Mechanics of axon growth and damage: A systematic review of computational models.

Normal axon development depends on the action of mechanical forces both generated within the cytoskeleton and outside the cell, but forces of large magnitude or rate cause damage instead. Computational models aid scientists in studying the role of mechanical forces in axon growth and damage. These studies use simulations to evaluate how different sources of force generation within the cytoskeleton interact with each other to regulate axon elongation and retraction. Furthermore, mathematical models can help optimize externally applied tension to promote axon growth without causing damage. Finally, scientists also use simulations of axon damage to investigate how forces are distributed among different components of the axon and how the tissue surrounding an axon influences its susceptibility to injury. In this review, we discuss how computational studies complement experimental studies in the areas of axon growth, regeneration, and damage.

Minimal Design of the Elephant Trunk as an Active Filament.

One of the key problems in active materials is the control of shape through actuation. A fascinating example of such control is the elephant trunk, a long, muscular, and extremely dexterous organ with multiple vital functions. The elephant trunk is an object of fascination for biologists, physicists, and children alike. Its versatility relies on the intricate interplay of multiple unique physical mechanisms and biological design principles. Here, we explore these principles using the theory of active filaments and build, theoretically, computationally, and experimentally, a minimal model that explains and accomplishes some of the spectacular features of the elephant trunk.

Image-based axon model highlights heterogeneity in initiation of damage.

Head injury simulations predict the occurrence of traumatic brain injury by placing a threshold on the calculated strains for axon tracts within the brain. However, a current roadblock to accurate injury prediction is the selection of an appropriate axon damage threshold. While several computational studies have used models of the axon cytoskeleton to investigate damage initiation, these models all employ an idealized, homogeneous axonal geometry. This homogeneous geometry with regularly spaced microtubules, evenly distributed throughout the model, overestimates axon strength because, in reality, the axon cytoskeleton is heterogeneous. In the heterogeneous cytoskeleton, the weakest cross section determines the initiation of failure, but these weak spots are not present in a homogeneous model. Addressing one source of heterogeneity in the axon cytoskeleton, we present a new semiautomated image analysis pipeline for using serial-section transmission electron micrographs to reconstruct the microtubule geometry of an axon. The image analysis procedure locates microtubules within the images, traces them throughout the image stack, and reconstructs the microtubule structure as a finite element mesh. We demonstrate the image analysis approach using a C. elegans touch receptor neuron due to the availability of high-quality serial-section transmission electron micrograph data sets. The results of the analysis highlight the heterogeneity of the microtubule structure in the spatial variation of both microtubule number and length. Simulations comparing this image-based geometry with homogeneous geometries show that structural heterogeneity in the image-based model creates significant spatial variation in deformation. The homogeneous geometries, on the other hand, deform more uniformly. Since no single homogeneous model can replicate the mechanical behavior of the image-based model, our results argue that heterogeneity in axon microtubule geometry should be considered in determining accurate axon failure thresholds.

Protein-protein interactions in neurodegenerative diseases: A conspiracy theory.

Neurodegenerative diseases such as Alzheimer's or Parkinson's are associated with the prion-like propagation and aggregation of toxic proteins. A long standing hypothesis that amyloid-beta drives Alzheimer's disease has proven the subject of contemporary controversy; leading to new research in both the role of tau protein and its interaction with amyloid-beta. Conversely, recent work in mathematical modeling has demonstrated the relevance of nonlinear reaction-diffusion type equations to capture essential features of the disease. Such approaches have been further simplified, to network-based models, and offer researchers a powerful set of computationally tractable tools with which to investigate neurodegenerative disease dynamics. Here, we propose a novel, coupled network-based model for a two-protein system that includes an enzymatic interaction term alongside a simple model of aggregate transneuronal damage. We apply this theoretical model to test the possible interactions between tau proteins and amyloid-beta and study the resulting coupled behavior between toxic protein clearance and proteopathic phenomenology. Our analysis reveals ways in which amyloid-beta and tau proteins may conspire with each other to enhance the nucleation and propagation of different diseases, thus shedding new light on the importance of protein clearance and protein interaction mechanisms in prion-like models of neurodegenerative disease.

