Single-Cell Reprogramming of Mouse Embryo Development Through a Critical Transition State

Our work dealing with the temporal development of the genome-expression profile in single-cell mouse early embryo indicated that reprogramming occurs via a critical transition state, where the critical-regulation pattern of the zygote state disappears. In this report, we unveil the detailed mechanism of how the dynamic interaction of thermodynamic states (critical states) enables the genome system to pass through the critical transition state to achieve genome reprogramming. Self-organized criticality (SOC) control of overall expression provides a snapshot of self-organization and explains the coexistence of critical states at a certain experimental time point. The time-development of self-organization is dynamically modulated by exchanges in expression flux between critical states through the cell nucleus milieu, where sequential global perturbations involving activation-inhibition of multiple critical states occur from the early state to the late 2-cell state. Two cyclic fluxes act as feedback flow and generate critical-state coherent oscillatory dynamics. Dynamic perturbation of these cyclic flows due to vivid activation of the ensemble of low-variance expression (sub-critical state) genes allows the genome system to overcome a transition state during reprogramming. Our findings imply that a universal mechanism of long-term global RNA oscillation underlies autonomous SOC control, and the critical gene ensemble at a critical point (CP) drives genome reprogramming. Unveiling the corresponding molecular players will be essential to understand single-cell reprogramming.


Introduction
In mammalian embryo development, a large number of molecular-level epigenetic studies [1][2][3] revealed the occurrence of stunning global epigenetic modifications on chromatins (DNA + histones) associated with reprogramming processes in mammalian embryo development. However, the genome-wide principle that drives such extremely complex epigenetic modifications is still unknown.
In our previous studies, based upon transcriptome experimental data for seven distinct cell fates [4], we recognized that a self-organized critical transition (SOC) in whole-genome expression plays an essential role in the change of the genome expression state at both the population and single-cell levels (see Methods, for more details [4][5][6]).
Essential points of SOC control of overall expression can be summarized as: i) SOC control of overall expression represents self-organization of the coexisting critical states (distinct response expression domains) through a critical transition. Temporal variance expression (normalized root mean square root fluctuation: nrmsf; see Methods) acts as an order parameter in self-organization. Distinct critical states can be observed in a transitional behavior of ensemble expression profile (e.g., bimodality coefficient) or bifurcation of ensemble expression state (e.g., probability density profile) according to nrmsf. Coherent behaviors emerge from stochastic expression within critical states (coherent-stochastic behaviors; see non-equilibrium statistical mechanism underpinning the spontaneous emergence of order out of disorder [7]) as the collective behaviors of groups with more than around 50 genes (meanfield approach) [6,8,9]. ii) Self-organization based on SOC occurs through distinguished critical behaviors: sandpile-SOC control of overall expression can be seen in an emergent level of collective behaviors.
The presence of a 'collective organization layer' in gene expression, while a naturally encompassed by the existence of 'tissue specific gene expression profiles' across the whole genome, seems at odds with the major part of biological experimentation strictly focusing on gene-specific rules. The opposition 'microscopic rules' versus 'collective behavior', is a classical issue in science: it is worth stressing the presence of microscopic laws does not falsify the statistical approach (that in this condition could be considered as a surrogate of a more 'fundamental' microscopic approach) while the statistical 'emergent' way can hold even in absence of microscopic deterministic rules [11]. In this last case, the 'fundamental level' corresponds to the collective organization layer [11,12]. As for our specific case, a gene-bygene set of deterministic microscopic rules affecting thousands of gene products (some of which have very low concentration inside single cells and thus, undergoing high stochastic fluctuation) is expected to be both more unreliable and energy consuming than self-organized criticality that in turn is very clearly emerging as different scaling regimens along order parameter (Figure 1).   Recently, we demonstrated that essentially the same critical-state dynamics that we observed for cell differentiation processes [4,6] are also present in overall RNA expression in single-cell mouse embryo development, which is particularly relevant to give further proof of SOC control as a universal characteristic [13]. Overall RNA expression and its dynamics exhibit typical features (genome avalanche and sandpile type criticality) of self-organized criticality (SOC) control in mouse embryo development; Figure 1 shows a genome avalanche,

