The Squid Giant Axon: Mathematical Models

The voltage clamp results of Hodgkin and Huxley have been reanalyzed in terms of alternative mathematical models. The model used for the potassium conductance changes is similar to that of the HH model except that an empirical functional relationship replaces the fourth power Law used by HH and the twenty-fifth power law used by Cole and Moore. The model used for the sodium conductance changes involves the explicit use of one variable only rather than the two variables m and h of HH. The rise and fall of the sodium conductance during a depolarizing voltage clamp is obtained by specifying that this one variable satisfies a second order differential equation which results from the coupling of two first order equations. Not only can the adjustable parameters of these models be made to give good fit to the clamp conductance data but the models can also then be used to compute action potential curves. Theoretical interpretations can also be given to these mathematical models.     

Exploring effects of DNA methylation and gene expression on pan-cancer drug response by mathematical models

Since genetic alteration only accounts for 20%–30% in the drug effect-related factors, the role of epigenetic regulation mechanisms in drug response is gradually being valued. However, how epigenetic changes and abnormal gene expression affect the chemotherapy response remains unclear. Therefore, we constructed a variety of mathematical models based on the integrated DNA methylation, gene expression, and anticancer drug response data of cancer cell lines from pan-cancer levels to identify genes whose DNA methylation is associated with drug response and then to assess the impact of epigenetic regulation of gene expression on the sensitivity of anticancer drugs. The innovation of the mathematical models lies in: Linear regression model is followed by logistic regression model, which greatly shortens the calculation time and ensures the reliability of results by considering the covariates. Second, reconstruction of prediction models based on multiple dataset partition methods not only evaluates the model stability but also optimizes the drug-gene pairs. For 368,520 drug-gene pairs with P < 0.05 in linear models, 999 candidate pairs with both AUC ≥ 0.8 and P < 0.05 were obtained by logistic regression models between drug response and DNA methylation. Then 931 drug-gene pairs with 45 drugs and 491 genes were optimized by model stability assessment. Integrating both DNA methylation and gene expression markedly increased predictive power for 732 drug-gene pairs where 598 drug-gene pairs including 44 drugs and 359 genes were prioritized. Several drug target genes were enriched in the modules of the drug-gene-weighted interaction network. Besides, for cancer driver genes such as EGFR, MET, and TET2, synergistic effects of DNA methylation and gene expression can predict certain anticancer drugs’ responses. In summary, we identified potential drug sensitivity-related markers from pan-cancer levels and concluded that synergistic regulation of DNA methylation and gene expression affect anticancer drug response.     

BCL-3 Attenuation of TNFA Expression Involves an Incoherent Feed-Forward Loop Regulated by Chromatin Structure

Induction of genes is rarely an isolated event; more typically occurring as part of a web of parallel interactions, or motifs, which act to refine and control gene expression. Here, we define an Incoherent Feed-forward Loop motif in which TNFα-induced NF-κB signalling activates expression of the TNFA gene itself and also controls synthesis of the negative regulator BCL-3. While sharing a common inductive signal, the two genes have distinct temporal expression profiles. Notably, while the TNFA gene promoter is primed to respond immediately to activated NF-κB in the nucleus, induction of BCL3 expression only occurs after a time delay of about 1h. We show that this time delay is defined by remodelling of the BCL3 gene promoter, which is required to activate gene expression, and characterise the chromatin delayed induction of BCL3 expression using mathematical models. The models show how a delay in inhibitor production effectively uncouples the rate of response to inflammatory cues from the final magnitude of inhibition. Hence, within this regulatory motif, a delayed (incoherent) feed-forward loop together with differential rates of TNFA (fast) and BCL3 (slow) mRNA turnover provide robust, pulsatile expression of TNFα . We propose that the structure of the BCL-3-dependent regulatory motif has a beneficial role in modulating expression dynamics and the inflammatory response while minimising the risk of pathological hyper-inflammation.     

