s24}}}+kS{m}_{re{2}_{s24}}^{n{p}_{re{2}_{s24}}}}\right)-{k}_{s{2}_{d}}\left[mHES1\right]\end{array}$ |

3 | $\frac{d\left[dime{r}_{HES1}\right]}{dt}={k}_{3}\xb7{\left[HES1\right]}^{2}-{k}_{3r}\xb7\left[dime{r}_{HES1}\right]$ |

4 | $\frac{d\left[ASCL1\right]}{dt}=k{g}_{re7}\xb7\left(1-\frac{{\left[HES1\right]}^{n{m}_{re{7}_{s3}}}}{{\left[HES1\right]}^{n{m}_{re{7}_{s3}}}+kS{m}_{re{7}_{s3}}^{n{m}_{re{7}_{s3}}}}\right)-kas{s}_{re8}\xb7\left[ASCL1\right]$ |

5 | $\frac{d\left[PTE{N}_{ca}\right]}{dt}=\frac{\left[HES1\right]\xb7kcat{p}_{re11}\xb7\left[PTE{N}_{ci}\right]}{k{M}_{re{11}_{s15}}+\left[PTE{N}_{ci}\right]}-\frac{\left[GSK3{B}_{ca}\right]\xb7kcat{p}_{re12}\xb7\left[PTE{N}_{ca}\right]}{k{M}_{re{12}_{s22}}+\left[PTE{N}_{ca}\right]}$ |

6 | $\frac{d\left[PTE{N}_{ci}\right]}{dt}=\frac{\left[GSK3{B}_{ca}\right]\xb7kcat{p}_{re12}\xb7\left[PTE{N}_{ca}\right]}{k{M}_{re{12}_{s22}}+\left[PTE{N}_{ca}\right]}-\frac{\left[HES1\right]\xb7kcat{p}_{re11}\xb7\left[PTE{N}_{ci}\right]}{k{M}_{re{11}_{s15}}+\left[PTE{N}_{ci}\right]}$ |

7 | $\frac{d\left[PI{P}_{ca}\right]}{dt}=\frac{\left[PI3{K}_{ca}\right]\xb7kcat{p}_{re14}\xb7\left[PI{P}_{ci}\right]}{k{M}_{re{14}_{s17}}+\left[PI{P}_{ci}\right]}-\frac{\left[PTE{N}_{ca}\right]\xb7kcat{p}_{re13}\xb7\left[PI{P}_{ca}\right]}{k{M}_{re{13}_{s16}}+\left[PI{P}_{ca}\right]}$ |

8 | $\frac{d\left[PI{P}_{ci}\right]}{dt}=\frac{\left[PTE{N}_{ca}\right]\xb7kcat{p}_{re13}\xb7\left[PI{P}_{ca}\right]}{k{M}_{re{13}_{s16}}+\left[PI{P}_{ca}\right]}-\frac{\left[PI3{K}_{ca}\right]\xb7kcat{p}_{re14}\xb7\left[PI{P}_{ci}\right]}{k{M}_{re{14}_{s17}}+\left[PI{P}_{ci}\right]}$ |

9 | $\frac{d\left[PI3{K}_{ca}\right]}{dt}=\frac{\left[aPK{C}_{ca}\right]\xb7kcat{p}_{re15}\xb7\left[PI3{K}_{ci}\right]}{k{M}_{re{15}_{s21}}+\left[PI3{K}_{ci}\right]}-\frac{V{p}_{re17}\xb7\left[PI3{K}_{ca}\right]}{k{M}_{re{17}_{s20}}+\left[PI3{K}_{ca}\right]}$ |

10 | $\frac{d\left[PI3{K}_{ci}\right]}{dt}=\frac{V{p}_{re17}\xb7\left[PI3{K}_{ca}\right]}{k{M}_{re{17}_{s20}}+\left[PI3{K}_{ca}\right]}-\frac{\left[aPK{C}_{ca}\right]\xb7kcat{p}_{re15}\xb7\left[PI3{K}_{ci}\right]}{k{M}_{re{15}_{s21}}+\left[PI3{K}_{ci}\right]}$ |

11 | $\frac{d\left[aPK{C}_{ca}\right]}{dt}=\frac{\left[PI{P}_{ca}\right]\xb7kcat{p}_{re16}\xb7\left[aPK{C}_{ci}\right]}{k{M}_{re{16}_{s19}}+\left[aPK{C}_{ci}\right]}-\frac{V{p}_{re18}\xb7\left[aPK{C}_{ca}\right]}{k{M}_{re{18}_{s18}}+\left[aPK{C}_{ca}\right]}$ |

