Quantitative analysis of tumour spheroid structure.

Tumour spheroids are common in vitro experimental models of avascular tumour growth. Compared with traditional two-dimensional culture, tumour spheroids more closely mimic the avascular tumour microenvironment where spatial differences in nutrient availability strongly influence growth. We show that spheroids initiated using significantly different numbers of cells grow to similar limiting sizes, suggesting that avascular tumours have a limiting structure; in agreement with untested predictions of classical mathematical models of tumour spheroids. We develop a novel mathematical and statistical framework to study the structure of tumour spheroids seeded from cells transduced with fluorescent cell cycle indicators, enabling us to discriminate between arrested and cycling cells and identify an arrested region. Our analysis shows that transient spheroid structure is independent of initial spheroid size, and the limiting structure can be independent of seeding density. Standard experimental protocols compare spheroid size as a function of time; however, our analysis suggests that comparing spheroid structure as a function of overall size produces results that are relatively insensitive to variability in spheroid size. Our experimental observations are made using two melanoma cell lines, but our modelling framework applies across a wide range of spheroid culture conditions and cell lines.

Designing combination therapies using multiple optimal controls.

Strategic management of populations of interacting biological species routinely requires interventions combining multiple treatments or therapies. This is important in key research areas such as ecology, epidemiology, wound healing and oncology. Despite the well developed theory and techniques for determining single optimal controls, there is limited practical guidance supporting implementation of combination therapies. In this work we use optimal control theory to calculate optimal strategies for applying combination therapies to a model of acute myeloid leukaemia. We present a versatile framework to systematically explore the trade-offs that arise in designing combination therapy protocols using optimal control. We consider various combinations of continuous and bang-bang (discrete) controls, and we investigate how the control dynamics interact and respond to changes in the weighting and form of the pay-off characterising optimality. We demonstrate that the optimal controls respond non-linearly to treatment strength and control parameters, due to the interactions between species. We discuss challenges in appropriately characterising optimality in a multiple control setting and provide practical guidance for applying multiple optimal controls. Code used in this work to implement multiple optimal controls is available on GitHub.

Mathematical Models of Cancer Cell Plasticity.

Quantitative modelling is increasingly important in cancer research, helping to integrate myriad diverse experimental data into coherent pictures of the disease and able to discriminate between competing hypotheses or suggest specific experimental lines of enquiry and new approaches to therapy. Here, we review a diverse set of mathematical models of cancer cell plasticity (a process in which, through genetic and epigenetic changes, cancer cells survive in hostile environments and migrate to more favourable environments, respectively), tumour growth, and invasion. Quantitative models can help to elucidate the complex biological mechanisms of cancer cell plasticity. In this review, we discuss models of plasticity, tumour progression, and metastasis under three broadly conceived mathematical modelling techniques: discrete, continuum, and hybrid, each with advantages and disadvantages. An emerging theme from the predictions of many of these models is that cell escape from the tumour microenvironment (TME) is encouraged by a combination of physiological stress locally (e.g., hypoxia), external stresses (e.g., the presence of immune cells), and interactions with the extracellular matrix. We also discuss the value of mathematical modelling for understanding cancer more generally.

Optimal control of acute myeloid leukaemia.

Acute myeloid leukaemia (AML) is a blood cancer affecting haematopoietic stem cells. AML is routinely treated with chemotherapy, and so it is of great interest to develop optimal chemotherapy treatment strategies. In this work, we incorporate an immune response into a stem cell model of AML, since we find that previous models lacking an immune response are inappropriate for deriving optimal control strategies. Using optimal control theory, we produce continuous controls and bang-bang controls, corresponding to a range of objectives and parameter choices. Through example calculations, we provide a practical approach to applying optimal control using Pontryagin's Maximum Principle. In particular, we describe and explore factors that have a profound influence on numerical convergence. We find that the convergence behaviour is sensitive to the method of control updating, the nature of the control, and to the relative weighting of terms in the objective function. All codes we use to implement optimal control are made available.

Adsorption of Collagen-like Peptides onto Gold Nanosurfaces.