Classifying Drugs by their Arrhythmogenic Risk Using Machine Learning.

All medications have adverse effects. Among the most serious of these are cardiac arrhythmias. Current paradigms for drug safety evaluation are costly, lengthy, conservative, and impede efficient drug development. Here, we combine multiscale experiment and simulation, high-performance computing, and machine learning to create a risk estimator to stratify new and existing drugs according to their proarrhythmic potential. We capitalize on recent developments in machine learning and integrate information across 10 orders of magnitude in space and time to provide a holistic picture of the effects of drugs, either individually or in combination with other drugs. We show, both experimentally and computationally, that drug-induced arrhythmias are dominated by the interplay between two currents with opposing effects: the rapid delayed rectifier potassium current and the L-type calcium current. Using Gaussian process classification, we create a classifier that stratifies drugs into safe and arrhythmic domains for any combinations of these two currents. We demonstrate that our classifier correctly identifies the risk categories of 22 common drugs exclusively on the basis of their concentrations at 50% current block. Our new risk assessment tool explains under which conditions blocking the L-type calcium current can delay or even entirely suppress arrhythmogenic events. Using machine learning in drug safety evaluation can provide a more accurate and comprehensive mechanistic assessment of the proarrhythmic potential of new drugs. Our study paves the way toward establishing science-based criteria to accelerate drug development, design safer drugs, and reduce heart rhythm disorders.

Neuronal Oscillations on Evolving Networks: Dynamics, Damage, Degradation, Decline, Dementia, and Death.

Neurodegenerative diseases, such as Alzheimer's or Parkinson's disease, show characteristic degradation of structural brain networks. This degradation eventually leads to changes in the network dynamics and degradation of cognitive functions. Here, we model the progression in terms of coupled physical processes: The accumulation of toxic proteins, given by a nonlinear reaction-diffusion transport process, yields an evolving brain connectome characterized by weighted edges on which a neuronal-mass model evolves. The progression of the brain functions can be tested by simulating the resting-state activity on the evolving brain network. We show that while the evolution of edge weights plays a minor role in the overall progression of the disease, dynamic biomarkers predict a transition over a period of 10 years associated with strong cognitive decline.

Spatially-extended nucleation-aggregation-fragmentation models for the dynamics of prion-like neurodegenerative protein-spreading in the brain and its connectome.

The prion-like hypothesis of neurodegenerative diseases states that the accumulation of misfolded proteins in the form of aggregates is responsible for tissue death and its associated neurodegenerative pathology and cognitive decline. Some disease-specific misfolded proteins can interact with healthy proteins to form long chains that are transported through the brain along axonal pathways. Since aggregates of different sizes have different transport properties and toxicity, it is important to follow independently their evolution in space and time. Here, we model the spreading and propagation of aggregates of misfolded proteins in the brain using the general Smoluchowski theory of nucleation, aggregation, and fragmentation. The transport processes considered here are either anisotropic diffusion along axonal bundles or discrete Laplacian transport along a network. In particular, we model the spreading and aggregation of both amyloid-ß and τ molecules in the brain connectome. We show that these two models lead to different size distributions and different propagation along the network. A detailed analysis of these two models also reveals the existence of four different stages with different dynamics and invasive properties.

Visualizing the invisible: The effect of asymptomatic transmission on the outbreak dynamics of COVID-19.

