Mathematical Modeling of Photo- and Thermomorphogenesis in Plants.

Increased day lengths and warm conditions inversely affect plant growth by directly modulating nuclear phyB, ELF3, and COP1 levels. Quantitative measures of the hypocotyl length have been key to gaining a deeper understanding of this complex regulatory network, while similar quantitative data are the foundation for many studies in plant biology. Here, we explore the application of mathematical modeling, specifically ordinary differential equations (ODEs), to understand plant responses to these environmental cues. We provide a comprehensive guide to constructing, simulating, and fitting these models to data, using the law of mass action to study the evolution of molecular species. The fundamental principles of these models are introduced, highlighting their utility in deciphering complex plant physiological interactions and testing hypotheses. This brief introduction will not allow experimentalists without a mathematical background to run their own simulations overnight, but it will help them grasp modeling principles and communicate with more theory-inclined colleagues.

Modeling of Reactive Sputtering-History and Development.

This work critically reviews the evolution of reactive sputtering modeling that has taken place over the last 50 years. The review summarizes the main features of the deposition of simple metal compound films (nitrides, oxides, oxynitrides, carbides, etc.) that were experimentally found by different researchers. The above features include significant non-linearity and hysteresis. At the beginning of the 1970s, specific chemisorption models were proposed. These models were based on the assumption that a compound film was formed on the target due to chemisorption. Their development led to the appearance of the general isothermal chemisorption model, which was supplemented by the processes on the surfaces of the vacuum chamber wall and the substrate. The model has undergone numerous transformations for application to various problems of reactive sputtering. At the next step in the development of modeling, the reactive sputtering deposition (RSD) model was proposed, which was based on the implantation of reactive gas molecules into the target, bulk chemical reaction, chemisorption, and the "knock-on effect". Another direction of the modeling development is represented by the nonisothermal physicochemical model, in which the Langmuir isotherm and the law of mass action are used. Various modifications of this model allowed describing reactive sputtering processes in more complex cases when the sputtering unit included a hot target or a sandwich one.

The Alchemist, Metal-Divider and Transmuter Carl F. Wenzel and his 1776 Award from the Royal Danish Academy of Sciences through Professor C.†G. Kratzenstein.

C. F. Wenzel was a chemist and an alchemist. He had deep knowledge of acids, bases and salts, and he was credited with the first formulation of the Law of Mass Action. Yet he was also an alchemist, who on the eve of the Chemical Revolution published his beliefs in transmutation and in the division of metals into their constituents, for which he was rewarded with the gold medal of the Royal Danish Academy of the Sciences. His promoter, Professor C. G. Kratzenstein, was himself a believer in transmutation, even if he voiced some reservations.

Strong Base Anion Exchange Selectivity of Nine Perfluoroalkyl Chemicals Relevant to Drinking Water.

Selectivity with respect to chloride (KPFAS∕C) was determined for nine drinking water relevant perfluoroalkyl and polyfluoroalkyl substances (PFAS): perfluoro-2-propoxypropanoic acid (GenX), five perfluoroalkyl carboxylic acids (PFCAs), and three perfluoroalkyl sulfonic acids (PFSAs). Three single-use strong base anion exchange gel resins were investigated, targeting drinking water relevant equilibrium PFAS liquid concentrations (≤500 ng/L). Except for the longest carbon chain PFCA (perfluorodecanoic acid) and PFSA (perfluorooctanesulfonic acid) studied, PFAS followed traditional ion exchange theory (law of mass action), including increasing equilibrium PFAS liquid concentrations with increasing equilibrium chloride liquid concentrations. Overall, KPFAS∕C values were (i) similar among resins for a given PFAS, (ii) 1-5 orders of magnitude greater than the selectivity of inorganic anions (e.g., nitrate) previously studied, (iii) 2 orders of magnitude greater for the same carbon chain length PFSA versus PFCA, (iv) found to proportionally increase with carbon chain length for both PFSAs and PFCAs, and (v) similar for GenX and perfluorohexanoic acid (six-carbon PFCA). A multisolute competition experiment demonstrated binary isotherm-determined KPFAS∕C values could be applied to simulate a multisolute system, extending work previously done with only inorganic anions to PFAS. Ultimately, estimated KPFAS∕C values allow future extension and validation of an open-source anion exchange column model to PFAS.

Free Drug Theory - No Longer Just a Hypothesis?

