Globally and Locally Identifiable.

The adjectives globally and locally are also used in the literature to define some of these notions. Globally structurally identifiable (SI) simply means uniquely structurally identifiable. Locally structurally identifiable or just identifiable means structurally identifiable, but not necessarily uniquely. Distinct degrees of identifiability, or lack thereof, are illustrated in Fig. 1.

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Figure 1. The distinct degrees of structural identifiability SI (or unidentifiability) of the parameters and parameter combinations of a dynamic system model.

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Also, unidentifiable parameters – although they have an uncountably infinite number of possible solutions – are usually implicitly constrained by the model structure, Relationships (1), which define the interval parameter bounds noted in Definition 4 above. This means they cannot be given any value for, say, model simulation purposes, as explained further in the next paragraph and illustrated in Example 2 below.

Identifiable Parameter Combinations and Their Importance.

If a model is structurally unID (i.e. if some p_i are not SI), there always exist identifiable combinations of parameters (abbrev: combos), and each such combination can have one or a finite number of solutions. Finding explicit forms of identifiable combinations of parameters in a structurally unID model can be important in model quantification and applications. Combos are structural invariants of the model and – for many applications – they can provide a simpler input-output equivalent structure, i.e. with the same input-output properties [8], e.g. for simply simulating the model. These structural invariant combos also can provide interval bounds for unID parameters, the substance of identifiability definition 4 above [7].

Explicit expressions for SI combos in a model can be quite useful, particularly for large nonlinear models that involve tens of state variables and scores or hundreds of parameters – e.g. large protein network models – but also for not-so-large dimensional models. Input-output equivalent models expressed only in terms of SI parameters and parameter combos are simpler and therefore easier to implement, quantify and analyze. Explicit parameter combinations also can be used to reveal parameters that might be fixed, or otherwise established independently– an experiment design application – in order to improve identifiability. The algebra, however, can be tedious if accomplished analytically, even for the simplest models. Motivation for automation of this analysis by machine computation is illustrated in the first three examples below – which also illustrate some important results about nonuniqueness of SI parameter solutions – further motivating computation of SI parameter combinations. Explicit expressions for SI combos also can be used to help reparameterize a model i.e. fully recast a model in terms of identifiable parameters and parameter combinations, as illustrated in Example 2 below.

The differential algebra-based algorithms reported by Meshkat and coworkers [4]–[6] are focused here on computing SI parameter combinations symbolically, in a new computer program COMBOS, providing SI parameter combinations explicitly. COMBOS – described in a subsequent section – is implemented as a user-friendly web application (app), accessible online at: http://biocyb1.cs.ucla.edu/combos.

Three other published computational methods for SI analysis of nonlinear ODE models are notable for giving useful information in the unidentifiable case. These include: DAISY, a differential algebra-based method capable of global identifiability solutions [9]. The second method is a generating series approach, called GENSSI – also capable of global identifiability solutions [10]. The third is a non-parametric bootstrap-based algorithm for local identifiability analysis, based on the method of mean optimal transformations [11]. For unidentifiable models, the second and third methods can isolate “groups of functionally related parameters, so that the modeler can determine which parameters must be fixed in order to improve identifiability” [11], but they do not determine algebraic relationships among the parameters. DAISY takes identifiability analysis a step further, and does provide algebraic relationships among the parameters, in a form that can sometimes be manipulated to find SI parameter combinations – with varying levels of additional effort – as demonstrated in Examples 1, 3, 5, 6, 8 and 9 below. In contrast, COMBOS completes the task and yields algebraic relationships among SI parameters explicitly.

We remark that Examples 1–3 utilize relatively simple linear compartmental ODE models to both illustrate the complexity of the identifiability problems addressed and motivate automated COMBOS solutions for more dynamically complex nonlinear models. These linear examples have analytic solutions and are solvable by classical methods [1], [2], but not without substantial algebraic analysis for achieving solutions – as noted in the examples.

Example 1: Unidentifiable 2-Compartment Model (Fig. 2).

