Approximate Bayesian computation reveals the importance of repeated measurements for parameterising cell-based models of growing tissues

Bayesian parameter inference for epithelial mechanics

Cell-based mechanical models, such as the Cell Vertex Model (CVM), have proven useful for studying the mechanical control of epithelial tissue dynamics. We recently developed a statistical method called image-based parameter inference for formulating CVM model functions and estimating their parameters from image data of epithelial tissues. In this study, we employed Bayesian statistics to improve the utility and flexibility of image-based parameter inference. Tests on synthetic data confirmed that both our non-hierarchical and hierarchical Bayesian models provide accurate estimates of model parameters. By applying this method to Drosophila wings, we demonstrated that the reliability of parameter estimation is closely linked to the mechanical anisotropies present in the tissue. Moreover, we revealed that the cortical elasticity term is dispensable for explaining force-shape correlations in vivo. We anticipate that the flexibility of the Bayesian statistical framework will facilitate the integration of various types of information, thereby contributing to the quantitative dissection of the mechanical control of tissue dynamics.

An Approximate Bayesian Computation Approach for Embryonic Astrocyte Migration Model Reduction

During embryonic development of the retina of the eye, astrocytes, a type of glial cell, migrate over the retinal surface and form a dynamic mesh. This mesh then serves as scaffolding for blood vessels to form the retinal vasculature network that supplies oxygen and nutrients to the inner portion of the retina. Astrocyte spreading proceeds in a radially symmetric manner over the retinal surface. Additionally, astrocytes mature from astrocyte precursor cells (APCs) to immature perinatal astrocytes (IPAs) during this embryonic stage. We extend a previously-developed continuum model that describes tension-driven migration and oxygen and growth factor influenced proliferation and differentiation. Comparing numerical simulations to experimental data, we identify model equation components that can be removed via model reduction using approximate Bayesian computation (ABC). Our results verify experimental studies indicating that the choroid oxygen supply plays a negligible role in promoting differentiation of APCs into IPAs and in promoting IPA proliferation, and the hyaloid artery oxygen supply and APC apoptosis play negligible roles in astrocyte spreading and differentiation.

Reverse engineering morphogenesis through Bayesian optimization of physics-based models

Morphogenetic programs coordinate cell signaling and mechanical interactions to shape organs. In systems and synthetic biology, a key challenge is determining optimal cellular interactions for predicting organ shape, size, and function. Physics-based models defining the subcellular force distribution facilitate this, but it is challenging to calibrate parameters in these models from data. To solve this inverse problem, we created a Bayesian optimization framework to determine the optimal cellular force distribution such that the predicted organ shapes match the experimentally observed organ shapes. This integrative framework employs Gaussian Process Regression, a non-parametric kernel-based probabilistic machine learning modeling paradigm, to learn the mapping functions relating to the morphogenetic programs that maintain the final organ shape. We calibrated and tested the method on Drosophila wing imaginal discs to study mechanisms that regulate epithelial processes ranging from development to cancer. The parameter estimation framework successfully infers the underlying changes in core parameters needed to match simulation data with imaging data of wing discs perturbed with collagenase. The computational pipeline identifies distinct parameter sets mimicking wild-type shapes. It enables a global sensitivity analysis to support the regulation of actomyosin contractility and basal ECM stiffness to generate and maintain the curved shape of the wing imaginal disc. The optimization framework, combined with experimental imaging, identified that Piezo, a mechanosensitive ion channel, impacts fold formation by regulating the apical-basal balance of actomyosin contractility and elasticity of ECM. This workflow is extensible toward reverse-engineering morphogenesis across organ systems and for real-time control of complex multicellular systems.

