International Journal of
Molecular Sciences
Review
Synchronization between Attractors: Genomic Mechanism of
Cell-Fate Change
Masa Tsuchiya 1, *, Paul Brazhnik 2 , Mariano Bizzarri 3
1
2
3
4
*
and Alessandro Giuliani 4, *
SEIKO Life Science Laboratory, SEIKO Research Institute for Education, Osaka 540-6591, Japan
Academy of Integrated Science, Virginia Tech, Blacksburg, VA 24061, USA; brazhnik@vt.edu
Systems Biology Group, Department of Experimental Medicine, University La Sapienza, 00163 Roma, Italy;
mariano.bizzarri@uniroma1.it
Environment and Health Department, Istituto Superiore di Sanitá, 00161 Rome, Italy
Correspondence: tsuchiya.masa@gmail.com (M.T.); alessandro.giuliani@iss.it (A.G.)
Abstract: Herein, we provide a brief overview of complex systems theory approaches to investigate
the genomic mechanism of cell-fate changes. Cell trajectories across the epigenetic landscape, whether
in development, environmental responses, or disease progression, are controlled by extensively coordinated genome-wide gene expression changes. The elucidation of the mechanisms underlying these
coherent expression changes is of fundamental importance in cell biology and for paving the road to
new therapeutic approaches. In previous studies, we pointed at dynamic criticality as a plausible characteristic of genome-wide transition dynamics guiding cell fate. Whole-genome expression develops
an engine-like organization (genome engine) in order to establish an autonomous dynamical system,
capable of both homeostasis and transition behaviors. A critical set of genes behaves as a critical
point (CP) that serves as the organizing center of cell-fate change. When the system is pushed away
from homeostasis, the state change that occurs at the CP makes local perturbation spread over the
genome, demonstrating self-organized critical (SOC) control of genome expression. Oscillating-Mode
genes (which normally keep genome expression on pace with microenvironment fluctuations), when
in the presence of an effective perturbative stimulus, drive the dynamics of synchronization, and thus
guide the cell-fate transition.
Citation: Tsuchiya, M.; Brazhnik, P.;
Bizzarri, M.; Giuliani, A.
Synchronization between Attractors:
Keywords: genome expression; cell-fate decision; reversion of cancer; self-organized criticality (SOC);
critical point; genome engine; genome attractor; biological regulation; transition theory
Genomic Mechanism of Cell-Fate
Change. Int. J. Mol. Sci. 2023, 24,
11603. https://doi.org/10.3390/
ijms241411603
Academic Editors: Peter J.K. Kuppen,
Carmine Stolfi and Massimo Nabissi
Received: 26 June 2023
Revised: 12 July 2023
Accepted: 14 July 2023
Published: 18 July 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1. Introduction
The actions of a few master regulators of gene expression [1] have been shown to
guide cell-fate changes, where a cascade of events, starting from a small number of transcription factors, either induces or suppresses the coordinated expression of thousands
of genes [2]. This global coordination suggests that major modifications in cell function
at different organizational layers drive cell-fate transitions. However, the mechanistic
interpretation of these changes as simple straightforward outcomes of gene–gene interactions must be reconciled with biological constraints, like the complex architecture of cell
genome organization, the lack of a sufficient number of regulatory molecules [3–5], and
microenvironment-dependent factors [6]. Therefore, in order to properly appreciate this
complexity, we should apply additional organizational concepts toward understanding
the cell-fate transition process from a systems biology perspective. In order to elucidate
the mechanism of how the transition to a different global mode of expression occurs at
the whole-genome scale, we consider genome expression as an integrated dynamical system, and not as the result of coarse graining on a multiplicity of local regulations; we
also further exploit basic principles of dynamical systems theory and statistical mechanics. In non-equilibrium statistical mechanics, Self-Organized Criticality (classical SOC:
c-SOC) was proposed as a general theory with which to accommodate self-organization
Int. J. Mol. Sci. 2023, 24, 11603. https://doi.org/10.3390/ijms241411603
https://www.mdpi.com/journal/ijms
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and emergent order in thermodynamically open systems. Through considerable research
over the past several decades, c-SOC has become an important topic in many scientific
research domains, such as brain research and social science [7,8]. In genomics, our own
findings [9–16] revealed that self-organized criticality (SOC) is also a plausible, physically
motivated, candidate framework for massive, coordinated gene expression regulation.
The occurrence of SOC in cell-fate changes has been demonstrated using a mean-field
approach that exploits the average characteristics of groups of genes instead of dealing
with noisy expressions of individual genes. This technique is grounded in the existence
of the Coherent Stochastic Behavior (CSB; see more in [9–16]) of gene expressions within
sufficiently large gene groups. While individual genes within a given group display
stochasticity, the group’s average expressions (centers of mass, CMs) quickly converge
to certain values after the size of the group becomes sufficiently large. The genomic
mechanism of cell-fate transitions has been verified at both the cell-population and singlecell levels, including those of HRG-Stimulated MCF-7 cells [17,18], atRA- and DMSOstimulated HL-60 cells [19], Th17 cell differentiation from Th0 cells [20], and mouse and
human early embryonic development [21,22].
We demonstrated that whole-genome expression is dynamically self-organized,
through emergent critical behaviors (dynamic criticality), into a few distinct expression
domains (critical states). The coexistence of SOC critical states has been confirmed through
multivariate analysis of gene expression profiles [14].
A specific set of genes (grouped within the “critical point”, CP), displays a singular
behavior. The CP emerges at the boundary between critical states and promotes the spread
of the perturbation in gene expression across the entire genome. At critical transition, an
abrupt change in the expression level of CP genes provokes a sort of ‘genome avalanche’
that, through a domino effect, affects the entire gene expression pattern. However, the SOC
approach we adopted significantly differs from the classical one (c-SOC), which considers
the phase transition from one global critical state to another (i.e., subcritical to supercritical
genome state transition) as the genome is approaching a critical point (attractor). The set
of genes forming the critical point (CP) in our SOC models initiates critical behavior, and
the CP state change guides a global transition in the genome, whereas the CP in c-SOC
corresponds to a specific state of the genome, and thus provides fundamentally different
findings from those of our SOC. Moreover, at odds with c-SOC, the different states are
contemporaneously present in the genome system in our SOC approach.
Below, we list several essential key points that are important for understanding the
self-organized critical (SOC) control of genome expression in cell-fate transitions (see more
in [13,15].
(1)
(2)
(3)
(4)
(5)
The self-organization of genome expression becomes evident by means of nrmsf
(Normalized Root Mean Square Fluctuation) a metric parameter quantifying the
genes’ temporal variability, which uncovers distinct response domains (critical states).
Coherent Stochastic Behavior (CSB) occurs within sufficiently large gene groups. This
is a consequence of the law of large numbers, which allows for the regularization
of the noise present at the single-gene level. This implies that the center of mass
(CM) of an ensemble of stochastic gene expressions acts as an attractor: the CMs of
whole-genome expression and the critical states act as the genome attractor (GA) and
local critical state attractors, respectively.
The autonomous control of fluxes between critical-state attractors gives rise to an
open thermodynamic engine-like system (genome engine).
A specific set of critical genes, identified by the metric parameter (nmrsf ), behaves as
a critical point (CP). Genes within the CP display extremely high temporal variability
and transmit their fluctuations across the genome. The state change in the CP gives
rise to an expression change wave spreading across the entire genome that allows
cell-fate change.
The initiation of genomic transition results in a synchronization between the CP and
GA specifying the cell fate.
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(6)
(7)
At the tipping point (critical transition), the switching of cyclic flow within the genome
engine coherently enhances or suppresses the genome engine by revealing fluctuations
in critical states.
The aforementioned dynamical perspective acquires a material basis in terms of
Peri-centromeric Domains (PADs) bursting at the transition [14,23].
Here, we show how the synchronization dynamics between the CP and GA are
modulated by Oscillating-Mode (OM) genes to guide the cell-fate critical transition.
At any layer of biological organizational hierarchy, from protein molecules to ecosystems, continuous oscillations occur within the system configuration around its typical state
(be it a native protein structure or an abundance of species in an ecological system). These
oscillations are due to the system’s responses to continuous challenges from environmental
variations [24]. The adaptation to these challenges, in order to maintain global homeostasis
(i.e., the resilience of the system’s attractor), is the task of so-called Oscillating-Mode (OM)
genes, which display elevated temporal variance, mirroring such an endless adaptive
process. The identification of such OM genes could provide novel targets for therapeutic
interventions, particularly for those aimed at promoting phenotypic reversion, as already
observed [25]. Finally, when OM genes’ coordinated expression exceeds a given threshold,
the CP synchronizes with the GA, thus driving genome expression transition. Here, we
will give an account of the mechanism by which this transition (initiated by the same OM
genes in charge of homeostasis) happens.
The paper is organized as follows: Section 2 describes how, at the critical transition, OM
genes dynamically modulate the synchronization of the CP and GA; this was conducted
through the application of the Expression Flux Analysis (EFA: [15,16]) on time series
genome expression data. Section 3 provides a potential mechanism for the synchronization
between the CP and GA, and Section 4 gives a discussion of the biological basis of a potential
unified SOC mechanism for the reversion of cancer cell-fate change, while Section 5 reports
general conclusions.
2. Synchronization Dynamics between the CP and GA Drive Critical Cell-Fate
Transitions through the Action of Oscillating-Mode Genes
In our previous works, we identified the variance in gene expression as an important
metric for characterizing the dynamics of genome expression rearrangement. The degree
of nrmsf (Normalized Root Mean Square Fluctuation) is the metric parameter for the
self-organization of genome expression [15,16]. When, at each experimental time, gene
expressions are sorted according to the value of nrmsf, from high to low, and arranged in
groups (with the number of genes in each group n > 50), the Coherent Stochastic Behavior
(CSB) becomes evident. In each group, the average expression quickly converges to a
specific value as the size of the group becomes sufficiently large, according to the law of
large numbers. Within those clusters, a particular set of genes displays a bimodal singular
behavior that acts as a critical point (CP).
