# Self-Organization of Genome Expression from Embryo to Terminal Cell Fate: Single-Cell Statistical Mechanics of Biological Regulation

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## Abstract

**:**

## 1. Introduction

## 2. Results

## 3. Discussion

#### 3.1. The “Phase Transition”

#### 3.2. The Biological Mechanisms of Sandpile-Type Criticality

#### 3.3. Regarding Backward Reprogramming

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A.

#### Appendix A.1. Biological Datasets

- Human: oocyte (m = 3), zygote (m = 3), 2-cell (m = 6), 4-cell (m = 12), 8-cell (m = 20), morula (m = 16) and blastocyst (m = 30),
- Mouse: zygote (m = 4), early 2-cell (m = 8), middle 2-cell (m = 12), late 2-cell (m = 10), 4-cell (m = 14), 8-cell (m = 28), morula (m = 50), early blastocyst (m = 43), middle blastocyst (m = 60) and late blastocyst (m = 30), where m is the total number of single cells.

_{0}= 0, t

_{1}= 1, 3, 6, 9, 12, 16, 24, t

_{T}

_{=8}= 48 h. For each time point, sample numbers are as follows: GSM1004869-SL2653 (t = 0 h); GSM1004941-SL1851 (t = 1 h); GSM1004943-SL1852 (t = 3 h); GSM1005002-SL1853 (t = 6 h); GSM1005003-SL1854 (t = 9 h); GSM1004934-SL1855 (t = 12 h); GSM1004935,6,7-SL1856, SL8353, SL8355 (t = 16 h; average of three data); GSM1004942-SL1857 (t = 24 h); GSM1004960-SL1858 (t = 48 h).

#### Appendix A.2. SOC Control Mechanism of the Cell-Fate Change

_{i}}:

_{i}is the rmsf value of the i-th RNA expression, which is expressed as ε

_{i}(s

_{j}) at a specific cell state s

_{j}(e.g., in mouse, S = 10: s

_{1}= zygote, early 2-cell, middle 2-cell, late 2-cell, 4-cell, 8-cell, morula, early blastocyst, middle blastocyst and s

_{10}= late blastocyst), and $\u27e8{\epsilon}_{i}\u27e9$ is its expression average over the number of cell states. Regarding self-organization with critical dynamics, we investigated averaging behaviors in nrmsf and in the fold-change in expression, where we observed a stochastic resonance effect in the terminal cell-fate process. For methodological details, refer to our previous works [2,3].

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**Figure 1.**Glossary and strategy of analysis: An extremely schematic view of the data analysis strategy (top) used to reveal SOC gene-expression regulation together with a glossary of the principal features of self-organization behavior highlighted by previous analyses in several biological processes (bottom); scaling divergent behavior (e.g., Figure 1A in [1]; Figure 5F in [2]; Figure 4 in [3]; Figure 1A in [4]); co-existence of critical states (Figures 4, 5 and 8 in [1]; Figures 1 and 2C in [2]; Figure 9 in [3]; Figure 3A in [4]); barcode genes (Figures 8 and 9 in [2]); self-similar profile transitions (Figure 3 in [2]; Figure 6 in [3]); coherent-stochastic behaviors (Figures 7 and 9 in [1]; Figures 5A and 6 in [2]; Figures 10 and 11 in [3]; Figure 3 in [4]); sandpile-type criticality (Figure 2D in [2]; Figures 3A and 4 in [3]); SOC control landscape (Figure 8 in [3]; Figure 2 in [4]); timing of reprogramming (Figures 5, 7B–D and S1 in [3]; Figure 2 in [4]); genome engine mechanisms (Figures 12 and 13 and the Discussion in [3]; Figures 5–7 in [4]).

**Figure 3.**Scaling-divergent behaviors and critical points: (

**A**) First column: Genome avalanches, scaling-divergent behaviors in overall expression, as important features of the SOC control of overall expression are evident in the log-log plot of average behaviors: mouse (first row), human (second) embryo development and Th17 immune cell differentiation (third). Second column: Correlation distance, expressed as (1 − r), where r is the Pearson correlation coefficient between the gene-expression profiles in the zygote and other development states (mouse: first row and human: second row) and between t = 0 and other time points (t