Data-driven modelling makes quantitative predictions regarding bacteria surface motility

In this work, we quantitatively compare computer simulations and existing cell tracking data of P. aeruginosa surface motility in order to analyse the underlying motility mechanism. We present a three dimensional twitching motility model, that simulates the extension, retraction and surface association of individual Type IV Pili (TFP), and is informed by recent experimental observations of TFP. Sensitivity analysis is implemented to minimise the number of model parameters, and quantitative estimates for the remaining parameters are inferred from tracking data by approximate Bayesian computation. We argue that the motility mechanism is highly sensitive to experimental conditions. We predict a TFP retraction speed for the tracking data we study that is in a good agreement with experimental results obtained under very similar conditions. Furthermore, we examine whether estimates for biologically important parameters, whose direct experimental determination is challenging, can be inferred directly from tracking data. One example is the width of the distribution of TFP on the bacteria body. We predict that the TFP are broadly distributed over the bacteria pole in both walking and crawling motility types. Moreover, we identified specific configurations of TFP that lead to transitions between walking and crawling states.

Choice of friction coefficient deeply affects tissue behaviour in stochastic epithelial vertex models

To understand the mechanisms that coordinate the formation of biological tissues, the use of numerical implementations is necessary. The complexity of such models involves many assumptions and parameter choices that result in unpredictable consequences, obstructing the comparison with experimental data. Here, we focus on vertex models, a family of spatial models used extensively to simulate the dynamics of epithelial tissues. Usually, in the literature, the choice of the friction coefficient is not addressed using quasi-static deformation arguments that generally do not apply to realistic scenarios. In this manuscript, we discuss the role that the choice of friction coefficient has on the relaxation times and consequently in the conditions of cell cycle progression and division. We explore the effects that these changes have on the morphology, growth rate and topological transitions of the tissue dynamics. These results provide a deeper understanding of the role that an accurate mechanical description plays in the use of vertex models as inference tools. This article is part of a discussion meeting issue 'Causes and consequences of stochastic processes in development and disease'.

Mutation of p53 increases the competitive ability of pluripotent stem cells

During development, the rate of tissue growth is determined by the relative balance of cell division and cell death. Cell competition is a fitness quality-control mechanism that contributes to this balance by eliminating viable cells that are less fit than their neighbours. The mutations that confer cells with a competitive advantage and the dynamics of the interactions between winner and loser cells are not well understood. Here, we show that embryonic cells lacking the tumour suppressor p53 are 'super-competitors' that eliminate their wild-type neighbours through the direct induction of apoptosis. This elimination is context dependent, as it does not occur when cells are pluripotent and it is triggered by the onset of differentiation. Furthermore, by combining mathematical modelling and cell-based assays we show that the elimination of wild-type cells is not through competition for space or nutrients, but instead is mediated by short-range interactions that are dependent on the local cell neighbourhood. This highlights the importance of the local cell neighbourhood and the competitive interactions within this neighbourhood for the regulation of proliferation during early embryonic development.

Free and Interfacial Boundaries in Individual-Based Models of Multicellular Biological systems

Coordination of cell behaviour is key to a myriad of biological processes including tissue morphogenesis, wound healing, and tumour growth. As such, individual-based computational models, which explicitly describe inter-cellular interactions, are commonly used to model collective cell dynamics. However, when using individual-based models, it is unclear how descriptions of cell boundaries affect overall population dynamics. In order to investigate this we define three cell boundary descriptions of varying complexities for each of three widely used off-lattice individual-based models: overlapping spheres, Voronoi tessellation, and vertex models. We apply our models to multiple biological scenarios to investigate how cell boundary description can influence tissue-scale behaviour. We find that the Voronoi tessellation model is most sensitive to changes in the cell boundary description with basic models being inappropriate in many cases. The timescale of tissue evolution when using an overlapping spheres model is coupled to the boundary description. The vertex model is demonstrated to be the most stable to changes in boundary description, though still exhibits timescale sensitivity. When using individual-based computational models one should carefully consider how cell boundaries are defined. To inform future work, we provide an exploration of common individual-based models and cell boundary descriptions in frequently studied biological scenarios and discuss their benefits and disadvantages.