The main difference between c-SOC and our model of SOC is that, in the latter, wholegenome expression is coordinated by the critical set of genes (CP genes) behaving as the
critical point (CP). In our model, whole-genome expression self-organizes into coexistent
local critical states (distinct critical response domains), with the CP serving as a boundary
between critical states (see the differences between a population of cells and a single
cell in [13]). The presence of multiple local organizations of the system corresponds to a
“chimera states” structure [26] and adds biological realism to c-SOC.
At the critical transition, an inversion of the bimodal singular behaviors at the CP
occurs, (Figure 3; refer to updated descriptions of SOC methods in [Tsuchiya, M., et al.,
2023] and more results in [13]), where the ON or OFF state change occurs within the
CP, therefore determining the “collective” activation/inactivation of critical genes. The
state change within the CP happens according to the change in critical behaviors, e.g., the
disappearance of sandpile-type critical behaviors and switching bimodal behaviors (refer
to [13]). Thus emerges the following pivotal question:
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How does the genome avalanche (global expression change) at the critical transition occur
in accordance with the state change within the CP?
The CP partially overlaps with the GA (see Figures 3C and 1). In our study on HRGStimulated MCF-7 cells [15], we observed that the CP and GA synchronize at the critical
transition (Figure 2; refer to the potential biophysical reason for the synchronization in
Section 3). Noticeably, in control experiments, the EGF stimulus (that is not able to induce
differentiation) does not provoke a critical transition in MCF-7 cells, and synchronization
of the CP and GA does not happen (Figure 2B). On the contrary, in the case of HRG
(efficient stimulus) at the critical transition, the synchronization between the CP and GA
does occur (Figure 3). This synchronization induces a sequential change in the CP, which,
in turn, determines a significant state change (genome avalanche: Figure 2B). The timing
of the critical transition occurs at 15–20 min (ON to OFF for HRG-Stimulated MCF-7
cells). Furthermore, the timing of the critical transition coincides with that of the switching
genome engine for HRG-MCF-7 cells (see Section 2.2).
Figure 1. CP and GA. (A) Top left panel. In HRG-Stimulated MCF-7 cells, the same kind of sandpiletype critical behavior is displayed by different size (n) groupings (Here, the kth group vector is
denoted as xk (tj ), k = 1, 2, . . . , K; K = 20 groups corresponding to black dots, 30—blue, and 40—red;
groups contain 1113, 742, and 556 mRNA species, respectively). This suggests the existence of scaling
behaviors (i.e., renormalization) due to CSB. Top right panel: Based on groupings by the degree of
𝑙𝑛〈𝑔 10 min 〉
−
−
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nrmsf, the logarithm plot of the average nrmsf, ln<nrmsf >, vs. CM of the group, lnh gk (10 min)i, at
t = 10 min shows that the summit in sandpile critical behavior corresponds to ln<nrmsf >CP ~−2.5
to −2.6 (< . . . > represents the ensemble average). Bottom right panel: CM grouping (ck (tj )) according
to the degree of nrmsf reveals that the CP can be considered a nearly fixed point (time independent)
and exists in the vicinity of the genome attractor GA (see more in Figure 1 in [13]). (B) HRG- and
(C) EGF-Stimulated MCF-7 cancer cells. These plots show that the CP exists near the GA. The different
colors of the dots represent responses at different experimental times.
Figure 2. Synchronization of the CP and GA at Critical Transition with global avalanches. Panel
(A) (left): In HRG-Stimulated MCF-7 cells, noticeable synchronization of the CP and GA occurs
at 10–30 min and 2–4 h. As for the 10–30 min period, global avalanches occur: at 15–20 min,
the critical transition happens with ON–OFF switched in the CP (CP: ON at 15 min; CP: OFF at
20 min), where the CP timing of ON–OFF coincides with that of the upregulated and downregulated
whole expression, respectively (panel (B)). As for 2–4 h, cell-fate change occurs after 3 h [23,27]
with CP being ON. Panel (C): In EGF-Stimulated MCF-7 cells (cell proliferation without cell-fate
change [17,18]), synchronization of the CP and GA does not occur, and, thus, no global avalanche
happens within the same time window. In panel (A), CP responses are amplified seven times.
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Figure 3. Critical Behaviors at CP in HRG-Stimulated MCF-7 cells. In both panels (A,B), gene
expressions are sorted according to their nrmsf value, from high to low, and then grouped into
30 groups with an equal number of 742 elements. (A) Dots of different colors represent expression
responses at different experimental time points, and the x- and y-axes are natural logs of (average)
group expressions and corresponding nrmsf values, respectively. Low-variance genes (ln<nrmsf >
~−2.55), are located below the CP, and the scaling behavior is evident. Low-variance genes (subcritical
state) behave collectively, whereas genes above the CP, i.e., genes with high variance (supercritical
state), exhibit divergent behavior. (B) Along the metric parameter, genes around the CP region (group
average; black dots: 0–15 min; red dots: 0–20 min) exhibit an inversion of bimodal critical behaviors
(red dashed arrow for upregulation; black one for down-regulation) at 15–20 min; the x-axis here
represents the ratio of CM values of the group. (C) Whole expressions are sorted and grouped based
on the degree of temporal expression change (an expression group vector is denoted as x(tj ) at t = tj ).