_{j}) (Th17). This distance corresponds to the relative change in the expression profile on the whole-genome scale. The results show that there is a difference in scaling-divergent behaviors: in mouse and human embryo, correlation behaviors significantly change at low and middle nrmsf, respectively, whereas in terminal Th17 cell, significant change occurs at high nrmsf. Third column: Euclidean distance from the initial-state response (zygote state for embryo development and response at t = 0 for Th17 cell) shows that two distinct biological processes (reprogramming in early embryo development versus immune cell differentiation) show opposite scaling-divergent behaviors. Scaling behavior (i.e., constant behavior in Euclidean distance) occurs in the ensemble of high-variance RNA expression (region of high nrmsf) in early embryo development, and divergent behavior occurs in the ensemble of low-variance RNA expression (region of low nrmsf: sub-critical state), whereas the T cell terminal cell fate (single cell) has opposite behaviors. Log-log plots represent the natural logarithm of the group average (< >) of expression (x-axis) and nrmsf (y-axis) (n = 485 (mouse), 475 (human), and 375 (Th17) for each dot), where overall expression is sorted and grouped (35 groups) according to the degree of nrmsf (Appendix A); (

**B**) Linear regression of scaling regions in the scaling-divergent behaviors; mouse embryo development (upper row) and Th17 cell differentiation (lower row); (

**C**) critical point: in mouse embryo development, a summit (CP) of sandpile criticality (middle panel) corresponds to a tipping point of transitional behavior of the bimodality coefficient (right) and, furthermore, to the intersection of linear regressions (left) (see more in [4]). This suggests that the CP is fixed during early mouse embryo development. In Th17 cell differentiation, the CP corresponds to the onset of divergent behaviors, which is also fixed in single-cell differentiation (see (

**B**)).

**Figure 4.**Timing of the genome-state change on the SOC control landscape in a single cell: A change in critical dynamics through sandpile-type criticality (diverging up- and down-regulation at the CP around ln(<nrmsf>) ~ −5.5), which affects the entire genome-expression dynamics (see details in [4]), appears in the change in overall expression (e.g., fold-change) between different time points. Thus, the erasure of sandpile-type criticality in the zygote state points to the timing of a genome-state change in mouse embryo development. This erasure of zygote criticality (upper row: development of initial-state criticality) occurs after the late 2-cell state to reveal a stochastic expression pattern as a linear correlative behavior (refer to [3]). The near-transition point (see a schematic picture of the SOC landscape: top panel) occurs at the middle-late 2-cell states (lower row: development of neighboring criticality), at which sandpile-type criticality disappears and thereafter recovers. Plots reveal the existence of an SOC control landscape and a transition state at around the middle-later 2-cell states. The x- and y-axes represent the fold-change in expression and the group average expression. A detailed mechanism of how reprogramming occurs via the interaction of distinct coherent expression states is given in [4], where the collective behavior of stochastic low-variance RNA expression as a generator of autonomous SOC control guides the reprogramming of mouse embryo development.

**Figure 5.**Critical transition revealed through changes in the Pearson correlation between the zygote and cell developed states: (

**A**) Pearson correlation for the developed cell state with the zygote exhibits a critical transition as a tangent hyperbolic function, $0.59-0.44tanh(0.78x-2.5)$, (p < 10

^{−3}) (black dash: mouse embryo; red: human embryo) with between-whole expression profiles (right panel): (I) zygote vs. early 2-cell state; (II) zygote vs. late 2-cell state; (III) zygote vs. early blastocyst; (IV: the plot shows that a phase transition occurs at the inflection point (zero second derivative of the tangent hyperbolic function); there is a phase difference between the 4-cell and 8-cell states for human and between the middle and late 2-cell states for mouse. Since there are zero values in RNA expression (Reads Per Kilobase Mapped (RPKM) values), before taking the natural log of expression, we add a value of 1 to all RPKM values for between-whole expression profiles. ε (cell state) represents whole RNA expression at a specific cell state. (

**B**) Temporal development of the Pearson correlation of whole expression at t = 0 h. It follows 0.67 + 1/(3.10 + x) (p < 10

^{−8}; red dashed line) without any inflection (i.e., no phase difference as in embryo development). The (negative) derivatives of Pearson correlation for (

**A**,

**B**) are taken from the fitting functions.

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**MDPI and ACS Style**

Giuliani, A.; Tsuchiya, M.; Yoshikawa, K.
Self-Organization of Genome Expression from Embryo to Terminal Cell Fate: Single-Cell Statistical Mechanics of Biological Regulation. *Entropy* **2018**, *20*, 13.
https://doi.org/10.3390/e20010013

**AMA Style**

Giuliani A, Tsuchiya M, Yoshikawa K.
Self-Organization of Genome Expression from Embryo to Terminal Cell Fate: Single-Cell Statistical Mechanics of Biological Regulation. *Entropy*. 2018; 20(1):13.
https://doi.org/10.3390/e20010013

**Chicago/Turabian Style**

Giuliani, Alessandro, Masa Tsuchiya, and Kenichi Yoshikawa.
2018. "Self-Organization of Genome Expression from Embryo to Terminal Cell Fate: Single-Cell Statistical Mechanics of Biological Regulation" *Entropy* 20, no. 1: 13.
https://doi.org/10.3390/e20010013