Biological Data Sets

We analyzed mammalian transcriptome experimental data for seven distinct cell fates in different tissues:

1. Cell population: Microarray data of the activation of ErbB receptor ligands in human breast cancer MCF-7 cells by EGF and HRG; Gene Expression Omnibus (GEO) ID: GSE13009 (N = 22277 mRNAs; experimental details in [21]), which has 18 time points: t₀ = 0, t₁ = 10, 15, 20, 30, 45, 60, 90min, 2, 3, 4, 6, 8, 12, 24, 36, 48, t_{T = 17} = 72h,
2. Cell population: Microarray data of the induction of terminal differentiation in human leukemia HL-60 cells by DMSO and atRA; GEO ID: GSE14500 (N = 12625 mRNAs; details in [22]), which has 13 time points: t₀ = 0, t₁ = 2, 4, 8, 12, 18, 24, 48, 72, 96, 120, 144, t_{T = 12} = 168h,
3. Single cell: RNA-Seq data of T helper 17 cell differentiation from mouse naive CD4+ T cells in RPKM values (Reads Per Kilobase Mapped), where Th17 cells are cultured with anti-IL4, anti-IFNγ, IL-6 and TGF-β (details in [23]); GEO ID: GSE40918 (mouse: N = 22281 RNAs), which has 9 time points: t₀ = 0, t₁ = 1,3,6,9,12,16,24, t_{T = 8} = 48h,
4. Single cell: RNA-Seq data of early embryonic development in human and mouse developmental stages in RPKM values; GEO ID: GSE36552 (human: N = 20286 RNAs) and GSE45719 (mouse: N = 22957 RNAs) (experimental details in [24]) and [25], respectively).

We analyzed 7 human and 10 mouse embryonic developmental stages:

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.

For microarray data, the Robust Multichip Average (RMA) was used to normalize expression data for further background adjustment and to reduce false positives [26–28], whereas for RNA-Seq data, RNAs that had zero RPKM values over all of the time points were excluded. Random real numbers in the interval [0–1] generated from a uniform distribution were added to all expression values for the natural logarithm. This procedure avoids the divergence of zero values in the logarithm. The robust mean-field behavior through the grouping of expression (see section II) was checked by multiplying the random number by a positive constant, a (a< 10), and thus, we set a = 1. Note: The addition of large random noise (a>>10) destroys the sandpile CP.

Emergent Properties of SOC in the Mean-Field Approach

In this report, we examine whether characteristic behavior at a critical point (CP) is present in overall expression to investigate the occurrence of SOC in various cell fates. We briefly summarize below how SOC in overall gene expression was elucidated in our previous studies [10,11].

1. Global and local genetic responses through mean-field approaches: Our approach was based on an analysis of transcriptome data by means of the grouping (gene ensembles) of gene expression (mean-field approach) characterized by the amount of change in time to reveal the coexistence of local and global gene regulations in overall gene expression [29,30]. A global expression response emerges in the collective behavior of low- and intermediate-variance gene expression, which, in many expression studies, is cut-off from the whole gene expression by an artificial threshold, whereas a local response represents the genetic activity of high-variance gene expression as elucidated in molecular biology.
2. Self-organized criticality as an organizing principle of genome expression: To understand the fundamental mechanism/principle for the robust coexistence of global and local gene regulation, and further, the role of the global gene expression response, we elucidated a self-organized criticality (SOC) principle of genome expression that could account for global gene regulation [11]. In self-organization, the temporal variance of expression, nrmsf (normalized root mean square fluctuation), acts as an order parameter to self-organize whole gene expression into distinct expression domains (distinct expression profiles) defined as critical states, where nrmsf is defined by dividing rmsf by the maximum of overall {rmsf_i}:
   where rmsf_i is the rmsf value of the i^th expression (mRNA or RNA), which is expressed as ε_i(t_j) at t = t_j (j = 0,1,..,T) and ⟨ε_i⟩ is its temporal average (note: we use an overbar for a temporal average when ensemble and temporal averages are needed to distinguish).
3. Coherent-stochastic behaviors in critical states: Coherent expression in each critical state emerges when the number of stochastic expressions is more than 50 [11] (called coherent-stochastic behavior: CSB). The bifurcation—annihilation of the ensemble of coherent gene expression determines the boundary of critical states (Fig 5 in [10]).
4. Biophysical reason for specific groupings: Coherent dynamics such as coherent oscillation emerge in the change in expression (e.g., fold change) (refer to Fig 7 in [10]; static and dynamic domains correspond to sub- and super-critical states, respectively), but not in expression itself. Furthermore, the change in expression but not expression itself, is amplified in a critical state (Fig 6B in [11])). This indicates stochastic resonance effect in the change in expression. Thus, regarding self-organization, we investigated averaging behaviors in nrmsf (order parameter) and the change in expression.
5. Characteristic properties of SOC: Distinctive critical behaviors emerge at a critical point in the averaging of two observables: the fold change in expression and nrmsf. Importantly, different critical behaviors (i.e., criticality) that occur at the same CP can originate from different averaging behaviors (MCF-7 cells [11]), which confirms that the mean-fields are not a statistical artifact. We call these sandpile-type transitional behavior and scaling-divergent behaviors at a critical point (CP) summarized as follows:
   1. Sandpile-type transitional behavior is based on the grouping of expression into k groups with an equal number of n elements according to the fold change in expression. A sandpile-type transition is the most common in SOC [31]. As n increased, the average value of a group (a mean-field) converges, and an ensemble of averages exhibits a sandpile profile. Good convergence in the group is obtained above n = 50 (Fig 2 in [11]), which stems from CSB. The top of the sandpile is a critical point (CP), where the CP usually exists at around a zero-fold change (null change in expression), and up- and down-regulated expression is balanced between different time points. This indicates that the critical behavior occurs through a ‘flattened expression energy profile’. This flatness suggests that the strength of the correlation tends to increase with the size of the system ensemble. As noted, a critical point can exist away from a zero-fold change through erasure of the initial-state criticality. A sandpile-type critical behavior shows that, as the distance from the CP (summit of the sandpile) increases, two different regulatory behaviors, which represent up-regulation and down-regulation, respectively, diverge. Furthermore, in the vicinity of the CP according to nrmsf, in terms of coherent expression, self-similar bifurcation of overall expression occurs to show transitional behavior (Figs 1, 3A in [11]). Thus, since a critical behavior and a critical transition occur at the CP, we can characterize it as a sandpile-type transition.
   2. Scaling-divergent behavior (genomic avalanche) is based on the grouping of expression according to nrmsf: a nonlinear correlation trend between the ensemble averages of nrmsf and mRNA expression at a fixed time point. Originally, we called this the DEAB (Dynamic Emergent Averaging Behavior) of expression [10], which has both linear (scaling) and divergent domains in a log-log plot. In the scaling domain, the quantitative relationship between the ensemble averages of nrmsf and mRNA expression, <nrmsf> and <ε>, is shown in terms of power-law scaling behavior, where higher <nrmsf> corresponds to higher <ε>:

Such scaling is lost at the CP in the MCF-7 cell fates. This shows that 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, i.e., critical states. In S1 File, we address the genuineness of the power-law scaling and the existence of collective behavior of gene expression in power-law scaling.

Chromosomes exhibit a fractal organization; thus, the power law behavior may reveal a quantitative relation between the aggregation state of chromatin through nrmsf and the average expression of an ensemble of genes. The entity of gene expression likely scales with the topology-associated chromatin domains (TADs), such that the degree of nrmsf should be related to the physical plasticity of genomic DNA and the high-order chromatin structure.  

Dynamic Flux Analysis for an Open Thermodynamic Genomic System

Sloppiness in SOC control reveals that genome expression is self-organized into a few critical states through a critical transition. The emergent coherent-stochastic behavior (CSB) in a critical state corresponds to the scalar dynamics of its center of mass (CM), X(t_j), where X ∈ {Super,Near,Sub}: Super, Near and Sub represent the corresponding critical states (section VI). Thus, the dynamics of X(t_j) are determined by the change in the one-dimensional effective force acting on the CM. We consider the effective force as a net flux of incoming flux from the past to the present and outgoing flux from the present to the future. Based on this concept of flux, it becomes possible to evaluate the dynamical change in the genetic system in terms of flux among critical states through the environment.

