Stem cell differentiation as a many-body problem

Abstract

Stem cell differentiation has been viewed as coming from transitions between attractors on an epigenetic landscape that governs the dynamics of a regulatory network involving many genes. Rigorous definition of such a landscape is made possible by the realization that gene regulation is stochastic, owing to the small copy number of the transcription factors that regulate gene expression and because of the single-molecule nature of the gene itself. We develop an approximation that allows the quantitative construction of the epigenetic landscape for large realistic model networks. Applying this approach to the network for embryonic stem cell development explains many experimental observations, including the heterogeneous distribution of the transcription factor Nanog and its role in safeguarding the stem cell pluripotency, which can be understood by finding stable steady-state attractors and the most probable transition paths between those attractors. We also demonstrate that the switching rate between attractors can be significantly influenced by the gene expression noise arising from the fluctuations of DNA occupancy when binding to a specific DNA site is slow.

Keywords: gene network; master equation; most probable path.

Fig. 1.

A schematic gene regulatory network for the embryonic stem cell. (A) Schematics for the detailed chemical reactions included in modeling the mutual repression between genes Oct4 and Cdx2. Inset simplifies the reactions into a diagram that is useful for illustrating the topology of complex networks. (B) Topology of the stem cell gene network. Each node represents either a transcription factor (single word) or a protein complex (two words) involved in early stem cell differentiation. Arrows that end at transcription factors represent transcriptional regulations with the mechanism shown in A. The green color is used for activation and the red color indicates repression. Arrows targeting protein complexes emerge from the monomers involved in forming the complex. Gene markers found in cells being in pluripotent states are colored in red and orange, and those found for differentiated state phenotypes are colored in blue.

Fig. 2.

Steady-state solutions for the embryonic stem cell network. (A) Gene expression levels in the steady-state solutions identified for the network when its dynamics are modeled in the adiabatic limit. (B) Dependence of the number of steady-state solutions, $N_{\text{ss}}$, on the DNA-unbinding rate f. (C) The probability distribution of Nanog expression at different DNA-unbinding rates.

Fig. 3.

The switching mechanism between steady states inferred from the most probable transition pathway. (A) Number of protein molecules along the most probable transition path from the stem cell steady-state SC1 to the primitive endoderm steady-state PE. The x axis indicates the progression along the transition and the y axis refers to different transcription factors. (B) The most probable differentiation pathway as in A plotted through the number of Nanog and Gata6 protein molecules. The steady-state solutions are shown as red circles. (C) Contributions to the action from individual genes along the most probable differentiation pathway from the steady-state SC1 to SC2 (Upper) and from the steady-state SC2 to PE (Lower).

Fig. 4.

The effect of DNA-binding kinetics on the switching pathway and switching rate between steady states. (A) Comparison of the most probable differentiation pathways from the steady-state SC1 to PE calculated at several different DNA-unbinding rates. The blue line is the same as the adiabatic result also shown in Fig. 3B. (B) Dependence of the transition action on the DNA-unbinding rate f. The black solid line is a numerical fit to the calculated actions (red circles), using the expression $a/\left(1/f+b\right)$, with a and b as fitting parameters. The dashed line indicates the action value obtained in the adiabatic limit.