Tri-layer wrinkling as a mechanism for anchoring center initiation in the developing cerebellum

Cell division angle predicts the level of tissue mechanics that tune the amount of cerebellar folding

Modeling has led to proposals that the amount of neural tissue folding is set by the level of differential expansion between tissue layers and that the wavelength is set by the thickness of the outer layer. Here, we used inbred mouse strains with distinct amounts of cerebellar folding to investigate these predictions. We identified a distinct critical period during which the folding amount diverges between the two strains. In this period, regional changes in the level of differential expansion between the external granule layer (EGL) and underlying core correlate with the folding amount in each strain. Additionally, the thickness of the EGL varies regionally during the critical period alongside corresponding changes in wavelength. The number of SHH-expressing Purkinje cells predicts the folding amount, but the proliferation rate in the EGL is the same between the strains. However, regional changes in the cell division angle within the EGL predicts both the tangential expansion and the thickness of the EGL. Cell division angle is likely a tunable mechanism whereby both the level of differential expansion along the perimeter and the thickness of the EGL are regionally tuned to set the amount and wavelength of folding.

Cell division angle regulates the tissue mechanics and tunes the amount of cerebellar folding

Modeling has proposed that the amount of neural tissue folding is set by the level of differential-expansion between tissue layers and that the wavelength is set by the thickness of the outer layer. Here we used inbred mouse strains with distinct amounts of cerebellar folding to investigate these predictions. We identified a critical period where the folding amount diverges between the strains. In this period, regional changes in the level of differential-expansion between the external granule layer (EGL) and underlying core correlate with the folding amount in each strain. Additionally, the thickness of the EGL is regionally adjusted during the critical period alongside corresponding changes in wavelength. While the number of SHH-expressing Purkinje cells predicts the folding amount, the proliferation rate in the EGL is the same between the strains. However, regional changes in the cell division angle within the EGL predicts both the tangential-expansion and thickness of the EGL. Cell division angle is likely a tunable mechanism whereby both the level of differential-expansion and thickness of the EGL are regionally tuned to set the amount and wavelength of folding.

Cerebellum lineage allocation, morphogenesis and repair: impact of interplay amongst cells

The cerebellum has a simple cytoarchitecture consisting of a folded cortex with three cell layers that surrounds a nuclear structure housing the output neurons. The excitatory neurons are generated from a unique progenitor zone, the rhombic lip, whereas the inhibitory neurons and astrocytes are generated from the ventricular zone. The growth phase of the cerebellum is driven by lineage-restricted progenitor populations derived from each zone. Research during the past decade has uncovered the importance of cell-to-cell communication between the lineages through largely unknown signaling mechanisms for regulating the scaling of cell numbers and cell plasticity during mouse development and following injury in the neonatal (P0-P14) cerebellum. This Review focuses on how the interplay between cell types is key to morphogenesis, production of robust neural circuits and replenishment of cells after injury, and ends with a discussion of the implications of the greater complexity of the human cerebellar progenitor zones for development and disease.

Exploring the potential of transfer learning for metamodels of heterogeneous material deformation

From the nano-scale to the macro-scale, biological tissue is spatially heterogeneous. Even when tissue behavior is well understood, the exact subject specific spatial distribution of material properties is often unknown. And, when developing computational models of biological tissue, it is usually prohibitively computationally expensive to simulate every plausible spatial distribution of material properties for each problem of interest. Therefore, one of the major challenges in developing accurate computational models of biological tissue is capturing the potential effects of this spatial heterogeneity. Recently, machine learning based metamodels have gained popularity as a computationally tractable way to overcome this problem because they can make predictions based on a limited number of direct simulation runs. These metamodels are promising, but they often still require a high number of direct simulations to achieve an acceptable performance. Here we show that transfer learning, a strategy where knowledge gained while solving one problem is transferred to solving a different but related problem, can help overcome this limitation. Critically, transfer learning can be used to leverage both low-fidelity simulation data and simulation data that is the outcome of solving a different but related mechanical problem. In this paper, we extend Mechanical MNIST, our open source benchmark dataset of heterogeneous material undergoing large deformation, to include a selection of low-fidelity simulation results that require ≈ 2 - 4 orders of magnitude less CPU time to run. Then, we show that transferring the knowledge stored in metamodels trained on these low-fidelity simulation results can vastly improve the performance of metamodels used to predict the results of high-fidelity simulations. In the most dramatic examples, metamodels trained on 100 high fidelity simulations but pre-trained on 60,000 low-fidelity simulations achieves nearly the same test error as metamodels trained on 60,000 high-fidelity simulations (1 - 1.5% mean absolute percent error). In addition, we show that transfer learning is an effective method for leveraging data from different load cases, and for leveraging low-fidelity two-dimensional simulations to predict the outcomes of high-fidelity three-dimensional simulations. Looking forward, we anticipate that transfer learning will enable us to better capture the influence of tissue spatial heterogeneity on the mechanical behavior of biological materials across multiple different domains.

