Interactions Between the Immune System and Cancer: A Brief Review of Non-spatial Mathematical Models

We briefly review spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system. We begin with the simplest (single equation) models for tumor growth and proceed to consider greater immunological detail (and correspondingly more equations) in steps. This approach allows us to clarify the necessity for expanding the complexity of models in order to capture the biological mechanisms we wish to understand. We conclude by discussing some unsolved problems in the mathematical modeling of cancer-immune system interactions.     

Binding Interactions of Antagonists with 5‐Hydroxytryptamine3A Receptor Models

Abstract

Homology modeling was performed on the N‐terminal extracellular regions of human, mouse, and guinea pig 5‐hydroxytryptamine type 3A receptors (5‐HT₃R) based on the 24% sequence homology with and on the crystal structure of the snail acetylcholine binding protein (AChBP). Docking of 5‐HT₃ antagonists granisetron, tropisetron, ondansetron, dolasetron ('setrons), and (+)‐tubocurarine suggests an aromatic binding cleft behind a hydrophilic vestibule. Several intra‐ and interface interactions, H‐bonds, and salt bridges stabilize the pentameric structure and the binding cleft. The planar rings of antagonists are intercalated between aromatic side‐chains (W183‐Y234, Y143‐Y153). S227 donates H‐bonds to the carbonyl groups of 'setrons. The tertiary ammonium ions interact with E236, N128 or E129, and/or W90 (cation‐π interaction). This offers a molecular explanation of the pharmacophore models of 5‐HT₃R antagonists. Docking artifacts suggest some ambiguities in the binding loops A and C of the 5‐HT_3AR models. Lower potencies of (+)‐tubocurarine for human, and those of tropisetron for guinea pig 5‐HT_3ARs can be attributed to steric differences of I/S230 in the binding cleft and to distinct binding interactions with E229 and S227, respectively. Ligand binding interferes with crucial intra‐ and interface interactions along the binding cleft.     

On a Class of Deterministic Population Models with Stochastic Foundation

Generalising a site-based stochastic model due to Royama, Solé et al. and Sumpter et al., we investigate competition in a single species with discrete, non-overlapping generations. We show that the deterministic limit of the dynamics depends on a few easily interpretable parameters only. Further, we discuss qualitative properties and limit sets of the corresponding difference equations, and we relate these to modes of competition. Moreover, a detailed analysis of stochastic effects in some relevant scenarios indicates that the behaviour of the stochastic model is very sensitive to further details of the model.     

On the Modelling of Biological Patterns with Mechanochemical Models: Insights from Analysis and Computation

The diversity of biological form is generated by a relatively small number of underlying mechanisms. Consequently, mathematical and computational modelling can, and does, provide insight into how cellular level interactions ultimately give rise to higher level structure. Given cells respond to mechanical stimuli, it is therefore important to consider the effects of these responses within biological self-organisation models. Here, we consider the self-organisation properties of a mechanochemical model previously developed by three of the authors in Acta Biomater. 4, 613–621 (2008), which is capable of reproducing the behaviour of a population of cells cultured on an elastic substrate in response to a variety of stimuli. In particular, we examine the conditions under which stable spatial patterns can emerge with this model, focusing on the influence of mechanical stimuli and the interplay of non-local phenomena. To this end, we have performed a linear stability analysis and numerical simulations based on a mixed finite element formulation, which have allowed us to study the dynamical behaviour of the system in terms of the qualitative shape of the dispersion relation. We show that the consideration of mechanotaxis, namely changes in migration speeds and directions in response to mechanical stimuli alters the conditions for pattern formation in a singular manner. Furthermore without non-local effects, responses to mechanical stimuli are observed to result in dispersion relations with positive growth rates at arbitrarily large wavenumbers, in turn yielding heterogeneity at the cellular level in model predictions. This highlights the sensitivity and necessity of non-local effects in mechanically influenced biological pattern formation models and the ultimate failure of the continuum approximation in their absence.     

Global Stability for a Class of Virus Models with Cytotoxic T Lymphocyte Immune Response and Antigenic Variation

We study the global stability of a class of models for in-vivo virus dynamics that take into account the Cytotoxic T Lymphocyte immune response and display antigenic variation. This class includes a number of models that have been extensively used to model HIV dynamics. We show that models in this class are globally asymptotically stable, under mild hypothesis, by using appropriate Lyapunov functions. We also characterise the stable equilibrium points for the entire biologically relevant parameter range. As a by-product, we are able to determine what is the diversity of the persistent strains.     

