Exploration of the intercellular heterogeneity of topotecan uptake into human breast cancer cells through compartmental modelling

A mathematical multi-cell model for the in vitro kinetics of the anti-cancer agent topotecan (TPT) following administration into a culture medium containing a population of human breast cancer cells (MCF-7 cell line) is described. This non-linear compartmental model is an extension of an earlier single-cell type model and has been validated using experimental data obtained using two-photon laser scanning microscopy (TPLSM). A structural identifiability analysis is performed prior to parameter estimation to test whether the unknown parameters within the model are uniquely determined by the model outputs. The full model has 43 compartments, with 107 unknown parameters, and it was found that the structural identifiability result could not be established even when using the latest version of the symbolic computation software Mathematica. However, by assuming that a priori knowledge is available for certain parameters, it was possible to reduce the number of parameters to 81, and it was found that this (Stage Two) model was globally (uniquely) structurally identifiable. The identifiability analysis demonstrated how valuable symbolic computation is in this context, as the analysis is far too lengthy and difficult to be performed by hand.     

Sensitive high-performance liquid chromatographic fluorescence assay for the quantitation of topotecan (SKF 104864-A) and its lactone ring-opened product (hydroxy acid) in human plasma and urine

A sensitive reversed-phase high-performance liquid chromatographic fluorescence method is described for the simultaneous determination of topotecan (I) and the hydrolysed lactone ring-opened product hydroxy acid (II) in plasma and for the determination of I in urine. To 250 μl of plasma, a 750-μl volume of cold methanol was added to stabilize the pH-dependent conversion of I into II. In plasma, the lower limit of quantitation (LLQ) for both compounds was 0.10 ng/ml. The between-day variation for I at the LLQ was 7.1% and for II was 5.5%. Prior to injection, urine samples were acidified with orthophosphoric acid and diluted with phosphate-buffered saline (PBS). In urine, the calibration curve for I was linear in the range of 10 to 250 ng/ml and the LLQ was 10 ng/ml. The assay was developed to enable pharmacological analysis of I, in on-going phase I and II studies, in patients with solid tumors.     

A mathematical model for dorsal closure

During embryogenesis, drosophila embryos undergo epithelial folding and unfolding, which leads to a hole in the dorsal epidermis, transiently covered by an extraembryonic tissue called the amnioserosa. Dorsal closure (DC) consists of the migration of lateral epidermis towards the midline, covering the amnioserosa. It has been extensively studied since numerous physical mechanisms and signaling pathways present in DC are conserved in other morphogenetic events and wound healing in many other species (including vertebrates).We present here a simple mathematical model for DC that involves a reduced number of parameters directly linked to the intensity of the forces in the presence and which is applicable to a wide range of geometries of the leading edge (LE). This model is a natural generalization of the very interesting model proposed in Hutson et al. (2003). Being based on an ordinary differential equation (ODE) approach, the previous model had the advantage of being even simpler, but this restricted significantly the variety of geometries that could be considered and thus the number of modified dorsal closures that could be studied.A partial differential equation (PDE) approach, as the one developed here, allows considering much more general situations that show up in genetically or physically perturbed embryos and whose study will be essential for a proper understanding of the different components of the DC process. Even for native embryos, our model has the advantage of being applicable since an early stages of DC when there is no antero-posterior symmetry (approximately verified only in the late phases of DC).We validate our model in a native setting and also test it further in embryos where the zipping force is perturbed through the expression of spastin (a microtubule severing protein). We obtain variations of the force coefficients that are consistent with what was previously described for this setting.     

A mathematical model of in vitro cancer cell growth and treatment with the antimitotic agent curacin A

A mathematical model of cancer cell growth and response to treatment with the experimental antimitotic agent curacin A is presented. Rate parameters for the untreated growth of MCF-7/LY2 breast cancer and A2780 ovarian cell lines are determined from in vitro growth studies. Subsequent growth studies following treatments with 2.5, 25 and 50 nanomolar (nM), concentrations of curacin A are used to determine effects on the cell cycle and cell viability. The model's system of ordinary differential equations yields an approximate analytical solution which predicts the minimum concentration necessary to prevent growth. The model shows that cell growth is arrested when the apoptotic rate is greater than the mitotic rate and that the S-phase transition rate acts to amplify this effect. Analysis of the data suggests that curacin A is rapidly absorbed into both cell lines causing an increase in the S-phase transition and a decrease in the M-phase transition. The model also indicates that the rate of apoptosis remains virtually constant for MCF-7/LY2 while that of A2780 increases 38% at 2.5 nM and 59% at 50 nM as compared to the untreated apoptotic rate.     

