Dynamical compensation and structural identifiability: analysis, implications, and reconciliation

The concept of dynamical compensation has been recently introduced to describe the ability of a biological system to keep its output dynamics unchanged in the face of varying parameters. Here we show that, according to its original definition, dynamical compensation is equivalent to lack of structural identifiability. This is relevant if model parameters need to be estimated, which is often the case in biological modelling. This realization prompts us to warn that care should we taken when using an unidentifiable model to extract biological insight: the estimated values of structurally unidentifiable parameters are meaningless, and model predictions about unmeasured state variables can be wrong. Taking this into account, we explore alternative definitions of dynamical compensation that do not necessarily imply structural unidentifiability. Accordingly, we show different ways in which a model can be made identifiable while exhibiting dynamical compensation. Our analyses enable the use of the new concept of dynamical compensation in the context of parameter identification, and reconcile it with the desirable property of structural identifiability

Observability and Structural Identifiability of Nonlinear Biological Systems

Observability is a modelling property that describes the possibility of inferring the internal state of a system from observations of its output. A related property, structural identifiability, refers to the theoretical possibility of determining the parameter values from the output. In fact, structural identifiability becomes a particular case of observability if the parameters are considered as constant state variables. It is possible to simultaneously analyse the observability and structural identifiability of a model using the conceptual tools of differential geometry. Many complex biological processes can be described by systems of nonlinear ordinary differential equations, and can therefore be analysed with this approach. The purpose of this review article is threefold: (I) to serve as a tutorial on observability and structural identifiability of nonlinear systems, using the differential geometry approach for their analysis; (II) to review recent advances in the field; and (III) to identify open problems and suggest new avenues for research in this area.Comment: Accepted for publication in the special issue "Computational Methods for Identification and Modelling of Complex Biological Systems" of Complexit

Input-dependent structural identifiability of nonlinear systems

A dynamic model is structurally identifiable if it is possible to infer its unknown parameters by observing its output. Structural identifiability depends on the system dynamics, output, and input, as well as on the specific values of initial conditions and parameters. Here we present a symbolic method that characterizes the input that a model requires to be structurally identifiable. It determines which derivatives must be non-zero in order to have a sufficiently exciting input. Our approach considers structural identifiability as a generalization of nonlinear observability and incorporates extended Lie derivatives. The methodology assesses structural identifiability for time-varying inputs and, additionally, it can be used to determine the input profile that is required to make the parameters structurally locally identifiable. Furthermore, it is sometimes possible to replace an experiment with time-varying input with multiple experiments with constant inputs. We implement the resulting method as a MATLAB toolbox named STRIKE-GOLDD2. This tool can assist in the design of new experiments for the purpose of parameter estimation

AutoRepar: a method to obtain identifiable and observable reparameterizations of dynamic models with mechanistic insights

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGMechanistic dynamic models of biological systems allow for a quantitative and systematic interpretation of data and the generation of testable hypotheses. However, these models are often over-parameterized, leading to nonidentifiability and nonobservability, that is, the impossibility of inferring their parameters and state variables. The lack of structural identifiability and observability (SIO) compromises a model's ability to make predictions and provide insight. Here we present a methodology, AutoRepar, that corrects SIO deficiencies of nonlinear ODE models automatically, yielding reparameterized models that are structurally identifiable and observable. The reparameterization preserves the mechanistic meaning of selected variables, and has the exact same dynamics and input-output mapping as the original model. We implement AutoRepar as an extension of the STRIKE-GOLDD software toolbox for SIO analysis, applying it to several models from the literature to demonstrate its ability to repair their structural deficiencies. AutoRepar increases the applicability of mechanistic models, enabling them to provide reliable information about their parameters and dynamics.Consejo Superior de Investigaciones Científicas https://doi.org/10.13039/501100003339 | Ref. PIE 202070E062MCIN/AEI/10.13039/501100011033 | Ref. RYC-2019-027537-IMCIN/AEI/10.13039/501100011033 | Ref. PID2020-113992RA-I00MCIN/AEI/ 10.13039/501100011033 | Ref. PID2020-117271RB-C2MCIN/AEI/ 10.13039/501100011033 | Ref. DPI2017-82896-C2-2-RXunta de Galicia | Ref. ED431F 2021/00

An analysis of P\mathbb{P}-invariance and dynamical compensation properties from a control perspective

Dynamical compensation (DC) provides robustness to parameter fluctuations. As an example, DC enable control of the functional mass of endocrine or neuronal tissue essential for controlling blood glucose by insulin through a nonlinear feedback loop. Researchers have shown that DC is related to structural unidentifiability and $\mathbb{P}$-invariance property, and $\mathbb{P}$-invariance property is a sufficient and necessary condition for the DC property. In this article, we discuss DC and $\mathbb{P}$-invariancy from an adaptive control perspective. An adaptive controller is a self-tuning controller used to compensate for changes in a dynamical system. To design an adaptive controller with the DC property, it is easier to start with a two-dimensional dynamical model. We introduce a simplified system of ordinary differential equations (ODEs) with the DC property and extend it to a general form. The value of the ideal adaptive control lies in developing methods to synthesize DC to variations in multiple parameters. Then we investigate the stability of the system with time-varying input and disturbance signals, with a focus on the system's $\mathbb{P}$-invariance properties. This study provides phase portraits and step-like response graphs to visualize the system's behavior and stability properties

