An approach to multiscale modelling with graph grammars

Background and Aims

Functional–structural plant models (FSPMs) simulate biological processes at different spatial scales. Methods exist for multiscale data representation and modification, but the advantages of using multiple scales in the dynamic aspects of FSPMs remain unclear. Results from multiscale models in various other areas of science that share fundamental modelling issues with FSPMs suggest that potential advantages do exist, and this study therefore aims to introduce an approach to multiscale modelling in FSPMs.

Methods

A three-part graph data structure and grammar is revisited, and presented with a conceptual framework for multiscale modelling. The framework is used for identifying roles, categorizing and describing scale-to-scale interactions, thus allowing alternative approaches to model development as opposed to correlation-based modelling at a single scale. Reverse information flow (from macro- to micro-scale) is catered for in the framework. The methods are implemented within the programming language XL.

Key Results

Three example models are implemented using the proposed multiscale graph model and framework. The first illustrates the fundamental usage of the graph data structure and grammar, the second uses probabilistic modelling for organs at the fine scale in order to derive crown growth, and the third combines multiscale plant topology with ozone trends and metabolic network simulations in order to model juvenile beech stands under exposure to a toxic trace gas.

Conclusions

The graph data structure supports data representation and grammar operations at multiple scales. The results demonstrate that multiscale modelling is a viable method in FSPM and an alternative to correlation-based modelling. Advantages and disadvantages of multiscale modelling are illustrated by comparisons with single-scale implementations, leading to motivations for further research in sensitivity analysis and run-time efficiency for these models.     

Stress and strain localization in stretched collagenous tissues via a multiscale modelling approach

Mechanobiology of cells in soft collagenous tissues is highly affected by both tissue response at the macroscale and stress/strain localization mechanisms due to features at lower scales. In this paper, the macroscale mechanical behaviour of soft collagenous tissues is modelled by a three-level multiscale approach, based on a multi-step homogenisation technique from nanoscale up to the macroscale. Nanoscale effects, related to both intermolecular cross-links and collagen mechanics, are accounted for, together with geometric nonlinearities at the microscale. Moreover, an effective submodelling procedure is conceived in order to evaluate the local stress and strain fields at the microscale, which is around and within cells. Numerical results, obtained by using an incremental finite element formulation and addressing stretched tendinous tissues, prove consistency and accuracy of the model at both macroscale and microscale, confirming also the effectiveness of the multiscale modelling concept for successfully analysing physiopathological processes in biological tissues.     

A multiscale hybrid approach for vasculogenesis and related potential blocking therapies

Solid tumors must recruit and form new blood vessels for maintenance, growth and detachments of metastases. Discovering drugs that block malignant angiogenesis is thus an important approach in cancer treatment and has given rise to multiple in vitro and in silico models. The present hybrid individual cell-based model incorporates some underlying biochemical events relating more closely the classical Cellular Potts Model (CPM) parameters to subcellular mechanisms and to the activation of specific signaling pathways. The model spans the three fundamental biological levels: at the extracellular level a continuous model describes secretion, diffusion, uptake and decay of the autocrine VEGF; at the cellular level, an extended lattice CPM, based on a system energy reduction, reproduces cell dynamics such as migration, adhesion and chemotaxis; at the subcellular level, a set of reaction-diffusion equations describes a simplified VEGF-induced calcium-dependent intracellular pathway. The results agree with the known interplay between calcium signals and VEGF dynamics and with their role in malignant vasculogenesis. Moreover, the analysis of the link between the microscopic subcellular dynamics and the macroscopic cell behaviors confirms the efficiency of some pharmacological interventions that are currently in use and, more interestingly, proposes some new therapeutic approaches, that are counter-intuitive but potentially effective.     

A model for glucose-induced wave propagation in pancreatic islets of Langerhans

A reaction-diffusion type model is constructed, describing the spatio-temporal dynamics of the basic intracellular variables assumed to be involved in the initiation of the insulin secretion process by beta -cells in the pancreatic islets of Langerhans. The model includes equations for the electric membrane potential of the cells, with respective kinetics for ionic currents, for concentrations of both free and stored intracellular calcium, and for the intra- and extracellular concentrations of glucose. An empirical expression connecting the equation for the intracellular glucose concentration to the electrical equation is introduced. The model reproduces the events observed in experiments in vitro upon external glucose application to the islets of Langerhans, such as usual bursting oscillations of the membrane potential and corresponding oscillations of the intracellular calcium concentration. It also allows simulation of electric wave propagation through the islet, initiated by the spatial gradient of glucose concentration within the islet. The gradient emerges due to glucose diffusing into the islets from the external medium, being high at the edges. The latter results show that glucose diffusion presents a means for wave initiation in the islets, which supports our previous assumption (Aslanidi et al., 2001).     