A mathematical model for the batch reactor kinetics of algae growth

A mathematical model based on the Einstein law of photochemical equivalence is proposed to describe the batch growth of unicellular algae. The model was applied in an integrated form to cell concentration versus growth time data taken over an extended range of cell concentrations which include both the regions of “exponential” and “linear” growth. It is shown that a certain function of cell concentration contained in the integrated form of the model is linearly dependent on the growth time over both the “exponential” and “linear” growth regions.     

Anticancer activities of genistein‐topotecan combination in prostate cancer cells

Prostate cancer is one of the leading causes of death in men aged 40 to 55. Genistein isoflavone (4′, 5′, 7‐trihydroxyisoflavone) is a dietary phytochemical with demonstrated anti‐tumour activities in a variety of cancers. Topotecan Hydrochloride (Hycamtin) is an FDA‐approved chemotherapy drug, primarily used for secondary treatment of ovarian, cervical and small cell lung cancers. This study was to demonstrate the potential anticancer efficacy of genistein‐topotecan combination in LNCaP prostate cancer cells and the mechanism of the combination treatment. The LNCaP cells were grown in complete RPMI medium, and cultured at 37°C, 5% CO₂ for 24–48 hrs to achieve 70–90% confluency. The cells were treated with varying concentrations of genistein, topotecan and genistein‐topotecan combination and incubated for 24 hrs. The treated cells were assayed for (i) post‐treatment sensitivity using MTT assay and DNA fragmentation, (ii) treatment‐induced apoptosis using caspase‐3 and ‐9 binding assays and (iii) treatment‐induced ROS generation levels. The overall data indicated that (i) both genistein and topotecan induce cellular death in LNCaP cells, (ii) genistein‐topotecan combination was significantly more efficacious in reducing LNCaP cell viability compared with either genistein or topotecan alone, (iii) in all cases, cell death was primarily through apoptosis, via the activation of caspase‐3 and ‐9, which are involved in the intrinsic pathway, (iv) ROS generation levels increased significantly with the genistein‐topotecan combination treatment. Treatments involving genistein‐topotecan combination may prove to be an attractive alternative phytotherapy or adjuvant therapy for prostate cancer.     

Invertibility of the TKF model of sequence evolution

We consider character sequences evolving on a phylogenetic tree under the TKF91 model. We show that as the sequence lengths tend to infinity the topology of the phylogenetic tree and the edge lengths are determined by any one of (a) the alignment of sequences (b) the collection of sequence lengths. We also show that the probability of any homology structure on a collection of sequences related by a TKF91 process on a tree is independent of the root location.     

A discrete mathematical model of unlabelled granulocyte kinetics

The equations used in formulating the continuous model of granulocyte kinetics developed by O'Fallon et al. (1971) were analyzed to see if they could be altered to simulate a feedback mechanism operating on the production and development of granulocytes. After extensive study and modification of the continuous model, it was found that a discrete model based on a Leslie matrix procedure was more effective for simulating the feedback system. This discrete model was used to show experimentally, from a mathematical view point, that a feedback mechanism of some kind must be operating on the production and development of granulocytes. Further, the discrete model was subjected to preliminary tests (simultaneous and cascading feedback) to demonstrate that it has the capability of responding to feedback control.This work was completed while the first author was at North Carolina State University at Raleigh, Department of Statistics, Biomathematics Division, part of it under grant number 5T1 GM 678-15, National Institutes of Health, and part of it under grant number 5 F32 CA05964-02 from the National Cancer Institute     

A mathematical model for insulin kinetics. III. Sensitivity analysis of the model

A non-linear mathematical model involving four variables and several constants incorporating beta-cell kinetics, a glucose-insulin feedback system and a gastrointestinal absorption term had been applied in earlier papers to various forms of diabetes mellitus. In this paper, we examine the response of the system to variations in the parameters and to initial conditions using sensitivity analysis. It is found that such a method leads to results that are consistent with clinical findings. Further, it is suggested that such an analysis could help in making some predictions regarding future directions in the therapy of diabetes mellitus.     

A stochastic formulation of the gompertzian growth model for in vitro bactericidal kinetics: parameter estimation and extinction probability

Time-kill curves have frequently been employed to study the antimicrobial effects of antibiotics. The relevance of pharmacodynamic modeling to these investigations has been emphasized in many studies of bactericidal kinetics. Stochastic models are needed that take into account the randomness of the mechanisms of both bacterial growth and bacteria-drug interactions. However, most of the models currently used to describe antibiotic activity against microorganisms are deterministic. In this paper we examine a stochastic differential equation representing a stochastic version of a pharmacodynamic model of bacterial growth undergoing random fluctuations, and derive its solution, mean value and covariance structure. An explicit likelihood function is obtained both when the process is observed continuously over a period of time and when data is sampled at time points, as is the custom in these experimental conditions. Some asymptotic properties of the maximum likelihood estimators for the model parameters are discussed. The model is applied to analyze in vitro time-kill data and to estimate model parameters; the probability of the bacterial population size dropping below some critical threshold is also evaluated. Finally, the relationship between bacterial extinction probability and the pharmacodynamic parameters estimated is discussed.     