On the dangers of making scientific models ontologically independent: taking Richard Levins’ warnings seriously

Levins and Lewontin have contributed significantly to our philosophical understanding of the structures, processes, and purposes of biological mathematical theorizing and modeling. Here I explore their separate and joint pleas to avoid making abstract and ideal scientific models ontologically independent by confusing or conflating our scientific models and the world. I differentiate two views of theorizing and modeling, orthodox and dialectical, in order to examine Levins and Lewontin’s, among others, advocacy of the latter view. I compare the positions of these two views with respect to four points regarding ontological assumptions: (1) the origin of ontological assumptions, (2) the relation of such assumptions to the formal models of the same theory, (3) their use in integrating and negotiating different formal models of distinct theories, and (4) their employment in explanatory activity. Dialectical is here used in both its Hegelian–Marxist sense of opposition and tension between alternative positions and in its Platonic sense of dialogue between advocates of distinct theories. I investigate three case studies, from Levins and Lewontin as well as from a recent paper of mine, that show the relevance and power of the dialectical understanding of theorizing and modeling.     

Conditions of appearance of local response and action potential in a simple mathematical model of nerve

A simple mathematical model of a nerve fibre is proposed. According to the model the current-voltage relation for the peak Na-current is represented as a broken line. It is shown that within the potential range, where the differential (slope) Na-conductance isnegative but its absolutevalue is smaller than the membrane conductance, the fibre gives a subthreshold local response. Within the range, where differential Na-conductance is negative and has absolute value larger than the membrane conductance, the action potential arises, the rise being independent on the strength of the stimulus and conditioned only by the membrane parameters. A simple formula is obtained which describes the relation between the strength of the threshold stimulus, I, and its duration, t₀. There are two parameters in this formula, which can be measured independently: the time constant of the membrane and the lag-period in the Na-current. Insertion of these parameters into the formula allows one to construct a theoretical I-t₀ curve in good agreement with experiment.     

Gene Circuit Analysis of the Terminal Gap Gene huckebein

The early embryo of Drosophila melanogaster provides a powerful model system to study the role of genes in pattern formation. The gap gene network constitutes the first zygotic regulatory tier in the hierarchy of the segmentation genes involved in specifying the position of body segments. Here, we use an integrative, systems-level approach to investigate the regulatory effect of the terminal gap gene huckebein (hkb) on gap gene expression. We present quantitative expression data for the Hkb protein, which enable us to include hkb in gap gene circuit models. Gap gene circuits are mathematical models of gene networks used as computational tools to extract regulatory information from spatial expression data. This is achieved by fitting the model to gap gene expression patterns, in order to obtain estimates for regulatory parameters which predict a specific network topology. We show how considering variability in the data combined with analysis of parameter determinability significantly improves the biological relevance and consistency of the approach. Our models are in agreement with earlier results, which they extend in two important respects: First, we show that Hkb is involved in the regulation of the posterior hunchback (hb) domain, but does not have any other essential function. Specifically, Hkb is required for the anterior shift in the posterior border of this domain, which is now reproduced correctly in our models. Second, gap gene circuits presented here are able to reproduce mutants of terminal gap genes, while previously published models were unable to reproduce any null mutants correctly. As a consequence, our models now capture the expression dynamics of all posterior gap genes and some variational properties of the system correctly. This is an important step towards a better, quantitative understanding of the developmental and evolutionary dynamics of the gap gene network.     

Modelling and validation of Lactobacillus plantarum fermentations in cereal-based media with different sugar concentrations and buffering capacities

An unstructured mathematical model is proposed to describe the fermentation kinetics of growth, lactic acid production, pH and sugar consumption by Lactobacillus plantarum as a function of the buffering capacity and initial glucose concentration of the culture media. Initially the experimental data of L. plantarum fermentations in synthetic media with different buffering capacity and glucose were fitted to a set of primary models. Later the parameters obtained from these models were used to establish mathematical relationships with the independent variables tested. The models were validated with 6 fermentations of L. plantarum in different cereal-based media. In most cases the proposed models adequately describe the biochemical changes taking place during fermentation and are a promising approach for the formulation of cereal-based probiotic foods.     

MiRNA Expression May Account for Chronic but Not for Acute Regulation of mRNA Expression in Human Thyroid Tumor Models

Background

For thyroid tumorigenesis, two main human in vitro models are available: primary cultures of human thyrocytes treated with TSH or EGF/serum as models for autonomous adenomas (AA) or papillary thyroid carcinomas (PTC) respectively, and human thyroid tumor derived cell lines. Previous works of our group have assessed properties of those models, with a special emphasis on mRNA regulations. It is often assumed that miRNA may be one of the primary events inducing these mRNA regulations.

