Modeling

Investigation of existing literature related to the Gluc-o-Gone project resulted in the discovery of several papers which directly describe that which the uOttawa iGEM team worked on completing. Thus, we present the most relevant modelling done to date which describes our system.

Background

Two different regimes for bacterial growth in a reactor setting are typically investigated; chemostat and plug-flow ([1]-[5]).
In the chemostat regime, a feed stream feeds a reactor vessel which is assumed to be well-mixed and an effluent stream drains the vessel at a rate which keeps the volume constant [7]. This model type is highly effective for bioreactors in which an axial or radial flow impeller is employed, assuring well-mixed, homogenous conditions and it is therefore well-studied. Typically, chemostat models for two competing bacterial strains growth-limited conditions result in washout of one of the species [1]. However, chemostat models are not the most effective for modelling bacterial growth in the gut, as it is neither well-mixed nor of constant volume, which are the two key properties on which chemostat modelling is based.
In the plug-flow regime, it is assumed that the velocity profile of the fluid moving through an infinitesimal cylindrical volume (as in a pipe) is constant over the cross-sectional area. This is not precisely true, as the velocity profile of a fluid is typically parabolic under laminar conditions and damped parabolic under turbulent conditions [7]. Bacterial growth models in plug-flow regimes are of importance as they can predict the ability of a bacterial strain to colonize and foul pipes connecting bioreactors.

Our System

Flow in the gut is neither constant nor laminar in nature; typical human diets do not allow for constant inflow or outflow streams, and peristalsis in the intestines creates turbulent flow due to continual expansion and contraction. However, the best model for Gluc-o-Gone is that of two-strain bacterial competition and wall-growth in a plug-flow regime. In this case, the intestines are approximated by a three dimensional cylinder filled with nutrient-rich liquid. Bacteria are assumed to be either planktonic or wall-attached and the biofilm formed by the bacteria is assumed to have zero thickness so as not to disturb flow. This reduces the complexities posed by the turbulent, batch-like nature of a real intestine that would render it nearly impossible to model.
The model that best fits this scenario is given by Jones and Smith.In their model, Jones and Smith examine the effects of adding an invading species into a plug flow reactor that is already colonized by one strain of bacteria under growth-limiting conditions.
They assume limited wall space, which may not apply to our system in particular, as the actual area occupied by gut bacteria is not well known (citation). However, including this factor sets an upper bound to wall growth in total and therefore should not affect the outcome of applying the model to our system, as long as this parameter is correctly chosen. It was also assumed that both bacterial strains would be competing for the same nutrients, which is valid for our system, the nutrient in question being glucose.
Given growth rates based on Monod functions and sloughing-off rates based on wall space occupied, Jones and Smith developed a system of six parabolic partial differential equations, which give a solution in which:

Two strains, differing only in their nutrient uptake characteristics, are seen to establish a stable mixed culture steady state in which the dominant strain is the sole occupant of an upstream band of the reactor while the subordinate strain is essentially the sole occupant of an adjoining band near the upstream end, below which the nutrient density is too low to support either organism.

This solution is, of course, desirable for Gluc-o-Gone, as it would allow for the presence of our strain in the gut, with minimal interference with the already-present bacterial strains. However, it is very important to note that their conclusion is based on differences only in the ability of either strain to take up nutrients and on E. coli motility in the gut.
It is presumed that L. plantarum differsfrom E. coli in nearly all the parameters used to generate this model. To the best of our knowledge, no model to date has been used to describe the interactions between organisms when this is the case. Therefore, in order to effectively model Gluc-o-Gone, further experimentation is required to generate the necessary parameters.
Finally, as discussed by Stephanopoulos and Lapidus, plasmid-bearing bacterial strains are at a disadvantage in a chemostat-based model of bacterial competition, when compared to strains without plasmids. We predict that our strain would be at a higher disadvantage than discussed in this model, as the cellulose synthase genes directly reduce the metabolism of the plantarum strains by removing sugar that would normally be available to it.

Reference

[1] Stemmons, E. D. and Smith, H. L. Competition in a chemostat with wall attachment. SIAM Journal of Applied Math. 61 (2000) pp. 567-595

[2] Jones, D. A., Smith, H. Microbial competition for nutrient and wall sites. SIAM Journal of Applied Math. 60 (200) pp. 1576-1600

[3] Jones, D. A., Smith, H. et al. Bacterial wall attachment in a flow reactor. SIAM Journal of Applied Math. 62 (2002) pp.1728-1771

[4] Ballyk, M. M., Jones, D. A., and Smith, H. Microbial Competition in Reactors with Wall Attachment: A Mathematical Comparison of Chemostat and Plug Flow Models. Microbial Ecology. 41 (2001) pp. 210-221

[5] Boldin, B. Introducing a population into a steady community: the critical case, the center manifold and the direction of bifurcation. SIAM Journal of Applied Math. 66 (2006) 1424-1453

[6] Strandber, P. E. The Chemostat. Univeristy of Linkoping Press. (2003)

Microbial Competition in Reactors with Wall Attachment A Mathematical comparison of Chemostat and Plug Flow Models

MICROBIAL COMPETITION FOR NUTRIENT AND WALL SITES

INTRODUCING A POPULATION INTO A STEADY COMMUNITY

Competition in a Chemostat with Wall Attachment

Chemostat-dynamics-of-plasmid-bearing,-plasmid-free-mixed-recombinant-cultures_1988_Chemical-Engineering-Science

BACTERIAL WALL ATTACHMENT IN A FLOW REACTOR

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