A plot of expression temporal changes for each group in log-log space reveals a sandpile-type critical
behavior (solid black dots: CM values of the group) at the critical point (the sumit of black dots),
where the CP in terms of the ensemble (group) average (< . . . >) occurs at the near-zero change (x = 0;
null expression change, i.e., the ratio of CM values is equal to one). The solid green dot represents the
CM value of whole expression, i.e., genome attractor (GA).
Looking at Figure 3C, it is worth noting how small expression changes near the CP
diverge to either up- or downregulating processes, consistent with a sandpile-type criticality.
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The CP exists nearby the genome attractor (GA: solid green cycle). At the critical transition,
a change in the center of mass of the whole expression (solid green: GA), from positive
at 0–15 min to a negative change at 0–20, denotes a change in the whole expression from
up- to downregulation (Figure 1). Orange dots represent single mRNA expressions in the
background between different time points. A detailed description of the methods for SOC
analysis is provided in [15,16].
In Figure 2, it is worth noting that, in the absence of a critical transition, (EGFStimulated cells) the oscillatory behavior is absent. In fact, EGF (Epidermal Growth Factor)
is unable to initiate the differentiation process, at odds with HRG (Heregulin) stimulus,
which ignites a cascade of events resulting in cell differentiation.
2.1. Genome Avalanche to MCF-7 Cancer Genome: Spatio-Temporal Behavior of the
Critical Transition
The spread of the differentiation stimulus across different chromosomes makes any
portion of the CP belonging to a chromosome follow almost the same temporal expression
response (see Figure 2.7 in [15]). At the critical transition, the average expression of
each chromosome synchronizes with its CP-related genes. This implies that CSB can
also be detected at the chromosome level, where the stochastic gene expression within a
chromosome converges to its CM.
It is worth noting the existence of different levels of synchronization of chromosomes
with the global CP (Figure 4A,B; see more in [15]). The clustering of synchronized and unsynchronized chromosomes (Figure 4B) shows how and where the critical transition spreads
across the genome. The GA is located at the branching of synchronized chromosomes and
the self-flux of the GA travels along two paths: one from the GA to chromosome 22, and the
other from the GA to chromosome 17. Here, self-flux corresponds to the second-order time
difference of the (group) average gene expression and represents an “effective force”. This
force can be interpreted as a sort of “biophysical energy” transferred to the chromosome
when its sign is positive and released from the chromosome when its sign is negative (see
more in [16]). These paths reveal the spatio-temporal behavior of the critical transition with
an amplified change in the GA (“energy transfer”) in the chromosome territories. Therefore,
synchronization of the GA with the CP guides a genome-wide avalanche (domino effect)
allowing a synchronized wave to spread through the chromosomes (Figure 4B).
2.2. Oscillating-Mode Genes Acting as Modulators of the CP and GA Synchronization:
Determination of the Timing of Cell-Fate Change
The application of Principal Component Analysis (PCA) to temporal gene expressions [14] allows for the detection of the decoupling of a set of specific genes from the global
set of genome expressions at the onset of transition. In PCA terms, this corresponds to the
drastic increase in variance explained by minor (by construction independent from the main
flux of variation) components. The presence of a strong, cell-type-specific gene expression
invariant profile (the cell-type attractor; Figure 5) generates a first principal component
(PC1) that explains a major part of the data set variance. The main orthogonal direction,
with respect to PC1, is PC2 (second principal component), which corresponds to the main
direction of motion around the cell-type attractor. Genes with maximal temporal variation
around their equilibrium position drive PC2. We call these genes Oscillating-Mode (OM)
genes. The continuous oscillations (mirrored by PC2 and other very minor components)
around the ideal profile (PC1) are responsible for the homeostatic adaptation to minor
environmental perturbations (small avalanches, in the SOC terminology). These oscillations
reflect the apparent conundrum of sandpile SOC regulation, in which the attractor state
is a critical state characterized by small avalanches functional to the stability of the gene
expression profile when exposed to continuous small environmental perturbations [28,29].
The CP formalism, as described in Section 2, is coherent with the PCA approach in
consideration of the fact that (local) supercritical genes correspond to the main drivers of
minor components that, at CP, transmit their activation to the core (GA). The CP, determined
by the metric parameter (nrmsf ) of self-organization, serves as the organizing center of
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cell-fate change; its activation (or deactivation) makes local perturbation spread over the
entire genome. It is worth noting that the specific signatures of SOC (refer to Figure 3) can
be recognized in all of the above-described cases.