B) Correlation distance
i.e., scaling-divergent behavior in a log-log-scale plot between expression and the temporal variance of expression (nrmsf). Figure 2 shows that sandpile-type criticality (critical behavior) of the zygote state survives at the early 2-cell state and disappears after the middle 2-cell state to reach a stochastic pattern in the 4-cell state (linear pattern revealed in randomly shuffled overall expression: S2 Fig. in [4]).
Importantly, genome avalanches (Figure 1) reveal that the regions of scaling and divergent regions in terms of nrmsf are opposite between embryo development and cell differentiation at a single-cell level. Since nrmsf is an order parameter [4][5][6], this reveals the opposite order of self-organization; note that similar scaling-divergent order for cell differentiation is also observed at a cell-population level in cancer cells [4,6]. Furthermore, since chromosomes exhibit fractal organization [14,15], the power law behavior may reveal a pattern, in which SOC control (sandpile-type criticality) disappears (see more in [4] and [13]).
In this report, we investigate the self-organizing dynamics of whole RNA expression in mouse embryo development [16] to unveil how genome reprogramming occurs by passing through a critical transition state. This is elucidated by a dynamic expression flux analysis (quantitative evaluation of perturbation based on self-organization through SOC).
The results of the flux dynamics suggest that the critical gene ensemble of the critical point plays an essential role in reprogramming. Our results suggest that the SOC control mechanism of genome dynamics is rather universal among several distinct biological processes [4]. We think our findings may provide a universal classification scheme for phenomena with far-from-equilibrium phase transitions, which has been missing in past studies [17]. .

Critical States
As has been demonstrated in cell differentiation [4][5][6], mouse embryo overall RNA expression is self-organized into distinct response domains (critical states). This can be seen as a transitional change in expression profile of groups of genes according to nrmsf (Figure 3A),     A) Single cell mouse embryo:

C) Intersection
overall expression: genome avalanche and sandpile-type criticality (Figures 1A and 2). This collective control in early embryo development points, notably, to the fact that bewildering epigenomic reprogramming molecular processes are regulated by a few hidden control parameters through SOC in terms of collective behaviors even at a single-cell level (see [4] for examples of control at the cell-population level).
Regarding a critical point (CP) in single-cell Th17 cell differentiation and mouse early embryo development, Figures 1B and 4B,C suggests that a single critical point (CP) may exist in the range of ln<nrmsf> from -5.5 to -6.0 for both processes: the onset of the divergence from the scaling occurs at around ln<nrmsf> ~ -6.0 for Th17 cell differentiation ( Figures   1B,4B), whereas linear regressions in scaling regions for mouse early embryo possesses intersection at around ln<nrmsf>= -5.5 ~ -6.0 (a critical point as in Figures 2, 3A).

B) Expression Flux Dynamics Representing the Exchange of Genetic Activity
Here, we elucidate a statistical mechanism to explain how a single mouse embryo cell achieves genome reprogramming by passing through the transition state (Figure 2).
where ΔP is the change in momentum with a unit mass (i.e., the impulse: The interaction flux of a critical state, X with respect to another critical state or the environment (milieu) Y (Figure 5B) can be defined as:  to form a strong coupling between them [6]. This suggests that the two cyclic fluxes act as feedback flow for the change in criticality to generate coherent oscillatory dynamics of critical states. This dynamic model also provides a mechanism for long-term global RNA oscillation underlying autonomous SOC control generated by the sub-critical state [4,6,19].
The formation of a dominant cyclic flux between a source and a sink provides a genome-engine metaphor for SOC control mechanisms to describe how expression flux is transmitted among critical states: the sub-critical state as a 'large piston' for short move and the super-critical state as a 'small piston' for large move with an 'ignition switch' (near-critical state with a critical point) are connected through a dominant cyclic state flux as a 'camshaft', resulted in anti-phase dynamics of two piston movements (refer to Discussion in [4]). This suggests that the genome engine, may be a Universal mechanism in the gene expression regulation of mammalian cells.