Understanding the outbreak dynamics of the COVID-19 pandemic has important implications for successful containment and mitigation strategies. Recent studies suggest that the population prevalence of SARS-CoV-2 antibodies, a proxy for the number of asymptomatic cases, could be an order of magnitude larger than expected from the number of reported symptomatic cases. Knowing the precise prevalence and contagiousness of asymptomatic transmission is critical to estimate the overall dimension and pandemic potential of COVID-19. However, at this stage, the effect of the asymptomatic population, its size, and its outbreak dynamics remain largely unknown. Here we use reported symptomatic case data in conjunction with antibody seroprevalence studies, a mathematical epidemiology model, and a Bayesian framework to infer the epidemiological characteristics of COVID-19. Our model computes, in real time, the time-varying contact rate of the outbreak, and projects the temporal evolution and credible intervals of the effective reproduction number and the symptomatic, asymptomatic, and recovered populations. Our study quantifies the sensitivity of the outbreak dynamics of COVID-19 to three parameters: the effective reproduction number, the ratio between the symptomatic and asymptomatic populations, and the infectious periods of both groups. For nine distinct locations, our model estimates the fraction of the population that has been infected and recovered by Jun 15, 2020 to 24.15% (95% CI: 20.48%-28.14%) for Heinsberg (NRW, Germany), 2.40% (95% CI: 2.09%-2.76%) for Ada County (ID, USA), 46.19% (95% CI: 45.81%-46.60%) for New York City (NY, USA), 11.26% (95% CI: 7.21%-16.03%) for Santa Clara County (CA, USA), 3.09% (95% CI: 2.27%-4.03%) for Denmark, 12.35% (95% CI: 10.03%-15.18%) for Geneva Canton (Switzerland), 5.24% (95% CI: 4.84%-5.70%) for the Netherlands, 1.53% (95% CI: 0.76%-2.62%) for Rio Grande do Sul (Brazil), and 5.32% (95% CI: 4.77%-5.93%) for Belgium. Our method traces the initial outbreak date in Santa Clara County back to January 20, 2020 (95% CI: December 29, 2019-February 13, 2020). Our results could significantly change our understanding and management of the COVID-19 pandemic: A large asymptomatic population will make isolation, containment, and tracing of individual cases challenging. Instead, managing community transmission through increasing population awareness, promoting physical distancing, and encouraging behavioral changes could become more relevant.

Passive Stretch Induces Structural and Functional Maturation of Engineered Heart Muscle as Predicted by Computational Modeling.

The ability to differentiate human pluripotent stem cells (hPSCs) into cardiomyocytes (CMs) makes them an attractive source for repairing injured myocardium, disease modeling, and drug testing. Although current differentiation protocols yield hPSC-CMs to >90% efficiency, hPSC-CMs exhibit immature characteristics. With the goal of overcoming this limitation, we tested the effects of varying passive stretch on engineered heart muscle (EHM) structural and functional maturation, guided by computational modeling. Human embryonic stem cells (hESCs, H7 line) or human induced pluripotent stem cells (IMR-90 line) were differentiated to hPSC-derived cardiomyocytes (hPSC-CMs) in vitro using a small molecule based protocol. hPSC-CMs were characterized by troponin+ flow cytometry as well as electrophysiological measurements. Afterwards, 1.2 × 106 hPSC-CMs were mixed with 0.4 × 106 human fibroblasts (IMR-90 line) (3:1 ratio) and type-I collagen. The blend was cast into custom-made 12-mm long polydimethylsiloxane reservoirs to vary nominal passive stretch of EHMs to 5, 7, or 9 mm. EHM characteristics were monitored for up to 50 days, with EHMs having a passive stretch of 7 mm giving the most consistent formation. Based on our initial macroscopic observations of EHM formation, we created a computational model that predicts the stress distribution throughout EHMs, which is a function of cellular composition, cellular ratio, and geometry. Based on this predictive modeling, we show cell alignment by immunohistochemistry and coordinated calcium waves by calcium imaging. Furthermore, coordinated calcium waves and mechanical contractions were apparent throughout entire EHMs. The stiffness and active forces of hPSC-derived EHMs are comparable with rat neonatal cardiomyocyte-derived EHMs. Three-dimensional EHMs display increased expression of mature cardiomyocyte genes including sarcomeric protein troponin-T, calcium and potassium ion channels, ß-adrenergic receptors, and t-tubule protein caveolin-3. Passive stretch affects the structural and functional maturation of EHMs. Based on our predictive computational modeling, we show how to optimize cell alignment and calcium dynamics within EHMs. These findings provide a basis for the rational design of EHMs, which enables future scale-up productions for clinical use in cardiovascular tissue engineering. Stem Cells 2018;36:265-277.

Understanding the mechanical link between oriented cell division and cerebellar morphogenesis.

