Team:Cornell/project/drylab/modeling/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 is 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 concentration, 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.

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. Modeled current output of the S.A.F.E. B.E.T. sensor is plotted as a function of time for input functions corresponding to varying river analyte concentrations—with oscillatory periods of 90 days (blue), 10 days (green) and 1 day (black). As in Fig. 1, for analyte changes on the order of 90 days, the sensor produces a fine-tuned output corresponding to analyte concentrations within its dynamic range. As the frequency of analyte variation increases, the current response approaches that expected from the time-averaged concentration (pink dash).