"What I cannot create I do not understand."
- Richard Feynmann

Reaction-diffusion Models

Goals

While single-cell models are useful for correctly implementing and debugging chemical reaction models, we need to use a reaction-diffusion model to fully understand our system's functionality. Through this modeling, we aim to implement and explore

the diffusion of AHL and lactate through the well
the relative accessibilities of bound and unbound E. coli to lactate produced by the target cell
testing the self-activation time of the system in different well volumes
determining the effects of flushing the well with a flow of water

Overview

The results from these simulations show that due to our experimental environment, the production of an AHL gradient oriented towards the outer edge of the well (i.e. higher concentration near the target cell) is not possible due to the fast diffusion of AHL. However, the greater capture of lactate by cells bound to the target relative to free-floating cells creates a concentration gradient of bound lactate and LuxR which translates to a difference in GFP signal between cells in the two locations. Combinatorially testing cases where E. coli cells bind or float around high and low lactate producers shows that our system acts as an AND gate of the two cancer markers, with E. coli bound to a high lactate producer fluorescing at orders of magnitude greater strength.

The "doughnut" model

Introduction

After implementing the single-cell models in the COMSOL multiphysics software package in an analogous geometry as a control, a more accurate geometry was constructed. Preliminary simulations were done in a geometry where a central circle representing the target cell was surrounded with smaller circles representing bound E. coli and circles uniformly distributed in the remaining space represented unbound E. coli. It was quickly realized, however, that this would not be an accurate representation of our test due to the rapid growth of the E. coli population. To address this, the discrete circles were abstracted to connected spaces representing the union of a collection of cells to allow for population growth to be simulated by modification of the ODE system over time. Owing to the shape of the layer of E. coli bound to the target cell, this geometry is referred to as the "doughnut" model. The AHL and lactate modules were implemented and tested separately before being combined into a full model.

All mathematical derivations are done in general, with values for variables being defined in the definition of our model geometry and assumptions, and in the parameters page.

Abstraction of discrete E. coli to a connected space

Figure 1: "Doughnut" model for our system.

The well in which our reaction takes place is a box of length 100μm. In the middle of the well, we have a circle representing our target cell surrounded by a ring (called the doughnut) of width equal to the diameter of an E. coli cell representing the union of all bound E. coli cells (see Figure 1). Finally, the remaining space represents both the medium and free-floating E. coli cells, which we refer to as the bulk. All species are able to diffuse freely through this medium and rates for our reactions are adjusted based on the size of the E. coli population at a given time.

Assumptions

One non-dividing target cell per well
Uniform distribution of cells in their respective spaces
A cell unbinding from the target is immediately replaced, so cell density in the doughnut is constant and maximal
Logistic growth of E. coli in the bulk with a doubling time of 30 minutes
E. coli take up lactate, but do not export it

Lactate Module

Lactate uptake by cells

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Lactate captured from a cancer cell by E. coli in the doughnut and the bulk

Since the simulations performed to characterize the lactate module make assumptions that are too stringent in the context of our testing environment. High rates of lactate production by the cancer cell do not readily translate to equivalent intracellular concentrations in the E. coli cells. In particular, those cells within the doughnut domain will capture a greater fraction of the excreted lactate due to its higher apparent concentration immediately around the target cell. Therefore, a simulation of the diffusion of lactate through the well and its capture by E. coli will provide a better idea of the input for our sensor.

Simulations of this effect show that, indeed, E. coli within the doughnut take up a concentration of lactate two orders of magnitude greater than the bulk.

AHL Module

Diffusion of AHL and self-activation time with logistic population growth

Suppose we start with an initial E. coli population corresponding to an optical density of $OD_{600}=0.1$ and a carrying capacity corresponding to $OD_{600}=2$, with a doubling time of 30 minutes.

Diffusion of AHL and self-activation time with constant population

AHL concentration per cell

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Activation times

For these simulations, it was assumed that all cells contain a constant amount of LuxR corresponding to the maximal concentration determined by the lactate module to study the effects of cell colocalization. Despite the orders-of-magnitude difference in the diffusion coefficients of AHL through water and across cell membranes, it is apparent that on a time scale of minutes, AHL is able to diffuse across the entire well. The small difference in concentrations between the wells with binding cells and floating wells can be attributed to a ~10% difference in total cell counts between these wells.

