General quasi-equilibrium multivalent binding model to study diverse and complex drug-receptor interactions of biologics.

Pharmacokinetics and pharmacodynamics of many biologics are influenced by their complex binding to biological receptors. Biologics consist of diverse groups of molecules with different binding kinetics to its receptors including IgG with simple one-to-one drug receptor bindings, bispecific antibody (BsAb) that binds to two different receptors, and antibodies that can bind to six or more identical receptors. As the binding process is typically much faster than elimination (or internalization) and distribution processes, quasi-equilibrium (QE) binding models are commonly used to describe drug-receptor binding kinetics of biologics. However, no general QE modeling framework is available to describe complex binding kinetics for diverse classes of biologics. In this paper, we describe novel approaches of using differential algebraic equations (DAE) to solve three QE multivalent drug-receptor binding (QEMB) models. The first example describes the binding kinetics of three-body equilibria of BsAb that binds to 2 different receptors for trimer formation. The second example models an engineered IgG variant (Multabody) that can bind to 24 identical target receptors. The third example describes an IgG with modified neonatal Fc receptor (FcRn) binding affinity that competes for the same FcRn receptor as endogenous IgG. The model parameter estimates were obtained by fitting the model to all data simultaneously. The models allowed us to study potential roles of cooperative binding on bell-shaped drug exposure-response relationships of BsAb, and concentration-depended distribution of different drug-receptor complexes for Multabody. This DAE-based QEMB model platform can serve as an important tool to better understand complex binding kinetics of diverse classes of biologics.

Target-mediated drug disposition model for drugs with N > 2 binding sites that bind to a target with one binding site.

The paper extended the TMDD model to drugs with more than two (N > 2) identical binding sites (N-to-one TMDD). The quasi-steady-state (N-to-one QSS), quasi-equilibrium (N-to-one QE), irreversible binding (N-to-one IB), and Michaelis-Menten (N-to-one MM) approximations of the model were derived. To illustrate properties of new equations and approximations, N = 4 case was investigated numerically. Using simulations, the N-to-one QSS approximation was compared with the full N-to-one TMDD model. As expected, and similarly to the standard TMDD for monoclonal antibodies (mAb), N-to-one QSS predictions were nearly identical to N-to-one TMDD predictions, except for times of fast changes following initiation of dosing, when equilibrium has not yet been reached. Predictions for mAbs with soluble targets (slow elimination of the complex) were simulated from the full 4-to-one TMDD model and were fitted to the 4-to-one TMDD model and to its QSS approximation. It was demonstrated that the 4-to-one QSS model provided nearly identical description of not only the observed (simulated) total drug and total target concentrations, but also unobserved concentrations of the free drug, free target, and drug-target complexes. For mAb with a membrane-bound target, the 4-to-one MM approximation adequately described the data. The 4-to-one QSS approximation converged 8 times faster than the full 4-to-one TMDD.

Quasi-Equilibrium States and Phase Transitions in Biological Evolution.

We developed a macroscopic description of the evolutionary dynamics by following the temporal dynamics of the total Shannon entropy of sequences, denoted by S, and the average Hamming distance between them, denoted by H. We argue that a biological system can persist in the so-called quasi-equilibrium state for an extended period, characterized by strong correlations between S and H, before undergoing a phase transition to another quasi-equilibrium state. To demonstrate the results, we conducted a statistical analysis of SARS-CoV-2 data from the United Kingdom during the period between March 2020 and December 2023. From a purely theoretical perspective, this allowed us to systematically study various types of phase transitions described by a discontinuous change in the thermodynamic parameters. From a more-practical point of view, the analysis can be used, for example, as an early warning system for pandemics.

Non-Normalizable Quasi-Equilibrium Solution of the Fokker-Planck Equation for Nonconfining Fields.

We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep as compared to the temperature, physical observables, like the mean square displacement, are essentially time-independent over a long time interval, the stagnation epoch. However, the standard Boltzmann-Gibbs (BG) distribution is non-normalizable, given that the usual partition function is divergent. For this regime, we have previously shown that a regularization of BG statistics allows for the prediction of the values of dynamical and thermodynamical observables in the non-normalizable quasi-equilibrium state. In this work, based on the eigenfunction expansion of the time-dependent solution of the associated Fokker-Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, being valid for times that are long, but still short as compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is an error function, which is shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker-Planck equation in hand, we show the validity of the regularized BG statistics and how it perfectly describes the time-independent regime though the quasi-stationary state is non-normalizable.

