4. Statistical models for the quantification and analysis of cellular SAC phe-
4.5. Summary and Discussion
4. Statistical models for the quantification and analysis of cellular SAC phenotypes
0 4 8 12 16
time [h]
65% Mad2
30% Mad3 60% Mad3 120% Mad3
censoring time prometaphase length
Figure 4.7.: Data for three strains with altered protein amount and Mad3 in a 65% Mad2 P188 background.
4.5. Summary and Discussion
0 0.5 1
0 0.5 1
0 50 100
0 50 100
% Mad2
% Mad3
WT fraction
WT fraction wild type fraction
0 0.5 1
0 0.5 1
0 50 100
0 50 100
% Mad3 WT fraction
WT fraction
% Mad2 0
0.5 1
A model for independent perturbation effects
wild type fraction
0 0.5 1
B model for interacting perturbation effects
65% Mad2 + 120% Mad3 65% Mad2 + 60% Mad3 65% Mad2 + 30% Mad3
double perturbations: model
Mad2 abundance
Mad3 abundance
Mad2 abundance
Mad3 abundance WT
fraction
WT fraction
data
BIC = 10.18 BIC = -3.95
Figure 4.8.: Data-driven hypotheses testing of SAC perturbation response using MEMO.
Comparison of model agreement for independent (A) and cooperative (B) perturbation ef-fects of Mad2 and Mad3 with the measured fraction of wild type-like cells in strains with double perturbations. Crosses indicate data from three double perturbation strains. Curves show the wild type fraction in cut planes through the Mad2-Mad3 plane (as indicated by the same colour) as computed from the respective model. The model in (A) reflects in-dependent effects of Mad2 and Mad3 perturbations by just multiplying the models for the individual Mad2 and Mad3 perturbations from Figure 4.6. The model in (B) considers co-operative effects of perturbations in Mad2 and Mad3 by modelling the threshold parameters K in each of the two Hill-type functions to be inversely proportional to the amount of the other protein. As indicated by the lower BIC of the model in B, a cooperative influence of both proteins on the wild type fraction is more likely. Figure adapted from Geissenet al.
(2016).
4. Statistical models for the quantification and analysis of cellular SAC phenotypes
functions encode hypotheses and can be derived from measurement data or mechanistic mod-els, such as for example ordinary differential equation (ODE) constrained mixture models (ODE-MM) as described in Hasenaueret al.(2014a). ODE-MMs use mechanistic models of single cell behaviour and subpopulation structure to integrate data collected under different experimental conditions (Hasenaueret al., 2014b; Thomaset al., 2014), and could be used to reconstruct differences between subpopulations. MEMO provides an extension to ODE-MM as censored data can be studied and knowledge about the signalling pathway is not required.
This renders MEMO more flexible and easier to use for explorative data analysis. Table 4.5 provides a comparison of the features of MEMO, the modelling packages FLAME (Pyneet al.
(2009)), JCM (Pyne et al. (2014)), BayesFlow (Johnsson et al.(2016)), and the algorithms presented in Lee & Scott (2012).
In Section 4.4.1 we have addressed Problem 4.2. We have characterized the subpopulation structures of the different yeast strains regarding SAC functionality. The statistical modelling of variability in the prometaphase datasets of several strains has revealed that these strains contain cells with two different cellular phenotypes regarding the functionality of the SAC.
One subpopulation of cells keeps the phenotype of wild type cells, despite the altered amount of protein. A second subpopulation displays a phenotype of impaired SAC functionality.
Note that this effect is clearly not an artefact of inhomogeneous efficacy of the experimental manipulation, since the treatment is on the genetic level. Each cell has the altered promoter and therefore expresses the respective protein in a different amount as the wild type. Since all cells in a strain are genetically identical, there has to exist non-genetic variability to an extend that suffices to result in these quantitative differences in the cellular phenotype. The results obtained in this chapter can be used for subsequent mechanistic modelling, as we demonstrate in Chapter 5.
Cell-to-cell variability in SAC signalling has been reported before but not quantified beyond sample statistics. Moreover, potential subpopulations with different phenotypes are present in the data of other studies that perturbed SAC signalling (Morrowet al., 2005; Saurinet al., 2011; Thoma et al., 2009). However, the interpretation of these data is difficult because it cannot be ruled out that in these studies the population split is an artefact of the experimen-tal treatment, such as incomplete RNAi knock down. While Morrow et al. (2005) attribute the population split they observed to an incomplete knock-down in RNAi treated cells, other publications show bimodal patterns in SAC arrest but do not comment on the potential sub-populations at all (Saurinet al., 2011; Thomaet al., 2009).
In Section 4.4.2 we have addressed Problem 4.3 and quantified the sensitivity of SAC func-tionality with respect to Mad2 and Mad3. We gained the fraction of the subpopulation with wild type-like SAC phenotype as a function of the relative protein concentration. The de-pendency on Mad2 is well-approximated by a Hill-function with a Hill-coefficient of 12.3.
Therefore the sensitivity towards changes in Mad2 is highly ultrasensitive. Sensitivity with respect to changes in Mad3 has a Hill-coefficient of 2.1 and is therefore more graded. The model predicts a population split for a wide range of relative Mad3 amounts.
In Section 4.4.3 we have addressed Problem 4.4 and assessed the mode of interaction of Mad2 and Mad3 in promoting a functional SAC. We have presented findings that indicate that Mad2 and Mad3 act cooperatively. A simultaneous reduction of both proteins has a more pronounced effect as it would be predicted for independent effects.
50
4.5. Summary and Discussion
Properties FLAME JCM Lee & Scott BayesFlow MEMO
Censoring
left - - X - X
right - - X - X
interval - - - - X
distributed - - - - X
fitted - - - - X
Truncation
left - - X -
-right - - X -
-fitted - - - -
-Distributions
normal - - X X4 X
log-normal - - - - X
gamma - - - - X
skew-normal X - - -
-t X X - -
-skew-t X X - -
-Johnson SU - - - - X
Data dimension uni-variate −1 X X X X
multi-variate X X ∼2 X
-Multi-sample fitting X X - X X
Multi-experiment fitting - −3 - - X
Simultaneous analysis of all data - - - - X
Automated model selection - - - - X
Table 4.5.:Comparison of the features of MEMO and other packages.1The methods used in FLAME allow for the analysis of univariate data, the implementation does however yield an error with the version available on January 20, 2016. 2The methods used by Lee &
Scott (2012) allows for the analysis of multi-variate data, the comments in the code state however that it is only correctly implemented for uni-directional sampling in each coordi-nate. In the README it is furthermore stated, that the current implementation considers that the truncation is only on the first coordinate. 3JCM exploit prior knowledge of the subpopulation structure to perform the inter-condition matching, In general this will not be available. Furthermore, JCM does not allow for a description of the underlying mech-anisms and hypothesis testing. 4 Skewed and/or heavy tailed distributions are handled by merging of Gaussian components into super components.