Size-structured populations with growth fluctuations: Feynman--Kac formula and decoupling

Abstract

We study a size-structured population model in which individual cells grow at a rate determined by a fluctuating internal variable (e.g., gene expression levels). Many previous models of phenotypically heterogeneous populations can be viewed as special cases of this model, and it has previously been observed that the internal variable decouples from cell size under certain conditions. In this work, we generalize these results and connect them to the Feynman-Kac formula, which yields relationships between the lineage dynamics and population distribution in branching processes. To this end, we derive conditions for decoupling, both in the lineage and population ensemble. When decoupling occurs in both ensembles, the size dynamics can be transformed, via a random time change, into a growth-homogeneous process, and expectations can be evaluated through an exponential tilting procedure that follows from the Feynman-Kac formula. We further characterize weaker, ensemble-specific forms of decoupling that hold in either the lineage or the population ensemble, but not both. We provide a more general interpretation of tilted expectations in terms of the mass-weighted phenotype distribution

Figures (4)

• Figure 1: (A) A diagram of the model. A cell has size ($Y$) and growth phenotypes ($X$). Division occurs at a rate that depends on both. We have omitted the explicit dependence on the initial size in this figure for simplicity. (B) A simulation of our model in the case when $X$ is an OU process, showing the evolution of $X$ (blue) and size $Y$ (red). (C) A diagram of a growing population showing the distinction between the lineage and population distributions.
• Figure 2: (left) Simulations of three models illustrating examples of (top to bottom) (\ref{['dec:SD']}), (\ref{['dec:WD']}), and no decoupling. For all models $d=1$ and the growth rate function is $\lambda = 1 + x$. For (\ref{['dec:SD']}) we have simulated an OU process for the ${\boldsymbol X}$ dynamics with $\theta = 1$, $\sigma^2 =0.01$, $\alpha=1/2$, $\sigma_Y = 0.01$. For (\ref{['dec:WD']}) we have simulated a model with $\mathcal{L} = 0$ and $r({\boldsymbol x}|{\boldsymbol x}')$ is Normal distribution with mean $0$ and standard deviation $\sigma_x = 0.1$. The last model combines the OU dynamics with this division kernel. (middle) The convergence of the empirical correlation coefficients between size variables and $X$ in the lineage and (right) population distribution. The dashed lines show the running average of the correlations and the shaded area shows $\pm$ one standard deviation.
• Figure 3: (A) The time change converts the growth phenotype independent process $\tilde{{\boldsymbol Y}}(T)$ to ${\boldsymbol Y}(t)$. (B) The top panel shows the original size dynamics, while the right panel shows the same process when the cell size is plotted as a function of $T(t)$ rather than $t$.
• Figure 4: (left) Estimated phenotypic average $\hat{f}_p(t)$ compared to theoretical expectations $E_{\rm p}[X]$ and $\mathbb E_{\ell}[X]$, with cumulative time average shown in blue. (right) Error showing deviation of time-averaged estimate from true value and standard error. Results from $m= 100$ independent lineage simulations of OU with $T = 1000$ with $\sigma^2= 0.01$, $\theta =1$ and $\sigma_Y = 0.05$.

Theorems & Definitions (1)

• Theorem 1