The molecular behavior of proteins in the presence of inorganic surfaces is of fundamental biological significance. Examples include extracellular matrix proteins interacting with gold nanoparticles and metallic implant biomaterials, such as titanium and stainless steels. Uncharged inorganic surfaces that interact strongly with the solution phase (hydrophilic surfaces) have been commonly used in disease treatments. A deep understanding of the molecular behavior of body proteins in the presence of hydrophilic surfaces is important in terms of clinical applications. However, the adsorption mechanism of proteins onto hydrophilic surfaces remains not fully understood. Here, comprehensive molecular dynamics simulations are carried out to study the molecular response of a human collagen molecule segment (CMS) to the presence of a planar gold surface (AuNS) in explicit solvent, aiming to unravel the adsorption mechanism of proteins onto hydrophilic surfaces. The results demonstrate that in the presence of AuNS, the CMS first biasedly diffuses toward AuNS, followed by anchoring to the gold surface, and finally adsorbs stepwise onto AuNS, where the protein adjusts its structure to maximize the interaction with AuNS. We conclude that adsorption of proteins onto hydrophilic surfaces adheres to three steps, namely, biased diffusion, anchoring, and stepwise adsorption accompanied by structural adaptation. The obtained adsorption mechanism provides insights into the development of inorganic surfaces for biomedical and therapeutic applications.

Interaction of gold nanosurfaces/nanoparticles with collagen-like peptides.

Nanotechnology has quickly emerged as a promising research field with potential effects in disease treatments. For example, gold nanoparticles (AuNPs) have been extensively used in diagnostics and therapeutics. When administrated into human tissues, AuNPs first encounter extracellular matrix (ECM) molecules. Amongst all the ECM components, collagen is the main tension-resisting constituent, whose biofunctional and mechanical properties are strongly dependent on its hierarchical structure. Therefore, an in-depth understanding of the structural response of collagen to the presence of gold nanosurfaces (AuNS) and AuNPs is crucial in terms of clinical applications of AuNPs. However, detailed understanding of the molecular-level and atomic-level interaction between AuNS/AuNPs and collagen in the ECM is elusive. In this study, comprehensive molecular dynamics (MD) simulations have been performed to investigate the molecular behaviour of a collagen molecule segment (CMS) in the presence of AuNS/AuNPs in explicit water, aiming to explore the interaction of AuNS/AuNPs with collagen triple helices at the molecular and atomic levels. The results show that the CMS forms a rapid association with AuNS/AuNPs and undergoes a severe unfolding upon adsorption on AuNS/AuNPs, indicating an unfolding propensity of gold surfaces. We conclude that collagen triple helices unfold readily on AuNS and bare AuNPs, due to the interaction of gold surfaces with the protein backbone. The revealed clear unfolding nature and the unravelled atomic-level unfolding mechanism of collagen triple helices onto AuNPs contribute to the development of AuNPs for biomedical and therapeutic applications, and the design of gold-binding proteins.

From ionic to cellular variability in human atrial myocytes: an integrative computational and experimental study.

Variability refers to differences in physiological function between individuals, which may translate into different disease susceptibility and treatment efficacy. Experiments in human cardiomyocytes face wide variability and restricted tissue access; under these conditions, computational models are a useful complementary tool. We conducted a computational and experimental investigation in cardiomyocytes isolated from samples of the right atrial appendage of patients undergoing cardiac surgery to evaluate the impact of variability in action potentials (APs) and subcellular ionic densities on Ca2+ transient dynamics. Results showed that 1) variability in APs and ionic densities is large, even within an apparently homogenous patient cohort, and translates into ±100% variation in ionic conductances; 2) experimentally calibrated populations of models with wide variations in ionic densities yield APs overlapping with those obtained experimentally, even if AP characteristics of the original generic model differed significantly from experimental APs; 3) model calibration with AP recordings restricts the variability in ionic densities affecting upstroke and resting potential, but redundancy in repolarization currents admits substantial variability in ionic densities; and 4) model populations constrained with experimental APs and ionic densities exhibit three Ca2+ transient phenotypes, differing in intracellular Ca2+ handling and Na+/Ca2+ membrane extrusion. These findings advance our understanding of the impact of variability in human atrial electrophysiology. NEW & NOTEWORTHY Variability in human atrial electrophysiology is investigated by integrating for the first time cellular-level and ion channel recordings in computational electrophysiological models. Ion channel calibration restricts current densities but not cellular phenotypic variability. Reduced Na+/Ca2+ exchanger is identified as a primary mechanism underlying diastolic Ca2+ fluctuations in human atrial myocytes.

Environmental factors in breast cancer invasion: a mathematical modelling review.

This review presents a brief overview of breast cancer, focussing on its heterogeneity and the role of mathematical modelling and simulation in teasing apart the underlying biophysical processes. Following a brief overview of the main known pathophysiological features of ductal carcinoma, attention is paid to differential equation-based models (both deterministic and stochastic), agent-based modelling, multi-scale modelling, lattice-based models and image-driven modelling. A number of vignettes are presented where these modelling approaches have elucidated novel aspects of breast cancer dynamics, and we conclude by offering some perspectives on the role mathematical modelling can play in understanding breast cancer development, invasion and treatment therapies.

