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Contents 1 Introduction 1.1 Motivation for modeling 1.2 The questions of interest 2 Proof of principle: qualitative analysis of system's behavior 2.1 Exponential growth of cellular populations: analytic analysis of an ODE model 2.2 Potential macroscopic effect of spatial aspects of Dap diffusion: cellular automaton on a grid 2.3 Potential macroscopic effect of stochastic and spatial aspects of Dap diffusion and cell growth: cellular automaton in 3D 3 Assessing robustness and optimizing system's behavior: quantitative analysis 3.1 Problem description 3.2 Assessing robustness of two potential designs: numerical simulations of ODE models 3.3 Assessing tunability of two potential designs: numerical simulations of ODE models 3.4 Potential macroscopic effect of stochastic phenomena: stochastic simulations with Gillespie algorithm 4 Summary 5 Appendix 5.1 Tools description 5.1.1 Biocham 5.1.2 MGS 5.2 Models, Initial Conditions Files and Sources

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 Blah[add link to corresponding section].

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 Blah[add link to corresponding section]. 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

We want with this work to characterize the diffusion of the DAP and the effect on the cells. We have a lawn of bacteria with germinal cells and some somatic cells.

Potential macroscopic effect of stochastic and spatial aspects of Dap diffusion and cell growth: cellular automaton in 3D

We try with this model to see the effect of DAP on the cells. 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 simple 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. This approach differs form the precedents one by the level of description of the model and the numerical analysis done on the model.

Potential macroscopic effect of stochastic phenomena: stochastic simulations with Gillespie algorithm

Summary

The goal of our modeling work was to test our design, mainly to identify potential flaws of the system at early developmental stages.

In part II, we showed that the system can present – at least qualitatively – the desired behaviour: an exponential growth of the two populations of coexisting cellular types.

In part III, our results indicated that the system’s behavior should be reasonably robust, and provided arguments in favour of the design having a negative regulation of recombinase expression by Dap (increased robustness and tunability).

In all cases, models incorporating additional details, related to space and/or stochasticity, indicated that these phenomena should not affect the global behavior of the system. So previous conclusions, obtained using deterministic models, should remain valid despite the fact that we neglected noise- and space-related issues.

We would like to stress here that these results should be taken with great care, given the extreme simplicity of our models and the lack of data to provide information on parameter values and initial conditions. But still, globally…

…all these results corroborate our initial design.

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.

Models, Initial Conditions Files and Sources