A steady state is a condition in which the state variables of a model—such as species concentrations—do not change over time. Mathematically, this is expressed by setting the differential equations describing the time evolution of the metabolic system to zero. This creates a system of algebraic nonlinear equations. COPASI can use multiple strategies and numerical methods to solve them.
Internally, all calculations operate on particle numbers and particle number rates, rather than concentrations. The reduced model (Deterministic Interpretation of the Model) is used for these computations. The Jacobian (utilized in the Newton method and for computing eigenvalues) is calculated using finite differences. Eigenvalues of the Jacobian are computed using LAPACK.
Use Newton
Boolean value. Determines whether to use the damped Newton method
to solve the nonlinear algebraic equations defining the steady state.
The initial concentrations set by the user are used as the starting guess.
Default: true (use Newton).
The damped Newton method is a modified version of the classical Newton method, which iteratively refines a guess until the residual error is below a threshold. If too many iterations are needed without finding an acceptable solution, the method halts with no result.
The classic Newton iteration is: \(x_i = x_{i-1} - \frac{f(x_{i-1})}{f'(x_{i-1})}\) In the damped method, if $x_{i-1}$ has a larger residual error than $x_i$, the following is used: \(x_i = x_{i-1} - \frac{f(x_{i-1})}{f'(x_{i-1})} \cdot 2^{-n}, \quad n = 0, \ldots, 32\) The first value providing a smaller residual error is accepted. If none is found, the procedure stops (local minimum).
Use Integration
Boolean value. Determines whether to use the
deterministic ODE solver
to simulate the system forward in time until a steady state is reached.
If a steady state is not reached within $10^{10}$ units of time, the
method stops with no solution. If “Use Newton” is set, the Newton method
is attempted at each intermediate time point. Default: true (use integration).
Use Back Integration
Boolean value. Specifies whether to use the deterministic ODE solver to
simulate the system backward in time until a steady state is reached.
The stopping criteria are the same as above. If “Use Newton” is set,
the Newton method is attempted at each step. Value of true means back
integration is used.
Accept Negative Concentrations
Boolean value. Determines whether steady states yielding negative
concentrations are acceptable. true allows negative concentrations;
false (default) discards such results.
Iteration Limit
Positive integer. Sets the maximum number of iterations for the damped
Newton method before it reports failure. Default: 50.
Derivation Factor
Numeric value. Step size used for numerical estimation of
$f’(x_{i-1})$. Default: 0.001.
Resolution Positive numeric value. Specifies the threshold for considering the system to be at steady state—if the absolute change in every state variable is below this value, the system is at steady state. Default: $10^{-9}$.
Note: Although the calculation internally uses particle numbers, this value is interpreted as a concentration (heuristically). Modelers should check that the default is appropriate for their model units.
Target Criterion Specifies the acceptance criterion for the steady state. Options include:
