LLRQ - Log-Linear Reaction Quotient Dynamics

A Python package for analyzing chemical reaction networks using the log-linear reaction quotient dynamics framework described in Diamond (2025) "Log-Linear Reaction Quotient Dynamics" (arXiv:2508.18523).

Overview

This framework introduces a novel approach to modeling chemical reaction networks where reaction quotients (Q) evolve exponentially toward equilibrium when viewed on a logarithmic scale. Unlike traditional mass action kinetics, this yields analytically tractable linear dynamics in log-space.

Key Equation

For reaction networks, the dynamics follow:

d/dt ln Q = -K ln(Q/Keq) + u(t)

Where:

• Q: Vector of reaction quotients measuring distance from equilibrium
• Keq: Vector of equilibrium constants
• K: Relaxation rate matrix
• u(t): External drive vector (e.g., ATP/ADP ratios)

Framework Advantages

1. Analytical Solutions: Exact solutions exist for arbitrary network topologies
2. Thermodynamic Integration: Automatic incorporation of constraints via ΔG = RT ln(Q/Keq)
3. Decoupled Dynamics: Conservation laws separate from reaction quotient evolution
4. Linear Control: External energy sources couple linearly to dynamics
5. Tractable Analysis: Decades of linear systems theory become applicable

Installation

From PyPI (when available)

pip install llrq

From source

git clone <repository-url>
cd llrq
pip install -e .

Requirements

• Python ≥3.8
• numpy ≥1.20.0
• scipy ≥1.7.0
• matplotlib ≥3.3.0
• python-libsbml ≥5.19.0
• tellurium ≥2.2.0
• cvxpy ≥1.7.2

Quick Start

Simple Reaction Example

import llrq
import numpy as np

# Create a simple A ⇌ B reaction
network, dynamics, solver, visualizer = llrq.simple_reaction(
    reactant_species="A",
    product_species="B",
    equilibrium_constant=2.0,
    relaxation_rate=1.0,
    initial_concentrations={"A": 1.0, "B": 0.1}
)

# Solve the dynamics
solution = solver.solve(
    initial_conditions={"A": 1.0, "B": 0.1},
    t_span=(0, 10),
    method='analytical'
)

# Visualize results
fig = visualizer.plot_dynamics(solution)

SBML Model Example

import llrq

# Load SBML model
network, dynamics, solver, visualizer = llrq.from_sbml(
    'model.xml',
    equilibrium_constants=np.array([2.0, 1.5, 0.8]),  # Keq for each reaction
    relaxation_matrix=np.diag([1.0, 0.5, 1.2])        # K matrix (diagonal)
)

# Solve with external drive
def atp_drive(t):
    return np.array([0.5, 0.0, -0.2])  # Drive for each reaction

dynamics.external_drive = atp_drive
solution = solver.solve(
    initial_conditions=network.get_initial_concentrations(),
    t_span=(0, 20)
)

# Plot results
fig = visualizer.plot_dynamics(solution)

API Reference

Core Classes

• SBMLParser: Parse SBML models and extract network information
• ReactionNetwork: Represent reaction networks with stoichiometry and species
• LLRQDynamics: Implement log-linear dynamics system
• LLRQSolver: Solve dynamics with analytical and numerical methods
• LLRQVisualizer: Create publication-quality plots

Control Systems

• LLRQController: Base controller class with LQR optimal control
• CVXController: Advanced optimization-based control using CVXPY
• FrequencySpaceController: Frequency-domain control for sinusoidal inputs
• ThermodynamicAccountant: Entropy production calculations and energy balance

Convenience Functions

• llrq.from_sbml(): Load SBML model and create complete LLRQ system
• llrq.simple_reaction(): Create simple A ⇌ B reaction system
• llrq.simulate_to_target(): One-line controlled simulation to target state
• llrq.compare_control_methods(): Compare LLRQ vs mass action control performance
• llrq.create_entropy_aware_cvx_controller(): Create entropy-aware optimization controller

Examples

See the examples/ directory for complete working examples:

Basic Usage

• simple_example.py: Basic A ⇌ B reaction with visualization
• linear_vs_mass_action_simple.py: Compare LLRQ approximation with mass action
• mass_action_example.py: Mass action kinetics integration

Control Systems

• lqr_complete_example.py: Linear Quadratic Regulator control design
• cvx_sparse_control.py: Sparse control using L1 regularization
• cvx_constrained_control.py: Control with constraints and bounds
• cvx_custom_objective.py: Custom optimization objectives
• frequency_space_control.py: Frequency-domain control design
• frequency_space_simulation.py: Sinusoidal control simulation

Advanced Features

• entropy_production_demo.py: Thermodynamic entropy calculations
• entropy_aware_control.py: Control design with entropy constraints
• frequency_entropy_control.py: Frequency-domain entropy optimization
• integrated_control_demo.py: Complete control workflow demonstration

Mathematical Framework

Single Reaction Dynamics

For a single reaction with external drive:

d/dt ln Q = -k ln(Q/Keq) + u(t)

Analytical solution (constant u):

Q(t) = Keq * exp[(ln(Q₀/Keq) - u/k) * exp(-kt) + u/k]

Multiple Reactions

Vector form for reaction networks:

d/dt ln Q = -K ln(Q/Keq) + u(t)

Matrix exponential solution (constant u):

ln Q(t) = exp(-Kt) * [ln(Q₀/Keq) - K⁻¹u] + K⁻¹u + ln Keq

Connection to Mass Action

For single reaction A ⇌ B with mass action rates kf, kr:

• Equilibrium constant: Keq = kf/kr
• Relaxation rate: k = kr(1 + Keq) (ensures agreement near equilibrium)

Control Theory Integration

The linear dynamics enable direct application of control theory:

LQR Control: For quadratic cost J = ∫[x^T Q x + u^T R u]dt:

u*(t) = -R⁻¹B^T P x(t)

where P solves the Riccati equation: PA + A^T P - PBR⁻¹B^T P + Q = 0

Frequency Response: Transfer function H(s) = (K + sI)⁻¹B enables:

• Bode plot analysis for stability margins
• Optimal sinusoidal control design
• Frequency-domain entropy optimization

Thermodynamic Constraints: Entropy production rate:

dS/dt = x^T K x + x^T u

enables entropy-aware control design balancing performance and thermodynamic cost.

Applications

This framework enables:

Traditional Applications

• Metabolic Engineering: Optimize pathway design using K as design variable
• Drug Discovery: Predict drug effects throughout metabolic networks
• Systems Medicine: Classify metabolic disorders via eigenvalue analysis

Advanced Control Applications

• Optimal Control: LQR and model predictive control for metabolic regulation
• Sparse Control: L1-regularized control for minimal intervention strategies
• Constrained Optimization: Control with resource limits and safety constraints
• Frequency-Domain Design: Sinusoidal control for periodic metabolic cycles
• Thermodynamic Control: Entropy-aware control balancing performance and energy cost
• Robust Control: Uncertainty-aware control for model variations
• Real-Time Control: Adaptive control with online parameter estimation

Citation

If you use this package in research, please cite:

@article{diamond2025loglinear,
  title={Log-Linear Reaction Quotient Dynamics},
  author={Diamond, Steven},
  journal={arXiv preprint arXiv:2508.18523},
  year={2025}
}

License

Licensed under the Apache License 2.0. See LICENSE file for details.

Contact

• Author: Steven Diamond
• Email: steven@gridmatic.com
• Paper: arXiv:2508.18523