Optimal Proliferation and Differentiation of Chlamydia Trachomatis

Abstract

Chlamydia trachomatis is a bacterium that causes eye infection and blindness in humans. It has an unusual life cycle involving two developmental forms. Within a cytoplasmic inclusion, the reticulate body (RB) repeatedly divides by binary fission and asynchronously differentiates into the infectious elementary body (EB). Upon the death of the mammalian cell that host many such inclusions, only the EB form of the bacteria survive and proceed to infect other cells. Given the bacteria’s fast spreading infection, conventional wisdom would have the few initial EB turn into RB, divide and proliferate first, and then eventually start converting in order to maximize the terminal EB population upon host cell lysis. Several biological processes are seen as possible mechanisms for implementing such a conversion strategy. However, the optimality of an instinctual strategy with a period of proliferate without conversion prior to the onset of differentiation has never been substantiated theoretically or justified mathematically. This paper formulates three relatively simple models that capture the essential features of the Chlamydia life cycle. When the initial infection is caused by the endocytosis of a small EB population well below the carrying capacity of the host cell, the Maximum Principle requires for these models an optimal conversion strategy that confirms and rigorously justifies the prevailing view of no conversion at the early stage of the host cell infection. However, the conventional supposition is found to be inappropriate for an initial EB (-to-RB) population near or above the carrying capacity. Previously suggested and new biological mechanisms are examined for their role in implementing the different optimal conversion strategies associated with models investigated herein.

Stochastic analysis of optimal production of infectious progeny in Chlamydia

“…The optimal conversion strategy of this second model shows that maximum terminal EB population is in fact attained by a bang-bang conversion strategy. This result of the models in [5] is remarkable not only for its agreement with the empirical finding of a period of no conversion at the start of a new life cycle reported in [3], but also for buttressing the proposition that the conversion holiday is a consequence of the Darwinian force at work for the optimal proliferation and survival of the bacteria.…”

“…It has been targeted for elimination by the World Health Organization. (For references on data related to this infectious disease, see [5,3] and references cited therein)…”

“…It is our conjecture that such an unusual life cycle is the consequence of the usual Darwinian force at work to maximize the spread of the infectious disease. In our first effort test this proposition, we worked with deterministic optimal control models for a prescribed (known) terminal time for the life cycle and investigated consequences of several proliferation and conversion scenarios [5]. However, empirical data in [3] show variations of life cycle features such as terminal time and initiation of the RB-to-EB conversion.…”

“…We begin with a simple birth and death process model for the evolution of the RB and EB populations that focuses on the complementary choice of division of RB (birth) and conversion of RB to EB (death) over a finite time  before the host cell lyses. A more realistic feature (than a prescribed terminal time for the models in [5]) toward the same goal of determining the maximal terminal infectious EB population is a criterion for setting the terminal time of the proliferation-conversion cycle based on the empirical findings of [3]. In the first model, we consider only time invariant probabilities for division and RB-to-EB conversion.…”

“…, respectively, as they allow for direct comparison with the results for the deterministic optimal control problem of [5,7]. In terms of the new notations, we revert to the original un-normalized quantities and state the optimization problem as…”