Quantifying the Waddington landscape and biological paths for development and differentiation

Abstract

We developed a theoretical framework to prove the existence and quantify the Waddington landscape as well as chreode-biological paths for development and differentiation. The cells can have states with the higher probability ones giving the different cell types. Different cell types correspond to different basins of attractions of the probability landscape. We study how the cells develop from undifferentiated cells to differentiated cells from landscape perspectives. We quantified the Waddington landscape through construction of underlying probability landscape for cell development. We show the developmental process proceeds as moving from undifferentiated to the differentiated basins of attractions. The barrier height of the basins of attractions correlates with the escape time that determines the stability of cell types. We show that the developmental process can be quantitatively described and uncovered by the biological paths on the quantified Waddington landscape from undifferentiated to the differentiated cells. We found the dynamics of the developmental process is controlled by a combination of the gradient and curl force on the landscape. The biological paths often do not follow the steepest descent path on the landscape. The landscape framework also quantifies the possibility of reverse differentiation process such as cell reprogramming from differentiated cells back to the original stem cell. We show that the biological path of reverse differentiation is irreversible and different from the one for differentiation process. We found that the developmental process described by the underlying landscape and the associated biological paths is relatively stable and robust against the influences of environmental perturbations.

Fig. 1.

The original artistical picture of Waddington epigenetic landscape (1).

Fig. 2.

(A) The illustration of cell fate decision module. (B) The landscape of the development at different stages or different paramenters. (C) Escape time under certain fluctuations versus the parameter a and the barrier height from the central undifferentiated state to differentiated state versus the parameter a. (a = a₁ = a₂, b₁ = b₂ = 1, k₁ = k₂ = 1, and S = 0.5, n = 4).

Fig. 3.

(A) Two dimensional illustration of dominant kinetic path and flux between three basins of attraction in gene network. (B) Three dimensional illustration of dominant kinetic path and flux between three basins of attraction in gene network.

Fig. 4.

(A) Barrier heights BH from side (center) to center (side) basin with solid line (dotted line) versus fluctuations via diffusion coefficients, D. (B) Logarithm of escape time τ from center (side minimum) versus barrier height, BH. (C) RR_path from from center (side) to side (center) versus D. RR_path represents the ratio of the weights between the path at fluctuation strength D compared with the low fluctuation strength D₀ = 0.01. (D) RR_path from versus the barrier height BH.

Fig. 5.

The quantified Waddington developmental landscape and pathways (a = a₁ = a₂, b₁ = b₂ = 1, k₁ = k₂ = 1, S = 0.5, n = 4, and λ = 0.01).