Delayed self-regulation and time-dependent chemical drive leads to novel states in epigenetic landscapes

Abstract

The epigenetic pathway of a cell as it differentiates from a stem cell state to a mature lineage-committed one has been historically understood in terms of Waddington's landscape, consisting of hills and valleys. The smooth top and valley-strewn bottom of the hill represent their undifferentiated and differentiated states, respectively. Although mathematical ideas rooted in nonlinear dynamics and bifurcation theory have been used to quantify this picture, the importance of time delays arising from multistep chemical reactions or cellular shape transformations have been ignored so far. We argue that this feature is crucial in understanding cell differentiation and explore the role of time delay in a model of a single-gene regulatory circuit. We show that the interplay of time-dependent drive and delay introduces a new regime where the system shows sustained oscillations between the two admissible steady states. We interpret these results in the light of recent perplexing experiments on inducing the pluripotent state in mouse somatic cells. We also comment on how such an oscillatory state can provide a framework for understanding more general feedback circuits in cell development.

Keywords: epigenetics; gene regulatory networks; mathematical modelling.

Figure 1.

Cell differentiation in single-gene regulatory network with delay. Somatic (x = 0), induced pluripotent (x ≈ 2), and Area 51 cells in a single-gene regulatory circuit. (a) Steady-state values for equation (2.2) without drive or delay (α₀ = 0, d = 0). Depending on the initial value x(t = 0), the somatic (solid line (red)) and the iPS cells (dash-dotted line (blue)) are stable. The unstable state x = 1 (dashed line (green)) is also shown. If the initial state x(t = 0) has a value infinitesimally above the unstable state x = 1, the system transitions to the pluripotent state (+ points), while if x(t = 0) has an infinitesimally smaller value than x = 1 the system transitions to the somatic state (× points). (b) Corresponding steady states with a non-zero drive (α₀ = 0.5), a decay constant β = 0.5, and the coefficient of self-promotion α₁ = 1.0. Depending on the duration d = 2 (solid line (red)) the somatic, or d = 3 (dash-dotted line (blue)) iPS cells are chosen. (c) Shows x(t) versus t corresponding to equation (2.2) for a delay of τ = 500 and for drive d = 10 (solid line (red)), and d = 1000 (dashed line (blue)) indicating stability of somatic and iPS states. (d) Shows x(t) versus t for d = 500 with sustained fluctuations between the iPS and somatic states. (Online version in colour.)

Figure 2.

‘Area 51’ oscillations as a function of parameters. The presence of the oscillatory state for different values of the parameters α₀, α₁, that characterize the single-gene expression kinetics, the driving time d, delay time τ and the order of the Hill function n. (a) Presence of ‘Area 51’ states for parameter values n = 5, τ = 500, d = 300, α₀ = 0.5 and α₁ = 2.0. Changing the parameters α₀ = 0.6 and α₁ = 1.0, while keeping n, τ and d unchanged also shows oscillations as in (b). For a choice of parameters α₀ = 0.5, α₁ = 1.0, and n = 6 while keeping parameters τ and d same as (a) shows oscillations with accessible short lived intermediate states that lie between pluripotent and somatic fixed points. This is shown in (c). In (d) by changing the driving time to a lower value d = 100 and n = 5 while holding all other parameters same as in (c) the duration of time spent in the somatic state can be increased. (Online version in colour.)

Figure 3.

Intermediate states in cellular reprogramming. Fluctuations in the ‘Area 51’ region as a combined result of time-dependent drive d and delay τ for d = τ = 500. (a) Sustained oscillations for the parameters of figure 1d. (b,c) Indicate the oscillations in the transient (500 ≤ t ≤ 540) and sustained oscillatory (7500 ≤ t ≤ 7650) regions. (Online version in colour.)

Figure 4.

Phase diagram showing regions where somatic and pluripotent states are stable as a function of the delay time τ. The phase boundaries indicating point of no return (circles (blue) and dashed line), d_PNR, and those committed to the pluripotent state (triangles (red) and solid line), d_CPS are indicated. The region between the two states marks the region when the cell fate attains neither fixed point, but oscillates indefinitely, termed ‘Area 51’ [10]. (Online version in colour.)

Figure 5.

Phase diagram of the expression level of the gene x(t) as a function of its delayed response x(t + τ). Limit cycle behaviour for the genetic circuit for delay parameters d = 500 for the chemical drive and τ = 500 for the positive chemical feedback in equation (2.2).