We have developed Simulation-based Reconstructed Diffusion (SbRD) to determine diffusion coefficients corrected for confinement effects and for the bias introduced by two-dimensional models describing a three-dimensional motion. We validate the method on simulated diffusion data in three-dimensional cell-shaped compartments. We use SbRD, combined with a new cell detection method, to determine the diffusion coefficients of a set of native proteins in Escherichia coli. We observe slower diffusion at the cell poles than in the nucleoid region of exponentially growing cells, which is independent of the presence of polysomes. Furthermore, we show that the newly formed pole of dividing cells exhibits a faster diffusion than the old one. We hypothesize that the observed slowdown at the cell poles is caused by the accumulation of aggregated or damaged proteins, and that the effect is asymmetric due to cell aging.

Connectomes represent comprehensive descriptions of neural connections in a nervous system to better understand and model central brain function and peripheral processing of afferent and efferent neural signals. Connectomes can be considered as a distinctive and necessary structural component alongside glial, vascular, neurochemical, and metabolic networks of the nervous systems of higher organisms that are required for the control of body functions and interaction with the environment. They are carriers of functional phenomena such as planning behavior and cognition, which are based on the processing of highly dynamic neural signaling patterns. In this study, we examine more detailed connectomes with edge weighting and orientation properties, in which reciprocal neuronal connections are also considered. Diffusion processes are a further necessary condition for generating dynamic bioelectric patterns in connectomes. Based on our precise connectome data, we investigate different diffusion-reaction models to study the propagation of dynamic concentration patterns in control and lesioned connectomes. Therefore, differential equations for modeling diffusion were combined with well-known reaction terms to allow the use of connection weights, connectivity orientation and spatial distances. Three reaction-diffusion systems Gray-Scott, Gierer-Meinhardt and Mimura-Murray were investigated. For this purpose, implicit solvers were implemented in a numerically stable reaction-diffusion system within the framework of neuroVIISAS. The implemented reaction-diffusion systems were applied to a subconnectome which shapes the mechanosensitive pathway that is strongly affected in the multiple sclerosis demyelination disease. It was found that demyelination modeling by connectivity weight modulation changes the oscillations of the target region, i.e. the primary somatosensory cortex, of the mechanosensitive pathway. In conclusion, a new application of reaction-diffusion systems to weighted and directed connectomes has been realized. Because the implementation was realized in the neuroVIISAS framework many possibilities for the study of dynamic reaction-diffusion processes in empirical connectomes as well as specific randomized network models are available now.

Recorded potentials in the extracellular space (ECS) of the brain is a standard measure of population activity in neural tissue. Computational models that simulate the relationship between the ECS potential and its underlying neurophysiological processes are commonly used in the interpretation of such measurements. Standard methods, such as volume-conductor theory and current-source density theory, assume that diffusion has a negligible effect on the ECS potential, at least in the range of frequencies picked up by most recording systems. This assumption remains to be verified. We here present a hybrid simulation framework that accounts for diffusive effects on the ECS potential. The framework uses (1) the NEURON simulator to compute the activity and ionic output currents from multicompartmental neuron models, and (2) the electrodiffusive Kirchhoff-Nernst-Planck framework to simulate the resulting dynamics of the potential and ion concentrations in the ECS, accounting for the effect of electrical migration as well as diffusion. Using this framework, we explore the effect that ECS diffusion has on the electrical potential surrounding a small population of 10 pyramidal neurons. The neural model was tuned so that simulations over ∼100 seconds of biological time led to shifts in ECS concentrations by a few millimolars, similar to what has been seen in experiments. By comparing simulations where ECS diffusion was absent with simulations where ECS diffusion was included, we made the following key findings: (i) ECS diffusion shifted the local potential by up to ∼0.2 mV. (ii) The power spectral density (PSD) of the diffusion-evoked potential shifts followed a 1/f2 power law. (iii) Diffusion effects dominated the PSD of the ECS potential for frequencies up to several hertz. In scenarios with large, but physiologically realistic ECS concentration gradients, diffusion was thus found to affect the ECS potential well within the frequency range picked up in experimental recordings.