12 | $\frac{d\left[aPK{C}_{ci}\right]}{dt}=\frac{V{p}_{re18}\xb7\left[aPK{C}_{ca}\right]}{k{M}_{re{18}_{s18}}+\left[aPK{C}_{ca}\right]}-\frac{\left[PI{P}_{ca}\right]\xb7kcat{p}_{re16}\xb7\left[aPK{C}_{ci}\right]}{k{M}_{re{16}_{s19}}+\left[aPK{C}_{ci}\right]}$ |

13 | $\frac{d\left[GSK3{B}_{ca}\right]}{dt}=\frac{V{p}_{re20}\xb7\left[GSK3{B}_{ci}\right]}{k{M}_{re{20}_{s23}}+\left[GSK3{B}_{ci}\right]}-\frac{\left[aPK{C}_{ca}\right]\xb7kcat{p}_{re19}\xb7\left[GSK3{B}_{ca}\right]}{k{M}_{re{19}_{s24}}+\left[GSK3{B}_{ca}\right]}$ |

14 | $\frac{d\left[GSK3{B}_{ci}\right]}{dt}=\frac{\left[aPK{C}_{ca}\right]\xb7kcat{p}_{re19}\xb7\left[GSK3{B}_{ca}\right]}{k{M}_{re{19}_{s24}}+\left[GSK3{B}_{ca}\right]}-\frac{V{p}_{re20}\xb7\left[GSK3{B}_{ci}\right]}{k{M}_{re{20}_{s23}}+\left[GSK3{B}_{ci}\right]}$ |

Parameter (unit) | 2.5 h period | Minimum | Maximum | Parameter Description |
---|---|---|---|---|

${k}_{s{2}_{d}}$(h^{−1}) | 0.99 | 0.69 | 2.45 | Degradation rate constant of reaction 3 for substrate s3 |

${k}_{s{3}_{d}}$(h^{−1}) | 1.29 | 1.03 | 2.28 | Degradation rate constant of reaction 4 for substrate s2 |

${k}_{3}$ (µM·h^{−1}) | 0.4074 | 0.37 | 0.65 | Dimerization rate constant of reaction 5 |

${k}_{3r}$ (h^{−1}) | 2.3 | 0.61 | 2.49 | Dissociation rate constant of reaction 6 |

$kas{s}_{re8}$(h^{−1}) | 31.2 | <0.001 | >100 | Degradation rate constant of reaction 8 for substrate s5 |

$kcat{p}_{re11}$ (h^{−1}) | 141.6 | 108 | 146 | Turnover number of reaction 11 |

$kcat{p}_{re12}$ (h^{−1}) | 132.6 | 130 | 174 | Turnover number of reaction 12 |

$kcat{p}_{re13}$ (h^{−1}) | 209.4 | 162 | 216 | Turnover number of reaction 13 |

$kcat{p}_{re14}$(h^{−1}) | 132 | 128 | 173 | Turnover number of reaction 14 |

$kcat{p}_{re15}$(h^{−1}) | 132 | 128 | 173 | Turnover number of reaction 15 |

$kcat{p}_{re16}$ (h^{−1}) | 174 | 171 | 238 | Turnover number of reaction 16 |

$kcat{p}_{re19}$ (h^{−1}) | 132 | 95 | 183 | Turnover number of reaction 19 |

$k{g}_{1}$ (µM·h^{−1}) | 361.2 | 183 | 526 | Maximal transcription rate of reaction 2 |

$k{g}_{2}$ (µM·h^{−1}) | 25.74 | 13.1 | 31.6 | Mass action constant of reaction 1 |

$k{g}_{re7}$ (µM·h^{−1}) | 10.86 | <0.001 | >100 | Maximal transcription rate of reaction 7 |

$k{M}_{re{11}_{s15}}$ (µM) | 50.0 | 48.9 | 65.5 | Michaelis–Menten constant of reaction 11 for substrate s15 |

$k{M}_{re{12}_{s22}}$ (µM) | 1.62 | 1.26 | 1.66 | Michaelis–Menten constant of reaction 12 for substrate s22 |

$k{M}_{re{13}_{s16}}$ (µM) | 0.21 | 0.21 | 0.29 | Michaelis–Menten constant of reaction 13 for substrate s16 |

