Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells

Adherent cells exert traction forces on to their environment, which allows them to migrate, to maintain tissue integrity, and to form complex multicellular structures. This traction can be measured in a perturbation-free manner with traction force microscopy (TFM). In TFM, traction is usually calculated via the solution of a linear system, which is complicated by undersampled input data, acquisition noise, and large condition numbers for some methods. Therefore, standard TFM algorithms either employ data filtering or regularization. However, these approaches require a manual selection of filter- or regularization parameters and consequently exhibit a substantial degree of subjectiveness. This shortcoming is particularly serious when cells in different conditions are to be compared because optimal noise suppression needs to be adapted for every situation, which invariably results in systematic errors. Here, we systematically test the performance of new methods from computer vision and Bayesian inference for solving the inverse problem in TFM. We compare two classical schemes, L1- and L2-regularization, with three previously untested schemes, namely Elastic Net regularization, Proximal Gradient Lasso, and Proximal Gradient Elastic Net. Overall, we find that Elastic Net regularization, which combines L1 and L2 regularization, outperforms all other methods with regard to accuracy of traction reconstruction. Next, we develop two methods, Bayesian L2 regularization and Advanced Bayesian L2 regularization, for automatic, optimal L2 regularization. Using artificial data and experimental data, we show that these methods enable robust reconstruction of traction without requiring a difficult selection of regularization parameters specifically for each data set. Thus, Bayesian methods can mitigate the considerable uncertainty inherent in comparing cellular traction forces

Advanced data analysis for traction force microscopy and data-driven discovery of physical equations

The plummeting cost of collecting and storing data and the increasingly available computational power in the last decade have led to the emergence of new data analysis approaches in various scientific fields. Frequently, the new statistical methodology is employed for analyzing data involving incomplete or unknown information. In this thesis, new statistical approaches are developed for improving the accuracy of traction force microscopy (TFM) and data-driven discovery of physical equations. TFM is a versatile method for the reconstruction of a spatial image of the traction forces exerted by cells on elastic gel substrates. The traction force field is calculated from a linear mechanical model connecting the measured substrate displacements with the sought-for cell-generated stresses in real or Fourier space, which is an inverse and ill-posed problem. This inverse problem is commonly solved making use of regularization methods. Here, we systematically test the performance of new regularization methods and Bayesian inference for quantifying the parameter uncertainty in TFM. We compare two classical schemes, L1- and L2-regularization with three previously untested schemes, namely Elastic Net regularization, Proximal Gradient Lasso, and Proximal Gradient Elastic Net. We find that Elastic Net regularization, which combines L1 and L2 regularization, outperforms all other methods with regard to accuracy of traction reconstruction. Next, we develop two methods, Bayesian L2 regularization and Advanced Bayesian L2 regularization, for automatic, optimal L2 regularization. We further combine the Bayesian L2 regularization with the computational speed of Fast Fourier Transform algorithms to develop a fully automated method for noise reduction and robust, standardized traction-force reconstruction that we call Bayesian Fourier transform traction cytometry (BFTTC). This method is made freely available as a software package with graphical user-interface for intuitive usage. Using synthetic data and experimental data, we show that these Bayesian methods enable robust reconstruction of traction without requiring a difficult selection of regularization parameters specifically for each data set. Next, we employ our methodology developed for the solution of inverse problems for automated, data-driven discovery of ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). To find the equations governing a measured time-dependent process, we construct dictionaries of non-linear candidate equations. These candidate equations are evaluated using the measured data. With this approach, one can construct a likelihood function for the candidate equations. Optimization yields a linear, inverse problem which is to be solved under a sparsity constraint. We combine Bayesian compressive sensing using Laplace priors with automated thresholding to develop a new approach, namely automatic threshold sparse Bayesian learning (ATSBL). ATSBL is a robust method to identify ODEs, PDEs, and SDEs involving Gaussian noise, which is also referred to as type I noise. We extensively test the method with synthetic datasets describing physical processes. For SDEs, we combine data-driven inference using ATSBL with a novel entropy-based heuristic for discarding data points with high uncertainty. Finally, we develop an automatic iterative sampling optimization technique akin to Umbrella sampling. Therewith, we demonstrate that data-driven inference of SDEs can be substantially improved through feedback during the inference process if the stochastic process under investigation can be manipulated either experimentally or in simulations

