Dynamic compensation, parameter identifiability, and equivariances

Ideal adaptive control in biological systems: an analysis of P -invariance and dynamical compensation properties

BACKGROUND

Dynamical compensation (DC) provides robustness to parameter fluctuations. As an example, DC enables 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 the structural unidentifiability and the P -invariance property. The P -invariance property is a sufficient and necessary condition for the DC property. DC has been seen in systems with at least three dimensions. In this article, we discuss DC and P -invariance from an adaptive control perspective. An adaptive controller automatically adjusts its parameters to optimise performance, maintain stability, and deal with uncertainties in a system.

RESULTS

We initiate our analysis by introducing a simplified two-dimensional dynamical model with DC, fostering experimentation and understanding of the system's behavior. We explore the system's behavior with time-varying input and disturbance signals, with a focus on illustrating the system's P -invariance properties in phase portraits and step-like response graphs.

CONCLUSIONS

We show that DC can be seen as a case of ideal adaptive control since the system is invariant to the compensated parameter.

A calibration and uncertainty quantification analysis of classical, fractional and multiscale logistic models of tumour growth

BACKGROUND AND OBJECTIVE

The validation of mathematical models of tumour growth is frequently hampered by the lack of sufficient experimental data, resulting in qualitative rather than quantitative studies. Recent approaches to this problem have attempted to extract information about tumour growth by integrating multiscale experimental measurements, such as longitudinal cell counts and gene expression data. In the present study, we investigated the performance of several mathematical models of tumour growth, including classical logistic, fractional and novel multiscale models, in terms of quantifying in-vitro tumour growth in the presence and absence of therapy. We further examined the effect of genes associated with changes in chemosensitivity in cell death rates.

METHODS

The multiscale expansion of logistic growth models was performed by coupling gene expression profiles to the cell death rates. State-of-the-art Bayesian inference, likelihood maximisation and uncertainty quantification techniques allowed a thorough evaluation of model performance.

RESULTS

The results suggest that the classical single-cell population model (SCPM) was the best fit for the untreated and low-dose treatment conditions, while the multiscale model with a cell death rate symmetric with the expression profile of OCT4 (Sym-SCPM) yielded the best fit for the high-dose treatment data. Further identifiability analysis showed that the multiscale model was both structurally and practically identifiable under the condition of known OCT4 expression profiles.

CONCLUSIONS

Overall, the present study demonstrates that model performance can be improved by incorporating multiscale measurements of tumour growth when high-dose treatment is involved.

Symmetries in Dynamic Models of Biological Systems: Mathematical Foundations and Implications

Symmetries are ubiquitous in nature. Almost all organisms have some kind of “symmetry”, meaning that their shape does not change under some geometric transformation. This geometrical concept of symmetry is intuitive and easy to recognize. On the other hand, the behavior of many biological systems over time can be described with ordinary differential equations. These dynamic models may also possess “symmetries”, meaning that the time courses of some variables remain invariant under certain transformations. Unlike the previously mentioned symmetries, the ones present in dynamic models are not geometric, but infinitesimal transformations. These mathematical symmetries can be related to certain features of the system’s dynamic behavior, such as robustness or adaptation capabilities. However, they can also arise from questionable modeling choices, which may lead to non-identifiability and non-observability. This paper provides an overview of the types of symmetries that appear in dynamic models, the mathematical tools available for their analyses, the ways in which they are related to system properties, and the implications for biological modeling.

Integrating transcriptomics and bulk time course data into a mathematical framework to describe and predict therapeutic resistance in cancer

A significant challenge in the field of biomedicine is the development of methods to integrate the multitude of dispersed data sets into comprehensive frameworks to be used to generate optimal clinical decisions. Recent technological advances in single cell analysis allow for high-dimensional molecular characterization of cells and populations, but to date, few mathematical models have attempted to integrate measurements from the single cell scale with other types of longitudinal data. Here, we present a framework that actionizes static outputs from a machine learning model and leverages these as measurements of state variables in a dynamic model of treatment response. We apply this framework to breast cancer cells to integrate single cell transcriptomic data with longitudinal bulk cell population (bulk time course) data. We demonstrate that the explicit inclusion of the phenotypic composition estimate, derived from single cell RNA-sequencing data (scRNA-seq), improves accuracy in the prediction of new treatments with a concordance correlation coefficient (CCC) of 0.92 compared to a prediction accuracy of CCC = 0.64 when fitting on longitudinal bulk cell population data alone. To our knowledge, this is the first work that explicitly integrates single cell clonally-resolved transcriptome datasets with bulk time-course data to jointly calibrate a mathematical model of drug resistance dynamics. We anticipate this approach to be a first step that demonstrates the feasibility of incorporating multiple data types into mathematical models to develop optimized treatment regimens from data.

