# A Novel Framework for Parameter and State Estimation of Multicellular Systems Using Gaussian Mixture Approximations

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Modeling of the Multicellular Dynamics

#### 2.2. Observability of Multicellular Systems Dynamics

#### 2.3. Estimator Design

- (I)
- Prediction step, a priori state and error covariance estimates:$$\begin{array}{c}\hfill {\widehat{\mathbf{x}}}_{k}^{-}={F}_{\mathrm{d}is}{\widehat{\mathbf{x}}}_{k-1}^{+}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}{P}_{k}^{-}={F}_{\mathrm{d}is}{P}_{k-1}^{+}{F}_{\mathrm{d}is}^{T}+V\end{array}$$
- (II)
- Computation of the estimator gain:$$\begin{array}{c}\hfill {K}_{k}={P}_{k}^{-}{H}_{\mathrm{d}is}^{T}{\left({H}_{\mathrm{d}is}{P}_{k}^{-}{H}_{\mathrm{d}is}^{T}+W\right)}^{-1}\end{array}$$
- (III)
- Correction using current measurement ${\mathbf{y}}_{k}$, posterior estimates:$$\begin{array}{c}\hfill {\widehat{\mathbf{x}}}_{k}^{+}={\widehat{\mathbf{x}}}_{k}^{-}+{K}_{k}\left({\mathbf{y}}_{k}-{H}_{\mathrm{d}is}{\widehat{\mathbf{x}}}_{k}^{-}\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{P}_{k}^{+}=\left(I-{K}_{k}{H}_{\mathrm{d}is}\right){P}_{k}^{-}\end{array}$$

- (I)
- Initialization: A set of ${N}_{P}$ initial estimates (particles) is drawn from the initial state PDF:$$\begin{array}{c}\hfill {\widehat{\mathbf{x}}}_{0,j}^{+}\sim p\left({\mathbf{x}}_{0}\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}j=1,\dots ,{N}_{P}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$Subsequently, the following steps are executed recursively for $k>0$:
- (II)
- Propagation: Each particle is propagated from ${t}_{k-1}$ to ${t}_{k}$ with Equation (12):$$\begin{array}{c}\hfill {\widehat{\mathbf{x}}}_{k,j}^{-}=\underset{{t}_{k-1}}{\overset{{t}_{k}}{\int}}\mathbf{f}>({\mathbf{x}}_{k-1,j}^{+}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\mathbf{v}}_{k-1})\mathrm{d}t\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}{\widehat{\mathbf{y}}}_{k,j}^{-}=\mathbf{h}>({\mathbf{x}}_{k,j}^{-}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\mathbf{w}}_{k-1})\end{array}$$
- (III)
- Weighting: The relative likelihood ${q}_{j}$ of each particle conditioned on the current measurement is computed by evaluating $p>({\mathbf{y}}_{k,j}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}{\mathbf{x}}_{k,j}^{-})$ using Equation (13). Furthermore, all weights are normalized.$$\begin{array}{c}\hfill {q}_{j}={q}_{j}{\left(\sum _{j=1}^{{N}_{P}}{q}_{j}\right)}^{-1}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$
- (IV)
- Regularization and resampling: The posterior state PDF $p>({\mathbf{x}}_{k}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}{\mathbf{y}}_{k})$ is approximated by a sum of weighted kernel functions, and ${N}_{P}$ a posteriori particles are generated from this PDF.$$\begin{array}{c}\hfill p>({\mathbf{x}}_{k}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}{\mathbf{y}}_{k})=\sum _{j=1}^{{N}_{P}}{q}_{j}K>({\widehat{\mathbf{x}}}_{k,j}^{+})\phantom{\rule{0.166667em}{0ex}}.\end{array}$$From the posterior PDF, any desired statistical measures, e.g., mean and covariance, can be determined.