1. Self-flux.

The effective force is deduced as the decomposition of IN flux from the past (t_j-1) to the current time (t_j) and OUT flux from the current time (t_j) to a future time (t_j+1), so that the effective force as a net self-flux, f (X(t_j)) from its temporal average, can be written as (1) where ΔP is the change in momentum with a unit mass (i.e., the impulse: FΔt = ΔP) for a time difference: Δt = tj+1—t_j-1, and Δt_j = tj—t_j-1; t_j is the natural log of the j^th experimental time point (refer to biological data sets); the CM of a critical state is with the natural log of the i^th expression ε_i(t_j), ln (ε_i(t_j)) at t = t_j; the temporal average of the net self-flux, <f(X)> = <INflux>—<OUTflux>, and N is the number of expressions in a critical state (Table 1).

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Table 1. Averaged Critical States.

https://doi.org/10.1371/journal.pone.0167912.t001

As noted, the negative force, f (X(t_j)), is taken such that a linear term in the nonlinear dynamics of X(t_j) corresponds to a force under a harmonic potential energy. The sign of the net self-flux shows the net incoming force to X(t_j) (X< =): net IN flux for f(X(t_j))> 0, and net outgoing force from X(t_j) (X = >): net OUT flux for f(X(t_j))< 0, where the net IN (OUT) flux corresponds to activation (deactivation) flux (force), respectively.

4. The average flux balance.

If we take a temporal average of Eqs (1) or (3) and Eq (2) for the data for both MCF-7 and HL-60 cells, we obtain the average flux balances of a critical state (Table 2), X, and its interaction with other states or the environment, respectively: (4)

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Table 2. Average Interaction Fluxes: Arrow shows the direction of flux.

https://doi.org/10.1371/journal.pone.0167912.t002

I. Perturbation of SOC control and the Genome-State Change

We investigated the occurrence of critical transitions in distinct cell types. First, we examined whether the sandpile-type critical transition (see Methods) occurs around the zero-fold change (i.e., null change in expression) between different time points. The critical point (CP), at the top of the sandpile, corresponds to a group of genes that show almost no (average) change in expression. Next, we assessed whether or not the CP is a fixed point in time by evaluating if the average nrmsf value of the CP group changes over time. The basic hypothesis that justifies the choice of nrmsf as a metric for evaluating gene-expression dynamics is that gene expression groups scale with topology-associated chromatin domains (TADs) [32–36]. The degree of gene expression normalized fluctuation is presented with respect to chromatin remodeling, i.e., nrmsf is expected to be associated with the physical plasticity of genomic DNA and the high-order chromatin structure [11]. Thus, we can expect that, as the flexibility of a given genome patch increases, so should the nrmsf of the corresponding genes. To confirm this further, the scaling-divergent behaviors (Fig 4 and Methods) of nrmsf and average expression, another important feature of SOC, may show a quantitative relation with the aggregation state of chromatin. In S1 File, we show that gene expression exhibits collective behavior (as coherent-stochastic behavior: CSB) in the power-law scaling through interactions among genes.

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Fig 4. Time-development of the characteristic behaviors of SOC.

A) MCF-7 cells and B) HL-60 cells. At each experimental time point, t_j (A: 18 time points and B: 13 points; Methods), the (ensemble) average nrmsf value of the CP at t = t_j, <nrmsf(CP(t_j))> is evaluated at the sandpile-type critical point (top of the sandpile: right panels). In the top center panels, <nrmsf(CP(t_j))>, is plotted against the natural log of t_j (A: min and B: hr). Error bar represents the sensitivity of <nrmsf(CP(t_j))> around the CP(t_j), where the bar length corresponds to the change in nrmsf in the x-coordinate (fold-change in expression) from x(CP(t_j))—d to x(CP(t_j)) + d for a given d (A: d = 0.005; B: d = 0.01; due to much more mRNAs in MCF-7 cells). Temporal averages of <nrmsf(CP(t_j))> are A) _HRG = 0.094 and _EGF = 0.081, and B) _{DMSO, atRA} = 0.078, where an overbar represents temporal average. Note: An overbar for a temporal average is used when ensemble and temporal averages are needed to distinguish. A) MCF-7 cells: The temporal trends of <nrmsf(CP(t_j))> are different for HRG and EGF. The onset of scaling divergence (left panels: second and third rows) occurs at around (black dashed line), and reflect the onset of a ‘genome avalanche’ (Methods). B) HL-60 cells: The trends of <nrmsf(CP(t_j))> for the responses to both DMSO (black line) and atRA (blue) seem to be similar after 18h (i.e., global perturbation; see section VI). The scaling-divergent behaviors for both DMSO and atRA reveal the collapse of autonomous bistable switch (ABS [10,11]) exhibited by the mass of groups in the scaling region (black solid cycles) for both DMSO and atRA. The onset of divergent behavior does not occur around the CP ( = 0.078), but rather is extended from the CP (see section III). The power law of scaling behavior in the form of 1- <nrmsf> = α<ε>^-β is: A) α = 1.29 & β = 0.172 (p< 10⁻¹⁰) for HRG, and α = 1.26 & β = 0.157 (p<10⁻⁹) for EGF; B) α = 1.60 & β = 0.301 (p< 10⁻⁶) for DMSO, and α = 1.42 & β = 0.232 (p< 10⁻⁶) for atRA. Each dot (different time points are shown by colors) represents an average value of A) n = 742 mRNAs for MCF-7 cells, and B) n = 505 mRNAs for HL-60 cells.

https://doi.org/10.1371/journal.pone.0167912.g004

A transcriptome analysis based on a mean-field (grouping) approach (Methods) reveals that sandpile transitions occur and the position of the CP exhibits time-dependence in terms of nrmsf, which reflects the temporal development of SOC, i.e., the CP is not a fixed point (Fig 4). Interestingly, the CP of the initial state disappears over time, suggesting the occurrence of a crucial change in the genome state (Fig 5). Regarding critical transitions around the CP (Fig 6), different types of dynamical bifurcation or annihilation of a characteristic coherent expression state occur around the CP in different cell models:

1. Unimodal-flattened-bimodal transition for MCF-7 cell-HRG and -EGF models, and
2. Unimodal-flattened-unimodal transition for HL-60 cell-atRA and -DMSO.

Both models point to symmetry breaking in the expression profile.

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Fig 5. Genome-state change revealed through erasure of the initial-state criticality in overall expression (cell population level): The grouping of overall expression at t = t_j (j ≠ 0) according to the fold change in expression from the initial overall expression (t = 0) shows how the initial-state critical behavior is erased over time, i.e., a sandpile profile in overall expression is destroyed at t = t_j from t = 0.

This event points to the time when the genome-state change occurs. On the x-axis, ln(<ε(t)/ε(0h)>) (t: min or hr) represents the natural log of the ensemble average (< >) of the fold change in expression, ε(t)/ε(t = 0h), and on the y-axis, ln(<ε(t)>) represents the natural log of the ensemble average of expression, <ε(t)>. A) MCF-7 cells: In HRG-stimulated cells (black), the divergent behavior in up-regulation is no longer observed, and the same shape is apparent after 3h. This suggests that the genome-state change occurs at 3h (red) through erasure of the initial-state up-regulation process (partial erasure). In contrast, in EGF-stimulated cells (blue), almost the same sandpile profile remains for up to 36h, which suggests that no genome-state change occurs (see the local perturbation in Fig 14). B) HL-60 cells: A CP is erased at 24h (red) and 48h (red) in DMSO- (black) and atRA-stimulated (blue) cells through the disappearance of divergent behaviors in both up- and down-regulation of the initial state (full erasure), respectively. Furthermore, these plots suggest that the epigenomic states in DMSO- and atRA-stimulated cells become the same after 48h. Note: The Pearson correlations of overall expression between different time points are near-unity (Fig 1A). Each dot represents the average value of A) n = 742 mRNAs for MCF-7 cells, and B) n = 505 mRNAs for HL-60 cells.

https://doi.org/10.1371/journal.pone.0167912.g005

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Fig 6. The self-similar bifurcation or annihilation of a characteristic coherent expression state in the vicinity of a critical point in different cell types.