D) Change in Criticality: A Global Impact on Whole Genome-Expression System
Here, we further clarify the SOC control mechanism of the reprogramming of singlecell embryo development through the breakdown of initial-state.
The reprogramming occurs at the middle-late 2 cell states: the zygote-state SOC control of overall gene expression (i.e., initial-state global gene expression regulation mechanism) is destroyed through the erasure of the zygote-state criticality. Figure 5D shows that the mutual

Transition State
The erasure of initial-state criticality (e.g., in the zygote state) points to the onset of genome reprogramming after the middle 2-cell state. As noted, an initial state can be the early 2-cell state instead of the zygote state; this independent choice of the initial state further confirms the timing of the genome-state change [4].  (Figure 2; more in [4]), which is supported by sequential Interaction flux dynamics (Figures 5B, 7) describes the erasure as a thermodynamical event that passes through a critical transition state, which shows how Therefore, two major global perturbations, which involve the activation-inhibition of multiple critical states, occur between the early and middle 2-cell states, and between the middle and late 2-cell states during genome reprogramming. This global perturbation event is clearly seen in the net kinetic energy flux [4] in a critical state ( Figure 5C): where the kinetic energy of the CM for the critical state with unit mass at s = sj is defined as

III. Discussion and Conclusion
Thermodynamics allows for incredibly precise predictions thanks to the 'generality of its premises' (Einstein's words [20]), this allowed us to grasp the essentials by skipping the (largely unknown) detailed biological mechanisms and focusing on the phenomenology of the genome expression global changes. The erasure of epigenetic marks (refer to the Discussion in [4]) is consistent with thermodynamics perspective beside its actual mechanism.
The time-development of sandpile criticality reveals the existence of a transition state which in turn suggests that programming of mouse embryo occurs through the transition state.
As a proof of this concept, it was demonstrated that EGF-stimulated MCF-7 cells do not erase sandpile criticality (see Fig. 5A in [4]), i.e., no genome-state change (consistent with the experiment no cell-differentiation occurs [21]). This shows that the event of single-cell programming is related to overcoming of the transition state [22,23], which is a typical event in thermodynamic reaction mechanism. Moreover, the timing of the reprogramming does not depend on the selection of an initial cell state. The result helps us to obtain a quantitative appreciation of the still largely qualitative notion of the epigenetic landscape.
Intriguingly, our statistical thermodynamics approach reveals that the collective behavior of an ensemble of low-variance RNA expression (sub-critical state), which shows only marginal changes in expression and consequently are considered to be devoid of any interest, guides the genome to pass through the transition state (erasure of initial-state sandpiletype criticality) to reprogram the mouse embryo. The sub-critical state gene ensembles act as a driving force to transmit their potentiality, or energy of coherent transcription fluctuations, to high-variance genes (the genome-engine mechanism [4]). We observed also this generator role of the sub-critical state in cell differentiation [4] (MCF-7 human cancer cells and HL-60 human promyelocytic leukemia cells). Thus, genome-engine mechanism may provide a universal SOC control mechanism in the genome system.
Sandpile-type critical point (CP) exists around the edge between the near-and supercritical states in the mouse genome. The erasure of sandpile-type criticality through embryo development induces dynamic change in interaction flux dynamics between near-and subcritical states, which determines the activation-inhibition dynamics of the sub-critical state.
The sub-critical state generates autonomous SOC control of overall expression, and thus the critical dynamics (temporal change in criticality) affect the entire genome expression system.
This suggests that a critical gene ensemble of sandpile-type criticality (i.e., critical point) should exist to affect the entire genome expression. Therefore, elucidation of the molecular mechanism that guides the genome system through a transition state is expected to unveil molecular clues as to how a single cell can succeed or fail at reprogramming, such as in an iPS single cell.
The establishment of a material basis for the observed phenomenology by unveiling the molecular mechanism on the criticality of gene ensemble is awaited as the next study, which may lead us toward a comprehensive understanding of single-cell reprogramming.

Biological Data Sets:
We analyzed the following mammalian RNA-Seq data: The addition of large random noise (a>>10) destroys the sandpile CP.

