sign_vector_conditions.robustness

Robustness of existence and uniqueness of equilibria

Let us consider the following matrices:

sage: S = matrix([[1, 0, 1, 0], [0, 0, 0, 1]])
sage: S
[1 0 1 0]
[0 0 0 1]
sage: St = matrix([[1, 0, 1, 1], [0, 1, 0, -1]])
sage: St
[ 1  0  1  1]
[ 0  1  0 -1]

To check, whether the corresponding chemical reaction network has a unique equilibrium for all rate constants and all small perturbations of St, we consider the topes of the corresponding oriented matroids:

sage: from sign_vectors.oriented_matroids import *
sage: topes_from_matrix(S, dual=False)
{(+0+-), (+0++), (-0--), (-0-+)}
sage: topes_from_matrix(St, dual=False)
{(---+), (-+--), (++++), (----), (+-++), (+++-)}

One can see that for every tope X of the oriented matroid corresponding to S there is a tope Y corresponding to St such that X conforms to Y. Therefore, the exponential map is a diffeomorphism for all c > 0 and all small perturbations of St. The package offers a function that checks this condition directly:

sage: from sign_vector_conditions import *
sage: condition_closure_sign_vectors(S, St)
True

There is an equivalent condition. To verify it, we compute the maximal minors of the two matrices:

sage: S.minors(2)
[0, 0, 1, 0, 0, 1]
sage: St.minors(2)
[1, 0, -1, -1, -1, -1]

From the output, we see whenever a minor of S is nonzero, the corresponding minor of St has the same sign. Hence, this condition is fulfilled. This condition can also be checked directly with the package:

sage: condition_closure_minors(S, St)
True

Now, we consider matrices with variables:

sage: var('a, b, c')
(a, b, c)
sage: S = matrix([[c, 1, c]])
sage: S
[c 1 c]
sage: St = matrix([[a, b, -1]])
sage: St
[ a  b -1]

We cannot check the first condition since there are variables in S and St. Therefore, we want to obtain equations on the variables a, b, c such that this condition is satisfied. First, we compute the minors of the matrices:

sage: S.minors(1)
[c, 1, c]
sage: St.minors(1)
[a, b, -1]

The function from the package supports symbolic matrices as input. In this case, we obtain the following equations on the variables:

sage: condition_closure_minors(S, St) # random
[{-b > 0, c == 0},
 {-b < 0, c == 0},
 {-b > 0, c > 0, -a*c > 0},
 {-b < 0, c < 0, -a*c < 0}]

Thus, there are four possibilities to set the variables: From the first two sets of conditions, we see that the closure condition is satisfied if c is zero and b is nonzero. The closure condition is also satisfied if a and b are negative and c is positive or if a and b are positive and c is negative.

We can also apply the built-in function solve_ineq to the resulting sets of inequalities. For instance, the last set can be equivalently written as:

sage: solve_ineq(list(condition_closure_minors(S, St)[3])) # random
[[c < 0, 0 < b, a < 0]]

Functions

condition_closure_minors(...)

Closure condition for robustness using maximal maximal minors.

condition_closure_sign_vectors(...)

Closure condition for robustness using sign vectors.
sign_vector_conditions.robustness.condition_closure_minors(stoichiometric_matrix, kinetic_order_matrix)

Closure condition for robustness using maximal maximal minors.

INPUT:

  • stoichiometric_matrix – a matrix with maximal rank

  • kinetic_order_matrix – a matrix with maximal rank

OUTPUT: Return whether the closure condition for robustness regarding small perturbations is satisfied. If the result depends on variables, a list of sets is returned. The condition holds if the inequalities in (at least) one of these sets are satisfied.

Note

The matrices need to have the same rank and number of columns. Otherwise, a ValueError is raised.

sign_vector_conditions.robustness.condition_closure_sign_vectors(stoichiometric_matrix, kinetic_order_matrix) → bool

Closure condition for robustness using sign vectors.

INPUT:

  • stoichiometric_matrix – a matrix

  • kinetic_order_matrix – a matrix

OUTPUT: Return whether the closure condition for robustness regarding small perturbations is satisfied.

Note

This implementation is inefficient and should not be used for large examples. Instead, use condition_closure_minors().