Chemotaxis

Motivation

To stay true to its title 'pathogen-hunting microbe', BaKillus should ideally be able to swim towards its prey via chemotaxis. However, the ability to react to an AIP- or CSP-gradient requires the introduction of a chimeric receptor into Bacillus subtilis. Chemotaxis receptors are quite complex and can, for example, take different methylation states to allow for attenuation of the chemotactic response. This complexity puts the construction of a custom receptor beyond the scope of an iGEM project.

To evaluate how a hypothetical chemotactic BaKillis would act in an experimental setting, we developed a model to simulate the movement of chemotactic hunters in response to an attractant gradient produced by a growing colony of prey bacteria.

Overview

Our multi-scale hybrid model essentially consists of 3 components: growth of prey bacteria (algebraic model), diffusion of attractant molecules (spatial partial differential equation) and chemotactic behavior of the hunter bacteria (stochastic spatial agent model coupled to differential equations for internal state with a stochastic driving force). All these processes take place in a defined geometric space (representing, for example, a microfluidic device). The experimental scenario we wish to capture in our model is the following: a colony of prey bacteria is spotted onto the arena, growing in place and producing attractant molecules. A fixed population of hunter bacteria is placed into the arena as well. Their reaction (biased movement) to the building attractant gradient is what we seek to observe.

Growth of prey

The prey bacteria will grow on a population level, according to a logistic growth curve with explicit solutions (ref!):

\[ y = ln \frac{N}{N_0} = \frac{A}{1 + e^{\frac{4\mu_m}{A} (\lambda - t) + 2}}\]

Here, \(N_0\) is the population size at \(t = 0\), \(A = ln \frac{N_\infty}{N_0}\) is the asymptote, \(\mu_m\) is the growth rate and \(\lambda\) the duration of the lag-phase.

Diffusion of attractants

The concentration of attractant molecules (\(c\)) in space changes according to a partial differential equation comprising diffusion, degradation and production by the prey bacteria.

\[ \frac{\partial c}{\partial t} = D\Delta c + p c_{prey} - a c\]

\(D\) is the diffusion coefficient, \(a\) is the degradation rate, \(p\) the rate of production and \(c_{prey}\) the concentration of prey/producer bacteria.

Chemotaxis

The chemotaxis of the hunter bacteria (BaKillus) is represented by an agent-based model adapted from [Matthäus ref]. Each bacterium has its own methylation and phosphorylation states of the chemotactic pathway which change according to a system of ordinary differential equations in response to attractant concentrations. In a classical 2-state model of bacterial chemotaxis, each bacterium will be in either running or tumbling state at each time point. In a running step, bacteria will swim forward in a straight line whereas in a tumbling step, they will change their direction by a random angle picked from a gamma distribution.

It should be noted that the model used here actually describes chemotaxis of E. coli. While the wiring of the chemotaxis pathway differs slightly, the same behavior is expected for B. subtilis . Therefore, we opted for the use of the better studied E. coli-model.

Implementation