Single-Cell Genome Dynamics in Early Embryo Development: A Statistical Thermodynamics Approach

A statistical thermodynamics approach to the temporal development of biological regulation provides a phenomenological description of the dynamical behavior of genome expression in terms of autonomous self-organization with a critical transition (Self-Organized Criticality: SOC). In early mouse embryo development, the dynamical change in the self-organization of overall expression determines how and when reprogramming of the genome-expression state occurs. Reprogramming occurs via a transition state (climbing over an epigenetic landscape), where the critical-regulation pattern of the zygote state disappears. A critical transition is well captured in terms of the bimodality of expression ensembles, which reflects distinct thermodynamic states (critical states). These critical states exhibit a genome avalanche pattern: competition between order (scaling) and disorder (divergence) around a critical point. The genome avalanche in mouse embryo development, which is committed to erase a previous ordered state, reveals that the direction of early embryo single-cell development traverses the same steps as in differentiation, but in the opposite order of self-organization.


Introduction
A fundamental issue in bioscience is to understand the mechanism that underlies the dynamic control of genome-wide expression through the complex temporal-spatial selforganization of the whole genome to regulate changes autonomously in the cell fate.
Our recent studies on genome-scale expression dynamics in the cell-fate change demonstrated the possibility of a statistical thermodynamic analysis [1][2][3]. The thermodynamics phenomenological character is at the basis of Albert Einstein's famous statement 'A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended is its area of applicability. Therefore, the deep impression which classical thermodynamics made upon me. It is the only physical theory of universal content which I am convinced that, within the framework of applicability of its basic concepts, it will be never overthrown' [4].
An open-thermodynamic approach clearly reveals how and when the change in a genome-expression state occurs as a thermodynamic event in the disappearance of critical phenomena that arise from an initial state. Furthermore, a detailed global perturbation mechanism for the self-organization of overall expression was elucidated for cell differentiation processes. This demonstration of statistical thermodynamics in the genome [5] is particularly crucial in biology, where the extreme complexity of the system rules out any current strict mechanistic approach that considers all of the microscopic regulations involved.
The active process that controls global gene expression in the case of cell fate determination, involves a particular kind of self-organization, called Self-Organized Criticality (SOC) [2,[6][7][8], which drives the state transitions of gene expression via the mutual interaction among three different gene expression states; 'critical', 'near critical' and 'subcritical'. Bimodality in gene expression levels is a clear signature of the self-organizing critical transition that is organized into distinct critical states (Figure 1). Each gene belongs to one of these three states based on the relative variation of its expression value. This allows for a phenomenological (and thus independent of any mechanistic hypothesis) description of the expression dynamics at the global level.
In this report, we investigate whether essentially the same critical-state dynamics we observed for cell differentiation processes [2,3] does exist in overall RNA expression of single-cell mouse embryo development. This case is particularly relevant to give a further proof of SOC control as universal character.

Results
Early embryo development involves a crucial step in global gene expression control: erasure of the initial maternal-only imprinting' of gene regulation to start a new global control that is driven by the new genetic set-up. In the zygote, the new diploid genome is still inactive in a chromatin fully condensed state, and the metabolism is orchestrated by maternal RNA molecules. Thus, for further development, it is important to erase the previous control to allow for a new start based on the new maternal/paternal mixed genome. In some sense, this 'erasure' path can be considered to be a transformation that proceeds in the direction opposite that for acquisition of a terminal cell fate.

Sub-critical
Nearcritical Super-critical CP exists  RNA species. This is consistent with the fact that, in the early stages of development, the embryo must eliminate the maternal-only imprinting' of the zygote to 'start-from-scratch' with a new developmental program driven by the maternal/paternal genome pattern [9]. In contrast, the path toward a terminal cell fate is a path toward increasing order with respect to the undifferentiated state [10].
These findings are purely phenomenological at the present and await the discovery of mechanistic microscopic bases. However, our results suggest that a single-cell statistical ln(<ε (2-cell(E))>)
Regarding backward single cell reprogramming such as an induced pluripotent stem (iPS) cell from a somatic cell, a stochastic model [11,12]

SOC Control Mechanism of the Cell-fate Change
We previously demonstrated that the critical dynamics of self-organizing wholegenome expression (self-organized criticality: SOC) play essential roles in determining the cell-fate change in distinct biological processes at both the single-cell and population (tissue) levels [3].
In SOC-controlled self-organization (in terminal cell differentiation), nrmsf (normalized root mean square fluctuation) acts as an order parameter for the self-organization of whole gene expression [2,3]. We can examine if nrmsf acts as an order parameter in early embryo development, which is determined by dividing rmsf by the maximum overall {rmsf i }: where rmsf i is the rmsf value of the i th RNA expression, which is expressed as For methodological details, refer to our previous works [2,3].