Homogenization of Large-Scale Movement Models in Ecology

A difficulty in using diffusion models to predict large scale animal population dispersal is that individuals move differently based on local information (as opposed to gradients) in differing habitat types. This can be accommodated by using ecological diffusion. However, real environments are often spatially complex, limiting application of a direct approach. Homogenization for partial differential equations has long been applied to Fickian diffusion (in which average individual movement is organized along gradients of habitat and population density). We derive a homogenization procedure for ecological diffusion and apply it to a simple model for chronic wasting disease in mule deer. Homogenization allows us to determine the impact of small scale (10–100 m) habitat variability on large scale (10–100 km) movement. The procedure generates asymptotic equations for solutions on the large scale with parameters defined by small-scale variation. The simplicity of this homogenization procedure is striking when compared to the multi-dimensional homogenization procedure for Fickian diffusion,and the method will be equally straightforward for more complex models.     

Are Existing Global Scenarios Consistent with Ecological Feedbacks?

Scenarios can help planners and decision makers to think through uncertainties about the future and make decisions that are robust to a variety of possible outcomes. To develop useful scenarios we need to understand the main processes of relevance to the system of interest. Ecological processes, and the feedbacks that they can create between human actions and human well-being, are thought to be important for human societies. Current uncertainties over the long-term resilience of ecosystems and the substitutability of ecosystem goods and services can be translated into three alternative realities: ecosystems may be relatively brittle, relatively resilient, or largely irrelevant. Although these extremes are only rough characterizations of reality, they help us to focus our thinking about the possible outcomes of interactions between humans and the rest of the biosphere. Existing global scenarios can be categorized into a small number of families based on shared themes and assumptions about the future. Considering the internal consistency of four of the main scenario families in relation to the three alternative ecological realities suggests that all existing scenarios make strong, implicit assumptions about the resilience of ecosystems. After a detailed discussion of individual examples, we present a synthesis of the incorporation of ecology in existing scenarios. All current scenarios are inconsistent with at least one possible property of ecosystems and their likely interaction with society. The interrelationships between ecological reality, human views of ecosystems, and social responses to actual and perceived ecological change are complex. For the Millennium Ecosystem Assessment and future scenario exercises, we recommend that essential ecological assumptions should be made explicit to ensure that the details of each scenario are consistent with both the perceived and the actual degree of resilience of ecosystems.     

Deriving Macroscopic Myocardial Conductivities by Homogenization of Microscopic Models

We derive the values for the intracellular and extracellular conductivities needed for bidomain simulations of cardiac electrophysiology using homogenization of partial differential equations. In our model, cardiac myocytes are rectangular prisms and gap junctions appear in a distributed manner as flux boundary conditions for Laplace’s equation. Using directly measurable microproperties such as cellular dimensions and end-to-end and side-to-side gap junction coupling strengths, we inexpensively obtain effective conductivities close to those given by simulations with a detailed cyto-architecture (Stinstra et al. in Ann. Biomed. Eng. 33:1743–1751, 2005). This model provides a convenient framework for studying the effect on conductivities of aligned vs. brick-like arrangements of cells and the effect of different distributions of gap junctions along the myocyte membranes.     

State-dependent Lipid Interactions with the A2a Receptor Revealed by MD Simulations Using In Vivo-Mimetic Membranes

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Interactions of γ-L-glutamyltaurine with excitatory aminoacidergic neurotransmission

The in vitro effects of -L-glutamyltaurine on different stages of excitatory aminoacidergic neurotransmission were tested with -D-glutamyltaurine as reference. -L-Glutamyltaurine enhanced the K⁺-stimulated release of [³H]glutamate from cerebral cortical slices (25% at 0.1 mM) and slightly inhibited the uptake by crude brain synaptosomal preparations (about 10% at 1 mM). -L-Glutamyltaurine was also a weak displacer of glutamate and its agonists from their binding sites in brain synaptic membrane preparations, being, however, less selective to quisqualate (QA) sites than -D-glutamyltaurine. The basal influx of Ca²⁺ into cultured cerebellar granular cells was not affected by 1 mM -L-glutamyltaurine, but the glutamate- and its agonist-activated influx was significantly inhibited in low-Mg²⁺ (0.1 mM) and Mg²⁺-free media. The glutamate-evoked increase in free intracellular Ca²⁺ and the kainate-activated formation of cGMP in cerebellar slices were both markedly inhibited by 0.1 mM -L-giutamyltaurine. We propose that -L-glutamyltaurine may act as endogenous modulator in excitatory aminoacidergic neurotransmission.     

