Mathematical Modeling
Mathematical modeling is an essential part of our iGEM project because it allows us to go beyond raw data and extract meaningful insights about how our biosensor works. By fitting equations to our experimental results, we can describe sensor behavior in a precise, quantitative way. This enables us to compare performance across conditions, predict sensor responses to new samples, and determine the optimal way to use our biosensor in the field.
In our project, modeling had three major goals:- Understand the time course of fluorescence to identify the optimal detection window.
- Quantify the dose–response curve to measure sensitivity and define thresholds.
- Calculate key performance indicators (KPIs) such as detection limits, dynamic range, and response time.
Together, these models support our ultimate aim: to create a biosensor that is simple, reliable, and useful for farmers and families testing rice for arsenic contamination.
Time-Series Analysis of Biosensor FluorescenceIn addition to studying endpoint dose–response, we examined how the fluorescence signal changed over time. Time-series analysis helps us determine the response time needed for a reliable readout, understand the kinetic behavior of the biosensor, and identify the optimal detection window. This complements the dose–response analysis, which focuses on sensitivity at fixed times.
Mathematical ModelWe modeled the fluorescence over time with the equation:
Where:
- F₀: baseline fluorescence (accounts for leakage)
- A: amplitude of the activation burst caused by arsenic
- k: rate constant for the rise of the signal
- γ: decay constant, reflecting resource depletion or dye unbinding
This model captures the burst–plateau–decay shape observed in the data.
Biochemical InterpretationThe general shape of the time-series curves reflects the underlying biology of the system: Burst phase: Strong transcription initially drives Broccoli aptamer fluorescence. Repression at 0 ppb: ArsR quickly binds the promoter, shutting down expression after the initial burst.
Activation at 5–25 ppb: Arsenic binds ArsR, reducing repression and sustaining transcription. Decline phase: After ~45 minutes, fluorescence plateaus or falls as resources run out and arsenic toxicity builds.
This explains why the 20–30 minute range provides the clearest distinction between arsenic concentrations which we will see in the response-curve analysis later.
Fitted ParametersWe fitted the model to time-series data at 0, 5, 10, and 25 ppb. The table below shows the parameter estimates:
These parameters provide biological insights: higher arsenic concentrations increased amplitude (A) and slowed decay (lower γ), producing stronger and longer-lasting fluorescence.
Comparison of Time-Series Traces
This plot overlays fluorescence at different ppb concentration levels. It shows the burst, plateau, and decay phases, and demonstrates how arsenic concentrations affect the shape and magnitude of the signal.
Dose–Response Analysis for Arsenic Biosensor
Purpose of the Dose–Response ModelAs a team, we want our biosensor to give a clear, trustworthy answer: is rice safe or unsafe at the FDA’s infant rice threshold of 100 ppb arsenic? To achieve this, we modeled the relationship between arsenic concentration (ppb) and sensor fluorescence. This allows us to:
- Translate fluorescence → ppb concentration for the phone app
- Select the best time window to read the test strip
- Calculate Key Performance Indicators (KPIs) such as the limit of detection (LOD), dynamic range, and signal-to-noise ratio (SNR)
The 4-Parameter Logistic (4PL) Model
Where:
- A₁: minimum plateau (baseline signal at very low arsenic)
- A₂: maximum plateau (upper limit when the sensor is fully activated)
- K: midpoint concentration (EC50), showing system sensitivity
- n: slope (Hill coefficient), describing how sharply the sensor responds to arsenic
Leakage and Its Treatment in Modeling
One challenge we faced in our biosensor development was fluorescence leakage at 0 ppb arsenic. Even in the absence of arsenic, the system sometimes produced measurable signal due to incomplete repression by ArsR or other sources of background activation. Leakage reduces the contrast between the negative control (0 ppb) and the lowest positive concentrations (e.g., 5 ppb), which is critical for detecting low levels of contamination.
In our modeling process, we addressed leakage in two main ways:
Background subtraction: Each timepoint included dedicated background wells, and their average readings were subtracted from the fluorescence of all test wells. This reduced noise caused by non-specific fluorescence.
Curve fitting with flexible minimum plateau (A₁): In the 4-parameter logistic (4PL) model, the minimum asymptote (A₁) was not fixed at zero. Instead, it was estimated from the data, allowing the model to account for leakage at 0 ppb. This ensured that the dose–response curve accurately reflected the real baseline of the system rather than assuming perfect repression.
