The E. coli transcriptional regulatory network and its spatial embedding.

Usually complex networks are studied as graphs consisting of nodes whose spatial arrangement is of no significance. Several real biological networks are, however, embedded in space. In this paper we study the transcription regulatory network (TRN) of E. coli as a spatially embedded network. The embedding space of this network is the circular E. coli chromosome, i.e. it is practically one dimensional. However, the TRN itself is a high-dimensional network due to the existence of an adequate number of long-range connections. We find that nodes in short topological distance l = 1, 2 tend, on average, to be in shorter spatial distances r indicating an abundance of short-range connections as well. Community analysis of the TRN reveals the interesting fact that highly interconnected subnets consist of nodes that tend to be in spatial proximity on the circular chromosome. We also find indications that for certain transcriptional aspects of the E. coli it is advantageous to treat the circular genome as two line segments starting from the OriC and ending to Ter.

Monte Carlo simulations in drug release.

We present methods based on simple sampling Monte Carlo simulations that are used in the study of controlled drug release from devices of various shapes and characteristics. The manuscript is part of a special tribute issue for Prof. Panos Macheras and we have chosen applications of the Monte Carlo method in the field of drug release that were pioneered by him and his research group. Thus, we focus on the investigation of diffusion based release and we present methods that go beyond the application of the classical fickian diffusion equation. We describe methods that have proven to be effective in illuminating the profound effects of the substrate heterogeneity on the drug release profiles and demonstrate some of the most powerful applications of agent based simulations and numerical methods in the field of pharmacokinetics.

A minimal model for gene expression dynamics of bacterial type II toxin-antitoxin systems.

Toxin-antitoxin (TA) modules are part of most bacteria's regulatory machinery for stress responses and general aspects of their physiology. Due to the interplay of a long-lived toxin with a short-lived antitoxin, TA modules have also become systems of interest for mathematical modelling. Here we resort to previous modelling efforts and extract from these a minimal model of type II TA system dynamics on a timescale of hours, which can be used to describe time courses derived from gene expression data of TA pairs. We show that this model provides a good quantitative description of TA dynamics for the 11 TA pairs under investigation here, while simpler models do not. Our study brings together aspects of Biophysics with its focus on mathematical modelling and Computational Systems Biology with its focus on the quantitative interpretation of 'omics' data. This mechanistic model serves as a generic transformation of time course information into kinetic parameters. The resulting parameter vector can, in turn, be mechanistically interpreted. We expect that TA pairs with similar mechanisms are characterized by similar vectors of kinetic parameters, allowing us to hypothesize on the mode of action for TA pairs still under discussion.

A fractal kinetics SI model can explain the dynamics of COVID-19 epidemics.

The COVID-19 pandemic has already had a shocking impact on the lives of everybody on the planet. Here, we present a modification of the classical SI model, the Fractal Kinetics SI model which is in excellent agreement with the disease outbreak data available from the World Health Organization. The fractal kinetic approach that we propose here originates from chemical kinetics and has successfully been used in the past to describe reaction dynamics when imperfect mixing and segregation of the reactants is important and affects the dynamics of the reaction. The model introduces a novel epidemiological parameter, the "fractal" exponent h which is introduced in order to account for the self-organization of the societies against the pandemic through social distancing, lockdowns and flight restrictions.

Chromosomal origin of replication coordinates logically distinct types of bacterial genetic regulation.

For a long time it has been hypothesized that bacterial gene regulation involves an intricate interplay of the transcriptional regulatory network (TRN) and the spatial organization of genes in the chromosome. Here we explore this hypothesis both on a structural and on a functional level. On the structural level, we study the TRN as a spatially embedded network. On the functional level, we analyze gene expression patterns from a network perspective ("digital control"), as well as from the perspective of the spatial organization of the chromosome ("analog control"). Our structural analysis reveals the outstanding relevance of the symmetry axis defined by the origin (Ori) and terminus (Ter) of replication for the network embedding and, thus, suggests the co-evolution of two regulatory infrastructures, namely the transcriptional regulatory network and the spatial arrangement of genes on the chromosome, to optimize the cross-talk between two fundamental biological processes: genomic expression and replication. This observation is confirmed by the functional analysis based on the differential gene expression patterns of more than 4000 pairs of microarray and RNA-Seq datasets for E. coli from the Colombos Database using complex network and machine learning methods. This large-scale analysis supports the notion that two logically distinct types of genetic control are cooperating to regulate gene expression in a complementary manner. Moreover, we find that the position of the gene relative to the Ori is a feature of very high predictive value for gene expression, indicating that the Ori-Ter symmetry axis coordinates the action of distinct genetic control mechanisms.

On the unphysical hypotheses in pharmacokinetics and oral drug absorption: Time to utilize instantaneous rate coefficients instead of rate constants.

