Revision as of 22:21, 15 October 2018

1. Introduction

Our goal in the dry lab this year was to create four different models that allow our wet lab team to characterize their results, and allow future researchers to benchmark their results creating standard measures in the field of cellular flotation. First, we created a generic differential bioreactor model that allowed our team to predict the effectiveness of our E. coli cells to clean waste-waters if coupled with any surface binding method. We performed a complete sensitivity analysis on this model to allow future researchers to reuse this model with completely different parameters, strains of bacteria and object of waste. Then we created an algorithm that can track cellular flotation from frame to frame, and characterize exactly how the cells float; previously, we could only tell whether they floated or not. This coupled with our ODE buoyancy model allows us to define a maximum carrying capacity for each strain. Both of these models allowed our team to benchmark their results and will allow future researchers to quantify the performance of flotation as well.

Note: In the very end of this paper, we included a nomenclature defining all the variables used.

A Differential Bioreactor Model

2.1 Goals

Explore a possible application for our genetically engineered E. coli biomass that utilizes flotation.
Develop a generic bioreactor that can be reused in many different conditions and for a variety of purposes.

Mathematical Formulation

Concepts

Monod Equation
In order to run a bioreactor, we need to understand at what rate the E. coli grow, and how they respond to environmental conditions. Bacterial growth can be separated into four phases:

The Lag Phase: Little growth is observed.
The Exponential Phase: Exponential growth is observed after the cells get used to their environment, and so long as the limiting substrate is still in surplus.
The Stationary Phase: No growth is observed, death rate and growth rate are approximately equal.
Death Phase: There is no more substrate, thus the population dies.

If N is the concentration of biomass population, then the exponential growth phase is given by:
Ṅ = μN
where μ is the specific growth rate, the “rate of increase of cell concentrations per unit cell concentration” (h^− 1) . μ depends on the substrate concentration; the relation is given by the Monod equation:
$$\mu(S) = \mu_m \frac{S}{K_S + S}$$
where S is the substrate concentration (in mol L^− 1) , and K_S is the Monod Constant (in mol L^− 1). As substrate is consumed, more biomass is being created at the rate of:
$$\frac{dN}{dS} = -\gamma \frac{dS}{dt}$$
where t is time. γ should be interpreted as “the ratio of the mass of cells formed to the mass of substrate consumed” .

The Model

Call the ’Main Reactor’ (M.R) and the ’Flotation Tank’ (F.T). Let X_u be the concentration of E. coli (in mol L^− 1), X_w the concentration of particle bound to the surface of the E. coli (in mol L^− 1). Below we will outline the steps of the bioreactor seen in Figure 1.

All the substrate, waste and cellular population is homogeneously distributed between the F.T and the M.R. Initially there is no particle bound to any of the cells, i.e X_w = 0.
When activated, the bioreactor begins to pump solution from the M.R to the F.T at rate D, this is represented by D(N + X_u + X_w + S).
In the F.T, the cells are floating on the surface, and are scraped off and removed. Thus only X_u and S remain within the solution, which is pumped back into M.R at rate D. In practice, a cellular filter alike the one used for skimming could be placed at the exit of the F.T to capture any dead cells.
Within M.R the unbound particle binds to the E. coli with some rate constant α, and detaches with some rate β (equilibrium process).
This process operates until the particle concentration is as low as desired.

Bioreactor Assumptions

Our cell surface engineering technique has specificity to only one substance in the waste-water (i.e. the substance desired for removal).
We will only observe exponential bacterial growth within the reactor, i.e. the biomass has been cultured to sufficiently large optical density before being introduced into the bioreactor.
Suspended particles attach to the surface of our E. coli at a rate proportional to the particle concentration and the ratio of unused area available on the surface.
Bound particles detach from the surface of our E. coli back into the waste-water at a rate proportional to the particle concentration bound to the E. coli.
Once the E. coli that are pumped out of the main chamber end up in the flotation tank, they are immediately removed – no more growth nor unbinding of particle occurs.
We ignore all the spatial properties of pipes between the main reactor and flotation tank, as soon as part of the solution leaves the main take, it immediately arrives in the flotation tank, and vice versa.
The pipe leading into the flotation tank sprinkles the solution over the flotation tank so lightly that the E. coli cells do not sink into the tank – they float on the surface, allowing for immediate removal.
The solution within the main reactor is continuously mixed; it can be considered a homogeneous solution).
All the E. coli in the flotation tank float.
A higher dilution rate (D) does not affect the detachment rate.

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