Rabbit-specific computational modelling of ventricular cell electrophysiology: Using populations of models to explore variability in the response to ischemia.

Computational modelling, combined with experimental investigations, is a powerful method for investigating complex cardiac electrophysiological behaviour. The use of rabbit-specific models, due to the similarities of cardiac electrophysiology in this species with human, is especially prevalent. In this paper, we first briefly review rabbit-specific computational modelling of ventricular cell electrophysiology, multi-cellular simulations including cellular heterogeneity, and acute ischemia. This mini-review is followed by an original computational investigation of variability in the electrophysiological response of two experimentally-calibrated populations of rabbit-specific ventricular myocyte action potential models to acute ischemia. We performed a systematic exploration of the response of the model populations to varying degrees of ischemia and individual ischemic parameters, to investigate their individual and combined effects on action potential duration and refractoriness. This revealed complex interactions between model population variability and ischemic factors, which combined to enhance variability during ischemia. This represents an important step towards an improved understanding of the role that physiological variability may play in electrophysiological alterations during acute ischemia.

Stochastic linear multistep methods for the simulation of chemical kinetics.

In this paper, we introduce the Stochastic Adams-Bashforth (SAB) and Stochastic Adams-Moulton (SAM) methods as an extension of the τ-leaping framework to past information. Using the Θ-trapezoidal τ-leap method of weak order two as a starting procedure, we show that the k-step SAB method with k ≥ 3 is order three in the mean and correlation, while a predictor-corrector implementation of the SAM method is weak order three in the mean but only order one in the correlation. These convergence results have been derived analytically for linear problems and successfully tested numerically for both linear and non-linear systems. A series of additional examples have been implemented in order to demonstrate the efficacy of this approach.

Stochastic dynamics of interacting haematopoietic stem cell niche lineages.

Since we still know very little about stem cells in their natural environment, it is useful to explore their dynamics through modelling and simulation, as well as experimentally. Most models of stem cell systems are based on deterministic differential equations that ignore the natural heterogeneity of stem cell populations. This is not appropriate at the level of individual cells and niches, when randomness is more likely to affect dynamics. In this paper, we introduce a fast stochastic method for simulating a metapopulation of stem cell niche lineages, that is, many sub-populations that together form a heterogeneous metapopulation, over time. By selecting the common limiting timestep, our method ensures that the entire metapopulation is simulated synchronously. This is important, as it allows us to introduce interactions between separate niche lineages, which would otherwise be impossible. We expand our method to enable the coupling of many lineages into niche groups, where differentiated cells are pooled within each niche group. Using this method, we explore the dynamics of the haematopoietic system from a demand control system perspective. We find that coupling together niche lineages allows the organism to regulate blood cell numbers as closely as possible to the homeostatic optimum. Furthermore, coupled lineages respond better than uncoupled ones to random perturbations, here the loss of some myeloid cells. This could imply that it is advantageous for an organism to connect together its niche lineages into groups. Our results suggest that a potential fruitful empirical direction will be to understand how stem cell descendants communicate with the niche and how cancer may arise as a result of a failure of such communication.

Population of computational rabbit-specific ventricular action potential models for investigating sources of variability in cellular repolarisation.

Variability is observed at all levels of cardiac electrophysiology. Yet, the underlying causes and importance of this variability are generally unknown, and difficult to investigate with current experimental techniques. The aim of the present study was to generate populations of computational ventricular action potential models that reproduce experimentally observed intercellular variability of repolarisation (represented by action potential duration) and to identify its potential causes. A systematic exploration of the effects of simultaneously varying the magnitude of six transmembrane current conductances (transient outward, rapid and slow delayed rectifier K(+), inward rectifying K(+), L-type Ca(2+), and Na(+)/K(+) pump currents) in two rabbit-specific ventricular action potential models (Shannon et al. and Mahajan et al.) at multiple cycle lengths (400, 600, 1,000 ms) was performed. This was accomplished with distributed computing software specialised for multi-dimensional parameter sweeps and grid execution. An initial population of 15,625 parameter sets was generated for both models at each cycle length. Action potential durations of these populations were compared to experimentally derived ranges for rabbit ventricular myocytes. 1,352 parameter sets for the Shannon model and 779 parameter sets for the Mahajan model yielded action potential duration within the experimental range, demonstrating that a wide array of ionic conductance values can be used to simulate a physiological rabbit ventricular action potential. Furthermore, by using clutter-based dimension reordering, a technique that allows visualisation of multi-dimensional spaces in two dimensions, the interaction of current conductances and their relative importance to the ventricular action potential at different cycle lengths were revealed. Overall, this work represents an important step towards a better understanding of the role that variability in current conductances may play in experimentally observed intercellular variability of rabbit ventricular action potential repolarisation.

