# Elucidating Cellular Population Dynamics by Molecular Density Function Perturbations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 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

- Cahan, P.; Daley, G.Q. Origins and implications of pluripotent stem cell variability and heterogeneity. Nat. Rev. Mol. Cell Biol.
**2013**, 14, 357–368. [Google Scholar] [CrossRef] [PubMed] - Flusberg, D.A.; Sorger, P.K. Surviving apoptosis: Life-death signaling in single cells. Trends Cell Biol.
**2015**, 25, 446–458. [Google Scholar] [CrossRef] [PubMed] - Xia, X.; Owen, M.S.; Lee, R.E.C.; Gaudet, S. Cell-to-cell variability in cell death: Can systems biology help us make sense of it all? Cell Death Dis.
**2014**, 5, e1261. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Golding, I.; Paulsson, J.; Zawilski, S.M.; Cox, E.C. Real-time kinetics of gene activity in individual bacteria. Cell
**2005**, 123, 1025–1036. [Google Scholar] [CrossRef] [PubMed] - Raj, A.; Peskin, C.S.; Tranchina, D.; Vargas, D.Y.; Tyagi, S. Stochastic mRNA synthesis in mammalian cells. PLoS Biol.
**2006**, 4, e309. [Google Scholar] [CrossRef] [PubMed] - Raj, A.; van Oudenaarden, A. Nature, nurture, or chance: Stochastic gene expression and its consequences. Cell
**2008**, 135, 216–226. [Google Scholar] [CrossRef] [PubMed] - Jia, G.; Stephanopoulos, G.; Gunawan, R. Ensemble kinetic modeling of kinetic metabolic networks from dynamics metabolic profiles. Metabolites
**2012**, 2, 891–912. [Google Scholar] [CrossRef] [PubMed] - Stamakis, M. Cell population balance and hybrid modeling of population dynamics for a single gene with feedback. Comput. Chem. Eng.
**2013**, 53, 25–34. [Google Scholar] [CrossRef] - Hasenauer, J.; Hasenauer, C.; Hucho, T.; Theis, F.J. ODE constrained mixture modelling: A method for unraveling subpopulation structures and dynamics. PLoS Comput. Biol.
**2014**, 10, e1003686. [Google Scholar] [CrossRef] [PubMed] - Henson, M.A. Dynamic modeling of microbial cell populations. Curr. Opin. Biotechnol.
**2003**, 14, 460–467. [Google Scholar] [CrossRef] - Hasty, J.; Pradines, J.; Dolnik, M.; Collins, J.J. Noise-based switches and amplifiers for gene expression. Proc. Natl. Acad. Sci. USA
**2000**, 97, 2075–2080. [Google Scholar] [CrossRef] [PubMed] - Manninen, T.; Linne, M.L.; Ruohonen, K. Developing Ito stochastic differential equation models for neuronal signal transduction pathways. Comput. Biol. Chem.
**2006**, 30, 280–291. [Google Scholar] - Samoilov, M.S.; Arkin, A.P. Deviant effects in molecular reaction pathways. Nat. Biotechnol.
**2006**, 24, 1235–1240. [Google Scholar] [CrossRef] [PubMed] - Shahrezaei, V.; Swain, P.S. Analytical distributions for stochastic gene expression. Proc. Natl. Acad. Sci. USA
**2008**, 105, 17256–17261. [Google Scholar] [CrossRef] [PubMed] - Poovathingal, S.K.; Gruber, J.; Halliwell, B.; Gunawan, R. Stochastic drift in mitochondrial DNA point mutations: A novel perspective ex silico. PLoS Comput. Biol.
**2009**, 5, e1000572. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Saltelli, A.; Ratto, M.; Andres, T.; Campolongo, F.; Cariboni, J.; Gatelli, D.; Saisana, M.; Tarantola, S. Global Sensitivity Analysis. The Primer; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Kucherenko, S.; Rodriguez-Fernandez, M.; Pantelides, C.; Shah, N. Monte Carlo evaluation of derivative-based global sensitivity measures. Reliab. Eng. Syst. Saf.
**2009**, 94, 1135–1148. [Google Scholar] [CrossRef] - Hafner, M.; Koeppl, H.; Hasler, M.; Wagner, A. “Glocal” Robustness Analysis and Model Discrimination for Circadian Oscillators. PLoS Comput. Biol.
**2009**, 5, e1000534. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Marino, S.; Hogue, I.B.; Ray, C.J.; Kirschner, D.E. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol.
**2008**, 254, 178–196. [Google Scholar] [CrossRef] [PubMed] - Gunawan, R.; Cao, Y.; Petzold, L.; Doyle, F.J., III. Sensitivity analysis of discrete stochastic systems. Biophys. J.