By explicitly modeling leakage, we ensured that our biosensor’s limit of detection and signal-to-noise ratio were calculated in a realistic way. This strengthens our conclusions and helps future teams refine promoter and repressor designs to further reduce leakage in similar biosensor systems.
Comparison of Read TimesWe fitted the dose–response model at five specific times: 5 min, 10 min, 30 min, 60 min, and 90 min. This helps us see how the sensor develops over time.
What we observed:
- 5–10 min: Signals are still forming. Responses are noisy, SNR is low, and the curves are unreliable.
- 30 min: A clear S-shaped curve emerges with good separation.
- 60–90 min: The system begins to plateau and drift, with less distinction between concentrations.
Dose–Response at 5 min
Dose–Response at 10 min
Dose–Response at 30 min
Dose–Response at 60 min
Dose–Response at 90 min
Optimal Detection Window
We systematically tested every minute from 5 to 90 and scored each by combining fit quality (R²) with signal-to-noise at 100 ppb. The optimal detection window occurred in the 20–30 min range, where the biosensor gave its strongest, most reliable separation between 0 and 100 ppb.
Biologically, this matches what we observed in the lab: an initial burst of transcription that provides a strong signal before resource limitations set in. Analytically, this is where the dynamic range and SNR are highest, making it the best time to lock in the readout for the field kit.
Refined 25-Minute Dose–Response Fit
When we examined the 25-minute data more closely, we found that the logistic fit produced an extremely steep slope (n ≈ 20), which made some KPIs unstable. To make the model more realistic, we constrained the slope parameter to a maximum of n = 5. This produced a stable and biologically interpretable curve.
With the constrained fit at 25 minutes:
- A₁ ≈ 33,358 (minimum baseline)
- A₂ ≈ 1,981 (maximum response)
- K ≈ 7.4 ppb (EC50, midpoint concentration)
- n ≈ 5.0 (slope at the upper bound)
- R² ≈ 0.919 (fit quality)
This refined model supports the conclusion that 25 minutes is the optimal detection window: signals are clear, the model is interpretable, and key performance indicators are well defined.
Key Performance Indicators (KPIs)
To evaluate the effectiveness of our biosensor, we identified a set of Key Performance Indicators (KPIs). These KPIs help us measure how sensitive, accurate, and reliable the sensor is for detecting arsenic in rice samples. They also allow us to compare different designs and operating conditions objectively.
Limit of Detection (LOD)The Limit of Detection (LOD) tells us the smallest amount of arsenic that our biosensor can reliably detect above the background noise. We calculate it as the fluorescence signal of the 0 ppb blank plus three times its standard deviation (Blank + 3σ). In context, the LOD is important because it determines whether our test can detect arsenic at levels relevant to food safety. Our models suggest the biosensor can detect concentrations well below the FDA’s 100 ppb safety threshold, which is a promising result.
Dynamic RangeThe dynamic range describes the range of arsenic concentrations where the biosensor gives a useful, measurable response. For our system, the optimal dynamic range was between 5 and 100 ppb arsenic, which aligns well with the contamination levels reported in rice products. This range ensures that the sensor is most effective at detecting concentrations that matter for food safety.
Response TimeResponse time refers to how long it takes before the biosensor produces a clear and reliable signal. From our dose–response time course, we determined that the optimal detection window is around 25 minutes. This is fast enough to be practical in real-world settings, such as farmers or families testing rice in the kitchen, while still allowing the system to generate a strong and interpretable signal.
Signal-to-Noise Ratio (SNR)The Signal-to-Noise Ratio (SNR) compares the strength of the fluorescence signal at a given arsenic concentration (e.g., 100 ppb) to the variability of the blank (0 ppb). A higher SNR means that the biosensor can more clearly distinguish arsenic-containing samples from arsenic-free samples. At 25 minutes, the SNR is maximized, making this the best time to lock in a test result.
ConclusionBy analyzing KPIs such as LOD, dynamic range, response time, and SNR, we confirmed that our biosensor performs best in the 20–30 minute window, with 25 minutes being the optimal detection time. These indicators show that the system is sensitive enough to detect arsenic at relevant levels, produces results quickly, and can be trusted to distinguish safe from unsafe rice samples. This KPI analysis supports our project’s goal of developing a simple, reliable, and community-friendly biosensor.
