How and why to build a mathematical model: A case study using prion aggregation

Supercritical Carbon Dioxide Extraction of Coumarins from the Aerial Parts of Pterocaulon polystachyum

Pterocaulon polystachyum is a species of pharmacological interest for providing volatile and non-volatile extracts with antifungal and amebicidal properties. The biological activities of non-volatile extracts may be related to the presence of coumarins, a promising group of secondary metabolites. In the present study, leaves and inflorescences previously used for the extraction of essential oils instead of being disposed of were subjected to extraction with supercritical CO2 after pretreatment with microwaves. An experimental design was followed to seek the best extraction condition with the objective function being the maximum total extract. Pressure and temperature were statistically significant factors, and the optimal extraction condition was 240 bar, 60 °C, and pretreatment at 30 °C. The applied mathematical models showed good adherence to the experimental data. The extracts obtained by supercritical CO2 were analyzed and the presence of coumarins was confirmed. The extract investigated for cytotoxicity against bladder tumor cells (T24) exhibited significant reduction in cell viability at concentrations between 6 and 12 μg/mL. The introduction of green technology, supercritical extraction, in the exploration of P. polystachyum as a source of coumarins represents a paradigm shift with regard to previous studies carried out with this species, which used organic solvents. Furthermore, the concept of circular bioeconomy was applied, i.e., the raw material used was the residue of a steam-distillation process. Therefore, the approach used here is in line with the sustainable exploitation of native plants to obtain extracts rich in coumarins with cytotoxic potential against cancer cells.

MIL-CELL: a tool for multi-scale simulation of yeast replication and prion transmission

The single-celled baker's yeast, Saccharomyces cerevisiae, can sustain a number of amyloid-based prions, the three most prominent examples being [URE3], [PSI+], and [PIN+]. In the laboratory, haploid S. cerevisiae cells of a single mating type can acquire an amyloid prion in one of two ways (i) spontaneous nucleation of the prion within the yeast cell, and (ii) receipt via mother-to-daughter transmission during the cell division cycle. Similarly, prions can be lost due to (i) dissolution of the prion amyloid by its breakage into non-amyloid monomeric units, or (ii) preferential donation/retention of prions between the mother and daughter during cell division. Here we present a computational tool (Monitoring Induction and Loss of prions in Cells; MIL-CELL) for modelling these four general processes using a multiscale approach describing both spatial and kinetic aspects of the yeast life cycle and the amyloid-prion behavior. We describe the workings of the model, assumptions upon which it is based and some interesting simulation results pertaining to the wave-like spread of the epigenetic prion elements through the yeast population. MIL-CELL is provided as a stand-alone GUI executable program for free download with the paper. MIL-CELL is equipped with a relational database allowing all simulated properties to be searched, collated and graphed. Its ability to incorporate variation in heritable properties means MIL-CELL is also capable of simulating loss of the isogenic nature of a cell population over time. The capability to monitor both chronological and reproductive age also makes MIL-CELL potentially useful in studies of cell aging.

Different Trajectories for Diabetes Mellitus Onset and Recovery According to the Centralized Aerobic-Anaerobic Energy Balance Compensation Theory

We recently reported that the restoration of cervical vertebral arterial blood flow access (measured as systolic peak (PS)) to the rhomboid fossa leads to the recovery of the HbA1c level in the case of patients with a pre-Diabetes Mellitus (pre-DM) condition. The theory of centralized aerobic-anaerobic energy balance compensation (TCAAEBC) provides a successful theoretical explanation for this observation. It considers the human body as a dissipative structure. Reported connections between arterial hypertension (AHT) and the level of HbA1c are linked through OABFRH. According to the TCAAEBC, this delivers incorrect information about blood oxygen availability to the cerebellum. The restoration of PS normalizes AHT in 5-6 weeks and HbA1c in 12-13 weeks. In the current study, we demonstrate the model which fits the obtained experimental data. According to the model, pathways of onset and recovery from pre-DM are different. The consequence of these differences is discussed. The great significance of the TCAAEBC for medical practice forces the creation of an appropriate mathematical model, but the required adjustment of the model needs experimental data which can only be obtained from an animal model(s). The essential part of this study is devoted to the analysis of the advantages and disadvantages of widely available common mammalian models for TCAAEBC cases.