A coherent expression state (CES), which contains approximately 1000 expression points, is bifurcated or annihilated around the CP through nrmsf grouping. The average nrmsf of an expression group, <nrmsf>, is evaluated for a variable range: x + 0.01^.(m-1) <nrmsf< x + 0.01^.m (integer, m≤10 for each x = 0.7, 0.8 and 0.9). The probability density function (PDF) in the regulatory space (z-axis: probability density) around <nrmsf(CP)> shows. A) MCF-7 cells: In the HRG response at 15-20min, around <nrmsf(CP)> (<nrmsf(CP)>_HRG = 0.0948 and <nrmsf(CP)>_EGF = 0.0809; Fig 4A), the PDF exhibits a bimodal-flattened-bimodal transition, in which a bimodal profile points to the existence of two CESs: one represents a low-expression state (LES) and the other represents a high-expression state (HES); the valley defines the boundary between low and high expression [10]. In the EGF response (second row), above <nrmsf(CP)>, a low-expression state (LES) is annihilated and only a high-expression state (HES) exists, i.e., a unimodal-bimodal transition is present. In the HRG response, at a time period other than 15-20min, a unimodal-bimodal transition occurs [10]. This result shows the self-similar symmetry-breaking event to the overall expression for MCF-7 cells, even at a transient change in the HRG response: a bimodal-flattened-bimodal transition at 15–20 min. B) HL-60 cells: A unimodal-flattened-unimodal transition occurs at 18-24h (pseudo-3-dimensional PDF plots of Fig 3B) for DMSO and at 0-2h for atRA, which again reveals the self-similarity to the overall expression for HL-60 cells (section III).

https://doi.org/10.1371/journal.pone.0167912.g006

These results offer the following insights:

1. SOC control in different cell models: The self-similarity of symmetry breaking around the CP suggests the existence of critical states (distinct transcription response domains) in different cell models (see section III). Furthermore, the self-similarity in the overall expression suggests the occurrence of i) a unimodal-flattened-bimodal transition for MCF-7 cell fates, and ii) a unimodal-flattened-unimodal transition for HL-60 cell fates, which will be shown to be robust over time (section III), except for a transient change (see Fig 6A, an example in the HRG response: a bimodal-bimodal transition at 15–20 min due to a bifurcation in the unimodal profile; refer also to Fig 7B in [10]). The flattened distribution shows that the degree of cooperation in expression regulation, i.e., the strength of the correlation around the CP, tends to increase with the size of the system ensemble, as expected in the critical dynamics of biological systems (see more in subsection (iv) of Discussion). The unimodal-unimodal transition in HL-60 cell fates stems from the fact that a pair of two coherent expression states (a bimodal expression profile as an autonomous bistable switch [10,11]) collapses to a single coherent state in the scaling region (low nrmsf region: Fig 4B).
2. Perturbation of SOC control: The temporal development of SOC control reflects the presence of dynamic changes in critical states in terms of both exchanging of genes between critical states and changes in the expression profile. This implies the perturbation of SOC control through the interaction between critical states (see sections V and VI). Interestingly, regarding HL-60 cell fates, at 12-18h, a pulse-like global perturbations involving the regulation of critical states occur for the responses to both DMSO and atRA (section VI). After 18h, the temporal trends of the CP(t_j) for DMSO and atRA in terms of nrmsf become similar (Fig 4B). Note that the bifurcation-annihilation events of CES around the CP for both DMSO and atRA at 24-48h become almost identical (data not shown). This shows that the dissipation of the stressor-specific perturbation in HL-60 cells drives the cell population toward the same attractor state [22,37].
3. Genome-state change: Critical dynamics appear in the change in expression between different time points. Thus, in the change in expression at t₀-t_j (t₀<t_j), the erasure of the criticality of the initial state (Fig 5) at t = t_j indicates that a genome-state change occurs: at 3h in HRG-stimulated MCF-7 cells, and at 24h and 48h in DMSO- and atRA-stimulated HL-60 cells. These HL-60 genome-state changes further confirm that both DMSO- and atRA-stimulated HL-60 cells converge toward the same global gene-expression profile at 48h. Erasure of the initial-state critical behavior occurs in different cell types; divergent behavior in up-regulation (a partial erasure of criticality) disappears in HRG-stimulated MCF-7 cells, whereas the full erasure of criticality occurs in both DMSO- and atRA-stimulated HL-60 cells. As demonstrated in section VI, these genome-state changes occur after dissipative pulse-like global perturbations in SOC control (at 12-18h for HL-60 cells and at 15-20min for HRG-stimulated MCF-7 cells). In contrast, as a proof of concept, the genome-state change does not occur in EGF-stimulated MCF-7 cell (Fig 5; refer also to local perturbation; section VI), which is consistent with cell proliferation (and the absence of differentiation) in the EGF response [38,39].

Note: The independence of the choice of the initial state at t = t₀ for the breakdown of sandpile type criticality at t = t_b (condition: t₀<t_b) further confirms the timing of the genome-state change. We observed that the time point (t₀) of the initial state is earlier or equal to the onset of the pulse-like global perturbation (12-18h) for DMSO-stimulated HL-60 cells (t₀≤12h; see S1 Fig); after the global perturbation, this independence breaks (i.e., the breakdown of sandpile type criticality does not occur), suggesting that a pulse-like global perturbation directly concerns with the first stage of cell-fate determination (process for autonomous terminal differentiation; see subsection (ii) in Discussion). The changes in the genome, thus, reveal two critical events: a pulse-like global perturbation and erasure of the initial-state criticality that determine the cell-fate change (see subsections (i)-(iii) in Discussion).

To dig deeper into these findings, in the following sections, we investigate the transcriptome SOC control in embryonic development at a single-cell level based on next-generation RNA sequencing data, and address mechanism of the genome-state change in terminal cell fates revealed through the perturbation of genome-wide self-organization.

II. Control of Single Zygotic Cell Embryonic Development and T helper 17 Cell Differentiation Through Self-Organized Criticality

The RNA-Seq approach allows us to check the tenability of SOC control at a single-cell level. Here we analyze RNA-Seq data related to immune cell differentiation, and human and mouse embryonic development.

Fig 7A shows Pearson correlations between gene expression profiles related to the zygote and early embryo single-cell states. We previously demonstrated (see Fig 1) that the between-profiles Pearson correlation follows a tangent hyperbolic function with an increase in the number of genes. A similar transitional behavior (the presence of a critical transition) is observed in both human and mouse early embryo development. This transition takes place between the 4-cell and 8-cell states in human and between the middle and late 2-cell states in mouse.

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Fig 7.

Genome-state change revealed through erasure of the initial-state critical point in overall expression (single cell level): ‬‬‬‬‬‬‬Transcriptome RNA-Seq data (RPKM) analysis for A-C) human and mouse embryo development, and D) T helper 17 cell differentiation. A) In both human and mouse embryos (red: human; blue; mouse; refer to development stages in Methods), a critical transition is seen in the development of the overall expression correlation between the cell state and the zygote single-cell stage, with a change from perfect to low (stochastic) correlation: the Pearson correlation for the cell state with the zygote follows a tangent hyperbolic function: a − b ⋅ tanh(c ⋅ x − d), where x represents the cell state with a = 0.59, b = 0.44 c = 0.78 and d = 2.5 (p< 10⁻³) for human (red dashed line), and a = 0.66, b = 0.34 c = 0.90 and d = 3.1 (p< 10⁻²) for mouse (blue). The (negative) first derivative of the tangent hyperbolic function, -dr/dx, exhibits an inflection point (zero second derivative), indicating that 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 (inset); a phase transition occurs at the inflection point. Notably, the development of the SOC control in the development of the sandpile-type transitional behaviors from the zygote stage (30 groups; n: number of RNAs in a group: Methods) is consistent with this correlation transition: B) In human, a drastic change in criticality occurs after the 8-cell state; a sandpile-type CP (at the top of the sandpile) disappears, and thereafter there are no critical points. This shows that the zygote SOC control in overall expression (i.e., zygote self-organization through criticality) is destroyed after the 8-cell state, which indicates that the memory of the initial stage of embryogenesis is lost through a stochastic pattern as in the linear correlation trend (refer to the random expression matrix in S1 Fig or to Fig 3D). The results suggest that reprogramming (massive change in expression) of the genome occurs after the 8-cell state. C) In mouse, a sandpile-type CP disappears right after the middle stage of the 2-cell state and thereafter a stochastic linear pattern occurs, which suggests that reprogramming of the genome after the middle stage of the 2-cell state destroys the SOC zygote control. D) In Th17 cell differentiation, a sandpile-type CP disappears after 3h through a stochastic linear pattern. Therefore, the plot suggests that the genome-state change occurs at around 3-6h in a single Th17 cell. ‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬

https://doi.org/10.1371/journal.pone.0167912.g007

Notably, the correlation transition corresponds to the onset of the breakdown of SOC control in early embryo development according to the number of cell states starting from the zygote single-cell state. In both human and mouse embryos, the maternal SOC zygote controls break down through cell-state development, and the overall expression becomes stochastic. In human, the maternal SOC zygote control survives until the 8-cell state stage. After the morula state (Fig 7B), no sandpile-type critical point exists, and the overall gene-expression profile is fully stochastic compared with the overall expression in the zygote (see S2 Fig), which indicates that the memory of genome expression in the zygote is lost in the morula state. In contrast, in the mouse embryo (Fig 7C), the maternal SOC controls survive from the zygote to the 2-cell state (middle stage).