Methods

The purpose of this study was to investigate the representativity of those models to study microRNA regulations and their relation with mRNA expression. To achieve this aim, the miRNA expressions profiles of primary cultures treated with TSH or EGF/serum and of 6 thyroid cancer cell lines were compared to the expression profiles of 35 tumor tissues obtained by microarrays.

Results

Our data on primary cultures have shown that the TSH or EGF/serum treatment did not greatly modify the microRNA expression profiles, which is contrary to what is observed for mRNA expression profiles, although they still evolved differently according to the treatment. The analysis of miRNA and mRNA expressions profiles in the cell lines has shown that they have evolved into a common, dedifferentiated phenotype, closer to ATC than to the tumors they are derived from.

Conclusions

Long-terms TSH or EGF/serum treatments do not mimic AA or PTC respectively in terms of miRNA expression as they do for mRNA, suggesting that the regulations of mRNA expression induced by these physiological agents occur independently of miRNA. The general patterns of miRNA expression in the cell lines suggest that they represent a useful model for undifferentiated thyroid cancer. Mirna probably do not mediate the rapid changes in gene expression in rapid cell biology regulation.     

On the Use of Bone Remodelling Models to Estimate the Density Distribution of Bones. Uniqueness of the Solution

Bone remodelling models are widely used in a phenomenological manner to estimate numerically the distribution of apparent density in bones from the loads they are daily subjected to. These simulations start from an arbitrary initial distribution, usually homogeneous, and the density changes locally until a bone remodelling equilibrium is achieved. The bone response to mechanical stimulus is traditionally formulated with a mathematical relation that considers the existence of a range of stimulus, called dead or lazy zone, for which no net bone mass change occurs. Implementing a relation like that leads to different solutions depending on the starting density. The non-uniqueness of the solution has been shown in this paper using two different bone remodelling models: one isotropic and another anisotropic. It has also been shown that the problem of non-uniqueness is only mitigated by removing the dead zone, but it is not completely solved unless the bone formation and bone resorption rates are limited to certain maximum values.     

Mathematical Models of the Kinetics of the Cultivation of Microorganisms

Various equations of mathematical models for the kinetics of the development of various biological processes were obtained on the basis of the generalized differential equation of biomass growth. Aerobic periodic cultivation of the yeast Saccharomyces cerevisiae was carried out to provide a comparative evaluation of advantages and disadvantages of four types of mathematical models. It is shown that the exponential model is a particular solution to the generalized differential equation. The developed mathematical model can be used to predict the course of biological processes in time and can serve as a tool for a computational experiment in order to clarify the dependence of the rate of a biological process on changes in certain parameters that affect the development of cells.     

Occurrence and Treatment of Bone Atrophic Non-Unions Investigated by an Integrative Approach

Recently developed atrophic non-union models are a good representation of the clinical situation in which many non-unions develop. Based on previous experimental studies with these atrophic non-union models, it was hypothesized that in order to obtain successful fracture healing, blood vessels, growth factors, and (proliferative) precursor cells all need to be present in the callus at the same time. This study uses a combined in vivo-in silico approach to investigate these different aspects (vasculature, growth factors, cell proliferation). The mathematical model, initially developed for the study of normal fracture healing, is able to capture essential aspects of the in vivo atrophic non-union model despite a number of deviations that are mainly due to simplifications in the in silico model. The mathematical model is subsequently used to test possible treatment strategies for atrophic non-unions (i.e. cell transplant at post-osteotomy, week 3). Preliminary in vivo experiments corroborate the numerical predictions. Finally, the mathematical model is applied to explain experimental observations and identify potentially crucial steps in the treatments and can thereby be used to optimize experimental and clinical studies in this area. This study demonstrates the potential of the combined in silico-in vivo approach and its clinical implications for the early treatment of patients with problematic fractures.     

An on-lattice agent-based Monte Carlo model simulating the growth kinetics of multicellular tumor spheroids