This model has been used to represent the dynamics of numerous physiological systems, e.g. distribution and turnover of labeled cholesterol in humans, with impulsive injection and concentration measurements of tracer in compartment 1 [12].(2)

By definition, k_ij ≥0 for i ≠ j. We assume k_ij >0 (an additional constraint relation), but there are no known relationships among the k_ij, for i ≠ j. This model has five unknown nonzero parameters: k₀₁, k₀₂, k₁₂, k₂₁, and V₁. The right-hand side equalities in the first two equations in (2) are simplified by defining the compartment turnover rates: - k₁₁ ≡ k₀₁+ k₂₁ and - k₂₂ ≡ k₁₂+ k₀₂, which we use in the second part of this example below. The input-output transfer function for this model H₁₁(s) = Y(s)/U(s) embodies all information about the parameters that can be obtained from ideal input-output data. H₁₁(s) is evaluated using Laplace transform analysis of the ODEs in (2):(3)

The α and β coefficients of the numerator and denominator polynomials in (3) can always be evaluated uniquely from input/output data (a well-known algebraic property of polynomials) and these are:(4)

The α and β coefficients are thus SI parameter combinations (combos). It is also clear from (4) that V₁, , and the product (from equation in (4)) are another set of four SI combinations of this model [1]. In either case, there are five unknown k_ijs in Eqs. (4) that cannot be all found from only these combo equations, and hence the model is structurally unID. Only V₁ can be evaluated uniquely, from Eq. (4).

If it is known that k₀₁ ≡ 0, a very commonly used different 2-compartment kinetic model – e.g. for the disposition of drugs [13] or hormones [14] – we have four unknowns and a unique solution is available from Eqs. (4), i.e. the model is uniquely SI, because: , , work and . Similarly, if only k₀₂ ≡ 0, we obtain a (different) model – used even more often to describe drug kinetics, e.g. [15], [16] – and we again have a unique solution for the remaining four parameters [1].

A binary nature for the identifiability (SI) question has been illustrated in this example. The model is either uniquely identifiable or it is not identifiable at all, depending on prior knowledge of certain parameters. However, SI of all parameters does not generally imply a unique solution set, as illustrated in the next example.

Example 2: Unidentifiable 2-Compartment Model Reparameterized into Uniquely SI Models.

We reparameterize the unidentifiable 2-compartment model in Fig. 2 in two ways here, both in terms of the SI parameter combinations. Redefining (scaling) the unmeasured state variable x₂ in Eq. (2) as reparameterizes the model with all coefficients as the uniquely SI parameter combos: k₁₁, k₂₂, V₁ and k₁₂k₂₁.(5)

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Figure 2. A SI 2-compartment model, with both input and output in compartment 1.

All k_ij >0 are unidentifiable; but volume V₁ and the three parameter combinations (combos) , - k₁₁ ≡ k₀₁+ k₂₁ and - k₂₂ ≡ k₁₂+ k₀₂ are uniquely SI.

https://doi.org/10.1371/journal.pone.0110261.g002

This model form can be readily simulated. But, it is not a compartmental structure, with balanced exchange fluxes between two compartments and nonzero flux leaks from both.

To preserve the original compartmental structure, we can add and subtract from the ODE and also add and subtract k₁₂k_21×1 from the ODE, as in the alternative re-parameterized model:(6)

The compartmental model structure in Fig. 3 also has parameters made up only of ID combos. It does, however, have a restricted region of validity, because the leaks in the transformed model must retain their nonnegativity, i.e. or and , as noted in definition 4 above.

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Figure 3. The unidentifiable 2-compartment model of Figure 2 is reparameterized here into a uniquely SI equivalent model, with the same structure, and with new parameters evaluated and designated in terms of SI combos.

https://doi.org/10.1371/journal.pone.0110261.g003

Example 3: Some 3-Compartment Models: Combos and Ambiguities.

The model in Fig. 4 is described by:

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Figure 4. A 3-compartment unidentifiable model with both input and output in compartment 1.