Reverse engineering morphogenesis through Bayesian optimization of physics-based models

Morphogenetic programs direct the cell signaling and nonlinear mechanical interactions between multiple cell types and tissue layers to define organ shape and size. A key challenge for systems and synthetic biology is determining optimal combinations of intra- and inter-cellular interactions to predict an organ's shape, size, and function. Physics-based mechanistic models that define the subcellular force distribution facilitate this, but it is extremely challenging to calibrate parameters in these models from data. To solve this inverse problem, we created a Bayesian optimization framework to determine the optimal cellular force distribution such that the predicted organ shapes match the desired organ shapes observed within the experimental imaging data. This integrative framework employs Gaussian Process Regression (GPR), a non-parametric kernel-based probabilistic machine learning modeling paradigm, to learn the mapping functions relating to the morphogenetic programs that generate and maintain the final organ shape. We calibrated and tested the method on cross-sections of Drosophila wing imaginal discs, a highly informative model organ system, to study mechanisms that regulate epithelial processes that range from development to cancer. As a specific test case, the parameter estimation framework successfully infers the underlying changes in core parameters needed to match simulation data with time series imaging data of wing discs perturbed with collagenase. Unexpectedly, the framework also identifies multiple distinct parameter sets that generate shapes similar to wild-type organ shapes. This platform enables an efficient, global sensitivity analysis to support the necessity of both actomyosin contractility and basal ECM stiffness to generate and maintain the curved shape of the wing imaginal disc. The optimization framework, combined with fixed tissue imaging, identified that Piezo, a mechanosensitive ion channel, impacts fold formation by regulating the apical-basal balance of actomyosin contractility and elasticity of ECM. This framework is extensible toward reverse-engineering the morphogenesis of any organ system and can be utilized in real-time control of complex multicellular systems.

Cell cycle dynamics control fluidity of the developing mouse neuroepithelium

As developing tissues grow in size and undergo morphogenetic changes, their material properties may be altered. Such changes result from tension dynamics at cell contacts or cellular jamming. Yet, in many cases, the cellular mechanisms controlling the physical state of growing tissues are unclear. We found that at early developmental stages, the epithelium in the developing mouse spinal cord maintains both high junctional tension and high fluidity. This is achieved via a mechanism in which interkinetic nuclear movements generate cell area dynamics that drive extensive cell rearrangements. Over time, the cell proliferation rate declines, effectively solidifying the tissue. Thus, unlike well-studied jamming transitions, the solidification uncovered here resembles a glass transition that depends on the dynamical stresses generated by proliferation and differentiation. Our finding that the fluidity of developing epithelia is linked to interkinetic nuclear movements and the dynamics of growth is likely to be relevant to multiple developing tissues.

Development of a scoring function for comparing simulated and experimental tumor spheroids

Progress continues in the field of cancer biology, yet much remains to be unveiled regarding the mechanisms of cancer invasion. In particular, complex biophysical mechanisms enable a tumor to remodel the surrounding extracellular matrix (ECM), allowing cells to invade alone or collectively. Tumor spheroids cultured in collagen represent a simplified, reproducible 3D model system, which is sufficiently complex to recapitulate the evolving organization of cells and interaction with the ECM that occur during invasion. Recent experimental approaches enable high resolution imaging and quantification of the internal structure of invading tumor spheroids. Concurrently, computational modeling enables simulations of complex multicellular aggregates based on first principles. The comparison between real and simulated spheroids represents a way to fully exploit both data sources, but remains a challenge. We hypothesize that comparing any two spheroids requires first the extraction of basic features from the raw data, and second the definition of key metrics to match such features. Here, we present a novel method to compare spatial features of spheroids in 3D. To do so, we define and extract features from spheroid point cloud data, which we simulated using Cells in Silico (CiS), a high-performance framework for large-scale tissue modeling previously developed by us. We then define metrics to compare features between individual spheroids, and combine all metrics into an overall deviation score. Finally, we use our features to compare experimental data on invading spheroids in increasing collagen densities. We propose that our approach represents the basis for defining improved metrics to compare large 3D data sets. Moving forward, this approach will enable the detailed analysis of spheroids of any origin, one application of which is informing in silico spheroids based on their in vitro counterparts. This will enable both basic and applied researchers to close the loop between modeling and experiments in cancer research.

