Template:Greece/PSP

Quorum​ ​Sensing

Motivation

Can​ ​we​ ​predict​ ​if​ ​and​ ​when​ ​our​ ​engineered​ ​bacteria​ ​will​ ​invade​ ​their​ ​surrounding​ ​tissue​ ​in​ ​order​ ​to​ ​transfer​ ​the​ ​classifier​ ​plasmids?​ ​​What​ ​can​ ​cause​ ​our bacteria​ ​to​ ​stop​ ​growing​ ​and​ ​invade​ ​their​

Overview

ial​ ​growth​ ​phase,​ ​but​​ ​a​ ​small​ ​initial​ ​population​ ​​colonizing​ ​​colonizes​​ ​a​ ​new​ ​environment (thus​ ​requiring​ ​a​ ​custom​ ​​growth​ ​model​)​ ​and​ ​instead​ ​of​ ​developing​ ​in​ ​the​ ​full​ ​volume​ ​of​ ​a​ ​liquid​ ​medium,​ ​it​ ​colonizes​ ​the​ ​solid,​ ​permeable​ ​surface​ ​of​ ​the colon​ ​epithelium​ ​(thus​ ​requiring​ ​the​ ​simulation​ ​of​ ​​diffusion​). This​ ​model​ ​therefore​ ​comprises​ ​3​ ​parts:

• 1.A​ ​network​ ​of​ ​chemical​ ​reactions​ ​in​ ​fixed-volume​,​​ ​well-stirred​ ​conditions​ ​that​ ​model​ ​the​ ​production​ ​and​ ​consumption​ ​of​ ​AHL​ ​as​ ​well​ ​as​ ​the​ ​systems regulatory​ ​elements.
• 2.A​ ​custom​ ​growth​ ​model​ ​that​ ​evolves​ ​an​ ​initial​ ​inoculation​ ​to​ ​the​ ​environment's​ ​assumed​ ​carrying​ ​capacity
• 3.A​ ​diffusive​ ​model​ ​for​ ​the​ ​evolution​ ​of​ ​AHL​ ​spatial​ ​distribution

The​ ​“spatial​ ​equilibrium​ ​model”​ ​is​ ​built​ ​of​ ​the​ ​first​ ​2​ ​parts,​ ​while​ ​the​ ​“diffusive​ ​model”​ ​incorporates​ ​the​ ​last.​ ​Previous​ ​models​ ​of​ ​quorum​ ​sensing​ ​that​ ​we​ ​have studied​ ​only​ ​supply​ ​the​ ​first​ ​part.

Availability

The​ ​accompanying​ ​code​ ​is​ ​available​ ​in​ ​the​ ​designated​ ​GitHub​ ​repository: https://github.com/igem-greece-2017

Spatial equilibrium model

Chemical interaction model description

Quorum sensing network. The arrows imply chemical reactions.

the protein LuxI, which produces AHL, and LuxR, which binds it and then, activated, goes on to induce the “DNA”. The “DNA” species refers to plasmids carrying the Lux regulatory system. Its induction by (LuxR.AHL)₂ marks the off->on transition; when most of the DNA is in the (uninduced) “DNA' form, the switch is off; when most of it is in the “DNA.(LuxR.AHL)₂” form, the switch is on.

Dilution mechanism that keeps the growth rate and the cell density constant. These are the conditions modelled in [1].

We built this first model to better understand the constraints and basic properties of the physical system, before going into greater depth in the following sections. A caveat to our extrapolation is that cellular metabolism could be significantly altered when cells enter a stationary growth phase, impacting the core QS functionality. We keep this in mind, but have found no way to account for it.

In the following section, we present exploratory simulations of the system's behaviour.

Results (no custom growth model)

The growth model impacts the QS system greatly. As is evident in [figure 7 vs 8], while the growth rate is high (r=1.5) quorum sensing is difficult to achieve. This corroborates the result in [figure 3]. As the volumes increase and the growth curve remains the same, more AHL has to be produced to achieve the same concentration, which takes more time. At an extreme volume of $\SI{5}{\milli\liter$, in [figure 9] QS still happens, but much later, at 40 hours. This volume is significant, because it is the volume of a small petri dish, which we would like to simulate with the diffusive model to compare the results.

Diffusive model

Chemical interaction model description

We've modelled a bacterial population in a well-stirred liquid culture so far. Without the “growth model”, we either model a stationary population or one that grows at a steady rate, its density maintained constant by compensating dilution. With the growth model, an initial inoculation grows to the environment's carrying capacity, modelling a bacterial colonization of a new environment.

Bacteria growing on a surface are packed very closely together, but the AHL they produce is free to leave their immediate surroundings and diffuse into the surrounding area. Diffusion is a well-described physical phenomenon and this model aims to couple the diffusive process with the chemical interactions of AHL inside millions of independent bacterial cells that are geometrically defined. There doesn't seem to be any case of chemical gradients affecting diffusion of AHL, therefore our model is concerned only with its concentration.

The primary goal was to simulate bacteria growing on the surface of an agar plate, as these conditions are easy to recreate experimentally and thus provide verification to our model. Our collaboration with iGEM Columbia was meant to enable these experiments, but unfortunately material shortages only allowed us to experiment in liquid cultures.

