A note on interaction and preimplantation development stages

Having obtained observations of gene expression profiles from the literature, we estimated autoregressive models of mouse preimplantation development and observed interaction among stages. Gene expression profile at the earliest stage is not generally significant, but for observations where profiles at 8-cell embryo and morula stages are relatively great, the effect of unfertilized egg is significantly positive.


INTRODUCTION
Preimplantation development is important to reproductive biology applications including livestock breeding and treatments that repair defective tissues (Hamatani et al., 2004;Wang and Dey, 2006;Chakrabarty et al., 2007), and developmental biology may be studied with genomics (Ko, 2001), Canestro et al. (2007).More specifically we can learn from changes in gene expression during preimplantation development in the mouse (Tanaka et al., 2000;Zeng and Schultz, 2003;Wobus and Boheler 2005).
This research note describes statistical analysis and results of global gene expression profiles at preimplantation development stages.Regression models of stage profiles were estimated as linear functions of earlier stage profiles, and investigation of insignificant variables led us to discover significant interaction between preimplan-tation development stages.This interaction is important to understanding temporal changes during preimplantation development.

METHODOLOGY
We obtained 21,940 observations of gene expression profiles across the following stages of preimplantation development and assigned to them variables names (in parentheses): unfertilized egg (a), fertilized egg (b), 2-cell embryo (c), 4-cell embryo (d), 8-cell embryo (e), morula (f), and blastocyst (g).The raw data are Supplemental Data for Hamatani et al. (2004): "Embryos were collected from super-ovulated C57BL/6J mice by the standard method.MRNAs were extracted using a Quickprep micro

RESULTS
Preimplantation development stages up to and including the morula enjoyed consistently significant relationships with previous stages in the main effects models of global gene expression profiles.Table 1 shows coefficient values of the first five linear regression models that did not require investigation into interaction (see Responses b through f), and the final model of blastocyst which is not a significant function of the gene expression profile at the unfertilized egg stage (see Coefficient a).
Based on the fact that we cannot reject the hypothesis that co-efficient a = zero in a model of the blastocyst stage, we feel obliged to consider interaction with the full second-order model.While the unfertilized egg stage may not be generally significant to the blastocyst stage, significance may be found for relatively great and/or small gene expression profiles at other stages.In the full second-order model of blastocyst stage the unfertilized eggs stage interacts significantly with 8-cell embryo (e), and morula (f).To further investigate interaction, we used k-means clustering, separated observations according to gene expression profiles at the 8-cell embryo and morula stages, and refit models of the blastocyst stage separately for each cluster.
For observations, where gene expression profiles at 8cell embryo and morula stages are relatively great the effect of unfertilized egg was positively significantly.For observations where gene expression profiles at 8-cell embryo and morula stages are relatively small, the effect of unfertilized egg was negatively significant.In order to make the distinction between relatively great and small gene expression profiles at 8-cell embryo (e) and morula (f) stages, we fit a linear probability model of the binary cluster variable as a function of e and f: Figure 1 of fitted values versus observations of the cluster confirms that a cut-point of 0.5 is reasonable.In other words, whenever the fitted value associated with a new observation and Equation 1 is greater than 0.5 one should assume the effect of unfertilized egg on blastocyst is positively significant, and an appropriate model for prediction of temporal changes.When the fitted value associated with a new observation and Equation 1 is less than 0.5 it is assumed that the effect of unfertilized egg on blastocyst is negatively significant, and an appropriate model.
We refit separate models of blastocyst according to the 0.5 rule described above and verified the interaction.Table 2 shows significant independent variable coefficient values for the separate models.They are consistent but for the opposite signs associated with unfertilized egg.

DISCUSSION
Articles which have used statistical analysis to study preimplantation development include studies on the early stages of the mouse embryo (Fleming and Pickering, 1985).Jurisicova et al. (1996) investigated the association between expression in human embryos and factors known to influence in vitro fertilization pregnancy outcomes.Takai et al. (2000) explained how prenatal exposure to an estrogenic compound alters postnatal development.Cervero et al. (2004) studied body weight and reproductive function during human embryonic preimplantation development.
We found a textbook case of interaction among preimplantation development phases.Gene expression profile at the earliest stage is not generally significant in a main effects model of the final stage.However, for observations where gene expression profiles at 8-cell embryo and morula stages are relatively great (small), the effect of unfertilized egg is positive (negative) E (cluster) = -1.547+ 0.378e + 0.274f (1)

Table 1 .
Independent variable coefficient values versus models of preimplantation stages.