A mathematical model for germinal centre kinetics and affinity maturation

We present a mathematical model which reproduces experimental data on the germinal centre (GC) kinetics of the primed primary immune response and on affinity maturation observed during the reaction. We show that antigen masking by antibodies which are produced by emerging plasma cells can drive affinity maturation and provide a feedback mechanism by which the reaction is stable against variations in the initial antigen amount over several orders of magnitude. This provides a possible answer to the long-standing question of the role of antigen reduction in driving affinity maturation. By comparing model predictions with experimental results, we propose that the selection probability of centrocytes and the recycling probability of selected centrocytes are not constant but vary during the GC reaction with respect to time. It is shown that the efficiency of affinity maturation is highest if clones with an affinity for the antigen well above the average affinity in the GC leave the GC for either the memory or plasma cell pool. It is further shown that termination of somatic hypermutation several days before the end of the germinal centre reaction is beneficial for affinity maturation. The impact on affinity maturation of simultaneous initiation of memory cell formation and somatic hypermutation vs. delayed initiation of memory cell formation is discussed.     

A mathematical model of kinetics of the biosynthesis of tetracyclines

Summary A mathematical model simulating the behaviour or Streptomyces aureofaciens in batch culture under conditions when tetracyclines are synthesized in excessive amounts has been formulated. The response of the mathematical model to the experimental conditions applied corresponds with data obtained in the experiments. The mathematical model demonstrated that the level of tetracycline production is determined during the period of culture growth beginning with exhaustion of inorganic phosphate from the medium and ending with inhibition of the synthesis of enzymes caused by the synthesized tetracyclines. Further tetracycline synthesis is then proportional to the amount of enzymes synthesized in this interval.List of symbols E Activity of ACT-oxygenase (10×nkat/g) - P Product concentration (mg/l) - k ₁-k ₆ Rate constants - K _S Saturation constant (g sugar/l) - K _I1 Inhibition constant (mg product/l) - K _I2 Inhibition constant (mM phosphate/l) - K _I3 Inhibition constant (mg product/l) - S ₁ Substrate sucrose (g sugar/l) - S ₂ Substrate concentration — phosphate (mM/l) - r _P Specific rate of product formation (mg product/g · h) - r _E Specific rate of enzyme synthesis (10×nkat/g² · h), Expressed by activity units - t Cultivation time (hour) - X Biomass dry weight (g/l) - Y _S/X Yield coefficient - Specific growth rate (h^-1)     

A mechanistic mathematical model for the catalytic action of glutathione peroxidase

Abstract

Glutathione peroxidase (GPx) is a well-known seleno-enzyme that protects cells from oxidative stress (e.g., lipid peroxidation and oxidation of other cellular proteins and macromolecules), by catalyzing the reduction of harmful peroxides (e.g., hydrogen peroxide: H₂O₂) with reduced glutathione (GSH). However, the catalytic mechanism of GPx kinetics is not well characterized in terms of a mathematical model. We developed here a mechanistic mathematical model of GPx kinetics by considering a unified catalytic scheme and estimated the unknown model parameters based on different experimental data from the literature on the kinetics of the enzyme. The model predictions are consistent with the consensus that GPx operates via a ping-pong mechanism. The unified catalytic scheme proposed here for GPx kinetics clarifies various anomalies, such as what are the individual steps in the catalytic scheme by estimating their associated rate constant values and a plausible rationale for the contradicting experimental results. The developed model presents a unique opportunity to understand the effects of pH and product GSSG on the GPx activity under both physiological and pathophysiological conditions. Although model parameters related to the product GSSG were not identifiable due to lack of product-inhibition data, the preliminary model simulations with the assumed range of parameters show that the inhibition by the product GSSG is negligible, consistent with what is known in the literature. In addition, the model is able to simulate the bi-modal behavior of the GPx activity with respect to pH with the pH-range for maximal GPx activity decreasing significantly as the GSH levels decrease and H₂O₂ levels increase (characteristics of oxidative stress). The model provides a key component for an integrated model of H₂O₂ balance under normal and oxidative stress conditions.     