At the critical transition for HRG-Stimulated MCF-7 cells (15–20 min), PC2 genes
are clearly activated (Figure 5), with a corresponding increase in variance explained by
PC2. On the contrary, largely invariant genes have near-zero scores on minor components
and correspond to the core of the sandpile. These genes constitute the CM of the system
(genome attractor: GA) and are put into motion only when the transition is activated
by a domino effect summing up the small avalanches. This process ends up in a new
attractor state [14,23].
Therefore, we investigate how high PC2 score genes (|PC2| > 2: OM genes) dynamically modulate the synchronization of the CP and GP at the critical transition (threshold
value stems from the fact that principal components are expressed in terms of z-scores, i.e.,
with zero mean and unit standard deviation). The temporal average of the flux network
between the OMG–CP–GA network (OMG stands for OM genes) reveals that external flux
outside of the OMG–CP–GA network is relatively small (Figure 6), which indicates that
INcoming and OUTgoing fluxes in the network are almost balanced (i.e., forming a weakly
non-equilibrium thermodynamic system) for MCF-7 cell response. Hence, we investigate
fluctuations from the temporal average of the OMG–CP–GA network in order to uncover
how OM genes temporally regulate the synchronization between the CP and GA. It is
worth noting that the increase in the variance in minor components corresponding to OM
gene hyper-activation happens with the same timing as for the genome engine switching.
Figure 4. Cont.
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Figure 4. (A). At the critical transition (10–15–20 min), chromosomes are either synchronized or
unsynchronized with the CP dynamics (black lines in small plots). Small plots show examples of
synchronized (9, 17, 22: blue lines) and unsynchronized (1, 5, 8: red lines) chromosomes’ self-fluxes
with the CP dynamics (x-axis: In the background plot (large plot), the x-ax is represents change in
self-flux; y-axis: self-flux). The numerical values of the self-flux of chromosome 9 and 17 CPs are
magnified by three and five times, respectively, and the same holds for chromosomes 1, 5, and 17.
In the background plot (large plot), the x-axis represents change in self flux, and the y-axis shows
a temporal Pearson correlation in the dynamics of the GA. (B) The GA is located at the edge of
two paths (chromosomes 9 and 7), where the self-flux of the GA is amplified through them (i.e.,
genome avalanche). This reveals the spatio-temporal dynamics of the critical transition through the
synchronized chromosomes. The x- and y-axes use the same scale of (A).
As demonstrated in Figure 2A, the HRG-Stimulated MCF-7 cell response shows that
the synchronization in self-flux (effective force) between the CP and GA occurs at the
10–30 min period and 2–4 h period windows. In the former case, small fluctuations in
local critical-state attractors coherently switch the genome engine (critical-state attractor
network; Figure 6) from suppression to enhancement (see Figure 7A). This non-linear
behavior of small fluctuations plays a crucial role in the genome engine to switch it to the
cell-fate-guiding genomic transition. On the other hand, for the temporal expression flux in
the OMG–CP–GA network, Figure 7B demonstrates that OM genes emit large fluxes to both
CP and GA at 15 min. However, the CP and GA actually receive relatively small fluxes from
OM genes and a large flux (obtaining free energy or entropy) from the external flux, i.e.,
outside of the OMG–CP–GA network. These non-linear behaviors clearly demonstrate the
broken mass-action law of the non-equilibrium process in the genome and send a large flux
back to the OM genes. Similar INcoming and OUTgoing flux behaviors in the CP and GA
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exhibit synchronization of the CP and GA during that 20–30 min. Internal interaction fluxes
are negligible this time, so that the lack of cyclic flux between the CP and GA eliminates
anti-phase correlation behaviors. This behavior is reminiscent of the crucial role played by
the microenvironment (external to the network) on the differentiation process.
Figure 5. For HRG-Stimulated MCF-7 cells, Pearson’s correlation between gene expression profiles
at different time points reveals the occurrence of the critical transition at 15–20 min, around the
CM (gene expressions: back solid dots; CM of whole expression: yellow dot), corresponding to
the decrease in Pearson r and, visually, to the increase in the scattering of points from the diagonal
identity line. Axes (black lines) for the first (PC1) and second (PC2) principal components are shown.
At the critical transition, the genes that are most loaded on PC2 (which are oscillating around the
PC1 invariant profile) exhibit maximum departure from the invariant profile, corresponding to a net
increase in the proportion of variance explained by the second component.
Figure 6. Temporal Average of the Flux Network for HRG-Stimulated CP, GA, and OM Genes. MCF-7
cancer cell response; left panel: The diagram demonstrates the genome engine mechanism (see more
in [15]). Subcritical state interactions generate a dominant flux through which to establish the genome
engine mechanism. The subcritical state acts as a “large piston” for small movements (low-variance
expression), while the supercritical state behaves as a “small piston” for large movements (highvariance expression). A sort of “ignition switch” (critical point) connects the different states through a
dominant cyclic state flux (which we can equate to a ‘camshaft’), resulting in the anti-phase dynamics
−
of the two pistons. Numerical values show average between-state expression fluxes based on a
−
2
10 scale. Right Panel: Temporal average of the flux network for OMGs (Oscillating-Mode genes:
|PC2| > 2.0), the CP, and the GA. The external flux for both the genome engine and the CP–GA–
OMG network is relatively small, such that INcoming and OUTgoing fluxes are almost balanced (i.e.,
weakly non-equilibrium system).
