Stochastic Chlamydia Dynamics and Optimal Spread

Abstract

Chlamydia trachomatis is an important bacterial pathogen that has an unusual developmental switch from a dividing form (reticulate body or RB) to an infectious form (elementary body or EB). RBs replicate by binary fission within an infected host cell, but there is a delay before RBs convert into EBs for spread to a new host cell. We developed stochastic optimal control models of the Chlamydia developmental cycle to examine factors that control the number of EBs produced. These factors included the probability and timing of conversion, and the duration of the developmental cycle before the host cell lyses. Our mathematical analysis shows that the observed delay in RB-to-EB conversion is important for maximizing EB production by the end of the intracellular infection.

Appendices

Appendix

A: Summary and Description of the Six Relevant Models

As the present work is the third report of our work to examine the development of C. trachomatis and its bio-theoretic foundation, we summarize below all the models involved in the discussion herein, both new and previously analyzed, and clarify their relation to each other.

• The GD-Model: The data of Lee et al. (2018) reveal a complex developmental cycle that features repeated divisions of an RB form of the bacterium and the conversion of RB to the EB form that survives host cell lysis to infect other human cells and spread the bacteria. A probabilistic model with gamma distributions assigned to the elapsed times between different state transitions, known as the GD-Model herein, was formulated in Lee et al. (2018) to show the empirical findings can be faithfully replicated by a mathematical model based on a few simple theoretical ingredients.

• The PT-Model: The GD-Model reproduces the features of the chlamydial development observed but does not provide a bio-theoretic basis for these features. To address the question what role natural selection plays in these features, a deterministic PT-Model for the evolution of the two (lumped) C. trachomatis populations in a cytoplasmic inclusion was formulated in Wan and Enciso (2017) to focus on the alternative cell fate choices of an RB division into two RBs or an RB conversion into an EB. The resulting RB-to-EB conversion strategy that maximizes the EB population at a known terminal time (without any built-in mechanism to induce the observed conversion holiday at the start of the developmental cycle) is qualitatively the same as that found empirically in Lee et al. (2018).

• The BD-Model: The variability of the many features of the chlamydial development among the collection of infected cells examined necessitates formulation and analysis of probabilistic models. The alternative choice of division and conversion is captured by a simple birth and death process model (the BD-Model) for the RB and EB populations in Sect. 2. This BD-Model provides a stepping stone to our two approaches to the optimal control problem for our posit that the development cycle is the optimal strategy for maximizing the spread of the bacterial infection.

• The RT-Model: With data available for the lysis time of the infected cells examined, one approach is to treat T as a random variable with a probability distribution (or a density function) estimated from the data. This leads to the stochastic optimal control RT-Model of Sect. 4.

• The WS-Model: An alternative approach assumes the variability of lysis time among host cells to be a consequence of the uncertain environment experienced by different host cells and the resulting in random elapsed times transitioning from an RBs to two smaller RBs (after a division) or to an EB (through a conversion) captured by the BD-Model. The lysis time for each host cell in this WS-Model is, as seen from Sect. 3, determined by a threshold condition of a weighted sum of terminal RB and EB populations.

• The MP-Model: The highly idealized 2-form models are too coarse-grained for matching with available developmental data. A qualitative difference is observed between the theoretical result and the data reported in Lee et al. (2018) on the post conversion holiday growth of the RB population. The discrepancy is removed by a four-form model, the MP-Model of Sect. 6.

B: Method of Characteristics for Generating Functions

The characteristic ODE of the first order PDE for $G(x,\tau )$ are

$$\begin{aligned} \frac{\mathrm{d}x}{\mathrm{d}\tau }=-\rho +(1+\rho )x-x^{2}, \ \ \ \ \ \ \ \ \frac{\mathrm{d}G}{\mathrm{d}\tau } =0. \end{aligned}$$

