Mathematical Models of Chiral Symmetry-breaking - A Review of General Theories, and Adiabatic Approximations of the APED System.

We review the literature surrounding chiral symmetry-breaking in chemical systems, with a focus on understanding the mathematical models underlying these chemical processes. We comment in particular on the toy model of Sandars, Viedma's crystal grinding systems and the APED model. We include a few new results based on asymptotic analysis of the APED system.

Breather modes of fully nonlinear mass-in-mass chains.

We propose a model for a chain of particles coupled by nonlinear springs in which each mass has an internal mass and all interactions are assumed to be nonlinear. We show how to construct an asymptotic solution of this system using multiple timescales, the systematic solution of coupled equations by repeated application of a consistency condition. Our results show that for some combinations of nonlinearity the dynamics are governed by the NLS as in the more usual mass-in-mass chains with linear interactions between inner and outer masses. However, when both nonlinearities have quadratic components, we show that the asymptotic reduction results in a Ginzburg-Landau equation instead of NLS.

Analysis of phenotype-genotype associations using genomic informational field theory (GIFT).

We show how field- and information theory can be used to quantify the relationship between genotype and phenotype in cases where phenotype is a continuous variable. Given a sample population of phenotype measurements, from various known genotypes, we show how the ordering of phenotype data can lead to quantification of the effect of genotype. This method does not assume that the data has a Gaussian distribution, it is particularly effective at extracting weak and unusual dependencies of genotype on phenotype. However, in cases where data has a special form, (eg Gaussian), we observe that the effective phenotype field has a special form. We use asymptotic analysis to solve both the forward and reverse formulations of the problem. We show how p-values can be calculated so that the significance of correlation between phenotype and genotype can be quantified. This provides a significant generalisation of the traditional methods used in genome-wide association studies GWAS. We derive a field-strength which can be used to deduce how the correlations between genotype and phenotype, and their impact on the distribution of phenotypes.

Mathematical modelling of earlier stages of COVID-19 transmission dynamics in Ghana.

In late 2019, a novel coronavirus, the SARS-CoV-2 outbreak was identified in Wuhan, China and later spread to every corner of the globe. Whilst the number of infection-induced deaths in Ghana, West Africa are minimal when compared with the rest of the world, the impact on the local health service is still significant. Compartmental models are a useful framework for investigating transmission of diseases in societies. To understand how the infection will spread and how to limit the outbreak. We have developed a modified SEIR compartmental model with nine compartments (CoVCom9) to describe the dynamics of SARS-CoV-2 transmission in Ghana. We have carried out a detailed mathematical analysis of the CoVCom9, including the derivation of the basic reproduction number, R 0 . In particular, we have shown that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 via a candidate Lyapunov function. Using the SARS-CoV-2 reported data for confirmed-positive cases and deaths from March 13 to August 10, 2020, we have parametrised the CoVCom9 model. The results of this fit show good agreement with data. We used Latin hypercube sampling-rank correlation coefficient (LHS-PRCC) to investigate the uncertainty and sensitivity of R 0 since the results derived are significant in controlling the spread of SARS-CoV-2. We estimate that over this five month period, the basic reproduction number is given by R 0 = 3 . 110 , with the 95% confidence interval being 2 . 042 ≤ R 0 ≤ 3 . 240 , and the mean value being R 0 = 2 . 623 . Of the 32 parameters in the model, we find that just six have a significant influence on R 0 , these include the rate of testing, where an increasing testing rate contributes to the reduction of R 0 .

Stochastic fractal and Noether's theorem.

We consider the binary fragmentation problem in which, at any breakup event, one of the daughter segments either survives with probability p or disappears with probability 1-p. It describes a stochastic dyadic Cantor set that evolves in time, and eventually becomes a fractal. We investigate this phenomenon, through analytical methods and Monte Carlo simulation, for a generic class of models, where segment breakup points follow a symmetric beta distribution with shape parameter α, which also determines the fragmentation rate. For a fractal dimension d_{f}, we find that the d_{f} th moment M_{d_{f}} is a conserved quantity, independent of p and α. While the scaling exponents do not depend on p, the self-similar distribution shows a weak p dependence. We use the idea of data collapse-a consequence of dynamical scaling symmetry-to demonstrate that the system exhibits self-similarity. In an attempt to connect the symmetry with the conserved quantity, we reinterpret the fragmentation equation as the continuity equation of a Euclidean quantum-mechanical system. Surprisingly, the Noether charge corresponding to dynamical scaling is trivial, while M_{d_{f}} relates to a purely mathematical symmetry: Quantum-mechanical phase rotation in Euclidean time.

Mathematical modelling of telomere length dynamics.

Telomeres are repetitive DNA sequences located at the ends of chromosomes. During cell division, an incomplete copy of each chromosome's DNA is made, causing telomeres to shorten on successive generations. When a threshold length is reached replication ceases and the cell becomes 'senescent'. In this paper, we consider populations of telomeres and, from discrete models, we derive partial differential equations which describe how the distribution of telomere lengths evolves over many generations. We initially consider a population of cells each containing just a single telomere. We use continuum models to compare the effects of various mechanisms of telomere shortening and rates of cell division during normal ageing. For example, the rate (or probability) of cell replication may be fixed or it may decrease as the telomeres shorten. Furthermore, the length of telomere lost on each replication may be constant, or may decrease as the telomeres shorten. Where possible, explicit solutions for the evolution of the distribution of telomere lengths are presented. In other cases, expressions for the mean of the distribution are derived. We extend the models to describe cell populations in which each cell contains a distinct subpopulation of chromosomes. As for the simpler models, constant telomere shortening leads to a linear reduction in telomere length over time, whereas length-dependent shortening results in initially rapid telomere length reduction, slowing at later times. Our analysis also reveals that constant telomere loss leads to a Gaussian (normal) distribution of telomere lengths, whereas length-dependent loss leads to a log-normal distribution. We show that stochastic models, which include a replication probability, also lead to telomere length distributions which are skewed.

Effects of competition between random sequential nucleation of point-sized seeds and island growth by adsorption of finite-sized grains.

We study random sequential adsorption of particles from a pool onto a one-dimensional substrate following ballistic deposition rules with separate nucleation and growth processes occurring simultaneously. Nucleation describes the formation of point-sized seeds, and after a seed is sown, it acts as an attractor and grows in size by the addition of grains of a fixed size. At each time step either an already-nucleated seed can increase in size, or a new seed may be nucleated. We incorporate a parameter m to describe the relative rates of growth to nucleation. We solve the model analytically to obtain a gap size distribution function and a general expression for the jamming coverage as a function of m. We show that the jamming coverage θ(m) reaches its maximum value of θ(m)=1 in the limit m→∞ following a power-law θ(∞)-θ(m)∼Km^{-1/2} for some constant K. We also perform an extensive Monte Carlo simulation and find good agreement between analytic and numerical results.

The Effects of Insulin Resistance on Individual Tissues: An Application of a Mathematical Model of Metabolism in Humans.

Whilst the human body expends energy constantly, the human diet consists of a mix of carbohydrates and fats delivered in a discontinuous manner. To deal with this sporadic supply of energy, there are transport, storage and utilisation mechanisms, for both carbohydrates and fats, around all tissues of the body. Insulin-resistant states such as type 2 diabetes and obesity are characterised by reduced efficiency of these mechanisms. Exactly how these insulin-resistant states develop, for example whether there is an order in which tissues become insulin resistant, is an active area of research with the hope of gaining a better overall understanding of insulin resistance. In this paper, we use a previously derived system of 12 first-order coupled differential equations that describe the transport between, and storage in, different tissues of the human body. We briefly revisit the derivation of the model before parametrising the model to account for insulin resistance. We then solve the model numerically, separately simulating each individual tissue as insulin resistant, and discuss and compare these results, drawing three main conclusions. The implications of these results are in accordance with biological intuition. First, insulin resistance in a tissue creates a knock-on effect on the other tissues in the body, whereby they attempt to compensate for the reduced efficiency of the insulin-resistant tissue. Second, insulin resistance causes a fatty liver, and the insulin resistance of tissues other than the liver can cause fat to accumulate in the liver. Finally, although insulin resistance in individual tissues can cause slightly reduced skeletal muscle metabolic flexibility, it is when the whole body is insulin resistant that the biggest effect on skeletal muscle flexibility is seen.

Mathematical modelling of hepatic lipid metabolism.

The aim of this paper is to develop a mathematical model capable of simulating the metabolic response to a variety of mixed meals in fed and fasted conditions with particular emphasis placed on the hepatic triglyceride element of the model. Model validation is carried out using experimental data for the ingestion of three mixed composition meals over a 24-h period. Comparison with experimental data suggests the model predicts key plasma lipids accurately given a prescribed insulin profile. One counter-intuitive observation to arise from simulations is that liver triglyceride initially decreases when a high fat meal is ingested, a phenomenon potentially explained by the carbohydrate portion of the meal raising plasma insulin.

Stochastic simulations of normal aging and Werner's syndrome.

Human cells typically consist of 23 pairs of chromosomes. Telomeres are repetitive sequences of DNA located at the ends of chromosomes. During cell replication, a number of basepairs are lost from the end of the chromosome and this shortening restricts the number of divisions that a cell can complete before it becomes senescent, or non-replicative. In this paper, we use Monte Carlo simulations to form a stochastic model of telomere shortening to investigate how telomere shortening affects normal aging. Using this model, we study various hypotheses for the way in which shortening occurs by comparing their impact on aging at the chromosome and cell levels. We consider different types of length-dependent loss and replication probabilities to describe these processes. After analyzing a simple model for a population of independent chromosomes, we simulate a population of cells in which each cell has 46 chromosomes and the shortest telomere governs the replicative potential of the cell. We generalize these simulations to Werner's syndrome, a condition in which large sections of DNA are removed during cell division and, amongst other conditions, results in rapid aging. Since the mechanisms governing the loss of additional basepairs are not known, we use our model to simulate a variety of possible forms for the rate at which additional telomeres are lost per replication and several expressions for how the probability of cell division depends on telomere length. As well as the evolution of the mean telomere length, we consider the standard deviation and the shape of the distribution. We compare our results with a variety of data from the literature, covering both experimental data and previous models. We find good agreement for the evolution of telomere length when plotted against population doubling.

Modelling the regulation of telomere length: the effects of telomerase and G-quadruplex stabilising drugs.

Telomeres are guanine-rich sequences at the end of chromosomes which shorten during each replication event and trigger cell cycle arrest and/or controlled death (apoptosis) when reaching a threshold length. The enzyme telomerase replenishes the ends of telomeres and thus prolongs the life span of cells, but also causes cellular immortalisation in human cancer. G-quadruplex (G4) stabilising drugs are a potential anticancer treatment which work by changing the molecular structure of telomeres to inhibit the activity of telomerase. We investigate the dynamics of telomere length in different conformational states, namely t-loops, G-quadruplex structures and those being elongated by telomerase. By formulating deterministic differential equation models we study the effects of various levels of both telomerase and concentrations of a G4-stabilising drug on the distribution of telomere lengths, and analyse how these effects evolve over large numbers of cell generations. As well as calculating numerical solutions, we use quasicontinuum methods to approximate the behaviour of the system over time, and predict the shape of the telomere length distribution. We find those telomerase and G4-concentrations where telomere length maintenance is successfully regulated. Excessively high levels of telomerase lead to continuous telomere lengthening, whereas large concentrations of the drug lead to progressive telomere erosion. Furthermore, our models predict a positively skewed distribution of telomere lengths, that is, telomeres accumulate over lengths shorter than the mean telomere length at equilibrium. Our model results for telomere length distributions of telomerase-positive cells in drug-free assays are in good agreement with the limited amount of experimental data available.

Theory, modelling and simulation in origins of life studies.

Origins of life studies represent an exciting and highly multidisciplinary research field. In this review we focus on the contributions made by theory, modelling and simulation to addressing fundamental issues in the domain and the advances these approaches have helped to make in the field. Theoretical approaches will continue to make a major impact at the "systems chemistry" level based on the analysis of the remarkable properties of nonlinear catalytic chemical reaction networks, which arise due to the auto-catalytic and cross-catalytic nature of so many of the putative processes associated with self-replication and self-reproduction. In this way, we describe inter alia nonlinear kinetic models of RNA replication within a primordial Darwinian soup, the origins of homochirality and homochiral polymerization. We then discuss state-of-the-art computationally-based molecular modelling techniques that are currently being deployed to investigate various scenarios relevant to the origins of life.

The effects of a telomere destabilizing agent on cancer cell-cycle dynamics--integrated modelling and experiments.

The pentacyclic acridinium salt RHPS4 displays anti-tumour properties in vitro as well as in vivo and is potentially cell-cycle specific. We have collected experimental data and formulated a compartmental model using ordinary differential equations to investigate how the compound affects cells in each stage of the cell cycle. In addition to a control case in which no drug was used, we treated colorectal cancer cells with three different concentrations of the drug and fitted simulations from our models to experimental observations. We found that RHPS4 caused a concentration-dependent, marked cell death in treated cells, which is best modelled by allowing the rate parameters corresponding to cell death to be sigmoidal functions of time. We have shown that the model is "identifiable", meaning that, at least in principle, the parameter values can be determined from observable quantities. We find that at low concentrations RHPS4 primarily affects the cells in the G(2)/M phase, and that the drug has a delayed effect with the delay decreasing at larger doses. Since the drug diffuses into the nucleus, the observed delayed effect of the compound is unexpected and is a novel finding of our research into this compound.

Evidence that ACN1 (acetate non-utilizing 1) prevents carbon leakage from peroxisomes during lipid mobilization in Arabidopsis seedlings.

ACN1 (acetate non-utilizing 1) is a short-chain acyl-CoA synthetase which recycles free acetate to acetyl-CoA in peroxisomes of Arabidopsis. Pulse-chase [2-(13)C]acetate feeding of the mutant acn1-2 revealed that acetate accumulation and assimilation were no different to that of wild-type, Col-7. However, the lack of acn1-2 led to a decrease of nearly 50% in (13)C-labelling of glutamine, a major carbon sink in seedlings, and large decreases in primary metabolite levels. In contrast, acetyl-CoA levels were higher in acn1-2 compared with Col-7. The disappearance of eicosenoic acid was slightly delayed in acn1-2 indicating only a small effect on the rate of lipid breakdown. A comparison of transcript levels in acn1-2 and Col-7 showed that induced genes included a number of metabolic genes and also a large number of signalling-related genes. Genes repressed in the mutant were represented primarily by embryogenesis-related genes. Transcript levels of glyoxylate cycle genes also were lower in acn1-2 than in Col-7. We conclude that deficiency in peroxisomal acetate assimilation comprises only a small proportion of total acetate use, but this affects both primary metabolism and gene expression. We discuss the possibility that ACN1 safeguards against the loss of carbon as acetate from peroxisomes during lipid mobilization.

Mathematical models of the homochiralisation of crystals by grinding.

We review the existing mathematical models which describe physicochemical mechanisms capable of producing a symmetry-breaking transition to a state in which one chirality dominates the other. A new model is proposed, with the aim of elucidating the fundamental processes at work in the crystal grinding systems of Viedma (Phys Rev Lett 94:065504, 2005) and Noorduin (J Am Chem Soc 130:1158-1159, 2008). We simplify the model as far as possible to uncover the fundamental competitive process which causes the symmetry-breaking, and analyse other simplifications which might be expected to show symmetry-breaking.

Modelling the peroxisomal carbon leak during lipid mobilization in Arabidopsis.

