# Analysis of Single-Cell Data : ODE Constrained Mixture by Carolin Loos By Carolin Loos

Carolin bogs introduces novel techniques for the research of single-cell info. either techniques can be utilized to review mobile heterogeneity and as a result develop a holistic figuring out of organic tactics. the 1st approach, ODE restricted mix modeling, permits the identity of subpopulation buildings and assets of variability in single-cell photograph info. the second one procedure estimates parameters of single-cell time-lapse facts 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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Additional info for Analysis of Single-Cell Data : ODE Constrained Mixture Modeling and Approximate Bayesian Computation

Example text

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 ) . (my,i my,j )2 +1 Another parametrization is given by assuming that the mean obtained by the MEs describes the median of the log-normal distribution. 7). 3 Robust Computation of Mixture Probabilities When using mixture models, the calculation of the likelihood is generally unstable (Problem 5).

Additionally, we demonstrate and tackle the numerically instability of the likelihood calculation arising due to the use of mixture probabilities (Problem 5). 5) e,k,j s=1 with x˙ es = f (xes , ψ es , ue ) , xes (0) = x0 (ψ es , ue ) , ϕes = h (xes , ψ es , ue ) . 10: Scatterplot and marginals of measurands A and B with (A) positive correlation and (B) negative correlation. The measurements are given in arbitrary units. 4 Simultaneous Analysis of Multivariate Measurements For a simpler notation, we further neglect the indices e, k and j, corresponding to the experiment, time point and single-cell, respectively.

K2 + k3 )2 Moment Equations for The Conversion Process with Variability in Parameters The MEs for the conversion process with accounting for additional variability in the parameters are presented in the following. Given the means μki and standard deviations σki of the parameters for i = 1, 2, 3, we obtain dmB = (mB − 1)(mk3 − mk1 ) + mB mk2 − CB,k1 + CB,k2 + CB,k3 , dt dmki = 0 for i = 1, 2, 3 , dt (mB − 1)(mk3 − mk1 ) + mB mk2 − CB,k1 + CB,k2 + CB,k3 dCB,B = + dt Ω 2(CB,k3 − CB,k1 )(mB − 1) + 2CB,k2 mB + 2CB,B (mk3 + mk2 − mk1 ) , dCB,k1 dt dCB,k2 dt dCB,k3 dt dCki ,kj dt = CB,k1 (mk2 + mk3 − mk1 ) + Ck1 ,k2 mB − Ck1 ,k1 (mB − 1) + Ck1 ,k3 mB − 1 , = CB,k2 (mk2 + mk3 − mk1 ) + Ck2 ,k2 mB − Ck1 ,k2 (mB − 1) + Ck2 ,k3 mB − 1 , = CB,k3 (mk2 + mk3 − mk1 ) + Ck2 ,k3 mB − Ck1 ,k3 (mB − 1) + Ck3 ,k3 mB − 1 , = 0 for i, j = 1, 2, 3. 