The Riccati equation for $x(\tau )$ can be written as

$$\begin{aligned} \frac{\mathrm{d}x}{\mathrm{d}\tau }=(1-x)(x-\rho ). \end{aligned}$$

with $x_{p}(\tau )=1$ as a particular solution. The decomposition

$$\begin{aligned} x(\tau )=1+\frac{1}{z(\tau )} \end{aligned}$$

transforms the Riccati equation into a linear ODE

$$\begin{aligned} \frac{\mathrm{d}z}{\mathrm{d}\tau }-\left( 1-\rho \right) z=1. \end{aligned}$$

with an exact solution

$$\begin{aligned} z(\tau )=e^{-f(\tau )}\left\{ \frac{1}{x_{0}-1}+I(\tau )\right\} \end{aligned}$$

where as given in (9)

$$\begin{aligned} f(\tau )=-\tau +\int _{0}^{\tau }\rho (\xi )\mathrm{d}\xi , \ \ \ \ \ \ \ I(\tau )=\int _{0}^{\tau }e^{f(\zeta )}\mathrm{d}\zeta , \end{aligned}$$

and $x_{0}$ is a constant of integration. The corresponding solution for $x(\tau )$ is

$$\begin{aligned} x(\tau )=1+\frac{1}{z(\tau )}=1+\frac{\left( x_{0}-1\right) e^{f(\tau )}}{1+\left( x_{0}-1\right) \ I(\tau )} \end{aligned}$$

(85)

with

$$\begin{aligned} x(0)=x_{0}. \end{aligned}$$

The solution for the other characteristic ODE is

$$\begin{aligned} G(x,\tau )=G_{0}=x_{0}^{N} \end{aligned}$$

(86)

since there are exactly N RB units initially so that

$$\begin{aligned} \left[ G(x,\tau )\right] _{\tau =0}=G(x_{0},0) =\sum _{k=0}^{\infty }P_{k}(0)x_{0}^{k}=x_{0}^{N}. \end{aligned}$$

To complete the solution, we solve (85) for $x_{0}$ in terms of x and $\tau$ to get

$$\begin{aligned} x_{0}=1+\frac{x-1}{e^{f(\tau )}-(x-1)I(\tau )} \end{aligned}$$

(87)

Upon using this expression for $x_{0}$ in (86), we obtain

$$\begin{aligned} G(x,\tau )=x_{0}^{N}=\left\{ 1+\frac{x-1}{e^{f(\tau )}-(x-1)I(\tau )}\right\} ^{N}. \end{aligned}$$

(88)

C: Proof of Proposition 4

Proof

The optimal control $u_{op}(t)$ must be the lower corner control 0 at least in a small interval $(t_{0},t_{s}]$ adjacent to the switch point. If not and $u_{op}(t)=u_{\max }$ for $0\le t_{0}<t\le t_{s},$ then

$$\begin{aligned} \lambda ^{\prime }(t_{s})=u_{\alpha }\lambda (t_{s})-u_{\max }=-\alpha <0 \end{aligned}$$

so that $\lambda (t)$ is a decreasing function of t in some small neighborhood of $t_{s}$. With $1-\lambda \left( t\right) \le 0$ for $t\le t_{s}$, the upper corner control $u_{\max }$ does not maximize the Hamiltonian at least for t in that neighborhood; hence, $u(t)=u_{\max }$ is not optimal there. Since the singular solution does not apply, we are left with the only option of $u_{op}(t)=0$ in that neighborhood. In that case, the adjoint DE (31) and the continuity of the adjoint function require

$$\begin{aligned} \lambda ^{\prime }=-\alpha \lambda _{.}\ \ \ \ \ \lambda (t_{s})=1 \end{aligned}$$

and therewith

$$\begin{aligned} \lambda (t)=e^{-\alpha (t-t_{s})},\ \ \ \ \ (t\le t_{s}). \end{aligned}$$

With this, the Hamiltonian,

$$\begin{aligned} H(u)=\alpha \lambda (t)R(t)+\left[ 1-e^{-\alpha (t-t_{s})}\right] u(t)R(t),\ \ \ \ \ (t\le t_{s}), \end{aligned}$$

(89)

is maximized by the lower corner control $u_{op}(t)=0$ for all t in the interval $[0,t_{s})$. $\square$