Local fluctuations of genetic processes defined on two time scales, with applications to effective size estimation.

In this paper we develop a general framework for how the genetic composition of a structured population with strong migration between its subunits, evolves over time. The dynamics is described in terms of a vector-valued Markov process of allele, genotype or haplotype frequencies that varies on two time scales. The more rapid changes are random fluctuations in terms of a multivariate autoregressive process, around a quasi equilibrium fix point, whereas the fix point itself varies more slowly according to a diffusion process, along a lower-dimensional subspace. Under mild regularity conditions, the fluctuations have a magnitude inversely proportional to the square root of the population size N, and hence can be used to estimate a broad class of genetically effective population sizes Ne, with genetic data from one time point only. In this way we are able to unify a number of existing notions of effective size, as well as proposing new ones, for instance one that quantifies the extent to which genotype frequencies fluctuate around Hardy-Weinberg equilibrium.

Mathematical description of drug-target interactions: application to biologics that bind to targets with two binding sites.

The emerging discipline of mathematical pharmacology occupies the space between advanced pharmacometrics and systems biology. A characteristic feature of the approach is application of advance mathematical methods to study the behavior of biological systems as described by mathematical (most often differential) equations. One of the early application of mathematical pharmacology (that was not called this name at the time) was formulation and investigation of the target-mediated drug disposition (TMDD) model and its approximations. The model was shown to be remarkably successful, not only in describing the observed data for drug-target interactions, but also in advancing the qualitative and quantitative understanding of those interactions and their role in pharmacokinetic and pharmacodynamic properties of biologics. The TMDD model in its original formulation describes the interaction of the drug that has one binding site with the target that also has only one binding site. Following the framework developed earlier for drugs with one-to-one binding, this work aims to describe a rigorous approach for working with similar systems and to apply it to drugs that bind to targets with two binding sites. The quasi-steady-state, quasi-equilibrium, irreversible binding, and Michaelis-Menten approximations of the model are also derived. These equations can be used, in particular, to predict concentrations of the partially bound target (RC). This could be clinically important if RC remains active and has slow internalization rate. In this case, introduction of the drug aimed to suppress target activity may lead to the opposite effect due to RC accumulation.

Apparent activation energies of protein-protein complex dissociation in the gas-phase determined by electrospray mass spectrometry.

We have developed a method to determine apparent activation energies of dissociation for ionized protein-protein complexes in the gas phase using electrospray ionization mass spectrometry following the Rice-Ramsperger-Kassel-Marcus quasi-equilibrium theory. Protein-protein complexes were formed in solution, transferred into the gas phase, and separated from excess free protein by ion mobility filtering. Afterwards, complex disassembly was initiated by collision-induced dissociation with step-wise increasing energies. Relative intensities of ion signals were used to calculate apparent activation energies of dissociation in the gas phase by applying linear free energy relations. The method was developed using streptavidin tetramers. Experimentally determined apparent gas-phase activation energies for dissociation ([Formula: see text]) of complexes consisting of Fc parts from immunoglobulins (IgG-Fc) and three closely related protein G' variants (IgG-Fc•protein G'e, IgG-Fc•protein G'f, and IgG-Fc•protein G'g) show the same order of stabilities as can be inferred from their in-solution binding constants. Differences in stabilities between the protein-protein complexes correspond to single amino acid residue exchanges in the IgG-binding regions of the protein G' variants. Graphical abstract Electrospray mass spectrometry and collision-induced dissociation delivers apparent activation energies and supramolecular bond force constants of protein-protein complexes in the gas phase.

Target-mediated drug disposition model for drugs with two binding sites that bind to a target with one binding site.

The paper extended the TMDD model to drugs with two identical binding sites (2-1 TMDD). The quasi-steady-state (2-1 QSS), quasi-equilibrium (2-1 QE), irreversible binding (2-1 IB), and Michaelis-Menten (2-1 MM) approximations of the model were derived. Using simulations, the 2-1 QSS approximation was compared with the full 2-1 TMDD model. As expected and similarly to the standard TMDD for monoclonal antibodies (mAb), 2-1 QSS predictions were nearly identical to 2-1 TMDD predictions, except for times of fast changes following initiation of dosing, when equilibrium has not yet been reached. To illustrate properties of new equations and approximations, several variations of population PK data for mAbs with soluble (slow elimination of the complex) or membrane-bound (fast elimination of the complex) targets were simulated from a full 2-1 TMDD model and fitted to 2-1 TMDD models, to its approximations, and to the standard (1-1) QSS model. For a mAb with a soluble target, it was demonstrated that the 2-1 QSS model provided nearly identical description of the observed (simulated) free drug and total target concentrations, although there was some minor bias in predictions of unobserved free target concentrations. The standard QSS approximation also provided a good description of the observed data, but was not able to distinguish between free drug concentrations (with no target attached and both binding site free) and partially bound drug concentrations (with one of the binding sites occupied by the target). For a mAb with a membrane-bound target, the 2-1 MM approximation adequately described the data. The 2-1 QSS approximation converged 10 times faster than the full 2-1 TMDD, and its run time was comparable with the standard QSS model.