The breakdown of early SOC zygote control in overall expression indicates that significant global perturbation (refer to section VI) occurs to destroy the SOC zygote control in early embryo development. The human scenario may be explained by the known fact that the genome of the human embryo is not expressed until the 4–8 cell stage, which suggests that there is no apparent significant perturbation as reprogramming in early human embryo development (see more in Discussion). The timing of reprogramming is further confirmed by the independence of the breakdown of criticality from the choice of the initial cell-state (see S1 Fig).

Along similar lines of reasoning, T helper 17 (Th17) cell differentiation shows that initial SOC control (t = 0) is destroyed after 3h (Fig 7D), which reveals that the Th17 genome-state changes at around 3-6h after the induction. Sandpile criticality emerges again after 6h in Th17 cell differentiation (see S2 Fig). The embryo and immune cell results confirm the presence of specific SOC controls, not only in large cell populations, but also at a single-cell level.

Next, we examine the development of SOC control between sequential cell states. As shown in Fig 8, from the zygote to the morula state in mouse, one sandpile-type critical regulation transitions to another through a non-critical transition of regulation (absence of critical behaviors: non-SOC control) in the middle-late 2-cell state. This confirms that reprogramming of the early mouse embryo cells from the zygote destroys SOC control to initiate self-organization in the new embryonal genome at the late 2-cell state, which exhibits a stochastic overall expression pattern (Fig 7C). Thereafter, the SOC control again takes over embryo development. Non-SOC control exhibits almost a linear behavior, which is a characteristic of randomized expression (see S2 Fig and refer to the similar linear behavior in Fig 3D in a different cell type).

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Fig 8. The SOC control landscape as revealed by a sandpile-to-sandpile transition in mouse embryo development.

The development of a sandpile transitional behavior between sequential cell states suggests the existence of an SOC control landscape (first row: schematic picture of a valley-ridge-valley transition; x-axis: cell state; y-axis: degree of SOC control). The second and third rows show that a sandpile-to-sandpile transition occurs in mouse embryo development from the zygote single-cell stage to the morula cell state: I: a sandpile (i.e., critical transition) develops from the zygote single-cell stage to the early 2-cell state = > II: a sandpile is destroyed from the middle to the late 2-cell state, which exhibits stochastic expression (i.e., no critical transition; refer to the random mouse expression matrix in S2 Fig) = > III: a sandpile again develops from the 8-cell state to the morula state. These results show that a significant perturbation (reprogramming) in self-organization occurs from the middle stage to the late 2-cell stage through a stochastic overall expression (refer to Fig 7). Note: Qualitatively, there is a high degree of SOC control for a well-developed shape of the sandpile-type transition (SOC control), an intermediate degree of SOC control for a weakened (broken) sandpile, and a low degree for non-SOC control. The latter is due to stochastic expression. The linear behavior (absence of a critical point) in mouse embryo development is also reflected in the low Pearson correlation (r ~0.21 after the 8-cell state) (Fig 7A and 7C).

https://doi.org/10.1371/journal.pone.0167912.g008

The transition of SOC control through non-SOC control suggests that an SOC-control “landscape”, i.e., a valley (SOC control)—ridge (non SOC control)—valley (SOC control), is seen in early mouse embryo development. The genome expression dynamics in the early embryo, through the development of SOC control, are consistent with the ‘epigenetic landscape’ frame, in the broad terms of the global activation-deactivation dynamics of the genome generally consistent with the DNA de-methylation-methylation landscape [40].

The onset of the genome-state change (at the breakdown of initial-state SOC control) exhibits a clear difference between single cells and a cell population (Fig 5): cell populations do not exhibit a stochastic pattern, in contrast to single cells in early human and mouse embryonic development (Fig 7 and S2 Fig). The stochastic pattern is confirmed by low Pearson correlation between the zygote and early embryo single-cell states at the onset (r< 0.5; Fig 7A), whereas in cell populations, the Pearson correlation for overall expression at any different time points is close to unity (Fig 1A and Fig 2B).

This result indicates that there is a critical transition at the genome-state change in the ensemble of cells from single-cell stochastic to highly correlated cell-population behavior (i.e., emergent criticality-induced complexity matching [12]) in overall expression—the emergent layer of a relevant collective regulation starting from a given minimal threshold number of cells [11]. The near-stochastic pattern (Fig 7D) of helper Tell 17 cell differentiation at the onset (single cell) confirms such a transition. The elucidation of the statistical mechanism [41] of the emergent layer of collective regulation in a cell population may explain how coherent oscillation of a critical state in the ensemble of stochastic expression (coherent-stochastic oscillation) [10,11] emerges in interacting cell ensembles.

III. Distinct Time-Averaged Critical States in Terminal Cell Fates

The self-similarity around the CP (Fig 6) to overall expression highlights three distinct distribution patterns of gene expression relative to different critical states:

1. A unimodal profile corresponding to high-variance expression for a super-critical state, which belongs to a flexible genomic compartment for dominant molecular transcriptional activity.
2. A flattened unimodal profile (intermediate-variance expression) for a near-critical state, corresponding to an equilibrated genomic compartment, where the critical transition emerges.
3. A bimodal profile for HRG and EGF responses in MCF-7 cells or a unimodal profile for the DMSO and atRA responses in HL-60 cells for a sub-critical state (low-variance expression). The sub-critical state is the compartment where the ensemble behavior of the genomic DNA structural phase transitions is expected to play a dominant role in the expression dynamics. In the HRG response, a subset of consecutive genes pertaining to the same critical state (called barcode genes) on chromosomes, spanning from kbp to Mbp, has been shown to be a suitable material basis for the coordination of phase-transitional behaviors (refer to Fig 8A in [11]).

The presence of different distributions of a sub-critical state points to different forms of SOC in biological processes with a varying sub-critical state, with regard to the bimodal character of the corresponding expression profile.

In the next section, we will show the existence of the expression flux flow between critical states, which induces temporal fluctuation of the critical point as shown in Fig 4. For evidence of such flow, we need to focus on the average critical state, and then show how perturbation from this average generates activation/inactivation fluxes across the critical states.

The self-similarity of the symmetry breaking around the CP suggests that critical states have distinct profiles. The degree of nrmsf acts as the order parameter for distinct critical states in mRNA expression [10,11]. Thus, to develop a sensible mean-field approach, we estimate bimodality coefficients along nrmsf by the following steps (Fig 9):

1. Sort and group the whole mRNA expression according to the degree of nrmsf. The nrmsf grouping is made at a given sequence of discrete values of nrmsf, and
2. Evaluate the corresponding temporal average of the bimodality coefficient over time to examine if the mean field (behavior of averages of groups) shows any distinctive behavior to distinguish critical states.

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Fig 9.

Critical states revealed through distinct functional behaviors of the bimodality coefficient: A) MCF-7 cells, B) HL-60 cells and C) Random DMSO expression matrix (see Fig 3D). The left panels (A and B) show the frequency distribution of expression according to the degree of nrmsf. The center panels show the temporal average of Sarle's bimodality coefficient, <b_i>, of the i^th group over time; the value 5/9 represents the threshold between the unimodal (below 5/9) and bimodal or multimodal distributions (above 5/9). The grouping of expression is made at a specific sequence of discrete values of nrmsf (x_i: nrmsf_i = i/100; i: integers) with a fixed range: x_i—k^.d < x_i < x_i + k^.d. The values of k and d are set to be k = 150 and d = 0.0001 for MCF-7 and HL-60, and k = 100 and d = 0.001 for a random matrix based on the convergence of the bimodality coefficient. The convergence of the difference in bimodality coefficients at x_i with an increase in k (i.e., as the number of elements in a group increases) is shown between the next neighbors, <b_i(k;x_i)>—<b_i(k-1;x_i)> for HL- 60 cells in the right panels of B). The 6 colored dots represent the convergent behaviors of different nrmsf points. The behavior of the time average of the bimodality coefficient exhibits A) Tangent hyperbolic function, b_i = a − tanh(b + c⟨nrmsf⟩_i); a = 1.38 and 1.35; b = 0.123 and 0.301; c = 13.8 and 10.2 for HRG (p< 10⁻⁴) and EGF (p< 10⁻¹⁰), respectively, B) Heaviside step function-like transitions for HL-60 cell fates, and C) No transition for a random DMSO expression matrix, which importantly reveals that random noises through the formation of a Gaussian distribution destroy a sandpile critical behavior. Based on these distinct behaviors, we can determine the boundaries of averaged critical states (Table 1; see section III): A) Critical states are defined by two points: the average CP, , the onset of a genome avalanche (Fig 4A), for the upper boundary of the sub-critical state, and the point where the change in the bimodality coefficient, Δ<b_i>, reaches zero for the lower boundary of the super-critical state; the near-critical state is between them, and B) Step function-like transitions reveal the boundaries of averaged critical states, where the near-critical state corresponding to the transitional region separates the other states.

https://doi.org/10.1371/journal.pone.0167912.g009

Fig 9 clearly reveals that, in the overall expression, the mean-field behavior of the average bimodality coefficient confirms the unimodal-bimodal transition of MCF-7 cells and the unimodal-unimodal transition of HL-60 cells. Notably, the mean-field behaviors of bimodality coefficients follow tangent hyperbolic functions (Fig 9A) for MCF-7 cells and Heaviside-like step functions for HL-60 cells (Fig 9B). The distinct time-average behaviors between different cell types further support cell type-specific SOC control.