$k{M}_{re{14}_{s17}}$ (µM) | 28.4 | 21.6 | 28.9 | Michaelis–Menten constant of reaction 14 for substrate s17 |

$k{M}_{re{15}_{s21}}$ (µM) | 12.7 | 9.4 | 13.1 | Michaelis–Menten constant of reaction 15 for substrate s21 |

$k{M}_{re{16}_{s19}}$ (µM) | 0.45 | 0.22 | 0.49 | Michaelis–Menten constant of reaction 16 for substrate s19 |

$k{M}_{re{17}_{s20}}$ (µM) | 1.2 | 1.18 | 1.61 | Michaelis–Menten constant of reaction 17 for substrate s20 |

$k{M}_{re{18}_{s18}}$ (µM) | 0.91 | 0.89 | 1.52 | Michaelis–Menten constant of reaction 18 for substrate s18 |

$k{M}_{re{19}_{s24}}$ (µM) | 9.0 | 6.2 | 12.8 | Michaelis–Menten constant of reaction 19 for substrate s24 |

$k{M}_{re{20}_{s23}}$ (µM) | 0.62 | 0.38 | 0.88 | Michaelis–Menten constant of reaction 20 for substrate s23 |

$kS{m}_{re{2}_{s24}}$ (µM) | 0.04 | 0.029 | 0.048 | Half-maximal inhibitory concentration of substrate s24 in reaction 2 |

$kS{m}_{re{2}_{s9}}$ (µM) | 0.0023 | <0.001 | 0.0025 | Half-maximal inhibitory concentration of substrate s9 in reaction 2 |

$kS{m}_{re{7}_{sa}}$ (µM) | 0.116 | <0.001 | >100 | Half-maximal inhibitory concentration of substrate s3 in reaction 7 |

$k{S}_{{p}_{re{2}_{s11}}}$ (µM) | 2.5 | 2.18 | 3.17 | Half-maximal effective concentration of substrate s24 in reaction 2 |

${V}_{{p}_{re17}}$(µM·h^{−1}) | 88.2 | 67.9 | 90.9 | Maximal reaction rate constant of reaction 17 |

${V}_{{p}_{re18}}$ (µM·h^{−1}) | 14.52 | 10.7 | 14.9 | Maximal reaction rate constant of reaction 18 |

${V}_{{p}_{re20}}$ (µM·h^{−1}) | 24.0 | 17.9 | 33.3 | Maximal reaction rate constant of reaction 20 |

$n{m}_{re{2}_{s24}}$ | 2 | 2 | 2 | Inhibition coefficient of reaction 2 for substrate s24 |

$n{m}_{re{2}_{s9}}$ | 5 | 5 | >10 | Inhibition coefficient of reaction 2 for substrate s9 |

$n{m}_{re{7}_{s3}}$ | 2 | 1 | >10 | Inhibition coefficient of reaction 7 for substrate s3 |

$n{p}_{re{2}_{s11}}$ | 3 | 2 | 3 | Hill coefficient of reaction 2 for substrate s11 |

**Table 3.**Differential equations of a model with an incorporated inhibition of HES1 dimerization by Id2.

Equation No. | Equation |
---|---|

1’ | $\frac{d\left[HES1\right]}{dt}=k{g}_{2}\xb7\left[mHES1\right]-{k}_{s{3}_{d}}\xb7\left[HES1\right]-\frac{2\xb7{k}_{3}\xb7{\left[HES1\right]}^{2}}{kS{m}_{Id}}+2\xb7{k}_{3r}\xb7\left[dime{r}_{HES1}\right]$ |

3’ | $\frac{d\left[dime{r}_{HES1}\right]}{dt}=\frac{{k}_{3}\xb7{\left[HES1\right]}^{2}}{kS{m}_{Id}}-{k}_{3r}\xb7\left[dime{r}_{HES1}\right]$ |

Equation No. | Equation |
---|---|

2’ | $\begin{array}{ll}\frac{d\left[mHES1\right]}{dt}=& \frac{k{g}_{1}\xb7{\left[NOTCH\right]}^{n{p}_{re{2}_{s11}}}}{{\left[NOTCH\right]}^{n{p}_{re{2}_{s11}}}+kS{p}_{re{2}_{s11}}^{n{p}_{re{2}_{s11}}}}\xb71\\ & \left(1-\frac{{\left[GSK3{B}_{ca}\right]}^{n{p}_{re{2}_{s24}}}}{{\left[GSK3{B}_{ca}\right]}^{n{p}_{re{2}_{s24}}}+kS{m}_{re{2}_{s24}}^{n{p}_{re{2}_{s24}}}}\right)-{k}_{s{2}_{d}}\left[mHES1\right]\end{array}$ |

