An algebraic approach to challenges on identification problems
in systems biology
Mizuka Komatsu
Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japanm-komatsu@stu.kobe-u.ac.jp
Abstract
This talk is of twofold. A challenge appeared in identification problems, in particularunidentifiablility problems in parameter estimations, in the field of system biology is introduced with an example of PBPK models in the first section. In the second section, analgebraic and algebro-geometric approach to the challenge in a wider context is explained.
More precisely, a method to extract a geometric structure that are uniquely determined byobserved time series data and unidentifiable state-space models, followed by an underlying
theory including, e.g., differential algebra, is introduced. An idea of model reduction technique that preserves the geometric structures of original models in a sense, is also discussed
in the talk.
keywords: systems biology, parameter estimation, PBPK model, differential algebra, algebraic geometry, Gröbner basis, model reduction, generalized Lotka-Volterra equations1Introduction of challenges in identification problems insystems biology

In the field of systems biology, biological phenomena and their interactions are analysed integratedly. In such a field, biological systems are often modelled as and investigated throughmathematical models. One of common modelling approaches is by ordinary differential equations, ODEs, which are constructed based on biological knowledge. Partial differential equations,PDEs, are also one of major modelling approaches in such a field. However, considering possible constructions of PDEs based on refining compartment models, a class of ODEs, analyticalmethods for ODE models are considered to be still useful to the ones for PDEs. Based on this,our study focuses on ODE models appeared in systems biology. In particular, we consider suchmodels with unknown parameters to be estimated using observed time-series data, etc. Parameter estimation problems of such models, which is a type of identification problems of ODEs, arethe main topic in this talk.
In order to illustrate challenges in parameter estimation problems in systems biology, anexample based on physiologically-based pharmacokinetic models, so-called PBPK models is introduced. Roughly speaking, PBPK models describe drug responses in human bodies dynamically[1].
In the field of pharmacokinetics, such models are constructed and practically used for predicting
the time course of concentration of drugs in plasma and other sites. Although PBPK models
have their origin in such a field, their applications can be seen in other fields, e.g., [2], in which we
newly applied the model into immunology. PBPK model is constructed by compartments eachof which corresponds to an organ or a compound of organs. State variables of a PBPK model
correspond to the amount of the interests, for example, drugs in compartments. Some of model
parameters are assigned to the meaning of certain biological functions such as, e.g., clearance
rates at organs. Thus, not only the behaviors of state variables but also parameter values areof importance, which tend to be unknown. Using relevant time-series data, e.g., drug concentrations in plasma measured at several time points from subjects, such parameters, i.e., biologicalfunctions, are quantitatively estimated for the corresponding subjects. In such estimations, there
exist an intrinsic challenge; the parameters tend to be unidentifiable. In other words, parameters1cannot be uniquely determined from given data due to its insufficiency. Unidentifiablility of parameters causes difficulties in investigation of corresponding biological systems. For example, inthe context of immunology, such situation makes detection of immunological abnormalities fromestimated parameters difficult in spite of the desire. Besides, estimations conducted without
considering the unidentifiable property may overlook possible important considerations on thesystems as we pointed out in [4]. Due to experimental constraints, unidentifiability problems
often appear in systems biology, suggesting that the importance of approaches to dealing withunidentifiable models.

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Algebraic approaches to unidentifiable state-space models: the parameter variety

Based on the previous section, we consider an approach to unidentifiable models, especiallydescribed by ODE models with polynomial terms of the state and the input variables. In thisstudy, we focus on the fact that the structures determined by observed data and unidentifiable
models are uniquely determined as sets, which we call the parameter varieties, and then propose
a method to describe them explicitly. In this method, a certain input–output relationship ofthe models are derived using the Gröbner basis for an ideal corresponding to the model at first.
Then, the sets of parameters, each of which generates the given data, are described as sets ofconstraints in terms of parameters i.e. the algebraic varieties. Once the structure, i.e., thevariety is extracted, overlooking of the feasible parameters, which may lead to insufficient orinappropriate system considerations, would never occur.
We applied the proposed approach in the analysis of viral dynamics, which reveals an important fact on the efficacy of the drug that was missed in a previous study [3]. Fig. 1 shows
examples of estimated parameter varieties in parameter spaces of an unidentifiable model appeared in [3] given observed time-series data taken from two different subjects. See [4] for thedetails of the varieties. As can be seen in Fig. 1, our method captures feasible parameters exhaustively compared to conventional approaches [3], suggesting an applicability of our method
to, for example, classifications of subjects in terms of their varieties.
Technically speaking, ODEs we consider may not be able to be dealt with commutative
algebra naively since they contain derivatives of variables with respect to time. In fact, in orderto derive input-output equations from such models, derivative operations, which are not allowedin commutative algebra, for the models are required. Thus, our method appears to fall under
the umbrella of differential algebra, which is rather difficult compared to non-differential one.
However, thanks to state-space representations of the models, it is guaranteed that input-output
equations can be derived considering certain truncated differential ideals corresponding to themodels, which can be regarded as non-differential ideals. Theoretical details of our method will beexplained in the talk. In practice, the derivation of input-output equations relies on computationof the Gröbner basis, which implies its computational expensiveness. Considering this, in thetalk, we discuss an idea of a model reduction method preserving the parameter variety, which isbased on form-invariant transformations of generalized Lotka-Volterra equations.

Acknowledgments
We are grateful to Takaharu Yaguchi for collaboration throughout this work and Shinji Nakaoka
for discussions on the generalized Lotka-Volterra equations. This work was supported by JST
CREST JPMJCR1914 and JSPS KAKENHI Grant Numbers JP20J21185 and 20K11693.

2Figure 1: The parameter varieties of an unidentifiable model in two-dimensional parameter
spaces that describes viral dynamics [3] given time-series data of viral load. 2 − D and 3 − D
are labels that are assigned to subjects from which temporal measurements are taken. a1 , a2 , a3
are parameters in the model. The symbols on the varieties correspond to estimated parametersin the previous study [3].

References[1] S. A. Peters, “Phisiologically-based pharmacokinetic(PBPK) modeling and simulations: principles, methods, and applications in the pharamacoceutical industry,” Wiley (2011).
[2] M. Komatsu and T. Yaguchi, “Mathematical modeling of kinetics of antigens and antibodies
based on experiments toward analysis of the mechanism of allergies,” Transactions of theJapan Society for Industrial and Applied Mathematics, Vol. 28, pp. 162–204 (2018).
[3] A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J. Layden and A.
S. Perelson, “Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-α
therapy,” Science, Vol. 282, pp. 103–107 (1998).
[4] M. Komatsu and T. Yaguchi, “Method for estimating hidden structures determined byunidentifiable state-space models and time-series data based on the Gröbner basis,” arXiv.

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