In wells of volumes 1 nL, 10 nL, and 100 nL, the AHL concentration reaches a sufficient concentration to activate the system in 10 minutes, x minutes, and y minutes, respectively. Thus, a 100 nL well is sufficient to differentiate the two cases on our experimental timescale.

Nanoliter well with constant velocity flow

GFP per cell when a flow is applied

AHL per cell with flow

Although the machinery for allowing a flow of water through channels to flush the nanowells could not be put into practice due to time constraints, a simulation of this setup was done to shed light on the benefits of this system. Having a constant velocity flow in a channel with an open boundary with the well is equivalent to increasing the well's volume, thus, should result in similar effects.

The addition of a well flushing mechanism, as expected, reduces the concentration of AHL in the bulk

Full Model

AND-gate

Flushing the well with a flow

Conclusions

By modeling the diffusion of lactate through our well, we have demonstrated that the E. coli bound to their target cell, through active transport and capture by binding to LldR, reduce the accessible lactate concentration of the unbound cells, producing a concentration gradient of its output signal LuxR.

Simulations of the AHL module demonstrated that the leakiness of the LuxR promoter leads to the self-activation of this module within our experiment's timeframe. The degradation of AHL via the action of AiiA did not lead to a drastic change in this behavior, while the introduction of riboregulation to the LuxR promoter tied to LuxI suppressed self-activation sufficiently.

Model Details

Logistic growth of E. coli

Let us define $n_\text{bulk}:[0,t_\text{sim}]\longrightarrow \mathbb N$ as the function representing the size of the E. coli population in the bulk at time $t$, with an initial population size of $n_0$, a carrying capacity of $K$, and a growth rate of $R$. The closed-form equation of this function is then $$n_\text{bulk}(t) = \frac{n_0 K e^{Rt}}{K + n_0(e^{Rt}-1)}$$

If we denote the doubling time of the population by $t_2$, we can solve for the growth rate $R$ and define a new constant $g$ s.t. $$R = \frac{\log2 + \log(K+n_0) - \log(K-2n_0)}{t_2} =: \frac{\log g}{t_2}$$ When we plug this value back into the original equation, it simplifies to $$n_\text{bulk}(t) = \frac{n_0 K g^{\frac{t}{t_2}}}{K + n_0(g^{\frac{t}{t_2}}-1)}$$ Finally, if we define the time we end our experiment, denoted $t_\text{sim}$, as the time when the population reaches size $(1-\varepsilon)K$ for some small $\varepsilon>0$, then we can solve for $t_\text{sim}$ $$t_\text{sim} = \frac{1}{R}\log\frac{(K-n_0)(1-\varepsilon)}{n_0\varepsilon}$$

Carrying capacity

Some stuff here about the carrying capacity.

Diffusion and transport of chemical species

Under alkaline conditions, E. coli actively import lactate via a proton-motive symporter lldP.[] Thus, a cross-membrane transport reaction had to be implemented. Since diffusion across barriers is implemented with equal diffusion coefficients for both directions in COMSOL, the transport of lactate across the doughnut membrane was implemented by modeling two different states of unbound lactate. If we let our reference space be the interior of the doughnut, we can define two pseudo-species $Lact_\text{in}$ and $Lact_\text{out}$, denoting intracellular and extracellular lactate, respectively. $Lact_\text{out}$ is produced by the target cell and can diffuse freely though the medium and all membranes, while $Lact_\text{in}$ is converted irreversibly from $Lact_\text{out}$ and cannot diffuse out. $$ Lact_\text{out} \mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}^{k_{\mathrm{ext}}} Lact_\text{in} $$ Only the $Lact_\text{in}$ state can react with the other chemical species in the doughnut and only $Lact_\text{out}$ can react with other chemical species in the bulk. The value of $k_\text{ext}$ was estimated to be

Since AHL can freely diffuse across membranes, unbound AHL is only modeled as one state that can diffuse across all barriers. The effective diffusion coefficients of AHL and Lactate through the doughnut membrane $D_e$ were approximated relative to their coefficients in water $D_{aq}$ by the relation $$\frac{D_e}{D_{aq}}\approx 0.25$$ using the approximation proposed in a 2003 review by Stewart.

Correcting concentrations

Since E. coli are not fully packed into the domains in our model, a concentration correction has to be applied to the chemical species. If no correction is made for the strictly intracellular species, then the reported concentrations are single-cell. To compensate for the fact that AHL can diffuse, under the assumption that diffusion within a domain is instant, the production rate of

Team:ETH Zurich/Modeling/Reaction-diffusion