Cell-type switches induced by stochastic histone modification inheritance

Figure 1.  Illustration of gene regulation combines a positive feedback with stochastic histone modification at the enhancer region. Histone modifications alter the 50% effective concentration (EC50) of the toggle switch to regulate gene expression (see Formulations for details)

Figure 2.  The bifurcation diagrams with respect to $ K $ and $ u $. (A) The protein level at steady state as a function of $ K $. Blue dashed lines show the saddle-node bifurcations ($ K^* $ and $ K^{**} $). (B) The heat map representation of joint density distribution of the protein level and $ u $ for a typical trajectory under random inheritance of histone modifications and without the external noise. In the simulation, $ \varphi(u_k) = a + b (u_k - a) $ and $ a = 0.5, b = 0.8 $ (to be detailed in Section 3.2)

Figure 3.  Cell-type switches induced by external noise. (A) Sample dynamics of protein level in $ 10^5 $ cycles. In the simulation, $ K = 450 $, $ \sigma = 1.8 $. (B) The heat map representation of the switching frequency with respect to $ K $ and $ \sigma $. (C) A sample dynamics of $ \lambda_p $ with $ \sigma = 1.5 $. Inset is the probability distribution of $ \ln(\lambda_p) $. (D) The probability distribution of the steady state lifetimes with respect to the trajectory in (A)

Figure 4.  Dynamics of nucleosome states under random histone modification inheritance. (A) A sample dynamics of the nucleosome state along with cell cycles. (B) The distribution (blue) of the sequence $ \{u_k\} $ with three sets of parameters of $ a $ and $ b $, and the Beta distribution density function (red) (16) with shape parameters $ (\alpha, \beta) $ given by (17)

Figure 5.  Frequencies of cell-type switches with histone modification to affect the transcription activities. (A) The heat map representation of the switching frequency with respect to $ a $ and $ b $ without the external noise. (B) The switching frequency as a function of $ a $ with varying parameter $ b $. (C) The heat map representation of the switching frequency with respect to $ b $ and the noise strength $ \sigma $ (here $ a = 0.5 $). (D) The heat map representation of the switching frequency with respect to $ a $ and the noise strength $ \sigma $ (here $ b = 0.7 $)

Figure 6.  Dynamic equilibrium of cell regeneration. Figures show the probability distribution of the protein level at the end of cycle $ 0 $, $ 5 $, $ 10 $, $ 15 $, and $ 20 $. Results for 4 sample simulations ((A)-(D)) are shown, each with different initial conditions. Parameters are $ a = 0.5, b = 0.7 $

Figure 7.  Stationary distribution of proteins in a cell colony. (A)The probability distribution of the protein level at the end of $ 20 $ cycles with fixed $ b = 0.7 $ and varied $ a $: $ a = 0.4 $, $ a = 0.5 $, and $ a = 0.6 $. (B) The probability distribution of the protein level at the end of $ 20 $ cycles with fixed $ a = 0.5 $ and varied $ b $: $ b = 0.6 $, $ b = 0.7 $, and $ b = 0.8 $. We take $ \sigma = 0.5 $ in simulations

Figure 8.  Potential landscapes. (A)The potential landscape (23) as a function of protein level with $ K = 200 $, $ K = 450 $, and $ K = 750 $. In the simulation, $ \sigma = 0.1 $. (B)The Waddington landscape of colony evolution from single cells. The potential was obtained from simulations results of 30 independent trajectories. Parameters are $ a = 0.5, b = 0.7 $

Figure 9.  Dynamics of intermediate cell states. (A) A sample dynamics of transitions between three cell states based on the assumption (24). (B) The distribution of nucleosome state $ u_k $. (C) The distribution of the protein levels along the trajectory. In the simulation, we took $ m = 6 $, $ c = 0.25 $, $ d = 0.5 $, $ u_0 = 0.488 $, $ n = 2.5 $, $ \gamma_1 = 0.499 $, $ \gamma_2 = 6.471 $, $ \sigma = 0.5 $, and the other parameters are the same as those in Table 1