The cerebellum is a tightly folded structure located at the back of the head. Unlike the folds of the cerebrum, the folds of the cerebellum are aligned such that the external surface appears to be covered in parallel grooves. Experiments have shown that anchoring center initiation drives cerebellar foliation. However, the mechanism guiding the location of these anchoring centers, and subsequently cerebellar morphology, remains poorly understood. In particular, there is no definitive mechanistic explanation for the preferential emergence of parallel folds instead of an irregular folding pattern like in the cerebral cortex. Here we use mechanical modeling on the cellular and tissue scales to show that the oriented granule cell division observed in the experimental setting leads to the characteristic parallel folding pattern of the cerebellum. Specifically, we propose an agent-based model of cell clones, a strategy for propagating information from our in silico cell clones to the tissue scale, and an analytical solution backed by numerical results to understand how differential growth between the cerebellar layers drives geometric instability in three dimensional space on the tissue scale. This proposed mechanical model provides further insight into the process of anchoring center initiation and establishes a framework for future multiscale mechanical analysis of developing organs.

Revisiting the wrinkling of elastic bilayers I: linear analysis.

Wrinkling is a universal instability occurring in a wide variety of engineering and biological materials. It has been studied extensively for many different systems but a full description is still lacking. Here, we provide a systematic analysis of the wrinkling of a thin hyperelastic film over a substrate in plane strain using stream functions. For comparison, we assume that wrinkling is generated either by the isotropic growth of the film or by the lateral compression of the entire system. We perform an exhaustive linear analysis of the wrinkling problem for all stiffness ratios and under a variety of additional boundary and material effects. Namely, we consider the effect of added pressure, surface tension, an upper substrate and fibres. We obtain analytical estimates of the instability in the two asymptotic regimes of long and short wavelengths. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.

Machine learning in drug development: Characterizing the effect of 30 drugs on the QT interval using Gaussian process regression, sensitivity analysis, and uncertainty quantification.

Prolonged QT intervals are a major risk factor for ventricular arrhythmias and a leading cause of sudden cardiac death. Various drugs are known to trigger QT interval prolongation and increase the proarrhythmic potential. Yet, how precisely the action of drugs on the cellular level translates into QT interval prolongation on the whole organ level remains insufficiently understood. Here we use machine learning techniques to systematically characterize the effect of 30 common drugs on the QT interval. We combine information from high fidelity three-dimensional human heart simulations with low fidelity one-dimensional cable simulations to build a surrogate model for the QT interval using multi-fidelity Gaussian process regression. Once trained and cross-validated, we apply our surrogate model to perform sensitivity analysis and uncertainty quantification. Our sensitivity analysis suggests that compounds that block the rapid delayed rectifier potassium current I Kr have the greatest prolonging effect of the QT interval, and that blocking the L-type calcium current I CaL and late sodium current I NaL shortens the QT interval. Our uncertainty quantification allows us to propagate the experimental variability from individual block-concentration measurements into the QT interval and reveals that QT interval uncertainty is mainly driven by the variability in I Kr block. In a final validation study, we demonstrate an excellent agreement between our predicted QT interval changes and the changes observed in a randomized clinical trial for the drugs dofetilide, quinidine, ranolazine, and verapamil. We anticipate that both the machine learning methods and the results of this study will have great potential in the efficient development of safer drugs.

Microtubule Polymerization and Cross-Link Dynamics Explain Axonal Stiffness and Damage.

Axonal damage is a critical indicator for traumatic effects of physical impact to the brain. However, the precise mechanisms of axonal damage are still unclear. Here, we establish a mechanistic and highly dynamic model of the axon to explore the evolution of damage in response to physical forces. Our axon model consists of a bundle of dynamically polymerizing and depolymerizing microtubules connected by dynamically detaching and reattaching cross-links. Although the probability of cross-link attachment depends exclusively on thermal fluctuations, the probability of detachment increases in the presence of physical forces. We systematically probe the landscape of axonal stretch and stretch rate and characterize the overall axonal force, stiffness, and damage as a direct result of the interplay between microtubule and cross-link dynamics. Our simulations reveal that slow loading is dominated by cross-link dynamics, a net reduction of cross-links, and a gradual accumulation of damage, whereas fast loading is dominated by cross-link deformations, a rapid increase in stretch, and an immediate risk of rupture. Microtubule polymerization and depolymerization decrease the overall axonal stiffness, but do not affect the evolution of damage at timescales relevant to axonal failure. Our study explains different failure mechanisms in the axon as emergent properties of microtubule polymerization, cross-link dynamics, and physical forces. We anticipate that our model will provide insight into causal relations by which molecular mechanisms determine the timeline and severity of axon damage after a physical impact to the brain.