Systematic comparison of modeling fidelity levels and parameter inference settings applied to negative feedback gene regulation

Quantitative stochastic models of gene regulatory networks are important tools for studying cellular regulation. Such models can be formulated at many different levels of fidelity. A practical challenge is to determine what model fidelity to use in order to get accurate and representative results. The choice is important, because models of successively higher fidelity come at a rapidly increasing computational cost. In some situations, the level of detail is clearly motivated by the question under study. In many situations however, many model options could qualitatively agree with available data, depending on the amount of data and the nature of the observations. Here, an important distinction is whether we are interested in inferring the true (but unknown) physical parameters of the model or if it is sufficient to be able to capture and explain available data. The situation becomes complicated from a computational perspective because inference needs to be approximate. Most often it is based on likelihood-free Approximate Bayesian Computation (ABC) and here determining which summary statistics to use, as well as how much data is needed to reach the desired level of accuracy, are difficult tasks. Ultimately, all of these aspects-the model fidelity, the available data, and the numerical choices for inference-interplay in a complex manner. In this paper we develop a computational pipeline designed to systematically evaluate inference accuracy for a wide range of true known parameters. We then use it to explore inference settings for negative feedback gene regulation. In particular, we compare a detailed spatial stochastic model, a coarse-grained compartment-based multiscale model, and the standard well-mixed model, across several data-scenarios and for multiple numerical options for parameter inference. Practically speaking, this pipeline can be used as a preliminary step to guide modelers prior to gathering experimental data. By training Gaussian processes to approximate the distance function values, we are able to substantially reduce the computational cost of running the pipeline.

Image-based parameter inference for epithelial mechanics

Measuring mechanical parameters in tissues, such as the elastic modulus of cell-cell junctions, is essential to decipher the mechanical control of morphogenesis. However, their in vivo measurement is technically challenging. Here, we formulated an image-based statistical approach to estimate the mechanical parameters of epithelial cells. Candidate mechanical models are constructed based on force-cell shape correlations obtained from image data. Substitution of the model functions into force-balance equations at the cell vertex leads to an equation with respect to the parameters of the model, by which one can estimate the parameter values using a least-squares method. A test using synthetic data confirmed the accuracy of parameter estimation and model selection. By applying this method to Drosophila epithelial tissues, we found that the magnitude and orientation of feedback between the junction tension and shrinkage, which are determined by the spring constant of the junction, were correlated with the elevation of tension and myosin-II on shrinking junctions during cell rearrangement. Further, this method clarified how alterations in tissue polarity and stretching affect the anisotropy in tension parameters. Thus, our method provides a novel approach to uncovering the mechanisms governing epithelial morphogenesis.

Parameter estimation and uncertainty quantification using information geometry

In this work, we: (i) review likelihood-based inference for parameter estimation and the construction of confidence regions; and (ii) explore the use of techniques from information geometry, including geodesic curves and Riemann scalar curvature, to supplement typical techniques for uncertainty quantification, such as Bayesian methods, profile likelihood, asymptotic analysis and bootstrapping. These techniques from information geometry provide data-independent insights into uncertainty and identifiability, and can be used to inform data collection decisions. All code used in this work to implement the inference and information geometry techniques is available on GitHub.

Seven challenges in the multiscale modeling of multicellular tissues

The growth and dynamics of multicellular tissues involve tightly regulated and coordinated morphogenetic cell behaviors, such as shape changes, movement, and division, which are governed by subcellular machinery and involve coupling through short- and long-range signals. A key challenge in the fields of developmental biology, tissue engineering and regenerative medicine is to understand how relationships between scales produce emergent tissue-scale behaviors. Recent advances in molecular biology, live-imaging and ex vivo techniques have revolutionized our ability to study these processes experimentally. To fully leverage these techniques and obtain a more comprehensive understanding of the causal relationships underlying tissue dynamics, computational modeling approaches are increasingly spanning multiple spatial and temporal scales, and are coupling cell shape, growth, mechanics, and signaling. Yet such models remain challenging: modeling at each scale requires different areas of technical skills, while integration across scales necessitates the solution to novel mathematical and computational problems. This review aims to summarize recent progress in multiscale modeling of multicellular tissues and to highlight ongoing challenges associated with the construction, implementation, interrogation, and validation of such models. This article is categorized under: Reproductive System Diseases > Computational Models Metabolic Diseases > Computational Models Cancer > Computational Models.