The solvolysis of topotecan in alcohols and acetic anhydride

Six derivatives of 10-hydroxycamptothecin were prepared via solvolysis of topotecan in corresponding alcohols and acetic anhydride. We attributed the specific reactivity of topotecan to the internal hydrogen-bonding between 10-hydroxy and the nitrogen atom in position 9. As a result the reaction underwent through an intermediate ortho-quionomethlide species to reach equilibrium. Bioactivity screening data showed all products could potentially inhibit the proliferation of several cancer cell lines in vitro and a bigger size group in 9-position would be favorable for the anti-tumor activities observably.     

A mathematical model for the bacterial oxidation of a sulfide ore concentrate

The effect of dilution rate and feed solids concentration on the bacterial leaching of a pyrite/arsenopyrite ore concentrate was studied. A mathematical model was developed for the process based on the steady-state data collected over the range of dilution rates (20 to 110 h) and feed solids concentrations (6 to 18% w/v) studied. A modified Monod model with inhibition by arsenic was used to model bacterial ferrous ion oxidation rates. The model assumes that (i) pyrite and arsenopyrite leaching occurs solely by the action of ferric iron produced from the bacterial oxidation of ferrous iron and (ii) bacterial growth rates are proportional to ferrous ion oxidation rate. The equilibrium among the various ionic species present in the leach solution that are likely to have a significant effect on the bioleach process were included in the model. (c) 1994 John Wiley & Sons, Inc.     

A synergistic kinetics model for enzymatic cellulose hydrolysis compared to degree-of-synergism experimental results

It is demonstrated that a two-enzyme component synergistic model can account for the observation that the degree of synergism goes through a maximum as the total enzyme concentration is increased. The degree of synergism is low at low enzyme concentration because the extent of conversion is low and therefore the cellulose chain ends, present originally, are not exhausted; thus the action of the cellobiohydrolase (CBH) is not dependent on the chain ends generated by the endoglucanase (EG). The degree of synergism declines at high enzyme concentration due to saturation of adsorption sites with CBH, thus decreasing the generation of chain ends by EG. (c) 1993 John Wiley & Sons, Inc.     

A mathematical model of doxorubicin penetration through multicellular layers

Inadequate drug delivery to tumours is now recognised as a key factor that limits the efficacy of anticancer drugs. Extravasation and penetration of therapeutic agents through avascular tissue are critically important processes if sufficient drug is to be delivered to be therapeutic. The purpose of this study is to develop an in silico model that will simulate the transport of the clinically used cytotoxic drug doxorubicin across multicell layers (MCLs) in vitro. Three cell lines were employed: DLD1 (human colon carcinoma), MCF7 (human breast carcinoma) and NCI/ADR-Res (doxorubicin resistant and P-glycoprotein [Pgp] overexpressing ovarian cell line). Cells were cultured on transwell culture inserts to various thicknesses and doxorubicin at various concentrations (100 or 50 μM) was added to the top chamber. The concentration of drug appearing in the bottom chamber was determined as a function of time by HPLC-MS/MS. The rate of drug penetration was inversely proportional to the thickness of the MCL. The rate and extent of doxorubicin penetration was no different in the presence of NCI/ADR-Res cells expressing Pgp compared to MCF7 cells. A mathematical model based upon the premise that the transport of doxorubicin across cell membrane bilayers occurs by a passive “flip-flop” mechanism of the drug between two membrane leaflets was constructed. The mathematical model treats the transwell apparatus as a series of compartments and the MCL is treated as a series of cell layers, separated by small intercellular spaces. This model demonstrates good agreement between predicted and actual drug penetration in vitro and may be applied to the prediction of drug transport in vivo, potentially becoming a useful tool in the study of optimal chemotherapy regimes.     

A multisegmental cross-bridge kinetics model of the myofibril

Striated muscle is a mechanical system that develops force and generates power in serving vital activities in the body. Striated muscle is a complex biological system; a single mammalian muscle fibre contains up to hundred or even more myofibrils in parallel connected via an inter-myofibril filament network. In one single myofibril thousands of sarcomeres are lined up as a series of linear motors. We recently demonstrated that half-sarcomeres (hS) in a single myofibril operate non-uniformly. We outline a mathematical framework based on cross-bridge kinetics for the simulation of the force response and length change of individual hS in a myofibril. The model describes the muscle myofibril in contraction experiments under various conditions. The myofibril is modeled as a multisegmental mechanical system of hS models, which have active and viscoelastic properties. In the first approach, a two-state cross-bridge formalism relates the hS force to the chemical kinetics of ATP hydrolysis, as first described by Huxley [1957. Muscle structure and theories of contraction. Prog. Biophys. Mol. Biol. 7, 255-318]. Two possible types of biological variability are introduced and modeled. Numerical simulations of a myofibril composed of four to eight hS show a non-uniform hS length distribution and complex internal dynamics upon activation. We demonstrate that the steady-state approximation holds only in restricted time zones during activation. Simulations of myofibril contraction experiments that reproduce the classic steady-state force-length and force-velocity relationships, strictly constrained or “clamped” in either end-held isometric or isotonic contraction conditions, reveal a small but conspicuous effect of hS dynamics on force.     