In contrast, a randomly shuffled expression matrix (DMSO: HL-60 cells) does not exhibit any apparent transitional behavior (Fig 9C), which further confirms the existence of distinctive averaged critical states in both MCF-7 and HL-60 cells.

Two different behaviors in terms of the bimodality coefficient (see Table 1) are evident:

1. In MCF-7 cells, the averaged CP corresponds to the onset of scaling-divergent behavior and the unimodal-bimodal symmetry breaking of the expression profile (Fig 6A), so that the averaged sub-critical state (bimodal profile) is below the averaged CP (nrmsf< 0.094 for HRG and nrmsf< 0.081 for EGF; Fig 4). The super-critical state (unimodal profile) is above the point where the change in the bimodality coefficient reaches zero (nrmsf> 0.165 for HRG and nrmsf> 0.160 for EGF), and the near-critical state (flattened unimodal) is between them (Fig 9A).
2. In HL-60 cells, the transition of bimodality coefficients clearly distinguishes average critical states (Fig 9B). Notably, the averaged CP does not correspond to the onset of scaling-divergent behavior for either response (Fig 4B). In fact, the onset is extended: in the DMSO response, the scaling region extends to the upper boundary of the near-critical state, while in the atRA response, the scaling region extends to the upper boundary of the sub-critical state, where the averaged CP exists in the (averaged) sub-critical state. This is attributable to the collapse of autonomous bistable switch (ABS) of the sub-critical state, as discussed in the previous section.

In summary, the mean-field behavior of bimodality coefficients exhibits markedly different behaviors that can be used to distinguish averaged critical states.

IV. Coherent-Stochastic Behavior (CSB) in Critical States

Critical states display coherent-stochastic behavior (CSB), where coherent behavior emerges in ensembles of stochastic expression [11]. In Fig 10A and 10B, random sampling of the averaged critical states for both MCF-7 and HL-60 cells clearly shows that

1. The near-zero Pearson correlation between different randomly selected gene ensembles in the critical states reveals stochastic expression, and
2. There is a sharp damping in variability (Euclidean distance of single time points from the center of mass CM(t_j) of the critical states). This is a further confirmation that the CM(t_j) of the critical states represents their coherent dynamics.

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Fig 10.

Coherent-stochastic behaviors in critical states: A) MCF-7 cells and B) HL-60 cells. 200 random-number ensemble sets are created, where each set has n (variable) sorted numbers, which are randomly selected from an integer series {1,2,..,N} (N: total number of mRNAs in a critical state: Table 2). These random sets are used to create random gene ensembles at t = t_j (j^th experimental time points; Methods) from each critical state. 1) Left panels (A and B): For each n at t = t_j, Pearson correlations are evaluated between different random gene ensembles in the critical states by averaging over the 200 ensemble sets, and then, their temporal average over experimental time points are evaluated. This gives a near-zero Pearson correlation, consistent with the global stochastic character of microscopic transcriptional expression regulation in critical states. 2) Right panels (A and B): For each random gene ensemble, the Euclidean distance of single time points from the center of mass CM(t_j) of the critical states is evaluated by averaging over the 200 ensemble sets. The sharp damping of variability confirms that the emergent coherent dynamics of the critical states correspond to the dynamics of the CM(t_j).

https://doi.org/10.1371/journal.pone.0167912.g010

The emergent CSB of critical states through SOC control of the entire expression shows how a population of cells can overcome the problem of stochastic fluctuation in local gene-by-gene regulation. Moreover, the fact that CM represents the coherent dynamics of CSB tells us how macroscopic control can be tuned by just a few hidden parameters through SOC. This collective behavior emerges from intermingled processes involving the expression of more than 20,000 genes; this corresponds to the notion that, despite their apparent bewildering complexity, cell state/fate changes collapse to a few control parameters, and this ‘sloppiness’ [9] derives from a low effective dimensionality in the control parameter-space emerging from the coherent behavior of microscopic-level elements.

Furthermore, mRNA expression in a microarray reflects populations of millions of cells, so that the ensemble of expression at t = t_j represents a snapshot of time-dependent thermodynamic processes far from equilibrium, where the usual pillars of equilibrium thermodynamics such as ‘time-reversibility’ and ‘detailed balance’ break down, to reveal the characteristics of a dissipative or far-from equilibrium system.

V. Sub-Critical State as a Generator of Self-Organizing Global Gene Regulation

The temporal development of critical transitions in overall expression suggests that molecular stressors on MCF-7 and HL-60 cells induce the perturbation of self-organization through interactions between critical states. The emergent coherent-stochastic behavior (CSB) corresponds to the dynamics of the center of mass (CM) of critical states (refer also to Fig 6B in [11], which shows ON-OFF coherent oscillation of the sub-critical state with its CM). Hence, an understanding of the dynamics of the CM of critical states and their mutual interactions should provide insight into how the perturbation of self-organization in whole-mRNA expression evolves dynamically through perturbation.

Here, it would be useful to abstract the essence of the dynamics of critical states and their mutual interactions into a simple one-dimensional CM dynamical system: the CM of a critical state, X(t_j), is a scalar point, and thus, the dynamics of X(t_j) can be described in terms of the change in the one-dimensional effective force acting on the CM. From a thermodynamic point of view, this force produces work, and thus causes a change in the internal energy of critical states. Hence, we investigate the genome as an open thermodynamic system. The genome is considered to be surrounded by the intranuclear environment, where the expression flux represents the exchange of genetic energy or activity. This picture shows self-organized overall expression under environmental dynamic perturbations; the regulation of mRNA expression is managed through the mutual interaction between critical states and the external connection with the cell nucleus milieu.

To quantitatively designate such flux flow, we set up the effective force acting on the CM, f(X(t_j)) at t = t_j, where the expression of each gene is assigned to have an equal constant mass (set to unity). The impulse, FΔt, corresponds to the change in momentum ΔP and is proportional to the change in average velocity: v(t_j+1)—v(t_j). Since a consideration of the center of mass normalizes the number of genes being expressed in a critical state, we set the proportionality constant, i.e., the mass of the CM, to be unity. Thus, f(X(t_j)) = –F = (v(t_j)—v(t_j+1))/Δt, where Δt = tj+1—t_j-1, and the force is given a negative sign, such that a linear term in the nonlinear dynamics of X(t_j) corresponds to a force under a harmonic potential energy. The effective force, f(X(t_j)) can be decomposed into IN flux: incoming expression flux from the past t = t_j-1 to the present t = t_j, and OUT flux: outgoing expression flux from the present t = t_j to the future t = t_j+1 (Δt_j = tj—t_j-1):

We call the force, f(X(t_j)), the net self-flux of a critical state. The net self-flux, IN flux—OUT flux, has a positive sign for incoming force (net IN self-flux) and a negative sign for outgoing force (net OUT self-flux).

When we adopt this concept of expression flux, it becomes straightforward to define the interaction flux of a critical state X(t_j) with respect to another critical state or the environment Y: where, again, the first and second terms represent IN flux and OUT flux, respectively, and the net, IN flux- OUT flux, represents incoming (IN) interaction flux from Y for a positive sign and outgoing (OUT) interaction flux to Y for a negative sign. As noted, the interaction flux between critical states can be defined as when the number of gene expressions in a critical state is normalized, i.e., when we consider the CM. Due to the law of force, the net self-flux of a critical state is the summation of the interaction fluxes with other critical states and the environment (see Methods).

Next, we consider how the net IN (OUT) flux of a critical state, the effective force acting on the CM of a critical state, corresponds to the dynamics of its CM. Fig 11 clearly shows that the trend of the dynamics of the CM of a critical state follows its net self-flux dynamics, in that the CM is up- (down-) regulated for net IN (OUT) flux, where the CM is measured from its temporal average value. This implies that the respective temporal average values are the baselines for both flux and CM; this is further confirmed by the existence of an average flux balance in critical states, where the net average fluxes coming in and going out at each critical state are balanced (near-zero) (Methods).

