Mesoscale Modelling

Abstract

Increasingly, industrial materials are being designed to have structure on length scales of tens to thousands of nanometers. These structures are crucial to achieving a particular desired material property. Such structures, however, may depend on the underlying chemistry of the material for their existence. For example, a thousandfold increase in the ionic conductivity of a polymer blend may only occur in a narrow region of a hugely complex phase diagram, the location' of which region can be expected to depend on the molecular chemistry and physics from the monomer scale to the coil size.

Traditional Computational Chemistry has proved incapable of dealing with the length and time scales involved in the formation of these ‘Mesoscale’ structures. On the other hand, traditional Computational Physics has proved incapable of consistently incorporating the necessary chemical detail for modelling real industrial materials. In this paper we present two novel methods which successfully address both the chemistry and the physics of mesophase formation. The methods, described in detail, are MesoDyn and Dissipative Particle Dynamics (DPD).

Unlike phenomenological theories of materials, such as the Landau models which one finds in much of the computational physics literature, the two models mentioned incorporate molecular geometry and connectivity explicitly. We discuss each of the methods briefly.

We then give an overview of how these methods are being used in industry to optimise materials and processes. We discuss previous simulation results for triblock Pluronic surfactants in solution studied with MesoDyn, and for diblock copolymers studied with DPD, where the known experimental changes in morphology from micellar to hexagonal to bicontinuous to lamellar have been successfully reproduced. We also present new results for several systems, including binary and ternary blends, where the third component in the latter system is a diblock copolymer, which acts as a compatibiliser. We discuss the effects of changing solvent character on the material properties of these systems, as well as the effects of an externally imposed shear flow.     

Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling

In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-\(\upkappa \)B pathway. The second model is a force-based individual-based model of the development of a solid avascular tumour with specific application to tumour cords, i.e. a mass of cancer cells growing around a central blood vessel. In each case, we compare our computational simulation results with experimental data. In the final discussion section, we outline how to take the work forward through the development of a multiscale model focussed at the cell level. This would incorporate key intracellular signalling pathways associated with cancer within each cell (e.g. p53–Mdm2, NF-\(\upkappa \)B) and through the use of high-performance computing be capable of simulating up to \(10^9\) cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.     

Multi-Formalism Modelling and Simulation: Application to Cardiac Modelling

Cardiovascular modelling has been a major research subject for the last decade. Different cardiac models have been developed at a cellular level as well as at the whole organ level. Most of these models are defined by a comprehensive cellular modelling using continuous formalisms or by a tissue-level modelling often based on discrete formalisms. Nevertheless, both views still suffer from difficulties that reduce their clinical applications: the first approach requires heavy computational resources while the second one is not able to reproduce certain pathologies. This paper presents an original methodology trying to gather advantages from both approaches, by means of a hybrid model mixing discrete and continuous formalisms. This method has been applied to define a hybrid model of cardiac action potential propagation on a 2D grid of endocardial cells, combining cellular automata and a set of cells defined by the Beeler-Reuter model. For simulations under physiological and ischemic conditions, results show that the action potential propagation as well as electrogram reconstructions are consistent with clinical diagnosis. Finally, the advantage of the proposed approach is discussed within the frame of cardiac modelling and simulation.     

Modelling biocontrol of

The two haplotypes of Varroa destructor that have been identified as parasites of the Western honeybee (Apis mellifera L.) show disparate levels of virulence towards honeybee colonies. The Korea haplotype has been associated with severe colony mortality, whereas untreated colonies of European A. mellifera have survived long-term infestation by the Japan haplotype. The possible existence of a benign haplotype of V. destructor raises the prospect that it be used to “inoculate” colonies to provide biocontrol of the virulent haplotype. The feasibility of such a strategy was investigated using a mathematical model. Competition for resources during reproduction is known to reduce varroa mites’ reproduction rates as their infestation levels increase. Results from modelling suggested this density-dependent effect is sufficient for an established benign population to prevent the virulent population reaching destructive levels if a colony is subject to sporadic influxes of virulent mites. A colony faced with a continuous influx of mites could be protected if the proportion of virulent mites in the influx were below a threshold level (dependent on length of breeding season and intensity of influx). This condition might be achieved by “inoculating” neighbouring apiaries and controlling feral colonies in the vicinity. Decreased brood cell invasion rate by the benign haplotype decreased the threshold level. Any reproductive isolation between the benign and virulent haplotypes would cause further reproductive suppression, driving sporadic influxes of the virulent haplotype to extinction and conferring greater tolerance to a colony faced with a virulent influx. Increased colony resistance to varroa in the model was synergistic with the inoculation of colonies in the absence of reproductive isolation, but potentially antagonistic in its presence—although not to an extent that would preclude their joint use.     