Robustness and exploration between the interplay of the nonlinear co-dynamics HIV/AIDS and pneumonia model via fractional differential operators and a probabilistic approach.

In this article, we considered a nonlinear compartmental mathematical model that assesses the effect of treatment on the dynamics of HIV/AIDS and pneumonia (H/A-P) co-infection in a human population at different infection stages. Understanding the complexities of co-dynamics is now critically necessary as a consequence. The aim of this research is to construct a co-infection model of H/A-P in the context of fractional calculus operators, white noise and probability density functions, employing a rigorous biological investigation. By exhibiting that the system possesses non-negative and bounded global outcomes, it is shown that the approach is both mathematically and biologically practicable. The required conditions are derived, guaranteeing the eradication of the infection. Furthermore, adequate prerequisites are established, and the configuration is tested for the existence of an ergodic stationary distribution. For discovering the system's long-term behavior, a deterministic-probabilistic technique for modeling is designed and operated in MATLAB. By employing an extensive review, we hope that the previously mentioned approach improves and leads to mitigating the two diseases and their co-infections by examining a variety of behavioral trends, such as transitions to unpredictable procedures. In addition, the piecewise differential strategies are being outlined as having promising potential for scholars in a range of contexts because they empower them to include particular characteristics across multiple time frame phases. Such formulas can be strengthened via classical techniques, power law, exponential decay, generalized Mittag-Leffler kernels, probability density functions and random procedures. Furthermore, we get an accurate description of the probability density function encircling a quasi-equilibrium point if the effect of H/A-P minimizes the propagation of the co-dynamics. Consequently, scholars can obtain better outcomes when analyzing facts using random perturbations by implementing these strategies for challenging issues. Random perturbations in H/A-P co-infection are crucial in controlling the spread of an epidemic whenever the suggested circulation is steady and the amount of infection eliminated is closely correlated with the random perturbation level.

Theoretical and mathematical codynamics of nonlinear tuberculosis and COVID-19 model pertaining to fractional calculus and probabilistic approach.

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a novel virus known as coronavirus 2 (SARS-CoV-2) that affects the pulmonary structure and results in the coronavirus illness 2019 (COVID-19). Tuberculosis (TB) and COVID-19 codynamics have been documented in numerous nations. Understanding the complexities of codynamics is now critically necessary as a consequence. The aim of this research is to construct a co-infection model of TB and COVID-19 in the context of fractional calculus operators, white noise and probability density functions, employing a rigorous biological investigation. By exhibiting that the system possesses non-negative and bounded global outcomes, it is shown that the approach is both mathematically and biologically practicable. The required conditions are derived, guaranteeing the eradication of the infection. Sensitivity analysis and bifurcation of the submodel are also investigated with system parameters. Furthermore, existence and uniqueness results are established, and the configuration is tested for the existence of an ergodic stationary distribution. For discovering the system's long-term behavior, a deterministic-probabilistic technique for modeling is designed and operated in MATLAB. By employing an extensive review, we hope that the previously mentioned approach improves and leads to mitigating the two diseases and their co-infections by examining a variety of behavioral trends, such as transitions to unpredictable procedures. In addition, the piecewise differential strategies are being outlined as having promising potential for scholars in a range of contexts because they empower them to include particular characteristics across multiple time frame phases. Such formulas can be strengthened via classical technique, power-law, exponential decay, generalized Mittag-Leffler kernels, probability density functions and random procedures. Furthermore, we get an accurate description of the probability density function encircling a quasi-equilibrium point if the effect of TB and COVID-19 minimizes the propagation of the codynamics. Consequently, scholars can obtain better outcomes when analyzing facts using random perturbations by implementing these strategies for challenging issues. Random perturbations in TB and COVID-19 co-infection are crucial in controlling the spread of an epidemic whenever the suggested circulation is steady and the amount of infection eliminated is closely correlated with the random perturbation level.