Modeling the Axon as an Active Partner with the Growth Cone in Axonal Elongation.

Forces generated by the growth cone are vital for the proper development of the axon and thus brain function. Although recent experiments show that forces are generated along the axon, it is unknown whether the axon plays a direct role in controlling growth cone advance. Here, we use analytic and finite element modeling of microtubule dynamics and the activity of the molecular motors myosin and dynein to investigate mechanical force balance along the length of the axon and its effects on axonal outgrowth. Our modeling indicates that the paradoxical effects of stabilizing microtubules and the consequences of microtubule disassembly on axonal outgrowth can be explained by changes in the passive and active mechanical properties of axons. Our findings suggest that a full understanding of growth cone motility requires a consideration of the mechanical contributions of the axon. Our study not only has potential applications during neurodevelopment but might also help identify strategies to manipulate and promote axonal regrowth to treat neurodegeneration.

Multiphysics of Prionlike Diseases: Progression and Atrophy.

Many neurodegenerative diseases are related to the propagation and accumulation of toxic proteins throughout the brain. The lesions created by aggregates of these toxic proteins further lead to cell death and accelerated tissue atrophy. A striking feature of some of these diseases is their characteristic pattern and evolution, leading to well-codified disease stages visible to neuropathology and associated with various cognitive deficits and pathologies. Here, we simulate the anisotropic propagation and accumulation of toxic proteins in full brain geometry. We show that the same model with different initial seeding zones reproduces the characteristic evolution of different prionlike diseases. We also recover the expected evolution of the total toxic protein load. Finally, we couple our transport model to a mechanical atrophy model to obtain the typical degeneration patterns found in neurodegenerative diseases.

Symmetry Breaking in Wrinkling Patterns: Gyri Are Universally Thicker than Sulci.

Wrinkling instabilities appear in soft materials when a flat elastic layer on an elastic substrate is sufficiently stressed that it buckles with a wavy pattern to minimize the energy of the system. This instability is known to play an important role in engineering, but it also appears in many biological systems. In these systems, the stresses responsible for the wrinkling instability are often created through differential growth of the two layers. Beyond the instability, the upper and lower sides of the elastic layer are subject to different forces. This difference in forces leads to an interesting symmetry breaking whereby the thickness becomes larger at ridges than at valleys. Here we carry out an extensive analysis of this phenomenon by combining analytical, computational, and simple polymer experiments to show that symmetry breaking is a generic property of such systems. We apply our idea to the cortical folding of the brain for which it has been known for over a century that there is a thickness difference between gyri and sulci. An extensive analysis of hundreds of human brains reveals a systematic region-dependent thickness variation. Our results suggest that the evolving thickness patterns during brain development, similar to our polymer experiments, follow simple physics-based laws: Gyri are universally thicker than sulci.

Determining the Differential Effects of Stretch and Growth in Tissue-Expanded Skin: Combining Isogeometric Analysis and Continuum Mechanics in a Porcine Model.

BACKGROUND: The relative effects of skin growth and stretch during tissue expansion have not been studied. The authors use novel analytic techniques that allow calculation of these factors at any point of a skin patch. OBJECTIVE: The authors sought to determine how stretch and growth change with different expansion rates and to correlate these values with histologic and cellular changes in skin. MATERIALS AND METHODS: Two minipigs were implanted with a total of 5 tissue expanders under tattooed skin grids. One pig was expanded over 35 days and the second over 15 days. Isogeometric analysis allowed calculation of growth and stretch. Expanders with similar total deformation were compared between protocols. Regression analysis determined predictive effects of stretch and growth on histologic data from the second animal. RESULTS: Deformation was more attributable to stretch in rapid than in slow expansion (1.40 vs1.12, p < .001). Growth was higher in slow expansion than in rapid (1.52 vs 1.07, p < .001). Both growth and stretch predicted epidermal thickness, dermal thinning, and keratinocyte proliferation. Growth predicted vascularity. CONCLUSION: Isogeometric analysis allows determination of precise surface area changes for correlation to microscopic-level data. Using the model, the authors identified that skin deformation in rapid expansion is more attributable to stretch.

