GM Solver

Welcome to GM Solver, your accessible platform for exploring Growth Mechanics (GM) in cellular biology. Unlike traditional static models, GM Solver allows you to simulate the optimal dynamic behavior of self-replicating cells. It solves the "Equations of Optimal Motion" (EOM) derived from first principles, predicting how a cell should dynamically allocate resources to maximize fitness over time in changing environments. In this tutorial, you will learn how to use GM Solver’s intuitive interface to construct nonlinear dynamical models without writing any code. You will discover how to integrate kinetic data from the BRENDA enzyme database, define time-dependent environmental conditions (such as nutrient shifts or oscillations), and analyze the resulting dynamic trajectories. We will guide you through our interactive visualization tools, including time-resolved plots and dynamic metabolic pathway maps. GM Solver empowers experimental biologists, bioengineers, and students to investigate the fundamental principles of cellular growth. GM Solver is freely accessible to all users, promoting open science and collaboration within the research community.

Model Input

In GM Solver, getting started is seamless. Users can upload their model file in an open-source spreadsheet format (ODS) via the “Choose File” button. If you are starting from scratch, you can download a preformatted GM template using the “Download GM template” button (Figure 1). This template provides the correct structure for all required matrices—including Mass Fractions, Kinetics, and Environmental Functions—ensuring your model is ready for simulation.

Figure 1: Interface for uploading your GM model file and downloading the GM template.

The mass fraction matrix M is the core structural component of your model. It represents the distribution of mass for each reactant across the system's reactions. In the GM framework, mass conservation is a fundamental principle.

• Rows represent Metabolites. Rows starting with "x_" denote external reactants (nutrients in the medium).
• Columns represent Reactions.
• The last row, labeled "Protein", represents the total concentration of proteins.
• The last column, labeled "Ribosome", represents the synthesis machinery responsible for producing proteins.

Normalization: Unlike simple stoichiometric matrices, the entries in M are mass fractions. This means the sum of negative entries (reactants consumed) must equal -1, and the sum of positive entries (products formed) must equal +1 for each reaction column. This ensures that mass is strictly conserved in every step of the metabolic network.

Figure: Example of a Mass Fraction Matrix (M) for a cellular system in GM Solver.

Growth Mechanics relies on nonlinear enzyme kinetics to predict economic trade-offs. GM Solver organizes these parameters into specific matrices, which must align with the structure of your Mass Fraction matrix (M).

K Matrix (Michaelis Constants)

The K matrix defines the Michaelis constant (K_m) for each metabolite-reaction pair.

• Values describe the substrate concentration [g L⁻¹] at which the reaction rate is half-maximal.
• If specific data is unavailable, GM Solver applies a conservative default value of 0.1 g L⁻¹.

Figure: Example of the K Matrix (Michaelis Constants).

Turnover Numbers (kcat)

The kcat matrix defines the catalytic efficiency of the enzymes. In the GM framework, these values determine the "turnover times" (τ), linking metabolic flux to proteome allocation.

• kcat_f: Turnover numbers for the forward reaction [h⁻¹].
• kcat_b: Turnover numbers for the backward reaction [h⁻¹].

Figure: Example of the kcat Matrix containing forward and backward turnover numbers.

Regulatory Constants (KI and KA)

To capture complex regulation, you can define Inhibition (KI) and Activation (KA) constants. These matrices allow you to model how specific metabolites inhibit or activate enzymes, adding a layer of biological realism to the dynamic trajectory.

BRENDA Database Integration

Finding accurate kinetic parameters can be challenging. GM Solver integrates directly with the BRENDA enzyme database. An interactive table allows you to filter by organism, EC number, or substrate to retrieve experimentally validated K_m and kcat values, facilitating the construction of biologically accurate models.

Figure: Interactive BRENDA database table for retrieving enzyme kinetic parameters.

GM Solver allows you to incorporate experimental data and initialization parameters to refine the optimization process.

Amplitude Priors (Δq₀)

This vector sets the initial amplitudes for metabolite and protein oscillations. In the Growth Mechanics framework, these values initialize the solver by defining the departure from a steady state. They help the numerical integrator converge on the optimal dynamic trajectory.

Growth and Proteome Data (μ, φ)

You can optionally provide experimentally measured growth rates (μ) and proteome fractions (φ). When enabled, GM Solver can use this data to constrain the model or estimate kinetic parameters that are consistent with the observed macroscopic growth behavior.

Figure: Input interface for Amplitude Priors and experimental growth data.

A key feature of GM Solver is its ability to simulate growth in changing environments. Unlike static models, you define time-dependent functions for external nutrient concentrations, denoted as a(t).

Integration Settings and Cell Density

In the "rho" tab, you define the global simulation parameters:

• Cell Density (ρ): The total cell mass density [g L⁻¹], which is a fixed constraint in the GM framework.
• Simulation Time (T_end): The total duration of the simulation in hours.
• Step Size (ΔT): The time step [h] used for the numerical integration of the equations of motion.

Figure: Input interface for defining the simulation time duration and integration step size.

External Concentration Functions a(t)

For each external reactant (rows starting with "x_"), you assign a function describing its concentration over time. The available forms include:

Users can edit the coefficients for each function to customize the specific concentrations. The interface provides informative tooltips and renders the mathematical equation for your selection.

Crucially, a Preview Table (0–3 h) is generated to help you sanity-check values [g L⁻¹] before running full simulations, ensuring that the environmental conditions are biologically realistic.

Figure: Interface for defining time-dependent external concentrations a(t) and previewing the environment.

Solving the GM Equations of Optimal Motion requires a numerically consistent model. The “Check Model” feature is your first step before simulation.

Figure: The “Check Model” feature verifies structural and numerical integrity.

Validation Checks

The validator performs rigorous checks to ensure the solver can proceed:

• Structural Integrity: Ensures matrix dimensions match and all rows/columns are named.
• Mass Balance: Verifies that mass fraction columns sum to 0.
• Positivity: Checks that kinetic parameters (K_m, k_cat) and cell density (ρ) are positive.
• Environment Validity: Evaluates your defined a(t) functions over the entire simulation time (T_end) to ensure concentrations remain positive throughout the experiment.

Figure: A successfully validated model is ready for simulation.

Solving the Equations of Optimal Motion

Once validated, clicking “Run” initiates the solver. GM Solver uses the daspk solver (from the deSolve package) to integrate the differential-algebraic equations (DAE) that govern the optimal trajectory.

A progress bar tracks the integration in real-time, providing estimates of the remaining calculation time. For typical models, the optimization is highly efficient, often completing in seconds.

Figure: The progress bar tracks the integration of the optimal dynamic trajectory.