Team:Cornell/project/drylab/modeling/time response

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<b>Fig. 1. A sinusoidally oscillating concentration of analyte (with period, T, of 90 days) in the feed stream (green) is plotted alongside the modeled sensor output (blue). For analyte changes on this timescale, current output traverses the entire range of the outputs from basal current of 4 $\mu$A to saturating current just under 6 $\mu$A&#8212; giving a fine-tuned output corresponding to analyte concentrations within the dynamic range.     </b>

Revision as of 06:18, 26 October 2012

Time Response

How do our sensors respond to varying analyte concentration over time?

To make sense of data collected remotely, we must know the time it takes for our sensor to produce a signal in response to a change in analyte concentration in the sample stream. In other words, we need to know whether current output at any time can be taken to represent analyte concentration at that time , or whether our system has memory. From preliminary data presented for our arsenic sensor in our results page, we know the response time of the S.A.F.E. B.E.T. sensor is on the order of 1.5 days. That is, it takes nearly a day and a half for our system to fully respond upon a sudden change in analyte concentration in the reactor.

However, knowledge of the characteristic response time is insufficient to predict the dynamic behavior of our sensor when the concentration of analyte is variable in time. In other words, the response time of the sensor can tell us how long it takes our system to respond to a single perturbation in analyte, but further analysis is necessary in the prediction of the sensor’s response to continuously changing analyte concentration. For example, if changes in concentration occur slowly over time, our sensor may produce a signal that perfectly tracks the analyte concentration —with some lag. On the other hand, if changes occur quickly in relation to the response time of the sensor, we might expect an output corresponding to a time-average of the analyte concentration. Here, we model the current output of our sensor in response to oscillating concentration of analyte in order to predict the time-scales over which the S.A.F.E. B.E.T. sensor filters out fine changes in analyte concentration.

Problem Setup

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam sed nisl quis tellus convallis sagittis. Sed blandit metus at nulla mollis luctus. In eget turpis eros, eget lacinia sem. Curabitur ornare mauris nec lectus convallis vel ornare ligula viverra. Sed vel lectus mattis nisi auctor vestibulum vel nec ipsum. Integer pellentesque dolor lobortis elit viverra vitae vehicula dui cursus. Donec viverra, lectus eu faucibus rutrum, leo risus posuere justo, at rhoncus felis est in nibh. Pellentesque feugiat porta quam nec molestie. Morbi nunc dolor, consectetur in tempus in, hendrerit a augue. Phasellus ultrices volutpat diam vitae tincidunt. Mauris justo leo, blandit et tristique eu, lacinia in risus. Mauris elit eros, sollicitudin quis sodales quis, placerat a ante.

Nam ac pulvinar felis. Mauris vitae erat at orci semper aliquet vitae quis urna. Donec sit amet tortor porttitor diam bibendum viverra. Nam dui nulla, viverra sed lacinia lobortis, ullamcorper et neque. Etiam rhoncus nibh a lacus varius vehicula convallis mi rutrum. Vestibulum vel nunc sit amet ipsum feugiat consectetur. Nulla nec ante vitae dui tristique accumsan. Morbi felis est, ornare a vestibulum vel, vulputate a eros. Nulla facilisi.

Solution

Concentration of Analyte In Reactor
First, we need to solve for the concentration of analyte in the reactor as a function of time ($c(t)$) given an oscillating input of analyte ($[A](t)$)—with arbitrary frequency and amplitude—in the river. By performing a mass balance around the reactor (and ignoring any contribution from generation or consumption), we may write the following differential equation: $$\frac{dc}{dt} = [A](t)\cdot\frac{F}{V}-c\cdot\frac{F}{V}=D([A](t)-c),$$ $$ \mathrm{where} \ \ [A](t) = A_0\cdot\sin(2 \pi ft)+A_0,$$ and the dilution rate, $D$, equals 3.6 day$^{-1}$ for our system. Using $e^{Dt}$ as an integrating factor, we find $$c(t) = \frac{(A0 (D^2 \sin(2 \pi f t)+D^2-2 \pi D f \cos(2 \pi f t)+4 \pi^2 f^2))}{(D^2+4 \pi^2 f^2)}+k_1 e^{-D t}$$ $$ \mathrm{where} \ \ k_1 = - \frac{(A_0 (D^2-2 \pi D f+4 \pi^2))}{(D^2+4 \pi^2 f^2)} \ \ \mathrm{such\ that} \ c(0) =0 $$

Concentration of Analyte In Reactor
Now that we have an analytical solution for the concentration of analyte in the reactor over time, we wish to model the current response given the input function $c(t)$. To accomplish this, we model the time rate of change of current of our arsenic sensor using a Hill function [1] with a cooperativity coefficient of unity—fitting data presented in Fig. X of our Current Response characterization page—and lumping all transcriptional, translational, and post-translational processes: $$\frac{dI}{dt} = \frac{\beta_1 \cdot c(t)}{c_{1/2}+c(t)}+\beta_0 - \alpha I, $$ where $c(t)$ is as defined above; $\beta_1$ is 1.4 $\mu$A/day, $\beta_0$ is 2.8 $\mu$A/day, $c_{1/2}$, the half-saturating analyte concentration, is 100 $\mu$M, $\alpha$ = 0.69 day$^{-1}$; the saturating current output is $(\beta_1 + \beta_0)/\alpha$.

Using this ordinary differential equation, we may numerically solve for current as a function of time—given the input function derived above. The results of such numerical solutions for various input function—performed using the differential equation solver ode45 in MATLAB—are shown below.

Results: Time-Averaged Output for Rapidly Oscillating Analyte Concentrations

Fig. 1. A sinusoidally oscillating concentration of analyte (with period, T, of 90 days) in the feed stream (green) is plotted alongside the modeled sensor output (blue). For analyte changes on this timescale, current output traverses the entire range of the outputs from basal current of 4 $\mu$A to saturating current just under 6 $\mu$A— giving a fine-tuned output corresponding to analyte concentrations within the dynamic range.

Fig.2. Nam ac pulvinar felis. Mauris vitae erat at orci semper aliquet vitae quis urna. Donec sit amet tortor porttitor diam bibendum viverra. Nam dui nulla, viverra sed lacinia lobortis, ullamcorper et neque. Etiam rhoncus nibh a lacus varius vehicula convallis mi rutrum. Vestibulum vel nunc sit amet ipsum feugiat consectetur. Nulla nec ante vitae dui tristique accumsan. Morbi felis est, ornare a vestibulum vel, vulputate a eros. Nulla facilisi.

References

[1] Hill, A. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: 4–7.