# Team:Amsterdam/Modeling

## Modelling

#### Trying to find a match

### Overview

### Background

In our project we focussed on the stability in different ways. On a mathematical point of view there are also different definitions on stability. A set of differential equations can have stable steady states for example, but you could also argue that convergence to certain solutions is a measure for robustness and stability. Here we will also shine a light on these types of stability. These simulations influence the way we envision further applications.

### Aim

We want to answer questions about the dynamics of an engineered consortium, which will help in envisioning an archetypal final application. What will happen to the growth rate of the organisms during the cultivation? With what initial conditions and parameters will the ratio between the biomass of the different species converge to the same ratio? And if so, what will that ratio be? What is the influence of the light intensity on Synechocystis and how will this be influenced by the presence of a chemoheterotroph which blocks and scatters light? To answer these questions we created kinetic models consisting of ordinary differential equations (ODEs).

### Approach

We modelled the biomass per liter of Synechocystis and E. coli in chemostat (being the turbidostat a particular case of the chemostat in which the D = umax) as well as batch cultures. We created a set of ODEs and analyzed them with pydstool - a python tool, which can solve differential equations numerically. Model parameters were measured as accurate as possible in the wet lab specifically for this purpose, i.e. in silico simulations that increase and test our understanding of the underlying interactions.

### Results

We analyzed the ratio of biomass of the different cultures. We can see that in a lot of cases this ratio converges. Important parameters for what ratio the plots converge to seem to be the mus of the different species. The initial conditions of the ratios when grown in a batch seem not that important, as is also shown in the lab. The ratio we arrive at in the lab however, seem different. This seems to hint at wrongly chosen or inaccurately measured parameters, however we can still study the long term behavior.

### Connections

Sometimes modellers tend to be the lone wolfs in a project. We didn't want this to happen, so there are some clear connections between the tools we created with modelling and the wet lab. Initially the need to search for compounds which could be produced genetically stable, came from the wet lab, where we saw that most producing strains are unstable. Before we even started engineering *Synechocystis*, we wanted to find out whether we could produce a compound genetically stable. This is where the Stable Compound Generater comes in. We also needed to engineer an auxotroph in order to to use serial propagations of consortia in emulsions to find a more robust consortium. Both algorithms provided information which was really used in the lab.

### approach

#### Batch

Unlimited cell growth is exponential. The amount of biomass per time for an exponentially growing species can be given by the following ODE:

Herein is a the amount of biomass per liter and μ the growth rate normalized for biomass. One can easily verify that the solution of this differential equation is indeed exponential growth (a = ce^{μt} ). Now from experimental data it has been shown that limited growth on a substrate is a bit different. The normalized growth rate is then dependent on the concentration of substrate. According to the monod equation, the μ is dependent on [S] in
the following way:

Herein is μ_{max} the maximal growth rate (equal to the growth rate at unlimited growth), and [S] the concentration of substrate. kS is the concentration of [S] at a rate 1/2 μ_{max}. We also know this equation from enzyme kinetics as the Michaelis-Menten equation. In enzyme kinetics this equation is used to calculate the rates at which enzymes convert products. However the Michaelis-Menten equation is based on theoretical arguments, while Monod is based on experimental findings. According to Monod, the growth rate saturates as the concentration becomes higher (see figure 1).

In our consortium,

Herein is syn the amount of biomass of _{syn/s}.
It can be easily seen that such a relationship will not be stable if μmax,ec << μmax,cyn, but in most cases