Theoretical study of Beloussov's hyper-restoration hypothesis for mechanical regulation of morphogenesis

Socket Array Irregularities and Wing Membrane Distortions at the Eyespot Foci of Butterfly Wings Suggest Mechanical Signals for Color Pattern Determination

Eyespot foci on butterfly wings function as organizers of eyespot color patterns during development. Despite their importance, focal structures have not been examined in detail. Here, we microscopically examined scales, sockets, and the wing membrane in the butterfly eyespot foci of both expanded and unexpanded wings using the Blue Pansy butterfly Junonia orithya. Images from a high-resolution light microscope revealed that, although not always, eyespot foci had scales with disordered planar polarity. Scanning electron microscopy (SEM) images after scale removal revealed that the sockets were irregularly positioned and that the wing membrane was physically distorted as if the focal site were mechanically squeezed from the surroundings. Focal areas without eyespots also had socket array irregularities, but less frequently and less severely. Physical damage in the background area induced ectopic patterns with socket array irregularities and wing membrane distortions, similar to natural eyespot foci. These results suggest that either the process of determining an eyespot focus or the function of an eyespot organizer may be associated with wing-wide mechanics that physically disrupt socket cells, scale cells, and the wing membrane, supporting the physical distortion hypothesis of the induction model for color pattern determination in butterfly wings.

Macroevolution, differentiation trees, and the growth of coding systems

An open process of evolution of multicellular organisms is based on the rearrangement and growth of the program of differentiation that underlies biological morphogenesis. The maintenance of the final (adult) stable non-equilibrium state (stasis) of a developmental system determines the direction of the evolutionary process. This state is achieved via the sequence of differentiation events representable as differentiation trees. A special type of morphogenetic code, acting as a metacode governing gene expression, may include electromechanical signals appearing as differentiation waves. The excessive energy due to the incorporation of mitochondria in eukaryotic cells resulted not only in more active metabolism but also in establishing the differentiation code for interconnecting cells and forming tissues, which fueled the evolutionary process. The "invention" of "continuing differentiation" distinguishes multicellular eukaryotes from other organisms. The Janus-faced control, involving both top-down control by differentiation waves and bottom-up control via the mechanical consequences of cell differentiations, underlies the process of morphogenesis and results in the achievement of functional stable final states. Duplications of branches of the differentiation tree may be the basis for continuing differentiation and macroevolution, analogous to gene duplication permitting divergence of genes. Metamorphoses, if they are proven to be fusions of disparate species, may be classified according to the topology of fusions of two differentiation trees. In the process of unfolding of morphogenetic structures, microevolution can be defined as changes of the differentiation tree that preserve topology of the tree, while macroevolution represents any change that alters the topology of the differentiation tree.

Imaging Approaches and the Quantitative Analysis of Heart Development

Heart morphogenesis is a complex and dynamic process that has captivated researchers for almost a century. This process involves three main stages, during which the heart undergoes growth and folding on itself to form its common chambered shape. However, imaging heart development presents significant challenges due to the rapid and dynamic changes in heart morphology. Researchers have used different model organisms and developed various imaging techniques to obtain high-resolution images of heart development. Advanced imaging techniques have allowed the integration of multiscale live imaging approaches with genetic labeling, enabling the quantitative analysis of cardiac morphogenesis. Here, we discuss the various imaging techniques used to obtain high-resolution images of whole-heart development. We also review the mathematical approaches used to quantify cardiac morphogenesis from 3D and 3D+time images and to model its dynamics at the tissue and cellular levels.

Growth anisotropy of the extracellular matrix shapes a developing organ

Final organ size and shape result from volume expansion by growth and shape changes by contractility. Complex morphologies can also arise from differences in growth rate between tissues. We address here how differential growth guides the morphogenesis of the growing Drosophila wing imaginal disc. We report that 3D morphology results from elastic deformation due to differential growth anisotropy between the epithelial cell layer and its enveloping extracellular matrix (ECM). While the tissue layer grows in plane, growth of the bottom ECM occurs in 3D and is reduced in magnitude, thereby causing geometric frustration and tissue bending. The elasticity, growth anisotropy and morphogenesis of the organ are fully captured by a mechanical bilayer model. Moreover, differential expression of the Matrix metalloproteinase MMP2 controls growth anisotropy of the ECM envelope. This study shows that the ECM is a controllable mechanical constraint whose intrinsic growth anisotropy directs tissue morphogenesis in a developing organ.

Soft-Tissue Material Properties and Mechanogenetics during Cardiovascular Development

During embryonic development, changes in the cardiovascular microstructure and material properties are essential for an integrated biomechanical understanding. This knowledge also enables realistic predictive computational tools, specifically targeting the formation of congenital heart defects. Material characterization of cardiovascular embryonic tissue at consequent embryonic stages is critical to understand growth, remodeling, and hemodynamic functions. Two biomechanical loading modes, which are wall shear stress and blood pressure, are associated with distinct molecular pathways and govern vascular morphology through microstructural remodeling. Dynamic embryonic tissues have complex signaling networks integrated with mechanical factors such as stress, strain, and stiffness. While the multiscale interplay between the mechanical loading modes and microstructural changes has been studied in animal models, mechanical characterization of early embryonic cardiovascular tissue is challenging due to the miniature sample sizes and active/passive vascular components. Accordingly, this comparative review focuses on the embryonic material characterization of developing cardiovascular systems and attempts to classify it for different species and embryonic timepoints. Key cardiovascular components including the great vessels, ventricles, heart valves, and the umbilical cord arteries are covered. A state-of-the-art review of experimental techniques for embryonic material characterization is provided along with the two novel methods developed to measure the residual and von Mises stress distributions in avian embryonic vessels noninvasively, for the first time in the literature. As attempted in this review, the compilation of embryonic mechanical properties will also contribute to our understanding of the mature cardiovascular system and possibly lead to new microstructural and genetic interventions to correct abnormal development.

Inducing Biomechanical Heterogeneity in Brain Tumor Modeling by MR Elastography: Effects on Tumor Growth, Vascular Density and Delivery of Therapeutics

Simple Summary

Biomechanical forces aggravate brain tumor progression. In this study, magnetic resonance elastography (MRE) is employed to extract tissue biomechanical properties from five glioblastoma patients and a healthy subject, and data are incorporated in a mathematical model that simulates tumor growth. Mathematical modeling enables further understanding of glioblastoma development and allows patient-specific predictions for tumor vascularity and delivery of drugs. Incorporating MRE data results in a more realistic intratumoral distribution of mechanical stress and anisotropic tumor growth and a better description of subsequent events that are closely related to the development of stresses, including heterogeneity of the tumor vasculature and intrapatient variations in tumor perfusion and delivery of drugs.

