# Bilateral Feedback in Oscillator Model Is Required to Explain the Coupling Dynamics of Hes1 with the Cell Cycle

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modelling Strategy

#### 2.1. Phase Representation of Bcc Data

#### 2.2. Specifying the Coupled Oscillator Model

- ${\theta}_{\mathrm{Hes1}}=0$ will be considered the peak of an oscillatory Hes1 wave, and ${\theta}_{\mathrm{Hes1}}=\pi $ is the trough.
- ${\theta}_{\mathrm{C}.\mathrm{C}.}=0$ will denote the beginning and the end of a cell cycle, i.e., a cell division event.

#### 2.3. Numerical Implementation

#### 2.4. Model Optimisation Strategy

## 3. Results of Model Simulations and Comparisons to Biological Data

#### 3.1. The Uncoupled Scenario

#### 3.2. The Symmetric Interaction Strength Scenario

#### 3.3. The Unconstrained Interaction Strength Scenario

#### 3.4. Asymmetry in Interaction Strength Is Predictive of Elongation in Bcsc Data

#### 3.5. Cluster-Dependent Coupling Strength Points to Gene Expression and Cell Cycle Duration Differences

#### 3.6. Mathematical Analysis and Long-Term Behaviour of Our Model

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Unidirectional Interaction Constraints

**Figure A1.**Results of the solutions to the model Equation (5) with the constraint ${\kappa}_{\mathrm{C}.\mathrm{C}.}=0$ when ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}}$ is optimised to the data vector ${\mathbf{\Theta}}_{\mathrm{Hes1}}^{\left(i\right)},\phantom{\rule{3.33333pt}{0ex}}i=1,2,3$, frequency parameters are set as ${\omega}_{\mathrm{Hes1}}={24}^{-1}$ and ${\omega}_{\mathrm{C}.\mathrm{C}.}={27}^{-1}$. (

**A**–

**C**) Square torus diagram trajectories of solutions ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}}$, i = 1, 2, 3 superimposed over data trajectories from Figure 2B. (

**D**–

**F**) Psedo-time plots of rebuilt Hes1 dynamics found via $cos\left(\right)open="("\; close=")">{\theta}_{\mathrm{Hes1}}^{\left(i\right)}$ superimpsoed over mean traces from Figure 2A. In (

**A**–

**F**), the black lines denote the model results, coloured lines denote the data (red: cluster 1, blue: cluster 2 and green: cluster 3), and in (

**D**–

**F**), the shaded areas denote the standard deviations of the data. (

**G**) Bar chart of cell cycle duration predictions from the simulation for each cluster case.

**Figure A2.**Results of the solutions to the model Equation (5) with the constraint ${\kappa}_{\mathrm{Hes1}}=0$ when ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}}$ is optimised to the data vector ${\mathbf{\Theta}}_{\mathrm{Hes1}}^{\left(i\right)},\phantom{\rule{3.33333pt}{0ex}}i=1,2,3$, frequency parameters are set as ${\omega}_{\mathrm{Hes1}}={24}^{-1}$ and ${\omega}_{\mathrm{C}.\mathrm{C}.}={27}^{-1}$. (

**A**–

**C**) Square torus diagram trajectories of solutions ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}},\phantom{\rule{3.33333pt}{0ex}}\mathit{i}$ = 1, 2, 3 superimposed over data trajectories from Figure 2B. (

**D**–

**F**) Psedo-time plots of rebuilt Hes1 dynamics found via $cos\left(\right)open="("\; close=")">{\theta}_{\mathrm{Hes1}}^{\left(i\right)}$ superimposed over mean traces from Figure 2A. In (

**A**–

**F**), the black lines denote the model results, coloured lines denote the data (red: cluster 1, blue: cluster 2 and green: cluster 3), and in (

**D**–

**F**), the shaded areas denote the standard deviations of the data. (

**G**) Bar chart of cell cycle duration predictions from the simulation for each cluster case.

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**Figure 1.**Graphs and schematics representing the results in [26]. (