The Free Drug Hypothesis is a well-established concept within the scientific lexicon pervading many areas of Drug Discovery and Development, and yet it is poorly defined by virtue of many variations appearing in the literature. Clearly, unbound drug is in dynamic equilibrium with respect to absorption, distribution, metabolism, elimination, and indeed, interaction with the desired pharmacological target. Binding interactions be they specific (e.g. high affinity) or nonspecific (e.g. lower affinity/higher capacity) are governed by the same fundamental physicochemical tenets including Hill-Langmuir Isotherms, the Law of Mass Action and Drug Receptor Theory. With this in mind, it is time to recognise a more coherent version and consider it the Free Drug Theory and a hypothesis no longer. Today, we have the experimental and modelling capabilities, pharmacological knowledge, and an improved understanding of unbound drug distribution (e.g. Kpuu) to raise the bar on our understanding and analysis of experimental data. The burden of proof should be to rule out mechanistic possibilities and/or experimental error before jumping to the conclusion that any observations contradict these fundamentals.

Systems analysis shows that thermodynamic physiological and pharmacological fundamentals drive COVID-19 and response to treatment.

Why a systems analysis view of this pandemic? The current pandemic has inflicted almost unimaginable grief, sorrow, loss, and terror at a global scale. One of the great ironies with the COVID-19 pandemic, particularly early on, is counter intuitive. The speed at which specialized basic and clinical sciences described the details of the damage to humans in COVID-19 disease has been impressive. Equally, the development of vaccines in an amazingly short time interval has been extraordinary. However, what has been less well understood has been the fundamental elements that underpin the progression of COVID-19 in an individual and in populations. We have used systems analysis approaches with human physiology and pharmacology to explore the fundamental underpinnings of COVID-19 disease. Pharmacology powerfully captures the thermodynamic characteristics of molecular binding with an exogenous entity such as a virus and its consequences on the living processes well described by human physiology. Thus, we have documented the passage of SARS-CoV-2 from infection of a single cell to species jump, to tropism, variant emergence and widespread population infection. During the course of this review, the recurrent observation was the efficiency and simplicity of one critical function of this virus. The lethality of SARS-CoV-2 is due primarily to its ability to possess and use a variable surface for binding to a specific human target with high affinity. This binding liberates Gibbs free energy (GFE) such that it satisfies the criteria for thermodynamic spontaneity. Its binding is the prelude to human host cellular entry and replication by the appropriation of host cell constituent molecules that have been produced with a prior energy investment by the host cell. It is also a binding that permits viral tropism to lead to high levels of distribution across populations with newly formed virions. This thermodynamic spontaneity is repeated endlessly as infection of a single host cell spreads to bystander cells, to tissues, to humans in close proximity and then to global populations. The principal antagonism of this process comes from SARS-CoV-2 itself, with its relentless changing of its viral surface configuration, associated with the inevitable emergence of variants better configured to resist immune sequestration and importantly with a greater affinity for the host target and higher infectivity. The great value of this physiological and pharmacological perspective is that it reveals the fundamental thermodynamic underpinnings of SARS-CoV-2 infection.

Finding Liebig's law of the minimum.

Liebig's law of the minimum (LLM) is often used to interpret empirical biological growth data and model multiple substrates co-limited growth. However, its mechanistic foundation is rarely discussed, even though its validity has been questioned since its introduction in the 1820s. Here we first show that LLM is a crude approximation of the law of mass action, the state of art theory of biochemical reactions, and the LLM model is less accurate than two other approximations of the law of mass action: the synthesizing unit model and the additive model. We corroborate this conclusion using empirical data sets of algae and plants grown under two co-limiting substrates. Based on our analysis, we show that when growth is modeled directly as a function of substrate uptake, the LLM model improperly restricts the organism to be of fixed elemental stoichiometry, making it incapable of consistently resolving biological adaptation, ecological evolution, and community assembly. When growth is modeled as a function of the cellular nutrient quota, the LLM model may obtain good results at the risk of incorrect model parameters as compared to those inferred from the more accurate synthesizing unit model. However, biogeochemical models that implement these three formulations are needed to evaluate which formulation is acceptably accurate and their impacts on predicted long-term ecosystem dynamics. In particular, studies are needed that explore the extent to which parameter calibration can rescue model performance when the mechanistic representation of a biogeochemical process is known to be deficient.

