# Elucidating Cellular Population Dynamics by Molecular Density Function Perturbations

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## Abstract

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## 1. Introduction

## 2. Material and Methods

#### 2.1. Molecular Density Function Perturbation (MDFP) Analysis

**x**at time t given that the cell state is ${x}_{\tau}$ at time $\tau $ ($t\ge \tau $). In the following, we consider introducing a mean shift perturbation to the PDF at time $\tau $ to give:

**x**) with respect to a perturbation to $\delta {e}_{\mathrm{j}}$ on the state variable ${x}_{j}$, as follows:

#### 2.2. Green’s Function Matrix Analysis

## 3. Results

#### 3.1. TRAIL-Induced Cell Death Model in HeLa Cells

#### 3.2. GFM Analysis of TRAIL-Induced Cell Death

#### 3.3. MDFP Analysis of TRAIL-Induced Cell Death

#### 3.4. MDFP Analysis of Apoptotic and Non-Apoptotic HeLa Cells

## 4. Discussion

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**A heatmap of the molecular density function perturbation (MDFP) sensitivity coefficient. The x-axis represents the time $\tau $ at which the perturbation is introduced while the y-axis represents the observation time t. The MDFP coefficient in the heatmap is scaled such that the magnitude falls within ±1, and the scaling factor is reported in the lower right corner of the plot. The sensitivity values for $t<\tau $ are set to zero for causal systems.

**Figure 2.**Signal transduction pathway and model simulation of TRAIL (tumor necrosis factor-related apoptosis-inducing ligand)-induced apoptosis in HeLa cells. (

**a**) Signal transduction pathway of apoptosis. Type I pathway describes the activation of caspase-3 by caspase-8 while type II pathway describes a mitochondria-dependent activation of caspase-3. Active caspase-3 subsequently cleaves the substrate poly(ADP-ribose) polymerase (PARP) to produce cleaved poly(ADP-ribose) polymerase (cPARP). (

**b**) Model simulation of signal transduction pathway in response to TRAIL.

**Figure 3.**Green’s function matrix (GFM) analysis of cPARP activation by a constant TRAIL stimulus. (

**a**) cPARP activation follows a delayed switch-like trajectory in response to a constant TRAIL stimulus. (

**b**,

**c**) Ten largest GFM sensitivity coefficients of cPARP concentration (in magnitude) with respect to perturbations to the state variables in the network, as shown on the label of each subfigure. The x-axis gives the time of perturbation $\tau $ while the y-axis represents the time of observation $t$. Each heatmap is scaled to have values within ±1, using the scaling factor reported in the lower right corner of the plot. Panel (

**b**) shows the GFM sensitivity coefficients in the pre-MOMP phase (before 2.36 h). Panel (

**c**) shows the GFM sensitivity coefficients in the post-MOMP phase (after 2.36 h).

**Figure 4.**MDFP analysis of cPARP activation by a constant TRAIL stimulus. (

**a**) Time evolution of the distribution of cPARP concentration shows a switch-like behavior. (

**b**,

**c**) Ten largest MDFP coefficients of cPARP concentration (in magnitude) with respect to the perturbations to different state variables in the network. The x-axis gives the time of perturbation $\tau $ while the y-axis gives the time of observation $t$. Each heatmap is scaled to have values within ±1, using the scaling factor reported in the lower right corner of the plot. Panel (

**b**) shows the MDFP sensitivity coefficients pre-MOMP (until 1.76 h). Panel (

**c**) shows the MDFP sensitivity coefficients post-MOMP.

**Figure 5.**MDFP analysis of the final cleaved PARP levels in (

**a**,

**c**) apoptotic and (

**b**,

**d**) non-apoptotic cell subpopulations. (

**a**,

**b**) The level of cPARP normalized with respect to the total PARP level. The dashed lines (--) indicate the 1 and 99 percentiles of the cPARP levels, while the solid line (-) represents the median level. (

**c**,

**d**) Ten largest sensitivity coefficients in magnitude in apoptotic and non-apoptotic cells, respectively.

**Figure 6.**Validation of the MDFP sensitivity analysis of cPARP. A positive mean shift perturbation to pro-caspase-8 was given either at $\tau =0$ h (+) or at $\tau =2.14$ h ($\times $). Panel (

**a**) shows the mean; panel (

**b**) gives the median; and panel (

**c**) gives the standard deviation of the cPARP concentration. The unperturbed simulation is shown as solid lines ($-$).

**Figure 7.**Comparison of GFM and MDFP analyses. A positive mean shift perturbation was given either to pro-caspase-8 a$\tau =0$ h (+) or to mitochondrial open pores M* at $\tau =2.14$ h ($\times $). Panel (

**a**) shows the mean of the cPARP concentration distribution, panel (

**b**) gives the median, and panel (

**c**) gives the standard deviation. The unperturbed simulation is shown as solid lines ($-$).

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Perumal, T.M.; Gunawan, R.
Elucidating Cellular Population Dynamics by Molecular Density Function Perturbations. *Processes* **2018**, *6*, 9.
https://doi.org/10.3390/pr6020009

**AMA Style**

Perumal TM, Gunawan R.
Elucidating Cellular Population Dynamics by Molecular Density Function Perturbations. *Processes*. 2018; 6(2):9.
https://doi.org/10.3390/pr6020009

**Chicago/Turabian Style**

Perumal, Thanneer Malai, and Rudiyanto Gunawan.
2018. "Elucidating Cellular Population Dynamics by Molecular Density Function Perturbations" *Processes* 6, no. 2: 9.
https://doi.org/10.3390/pr6020009