Abstract

The purpose of this study is to develop a methodology that incorporates a more accurate assessment of tissue mechanical properties compared to current mathematical modeling by use of biomechanical data from magnetic resonance elastography. The elastography data were derived from five glioblastoma patients and a healthy subject and used in a model that simulates tumor growth, vascular changes due to mechanical stresses and delivery of therapeutic agents. The model investigates the effect of tumor-specific biomechanical properties on tumor anisotropic growth, vascular density heterogeneity and chemotherapy delivery. The results showed that including elastography data provides a more realistic distribution of the mechanical stresses in the tumor and induces anisotropic tumor growth. Solid stress distribution differs among patients, which, in turn, induces a distinct functional vascular density distribution—owing to the compression of tumor vessels—and intratumoral drug distribution for each patient. In conclusion, incorporating elastography data results in a more accurate calculation of intratumoral mechanical stresses and enables a better mathematical description of subsequent events, such as the heterogeneous development of the tumor vasculature and intrapatient variations in tumor perfusion and delivery of drugs.

Stability bounds of a delay visco-elastic rheological model with substrate friction

Cells and tissues exhibit sustained oscillatory deformations during remodelling, migration or embryogenesis. Although it has been shown that these oscillations correlate with intracellular biochemical signalling, the role of these oscillations is as yet unclear, and whether they may trigger drastic cell reorganisation events or instabilities remains unknown. Here, we present a rheological model that incorporates elastic, viscous and frictional components, and that is able to generate oscillatory response through a delay adaptive process of the rest-length. We analyse its stability as a function of the model parameters and deduce analytical bounds of the stable domain. While increasing values of the delay and remodelling rate render the model unstable, we also show that increasing friction with the substrate destabilises the oscillatory response. This fact was unexpected and still needs to be verified experimentally. Furthermore, we numerically verify that the extension of the model with non-linear deformation measures is able to generate sustained oscillations converging towards a limit cycle. We interpret this sustained regime in terms of non-linear time varying stiffness parameters that alternate between stable and unstable regions of the linear model. We also note that this limit cycle is not present in the linear model. We study the phase diagram and the bifurcations of the non-linear model, based on our conclusions on the linear one. Such dynamic analysis of the delay visco-elastic model in the presence of friction is absent in the literature for both linear and non-linear rheologies. Our work also shows how increasing values of some parameters such as delay and friction decrease its stability, while other parameters such as stiffness stabilise the oscillatory response.

Glomerular filtration and podocyte tensional homeostasis: importance of the minor type IV collagen network

The minor type IV collagen chain, which is a significant component of the glomerular basement membrane in healthy individuals, is known to assemble into large structures (supercoils) that may contribute to the mechanical stability of the collagen network and the glomerular basement membrane as a whole. The absence of the minor chain, as in Alport syndrome, leads to glomerular capillary demise and eventually to kidney failure. An important consideration in this problem is that the glomerular capillary wall must be strong enough to withstand the filtration pressure and porous enough to permit filtration at reasonable pressures. In this work, we propose a coupled feedback loop driven by filtration demand and tensional homeostasis of the podocytes forming the outer portion of the glomerular capillary wall. Briefly, the deposition of new collagen increases the stiffness of basement membrane, helping to stress shield the podocytes, but the new collagen also decreases the permeability of the basement membrane, requiring an increase in capillary transmural pressure drop to maintain filtration; the resulting increased pressure outweighs the increased glomerular basement membrane stiffness and puts a net greater stress demand on the podocytes. This idea is explored by developing a multiscale simulation of the capillary wall, in which a macroscopic (µm scale) continuum model is connected to a set of microscopic (nm scale) fiber network models representing the collagen network and the podocyte cytoskeleton. The model considers two cases: healthy remodeling, in which the presence of the minor chain allows the collagen volume fraction to be increased by thickening fibers, and Alport syndrome remodeling, in which the absence of the minor chain allows collagen volume fraction to be increased only by adding new fibers to the network. The permeability of the network is calculated based on previous models of flow through a fiber network, and it is updated for different fiber radii and volume fractions. The analysis shows that the minor chain allows a homeostatic balance to be achieved in terms of both filtration and cell tension. Absent the minor chain, there is a fundamental change in the relation between the two effects, and the system becomes unstable. This result suggests that mechanobiological or mechanoregulatory therapies may be possible for Alport syndrome and other minor chain collagen diseases of the kidney.

On the problem of biological form

Embryonic development, which inspired the first theories of biological form, was eventually excluded from the conceptual framework of the Modern Synthesis as irrelevant. A major question during the last decades has centred on understanding whether new advances in developmental biology are compatible with the standard view or whether they compel a new theory. Here, I argue that the answer to this question depends on which concept of morphogenesis is held. Morphogenesis can be conceived as (1) a chemically driven or (2) a mechanically driven process. According to the first option, genetic regulatory networks drive morphogenesis. According to the second, morphogenesis results from an invariant tendency of embryonic tissues to restore changes in mechanical stress. While chemically driven morphogenesis allows an extension of the standard view, mechanically driven morphogenesis would deeply transform it. Which of these hypotheses has wider explanatory power is unknown. At present, the problem of biological form remains unsolved.

The role of topology and mechanics in uniaxially growing cell networks

In biological systems, the growth of cells, tissues and organs is influenced by mechanical cues. Locally, cell growth leads to a mechanically heterogeneous environment as cells pull and push their neighbours in a cell network. Despite this local heterogeneity, at the tissue level, the cell network is remarkably robust, as it is not easily perturbed by changes in the mechanical environment or the network connectivity. Through a network model, we relate global tissue structure (i.e. the cell network topology) and local growth mechanisms (growth laws) to the overall tissue response. Within this framework, we investigate the two main mechanical growth laws that have been proposed: stress-driven or strain-driven growth. We show that in order to create a robust and stable tissue environment, networks with predominantly series connections are naturally driven by stress-driven growth, whereas networks with predominantly parallel connections are associated with strain-driven growth.