None of the k_ij >0 are SI, but volume V₁ and the five combos k₁₁, k₂₂, k₃₃, k₁₂k₂₁, k₁₃k₃₁ are uniquely SI. If , all 5 remaining k_ij become SI, but not uniquely – each has two feasible solutions.

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(7)This model has been used, for example, to represent the intra-and extravascular kinetics of thyroid hormone in response to an intravenous tracer dose (impulse δ(t)) of hormone in the euthyroid human, rat and sheep [17]. It has seven unknown parameters: k₀₂, k₀₃, k₁₂, k₁₃, k₂₁, k₃₁ and V₁, all assumed positive (the constraint relations). The measured output y(t) is the concentration of substance (tracer hormone in blood plasma) in compartment 1.

The transfer function is evaluated from Laplace transforms of Eqs. (7). To simplify the algebra (as in Example 1), we define the parameter combinations:(8)

Then, with much of the detailed algebra suppressed, we arrive at:(9)with coefficients evaluated as:

(10)As in Example 1, we know all six α and β coefficients (combo set 1) of the numerator and denominator polynomials of H₁₁(s) can be evaluated uniquely from (ideal) input-output data – Eqs, (10) – over any time interval, i.e. they are uniquely SI. The set of combos in Eq. (8), i.e. k₁₁, k₂₂, k₃₃, k₁₂k₂₁, k₁₃k₃₁, plus V₁ in Eq. (10), are also SI, but not all uniquely. Either set yields six k_ij equations in the seven unknown parameters. The volume V₁ is uniquely identifiable, in this case directly from Eq. (10), but none of the six k_ij can be determined uniquely from the remaining five equations in Eqs. (8) or (10). We remark that the structural symmetry of the model renders compartments 2 and 3 interchangeable in Fig. 4– because input and output are only in central compartment 1 in this example – the reason the model cannot be globally identifiable from this limited input-output pair. Indeed, additional algebraic analysis (not shown) indicates that combo k₁₁ is uniquely identifiable, while combos k₂₂, k₃₃, k₁₂k₂₁ and k₁₃k₃₁ are locally ID, each with exactly two solutions – results not easily gleaned by inspection or a less than rigorous analysis.

Example 4: Nonuniqueness of SI Parameters and Parameter Combinations.

If in Eq. (7) we have a different model, one that has been used e.g. to represent bilirubin kinetics [18]. Transfer function analysis yields six independent equations with six unknowns [1]. All parameters of this model are SI, but – as in the previous example – not uniquely, despite having 6 equations in 6 unknowns. There are two solutions, and both are physiologically feasible if no additional constraints are available a priori. The two sets of solutions for all unknown model parameters can be determined from the following algorithm, obtained by successive substitution into Eqs. (7) – (10) with k₀₃ ≡ 0, i.e.(11)

The equations marked with an asterisk each have two nontrivial combo solutions, because they all involve k₂₂, which itself has two. Additional information about the model or its parameters is needed to distinguish between these two solutions. Similar results (two solutions) are obtained if only k₀₂ ≡ 0, yet another variation of this 3-compartment model.

Unidentifiable models in Examples 3 and 4 here show how additional parameter information can render a model more identifiable, but not necessarily uniquely. The Example 3 model has only SI combos (and V₁) identifiable, with two distinct solutions for all but one combo, whereas fixing one parameter () in the same structural model in Example 4 renders all remaining individual k_ij parameters SI, but still with two distinct solutions.

Besides these complexities, lack of structural identifiability also poses a challenge for further quantitative analysis of these models. For example, without individual parameter or combo values, it cannot be simulated. This indeterminacy can be overcome by reparameterizing the model in terms of the identifiable combinations (combos) of parameters and computing solutions using combos in the ODEs – as illustrated in these motivational examples. Reparameterization, however, requires explicit knowledge of the SI combinations and – as also illustrated – the algebra for finding them can be quite tedious. Machine computation clearly is needed to facilitate the process.

Differential Algebra Algorithms & Extensions for COMBOS.

