Fitness response relation of a multitype age-structured population dynamics

We construct a pathwise formulation for a multitype age-structured population dynamics, which involves an age-dependent cell replication and transition of gene- or phenotypes. By employing the formulation, we derive a variational representation of the stationary population growth rate; the representation comprises a tradeoff relation between growth effects and a single-cell intrinsic dynamics described by a semi-Markov process. This variational representation leads to a response relation of the stationary population growth rate, in which statistics on a retrospective history work as the response coefficients. These results contribute to predicting and controlling growing populations based on experimentally observed cell-lineage information.

Figure 1

(a) A schematic illustration of single-cell tracing by the mother machine. (b) A schematic diagram of the stochastic laws of the single-cell dynamics. The interdivision interval and the type transition are governed by $\pi \left(\tau \right|x)$ and $\mathbb{T}\left(x|{\tau }^{\prime },x^{\prime }\right)$, respectively. (c) A schematic illustration of the transition dynamics of type $x\left(t\right)$ (the upper panel) and the age $a\left(t\right)$ (the lower panel). Upon a division event, the type transition occurs and the age resets to 0. Also, we denote an age of a mother cell at $i\mathrm{th}$ division event by ${\tau }_i$.

Figure 2

(a) A schematic illustration of the dynamics cytometer. (b), (c), (d) Schematic images of the cell death, division, and death upon division, respectively. (e) A schematic illustration of the lineage tree generated by the MTASP. Such a lineage tree can be observed by the the dynamics cytometer. The gray color denotes cell death. The dashed arrow represents a retrospective tracking.

Figure 3

(a) A schematic diagram of the two state model of the cells used in the simulation. The left panel shows the values of the state-switching probability and its stationary distribution: ${\mathrm{\Sigma }}_{x^{\prime }\in \mathrm{\Omega }}\mathbb{T}\left(x|x^{\prime }\right){\rho }^{\mathrm{s}\mathrm{t}}\left(x^{\prime }\right)={\rho }^{\mathrm{s}\mathrm{t}}\left(x\right)$. The right panel represents the distributions of the interdivision intervals and the death intervals for the two states of the cells. (b) A lineage tree obtained by the conducted simulation. The first ${10}^4$ cells are shown in the lineage tree. (c) A schematic diagram of the state-switching probability, the distribution of the interdivision interval, and that of the death interval for the retrospective process. The left panel shows the values of the transition probability for the retrospective process and the corresponding stationary distribution: ${\mathrm{\Sigma }}_{x^{\prime }\in \mathrm{\Omega }}{\mathbb{T}}_B\left(x|x^{\prime }\right){\rho }_B^{\mathrm{s}\mathrm{t}}\left(x^{\prime }\right)={\rho }_B^{\mathrm{s}\mathrm{t}}\left(x\right)$. In the right panel, the solid curves are calculated by the analytic expressions, Eq. (54), and the histograms are empirically obtained from a retrospective path from the lineage tree.

Figure 4

The responses of the stationary population growth rate and the response coefficients for the perturbations to $\pi \left(\tau |\mathcal{R}\right)$ (a), (c) and $\pi \left(\tau |\mathcal{B}\right)$ (b), (d). (a), (b) The actual responses of the growth rate (points) and the predicted responses (lines) respectively to the perturbations for the state $\mathcal{R}$ (a) and the state $\mathcal{B}$ (b). The parameters are perturbed around the same parameter values as in Eq. (50). (c), (d) The response coefficient and the changes in $\pi \left(\tau \right|x)$ induced by the parameter perturbations of $\alpha \left(\mathcal{R}\right)$ and $\beta \left(\mathcal{R}\right)$ (c) and $\alpha \left(\mathcal{B}\right)$ and $\beta \left(\mathcal{B}\right)$ (d).

Figure 5

The responses of the stationary population growth rate and the response coefficients for the perturbations to $\gamma \left(a,\mathcal{R}\right)$ (a), (c) and $\gamma \left(a,\mathcal{B}\right)$ (b), (d). (a), (b) The actual responses of the growth rate (points) and the predicted responses (lines) respectively to the perturbations for the state $\mathcal{R}$ (a) and the state $\mathcal{B}$ (b). The parameters are perturbed around the same parameter values as in Eq. (50). (c), (d) The response coefficient and the changes in $\gamma (a,x)$ induced by the parameter perturbations of ${\alpha }_{\gamma }\left(\mathcal{R}\right)$ and ${\beta }_{\gamma }\left(\mathcal{R}\right)$ (c) and ${\alpha }_{\gamma }\left(\mathcal{B}\right)$ and ${\beta }_{\gamma }\left(\mathcal{B}\right)$ (d).

Figure 6

The responses of the stationary population growth rate and the response coefficients for the perturbations to $\mathbb{T}$. (a), (b) The actual responses of the growth rate (points) and the predicted responses (lines) respectively to the perturbations of ${\theta }_{\mathcal{R}}$ (a) and ${\theta }_{\mathcal{B}}$ (b). (c) The changes in the value of the components in $\mathbb{T}\left(x\right|x^{\prime })$ as a function of either ${\theta }_{\mathcal{R}}$ or ${\theta }_{\mathcal{B}}$. (d) The response coefficient $j_B^{*}\left(x,x^{\prime }\right)$. (e) The changes in the components of $\mathbb{T}\left(x\right|x^{\prime })$ induced by the parameter perturbations of ${\theta }_{\mathcal{R}}$ or ${\theta }_{\mathcal{B}}$. The components not shown in the chart are zero.

Figure 7

A schematic illustration of the lineage tree ${\mathcal{T}}_T$. The root cell is one of the daughters generated from the mother cell with $\left({\tau }_0,x_0\right)$. The paths highlighted by the dark gray color represent surviving genealogical paths with history ${\left\{{\mathit{x}}_i\right\}}_{i=1}^{n+1}$. The lineage tree ${\mathcal{T}}_T$ highlighted by the light gray color has $N$ surviving genealogical paths. $P_T^{{\left\{{\mathit{x}}_i\right\}}_{i=1}^{n+1}}\left(N|{\tau }_0,x_0\right)$ represents the conditional probability that such a lineage tree ${\mathcal{T}}_T$ appears.