Model Description

We start with Fick's laws of diffusion [6]. In the simplest case of isotropic media without mass transport phenomena or external potentials, the driving force of diffusion is the concentration gradient and the diffusion coefficient is a constant real number. Thus, the general diffusion equation takes the form of the simpler heat equation:

$$\frac{\partial{\mathit{[AHL]}}}{\partial{t}} = D \nabla^2\mathit{[AHL]}$$

It is a parabolic partial differential equation (PDE) in space and time. To specify a solvable problem based on such an equation, many more ingredients are needed:

• a geometry
• initial conditions
• boundary conditions

To actually solve it, we furthermore need to select a solution algorithm, which requires its own ingredients.

If we allowed the diffusion coefficient to vary in space, we'd have a more general form of diffusion. Adding a production term $q$ to the right side, it becomes:

$$\frac{\partial{\mathit{[AHL]}}}{\partial{t}} = \nabla \cdot \left(D \nabla \mathit{[AHL]} \right) + q$$

Geometry

The first design decision is to express the problem's geometry. At a first glance at the task at hand, to model bacteria growing on an agar plate, one might assume a top-down 2D perspective, with the AHL diffusing across the surface away from the bacteria. The diffusion of AHL is inherently a 3D phenomenon though, and this perspective couldn't easily incorporate the effects of diffusion along the height of the agar gel. In the end, we decided to model the entire 3D volume of an agar plate, with the bacteria at the top of the agar. The agar forms a short cylinder (a disk), surrounded by plastic on 3 sides and air on top. The cylinder is in the order of millimetres in height and centimetres across. An E. Coli cell is about 1μm -- a huge difference in scale! This difference makes the problem quite difficult to solve in practice.

An important simplification at this point is to assume axial symmetry around the axis of the cylinder, thus making the problem tractable. We express the PDE in cylindrical coordinates; after eliminating the angular coefficients of the derivatives, we are left with:

$$\rho \frac{\partial{\mathit{[AHL]}}}{\partial{t}} = D \left(\frac{\partial}{\partial{\rho}}\left(\rho \frac{\partial{\mathit{[AHL]}}}{\partial{\rho}}\right) + \frac{\partial}{\partial{z}}\left(\rho \frac{\partial{\mathit{[AHL]}}}{\partial{z}}\right)\right) + \rho q$$

If we now transform $\rho \mapsto x$ & $z \mapsto y$, thus having:

$$x \frac{\partial{\mathit{[AHL]}}}{\partial{t}} = \nabla \cdot \left(xD \nabla \mathit{[AHL]} \right) + x q$$

This is identical to the diffusion equation above, where the time coefficient is $x$, the diffusion coefficient $xD$ and the production coefficient $xq$. Therefore, we'll solve this problem on a 2D vertical cross-section of the cylinder, whose solutions are the same as the initial equation on the full 3D cylinder. The final geometry is shown in [figure 10]. The bacteria are the small red rectangles shown in the zoomed-in image.

Boundary and initial conditions

Boundary conditions specify what happens to the concentration (Dirichlet BC), or to its gradient (Neumann BC), at the geometric boundaries. There's a lot of those: the edges of the agar, as well as the edges of the rectangles that represent a ring of bacteria. A Neumann BC takes the form $$\nabla \mathit{[AHL]} \cdot \hat{n} = q$$, where $q$ is the flux through the boundary. $\hat{n}$ is the unit vector normal to the boundary.

The edges of an agar plate are all reflecting boundaries, because AHL can't diffuse through them: plastic walls at the sides and bottom, air at the top. Thus, at the boundary $q=0$: no AHL goes through.

A boundary condition on the edge of the bacteria could simulate the semipermeable cell membrane, but unfortunately the current version of Matlab doesn't accommodate conditions on internal boundaries. To evaluate the significance of this restriction we've run a test simulation ([figure 11]).

The initial conditions are described by the [AHL] at each point in the geometry at $t=0$. Here, they are 0.

Solution algorithm

We solve the PDE with the finite element method (FEM). This method discretizes the continuous space into small elements by overlaying a mesh ([figure 10]) and transforming the continuous geometry into a graph. Each node is a dependent variable. The time-dependent PDE is transformed in this manner into a large system of ODEs, with 1 equation for each node (because the boundary conditions are Neumann).

Coupling AHL diffusion and localized chemical interactions

By and large the most interesting piece of this puzzle is how to couple the spatially-oblivious chemical interaction network described before with the diffusion. [AHL] has become a spatial field. Each ring/layer of bacteria (see footnote in [figure 10]) is an autonomous agent that interacts with the locally available AHL. A few things are evident: AHL concentrations at points on cytoplasms depend both on diffusion and on chemical interactions, and many more dependent variables are required, to store the concentration of every species at every bacterium.

A bacterium can be thought as a dynamic system with all the other chemical species as its state and local AHL as its input signal. The dynamic system can be seen as a function of the input and the previous state. Multiple dynamic systems can also be seen as a single function, because the function takes a point in space as input and knows which individual system to feed the input to. Since it can be seen as a function, it can readily be plugged into the equation above as the production coefficient - problem solved!