**2005**, 88, 2530–2540. [Google Scholar] [CrossRef] [PubMed] - Komorowski, M.; Costa, M.J.; Rand, D.A.; Stumpf, M.P.H. Sensitivity, robustness, and identifiability in stochastic chemical kinetics models. Proc. Natl. Acad. Sci. USA
**2011**, 108, 8645–8650. [Google Scholar] [CrossRef] [PubMed] - Plyasunov, S.; Arkin, A.P. Efficient stochastic sensitivity analysis of discrete event systems. J. Comput. Phys.
**2007**, 221, 724–738. [Google Scholar] [CrossRef] - Rathinam, M.; Sheppard, P.W.; Khammash, M. Efficient computation of parameter sensitivities of discrete stochastic chemical reaction networks. J. Chem. Phys.
**2010**, 132, 034103. [Google Scholar] [CrossRef] [PubMed] - Gunawan, R.; Jung, M.Y.L.; Braatz, R.D.; Seebauer, E.G. Parameter sensitivity analysis applied to modeling of transient enhanced diffusion and activation of Boron in Silicon. J. Electrochem. Soc.
**2003**, 150, G758–G765. [Google Scholar] [CrossRef] - Gunawan, R.; Doyle, F.J., III. Phase sensitivity analysis of circadian rhythm entrainment. J. Biol. Rhythms
**2007**, 22, 180–194. [Google Scholar] [CrossRef] [PubMed] - Ingalls, B. Sensitivity analysis: From model parameters to system behaviour. Essays Biochem.
**2008**, 45, 177–193. [Google Scholar] [CrossRef] [PubMed] - Zi, Z. Sensitivity analysis approaches applied to systems biology models. IET Syst. Biol.
**2011**, 5, 336–346. [Google Scholar] [CrossRef] [PubMed] - Perumal, T.M.; Wu, Y.; Gunawan, R. Dynamical analysis of cellular networks based on the Green’s function matrix. J. Theor. Biol.
**2009**, 261, 248–259. [Google Scholar] [CrossRef] [PubMed] - Perumal, T.M.; Gunawan, R. Understanding dynamics using sensitivity analysis: Caveat and solution. BMC Syst. Biol.
**2011**, 5, 41. [Google Scholar] [CrossRef] [PubMed] - Perumal, T.M.; Krishna, S.M.; Tallam, S.S.; Gunawan, R. Reduction of kinetic models using dynamic sensitivities. Comput. Chem. Eng.
**2013**, 56, 37–45. [Google Scholar] [CrossRef] - Spencer, S.L.; Gaudet, S.; Albeck, J.G.; Burke, J.M.; Sorger, P.K. Non-genetic origins of cell-to-cell variability in TRAIL-induced apoptosis. Nature
**2009**, 459, 428–432. [Google Scholar] [CrossRef] [PubMed] - Varma, A.; Morbidelli, M.; Wu, H. Parametric Sensitivity in Chemical Systems; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Peter, D.H. Kernel estimation of a distribution function. Commun. Stat. Theory Methods
**1985**, 14, 605–620. [Google Scholar] [CrossRef] - Niepel, M.; Spencer, S.L.; Sorger, P.K. Non-genetic cell-to-cell variability and the consequences for pharmacology. Curr. Opin. Chem. Biol.
**2009**, 13, 556–561. [Google Scholar] [CrossRef] [PubMed] - Albeck, J.G.; Burke, J.M.; Aldridge, B.B.; Zhang, M.; Lauffenburger, D.A.; Sorger, P.K. Quantitative analysis of pathways controlling extrinsic apoptosis in single cells. Mol. Cell
**2008**, 30, 11–25. [Google Scholar] [CrossRef] [PubMed] - Albeck, J.G.; Burke, J.M.; Spencer, S.L.; Lauffenburger, D.A.; Sorger, P.K. Modeling a snap-action, variable-delay switch controlling extrinsic cell death. PLoS Biol.
**2008**, 6, 2831–2852. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Roux, J.; Hafner, M.; Bandara, S.; Sims, J.J.; Hudson, H.; Chai, D.; Sorger, P.K. Fractional killing arises from cell-to-cell variability in overcoming a caspase activity threshold. Mol. Syst. Biol.
**2015**, 11, 803. [Google Scholar] [CrossRef] [PubMed] - Gaudet, S.; Spencer, S.L.; Chen, W.W.; Sorger, P.K. Exploring the contextual sensitivity of factors that determine cell-to-cell variability in receptor-mediated apoptosis. PLoS Comput. Biol.
**2012**, 8, e1002482. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**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 ($-$).

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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