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Fig 11. Dynamics of the center of mass (CM) of critical states compared with the net self-flux dynamics: Colored lines (red: super-critical; blue: near-critical; purple: sub-critical state) represent net self-fluxes of critical states from their temporal averages (effective force acting on the CM: Methods).

Black lines represent the dynamics of the CM of critical states from their temporal averages, which are increased three-fold for comparison to the corresponding net self-fluxes. The plots show that the net self-flux dynamics follow up- (down-) regulated CM dynamics, such that the sign of the net self-flux (i.e., IN and OUT) corresponds to activation (up-regulated flux) for positive responses and inactivation (down-regulated flux) for negative responses. The natural log of the experimental time points (MCF-7: minutes and HL-60: hours) is shown.

https://doi.org/10.1371/journal.pone.0167912.g011

Thus, we consider both temporal averaged expression flux among critical states and the fluctuation of expression flux from the average (flux dynamics; Methods), so that

• Averaged expression flux shows a temporal average expression flow among critical states through the environment, i.e., the characteristics of an open thermodynamic genomic system (“genome engine” in Discussion), and
• The flux dynamics represent fluctuation of the expression flux flow that is markedly different from the basic properties in equilibrium Brownian behavior under a detailed balance. Furthermore, the sign of the net self-flux (i.e., net IN or OUT) corresponds to the activation (up-regulation) of flux for positive responses and inactivation (down-regulation) for negative responses.

We examined the average flux network for the processes of MCF-7 and HL-60 cells. Fig 12A and Table 2 intriguingly reveal four distinct processes that share common features in their flux networks:

1. A dominant cyclic flux between super- and sub-critical states: Average net IN and OUT flux flows reveal how the internal interaction of critical states and external interaction with the cell nucleus environment interact: notably, the formation of robust average cyclic state-flux between super- and sub-critical states through the environment forms a dominant flux flow in the genomic system. This formation of the cyclic flux causes strong coupling between the super- and sub-critical states as revealed through the correlation analysis [11].
2. Sub-critical state as a source of internal fluxes through a dominant cyclic flux: The direction of the average flux in super- and sub-critical states reveals that the sub-critical state is the source of average flux, which distributes flux from the nucleus environment to the rest of the critical states, where the super-critical state is the sink to receive fluxes from the near- and sub-critical states. Importantly, this clearly shows that the sub-critical state, an ensemble of low-variance expression acts as a generator of perturbation in genome-wide self-organization.

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Fig 12. Genomic expression dynamics revealed through a flux analysis that includes crosstalk with the environment: A) Average values of interaction flux (colored arrows) for MCF-7 and HL-60 cells show that a sub-critical state acts as an internal ‘source’, where IN flux from the environment is distributed to other critical states.

In contrast, a super-critical state acts as an internal ‘sink’ that receives IN fluxes from other critical states, and the same amount of expression flux is sent to the environment, due to the average flux balance (Methods). Furthermore, the formation of a dominant cyclic state flux is revealed between super- and sub-critical states through the environment. The average interaction flux is represented as i-j: interaction flux of the i^th critical state with the j^th critical state (i, j = 1: super- (Super; red), 2: near- (Near; blue), 3: sub-critical state (Sub; purple)), and a colored arrow for an internal i-j interaction points in the direction of interaction with the relative amount of flux (see details in Table 2; positive and negative values represent incoming and outgoing flux, respectively, at a critical state; outline and base colors are based on the i^th critical and j^th critical states, respectively). E represents the internal nucleus environment. B) An early flux dynamic event in the HRG response resulting from the HRG interaction flux dynamics (see C) are shown. Interaction flux dynamics i< = j (or i = >j; color based on j) represent the interaction flux from the j^th critical state to the i^th critical state or vice versa. The interaction fluxes (see HRG in C; the flux direction changes at y = 0) align to suppress the cyclic state flux at 10min (the first point in C), where the interaction flux shows 1< = E, 1 = >2, and 1 = >3 at the super-critical state, 2< = E, 2< = 1, 2 = >3 at the near-critical state, and 3 = >E, 3< = 1, 3< = 2 at the sub-critical state. They then align to enhance the cyclic state flux at 45 min (5^th point in C). This change in the dynamic flux structure is due to the global perturbation at 15-20min (see Fig 14; section VI). At each node, the net flux (Fig 11; 10min: first point; 45min: 5th point) is indicated as IN for net incoming flux (y >0), OUT for outgoing flux (y <0), or Balance (y~0). Note: The average flux balance at each node is maintained, but not at individual time points. C) Notably, for MCF-7 cell fates (HRG and EGF), near-synchronous interaction flux dynamics are seen at sub-critical states. The plots further show that the overall patterns are similar between the same cell types, which again supports cell-type-specific SOC control. Dashed lines represent interaction flux dynamics for i-j (color based on j).

https://doi.org/10.1371/journal.pone.0167912.g012

In summary, the results regarding average flux reveal the roles of critical states in SOC control of overall expression, and provide a statistical mechanics picture of how global self-organization emerges through the interaction among critical states and the cell nucleus microenvironment.

VI. SOC Control Mechanism of Overall Gene Expression

The flux dynamics at critical states reveal the dynamic mechanism of SOC control in overall expression. We can elucidate how dynamic interaction among critical states and the environment perturbs the average flux flow in the genomic system, as follows:

2) SOC control mechanism of the genome-state change—role of the critical gene ensemble.

Here, we describe the SOC control mechanism of the genome-state change in terminal cell fates at the cell population level. The genome-state change occurs through the breakdown of initial-state criticality as follows:

Expression flux dynamics (Fig 13) show that the net self-flux of the sub- and super-critical states can be well-described in terms of the net interaction between sub- and near-critical states (Sub-Near), and between sub- and super-critical states (Sub-Super), respectively. Namely, the dynamic interactions of Sub-Near and Sub-Super determine the net self-flux of the sub- and super-critical states, respectively, which represents the effective driving forces acting on their CMs, and thus determines their coherent oscillatory dynamics (Fig 11; see Fig 6 in [11]). Biologically, the between-states interaction serves as the underlying basic mechanism of self-regulatory gene expression such as epigenetics through the incorporation of rich variety of transcriptional factors and non-coding RNA regulation to determine the critical-state coherent oscillatory behaviors.

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Fig 13.

SOC control mechanism of overall gene expression in terminal cell fates (MCF-7 and HL-60 cells): A) Flow regime shows how a dynamical change in criticality (critical behaviors) at the CP in terminal cell fates affects the entire genome (thick yellow arrow with a black dashed outline) as follows (Super: super-; Near: near-; Sub: sub-critical state): The dynamical change in criticality (critical behaviors) that originates from the CP at Near perturbs the net interaction flux (2–3 + 3–2; refer to Fig 12A) at Sub-Near, which determines the net self-flux of Sub (B: right panels), and thus directly perturbs the Sub-response. This induces a perturbation in the net interaction flux of Sub-Super (1–3+3–1), i.e., directly perturbs the Super-response (B: left panels). Furthermore, these perturbations on Sub and Super disturb the dominant cyclic flux between Sub and Super, which in turn has an impact on sustaining the critical dynamics (source: Sub and sink: Super); solid thick arrows represent average fluxes (Table 2). Note: the net self-flux of a critical state represents the effective driving force acting on CM, and thus dynamic interactions among states determine the coherent oscillatory dynamics of Sub and Super (Fig 11). This schematic picture also shows that the erasure of an initial-state criticality at the genome-state change reflects the destruction of the initial-state SOC control on the dynamical change in the entire genomic system (i.e., pruning procedure for regulating global gene expression at the initial state).

https://doi.org/10.1371/journal.pone.0167912.g013

This essential role of the interactions explains how the temporal change in criticality at the near-critical state, i.e., in expression of the critical gene ensemble of the CP, directly perturbs the sub-critical state (the generator of flux dynamics) through their mutual interaction, and the perturbation of this generator can spread over the entire system (Fig 13A).

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 (i.e., initial-state global gene expression regulation mechanism) is destroyed through the erasure of an initial-state criticality; this suggests that the critical gene ensemble of the CP plays a significant role in determining the cell-fate change. In other words, a statistical mechanical layer [41] for the dynamic control of genome-wide expression emerges in cells, where the critical gene ensemble ‘drives’ the fate of cell ensembles.

3) Global and local perturbations exist in the SOC control.