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Figure 7. Switching of the OMG–CP–GA flux network coincides with switching of the genome engine
for MCF-7 cancer cells. (A) Switching of the genome engine and (B) switching of the OMG–CP–GA
flux network. The flux network among OM genes (Oscillating-Mode genes: |PC2| > 2.0), the CP, and
the GA, where the OM genes modulate/generate the synchronization of the CP and GA. Switching
of the flux network occurs at the time when the genome engine is switched, (A). Furthermore, the
flux network reveals synchronization between the CP and GA, which demonstrates similar IN and
OUT flux behaviors in the CP and GA, while exhibiting non-linear behaviors (broken mass-action
law) and switching of the OMG–CP–GA network coinciding with those of the genome engine switch.
Flux represented by arrows with different colors are the same as Figure 6.
Furthermore, the timing of switching in the OMG–CP–GA network coincides with that
of the genome engine (Figure 7), where the OMGs suppress cyclic fluxes between the CP
and OM genes and between the GA and OM genes at 15 min and 30 min, their cyclic fluxes
enhanced to support the switching of the OMG–CP–GA network. The synchronization
of the CP and GA also occurs in the 2–4 h time window. The erasure of the initial-state
sandpile criticality (Figure 4D in [13]), which erases the initial memory of the CP genes,
occurs after 3 h. These results are consistent with the timing of the biological commitment of
differentiation at 15 min and late transcription activities at 3 h, leading to the determination
of the differentiation state, coupled with the suppression of proliferation and stopping the
genome boost through the ERK pathway (see more in [23,27]). Thus, we conclude that
cell-fate change, primed at 15–20 min, occurs after 3 h in HRG-Stimulated MCF-7 cells.
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3. Structural Bases of Genome Expression Dynamics
Here, we put forward a potential mechanism for the synchronization between the CP
and GA. The Yoshikawa group [30–34] revealed that the higher-order structure of DNA,
above the scale of several tens of kbps, undergoes a first-order phase transition between its
elongated and compact states. Accompanied by various physio-chemical environmental
changes, the free energy barrier on the first-order phase transition undergoes a drastic
decrease and tends to exhibit noticeable fluctuations when approaching criticality. As
Figure 3B shows, the bimodal singular behavior, nearby the CP, exhibits both activated
and suppressed expression states, and its inversion at the critical transition suggests the
formation of a phase segregation (bimodal domain of up- and downregulation) with a free
energy barrier. When the free energy barrier is lowered, the dynamics of the CP synchronize
to those of the GA to induce the cell-fate guiding critical transition.
The bimodal character of the free energy barrier between the CP and GA has a structural counterpart highlighted by a recent finding related to chromatin transitional dynamics,
regarding the spatial interaction of two chromatin types. Silencing the Pericentromeric
Associated Domains (PADs) that keep the chromatin in its collapsed heterochromatic state
allows for the transcription of euchromatin domains [23]. Thus, understanding the higherorder structural change in genome-sized DNA in relation to the mechanism of control
of several thousand genes is mandatory for comprehending gene expression controls in
cell-fate change. Recently, such spatial interaction at the basis of chromatin dynamics was
observed by Parmentier et al. [35], demonstrating how global chromatin decompaction,
and the consequent stochastic gene activation, constitutes the initial step toward fate
commitment in human hematopoietic cells.
In the case of MCF-7 cells perturbed by HRG, the global departure from gene expression identity (coming from the existence of GA) is maximal between 15 and 20 min. After
that, the effect of the perturbation vanishes and the system comes back to its ‘business-asusual’ behavior. By this time, it seems as though nothing relevant for the cell transition has
happened. Nevertheless, if we wait for a much longer time, we will notice that this ‘shaking’
was not without consequences. The initial ‘wave of change’ dissipates after 30 min (the
system goes back to its initial equilibrium gene expression profile), but the ‘memory’ of the
initial perturbation remains for around 1–1.5 h, due to the action of FOS, a proto-oncogene
of retroviral origin which plays a key role in both tumorigenesis and tumor suppression.
In particular, FOS downregulates MAPK/ERK pathway, a cascade of subsequent protein
activation that communicates signals from membrane receptors to the cell nucleus, causing
tumor growth. ERK activation is abated by 3 h; after this time, the system is committed to
leaving the neoplastic fate and entering the path leading to the terminally differentiated
state, evident after one week.
Can we link these gene expression results to events happening at the chromatin
organization level?