Estimating parameters of a stochastic cell invasion model with fluorescent cell cycle labelling using approximate Bayesian computation

We develop a parameter estimation method based on approximate Bayesian computation (ABC) for a stochastic cell invasion model using fluorescent cell cycle labelling with proliferation, migration and crowding effects. Previously, inference has been performed on a deterministic version of the model fitted to cell density data, and not all parameters were identifiable. Considering the stochastic model allows us to harness more features of experimental data, including cell trajectories and cell count data, which we show overcomes the parameter identifiability problem. We demonstrate that, while difficult to collect, cell trajectory data can provide more information about the parameters of the cell invasion model. To handle the intractability of the likelihood function of the stochastic model, we use an efficient ABC algorithm based on sequential Monte Carlo. Rcpp and MATLAB implementations of the simulation model and ABC algorithm used in this study are available at https://github.com/michaelcarr-stats/FUCCI.

A practical guide to pseudo-marginal methods for computational inference in systems biology

For many stochastic models of interest in systems biology, such as those describing biochemical reaction networks, exact quantification of parameter uncertainty through statistical inference is intractable. Likelihood-free computational inference techniques enable parameter inference when the likelihood function for the model is intractable but the generation of many sample paths is feasible through stochastic simulation of the forward problem. The most common likelihood-free method in systems biology is approximate Bayesian computation that accepts parameters that result in low discrepancy between stochastic simulations and measured data. However, it can be difficult to assess how the accuracy of the resulting inferences are affected by the choice of acceptance threshold and discrepancy function. The pseudo-marginal approach is an alternative likelihood-free inference method that utilises a Monte Carlo estimate of the likelihood function. This approach has several advantages, particularly in the context of noisy, partially observed, time-course data typical in biochemical reaction network studies. Specifically, the pseudo-marginal approach facilitates exact inference and uncertainty quantification, and may be efficiently combined with particle filters for low variance, high-accuracy likelihood estimation. In this review, we provide a practical introduction to the pseudo-marginal approach using inference for biochemical reaction networks as a series of case studies. Implementations of key algorithms and examples are provided using the Julia programming language; a high performance, open source programming language for scientific computing (https://github.com/davidwarne/Warne2019_GuideToPseudoMarginal).

Chaste: Cancer, Heart and Soft Tissue Environment

Chaste (Cancer, Heart And Soft Tissue Environment) is an open source simulation package for the numerical solution of mathematical models arising in physiology and biology. To date, Chaste development has been driven primarily by applications that include continuum modelling of cardiac electrophysiology ('Cardiac Chaste'), discrete cell-based modelling of soft tissues ('Cell-based Chaste'), and modelling of ventilation in lungs ('Lung Chaste'). Cardiac Chaste addresses the need for a high-performance, generic, and verified simulation framework for cardiac electrophysiology that is freely available to the scientific community. Cardiac chaste provides a software package capable of realistic heart simulations that is efficient, rigorously tested, and runs on HPC platforms. Cell-based Chaste addresses the need for efficient and verified implementations of cell-based modelling frameworks, providing a set of extensible tools for simulating biological tissues. Computational modelling, along with live imaging techniques, plays an important role in understanding the processes of tissue growth and repair. A wide range of cell-based modelling frameworks have been developed that have each been successfully applied in a range of biological applications. Cell-based Chaste includes implementations of the cellular automaton model, the cellular Potts model, cell-centre models with cell representations as overlapping spheres or Voronoi tessellations, and the vertex model. Lung Chaste addresses the need for a novel, generic and efficient lung modelling software package that is both tested and verified. It aims to couple biophysically-detailed models of airway mechanics with organ-scale ventilation models in a package that is freely available to the scientific community. Chaste is designed to be modular and extensible, providing libraries for common scientific computing infrastructure such as linear algebra operations, finite element meshes, and ordinary and partial differential equation solvers. This infrastructure is used by libraries for specific applications, such as continuum mechanics, cardiac models, and cell-based models. The software engineering techniques used to develop Chaste are intended to ensure code quality, re-usability and reliability. Primary applications of the software include cardiac and respiratory physiology, cancer and developmental biology.