Anticooperative Nearest-Neighbor Interactions between Residues in Unfolded Peptides and Proteins

Growing evidence suggests that the conformational distributions of amino acid residues in unfolded peptides and proteins depend on the nature of the nearest neighbors. To explore whether the underlying interactions would lead to a breakdown of the isolated pair hypothesis of the classical random coil model, we further analyzed the conformational propensities that were recently obtained for the two guest residues (x,y) of GxyG tetrapeptides. We constructed a statistical thermodynamics model that allows for cooperative as well as for anticooperative interactions between adjacent residues adopting either a polyproline II or a β-strand conformation. Our analysis reveals that the nearest-neighbor interactions between most of the central residues in the investigated GxyG peptides are anticooperative. Interaction Gibbs energies are rather large at high temperatures (350 K), at which point many proteins undergo thermal unfolding. At room temperature, these interaction energies are less pronounced. We used the obtained interaction parameter in a Zimm-Bragg/Ising-type approach to calculate the temperature dependence of the ultraviolet circular dichroism (CD) of the MAX3 peptide, which is predominantly built by KV repeats. The agreement between simulation and experimental data was found to be satisfactory. Finally, we analyzed the temperature dependence of the CD and ³J(H^NH^α) parameters of the amyloid β_1–9 fragment. The results of this analysis and a more qualitative consideration of the temperature dependence of denatured proteins probed by CD spectroscopy further corroborate the dominance of anticooperative nearest-neighbor interactions. Generally, our results show that unfolded peptides—and most likely also proteins—exhibit some similarity with antiferromagnetic systems.     

Interactions of β-lactoglobulin with serotonin and arachidonyl serotonin

β‐Lactoglobulin (β‐LG) is a lipocalin, which is the major whey protein of cow's milk and the milk of other mammals. However, it is absent from human milk. The biological function of β‐LG is not clear, but its potential role in carrying fatty acids through the digestive tract has been suggested. β‐LG has been found in complexes with lipids such as butyric and oleic acids and has a high affinity for a wide variety of compounds. Serotonin (5‐hydroxytryptamine, 5‐HT), an important compound found in animals and plants, has various functions, including the regulation of mood, appetite, sleep, muscle contraction, and some cognitive functions such as memory and learning. In this study, the interaction of serotonin and one of its derivatives, arachidonyl serotonin (AA‐5HT), with β‐LG was investigated using circular dichroism (CD) and fluorescence intensity measurements. These two ligands interact with β‐LG forming equimolar complexes. The binding constant for the serotonin/β‐LG interaction is between 10⁵ and 10⁶ M⁻¹, whereas for the AA‐5HT/β‐LG complex it is between 10⁴ and 10⁵ M⁻¹ as determined by measurements of either protein or ligand fluorescence. The observed binding affinities were higher in hydroethanolic media (25% EtOH). The interactions between serotonin/β‐LG and AA‐5HT/β‐LG may compete with self‐association (micellization) of both the ligand and the protein. According to far‐ and near‐UV CD results, these ligands have no apparent influence on β‐LG secondary structure, however they partially destabilize its tertiary structure. Their binding by β‐LG may be one of the peripheral mechanisms of the regulation of the content of serotonin and its derivatives in the bowel of milk‐fed animals. © 2011 Wiley Periodicals, Inc. Biopolymers 95: 871–880, 2011.     

E.?coli Outer Membrane and Interactions with OmpLA

The outer membrane of Gram-negative bacteria is a unique asymmetric lipid bilayer composed of phospholipids (PLs) in the inner leaflet and lipopolysaccharides (LPSs) in the outer leaflet. Its function as a selective barrier is crucial for the survival of bacteria in many distinct environments, and it also renders Gram-negative bacteria more resistant to antibiotics than their Gram-positive counterparts. Here, we report the structural properties of a model of the Escherichia coli outer membrane and its interaction with outer membrane phospholipase A (OmpLA) utilizing molecular dynamics simulations. Our results reveal that given the lipid composition used here, the hydrophobic thickness of the outer membrane is ∼3 Å thinner than the corresponding PL bilayer, mainly because of the thinner LPS leaflet. Further thinning in the vicinity of OmpLA is observed due to hydrophobic matching. The particular shape of the OmpLA barrel induces various interactions between LPS and PL leaflets, resulting in asymmetric thinning around the protein. The interaction between OmpLA extracellular loops and LPS (headgroups and core oligosaccharides) stabilizes the loop conformation with reduced dynamics, which leads to secondary structure variation and loop displacement compared to that in a DLPC bilayer. In addition, we demonstrate that the LPS/PL ratios in asymmetric bilayers can be reliably estimated by the per-lipid surface area of each lipid type, and there is no statistical difference in the overall membrane structure for the outer membranes with one more or less LPS in the outer leaflet, although individual lipid properties vary slightly.