Joining and decomposing reaction networks

In systems and synthetic biology, much research has focused on the behavior and design of single pathways, while, more recently, experimental efforts have focused on how cross-talk (coupling two or more pathways) or inhibiting molecular function (isolating one part of the pathway) affects systems-level behavior. However, the theory for tackling these larger systems in general has lagged behind. Here, we analyze how joining networks (e.g., cross-talk) or decomposing networks (e.g., inhibition or knock-outs) affects three properties that reaction networks may possess---identifiability (recoverability of parameter values from data), steady-state invariants (relationships among species concentrations at steady state, used in model selection), and multistationarity (capacity for multiple steady states, which correspond to multiple cell decisions). Specifically, we prove results that clarify, for a network obtained by joining two smaller networks, how properties of the smaller networks can be inferred from or can imply similar properties of the original network. Our proofs use techniques from computational algebraic geometry, including elimination theory and differential algebra.Comment: 44 pages; extensive revision in response to referee comment

Identification of neutral biochemical network models from time series data

<p>Abstract</p> <p>Background</p> <p>The major difficulty in modeling biological systems from multivariate time series is the identification of parameter sets that endow a model with dynamical behaviors sufficiently similar to the experimental data. Directly related to this parameter estimation issue is the task of identifying the structure and regulation of ill-characterized systems. Both tasks are simplified if the mathematical model is canonical, <it>i.e</it>., if it is constructed according to strict guidelines.</p> <p>Results</p> <p>In this report, we propose a method for the identification of admissible parameter sets of canonical S-systems from biological time series. The method is based on a Monte Carlo process that is combined with an improved version of our previous parameter optimization algorithm. The method maps the parameter space into the network space, which characterizes the connectivity among components, by creating an ensemble of decoupled S-system models that imitate the dynamical behavior of the time series with sufficient accuracy. The concept of sloppiness is revisited in the context of these S-system models with an exploration not only of different parameter sets that produce similar dynamical behaviors but also different network topologies that yield dynamical similarity.</p> <p>Conclusion</p> <p>The proposed parameter estimation methodology was applied to actual time series data from the glycolytic pathway of the bacterium <it>Lactococcus lactis </it>and led to ensembles of models with different network topologies. In parallel, the parameter optimization algorithm was applied to the same dynamical data upon imposing a pre-specified network topology derived from prior biological knowledge, and the results from both strategies were compared. The results suggest that the proposed method may serve as a powerful exploration tool for testing hypotheses and the design of new experiments.</p

Modeling and identification of a gene regulatory network programming erythropoiesis (1)

The development of mature blood cells of distinct lineages from the hematopoietic stem cells (hematopoiesis) involves a progressive restriction of differentiation potential and the establishment of lineage-speci&#64257;c gene expression profiles. The establishment of these profiles relies on lineage-speci&#64257;c transcription factors to modulate the expression of their target genes. This work is embedded in a wider ErasmusMC/CWI collaboration that develops the informatics and mathematics to underpin studies on gene expression regulation by mapping and analyzing the regulatory pathways and networks of transcription factors that control cellular functions (so called 'Gene Regulatory Networks' or 'GRNs'). This project is concerned with the mathematical part and concentrates on a GRN central to erythropoiesis. Among the many housekeeping and tissue-speci&#64257;c genes involved in the differentiation and the commitment of hematopoietic stem cells to erythrocytes (erythropoiesis), we focus on a small pool of genes (Gata-1, Gata-2, Pu.1, EKLF, FOG-1, alpha/beta-globin) known to be critically involved in an intricate but well-less investigated regulatory circuit. Based on the regulatory interactions in the GRN we have developed models in the form of a system to account for the dynamics of gene expression and regulation involved in this process. Because of the lack of information about a signi&#64257;cant number of model parameters, our focus is on system identi&#64257;cation. In this first report some preliminary results are presented based on synthetic data. However, time series of the levels of all relevant mRNA’s are available from micro-array analysis of G1E cells, a murine cell line which recapitulates erythropoiesis. In the follow-up report a detailed account will be given of the parameter estimation and identifiability analysis with respect to these data. This will eventually allow for a thorough evaluation of the role of various characterized as well as hypothetical regulatory mechanisms. In depth characterization of the necessary expression patterns and gene regulatory interactions responsible for the the set of commitments all along the erythroid lineage is essential to gain fundamental insight into the behaviour of these complex networks and to design further experiments. Ultimately, this may lead to ways to rescue erythroid differentiation in several anemic diseases

A simple model to control growth rate of synthetic E. coli during the exponential phase: model analysis and parameter estimation

International audienceWe develop and analyze a model of a minimal synthetic gene circuit, that describes part of the gene expression machinery in Escherichia coli, and enables the control of the growth rate of the cells during the exponential phase. This model is a piecewise non-linear system with two variables (the concentrations of two gene products) and an input (an inducer). We study the qualitative dynamics of the model and the bifurcation diagram with respect to the input. Moreover, an analytic expression of the growth rate during the exponential phase as function of the input is derived. A relevant problem is that of identi ability of the parameters of this expression supposing noisy measurements of exponential growth rate. We present such an identi ability study that we validate in silico with synthetic measurements