The huge size of the DNA molecule, compared to the size of a cell nucleus, forces
this molecule to stay in an extremely collapsed compact state that must be selectively
unfolded in order to open the way for a drastic change in gene expression. Figure 8
depicts the link between phenomenology (gene expression) and the reorganization of
chromatin (change in number and size of Pericentromere-Associated Domains, PADs). The
centromere of chromosomes is rich in so-called ‘satellite repeats’, which are very repetitive
DNA sequences condensed as heterochromatin. This pericentromeric heterochromatin
acts as a repressive environment in the nucleus. These regions are visible as ‘knots’ via
microscopy and are called PADs. The changes in both PAD size and number correspond
to a change in transcription activity through the folding/unfolding of chromatin, thus
changing the accessibility of chromatin patches to gene expression. Control cells (ST, black
squares), except from a few outliers, have a constant number of PADs, mainly concentrated
in a narrow area range (between two and four squared micrometers). At the critical point
(15 min after HRG, orange circles), the situation is completely different: the PADs are
smaller and much more numerous, with a strong negative correlation between dimension
Int. J. Mol. Sci. 2023, 24, 11603
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and number, pointing to a dramatic unfolding of chromatin through the splitting of large
PADs into smaller ones.
Figure 8. The x-axis represents the size (area) of PADs, and the y-axis represents their per-nucleus
average numbers. Black dots correspond to a non-critical condition, while yellow dots refer to HRG
stimulus at the transition point (15 min). The distribution of size and number of PADs follows an
approximate 1/f scaling (hyperbolic negative relation between size and number, top panel, with a
scaling exponent equal to -0.67 and an R2 = 0.56) that is a hallmark of the dynamic fusion splitting
of PADs, typical of the ‘edge-of-chaos’ condition. The inset of the figure (bottom panel) reports the
normalized standard deviation of expression (nrmsf ) for the 22,277 genes (x-axis) together with the
total number of messenger mRNAs (y-axis), which is a crude but effective index of transcription activity. It is immediate to note both the global similarity of the two distributions and the corresponding
position of the 15 min critical point. Both viewpoints are consistent in indicating the commitment of
differentiation at 15 min, suggesting the unfolding of chromatin as the main driver of the process.
This unfolding is registered by the increase in number and shrinking in size of PADs, signaling
a drastic remodeling of chromatin at the critical transition and allowing for a global change in
genome expression [23].
4. Discussion
Critical transitions in cancer have been observed in response to both internal and
external cues. Indeed, cancer can be stimulated either to acquire new malignant traits
or to revert to a non-invasive, physiological phenotype [25]. Destabilization of the interplay among cells and their microenvironment promotes a critical state transition—by
Int. J. Mol. Sci. 2023, 24, 11603
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analogy with those observed in physical systems—that can be driven by a number of
specific cues to access new attractor states (i.e., phenotypic configurations). This reversion
is associated with the modulation of a set of “critical genes”, which control a few numbers of key pathways (i.e., epithelial–mesenchymal transition [EMT], migration, invasiveness). The reversion of EMT should be viewed as an earlier and inescapable event during
phenotypic reversion [36,37].
During such a reversion, cancer cells re-express several previously repressed genes,
including those involved in cell plasticity (Oct4, SOX-2, and Nanog), while genes (PSEN1,
Notch-1) and proteins (SNAI1, αSMA, N-cadherin) supporting epithelial–mesenchymal
transition (EMT) are downregulated [38]. Overall, a general destabilization of the chromatin
organization occurs, enabling cells to regain a pluripotent state, from which they can later
be forced into a new, differentiated phenotype. Indeed, epigenetic changes associated with
tumor reversion require several preliminary events before critical enzymatic pathways
can be inhibited or enhanced to support the transition toward a new attractor. Specifically,
chromatin should be “opened” and drastically remodeled, eventually through cytoskeleton
(CSK) and nucleoskeleton (NSK) reshaping, in order to facilitate access to gene regulatory
regions [39]. Both CSK and NSK are finely tuned transducers of mechanical and biophysical forces modulating gene expression. It is worth noting that some reverting factors can
trigger tumor reversion in cancer cells by drastically rewiring the cytoskeletal architecture [40]. Furthermore, a number of oocyte/embryo components—including polymerase
II—facilitate the rewiring of genome expression by speeding up the rate of transcription,
promoting chromatin “de-condensation” so as to access previously quiescent segments of
the genome [41], and, finally, enabling a “global reversal” of DNA methylation to make
somatic nuclei closely resemble those of stem cells [42].
However, recovering a stem cell-like profile would be, in principle, incompatible
with tumor reversion, since “uncontrolled plasticity”—as shown by induced pluripotent cells—can give rise to tumor development in a permissive milieu [43]. The attractor’s destabilization and subsequent “reprogramming” into a non-malignant phenotype
could probably not rely solely on internal mechanisms, and may likely require the presence of tissue-dependent constraints that will ultimately drive the process toward a
specific direction [44].
For this reason, the application of SOC analysis to time series genome expression
data is crucial for identifying those critical transition points in which cells/tissues show
high sensitivity to even weak control actors [45]. It is precisely at these points that the
system can be effectively “displaced” from its quasi-stable state, and then “driven” towards
different/alternative cell-fate commitments.