Quasi-equilibrium state based quantification of biological macromolecules in single-molecule localization microscopy.

The stoichiometry of molecular components within supramolecular biological complexes is often an important property to understand their biological functioning, particularly within their native environment. While there are well established methods to determine stoichiometryin vitro, it is presently challenging to precisely quantify this propertyin vivo,especially with single molecule resolution that is needed for the characterization stoichiometry heterogeneity. Previous work has shown that optical microscopy can provide some information to this end, but it can be challenging to obtain highly precise measurements at higher densities of fluorophores. Here we provide a simple approach using already established procedures in single-molecule localization microscopy (SMLM) to enable precise quantification of stoichiometry within individual complexes regardless of the density of fluorophores. We show that by focusing on the number of fluorophore detections accumulated during the quasi equilibrium-state of this process, this method yields a 50-fold improvement in precision over values obtained from images with higher densities of active fluorophores. Further, we show that our method yields more correct estimates of stoichiometry with nuclear pore complexes and is easily adaptable to quantify the DNA content with nanodomains of chromatin within individual chromosomes inside cells. Thus, we envision that this straightforward method may become a common approach by which SMLM can be routinely employed for the accurate quantification of subunit stoichiometry within individual complexes within cells.

Demystification of entangled Mass Action Law.

Recently, Gorban (2021) analysed some kinetic paradoxes of the transition state theory and proposed its revision that gave the "entangled mass action law", in which new reactions were generated as an addition to the reaction mechanism under consideration. These paradoxes arose due to the assumption of quasi-equilibrium between reactants and transition states. In this paper, we provided a brief introduction to this theory, demonstrating how the entangled mass action law equations can be derived in the framework of the standard quasi steady state approximation in combination with the quasi-equilibrium generalized mass action law for an auxiliary reaction network including reactants and intermediates. We also proved the basic physical property (positivity) for these new equations, which was not obvious in the original approach.

Kinetics of the enzyme titration process by reversible modifiers.

The effect of reversible modifiers on the initial rate of enzyme catalysed reactions has been investigated in a quasi-equilibrium approximation using the general modifier mechanism of Botts and Morales. It has been shown that, when investigating the dependence of the initial rate on the modifier concentration at a fixed substrate concentration, the kinetics of enzyme titration by reversible modifiers can generally be described using two kinetic constants. Just as the dependence of the initial rate on the substrate concentration (at a fixed modifier concentration) is described using two kinetic constants: the Michaelis constant Km and the limiting rate Vm. Only one constant M50 is needed to describe the kinetics of linear inhibition, and in the case of nonlinear inhibition and activation, along with M50 the constant QM is also needed. Knowing the values of the constants M50 and QM, it is possible to unambiguously determine the modification efficiency, that is, to calculate how many times the initial rate of the enzyme catalysed reaction will change when a certain modifier concentration is added to the incubation medium. The properties of these fundamental constants have been analysed in detail and the dependence of these constants on other parameters of the Botts-Morales model have been shown. Equations describing the dependence of relative reaction rates on the modifier concentration using these kinetic constants are presented. Various ways of linearising these equations for calculating the kinetic constants M50 and QM from experimental data are also presented.

An Extended Model Including Target Turnover, Ligand-Target Complex Kinetics, and Binding Properties to Describe Drug-Receptor Interactions.

Since the beginning of this century, target-mediated drug disposition has become a central concept in modeling drug action in drug development. It combines a range of processes, such as turnover, protein binding, internalization, and non-specific elimination, and often serves as a nucleus of more complex pharmacokinetic models. It is simple enough to comprehend but complex enough to be able to describe a wide range of phenomena and data sets. However, the complexity comes at a price: many parameters. In this chapter, we present an overview of the temporal development of the compounds involved after different types of drug doses and offer convenient handles for dissecting data sets in a sophisticated manner in order to estimate the values of these parameters, such as rate constants and pertinent concentrations.

Henri-Michaelis-Menten kinetics of reversible enzymic reactions, and the determination of rate constants from kinetic constants.

The Michaelis constants derived for two reversible uni-reactant - uni-product reaction models, given originally by Haldane, are corrected. In the direction starting with the reactant having the lower binding constant, the steady state is one in which the enzyme-reactant intermediate has a concentration approximating the final equilibrium concentration., Consequently, the Haldane relationship is generally invalid, and kinetic analyses to validate the use of the kinetic constant ratio (kcat/Km) as a measure of specificity are also generally invalid. This correction of Michaelis constants is pertinent to attempts to back calculate rate constants from experimental values: the Michaelis constant used must be correct.