Generalized indicator-based determination of solution pH.

pH indicators can be used both fast responsive as well as long-term stable sensors. They have been extensively used for monitoring pH changes in fast kinetic reactions as well as slowly changing pH in oceanic waters. If the pH range that needs to be covered is narrow it is possible to use only one indicator of appropriate protonation constant; otherwise, mixtures of two or more indicators are used for monitoring pH values covering a broad range of pH. In this paper we presented a new methodology for determining pH of solutions using mixtures of pH indicators. The pH calculation is based on the strict application of the basic laws of mass action and mass conservation. The proposed method was evaluated by the successful determination of the pH values of solutions containing three indicators (neutral red, phenol red (two different protonation constants), and methyl orange) covering a wide range of pH values from 0.5 to 9. The method was also applied for rapid monitoring of pH changes in stopped-flow measurements, investigating the reactions of CO2 in aqueous amine solutions.

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

Biological systems are inherently complex, and the increasing level of detail with which we are able to experimentally probe such systems continually reveals new complexity. Fortunately, mathematical models are uniquely positioned to provide a tool suitable for rigorous analysis, hypothesis generation, and connecting results from isolated in vitro experiments with results from in vivo and whole-organism studies. However, developing useful mathematical models is challenging because of the often different domains of knowledge required in both math and biology. In this work, we endeavor to provide a useful guide for researchers interested in incorporating mathematical modeling into their scientific process. We advocate for the use of conceptual diagrams as a starting place to anchor researchers from both domains. These diagrams are useful for simplifying the biological process in question and distinguishing the essential components. Not only do they serve as the basis for developing a variety of mathematical models, but they ensure that any mathematical formulation of the biological system is led primarily by scientific questions. We provide a specific example of this process from our own work in studying prion aggregation to show the power of mathematical models to synergistically interact with experiments and push forward biological understanding. Choosing the most suitable model also depends on many different factors, and we consider how to make these choices based on different scales of biological organization and available data. We close by discussing the many opportunities that abound for both experimentalists and modelers to take advantage of collaborative work in this field.

Homeopathy and law of Similars

This paper explains the elementary law of homeopathy, the Law of Similars, on the bases of thermodynamic aspects by means of the chemical thermodynamic. Le Chatelier principle was used, to explain the re-establishment of starting biochemical equilibrium compartmentalized in individual human cells of an ill person consuming the remedy, to clarify the Law of Similars. In addition, the application of the Law of mass action during the re-establishment of the initial equilibrium in an ill person when digesting the remedy exposed the law of Similars as the strongest outcome of homeopathy

Biological Pathway Specificity in the Cell-Does Molecular Diversity Matter?

Biology arises from the crowded molecular environment of the cell, rendering it a challenge to understand biological pathways based on the reductionist, low-concentration in vitro conditions generally employed for mechanistic studies. Recent evidence suggests that low-affinity interactions between cellular biopolymers abound, with still poorly defined effects on the complex interaction networks that lead to the emergent properties and plasticity of life. Mass-action considerations are used here to underscore that the sheer number of weak interactions expected from the complex mixture of cellular components significantly shapes biological pathway specificity. In particular, on-pathway-i.e., "functional"-become those interactions thermodynamically and kinetically stable enough to survive the incessant onslaught of the many off-pathway ("nonfunctional") interactions. Consequently, to better understand the molecular biology of the cell a further paradigm shift is needed toward mechanistic experimental and computational approaches that probe intracellular diversity and complexity more directly. Also see the video abstract here https://youtu.be/T19X_zYaBzg.

Activity-based analysis of potentiometric pH titrations.

The discrepancy between concentrations and activities is a predicament well known to the analytical chemist. Because of the difficulty of determining activity coefficients, the standard technique for quantitative equilibrium studies is to work under a particular 'constant ionic strength' by adding an excess of an inert salt. Under such conditions, activity coefficients are approximately constant and can be taken into the equilibrium constants which are defined for the chosen ionic strength (I). Here we propose a fundamentally different approach. Throughout the numerical analysis of the titration data, activity coefficients for all individual species are approximated by well-known equations based on the work of Debye-Hückel. The computational analysis of the measurements strictly obeys the law of mass conservation and obeys the law of mass action only approximately. The main novelty is that now the addition of inert salts is no longer required and measurements are done at minimal I. Consequently, the thermodynamic equilibrium constants are now determined much more robustly based on experiments taken at low I. The approach has been tested and validated with the two very well investigated 3-protic phosphoric and citric acids. In summary: the technique of artificially keeping ionic strength constant has been replaced by improved computational analysis.

The different faces of mass action in virus assembly.