Detailed explanation and proofs of the underlying theory are given in [4]–[6]. In brief, the model ODEs are transformed algorithmically into equations with no state variables, only parameters, inputs and outputs and their derivative terms, using a differential elimination method [6]. The resulting equivalent model is called the input-output map 0, a subset of a complete characteristic set. The mapping from the P parameters p to the coefficients c(p) in these input-output equations is tested for uniqueness (distinctness), to determine identifiability, by computing a Gröbner basis for these equations. If this mapping is finite-to-one, the model is identifiable and no additional Gröbner bases are needed and no more are computed in COMBOS – yielding relatively quick solutions. If this coefficient mapping is infinite-to-one, the model is unidentifiable and – in COMBOS – identifiable parameter combinations are computed by finding additional Gröbner bases for the system of equations c(p) = c(p*). To speed up computation time, a numerical value is chosen for p*. There are P! possible Gröbner bases for the system of equations c(p) = c(p*) for a parameter vector p [4], which can be computationally prohibitive for large P. In our earlier work, we generally did not need to find more than P different Gröbner bases: trial-and-error randomization of the ordering of the parameter vector p followed by the computation of Gröbner bases corresponding to P shifts of the parameter vector p generated a sufficient number of SI parameter combinations. This worked well enough for small to medium size models, but it does not guarantee that enough combinations would be available for all models. Algorithmic refinements were needed to automate this process in the COMBOS implementation, and this entailed a computability compromise, consistent with algorithmic goals.

The main overall goal is to find simple parameter combinations. The simplest identifiable parameter combinations (in terms of polynomial degree and number of terms) among these Gröbner bases is determined by searching them for elements and factors of a certain form, which we call “decoupled” terms [4]. We then add these identifiable parameter combinations to the set of the original coefficients of the input-output equations and determine the simplest algebraically independent set among these. This procedure may not generate the absolute simplest set of identifiable parameter combinations, which would require checking of all P! possible Gröbner bases – not practical with current software tools.

Computationally tractable automation of parameter combination selection in COMBOS was implemented by systematic and exhaustive ordering and reordering of Gröbner bases, and repeating this process two additonal times – each with a different ordering – to expand the search to a larger subset of the P! possible Gröbner bases for the system of equations c(p) = c(p*). 3P was chosen heuristically because it generated a sufficient number of simple identifiable parameter combinations in all example runs of unidentifiable models with as many as 10 parameters.

These algorithmic refinements are implemented in COMBOS as follows. P shifts of parameter vector p are made and Gröbner bases are computed for each of these. The parameter vector is then reordered and Gröbner bases corresponding to P shifts of this reordered parameter vector are computed. Shift means transforming (p₁, …, p_P) to (p₂, …, p_P, p₁). This gives each parameter a ranking that eliminates it last. Reordering means transforming (p₁, …, p_P) to (p₁, p₃, …, p_P-1, p₂, p₄, …, p_P) if P is even; and (p₁, p₃, …, p_P, p₂, p₄, …, p_P-1) is P if odd. This gives each parameter a “new neighbor”, so new combinations can possibly be found. This process is repeated one more time, for a total of 3P rank orderings.

Syntax & Naming Conventions for Specifying the Model.

Equations are entered using ASCIIMathML library notation (http://www1.chapman.edu/~jipsen/mathml/asciimath.html), chosen over its MathML and LaTeX counterparts for its simplicity, reduced character count and thus usability. Table 1 lists the naming guidelines for ODE equations of the form , with initial conditions (ICs): x(0, p) and outputs y = g(x, p). Users must adhere to this common standardized notation.

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Table 1. Syntax & Naming Conventions for Specifying the Model.

https://doi.org/10.1371/journal.pone.0110261.t001

Numbering of state variables, inputs and outputs MUST start from 1 and be monotonically increasing. This means the numerical subscript of an input or output does not necessarily correspond to that associated with a state variable (or compartment). For instance, a system with a single output of/from state variable (compartment) 2 is written ; or input u₁ can be in any ODE, not just the first one, etc.