D: Proof of Proposition 7

Proof

With the Euler BC $\lambda _{g}(T_{2})=0$, the Hamiltonian reduces to

$$\begin{aligned} H_{s}(T_{2})=u(T_{2})R_{g}(T_{2})F_{c}(T_{2}) \end{aligned}$$

• (i) For the general case with $F_{c}(T_{2})>0$, $H_{s}(T_{2})$ is maximized by the upper corner control so that $u_{op}(T_{2})=u_{\max }$. Given

  $$\begin{aligned} \lambda _{g}(T_{2})=0,\ \ \ \ \lambda _{g}^{\prime }(T_{2}) =-u_{\max }F_{c}(T_{2})<0, \end{aligned}$$

  we have $\lambda _{g}(t)\ge 0$ but, by continuity, $\lambda _{g}(t)<F_{c}(t)$ for some interval $(t_{s},T_{2}]$ adjacent to $T_{2}$. It follows from $R(t)>0$ that $u_{op}(t)=u_{\max }$ at least in $(t_{s},T_{2}]$ with $t_{s}$ being the root of (66) nearest to (but still <) $T_{2}$.

• (ii) For the case $F_{c}(T_{2})=0$ (and $F_{c}(t)>0$ for $t<T_{2}$), we have

  $$\begin{aligned} \left[ \left\{ \lambda _{g}-F_{c}\right\} ^{\prime }\right] _{t=T_{2}}= & {} u_{\alpha }\left[ \lambda _{g}-F_{c}\right] _{t=T_{2}} -\alpha F_{c}(T_{2})+f_{T}(T_{2}) \\= & {} u_{\alpha }\left\{ \lambda _{g}-F_{c}\right\} _{t=T_{2}}+f_{T}(T_{2}) \ge u_{\alpha }\left\{ \lambda _{g}-F_{c}\right\} _{t=T_{2}} \end{aligned}$$

  so that $\left[ F_{c}(t)-\lambda _{g}(t)\right]$ is a decreasing function of t but remains $>0$ at least in some interval $(t_{s},T_{2}]$. It follows from (61) that we have $u_{op}(t)=u_{\max }$ at least in $(t_{s},T_{2}]$ with $t_{s}$ being the root of (66) nearest to (but still <) $T_{2}$.

$\square$

E: Proof of Proposition 8

Proof

We have already $u_{op}(t)=u_{\max }$ for $t_{s}<t\le T_{2}$ from Proposition 7 where the switch point $t_{s}$ is the root of (70) nearest to $T_{2}$. We also learned from the development prior to this proposition that $u_{op}(t)$ cannot be $u_{\max }$ or the singular solution for $t\lesssim t_{s}$.

For the lower corner control in that range, the corresponding adjoint function, denoted by $\lambda _{\ell }(t)$, is determined by

$$\begin{aligned} \lambda _{\ell }^{\prime }=-\alpha \lambda _{\ell }, \ \ \ \ \ \lambda _{\ell }(t_{s})=F_{c}(t_{s}). \end{aligned}$$

The exact solution is

$$\begin{aligned} \lambda _{\ell }(t)=F_{c}(t_{s})e^{-\alpha (t-t_{s})}. \end{aligned}$$

• (i) For $t_{s}\le T_{1}$, we have $F_{c}(t)=1$ and $\lambda _{\ell }(t)=e^{\alpha (t_{s}-t)}$ so that $1-e^{\alpha (t_{s}-t)}<0$ for all $t<t_{s}$ and

  $$\begin{aligned} H_{s}(t)=u(t)R(t)\left[ 1-e^{\alpha (t_{s}-t)}\right] +\alpha R(t)e^{\alpha (t_{s}-t)}, \end{aligned}$$

  is maximized by $u_{op}(t)=0$ there.

• (ii) For $t_{s}>T_{1}$, $F_{c}(t)$ decreases only linearly with increasing t while $\lambda _{\ell }(t)$ decay exponentially with $\lambda _{\ell }(t_{s})=F_{c}(t_{s})$ so that $u_{op}(t)=0$ is also optimal for $t<t_{s}$.

$\square$