Applied Stochastic System Modeling

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A second objective is to present new methods for solving such problems, i. The scope of the journal is now broadened both in supporting topics and in appropriate methodology. Topics to be added include managerial processes, reliability, quality control, data analysis and data mining. New methodologies include wavelets, Markov-chain Monte Carlo methods and spatial statistics. View all. All Articles RSS.

Register Now. Forgotten password? Only very small deviations from the reference data narrow the range of feasible parameters sufficiently down. This implies that already small experimental inaccuracies may have a strong effect on identified parameter values. Because of the low identifiability of parameters PY3 and DY3 these were excluded from distance calculations in a,c. This measure somewhat increases the error-distance slope for methods, for which PY3 and DY3 were variable b,d.

Results based on all parameters are shown in Supplementary Figure 12c-e.

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The better performance of Rates SCT can evidently be attributed to the additional stratification normalisation step that in effect separates the overall identification task into smaller sub-problems—here, one for each number of molecules Supplementary Methods 9. This appears especially useful for regions with relatively few events that are otherwise outnumbered in the global likelihood Supplementary Methods 8.

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Supplementary Figure 13 reports on additional results of local Hessian sensitivity analysis, indicating that for Freq 1D and Freq 2D the difference between production and degradation rates is the single most important feature. The results of this section suggest that Rates SCT id, Rates SCT and Freq 2D are associated with the lowest bias and dispersion and thus appear as the most accurate identification methods. Accordingly, single-cell techniques that have increasingly become available in recent years can actually be expected to become a powerful tool for system identification.

The problems of systems identification described above have practical importance specifically for human biological system intervention. In the following, we illustrate this issue by virtual treatment experiments. In Fig.

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Analogous results for two additional regulation types are presented in Supplementary Figure Comparison of parameter identification methods. Freq 1D and Freq 2D are naturally on the same scale. For illustration purposes, the minimum error of Rates SCT was not adjusted to be zero. Red horizontal lines indicate true parameter values. Generally, the highest parameter variability is observed for the respective 3rd y -set points of production PY3 and degradation DY3 rates corresponding to regions scarcely populated during the entire time course.

Thus, treatment response can substantially depend on how precisely regulation is implemented in different cell types or individual patients. This finding clearly argues in favour of system identification being performed without an a priori chosen kinetic model in the first place. We applied stochastic system identification to the flow cytometry data of Kashiwagi et al.

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Our analysis demonstrated that many different regulation types were consistent with these data, i. This in turn implies that system identification results are sensitive to experimental imprecision of the same order. To assess whether other data types and evaluation methods can amend this situation, we generated diverse semi-empirical synthetic data sets. The corresponding method screen indicated that primarily rate reconstruction from single-cell tracking and, secondary, frequency distributions derived from single-cell metabolic labelling are the most accurate approaches for parameter identification.

In contrast, the use of population-averaged rates and likelihoods predicted by single-cell tracking data performed less favourably.

The above results were obtained by application of quasi unconstrained piecewise-linear rate functions that generally differ from commonly employed expression models such as classical birth and death processes or Michaelis—Menten type kinetics. In particular, degradation rates are still widely assumed to be linear. In contrast, our reanalysis of the metabolic labelling data of Miller et al. Correlation between trajectories of degradation rate and gene expression showed a quasi-random, sign-balanced distribution across genes, while wholly positive correlation would be expected if degradation was indeed linear i.

A number of factors can influence molecule degradation, ranging from priming by adaptor proteins, 23 , 24 cooperative or stepwise degradation, 1 , 9 , 33 storage in granules and P-bodies, 25 , 31 effects of regulatory RNAs and RNA-binding proteins 26 , 30 , 31 to overall resource limitation. Notably, regulated degradation can also impact the assessment of transcriptional and translational noise. Employing more versatile identification approaches like the one presented in this study explores wider regulatory possibilities that more likely include the true regulation type as long as data overfitting can be excluded.

Here, we counteracted overfitting of noisy data by smoothing the fluorescence intensity profiles and limited model over-parameterisation by considering only two or three parameters per rate. For production and degradation rates, a minimum of three parameters was necessary to test whether non-monotonous rate functions, specifically those with a minimum related to noise-driven dynamics , could be consistent with experiment.

Occasionally, we observed pronounced rate changes in sparsely populated regions at large molecule numbers. This might be indicative of low parameter identifiability and could, if desired, be amended by implementing smooth function variation depending on data density. We do not oppose the common preference for the simplest model.

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Our reanalysis of the metabolic labelling data of Miller et al. Evidently, system identification can best be tackled by appropriate measurement and evaluation techniques, like single-cell tracking reaction event monitoring , that allow precise parameter identification of basic models. We propose that rate functions successfully identified by rather flexible identification approaches can subsequently be represented by more specialised mappings to reduce the number of parameters as needed.

This two-step procedure largely decouples the problems of true and apparent regulatory equivalence from those of relative parameter insensitivity as conferred by mathematical functions Introduction. Nevertheless, the number of effectively unconstrained independent parameters needs to be kept as low as possible to enable quasi exhaustive parameter searches, for which we provided an expert monitoring method.

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The starting point and a basic motivation of this study was to further explore our previous hypothesis that noise regulation can be a major driving force of cell dynamics. This idea was later also termed noise-controlled cell regulation by Pujadas and Feinberg. This principle was similarly implemented in our novel noise-driven optimisation NDO method showing very good performance.

To capture noise regulation on top of deterministic growth processes in highly proliferative cells like E. Surprisingly, we found indications of the possibility that noise-dominated regulation may indeed be favoured in proliferating cell populations. Corresponding changes in the noise content of cellular regulation could also be implied in other processes involving variation of proliferation intensity, like batch or fed-batch cell culture, eukaryotic cell differentiation 45 or cancer.

The general need to identify production and degradation rates is illustrated by our virtual treatment experiments. Evidently, the influence of different regulation types will be more prominent in real-world multi-dimensional settings requiring considerable future efforts to develop clear-cut and effective experimental and mathematical methods for biological systems identification.

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We aimed at modelling fluorescent protein frequency distributions F matching experimental cytometry data listing cell counts per binned fluorescent intensity value. For this purpose, we employed a rate equation that accounts for cell proliferation using the multi-phase cell cycle model of Leon et al. For comparison with experiment, F is marginalised across cell cycle phases, i. Notably, equation 2 is similar to a chemical master equation CME.

Hence, we use the term rate equation for clarity. Indeed, equation 2 can be derived from an extended CME also accounting for the number of cell divisions Supplementary Methods 1. In this context, it describes the dynamics of the mean number of cells that harbour n molecules and proceed in cell cycle phase i , irrespective of cell division history. For zero growth, the number of cells is constant and equation 2 becomes equivalent to a one-dimensional CME for the probability distribution p n.

The FP-terms in 3 are then evaluated using the results of 2. We link the regulated noise term DB FP x with noise-driven dynamics 3 and generally excluded boundaries in actual function evaluations. Nevertheless, according to the time derivative of the total number of molecules T Supplementary Methods 7. Without loss of generality, A x , B x and the FP-terms in 3 can also be applied to growing cell populations and multiple cell cycle phases.