2″ | $\begin{array}{ll}\frac{d\left[mHES1\right]}{dt}=& \frac{k{g}_{1}\xb7{\left[NOTCH\right]}^{n{p}_{re{2}_{s11}}}}{{\left[NOTCH\right]}^{n{p}_{re{2}_{s11}}}+kS{p}_{re{2}_{s11}}^{n{p}_{re{2}_{s11}}}}\xb7\left(1-\frac{{\left[dime{r}_{HES1}\right]}^{n{p}_{re{2}_{s9}}}}{{\left[dime{r}_{HES1}\right]}^{n{p}_{re{2}_{s9}}}+kS{m}_{re{2}_{s9}}^{n{p}_{re{2}_{s9}}}}\right)\\ & 1-{k}_{s{2}_{d}}\left[mHES1\right]\end{array}$ |

5’ | $\frac{d\left[PTE{N}_{ca}\right]}{dt}=\frac{\left[HES1\right]\xb7kcat{p}_{re11}\xb7\left[PTE{N}_{ci}\right]}{k{M}_{re{11}_{s15}}+\left[PTE{N}_{ci}\right]}-\frac{1\xb7kcat{p}_{re12}\xb7\left[PTE{N}_{ca}\right]}{k{M}_{re{12}_{s22}}+\left[PTE{N}_{ca}\right]}$ |

6’ | $\frac{d\left[PTE{N}_{ci}\right]}{dt}=\frac{1\xb7kcat{p}_{re12}\xb7\left[PTE{N}_{ca}\right]}{k{M}_{re{12}_{s22}}+\left[PTE{N}_{ca}\right]}-\frac{\left[HES1\right]\xb7kcat{p}_{re11}\xb7\left[PTE{N}_{ci}\right]}{k{M}_{re{11}_{s15}}+\left[PTE{N}_{ci}\right]}$ |

7’ | $\frac{d\left[PI{P}_{ca}\right]}{dt}=\frac{1\xb7kcat{p}_{re14}\xb7\left[PI{P}_{ci}\right]}{k{M}_{re{14}_{s17}}+\left[PI{P}_{ci}\right]}-\frac{\left[PI{P}_{ca}\right]\xb7kcat{p}_{re13}\xb7\left[PI{P}_{ca}\right]}{k{M}_{re{13}_{s16}}+\left[PI{P}_{ca}\right]}$ |

8’ | $\frac{d\left[PI{P}_{ca}\right]}{dt}=\frac{1\xb7kcat{p}_{re14}\xb7\left[PI{P}_{ci}\right]}{k{M}_{re{14}_{s17}}+\left[PI{P}_{ci}\right]}-\frac{\left[PI{P}_{ca}\right]\xb7kcat{p}_{re13}\xb7\left[PI{P}_{ca}\right]}{k{M}_{re{13}_{s16}}+\left[PI{P}_{ca}\right]}$ |

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**MDPI and ACS Style**

Iwasaki, T.; Takiguchi, R.; Hiraiwa, T.; Yamada, T.G.; Yamazaki, K.; Hiroi, N.F.; Funahashi, A.
Neural Differentiation Dynamics Controlled by Multiple Feedback Loops in a Comprehensive Molecular Interaction Network. *Processes* **2020**, *8*, 166.
https://doi.org/10.3390/pr8020166

**AMA Style**

Iwasaki T, Takiguchi R, Hiraiwa T, Yamada TG, Yamazaki K, Hiroi NF, Funahashi A.
Neural Differentiation Dynamics Controlled by Multiple Feedback Loops in a Comprehensive Molecular Interaction Network. *Processes*. 2020; 8(2):166.
https://doi.org/10.3390/pr8020166

**Chicago/Turabian Style**

Iwasaki, Tsuyoshi, Ryo Takiguchi, Takumi Hiraiwa, Takahiro G. Yamada, Kazuto Yamazaki, Noriko F. Hiroi, and Akira Funahashi.
2020. "Neural Differentiation Dynamics Controlled by Multiple Feedback Loops in a Comprehensive Molecular Interaction Network" *Processes* 8, no. 2: 166.
https://doi.org/10.3390/pr8020166