Microscale Measurements of Cell and Tissue Mechanics in Three Dimensions

Two-dimensional (2D) studies have revealed that mechanical forces drive cell migration and can feedback to regulate proliferation, differentiation and the synthesis/remodeling of extracellular matrix (ECM) proteins. Whether these observations can be translated to clinical settings or be utilized for tissue engineering will depend critically on our ability to translate these findings into physiologically relevant three-dimensional (3D) environments. The general goal of this dissertation has been to develop and apply new technologies capable of extending studies of cell and tissue mechanics into 3D environments. In the first project, we measured both shear and normal traction forces exerted by cells cultured on planar substrates. We observed that focal adhesions serve as pivots about which cells generate rotational moments. In the second project, we combined enzymatically degradable synthetic hydrogels with finite element models to measure the mechanical tractions exerted by cells fully encapsulated within 3D matrices. We found that cells reach out thin protrusions and pull back inward towards the cell body with the highest forces at the tip. Cellular extensions that were invading into the surrounding matrix displayed a strong inward force 10-15 microns behind the leading tip, suggesting that growing extensions may establish a contractile waypoint, before invading further. To study the forces cells exert during tissue remodeling, we utilized photolithograpy to generate arrays of microtissues consisting of cells encapsulated in 3D collagen matrices. Microcantilevers were used to constrain the remodeling of the collagen gel and to report the forces generated during this process. We used this technique to explore the effects of boundary stiffness and matrix density within model tendon and cardiac tissues. Finally, we combined this system with a Foerster radius energy transfer (FRET) based biosensor of fibronectin conformation to reveal how tissue geometry and cell-genereated tractions cooperate to pattern matrix conformation during tissue remodeling. Together, these studies highlight novel approaches to understand the nature of cell-ECM interactions in 3D matrices. Such mechanical insights will help us to understand how physical forces drive cell migration and behavior within physiologically relevant environments

Reconstruction and Simulation of Cellular Traction Forces

Biological cells are able to sense the stiffness, geometry and topography of their environment and sensitively respond to it. For this purpose, they actively apply contractile forces to the extracellular space, which can be determined by traction force microscopy. Thereby cells are cultured on elastically deformable substrates and cellular traction patterns are quanti- tatively reconstructed from measured substrate deformations, by solving the inverse elastic problem. In this thesis we investigate the influence of environmental topography to cellular force generation and the distribution of intracellular tension. For this purpose, we reconstruct traction forces on wavy elastic substrates, using a novel technique based on finite element methods. In order to relate forces to single cell-matrix contacts and different structures of the cytoskeleton, we then introduce another novel variant of traction force microscopy, which introduces cell contraction modeling into the process of cellular traction reconstruction. This approach is robust against experimental noise and does not need regularisation. We apply this method to experimental data to demonstrate that different types of actin fibers in the cell statistically show different contractilities. We complete our investigation by simulation studies considering cell colonies and single cells as thermoelastically contracting continuum coupled to an elastic substrate. In particular we examined the effect of geometry on cellular behavior in collective cell migration and tissue invasion during tumor metastasis

Leveraging elasticity theory to calculate cell forces: From analytical insights to machine learning

Living cells possess capabilities to detect and respond to mechanical features of their surroundings. In traction force microscopy, the traction of cells on an elastic substrate is made visible by observing substrate deformation as measured by the movement of embedded marker beads. Describing the substrates by means of elasticity theory, we can calculate the adhesive forces, improving our understanding of cellular function and behavior. In this dissertation, I combine analytical solutions with numerical methods and machine learning techniques to improve traction prediction in a range of experimental applications. I describe how to include the normal traction component in regularization-based Fourier approaches, which I apply to experimental data. I compare the dominant strategies for traction reconstruction, the direct method and inverse, regularization-based approaches and find, that the latter are more precise while the former is more stress resilient to noise. I find that a point-force based reconstruction can be used to study the force balance evolution in response to microneedle pulling showing a transition from a dipolar into a monopolar force arrangement. Finally, I show how a conditional invertible neural network not only reconstructs adhesive areas more localized, but also reveals spatial correlations and variations in reliability of traction reconstructions

Forces and Flow of Contractile Networks

Biological cells use contractile networks of cross-linked semiflexible biopolymers, the so-called actin cytoskeleton, to control their shapes and to probe the mechanical properties of their environment. These processes are essential for cell survival and function. In this thesis we present a general framework to model two-dimensional contractile networks embedded in either two- or three-dimensional space. A surface representation with triangles and edges allows us to explicitly address the heterogeneity of biopolymer networks. In adherent cells, thick polymer bundles called stress fibers strongly influence cellular mechanics. We establish methods to assess their contribution to traction force generation, intracellular force balance, and intracellular flow from experimental data. Further, we develop a theory for the excitable nature of the cell cortex, which is a thin polymer layer lining the inner side of the cell membrane, and show how it is related to global cell shape changes

Nuclear Connectivity in Mesenchymal Stem Cell Differentiation and its Role in Mechanotransduction

Mechanical forces transmitted through the cellular microenvironment are critically important for tissue development, homeostasis, and degeneration, particularly for tissues in the musculoskeletal system that play primary load bearing roles. When these tissues degenerate, their limited healing capacity has lead many to propose tissue engineering strategies in order to provide functional replacements. While native tissue structural and mechanical benchmarks are commonly used to quantify and validate outcomes for tissue engineering and to refine experimental approaches, these same benchmarks have not yet been extended to the cellular and sub-cellular level. This is true despite the fact that it is at this length scale where tissue specific mechanotransduction occurs. With this motivation, and building on an established biomaterial framework for engineering fibrocartilagenous tissues, this thesis further refines macro-scale functional benchmarks and defines new micro-scale benchmarks that are operative at the cellular and sub-cellular level. In doing so, we identify a number of structural attributes of the cellular cytoskeleton and its attachments to the nucleus that are important for mechanotransduction. In normal functioning tissues, maintenance of phenotype in mechanically active microenvironments requires appropriately tuned mechanotransduction machinery. Understanding how these sub-cellular mechanoactive species (including the cytoskeleton, the nucleus, and nuclear connectivity) change with differentiation and development, as well as their role in mechanical signaling, will be of primary importance for the success of tissue engineering strategies