Differential algebra algorithms currently handle only polynomial and rational function terms for SI analysis. Terms with trigonometric or transcendental functions, e.g. sin, cos, ln – as well as constants (e.g. pi, e) – are not recognized as such by the parser and would be incorrectly treated as parameters if encountered.

Initial Condition Handling in COMBOS.

Initial conditions may or may not be known for a given state variable in a model, and knowledge of initial conditions can change the resulting SI of the parameters or parameter combinations. The interface provides a convenient way to enter known ICs. Known initial conditions are usually numbers, but – importantly – COMBOS also accommodates functions (combinations) of the parameters, which can be useful in some applications, e.g. when ICs represent nonzero steady state functions of parameters to be estimated from transient response data initiated from steady state (e.g, see [20]). When not known, initial conditions are simply treated as generic by default. This means that identifiability properties obtained without initial conditions hold true almost everywhere, i.e. except on a “thin set,” – a set of measure zero [21]. This means there might be particular initial conditions that yield different identifiability properties [21]. These are called inaccessible states that – under very particular circumstances – lead to unidentifiability of parameters [21]. The COMBOS algorithm presently handles inaccessible initial states for models that have one or more constant ODE solution trajectories, i.e. , which renders the model reducible. These constant solutions are incorporated into the algorithm for finding the input-output (I-O) equations, 0, and change identifiability properties of the model, as illustrated in Example 8. Otherwise the system is assumed accessible from all known initial conditions.

For each state variable, a subfield is available for entering the corresponding initial condition, assumed unknown (generic) and hidden from view at the start, so as not to obscure the view of the overall interface and thus its usability. Visibility of the subfield can be manually toggled so that it is displayed only if there are known initial values to enter. For the copy/paste method, users need only list values for known initial conditions.

For each known initial condition specified, the program checks that none are functions of undefined state variables, inputs, outputs or parameters.

Comparisons with DAISY.

DAISY (Differential Algebra for Idenfiability of SYstems: http://www.dei.unipd.it/~pia) is a freely downloadable computer algebra software package for analyzing parameter SI [9]. Users include an input file with the mathematical model in the syntax required by DAISY and run the program using a command line interface. Results provide solutions to the system of parameter-coefficient mapping equations c(p) = c(p*) at a pseudo-random point p*. In the case of local identifiability, users can then count the number of solutions; and in the case of unidentifiability, they can sometimes algebraically manipulate the solution to get identifiable parameter combinations [4], [5]. In some cases (see Examples 5, 6, 8, and 9), this can be readily done by moving all parameters in DAISY solutions to one side of the equations, via addition/subtraction or multiplication/division, but not in general. Even for very simple examples such as the 2- and 3-compartment models of Examples 1 and 3 above, this trick of moving all the parameters to one side is not so easy, as we now show.

For Example 1– with p = (k_01, k_02, k_12, k_21, V₁) – DAISY code generated the pseudo-random point p* = (8, 7, 13, 12, 3) for which it provides precisely the following solution:

While the SI combination k₀₂+ k₁₂ is readily obtained from the third equation, by simple rearrangement, the combos k₁₂ k₂₁ and k₀₁+ k₂₁ are not so easy to unravel. In particular, k₀₁+ k₂₁ requires several steps of algebraic manipulation, not so obvious to users unfamiliar with compartmental model equations. The 3-compartment model of Example 3 requires similar algebraic manipulations to obtain all SI parameter combinations. In contrast, COMBOS finds all SI combinations explicitly for these examples, in a matter of seconds.

To further evaluate COMBOS functionality and accuracy, we next compared the results of COMBOS SI analyses with the example solutions computed using DAISY reported in four papers [9], [19], [21], [22]. Both COMBOS and DAISY provided identical results about SI for all of these examples, which included a mix of both linear and nonlinear models up to dimensionality four. We present and discuss only four examples that differed in some respects here, with all models presented in COMBOS math syntax.

Example 5.

The following 3-compartmental model example [19] has one input and one output, tested with and without initial conditions:(12)

With no initial conditions given, both DAISY and COMBOS indicate the model is unidentifiable. COMBOS, however, additionally indicates explicitly that and combination are uniquely identifiable; and and are locally identifiable, with 2 solutions. With additional effort, this also can be derived from the DAISY results, with some additional cross-multiplication computations. As noted in [4], [5], this manipulation does not work for some notable unidentifiable models.

