Team:Valencia/Modeling

pH variations with light

We have successfully simulated the behaviour of pH changes in response to irradiation on cyanobacteria Synechocystis sp. PCC 6803, as you can see by this graph:

Why is this important?

A model is not good or wrong, is more or less useful. Our model apart from making us think about the causes and relationships among the effects involved in this mechanism, has help us understand some results of the pH variations in our pH-stat.

How have done this?

First of all, we have to think on the physics of the behaviour we wanted to describe. We have light arriving upon a cyanobacteria and pH changing more or less proportionately to the time of irradiation.

Thus, we have a light input and a pH variation output.

Now we need a transfer function that giving a light input delivers and output such as the one observed. To that purpose we can observe that the response (pH variation) behaves like a first order system with a lagging time.

We work on the frequency plane (using Laplace transforms). Thus, transfer function of a first order system with a lagging time is such as:

Using the transfer function of the system we can get the response such as:

y(t) = pH - pH₀
y(s)= L [y(t)] where L[] function is Laplace transform
y(s) = G(s)·u(s)

With the experimental data we have obtained the parameters values:

And with them we can adjust almost perfectly the experimental upwards dynamics:

We assume that the dynamics downwards should have the same underlying physics, so parameters should remain the same, we obtain a theoretical simulation and tallies real data:

As we can see, they are pretty much equivalent.
Anyway, let's consider now more than one peak. If we have different pulses of light, it's straight ofrward to think that we'll have different peaks of pH variations, but, if we look closer to the time series, we observe that gain response is weaker as time goes by and the basal value follows a slow, but constant upward tendency.

In order to model this behaviour we'll use two functions:
On one hand a buffering function that will multiply the response function

On the other hand a function that models appropriately the increase of a basal point is the following one and should be added to the response function.

As we have done before, we can calculate the following parameters:

With them, we can get the final equations:
y(s) = G(s)·u(s)
y(s)= L^-1[y(s)] (reversed Laplace transform)
Y(t) = y(t)·A(t)+B(t)
pH=Y(t) + pH₀

Function y(t) is not easy to describe as it is a stepwise function depending of the time of the simulation, it is like:

Now, just run the set of equations and plot the results accordingly and you have the graph:

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