Mutation of the ACN1 (acetate non-utilizing 1) locus of Arabidopsis results in altered acetate assimilation into gluconeogenic sugars and anapleurotic amino acids and leads to an overall depression in primary metabolite levels by approx. 50% during seedling development. Levels of acetyl-CoA were higher in acn1 compared with wild-type, which is counterintuitive to the activity of ACN1 as a peroxisomal acetyl-CoA synthetase. We hypothesize that ACN1 recycles free acetate to acetyl-CoA within peroxisomes in order that carbon remains fed into the glyoxylate cycle. When ACN1 is not present, carbon in the form of acetate can leak out of peroxisomes and is reactivated to acetyl-CoA within the cytosol. Kinetic models incorporating estimates of carbon input and pathway dynamics from a variety of literature sources have proven useful in explaining how ACN1 may prevent the carbon leak and even contribute to the control of peroxisomal carbon metabolism.

Nonlinear breathing modes at a defect site in DNA.

Molecular-dynamics simulations of a normal DNA duplex show that breathing events typically occur on the microsecond time scale. This paper analyzes a 12 base pairs DNA duplex containing the "rogue" base difluorotoluene (F) in place of a thymine base (T), for which the breathing events occur on the nanosecond time scale. Starting from a nonlinear Klein-Gordon lattice model and adding noise and damping, we obtain a mesoscopic model of the DNA duplex close to that observed in experiments and all-atom molecular dynamics simulations. The mesoscopic model is calibrated to data from the all-atom molecular dynamics package AMBER for a variety of twist angles of the DNA duplex. Defects are considered in the interchain interactions as well as in the along-chain interactions. This paper also discusses the role of the fluctuation-dissipation relations in the derivation of reduced (mesoscopic) models, the differences between the potential of mean force and the potential energies used in Klein-Gordon lattices, and how breathing can be viewed as competition between the along-chain elastic energy and the interchain binding energy.

Sequence selection during copolymerization.

We analyze a simple model for the copolymerization of two chemically distinct monomers that displays a wide variety of product sequence compositions depending on the rate coefficients for the stepwise addition of each monomer to a growing polymer chain. We show that in certain regions of parameter space the copolymer sequence composition depends sensitively on these parameter values, and we compute the information content of the resultant sequences. Finally, we show how the model may be used to interpret experimental data on copolymerization reactions.

Symmetry-breaking in chiral polymerisation.

We propose a model for chiral polymerisation and investigate its symmetric and asymmetric solutions. The model has a source species which decays into left- and right-handed types of monomer, each of which can polymerise to form homochiral chains; these chains are susceptible to 'poisoning' by the opposite-handed monomer. Homochiral polymers are assumed to influence the proportion of each type of monomer formed from the precursor. We show that for certain parameter values a positive feedback mechanism makes the symmetric steady-state solution unstable. The kinetics of polymer formation are then analysed in the case where the system starts from zero concentrations of monomers and chains. We show that following a long induction time, extremely large concentrations of polymers are formed for a short time, during this time an asymmetry introduced into the system by a random external perturbation may be massively amplified. The system then approaches one of the steady-state solutions described above.

Nonlinear breathing modes due to a defect in a DNA chain.

Recent molecular-dynamics simulations of a DNA duplex containing the 'rogue' base difluorotoluene (F) in place of a thymine (T) base show that breathing events can occur on the nanosecond time-scale, whereas breathing events in a normal DNA duplex take place on the microsecond time-scale. The main aim of this paper is to analyse a nonlinear Klein-Gordon lattice model of the DNA duplex, including both nonlinear interactions between opposing bases and a defect in the interaction at one lattice site, each of which can cause localization of energy. Solutions for a breather mode either side of the defect are derived using multiple-scales asymptotics and are pieced together across the defect to form a solution which includes the effects of the nonlinearity and the defect. We consider defects in the inter-chain interactions and in the along-chain interactions. In most cases we find in-phase breather modes and/or out-of-phase breather modes, with one case displaying a shifted mode.