## 3. Results and Discussion

#### 3.1. Comments on the Computational Implementation

#### 3.2. Linear Dynamics: Protein Expression

#### 3.3. Nonlinear Dynamics: Intracellular Oscillator Modeled by Lotka–Volterra Dynamics

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

NDF | Number density distribution function |

ODE | Ordinary differential equation |

PBE | Population balance equation |

PBM | Population balance model |

PDE | Partial differential equation |

Probability density function | |

SDE | Stochastic differential equation |

## References

- Herberg, M.; Glauche, I.; Zerjatke, T.; Winzi, M.; Buchholz, F.; Roeder, I. Dissecting mechanisms of mouse embryonic stem cells heterogeneity through a model-based analysis of transcription factor dynamics. J. R. Soc. Interface
**2016**, 13, 201160167. [Google Scholar] [CrossRef] [PubMed] - Müller, T.; Dürr, R.; Isken, B.; Schulze-Horsel, J.; Reichl, U.; Kienle, A. Distributed modeling of human influenza a virus-host cell interactions during vaccine production. Biotechnol. Bioeng.
**2013**, 110, 2252–2266. [Google Scholar] [CrossRef] [PubMed] - Tapia, F.; Vázquez-Ramírez, D.; Genzel, Y.; Reichl, U. Bioreactors for high cell density and continuous multi-stage cultivations: Options for process intensification in cell culture-based viral vaccine production. Appl. Microbiol. Biotechnol.
**2016**, 100, 2121–2132. [Google Scholar] [CrossRef] [PubMed] - Franz, A.; Dürr, R.; Kienle, A. Population Balance Modeling of Biopolymer Production in Cellular Systems. IFAC Proc. Vol.
**2014**, 47, 1705–1710. [Google Scholar] [CrossRef] - Nopens, I.; Torfs, E.; Ducoste, J.; Vanrolleghem, P.; Gernaey, K. Population balance models: A useful complementary modelling framework for future WWTP modelling. Water Sci. Technol.
**2015**, 71, 159–167. [Google Scholar] [CrossRef] - Pigou, M.; Morchain, J. Investigating the interactions between physical and biological heterogeneities in bioreactors using compartment, population balance and metabolic models. Chem. Eng. Sci.
**2015**, 126, 267–282. [Google Scholar] [CrossRef] [Green Version] - Liou, J.; Fredrickson, A.G.; Srienc, F. Selective synchronization of Tetrahymena pyriformis cell populations and cell growth kinetics during the cell cycle. Biotechnol. Prog.
**1998**, 14, 450–456. [Google Scholar] [CrossRef] [PubMed] - Müller, S.; Harms, H.; Bley, T. Origin and analysis of microbial population heterogeneity in bioprocesses. Curr. Opin. Biotechnol.
**2010**, 21, 100–113. [Google Scholar] [CrossRef] [PubMed] - Binder, D.; Drepper, T.; Jaeger, K.E.; Delvigne, F.; Wiechert, W.; Kohlheyer, D.; Grünberger, A. Homogenizing bacterial cell factories: Analysis and engineering of phenotypic heterogeneity. Metab. Eng.
**2017**, 42, 145–156. [Google Scholar] [CrossRef] [PubMed] - de Vargas Roditi, L.; Claassen, M. Computational and experimental single cell biology techniques for the definition of cell type heterogeneity, interplay and intracellular dynamics. Curr. Opin. Biotechnol.
**2015**, 34, 9–15. [Google Scholar] [CrossRef] [PubMed] - Hasenauer, J.; Waldherr, S.; Doszczak, M.; Radde, N.; Scheurich, P.; Allgöwer, F. Identification of models of heterogeneous cell populations from population snapshot data. BMC Bioinform.
**2011**, 12, 125. [Google Scholar] [CrossRef] [PubMed] - Natarajan, A.; Srienc, F. Glucose uptake rates of single E. coli cells grown in glucose-limited chemostat cultures. J. Microbiol. Methods
**2000**, 42, 87–96. [Google Scholar] [CrossRef] - Simon, D. Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2006. [Google Scholar]
- Fredrickson, A.G.; Ramkrishna, D.; Tsuchiya, H.M. Statistics and dynamics of procaryotic cell populations. Math. Biosci.
**1967**, 1, 327–374. [Google Scholar] [CrossRef] - Mangold, M. Use of a Kalman filter to reconstruct particle size distributions from FBRM measurements. Chem. Eng. Sci.
**2012**, 70, 99–108. [Google Scholar] [CrossRef] - McLachlan, G.; Peel, D. Finite Mixture Models; Wiley Series in Probability And Statistics; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2000. [Google Scholar]
- Slack, M.D.; Martinez, E.D.; Wu, L.F.; Altschuler, S.J. Characterizing heterogeneous cellular responses to perturbations. Proc. Natl. Acad. Sci. USA
**2008**, 105, 19306–19311. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Altschuler, S.J.; Wu, L.F. Cellular Heterogeneity: Do Differences Make a Difference? Cell
**2010**, 141, 559–563. [Google Scholar] [CrossRef] [PubMed] - 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] - Dürr, R.; Müller, T.; Duvigneau, S.; Kienle, A. An efficient approximate moment method for multi-dimensional population balance models—Application to virus replication in multi-cellular systems. Chem. Eng. Sci.
**2017**, 160, 321–334. [Google Scholar] [CrossRef] - Ramkrishna, D. Population Balances: Theory and Applications to Particulate Systems in Engineering; Academic Press: San Diego, CA, USA, 2000. [Google Scholar]
- Luenberger, D. An introduction to observers. IEEE Trans. Autom. Control
**1971**, 16, 596–602. [Google Scholar] [CrossRef] - Zeitz, M. Observability canonical (phase-variable) form for non-linear time-variable systems. Int. J. Syst. Sci.
**1984**, 15, 949–958. [Google Scholar] [CrossRef] - Blanke, M.; Kinnaert, M.; Lunze, J.; Staroswiecki, M.; Schröder, J. Diagnosis and Fault-Tolerant Control, 3rd ed.; Springer-Verlag: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
- Liu, Y.Y.; Slotine, J.J.; Barabási, A.L. Observability of complex systems. Proc. Natl. Acad. Sci. USA
**2013**, 110, 2460–2465. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mangold, M.; Bück, A.; Schenkendorf, R.; Steyer, C.; Voigt, A.; Sundmacher, K. Two state estimators for the barium sulfate precipitation in a semi-batch reactor. Chem. Eng. Sci.
**2009**, 64, 646–660. [Google Scholar] [CrossRef] - Zeng, S.; Waldherr, S.; Ebenbauer, C.; Allgöwer, F. Ensemble Observability of Linear Systems. IEEE Trans. Autom. Control
**2016**, 61, 1452–1465. [Google Scholar] [CrossRef] - Wang, X.; Li, T.; Sun, S.; Corchado, J.M. A Survey of Recent Advances in Particle Filters and Remaining Challenges for Multitarget Tracking. Sensors
**2017**, 17, 2707. [Google Scholar] [CrossRef] [PubMed] - Gerards, A. Chapter 3: Matching. In Network Models; Elsevier: Amsterdam, The Netherlands, 1995; Volume 7, pp. 135–224. [Google Scholar]
- Higham, D.J. An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations. SIAM Rev.
**2001**, 43, 525–546. [Google Scholar] [CrossRef] [Green Version] - Drengstig, T.; Ni, X.Y.; Thorsen, K.; Jolma, I.W.; Ruoff, P. Robust Adaptation and Homeostasis by Autocatalysis. J. Phys. Chem. B
**2012**, 116, 5355–5363. [Google Scholar] [CrossRef] [PubMed] - Isensee, J.; Diskar, M.; Waldherr, S.; Buschow, R.; Hasenauer, J.; Prinz, A.; Allgöwer, F.; Herberg, F.W.; Hucho, T. Pain modulators regulate the dynamics of PKA-RII phosphorylation in subgroups of sensory neurons. J. Cell Sci.
**2014**, 127, 216–229. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**Left**) Two-dimensional cell distribution; only individual size is measurable, but not growth rate; (

