Smoothing in linear multicompartment biological processes subject to stochastic input

Many physical and biological systems rely on the progression of material through multiple independent stages. In viral replication, for example, virions enter a cell to undergo a complex process comprising several disparate stages before the eventual accumulation and release of replicated virions. While such systems may have some control over the internal dynamics that make up this progression, a challenge for many is to regulate behavior under what are often highly variable external environments acting as system inputs. In this work, we study a simple analog of this problem through a linear multicompartment model subject to a stochastic input in the form of a mean-reverting Ornstein-Uhlenbeck process, a type of Gaussian process. By expressing the system as a multidimensional Gaussian process, we derive several closed-form analytical results relating to the covariances and autocorrelations of the system, quantifying the smoothing effect discrete compartments afford multicompartment systems. Semianalytical results demonstrate that feedback and feedforward loops can enhance system robustness, and simulation results probe the intractable problem of the first passage time distribution, which has specific relevance to eventual cell lysis in the viral replication cycle. Finally, we demonstrate that the smoothing seen in the process is a consequence of the discreteness of the system, and does not manifest in systems with continuous transport. While we make progress through analysis of a simple linear problem, many of our insights are applicable more generally, and our work enables future analysis into multicompartment processes subject to stochastic inputs.

Figure 1

Multicompartment model of viral replication subject to stochastic input. (a) The general viral replication cycle. Virions enter a cell and progress through a multistage process, before accumulating following repackaging. (b) We study a linear analog of the virus problem, namely a system subject to random input, $I\left(t\right)$, modeled as an Ornstein-Uhlenbeck process with mean $\mu$, reversion strength $\theta$, and noise magnitude $\sigma$. A realization of the input is shown in panel (c). Material then progresses through $n$ compartments at constant rate $k$. (d) A realization of the system in panel (b), where $X_{\nu }\left(t\right)$ models the concentration of material in each compartment $\nu =1,2,\cdots ,6$. It is evident that passage through the compartments has a smoothing effect.

Figure 2

Variance and autocorrelation function for the linear multicompartment model. (a) Stationary standard deviation as a function of compartment number [compartment $\nu =0$ refers to $I\left(t\right)$]. Shown are mean $\pm$ std from numerical simulations constructed from 10 replicates of 100 simulations (gray), and the corresponding analytical solution (color). (b) Simulated and analytical ACFs for compartment $\nu =3$. (c) Second derivative of the ACF at $\ell =0$ for $\theta =k=1$ calculated directly from Eq. (14) (diamonds), and using the analytical expression in Eq. (16) (solid). (d–f) Analytical ACFs for $I\left(t\right)$ (gray dashed) and $X_{\nu }\left(t\right)$ (color of increasing brightness for $\nu =1,2,\cdots ,6$). Unless otherwise stated, the other parameters are fixed at $\theta =\mu =k=1$, and $\sigma =0.5$.

Figure 3

Adjustment to stationary variance due to intrinsic noise. Stationary standard deviation as a function of compartment number for (colored curves) systems of various system sizes. The $V\to \infty$ limit (gray solid) corresponds to Eq. (13). Also shown are mean $\pm$ std from numerical simulations constructed from 20 replicates of 200 simulations (gray), and asymptotic approximations constructed using Stirling's formula (black dashed). Other parameters are fixed at $\theta =\mu =k=1$, and $\sigma =0.5$.

Figure 4

First passage time distributions for the linear multicompartment model. (a), (d) Realizations of a three compartment system initiated using (a) the fixed initial condition and (d) the partially fixed initial condition. Solutions are terminated at $t=\tau :X_3\left(\tau \right)>a$, yielding $\tau$ as the FPT. (b), (e) Distribution function for the FPT constructed from (color) 1000 realizations of the SDE and (dashed black) a finite difference solution to Eq. (21). (c), (f) Mean, 2.5% quantile, and 97.5% quantile for the FPT distribution constructed from (gray) 1000 realizations of the SDE and (black) a finite difference solution to Eq. (21). Shown in red-dashed is an approximation to the mean FPT constructed by scaling the FPT for $\nu =1$ based on matching the second-derivative of the autocorrelation function [Eq. (16)]. The barrier for each compartment is located at $a=\tilde{a}{\sigma }_{\nu }$ where $\tilde{a}=1$. Other parameters are fixed at $\theta =\mu =k=1$, and $\sigma =0.5$.

Figure 5

Cumulative production at the first passage time. We investigate the cumulative production at the first passage time, denoted by $A\left(\tau \right)={\int }_0^{\tau }k{X}_6\left(t\right)dt$, for a six compartment system. (a) Relationship between $\tau$ and $A\left(\tau \right)$ based on 1000 replicates of the SDE with $\tilde{a}=k=1$. We show both a scatter plot of the joint density and kernel density estimates constructed from the marginals. (b), (c) Mean, 2.5% quantile, and 97.5% quantile of $A\left(\tau \right)$ constructed from 1000 replicates of the SDE as (b) $\tilde{a}$ is varied, and (c) $k$ is varied.

Figure 6

System with nonlocal feedback. We investigate the effect of a nonlocal feedback (or feedforward) of magnitude $\varepsilon$ from compartment $n$ to compartment $m$ on (b,c) the stationary standard deviation of $X_6\left(t\right)$, denoted ${\sigma }_6$, and (d,e) the magnitude of the ACF curvature, denoted $|{\rho }_6^{″}\left(0\right)|$. In all cases, a reduction in each statistic corresponds to a potentially more robust system.

Figure 7

Material transport through a system approaching the continuum limit. (a) Compartments in the discrete system are divided into subcompartments of “length” $\mathrm{\Delta }$, while the “transfer density” is kept fixed. (b)–(d) Solutions of the discrete system for decreasing $\mathrm{\Delta }$. All systems are subject to identical input $I\left(t\right)$ (gray). In panel (d), the shifted input $I(t-10)$ is shown at $\nu =10$ for comparison with the transported concentrations. Other parameters are fixed: $\theta =\mu =k=1$ and $\sigma =0.5$. (e) Solution to the continuum limit approximation [Eq. (27)] for $\mathrm{\Delta }=1$.