5. Conclusions
Through this review paper, focusing on MCF-7 cancer cells as an example, we described the putative genomic mechanism of cell-fate change [15] that has been supported
by recent chromatin experimentation [23], as well as multivariate statistical analysis [14].
Our studies [13,15] revealed how the complex and dynamically evolving molecular
networks found in biological systems can give rise to globally coherent orchestrated responses. We have demonstrated that the identified critical point (CP) in gene expressions
serves as the organizing center of cell-fate change, and its activation or deactivation causes
the spread of local perturbation over the genome, affecting the genome attractor (GA). Cellfate change asks for massive control of whole-genome expression at the critical transition
through the synchronization of the CP and GA. Expression Flux Analysis (EFA: 15,16])
demonstrates that this synchronization happens through the modulation of OscillatingMode (OM) genes. These genes are normally in charge of genome homeostasis, to keep
gene expression levels on pace with microenvironment fluctuations. The critical transition
occurs to provoke cell-fate change when the perturbation entity on OM genes exceeds a
certain threshold. This model can be viewed as a sort of ‘domino-effect’, in which small
avalanches, initially confined in the periphery of the system, invade the core (GA) of the
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system, putting ‘in motion’ the genes normally preserved by microenvironment vagaries.
Right before the critical transition, large-expression flux is either emitted to or received from
the CP. This is a signature that oscillation exceeds the threshold value, allowing CP–GA
synchronization, thus acquiring the possibility of driving the state transition.
The SOC analysis provides further understanding of the critical transition spatiotemporal behavior within the cancer genome. The CP genes spread over all chromosomes,
where their averaging behavior of the CP portion exhibits similar dynamic behaviors to
the overall CP dynamics. It is worth noting that there exist both unsynchronized and
synchronized chromosomes; a CM of the gene expression in a chromosome synchronizes
dynamically to that of the CP over experimental time points (high temporal Pearson
correlation). At the critical transition, a change in the GA spreads over the synchronized
chromosomes via the amplification of the temporal change in the GA. This is consistent
with the spatio-temporal behavior of the critical transition, demonstrating an avalanching
process which exhibits domino effect in the cancer genome.
The proposed mechanism represents a unifying step toward a time-evolutional transition theory of biological regulations, especially for the development of possible control
strategies for cancer cell fates. The suggested SOC model has a material counterpart made
evident by the temporal correlation between the transition in gene expression and chromatin folding/unfolding dynamics that make the regulation of large ensembles of genes
physically feasible.
The search for effective ways to induce cancer reversion is probably the most ambitious
goal to which the results described in this review could be advantageously applied. In
any case, the proposed approach is not a singular exploit. Many statistical mechanics
approaches to gene expression regulation have recently flourished, and they are essentially
convergent with our SOC model (see, for example, [35,46,47]). Of particular interest are
the multiscale approaches that combine the gene expression cell-based dynamics toward
attractor states and the dynamics of cell–cell interactions, giving rise to a complex tissue
architecture [48]. All in all, we are convinced that cell biology is entering into a new
era, which will foster a fruitful interaction between the gene-centric perspective of recent
decades with physically oriented integrative approaches to biological systems considered
according to their proper nature of interaction networks.
Author Contributions: M.T. developed the genome-engine model and SOC analysis, A.G. conducted
the multivariate analysis and the statistical formalization, P.B. critically revised the entire structure of
the paper, M.B. conceptualized the biological meaning of the results and put the dynamical analysis
in the context of cancer reversion. All authors contributed equally to the writing of the manuscript.
All authors have read and agreed to the published version of the manuscript.
Funding: This work received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Microarray data of the activation of ErbB receptor ligands in MCF7 human breast cancer cells stimulated by EGF and HRG; Gene Expression Omnibus (GEO) ID:
GSE13009.
Acknowledgments: M.T. sincerely thanks Kenichi Yoshikawa for his support on the cell-fate project
over the years, as well as the following institution and individuals who helped complete this research
project: the SEIKO Life Science Laboratory, Osaka, Japan, his family (particularly his daughters,
Kimiko and Kazumi Tsuchiya, and Harry Taylor), and D. Ikeda.
Conflicts of Interest: The authors declare no conflict of interest.
Int. J. Mol. Sci. 2023, 24, 11603
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Abbreviations
CM: Center of Mass. CP: Critical Point. CSB: Coherent Stochastic Behavior. CSK: Cytoskeleton. DMSO: di-Methyl Sulfoxide. EFA: Expression Flux Analysis. EGF: Epidermal
Growth Factor. EMT: Epithelial–Mesenchymal Transition. GA: Genome Attractor. HRG:
Heregulin. SOC (c-SOC): Self-Organized Criticality (classical Self-Organized Criticality).
nmrsf : Normalized Root Mean Square Fluctuation. NSK: Nucleoskeleton. OM (OMG):
Oscillating Mode (Oscillating-Mode Genes). PAD: Pericentromere-Associated Domain.
PCA: Principal Component Analysis.
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