Mechanical Compression Regulates Brain Cancer Cell Migration Through MEK1/Erk1 Pathway Activation and GDF15 Expression

Mechanical compression is a common abnormality of brain tumors that has been shown to be responsible for the severe neurological defects of brain cancer patients representing a negative prognostic factor. Indeed, it is of note that patients that undergo resection exhibited higher survival rates than those subjected to biopsy only, suggesting that compressive forces generated during brain tumor growth play a key role in tumor progression. Despite the importance of mechanical compression in brain tumors, there is a lack of studies examining its direct effects on brain cancer cells and the mechanisms involved. In the present study, we used two brain cancer cell lines with distinct metastatic potential, the less aggressive H4 and the highly aggressive A172 cell lines, in order to study the effect of compression on their proliferative and migratory ability. Specifically, we used multicellular tumor spheroids (MCS) embedded in agarose matrix to show that compression strongly impaired their growth. Using mathematical modeling, we estimated the levels of compressive stress generated during the growth of brain MCS and then we applied the respective stress levels on brain cancer cell monolayers using our previously established transmembrane pressure device. By performing a scratch assay, we found that compression strongly induced the migration of the less aggressive H4 cells, while a less pronounced effect was observed for A172 cells. Analysis of the gene expression profile of both cell lines revealed that GDF15 and small GTPases are strongly regulated by mechanical compression, while GDF15 was further shown to be necessary for cells to migrate under compression. Through a phospho-proteomic screening, we further found that compressive stimulus is transmitted through the MEK1/Erk1 signaling pathway, which is also necessary for the migration of brain cancer cells. Finally, our results gave the first indication that GDF15 could regulate and being regulated by MEK1/Erk1 signaling pathway in order to facilitate the compression-induced brain cancer cell migration, rendering them along with small GTPases as potential targets for future anti-metastatic therapeutic innovations to treat brain tumors.

Are Homeostatic States Stable? Dynamical Stability in Morphoelasticity

Biological growth is often driven by mechanical cues, such as changes in external pressure or tensile loading. Moreover, it is well known that many living tissues actively maintain a preferred level of mechanical internal stress, called the mechanical homeostasis. The tissue-level feedback mechanism by which changes in the local mechanical stresses affect growth is called a growth law within the theory of morphoelasticity, a theory for understanding the coupling between mechanics and geometry in growing and evolving biological materials. This coupling between growth and mechanics occurs naturally in macroscopic tubular structures, which are common in biology (e.g., arteries, plant stems, airways). We study a continuous tubular system with spatially heterogeneous residual stress via a novel discretization approach which allows us to obtain precise results about the stability of equilibrium states of the homeostasis-driven growing dynamical system. This method allows us to show explicitly that the stability of the homeostatic state depends nontrivially on the anisotropy of the growth response. The key role of anisotropy may provide a foundation for experimental testing of homeostasis-driven growth laws.

Empirically Determined Vascular Smooth Muscle Cell Mechano-Adaptation Law

Cardiovascular disease can alter the mechanical environment of the vascular system, leading to mechano-adaptive growth and remodeling. Predictive models of arterial mechano-adaptation could improve patient treatments and outcomes in cardiovascular disease. Vessel-scale mechano-adaptation includes remodeling of both the cells and extracellular matrix. Here, we aimed to experimentally measure and characterize a phenomenological mechano-adaptation law for vascular smooth muscle cells (VSMCs) within an artery. To do this, we developed a highly controlled and reproducible system for applying a chronic step-change in strain to individual VSMCs with in vivo like architecture and tracked the temporal cellular stress evolution. We found that a simple linear growth law was able to capture the dynamic stress evolution of VSMCs in response to this mechanical perturbation. These results provide an initial framework for development of clinically relevant models of vascular remodeling that include VSMC adaptation.

Experimental measurements and mathematical modeling towards quantification of brain swelling stress

Traumatic brain injury results in brain tissue swelling which can be a life threatening condition due to skull confinement. While previous efforts successfully measured the exhibited volume change in brain tissue swelling, no data exist to provide information about the exhibited stresses. In this study, confined compression mechanical testing was employed to measure swelling stress in murine brain tissue samples by varying the ionic concentration of the bathing solutions. Subsequently, computer simulations of the experimental protocol were employed to confirm a triphasic mathematical model describing the effect and provide insights into the experimental data. We measured the swelling stress to be in the range of 1.2-6.7kPa (9.0-50.2mmHg) depending on the ionic strength of the bathing solution, while a good correspondence was demonstrated among the experimentally measured and simulated responses. Furthermore, the mathematical model featured the osmotic pressure as the primary contributor to the swelling stress, while a parametric analysis showed that the densities of the intracellular fixed charges and of the non-permeable solutes significantly affect the swelling stress.

The Solid Mechanics of Cancer and Strategies for Improved Therapy

Tumor progression and response to treatment is determined in large part by the generation of mechanical stresses that stem from both the solid and the fluid phase of the tumor. Furthermore, elevated solid stress levels can regulate fluid stresses by compressing intratumoral blood and lymphatic vessels. Blood vessel compression reduces tumor perfusion, while compression of lymphatic vessels hinders the ability of the tumor to drain excessive fluid from its interstitial space contributing to the uniform elevation of the interstitial fluid pressure. Hypoperfusion and interstitial hypertension pose major barriers to the systemic administration of chemotherapeutic agents and nanomedicines to tumors, reducing treatment efficacies. Hypoperfusion can also create a hypoxic and acidic tumor microenvironment that promotes tumor progression and metastasis. Hence, alleviation of intratumoral solid stress levels can decompress tumor vessels and restore perfusion and interstitial fluid pressure. In this review, three major types of tissue level solid stresses involved in tumor growth, namely stress exerted externally on the tumor by the host tissue, swelling stress, and residual stress, are discussed separately and details are provided regarding their causes, magnitudes, and remedies. Subsequently, evidence of how stress-alleviating drugs could be used in combination with chemotherapy to improve treatment efficacy is presented, highlighting the potential of stress-alleviation strategies to enhance cancer therapy. Finally, a continuum-level, mathematical framework to incorporate these types of solid stress is outlined.

Biphasic modeling of brain tumor biomechanics and response to radiation treatment

Biomechanical forces are central in tumor progression and response to treatment. This becomes more important in brain cancers where tumors are surrounded by tissues with different mechanical properties. Existing mathematical models ignore direct mechanical interactions of the tumor with the normal brain. Here, we developed a clinically relevant model, which predicts tumor growth accounting directly for mechanical interactions. A three-dimensional model of the gray and white matter and the cerebrospinal fluid was constructed from magnetic resonance images of a normal brain. Subsequently, a biphasic tissue growth theory for an initial tumor seed was employed, incorporating the effects of radiotherapy. Additionally, three different sets of brain tissue properties taken from the literature were used to investigate their effect on tumor growth. Results show the evolution of solid stress and interstitial fluid pressure within the tumor and the normal brain. Heterogeneous distribution of the solid stress exerted on the tumor resulted in a 35% spatial variation in cancer cell proliferation. Interestingly, the model predicted that distant from the tumor, normal tissues still undergo significant deformations while it was found that intratumoral fluid pressure is elevated. Our predictions relate to clinical symptoms of brain cancers and present useful tools for therapy planning.

Remodeling of extracellular matrix due to solid stress accumulation during tumor growth

Solid stresses emerge as the expanding tumor displaces and deforms the surrounding normal tissue, and also as a result of intratumoral component interplay. Among other things, solid stresses are known to induce extensive extracellular matrix synthesis and reorganization. In this study, we developed a mathematical model of tumor growth that distinguishes the contribution to stress generation by collagenous and non-collagenous tumor structural components, and also investigates collagen fiber remodeling exclusively due to solid stress. To this end, we initially conducted in vivo experiments using an orthotopic mouse model for breast cancer to monitor primary tumor growth and derive the mechanical properties of the tumor. Subsequently, we fitted the mathematical model to experimental data to determine values of the model parameters. According to the model, intratumoral solid stress is compressive, whereas extratumoral stress in the tumor vicinity is compressive in the radial direction and tensile in the periphery. Furthermore, collagen fibers engaged in stress generation only in the peritumoral region, and not in the interior where they were slackened due to the compressive stress state. Peritumoral fibers were driven away from the radial direction, tended to realign tangent to the tumor-host interface, and were also significantly stretched by tensile circumferential stresses. By means of this remodeling, the model predicts that the tumor is enveloped by a progressively thickening capsule of collagen fibers. This prediction is consistent with long-standing observations of tumor encapsulation and histologic sections that we performed, and it further corroborates the expansive growth hypothesis for the capsule formation.

Stress-mediated progression of solid tumors: effect of mechanical stress on tissue oxygenation, cancer cell proliferation, and drug delivery

Oxygen supply plays a central role in cancer cell proliferation. While vascular density increases at the early stages of carcinogenesis, mechanical solid stresses developed during growth compress tumor blood vessels and, thus, drastically reduce not only the supply of oxygen, but also the delivery of drugs at inner tumor regions. Among other effects, hypoxia and reduced drug delivery compromise the efficacy of radiation and chemo/nanotherapy, respectively. In the present study, we developed a mathematical model of tumor growth to investigate the interconnections among tumor oxygenation that supports cancer cell proliferation, the heterogeneous accumulation of mechanical stresses owing to tumor growth, the non-uniform compression of intratumoral blood vessels due to the mechanical stresses, and the insufficient delivery of oxygen and therapeutic agents because of vessel compression. We found that the high vascular density and increased cancer cell proliferation often observed in the periphery compared to the interior of a tumor can be attributed to heterogeneous solid stress accumulation. Highly vascularized peripheral regions are also associated with greater oxygenation compared with the compressed, less vascularized inner regions. We also modeled the delivery of drugs of two distinct sizes, namely chemotherapy and nanomedicine. Model predictions suggest that drug delivery is affected negatively by vessel compression independently of the size of the therapeutic agent. Finally, we demonstrated the applicability of our model to actual geometries, employing a breast tumor model derived from MR images.

Elasticity-based targeted growth models of morphogenesis

Embryonic tissue mechanics play an important role in regulating morphogenesis during organ formation, both in a bottom-up sense, where changes in gene expression drive mechanical shape changes, and in a top-down sense, where perturbations in tissue mechanics feed back to drive changes in gene expression. In growing tissues that can generate internal forces and have complex geometries, like those in the embryo, it can often be difficult to empirically determine the mechanical state of the tissue, let alone the relationships between gene expression and mechanical behavior. Mathematical models can be used to fill this gap. Here, we discuss elasticity-based models for growing tissues with a specific focus on targeted growth in embryonic tissues.

Mechanical effects of the surface ectoderm on optic vesicle morphogenesis in the chick embryo

Precise shaping of the eye is crucial for proper vision. Here, we use experiments on chick embryos along with computational models to examine the mechanical factors involved in the formation of the optic vesicles (OVs), which grow outward from the forebrain of the early embryo. First, mechanical dissections were used to remove the surface ectoderm (SE), a membrane that contacts the outer surfaces of the OVs. Principal components analysis of OV shapes suggests that the SE exerts asymmetric loads that cause the OVs to flatten and shear caudally during the earliest stages of eye development and later to bend in the caudal and dorsal directions. These deformations cause the initially spherical OVs to become pear-shaped. Exposure to the myosin II inhibitor blebbistatin reduced these effects, suggesting that cytoskeletal contraction controls OV shape by regulating tension in the SE. To test the physical plausibility of these interpretations, we developed 2-D finite-element models for frontal and transverse cross-sections of the forebrain, including frictionless contact between the SE and OVs. With geometric data used to specify differential growth in the OVs, these models were used to simulate each experiment (control, SE removed, no contraction). For each case, the predicted shape of the OV agrees reasonably well with experiments. The results of this study indicate that differential growth in the OV and external pressure exerted by the SE are sufficient to cause the global changes in OV shape observed during the earliest stages of eye development.

Evolution of osmotic pressure in solid tumors

The mechanical microenvironment of solid tumors includes both fluid and solid stresses. These stresses play a crucial role in cancer progression and treatment and have been analyzed rigorously both mathematically and experimentally. The magnitude and spatial distribution of osmotic pressures in tumors, however, cannot be measured experimentally and to our knowledge there is no mathematical model to calculate osmotic pressures in the tumor interstitial space. In this study, we developed a triphasic biomechanical model of tumor growth taking into account not only the solid and fluid phase of a tumor, but also the transport of cations and anions, as well as the fixed charges at the surface of the glycosaminoglycan chains. Our model predicts that the osmotic pressure is negligible compared to the interstitial fluid pressure for values of glycosaminoglycans (GAGs) taken from the literature for sarcomas, melanomas and adenocarcinomas. Furthermore, our results suggest that an increase in the hydraulic conductivity of the tumor, increases considerably the intratumoral concentration of free ions and thus, the osmotic pressure but it does not reach the levels of the interstitial fluid pressure.

Bending and twisting the embryonic heart: a computational model for c-looping based on realistic geometry

The morphogenetic process of cardiac looping transforms the straight heart tube into a curved tube that resembles the shape of the future four-chambered heart. Although great progress has been made in identifying the molecular and genetic factors involved in looping, the physical mechanisms that drive this process have remained poorly understood. Recent work, however, has shed new light on this complicated problem. After briefly reviewing the current state of knowledge, we propose a relatively comprehensive hypothesis for the mechanics of the first phase of looping, termed c-looping, as the straight heart tube deforms into a c-shaped tube. According to this hypothesis, differential hypertrophic growth in the myocardium supplies the main forces that cause the heart tube to bend ventrally, while regional growth and cytoskeletal contraction in the omphalomesenteric veins (primitive atria) and compressive loads exerted by the splanchnopleuric membrane drive rightward torsion. A computational model based on realistic embryonic heart geometry is used to test the physical plausibility of this hypothesis. The behavior of the model is in reasonable agreement with available experimental data from control and perturbed embryos, offering support for our hypothesis. The results also suggest, however, that several other mechanisms contribute secondarily to normal looping, and we speculate that these mechanisms play backup roles when looping is perturbed. Finally, some outstanding questions are discussed for future study.

Role of constitutive behavior and tumor-host mechanical interactions in the state of stress and growth of solid tumors

Mechanical forces play a crucial role in tumor patho-physiology. Compression of cancer cells inhibits their proliferation rate, induces apoptosis and enhances their invasive and metastatic potential. Additionally, compression of intratumor blood vessels reduces the supply of oxygen, nutrients and drugs, affecting tumor progression and treatment. Despite the great importance of the mechanical microenvironment to the pathology of cancer, there are limited studies for the constitutive modeling and the mechanical properties of tumors and on how these parameters affect tumor growth. Also, the contribution of the host tissue to the growth and state of stress of the tumor remains unclear. To this end, we performed unconfined compression experiments in two tumor types and found that the experimental stress-strain response is better fitted to an exponential constitutive equation compared to the widely used neo-Hookean and Blatz-Ko models. Subsequently, we incorporated the constitutive equations along with the corresponding values of the mechanical properties - calculated by the fit - to a biomechanical model of tumor growth. Interestingly, we found that the evolution of stress and the growth rate of the tumor are independent from the selection of the constitutive equation, but depend strongly on the mechanical interactions with the surrounding host tissue. Particularly, model predictions - in agreement with experimental studies - suggest that the stiffness of solid tumors should exceed a critical value compared with that of the surrounding tissue in order to be able to displace the tissue and grow in size. With the use of the model, we estimated this critical value to be on the order of 1.5. Our results suggest that the direct effect of solid stress on tumor growth involves not only the inhibitory effect of stress on cancer cell proliferation and the induction of apoptosis, but also the resistance of the surrounding tissue to tumor expansion.

Physical models of mesoderm invagination in Drosophila embryo

The invagination of the mesoderm in the Drosophila melanogaster embryo is an intensely studied example of epithelial folding. Several theoretical studies have explored the conditions and mechanisms needed to reproduce the formation of the invagination in silico. Here we discuss the aspects of epithelial folding captured by these studies, and compare the questions addressed, the approaches used, and the answers provided.

Morphogenesis can be driven by properly parametrised mechanical feedback

A fundamental problem of morphogenesis is whether it presents itself as a succession of links that are each driven by its own specific cause-effect relationship, or whether all of the links can be embraced by a common law that is possible to formulate in physical terms. We suggest that a common biophysical background for most, if not all, morphogenetic processes is based upon feedback between mechanical stresses (MS) that are imposed to a given part of a developing embryo by its other parts and MS that are actively generated within that part. The latter are directed toward hyper-restoration (restoration with an overshoot) of the initial MS values. We show that under mechanical constraints imposed by other parts, these tendencies drive forth development. To provide specificity for morphogenetic reactions, this feedback should be modulated by long-term parameters and/or initial conditions that are set up by genetic factors. The experimental and model data related to this concept are reviewed.

Computational and experimental study of the mechanics of embryonic wound healing

Wounds in the embryo show a remarkable ability to heal quickly without leaving a scar. Previous studies have found that an actomyosin ring (purse string) forms around the wound perimeter and contracts to close the wound over the course of several dozens of minutes. Here, we report experiments that reveal an even faster mechanism which remarkably closes wounds by more than 50% within the first 30s. Circular and elliptical wounds (~100μm in size) were made in the blastoderm of early chick embryos and allowed to heal, with wound area and shape characterized as functions of time. The closure rate displayed a biphasic behavior, with rapid constriction lasting about a minute, followed by a period of more gradual closure to complete healing. Fluorescent staining suggests that both healing phases are driven by actomyosin contraction, with relatively rapid contraction of fibers at cell borders within a relatively thick ring of tissue (several cells wide) around the wound followed by slower contraction of a thin supracellular actomyosin ring along the margin, consistent with a purse string mechanism. Finite-element modeling showed that this idea is biophysically plausible, with relatively isotropic contraction within the thick ring giving way to tangential contraction in the thin ring. In addition, consistent with experimental results, simulated elliptical wounds heal with little change in aspect ratio, and decreased membrane tension can cause these wounds to open briefly before going on to heal. These results provide new insight into the healing mechanism in embryonic epithelia.

Apical constriction initiates new bud formation during monopodial branching of the embryonic chicken lung

Branching morphogenesis sculpts the airway epithelium of the lung into a tree-like structure to conduct air and promote gas exchange after birth. In the avian lung, a series of buds emerges from the dorsal surface of the primary bronchus via monopodial branching to form the conducting airways; anatomically, these buds are similar to those formed by domain branching in the mammalian lung. Here, we show that monopodial branching is initiated by apical constriction of the airway epithelium, and not by differential cell proliferation, using computational modeling and quantitative imaging of embryonic chicken lung explants. Both filamentous actin and phosphorylated myosin light chain were enriched at the apical surface of the airway epithelium during monopodial branching. Consistently, inhibiting actomyosin contractility prevented apical constriction and blocked branch initiation. Although cell proliferation was enhanced along the dorsal and ventral aspects of the primary bronchus, especially before branch formation, inhibiting proliferation had no effect on the initiation of branches. To test whether the physical forces from apical constriction alone are sufficient to drive the formation of new buds, we constructed a nonlinear, three-dimensional finite element model of the airway epithelium and used it to simulate apical constriction and proliferation in the primary bronchus. Our results suggest that, consistent with the experimental results, apical constriction is sufficient to drive the early stages of monopodial branching whereas cell proliferation is dispensable. We propose that initial folding of the airway epithelium is driven primarily by apical constriction during monopodial branching of the avian lung.