This work aims to explore the unphysical assumptions associated with i) the homogeneity of the well mixed compartments of pharmacokinetics and ii) the diffusion limited model of drug dissolution. To this end, we i) tested the homogeneity hypothesis using Monte Carlo simulations for a reaction and a diffusional process that take place in Euclidean and fractal media, ii) re-considered the flip-flop kinetics assuming that the absorption rate for a one-compartment model is governed by an instantaneous rate coefficient instead of a rate constant, and, iii) re-considered the extent of drug absorption as a function of dose using an in vivo reaction limited model of drug dissolution with integer and non-integer stoichiometry values. We found that drug diffusional processes and reactions are slowed down in heterogeneous media and the environmental heterogeneity leads to increased fluctuations of the measurable quantities. Highly variable experimental literature data with measurements in intrathecal space and gastrointestinal fluids were explained accordingly. Next, by applying power law and Weibull input functions to a one-compartment model of disposition we show that the shape of concentration-time curves is highly dependent on the time exponent of the input functions. Realistic examples based on PK data of three compounds known to exhibit flip-flop kinetics are analyzed. The need to use time dependent coefficients instead of rate constants in PBPK modeling and virtual bioequivalence is underlined. Finally, the shape of the fraction absorbed as a function of dose plots, using an in vivo reaction limited model of drug dissolution were found to be dependent on the stoichiometry value and the solubility of drug. Ascending and descending limbs were observed for the higher stoichiometries (2.0 and 1.5) with the low solubility drug. In contrast, for the more soluble drug, a continuous increase of fraction absorbed as a function of dose is observed when the higher stoichiometries are used (2.0 and 1.5). For both drugs, the fraction absorbed for the lower values of stoichiometry (0.7 and 1.0) exhibit a non-dependency on dose profile. Our results give an insight into the complex picture of in vivo drug dissolution since diffusion-limited and reaction-limited processes seem to operate under in vivo conditions concurrently.

Monte Carlo simulations of drug release from matrices with periodic layers of high and low diffusivity.

We have studied drug release from matrices with periodic layers of high and low diffusivity using Monte Carlo simulations. Despite the fact, that the differential equations relevant to this process have a form that is quite different from the classical diffusion equation with constant diffusion coefficient, we have found that the Weibull model continues to describe the release process as well as in the case of the "classical" diffusion controlled drug release. We examine the similarities and differences between release from matrices with periodic layers and matrices with random mixtures of high and low diffusivity area and show that the periodic geometrical arrangement of the low diffusivity areas has an influence in the release profile which is negligible for low diffusivity ratios, but becomes important in the case of high diffusivity ratios and for intermediate values of the periodic "length". Such an arrangement in periodic layers leads to Weibull exponent a which are lower than those of the corresponding random arrangement and exponents b which are higher than those of the random case.

On the dilemma of fractal or fractional kinetics in drug release studies: A comparison between Weibull and Mittag-Leffler functions.

We compare two of the most successful models for the description and analysis of drug release data. The fractal kinetics approach leading to release profiles described by a Weibull function and the fractional kinetics approach leading to release profiles described by a Mittag-Leffler function. We used Monte Carlo simulations to generate artificial release data from euclidean and fractal substrates. We have also used real release data from the literature and found that both models are capable in describing release data up to roughly 85% of the release. For larger times both models systematically overestimate the number of particles remaining in the release device.

Monte Carlo simulations for the study of drug release from matrices with high and low diffusivity areas.

We use Monte Carlo simulations in order to study diffusion controlled drug release from matrices consisting of random mixtures of high and low diffusivity areas (random mixing), and from matrices covered by a thin film of low diffusivity (ordered mixing). We compared our results with the Weibull model for drug release and found that it provides an adequate description of the release process in all cases of random mixing and most cases of ordered mixing. We have studied the dependence of the Weibull parameters on the diffusion coefficient and, in most cases, found a rather simple linear dependence. Moreover, our results indicate that a device covered by a thin film with diffusion coefficient three orders of magnitude lower that the coefficient of the rest of the device, will release drug at constant rate for most of the release process. This last result may have considerable practical applications.

On the use of the Weibull function for the discernment of drug release mechanisms.

Previous findings from our group based on Monte Carlo simulations indicated that Fickian drug release from Euclidian or fractal matrices can be described with the Weibull function. In this study, the entire drug release kinetics of various published data and experimental data from commercial or prepared controlled release formulations of diltiazem and diclofenac are analyzed using the Weibull function. The exponent of time b of the Weibull function is linearly related to the exponent n of the power law derived from the analysis of the first 60% of the release curves. The value of the exponent b is an indicator of the mechanism of transport of a drug through the polymer matrix. Estimates for b< or =0.75 indicate Fickian diffusion in either fractal or Euclidian spaces while a combined mechanism (Fickian diffusion and Case II transport) is associated with b values in the range 0.75

Modeling and Monte Carlo simulations in oral drug absorption.

