Multi-Scale stochastic model for a defense system against Bacteriophage Infection.

Contents

1. Molecular Scale
   1. Wild Type Model
   2. Kamikaze Model
   3. The Burst Size Distribution
2. Population Scale
   
   1. Molecular-Population integration using Cellular Automata
      
   2. Mathematical Model using Delay Differential Equations
      
   3. Agent Based Simulation Applet

Summary : Modelling the Defense System

Bacteriophage infection is a complicated process, once the virus has infected it steals the translation machinery of its host. Bacterium ribosomes synthesize virus proteins and virus assembly takes place. Beyond a phage and time threshold bacterium can’t take it anymore and explodes setting free the newly synthesized bacteriophages. Let’s take a look at the big picture: biochemical reactions taking place inside infected bacterium, new synthesized phage for each infected bacterium, other bacteria get infected, infection propagation. We need to approach this problem in a multi-scale fashion: molecular scale and population scale. We designed and implemented a Stochastic Molecular Model for the essential reactions involved in the infection process: T7’s DNA insertion, transcription, translation, capsid assembly, etc. to create a Wild Type Simulation. Then we added the toxins to the model to simulate the dynamics of the kamikaze system.

With a fairly big number of simulations we are going to generate Probability Distributions for the number of molecules for each metabolite as a function of time. We are particularly interested in the Burst-Size Distribution (BSD); the burst-size is the number of phages an infected cell will produce. Once we have the BSD we are ready for the Spatial Population Model. The kamikaze system we designed is meant to increase the probability that the population as a whole survive an infection process. We make infected-E. Coli commit suicide for the benefit of the population. In case suicide wasn’t altruistic enough we thought an alarm system might be useful. Once a bacterium is infected it will use AHL to communicate the message that phages are near, advised bacteria will produce antisense RNA against T7’s DNA polymerase. To simulate the population scale dynamics we used two different approaches:
We solved the system of Ordinary Differential Equations (ODE’s) described in Beretta[1] and We designed and implemented a Cellular Automaton (CA) to approach the spatial dynamics. Using the CA we simulate:

• a) Bacteria’s duplication, movement, infection and lysis.
• b) Quorum Sensing and T7 Diffusion.
• c) The alarm system.

So let’s put all together! Events occurring in the CA are stochastic processes. The attributes of the bacteria in the CA are [http://en.wikipedia.org/wiki/Random_variable random variables] with and associated Probability Distribution. We have distributions from literature and distributions generated by our simulations. So, for instance, when a bacterium gets infected we sample the Burst-Size Distribution, when a bacterium duplicate we sample the Duplication Time Distribution to assign lifetime to the newborn bacteria and so on. Sampling the distributions is the link between kinetic and population simulations: Random Variables in the population simulations take values from the distributions generated by the Molecular Simulations and voila, now we have the big picture.

Bacteria-phage interaction essentially is a fight for survival between two populations. Although we modified E. coli at the molecular level to prevent the replication of T7 and T3 , our ultimate goal is that E. coli can contend against infection at population level. For this reason we decided to simulate, at population level, phage infection and the effectiveness of our genetic circuit.

For this purpose, we use three different approaches:

• Modelling population dynamics with a system of differential equations.
• Spatial agent-based model.
• Modeling bacteria behaviour and Bacteriophage infection using Cellular Automata.