Modelling hyphal networks

The indeterminate growth habit of fungal mycelial can produce massive organisms spanning kilometres, whereas the hypha, the modular building block of these structures, is only a few microns in diameter. The qualitative and quantitative relationship between these scales is difficult to establish using experimental methods alone and a large number of mathematical models have been constructed to assist in the investigation of the multi-scale form and function of filamentous fungi. Many such models operate at the colony-scale, representing the hyphal network as either a regular lattice or as a geometrically-unconstrained structure that changes according to a minimal set of specified rules focussed on the fundamental processes responsible for growth and function. In this review we discuss the historical development and recent applications of such models and suggest some future directions.     

Modelling malaria pathogenesis

Almost 20 years after the development of models of malaria pathogenesis began, we are beyond the 'proof-of-concept' phase and these models are no longer abstract mathematical exercises. They have refined our knowledge of within-host processes, and have brought insights that could not easily have been obtained from experimentation alone. There is much potential that remains to be realized, however, both in terms of informing the design of interventions and health policy, and in terms of addressing lingering questions about the basic biology of malaria. Recent research has begun to iterate theory and data in a much more comprehensive way, and the use of statistical techniques for model fitting and comparison offers a promising approach for providing a quantitative understanding of the pathogenesis of such a complex disease.     

Modelling semantic transparency

We present models of semantic transparency in which the perceived transparency of English noun–noun compounds, and of their constituent words, is predicted on the basis of the expectedness of their semantic structure. We show that such compounds are perceived as more transparent when the first noun is more frequent, hence more expected, in the language generally; when the compound semantic relation is more frequent, hence more expected, in association with the first noun; and when the second noun is more productive, hence more expected, as the second element of a noun–noun compound. Taken together, our models of compound and constituent transparency lead us to two conclusions. Firstly, although compound transparency is a function of the transparencies of the constituents, the two constituents differ in the nature of their contribution. Secondly, since all the significant predictors in our models of compound transparency are also known predictors of processing speed, perceived transparency may itself be a reflex of ease of processing.     

Modelling microtubule patterns

The cellular cytoskeleton is well studied in terms of its biological and physical properties, making it an attractive subject for systems approaches. Here, we describe the experimental and theoretical strategies used to study the collective behaviour of microtubules and motors. We illustrate how this led to the beginning of an understanding of dynamic cellular patterns that have precise functions.     

Modelling molecular structures

This paper gives a guided tour of the teaching programme on molecular modelling offered at the University of Manchester Institute of Science and Technology (UMIST) for first and second-year students. The article describes theory (mini-lectures and self-study projects), and problems that students are expected to tackle at the computer.     

Modelling of biofilms

A mixed-culture biofilm (MCB) model is available which describes the progression of biofilm thickness and the spatial distribution and development in time of dissolved and paniculate components in the biofilm. The MCB model is able to predict the physico-chemical conditions at the interface between the biofilm and the solid surface, on which the biofilm grows, as a function of the conditions in the bulk fluid, the microbial composition of the biofilm, and the transport and transformation processes which take place in the biofilm. The mass balance equations of the MCB model are generally valid and can be applied to almost any microbial system if its kinetics and stoichiometry can be provided. AQUASIM is a new computer program for the identification and simulation of aquatic systems. The program solves the equations of the MCB model. It has a window-type user interface and includes routines for simulation, sensitivity analysis, automatic parameter estimation and data fitting. The MCB model has been developed and is primarily used in the field of waste water treatment. However, under certain conditions and with some additional simplifications this model can also be used for the investigation of biofouling and biocorrosion problems. The possibilities and limitations of the application of the MCB model and of AQUASIM to this type of problem are briefly discussed.     