A group theoretic approach to model comparison with simplicial representations

The complexity of biological systems, and the increasingly large amount of associated experimental data, necessitates that we develop mathematical models to further our understanding of these systems. Because biological systems are generally not well understood, most mathematical models of these systems are based on experimental data, resulting in a seemingly heterogeneous collection of models that ostensibly represent the same system. To understand the system we therefore need to understand how the different models are related to each other, with a view to obtaining a unified mathematical description. This goal is complicated by the fact that a number of distinct mathematical formalisms may be employed to represent the same system, making direct comparison of the models very difficult. A methodology for comparing mathematical models based on their underlying conceptual structure is therefore required. In previous work we developed an appropriate framework for model comparison where we represent models, specifically the conceptual structure of the models, as labelled simplicial complexes and compare them with the two general methodologies of comparison by distance and comparison by equivalence. In this article we continue the development of our model comparison methodology in two directions. First, we present a rigorous and automatable methodology for the core process of comparison by equivalence, namely determining the vertices in a simplicial representation, corresponding to model components, that are conceptually related and the identification of these vertices via simplicial operations. Our methodology is based on considerations of vertex symmetry in the simplicial representation, for which we develop the required mathematical theory of group actions on simplicial complexes. This methodology greatly simplifies and expedites the process of determining model equivalence. Second, we provide an alternative mathematical framework for our model-comparison methodology by representing models as groups, which allows for the direct application of group-theoretic techniques within our model-comparison methodology.

Understanding fibrosis pathogenesis via modeling macrophage-fibroblast interplay in immune-metabolic context

Fibrosis is a progressive biological condition, leading to organ dysfunction in various clinical settings. Although fibroblasts and macrophages are known as key cellular players for fibrosis development, a comprehensive functional model that considers their interaction in the metabolic/immunologic context of fibrotic tissue has not been set up. Here we show, by transcriptome-based mathematical modeling in an in vitro system that represents macrophage-fibroblast interplay and reflects the functional effects of inflammation, hypoxia and the adaptive immune context, that irreversible fibrosis development is associated with specific combinations of metabolic and inflammatory cues. The in vitro signatures are in good alignment with transcriptomic profiles generated on laser captured glomeruli and cortical tubule-interstitial area, isolated from human transplanted kidneys with advanced stages of glomerulosclerosis and interstitial fibrosis/tubular atrophy, two clinically relevant conditions associated with organ failure in renal allografts. The model we describe here is validated on tissue based quantitative immune-phenotyping of biopsies from transplanted kidneys, demonstrating its feasibility. We conclude that the combination of in vitro and in silico modeling represents a powerful systems medicine approach to dissect fibrosis pathogenesis, applicable to specific pathological conditions, and develop coordinated targeted approaches.

Mathematical Modeling of a Supramolecular Assembly for Pyrophosphate Sensing

The power of sensing molecules is often characterized in part by determining their thermodynamic/dynamic properties, in particular the binding constant of a guest to a host. In many studies, traditional nonlinear regression analysis has been used to determine the binding constants, which cannot be applied to complex systems and limits the reliability of such calculations. Supramolecular sensor systems include many interactions that make such chemical systems complicated. The challenges in creating sensing molecules can be significantly decreased through the availability of detailed mathematical models of such systems. Here, we propose uncovering accurate thermodynamic parameters of chemical reactions using better-defined mathematical modeling-fitting analysis is the key to understanding molecular assemblies and developing new bio/sensing agents. The supramolecular example we chose for this investigation is a self-assembled sensor consists of a synthesized receptor, DPA (DPA = dipicolylamine)-appended phenylboronic acid (1) in combination with Zn2+(1.Zn) that forms various assemblies with a fluorophore like alizarin red S (ARS). The self-assemblies can detect multi-phosphates like pyrophosphate (PPi) in aqueous solutions. We developed a mathematical model for the simultaneous quantitative analysis of twenty-seven intertwined interactions and reactions between the sensor (1.Zn-ARS) and the target (PPi) for the first time, relying on the Newton-Raphson algorithm. Through analyzing simulated potentiometric titration data, we describe the concurrent determination of thermodynamic parameters of the different guest-host bindings. Various values of temperatures, initial concentrations, and starting pHs were considered to predict the required measurement conditions for thermodynamic studies. Accordingly, we determined the species concentrations of different host-guest bindings in a generalized way. This way, the binding capabilities of a set of species can be quantitatively examined to systematically measure the power of the sensing system. This study shows analyzing supramolecular self-assemblies with solid mathematical models has a high potential for a better understanding of molecular interactions within complex chemical networks and developing new sensors with better sensing effects for bio-purposes.

Analysis of steady-state carbon tracer experiments using akaike information criteria

INTRODUCTION

Carbon isotope tracers have been used to determine relative rates of tricarboxylic acid cycle (TCA) cycle pathways since the 1950s. Steady-state experimental data are typically fit to a single mathematical model of metabolism to determine metabolic fluxes. Whether the chosen model is appropriate for the biological system has generally not been evaluated systematically. An overly-simple model omits known pathways while an overly-complex model may produce incorrect results due to overfitting.