With all initial conditions given, both DAISY and COMBOS indicate the model is locally identifiable. COMBOS specifically provides additional SI results: the four locally SI parameters have two solutions (established in 10 seconds runtime). This result also can be attained from DAISY results by counting solutions in the output.

Example 6.

The following nonlinear polynomial 4-compartmental HIV/AIDS model describes the dynamics of uninfected, latently infected and actively infected cells, along with the free virus particles [22]:(13)

With no initial conditions given, both DAISY and COMBOS find uniquely identifiable and locally identifiable. COMBOS results additionally indicate that have three local solutions; combination is uniquely identifiable; and are locally identifiable with 3 solutions (runtime: 90 seconds). Again, as noted above, this result can be obtained by additional algebraic manipulation of DAISY results.

Example 7.

This example, from [9], is a 4-compartmental mammillary compartment model with one input and one output, tested with and without initial conditions:(14)

With no initial conditions given, both DAISY and COMBOS found the model locally identifiable. COMBOS results explicitly indicated that parameter is uniquely identifiable, with the others all locally identifiable – with 3 solutions each. DAISY produces all local solutions, but you have to locate and count them in the output. When at least two initial conditions are known among , both DAISY and COMBOS results indicate that the model becomes globally identifiable. All COMBOS runs completed in less than 1 minute.

Example 8.

This simple nonlinear model [21] has one input and two outputs:(15)

The authors of [21] use this example to illustrate how DAISY algorithms handle special initial conditions (ICs), i.e. ICs that yield SI parameter solutions valid almost everywhere. They note that, for partially null (particular) initial conditions , i.e. {0, x₂, 0}, this model is reducible to . This means that for this very particular initial state, the model is inaccessible from the outputs, for any , meaning the set of states reachable from these initial conditions is empty. It is noteworthy that this particular IC is an unstable equilibrium point of the system, i.e. , making it an unlikely IC in practice. In any case, all parameters are clearly unidentifiable – without machine computation. The authors of [21] correctly analyze this special case using DAISY; and also show that for all other initial conditions DAISY finds uniquely identifiable.

The COMBOS algorithm also finds all parameters unidentifiable for these particular initial conditions, using a slightly different approach. As noted earlier, COMBOS specifically checks for constant solution trajectories for the initial value problem and incorporates them into the algorithm for finding the input-output equations. In this example, the constant solutions effectively reduce the dimensionality of the input-output relations, thus leading to unidentifiability of the system.

With no initial conditions provided, COMBOS explicitly finds p₂ and combination p₁p₃ uniquely SI. For this simple model, a little extra effort yields the same result from DAISY output, manually extracting p₁p₃ from the parameter solution for p₁, by cross-multiplication.

All COMBOS runs were completed in about 5 seconds. We remark that, for unidentifiable models in general, DAISY runtimes can be substantially shorter than COMBOS runtimes, because multiple Gröbner Bases are computed in COMBOS in search of identifiable combinations in simplest and explicit form.

Benchmarking COMBOS and New Identifiability Results.

The dimensionality of models COMBOS can handle is limited by the computational capacity of the underlying symbolic algebraic program, currently Maxima. The first test-group below includes five generic linear models of increasing order n, analyzed for SI under generic initial conditions. Most of the SI parameter combinations results for these models have not previously been published. These examples are then followed by SI analysis of several nonlinear systems biology models. The intention is not to give a hard upper bound on the dimensionality that COMBOS can manage, but to show – using systematically more complex examples – that the success of COMBOS is highly dependent on the complexity as well as the size of the model.

Test-Model 1 is a linear ODE system with a full matrix of parameters, , all states measured, one input – in the equation only – and generic ICs:(16)

Results were generated up to dimension n = 7, with all parameters found globally identifiable in 3 to 20 secs. COMBOS currently has a default limit of 60 parameters. For n = 8, Eqs. (16) have 64 parameters, so computations did not extend beyond n = 7. It can be shown by induction that all parameters are uniquely SI for any n.