A 2020 view of tension-based cortical morphogenesis

Brain structures change shape dramatically during development. Elucidating the mechanisms of morphogenesis provides insights relevant to understanding brain function in health and disease. The tension-based morphogenesis (TBM) hypothesis posits that mechanical tension along axons, dendrites, and glial processes contributes to many aspects of central nervous system morphogenesis. Since TBM was proposed in 1997, extensive evidence supports a role for tension in diverse cellular phenomena, but tension’s role in cortical folding has been controversial. An extensively revised version of the TBM model for cerebral cortex addresses limitations of the original model, incorporates new features, and can be tested by many experimental approaches. For cerebellar cortex, a revised model accounts for many aspects of its development and adult architecture.

Mechanical tension along the length of axons, dendrites, and glial processes has been proposed as a major contributor to morphogenesis throughout the nervous system [D. C. Van Essen, Nature 385, 313–318 (1997)]. Tension-based morphogenesis (TBM) is a conceptually simple and general hypothesis based on physical forces that help shape all living things. Moreover, if each axon and dendrite strive to shorten while preserving connectivity, aggregate wiring length would remain low. TBM can explain key aspects of how the cerebral and cerebellar cortices remain thin, expand in surface area, and acquire their distinctive folds. This article reviews progress since 1997 relevant to TBM and other candidate morphogenetic mechanisms. At a cellular level, studies of diverse cell types in vitro and in vivo demonstrate that tension plays a major role in many developmental events. At a tissue level, I propose a differential expansion sandwich plus (DES+) revision to the original TBM model for cerebral cortical expansion and folding. It invokes tangential tension and “sulcal zipping” forces along the outer cortical margin as well as tension in the white matter core, together competing against radially biased tension in the cortical gray matter. Evidence for and against the DES+ model is discussed, and experiments are proposed to address key tenets of the DES+ model. For cerebellar cortex, a cerebellar multilayer sandwich (CMS) model is proposed that can account for many distinctive features, including its unique, accordion-like folding in the adult, and experiments are proposed to address its specific tenets.

Folding drives cortical thickness variations

The cortical thickness is a characteristic biomarker for a wide variety of neurological disorders. While the structural organization of the cerebral cortex is tightly regulated and evolutionarily preserved, its thickness varies widely between 1.5 and 4.5 mm across the healthy adult human brain. It remains unclear whether these thickness variations are a cause or consequence of cortical development. Recent studies suggest that cortical thickness variations are primarily a result of genetic effects. Previous studies showed that a simple homogeneous bilayered system with a growing layer on an elastic substrate undergoes a unique symmetry breaking into a spatially heterogeneous system with discrete gyri and sulci. Here, we expand on that work to explore the evolution of cortical thickness variations over time to support our finding that cortical pattern formation and thickness variations can be explained - at least in part - by the physical forces that emerge during cortical folding. Strikingly, as growth progresses, the developing gyri universally thicken and the sulci thin, even in the complete absence of regional information. Using magnetic resonance images, we demonstrate that these naturally emerging thickness variations agree with the cortical folding pattern in n = 9 healthy adult human brains, in n = 564 healthy human brains ages 7-64, and in n = 73 infant brains scanned at birth, and at ages one and two. Additionally, we show that cortical organoids develop similar patterns throughout their growth. Our results suggest that genetic, geometric, and physical events during brain development are closely interrelated. Understanding regional and temporal variations in cortical thickness can provide insight into the evolution and causative factors of neurological disorders, inform the diagnosis of neurological conditions, and assess the efficacy of treatment options.