**A**) Schematic illustrating that cell division at a peak in Hes1 yields regular and consistent cell cycle durations (

**top panel**) whereas division at a trough gives rise to longer cell cycles (middle panel). Schematic of cyclins (cell cycle proteins) that rise and fall in abundance once every cell cycle (

**bottom panel**). (

**B**–

**D**): There exist three distinct mitosis to mitosis dynamic behaviours of normalized (z-scored) Hes1 expression: Cluster 1 with high-low-high, Cluster 2 with medium-low-high and Cluster 3 with low-high-low-high dynamics (red, blue, green lines, respectively, shading is ± one standard deviation). (

**E**) The momentary position within the oscillation (i.e., the phase that we define in Section 2) at the beginning of the cell cycle of each cell (total n = 164) is plotted as a polar histogram and separated by cluster classification. Means are marked by radial lines. (

**F**) Bar chart of mean cell cycle duration of breast cancer cells per cluster. The cells whose Hes1 expression dynamics are classified as Cluster 3 show significantly (p < 0.0001) longer cell cycles than those in Clusters 1 or 2.

**Figure 2.**Developing the phase progression data. (

**A**) Mean traces of all three clustered, normalised (z-scored) Hes1 traces in pseudo-time. (

**B**) A uniformly spaced cell cycle oscillator vector ${\mathbf{\Theta}}_{\mathrm{C}.\mathrm{C}.}\in [0,2\pi )$ on the x-axis plotted against the Hes1 oscillator phase reconstruction ${\mathbf{\Theta}}_{\mathrm{Hes1}}$ of traces from (

**A**) found via application of the Hilbert transform forming phase space trajectories on a square torus diagram. (

**C**) Alternative view of phase progression between our data vectors ${\mathbf{\Theta}}_{\mathrm{Hes1}}$ and ${\mathbf{\Theta}}_{\mathrm{C}.\mathrm{C}.}$. Both oscillators are represented as points on a circle (coloured dots are the Hes1 and black dots are the cell cycle oscillators), at five snapshotted times within the pseudo-timed cell cycle.

**Figure 3.**Schematic illustrating our method of developing model simulations with an arbitrary example trajectory.

**Figure 4.**Results of the solutions to model Equation (5) with the constraint ${\kappa}_{\mathrm{Hes1}}={\kappa}_{\mathrm{C}.\mathrm{C}.}=0$ when ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}}$ is optimised to the data vector ${\mathbf{\Theta}}_{\mathrm{Hes1}}^{\left(i\right)},\phantom{\rule{3.33333pt}{0ex}}i=1,2,3$, the frequency parameters are set as ${\omega}_{\mathrm{Hes1}}={24}^{-1}$ and ${\omega}_{\mathrm{C}.\mathrm{C}.}={27}^{-1}$. (

**A**–

**C**) Square torus diagram trajectories of solutions ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}},$

**i**= 1, 2, 3 superimposed over data trajectories from Figure 2B. (

**D**–

**F**) Pseudo-time plots of rebuilt Hes1 dynamics found via $cos\left(\right)open="("\; close=")">{\theta}_{\mathrm{Hes1}}^{\left(i\right)}$ i = 1, 2, 3 superimposed over mean traces from Figure 2A. In (

**A**–

**F**), the black lines denote the model results, coloured lines denote the data (red: cluster 1, blue: cluster 2 and green: cluster 3) and in (

**D**–

**F**) the shaded areas denote the standard deviations of the data. (

**G**) Bar chart of cell cycle duration predictions from the simulation for each cluster case.

**Figure 5.**Results of the solutions to model (5) with the constraint ${\kappa}_{\mathrm{Hes1}}={\kappa}_{\mathrm{C}.\mathrm{C}.}$ when ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}}$ is optimised to the data vector ${\mathbf{\Theta}}_{\mathrm{Hes1}}^{\left(i\right)},\phantom{\rule{3.33333pt}{0ex}}i=1,2,3$, frequency parameters are set as ${\omega}_{\mathrm{Hes1}}={24}^{-1}$ and ${\omega}_{\mathrm{C}.\mathrm{C}.}={27}^{-1}$. (