Physiology-based model of cell viscoelasticity

The measured viscoelastic properties of biological tissues is the result of the passive and active response of the cells. We propose an evolution law of the remodeling process in the cytoskeleton which is able to mimic the viscous properties of biological cellular tissues. Our model is based on dynamical changes of the resting length. We show that under the small strain regime, the linear rheology models are recovered, with the relaxation time being replaced by the cell resistance to remodel. We implement the one-dimensional model into network systems of two and three dimensions, and show that the same conclusions may be drawn for those systems.

Mathematical modelling in developmental biology

In recent decades, molecular and cellular biology has benefited from numerous fascinating developments in experimental technique, generating an overwhelming amount of data on various biological objects and processes. This, in turn, has led biologists to look for appropriate tools to facilitate systematic analysis of data. Thus, the need for mathematical techniques, which can be used to aid the classification and understanding of this ever-growing body of experimental data, is more profound now than ever before. Mathematical modelling is becoming increasingly integrated into biological studies in general and into developmental biology particularly. This review outlines some achievements of mathematics as applied to developmental biology and demonstrates the mathematical formulation of basic principles driving morphogenesis. We begin by describing a mathematical formalism used to analyse the formation and scaling of morphogen gradients. Then we address a problem of interplay between the dynamics of morphogen gradients and movement of cells, referring to mathematical models of gastrulation in the chick embryo. In the last section, we give an overview of various mathematical models used in the study of the developmental cycle of Dictyostelium discoideum, which is probably the best example of successful mathematical modelling in developmental biology.

Regional differences in actomyosin contraction shape the primary vesicles in the embryonic chicken brain

In the early embryo, the brain initially forms as a relatively straight, cylindrical epithelial tube composed of neural stem cells. The brain tube then divides into three primary vesicles (forebrain, midbrain, hindbrain), as well as a series of bulges (rhombomeres) in the hindbrain. The boundaries between these subdivisions have been well studied as regions of differential gene expression, but the morphogenetic mechanisms that generate these constrictions are not well understood. Here, we show that regional variations in actomyosin-based contractility play a major role in vesicle formation in the embryonic chicken brain. In particular, boundaries did not form in brains exposed to the nonmuscle myosin II inhibitor blebbistatin, whereas increasing contractile force using calyculin or ATP deepened boundaries considerably. Tissue staining showed that contraction likely occurs at the inner part of the wall, as F-actin and phosphorylated myosin are concentrated at the apical side. However, relatively little actin and myosin was found in rhombomere boundaries. To determine the specific physical mechanisms that drive vesicle formation, we developed a finite-element model for the brain tube. Regional apical contraction was simulated in the model, with contractile anisotropy and strength estimated from contractile protein distributions and measurements of cell shapes. The model shows that a combination of circumferential contraction in the boundary regions and relatively isotropic contraction between boundaries can generate realistic morphologies for the primary vesicles. In contrast, rhombomere formation likely involves longitudinal contraction between boundaries. Further simulations suggest that these different mechanisms are dictated by regional differences in initial morphology and the need to withstand cerebrospinal fluid pressure. This study provides a new understanding of early brain morphogenesis.

Epithelial machines of morphogenesis and their potential application in organ assembly and tissue engineering

Sheets of embryonic epithelial cells coordinate their efforts to create diverse tissue structures such as pits, grooves, tubes, and capsules that lead to organ formation. Such cells can use a number of cell behaviors including contractility, proliferation, and directed movement to create these structures. By contrast, tissue engineers and researchers in regenerative medicine seeking to produce organs for repair or replacement therapy can combine cells with synthetic polymeric scaffolds. Tissue engineers try to achieve these goals by shaping scaffold geometry in such a way that cells embedded within these scaffold self-assemble to form a tissue, for instance aligning to synthetic fibers, and assembling native extracellular matrix to form the desired tissue-like structure. Although self-assembly is a dominant process that guides tissue assembly both within the embryo and within artificial tissue constructs, we know little about these critical processes. Here, we compare and contrast strategies of tissue assembly used by embryos to those used by engineers during epithelial morphogenesis and highlight opportunities for future applications of developmental biology in the field of tissue engineering.

Computational models for mechanics of morphogenesis

In the developing embryo, tissues differentiate, deform, and move in an orchestrated manner to generate various biological shapes driven by the complex interplay between genetic, epigenetic, and environmental factors. Mechanics plays a key role in regulating and controlling morphogenesis, and quantitative models help us understand how various mechanical forces combine to shape the embryo. Models allow for the quantitative, unbiased testing of physical mechanisms, and when used appropriately, can motivate new experimentaldirections. This knowledge benefits biomedical researchers who aim to prevent and treat congenital malformations, as well as engineers working to create replacement tissues in the laboratory. In this review, we first give an overview of fundamental mechanical theories for morphogenesis, and then focus on models for specific processes, including pattern formation, gastrulation, neurulation, organogenesis, and wound healing. The role of mechanical feedback in development is also discussed. Finally, some perspectives aregiven on the emerging challenges in morphomechanics and mechanobiology.

Computational simulation of hemodynamic-driven growth and remodeling of embryonic atrioventricular valves

Embryonic heart valves develop under continuous and demanding hemodynamic loading. The particular contributions of fluid pressure and shear tractions in valve morphogenesis are difficult to decouple experimentally. To better understand how fluid loads could direct valve formation, we developed a computational model of avian embryonic atrioventricular (AV) valve (cushion) growth and remodeling using experimentally derived parameters for the blood flow and the cushion stiffness. Through an iterative scheme, we first solved the fluid loads on the axisymmetric AV canal and cushion model geometry. We then applied the fluid loads to the cushion and integrated the evolution equations to determine the growth and remodeling. After a set time of growth, we updated the fluid domain to reflect the change in cushion geometry and resolved for the fluid forces. The rate of growth and remodeling was assumed to be a function of the difference between the current stress and an isotropic homeostatic stress state. The magnitude of the homeostatic stress modulated the rate of volume addition during the evolution. We found that the pressure distribution on the AV cushion was sufficient to generate leaflet-like elongation in the direction of flow, through inducing tissue resorption on the inflow side of cushion and expansion on the outflow side. Conversely, shear tractions minimally altered tissue volume, but regulated the remodeling of tissue near the cushion surface, particular at the leading edge. Significant shear and circumferential residual stresses developed as the cushion evolved. This model offers insight into how natural and perturbed mechanical environments may direct AV valvulogenesis and provides an initial framework on which to incorporate more mechano-biological details.

