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simbio.complexstep.abs

abs function for SimBiology local sensitivity analysis

Description

example

M = simbio.complexstep.abs(X) returns M, where M = abs(real(X)) + 1i*imag(X). simbio.complexstep.abs is equivalent to abs when the input is real.

  • When you have a SimBiology model with a custom function that calls abs and you are performing local sensitivity analysis on the model, replace abs with simbio.complexstep.abs in your custom function.

  • You do not need to update SimBiology expressions (such as reaction rates or rules) that directly call abs. SimBiology automatically replaces abs with simbio.complexstep.abs whenever:

  • simbio.complexstep.abs is not differentiable when the real part of the input is 0 [1]. For more information, see Local Sensitivity Analysis (LSA).

Examples

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This example shows how to use replacements for abs, min, and max in some custom functions so that the model becomes compatible with local sensitivity analysis (LSA). The replacement functions are simbio.complexstep.abs, simbio.complexstep.min, and simbio.complexstep.max. Specifically, one of the custom functions used by this example computes the net amount of a drug species moved between two compartments. The other custom function sets the thresholds for the forward and reverse reaction fluxes.

Create a model.

model       = sbiomodel("model");
c1          = addcompartment(model,"c1");
c2          = addcompartment(model,"c2");
s1          = addspecies(c1,"Drug",2);
s2          = addspecies(c2,"Drug");
netFlux     = addspecies(c1,"netFlux");
reaction    = addreaction(model,"c1.Drug <-> c2.Drug");
kf          = addparameter(model,"kf",1.0);
kr          = addparameter(model,"kr",1.5);
fluxMin     = addparameter(model,"fluxMin",0.1);
fluxMax     = addparameter(model,"fluxMax",10);

Define the net amount of drug species moved between two compartments using a custom function calculateNetFlux based on the constrained forward and reverse reaction fluxes, which are defined later.

rule = addrule(model,"netFlux = calculateNetFlux(boundedForwardFlux,boundedReverseFlux)","rate");

calculateNetFlux uses simbio.complexstep.abs, and the function is already saved in the provided file named calculateNetFlux.m.

type calculateNetFlux.m
function netFlux = calculateNetAmount(forwardFlux,reverseFlux)
    netFlux   = simbio.complexstep.abs(forwardFlux-reverseFlux);
end

Define the forward and reverse fluxes of the reaction. Set the thresholds on the fluxes using the imposeBounds custom function.

boundedForwardFlux = addparameter(model,"boundedForwardFlux","Constant",false);
boundedReverseFlux = addparameter(model,"boundedReverseFlux","Constant",false);
forwardFlux        = addparameter(model,"forwardFlux","Constant",false);
reverseFlux        = addparameter(model,"reverseFlux","Constant",false);

forwardFlux        = addrule(model,"forwardFlux = kf*c1.Drug","repeatedAssignment");
reverseFlux        = addrule(model,"reverseFlux = kr*c2.Drug","repeatedAssignment");
boundedForwardFlux = addrule(model,"boundedForwardFlux = imposeBounds(forwardFlux,fluxMin,fluxMax)","repeatedAssignment");
boundedReverseFlux = addrule(model,"boundedReverseFlux = imposeBounds(reverseFlux,fluxMin,fluxMax)","repeatedAssignment");
reaction.ReactionRate = "boundedForwardFlux - boundedReverseFlux";

imposeBounds uses simbio.complexstep.min and simbio.complexstep.max to set the lower and upper limits for the reaction flux.

type imposeBounds.m
function boundedFlux = imposeBounds(fluxInput,fluxMin,fluxMax)
    fm             = simbio.complexstep.max(fluxMin,fluxInput);
    boundedFlux    = simbio.complexstep.min(fluxMax,fm);
end

Enable local sensitivity analysis.

configset = getconfigset(model);
configset.RuntimeOptions.StatesToLog        = "netFlux";
configset.SolverOptions.SensitivityAnalysis = true;
sensitivityOptions          = configset.SensitivityAnalysisOptions;
sensitivityOptions.Inputs   = [kf,kr];
sensitivityOptions.Outputs  = netFlux;

Temporarily disable the warning about unsupported functions. The warning is safe to ignore.

warnState  = warning("off","SimBiology:senscsverify:UnsupportedFunction");
cleanupobj = onCleanup(@()warning(warnState));

Simulate the model and perform sensitivity analysis.

simdata = sbiosimulate(model);
sbioplot(simdata);

Figure contains an axes object. The axes object with title States versus Time contains 3 objects of type line. These objects represent netFlux, d[netFlux]/d[kf], d[netFlux]/d[kr].

delete(cleanupobj);

Input Arguments

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Input array, specified as a numeric scalar, vector, matrix, or multidimensional array.

Data Types: double
Complex Number Support: Yes

Output Arguments

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Output array, returned as a numeric scalar, vector, matrix, or multidimensional array. The size of the output array is the same as the input array.

  • For a complex number x with a nonnegative real part, simbio.complexstep.abs(x) = x.

  • For a complex number x with a negative real part, simbio.complexstep.abs(x) = -real(x) + 1i*imag(X).

References

[1] Martins, Joaquim, Ilan Kroo, and Juan Alonso. “An Automated Method for Sensitivity Analysis Using Complex Variables.” In 38th Aerospace Sciences Meeting and Exhibit. Reno, NV, U.S.A.: American Institute of Aeronautics and Astronautics, 2000. https://doi.org/10.2514/6.2000-689.

Version History

Introduced in R2022b