Team:Michigan/Modeling/

From 2014.igem.org

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We wanted to capture this balance between simple assumptions and an effective model. We began with a conceptual representation of what we had hypothesized our system to be: a three compartment model in which the protein was synthesized directly into the cytosol, transferred to the periplasm in a first order reaction, from there transferred to the media outside the cell in a first order reaction, with the possibility for reverse reactions to occur at different rates.

Here we defined the direction of our rate constants and added a degradation term D[M] to the media in order to allow for a steady state to be reached. Notice that in this model the volume of the media is infinite, while the volume of the Cytosol and Periplasm are constant.

This illustrates the directions in which our protein flows and allows us to mathematically model our system as a set of differential equations through applying the law of mass action. The law of mass action dictates that the mass transport rate of a component across a compartment boundary is directly proportional to its concentration in the originating compartment. Mass action equations were developed for the rates of each process and were then combined to form a quantitative model consisting of dynamic equations for each compartment concentration. The rate constants account for the directional flow of mass across the inner and outer membranes that separate the cytosol from the periplasm and the periplasm from the media. Furthermore, included is also the degradation of protein concentration outside of the cell.

Using specific techniques of integration and simplifying for our initial conditions, we arrived at these equations for the respective concentrations.

In order to successfully analyze our model, we needed to rid all of the above equations of their dependence on external concentrations. In other words, we needed to reduce the equations to be, only, in terms of the equation parameters, i.e. kn, D[M], S and components of time.

In order to obtain reliable data and to calibrate our model, we needed to evaluate it for each compartment at their steady states. The steady state occurs when the rate of change of concentration equals zero. These are derived from our Equations 1, 2, & 3.

Through extensive algebraic manipulation and substitution, we were able to reduce the many parameters seen in Equations 5,6 & 7 to these simple “napkin” equations. Notice that the relationships imply exponential decay, which implies that the concentrations will slowly reach a steady state, which is intuitive to a molecule diffusing into a non-infinite fluid.

In order to make the model usable and able to be calibrated, we reduced all parameters to measurable quantities. In doing so, we can then use experimental data to find a quantity for k1 which relates to the steady state concentrations of the respective compartments. We can then use an arbitrary time point to find a corresponding concentration of each compartment, which is all we would need for complete calibration of our system.

In doing so we modify our “napkin” equations to include values we can find experimentally.

Unfortunately this is wrong because of the P and C being functions of time, assuming that P and C are constant gets rid of this issue whilst retaining the general solution for media concentration. We can assume that the concentration of protein in the cytoplasm and periplasm quickly reach a steady state.

We proceeded to make an model using MATLAB’s SimBiology program. SimBiology uses its own system of differential equations based on what we define our system to be, independent of our analytical model. It then created a time course of the different compartment concentrations that allowed us to examine assumptions in our model. Our instructions to SimBiology were to make a 3 compartment system using mass action physics. This includes first order reactions of transferring molecules from compartment to compartment. After plugging in estimated rates of reactions the following graph was obtained.

To simplify our model, we assumed that the mass of cells does not change and specified that the volume occupied by cells was significantly smaller than the volume of the media. We said that this ratio was 1:100. The resulting graph seen above validates our assumption that the Cytosol and Periplasm will reach a biological steady state at time t = t(critical) while the concentration in the media continues to increase. Our analytical model assumes that t(0)=t(critical).
We also modeled a one compartment, and two compartment system. Below is a summary table of all pertinent equations for comparison between systems.