We seek to elucidate the role of macromolecular crowding in transcription and translation. It is well known that stochasticity in gene expression can lead to differential gene expression and heterogeneity in a cell population. Recent experimental observations by Tan et al. have improved our understanding of the functional role of macromolecular crowding. It can be inferred from their observations that macromolecular crowding can lead to robustness in gene expression, resulting in a more homogeneous cell population. We introduce a spatial stochastic model to provide insight into this process. Our results show that macromolecular crowding reduces noise (as measured by the kurtosis of the mRNA distribution) in a cell population by limiting the diffusion of transcription factors (i.e. removing the unstable intermediate states), and that crowding by large molecules reduces noise more efficiently than crowding by small molecules. Finally, our simulation results provide evidence that the local variation in chromatin density as well as the total volume exclusion of the chromatin in the nucleus can induce a homogenous cell population.

In heme proteins, the efficient transport of ligands such as NO or O2 to the binding site is achieved via ligand migration networks. A quantitative assessment of ligand diffusion in these networks is thus essential for a better understanding of the function of these proteins. For this, Xe migration in truncated hemoglobin N (trHbN) of Mycobacterium Tuberculosis was studied using molecular dynamics simulations. Transitions between pockets of the migration network and intra-pocket relaxation occur on similar time scales (10 ps and 20 ps), consistent with low free energy barriers (1-2 kcal/mol). Depending on the pocket from where Xe enters a particular transition, the conformation of the side chains lining the transition region differs which highlights the coupling between ligand and protein degrees of freedom. Furthermore, comparison of transition probabilities shows that Xe migration in trHbN is a non-Markovian process. Memory effects arise due to protein rearrangements and coupled dynamics as Xe moves through it.

Biomineralization is the process by which organisms use minerals to harden their tissues and provide them with physical support. Biomineralizing cells concentrate the mineral in vesicles that they secret into a dedicated compartment where crystallization occurs. The dynamics of vesicle motion and the molecular mechanisms that control it, are not well understood. Sea urchin larval skeletogenesis provides an excellent platform for investigating the kinetics of mineral-bearing vesicles. Here we used lattice light-sheet microscopy to study the three-dimensional (3D) dynamics of calcium-bearing vesicles in the cells of normal sea urchin embryos and of embryos where skeletogenesis is blocked through the inhibition of Vascular Endothelial Growth Factor Receptor (VEGFR). We developed computational tools for displaying 3D-volumetric movies and for automatically quantifying vesicle dynamics. Our findings imply that calcium vesicles perform an active diffusion motion in both, calcifying (skeletogenic) and non-calcifying (ectodermal) cells of the embryo. The diffusion coefficient and vesicle speed are larger in the mesenchymal skeletogenic cells compared to the epithelial ectodermal cells. These differences are possibly due to the distinct mechanical properties of the two tissues, demonstrated by the enhanced f-actin accumulation and myosinII activity in the ectodermal cells compared to the skeletogenic cells. Vesicle motion is not directed toward the biomineralization compartment, but the vesicles slow down when they approach it, and probably bind for mineral deposition. VEGFR inhibition leads to an increase of vesicle volume but hardly changes vesicle kinetics and doesn't affect f-actin accumulation and myosinII activity. Thus, calcium vesicles perform an active diffusion motion in the cells of the sea urchin embryo, with diffusion length and speed that inversely correlate with the strength of the actomyosin network. Overall, our studies provide an unprecedented view of calcium vesicle 3D-dynamics and point toward cytoskeleton remodeling as an important effector of the motion of mineral-bearing vesicles.

Loss of mitochondrial function is a fundamental determinant of cell injury and death. In heart cells under metabolic stress, we have previously described how the abrupt collapse or oscillation of the mitochondrial energy state is synchronized across the mitochondrial network by local interactions dependent upon reactive oxygen species (ROS). Here, we develop a mathematical model of ROS-induced ROS release (RIRR) based on reaction-diffusion (RD-RIRR) in one- and two-dimensional mitochondrial networks. The nodes of the RD-RIRR network are comprised of models of individual mitochondria that include a mechanism of ROS-dependent oscillation based on the interplay between ROS production, transport, and scavenging; and incorporating the tricarboxylic acid (TCA) cycle, oxidative phosphorylation, and Ca(2+) handling. Local mitochondrial interaction is mediated by superoxide (O2.-) diffusion and the O2.(-)-dependent activation of an inner membrane anion channel (IMAC). In a 2D network composed of 500 mitochondria, model simulations reveal DeltaPsi(m) depolarization waves similar to those observed when isolated guinea pig cardiomyocytes are subjected to a localized laser-flash or antioxidant depletion. The sensitivity of the propagation rate of the depolarization wave to O(2.-) diffusion, production, and scavenging in the reaction-diffusion model is similar to that observed experimentally. In addition, we present novel experimental evidence, obtained in permeabilized cardiomyocytes, confirming that DeltaPsi(m) depolarization is mediated specifically by O2.-). The present work demonstrates that the observed emergent macroscopic properties of the mitochondrial network can be reproduced in a reaction-diffusion model of RIRR. Moreover, the findings have uncovered a novel aspect of the synchronization mechanism, which is that clusters of mitochondria that are oscillating can entrain mitochondria that would otherwise display stable dynamics. The work identifies the fundamental mechanisms leading from the failure of individual organelles to the whole cell, thus it has important implications for understanding cell death during the progression of heart disease.

Nitric oxide (NO) is a gaseous signaling molecule that plays an important role in neurovascular coupling. NO produced by neurons diffuses into the smooth muscle surrounding cerebral arterioles, driving vasodilation. However, the rate of NO degradation in hemoglobin is orders of magnitude higher than in brain tissue, though how this might impact NO signaling dynamics is not completely understood. We used simulations to investigate how the spatial and temporal patterns of NO generation and degradation impacted dilation of a penetrating arteriole in cortex. We found that the spatial location of NO production and the size of the vessel both played an important role in determining its responsiveness to NO. The much higher rate of NO degradation and scavenging of NO in the blood relative to the tissue drove emergent vascular dynamics. Large vasodilation events could be followed by post-stimulus constrictions driven by the increased degradation of NO by the blood, and vasomotion-like 0.1-0.3 Hz oscillations could also be generated. We found that these dynamics could be enhanced by elevation of free hemoglobin in the plasma, which occurs in diseases such as malaria and sickle cell anemia, or following blood transfusions. Finally, we show that changes in blood flow during hypoxia or hyperoxia could be explained by altered NO degradation in the parenchyma. Our simulations suggest that many common vascular dynamics may be emergent phenomena generated by NO degradation by the blood or parenchyma.

The dynamics of inheritance of histones and their associated modifications across cell divisions can have major consequences on maintenance of the cellular epigenomic state. Recent experiments contradict the long-held notion that histone inheritance during replication is always local, suggesting that active and repressed regions of the genome exhibit fundamentally different histone dynamics independent of transcription-coupled turnover. Here we develop a stochastic model of histone dynamics at the replication fork and demonstrate that differential diffusivity of histones in active versus repressed chromatin is sufficient to quantitatively explain these recent experiments. Further, we use the model to predict patterns in histone mark similarity between pairs of genomic loci that should be developed as a result of diffusion, but cannot originate from either PRC2 mediated mark spreading or transcriptional processes. Interestingly, using a combination of CHIP-seq, replication timing and Hi-C datasets we demonstrate that all the computationally predicted patterns are consistently observed for both active and repressive histone marks in two different cell lines. While direct evidence for histone diffusion remains controversial, our results suggest that dislodged histones in euchromatin and facultative heterochromatin may exhibit some level of diffusion within "Diffusion-Accessible-Domains" (DADs), leading to redistribution of epigenetic marks within and across chromosomes. Preservation of the epigenomic state across cell divisions therefore might be achieved not by passing on strict positional information of histone marks, but by maintaining the marks in somewhat larger DADs of the genome.

Integrating reference datasets (e.g. from high-throughput experiments) with unstructured and manually-assembled information (e.g. notes or comments from individual researchers) has the potential to tailor bioinformatic analyses to specific needs and to lead to new insights. However, developing bespoke analysis pipelines from scratch is time-consuming, and general tools for exploring such heterogeneous data are not available. We argue that by treating all data as text, a knowledge-base can accommodate a range of bioinformatic data types and applications. We show that a database coupled to nearest-neighbor algorithms can address common tasks such as gene-set analysis as well as specific tasks such as ontology translation. We further show that a mathematical transformation motivated by diffusion can be effective for exploration across heterogeneous datasets. Diffusion enables the knowledge-base to begin with a sparse query, impute more features, and find matches that would otherwise remain hidden. This can be used, for example, to map multi-modal queries consisting of gene symbols and phenotypes to descriptions of diseases. Diffusion also enables user-driven learning: when the knowledge-base cannot provide satisfactory search results in the first instance, users can improve the results in real-time by adding domain-specific knowledge. User-driven learning has implications for data management, integration, and curation.

Genetically encoded biosensors based on fluorescence resonance energy transfer (FRET) have been widely applied to visualize the molecular activity in live cells with high spatiotemporal resolution. However, the rapid diffusion of biosensor proteins hinders a precise reconstruction of the actual molecular activation map. Based on fluorescence recovery after photobleaching (FRAP) experiments, we have developed a finite element (FE) method to analyze, simulate, and subtract the diffusion effect of mobile biosensors. This method has been applied to analyze the mobility of Src FRET biosensors engineered to reside at different subcompartments in live cells. The results indicate that the Src biosensor located in the cytoplasm moves 4-8 folds faster (0.93+/-0.06 microm(2)/sec) than those anchored on different compartments in plasma membrane (at lipid raft: 0.11+/-0.01 microm(2)/sec and outside: 0.18+/-0.02 microm(2)/sec). The mobility of biosensor at lipid rafts is slower than that outside of lipid rafts and is dominated by two-dimensional diffusion. When this diffusion effect was subtracted from the FRET ratio images, high Src activity at lipid rafts was observed at clustered regions proximal to the cell periphery, which remained relatively stationary upon epidermal growth factor (EGF) stimulation. This result suggests that EGF induced a Src activation at lipid rafts with well-coordinated spatiotemporal patterns. Our FE-based method also provides an integrated platform of image analysis for studying molecular mobility and reconstructing the spatiotemporal activation maps of signaling molecules in live cells.

The analysis of single-cell genomics data presents several statistical challenges, and extensive efforts have been made to produce methods for the analysis of this data that impute missing values, address sampling issues and quantify and correct for noise. In spite of such efforts, no consensus on best practices has been established and all current approaches vary substantially based on the available data and empirical tests. The k-Nearest Neighbor Graph (kNN-G) is often used to infer the identities of, and relationships between, cells and is the basis of many widely used dimensionality-reduction and projection methods. The kNN-G has also been the basis for imputation methods using, e.g., neighbor averaging and graph diffusion. However, due to the lack of an agreed-upon optimal objective function for choosing hyperparameters, these methods tend to oversmooth data, thereby resulting in a loss of information with regard to cell identity and the specific gene-to-gene patterns underlying regulatory mechanisms. In this paper, we investigate the tuning of kNN- and diffusion-based denoising methods with a novel non-stochastic method for optimally preserving biologically relevant informative variance in single-cell data. The framework, Denoising Expression data with a Weighted Affinity Kernel and Self-Supervision (DEWÄKSS), uses a self-supervised technique to tune its parameters. We demonstrate that denoising with optimal parameters selected by our objective function (i) is robust to preprocessing methods using data from established benchmarks, (ii) disentangles cellular identity and maintains robust clusters over dimension-reduction methods, (iii) maintains variance along several expression dimensions, unlike previous heuristic-based methods that tend to oversmooth data variance, and (iv) rarely involves diffusion but rather uses a fixed weighted kNN graph for denoising. Together, these findings provide a new understanding of kNN- and diffusion-based denoising methods. Code and example data for DEWÄKSS is available at https://gitlab.com/Xparx/dewakss/-/tree/Tjarnberg2020branch.

Stochastic channel gating is the major source of intrinsic neuronal noise whose functional consequences at the microcircuit- and network-levels have been only partly explored. A systematic study of this channel noise in large ensembles of biophysically detailed model neurons calls for the availability of fast numerical methods. In fact, exact techniques employ the microscopic simulation of the random opening and closing of individual ion channels, usually based on Markov models, whose computational loads are prohibitive for next generation massive computer models of the brain. In this work, we operatively define a procedure for translating any Markov model describing voltage- or ligand-gated membrane ion-conductances into an effective stochastic version, whose computer simulation is efficient, without compromising accuracy. Our approximation is based on an improved Langevin-like approach, which employs stochastic differential equations and no Montecarlo methods. As opposed to an earlier proposal recently debated in the literature, our approximation reproduces accurately the statistical properties of the exact microscopic simulations, under a variety of conditions, from spontaneous to evoked response features. In addition, our method is not restricted to the Hodgkin-Huxley sodium and potassium currents and is general for a variety of voltage- and ligand-gated ion currents. As a by-product, the analysis of the properties emerging in exact Markov schemes by standard probability calculus enables us for the first time to analytically identify the sources of inaccuracy of the previous proposal, while providing solid ground for its modification and improvement we present here.

Recent advances in electron microscopy have enabled the imaging of single cells in 3D at nanometer length scale resolutions. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. Enabling such simulations requires watertight meshing of electron micrograph images into 3D volume meshes, which can then form the basis of computer simulations of such processes using numerical techniques such as the finite element method. In this paper, we describe the use of our recently rewritten mesh processing software, GAMer 2, to bridge the gap between poorly conditioned meshes generated from segmented micrographs and boundary marked tetrahedral meshes which are compatible with simulation. We demonstrate the application of a workflow using GAMer 2 to a series of electron micrographs of neuronal dendrite morphology explored at three different length scales and show that the resulting meshes are suitable for finite element simulations. This work is an important step towards making physical simulations of biological processes in realistic geometries routine. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery at the interface of geometry and cellular processes. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open.

We introduce a Stochastic Reaction-Diffusion-Dynamics Model (SRDDM) for simulations of cellular mechanochemical processes with high spatial and temporal resolution. The SRDDM is mapped into the CellDynaMo package, which couples the spatially inhomogeneous reaction-diffusion master equation to account for biochemical reactions and molecular transport within the Langevin Dynamics (LD) framework to describe dynamic mechanical processes. This computational infrastructure allows the simulation of hours of molecular machine dynamics in reasonable wall-clock time. We apply SRDDM to test performance of the Search-and-Capture of mitotic spindle assembly by simulating, in three spatial dimensions, dynamic instability of elastic microtubules anchored in two centrosomes, movement and deformations of geometrically realistic centromeres with flexible kinetochores and chromosome arms. Furthermore, the SRDDM describes the mechanics and kinetics of Ndc80 linkers mediating transient attachments of microtubules to the chromosomal kinetochores. The rates of these attachments and detachments depend upon phosphorylation states of the Ndc80 linkers, which are regulated in the model by explicitly accounting for the reactions of Aurora A and B kinase enzymes undergoing restricted diffusion. We find that there is an optimal rate of microtubule-kinetochore detachments which maximizes the accuracy of the chromosome connections, that adding chromosome arms to kinetochores improve the accuracy by slowing down chromosome movements, that Aurora A and kinetochore deformations have a small positive effect on the attachment accuracy, and that thermal fluctuations of the microtubules increase the rates of kinetochore capture and also improve the accuracy of spindle assembly.

This study presents a modelling framework in which information on muscle fiber direction and orientation during contraction is derived from diffusion tensor imaging (DTI) and incorporated in a computational model of the surface electromyographic (EMG) signal. The proposed model makes use of the principle of reciprocity to simultaneously calculate the electric potentials produced at the recording electrode by charges distributed along an arbitrary number of muscle fibers within the muscle, allowing for a computationally efficient evaluation of extracellular motor unit action potentials. The approach is applied to the complex architecture of the first dorsal interosseous (FDI) muscle of the hand to simulate EMG during index finger flexion and abduction. Using diffusion tensor imaging methods, the results show how muscle fiber orientation and curvature in this intrinsic hand muscle change during flexion and abduction. Incorporation of anatomically accurate muscle architecture and other hand tissue morphologies enables the model to capture variations in extracellular action potential waveform shape across the motor unit population and to predict experimentally observed differences in EMG signal features when switching from index finger abduction to flexion. The simulation results illustrate how structural and electrical properties of the tissues comprising the volume conductor, in combination with fiber direction and curvature, shape the detected action potentials. Using the model, the relative contribution of motor units of different sizes located throughout the muscle under both conditions is examined, yielding a prediction of the detection profile of the surface EMG electrode array over the muscle cross-section.