Selection for Divergent Reproductive Investment Affects Neuron Size and Foliation in the Cerebellum

The cerebellum has a highly conserved internal circuitry, but varies greatly in size and morphology within and across species. Despite this variation, the underlying volumetric changes among the layers of the cerebellar cortex or their association with Purkinje cell numbers and sizes is poorly understood. Here, we examine intraspecific scaling relationships and variation in the quantitative neuroanatomy of the cerebellum in Japanese quail (Coturnix japonica) selected for high or low reproductive investment. As predicted by the circuitry of the cerebellum, the volumes of the constituent layers of the cerebellar cortex were strongly and positively correlated with one another and with total cerebellar volume. The number of Purkinje cells also significantly and positively co-varied with total cerebellar volume and the molecular layer, but not the granule cell layer or white matter volumes. Purkinje cell size and cerebellar foliation did not significantly covary with any cerebellar measures, but differed significantly between the selection lines. Males and females from the high-investment lines had smaller Purkinje cells than males and females from the low-investment lines and males from the high-investment lines had less folded cerebella than quail from the low-investment lines. These results suggest that within species, the layers of the cerebellum increase in a coordinated fashion, but Purkinje cell size and cerebellar foliation do not increase proportionally with overall cerebellum size. In contrast, selection for differential reproductive investment affects Purkinje cell size and cerebellar foliation, but not other quantitative measures of cerebellar anatomy.

Full Electrostatic Control of Nanomechanical Buckling

Buckling of mechanical structures results in bistable states with spatial separation, a feature desirable for sensing, shape configuration, and mechanical computation. Although different approaches have been developed to access buckling at microscopic scales, such as heating or prestressing beams, little attention has been paid so far to dynamically control all the parameters critical for the bifurcation-the compressive stress and the lateral force on the beam. Here, we develop an all-electrostatic architecture to control the compressive force, as well as the direction and amount of buckling, without significant heat generation on micro- or nanostructures. With this architecture, we demonstrated fundamental aspects of device function and dynamics. By applying voltages at any of the digital electronics standards, we have controlled the direction of buckling. Lateral deflections as large as 12% of the beam length were achieved. By modulating the compressive stress and lateral electrostatic force acting on the beam, we tuned the potential energy barrier between the postbifurcation stable states and characterized snap-through transitions between these states. The proposed architecture opens avenues for further studies in actuators, shape-shifting devices, thermodynamics of information, and dynamical chaos.

Advanced 4D Bioprinting Technologies for Brain Tissue Modeling and Study

Although the process by which the cortical tissues of the brain fold has been the subject of considerable study and debate over the past few decades, a single mechanistic description of the phenomenon has yet to be fully accepted. Rather, two competing explanations of cortical folding have arisen in recent years; known as the axonal tension and the differential tangential expansion models. In the present review, these two models are introduced by analyzing the computational, theoretical, materials-based, and cell studies which have yielded them. Then Four-dimensional bioprinting is presented as a powerful technology which can not only be used to test both models of cortical folding de novo, but can also be used to explore the reciprocal effects that folding associated mechanical stresses may have on neural development. Therein, the fabrication of "smart" tissue models which can accurately simulate the in vivo folding process and recapitulate physiologically relevant stresses are introduced. We also provide a general description of both cortical neurobiology as well as the cellular basis of cortical folding. Our discussion also entails an overview of both 3D and 4D bioprinting technologies, as well as a brief commentary on recent advancements in printed central nervous system tissue engineering.

Cerebellar folding is initiated by mechanical constraints on a fluid-like layer without a cellular pre-pattern

Models based in differential expansion of elastic material, axonal constraints, directed growth, or multi-phasic combinations have been proposed to explain brain folding. However, the cellular and physical processes present during folding have not been defined. We used the murine cerebellum to challenge folding models with in vivo data. We show that at folding initiation differential expansion is created by the outer layer of proliferating progenitors expanding faster than the core. However, the stiffness differential, compressive forces, and emergent thickness variations required by elastic material models are not present. We find that folding occurs without an obvious cellular pre-pattern, that the outer layer expansion is uniform and fluid-like, and that the cerebellum is under radial and circumferential constraints. Lastly, we find that a multi-phase model incorporating differential expansion of a fluid outer layer and radial and circumferential constraints approximates the in vivo shape evolution observed during initiation of cerebellar folding.

