By Carolin Loos

ISBN-10: 3658132337

ISBN-13: 9783658132330

ISBN-10: 3658132345

ISBN-13: 9783658132347

Carolin bogs introduces novel techniques for the research of single-cell info. either ways can be utilized to check mobile heterogeneity and hence strengthen a holistic realizing of organic tactics. the 1st process, ODE limited combination modeling, permits the identity of subpopulation buildings and assets of variability in single-cell photo facts. the second one process estimates parameters of single-cell time-lapse info utilizing approximate Bayesian computation and is ready to take advantage of the temporal cross-correlation of the information in addition to lineage details.

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**Example text**

For simplicity, we further neglect the subpopulation index s. 3. g. 6) or multiplicative log-normally distributed measurement noise ∼ log N (0, Γ) . 7) The entries of Γ are incorporated in the parameter vector. The moments of the measurand without measurement noise are denoted by my = (my,1 , . . , my,d ) and Cy and can be calculated from x. 2 Multivariate Mixture of Normal and Log-Normal Distributions In this thesis we focus on the mixture of d-dimensional multivariate normal distributions N (y|μ, Σ) = 1 1 1 d (2π) 2 det (Σ) 2 e− 2 (y−μ) T Σ−1 (y−μ) , and multivariate log-normal distributions logN (y|μ, Σ) = 1 d 2 (2π) det (Σ) with mixture parameters ϕ = (μ, Σ).

1. 2 Problem Statement Despite the successes achieved using ODE-MM, there are several open issues. So far, only the means of the subpopulations are described by ODEs and the variances are treated as additional parameters. If many time points and experimental conditions are observed, the unknown variances increase the dimension of the parameter space signiﬁcantly. As the predictive power of a model generally decreases with its complexity, it is desirable to reduce the number of its parameters. Moreover, RREs provide only a description of the averaged behavior of a subpopulation.

Mixture of Multivariate Log-Normal Distributions For the log-normal distribution we use two diﬀerent parametrizations. If the mean obtained by the MEs describes the mean of the log-normal distribution, we use 1 my,i = eμi + 2 Σii , 1 Cy,ij = eμi +μj + 2 (Σii +Σjj ) (eΣij − 1) , for i, j = 1, . . , d. This yields the mixture parameters 1 μi = log(my,i ) − Σii , 2 Cy,ij + 1) , Σij = log( my,i my,j and their derivatives ∂μi 1 ∂my,i 1 ∂Σii = − , ∂θ my,i ∂θ 2 ∂θ ∂Σij = ∂θ = ∂ 1 Cy,ij my,i my,j 1 Cy,ij my,i my,j +1 Cy,ij my,i my,j ∂θ ∂my,i y,j my,i my,j ∂C∂θy,ij − Cy,ij (my,i ∂m ∂θ + my,j ∂θ ) .

### Analysis of Single-Cell Data : ODE Constrained Mixture Modeling and Approximate Bayesian Computation by Carolin Loos

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