Guilt-by-association codifies the empirical observation that a gene's function is informed by its neighborhood in a biological network. This would imply that when a gene's network context is altered, for instance in disease condition, so could be the gene's function. Although context-specific changes in biological networks have been explored, the potential changes they may induce on the functional roles of genes are yet to be characterized. Here we analyze, for the first time, the network-induced potential functional changes in breast cancer. Using transcriptomic samples for 1047 breast tumors and 110 healthy breast tissues from TCGA, we derive sample-specific protein interaction networks and assign sample-specific functions to genes via a diffusion strategy. Testing for significant changes in the inferred functions between normal and cancer samples, we find several functions to have significantly gained or lost genes in cancer, not due to differential expression of genes known to perform the function, but rather due to changes in the network topology. Our predicted functional changes are supported by mutational and copy number profiles in breast cancers. Our diffusion-based functional assignment provides a novel characterization of a tumor that is complementary to the standard approach based on functional annotation alone. Importantly, this characterization is effective in predicting patient survival, as well as in predicting several known histopathological subtypes of breast cancer.

After endocytic uptake, influenza viruses transit early endosomal compartments and eventually reach late endosomes. There, the viral glycoprotein hemagglutinin (HA) triggers fusion between endosomal and viral membrane, a critical step that leads to release of the viral segmented genome destined to reach the cell nucleus. Endosomal maturation is a complex process involving acidification of the endosomal lumen as well as endosome motility along microtubules. While the pH drop is clearly critical for the conformational change and membrane fusion activity of HA, the effect of intracellular transport dynamics on the progress of infection remains largely unclear. In this study, we developed a comprehensive mathematical model accounting for the first steps of influenza virus infection. We calibrated our model with experimental data and challenged its predictions using recombinant viruses with altered pH sensitivity of HA. We identified the time point of virus-endosome fusion and thereby the diffusion distance of the released viral genome to the nucleus as a critical bottleneck for efficient virus infection. Further, we concluded and supported experimentally that the viral RNA is subjected to cytosolic degradation strongly limiting the probability of a successful genome import into the nucleus.

Most viruses have evolved strategies for preventing interferon (IFN) secretion and evading innate immunity. Recent work has shown that viral shutdown of IFN secretion can be viewed as a social trait, since the ability of a given virus to evade IFN-mediated immunity depends on the phenotype of neighbor viruses. Following this idea, we investigate the role of spatial structure in the evolution of innate immunity evasion. For this, we model IFN signaling and viral spread using a spatially explicit approximation that combines a diffusion-reaction model and cellular automaton. Our results indicate that the benefits of preventing IFN secretion for a virus are strongly determined by spatial structure through paracrine IFN signaling. Therefore, innate immunity evasion can evolve as a cooperative or even altruistic trait based on indirect fitness effects that IFN shutdown exerts on other members of the viral population. We identify key factors determining whether evasion from IFN-mediated immunity should evolve, such as population bottlenecks occurring during viral transmission, the relative speed of cellular infection and IFN secretion, and the diffusion properties of the medium.

The earliest models for how morphogen gradients guide embryonic patterning failed to account for experimental observations of temporal refinement in gene expression domains. Following theoretical and experimental work in this area, dynamic positional information has emerged as a conceptual framework to discuss how cells process spatiotemporal inputs into downstream patterns. Here, we show that diffusion determines the mathematical means by which bistable gene expression boundaries shift over time, and therefore how cells interpret positional information conferred from morphogen concentration. First, we introduce a metric for assessing reproducibility in boundary placement or precision in systems where gene products do not diffuse, but where morphogen concentrations are permitted to change in time. We show that the dynamics of the gradient affect the sensitivity of the final pattern to variation in initial conditions, with slower gradients reducing the sensitivity. Second, we allow gene products to diffuse and consider gene expression boundaries as propagating wavefronts with velocity modulated by local morphogen concentration. We harness this perspective to approximate a PDE model as an ODE that captures the position of the boundary in time, and demonstrate the approach with a preexisting model for Hunchback patterning in fruit fly embryos. We then propose a design that employs antiparallel morphogen gradients to achieve accurate boundary placement that is robust to scaling. Throughout our work we draw attention to tradeoffs among initial conditions, boundary positioning, and the relative timescales of network and gradient evolution. We conclude by suggesting that mathematical theory should serve to clarify not just our quantitative, but also our intuitive understanding of patterning processes.