Multiplexing information flow through dynamic signalling systems

We consider how a signalling system can act as an information hub by multiplexing information arising from multiple signals. We formally define multiplexing, mathematically characterise which systems can multiplex and how well they can do it. While the results of this paper are theoretical, to motivate the idea of multiplexing, we provide experimental evidence that tentatively suggests that the NF-κB transcription factor can multiplex information about changes in multiple signals. We believe that our theoretical results may resolve the apparent paradox of how a system like NF-κB that regulates cell fate and inflammatory signalling in response to diverse stimuli can appear to have the low information carrying capacity suggested by recent studies on scalar signals. In carrying out our study, we introduce new methods for the analysis of large, nonlinear stochastic dynamic models, and develop computational algorithms that facilitate the calculation of fundamental constructs of information theory such as Kullback–Leibler divergences and sensitivity matrices, and link these methods to a new theory about multiplexing information. We show that many current models such as those of the NF-κB system cannot multiplex effectively and provide models that overcome this limitation using post-transcriptional modifications.

Cells use signalling systems to pass on information arising from their ever-changing environment to their processing units. These biochemical networks regulate the transmission of multiple signals within the noisy and complex cellular environment, controlling whether to turn on or off processes of cell defence, death, division, and others. The question of how they actually achieve that becomes particularly critical given that many diseases occur when signalling systems malfunction. In this paper, we develop methodology and computational tools for simulating, measuring and analysing the ability of signalling systems to transmit multi-dimensional signals. We specifically focus on the capacity of signalling systems to simultaneously transmit multiple signals, such as temperature changes, presence and concentration of cytokines, viral and bacterial pathogens or drugs, through a single noisy, dynamic signalling system. We argue that a signalling system can act as an information hub, sending information in a multiplexed fashion rather similar to the way in which telecommunications networks send multiple signals over a shared medium by combining them into one.

A scalable method for parameter identification in kinetic models of metabolism using steady-state data

MOTIVATION

In kinetic models of metabolism, the parameter values determine the dynamic behaviour predicted by these models. Estimating parameters from in vivo experimental data require the parameters to be structurally identifiable, and the data to be informative enough to estimate these parameters. Existing methods to determine the structural identifiability of parameters in kinetic models of metabolism can only be applied to models of small metabolic networks due to their computational complexity. Additionally, a priori experimental design, a necessity to obtain informative data for parameter estimation, also does not account for using steady-state data to estimate parameters in kinetic models.

RESULTS

Here, we present a scalable methodology to structurally identify parameters for each flux in a kinetic model of metabolism based on the availability of steady-state data. In doing so, we also address the issue of determining the number and nature of experiments for generating steady-state data to estimate these parameters. By using a small metabolic network as an example, we show that most parameters in fluxes expressed by mechanistic enzyme kinetic rate laws can be identified using steady-state data, and the steady-state data required for their estimation can be obtained from selective experiments involving both substrate and enzyme level perturbations. The methodology can be used in combination with other identifiability and experimental design algorithms that use dynamic data to determine the most informative experiments requiring the least resources to perform.

AVAILABILITY AND IMPLEMENTATION

https://github.com/LMSE/ident.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Mathematical Details on a Cancer Resistance Model

One of the most important factors limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. We have recently introduced a framework for quantifying the effects of induced and non-induced resistance to cancer chemotherapy (Greene et al., 2018a, 2019). In this work, we expound on the details relating to an optimal control problem outlined in Greene et al. (2018a). The control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie algebraic techniques. A structural identifiability analysis is also presented, demonstrating that patient-specific parameters may be measured and thus utilized in the design of optimal therapies prior to the commencement of therapy. For completeness, a detailed analysis of existence results is also included.

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.