PurposeTo develop an on-lattice agent-based model describing the growth of multicellular tumor spheroids using simple Monte Carlo tools.MethodsCells are situated on the vertices of a cubic grid. Different cell states (proliferative, hypoxic or dead) and cell evolution rules, driven by 10 parameters, and the effects of the culture medium are included. About twenty spheroids of MCF-7 human breast cancer were cultivated and the experimental data were used for tuning the model parameters.ResultsSimulated spheroids showed adequate sizes of the necrotic nuclei and of the hypoxic and proliferative cell phases as a function of the growth time, mimicking the overall characteristics of the experimental spheroids. The relation between the radii of the necrotic nucleus and the whole spheroid obtained in the simulations was similar to the experimental one and the number of cells, as a function of the spheroid volume, was well reproduced. The statistical variability of the Monte Carlo model described the whole volume range observed for the experimental spheroids. Assuming that the model parameters vary within Gaussian distributions it was obtained a sample of spheroids that reproduced much better the experimental findings.ConclusionsThe model developed allows describing the growth of in vitro multicellular spheroids and the experimental variability can be well reproduced. Its flexibility permits to vary both the agents involved and the rules that govern the spheroid growth. More general situations, such as, e. g., tumor vascularization, radiotherapy effects on solid tumors, or the validity of the tumor growth mathematical models can be studied.     

A topological exchange model for cell self-sorting.

A new model of cell sorting based on a geometric representation (trivalent map) and its associated structure (triangulated graph) is presented. This model varies substantially from existing mathematical models in that individual cells are represented by variable n-gons, n ? 3, whereas in current models cells are considered to be fixed (non-deformable) geometric, i.e. hexagonal or square objects. An exchange mechanism which operates on this graph is investigated and possible drivers are discussed.     

Novel Cancer Chemotherapy Hits by Molecular Topology: Dual Akt and Beta-Catenin Inhibitors

Background and Purpose

Colorectal and prostate cancers are two of the most common types and cause of a high rate of deaths worldwide. Therefore, any strategy to stop or at least slacken the development and progression of malignant cells is an important therapeutic choice. The aim of the present work is the identification of novel cancer chemotherapy agents. Nowadays, many different drug discovery approaches are available, but this paper focuses on Molecular Topology, which has already demonstrated its extraordinary efficacy in this field, particularly in the identification of new hit and lead compounds against cancer. This methodology uses the graph theoretical formalism to numerically characterize molecular structures through the so called topological indices. Once obtained a specific framework, it allows the construction of complex mathematical models that can be used to predict physical, chemical or biological properties of compounds. In addition, Molecular Topology is highly efficient in selecting and designing new hit and lead drugs. According to the aforementioned, Molecular Topology has been applied here for the construction of specific Akt/mTOR and β-catenin inhibition mathematical models in order to identify and select novel antitumor agents.

Experimental Approach

Based on the results obtained by the selected mathematical models, six novel potential inhibitors of the Akt/mTOR and β-catenin pathways were identified. These compounds were then tested in vitro to confirm their biological activity.

Conclusion and Implications

Five of the selected compounds, CAS n° 256378-54-8 (Inhibitor n°1), 663203-38-1 (Inhibitor n°2), 247079-73-8 (Inhibitor n°3), 689769-86-6 (Inhibitor n°4) and 431925-096 (Inhibitor n°6) gave positive responses and resulted to be active for Akt/mTOR and/or β-catenin inhibition. This study confirms once again the Molecular Topology’s reliability and efficacy to find out novel drugs in the field of cancer.     

Effect of photoperiod on growth of the plants,and sesamin content and CYP81Q1 gene expression in the leaves of sesame (Sesamum indicum L.)

Sesamin is a major lignan constituent of sesame (Sesamum indicum) seed and considered responsible for a number of beneficial human health effects. We previously reported that sesamin is present in sesame leaves, and proposed use of sesame leaves as a sesamin-containing material. This study focused on the possibility that both leaf yield and sesamin content would be increased with increasing photoperiod. Additionally, it was hypothesized that sesamin content would be affected by photoperiod in relation to CYP81Q1 gene expression. We thus investigated the effect of photoperiod on growth and leaf sesamin content in relation to CYP81Q1 gene expression to confirm our hypothesis. Under short-day (SD) condition, increase of leaf area was suppressed due to the phase transition from vegetative to reproductive growth, which resulted in reduction of leaf yield. Under long-day (LD) conditions, vegetative growth was continued, and both leaf area and yield increased as photoperiod increased up to 24 h (continuous light). Sesamin accumulated particularly in the leaves of plants grown under a 24-h photoperiod for 4 weeks. High expression level of the CYP81Q1 gene in those plants indicates that photoperiod-dependent differences in leaf sesamin content correlate with differences in CYP81Q1 gene expression levels. We conclude that cultivation under continuous light enables high-yield production of sesame leaves containing distinctively high levels of sesamin.