Test-Model 2 is the same linear system, with the same input, but one less output, i.e. states measured:(17)

For this model, it is well-known that p₁₁ is uniquely SI for n = 1. For n >1, all models are unidentifiable. For n = 2, p₁₁, p₂₂ and combo p₁₂p₂₁ are uniquely SI. For n >2, results are more complicated. For example, for n = 3, p₁₁, p₂₁ and combos p₂₂+ p₃₃, p₂₃/p₁₃, p₂₂ – (p₁₂ p₂₃)/p₁₃,

− (p₁₂ p₂₃ p₃₃)/p₁₃+ p₂₃ p₃₂ – (p₁₂p₂₂ p₂₃)/p₁₃+ and.

− (p₁₁ p₂₃ p₃₃)/p₁₃+ p₂₃ p₃₁ – (p₁₁ p₂₂ p₂₃)/p₁₃+ p₂₁p₂₂ are uniquely SI, a new result.

Test-Models 3, 4 and 5 are variations of single-input Test-Model 2– each with different outputs defined. Test-Model 3 has a single output sampled in the same compartment as the input: . Test-Model 4 has a single output sampled in a different compartment from the input:. Test-Model 5 has two outputs sampled in separate compartments, with one sample in the same compartment as the input: and the second in the n^th compartment .

COMBOS results were generated up to dimension for Test-Models 2 thru 5. Computations did not complete for n>3 because the models were too complicated for the Gröbner basis solvers in Maxima. All were found unidentifiable overall, in 3 to 90 seconds runtime. As already noted, some SI analysis results for the simplest of these models are consistent with known or published identifiability properties, but most parameter combination results are new and not so simple – as given below.

For Test-Model 3, n = 1: p₁₁ is uniquely SI. For n = 2: p₁₁, p₂₂ and combo product p₁₂ p₂₁ are uniquely SI. For n = 3: p₁₁ and combos p₃₃+ p₂₂, p₁₃p₃₁+ p₁₂ p₂₁, p₂₂ p₃₃− p₂₃ p₃₂ and.

p₁₁ p₂₂ p₃₃ – p₁₂ p₂₁ p₃₃− p₁₁p₂₃ p₃₂+ p₁₃ p₂₁ p₃₂+ p₁₂ p₂₃ p₃₁− p₁₃ p₂₂ p₃₁ are uniquely SI.

For Test-Model 4, n = 1: p₁₁ is uniquely SI. For n = 2: p₂₁ and combos p₂₂+ p₁₁, p₁₁ p₂₂− p₁₂ p₂₁ are uniquely SI. For n = 3, p₃₁ and combos p₃₃+ p₂₂+ p₁₁,

p₂₁ p₃₂− p₂₂ p₃₁, – p₂₂ p₃₃− p₁₁ p₃₃+ p₂₃ p₃₂+ p₁₃p₃₁ – p₁₁ p₂₂+ p₁₂ p₂₁ and.

p₁₁ p₂₂ p₃₃− p₁₂ p₂₁ p₃₃− p₁₁ p₂₃ p₃₂+ p₁₃ p₂₁ p₃₂+ p₁₂ p₂₃ p₃₁₋ p₁₃ p₂₂p₃₁ are uniquely SI.

For Test-Model 5, n = 1, p₁₁ is uniquely SI. For n = 2, p_11, p₁₂, p₂₁ and p₂₂ are uniquely SI. For n = 3, p_11, p₃₁ and combos: p₃₃+ p₂₂, p_32/p₁₂, p₃₃− (p₁₃p₃₂)/p₁₂, − p₁₃p₃₃+ (p₃₂)/p₁₂− p₁₂p₂₃+ p₁₃p₂₂ and (p₁₁p₁₃p₃₂)/p₁₂− p₁₃p₃₁+ p₁₁p₂₂− p₁₂p₂₁ are uniquely SI.