… solved, but for the ODE solver which fails miserably at such a convoluted, nested system of equations! The alternative that has been successfully implemented is to augment the system of node equations. To the equations produced by the spatial discretization a new set of equations for every bacterium involved is added.

The bacteria expect a single value for the concentration of AHL in the cytoplasm, but as can be seen in [figure 10] to each bacterium correspond 6 mesh nodes. For simplicity, the value of [AHL] that the bacteria see is the mean of those nodes.

The d[AHL] produced from the bacterial equations also has to be distributed correctly on the mesh nodes. What we want to simulate when distributing the bacterial output is a constant source on the entire cytoplasm. There is a certain complexity in how the finite element method formulates the node equations and the nodes are not all equivalent, so we can't simply split the d[AHL] into many parts and give one to each node. Instead, we rely on the FEM algorithm to solve the spatial distribution problem for us by assigning to the bacterial geometries a constant production coefficient of 1, then hijack it by multiplying the resulting node increment coefficients by the d[AHL] produced by the bacteria.

Much is assumed or reverse-engineered in order to arrive at this coupling mechanism. To verify that the model is still on track, we run test simulations with a single bacterium and a small surrounding space. The results should be similar to (but not exactly the same with) the non-diffusive model. Indeed, there is close agreement ([figure 11]).

Left: The geometry on which the diffusion PDE is solved. It represents an axisymmetric 3D cylindrical geometry: an agar plate. The left side is the cylinder’s axis, the right side is the rim. The top is the cylinder’s surface. On the top near the axis there are some bacteria. Since this perspective is a cross-section of the agar plate, the bacteria actually occupy a small disk on the surface of the agar near the axis (every shape in this geometry should be rotated around the axis to imagine its 3D representation). Right: Each red rectangle represents an E. Coli cell. The bacteria are organized in orderly rings and layers with no spaces between them (maximum density). In this case, there are 600 rings of bacteria and 4 layers. [FOOTNOTE]{Due to constraints with the mesh generation, this isn’t exactly the case. The reality is more complicated, but it simulates bacteria packed closely together. Notice that each red rectangle has a blue rectangle next to it. Only 1 in 3 red rectangles actually interacts with the AHL, the rest is inert geometry. Thus, 1 cell covers the space of 6 rectangles on the same layer, plus 6 more on the layer below. The cell’s AHL output is multiplied by the number of bacteria it replaces, thus in fact concentrating the production of this entire region on 1 cell. This should be a slight source of error though, because the diffusion coefficient is large.} The blue lines are the mesh, the solver’s spatial discretization. Observe how the mesh around the bacteria is very orderly, but also rather coarse (compared to the feature size). The loss in accuracy in this area is intentional: our model inherently can’t resolve concentration differences inside each bacterium’s cytoplasm, therefore a finer mesh would not provide extra information, only modelling artifacts -- and much more computation time!

The spatial equilibrium model for 1 bacterium in a $\SI{2.12e-4}{\nano\liter}$ volume.	Same conditions, but simulated with the diffusive model. A diffusive barrier simulates the cell wall. Good agreement with the equilibrium model.	Diffusive model without the cell wall. Again, quite similar to the case with the wall, but much simpler to scale up. This bacterial model is used in the larger geometries.

Implementing Growth

Growth requires adding new bacteria to the geometry, but the finite element method doesn't accommodate such changes. Consequently, the solution has to be stopped and restarted every time a new bacterium is added. This would be computationally prohibitive.

Instead, the complete growth curve is precalculated and then quantized adaptively to levels corresponding to adding many rings of bacteria at the same time, possibly adding millions of new bacteria at each growth step ([figure 12]). The final number of bacteria generally depends on the nutrients provided by the growth medium and, for an agar plate with few added nutrients, is expected to be around $10^8.9$ bacteria [2]. To further mimic the way bacterial colonies grow, once there are enough bacteria the older cells in the center die.

Bacterial growth constantly dilutes the cytoplasm - a process which heavily affects QS ([figure 3]). Here, the growth model is implemented in large discrete steps. In each step, the ratio of existing to new bacteria is calculated and a dilution performed on the existing ones, in order to keep the quantity of every chemical species constant during the growth step, apart from the DNA, which cancels its dilution with replication. This discrete dilution event creates minor artifacts on the simulations that manifest as discontinuities on the graphs.

Results

The basic question that we want to answer at this point is if and when will the bacterial colony exhibit QS behavior on an agar disk - faster or later than in a liquid culture with the same number of bacteria? We present 2 simulations that differ in scale. The first is a tiny agar disk measuring 3.4mm across and .551mm in depth, for a volume of 5μL ([figure 13]). The second is much larger in scope and computational effort: agar in a small petri dish 34mm in diameter and 5.51mm in depth. The larger scope allows direct experimental evaluation of the model.

The tiny disk is inoculated with 7e4 bacteria, which gradually grow to a final size of 8.12e6 bacteria over 11 growth steps (see the [video][here]). Quorum sensing is indeed triggered early under these conditions, at 11 hours ([figure 14][here]).