The spontaneous encapsulation of genomic and non-genomic polyanions by coat proteins of simple icosahedral viruses is driven, in the first instance, by electrostatic interactions with polycationic RNA binding domains on these proteins. The efficiency with which the polyanions can be encapsulated in vitro, and presumably also in vivo, must in addition be governed by the loss of translational and mixing entropy associated with co-assembly, at least if this co-assembly constitutes a reversible process. These forms of entropy counteract the impact of attractive interactions between the constituents and hence they counteract complexation. By invoking mass action-type arguments and a simple model describing electrostatic interactions, we show how these forms of entropy might settle the competition between negatively charged polymers of different molecular weights for co-assembly with the coat proteins. In direct competition, mass action turns out to strongly work against the encapsulation of RNAs that are significantly shorter, which is typically the case for non-viral (host) RNAs. We also find that coat proteins favor forming virus particles over nonspecific binding to other proteins in the cytosol even if these are present in vast excess. Our results rationalize a number of recent in vitro co-assembly experiments showing that short polyanions are less effective at attracting virus coat proteins to form virus-like particles than long ones do, even if both are present at equal weight concentrations in the assembly mixture.

[Anomalous Kinetics of Amyloidogenesis Suggest a Competition between Oligomers and Fibrils].

Meisl et al. have recently observed an anomalous dependence of the amyloid formation rate on the protein concentration. A novel mechanism of fibril growth has been proposed by Meisl et al. to explain the abnormality; it consists in the fibril-catalyzed initiation of fibril formation with saturation of catalytic sites at high concentrations of substrates. Our article describes an alternative explanation of the anomalous kinetics, assuming that the formation of metastable oligomers competes with fibril formation by decreasing the concentration of free monomers. Oligomers are indeed observed in the course of amyloid formation, but are usually considered as seeds of amyloid fibrils rather as their competitors. However, the oligomers visually detectable by electron microscopy were shown to be close in size to those that can be derived from the anomalous dependence of the amyloid growth rate on the protein concentration, given that the anomaly results from competition between oligomer formation and amyloidogenesis.

Characterization of Kinetic Binding Properties of Unlabeled Ligands via a Preincubation Endpoint Binding Approach.

The dissociation rates of unlabeled drugs have been well studied by kinetic binding analyses. Since kinetic assays are laborious, we developed a simple method to determine the kinetic binding parameters of unlabeled competitors by a preincubation endpoint assay. The probe binding after preincubation of a competitor can be described by a single equation as a function of time. Simulations using the equation revealed the degree of IC50 change induced by preincubation of a competitor depended on the dissociation rate koff of the competitor but not on the association rate kon To validate the model, an in vitro binding assay was performed using a smoothened receptor (SMO) and [(3)H]TAK-441, a SMO antagonist. The equilibrium dissociation constants (KI) and koff of SMO antagonists determined by globally fitting the model to the concentration-response curves obtained with and without 24 h preincubation correlated well with those determined by other methods. This approach could be useful for early-stage optimization of drug candidates by enabling determination of binding kinetics in a high-throughput manner because it does not require kinetic measurements, an intermediate washout step during the reaction, or prior determination of competitors' KI values.

Cato Guldberg and Peter Waage, the history of the Law of Mass Action, and its relevance to clinical pharmacology.

We have traced the historical link between the Law of Mass Action and clinical pharmacology. The Law evolved from the work of the French chemist Claude Louis Berthollet, was first formulated by Cato Guldberg and Peter Waage in 1864 and later clarified by the Dutch chemist Jacobus van 't Hoff in 1877. It has profoundly influenced our qualitative and quantitative understanding of a number of physiological and pharmacological phenomena. According to the Law of Mass Action, the velocity of a chemical reaction depends on the concentrations of the reactants. At equilibrium the concentrations of the chemicals involved bear a constant relation to each other, described by the equilibrium constant, K. The Law of Mass Action is relevant to various physiological and pharmacological concepts, including concentration-effect curves, dose-response curves, and ligand-receptor binding curves, all of which are important in describing the pharmacological actions of medications, the Langmuir adsorption isotherm, which describes the binding of medications to proteins, activation curves for transmembrane ion transport, enzyme inhibition and the Henderson-Hasselbalch equation, which describes the relation between pH, as a measure of acidity and the concentrations of the contributory acids and bases. Guldberg and Waage recognized the importance of dynamic equilibrium, while others failed to do so. Their ideas, over 150 years old, are embedded in and still relevant to clinical pharmacology. Here we explain the ideas and in a subsequent paper show how they are relevant to understanding adverse drug reactions.