Dimensional, Geometrical, and Physical Constraints in Skull Growth.

After birth, the skull grows and remodels in close synchrony with the brain to allow for an increase in intracranial volume. Increase in skull area is provided primarily by bone accretion at the sutures. Additional remodeling, to allow for a change in curvatures, occurs by resorption on the inner surface of the bone plates and accretion on their outer surfaces. When a suture fuses too early, normal skull growth is disrupted, leading to a deformed final skull shape. The leading theory assumes that the main stimulus for skull growth is provided by mechanical stresses. Based on these ideas, we first discuss the dimensional, geometrical, and kinematic synchrony between brain, skull, and suture growth. Second, we present two mechanical models for skull growth that account for growth at the sutures and explain the various observed dysmorphologies. These models demonstrate the particular role of physical and geometrical constraints taking place in skull growth.

The importance of mechano-electrical feedback and inertia in cardiac electromechanics.

In the past years, a number cardiac electromechanics models have been developed to better understand the excitation-contraction behavior of the heart. However, there is no agreement on whether inertial forces play a role in this system. In this study, we assess the influence of mass in electromechanical simulations, using a fully coupled finite element model. We include the effect of mechano-electrical feedback via stretch activated currents. We compare five different models: electrophysiology, electromechanics, electromechanics with mechano-electrical feedback, electromechanics with mass, and electromechanics with mass and mechano-electrical feedback. We simulate normal conduction to study conduction velocity and spiral waves to study fibrillation. During normal conduction, mass in conjunction with mechano-electrical feedback increased the conduction velocity by 8.12% in comparison to the plain electrophysiology case. During the generation of a spiral wave, mass and mechano-electrical feedback generated secondary wavefronts, which were not present in any other model. These secondary wavefronts were initiated in tensile stretch regions that induced electrical currents. We expect that this study will help the research community to better understand the importance of mechanoelectrical feedback and inertia in cardiac electromechanics.

Segmental aortic stiffening contributes to experimental abdominal aortic aneurysm development.

BACKGROUND: Stiffening of the aortic wall is a phenomenon consistently observed in age and in abdominal aortic aneurysm (AAA). However, its role in AAA pathophysiology is largely undefined. METHODS AND RESULTS: Using an established murine elastase-induced AAA model, we demonstrate that segmental aortic stiffening precedes aneurysm growth. Finite-element analysis reveals that early stiffening of the aneurysm-prone aortic segment leads to axial (longitudinal) wall stress generated by cyclic (systolic) tethering of adjacent, more compliant wall segments. Interventional stiffening of AAA-adjacent aortic segments (via external application of surgical adhesive) significantly reduces aneurysm growth. These changes correlate with the reduced segmental stiffness of the AAA-prone aorta (attributable to equalized stiffness in adjacent segments), reduced axial wall stress, decreased production of reactive oxygen species, attenuated elastin breakdown, and decreased expression of inflammatory cytokines and macrophage infiltration, and attenuated apoptosis within the aortic wall, as well. Cyclic pressurization of segmentally stiffened aortic segments ex vivo increases the expression of genes related to inflammation and extracellular matrix remodeling. Finally, human ultrasound studies reveal that aging, a significant AAA risk factor, is accompanied by segmental infrarenal aortic stiffening. CONCLUSIONS: The present study introduces the novel concept of segmental aortic stiffening as an early pathomechanism generating aortic wall stress and triggering aneurysmal growth, thereby delineating potential underlying molecular mechanisms and therapeutic targets. In addition, monitoring segmental aortic stiffening may aid the identification of patients at risk for AAA.