Fluctuation and noise propagation in phenotypic transition cascades of clonal populations

Quantitative modeling of fluctuations of each phenotype is a crucial step towards a fundamental understanding of noise propagation through various phenotypic transition cascades. The theoretical formulas for noise propagation in various phenotypic transition cascades are derived by using the linear noise approximation of master equation and the logarithmic gain. By virtue of the theoretical formulas, we study the noise propagation in bidirectional and unidirectional phenotypic transition cascades, respectively. It is found that noise propagation in these two phenotypic transition cascades evidently differs: In the bidirectional cascade, a systemic random environment is provided by a correlated global component. The total noise of each phenotype is mainly determined by the intrinsic noise and the transmitted noise from other phenotypes. The intrinsic noise enlarged by interconversion through an added part shows a novel noise propagation mechanism. However, in the unidirectional cascade, the random environment of each downstream phenotype is provided by upstream phenotypes. The total noise of each downstream phenotype is mainly determined by the transmitted noises from upstream phenotypes. The intrinsic noise and the conversion noise can propagate in both bidirectional and unidirectional phenotypic transition cascades.

Figure 1

The bidirectional phenotypic transition cascade. A schematic diagram of transitions between species A and species B in bacterial community.

Figure 2

(a) Time courses of species A and B with the initial conditions $N_1\left(0\right)=N_2\left(0\right)=20$ at MGR $g_1=0.1$. (b) Stationary populations of species A and B as a function of MGR $g_1$ as ITR $s_1=0.001$: Lines are theoretical predictions with Eqs. (24) and (25), and solid markers are from simulations using the Gillespie method [39]. The other parameter values are given in the text. All the parameters are dimensionless.

Figure 3

Effects of MGR $g_1$ on (a) the normalized covariance $V_{12}$ and (b) the normalized variations $V_{ii}(i=1,2)$ as ITR $s_1=0.001$. Lines are theoretical predictions with the definition of $V_{ij}(i,j=1,2)$, in which the detailed expressions of $\mathbf{C}$ are given by Eqs. (26) and (28). Solid markers are from simulations using the Gillespie method [39]. The other parameter values are given in the text. All the parameters are dimensionless.

Figure 4

Effects of MGR $g_1$ on noise propagations in bacterial community, $s_1=0.001$. All the parameters are dimensionless.

Figure 5

Stationary populations of species A and B as a function of ITR $s_1$ as MGR $g_1=0.121$. All the parameters are dimensionless.

Figure 6

Effects of ITR $s_1$ on (a) the normalized covariance $V_{12}$ and (b) the normalized variations $V_{ii}(i=1,2)$ as MGR $g_1=0.121$. Lines are theoretical predictions with the definition of $V_{ij}(i,j=1,2)$, in which the detailed expressions of $\mathbf{C}$ are given by Eqs. (26) and (28). Solid markers are from simulations using the Gillespie method [39]. The other parameter values are given in the text. All the parameters are dimensionless.

Figure 7

Effects of ITR $s_1$ on noise propagations in bacterial community, $g_1=0.121$. All the parameters are dimensionless.

Figure 8

The unidirectional phenotypic transition cascade. A schematic diagram of differentiations from SCs to TACs and then to FDCs in a colonic crypt.

Figure 9

(a) Time courses of SCs, TACs, and FDCs with the initial conditions $N_1\left(0\right)=N_2\left(0\right)=N_3\left(0\right)=100$ at ITR $s_1=0.25$. (b) Stationary populations of SCs, TACs, and FDCs as a function of ITR $s_1$ as MGR $g_1=0.69$: Lines are theoretical predictions with Eqs. (44, 45, 46) and solid markers are from simulations using the Gillespie method [39]. The other parameter values are given in the text. All the parameters are measured in ${\mathrm{hours}}^{-1}$.

Figure 10

Effects of ITR $s_1$ on (a) the normalized covariance $V_{ij}$ and (b) the normalized variations $V_{ii}$ as MGR $g_1=0.69$. Lines are theoretical predictions with the definition of $V_{ij}(i,j=1,2,3)$, in which the detailed expressions of $\mathbf{C}$ are given by Eqs. (47, 48, 49, 50, 51, 52). Solid markers are from simulations using the Gillespie method [39]. The other parameter values are given in the text. All the parameters are measured in ${\mathrm{hours}}^{-1}$.

Figure 11

Effects of ITR $s_1$ on noise propagations in cell states cascade in colonic crypt, $g_1=0.69$. All the parameters are measured in ${\mathrm{hours}}^{-1}$.

Figure 12

Stationary populations of SCs, TACs, and FDCs as a function of MGR $g_1$ as ITR $s_1=0.1$. All the parameters are measured in ${\mathrm{hours}}^{-1}$.

Figure 13

Effects of MGR $g_1$ on (a) the normalized covariance $V_{ij}$ and (b) the normalized variations $V_{ii}$ as ITR $s_1=0.1$. Lines are theoretical predictions with the definition of $V_{ij}(i,j=1,2,3)$, in which the detailed expressions of $\mathbf{C}$ are given by Eqs. (47, 48, 49, 50, 51, 52). Solid markers are from simulations using the Gillespie method [39]. The other parameter values are given in the text. All the parameters are measured in ${\mathrm{hours}}^{-1}$.

Figure 14

Effects of MGR $g_1$ on noise propagations in cell states cascade in colonic crypt, $s_1=0.1$. All the parameters are measured in ${\mathrm{hours}}^{-1}$.