Our mathematical model describes the spatiotemporal dynamics of the LOGIC biological computing system. We developed a comprehensive framework combining reaction-diffusion equations, cellular response dynamics, and signal processing to predict system behavior and optimize design parameters.
Model
Overview
Fundamental Equations
Signal Molecule Diffusion
The concentration of AHL molecules in the culture medium follows the reaction-diffusion equation:
\[\frac{\partial C}{\partial t} = D\nabla^2C - \lambda C + S(x,y,t)\]
Where $C$ represents AHL concentration, $D$ is the diffusion coefficient, $\lambda$ denotes the decay rate, and $S(x,y,t)$ is the source term representing AHL release.
For numerical computation, we discretize using the finite difference method:
\[C_{i,j}^{n+1} = C_{i,j}^n + \Delta t \left[ D\frac{C_{i+1,j}^n + C_{i-1,j}^n + C_{i,j+1}^n + C_{i,j-1}^n - 4C_{i,j}^n}{\Delta x^2} - \lambda C_{i,j}^n \right]\]
Cellular Response Dynamics
The production of output molecules by engineered cells follows Hill kinetics:
\[\frac{d[Output]}{dt} = \alpha \frac{[AHL]^n}{K_d^n + [AHL]^n} - \beta[Output]\]
Where $\alpha$ represents the maximum production rate, $K_d$ is the dissociation constant, $n$ denotes the Hill coefficient, and $\beta$ is the degradation rate.
Logic Gate Transfer Functions
Different cell types implement distinct logical operations through varied response curves:
AND Gate (High-pass filter):
\[Output = \begin{cases} 1 & \text{if } [AHL_{total}] > \theta_{high} \\ 0 & \text{otherwise} \end{cases}\]
XOR Gate (Band-pass filter):
\[Output = \begin{cases} 1 & \text{if } \theta_{low} < [AHL_{total}] < \theta_{high} \\ 0 & \text{otherwise} \end{cases}\]
Amplifier:
\[Output = \min(\gamma \cdot [AHL_{input}], Output_{max})\]
Multi-molecule System
For the complete system with multiple orthogonal AHL molecules, we solve coupled PDEs:
\[\frac{\partial C_k}{\partial t} = D_k\nabla^2C_k - \lambda_k C_k + \sum_i S_{k,i}(x_i,y_i,t)\]
Where $k$ indexes different AHL types.
Parameter Estimation
Diffusion coefficients were estimated based on molecular weight and medium properties, with larger molecules exhibiting slower diffusion rates. Decay rates account for both hydrolysis and enzymatic degradation, showing temperature dependence following Arrhenius kinetics. Response thresholds were determined through iterative parameter fitting to experimental data.
Timing Analysis
Signal Propagation Time
The time for signal to reach threshold concentration at distance $r$:
\[t_{signal} = \frac{r^2}{4D} + \frac{1}{\lambda}\ln\left(\frac{C_0}{C_{threshold}}\right)\]
Optimal Input Timing
For cascaded full adders, the second input should be applied after:
\[t_{optimal} = t_{diffusion} + t_{response} + t_{carry}\]
Validation Strategy
Concentration Verification
Using Beer-Lambert law for optical density measurements:
\[A = \epsilon \cdot l \cdot C\]
Relating pixel intensity to concentration:
\[I_{pixel} \propto e^{-A} = e^{-\epsilon l C}\]
GFP Quantification
Fluorescence intensity relates to GFP concentration:
\[F = \phi \cdot I_{excitation} \cdot \epsilon_{GFP} \cdot [GFP]\]
System Performance Metrics
\[Accuracy = \frac{N_{correct}}{N_{total}} \times 100\%\]
\[SNR = 20\log_{10}\left(\frac{Signal_{ON}}{Signal_{OFF}}\right)\]
\[\chi_{ij} = \frac{Response_i(AHL_j)}{Response_i(AHL_i)}\]
Optimization Results
Through systematic parameter sweeps, we identified optimal configurations for component spacing, input concentrations, temperature conditions, and pH ranges that maximize system performance while maintaining biological viability.
Future Improvements
Our modeling framework can be extended to incorporate stochastic effects, three-dimensional diffusion patterns, adaptive control mechanisms, and machine learning approaches for parameter optimization.