Evaluating an interpolation approach for modelling spatial variability in pest development

Air temperatures estimated by partial thin plate spline interpolation, or from the 'nearest station' (Voronoi polygon method), were used to model the phenology of three pests of horticultural crops throughout England and Wales. Temperatures for a particularly hot (1976) and a particularly cold (1986) year were interpolated to a grid resolution of 1 km. Estimates were made of the timing of spring emergence (Cecidophyopsis ribis (Westwood)), the maximum number of generations completed during the summer (Plutella xylostella (Linnaeus)) and the numbers of days when mating was possible (Merodon equestris (Fabricius)). The relative accuracy of the two temperature estimation methods was compared using jack-knife cross-validation. For C. ribis and P. xylostella, modelling with interpolated temperature input data was more accurate than using data from the 'nearest station'. Of the three phenology models used, the one that relied on an activity threshold (M. equestris) was the most sensitive to both types of input data. Spatial variability in the activity of M. equestrisadults was investigated in the two main areas (south-west peninsula and Lincolnshire) where its host crop (Narcissus) is grown. Modelling at cruder scales (up to 25*25 km) masked local variation, but the degree to which this was important varied from region to region and over time, as did the geography of the variability itself. The results indicate that interpolated data, computed to a resolution of 1 km using the UK synoptic network, have the potential for wider use within agricultural decision support systems for horticultural crops.     

Mathematical model of activation wave propagation in an isolated cardiomyocyte

In order to describe spontaneous wave-like contractions of a single isolated cardiomyocyte a mathematical model is proposed, which relates this phenomenon to propagation of calcium ion concentration wave along the cell. Free diffusion of Ca2+ ions as well as their reversible binding to regulatory proteins in contractile apparatus, Ca2+ accumulation in sarcoplasmic reticulum, and Ca-induced Ca2+ release are included in the governing equations. The model agrees with some observations. It predicts also some effects which may by a subject of future experimental research.     

A trait-based approach for modelling microbial litter decomposition

Trait-based models are an emerging tool in ecology with the potential to link community dynamics, environmental responses and ecosystem processes. These models represent complex communities by defining taxa with trait combinations derived from prior distributions that may be constrained by trade-offs. Herein I develop a model that links microbial community composition with physiological and enzymatic traits to predict litter decomposition rates. This approach allows for trade-offs among traits that represent alternative microbial strategies for resource acquisition. The model predicts that optimal strategies depend on the level of enzyme production in the whole community, which determines resource availability and decomposition rates. There is also evidence for facilitation and competition among microbial taxa that co-occur on decomposing litter. These interactions vary with community investment in extracellular enzyme production and the magnitude of trade-offs affecting enzyme biochemical traits. The model accounted for 69% of the variation in decomposition rates of 15 Hawaiian litter types and up to 26% of the variation in enzyme activities. By explicitly representing diversity, trait-based models can predict ecosystem processes based on functional trait distributions in a community. The model developed herein illustrates that traits influencing microbial enzyme production are some of the key controls on litter decomposition rates.     

Nonlinear pulse wave reflection at an arterial stenosis

A simple model is presented to analyze the effect of stenoses of different severities in a long elastic tube or artery on the pressure and flow-rate wave forms incident upon them. Wave propagation in the undisturbed tube is taken to be linear; nonlinearity arises from the quadratic dependence of stenosis pressure drop on flow rate. Before the model can be applied in practice, important physiological questions must be answered; e.g.: (a) Can the incident wave form and mean proximal pressure be regarded as given input? (b) is the mean flow rate given, or does the peripheral resistance remain constant? Results are given on the assumption that the answer to (a) is yes. The principal conclusion is that the input impedance spectrum of a stenosed artery depends strongly on the incident wave form, as well as on the severity of the stenosis and on the distance from it at which measurements are made. There is good qualitative agreement with the results of experiments and of other models.     