Mechanical instability and interfacial energy drive biofilm morphogenesis

Surface-attached bacterial communities called biofilms display a diversity of morphologies. Although structural and regulatory components required for biofilm formation are known, it is not understood how these essential constituents promote biofilm surface morphology. Here, using Vibrio cholerae as our model system, we combine mechanical measurements, theory and simulation, quantitative image analyses, surface energy characterizations, and mutagenesis to show that mechanical instabilities, including wrinkling and delamination, underlie the morphogenesis program of growing biofilms. We also identify interfacial energy as a key driving force for mechanomorphogenesis because it dictates the generation of new and the annihilation of existing interfaces. Finally, we discover feedback between mechanomorphogenesis and biofilm expansion, which shapes the overall biofilm contour. The morphogenesis principles that we discover in bacterial biofilms, which rely on mechanical instabilities and interfacial energies, should be generally applicable to morphogenesis processes in tissues in higher organisms.

Engineers have long studied how mechanical instabilities cause patterns to form in inanimate materials, and recently more attention has been given to how such forces affect biological systems. For example, stresses can build up within a tissue if one layer grows faster than an adjacent layer. The tissue can release this stress by wrinkling, folding or creasing.

Though ancient and single-celled, bacteria can also develop spectacular patterns when they exist in the lifestyle known as a biofilm: a community of cells adhered to a surface. But do mechanical instabilities drive the patterns seen in biofilms?

To investigate, Yan, Fei, Mao et al. grew biofilms of the bacterium called Vibrio cholerae – which causes the disease cholera – on solid, non-growing ‘substrates’. This work revealed that as the biofilms grow, their expansion is constrained by the substrate, and this situation generates mechanical stresses. To release the stresses, the biofilm initially folds to form wrinkles. Later, as the biofilm expands further, small parts of it detach from the substrate to form blisters. The same forces that keep water droplets spherical (known as interfacial forces) dictate how the blisters evolve, interact, and eventually shape the expanding biofilm. Using these principles, Yan et al. could engineer the biofilm into desired shapes.

Collectively, the results presented by Yan et al. connect the shape of the biofilm surface with its material properties, in particular its stiffness. Understanding this relationship could help researchers to develop new ways to remove harmful biofilms, such as those that cause disease or that damage underwater structures. The stiffness of biofilms is already known to affect how well bacteria can resist antibiotics. Future studies could look for new genes or compounds that change the material properties of a biofilm, thereby altering the biofilm surface.

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.

Photocontrollable Wrinkle Morphology Evolution on Azo-Based Multilayers for Hierarchical Surface Micropatterns Fabrication

Inspired by nature, comprehensive understanding and ingenious utilization of the self-organized wrinkling behaviors of the sandwiched multilayer bonded on substrates are important for engineering and/or functional laminated devices design. Herein, we report a facile and effective strategy to regulate the wrinkles morphology evolution and the resultant hierarchical surface micropatterns on azobenzene-based laminated multilayers by visible-light irradiation. Revealed by systematic experiments, the photocontrolled dynamic wrinkle evolutions are triggered by the reversible photoisomerization of azobenzene in the top azopolymer film and are strongly dependent on the intermediate photoinert layers (e.g., polystyrene and oxygen plasma-induced SiO _x layer) with the wrinkle-reinforcing effect or the stress relaxation acceleration effect. Interestingly, large-area well-defined hierarchical surface wrinkle patterns could be fabricated on the multilayers upon selective exposure. In the unexposed region, the wrinkles evolved into highly oriented patterns, whereas in the exposed region, they were fully erased or evolved into smaller-wavelength wrinkles. This study not only sheds light on the morphological evolution of the wrinkling laminated composites in engineering and nature but also paves a new avenue to conveniently and controllably realize the hierarchical stimulus-responsive surface patterns.

Nonlinear and nonlocal elasticity in coarse-grained differential-tension models of epithelia

The shapes of epithelial tissues result from a complex interplay of contractile forces in the cytoskeleta of the cells in the tissue and adhesion forces between them. A host of discrete, cell-based models describe these forces by assigning different surface tensions to the apical, basal, and lateral sides of the cells. These differential-tension models have been used to describe the deformations of epithelia in different living systems, but the underlying continuum mechanics at the scale of the epithelium are still unclear. Here, we derive a continuum theory for a simple differential-tension model of a two-dimensional epithelial monolayer and study the buckling of this epithelium under imposed compression. The analysis reveals how the cell-level properties encoded in the differential-tension model lead to linear and nonlinear elastic as well as nonlocal, nonelastic behavior at the continuum level.