A model for the effects of primary substrates on the kinetics of reductive dehalogenation

A kinetic model that describes substrate interactions during reductive dehalogenation reactions is developed. This model describes how the concentrations of primary electron-donor and -acceptor substrates affect the rates of reductive dehalogenation reactions. A basic model, which considers only exogenous electron-donor and -acceptor substrates, illustrates the fundamental interactions that affect reductive dehalogenation reaction kinetics. Because this basic model cannot accurately describe important phenomena, such as reductive dehalogenation that occurs in the absence of exogenous electron donors, it is expanded to include an endogenous electron donor and additional electron acceptor reactions. This general model more accurately reflects the behavior that has been observed for reductive dehalogenation reactions. Under most conditions, primary electron-donor substrates stimulate the reductive dehalogenation rate, while primary electron acceptors reduce the reaction rate. The effects of primary substrates are incorporated into the kinetic parameters for a Monod-like rate expression. The apparent maximum rate of reductive dehalogenation (q _{m, ap} ) and the apparent half-saturation concentration (K _ap ) increase as the electron donor concentration increases. The electron-acceptor concentration does not affect q _{m, ap} , but K _ap is directly proportional to its concentration.Definitions for model parameters RX halogenated aliphatic substrate - E-M ⁿ reduced dehalogenase - E-M ⁿ⁺² oxidized dehalogenase - [E-M ⁿ ] steady-state concentration of the reduced dehalogenase (moles of reduced dehalogenase per unit volume) - [E-M ⁿ⁺²] steady-state concentration of the oxidized dehalogenase (moles of reduced dehalogenase per unit volume) - DH₂ primary exogenous electron-donor substrate - A primary exogenous electron-acceptor substrate - A2 second primary exogenous electron-acceptor substrate - X biomass concentration (biomass per unit volume) - f fraction of biomass that is comprised of the dehalogenase (moles of dehalogenase per unit biomass) - stoichiometric coefficient for the reductive dehalogenation reaction (moles of dehalogenase oxidized per mole of halogenated substrate reduced) - stoichiometric coefficient for oxidation of the primary electron donor (moles of dehalogenase reduced per mole of donor oxidized) - stoichiometric coefficient for oxidation of the endogenous electron donor (moles of dehalogenase reduced per unit biomass oxidized) - stoichiometric coefficient for reduction of the primary electron acceptor (moles of dehalogenase oxidized per mole of acceptor reduced) - stoichiometric coefficient for reduction of the second electron acceptor (moles of dehalogenase oxidized per mole of acceptor reduced) - r _RX rate of the reductive dehalogenation reaction (moles of halogenated substrate reduced per unit volume per unit time) - r _d1 rate of oxidation of the primary exogenous electron donor (moles of donor oxidized per unit volume per unit time) - r _d2 rate of oxidation of the endogenous electron donor (biomass oxidized per unit volume per unit time) - r _a1 rate of reduction of the primary exogenous electron acceptor (moles of acceptor reduced per unit volume per unit time) - r _a2 rate of reduction of the second primary electron acceptor (moles of acceptor reduced per unit volume per unit time) - k _RX mixed second-order rate coefficient for the reductive dehalogenation reaction (volume per mole dehalogenase per unit time) - k _d1 mixed-second-order rate coefficient for oxidation of the primary electron donor (volume per mole dehalogenase per unit time) - k _d2 mixed-second-order rate coefficient for oxidation of the endogenous electron donor (volume per mole dehalogenase per unit time) - b first-order biomass decay coefficient (biomass oxidized per unit biomass per unit time) - k _a1 mixed-second-order rate coefficient for reduction of the primary electron acceptor (volume per mole dehalogenase per unit time) - k _a2 mixed-second-order rate coefficient for reduction of the second primary electron acceptor (volume per mole dehalogenase per unit time) - q _m,ap apparent maximum specific rate of reductive dehalogenation (moles of RX per unit biomass per unit time) - K _ap apparent half-saturation concentration for the halogenated aliphatic substrate (moles of RX per unit volume) - k _ap apparent pseudo-first-order rate coefficient for reductive dehalogenation (volume per unit biomass per unit time)