Drug dissolution, release and uptake are the principal components of oral drug absorption. All these processes take place in the complex milieu of the gastrointestinal tract and they are influenced by physiological (e.g. intestinal pH, transit time) and physicochemical factors (e.g. dose, particle size, solubility, permeability). Due to the enormous complexity issues involved, the models developed for drug dissolution and release attempt to capture their heterogeneous features. Hence, Monte Carlo simulations and population methods have been utilized since both dissolution and release processes are considered as time evolution of a population of drug molecules moving irreversibly from the solid state to the solution. Additionally, mathematical models have been proposed to determine the effect of the physicochemical properties, solubility/dose ratio and permeability on the extent of absorption for regulatory purposes, e.g. biopharmaceutics classification. The regulatory oriented approaches are based on the tube model of the intestinal lumen and apart from the drug's physicochemical properties, take into account the formulation parameters the dose and the particle size.

Analysis of Case II drug transport with radial and axial release from cylinders.

Analysis is presented for Case II drug transport with axial and radial release from cylinders. The previously reported [J. Control Release 5 (1987) 37] relationships for radial release from films and slabs are special cases of the general solution derived in this study. The widely used exponential relation M(t)/M(infinity) = kt(n) describes nicely the first 60% of the fractional release curve when Case II drug transport with axial and radial release from cylinders is operating.

Explosive site percolation and finite-size hysteresis.

We report the critical point for site percolation for the "explosive" type for two-dimensional square lattices using Monte Carlo simulations and compare it to the classical well-known percolation. We use similar algorithms as have been recently reported for bond percolation and networks. We calculate the explosive site percolation threshold as p(c) = 0.695 and we find evidence that explosive site percolation surprisingly may belong to a different universality class than bond percolation on lattices, providing that the transitions (a) are continuous and (b) obey the conventional finite size scaling forms. Finally, we study and compare the direct and reverse processes, showing that while the reverse process is different from the direct process for finite size systems, the two cases become equivalent in the thermodynamic limit of large L.

Monte Carlo simulations and fractional kinetics considerations for the Higuchi equation.

We highlight some physical and mathematical aspects relevant to the derivation and use of the Higuchi equation. More specifically, the application of the Higuchi equation to different geometries is discussed and Monte Carlo simulations to verify the validity of Higuchi law in one and two dimensions, as well as the derivation of the Higuchi equation under alternative boundary conditions making use of fractional calculus, are presented.

A reappraisal of drug release laws using Monte Carlo simulations: the prevalence of the Weibull function.

PURPOSE: To verify the Higuchi law and study the drug release from cylindrical and spherical matrices by means of Monte Carlo computer simulation. METHOD: A one-dimensional matrix, based on the theoretical assumptions of the derivation of the Higuchi law, was simulated and its time evolution was monitored. Cylindrical and spherical three-dimensional lattices were simulated with sites at the boundary of the lattice having been denoted as leak sites. Particles were allowed to move inside it using the random walk model. Excluded volume interactions between the particles was assumed. We have monitored the system time evolution for different lattice sizes and different initial particle concentrations. RESULTS: The Higuchi law was verified using the Monte Carlo technique in a one-dimensional lattice. It was found that Fickian drug release from cylindrical matrices can be approximated nicely with the Weibull function. A simple linear relation between the Weibull function parameters and the specific surface of the system was found. CONCLUSIONS: Drug release from a matrix, as a result of a diffusion process assuming excluded volume interactions between the drug molecules, can be described using a Weibull function. This model, although approximate and semiempirical, has the benefit of providing a simple physical connection between the model parameters and the system geometry, which was something missing from other semiempirical models.

Michaelis-Menten kinetics under spatially constrained conditions: application to mibefradil pharmacokinetics.

Two different approaches were used to study the kinetics of the enzymatic reaction under heterogeneous conditions to interpret the unusual nonlinear pharmacokinetics of mibefradil. Firstly, a detailed model based on the kinetic differential equations is proposed to study the enzymatic reaction under spatial constraints and in vivo conditions. Secondly, Monte Carlo simulations of the enzyme reaction in a two-dimensional square lattice, placing special emphasis on the input and output of the substrate were applied to mimic in vivo conditions. Both the mathematical model and the Monte Carlo simulations for the enzymatic reaction reproduced the classical Michaelis-Menten (MM) kinetics in homogeneous media and unusual kinetics in fractal media. Based on these findings, a time-dependent version of the classic MM equation was developed for the rate of change of the substrate concentration in disordered media and was successfully used to describe the experimental plasma concentration-time data of mibefradil and derive estimates for the model parameters. The unusual nonlinear pharmacokinetics of mibefradil originates from the heterogeneous conditions in the reaction space of the enzymatic reaction. The modified MM equation can describe the pharmacokinetics of mibefradil as it is able to capture the heterogeneity of the enzymatic reaction in disordered media.