Normalized Root Mean Square Fluctuation (nrmsf):
Nrmsf is defined by dividing rmsf (root mean square fluctuation) by the maximum of overall {rmsfi}: , where rmsfi is the rmsf value of the i th RNA expression, which is expressed as εi(sj) at a specific cell state sj (in mouse, S = 10: s1 = zygote, early 2-cell, middle 2-cell, late 2-cell, 4-cell, 8-cell, morula, early blastocyst, middle blastocyst and s10 = late blastocyst), and 〈 〉 is its expression average over the number of cell states. Note: nrmsf is time-independent variable and an order parameter for self-organization of genome expression as demonstrated in our previous works [4][5][6].

Bimodality Coefficient:
Sarle's bimodality coefficient for a finite sample (b) [26] is given by where n is the number of items in the sample, g is the sample skewness and k is the sample excess kurtosis.
It is worth noting that gene expression in single cell RNA-Seq data we used in the present paper do have many zero values. This is not an artifact but a consequence of the 'toggle-switch' mechanism of gene regulation [27] and are essential for the understanding of regulation mechanisms. On a purely computational perspective, zeroes alter the 'standard values' of the bimodality coefficient (b= 5/9 ~ 0.56 indicates a bimodal or multimodal transition from a unimodal profile), this implies we must limit ourselves to consider bimodality coefficient a step-like behavior as the signature of a transition ( Figure 3A) without entering in the details of their actual values.

SOC Control Mechanism of Overall Expression:
A self-organized critical transition (SOC) in whole-genome expression plays an essential role in the change of the genome expression state -SOC control of overall expression at both the population and single-cell levels [4][5][6]. The basic findings of these studies can be summarized as: i). SOC of overall expression does not correspond to a phase transition from one critical state to another. Instead, it represents self-organization of the coexisting critical states through a critical transition, i.e., SOC consolidates critical states into a genome expression system (called 'SOC control of overall expression') accordingly to temporal expression variance (nrmsf).
Nrmsf acts as an order parameter in self-organization. In critical states, distinct coherent (collective) behaviors emerge in ensembles of stochastic expression (coherent-stochastic behavior), where coherent dynamics of high-variance gene expression (super-critical state) is anti-phase to that of low-variance gene expression (sub-critical state).
ii). The characteristics of the self-organization through SOC become apparent only in the collective behaviors of groups with an average of more than around 50 genes (mean-field approach). The same value of 50 genes (or around this value) as the threshold for the onset of coherent ensemble behavior was previously recognized in a completely different context and by different analytical techniques [6,8,9]. This effect is clearly a statistical one, but not a 'simple statistical one' in the sense it is a proxy of the underlying gene regulation network (GRN).
iii). Self-organization occurs through distinguished critical behaviors: sandpile-type criticality and scaling-divergent behavior: a) Sandpile-type critical behavior (criticality) based on the grouping of expression according to the fold change in expression. A summit of the sandpile represents the critical point (CP); as the distance from the CP increases, the divergence of two different regulatory behaviors occurs, which represent up-regulation and down-regulation, respectively. Furthermore, in the vicinity of the CP according to nrmsf, in terms of coherent expression, self-similar bifurcation of overall expression occurs to show a unimodal-bimodal transition ( Figures. 1, 3A in [6]: MCF-7 cancer cells) and a step function like transition ( Figure 3A in [4]: HL-60 cancer cells); these indicates the existence of cell-type specific transitions. Thus, since a critical behavior and a critical transition occur at the CP, we can characterize it as a sandpile-type transition.
b) Scaling-divergent behavior (genomic avalanche) based on the grouping of expression according to nrmsf: a nonlinear correlation trend between the ensemble averages of nrmsf and gene expression at each time point, which has both linear (scaling) and divergent domains in a log-log plot; the onset of divergence occurs at the CP: order (scaling) and disorder (divergence) are balanced at the CP (), which presents a genomic avalanche. The scaling-divergent behavior reflects the co-existence of distinct response domains (critical states) in overall expression. Distinct critical behaviors from different averaging behaviors occur (numerically) at around the same CP.
The occurrence of a temporal change in criticality directly affects self-organization in the entire genomic system (Figure 13 in [4]). The genome-state change (cell-fate change in the genome) occurs in such a way that the initial-state SOC control of overall gene expression -initial-state global gene expression regulation mechanism is destroyed through the erasure of an initial-state criticality.