Buckling without Bending: A New Paradigm in Morphogenesis

A curious feature of organ and organoid morphogenesis is that in certain cases, spatial oscillations in the thickness of the growing "film" are out of phase with the deformation of the slower-growing "substrate," while in other cases, the oscillations are in phase. The former cannot be explained by elastic bilayer instability, and contradict the notion that there is a universal mechanism by which brains, intestines, teeth, and other organs develop surface wrinkles and folds. Inspired by the microstructure of the embryonic cerebellum, we develop a new model of 2D morphogenesis in which system-spanning elastic fibers endow the organ with a preferred radius, while a separate fiber network resides in the otherwise fluidlike film at the outer edge of the organ and resists thickness gradients thereof. The tendency of the film to uniformly thicken or thin is described via a "growth potential." Several features of cerebellum, +blebbistatin organoid, and retinal fovea morphogenesis, including out-of-phase behavior and a film thickness amplitude that is comparable to the radius amplitude, are readily explained by our simple analytical model, as may be an observed scale invariance in the number of folds in the cerebellum. We also study a nonlinear variant of the model, propose further biological and bioinspired applications, and address how our model is and is not unique to the developing nervous system.

Modeling mechanical inhomogeneities in small populations of proliferating monolayers and spheroids

Understanding the mechanical behavior of multicellular monolayers and spheroids is fundamental to tissue culture, organism development, and the early stages of tumor growth. Proliferating cells in monolayers and spheroids experience mechanical forces as they grow and divide and local inhomogeneities in the mechanical microenvironment can cause individual cells within the multicellular system to grow and divide at different rates. This differential growth, combined with cell division and reorganization, leads to residual stress. Multiple different modeling approaches have been taken to understand and predict the residual stresses that arise in growing multicellular systems, particularly tumor spheroids. Here, we show that by using a mechanically robust agent-based model constructed with the peridynamic framework, we gain a better understanding of residual stresses in multicellular systems as they grow from a single cell. In particular, we focus on small populations of cells (1-100 s) where population behavior is highly stochastic and prior investigation has been limited. We compare the average strain energy density of cells in monolayers and spheroids using different growth and division rules and find that, on average, cells in spheroids have a higher strain energy density than cells in monolayers. We also find that cells in the interior of a growing spheroid are, on average, in compression. Finally, we demonstrate the importance of accounting for stochastic fluctuations in the mechanical environment, particularly when the cellular response to mechanical cues is nonlinear. The results presented here serve as a starting point for both further investigation with agent-based models, and for the incorporation of major findings from agent-based models into continuum scale models when explicit representation of individual cells is not computationally feasible.

Protection of cortex by overlying meninges tissue during dynamic indentation of the adolescent brain

Traumatic brain injury (TBI) has become a recent focus of biomedical research with a growing international effort targeting material characterization of brain tissue and simulations of trauma using computer models of the head and brain to try to elucidate the mechanisms and pathogenesis of TBI. The meninges, a collagenous protective tri-layer, which encloses the entire brain and spinal cord has been largely overlooked in these material characterization studies. This has resulted in a lack of accurate constitutive data for the cranial meninges, particularly under dynamic conditions such as those experienced during head impacts. The work presented here addresses this lack of data by providing for the first time, in situ large deformation material properties of the porcine dura-arachnoid mater composite under dynamic indentation. It is demonstrated that this tissue is substantially stiffer (shear modulus, μ=19.10±8.55kPa) and relaxes at a slower rate (τ₁=0.034±0.008s, τ₂=0.336±0.077s) than the underlying brain tissue (μ=6.97±2.26kPa, τ₁=0.021±0.007s, τ₂=0.199±0.036s), reducing the magnitudes of stress by 250% and 65% for strains that arise during indentation-type deformations in adolescent brains.

STATEMENT OF SIGNIFICANCE

We present the first mechanical analysis of the protective capacity of the cranial meninges using in situ micro-indentation techniques. Force-relaxation tests are performed on in situ meninges and cortex tissue, under large strain dynamic micro-indentation. A quasi-linear viscoelastic model is used subsequently, providing time-dependent mechanical properties of these neural tissues under loading conditions comparable to what is experienced in TBI. The reported data highlights the large differences in mechanical properties between these two tissues. Finite element simulations of the indentation experiments are also performed to investigate the protective capacity of the meninges. These simulations show that the meninges protect the underlying brain tissue by reducing the overall magnitude of stress by 250% and up to 65% for strains.