Ten-year trends in vertical distribution of radiocesium in Fukushima forest soils, Japan.

To elucidate interannual changes in the vertical distribution of 137Cs in forest ecosystems contaminated by the Fukushima Dai-ichi Nuclear Power Plant accident, we investigated 137Cs inventories in forest soils (both organic and mineral soil horizons) at 10 sampling plots with different 137Cs deposition levels and dominant species for up to 10 years after the accident. We examined the temporal variation of the 137Cs inventories by depth with exponential regression models (assuming that the transition and partitioning of 137Cs are still active) and exponential offset regression models (assuming a shift to a stable 137Cs distribution, defined as the "quasi-equilibrium steady-state" in the Chernobyl accident). In the organic horizon, the 137Cs inventories were exponentially decreasing, and it might take more time to converge in the quasi-equilibrium steady-state at most plots. In the mineral soil horizon, most of 137Cs was found in the surface layer of the mineral soil horizon (0-5 cm). In this layer, the inventories first increased and then become relatively constant, and the exponential offset model was selected at most plots, suggesting entry into the quasi-equilibrium steady-state over the observation period. Although we also observed exponentially increasing trends in a lower layer (5-10 cm) of the mineral soil horizon, there was no clear increasing or decreasing trend of 137Cs inventory in the deeper mineral soil layers (10-15 and 15-20 cm). Our calculation of the relaxation depth and migration center revealed that downward migration of 137Cs is not significant in terms of the overall 137Cs distribution in the mineral soil horizon over 10 years.

Investigation of Lithium Ion Diffusion of Graphite Anode by the Galvanostatic Intermittent Titration Technique.

Graphite is used as a state-of-the-art anode in commercial lithium-ion batteries (LIBs) due to its highly reversible lithium-ion storage capability and low electrode potential. However, graphite anodes exhibit sluggish diffusion kinetics for lithium-ion intercalation/deintercalation, thus limiting the rate capability of commercial LIBs. In order to determine the lithium-ion diffusion coefficient of commercial graphite anodes, we employed a galvanostatic intermittent titration technique (GITT) to quantify the quasi-equilibrium open circuit potential and diffusion coefficient as a function of lithium-ion concentration and potential for a commercial graphite electrode. Three plateaus are observed in the quasi-equilibrium open circuit potential curves, which are indicative of a mixed phase upon lithium-ion intercalation/deintercalation. The obtained diffusion coefficients tend to increase with increasing lithium concentration and exhibit an insignificant difference between charge and discharge conditions. This study reveals that the diffusion coefficient of graphite obtained with the GITT (1 × 10-11 cm2/s to 4 × 10-10 cm2/s) is in reasonable agreement with literature values obtained from electrochemical impedance spectroscopy. The GITT is comparatively simple and direct and therefore enables systematic measurements of ion intercalation/deintercalation diffusion coefficients for secondary ion battery materials.

An LC-MS/MS approach to assess total and free protein target in the serum of cynomolgus monkey.

AIM: Develop LC-MS/MS-based assays to measure total and free complement C5 in cynomolgus monkey serum as a target engagement biomarker for pharmacokinetic/pharmacodynamic correlation study. Materials & methods/results: The C5-specific signature peptide derived from pellet digestion of serum proteins with and without prior immunodepletion of the drug-bound C5 by protein A beads was quantified to assess free and total C5 levels, respectively. Conditions for immunodepletion by protein A were optimized to ensure complete depletion of IgGs (and drug-bound C5). The effect of sample dilution on drug-target dissociation and thus free C5 measurement was evaluated by applying a mathematical simulation. CONCLUSION: The procedure described here allows for the assessment of protein target engagement, aiding in pharmacokinetic/pharmacodynamic correlation analysis and human dose projection.

An alternative extragradient projection method for quasi-equilibrium problems.

For the quasi-equilibrium problem where the players' costs and their strategies both depend on the rival's decisions, an alternative extragradient projection method for solving it is designed. Different from the classical extragradient projection method whose generated sequence has the contraction property with respect to the solution set, the newly designed method possesses an expansion property with respect to a given initial point. The global convergence of the method is established under the assumptions of pseudomonotonicity of the equilibrium function and of continuity of the underlying multi-valued mapping. Furthermore, we show that the generated sequence converges to the nearest point in the solution set to the initial point. Numerical experiments show the efficiency of the method.