Not just inductive: a crucial mechanical role for the endoderm during heart tube assembly

The heart is the first functioning organ to form during development. During gastrulation, the cardiac progenitors reside in the lateral plate mesoderm but maintain close contact with the underlying endoderm. In amniotes, these bilateral heart fields are initially organized as a pair of flat epithelia that move towards the embryonic midline and fuse above the anterior intestinal portal (AIP) to form the heart tube. This medial motion is typically attributed to active mesodermal migration over the underlying endoderm. In this model, the role of the endoderm is twofold: to serve as a mechanically passive substrate for the crawling mesoderm and to secrete various growth factors necessary for cardiac specification and differentiation. Here, using computational modeling and experiments on chick embryos, we present evidence supporting an active mechanical role for the endoderm during heart tube assembly. Label-tracking experiments suggest that active endodermal shortening around the AIP accounts for most of the heart field motion towards the midline. Results indicate that this shortening is driven by cytoskeletal contraction, as exposure to the myosin-II inhibitor blebbistatin arrested any shortening and also decreased both tissue stiffness (measured by microindentation) and mechanical tension (measured by cutting experiments). In addition, blebbistatin treatment often resulted in cardia bifida and abnormal foregut morphogenesis. Moreover, finite element simulations of our cutting experiments suggest that the endoderm (not the mesoderm) is the primary contractile tissue layer during this process. Taken together, these results indicate that contraction of the endoderm actively pulls the heart fields towards the embryonic midline, where they fuse to form the heart tube.

Extracellular matrix and the mechanics of large artery development

The large, elastic arteries, as their name suggests, provide elastic distention and recoil during the cardiac cycle in vertebrate animals. The arteries are distended from the pressure of ejecting blood during the active contraction of the left ventricle (LV) during systole and recoil to their original dimensions during relaxation of the LV during diastole. The cyclic distension occurs with minimal energy loss, due to the elastic properties of one of the major structural extracellular matrix (ECM) components, elastin. The maximum distension is limited to prevent damage to the artery by another major ECM component, collagen. The mix of ECM components in the wall largely determines the passive mechanical behavior of the arteries and the subsequent load on the heart during systole. While much research has focused on initial artery formation, there has been less attention on the continuing development of the artery to produce the mature composite wall complete with endothelial cells (ECs), smooth muscle cells (SMCs), and the necessary mix of ECM components for proper cardiovascular function. This review focuses on the physiology of large artery development, including SMC differentiation and ECM production. The effects of hemodynamic forces and ECM deposition on the evolving arterial structure and function are discussed. Human diseases and mouse models with genetic mutations in ECM proteins that affect large artery development are summarized. A review of constitutive models and growth and remodeling theories is presented, along with future directions to improve understanding of ECM and the mechanics of large artery development.

On integrating experimental and theoretical models to determine physical mechanisms of morphogenesis

Researchers in developmental biology are increasingly recognizing the value of theoretical models in studies of morphogenesis. However, creating and testing realistic quantitative models for morphogenetic processes can be an extremely challenging task. The focus of this paper is on models for the mechanics of morphogenesis. Models for these problems often must include large changes in geometry, leading to highly nonlinear problems with the possibility of multiple solutions that must be sorted out using experimental data. Here, we illustrate our approach to these problems using the specific example of head fold formation in the early chick embryo. The interplay between experimental and theoretical results is emphasized throughout, as the model is gradually refined. Some of the limitations inherent in theoretical/computational modeling of biological systems are also discussed.

Mechano-electrical vibrations of microtubules--link to subcellular morphology

Spontaneous mechanical oscillations were predicted and experimentally proven on almost every level of cellular structure. Besides morphogenetic potential of oscillatory mechanical force, oscillations may drive vibrations of electrically polar structures or these structures themselves may oscillate on their own natural frequencies. Vibrations of electric charge will generate oscillating electric field, role of which in morphogenesis is discussed in this paper. This idea is demonstrated in silico on the conformation of two growing microtubules.

A potential role for differential contractility in early brain development and evolution

Differences in brain structure between species have long fascinated evolutionary biologists. Understanding how these differences arise requires knowing how they are generated in the embryo. Growing evidence in the field of evolutionary developmental biology (evo-devo) suggests that morphological differences between species result largely from changes in the spatiotemporal regulation of gene expression during development. Corresponding changes in functional cellular behaviors (morphogenetic mechanisms) are only beginning to be explored, however. Here we show that spatiotemporal patterns of tissue contractility are sufficient to explain differences in morphology of the early embryonic brain between disparate species. We found that enhancing cytoskeletal contraction in the embryonic chick brain with calyculin A alters the distribution of contractile proteins on the apical side of the neuroepithelium and changes relatively round cross-sections of the tubular brain into shapes resembling triangles, diamonds, and narrow slits. These perturbed shapes, as well as overall brain morphology, are remarkably similar to those of corresponding sections normally found in species such as zebrafish and Xenopus laevis (frog). Tissue staining revealed relatively strong concentration of F-actin at vertices of hyper-contracted cross-sections, and a finite element model shows that local contraction in these regions can convert circular sections into the observed shapes. Another model suggests that these variations in contractility depend on the initial geometry of the brain tube, as localized contraction may be needed to open the initially closed lumen in normal zebrafish and Xenopus brains, whereas this contractile machinery is not necessary in chick brains, which are already open when first created. We conclude that interspecies differences in cytoskeletal contraction may play a larger role in generating differences in morphology, and at much earlier developmental stages, in the brain than previously appreciated. This study is a step toward uncovering the underlying morphomechanical mechanisms that regulate how neural phenotypic differences arise between species.

Physics and the canalization of morphogenesis: a grand challenge in organismal biology

Morphogenesis takes place against a background of organism-to-organism and environmental variation. Therefore, fundamental questions in the study of morphogenesis include: How are the mechanical processes of tissue movement and deformation affected by that variability, and in turn, how do the mechanic of the system modulate phenotypic variation? We highlight a few key factors, including environmental temperature, embryo size and environmental chemistry that might perturb the mechanics of morphogenesis in natural populations. Then we discuss several ways in which mechanics-including feedback from mechanical cues-might influence intra-specific variation in morphogenesis. To understand morphogenesis it will be necessary to consider whole-organism, environment and evolutionary scales because these larger scales present the challenges that developmental mechanisms have evolved to cope with. Studying the variation organisms express and the variation organisms experience will aid in deciphering the causes of birth defects.