Wrinkling instabilities in soft bilayered systems

Wrinkling phenomena control the surface morphology of many technical and biological systems. While primary wrinkling has been extensively studied, experimentally, analytically and computationally, higher-order instabilities remain insufficiently understood, especially in systems with stiffness contrasts well below 100. Here, we use the model system of an elastomeric bilayer to experimentally characterize primary and secondary wrinkling at moderate stiffness contrasts. We systematically vary the film thickness and substrate prestretch to explore which parameters modulate the emergence of secondary instabilities, including period-doubling, period-tripling and wrinkle-to-fold transitions. Our experiments suggest that period-doubling is the favourable secondary instability mode and that period-tripling can emerge under disturbed boundary conditions. High substrate prestretch can suppress period-doubling and primary wrinkles immediately transform into folds. We combine analytical models with computational simulations to predict the onset of primary wrinkling, the post-buckling behaviour, secondary bifurcations and the wrinkle-to-fold transition. Understanding the mechanisms of pattern selection and identifying the critical control parameters of wrinkling will allow us to fabricate smart surfaces with tunable properties and to control undesired surface patterns like in the asthmatic airway.This article is part of the themed issue 'Patterning through instabilities in complex media: theory and applications.'

The mechanical importance of myelination in the central nervous system

Neurons in the central nervous system are surrounded and cross-linked by myelin, a fatty white substance that wraps around axons to create an electrically insulating layer. The electrical function of myelin is widely recognized; yet, its mechanical importance remains underestimated. Here we combined nanoindentation testing and histological staining to correlate brain stiffness to the degree of myelination in immature, pre-natal brains and mature, post-natal brains. We found that both gray and white matter tissue stiffened significantly (p≪0.001) upon maturation: the gray matter stiffness doubled from 0.31±0.20kPa pre-natally to 0.68±0.20kPa post-natally; the white matter stiffness tripled from 0.45±0.18kPa pre-natally to 1.33±0.64kPa post-natally. At the same time, the white matter myelin content increased significantly (p≪0.001) from 58±2% to 74±9%. White matter stiffness and myelin content were correlated with a Pearson correlation coefficient of ρ=0.92 (p≪0.001). Our study suggests that myelin is not only important to ensure smooth electrical signal propagation in neurons, but also to protect neurons against physical forces and provide a strong microstructural network that stiffens the white matter tissue as a whole. Our results suggest that brain tissue stiffness could serve as a biomarker for multiple sclerosis and other forms of demyelinating disorders. Understanding how tissue maturation translates into changes in mechanical properties and knowing the precise brain stiffness at different stages of life has important medical implications in development, aging, and neurodegeneration.

Quantifying the relationship between cell division angle and morphogenesis through computational modeling

When biological cells divide, they divide on a given angle. It has been shown experimentally that the orientation of cell division angle for a single cell can be described by a probability density function. However, the way in which the probability density function underlying cell division orientation influences population or tissue scale morphogenesis is unknown. Here we show that a computational approach, with thousands of stochastic simulations modeling growth and division of a population of cells, can be used to investigate this unknown. In this paper we examine two potential forms of the probability density function: a wrapped normal distribution and a binomial distribution. Our results demonstrate that for the wrapped normal distribution the standard deviation of the division angle, potentially interpreted as biological noise, controls the degree of tissue scale anisotropy. For the binomial distribution, we demonstrate a mechanism by which direction and degree of tissue scale anisotropy can be tuned via the probability of each division angle. We anticipate that the method presented in this paper and the results of these simulations will be a starting point for further investigation of this topic.

Sub-100 nm wrinkling of polydimethylsiloxane by double frontal oxidation

We demonstrate nanoscale wrinkling on polydimethylsiloxane (PDMS) at sub-100 nm length scales via a (double) frontal surface oxidation coupled with a mechanical compression. The kinetics of the glassy skin propagation is resolved by neutron and X-ray reflectivity, and atomic force microscopy, combined with mechanical wrinkling experiments to evaluate the resulting pattern formation. In conventional PDMS surface oxidation, the smallest wrinkling patterns attainable have an intrinsic lower wavelength limit due to the coupling of skin formation and front propagation at fixed strain ε_prestrain, whose maximum is, in turn, set by material failure. However, combining two different oxidative processes, ultra-violet ozonolysis followed by air plasma exposure, we break this limit by fabricating trilayer laminates with excellent interfacial properties and a sequence of moduli and layer thicknesses able to trivially reduce the surface topography to sub-100 nm dimensions. This method provides a powerful, yet simple, non-lithographic approach to extend surface patterning from visible to the deep UV range.