pH variations with light
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 - pH
0
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.
--imatge llarga--
In order to model this behaviour we'll use two functions:
On one hand a buffering function that will multiply the response function
--imatge A(t)--
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.
--imatge B(t)--
As we have done before, we can calculate the following parameters:
--imatge tres param--
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 t describe as it is a stepwise function depending of the time of the simulation, it is like:
--imatge y(t)--
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