SIAN-Julia

The current stable version of SIAN.jl can be installed via the following command:

>] add "SIAN"
> using SIAN

The installation from source is possible via GitHub and SSH or HTTPS:

> ]add https://github.com/alexeyovchinnikov/SIAN-Julia.git

or

> ]add git@github.com:alexeyovchinnikov/SIAN-Julia.git

In this example we would like to consider the following simple non-linear ODE system:

x1'(t) = r1 * x1(t) * (1 - x1(t) / k1 + x2(t) / k2),
x2'(t) = r2 * x2(t) * (1 - x1(t) / k1 + x2(t) / k2),
y(t) = x1(t)

To this end, we can run:

using SIAN

ode = @ODEmodel(
    x1'(t) = r1 * x1(t) * (1 - x1(t) / k1 + x2(t) / k2),
    x2'(t) = r2 * x2(t) * (1 - x1(t) / k1 + x2(t) / k2),
    y(t) = x1(t)
);

output = identifiability_ode(ode, get_parameters(ode));

The last command prints the following:

Solving the problem
Constructing the maximal system
Truncating
Assessing local identifiability
Locally identifiable parameters: [r1, k1, r2, x1]
Not identifiable parameters:     [k2, x2]
Randomizing
GB computation
Remainder computation

=== Summary ===
Globally identifiable parameters:                 [x1, k1, r1, r2]
Locally but not globally identifiable parameters: []
Not identifiable parameters:                      [k2, x2]
===============

Once the module is imported via using SIAN, the following functions are available immediately via the export of the module: @ODEmodel, identifiability_ode, get_parameters.

Other SIAN functions are available via prefix call, such as SIAN.<any_function_name>.

We recommend checking the examples folder to get started with using SIAN, see this readme file.

The algorithm is based on the following papers:

• Global Identifiability of Differential Models (Communications on Pure and Applied Mathematics, Volume 73, Issue 9, Pages 1831-1879, 2020.)
• SIAN: software for structural identifiability analysis of ODE models (Bioinformatics, Volume 35, Issue 16, Pages 2873–2874, 2019)

The original Maple implementation is located here.

The main function "identifiability_ode" has two required arguments:

1. an ODE model (created by the @ODEmodel macros)
2. array of parameters for which the identifiability analysis is requested

and three optional keys:

1. p, which is the probability of correctness, with the default value p = 0.99,
2. p_mod, which is a prime number and is the characteristic of the field over which the computation of Groebner basis will occur. If p = 0 (the default value), the computation will be over the rational numbers. If p > 0, then the computation will be over Z/pZ. The current limit for Groebner basis modular arithmetic implementation suggests that prime numbers bigger than 2^29 - 3 are not to be used. When using p_mod>0, the same probability of correctness is no longer guaranteed and the program will raise a warning message in that case.

The function get_parameters has one required argument, an ODE model (created by the @ODEmodel macros), and one optional key, initial_conditions. If the key is set to true (the default value), then the function will return the set of all parameters and state variables. If the key is not set to true, then the function will return the set of all parameters.

The function "generate_replica" has two required arguments:

1. an ODE model (created by the @ODEmodel macros)
2. an integer r

and returns the r-fold replica of the ODE model (the state, output, and input variables are replicated and the parameters are not replicated). This function can be used to check the r-experiment identifiability of the parameters.

The folder examples/ contains examples of using this.

The folder without-macros/ contains an earlier version on this implementation that did not use the macros.

If an ODE model has been entered with parameters for some of which it is desirable to further specify their values, this can be done using the SetParameterValues function, which accepts:

1. an ODE model (created by the @ODEmodel macros)
2. a dictionary (or ordered dictionary) of values such as (taken from the NFkB.jl example)

OrderedDict(

    a1 => Nemo.QQ(1, 2),   
    
    a2 => Nemo.QQ(1, 5),
    
    a3 => Nemo.QQ(1),
    
    c_1a => Nemo.QQ(5, 10^(7)),
    
    c_2a => Nemo.QQ(0),
    
    c_5a => Nemo.QQ(1, 10^(4)),
    
    c_6a => Nemo.QQ(2, 10^(5)),
    
    c1 => Nemo.QQ(5, 10^(7)),
    
    c2 => Nemo.QQ(0),
    
    c3 => Nemo.QQ(4, 10^(4)),
    
    c4 => Nemo.QQ(1, 2),
    
    kv => Nemo.QQ(5),
    
    e_1a => Nemo.QQ(5, 10^(4)),
    
    c_1c => Nemo.QQ(5, 10^(7)),
    
    c_2c => Nemo.QQ(0),
    
    c_3c => Nemo.QQ(4, 10^(4))

)

for instance, to specify that a1 is 1/2, etc.