Team:Amsterdam/Modeling

iGEM Amsterdam 2015

Algorithms

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:


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 = ceμt ). 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:


Monod equation

Herein is μmax 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 μmax. 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).


ODE

In our consortium, E. coli grows limited on acetate. So now we know how μ depends on the concentration of acetate. We now need to model the concentration of acetate. We assume the synthesis of acetate is growth coupled and depends linearly on the growth of Synechocystis. We also know that the maximal uptake rate of acetate by E. coli is dependent 1/y herein is y the yield of E. coli on acetate in is in milligram Dry Weight E. coli per millimole acetate per liter. The uptake of substrate per unit time is also in a saturable way dependent on the concentration of S, exactly the same way the growth speed is dependent on the the concentration of substrate. The synthesis of the substrate is growth coupled and is dependent on the amount of biomass of Synechocystis formed in time. We arrive at the following differential equations:


ODE

Herein is syn the amount of biomass of Synechocystis and ec the amount of biomass of E. coli. ys is the substrate yield of Synechocystis. Since in this model the substrate is only formed when Synechocystis forms biomass, there is a constant amount of substrate formed. The yield is usually expressed in 5 gram dry weight per mole substrate used. In this case we mean gram dry weight mole substrate per formed. So to find the amount of substrate that is formed per amount of biomass that is formed we simply take 1/ysyn/s. It can be easily seen that such a relationship will not be stable if μmax,ec << μmax,cyn, but in most cases E. coli has a much higher growth rate. In this model, Synechocystis is not dependent on E. coli. There are two ways in which Synechocystis may be dependent on E. coli, which we have explored. Firstly, E. coli may produce a substrate Synechocystis grows on, as is the case with the auxotrophic Synechocystis. Secondly, E. coli may decrease the light intensity in the culture, in this way slowing down the growth of Synechocystis. The growth rate of Synechocystis can only be limited by one of these two processes and it will always be limited by the process that slows it the most. Either μsyn is lower than μmax,syn because there is a photon shortage, but then the amount of substrate available at that growth rate would be enough, or the amount of substrate is limiting, but then the amount of photons available would also be enough for that given growth rate. So actually the growth rate of Synechocystis would be