The law of mass action and the pharmacological concentration-effect curve: resolving the paradox of apparently non-dose-related adverse drug reactions.

BACKGROUND: Adverse drug reactions are sometimes described as being 'non-dose-related' because no relationship has been found between increasing doses and either the intensity of the response or the proportion of individuals in whom the response occurs; furthermore, hypersensitivity reactions are often regarded as being non-dose-related, even if different doses have not been studied. However, the law of mass action implies that all pharmacological effects are concentration related and should increase in intensity with increasing dose. We set out to explain this paradox. METHODS: We searched for published adverse drug reactions that have been described as non-dose-related and analysed them. RESULTS: We identified four categories of explanations that resolve the paradox: (i) the reaction is not real; it may have occurred by chance or there may be methodological problems, such as bias or confounding factors; (ii) the dose-response curve for the adverse effect reaches a maximum at doses lower than were studied (i.e. a hypersusceptibility reaction); this underpins the use of test doses to predict the possibility of an adverse reaction at therapeutic doses; (iii) susceptibility to the adverse reaction differs widely among individuals; and (iv) imprecision or inaccuracy in the measurement of either dose or effect obscures dose responsiveness. This last explanation encompasses: (a) reactions related to cumulative dose; (b) dissociation between dose and concentration through saturable pharmacokinetics; and (c) variability in the measurement of the effect. CONCLUSIONS AND IMPLICATIONS: If an adverse drug reaction appears to be non-dose-related, the reasons should be sought, having these mechanisms in mind.

Modelling the growth of plants with a uniform growth logistics.

BACKGROUND AND AIMS: The increment model has previously been used to describe the growth of plants in general. Here, we examine how the same logistics enables the development of different superstructures. METHODS: Data from the literature are analyzed with the increment model. Increments are growth-invariant molecular clusters, treated as heuristic particles. This approach formulates the law of mass action for multi-component systems, describing the general properties of superstructures which are optimized via relaxation processes. RESULTS: The daily growth patterns of hypocotyls can be reproduced implying predetermined growth invariant model parameters. In various species, the coordinated formation and death of fine roots are modeled successfully. Their biphasic annual growth follows distinct morphological programs but both use the same logistics. In tropical forests, distributions of the diameter in breast height of trees of different species adhere to the same pattern. Beyond structural fluctuations, competition and cooperation within and between the species may drive optimization. CONCLUSION: All superstructures of plants examined so far could be reproduced with our approach. With genetically encoded growth-invariant model parameters (interaction with the environment included) perfect morphological development runs embedded in the uniform logistics of the increment model.

Bioenergetic medicine.

Here we discuss a specific therapeutic strategy we call 'bioenergetic medicine'. Bioenergetic medicine refers to the manipulation of bioenergetic fluxes to positively affect health. Bioenergetic medicine approaches rely heavily on the law of mass action, and impact systems that monitor and respond to the manipulated flux. Since classically defined energy metabolism pathways intersect and intertwine, targeting one flux also tends to change other fluxes, which complicates treatment design. Such indirect effects, fortunately, are to some extent predictable, and from a therapeutic perspective may also be desirable. Bioenergetic medicine-based interventions already exist for some diseases, and because bioenergetic medicine interventions are presently feasible, new approaches to treat certain conditions, including some neurodegenerative conditions and cancers, are beginning to transition from the laboratory to the clinic.

Algebraic Statistical Model for Biochemical Network Dynamics Inference.

With modern molecular quantification methods, like, for instance, high throughput sequencing, biologists may perform multiple complex experiments and collect longitudinal data on RNA and DNA concentrations. Such data may be then used to infer cellular level interactions between the molecular entities of interest. One method which formalizes such inference is the stoichiometric algebraic statistical model (SASM) of [2] which allows to analyze the so-called conic (or single source) networks. Despite its intuitive appeal, up until now the SASM has been only heuristically studied on few simple examples. The current paper provides a more formal mathematical treatment of the SASM, expanding the original model to a wider class of reaction systems decomposable into multiple conic subnetworks. In particular, it is proved here that on such networks the SASM enjoys the so-called sparsistency property, that is, it asymptotically (with the number of observed network trajectories) discards the false interactions by setting their reaction rates to zero. For illustration, we apply the extended SASM to in silico data from a generic decomposable network as well as to biological data from an experimental search for a possible transcription factor for the heat shock protein 70 (Hsp70) in the zebrafish retina.