Mathematical deconvolution of CAR T-cell proliferation and exhaustion from real-time killing assay data

Chimeric antigen receptor (CAR) T-cell therapy has shown promise in the treatment of haematological cancers and is currently being investigated for solid tumours, including high-grade glioma brain tumours. There is a desperate need to quantitatively study the factors that contribute to the efficacy of CAR T-cell therapy in solid tumours. In this work, we use a mathematical model of predator-prey dynamics to explore the kinetics of CAR T-cell killing in glioma: the Chimeric Antigen Receptor T-cell treatment Response in GliOma (CARRGO) model. The model includes rates of cancer cell proliferation, CAR T-cell killing, proliferation, exhaustion, and persistence. We use patient-derived and engineered cancer cell lines with an in vitro real-time cell analyser to parametrize the CARRGO model. We observe that CAR T-cell dose correlates inversely with the killing rate and correlates directly with the net rate of proliferation and exhaustion. This suggests that at a lower dose of CAR T-cells, individual T-cells kill more cancer cells but become more exhausted when compared with higher doses. Furthermore, the exhaustion rate was observed to increase significantly with tumour growth rate and was dependent on level of antigen expression. The CARRGO model highlights nonlinear dynamics involved in CAR T-cell therapy and provides novel insights into the kinetics of CAR T-cell killing. The model suggests that CAR T-cell treatment may be tailored to individual tumour characteristics including tumour growth rate and antigen level to maximize therapeutic benefit.

Parameter estimation and identifiability in a neural population model for electro-cortical activity

Electroencephalography (EEG) provides a non-invasive measure of brain electrical activity. Neural population models, where large numbers of interacting neurons are considered collectively as a macroscopic system, have long been used to understand features in EEG signals. By tuning dozens of input parameters describing the excitatory and inhibitory neuron populations, these models can reproduce prominent features of the EEG such as the alpha-rhythm. However, the inverse problem, of directly estimating the parameters from fits to EEG data, remains unsolved. Solving this multi-parameter non-linear fitting problem will potentially provide a real-time method for characterizing average neuronal properties in human subjects. Here we perform unbiased fits of a 22-parameter neural population model to EEG data from 82 individuals, using both particle swarm optimization and Markov chain Monte Carlo sampling. We estimate how much is learned about individual parameters by computing Kullback-Leibler divergences between posterior and prior distributions for each parameter. Results indicate that only a single parameter, that determining the dynamics of inhibitory synaptic activity, is directly identifiable, while other parameters have large, though correlated, uncertainties. We show that the eigenvalues of the Fisher information matrix are roughly uniformly spaced over a log scale, indicating that the model is sloppy, like many of the regulatory network models in systems biology. These eigenvalues indicate that the system can be modeled with a low effective dimensionality, with inhibitory synaptic activity being prominent in driving system behavior.

Dynamical compensation and structural identifiability of biological models: 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. However, the original definition of dynamical compensation amounts to lack of structural identifiability. This is relevant if model parameters need to be estimated, as is often the case in biological modelling. 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.

A robust behaviour is a desirable feature in many biological systems. The study of mechanisms capable of maintaining the transient response unchanged despite environmental disturbances has recently motivated the introduction of a new concept: Dynamical Compensation (DC). However, the original definition of DC with respect to a parameter amounts to structural unidentifiability of that parameter, which means that it cannot be estimated by measuring the model output. Since most biological models have unknown parameters that need to be estimated, DC can be considered a negative property for the purpose of model identification. In this paper we reconcile these two conflicting views by proposing a new definition of DC that captures its intended biological meaning (i.e. robustness, which should be a systemic property, intrinsic to the dynamics) while making it distinct from structural unidentifiability (which is a modelling property that depends on decisions made by the modeller, such as the choice of model outputs or unknown parameters, and on experimental constraints). Our definition enables a model to have DC with respect to a structurally identifiable parameter, thus increasing the applicability of the concept.

Analysis of network motifs in cellular regulation: Structural similarities, input-output relations and signal integration

Much of the complexity of regulatory networks derives from the necessity to integrate multiple signals and to avoid malfunction due to cross-talk or harmful perturbations. Hence, one may expect that the input-output behavior of larger networks is not necessarily more complex than that of smaller network motifs which suggests that both can, under certain conditions, be described by similar equations. In this review, we illustrate this approach by discussing the similarities that exist in the steady state descriptions of a simple bimolecular reaction, covalent modification cycles and bacterial two-component systems. Interestingly, in all three systems fundamental input-output characteristics such as thresholds, ultrasensitivity or concentration robustness are described by structurally similar equations. Depending on the system the meaning of the parameters can differ ranging from protein concentrations and affinity constants to complex parameter combinations which allows for a quantitative understanding of signal integration in these systems. We argue that this approach may also be extended to larger regulatory networks.