Stress wave propagation in bone

The traveling wave characteristics for a single compressive pulse were studied in fresh and embalmed human long bones. The stress wave was generated by the longitudinal impact of a steel ball on one end of a bone and was monitored by bonded strain gages. The dynamic properties, namely velocity, attenuation coefficient and dispersion were correlated with the mineral density, porosity, and cross-sectional area of the specimens. Statistically significant relationships were found between several of these parameters. These stress wave propagation characteristics are important for our understanding of the dynamic loading of bone and they may also provide a basis for the development of noninvasive techniques for studies of diseased or fractured bone.     

A hybrid representation approach for modelling complex dynamic bioprocesses

This paper considers the use of hybrid models to represent the dynamic behaviour of biotechnological processes. Each hybrid model consists of a set of non linear differential equations and a neural model. The set of differential equations attempts to describe as much as possible the phenomenology of the process whereas neural networks model predict some key parameters that are an essential part of the phenomenological model. The neural model is obtained indirectly, that is, using the prediction errors of one or more state variables to adjust its weights instead of successive presentations of input-output data of the neural network. This approach allows to use actual measurements to derive a suitable neural model that not only represents the variation of some key parameters but it is also able to partly include dynamic behaviour unaccounted for by the phenomenological model. The approach is described in detail using three test cases: (1) the fermentation of glucose to gluconic acid by the micro-organism Pseudomonas ovalis, (2) the growth of filamentous fungi in a solid state fermenter, and (3) the propagation of filamentous fungi growing on a 2-D solid substrate. Results for the three applications clearly demon- strate that using a hybrid model is a viable alternative for modelling complex biotechnological bioprocesses.     

A novel numerical modelling approach for keratoplasty eye procedure

Biomechanics and Modeling in Mechanobiology - Objective of the work is to investigate stress and deformation that conrneal tissue and donor graft undergo during endothelial keratoplasty. In order...     

Mathematical modelling of flow through an irregular arterial stenosis.

A mathematical model of flow through an irregular arterial stenosis is developed. The model is two-dimensional and axi-symmetric with the stenosis outline obtained from a three-dimensional casting of a mildly stenosed artery. Agreement between modelled and experimental pressure drops (obtained from an axi-symmetric machined stenosis with the same profile) is excellent. Results are also obtained for a smooth stenosis model, similar to that used for most mathematical modelling studies. This model overestimates the pressure drop across the stenosis, as well as the wall shear stress and separation Reynolds number. Also, the smooth model predicts one instead of three recirculation zones present in the irregular model. The original stenosis is modified to increase the severity from 48 and 87% areal occlusion, while maintaining the same general shape. This has the effect of increasing the pressure drop by an order of magnitude and decreasing the number of recirculation zones to one, with a lower separation Reynolds number.     

A multiscale computational model of arterial growth and remodeling including Notch signaling

Blood vessels grow and remodel in response to mechanical stimuli. Many computational models capture this process phenomenologically, by assuming stress homeostasis, but this approach cannot unravel the underlying cellular mechanisms. Mechano-sensitive Notch signaling is well-known to be key in vascular development and homeostasis. Here, we present a multiscale framework coupling a constrained mixture model, capturing the mechanics and turnover of arterial constituents, to a cell–cell signaling model, describing Notch signaling dynamics among vascular smooth muscle cells (SMCs) as influenced by mechanical stimuli. Tissue turnover was regulated by both Notch activity, informed by in vitro data, and a phenomenological contribution, accounting for mechanisms other than Notch. This novel framework predicted changes in wall thickness and arterial composition in response to hypertension similar to previous in vivo data. The simulations suggested that Notch contributes to arterial growth in hypertension mainly by promoting SMC proliferation, while other mechanisms are needed to fully capture remodeling. The results also indicated that interventions to Notch, such as external Jagged ligands, can alter both the geometry and composition of hypertensive vessels, especially in the short term. Overall, our model enables a deeper analysis of the role of Notch and Notch interventions in arterial growth and remodeling and could be adopted to investigate therapeutic strategies and optimize vascular regeneration protocols.

     

A segment alignment approach to protein comparison

MOTIVATION: Local structure segments (LSSs) are small structural units shared by unrelated proteins. They are extensively used in protein structure comparison, and predicted LSSs (PLSSs) are used very successfully in ab initio folding simulations. However, predicted or real LSSs are rarely exploited by protein sequence comparison programs that are based on position-by-position alignments. RESULTS: We developed a SEgment Alignment algorithm (SEA) to compare proteins described as a collection of predicted local structure segments (PLSSs), which is equivalent to an unweighted graph (network). Any specific structure, real or predicted corresponds to a specific path in this network. SEA then uses a network matching approach to find two most similar paths in networks representing two proteins. SEA explores the uncertainty and diversity of predicted local structure information to search for a globally optimal solution. It simultaneously solves two related problems: the alignment of two proteins and the local structure prediction for each of them. On a benchmark of protein pairs with low sequence similarity, we show that application of the SEA algorithm improves alignment quality as compared to FFAS profile-profile alignment, and in some cases SEA alignments can match the structural alignments, a feat previously impossible for any sequence based alignment methods.     

A multiscale approach to the elastic moduli of biomembrane networks

We develop equilibrium fluctuation formulae for the isothermal elastic moduli of discrete biomembrane models at different scales. We account for the coupling of large stretching and bending strains of triangulated network models endowed with harmonic and dihedral angle potentials, on the basis of the discrete-continuum approach presented in Schmidt and Fraternali (J Mech Phys Solids 60:172-180, 2012). We test the proposed equilibrium fluctuation formulae with reference to a coarse-grained molecular dynamics model of the red blood cell (RBC) membrane (Marcelli et al. in Biophys J 89:2473-2480, 2005; Hale et al. in Soft Matter 5:3603-3606, 2009), employing a local maximum-entropy regularization of the fluctuating configurations (Fraternali et al. in J Comput Phys 231:528-540, 2012). We obtain information about membrane stiffening/softening due to stretching, curvature, and microscopic undulations of the RBC model. We detect local dependence of the elastic moduli over the RBC membrane, establishing comparisons between the present theory and different approaches available in the literature.     

An object-oriented modelling framework for the arterial wall

An object-oriented modelling framework for the arterial wall is presented. The novelty of the framework is the possibility to generate customisable artery models, taking advantage of imaging technology. In our knowledge, this is the first object-oriented modelling framework for the arterial wall. Existing models do not allow close structural mapping with arterial microstructure as in the object-oriented framework. In the implemented model, passive behaviour of the arterial wall was considered and the tunica adventitia was the objective system. As verification, a model of an arterial segment was generated. In order to simulate its deformation, a matrix structural mechanics simulator was implemented. Two simulations were conducted, one for an axial loading test and other for a pressure–volume test. Each simulation began with a sensitivity analysis in order to determinate the best parameter combination and to compare the results with analogue controls. In both cases, the simulated results closely reproduced qualitatively and quantitatively the analogue control plots.     

The velocity of the arterial pulse wave: a viscous-fluid shock wave in an elastic tube

Background

The arterial pulse is a viscous-fluid shock wave that is initiated by blood ejected from the heart. This wave travels away from the heart at a speed termed the pulse wave velocity (PWV). The PWV increases during the course of a number of diseases, and this increase is often attributed to arterial stiffness. As the pulse wave approaches a point in an artery, the pressure rises as does the pressure gradient. This pressure gradient increases the rate of blood flow ahead of the wave. The rate of blood flow ahead of the wave decreases with distance because the pressure gradient also decreases with distance ahead of the wave. Consequently, the amount of blood per unit length in a segment of an artery increases ahead of the wave, and this increase stretches the wall of the artery. As a result, the tension in the wall increases, and this results in an increase in the pressure of blood in the artery.

Methods

An expression for the PWV is derived from an equation describing the flow-pressure coupling (FPC) for a pulse wave in an incompressible, viscous fluid in an elastic tube. The initial increase in force of the fluid in the tube is described by an increasing exponential function of time. The relationship between force gradient and fluid flow is approximated by an expression known to hold for a rigid tube.

Results

For large arteries, the PWV derived by this method agrees with the Korteweg-Moens equation for the PWV in a non-viscous fluid. For small arteries, the PWV is approximately proportional to the Korteweg-Moens velocity divided by the radius of the artery. The PWV in small arteries is also predicted to increase when the specific rate of increase in pressure as a function of time decreases. This rate decreases with increasing myocardial ischemia, suggesting an explanation for the observation that an increase in the PWV is a predictor of future myocardial infarction. The derivation of the equation for the PWV that has been used for more than fifty years is analyzed and shown to yield predictions that do not appear to be correct.

Conclusion

Contrary to the theory used for more than fifty years to predict the PWV, it speeds up as arteries become smaller and smaller. Furthermore, an increase in the PWV in some cases may be due to decreasing force of myocardial contraction rather than arterial stiffness.