Determination of somatic and cancer stem cell self-renewing symmetric division rate using sphere assays.

Representing a renewable source for cell replacement, neural stem cells have received substantial attention in recent years. The neurosphere assay represents a method to detect the presence of neural stem cells, however owing to a deficiency of specific and definitive markers to identify them, their quantification and the rate they expand is still indefinite. Here we propose a mathematical interpretation of the neurosphere assay allowing actual measurement of neural stem cell symmetric division frequency. The algorithm of the modeling demonstrates a direct correlation between the overall cell fold expansion over time measured in the sphere assay and the rate stem cells expand via symmetric division. The model offers a methodology to evaluate specifically the effect of diseases and treatments on neural stem cell activity and function. Not only providing new insights in the evaluation of the kinetic features of neural stem cells, our modeling further contemplates cancer biology as cancer stem-like cells have been suggested to maintain tumor growth as somatic stem cells maintain tissue homeostasis. Indeed, tumor stem cell's resistance to therapy makes these cells a necessary target for effective treatment. The neurosphere assay mathematical model presented here allows the assessment of the rate malignant stem-like cells expand via symmetric division and the evaluation of the effects of therapeutics on the self-renewal and proliferative activity of this clinically relevant population that drive tumor growth and recurrence.

Comparison of continuous and discrete stochastic ion channel models.

The stochastic behaviour of ion channels can be described by a discrete model or by an approximate continuous approach. While the discrete approach is exact, it is also less computationally efficient, and so the continuous model is often the method of choice since it allows for incorporation into a multiscale environment. However, in recent years the accuracy of the stochastic continuous approach for calculating statistics of certain quantities in the Hodgkin-Huxley model has come into question. In this paper, we show that by correct formulation of the continuous model, the first two moments in the number of open sodium and potassium channels in the Hodgkin-Huxley model, calculated under voltage clamp conditions using the continuous approach are in good agreement with those obtained from the discrete model.

Fast stochastic simulation of biochemical reaction systems by alternative formulations of the chemical Langevin equation.

The Chemical Langevin Equation (CLE), which is a stochastic differential equation driven by a multidimensional Wiener process, acts as a bridge between the discrete stochastic simulation algorithm and the deterministic reaction rate equation when simulating (bio)chemical kinetics. The CLE model is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. The contribution of this work is that we observe and explore that the CLE is not a single equation, but a parametric family of equations, all of which give the same finite-dimensional distribution of the variables. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation, which is just the rank of the stoichiometric matrix. On the practical side, we show that in the case where there are m(1) pairs of reversible reactions and m(2) irreversible reactions there is another, simple formulation of the CLE with only m(1) + m(2) Wiener processes, whereas the standard approach uses 2(m(1) + m(2)). We demonstrate that there are considerable computational savings when using this latter formulation. Such transformations of the CLE do not cause a loss of accuracy and are therefore distinct from model reduction techniques. We illustrate our findings by considering alternative formulations of the CLE for a human ether a-go-go related gene ion channel model and the Goldbeter-Koshland switch.

A Bayesian network model of proteins' association with promyelocytic leukemia (PML) nuclear bodies.

The modularity that nuclear organization brings has the potential to explain the function of aggregates of proteins and RNA. Promyelocytic leukemia nuclear bodies are implicated in important regulatory processes. To understand the complement of proteins associated with these intra-nuclear bodies, we construct a Bayesian network model that integrates sequence and protein-protein interaction data. The model predicts association with promyelocytic leukemia nuclear bodies accurately when interaction data is available. At a false positive rate of 10%, the true positive rate is almost 50%, indicated by an independent nuclear proteome reference set. The model provides strong support for further expanding the protein complement with several important regulators and a richer functional repertoire. Using special support vector machine (SVM)-nodes (equipped with string kernels), the Bayesian network is also able to produce predictions on the basis of sequence only, with an accuracy superior to that of baseline models. Supplementary Material is available online at www.liebertonline.com.

Effects of intrinsic and extrinsic noise can accelerate juxtacrine pattern formation.

Epithelial pattern formation is an important phenomenon that, for example, has roles in embryogenesis, development and wound-healing. The ligand Epithelial Growth Factor (EGF) and its receptor EGF-R, constitute a system that forms lateral induction patterns by juxtacrine signalling-binding of membrane-bound ligands to receptors on neighbouring cells. Owen et al. developed a generic ordinary differential equation model of juxtacrine lateral induction that exhibits stable patterning under some conditions. The model predicts relatively slow pattern formation. We examine here the effects of both intrinsic and extrinsic cellular noise arising from the stochastic treatment of this model, and show that this noise could have an accelerating effect on the patterning process.

Predicting disulfide connectivity from protein sequence using multiple sequence feature vectors and secondary structure.

MOTIVATION: Disulfide bonds are primary covalent crosslinks between two cysteine residues in proteins that play critical roles in stabilizing the protein structures and are commonly found in extracy-toplasmatic or secreted proteins. In protein folding prediction, the localization of disulfide bonds can greatly reduce the search in conformational space. Therefore, there is a great need to develop computational methods capable of accurately predicting disulfide connectivity patterns in proteins that could have potentially important applications. RESULTS: We have developed a novel method to predict disulfide connectivity patterns from protein primary sequence, using a support vector regression (SVR) approach based on multiple sequence feature vectors and predicted secondary structure by the PSIPRED program. The results indicate that our method could achieve a prediction accuracy of 74.4% and 77.9%, respectively, when averaged on proteins with two to five disulfide bridges using 4-fold cross-validation, measured on the protein and cysteine pair on a well-defined non-homologous dataset. We assessed the effects of different sequence encoding schemes on the prediction performance of disulfide connectivity. It has been shown that the sequence encoding scheme based on multiple sequence feature vectors coupled with predicted secondary structure can significantly improve the prediction accuracy, thus enabling our method to outperform most of other currently available predictors. Our work provides a complementary approach to the current algorithms that should be useful in computationally assigning disulfide connectivity patterns and helps in the annotation of protein sequences generated by large-scale whole-genome projects. AVAILABILITY: The prediction web server and Supplementary Material are accessible at http://foo.maths.uq.edu.au/~huber/disulfide

Exploring synonymous codon usage preferences of disulfide-bonded and non-disulfide bonded cysteines in the E. coli genome.

High-quality data about protein structures and their gene sequences are essential to the understanding of the relationship between protein folding and protein coding sequences. Firstly we constructed the EcoPDB database, which is a high-quality database of Escherichia coli genes and their corresponding PDB structures. Based on EcoPDB, we presented a novel approach based on information theory to investigate the correlation between cysteine synonymous codon usages and local amino acids flanking cysteines, the correlation between cysteine synonymous codon usages and synonymous codon usages of local amino acids flanking cysteines, as well as the correlation between cysteine synonymous codon usages and the disulfide bonding states of cysteines in the E. coli genome. The results indicate that the nearest neighboring residues and their synonymous codons of the C-terminus have the greatest influence on the usages of the synonymous codons of cysteines and the usage of the synonymous codons has a specific correlation with the disulfide bond formation of cysteines in proteins. The correlations may result from the regulation mechanism of protein structures at gene sequence level and reflect the biological function restriction that cysteines pair to form disulfide bonds. The results may also be helpful in identifying residues that are important for synonymous codon selection of cysteines to introduce disulfide bridges in protein engineering and molecular biology. The approach presented in this paper can also be utilized as a complementary computational method and be applicable to analyse the synonymous codon usages in other model organisms.

Protein contact prediction using patterns of correlation.

We describe a new method for using neural networks to predict residue contact pairs in a protein. The main inputs to the neural network are a set of 25 measures of correlated mutation between all pairs of residues in two "windows" of size 5 centered on the residues of interest. While the individual pair-wise correlations are a relatively weak predictor of contact, by training the network on windows of correlation the accuracy of prediction is significantly improved. The neural network is trained on a set of 100 proteins and then tested on a disjoint set of 1033 proteins of known structure. An average predictive accuracy of 21.7% is obtained taking the best L/2 predictions for each protein, where L is the sequence length. Taking the best L/10 predictions gives an average accuracy of 30.7%. The predictor is also tested on a set of 59 proteins from the CASP5 experiment. The accuracy is found to be relatively consistent across different sequence lengths, but to vary widely according to the secondary structure. Predictive accuracy is also found to improve by using multiple sequence alignments containing many sequences to calculate the correlations.