**right**) model-based online estimation.

**Figure 2.**Benchmark 1: protein expression, ${N}_{GMD}=2$; reconstruction of integral quantities of GMDs using Kalman filters and the Hungarian algorithm for data association.

**Figure 3.**Benchmark 2: Lotka–Volterra dynamics, ${N}_{GMD}=2$; reconstruction of integral quantities of GMDs using particle filter: modes of the PDF (red) represent estimates of the GMDs (only each second measurement (blue) is depicted to improve visibility).

Parameter | Value |
---|---|

k | 2 |

d | $0.01$ |

V | ${10}^{-2}\xb7\mathrm{d}iag\left([2.1,\phantom{\rule{0.166667em}{0ex}}2.73,\phantom{\rule{0.166667em}{0ex}}2.11,\phantom{\rule{0.166667em}{0ex}}4.2,\phantom{\rule{0.166667em}{0ex}}0.04,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}0.04,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}0.01]\right)$ |

W | ${10}^{-3}\xb7\mathrm{d}iag\left([1,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}0]\right)$ |

Parameter | Value | Parameter | Value |
---|---|---|---|

${k}_{1}$ | 0.8 | V | ${10}^{-4}\xb7\mathrm{d}iag\left([0.6,\phantom{\rule{0.166667em}{0ex}}0.05,\phantom{\rule{0.166667em}{0ex}}1.2,\phantom{\rule{0.166667em}{0ex}}{10}^{-3},\phantom{\rule{0.166667em}{0ex}}5\times {10}^{-4},\phantom{\rule{0.166667em}{0ex}}{10}^{-4}]\right)$ |

${k}_{2}$ | 1.2 | W | ${10}^{-4}\xb7\mathrm{d}iag\left([0.6,\phantom{\rule{0.166667em}{0ex}}1.2,\phantom{\rule{0.166667em}{0ex}}5\times {10}^{-4}]\right)$ |

${k}_{3}$ | 0.4 | ${N}_{P}$ | 200 |

${k}_{4}$ | 0.5 | $K\left(\mathbf{x}\right)$ | $\mathcal{N}(\mathbf{x},\phantom{\rule{0.166667em}{0ex}}0.01\xb7V)$ |

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**MDPI and ACS Style**

Dürr, R.; Waldherr, S.
A Novel Framework for Parameter and State Estimation of Multicellular Systems Using Gaussian Mixture Approximations. *Processes* **2018**, *6*, 187.
https://doi.org/10.3390/pr6100187

**AMA Style**

Dürr R, Waldherr S.
A Novel Framework for Parameter and State Estimation of Multicellular Systems Using Gaussian Mixture Approximations. *Processes*. 2018; 6(10):187.
https://doi.org/10.3390/pr6100187

**Chicago/Turabian Style**

Dürr, Robert, and Steffen Waldherr.
2018. "A Novel Framework for Parameter and State Estimation of Multicellular Systems Using Gaussian Mixture Approximations" *Processes* 6, no. 10: 187.
https://doi.org/10.3390/pr6100187