A palmitoyltransferase Approximated gene Bm-app regulates wing development in Bombyx mori

The silkworm Bombyx mori is an important lepidopteran model insect in which many kinds of natural mutants have been identified. However, molecular mechanisms of most of these mutants remain to be explored. Here we report the identification of a gene Bm-app is responsible for the silkworm minute wing (mw) mutation which exhibits exceedingly small wings during pupal and adult stages. Compared with the wild type silkworm, relative messenger RNA expression of Bm-app is significantly decreased in the u11 mutant strain which shows mw phenotype. A 10 bp insertion in the putative promoter region of the Bm-app gene in mw mutant strain was identified and the dual luciferase assay revealed that this insertion decreased Bm-app promoter activity. Furthermore, clustered regularly interspaced short palindromic repeats/RNA-guided Cas9 nucleases-mediated depletion of the Bm-app induced similar wing defects which appeared in the mw mutant, demonstrating that Bm-app controls wing development in B. mori. Bm-app encodes a palmitoyltransferase and is responsible for the palmitoylation of selected cytoplasmic proteins, indicating that it is required for cell mitosis and growth during wing development. We also discuss the possibility that Bm-app regulates wing development through the Hippo signaling pathway in B. mori.

Neuronal differentiation influences progenitor arrangement in the vertebrate neuroepithelium

Cell division, movement and differentiation contribute to pattern formation in developing tissues. This is the case in the vertebrate neural tube, in which neurons differentiate in a characteristic pattern from a highly dynamic proliferating pseudostratified epithelium. To investigate how progenitor proliferation and differentiation affect cell arrangement and growth of the neural tube, we used experimental measurements to develop a mechanical model of the apical surface of the neuroepithelium that incorporates the effect of interkinetic nuclear movement and spatially varying rates of neuronal differentiation. Simulations predict that tissue growth and the shape of lineage-related clones of cells differ with the rate of differentiation. Growth is isotropic in regions of high differentiation, but dorsoventrally biased in regions of low differentiation. This is consistent with experimental observations. The absence of directional signalling in the simulations indicates that global mechanical constraints are sufficient to explain the observed differences in anisotropy. This provides insight into how the tissue growth rate affects cell dynamics and growth anisotropy and opens up possibilities to study the coupling between mechanics, pattern formation and growth in the neural tube.

Summary: A mechanical model of the vertebrate neuroepithelium, based on experimental observations, suggests that the rate of neuronal differentiation influences tissue growth and the shape of lineage-related clones.

Mechanical characterization of disordered and anisotropic cellular monolayers

We consider a cellular monolayer, described using a vertex-based model, for which cells form a spatially disordered array of convex polygons that tile the plane. Equilibrium cell configurations are assumed to minimize a global energy defined in terms of cell areas and perimeters; energy is dissipated via dynamic area and length changes, as well as cell neighbor exchanges. The model captures our observations of an epithelium from a Xenopus embryo showing that uniaxial stretching induces spatial ordering, with cells under net tension (compression) tending to align with (against) the direction of stretch, but with the stress remaining heterogeneous at the single-cell level. We use the vertex model to derive the linearized relation between tissue-level stress, strain, and strain rate about a deformed base state, which can be used to characterize the tissue’s anisotropic mechanical properties; expressions for viscoelastic tissue moduli are given as direct sums over cells. When the base state is isotropic, the model predicts that tissue properties can be tuned to a regime with high elastic shear resistance but low resistance to area changes, or vice versa.