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

Contents

1. Summary : Modelling the Defense System
2. Motivation
3. Molecular Scale
   1. Wild Type Model
   2. Kamikaze Model
   3. The Burst Size Distribution
4. Population Scale
   
   1. Multi-Scale 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 gets infected it will produce 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 a system of Delay Differential Equations (DDE’s) described in Beretta (2001) 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.

Motivation

The output of biological systems is the sum of the output of many equally complex sub-systems. If we try to model an organism as a deterministic physical state we will be unable to describe such a complex system because of the lack of detailed knowledge. Physical scientists are curious to know whether the present techniques of physical sciences are sufficient to explain biological phenomena. The model of biological systems via stochastic processes allows the incorporation of effects of secondary factors for which a detailed knowledge is missing. The truth is that we observe in nature distributions for phenotypes instead of punctual values (e.g. Cell diameter, human height, cell lifetime, number of offspring in animals and so on).

The masterful book published in 1926 (d’Herelle 1926) describe the three-step process of the life history of bacteriophage virus. During the next decades there was a lot of effort trying to describe the basic characteristics of the intracellular dynamics of the infection process. Nowadays we have understood some of the basic reactions and processes that take place inside the cell, from the moment the phage insert its DNA to the moment the bacterium lyses.

For the chemical reactions inside the cell deterministic models using ODE’s have shown to be accurate in some cases, in other cases stochastic approaches are used to take into account the small number of some molecules inside the cell. In the first steps of the infection process there are some molecules that actually have small numbers. Random fluctuations in the first moments of infection can propagate in time and cause larger fluctuations for the number of molecules of a specific specie.

Taking into account the above considerations we decided to implement a stochastic approach for the intracellular simulations. Using this approach we will get insight in the variability of the phenotypes involved in phage development.

At the population scale we need to model spatial and temporal dynamics. Events like infection are stochastic and depend upon many variables. We incorporated the intracellular simulations in the population scale by sampling the distributions mentioned above.

By using a multi-scale model we simulated observed behaviour but we can also make predictions about the system as a whole. Previous attempts to model T7 life cycle were focused only in the intracellular scale but failed to incorporate the population dynamics [1] ; population models didn’t take into account intracellular dynamics [5][6] . Moreover opulation models take burst-size as a constant value taken from literature, this is unrealistic since the reported values for the burst-size have a lot of variance . Our model takes into account the random fluctuations in the system so we can simulate the experimental data and distributions.

Putting all together we get a model that takes into account the previous mentioned processes and incorporates the 2 scales at which the infection process takes place.

Simulations results are in good agreement with existing experimental data. Thanks to the structure and design of the model this can be easily modified in order to simulate infection dynamics for different bacteria and phages. Furthermore, our Molecular model can be used as a reliable tool for sampling biomolecules distributions involved in phage infection processes.

References

[1] Drew Endy, Deyu Kong, John Yin. 1996. Intracellular Kinetics of a Growing Virus: A Genetically Structured Simulation for Bacteriophage T7
[2] S. Goel. Stochastic Models in Biology. 2003
[3] Lingchong You, Patrick F. Suthers, and John Yin. 2002. Effects of Escherichia coli Physiology on Growth of Phage T7 In Vivo and In Silico
[4]Watson-Stent-Cairns, Phage and The Origin of Molecular biology. 1992 [5]Yin, Evolution of Bacteriophage T7 in a growing plate. 1992. [6]Yin, Replication of viruses in a growing plaque: a reaction-diffusion model.