Team:Amsterdam/Modeling
Modelling and simulations
Trying to find a match
Overview
Background
In our project we focussed on the stability in different ways. In mathematical sense there are also different ways of stability. Steady states in differential equations may be stable or not. But a set of differential equations may also converge to an interesting solution. This convergence may also be called stability.
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
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We show that consortia may show a robustness to the initial conditions. We looked at behaviour in different environments. We show simulations of these models.
Connections
To create models as accurate as possible, we measured these in wet lab. Here the results of the physiology
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 = 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:
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).
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:
Reliance on each other
Auxotrophs are organisms which need a certain compound which they can take up from their environment to be able to grow. They can be very useful in synthetic biology, as an auxotrophy can regulate the growth of a certain organism. It is also useful synthetic consortium, as it can create a dependency of one species on another. In our project we wanted to make the chemoheterotroph dependent on the cyanobacterium by the carbon compound the cyanobacteria produces, because this will be part of the flux of CO2 into product. However, we also wanted to create dependency of the cyanobacterium on the chemoheterotroph, because it can stabilize a consortium if, for example, the chemoheterotoph has a lower growth rate than the photoautotroph on the carbon compound. Other important reasons are that this dependency will decrease the risk of the cyanobacterium surviving on its own in the environment in case of an outbreak. Some chemoheterotrophs are very efficient in the production of a certain compound. If the photoautotroph does not have to synthesize the compound itself, it may increase the growth rate of the photoautotroph. This increases the rate at which carbon is fixated and carbon compound is produced, which in its turn benefits the chemoheterotroph. It is also suggested that this interdependency may create a more robust system. So the second question that we had in the lab is how can we create an auxotroph? To answer this question we also created an algorithm, the Auxotrophy Sniper.