# Paris/Modeling

### From 2007.igem.org

## Contents |

# 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*.

### Questions of Interest

The first, most obvious question deals with proving the *feasibility* of our system ( Section 2). 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 this 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:

- A SMB with a fixed recombination rate
- A SMB with a recombination rate depending on DAP starvation (feedback)

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 and tunability ( Section 3.1). 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 3.2).

# Proof of Principle: Qualitative Analysis of System's Behavior

In this section, we develop models to test the feasibility of our system. We focus on a simple, essential qualitative property: the growth of the two coexisting cell types. This property is investigated under various modeling assumptions.

### 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

In this section, we aim at considering SMB as a dynamical system with a dynamical structure and studying the impact of the cells organization on the future of the population. In order to achieve this goal, we have developed a mechanistic model based on a masses/springs system, that will allow cell to divide and die.

# Assessing Robustness and Optimizing System's Behavior: Quantitative Analysis

In this section, we focus on more quantitative properties of system's behavior: robustness and optimization capabilities. Two slightly different designs are compared.

### Assessing robustness and tunability of two potential designs: numerical simulations of ODE models

In section Design process two designs have been proposed. The only difference is that one of them incorporates a negative feedback of cre recombinase by dap. We developed simple models to evaluate the relative benefits of both designs in term of robustness and optimization capabilities.

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

In this last part, we are developing a stochastic simulation of the microscopic model. The major contribution is to handle in a stochastic context a dynamic and heterogeneous population of bacteria. We were able to achieve this goal by proposing an optimized extension of the 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 2, 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 3, 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. This result is not surprising. Intuitively, even if SMB system is heterogeneous (because it is composed of two distinct populations of cells), the spatial distributions of both cells types are uniform; thus, the system evolution is space independent. Nevertheless, we were interested in checking the property using models and simulations. In fact this led us to use and develop original techniques (like Delaunay triangulation or extended to nested membranes systems Gillespie algorithm). From a general point of view, developing such techniques is also of great interest for synthetic biology. Following the concepts of decoupling and abstraction that characterize biosynthetic developments, we have to ideally validate and study designs before constructing them physically. These validations appear at each step of a standard development, at the levels of systems, devices and bricks designing. We tried to follow and contribute to this methodology by considering at first phenomenological and global models, and molecules scaled models at last.

Getting back to SMB, 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

BIOCHAM is a programming environment for modeling biochemical systems, making simulations and querying the model in temporal logic.

#### 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.