**A**–

**C**) Square torus diagram trajectories of solutions ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}}$, i = 1, 2, 3 superimposed over data trajectories from Figure 2B. (

**D**–

**F**) Pseudo-time plots of rebuilt Hes1 dynamics found via $cos\left(\right)open="("\; close=")">{\theta}_{\mathrm{Hes1}}^{\left(i\right)}$ superimposed over mean traces from Figure 2A. In (

**A**–

**F**), the black lines denote the model results, coloured lines denote the data (red: cluster 1, blue: cluster 2 and green: cluster 3), and in (

**D**–

**F**), the shaded areas denote the standard deviations of the data. (

**G**) Bar chart of cell cycle duration predictions from the simulation for each cluster case.

**Figure 6.**Results of the solutions to model (5) with no parameter constraints when ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}}$ is optimised to the data vector ${\mathbf{\Theta}}_{\mathrm{Hes1}}^{\left(i\right)},\phantom{\rule{3.33333pt}{0ex}}i=1,2,3$, frequency parameters are set as ${\omega}_{\mathrm{Hes1}}={24}^{-1}$ and ${\omega}_{\mathrm{C}.\mathrm{C}.}={27}^{-1}$. (

**A**–

**C**) Square torus diagram trajectories of solutions ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}},$

**i**= 1, 2, 3 superimposed over data trajectories from Figure 2B. (

**D**–

**F**) Pseudo-time plots of rebuilt Hes1 dynamics found via $cos\left(\right)open="("\; close=")">{\theta}_{\mathrm{Hes1}}^{\left(i\right)}$, i = 1, 2, 3 superimposed over mean traces from Figure 2A. In (

**A**–

**F**), the black lines denote the model results, coloured lines denote the data (red: cluster 1, blue: cluster 2 and green: cluster 3), and in (

**D**–

**F**), the shaded areas denote the standard deviations of the data. (

**G**) Bar chart of cell cycle duration predictions from the simulation for each cluster case.

**Figure 7.**Bar chart summarising cell cycle simulation times for each parameter constraint case. Taken from (

**F**) in Figure 1 and (

**G**) in Figure 4, Figure 5 and Figure 6, Figure A1 and Figure A2. Bar charts have been normalised by dividing all three by the cluster 1 simulation time value for each constraint. Strategy 5 has been highlighted since it most closely matches the “data”.

**Figure 8.**Heatmaps of two-dimensional ${\kappa}_{\mathrm{C}.\mathrm{C}.}\times {\kappa}_{\mathrm{Hes1}}$ parameter space with the colour representing the residual value (Euclidean distance) between solution vector ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}}$ and data vector ${\mathbf{\Theta}}_{\mathrm{Hes1}}^{\left(i\right)}$ where $i=1,2,3$ in panels (

**A–C**) respectively, in the case with no parameter constraints. The black

**x**marks the point in this space where the residual is the lowest for each cluster, i.e., the coupling strength parameter values that yielded the best fit to the data and thus those used to produce the results in Figure 6.

**Figure 9.**(

**A**–

**C**) Long-term solutions of Equation (5) with optimal parameter values found in Figure 8 for clusters 1, 2 and 3, respectively. (

**D**) Two-dimensional ${\kappa}_{\mathrm{C}.\mathrm{C}.}\times {\kappa}_{\mathrm{Hes1}}$ parameter space with optimal parameter values found in Figure 8 for clusters 1, 2 and 3 plotted as a red, blue and green

**x**, respectively. The yellow region represents parameters where the condition of Equation (7) is met and when trajectories will asymptotically reach a stable phase-locked solution. The blue region represents the region where Equation (7) is not met, and quasiperiodicity will occur.