So far, we have applied the expression flux concept to the effective force acting on the CM of a critical state, X(t_j). This concept can also be extended to define kinetic energy self-flux for the CM of a critical state. The kinetic energy of the CM with unit mass at t = t_j is defined as 1/2^.v(t_j)², such that the net kinetic energy self-flux, K(X(t_j)) at t = t_j, from its temporal average, < K(X) >, can be defined as

We investigate when a significant kinetic energy self-flux occurs to better understand the global perturbation in the SOC control. Fig 14 shows that global perturbations are more evident in the net kinetic energy self-flux dynamics than in the effective force (Fig 11):

1. MCF-7: Global perturbation involving a change in net kinetic energy flux in critical states occurs only at 10-30min, where a pulse-like transition occurs from outgoing kinetic energy flux to incoming flux at 15-20min. The occurrence of this transition is confirmed by a pulse-like maximal response in Pearson autocorrelation (P(t_j;t_j+1) at 15-20min: see Fig 4A in [11]). Dissipation of the kinetic energy is evident in the HRG response, which quickly ends at 30min. In contrast, in the EGF response, only vivid flux activity in the super-critical state is apparent until t = 36h, which demonstrates local perturbation in the EGF response. Thus, global and local perturbations differentiate the cell fate between the responses to HRG and EGF: the dissipative pulse-like global perturbation in the HRG response at 15-20min leads to the genome-state change at 3h (Fig 5A), whereas the local perturbation in the EGF response does not induce the state change (see section I). Furthermore, this global perturbation (see also paragraph 1) above) suggests the existence of a novel primary signaling transduction mechanism or a biophysical mechanism which can induce the inactivation of gene expression in the near- and sub-critical states (a majority of expression: mostly low-variance gene expression) within a very short time. This global inactivation mechanism that is associated with more than 10,000 of low-variance mRNAs in a coordinated manner should be causally related to the corresponding changes in the higher-order structure of chromatin (as discussed in the literature from a theoretical perspective [42–44] and a biological perspective [45,46]).
2. HL-60: In the response to DMSO, a clear pulse-like global perturbation occurs at 12-18h. In the response to atRA, the first significant global perturbation occurs early (2-4h) and a second smaller global perturbation occurs at 12-18h, which can be confirmed by a change in the effective force (Fig 11). They both show distinct dissipative oscillatory behavior of kinetic energy flux dynamics. Again, these global perturbations occur before the genome-state changes (24h for DMSO and 48h for atRA; Fig 5B). As shown in S1 Fig, a pulse-like global perturbation may relate to process for the autonomous terminal differentiation (the first stage of cell-fate determination; see more in subsection (ii) in Discussion).

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Fig 14. Local and global perturbations in self-organizing genome-wide expression: The kinetic energy self-flux dynamics (y-axis) for the CM of a critical state (see section VI) exhibit clear energy-dissipative behavior.

Notably, the results show the occurrence of global and local perturbations of self-organization. Pulse-like global perturbations show a transition from IN to OUT net kinetic energy flux or vice versa (IN-OUT switching) in more than one critical state: at 15-20min, HRG; at 12-18h, DMSO; and at 2-4h (significant) and 12-18h, atRA. In contrast, local perturbation is observed in the EGF response for up to 36h: there is only a marked response in the super-critical state, and almost no response in the other states (i.e., the dynamics of the CM of critical states are localized around their average: Fig 11). The results suggest that the global and local perturbations differentiate MCF-7 cell fates, whereas global perturbations drive the state change in HL-60 cells (see section I).

https://doi.org/10.1371/journal.pone.0167912.g014

(i) Biological Interpretations of the Global and Local Perturbations of SOC Control in MCF-7 Cells

Regarding our present finding of the global perturbation in the SOC control in MCF-7 cell-HRG dynamics, a study of the early gene response [38] established that EGF and HRG induced a transient and sustained dose-dependent phosphorylation of ErbB receptors, respectively, followed by similar transient and sustained activation kinetics of Akt and ERK.

Following ERK and Akt activation from 5-10min, the ligand-oriented biphasic induction of early transcription key proteins of the AP-1 complex (c-FOS, c-MYC, c-JUN, and FRA-1) took place: high for HRG and negligible for EGF. The proteins of the AP-1 complex are non-specific stress-responders supported by the phosphorylation of ERK in a positive feedback loop. In addition, the key reprogramming transcription factor c-MYC (the protein of which peaked at 60min, as confirmed in JE’s laboratory at Latvian Biomedical Research & Study Centre) can amplify the transcription of thousands of active and initiated genes [48] or direct-indirect targets [49] and modify chromatin by recruiting histone acetyltransferase [50]. Further Saeki et al. [21] revealed that after HRG the early sustained by ERK activation of AP-1, c-FOS in particular, induces, the sequential activation of late transcription factors EGR4, FOSL-1, FHL2, and DIPA peaking at 3h. In turn, those begin to down-regulate an ERK proliferative pathway by a negative feedback loop. This allowed differentiation to occur (differentiation needs suppression of proliferation).

Thus, the continuity of the biological relay of the HRG-induced early (pre-committing) and late transcription activities leads to a commitment of differentiation from 3h (cell-fate change), which is necessarily coupled to the suppression of proliferation and stops the genome boost by the ERK pathway. Neither a sustained ERK dependent positive feed-back loop, nor the following negative feed-back is achieved in the case of EGF. Subsequently, these cells did not differentiate but continued proliferation.

In the corresponding expression data, we observed, after HRG, a powerful genome engine causing a pulse-like global perturbation (15-20min) as pre-committing and erasure of the initial-state critical transition to induce the genome-state change at 3h, which was not observed after treatment with EGF, in which case only local perturbation (i.e., only vivid activity of the super-critical state) was observed.

It is important to note that the sub-critical state (low-variance expression: a majority of mRNA expressions; Table 1) generates the global perturbation in the HRG response (section V). The dynamic control of gene expression in the sub-critical state is expected from the cooperative ensemble behaviors of genomic DNA structural phase transitions (see (v) below) through interaction with environmental small molecules, which has not been considered in previous biological studies. Thus, a true biological picture for MCF-7 cells may be obtained by deciphering the biological functions of genes in the sub-critical state in a coordinated (rather than an individual) manner.

(ii) Global Perturbations of SOC Control in HL-60 Cells Committed to Differentiation

The general mechanisms of the commitment to differentiation are not yet well understood. Developmental biologists usually discriminate the two phases into (1) reversible, with the capability of autonomous differentiation; and (2) essentially irreversible [51]. A study in an HL-60-DMSO cell model [52] found that a minimum induction time of 12h was needed for cells to commit to differentiation. In turn, Tsiftsoglou et al. [53] found that exposure to differentiation inducers for only 8 to 18h, which is much shorter than the duration of a single generation, is needed to provide commitment for autonomous terminal differentiation.

Consistent with these findings, we revealed that, at 12-18h, global perturbation involving critical states is observed for the responses to both DMSO and atRA. The induction of differentiation for both inducers is different in the sense that, in contrast to DMSO, which induces the development of macrophages/monocytes, treatment with atRA leads to segmented neutrophils [53]. Interestingly, our analysis also shows that the achievement of cell-fate determination at 24h for DMSO and at 48h for atRA (Fig 5B) occurs in these two models in different ways, although they converge at the same final state at 48h.

In particular, we observed early global genome perturbation in the response to atRA (at 0-4h), which was not seen in the DMSO model. This may be due to a difference in Ca²⁺ influx. Calcium release from the ER combined with capacitative calcium influx from the extracellular space leads to markedly increased cytosolic calcium levels and is involved in cell activation [54] to control key cell-fate processes, which include fertilization, early embryogenesis [55], and differentiation [56]. The amplitude and duration of the Ca²⁺ response are decoded by downstream effectors to specify cell fates [57]. Yen et al. [58] and others have shown that pre-committed HL-60 cells display early cytosolic Ca²⁺ influx. Moreover, intracellular calcium pump expression is modulated during myeloid differentiation of HL-60 cells in a lineage-specific manner, with higher actual flux in the atRA response [59].

(iii) SOC Control in Human and Mouse Related to the Developmental Oocyte-to-Embryo Transition

Fertilized mature oocytes change their state to become developing embryos. This process implies a global restructuring of gene expression. The transition period is dependent on the switch from the use of maternally prepared stable gene transcripts to the initiation of proper embryonic genome transcription activity. It has been firmly established that, in mice, the major embryo genome activation occurs at the two-cell stage (precisely between the mid and late 2-cell states), while in humans this change occurs between the 4- and 8-cell stages [60]. We detected these time intervals, which differ for mouse and human, in the development of SOC control: from sandpile-type critical transitions to stochastic expression distributions (Fig 7A and 7B). Reprogramming of the genome destroys the SOC control of the initial stage of embryogenesis. Developmental studies by Wennekamp et al. in Hiiragi’s group [61] revealed the onset of symmetry breaking between cells in the early embryo and consequently the specification of distinct cell lineages strictly consistent with our model. In addition, what about the physical state of chromatin during these developmental steps?