Modelling biological rhythms

With our growing awareness of the complexity underlying biological phenomena, our need for computational models becomes increasingly apparent. Due to their properties, biological clocks have always lent themselves to computational modelling. Their capacity to oscillate without dampening - even when deprived of all rhythmic environmental information - required the hypothesis of an endogenous oscillator. The notion of a 'clock' provided a conceptual model of this system well before the dynamics of circadian oscillators were probed by computational modelling. With growing insight into the molecular basis of circadian rhythmicity, computational models became more concrete and quantitative. Here, we review the history of modelling circadian oscillators and establish a taxonomy of the modelling world to put the large body of circadian modelling literature into context. Finally, we assess the predictive power of circadian modelling and its success in creating new hypotheses.     

Modelling cartilage mechanobiology

The growth, maintenance and ossification of cartilage are fundamental to skeletal development and are regulated throughout life by the mechanical cues that are imposed by physical activities. Finite element computer analyses have been used to study the role of local tissue mechanics on endochondral ossification patterns, skeletal morphology and articular cartilage thickness distributions. Using single-phase continuum material representations of cartilage, the results have indicated that local intermittent hydrostatic pressure promotes cartilage maintenance. Cyclic tensile strains (or shear), however, promote cartilage growth and ossification. Because single-phase material models cannot capture fluid exudation in articular cartilage, poroelastic (or biphasic) solid/fluid models are often implemented to study joint mechanics. In the middle and deep layers of articular cartilage where poroelastic analyses predict little fluid exudation, the cartilage phenotype is maintained by cyclic fluid pressure (consistent with the single-phase theory). In superficial articular layers the chondrocytes are exposed to tangential tensile strain in addition to the high fluid pressure. Furthermore, there is fluid exudation and matrix consolidation, leading to cell 'flattening'. As a result, the superficial layer assumes an altered, more fibrous phenotype. These computer model predictions of cartilage mechanobiology are consistent with results of in vitro cell and tissue and molecular biology experiments.     

Modelling bacterial speciation

A central problem in understanding bacterial speciation is how clusters of closely related strains emerge and persist in the face of recombination. We use a neutral Fisher-Wright model in which genotypes, defined by the alleles at 140 house-keeping loci, change in each generation by mutation or recombination, and examine conditions in which an initially uniform population gives rise to resolved clusters. Where recombination occurs at equal frequency between all members of the population, we observe a transition between clonal structure and sexual structure as the rate of recombination increases. In the clonal situation, clearly resolved clusters are regularly formed, break up or go extinct. In the sexual situation, the formation of distinct clusters is prevented by the cohesive force of recombination. Where the rate of recombination is a declining log-linear function of the genetic distance between the donor and recipient strain, distinct clusters emerge even with high rates of recombination. These clusters arise in the absence of selection, and have many of the properties of species, with high recombination rates and thus sexual cohesion within clusters and low rates between clusters. Distance-scaled recombination can thus lead to a population splitting into distinct genotypic clusters, a process that mimics sympatric speciation. However, empirical estimates of the relationship between sequence divergence and recombination rate indicate that the decline in recombination is an insufficiently steep function of genetic distance to generate species in nature under neutral drift, and thus that other mechanisms should be invoked to explain speciation in the presence of recombination.     

Modelling Mediterranean pasture dynamies

Pasture vegetation in an open woodland of Quercus rotundifolia subjected to periodic ploughing was sampled in spring during 8 consecutive years. The frequency of herbaceous species was recorded in a total of 69 permanent plots located on 5 adjacent sites with similar lithology, slope and orientation but differing in age since previous ploughing.Vegetation dynamics expressed as trajectories of permanent plots in a non-metric multidimensional scaling space has been modelled in terms of evironmental variables. By fitting a generalized linear model, the dynamics are shown to be related to years since last ploughing, geographical location of plots, total annual rainfall and November rainfall. Meteorological patterns of the sampling period are also described.Abbreviations GLM = Generalized Linear Model - NMDS = Non-metric Multi-Dimensional Scaling - UPGMA = Unweighted Pair-Group Method using Arithmetic Averages