Structural identifiability concerns finding which unknown parameters of a model can be quantified from given input-output data. Many linear ODE models, used in systems biology and pharmacokinetics, are unidentifiable, which means that parameters can take on an infinite number of values and yet yield the same input-output data. We use commutative algebra and graph theory to study a particular class of unidentifiable models and find conditions to obtain identifiable scaling reparametrizations of these models. Our main result is that the existence of an identifiable scaling reparametrization is equivalent to the existence of a scaling reparametrization by monomial functions. We provide an algorithm for finding these reparametrizations when they exist and partial results beginning to classify graphs which possess an identifiable scaling reparametrization.

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Identifiability concerns finding which unknown parameters of a model can be estimated, uniquely or otherwise, from given input-output data. If some subset of the parameters of a model cannot be determined given input-output data, then we say the model is unidentifiable. In this work, we study linear compartment models, which are a class of biological models commonly used in pharmacokinetics, physiology, and ecology. In past work, we used commutative algebra and graph theory to identify a class of linear compartment models that we call identifiable cycle models, which are unidentifiable but have the simplest possible identifiable functions (so-called monomial cycles). Here we show how to modify identifiable cycle models by adding inputs, adding outputs, or removing leaks, in such a way that we obtain an identifiable model. We also prove a constructive result on how to combine identifiable models, each corresponding to strongly connected graphs, into a larger identifiable model. We apply these theoretical results to several real-world biological models from physiology, cell biology, and ecology.

Parameter identifiability problems can plague biomodelers when they reach the quantification stage of development, even for relatively simple models. Structural identifiability (SI) is the primary question, usually understood as knowing which of P unknown biomodel parameters p 1,…, pi,…, pP are-and which are not-quantifiable in principle from particular input-output (I-O) biodata. It is not widely appreciated that the same database also can provide quantitative information about the structurally unidentifiable (not quantifiable) subset, in the form of explicit algebraic relationships among unidentifiable pi. Importantly, this is a first step toward finding what else is needed to quantify particular unidentifiable parameters of interest from new I–O experiments. We further develop, implement and exemplify novel algorithms that address and solve the SI problem for a practical class of ordinary differential equation (ODE) systems biology models, as a user-friendly and universally-accessible web application (app)–COMBOS. Users provide the structural ODE and output measurement models in one of two standard forms to a remote server via their web browser. COMBOS provides a list of uniquely and non-uniquely SI model parameters, and–importantly-the combinations of parameters not individually SI. If non-uniquely SI, it also provides the maximum number of different solutions, with important practical implications. The behind-the-scenes symbolic differential algebra algorithms are based on computing Gröbner bases of model attributes established after some algebraic transformations, using the computer-algebra system Maxima. COMBOS was developed for facile instructional and research use as well as modeling. We use it in the classroom to illustrate SI analysis; and have simplified complex models of tumor suppressor p53 and hormone regulation, based on explicit computation of parameter combinations. It’s illustrated and validated here for models of moderate complexity, with and without initial conditions. Built-in examples include unidentifiable 2 to 4-compartment and HIV dynamics models.

This work focuses on the question of how identifiability of a mathematical model, that is, whether parameters can be recovered from data, is related to identifiability of its submodels. We look specifically at linear compartmental models and investigate when identifiability is preserved after adding or removing model components. In particular, we examine whether identifiability is preserved when an input, output, edge, or leak is added or deleted. Our approach, via differential algebra, is to analyze specific input-output equations of a model and the Jacobian of the associated coefficient map. We clarify a prior determinantal formula for these equations, and then use it to prove that, under some hypotheses, a model's input-output equations can be understood in terms of certain submodels we call "output-reachable". Our proofs use algebraic and combinatorial techniques.