The erasure of paternal imprinting by DNA 5-methylcytosine de-methylation and hydroxymethylation as part of epigenetic reprogramming occurs in the embryo, which allows significant decompaction of the repressive heterochromatin and an increase in the flexibility of the transcribing part of chromatin [62]. Detailed studies of the DNA methylation landscape in human embryo by Guo and colleagues [40] revealed a decrease in the level of methylation of gene promoter regions from the zygote to the 2-cell stage, which would erase oocyte imprinting. The strength of this correlation increases gradually until it becomes particularly strong after human embryonic genome activation (full reprogramming) at the 8-cell stage [63]. DNA de-methylation unpacks repressive heterochromatin, which manifests in the dispersal and spatio-temporal reorganization of pericentric heterochromatin as an important step in embryonic genome reprogramming [64]. This synchronization of the methylome to confer maximum physical decompaction and flexibility to chromatin and the reprogramming of transcriptome activity for totipotency support the feasibility of SOC, which was determined here by an independent method.

In addition, transposable elements, which are usually nested and epigenetically silenced in the regions of hypermethylated constitutive heterochromatin, also become temporarily activated during the oocyte-to-embryo transition in early embryogenesis [65]. In turn, in human, a peak in SINE activation coincides with the 8-cell stage [40]. The significance of transient retrotransposon activation in embryogenesis, as has been suggested [62], may be that the thousands of endogenous retro-elements in the mouse genome provide potential scope for large-scale coordinated epigenetic fluctuations (further harnessed by piRNA along with de novo DNA methylation). In other words, this should create a necessary critical level of transcriptional noise as a thermodynamic prerequisite for the non-linear genome-expression transition using SOC.

(iv) Extended Concept of Self-Organized Criticality in the Cell-Fate Decision

The recent success with induced pluripotent stem cells (iPSCs) [1] is a remarkable breakthrough not only for possible manipulation of the cell fate through somatic genome reprogramming, but also for understanding the mechanisms of both development and disease.

However, it is still a daunting challenge to elucidate the mechanism of how the transition to a different mode of global gene expression involving the entire genome through a few reprogramming stimuli occurs to achieve the self-control of on/off switching for thousands of functionally unique heterogeneous genes in a small and highly packed cell nucleus.

A fundamental issue is to elucidate the mechanism of the self-organization at the ‘whole genome’ level of gene-expression regulation that is responsible for massive changes in the expression profile through genome reprogramming. To explain this mechanism of massive state changes, self-organized criticality (SOC) has been found to be one of the most important discoveries in statistical mechanics. SOC was proposed as a general theory of complexity to describe self-organization and emergent order in non-equilibrium systems (thermodynamically open systems). However, despite considerable research over the past several decades, a universal classification has not yet been developed to construct a general mathematical formulation of SOC [20]. Useful background information on SOC is found in [20, 66–72].

Recently, the concept of SOC has been used (and extended) to propose a conceptual model of the cell-fate decision (critical-like self-organization or rapid SOC) through the extension of minimalistic models of cellular behavior [73,74]. A basic principle for the cell-fate decision-making model (a coarse-grained model, the same as in our approach) is that gene regulatory networks adopt an exploratory process, where diverse cell-fate options are first generated by the priming of various transcriptional programs, and then a cell-fate gene module is selectively amplified as the network system approaches a critical state [75,76]; these review articles also present a useful survey of studies on self-organization in biological systems.

Self-organizing critical dynamics of this type are possible at the edge between order and chaos and are often accompanied by the generation of exotic patterns. Self-organization is considered to occur at the edge of chaos [77–80] through a phase transition from a subcritical domain to a supercritical domain: the stochastic perturbations initially propagate locally (i.e., in a sub-critical state), but due to the particularity of the disturbance, the perturbation can spread over the entire system in an autocatalytic manner (into a super-critical state) and thus, global collective behavior for self-organization develops as the system approaches its critical point.

We checked the above paradigm in different cases of cell-state transitions at both the population and single-cell levels, and interpreted the observed changes in gene expression in terms of critical-like self-organization. A rigorous statistical-mechanical analysis of the time-development of perturbation in self-organization (see sections IV-VI) revealed the relevant features, however, these were somewhat different from classical SOC models.

We found that:

1. SOC control in overall expression exists at both the population and single-cell levels. In the cell-fate change at the terminal phase (determination of differentiation for the cell population), SOC does not correspond to a phase transition in the overall expression from one critical state to another. Instead, it represents a self-organization of the coexisting critical states through a critical transition.
2. The timing of the genome-state change (i.e., cell-fate change in the genome) is determined through erasure of the initial-state criticality at both the population and single-cell levels. This suggests the existence of specific molecular-physical routes for the erasure of critical dynamics for the cell-fate decision. The cell-fate change (commitment to cell differentiation) in terminal cell fates occurs at the end of dissipative global perturbation in self-organization: first stage by means of a pulse-like global perturbation and commitment stage through erasure of the initial-state criticality.
3. Dynamic interactions between critical states determine the critical-state coherent dynamics (coherent-stochastic oscillation [11]). The occurrence of a temporal change in criticality perturbs this between-states interaction, which directly affects the entire genomic system. Thus, the erasure of an initial-state criticality at the genome-state change reflects the destruction of the initial-state SOC control on the dynamical change in the entire genomic system (i.e., pruning procedure for regulating global gene expression at the initial state).
4. The sub-critical state (ensemble of low-variance gene expressions) sustains critical dynamics in SOC control of terminal cell fates. Furthermore, a sub-critical state forms a robust cyclic state-flux with a super-critical state (ensemble of high-variance gene expressions) through the cell nuclear environment. The results show that there is no fine-tuning of an external driving parameter to maintain critical dynamics in SOC control.

It has been pointed out that fine-tuning of a driving parameter for self-organization in classical SOC generates a controversy regarding the real meaning of self-organization (see more in [75]).

However, despite these differences, the occurrence of global and local perturbations in cell-fate decision processes suggests that there may be another layer of a macro-state (genome state) composed of distinct micro-critical states (found by us). After a pulse-like global perturbation occurs in multiple micro-states through the erasure of initial-state criticality, the genome state transitions to be super-critical to guide the cell-fate change (HRG-stimulated MCF-7 cells, and DMSO- and atRA-stimulated HL-60 cells). On the other hand, the genome state of EGF- stimulated MCF-7 cells remains sub-critical (no cell-fate change) because the local perturbation only induces the activation of a micro-critical state. Further studies on this matter are needed to clarify the underlying fundamental mechanism, and the development of a theoretical foundation for the SOC control mechanism as revealed in our findings is needed.

(v) Mechanism of the ‘Genome Engine’ in Self-Organization and Genome Computing

In the genome engine, the role of the sub-critical state as a generator leads to a new hypothesis that the genomic compartment, spanning from kbp to Mbp, which produces low-variance gene expression may be the mechanical material basis for the generation of global perturbation, where the coordinated ensemble behavior of genomic DNA structural phase transitions through interactions with environmental molecules plays a dominant role in the expression dynamics [81,82]. In HRG-stimulated MCF-7 cell differentiation, a subset of consecutive genes pertaining to the same critical state (called barcode genes from kbp to Mbp; see Fig 8A in [11]) on chromosomes has been shown to be a suitable material basis for the coordination of phase-transitional behaviors. The critical transition of barcode genes was shown to follow the sandpile model as well as genome avalanche behavior. This indicated that there is a non-trivial similarity through SOC between the coherent-stochastic network of genomic DNA transitions and the on-off nerve-firing in neuronal networks [72]. Thus, a potential function of the genome engine may be asynchronous parallel computing, and coherent-stochastic networks based on the on/off switching of sub-critical barcode genes may act as rewritable self-organized memory in genome computing.

Here we have highlighted how the genome engine, i.e., the global response, is driven by the oscillatory behavior of a sub-critical state, i.e., of genes for which the change in expression is quantitatively minor but strongly coherent. This coherence appears when we consider ensembles of genes with n (number of genes) greater than 50 [11]; this behavior is consistent with the particular organization of chromatin into topology-associated domains. The relative flexibility of each domain is probably associated with the changes in expression.

The demonstration of a ‘genetic energy flux dynamics’ across different critical states tells us that chromatin is traversed by coherent waves of condensation/de-condensation, analogous to the allosteric signals in protein molecules. The possibility of controlling such signal transmission through control of the higher-order structure of genomic DNA raises the possibility of very intriguing applications, such as in the much more mature case of allosteric drugs [83,84].