Example 9: SI Analysis and Reparameterization of a Minimal Viral Disease Dynamics Model.

Equations for a classic viral disease model [23] are rewritten in original form on the left (Target cells T, Infected cells I, Virions V) and in COMBOS form for analysis on the right below. This model has eight unknown parameters:. The (constant) input s ≡ p₁u(t) ≡ p₁1(t) is a step function of unknown magnitude p₁.(18)

A slightly simpler version of model (with s, ε and equal zero) was analyzed in [24] for identifiability of parameters p₂, p₅, p₇ and p₈, also using a differential algebra approach. Parameters p₂, p₅ and p₈ were found to be the only parameters identifiable. Analysis using COMBOS online was in agreement, and also provided potentially useful supplemental SI information: p₂ and the combos p₃p₄ and p₁p₆p₇ are uniquely identifiable, and p₅ and p₈ are both locally identifiable, each with two solutions. Computation time was ∼10 seconds. Analysis using DAISY provided the same results after algebraically manipulating the parameter solutions.

Using the COMBOS results, the input-output equivalent reparameterized SI model (with parameters p) is readily written by scaling x₁ by p₆p₇ and x₂ by 1/p₁. i.e. . The resulting ODE model is:(19)

The structure is identical to the original, but with a new and equivalent set of (structural invariant) parameters: p₂, p₅, p₈ and combos p₃p₄ and p₁p₆p₇.

Example 10: Tumor-Suppressor p53 Dynamics Model – Full Nonlinear (NL) Model.

This NL model has 4 ODEs, 4 outputs and 23 unknown parameters [1]. The equations in COMBOS ASCIIMathML format are:

COMBOS, reports all parameters except p22 uniquely SI, with p22 locally SI with four solutions. Since this model is at least locally SI, only one Gröbner basis computation is required. Identifiability results are achieved in about 22 minutes, due to the large number (23) of parameters. However, it is clear that p22, with four possible solutions, has only one nonnegative real – and therefore feasible – solution, because p22 appears only in the single quartic p22∧4 term – in the fourth equation. Therefore this model is globally identifiable.

We remark that, whereas the number of nonnegative real solutions is explicitly determined for this particular model, the current implementation of COMBOS does not return a number of nonnegative real solutions in general, because only a single numerical point for p* is tested when solving the equations c(p) = c(p*). Testing a different numerical point for p* might lead to a different number of nonnegative real solutions.

Example 11: Model Reduction Using COMBOS: Simplifying the p53 Dynamics Model for Facilitating Quantification.

The nonlinearities (NL) in this model are of three types: several simple Michaelis-Menten (M-M) functions – of the form: , a fourth-order hill function, and products of state variables. For the purpose of model quantification, simplifications of NL terms can be helpful – and were helpful – in quantifying this model from four sets of data corresponding to the output equations above, in [1]. The M–M functions, in particular, are approximately linear (first-order ∼Ax) for low substrate values and constant (zero-order combo A/B) for large substrate values. Either of these circumstances can in principle occur during transient stimulation of the p53 regulation system by stressor signals and either would therefore be better represented by simpler corresponding M–M terms in the model. In other words, if the substrate signal in an M–M function is very large during a transient, the M–M function is approximately constant. If it is very small, the M–M function is approximately linear. In either case, the number of parameters that need to be estimated is reduced by one.

We systematically screened each of the NL M–M terms in each of the four p53 ODEs, simplifying them one at a time in the above manner, and used COMBOS to delineate the identity of the SI parameters and parameter combinations (and their number) resulting from each simplification. For example, the first full ODE:(20)is reduced to the following for (kinase signal intensity) u <<p₇, showing the combo p₆/p₇ to be SI:

For u >>p₇, Eq. (20) is reduced to the following ODE, with ≡ p₄(1+p₆) SI:

Preliminary quantification of the full (and SI) model from experimental data [25] provided initial parameter estimates consistent with inequality u >>p₇ and others like it in the four equations. This led to several simplifications, thereby eventually providing more precise parameter estimates for the smaller number of them remaining [1].