OBJECTIVES

The objectives were to develop and study a method that systematically evaluates multiple TCA cycle mathematical models as part of the fitting process.

METHODS

The problem of choosing overly-simple or overly-complex models was approached by developing software that automatically explores all possible combinations of flux through pyruvate dehydrogenase, pyruvate kinase, pyruvate carboxylase and anaplerosis at propionyl-CoA carboxylase, and equivalent pathways, all relative to TCA cycle flux. Typical TCA cycle metabolic tracer experiments that use ¹³C nuclear magnetic resonance for detection and quantification of ¹³C-enriched glutamate products were simulated and analyzed. By evaluating the multiple model fits with both the conventional sum-of-squares residual error (SSRE) and the Akaike Information Criterion (AIC), the software helps the investigator understand the interaction between model complexity and goodness of fit.

RESULTS

When fitting alternative models of the TCA cycle metabolism, the SSRE may identify more than one model that fits the data well. Among those models, the AIC provides guidance as to which is the simplest of the candidate models is sufficient to describe the observed data. However under some conditions, AIC used alone inappropriately discriminates against necessary metabolic complexity.

CONCLUSION

In combination, the SSRE and AIC help the investigator identify the model that best describes the metabolism of a biological system.

A Systematic Review of the Efforts and Hindrances of Modeling and Simulation of CAR T-cell Therapy

Chimeric antigen receptor (CAR) T-cell therapy is an immunotherapy that has recently become highly instrumental in the fight against life-threatening diseases. A variety of modeling and computational simulation efforts have addressed different aspects of CAR T-cell therapy, including T-cell activation, T- and malignant cell population dynamics, therapeutic cost-effectiveness strategies, and patient survival. In this article, we present a systematic review of those efforts, including mathematical, statistical, and stochastic models employing a wide range of algorithms, from differential equations to machine learning. To the best of our knowledge, this is the first review of all such models studying CAR T-cell therapy. In this review, we provide a detailed summary of the strengths, limitations, methodology, data used, and data gap in currently published models. This information may help in designing and building better models for enhanced prediction and assessment of the benefit-risk balance associated with novel CAR T-cell therapies, as well as with the data need for building such models.

Lost in modelling and simulation?

Over the past few decades, physiologically-based pharmacokinetic modelling (PBPK) has been anticipated to be a powerful tool to improve the productivity of drug discovery and development. However, recently, multiple systematic evaluation studies independently suggested that the predictive power of current oral absorption (OA) PBPK models needs significant improvement. There is some disagreement between the industry and regulators about the credibility of OA PBPK modelling. Recently, the editorial board of AMDET&DMPK has announced the policy for the articles related to PBPK modelling (Modelling and simulation ethics). In this feature article, the background of this policy is explained: (1) Requirements for scientific writing of PBPK modelling, (2) Scientific literacy for PBPK modelling, and (3) Middle-out approaches. PBPK models are a useful tool if used correctly. This article will hopefully help advance the science of OA PBPK models.

The Art and Science of Building a Computational Model to Understand Hemostasis

Computational models of various facets of hemostasis and thrombosis have increased substantially in the last decade. These models have the potential to make predictions that can uncover new mechanisms within the complex dynamics of thrombus formation. However, these predictions are only as good as the data and assumptions they are built upon, and therefore model building requires intimate coupling with experiments. The objective of this article is to guide the reader through how a computational model is built and how it can inform and be refined by experiments. This is accomplished by answering six questions facing the model builder: (1) Why make a model? (2) What kind of model should be built? (3) How is the model built? (4) Is the model a “good” model? (5) Do we believe the model? (6) Is the model useful? These questions are answered in the context of a model of thrombus formation that has been successfully applied to understanding the interplay between blood flow, platelet deposition, and coagulation and in identifying potential modifiers of thrombin generation in hemophilia A.

Towards Generative Design of Computationally Efficient Mathematical Models with Evolutionary Learning

In this paper, we describe the concept of generative design approach applied to the automated evolutionary learning of mathematical models in a computationally efficient way. To formalize the problems of models' design and co-design, the generalized formulation of the modeling workflow is proposed. A parallelized evolutionary learning approach for the identification of model structure is described for the equation-based model and composite machine learning models. Moreover, the involvement of the performance models in the design process is analyzed. A set of experiments with various models and computational resources is conducted to verify different aspects of the proposed approach.