Consistent formulation of the growth process at the kinematic and constitutive level for soft tissues composed of multiple constituents

Previous studies have investigated the possibilities of modelling the change in volume and change in density of biomaterials. This can be modelled at the constitutive or the kinematic level. This work introduces a consistent formulation at the kinematic and constitutive level for growth processes. Most biomaterials consist of many constituents and can be approximated as being incompressible. These two conditions (many constituents and incompressibility) suggest a straightforward implementation in the context of the finite element (FE) method which could now be validated more easily against histological measurements. Its key characteristic variable is the normalised partial mass change. Using the concept of homeostatic equilibrium, we suggest two complementary growth laws in which the evolution of the normalised partial mass change is governed by an ordinary differential equation in terms of either the Piola-Kirchhoff stress or the Green-Lagrange strain. We combine this approach with the classical incompatibility condition and illustrate its algorithmic implementation within a fully nonlinear FE approach. This approach is first illustrated for a simple uniaxial tension and extension test for pure volume change and pure density change and is validated against previous numerical results. Finally, a physiologically based example of a two-phase model is presented which is a combination of volume and density changes. It can be concluded that the effect of hyper-restoration may be due to the systemic effect of degradation and adaptation of given constituents.

Perspectives on biological growth and remodeling

The continuum mechanical treatment of biological growth and remodeling has attracted considerable attention over the past fifteen years. Many aspects of these problems are now well-understood, yet there remain areas in need of significant development from the standpoint of experiments, theory, and computation. In this perspective paper we review the state of the field and highlight open questions, challenges, and avenues for further development.

A POROELASTIC MODEL FOR CELL CRAWLING INCLUDING MECHANICAL COUPLING BETWEEN CYTOSKELETAL CONTRACTION AND ACTIN POLYMERIZATION

Much is known about the biophysical mechanisms involved in cell crawling, but how these processes are coordinated to produce directed motion is not well understood. Here, we propose a new hypothesis whereby local cytoskeletal contraction generates fluid flow through the lamellipodium, with the pressure at the front of the cell facilitating actin polymerization which pushes the leading edge forward. The contraction, in turn, is regulated by stress in the cytoskeleton. To test this hypothesis, finite element models for a crawling cell are presented. These models are based on nonlinear poroelasticity theory, modified to include the effects of active contraction and growth, which are regulated by mechanical feedback laws. Results from the models agree reasonably well with published experimental data for cell speed, actin flow, and cytoskeletal deformation in migrating fish epidermal keratocytes. The models also suggest that oscillations can occur for certain ranges of parameter values.

Surprisingly simple mechanical behavior of a complex embryonic tissue

BACKGROUND

Previous studies suggest that mechanical feedback could coordinate morphogenetic events in embryos. Furthermore, embryonic tissues have complex structure and composition and undergo large deformations during morphogenesis. Hence we expect highly non-linear and loading-rate dependent tissue mechanical properties in embryos.

METHODOLOGY/PRINCIPAL FINDINGS

We used micro-aspiration to test whether a simple linear viscoelastic model was sufficient to describe the mechanical behavior of gastrula stage Xenopus laevis embryonic tissue in vivo. We tested whether these embryonic tissues change their mechanical properties in response to mechanical stimuli but found no evidence of changes in the viscoelastic properties of the tissue in response to stress or stress application rate. We used this model to test hypotheses about the pattern of force generation during electrically induced tissue contractions. The dependence of contractions on suction pressure was most consistent with apical tension, and was inconsistent with isotropic contraction. Finally, stiffer clutches generated stronger contractions, suggesting that force generation and stiffness may be coupled in the embryo.

CONCLUSIONS/SIGNIFICANCE

The mechanical behavior of a complex, active embryonic tissue can be surprisingly well described by a simple linear viscoelastic model with power law creep compliance, even at high deformations. We found no evidence of mechanical feedback in this system. Together these results show that very simple mechanical models can be useful in describing embryo mechanics.

Mechanics of head fold formation: investigating tissue-level forces during early development

During its earliest stages, the avian embryo is approximately planar. Through a complex series of folds, this flat geometry is transformed into the intricate three-dimensional structure of the developing organism. Formation of the head fold (HF) is the first step in this cascading sequence of out-of-plane tissue folds. The HF establishes the anterior extent of the embryo and initiates heart, foregut and brain development. Here, we use a combination of computational modeling and experiments to determine the physical forces that drive HF formation. Using chick embryos cultured ex ovo, we measured: (1) changes in tissue morphology in living embryos using optical coherence tomography (OCT); (2) morphogenetic strains (deformations) through the tracking of tissue labels; and (3) regional tissue stresses using changes in the geometry of circular wounds punched through the blastoderm. To determine the physical mechanisms that generate the HF, we created a three-dimensional computational model of the early embryo, consisting of pseudoelastic plates representing the blastoderm and vitelline membrane. Based on previous experimental findings, we simulated the following morphogenetic mechanisms: (1) convergent extension in the neural plate (NP); (2) cell wedging along the anterior NP border; and (3) autonomous in-plane deformations outside the NP. Our numerical predictions agree relatively well with the observed morphology, as well as with our measured stress and strain distributions. The model also predicts the abnormal tissue geometries produced when development is mechanically perturbed. Taken together, the results suggest that the proposed morphogenetic mechanisms provide the main tissue-level forces that drive HF formation.

Axons pull on the brain, but tension does not drive cortical folding

During human brain development, the cerebral cortex undergoes substantial folding, leading to its characteristic highly convoluted form. Folding is necessary to accommodate the expansion of the cerebral cortex; abnormal cortical folding is linked to various neurological disorders, including schizophrenia, epilepsy, autism, and mental retardation. Although this process requires mechanical forces, the specific force-generating mechanisms that drive folding remain unclear. The two most widely accepted hypotheses are as follows: (1) Folding is caused by differential growth of the cortex and (2) folding is caused by mechanical tension generated in axons. Direct evidence supporting either theory, however, is lacking. Here we show that axons are indeed under considerable tension in the developing ferret brain, but the patterns of tissue stress are not consistent with a causal role for axonal tension. In particular, microdissection assays reveal that significant tension exists along axons aligned circumferentially in subcortical white matter tracts, as well as those aligned radially inside developing gyri (outward folds). Contrary to previous speculation, however, axonal tension is not directed across developing gyri, suggesting that axon tension does not drive folding. On the other hand, using computational (finite element) models, we show that differential cortical growth accompanied by remodeling of the subplate leads to outward folds and stress fields that are consistent with our microdissection experiments, supporting a mechanism involving differential growth. Local perturbations, such as temporal differences in the initiation of cortical growth, can ensure consistent folding patterns. This study shows that a combination of experimental and computational mechanics can be used to evaluate competing hypotheses of morphogenesis, and illuminate the biomechanics of cortical folding.