# Differential Equations

We use a mathematical model to check if the design works before starting to work at the laboratory. To develop this model the first step is to create a deterministic model, which is based on differential equations. This type of model describes the mean behavior for each of the substances in the synthetic circuit over time. However, it does not take into account the probabilities involved in each of the events described, the population interactions and the noise of the system.

The deterministic model is based on the law of mass conservation and expresses the inputs and outputs of the system with expressions that use the law of mass action and known models like Hill's or Michaelis Menten.

Below are shown and explained the differential equations that allowed us to work in both projects.

### Glucocorticoid Sensor

Below it is shown differential equations for the glucocorticoid sensor based on the circuit shown in the project description. However, before describing the modeled system, it is necessary to know all the symbols that will be used in the documents and the simulation. The following table contains all the substances involved in the processes and the constants required for the simulation.

### Nickel removal system

As in the Glucocorticoid Sensor project the differential equations were based on the circuit shown in the project description . Before explaining the equations ,it is necessary to know all the symbols that will be use in the documents and the simulation. The following table contains all the substances involved in the processes and the constants required for the simulation.

As you can see there are many constants that define the system, so the next step in the mathematical model is to fin those parameters

### Results

After running the scripts for deterministic model if the Nickel removal system and the parameters found in the literature the results showed that after 15 minutes approximately the system is stable and the bacteria introduced most of the nickel leaving the outside concentration of nickel under the maximum limit of nickel in water: