Paris/Modeling
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Introduction
Motivation for modeling
We want to construct a multicellular bacterial organism made of two co-existing cell types. Brainstorming resulted in proposing the following system:
- so-called soma cells, that produce a metabolite (DAP), and will not be able to divide,
- so-called germ cells that are able to grow, but only in presence of (sufficient quantities of ) DAP, and are able to differentiate into soma cells.
Informal reasoning indicate that this design should be correct, in the sense that it leads to an exponential growth of the two coexisting cell types. However, before actually constructing this system, we would like to assess the quality of our design using modeling approaches. Since different questions needed to be answered, we developed different types of models, each adapted to a particular problem.
The questions of interest
The first, most obvious question deals with proving the feasibility of our system (Section II). In our case, this amounts to check that the system presents a very simple, qualitative behavior: the two cell populations grow! Accordingly, we tested whether this property holds under various modeling assumptions.
The simplest model that we have considered is a phenomenological ODE model (Section 2.1). Being very simple, analytic analysis is possible and the stability of equilibria can be investigated under mild assumptions on parameters.
However, a number of phenomena that might play an important role are neglected in the previous model. By assuming that cellular and molecular concentrations evolve continuously and that the solution is well-mixed, noise and space-related issues may have been overlooked. To test whether these phenomena may affect the qualitative behavior of the system (i.e. growth), we developed two models, one focusing on spatial aspects of Dap diffusion on cell differentiation (Section 2.2), the other incorporating dynamical aspects of cell spatial organization (Section 2.3). These results are rather general, in the sense that the level of abstraction of these models does not allow to distinguish between the two slightly different designs proposed in Section Design process.
In addition to feasibility, robustness and tunability of the system are also of prime interest. More precisely, we would like to find an objective criteria to discriminate between the two competing designs proposed in Section Design process. The two designs differ by the presence or absence of a negative regulation of cre recombinase expression by Dap. To address this problem, we developed two numerical ODE models and investigated their relative robustness (Section 3.1) and tunability (Section 3.2). Finally, it is also important to check that stochastic phenomena that are neglected in ODE models do not affect the macroscopic behavior. Stated differently we checked whether the deterministic models and their stochastic counterparts present globally the same behavior (Section 3.3).
Proof of principle: qualitative analysis of system's behavior
Those models are at macroscopic scale. They are focused on the evolution of the population, with global rules avoiding description of all the microscopic mechanisms. We present tree different works, with different approaches (ODE, automaton)...
Exponential growth of cellular populations: analytic analysis of an ODE model
We present here a theoretical approach based on population dynamics. We consider here the case of a well mixed, homogeneous, culture of the SMB organism, i.e. there is no space in this analysis and we follow only the variation of the different cell lines concentrations in the culture volume.
Potential macroscopic effect of spatial aspects of Dap diffusion: cellular automaton on a grid
In this part of our work, we aim at characterizing the diffusion of the DAP and the effect on the cells differentiation. This study consists in observing by simulation, the diffusion of DAP in a lawn of germ cells with some isolated somatic cells using a cellular automaton.
Potential macroscopic effect of stochastic and spatial aspects of Dap diffusion and cell growth: cellular automaton in 3D
We try with this work to characterize the effect on the cells, of the DAP diffusion in a free space where cells can divide or die.
We have a growing culture with germinal cells and somatic cells.
We want to see if we can have different kinds of evolution for our cells.
as we can see in the DAP diffusion automaton the diffusion mechanism and the effect on differentiation can be describe more accurately, so for the moment we just ignore the diffusion putting a black box on it and just focused on the total number of DAP entities.
Assessing robustness and optimizing system's behavior: quantitative analysis
Problem description
Assessing robustness of two potential designs: numerical simulations of ODE models
Assessing tunability of two potential designs: numerical simulations of ODE models
This model aims at describing the dynamic evolution of populations of germen and soma type bacteria. It is based on a set of differential equations describing DAP synthesis, DAP transport, differentiation of germen bacteria into soma and bacteria death.
Potential macroscopic effect of stochastic phenomena: stochastic simulations with Gillespie algorithm
Summary
Appendix
Tools description
For our simulations we used unusual tools, Biocham and MGS. Thanks to their specificities and capacities, we were able to simulate easily the mechanisms that we wanted to focus on.
Biocham
MGS
MGS is an experimental programming language developed at the university of Evry and dedicated to the modeling and the simulation of dynamical systems with a dynamical structure. We briefly present in this section the philosophy of MGS programming.