**Figure 10.**Torus diagram trajectories for one simulation of Equation (5) at uniformly spaced ${\theta}_{\mathrm{Hes1}}\left({t}_{0}\right)$ initial conditions with optimal parameters for cluster 1 in panel (

**A**), cluster 2 in panel (

**B**) and cluster 3 in panel (

**C**), as stated in Table 2. Red, blue and green trajectories in panels (

**A**–

**C**) here are the same as the black trajectories in Figure 6A–C, respectively.

**Table 1.**Table of residual comparisons for each modelling strategy 1 to 5. Values are the minimised R values found by fitting the pseudo-timed model solution ${\mathbf{\theta}}_{\mathrm{Hes1}}^{\mathbf{\left(}\mathit{i}\mathbf{\right)}}$ to ${\mathbf{\Theta}}_{\mathrm{Hes1}}^{\left(i\right)}$ for $i=1,2,3$ within the given parameter constraints. Highlighted in cyan is the strategy with the lowest summed residuals across clusters.

Modelling Strategy | Coupling Strength Parameter Constraint | Cluster 1 | Cluster 2 | Cluster 3 | Sum |
---|---|---|---|---|---|

1 | ${\kappa}_{\mathrm{Hes1}}=0,\phantom{\rule{3.33333pt}{0ex}}{\kappa}_{\mathrm{C}.\mathrm{C}.}=0$ | 12.1199 | 23.5206 | 6.8118 | 42.4523 |

2 | ${\kappa}_{\mathrm{Hes1}}={\kappa}_{\mathrm{C}.\mathrm{C}.}$ | 8.9520 | 3.0552 | 2.9236 | 14.9308 |

3 | ${\kappa}_{\mathrm{C}.\mathrm{C}.}=0$ | 9.9229 | 1.7894 | 3.0774 | 14.7897 |

4 | ${\kappa}_{\mathrm{Hes1}}=0$ | 8.8340 | 7.3753 | 2.8757 | 19.085 |

5 | Optimal (no constraints) | 8.8352 | 1.7967 | 2.8702 | 13.5021 |

**Table 2.**Table summarizing the optimal ${\kappa}_{\mathrm{Hes1}}$ and ${\kappa}_{\mathrm{C}.\mathrm{C}.}$ values for the model Equation (5) for each cluster case.

Cluster | 1 | 2 | 3 |
---|---|---|---|

Initial Hes1 phase (1 d.p.) | 6.1 | 2.0 | 3.4 |

${\kappa}_{\mathrm{Hes1}}$ | 0 | 0.86 | 0.01 |

${\kappa}_{\mathrm{C}.\mathrm{C}.}$ | 0.99 | 0 | 0.13 |

${\omega}_{\mathrm{Hes1}}$ | 0.0417 | 0.0417 | 0.0417 |

${\omega}_{\mathrm{C}.\mathrm{C}.}$ | 0.0370 | 0.0370 | 0.0370 |

Condition (7) met? | Yes | Yes | No |

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## Share and Cite

**MDPI and ACS Style**

Rowntree, A.; Sabherwal, N.; Papalopulu, N.
Bilateral Feedback in Oscillator Model Is Required to Explain the Coupling Dynamics of Hes1 with the Cell Cycle. *Mathematics* **2022**, *10*, 2323.
https://doi.org/10.3390/math10132323

**AMA Style**

Rowntree A, Sabherwal N, Papalopulu N.
Bilateral Feedback in Oscillator Model Is Required to Explain the Coupling Dynamics of Hes1 with the Cell Cycle. *Mathematics*. 2022; 10(13):2323.
https://doi.org/10.3390/math10132323

**Chicago/Turabian Style**

Rowntree, Andrew, Nitin Sabherwal, and Nancy Papalopulu.
2022. "Bilateral Feedback in Oscillator Model Is Required to Explain the Coupling Dynamics of Hes1 with the Cell Cycle" *Mathematics* 10, no. 13: 2323.
https://doi.org/10.3390/math10132323