Introduction

This model is to better illustrate the effect of the whole imaging system. We hope to elucidate the substantial effect which the motion control system of the C. Crescentus has upon our imaging system:

Does the motion control system of flagellum and holdfast indeed improve the resolution of the expected image?

Given an input light array, i.e. the expected image, we in detail simulate our imaging system with the host E. Coli and C. Crescentus respectively, and give a prediction of their respective output image. We draw a comparison between the resolution of their output to see whether the motion control system in the C.Crescentus indeed works in the imaging setting.

In practice, our model can serve as a tool to predict the output image. The user can upload their own photo, our model will tell you how our Cimage would perform.

Methods

We do the simulation of the whole Cimage system in an agent based fashion. Each cell is considered an agent, who can do the following:

• Express fluorescent proteins with respect to the input light array
• Motion in a biased random walk
• Division stochastically

We keep track of the following of each agents:

• Position, in terms of x coordinates and y coordinates in the imaging plane
• Motion state, including
  • their flagellum state, clockwise or counter-clockwise rotating
  • the time left for the flagellum state switch, from clockwise to counter-clockwise or converse
  • their velocity, both direction and magnitude
• Division state, the time left for the next division

The model is as follows:

Fluorescence Expression

Since we keep track of the position of each agent, at any time, we know exactly where they lie. For convenience, suppose at time t, and agent x is being considered. Since the input light array is known and assumed constant, we know the exact light intensity at the position where x lies. Given the light environment, our compartmental kinetic modeling can tell us how this light condition will affect the protein concentrations within the agent x. We assume that within a short time , the light condition doesn’t change much. Using the kinetic model, we can calculate the concentrations of fluorescence proteins at the time . Through an iteration of above process, we can predict the amount of fluorescence in the cell x at any time after the time t.

Swimming rule: Motion control

Due to the work by Fan Bai, E. coli switches the direction of rotation in a stochastic way. We assume the following: 1. E. coli stays in either of the two motion states, clockwise rotating which corresponds to swimming in a specific direction and counter-clockwise rotating which corresponds to cell tumbling, and switches alternatively between the two states. 2. The time E. coli spends staying in either states follows an exponential distribution with respective means. 3. In the swimming state, the bacteria swims straight in a linear direction; in the tumbling state, the bacteria changes the direction of swimming randomly using a uniform distribution.

As for the C. Crescentus, we assume that

1. They swim in a similar way like the E. coli, except that the magnitude and the bias of C.Crescentus is influenced by the light regulated DgrA,DgrB and HfiA concentrations in the C.Crescentus Motion control system. The bias refers to the ratio of the time C.C spends swimming over the time it spends on tumbling.
2. The way in which the Proteins DgrA,DgrB and HfiA influences the rotation remains in detail unclear. We simply apply a mathematically negative influence from the related proteins to the magnitude of the velocity and the bias of states.

Division

Time between successive mitosis can be modeled using an Erlang distribution

The Erlang distribution with factor k is the sum of k exponential distributions with same mean. The use of this distribution is motivated by considering that biologically, a cell has to finish several elementary biological processes (such as replicating all its chromosomes) before being able to divide. Assuming (with over-simplification) that each of these processes has the same mean duration and follows an exponential law, as commonly assumed for Poisson processes, the overall time between two mitosis events will follow an Erlang distribution. (Ref : Drasdo 2012)

Plasmids repartition occurring at mitosis can be represented by a normal distribution

Simulation

Realization of the Cimager system is simulated using the Gillespie Algorithm implemented in Matlab by our team.

Results