Fig 1.
Physiological and pathological cell fate reprogramming: A mathematical approach.
Reprogramming-like phenomena in response to damage signalling may constitute a reparative route through which human tissues respond to injury, stress, and disease via induction of a transient acquisition of epigenetic plasticity and phenotypic malleability. However, tissue regeneration/rejuvenation should involve not only the transient epigenetic reprogramming of differentiated cells, but also the committed re-acquisition of the original or alternative committed cell fate. Chronic or unrestrained epigenetic plasticity would drive ageing/cancer phenotypes by impairing the repair or the replacement of damaged cells; such uncontrolled phenomena of in vivo reprogramming might also generate cancer-like cellular states. Accordingly, we now know that chronic senescence-associated inflammatory signalling (SAIS) might lock cells in highly plastic epigenetic states disabled for reparative differentiation and prone to malignant transformation. We herein introduce a first-in-class stochastic, multiscale reduction method of combined epigenetic regulation (ER)-gene regulatory network (GRN) to mathematically model and computationally simulate how ER heterogeneity regulates the entry-exit mechanisms and kinetics of physiological and pathological cell fate reprogramming.
Fig 2.
Schematic reprentation of the ER-GRN model and its multiscale reduction.
(a): Gene regulatory network (GRN) of two self-activating, mutually-inhibitory genes with epigenetic regulation (ER). In the GRN model, the gene product (single circle, denoted by Xi in S2 Table) is its own transcription factor which, upon dimerisation (two joint circles), binds the promoter region of the gene thus triggering gene transcription. The transition rates corresponding to this GRN are given in S2 Table. For simplicity, we use an effective model in which the formation of the dimer and binding to the promoter region is taken into account in a single reaction, and the resulting number of promoter sites bound by two transcription factors is denoted by Xij (see S2 Table). Furthermore, depending on whether the epigenetic state is open (i.e. predominantly acetylated (A)) or closed (i.e. predominantly methylated (M)) the promoter region of the gene is accessible or inaccessible to the transcription factor, respectively. (b): Schematic representation of the time separation structure of the multiscale method developed to simulate the ER-GRN system. See text and S1 Text for more details.
Fig 3.
Phase diagram of the two-gene system, Eqs (5) and (6).
Vertical blue (horizontal green) hatching denotes regions where the pluripotency (differentiated) state is stable. Diagonal pink hatching denotes regions where the undecided state is stable. Regions of the phase diagram where different hatchings overlap correspond to regions of bistability or tristability. In the labels in the plot, P stands for pluripotency, D stands for differentiation and U for undecided. This phase diagram was obtained using the methodology formulated in [57]. Parameter values: ω11 = ω21 = 4.0. Other parameter values as per S7 Table.
Fig 4.
Scatter plots showing heterogeneity in the behaviour of bistable (a)differentiation ER systems (DERSs) and (b)pluripotency ER systems (PERSs).
The vertical axis corresponds to the average opening time and the horizontal axis, to the average closing time. Each dot in plot (a) represents a DERS within the ensemble (see Section ER-system ensemble generation and parameter sensitivity analysis). We analyse a total of 90 DERS parameter sets and 100 PERSs. The red cluster includes 31 sets, the green cluster contains 13 sets, and the blue cluster has 46 sets. Different colours and black lines show the three clusters resulting from a k-means analysis discussed in Sections Co-factor heterogeneity gives rise to both pluripotency-locked and differentiation-primed states and Analysis of ensemble heterogeneity. Dots in plot (b) represent PERSs within the ensemble defined in Section ER-system ensemble generation and parameter sensitivity analysis.
Table 1.
Minimum action values, , corresponding to the optimal escape paths shown in S4 Fig (for details, see Section Transitions between ER states: Minimum action path approach and Section Co-factor heterogeneity gives rise to both pluripotency-locked and differentiation-primed states for details).
Parameter values are given in S5 Table.
Fig 5.
Empirical CDFs for the whole ensemble of DERS parameter sets (magenta lines) for those c1j significantly different between clusters.
This ensemble has been generated according to the methodology explained in Section ER-system ensemble generation and parameter sensitivity analysis (see also [16]). We also show the partial empirical CDFs corresponding to each of the clusters from Fig 4(a) (red, green, and blue lines). For reference, we also show the CDF for a uniform distribution (black line).
Fig 6.
Empirical CDFs for the DERS parameter sets within the blue cluster for those cij significantly different.
This ensemble has been generated according to the methodology explained in Section ER-system ensemble generation and parameter sensitivity analysis (see also [16]). The DERSs within the blue cluster have been divided into two subsets: those such that (SC-locked, blue lines) and those such that
(non-SC-locked, orange lines), with T = 0.7. For comparison, we plot the CDFs of the whole DERS ensemble (magenta lines), and, for guidance the CDF corresponding to a uniformly distributed random variable (black lines).
Fig 7.
Effect of the different reprogramming strategies of blue cluster DERSs, as evaluated in terms of the statistics of the differentiation time (τD).
(a) Two step reprogramming is illustrated by the green square (first step), which finally becomes the red square (second step). One step reprogramming is depicted as the red diamond (see Section Ensemble-based strategies for unlocking resilient pluripotency for details). (b) Comparison of τD for the original DERS and the ones resulting from the reprogramming strategies. We consider a base-line scenario where the number of HMEs is exactly equal to average, i.e. eHDM = eHDAC = Z. We then compare the simulation results obtained for different scenarios regarding the different strategies to the base-line scenario. Parameter values: Z = 5 and Y = 15. Other parameter values given in S6 Table.
Fig 8.
Plots showing the effect of the variation of HDM and HDAC on the statistics of the differentiation time (τD).
We consider a base-line scenario where the number of HMEs is exactly equal to average, i.e. eHDM = eHDAC = Z. We then compare the simulation results obtained for different scenarios regarding the abundance of HDM and HDAC to the base-line scenario, i.e. by changing the values of (eHDM, eHDAC). Parameter values: Z = 5 and Y = 15. Other parameter values given in S5 Table.
Fig 9.
Strategies to unlock pluripotent stem-like states in ageing and cancer.
Epigenetic regulation heterogeneity of differentiation genes (DERSs), but not that of pluripotency genes (PERSs), was predominantly in charge of the entry and exit decisions of the pluripotent stem-like states (blue). The application of the hybrid numerical method validated the likelihood of epigenetic heterogeneity-based strategies capable of unlocking and directing the transit from differentiation-refractory to differentiation-primed (red) epistates via kinetics changes in epigenetic factors. (Note: The epigenetic parameters regulating the entry into robust epi-states throughout the entire ER-GRN system revealed a regime of tristability in which pluripotent stem-like (blue) and differentiated (red) steady-states coexisted with a third indecisive (green) state). (R: Recruited; U: Unrecruited).